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PROPERTY
DEPARTMENT
MACHTNE DESICN
SIBLEY SCHOOL
CORNELL UNIS/ERSITY
RECEIVED ~~THE
CONSTRUCTOR
Professor at the Royal Technical High School at Berlin, Royal Privy Couneillor,
Member of the Royal Technical Deputation,
Corresponding Member of the Institute of Lombardy and of the Swedish Technical Soeiety,
Foreign Member of the Royal Academy of Sciences of Stockholra,
Honorary Member of the Technical Societies of Riga and Erfurt,
of the Technical Soeiety of Frankfurt a M., of the Soeiety of Arts of Geneva,
of the Flora Soeiety of Cologne. of the American Philosophical Soeiety
and of the American Soeiety of Mechanical Engineers.
A
OF
MACHINE DESIGN
BY
WlTH PORTRAIT AND OVER 1200 ILLUSTRATIONS
AUTHORIZED TRAHiSLATION
COMPLETE AND UNABRIDGED
FROM THE FOURTH ENLARGED GeRMAN EDIT10N
BY
HENRY HARRISON SUPLEE, B. Sc„
Member of the American Soeiety of Mechanical Engineers,
Member of the Franklin Institute
H, H. SUPLEE
WEST CHELTEN AVENUE
1894Copyright, 1890, by John M. Davis.
Copyright, 1893, by Henry Harrison Suplee.
Bntered at Stationers Hali.Translator’s Preface.
In presenting to the engineering profession of England and America this translation of
Reuleauxs Constructor, a few prefatory remarks may be permitted. Although the. first edition
of the German work appeared as long ago as i86i,and translations have been made into French,
Swedish and Russian, no English translation has hitherto been made, notwithstanding the fact
that repeated editions and enlargements of the original German work have appeared.
The translation‘here given, therefore, is the first presentation to English speaking engi-
neers of a work which during the past thirty years has acquired the highest reputation over ali
Europe, and is so well known to German reading engineers and students in this country that no
excuse is needed for its present appearance.
The freedom with which the author has drawn from English and American sources as well
asfrom Continental practice gives the work a value not found in other treatises upon machine de-
sign, while the vast improvement which has been made by the introduction of the kinematic
analysis and the resulting classification of the details of the subject, cannot fail to appeal to the
instructor as well as to the practising engineer.
The translation has been made from the Fourth Enlarged German Edition of 1889, the
last which has appeared in the original, and is complete and unabridged in every respect. The
introduction to this edition is especially worthy of note, as it contains the author’s summary of the
principies set forth in his larger work on Theoretical Kinematics,* and the more so as it includes
a brief glance at the stili wider subject included in his work on Applied Kinematics, as yet
unpublished in Germany, and embodying a mass of manuscript which it is trusted will at no
distant day be given to the public.
The work of translation has been done with the especial sanction and exclusive authoriza-
tion of Prof. Reuleaux, by whom also the portrait and special introduction to the American
edition have been furnished.
The transformation of the notation of the work from the metric system to the English
values has involved much labor and while it is too much to expect entire freedom from errors, not-
withstanding the care which has been given to this portion of the work, it is trusted that but few
errors will be found. It is especially requested that any corrections which may be found neces-
sary will kindly be sent to the translator for future use.
HENRY HARRISON SUPLEE.
Philadelphia, September, 1893.
* It is to be regretted that Prof. Kennedy’s translation of this valuable work is now out of print, and it is hoped that a
new edition may be issued.AUTHOR'S INTRODUCTION TO THE AMERICAN EDITIOR
The present translation of the Constructor places my book before a large circle of
readers who have been practically active and energetic in the development of machine design,
for no one of the technical professions has been followed by the English-speaking race with
more activity and success than that of the construction of machinery. I therefore take pleasure
in prefacing this book with a few words of special introduction.
During the series of years in which my Constructor has grown from a small beginning
to a large volume, the practice of machine construction has also been continuously developing,
so that in every new edition changes and additions have been necessary. Much new matter has
been added in this edition to the theoretical portion ; first, in the section on Graphical Statics,
enabling many numerical calculations to be dispensed with, using in their places graphical meth-
ods ; second, by the introduction of the methods of Kinematics, or the Science of controlled
movements, a Science which reduces the apparently inexhaustible complexity of machine forms
to a few simple and fundamental principies, the command of which may be of extraordinary
value to the engineer. I am stili constantly engaged with the subject of Kinematics, especially
with its practical applications, but on account of the pressure of other occupations I have not as
yet been able to carry out my intention of treating this portion of the subject in a separate work,
corresponding to my work on Theoretical Kinematics. The work already published on this
subject I have therefore characterized as an “ outline ” of a theory of machines.” *
The simplification of the conceptions concerning machines to which these kinematical
sttidies led me, was of such importance that I have introduced the kinematical treatment into
the Constructor in various places, especially in the latter portion of the book. Even where no
special reference has been made to it, the theory has been followed, although the proof has been
omitted in order to avoid burdening the non-theoretical reader with details not absolutely neces-
sary for the practical application. It is in this manner that kinematical axioms have been intro-
duced into Chapter XVIII., where the subject of ratchets is treated. These were formerly
considered as devices of only minor importance, but the application of kinematical investigation
reveals the fact that they are of the very greatest importance, occupying a position in machine
construction superior to that of any other element or combination, and this notwithstanding the
apparent simplicity and almost insignificant appearance of the original contrivance. A similar
treatment has been given to the subject of Pressure organs, Chapter XXIII. Hitherto fluids,
such as water, steam, gas, etc., have been considered as somethlng apart from the machine, not
belonging to it, but rather introduced from the outside. The idea that fluids, broadly considered,
are but the exact opposites of tension organs, such as ropes, chains, belts, wire cables, etc., is
wholly contrary to earlier conceptions, and yet it is just this introduction of the kinematic method
which has led to an unexpected insight and very great simplification. An illustration of this is
seen in the manner in which valves for pressure organs are treated as ratchets. In Chapter
XVIII. ratchets formed of rigid elements only are considered, but the principies there deduced
are applied in Chapters XXIII. to XXVI. to fluid elements with most satisfactory results. Since
* F. Reuleaux, The Kinematics of Machinery. Outlines of a Theory of Machines. Translated and edited by Alex.
B. W. Kennedy, C. E., London. Macmillan & Co., 1876.INTRODUCTION.
v
the kinematic analysis has shown that such devices as pneumatic tubes, canal locks, and the like,
both aficient and modern, belong to precisely the same class of constrained combinations as
steam engines and, water wheels, the whole subject has been condensed and simplified in a man-
ner not possible under the earlier conceptions. The value of the kinematic method is evident in
in Section 333, where fifty different combinations of pressure organs aregathered together under
a few and simple fundamental principies. Another instance is shown near the end of the book
in the discussion of what I have called “ Fluid valves.” From the time of Hero of Alexandria
down to the present day, these fluid valves have been used in what is now seen to be a continu-
ous series of applications of a simple kinematical principle. These important simplifications will
both excuse and justify the wide departure from previous conceptions which characterizes the
latter part of the volume.
In regard to the other and principal object of the work, namely, the treatment of the
practical construction of machine details, this has not been as consistently and fully revised as I
had intended and desired ; chiefly owing to the long delay in the completion of the last edition.
In my lectures I have been able to follow the the technical advances which have been made in
the detailed construction of bearings, levers, cranks, connecting rods, etc., and discuss them
accordingly, but in the book itself many of these subjects stili appear in the older dress. For
these imperfections the kind indulgence of the reader is requested, and in the next edition an
earnest endeavor will be made to bring these subjects up to date.
To Mr. Henry Harrison Suplee, to whom I have given the exclusive right of translation,
I take this opportunity to express my particular appreciation of the great care and extraordinary
accuracy which he has displayed in the production of this English version, and also my gratifica-
tion at the care which has been given to the printing and the reproduction of the illustrations.
Mr. Suplee has recalculated and transformed all the formulae and numerous tables into the
English system of measurements, and also reworked all the examples, and has shown in this
portion of the work a patience that deserves especial recognition. It is a matter of regret that
the time has not yet arrived for the general acceptance of the metric system in England and
America, and until such time comes tedious transformations of this sort will often be necessary
and will merit our gratitude.
I can only add that it is my earnest desire that the friendly acceptance of my book by
English speaking engineers may correspond to the magnitude of the labor which has been
expended in the preparation of this translation,
F. REULEAUX,
Honorary Member, American Society of Mechanical Engineers.
Berlin, February. 1803.INTRODUCTION TO THE FOURTH GERMAN EdITION.
The fourth enlarged edition of the Constructor is
presented to its readers much later than I had hoped.
As some excuse for the delay I plead the great labor
involved in the re-arrangement of more than half the
book. As already explained, it has been my intention
to re-arrange the matter upon a kinematical basis.
It was not, however, entirely due to this re-arrange-
ment that the work was delayed, but also to the fact
that nearly one-half of the work had to be re-written.
In many places I found almost everything lacking to
make what I had previously determined upon, namely,
a complete and consistent whole, and much more was
needed than 1 had imagined. In addition to these
shortcomings the spirit of invention has been more ac-
tive than ever during the past few years and advanced
at such a rapid rate that I could by no means overtake
it. It is hoped that these conditions may be accepted
as at least a partial excuse for the delay and for the
shortcomings of the work.
The first point to which I desire to call attention in
the new matter is the subject of Ratchets, which upon
closer examination will be found to be the most im-
portant of all forms of driving mechanism. This sub-
ject has not until now been treated as an element of
construction, it having been apparently overlooked that
those forms of driving mechanism in which pawls and
ratchet wheels form a part, are in reality a most
important and prolific class. Special forms have in-
deed been treated mainly as checking devices but
without any attempt to indicate the general principies,
or wide extent of the construction. Locks, in spite of
their universal use and of the high order of inventive
talent devoted to them, have had no analytical treat-
ment, but have been relegated to the domain of tech-
nology rather by accident than otherwise, and from
Prechtl to Karmarsch and his followers, have been giv-
en an intellig^nt but by no means fundamental treat-
ment. Gun locks, although having a similar name to
door locks, have a very different construction, but have
found no resting place in technical literature. It has
often been observed that while we place in the hands
of our soldiers the modern rifles and cannon, there is
no place in the head for them, either in machine shop
training, in machine design, in applied mechanics or
in technology, or indeed anywhere. In §252 I have
placed them in that class which I have termed Locking
Ratchets where they fall into their proper place as
members of the great division of ratchet gearing.
The safety devices for elevators and hoisting machines,
—Checking Ratchets, I have termed them—have been
entirely overlooked; books have been written about
them, catalogues and price lists issued, but the funda-
mental principle of their construction quite overlooked.
As for escapements of clocks and watches, these have
been sent hither and thither, now in mechanical text
books, now in kinematics, now in applied mechanics,
again in encyclopaedias, where their fundamental prin-
ciple has been entirely lost, their intimate relation to
ratchet mechanism being hardly noticed. They will
here be found classified in their proper place in §258.
Many of the readers of previous editions may shake
their heads at this statement, but an examination of
the fourth edition will show how the action of the pis-
ton engine is similar in principle to a watch escape-
ment, the action of the slide valve being practically
identical with the anchor of the escapement, see §§324,
325, pp. 228-232. It has only been by more recent in-
vestigations that I have become convinced of the*
relations of these various forms of escapements. The
correctness of this position will be confirmed by
comparing the the pneumatic postal tubes, canal locks,
sluices, hydraulic cranes and numerous other hydraulic
devices, hydraulic riveting machines, and all the many
kinds of direct-acting steam pumps; these and many
others, when considered from the present point of view,
arrange themselves in a complete and orderly manner
as true escapements. The similarity is especially well
marked in the case of a deep mine pump, of which
the successive puffs of the exhaust are not infrequently
used by neighboring dwellers to indicate intervals of
time; the steam end practically as well as theoretically
becoming a time-piece. Nay, more : I am convinced
that it is not a pure accident that throughout the cen-
turies in which the delicate clock escapement has been
known, the steam engine has so slowly developed ;
for although both the clock and the engine are in prin-
ciple escapements, yet in the clock there is an escape-
ment of precision, and in the steam engine an
escapement of force, * but both devices are theoretically
a solution of the same problem. Closely allied to the
steam engine are the various water pressure engines,
and water pumps, which as I have shown in §319, are
truly continuous ratchet trains. From the ratchet to
the escapement, however, what a long, long gap !
The water pump and hydraulic pressure engine differ
from each other only in the different motion and
action of the valves—and yet the inventive genius of
♦In my Theoretical Kinematics, I have considered the steam engine as a
reciprocatingrunning ratchet tram, but I have since perceived this classifica-
tion to be incorreci and therefore desire to emphasize its proper classification
here.INTRODUCTION
vii
mankind required over*two thousand years to make
that little step, (see § 325). How important, then, to
make this fundamental connection ciear!
Another important, and hitherto neglected subject,
is that of the more recent steering devices, which move
in either direction, or remain at rest, as required. This
principle has found many applications in power steer-
ing gear for vessels, and has even made possible the
solution of the difiicult problem of guiding the auto-
matic fish-torpedo at a determinate depth. It is not
surprising that uncertainty should exist as to the theo-
retical classification of these devices. I have, how-
ever, shown that they are properly considered as
escapements, and, in fact, as escapements of a special
kind, which I have termed “ adjustable ” escapements.
Such adjustable escapements of rigid construction are
shown in § 259, and those constructed with pistons
and fluids, in § 329.
The chapter upon Ratchet Gearing is not only en-
tirely new, but it has also involved a new and more
elaborate treatment of many subjects discussed in
earlier chapters of the book. These I here only name :
Screw thread systems in Chapter IV.; Thrust-bearings
for screw propeller shafts ; Columns ; Long distance
shafting transmission, etc., in § 351 ; Couplings, Fric-
tion gearing; Transmission of motion by toothed gear-
ing (p. 128); Spiral gears (p. 141) ; Globoid gearing
(p. 142), Proportions of gearing, (§ 2 26-§ 228). Rat-
chet wheels are treated in a similar manner to spur
gear wheels, to which they bear a close relation,
(§ 246).
From this point the book takes a fresh start, with
the discussion of another species of machine elements,
namely, Tension organs, as I have termed them,
(Chapters XIX. to XXII). While the elements pre-
viously considered approximate so closely to rigidity
that they may properly be termed rigid elements, those
which follow possess the peculiarity that they are only
adapted to resist tension; these elements include
cords, ropes, wire, bands, belts, chains, etc. In § 262
it is shown how these are used in connection with
other elements in three distinet ways, as for 4 ‘guiding, ”
for “ winding,” and for “driving.” An examination
of pages 182 to 176 will make the importance of this
subject e vident, and shows its scope to be far greater
than might at first have been expected. The important
distinction between the functions of ‘4 driving ” and
.“guiding” is shown in the discession of the differen-
tial tackle and the ordinary system in connection with
Fig 813, (P* 176).
In discussing Cord Friction (§ 264) I have attempt-
ed to show by a graphical representation relations not
otherwise easy to make ciear. In § 268 I have called
attention to some points which should be considered
in connection with stiffness of ropes. • The subject of
pocketed sheaves has been treated in connection with
chaiins, and also the chain system of boat propulsion.
In the chapter upon Belt Transmission, is intro-
duced a new subject and one which appears to me of
great importance, and which I have called “ Specific
Capacity.” By its use it is possible to facilitate very
greatly the calculations of Belting, Rope Transmission,
Water Transmission and even Shafting, and bring them
to a comparable basis, (see § 349 and § 351).
The discussions of Hemp and Cotton Rope trans-
mission are both new, and that of Wire Rope greatly
enlarged over previous editions. By the introduction
of the subject of the “ mean deflection ” (p. 198) and
the diagram (Fig. 884) the question of the deflection is
greatly simplified, and a graphical solution is also
given. Transmission with inclined cables, which in
previous editions was only given an approximate
solution, unsuited for long spans, is here accurately
discussed (assuming the catenary as a parabola) and
extended to long stretehes of cable. This has been
done in view of the use of rope transmissions and
telegraph cables over valleys, etc.
Next follows my system of “ Ring Transmission ”
by wire rope. This offers great advantages over the
previous system of line transmission, and has met
with much success in Germany, Austria, and Switzer-
land, as well as in America ; and further discussion of
it will be given hereafter. The use of chain transmis-
sion in mines, both in Germany and elsewhere, is dis-
cussed. The subject of brakes brings the book to
another point where a fresh start must be made.
The third group of machine elements includes
those called “Pressure Organs,” and those are treated
in Chapters XXIII. to XXVI. These are directly opposed
to tension organs, since they are only capable of resist-
ing compression, and include not only fluids, both
liquid and gaseous, but also granular materials, etc.
(§ 3°8)-
Although these elements have been primarily ar-
ranged in a manner adapted for a practical hand book,
I believe that my theoretical treatment of the subject
will also find acceptance, and hence have here included
the essentials of the theory also (see § 319). Pressure
organs are serviceable not only in machines, but also
for the transmission of force and motion ; by them we
can control the motion of a force in a determinate
path and with a determinate velocity quite as well as
with rigid elements, and indeed upon closer inspection
we perceive that pressure organs are used in nearly ali
the most important prime movers, (steam engines and
hydraulic motors), and hence they are surely entitled
to be classed among machine elements. The extent
to which this conception facilitates the subject of ma-
chine construction will be seen by an examination of
the latter part of this volume.
I have thought it advisable to give also at this time
a general review of the resuit of my labors in the
field of Kinematics. These have been fully and thor-
oughly given in my lectures for the past twenty-five
years, and are therefore not new to my immediateINTRODUCTION
viii
f
pupils, while the publication of my Theoretical Kine-
matics has placed the the theory before a larger circle
of scientific readers. I cannot assume, however, that
the readers of this practical hand book are ali familiar
with the above mentioned work, and I therefore give
the following abstract, covering the most important
portions of my treatment of the subject.
Motion and the effects which are dependent upon
motion form the subject of the study of Scientific Me-
chanics ; and hence to it belongs properly the problems
of motion in machinery. The motions in a machine,
however, may be distinguished from others in that
they can be treated independently of the material
parts of the machine, and of the forces acting upon
them. The important bearing which this separation
gives to the subject of machine construction was per-
ceived about one hundred years ago, but has made
small progress during the century and has only re-
cently been taken up [10-23].*
I took up this subject in 1862, laying down the
principies in my lectures; in 1864 first propounded
them publicly before the convention of the Swiss
‘‘ Naturforscher " and their German guests ; first pub-
lished them serially in the Berliner Verhandlungen in
1865, and finally in 1872-75 published my book en-
titled “ Theoretische Kinematik”
The modern discussion of these principies begins
with the publication, by the celebrated physicist Am-
pere, in 1830, of his Essai sur la Philosophie des Sci-
ences, in which he gave the subject the distincti ve
name Kinematics (Cinematicque), which name is well
derived from the Greek kineo, to drive, to constrain,
since it treats of constrained or controlled motions.
I have defined the term Kinematics [40] as “the
study of those arrangements of the machine by which
the mutual motions of its parts. considered as changes
of position are determined. ” This I have divided into
to parts : “Theoretical" and “Applied" Kinematics,
the former treating of the general and fundamental
principies, and the latter of their practical applications.
a. Theoretical Kinematics.
It is this branch of the subject which is treated in
my well-known book ‘ ‘ The Kinematics of Machinery.'*
The following is a condensed analysis of the treatment
there expanded at greater length :
1.	A material system having motion within itself,
I call a machinal system, as may be determined ac-
cording as the motion is constrained or not [32].
2.	Motions can only be constrained by forces.
These forces differ in the two systems, since in the pure
machinal system sensible and latent forces enter into
equilibrium with each other, while in the pure cosmi-
cal system sensible forces enter into equilibrium with
sensible forces, [33]. It therefore follows that the
♦ The numbers in brackets referto the pages of Reuleaux’s “ Kinematics
of Machinery. Translated by Prof. A. B. W. Kennedy. London, Macmiilan
& Co., 1876.
two systems can not always be accurately determined
[34].* The terms “latent " and “sensible ” are here
used in a similar sense as in thermal physics. Latent
forces are those which exert internal resistance to de-
formation of a body under the action of external
forces ; sensible forces are those which act upon the
body from without [33].
3.	The motions of the machine can be logically
controlled according to a predetermined conception,
since the action of all external forces which do not
tend to produce the desired end can be opposed and
neutralized by latent forces [35].
4.	From the preceding follows the definition of a
machine :—
A machine is a combination of resistent bodies so
arranged that by their means the mechanical forces of
nature can be compelled to do work accompanied by cer-
tain determinate motions [35, 50, 203].
5.	If we consider the machine to be made of rigid
materials and neglect its mass, we need only take into
account geometrical considerations [42]. If a body
A, by means of latent forces, is to be prevented from
being put in motion by any external forces (case 3), it
must be held in a stationary position by at least one
body B. The body B, then acts as the envelope of A,
and conversely A is the envelope of B, the relation
being a reciprocal one. There are also reciprocal
envelope forms possible between the bodies A and B
ior a relative motion, which shall exclude all other
relative motions [43]. Such a pair of bodies, I have
called a kinematic pair of elements and a machine
consists solely of bodies which thus correspond, pair-
wise, reciprocally [43J.
6.	In order to obtain a determinate motion in a
given space by means of a kinematic pair of elements,
one of the elements of the pair must be held at rest
with regard to the given portion of space under con-
sideration. The relative motion of the moving ele-
ment to the fixed one will then be that of absolute
motion, so far as the given portion of space is con-
cemed [43].
7.	The choice between the two elements as to
which shall be stationary and which movable is not
limited; the substitution of the fixed for the moving
element I have called the inversion of the pair [93].
8.	The control which can be exercised over a de-
terminate motion in this manner is not mathematically
exact but only approximate (case 5) because the latent
forces of bodies can only be brought into action by
their deformation. If, however, the elements are
made of materials which possess a high degree of
resistance and are given proper dimensions (machine
construction) the deformation can be kept within such
small limits as to be practically insignificant, and the
resuit considered as determinate [33]. (Compare cases
46 to 49, below).
_______ *?
* The internal forces of a moving system form the subject of d’ Alem-
bert’s principle.INTRODUCTION.
IX
9.	Each element of a kinematic pair may be rig-
idly combined with an element of another similar pair
without interfering with the relative motion of the
separate pairs. In this manner a large number of pairs
of elements may be arranged in a series, so that each
element of a pair is firmly connected with an element
of another pair. Such a series of pairs of elements re-
turns upon itself, resembling a chain [46], consisting
of links connected together. I have called such a
series a kinematic chain, and the body which is formed
by the junction of the elements of two different pairs
is a link of the kinematic chain [46]. There are
therefore as many links as there are pairs of kinematic
elements.
10.	A kinematic chain may close or retum upon
itself in various ways; among these is one in which
every alteration in the position of a link relatively to
the one next to it is accompanied by an alteration in
the position of every other link relatively to the first
[46]. In such a chain each link has only a single
relative motion with regard to every other link. Such
a kinematic chain I call a constrained closed—or sim-
ply a closed chain [46].
11.	A constrained closed kinematic chain compels
a definite determinate motion in a given portion of
space when one link of the chain is fixed, with regard
to this given portion of space. A closed kinematic
chain of which one link is thus made stationary, is
called a mechanism [47].
12.	A constrained closed kinematic chain, there-
fore, can be formed into a mechanism in as many
ways as it has links [47]. The substitution of the
stationary link of a kinematic chain for another link I
have called its inversion.
13.	A kinematic chain may have so few members
and be closed in such a manner that the links can have
no motion relative to each other, and that the pairs
themselves do not have their own motion. This I
have termed fixed closure [485].
14.	The manner of closure of the chain can be
chosen so that adjoining links can have more than one
relative motion. This I have called unconstrained
closure [485].
15.	A kinematic chain in which a series of pairs of
elements are arranged in the stated manner, but of
which the first and last elements are not connected, I
have called an open chain.
16.	Kinematic chains of the kinds above men-
tloned can be combined with each other, forming con-
structions which may be called compound chains.
These may have constrained, unconstrained, or fixed
closure or may be open chains. The same conditions
exist for these as for the previously described chains,
which may for sake of distinction, be called simple
chains.
17.	From the preceding we may give the follow-
ing general definition of a mechanism, as follows :
A mechanism is a closed kinematic chain of which
one link is fixed : this chain is compound or simple
and consists of kinematic pairs of elements ; these
carry the envelopes required for the motion which the
bodies in contact must have, and by these all motions
other than those desired in the mechanism are pre-
vented [50].
18.	From all that has preceded, it is apparent ihat
the * investigation of the motions in machinery is a
subject which is based in great part upon geometry.
This has been treated as a separate subject of Phoron-
omy, or the study of geometrical motion. The most
important principies of this subject I have treated in
Chapter II. of my “ Theoretical Kinematics, ” with
applications to constrained as well as cosmical mo-
tions [56 to 85]. It is there shown that all relative
motion can be considered as that of a pair of ruled
surfaces, so that the motion is reduced to a rolling of
the two ruled surfaces upon each other, and under cer-
tain circumstances with a simultaneous endlong slid-
ing upon each other of the generators which are in
contact. These rolling surfaces, for which previously
no special name had been used, I have called axoids,
the combined sliding and rolling motion being termed
twisting. When rolling motion is absent only sliding
remains, when on the contrary, the sliding is omitted
only the motion of rolling remains. In the latter case
certain sections through the axoids give curves which
twist upon each other, or roll with a cross sliding
action. The combined points of these curves form
centres of rotation or poles about which, as instanta-
neous centres, both bodies turn. These centres or
poles travel in the paths of the aforesaid curves
whence the latter maybe called pole-paths (Polbahnen)
or centroids. * The study of axoids and centroids will
greatly extend the range of phoronomic researches.
19.	In order to pass from the general principies to
the special applications of kinematics, further consid-
eration must be given to the elementary pairs, The
simplest form must necessarily be that in which the
corresponding envelopes actually surround, one the
other, and such I have called a closed pair. Of this
there are but three forms : 1, the twisting pair (screw
and nut); 2, the turning pair (pin and collar); 3, the
sliding pair (full and open prism, or prism pair [91].
The two latter may be considered as particular cases
of the first. In all three no change in the character of
the motion is caused by inversion (case 7).
20.	In a pair of elements it is not always neces-
sary to use all of both envelope forms. The question
of the minimum number of points necessary to insure
resistence to disturning forces, I have discussed in § 17
of my Kinematics, under the title : “The Necessary
and Sufficient Restraint of Elements.''
♦The term “centroid,” due to Prof. ClifFord, and used by Prof. Kennedj
in his translation of the “ Kinematics,” will be hereafter used as the transla*
tion of Polbahn.—TrAns.X
INTRODUCTION.
21.	We have thus far omitted from consideratiori
such elementary pairs as are not closed. These pos-
sess the general property of giving a change in the
character of the motion when they are inverted. I
have called them “higher” pairs of elements [115],
and conversely the closed pairs may be termed “ low-
er ” pairs. It is only in special cases that no change
occurs in the character of the motion by inversion of
higher pairs. A series of higher pairs, for the most
part entirely new, has been discussed in § 21 of my
Kinematics.
22.	I have given (§ 30 to § 39) seven geometric
methods of determining the restraining bodies for
higher pairs, many of which were already known,
but which were then for the first time grouped into one
general system.
23.	Incomplete pairs [169] are those which are
not entirely closed by the latent forces, but are partly
closed in some other manner. Examples of these are
half-journal bearings, in which the weight of the parts
is used to keep the journal down in its bearing ; knife-
bearings for scale beams, the V bearings for the beds
of planing machines, etc. Pairs may also be closed
by the action of springs or other external forces. The
closure of a pair of elements in this manner I have
termed “force closure.” This form of closure can
only be used when the disturbing forces are not suffi-
ciently great to overcome the closing force.
24.	Force closure also finds application in higher
pairs of elements. An important example is found in
the driving wheels of locomotive engines, and another
stili more important, in the axoid rolling action of
friction wheels. (See Chapter XVI. of this volume.)
25.	The application of force closure can be car-
ried stili further. By its application we are enabled to
utilize two classes of elements which are only capa-
ble of opposing resistance in one direction (case 8).
These are what I have called “ tension organs ” and
“pressure organs,” (see § 261 and § 308 of this vol-
ume). These I have grouped together as “ flectional
kinematic elements [173]. They include a long series
of most useful machines, such as belt and rope trans-
mission Systems, pumps, water-wheels, etc., all in-
volving the principies of force closure.
26.	Force closure may be used in a dynamical
as well as in a statical manner, as in the case of an
engine crank which is carried over the dead centre by
the action of the fly wheel [186].
27.	In such cases the closure may also be effected
by means of another kinematic chain used in combi-
nation with the first [178]. This I have called chain
closure. An example is found in a double engine with
cranks at right angles.
28.	The preferable form of chain-closure is that in
which similar elements are employed. This occurs
(case 25) when one force-closing chain is used in con-
nection with another of the same kind, the two being
so combined that each supplies the necessary closing
force for the other ; whence it follows that the sensible
and latent forces in the two chains counteract each
other in the same manner as if they were composed of
rigid elements. [§ 44, “ Complete Kinematic Closure
of the Flectional Elements. ”] Examples of this are
found in the ordinary belt transmission, and in the so-
called “water rod.” By means of this method of
closure, which is destined to be much more widelv
used than heretofore, the applications of flectional
elements have been greatly extended for purposes to
which rigid elements are not adapted, such as the
transmission of force in a path of constantly changing
direction, as in the use of high pressure water trans-
mission systems through pipe conductors.
29.	Finally a kinematic chain may be closed by
the application of springs [176]. These may be so
constructed as to oppose resistance in a number of
chosen directions, but not in all directions ; e. g., both
tension and compression, also bending in one plane, but
not in a second plane at right angles to the first. This
latter condition is seen in the case of flat or piate
springs, also in#the piate link shown in Fig. 507 of this
book, where the spring acts as a substitute for a pin
connection. In the piate link the force closure and
complete kinematic closure are replaced by chain
closure. Another example is found in the Emery
Scale, Fig. 789c.
30.	The pairing of flectional with rigid elements
may be assumed, a priori’ to be practicable in the same
manner as that of rigid elements [544].
31.	If the principies of investigation, however
they may be set forth, are correctly based, they should
when applied to the historical development of ma-
chines, shed a light upon the whole subject from the
rude attempt at invention to the highest attainments of
mechanical ingenuity. This subject I have discussed
as a “Sketch of the History of Machine Develop-
ment, [201 to 246], in which the substitution of pair
closure for force closure is made most apparent.
32.	In order to facilitate the elucidation of the
action of machinery, and to abridge the labor of
the application of the preceding methods, it became
necessary to devise a system of kinematic notation.
This is given in Chapter VII., pages 247-273, of
my Kinematics of Machinery. The elements are
designated by capital letters, of which twelve are
required, and the relations of these are indicated by
auxiliary symbols derived for the most part from
those already used in mathematical notation. For
the symbolical representation of the kinematic chain
I have also introduced the conception of an order
in which each pair in the chain is numbered from
1 upwards, and the links represented by the small let-
ters from a onwards [270-273]. The pair and the link
at which the numbering and lettering is to begin may
be agreed upon previously, as well as the^ direction in
which they are to proceed. The link between the
pairs 1 and 2 will then be indicated by a ; that betweenINTROR UCTION
2 and 3, by b, etc. For instance, the connecting rod
and crank device, shown in Fig. 1022 of this book, is
a
indicated by the formula (C3" P-4-) 7- Translated, this
means that the kinematic chain of the mechanism con-
sists of three parallel, closed cylinder pairs, and one
closed prism pair at right angles to them ; that it con-
tains four links, which I have called the crank, the
coupler, the slide and the link, and designated by a,
b, c, and d; that this chain is converted into a mech-
anism by the link d being held fast; that the right line
from the centre of the bearing 2 to the end of the
coupler b (the connecting rod) moves around the axis
1 of the crank shaft, and that the crank a is driven, by
means of the coupler b, by the slide (cross-head) c.
This is certainly expressing very much by means of
very few symbols, dispensing with long and compre-
hensive definition. According to case 12, this chain
can be converted into three other kinds of machines,
symbolically indicated by : (C3" P-1-)*, (Cf P-1-)1*, etc.
These symbols have as yet been used but little by
practical designers, but those who have made use of
them have found them brief and accurate both for
writing and for descriotion otherwise requiring much
longer explanation.
33.	The application of the system of symbols
leads to what I have termed “ Kinematic Analysis,”
[Chapter VIII.] The application of this analysis to
the so-called “mechanical powers,” [275-283] leads to
interesting conclusions, this is also the case with the
cylindric crank chain [283-341], which taken in con-
nection with Chapter V., yields a wealth of valuable
results.
34.	This is followed by an analysis of “chamber-
crank ” trains, Chapter IX. In this, it is shown that
upwards of a hundred pressure organ machines,
hitherto considered as separate inventions, have a sys-
tematic relationship dependent largely upon kinematic
inversion ; and a number of difficulties are cleared up.
35.	In Chapter X. the subject of the so-called
“ chamber-wheelr trains is analyzed; the principies of
which I had previously investigated in 1868.
36.	Finally, in Chapter XI., is given the Analysis
of the Constructive Elements of Machinery, including
a brief investigation of ratchet mechanisms. At the
time this portion was written my investigation of that
subject, however, had not been carried to any great
extent, and in the present volume for the first time have
I set forth the extraordinary and varied importance of
ratchet mechanism.
37.	To this subject is added an analysis of the
complete machine [486-526], in which the striet limits
of theoretical kinematies are frequently overstepped and
encroachments made upon the domain of applied kine-
maties. The older ideas of the “receptor" the ‘‘com-
municator, ” and the “ tool ” are examined and rejected
and machines classified as “ place-changing ” and
‘ ‘ form-changing ” machines. This classification will
xi
be found to possess a decided value and will be referred
to agam. (Cases 42 to 49.)
38.	Kinematic Analysis has as a necessary coun-
terpart Kinematic Synthesis. This has been already
seen (cases 19, 21, 30) in the application of pairs,
chains and mechanisms to given machinal purposes.
Kinematic synthesis may also be called a theory of the
invention of mechanisms. This it can only be, how-
ever, in a limited sense. It can in no case enable the
genius of the inventor to be dispensed with, but by the
aid of this theory his scope can be greatly extended.
The application of synthesis to problems which have
already been solved may also point the way to the so-
lution of others as yet undetermined.
In discussing this synthesis, I have grouped the
pairs of kinematic elements into 21 orders [538-544] by
means of which the determination of the greater num-
ber of kinematic chains and dependent mechanisms
may be made ; also eight classes of simple chains.
The application of synthesis may be made in two
forms, the direct and the indirect, and these again into
general and special synthesis. Of these the indirect
synthesis is the most useful [529]. It is my expecta-
tion that this theoretical exposition of the subject,
which I cannot expect to extend further, but by means
of which I have been able to devise a number of new
mechanisms, may find many successful applications
by others.
b. Applied Kinematies.
39.	Applied Kinematies is not so much to be con-
sidered as standing in opposition to theoretical kine-
maticb as it is included in it. In fact, applied kinematies
has existed as a study for a long time, as in the treatise
of Monge, without the existence of any theoretic
foundation. That such a treatment of kinematies may
be very useful for a time is readily admissible, but an
ex post facto theoretical dtecussion may seem of little
value to the practical man. Indeed my highly es-
teemed former preceptor, Redtenbacher, considered an
actual theoretical treatment of the movements of ma-
chinery to be an impossibility.
Under these circumstances I did not feel inclined
to follow the ‘ ‘ Theoretical Kinematies ” hastily with a
treatise on the applied Science. For this purpose it
was not possible to arrange ali the various forms of
machines under the new classification hurriedly and
properly in permanent form. Notwithstanding the
simplicity of the preceding system, its application de-
veloped many difficulties and required a succession of
researches with which even my immediate pupils are
not fully acquainted. A not inexcusable impatience
on their part has led me to have my investigations in
applied kinematies multiplied for a limited circulation
although the matter was incomplete. I gave this per-
mission reluctantly and with the condition that only
a limited number of impressions, to be considered as
“ manuscript, ” should be circulated. In this mannerXll
INTR OD UCTION
four parts of the work have appeared, the last consist-
ing entirely of the applicatiori of the symbols to lecture
room models. The resuit of such premature publica-
tion cannot always be foreseen by those who have
urged it, but for the misunderstandings which have
arisen from this source I can only express my regret.
In the meantime I have since 1882 been engaged
in the partial applicatiori of the principies of kinemat-
ics to this book in such a manner as to avoid burden-
ing the reader with theoretical matter, which would be
contrary to the purpose of the work. The most im-
portant subjects to which the kinematic method has
been applied are here briefly noted.
40.	With the great extension of modern mechani-
cal engineering we find that the various mechanisms,
(the number of which as we have seen is not great),
are given a great variety of applications. It is the
object of applied kinematics to furnish a ciear distinc-
tion between the various methods of practical applica-
tion. It is apparent that the preceding analysis does
not extend to this point, since it does not include the
subject of the method of constraint, but only treats of
the combination of the elementary parts which are
involved. We may therefore properly term it the
Elementary Kinematic Analysis. As a counterpart for
this in applied kinematics we may place the subject of
another analysis which relates to the conditions of
motion in a given train, and which may be called
Train Analysis, or the Analysis of Trains. This anal-
ysis is not intended to solve anew the construction of
the various trains, but rather to elucidate clearly their
method of action ; a train consisting of a closed group
of elements and bearing thesame relation to a machine
as an atom does to a molecule.
41.	Train analysis does not admit of an arrange-
ment logically similar to the elementary analysis, but
possesses a new and different order. This is due to the
fact that the elements of which trains are composed
occur only in pairs, while the trains of which machines
are composed are considered singly. In Vose’s pump,
for example, Fig. 979a, there are two ratchet trains
combined in one machine, while in Downton’s pump,
Fig. 979c, there are three trains.
42.	The various methods of tain action may be
divided into four principal kinematic divisions, viz.:
Guiding, Storing, Driving, and Forming, §333.* The
first three divisions are “ Place-changing ” and the last
is “ Form-changing.”
43.	Various forms of guiding devices may be
mentioned; linkages by means of which curved paths
are obtained, parallel and straight line motions, also
“position motions,” as I have termed those by means
of which a System of points may be transferred to
another position parallel to the first. Guiding devices
can be constructed from kinematic chains of every
* In § 333, the secondof these has been translated “ Supporting/* and the
English laneuaee lacks a suitable equivalent for “ Haltung, but in a corres-
pondence with the author, the above has been adopted.—Trans.
kind. It was by means of examples with chains for
this purpose that the general conditions of motion in
theoretical kinematics were illustrated, and the same
conditions belong also to applied kinematics.
34. Storing includes those especial machine organs
by means of which work can be accumulated and the
supply drawn upon for later use. This, until now has
not been considered as a special mechanical concep-
tion, aithough it has had numerous applications. Stor-
age of power may be accomplished in three quite
different ways.
a.	By means of rigid elements, this being statical
or dynamical. Examples of statical storage are found
in elevated weights, compressed springs, etc., and of
dynamical storage in fly wheels, or pendulums. One
of the oldest forms of dynamical storage is the old-
fashioned spindle [216].
b.	By means of tension organs, acting by winding
the tension organ upon a drum or pulley. Examples
are seen in tower clocks, etc.
c.	By means of pressure organs. These are the
most frequently used, and examples include tanks for
water, oil, gas, air, steam, also hydraulic accumulators.
45.	Driving. In this term I include the transmis-
sion of motion within a single train and also from one
system to another. As “guiding ” includes the control
of the path of a point, “driving” considers the control
of the velocities of various points in their paths. Ex-
amples in this branch of applied kinematics are those
which take into consideration the velocity of the var-
ious parts of a mechanism. (See the close of Case 38).
46.	Forming, includes the working of materialsby
means of machine tools. This fourth division is the
richest of ali, and offers the widest range to the genius
of invention. This operation takes place by the action
of the tool upon the material, or as I have called it,
the “ work piece ” [495]. In form changing machines,
the work-piece is a part or the whole of a kinematic
link, and is paired or chained with the tool by so ar-
ranging the latter that it itself changes the original
form of the work-piece into that of the envelope cor-
responding to the motion in the pair or linkage em-
ployed [495]. We can distinguish between three forms
in which this action can occur.
a.	The tool is hard and operates by cutting the
material from the work-piece which lies without the
envelope of the desired form. Examples are found in
lathes, planers, grinding machines, etc.
b.	The tool is of high resistance so as to be able
to maintain its form, but does not act by cutting, but
by pressure upon the yielding work-piece. It follows
that the material which lies outside of the desired form
is forced into another part of the work-piece without
being removed from it. Examples are found in coin-
ing presses, rolling mills, wire drawing benches, etc.
c.	The tool and the work-piece are both alike
yielding, and act each upon the other, each being theINTRODUCTION
Xlll
tool for the other piece. Examples are the various
kinds of spinning, weaving, and other textile machin-
ery. Ali three forms are described in this volume,
many examples being given among the pressure
organs.
47.	It may appear from the preceding as if the
theory of the action of the tool breaks through the
logical arrangement given in the theoretical kinematics,
since in Case a, one of the elements, the tool, cuts
away and destroys its partner because it is enough
harder to cut it. We must here distinguish between
yielding and unyielding elements. This looks like a
return to empiricism. The defect in the logic, how-
is only apparent. All elementary pairs without excep-
tion involve the idea that both of the partners evoke
the latent forces by the action of deformation; and at
the same time the friction between the moving parts
induces wear. Applied mechanics takes friction into
account in considering elementary pairs and investi-
gates and provides for the consequent wear. The
machine constructor endeavors by all means within
his power to reduce the alteration of form at points
where it is not desired, but where it is the end to-be
accomplished he takes every opportunity to increase it.
The form-changing action which occurs between the
tool and the work-piece differs in degree only and not in
kind, from the action taking place between the elements
of every other pair in the machine [5^3].
48.	A similar idea may arise in connection with
the method of form-changing given above under (b),
in which an alteration of form takes place without an
actual removal of any of the particles. In this case the
the correspondence of the kinematic to the mechanical
action is evident. In case 8, as already noted, the de-
formation which takes place in non-rigid bodies makes
it only practicable to obtain approximate Solutions.
This only involves a quantitative, and not a qualita-
tive distinction [502].
Examples of this occur in the construction of in-
struments of precision. It is not possible to construet
even a simple cylindrica 1 pair (case 19) such as a cen-
tre for a theodolite, or for an astronomical telescope,
entirely free from error. By the use of a variety of
methods the errors are kept as small as possible, and
then by other methods, nearly always kinematic, the
residual errors are determined and the proper correc-
tions made.
49.	In other instances the designer may utilize the
elastic yielding of the members of a kinematic chain,
as for instance in the method of Adolph Hirn, by
which the springing of the beam of a steam engine is
used to produce the indicator diagram of the steam
pressure ; or the torsional deflection of a large shaft to
measure the power transmitted. *
This method is also found in Gidding's device for
measuring valve friction (p. 285), and also in the
Emery scale, in which a very small deflection of a
diaphragm measures accurately weights of many tons.
Although in many instances the deformations of
material may be neglected, yet we should never per-
mit ourselves to forget that they have been neglected.
Otherwise important errors may creep into theoretical
deductions, as well as in practical construction. This
subject of the yielding of materials is receiving more
attention at present than formerly.
50.	The “order” of a system of transmission is
a subject of importance since there are several meth-
ods by which the various parts may be kinematically
arranged. I have applied the term “order” to the
method of arrangement, and distinguish between three
different methods.
a. “Series Order.” This “ order” exists when a
number of transmissions are arranged in series, so
that each acts upon the following one. If in a single
machine, two, three, four or “ n ” transmissions are
thus arranged in series, I call the whole a system of
the second, third, fourth or nth order. Examples are
found in Figs. 766, 767.
A transmission can return upon itself. This I have
called a “ring” system of transmission. (See p. 208).
This return to the original must always occur in the
kinematic chaici of any mechanism since the elements
exist only in the relation of pairing (Case 5). In the
system under consideration (Case 41), the groups of
elements follow each other in a series, or line as it may
be termed, whence I have termed such a series a
“ line ” transmission (p. 257). Ring transmission may
also be combined with line transmission, the line being
divided into two or more parts. An example of the
first kind is seen on page 229, in which the pump
mechanism is combined with the steam mechanism, as
a line with a ring system. An intermediate form be-
tween ring and line transmission is referred to on page
208.
b. Combined Order. By this title is meant a com-
bination of transmissions in which each transmission
is connected to the next, but in which any one can be
stopped without stopping the others. An example of
this is shown in the ring transmission in Fig. 917.
Under certain circumstances a number of the driven
pulleys Tv T2, Ts - - Tn, may be allowed to run
empty, in which case they become merely supporting
sheaves (Case 43) ; as soon, however, as any load is
thrown on any of them; the entire system is influenced
by the increased stress upon the rope.
Another example of “compoond” order, is the
multiple expansion steam engine. Here each engine
of the compound, triple, or multiple expansion engine
, may be considered singly as a separate chain, and the
entire machine as a series of transmissions. Each en-
gine, Tv Tz, T3, etc., exerts an influence upon the
action of the others, but is not indispensable to their
action, as would be the case if arranged in ‘ ‘ series ”
order. Compound, Triple, Quadruple expansion en-
* See Bcrliner Verhandlungen.XIV
INTRODUCTIOK
gines are therefore, respectively of the second, third,
and fourth order, but should also be considered to be-
longto the class of “Compound order.”
c.	“Parallel Order.” This arrangement is the
oldest and the one which occurs most frequently. It
occurs when a number of different machines are all
driven from one and the same transmission, this being
the usual arrangement in manufacturing establish-
ments. Any of the machines can be stopped or
started independently of the others without affecting
the motion, a suitable regulator being assumed. This
principle may also be applied to the motors by which
the transmission is driven, automatic couplings, such
as shown on page ioi, being used. A “parallel”
order occurs in rope transmission when a number of
ropes are used on the same pulley ; another instance
is that of a train which is pulled by two locomotive
engines.
The three different £ ‘ orders ” are not always sharply
defined, but the distinction will be found of material
assistance in the study of transmissions. An example
in which all three “orders” are used is found in the
engine shown in diagram in Fig. 1023. Here the cyl-
inder, piston, valve and steam form an escapement :
the connection c 1 r being driven, and in turn operating
a second rt 1± b, and thence the valve. These three
transmissions therefore form a “series” order, this
also returning to itself and being thus a ring system,
and of the third order. The fly-wheel and its bearings
form a dynamical power storage system, absorbing
and giving out power in response to the irregularities
of the action of the piston, this being of the i * com-
pound ” order. Frequently such an engine is made
with an additional cut-off valve gear, with governor,
also of ‘ ‘ compound ” order, also possibly a feed pump,
(•“ parallel ” order) and the engine usually drives an
bxtensive transmission system by which a number of
machines are operated (“ parallel ” order).
In § 260 is shown the manner in which physical
and Chemical trains are arranged in series, the action
of heat, of gases and electricity being considered ;
the steam engine being the most notable example.
51. The magnitude of the exponent of the order
of any train has an important influence upon the hurt-
ful resistance of a machine, especially in a series order
of a high degree. In such cases the injurious resist-
ance increases at least directly as the exponent, and
frequently more rapidly. It is therefore important in
machine design to keep the degree of the order as low
as practicable. In the system of pneumatic clocks of
Mayrhofer (p. 171) the mechanism for several years
was as high as the 17th order, but the degree subse-
quently reduced to the 8th order. It may safely be
affirmed that the simplicity of a machine may be
measured by the closeness which the exponent of its
order approaches unity. Examples are found in the
Giffard injector, in which the guiding and driving
mechanisms are united in one, and exponent becomes
=1 ; the same is true of Siemens Geyser pump,
Fig. 971 a.	The apparatus of Morrison & Ingram,
Fig. 1181, is a device of the 2nd order, which acts by
a combination of guiding and driving.
52. The preceding pages ha ve shown that applied
kinematics, by means of the separation of the con-
trolled motion into the forms of Guiding, Storing,
Driving and Forming, and by means of the division of
the various ‘ ‘ orders, ” has enabled the machine prob-
lem to be solved as a whole. Theoretical kinematics
has assisted in this solution by enabling the various
problems to be investigated in a purely scientific man-
ner. Without such a theoretical investigation, a sys-
tem of applied kinematics would be an impossibility.
At the same time practical instruction must be given
by actual daily work as well. A ciear understanding
of the principies of the applied Science cannot but be
useful to the practical man, and as I believe, welcome
also.
The fundamental principies of machine construc-
tion as I have sought to lay them down in the preced-
ing pages, coincide in many points with the practical
methods already in use. The practical mechanic is
well acquainted with crank trains, gear trains, and
the like, or if he is not familiar with them he is readily
taught, but in combining these and arranging them so
as to act upon each other the theory comes into play
and shows clearly the best arrangement for the end in
view. This is well shown in the case of the various
valve gears, which have been in fact developed inde-
pendently, instead of being the resuit of a theoretical
analysis of various combinations of kinematic chains.
The application of the kinematic analysis will facilitate
work of this sort, making it clearer and simpler the
more fully the fundamental principies are understoocL
For this reason I have introduced the kinematic princi-
pies into this work, not to reduce invention to an art to
be taught, but rather to bring the principies of Science
to its assistance.
I am ready to admit that the general view of theo-
retical kinematics which I have placed before the prac-
tical man, may not be accepted without further proof
being demanded. It may be considered only as an
ingenious form of theorizing, of but little practical
value. For the present I must ask my readers*to prove
by the test of practical application how far the princi-
pies of kinematics may be made of genuine practical
value.
The principies included in cases 40 to 51 are
practically applied in the latter half of this volume.
The application of the analysis to thesubject of ratchet
gearing has produced an extensive series of results.
Storage is clearly shown to be a form of ratchet gear;
the discussion of the degree of “ order” of ratchet
trains will also, I believe, be found very useful. In
the discussion of pressure organs (Chapter^XXIII. and
foliowing) the subject of storage is highly developed.
The notion of the two divisions of guiding, and driv-xiv
INTR OD UC TION
xv
ing will also be found most useful. In like manner
the methods of analysis as applied to ratchet trains, are
found capable of equally prolific results when applied
to pressure organ trains, not, to my knowledge, oth-
erwise attainable. The great number of applications in
this direction will be seen in § 333, these being the re-
suit of the application of the theory sketched under
Case 46, above.
SinGe the subject of friction was considered in con-
nection with rigid elements, it was also necessary to
to take into account this resistance to the motion of
fluids (§ 340), as also the loss of heat in steam pipes
(briefly discussed in § 338). In § 362 the very import-
ant subject of boiler design is only generally consid-
ered.
The closing chapter relates to valves. These are
treated as ratchets, not omy from the theoretica!
standpoint, but also practically, and much more fully
than in previous editions. The section on “fluid
valves ” will, I trust, be found of use to the practical
man, as a subject worthy of further investigation.
In closing, I may refer to the increasing size of this
volume. In spite of my earnest efforts, it has not
been possible to reduce its bulk. In many places
evidence will be found of attempts at condensation, but
nevertheless the work can hardly be called properly a
“hand book” any longer. When discussing purely
technical matters I can be brief, but in a practical work,
it is above all things necessary to be ciear and intelli-
gible. In this I have endeavored to be guided by the
dictum of Boileau : “ Un ouvrage ne doit point para 'ire
trop travaille, mais il ne saurait etre irop travailleJ’
Funchal, February, 1889.
F. REULEAUX.TABLE OP CONTEXTS
SECTION I.
STRENGTH OF MATERIALS.
Introductory....................... i
Co-efficients of Resistance......' i
Resistance to Tension and Compres-
sion........................... 2
Bodies of Uniform Strength......... 2
Resistance to Shearing............. 2
Resistance to Bending.............. 2
Table of Sections.................. 5
Value of the Quantity S............ 8
Sections of Uniform	Resistance.... 8
Bodies of Uniform Resistance to
Bending........................ 8
Resistance to Shearing in the Neu-
tral Plane................... 10
Beams with a Comraon Load........	11
Resistance to Torsion............ 11
Polar Momentof Inertia and Section
Modulus...................... 11
Bodies of Uniform Resistance to
Torsion...................... 13
Resistance to Buckling............. 13
Oolumns of Uniform Resistance....	13
Compound Stresses................ 13
Resistance of Walls of Vessels...	15
Calculation of Springs........... 18
SECTION II.
THE ELEMENTS OF GRAPHOSTATICS.
Introductory..................... 22
Multiplication by Lines.......... 22
Division by Lines.... ........... 23
Multiplication and Division Com-
bined........................ 23
Areaof Triangles................. 23
Area of Quadrilateral Figures....	23
Areaof Polygons.................. 24
Graphical Calculation of Powers--	24
Powers of Trigonometrical Func-
tions........................ 25
Extraction of Roots............... 26
Addition and Subtraction of Forces. 26
Isolated forces in One Plane—Cord
Polygon....................... 26
Equilibrium of External Forces of
Cord Polygon................. 27
Equilibrium of Interna! Forces of
Cord Polygon.................  28
Resultant of Isolated Forces in One
Plane......................... 29
Conditions of Equilibrium of Isolat-
ed Forces..................... 29
Force Couples..................... 29
Equilibrium between Three Parallel
Forces........................ 30
Resultant of Several Parallel Forces 31
Decomposition of Forces........... 31
Uniformly Distributed Parallel
Forces........................ 32
Twisting and Bending Movements..	33
Determination of Centre of Gravity 33
Resultant of Load on Water Wheel. 34
Force Plans for Framed Structures. 35
Force Plans for Roof Trusses.....	36
Graphical Determination of Wind
Stresses...................... 37
Force Plans for Framed Beams-----	38
Remarks........................... 38
SECTION III.
THE CONSTRUCTION OF MACHINE
ELEMENTS.
Introductory...................... 39
CHAPTER I.
RIVETING.
Rivets.............................. 39
Strength of Riveted Joints.......... 40
Table and Proportional Scale.....	40
Riveting disposed in Groups......... 40
Steam Boiler Riveting............... 42
Table for Boiler Riveting........... 42
Table of Weights of SheetMetal...	43
Especial Forms of Riveted Joints... 43
CHAPTER II.
HOOPING.
Hooping by Shrinkage................ 45
Cold Hooping........................ 45
Examplesof Forced Connections.. .	46
Dimensions of Rings for Cold Fore-
ing.......................... 47
CHAPTER III.
KEYING.
Keyed Connections................... 47
Cross Keyed Connections............. 48
Longitudinal Keys................... 48
Edge Keys........................... 49
Methods of Keying Screw Propel-
lers............................ 49
Unloaded Keys....................... 49
Methods of Securing Keys............ 50
CHAPTER IV.
BOLTS AND SCREWS.
Geometrical Construction of Screw
Thread.......................... 50
Whitworth Screw System.............. 51
Sellers’ Screw Thread System....	52
Metrical Screw Systems.............. 52
New Systems......................... 53
Nuts, Washers and Bolt Heads....	54
Table for Metrical Bolts and Nuts -.	55
Weight of Round Iron................ 55
Special Forms of Bolts.............. 55
Wrenches............................ 56
Nut Locks........................... 56
Special Forms of Screw Threads. ..	58
Screw Connections, Flange Joints..	59
Unloaded Bolt Connections.,.....	60
CHAPTER V.
JOURNALS.
Various Kinds of Journals........... 60
A.—LATERAL JOURNALS.
Overhung Journals............... 61
Example of Table of Journals....	62
Neck Journals.. .................... 62
Fork Journals....................... 63
Multiple Journals................... 63
Half Journal........................ 64
Friction of Journals................ 64
B.—THRUST BEARINGS.
Proportions of Pivots............... 65
Friction of Flat Pivot Bearings-	66
Collar Thread Bearings.............. 66
Multiple Collar Thread Bearings...	66
Compound Link as Thrust Bearing.	67
Attachment of Journals.............. 67
CHAPTER VI.
BEARINGS.
Design and Proportion........• • •	68
A,—LATERAL BEARINGS.
Pillow Blocks....................... 68
Proportional Scale for Pillow Blocks 68
Various Forms of Journal Boxes... 69
Narrow Bases—Large Pillow Blocks 69
Adjustable Pillow Blocks........... 70
Bearings with Three Part Boxes__	70
Wall Bearings...................... 71
Yoke Bearings...................... 72
Wall Brackets...................... 72
Hangers............................ 73
Adjustable Hangers................. 74
Special Forms of Bearings.......... 74
B.—THRUST BEARINGS.
Step Bearings...................... 75
Wall Step Bearings................. 75
Independent Step Bearings.......... 76
Thrust Bearings with Wooden Sur-
faces.......................... 76
Multiple Collar Bearings........... 77
Examples of Thrust Bearings.....	78
CHAPTER VII.
SUPPORTS FOR BEARINGS.
General Considerations............. 79
Simple Supports.................... 79
Multiple Supports for Bearings.. 80
Calculation for Iron Columns....	82
Forms for Iron Columns........, ,..	84
CHAPTER VIII.
AXLES.
Various Kinds of Axles............. 85
A.—AXLE% WITH CIRCULAR SECTION.
Simple Symetrical Axles............ 85
Non-Symmetrical Simple Axles.... 86
Graphical Calculation of Simple
Loaded Axles................... 86
Proof Diagrams.................. 87
Axles Loaded at Two Points...... 78
Railway Axles, Crane Pillars-......	88
Axles with Three or More Bearings.	89
Axles with Inclined Loads.......... 9°
B.—AXLES WITH COMBINED SECTION.
Annular Section..................   9°
Axles with Cruciform Section....	90
Modified Ribbed Axle............ 91
Compound Axles for Water Wheels. 91
Construction of Rib Profiles....	91
Wooden Axles...............«.... 92
CHAPTER IX.
SHAFTING.
Calculation s f or Cy lin drical Shaf ting 92
Wrought Iron Shafting.............. 93
Line Shaf ting..................... 93
Determination of the Angle of Tor-
sion......................... 93
Journals for Shafting—Round Rolled
Shafting....................- 94
Combined Sections—Wooden Shaft-
ing........................;. 94
Shafting Subjected to Deflection,..	94
CHAPTER X.
COUPLINGS.
Various Kinds of Couplings.	95
I.	Rigid Couplings........... 95
II.	Flexible Couplings........ 96
Various Kinds of Flexible Couplings 96
Couplings for Lengthwise and Par-
allel Motion............*.......... 96
Jointed Couplings.................. 97
III.	Clutch Couplings.......... 98
Toothed Clutch Couplings........... 98TABLE OF CONTENTS.
XVII
Friction Clutches............ 99
Automatic Couplings.......... ioi
CHAPTER XI.
SIMPLE LEVERS.
Journals for Levers ............ ioi
Cast Iron Rock Arms............. 102
Rock Arm Shafts ...............  102
Lever Arms for Rectangular Sec-
tion....................... 103
Lever Arms for Combined Section.. 103
CHAPTER XII.
CRANKS.
Various Kinds of Cranks.........104
Single Wrought Iron Cranks..... 104
Graphostatic Calculation of Single
Crank...................... 104
Cast Iron Cranks............... 105
Return Cranks.................. 105
Graphostatic Calculation of Return
Crank...................... 105
Simple Crank Axle.............. 106
Multiple Crank Shafts.......... 107
Locomotive Axles............... i°7
Hand Cranks.................... 109
CHAPTER XIII.
COMBINED LEVERS.
Various Kinds of Combined Levers. 110
Walking Beams.................
Scale Beams...................
CHAPTER XIV.
CONNECTING RODS.
Various Parts of Connecting Rods.
Connections for Overhung Crank
Pins......................
Stub Ends for Fork Journals...
Connections for Neck Journals.
Round Connecting Rods.........
Rods for Rectangular Section......
Channeled and Ribbed Connecting
Rods......................
Cast and Wrought Iron Rods.
CHAPTER XV.
CROSS HEADS.
Various Kinds of Cross Heads..
Free Cross Heads..............
Cross Heads for Link Connections.. 119
Cross Heads for Guides........ 119
Guides and Guide Bars......... 121
CHAPTER XVI.
no
in
112
112
114
114
116
116
117
118
118
119
FRICTION WHEELS.
Classification of Wheels...• • • • •
The Two Applications of Friction
Wheels......................
Friction Wheels for Parallel Axes..
Friction Wheels for Inclined Axes..
Wedge Friction Wheels.........
Special Applications of Friction
Wheels......................
Roller Bearings..................
CHAPTER XVII.
122
123
123
124
125
126
126
TOOTHED GEARING.
Classification of Gear Wheels..... 127
A.	The Construction of Spur
Teeth...............
General Considerations............. 128
Pitch Radius, Circumferential Di Vis-
ion .......................... 128
Table of Radii of Pitch Circles.... 128
General Solution of Tooth Outlines 129
The Aetion of Gear Teeth........... 129
The Cycloidal Curves............... 13°
The Generation of Cycloidal Curves 130
Tooth Outlines of Circular Ares--- 131
Evolute Teeth for Interchangeable
Gears........................ 13*
Pin Teeth.......................... 132
Disc Wheels with Pin Teeth........ 133
Mixed Tooth Outlines, Thumb Teeth 133
Tooth Friction in Spur Gearing---- 134
General Remarks.................... *35
B.	Conical Gear Wheels.....
General Considerations.......... 135
Construction Circles for Bevel Gears 135
The Plane Gear Wheel............ 136
C.	Hyperboloidal Gear Wheels
Base Figures for Hyperboloidal
Teeth for Hyperboloidal Gears... 138
D.	Spiral Gears...........
Cylindrical Spiral Gears........ 138
Approximately Cylindrical Spiral
Gears....................... 139
Spiral Gear Teeth and their Friction 140
Spiral Bevel Gears.............. 141
Globoid Spiral Gears............ 142
XE. Calculation of Pitch and
Force of Gearing....
Pitch of Gear Wheels, Tooth Sec-
tion......................... 144
Pitch and Face of Hoisting Gears.. 144
Table of Cast Iron Hoisting Gears. 145
Pitch and Face of Gearing for Trans
mission..................... 145
Examples and Comments........... 147
F. Dimensions of Gear Wheels
TheRim........................   147
The Arms of Gear Wheels......... 149
Table of Gear Wheel Arms........ 149
Gear Wheel Hubs................. 150
Weight of Gear Wheels........... 150
CHAPTER XVIII.
RATCHET GEARING.
Classification of Ratchet Gearing... 150
Toothed Running Ratchet Gears... 150
The Thrust upon the Pawl........ 152
The Sliding Flanks.............. 153
Spring Ratchets, Quadrants...... 153
Methods of Securing Pawls, Silent
Ratchets..................   153
Special Forms of Ratchet Wheels.. 154
Multiple Ratchets............... 154
Step Ratchets................... 155
Stationary Ratchets............. 156
Ratchets of Precision........... 157
General Form of Toothed Ratchets. 158
Dimensions of Parts of Ratchet
Gearing..................... 158
Running Friction Ratchets....... 158
Release of Friction Pawls....... 161
Stationary Friction Ratchets.... 161
Releasing Ratchets.............. 162
Checking Ratchets............... 163
Continuous Running Ratchets----- 164
Continuous Ratchets with Locking
Teeth....................... 165
Locking Ratchets................ 166
Escapements, Their Varieties.... 167
Uniform Escapements............. 167
Periodical Fscapements.......... 169
Adjustable Escapements.......... 170
General Remarks upon Ratchet
Mechanism................... 171
CHAPTER XIX.
TENSION ORGANS CONSIDERED AS MACHINE
ELEMENTS.
Various Kinds of Tension Organs.. 172
Methods of Application........... 172
Technological Applications of Ten-
sion Organs................... 177
Cord Friction.................... 177
Ropes of Organic Fibres.......... 178
Wire Rope........................ 179
Weight of Wire Rope and its Influ-
ence......................... 180
Stiffness of Ropes............... 181
Rope Connections and Buffers..... 181
Stationary Chains................ 182
Running Chains................... 182
Calculations for Chains.......... 183
Weight of Chain.................. 183
Chain Couplings ................. 184
Chain Drums and Sheaves.......... 185
Ratchet Tension Organs........... 185
CHAPTER XX.
BELTING.
Self-Guiding Belting............. 186
Guide Pulleys for Belting........ 186
Fast and Loose Pulleys........... 188
Cone Pulleys.................... 189
Cross Section and Capacity of Belts. 190
Examples of Belt Transmission.... 191
Belt Connections................. 191
Proportions of Pulleys.......... 193
Efficiency of Belting........... 194
CHAPTER XXI.
ROPE TRANSMISSION.
Various Kinds of Rope Transmis-
sion........................ 194
A.	Hemp Rope Transmission. 194
Specific Capacity, Cross Section of
Rope....................... 195
Sources of Loss in Hemp Rope
Transmission............... 195
Pressure and Wear on Hemp Rope. 196
B.	Cotton Rope Transmission. 196
C.	Wire Rope Transmission.... 196
Specific Capacity, Cross Section of
Rope....................... 196
Influence of Pulley Diameter..... 197
Deflecti on of Wire Ropes....... 198
Tightening Driving Ropes........ 200
Short Span Cable Transmission---- 200
Transmission with Inclined Cable.. 200
Construction of the Rope Curve.. •. 202
Arrangement of Pulleys.......... 202
Construction of Rope Pulleys..... 202
Construction of Pulley Stations.• 204
Efficiency of Rope Transmission.. . 205
Reuleaux’s System of Rope Trans-
mission ........................ 206
CHAPTER XXII.
CHAIN TRANSMISSION. STRAP BRAKES.
Specific Capacity of Driving Chains 211
Efficiency of Chain Transmission... 213
Intermediate Stations for Transmis-
sion ..........................  213
Strap Brakes.................... 214
Internal Strap Brakes........... 316
CHAPTER XXIII.
PRESSURE ORGANS CONSIDERED AS MA-
CHINE ELEMENTS.
Various Kinds of Pressure Organs. 216
Methods of Using Pressure Orgaus. 216
Guiding by Pressure Organs...... 216
Guide Mechanism for Pressure Or-
gans......................... 217
Reservoirs for Pressure Organs--218
Motors for Pressure Organs...... 219
A.	Running Mechanism for
Pressure Organs.......
Running Mechanism operated by
Weight...................... 219
Running Mechanism Operated by
Impact...................... 219
Running Mechanism Operated
against Gravity.. .........  221
Running Mechanism in whieh the
Motor is Propelled.......... 222
B.	Ratchet Mechanism for Pres-
sure Organs...........
Fluid Running Ratchet Trains....223
Fluid Trains with Stationary Rat-
chets. ... .................. 225
Escapements for Pressure Organs.. 226
A.	Unperiodic Escapements for
Pressure Organs.......
Fluid Escapements for Transporta-
tion ........................ 227
Hydraulic Tools................  228
Pressure Escapements for Moving
Liquids...................... 228
B.	Periodical Pressure Escape-
ments ................
Pumping Machinery............... 229
Fluid Transmission at Long Dis-
tance....................... 233
Rotative Pressure Engines....... 233
Valve Gears for Rotative Engines.. 234
C.	Adjustable Power Escape-
ments ................
Adjustable Pump Gears............236xviii
Adjustable Gears for Rotati ve Mo-
tors........................ 237
D. Escapements for Measure-
ment of Vol a.ne......
Running Measuring Devices....... 239
Escapements for Measurement of
Fluids.....................  239
Technological Applications of Pres-
sure Organs................. 240
CHAPTER XXIV.
CONDUCTORS FOR PRESSURE ORGANS.
Formulae for Cast Iron Pipes.... 242
Weights of Cast Iron Pipes...... 242
Pipes for High Pressures........ 242
Wrought Iron and Steel Pipes.... 243
Steam Pipes..................... 245
Pipes of Copper and other Metals.. 246
Resistance to Flow in Pipes..... 246
Methods of Connecting Cast Iron
Pipes.....................   248
Connections for Wrought Iron and
Steel Pipes................. 249
Connections for Pipes of Lead and
other Metals.................251
Flexible Pipes ................. • 252
Pistons......................... 252
Plungers and Stuffing Boxes..... 253
TABLE OF CONTENTS.
pistons with Valves............... 255
Piston Rods.....................   255
Specific Capacity of Pressure Trans-
mission systems............... 255
Ring System of Distribution, with
Pipe Conductors............... 256
Specific Capacity of Shafting..... 257
Specific Value of Long Distance
Transmissions................. 259
CHAPTER XXV.
RESERVOIRS FOR PRESSURE ORGANS.
Various Kinds of Reservoirs....... 260
Cast Iron Tanks .................. 260
Riveted Tanks .................... 260
Tanks with Concave Bottoms........ 262
Combination Forms for Tanks....... 264
High Pressure Reservoirs, or Ac-
cumulators.................... 264
Steam Boilers..................... 265
Boiler Details subjected to Intemal
Pressure...................... 266
Boiler Flues subjected to External
Pressure...................... 269
Future Possibilities in Steam Boiler
Construction.................. 270
Reservoirs for Air and Gas........ 272
Other forms of Storage Reservoirs. 273
CHAPTER XXVI.
RATCHETS FOR PRESSURE ORGANS, OR
VALVES.
The two Divisions of Valves...... 273
A.	Lift Valves.............  274
Hinged or Flap Valves............ 274
Round Self-Acting Valves......... 275
Unbalanced Pressure on Lift Valves 277
Closing Pressure of Self-Acting
Valves......................  27S
Mechanically Actuated Pump Valves 278
Valves with Spiral Movement...... 279
Balanced Valves ................. 279
B.	Sliding Valves...........
Rotary Valves and Cocks.......... 281
Gate Valves for Open and Closed
Conductors................... 282
Slide Valves....................  282
Balanced Slide Valves............ 285
Fluid Valves..................... 287
Stationary Valves................ 289
Stationary Machine Elements in
General...................... 289
SECTION IV.
Mathematical Tables.......... 291-301
ERRATA.
Page 14, Case IV., first panel of table should read P=4 7t2 ijf-
Page 15, line 13 from bottom, second column, omit words	thick.”
Page 53, line 31 from bottom, second column, read ‘‘ interpolated diameter/' instead of ‘4 interpolated meter. ”
Page 61, formula (89) substitute P, instead of p.
Page 61, line 11 from top, second column, after ‘‘ Proportions of Journals/' insert the formula number (93).
Page 63, line 39 from top, first column, after “ Formula for Fork Journals" insert the formula number (98).
Page 64, the formulse on lines 12 and 14, of § 96, should be numbered respectively (99) and (100).
Page 64, line 33 from top, second column, for “Prowny ” read “Prony.”
Page 89, line 17 from bottom, first column, for 85 mm.=8%" read 85 mm. =3%"-
Page 89, illustration at the bottom of second column, the diagram to the left should be Fig. 409, and that to
the right, Fig. 410.
Page 97, line 16 from bottom, second column, for <£drawn ’’ read “driven.”
Page 103, the last formula on first column should be numbered (154) instead of (155).
Page 144, formula at bottom of first column, the cube root sign applies to the whole of the second member
and not to the numerator only, as printed.
Page 175, line 17 from bottom. second column, for Harturcn, read Hartwich.
Page 195, line 29 from top, second column, for “can only be given by indeterminate results,” read “ can
only give approximate results. ”
Page 206, title of § 301 read Reuleauxs instead of Reuleux s.
Page 255, example in second column, for 4 in. stroke, read 40 in. stroke.
Page 263. The following revisions of formulae (385) and (386) have been commumcated by Prof. Reuleaux
and should be inserted :
-■r!	I
•(385)
•(386)THE CONSTRUCTOR:
A HAND-BOOK OF MACHINE DESIGN.
BY F\ REULBAUX.
Section I.— STRENGTH OF MATERIALS.
11.
Introductory.
The study of the strength of materials ultimately depends upon
the question of the resistance which rigid bodies oppose to the
operation of forces, and the following definitions are to be
noted :
SuperFiciae Pressure is the pressure upon a unit of surface.
Tensiee Strength is the resistance per unit of surface,
which the molecular fibres oppose to separation.
Modueus oe Resistance is the strain which corresponds to
the limit of elasticity, compression and extension, each hav-
ing a corresponding modulus.
Modueus of Rupture is the strain at which the molecular
fibres cease to hold together.
# Modueus of Beasticity is the measure of the elastic exten-
sion of a material, and is the force by which a prismatic body
would be extended to its own length, supposing such extension
were possible.
Theoreticae Resistance is the force which, when applied to
any body, either as tension, compression, torsion or flexure,
will produce in those fibres which are strained to the greatest
extent a tension equal to the modulus of resistance ; or, in
other words, it is the load which strains a body to its limit of
elasticity.
The Practicae Resistance often improperly termed merely
Resistance, is a definite but arbitrary working strain to which a
body may be subjected within the limits of elasticity.
The CoefficienT of Safety is the ratio between the theo-
retical resistance and the actual load, or, what amounts to the
same thing, the ratio between the elastic limit and the actual
tension of the fibres.
The Breaking Eoad is that load which causes in those fibres
which are subjected to the greatest strain, a tension equal to
the modulus of rupture ; in every case this is equal to the force
necessary to tear, crush, shear, twist, break, or otherwise de-
form a body.
The Factor of Safety is the ratio between the breaking load
and the actual load.
As a general rule, for machine construction, the Coefficient
of Safety may be taken as double that which is used for con-
struction subjected to statical forces. Circumstances may also
require it to be taken as either greater or less than the custom-
ary value, sometimes even narrower than is permitted for stati-
cal forces. Care must be taken never to permit a material to be
strained in use to its theoretical resistance ; although, indeed,
there are some materials, such as wrought iron, which have
been strained slightly beyond the limit of elasticity, without re-
ducing the breaking load, or causing any apparent injury. (See
?2.)
The determination of the breaking load, and consequently
the use of the modulus of rupture, is limited to those cases in
which the actual breaking of the structure must be considered ;
but for the actual calculations of working machinery the modu-
lus of resistance, or limit of elasticity is of primary importance.
§ 2.
COEFFICIENTS OF RESISTANCE.*
The coefficients given in the following table are selected as the
mean of tnany experiments upon the various materials named.
Under the title “Wood ” is given an average value from ex-
periments made with oak, beech, fir and ash. Those materials
which show the greatest difference between the modulus of
rupture and the limit of elasticity also possess in the highest
degree the property of toughness.
. * Throughout the original work all dimensions and quantities are given
in the metric System, but these have been transformed into English units
forEnglish readers, except in the following table, where both are given.
. F________ ,»n/uSuL iron snow tnat a strain beyond
the limit of elasticity, if not carried too far, although it wTill
cause a permanent deformation, will not lower the modulus of
elasticity, but will raise the modulus of resistance.
For example, a rod of wrought iron, subjected to a tensile
strain of 28,400 lbs. per square inch, was subsequently found to
have its limit of elasticity raised from 21,300 lbs. to 28,400 lbs.
(This property is utilized in drawing wire).
Tenacity is a particularly desirable property for a material
of construction, and it may generally be approximately meas-
ured by the ratios K : T and Ki : Tt.
If the rod above referred to be subject to compression it
will return to its former limit of elasticity.
Tabee of Coefficients.*
Material.
Wrought Ircn
Iron Wire . . .
Sheet Iron
Cast Iron .
Spring Steel (hardened)
Copper (hammered)
Copper Wire ....
Brass Wire............
Bell Metal (bronze) . . .
Phosphor Bronze ....
Aich Metal............
Eead..................
Wood.................
Hemp Rope (new) . . .
Hemp Rope (old) ....
Belting...............
Granite...............
Eimestone .......
Quartz................
Sandstone.............
Brick.................
Eimestone Masonry . .
Sandstone Masonry . .
Brickwork.............
Modulus	Modulus of Re- sistance.	
of Elasticity. E. '	Tension. T.	Com- pres- sion. Ti.
20,000	15	15
28,400,000	21,300	21,300
20,000	30	
28,400,000	42,600	
17,000 24,140,000	—	
10,000	7-5	15
14,200,000	10,650	21,300
20,000	50 to 70	
28,400,000	70 to 90,000	
. 20,000	25	
28,400,000	35,5oo	
1 30,000	65 to 150	
42,600,000	90 to 200,000	
11,000 15,620,000	2-5	
	3,550	
13,000	12	
18,460,000	17,040			
6,500	4-8	
9,230,000	* 6,816	
10,000	13	
14,200,000	18,460	
3,200	9	
4,544,000	12,780	
		15	
500	21.300 T5 21.300 1	
710,000	1,420	
11,000 1,562,000	2	1.8
	2,840	2,556
250 (?)	5 (?)	
355,ooo	7,100	
5°(?)	i(?)	
71,000	1,420	
15 to 20	1.6	
20 to 30,000	2,272	
—	—	
—	—	
	—	—
—			
—	%		
—	—	—
_	—	
Modulus of
Rupture.
The upper figures are kilogrammes per
rer figures are pounds per square inch.
Ten-
sion.
K.
40
56.800
70
99,400
32
45,440
11
15,620
80
113.600
80
113.600
100
142.000
30
42,600
40
46.800
12
17.040
50
71.000
*3
18,460
36
51 120
75
06,-500
*-3
1,846
9
12,780
12
17.040
5
7,100
2.9
4,118
Com-
pres-
sion.
Kl
31,240
63
38,46o
99,400
110
156,200
5
7.100
5
7.100
8
11,360
5
7.100
12
17,040
7
9,940
0.6
85*
5
7.100
i-5
2,130
0.4
568
square millimetre, and the2
THE CONSTRUCTOR.
2 3-
Resistance to Tension and Compression.
A body is said to be under tension wben the action of a force
P, tends to extend it in the direction of its length. When the
force acts in the opposite direction the body is said to be under
compression ; but when the length is great in proportion to the
cross section, a combined action occurs. (See $ 16.)
het q be the cross section of the member : .S, the strain due to
the action of the force P ; then if we neglect the weight of the
material we have :
P=Sq	(I)
♦Example. A rafter exerts a horizontal thrust of 22,000 pounds, which
must be borne by a rod of circular cross section. If we make .S1 = 7100
pounds we have for the diameter of the rod d,
Sq — 7100—d2 =* 22,000
4
from which d = 1.98" say 2".
The principal action which the application of a force to a
member produces is the consequent elongati011 or compression.
A prismatical body subjected to the action of a force P’ will
have its original length l increased by in amount A, determined
by the formula
iL—JL	(2)
l ~ E
and this holds good as long as .S* is not greater than the modu-
lus of Resistance for tension T. This relation is also true for
compression, in which case the limit depends upou the modu-
lus of resistance Tx for compression.
Example. Suppose the rod, whose diameter was determined in the pre-
ceding example, to have a length of 114 ft. 10 in. or 1378 inches, its elonga-
tion under those conditions would be
^	1378 X 7100
28,400,000
0.3445 in. say—in.
32
The preceding formula (2) is a fundamental one, and upon it
is constructed the whole systematical study of the strength of
materials.
Formula (1) is of use when a section is strained beyond the
limit of elasticity, as by it we may determine the force required
to rend or crush a material, using the proper Modulus of Rup-
ture.
Example. The force necessary to pull the above given rod
asunder is
SHAPE.	EQUATION. REMARKS	
	II ^ | ^5	P, is distributed un* iformly throughout the whole length of thefigure. Cross sec- tion circular. Profile parabolic. Approxi- mate form, a trun- cated cone with end diameter = — 2
	II	
	y d 1	P, is uniformly de- creased from above downward. Cross section circular. Form conical.
j	y r-k 	X	The body is strain- ed by its own weight, y being the weight of a unit of volume. The cross section in- creases with the in- creasing load in the logarithmic propor- tion given.
	P js q=5^ | e = 2.718 = Base of natural logarithms. log q — log ^ + o-434^	
«5-
P=K q
TT
or	P= 56,800 X (2)2 — = 178,442 lbs.
2 4-
Bodies of Uniform Strength.
By bodies of uniform strength are meant those in which the
shape is so made that the cross sections at various points are
subjected to the same strain S, and consequently a proportion-
ally economical distribution of material secured.
Such forms are not often employed in practice, although ap-
proximate shapes may often be adopted, but they serve in many
cases to determine the general style of a structure, and give it
the effect of proportional strength without adhering too closely
to the exact form. These forms will be found of value to the
designer for both reasons : principally as a guide to the style of
his work rather than for close determinations of economy.
If a designer has become thoroughly familiar with the resist-
ing capacity of various shapes, and can keep them so clearly in
his mind that he can perceive the general form of the proper
curve to be used in any particular case, he will be able to pro-
duce, with an artistic freedom, designs which will approach the
shapes indicated by mathematical analysis.
The following forms are alike suitable for tension and com-
pression. As examples of their practical use, the first two are
applicable to cast columns, and the third is suitable for chim-
neys of masonry as well as for high piers of bridges and via-
ducts.
♦In all cases the quantities given in the original examples have been
converted into their English equivalents, which will account for the ua*
usual quantities chosen. (Trans.).
RESISTANCE TO shearing.
A body is said to be subjected to a shearing strain in any
cross section when the distorting force acts in the plane of that
cross section.
Let q, be the sectional area, and 5, the force acting upon it,
so that we have as in the case of tensile and compressive
strains
P=Sq	(3)
The limit of elasticity will be reached when .S==§ of the lesser
of the two Moduli of Resistance of the material, in the case of
wrought iron, where T =T1 = 21,300, S = 17,040 lbs. while for
cast iron T<Tj and = 10,650 and S = 8,520. In this case the
maximum strain is not in the plane of the cross section but at an
angle of 450 with it, and is f the value of S. The deformation
which two surfaces suffer under a shearing strain is very slight
within the limits of elasticity, but it is noticeable in the case of
torsional distortion where numerous sections under shearing
strain adjoin each other.
Equation (3) holds good for cases in which the division of
the adjoining surfaces occurs, such as shearing, planing, bor-
ing, in fact, the work of the entire class of machines which act
by the removal of a portion of the material upon which they
are working.
The strain of separation S, in this case is somewhat less
than the modulus of rupture for tension (K). This is due to
the fact that in the case of shearing forces both K, ani
come into action. For the calculation of such machines a
coefficient of rupture equal to 1.1 K may be taken.
2 6.
Resistance to Bending.
Elasticity and Strength of Flexure.
A prismatic body upon which the external forces act in a
direction at right angles to the axis of the prism, is said to beTHE CONSTRUCTOR.
3
subject to strains of flexure. As long as the elastic limit is not
overstepped there will exist an equilibrium in each normal
section of the prism between the moments of the external
forces on the one hand and the moments of the opposing
internal forces on the other hand, both acting about the neu-
tral axis of the given section. This may be considered as a'
sort of equator of each section since it passes through its cen-
tre of gravity at right angles to the plane in which the bending
takes place. It thus divides the section into two portions, in
one of which all the fibres which are parallel to the axis of the
prism are subiected to a tension proportional to their distance
from the neutral axis (the tension side of the section), while in
the other portion the corresponding fibres are subjected to
compression in a like proportion (the compression side of the
section). It follows that fibres which are at equal distances
from the neutral axis will be deformed to the same extent.
The resistance to bending is therefore a combination of the
resistances to tension and to compression, both acting in a
peculiar manner, that is, in rotation about an axis.
Let:
31— the statical moment of a cross section subjected to a
bending force, taken with reference to the neutral axis
of the section, that is, the so-called moment of force.
/= the moment of inertia of the section with regard to
the neutral axis.
tf=the distance from the neutral axis of the fibres which
are subjected to greatest tension or compression;
7. ethose which are farthest from the neutral axis.
*S=the corresponding strain in these fibres, then :
M=S—
a
(4)
The product ~ is the statical moment of the entire collec-
tion of fibres of the cross section with reference to the neutral
axis and is the strain moment of the section under considera-
tion. For any prismatic shape subjected to a bending force P\
whose lever arm is x, we may put 3f=Px for each cross sec-
tion, and for each section this will have a different value. ^ The
section at which the product Px has its greatest value is the
section of danger, and the bending force P\ which produces in
it the strain S, is the resisting strength of the shape for the
strain S, so that in this case we have
SJ
P=-
Xm&
(5)
in which xm, is that value of x, which makes Px a maxi-
mum.
The axis passing through the centres of gravity of succes-
sive cross sections of a figure is not subject to any change
of length under flexure, but is only curved, and to find its
radius of curvature under any load we have the formula
(6)
This curve is called the elastic curve, and is determined by
thegeneral equation
d2y = M_
dT* JE	(7)
> In the following tables are given the values of the quan-
tities for calculations of flexure under the various conditions
shown in the figures, being :
The Bending Moment 3f, for any point x,
The Bending Load Py according to formula (5),
The Co-ordinates xy y, for the elastic curve,
The value,/, of the abscissa^, at the point of application of
the force as shown in Figures I to VI, and the value of the
greatest deflection / in the cases shown in Figures VII to
XIII.
In all the examples the weight of the beam itself has been
neglected, as this may usually be done in machine construc-
tion, although not in bridge work.
Figures VII to X are suitable for the latter purpose, as
in them the weight of the beam may readily be taken into
account.
In Figures XI and XII may be seen how unequal distri-
bution of the load affects the sustaining power, as a beam
loaded like Fig. XI or XII has 1 times the sustaining power
of one loaded like VII or VIII, or for the same load a corres-
pondingly reduced deflection. These are important considera-
tions in connection writh the distribution of weights in buildings.
The distribution in XIII is also unfavorable, as it has only ^
the sustaining power of VIII, with a greater deflection.
It is to be noticed that the deflection f increases as the third
power of the length, and that it varies greatly under the various
conditions given.4
THE CONSTRUCTOR.
Example.
Bending Moment M.	Sustaining Power P.	Equation of Elastic Curve.	Deflection yi	Remarks.
	P= 8 l a	P P [-.r* 4 /£ 16 Li» 3	/-JLJL JE 191	Weakest Sections ac B and C.
For A B M = Pe	SJ ca	y=*f- P + y/ P2 _ ^2 + i. ) in which r-H Pc	. P P a J J E 8 l	Weakest Section at an in- determinate point be- tween A and B.
M= Lll 2 /	p=*EL la	P P \X J_X41 JE 6 L / “ 4 /4J	f=±- ‘2 ■ je a	Weakest Section at B.
	P = 8 SJ- l a	[£._ .i + i41 JE24 L/	f_ P % JE 384	Weakest Section in the Middle.
—¥(M)	P= 8 l a	_ P /3 r .r ^ ^ /i? 48 L/ 3 /3 + /4J	/=JC±. JE 192	Weakest Section at C. Greatest Deflection wheo + '/«) Reaction at X = % P Turning point x = % l
»-"(1-1*4)		, _ * r~ - =■ - + -i ^ yA 24 |yj p p J	y= —— J JE 384	Weakest Section at B, Point of reversal at
, r P X X* M 		 3 *		P P fx 1 jrB“| ^ JE 12 [/ 5 /5j	J JE 15	Weakest Section at 2?.
»-'Hi-j*’i4)	p-^41 la	P /3 ["3 .r ^ 2 jr5"! ^ ^ 12 L8 / k/3 + i /5 J	P 3P J JE 320	Weakest Section in the Middle.
	P-*6^~ l a	P P [5X -r3 , 2 jr^l “/.E 12 L» / ^ + 5 t& J	P P f~~JE 60	Weakest Section in the Middle.
XIV. In the case of a beam supported upon two symmetrically placed supports A and B, and carrying a
P X f X	c\
uniformly distributed load Py we bave for the bending moment M = ——	— 1 + ~ J.
The supporting power varies according to the position of the supports, and also with the relation of c to /;
it will become a maximum when c = 0.207 l j^that is, l ~THE CONSTRUCTOR.
5
The supporting power will tnen approximate to
p=x1ii'
or nearly six times as great as in case VIII, showing the ad-
vantage of this method of support. The weakest sections are
at Ay B and C.
2 7-
Table of Sections.
j
The value of— in equation (4) depends almost entirely upon
the shape of the cross section of the beam, and this we shall
hereafter call the Section Modulus.
The following table shows a large number of sections in use
for vaiious purposes, and gives the corresponding values of the
following quantities:
The equatorial moment of inertia J, for the neutral axis,
shown in the figures by the dotted line.
The greatest distance a, of the fibres under tension and com-
pression, or their separate values a', and a/','when the section
is not symmetrical about both axes.
The equatorial section modulus Z = for which two values
are given, when a', and a", are different; and
The sectional area of the figure, which will be found of
Service in calculating weights.
To determine the value of a, experimentally or graphically,
a model of the section may be cut out of cardboard, and its
centrc cf gravity found by balancing on knife edges, or else the
graphostatic method given in § 46 may be employed.
The following example will serve to show the application of
the table:
Example. Required the moment of inertia of a circular section 4 inches
in diameter. Acccrding to No. XX in the table:
/ ■= 7- d^ — 0.0491	— 0.0491 X 256= 12.5706
04
By making various combinations of the forms given in the
tables other sections may be obtained to which the same
formulae will apply. As an example, the Section No. VIII may
also be used for a rectangular tube, and No. XI for an E shaped
section.
It is a matter of some importance for the designer to keep in
mind some general conclusions, which may be drawn from the
tables as to the influence of various shapes upon the strength.
It will be plainly seen that the depth of a section is the dimen-
sion which has the greatest influence upon the strength, and
also that those portions of the section which are furthest re-
moved from the neutral axis are of the most Service.
It is upon this point that the peculiar strengthening effect of
ribs depends, and which makes their use so advantageous in
cast iron constructions. These ribs do not act so much by the
mere strength of their own cross section as by the fact that
they strengthen those portions wdiich are furthest from the
neutral axis. This is a feature to be carefully wratched, and its
importance may be made ciear by an example.
If we take a section of the form given in No. XV., and
make its dimensions as follows : b = 8 b\, h = 12 b\, hi, = 11
b\ (Fig. J, 2 9) and then divide it into two rectangular parts by
a horizontal section, we have for the modulus of each section :
~* b'. = 2oy6 V and 8 b*
6 6
which, together, give 21.5 ^l8.
The same material, when taken as a whole, in a single sec-
tion (see § 9) would have a modulus Z = 34.8 £i3, so that it
has more than \ x/2 times the resistance of its separated por-
tions, and as a matter of fact the right angle rib or T head is
about ten times the value in that connection than if taken by
itself. This is also found in a stili higher degree in sections of
other shapes.
SECTION TABLE.6
THE CONSTRUCTOR.
SECTION TABLE—(
Moment of Inertia J.		Distance <2.	Section Modulus Z.	* Area F.
1 + 6^ 64 ~ 0,638 34		0.924 3	0.677	2.82832
bh*—{b — bx) Aj3		h	b h* — (6 — <H)	3A — (3 — b$hx
12		2	6 h	
3 (/zS-AjS) +^(^3_A,3)		h	b (^3 — Aj3) + bx (Aj3 — /z23)	
12		2	6h	
b h* + ^^3 12		A 2	b h? + bx hf 6 h	3 A -+■ bx h\
3£3_ (3 — ^0)^3 + ^^8		h	bk* — (b — b2) + bx A2s	bh — (3—32) -i-3i
12		2	bh	
3 A3 _|_ (/^ _ £) +(A — Ai)		h_	bh3 + (Aj — 3) hx3 + (A — Aj) 33	33 + (31-3)31+(^-31)3
12		2	bh	
bh* 36		, A « =— 3 “" = 7*	z- 12 z„ = 3^ 24	bh 2
^» + 4^^ +^2 36(3 + ^)		_ 3 + 2 ^ h b + 3 _ 2 £ + h ^ -Mi 3	7/ _ + 4 3 <5i + ^i2ro «(* + **.) 7tt _ ^ + 4 £ + ^12_ 12 (2 b + 3!) h	2
			 3 ^2“ + ^1 ^1 + ^2)	Z' = Z.	
i-[^(a-3-/S)+^(/3 + a"3)]		2 [£ h — (b — 3j) /*i] a" = h — a'	Z" = ./ an	32^2 + 3^2
j +^(/3+^) +^(a"3.	-ir3)]	Determined graphically or by experiment.	z- l a' Z'=l a''	*(*-/)+*i(Ahr> + h{a"-e)
J (*'3 -/3) + <*1(/3+*3-*’3_£5) + ^ (a*3		Determined graphically or by experiment	Z’= Z a' z" — L a”	b {*’-/) + b\(f+g-i—k)+b2{a"—£ )THE CONSTRUCTOR.
7
SECTION TABLE—(Continued).8
THE CONSTRUCTOR.
?8.
VaEUE OF THE QUANTITY S.
The limit of elasticity in a deflected beam, both on the ten-
sion and compression sides, will be reached when their respec-
tive strains X become equal to the modulus of resistance. It is
therefore of great importance to select such a value for 6*, that
the modulus of resistance may not be reached on either side.
These conditions will be met for sections wdiich are symmet-
rical about two axes, by taking the lesser of the two values of
Sy as in the case of cast iron, the modulus fortension sliould be
used.
In those sections in wdiich d', differs from a", the first thing
to be determmed is the position of the tension and compres-
sion sides. Let
a = the greatest distance from the neutral axis on the tension
side.
a\ — the greatest distance on the compression side,
T = the modulus for tension.
7i = the modulus for compression,
M — the statical moment of the bending force,
m = the coefficient of safety, so that for double, triple,
safety, etc., m = 2, or 3,
Then we may take :
a	T	T	J
When—-> — then M =----------
ax	7\	m	a
a	T	7i	J
When —<T •— then M =--------
a\	T\	7U	ai
a T
When—* = then M =
ax Ti
ZZor * /
m a 7)i a\
Example.
T
For Cast Iron — =
Ti
Taking the parabolic section No. XXIV. a=$ h, ax f h.
This gives —= §, so that -E—	-E, and for S, we hav* TC or.
Cl\	CL\	tft
10,650	IO.65O	...
—5— and M = ---------— 0.114 bfi2.
m	m
With wrought iron, in which 7'= Tx no investigation is necessary.
Sections of Uniform Resistance.
b In order to use the material to the greatest advantage to re-
sist bending strains, it is necessary to pay especial attention to
its distribution, particularly in those portions which are furthest
from the neutral axis. The best economy is attained in this
matter when the section is shaped so that the strains on both
tension and compression sides shall reach the elastic limit
simul taneously.
For this purpose it is necessary to make
E -T.
a\ T
Sections which are thus proportioned are known as Sections of
Uniform Resistance. Wrought iron sections which are sym-
metrical about two axes fulfT these conditions, since T = Tv
For cast iron, wThen the bending strain is exerted constantly in
one direction, it is best to make ax — 2 a, for Tx = 2 T.
Taking these conditions into considerat ion, the following
sections (Figs. 1, 2, 3) have been drawn, in which b and bx may
have any desired proportion to each other :
J =	278 bk	44o£*	992<54
z =	34-86’	55	02.4^
F =	I9^2	25P	40.8 b2
0 =	I	0.97	.04
The tension side is nearest to the neutral axis. Since the
section modulus is determined from the value of JXisahvays
ai
to be taken as —L. F is the area of the section, and <f> is the
7)1
proportional economy of material, the cross section of Fig. 1
being taken as unity.
The value of y> may be determined thus :
_ ft Mi f	(8)
in which the sub-numbered letters belong to the required sec-
tion and the un-numbered letters to the given section whose
economy is to be taken as unity. In this equation F— fib2,
Z = ab* and .S is taken equal to except when the ratio of a
to ax is not the same for both sections. It will be seen from
an examination of (8) that a slight variation from the exact
proportions is not very material. When the bending force acts
alternately in opposite directions, so that the strains are re-
versed, the sections which are =ymmetrical about two axes are
the best for cast iron as wcii as for other materials, and the
smaller value for Sx should always be taken under such circum-
stances. If the force is constantly changing its direction, so
that the neutral axi passes through the centre of gravity, the
most economical section is that of a circular riug, its resistance
being greater than the cruciform or star-shaped sections, silch
as X., XII. and XX'/,, Table $ 7, since there is in the former
case a constant prop jrtion of the section and the greatest dis-
tance from the plane of the bending.
Example. A projecting beam of cast iron loaded as in Nc. I., § 6, carries
a weight P-~ 5,500 pounds at its extremity, the length being 78.75 inches.
Taking the cross section of the shape Fig 2, we have by equation (4):
M = SZi
M = 5,500 X 78.75* z = 55^
To obtain double security we take 5T = 2I’^— = 10,650. This gives:
5,500 X 78.75 = 10,650 X 55^3
t = 7 5,500 X 78.75 = o.9»
v 10,650 x 55
The sectional area will then be 25 (0.9)2 = 20.25 sq. in., as de-
termined by the constant given for the section Fig. 2. If the
security be taken at i£,
21,300
S =	— = 14,200.
This gives a lighter beam, and according to equation (8) its
weight would be ^	? = 0.825 of the preceding.
2 10.
Bodies of Uniform Resistance to Bending.
A body is said to offer uniform resistance to bending when its
shape is so chosen that in ‘all sections of its length the maxi-
mum strain, S, for tension or compression has the same value,
and the general forni of equation (4) for such bodies is
= Constant.	(9)
Bodies shaped so as to oppose a uniform resistance to bending
are frequently used in machiue construction, approximations to
the exact forms being often adopted, examples having already
been shown in § 4. A variety of such shapes are given in the
following table.
The deflection in bodies of uniform resistance is of necessity
greater than in prismatic bodies of the same strength. In many
of the examples of the following table the deflection, f, is
given, and in I. it is double, and in V. ij times what it would
be in prismatic bodies similarly loaded.
The elastic line for the following bodies, when exactly formed,
is determined from the following equation :
d2y __ Mp	do	(10)
dx2	E Jo	cix
in which
M0 == the moment of the bending force for any given sec-
tion,
J0 = its moment of inertia,
d0 = its greatest fibre distance,
dx = the greatest fibre distance on the same side as d0 for
any other section at a point x.
For the radius of curvature, p, of the elastic curve at a point
whose co-ordinates are x, y, we have :
P = AA a*	(”)
Mo CLo
w^hich value is constant, and represents a circular arc when
dx = d0 ; that is, when the section is of uniform height at all
points, as in V., X., XIV.THE CONSTRUCTOR.
9
No.	Form.	Application of Load.	Equation.	Sustaining Power.	Volume.	Remarks.
			For rectangular section			Deflection of the free end; y 2 Pt* " 3/°^; j _b h*
I.	—i—		z y- x b~A* = i 9	p S b A? 61	f ikl	
			In Cases I. and II.,			12
II.			y * a y / Parabolic truncated wedge.	p_S b A2 6 /	A bhl 3	The elastic curve is a parabola.
III		vd .2 V 1’ « £	Approximation to Form I., Truncated wedge.	p_Sbh* “ 61	3. bhl 4	Weakest section at the base.
IV.	Es?1'	V. o t-I U a u .2 c§	Approximation to Form 11., Truncated wedge.	p__S b A* 6 1	A bAl 4	In normal conditions the elastic curve bisects the angle of the wedge.
V.	rni	8 «J 4> £ £ O -O O	y-h; z X b = 7 Normal wedge.	p	S b h* 61	-L bhl 2	Elastic curve a circular arc. y_ 1 2 J0 E ; jo-tR 12
	i—	£X & t/I	z b 7 = ~h '			0 r—
VI.	is=*l	The load i	y 2>f~A~ h y / Cubic parabolic truncated pyra- mid.	p^S b A2 61	A bhl 5	y 3 f x Equation — — A] - is of great- A 1 l est importance when all the sec* tions are similar.
VII			Approximation to Form VI., Truncated Pyramid.	p_Sb h* 61	2 bAl 27	Weakest section at the base.
	Lr— "Tj*					
VIII.			Truncated cone; approximating to the form given by equa- tion h y	P^ S TT cfi 32 l	1* 7Tld* xo8	To obtain the same strength as in Forms I. to VII., make d _ 3 Ti y 37T A
	:		For rectangular section, z y* x2			
IX.	re#!7	vo cw «H O 4) •s H	b h* fi , y x Wedge.	p S b A 2 3'	A bhl 2	A very useful form when the sharp end is removed.
X.	Itl	ded, as in Case VII., ir	y — ^ • ■* Sf z / V 1 Parabolic-sided wedge.	p_Sbh* 3*	JL bhl 3	Elastic curve a circular arc. /= 1 PlZ 4 Jo E' Jo— Uti 12
XI.		ri JD S* 0 E «2 P	z b^ y h y fi Truncated pyramid on semi- | cubic parabola. i 		p==a S b A* 3/	A ihi 7 .	Applicable to stone brackets, etc., in architecture.IO
THE CONSTRUCTOR.
No.
XII.
XIII.
XIV.
XV.
Form.
Application
of
Load.
U
fc£
Equation.	Sustaining Power.	Volume.	Remarks.
Approximation to Form XI.	p_ S b k3	llbhl	Weakest section at the base.
	3 l	27	
For rectangular section			
z y- jc3 Vh *= 1* ; y- 			p_S b h*	L bhi	Fundamental shape for archi-
	2 /	5	tecture.
1 1 k*			
Wedge, on semi-cubic parabola.			
y — h\			Elastic curve a circular arc.
7-/r	p== S b h*	L bhl	, 2 Pl 3 6 J0 E'
/ T b	2 l	4	r bh3
Sides on cubic parabola.			12
z b y~ h V X			
	p== S b h*	JL bhl	Value depends upon the sim-
h /	2 l	3	plicity of the form.
Pyramid.			
The preceding are only a few of the simpler forms which
may be used, and it would be easy to multiply examples.
By altering the breadth, or height, the relations become
more or less complicated, as the case may be.
For instance, in Case I., which is based on the parabola
j- = ^£*, it may be made the biquadratic parabola, —
etc. Combination sections give rise to new forms, and a great
number of combinations may be made. Examples will be
found in the chapter on axles and shafts.
The following discussion of springs will also give some in-
stances of special forms, in which the neutral surface is irregular.
§ ii.
Resistance to Shearing in the; Neutrae Peane.
Since in a deflected beam there is on the tension side a con-
tinual tension, and on the compression side a continual com-
pression of the respective fibres, it follows that the neutral
plane is subjected to a shearing action, and this must not be
neglected in determining the width of the beam.
The lower limit permissible is indeed a matter not likely to
be reached, but at the same time it should be investigated.
Calling the least permissible width Zo, and the mean force on
either side of a given section R, then in order that the shearing
strain at the neutral plane shall not exceed a value So, we must
have :
Zo>* U	M
So 2/
in which So should in no case exceed f of the lesser modulus
of resistance of the material under consideration (see | 5). Jy
as before, is the moment of inertia of the section, i.e., the
summation of the products of the elements of the section by
the square of their distances from the neutral plane, while V is
the statical moment of the section, i.e.y the summation of the
products of the elements of the section by their distances from
the neutral plane.
For the rectangular section No. I., Table (§ 7),
U=b—,
4
and for the double T section, No. VIII.,
u= bJi* — (b — bY) hi2
4
R is to be chosen according to the case under consideration,
as, for example, in No. II. Table ($ 6) for all sections between B
p
and Cy it is equal to the reaction—, etc.
Equation (14) is not so much used to determine a value for
Zoy as to find out in any case whether the breadth of the neu-
tral plane has been taken too small. As a matter of fact, this
is a question which very seldom arises in ordinary construc-
tions, especially in mackine construction.
4
If in (14) we give zo any desired value, and make So = ^ S
we obtain
s=AiL _u
4 Zo 2/
and substituting this in equation (4) we get:
M_ 5 U	(15)
R ~
M
8 zo a
—? is the lever arm of the force R; this we may call A►
U: Zo a contains one of the height dimensions of the section ;
hence equation (15) expresses a relation between two dimen-
sions of the body under consideration. For a simple rectangu-
lar cross section, taking the value of Uy given above, in which
Zo =	b, and	a = — ; JL = ii.
2 A 5
A greater value for h must not be taken if we do not wish the
shearing strain to exceed the extension or compression in the
tension and compression sides of the beam. These considera-
tions are often of importance for the dauger section, as, for ex-
ample, in No. II., Table (§6) for the point B. In this case
1	/ = 8
A = — and we make —	— This limit of height, however,
2	/*	5
is so great that it is very rarely reached in practice.
The most important application of this principle is found in
the case of notched beams of wood, such as often occur in
building construction. In such cases the resistance of the neu-
tral plane is often very much reduced by the cutting of the
notches, sometimes to one-half what it would be in the solid
h
beam, and making a corresponding reduction in the value of ~r
For the double T section we have :
h
A
16
[>-«-7 ■)(*)']
If the brackets in the denominator contain an improper
fraction the value of JL- will approach the upper limit, but lor
ATHE CONSTRUCTOR.
ii
ali ordinary cases this value is very great. The nearest ap-
proach to this shearing action probably occurs in T beams
where the flange joins the web, but examples are very rare.
2 12.
Beams with a Common Load.
When two prismatic beams are United in the middle, and at
that point subjected to a force P, the beams being supported at
the ends, they will both be deflected, and the sum of their re-
actions P/) and P'\ enter into the support of P.
The double reactions are found from the formula in Table
(2 6), No. II., column 2, as follows :
P'	J' E' l" 3
p/7	Jff E" T73”*
and since
we get
p/ _ 4§L1' and P"
r - 4 a' l'
4 a" l"
_S'
S'
: _ e — f — v	(16)
' — E"	*
If the two beams are of the same material (E7 = E")t to ob-
tain equal security, the product—( — ^____T
a" \l// J — A*
If the beams are not the same length then a' = an > i.e., the
heights must be the same unless the breadths are equal to each
ether.
FiG 4-
a' _ /r_y =/!_V = jL
a" “ \ r / M J 9
(17)
a — the distance of the farthest elements of the section
from the centre of gravity,
X = the shearing strain in the elements at a distance a, then
d/=
a
If the body is of a uniform section, then -If is constant. Now
a
if A be the lever arm of the rotating force P, for a moment My
the weakest or danger section will be that for which M is a
maximum, and for it we have
p — Jp
Ani CL
(18)
Example. A cast iron support shaped like a cross. Fig. 4, must support a
weight, P, at the intersection. The lengths of the arras are to each other
as 3 : 2. In order to obtain equal security in the four arms, which are of
prismatic shape. we have from (16)
Hence the cross section of the short arms must be to that of the long arms
as 4 : 9, and if the arms w*ere of the same section the supporting power of
the short arms would be to that of the long arms as 9 : 4.
It also follows from the preceding, that rectangular sheet
metal plates carrying a uniformly distributed load are stronger
parallel to their shorter axis than parallel to the longer axis.
For given loads and materials formula (16) may be used to
govern the choice of dimensions and the relations of length to
breadth.
For beams of cast or wrought iron resting upon each other,
a suitable proportion may be secured by taking the sum of their
several supporting powers as the supporting power of the
combination. This is often a matter for consideration in
strengthening existing structures.
§ 13.
Resistance to Torsion.
Resisting Power and Angle of Rotation.
A prismatic body which is subjected to the action of a force
couple tending to rotate it about its geometric axis, opposes to
such action its Resistance to Torsion. Under these conditions
the elements in a normal section are subjected to a shearing
strain, and until the elastic limit is reached there exists an
equilibrium between the external rotating forces on the one
hand and the strain moments of the various elements of the
section on the other hand ; both being taken with regard to the
polar axis of the centre of gravity of the section, i. <?., the axis
passing through the centre of gravity of the section and at
right angles to it. Resistance to torsion may properly be con-
sidered a higher species of resistance to shearing, to which it
bears the same relation that resistance to bending holds to ten-
sile and compressive strength.
M = the statical moment for any given section of the
rotating force,
Jp = the polar moment of inertia of the section, i. c., its
moment of inertia taken with regard to its polar
axis (see 2 14),
in which Am> is that value of A, which gives M, a maximum.
The limit of elasticity is reached, as in the case of shearing
action, when .S = ~ of the lesser of the moduli of resistance
for tension or compression (see 2 5). This is plainly visible by
a comparison between the action of bending and twisting.
The relative rotation which two sections of a prism at a given
distance apart make with each other is called the angle of tor-
sion. It is represented by the letter ; and for two sections
separated by a distance x, we have in general ternis :
i±~JL	(i9)
dx JpG
in which G is the modulus of torsion for the material used, and
is equal to ~ of the modulus of elasticity E.
In the following table will be found the values for :
The moment A/, at a given point nr, of the prism,
The force P, according to formula (18), and
The torsional deflection in terms of angular measure, or in
other words, the angle of torsion
These quantities are given for a variety of cases, as shown in
the cuts, and from them total moment, PR, of the twisting
force may be determined. In case IV., 6* is the point of appli-
cation at which the collected forces, with a lever arm R, would
act, if concentrated to produce an equivalent resuit to the sum
of the separate eiforts, l0 being the distance of the point S from
the immovable end of the prism.
Questions relating to torsion are of varying importance in
machine construction, and come especially into consideration in
calculations relating to springs. Case IV. illustrates the condi-
tions which occur in determination of mill shafting. Cases V.
and VI. occur in machine framing.
? 14.
Potar Moment of Inertia and Section Modueus.
The polar moment of inertia, Jpy is easily determined, since
we have
Jp=J, + A (20)
in which Jx and J2 are the equatorial moments of inertia taken
with regard to two axes at right angles to each other, and whose
values are given for a variety of sections in the table of (2 7).
From this may be obtained the polar section modulus = Zp
for use in the preceding cases. An exception must be made for
those sections in which we have not Jx — J2t as in cases III.,
VII., XII., XX., XXV., etc., 2 7. For these it will be necessary
to make a special correction in the values of Jp	= Zpt to
provide for the warped surface which is assumed by the section
under a heavy torsional strain.
For a rectangle, which is a section of frequeut occurrence in
machine design, the corrected value of Jp and Zp—^~ is given
in the following table, while for the circle and the square no
corrections are necessary for the values obtained from equa-
tion 20.
Example. A cylindrical prism of wrought iron is subjected to a torsional
strain applied as in case I. of the following table. The force P=-= 1,000 lbs.,
and the lever arm R = 24" ; while the bar is 4" in diameter and 48" long.
These quantities give for S, the strain at the circumference
S= — RR =
Jp
16
i6_ rR
7T d3
1,000 X 24
= 1,909 lbs.
$ - •
3.1416	64
and to get the angle of torsion we substitute this value in the formula:
_S_ l
G
=	T»9°9	48
11,360,000	2
which corresponds to an angle of about o° 14'.
0.00412
THE CONSTRUCTOR.
Moment M.	Twisting Force P.	Angle of Torsion #.	Remarks.
M = P R for ali points between A, and B.	p S Jp aR	Jp G S l “ G a	All sections between A, and i>, are equally strong.
PR*	p_SJp a R	* = L*L*I 2 Jp G 1	5 / 2	G a	Weakest section at B.
M^PR — /a P R = the collected moment of all the twisting forces.	p_SJp aR	* - 1 P*J 3 Jp G 1 cLL 3 G a	Twisting forces decrease uni- formlyfrom B, toA. Weak- est section at B.
M = the sum of the moments within x.	p	S Jp aR	5 lo ~ G a	General form of Cases I., IL and III. Weakest section at B. The value of # in III. will be reached in IV., when lo = —• 3
In the portion c: M= P RCj In the portion C\: m=prL l	When Ci <^c p^SJp l a R c	6*1^* ^ p cH M^l^ 11 ll	The shorter portion C\ is the weaker.
M-PR(~r	P= 2 $Jp aR	87pC 1 S / “ 8 G a	Weakest points at A\ and B.
If we wish to reduce d, so that S shall be equal to one-half the modulus of
Tesistance for torsion, i. e.y — — • —— • 21,300 = 8,520 lbs., we make
2	5 _____________
d = & 16 pj? = ^l6 X 1,000 X 24 = ,.42"
7T 5"	3.1416X8,520
or about 2*^ inches.
In this case the angle of torsion would be
— . —4—- = 0.0288"
11,360,000	1.25
which gives an angle of about i° 39".
SECTION TABLE.
No.	Section.		Polar Moment of Inertia ^/p.	Polar Section Modulus, ^ Jp
I.		► L 1	-S-* 32	-4-0 10
				
No.	Section.		Polar Moment of Inertia^.	Polar Section Modulus, 7 «/i?
				
II.	__ L		£4	&
			6	3\/ 2
	u	-—i	*i		
				£2 /*2
III.	1	214 L,—4-J	I J3A3 3 J2 + /fi	3/^+ //2 Approximately P- /fi •9 	i	T- 3 (o.4<5 -f 0.96A)THE CONSTRUCTOR.
*3
§ 15-
Bodies of Uniform Resistance to Torsion.
In order to make a body of uniform resistance to torsion it is
necessary to take such sectional areas at various points as shall
make in equation (17), 5a constant, and also to take
= constant.	(21)
Jp
In case I. of the table in \ 13, for all sections M = PR, and
bence in this case the body should be prismatic in shape. For
cases II. and III. the necessary formulae are given in the follow-
ing table. For such bodies the angle of torsion is greater than
for those of prismatic shape. The angle for each is given in
the table, and is derived from the following :
dT- = TLr	(22)
d X	Jx Cr
m which Jx is the polar moment of inertia for the section taken
at the point x.
Form.
Applica.-! Equation an(j Angle of Torsion.
I'-	Case II., \ 13.
1	
1——J	
i"-	
	cn
	
BLiissBlfaiip if	d
	V 1n
1J	c$
1	
Circular section
& dL ; P R=
l
d =
S—
16
-I-
Approximate form =
cated cone, with
^±.d.
3
= a trun-
extremity
Circular section
H;PX
r —	/2
5	/
S— d*\
16
* = V
Approximate form = a trun-
cated cone, with extremity
d
For other bodies of uniform resistance to torsion, see Torsion
Springs (J 20).
§ 16.
Resistance to Buckung.
Combined Bending and Compressive Strains.
A prismatic body is subjected to combined bending and com*
pressive stresses, to which it yields by buckling, when its di-
ameter is comparatively small in comparison with its length-
Under these conditions a compression applied in the direction
of the axis is opposed, both by the resistance of the body to
compression and also to bending, with this difference, that in
this case the lever arm of the bending force is not the abscissa,
but the ordinate of the elastic curve. From this it follows that
(neglecting some very small elements) any compressive force
Py capable of producing a bending, would do so even up to the
breaking point, provided that the laws of perfect elasticity held
good until rupture occurred. This would only be true if the
theoretical resistance and the breaking load were the same, and
the elasticity of the prism held them in equilibrium until the
final yielding of the point of application of the force P oc-
curred.
In the following table (p. 14) the principal formulae are given
for a number of the most important applications of these buck-
ling stresses. In the table
P = the modulus of elasticity of the material assumed to be
of prismatic shape;
J = the least moment of inertia of its section taken with ref-
erence to a line of gravity, for example, in a rectangle of
which the greater side is b and the lesser side h} according to \ 7,
h b3
12
It may be remarked that the valuable experimental researches
of Hodgkinson, as given in his rules, show a somewhat smaller
breaking load than the formulae in the table; this, however,
does not detract from the value of the latter, since these are only
strictly correct for perfectly elastic bodies, but at the same
time they will be found practically reliable if the force P is not
permitted to exceed a definite proportion of the breaking load.
Different materials demand a different factor of safety. For
cast iron, % to x/e the breaking load, or less, and for wrought
iron the same, and for wood ^ to T^, or y1^, should be the limit.
These inequalities often arise from the fact that it is not
always easy to determine which of the applications of the
table really meets the case in question. In order to determine
the actual security from rupture, it is often necessary to make a
comparison with other existing strains. From this standpoint
the ratios of diameter to length in the following table have
been determined in order that the resistance to compression and
to buckling may be as nearly alike as possible.
In Hodgkinson’s experiments it was showm that columns
standing upon flat bases were nearly as strong as those which
were firmly fixed at one end.
In the third section many applications of these formulae will
be given.
2 17-
Coeumns of Uniform Resistance.
Columns subjected to combined compressive and buckling
stresses are said to be of uniform resistance when its various
sections are so proportioned that a very small degree of buck-
ling will produce the same strain in each section.
For case II. of the preceding table, when the section is circu-
lar, the following formulae (by Redtenbacher) may be used :
This may be separated into a double equation by making ;
which gives:
——— = -i- (2 (p — sin 2 <p)
From these equafions a limiting
curve may readily be found, whose
abscissas are those of a cycloid, and
whose ordinates are those of a
sinoide, and which may be called a
cycloidal sinoide. A method of
drawing this curve is given here-
after, in the discussion of connect-
ing rods, and the approximate shape
is also shown in the second form of
Fig. 5, in which the outline is a
circular curve, or at least a line of
very slight curvature. The strength
of these columns may be taken as
^ that of a cylindrical column of a
diameter h and length /.
I 18.
Compound Stresses.
It very often occurs that a variety
of forces act upon a body at the
same time and in a variety of ways, so that, for instance, a sec-
tion te subjected at the same time to tension and bending, or to
torsion and bending, etc.
The resistance and the maximum strains are then to be de-
termined in a different manner, according to circumstances.
In the following table ip. 15) are given the principal formula*
for some of the more commonly occurring cases.
Uet:
= the greatest strain at the weakest section ;
Z = the section modulus at the weakest section, which
latter is indicated at B in the figures;
F == the area of the section ;
J = its moment of inertia (§ 7);
Mb = a bending moment;
Md = a twisting moment;
Mi = an ideal moment, so that
(Mb)i = an ideal bending moment, and
\Md)i = an ideal twisting moment.
An examination of these formulae will show that in many
cases the combination of strains is a matter of importance.H
THE CONSTRUCTOR.
BUCKLING STRAINS.
						The column is	to be considered under compression, when:	
No.	Application.			Theoretical Support- ing Power.	Remarks.	For circular section	For rectangular section	
						—j- is less than d	——- (3 = lesser side). 0	Materi al.
			c			5	sX	Cast iron.
I.		I ;	l 1 ii • ■ j	p_ JE 4 &	Fretly loaded post. The lower end fixed. Weak- est section at the point of attachment.		14	Wrought iron.
			1 •			6	8	Wood.
			i.					
	WMMl							
		1	p					
		fi				xo	11M	Cast iron
II.	1		;J	nS IA /2	Post free at both ends, but kept in the line of the axis. Weakest sec- tion in the middle.	24	28	Wrought iron.
		v				11 lA	13H	Wood.
		\	i	%				
	RHW^I							
	f i		P			14	x6	Cast iron.
III.	ij j			P = 2 7r2 IA /2	Post fixed below, and held in the line of the axis above.	33	38	Wrought iron.
	1 j					x6	x9	Wood.
	i I							
•								
								
			P			20	23	Cast iron.
IV.	t			4	Post fixed at both ends in the line of the axis. Weak points at the ends and in the mid- dle.	48	56	Wrought iron.
	i	i j				23	27	Wood.
								
								
For example, in case I., if R = —, i. e.y if the load is hung at
2
the edge of the section, P= and hence is only one-fourth as
4
great as it would be if applied centrally. If the section is circular
S—cP
(d)y we have P-
i + 8t
, making R = — ; P = — — d2,
2	5	4
sion for the lever arm of the force P for each case. This can
generally be readily determined graphically, and so determined
just like any case of ordinary bending.
For example, in case II., if a = 450, we have cos a = sin a =
0.707 for the value of h, and the section at B is to be calculated
as if acted upon by a force P, with a lever arm 0.707 l (the pro-
h
jection of l on the plane of attachment) + 0.707 E.
6
In case I., making R = of for a circular section (M£)i = R~ -
8
and the sustaining power is stili less than with a rectangular
section. Case III. is derived from I. and II., and may be
changed into either by making a, or R = o.
The so-called ideal moments are especially useful in these
calculations. It will be noticed that in the case of elliptical
and rectangular sections, h is taken in the plane of bending.
These dimensions being known in advance, since the choice of
profile is frequently permitted, it is possible by the use of the
ideal moments to consider the question of combined strains,
since the quantity in the parenthesis to the right is the expres-
and substituting S — (P, we get P = S — d2, as we should since
32	4
the stress is now purely tensile.
In this case
~ is the lever arm
o
which, if acting with a bending force P, would produce a strain
of the same amount as that in the line of the axis. This is
only rigidly exact when the shearing action which occurs in
bending is neglected. Many useful applications of cases IV.
and V. are found in discussing axles and shafts.THE CONSTRUCTOR.
iS
COMPOUND STRAINS.
No.
II.
III.
IV.
V.
Application.
Sustaining Power.
P =
SF
1 + R Z
for rectangular.section, {bh)
r	Sbh
, P
1 + 6 -T-
P =
5P
COS a + -g- l sin a
for rectangular section, {bh)
P=—
cos a -j- 6 ~ stn a
IP = -
SF
P =
cos a -j—(/ sin a -f R cos a)
for rectangular section, (bh)
S bh
* 1 , ■ , R \
cos a -J- 6 —^—{sin a -)—j~cos a)
P =
SZ
-l-/ + -|-vA?+jP
Pl is a bending moment Mb,
/>P is a twisting moment Md.
P =
52
A/r + Mg + 2	A/2 cos~i
in which Mx is the bending moment
of Pi, Afr that of P2‘
Ideal Moments.
Ideal bending moment for stress 5:	= P ^P -|—
For circular section (d):
(^)< =/>(*+4-)
For elliptical section {bh):	For rectangular section (bh):
(»),-*»(*+4-) j (»),-^(j?+4-)
Ideal bending moment for stress 5:
{Mb ) —

For circular section (a?) :
^ A/& ^ i — P sin a -j-
-(-cos a)
For elliptical section (bh):
^ Mb ^ i — P sin a +
h \
—g— cos aj
For rectangular section {bh):
^ Mb Ji = P sin a -j-
Pccsa)
Ideal bending moment: ^ Mb 'ji = P ^P cos a + l sin a + -E- cos .
For circular section {d):
{^Mb 'ji = P [fi cos a -f
. .	, d \
l sin a d--— cos a J
For elliptical section {bh):
^ Mb = P ^P cos a +
7 .	,	h \
/ sin a H---— cos a J
For reotangular section (bh) :
^ Mb 'ji = P ^P cos a +
, .	h \
l sin a d---— cos a J
Ideal bending moment for the stress 5:
Ideal twisting moment
{m)*_ -i~Mb +-f-|/r^i2 +
lent s
{Mt)i----|~Jlft + T1
+ Mi
Ideal bending moment for the stress 5:
(*>)*- ^
Mi2 d- M^ -f a M\ M*i cos a
In cases IV. and V. it is supposed that the section is arranged in four symmetrical
portions about two lines of gravity, perpendicular to each other.
! 19-
Resistance of Waees of vessels.
*Boilers, Cylinders, Etc.
The following table will serve to- determine the resistance of
the walls of cylindrical vessels subjected to pressure for the
cases which usually occur in practice.
The theory of resistance under these conditions is not fully
settled, especially in the case of comparatively thin shells sub-
jected to external pressure, for which the corresponding formulae
do not give satisfactory results.
In the following cases, let:
p = the unbalanced pressure upon the walls of the vessel ;
kS* = the maximum stress for the material used ;
E = the modulus of elasticity of the material:
r == the radius of the vessel;
d = the thickness of the walls.
Although only approximate, the formulae for cases I. and II.
hold good up to the limit of rupture.
Examples: i. Given a wroughi irem cylinder, 40 inches diameter,
thick, with a stress upon tl^e material of 11,500 lbs. Under conditions of
case I., the intemal pressure permissible would be
w---- \
2.	A spherical vessel of the diameter and thickness given above, according
to case II., would have a safe resistance
0.375
P— 23,000----= 431 lbs.
20
3.	A piate held as in case IV., 40" dia., thick, and a pressure of 212 lbs.,
with a maximum stress 5 = 11,500, would have a thickness
8 = 20 l/ - l/ 212 - r= 20 X 0.816 X 0.136 — 2.22"
y 3 y 11,500
or about 2% inebes.
The deflection f \ which a circular piate gives under a force p,
may be determined, according to Grashof, by the formula for
case III. :
JL — A-(±lY p
6 — 6 \ 6 J E
(24)
and for case IV. :
(/^-0
P== 11,500
* 11,500 x 0.0185 = 212 lbs.
/=
2 25
4(1)
imple 3 prec<
. (js-Y. —
\2.25 /	28,
E
(25)
Example: The piate of Example 3 preceding, with a value of E = 28,400,-
000, would have a deflection of
212
400,000 *
>.0175"16
THE CONSTRUCTOR.
RESISTANCE TO PRESSURE.
No.	Application.	Pressure p.	Thickness S.
			*
	/JRO i		
.5 O		>=s(/■+“-»)	4-4 (■+-$:)
		-	
g			
xt 0. Ul		r	8 p r 2 S
V cS S			
C §	■P		| 0» II
V	rw^.			
			
S 'O s 3 0			T-/-r/+
>	Pl:		
For vessels whose walls are required to be made very thick,
as in the case of the cylinders and pumps of hydraulic presses
or for cannon, etc., the preceding formulae do not apply. Under
these conditions the relative radial distances of the various por-
tions of the thickness of the metal vary greatly, and their
relation has an important influence upon the resistance. It
is the relation which exists between the various stresses at
different points which governs the various formulae for the
thickness of the walls, which are given below. Brix calculates
the stresses at different points on the radius upon the supposi-
tion that the internal diameter is not altered by the pressure ;
Barlow admits such an alteration by pressure that the area of the
annular section of metal is not reduced ; Lame makes neither
of these assumptions, but calculates very closely the changes
in the various stresses which are caused by the internal pressure
at each point, and in this way has obtained the most reliable data
as to the real behavior of the particles of the material, accord-
ing to the modera theory. The resuits of the three theories are
given in the following table :
Quantities.		Brix.	Barlow.	hame.
u	r	t .Slog nat eS— 1		(r+ sy-r*
<v a -	p — 6			^ {r +. i)! + ^ tf S+p
O	r ~	es	s-p	V S-P 1
	•-i	2 S —	2 5	.„(r+ i)*-**
£ 1		r 1 P	1 + s p &	(r+ ^)3-h 2 1* *f2{s+pi
to	r	2 5	2 S 	p	V '2 s-p 1
For the stress S' in an annular ring lying between the radii
r' and r, Lame gives
'-f [■+(*)>*[■-(,)]
If r* is the external radius of the vessel, so that r' = r+&,
we have:
S' :
r(	^4-y	!+Il	p (	‘+4)s	— 1
L	(^4	r J	2	(*+4	r
(26)
gr if we put (i -f = fi
S' =
j-l p — I
Example : If 8 = r, that is, /i = 2, S' =
ceding formulae, taking p =
5
8
S, we have
5-----o~ and as in the pre*
-5—=
40
-S.
S — ^ -
8	40	5
This shows that the material is not used in an economical manner in ves-
sels with excessively thick walls.
All three theories admit that the inner portion of the wall is
strained the most, and hence it is for the inner wall that
should be chosen. The formulae of Lame, as well as those of
Barlow, show that beyond certain limits an increase in the
thickness is not attended with any increase of strength. With
a given resisting power S, this limit will be reached when p = S;
the theoretical resistance will be attained when p = the modu-
lus of resistance of the material. At this point the internal
pressure begins to stretch the inner fibres of the walls, and any
increase in strain will cause rupture. The theoretical limit in
this case is reached when p = T> which is
For Cast Iron = 10,650 lbs.
“ Wrought Iron = 21,300 “
“ Cast Steel = 36,000 “
Lack of homogeneity in the material may cause the danger
pressure to be reached far within these limits, the material
breaking without previously stretching.
Since stresses exceeding 36,000 pounds are reached in guns of
large calibre, it is evident that ordinary bronze is unsuitable for
such conditions, and even homogeneous Steel is often unequal
to the pressure. The erosion of the chamber in the case of
ordinary bronze cannon also acts to weaken the inner ring of
material, and must be considered as a Chemical deteriorating
action.
Various methods have been devised for strengthening guns
by giving the various layers different tensions. Of these
methods the principal is that of hooping. The principal resuit
of this construction is to produce a compression in the inner
layer. The pressure of the gases of explosion must then first
overcome this compression and restore the normal condition
before it can produce any extension of the fibres, and as a resuit
a much higher degree of resistance is secured than when the
metal is left in its normal condition.
The calculations of the resistance of hooped guns offer many
difficulties. If we have not only the inner pressure, but also
the outer pressure, p', to consider, we may take the following
formula, after Lame:
s+p

p+zp'
(27)
Putting 1 +
we have :
= p, as before, and solving with regard to pt

(28)
in which 6* will become less with regard to /, the greater p' be-
comes.
'	In the case of hooped guns
p' is not constant and invari-
ble, but depends upon the
effect which the internal pres-
sure/ has through the walls
upon the hoops.
Referring to Fig. 6, let it
first be considered that under
normal conditions the inner
ring is under no strain, that
is, / = <?, and also S\ = S/\
= o.
Now when the inner pres-
sure p becomes sensible while
the external pressure p" = o,
or at least may be neglected,
then the layer at r' will be-
come extended, and the ten-
sions will be Sp = S/. The
stress Sp in the inner side of
the hoop reacts with a pres-THE CONSTRUCTOR.
17
sure/', and substituting this in Lame’s formula, making 1 +
= p't will give
fi'
— c /1—
p' = S,
(29)
2	+ 1
Making S' = S/ = 5/ and substituting this value of p' in (28),
gives
p = S
P— 1
4-2 5/
/*'2 4- 1	4- 1
^24- 1
According to (26), S' is dependent upon p and S, and by sub-
stituting and transforming, we get
p = s£
11* fi'1 ---- I
(■+T-),(-+^r-
+ I (^-r)2 (I + -^_)a +1
= 5
(30)
In this case the stress .S upon the inner ring is always greater
than p, but the ratio approaches much nearer to unity than
before, as the following table shows :
'Wher		We have			And also		
6	fi'			/	5	5'	5'
r		/1	c	5	P	P	5
1	0	2	I	0.600	1.667	O.667	0.400
1	0.5	2	1.5	0.800	1.250	0.406	0.325
2	1	3	2	0.905	1.057	0.143	o.i35
It will be seen that the mere hooping of a gun with a ring of
the same material as the inner tube adds very materially to its
strength. If, however, the ring is forced on in any manner so
as to produce an initial strain p' upon the tube, a stili greater
advantage will be the resuit.
If we insert the value of p' from (29) into formula (28), we have
p = S
V2— I
/i2H- 1
425/
fi'2 — 1	fi2
1
(31)
In this formula S/ is partially a function of /, and also de-
pends partly upon the extent to wdiich the tube reacts. This
latter condition exerts a most important influence upon the
strength, as we shall see hereafter.
If wre assume that the hoop is under such an initial strain
that, for the maximum value of /, the value of S2' = 6* (which
is doubtless the most desirable condition), we shall then obtain
from (31)
JL
5
p1— 1
p*+ 1
+ 2
p” + 1 u2 4-1
If fi = r, fi' = fi, =	r', then we have u = 2, fi' =
(32)
and
this gives
A =	+
5	5 ^
13
79
65
This shows Sto be less in value'than p, or in other words, it
is possible to permit the internal pressure p to exceed the mod-
ulus of resistance without overstraining the material. It is also
evident that by encircling the hoops wTith additional hoops, this
principle may be extended stili further, and the ratio between
p and S stili further increased.
If the material of the gun be taken as ordinary cast Steel,
with a.modulus of resistance of 36,000 lbs., the pressure of the
gases of explosion could not be permitted to exceed 43,000 lbs.,
without causing a permanent deformation of the bore. Recent
experiments, however, have shown somewhat greater figures
than the above.
Some of the later tests in England
have shown pressures of 25.8 tons on
the square inch, although this pres-
sure is considered by some engineers
to be rather too high to be safe. It
is quite possible that in this case the
modulus of resistance of the material
exceeded that given above ; or the
interior tube may have been hard-
ened, which, if properly done, is de-
cidedly advantageous.
The compression exerted upon a
cylindrical tube by an external pres-
sure, as in Fig. 7, may be determined by an application
P'
mula (27). If we assume the internal pressure/, to be =o, or
at least so small as to be neglected, we get:
(54-2/0 n2 = S
from which :N
The minus sign indicates the change from tension to com-
pression. When the internal pressure = 0, the stress in the ex-
ternal fibres is :
S' = ?2i2+2
2	fi1
which g^ves :
S' =
—p'
j£2_+_i
I
(34)
This value is less than the preceding; for by division we ob-
taim the ratic :
S'  1 p2 4-1
S	2 p2
(35)
which can only = 1, when fi = o. Hence for external pressure
the greatest stress is always on the innermost fibres.
If, for example, fi = r, and hence fi = 2, then the stress in the
inner wall of the tube will be S—-----p' and S' = — —p'} so
5	3	3
that S' =~S. This is a greater proportion than when the
pressure is from within, as under these circumstances according
to formula (30) 5 = -- /, only.
It is not uncommon in machine constfuction to strengthen
hubs and other parts of machinery by forcing on hoops or rings,
and the calculations relating to such construction are closely
allied to the preceding. The following case will serve to illus-
trate.
In Fig. 8 is a ring B, which is to be forced on to the cylindri-
cal shaft A. The following applies either to shrinking, or to
cold forcing. Before the operation the radius of the shaft is rlt
and the radius of the hole in the ring r2) while afterwards they
both have the same radius r.
Under these conditions the shaft B will be subjected to a uni-
formly distributed compression Sv while the inner surface of the
ring will be under a similar tension S2. Taking the correspond-
ing moduli of elasticity E1} and E2f we have from formula (2) :
^1 ___ ri r	S2 __ r — r2
Ei	r\	E2~~ r2
Adding these together, we get :
It is most important for the designer to know the best values
for rY and r2.
If we call Tp = —---we havei8
THE CONSTRUCTOR.
6*! and 5*2 are dependent upon each other, and their relation
is expressed by Laniet formula :
Si — S2
r — i
P2 + i


which may be abbreviated by putting
^ = S3 p
This gives:
* =
^1 _j_	^1	^2 i ^2 P
E\~_P E2 _ E2 Ex
(37)
_______________SjP
E,
The difference between the value of the denominator and
unity is so slight that in practice it may be neglected, and for a
practical and useful formula we have
V>:
■^1 i *^1	__ ^2 i *^2 P
'	„ ZT	ZT ' ZT
E1 P E2
El
(38)
In this formula we have for the following:
d
= 0.5
0.6
0.7	0.8
i.5
2.0
3-o
p = 0.385 0.438 0.486 0.528 0.600 0.724 0.800 0.882
We also have from equation (38):
t -g»
(39)
5, = —--------- and S, =
14. Ai	, _L ClI
+	E2 p + E,
This value of V; is generally so small that great care is neces-
sary, in turning and boring, to secure the correct sizes for rx
and rv
Example : With a wrought iron shaft and a cast iron hub we have
Ex = 28,400,000'; Eo *—• 14,200,000.
If 8 = 2r, then p = o.8 by the table above ; and we may also assume that the
stress, 5», in the interior of the ring due to the forcing should not exceed 7200 lbs.*
This gives from equation (38)
,	7200	. 7200 X 0 8	1
d/ = —--------— -l-	-------- =	- «=■ .00071,
14,200,000	28,400,000	1408
or, in other words, the increased diameter of the shaft over that of the hole must be
0.00071 times its own diameter.
If we make 1It = -r—, we shall have a stress in the ring of
600
600
X 14,200,000
14,200,000 X 0.8
* 16,950,
1 +
28,400,000
©r nearly 17,000 pounds, which would be too great for the ring to stand.
• ? 20.
The Caecueation of Springs.
The materials used in machine construction are all more or
less elastic and yielding, so that it is only by a judicious dis-
position and proportioning that we are able to avoid an injuri-
ous deformation of their parts when subjected to the action of
external forces. Indeed, it is the principal aim of the construc-
tive engineer to keep the various forms of distortion, such as
extension, compression, beuding and twisting, within as narrow
limits as possible. I11 the case of springs, however, it is sought
to utilize this property of elasticity for a variety of purposes ;
such as to modify shocks, as in the case of buffers and car
springs, or as a source of motive power in clocks and w^atches ;
or in cushions, mattresses, etc.
All bodies which will permit great alterations of form within
the elastic limit may properly come under the designation of
springs.
The only .substances which are of Service for springs under
the action of tension and compression are those which are soft
and readily compressible, such as rubber; while the more rigid
materials, such as wood and the metals, are used in flexure, or
in torsion.
I11 the following table is given a number of forms of the most
usual springs, both for bending and torsion, with their respec-
tive properties.
Next to elasticity, the property of a spring to be considered
is the economy of material, both 011 account of cost and space
occupied. I11 order to make it possible to compare different
springs in this respect, the relative volume is given in the last
column of the table, for the same load and application in the
different cases, the volume of the triangular spring being taken
as unity.
In all the formulae of the table we have
E = the modulus of elasticity,
2
G = the modulus of torsion =— E, (see § 13).
The coefRcients for the resistance of the materials used in
springs will be found in $ 2. It must not be forgotten that for
materials used in torsion, to obtain the same security as when
used in flexure, the permissible stress .S should be f its usual
value (see § 5). The formulae are intended only to be used
when the force P is applied as shown in the figures.
The volume V of any form of spring is according to the
formula:
V=C.(P.f)^i	(40)
in which C is a constant depending upon the form of the spring ;
while Pf is the product of the load into the deflection, or the
so-called work of the spring. This shows the interesting fact
that springs of the same general form and same material are
always of the same weight for the same work, without regard to
the actual length or proportion of dimensions.
No.
Form.
Name.
Supporting Power.
T.
Rectangular
Spring.
6 /
II.
Simple
Triangular
Spring.
P
S_ bW
6 l
111
Compound
Triangular
Spring.
6 /
i — No. of plates.
Deflection.	Elasticity.	I Relative | Volume.	Remarks.
			An approxi- mation to
^ fi Pl* f~ 6 Ebh3	f s l	3_	y h v /
	l E h	2	will be secured by making the
			end = 	 h. 3
Pl3 Jt~6 Ebh*	-h* 11 fo|co	1	Body of uni- form resistance to bending. In practice the end is made somewhat thick- er.
Pl3 f~6EibA3	/ S l l Ah	1	This is equiv- alent to a sim- ple triangular spring, with a base = i b, as shown by the dotted lines.THE CONSTRUCTOR.
'9
No.	Form.		Name.	Supporting Power.	Deflection.	Elasticity.	I Relative | Volume	Remarks.
rv.		!SPikh LiM'J 1 Bl ijj f	Flat Spiral Spring.	5	b» 6	R	^ k N 11 *< 11 s	/* S / R 2 E h	X	/ == the devel- oped length of the spiral. All three forms of uniform re- sistance. The vaiue ~- is the a-gle of rotation pro- duced by the load P.
V.		xw^BSSmi ii\P	Flat Helical t Spring.	S bh* P~ 6~ R	, „ „	/ S l R 2 E h	1	
r VI. VII.		i HlSIii 1 H| p	Round Helical Spring.	„ Sir a3 P~ 32 R	, 64 ^	/ S l R 2 E d	4 3	
		■■H wd iUlA ' ' ' MflSMi	~*	™	Simple Round Torsion Spring.	_ rr d3 P~~S^6~R	§1* »,|y> «I* 1 § 1	f SI R “ 2 G d	5^ 12	Cases VII. to X. are bodies of uniform resist- ance to torsion.
VIII.		mmm mmm ifiiiiiii rn& KltA >i|i^—L^=iSi5w! \ mW% \ «f iiSiiB 	i		Simple Flat Torsion Spring.	n S b*h* 3 R y/& + h* Approximately when h > b, S b* as iv 3 (0.43 4- 0.96h)	. „0 «w S-RV-i-q ■ & + m 33 33	f s R G /s/ 32 4- 32 bh	5 8	Springs of the form of VII. and VIII. may also be combined into compound forms.
IX.		d	Helical Spring of Round Wire	_ „ IT d3 P==S~rt~R	32 PFC-l ^ ir Gd4	/ s l R “ 2 G d	_5^ X2	In cases IX. to XII. l is al- ways the devel- oped length of the spring.
X.			Helical Spring of Flat Wire.	S b*h* P i Ts Approximately when h>b, S b*h* p =		 J? 3 (0.43 + 0.96A)	, PJSV * 32 4- h* b3h3	f s R “ G	Fs 8 5 6	It is immate- rial whether the breadth of the piate is parallel, normal, or ob- lique to the axis.
						ly/ b* 4. 32 bh		
XI.		Wm v mmmmw ** .... \iy8K v	Conical Spring of Round Wire	7T rt3	Approximately 16 /V?2/ ir £d*	<o|u II s|^		Here, as ia case XII., also the spring is measured to the apex of the cone. The weakest point is at B.
XII.		HBK v> ^ imi MMrW*am, p	Flat Volute Spring.	r 5 *** 3 ^ v/^a + ^2 Approximately when h> b, n S »» R 3 (0.43 -f 0.96A)	Approximately 3 PR2/ 324-^2 /r= 3 G 33A3	S 1 5 R x 2 G	_5_ 4	By making a gradual reduc- tion in the vaiue of 3, from B to the end, this may be made a form of uniform. resistance.
		fe				ly/b2 4- A2		
		Spf':iS^^®r				bh		20
THE CONSTRUCTOR.
The quotient shows that a small modulus of elasticity,
when combined with a high modulus of resistance, indicates
the best material for the construction of springs. According to
the table in § 2, we have :
Hardened and tempered steel
Ordinary steel (not hardened)
Brass
Wood
* 42,600,000
(90,000 )2
28.400.000
~- (35.500)2
9.230.000
(6,8i6.‘2
1.562.000
(2,84o)2
.00052
.00223
.01986
.01936
This shows that hardened and tempered steel is theoretically
the best material of springs. It is also worthy of note that in
all the examples given, the deflection is proportioned to the
load. It follows from this fact that the time of vibration which
any of these loaded springs possesses, is of the so-called “sim-
ple” character, of the same nature as that of a pendulum.
Neglecting the weight of the spring itself, we have for the vibra-
tion of a loaded spring the same rate as that of a simple
m&thematical pendulum of a length equal to the deflection of
the spring/, which is
\fjL (41)
in which g is the acceleration of gravity = 32.2 ft.
Examples on the theory of springs: i. Given a simple triangular spring, as in case
II.f for a load P — iio lbs., and a deflection f— 0.78". Taking the material as cast
Steel, with E = 42,600,000, and making S, the greatest permissible stress, = 56,800
theri
lbs., and also taking the length l = 15.75", we then have
/	S l	0.78 _ _56,8oo
l	E * h	15-75	42,600,000
from which
__ 56,800 X 15-75 X 15-75 _	„
0.78 X 42,600,000	°-424
Substituting this in the formula'
,	, pP .	. PZ*
f— 6 -^775- °r 3 = 6-
15-75
h
Ebh3
Efh3
we get:
6 X no X G5-75)3
1.018"
The volume Y
42,600,000 X 0.78 X (0.424)3
bhl 1.018 X Q-424 X 15-75
Example 2 : If we keep the same conditions, but make the length xi.8" we shall
have
* 0.238"
The volume in this case :
__ 56,800 X 11.8 X 11.8
0.78 X 42,600,000
„_____6 X no X (n-8)3
42,600,000 X 0.78 X (0-238)3
= is	_2-42X 0.238 x 11.8
= 3-39 cu. in., thuscon-
firmmg the remarks on formula (40) by showing that the volume depends upon the
load and the deflection, and is independent of the proportional dimensions.
Example 3: Let us now suppose the same conditions to be applied to a helical
spring such as No. IX , also made of cast steel Since this is a torsion spring, in
order to obtain the same security we must make S = —of its preceding value or
—• 56,800 = 45,440; and the wire may be taken as 0.24 in diameter.
We then have from the table
P=S "
from which we get
R
16
d3
. 45,440
16
(o-24)8
R
45,440 X 3.1416 X (0-24)3 __
16 X IIO	1,1
The length l is obtained from column (6).
f _ _ S /	_ , /Gd
2 RS
in which G = — E =
5
17,040,000.
= 31 3"
/== a78 X 17,040,000 X 0-24
2 X 1.121 X 45,440J
This would make the number of coils
/= H _____________3T«3	=
2TR 2 X * X I.I2I 4'4’
fecafen r^d PrefaTed' *“* < the wi« «“*
The volume V\ -
4
This gives the ratio =
= 3i-3 X 0.7854 X (0.24)2
x-4i6________ 5
3-4
= 0.416 or -
1.416 cu. in.
as given in the table.
Example 4: Torsion springs have recently been applied to railwav cars in the
form shown in Fig. 9, which is the design of an American, Mr. Dudley The I |
shaped spring is bent at the ends into two elbows, A B, which are attarh^l kUiT
,0 a block which rests on the axle box. A saddle, A, tWn.mk th^ oad of tL cS
to the spring, while the other end is supported at Cby a hook.
In order to determine the stress 5, in one branch A C, of such a spring, let us takfr
the diameter d = 1.14", and the lever arm, R, which is the horizonta! projection of
A B, as 4". The load on the spring is one-fourth the load on the car, 22,000 lbs. -f-
one-fourth the weight of the car itself, 18,000, and one-half of this is borne by each
branch of the spring, making the load at the end of the lever R in this case to be
5,000 lbs.
In the preceding table, under case VII., column 4,
o_ 16 PR 16 5000 x 4
------------- -------------rr- = 68,750
7V O3	7r	(1.14)3
If this spring is Eiafle of Sheffield steel which has a modulus ol elasticity E —
24,140,000, then tne modulus of torsion G = — E = 9,656,000.
If the length l —	ths deflection, according to column 6 in the table, will be
, 2PSi _ 2. X 4 X 68,750 X 33 5	, „
^ Gd	oco X 1.14
The above describud Spring weigfts 24-2 pounds, which is about of its gross
ioaa, or about of its netloacU
Fig. 9.
A double armed piate spring of the form No. III., to have the same supporting
po wer would weigh about a hundred pounds, or its gross load and ^ its net load.
As long ago as 1857 I called attention to the superior economy of torsion springs
over piate springs for railway use. The principal reason for the tardiness of railway
men in appreciating this fact may have been partly due to the difficulty of securing^
a proper temper in the round steel, although this seems to have been entirely over-
come in the case of the Dudley Spring.
In the little pamphlet on “ The Construction and Calculations of Springs," which,
I published at that date, the comparative weight of the torsion springs VII. and IX.,
and the triangular piate springs II. and III., is given as instead of Ts5 as in the-
preceding table, but the latter is shown to be more nearly correct in practice.
Fig. io.
In Figure 10 is shown the manner in which a helical spring may be applied to the
bearing of a goods wagon in the place of a piate spring. The box is guided in the
frame B, B, and the spring D is interposed between the sili A of the wagon and the
journal box C. The form of the spring is a single helix with the lower end flattened
for about of a tum in order to give a fair bearing in the cap E of the box. The
upper end screws for about 1 x/2 turns into the cap F, where it is elamped by the screw
G after the load has been equalized, and in this way any desired adjustment may be
secured.
An example will probably be the best method of showing the manner of calculat-
ing such a spring.
An ordinary German four wheeled goods wagon weighs about 11,000 pounds, and
carries about 22.000 pounds load. This gives about 8,250 pounds to be supported by
each spring. We will assume a deflection of 1^", with a permissible fibre stress
of 68,000 lbs., and take G = 9,656.000, as before.
Since it is desirable to use such diameters of spring steel as correspond to commer-
cial sizes, it is better to select a diameter d for the st6el, and deduce a corresponding.
radius R for the helix, according to the formula for case IX., coi. 4, page 64,
R = S
7T d3
l6~PTHE CONSTRUCTOR.
21
We will take the successive cases in which the diameter d of the Steel is i", iTy',
and i^".
Now for these respective values we must select such a number of coils, n, that
•tvith a load P = 8,250 lbs., we shall get a compression/— 1.75".
We have for n the fact that 27rRn = the length l of the uncoiled spring. Substi-
•tuting this in the formula for/, case IX., we get
.S 27rR^n
rom which
/G _d_
n 47T.S ' R*
Now the least possible distance between the cap F and the Socket E is nd -f r,
-and we must also provide for a space <r between the coils, say 0.3". This gives tne
-distance j from centre to centre of coils of the unloadea spring
nd -\-f + na
s --------------
n
The total height of the spring, however, will be greater than ns by 1.5$ + d, since
a l/2 coils enter into the cap F, and one-half the diamater of the Steel in the last coii
Jnust be added both top and bottom. Adding to this the thickness 8 of the cap and
*he socket, as weobtain the entire height occupied by the spring and its fittings.
Of course this height is limited by the space at our disposal between A and C, and
in this case it may be taken at 14 inches, and in any case it will generally be depen-
■dent upon other circumstances. Substituting these values in the several formulae, as
?given above, and tabulating the results, we have:
d =	1"	itb"	P/s"	*rV'
R =	1.619	1.94	2.30	2.71
or say ==	iH	2.00	2	2^
h =	7-5	5-6	4.18	3-19
11 d =	7-5	5-95	4.70	378
na =	2.25	1.68	1.25	0.96
/ =	J-75	i-75	i-75	i-75
n{d +&) +/=	11.50	9-38	7.70	6.49
j =	i-53	1.67	1.84	2.03
or say =				2.00
1.5* =	2^	215	m	3.00
26 =	1.00	1.00	1.00	1.00
Total height =	15.75	I3.82	12.45	11-49
or about =	I5&	13%	12^	
It will be seen that the first size is too high for the space at our disposal, but that
"the others may be used as circumstances may dictate, ali fcur springs ha vir,g the
^same compression and stress upon the material. The simplicity of this construction
is noteworthy, and the economy of material is noticeable. For passenger coaches,
where more elastic springs are desirable,/ may be made from 2 to 3 inches, and a
•combination of several springs may be used.
The use of vulcanized rubber for springs and buffers is now
quite general, usually iu the shape of rings or collars between
plates of iron. The resistance of rubber to distortion has not
yet been fully investigated in an experimental manner, but the
following examination of buffer springs may be of Service ; the
•data being from the valuable researches of Chief Fngineer
Werder, of the establishment of Klett & Co., of Nuremberg.
The usual sections of buffer rings are shown in Figs. n and
12, in which one of the plates carries an annular projection, and
i:he other a corresponding depression, between which the springs
-are held, and lateral motion in the buffer case prevented.
When the ring of rubber is under compression its volume is
nnchanged ; the cross section is reduced, but the diameter of the
xing is increased proportionally. The elernents wrhich are sub-
jected to the greatest strain lie at the circumference E, and are
uuder tension, as is proved by the cracks which appear when
the limit of elasticity is exceeded.
The limit of elasticity will be reached by a load of about 700
pounds to the square inch of original cross section normal to
the axis ; or, in other wTords, a modulus of compression T — 700
may be taken. This modulus is slightly higher for rubber of
the lightest specific gravity (about 780 lbs.), and less for a heavier
specific gravity (about 640 lbs.). The specific gravity y, which
depends upon the proportion of sulphur, varies from x for the
lightest, to 1.15, or eveu 1.32, for the heaviest.
When the elastic limit is reached, the middle section E E of
the ring is double its original area, while the periphery is
f the original periphery A B C D.
The compressibility within the elastic limit is dependent upon
the quality of the rubber, and may be approximately determined
by the following empirical formula :
a = 0.026	W
in which : X = the extent to which the spring is compressed by
a load P\ l = the original thickness of the spring; q = the
original cross section in a plane normal to the axis ; and y = the
specific gravity of the material.
Fig. n.	Fig. 12.
Example: A buffer spring similar in shape to Fig. ii has an outside diameter of
5^", and an inside diameter of 2^|", which gives an area of cross section q of 18.07
square inches. The thickness of the uncompressed ring l =	and its specific
gravity y = 1. The load to be supported is 5500 pounds, which corresponds to a
p
pressure per square inch -= 305 pounds,- which is well within the elastic limit.
<1
According to (42) the compression will be :
A = 0.026 X 1-375 >/ 305 = 0.624, say
The Belgian engineer, Stevart, has also made extensive re-
searches upon the subject of the resistance of rubber. These
experiments appear to confirm the opinion that any change of
shape is unaccompanied by any change of volume, and that
rubber is practically as incompressible as water. The experi-
ments on tension gave a modulus of elasticity of 119.28 lbs. In
regard to compression Stevart deduced a formula similar to the
preceding:
73Ta= /" aP+f‘
in which a is a coefficient dependent upon the form of the
spring, and determined experimentally. In the case of locomo-
tive buffers, which are composed of several rings, the compres-
sion of each ring should be computed separately, and their sum
taken.
Rubber springs are used very extensively, but the principal
objection to them is that the material gradually loses its elas-
ticity and becomes a hard, unyielding mass. It has been found
that this is largely due to friction betwTeen the rings and their
cases, and great care should be taken that boxes for rubber
springs should have ample allowance made for the increased
diameter of the spring when compressed.
It is a matter of importance to choose such a shape for a
rubber spring that it shall not have a tendency to form puckers
in the edge when it is under pressure. This is shown in Fig.
12, where the slight concavity in the edge would soon develop
a crack when compressed, as at TT. while the shape in Fig. 11
has no such tendency.22
THE CONSTRUCTOR.
SECTION II.
THE ELEMENTS OF GRAPHOSTATICS.
? 21.
INTRODUCTORY.
The equilibrium of forces may be very clearly shown by the
graphical method, since it is possible to show the direction, ex-
tent and position of any force by a right line. The direction
of a force is determined by the angle which its representative
line makes with the horizontal axis of co-ordinates ; the length
of the line gives the absolute amount of force exerted, the alge-
braic character of the force (plus or minus) being indicated by
arrow-heads ; while the position of the line in the system of co-
ordinates, makes it possible to show any constants which
may occur in the equation of any right line. This repre-
sentation of forces by means of geometric magnitudes makes it
possible to solve problems in statics entirely by means of
geometrical constructions, and in many cases this will be found
simpler and more convenient than the use of algebraic analysis,
especially in those cases in which the values to be determined
are tkemselves geometrical quantities, and therefore are to be
drawn when found. The various details of the method have
been arranged and collected into a system which has been called
Graphical Statics, or, as we have here termed it, Graphostatics.*
This method is of especial value in the study of Machine De-
sign, and in the following sections of this work many applica-
tions of it will be found. It is for this reason that the following
brief exposition of the leading principies of the method have
been here grouped together.
There is a distinction to be made between Graphostatics,
properly so-called, and the mere graphical calculations of simple
values, considered merely as magnitudes. This is more properly
to be considered as graphical arithmetic, or Arithmography.f
In the following pages this branch of the subject is not very
fully discussed, only an outline of its application to pure arith-
metic being given. It will be found, however, to be a subject
of much use to the mechanic, as many examples of its applica-
tion in future pages will show.
I 22.
Muetipeic a/tion by Lines.
In graphical calculations dimensions are taken with the
dividers and scale, and any convenient unit may be selected,
such as the inch, millimetre, decimeter, square foot, cubic foot,
nnit of velocity, unit of money, etc , etc. It is readily apparent
that the operations of addition and subtraction may be per-
formed by simply marking off the various values upon any Ime.
The operation of multiplication is not quite so simple, and a
brief explanation may be necessary.
In all cases it is of course necessary that the same unit must
be chosen for all the quantities involved. and this holds good for
multiplication as well, and the same unit must be used to meas-
nre the resuit as has been chosen to express the original quanti-
ties. If, now, we wish to multiply two lines, a and b, together,
or, more correctly, to multiply a line of the length a by a line of
the length b} we must find a line x, which will contain our
chosen unit, a X b times. This is a simple operation, and may
be performed in several ways by means of similar triangles.
I.	Draw O E, Fig. 13, horizontal, making its length equal to
imity; erect at E a perpendicular, and intersect this from O with
O B= b. Lay off OA = a, and from A draw a parallel to E B%
intersecting O B produced at C. Then O C will be the desired
product x. That is to say, = ~q~^ and since O E = 1, we
* See Culmann, “Graphical Statics,” Ziirich, 1866.
f See “ Principies of Graphical Arithmetic,” by Dr. Eggers, Schaffhausen, 1865;
also Schlesinger, “ Power Curves,” in Journal of the Austrian Society of Engineers
and Architects, 1866; also E. Stamm, “ Graphical Calculus,” Proc. Royal Inst.,
Lombardy, Vol. VI.
have x = This solution requires that one factor, b, shall
be greater than unity.
II.	Fig. 14. A modification of the preceding may be made
by drawing E B inclined, instead of perpendicular to O E, in.
which case both factors may be less than unity.
III.	We may make, as in Fig. 15, O E and O B as before,.
produce O A = a, and draw A C, so that the angle O A C~
O E B, so that A C will be the anti-parallel to E B. Then O O
will be the desired product x, since the triangles O E B and
O A C are similar. This anti-parallelism is shown by the fact
that O Ef =0 E, O B' = O B, and AC is parallel to E' B'.
If the triangle B E' B' is rotated to the right about an axis,
passing through B B/i the two triangles, B B' Ef and B B' E>
will form a parallelogram ; hence the term anti-parallel. This
construction is most convenient when E B is perpendicular to-
O E, which can only occur when b is greater than 1.
IV.	We may make, as in Fig. 16, O E = unity, lay out ort
O E the factor O A = a, and erect a perpendicular or inclined
line at Ey in which E B=b; then draw through A a parallel
to E B, and this latter line will intersect O B, prolonged so that
A C=x, since C A : O A = B E : O Ey or x : a= b : i; a and
being either greater or less than 1. Now make E BY — bly and
draw O Bx to intersect CA, prolonged at Clt then A C1 = xl9
the product of a and blt and C C1 will be the product of a into
B Bv or:
x + xx = a (b + bji
Of course, the factor 3, which is to be multiplied by a, may-
extend on both sides of the base line O E, and the desired pro*
duct, a b =sx, will then be the distance on the parallel to by
which is included between the two lines drawn from O through
the extremities of b.
V.	In Fig. iy O E — unity, E B = the factor b, and O B any-
value, so that O B <^0 E + E B. Lay out O A on O B, making^
it = the factor at and draw from A an anti-parallel to E B (see
III.), then A C will = x. For A C: O A = B E : O E, or
x : a = b : i, and a and b may both be less than i.
VI.	Again, we may make Fig. 18 O E == i, erect a perpen-
dicular at Ef make E A = a, E B = b, join O with B, draw
B B/ normal to O B, and draw from A a parallel to B B' then
will E C = the desired product x. For we have E C: E A =-
B E : OE orx:a = b : i.
It often occurs in designing that we have already a diagram
drawn which may serve for a portion of the construction, and
in such cases the following methods m£y be found convenient.
VII.	Fig. 19. O A = a and B' B — b are either at right
angles or inclined to each other, so that Bf falis between O and
A. Lay out on O A the unit O E, join B with E, and draw
from A a parallel’to B Et and from the point Ct where it inter-THE CONSTRUCTOR.
2S
sects O Cy draw C C/ parallel to B B/, then C C' will = x. for
we have C (? : O A -=B B' : O E or x \a — b : r.
VIII.	Fig. 20. Given as before, OA = a and B B' = b, either
perpendicular or inclined to O A. Draw O E parallel to B B/t
and equal in length to unity ; join E to A> and draw from B a
parallel to E A. This will cut off on O A prolonged the dis-
tance BC=xt for B' C : B' B= OA : O E or x : b = a : i.
IX.	Fig. 21. Given A A' = a and B B' — b perpendicular.
Draw A B, and prolong it until it intersects at E a line drawn
parallel to A A' at a distance O E = i. Join E A/, and draw
from B a parallel to it, cutting A A' at C> then will A C= x;
for A C: C B = A A/ : A/ Ef and A C: B/ B = A A/ : E O,
or x : b = a : i.
X.	Fig. 22. Given A A' — a and B O = b, perpendicular
to A A/. Open the dividers to O E = i, and intersect A A' at
E. Draw from A' a parallel to O E, and from A a normal, the
o
two lines intersecting at C\ then AC— the desired product x.
For, since the angle C A Af — B O E, we have A C : A Af —
O B : O E, or x : a — b : i. The line A A' is in this case pro-
jected upon a perpendicular to O E, or is what is called the
anti-projection of A A' to O E*
XI.	Fig. 23. When a and b intersect each other at right
angles, as in the figure where A A' — a and B O — b, then
draw from B a parallel to A A\ and mark off with the dividers
from O, O E — 1. Draw A' C parallel to O E, and A C normal
to A' Cy then A C — xy for since the angles at E and A' are
equal, we have A C : A Af — O B \ O E, or x \a = b w.
The continuous multiplication of several factors may be
accomplished by combinmg the preceding methods in various
ways.
Suppose we desire to obtain the product of three lines, a, by c,
we may first find, according to I, the product xx — a b (Fig. 24),
transfer O C= a b down to O O upon O Ay draw from O the
line O D = Cy erect from C' a perpendicular, and prolong O D
to Fy and O /'will be the desired product, x — ab c.
Or we may make, as in Fig. 25, after having found O C/ = a by
draw ED — c (Case IV.), and prolong O D until it intersects at
F a perpendicular from C', when O F— x.
I 23.
of a and by times. From the previous examples we may de-
rive the following methods of division.
I.	Fig. 26. Make O E = unity, erect at E a perpendicular
or inclined line, intersect it with the divisor O B = b, prolong
O By and make O A — the dividend a. Draw from A a paral-
lel to B Ey and its intersection with O E prolonged will give
the quotient x. For we have O C : O E = O A : O B, that is,
x : 1 = a : by or x — TL.
0
II.	Fig. 27. Make O E — unity, also lay off on O E the dis-
tance O B = the divisor by erect at B a perpendicular, and in-
tersect it from O with O A— the dividend a. A perpendicular,
erected from E, will then intersect O A at C, and O C — x, for
we have again O C: O E — O A : O B or x : 1 = a : b.
III.	Fig. 28. Make O B= the divisor b; on O B lay off O E'
= 1; at B erect a perpendicular AB — the dividend a ; join
O A. Frect at E a perpendicular, and it will intersect O A at
C. Then E C = x, for E C : O E — A B : O B, or x : 1 —
a : b.
§ 24.
Mui/tipijcation and Division Combined.
When it is desired to multiply a number a into a fraction
the operation really consists in multiplying a by b, and dividing
the product a X b by cy in order to obtain the resuit x. If we
recollect that for x — we may write x 1 a = b : e, we wTill
see at once how the combined operation may be performed by
making the distance O E equal to the denominator Cy instead
of unity, as heretofore. We will then be multiplying the line a
by the ratio instead of . The following illustrations will
make the operation ciear.	^
I. In order to multiply a quantity a by a fraction—, we make,
in Fig. 29, O A = a, lay off on O A, O E == ct erect at E a
perpendicular, and intersect it at B, with a distance from O
equal to b; then prolong O B until it intersects at C a line
drawn from A} parallel to E B. Then O C will equal Xy for
we have O C: O B — O A : O Et or x : b = a : c, or x —
ab	c
II. If we wish to find the product	we make, Fig. 30, OA
= at and make the distance 0 E — twice the unit of measure-
ment, draw E B — b perpendicular to O E; draw a line from A
parallel to E By and prolong O B until it intersects this last
line at C. Then A C will be the desired product xy for
A C: O A = B E : O Ey or x : a = b : 2, or x — —.
2
These methods, which may be extended much in the same
manner as the various methods of multiplication given in § 22,
will be found of great Service in the graphica! calculations of
areas, as we shall see.
? 25.
Division by Lines.
Area ob Triangdes.
Division may readily be accomplished by reversmg the Since the area of a triangle is equal to the half-product of its
methods employed for multiplication. To divide a line a by a base and altitude, it is readily calculated by the method given
line by we must find a third line xy which must contain the unit in the preceding section.
I. Fig. 31. Selecting the side O B — b cf the given triangle
O A B as a base, which gives the perpendicular A A' — tli^
* See Culmann's “ Graphical Statics.124
THE CONSTRUCTOR.
height h, although this line need not be drawn, we mark off the III. Fig. 37. The diagonal A C=b divides the figure A B CO
distance O E = 2 units (inches, decimeters, etc.), and draw into two triangles, the sum of whose heignts = O 0/y which is
from B a line B Cy parallel to an imaginary line A E. This
line B C will intersect the side O A prolonged at C, and a per-
pendicular dropped from C to O B, will give C O — ---- = the
desired area f (see VII., 8 22, and II., 8 24).
II.	Fig. 32. From the end of the base line O B draw the
perpendicular O E — 2 units, draw the altitude A A' ; also draw
from A the line A C parallel to E B. This will cut off on the
base line the distance A' Cy which is the product f = —.
(§ 22, VIII., and ? 24, II.)	2
III.	Fig. 33. Prolong the base line B C and the side B A
until the vertical distance between them O E — 2 units. Join
Fig. 33.	Fig. 34.
E to Cy and draw from A a line parallel to E Cy intersecting
the base at	D,and B	D = — = /. (§ 22. IX., and § 24, II.)
IV.	Fig. 34. From the vertex O, with the dividers open a
distance equal to 2 units, intersect the base at Ey and make the
anti-projection of the base AB by drawing B C parallel to O Ey
and A C normal to B C. Then AC= the product of the base
by and one-half the altitude O 0/ = hf and hence is the desired
area f of the triangle. (8 22, X., and 8 24, II.)
If the unit is taken as one inch, the value of the area f will
be given in square inches, or if a decimeter is taken as the unit,
the area will be in square decimeters, etc.
If we find f= I", the area of the triangle is seven-eighths of
a square inch ; or if it measures 72 millimeters, the area would
be 0.72 square decimeters, or 0.70 X 10,000 = 7200 sq. mm.
8 26.
Area of Quadrieaterae Figures.
In determining the area of a quadrilateral figure, it is either
©btained directly, as in the case of a parallelogram ; or it may
b
Fig. 35.	Fig. 36.
be separated into triangles, which are measured separately ; or
the figure may be reduced to its equivalent triangle.
I.	Required the area of the parallelogram A B C Oy Fig. 35.
Taking the side O A as a base line, lay off O E — unity, and
erect the perpendicular E E' — h. Prolong O E until it inter-
sects a perpendicular from A at Dy and the distance A D will
be the area o i f=b h. (8 22, IV.)
II.	The quadrilateral figure A B C Oy Fig. 36, may readily
be replaced by a triangle of equal area by drawing the line O A/
parallel to the diagonal O By for sin ce the triangle O A' B is
equal in area to O B C, we have the area of the triangle O A' A
is equal to the area of the figure A B C O. Nowr, according to
IV., 8 25, we make O E — 2, and draw A D, the anti-projection
of A A' and A D — /y the desired area.
the anti-projection which O B makes on A C. The multiplica-
tion of O O' by — may be made according to XI., 8 22, and
II., 8 24. Draw O' B E parallel to A Cy making O E = 2, also
draw A D parallel to E O, and C D normal to A Dy then C D
=f — the area of A B C O.
IV. Fig. 38. The figure A B C O may be converted into a
triangle wrhose altitude = 2, when the base will be equal to the
product From O describe an arc with a radius O E — 2,
and draw a tangent passing through an angle of the figure at B,
opposite the angle O. From the other two angles, A and Cy
draw lines parallel to the diagonal O B, intersecting the tangent
at A' and C'. A' C/ will then be the base of a triangle whose
altitude = 2, and whose area is the same as the figure A B C Oy
and the area f = A' C/. Many similar methods may be deduced
from the preceding examples.
\ 27-
Area of Poeygons.
The area of a polygon is measured by reducing it to its
equivalent triangle. This may be done in the following manner :
From the angle O of the polygon O A B C D Ey Fig. 39, draw
a diagonal O B to the next angle but one, and then from the
Fig. 39.	Fig. 40.
intermediate angle A draw A B' parallel to O B, prolonging
the third side B C to B'. If we join O B/, we have the triangle
O B B/ — O B Ay and hence the figure O Bf C D E will have
the same area as the original figure, but will have one less side.
Then join O Cy and draw B' C' parallel to O Cy and so we may
proceed until wTe have obtained a triangle O C' D' of equivalent
area to the original figure, and wrhose area may be determined
by any of the preceding methods.
Regular polygons, such as the hexagon, Fig. 40, only require
half the operation to be performed, and then the area measured
as a parallelogram.
8 28.
Graphicae Caecueation of Powers.
A line a2 raised to the n*h power, really means the determina-
tion of a line x whose length shall contain the unit of measure-
ment a11 times. The following methods are applicable when a
is a positive or negative whole number, and the process is really
a repeated application of the multiplication of a by a. As in
the previous cases, this operation may be performed in various
ways.
I. (See 8 22, I.) In Fig. 41 make O E = unity, erect at E
a perpendicular, and intersect it at Al wdth the distance OAl = ay
the original factor. Carrying this distance O Ax dowTn to Bly
and erecting a perpendicular at Bly we get O A2 = a2 (see I.,
8 22). This again carried dowm to B2, and a perpendicular
erected at B2, gives O A3 == a3, and so O A4 = a1, OA^ = ahy etc.
If we lay off O Bmy equal to any pow^er of a, say am, and erect
perpendicular at Bm, the intersection wdth O Aj prolonged wfill
give the value of am + *. Again, if we drop a perpendicular
from the end point Am + 1 of any powrer of a to the axis O Ey
it will cut off a distance O Bmy which will be the next lesser
powrer of a (see I., 8 23).THE CONSTRUCTOR.
25
The perpendicular Ax E, from Ax upon O E, gives the first
power ax. If we now make O A0 = O E, and drop the perpen-
Am+l
dicular Ao B—i,we ha ve OB—i = a~x, which = —, which is
the reciproca! of O Ax; in the same manner we get O B — 2 =
II.	By combining the methods of multiplication I. and III.
of § 22, the following method for powers is derived. In Fig. 42,
make OE = 1, O Ax = a, E Ax perpendicular to O E, and draw
from Ax a perpendicular to O Ax, cutting O E at A2; then OA2
= dl. From A2 a perpendicular to the base will give A3 and
OAb = a% ; another perpendicular to O A3 gives A± and OA± =
a4, and this may be continued indefinitely for positive powers of
a. By working backward from E, we get O A —1 as the recipro-
calof a, O A— 2 = and so on for negative powers of a.
Both the preceding methods assume that a is greater than 1;
the following may be used when a is less than 1:
III.	In Fig. 43 make O E — 1, and draw O A — a at such an
angle that A E is perpendicular to OA. Erect the perpendicu-
lar E 1, and continue with the alternate perpendiculars 1 2, 2 3,
3 4, etc., and we have : O 1 = —, 0 2 = O 3 = etc.
0	*	a	a2 0 a6
Working to the left from E in a similar manner, we get
O — 2 = a2, O — 3 — as, O — 4 = a4, etc., the positive powers
being to the left, and the negative powers to the right.
The zigzag lines which are thus drawn back and forth between
the two axes have a relation to the powers of a which may be
utilized in the following manner:
IV.	Make, in Fig. 44, O E = 1, O A = a, and the angle
O A E = 90*; also OB at right angles to O E, and prolong EA
to B. Now draw the alternate perpendiculars as before, and we
have the following values: O A — ay A 2 = a\ 2 3 = a3, etc.,
also OE= a°, E — 1 ~ a~l = —, — 1 — 2 = —etc.
a	a1
V.	Fig. 45. Make O E = 1, and describe upon it as a diam-
eter a semicircle, make O 1 = af and from 1 drop a perpendicu-
lar 1 2 upon O E, then O 2 = a2 (see Problem III. of this section).
With O 2 as a radius from O, describe an arc, and from its inter-
section with the circumference drop the perpendicular 2 4, and
O 4 = a4, and by continuing in the same manner, we get 08 =
a8, O 16 = a>6, etc. The intersection 3 of the radius O 1 with
the perpendicular 2 4 is, at a distance from O, equal to a*. For
we have : O 3 : O 1 = O 4 : O 2; or, O 3 : a = a4 : a2, that is,
O 3 = a*.
In this way we may prove that each line drawn from O to the
upper extremity of the successive perpendiculars on O E> inter-
sects the following perpendicular at a distance from O equal to
the next less power of a. This provides a method of obtaining
the intermediate powers of a by merely drawing radii and per-
pendiculars. Each newly-found power gives a radius for a suc-
ceeding one, and the operation may be continued indefinitely,
as shown in the diagram.
VI.	The following method is suitable for any given value of
a, whether greater or less than 1. In Fig. 46 make O E = 1
on the axis X O X, erect a perpendicular at O, Y O Y, and
mark off O A = a. Join A E, and draw A 2 normal to EA,
and it will cut off on the axis of X, a distance O 2 = a'1; then
draw 2 3 at right angles to A 2, and we get on the axis of Y,
Fig. 46.
O 3 == a3, and on the axis of Xy O 4 = a4, thus getting the even,
positive powers of a on the axis of X, and the odd powers on
the axis of Y. By carrying the spiral in the other direction we
get the negative powers in a similar manner. Joining A E, we
have O E = a° = 1; from that wTe get O — 1 = —, and in a
. .	11	^
similar manner—etc. (See § 22, VI.) This method is
a2 a3	'
very suitable for showing a succession of powers in a single dia-
gram.
i 29*
Powers of the Trigonometricae Functions.
The methods already given for the determination of the
powers of numbers are also applicable to the powers of the trigo-
nometrical functions with but slight modifications.
I.	Powers of Sines and Cosines. Fig. 47. Make O E = 1;
the angle E O A = <p, the angle the powers of whose functions
are to be determined, E A being at right angles to O A. Draw
also the alternate perpendiculars A 2, 2 3, 3 4, etc., and E — 1,
— 1 — 2, etc. Then O A = cos <j>, O 2 = cos2 <p, O 3 = cos3 <j>,
O 4 — cos4 $; O — 1 =	, O — 2 = —J—t etc.
cos <j>	cos 2<p
By drawing the alternate perpendiculars A //.,//. ///., III. IV.;
O — /, — /, — IIy etc., we also get A E = sin (j>, A II = sin2
II, III = sin» III, IV = sin* <p, O — / = -N-, —/—//=,
T	sin (p
etc-
smJ ^
II.	Powers of Tangents and Cotangents. Fig. 48. Make
EO — 1, and O E A = <j>. Draw from A the spiral of perpen-
diculars as in V., $ 28, and we get the following values : O A =
tan <p} O 1 = tan2 $>, A 3 = tan3 $>, etc. O £ = 1 = tan o26
THE CONSTRUCTOR.
Fig. 47-	Fig. 48.
0 — 1 = cot $, <9 — 2 = cot2 0, etc. This method shows very
clearly the convergence and divergence according to the sign of
the power under consideration.
I 30.
Extraction of Roots.
The extraction of the square root is readily performed by the
graphical method, as will be seen at once when it is remembered
that v/flis a mean proportional between a and 1. The previ-
ously described methods for powers also suggest methods fo**
______c
the extraction of roots, and the three following cases will suffice:
I.	In Fig. 49 make O E — 1, O A = a, describe a semicircle
on O A, erect a perpendicular at E, intersecting the circumfer-
ence at Cy and join O C, then O C — x — \/ a (see $ 28). In
this case a > 1, but in the following case a < 1.
II.	Fig. 50. Make O E = 1, O A — a, describe a semicircle
on O E, erect a perpendicular at A, and join O C, then wrill
O C = x = \/ a.
III.	Fig. 51. Make O E — 1, and mark off on O E prolonged
E A — a, draw on O A a. semicircle, and erect a perpendicular
at E, intersecting the circumference at C; then will E C =
x — >/a.
The extraction of the fourth root may be performed by re-
peating the method for square root. The graphical extraction
of the cube root, fifth root, etc., is not so simple. Culmann uses
for this purpose the logarithmic spiral, and Schlesinger con-
structs a curve according to the method in $ 28, but the advan-
tages are not sufficient to warrant a further examinatiori. of the
subject at this point.
Addition and Subtraction of Forces.
In ali the preceding operations we have only considered the
lines to represent absolute quantities, and paid little or no atten-
tion to their direction or position in the plane of the diagram.
The principal advantages of the graphical method are those
which are connected with probiems relating to the equilibrium
of forces, and it is the application of the preceding methods of
graphical arithmetjc to the calculation of forces which really
constitutes the method of graphostatics.
When several forces are acting upon the same point, their
resultant may be obtained by the addition of the lines repre-
senting the forces when projected upon the co-ordinate axes.
This addition of the projection of forces is known as graphical
addition. This addition is performed by placing the lines repre-
senting the forces end to end, forming a polygon, care being
taken to avoid repeating any of the lines. If the forces,
1, 2, 3, 4, 5, 6, Fig. 52, acting at O are in equilibrium, the sum
of their projections wTill equal zero, and the polygon formed by
the lines, as shown in Fig. 56, will close. The figure thus con-
Structed is called a force polygon. It is immaterial as to the
order in which the lines are taken, as in Fig. 53 the resuit is the
same whether taken in the order, 1, 2, 3, 4, 5, 6, or 1, 3, 4, 6, 5, 2,
although the shape of the polygon will be different. As in
arithmetic, the graphical subtraction of forces is the reverse of
addition, and practically amounts to a separation of the sides of
the force polygon into their respective forces. In graphostatics
the forces are all taken in one plane, by projecting upon the
plane of the diagram those forces which may be without it. The
Fig. 52.
preceding method of addition and subtraction of lines, which
here represent forces, but which may be taken to represent any-
thing, is called geometrical addition and subtraction. They-
bear the same relation to geometrical multiplication and divi-
sion as the corresponding arithmographical methods do to each
other. These little used methods, which are of the greatest in-
terest to geometers, we cannot discuss here.
832.
Resuetant of Severae Forces.
In the preceding section we assumed that the given forces
held each other in equilibrium, from which it followed that the
diagram formed by the lines representing the forces returned to
the starting-point and formed a closed polygon. If, however,
the force polygon for a group of forces, such, for example, as
the forces 1 to 5, Fig. 54, does not close, it follows that equi-
librium does not exist at the point O. In order to obtain equi-
librium it is necessary to apply a force 6 to the same point,
whose direction and extent correspond to the line 5 6 of the
polygon. This is the force necessary to bring the other forces
into a state of equilibrium, and from it wre also obtain a result-
ant force E, which is given in direction and absolute extent by
the closing line of the polygon, but acts as an expression of the
algebraic sum of the other forces, as shown by the arrow-head.
From this it follows that in every closed force polygon each
single force represents the resultant of all the others in absolute
extent and direction, except that the resultant tends to produce
motion in an opposite direction from the corresponding force in
the polygon. In an unclosed polygon the line necessary to close
the figure gives the direction and extent of the resultant of the
other forces, always tending to produce motion opposed to the
closing force.
For example, in Fig. 54, A2 is the resultant for 1 and 2, and in
a similar manner the resultant for any of the other forces in
combination may be found.
The method of representing the properties of forces by lines
is also applicable to other quantities which possess the attributes
of magnitude and direction, such as velocities ; also to the deter-
mination of the path of the line which passes through the cen-
tres of gravity of the stones of a vault, for instance ; and in a
figurative sense it may be applied to scientific discussions, in
which the final resuit acts as a closing line to the force polygon
of argument.
\ 33-
Isoeated Forces in One Pe ane. Cord Poeygon.
If lines which represent forces, and hold a body in equilibrium,
do not intersect in one point, a condition which frequently
occurs, but have a number of intersections from n to	(;/ — 1)
in number, the foregoing solution can no longer be used ; but at
the same time this more complicated case may readily be re-
duced to the simpler form.
For this purpose we assume the existence of a System of rigid,
straight lines which, extending from each force to the next,
form a polygon capable of resisting both tension and compres-
sion in the direction of its sides, and in %hich each single force
is in equilibrium with the two forces which act along the sides
intersecting it. A polygon formed in this manner is called aTHE CONSTRUCTOR.
2 7
Cord Polygon, or in arch construction a thrust line, because ali
tbe sides are in compression, and in general such a figure may
be called a link polygon.
The angles of the cord polygon are called “knots.” The
liyk polygon m?y be used for the investigation of forces accor-
ding to the preceding methods, when at each knot there exists
Fio. 55-
an equilibrium between the external forces and the stresses in
the sides of the polygon ; for example, when the forces SV2 and
S2.3t at the knot K2, ha ve a resultant equal and opposed to P2,
in extent and direction. The forces in the sides of the polygon
may be called the internal forces of the link polygon. We have,
then, for any given case two sets of forces to investigate :
(1) the external forces, (2) the internal forces.
since for each set there exists an equilibrium.
I 34-
EQUIU3RIUM OF THE EXTERNAE FORCES OF THE CORD
POEYGON.
If we take the forces P1 and P2, find their resultant, combine
this with P3, find a second resultant, combine it with P4, etc.,
we will find that in order to obtain equilibrium, the resultant
with the next to the last force Pn — 1, of the polygon, will be
equal and opposed to the closing force Pn. This holds good so
long as the direction and extent of the forces remains unchanged.
From this it follows that the co-ordinate distances of the point
of application of any force may be made equal to zeiro, without
affecting the equilibrium of the external forces. The combina-
tion of these latter forces may then be effected in the same
manner as if they acted at a single point. In this way the force
Fig. 56.
polygon can be used to determine the equilibrium of several
mdependently-acting forces. If equilibrium exists, the polygon
closes, and if it does not close, it shows the extent and direction
of the force necessary to maintain equilibrium. It is practica-
ble in this way to determine two unknown quantities in a force
polygon. These may also refer to two forces, and may be either
direction or extent, or, as sometimes occurs in practice, the
direction of one force and the extent of the other.
The following cases will serve to illustrate :
I.	Both directions given. In Fig. 56 we have the directions
of the force lines 4 5' and A 6', and by their intersection at 5, we
determine at once their length 4 5 and A 5. If their directions
are interchangeable we have two Solutions possible, the second
giving the directions A VI\ and 4 V'y and hence the forces
A VI and 4 V.
II.	The extent of both forces given. Fig. 57. With the dis-
tances equal to the extent of the two forces, we describe circular
ares from A and 4, and the intersection of these ares determines
the direction of the forces. Since the ares intersect at two points,
two Solutions follow, giving the lines A 5, 4 5, and A Vy 4 V.
III.	The direction of one force and the extent of the other
given. In Fig. 58 let the line 4 5 be the given direction of one
force. With a radius .<4 5, equal to the extent of the other force,
describe the arc shown by the dotted curve, and the two inter-
sections give two Solutions of the problem, as in cases I. and II.
If the arc failed to intersect the line at ali, it would prove the
case to be impossible.
'Fig. 59.	Fig. 60.
The following examples will show the practical applications
of the preceding principies :
Example I. A erane ABC, Fig. 59, carries a load L at A ; it is of a cylindrical
shape at B, and held in position by a roller bearing, and at C there is also a pivot
step. Required the forces Px and P2 at B and C. The centre of gravity of the erane
itself is at S, and its weight is equal to G.*
Both L and G act in a vertical direction, and the force at Plt if the bearing is
smooth and we neglect its friction, acts in a horizontal direction. Combining G and
L into one force Q= G 4- L, the position of whose resultant is T Q, we have the
intersection O of a vertical through T Q, with a horizontal through Px as a point in
the direction of the line of the force P.2. This force must also act through the centre
of the pivot C, since this is restrained from Iateral motion by its bearing. This gives
C O for the direction of the force P2. We can now draw the force polygon, Fig. 60,
drawing L + G vertical, G P\ parallel to O P\, and P* Px parallel to O C. This
determines the extent of both P\ P2, and by further analysis the entire load on the
pivot C may be found.
i
Fig. 61.	Fig. 62.
Example II. A erane constructed as shown ii» Fig. 61 carries a similar load to the
preceding. ^ It is arranged with a cylindrical bearing at B, and at C there is a conical
roller bearing upon a conical surface pn the base of the column, the axes of both
cones intersecting in the middle of the bearing B.
We have, as before, the mean load Q — L + G ; we also have the direction of the
pressure Px, as it must be normal to the surface of the cone at the point of contact.
The intersection of Px and Q determines O, and a line from O through the centre
of the bearing B gives the direction of P2. The force polygon can now be drawn, as
shown in Fig. 62, and by making the vertical equal to Q, and the other two sides
parallel to P\ and P2, we determine at once the extent of the two latter forces. The
vertical component of P2 will, in this case, be less than the total load Q, while in the
previous example they were alike. Hence it follows that the conical roller supports
a portion of the load.
Fig. 64.
Example III. The erane shown in Fig. 63 is similar in construction to the pre-
ceding one, except that the axes cf the conical rollers intersect at a point D below
the bearing B. If we now draw C O normal to the surface of contact C D, to the
point O, and construet the force polygon, Fig. 64, we see that the change in the posi-
tion of the apex of the cone D causes the force Po to act from below instead of from
above, as in the previous case. It will therefore be necessary to provide the bearing
B with a collar to oppose the upward pressure, f
♦ In ordinary wharf cranes the value of G, which mainly depends upon the capa-
city and overhang of the erane, may be taken at ^ to J the load.
f This defect may be seen in numerous existing examples of erane construction.
In a case which came under the author’s observation, a erane intended to have a
capacity of thirty tons gave way under a load of only about twenty tons, because the
proper provision was not made for the direction of a force upon a bearing.28
THE CONSTRUCTOR.
Example I V. Three forces of 70, 50 and 80 pounds act, as shown in Fig. 65, upon
a body AB in such a manner that their resultant passes through the point A. Two
other forces of 95 and 60 pounds also act upon the point A, and hold the preceding
forces in equilibrium. Required the angles which the latter forces make with the
former.
Lay out the forces of 70, 50 and 80 pounds, as shown from C to D in Fig. 66, by the
heavy lines, then describe from C and D circular ares with radii of 60 and 95 respec-
tively, and thus obtain the intersections E E' or F F, which, when juined with C
and Z>, complete the force polygon. Both Solutions are given in the diagram.
85
Example V. An obelisk isto be raised upon its base, Fig. 69, by turning it about
the angle A, the lifting force to be applied in a given direction at the apex B.
Required the direction to be given to a force Ps of given extent, applied to the
point A, in order that the base shall only be subjected to vertical pressure. Draw a
vertical line through the centre of gravity S of the obelisk, intersecting the direction
of the force P\ at O. A line from O through A will then give the direction of the
resultant of the two forces. This resultant is now to be resolved into a vertical com-
ponent P2, and a force Pz of given extent but undetermined direction. To deter-
tnine the direction we draw, as in Fig. 68, C Q and Q Plt and erect a perpendicular
through /V From Cwith a radius equivalent to P3, describe an arc, intersecting
the vertical at D and D', showing that two Solutions are possible—one giving P2 the
value P\ D and P3, the direction D C; the other giving P2 the value P\ D\ and
the direction Dx C.
If P3 should just equa! the perpendicular distance from Cto Px P2, then but one
solution exists. The two results for the example given are shown in Fig. 67 at A P3
and A P3.
Bxamples of this character seldom occur in actual practice.
2 35.
Equilibrium of Internal Forces in the Cord Polygon.
As already stated, we mean by the internal forces of the cord
or link polygon the tension or compression which may exist in
the different sides of the figure, as shown at SV2, ^2.3, etc., Fig.
69. These forces are of such an extent
1	that they hold each other in equilibrium at
the knots Kx K2 Kz, etc. Any two of these,
for example, >Si.2, S2.3, may be determined
from their resultant P2i when either their
^direction, their magnitude, or one direction
and one magnitude are given (see § 34).
This is done in the following manner:
Construet the force polygon, Fig. 70, of the
external forces Px, A. p» which, if equilib-
rium exists, will form a closed figure. From
Fig. 70.
the extremities of the sides corresponding to the force P2
draw two lines parallel to the sides Sx.2y S2.3, intersecting
at O; then the length of the lines Ol and 02 will represent
the magnitude of the stresses in the sides SV2, S2.3. In a
like manner we may draw lines connecting the several cor-
ners of the polygon, Fig. 70, with the pole O, and deter-
anine all the internal forces of the link polygon, both in mag-
nitude and direction; so that when the external forces are
known, and also the direction of two of the internal forces,
the direction and magnitude of the others can be determined.
This assists greatly in the construction of the link polygon, for
by selecting one knot and determining the pole O, the sides of
the link polygon can be drawn parallel to the respective rays.
The actual lengths of the sides of the link polygon are deter-
mined by the positions of the lines of the external forces, from
which the positions of the internal forces are also determined.
The cord polygon will vary in its form according to the choice
of a starting-point from which it is drawn. In Fig. 69 two
forms are shown in dotted lines within the cross-hatched figure,
their sides being parallel to those of the first polygon. Another
solution of the same problem (the combination of the external
forces into a link polygon) may be obtained by an application
of the double solution of Case I., \ 34.
In Fig. 72 we ha ve the directions Sx.2 and S2.3 drawn from the
etxtremity of the force P2} giving a new cord polygon, Fig. 71,
of a very different form from the preceding one, which is also
included in Fig. 71 for purposes of comparison. With the ex-
ception of the first two sides, we have an entirely difierentTHE CONSTRUCTOR.
29
The cord or link polygon, when taken in connection with the
force polygon, forms what has been termed the graphical plan
of forces. In most cases the entire subject can be discussed by
the construction of one figure which may then be called the
Force-pan, and of which examples are given in \ 48.
? 36.
Resultant of Isolated Forces in One Plane.
If we assume two of the sides of a cord polygon to be divided,
and insert at the points of division forces corresponding to the
stresses in the divided sides, the equilibrium will remain undis-
turbed, as, for instance, in Fig. 73, the sides Kx E6, and Kx
are cut and sustained. It will then be evident that the resultant
of the forces, Sv6 and S4.5, either on the right or the left will
hold the remainder of the forces of the polygon in equilibrium*
The position of this resultant force is determined by prolonging
the sides until they intersect at M. The direction and extent
of this resultant is determined in the force polygon, Fig. 74, by
the diagonal 4.6, which is the closing line of the forces Sx.6 =
Oe, and ^4.5 = Oq. This force is also on the one side the
resultant of the forcee P$ and P6, and on the other side, of the
forces Plt P2, /3 and P±.
In general it tnay be stcrfed that the point of intersection of
any two prolonged sides of the polygon is a poi7itof the resultant
of all the extsntal forces beyond ihese sides, from which the
direction and extent of said resultant may be determined.
This principle is of great utility, as many examples will hete*
after illustrate. By reversing the above rule, the cord and force
polygons may also be used for the decomposition of forces, as
well as their resolution. For instance, if it is desired to decom-
pose the force 4.6 into two others, P-„ and P6, of given direction,
draw one of them (for example, PG) in the cord polygon until
it intersects 4.6 in the point N, and through this point draw P*>,
parallel to the side 4.5 of the force polygon. The first chosen
line, K& N, may be drawn either forwards or backwards on M N,
without disturbing the equilibrium.
3 37-
CONDITIONS OF EQUILIBRIUM FOR ISOLATED FORCES IN
One Plane.
In the preceding discussions it has been assumed that the
forces whose equilibrium has been investigated were so situated
that equilibrium really existed, so that according to the rule in
the preceding paragraph it would be possible to reduce them to
Fig. 75-
two equal and opposing forces. This is, however, not neces-
sarily the case when the force polygon is a closed figure, but it
must follow when the cord polygon is also a closed figure, i. e.,
the actual positions of the forces must also be taken into
account. If the positions are not correctly taken, the cord
polygon will show what modification must be made in order to
secure equilibrium and avoid the formation of rotating couples ;
which will be discussed in the next section. For this purpose
one of the forces should be left to be determined in position a
the last.
I. Eet this force be Fig. 75- Its magnitude is known, and
its direction is parallel to the given line Z Z. After construct-
ing the force polygon, Fig. 76, choose a pole O, and draw the
rays to the angles from 1 to 6, so that Kx K2 is parallel to 1 O,
K2 K3 to 2 O, Kz R4 to 3 O, etc., until Kh K3 is reached. Then
the closing line of the cord polygon must have the direction
6 (9, and must also pass through Kx. This determines its posi-
tion entirely, and its intersection R6 with Kh is a point of
the force P6, which is now drawn parallel to 5.6.
If the final force is not given either in direction or magnitude,
it ma)T be determined from the direction and position of the
other forces as follows :
II. Eet the yet indeterminate force be P6, Fig. 77, while we
have given the direction of the force Plf which is Kx Plt and its
position Kv We can draw the force polygon from the points
y to 5. while from the point 1 we have only given the direction
A I. The cord polygon may also be commenced by starting
from Rlf and continuing through the points R2> Af3, KRb and
K'h. We may then select any direction for its closing side Kx Ly
and its intersection with Rs Kfh will be a point in the line of
the desired force P6. In order to determine its magnitude and
direction, draw, in Fig. 78, O 6 parallel to Kx L, and join the
point 5 with the point 6, when the line 5.6 will give the desired
magnitude and direction of the force Px.
«38.
Force Couples.
When a plane figure is subjected to the action of forces in
couples, acting in its plane in such a manner that, while equal
in magnitude and opposite in direction, they fall upon parallel
lines, and do not oppose each other in the same straight line,
the force polygon will close without necessarily proving the
existence of equilibrium in the figure.
The conditions which obtain under these circumstances may
be examined as follows :
The forces Px P3 and P2 P±, Fig. 79, form a closed force poly-
gon 1, 2, 3, 4, Fig. 80, but at the same time equilibrium does not30
THE CONSTRUCTOR.
exist in the figure, but instead, a tendency to rotate about a
common point with a statical moment which is equal to the sum
of the moments of the couples (Px — P3) and (P2 — P4). In
order to secure equilibrium it is necessary to introduce an addi-
tioual couple (P5 — P6), whose tendency shall be to cause a
rotation in the opposite direction, and whose moment shall be
equal to the combined moments of the previous couples, and
whose direction shall be parallel to the lines V V'and VI VI,
Fig. 82.
Fet us take, Fig. 81, the force polygon A 1, 2, 3, 4. This is
not vet complete, for we stili lack the forces 5 and 6. We know
that they must act through A, in opposition to the other couples,
but their magnitude is net yet determined.
As already said, the two forces must be equal and parallel in
order to be in equilibrium with the other couples, and only two
forces can fulfill the conditions. Their direction is given, and
can be laid off as at A Z. We choose any pole O, and join the
rays O A, O 1, O 2, O 3, O 4 (= O A), and can then proceed to
construet the cord polygon, Fig. 82.
For this we have lines of direction II, IIII, etc., up to VI
VI, given from Fig. 79. Starting from any point Kx on II, we
draw lines parallel to the rays O A and O 1 (their resultant
being the force Px) until they intersect VI VI in K3, and IIII
in K2; then draw K2 R\ parallel to O A, intersecting IIIIII at
A 3, K3 I\4 parallel to O 3 until it intersects IVIV, and K4 Kb
parallel to O 4, intersecting V V at Kh. Only the closing line
of the cord polygon is now lacking, as 't is the line joining Kh
with H6. which latter point has already been determined. We
can now (see \ 37, II.) draw the ray Oh parallel to Kh K6, com-
pleting the force polygon and the line A 5, will give the magni-
tudes of Pb and P6. The path around the force polygon may
be taken as A 1, 2, 3, 4, 5 A, the sides 4.5 and 5 A being sup-
posed to make au infinitely small angle with each other.
The previous examples upon the force and cord polygon serve
to show how geometrical addition and subtraction may be used
to determine the equilibrium of diverging forces in one plane.
Forces acting in intersecting or parallel planes may be examined
in the sanie manner, and in rnany cases without a great degree
of complication, as some follownng examples will illustrate. It
is not intended, liowever, to undertake a general discussion of
the subject here, but rather proceed at once to practical appli-
cations of the special case of parallel forces.
2 39-
Equilibrium Between Three Parallel Forces.
In discussing the equilibrium between parallel forces, we may
use purely arithmetical methods, or use geometrical addition
and subtraction (force and cord polygons), as may be found
most convenient.
The present problem may be stated as that in which a force Q
acts upon a body, and is to be held in equilibrium by two un-
r

Fig. 83.
known forces, Px and P2, acting parallel to it and to each other.
Drawing the line ABC, Fig. 83, normal to the given direc-
tion of the forces, wre must have, in the existence of equilibrium,
Px. A B = P2. B C, or Px ax = P2 a2, and also Px + Pt = Q.
In order to determine Px = P2 graphically, we may follow
the method in \ 24, and in Fig. 84 make O E = the divisor alt
O A = the factor a2, and taking £ B to represent temporarily
the force P2, draw A C parallel to E B, which gives the propor-
tional value of Px. By placing the triangle C A O in the dotted
position O' B A'f we have A/ E = Px + P2 = Q. This gives
a figure in a form well suited for application to Fig. 83, as will
be shown in the following examples :
I.	In Fig. 85 draw A D equal in value to Q, join D with the
third point of application C, and prolong Q until it intersects
at Ea line drawn through D parallel to A C. Then will we
have the following relations, B E = Pv E F = P2. In Fig. 86
is shown a similar case, but with Q inclined to A B C, and in
Fig. 87 Q is beyond A C
II.	By resolving the force Q into two components applied at
the points A and C\ Figs. 88, 89, 90, we obtain inclined forces
whose components parallel to Q are the desired values for Px
and P2, while the components which are parallel to A B C neu-
tralize each other. In all three figures B F— Px and F D = P2.
III.	By constructing the force polygon, making A D — Q,
and using any pole O, Figs. 91, 92, 93, and drawing the sides of
the link polygon, so that A b is parallel to A O, b c parallel to
Fig. 91.	Fig. 92.	Fig. 93.
D O, and joining the closing line c A, the parallel to the latter
in the force polygon O E will give E A — Px, and D E — P2.
If it is desired to make the closing line fall upon A B C, or
lie parallel to it, the cord polygon A b C must be first drawn,
and the pole O, determined by the intersection with A B of a
line D O parallel to b C, D A having first been drawn equal to
Q, O E may then be drawn parallel to A b, and we have
E A = Px, and E D = P2.
In these cases Q is equal in magnitude to the resultant of Px
and P2, and opposed to them in direction. If Q is to be deter-
mined when Px and P2 are given, similar methods to the fore-
going are to be followed.
Returning to the diagram OEA C B, Fig. 94, which we have
already used in case F, we construet the triangles C A O and
B A' O', and draw B' C parallel to O A ; O' C' and O B' par-
allel to A' B, giving B' B = ax, B C' = a2, B/ O = P2i O/ Cf
— P\'
From this we obtain the following Solutions :
IV.	Transfer one of the forces to the opposite side of A C,
r8
Fig. 98.
Figs. 95, 96, so that A D — P2 and E C = Px, join D to E, andTHE CONSTRUCTOR.
3i
the line D E will intersect A 0 at B, which will be the point
of application of the resultant Qy whose magnitude = E D'
— P1 4 P2> since D D' is drawn parallel to A C.
In Fig. 96 Px and P2 act in opposite directions, and their alge-
braic sum D' E must be taken, and, as shown, the resultant Q
acts beyond A C.
V.	The method shown in Fig. 97 follows from (II) :
From the extremity a of a A = P1 draw a line A/ a of any
length, making it parallel to AC. In a similar manner draw
c C' from the extremity of c C — P2. Draw A/ A and C' Cy
prolonging them until they meet at F, which latter will be a
point in the line of the resultant E B, and the value of 0
will be P1 4 P2, which is also the resultant of D E = C' C and
E F— A' A.
VI.	Following the method in (III), we may proceed as fol-
lows, Fig. 98 : Make D E = P2y E A = PXy choose a pole <9,
and join the closing line O E of the force polygon. Draw A c
parallel to E O, c b parallel to O D, and A b parallel to (or, as
in this case, the prolongation of) A O, and the intersection b
will be a point in the line of the resultant Q, whose magnitude
= D A.
\ 4°-
Resuetant of Severae Paraeeee Forces.
When we have a number of parallel forces QXy Q2y 03, Q4,
acting upon a body in given positions in one plane, we can
determine their resultant by a combination of the preceding
methods, resolving them in pairs until ali are combined.
I. In order to combine the forces Qx to Qiy intersecting a
common normal A F, Fig. 99, we first combine Qx and Q2 by
transposition, as in Fig. 96, and obtain the resultant, Qx 4 Q2
= b c. This may then be combined with 03, giving d d' =
Qi + 02 + 03» and this resuit with 04, which finally gives the
resultant, Q = Qx 4 02 4 03+4 passing through M. This solu-
tion is one which is sometimes desirable in machine construc-
tion, as, for example, in the distribution of the weight of a loco-
motive engine upon the various axles. The method of deter-
mining the resultant of several parallel forces in this w^ay by
the successive combination of pairs is very tedious and of limited
application, and the method given below of using the force and
cord polygons is much simpler.
Fig. i 00.
II. Fig. 100. Form the force polygon of the given forces Qx
to 06, by laying off lines successi vel y from A, equal in length
to the magnitudes of the several forces A 1,2, 3, 4, 5, 6, as
shown in the left of the figure. The magnitude of the resultant
will then be equal to the length of the closing line £ A.
To determine its position, proceed as followrs : Select any
point beside the line A 5, as a pole O, and join the rays O Ay
O I, O 2, O 3, etc. Starting from a point b under QXy draw b b'
parallel to A O, and b c parallel to 1 O, and continue by drawing
cd parallel to 2 (9, d e parallel to 3 (9, etc., and finally reaching
the closing line of the polygon g g' parallel to O 6, intersecting
b b' at q, which determines the position of the resultant Q (see
2 35)-
The method shown in (£ 36) may also easily be applied to the
resolution of such forces, as in Fig. 100, the intersection of d c,
prolonged to c' y gives the position of the resultant of Qx and Q2,
and its magnitude is showm at A . 2 in the force polygon, and in
a like manner e' is the position of the resultant of Q± and 05.
, ? 4i-
Decomposition of Forces into Two or More Paraeeee
Forces.
The methods of resolving forces by means of the cord poly-
gon will also serve for their decomposition.
If, for example, in any portion of a cord polygon a q b c d,
Fig. 101, it is desired to substitute for a force Qy two forces Qx
and Q2 passing through e and fy we have only to join the points
e and f to obtain the form of cord polygon for the new forces,
a
aJ'
Fig. ioi.	Fig. 102.
and determine their relative magnitudes by drawing 01 parallel
to ^y*in the force polygon below7. If the required force Qx and
02 both lie on the same side of 0, Fig. 102, the solution is
similar. We now prolong a q to its intersection e writh Qlt and
join e f. Also mark the intersection of Qx with q by and Q2 with
q a. In the force polygon below we have Qx = A 1, 02 = 1 . 2,
Fig.103.
Fig. 103, whose load is to be opposed by reactions Px and P2 at
A and Gy wTe may first determine a resultant 0 of all the forces,
as in (§ 40), and then decompose this into values for Px and P2
by the method just given. We also omit the determination of
Q altogether, and proceed to determine Px and P2 directly as
followrs:
Choose any pole (9, and form the force polygon K1 . 2 .... 5 (9,
and construet the cord polygon, making its sides parallel to
their respective ra}7s, and draw b a parallel to K O and f gy par-
allel to O 5, their intersections with the lines of the forces Px
and P2 being a and g. Join ag, which will be the closing line of
the polygon, and its parallel 06 in the force polygon gives P2 =
5. 6 and Px = 6.7. If the sides a b and f g of the cord polygon
are prolonged in the other direction w7e obtain a' and gfy giving,
how7ever, the same resuit, since a' g' is parallel to a g. The
cord polygon would then be the figure a' g' mbde e f 111 a', and
m indicates the position of the resultant of the forces Qx to
or of Px and P2.
When a loaded beam is supported by three or more bearings
it is necessary to take into account the resistance of the beam
itself wTith some degree of accuracy, or else the problem be-
comes indeterminate. This indeterminate character may, how-
ever, be eliminated by the introduction of an equalizing lever.32
THE CONSTR CCTOR.
Suppose we have, Fig. 104, a beam B CD} the resultant Q of
whose entire load acts at AIf and is opposed by the reactions of
three supports at PXf P2t P3) at right angles through the points
Bf C and D.
We may now assume, temporarily, an approximate ratio be-
tween two of the forces, e.g.f P1 and P2, and permit them to act
at the extremities of an equalizing beam Bl Cly which in tum
supports the main beam at E Ex; making the ratio of Ex C\ : Ex Bx
the same as has been chosen for Px : P2. Nowdecompose Qinto
the components acting at E and D by means of the coiu and
Fig. 104.
force polygons e nt d and A O 12. This gives A 1 = Qy 1.2 =
P3) 2 A = Px + P2y which last sum may be then divided between
Bx and Cx by any of the above methods. p
Fach different approximation of the ratio will give a dif-
P1
ferent value for P3 If Px and P2 are made equal to each other,
E will be in the middle of B Cy and the equalizing .lever will be
of equal arms. The distribution of the load of locomotives and
cars upon their spring is usually made with such equalizing
levers.
If the load is to be supported upon more than three or four
points it will be necessary to use several equalizing levers, and
Fig. 105.
examples of this will be found in some locomotives. If, for
example, we suppose My Fig. 105, to be the point of application
of the total load Q of a locomotive, supported upon three axles
B C D in such a manner that the weight shall be transmitted to
the axles through the springs as shown, and also that the
weights upon the wheels C and D shall bear a determinate rela-
tion to each other. This can be accomplished by the use of
three springs and one equalizing lever upon each frame of the
locomotive, the whole weight being thus supported upon eight
points. Taking the relation between the forces P2 and P3=p :qf
we erect a perpendicular E ey whose distance from the axle C
and D is in the proportion q : p. From any point e' on this
line draw lines to the bearing points of the wheels upon the
rails, and any horizontal line will intersect these inclosed lines
in points which will give the proportional length of arms c c d
for the equalizing lever. The distances of the points c and d
from the verticals through C and D give the length of the arms
of the springs cx c2 and dx d2. These springs are made with
arms of equal stiffness, since they are to support equal loads at
both ends. For any chosen ratio p : q, and given distance be-
tween the axles, the actual length of the equalizing lever will
not affect the ratio of Px to the sum P2 -|- P3) as an inspection
of the cord polygon b m c d will show.
The springs which are attached to the ends of the equalizing
lever must, of course, be made of sufficient stiffness to support
the load which is thrown upon them, and the length of the sup-
ports and their proportions chosen according to the previously
determined distribution of the weight.
Many similar examples to the preceding might be given, as
they are of frequent occurrence in practice. The two springs
•which are attached to the equalizing lever may be replaced by a
{Single spring, as in Fig. 106. In this case the axes C C are con-
nected rigidly to the lever b e cy and the lever itself rests upon
a Spring bx ex cx, whose extremities are fastened to the frame
The arms bx ex and cx ex of the spring are of unequal lengtn,
and have the same relation p : q as that which exists between
the arms of the lever b e c. If the arms of the lever are not
properly proportioned, or if any error has been made in the dis*
tribution of the load, it will be made apparent by the inciiaed
position which will be assumed by the equalizing lever.
i 42.
Uniformi/v Distributed Paraeeee Forces.
When a beam is subjected to a uniformly distributed load, the
force and cord polygons cannot be determined by the preceding
methods, since in such cases the cord polygon becomes a figure
of curved outline. The character of the curve may be deter-
mined in the following manner: If we assume the load to be
concentrated at a number of equidistant points, as in 1, 2,---9,
Fig. 107, and construet the cord polygon for these conditions, it
will be evident that the sides a AI and b c will intersect midway
between 1 a and 2 by and also midway between a b'y since the
forces 1 and 2 are equal to each other. In the same manner c d
and a AI intersect midway between 3 c and 1 a, which is also in
the line of 2 bf that is, at 6', and likewise d e and a AI intersect
midway between b' and c'. In this way it may be shown that
the intersections of the prolonged sides of the polygon from
a AI to i AI are at equal distances from each other. This indi-
cates a known property of the parabola wffiose vertex lies on
Ej ai
line E AI, and whose abscissa e E = —-—. This parabola is
the form assumed by the cord polygon when the load is uni-
formly distributed, as was previously assumed. If wTe note that
the triangle A AI B represents the entire load collected at Ey it
•will readily be seen how the curve may be drawn in any case.
If the chord A E B is inclined, as shown in Fig. 108, the divi-
sions of A M and M B will be equal in number, but the divi-
sions of A AI will be of different size from those of AI B. The
point e lies in the middle of E AIy but is not the vertex of the
parabola.
Link polygons which assume the form of curves may also be
used to show the effect of moving loads, and are then the figures
which are contained within the successive sides of a regular
polygon. Many examples are to be found in the case of railway
bridges, traveling cranes, engine guide bars, etc.
§ 43-
The investigation of the action of parallel forces, such as Qx
to Qi and Px P2y Fig. 109, whose direction is normal to a beam,
requires a knowledge of the statical moments of the external
forces. These can best be obtained by use of the force and cord
polygons. After constructing the force polygon A O 4, and cord
polygon ab cd e fy let it be required to find the statical moment
for any point ^ upon the beam. This moment is the product of
the resultant of all the forces upon one side or the other of the
line ^ Sx into the lever arm l of this resultant from 6* Sv
The magnitude of this resultant is obtained from the distance
h i = i. 5 in the force polygon, cut off by the rays O 1 and O 5,THE CONSTRUCTOR.
33
which are parallel to b c and f a, and its point of application is
determined by prolonging these sides until they intersect at g.
By drawing the perpendicular g go., the lever arm / of the re-
sultant P — h i is determined, for the force acting at the point
S, and hence we have M — P l.
This multiplication may also be performed graphically. By
drawing the perpendicular O k in the force polygon, we obtain
the altitude of the triangle O h i from the base h i, and this tri-
angle is similar to the triangle g s s0% whose altitude is l. Call
in O k = H and s sa = t, we have
P:H=t:l,
or M=Pl = Ht.
This proves that the statical moment at anypoint in a beam is
proporiional to the corresponding ordinate of the cord polygon,
parallel to the direction of the force s, since H is a constant.
By making H equal to unity the conditions becotne similar to
Case I., \ 22, in graphical multiplication, and the moment M
becomes equal to the ordinate t. It is not necessary to deter-
mine the position of the point of application g of the resultant,
since it is the relation between the statical moments which is
most desirable, whether H be chosen as a unit or not. This
property of the cord polygon for parallel forces is most useful,
and an example may be found in the case of axles.
For such cases as for many others, it is most useful, since no
modification of the diagram is necessary, the moments being
found by the same construction which is required for the deter-
mination of the forces. It is often convenient in practice to
cover the figure containing the moment ordinates with section
lining or with a light tint of color.
2 44.
Composition and Decomposition of Staticae Moments.
As shown in the preceding section, statical moments may be
shown by means of lines of definite length and position in the
same manner as simple forces. When two statical moments act
in the same or in different directions, they may be combined by
means of graphical addition in the same manner as has already
T
Fig. i io.
been shown in \ 31. Ii A B C and A D C, Fig. 110, represent
the cord polygons for two sets of parallel forces which act nor-
mal to the axis of a revolving body A C, in the directions A' B?
and A' D' we have the following method : For a point .S on the
axis of the body we have the triangle Tx S T/, in which the
angle 0 = Br A' D' and Tx Td = 5* T= t for the desired
moment. The combination of the cord polygons ABC and
A D C, which may be called the moment surfaces, will give the
resultant moment surface A T U C. The sides A 7 and C U
are here straight lines, while T Uf is a curve, in most cases a
hyperbola. In actual practice the straight line joining T and U
may be used with but little error, and its detailed construction
is unnecessary.
By a reversal of the above construction it is possible to decom-
pose any given statical moment t into two others, tx and i2} if
their directions be given.
2 45-
Twisting Moments and their Graphicae Combination
with Bending Moments.
Next in importance to bending moments, and often acting in
combination with them, are twisting moments. In Fig. m let
A B C D be the axis of a rotating body, subjected to bending
forces at B C} and supported at A and D; the force polygon
being represented at A O 2 and the moment-surface at A b cD,
and let the portion between B and Cbe subjected to a twisting
moment P. Ry and the moment-surface of the latter be required.
According to $ 43, and the method of multiplication given in
Rule I., I 22, we find a line corresponding in value to P P by
laying off in the force polygon A p = P, joining the ray O py
prolonging O A and O p, and drawing q r parallel to A p at a
distance equal to R, giving a length q*r equal to P R upon the
same scale used for the polygon A b c D. The moment-surface
for the twisting between B and C will then be included in the
rectangle B Cvu, whose altitude B u — Cv=qr. In common
practice itis desirable to convert this torsion surface into one
representing equivalent bending moment. This may be done
by taking a proportional value which shall give the same
security as the bending moment. It has been shown in \ iS
that the latter is equal to -|- the twisting moment. We may
them make B ux= C vx = ~~ B u, in order to obtain the mo-
ment-surface of the bending moment between B and C, wThich
may be measured upon the same scale as A b c d.
If we wish to combine this with the given bending moment
we may do so graphically by first using the formula IV. of the
table of l 18, p. 60, in which the ideal bending moment for the
combined action of a twisting moment Md and a bending mo-
ment Mb is :	______4______
Mi =	Mb + ^ + Md1.
In this case we make B b1 = -f- B b, = C c, E e =
JL E e, etc., rotate B ult C vl and E wlt down upon A D, and
add the hypoteneuses bx u/, v/, ex w/ to the lines b blt	e ex.
The combined length of these lines gives the length for the
ordinates at B,Cand D,from which the resultmg ideal cord
polygon	B b b' e' c' c D may be constructed.
v 2 46.
Determination of thf Centre of Gravity by means
of the Force Pean.
The position of the centre of gravity of a plane figure may
often be very conveniently determined by means of the force
plan. This may be done by dividing the figure into a number
of strips of uniform width, such that their area may be con-
sidered as proportional to their middle ordinate, constructing34
THE CONSTRUCTOR.
the force and cord polygons, and taking the line of the resultant
as a line of gravity. If the figure is not symmetrical, it will be
necessary to divide the figure again in another direction and
determine another line of gravity, when the position of the
centre of gravity will be found at the intersection of the two
lines. For figures of simple form larger determinate sections
may be taken instead of strips, their area determined in any
convenient manner, and the diagram constructed accordingly.
Suppose, for example, that it is required to determine the
position of the centre of gravity of the T-shaped section shown
in Fig. ii2. The figiire is symmetrical about the axis Y Y, so
that the centre of gravity must lie somewhere in that line. We
may divide the figure into the rectangular b X c, bx X o1 and
b2 X g, which we will call respectively the areas i, 2 and 3.
We have also given c = 1.5 b2 and cx = b2. This gives the
three forces as 1.5 —

and —, which are then laid off at
g_
2 ' 2*2
A 1 2 3, a pole O selected, and K/ Kx drawn parallel to O A,
Kx K2 parallel to O 1, K2 K% parallel to O 2, K3 Kf parallel to
O 3, when the intersection of the sides K\ Kf and K3 K3' at M
gives a point on the line of gravity M M', wdiose intersection
5* with the axis Y Y is the centre of gravity of the figure.
§ 47-
Resultant oe the Load on a Water Wheel.
It is very important in designing a water wheel to be able to
determine the position of the resultant of the w7ater acting upon
it, and the method of doing so wdll furnish an excellent illus-
tration of the application of the principies of the preceding
sections.
Fig. 113.
In the breast wheel, which is shown in Fig. 113, there are ten
buckets in the half section, the third from the top being the
first to receive a charge of wrater, the amount being eStimated
from a previously determined coefficient. The level of the
water in the succeeding buckets may be considered as horizon-
tal, and the discharge from the buckets is prevented by the cul-
vert K L, so that if we neglect the leakage arouud the edges of
the culvert we may count that all the buckets from No. III. to
No. X. contain the same load of water, acting in each case as if
its weight were concentrated at the centre of gravity of each of
the respective prisms of water. Bucket No. XI. we may con-
sider as entirely empty.
I.	Determination of the culvert arc K L. The contents of a
bucket section are determined by the cross section contained
between two adjoining bucket divisions prolonged, as govemed
by the coefficient of charge, = 0.04. Now, in bucket I. lay off
k i = 0.4 of the bucket spacing, and draw l m radial; then the
section k l m n will represent a bucket charge. In bucket II.
its figure assumes the shape r p u t, in which the angle t is the
beginning of the scoop of the bucket r t, and k tu will be equal
to the desired culvert angle K M L, and u t M will be equal to
the complement N M K.
The construction is as followTs : In the right angled triangle
o p q make o p — the middle breadth of the figure k l m n, and
also make p q — 2.1 m \ then transform this triangle into one of
equal area, r p s (by drawing o s parallel to rq, and joining rs,
see § 25. Join s t, and draw r u parallel to it, and join u and t,
and if we neglect the curvature of p u, w7e may consider the
quadrilateral r p ut as the form of a filled bucket just at the
moment of discharge. This requires the angle K M N=ut My,
but owing to the splashing of the wrater, the culvert is raised as
high as/.
II.	Determination of the water level in the various buckets.
We will begin with bucket IV. Here the figure r p t uis again
drawn, and the line t' v, and its parallel w u', determined ex-
perimentally, so that the diagonal w v shall be horizontal, which
may readily be done after a few trials. Proceed in the same
manner with buckets V., VI. and VII.
In bucket III. the figure r p u t is first converted into the
quadrilateral with the upper line p x, and this then into the
pentagonal figure, with the level upper liney z.
In bucket VIII. we first get the figure with the upper linep1 xlt
and then from this the figure with the level upper line y\ zx.
Proceed in the same manner for buckets IX. and X.
III.	Force plan for the water load. Now determine the cen-
tre of gravity for each loaded bucket, and also lay off the force
polygon A O 8 for these eight forces. From this constant the
link polygon d b e a/g hi, according to the methods previously
given, and i, will be a point in the resultant of all the forces.
It is to be noticed that the centres C and D fall so nearly in the
same line that their forces have been united, so that i d is par-
allel to A O, d b to O 2, and the intermediate parallel to O 1
omitted.
Suppose, now7, that it is desired to determine completelv the
position of the centre of gravity P, of all the prisms of water.
Draw7 through A, B, C, etc., horizontal lines, assume the force
polygon A 08, to be tumed around 90°, and draw a second cord
polygon ; or, what is shorter, draw the second polygon wdth its
sides normal to the rays of the force polygon, giving the figure
a'b'c'd'e'f'1--i/. A horizontal through i will then intersect
the vertical which was previously determined, and so fix the
position P, of the centre of gravity of the entire mass of wTater.
By taking the buckets in a different position, a slight difference
in the position of i P, may be found ; but in most cases the
deviation will be very slight.
,§43.
Force Plans for Framed Structures.
Framed structures are of very general application wffierever
loads are to be supported, and their discussion may be classified
as a system by itself, while their use extends from the simple
trussed beam to the bridge and roof truss; also for wTalking
beams and many other uses.
The tensile and compressive stresses in these various forms
may readily be examined by means of the force plan, w7hich
consists of both the force and cord polygons and their modifica-
tions. The subsequent examples will serve to illustrate the
Principal cases. In all of these cases it is assumed that at the
knots-f. e., at the points wffiere several members meet,—a joint
is supposed to exist; or at least no account is taken of the re-
sistance to bending at the knots.
In order to form such a plan for any given construction, it is
necessary first to determine the division and direction of the
forces, and then, beginning at one of the external forces and
laying off its direction and magnitude to the next knot, com-
bining it there with the external forces at that point, laying off
the resultant to the next bend, etc. Upon such combinations
of force triangles or quadrangles the force plan is constructed.
If it is desired to determine the directions of the components
of a given or determined force, the principies laid dowm in § 32
must be borne in mind. These may be generally expressed in
the followdng rules :
If one force is to be separated into two or more forces, its di-
rection is to be reversed and it is to be made the closing line
S' in the paths of the other forces.
If two or more give?i forces are to be conbined with two or
more other forces, the force polygon will consist of the given
forces and their closing line S.THE CONSTRUCTOR.
35
The first rule is only a special case under the second or
general rule, since the single force may be considered as an un-
closed force polygon whose closing line passes backward over
the same path to the starting point.
Fig. i 14.
Fig. i i 5.
In the investigation of each member in a frame without error,
it is best to assume the member to be cut, and to determine the
external forces at each section which oppose the internal forces;
the direction of the forces may then also be determined with
precision.
2 49-
Force Pean$ for Framed Structures.
I.	Simple Trussed Beams. Fig. 116. The beam ABC is
supposed to carry at B a load equal to 2 P, acting in a direction
normal to A C, and to be supported at A and C. Since A B
= B C, the reaction at each support is equal to P. It is then
required to determine the stresses upon the various members
from I to 5, as marked in the figure.
Referringto the diagram marked a, let a b be the reaction P\
•which acts upward at A. We now ha ve to construet a diagram
of the internax lorces acting in A B and AD. To simplify mat-
ters, we will give these forces the same numbers as their corre-
sponding members; drawing 1 parallel to A B, and 2 parallel
to A D. The direction of the force P\ in the closing line of the
force-triangle, determines the direction in the other two sides, as
shown by the arrows, by the lines 1 and 2. (See § 48.) In this
case there will be compression in AB and tension in A D.
In order to show this clearly, in all the following strain dia-
grams the forces acting compressively in struts or pests will be
indicated by double lines, while all tension members, links or
rods will be shown by single lines.*
Following out this idea, we shall, in the following illustrations,
show all struts or compression members in the construction
drawings as having a measurable thickness, as if made of wood,
while the tension members will be represented by simple lines,
although this is not intended to indicate any limit as to the
choice of materials.
For the knot at B we make a b c — 2 P, and, following in the
direction d a c (because the thrust is from A towards B), and
join the closing lines 3 and 4, both of which represent compres-
sion. The combination of 2 and 3 determines 5, which is ten-
sion. This gives an entirely symmetrical plan, which was to be
Fig. i i 7.
expected from the symmetrical form of the structure, and an
investigation of one-half is practically sufficient.
If the load 2 P is taken as uniformly distributed over the
entire distance ABC, instead of being concentrated at B, the
p
reactions at A and B will each be equal to , and the load at
B = P, so that y2 of the load on AB and B C is referred to the
knots A, B and C. From these conditions we obtain the force
plan b, which is geometrically similar to the other, but only half
as large.
This distinction has been suggested by Culmann.
II.	Double-trussed Beam (much used for constructions of all
sizes). Fig. 117. In this case take vertical forces P\ at B and
C, and corresponding vertical reactions at A and D. In the
first force plan, a is drawn equal to P, and 1 and 2 parallel re-
spectively to A B and A E, thus determining the forces 1 and
2 ; 1, being compression and 2, tension. Lines now drawn par-
allel to B E and E F, determine the compression in 3, and the
tension in 5, while the compression at 4 is the closing line of
3, 1, and P; and the other half of the diagram is similar. If the
vertical forces at A and B are not of the satne magnitude,
which is often the case in practice, the structure should be
strengthened by the introduction of the diagonals E C and
B E
The second diagram shows the construction in this case. Let
P1 = aY bx be the force acting at A, and P2 = a2 c2 at B. Draw
a vertical line from 1 to a horizontal through Clf which gives
the length 3, of the vertical force at B, and by drawing the
dotted diagonal line their resultant is found. If any of the ten-
sion members are omitted the framework will tend to take an in-
clined position until the various parts are at such an angle with
each other that both constructions will give the same value for
3. For this reason it is best in nearly every case to use the di-
agonal counterbraces.
III.	Triple Trussed Beam. Fig. 118. The uniformly distrib-
uted load upon the framework gives the following distribution of
forces. The force 3 P = a b c is first decomposed in 2 and 1,
or ce and e a ; then 1, is connected to a b = 2 P, by the line b e,
and this latter decomposed into 3 and 4 or e f, and f b ; 2
and 3 are now joined by f c, and the components at 5 and 6
or f g and^ c found. Since 6 and 10 are equal to each other,
we may draw c h parallel to G H, and equal to c g, which gives
g h = 7 ; the rest of the force plan is similar to the first half.
IV.	“ Another form of Triple Trussed Beam is shown in Fig.
119. The space between B and C is twice as great as between A
and B, and the uniformly distributed load is equal to 12 P, act-
ing at the various knots as shown in the figure.
In the force plan, make a b c = 5 P, and draw parallel to 1
and 2, the lines a e and e c; then join 1 with 3 p (for the knot
at B), and decompose into 3 and 4, or e y*and f b. Now com-
bine 2 with 3, giving c f, and draw 5 and 6 parallel to P P and
F G, respectively. This case differs from the preceding, in that
5 is now compression instead of tension. The equality of the
forces 6 and 10 gives g h = 7, and the similar half of the dia-
gram need not be drawn.
V.	Multiple Trussed Beam. Fig. 120. The beam A J is
divided into eight equal parts, which are represented as being
uniformly loaded, the load at each knot being shown in the
figure. In constructing the force plan we make a e = 7 P, and
by drawing the lines parallel to I and 2, we obtain a f and f e ;
then lay off a b — 2 P, and join the resultant b f. This decom-
poses into 3 and 4, or f g and^ b. The forces 2 and 3 combine
to give the resultant g e, which, by drawing lines parallel to
K C and K L, gives g h and h e for the values of 5 and 6. We
now find that to proceed further we have three forces of given
direction only, and since this is indeterminate, we must obtain
one magnitude as well. This, for example, may be done for
the force 7 as follows : The strut C L sustains the vertical com-
ponents of 5 and 9, as well as its own direct load 2 P. Now 5
and 9 are equal to each other, since they are placed symmetri-
cally, and carry equal loads from the struts B K and K M.
Hence in the force plan we may make h i, which represents the36
THE CONSTRUCTOR.
force 7, equal to twice the projection of 5 upon Ilie vertical
+ ,2 P-. This we can now combine with 6 — h ey giving i e,
which in turn decomposes into i m and m e, or io and n.
Returning to the knot Cy we may now take the line h i, and by
drawing parallels to CLy C M and C D, obtain the figure h i k c.
which determines the forces 8 and 9. In the same manner pro-
ceed from 12 to 15, which will complete the half plan. It may
be noted that the principal beam^ yissubjected to a uniform
compression throughout its entire length.
The force plan will, of course, be modified by various dis-
tributious of the load, as in the case of simple beams, as shown
in cases XII. and XIII., \ 6.
2 50.
Force Peans for Roof Trusses.
Roof trusses furnish many and varied examples of frame-
work.* In the following examples a uniformly distributed
2P
vertical load is assumed, so that the burden upon any portion
of a rafter may be considered as proportional to the length of
that portion.
I.	Roof with Simple Principals. Fig. 121. A uniform load
2 P upon each half gives as the external forces P, 2 P and P at
Ay B and C. Lay off in the force plan a b = P, and draw a c
and b c parallel to A B and A Cy determining the forces 1 and
2 ; I being compression and 2 tension. Then draw the vertical
c ey and also draw b e parallel to C D, thus giving both 3 and 5,
and the diagram is completed by drawing d e.
II.	Roof with Single-Trussed Principals. Fig. 122. This
form is similar to the preceding, w ith the addition of the struts
C E and C F. The distance A E is to E B, as 3 is to 2 ; and
the loads upon the respe 'dive portions are 6 P and 4 P, which
give the forces at the various knots as shown in the figure.
Make ac in the force plan equal to 7 P\ and by drawing lines
paraliel to A E and A Cy obtain the forces 1 and 2, or ad and
d c; then combine 1 with 5 P = a b, and decompose the dotted
* Many subjects for Graphical Analysis may be found in Ritter’s “ Roof
and Briclge Construction,” Hannover, 1863, in which the forces in the vari-
ous members will also be found carefully determined numerically, thus
affording convenient proof.
resultant into d e and e b respectively parallel to E C and E Br
giving the forces 3 and 4, both being compression. By repeat-
ing 2 and 3, in drawing 7 and 8, we obtain the figure ede fg9
in which c g gives 5. This latter force might also have beea
2P
Fig. 123.
found by combining 4 and 4 Py and decomposing the resultant
by lines parallel to B C and B Ff an illustration of the various
methods in w7liich the force plan may be used.
III.	Another form, with Single Trussed Principals. Fig. 123.
This roof is similar to the preceding except that the struts E
C and C F are placed horizontally. In this case A E = E By
and the external forces at A and D are both equal to 3 P.
The forces from a to c in the force plan are determined as be-
fore, giving d a and c d for the forces 1 and 2, and the com-
bination of 1 with 2 P gives the resultant d b, from which the
thrusts 3 and 4, or de and e by are obtained. The value of 1
is the same as 3, and 8 is the same as 2 ; wiiile 5 is the closing
line oi c d e d fy or of c d f The force 5 must also be the com-
bination of the equal forces 4 and 6 with 2 P, which the dia-
gram shows to be the case. If the rod C B is omitted, as is
frequently done, the strut E C Fy if there is no joint at Cy will,
oppose its resistance to bending to the force 5 ; but there will
be a tendency to rise at the apex B, if the fastening be not made
sufficiently strong.
IV.	Third Roof with Single Trussed Principals. Fig. 124.
In this form of truss, frequently known as the Belgian or
French truss, the single vertical rod of the preceding form is
replaced by a triangle B C D. The struts are placed in the
middle of the rafters and the external forces are distributed as
shown in the figure. In the force plan a b c = 3 Py and 1, and
2 are determined as before. By the decomposition of the re-
sultant of i and 2 Py wre obtain the forces 3 and 4, or d e and
b ey and from the resultant e cy of the forces 2 and 3, we get the
tensions 5 and 6, in c f and e f. The second half of the dia-
gram is the symmetrical counterpart of the first.
V.	Roof Truss with Double Trussed Principals. Fig. 125.
This construction does not differ greatly from that shown in
Fig. 124, except that the struts employed to strengthen theTHE CONSTRUCTOR.
37
rafters are divided into two. The spaces are equal to each
other and the load uniformly distributed. As shown in the
figure this gives a reaction of 5 P, or A and D. In the force
plan a d = 5 P, and lines parallel to A E and A C drawn, de-
termining the forces 1 and 2, or d ^and e a. We then combine
£ a with a b — 2 P, and decompose the dotted resultant e b,
into the thrusts e f and f b, or 3 and 4, by drawing these lines
parallel to E C and E F. Again we take the resultant of the
forces 4 and 2 P, and decompose it into 5 and 6, or f g and
g c, which brings us to the middle of the symmetrical figure.
The force 7 is the resultant of 6, and its counterpart 8, and the
load 2 P, and the half of this force is therefore equal to the pro-
jection of 6 upon the vertical, less Pt or in the diagram, to d h.
VI. English Roof Truss, with Multiple Trussed Principals.
Fig. 126. Here we have inclined struts, with vertical tie rods.
The load is again uniformly distributed, each space bearing the
load of 2 P. The reactions at A and D are each = 3 P. In
the force plan we have ab-\-bc-{-cd-\-de = 3 X 2 P -f- P —
7 P, which gives the length of a e. The forces 1 and 2 are
found by drawingy# and e f, parallel to A E and A L. Now
consider 1 as combined with a b — 2 P, and the resultant f b,
decomposed into f g and^ bf giving the forces 3 and 4 ; again,
combine 2 and 3, and then decompose the resultant g e, into 5
and 6, or g h and h e, by drawing these latter parallel to L F
and L M. In this manner we continue until we reach 12, or
l dy which we then project upon the vertical. Now taking from
d m, one-half the load P — d e, we have m e for one-half the
stress on the middle rod B C. The remaining half of the force
plan is similar.
VII. Polygonal or Sickel Shaped Roof Truss. Fig. 127.
This roof may be considered as a modification of the preceding
form, and is used for higher and wider spans. It is hardly
proper to assume that the load is here uniformly distributed
-even if the spaces are equal, for in the case of snow, much less
weight would be carried by the steep portions AB or G H,
than by the flatter surfaces C D or D E- We must therefore
cstimate the forces Plt P2, P3, acting as B, C, Dy Ey E, G, and
make the reactions at A and B equal to Q — Px -f- P2 4- B3.
In the force plan a b = Px, b c = P2, c d =P3, and a d = Q,
which is first decomposed into 1 and 2, by drawing e a and d e
parallel to A B and A J. Then combining 1 with Plt and de-
composing the resultant, as before, we get 3 and 4, or e f and
f b. Having 2 and 3, we get in like manner 5 and 6, or g f
and d g; then combining 4 and 5 with P2y and decomposing
with parallels to CA7 and C D, we obtain the forces 8 and 9,
and so proceed until we reach 12, which is the middle of the
symmetrical figure. The members K A, D A, E L, and M L
are ali subject to tension.
2 5i-
The Graphicae Determination of Wind Stresses.
In designing large and important roof trusses it is important
to investigate the stresses due to wind pressure, as well as those
due to the weight of the roof and of snow, and indeed, in some
cases, the resistance to wind is the most important of ali.
As an illustration of the applicability of the graphical method
to the determination of wind stresses, we will take the English
roof truss, Fig. 126, whose conditions under a vertical load have
already been examined, and consider it as also subjected to a
wind stress IV, as shown in Fig. 128.
We have first to determine the forces Ql and Q2, acting at the
points A and D. The wind pressure will be taken as acting on
the surface of the roof from A to B. Let IV be the resultant of
the entire wind pressure acting normal to AB, and let P be the
total vertical load upon that half of the truss. By combining
these two forces we obtain the direction ES of their resultant,
and also its magnitude, which we then lay off on the force plan
at c cv Upon the other half of the truss we have only the verti-
cal load, which may be considered as acting at J’, and equal in
magnitude to P. By prolonging its direction until it intersects
the previously determined line at S, we have at 5 a point in the
resultant of the entire load upon the roof, including wind pres-
sure. By making cx a2 in the force plan equal to P\ we have a c
for the direction of this resultant, which may then be laid off at
S T in the drawing. In order to determine the forces Qx and Q2
we must recollect that, according to \ 34, when we have two
closing forces to determine, we must also have at least two con-
ditions given. In this case, then, we must first find the direc-
tion of Qx and Q2.
The wind pressure produces a horizontal thrust which must
be met by the stability of the walls or columns upon which the
roof rests. In each case it must be determined whether this
horizontal thrust is borne equally or unequally by both sup-
ports, and in what proportion it is divided. To this end we
first find (according to f 39) the proportion of the vertical com-
ponent of the force a c, which comes upon each support (as
found by the intersection of 5 T, prolonged with A D), and
then combine these vertical forces with their respective hori-
zontal components. It often happens that ali the horizontal
thrust is borne by one of the supports, which it must of course
be prepared to resist. This often occurs in the case of railway
stations, and under such circumstances the direction of each force
must be determined separately. First prolong the vertical at
D downward until it intersects .S T\ and join the intersection
with A (the lines are only indicated in the figure). This gives
the direction of the force at A. We have now both the direc-
tion of the reaction at D and the direction of that at A. We
must also consider the distribution of the forces at the various
knots between A and B, and between B and D. We have for
the points between A and B the resultants between the propor-
tional parts of P and W, while from B to D we have simply the
proportional parts of P This gives at A the force Pl ; at E, E
and G, the force P2; at the peak, the force P3; at H, J and K\
P	P
the force P± = —, and at D, the vertical force Pb ==- .
4	8
Returning now to the force plan, we make cd — PY, d e =
ef—f g = P2, g h = P3, h i = i k = k l = P4, and l a = P5.
We now have finally the length b l, for the value of the reaction
Q2 at the point D, and a line (not shown) from b to d, gives the
magnitude of the force Qx acting at A.
The determination of the stresses in the various members can
now readily be made. The decomposition of b d by drawing
b m and m d parallel respectively to A E and A L, gives the
forces 1 and 2. We thus proceed until we reach the rod B C, or3«
THE CONSTRUCTOR.
No. 13, for which we get the tension r s — 13, by drawing the
vertical r s from r, until it intersects the line n s, drawn parallel
to B D. We then continue to determine the forces from 15 to
25, as already shown. The force plan shows that under these
conditions similarly placed struts are subjected to dissimilar
stresses. The determination of the stresses might have been
made in the reverse order, beginning with the triangle x b l,
which should give the same results, and which may be used to
prove the accuracy of the work. A proof is also made by the
accuracy with which the line w x drawn from wt parallel to
K O, intersects the point x, which was first determined by the
intersection of b x and l x. As a matter of fact, it will be found
to require careful drawdng in order to insure the closiug of the
diagram.
By comparing the last force plan with that found for the same
roof truss in Fig. 126 (the scale being the same), it will be seen
how greatly the wind stresses affect the structure. In order to
complete the calculation, a second plan should be drawn, as-
suming the wind to act also upon B D.
I 52.
Force Pians for Framed Beams.
Beams of various forms are often framed in various shapes
and made both of wrought and cast iron, and have rnany appli-
cations, such as walking beams for steam engines, for cranes,
arms, &c. A few examples will show the method of investiga-
tion for such cases.
I.	Projecting Frames with straight members, Fig. 130. The
load Pacts at A in a direction norinal to the axis of the frame,
which is supported at B and C. The force plan is constructed
as follows : Draw a b = P, and from its extremities draw a c
and b c parallel to 1 and 2, which gives the forces in those mem-
bers. Each of these is then decomposed into two other forces—
I into 3 and 4, 2 into 5 and 6, giving the triangles b e c and ad c.
The forces 3 and 5 are then combined and the resultant de-
composed into 7 and 8. To do this we transfer 5 = d c tof e,
and join the resultant f b, which can readily be separated into
7 and 8. We proceed in this manner for the remaining mem-
bers, and as the frame is symmetrical about the axis g c, only
Fig. 130.
one-half of the diagram need be completed. The lines g a and
b g, which are the final resultants of 15 with 17, and 16 writh 18,
are also the external forces at B and C\ the points of attach-
ment, provided that their direction be permitted to remain the
same.
II.	Double Loaded Frame, Fig. 132. In this case we have
the force Px acting downwards at A, and a force P2 acting up-
wards at D, while the points of attachment remain at B and C
as before. The members A B and A C are polygonal formed.
The force plan is drawn just as before, until the force 13 is
reached. At D the members are attached to each other at their
intersection, so that the force P2 acts upon both 15 and 16. At
this same point we have the action of the forces 12 and 13.
Now join the extremities of 12 and 13 by the dotted line shown,
and mark off the length of the force P2i which is subtracted,
because its action is upw7ard, thus obtaining the resultant of the
three forces. We can then draw 15 and 16 and proceed without
interruption to 20. Finally, wTe draw b f and e a, the external
forces at Qx and Q2, which hold the entire frame in equilibrium.
III.	Framed Boom for a Crane, Fig. 133. This figure is por-
tion of a framed arch which may be used for the projecting
boom of a large crane. At A and D we have the forces Fx and
P2, and at B and C} the external forces Qx and Q2. The force
plan is now required to determine the internal forces acting on
the various members of the structure. Before this can be done,
w*e must first determine the as yet unknown direction of the
force Q2. Prolong Px and P2 to their intersection at E> and by
drawing in the force plan, the triangle a b c, determine the di-
rection F E of their resultant; then prolong Qx until it inter-
sects E E at G, and join C G, which w ill be the required di-
rection of the force Q2. Completing the figure in the force
plan, wre have c d = Qx and d a = Q2. We now proceed from
Px = a b and lay off the forces 1 and 2, decompose 2 into 3
and 4 ; combine 3 and 1 and decompose their resultant, obtain-
ing 5 and 6. We thus proceed until we reach 12, which we
obtain by combining 9 and 8 and decomposing the resultant
into 11 and 12. We now have to combine 10 and 11 with P2r
and decompose the resultant into 13 and 14. We first transfer
the force 11 to e, making it equal to e f, in order to avoid the
confusion of lines, w hich w7ould occur if the construction w ere
made at a. Now drawing the path 11, 10, P2f we have the clos-
ing line cf \ wdiich decomposes into 13 and 14. We then have
15 and 16 from the resultant of 13 and 12, and finally, 17, as the
line joining 15 and i6wdth d, since 16 and 17 must have the
resultant a d = Q2. If the w7ork is correctly done, we will find
17 falis parallel to B C, which affords a convenient and valuable
proof for the w7hole work
? 53-
REMARKS.
The foregoing problems and methods will serve as general
examples of the various applications of Arithmography and
Graphostatics, and at the same time it must be noted that great
care and neatness are most essential in the use of the method.
It may be added that it is desirable to use as few letters and
figures as possible in designating the various lines ; a common
fault of beginners being the disfigurement of their work in this
respect. The necessary marks should be made quite small and
in faint pencil, so that they may be readily erased if so desired.
It is necessary also to be provided with the best grades of
pencils, well sharpened, a good draw7ing-board, reliable pro-
tractor scale, dividers, and flexible spline ; and it is the author’s
experience that these cannot be used too carefully. In order to
acquire facility in the methods and confidence in the results,
the beginner is advised to begin w7ith simple examples wdnch.
can be thoroughly understood, and practice upon these care-
fully.
By proceeding in this manner it will be possible to obtain a
skill and grasp of the graphical method which will enable the
student to use it freely for the solution of a great variety of
problems, and extend its scope far beyond the range of the ex-
amples which have been given.THE CONSTRUCTOR.
39
SECTION III.
THE CONSTRUCTION OF MACHINE ELEMENTS.
Introductory.
Under the title of “Machine Elements” we may consider
those single or grouped parts whick are employed to a greater
or less extent in ali forms of machinery. It is not practicable
to determine their nutnber, nor, indeed, is that a matter of im-
portance, since the selection of groups and details is not based
upon any positive or generally aceepted system. The following
selection of the constructive elements of machinery may be
found useful and convenient, which is the principal end to be
attained.
In the previous sections a number of general formulse have
beengiven, while in the cases which follow detailed examples
are selected. The dimensions and weights are expressed in
inches and pounds, except where otherwise distinctly stated;
velocities in feet per second ; and rotations in turns per minute.
The measure of force is the pound ; that of work in foot pounds
per minute, or for larger quantities in horse-power—(33,000 foot
pounds).*
CHAPTER I.
RIVET1N&.
I 54-
Rivets.
Rivets are priucipally used for joining sheet metals or other
flat shapes together for the construction of a variety of sheet
and framed structures. They may be considered as a funda-
mental machine element acting to transform detailed parts into
combinations.
In the illustrations various forms of rivets are shown. The
common wrought-iron rivet is shown in Fig. 132, with the
Fig. 132. Fig. 133. Fig. 134. Fig. 135. Fig. 136.
button head, while Fig. 133 shows the conical head generally
formed by hand riveting. The length of body required to form
the head varies from 1.3 to 1.7 times the diameter, according to
the completeness with which the rivet filis the hole. When the
head is formed by dies instead of the hand hammer, the shape
is usually conoidal or spherical, as in Figs. 134-135.
The slight bevel given to each end of the rivet, as shown in
Fig. 138, adds materially to its strength. The double conical
hole shown in Fig. 137 assists in uniting the plates, and this
shape may be produced by using in the punching machine a die
slightly larger than the diameter of the punch. This difference
has been experimentally determined for wrought-iron plates,
and is secured by making the hole in the die equal to the diam-
eter of the punch plus X the thickness of the piate.
In Fig. 136 is shown a form of couutersunk rivet used in
shipbuilding.
For bridge construction great care should be taken in the
choice of proportions. Figs. 137-139 show the proportions
adopted for the Dirschauer Bridge after the careful researches
of the engineer Kr iiger. Fig. 137 shows the normal rivet head,
and Figs. 138 and 139 the half and full countersunk heads.
Rivets up to 1 or i1/, inches in diameter may readily be closed
* These quantities are all given in metrical units in the original, but have
been transtormed in the text into English units. It must be remerabered
that the metrical horse-power (75 kgm.) is slightly smaller than the English
horse-power.—Trans.
with hammers of 8 to 10 pounds weight; but if the head is to
be formed in a swage or die, a heavier hammer, say 16 pounds
weight, is necessary.
The rate at which this work can be done by skilful riveters
per day, according to Molinos and Pronnier, is as follows:
Diameter of Rivet.	No. per	Day
............................   200	tO	250
X"..................................l8o	“	200
...............................IOO	“	125
1 "................................. 90	((	IOO
These figures are for horizontal bridge work ; on vertical
members about three-fourths these numbers may be taken.
Fig. 137.	Fig. 138.	Fig. 139.
Much higher rates are shown upon boiler riveting, as may be
seen from the following table, based upon observation of eleven
days’ work at the boiler works at Piedboeuf (Aachen) :
Diameter of Rivet.
No. per Day.
•	• 350
•	• 325
. . 300
. • 280
. . 260
. . 240
. . 220
. . 200
In cylindrical shells of more than three feet diameter these
rates may be increased ten per cent., while for awkward or diffi-
cult work ten per cent. reduction should be made. Each man
had the assistance of two strikers, one holder and one boy,
sizes less than requiring but one striker.
Hand riveting is now being largely superseded by machine
work. These machines possess the advantage of performing
the work much more rapidly, thus insuring a stronger joint, be-
sides which they are much more economical. Since their first
introduction at the time of the building of the Conway Bridge,
tjhey have been extensively used for bridge work, and with the
improvements which have been successively made they are
rapidly displacing hand riveting for boiler work.*
* Among modern riveting machines the hydraulic riveter of Tweddell
holds the nrst place. For a description the following references may serve:
Polyt. Zentral bl. 1874, p. 103 ; Engineering, Jau., 1875, P- 76 ; Sellers' Im-
proved TweddelFs Machine, Jour. Fratik. Inst., 1876, p. 305 ; TweddelFs
Machine on a Crane, Sci. American, 1876, p. 226 ; Small Tweddell Machine,
Revue Indust., 1876, p. 349.
Other forms of steam and hydraulic riveters well suited for boiler work
are :
Garforth’s Machine, Kronauer’s Zeich. III., Johnson’s Imp. Cyc., Pl. 42;
the excellent steam riveter of Gouin, see Molinos & Pronnier, Ponts Metal-
liques, p. 180 ; also the hydraulic riveter at Creusot (g^iving a pressure of 20
to 80 tons on the rivet, and closing 2 to 25 heads per minute), Revue Indust.,
1875, p. 349 ; also the very heavy machine of Kay & George (giving a pres-
sure of 120 tons on the rivet), Engineering, 1875, p. 223. An apparently very
ingenious machine is that of Allen, used especially for boiler riveting. In
this machine the frame and post are temporarily held together by a rod
operating in a very ingenious manner through one of he open rivet holes.
At the Philadelphia Centennial Exhibition this machine closed three or
more rivets per minute.40
THE CONSTRUCTOR.
I 55-
Strength of Riveted Joints.
Riveted joints are intended either to resist direct stresses (as
in bridges and similar structures), or to secure a tight joint
against moderate internal pressure (as in ships, gasholders,
etc.), or in most cases both these conditions are united (as in
the case of steam boilers). A distinction may then be made
between joints for strength and joints for tightness, the seams
of steam boilers standing midway between the two.

Fig. 141.
Fig. 142.
I
n
1 +
or for butt joint riveting:
a	7T / d V d
T = 2”iW + 7
which gives:
d	1
P= 1---— =------:---
1 + JLJL*
2 11 7T d
8 n 6
b^_
6
7r r d \2'
- -syr
= (0.5+0.56 /4-)
For butt joint riveting :
b'  5 a — d
6	8	n 6
b"
6
t(t)
=(0.5+0.79 /4-)4-
In both cases a good value of b, in practice, giving sufficient
room for rivet heads, will be secured by making :
r	. b	d
b = 1.5 d,or -j- = 1.5-j-
(47)
A point of interest is the superficial pressure />, which exists
between the body of the rivet and the cylindrical surface of the
rivet hole. If S2 is the stress in the punched piate wre have—
For lap riveted joints :
p	d
= 0-2 K ~<r
For butt riveted joints :
p	d
ir=a4,r^r
(48)
(49)
Joints for strength are either simple lap joints, Fig. 140, or
butt joints, Fig. 142, the latter coming into general use for
bridge work. The joint shown in Fig. 141, called a flap joint,
is also somewhat used for vertical tubes, chimney stacks, etc.,
the flap really being nothing but a narrow piate.
For any given thickness, 6, of piate it is impracticable to
make the riveted joint the Same strength as the piate itself, but
the ratio between the strength of the piate and the strength of
the joint can be made a maximum. This will best be attained,
with the assumption of a sufficient margin, when the strength
of the rivets and the strength of the remainder of the metal
betw^een the rivet holes are equal to each other, i. e., wdien they
reach their limit of elasticity at the same time. If the rivets
and piate are of the same material, we have, according to \ 5,
the stress in the cross section of the rivets as 0.8 that of the
piate. From this we derive the following formulae, in which
the friction of the joint is neglected, as being of uncertain
value:
Let—
<5 = the thickness of the piate,
d = the diameter of rivet,
a = the pitch of rivets, i. e.} the distance from centre to
centre of adjacent rivets,
n = the number of rows of rivets,
<p = the modulus of resistance of the joint, being the ratio
of the resistance of the joint to that of the full piate ;
then the highest ratio of resistance will be attained wThen we
have for lap joint riveting :
a * ( d V
-r=n-r\-r) +
which gives :
d
<> = 1--„ = —
The following table and scale will serve to reduce the numer-
ical labor of these calculatious :
256.
Table and Proportional Scale.
d 6 |	1.	0	1.5		2.0		2.5 ;	3-o		4.0	
—	1	2 i	1	2	1 i	2	1 l 2	1 |	2	1 j	2
a =	1.63	2.22	2.92 1	4-33	4-52	7-o4	6.4310.37	00 b\ 			14-33	I4*°7	24.14
. V c 8 ~	°-39	o*39	0.88 |	0.88	i-57	1-57	2.54, 2.54	3-53	3-53	6.28	6.28
2, b" 0, ~T~ ~	1.06	1.06	1.78	1.78	2.58	2.58	1 i 3-46 3-46	4-3i	4-31	6.48	6.48
^ 0 =	o-39	0-55	0.49	0.65	0.56	0.72	0.61 0.76,	0.65	0.79	0.72	0.83
	0.63	0.63	0.94	0.94	1.26	1.26	1-57 1-57 |	1.88	1.88	2.51	2.51
a ~8~ =	2.26	3-52	4*33	7-i5	7.04	12.05	IO.37 l8.2I	1+33	25.61	24.14	44.21
. b’ .5 8	0.79	0.79	0.96	0.96	3-14	3-M	4-91 j 4-91	7.07	7.07	12.56	12.56
£ 1 fi	1.29	1.29	2.20	2.20	3-24	3-24	. | 4-37 4-37	5.60	5.60	8.32	8.32
W </> =	0.56	0.72	0.65	0.79	0.72	0.83	0.76 0.86	0.79	0.90	0.83	0.9,
P s*	1.26	1.26	00 00	1.88	2.51	2.51	3-x4 3 H	3-77	3-77	5-03	5-Q3
(43)
(44)
The overlap of the piate is subjected both to shearing and
bending. For the former conditions, call the lap b', and for the
latter b", measuring in both cases from the centre of the rivets
to the edge of the joint. To obtain the same resistance in the
lap as in the perforated portion of the piate wre have—
For lap joint riveting :
b' ______ S a — d
T
(45)
(46)
In the proportional scale, Fig. 143, the principal values are
graphically shown. It will be seen that the higher ratios of
strength are not very practically obtained, for the large diameter
rivets are in convenient to handle. The advantages of lap joint
riveting are also shown. The objection to butt joint riveting,
which overrules its advantages, lies in the rapid increase in the
. d
value of p, as it will be seen that with a ratio ~^-= 3 the elas-
tic limit of v/rought iron is exceeded, the stress reaching 30,000
lbs., while the stress upon the metal between the holes is only
8,600 lbs. This explains the failure of riveted joints under
. d
variable tension loads. If the ratio of -j- = 2 is used, the
excessi ve stress in the rivet holes cannot occur. Fairbairn,
upon whose experimental researches these conclusions are
based, States that for riveted structures the diameter of rivet
may best be taken as equal to X ; but this conclusion is
not fully borne out by experience. The use of the value for the
lap b = 1.5 d is approximately correct at least for lap joint
riveting, as the dia^ram shows it to give a slight margin both
over the values of b, for shearing or for bending.
2 57.
Riveting Disposed in Groups.
If more than twro rows of rivets are to be used the efficiency
of the joint may be decidedly increased without using incon-
veniently large rivets by disposing the rivets in groups on
either side of a Central row, arranging them according to an
arithmetical series.
The numbers in adjacent rows may then be placed as follows :
1:2:1
1:2:312:1
1 : 2 : 3 : 4 : 3 : 2 : 1
Total 4
“	9
“ 16THE CONSTRUCTOR.
41
0.72
If now we assume that th.'> force P, upon each
O.SU strip between the dotted lines, is equally divided
„ „ _ among the rivets, we have for the efficiency of
0 03 the first row :
T-~T (4)+^4
(4)+£
—— = m —
d	5
(5o)
If the stress in the punched piate in the lines
I, II, III, IV, etc., Fig. 146 be called S], S1*
S™,	etc., we have :
P = S* (ina — d^j d
= [nia — 2d^j
- 6
And from this when S\ = S1* we have :
a	m2 -j- 1
(5i)
d	m
And upon the same supposition :
m* — 6 #
" m* — 3 *
(52)
s1”	ni2 —	3 .	s"
S\	m1 —	2 ’	Sl
			ni1 —10
			m'1 — 4 *
that is, the stresses at the lines III, IV, V, are
smaller than S* = S1*. The useful application
of this fact may be readily seen.
L,et us introduce :n (50):
-T = -jr=‘ 15or say 1.6 . . (53)
that is, we make the ratio d :	constant and =
1.6. For the modulus of the efficiency of the
joint 0, when the stress in the solid piate is ^i,
we have:
A=,
FIG. 143.
The following illustrations show examples of this form of
riveting, which may be termed Group Riveting, and is espec-
ially adapted to lap joints. The dotted lines show the limit of
the area including each group, and the spaces between the
.	,	1.11 in iv v. vi.vn.
S2 ~ m & m2 + 1
We also have for the pressure py on the rivets :
— p ...........................
(54)
ts:
(55)
		A ! 2,a
	•••£&■*•	
	<*> '°l°	
	010	
		
i 0 'Ao 10 ~4>* 0 I o^So	'A sia { N.		j j 6 't# 11 : 6tj 6 ; 6 Y»o ! |! i -0v-s
1 0 $ 0 10 00 j 0 6 0			i i ! 9 6? -1 M— j —T*~r- i |<j> ? t? i 4> i <i> ? 4> j ? 4» 9^
| 0 4> 0 ! 0 60			
| 0 <j> 0			J	Q-iLo.
^ m1 d d
Tabulating the results of the applications of these equations
to various groups we have :
m =	2	3	4	5
0	j
1	6
<b!
* i
9
l'.R
Fig. 144.
Fig. 145-
Fig. 146.
rivets in each longitudinal row are uniform. If m be the num-
ber of rivets in the m;ddle row of each group, then the total
number of rivets in each group is w2.*
a T	1.6		1.6 1.6	1.6
a V	2.50		3 33 4.25	5.20
a T	4.00		5.32 6.80	8.32
0 =	0.80		0.90 0.94	0.96
For joining narrow	plates the rivets may often be disposed in			
1 fo ^ 0 i 0 ^To			J/°°°	°«j°\
	J"		i 000	0 -0* 0
				°o°/-
Fig. 147.
Fig. 148.
* This follows because, £i + m J +	|" i +	— i)J (m — i) — m2.
double groups, while for the union of several plates, as in the
construction of piate girders, a number of groups may be em-42
THE CONSTRUCTOR.
ployed. In Fig. 147 a triple row is shown, and in Fig. 148 a
quadruple row, the joint in each case being made with a flap.
Besides the advantage which results from the disposition of
the rivets in groups in such cases, there is also a gain in making
the flap somewhat thicker than the plates to be joined. The
calculation in this case may be derived from the preceding
methods ; also see the latter part of $ 59.
§58.
Steam Boieer Riveting.
For the joints of steam boilers parallel riveting is generally
used. Iu this case the question of the tightness of the joint
The following table contains the collected results of the pre-
ceding formulae for the more commonly occurring proportions :
2 59-
Tabee and Proportionae Scaee for Steam Boieer
Riveting.
Fig. 149.
Fig. 150.
Fig. 151.
prevents a wide spacing of the rivets. For the sanie reason the
thinner plates require proportionally larger rivets than the
heavier plates, and the lap of the plates, and also of the rivet
heads, must be greater. The method of caulking is also to be
considered. The older method consisted in driving the caulk-
ing chisel into the perpendicular edge of the piate and forcing
the lower edge of the groove thus made down upon the lower
piate, as shown in Fig. 149. The modern method, shown in
Fig. 150, requires the piate to be planed on the edge to an angle,
which can then be caulked without grooving. The
angle of bevel should be about i8^°, or about 1 in 3.
The method of the American, Connery, consists in the
use of a round nosed caulking tool, Fig. 151, which is
much less liable to injure the lower piate than the
sharp chisels of the previous methods ; the action ex-
tends farther into the piate also.
The consideration of the tightness of the joint com-
pels a modification of the theoretical treatment of the
proportions of boiler seams, based upon practical ex-
perience.
According to Lemaitre the following proportions
are suitable for single riveted joints :
		Round Head.		Conical Head.			ti .s t! >	iveting.	Modulus of Efficiency.			1-1 rt
5	d	Height o.6d.	•** 00 rt s	P w°	>4 rt P	fcJO c	S 'Eb .S ‘S	Double Ri	0'	0"	02" !	% 0 ‘v.H
X	Xs	X	%	A	X	X	iX	1%	.0.66	0.51	0.60	2
A	A	X	X		X	1	iX	2%	0.66	0.51	0.60	4X
X	A	A	1.00	A	i'/b	iX	iX	2X	0.63	0.48	0-59	6X
A	X	X	1 y%	X	iX	lX	iX	2%	0.61	0.48	o-59	13X
x	11 IT	A	iX	A	iX	2	iX	2 y»	0.60	0.47	o.59	20
A	t!	X	1X	X	iX	2 X	2	3/s	0.60	0.47	o*59	29
X	it	A'	1x	X	I x	2X	2X	3X	o.59	0.47	o.59	36X
A	1	X		H	2.00	2X	2 X	3X	0.58	0.47	o.59	50X
X	iA	ii	2.00	7A	2'/i	3	*x	4'/s	0.58	0.47	o.59	66'
d = 1.5^ + 0.16"
a = 2d + o-4/7
b = \.$d
!...........
Double riveted joints are also much used for steam
boilers, especially for the longitudinal seams, while
single riveting is used for the circumferential joints,
since the stress upon the longitudinal joints is much
greater than upon the circumferential joints.
For double riveted joints, that is, for riveting in two
parallel rows, we have for the pitch a2 of the rivets in
each row:
a2 — $d + 0.78" .
while the space between the two rows may be taken
as equal to the previous value of a, or 2d -f- o.^/f.
In some cases this value is used for the pitch of both
rows (see Fig. 153).
We have previously taken the modulus of efficiency
0, so that the rivets and the perforated piate have not
the same degree of security. The values of 0, for the
rivets and for the piate should therefore be determined
separately, and the smaller value taken for that of the
completed joint. Let :
07 = the modulus for the perforated piate,
<p// = that of the rivets,
then according to the previous formulae we have :
a — d
7T	d2
—- n -—r-
5
For the pressure p upon the body of the rivet we $
have finally, both for single and double joints,
P
S.,
£-tO-+*9--
The stress S2 in the perforated plates of boiler shells *
is not generally ^ermitted to exceed 5000 to 6000 0
pounds per square inch of cross section.
Fig. 152
5555555555B555555J5555555555 5555555555555THE CONSTRUCTOR.
43
The rivet length is == 2(5 -|~ i -7^, upon the assumption that
both plates are of the thickness 4, and this length gives an
ample allowance for the full clearance of the rivet holes (see
$ 54). The last column is of Service in making estimates of
weights.
Fig. 152 is a graphical presentation of the principal results
of the preceding formulae. It will be noticed that for single
riveting the modulus for the rivets, is always less than the
modulus (p' for the perforated piate, and is nearly always less
than y2. It foliows that for single riveted joints of steam
boilers we should never assume a greater strength than one-half
that of the solid piate. By the adoption of double riveting,
while retaining the same pitch, a — 'id -j- 0.4", we ought to
obtain, according to the formula of § 58, a value of </>//, twice
as great, which in the case of very light plates would exceed
unity. In that case, however, the value of <p' is the lesser, and
it determines the efficiency of the joint, so that the only gain
due to the double riveting in that case is the increase in the
value of <pf. If, however, we choose for double riveting the
pitch a2, as given from equation (57), both <pf and <p" will be in-
creased in value. The lesser of the two moduli is that for the
rivets, and its value is obtained from
d*_
at5
(60)
Its value lies between 0.75 and 0.59, and is shown in the table
and diagram. The pressure p in all cases remains within prac-
tical limits.
In Fig. 153 is shown double riveting in which the pitch kept
equal to 2d -f- 0.4", while in Fig. 154 the pitch is made equal to
3d -f- 0.78'7 for rivets in the same row, while the diagonal dis-
tance between rivets of the two rows is the same as for single
riveting. For a flap joint such as is shown in Fig. 155, we have
a combination of parallel and group riveting. This method is
used in Germany for steam boilers, but is little used in America,
if at all. The flap is placed on the inside of the boiler shell,
and the flap seams have only half as many rivets as the main
seam, but of the same diameter. The objection that the inner
edge of the main seam is made inaccessible is counterbalanced
by the increase in strength. We have
<f>"
0.3 7r d* 1 2
a4
the lesser of which will be found to exceed the value obtained
for ordinary double riveted joints.*
Example. 8 = XY'* d — £6", a — -zd + 0.4" = 1.65", say 1%" ; <f>' = 3 3 „
= 0.81.	4>" = °'3 *	°~3^1 = 0.71. It may be remarked that American prac-
1.65X03125
tice gives wider pitches than are generally used in Europe.f
* The two rivets which lie between any pair of rivets in the main joint
each bear a stress of £ P, and the rivets of the main joint also sustain J P.
The flap does not transmit any stress to the rivets of the main seam For
the stress on rivets P = £ nd- &3. for the solid piate P= S\ 2aS. The modu-
lus <£", taking S3 for shearing stress = —is found from <f>" =	-
f o 3	2ao
=	—• For the main seam we obtain <f>', from P= § (2a — 2d) 8 S"z =
S\ 2aS, whence <f> =	, which is greater than ~ a —.
1	2 a	2 a
+ For example, in single riveting piate and	rivets have iji" pitch
(the formula would give about 1^"); ^6" piate and	livets have a pitch of
2 ¥%' (the formula would give about 2^5 0 ; iu double riveting, for yi" plates
with yj' rivets, the pitch would be 3^", while the formula would give
about 3".
Three rows of rivets are used in this form of joint, and the
outside rows of wide pitch make this method more trouble-
some of execution than the group riveting shown in Fig. 144,
which has a modulus of 0.80. This is a point which should be
bome in mind.
The joints of gasometers exhibit but little variet)T in plates or
rivets. The rivets are usuallv about to T5// in diameter and
1" pitch, with a lap at the joint of about , the rivets being
closed cold and the joints caulked with red lead.
Tabtf of the Weight of Sheet Metae.
Thickness
Weight in Pounds per Square Foot.
in Inches.	Wro’t Iron	Cast Iron.	Brass. j	Copper.	Lead. j	Zinc.
* i	2.53	2-34	2.73	2.89 |	3-7i	2.34
	5*05	4.69	5-47	5.78 1	7.42	4.69
h	7.58	7-°3 i	8.20	8.67	11.13	7-03
X	10.10	9-38 j	10.94	II.56	14.83	9-38
	12.63	11.72	13-87	14-45	18.54	11.72
Vi	I5-16	14.06 j	16 41	17-34	22.25	14.06
T5	17.68	16.41	19.14	20.23	25.96	16.41
X	20.21	18.75	21.88	23-13	29.67	18.75
A	22.73	21.09	24.61	26.02	33-38	21.09
X	25.27	23-44	27-34	28.91	37-°8	23-44
H	27.79	25.78	3«.°8	31.80	40.79	25.78
X	30-31	28.13	32.81	34.69	44-50	28.13
11	32.84	1 30.47	35-55	37-58	48.21	30.47
X	35-37	32.81	38.28	40.47	51.92	32.81
H	37-9°	j 35.i6	41.02	43-36	55-93	35-16
I	40.42	| 37-50	43-75	46.25	59-33	37-50
: 61.
Speciae Forms of Riveted Joints.
Junction o/Several Plates—In Fig. 156 is shown the junction
of three plates. In this case the corner of sheet No. 2 is bev-
eled off and No. 1 worked down over the lap.
In Fig. 157 the junction of four plates is shown. Here the
angles of sheets Nos. 2 and 3 are beveled and Nos. 1 and 4 are
left unaltered. In the construction of steam boilers the shell
may be formed either in cylindrical sections, as shown in Fig.
158, or in sections of a conical shape, the taper of all the sec-
tions bearing the same relation to the direction of the flame a9
shown in Fig. 159. This latter method requires that a slight
curvature should be given to the sheets in order to secure the
required taper. The determination of the taper and curvature
of the sheets and lines for the rivet holes may be made in the
following manner :44
THE CONSTRUCTOR.
Let—
D — the diameter of the shell, as in Fig. 159,
B a=s the breadth of the sheet, Fig. 160, on a circumferen-
tial seam,
L = the length of the sheet between pitch lines of rivets,
f — the versed-sine of the arc B ; we then have :
/ _ ,/ B B
6 ~ A
• (61)
Example. In a riveted tube where each section is made of an entire sheet
we have B = irD. If the breadth B is twice the length Z., we have
-j- = 0.7854 X 2 = 1.5708,
so that f will be a little greater than 1% times the thickness of the piate.
In arranging the junction of sheets when the flap joint is em-
ployed, care must be taken to avoid complicated intersections.
This is best accomplished by making the flaps on the longitud-
inal and circumferential seams come on opposite sides of the
plates. Where the flaps are both on the sanie side, they are
somgtimes let into each other.
Reinforcernent of Plates.—This may often be done very
readily by the use of angle and T iron. In Fig. 161 is shown
Fig. i 61.
Fig. 162.
«L
4
.c
K-bl-18-r» $ —
Fig. 163.
3.1 ^
an internal angle iron, and in Fig. 162 an external, and in Fig.
163 a simple T iron. The proportions for angle iron given by
Redtenbacher are as follows :
h = height of angle arm,
<5 = thickness.
h = 4.5 <5 + 1".
For T iron hx = the base = 8 6 -f 2", and the height of the
rib =	practice a great variety of proportions are made
to suit all possible cases, examples of which may be found in
the illustrated catalogues of the mills where they are rolled.
Fig. 164.	Fig. 165.
Fig. 166.
The strengthening of parallel plates which are near together
is best done by the use of staybolts. In Figs. 164 and 165 is
shown a copper staybolt after and before riveting, this form
being used in locomotive fire boxes and marine boilers. The
Central hole affords a warning of the corrosion or w?eakening of
the bolt by the escape of steam. It is best to remove the screw
thread from the projecting portions before riveting over the
heads. Fig. 166 shows a form of iron staybolt for the same
purpose. The short piece of tube between the plates prevents
them from being drawm out of shape w?hile riveting, and the
opening permits a free circulation of wTater. The bolt is pro-
tected from corrosion by being incased in copper. Screw stay-
bolts are now often made of soft wrought iron or mild steel,
but copper bolts are stili preferred by many.
Fig. 167. Fig. 168.	Fig. 169.	Fig. 170.
Construction of Angles (Figs. 167-170).—Angle junctions in
riveted work are made either by flanging the piate or by the
use of angle iron. In Fig. 167 the flange is turned inward, and
in Fig. 168 it is turned outward. In these cases h is made the
same as for angle iron of the same thickness. Figs. 169 and
170 show the use of internal and external angle iron.
Fig. i71.	Fig. 172.
Construction of Solid Angles.—These are the most difficult
fornis of riveted work, and may be made in several manners,
the most important being shown in the illustrations. In Fig.
171 the vertical angle is made as in Fig. 167, and the horizontal
angles as in Fig. 169, sheet No. 2 being beveled under the angle
iron. In Fig. 172 all three angles are made as in Fig. 169, the
Fig. 173.	Fig. 174.
vertical angle iron being cut and bent over the horizontal. In
Fi g. 173 the angles are all made as in Fig. 169, but the angle
irons are welded together at their junction. This makes au ex-
cellent piece of work, but is difficult and expensive, and re-
quires firm support for the work, and is only applicable for
important constructions. In Fig. 174 the vertical angle is made
like Fig. 169, while the lower joint is made as in Fig. 170, mak-
ing simple and substantial corner.THE CONSTRUCTOR.
45
CHAPTER II.
HOOPING.
§ 62.
Hooping by Shrinkage.
The use of hoops or bands is a very efficient method of unit-
ing sorne combinations of machine elements, and also for
strengthening existing combinations. The hoops or bands are
arranged so as to encircle the portions to be united, and caused
to exert sufficient pressure upon them to create such friction
between the surfaces as to prevent auy relative motion. It fol-
io ws that the material in the band is subjected to tension while
the parts which are held together are under compression. The
bodies to be hooped are nearly always either cylindrical or
conical in shape.
The pressure required to secure the hoops may be obtained
either by shrinkage, a method formerly used very extensively,
or by cold pressure, a modification being described in the latter
part of $ 64.
The elongation which is produced by elevating the tempera-
ture to a red heat may be taken for Steel and wrought iron at
about while to keep within the limits of elasticity the re-
sistance to contraction should be, for
Cast or wrought iron................T¥Vo
Cast Steel.................... . .	^0
Hence the allowance for shrinkage to be made in boring for a
cast iron hub to fit over an unyielding centre should not be
greater than	and is best made from	t° rsW» especi-
ally if the centre is very heavy. The ring can then be fitted to
its place when at a dull red heat. For wrought iron or steel
rings, su'ch as wheel centres, such precautions are not so essen-
tial, since these materials permit of a slight extension without
injury (see §2). If the centre possesses but very slight yield-
ing elasticity, there may be danger, however, that the contrac-
tion due to excessive cold may overstrain the material.
.....—---------
FiG. i 75-
When wrought iron bands are to be used to secure iron jour-
nals to wooden shafting, as shown in Fig. 175, the end of the
shaft is made slightly conical, so that the bands, being raised
only to a dull red heat, may be driven on with the hammer.
The rings may be forged tapering, but the taper may be also
readily produced by Clerk’s method by repeated heating and
cooling.* The red hot ring is immersed in the cooling tub for
f 1	<•-				“i	r		
i	1		N26
♦ h .i...			. \
Fig. 176.
about half its axial height. The rapid contraction of one por-
tion of the ring deflects the warmer portion towards the centre,
and by repeating the process the taper may be produced to
almost any extent which may be required.
The following experiments, made in the Royal Technical
Academy, will serve to illustrate the process. The ring shown
in Fig. 176 had the following original dimensions :
„ = 5^", * =	£> = 8^".
After the first immersion the contraction was
u	second	u	ii	A
it	third	(i	(i	T<J
(i	fourth	Ii	ii	■?v
	fifth	ii	ii	H
<<	sixth	it	ii	it
* See Procedings of the Royal Society, March, 1873. Civilingenieur, Vol.
X., 1864, p. 238.
After the last immersion the dimensions were found to be D =
7xt// at the upper edge, and at the lower edge = 8l\//.
A method of connecting two flat bars by shrinking on a hoop
is shown in Fig. 177, and has been used at Seraing with good
results.
The hubs of gear wheels or revolving cylinders are advanta-
geously strengthened by bands if they are cast in several parts,
as in this way they are firmly united into a compact whole.*
Cobd Hooping.
In the place of shrinking bands to their places, the more re-
cent method of forcing them on cold has come into use for
bands of moderate size, such as for hubs of wheels, cranks and
levers. In this case the ring and its seat are both made truly
cylindrical, with merely a slight bevel for entrance, and then
by means of a press forced together.-f The difference in diam-
eter between the ring and hub is very small, and may be calcu-
lated as described in \ 19.
An investigation of the force required to push a ring on may
be found desirable. The force which is necessary to press a
cylindrical pin into a hub by continuous motion may be taken
as nearly proportional to rate of progress, since it has to over-
come the resistance of sliding friction between the surfaces.
The pressure p, per unit of surface, is equal to the initial radial
stress Slt which exists upon the pin. If we make r the radius
of the pin, l the length of the hole,/the co-efficient of friction,
we have for a maximum value of the forcing pressure Q :
Q = 2. r t? IS\f.............(62)
Taking f = 0.2, as indicated by experience shown in the fol-
lowing cases, we have
f-s-Mi......................«•*
For the tangential stress S2i in the metal of the ring, we obtain
from formula 37, § 19 :
...................................(64)
Sl p
And taking the thickness of the metal of the ring as d, we get
the value of p :

(-0
(65)
— +1
This gives for the following ratios of thickness to radius, cor-
responding numerical values :
6
= 1.30
0.55	0.60	0.65	0.70	0-75
0.415	0.438	0.463	0.486	0.508
0.85	0.90	1.00	I.IO	1.20
0.548	0.566	0.600	0.630	0.658
1.40	1.50	1.60	1.70	1.80
0.704	0.724	0.744	o-759	0.774
1.90	2.00
The following table shows examples of the practice of many
of the leading Continental railways. In the table, 2r = diam-
eter, l = length, 6 = thickness of hub, Q — total forcing
pressure ; also, W. I. = wrought iron, C. I. = cast iron, S =
steel, C. S. = cast Steel, B. S. = Bessemer Steel.
* For an account of the strengthening of a piston head by shrinking on
bands, see the Berliner Verhandlungen, 1876, Sheet XVI.
f Sometimes these surfaces are made slightly conical, such being the case
in Nos. 15 to 17 of the following examples.No.1
I
2
3 I
4 I
5
6
7
8
9
io
ii
12
13
14
15
16
17
18
*9
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
5i
52
THE CONSTRUCTOR.
5 64.
HXAMPIyES oe Forced Coxxectioxs.
DESCRIPTION.	Dimexsions. Inches. \			1 Materiae-		Q	REMARKS.
	2r	l '	: 6	Hub.	Axle. 1 !	Pounds.	
EASTERN RAIEWAY OF PRUSSIA. Locomotive Driving and Coupled Wheels,	7lA	7 A	-515 01 ?	W. I. 1	c.s.	160,600	With Key. Data as given.
Locomotive Trailing Wheels,		7 A	6U	3	! W. I. i	I c.s.	160,600	No Key. Data furnished.
Tender Wheels,		5/2	6}.\	2 A	W. I. 1	1 c.s.	118,800	No Key. Data furnished.
Car Wheels, with Spokes,		5 A	9%	2T6	W. I.	c.s.	110,000	No Key. Data furnished.
Steel Piate Wheels, for Cars,		5 A	rA	2	c.s.	c.s.	* 110,000	No Key. Data furnished.
Steel Piate Wheels, for Locomotives, . .	7)4	7	2tV	c.s.	c.s.	154,000	No Key. Data furnished.
UPPER SIEESIAX RAIEWAY.							
Locomotive Driving Wheels,		7ti	i	3H	W. I.	c.s.	220,000 to 330,000	With Key. Measured Dimensions.
Locomotive Trailing Wheels,			7/i	3 }i	W. I.	c.s.	165,000 to 220,000	With Key. Measured Dimensions.
Tender Wheels,		5xi	>- 5 /To	33A	W. I.	c.s.	110,000 to 132,000	No Key. Measured Dimensions.
Steel Piate Car Wheels,		5 A	6 H	iA	c.s.	c.s.	110,000 to 132,000	No Key. Measured Dimensions.
Wrought Iron Spoked Car Wheels, . . .	5/4	7A	2 A	w. I.	c. s.	132,000 to 154,000	No Key. Measured Dimensions.
HAXOVERIAX STATE RAIEWAY.							
Locomotive Driving and Coupled Wheels,	7%	7	3lA	W. I.	c.s.	165,000 to 176,000	With Key. Data furnished.
Locomotive Trailing Wheels,		6)4	614	274	W. I.	c. s.	143,000 to 154,000	Without Key. Data furnished.
Standard Car Wheel,		5/4	S	174	W. I.	c.s.	88,000 to 110,000	Without Key. Data furnished.
magdeburg-haeberstadt r. r. Car Wheels,		5/4	7 7A	2A	W. I.		110,000 to 132,000	No Key. Measured Dimensions.
Car Wheels,		5/4	s M	A A	c.s.		110,000 to 132,000	No Key. Measured Dimensions.
Locomotive Wheels,		7/i	W	3%	w. I.		176,000 to 198,000	No Key. Measured Dimensions.
SAARBRUCK RAIEWAY.							
Locomotive Driving and Coupled Wheels,	7	7)4		C. I.	c.s.	137,940	With Key. Measured Dimensions.
Locomotive Driving and Coupled Wheels,	7	8	2A	W. I.	c.s.	247,588	With Key. Measured Dimensions.
Locomotive Driving and Coupled Wheels,	7 H	715j	3U	W. I.	c.s.	247,588	With Key. Measured Dimensions.
Locomotive Trailing Wheels,		6	6	2i^	W. I.	c.s.	198,000	No Key. Measured Dimensions.
Tender Wheels,		5X	6j4		c. I.	W. I.	136,840	No Key. Measured Dimensions.
Tender Wheels,		5t!	7/s	2X	w. I.	W. I.	168,080	No Key. Measured Dimensions.
Tender Wheels,		5rt	5-H	2	W. I.	c.s.	^98,000	No Key. Measured Dimensions.
Standard Car Wheels,		5 '4	7%	2 A	W. I.	c.s.	165,000 to 193,600	No Key. Measured Dimensions.
Freight Car Wheels,		5 A	8A	4 A	c. I.	W. I.	165,000	No Key. Measured Dimensions.
Coal Car Wheels,		5%	7%	2A	W. I.	W. I.	193,600	No Key. Measured Dimensions.
Passenger Car Wheels,		4/i	7lA	3/s	C. I.	W. I.	110,000 to 165,000	No Key. Measured Dimensions.
RIGA—DUNABERG RAIEWAY.					i		
Driving and Coupled Wheels (Stephenson),	7	7%	3U	W. I.	w. 1.1	90,200	With Key. Measured Dimensions.
Locomotive Trailing Wheels “	(>'/2	6yi	2 u	W. I.	W. I.	90,200	With Key. Measured Dimensions.
Tender Wheels “	5A	7	3	W, I,	W. I.	85,800	With Key. Measured Dimensions.
Driving, Coupled & Trailing Wheels (Borsig),	6%	6%	3%	w. i.	W. I.	90,200	With Key. Measured Dimensions.
Tender Wheels (Borsig),		5rs	7		W. I.	W. I.	85,800	With Key. Measured Dimensions.
Passenger Car Wheels (Ashbury), ....	4/4	6 Vz	2 H	W. i.	W. I.	68,200	With Key. Measured Dimensions.
Freight Car Wheels (Zypenl, ......	5 'A	10 X		W. I.	W. I.	77,000	No Key. Measured Dimensions.
Freight Car Wheels (Zypen),	1	5'A	8 %		w. I.	B. s.	88,000	No Key. Measured Dimensions.
BORSIG EOCOMOTIVE WORKS. |							
Locomotive Trailing and Tender Wheels,	6to8	6J-7	2^-3	W. I.	W.l.orS.	155,000 to 220,000	With Key. Measured.
Locomotive Driving and Coupled Wheels,	6J-8	7to8	3to4	W. I.	W.l.orS.i	220,000 to 330,000	With Key. Measured.
Crank Pins,	i |	4to6	7to8	2-2\	W. I.	W.l.orS.	110,000 to 165,000	No Key.
wohekr eocomotive WORKS. j					i		
Locomotive Driving and Coupled Wheels,	7%	7 H	iX	W. I.	W.l.orS.	220,000	With Key. Measured.
Locomotive Trailing Wheels,		7lA		3	W. I.	W.l.orS.	220,000	No Key. Measured.
Tender Wheels,			5X	6 %	9 9 2t^	w. I.	W.l.orS.	132,000	No Key. Measured.
NORTHERN RAIEWAY OF FRANCE.							
Locomotive Wheels (Stephenson), . . .						176,000	Data furnished.
Locomotive Wheels (Clapeyron), ....						176,000	Data furnished.
Tender Wheels, with strong hubs, . . .						176,000	Data furnished.
Crank Pins,							33,000	Data furnished.
paris-eyons-mediterranean r. r.							
Locomotive Driving Wheels,		. .	. .	. ,	W. I.	W. I.	77,000 to 88,000	With Key.
Locomotive Trailing Wheels,		. .	• •	. .	W. I.	W. I.	55,000 to 66,000	No Key.
Tender Wheels,					W. I.		55,000 to 66,000	No Key.
Car Wheels,							39,600 to 48,400	No Key.
Car Wheels,						W.l.orS.	22,000 66,000	No Key.
Crank Pins,										No Key.THE CONSTRUCTOR.
47
From example No. 12 we obtain in formula (63) the value
Si =
5 x 176,000 _
7-5 X 7T X 7
: 5,336 lbs.
According to (65) p = 0.53, and substituting these values in (64)
gives S2 = 10,679 lbs.
From example No. 10 we have :
■Si =
5 x 132,000
5.125 X 77 X 7*3I25
also p = 0.44, and hence S2 = 12,734 lbs.
From example No. 37 we have :
= 5,603 lbs.
5 X 220,000
7-5 X ~ X 6.7
6,979 lbs.
also p = 0.526, giving S2 = 13,250 lbs.
From eyample No. 16, taking Q = 132,000 lbs., wTe get 5i =
4867 lbs.; p = 0.77 ; 52 = 6320 lbs.; and in No. 17, we have S1 =
6617 lbs.; p = 0.569, and 52= 11,629 lbs., neither of which are
excessive.
The force required to force a hub ofF an axle upon which it
has been pressed, is not materially different from the force with
which it was pushed on. The bore of such a hub may also be
reduced when necessary by forcing rings upon it. Such rings,
when used for car wheel hubs, are usually made of rectangular
cross sections, the diameter ranging from 2//Xi//, to i^^X
etc.
An inspection of the table will show that there is a tendency
towards increasing pressures. For car wheels, where until quite
recently, pressures of 60,000 to 90,000 pounds were used, we
now find 80,000 to 110,000 pounds not infrequently ; while for
locomotive wheels, over 200,000 pounds is the rule.
Midway between the methods of shrinkage, and of cold forc-
ing comes the lesser used method of expansion by use of boiling
water. This system secures a much more uniform action of the
temperature than is practicable with a red heat, and has been
used with excellent results upon the Russian railways for fitting
tires to piate wheels. The tires are suspended by a erane, in a
tank of water which iskept at the boiling temperature by a jet
of steam (the allowance for expansion being a little less than
of an inch to the foot of diameter. An immersion of 10 to
15 minutes is required to obtain the desired expansion. Three
workmen can in this manner fit 12 to 14 tires per day of eleven
hours. This method may also be found applicable to the fitting
of hubs.
65-
Dimensions of Rings for Cold Forcing.
Since the forms of the various hubs may be taken as cylin-
drical in nearly every case, the stress may be calculated by the
formulae already given. It is, however, desirable to present these
in such form that they may be used to determine the thickness
ofhub which,wThen forcedon cold, shall resist a determinate force.
In (62) instead of the radial stress Slt substitute the tangential
stress 52, giving Q = 2 7r r l fS2p, which combined with (65)
gives:
J2 tt r lf S2+ Q _
M 2 7T r l/S2 — Q
(66)
In this, Q is the maximum force which the hub can oppose to
tuming, at the diameter of the fit. If we take the moment
of the force tending to rotate the wheel as P P, we must have
Q r>PP.	vtill then be the factor of resistance against
slipping in any such case. This mode of attachment is then
only practicable when 2 n r l f S2^> Q. By choosing different
values for 52, and Q, various thicknesses for the metal of the
hub may be obtained.
Example. The following data are taken from Borsig’s Express Locomo-
tive at the Vienna Exposition ; Two pairs of coupled driving wheels of
38.19" radius, without keys; bore of cylinders 17"; steam pressure 147
pounds; crank radius R = 10". If we suppose the entire force upon the
piston to act upon a single wheel, we have :
P R = (17)2 X 0.7854 X 147 X 10 = 33,366 X 10
■ = 86,440 lbs.
The bore of the wheel is 7.72" hence r — 3.86''' while l — 7.87". This gives
P R _ 333.660
r ~	3.86
^ The moment 333,660 is that which the friction of the wheel upon the axle
should be able to resist without slipping. Hence it follows that Q must
neessarilybe greater than 86,440. If we.take a value of Q= 154,000 lbs-,
thus giving ample margin against slipping, and also use a wrought iron
hub, making S2 = 7120 lbs., taking f = 0.2 as before:
S_
r

2 w X 3-S6 X 7-87 X 0.2 X 7120	154»000
2 jr X 3-86 X 7-87 X 0.2 X 7120 — 154,000

425.656
117.656
= 0.92
hence	S = 3.86" X 0.92 = 3.55"
The actual thickness of the hub was 3.54"*
The ring form is not the only form of construction which
may be used for joining members by forcing, since other forms
may also be used. An example may be found in Erhardt’s
flange joint, Fig. 178*. In this case clamps of hardened Steel
are used to create the pressure. These clamps serve to press
the light flanges together, and they may be forced on by use of
a screw clamp or other sui table press. Tests of such joints
under steam, pneumatic and hydraulic pressure have shown the
joint to be tight and serviceable.
The system of forced connections has grown into extensive
use, and appears to be applicable to many forms of construc-
tion, and it is to be hoped that the forcing press, for which the
firm of Schaeffer & Budenberg have made suitable pressure
gauges, may be found an indispensable tool in ali large
workshops.
CHAPTER III.
KE YING.
\ 66.
Keyed Connections.
The simplest form of keyed connection consists of three
parts, viz. : the two parts to 'be connected, and the key itself.
The key is made with a slight amount of bevel on both sides,
or a greater angle on one side, according to the manner in
Fig. 179.
Fig. iSo.
which the connection is made. The trigonometrical tangent
of this angle is called the draft of the key. In Figs. 179 and
180 are shown both forms of keys. For the latter form wre will
assume that both sides have the same angle.
Let:
a = the angle of draft,
P= the force to be transmitted,
Q = the driving force upon the key, normal to P\
Q'= the opposing force, tending to drive out the key,
f = tg d, the co-efficient of friction between the surfaces
of the three parts.
For keys with draft upon one side, we have :
0 = /)2/^( a-f 2f)	\
Q'= P2 tg(2 0 — a) ; ...........
In order that Q' should not be negative and the key come out
of itself, we must have a < 2 (p. Forf — O. 1, this gives t g a <
(67)
* Royal Prussian Patent, May 23, 1876. No. 159. Illustrated by drawing,
model and description.48
THE CONSTRUCTOR.
For keys with drafton both sides we have approximately :
Q = P 2 tg («+*)')
\..................(68).
Q'=P2tg (0—0) J
In this case it is necessary to keep below the full value off for
each edge of the key in order that the connection may not force
itself apart. The total draft will be found to have approximate-
ly the same minimum value as in the previous case.
In practice it has been found that keys which have shown en-
durance and resistance under load, have been made with a total
draft of 3^, and even or less, while others made with
TV or sometimes l/e are less secure.
The load upon a key may act in three different manners each
of which may agam be positive or negative. In the first, the
load acts normal to the base of the wedge, as at QK, Fig. 181,
or as Py in Fig, 179 and 180 ; and for this form, the term Cross
Key may be used. The second case occurs when the load acts
normal to the plane KH Qy as K Ly in Fig. 181, which may be
called a Longitudinal Key. The third case is that in which the
force acts normal to the plane Q K Ly as K Ht Fig. 181, which
may be called a wedge key.
§ 67.
Cross Keyed Connections.
In Fig. 182 we have an example of a cross keyed connection.
The rod and the key are both of wrought iron, the boss is cast
iron. The stresses for a given force P upon the rod are : the
bending stress upon the key, as in Case VIII. § (6) (Stress 6\);
the shearing stress between the key and the inner edge of the
Fig. 182. Fig. 183. Fig. 184. Fig. 185.
boss (Stress .S2), and the tension upon the segment shaped sec-
tions of the rod on both sides of the mortise for the key (Stress
6*3). If, according to § 2, we make S2 = 0.8 Slt and S1=S3f we
have:
5	d
3 = o. 267 dy or say —
If we make hl=o.S dy h2 = d, 6—0.5 dt we shall have good
practical proportions. Iu Fig. 183 we have two wrought iron
rods coupled by wrought iron keys. In this case a wrought
iron sleeve is used, whose thickness <5 = 0.25 d. Fig. 184 shows
a form similarto Fig. 182, except that the key passes below the
boss, instead of going through it, while in Fig. 185 the key is
let into the side of the rod.
The pressure upon the base surface of the key in the case of
Fig. 182 may be taken as :
_ P _ (0.7854 d* — bd)Sz
p ~ b d bd
which gives p = 2.14 S3..................................(70)
uite a high enough value, especially if we take, in Fig. 183,
= 0.25 d. The pressure becomes yet higher for the method
shown in Fig. 185, for which case the value of S3 shonld not be
taken too great. If the connection is intended to be taken
apart frequently, the value of p should not be allowed to be too
great. This may be accomplished either by reducing the value
of S3, or by providing an increased cross section of metal about
the mortise for the key, or by extending the surface by means
of cotters or gibs, as shown in Fig. 186. The key may then be
made smaller than already given above. The forms of keyed
connection shown are usecl for example in the rod connections
of water wheels, and in similar cases.
Fig. 187.
In Fig. 1S7 is showm a method of keying a foundation bolt.
The gibs or cotters are used to increase the strength. Fol-
lowing the calculations of $ 12, the depth of the three pieces
may be made alike in the middle. The anchor piate in the
foundation masonry should be arranged so as to give access to
a nut on the lower end of the bolt, and this can be tightened
by hand until the bearing is thrown upon the key, and the
driving in of the latter binds all the parts firmly together.
§68.
I^ONGITUDINAE KEYS.
Keys of this class are principally used to secure the hubs of
wheels to their shafts or axles. For this purpose they may be
considered as divided into three classes, as follows :
Concave, or hollowed keys, Fig. 188, 1 ;
Flat Surface keys, Fig. 188, 2, 4, 5,
and Recessed keys, Fig. 188, 3.
The Concave key is only suitable for constructions involving
small resistance, and acts merely by the friction due to the
pressure which it causes. The flat surface key is capable of
i	2	3	4	5
resisting much greater force and vibration, and when used in
the multiple manner shown in 4 and 5, it makes a secure and
efficient fastening. The recessed key, shown in 3, affords a
very secure method of fastening hubs to shafts to which they
have been closely fitted, and is simply and readily made. Keys
of this kind are also used as an additional precautionary fasten-
ing for hubs which have been forced on.
In determining the dimeusions of keys it will be found most
convenient to use empirical methods, except in cases of great
vibration ; the following formulae will be found to cover the
usual range of work. The material for the keyis taken as
Steel, and distinction is also made between cases in which the
hub is subjected merely to endlong pressure, and those where
torsional stresses exist. The former may be called draft-keys,
the latter torsion keys.
If we call the diameter of the shaft Dy the breadth of the
key Sy and the middle depth of the key Sx we have :
for Draft keys, ^ = o.24// H——; Sx = 0.16" + —
for Torsion keys, 5*= 0.16" +	$1 = 016'' + ~~
The taper of such keys is made aboutTHE CONSTRUCTOR.
49
For the more commonly occurring diameters we have the
following proportions :
= 1"	2"	i"	4V 5"	6" 7*	8"	9"	10"
			For Draft Keys :				
= 1"	i "	1"	W' 1"	ii" 1}"	if"	1V'	ii"
L=i"	N'	tV'	1// 9// 5 To	i" i"	13// 1 6	V'	1"
			For Torston Keys.				
= |"	H'	a// 4	1" ixy'	if" i*"	ij// 2//		2i"
	 1// l	 4	3// 8	¥'	A" H	i" i"	I7/	1*	IlV
For shafts of
5= IL, 5,=
3	5
less diameter than i"y we may make
If several keys are to be used, they may
be made the same dimensions as single keys. For hubs which
have been forced on, and hence would be secure without any
key, the dimensions for draft keys may be used.
§69.
Edge Keys.
When ihe pressure upon a key acts at right angles to the
plane of its height, the difference between the positive and
negative direction of the forces is readily distinguishable.
\
Fig. 189.	Fig. 190
When the pressure acts as in Fig. 189, the combination is inse-
cure, since the only binding action of the key is that due to the
pressure, and consequ-nt friction between the parts. If the
base of the key is rough, and the inclined face smooth, the
action of a force in the direction H', tends to tighten the parts
together. An application of this action is shown in the curved
key of Kernaul, shown in Fig. 190. When the hub is rotated
in the direction of the arrow, the action is the same as that of
the force H', in the preceding case, and the shaft is firmly
grasped. A couutersunk screw at a, is used to tighten the key,
and a similar one at by to loosen it. This principle will be dis-
cussed later, under the subject of couplings.
§ 7o.
Methods oe Keying Screw Propeeeers.
In securing the propellers of steamships the greatest care
must be observed in the methods employed, and in their appli-
cation. In Fig. 191 is shown Rennie’s method of securing one
of the blades of a Griffith’s double bladed propeller. In this>
case a rectangular key is used, passing through a cylindrical pin
which is cast in one piece with the blade and which is in turn
held firmly by the four smaller keys shown. These latter keys
are held in their places by caps secured firmly by jam nuts. (See
§71.) The blade and hub are both of bronze.
Example. In a propeller by Penn & Son, d = 15", & = 7^5", b = 2^".	*
Fig. 192. This shows a method used by Maudslay, Sons and
Field, Ravenhill & Hodgson, and others. Two rectangular
keys, passing through the hub of the propeller boss, and re-
cessed into the metal of the shaft, act at the same time to receive
the thrust of the screw and to prevent rotation upon the shaft.
In this case the hub is made of bronze.
Example. In the “ Eord Warden,” the middle diameter of d= 19", / = 52",
h = 8^", b = 3 yk" ] in the “Eord Clyde,” d	1= 54", h = 10", £ = 3".
Fig. 193. This shows a method of using two longitudinal
keys. The hub is bored with a quick taper, and a heavy bronze
nut holds the hub upon the cone, while the longitudinal keys
resist the action of torsion.
Example. In the “ Minotaur,” engined by Penn & Son, the mean diame-
ter d = 18^", l = 48",	3' •
The ordinary rectangular keys are also used to secure screw
propellers, as well as special forms of fastenings.*
? 7i-
Uneoaded Keys.
The force P, which under ordinary conditions bears upon a
key, may by various methods be supported by other means;
the key in such a case may be said to be unloaded. Such con-
structions offer a much greater security, and permit much
lighter keys, than the methods previously described. A few
examples will serve to illustrate.
* See N. P. Burgh, Modera Screw Propulsion, London, 1869.50
THE CONSTRUCTOR.
Fig. 194. This shows a form of connection used for mine
pump rods; the interlocked uotches receive the load of tension
upon the rod, and the hollow key only serves to bind the parts
together without itself supporting any of the weight. Fig.
195 shows Wiedenbruck^ rod connection.* The hub is made
Fig. 194.	Fig. 195.	Fig. 196.
in halves and the reversed conical seats receive the load. In
Fig. 196 is shown a connection for two intersecting plates ; by
Bayliss.f
The method of keying shown in Fig. 192, Hy may be made
quite secure by relieving the key from the load, and examples
of this form are often found.
? 72-
Methods of Securing Keys.
In order that a key may not back out under its load, the
angle of taper should be less than 4^, or if it is symmetrical in
form, each side should be less than -j1^, providing the co-effi-
cient of friction is equal to y1^. Even when the taper is made
Fig. 197.
Fig. 198.
Fig. 199.
Fig. 200. Fig. 201.
less than this, however, keys are very apt to become loose if
they are subjected to much vibration, and to sudden and irregu-
lar changes of load. In order to provide against such emer-
gencies, and also in order to permit the use of greater taper,
various methods of securing keys are employed.
The simplest method consists in splitting and spreading the
small end of the key, and for some purposes this is sufficient.
In order to prevent a key from flying back, or jumping out, the
projecting end may be drilled and fitted with a split pin. For
the keys used in connecting rod ends various methods are used,
examples of which will be found ki the following figures. Figs.
197, 198, and 199 use screws under tension. When these are
used in locomotives or mariue en-
gines, the screw is again secured by
the use of a jam-nut. Fig. 200 is
used with a set-screw, the point of
which bears in a shallow channel in
the side of the key, so that if the
pressure of the set-screw is unable
to hold the key, it will at least keep
it from flying out. The channel is
also of Service in preventing the
point of the set-screw from marring
the finished surface of the key. Fig.
201 shows a form of screw-clamp.
This clamps the key by drawung the two blocks tightly against
its sides ; the screw passes through a slot in the key.
Fig. 202 shows the method employed in securing the key
used in the form of fastening for screw propellers, used by
Maudslay, and shown in Fig. 192. A small block is bolted fast
to the projecting end of the key, and a bronze cap is screwed
down over all.
Fig. 202.
♦German Patent, No. 1507. See also patent No. 510 of H. Rademacher for
improved rod connections.
+ See Pract. Mech. Journal, Vol. III, 3d Series. P. 342.
CHAPTER IV.
BOLTS AND SCREIVS.
I 73-
Geometricae Construction oe the Screw Thread.
Screws are used in machine construction to produce three
kinds of effects, viz.: for clamping or joining parts together,
for producing pressure, and for the transmission of motion. We
shall now only consider the first two classes. Screws may be
classified with regard to the shape of the cross-section of their
threads into:
Triangular or V,
Square or Rectangular,
and Trapezoidal.
All these forms belong practically to the so-called axial screw
thread surfaces.* By this is meant the sur/ace which is described
by a right line ABC\ Fig. 203, when one of its points remains
upon a directrix, in this case the axis O D, of the screw, while
the generating line itself maintains a constant angle a, with the
axis, proportional to the advance wffiich the directing point
makes upon the axis. The angle a is called the angle of
advance, and its complement B, is the base angle of the screw
thread. These are either V, or square, according as the angle a
0
Fig. 203.
is an acute or right angle. The normal cylinder upon the axis
O B>y upon which the screw thread is described, is called the
pitch cylinder. This cylinder is supposed to pass through the
threads of the screw at such a point that two adjoining sections
bear the relation of screw and nut.
The space passed over by the directing point during one rota-
tion around the axis is called the pitch of the screw, and will
hereafter be designated by the letter s; and the angle which a
line tangent to the screw-thread at any point makes with the
base of the pitch cylinder, is called the pitch angle, called d.
From this it will be seen that threads described upon concentric
cylinders may have the same pitch, but different pitch angles.
The area of a V screw thread may be taken as equal to the
sum of the surfaces the two halves of the thread, opened out to
an angle of 180°. For rectangular threads the area is simply
that of the corresponding simple surfaces. For trapezoidal
threads the area is equal to the sum of the inclined and parallel
surfaces (see § 86).
? 74-
Dimensions of V Screw Threads.
For any given force Py acting parallel to the axis of a screw,
the resistance of the metal of the body of the screw may be
determined according to Case I., page--, but the stress may
instead be taken as a simple case of tension if the value of S,
be not made too great. If we take for wrought iron, 6*= 3600,
and let dx be the diameter at the base of the thread, we have :
*/1==o.02 >/ P 1
F—2750 d\ J
(72)
The nut is generally hexagonal, but is sometimes made square ;
we will here limit ourselves to the former shape. The thickness
of the nut is usually made equal to the outside diameter d, of
the screw. This makes it much stronger than the threads of the
screw ; f but the depth is desirable, as it distributes the pressure
over a greater area of screw thread. For the superficial pres-
*See the article : Conceming some Properties of Screw Threads. Berliner
Verhandlung, 1878, p. 16.
f Recent investigations made at the Stevens Institute at Hoboken, show
that the resistance of the thread is reached when the thickness of the nut
-s made 0.45 to 0.4 d. See Railroad Gazette, 1877, November, p. 483.THE CONSTRUCTOR.
51
«ure p, we have for a depth of thread t and n threads in the nut,
both for V, and square threads :
(73)
(74)
s 1 <1 r t /1 \1
^	4 ’ n tj^1 3 d + ^	J '
Introducing the pitch S> and making n s = d, ^
In both equations the third member may be neglected.*
The value of p should not be permitted to exceed 1440 lbs.
If n = 8, and -y- = 12, we have, taking Sy as above,
p = 3600 X f (1 — i + tti) or about = 1000 lbs.
In the consideration of this subject, the friction of a screw
should not be neglected.
Let:
Q = the force acting at the mean diameter of a screw,
normal to the plane of the axis ;
6/ = the pitch angle of the thread at the mean diameter,
f—tg§y the coefficient of friction ;
we then have, in order to overcome the thread friction, due to
the force P, for square thread :
f 4- tg P
Q = p[~ftg <5r ~	p (* + *)
or for the resistance:
f— tg 6'
Q' = P 1+ ftgv = P-J/)
while for V threads, we have :
f dk tg V
Q = P—------~—
(75)
in which f'
i^FftgV
J
: Ptg ir ± V) .
(76)
COS P
In order to overcome the friction on the base of the nut also,
the value of Q must be more than twice as great. For tg 6\
we may take then tg S. This value is usually so small that the
friction often cannot resist the load, and the value of Q' becomes
negative.
I 75-
The Whitworth Screw System.
By a system of screw threads is meant a collection of rules or
formulae by which the profile of thread, pitch, diameter, and
other details of screws and nuts may be determined. Such a
system was first formulated by Whitworth in 1841, and since
that time the subject has been more and more studied, until it
is now considered one of the greatest importance,| especially in
regard to the metric system.
; d
Fig. 204.
A united opinion on the subject has not yet been reached.
Many weighty reasons have been advanced for the introduction
*p=m p n t {d—/) n, hencep — S	d^ : n n(1--------d2 = —— -P-*
4	a \	& J	4 w
: 1---* which, by neglecting	+» etc.,
gives the above resuit. In this case, p is the pressure upon the projected
area of the screw thread.
f See The Metrical Screw System, etc., by a Committee of the Society ot
■German Engineers. Berlin, Gartner, 1876.
of the Whitworth system into Germany, while others, equally
strong, have been advanced for the metric system.
The Whitworth system takes for the form of a section of a
screw thread an isosceles triangle whose base is equak to the
pitch s, and whose angle at the apex = 550, froni which the
height t0 = 0.96 s. The thread is rounded at top and bottom to
an amount equal to J /0, so that the working depth / = § t0 =
0.64 s. The values of the pitch 5 were given by Whitworth in
a table which extended to i/7.* The use of this system developed
some deficiencies, among others the difficulty of originating the
cross-section of thread, and the gradation of diameters. The
original table of diameters was not altogether satisfactory to
Whitworth himself, and in 1857 he extended the old table by a
new one, which, since that time, has been known in Bngland
as the Standard system for bolts and nuts.f In Germany, how-
ever, the whole subject is yet under active discussion.
The following table gives the old and new scales, the values
of d and s being in English inches. The values for T5F" and
rV7 are only given approximately.
Whitworth’s Screw Thread Scaees.
New Scale.	Old Scale	1	New Scale. ]	Old Scale.	I
d.	d. .		♦ d.	d.	
O.IOO		48	I.125		7
O.125	X	40	I.250	1K	7
0.150		32	1-375	1H	6
0.175		24	1.500	r/z	6
0.200		24	1.625	1 H	5
0.225		24	i.75o		5
0.250	X	20	1.875	1 y%	4 X
0.275		20	2.000	2	4X
0.300	*	**	2.125		4 X
0*325		18	2.250		4
0*350		18	2.375	r/i	4
0*375	X	l6	2.500	r/z	4
0.400		l6	2.675	2 H	4
0.425	«	14	2.750	2^	3X
0450	rs-	14	2.875	2%	3X
o.475		14	3.000	3	3X
0.500	X	12	3 250	3K	3X
0.525		12	3*500	3 %	3X
0*550		12	3-750	iK	3
0-575		12	4.000	4	3
0.600		12	4.250	4 'A	2^
0.625	X	II	4500		2^
0.650		II	4.750	4%	2X
0.675		II	5.000	5	2X
0.700		II	5.250	5%	2 yk
0.750	X	IO	5-5oo	5'A	2#
0.800		IO	5.750	sU	2/2
0.875	X	9	6.000	6	2/2
0.900		9			
1.000	1	8			
Whitworth’s Pipe Thread Scate-
d=X	X	yk	XX 1	IX	iX *X 2
n = 28	19	19	14 14 II	II	II II II
The regularity of the progression might be improved upon.
This may be more clearly illustrated in the following diagrams.
The greatest irregularity lies between the sizes from to
2\ and the gradation of diameters is also uneven. The
cause for this lies in the system of measurement used. Whit-
worth evidently perceived the desirability of introducing a
decimal notation, but also wished to retain the fractional divi-
sions in halves, quarters, eighths, &c.; this has partly been
secured, neglecting sixteenths, by having the gradation based
upon fortieths, and their combinations as shown in Fig. 206.
For the pressure p, we have from (74), taking t — 0.64 s :
'5r= 4 xo.64 £1 ~ l'9~T + °'* 4
If we make S = 3555 lbs. we have for d = o.i", 3//r, and 6//,
the values of p = 938 lbs., 1152 lbs. and 1209 lbs. For t g 6, we
have, when d = o.i",	, and 6", the values 0.0663, 0*°3°3> and
0.0212.
* Briggs stated the relation of pitch and diameter of the Whitworth system
to be approximately:—
s = 0.1075 d — 0.0075 d2 + 0.024.
fSee Eng. and Arch. Journal, 1857, P- 262 ; 1858, p. 48 ; also Shelly, Work»
shop Appliances. Tondon, 1876, p. 102.52
THE CONSTRUCTOR.
. ? 76.
Seeeers’ Screw Thread System.
Tlie confusion in the use of screw threads having become
very troublesome in the United States, Mr. William Sellers
brought before the Franklin Institute, in 1864, a System which
he proposed for general use.* A committee of the Institute
reported upon the System in December of the same year, and
recommended its general adoption. f This System is now
Fig, 207.
generally known as the Sellers System. The profile of this
thread is shown in Fig. 207. The thread angle 2 (i = 6o° ; the
depth t = 0.75 tQ = 0.65 s. The pitch is determined by the
formula .S= 0.24 y/d + 0.625 — 0.175, the resuit, as with Whit-
worth’s system, being so modified that the number of threads
per inch shall be a whole number.
The following table gives the adopted number of threads per
inch for various diameters :
	A	y%	A	X	A	H	X 7A
i = 20 s	18	16	14	13	12	II	10 9
d=i'/$	iX			i#		I#	2
E = 7 s	7	6	6	5X	5	5	4X
d=.i%			3	3X	3 X	3X	4
f=4*	4	4	3X	3'A	3X	3	3
d = 5	5lX	5X	5X	6			
T =	2'A	2X	2 y	2X			
The Sellers System compares very favorably with the Whit-
worth System, and notwithstanding the difference in profile, it
gives almost the same depth of thread. The angle is very con-
venient, and the simplicity of the profile is such that a suitable
tool may easily be made and used in the shop. These facts
explain the rapid introduction of the system in America. The
progression of the pitch is also more uniform than in Whit-
worth’s System ; and the uncertainty about the thread of the
screw, which was always a stumbling block in the original
Whitworth Scale, is avoided. The values for and T^// are
retained as in the Whitworth Scale of 1857, and T9^7/ is also pro-
vided for, so that the requirements of the English system of
measurements are fully met, up to 2//.
? 77-
Metricae Screw Systems.
Recognizing the advantages which have followed from the
introduction of the Whitworth System, various attempts have
been made to devise a system of screw-threads which shall be
adapted to the metric system of measurements. The following
fourteen Systems have been suggested :
Armengaud, Redtenbacher, Paris-Eyons-Mediterranean R. R.,
Northern Railway of France, J. F. Cail, the French Navy,
Bodmer, two Systems proposed by Ducommun, of Mulhouse.
Alsace ; the Engineering Society of Mulhouse, Reishauer &
Bluntschli, of Zurich; the Pfalz-Saarbriick Society of German
Engineers, and two Systems of Delisle.
The formulae and tables given in the previous editions of this
work have also been spoken of as systems, but they are not en-
titled to any such position, as they were merely adaptations of
the Whitworth system. The number of proposed systems may
be taken as an indication of the difficulty of the task. Indeed,
it is only by very carefully weighing the respective merits of
the various pians, that it is possible to say which is the best.
The following requirements should be kept in mind as essentials
in considering any system :
1.	The profile of the thread should be such as may be readily made with
requisite accuracy. In this respect Whitworth’s system is deficient, and
the profile of the Sellers thread is to be preferred.
2.	The pitch should be given, so far as possible, directly by the formula,
without requiring any modification of its resuIts. Both Sellers and Whit-
wotth are deficient in this point, since they both modify the results of their
formula.*
3.	The gradation of bolt diameters should be so disposed that fractions of
millimeters should not occur in diameters, and that their gradation shoulcL
conflict as little as possible with the decimal system.
All three of these requirements should be attained within the
limits of generally used sizes, and should at least extend to
bolts of 80 mm. in diameter. The last three systems, viz.: the
Pfalz-Saarbriick system and the two of Delisle, are the only
ones which appear to have considered these points, and these
wre shall examine somewhat in detail.
£78.
Metricae Screw Thread Systems.
Deeisee /, Pfaez-Saarbruck and Deeisee II.
The following three diagrams show the gradation of pitch and
diameter for the three systems, the ordinates representing the
pitch being shown 011 five times the scale of the bolt diameters,
and the values being also given for d and s in the annexed tables.
In the first two,cases the profile of thread is exactly the same as
in Sellers’ system, while in the third, the base angle is made
26° 34'. This has been chosen for the purpose of making the
theoretical height of the triangle of the thread equal to E The
thread is flattened as in the Sellers system.
All three of these systems are marked by simplicity and in-
telligibility. These features have been attained by abandoning
the idea of representing the relation of 5 to d by a single equa-
tion (such as that of a parabola), and using two or more equa-
tions of straight lines. A noticeable irregularity exists in the
Pfalz-Saarbriick system between the diameters of 26 and 28 mm.,
indicating that a somewhat finer pitch is used in proportion to
the diameter below 26 mm.
The second system of Delisle is rather simpler than the first ;
there is also an important difference in the angle of thread, as
will be seen subsequently.
* In the old Whitworth scale all 33 values were modified ; in the Sellers sys-
tem this is done with 31 out of 34 sizes.
♦Journal of the Franklin Institute, 1864, Vol. 47, p. 344.
f Journal of the Franklin Institute, 1865, Vol. 49, p. 53.THE CONSTRUCTOR.
53
DEEISEE /. FlG. 208.
0.8
5 j 6 j 7 I S {io 112 114 j16
i.o’ii.2 i.4!i 6!i.8 2.0'2.2 2.4
Illi!
18 I 20
2.6:2.8
d = 24 28 32 36 40 48
S =-- |3-2j3 6j4.oj44 4.8j5.2
fElZLi??.
5*6|6.o 6.4 6.8
Diameter and pitch both in millimeters.
For any interpolated diameter the next lesser
Deeisee //. Fig. 210.			
d = 5 =	6 1.0	8 1 10 112 j 14 j 16 ! 18 j 20 j 24 1.21.4 1.6,1.8,2.0 2.212.4 2.8 1 1 l 1 ! 1 1 i	
d = S —	28 32 3^|'4£. 48|56 64 3.2'3.6 4.0 4.4 4-8 5-2 5-6 l .1 i 1 1 1		72J80 6.0 6.4 I
For any interpolated meter the next greater
ordinate is to be taken, as for example d = 60.*
In all three Systems the superficial pressure is quite satisfac-
t:ory. Aceording to formula (74), taking 5 = 3600 we obtain for
values of p—
Delisle I..........S600 to 11,500 Ibs.
Pfalz-Saarbriick . . 8600 to n,ooo “
Delisle II ... . 7600 to 10,000 “
2 79-
New Systems.
A thorough investigation of the proposed systems of the Ger-
iman Society of Engineers failed to produce any definite results,
and the whole subject of a metrical screw thread system is stili
xmsettled. For this reason it has been thought advisable to of-
fer a further discussion of the problem.*
It might seem a shorter plan to adopt some one of the three
preceding systems, yet they all seem capable of improvement.
* This is especially necessary for use in technical instruction, which will
afiford the surest method of introducing a metric screw thread system into
practical use. The advocates of the Whitworth system urge the desirability
of an International Standard, in view of the widespread use of the
American system, which is indeed alreadyin use to some extent m Ger-
many. In this case the conflict between the two systems of measurement
lias been met by proposing to take for any dimension in English units the
next higher dimension in millimetres. Such a system would be impractic-
able for educational purposes and would lead to many errors m actual
practice. It also seetns only to be practicable for the old Whitworth scale,
and for the new scale, with its close divisions, its apphcation would be im-
possible. A comparison between the preceding diagrams will show that a
close adherence to the Whitworth system would resuit in a complication of
^iimensions which would be most undesirable.
The subject will bear further investigation in two main points
one being the gradation of diameters and the other the profile
of thread. The actual diameters and their gradation are of
more practical importance than the gradation of threads. This
is shown by the fact that the Whitworth profile has long been
in use with the bolt diameters taken in Prussian inches, and
more recently with dimensions in millimetres with Whitworth
profile. One of the first requisites of such a series is that the
diameters should follow the decimal divisions (see the third
condition of £ 77). This point is not met by the preceding sys-
tems, since they lack the natural divisions 30, 50, 60 and 70.
The removal of this objection introduces a new difficulty, but
not an iusuperable one.
The critical feature of the screw thread system is really the
relation which the diameter bears to the profile. A thread should
not be said to be coarse or fine, implying the ratio s: d, but
rather should the depth of thread be considered, or the rati o t: d.
This can best be illustrated by an example :
If we select two equal sizes fromthe systems Delisle I and 11,
we shall find that for the same pitch the threads are not alike.
If d= 60 mm. we shall have (see the dotted lines in Fig. 208
and 210) in both cases s = 5.6, hence the angle of thread is the
same.
The working depth ty however, is :
in I \t-=.%t0 =0.65 5 = 3.64 mm.
in II: t=. }(t0 = 0.75 5 = 4.20 mm.
*In both his systems Delisle has provided for the interpolation of inter-
mediate diameters, but these have been omitted frorn the diagrams and
tables to avoid obscurity.54
THE CONSTRUCTOR.
This gives for the diameter of the bolt at the bottom of tbe
thread
in I : dl = 57.70—cross section 2182 sq. mm.
in II: <7!= 51.60— “	“	2091 “	“
which shows a difference in resistauce of about 5 per cent. be-
tween the two bolts, the second having the coarser thread. We
see here that a choice of the relation of s to d affects the pro-
file of thread, and it is this which led Delisle to suggest two
Systems.
Whether the angle of 530 87 is preferable to the Sellers angle
of 6o° is uncertain. Among the preceding systems may be
noted two for the latter, fi ve for the former, and three for
Fig. 21 i.
and hence
stili smaller angles ; and if the choice be given, it seems rather
better to go below the Whitworth angle of 550 than above it.
We prefer the angle as shown in Fig. 211 :
2 /?=53° 8' or t0 =s \
t = to = S	)
For sizes of d from 4 to 40 mm. the pitch may be
s = 0.4 + 0.1 d	........
and for sizes of d, from 40 to 80 mm. and over*
s = 2	0.06 d	........
with the following series of diameters :
(77)
v
(78)
■ (79)
4	5	6	7	8	9	10
12	14	l6	l8	20	22	24	26	28	30	32
36	40	.	45	50	60	70	80
Formula (78) is the same as in Delisle II, from 6 to 40 mm.
Interpolation for intermediate diameters seems unnecessary ;
Fig. 212.
should it be done, however, the formula should not be departed
from, since the values in the second and third groups above
will give round numbers, and offer no difficulty for their pro-
duction on the screw cutting lathe. If it is stili desired to use
the angle of 6o°, and yet retain the other proportions, we may
take
for d = 4 to 8, s' — 0.2 d (as in Delisle I) 1
for d — 8 to 40, s/ = 0.8 -f- o. 1 d (as in Delisle I > . . (80)
for d — 40 to 80, sf = 1.6 + 0.08 d	J
in which arrangements the sizes 30, 45, 50, 60, 70 remain in the
series, which may also be extended above 80 mm. The two
pians may becompared to Fig. 212, in which the formulae are re-
spectively applied to a diameter of 80 mm. The radii to the
♦For sizes over 80 mm. we have not yet established relations. If we take
d = 150 mm. which is about as high as Whitworth or Sellers have gone, $ =
11 mm., which seems a good proportion. See g 87.
bottom of thread r\ and rlf are almost identical, as are alsa
working depths, although the profiles differ, as shown by the
triangles ABC and D E F. Instead of numbering the sizes
arbitrarily, it seems preferable to use the bolt diameter for the
number. Screw No. 20 would then stand for d— 20 mm., No.
4 for d =4 mm. Any establishment could omit numbers not
desired without impairing the system, while for fine work
smaller numbers could readily be added.
I 80.
Nuts, Washers and Boet Heads.
The thickness of metal in a nut bears a close relation both to
the depth of thread t, and to the pitch s. It is desirable that
the formula to be used should give the dimensions readily in
order to avoid the necessity of approximating.
Fig. 213.
For the diameter D, of the inscribed circle of the hexagon we
may take for finished nuts :
D = .04 -f” d -f- 0.5 s ............(81)
The maximum pressure upon the base of the nut in this case
(for d = 3/a) = about 2400 lbs. per square inch. Unfinished
nuts are made somewhat heavier, and lor them we have
Dl = 0.14" + d + $s ...............(82)
Fig. 214.	Fig. 215.	Fig. 216.
The use of the washer insures a better bearing for the nut in
case the surface is not true. Its dimensions may be taken as
diameter = U=d + 10 s
thickness = a = ^ s
4
Bolt heads are often made square, but are preferable hexagon-
al, and for them we may take D and Dlf the same as for nuts,
and the height h = 0.7 d. Fig. 213.
For finished nuts the upper surface may be finished with a
bevel of a frustum of a cone whose base = D, and a base angle
of 30°, Fig. 214, or as a portion of a sphere with a radius of f
D, Fig. 215, while unfinished nuts have the corners beveled ofF
above and below, as shown in Fig. 216.THE CONSTRUCTOR.
55
\ Si.
Tabide and Proportionae Scabe: for
Metricae Boets and NUTS.;f
The preceding table contains a summary of
the preceding discussion, and Fig. 217 is a dia-
gratn in which the relations of the parts are
shown graphically. The value for s is shown on
a five-fold scale. The dotted line gives the value
for t7 of formula (So).
The diagram Fig. 217 shows the pitch of
thread and the pressure upon a unit of area, for
the dimensions of nuts and bolt heads for the
preceding metric screw thread System for diam-
eters from 4 to 80 mm.
The pitch is shown fi ve times full scale (line
E) and ten times full scale (line F); the bolt
diameter in its actual size (line/)), ali measured
from the base line A. The line B is 1 mm.
from Z), and C is 4 mm. from D while the dis-
tance between A and G is 0.7 d.
The various details may be sumtned up as
foliows :
Between A and E = the fivefold pitch,
“	E and B =	dia.	of	finished nut,
“	E and C =	dia.	of	rough nut,
‘ ‘	F and D =	dia.	of	washer,
“	A and G =	height	of bolt head.
The tangent of the pitch angle ranges from
0.064 to 0.047, and the pressure per sq. mm. on
the thread, from 0.46 to 0.67 kilogrammes.
? 82.
%	Weight of Round Iron.
The weights in the following table are given
by the Formula
G = 2.617 h2,
the bars being one foot long and d = diameter
in inches. For cast iron, multiply the values
in the taole by o 93 and for bronze by 1.092. A
hexagonal rod whose inscribed diameter = d is
1.103 time the wreight of a bar of wrought iron
of the same diameter.
Boets and Nuts. (Metric svstem).
Bolt Dia. d	i Pitch. i	! Depth of ! Thread.	i Bottom Dia. of | Bolt.	; Nut.		Washer.		! Bolt Head	Load. P
mm.	1 j !	1 *	! 4	! D	i A	! U j	ti	n	kilos.
4	| 0.8	0.60	2.80	i 9	—	12	I	3	l6
5	1 0-9	0.68	| 3.65	I 10.5	—	: 14	I	35	27
6	1.0	0 75	4.50	12	—	! l6	I	4	41
7	1 1.1	083	i 5-35	! 13.5	—	18	15	5	i 57
8	1.2	0.90	( 6.20	: 15	1 —	; 20	1-5	6	i 77
9	i J-3	0.98	7-05	| 16.5	; —	j 22	i-5	6	1 99
IO	I 1-4	1.05	7.90	! 18	1 21	i 24	1-5	7	! 125
12	1 1.6	1.20	9.60	1 21	! 24	! 28i	2	8	184
14	! 1.8	i-35	11.30	24	i 27	i 32	2	10	255
16	2.0	1.50	13.00	27	| 30	! 36	2	11	338
18	2.2	1.65	14.70	30	33	i 4°	3	13	432
20	: 2.4	1.80	16.40	33	36	; 44!	3	14	538
22	! 2.6	i-95	18.10	36	39	48!	3	15 ;	655
24	2.8 ^	2.10	19.80	39	42	52!	3	17	784
26	3-o'	2.25	21.50	42	45	56	4	18	841
28	3-2	2.40	23.20	45	48	6o|	4	20	1076
3°	34	2.55	24.90	48	5i	64!	4	21	1240
32	3-6	2.70	26.60	5i	54	68	4	22	1415
36	4.0	3.00	30.00	57	60	76	5	25	1800
40	4.4	3.30	3340	63	66	84:	5	28 !	2231
45	4-7	3-53	37.95	1 70	73	92	6	32	2880
50	5-o	3-75	42.50 ;	! 76	79	DX3	6	35 |	3613
60	5.6	4.20	51.60	i 89	92	Il6	7	42 1	5325
70	6.2	465	60.70	102	105	132	7	49	7369
80	6.8	510	! 69.80	115	118	148	8	56	9744
Weight of Wrought Iron Rods. One Foot Fong.
j d	G	i d	1 G	j d	j G !
X	.163		3-68	2 / S	11.82
fV	•255	|iX	t 4-09	2 UT	12.50 1
H	.368	!iA	I 4-50		!3-25 i
t%	.500	jiX	1 4-94	2l\	13-95 i
H	.654	r*	i 536	2^	14.76 j
f/	.826	kx	! 5-89	1 2^	I5-54
H	1.02		j 6.39	j 2K	16.36
11	1.23		6.91	2rV	17.14
H 1	1.47	‘iH	7-43	2%	18.03
1 S i TU !	1.72	jU	8.01	2ri	19.11
X	2.0	‘Ixf	8-57	2^	T9*79
II ;	2.29	17A	9.20		20 61
1 !	2.61	iiH	9-79	2%	21.63
Uj;	2.94	2	10.47	0 1 5 ! 2tt 1	22.52
1 'A :	3.3i	2 i'V	11.02	3 ’	23.56
§83.
Speciae Forms of Boets.
Instead of being made with square or hexagon heads, bolts
are sometimes fitted with special heads, instances of which are
shown in Figs. 218 to 222 ; the last being countersunk. These
are all furnished with means to prevent the bolt from turning
when the nut is operated.
♦This table has been kept in the metric sj^stem for obvious reasons. Trans*56
THE CONSTRUCTOR.
In Fig. 223 is shown an anchor bolt with cast iron piate for
brickwork. the bolt being inserted from above and locked by
being turned 90°. The area of the avichor piate should in no
case be less ihan 100 dj2
In Fig 224 is shown a form of anchor bolt for masonry with
a cast :: *i washer, secured by a key. The washer should be
not less than 25 d x2 in area. Such plates are often made of
wrought iron.
Fig. 218. Fig. 219. Fig. 220. Fig. 221. Fig. 222.
In Figs. 225 and 226 are shown bolts secured by cross keys and
side keys. I11 these two figures the nuts are shown in different
positions, the latter being the more convenient to use the pro-
portions shown in Figs. 214 to 216. Figs. 227 and 228 are fornis
of stud bolts. Fig. 229 is a cap screw*. For small work these
cap screws are often made with cylindrical heads with slots for
use with a screw-driver.
Fig. 225. Fig. 226. Fig. 227. Fig. 228. Fig. 229.
?84.
Wrenches.
A wrench is a lever adapted to tighten and loosen nuts and
bolt heads. The simple wrench, shown in Fig. 230, in two
forms, consists of a flat or round handle fitted to the shape of
the nut, the dimensions being based upon the unit D, wdiich is
the diameter of the nut as given in formula (81) . The double
wrench, Fig. 231, is arranged toreceive nuts of different sizes at
the opposite ends of the handle. If the ends are inclined so as
to bring the corners of the hexagon at 150 and 450 wdth the
axis of the handle the wrench will be able to operate in con-
tracted spaces by ^ revolution of the nut.*
Nut Locks.
For bolts made according to the preceding proportions, the
angles of pitch are not steep enough to allowT the pressure in
the directioii of the axis of the bolt to overcome the resistance
of friction and cause backw7hrd rotation. If, howTever, there is
much vibration, lost motion ma}T appear and gradually cause
the connection to w^ork loose. This is true of foundation bolts
as well as of those in moving parts of machinery and in loco-
motive and marine engines. For these and similar cases it is
necessary to have some method of securing the bolt or nut
from coming loose, and a variety of such nut locks are here
shown.
Fig. 232.	Fig. 233.	Fig. 234.
One of the oldest and most useful forms is the jam nut, Fig.
232. Both nuts should be truly faced so that they will bear
fairly upon each other. The thin nut is frequently placed
under the thicker oue, but this is immaterial since a nut of a*
thickness of 0.45 to 0.4^ is as strong as the bolt thread. The
security obtained by the use of the jam nut is not very great,
and the form wdth right and left hand thread, as showm in Fig.
244, is to be preferred when greater security is essential. In
Fig. 233 is showm a split pin, often used in connection wrth a
jam nut. Fig. 234 sliowTs an arrangement wdth a key upon the
nut, making a very convenient and secure combination. In
the three preceding cases the action is such as to tighten the
nut upon the thread. The three following methods are made
to hold by fastening the nut orbolt, or both, to the parts which
they are intended to hold togetlier. Fig. 235 is used in the
spring hangers of Borsig’s locomotives, Fig. 236 011 an oil cup
lid, and Fig. 237 011 a set screw for a connecting rod end,
arranged to lock at any 1-12 part of a turn.
In the followung methods the nut is held from turning by be-
ing locked to one of the stationary pieces, the bolt itself being
secured in a similar manner. The form shown in Fig. 238 is
used for bearing cap bolts, the support at "the middle of the
* This idea is due to Proell.THE CONSTRUCTOR.
57
split pin keeping it from bending. The method shown in Fig.
239	is used for the bolts in a steam piston, while that in Fig.
240	is for a bearing cap. The latter form is arranged by means
of the sever notches, to lock at every ^ of a turn, while the
other two require l/e of a tnrn between successive positions.
I
EiG. 235.	Fig. 236.	Fig. 237.
Fig. 241 shows a device for securing the nuts of stuffing box
bolts as applied to locomotive engines. The ratchet wheels are
attached to the nuts, and similar notched nuts may be used to
advantage in many places.
Fig. 238.
Fig. 239.
Fig. 240.
A method of securing the bolts for locomotive springs, used
by Borsig, is shown in Fig. 242. The tension of the spring
keeps the bolt from turning, and the cap which secures the nut
is fitted to the end of the bolt as shown ; this locks for every ye
of a turn. Fig. 243, shows a nut arranged to be locked by a
set screw. This method, used by Penn, is a very useful form
Fig. 241.
for bearings, spring hangers, and other situations, since it per-
mits any fraction of a turn to be made. The nut, in this case,
should be a little thicker than usual in order that the lower
cylindrical portion may not be too weak. The diameter Dlf is
in this case taken from formula (82). The small set screw
should be made of Steel and hardened. This form of nut lock
is especially useful on marine engines.
Fig. 242.
Fig. 243.
A different ciass of nut locks depends for its action upon the
introduction of an elastic resistance between the bolt and the
nut.* The elastic washer of Pagel and similar devices have
found many applications. Parsons’ bolts belong to this ciass. |
* See Dudewig Nut Eocking Devices. Bavarian Industrial and Technical
Journal, 1870, pp. 17, 144, 283; also Journal of the Society of German
Engineers.
f Engineer, Julv, 1867, p. 16; Nov., p. 391; Engineering, 1867, Nov., p. 411 ;
Railroad Journal', 1868, pp. 77, 117.
In this form the body of the bolt is fluted, so that the cross
section is reduced to about the same area as that of the bolt at
the base of the thread. This increases the elasticity of the bolt
and enables the nut to be tightened so that it is much less likely
to come loose. Fig. 244 shows a modification of this form
used by Gerber for bridge connections. The security is stili
further increased by the use of a left hand jam nut. Instead of
being fluted, the body of the bolt may be flattened on four sides,
or the reduction of area may be obtained by drilling a hole
into the bolt from the head to the beginning of the thread.
One of the most important instances of screw fastenings may
be found in the construction of built-up screw propellers. in
which the blades of the screw are bolted fast to the hub, a con-
nection requiring the greatest strength and security. Fig. 245
shows the base of such a propeller blade, from the same
example as shown in Fig. 192. The flange of the blade issecur-
ed to the hub by sixteen cap-screws. Four set screws serve to
provide a small adjustment of the blade within the range of
motion of the oval bolt holes.
Ali of the cap screws are secured. Fig. 246 shows the method
adopted by Penn. The bolts, which in the case of the Minotaur
are	diameter, have a common ring washer under the
heads. When the bolts have been screwed up as tightly as pos-
sible, a ratchet washer with hexagonal hole is slipped over
each bolt head. These ratchet washers are prevented from
turning by the introduction of small locking pieces which are
bolted fast to the large ring washer, being held down by the
thin nuts shown. The ratchet washers have 11 teeth, and
hence each bolt may be locked at "it part of a turn. Fig. 247
shows a method by Maudslay. Here each pair of bolts is held
by a flat key which permits iocking at ^ part of a revolution.58
THE CONSTRUCTOR.
A continuous washer ring is not used with this method, but one
washer is put under each pair of bolt keads, to which tke lock-
ing key is bolted. Another method by Maudslay is shown in
Fig. 248. A double washer is placed under two adjacent bolt
heads, and each bolt is locked by a small block held against
one of the faces of the bolt head by a small bolt. Three bolt
holes situated 40° apart are tapped in the washer for each block,
thus giviug an adjustment of tV of a turn. The method by
Penn gives the best opportunity for adjustment.
{ 86.
Speciae Forms of Screw Threads.
Screw threads of square or trapezoidal sectiou may be used
for bolts, but in their use it is desirable to use a deeper nut in
order to secure a sufficient number of threads in the nut to keep
the pressure per square inch 011 the thread within the pre-
scribed limits. Trapezoidal threads are well suited for bolts,
since the relation between s and d permits the use of the same
proportions as those given for y threads in Fig. 211. In fact
the thread in Fig. 250 may be given the same proportions as
that in Fig. 211, for depth i, and pitch s, inaking the angles
respectively equal to o° 011 one side and 450 on the other.
These fornis of screw-threads are priucipally used for screw-
presses and for similar uses.
For such screws the diameter dlt at the bottom of thread,
is generally determined from formula (72). If, however, it is
desired to make the diameter dl a mimimum, we must consider
the pressure to act only on one side of the thread in the nut
and then take the pressure per square inch at double the previ-
ous allowance, or 1 = 7110 lbs. We then have,
dx = 0.0134 v/ P )
P= 5568dl j
(84)
The depth of thread, both for square and trapezoidal threads, is,
d_ __
~~ 10 “	8
and for square threads—
_
5 ~~
and for trapezoidal threads—
2
A
4

d=-
(85)
15	& .
Formula (84) is applicable to screws of locomotiye springs,
since in this case the conditions are well complied with.
In order that the nut may not wear or grind out, the working
pressure on the threads should not exceed say 700 lbs. per square
inch. These conditions will obtain, according to (73), when the
nnmber of threads n, in the cast iron or bronze nut is not less
than
n =0.00035
or,
n — 0.0014
(-C)
4 (-4)
If t =---d, we have
10
= 0.00245 ^=0.00312
(86)
(87)
Example. For a pressure of 55,000 lbs., we have, under the preceding for-
mulae, from (84) the diameter at the bottom of the thread
d1 = 0.0134 v/ P = 0.0134 X 234.5 = 3.14"
The depth of thread, from (85)= -™L = o.392//, which gives
d = 3.92", or about 3—i-'7.
10
From (87) wehave, making 5 = 7710 lbs , the mimimum num-
ber of threads in the nut n — 00245	17.4 which gives for the
height ofthe nut for square thread h — ns = i7.4X-785 = 13.65",
while for trapezoidal thread h = 17.4 X .523 = 9. i/r.
In many cases the diameter of such screws is made greater
than the normal diameter indicated in the preceding discussion
for the given load. Such screws may be called enlarged screws,
as compared with the normal dimensions as previously deter-
mined. For such screws the same cross section of thread and
the same height of nut may be given as for the normal screw
of the same load, in which case the wear will practically be the
same for both examples. Knlarged screws are frequently used
for presses, where the diameter must be made greater than indi-
cated by formula (84) in order to resist bending stresses.
I 87.
Screw Connections, Feange Joints.
In screwed connections a distinction may be made as to
whether the force acts parallel to the direction of the axis, or
at right angles to it. The latter condition, which produces
shearing stresses, is shown in the examples given in Figs. 251,
252 and 253. If we take df as the diameter of the rod through
which the force acts, wTe may call d', the bolt diameter, and
Fig. 255.
then determine their relation for various cases. In Fig. 251,
d'—d; in Fig. 252, d'— 1.4 d ; in Fig. 253, d\=d ; theincreased
diameter for Fig. 252, being given because it is possible in that
case for the load to act so unequally that the greater portion
may pass through one of the rods. Fig 254 shows a tumbuckleTHE	CONSTRU
59
with right and left hand thread. In this it is desirable to make
the nut somewhat deeper than d, as shown. A form of junc-
tion piece for a point where four members meet is shown in
Fig. 255. Sueh examples as the preceding are of frequent oc-
«urrence in bridge.and roof construction.*
Bolt connections which bring shearing stresses upon the
bolts are of frequent occurrence in bridges built with pin-con-
nections, the general method in use in America. These designs
exhibit very fully the substitution of bolt or pin-connections
for riveting, and the method has been carried to great perfection.
Some examples are here given. Figs. 256 and 257 show an in-
tersection of several members of the bridge over the Ohio, at
Cincinnati. The top chord and the posts are double, and are
chord by means of a bolt passing entirely through the beams
and threaded at both ends. The nut on the left end is in the
form of a fork to receive the ends of the braces, while the right
hand end is fitted with a thin octagonal nut. The ends of the
braces are held by a bolt passing through the fork, with a nut
at each end. The pins are carefully turned and closely fitted ;f
after years of Service they show no signs of looseness.J The
proportions are such that stress on the bolts does not exceed
* Other good examples of similar work in roof construction may be found
in E. Brandfs “Iron Constructions,” Berlin, Ernst and Korn, 1871, zd
Edition.	,	.	.	,
t It is well known that variations in temperature dunng the bonng of the
holes for the pins in the eye bars may make sufficient difference to mater-
ially affect the fit. This has been overcome by the use of a double boring
machine which the author saw at work in the notable bridge works at
Phoenix ville, whereby both ends are bored simultaneously, the distance
being gauged by a wrought iron jig bar, which varied in length to the same
extent as the eye-bars themselves.
t See H. Fontaiue. “ Vlndustrie des Etats Unis,” Pans,Baudry, 1878. Rol-
ler, Highway Bridge’s New York, Wiley, 1878.
15,000. lbs, in most cases not more than 10,000 to 12,000 lbs. The
e connection of the posts to the chords (in the illustration the riv-
ets are omitted) is both simple and strong. The posts are provided
with cast iron ends, which are fitted with square projections en-
tering into the tops of the posts ; in these capitals are wrought
iron dowrel pins which pass through the lower angle iron and
lower plates of the top chord. The diameter d} of the main
bolts varies from 4 to 5^ or 6 inches or even heavier, according
to the load. Their dimensions are based upon as bearing stress
of 8000 lbs., while the diagonal braces and the lower chord are
proportioned upon a tensile stress of 10,000 lbs. (a ratio of 0.8,
see §5). The compressi ve stress in the top chord is about 8,500
lbs., and in the posts, owing to the bending action, only about
5000 lbs.
Fig. 258 shows an intersection on the lower chord of the
Niagara railway bridge (9 spans over a total width of stream of
about 1900 feet). In this case the posts and top chord are
made of the ingenious Phoenix columu of quadrant iron. The
illustration especially shows the method by which the cross
beams are connected to the longitudinal members. In this
case the stress in the body of the screw bolts is about 8000 lbs.,
rather more than given for press screws in § 86. A cast iron
base, through which the large pin bolt passes receives the
thrust of the post, and to it the cross beams of I shape are
bolted. On these cross beams are wooden stringers to w7hich
the roadway is secured.
It will be noticed that these examples of bolt work far ex-
ceed the limit of size set by the Society of German Kngineers
for bolt dimensions, viz., 80 mm. or 3X3^//. Should such
sizes be necessary the formulae in § 79 should be reconsidered.
Fig. 259.	Fig 260.	Fjg. 261.
In uniting the various parts of iron constructions, flange
joints are very frequently used. These are made in a great
variety of forms for various conditions. The following figures
show some examples of corner junctions with flanges. Fig.
259 shows three external flanges, with a dished base. Fig. 260,
also three external flanges, with an external plinth on the base.
Fig. 261 shows one external flange, and two which are half.
external and half internal. Fig. 262 has three half external
flanges and a base as in Fig. 260. Fig. 263 has also three half
external flanges, and Fig. 264 two external and one half-
external flange. The last three examples produce a more
pleasing external appearance than the preceding forms. # If the
form shown in Fig. 262 is used, with the flanges all turned
inward, the bolts cannot be unscrewed from without.
Proportions for flange joints are shown in Fig. 265, the bolt
diameter df being obtained from the thickness of metal d. The
distance between bolts is usually to 3 D} D being the
width of the nut across the flat dimension. The width of
flange is given in the illustration for metric sizes = 10 mm. +
2.8 d = y%" -f 2.8 d. _
If the flanges are finished on the planing machine, a ledge is
left for finishing, as shown on the left of Fig. 265, in order that
a fair bearing may be secured. Flange joints which are to
be bolted together wTithout finishing are made as shown in Fig.
266, with a gasket of some form of elastic packing. Such
flanges are sometimes made for vessels with very thin walls, and
on the left of Fig. 266 is shown the method of assembling a
cylindrical vessel, such as a water tank. The base has internal
flanges for the bottom pieces, with an external flange for the
connection to the body. By turning the flanges of the bottom
inward a flat exterior base is obtained, well adapted to sustain
the load of the water. The walls are very light,o = only about
the bolts are diameter, and their distance from centre
to centre, in the base, 13.5 d, and in the vertical joints of the
walls 15 d, and in the circumferential joints 20 d.6o
THE CONSTRUCTOR.
i 88.
UNI^OADED BOUT CONNECTIOXS.
Various methods have been adopted to relieve bolts, in a
connection, from the direct stresses due to the load, inuch in
the same manner as has been described in § 71, for keyed con-
nections. In Figs 267 and 268 are shown methods of notching
two plates together. The bolts are relieved from the action of
tensile or compressive stresses which act normal to the direction
of the tongue and groove.
Fig. 269 shows a method of constructing a prismatic intersec-
tion so as to relieve the bolts from transverse stresses ; while
* {
FiG. 265.
Fig. 270 shows a very convenient and usefnl form in which the
projections on each piece lip over the other, greatly increasing
the secnrity of the connection. The bar may be made of wrought
iron and the fitting should be made to conform carefully to the
position of the bolt holes.
If the parts are large they are often both made of cast iron,
and in some cases a turned dowel is let into both parts. The
coustructions shown in Figs. 269, 270, are used in the frame-
work of large water-wheels, in wThich case the lower piece is
made flat, thickened wherever it may be found necessary.
In many cases the lateral stresses are not great, while at the
same time it is not desired that the bolts shall be made to fit
closely. In such positions dowel pins are frequently used,
being made of steel and fitted to reamed holes:
An .example of bolt connection of large proportions, in which
the lateral stresses are relieved, is shown in Fig. 271.
This is taken from the bridge over the Mississippi at St.
Louis, and shows the bearing of the end of the lower member
of oue of the arches, which are composed of steel tubes. There
are four such bearings at the end of each arch, or 24 bearings in
all. The shoe to which the end of the tube is fitted is made of
wrought iron, and the sole piate, of cast iron. Three bolts pass
through both plates, the diameter d at the thread being
and in the body	The shoe is tongued into the sole piate
Fig. 269.	Fig. 270.
and the latter is supported by the masonry of the pier. The
hole across the shoe is for the receptiou of the bolt by which
the adjoining bearings are braced together.
CHAPTER V.
TOURNALS.
Various Kinds of Journaus.
i 89.
Journals are made for the purpose of permitting parts of
machinery to rotate about a geometrical axis and hence they are
necessarily round, and their use involves some form of bearing
or box for a support.
A journal may be subjected to pressure upon its side, or rather,
normal to its axis ; or the pressure may act lengthwise, in the
direction of the axis. This gives us the twTo divisions :
1,	Lateral journals.
2,	End, or thrust journals.
In the calculatious relating to these, both the questions of
strength and of friction must be considered. In machine con-
struction many forms of journals are employed, the most
important of wThich will be here considered.THE CONSTRUCTOR.
61
A. LA TERAL JOURNALS.
OVFRHUNG JOURXAIvS.
\ 90
A lateral journal which is connected on one side only to the
member to which it belongs is said to be overhung. Such jour-
nals are usually made cylindrical, as in Fig 272, with a collar
at the outer end, the height of the shoulder e above the diame-
ter d being—
e = yi" -f 0.07 d....................(88)
If the lateral pressure = P> the length of the journal = 7,
occur, but while fair results are obtained with the smaller values,
the increased value of a secures greater durability. Good
lubrication is of the highest importance, and especially a good
distribution of the lubricant over the bearing surfaces.
For bronze bearings under favorable conditions when the
pressure is constantly in one direction, a may be taken = 75,
while if the direction of pressure is periodically reversed, a may
be taken =3 iy>.
The following table will give the general proportions for
lateral journals :
PROPORTIONS OF JOURNALS.
I
ElG. 272.
and the permissible stress at the root of the journal = S, we
have for the diameter to resist the pressure

The ratio of l: d, determines the superficial pressure betw^een
journal and bearing. In ordinary circumstances the pressure per
p
unit of area on the lower half of the bearing is^ = — . When
the journal is revolving, this pressure is not the same at all
points, but has at the base lifie a value = p° = — py and at
7T
4
any angle /J, from the base line, a value p' = — p cos /?. Since
the relation between p° and p is constant, we may use the lat-
ter value for all purposes of calculation.
For any' given value of p, we have from the preceding :
7= -UT:.................................<s°>
In order that the wear may not be too great at high rotati ve
velocities, it is advisable to take somewhat less than the max-
imum value given above, and it may be made proportional to ?/,
the number of revolutions per minute, or :
d
7T ^
—z-----«
16 a
(91)
in which a is a constant, dependent upon material and lubrica-
tion. By combining (91) with (89) we get:
if «A
T /r S a
..........(92)
These four equations should be applied and the greater values
of d, and -j- used.
The maximum value of p —- .
a
For the value of the constant, the folloyjing considerations
obtain. If the pressure on the journal acts constantly in the
same direction it produces a higher superficial pressure on the
lubricant than wThen, for example, the pressure is reversed
frequently, as in a steam-engine crank pin. In the latter case
there exists a kind of pumping action between the journal and
bearing, which constantly drawsthe oilintothe bearing surfaces,
keeping them lubricated so that a higher value of p may be
taken than when the pressure is acting continuously in one
direction. Such bearings, however, are frequently subjected to
violent thrusts and shocks, so that a lower value of’ ►S' should be
taken than with journals in which the directions of pressure is
constant. For journals which only make a partial revolution,
much higher pressure may be permitted, than for revolving
journals. The former may be classed as journals at rest, as dis-
tinguished from running journals. The constant a in equation
(91) must be determined from practical considerations. It will
be found that in practice, wide variations in the value of a
Constant Pressure.
		Wrought Iron.	Cast Iron.	Steel.
C/5 13 s	Po =	8500	4260	14,220
3 0 1	H-s = /	8500	4260	14,220
% s	d ~	0-5	0.5	o-5
* j2 Eo	d =	O.O^y/P	0.0248.V/P	0.0135^
Intermittent Pressu re.
			Wrought Iron.	Cast Iron.	Steel.
<A 13 5 u	po	=	8500	4260	14,220
3 O •—>	s	=	8500	4260	14,220
.5 • > 0 S	l d	=	0-5	0.5	o-5
* 55	d	—	o.oi7v/p	o-0245\/p	°-oi35\/p
				„onstant Pressure	
			Wrought Iron.	Cast Iron.	Steel.
<A	P	=	700	360	700
S 0 3 >0 O t-i	S	=	8500	4260	14,220
“VII 1 'E ^ c ^	l d	=	1.5	T-5	1.94
3 &	d	=	0.03 </p	0.04 Is/p	0.027 s/p
Intermittent Pressur
Wrought Iron.	Cast Iron.	Steel.
r
Ifl 13 E 0	p =	1422	700	1422
3 10 O N-l	5 =	7000 *	3500	11,840
5 vii ■ ‘S > s ^	l d “	1.0	1.0	1-3
3 P4	d =	0.027y/p	o-o$7\/r	0.024 v/>
Constant Pressure.
			Wrought Iron.	Stee .
	r*			
	a	=	75	75
3 0 0 10	S	=	8500	14,220
i Air W ^	l ~d	=	0.13 >/n	0.17 v/»
i* s	d	=	0.0244 y/"-^-v/p	0,019 -Jvf
			Intermittent Pressure.	
			Wrought Iron.	Steel.
t/5 13 £3	a	==	150	150
3 0 D IO ^ M	S	=	7000	11,840
1 All ^	l d	=	0.08v/n	0.10^
10 g			°-0273 U fs/p	
■s» £	d	=		H N V «S O d
If n > 150, the ratio of l: d, is first approximatad and the
value substituted in the last formulas of the table.62
THE CONSTRUCTOR.
For hollow journals the following proportions may be adopted.
Let d0 = the external and dx the interual diameter of the
equivalent solid journal, f =	, we have :
d0 == _________i
d 3 (~	T*.....................(94
y' I — 'V'
the length of both solid and hollow journals being the same.
Fig. 273.
If, however, the ratio of diameter to length is to be the same
then
d0 _ ___________1_______
d	f " 7«.(95)
y 1 — if
from which the following series is obtained.
dl:	o*- II •€- II	0.4	05	0.6	0.7	o-75	0.8
1:	I 	 ^4 =	I.OI	1.02	1.05	I.IO	1.14	1.19
1 : \/	i — V>4 =	I.OI	I.93	1.06	115	1.21	1.30
In both cases there exists a smaller superficial pressure for
the hollow journal than for the solid one. A common ratio of
internal to external diameter is 0.6, and such journals were fre-
quently used in cast iron work and are again being used in con-
nection with hollow Steel shafting and axles.
Bronze boxes or their substitutes, such as white metal or
other combinations, belong more especially to the subject of
bearings 96), and their use permits a higher superficial pres-
sure without creating an excessive increase in the coefficient of
friction. For moderate speeds, boxes of cast iron give results
which are as satisfactory as can be obtained with bronze. This
is especially the case with machines which are actuated by hand.
For heavier or continuous Service cast iron boxes are only suit-
able when the pressure is not great, and examples of such
bearings will be given in a later chapter. Bearings of wood
may be operated satisfactorily at a pressure double that which is
used with bronze, if the journal runs in water, or is kept wet.
For heavy mill shafting making from 60 to 80 revolutions per
minute, wooden bearings lubricated with grease are often used.
For mill spindles, boxes with bearings of willow wood are
sometimes used with good results. In this case the speed some-
times exceeds 100 revolutions per minute, but the pressures are.
light.
a pressure of nearly 1000 lbs. per square inch, which appears to
be too great; and in actual practice these journals are obliged
to be kept cool with water.
In actual practice there is very little uniformity in the pro-
portions of journals. Sometimes the distinction between con-
stant and alternate action of load is considered but often it is
neglected. In the case of locomotive crank pins, for example,
p is frequently as high as 1500 to 3000 pounds per square inch,
and on the cross head pin, as high as 4500 pounds. On the
other hand quite low values of p are sometimes found on the
crank pins of marine engines. t In ali cases careful lubrication
is of the utmost importance. When the number of revolutions
is very great the length of the journal should be made greater
than is given above.
Table of Journals.
Value of P.
Direchon of Load Constant. Direction of Load Varying.
d	e	Wrt. Iron i /	Castlron / pr-1-5	Steel I / — = 1.94 i d		Wrt. Iron!c’t Iron l l ~d~ * j~d=I 1		Steel / T”*-3
1	0.20	1121	554	1419		1419	724	1833
1.25	0 20	1752	866	2217		2217	1113	2188
-.50	0.25	2523	1247	3193		3*93	1629	4*24
1-75	0 25	3434	1698	4346		4346	2218	5*63
2.00	0.28	4485	2218	5677		5677	2896	733*
2 25	0.28	5677	2807	6870		6870	3666	9278
2.50	0.32	7009	3465	8871		8871	4526	i*455
•75	0.32	8481	4i93	10734		io734	5476	13861
3-oo	0 32	10093	4989	12774		12774	65*7	16495
325	0.36	11845	5856	14992		14992	/649	19359
3-5o	0.40	13738	6792	17387		*7387	8870	22452
4.00	0.40	17943	8870	22709		22709		29325
4-25	0.40	20256	10014	25637		25637		33106
4-50	044	22709	11227	28742		28742		37”5
4.75	0.46	25303	12509	32025		32025		4*353
5-0 >	0 48	28036	13860	35484		35484		45821
5-5o	0.50	33924	16771	42935		42935		55443
6.00	0.52	40373	*9959	51096		51096		65982
6.50	0.60	4738-	23424	59967		59967		79260
7.00	0.62	5495	27167	69548		69548		89809
7-5°	0.64	63082	31187	79838		79838		103097
8.00	0.68	71773	34483	90868		90868		117301
8.50	0.72	81025	40058	102520		102520		j32422
9.00	0.74	90838	44909	1149*5		1149*5		148460
9.50	0.76	101212	50037	128097		128097		165413
1000	0.80	112141	55443	14*935		14*935		183284
10.50	0.85	123641	61126	156483		156483		202070
11.00	0.90	135696	67087	171741		i7*74i		221773
11.50	0.92	148313	73324	187709		187709		242394
12.00	0.95	161489	79838	204386		204386		263934
4. Example. An axle on a railway carriage makes from 200 to3oo revolu-
tions per minute ; n may taken =270, and from'(93) wehave -^- = 0.13%/:270
'== 2.T4. In practice the ratio is made from 1.8 to 2.0. The journals of fan
blowers are often operated at more than 1200 revolutions*; hence we get, in
such cases -^- = 0.13 \/1200 =4.5, or for Steel --- = 0.17	1200=5.9. The
blowers made by Sturtevant, of Boston, have Steel shafts, with the journals
5 to 6 diameters'in length.
2 92*
2 91*
Examples and Tables of Journals.
In the following tables are collected the results of the for-
mulae (93) in which the number of revolutions of the journal is
assumed to be not greater than 150.
1.	Example. a water wheel wreighing 66,000 pounds carries a load of 212
cubic feet of water. The axis of the wheel is oi cast iron, and the load is
equally distributed between the twTo journals, giving a load upon each
journal of 33,000 4- 6605 = 39,605 lbs. The nearest value to this in the table
is 40,058 lbs., Tyhicli would give a diameter of 8% inches, and a length of
12^ inches.
2.	Example. A wrought iron shaft for a similar load, but subjected to
alternating action, should have, according to the table, a diameter of about
5%", and the same length.
If in cast Steel, with alternate action, diameter should be about 4% inches,
and length of 4-75 X *-3 *= 6.175".
3.	Example.* The centres of the walking beam of the water engine at
Bleyberg in Belgium each bear a load of 309,210 lbs. The journals are hol-
low, with a ratio of external to internal diameter of 0.5. We have from
<93) and (94)	_____
do = 1.02 X 0.043	309,210 = 24.38"
and a length	/® = 24.38 X 1.5 = 36.57"
which gives a pressure of about 350 pounds per square inch of projected area.
The actual dimensions of these journals are dQ= 19%", /0=i8//,
which gives a stress at the base of the journal of a little over
4000lbs., but the actual bearingis only 15&" long, which gives
* Portfeuille de John Cockerill, I. p. 189.
Neck Journals.
When a journal is placed between two loaded parts of a shaft,
as shown in Fig. 274, it is called a Neck Journal.
In such cases the diameter d/ is dependent upon other condi-
tions than those of mere pressure. In order that the wear
f See Marks, “ Crank Pins and Journals,” Philadelphia, Kildare, 1878,
where the following values of p are given : Swatara, 400 ; Saco, 412 ; Wamp-
anoag, 725; Wabash, 470. The third of these engines had a cylinder 100"
diameter, and crank pin 16" dia., 27" long, and the stress in the preceding
cases was respectively, 4039, 3071, 10,537, and 2745 lbs.THE CONSTRUCTOR.
<53
may not be greater than in the caseof overhung journals, the
conditions of speed, lubrication, bearing metal, being the same,
the length should not be made less than the corresponding
end journal. If it is practicable to make the length greater, it
may be done to advantage, and the weai thereby greatly re-
duced * In many cases, however, the lack of space limits the
length, as for example, in the case of crank axles for inside
connected locomotives. Such journals are properly considered
merely as enlarged end journals.
For hollow journals of this type formula (94) may be used.
1. Example. Borsig’s Express Locomotive at the Vienna' Exposition.f
The journal of the rear drawing axle of Steel was loaded with 13,200 lbs.
d' = jyb", V = 7T5S " According to the table the ^corresponding journal is
given asd = 3%", / = 3.125 X 1-94 = 6.1", and p =-I3,2Q^— = 692.4 lbs.
3.125 X 6.1
In this case V is much greater than /, and for the given values of /', and a
13,200
Intermittent Pressure.
Wrought Iron. Cast Iron.
P
s
d
1422
7110
711
3550
Steel.
1422
11,845
3-5
d —	0.0185 v/ P	0.026 %/ P 0.0158%/ P
High Speed journals of this sort are seldom used, and need
not be considered here. It will be noticed that these Fork
wehave p =
= 253.3 lbs.
7-125 X 7-3125
while if V = /, the pressure p = —13,200^
* 303 lbs.
7.125 X 6.1
2. Example. In the same locomotive the forward axle carried the crank
pin journal upon which the entire force of the piston was exerted. The
total pressure on the piston was 32,120 lbs., and the dimensions of the pin
were d’ = 4Vs", I = 4/4"-
The corresponding values from the table of the preceding
secti on give d —	> / = 4,25 X i-3 = 5/^///> = about 1400 lbs.
The actual value of py for the sizes used is ———---------= 1730
4.125 X 4-5
lbs; In this case l' is less than /, on account of lack of room,
which accounts for the increase in superficial pressure.
2 93-
Fork Journals.
A Neck Journal which is held at both ends in a yoke or fork,
as shown in Fig. 275, may be called a Fork Journal. Such
journals may safely be made of lesser diameter than those which
are overhung. If we let P— the load, /= length, and d. —
diameter, and s, the maximum permissible stress, we have from
case VIII. I 6,

and if, as in the beginning of ? 90, we putp = —^>o
J_	Jf5 _ T _S
d» \p
(96)
(97)
Proceeding as in \ 90 we obtain the following collection of
proportions.
Formula: for Fork Journals.
Constant Ptessure.
Fig. 277.
Journals are comparatively small in diameter and of greater
length ratio than the preceding forms.
Example. A Fork Journal of wrought iron bears a load P= 4400 lbs., act-
ing constantly in one direction and revolves at a moderate speed. We have
then d=0.0212 %/44°° == 1.4", / — 1.4" X 3 = 4-2"- For an overhung journal
under similar conditions we have, from the table of § 91, d = 2", / = 3". The
product of the length and diameter is approximately the same in both
cases. If the length 4.2" is found inconveniently long, it may be diminished,
providing d be proportionally increased. The strength will then be un
necessanly increased and the resistance of friction somewhat greater.
These are only examples of the many variations which are to be met among
the many conditions of practice.
2 94-
Multiple Journals.
In some cases the resistance of friction becomes so great that
a modification of the fork journal is resorted to in order to re-
duce it within practical limits. Such an arrangement is shown
in Fig. 276, which may be called a multiple journal. If we as-
sume the load to be equally distributed among the plates, this
		Wroughtflron.	Cast Iron.	Steel.	
	' po	= 8500 '	4250	14,220	
bJD H «	s	= 8500	4250	14,220	
& § s 1- 4 K. S 1 > O S C/2	l d	= I	1	I	
t	d	= 0.01 2 I %/ P	0.0171 C p	0.0095 %/ P	
Intermittent Pressure.
				Wrought Iron.	Cast Iron.	Steel.
		P°	=	8500	4250	14,220
.5 % i		S	=	8500	4250	14,220
s 1 J		l				
£ 0 0 •“»		d	=	I	I	I
w *						
1	.	d	=	0.0121 %/ p	O.OI71 %/ P	0.0095%/ P
				Constant Pressure.		
				Wrought Iron.	Cast Iron	Steel.
	r	P	=	711	355	711
£		S	=	8500	4250	14,220
d
Fig. 276.
arrangement* may be considered as a series of fork journals. If the
number of members on each side be taken = A' each pair will
support a portion of the load Pt and d wTill be ^ times
as large as would be required for a simple fork journal.
If K =	2345678
We have^/"j-= 0.7 0.57 0.5 0.45 0.41 0.38 0.35
Journals of this kind are generally of the slow-moving class,
with a length ratio = i. The total length of journal is the =
2 K d. Journals of this sort will be found is some varieties of
chain links, of which examples wTill be given later. *
=	o.o2i2%/ P 0.029%/ P 0.0185%/ P
* See § 109.
t See Berliner Verhandlung, 1874, p. 389.
* Joints of this kind may sometimes be subjected satisfactorily to a greater
pressure than the calculation would indicate. Engineer Vollhering has
used such a joint in a System of levers to operate a heavy drawbridge. In
this case the load was about 95,000 lbs. K= 10, the thickness of each piate
d = iT\, both plates and journal being of Steel.64
THE CONSTRUCTOR.
\ 95-
Hatf Journai,.
In those cases in which the reduction of tke moment of fric-
tion is of great importance, the length of a journal may be
somewhat increased, if the bearing surface is limited to one-
half the circumference, as shown in Fig. 277, which shows such
a bearing, the load acting constantly in one direction and the
movement extending only through a small angle. In such
cases it is desirable to have a small supplementary journal as
shown in the figure, in order to meet unexpected lateral pres-
sures. In such half journals, provided the unused side of the
materi al is proportionally increased, d is inde pendent of P,
and p only is to be considered. We have for
Wrought Iron. Cast :ron.	Steel.
Po	—	8500	4250	14,220
P	=	6700	3340	II,l6o
Exainple : For a pressure 220,000 Ibs., acting in a constant direction
upon a slow moving journal of wrought iron, we have from (93) d — 0.017
\/220,000 = 7,97", say S", and / = 4"; for a fork journal, according to (98)
d = 0.0121 \/220,000 = 5.67", and l is the same; for a multiple bearing with
eight parts on a side d = 0.35 X 5.67 = 1.98", say 2", and a total length 7 = 2
X 16 -== 32". If now wetake ior a half journal the same conditions and make
d === 2", we get /==2X8= 16". We may, however, make d — 1.5", in which
case l t.2s " X 16 = 21 28". The journal friction will in this case be £ that
of the overhung journal.	that of the fork journal, § that of the multiple
bearing journal, which latter is nearly 60 per cent. longer.
An application of this form of journal will be seen in Fossey’s
Coupling. Woolf has also used it on the cast iron crosshead of
a large pumping engine.*
The principle of the half journal may be seen carried to its
extreme limit in the knife edge bearings of weighing machine
in which the friction is reduced to a minimum. Thesuperficial
pressure upon these very small surfaces is correspondingly
high, ranging from 15,000 to 150,000 lbs. per square inch. The
hardened Steel edges and bearings seem to be able to stand these
pressures without injury.f
§ 96*
Friction of Journaes.
New journals show greater frictional resistance than those
which have worn to a good bearing.
At first the journal only comes in contact with the m^tal of
the bearing in a limited number of spots until after a moderate
amount of wear the superficial pressure is distributed over the
projected area of the bearing, giving the value of p, as indicated
in l 904
For a diameter d, and load Py for a cylindrical journal, whose
coefficient of friction = fy we have for the initial force Fy which
the resistance of friction holds in equilibrium,
for new, unworn journals
F=—/P,
2
and for smoothly worn journals
R— -_-fP
8
The reduction in frictional resistance is equal to — ; or about
7r2
0.81 times less in a smoothly worn bearing than in a new one.
The actual value of F is, however, greatly dependent on f,
This, however, is not only dependent on the lubrication and
conditi on of surfaces, as according to the theories of Morin and
Coulomb, but also upon the superficial pressure py and speed of
rubbing surfaces v. f
Additional researches upon this subject are yet greatly to be
desired. II
♦ See Tredgold, “Cornish Pumping Engines.”
f In large track scales, pressures as high as 425,000 lbs. per square inch
are found upon bearings less than 3y' wide. The knife edges on the large
Werder Testmg Machine at the Royal Technical Academy are 360 mm. long,
and sustain a maximum pressure of 100,000 kilograms, or 277.8 kg. per mm ,
or at i mm., in width is equal to 556.6 kilograms per square millimetre, or
810,000 lbs. per square inch, and this pressure has been sustained without
apparent injury.
t See Reye, Theorie der Zapfenreibung, Civ. Ing* VT., 1860, p. 235, also
Grove, Trag-und Stuzzapfen, Mitth. d. Gen. Vereins fur Hannover, 1876.
\ See Hirn, Studes sur les frottements medints, Bulletin von Miilhausen,
1854. p. 188, also the researches of Reuuie, Sella, Bochet, and others.
| Engineer, Nov., 1873, p. 312, contains a brief, but valuable discussion
upon the action of railway axles in their actual conditions of operation.
The following abstract gives the results:
The brasses were all ,poured from the same crucible and consisted of a
Rennie’s experiments with cast iron journals in bronze bear
ings, with copious lubrications :
When p = 3-2	175	315	492	668	739
f — 0.1570.225 0.215 0.222 0.234 0.234
no account being taken of vy in these experiments.
Hirn experimented with cast iron on bronze with full lubri-
cation, the value of v being equal to 335 feet per minute:
When p = 3	5.26	7.54	9.71	12
f— 0.0376	0.0211	0.0226 0.01990.0183
and these experiments showTed that for small values of py f
diminishes as p increases.
Hirn also found that if p remained constant, and equal to 12
lbs., that when
^ = 92	164	184	275	327	335	367
/=0.0086 0.0121 0.012S 0.0165 0.0181 0.0183 0.0191
thus being at all times quite small, but stili constantly increas-
ing with the increase of velocity.
Morin’s researches gave with pressures of 14 to 20 pounds
per square inch, values of fy from 0.05 to 0.11 for journals lubri-
cated with oil, and from 0.08 to 0.16 when lubricated with
grease.
The following results w7ere obtained at the Royal Technical
Academy from experiments after Morin, upon Clair’s apparatus.
The journal was of wrrought iron in brass bearings, freely lubri-
cated with oil.
First Test.	Second Te st,
Bearing Surface . ..... . 12.800 sq. mm.	128 sq. mm
Total pressure P.............16.5 kilo	16.5	kilo
Pressure pr. sq. mm..........0.00129 kilo	0.129	“
Observed friction............1.25 kilo	2.65	“
Coefficient/*................0.076	0.160
The author’s experiments with an apparatus resembling a
Prowny brake with surfaces of wrought iron on bronze with
good lubrication and velocities of 30 to 35 feet per minute, gave
the following results :
P— 50	122	192	335	484	624	711
/“=0.090 0.087 0.095 o. 118 0.171 0.184 0.180
Here the value of f was doubled, while p increased 15 times.
If p remained constant and equal to 470 lbs. we have
for	^ = 079	14.17	34.64	55.1
/*=0.222 0.210 0 191 0.167
In this case the coefficient of friction diminishes for an increase
in the value of v, contrary to the results in Hirn’s observations,
the value of p being above 40 times greater than Hirn used.
These latter results appear to be more in accordance with
Morin’s, in that the friction of rest is greater than the friction
of motion, and hence for small velocities the friction should be
greater than with higher velocities. This law appears to hold
good only between certain limits for v, either side of which J
increases for increasing velocity. Him’s researches lay beyond
these limits. Those of the author are only preliminary to a
fuller series of observations.
The following table give some results of the wear on boxes
of various kinds in railway Service :
Kind of Alloy.fl
j Distance, Km. for
i a wear of 1 kilo-
gramme from 4 boxes.
Wear on 4 boxes in
grammes for 1000
Kilometres.
1.	Gun Metal 83 Cu. 17 Sn. . .
2.	“	“ 82 Cu. 18 Sn. . .
3.	White Metal 3 Cu. 90 Sn. 7 Sb.
4.	“ 5 Cu. 85 Sn. xoSb
5.	Lead Composit’n 84 Pb. 16Sb
6.	Phosphorbronze.........
7.	Parsons’ White Brass .	. .
8.	Dewrance’s Babbit Metal .
Kilometres.
90,390
99,900
72.280
88,145
81.280
429,200
385,275
637,679
Grammes.
11.06
10.01
13.83
U-34
12.30
2-33
2.60
i-57
mixture of 7 parts copper and 1 part tin. They all worked under the same
car and all had the same lubrication. In running 28000 miles the losses were
as follows:
Boxes.
loss •= 5 lbs.
“ =3
— 2^ “
Taking the journal load as x 1,000 lbs., the value of p in [the three cases is
612, 554 and 427 lbs.
Journals.	
1. rf=3h	loss == 3y'
2. “ * 3h “ - 6*.	
3- “3ft =7",	—* rhrs
% Nos. 1 to 6 are from the work of Dr. Kunzel on Bronze bearings, Dresden,
1875. The others are from The Engineer, Vol. 41,1876, pp. 4 and 31, all be-
ing given in metric quantities as readily comparable.THE CONSTRUCTOR.
65
B. THRUST
\ 97-
Proportions of Pivots.
A thrust bearing which is formed on the end of a shaft and
bears the pressure upon its sectional area, is termed a pivot.
For ordinary cases these are made in the form shown in Fig.
178. The pressure p is uniformly distributed over the area of
the end of the shaft, and the velocity is proportional to the dis-
tance p of any given element from the centre. A small oil
chamber of a radius rx is formed in the middle of the bearing.
If the outer radius is r0, we ha ve
0.5	p{n
fi'=—p
and for the elements on the outside radius
0-5P Ti+r« )
A= n
In the formulae for a uniformly distributed pressure p, we
have taken rx = J rQ and the two diametral oil channels are
made of a width == tV d. We then have for a given load P:
P=8i6pd*	..........(101)
In order that there may not be too much wear for fast run.
a
ning bearings (see \ 90) we may take p =	and have for high
speed pivots :
P= 816 2 —
n
(102)
Alternating pressures do not occur in these bearings and
need not be considered. The value of a may be taken for
wrought iron on bronze as = 75.
Bearings of lignum vitae running in wTater may bear loads of
1500 pounds per square inch even at high speeds.*
The following formulae and tables will serve for the propor-
tions for end pivots :
Formui^E for Pivots.............(103)
Wro’t Iron or Steel Cast Iron Iron or Steel
on	on	on
Bronze.	Bronze. Tignum Vitae.
f p= 1422	700	---
Slow moving Pivots j d_ a035 ^/p0.05 ------------
\p= 700	350	1422
n °r < 150	j </= 0.05	o. 07 v//> 0.035
(a = 75	--- /=1422
n>J5°	I d = 0.004 v/Pn ----- </=0.035 y/p
Fi.at Pivots.
d=	0.03S \Cp	0.05 Vp	°-°7 \/P
1	8l6	398	204
1-25	1275	622	3i9
1.50	1836	895	459
1.75	2500	1219	625
2.00	3265	1592	816
2.25	4132	2016	1033
2.50	5102	2483	1275
2-75	6173	3011	1543
3.00	7347	3494	1836
3-25	8622	4205	2155
3-50	IOOOO	4877	2500
3-75	11479	5599	2869
4.00	13061	6370	3265
425	14745	7192	3686
4-50	16530	8063	4l32
4*75	18418	8983	4604
5.00	20498	9954	5102
5-25	22140	10974	5535
5-50	24694	12044	6673
5-75	26990	13164	6747
6.00	29388	14334	7344
6.25	31890	15630	7972
6.50	34490	16900	8623
6.75	37190	18220	9298
7.00	41690	19600	IOOOO
* Penn has used lignum vitae bearings with pressures of p = 7000 to 8000
pounds. (See Burgh.)
Example 1. A erane in the harbor of Cherbourg carries a load of 33,000 lbs.
on an end pivot 6f'' diameter. Adding its own weight of 6600 lbs. gives a
value P= 39,600 lbs. This is a slow moving pivot and we have from the
table for this load a diameter between 6$" and 7.00". A similar erane of
4000 pounds weight and 20,000 pounds load has d = 6fg", while the table
would give about 5^":
Example 2. A driving shaft making 100 revolutions per minute, with a
load of 2200 lbs., should have, by the table, a diameter 01about 2§".
Example 3. A turbine, making 200 revolutions per minute and 3080 lbs
load, should have a step, according to (103), of 0.004 s/Pn = 0.004 \/3080 X 200
= 3i".
The length of journals in the case of such pivots is usually
made from 1 to 1.5 d, its value being sufficiently great to pro-
vide for the lateral pressure.
Fig. 278.	Fig. 280.
There is a general tendency in machine practice to use
smaller diameters for pivot bearings,! in order to reduce the
resistance of friction.
In order to reduce the effect of higher speeds upon pivots
bearing heavy pressure a series of disks is often used. If, in
Fig. 279, the number of plates between the end of the spindle
and the step is 1, 2, 3, 4, ... i, we have for the proportion of
tums between each pair of surfaces	lA,	—*— times n
l+i
This device has been used for steps of turbines, mill spindles,
etc., by Escher, Wyss & Co., Reiter and others. But few ex-
amples now remain of this firm for the thrust bearings of
screw propeller shafts; the disks bound together and were
Fig. 279.
overheated and injured. So far as experience indicates, such
thrust bearings are capable of standing pressures of 1400 pounds
per square inch or even more. The important point to be con-
sidered is, therefore, the reduction of the superficial pressure p.
The use of other materials than iron, wood or bronze, and
their substitutes, such as white metal, Babbitt metal, etc., has
often been attempted. The subject of wooden bearings will be
considered hereafter. Besides the use of hardened Steel, which
is of small value for great pressures, such bearings have also
been made of stone, glass,! or hard burned clay,§ but none of
f At the establishment of Gruson, in Madgeburg, a boring mill is made
with cast iron spindle in cast iron bearings, with a superficial pressure of
more than 20,000 pounds, without ill results.
X Bearings of glass have been used for more than twelveyears at the works
of F-Acker & Co., at Graggenan, near Rastatt. These bearings are very
durable and cheap and require but little lubrication.
g Shown at the Exposition of 1867, by Eeoni, of Eondon, with good results.66
THE CONSTRUCTOR.
these materials have come into general use. Girard used a
pump to keep a film of water between the friction surfaces, and
after deducting the power to operate the pump showed a very
light resistance.* A similar device was shown by Girard at the
Exposition of 1867, i11 which the water jet was operated by a
blast of air. This apparatus was rather of the nature of a
scientific apparatus, than as a practical application. There
were also exhibited journals which ran in bearings in which
water was inclosed.f The experience of general practice, how-
ever, shows that the ordinary forms are sufficient, without re-
quiring the use of any of these complicated devices.
i 98.
Friction op Feat Pivot Bearings.
If a flat pivot bearing with annular bearing surface, as in Fig
278, has an inner radius rlf and an outer radius r0 with a load
R\ we have for the tangential frictional resistance
v3
F = —fP
-00’
(103)
in which f is the coefficient of friction. For rapidly running
pivots we have
F

1 -f
?-v
'0 /
(104)
ll„ ii			11
w	1		D
1		M	
Fig. 281.
Fig. 282.
Fig. 283.
what higher velocity. For this reason such bearings are not to
be recommended when high values of P must be carried. The
resistance of friction may be calculated by the formulae of the
preceding section.
I 100.
Muetipee Coeear Thrust Bearings
Thrust bearings are frequently made with a row of collars on
the shaft as shown in Figs. 281-283. If the collars are similar
the pressure may.be taken as distributed uniformly among
them. If f be a constant value we have for m collars but —
m
part of the value is given by (104) for each collar, although the
total frictional resistance will be the same, being the sum of
the resistances of ali the collars. Nevertheless the results of
experience, especially with screw propeller shafts, shows the
necessity of making m large in order to keep the pressure p as
small as possible. This is due to the fact that heavily loaded
shafts give, according to (104), so great frictional resistance as
to case excessi ve heating and consequent injury, and experi-
mental researches have shown the value in reducing p, and
consequently /. The best values of p lie between 40 to 80
pounds per square inch. When bearing of this kind is placed
at the end of a shaft it may be reduced in diameter as shown in
Fig. 283, and in such cases p may be made somewhat greater,
even as high as 350 pounds, but in such a case there is a great
tendency to heat.
Example 1. Screw Propeller Engine, by Indret. Thrust on shaft 39,600
lbs., *= 55 lbs. zri~D= 15". Breadth of collars = b =- r0 — rx = 2". Num-
ber of collars M= 9.
The second value is rather less than the first, since, from the
previous proportions rx = J r0, which gives for running pivots
F= \ f P\ and the ratio of the two values is as 7 to 6, while if
rx = Oy it is as 4 to 3. For values of /see § 96.
Example. Inthecraneof example 1, §97,	39,600 lbs. r0 = 3^".
/•=-0.15. This gives in (104)
F=*0.075^ X 39,600 = 39^0 lbs.
The force required to overcome this resistance, if acting at a lever arm 40
inches from the axis would be—
2***^-3» lbs.
40
? 99-
Coeear Thrust Bearings.
The use of collars to receive thrusts on hocizontal bearings is
similar to such use on vertical shafts, and a form is shown in
Fig. 280. In this case the inner diameter 2rx cannot be less
than the diameter D of the shaft. It is best to make it suffi-
ciently greater to permit a small oil channel to be used as
shown in the figure, and oil ways should also be cut in the
bearing surface.
I
Here / =
39600
= 40 lbs.
9*r X 17 X 2
The velocity v, at a radius r0= 275 ft. This gives in (104) taking/=oi.
F= — 39600 (1 +—) = 3465 lbs.
2	\	9.5/
and the friction horse power
Hp_3465X2Z5	h. P.
33000
Example 2. Turbines on the RhineatSchaffhausen. P=- 30800 lbs. « =
48. 2 r\ = D — 9". Collar width b, =«= r<> — rx ==	*n == 9.
This gives / =
30800
9 *r X 10,625 X 1.625
= 63 lbs.
v —133 feet.
f=-TL 30800/1 -i—?—)=2664.
2	V 12.25 t
P.
33OOO
Example 3. Girard Turbine at Geneva.J
P= 33,000 lbs. n = 16, 2 rx = D = 9.8''
b = rQ — n 1 = w = 12
This gives p =----7-^—°^------=«57 lbs.
12 w X xi.i75 X 1-375
v = 46.7 ft. From (104) we get 2970 and the friction horse-power is
h.p.-3£X±1—*h.p.
33,000
Example 4. Eangdon lays down the rule that for collar thrust bearings
of screw propeller engines' there should be % square inch of surface for
every indicated horse-power.g If A’ —- the horse power and c the velocity of
tb.e ship
Pc
N —-----
33,000
33,000 x P 44,000
This gives p =
0.75 Pc
add il c == 1000 « ser minute, p = 44 lbs.
Example 5. A large centrifugal machine by Eangen & Sons, In Cologne,
ha9 a collar step of the following proportions :
/>== 4400 lbs. n = 800, 2rj= 1", 2 r0 = 1.57" m = 11
= 366 lbs.
If r0 — rY is made fhe same as before, good proportions will
be obtained, although the rubbing surfaces will move at a some-
* Ii v (0.78^ —0.5*)
which is an excessive pressure, liable to cause heating, and demanding
most careful lubrication. In this case v = 275 and takmg/— 0.1 we get as
before
F= 260 and H. P. =———	3 p-
33,cxo
♦See Armengand, “ Vignole des Mecaniciens,” p. 139.
t Exhibited by Jouffray. See Armengand “ Progrfes de Tindustrie k 1’exp.
tiniverselle,” Vol. I, Pl. 8.
X Oppermann, Portefeuille econ des machines, Vol. 17. Also Engineering,
1872, Vol. 14, P- 238-
l See Burgh.THE CONSTRUCTOR.
67
In ali these examples the co-efficient of friction/ has been
teken = 0.1, and for the moderate pressure of the first three ex-
amples a lower value might have been taken. The examples
will suffice to show the importance of the selection of a suitable
value for/, and other cases will be examined in § 122.
? 101.
The Compound Eink as a Thrust Bearing-.
In the previously examined cases it has been the object of
the various pians to reduce the journal friction to a minimum,
but there are sometimes occasions in which it is desired to give
a journal a definite amount of frictional resistance, without
danger of its sticking fast, so that it may be rotated with a
moderate force, and may also be readily clamped in any desired
position. This may be accomplished, for example, by a thrust
journal made in the form of a truncated cone. If the radii of
the large and small ends are respectively rQ and and the half
angle a, we have for the force Fy instead of (104),
F=S-	-P— (1+^.(105)
2 sui a \ r0 J
and by varying the angle a, may give any desired value to F.*
Very acute pivots sometimes bind in an injurious manner,
and hence the increase of /^cannot be carried to an extreme in
this way. Clamping of this sort may better be accomplished
by the use 'of compound bearing surfaces, so arranged as to
press on each other, as shown in Fig. 284. Each piate then
transmits the axial pressure to the next. If m is the number of
contact surfaces, the friction at the radius r0 of the bearing is
found by an analogous equation to (104),
F= + ...................(106)
Example. Eet F — P, and let f= 0.1
20
m = —--------whence, if r1= % r0, m = 13
1 + ^
ro
This arrangement has been used by the writer with success in
many parts of machines where a clamp was desired. Fcrmerly
the joints of dividers were made with four plates at the pivot.
I 102.
Attachment oe Journaes.
When a journal cannot be made in one pi&ce with the rest of
the shaft, various methods of attachment may be used; such
devices are mainly necessary in fitting iron iournals to wooden
shafts, as for water-wheels.
* Applications of this principle may be seen irTthe spindles of astro-
nomical and surveying instruments. Formula (105) may be used to de-
termine the friction of stop-cocks.	j,
In Fig. 285 is shown a form of attachment in which a cross
anchor piece is forged on the shank of the journal, and a slot
mortised in the end of the shaft to receive it. After the journal
Fig. 285.	Fig. 286.
is in place it is clamped by driving on the previously heated
metal bands (see § 62). The angle of taper is Fig. 286 is a
very good form in which the shank of the journal is keyed in
Fig. 287.	Fig. 2S8.
place. In Fig. 287 is shown a cast iron journal with two wings,
arranged to be driven in, and Fig. 288 shows the proportions of
the same when four wings are used. If three wings are desired
their thickness may be made equal to T% d.
Fig. 289.
Fig. 289 shows a form in which the four wings are surrounded
by a conical shell, which is held in place by bolts and anchor
plates. The shell is sometimes made with keyways cast in it to
act as a centre for the hub of a gear wheel.
d
Fig. 290.
Fig. 290 shows a very practical form. The journal is cast on a
piate strengthened by heavy cross arms, and a wrought iron
nng is shrunk on, while the whole is fastened to the shaft by
the four bolts, whose nuts are let into the wood, as shown.68
THE CONSTRUCTOR.
CHAPTER VI.
BEARINGS.
I 103.
Design and Proportion.
construction of bearings, and the foliowing example will show
its use : *
The poles O, Ov 02, Fig. 294, are used for the journal diam-
eter d; the poles P> Px and P2, for those dimensions which de-
pend on the modulus d1= 1.15 d -f 0.4". This gives dx=09
The mechanieal devices by which the journals of shafts and
axles are carried are called bearings. A complete bearing may
be divided into three portions: 1, the boxes ; 2, the body or
frame; 3, the connecting parts.
The various forms may be divided according to their uses
into the two main classes :
A.	Bearings for Lateral journals or Lateral Bearings.
B.	Bearings for end-long pressure or Thrust Bearings.
Under these classes the principal distinction is to be made as
to the side on which the bearing is to be supported. If we
suppose the journal to be inclosed in a cube 1.2 . . . 8, Figs.
291, 292, we have for lateral bearings
A Pillow Block, when the base lies in 1, 2, 3,
A Wall Bearing,	“	“	“	“	1-8 or 2-8,
A Front Bearing,	“	“	“	“	1-6 or 4-7,
AHanger,	“	"	“	“	5-7.
For Thrust Bearings we may have Foot	Step Bearings, Wall
Step Bearings, or Hanging Step Bearings.
Especial care is to be taken for the equalization of wTear and
for efficient lubrication, and these points affect mainly the
boxes.
The examples which follow have only been selected from the
vast number of forms to show typical cases.
The dimensions are based upon a proportional scale. As the
unit for the thickness of the brasses we have e =0.07 d -f-
d being the bore of the boxes, and volues of e are given in the
second column of the table in g 91. The modulus for the body
of the bearings is :
^ = 1.15*/+0.4".............*...........(107)
ZA.—LA TERAL BEARJNGS
§ I04.
Pillow Block
In Fig. 293 is shown a form of pillow biock sui table for jour-
nals from	to 8//. The proportions of the body and cap
are based on the modulus d1 (see 107), with the exception of
the oil cup on the cap, wThich would then be rather too large for
small bearings, in which it is made in length equal to the
width of the cap, and in width equal to 0.7 dv
The length of the boxes is dependent upon the length of the
journal, which, as discussed in § 90, may be 1.5 dy 2 d, etc.
For the form shown a good proportion is 1=2 d> the projecting
portion of the boxes being governed by the proportion of
length to diameter adopted.
The bolts for the base piate are made somewhat heavier than
those for the cap, as they are screwed up much tighter, and
they are often made with special heads to fit a separate sole
piate as shown in Fig. 294. The ends of the base are given a
bevel in order to permit the use of side keys. The coring out
of the sole piate reduces its weigh and also simplifies the rna-
chine work. The spaces between the cap and the body of the
bearing are filled with slips of wood so that the cap bolts may
be tightened without binding the shaft. In cases where the
load is great, the pressure alternating, the joint is closely fitted
without spaces, and if wrear in the journal is to be taken up the
surfaces are filed down.
§ '°5-
Proportional Scale for Pillow Blocks.
The proportional scale may be used to great advantage in the
Fig. 293;
when d = -i—-— = 0.34" hence P must be placed when
115
the vertical space betwreen the rays Oa and Ob is equal to
— o-34//. The intersection of the rays from O and from P> by the
ordinates I, II, etc., give the dimensions of the corresponding
sizes. The dimensions of the boxes must be obtained from
another pole, as they depend upon another modulus. This
modulus is ^ = .07^-f- and becomes = O, when d = ——^
=— 1.78r/. The poles E and Elt therefore, are placed on the
vertical line on which the distance a' b' equals 1 •78//. For
the oil cup in the cap the width is :
0.25 dj + o.4// = o.4// + 0.25 (0.4" -f- 1.15 d)=z
= 0.29 d -f 0.5
= 4.16 (.07 d -f y%") = 4.16 e
Hence E is also the pole for the oil cup.
♦The firm of Escher, Wyss & Co, in Zurich, have used the proportional
scale very well for designing bearings, both in determining the geometncal
proportions throughout and also by the excellent method of a single pole.THE CONSTRUCTOR.
69
i 106.
Various Forms of Journal Boxes.
It is often found convenient to give the boxes of a pillow-
lbiock other forms than those of the preceding illustrations, as
for example octagonal, as in Fig. 295, or cylindrical, as in Figs.
296 and 297. The last two forms are suitable for bearings in
lathe headstocks, and in such cases the boxes are kept from
Fig. 295.	Fig. 296.	Fig. 297.
the form shown in Fig. 294, \ 105, so that the base may be re>
moved from the base piate when necessary without disturbing
slipping out of place by the flanges whose width is 2^, as shown
in Fig. 296, or by projecting pins, Fig. 297, fitting into recesses
in the base and cap. Each of these forms has its advantages
and objections, and it is hardly possible to decide which form is
the most desirable, special conditions being generally present.
The modifications in the base and cap to admit the forms shown
in Figs. 296 and 297 are readily made without requiring detailed
instructions.
Boxes in which white metal or similar compositions are used
require special construction, since these materials are not strong
enough to resist the stresses with the same security as solid
bronze boxes ; for such bearings a cast-iron or bronze shell is
made, in 'which a lining of the softer metal can be poured, as in
Fig. 298. In such cases the shell should be cleaned with acid
and tinned before pouring the lining metal.
Boxes of lignum vitae (see U 97 ~117) must be made of
simple shape. A convenient shape is shown in Fig. 299,
which the general form of the bearing may be made.
In America examples are often found of bearings in which
the soft metal is run directly into recesses in the base and
cap. Fig. 300 shows such a bearing as made for the journals of
fan-blowers and shafting, by Sturtevant, of Boston. The base is
hollow^ed out to serve as an oil chamber, and the oil is fed to
the jouraal by a wick. The details are shown in Fig. 301. These
journals are made very long (/ = 4d), and hence the superficial
pressure is small.
I io7-
Narrow Base Bearings. Large Pieeow Blocks.
It is often desirable, when space is limited, to make bearings
with narrow bases, and this may be done by making the cap-
bolts with collars as shown in Fig. 321, and also Fig. 312. This
permits the holding down bolts to be dispensed with, and space
saved. Such collared bolts are also used for pillow blocks,
which are subjected to both upward and downward stresses,
since the boxes are firmly bound together (see § 88). Fig. 302
shows a form of pillow block for journals of 8 to 12 inches in
diameter. It is made with four cap bolts and four base bolts,
by which it is secured to the base piate. The base bolts are of
the solidity of the latter. The body of the pillow7 block is cored
out to a greater extent than in the previous form, and w7hen
Fig. 302.
the journal is used for a crank shaft, or is subjected to jarring
strains, the cap bolts should be provided with jam nuts, or some
of the other forms of security, such as is shown in § 85.
3 108.
Pillow Block with Adjustable Bearing.
In many cases it is only necessary to adjust the height of
pillow7 blocks from time to time by inserting liners beneath the
Fig. 303.
base, but in some situations it is desirable to provide a special
means of obtaining such an adjustment. In Fig. 303 is shown
such an adjustable bearing for use in screw propeller shafts.
The body of the bearing is not bolted down, but rests solely
by its w7eight upon the wTedge System, by means of wThich it can
be raised or low7ered as may be found necessary. The upper
box is provided with flanges through which the cap bolts (omit-
ted in the illustration) pass. The lower box is lined with wrhite
metal, which is poured into the recessed bearing.1 o
THE CONSTRUCTOR.
i 109.
Adjustabee Pieeow Beocks.
Many attempts have been made to arrange the boxes in a
pillow block so that they may be self-adjusting and so adapt
themselves to various positions, which thejournal may assume
and secure for it at ali times a full bearing and support.* * * § For
this purpose, among other methods, the plan has been adopted
of making the boxes with Central spherical portion fitting into
corresponding recesses in the body of the pillow block. This
forni of bearing has been widely introduced in America by
Messrs. Wm. Sellers & Co., and adapted to a great variety of
positions.
Sellers has always urged the desirabiiity of the principle of
keeping the pressure between journal and bearing at a mini-
mum, f This practice permits the use of cast iron boxes, for
which a pressure of not more than 15 pounds per square inch is
used.J
The use of moderate superficial pressures is raost practicable
in the case of bearings for line shafting in which the propor-
tions may be made such as to give but light pressure. This ad-
vantage will be seen 011 reference to § 92.g
Fig. 304 shows Sellers’ forni of pillow block. The cast iron
boxes are made with a spherical enlargement in the middle,
which is held between corresponding recesses in the cap and
base. The boxes are prevented from revolving by the hollows
Fig. 304.
in the sides which receive the bodies of the cap bolts. Three
openings are made for oil or grease and two drip cups, which
are cast on the base piate, serve to receive the superfluous oil. j|
The modulus upon which the proportions of this bearing are
based, is not that given in (107), but the following :*[
Di = i.\d + o. '	............(108)
The length of the boxes =4 d. The shape adopted by Sel-
lers shows the care in modelling which is characteristic of the
American designs of engineers. The Sellers’ bearings have
been used to a considerable extent in Germany.
* Various designs have been made by Bodmer at Manchester, Schonherr
at Chemnitz, Stehelin at Thann, and Zimmertnan at Karlsruhe.
fTreatise on Machine tools, etc., as made by W. Sellers & Co., Philadel-
phia, Lippincott, 1873, p. 161.
t As an example of the performance of these cast iron bearings Sellers
cites a bearing which had been in Service forsixteen years and in which the
lovrer box was not yet worn to a polish over its entire surface. The shaft
made 50 revolutions per minute and was 4% in. diameter, and carried near
the bearing a 72 in. pulley, of 20 in. face, transmitting 52 horse-power. In
other examples it isshown that after a year’s use the tool marks were stili
visible. The small superficial pressure does not force the oil out, and
hence the journal is carried on a film of lubricant The consumption of
oil is very small, and Messrs. Sellers state that a shaft making 120 revolu-
tions per minute consumed but 2l/2 ounces of oil in six months.
§ As an example of the impracticable results which would follow from au
attempt to obtain such light pressures to overhung joumals, we may take
Sellers' value of 15 pounds per square inch and apply it to an example. If
P= 17,600 lbs., we have for a wrought iron shaft with a constant direction to
the pressure, from the table in § 91, d = 4", 1=6". If p = 15 lbs. we have
from formula (90) T- =11.9 and from (89) we have = it.2w;hence 2 =.
d
X1.2 x 11.9 = 133" ! !
g Sellers recommends a mixture of tallow and oil, which becomes more
liquid should the bearing grow warm.
See Berliner Verhandlungen, i876t'p. 89.
Another form of adjustable pillow block is showrn in Fig.
305. This is used by Sturtevant in some of his fan blowers. In
this case the ratio of / to d is very great (see example 4, $ 91).
The adjustability is obtained by pivoting the bearing A upon a
i	'/////////////zv////.
‘	i
Fig. 305*
cross bolt By which passes through the cheeks of the pedestal
also ; the latter being adjustable about the axis BC. The bear-
ing is lined with white metal, and the end thrust is taken up by
a block of lignum vitae. If au adjustment in the direction A A
is required, the bolt C may be loosened and the required move-
ment made. The provision for lubrication is especially note-
worthy both in the manner of supply and in the collection of
the overflow.
2 110.
Bearings with Three-Part Boxes.
In horizontal steam engines and in similar Service, the
pressure upon thejournal is thrown first on one side and then
on the other, while at the same time there is a constant vertical
pressure, such for instance as is due to the weight of a fly
wheel. Attempts to remedy the tendency to overwear by mak-
ing the boxes inclined, have proved but a partial remedy, and the
best method of construction in such cases is to make the box
in three parts, one of which receives the constant vertical
pressure, while the other two provide for the backward and for-
ward thrust. Such a bearing is shown in Fig. 306. The modu-
lus dx =.1.15 d 4- 0.4". The bottom box rests on two wedges
which are tapped with screw threads and can be adjusted and
locked at any desired point by the bolts shown. The side boxes
are each held up by two Steel set screws ; a wrought iron piate
being interposed between the screws and the boxes. If it be-THE CONSTRUCTOR.
7r
comes necessary to remove the side boxes the cap is first taken
ofF, and the iron plates taken out, when the boxes can be sepa-
rated far enough from the shaft to permit their removal without
those cases in which an alternating up and down pressure is
combined with a constant lateral pressure. The latter would
not be provided for in an ordinary pillow block, but here it
Fig. 306.
interference with the shaft. The body of the bearing is in
creased in width in order to provide for the kincreased lateral
pressure.
Fig. 307.
Another three-part bearing * is shown in Fig. 307. In this
case there is no vertical adjustment to the lower box—and if
necessary it must be raised by packing underneath. The side
boxes are set up by wedges which are adjusted by set screws
through the cap. Each wedge carries a screw on its upper end,
and the nuts for these screws are fitted so as to revolve in the
cap, being turned by a wrench on the hexagonal head, and
then clamped in position by the thin jam nut shown. The
heavy inclined ribs stiffen the body of the bearing to resist the
stock and thrust of the piston. It is often convenient (as in
the case of the original of the figure) to cast the body of the
bearing in one piece with the bed piate of the engine.
A third, and simple form of three-part bearing (by Schultz
Brothers in Mayence) is shown in Fig. 308. It is suitable for
* From a steam engine by the Soc. Fives- Lille in Paris.
Fig. 308.
is taken up by the small side box. This form is suited for
small vertical engines in which the pull of the belt is toward
one side.
3 ni. •
PFDESTAIv Bearings.
Bearings which are not placed directly upon a base piate, but
are raised upon feet or pedestal are called pedestal bearings
Fig. 309.
That shown in Fig. 309 is simiiar to the one in Fig. 293, placed
upon a pedestal. Such pedestals vary greatly both in form and
height. The width of the foot is made equal to the height of
the journal in the form shown, which gives the base and the
legs a sufiiciently slender appearance.
8 na.
Waee Bearings.
The wall bearing shown in Fig. 310 is the same as shown in
Fig. 293> with the addition of the bracket. The base here is
placed at right angles to the joint in the boxes and parallel to
the axis of the bearing, the whole being made in the bracket
form shown.
The cap and the boxes are of the same form and proportions
as for a pillow block for the same size journal. The bolts may
either be tapped into the body of the bearing, or made as stud
bolts, usiug the forms shown in Figs. 225 and 226 § 83, with
key.
For larger sizes the opening in the piate should be surrounded
with a rib of a thickness 0.1^ and width = 0.4^, the latter
being measuredin the direction of the axis of the journal.
Fig. 311 shows an adjustable wall bearing by Sellers. In this
case the cast iron boxes are somewhat lighter than for pillow
blocks and are made with a cylindrical cross piece in the
middle, in which the spherical seats are placed. The especial
feature is the methodby which the vertical adjustment is made.
The two plugs which support the boxes have cast upon them a
very shallow screw thread, and the nuts in the sockets have
also their threads cast in them. The thread only extends along72
THE CONSTRUCTOR.
a portion of the length of the plugs as shown, in order to per-
mit securing them in position. This is done by the two self
screws which clamp them firmly in their places.
The opening through the upper plug gives access for the tube
of a lubricator.
Fig. 310.
The projection from the wall a is made constant for bear-
ings for journals 2" to	in diameter and equals 6//. The ele-
gance of the form is noticeable in the principal elevation and
also in the horizontal section.
I U3-
Yokk Bearings.
The bearings used on vertical shafts may be considered as a
variety of wall bearings. In situations where the space is lim-
ited the forms shown are not always convenient, the first, be-
cause it is not symmetrically disposed about the parting of the
Fig. 31 i.
boxes, and the second, because of the space it requires. For
this Service a compact, symmetrical bearing, whose base is at
right angles to the parting of the boxes, is often very desirable.
Such a construction is shown in Fig. 312, and may be called a
Yoke Bearing. In this case the cap and body together form a
rectangular yoke, in which the bronze boxes are placed in a
transverse direction. In the illustration the wear can only be
taken up in one direction, but if itis desired in both directions
the cast iron block on the right may be replaced by a wedge as
shown on the left.
By removing the cap, the wedge and the block can be easi-
ly removed and the shaft moved sideways to a sufficient extent
topermit the removal ofthe boxes. The cap bolts are provided
with collars forged upon them and serve also to fasten the bear-
ing in place. The modulus for the dimensions is the same as
(107), dx == J.15 d + 0.4".*
Fig. 312.
g 114.
Waee Brackets.
In Fig. 313 is shown a form of bearing similar to Fig. 293,
which may be called a wall bracket bearing. The cap bolts
are inserted from below, which permits their ready removal and
replacement. If only two bolts are used in the wall piate, it is
Fig. 313-
desirable that it should be held from lateral motion between
wedges, and should also be firmly secured against vertical mo-
* For such a Yoke Bearing, see Engineers’ and Machinists’ Assistant,
London, 1854, Pl. I.THE CONSTRUCTOR.
73
tions by some of the methods given in the folio wing chapter.
Where it is not practicable to secure it in this ruanner, four
bolts should be used.
Another form of wall bracket is shown in Fig. 314. It is sim-
ilar to the Yoke Bearing, and can often be of Service, as for
example in Fig. 350, \ 126, although itis not of as general ap-
plication as the preceding form. The bolts for the cap are
made with heads, of the ordinary cap screw form.
Various other wall and bracket bearings may be made by
combination of a wall piate and pillow block in different po-
sitions, and these may be grouped in the general class of Arm
Bearings, each form being govemed by the conditions of the
special case under consideration.
115.
Hangers.
According to the definition in § 103 a pillow block by inver-
sion becomes ahanger, the pressure of the journal fallingupon
the cap box. If the journal is one of wrought iron proportioned
to bear the loads given in §91, the bolts for the cap and base
piate will not be strong enough if determined from the same
Fig. 315*
unit of proportion as already given for such bearings. This is
also true for the cap, and feet of the base.
For this Service, good dimensions may be obtained by using
for theboxes theprevious modulus dx = 1.15 d -f o.\", and also
E as before, and for ali other portions the special modulus,
£>[' = 1.75 d + 0.4"	............(109)
If a pillow block is to be used as a hanger for a neck journal,
the cap bolts should be increased tosuch size as would be given
by the use of formula (109), in which d is the diameter of the
neck journal corresponding to an equivalent end journal.
Example : A load of 17,600 lbs. would give, according to the
table in §91 for a wTOught iron journal a diameter of about 4
inches. If this load is carried on the cap of the bearing we use
the modulus,
£>[' = 1.75 d + 0.4" = 1-75 x 4" + 0.4" = 7-4"
This gives for the diameter of the cap bolts 7.4 x 0.2 = 1.48^
say 1	A neck journal of 6diameter to bear the same
load would have for its normal unit d = 1.15 x 6.75 -f- 0.4 =
8.i5//, which is greater than the preceding value andhencemay
be used safely, even should the full load be carried by the cap.
Sellers makes a short hanger which resembles in form and
dimer^ions the corresponding size pillow block, with the boxes
turned 180° and the drip cups cast on the cap instead of the
base. In most cases, howTever, a greater distance is required be-
tween the shaft and the base piate for hangers than is given in
pillow blocks, for which reason they are best considered as a
separate form of construction.
The hanger shown in Fig. 315 is called, from its form, a
Ribbed Hanger. The boxes are carried in the hook-shaped por-
tion below, their form bei ng the same as we have already
shown. The cap is secured with a key and clamped in the de-
sired position by the bolt shown.
For journals of less than 2 inches diameter, butone bolt need
be used in each foot, and in such case their diameter is = 0.3 dl9
the bosses on the piate be altered to correspond.

Fig. 316.
In the Post Hanger,'Fig. 316, the general arrangement is the
same as in the preceding form, the principal difference being in
the frame. The column is made hollow and its internal diam-
eter = 0.55 dv For the larger sizes four bolt holes are made in
the base piate, as shown in Fig. 315.
Hangers are not generally bolted directly to the ceiling
beams, but to strong pieces, or intermediate timbers, and by
Fio. 317.	Fig. 318.
varying the thickness of these pieces any desired amount of
drop may be obtained. If the variation is too great to be se-
cured in this manner a different depth hanger must be used.
If the building is of so-called fire-proof construction, with74
THE CONSTRUCTOR.
ceilings of iroii beatns and brick arches, the form of tke base
of the hanger must be correspondingly modified. A practical
method is shown in Fig. 317, in which hook bolts are used. The
bolts, which are four in number, pass through sockets cast in
the base of the hanger, and their method of attachment avoids
weakening the beam. The base of the hanger is made with
ledges which fit over the edge of the beam and permit tiie use
of wedges on each side.
The form shown in Fig. 318, which is due to Fairbairn is in-
tended to bring the shaft parallel to the beam, while the pre-
vious form carries the shaft at right angles to the beams. The
attachment of the hanger both to the beam and the arch makes
a very secure fastening, but the inaccessibility of the bolt head
is an objection. In this case also the beam is not weakened by
drilling, hook bolts and keys being used, as in the previous
case.
Adjustabee Hangers.
RThe most generally used of the Sellers’ adjustable bearings is
the^hanger shown in Fig. 319. x Thel method of holding and
Fig. 320.
adjusting the boxes by means of screw plugs is the same as
shown in the wall bearings, Fig. 311. Especially to be noted is
the attachment of the drip cup, which may be easily removed
by withdrawing the small pin with enlarged ends.
The drop, or distance from base to centre of shaft, = a —
3.5 dyf in the illustration, but in some cases it must be made
greater. These hangers, like ali of Sellers’ bearings, show
very careful modeling and proportioning, which the small size
of the illustrations can only imperfectly show.
In Fig. 320 is shown Sellers’ countershaft hanger. In this
form the shaft isput in place from the side, and the amount of
wear in the boxes is so slight that they are made solid, instead
of in halves. The cap—which is secured by a bolt, holds the
box in place, and the drip cup is cast in one piece with the
body of the hanger and provision is made for a drip cock to
remove the waste oil.
The illustration shows also the arm for carrying the belt
shifter.
Sturtevant uses ball and Socket hangers also for the counter-
shafts of his fan blowers. These are somewhat different from
the preceding. Fig. 321 shows the boxes in perspective and in
cross section. The section shows the white metal lining and
also the arrangement of double oil chambers, which, by means
Fig. 321.	Fig. 322.
of wicking, keep the jouinal lubricated.^ The outer ends of the
box casting are formed into drip channels, and also receive the
shoulders on the shaft. These shoulders, as shown in Fig.
322, run freely in the boxes without contact. The joumal as
showm is on the end of the shaft, and the pressure is so small
that the wTear is inappreciable.
i 117.
Speciae Forms of Bearings.
In propeller shafts where the screw is arranged to be lifted it
is necessary to desigu bearings which are to be entirely im-
mersed in water. Penn’s practice is to line such bearings with
wood, which has proved especially satisfactory. In Fig. 323, is
given an illustration of such a bearing as constructed by
Ravenhill & Hodgson, the diameter of the shaft being about
19 inches. The body of the bearing is of bronze, the boxes are
of cylindrical section fitted with strips of lignum vitae set in a
Fig. 323.
special lining metal. The pin, projecting from the bottom,
enters into a corresponding recess in the stern frame, when the
screw is lowered into place.
On the Prussian State Railway there have recently been
adopted two Standard forms of bearings for use under cars—one
form being for bronze, the other for white metal boxes. InTHE CONSTRUCTOR.
7 5
Fig, 324 is shown details in partial section of the latter form,
with a few dimensions. The bearing is made in two principal
parts, the body and the lower portion, both being provided
with oil chambers having openings and covers to keep out the
dust. The joint between the two parts is in the horizontal
plane passing through the axis of the journal, the parts being
kept in position by three dowel pins. A wrought iron yoke
holds the lower portion up to the body of the bearing by means
of the bolt shown, the head being secured by the internal hex-
agonal socket shown.
The white metal lining is cast in the body of the box by be-
ing poured upon the journal. The inner end of the journal is
provided with a wooden dust guard packed with a ring of felt.
As will be seen, lubrication is provided both above and be-
low. The upper chamber contains wicking and affords a means
of prompt and copious lubrication in case the journal grows
hot. The principal source of lubrication, however, is from be-
low, the oil being wiped upon the journal by a brush, which is
fed with oil by a wick reaching into the chamber below. The
oil brush is shown, with its spring holders in the lower right
hand corner of the illustration.
In order to permit the boxes to adjust themselves to the
journal when the axle assumes an inclined position with re-
gard to the bearing a certain amount of play is given, as is
shown in the plan view, where the ledges cast upon the
bearing are made parallel for a short distance and then diverge
from below upward from a width of 34 mm. to 42 mm.
Ali the dimensions in Fig. 324 are in millimeters, as this is a
Standard Prussian railway journal box.
This construction is undoubtedly well adapted to meet the
requirements, but it is a question whether the results might not
be attained by simpler means.*
The second form of Standard bearing of the Prussian Railways
differs from the first mainly in the boxes. These are cast of
bronze with semi-cylindrical projections on the track, which
enter into corresponding recesses in the bearing, and permit
the boxes to adjust themselves to the journal.
The guides for the bearings are given an amount of play
similar to the previous form, and there is no change in the de-
tails of the lower portion.
Fig. 325 shows a form of American axle bearing. This is sim-
ilar to the older pattern designed by Lightner f It isonly ar-
ranged for lubrication from below and is designed so as to per-
mit a box to be removed and replaced in the shortest possible
time. The body is of very simple form and is cast in one piece
and a large opening and lid renders it readily accessible from
without. The box is made of bronze, and between it and the
* This question of railway journal boxes is an instructive example of the
importance of constructive siraplicity as applied to machine elements. Since
in the jear 1877 in Prussia there were in use 315,000 axles or over 630,000
boxes. The cost of these represents an investraent upon which every
penny economized in construction foots up an important total.
t See Heusinger, Schmiervorrichtungen (Lubrication), Wiesbaden, 1864,
p. 88.
body of the bearing is a filling block, somewhat similar to that
used in the bearing shown in Fig. 312, arranged so that its re-
moval facilitates the changing of boxes.
This filling block, which is sometimes rounded on top to
provide adjustment, is held between twosmall projections, but
can easily be removed when the pressure is removed by use of
a liftingjack. The change of boxes can be effected in a few
minutes.
A brush or pad for distributing the oil is not used, but instead
the vacant space in the bearing is packed with waste, which
feeds the oil to the journal. This form of journal box has
proved very efficient in Service.*
B. THRUST BEARINGS.
§ H8.
Step Bearings.
In Fig. 326 is shown a form of step bearing for vertical shaft.
The bearing piece orstep proper is made with very obtuse
Fig. 326.
point on the under side in order that it may be able to adjust
itself to the shaft. In order to provide for adjustment in the
position of the bearing the bolt holes in the base piate are
elougated in a cross-wise direction, while those in the bearing
are elongated length-wise, thus permitting adjustment in any
direction.
§ 119-
Waee Step Bearings.
The following is a modified form of step bearings, and is in-
tended to be used with the wall piate supported on a key be-
neath its lower edge ; this key may be made = 0.8 dx deep, so
* A Standard axle and journal box were adopted in the United States in 1873,
and at that time there were over 1,200,000 axles in Service.76
THE CONSTRUCTOR.
that by its removal the bearing may be taken from under the
journal, without removing the shaft from its place.
Fig. 327.
The recess in the step piate serves an oil chamber; end-
long wear may be taken up very conveniently by the adjust-
ment provided by the set screw.
2 120.
Independent Step Bearings.
In many cases, as in examples by Belgian designers, the
lower bearing of a vertical shaft is divided into two independent
parts, a pure lateral bearing and a pure thrust bearing. For
the lateral bearing may be used a pillow block or yoke bearing
of one of the forms already described, while the vertical thrust
is taken by a simple step quite close to the preceding bearing.
This makes the step bearing readily accessible and also readily
adjustable in the direction of wear.
The following example is selected from among a number of
such bearings.
Fig. 328.
The step itself is made of bronze. This is carried on the
bluntly coned head of the stout set screw, a Steel piate being
interposed, while the prismatic form of the screw head pre-
vents rotation of the step. The screw itself is kept from mov-
ing by Penn’s method within the bearing, and the whole is
bolted down to a base piate. The modulus for the dimensions
is the same as before. An application of this form is shown in
? 126,
4 121.
Thrust Bearings with Wooden Surfaces.
For bearings which are operated wet, the use of Lignum
Vitae has been found to give the best results. The wood is
inserted in a similar manner to that shown in § 117, the pieces
being made in the form of plugs. In Fig. 329 is shown the
step of a screw propeller shaft of this
type. The plugs are inserted in a
bronze piate, and the end of the shaft
faced with bronze.
A bearing of this form on the
“Orontes” had 37 plugs each
diameter, and on 50 H. P. nominal
English gunboats the thrust plates
have 7 plugs each 2/7 diameter. Both
these examples are bv James Watt
& Co.
Collar bearings with surfaces of
wood are often made; these should
be always wdrked under w^ater. Penn,
to whom the introduction of such
wooden bearing surfaces is mainly due,
has especially used them in various
bearings in the length of a screw pro-
peller shaft, the lower half of the shaft
running in a water trough. The usual
construction of the thrust ring between the hub of the screw
propeller and the stern post is shown in Fig. 330. A is the
shaft with a bronze sleeve fitting into the wooden lining of the
hole through the stern tube; B is the hub of the screw pro-
Fig. 329.
peller; Cf the thrust ring with its wooden plugs ; D is the
nozzle on the end of the stern tube showing the stiffening ribs
which assist in receiving the thrust. The parts B, C} D and
E are of bronze.
Fig. 331 sLows a form of thrust ring used on the imperial
steamships “Kaiser,” “Friedrich Karl,” “Preussen,” “ Vineta,”
“Freya/' “Ariadne,” “Nautilus” and “Cyklop.” The ring
is made in halves, and can readily be removed and replaced.THE CONSTRUCTOR.
77
The two axial projections enter into recesses in the flange on
the end of the tube, and prevent the thrust ring frora revolving.
The dimensions of the wooden bearing surfaces on the various
ships above named are approximately as given in the following
table :
	; Kaiser. 1	I Friedr. Karl.	Preussen.	Vineta.	Freya.	Ariad- ne.	Nauti- lus.	Cyklop
D' y 1 Surface | sq. ft.	j 28" 1 6)4" | 4.078	2A3X", O.74O :	26^" 7'A" 2.629	16%" 3'X" i 1.188	~ M •ooX£ N.	'9'X" aX" \ 1.840	i°lX" 2tV' 0.476 i	8^ 0.238
In the “Wasp” the thrust ring is made with 6 sectors of 3.186
sq. ft. surface, in the “Leipzig” there are 80 small sectors with
a total surface of 2.422 sq. ft. The use of such thrust rings filled
wuth blocks of lignum vitae has been most successful in vessels
of the German navy, and the wear on the wood has been so
slight that renewal is rarely necessary.
I 122.
Mui/tipte Cottar Bearings.
For thrust bearings which are subjected to heavy Service, the
multiple collar bearing is most valuable. These are very gen-
erally used to receive the thrust of screw propellers, but are
also used in other situations, as, for example, large turbines,
also centrifugal machines of great size and weight. such as are
used in sugar refineries. The forms wThich may be given to
these bearings are quite varied; but in every case the most
important consideration is the pressure to wThich the various
surfaces are subjected.
For pillow blocks in which the shaft is made with several
collars, the boxes may be cast in bronze wTith internal collars

FiG. 332.
as shown in Fig. 332.* For larger dimensions, the boxes may
be strengthened by ring shaped ribs, let into recesses in the
cap and body of the bearing.
take-up of the wear upon the rings. The cross section in upper
right hand portion of the illustration shows the construction
and application of the bronze rings. The arrangement provides
for a constant distribution of grease, thus preventing the rust-
ing of the journal by the application of water for cooling.
Fig. 334.
0	^_______JO/ ^ ouGvni a. uirust Dearmg by Penn, as
used on the “ Kaiser.” Here the bearing surfaces are made in
separate rings of stili simpler form than the preceding. These
ExampleThe thrust bearing on the “ City of Richmond,'’ built by Todd
& MacGregor, of Glasgow, from the designs of JafFrey.f has 12 rings f inside
diameter, 19"; outside diameter, 23"; total length of the bearing, 43%". The
boxes are strengthened by three ribs of depth by 4" wide. The engines
indicate 3340 H. P., and the speed of the vessel is about 1342 feet per minute
James Watt & Co. make the boxes free in the bearing, and
support them by set screws at the ends, as shown in Fig. 23?.
On the “Medusa” and
“Triton ” four set screw^s
are used in each flange,
the shaft being 7// diam-
eter, with six rings. In
the “Jason,” by the same
^ firm, there are six set
screws in each flange, the
shaft being 12" diameter,
with eight rings.J
FlG. 333.	Boxes of cast iron lined
with white metal are
sometimes used by various makers, as, for example, in the
“Mooltan” by Day & Co., in which the shaft is 13X" diameter,
and has twelve rings. The design shown in Fig. 334, which is
a French pattern, uses an adjustable bearing lined witk white
metal.
In Fig. 335 is shown a form of thrust bearing in which the
rings are made of bronze separately, and fitted to the body and
cap. This form is the design of Ravenhill & Hodgson. Espe-
cially to be noted is the arrangement of bolts. These are in
two sets, the first securing the body of the bearing to the sole
piate, and the second being the cap bolts. The ledge or tongue
which is let into the sole piate is arranged with a space as
shown on the left, in which a key is fitted to provide for the
* See Armengaud, Vignole des Mecaniciens. Pl. 13, Fig. 32.
f Engineering, May, 1875, p. 403.	t See Burgh.

rings, which are made of bronze, are in halves for convenience
of construction. In the “Kaiser” d is equal to 18 Yzf\ and there
are eight rings on the shaft and in the bearing. The six bolts
-s,io.............—4
Fig. 336.
are arranged so as to act both as cap bolts and fastenings for
the bearing. The adjustment for wear is similar to the pre-
ceding case. The dimensions are based on the same modulus
as already given, viz. : dt = 1.15 d-\~ o.4//.
A most noticeable form of thrust bearing is that of Maudslay,7«
THE CONSTRUCTOR.
shown in Figs. 338 to 340, as used on the “Elizabeth.” For
each collar on the shaft there is provided a separate ring and
support, with means for ample
lubrication. The bearing rings
are made of horse,shoe form,
and are of cast iron lined with
white metal. The collars on
the shaft dip into an oil
trough. They are also pro-
vided with oil cups above, so
that as in the case of the car
axle box previously described,
lnbrication is supplied both
above and below. Each ring
may be adjusted by its own
set screws, or all can be ad-
justed together. The propor-
tions are all based upon the
previous modulus, dx — 1.15 d
-ho.4", and the shape and
dimensions give an excellent
appearance. In the “ Eliza-
beth ” the shaft is i2^/r diameter.
---------1,0__________
Fig. 338.
k 123-
Examples of Thrust Bearings.
The following examples are taken from twelve of the most
important vessels of the German navy, the data being furnished
to the author with the approval and authority of the Chief
of Admiralty. The power and speed of the engines and the
velocity of the vessel are all most important data, and are
obtained from ofiicial tests. From these may be obtained, as
in § roo, the maximum pressure upon the thrust bearing sur-
faces. It is important to observe that in only two cases out of
the twelve was a thrust ring used between the stern post and
propeller hub. The elasticity of the hull of the ship may some-
times cause the entire force to be thrown on the thrust bearing,
and at other times much may be taken by the thrust ring. The
data given in the table will also be found valuable for other
purposes.
EXAMPLES OF THRUST BEARINGS.
No.	Name OF	Builder of	u •og 1» !> 0 rt n. u M '•8 %	<u ti tr. 5 8.2 O u — v	f Propeller Shaft.	s* .2s 0 ^	5« ai |? o-C (j a o«	u- 0 3S S a	v j »2 4» fi fi	V- O K .d co G 0 73 “ C en	Jh QJ VI §| cO © .5u M fi fi 0	a is	2 c «•2 sfi t/j 4»	ai M u cO 2
	Vessel	Engines.	C u M O w	*u n. ►*. 4)	d s	« p.	•*a	co v	‘S<0 4j * n-S	»- s wS	5 41 §£ oX CQ-Q	fi »4	w a c* 0	£
												Oil		No
1	Armored Frigate Konig Wilhelm.	Mandslay Sons & Field, Dondon.	8325	M91	18"	63.86	6	Anti- mony.	8.467	24tf"	18"	Bearing cooled with	Worked well.	thrust ring instem
												Water.	Ran warm	post. Thrust
2	Armored Frigate Kaiser.	John Penn & Sons, Greenwich.	7803.3	H57	iS"	77.00	&	Bronze.	7.104	23"	181V'	Ditto.	before the thrust ring	ring in stem
													was applied.	post, i 121.
		(Societe des Forges et ) •< Chantiers de la Mediter- >											Made with- out thrust	
3	Armored Frigate Friedrich Karl.		3503	1328	15"	61.82	11=18%" 1=2	White Metal.	8.004	18%"	*5xA”	Ditto.	ring and ran warm. Since	Ditto.
		(ranee, Marseilles. )											its applica- tion, works	
														
	Armored Frigate	(Stettiner Maschinenbau											well.	
4	Preussen.	< Aktiengesellschaft Vulkan > (in Bredow bei Stettin. j	4386.7	1408	16K"	64.5	8	Bronze.	5.371	20		Ditto.	1 Worked well	Ditto.
	Decked Corvette												Ditto.	Ditto.
5	Eeipzig.	Ditto	3519 3	1437	16"	72.4	8	BrOnze.	4.816	*9 Vs"	: 16"	Ditto.		
	Decked Corvette	John Penn & Sons,									1		Ditto.	Ditto.
6	Vineta.	Greenwich.	1359-3	1120	10^"	67.9	6	Bronze.	1.489	12^"	io>6"	Ditto.		
	Decked Corvette	(Markisch-Schlesische Ma-")											Ditto.	Ditto.
		■4 schinenbau, und Hiitten >	2598.8	1557		82.52	8	Bronze.	2.528	15"	12 X”	Ditto.		
7	Freya.	(Aktiengesellschaft. J											Ran warmed	
												Ditto.	first, after-	Ditto.
8	Decked Corvette	Ditto.	1726.9	1282		80.24	7	Bronze.	3-391	14 H"	uX"		wards work-	
	Ariadne.												ed well.	
	Decked Corvette	Mazeline & Co.,				62.09		Anti-				Ditto.	Worked	No
9	Augusta.	Havre.	1127	1245	n"		11	mony.	5.177	I4iV'	«K"		well.	thrust ring. Fitted
IO	Gunboat	Moller & Hollberg						Anti- mony.						
	Nautilus.	in Grabow.	504.2	1047	7X”	109.30	6		I-I59	9%"	7X"	Ditto.	Ditto.	with
	Gunboat Cyklop.	(Stettiner Maschinenbau J												thrust
11		•< Aktiengesellschaft Vulkan > (inBredow bei Stettin. J	245-4	894	sW'	143.89	4	Lignum Vitae.	0.496	7 H”	5%"	Ditto.	Ditto.	ring. Ditto.
13	ArmoredGunboat Wespe.	/Aktiengesellschaft, Weser) (in Bremen. J	799*7	1054	6K"	138.85	1=10 8=9%"	Bronze.	1.728	9%"	7%”	Ditto.	Ditto.	Ditto.CHAPTER VII.
SUPPORTS FOR BEARINGS.
THE CONSTRUCTOR.
\ 125.
SlMPEE SUPPORTS.
79
1124.
General Considerations.
The function of a support for one or more bearings is to hold
theni in a firm and delinite position with regard to the frame
or other parts of a machine. Such supports are nearly always
made of cast iron, and in the following treatment of the subject
this material is the only one considered.
Simple supports are those which are intended to hold but
one bearing, in distinction from those supports which are ar-
ranged to receive several. In both cases the following consid-
erations should be observed as closely as may be, when, as is
usually the case, the shafts wrhich the bearings carry are litted
with gear wlieels which should be near the bearings.
1.	The bearings should be as near to the hubs of the gear
wheels as practicable.
2.	The pressure upon the journal should, in no case, act in
the direction of the joint between the boxes.
3.	The support for the boxes should be so arranged as to
allow the easy removal of shafts and gear wheels.
4.	The number of bearing surfaces should be made as few as
possible, and ali finished surfaces should be capable of being
finished at one setting on the planing machine.
5- Whenever possible, and especially in situations of difficult
access, the bearings should be so disposed that the boxes may
be removed and renewed without involving the removal of the
shafts from their position.
A simple Support for a single pillow block is shown in Fig. 341.
It is intended lor a bearing such as is showm in $ 107; hence
the upper portion is made correspondingly narrow. The two legs
which form the main portions are reinforced by a cross girth,
D E. The position of the points D and E may always be well
placed by observing the following method: Taking the total
keight A B as a diameter, draw from the centre E a semi-circle
A G B; take the middle point of the are A G B at G ; join B G,
and proloug it, making GH=AF; then join H to A, and
draw G C parallel to HA, and A C is the height from the base
to the cross girth. The dimensions of the various parts are
dependent upon the pressure on the bearing, and must usually
be governed by the dimensions of the pillow block and by the
judgment of the designer. In order to meet the requirements
of Rule 5 of the preceding sectioh, there should be under the
pillow block a removable piate, which may be given a thick-
ness of 0.3 dv
Fig. 342 is a similar form of support suitable for heavier di-
mensions.
Fig. 343 is a support for a wall bearing. This is arranged to
be built into the wall, and forms an opening through which
the shaft can pass, and resembling what a builder calls a bull’s
eye window. The pressure of the journal is received by the
Fig. 343-
bracket bearing, which is supported on the key beneath, and
can be removed without disturbing the shaft. One point which
should not be overlooked is the bearing piate in the wall,
shown in tangential dotted lines below the cylinder. The di-
mensions in the illustration are based on the modulus d1 of the
bearing.
Fig. 344.
A wall bracket support is shown in Fig. 344. This is intended
to carry a pillow block, and the T slot for the bolt heads ena-
bles the distancfe of the bearing from the wall to be adjusted.
This form may be used for bearings of various sizes. A simpler
and lighter form of bracket is shown in Fig. 345. This is
merely an arm attached to a wall and adapted for a horizontal
shaft.
Frequently the joint between the base of a bearing support
and its foundation is made with cernent. When this is done,
the base is adjusted to its position, restingupon wedges/and the
joint being closed with clay, the liquid cernent is run in ; this8o
THE CONSTRUCTOR.
will harden in a few days so that the wedges may be driven out
and the bolts fully tightened.
Fig. 345.
§ 126.
Multiple Supports for Beari ngs.
Z Fig. 346 represents a bridge support. The vertical shaft A B
comes from below, as for example, from a turbine, and trans-
mits its motion to the horizontal shaft C D. The journal pres-
sure acts at E, at right angles to the plane of the two shafts,
Fig. 346.
and at F it acts in an inclined direction downward, both from
the pressure of the gear teeth, and also because of the weight
of the wheels and shafts. These pressures are best received at
E, by a yoke bearing as shown in $ 113, and at E, by a bracket
bearing, \ 114, supported on an adjusting key.
Fig. 347 shows a support for a step-bearing. Here the hori-
zontal shaft A B runs in a bracket bearing at Cf and transmits
motion to a vertical shaft which is supported at D, by a step-
Fig. 347-
bearing, ? 119. The latter, as the illustration partially shows,
is carried on an adjusting key in such a manner that it can
readily be removed from below. The bridge which carries the
step-bearing is bolted to the box-shaped base and the nuts for
the foundation bolts are placed inside the base.
Another form for similar Service is shown in Fig. 348. The
shaft A C\ for the large gear-wheel terminates in the support
and is provided with a small bracket bearing at C. On account
of the position of the wheel, this is not very accessible. The
bearings for tlm vertical shaft D E F, are intended to be of the
form described in \ 120, a yoke bearing being fitted into a space
cast in the upper part of the frame at E, while an independent
Fig. 348.
step at .Fis used similar to that shown in Fig. 328. The upper
part of the frame is made cir-
cular in shape, so that a cast-
iron cover may be placed over
the pinion, as shown in the
dotted lines. The base piate
is held down to the stone
foundation by four bolts; two
of the bolts pass through the
columns, as shown in the illus-
trations, and so bind the two
plates firmly together. The
Fig. 349.	plan view shows how the col-
; ums are keyed into the entab-
lature. The base of the columns are let into the base piate as
shown in Fig. 349, and an iron cernent is used.
Fig. 350.THE	CONSTRU,
81
, F|S'358 shows a wall frame for four bearings. A horizontal
shaft A B,is to transnnt motion to the vertical shaft	and
two horizontal shafts i? and F, by means of beveTgears At l
In Fig. 350 is shown a support for two vertical shafts, A and
B, the motion being transmitted from one to the other by
means of spur gears. The shaft A, for instance, may be thatof
a turbine wheel, and B, the main driving shaft of the mill.*
At A there is a bracket bearing such as shown in Fig. 314, and-
at B a step bearing, with a removable block beneath it, so that
the bearing may be removed or examined without removing the
wheel or shaft.
Fig. 351 shows a frame for a vertical shaft A B, which trans-
mits its motion to a horizontal shaft D E. At C is a yoke bear-
ing and at E a bracket-bearing. The horizontal bevel gear is
inclosed in the semi-circular frame, so that a cover may easily
be adapted, as in the previous case. The removal of the vertical
shaft is not quite so convenient in this form as in some others,
but presents no serious difficulty. In some cases the lower part
of the frame is entirely closed and the shaft inclosed in a sort
of pilaster, to avoid accidents.
For a shaft running parallel to a wall, as at A B. Fig. 352,
and transmitting its motion to one D E, at iight angles, the
frame shown in the illustration is suitable. The bearing for the
main shaft at C may be a pillow-block, while a bracket bearing
is suitable at F. The distance of the pillow-block from the wall
is adjustable (as in Fig. 344). If the gears are equal in size
the form may be as shown in plan in Fig. 353. In this case the
Journal at Cruns in a bracket bearing. If the construction is
Fig. 353*	Fig. 354-
intended to fit in the corner of a building, the frame is modi-
fied as shown in Fig. 354; the bearings at G and H are then
the same. Both these forms are shown in Fig. 355 and 356 in
pseudo-perspective.
Very often a main overhead driving shaft is required to trans-
mit motion both to horizontal and vertical shafts from one
point, and tbe combination of Fig. 357 is an example. Here
the frame-work is made a portion of one of the columns of the
building and is really simple in construction ; at A should be
used a bracket like Fig. 313 ; at and wall brackets lil»
3*®» and at d, a step bearing like Fig. 327.
Fig. 351.
* Such a frame is used in a spinning-mill at Chur, the frame and one-half of the
iarge gear-wheel being in. an archway in the large end wall of the building.
is a bracket, and at C a step bracket, as in Fig «•
bearings at E and F are wall-brackets, like Fig. 31 c
while the82
THE CONSTRUCTOR.
By a proper choice of Journal diameters and clearances the
seats for the four bearings may be brought into one plane,
and the other conditions of $ 124 readily complied with.
An examination of the fundamental principies of construction
of supports for bearings will shovv that all fornis may be repre-
sented by a rigid piece adapted to hold in fixed relation two or
more revolving bodies, in such nianuer asto permit the applica-
tion of the various details of construction such as boxes, caps,
bolts, etc. It is often desirable to sketch out in the first place
a general scheme of the construction in order that the direction
and manner of resistances and arrangement of parts may be
examined more readily. The frame shown in Fig. 350 is simi-
lar to the elementary shape of Fig. 359, which resembles a sim-
ple connecting rod; which indeed the base piate really is, the
Fig. 359.
variations being due to the especial conditions and not to any
fundamental difference. The bridge frame, Fig. 346, is in ele-
mentary form Fig. 360. The step supports of Fig. 347 and 348
may be shown in principle either in Figs. 360 or 361, since in
these elementary schemes a bearing may be shown either by
Fig. 360.
Fig. 361.
Fig. 362.
the journals or the reverse. The four-fold bearing support just
described may be sketched in Fig. 362.
To show how these elementary sketches may serve, the fol-
lowiug application to one of L,emielle’s ventilators will indicate.
Fig. 363.
Here, Fig. 363, nine bearings are to be supported. Three of
these are for the drum, wThich is fast to the driving crank ; it is
carried by the two neck bearings at A and B, and the thrust
bearing at C. The six bearings at D, E, F, and G. H, /, are
for the rods of the buckets; the supports for all of these are
then the beams Al Aly the masonry, and the cranked rod
B, /, D, C
ing stresses ; it is therefore important to allow sufficient latitude
in the calculations to provide for variations in the manner of
application of the load.
The various methods of application may be treated as indi-
cated in the follownng illustrations, Fig. 364, which show the
three Cases II, III, and IV, of $ 16. The first shows a column
Fig. 364.
hinged at both ends, the second is hinged at one end, while the
third is rigidly held at both ends. The breaking loads of the
respective forms are :
7r
a
*J_E
r‘
2 7r
c
4 71
n
the columns being of prismatic form and of a height l; J being
the moment of inertia of the cross section and E the modulus
of elasticity of the material; l being taken in inches. As al-
ready stated in \ 16, experiment has shown that columns w7hose
ends are faced off square and true fall under Case c, even though
not held at the ends. If, therefore, a load smaller than that in-
dicated for Case a, be chosen for all cases, security will be as-
sured, even should both ends of the column be jointed.*
We may therefore take for the greatest permissible load in the
direction of the axis :
P = °-4 *	= 3-94	...........(109)
If d is the diameter for a solid circular cross section, we have
for cast iron, in which E — 14,200,000.
P— 2,750,000 d = 0.0245 ^ 1	^p	. . (no)
For Wrought Iron, E = 28,400,000. This gives
5,soo.coo^p, 0.0206 l	(m)
Example x. For a load P — 33,000 lbs., a solid cast iron column 157.5 in.
high, the diameter d — 0.0245	P —4.15'', or about 4Under
thesame conditions a wrought iron column would be 2%” diameter.
An inspection of the formula shows that the shorter l becomes,
the smaller is the value of d. The cross section must, however,
never be allowed to become so small that the limit of permis-
sible stress shall be passed.
In order that the stress upon the cross section shall not exceed
8500 lbs. for either cast or wrought iron (their modulus for com-
pression in either case being 21,300 lbs.), d should in no case be
taken as less than
d = 0.0122
or the load should not be greater than
P— 6397 d2
• • • (H2)
? I27-
Caucuuations for Iron Coeumns.
The calculation of the proportions of iron columns often be-
comes necessary in machine construction, for besides serving
merely as portions of building construction they are often coin-
bined with machine details, and also enter into the design of
framework as supports and similar relations. Their considera-
tion in this place is therefore appropriate.
Iron columns are generally considered as being subjected to
stresses of compression, and, also within certain limits, to bend-
The following table for round solid cast iron posts is calculated
from formulas (no) and (112), and gives the loads wThich may
safely be put upon columns of the respective heights and diame-
ters given.
The quantities marked with an asterisk are calculated from
formula (112) and are a marked reduction upon the loads other-
wrise obtained.
j E
* Drewitz has tested cast iron columns with a load equal to ir 2 y^-without
observing perceptible alteration. E)rbkam’s Banzeibung, V., p. 534.
THE CONSTRUCTOR.
83
STRENGTH OF SOEID CAST IRON COEUMNS.
d	1=8 ft.	10 ft.	i 12 ft.	14 ft.	16 ft.	18 ft.
1 in.	297	191	132	97	75	59
	i,5°4	994	671	493	377	298
2	4,753	3>°55	2,122	i,559	i,i93	942
	11,600	7,460	5,j8o	3,806	2,914	2,302
3	24,060	15,830	10,740	7,892	6,043	4,774
3/z	44,470	28,660	19,90°	14,620	11,200	8,845
4	76,040	48,890	33,950	24,950	19,100	•5,09°
4%	121,800	78,310	54,380	39,95°	30,590	24,170
5	154,4°°*	119,300 :	82,890	61,460	47,060	37,i8o
5/4	186,900*	174,7001	121,400	89,160	68,260	53,93°
6	222,400*	222,400*	171,900	126,250	96,680	76,400
Example 3. In a barracks in Berlin are hollow columns of 142 inches
height, bearing loads of 37.180 lbs. These are made of diameter d0 =
According to (115) this should give the internal diameter :
dx = 6.1875 ^ 1 — 0.00000036	= 5.88"
According to (116) we have :
d\ — 6.1875 sj, -o.cooI5 JZdN = 5.7I>'
This would give a thickness of metal of about yj'. The empirical thickness
for such a column is about -Hi", and the actual internal diameter was 4%".
Example 4. A cast iron column of 185 inches height and 9% inches outside
diameter has to bear a load of 275,000 lbs., and was made with an internal
diameter of 6% inches. According to (115), for direct resistance to thrust we
get:
Hollow Colum7is.-Ca.st iron columns aregener-
erally made hollow. The dimensions in this case
may readily be determined from the formulae for
solid columns.
If the external diameter is dQ7 the internal di-
ameter dlt and the diameter of a solid column of
«qual strength, d, we have
d0_	1_______
d\ — 9-25 ^ 1 — 0.00000036
but according to (116):
d
275,000 x (185)2
(9-25)4
j	I	275,000
h = 9-25 \J 1 —0.00015 -------— = 6
>	(9.25)2
or very near the actual dimensions.
.65"

ii-"”*-
Fig. 365.
* * * (113)
^L = 1
The ratio of internal to external diameter If- = ip is conve-
ho
niently made 0.7 to 0.8. We have for :
=	0.5	0.6	0.7	0.75	0.8	0.85	0.9
. = 1.016 1.035 1.07 I.IO 1.14	I*2° I*3I
o-95
1.52
The limits of stress fall within the formula for compression
and the above results are close approximations. It is to be ob-
served that d0 should in no case be taken less than :
0.0122
d0= "

V1-
ip2
or the load greater than
P= 6397 do2 (1—
These examples show how important it is, to take ali the
conditions into account, in order to avoid errors,- and a careful
examination of the circumstances attending each case should
always be considered.
Fluted Columns.—The cruciform section may serve as an ex-
a	ample of such columns. The thickness and
breadth, b and h, of the ribs may be determ-
4 ined by comparison with the diameter d, of an
'mtsgfak equivalent round solid column by making :
t=^(t)3=°-59(t)s- • • (II7)
from whlch the approximate thickness b, for
any breadth h, may be obtained. In order to
keep within safe limits the cross section should
not be less than :
or the load more than
P— 17000 b h
Example 5. To substitute a cruciform column for the solid one of Ex-
ample 1, we may take h = i.srf = 1.5 X 4-I5 = 6 225".
We then have from (117).
(H4)
\ 6.225/
O.72"
We have for:
d0
b - 4-15 X 0.59
The safe load according to (118) would be :
P= 17,000 X 6.228 X 0.72 = 76,200 lbs.
For a direct calculation of b and h we may use the following :
30 Pl2

Pl2
s]1
0.6	0.7	0.75 0.8	0.85	0.9	0.95	14,220,000 7T 2 h 3 and hence:	’ J ’ h2 ►
0.64	0.51	0.44 0.36	0.28	0.19	0.10	P= 4,762,000	6/13
1.25	1.40	1.51 1.67	1.89	2.29	3.20		i*
— 1p‘z
Example 2. The solid column of the preceding example to support a load
of 33,000 pounds was fouud to be 4.15'' diameter, and for a ratio 01 diameters
of 0.8 for a hollow column for the same load we have cLq — 1.14 X 4.15 =
4-73", say 4^", and the internal diameter dx = 0.8 X 4-73 = 3-78" say 3^",
giving a thickness of metal of x/2 inch. Substituting these values in (114) we
have for the greatest safe load, P= 6397 X (4-73)2 X 0.36 = 51,530 lbs. This
shows the dimensions obtained above to be amply strong, if the walls of the
column are cast of uniform thickness. If the ratio —y- had been taken as 0.7
«0
we should have obtained, from (114) d0 — 4.44", and d\ = 3.10, giving a thick-
ness of metal of 0.67".
In practice it is often necessary to work to a given external
diameter d0, in which case, for cast iron, the internal diameter
dly may be found from :
4I	c Pl'1
dx = du \ 1 -0.000,000,36
and the load
P= 2,750,000
dlczAt-
/2
(115)
in which P is the difference in supporting capacity between two
solid columns of the diameters dQ and dx respectively.
It is necessary also in this case to observe that P should not
be greater than P= 6397 (d02 — dx2)
and dx not greater than	(
p >.................(Il6)
1 o.ooois — J
in order that satisfactory castings may be produced.
. (u9)
Care should be taken that the load does not exceed the limit
given by (118).
Example 6 In the new building of the sugar refinery of Waghausel, built
in 1859-60, are columns of cruciform section.
Those m the basement bear a load of 264,000 lbs., and are 78.74" high, the
ribs being 2" X MiV'- According to (119) these posts should sustain a load of
2 X (14.1875)3
P 4,762,000-——^-= 4,386,000 lbs.
According to (118)
P = 17,000 X 2 X 14-1875 — 482,300 lbs.
which is much more than the actual load.
Columns of Angle and T Iron.—These are much used in
bridge trusses, especially in America. (See \ 87). The vertical
posts may be considered as columns with jointed ends. Case I,
Fig. 364, and the upper chord is in compression and may be con-
sidered as Case III, Fig. 364. The following figures show many
of the forms, in Section, which may be used for this purpose.
DUHHIIIHH +
Fig 367.
The first is the column of the Phoenix Bridge Works at Phcenix-
ville, Pennsylvania. This is shown made of four segments, but
six or more are used. This form may be strengthened by rivet-84
THE CONSTRUCTOR.
ing flat iron between the joints of the segments. The four fol-
lowing sections are from the Keystone Bridge Works.
The sectional distribution of material should be chosen so
that the (equatorial) moment of inertia on both the principal
axes are the same (see { 7). The fifth section shows a double
T iron, in the middle in dotted lines. This is used in bridge
chords, where two or more such shapes are sometimes intro-
duced. The last form is a combination of four pieces of angle
iron recently used for pump rods in mine shafts. The resistance
to thrust is here dependent upon the distance between the
guides of the rod.
Grouped Colurnus.—It is sometimes a question whether, in
the support of very important loads, as well as for economy of
material, it is not best to use two or three columns instead of
one. If we let m be the number used, instead of one, wTe have,
for the supposition that the columns are in compression, the re-
lation for similar sections.
V' = \/mV.....................(120)
This shows that grouped columns use \/m times as much
material as a single column. It is also economy of material to
use a small number of heavily loaded columns to sustain a given
load.
Example 7. This subject may also be treated by the aid of the preceding
table. If we have a load of 2800 lbs. upon a column 18 feet high, the diam-
eter for a solid round column would be 2%", while for four columns of 2
inches diameter we have 4 X 740 = 2960, or about the same. The cross sec-
tions are to each other as 4 X (2)2 :: (2.75)2, or as 16 : 7 56, or y/4:1.
Variations in the height of columns affect the economy of
material, other things being equal, to a marked degree, since
the resistance to compression varies directly as the height (/).
It is sometimes desirable to make a column in sev-
eral portions, when a proportional reduction in
height can thereby be secured. The triple Central
core of the column shown in Fig. 368, is an ex-
ample and is a form often used by architects in
connection with columns of brickwork.* This is
not as effective as a single column, since the volume
ratio is x/z i. e.t yi ^ 3 = 0.866.
In conclusion it must be remarked that the col-
umns which are used in machine construction are
usually made much heavier than the preceding
calculations indicate. This is due to the fact that
such columns are often subjected to bending and
tensional stresses, as well as to much vibration and
the additional material is needed to meet these con-
ditions. Columns of cast iron which are subjected
to tension, as in the framing of vertical engines,
should be made at least double the section given by (112), (114),
(116), and (118). The security is also made greater m the case
of buildings, as the resuit in Example 6 shows.
2 128.
Forms for Iron Cotum ns.
The columns which are used in machine construction must be
held down to the iron base plates of the machines, or if used in
connection with building construction are secured to foundations
of masonry. Heavily loaded columns are often placed upon
foundation stones with only a sheet of lead beneath, and no
fastenmg, but otherwise some form of anchorage must be used.
Fig. 368.
Fig. 369.	Fig. 370.	Fig. 371.
The illustrations show three forms of fastening. In each case
the sole piate is placed beneath the pavement. In the first case
a special form of sole piate is held down to the masonry by an
anchor bolt; in the second the flange which is cast on the
column is bolted to the keys shown ; the third construction (by
* For example, the columns in the vestibule of the theatre at Carlsruhe.
Borsig) is arranged with a short cylinder bolted to the faced sole
piate and made so as to give a space in which melted lead may
be poured after the column is set in its exact position. A hole
is left in the side of the column to admit the melted metal. The
portion of the base of the column which shows above the pave-
ment is made to conform to the general style of the building.
In Fig. 369 a simple moulding is used between the plinth and
shaft; in Fig. 370 a bead is added; and in Fig. 371 a double
moulding of more elaborate outline is used.
1	1
Fig. 372.
The capitals of such columns are made in many varied forms.
Fig. 372 shows, in section and elevation, a capital arranged to
carry a beam and also to support the base of the column of the
floor above. A recess in the top of the column receives the
main beam, and aflfords a good place for a joint. If iron beams
are used, this recess is made proportionately narrower. The base
of the upper column is securely bolted down as shown.*
i
Fig. 373-
Fig. 375-
The capitals of iron columns afford much opportunity for ef-
fective decoration, which in many cases is neglected, although
comparatively easy of execution. For the lower columns of
heavy buildings the simple cubic capital so often found in Ro-
manesque buildings is most suitable, and a good example is
shown in Fig. 373.f
Fig. 374-
A somewThat lighter form is shown in Fig. 374, and for some
situations the various Gothic capitals are suitable, Fig. 375. In
* Other forms will be lound in Brandfs Eisenkonstruktion, Berlin, 1865.
f Shown among other places in the Osten. Tloyd, in Trieste, and in the
Arsenal at Vienna.THE CONSTRUCTOR.
ali three examples the pattern making and moulding is not dif-
ficult. The form most used in machine construction is shown
in Fig. 376, being something between the Roman Dorie and the
Tuscan orders, and having an echinus beneath the cap piate,
and an astragal bead around the column ashort distance below.
By varying the distance of the latter from the former the effect
can be modified for tali er or shorter columns.
The heavier form of the Grecian Dorie is unsuitable for ma-
chine construction and is seldom used. More appropriate is
the modified Corinthian capital shown in Fig. 377. The top is
a cornice of overhanging leaves, terrainating in an astragal on
the shaft. By omitting the ornament the same form may be re-
tained, as shown in the right hand half of the illustration, and
also in Fig. 348. The fluting of the column is by no means ob-
jectionable, at least in Germany. The fluted capital is readily
cast by being made in a core box.
Fig. 378 shows a capital of Renaissance form with octagonal
abacus, well suited for slender columns.
The support of beams, either iron or wooden, is best accom-
plished by the introduction of a piate between the column and
the beam, and this may be treated simply, yet in harmony with
the style of the rest of the work. Fig. 379 shows such a sup-
port on the cubic capital already shown, and is adapted for very
in which the solidity and substantial character of
construction is well shown.
85
this form of
Fig. 382.
Fic. 381
heavy construction. Fig. 380 shows a lighter capital, in which
the support for the beam is made of a box form; Fig. 381 is a
stili lighter design. This illustration also shows the effect of a
high stylobate orbase moulding, suitable for tali slender columns.
As shown in this example, such bases are usually made octagonal
in section, which approaches the Gothic style, but they are fre-
-quently made round. As in architecture, the columns are usu-
<ally made tapering from below upward, the upper diameter be-
ing 0.8 to 0.7 that at the base. Fig. 382 shows a more elaborate
form of capital and bearer.
Columns of cruciform section, already referred to, are often
used in the construction of industrial establishments. They are
sometimes to be preferred to hollow columns, since the latter
are often cast of such unequal thickness as to be unreliable.
Figs. 383 to 385 show such a column. Fig. 383 is from the Rail-
way of St. Germain. Here the flutings extend from top to bot-
tom and the column is swelled slightly in the middle. The form
shown in Fig. 384, from the Tobacco Factory at Strasburg, is
more elegant in its appearance. Here a rectangular base is used
for the lower floor, but is omitted above. The method of
connecting the base and column, as well as the connection
between capital, beam, and column above, is shown in Fig. 385 ;
Fig. 383.
Fig. 385.
----
These examples will serve at least to show the variety of
forms of columns which may be used, and the manner in which
a little ornament maybe introduced into machine construction.
CHAPTER VIII.
AXLES.
? 129.
VARIOUS KlNDS OF AXEES.
Axles may be considered as beams which carry revolving or
oscillating loads, and hence are provided with journals at certain
portions of their lengths; they may be subjected to deflection
or to compression, as in the cases of journals already discussed,
according as the load act normal or parallel to the axis. Axles
which are subjected only to thrust, are not often found, the far
greater portion being those bearing deflecting loads, although
many are under combined stresses.
These may be divided into two classes : those which have the
load applied at but one place, and those in which several loads
are borne at various points. The first are called Simple-loaded
Axles ; the sec9nd, Multiple-loaded Axles.
The sections of axles of cast or wrought iron may be either
circular or varied, and this gives rise to another subdivision in
the calculations. The methods of graphostaties are especially
applicable to the subject of axles, and in the following pages
both numerical and graphical Solutions will be discussed.
A. AXLES WITH CIRCULAR SECTION.
8130-
Simpee Symmetricae Axees.
Fig. 386.86
THE CONSTRUCTOR.
The load Q is in this case applied normal to the direction of
the axis, midway between the two journals, upon a seat for a
hub, as shown in Fig. 386. The portion between the hub-seat
and journals is called the shank of the axle. The journals are
proportioned according to the methods given in Chapter V,
taking P= % Q, and the axle then proportioned so as to give
approximately the same strength as the journals throughout.
Let:
d — diameter, l = length of journals,
e = height of shoulder or collar,
D = diameter of middle, or hub-seat, b — its breadth,
Df — diameter of shank at the junction with Dy
e/ = y2 (D — D') the shoulder at the latter junction,
a = the length of shank,
then we have :
D'
d
a— 0.5 b
T
U—jU—*
The cubic parabola is shown drawn in the shank a2, being the
shape for uniform resistance in this case, and we have for the
diameter 6, at the root of the hub-seat:
(121)
This will give the axle the same security as the journal, so
that the approximate stress will be 8500 lbs. for wrought
iron, and 4260 lbs. for cast iron. If a higher or lower stress is
desired, the journal should be proportioned for the desired
stress, and the corresponding dimensions for the axle deduced.
Fig. 387.
The strongest form for the shank of an axle is that of a cubic
parabola (see § 10, No. VI), and the student will find this a valu-
able subject for investigation. In practice it is made a portion
of a truncated cone whose larger diameter = D', and thesmaller
diameter = d= 2 e. The value of e/ should only be made large
enough to provide sufficient depth for a keyway.
Non-Symmetricae Simpee AxeES.
If the two shank portions of a simple loaded axle are of un-
equal length, as in Fig. 387, the load on the two journals d1 and
d2 will be unequally divided, and we have the proportion
P1     P'2   fb____________	/122^
Q	al + a%9 Q	«1 + a%’ P2 ax	'
The hub-seat divides the axle into two parts, each of which
xnay be considered as the half of a symmetrical axle, and so the
6
~d2

(!24)
See (§ 10, Case VI, Remark).
Example — Eet the load Q. acting in one direction, be 14,520 pounds, ax =
47.25", «22 = 23.625'', £=13", material cast iron, number of revolutions, n =
150.
We have P\ — 0.5 Q= 7,260 lbs. /33=i.5 <0=21,780 lbs. From the table of
§ 91 we have dx = about 3%", d2 — about 6%", also /x = 5%", h =	We then.
have:
23.625 ^	say
> 4.6875
D = 6.25 ,
h — y/(5-625)2 + (9-S75)2 — 10.93, say 11".
<5-
5 — 6.25 Jl__________= 6.97, say 7".
^ 9-375
i !32.
Graphicae Caeceteation oe Simpee-Loaded Axees.
The determination of the forces actingupon the journals may
be made according to the methods given in Cases I to V, of § 39.
In a similar manner the cord polygon may be employed as in
'i 43 and \ 44 to determine the statical moments of the parallel
forces at various points, and the polygon so constructed may be
called the surface of moments. The simple method about to
be given will serve as a general graphical solution of the problem^
/. The Load Acis Normal to the Axis.
-	Jg				d t-
r*	D		I	i / |Ct ,/
\			y	
Fig. 388.
Fig. 389.	Fig. 390.
(a). Hub and Load between the Journals.—Draw the line A Cy
equal in length to the distance between centres of journals, and
upon it construet any triangle ABC. whose apex lies on the
line of the load Q. Draw A 3 normal to A C\ making A 3 = Q;
draw 3 . O parallel to B C\ and 2 . O parallel to A C; then A . 2
= Z5!, 2.3 = P2. By dropping the perpendiculars from the ends^
of the hub-seat we may divide Q into two forces Qx and Q2>
shown in the force polygon by O b, parallel to B1 Z?2 5 givin£T
A b==Q1. b . 3 = Qr The vertical ordinate /, at any point of the
surface of moments is proportional to the statical moment My
at its point of intersection with the axis, as for example the or-
dinate tlf at the base of the journal for Pv We have in any case 1
(■24)
.i	32	JLf■ //3 		_32_
	y	c 7T S	r lvly- ai —	T S
and hence:			
	i-1	"-r 11 u 0 «iV	'JT h
whole proportioned. The value of D' is determined for each
shank, and the greater value taken for both sides. If ax —a2
the axle becomes symmetrical.
If the seat for the load Q does not lie between the two jour-
nals, but projects, as in Fig. 389 (a2 becoming negative), the load
is said to overhang, and the journal D becomes a neck journal
(see ? 92).
We have for the relations of forces :
E=:«2 A _	<h +	<hE = a,  .
Q	ci\ ’ Qa\	* T	ai +
The diameter of the journal dx is first determined, then a
diameter for an assumed journal d2 for the loaded point, and
then a value D for a neck journal, taking for D the greater of
the values for D' and Df/, as given for the two ends by formula
(121), the length of journal being made /, = ^ lx + /22.
from which y can readily be obtained.*
(b). The Hub-Seat between the Journals and the Load Over-
hung, Fig. 390.—Draw the line A C. parallel to the axis, con-
struet a triangle with the points A> B and C on the lines of the
directioris of the forces, drop a perpendicular from the point Dy
where D d= Q, make O . 1 parallel to A Ct and equal to C D,
make A . 1.3 normal to A C\ also O . 3 parallel to C B, and 1
. 3 will = Q, A 1 = Plt 3 . A = P2. The force Q is decomposed
into two forces at the ends of the hubs, and by dropping thf.
perpendiculars, the points Cx and C2 are determined, and Oe
drawn parallel to Cx C2t giving the values c . 3 and 1 . C for the
* If it is desired to determine a series of values of t, beginning from tx, it
may readily be done by using a table of cube roots of numbers such as are
given at the end of this volume; if the greatest value of y is the starting:
oint, the table of cube roots of decimal numbers is useful, the space being:
ivided into ten parts and the outline laid off correspondingly.THE CONSTRUCTOR.
*7
forces at Cx and C2 respectively. The diagram shows that at a
point within the hub-seat the stresses are reversed and the bend-
ing moment is zero.
Fig. 391.
Fig. 392*
(c). Overhung Axle with Load Outsidethe Joumals, Fig. 391
—Construet the triangle A B Cy as in thepreceding case (b), and
place D so that Dd=Q, draw A . 3 normal to A C\ make O .
2= CD, and parallel to A Cand draw O . 3 parallel to C B,
and we have again A . 2 = Ply 3 . A = P2. Divide Q into Cx
and C2 and make Oc parallel to Cx C2, giving c . 3 and 2 . c for
the forces at Cx and C2. The journal at B beiug uniformly loaded,
its moment surface is outlined by a parabolic curve (see \ 42).
Fig. 393-
FiG. 394-
S*J-
(d). Overhung Axle, with Load between Joumals, Fig. 392.—
'onstruct the triangle A B C as in case (a), divide Q into Bx
nd Blt which gives the polygon A C, Bx B2 (which is equiva-
mt to the other one A Cy B± B3). In the force polygon, 1 . 3
= Q, 2 . I = Ply 3.2 = P2y and by making Ol parallel to B2 Bx
?e get b . 3 and 1 . b for the forces on Bx B3 and B2 B±*
Fig. 395-
II. The Load Acis Inclined to the Axis, Fig. 393.
The construction is similar to L, except that the force and cord
polygons are inclined according to the direction of Q. The
vertical projections a A, and 3 . ^give the journal pressures Px
and P2; the horizontal component of Q gives the axial thrust.
Another example is given the case of a rod worked from an
overhung arm, as in some forms of locomotive feed oumos, Fig.
395- The directions are here periodically reversed, and the re-
lations of the points continually changing.
III The Load Acts Parallel to the Axis, Fig. 394.
We here have two couples : one consisting of the two equal
journal pressures, and the other of the two pressures at the ends
of the hub-seat (see g38). Draw the lines A Bx and C B2 par-
allel to each other, and intersecting the perpendiculars dropped
*The conditions < f this case, but with very light stresses, are found in the
spindles of the American Ring Spinning Frame.
from the ends of the hub, join Bx with B2 and the surface of
moments is A Bx B2 C. To find the forces, prolong Q from B
until it intersects Cb, join it to the middle of the other journal,
make qb — Qy and drop the perpendicular q a, which is equal to
Fig. 396.
P. Make A 1 = Py draw 1 . O parallel to A Cy and O . 2 paral-
lel to B2 B1% then 1 . 2 is the force at bx and 2 . 1 that at b2. If
the hub should overhang, as in the case of a screw propeller*
Fig. 396, the diagram takes the form A B Cx C2.
? 133-
Proof Diagrams.
In order to calculate the resistance of a given axle to bending
it is necessary to know the section modulus at various points.
If all the sections are circular the moduli vary as the third power
of the diameter. Hence the various diameters are to be cubed.
Y
Fig. 397-
This may readily be done graphically by the method given in
\ 28. In order to compare such a diagram with one of the sur-
face of momen ts asjust discussed, it is necessary to construet
them to the same scale. For this purpose take the origin O of
the two axes X and Y, and make O a equal to the diameter (or
semi-diameter) of the shank of the axle, and lay off, below the
corresponding value Ob of its ordinate tx, drawT on al a semi-
circle a c by draw a e normal to a cy and taking O e as unity wre
have Ob = (Oa)3. Make O. 1 =y, O .2=y2, &c., and draw
the moments to the axes of X and Yy as 1, i/, I., 2, 2/, II., &c.,
and we have OI, OII, as the desired values of yx3, y23 . . .
which correspond to those of the principal diagram.
Such proof diagrams are very convenient to show what ap-
proximations may be made, and to detect possible errors in cal-
culation, and shows at once any deficiency in security, since the
relation of the actual stresses to the desired constant stress is
that of the ordinates of the proof diagram to those of the theo-
retical surface of moments. This numerical series may beplotted
in a curve, called the stress curve. By combining the theoreti-
cal diagram with the proof diagram on an exaggerated scale, as
shown in the illustration, the unit can be chosen to a greater
advantage.
i 134.
Axfes Loaded at Two Points.
In an axle loaded at two points, as in Fig. 398, the end por-
tions are called the shanks and the middle part the shaft. If
Qx and Q2 are the loads, ^ the length of shaft, we have for the
journal pressures

Qi	&i + s + a?t	Q%	+ -y + a2 x 5
If wre take the diameters corresponding to these pressures as88
THE CONSTRUCTOR.
dx and d2, and also kave the shanks ax and a2 given, we may de-
terinine next tke diameters at Dx and D2 at the points of appli-
cation of the loads Qx and Q2.
To find the diameters of the shaft at various points we have,
taking y for the diameter at any point distant x from the load
point Qx:
4=^I + ^(1-^)....................(,26)
an equation which gives the shaft the outline of a double cubic
parabola, which in practice may be replaced by two straight
lines, giving the shape a truncated cone.
The two seats for hubs are formed so as to give shoulders for
keyway, and have a determinate breadth b, governed by the
piece to be carried. In many cases such axles are symmetrical
and the two loads are equal to each other, hence ax = a2, Qx =
Q2• We then have Px = P2= Q\ = Q2 and y — Dy the shaft
being cylindrical. This is also the case when Px ax z=P2a2.
The graphical solution of the preceding problem is as readily
made as in the case of single loads. If we draw normals to the
axis A Dy Fig. 399, corresponding to the given loads Qx and Q2,
also draw A a, make a 1 = Qx and 1.2 = Q2y choose a pole O
and draw the rays Oa, O 1, O 2, proloug a O to its intersection
b with the line of the force QXy make b c parallel to 1 Oy cd par-
allel to 2 O and join d with'#. Draw O 3 parallel to d a in the
force polygon and we have 2 . 3 = P2, and 3 a = Px and a b c d
the surface of moments whose vertical ordinates t may be used
to determine their corresponding diameters of the axle as in I,
3 132-
The intersection e, of a b, and d c prolonged determines the
position E e of the resultant of Qx and Q2. If E e is desired at
once, as in the method given in § 40, theprevious case (£ 132, I)
is applicable since the direction of the line a d can be chosen at
will.
If one load acts beyond the bearings, Fig. 400, the reversal
point in the elastic line will appear as before; this occurs when
the resultant of Qx and Q2 falis between A and D (see \ 132, I).
The above mentioned shearing stress is given by 1 .3.
If the resultant of Qx and Q2 falis outside both journals, Fig.
401,	there will be no reversal, the force Px having the same di-
rection as Qx and Q2; in other respects the procedure is the sanie
as before.
Finally the resultant may just equal the force at Dy as in Fig.
402.	I11 this case there will be no bending stress in portion A
By which in the previous case was quite small; the two lines of
the surface of moments fall together. The shank A B and the
journal at A may therefore be made very light, unless other
forces than those already considered act upon them.
The decomposition of the forces acting upon the hub-seat de-
pend upon its breadth and the treatment is similar to $ 132.
Other variations may occur in the relations of the loads and
journals, but the preceding examples will suffice.
I I35-
Inceined Double Loaded Axees, Raieway Axees, Crane
PlEEARS.
The previous methods are almost as easily applied when the
loads act in an inclined direction. The inclined action is caused
by various conditions, and as an example we will consider rail-
way axles.
Besides the vertical load Q at the centre of gravity Sy of the
car, Fig. 403, there are forces due to centrifugal action and flexi-
bility, which produce a horizontal force which Scheffler, accord-
ing to Wohler’s researches, places at 04 Q* so that there is an
inclined resultant B, acting upon the axle. Since the value
0.4 Q was obtained by means of measurements on cars during
long runs, it includes the action of the elevation of the outer
rail in passing curves. This force R is also acting on the wTheel
flanges at Kx and K2 as well as at the journals A and D. It
must be noted that the wheel K2 opposed to H can only resist
forces acting normal to the coned face so that the angle L K2 Sf
should be made = 90°.
The points of intersection B and C of the wheel forces on the
axle give the positions for the verticals Qly Q2, and the horizon-
tal pressure may be neglected in determiuing the axle loads,
these being Px and P2. From these the journal diameters are
found and the greater taken for both.
Then from the point of application E of the resultant R let
fall a perpendicular E e and draw the triangle ad ey prolong the
directions of Qx and Q2 to b and cy and join b and c by a straight
line. Then drop perpendiculars from B' B", O C//i to b' b'\
c' c" and join these latter, and a b' b" c' c" d is the cord poly-
gon for the given conditions. The ordinate t serves to deter-
mine the diameter for any journal diameter dXi and the ordinate
t2 gives the root of the journal
The direction Kx B is readily determined as follows : Choose
any point on the line of R, as for example, E, join it wTith Kx
and K2 and decompose R = E r, Fig. 404, along the directions
E Kx and E K2 into E k2 and k2r=. E kXy draw k2ly horizontal
and E l parallel to the given direction K2 S/; then l E is the
force at K2y and r l that at KXy whose direction is sought, while
E k2 and k2l are the inner forces at the corner K2 of the cord
polygon E K2 Kly and in equilibrium with the force of the known
direction K2 S'.
Since the horizontal force //acts either to the right or left,
the larger side of the polygon ass/ b" b' must be used for both
halves of the axle, as shown in the dotted lines. The cord poly-
gon for the direct vertical load should also be drawn, and if it
gives a greater ordinate for the shaft than s s7, it should be used,
the diameter of the shaft generally being smallestin the middle.
Axles of railway cars make from 250 to 300 revolutions per
minute. For wrought iron the journals are generally made two
diameters in length. In passing around curves these journals
are subjected to considerable endlong pressure. The shoulder
e is generally made = \ d to £ d, and heavier than usual in ordi-
nary cases.
* Ad. Scheffler, Railway Axles. Braunschweig.THE CONSTRUCTOR.
89
In many countries Standard proportions for axles have been
adopted. Those of the Prussian railways are as follows, the di-
mensions depending upon the value of Q, wiiich is the total
load 011 each axle.*
Q =	3800 kilogrammes. D = 100 mm.	d =	65 mm.
“=	5500	“	“	115	“	“	75	“
“ =	8000	“	“	130	“	“	85	“
“ =	IOOOQ	“	“	I4O	“	“	95	“
The journal length / varies between	to	d, according to
judgment. These proportions are for wrought iron ; if Steel is
used Q may be increased by 20 per cent. For iron axles the
pressure upon any one journal should not exceed % Q. These
figures give a stress of 6.4 to 8.3 kilogrammes per square milli-
metre or 9000 to 12000 lbs., and the pressure p from 0.30 to 0.41
kilo. or 326 to 593 lbs.
In Fig. 405 is shown a Steel axle for the Royal Eastem Rail-
way, with its wheels, all dimensions being in millimetres.
In England a Standard axle has been adopted as shown in
Pig. 406, f and the Standard American axle is similar.J The
value of Q in this case is about 22000 lbs. In France there has
been no general Standard adopted, but the various roads have
adopted forms—for regular use. The Paris-Lyons-Mediterranean
Railway has eight forms. The form No. 8 has d =8 5 mm. (8^/7),
/=170 mm. (6^//)f length between centres of joumals= 1925
mm. (7diameter of hub-seat = 125 mm. (4W')* diameter
of the axle in the middle = 105 mm. (4j^//).
y
Crane pillars maybe considered as axles subjected to inclined
stresses, as the following example will show. The crane shown
in Fig. 407 is subject to the load Z, and also its own weight G,
and the resultant of these is at Q (see examples in § 34).
At A and B are bearings, and the pillar isheld in a base piate
C D, the piate being secured at E F. In order to determine
rces at E and F, construet the cord polygon efg, and force
1 = Q, 1 - e = Ox the force at Et
the forces __________,
polygon e 2 1 0, in which 2
* These dimensions are given in the metric system as representing Conti-
nental practice.
t See Engineer, Nov., 1873.
t See Engineer, June, 1873. The M. C. B. Standard varies slightly from the
«bove (Trans.)
e . 2 the force Q2 at E. All three external loads act parallel to
the axis, so that we can use the method shown in Fig. 394. In
the diagram to the right we make qx q2 = Q, and q2 q3 parallel to
A qx normal to A B. These lines then represent the horizontal
forces Px and P2 at A and B.
The bearing at A carries the entire vertical load, and hence
we have at A the inclined resultant P/ of Q and Px. We now
draw Cfx, normal to A Cyf2fx = Q, draw fxD and also f2f3
parallel to Cfx> then f2f3 will give the magnitude of a force act-
ing right at D and left at C. In a similar manner, draw ex e2 =
Q2, and draw ex Dy and make e2 ez parallel to ex C, and e2 e3 will
be the magnitude of a force acting left at C and right at D.
We therefore have P3 =/2/3 -f e3 e2 and P4 = e2e3-\-/3/2. The
vertical pressure of the pillar itself is all taken at hence we
get for its vertical component Q =f2fx — ex e2., which combined
with P4 gives the resultant P4. This is proved by the intersection
of P/ and P/ at S must fall on the line of the resultant of P%
and P3.
If we neglect the compression in the direction of the axis,
we may now draw the force polygon a 2 3 O of the forces PXy
P^ P3, PA1 as shown at the left of Fig. 407, and thus obtain the
surface of moments a b c d.
Fig. 408.
A crane with swivel column, to wiiich the jib or boom is
rigidly attached, may be examined as shown in Fig. 408. The
position of Q = L -|- G is taken as before, making qx q% repre-
sent Q, draw A qx normal to the axis, join qx D and draw q2 q%
parallel to A qx till it intersects with qx D. We then have q3 q2
for the horizontal force Px at A, and q2 q3 the corresponding
horizontal PK at D.
The step bearing at D will be subjected to an inclined thrust,
the resultant of Q and P4.
In a similar manner we obtain the horizontal forces P2 and P3
equal and opposite, and acting at B and C, and the resultant of
the force at B with Q gives the inclined force due to the rod B
E. The four horizontal forces have the same action as the load
on the axle in Fig. 394. We may thus obtain the surface of
moments a b c d, wiiich shows a zero point for bending moments
betwreen B and D, and also indicates a forward bending above
and a backwrard below’. In the force polygon 2 a — P2f a 2 = P3i
2 1 = P± and 12 = PV
Axees with Three or more Bearings.
The number of bearings for an axle is often as great as four.
In such a case the forces and moments may be found as follows :90
THE CONSTRUCTOR.
Starting at a, Fig. 409, with the given forces 1 to 5, \ve form
the force polygon ah O, and, according to \ 40, the link poly-
gon a b c d efgy and join the closing line g a, parallel to O 6,
in the force polygon ; giving 5 . 6 = the force P3 at Gy 6 . a =
the force Px at A. From Px and P3 the journals dx and d2 may
be determined, and the ordinates of the cord polygon give the
means of obtaining the axle diameter as before.
# The intersection gt of ab and f e, prolonged, is a point of the
line of direction G g9 of the resultant of the forces 1 to 4. If it
is desired to find the successive resultants of the various forces
as they are combined (see g 40), it will be found convenient to
choose O, so that ay*will be parallel to A F. The inclined link
polygon may also be transferred to a closing line parallel to
A F.
If the shanks of the axle overhang the journals, as in Fig.
410, the procedure is similar to the preceding. Beginning at
the point ay the force polygon a 5 O is constructed, and the first
side of the cord polygon b a, drawn to the line of the first force,
the second to the line C c> of the second force, and so on to the
closing line eb. The first and uth line of the cord polygon in-
tersect as before on the line H h of the resultant. Variations
on these examples may occur, as when the loads act in inclined
directions, or opposed to each other, the methods being similar
in all cases.
2137.
Axi,es with Inceined Loads.
The analytical investigation of axles becomes more diflicult
when, as in Fig. 411, the loads act in different planes, but the
graphical method is readily applied. The force polygons A Ox
Fig. 41 i.
i, and D 02 2, are constructed for the forces Qx and Q2i re-
spectively, Fig. 412, the polar distances G Ox and H 02 being
made equal to each other, so that the closing lines of the two
cord polygons A b' D, and A c" Dy coincide in A D. Then
construet the second cord polygon with the inclined ordinates
B B" — B b", C C" = C c”, &c., making the angle u with the
force plane of the ordinates of the first polygon, and inclined
backwards as drawn. Then make B b — B" b', C c~ C" c\
E e — E" e', & c., and draw the cord polygon A b e f c Dy from
tyhich can be obtained (according to g 44) the bending moments
C"
for the axle. The line b e f cis, a curve (hyperbola), A b and
c D are straight lines. Draw Ox O/ parallel to A 1, 02 O/ par-
allel to D 2, and drop the perpendiculars O/ 1 and 02/ F, and
A I wdll be the force 011 the journal PXf and D K that at p»
measured on the scale of the force polygon. Their directions
are determined by combining A G with H 2, and D H with G
1 at the angle u.
B. AXLES WITH COMBINED SECTI ON.
\ 138.
Annuear Section.
If it is desired to make an axle with annular section, or in
other words, a tubular axle, the journals should first be calcu-
lated, according to the method given in § 90 for tubular journals,
and then, retaining the same proportional thickness, determine
the dimensions of the other parts in the same manner as for
solid axles. The most commonly used ratio of internal to ex-
ternal diameters is o . 6. Instead of doing this, all the dimen-
sions for a solid axle may be determined, and then having chosen
a ratio for diameters, increase all the sizes according to formula
(95). See also g 141.
2 >39-
Axees with Cruciform Section.
In cases where axles are made of cast iron the cruciform sec-
tion, with circular centre and four ribs, is sometimes used. The
shanks are then usually made of the ordinary conoidal form,
Fig. 413, and in some cases the ribs gradually swell into a junc-
tion at the ends with the Central core, Fig. 414.
Fig. 413-4X4.
In designing such an axle, first proceed as if drawing a solid
circular section as shown by the dotted lines, of the diameter
corresponding to the'portion K when the ribs join the head.
Then for any point (x) of the shaft :
y = the diameter of the assumed round axle, or equivalent
conoid,
h = height of ribs ;
b — thickness of ribs;
k = diameter of core;
and the proportions are obtained from the following formula ?
f-V(T)-+r.KTX-vhE-itt
(127)
This formula serves for the pure cruciform section, without
core by making k — b.
The results vary so slightly when k = 0.2 h, that the follow-
ing table may be used for both sections :THE CONSTRUCTOR.
9i
Example 1.—Simple Cruciform Section.—If the height of the ribs at any
point is made double the diameter y, of the ideal conoid, we have in the
third line of the table, first and last columns, the thickness of rib £ = 0.07 h.
Example 2 —Suppose a core to be used and at any given place h — 1.5yt
and k — 0.6 h, we have, according to line 8, columns 6 and 1, b= 0.12 of the
height h at the same place.
FlG. 415*
COMPOUND AXEES FOR WaTER WHEEES. 4
In Fig. 417 is chown an axle for a water-wheel, made of cast
and wrought iron. This was made to replace a broken axle of
wrought iron, for a wheel 32.8 feet (10 m.) diameter, 19.68 feet
(6 m.) in width.* The load is carried at four points, as shown,
Fig. 416.
giving a total of 82,104 lbs.f The shaft consists of a drum of
sheet iron y%" thick and 44" outside diameter, made in three
sections riveted to the Central spiders of the wheel. The two
journals are fitted to the cast iron heads with a slight taper, the
ends being prolonged into the middle of the drum, wThere they
are drawn together by a right and left hand nut. The journals
We may make b, constant and determine k, or let % be con-
stant and b vary. The latter case is shown in Fig. 415. Here
the shanks are also cruciform in section, and the hub-seats are
mac^e to receive keys, as shown in both sections, and the Central
one is strengthened by transverse ribs. A small auxiliary jour-
nal is shown at the end of the main journal, and is very useful
in erection.
i 140.
Modified Ribbed Axee.
For heavily loaded axles the form shown in Fig. 416 is suit-
able, the ribs being provided with flanges along the edge. Fair-
bairn has used such axles for water-wheels, and Rieter& Co., of
Winterthur have made them for the same purpose. The pro-
portions are determined by taking the diameter y, of an ideal
shaft of circular section, and calculating h, as before. We may
then make the flange thickness c=b, the thickness of the ribs,
and then the flange breadth bx is obtained from the formula :
B	C	E
35,39*-	«1,924.	11,924.	22,858.
Fig. 417.
are diameter and w" long. The circumferential joints in
the drum are strengthened by pieces of angle iron as shown.
The stress in the shell of the drum is only 3100 lbs., and on the
riveting about 6400 lbs.
1142.
3*			/A Y
16	V h	/ h	
	6 (	A )*—12 (	A)»
	V	h > '	h '
(128)
from which the following table has been calculated .
CONSTRUCTION OF RlB PROFIEES.
In drawing the curved outline of ribs such as shown in the
preceding designs, the following methods may be employed. In
the various diagrams AB is the geometric axis of the piece, A1
the highest point of the curve, and K the lowest point, these
botli being already determined.
1. Circular Arc.—This can only be used to advantage when
on such a small scale that it can be drawn with compasses or
trammel.
2.	Earabola—TtraNi S D and C K parallel to A B, divide S1
D into any number of equal parts, as for example, six parts,
and divide D k into the same number. Drop perpendiculars
from I, II, III, &c., join the lines 1, S 2, 613, &c., and the in-
tersections of these with the perpendiculars I, II, III, &c., will
be points in the parabola.
3.	Sinoide.—Draw 6* D and C K parallel to AB; with a radius
A S draw a circle about A ; divide the arc .S E, cut off between
S D and C K into six, or any number of parts ; draw from the
points of division, lines parallel to A B, and from I, II, III, &c.,
perpendiculars to A B, and the intersections will give points in
the sinoide.
Annales du Genie Cwil, 1866 and 1872. S ^ d“ Rh°ne’ at G—
fSee diagram m Fig. 409, where the loads are in this proportion.92
THE CONSTRUCTOR.
4. E Iastie Line.—By bending an elastic rod of uniform pris-
matic cross section, keepiug it upon the points Elt S, and K?,
the elastic curve may be drawn directly from the rod, using it
as a ruler. For large sizes the rod may be %" to 1 ]i" thick, and
kept under water : for smaller sizes, about thick is sufficient.
Let:
P= the force acting to rotate the shaft;
R — the lever arm at which it acts;
N = the horse power transmitted;
n = the number of revolutions per minute;
</ = the diameter of the shaft;
L = the length of shaft in feet;
= the angle of torsion in degrees;
.S=the fibre stress at the circumference;
G = the modulus of torsion of the material = § of the modu-
lus of elasticity.
We then have for strength :
-PR..............(129)
and for stiffness:
5. Cardiode.—The following method may be used for drawing
the curve directly in the pattern loft. A wooden template S/ K
E C (Fig. 420) is made, in which E C and E S' are straight
edges, and (? S" = C S, and C E — C K. Guide points are
placed at C and E, and the edge C E kept against the point Cf
and the edge S' E against the point K. The point G' of the
Fig. 420.
\2.L 360
27T R
(>3°)
In order to have the same security for the shafting as already
given to journals the value of .S should be only 4 the fibre stress
(see § 5), but in actual practice the stress is taken the same as
for journals, viz.: for wrought iron .S = 8500 lbs., and for cast
iron ^=4250 lbs.
This gives the following results for strength :
For wTrought iron shafts,
</ = 0.091	PR= 3.33 f]—.....(131)
For cast iron shafts,
</=0.114 \3/ P R = 4.20	^............... (132)
n
template will then describe a cardiode curve and by attaching a
pencil point at S' it may be drawn directly for pattern makers’
use.
The most convenient method in practice is to obtain a few
points by (2) or (3), and then join them by a flexible spline or
ruler.
? 143-
Wooden Axees.
For some water-wheels axles of oak are stili used, and these
are made polygonal in section. They are made prismatic, the
diameter being at all points equal to that necessary at the point
of greatest stress, and the methods of attaching journals are
shown in § 102. The diameter may be obtained by multiplying
the diameter for cast iron by 1.55 (the cube root of the ratio of
the modulus of cast iron and oak).	,
This must be the full actual diameter, as it is sometimes
weakened by mortises cut for the arms of the wheel. Should
this give a less diameter than required fortheattachment of the
journals, the diameter at the latter point must be taken for the
whole axle. The choice between iron and wooden axles must
be governed entirely by local reasons of cost and convenience.
Example.—A water-wheel axle with shauks 106.25" long is loaded so that
the journals of cast iron require to be 3%" diameter and 5%" long. Accord-
ing to the formula in § 130 we have
=3 5 \j
106.25
2.625
For corresponding strength in wood, the axle should be at a minimum:
U =|i2 X 1-55 = 18.6".
CHAPTER IX.
,	SHAFTING.
I 144.
Caecueations for Cyeindricae Shafting.
In machine construction those axles which are used to trans-
mit twisting moments are called shafting.
In order to fulfill this purpose two requirements must be met:
first, the ultimate strength must be sufficiently great, and se-
cond, the torsional spring must be kept within proper limits.
In actual practice, shafting is subjected not only to torsional
stresses, but also to bending due to the weight and pressure of
gears, pulleys, levers, etc., which are carried. These latter in-
fluences wdll not be considered at first, and the calculations
made only for round, solid wrought- and cast-iron shafting.
In taking the torsion of shafting into cousideration the great-
est allowable twist in degrees should not be over 0.075° Per foot
in length of shafting, that is #° = 0.075 A which gives for stiff-
ness against torsion:
/=0.3/71=4.7^.................(133)
n
and for cast iron .shafts :
d = °-357 </~P~R = 5.63 A......(134)
n
The quotient of effect — is obtained from the relation to the
n
statical moment P R as follows :
PR^ 33°°QX 12
2 7r
N
n
= 63.020
N
n
(*35)
From these formulae the following table for round wrought
iron shafts has been calculated. An inspection of the table will
show that it is quite possible for a shaft to be strong enough to
resist permanent deformation and yet be so light as to be liable
to spring under its load. For example, a shaft 26 feet long, with
a twisting force of 220 lbs. applied at one end, and acting with a
lever arm of 20/r, gives a turning moment P R = 4400 inch lbs.,
which would require a shaft only 1 h inches diameter (seecolumn
2). This, however, wrould permit far too much torsion, and in
order that the angular deflection should not exceed the limit of
0.075° per foot, a corresponding value of P R in column 4, must
be found, and against it in column 1 will be given the diameter,
in this case about 2§//; which, by comparison with column 2,
gives about five-fold strength.
For short shafts this examiuation of angular deflection is un-
necessary, as for example, in the short lengths between two
gear wheels, for here the value of # will be small enough in any
case. With longer shafts, and in all special constructions, it is
important to consider the angular deflection and keep it within
the given limit.
For shafting of cast iron the same table may be used by tak-
N
ing double the values for P R, or for —
n
For Steel shafts, whoee modulus of resistance is f greater than
wrought iron, the diameters in both cases may be taken as
■v^o.6, that is, 0.84 times that of correspondingly loaded wrought
iron shafts.
Shafting which is subjected to sudden and violent shocks, as
in rolling mills, etc., must be made much stronger than the pre-
ceding formulas require, and these must be classed wfith the
special cases which occur in every branch of construction.
Examble 1. A erane chain carrying a load of 5940 lbs. is operated bv a
drum 01 7.3 inches radius, measured to the centre of the chain: required theTHE CONSTRUCTOR.
93
diameter of the shaft to resist the torsion thus brought upon it. Here PR =
43,362, which by reieretice to column 2 gives a diameter between 3 and 33^
inches, say 3yk". This would require to be somewhat increased for bending
stress, for which see § 150.
Example 2. A turbine delivers 92 horse power to a wrought iron shaft,
making 114 revolutions per minute, and of a length = %% teet. In this case
-— = 0.807, which, by column 2 in thetable, would require about 2>V\ inches
n
diameter. If the deflection is not to exceed 0.0750 per foot, we have, in coi-
umn 4, a value of— = 0.803, which gives a diameter of 4^", and with this
diameter the angle of torsion would be 8.5 X 0 075 = 0.65°. A similar case in
Sractice has a shaft diameter of 5%", which gives a stili smaller angular de-
ection.
1 145-
Wrought Iron Shafting.
d	FOR STRENGTH.		FOR STIFFNESS (Torsional)	
	PR	N n	i PR	>>
I	1,327	0.021	123	0.0019
	2,59*	0.052	3°!	0.0048
'X	4,479	0.071	625	0.0099
\	7,H2	0.114	i,,57	0.0183
2	10,616	0.168	i,975	0.0313
p/i	i5,H5	0.239	3,164	0.0502
2^	20,730	0.329	4,822	0.0765
	27,600	0.43S	7,061	0.1120
3	35,830	0.568	10,000	0.1587
3%	56,890	0.902	18,520	0.2941
4	84,930	1.347	31,600	O.5015
4%	120,900	1.919	50,620	O.8032
5	165,800	2.632	77,160	I.224O
5/4	220,800	3-5°3	11,000	1.7920
6	286,600	4.548	160,000	2.5390
6)4	364,400	5-784	220,300	3.4960
7	455,200	7.222	296,400	4.7040
7)4	559,800	8.883	390,600	6.2000
8	679,400	10.780	505,700	8.0240
8)4	815,000	12.930	644,400	io. 2300
9	967,400	15.350	810,000	12.8600
yA	1,138,000	I8.O5O	982,700	15.6000
10	1,327,000	21.050	1,230,000	19.5900
io)4	1,536,000	24.380	1,501,000	23.8100
11	1,766,000	28.020	1,808,000	28.6800
H>2	2,018,000	32.020	2,159,000	34.2600
12	2,293,000	36.390	2,560,000	40.6200
1146.
Line Shafting.
bearing no definite relation to the actual power. In most cases,
however, the use of the formulas above given for stiffness, with
a slight increase for very long shafts, will give satisfactory re-
sults.
A few examples will serve to illustrate the manner in which
the methods given may be applied, and the remarks which have
been made should be borne in mind in connection with the ap-
plication.
Example 1. The screw shaft of a large war ship is driven by twocylinders,
each exerting a total pressure of 176,000 pounds, on cranks of 21.75" radius,
situated at right augles to each other The shaft is of wrought iron, and
between the crank shaft and the propeller it is 72 feet long, by 15" diameter.
Calculating this for strength by formula (131) we have :
PR = 2y/ 0.5 X 176,000 X 21.75 = 5,412,000
d — 0.091	5,412,000 = 15.98", say 16".
If it is desired that the torsional deflection shall not exceed 0.075° per foot
of length, formula (133) must be used, giving:
d = 0.3 <$/5,412,000 = 14.47".
This is somewhat less than the previous dimensions, and consequently
the deflection will be less than 72 X .075 = 5.40.
Example 2. In the mills at Saltaire there is a cast iron driving shaft mak-
ing 92 revolutions per minute, and transmitting 300 horse power, the diame-
ter being 10 inches. According to formula (134) the diameter would be :
<* = 5.63 \/— = 7-56'',
v 92
so that the actual shaft is § stronger, and the other shafts in the mill are
proportionally heavy.
Example 3. In the rolling mill at Rheinfall is a line of wrought iron shaft-
ing, 223 feet long, transmitting 120 horse power. The speed is 95 revolutions,
giving the ratio -^- = ^^- = 1.263. The diameter for strength, as given
n 95
from column 3 ot the table, would be about 3%". The actual sizes are 3%"
in thejournals. and 4" in the body. The corresponding fibre stresses are
7,397 lbs. in thejournals, and 6,541 Ibs. in the body of the shaft. According
to the formula of Fairbairn, who designed the mills at Saltaire, this shaft
would have been made
d= 7.4 \J— = 8.oo",
n
or nearly eight times stronger than was actually used.
Example 4. In the spinning mill at Logelbach there is a cast iron shaft
8%" diameter, making 27 revolutions per minute and transmitting 140 horse
power by actual measurement The ratio	— 5.19.
n 27
Taking the double value in the table, since the material is cast iron, we
find in column 5, that d—S%. The diameter, for strength only, would be
found by column 3 to be about 7%"'.
Example 5. In the same mill is a line of cast iron shafting, 84 feet long,
transmitting 270 horse power, and making 50 revolutions, hence — = 5.4.
n
Thejournals are diameter, and the body of the shaft is the section
shown in Fig. 413, and its section approximates to that of a cylindrical shaft
of 8%" diameter. For such conditions the table gives in column 3, taking
double the value of—, weget^ = 8". The diameter 6%,f in the joumals
gives a fibre stress of about 5,200 lbs. From the length of the shaft it is ad-
visable to take the diameter for stiffness, which we get from the value cor-
N
responding to 2 — = 10.8 in column 5, which gives d=S%"t which is quite
n
close to the actual dimensions.
In the previous discussion we have assumed that the bending
forces upon shafting might be neglected. As a matter of fact,
this is rarely the case, only occurringwhen the turning moments
are those due to a simple force couple. Nearly all the shafting
used for power transmission is subjected to bending stresses due
to belt pull, pressure of gear teeth, weight of gears and pulleys,
and to take all of these into consideration would make a very
complicated calculation.
In most cases ample strength will be given by taking the
diameters according to the formulas (133) or (134)- As already
shown, these give ample strength, so that any ordinary bending
Stresses are provided for. These give reduced diameters for the
higher speeds, shafting for high speed machinery running at
120, 140 or even 200 revolutions per minute.
First movers run a lower speed and are proportionally heavier,
and the line shafting generally is gradually reduced in diameter
in the successive ascending floors of a building. Such line shaft-
ing is only occasionally made of cast iron, when moderate power
is to be transmitted.
The practice in the proportion of shaft diameters is not alto-
gether consistent. In many cases very high stresses are per-
mitted, as in the case of locomotives, in which stresses of 12,000
to 15,000 lbs. are borne by wrought iron cranked axles; shafts
of screw propeller engines usually carry 7,000 to 8,500 lbs., while
in many mstances the stresses upon line shafting are very light,
when the high rotative speed is taken into consideration. This
is particularly the case in England, the shafting running at
higher speeds with a proportional reducti on in diameter. The
greatest difficulty to be encountered lies in the fact that the
forces are rarely given with sufficient accuracy, the so-called
“nominal ” horse power which a shaft is supposed to transmit
? 147-
DETERMINATION OF THE ANGEE OF TORSION.
In a cylindrical shaft of a diameter d, which transmits a stati-
cal moment P R, throughout its length L, the modulus of tor-
sion of the material being G, we have from No.I, § 14, the angle
of torsion.
_ j?. A
Jp G G a

32 . 360 PR . 12 L __ 360 S 12 L
27r2 . d* . G	7r G d
which for wrought iron, in which G — 11,386,000 gives
i5° = 0.00062
PRL
: 0.0001208 S -
d4	d
For cast iron these values are doubled, giving
= 0.00124
PRL
d*
: O.OOO2436 S--
d
(136)
(r37)
(137)
Here L is taken in feet and is the stress at the point of ap-
plication on the shaft. It will be noticed that the angle # can
be determined very readily when S is known. It must be re-
membered that d and 6* are closely related, and that the value
of d depends upon the value taken for S.
Various applications of twisting moments ma^ be reduced to
a single one for use in the formulas, by classification under some
one of the followdng heads, taking the value for L as foliow
(aee § 13, \ 14);94
THE CONSTRUCTOR.
(a)	. L— the whole length of the shaft, in feet, when the force
is applied at one end and transmitted to the other.
(b)	. L = half the length of the shaft when the twisting forces
are applied over the entire length uniformly.
(c)	. L = one-third the length of the shaft when the twisting
forces diminish uniformly from one end to the other
of the shaft, as in § 14, case III.
(d)	. In general, the distauce of the point of application of a
collected number of twisting forces distributed in
any manner along the length of the shaft, may be
found by multiplying the power applied at each
point (in horse power), by itsdistance from the end
of the shaft, adding the several products together
and dividing by the total horse power transmitted.
The methods may be illustrated by the same examples which
were given in the preceding seetion.
Example The screw propeller shaft given in the previous Example 1,
gives the following data :
,	5 = 8,200 lbs., d = 15", Z, = 72 feet.
According to (137) we have
# = 0.0001218
8200 X 72
15
= 4-75°.
This will be reduced to T70 this value, or 3)^° when either of the cranks is
on the dead centre.
Example 2. The line of shafdng given in Example 3, of the preceding sec-
tion, is made of two diameters in the bearings and in the body, and these
raust be combined. The bearings may be taken at 4 inches long each, and
are 32 in number. We have then
r132.0.33 * 7397\	, /223—10)6541 \~\
Lv 3.875 )	+ \ 4 ;J
= 0.0001208 (20158 4- 348310) = 44^" !!
a deflection which must be very marked, with variable loads, and entirely
inadmissible with fine machinery.
Example 3. If the preceding shaft had been made 8 inches diameter, as by
Fairbairn’s formula, we have for --------= 1.263.
$° = 0.00062
62500 X 1.263 X 223
—w
= 2.67°.
Example 4. In the twine factory at Schaff hausen there is a shaft made of
Bessemer Steel. The length is 489 feet, and it transmits 200 horse power
from the Rhine up the bank and an angle of 230. The diameter is 4fg",
N— 200, n = 12j. This gives S= 4756, and if we take the modulus of elasti-
city of the Steel the same as wrought irou, we have
= 0.00012.8	= 58.34°.
Example 5. A shaft 164 feet long, and of a constant diameter, transmits 70
horse power at 100 revolutions. The power is taken off by a number of ma-
N
chines, ranged at uniform distances apart. According to the table for ■— =
0.7 the diameter should be about 4T4". In determiningthe torsion the value
of L is taken at one-half the length of the line (case b) giving :
$° = 0.00062	— nbont 6%°.
(4
Since the formula is based on an angular deflection of 0.750 per foot, we
might have obtained direct, 82 X o.75 = 6.35°, or nearly the same value.
If in any case the calculated deflection appears too great, the diameter of
the shaft inust be increased, and since the denominator of the equation is
the fourth power of the shaft diameter, a slight increase in its value effects
a marked reduction in the deflection.
Example 6. If the angle in the preceding example is desired to be reduced
to one-half its value, the diameter must be multiplied by 2 or by 1.189,
hence d = 4.25 X 1.189 = 5 inches.
2 148.
JOURNATS FOR SHAFTING. ROUND RODITO SHAFTING.
The journals on shafting are either end journals, and treated
as already shown, or, as in most cases, necked journals, and the
length of bearing made as given in § 92. For line shafting the
special determination of journals is unuecessary. Unless there
is some apparent reason for a special determination of the jour-
nals (as in the case of locomotives), the jourual length l is usu-
ally taken quite large, as faf, 2d, (see § 109 et seq.), care be-
ing taken to iusure proper support of the journals in the hangers.
The Kirkstall Forge Companv, ofLeeds, have produced shaft-
ing which is rolled round and requires no turning. The round
finish is given by the action of plane discs whose geometric axes
are horizontal and parallel, about eight inches apart, and revolve
rapidly. (See \ 195.) The discs are placed so as to act upon
the bar as it leaves the rolls, and are cooled with water, and
their action produces a true cylindrical form to the shafting,
and gives it a highly finished surface, so that it is at once ready
for use without being turned in a lathe. By this process the
modulus of resistance is also increased nearly 20 per cent. over
that of shafting rolled in the ordiuary manner, as shown by tests
by Kirkaldy, and given in the catalogue of the Kirkstall Forge
Company, for the Melbourne Exposition. This feature is not of
as much impoitance as at first appears, although it is of some
vaiue.
The absence of turning is also of advantage, and the increas-
ing use of this shafting is doubtless due to both causes. The
priucipal objection to it lies in the fact that the hard outer skin
canuot be disturbed without affecting the truth of its form.
Keyways cut in it invariably cause springing.
Some of the modern methods of securing pulleys without
cutting keyways may be used to avoid this difficulty. The jour-
nals and wheel seats on this kind of shafting do not require
turning.
2 549.
Combined Sections. Wooden Shafting.
The dimensious for shafting of various combined sections
(tubular, cruciform, fluted) are determined by finding the size
for round shafting of the same material, and then deducing the
dimensious of the desired seetion in the same manner as given
for axles in §§ 138 to 142.
Axles of wood (generally oak) are made of polygonal seetion
described about a circle not less than 1.75 times the diameter of
a cast iron shaft for the same work, this being the fourth root
of the ratio of the moduli of elasticity of the two materials.
Wooden shafting is now seldom used.
2 i5°-
Shafting subjected to Defdection.
Shafting is often loaded in such a manner as to be subjected
to bending stresses, and as already seen, this is the most general
condition in which it is used. Under these circumstances the
combined resistance must be taken into consideration. This is
most conveniently done by assuming an ideal bending moment
(see l 18).
Uet Md be the twisting moment for a given shaft seetion,
Mb be the bending moment for the same seetion ;
then the ideal bending moment combining them both will be:
( Mb),. =yiMb + yiV Md* + Mi .... (139)
This formula may be simplified for numerical calculations by
Poncelet’s theorem, approximately :
When Mb >	Md take (.7/^. =0.975	+ 0.25	. . (140)
and when Md > Mb take (Mb')i =0.625 Mb + 0.6 Md . . (141)
An examination will be made, first by the analytical, and then
by the graphical method.
I. Analytical Method.—The axle or shaft ABC, shown in
Fig. 421, carries a gear wheel R at Cf which acts tangentially to
rotate the shaft with a moment Md = Q R, and also acts to bend
the shaft with a force whose reactions are parallel to Q, and are
P\ = S- at A, and P2 = —at B. The greatest stress is
a-\- s	a-\- s
at Cy for there both bending moments are at their maximum
p p
Mb — —- = —-y hence calculation should be made for this point.
Example.—Let Q = 5500 lbs. R = 11^", a = 19%", s = 78%", then
P\ = Q= 0 8 Q = 4400 lbs.
/o = -q—- Q — 0.2 Q — 1100 lbs.
98.50 **	**
Also	Md = 5500 X n.75 = 64,625.
Mb = 4400 X 19-75 = 86,900.
Hence	Mb	and formula (140) is used.
We have {Mb)i = 0.975 X 86,900 + 0.25 X 64,625 = 84,727 + 13,656 =98,383" lbs.
From this the diameter at C can be calculated. If the shaft is of cast iron
with cruciform seetion, we have for the diameter Z>,

and taking N = 4250 we have
D =
J
98,383 X 32
4250 n
= 6K".
The journal at A is found in the table of §91, column 4, to be about 2^".
For the neck journal at B, we have from the table of| 145, taking thedouble
value for cast iron, d2 = 4^".THE CONSTRUCTOR.
Graphical Method.—The same example may be solved graphi-
cally. In Fig. 422, with a horizontal closing line, construet the
link polygon a b c, for the bending moments, and the force
polygon a 10, giving the forces Px and P2t and also a c c', the
surface of moments for the shank A C.
The moment Md is yet to be determined. In the force poly-
gon with a distance R from the pole O, draw a vertical ordinate;
this will be Md. Eay its value off at c' clf and bbx, and Y of
these values give cf c0 b0 b for the parallelogram of torsion for
the shank C B.
The combination of the bending and twisting moments may
then be made by formula (139)- Make cc2—y%cc' and join c.
b, then at any point of the polygon, as for example at f the
distance ff2 = Y% ff. Now transfer c' cQ to a b, at c' cQ'; then
will the hypotenuse of the triangle c2 c' cQ' divided by c2 c0' =
^(H c c') 2 + ( H c\ c')2y all<^ sum	^o/== c C2 + c2 c3 the
desired moment (Mb)i for the point C. In the same manner we
obtain ff%-\-TfY—fT + /2/3the moment (Mb)i for the point
F. The line c3f3 b0 is a curve (hyperbola) wdiich may be taken
approximately with sufficient accuracy as a straight line c3 bo2.
The various dimensions may be obtained from the polygon a c
b bQ c3 c' in a similar manner as shown in the discussion of axles.
Other discussions of this subject will be given when consid-
ering rock shafts and crank axles.
CHAPTER X.
CO UPLINGS.
8 I5I-
Various Kinds of Coupuings.
The devices by means of which the different lengths of shaft-
ing are connected together so that the motion may be trans-
mitted from one piece to the next, are called couplings.
They may be classed as follows:
1.	Rigid Couplings.
2.	Flexible Couplings.
3.	Releasing, or Clutch Couplings.
The first class includes the various forms of coupling for line
shafting and the like, in which the coupling and the coupled
portions have the same geometric axis. Flexible couplings are
those which permit more or less change in the relative position
of the coupled shafts; while clutch couplings are constructed so
as to be thrown in and out of engagement, usually when the
parts are in motion. These three classes are all shown in vari-
ous forms in the following examples :
8 !52-
/. Rigid Couplings.
Rigid couplings may be made either in a single piece, or in
several parts. Of the first sort is the so-called Muff •Couplings,
Fig. 423. The muff is fitted over both pieces of shafting, and
a single key binds the parts all firmiy together.
In giving the proportions of the various parts of the followring
couplings, we may take for a unit or modulus the thickness d of
the hub, making its value equal to :
d	2 "
<5 = —+ 4......................(142)
3	16
dTbeing the diameter of the shaft, wffiether of wrought or cast
iron. The dimensions of the key may be taken as given in \ 68,
Formula (71) for torsion keys.
95
More recently, in exposed situations, the projecting end of the
key is covered with a cap, in order to avoid accidents.
The form of coupling shown in Figs. 189 and 190, \ 69, looks
very practical, but the test of prolonged use will be necessary to
demonstrate its merits.
Fig. 423-
The simplest two-part coupling is the wrell-known piate coup-
ling, Fig. 424, and its form permits the nuts and oolt-heads to
be kept below the projecting flanges, and thus out of the way.
The number of bolts in a piate coupling i = 0.8 d -f- 2. The
diameter d of the bolts should be 0.125 d 4- tV7* which gives a
strength pioportional to that of a shaft calculated by formula
(133)> or if d is determined from formula (133) the bolt will bc
strong enough *

Fig. 424.
Piate couplings are extensively used in Eugland and Germany,
although they are being superseded by later forms. Their
strength has caused them to be used for coupling the lengths
of screw propeller shafts, and in this case the plates are forged
on the shafts, thus dispensing with the use of a key, Fig. 425.
This form was introduced by Langdon in 1852, and is in general
use, 4 to 6 bolts being used.
Examples: The following cases will serve to give the proportio», of such
piate couplings in exeeuted designs.
Jason, James Watt & Co., d = 12", D = 24", d\ = 3", b = 6", i = 4.
Warrior, John Penn & Son, d = 17", D = 37'', d\ — 4'', b — 10, i = 6„
Vessel by Ravenhill & Hodgson, d — 12", D = 25, d\ = 3", b = 6'', i = 4.
Fig. 426 shows a clamp coupling divided iuto tw7o parts longi-
tudinally. This form is provided with two keys, and the man-
ner in which it is bolted together. If it is desired to clamp the
Fig. 426
shafts together endwise, the small circumferential grooves and
lips may be used as shown. Such grooves may be used in depth
equal to 0.01 d -j- tV//* rnay be omitted where endlong clarup-
ing is unnecessary. If lock nuts are used on the bolts the main
* The dimensions in Fig. 424 are correct for English measurements, except
the bolt diameter, which is as given above, and the distance from hub to in-
side of flange, which should be 2.6 d 4- H".96
THE CONSTRUCTOR.
nuts may be counter-sunk as shown in the illustration. The
nuinber of bolts is = 2, 4 or 6, rarely more, and of diameter as
follows:
*i-
2,	4,	6 or more.
d , 3 " d	11" d	5"
6 + 8	7	32 ’ 8	16 "
Example : For a shaft 2§" diameter, with a coupling fitted with two bolts
the diameter dx = 0.77", say tg", for four bolts d\ = 0.72", say	forsix bolts
di = 0.67, say H".
This form of coupling has beeu made with bolts with differ-
ential thread passing through both parts and giving increased
clamping force.*
In England Butler’s cone coupling has been used, and was
designed for use with the cold rolled shafting described in § 148.
It is similar in construction to Sellers’, the three bolts being re-
placed by a single concentric screw thread and nut at each end.
The key which Sellers uses is omitted in Butler’s coupling, the
shafts being held only by the clamping action of the cones.
In the United States Cresson’s coupling is also much used.
Its construction is shown in Fig. 429. The clamping surfaces
are cast in one with the outer shell, and forced upon the shafts
by means of the taperiug screws. This coupling possesses the
same advantage as does Sellers’, in being adapted to shafts of
slightly unequal diameters.
Fig. 427.
The cone coupling shown in Fig. 427 is the design of the
author, and is a modification of the preceding form. The keys
are cast in one with the halves of the inner cone, and are planed
to fit the keyways in the shafts. The cone is made wTith a taper
of on a side, which will hold the parts securely when driven
on, without any other fastening. If there is much vibration,
however, it is advisable to have a screw thread cut on the inner
cones as shown, and the outer shell tightened by a spanner. I11
most ordinary cases the screw may be omitted, and a small Steel
countersunk set screw tapped into each side of the shell to
clamp the inner cone. If endless motion need not be considered
the circumferential grooves may be omitted.
With couplings for shafts larger than 2the bearing sur-
faces should be recessed to reduce the amount of finishing.
//. Flexible Coupiings.
\ »53-
Various Kinds of Fbexibee Coupiings.
Couplings which, permit the shafts to change their positions
may be required to meet three different conditions. The motion
of the shafts may be :
(a) Ivengthwise, or in the direction of the axis.
(£) Crosswise.
(c) Angular, the shafts being inclined to each other.
In some cases two, or even all three, of these conditions may
be present. In the first place the axes of the two shafts coin-
eide; in the second case they are parallel to each other ; in the
third case they intersect, while in the combination of (£) and (c)
the axes pass each other. All these cases occur in actual prae-
tice and meet with useful applications.
i iS4-
COUPUNGS FOR LFNGTHWISE AND PARAEEEE MOTIONS.
Endlong motion of the coupled shafts may be provided for by
giving various prismatic fornis to the parts of the couplings.
As an example, Sharp’s Coupling, Fig. 430, may serve. This
permits but slight movement lengthwise, and also a little angu-
lar displacement, and is therefore suitable for positions where

i..........AU....
Fig. 430.
In America Sellers has introduced a clamp coupling in which
•two cones are opposed to each other and drawn together by
three bolts, the whole being inclosed in a cyliudrical shell bored
out to fit the cones as shown in Fig. 428. The cones are cut
through on one side so that they are compressed by the action
the bearings cannot be accurately placed. In some recent ex-
amples of this form of coupling, one half is first made and the
second cast upon it, and in this case the outer recess is omitted.
I11 some French screw steamships in which the screw is
arranged to be lifted, one of the shaft couplings is arranged so
that sufficient endlong motion may be obtained to permit the
end of the shaft to be withdrawn from the hub of the screw7. i
Fig. 431.
of the bolts. A key is let into the cones and shafts diametri-
cally opposite the cut in the cones. An especial advantage
which results from the double cone construction lies in the fact
that it is not necessary for the two shafts to be exactly the same
diameter/j*
* Ruggles’ Coupling, Pract. Mech.Jour., 1866, p. 185. In Fig. 426 there are
two dimensions which require transforming: for 20 + 2.6 d, use l” + 2.6 d,
and for 10 -j- 1.3 d use f" ■+■ 1.3 d.
f Among other tests the Sellers Coupling stood the following: Two shafts
io feet long were fitted in three hangers, the middle hanger next the coup-
ling being set 1^" out of line, and after several weeks’ time, at 250 revolutione
per minute, the coupling remained intact.
For shafts in which the displacement is crosswise, the axes of
the shafts remaining parallel, Oldhanrs Coupling, Fig. 431, is
most applicable. This consists of two end pieces and one inter-
mediate piece. The latter has a prismatic tongue upon each
side, the two being placed at 90° with each other, and fitting
into corresponding grooves in the end plates, which latter are
keyed on to the shafts. If the two axes of the shafts coincide
t Among various interesting examples of such couplings may be 'men-
tioned those described in ArmengancTs “Vignoledes Mecaniciens,” Paris*
1863, Piate 9; also Eedieu, App. a vapeur de navigation, Paris, 1862, and Or-
tolan, Mach. & vapeur marines, Paris, 1859.THE CONSTRUCTOR.
97
at O, the tongues and grooves have no sliding action upon each
other. ^ If one of the axes is moved parallel to itself, say to P’
the middle of the intermediate piece will describe a circle
O Q P Q' of the diameter O P— the distance between the axes,
making two revolutions for every revolution of the shafts, the
other points of the disc describe cardiode paths. The velocity
ratio remains constant.
Another form of coupling for the purpose consists of two
cranks connected by a short drag-link to permit the necessary
movement. This is frequently used in connecting engine shafts.
? »55-
JOINTED COUPIJNGS.
The best known of ali the flexible couplings is the Universal
joint, known also as Hooke's Coupling, and also as Cardan s
Coupling.* This form of coupling permits the connection of
inclined axes within certain limits. It consists broadly of two
end pieces, and a middle piece, the latter containing two pairs
of journals placed at right angles with each other in the form
of a cross, each pair fitting into journals on one and the other
of the end pieces respectively.
The rate of transmission of motion is not uniform, and is
dependent upon the angle of inclination a of the two shafts,
the angular rotations o and of the driving and driven shafts
being expressed in the followdng ratio :
tan
------ — cos a
tan o
(>43)
which gives a periodical variation whose period is i8o°. The
following table gives the values of fi^r successive values of w,
for various angles a :
(J	II M O 0	20°	GJ 0 0	0 0 ■''t
3°°	29° 38'	28° 29'	26° 34'	230 51'
45°	44° 34'	430 12'	40° 54'	37° 27'
6o°	59° 34/	58° 26'	56° 22'	53° °4/
90°	9°°	90°	90°	900
120°	120° 2b/	121° 34'	1230 38'	126° 56'
135°	135° 26'	136° 4S/	1390 o6/	1420 33/
150°	I5O0 22	151° 3i'	1530 26'	156° OI°
1800	0 O 00	1800	1800	1800
For small values of a the variation is unimportant. For the
angular velocities a, and ult we have the relation :
___	cos a
o) i — sin2 o) sin2 a
(144)
which gives a maximum-------- and a minimum cos a. These
0	cos a
variations in velocity may be neglected when the moving masses
are inconsiderable and the angle a is small f
The detailed construction of the coupling admits of great
variations. Fig. 432 shows a form with cast-iron end pieces and
wrought iron middle piece. The relation — is varied also. The
Journal diameter d2 is determined by the methods already given,
and from the moment of rotation {PP) the journal pressure P2
\/ ( p r>\
may be taken with sufficient accuracy as = ———. The dis-
tance a should be made greater as the angle a is increased, a
being made quite in the illustration. The joints in the boxes
* If not the original inventor of the Universal Joint, the Italian, Cardan,
wasthe firstto describe it (1501-1576), and the Enghshman, Hooke (1635-1702),
first applied it for the transmission of rotary motion.
f In the table the values of to are so placed that when to = O, the cross
journals of the coupling lie in the plane of the shafts.
should be made in the plane of the shafts, not at right angles
to them, in order to provide for the wear.
Universal joints are used to good advantage in screw propeller
shafts in order to provide for the flexure due to the elasticity of
the hull of the vessel. In such cases two universal joints are
used on a shaft. A coupling for such Service is shown in Fig.
433. Here all these pieces are forgings, one end piece being
forged solid with the shaft. The middle piece is formed of a
double ring, the bearings being held between the two parts,
Fig. 433-
while the journals are secured to the end pieces. No provision
is made for taking up the wear upon the journal boxes, as the
angle a is so small that the wear is very slight. The length /2
of the journals need not be great, about 1 to 1.25 d2y and since
P can be kept small, the dimensions of the entire coupling may
be kept within reasonable limits.
Another form of universal joint is shown in Fig. 434. Here
the cross journals are made in the form of through bolts, pass-
ing through both end and middle pieces. This requires a slight
modification in the form, since the axes of the two bolts cannot
intersect each other. A slight error in motion follows, causing
Fig. 434»
a very small endlong motion with each revolution in the coupled
shafts, but this is generally insignificant. This form is suitable
for agricultural machinery, horse powers, etc. In the illustra-
tion a proportional scale is given. The modulus is as before,
<5 = i d -(- yV'. The pole P is therefore taken, so that 4 d +
— o, or so that d — — TV/.
The irregularity in motion shown in formula (144) is generally
of little consequence, and need not be considered except in
cases requiring geometrical exactness, as in the connections for
large tower clocks, or in cases where large masses are drawn at
high velocities, as in threshing tnachines. The variation can
be obviated by the use of a double universal joint, which con-
sists of two simple couplings. If, Fig. 435 a, the driving shaft
A is coupled to B by means of a short intermediate shaft Cy
the connections being made by two similar couplings of the
same angles, then the motion of A will be transmitted to B
without variation. In this case the driven shaft may be given
several poeitions with regard to A, being placed at B, making
the angle 2 a, or at B', parallel to A, or at A>// on the surface-
of a cone of half the angle a, which is made with the inter-
mediate axis C. The two universal joints are similarly placed
when the cross axes belonging respectively to A and B lie in
the same plane as the shaft C at the same time. In the posi-
tions B and B' all three shafts lie in the same plane, but not in
position B", In the last case A and B intersect.98
THE CONSTRUCTOR.
Cx and C2 connected on both sides. Clemens has used this form
with the angle 2 a = 90°. The doubling of the parts has the
Fig. 435.
If the cross axes are not placed similarly, but, for example,
%at 90°, as in Fig. 435 b, the variations of motion are increased,
and we have.
tan = tan u cos2 a, in which u and ox stand for A and B.
If a — 30°, we have for u — 450, tan = (b >/ 3 )2 = 0.75, hence
Wj = 36° 54/ instead of 40° 54', as in the table above.
A concealed forni of universal joiut is that used in rolling
mills, the cross being formed upon the end of the roller.
Fig. 436 shows a form of link coupling designed by the
author. A is the driving, and B the driven shaft, the arm C
being jointed at 3 by journals at right angles to B9 while at 2 it
slides in a bearing rigidly attached to A. The axis B makes
with A an angle a, and the piece 3 C 2 makes with A the
angle 90° — /3. For the relation between the angular rotation
ox and of the shafts A and Bf which are supposed to revolve
in fixed bearings, we have :
or
tan ox = tan «4
C
cos a
sin a tan /3 \ '
COS J
tan
sin Wj
cos a cos o1 — sin a tan /3
■ ■ (I4S)
In this case the transmission of motion is much more irregtilar
than with the universal joint, since the angular velocity ratio
between
1	and	C°S °
cos a =F sin a tan (3	1 -j- sin2 a tan /3
fluctuates. This coupling is really only a modification of the
universal joint, inclined to such an extent that the fork of the
shaft A stands at right angles to the axis. (Compare formula
(144) with (145)). By combining two such couplings symmetri-
cally with each other, as in Fig. 437, the motion will be uni-
formly transmitted. The two sleeves at 3 and 4 are formed in
Fig. 437-
one piece, their axes making the angle 2 /3 with each other. In
practice it is better to make these parts in the form of journals
and place the sleeves in the Cx and C2, Fig. 438. These parts
are also prolonged beyond the shafts in order to counterbalance
the weight. The pieces 3 and 4 can be bolted firmly together,
since their relative position to each other is constant. It must
be observed that a must not exceed 90 — /3, otherwise a dead
point wTill occur.
The parts 3 and 4 may also be connected by ball joints 3' 4/,
Fig. 439, in which case the device becomes the coupling of
Clemens.* Here the counter-weights are omitted, and the parts
* Clemens’ Angular Shaft Coupling, U. S. Patent, Nov. 10, 1869.
objection that it requires inore accurate fitting of the parts than
where only one side is connected. If the axis B is placed par-
allel to A, as at B7 in Fig. 437, the rate of motion will be very
irregular, for a=/3 — 30° as in the illustration, the velocity
ratio will vary between l/2 and 2.
i
Fig. 440.
In many screw vessels a simple form of flexible coupling is
used, suited for slight angular variations. In Fig. 440 this is
showm, and it will be seen to give slight flexibility similar to the
universal joint, and sufficient for many cases. A bearing should
be placed back of the coupling on each shaft.
III. Clutch Couplings.
2 156.
Toothed Ceutch Coupungs.
Couplings of this form may be distinguished by their method
of engagement, the clutch surfaces entering in and out of en-
gagement axially, radially or inclined.
Fig. 441.
The oldest form of clutch coupling, and one of the most
widely used, is that shown in Fig. 441. Here the engagement is
axial. The modulus for the proportions is the same as before,
6 = d -[-	; and an approximation to the number of teeth
may be given by making z = 1 -\- 0.6 d. The clutch is thrown
in and out of gear by a lever which works in the groove in the
portion of the clutch on B. Examples of suitable lever forks
are shown in Fig. 442.
Fig. 443-
Various forms of clutch teeth are used. The forms in general
use are given in Fig. 443. The first form is adapted for motion
in either direction, but can only be operated when moving
slowly. The second form is more readily thrown into action,THE CONSTRUCTOR.
99
i)ut is adapted to transmit motion only in one direction. The
driving faces are inclined very slightly, from the normal to the
direction of motion, the angle not being enough to cause any
tendency to disengagement. In the third form the teeth are
Fig. 442.
more blunt in shape at the point, which adds to their strength
against breakage when subjected to shock. The last form is a
combination of the preceding varieties, and like the first, may
be driven backward. In spinning machinery, light couplings
with many fine teeth are used and operated at high speeds. In
5ome screw vessels in which there is no provision for raising the
screw, it is desirable to disconnect it when proceeding under
sail alone, and some form of clutch coupling is used. A very
simple form is the so-called “ cheese coupling,” used in English
vessels, Fig. 444. The hub of the propeller is provided with a
bearing on each side, and formed with a T projection fitting into
a corresponding recess in the heavy flange (or cheese) on
the shaft A. The propeller blades are secured to the hub as
already shown (Fig. 191).
i 157*
Friction Ci<utches.
Couplings in which one portion transmits motion to the other
portion by means of friction, are often especially applicable,
since by the mere removal of the frictional contact the parts are
disconnected, and when they are thrown into contact the driven
Fig. 445-
portion is put into motion gradually. By making friction coup-
lings of large diameter, they may be used to transmit propor-
tionally great rotative moments. In Fig. 445 is shown a friction
coupling used by Ramsbottom in rolling mill work * The part
A is firmly clamped between the wood-lined surfaces of B ; but
the parts may be arranged so as to slip if undue resistatrce is
encountered, thus making it a safety coupling. The modulus
as before is S = — +	.
3	16
*See Engineer, 1866, January, p. 44; also Genie Industrielle, Vol. 32, p. 101;
and an older form of this coupling is shown in Salzenberg, Vortr. p. 173.
Cone couplings are used also, in many forms. In the example
shown in Fig. 446 the driven portion A of the coupling carries
a gear wheel shown in the dotted lines, to which motion is to
be transmitted from the shaft. The two parts are forced into
engagement by the screw and hand wheel b. If the parts are
so arranged that the motion of the hand wheel b is in the same
direction as the rotation of the part B, when the latter is throwm
into engagement, it is only necessary to hold the wheel b sta-
tionary in order to throw the clutch out of gear. From the mean
radius R of the cone surface, and the angle of taper a, we have
for an axial pressure Q, for any circumferential force P:
^	sina ,	\ (PR) f sina .	\	.	„
Q = P[—^ + cosaJ = — 'f—^cosa) • (*46)
in which f is the coefficient of friction between the cone surfaces.
and {PR) is the statical moment tending to rotate the shaft.

Fig. 446.
The angle a should not be taken at less than io°, in order
the parts may not become wedged together; for iron on itonyf
may be taken at 0.15. In order to keep both P and Q as small
as possible, R should be made large, say between 3 and 6 d.
The relative motion of the screw and hand wheel is of course
dependent upon the radius of the wheel b} and upon the pitch 5
of the thread.
Example: A wrought iron shaft of a diameter d — 2 inches, making 50 rev-
olutionsper minute, would transmit, according to the table of g 145.0.0313 X
50 = 1.5 H. P., or have a statical moment PR = 1975 inch pounds. If the ra-
dius of the friction cone couple is equal to 5 d, or 10 inches, we have, accord-
ing to (147) Q =	^ -S-y — -f cos a ^. If a = io°, andy=o.i5 we have
Q=*9T- 5	+ 0.9848 ) = 423 lb*.
r Suppose the hand wheel to have a radius of 4 inches and the screw a pitch
of yp, we then have for the force tobe exertedon the rimof the hand wheel:
_	0.25 X 423 o . 1T._
,	Jr X 4	8'4lb‘-
For the transmission of moderate force the cone coupling, or
some of its various modifications, has very generally been used.*
Fig. 447.
Instead of a single pair of external and internal cones, a num-
ber of small elements may be employed. This form is shown
in Fig. 447. The general caleulations are made asabove, except
that the lever arm R of the friction must be reduced, and may
be taken with sufficient accuracy at a point distant from the
outer circumference equal toone-third the width of the grooved
frictional surface. The operating lever in this case need make
but very little movement, and the arrangement of a fork mounted
on an eccentric bearing, as shown in the illustration, may be
conveniently adopted.|
When a cone coupling is intended to be used for the trans-
mission of large forces, the apparatus for pressing the parts to-
gether may sometimes be so arranged that it is mounted on
* Many such applications will be found in the description of the Suez Ca-
nal; see Armengaud, Pubi. Ind., Vol. 17, Pl. 9.
fSee Armengaud, Vignole des Mecaniciens, Piate II.IGO
THE CONSTRUCTOR.
the shaft, revolving with it, without creating so much pressure
against the bearing. The fork and grooved collar shown in Fig.
447 is not suitable for heavy clutches on account of the excessive
collar friction, hence the pressure is better applied by means of
a screw mounted on one of the shafts, and this may be conve-
niently arrauged so as to draw both shafts firmly together. Sup-
pose the shaft to be 4 inches in diameter, we have from the pre-
ceding, R — 6 d— 24'', and an axial pressure =	-j-
0.9848^ = 2818 lbs. This endlong pressure, instead of creating
hurtful collar resistance, may be utilized by arranging the parts
as shown in Fig. 448, which shows a friction clutch coupling of
Fig. 448.
the author’s design. As shown in the section, the part A ex-
tends over the part B, and both parts are drawn together by the
action of the screw and hand wheel. The only modification in
the screw gear is that the screw is made large enough to permit
the shaft to be passed through it, the thread being thus cut upon
the hub of the part A. This coupling runs very smoothly. The
concentric channels should be arranged with clearance at the
bottoms of the grooves, as shown in the section, to provide for
fitting and wear. The modulus for the parts is the same as be-
fore, viz., J	.
3	16
In Fig. 449 is shown the cylinder friction clutch of Koechlin.
In this case the clutch movement takes place radially. The part
A is a hollow cylinder in which three internal clamp pieces are
Fig. 449.
fitted, each being provided with a bronze shoe. These are thrown
in and out of action by means of a sliding collar B', which
operates right and left hand screws by means of the lever b.
The clamps slide in radial grooves and the details are fully shown
in the illustration. The nuts for the right and left hand screws
can be closely adjusted and clamped by set screws, so that a
radial movement of less than -fa" is sufficient. There is no
danger of wedging the parts fast in this form of clutch, as may
be the case in cone clutches, as the elastic reaction of the cylin-
der assists in the direction to release the parts. At the same
time the screws prevent the coupling from releasing itself and
the axial pressure Q, upon the collar B'f can be transmitted so
that the screws need not have too quick a pitch.
If 5 is the pitch, b the length of lever a.rm,f the coefficient of
friction of the clamping pieces, we have for the transmission of
a given moment {PR), neglecting the friction of the screws,
2S P ~__________ s PR
27r b f	nb f R
• 047)
which gives a very small value for Q.
If the parts are so arranged that B is the driven part, there
will be no collar friction at B'y when the coupling is not in ac-
tion. When the shaft is vertical, a weight may be used instead
of a collar and lever, and by gradually lowering it the apparatus
may be started with very little shock. The first clutch made by
Koechlin w7as designed for the transmission of 30 H. P.* The
above value corresponds to a minimum value of R. The modu-
d
lus is the same as before : b —-U ^—
3	16
A very excellent form of this coupling was designed by Bod-
mer, independently of Koechlin,f and a simiiar arrangemen*
has been adapted to mill gearing with success.j
Fig. 450-
Cylinder couplings in which the clamping pieces are operated
by togglejoints are also made. An example is shown in Fig.
450, which is a clutch by Fossey, as applied to mintmachinery.§
This is a very compact design and is arranged with four clamps,
which have no bronze shoes. The toggle links are as wide as
the clamps and are fitted with half-journals to transmit the pres-
sure outwards, while to draw the clamps back, light through
bolts are used (see §95). If the toggle links make with the axis
an angle 90° -}- a, we have for the axial collar-pressure :
Ptan a_ P R tan a
_7	_
. . (148)
The angle a may be taken very small, since there is no danger
of clamping. The value may be as small as a = 20, or even i°.
Fig. 45r-
_	^	, Q 0.0*	1
For a = 13/° we have —------------= —
P 0.15	5
Another form of cylinder coupling using toggle levers, has
* Bulletin von Miilhausen, 1854, P- J38.
f See Fairbairn, Mills and Millwork, Vol. II., p. 92.
t See Uhland’s Prakt. Masch Konstrukteur, 1869, p. 97.
\ See Armengaud’s Pubi. Industrielle, Vol. XVII, Pl. 10.THE CONSTRUCTOR.
IOI
been designed by Garand.* Jackson uses hydraulic pressure to
force the clamps into contact.f Dohmen-Leblanc uses springs to
throw the toggles out of action.J Schurmann uses, instead of
the separate clamps, a ring, which is compressed externally; g
Napier also uses a ring, expanded from within. || Becker ar-
rauges the clamp blocks to be operated by centrifugal force.^f
These are only a few of various modifications of the cylinder
coupling.
A form of axial friction coupling which acts with very slight
pressure is the Weston clutch, made by Tangye.** This acts
upon the principle of multiple piate friction t^see § ioi), as is
shown in Fig. 451.
The wooden discs are engaged with the case, and the iron ones
with the shaft. In the form shown the plates are pressed to-
Fig. 452.
gether by the springs, and released by drawing back the collar
B and releasing the spring pressure. A larger example of
Weston’s clutch is shown in Fig. 452. A is a winding drum, B
the shaft driven by the engine. The outer disc C, and the inner
discs of the coupling are held apart by spiral springs, as shown
at a. A light pull on the cord c holds the drum stationary ; a
strong pull engages the clutch for wdnding; if the cord is left
slack the load on the drum runs backward.
? 158.
Automatic Coupeings.
When power is transmitted to a shaft from two different
sources, as from two independent engines, it is desirable to have
one or both of them connected by a coupling which will auto-
matically release or engage with the shaft, according to the dis-
tribution of work. If one motor tends to overrun it will then
be given more of the work, and so the resistance will be equal-
ized. Such a device is the coupling of Pcuyer-Quertier, gener-
ally known as Pouyer’s Coupling.
Fig. 453-
This is shown in Fig. 453. In this case the parts are so dis-
posed that the part A, which is driven by one source of motive
power, is loose on the shaft B. This part A may have gear
teeth upon its circumference, for example, or may have a gear
wheel mounted upon its hub, as shown by the dotted lines ; the
hub being bushed with bronze. Upon the shaft B a ratchet
wheel is keyed ; the pawls a, ay being upon Ay engage with the
* Dingler’s Polyt. Journal, Vol. 149, p. 22.
fDingler’s Polyt. Journal, Vol. 153, p. 251.
t German Patent, 16952.
3Zeitschr. d. Vereins d. Ing., Vol. V, 1861, p. 301.
f Kngineer, 1868, July, p. 64.
f German Patent, 7205.
**In the U. S. by the Yale & Towne Mfg. Co.!
teeth when A drives By but if B gains upon Ay or A stops while
B continues to move, the pawls are thrown out of action. The
direction of motion is shown by the arrow. The pawls are re-
leased by the action of th* friction bands bx and b21 which are
carried forward by friction upon By whenever B gains upon Ay \
the levers b throwing the pawls a out of gear. As soon as the
limit of travel of the levers by b, is reached, the friction bands
bly b2 slip upon B, being able to move no faster than A. When
the speed of A increases and gains upon By the pawls are again
thrown into gear and A is automatically coupled to the shaft.
In order that the pawls may not bind upon the ratchet teeth in
releasing it is necessary that the angle y, which the pawl makes
with the face of the ratchet tooth, must be less than the com-
plement of the angle of friction ; in this case y = 6o°. Pouyer
uses only one friction band and makes both pawdsengage at the
same time. In the illustration the ratchet wheel is made with an
odd number of teeth (13), and the pawls are placed so that a
movement of only yi the pitch will cause the parts to become
engaged. The above proportion of the angle of the teeth is
of importance, as otherwise the points of the teeth are apt to be
broken. The pawls also should be of hardened steel.
Fig. 454-
In Germany Uhlhom’s Coupling is used for similar Service, as
shown in Fig. 454. Here A is the part connected to the motor,
and B is fast to the driven shaft. A is an internal ratchet wheel
into w^hich the pawls b enter. The springs a serve to insure the
entrance of the pawls into the teeth, which engagement continues
so long as k drives B. If the speed of A is retarded, the pawls
are retractea as shown in the lower part of the figure. In this
case the springs act to keep them out of gear, being the reverse
action to that of an ordinary ratchet gear.
The pawls are fitted with half-journals (see g 95), and are held
in place by a piate ring, as shown. Uhlhorn originally used
only twTo ratchet teeth in Ay but increased the number afterwards
to four, so that the parts would engage in a movement of one-
fourth a revolution. It is better to use an odd number, as three,
and by proper spacing of the pawls thegreatest play wdllbe one-
half a space, or one-sixth a revolution with three teeth, as in
the case of Pouyer’s Couplings. B may be the driving part in-
stead of the driven, but in that case the direction of the arrow
must be reversed.
CHAPTER XI.
SIMPLE LEVERS.
1159-
JOURNAUS FOR LFVERS.
In machine design a simple lever, or rocker arm, is a lever
arm 'which is mounted upon an axle or shaft, at the end, about
which it moves, and carries a journal upon the other end. For
the proportion of the journal see Chapter V. The forms which
EiG. 455-
may be given to suchjournals are shown in Fig. 455, and are
single overhung, double, or forked. The manner of securing
the pin in the hub or the lever is most important. The pin
should not be driven in up to the shoulder on the taper, but
sufficient space left to insure that the fit is tight in the taper.
This clearance is shown plainly in the figure. The same resuit102
THE CONSTRUCTOR.
may be attained by counter-sinking the collar into the hub on
the lever. In the case of double overhanging pins, care should
be taken that the load is equally divided between the two sides,
so that the pressure upon each pin shall be equal to x/z P. In
the fork-ended lever the fit on both ends of the pin should be
portions of the same cone.
Ex ample i. For P = 4400 lbs., we have from the table in §90 for alternating
pressure and wrought iron journal, the diameter d — 0.027 ^/4400 = 1.8", and
the length the same. For Steel, we have d — 0.024 \/ 44°° — 1.6", and the
length / = 1.3 X 1.6 = 2.08". For a forked lever, a wrought iron pin with the
same load the diameter, according to (9S) would be d — 0.0185 y/ 44°° — 1.2",
and the length /= 3.5 X 1.2" = 4-i 2 3".
All levers are not subjected to alternating pressure, but have
the pressure constantly in one direction, as for example, the
beatns of single-acting pumping engines, etc. In such cases
larger journals are needed.
Example 2. A wrought iron journal for a forked lever, under constant
pressure of 4400 lbs., according to formula (98;, should have a diameter d =
0.0212 v/ 4400= 1.4", and length l— 3d— 4.2''. If the material had been cast
iron we should have had d = 0.29 \/4400 = 1.92", say 2", and / = 6". For steel
we have d — 0.0185 y/ 4400 = 1.2", l— 4 d = 4.8".
I'IG. 456.
i 160.
Cast Iron Rock Arms.
Rock arms may be either of cast or wrought iron. The hubs
for wrought iron arms are given in the preceding illustrations,
and in Fig. 456 are given some proportions for the various parts
of cast iron arms. A fork-ended arm is shown below, among
the walking beatus, or if the fork hub is on the main axle, see
the rules already given under Axles, Chapter VIII.
3 161.
Rock Arm Shafts.
The axle upon which a rock arm works is usually subject both
to bending and torsional stresses. The methods of calculation
for all important cases are given in Chapters Vili and IX. The
case which occurs most commonly is the overhung rock arm at
the end of a shaft, and this is here given a special examination.
If we have a, the distance between two planes normal to the
axis, and passing through the middle of the pin and the middle
of the bearing on the shaft, Fig. 457, there is an ideal bending
moment with a lever arm R, acting upon the bearing of the
shaft, for a load P on the pin equal to
= Pa' = p(^a +	• • • • (*S°)
See 3 150.
The lever arm a' is readily obtained graphically, as is shown
in the illusiration. For its numerical determination we have,
if R> a
a' = 0.625 a -f- 0.6 R
and if R < a
a' = 0.957 a + 0.25 R
The lever hub must be made strong enough when the shaft is
only subject to torsion, or when it is also subject to bending.
Fig. 457-
For wrought iron shafts wrought iron levers should be used, and
for cast iron shafts cast iron levers.
Let;
w — thickness of metal of hub,
a = length of hub,
D — the shaft diameter for the statical moment PR of a
lever of the same resistance, see (133) and (134).
for
w	1	1	1
	2	25	3
w ~D	= 0.45	0.42	0.40
• («52)
If a lever is to be fitted to a shaft of greater diameter than P>r
we first determine the imaginary value of and insert it in
(152). The same method is adopted if a cast iron lever is to be
used with a wrought iron shaft, and vice versa. The shape for
cast iron levers is given above, in Fig. 456.
Example 1. If the lever of Example 1, § 159 is made of wrought iron, and
is 24 inches long, its statical moment P R — 24 X 4400 = 105600 inch pounds.
This gives, from (131) D — 0.091 xK 105600 =4.3", and if wetake ■— — we
have from (152), w = 0.45 X 4-3" = 1.93"* * = J-93" X 2 = 3.86", say 3%".
The hub may also be calculated of such dimensions as to bv
strong enough to be forced on cold, and thus obtain sufficient
friction to hold without the use of a key (see 3 65, formula 66)^
The friction Q of the hub upon the shaft must then be
in wThich D' is the diameter of the shaft at the point where the
hub is fitted.
Example 2.—In the.case’ 01 the same leverj_as the preceding^example
PR _	105600
Dx — D and

=	49116
We may then take Q = 50,000 and let / = A = 3%" and S% = 10,650, and sub-
stituting in formula (66), we get:
V
*T X 4»3 x 3-875 x 0.2 x 10,650 4- 50,000
w X 4-3 X 3-875 X 0.2 X 10,650 — 50,000
161500
61500
1'= % X 0.63 =_o 31
The key is used as an extra precaution for security.
A special method of keying, especially adapted for the hubs
of levers and wheels, has been designed by engineer Peters. It
consists of two parallel Systems of keys, as shown in Fig. 458*THE CONSTRUCTOR.
103
The taper of the keys is ^. The arrangement shown at (a) is
preferable, as it weakens the hub less than (b). The angle a
may be taken = 135°, the thickness of keys ^ — tV D', andmean
width h = 2b.
The form (a) is especially suited for hubs which are made in
two parts.
Those hubs which are upon shafts subjected to bending, are
considered under the heading of Combined Levers, in Chapter
Lever Arms of Rectanguear Section.
The calculations of the dimensions of simple lever arms of
rectangular section are made upon the assumption that the force
Fig. 459-
gular arm, and h and b the corresponding terins to be found, as
in Fig. 461, we have
in which
b	1	
l>o “1	+ a	
a ~ {	fB	
	<6 J	L6t

(»S5)
E
These formulas permit a choice of the ratios and which
may be left to the judgment of the designer. In (155) the angle
Fig. 460.
P acts in a plane, passing through the middle of the arm, Fig.
459, and in a direction normal to the arm.
If we let
h = width of the arm at the axis,
b = thickness of the arm at the axis,
*S*=the maximum permissible stress,
b = 6
PR
Sh2
Taking 5 for wrought iron — 8500, and for cast iron = 4250,
we have
for wrought iron.
.	PR
b = 0.00072 ——
for cast iron.
P R
0.00144 —— . . (153)
These formulae are adapted for the determination of b, when
h has been selected, the latter being most conveniently chosen
with regard to.the other condition.
Exampie 1.—Eet P= 4400 lbs., R=24" for a lever arm of wrought iron, and
JV=7l/z" we have from (153);
b=0.00072
4400X24
(7.125)2
= itf"
If b is kept constant for the whole length of the arm, the
width at the small end may be 0.5/7, while if a constant ratio of
b : h is kept, the small end —y$h (see $ 10, Case III and VII).
If the force /Moes not act in the middle plane, as often oc-
curs, then there must exist a combined bending and twisting
stress on the arm. We may then derive a combined stress whose
bending moment will give an ideal arm R'.
If the plane in which the force P acts is distant from the mid-
dle of the arm by an amount c, we may make approximately,
(see \150) :
R'=ysr+ %%/^+T5
or	R' = 0.975	R -J-	0.25 c
if	R>cy
and	R' = 0.625	R -J-	0.6 c
is	Ry c.
R/ may be determined readily by the graphical method, Fig.
460. The third case shows the method for inclined arms.
Exampie 2.—In the case of the lever of the preceding exampie, let C—
15.75''. This gives R C and we have from (154) :
R’ = o-975 X 24 + 0.25 X 15- 75 =
This gives for b,
= 23.4	+	3-94	= 27.34''
b = 0.00072
4400X27.34
(7-I25)2
= i-7
n
Cast iron arms are sometimes made of cruciform section, see
Fig. 456, in which case the ribs may be neglected.
8163-
Lever Arms of Combined Section.
The sections shown in Fig. 461 are designed to secure an
economy of material. Their dimensions are readily determined
by first calculating a corresponding arm of rectangular section,
and then transforming it into an I section, or double II shape.
If h0 be the depth and bQ the breadth of the equivalent rectan-
irons of the third exampie in Fig. 461 have been neglected, and
may be considered as making up for the weakening of the rivet
holes. The following table gives a series of values for (155)
which will simpiify the calculations materially. The table will
also be found useful for other purposes, as all sorts of beams,
erane booms, etc.
Fig. 461.
i 161.
TabeE for Transforming Arm Sections.
Values of —7—
1 -j- a
C	B T = 2,5	3	35	4	4*5	5	6	7	8	10
6	0.50	o.43	0.38	0.33	0.30	0.27	0.23	0.20	0.18	0.14
7	0.52	o.45	0.40	0.35	0.32	0.29	0.25	0.21	0.19	0.15
8	0.54	0.47	0.42	0.37	0.34	031	0.26	0.23	0.20	0.16
9	0.56	0 49	O.44	o.39	0.36	o.33	0.28	0.24	0.22	0.18
10	0.58	0.51	O.46	0.41	o.37	0.34	0.29	0.26	0.23	0.19
11	0.60	0-53	O.48	0-43	0-39	0.36	0.31	0.27	0.24	0.20
12	0.62	0.55	0.50	0.44	0.41	0-37	0.32	0.29	0.26	0.21
14	0.64	0.58	0.52	0.47	0.44	0.40	0.35	0.21	0.28	0.23
16	0.67	0-60	0-55	0.50	0.47	o-43	0.38	o-34	0.30	0.25
18	0.69	0.63	0.57	0.52	0.49	0.46	, 0.40	0.36	0-33	0.27
20	0.71	0.65	0.60	o-55	0.52	0.48	i 0.42	0.38	o-34	0.29
22	0.73	0.67	0.62	0-57	0-53	0.50	1 0.45	0.40	0.37	0.31
24	0.75	0.68	O.64	0.59	0.56	0.52	! 0.47	0.42 ;	0.38	°-33
27	0.76	0.71	0.66	0.62	0.58	o.55	O.50	0-45	0.41	0-35
30	0.78	o.73	0.68	0.64	0.61	o.57	0.52	0.47	o.43	0*37
33	0.79	0-75	0.70	0.66	0.63	0.60	0.54	0.50	o.45	o-39
36	0.81	0.76	0.72	0.68	0-65,	0.61	O.56	0.52	0.48	0.41
40	0.83	0.78	0.74	0.70	0.67*	0.64	O.58	o-54	0.50	0.44
45	0.84	0.80	0.76	0.72	0.69	0.66	0.61	0-57	°-53	o.47
50	0.85	0.81	0.78	o.74	0.71	0.68	O.63	0-59	0.56	0.49
Exampie 1. A lever arm has a length R = 78.75'' and the journal pres-
sure at the end = P = 5500 pounds. It is to be of cast iron of doubie Tsec-
tion with a height hQ = 12yj'. According to (153) we have for a rectangu-
lar section
bQ — 0.00144
5500 X 78.75.
= 3-9".
(12.625)2
This is also so thick as to be impracticable, and hence the double ^sec-
tion may be compared. Here we may take c : n = 1: 12, B: b = 4, and we
0.44 and b = 0.44 bQ =■ 1.71", and the flange
get from the table-
1 4- a104
THE CONSTRUCTOR.
foreadth B = o. 44 b = 1.71 X 0.44 — 0.752, the web thickness = c —	— h =
= i.°5", all °f which are practical dimensions. It maybe found de-
sirable to have c — b, or any reasonable ratio <or B : b, and c : h be chosen.
Example 2. A wrought iron arm has been found to require b0 = 2%", n —
12y&". It is desired to make — =0.25 and in coluran 10 we find 0 25 opposite
bQ
h	12 62 5
—-— = 16. Hence b — 0.57" and B — 10 X 0.57 = 5 70" and c = —= 0.8".
CHAPTER XII.
CRANKS.
I 165.
Various Kinds of Cranks.
Cranks are thcse forms of simple levers which are so arranged
that they may, together with their various connections, make
entire and repeated revolutions about an axis. These may be
divided into the following four classes :
1.	Single Overhung Cranks.
2.	Return Cranks.
3.	Double Cranks, or Cranked Axles.
4.	Eccentrics.
These will be briefly treated in succession.
§ 166.
Singee Wrought Iron Cranks.
These cranks may be proportioned according to the rules
given for simple levers and rocker arms (§ 159 et seq). Fig.
462 shows the usual form; the arm tapers to two-thirds its base
dimensions both ways, aijd is made slightly convex on the back.
Fig^ 462.	Fig. 463.
The crank-pin is forced or driven in, and secured with a cap
bolt. Fig. 463 shows a crank forged in one piece. In this case
tne width of the arm at the base is determined by the necessary
amount of shoulder on the shaft. The proportions of the piu
are obtained from the rules in £ 159.
i 167.
Graphostatic Caecueations for Singee Overhung
Crank.
The crank is such an important detail of machine construc-
tion that it demands a most careful discussion, hence a grapho-
static investigation of the stresses in it is here given.
The Craiik Axle.—Ha.ving calculated d and /, draw the
skeleton diagram of the crank, that is, the neutral axis
A B C D E, Fig. 464, in which B C represents the axis of the
crank arm, which in this case lies normal to the axis of the
shaft, and is placed in its proportional distance from the centre
of the crank-pin A, and from the bearing D. Then lay off the
force Pfrom a, normal to E ax choose the pole O of the force
polygon (this being best placed upon a line passing through
the end of /^and parallel to the axis E a), draw the ray a d O,
and line d E} also the ray O Px parallel to d E ; then a d E will
represent the cord polygon for the bending which P produces
upon the axle a C E, and P Px represents the force upon the
journal Ey and Px a the force upon the journal D. Also make
a /''equal to the crank radius R, draw F G, and this latter will
be the twisting raoment (§ 140) which Pexerts upon the axis.
This moment M& may be combined with the bending moment
M b, to give for each point an ideal bending moment,
M, = | Mh + f v/jJTV + Mi(see § 45),
from which the polygon curve c' d' e' and surface of moments
Cc' d' e' E are obtained. From the latter, in combination with
the pin diameter d, and ordinate t of the base of the pin, the
diameter of the shaft may be obtained according to formula
(124).
The Crank Arm.—Prolong E a to aQ, and transfer the cord
polygon D a d to the base line B C, that is, make the angle
a0 B C= the angle D a d, and then will B a0 C be, with hori-
zontal ordinates, the surface of moments for the bending of the
crank arm due to the force P. Also make Cc0 — B b0 — Cc9
then will the horizontal ordinates of the torsion rectangle
B b0c0 C be the moments with which P acts to twist the crank
arm about the axis B C. This moment may again be combined
with the bending moment to give an ideal moment as before ;
(a0 a' = f a0 Cy draw B a\ make at any point H, the space
H i — f B b0, and make H h = h0 h' -f h* i) which gives the
surface of moments B b' h F C for the crank arm. From this
and from the diameter d and ordinate /, we cau construet the
conoidal form of the arm IK L AI, according to formula (124).
From this, again, the profile X T U V of an arm of rectangular
section may be derived, the width h being assumed for any
point and the corresponding thickness b obtained from the value
y of the conoid, according to the formula :
in using which, the second table 01 numbers at the close of this
work will be found useful. If the position of the axis B C does
not give satisfactory results, the operation must be repeated
with a better relation of parts. By proceeding in this mauner
the dimensions of a craiik and axle may be so determined that
they will be equal in strength to the pin upon which the power
is exerted.
In the preceding diagram the crank arm was taken as normal
to the axle. A slight inclination may be neglected, but if the
angle is greater, as shown in Fig. 460, it should be so considered
in the diagram. The procedure is then as foliows (Fig. 465):
The diagram for the crank shape is constructed as before, the
portion under a b being used onlv for the shank A B of the
crank-pin, and the portion under CE being combined as before
with the torsion moment, to obtain the surface of moments
Cc' d' e' E.
The crank arm is again subjected to bending and twisting;
the lever arm is now B' C, A Bf being made normal to the axis
Fig. 465.THE CONSTRUCTOR.
105
B C of the crank arm, the bending polygon being a portion of
the triangle C B' C', in which the angle at B' is equal to the
angle d a D. The twisting force acts with a lever arm A B' ;
its moment is obtained by drawing an ordinate at a' normal to
B C, B' a' being taken equal to B' A. The combination of
moments gives the surface of momen ts B b" c" C in same
manner, and of the same use as in the preceding case.
2 168.
Cast Iron Cranks.
The crank-pin is sometimes made spherical instead of cylin-
drical; such a one is shown in Fig. 466 on a cast iron crank.
The sphere will be of suitable diameter if described from the
middle of a normally proportioned overhung crank-pin without
making allowance for shoulder. The crank-pin is secured by
cold rivetting the end in place, an excellent method and one
often used. The I formed section can be proportioned by the
nse of the table in 2 164. When h is taken as equal to the hub
Fig. 466.
diameter, the cross section sometimes works out too light to be
suitable for casting, and in such cases it must be increased
according to judgment. Sometimes cast iron cranks are made
simply by laying out the proper hubs for the shaft and crank-
pin, and then joining them by an arm of rectangular section.
If it is desired to employ the graphostatic method, the dimen-
sions may first be determined for a wrought iron crank of rec-
tangular section, and then doubling the depth (see 2 162) for
cast iron, and obtaining the proportions for I formed section
according to 2 164.
1	169.
The Return Craxk.
A return crank is one which is formed upon the pin of an
ordinary overhung crank, returning back toward and having
rotation about the same crank shaft as the main crank. Fig.
467 shows a wrought iron return crank otherwise similar in con-
struction to the one shown in Fig. 463. Frequently the return
arm is 011 the same line as the main crank, as shown in the
illustration, but in many cases it is differently placed. The arm
and pin of the return crank are similar in shape and propor-
tions to au ordinary overhung crank. The arm of the main
crank demands no especial consideration, when, as is usually
the case, there is but little pressure 011 the pin of the return
crank. The main crank-pin must be determined separately. It
is subjected both to bending and to torsion. For this purpose
the formula (154) are to be used, remembering that when the
return crank is driven by the main crank the moment of the
return crank is greatest in the middle of the main crank-pin.
2	170.
Graphostatic Caecueation of the Return Crank.
The graphostatic diagram for a return crank, with both main
and return crank inclined to the axis of the crank shaft, is
shown in Fig. 468. The skeleton A BCDEFGHI is first
■drawn, the dimensions A B C E and E G being taken to cor-
respond with those ehosen to meet the requirements of the
cranks under consideration. The pressure 1 upon the return
crank-pin is here taken as opposed to the pressure 2 upon the
main crank-pin.
Force polygon.—After choosing a scale for the measurement
of the forces, the force polygon (on the right) can be drawn.
The line o to 1, measured upward, represents the pressure on
the return crank-pin ; O is the pole ehosen on a borizontal line
drawn through o, and the line 1 to 2 represents the pressure on
the main crank-pin, measured downwards. Draw the rays o O,
1 O, 2 O, also draw the line a d' parallel to 1 0, until it inter-
sects at d' the line dropped from D (the line of direction of the
i
Fig. 467.
force 2) ; draw d' g parallel to 2 O until it intersects a perpen-
dicular through G, the line of the force 3, which we know acts
upward, but the magnitude of which is yet undetermined. In
order to determine it, as well as the fourth force which acts at
//, join g with H> giving H a as the closing line which is hori-
zontal because we have ehosen the pole O on a horizontal
through o. Now draw in the force polygon O 3 parallel to Eg>
then the line 2 to 3 is the third force acting at G upward, and
the line 3 to o gives the downward force at H. Hence we have
the figure ad' kg H as the cord polygon of the system of forces.
At k is a zero point (see 2 132) and for convenience in showing
the figure it is preferable to turn the triangle k g H over to the
position k g' H. The cord polygon thus found will be of Ser-
vice in constructing the surface of moments, as will be seen
later. For the determination of the shank A B draw from A
on the pressure 1 the triangle a b b', whose ordinates will serve
to determine its dimensions.
Crank-Pin C E D.—This is subject to bending, as shown by
the surface of moments c d' e, and to twisting by the force 1
acting as a lever arm r = Cc— B b. In order to determine the
twisting moment, take a l = r, and draw the ordinate 11', this
latter will then be the desired moment, and the corresponding
surface a rectangle on c e. Combining this, as before, with the
trapezoid c d' e gives the surface of moments c E d" e' e. Should
it occur that the only pressure acting is that upon the returnio6
THE CONSTRUTOR.
crank-pin, the surface will be modified as follows : prolong the
line a d' to m', and taking this bending polygon, obtain the
corresponding surface of moments c' d" e, from which the
crank pin C D E can be proportioned. The minimum length l
of the crank-pin must be that due to the pressure 2, as given
before, for overhung journals.
Axle F G HI.—This is subjected to bending according to
the polygon F f g' H, and also to torsion by the moment of the
force 2 less that of the force 1. In order to find the first, we
choose in the force polygon a second pole 0upon a horizontal
passiug through the starting point of the force 2, returning the
same pole distance. Draw 2 O' and make d g" parallel to it,
make d n = C c — R} and we have in the ordinate n n' the de-
sired twisting moment. Make the abscissa of the ordinate at
a' — A a — R — r, and this ordinate will then be the moment
with which the force 1 twists the arm backward. Taking this
from n n' gives the height Ff' of the torsion rectangle
FI i' f' which we may combine with the bending surface in
the manner already given, and thus obtain the surface of mo-
ments Ff"g" h" i" I. Should the case occur in which the
force 1 becomes zero, as is the case at some points in steam
engines when the return crank operates the valve motion, we
have for a bending surface Ff0 g" H, and for a torsion surface
F F0 i /, which gives a surface of greater ordinates to be used.
Such a case is given in uni ttered dotted outline shown upon
the base FI. It is assumed that the portion HI is subjected
only to the action of a torsion couple, hence the polygon there
becomes a rectangle.
Return Crank Arm B C.—This is subjected to torsion by the
force 1, with an arm A A0 perpendicular to C B prolonged (its
moment being equal to the ordinate at a0 ), and to bending by
the arm A0 C, whose polygon is a triangle on C A0 and angle at
A0 equal to l a a0. The reduced surface is shown at C B c0 c".
Main Crank Arm E F.—This is subjected to bending for-
wards with a moment surface DQ F F", the angle at D0 being
equal to e d g", and to forward twisting with an arm D D0,
which is perpendicular to F F prolonged; it is also subjected
to backward bending by the force 1, with a surface E0FF',
and backward twisting by the arm A F0 normal to FF. The
combined bending moments give the surface E d0 e0 F'" F, and
the combined twisting moments the rectangular shown upon
F F, the combination of both resulting in the final surface
F e"' f"' F. Should the force 1 become zero the figure will
be increased to that shown by the dotted lines.
I 171-
The Simpee Crank Axee.
Crank axles may be divided into simple and multiple cranks.
A simple crank axle is shown in Fig. 469.
Fig. 469.
The analytical discussion of such a crank axle is such a com-
plicated matter, and the practical results are so readily obtained
with ali needful accuracy by the graphostatic rnethod, that the
latter is only given here. In Fig. 470 is shown a skeleton dia-
gram ABCDEFGHoi a crank axle with both arms in-
clined.
If we make the value of the force P, which acts upon the
crank pin, equal to Q when it acts in the direction K M, it will
be equal to	when the connecting rod is in any inclined
position K L\ <X being the angle of the rod with the axis
K M. For a constant force Q the pressure P will be a maxi-
mum when K L acts normal to L A/, and this is so nearly the
same as the value for the vertical position M L1 of the crank, or
— —— , that this latter may be taken for the graphical exami-
COS CA j
nation without a closer determination. The force at M is equal
to Pin magnitude and also parallel in direction, and at K is a
normal pressure, which is A= Q tan oc and is a maximum for
the position Kx Lx M. Hence we may safely assume that the
moments with which the crank arms and the axle are bent
attain the maximum at the same time, and are those due to the
force P. In the example the crank pin is at £, at ^ and 7/ are
bearings, at A is a couple by which the shaft is subjected to
torsion due to the force P acting with a lever arm P. ‘This
problem is very similar to the preceding, the portion H G
taking the place of the return crank, wdth the difference that
the force at H is variable and indeterminate, but is dependent
upon the pressure Pat E.
Force Polygon.—In order to make the closing line of the
polygon horizontal, draw the line B e' to any desired point ef
on the normal E e', join e' wTith H; then on any convenient
scale draw the force P, from O. in the diagram on the right, and
make o O parallel to H e' > 1 O parallel to B e' and O 2 normal
to P. Then the distance 1 to 2 is the upward force P2 acting at
P, and 2 to o the force P3 at H, 02 being the pole distance.
Axle Shank H G.—This is subjected to bending by the force
P3 at H. The triangle H G g is the surface of moments, and
the ordinates may be used to determine the dimensions of the
journal at H.
Axle Shank B C.—The surface of moments for bending is
the triangle B C c. In addition to the bending is the twisting
moment P R ; in order to determine this make O' 1 normal to
P and equal to O 2, and also make E0 e0 parallel to 02 and
equal to P, then o Fa is the desired moment, which laid ofF at
C c' and A a' and combined writh B C c in the manner already
described, gives the surface of moments A B C c" b" a".
Crank Pin D E F.—The surface for bending moments is the
figure d ff' e' d'. For twisting we have the force P3 at H,
with a lever arm of E e — R. Make H g = E e — P and the
ordinate g g' is the desired moment, which transferred to
f"' d"' and combined wdth the preceding surface gives the
surface d f f" e" d". The greatest ordinate e e" should be
used if the pin is to be cylindrical.
Crank Arm G F.—Draw E D0 parallel to H D' normal to
C D. We then have forward bending by the force P at D0 ;
backward bending by P3 acting at D'. The cord polygons for
these are, the triangle D0 C i (with C i — o H0 in the force
polygon, where H0 h0 = C D0). and D' C' i; which when
combined give the surface C i" i'" for the bending of the arm
D C. We also have a forward twisting from the force Pwith
the arm F D = k k0 in the force polygon, and the moment o k
acting backward from the force P3 with a lever arm H D' =
HI in the cord polygon and a moment II. The difference be-
tween these moments laid off at D d0 and C c0 and the resulting
torsion rectangle combined with the bending triangle gives the
surface C D IP, so that all five portions of the diagram now
have their moment surfaces determined. The rnethod of using
these for the determination of dimensions is the same as
before.THE CONSTRCTOR.
107
The figures show clearly the various stresses at the respective
portions of the crank and throw light upon the manner in which
breakages occur.
If both crank arms are normal to the axis, the solution is
greatly simplified, and the diagram assumes the form given in
Fig. 471. In this we have again ABCDEFGH as the skel-
eton, and at A a. torsion couple whose moment is equal to PR.
Force Polygon.—In this case the altitude e e' of the triangle
Be' H is taken as the measure of the force P. Bb" is made
equal to e e\ b" O drawn parallel to e' H, and O b made normal
to Bb" y thus giving b" b as the force P3 at H, b B that at B,
and O b is the corresponding pole distance.
Axle Shank HG.—This is only subjected to bending, and the
surface of moments is HGg.
1
1
Fig. 471.
Axle Shank A B C—This is subjected to bending, as indi-
cated by the triangle B C c, and also to torsion by a moment
PR. Make e' O' parallel to and equal to the pole distance
b <9, d^aw e'"p parallel to e' O' and equal in length to E e = R,
then e e'" is the desired twisting moment, giving for A C the
torsion rectangle whose altitude A' a' — B b"' = e e'". The
combination of bending and torsion moments gives the moment
surface ABC c'" br a.
Crank Pi?i D E F.—This is subjected to bending according to
the surface of moments CGgc, and to torsion by the force Ps
at H, with a lever arm R — C D = Hf, and a moment ff' —
Gf" = Cf"'. By combining the twisting and bending mo-
ments the surface C G g' e" c' is obtained, and for cylindrical
crank pins the rectangle of a height G g" = Cc" = e e" is to
be substituted for the irregular outline.
Crank Arm F. G.—This is subjected to bending by the force
P3 acting at G. The surface of moments is G PR the angle at
G being equal to f Hf' ; it is subject to torsion by the same
force acting with a lever arm H G, giving a moment G g — G h
= Fi. The combination of twisting and bending moments
gives the surface F G h' i'.
Crank Arm CD.—Here we have bending with the force Pf
and an already known moment e e"' = C k at C. Twisting is
due to the moment Cc — CDl'. For the combined mo-
ments these give the surface C D dk'.
For the same given distances of E from B and H the torsion
stresses on the crank arms are greater for arms normal to the
axis than for inclined arms, so that in the former case heavier
arms are required. The torsion in the crank arms grow7s less
and less the nearer the points C and G approach B and H,
which is a point to be considered in the interest of economy of
material. It is also to be noted that the total length of crank
axle F G H or D C B is less for inclined arms than for right-
angled cranks.
In many cases a crank axle is so situated that it is subjected
to torsion at either one end or the other. In such cases the dia-
gram should be constructed for both sets of conditions, and laid
upon each other, the greater value in ali cases being taken. Of
course, care must be taken to use the same pole distance and
same scale for mearuring forces in both cases. An example of
such a case is found in the paddle engines made by Penn, with
oscillating cylinders, the air purnp being worked from the mid-
dle of the crank piu. The conditions in this case are somewhat
different from the preceding, and may be examined with the
help of the following diagram (Fig. 472) :
Here we have the skeleton ABCDEFGH, and not taking
into account the force at £> the force couple gives by means of
the cord and force polygon the moment values B b = C c = G g
= H h, from which the following results are obtained :
Axle Shank A B C.—Pure torsion, which, converted into an
eqaivalent bending moment, gives Bb' = Cc/ — %Bb (see IV.,
§ 16, when Mb — O).
Axle Shank G H.—This is the same as the preceding, and
H h' — G gf = C c'.
Crank Pin DE F.—We have here the same twisting moment
as in the axle shanks D d — Ff— B b and Dd0= F/0= B bf.
Crank Arm C D.—We have in this portion a bending moment
of the magnitude Cc" = D d' = Ccf of w hich the plane stauds
normal to the plane of the surface of the crank arm. The sur-
face of moments is in this case equal to a rectangle of the height
B b — Cc.
Crank Arm F G.—In this case we have both torsion and
bending. The couple is decomposed at G into two parts, one
acting normal to the axis of the crank arm, and the other in the
direction of the arm. The first gives the torsion rectangle
G Ff"g"y the latter the bending rectangle F G i which com-
bined give the moment surface P' G g"'f"\ in which we again
have p	G i, p r =. f G g", p t = G g" — q$-\-qr.
Thus far w7e have proceeded as though there were no force
acting at E. When such exists, however, first determine the
bending and twisting moments as showm in Fig. 472, add or
subtract, according to direction, the twisting moments, taking
into account the position of the planes of bending action, and
finally combine the bending and twisting moments so found,
according to the method of Ca^se IV., \ 16. The amount of
work w7hich this investigation requires of the drawing-room of
any machine-shop is small compared with the importance of a
thorough determination of all the stresses which act upon such
a piece of w7ork as a crank shaft forging.
Fig. 473-
i J72-
Muftipfe Crank Shafts, Locomotive Axtfs.
One of the most important forms of crank axles made of
wrought iron or Steel is that used for locomotive engines. As
an example of this subject, the crank axle for an inside con-
nected locomotive is given in Fig. 473. In drawing the diagram
of moments it is necessary to take into account tne diameter of
the driving-wheels, as will be shown in Fig. 474. cx and C\ are
centres of the steam cylinders, A1 and A2 are the journals, and
Bx Dx and B2 D2 are the hubs of the respective driving-wheels.
The cranks at Cx and C2 are placed at right angles with each
other, taking the position w7hich the axle shows in Fig. 473. An
inspection of the figure show7s three distinet loads acting upon
the axle : 1, the pressure in the vertical plane due to the weight
of the locomotive and to the lateral action upon the wheelio8
THE CONSTRUCTOR.
flanges ; 2, the horizontal pressure of the piston against the
crank C2 opposed by a corresponding adhesion at the circumfer-
ence of the driving-wheels; 3, the oblique pressure of the con-
necting rod acting upon the crank Cx. Other small pressures,
such as those due to the eccentrics, etc., may be neglected.
Forces and Moments in the Vertical Plane.—Fig. 474- From
the point S0 of the height of the centre of gravity of the loco-
motive lay ofF the force Q, to represent that portion of the
weight which is borne by the axle under consideration. The
oscillations and action of centrifugal force upon curves also
produces a horizontal force //, which may be taken as equal to
0.4 Q. The resultant R of the two forces Q and H is the load
upon the axle. This may be decomposed into the pressures Px
and P2 upon the journal at Ax and A2, and into the pressures
Qx and Q2 upon the wheels at Ex and E2, which pressures, with
their reactions, produce the stresses on the axle. The forces Qx
and Q2 can be decomposed into two others referred to the wheel
hubs Bx Dx and B2 D2. This gives six vertical pressures acting
to bend the axle, viz. : 1, 2, 3 and 4 acting downward at Dx, AXi
A2 and D2, and 5 and 6 acting upward at B2 and Bx. From
these forces, by choosing any desired pole distance, the force
polygon Ey 4, O may be constructed, and also the cord poly-
gon or surface of moments dx ax a2 d2 b2 bXy and this surface gives
by its ordinates the proportional bending moments in the verti-
cal plane for each point in the axle ; this entire surface is desig-
nated by the letter V.
Forces and Moments in the Horizontal Plane.—Fig. 475. As
already shown in a preceding paragraph, the pressure P011 the
crank pin for the position L Mof the crank is somewhat greater
than the pressure P0 on the piston ; its moment of rotation about
p
the shaft is--— . R cos af which = P0 R, so that upon the as-
cos a
sumption that the wheel on the left slips on the rail, the other
one must oppose a resistance whose moment equals P0R and
p
the frictional resistance 3 at E2—P0 — Combining this force
3 at E2 and also the force 4 = P0, and the resistances 1 and 2 at
the journals, we are enabled to construet the force polygon
Ax 2 O and the corresponding cord potygon H for the horizon-
tal forces, as shown in the light sectional portion of the diagram.
The forces 1 and 2 are found bv taking the position of the re-
sultant of the two forces 3 and 4, as shown in the figure, and
decomposing their sum into the portions which would go re-
specti vely to Ax and A2y as shown by the construction given in
the dotted lines.
Forces and Moments in the Inclined Plane of the Connecting
Rod.—The force Q = 5 acts at Cx, making an angi e with the
horizontal equal to M KL. As shown in the illustration, this
may be decomposed into the two opposing forces 6 and 7 at Ax
and A2y and by taking the same pole distance as before to con-
struet the force polygon we obtain the cord polygon S, shown
by the dark section lining, and giving the surface of moments
for bending in the inclined plane of the connecting rod.
Combination of the Three Preceding Cord Polygons for Bend-
ing of the Axle.—Fig. 476. Since the three preceding sets of
forces are acting at the same time to produce bending in the
axle, it is necessary to combine the diagrams in order to obtain
the final resuit. For this purpose we can treat the respective
ordinates in the same manner as if they were forces, as in § 44.
Taking the successive points upon the axle, w^e construet the
corresponding ordinate polygons, whose closing lines give the
resulting moment both in direction and magnitude. One of
these ordinate polygons is showm in the upper portion of Fig.
474, to the left: it belongs to the point Cx. The vertical ordi-
nate Vin this case acts upward, the horizontal ordinate H con-
tinues toward the left, and the inclined ordinate 6*also continues
to the left, thus giving the resultant T as the line joining the
origin of V with the termination of S. We thus obtain for the
entire axle the surface of moments D2 Dx ax cx c2 a2 b2, which
gives the proportion of bending stresses of the axle, as distin-
guished from those of the crank arms.
The Torsional Moments for the Axle.—The position of crank
described above and selected for this investigation gives a tor-
sional moment only upon the crank to the left, and also one of
the magnitude PR upon the axle extending to the point Dv If
both cranks stand at an angle of 450 with the horizontal, there
will be produced in both end shanks Cx Dx and C2D.2 moments
equal to s/ 2 PR, or about 1.4 PR. Under these circumstances
the moments at the ends become Dxdx' = D2d2', while in the
body of the shaft Cx C2 we have the moment Cx cx' = C2c/ = PRt
always keeping the scale of forces and the pole distance the
same in all of the diagrams. It must be remembered that in
this position of the cranks the bending moments are somewhat
different from those shown in the preceding diagrams.
Combinatioji of Bending and Twisting Moments. — The
bending and twisting moments can now be combined accor-
ding to the formula of §45, and thus the surface of moments
D2 Dx dxbx . . . . d2/f obtained, by the help of which the
shanks Cx Dx and C2 D2 and body of the axle Cx C2 can be pro-
portioned, affer the diameter for any one of the ordinates, as,
for example, that at BxbXy has been determined. The half of
the diagram which gives the greatest ordinates should be used
for both halves of the axle.
Crank Pin at Cv—The two crank pins are treated separately
in Figs. 477 and 478, since the moments can be laid out more
conveniently in that way. For the pin FG at Cx we have, in
addition to the bending moments obtained from Fig. 476, and
shown by the surface FG cx, the combined forces on the left, up
to the point E, acting to twist the pin. The resultant of these
forces is yet to be found. The vertical forces are those shown
at 1, 2 and 6 of Fig. 474, their algebraic sum being shown at It
in Fig. 477. The horizontal force acting backwards is //, repre-
sented as 1, in Fig. 475. The inclined force acting downwardsTHE CONSTRUCTOR.
109
and backwards, shown at III, corresponds to the force 6, of
Fig. 475. The closing line (not shown) from III to CY would
give the resultant, and its horizontal component IV acts to twist
the crank pin FG, with a lever arm E F— R. In the force
polygon (above, on the left) we take a O to be the pole distance,
as before ; lay ofF IV downward from O, draw a IVe, make
af=R; then will f e, perpendicular from f, be the twisting
moment Fff. Combining this with the surface of bending mo-
ments F G cly we obtain the final surface FG c/.
Fig. 477.
Crank Arm E F.—The ordinate polygon Vl Hl Si Tx (on the
left) is constructed for the point E. The horizontal component
hx of the resultant Tx acts to twist the arm E F, Fd = hl; the
vertical component vx acts to produce a bending of the arm in
the plane of the diagram, Fb = vx ; also the force /Kacting at
E tends to bend the arm normal to the plane of the diagram,
with a moment b 3l = Fdx at F. The combination of the bend-
ing moments gives the surface E Fb' b", which, with the tor-
sion rectangle E F d, gives the final snrface E Fb"'.
Crank Arm G H.—The ordinate polygon V2H2 S2 T2 is con-
structed for the point H. The horizontal component h2 acts to
twist the arm G H, Hdx — h2; the vertical component v2 shows
the bending in the plane of the diagram, G bx — v2; also, the
force Pbends the arm normal to the plane of the diagram with
a moment PR=.fh, of the force polygon above, on the left, in
wrhich Og — P, af= R. Again, make b2 b3 =f h. The combi-
nation of the bending moments gives the surface G H bx' b2",
and the combined bending and twisting moments give the final
surface G H b2‘
Fig. 478.
Crank Pin K L.—Fig. 478. This crank pin is subjected to
the bending moments which act between M and J, and indicated
by the surface K L c2, obtained from Fig. 476. The collected
forces which act on the left of C2 tend to twist the pin. The
resultant of the forces 3, 4 and 5, Fig. 474, shown at V in Fig.
476, acts downward, the resultant (difference) of the forces 2
and 3, Fig. 475, and shown at VI, acts horizontally backward,
and the force 7, of Fig. 475, shown at VII, acts inclined back-
wards. The vertical component of the force polygon V, VI,
VI/, acts to produce twisting at M, remembering that the crank
J K is taken in the horizontal position. The moment of this
vertical component has the magnitude k k'. Also we have act-
ing to twist the pin the couple shown on the left (as discussed
in connection with Fig. 472) with a moment already determined
and shown at Cx cx in Fig. 476, and here laid off at K k, from
which, since the previously determined twisting moment kk'
acts in the opposite direction, we must subtract k k', giving
finally for the crank pin K L the twisting moment K k', which,
when combined with the bending moment, gives the surface
KLc2'.
Crank Arm J K.—This is subjected to twisting by the moment
AV=the vertical component v2 of the ordinate polygon
V3H3S3 T3. For bending in the vertical plane we have the
moment Kl— Kk, as already shown in Fig. 472 ; also in the
same manner and direction by the vertical component of the
forces Vy VI and VII with the moments b b2 at K (see the dia-
gram of these moments in the upper left portion of Fig. 477)»
It is subject to bending in the horizontal plane by the horizon-
tal component h3 of the ordinate polygon, the moment being
b bx. The combination of bending moments gives the surface
J K bx' b/, and the final combination with the twisting moment
K d gives the surface J K b2".
Crank Arm L M.—The twisting moment is L dx — the verti-
cal component of the ordinate polygon for the point M. The
bending moment L b3 — K k, also b3 b± due to the vertical force
at M, and also the bending moment b3 b5 — the horizontal com-
ponent hA of the ordinate polygon. The combination of bend-
ing moments gives the surface M L b/, and the final combina-
tion with the twisting moment gives the surface M L b3".
Of the four crank arms, J K is subjected to the greatest stress
at the pin, and G H at the axle. In practice, therefore, the
surfaces J K b2" and G H b2"' should be drawn upon each other
and the greatest ordinate used. The resulting dimensions, w ith
possibly slight modifications, should then be used for ali four
arms.
Although the construction of such a graphostatfo diagram of
moments involves some labor, the resuit is most satisfactory,
since by assuming a stress of say f the modulus of working
stress (about 17,500 lbs. for wrought iron, 25,000 lbs. for Steel)
the design can be properly proportioned without further care.
The calculations for locomotive axles with outside cranks is
similar to the preceding, although the diagrams are necessarily
somewdiat different, although laid out in the same general
manner.
Fig. 479.	Fig. 480.
§ i73.
Hand Cranks.
The chief peculiarity in a hand crank lies in the adaptation
of the crank pin to be operated by hand. In Fig. 479 is shown
a crank for two men, and in Fig. 480 for oue man. The dimen-
sions for the parts indicated by the letters are as follows :
For 2 men.
For 1 man.
R= 14" to 18"
V = 16" to 19"
ii"to if"
I27/ to l67/
I27/ tO I3"
iJ"to 1\"
The other dimensions figured in the illustrations are in milli-
metres. When placed at opposite ends of the same shaft, hand
cranks should be set at 120° with each other.
Fig. 481.	Fig. 482. Fig. 483. Fig. 484.
1174.
Kccentrics.
An eccentric is nothing more than a ciank in which (if the
crank arm is R and the shaft diameter D) the crank pin diam-
eter d' is made so great that it exceeds D 4- 2 R, or is greater
than the shaft and twice the throw. The simpler forms of eccen-
tric construction are shown iu the illustrations. The most prae-IIO
THE CONSTRUCTOR.
tical of these is that shown in Fig. 483, the flanges 011 the strap,
as shown in the section, serving to retain the oil and insure good
lubrication.
The breadth of the eccentric (properly the length of pin /) is
the same as that of the equivalent overhung journal subjected
to the same pressure ; for the depth of flange a we have
a = 1.5 e — 0.07 /4-0.2................(157)
from which the other dimensions can be determined as in the
illustrations.
For some forms of shafts with multiple cranks or other ob-
structions the eccentrics caunot be made as shown above, but
must be in halves, bolted together.
CHAPTER XIII.
COMBINED LEVERS.
\ 175-
Various Kinds of Combined Levers.
Two simple levers with the same hub form what is termed a
Combined Lever. When both arms have a coinmon centre line
they form a Beam, or so-called Walking Beam ; and when they
form an angle with each other they are called an Angle Beam,
or frequently a Bell Crank. The pressure Q, upon the axle of
an angle lever A O B, Fig. 485, is determined by the relatio**
Q = \/Pi 4- P1 — 2 Px P2 cos a
if P1 is the force acting at Af and P2 that at B} both acting at
right angles to their respective arms ; a being the angle between
the arms. This may be shown graphically by making P1= O B
and P3 — O A, when Q will = A B, the third side of the tri-
angle. If the forces J\ and P2 do not act at right angles to the
arms, the triangle must be constructed by drawdng lines from
O, normal to the directions of the forces.
The variety of combined levers is very great, and only a few
of the principal forms are here given.
a.	b.
Fig. 486.
§ 176.
Waeking Beams.
One of the principal forms of combined levers is the walking
beam, for use in some forms of steam engine. These are usually
made of cast iron, with journals and pins similar to those given
in Fig. 456 ; and other fornis of journals are also showm in the
following figures.
Fig. 486 a shows an ornamented beam-end, with the pin keyed
fast Fig. 486 b shows a beam-end with a bored cross-head and
pins combined, fitted on the tumed end of the beam and secured
by the pinned collar shown. This construction requires careful
fitting, and is somew7hat expensive.
Fig. 487 a. This is a fork journal; the fit is made with a very
slight taper, secured by cap bolt and large wTasher at one end.
The pin is kept from turning by a projection under the head,
let into the boss on the beam.
a	b
487.
Fig. 487 b. This is a spherical bearing with its shauk driven
into the end of the beam and keyed fast, this form giving great
freedom of motion to the connecting rod.
The diameters of pins are determined as already given in $ 90.
The load is to be considered as acting continuously or intermit-
tently, according as the engine is single or double acting.
Fig. 488 shows a form of beam w7hich has been extensively
used. In order to secure lateral stiffness, the beam centre should
not be made too short. A good proportion is that given in the
figure, in which the distance betw7een centres of bearings is
made equal to 6d 4-	The distance between centres of
journals for the ends of the beam is made from 4.6 d2 to 5.5 d2;
Fig. 488.
d2 being the journal diameter, as showm. The depth h of beam
in the middle must not be made less than
/t = 4d+.......(158)
in which d is the diameter of the beam centre, and A the half
length of the beam. If the tw7o arms are of unequal length
their mean should be taken.*
The curved outline of such beams is drawm according to the
methods given in £ 142, starting from the crowm of the beam to
the hub for the pins at the ends. The ribs in the middle of the
beam are given the same thickness, c, as the flange at the edges,
and the breadth of flange is showm in the plan at B (see § 163).
Another form of beam is showm in Fig. 489. This is made
double, and in such case each half is calculated separately. In
Fig. 490 is shown a section of such a double beam in w7hich the
parts are somewhat widely separated. The two plates are firmly
bolted together, the bolts passing through tubular sti ts, as
shown, and the parallel motion rods are hung between the two
parts of the beam.
i
Fig.
• In the United States much greater depth is given to beams of this sort,
sometimes 2 to 2% times that given by the formula. Skeleton beams with.
cast-iron centres and wrought-iron bands are also much used.THE CONSTRUCTOR.
iii
A beam of somewhat unusual form is shown in Fig 491, being
» portion of the hydraulic riveting machine of Mackay & Mc-
Oeorge, built by Rigg.* The beam centre is at A, the rivet die
st B, the hydraulic pressure is exerted by small and large cylin-
ders at D and C respectively. The water pressure is taken from
an accumulator and discharged into an outlet pipe placed some-
what higher than D. By means of a suitably arranged valve
Fig 489.
gear the high pressure water is first exerted upon the small cyl-
inder, and water from the discharge pipe delivered to the large
cylinder, thus closing the die upon the rivet at B. Then the
high pressure water is also delivered to the large cylinder,
jnaking a stili greater pressure upon the rivet, with practically
Fig. 490.
no expenditure of water, as that cylinder is already filled. The
pressure upon the rivet is 60 tons. The beam is made of a sec-
tion of uniform resistance (see § 9). At E is a short shear for
cutting beams, angle iron, etc. The distance B C is 12 feet.
Wrought iron beams are not uncommon, and for moderate
Fig. 492.
loads and dimensions are couveniently made in the double form,
as shown in Fig. 492. The depth h in the middle may be taken
at 0.8 times the value given by formula (158). For larger beams
of wrought iron, the girder form shown in Fig. 491 is to be pre-
ferred.
Another form of beam is the equalizing lever, used to distrib-
ute the weight among the springs (see Figs. 102 and 103, \ 41).
In Fig. 493 is shown a lever of wTrought iron for a heavy engin^
{the Prussian Standard freight engine). The length AB is 1180
mm. = 46J", and the connections at Ay O and B are made with
♦ See Engineering, March, 1875, p. 223.
bolts. Fig. 494 shows the form used on American locomotives.
The example is from a passenger engine, and extends between
Fig. 494.
the springs of the driving-wheels, being 7J feet long. At (9, A
and B are half journals, and the connections at A and B are
not rigid. The bearings are not on a straight line, as in the
G erman form, but the variation is trifling.
* For similar examples see E. Brauer’s “ Konstruktion der Waage ” (Scale
Construction), Weimar, Voss, 1880.112
THE CONSTRUCTOR.
Scale beams should show very little deflection under their
load. They are therefore made very deep in proportion to their
total section, and the stresses taken at 4250, 8500 and 14,220 lbs.
respectively for cast-iron, wrought-iron and Steel.
CHAPTER XIV.
CONNECTING RODS.
«178.
Various Parts of Connecting Rods.
Connecting rods are used in various forms for transmitting
the motion of various reciprocating parts of machines to levers,
beams or cranks, or vice versa. It is necessary to consider sepa-
rately the ends or heads which contain the bearings for the
crank and cross-head pins, from the body of the rod. The
dimensions and proportions of the ends are governed, to a
greater or less extent, by the dimensions of the bearings, the
latter being either forked, overhung or necked, and their size
determined by the pressure to which they are subjected.
? 179-
CONNECTIONS FOR OVERHUNG CRANK PlNS.
The strap and key connection shown in Fig. 496 is widely
used. The boxes are surrounded and drawn together by the
Fig. 496.
strap and key, and by driving up the latter they may be closed
together to take up wear. In determining the dimensions, the
boxes and their surrounding parts will be considered separately,
as in the case with other bearings. The unit or modulus for the
boxes is*
e = 0.07^ + 0.118"....................(159)
being the same as used or other bearings, d being the diameter
of pin.
e=-07<3+.<f8
Fig. 497-
Fig- 497 shows two views of the brasses, the dimensions of the
other parts being based on the followdng modulus :
dx — 0.0267 >/P+ 0.2"...................(160)
The breadth b may be made equal to 0.8 dlt or if the length of
the journal is made equal to its diameter b becomes = d — 2 e*
Exampie: If P= 7920 pounds alternating load, we have from (93) d — 2%",.
/ also = 2y&", and according to (166)
d\ = o 0267 \/7920 +0.2"
= (0.0267 X 88.9) + 0.2" = 2.57", say 2TV'
We also have e = (0.07 X 2.375) + 0.181 = 0.3". Also b = / — 2 e — 2.375 —
0.6 = 1.77. Applying the value of d\ to Fig. 496, we have the thickness of
strap = 2.57 X 0.2 = 0.514 on the sides, and 2.57 X o 3 = 0.76 on the end ; also-
the thickness of key = 2.57 X 0*22 = 0.56", and the other dimensions in a
similar manner.
The key must be given much less taper when it is used with-
out a set screw, as in the illustration, than when a set screw is
used. In the former case a total taper of ^ is used, and in the
latter J is safe.
!
Fig. 498.
The boxes are best made to bear closely together instead of
being set open, as shown in the figure, and better practice in
this respect is shown in Figs. 499 and 500. In this case the
boxes must be filed off to permit them to be closed up for wear.
An objection to the forni of strap end just shown is that the
continual keying wp of the boxes tends to shorten the rod. The
reverse action takes place with Sharp’s strap end, Fig. 498, the
action of keying up tending to lengthen the rod.
In Fig. 499 is shown a capped end of solid bronze, as made
by Penn. The two halves are fitted closely together, so that
the joint must be filed out to take up for wear, or else a num-
ber of thin slips of copper may be inserted in the joints and
removed one at a time, as may be found necessary. The diam-
eter 6 of the bolts must be made, so that they shall not be less,
measured at the base of the thread, than the value given by
formula (84). For V thread this is given by making
6 = 0.0142
and if square thread bolts are used they should be made slightly
larger. The stress on the material with these sizes will then be
between 7000 and 8000 pounds, which is not excessive. (Com-
pare Exampie 2, \ 182.)THE CONSTRUCTOR.
The nuts of these bolts are fitted with Penn’s locking device,
Fig. 243. For rods of large dimensions, such as are used on
heavy marine engines, the boxes are cored out in order to secure
economy of material.
Fig. 500.
of the rod less. The key itself is made flat on the side which
bears on the pressure block, in order that liners may be intro-
duced when necessary. The key is secured by the method
shown in Fig. 201. It will be noticed that the nut is set so deep
in the recess that a Socket wrench is required to turn it. This
is done in order that nothing may project beyond the dotted
clearance line.
In Fig. 501 is shown another solid rod end, much used on
locomotive engines. The boxes are made without flanges on
the back, so that they can readily be removed after taking out
the key. In this case there is no pressure block, but the box
upon which the key acts is given instead a thickness of 3 e'in-
stead of 2 <?. The method of securing the key is the same as
before. An oil box is formed on the upper side, and covered
wTith a brass lid attached by screws. The hole in the oil cup,
shown in the plan, is fitted with a tube and wick.
In both of these forms the tendency of the wear is to shorten
Fig. 502.
is combined with its own locking device. The boxes are made
of wrought iron lined with white metal, and an oil chamber is
formed in the one shown on the left
Fig. 503.
Fig- 5°3 shows an end of cast iron, also made solid, and with
the key acting to take up the wear from below, much as in the
design of Sharp, Fig. 498. Cast iron rods were formerly much
used on the parallel motion connections of beam engines.
the length of the rod, and if the reverse is desired, the key may
be placed behind the other box. These rod ends are designed
so that as much of the work as possible may be performed by
the lathe, and sharp angles and corners have everywhere been
avoided, as they tend to weaken the material.
Fig. 501.
A third form of solid end is shown in Fig. 502, and is designed
by Krauss, of Munich. It is intended to be made of Steel, and
is very compact and simple. The key is made in two parts, and
Fig. 499.
In Fig. 500 is shown a rod wTith a solid end, and is a very ex-
cellent form, and wTith proper tools, not too expensive to con-
struet. The boxes are made of bronze, lined wTith white metal
and turned on the outside. The movable box is fitted with a
wrought iron pressure block, wThich receives and transmits the
pressure of the key. The boxes are provided with small pro-
jections, which engage with corresponding recesses and prevent
them from turning.
The key-slot in the rod is made wTith semi-circular ends,
partly because the machine which forms the slot leaves it in
that condition, and also because this shape weakens the sectionTHE CONSTRUCTOR.
114
2180.
Stub Ends for Fork Journaes.
Fork journals designed according to tlie method previously
given, are made muck smaller in diameter than the correspond-
ing overhung journals. On this account the breadth b' cannot
be determined in the same proportion to the diameter of pin d,
as with overhung pins, as the pioportion will vary somewhat
for various conditions. In order to take this difference into
account, \ve may take for such rod ends, instead of the modulus
given in (160), the following :
in which b is the breadth corresponding to the length of the
liormal pin, and dl its modulus, calculated according to (160).
This enables us to use the proportions of all the preceding ex-
amples for fork journals as readily as for overhung crank pins.
The thickness of metal e in the boxes may be made the same as
before, using in every case the actual diameter d' of the jour-
nal. The formula (161) assumes the same material to be used
in both cases, and gives the rod end for the fork journal
approximately the same strength as one proportioned for a nor-
mal pin. It is not, however, possible to make an empirical
formula cover every case, and some examples will be found
much heavier, such as, for example, would give a modulus of
dx_(L
dx “ \ b' ’ d )
One of the portions whose dimensions will not bear much
reduction is the key, siuce it is subject to shearing action and
its limited surface must not be subjected to too great pressure.
Fig. 505.
Fig. 504.
In Fig. 504 is shown a solid wrought iron end, suitable for
forked journals as made at Seraing. In the plan shown in Fig.
505 the journal and fork are formed in the rod end, and the
bearing is made in the crosshead, as shown later in Fig. 540,
\ 189. Such rod ends have been used for locomotives by Polon-
ceau, and for marine engines by Humphrey. In these cases the
values of b' and //, must be chosen to suit the space at the dis-
posal of the designer in each instance.
For fork bearings in which there is but little angular move-
ment, as, for example, in valve-rod connections, the form shown
in Fig. 506 may be used. In this case the key and block press
upon the half bearing of the outer part of the portion B, as
shown. Such connections are sometimes also made by using a
flexible Steel piate, as shown in Fig. 507 ; and this form may be
called a piate link. This has been used in some forms of loco-
motives and in the old style Eangen Gas Engine.
Fig. 508.
In Fig. 508 is given an end for a fork journal, such as would
be a suitable one for the cross-head end of the rod shown in
Fig. 5°°- The boxes are made cylindrical and fitted with a
wrought-iron pressure block. The pressure of the key is trans-
mitted to the block by a bronze intermediate, but this arrauge-
ment of key involves much clearance space. The method of
securing the key is that shown in Fig. 200, and the whole de-
sign is well worked out.
Another form for the cross-head end of a rod is shown in Fig.
509, and this is well suited to be used with the form given in
Fig. 501, on the crank end, for locomotive use. The key is set
up by turning the screw^; the latter can also be secured at every
sixth of a revolution, by means of the arrangement of pin ai^d
washer as shown in Fig. 237.
2 181.
For this reason the dimensions of the key should in no case be
made less than those given for stub ends for overhung pins.
Example. Given an alternating pressure ol 7920 pounds, let it be required to de-
sign a strap end similar to Fig. 496. We have for an overhung pin according to (93),
d = 2%", and for a fork journal, according to (98), d* =	and the length /' of
the latter = 2 d! = 3^6". We then give the strap the proper breadth for the over-
hung pin, that is, d— 2 e, or 2.375 — 0.58 = 1.795* say i\§". Wethen have, for the
moduli, respectively; for the overhung pin, according to^6o), d\ = 2.56", and for
the fork journal, according to (161), dx' = 2.56	=2.56	=
2.16. From this we get for the thickness of the strap 2 16 X 0.2 = 0.432", say TV\
on the sides; and at the end, 1.16 X 0.3 = 0.648. or nearly U". They may be made
the same as for an overhung pin, giving a thickness of 2.56 X 0.22 = 0.56 , or Ty ,
and a depth of 2.56 X 02 — 0.512 at the small end.
Connections for Enlarged or Neck Journaes.
As shown in § 92, there is no definite relation between the
diameter d' of a neck journal and the diameter d of the corre-
sponding overhung journal; hence it is impracticable to use the
rules dependent upon the length of the overhung journal, which
have been given in the discussions of return cranks, crank axles
and eccentrics. It is, however, necessary to devise some method
of proportioning the rod ends for such cranks, and for this pur-
pose we may use the figures given for overhung crank pins, by
making a modification in the modulus according to formula
(161). In such cases we must remember to use the value of d'
in determining the unit e for the proportions of the boxes.IIS
THE CONSTRUCTOR.
£jrawf//?.--Suppose, instead of the fork journal of the preceding example, we
have a neck journal of a diameter d’ = 4and length V = 3^", with a stub end
like that shown in Fig. 496. We have d\ = 2.56", A = 1.8125", ^ = 2.375. We may
make the value of b' the same as for the corresponding overhung crank, or b' = ot
and we then obtain from (161)	= di ±1	— = 2.56 X 1.4x4 = 3.62", say 3^".
,	> 2-375
For the boxes we have e = 0.07 X 4-75 + 0.125 = 0.45", say T75".
In the foliowing examples are given modern designs for rod
Fig. 509.
Fig. 51 i.
Fig. 512.
ends for neck journals, and others may be obtained by modifi-
cations of the preceding forms.
Fig. 510 shows a solid end connection lor a spherical journal.
The sphere in this case is made 1.5 times the diameter of the
Fig. 510.
Figs. 513 and 514 show two forms of eccentric straps, both
intended to be made of bronze. The breadth b' is equal to /,
the length of the corresponding cast-iron journal (see $ 92).
If d — di = l — b = 2.375//, we have, if d' = 15.75,
b' = l = 2.375//, d\= 1.8 ^l'^2S == 5‘7l//‘ ^ameter>
of the bolts of these eccentric straps is determined from the
following:
6 = 0.33 dl + °.°6 d/............(162)
in which dx' is the modulus for a neck journal and dx the mod-
Fig. 513.	Fic. 514.
corresponding cylindrical journal, and an example of this form
may be seen on the beam in Fig. 487^. This gives —- = 1.5 ;
d
and if, as before, we make b' — b, we have d\ — dx y/1.5 =s
1.225 dv If, again, d = 2.375/7, we have d/ — S-56^, dl = 2.56",
and d\ = 2.56 X 1225	313", say 3J". The boxes are made
without side flanges, so that they can be removed by backing
out the key. The key may be arranged to be fitted above or
below the boxes, as may be desired. When used upon locomo-
tive engines, this form is sometimes strengthened as indicated
by the dotted lines.
For the connections of crank axles, return cranks and similar
situations it is necessary to use a form of rod end which can
be opened. The following forms are designed for this purpose,
being made with blocks which are firmly bolted in place, but
readily removable.
Fig. 511 shows a form similar to Fig. 500. The block is fitted
between two shoulders and also secured by two through bolts.
Fig. 512 shows a design by Krauss, in the same style as Fig. 502,
and used with it on a locomotive connection. The block is here
made of bronze, and also forms one-half of the bearing; it is
held in place by a through bolt, which is omitted in the draw-
ing. A cross-section is shown above, the offsets serving to keep
the block from twisting on the bolt. The gap between the boxes
is filled with slips of copper. The rod and bolt are both made
of Steel.
Fig. 515.	Fig. 516.
ulus for the corresponding overhung pin. If we take the value*
above given, d/ = 5.71// and d1 = i.87/, we get
6 = 0.33 X J-8 + °-°6 X 5.71 = 0.9866", say i7/.
If we make d/ = d and d/ = dl} we obtain from (162) the
same dimensions as on a capped and bolted rod end.n6
THE CONSTRUCTOR.
In Fig. 515 is shown a design for a cast-iron strap, with bronze
lining, althougk this latter may be omitted. The eccentric rod
is secured by means of a key, and if two eccentrics are placed
close side by side, the keys should be placed at 450 from the
position shown.
Fig. 516. This is a wrought-iron strap, also lined with bronze.
In this, as in the preceding example, the joints between the two
halves of the bronze lining are close, and those of the strap are
open, and by filing the ends the halves may be closed together
to provide for wear. Instead of forging the rod in one piece
with the lower strap, it may be made with a T head and bolted
fast, as shown by the dotted lines.
Example The eccentric straps on the engines of the “ Arizona,” 6600 H. P., by
ohn Elder & Co., of Glasgow, are made as in Fig. 515, but with the rods attached
y T heads, as described above. The diameter of eccentrics dl = 54", the breadth
/ = 5", and the shaft diameter = 225^".
rod is made with a forked end, and two bearings, its lateral
stiffness is thereby increased, and m may be made as low as 4*
If m = 20 we have for wrought iron or Steel, C = 0.0346.
* Example 1.—For a wrought iron connecting rod 118.11" long, acting under
a pressure of 31,680 pounds, taking m = 20, and C = 0.0346 we have a
diameter D = 0.0346 ^ 118.n\/31680 = 5".
This gives the diameter in the middle; it may be somewhat
reduced' at the ends, these latter being made of a diameter
= 0.7 Dy giving a cycloidal sinoide as in Fig. 5, formula (23).
The ends of the rod should be worked off into the body m such
a manner as notto make too abrupt a change of cross section.
This becomes more important in high speed engines. In the
case of locomotives there is sometimes a marked bending action
upon the rod, there being a so-called “ whip action ” at every
revolution of the crank, dependent upon the rotative velocity
§182.
Round Connecting Rods.
The body of a connecting rod may be made of wrought iron,
cast iron, steel, or even wood. In the latter case it is usually
only subject to tension.
If the rod is of circular cross section, of diameter D, and
the force of tension be P, we have the following relations :
Wrought Iron	D	= 0.0148
	VP	
Steel	D	-0.0117
	y/P	
Cast Iron	D	= 0.0212
	y/P	
Oak	D s/P	-0.0578
These give stresses of 5600, 9500, 2800 and 400 pounds respec-
tively, or above two-thirds the value given for ordinary condi-
tions.
These formulse may also be used for short rods which are
subjected to compression, but if the length Ly of the rod is so
great as to permit bending, the diameter must be made some-
what greater. From an examination of case II, ? 16, and also
ijj-2	7 pp _
§ 127, we should not permit P to be greater than ——, in
which J is the moment of inertia of the cross section of the
rod, and E the modulus of elasticity of the material employed.
In order to determine how small P must be, or rather how
large the co-efficient of safety my must be taken so that we
I	7T2 / E
shall have P — —-----------------, there are various conditions
m
to be considered; the requirements being almost as varied as
in the case of columns.
Leaving then the value of tny to be subsequently determined,
we have J=	and E = 28,400,000, for wrought iron and
64
steel, 14,200,000 for cast iron, and 1,562,000 for oak, and hence
the following formulae for the diameter of rod.
Wrought Iron or Steel D = 0.0164 & m ^ L y/P
Cast Iron	D = 0.0195	s/p ■ . . (164)
Wood	D = 0.034 ^m
We have for
m = 1.5 2	3	4	6	8	10 15	20	25 30	40 50 60
•y/m — i.ii 1.19 1.32 1.41 1.56 1.68 1.78 1.97 2.11 2.24 2.34 2.51 2.66 2.78
If we represent the entire co-efficient of ^P by C we
may write for the above formulae
y/p "
and may then determine values for C according to the degree
of security required. As already stated, there is a wide variety
of values of m to be deduced from practice. For statiouary
engines of moderate size we find m, very high, often 50 to 60.
These however are not to be taken as standards because they
are rarely designed for economy of material, but rather for per-
fection of action. For medium and large stationary engines
we find m from 5 to 25, probably averaging about 20. If the
Fig. 517-
and the weight of the rod. This action also occurs in a lesser
degree in slower running engines, and is greatest at a point
betwTeen the middle and the crank end of the rod. For this
reason it is sometimes thought desirable to make the greatest
diameter of rod, not at the middle, but somewhat nearer the
crank end, as shown in Fig. 517.
For moderate piston speeds this point need hardly be con-
sidered as it is amply provided for in the co-efficient of security,
but for high speeds and heavy ends it should be given due con-
sideration. In the high speed type of engines such as the
Porter Allen, the greatest strength of rod will be found at the
crank pin end. At the same time, as will be seen, the value of
my for high speed locomotive engines, is usually made small.
For marine engines, m is usually taken quite high, viz.: 30,
40, 60 or even 80, and the ratio —= proportionally smaller.
In such engines the rod is generally made proportional to the
cylinder diameter, being about 0.085 to 0.095 times the bore. It
must be remembered that in marine engines the stresses due to
flexure of the hull, and general lack of rigidity, demand a
higher co-efficient of security than for stationary engines.
Fig. 518.
In Fig. 518 is shown a rod for a screw propeller engine. The
body of this rod is truly cylindrical, and the ends are similar to
that shown in Fig. 500.
Example 2.—Det P — 94,600 lbs. L = 60''. Taking-, as before, m —= 20 we
have	______
D	.	J~~L~
—— = 0.0346 i/ ——
y/P T y/P
D =. 0.0346 s/ 94,600 | / —- ■■ — = 4.67".
T v/94,600
In a similar case, executed by Maudslay, the rod was made 6r/
in diameter, wffiich corresponds to a value m = 54.7. The
diameter 6y of the bolts in this case w-as 3//, and according to
the rule given for Fig. 499, they should be
i 183.
Rods of Rectangudar Section.
If it is desired to make the body of the rod rectangular in
cross section, it is first necessary to determine the diameter for
circular section by the methods of the preceding section, and
then determine the equivalent rectangular section.THE CONSTRUCTOR.
ii 7
Let:
h, be the larger side of the rectangle,
b, be the shorter side,
6, the diameter of the equivalent circular section at the
same point; then for a given value of hy we have ;
4-«i.......................................<■«>
and for a given value of b
A
6
-4'(i)’— <4)*
and for a given ratio A- ;
b	b
d ^ 16 h °'
88
^ h
(166)
(167)
from which we deduce the following table :
h S	b 6	h 6	b 6	h b	b 6
1.0	0.84	1.6	0.72	1.0	0.88
1.1	0.81	l7	0.70	1.25	0.83
1.2	0.79	1.8	0.69	1.50	0.79
1-3	0.77	2.0	0.67	i.75	0.76
1.4	0.75	2.2	0.65	2.00	o.74
i.5	o.73	2.4	0.63	2.5	0.74
If it is desired to calculate the rectangular section directly,
without reference to the equivalent circular section, we proceed
i	i
U —	1*-	>j
Fig. 519.
as before, using the least moment of inertia of the section,
J — h £3, and thus obtain for wrought iron or steel :
for any given value of b :
h = 0.0000000425 vi
for any given value of h :
P IS
b3
b = 0.0002 N m
— f PL2
V h
(168)
(169)
and for any given ratio of /z, to b:
h = 0.0144	J ^AA^ ^ L v/ P
For the last formula we have, when :
b
^ = 1.5 1.6 1.7 1.8 1.9 2.0 2.1
(170)
2.2 2.3	2.4	2.5
= 1.36 1.42 1.49 1.55 1.62 1.68 1.74 1.80 1.87 1.93 1.99
The most important application of flat connecting rods is upon
locomotive engines. In this case the co-efficient of security is
taken very low, i. e.y the rod is made as light as possible, in
order that the “ stored velocity ” may be kept small, and the
“whip” action reduced.
An examination of practical examples shows values of m,
from 2 to 1.5, taken at the middle of the rod. At the cross
Fig. 520.
head end the depth is reduced to 0.8, to 0.7 that at the middle,
and the depth at the crank end is that due to the taper thus
indicated. An example of such a rod is shown in Fig. 519.
Example 2.—Given in a locomotive P= 28,600 lbs. L = 72" —= 2.5. We
O
have, if m = 1.5, according to \ 182, \/ tn — x.i. hence h = 0.0144 X x.x X x.99
72 v/28,600 = 3.5" and b = 3.5 X 0.4 = 1.40 say i^".
The “ whip ” action before referred to, is much more powerful
in the parallel rods of locomotive engines than in the main
connecting rods. Such a parallel rod, or side coupling rod is
shown in Fig. 520. The keys for the boxes at each end of the
rod are placed on the same side of the boxes, so that their
action will not affect the distance between centres, providing
i
Fig. 521.
the wear is alike upon both ends, and for this reason it is
desirable also that both pins should be of the same length.
(See § 92.) In determining the cross section of such rods, it is
to be assumed that the resistance offered is the same for both
wheels. This means, that for two coupled wheels, one-half
the total driving force is exerted upon each ; for three wheels,
two-thirds the total force is exerted upon the first rod, and one-
third upon the second. At the same time it must not be for-
gotten that under certain circumstances one of the wheels may
slip. For this reason it is advisable to take a somewhat larger
value for m, than for the driving rods. It is, therefore, not
advisable to make m, less than 2, and if possible it should be
greater, at least for two coupled wheels. If this is done there
need be no fear that the rod will be excessively strained through.
slippage of wheels.
Example 3.—The locomotive of the preceding example has two pairs ot
coupled driving wheels. We have for the force transmitted through the
coupling rod, P —	= 14,300 lbs. The length L '= 8 ft. 4 in. = 100",
and we will take the ratio -	= 2.5 as before. Taking m — 2, we have
from (170) h = 0.0144 X 1.19 X i.Q9 100 \/ 14,300 = 3.73" say 3^". This
gives for b, 3.75 X 0.4 = 1	This corresponds closely with the proportions
used on Borsig’s locomotives. Other examples in practice give values of
tn, as 1.9, 2.xi, 2.8, etc.
A rod of mixed section, passing from circular into rect-
angular, is shown in Fig. 521, being the very elegant connecting
rod of the Porter-Allen engine. In the illustration L — 5 feet.
§ 184.
Channefed and Ribbed Connecting Rods.
Cast iron connecting rods are often made of cruciform or
ribbed section, much in the same manner as axles. In such
Fig. 522.
cases it is best to determine an ideal round rod, according to
Fig. 5, from which the desired section can be derived.
For any given case, let:
S = the diameter of the ideal rod,
n, and b, the width, and thickness, respectively,
then for any selected value of b,
</4 v(t y
+
from which we get the following table :
(171)
6	b	6	b	! 6	b	6	b	I 6	b
h	h	~h	h	~V	h	h	h	| UT '	h
0.643	O.IO	0.700	0.14	0.748	0.18	0.816	0.25	0.901	0.36
0-653	0.11	0.714	0.15	0.758	0.19	0.831	0.27	0.928	0.40
0.673	0.12	0.724	O.16	0.768	0 20	0.855	0.30	0.958	0.45
0.690	o-i3	0736	0.17	0.789	0.22	0.872	o.33	0.987	0.50
Example 1.—In Fig. 522, let A B C Z>, be the ideal round rod from which
to construet a cruciform section ; E F G H, is the width selected for the ribs,
the ratio oi^T. being, for example 1.5 We then have —f— = —~ = 0 667
st	h 1.5	"
and this value in column 1, ot the table gives for , something between
h
11, and 12. This gives b — 0.12 h = o 12 S T. If P Q — 1.4 p q, we have
_ S.- *= 0.7 and in columns 3 and 4 we find b = 0.14 P Q.118
THE CONSTRUCTOR.
For constructive reasons the / section is preferred for loco-
motive rods. Such a rod is shown iu Fig. 523. This is made
with a slight swell in the middle, but the scale of the drawing
is too small to allow it to appear.
Fig. 523.
Such rods are either made with straight or rounded pro file,
as showm in Fig. 524. Neglecting the rounding we have for the
least moment of inertia of the section,
J=&(2C£*+(h — 2c)P)
For convenience of calculation we may, as in § 163, assume a
Fig. 524.
rectangular section of a height h, and breadth bQ, and then have
T— *il + 2 X [ (4-T-1 ]............................(,72)
c	B
from which, when the ratios —., and are given, the nu-
h	0
merical values can be readily deduced.
Exatnple 2.—A coupling rod of /section, on a locomotive engine built by
Krauss & Co., has the following diraensions: h =*- 3 149", b = 0.39", B — 1.85'',
c = 06, L = 96.45, and P = 10,890 lbs. To determine the degree of security
mt we substitute these values in (172) and obtain :
b0= \]i + 2 -5- (4.7s — 1) = 1.325".
v	3-*49
We then have from (168)
60*k	=
0.0000000 425 P L?
= ________(i-335)3 X 3-M9__ _ x
0.0000000425 X 10,890 X. (96-5)2
The completed rod weighecTonly 125 pounds.
?i85.
Forms or Cast and Wrought Iron Rods.
In Figures 525 and 526, are shown comparative forms fora
round connecting rod of wrought iron, and a cast iron rod of
cruciform section. In the case of the cast iron rod, the fluted
b.
c
CHAPTER XV.
portion terminates in collars near each end, the lower part at
the crank end being made of flat rectangular section, enough
longer than the crank arm to insure the necessary clearance.
In Fig. 527 are shown some special forms for forked ends.
Fig. 527 a, is a very short fork, Fig. 527 b, is for a flat wrought
iron rod, and Fig. 527 c, is suitable for a long rod of cast iron.
The boxes on these rods may be well secured by strap and key
as in Sharp’s pattern, Fig. 498. In some cases connecting rods
are made in the forni of trussed frames, and the form of the
ends are governed by the form of cross head used. The latter
will be considered in the following chapter.
CROSS HEADS.
2 186.
Various Kinds or Cross Heads.
A cross head is that portion of a machine which makes the
connection between the vibrating rod and the piston rod or other
piece having a rectilinear motion. Cross heads are made with
various kinds of journals, either overhung, forked or double;
a	b
(
Fig. 528.
and in this respect are similar to the ends of levers, the differ-
ence being that the path is curved in the one case and straightTHE CONSTRUCTOR.
ll9
in the other. The path of a cross head is generally determined
either by some form of parallel motion, or by guides, or in some
cases only by the piston or other rod to which it may be at*
tached. This gives the following classification :
1.	Free Cross Heads,
2.	Cross Heads for L,ink Guides,
3.	Cross Heads for Sliding Guides ;
and this classification will be observed in the following discus-
sion.'
2187.
Free Cross Heads.
In Fig. 528, a and by are shown two forms suitable for small
free cross heads. These are made with double journals of
wrought iron. The diameter of the piston rod in the cross
Fig. 529.
head should not be less than dr A modification of this form is
showrn in Fig. 529. Satisfactory proportions will be obtained by
making the height h in the middle equal to
jp
h = 2.sdi + —.....(173)
!4
in which A is the length of arm ; also for the thickness by which
is uni form,
P A
5 = 0.00035——...................(174)
The curve of the profile may be made as shown in § 142;
Example i. Given the load P= 8800 lbs., and the length of arm A = 15.75"
for a cross head, as in Fig. 529. According to the table of § 90, we have d2 —
0.027 \/P = 0.027 \/4400 = 1.85".
We have from (173)
and from (174)
h = 2.5 X 1.85" + “ = 5-75",
*4
b =
0.00035
83ro x 15-75
(5.75)a
1.47", say i%".
The other dimensions as given in the figure are: Hub thickness 0.5	=
0.5 X 1.85" = 0.925", say 1"; depth of key = 0.67 X 1.85 = iH” ; thickness" of
key = 0.2 X 1.87" = %"•
Example 2. The engine of the steamship “ Ea Piata ” has steam cylinder
103" diameter, with a maximum steam pressure of 26 pounds per square
inch, giving a total pressure of about 217,000 pounds on the piston rod. The
length A is 68", and in the executed engine the builder, Napier, has made
h = 28", b = 7", d2 = 10", the length of Journal = 15", these latter agreeing
closely with those obtained from §01. The hub length was made == 30", and
hub thickness 5", with a bore of 10". According to the above formulae,
we get d2 = 8.75", h = 27", b =
Fig. 530.
2 188.
Cross Heads for Link Connections.
Cross heads which are intended to be guided by a system of
linkages or parallel motions are made with a pair of link jour-
nals in addition to the journals for the connecting rod, and the
former are generally made as prolongations of the latter.
In Fig. 530 is shown a wrought iron cross head for use upon
a beam engine in connection with a Watt parallel motion. The
unit upon which the dimensions are based is
dx — 0.026 s/ P + 0.2
(>75)
in which Pis the total load on the cross head. The same mod-
ulus serves for the simple proportions of the following cross
heads. The load P3 upon the link journals can be determined
from the load P2 of the rod journals by the following relations :
/3  sin a
P2	cos /3
(176)
in which a is the greatest angle which the connecting rod makes
with the axis of the piston rod, and /3 the angle which the link
Fig. 531.
makes with a normal to the axis of the piston rod wrhen a is a
maximum, the latter positiou being determined most readily
from the drawing.
Example. If the angle a at its maximum is 200, and the corresponding:
value of /3 = 150, we have
sin a 0.3420
i---- = —---= O.35.
COS jS 0.9659
hence P3 = 0.35 P2.
When the connecting rod acts directly upon a crank the angle a is usuallv
200 or more, but when the connection is to a beam it is seldom greater than nA
Another form of wrought iron cross head for link connections is shown in
Fig. 531. Thisfoim is especially convenient when occasion requires that
the piston rod be disconnected readily, and is especially adapted for direct-
acting steam engines.
21%.
Cross Heads for Guides.
Cross heads for use with guide bars are made iu mauy varied
forms for steam engines and pumps. The form is modified to a
great extent by the number and arrangement of the guide bars.
Fig. 532 shows a much used form of cross head for four guide
bars. If the engine runs constantly in the same direction, and
the pressure upon the piston acts always in the direction of its
Fig. 532.
motion or in the opposite direction, the pressure will be almost
entirely confined to one pair of guide surfaces, the other pair
only coming into action in the case of extraneous forces. If
the pressure acts scmetimes with the direction of motion and
sometimes against it, the resuit will be to cause the pressure on
the slides to alternate. In most steam engines the pressure
changes not only in direction but in magnitude, especially near
the end of the stroke. The slides should be made of a softer
material than the guide bars in ordei that the greater wear may
come upon those parts which are most easily replaced. In order
to reduce wear it is also desirable that the surface of each slide
should not be less than 2.5 P; P being the total pressure 011 the
piston in kilogrammes, and the area thus obtained being in
square millimetres. This is about equivalent to 0.0018 P; P be-
ing the total pressure in pounds, and the area given in square
inches. Many use double this area, or 0.0036 P, with corre-
sponding reduced wear on the parts. The pressure on the sur-120
THE CONSTRUCTOR.
face of the slides, with the ordinary ratio of connectiug rod to
crank arm, will then be about 120 pounds per square inch in the
first case and about 60 pounds in the second.
If we represent the superficial pressure, rubbing velocity and
coefficient of friction for slide and crank pin respectively by
Pn vi>	we have for the lineal wear per second : Ux =
ViPi v\f\ an(^ ^2 = ^2/2 ^2/2» in which fix and //2 are coefficients
due to the materials used. Some of these values vary at differ-
Fig.533.
ent portions of the [stroke. If, however, we take them at the
same instant, we have the ratio of wear for that point,
G	Pipif 1 "l>i
The point of maximum wear upon guides is near the middle
27x R n	7x dn
of the stroke, where z\ = 7-----and z\ = 7--------
’	1	60 x 12	2	60 x i2
Taking the values of // and U the same in both cases, we ob-
tain, by substitution in the preceding equation,
P\
pi 2
which gives an average ratio of about tV and taking p2 at 1420
pounds gives about 120 pounds for pv If we consider the pres-
sure on the pin to be alternating and that on the slides contin-
!
uous, p<i becomes ouly 710, making px about 60 pounds. If the
ratio of connecting rod to crank arm is unusually small, the
pressure Q on the slides at mid-stroke should be calculated, and
^ rx*
it may be taken as Q =---—*
The cross head shown in Fig. 533 is arranged for a fork jour-
nal, the latter being also in this case made spherical. The fork,
which is keyed to the piston rod, is intended to be made of
wrought iron ; should it be made, instead, of cast iron, the
thickness of the metal about the hub should be increased to
0.28^, and its length to 1.75 dv This form permits the slides
to be brought closer together than in the preceding design.
A very simple form of cross head for four-bar guides is used
on many American locomotives, as shown at a and b, Fig. 534.
For constructive reasons, to obtain the necessary clearance, this
form is sometimes made as at b, with the middle plane of the
guides above the axis of the piston rod. The cross head is of
cast iron, with the pin cast in, and finished by special rnachin-
ery. A similar form of cross head to that shown at a is used
on the Porter-Allen engine, except that a steel pin is inserted as
shown at c. The flattening of the top and bottom of the pin
serves to assist in the distribution of the lubricant.*
The area of slides in America is about that given by the fore-
going rule.
Example. A wood-burning passenger engine has cylinder 16" diameter,
at no pounds pressure, giving P — 22,110 lbs. The surface of each slide
measured 79 square inches, or about 22,110 X 0.0036 — 79.59 sq. in.
The forms of cross head shown are generally fitted with slides
of white metal or bronze, and in some instances bearing surfaces
of glass have given gooa results.
There is one form of marine engine which requires a special
form of cross head. This is the so-called back-acting engine,
in which the crank shaft is placed between the cylinder and the
cross head, and there are two piston rods, passing above and
Fig. 535-
below the shaft. There have been many varieties of this type
constructed. In Fig. 535 is shown a design by Maudslay. The
body of the cross head is formed like an axle, with two project-
ing bosses for the attachment of the piston rods. The distance
E is governed by the diameter of the crank shaft, and A by the
clearance space required for the crank arms. In this design the
slides are placed outside of the piston rods ; other builders, as
Ravenhill, place them between the rods and the journal d\
where, as will be seen, there is sufficient room. The lower por-
tion of the slides are made of bronze and fitted with adjusting
keys. The dimensions of the body are obtained by considering
it as an axle, remembering that the forces act to produce twist-
ing with the arm E as well as bending with the arm A. The
length i' is to be taken in connection with the diameter d/, so
as to keep the pressure on the journal within practical limits.
English practice in such construction gives pressures ranging
from 800 to 1800 pounds per square inch. The diameter d of the
threaded ends of the rods is the same as given for Fig. 499.
Fig. 536.
I11 Fig. 536 is shown Stephenson’s cross head. Here the
guides are brought so close together that each pair merge intc
one, and there are but two guide bars. The middle piece, ot
wrought iron, is made with two journals, for a forked connect-
ing rod. The slides are best made of bronze, the area being as
before =r 0.0036 P, except in the case of locomotives, where the
limited space often causes it to be reduced to 0.0018 P.
Another design for double guide bars is that of Borsig, shown
in Fig. 537. This contains a fork journal, whose projected area
l' X d' should not be made too small. Sometimes this is made
so small that the pressure reaches 3000 to 4000 pounds, and hot
bearings and cut boxes are apt to lollow. Judgment in this re-
spect is most important for all bearings. The slides are made
of cast iron, with bronze shoes, which are packed out with thin
slips of copper or zinc.
* This is also done on a horizontal engine built by Brown, of Wintherthur
See Engineering, Jan., 1880, p. 70.THE CONSTRUCTOR.
121
Fig. 538 shows a noteworthy form of cross head used on the
Western Railway of France. The body is of wrought iron, the
slides and piston rod connection are of Steel. The manner in
Fig. 537-
which the reverse taper of the rod is secured, by means of a key
and conical ^hell of Steel, is of peculiar interest. Since this is
a special construction a few dimensions are given in the figure
(in millimetres). The area of the slides does not appear to be
Fig. 538.
large. The rather complicated form of the head of the pin is
shown in the lower right hand corner of the illustration.
Fig. 539 shows a cross head of the so-called “slipper ” type,
for single guide bar. This is wTell adapted for situations in
which the direction of rotation is constant and the pressure al-
ways downward. In order to provide for possible lifting forces,
and to meet the reverse action of compression and inertia, the
beveled shoes are used, although a square shoulder is to be pre-
ferred. The area of slide should not be less than 0.0036 P} pref-
erably more.*
Another form of cross head for single guide is given in Fig.
540. This is from a marine engine by Humphreys, Tennant &
Co., and is intended to serve for pressure in either direction. In
this case the beariug is in the cross head, and the pin is intended
Fig. 539-
to be fast in the connecting rod, being attached as in Fig. 537.
The wear on the bearing is taken up by the removal of thin
slips of copper, originally placed in the vertical joint; and wear
upon the guide, by the msertion of similar slips between the
cross head and slide. The whole construction is applicable to
many situations. The middle portion is in this case made of
bronze, but may be of cast iron, when the bearing is lined with
* For a similar cross head, desigtied by Stroudley, for locomotive Service,
see Engineering, Feb., 1867, p. 65.
white metal. The modulus for the dimensions is the same as in
formula (160), and the bolt diameter <5, as in Fig. 499, using foi
d the diameter of the equivalent normal wrought iron overhung
pin.
A somewhat similar cross head has been designed by Napier
for use with the horizontal back-acting marine engine, Fig. 541.
This is intended to be used with a forked connecting rod. The
Fig. 540.
middle block is made of cast iron, and the distance B is kept as
small as possible, in order to reduce the size and weight. The
depth of arm h is determined as in Case I or II, § 6. The bolts,
whose diameter 6 is calculated as for Fig. 499, are secured by
jam nuts.
In Fig. 542 is another excellent design by Maudslay for simi-
lar Service. This is for an ordinary connecting rod, as in Fig.
518. The pin is formed in the crooked wrought iron piece which
also forms the arms. The thickness b' of the latter is deter-
miued from the corresponding moment after having selected
the depth hf which in this case is made equal to d'. The value
Fig. 541.
of d' is calculated as in the case of an axle. The screw diam-
eter d' is calculated as before, and should be made full}T as large
as the formula gives. The small lug on the lower part of the
right arm is for the attachment of the pnmp rod. Such attach-
ments are frequently made to the cross heads of marine engines,
of which this is a good example. O11 the left the slide is shown
in section. This is cast of bronze, with the channels shown
Fig. 542.
filled with white metal. The small shoe on the right, which is
secured by screws, can be removed, so that slips of thin copper
can be inserted to take up for wear. These last two cross heads,
although unusual in appearance, show how a difficult construc-
tive problem can be solved completely, and may be regarded as
types.
1190.
Guides and Guide Bars.
Guides are made of wrought iron, cast iron or steel. If the
entire pressure comes upon one guide, as in the dcsigns just de-
scribed, and the guide is supported only at the ends, which are122
THE CONSTRUCTOR.
separated by a distance == sx -f- s2f it must be calculated to resist
bending. Taking the crank at right angles to the guide, as the
most unfavorable positiou, and calling the pressure Q> and the
distances of the two points of support from the centre of the
Fig. 543-
cross head as sl and s2, Fig. 543, we have the bending momeut
of the bar = Q ——, and for the relation between the depth
sx +
and width of bar :
The permissible value of stress 5* for wrought iron or Steel
should be small, say 7000 pounds, in order that but little deflec-
tion shall occur. Any springing is especially hurtful in this
case, since it prevents the entire surface of the slides from bear-
ing fairly, and thus causes greatly increased pressure upon the
Fig. 544.
points which are in contact. Deflections of TV7 or more are
sometimes found, with corresponding irregular wear upon the
slides. This subject can be thoroughly investigated graphically
by taking the various positions of the load.
In Fig. 544 is shown a form of cast iron guides, intended to
receive pressure ouly upon the lower guide. This is only sub-
ject to compression, and. hence very little deflection can occur.
The sectional view on the left shows the disposition of the ma-
terial, and it will be noticed that the flanges on the cross head
are arranged so as to retain the oil. The upper guide is bolted
to the lower, and should the motion be reversed, throwing the
pressure on the upper guide, the bolts must be made proportion-
ally stronger.
A form of guides which is coming more and more into use for
stationary engines is that shown in Fig. 545. Here the flat
guide surfaces are replaced by portions of a cylinder. An espe-
cial advantage of this construction lies in the possibility of bor-
ing the guide surfaces in exact alignment with the cylinder.
Any twisting of the cross head is prevented by the connecting
rod and crank pin, or, if necessary, a tongue on the lower slide
may fit into a groove in the guide.
The cross head for such guides may be similar to Fig. 537, the
lower guide being adjusted by a key.
The single guide bar has been used in locomotive practice,
Fig. 546, which was shown both on American and Belgian en-
gines at the Paris Exposition of 1878. The guide is bolted to
the cylinder at C> and to the yoke at J. The cross head is a
simple modification of the form in Fig. 534 £. Engineer J. J.
Birckel has shown that there is a heavy lateral stress on such a
guide bar, due to the necessary end play in the driving axles,
and a wide bar is therefore necessary. He makes the width
b — 2% hy and makes
.	_ t '[gT/*
h — Const V -QL,
in which G is the weight of the parts subject to lateral vibra-
tion, Q the normal component of the piston pressure, L the
length of guide bar, and H the. distance from centre of bar to
centre of rod. In the case of a cylinder iW' diameter at 100
lbs. steam pressure, G — 8800 lbs., L — 51.2 and H = 7.5//*
the values obtained are : b = 8", h = 3//.
Fig. 547-
Fig. 547 is a cast iron guide for horizontal marine engine,
suitable for a cross head such as is shown in Fig. 540. This is
especially arranged to retain the lubricating oil, and as the
cross head moves between the positions i/ — 1 and 2 — 2/,
every stroke, it dips in the oil at each end and carries it over
the guide.
Example. THe steamship “ Arizona ” is fitted with single guide bars and
automatic lubrication. The pressure on one slide is 64,000 lbs., the area
being 47" X 27" = 1269 sq. in., or a pressure of about 50 pounds per inch.
CHAPTER XVI.
FRICTION IVHEELS.
\ «9i.
CIASSIFICATION OF WHEEES.
Wheels are used in many varied ways to transmit motion in
machine construction. They may be divided into two great
classes:
1.	Friction wheels,
2.	Gear wheels,
according as they transmit motion by frictional contact, or by
the engagement of gear teeth.
Each of these classes may again be divided into :
(a)	Direct acting, and
(b)	Indirect acting wheels,
according as the force is transmitted directly from one wheel to
another, or indirectly, by means of belt, cord, chain, or similar
device. This gives four divisions for consideration, as follows :
I.	Direct Acting Friction Wheels, or friction gearing, pure
and simple.
II.	Direct Acting Tooth Gearing, otherwise ralled simply.
gearing.
III.	Indirect Acting Friction Wheels, such as Pulleys, Cora
Wheels, &c.
IV.	Indirect Acting Tooth Gearing, such as Chain Wheels.
The first three fornis exhibit the greatest variety, and will be
given the first consideration.
The relative positiou of the axes has a most important influ-
ence upon the form of a pair of wheels. The positions may be
grouped as follows:
1.	The axes geometrically coincide,
2.	They are parallel,
3.	They intersect, at an angle,
4.	They are at an angle, by pass without intersecting.
This gives four groups under each of the preceding main divi-
sions.THE CONSTRUCTOR.
123
3 192.
The Two Applications of Friction Wheels.
Direct acting friction wheels may be used to accomplish either
one of two different functions and their construction varies ac-
cording to the use to which they are put.
The first application is that in which the wheels are pressed
together with sufficient force to prevent the surfaces from slip-
ping upon each other, under which circumstances the motion
of one wheel will be transmitted to the other.
The second application is that in which the so-called rolling
friction is so small that the wheels, when interposed between
two surfaces which are relatively in motion, act to reduce the
otherwise injurious frictional resistance.
Hence we see that friction wheels may be used :
(a) To transmit motion, and	>
(£) To reduce resistance.
The first application includes what may be called driving
friction wheels, or commonly simple friction wheels, and the
secbnd application includes all the various forms of friction
rollers, roller bearings, ball bearings, and the like. The two
kinds have also been termed friction wheels and anti-friction
wheels.
\ *93-
Friction Wheels for Paranto Axfs.
The surfaces of a pair of friction wheels in contact are almost
always of circular curvature, and when a pair of such wheels
roll freely upon each other the number of revolutions will bear
an inverse relation to the radii of the respective circles. This
ratio is called the velocity ratio of the wheels. If we call the
revolutions per minute of each wheel n for the driver and nx
for the driven wheel; and the corresponding radii R and Rlf
we have for the velocity ratio :
nx__R
n Rx
(178)
Friction wheels for parallel axes are made with cylindrical
surfaces. Fig. 548. In order that there shall be no slipping be-
tween the surfaces we must have a pressure Q, which, to transmit
a force P, at the periphery of the wheels, must not be less than
of about 28 pounds per inch of face width, or from 15 to 20
pounds for the other woods above mentioned.
This gives for maple face :
P 1180 (HP)
28 v
(180)
and a width 1 x/2 to 2 times greater for the other woods, H P
being the horse power transmitted, and v the circumferential
velocity in feet per minute. Substituting for v its equivalent
value, — we have
12
f 2414 . H P
b~ R n
(181)
Such wheels are made in practice up to 6 feet in diameter and
30 inches face, transmitting upwards of 60 horse power.
According to the experiments of Wicklin, the coefficient of
friction is about 0.30 to 0.32, from which the pressure of contact
must be Q = 3^ P. The ease with which these wheels can be
thrown out of gear is a very convenient feature.
Example 1. Let 10 H. P. be required to be transmitted by friction wheels,
the speed of shaft being 80 revolutions per minute, and a circumferential
velocity of 1180 feet per minute given. We get from (180) b —	. 10 = 10"
face, and from (181) R = 2^14	= 30". If the driven shaft is run 100 rev-
10 X 80
olutions per minute, the radius of its wheel will be R\ — 30" X 0.8 = 24".
Example 2. Required to transmit 1 H. P., the given value being n = 90,
n\ = 75, R — 12", R — 13.66". From (181) we Have
b =
2414
■
12 X 90
If pine is used, this should be doubled, giving b = 4^".
The method of construction of these wheels is as follows:
For large wheels, 4 to 10 feet in diameter, the rims are made
from 6 to 7 inches deep, built up of wooden segments i}( in.
to 2 in. thick, forming to tV the circumfereuce, and so placed
that the direction of the fibre shall follow the circumference of
0 = j..................... (>79)
f being the co-efficient of friction.
Ihe value of f for various materials may be taken as follows :
For Iron on Iron ...»..........0.10 to 0.30
“ Wood on Iron................0.10 to 0.60
“ Wood on Wood ....... 0.40 to 0.60
Friction driving is often very simple and practically effective
It had been almost neglected for general uses, when it was very
successfully applied in various forms of saw mill machinery.
This was especially the case in the lumber regions of America.*
The best results are obtained in practice from surfaces of
wood on iron, the wooden surface being preferably the driver,
so thatany stoppage on starting shall not wear hollows in the
softer material.f The rim is built up in such a manner as to
place the grain of the wood as nearly as possible in the direc-
tion of the circumference. The best wood for the purpose is
maple, but lindeu, poplar and pine have been used with good
results. Great care must be taken to make the wheels truly
cylindrical, and they should be keyed upon their axles and fin-
ished while running in their own proper bearings. Under these
conditions a wheel of maple can transmit a circumferential force
* See Wicklin, “Frictional Gearing,” Sci. Am., vol. 26, p. 227; also Apple-
ton’s “ Cyclopsedia of Mechanics;” vol. 2, p. 36; also Cooper’s “ Use of Belt-
inf Surfaces of compressed paper against iron are now in general use.—
Trans
the wheel as nearly as possible. These segments are firmly
clamped together and secured by bolts or nails. The actual
face is made about 2 in. narrower than the working face b. This
rim is then securely fastened to the arms, which are very strong
and made with feet or pads which are mortised into the rim and
both keyed and bolted fast. The number of arms varies from 6
to 8, and for very wide faces two sets are used; see Fig. 549.
An additional ring of wood is then put on each side, bringing
the width up to the full value of b, and these outer segments
are deeper than the others, so that the ends of the keys are en-
tirely covered; the completed wheel is then turned and finished
in place, as before stated.
Smaller wheels are built upon iron drums, the segments being
screwed together and clamped between the outer rims, Fig. 550.
Projections on the iron rim, let into wood, prevent the latter
from turning. The total thickness of rim is about 4 in. Care
must be taken that the wood is thoroughly dry.
The driven wheel of iron is made similar to a belt pulley, but
with a much stronger rim and more and heavier arms ; when a
wider face than 16 in. to 18 in., double arms are used. Both
wooden and iron wheels should be carefully balanced, in order
to avoid vibration.
An important and ingenious use of friction wheels is in con-
nection with a drop hammer, the wheels being used to raise the
drop. MerrilPs drop hammer, Fig. 551, is operated by tw o iron
friction wheels A and C, which together act upon the oak
plank B, to which the hammer drop is attached. The roller A
is the driven one, and its shaft runs in eccentric bearings on
each side, which are operated by levers D and press the parts to-124
THE CONSTRUCTOR.
gether. When the parts are in the position shown, the plank
and hammer are raised, and when the lever D is lifted, the
wheels separate and the hammer is allowed to drop. In some
B
similar desigus both rollers are driven, as in the hammer of
Hotchkiss and Stiles,* and also in the so-called “ Precision
Hammer,” of Hasse Sc Co., of Berlin.f
2 «94-
Friction Wheees for Inceined Axes.
When the axes are inclined to each other, the surfaces of the
wheels, unless they are very narrow, become portions of cones,
with a common apex at the intersection of the axes. Fig. 552.
Each pair of circles in the surfaces then roll together as if cyl-
indrical. Wheels of this sort may be constructed in a similar
Fig. 552.
and the linear velocity high, in order thatthe driving force may
be kept as small as practicable. The most convenient modifi-
cation of this form is that in which the angle B of the cone is
made 180°, when we obtain a pair of friction disks, Fig. 555.
The velocity ratio, when A is the driver and B the driven,
and x is the distance from the axis of a, is expressed by :
”«—x sin /? which =
n r	11 r
(«32)
when /3 == 180°. The change of velocity is expressed by the
line O JV. If B is the driver and A driven, we have
___L___, which
11 x sin (3
(183)
when /3 = 180; n being the number of revolutions of B. These
are the equations of an equilateral hyperbola; see Fig. 555.
When the value of x approaches near zero, the driving of A by
B becomes impracticable.*
In Fig. 556 is shown a form of variable speed gear in
which one disk is placed between two others. The disks Ax
and A2 revolve with the sanie velocity in opposite directions,
and the driven disk B is placed between. The velocity ratio
can be varied from o to — proportional to x.f The pressure
is applied at the ends of both horizontal shafts. This arrange-
manner to those described in the precediug section. In Fig.
553 are shown, at a and b, two sizes of conical wooden friction
wheels. The outer disk is placed with the libres in a radial
direction, but the others have the grain of the wood arranged as
nearly as possible circumferentially. These disks should be
most carefully fitted, glued and bolted together. Especially im-
portant is it that conical surfaces should be turned to the cor-
rect angle. The pressure is applied from the end of one of the
two shafts in such a mauner that the force may be applied or
removed at the thrust bearing.
The most extensive application of friction driving, both with
cylindrical and conical surfaces, is found in locomotive engines.
The high pressures necessarily used compel in this case the use
of iron or steel tires. The force Q here exceeds 6 tons.J
In some cases a combination of oue conical wheel and one
narrow wheel with rounded edge, as in Fig. 554, may be used for
the transmission of small powers. In this case both wheels are
made of iron. The pressure is easily applied to the disk wheel
B, and the mechanism is so arranged that it can be shifted
along its axis, so that a variable speed motiou is obtained. It
must be noted that in this form the surfaces in contact are ne-
cessarily very limited, and hence it is desirable, as in the case of
friction couplings, to have the diameters as large as possible,
* See Appleton’s “Cyclopaedia of Mechanics,” vol. 2, p. 85.
t German Patent 2685. In this hammer the lower part of the plank is re-
duced, and the whole design very ingeniously worked out.
X The surfaces in contact are sensibly flattened. Krauss’ experiments
showed that with a pressure of 12.000 pounds, a steel tire on an iron railgave
a surface of contact of 0.309 sq. in., and with a pressure of 8250 pounds, a
surface of 0.24 sq. in. In the Fontaine locomotive the pressure of contact
was about 8 tons on each wheel.
ment has been used for driving centrifugal machines, and more
recently for potters’ wheels, the control over the speed being
especially useful in the latter case, the position of the variable
disk being controlled by a treadle.
Another arrangement of disk friction wheels to produce a
variable speed is that of Rupp, shown in 557. A is the driver,
B the driven, and C the intermediate, the latter being ad-
justable on its axis. The variation is between the limits
-R
R
and
R
a—R
according to the rclation
nx x
11 a — x
which gives the equilateral hyperbola shown in Fig. 557, inter-
secting the axis of ordinates when x — o. Rupp recommends
especially that the intermediate wheel be made of a number of
* In the variable speed gear of Eecoeur (German. Patent 17,078) a loose disk
is filled iu the centre of A, so that if B approaches too near the centre the
motion ceases.
f See Berliner Verhandlung, 1866, p. 39. This arrangement has been used
especially for regulating the speed of cotton-spinning machinery.THE CONSTRUCTOR.
125
thin disks, ali loose upon the shaft. This does not appear to be
advantageous in view of formula (184), since there is a different
ratio for each disk, and hence some of them must slip.
A similar device is that of Barnhurst, Fig. 558, in which the
disk is placed between two cones.*
By making two of the disks fast on one shaft, and placing
the driving wheel between them, with sufficient clearance to
enable either to be brought in contact with the driver, the driven
shaft may be operated in either direction or allowed to remain
at rest, Fig. 559. Ax A2 are the driven, and B the driver. This
is ingeniously applied in Cheret’s Press, in which the screw of
the press is on the axis of B, and is turned in either direction
by the friction wheels.
1195-
Friction Wheeds with Inceined Axes not Intersecting.
In the case of friction wheels whose axes are rigidly held, and,
while inclined, do not intersect each other, there is always more
or less lateral slipping. The figures which, under these condi-
tions, exert a maximum amount of rolling action and a mini-
mum of slipping are a pair of hyperboloids of revolution (see
$218). If, however, the axes are so arranged as to permit
longitudinal motion, either with the bearings or in them, the
wheels will be relieved from slipping. Such an arrangement, by
Robertson, is shown in Fig. 560. f The disk A acts upon a cyl-
* See Engineer, June, 1880, p. 404; also H. Kdnig, German Patent No. 9365.
t See Engineer11867, p. 410, in which many interesting designs by Robert-
son are given.
inder B, the axis of which makes a small angle with that of A.
When the disk A is revolved, it rolls a helical path upon the
cylinder, and also moves in the direction of its axis. The angle
a corresponds to the angle of the screw thread. Robertson has
applied this device as a feed motion to a wood lathe. This ar-
FiG. 560.
rangement may also be reversed, A being held in its bearings,
and B, with its bearings, permitted to travel.
The same principle may be used with cones on disks, but
these devices appear to possess limited practical application.
Friction wheels, the axes of which coincide, are the same as
friction couplings.
\ 196.
Wedge Friction Wheeds.
Wedge friction wheels are those in which the cross section of
the rim is wedge-shaped. They were designed in Italy by Mi-
notto and in England by Robertson, and hence are known by
both names ; in both cases being applied to wheels with parallel
axes. Two forms of rim section are given in Fig, 561. In this
case the radial pressure Q is much less than with cylindrical
wheels, and for any wedge angle 6 it is equal to
Q=p
sin-------\- f cos
2 J
(185)
A disadvantage of this form is the fact that true rolling action
only takes place in one cylindrical section through each rim,
and hence there is much hurtful friction from the slippage at
other points ; this defect becomes less as the ratio of the wedge
depths £, kx to the radii B, Rx diminishes.* In order that the
k k
ratio and may be kept as small as possible without re-
ducing the surface of contact, the rim is made with multiple
grooves, as in the form on the right. The angle 6 is generally
made = 30°, although Robertson used much smaller angles.
Fig. 561.
These wheels grow warm and wear rapidly when operated con-
tinuously at high speeds. Minotto has also made especial ef-
forts to design bevel wedge friction wheels; he uses only one
groove, and adjusts the position so that wedge profile shall al-
ways act at the same point. Robertson makes the grooves non-
adjustable, as in spur wheels. Wedge friction driving has been
proposed for locomotive driving, and models made on this plan
have ascended steep grades ; the wear in this case comes mainly
upon the track.
Wedge friction wheels have been used in America for many
years on winding engines; and they are especially useful in
driving ship’s windlasses, on account of the ease with which
they can be thrown in and out of gear.f More recently wedge
friction wheels have been used by Gwynne and also by Weber
in Berlin, at high speeds, and apparently with good endurance,
* Hausen, in Dingler'sJournal, vol. 137, 1855, p. 1, shows that the actual
rolling circle is always on that portion of the wedge surface towards the
driving-wheel, and changes its position when the driver becomes the driven.
See also Ad. Ernst, in Zeitschr. d. V. deutscher Ingenieure, xxvi, p. 243.
f H. D. Andrews’ steam windlasses are made with wedge gear of from 4 to
12 grooves. The dia>meters of the friction wheels are as follows:
H. P.	Slow speed.	Fast speed.	Drum. Diam. Eength.	
5			8—26"	6"	27"
8		. . . 4-30"	8—26"	8"	27"
10		. . . 6—36"	12—30"	8"	30"
15			12—30"	8"	30"126
THE	CONSTRUC.
driving centrifugal pumps at 700 revolutions per minute. These
wheels are with single groove and wedge, the wedge being of
curved profile, and hence acting somewhat like the adjustable
device of Minotto.*
Single-groove friction wheels have also been used in America
for mill gearing.
Sellers has devised an ingenious form of wedge friction gear
for changing the rate of feed on engine lathes. This is com-
posed, Fig. 562, of two simple disks and a pair of very obtuse
cone plates, the latter being pressed together by springs. The
axis of the cone plates is movable, thus giving change of speeds.
The ratio of change is similar to Rupp’s gearing, formula (184).
2197-
Special Applications of Friction Wheels.
The previously stated condition of wedge friction wheels, that
there is but one line at which rolling action takes place, and that
slippiug occurs at ali other points of contact, is utilized in vari-
ous methods in machine design, as for example, in rolling mill
machinery.
In this case a third piece is driven, compressed and altered in
form between two friction rolling members. The rolls and the
metal may be considered as a train of friction gearing. In the
case of a piate mill, the piate may be considered as a pair of
friction wheels of infinitely great radii; this is also the case in
rolling bars. In a tire mill one surface is an internal and one
an external wheel, of variable radius. The three-high mill may
be similarly compared to a train of friction gears.
A very interestiug application is that referred to in § 148, as
in use at the Kirkstall Forge, and shown in Fig. 563. A and B
are plane friction disks. The round bar C passes between them,
slightly above the centre and partly rolling, partly sliding, re-
ceives both an endlong motion and a motion of revolution upon
its axis. The disks revolve in the same direction, and of the
opposed forces which tend to cause revolution of the bar those
which act in the portion of the disks between their axes, i. e.,
between the vertical dotted lines in the figure, preponderate,
and determine the direction in which the round bar revolves.
The horizontal components of the sliding forces at all portions
of the disks, act to carry the bar forward, so that it receives a
combined spiral motion and is at the same time rolled and
straightened. The earlier method of rolling round bars was by
means of semicircular grooves, but this does not give either as
round or as straight a product. Many similar examples in roll-
ing mill machinery will be found, resembling friction driving
gear.
In the same way, various forms of grinding mills are made
upon the principle of friction combinations, as in the case of the
Bogardus mills, with flat grinding disks, and also in the case of
grinding rollers, Fig. 564. Here the round trough A revolves,
* See Engineering, 1868, pp. 502, 593, and 1869, p. 353. Engineer Brauer,
assistant in the Royal Technical High School, ha^ attempted to adapt the
principle of the Weston Clutch (£157) to friction wheels. The wheels are
made of a number of thin plates, with rubber washers between them, and
a slight axial pressure is sufficient to cause them to grasp each other with
much friction. A description will be found in Berlin Verhandlung, 1877, p.
295-
and in it act the rollers Blf B2, and the width of face of the
rollers compels a sliding action, forward on the outer edge and
backward on the inner. The trough may be stationary and the
shaft a, carrying the rollers, revolve. Rollers with inclined
axes are also used for grinding, and a similar device has been
made for straightening round rods.
ROLLER Bearings.
Roller bearings, sometimes called anti-friction rollers, may be
used in either of two forms:
(a), in such manner that the rollers are carried in their own
bearings, the latter receiviug the load ;
(0), or in such a manner that the rollers are placed between
two moving surfaces and act with a rolling motion upon both of
them.
Roller bearings are used in connection with surfaces which are
flat, round, or even spiral. Examples of rollers upon cylindri-
cal surfaces are given in Fig. 565, in which a and b are forms
used on pillar cranes, and bx is the more general form of b. Roll-
ers are also used in axle bearings, and in heavy pulley blocks,
where indeed the sheaves themselves are a form of friction roller.
A form of roller bearing which is subject to very heavy loads
is that used to carry the ends of bridge beams and trusses, to
provide for expansion and contraction. These are made either
with round rollers, as at a, Fig. 566, or with double segments, as
at b.
For round, solid rollers, the load may approximately be in-
vestigated as follows :—Let l be the length, r the radius of each
roller, and P the load. This load wTill be carried by a surface of
a -width b, included in the angle (measured at the centre of the
roller) /2 = 2<p. We have for the relation of these elements :
P=E l r^- and 5= ^ P*
48	16 r
E being the modulus of elasticity, and ^ the fibre stress upon
the material.
Also:
5=0.83	5
and
It will be seen that for any given material the relation —
can be sc made as to keep the stress within practicable limits.THE CONSTRUCTOR.
127
These may be chosen as follows, botb surfaces being of the
same material;
Casi Iron.	Wrought Iron. Steel {Jiardened)
E= 14,220,000	28,440,000	42,660,000
P
___= 425 to 500	340 to 400	1000 to 1400
l r
X= 1 1,000 to 12,000	11,000 to 13,500	25,000 to 32,000
Example 1. The bridge over the Elbe at Hohnstorf has spans of 330 feet*
The bearings are made of cast iron of the form shown at b. The pressure is
792,000 pounds on six rollers, the dimensions of the latter being, l — 53",
r= 4,125".
Wehavetherefore —= —I3J,000_ = 603.8,
rl 53X4.125
hence 0 = 2]------------603.8 = 0.126.
^ 14,220,000	^
This gives for the breadth b of the contact surface under this load,
6 = 0 r = 4.125 X 0.126 = 0.522, and S = ^ 02 =	(0.126)2 = 14,280 lbs.
Exa7npie 2. Bridge over the Rhine at Wesel; span 125.7 feet, rollers and
bearings of hardened Steel. The load is 770,000 pounds on six rollers, as
shown at a, and /=27 75", r = 3.875". These values give —	1193; 0 = 0.11
and 3 = 0.43", and 5= 32.450 lbs.
Example3 Clifton Bridge at Niagara. The load of 171,600 lbs. is carried
upon 11 Steel rollers, on bearings of the same material, their dimensions be-
ing/=6.3" r — 0.6". This gives a high value for -^- = 4127; 0 = 0.17, hence
b = 0.102, and 5 = 74,210 lbs.
Ball bearings are frequently used instead of cylindrical bear-
ings, and for some forms of journals are niost convenient,
although the bearing surfaces being only points, they are not so
well adapted for heavy pressures.
A
i
1
i
i
i
i
i
i
A*.
Fig. 567.
A form of roller bearing used in agricultural machinery is that
of Cambon, shown in Fig. 567. The Steel ring, with semicircu-
lar groove is secured to the shaft, and in the groove, or globoid
ring, the Steel balls, 09 to 13 in number, are placed. These are
held in place by a corresponding external ring, made in halves.
The outer ring is held in the journal box. Cambon uses balls of
to i7/ in diameter, which are rolled in a mill of similar con-
struction to the bearing.
It may be remarked that when roller bearings are used for car
or wagon wheels we have a combination of friction wheels, since
the wheels themselves are properly friction rollers and the intro-
duction of rollers into the axle bearing makes the latter device
what may be termed friction wheels of the second order.
Another System of the higher order is the very ingenious ar-
rangement of planet rollers of Mechwart. In this apparatus,
which seems to have been so completely conceived by the in-
ventor as to be incapable of further improvement, friction rollers
are utilized to the fullest possible extent. The System is indi-
cated in the diagram Fig. 568. E, Rv R2, are the rollers of a
roller-mill. The axis a, of the roller R, is carried in a station-
ary bearing; the axes ^ and 6 are carried upon links by ax and b.2
a2, so that they may be moved to or from ay or may be pressed
against the latter. The length of these links is governed by a
screw adjustment. The rollers are pressed together by the ring-
roller R3, acting upon the planet roller r3 and rollers rx and r2,
the latter being loose upon the axes of the main rolls Rx and R2.
The planet roller r3 acts against the roller r, which is fast on the
axis of R. If it is desired to exert greater pressure upon the
rollers, the roller Rx is forced towards R3 by means of the lever
combination aj bx c1 dx, the lever dx cl being held in position by
a ratchet section, the position being changed as the rollers wear.
The ring roller R3 reduces very greatly the wear upon the jour-
nals of the grinding rollers, as it converts the greater part of
their journal friction into rolling friction. In order to equalize
the effect of the weight of the upper roller, the lower roller Rt
is counterbalanced by a weight, which, acting through the Sys-
tem of levers c2 d2 a2 exerts an upward pressure equal to the
combined weight of Rx and R2.
The above described apparatus is fitted to both ends of the
rollers. In order to provide for any slight inequality in diame-
ter between the opposite ends of the rollers, another adjustment
is provided. This consists of the lever a b c, to which the planet
roller is suspended by the link b a3. This permits the planet
roller to be forced into the narrower space between r and R3y by
means of the worm and worm sector shown at c. The ring R3
is a continuous steel forging, and the rollers are chilled castings.
The rollers R, Rl} R2 are geared together, the gears having
double spiral teeth, as shown in g 222.
Friction rollers are sometimes used in connection with toothed
gearing, an example of which will be seen later in Fig. 589.
Whitworth used rollers in the place of a nut in his screw planing
machiue, and they are used in worm gearing by Bourdon, and
the higher form of worm, the globoid (see $ 224) by Jensen, and
by Hawkins. Many applications are also found among instru-
ments of precision, notably Atwood’s Machine, and Amsler’s
Planimeter.
CHAPTER XVII.
TOOTHED GEARING.
S*99-
Ceassification of Gear Wheees.
The relative position of the axes of gear wheels governs their
general form, although not to so great an extent as in the case
of friction wheels. This is due to the fact that the geometric
shapes, which in the case of friction wheels form the actual
surfaces, are only theoretically used in the case of gear wheels
as forms upon which to design the teeth.
Gear wheels for parallel axes are called spur gears; their form
is based on the cylinder. Wheels for inclined intersecting axes
are based upon cones, and are termed conical gears, or more
commonly bevel gears. For inclined, non-intersecting axes, the128
THE CONSTRUCTOR.
base form is the liyperboloid, which name is also given to the
gears. For many applications of inclined axes the teeth are
made spiral, giving the various forms of spiral gears and worm
gears.
If the motion is to be transmitted at a uniform rate, the base
figures are solids of revolution (cyliuders, cones, hyperboloids);
the wheels themselves being round, while if the motion is not
to be transmitted uniformly the outlines will be irregnlar. In
the following discussion only round gear wheels will be con-
sidered.
A. THE CONSTRUCTION OF SPUR TEETH.
Accuracy in spacing is of especial importance in the change
gears of a lathe, as any error in a gear produces a corresponding
defect in the screw which is being cut. Such defects are stili
more apparent if the lathe is used for cutting spiral gears (see
$ 221, below). The smooth motion which such spiral gears are
intended to produce may thus be p^evented by irregular cutting.
The choice of tooth outline to be adopted, either for the entire
product of an establishment. or for any class of work, can only
be made after a careful consideration of all the conditions upon
which so inuch depends. These considerations will be taken
up and discussed in the following sections :
§ 200.
Generae Considerations.
The form of gear teeth may be so chosen that all gears of the
same pitch will work together. Wheels of this sort are called
interchangeable, while wheels which are not so made will ruu
only in pairs.
In each pair of round wheels there are two circles, struck
from the ceutres of the wheels, which have at each moment the
same linear velocity, and are called in general the ratio circles.
The particular ratio circles for a pair of spur gear wheels are
called their pitch circles. Upon these circumferences are laid
out the pitch divisions, i. e., the spacings from centre to centre
of the teeth.
The teeth themselves are prismatic in shape, the base of the
prism being the outline of the tooth. The portion of the tooth
which projects bej^ond the pitch cylinder is called the point of
the tooth, and that portion within the pitch cylinder is the base.
The surfaces of the point are called the faces of the tooth, and
the surfaces of the base, the flanks.
Fig. 569.
In spur gear teeth we also have, Fig. 569, the length /, the
breadth on face b, the tooth thickness d, the pitch being indi-
cated by t, the two latter being measured on the curve of the
pitch circle.
All the teeth in one and the same wheel are made of the same
thickness and same spacing, so that any tooth will fit into any
space. It follows from this that, the spaces being made of
suitable size to receive the teeth, that the inverse ratio of the
number of revolutions n and nx of a pair of wheels is equal to the
direct ratio of the respective numbers of teeth Z, and ZXi or:
Z_
(186)
This statement is equally true for circular and non-circular
wheels. It also holds good if the thickness of the teeth is dif-
ferent at different portions of the circumference, providing only
that care is taken that the spaces in the smaller gear come
around to meet their proper teeth each revolution. If, there-
fore, under these conditions we have the number of teeth given
for any case, we may consider the above relation as the funda-
mentalformula of the transmission of motion by toothed gear-
ing. This is rather to be considered as an inevitable principle
of construction rather thau a fine geometrical distinction. It
depends upon the primitive form of gear construction which has
been in use for centuries in the Orient, where no other care is
taken in the proportioning of gears except that they are large
enough and that the pin teeth are sufficiently strong.
No general principle can be laid down for the form of the
flanks of teeth. For round wheels, the ratio of the angular
velocities, i. e., that of the differentials of the simultaneous
2
angles of rotation and a must equal the ratio —. This affects
Zl	da
the flanks as being those surfaces upon which the ratio —
d a
depends.
The form of the teeth is of great importance. Especially ne-
cessary is it that the division of the pitch shall be accurately
made ; errors in the shape of the flanks are even less injurious
than errors in the dividing. Accurate spacing can only be ac-
complished by the use of suitable gear-cutting machinery, and
such machines are now in general use.*
* The machines most extensively in use in Germany are those of the Ber-
lin Anhalt Maschinenfabrik, in Berlin, and the Maschinenbauanstalt of
Briegleb, Hanieu & Co., in Gotha.
8 201.
Pitch Radius. Circumferentiae Division.
For any pitch /, and number of teeth Z for a round wheel we
have for the radius R of the pitch circle:
R Z
— = — =0.15916 Z
t 2 7C
which gives, according to formula (186)
R___ nx
Rx n
(i«7>
(188)
The radius obtained from formula (187) is never a whole num-
ber, because tt is an irrational number, so that R will always
contain a fraction if the pitch is a whole number. The follow-
ing table will facilitate the computation in such cases. If the
irrational .feature is to be kept out of the value of R, the length.
of the pitch divisions must not be made whole numbers, but
fractions or multiples of xr, and this method is used in many es-
tablishments. If we call the pitch = /, we have under this plan:
..................<“»>
This corresponds to the so-called “ diametrical pitch ” System
of Kngland and America.
Example. Suppose a wheel of 24 teeth and a pitch of 6 X 3.14 millimetres,
24
we have according to (187), for the radius R, of its pitch circle, R =	6=
72 mm, and if we have in English units a pitch of & X 3.14 for a wheel of30
3°	45	13
teeth, we have according to (189), R — — X 3 = ~r — 2—7-
2	10	10
A convenient instrument in this connection is a circumfer-
ence scale. This consists of a prismatic rule of wood or metal
upon which, for the metric System, a length of 314 millimetres
is laid off, and on a parallel line the same distance is divided
into 100 equal parts. Corresponding points on the two scales
will then have to each other the ratio 1 : xr. This scale is also
useful for the rectification of circles and circplar ares. Similar
scales may be prepared upon sixteenths, tenths or any subdivi-
sion of the inch.
In the following discussion both methods will be used, namely:
that in which the pitch is taken in rational numbers, thus mak-
ing the radius irrational; and that in which the pitch is made
rational in units of the circumference scale, and hence the radius
becomes rational. The following table is not to be confounded
with that of Donkin, * made according to the expression,
which gives the radius of the circumscribing circle of a regular
polygon of Z sides, each having a length equal to t. This latter
radius differs from the radius R above referred to for small
values of Z, and confusion in this respect has given rise to
numerous errors.
8 202.
Tabee of Radii of Pitch Circees.
Examples in the use of the following table. (Note. This
table was calculated for use with the metric System, in which
the pitch is generally taken in millimetres. It may, however,
be used equally well in English units, by taking the pitch in
sixteenths, in order to make the divisions sufficiently small.)
Example 1. A wheel of 63 teeth. and i§" pitch is to be made; required. the
radius of the pitch circle. The pitch is here 30 sixteenths, and we have at
the intersection of the columns for 60 and 3, the number 10.03, hence R =
10.03 X t = 10.03 X 30 = 300.9 sixteenths or 18.8" giving a diameter of 37.6''.
The table may also be used to determine the number of teeth
when the pitch is chosen and the radius given.
♦ See Salzeuburg’s Vortage, p 93, and others.THE CONSTRUCTOR.
129
Example 2. Given a wheel of 40 inches radius and i.6"pitch. This gives
R 40
---= - - = 25. The nearest value to this in the table is 24.99^5 at the inter-
/	10
section of 150, and 7, and hence 157 is the number of teeth.
When the radius and number of teeth are given the table
may be used to find the pitch.
Example 3. Given R = 15!" Z = 54.
tion of 50 and 4, the value of y = 8.59.
We find in the table at the intersec-
We then have t = -—= 1.83".
8-59
Z	O	1	2	3	4	5	^ I	7	8	9
0	0.00	0-159	0.318	0.477	0.637	0.796	0-955	1.114	1-273	1432
IO	1-59	1 75	1.91	2.07	2.23	2-39	2-55	2.71	2.86	3.02
20	3-i«	3-34	3-5°	3-66	3.82	3-98	4.14	4.30	4.46	4.62
30	477	4-93	5-°9	5-25	5-4i	5-57	5-73	5-89	6.05	6.21
40	6-37	6-53	6.68	6.84	7.00	7.16	7-32	7.48	7.64	7.80
50	7.96	8.12	8.28	8-44!	8.59	8.75	8.91	9.07	9-23	9-39
60	9-55	9.71	9.87	10.03	10.19	io-35	10.50	10.66	10.82	10.98
70	II.I4	11.30	n.46	11.62	11-78	11.94	12.10	12.25	12.41	12.57
80	12-73	12.89	•3-°5	13.21	13-37	13-53	13.69	13-85	14.01	14.16
90	14-32	14.48	14.64	14.80	14.96	15.12	15.28	15-44	15.60	15-76
IOO	15-92	16.07	16.23	16.39	| 16.55	16.71	16.87	1703	17.19	17-35
IIO	17.51	17.67	17-83	17.98	18.14	18.30	18.46	18.62	18.78	18.94
120	19.10	19.26	19.42	19.58	19-73	19.89	20.05	20.21	20.37	20.53
I30	20.69	20.85	21.01	21.17	21-33	21.49	21.65	21.80	21.96	22.12
I40	22.28	22.44	22.60	22.76	,22.92	23.08	23.24	2340	23-55	23-71
150	23-87	24.03	24.I9	24-35	24.51	24.67	24-83	24.99	25-15	25 31
l6o	25-46	25.62	25.78	25.94	26.10	26.26	26.42	26.58	26.74	26.90
170	27.06	27.21	27-37	27-53	27.69	27.85	28,01	28.17	i 28.33	28.49
l8o	28.65	28.81	28.97	29.13	29.28	29.44	29.60	29.76	29.92	30.08
I90	30.24	30.40	30.56	30.72	30.88	31.04	3I-I9	3J-35	31-51	31-67
200	3I-83	31-99	32->5	32-3i	32.47	32.63	32.79	32-95	33-1°	33-26
210	33-42	33-58	33-74	33-9°	34.06	34.22	34.38	34-54	34-7°	34-85
220	35-oi	35-17	35-33	35-49	35-65	35-81	35-97	36-13	36.29	36-45
230	36.61	36.76	36.92	37.08	37-24	37-40	37-56	37-72	37.88	38.04
24O	38.20	38.36	38-Si	38.67	38.83	38.99	39-15	39-31	39-47	39-63
250	39-79	39,95	4O.II	40.27	40.42	40.58	40.74	40.90	41.06	41.22
260	41.38	41-54	41.70	41.86	42.02	42.18	42.34	42.49	42.65	42.81
270	42.97	43-13	43-29	43-45	43-61	43-77	43-93!	44.09!	44.25	44.4O
280	44-56	44.72	44-88	45-04	45.20	45-36	45-52	45-68!	45-84	46.OO
29O	46.15	46.31	46.47	46.63!	46.79	46.95	47.11	47-27 |	47-43	47-59
l 203.
generat Sotution of Tooth Outtines.
In a pair of gear wheels, the two tooth outlines which work
together lie in a section at right angles to the axes of the wheels
and in the plane of this section the construction and action of
the teeth is to be considered. The so-called general solution of
tooth outlines is that by which, if a form of tooth be given for
one wheel, the proper form of tooth for the other wheel may be
drawn so that the motion will be transmitted with a uniform
velocity ratio. Several such Solutions will be given.
I. The AuthoPs First Solution. Fig. 570. Given the tooth
profile a Sb c, also the pitch circle Tf of the wheel <9,#and the
pitch circle Tx of the wheel 0X; required the tooth curve ax S
for the w heel Ox.
Place the given curve so that the point S, where it crosses the
pitch circle, lies on the line joining the centres O 0V thus mak-
ing S, a point common to both profiles. In order to find a
second point alt which shall work in contact with a point a,
draw a 1 normal to the given curve at ay make the arc S i/ =
arc ,S 1, and the distance 1 sx — S i/, and S sx = i/ 1. Then
with as a centre strike an arc with a radius Sx at and from i'y
an arc with a radius 1 a, and the intersection of these ares will
be the desired point ax of the required curve. For such points
as <r, where the normal to the curve does not intersect the
pitch circle the given pitch circles cannot be used, therefore if
these points are required the pitch circles must be transposed
(exaggerated in the figure). The curve thus found sometimes
assumes an impracticable form without being geometrically
incorrect.
II. Abridged Solution. (Poncelet.) Fig. 571. Mark off on
the pitch circle TXf the points slf tXi uxvx............., which
roll into contact with points s, t, u, v, .	. . . of the given
circle T, draw from slt tly uu &c., ares with radii respectively
equal in length to the normals to the given tooth outline vay u
cy etc., then will a curve drawn tangent to these ares be the re-
quired outline. The points sy ty u, v, should be taken close to-
gether. If the lengths of the normals va, u c, etc., are taken
backward from the points^, tlf uXi &c., instead of forward, the
outline for an internal gear tooth will be obtained for the wheel
Tv
Fig. 572.	Fig. 573.
III.	The Author's Second Solution. Fig. 572. The tooth out-
line ab c S d e is given, and its pitch circle Ty also the pitch
circle Tv Draw the normals a 1, b 2, c 3, &c., also draw from O,
as a centre, ares through a, by ct &c., and make Sl = ai, SII =
b 2y S III = 03, &c., and draw the curve I, II, III, S, IV, V, &c.;
this curve will be the path of the point of contact of the teeth,
and may be called the Tine of Action.
IV.	Theoretical Profile of the Flank. Fig. 573. In order to
obtain the necessary strength it is frequently desirable to make
the root of the tooth as thick as can be done without interfering
with the path of the face of the corresponding tooth of the other
gear. This path may be determined in the following manner.
Let a S b be the profile of the tooth for the wheel T, ax Sx bx that
for the wheel Tx, ax aQ the prolongation of the flank outline for
the latter tooth, and I 5* II the line of action between the limits
of the outside diameter circles K and Kx. Lay off from Sf on
both pitch circles the corresponding spaces 5 1, 1 2, 2 3, &c., 6*
i/, 1' 2'y 2' 3', &c. Take in the dividers successively S af 1 ay 2
ay 3 a} Slc., and describe ares from i', 2', 3/, &c., and the envel-
ope of these ares will give the path a ax gy or so-called theoreti-
cal profile of the flank. The actual profile of the flank axfx is
drawn tangent to the theoretical curve, to the point where it
crosses the clearance circle Fx. The theoretical curve is a pro-
longed or abridged cycloidal curve (see § 205). In the figure, in
which T is a straight line, or rack, the curve is an abridged
evolute.
3204.
The Action of Gear Teeth.
In solution III, of the preceding section, reference was made
to the line of action * of a pair of gear wheels, and this line
bears an important relation to the theory of the action of gear
wheels.
The line of action intersects the pitch circle at the same point
as the tooth profile and cuts the latter at right angles, so that
the tangent N N of the line of action (Fig. 572) is normal to the
tooth profile. Each point of action corresponds to a point of
contact of the teeth and also to a point of contact of each of the
pitch circles ; so that, for example, the point II of the line of ac-
tion corresponds to the point 2 on T, and 2/ on T'. That por-
tiou of the pitch circle between the pitch point of the line of
* First discussed in Moli & Reuleaux’s “ Konstruktionslehre fur den Ma-
schinenbau.'’130
THE CONSTRUCTOR.
actiou and the initial point of contact is called the rolling arc
for the given point. For example 5 2 is the rolling arc on Tt
for the point II, and S 2' on Tlf for the sanie point.
The sum of the rolling ares between the two extreme points
(arc i5+55, or aro 1' S + S 5') is called the arc of action,
and its length indicates the duration of the action of the given
pair of teeth, which is easily determined graphically. It depends
upon the length of that portion of the line of action which it is
desired to use. This is usually taken between the limits of the
•circles of the outside and the base of the teeth, w7hich gives in
Fig. 572 the line of action VI.
For any wheel of given tooth outline and piteh diameter there
is but one line of action, and for a given line of action but one
tooth profile. This latter can only be determined from the line
of action when the rolling ares for the piteh point of the line of
action are also given.
For cycloidal teeth the rolling arc is also the line of action
and for this reason the geometrical discussion is much simpli-
fied. In order that a pair of gear wheels should work properly
together, their lines of action should correspond and their roll-
ing ares be of equal length for homologous points of action.
By conforining to these conditions any number of gear wheels
may be made to operate with a given wheel. Such wheels are
said to be interchangeable or series wheels, since the common
line of action is symmetrically disposedon eachsideof the piteh
circle, as well as on each side of a radial line passing through
its piteh point.
The ray drawn from the piteh point through any point on the
line of action (as .S I, in Fig. 572) gives the direction of the
pressure between the teeth for that point.
\ 205-
The Cyceoidae Curves.
For the generation of tooth outlines for gears to be used in-
terchangeably in series, the cycloidal curves or those produced
by rolling circles are the best. When one circle rolls upon
another in the same plane without sliding, each point in any
radius describes a curve which is called either a common, ex-
tended or abridged cycloid, according as the point is situated
on the circuinference of the circle, or on a radial line without or
within the circumference.
The stationary circle is the base circle of the curve, and its
radius will be here indicated by R ; and the radius of the rolling
circle by r. If we consider either radius negative when it lies
within the other circle, and negative when it lies without, we
may distinguish the fi ve kinds of cycloidal curves w7hose radii
have the relation R, and r, in the following manner:
Base Circle.	Rolling Circle.	Corresponding Curve Na me.
	4- r	Epicycloid.
4- 00	+ r	Orthocycloid.*
— R	+ r	Hypocycloid.
+ R	d= co	Evolute of circle.
+ R	— r	Pericycloid.
The following properties are common to all fi ve curves:
1.	The normal to a?iy element of the curve passes through the
corresponding point of contact of the generating and base circles.
2.	The ceritre of curvature of any element of the curve is at
the interseclion of the normal with a right line which joins the
starting point of the curve with the centre of the base circle.
For the extended, or abridged cycloidal curves the starting point
is taken on the radius prolonged of the curve element, at right
angles to a normal to the curve at the point of contact of the
rolling circles.
♦The author gives this natue to the common cycloid because the latter
term properly includes the whole class.
Upon the first property depends the suitability of the cycloidal
curves for use as tooth outlines, and in the second lies the prac-
ticability of approximating them by circular ares.
\ 206.
The Generation of Cyceoidae Curves.
I. Exact Solution. Fig. 574. G is the base circle, Wthe roll-
ing circle, A the starting point of the curve. Lay off from A,
on G and Wy small ares of uniform spacing, and let a, and al
be two of the corresponding points of division. From Ay with
radius a, a1} strike an arc, and from a, with the chord A alt
another arc, and the intersection of the two ares at R, will be a
point in the curve.
This solution, which is shown in Fig. 574, both for external
and internal rolling, holds good for all five curves.
i
Fig. 575.
II. Abridged Solution. From the points 1, 2, 3, . . . a, with
radii equal to the corresponding chords of the rolling circle,
strike ares, which ares will include the entire curve with suffi-
cient accuracy if the points of division be taken sufficiently
close together.
Iu order to draw the extended or abridged curve, starting say
at B, determine first a point Pf on the ordinary curve, then drawr
from a, with a radius ax B, an arc, and from Py another with
radius A B, and the two ares will intersect in a point Q, of the
curve.
Or, draw through a3 a radius az b in the rolling circle and
through b, an arc b C, concentric with the base circle, and make
a.2 Ql — A b, then will Ql be the point in the curve for the roll-
ing of the arc A az upon A a2.
§ 207.
The Generation of Interchangeabee Teeth.
The tooth profile for interchangeable gears is generated in a
similar manner, both for external and internal gears, by using a
rolling circle of constant diameter for each piteh.
I. External Teeth. Fig. 575. Given the number of teeth Zy
and piteh /, or ratio — of the wheel. Make O S — R — — ~
7T	2 7T
1
2
and the radius rQ of the rolling circle IV = 0.875 iTHE CONSTRUCTOR.
C-r =« 2.75 — ; draw the outside circle of the teeth R, with a ra-
7r
dius = i? + 0.3 and the inside circle F, with radius = R —
0.4 t, and make the thickness of tooth = — /. Arc Sb = arc a b ;
40
arc S c = arc f c. S a, the face curve, is generated by the roll-
ing of IV upon 7 ; S i, the flank curve, by the rolling of JV in
T, ‘For pinions of eleven teeth, S i becomes a straight line and
radial. Pinions with as few as seven teeth can be made to work
on this System, for although the flanks are undercut, they are
stili within the limits of the theoretical dank profile (see § 203,
and Fig. 573, where a seven tooth pinion is shown with a rack
tooth). The backlash is t.
II. Internal Teeth. Fig. 576. The generation of internal
teeth is similar to the preceding. The radius of base circle is
— Rt and the length of tooth above and below the pitch circle
is 0.3 t, and 0.41, as before ; rQ = 0.875 ^=2.75 —, and the thick-
7T
ness of tooth = — t. The flank 5 a is generated by rolling W
4°
upon Ty and the face S iy by rolling W inside of 7.
In the case of a rack R = oc, 5 a and S i tflen become similar
portions of the common orthocycloid (see Fig. 573).
In teeth of this form the line of action coincides with the
rolling circles, the portion included being == arc b a + the cor-
responding arc bx ax of the opposing wheel, when both are ex*
ternal gears, and+ the arc c i for an internal gear working with
a spur gear. The duration of action ey varies betwreen 1.22 and
I.60.
§ 208.
Tooth Outeines of Circuear Arcs.
Instead of using the exact tooth outlines as generated by the
rolling circles, two circular ares may be used as a close approx-
imation (see \ 205).
Fig. 577. Draw the pitch circle Ty and outer and inner circles
K and F, as before, also the centres A/, and Mx of the rolling
circles W and Wx, which latter are in contact with each other
and with the pitch circle at S. Drawr the diameters B MD and
Bx Mx Dx in such manner that the angle B M S = angle Bx Mx
Sx = 30°, join B and B, with the prolonged line Cx B S BXy and
draw through D and Dx the lines O D and O DXCX\ then will
the iutersections at C and Cx with BXCS Cx be the required
centres of curvature for the ares a B b, and c Bx i.
Through C and Cx draw circles with O as a centre, and on
these circles the centres for all the teeth will be found, the arc
a B b being struck from Cy and c Bx i from Cx.
The radii of curvature p may be calculated from the following
ormula:
p	2Z+11	,
— = 0.4 s-----and
t	ZdzII
27 + II
(I)
= 1.42-
(190)
The plus sign gives the radius C B, for the face ( p a) and the
minus sign gives the flank radius Cx Bx, (p i ). The flank should
be joined to the bottom of the space by a small circular fillet.
131
Examph z.—Given Z= 63, t = 1.3125", we have for the radius for the face
of the teeth:
.,	126 -f 11
p = 1-3^S X 0.45 —-+------
r a	03 + 11
For the flank radius we have:
£126 — 11
= 1.093"
p t = 1*3125 X 0.45
• 1.306".
Also,
v x 22 + n
pa =o.4XM* —
Example 2.—Given Z = 11,	= 0.4. We have:
'0.85"
22 —11
P = O.4 X 1.42 ------ 00
r	II — II
hence the flanks are straight radial lines.
Example 3.—Given Z = 7, t = 2We have:
p = 2 X 0.4 —A"-11 = I>I2w
7 + 11	*
Also,
p t = 2 X 0.4 -
* — 0.6'.
The negative sign indicates the undercut flank. This is shown in Fig. 573
It is better to use the exact method given in $ 207, for wheels
with fewer than fifteen teeth, as the approximation becomes less
accurate for the lower numbers.
?209.
Evotute Teeth for Interchangeabee Gears.
Gear teeth may be given the evolute form, which curve is de-
veloped by unwrapping a line from a base circle, which is con-
centric with, and bears a definite relation to the pitch circle.
External and Internal Teeth. Fig. 578 and Fig. 579. Given
the number of teeth Zy and pitch /, or ratio — for the required
7T
wheel. Make O S = R = ~S~=z —Z (and draw the
2 7T	2	\ n J
outer and inner circles, giving the distances f — 0.4 i, k = 0.3 t
abofe and below the pitch circle, also make the thickness of the
tooth = — t.
40
Draw the line N S Nx at an angle of 75° with O Sf and it will
be tangent to the base circle Gy the radius of which = r = 0.966
R = 0.154 Z ty — 0.483 Z	If now we unwrap the line iV*
^ upon the circle G} from 5 outward to a, and inward to gy the
path a Sg of the point S will be the required tooth outline,
which for wheels of fewer than 55 teeth may be prolonged by a
radial line to reach the bottom circle.
The line of action is the straight line NNx; and extends from
Sbto S bx on the other gear, or in the internal gear to 5 c. To
determine the duration of contact e the pitch t can be carried to132
THE CONSTRUCTOR.
the base circle by drawing radii, aud the length measured.
For two equal wheels of 14 teeth, e is only a little greater than
unity ; it varies between 1 and 2.5.
Rack Teeth. Fig. 580. The profile a Si is straight and makes
an angle of 750 with the pitch line T. The angle 750 can readily
be laid off by using the drawing triangles of 450 and 30° to-
gether.
For low numbered pinions the base circle closely approaches
the pitch circle. This sometimes introduces an error into the
Fig. 581.
action. If the portion 5 B, of the line N Nlt which lies be-
tween the pitch and base circles, Fig. 581, is shorter than the
length of face of the opposing tooth, the point a will interfere
with the flank of the pinion tooth, as shown in the path a /g.
(See also Fig. 573.) In order to avoid this, the tooth to which
the point a belongs must not extend above the iine K' K'.
This exists for teeth made in the manner given, when Z ^28.
Another method of avoiding this difficulty is to round off the
tooth at a, and this is more frequently adopted in practice. An
important application of evolute teeth is shown in § 222.
§210.
Pin Teeth.
Teeth with radial flanks can always be generated by making
the inner rolling circle for each wheel equal in diameter to one-
half the pitch circle. This will give radial flanks and curved
faces to both gears, but wheels made on this System are not in-
terchangeable, and are therefore not practical for general ma-
chine construction. Such teeth are stili much used by watch-
makers on account of the ease with which they may be fitted
by filing.
If the diameter of the rolling circle is made greater than the
radius of the pitch circle a form of tooth is obtained which is
practicable, but which is comparatively little used.
If, in a single pair of wheels, the rolling circle be taken for
one wheel equal to the pitch circle, of the other wheel, we obtain
for the teeth of the wheel upon which the rolling is done, au
outline of cycloidal form, while the teeth of the other wheel be-
come mere points. In practice these points are the ceutres
about which pins are described and such gears are called pin-
tooth gears.
Extemal Pin-tooth Gearing. Fig. 582. The pins are circular
in section and in diameter equal to — the tooth profile for
4®
the wheel Rx is then a curve parallel to the path ,S a, described
by rolling the circle T on Tx. The arc S b = ab, and circles of
the diameter of the pin, struck from successive points of the
path 6* ay will outline the tooth profile c d, the flank d i being a
circular quadrant. The curve of action SI is limited by the
outer circle K\' at /, and is in all cases greater than /, generally
not less than 1.1 t. This gives the limit of tooth length k/ and
also determines kv If it is desired to construet the actual line
of action, the method of case III, \ 203, may be employed.
Fig. 583 shows a pinion of six pins gearing into a wheel of
24 teeth. The diameter of the pins is here made = —. The
3
flanks of the 24 tooth wheel are made radial with square cor-
ners in order to permit ready filing and finishing.
I
i
Internal Pin-tooth Gearing. Fig. 5S4. This is similar to the
preceding. The tooth profile c d is a parallel to the curve 5 i9
generated by rolling T in 7^, the arc S b = i b. S / is the line
of action and is made equal to, or greater than 1.1 t. The flank
da is made radial.
In Fig. 585 the pinion is made with the pin teeth and the spur
teeth are on the interna! gear. The profile c d is parallel to the
curve 5* a, generated by rolling T upon Tx; the arc S b — a b,
SI is the line of action, as above, and is made equal to, or
greater than 1.1 t; the flank d i is made radial.
If in Fig. 584 we make the radius Rx infinitely great, we ob-
tain a rack, and the tooth profile is a curve parallel to the com-
mon cycloid. If we make Ry in Fig. 585, infinitely great, we
obtain a common form of rack, with pin teeth.
Pin teeth have the practical advautage that they may readily
be turned in the lathe. They are especially adapted for situa-
tions where they are exposed to the weather, as in sluices, swing
bridges, wind-mills, etc. In such cases the pins are often made
of round bar iron, without being turned.
Fig. 586.
Double Pin Gearing. Fig. 586. If two gears on this System
are run together, one gear may be made with very few teeth,
and hence a great difference in velocity ratio obtained, with a
minimum distance between centres. In this case both pitch
circles become rolling circles. 6* ay the pinion face, is generated
by rolling 1 x on Ty the action extending on SI for the point .S
on the wheel T. S aly the gear tooth face, is generated by roll-
ing T on 7~i, the action extending on the line SII, for the point
5, on the wheel Tv S iy the flank profile, is made to conform
to the theoretical profile Salg1 (see case IV, § 203), and the
other flank is made in a similar manner from the theoretical
profile S a g. Such gears are sometimes used in hoisting ma-
chinery.THE CONSTRUCTOR.
133
i 211.
Disc Wheees with Pin Teeth.
It is not an essential requirement that the tooth profile shall
be in the immediate line of the pitch circles, as it can be placed
within or without to a greater or less extent. In such cases a
tooth system is obtained in which the teeth of one wheel pass
almost or entirely around those of the other wheel, and hence
there can be no so-called bottom circle to the latter teeth. Such
wheels are so constructed that the teeth are placed upon the
side or face of a disc, or shield, and are called disc wheels, or
“shield gearing.” *
For such wheels pin teeth are well adapted. Fig. 587 shows
a pair of such wheels arranged for external action, and Fig. 588
for internal action. One wheel of each pair is fitted with round
pin teeth, and the other has, in the first case, a tooth profile
parallel to an extended epicycloid, and in the second case par-
allel to an extended hypocycloid.
A peculiar form of disc gearing is shown in Fig. 589. In this
case R = Rlf Z = 2,	= 4. the round pins being on R. The
flanks of Rl are entirely within the pitch circle, and become
straight lines parallel to the straight line hypocycloid i. The
arc of action is about 2 t, and the backlash can be reduced al-
most to zero, the teeth on R being made as rollers.
j
i
Fig. 589.
If the distance between centres O Ol of a pair of wheels for
internal action remains constant, and the radius is increased,
they will overlap entirely, and the pitch circles will cease to ap-
pear as an element in the construction. The wheels will have
equal angular velocity and revolve in the same direction.
Such a pair of disc wheels is shown in Fig. 590. Both wheels
are made with pin roller teeth, the sum of the pin radii being
equal to the distance O Ov The pins are shown of equal diam-
eters, although they may be unequal, as shown in the dotted
lines. Such wheels may be called Parallel Gears, as two radii
which are parallel in one position remain parallel at all times.f
A second form of parallel gears is shown in Fig. 591. The
curve a b c is a circular arc, of radius d ay which includes four
segments of the lenticular shaped pins for the wheel Ox.
If the pair of parallel gears of Fig. 590 are placed on opposite
sides of an axis A Ax normal to two adjoining pins and parallel
to O 0lt the action of the wheels will be correct. In Fig. 592 is
shown such a pair of right angle wheels.
Such gear wheels have been described more than once,* but
are rarely used ; they are well adapted to transmit motion to the
hands of large tower clocks.
Fig. 590.
Fig. 591.
g 212.
Mixed Tooth Outeines. Thumb Teeth.
By combining the preceding forms of teeth, practical shapes
may often be made for special Service. The two following ex-
amples will illustrate :
Mixed Outline. Fig. 593. For the low numbered pinions
sometimes used in hoisting machinery, it is important that the
pinion teeth shall not be too much undercut, so as to avoid dif-
ficulty in making the gears. It is desirable that the flanks on
the pinion should be radial. In order to obtain sufficient dura-
tion of action, which for a three tooth pinion should not be less
than 1.15 /, the face curves of the teeth should be prolonged
Fig. 593-
until they intersect. The curve 5 a is an arc of an evolute
formed by unwrapping the pitch line Tx from the circle T; S i
is the radial flank, obtained by rolling the circle IV of radius =
Yz R in T; S ax is the theoretical profile for the tooth space
for the wheel T.
S a acts with the point 5 of the rack tooth over the path SII.
S ax is a cycloidal curve generated by rolling W on Tly and acts
over the path 6* / with the flank 6* i of the wheel T.
* Called Scudi Dentati in Zonca’s Teatro di Machine, Padua, 1621.
t This form of gearing was described and named by the author in Berlin * See Tom Richards’ Aide-memoire. 1848, I, p. 656. Willis’ Principies of
Verhandlung, 1875, p. 294.	Mechanism, ^851, p. 145, Laboulaye, Cinematique, 1854, p. 275.THE CONSTRUCTOR.
Fig. 594-
Thumb-shaped Teeth. By combining the evolute and epicy-
cloid, using the two curves for opposite sides of the same tooth
a profile of great strength is obtained. This form is of especial
Service for heavy driving when the motion is constantly in the
same directiou.* From the peculiar form these have been called
thumb-shaped teeth. The following proportions will be found
suitable for cases in ordinary practice.
Fig. 595-
The rack teeth are made straight on the one side, as already
showu for rack teeth on the evolute system. Applications for
teeth of this form are given in \ 226.
\ 2I3«
Tooth Friction in Spur Gearing.
The friction of spur gear teeth is mainly dependent upon the
form of the tooth outline, and may be investigated by consider-
ing the form, extent and position of the line of action. In most
cases the friction is proportional to the duration of action e. A
coefficient, dependent upon the position of the line of action
may be determined from e, and may be taken = y, when the
arc of action is equally divided on both sides of the Central po-
sition ; as in the case of epicycloidal teeth; and = 1, when, as
in many cases, such as pin tooth gearing, the arc of action is
entirely on one side of the centre; while for evolute teeth it
may be taken = that being about midway between the two
preceding fornis. The tooth friction is also greatly dependent
upon the number of teeth in both wheels, being proportional to
their harmonic mean, and it diminishes rapidly as the number
of teeth is increased.
If we make the coefficient of friction = f and take the num-
ber of teeth as Z, and Zx, we have for the percentage of loss pr
in tooth friction:
a. Epicycloidal Teeth.
b. Evolute Teeth.
C. Pin Teeth *
The value of the coefficient of friction f is in no case small,
even when the teeth are well lubricated, on account of the usual
high pressures; a usual value may be taken, f = 0.15, while for
new and dry wheels it reaches 0.20 to 0.25 and even higher.
The minus sign in the formula is to be used when one of the
wheels (Zx) is an internal gear.
Example 1. In a pair of epicycloidal gears, of seven teeth, the value of e =
1.225. Taking/= 0.15 we have according to (191 a) for the loss by tooth fric-
tion:
pr- 3.14 X 0.15 X-y-X^p = 0.0824, or about 8^ per cent.
Example 2. Epicycloidal Teeth. Z—Zx = 40, «= 1.44 and we get:
pr = 3.14 x 0.15 X X = 0.0169, or about 1.7 per cent.
Example 3. Epicycloidal Teeth. Z = 7, Z\ — — 60 (internal gear). c = 1.40
and we get:
pr = 3 14 X 0.15 (f — &)	= 4.2 per cent.
Example 4. Epicycloidal Teeth. Z— 7, Z\ = 00 (rack). e = 1.37
pr = 3.14=015 (}+o) =4.6 per cent.
Example 5. Pin-tooth Gearing. Z— 6, Z\ — 40. We have, as determined
by construction, as in Fig. 583, e = 1.166. Hence we get from (191 c):
pr = 3.14 X 0.15 (& + &) X 1.66 = 2.6 per cent.
Example 6. Evolute Teeth. Z= Zx = 40.	6 = 1.92. We have from (191 £):
Pr = 3.14 X 0.15 X —X -3 X i-9g_ _ 3 ^ per centor double that in Kxamp. 2.
4°	4
Fig. 594. Spur Gearing with Thumb-shaped Teeth. a Si and
ax S ix are profiles formed of epicycloidal curves, according to
the description in § 207, in which rQ = 0.875 ^ or 2-75 —•
7r
a' S' i' and a/ S/ t\ are evolute curves developed from base
circles with radii r' = 0.8 R, and r/ = 0.8 Rv giving an angle
of 530 (more accurately 530 8'). For wheels of iess than fifteen
teeth, as in the seven toothed pinion shown in Fig. 594, the
flanks must be modified as shown in $ 203, to avoid interference.
In Fig. 595 is shown a four-toothed pinion on this system,
working with a rack. 5* a and ix are made as before with rQ
= 0.875 ^ and ^ i and ax with r = y R ; the evolute curves
being generated as before with an angle of 530.
It will be seen that the tooth friction is least with epicycloidal
teeth and greatest for pin gearing; evolute teeth being midway
between.
The wear upon gear teeth is affected by other considerations
besides that of the coefficient of friction, the pressure of the
teeth upon each other, and the relative rubbing movement of
various portions of the profile also entering into the problem.
The wear is therefore not constant for a constant pressure, and
it is an error to assume, as is sometimes done, that the form of
evolute teeth is unaltered by wear. These teeth usually show
the greatest proportional alteration by wear, since the flank of
the tooth below the pitch circle has a very much less rubbing
movement than the portion of the opposing tooth which rubs
against it and hence the wear is unequal.
♦This form of mixed outline has been described by Willis in 1851; it was
revived by Gee in 1876 and used in practice ; he made the angle a greater
than here given, viz. 68°.
Approximately.THE CONSTRUCTOR.
135
The effect of this may frequently be observed in practice,
where the smaller of a pair 01 evolute gear wheels will be no-
ticed to be worn into deep hollows below the pitch circle.
The conclusions given above about the percentage of loss may
also be determined geometrically in the following manner :
Take the two portions of the tooth profles which work together
and divide each by the chord of the corresponding portion of the
litte of action, multiply each resuit by the ratio of the length of
its portion of the line of action to the entire length of the line
of action, and then multiply the sum of the two quotients by the
coejficient of friction.
The resuit will be the percentage of loss, pr. The chord re-
ferred to becomes the line of action itself in the case of evolute
teeth. This method serves also for pin teeth, and is very useful
for the designer, as the data can all be taken off the drawing
with the dividers.
I 214.
General Remarks on the Foregoing Methods.
Each of the preceding methods possesses its merits and dis-
advantages.
Epicycloidal Teeth. These possess the great advantage that
they will work together in any series with as few as seven teeth,
while for evolute teeth the lowest in series is 14 teeth, and in
no case fewer than 11. The loss from tooth friction is a mini-
mum with this form, and the wear less injurious to the shape of
the tooth. The minor objections which ha ve been raised are
that the double curve increases the difficulty of construction,
and that any variation of the distance between centre causes im-
perfect action to follow.
Evolute Teeth. The advantages of this form are that the
simple shape is readily made and that any variation of the dis-
tance between centres does not affect the action.
Against these must be set the fact that for low numbered
pinions the flanks must be altered to avoid interference, or the
tops of the teeth must be taken off. The fact that the distance
between centres may vary is rather an objection in many cases,
as the arc of action is reduced, and in transmission of heavy
power the shocks upon the teeth are liable to be increased.
Evolute teeth are well suited for interchangeable gears, if low
numbered pinions are not required (30 teeth being the minimum),
and where but small power is to be transmitted they are excel-
lently adapted. For wheels which run only in pairs, and hence
for bevel gears, this form is excellent, since it is so readily made.
See l 222.
Pin tooth gearing and the mixed outlines are only used for
special work, such as in hoisting machinery and the like, and
in such cases the wheels are often made of wrought iron or Steel.
Disc wheels have a very limited application, but in some spe-
cial fornis of mechanism they are very useful, and will be dis-
cussed further. See Chapter XVIII.
B. CONICA L GEAR IVHEELS.
1215.
General Considerations.
In the case of conical gear wheels, or as they are generally
termed, Bevel Gears, the working circles of a pair of gears which
run together, lie on the surfaces of a pair of cones, the apex of
each cone being at the intersection of the axes of rotation. In
such case the pitch circles are taken at the p base circles of the
respective cones, as 5"D, and S E, Fig. 596. The length of the
teeth is measured on the supplementary cone, to each base cone,
B being the supplementary cone for 6" D, and ^ C that for S
E, B C being at right angles to A S. The length of teeth is laid
off 011 B and S C, and the width of face on S A ; the tooth
thickness being spaced off on the pitch circle and all the teeth
converging to the point A.
The respective radii S D and S E of the two cones are found
by dividing the angle a of the axes, in such a manner that the
perpendiculars SD and SE let fall from 5 to the axes, bear the
same ratio to each other as do the numbers of teeth, or inverse-
ly as the number of revolutions; thus S D : S E = Z : Z1 =
nx : n. There are, therefore, two Solutions possible, according
as the pitch line S A is taken within the angle a, or in its sup-
plement; or what is the same thing, according to which angle
is taken as the angle of the axes. The difference between the
two consists in the fact that for a constant direction of revolu-
tion of the driving shaft the driven gear revolves in one direc-
tion for the first solution and in the opposite direction for the
second solution. One of the Solutions gives an intemal gear,
when nx : n < cos a.
If bevel gears are required to interchange (see \ 200) they
must not only be of the same pitch, but must also have the same
length of contact line, A S, Fig. 596. Since these conditions
are very infrequent, it follows that bevel gears are generally
only made to work in pairs. In practice it is found that a vari-
ation of less than 5 per cent. in the length of *he contact line
may be neglected. Gears of the same pitch and same angle of
C\
Fig. 596.
axes, but with a small variation of contact line, are called
“bastard gears.” A pair of right angled bevel gears of 80 and
45 teeth, might be altered in practice, if required, into bastard
gears of 80 (1 ±0.05), i. e., 84 to 76 teeth, wdiich wcruld work
with the other gear of 45 teeth.
| 216.
Construction Circles for Bevel Gears.
The geometrical figures which are formed by one cone rolling
upon an other, require that both cones should have a common
apex. The surface thus developed is called a spherical cycloid.
Of these there are five particular forms, as with the plane cy-
cloids, the latter being really those for a cone with an apex:
angle of 180°. The spherical cycloid is very similar in form to
the plane cycloid, as are also the corresponding evolutes; the
branches of the curves assuming a zig-zag form.*
The use of the spherical cycloid for the formation of bevel gear
teeth would involve many difficulties. In order to construet
such teeth, it is therefore common to use the method (first de-
vised by Tredgold) of auxiliary circles, based upon the supple-
mentary cones, and enabling the teeth to be laid out in a simi-
lar manner to those of spur gears. The auxiliary circles for the
bevel gears R and Rly Fig. 597, are those of the spur gears hav-
ing the same pitch, their radii being respectivel.y r and rlf the
elements B S and CS of the supplementary cones.
For any given angle a between the axes, the radius r, and
number of teeth 3, for the auxiliary circle can be determined
♦See Berliner Verhandlung. 1876, pp. 321, 449, Reuleaux, Development of
the Spherical Cycloid.136
THE CONSTRUCTOR.
from the radii R and Rv and tooth numbers Z and by the
following formula :
r  v/R* -j- R* -+-2 R Rx cos a
R	Rl R cos a
z  \/Z2 -f- Z\ 4" 2 Z Zx cos a
Z	Zx-\- Z cos a
If the axes are at right angles, we have
(192)
r __>/R2 -f- R^ z
~R	R, ’ ~Z
\Zz*+z*
Z1
Example.—A pair of bevel gears have 30 and 50 teeth, and an anglebetween
axes a = 6o°, hence cos a = and we have for the auxiliarv circle of the 30
.	v/302 + 502 + 2 . 30.50.05	, \/ 4900
tooth gear : * = 3o ^--5Q + 3Q ^ Q>5-----= 6 -----= 32-3, say 32.

For the 50 tooth gear we have also: zi = 50 —-	4900----_ g,
3° + s° . 0.5
From these numbers and the given pitch, the auxiliary circles
can be laid off and the teeth drawn.
Low tooth numbers are not available for bevel gears, since the
errors which are involved in the method of auxiliary circles be-
come disproportionately great. By using not fewer than 24
teeth for the bevel gear, a minimum of 28 for the auxiliary cir-
cle is obtained, and the evolute System can be used to advant-
age. This form of tooth is best adapted for this purpose, on
account of its simplicity of form, notwithstanding the minor
defects which have already been noticed.
The loss from tooth friction in bevel gears is approximately
equal to that of their corresponding auxiliary gears.
Fig. 598.
i 2I7-
The Plane Gear Wheel.
Internally toothed bevel gears are not used, 011 account of the
practical difficulties involved in their construction. There is,
howrever, an interesting form of gear wheel which lies interme-
diate between the external and internal fornis. If the numeri-
cal ratio between a pair of bevel gears is = cos a, one of the So-
lutions for the base cone gives for the latter a plane surface, 5*
£*, Fig- 598.
The supplementary cone in this case becomes a cylinder, and
the radius of the construction circle becomes infinitely great,
hence the tooth outlines are similar to those used for rack teeth.
If the evolute System is used the teeth are very simple, and the
plane gear in some cases becomes a very convenient form of
construction.
As already stated, the ratio is
§ = COS .....................(193)
-«i
jD
from which, if for example a = 6o°, we have —- = If the
^1
angular relation of the axes is given it follows that but one ve-
locity ratio can be obtained. This is determined from the angle
>'2, which is one-half the apex angle of the cone R2f and from
the ratio —? = sin y2.
R1
It is sometimes very convenient to arrange a plane gear so
that it may work with both of a pair of bevel wheels. This is
shown in Fig. 599» in wThich the gears R2, R3 have the semi-apex
angles y2, y3, and have their axes at right angles. We then have :
R2
-z = tan y2 = cot y3,
from which we obtain the following values:
—2 = tany2 = J-	i i f i | 2	3	4
y2 =	140 i8°3o/ 26°4o' s6°5o/ 450	63°20/ 7i°3o 76°
~ = sin y2 = 0.242 0.317 0.449 0.600 0.707 0.800 0.894 0.948 0.970
R1
Either of the wheels R2, ^3, can be used with the plane gear
A*! if the number of teeth have the ratio given by the value of
sin y2. Although this limits its application, yet the plane gear
is frequently found very useful for angular transmissions.*
C. HYPERBOLOIDAL GEAR WHEELS.
§218.
Base Figures for Hyperboloidal Wheels.
Hyperboloidal wheels are used to transmit motion between
inclined, non-intersecting axes. The figures upon which they
are based are hvperboloids of revolution haviug a common
generatrix. These may be determined in the following manner.
Fig. 600.
In Fig. 600 is given a projection normal to the line of shortest
distance between the twTo axes. The angle a is divided into two
parts /3 and pv in such a manner that the perpendiculars let fall
from any point Ay of the line 5 A, upon the two axes, shall be
inversely proportional to the revolutions of the gears. S A is
then the contact line of the hyperboloids ; A B = Rr and A C
♦The socalled “Universal Gears” of Prof. Bevlich, introduced in 1866,
should be considered as a variety of conical gears'in which the angle of the
axes may be conveniently varied. These may be used for axes of angles
varying from o° to 1800. As shown in the illustration, these wheels are
formed of globoids of the III Class (see§224), the meridians lorming the
teeth and spaces. They have found but limited applicatioQ. A model of
these gears is in the kinematic cabinet of the Royal Technical High School.THE CONSTRUCTOR.
13 7
— R\, are projections of the radii of the hyperboloids intersect-
ing at A. We have
R' _ sin P_ ”1 — z........................(194)
7?/ sin	« Zx
The actual radii i? and are yet to be determined, as well as
the radii S £> = r, and 5 E — rx of the gorge circles.
For the latter we have :
cos a
r _ tan /? __ ^. . .	(195)
r, tan	„
1 1 — cos °
"1
that is, r and have the same relation to each other as the por-
tions A F and A G of a perpendicular to the line of contact.
If we call the shortest perpendicular distance between the axes
= a, we have :
r
a
% f*
I -j----COS a
1	11	1 f nX*
-L 2 — cos a -f- ( — 1
nx	\ nx)
?\
a
, n\
14----- cos a
n
. ni	1
1 4- 2 — cos a 4-
n
. («96)
The radii R and R1 are hypotenuses for the triangles whose
sides are R'and r, R/ and (see the left of the figure) or:
R — UR'-‘ +
Rx = VR'?+r?
(197)
R' and R/being determined as above, when the distance 5 A
— lis given. For the angles /3 and & we have the general ex-
pressions:
tan /?
sin a
n .
----1- cos a
(198)
tan /?! =
«1
n
sin a
4- cos a
As in the case of bevel gears, two Solutions are possible ac-
cordin<r as the angle a, or its supplement, is taken in determin-
ing the line of contact S A, Fig. 601. The choice of solution
Fig. 601.
govems the direction of rotation of the dnven gear, and one of
the Solutions renders it practicable to make an internal gear ;
although this construction has been littleused, and has but little
practical value.
If the angle of the axes a = 90" we have
— = tana /3 = f —1...................(»99)
r.	\ n f
also:
r __ n? ^
a	n2 4- «12 9
t\
a	w2 4- «i2 9
tanjS==
n	J
(200)
In the construction of the wheels, corresponding zones are
chosen on the two hyperboloids. If the distance between the
axes is small, the zones lying in the gorge circles are generally
unsuitable, but when the distance is greater they may be used
and the figures approximated by truncated cones.
Fig. 602.
Example 1. a = 400, ~ = %, (see Example 1, in § 221), a = 4",
R*
We have -=~. =
1.266
0.5 4- COS 4Q(
2 -f cos 400	2.766
1+2 cos 400
Also tan 0 =
r ~	1+2X2 cos 40" -+- 4
r = 0.31398 X 4= 1-256",
rx = 4—1.256 = 2.744"
sin 400	_ 0.6428
2.766
— Q-4577,
2-532 a
8.064
= 0-31398,
2 -f COS 40°
Pi — 40° — 0 = 26° 55'.
If we take SA = / = 8" we have R’ — l sin 130 5' = 8 X 0.526368 = 1.81",
Rx — 8 sin 26° 55' = 8 X 0.452634 = 3.62"; finally
R = \/ (1.S1)2 -+- (1.256)2 = 2.2" and
Ri = \/	(3.62)2 + (2.744)2 = 4.54"
Example 2. a. — 900,
or say the number of teeth Z = 36, and
Z\ = 20; a = 0.75". We have from (197):
~ = ( j) = —■ — 3-24, and from (200)
a X 92
r 52+92
and r1 — -
25
o 75 X 81
= 0.573",
For 0, we have tan 0 = ~~ = 1.8, hence 0 = 6o° 5/, and 0j = 290 3'.
If we make R — 2", we have from (197):
R' = v/“= v/?2-
o.5732 = i-9l6",
and hence Ri', according to (194) is = f Rx' = 1.063" hence
1.0632 -+- o. 1772 = 1.078".
The appearance of such a pair of gears is shown in Fig. 602. According
to the table in § 202 the pitch for the larger gear is: t —- = --- = 0.25".
1 o 8	5'73	5'73
and for the smaller gear t\ =	— 0.339"
3.18
Example 3. a = 90°,	= 1, 0 = 45°, r—ri,R — Rx. In this case the hy-
perboloids become similar (see Fxample 4, § 221.)
Example 4. In the special case in which ^ = cos a, and the position ot
the contact lihe, which is determined by 0, lies in the supplement to a, so
that — = cos a, the base figures become, the one a normal cone and the
•K
other a plane hyperboloid, see Fig 603. This construction is similar to the
preceding forms of plane and bevel gears, and may be conveniently used to
work with a train of common bevel gears, although but few practical appli-
cations occur, partially owing to the fact that the prolonged axisof the bevelTHE CONSTRUCTOR.
138
gear passes through the plane gear. For a = 60", — = — % = — cos 6o° we
obtain the plane gear. We liave tan /3 = £ \/ 3, /3 = 300, tan = 00,	= 900.
Also — 7 = Shl 3°0 = 0.5 ; r — ot rx = a, ^ — R', Rx = \/R\ -+- a- = y/4 Ri +a2.
Aj sin 9®
If be negative and less than cos a we obtain an hyperboloidal internal
gear.
Fig. 603.
Rack teeth may also be constructed to work with hyperbo-
loidal gears. In this case the teeth of the rack are inclined
while the pinion becomes an ordinary cylindrical spur gear,
since in order to satisfy equation (195) with rx = 00, the angle
= and /3j = a, see Fig. 604. Applications of this construc-
tion may be found in various machine tools.
2 219.
Teeth for Hyperboloidal Gears.
The construction of the exact fornis for the teeth of hyper-
boloidal gears is a very difficult operation, and in practice an
approximation is used similar to that employed for bevel gears.
The method adopted is to determine the supplementary cone to
the hyperboloid used, and as in the case of bevel gears, use the
corresponding construction circle.
1
Fig. 605.
The apex //“(Fig. 605) is determined by drawing A H per-
pendicular to the generatrix 5* A, which, as before, is taken
parallel to the plane of the drawing. The teeth will be formed
with sufficient accuracy if two construction hyperboloids are
taken with the same angle of contact as the base hyperboloids,
according to the conditions in (198) and (199), and the teeth are
formed on the surfaces, which are described by the edges of the
construction hyperboloids upon the base hyperboloids.*
If it is desired to approximate to the hyperboloidal zone by
the use of a conical surface, the apex must be determined. Iu
this case the generatrix S A is rotated about the axis H S until
A falis on the point J of the circumference, wffien the new pro-
jection of the generatrix will pass through the apex M of the
cone.
The tooth friction of hyperboloidal gears is uecessarily great.
This will be considered later, in connection with the speed of
the rubbing surfaces, wffiich is similar to that of the spiral gears
which are tangent at the gorge circles (see \ 220.)
D. SPIRAL GEARS.
•2 220.
Cylindricae Spirae Gears.
Cylindrical spiral gears may be used in the same manner as
hyperboloidal gears for the transmission of motion between in-
clined axes, and in some cases possess advantages over the lat-
ter. There are a number of useful variations of spiral gears.
In Fig. 606 is shown a pair of wheels, A and B, both with lefl
hand spirals and corresponding tooth profiles. The pitch angles
y and are so chosen that at the point of contact the pitch cy-
linders have a common tangent, so that if a be the angle of in-
clination of the axes, y -j- a = 180°. If we indicate by v
and vx the circumferential velocity in the direction of the tan-
gent and normal respectively, we have :
ii
v
sin y , n, R sin y Z
~—— = whence — =--------------— — —
sin 7j	11 Rx sin yx Zx
. . . (201)
The normal pitches, 1 == t sin y, and ix = tx sin yx must be equal
to each other, whence — r— Sln ^1.
tx sin y
As indicated by the components of velocity vf and v/, there
is an end long sliding action of the teeth upon each other, with
a velocity :
d = v' + vx' = c (cot y + cot )q)......(202>
This sliding consumes power and causes wear, and will be at
a minimum when v' and vx' aae equally great, that is when
r=yi-
With regard to the choice of y and yx the conditions may be
so taken that the position of the coinciding tangents of the two
spirals shall be slightly before or slightly after the actual line of
contact, but as close as may be possible. This is similar to the
position of the line of contact of hyperboloidal gears (§ 218)
and may be stated as follows :
as also
R
cot y
cot yx
— + cos a
n
H .
---f cos a
”1
• • • (2°3>
sin ft	,
cot y =........................................(204)
?t .
— -f- cos a
* See Herrmamfs Weisbach's Mechanics, II. ed., III, 1, p. 418 et seq.THE CONSTRUCTOR.
139
fl
For a = 90° we have cot y = —. Such spiral wheels, when
Tt
the teeth are well made, transmit motion very smoothly, butthe
surface of working contact is very small. When the axes are
at right angles and the wheels the same size, it is often incon-
venient to use spiral gears on account of the large size required.
”1 _ .
We have from (203) —	=
—	= 3, whence 7 = 180 26', and yj = 710
— 9 an<^ from (204) cot y
34'. The sliding velocity is c' = c (3 -f 0.333) = 3$ c. The small value of the
angle y makes it undesirable to use the smaller gear as the driver. These
objectionable features are of increasing importance and for example, —-”1 -
= 5. and — IO we get = 25, and 100, and y about ii£°and5§°. The
n	/tj
difficulty of cutting the teeth on the lathe also increases, as may readily be
seen.
§ 221.
Approximateey Cyeindricae Spirae Gears.
If, of the preceding conditions, only those of formulae (201)
and t2°3) are strictly observed, the difficulties of construction
are much reduced and at the same time satisfactory wheels ob-
tained.
Three methods may be employed : (a) a slight modification
from the eorrect spiral form may be given to both wheels, (d)
one gear may be made a true spiral, and the variation ali thrown
y
FlG. 608.	FiG. 609.
into the other gear, or (c) the wear which is at first caused by
running the approximate forms together may be disregarded
until the parts have wom themselves into smooth action. From
these reasons a wfidelv varying practice in the construction of
spiral gears will be found. One of the most important applica
tions is that of the worm and worm wheel, Fig. 608. In this
case a == 90° and Z = 1, the teeth of the wheel Rx being inciined
at an angle y, with the edge of the wheel, whence tan y =—.—
2 t: R
= 0.15916 -L-. In the arrangement shown in Fig. 609, we have
R
a = 90 — y and the teeth on Rl are made parallel to the axis.
The pitch of the screw is here made — —1— for a pitch L of the
cos y
wheel. The velocity ratio of transmissiou, according to the
fundamental formula (186) is nx: n = Z : Zv or this case it
equals —
* In the illustration Xi = 30, which in (203) for a true spiral would require
Rx — 900 R, and y — 88.10.
In many cases the worm is made a true spiral and the conse-
quent wear disregarded, but in more careful work the method
(d) is adopted and the worm wheel cut with a hob, which makes
the proper modification in the shape of the teeth.
The friction between the worm and teeth of the worm wheel
is very great, as the thread slides entirely across the teeth. We
have for the coefficient of fnction y for the ratio between the
actual force P* and a force P acting at the same lever arm on
the screw, but free from frictional resistance, approximately :
I + /• 27TR
P> _	t
P ~ I ~ft
27T R
For/=0.16 we have practically
P' . R	,	,
E=1+T........................(2°5)
jr>
It follows that to obtain the minimum of frictional loss, —
t
must be made as small as practicable.
P'
Morin gives the rule R — 3 /, wdiich makes — = 4 ; Red-
P'
tenbacher makes R = 1.6 /, whence — = 2.6. If w7e make
P'	. .	R
R = t, we get — = 2, and this is as lowT as — can well be
made. In this case it will be seen that a higher efficiency than
50 per cent. cannot be obtained, and it is also apparent that the
worm must be the driver, since the resistance of friction would
just balance the reverse driving action. The ordinary tooth
friction and the journal friction must of course be added.
Fig. 610.	Fig. 611.	Fig. 612.
The tooth outlines for both worm and wheel are the same as
for a rack and gear wheel, taken on a longitudinal section
through the axis of the worm. The evolute tooth is especially
applicable, and Zx must not be less than 28 (§ 209). The surface
of contact is theoretically only a mathematical point, but in
practice there is a small flattened surface of contact, and if a
larger surface is desired the wheel must be cut with a hob of the
same form as the worm which is to wTork with it.
Wheels which have a contact bearing of a point only, may be
called precision-gears, as distinguished from pow7er-transmitting
gears. The difference, however, cannot be sharply maintained,
for as already shown, worm gearing is used for the transmission
of both large and small forces.
The possible variations of the pitch angle permit a great va-
riety of spiral gear combinations, as the following examples
show:
Example i. Given ~ = the perpendlcular distance between axes a —
R + Rx, and the angle between axes a — 400. If we make y = 6o°, we have
from (S22o)yx = 180 — 40 — 60 = 8o° (see Fig. 610), and from (201)	U 'yi
sin y n
,	sin 8o°	0.5X0.9848
= 4.	—;—----- =	--- = 0.5686, from which R and R\ mav be readily
2	sin 69°	0.8660	' •
determined. If we make a — 4" we have
Ri = -
1 +
1.5686
- = 2-55"
and R = 1^45". For Z — 20, Z\ — 40, the normal pitch t= t sin y —
2 X 7T X 1-45 X 0.866
2 7rT?siny
_
- = 0.272 X 1.45 = 0.394",
The circumferential pitch t =
°*394
sin y 0.866 ~ 0-454 ’o 9848
The sliding velocity c\ according to (202) — c (cot 6o° cot 8o°) = c (0.5774
+ 0.1763) = 0.7537.	. .
Example 2. In order to make c' a minimum, we may make y = yi —
180—-a _ 180 ^ 4° _ see gjj We then have	R\ — 2.666,
7? = x.333, r = 2 X " Xl^3 X_o_9397 . * ***, / - h	- 0.419, and
= 2 cot 700 X c — 0.728 c. It will be seen that the value of c' in Example 1
approached very closely to the minimum.
Q-394
0.400 .140
THE CONSTRUCTOR.
Exautple 3. If so desired we may make y = 900, when one wheel will be-
come an ordinary spur gear, Fig 612, and we nave yi = 180 — 40 — 90= 50°.
= 0.5 X 0.7660 = 0.383, Rx = 2.89", R — 1.11", t = 0.348 t = t, tx = 0.454",
c' = 0.8391 c.
If instead of a, the normal pitch r is given, as is generally the
case with hobbed worm wheels, we choose y and yx and then
Z 7
have R sin y =-----—, wbence :
2 7r
R =	Ri = -----	—...................(206)
2 7T sin y	2 7r sin yx
Both R and r may be given, when y must be determined, and
we have :
sin y — ...............................(207)
2 7T R
Fig. 613.	Fig. 614.	Fig. 615.
The following examples illustrate a variety of cases :
Example 4. a = 90° Z = Zx. The sliding to be a minimum, hence y = yx
=	= 450. The two wheels are similar, both being left hand or as in
Fig. 613, both right hand. The sliding velocity is c' = 2 eoi 45° X c = 2 e.
Example 5. In the arrangement shown in Fig. 614 there is added to the
right angled pair A B. a third wheel C, also right angled, when the wheels
A and C wnll revolve in opposite directions. The middle gear B reverses the
motion, as in the case of bevel gears.
Example 6. When a = o, the axes are parallel and a pair of spur gears
with spiral teeth is obtained, this forni being called Hooke s or White’s
gearing, Fig. 615. y and yx together include 1800, and one gear is left, and
the other right hand. In this case the teeth are formed in true spirals. I11
this case the sliding velocity c' — o. For the wear on this form of gear see
g 222. When a = o and y = o the wheels become spur gears.
Fig. 616.	Fig. 617.
Fig. 618.
Fig. 619.
If the other limit of spiral gears is reached some noteworthy
forms are obtained.
Example 7. a = 900, y — io°, yx = 8o°, Rx = 00. This gives a rack and screw,
Fig. 616. If yx = 900 and the teeth normal, y = io° and a = 8o°, and the teeth
of the rack correspond to the section of a nut. In the Sellers planing ma-
chine the rack teeth are placed at such an angle that the lateral pressure
just balances the opposing tooth friction.
E tam ple 8. R = Rx = 00. This gives two racks, sliding in each other, Fig.
618. We have, as before, vx : v — sin y: sin yx. If a = 900, as^ji Fig. 619, and
y : ^ = 450, we have v — vx. This construction is used in some forms of
boring machinery for cannon, and in screw cuttiug machiues.
Example 9. a = 900, yx= 900, also y = o, both radii of indefinite magnitude,
Fig. 620. This is the so-called revolving rack, used on governors and similar
apparatus in which endlong motion is to be transmitted from a revolving
piece. The velocity ratio of A to B = o.
Example 10. The worm, or endless screw, as already stated, is a form of
spiral gear wheel. These are two special forms of worm gear which although
seldom used, are of interest. There are the forms of internal gearing showm
in Figs. 621 and 622. In the former the worm wheel is the internal gear,
while the latter shows an internal worm, with external or spur worm-wheel.
Fig. 620.
Fig. 621.
2 222.
Fig. 622.
Spirae Gear Teeth and their Friction.
Spiral gears are cut in a similar manner to screws, the tool
being carried in the slide rest of an engine lathe, and set at the
proper angle. The pitch of the screw thread is :
s — 2tt R tan y,
and the travel of the rest is effected by proper change gears,
according to the selected values of y and
The tooth outline to be used is determined according to the
radius of curvature of the supplementary spiral, that is, to that
at right angles to the spiral to be cut. The radii of curvature
r and rx to be used are :
R _ Ri
sin2 y 1 1 sin2 yx
(208)
These give the radii for the construction circles to be used
with the pitch r ; the shape of the tool with which the teeth are
cut is then determined.
Example 1.
we have :
For the wheels of the first example in the preceding section ,
____i-45	_____„_____2.55
sin 2 6o°
l*93 » n = -r
sin 2 8o°
= 2.58".
If it is preferred to determine r, graphically from formula
(208) the method given in § 29 may be employed.
The frictional resistance of spiral gearing is often a matter of
much importance. If the frictional resistance is assumed to be
zero, we have for the relation of the force Papplied to the driv-
ing wheel, to the force Q delivered by the driven wTheel:
P  sin y
Q	sin yx
(209)
The ordinary tooth friction, which is the same as that of the
construction gears (see $ 213) to which must be added the fric-
tion due to the sliding of the teeth, whenever a is greater than
zero. The value of the latter friction is governed by the sliding
velocity cFor the calculation of the loss of useful effect we
may use the formula :
P sin y 1 sin (y -f-
-5-=-------——7---------jr..............(210)
R sin y sin (yx — <)	v 7
in which <f> is the angle of friction for the coefiicient J\ whence
tan <l> — f. For f — 0.16 we have ^ = 90.
Example 2. For the wheels in the preceding example we have
P* __ sin 8o° sin 69° _ 0.9848 X 0.9336
P	sin 6o° sin 710 ~~ 0.8660 X 0.9455 “ I*12'
To this must be added the ordinary friction of the equivalent
spur gears. Another source of loss is that due to the lateral
forces K and Kx, acting in the direction of the axes. For these
we have
R	K
— = cot (y -f 0),	= cot (yx — <p) . . . . (211)
Example 3. For the preceding gears we have K= P cot 69° = 0.3839
Kx — Q cot 71° = 0.3443 Q from which values, in connection wTith the known
dimensions of thejournals the corresponding resistances can be determined.
When a =0, that is, for parallel axes, the sliding action of the
teeth is zero, and the value of P in (2io"i is the same as P;
hence spiral gears for parallel axes work without the tooth fric-
tion due to lateral sliding, the ordinary tooth friction alone re-
maining, as well as the forces K and Kx.
* Brocofs Tables will be found of Service in arranging change gears,
(Calcul des Rouages par Approximation. Paris, 1862).THE CONSTRUCTOR.
141
The tooth friction may be reduced to a very small amount by
reducing the bearing surface of the teeth of one gear to a point
of contact, or practically to a knife edge. Such gears (devised
by Hooke) are only of use for purposes of precision, but in some
cases are found serviceable.*
Instead of the edge bearing, a rounded surface may be used,
with its highest part corresponding to the lineal bearing as al-
ready shown by Hooke and by Willis. The tooth outlines for
both gears are determined as usual, and then one or both pro-
files are redrawn within the original curves, Fig. 623, and the
modified outlines used to form the tooth spiral; teeth so con-
structed running nearly free from friction. In such cases the
length of flank/, and face k may be reduced as shown. Such
forms are more properly to be considered as screw thread pro-
fles than as gear teeth. Willis has shown that in both gears
the flanks may be made radial and the crown of the teeth semi-
circular, Fig. 624. Since such teeth are weakest at the base, it
is preferable to use a modified form of the evolute tooth, Fig.
625. This may be approximated to by using a circular arc of
smaller radius than B S = R cos a, the centre B' being taken
on the normal N N, through the point of contact.
The space s between teeth at the middle of the gear, is called
in the Westphalian shops the “spring” of the teeth. If it is
desired to approximate to the frictionless action of the teeth,
this “ spring ” must be slightly greater than the pitch.
For very large transmissions the gears may be made in two
parts. Fig. 628 shows a pair of such gears for a reversing roll-
ing mill by the Hagener Steel Works. The pitch diameter is
43-3//, the pitch	the face of each gear 20//, and the total
weight 24,200 pounds. The teeth are made with double reverse
angles on each gear, so that the conditions are the same when
running in either direction, and the whole is a masterpiece of
machine work in Steel.
1	i
U-----1)---->|
Fig. 626.
A similar form to the preceding gears is the so-called step-
gearing, Fig. 626, frequently used in planing machines (by
Shanks, Collier and others). The tooth profiles may be modi-
fied as above, to reduce friction, but the gradation 5 should be
as great or greater than the pitch t. Fewer than four sections
shouid not be used.
An objection to the use of spiral gears is the axial pressure
Af, this, however, can be eliminated by the use of double gears
of opposite inclination. Such gears have been known for a
long time (White, 1808) and for moderate Service, have been
frequently used, as in spinning machinery, tower clocks, etc.,
and more recently they have been applied to heavy work, nota-
bly for rolling mill gearing, both in Germany and America.
The pinions used in rolling mill work are made -with 9 to 16
teeth, with pitch diameters from 4" to 24" and over. Evolute
teeth are used, with a base angle from 62 to 69°. The face length
of the teeth is made about 0.22 /.
If the evolute curve is accurately made, the tooth contact is
practically the same as with ordinary spur gears, and the surfa-
ces of contact can readily be discerned, extending diagonally
across the teeth. When such a surface of wear is visible, of
course the teeth are not free from friction. Fig. 627 shows a
cast Steel pinion of ten teeth, for rolling mill Service. This gear
is cast in one piece wfith its shaft and coupling ends, although in
many cases the shaft is made separately.
* These gears have been used in physical apparatus by Breguet for speeds
exceeding 2000, or according to Haton, as high as 8000 revolutions per second
or 480,000 per minute.
\	223
Spirae Bevei, Gears.
Spirally formed teeth are sometimes used on bevei gears, and
in this case the distance a, between the axes becomes zero, wdiile
the angle a remains to be given. For the curvature of the teeth
it is best to use a conical spiral of constant pitch, the projection
of which on the base of the cone is an Archimedean spiral.
Frequent applications of such wheels are tobe found in spinning
machinery, and they are operated successfully at quite high ve-
locities.*
Fig. 629.
The same varieties may be made in bevei, as in spur gears,
and in Fig. 629 is shown a reverse spiral bevei gear of cast iron,
as made by Jackson & Co., at Manchester. Similar gears are
made of cast Steel by AsthSver & Co., at Annen in Westphalia.
Stepped teeth are also used in bevei gears, and in Fig. 630 is
showm such a wheel by A. Piat fils, of Paris.
* For a machine for the correct construction
gears, see Genie Industriel, Vol. XII, p. 255.
of the teeth of spiral bevei142
THE CONSTRUCTOR.
2 224.
Geoboid Spirae Gears.
If a circle is revolved about an axis A Ax coinciding with one
of its diameters, and at the sanie time a radius C S is moved
about the centre C> with an angular velocity proportional to that
of the circle itself, the circle will generate a sphere and the
point of the radius which is at the surface of the sphere will
trace a forni of spiral curve. This may be called a spherical
spiral,* and adjoining lines of the spiral on the same meridian
are equidistant.
A
A,
Fig. 631.	Fig. 632.
If the radius C S passes the axis of rotation, the new spiral
will intersect the one previously traced, as at Ax, Instead of a
mere radial line, may be substituted a point which at the same
time traces the outline of a tooth space, so that a spherical screw
thread is generated with which a spur gear will engage at any
point, Fig. 632. If the axes A and B are maintained in their
proper positions, the spiral when driven, will operate the gear
in the same manner as a worm and worm wheel, $ 221.
The practical value of this especial forni is extended by the
fact that the axis of rotation need not coincide with a diameter
of the circle. Under these conditions there may occur a num-
ber of forms of bodies of revolution bearing an affinity to the
sphere, and to which the writer has given the general name of
globoids. The corresponding spirals may be called globoid
spirals and the resulting gears, globoid spiral gear wheels. Many
of these may be made of practical use. (See Fig. 633.)
There are numerous fornis of globoids according to the posi-
tion which the describing circle holds to the priucipal axis. The
axis about which the radius C S turns is called the counter-axis.
It stands at right angles to the starting position of the describ-
ing circle, and either intersects the principal axis,or is inclined
to it without cutting it. We have then r, for the radius of the
describing circle ; a the shortest distance between the axes A
and C\ c the distance of the centre of the describing circle from
the plane of the principal axis, d the angle which the principal
axis makes with the plane of the describing circle, extending
from o° to 90°. This gives four classes of globoids, as follows:
I.	a — o, c = o.
II.	a = o, c chosen at will.
III.	a chosen at will, c = o,
IV.	a and c chosen at will.
A right globoid is one in which d = o, and wThen (5 is an acute
angle we get an inclined globoid.
The first class is represented by the globoid Fig. 634, giving a
symmetrical conical section ; if d = o we obtain the previously
described sphere.
* More properly a spherical cycloid, see \ 216; its kinematic axoids are
normal cones.
The second class gives the inclined globoid, Fig. 635, with
unsymmetrical conical sections, with regard to the equator, the
spiral being on the zone mantles. If d — o we obtain a sym-
metrical, cylindrical hollow section of a sphere, Fig. 636. The
spiral, when a — o, becomes a spherical cycloid. If d = 90° the
figure becomes a plane cone, or plane ring, and the curve be-
comes a plane cycloid.
We have the third class when <5 = o, and a r, giving a so-
called cylindrical ring, or right globoid ring, Fig. 637 a, and
when a < r, the apple shaped globoid, Fig. 637 b.
If d is au acute angle, the globoid is flattened, Fig. 638 ; the
globoid of Class I is the limiting case. The spiral curves are
globoidal cycloids, which become plane figures wrhen d = 90°,
and the globoid becomes a plane ring or plane cone.
The fourth class gives the highest forms, Fig. 639, in which
6 = 0, and we may have # > r, a = r, or a < r.
The inclined globoids of this class have forms, the limits of
which are found in those of the second class, Fig. 635. If d =
90° we have again the plane cone or plane ring*
# The practical applications of the globoid spiral gears are va-
ried, and are found mainly in right globoids of classes III and
* Two right globoid rings may unite to form a pair of machine elements
when the thickuess of one is made equal to the hole in the other, as in Fig.
a. The two parts then bear the relation to each other of Journal and bear-
ing, and are sirailar to a bajl joint. Each of the two elements describes by
the relative motion of any point a corresponding path cn the other member.
a.	b.	c.
These conditions are approximately found in a pair of chain links. Such a
pair may also be considered as a contracted form of universal joint, A B C,
Fig. b, the same relative motion existing between A and C. The same thing
is shown in a fractional form in Fig. c, when some method of Jiolding the
parts together, such as bands, etc., must be used. This latter resembles
closely the ball and socket joints of the human skeleton.THE CONSTRUCTOR.
143
TV. In the valve gear of Stephenson’s locomotives, Fig. 640, is
found a globoid worm of class III, using the middle part of the
globoid apple, Fig. 637 b> (a <^ r). In this case the reversing
lever B is really a part of an internal gear with a radius Rx =
the radius r of the describing circle.* In this case the internal
gear has but a single tooth, although more might be used.
Fig. 637.	Fig. 638.
It will be seen that the globoid forms can be used as internal
gears. This is shown in Fig. 641, which represents a worm
formed as a globoid screw. Its form is practically the same as
that of the hole in the right globoid ring, Fig. 637 a. The sec-
tion shown in the figure is of such length that it includes one-
fourth of the entire circumference of the worm wheel B, al-
though it could be extended so as to include almost one-half.
Fig. 639.
The most important point to be considered is the formation of
the teeth. Rx is agam made equal to r. Since the globoid is
used in the internal form, the two tooth profiles, on r and Rx ,
fall together. The sliding is in the plane of a normal section
through B and A Ax and not endlong, and hence the shape of
the teeth is absolute.
I
(Internal gear tooth, with R = Rx ). The teeth can be made
of straight profile in the worm wheel as well as in the worm.f
The production of the globoid worm in the lathe is not diffi-
cult. This form has been frequently used in recent work. The
advantages appear to be in the simple form of tooth and in the
completeness of the engagement.
* The worm and internal worm-wheel, Fig. 621, isanother example of the
preceding case.
f This form is described bv Smeaton as used in a dividingenginebyHind-
ley, see also Willis. Principies of Mechanism, ist edition, 1851, p. 163.
An interesting modification is that of Hawkins, Fig. 642* In
this case the wrheel B is composed of friction rollers of quite
large size and the friction is thereby greatly reduced. Instead
of there being only four teeth, as would at first appear, there
is in reality an ideal number of teeth, a condition refeired to in
the fundamental discuss:on in \ 200. If for every revolution of
the globoid screw, one tooth of the wTheel engages, there must
for each space formed between the rollers be 10 teeth to a quarter
revolution, so that instead of 4 teeth in B, there are 4 (1 4- iol
= 44 teeth.	'
The gearing used in Jensen’s Winch, Fig. 643, belongs to the
globoid class IV, of the form shown in Fig. 639. Usually in
this form a = r, although sometimes a < r, as in Fig. 639 c.
Rx is again made = r, and the internal globoid form used. The
ratio is so chosen that a slow motion can be converted into a
fast one, as may also be done with the form shown in Fig. 641
if the pitch of the worm is made sufficiently great. The use of
rollers instead of teeth makes a very satisfactory construction.f
* Hawkins’ Worm Gearing. Sci. Am. Supplement, No. 104, p. 1648.
f See Uhland’s Prakt. Masch. Konstrukteur, also Engiueer, Vol. 24, p. 493.144
THE CONSTRUCTOR.
If in tlie first two classes of globoids the supplementary axis
is removed an indefinite distance, the globoids become plane
surfaces, and the globoid screws thereby reach the limit. The
limiting case of Class III is the ordinary worm and worm wlieel,
and another form is Long’s spiral gearing, which also belongs
to Class III; a is chosen at will, c~o, d = o. The globoid be-
comes a plane cone and the globoid screw becomes an Archi-
median spiral. If R becomes indefinitely great we obtain a
disk with a spiral groove engaging with a rack, the middle sec-
tion having full tooth contact from top to bottom.* When this
is brought into Class IV, we obtain the Archimedian spiral in
its most general form, i. e., the evolute of a circle.
E. CAL.CULA 77 ON OF PITCH AND FA CE OF GEARING.
whence:
_ lja_ s_ z_
t ~ ^ C’ S'y Z'
(217)
and for the radii R and R' :
R' _ Z' tf
R ~ Zt

The value of C depends upon the ratio of the teeth, and upon
the value of >S for the material used. If we assume the latter to
be the same for both cases, the number of the teeth alone re-
mains to be considered. A reduction in the number of teeth
increases the pitch, according to (217) ; and according to (218)
reduces the radius.
§ 225.
Pitch of Gear Wheees. Tooth Section.
The dimensions of gear wheels must, for the same pressure
on the teeth, be increased to meet shock in proportion to the
increase in initial velocity. For slow running gears this action
can be neglected. We may in this respect, therefore, divide
gears into twro classes, viz. :
Hoisting Gears and Transmission Gears ; and includes under
the term hoisting gears all those having a linear velocity at the
pitch circle of not more than 100 feet per minute, and under
transmission gears all those running at a higher velocity.
For a pitch /, face b, length of teeth /, and base thickness of
tooth h, wre ha ve for a tooth pressure Pcorresponding to a stress
S, the general formula:
bt =
(212)
and for the proportions of length and thickness already adopted
we have:
bt =
(213)
This assumes that the resistance of the teeth is proportional
to their cross section, which is also equally true for those which
have the same ratio of b to t to each other, a condition which is
often of much Service in practice.
3 226.
Pitch and Face of Hoisting Gears.
For a hoisting gear of cast iron let:
(P R) =z the statical moment of the driving force,
Z = the number of teeth,
R — its previously determined pitch radius, in inches,
t == the pitch,
we have for the given dimensions :
	t l — =* 0.073 \ 7T	Upr) 1 Z	. . (214)
	t — = O.OI45	^ R *	• • (215)
the face b being made	b — 2 t. . .		. . (216)
These are intended to give a fibre stress S of about 4200
pounds. The actual stress is properly somewhat less, because
the thickness of the tooth at the base is usually more than t»
as assumed in (213).
P R
Since the value of —— is the same as the pressure P9 we can
R
use (215) in cases in which Ponly is given, as for rack teeth.
In discussing the preceding formulae, consideration must be
given to the elements which are usually given or selected in
practice.
Let t' and t be the pitch for two cases respectively, and Z and
Z' the number of teeth. Also let 6" and S/ be the stress at the
base of the teeth, and let the constant, 6 C) (4-)'—
in (213) is made equal to 16.8, be called C or C' ; we then have,
according to (214):
/=	2 7T C(PX) (4)
sz
♦ See Civil Engineer and Arch. Journal, July, 1852, also Dingler’s Journal,
Vol. 125. Weisbach, III, ist Ed., p. 449, ad Ed., III, 2, p. 87.
Exantple 1. Z = 11, Z? = 7, hence
f	n	3f
-j- = ^ -- = ^ t-57* = 1.16 ;
y — 2 tf = 1.16 b.

49
121
so that the 7 toothed gear will be about % as large as the 11 toothed gear, or
a 42 toothed gear for the same case would be about % as large as a 66 toothed
gear, and with 1.16 times greater width of face.
The constant <7, for a given series of gears, should be invari-
able, and for ordinary spur gears may be taken equal to 16.8, as
in (213). For the so-called “ thumb teeth,” (§ 212), the constant
may be much smaller, and hence permit an important reduction
in dimensions. The value of — for wheels of more than ten
teeth is not less than 0.7, and introducing this value we get (?
= 8.4, that is 0.5 C; hence ‘‘thumb shaped ” profiles are capable
of sustaining twice as great a load as the ordinary form.
Example 2. If, for a given moment (P R) the thumb profile is substituted
for the ordinary form, without reducing the number of teeth, the pitch may
be reduced in the proportion

= 79
t,
or about 0,8 times, with a proportional reduction in diameter and face.
If, however, the teeth are taken in the above ratio of it : 7, we would have
for the pitch,
and the radius R' ■■

— 0.89 /,
R 0.202 = 0.58 R.
The influence of the stress ^is always important, and it should
not be increased above the normal value for the given material,
wThich latter is usually cast iron. An increase of one-fourth in
the permissible stress would reduce the pitch and diameter only
7 per cent., but on the other hand it must be remembered that
too low a value of S causes an unnecessary increase in the size
and weight, not only of the gears but also of the bearings, frame
work and other parts of the machine. The value of 6* used
above, viz.: 4200 pounds, has been show in practice to give sat-
isfactory results, and there appears to be no good reason for any
great variation from it.
When the gears are made of wrought iron, as is sometimes
the case, S may be made much higher, and may indeed be taken
double, say 8400 pounds. This gives a reduction in t' in the
proportion of /^0.5 = 0.79 t.THE CONSTRUCTOR.
Example 3. For comparison between a wrought iron gear of 7 teeth of
thumb shaped outline, with a cast iijbn gear of 11 teeth of ordinary shape,
we have:
o*5 X 0.5 ^	= ^ ^ 0.101 = 0.47 R,
~ = t & o-393 = 0.7 A
and^ = 0.7 b.
In Fig. 644 the five cases given in the last three examples are
shown on the same scale, side by side. In order to indicate the
fact that the moment {P R) is the same in ali cases, the shaft
diameter has been shown. It will be apparent that there is no
. definite relation between the diameter of the shaft and the ra-
dius of a gear.
The invariability of the moment, which has been maintained
in the preceding examples, does not exist of the tooth pressure
/*upon the driven gear is again transmitted through a second
so-called compound gear. If the pinion of a radius R, driving
a gear Rf> compounds by a pinion R2 on the same shaft into a
rack R2/1 for example, with a given pressure P} we have from
(214)
t = Coust. VT>
whence
__ Jlj?£_ z_
t ^ R2	C S' Z'
(219)
This gives
R' = R
O
5'
But R2 = Z2t and R/ = Z/ t2‘', and from formula (215) :
	t2' = t2 s	]a s C‘ S'		
Hence we get:				
v _ ■	£	Jc' s	z/	z
t	^ C' S'‘	^ C' S'•	z;	Z'•
By selecting the number of teeth we may make
z/
Z*
z/ *
— and then obtain :
t
s!
c_ s_
c* &
(220)
and for the radii:
RZ
R
& JC' S
Z ** C' S'
(221)
Example 4. A rack with a tooth pressure P, gearing with an 11 toothed
pinion, is driven by a larger gear which again engages with an 11 toothed
pinion, Fig. 645, the teeth being of the usual shape, and the materiat cast
iron.
This is to be replaced by making all parts of wrought iron, and reducing
the number of teeth in the rack pinion to 4, as shown in § 212, all teeth being
also altered to the thumb-shaped form. We then have C' = 0.5 C, S' = 2 S,
and hence: t’ — y/ = % /, and E' = R -pj \/ % = R.
It will be noticed that in this case the ratio between the larger
gear and the pinion on the same shaft is such that in (217) and
(218) both are determined for the same moment (P R.)
Example 5. If, in order stili further to reduce the dimensions, steel is used
instead of wrought iron, thus p#rmitting a stress of 14,000 pounds, we have
t* = / \/	0.5 X 0.3 = 0.387 t, and R' = 0.4.^ R = 0.145 R, or about f R.
The proportion of the results of the last two examples is
shown in Fig. 645. The force P on the teeth of the rack is the
same in all three cases.
The statical moment on the main shaft is, however, reduced
with the reduction in R'y as is consequently that of the inter-
mediate shaft.
The advantages of steel as a material for gear wheels have al-
ready been referred to in § 222. Its greater strength enables
much lighter wheels to be used for the same Service, than with
cast iron. For a gear of cast iron and of steel, to act against
the same moment, all other things being equal, we have, taking
S = 14,000, and S-
t'
14200—, and also —-^0.3 :
t	R
about — in
3
favor of the steel.
This gives for the ratio of weight (A)5, that is 0.3, the same as
the ratio of .S to S', or say three to one. This advantage also
exists for transmission gearing, although not to the same extent.
If the velocity ratio in a compound train is comparatively
great, it is interesting to note that the most advantageous ratio
145
between gears lies between 1 : 9 and 1 : io, this giving a mini-
mum of shafts and of teeth.*
Tabde of Cast Iron Hoisting Gears.
t	£ II ft.	PR	t		{PR)
	R	~Z	7T	R	Z
lA	127	10	0.15	107	8.67
yi	200	20	0.20	190	20.56
%	287	35	O.25	297	40.16
7A	39i	55	0.30	42S	69.40
I	5i«	82	0.35	583	110.20
IX	798	160	O.4O	761	164.50
	1150	277	0.45	963	186.00
	«564	440	O.5O	1020	320.50
2	2044	658	0.60	1712	555.20
2X	3200	1284	O.70	2330	881.70
Example i. A force of ioo pounds is exerted on a hand crank of 15 inches
radius; what should be the pitch and face of a 10 toothed pinion for the fur-
ther transmission?
Here we have	■*—- = 150, and the nearest value in the table
Z	10
in the third column, is 160, which corresponds to a pitch of 1^ inches. The
face is = 2 t = 2*4 inches.
Example 2. A rack is to work with a pressure of 2000 pounds on the teeth.
This would give a pitch of about 2 inches, or as given in the 4th and sth
columns, a pitch t : 0.65 ir, which is practically the same, and the width of
face = 2 t. If the rack is made of wrought iron, we have t = 0.707 X 2 =
1,414", and the face = 2.8".
§ 228.
Pitch and Face of Gearing for Transmission.
The fibre stress S, which is exerted upon the teeth by the ac-
tion of a given force Pt should be taken smaller for transmis-
sion gears as the circumferential velocity v increases, since the
* If <f> be the total ratio, and k the number of pairs of gears, and the ratio
between each pair be x — ~ we have 4> = x . The total number of teeth
in the train, y = k {Z + Z') = k Z' (1 + x). Now k —	and the product of
the number of teeth and the number of pairs gives
(/ n Z’ (1 + x)
>	{Inxf-
Diffesentiating and making the differential coefficient equal to zero we
"(l JC)
get In x =  --——— which equation is satisfied by x = 9.19. For example
$ = 600, and the number of teeth in smallest pinion = 7. We have the fol-
iowing combinations:
(a)	<j> = 20 30, gives y = 7 (2 -j- 20 -f- 30) = 364, y k — 728.
(b)	(j) = 4.5.5.6, gives ^ = 7(4 + 4 + 5+ 5+6) = 168, yk = 672.
(c)	<f> — 6.10.10, give%y = 7 (3 + 6	10 + 10) = 2qg, y k — 609.
The last solution is the best, for although it requires more teeth than {b}
it has one less pair of gears, and for solution (a) the number of teeth, viz.r
210 is inconvemently great.146
THE CONSTRUCTOR.
dynamic action of shock and vibration also increases. For cast
iron we may take :
9,600,000
v -|- 2164
(222)
in which v is the lineal velocity in feet per minute. For steel
5 may be taken 3 times, and for wood ^ times the value thus
obtained. For cast iron we obtain, for :
termined, the choice of the number of teeth Z is unrestricted.
In such cases we have for the width of face b:
b
396,000
N_
~Zl
(226)
If we give to A the successive values 30,000, 25,000, 20,000,
15,000, 10,000 and 5,000, we get the following numerical rela-
tions :
Cast Iron :
Comtnon and Thiitnb Teeth.
Common Teeth. Thumb Teeth.
Z'— IOO I 200 I 40O I 600 I 800 I IOOO I 1500 I 2000 1 2500
5=4240 I 4060 I 3744 I 3473 I 3238 I 3034 I 2620 I 2302 I 2068
For Steel:
S— 14,112,13,520; 12,467! 11,565110,782 10,103 8725 766516886
And for Wood :
5 — 2544 | 2436 | 2246 | 20S3 | 1943 | 1820 | 1572 | 1381 | 1240
The velocity v may be obtained when n and R (the latter in
inches) are given, by the following formula :
2 7T R n	„ _	,
- =------------= °-5236 Rn..............(223)
The selection of a proper value for v will be discussed bclow.
It is also found that the breadth of face £ should increase w7ith
the increase of P. Tredgold States that the pressure per inch
. R
of face, that is — should not exceed 400 pounds. This, how-
ever, is not to be followed implicitly, since pressures as*high as
1400 pounds have been successfully used in practice. It is bet-
ter, however, to consider the question of wear froin the product
P .
of —£• into which should not exceed a predetermined maxi-
mum. It is found that if —r x n exceeds 67,000 the wear be-
b
comes excessive. In a pair of wheels where the teeth of both
are made of iron, the greatest wear comes upon the teeth of the
smaller wheel. I11 this case we may make
P n
—,— = not more than 28,000
b
(2~4)
and if possible it should be taken at less than this value. For
smaller forces this constant, which we may call the co-efficient
of wear and designate as A, may readily be made as low as
12,000, and even 6,000, without obtaining inconvenient dimen-
sions. When the teeth are of wood and iron the wear upon the
iron may be neglected, as the wear comes almost entirely upon
the wooden teeth. For wooden teeth the value of A should not
exceed 28,000, and is better made about 15,000 to 20,000.* It is
impossible to give exact values in such constructions, and it
must Ue left to the judgment of the designer as to how far it
may be advisable to depart from the values obtained from exist-
ing examples.
It must be remembered that the different values of A do not
appreciably affect the streugth, but rather control the rapidity
of wear. When sufficient space is available and a low value can
be given to the co-efficient of wear, it is advisable to do so; if
this cannot be done, the co-efficient which is selected will give
an indication of the proportional amount of wear which may be
-expected.
In cases where a number of wheels gear into one other wffieel,
it is better to take, iustead of the number of revolutions of the
common wheel, the number of tooth contacts, that is the pro-
duct of the revolutions and number of wheels in the group.
If R is given, as is often the case with water-wheels, fly-wTheels,
&c., Pis also known, and since A can be chosen we have, tak-
ing N to be the horse powrer transmitted :
__ Pn _____ 63,000 N
hence from (213) for ordinary teeth,
— _i6^ P _
~ sTb “ ~s~n~
....	(225)
and for thumb shaped teeth,
~ Sb
S.4A_
5 71
If, however, as occurs in many cases, R is not previously de-
* See case 10, in \ 229 seg.
,	P11	N N ,
b =-------=2.1 -— = 13.2-—; t-
30,000	R	Zt
504.000	___252,000
71 S '	71 5
25,000
A	N ,	420,000	,	210,000
'	I5*4 ijCj» ^	„ o * ^	.. c
'Zt'
.	P71	N 0W
b=ToTcr^R=^&-zt'
N
N
71 S
t = 33^000	^
71 S ’
71 S
168,0001
71 S
A	P™	A
b =-----=4*2-77 — 26.4
15000	R	Zt
252,000	126,0001
=~tTs~ ; 1 = ~anr
; (227)
JPn__
10,000
N
.N
68,000
6.3 -js = 39.6 ^^/e—; t' =
zt
. Pn	N	N
12-6r = 792 zi;
71 S
nS ’	nS i
84,000
71 5
42,000
For transmission gears the minimum number of teeth should
not be fewer than 20, in order that the unavoidable errors of
construction shall not cause excessive wear; for quick-running
gears it is desirable to have stili more teeth. The gear wheels
ou high speed turbines seldom have fewer than 40, and often as
many as 80 teeth. When wood and iron teeth are used, the
least wear is produced when the wooden teeth are on the driver,
because the action begins at the base of the tooth and passes
toward the point,while on the driven gear the action isreversed.
If desired a number of teeth Z can be calculated which will
give a desired ratio b : t. If we combine formulae (225) and (226)
we obtain the useful relation :
__396,000	712 52 N
Z~~i6.WaT* ^
(228)
This shows the important influence of A upon Z, and the ef-
fect of the number of teeth upon the wTear; also the important
relation of the tooth profile, since the constant 16.8 (or for
thumb teeth 8.4) appears in the second power. It is also seen
that Z is dependent on the square of «, and the square of 5,
other things being constant. These points indicate the methods
of obtaining the least stress.
The value ofis sometimes madeasgreatas5. Forwdder
faces and sometimes for narrower, the rim of the gear is made
of two adjoining parts.
Example 1.—A water wheel of 60 horse power, 26 feet, 3 inches in diameter,
moving with a velocity at the circumference of 256 feet per minute, is to be
provided with an internal gear wheel, the pitch circle being 16 inches less
radius than that of the water wheel, and gearing into a pinion which is to
make 40 revolutions per minute.
.	25^
We have:	n —------—2—- = 3-1
3.14 + 26.25
. «1 4o ,	-/	256(157.5—16	„	33000 X 60
and — = —; also V =---------------------= 230 ft per minute. P— —-------------
n 3-1	I57-5	230
— 8608 lbs. This gives a permissible stress 5 = 4100 lbs. nearly% We will
choose for the smaller wheel = 25,000, which gives ~ = —5,000 =
o	0	n-i
2~000. — 625 hence b — ~~ —	= 13We then have from (227) t —
40	62.5	62.5	v "
420,000	,,, _T ..	.	„	2 rr R 2 7T 141.5
------- — 2.56". We then have Z = —— = ------------— 347. If we make
40 X 4100	#	t	2.56
348 teeth the wheel may be divided into 12 segments of 29 teeth each. For
the driven wheel we have Z\ = — Z= — X 348 = 27, whence Ri = -7-^ 2‘5-
»40	2 ir
== 11".
Example 2.—A turbine water wheel of 100 horse power has a vertical shaft
making 96 revolutions per minute, and it is required to drive a horizontal
shaft at 144 revolutions, hence a pair of bevel gears are required. We will
select wooden and iron teeth, and let the wooden teeth be on the driver. We
will assume v to be between 1200 and 1400 feet per minute, which gives JS
= 1600, and make A = 25,000, also — = 3. We then have from (228) Z =
962 X 16002;THE CONSTRUCTOR.
147
Example 3.—In a given train of gearing, Fig. 646, in which tlie correspond-
ing wheels of both pairs are of the
same size, the force transmitted in
each case is inversely as the number
of revolutions. In order to have the
co-efficient of wear --7- alike in both
0
cases it is only necessary to make ali
the gears of the same face. An ex-
ample of this kind may be found in
the back gearing of many lathes.
Example 4.- Eet it be required to
construet a pair of durable gears of
wooden and iron teeth under the fol-
io wing conditions: N = 5, n = «1 =
60, and ^ = 2. We may make v =
500, which gives, from (222), S= 2160,
and as great durability is required
we will take A as low as 10,000. These values in (228) give :

396,000	602 X 21602 X 5 _
16.82 X 9,000-3 *
: 80.8
which we may call 80 teeth.
We have from (227)
= 1.167"
. = 2*33
60 X 2160
396,000	5
an	9000	* 80 X 1.167
or 2 t, as intended.
Example 5.—Eet N= 40, n = 30, ni — 50 for a pair of iron gears with teeth
of common form, and let = 2.5. If we make v = 300, .S = 3400 and we
take A — 25,000. This gives for the driver gear:
396,000
16.S2 X 25,000-3
502 X ^4°°2 X 4°	, „
-------rs 4I-5
say 42 teeth, and Z\ = — Z =70,
we have t —
and b = 2.5 / = 6.175"
420000
50 X 3400
= 2-47
If we choose the thumb-shaped teeth, and make — = 3.5 we get:
396,000	502 X 34002 X 40
8_42 X 25,000— '
say 120, and Z\ = 200.	/' =
3-5
210,000
50 X 3400
= 118
= 1.235", b = 4-32.
This gives smaller teeth, but larger radii than when the common form is
US\Vken Steel is used for gear wheels, special proportions are obtained. It is
not too much to say that the value of the co-efficient of wear A should be
taken twice as great as for cast iron. The stress S, however, may be taken
times that permissible for cast iron. Taking these points into considera-
tion in formula (228) we see that A would reduce the number of teeth by y8,
and 5would increase it by	»that is> about 11 times, so that the net in-
crease would be if the above values are accepted. It may therefore be
laid down as a rule that steel gears should have more teeth for the same Ser-
vice than cast iron gears. The ratio of face to piteh may be m&de quite
large, and in the case of double spiral gears (as Fig. 627) the ratio — is some-
times made as great as 7 or 8. If the formula for thumb teeth be used, in-
stead of the usual shape, the constant 16.8 will give satisfactory results. The
value obtained for the piteh is that for the normal piteh t = t sin y, but the
width of face is the actual width, as b, in Fig. 627, 2 b' in Fig. 628.
Example 6.—Suppose the wheels given in Example^ 5 to be made with
double spiral teeth of steel. We take A — 56,000, and — =6, also S= 12,800.
We then get:
396,000
8-42 X 560002
We have r
5q2 X 12,8002 X 40 _ g7
6
8.4 X 56,0°° =
also b —
12,800 X 50
396.000	40
56.000
If we take Zx ■
87 X 0.74
* 84, we get Z = 140 and b =	If s -
60" we have
Q-74
0.866 '
0.854".
sin 60
We may take t = 0.875", which gives r = 0.866 X 0.875 =» 0.75/' and
A = ±!_ .
r 0.757'
We have then finally Ri = R — 19-47"
-- 5*93, or nearly 6.
? 229.
Exampees and Comments.
The following examples taken from actual practice will be of
interest: (see Table on following page).
No. 1. Erom the driving gear of the main steam engine of
Fleming^ Spinning and Weaving Mill in Bombay. The toothed
fly-wheel is the driver, and the teeth are shrouded, as shown in
Fig. 651. The coefficient of wear for the driven gear seems high,
and does not indicate long endurance.
No. 2. A toothed fly wheel engaging with a pair of equal spur
gears ; 300 horse-power transmitted by each gear, makiug a
total of 600 horse-power.
The value for
Pn_
b
must therefore
be multiplied by 2; see last column of the table.
No. 3. This is from the air compressor for the atmospheric
Pn
railway of St. Germain (now abandoned.) —- — is evidently too
high, as would probably have become apparent had the gears
continued in operation.
P .
No. 4. — is very high, but the small number of revolutions
b
Pn
keeps the value of----within reasonable limits»
b
Nos. 5 and 6. These are from the great water wheel at Greenock.
The pressure at the rim is great, but the teeth have worn well
in practice, as might have been predicted from the moderate
values of
Pn
b
The value of the latter is almost the same for
No. 6 as for No. 5, hence the wear should be about the same for
both gears.
No. 7. The teeth in the smaller gear are thinner than those of
the large fly-wheel, hence the two values for N. Probably the
larger wheel was originally made with wooden teeth.
P n
No. 9. Notwithstanding the high pressure the value of----
is reasonably small. The stress upon the teeth is quite high, as
is also the case with No. 4, and lower stresses are to be recom-
mended.
No. 10. This is one of the most noteworthy examples of the
whole collection, on account of the very slight wear exhibited.
The wooden teeth on the large wheel, (the fly-wheel of the
steam engine of the Kelvindale Paper Mill at Glasgow) ran for
26 x/z years, for 20 hours per day, with a wear upon the teeth,
measured at the piteh circle, of only about x/% inch. For the
first half of this time the engine indicated 84 horse-power, at 38
revolutions. The teeth were lubricated twice a week with tale
and graphite. The long endurance is doubtless partially due to
the great care which the teeth received, they having been cut upon
the wheel in place, but also to the moderate co-efficient of wear.
No. 11. The teeth were found too small in practice, as is indi-
cated by the stress of 3000 pounds ; from formula (222) we ob-
tain S= 1734 pounds.
No. 12. Two gears with wooden teeth engage with a single
pinion on the screw propeller shaft. The teeth are in twro sets
of 4width of face each.
No. 13. Very high pressure, which must appear in the wear
upon the teeth ; apparently it should be difficult to keep them
in good condition, owing to the high value of —.
No. 15. These teeth appear weak, as has been shown by re-
peated breakages. The wear must be rapid, as indicated by the
high value of ~y~*
No. 17. These gears, (designed by Fairbairn) were intended
ultimately to transmit double the power at first given, in which
case the stress would reach over 4000 pounds, which is admissible
Pn
but the value of —^— would then become rather too high to in-
dicate very great endurance.
Pn
No. 20. The value of —^— seems too high for the wooden
teeth ; it is almost too great also for the iron teeth, and it must
be remembered that with wooden and iron teeth, the wear comes
almost entirely upon the wooden teeth.
No. 22. These gears are from an establishment which has used
hyperboloidal gears with much success for power transmission.
The angle of the axes is 90°. The use of wooden teeth upon the
driver is to be criticised, as tending to increase the liability to
wear.
F. THE DIMENSIONS OF GEAR WHEELS.
2 230.
The Rim.
The ring of metal upon which the teeth of a gear wheel are
placed is called the rim. For cast iron spur gears, the thickness
of the rim is given by the formula
S = 0.4 / -f- 0.125"...........(229)148
THE CONSTRUCTOR.
\ EXAMPLES OF TRANSMISSION GEARING.												
No.	N	n	R	Z	t	b	V	P	S f. —! REMARKS. 0 b j			
1	1000	36.67 114.8	120 38-25	144 46	5.25	— 24	2300 1 14,000 1' i		1 rg7 j 2I,39° i Iron and Iron. I i 66,970 Steam Engine.			
2	3°°	100	146.5 37	230 58	4.00	14	1900	5,100 j 1614		_ 2 x 9107 364 ■ —		Iron and Iron. 1 36,40° |		
3	270	60 12	19.6 ~9«"	_J9_ 95	6.25	20.6	616	I4>3°°	1848	694	! Iron and Iron. 8,330	
4	240	13-3	IIO	208 *~68~	3.125	16	766	10,200	3270	639	**>498 Iron and Iron. 28,110 | Transmission for No. 8.	
		44	33									
5	192	Jh33_ 15.H	400 35-25	7°4 62	3.6	15	280	22,240	7252	1483	T,972 | Iron and Iron. 22,450 j Water Wheel.	
6	192	15.14 5o	106 32 '	208 r	3.18	15	840	7,425	2275	495	7,494 Iron and Iron. 24,750 j Transmission for No. 5.	
7	140	30 55	JJU 32	J[3£_ 72	2.8	8.6	900	5,000	4266 4^35	581	£7,440 | iron and Iron. 31,970 Steam Engine.	
8	140	30 54.5	66.5 35-75	133 76	3	13 />	1045	4,350	3700	335	10,040 18,230	Iron and Iron. Steam Engine.
9	120	i.51 i3.3	291 33	560 80	3.125	15	240	16,230	5688	1082	_M\34 14,39°	Iron and Iron. Water Wheel.
IO	100	45 158.8	84.5 24	_i7JL 50	3	10	2000	1,635	924	163	7,357 8,175	Wood and lVon. Steam Engine.
11	90	26 80	85.4	228	2.375	5-9	1163	2,500	3000	424	11,010	Wood and Iron. Steam Engine.
			27.75	74							33,9°°	
12	82.5	54 83	55.i	114	3.1	2x4.75	1558	3,440	1848	362	I9,54°	Wood and Iron. Screw Steamship.
			35.8	74		“•75					2x30,040	
13	5o	_4-o_ 7.32	5°-4 27.5	96 52	3.25	10.6	104	I5>5°°	7536	1463	5,849 10,700	Iron and Iron. Water Wheel. 1
14	20	774 40	85.4 16.5	248 48	2.2	6.3	328	1,980	2420	3H	_f,433 12,570	Iron and Iron. 1 Water Wheel.
i BEVEL GEARS.												
Io	300	93	24-37 45-7	50	3.1	13	1187	8200	3270 3697	630	58,660	Iron and Iron. Turbine.
		50		93							31,540	
16	300	100	J_?7 26.7	__55__ 49	2.7	10	1576	6170	3840	617	61,700	Iron and Iron. Transmission for No. i.
		111.8									68,980	
17	240	44 44	42	75	3-5	18	968	8050	2133	447	19,670	Iron and Iron. Transmission for No. 3.
18	200	4i	59	98	3.8	11.8	1260	5157	2000	437	17,920 34,960	Wood and Iron. Turbine.
		80	30.1	50								
19 20	130	93 124	24.8	80 60	2.4	8	1523	2772	2276 2417	346	32,220 42,970	Wood and Iron. Turbine. ,
	100	_93_ 144.7	234 15	70	2.1	6.3	1140	2860	2985 3840	454	42,220	Wood and Iron.
				45							65,690	Turbine.
21	50	93	25.6	75	2.1	6.3	1236	13*3	1564 1848	208	!9,38o	Wood and Iron. Turbine.
		218	10.8	32							45,43°	
HYPERBOLOIDAL GEARS.												
22	16	72	21.6 *9~	68	!-996 i-993	5.9	812	640	924 1250	108	7,8io	Iron and Wood. Transmission. 	Si		
		81.6		60							8,851	THE CONSTRUCTOR.
• 149
middle or at one
for gears of fine
See Fig. 647. The rim is thickened in the
edge to — d, and also stiffened by a rib, and
5
Fig. 647.
? 231.
The Arms op Gear Wheees.
The arms of gear wheels are made according to the following
forms, dependent upon the kind of rim used.
pitch the section of the rim is curved, which harmonizes well
with arms of oval section. According to (229) a pitch of i//
would give a rim thickness d = 0.4" + 0.125" = 0.52$" or a
little over y2'\ and for a pitch of y2", b = o.325//.
6
For bevel gears of cast iron the rim is made — S thick at the
outer edge, and of the various forms shown in Fig. 648.
Fig. 648.
Forwooden teeth itis necessary to have a deeper and stronger
rim, the dimensions being dependent somewhat upon the
method of inserting the teeth. The proportions for spur gears
Fig. 649.
Fig. 652. Ribbed sections, which are made sometimes as
shown in the dotted lines as may be most convenient in mould-
ing, and oval sections, in which the thickness /? of the arm is
generally inade one-half the width /z. A good proportion for
the arms is obtained when their number A is made as follows :
From these we obtain the following :
A	=	3	4	5	6	7	8	10	12
Z\	r-	30	53	83	119	162	211	330	475
Z\	{%=	11	23	36	52	7i	93	146	209
Example.—For a gear wheel of 50 teeth and 2" pitch, we have Z y/ t =»
50 y/ 2 = 50 X 1.414 = 70 and this lies between 53 and 83; being nearer the
latter we give the wheel five arms. If the pitch had been and the same
number of teeth Z \/ t — 50 y/0.75 = 50 X 0.866 = 43.3 or between three
and four arms, the latter number being used in practice.
The width of arm /z, in the plane of the wheel is somewhat a
matter of judgment, but may suitably be made according to the
ratio h = 2 to 2.5 t, when the thickness /3 may be obtained from
the following formula:
Fig. 650.
are shown in Fig. 649, and for bevel gears in Fig. 650. Forvery
wide faces the wooden teeth are made in two pieces and a stay
bar cast in the mortise.
Small pinions are often cast solid, and wThen subjected to
heavy pressures are strengthened by shrouding, as shown in
Fig. 651, and sometimes this shrouding is turned down to the
pitch line.
FiG. 651.
For double spiral gears of steel (see \ 223) shrouding is to be
recommended, and is very generally used. The use of shroud-
ing especially assists in securing good steel castings, for the
great shrinkage of the steel, nearly two per cent., tends to pro-
duce warped and twisted castings.
Small pinions are sometimes cut from solid wrought iron, in
which case the shrouding must be omitted.
Should this formula give a thickness either too great or too
h
small for convenience in casting, another value for — must be
taken and the calculation repeated. The following table will
assist in this operation.
The taper of the arms may be made as follows : the ribs at
the rim are made slightly narrower than the breadth of face b,
and at the hub, equal to, or slightly greater than b. For arms
of oval section /z, may be made equal 2 t at the centre of the
wheel, tapering to % this width at the rim.
? 232.
TabeE of Gear Wheee Arms.
h t	Value of when b								
	►E.! II S|	9	12	16	20	25	30	35	40
1.50	0.20	0.28	o-37	0.50	0.62	0.78	°-93 j	1 1.08 |	1.24
I*75	0.16	0.21	0.27	o-37	0.46	°’57	O.69	0.80 |	O.9I
2.00	0.12	0.16	0.21	0.28	o-35	0.44	0-53	0.61	O.7O
2.25	0.10	0.12	0.17	0.22	0.28	0-35	0.41	0.48	o-55
2.50	0.08	0.10	0.13	0.18	0.22	0.28	0-34	°-39	I o-45
2-75	0.06	0 08	0 11	0.15	0.18	0.23	0.28 l	j °-32	| o-37
3.00	O.O5	0.07	0.09	0.12	0.16	0.19	0.23	0.27	O.3I
									150
THE CONSTRUCTOR.
Example.—Eet a wheel have 6 arras, and 120 teeth of 2 inch pitch, the face
being 4 inches. If \ve make h — 2 i at the centre of the wheel, we have
h	Z	B
—— =■ 2, and ^ - = 20, hence we get froiu the table —— = 0.35, and/3 = 4 X
o-35 =	If thisis considered too thick, we raay raake h = 2.25/, which
gives j3 = 4 X 0.2S ■= 1.12".
For gears with wooden teeth, and for the iron wheels gearing
with them, the dimensions of the arms may be made 0.8 times
that given by the preceding rules. If more accurate dimensions
are required, the best plan is to determine the pitch of the
equivalent iron teeth, and use this value in the calculations.
? 233.
Gear Wheee Hubs.
The hub for a gear wheel generally tapers slightly each way
from the arms to the end, the length L = — b, or somewhat
•	4
more for wheels of very large diameter, and the thickness of
metal about the bore is made w = 0.4 h -f- 0.4", iu which h is
the same as in the preceding section. In cases of much im-
portance reference should be made to formula (66), \ 65.
If the wheel is not to be secured by shrinkage the thickness of
metal at the ends of the hub may be made = 3/ w. The kev way
is cut the entire length of the hub, and for wheels which are
subjected to heavy Service the metal should be reinforced over
the key way. Instead of this, the hub may be strengthened by
wrought iron rings, forced on one or both ends. Such rings are
usually of rectangular cross section, the thickness being yz wf
and add greatly to the strength of the hub. See Chapter III.
| 161 to the end.
? 234.
Weight of Gear Wheees.
The approximate weight G of gear wheels proportioned ac-
cording to the preceding rules may be obtained from the fol-
lowing:
G = 0.0357 b /2 (6.25 Z -J- 0.04 Z2) ... . (233)
The following table will facilitate the application of the
Q
formula as it gives the value of —T— for the number of teeth
b r
which may be given, and the weight can at once be found by
multiplying the value in the table by b t1.
Z	0	2	4	6	8
20	5*04	5.6°	6.18	677	7-33
30	7-99	8.6l	9-24 |	9.89	10.52
40	11.09	11.90	12.59 !	13-30	14.02
5°	14.74	I5-48	16.23	17.00	17.77
60	1S.55	19 35	20.15	20.97	21.80
70	22.65	23.50	24.36	25.24	26.12
80	27.02	27.93	28.85	2979	30.73
90	31-69	32.66	3363	34.62	35-63
100	3663	37.67	38.70	39-75	40.81
120	47-40	48.54	49.69	50.85	52.03
140	59-3°	60.56	61.82	63.10	64.27
160	72-35	7373	75-10	76.39	77-9°
180	86.54	88.03	89.52	91.02	92.54
200	101.88	IO3.48	104.98	106.70	108.34
320	118.36	120.08	122.15	123.52	125.27
Example.—For a cast iron gear wheel, proportioned according to the fore-
going rules, with 50 teeth, 2'' pitch and 4" face, we have b fi — 16, and by the
table the multiplier for 50 teeth is 14.74, and the weight = 16 X 14-74 = 235 84
lbs., say 236 pounds. For a gear of 50 teeth, 1%" pitch and 2%" face, we have
b fi = 3.90625, which multiplied by 14.74 gives 57.62 pounds.
For bevel gears or for gears with wooden teeth and lighter
arms (as given at the end of § 232) the weights will run slightly
less than given by the table.
CHAPTER .XVIII.
RATCHET GEARING.
1235.
Ceassification of Ratchet Gearing.
Ratchet gearing may be considered as a modification or ex-
tension of wheel gearing. The object of ratchets is to check the
action of certain portions of a machine or train of mechanism
and so modify an otherwise continnous motion into some inter-
mittent form.
Ratchet gearing may be divided into two main divisions
according to the nature of the checking action. When the
movement of the checked member is impeded in only one
direction w7e have what may be called a Running Ratchet; and
when the movement is checked in both directions, a Stationary
Ratchet.
The distinction will be understood by reference to the accom-
panying illustrations, in which Fig. 653 shows a ratchet wheel
and pawl a b <r, the shape of teeth and pawl permitting motion
of the wheel in one direction, and hence forming a Running
Ratchet Gearing, while in Fig. 654 the rectangular notches and
pawl for a Stationary Ratchet Gearing. The lifting of the pawl
is called the release, and the falling into gear is called the en-
gagement of the ratchet gearing.
If the two members b and c are held, a becomes the intermit-
tent mover, while if a be held, the parts b and c possess the
intermittent action ; as for example, the sustaining pawl and
ratchet wheel of a common hoisting winch in the first case, and
the reverse lever and quadrant of a locomotive in the second
case.
Ratchet gearing is a portion of constructive mechanism which
will repay close investigation. For this purpose the following
six groups may be considered:
1.	Ratchets pure and simple, such as a ratchet wheel and
pawl for the mere prevention of rotation. Examples: the
ratchets of a wdndlass, or of the beam of a loom.
2.	Releasing Ratchets; those which act to release members
which are under stress. and which by such release are permitted
to perform and determi iate work. Examples: the pawls which
release the drop of a pile driver, the trigger of a gun, or the trip
valve gear of some steam engines.
3.	Checking Ratchets; those which arrest parts which are
already in continuous motion. Example; the safety check
ratchets upon elevators, and upon mine hoists.
4.	Continuous Ratchets; those in which a combination of
pawls acts to drive a member in a givfcn direction with practi-
cally a continuous motion. Examples: a ratchet-driven wind-
lass ; some forms of counters.
5.	Locking Ratchets; those which act to detain certain mem-
bers a fixed relation against the action of external forces
until released. Examples: some forms of car couplings and of
releasing shaft couplings, also the mechanism of locks.
6.	Escapements ; those forms which permit a member under
the action of an impelling force to make a regularly intermit-
tent motion in one direction. Example: the various forms of
clock and watch escapements.
By following this classification, the various principal funda-
mental forms may ^e briefly examined.
? 236.
Toothed Running Ratchet Gears.
In running ratchets, the direction of motion which is not
checked by the pawl is called the forward motion, and the re-
verse, the backward motion. The teeth on the ratchet wheel
must therefore be so shaped that when the pawl is in engage-
ment the backward motion only must be impeded. It is also
important that the form should be so chosen that the first ten-
dency toward a backward movement should act to produce an
engagement of the pawl with the teeth.
In determining the form of teeth, Fig. 655, we observe that
the most effective point upon the circumference of the wheel for
the action of the pawl is that at which the joining line 1.2 of
the centre of the wheel 1, with the point of the pawl 2, is at
right angles with the pawl radius 3.2. If we describe a circle
upon the diameter 1.3, or the distance between centres of wheel
and pawl, the intersections 2 and 2' with the pitch circle of the
ratchet wheel will give the two most advantageous points of
application. If the point 2 be selected, the attempted reverse
movement of the wheel will subject the pawl to compression,
while if 2' be chosen the pawl must be of the hook shape shown,
and will be subject to tension. If the teeth of the wheel are to
be of straight outline, the flanks should be radial. If a point ofTHE CONSTRUCTOR.
151
action 2X or 22, in front or behind 2 or 2/, be chosen, the
mechanism will be operative, but less advantageously than when
constructed as above, for the lever arm of the force-couple act-
ing upon the wheel will be less, and hence the pressure greater.
The angle of the flank, which will cause the direction of the
force upon the pawl to pass through the axis 3, is found by
erecting a perpendicular from 2X or 22 upon 2X. 3 or 22.3.
Fig. 656. .
%
It is not necessary to bevel the end of the pawl so that it shall
bear in but one point of the tooth, as it is not difficult to shape
the tooth profile so that the force P shall pass through the axis
3, when the pawl engages with the tooth. This is accomplished
by making the profile of the flank of the tooth a circular arc
struck from 3 as a centre, as in Fig. 656 a.
The same resuit will be attained by giving this curve to the
end of the pawl, and making the point of the tooth the bearing,
as at by or both pawl and tooth may be formed to the curve, as
we may call this form of tooth the “dead” ratchet tooth.
Other forms of teeth will be considered hereafter.
Internally-toothed ratchet wheels may also be made with the
pawls adapted to act either in tension or compression, as at 2
and 2', Fig. 657. The axis 3 may be within the wheel, Fig. 658,
in which case the above given conditions for the best position
of the point of action cannot be fulfilled.
If the radius of the ratchet wheel be made infinitely great we
have a ratchet rack, Fig. 659, in which a is a pawl acting in
compression, and b a form acting in tension.
An important application of the ratchet rack is shown in Fig.
660, which is the upper portion of the lifting frame for a scre\r
propeller.*
Fig. 660.
The two ratchet racks a, which support the frame as it is grad-
ually lifted, are in the middle plane of the ship, being fast to the
walls of the propeller well. In order to insure the engagement
of the pawls b b, they are held in gear by the loop springs of
rubber. The frame is raised and lowered by a rope tackle, the
sheaves of which are shown, the so-called “ c/ieese-coupling”
(see $ 156), permitting the propeller to be lifted, when its tongue
and groove are in the proper vertical position. The pawls are
held out of gear by means of lines, during the operation of
lowering. The frame and ratchet racks are both made of bronze.
The bent lever is another pawl which engages in a notch in a
blade of the propeller, and prevents it from revolving during
the operation of raising or lowering. There are two wooden
struts, the bronze shod ends of which can be seen on each side
just above the pawls bf their function being to hold the frame
firmly in its lowest position, w7hen the propeller is revolving.
Fig. 661.
Fig. 662.
Ratchet racks are also used extensively in connection with the
hoisting machinery in shafts of mines, etc.
at c, Since the force which acts upon the pawd has no tendency „ ,	.
to cause it to lift out of gear, when constructed as thus described sh*D^ne F'«' 3*3. »7, where one of the beanngs for the same propeller is152
THE CONSTRUCTOR.
Instead of giving the ratchet wheel an infinitely grea^t radius,,
the arm 2.3 of the pawl may be made infinitely long. This
simply means that the motion of the pawl is guided in a straight
line, in some form of slide. In Fig. 061 such an arrangement is
shown for a ratchet wheel, and in Fig. 662 for a ratchet rack,
such forms being not uncommon.
§ 237.
The Thrust upon the Pawe.
The condition that the thrust upon the pawl, in a ratchet gear-
ing, shall pass through the axis of the pawl, is not always ful-
filled, and in some cases it is impracticable to attain such a
relation of the parts. The mutual action of the pawl and ratchet
wheel upon each other must therefore always be considered.
If the flank of the tooth of a spur ratchet wheel (or a tangent
to the flank of the outline is curved) does not form a right angle
with the plane 2.3 of the pawl, there may exist, under some cir-
cumstances, a tendency to force the pawTl into the tooth, or in
other cases to throw it out of gear.
In Fig. 663 the various cases are examined. If at the point of
contact 2 a normal N Nx to the plane of the tooth flank be
drawn, this normal may bear one of three relations to the tri-
angle 1.2.3. The “ thrust-normal ” N Nx may fall without
the triangle, or within the triangle, or it may fall upon one of
the sides of the triangle.
If it falis upon 2 . 3, the thrust is neutral; if it falis upon 2.1,
the thrust is zero ; that is, there will be no action of the pawl
upon the wheel, or vice versa„ barring the action of friction.
The angle d between the line 2.30/ the pawl and the tangent
at 2, which is equal to the angle between the normal to 2.3,
and the “thrust-normal,” is called the angle of thrust. By
considering this in connection with the angle of friction 0 vari-
ous relations are obtained.
On the one part, the force applied will act to alter the posi-
tion of the pawl, either to or from the centre of the ratchet
wheel: on the other part, it will also act to move the ratchet
wheel forward and backward.
These relations are classified for various conditions in the fol-
lowing table, in which a force which acts to force the point 2
from 1 is called an “ outward ” action, and the reverse, an “ in-
ward” action.
For the so-called “dead ” ratchet tooth a =90°, case 1, hence
there is tendency neither to inward or outward movement.
The variations above given are, however, more or less used in
practice, and the table will be of Service in considering the
action in such cases. Some examples will be given here, and
numerous others may be found in subsequent illustrations.
In many cases it is desirable that the pawl should be held in
engagement with the tooth by the action of the impelling force,
as in Fig. 664, this falling under the fourth or sixth case. This
1
Fig. 664.
form of tooth insures the retention of the pawl in place after it
has once entered the tooth, and is sometimes used in hoisting
machinery wThen heavy loads are to be sustained; an applica-
tion is also found in Pouyer’s Coupling, Fig. 453, in which the
secure engagement of the pawls is an important point.
Another secure form of pawl is shown in Fig. 655.
I
Fig. 665.
In this case the wheel is made with pin-teeth. The pawl has
a forked end, the inner flank tending to produce an inward
movement, the outer flank, outward movement.
In this case, as in the preceding, the wheel must be turned
through a small angle before the pawl can be released.
ANGLE OF THRUST a = 90°.			
The Thrust Action is:	The Impelling Force:	Outward Movement:	Inward Movement:
1) neutral.	is without effect.	is without effect.	is without effect.
ANGLE OF THRUST 0 < 90° and > 90° — 0.			
2)	inward. 3)	outward.	is wdthout effect. is without effect.	produces reverse motion. produces forward motion.	produces forward motion. produces reverse motion.
ANGLE OF THRUST 0 <90° —<p and >0.			
4)	inward. 5)	outward.	produces inward movement. produces outward movement.	produces reverse motion. produced by impelling force.	produced by impelling force. produces reverse motion.
ANGLE OF THRUST <r<0.			
6)	inward 7)	outward.	produces inward movement. produces outward movement.	is without effect. produced by impelling force.	produced by impelling force. is without effect.
ANGLE OF THRUST 0 = 0.			
8) null.	produces inward movement.	produces friction only.	produces friction only.THE CONSTRUCTOR.
153
S238.
The Seiding Feanks.
Wehavediscussed theaction of the flanks of tooth and pawl
which work together during the thrust. It is obvious that
greater liberty is permitted in the form of the sliding flanks. It
is only necessary that the form shall be such that the forward
movement of the ratchet wheel shall lift the pawls properly out
of gear. .The forms fall under cases 4 to 7. The usual form is
the common zig-zag ratchet, but others are also used, as in
Figs. 666 and 667, in both of wThich the teeth are symmetrical.
FiG. 666.	Fig. 667.
If it is desired to have the end of the pawl symmetrical, as in
Fig. 667, this may be done, and the pawl may be reversed for a
reverse movement as shown in the dotted lines. This form is
used on the feed motion of some machine tools.
For some purposes it is desirable to form the thrust flank
upon which the impelling force acts, in the same manner as the
sliding flank, in which case the pawl must be held in gear by
some extraneous force capable of resisting the maximum im-
pelling force which it is desired shall act.
Such a form is shown in Fig. 668, which is similar to the nut-
locking device shown previously in Fig. 241.
? 239-
Spring Ratchets. Quadrants.
The form of ratchet last described possesses an especial prop-
erty, that is, the action of the spring tends to force the pawl
into the space as soon as the point is over the middle of the
tooth. This causes the pawl to spring into engagement, hence
the name spring ratchet, and this action causes an acceleration
of the motion either forwards or backwards as the pawl is forced
into the space. Applications of this form are found in repeating
watches, in which the wheel is star-shaped, and hence called
the star, while the pawl is called the star pin or springer.*
1
Fig. 669.
A modified form, Fig. 669, is used in Thomas’ Calculating
Machine. In this case the spring itself acts as the pawl, being
attached directly to the arm without joint, forming a piate link.
(See \ 180.)
Instead of using an entire ratchet wheel, a portion only need
be made, if the required movement is but small, and in some
cases reduced only to a single tooth, as in Fig. 670.
Fig. 670.	Fig. 671.
Sometimes the two members may oe made of similar form,
each working alternately upon the other, Fig, 671. Fxamples
of this are found in the valve gear of some Cornish engines.
These belong to the so-called “dead” ratchets, and are called,
more or less appropriately, quadrants, or sextants.
?. 240.
Methods of Securing Pawes. Sieent Ratchets.
The engagement of the pawl with the ratchet wheel is usually
secured by the weight of the pawl, sometimes assisted by addi-
tional weights, as in Fig. 659. This may also be accomplished
by means of a spring. It is desirable to give such springs but
little movement, and small frictional resistance. It should
therefore be placed near the axis 3, and is best placed in the
line 1 . 3, so that 3.4,5, shall line in the same straight line,
Fig. 672 a. If this cannot be conveniently done, it may at least
be made nearly so, as at b. A weak spring with much move-
ment may be seen below in Fig. 680, yet at the same time the
line 3 . 4 . 5, is only slightly varied from a straight line. In
spinning machinery spiral springs of steel are used, and rubber
springs have been used in propeller hoisting frames, Fig. 660.
In devices in which the pawls are sometimes above and some-
times below the wheel the springs are sometimes replaced by
using several pawls. This is shown in Fig. 673, which is Wil-
ber’s ratchet for use in lawn mowers.
Fig. 673.	Fig. 674.
Three pawls, with half journal, are here used, and as the axis
1, lies in a horizontal position some one of the pawls is always
in engagement by its weight. The movement of the teeth
under the pawl, and the dropping of the latter into the spaces
produces wear upon the parts, and to avoid this action various
devices have been made ; these being known as silent ratchets.
A very useful form of silent ratchet is shown in Fig. 674.
The pawl is made with a projectiou 5, which is connected to a
friction band d, which is carried upon a hub 4 on the ratchet
wheel. When the wheel begins to move forwards, the arm 4 . 5
lifts the pawl b out of gear. The lift of the pawl is limited by
the pins at 5. As the forward motion continues, the band slips
upon 4 ; if reverse movement begins the pawl is at once thrown
into gear. This is used in spinning mules, also in PouyeFs
coupling, Fig. 453, in which two pawls, each with its own de-
vice are used. The principle involved in this device is capable
of wide and useful application, as will be seen hereafter.
Another form of silent ratchet is shown in Uhlhorn’s coupling,
Fig. 454. In this case the pawls b, lie close agaiust the flanks
of the teeth. They are throwrn into gear again by auxiliary
ratchets, the spring pawls of which are not silent. These lift
the pawls b, through a small angle when the engagement is
♦ This is shown later among the releasing ratchets.154
THE CONSTRUCTOR.
completed by the self*closing action of the tooth flanks, Case
4 or 6, § 237.
Ratchet drills, etc., are often made with silent ratchets. Wil-
ber’s ratchet, Fig. 673, may be used for this purpose. If it is
placed so that the axis, 1, is vertical, the friction of the pawls
against the case will lead them into gear in the forward move-
ment and draw them out on the return movernent, the friction
in this.case taking the place of any operating gear for the
ratchets. Various other fornis of silent ratchets are in use.
2 241.
Speciar Forms of Ratchet Wheees.
In spur ratchet gears the axes 1 and 3 of the wheel and pawl
lie parallel to each other. These axes, however, may be placed
in the same manner as with gear wheels so that they are
inclined or intersect each other. A great variety of forms of
ratchet gearing may thus be made. The variations do not at
first appear as important as they really are, but this will appear
in the further discussiou.
A forin of ratchet for inclined axes is the crown ratchet, Fig.
675, which is used in capstans; the wheel, a, is stationary, and
the arm and pawl, b and c, revolve.
\
Fig. 675.
Fig. 676.
The forms shown in Fig. 676 and Fig. 677 are for non-inter-
secting axes, and use crown wheels also, and hence are called
crown ratchets.
Fig. 677.	Fig. 678.
By making the wheel, a, in the form of a plane wheel, and
substituting a bolt for the pawl, some useful modifications are
made. Fig. 678 shows a form of ratchet used on a wine press,
in which the bolt can readily be lifted out and placed in the
successive holes as the lever arm is moved backward and
forward.
The ordinary jaw clutch coupling, Fig. 441, is really only a
form of crown ratchet with bolt pawl. The portion on the shaft
A is the ratchet wheel, and the part fitted to slide on the shaft
B corresponds to the bolt b.
§ 242.
Muetipee Ratchets.
It is frequently desired to construet ratchet gearing so that
the minimum limit of movement shall be less than the piteh of
the teeth on the wheel. This is accomplished by using two or
more pawls acting at corresponding sub-divisions of the teeth.
Such multiple ratchets exhibit a wide variety of forms and find
tnany useful applications, and in many cases their true nature
is not fully understood.
Fig. 679 is a multiple ratchet of common form, with three
pawls, in which the pawls are set a distance from each other
Fig. 679.
equal to f of the piteh. From this arrangement the wheel can
be moved spaces equal to
'A> %> 1,	'2A> etc.,
of the piteh, that is, through ]/$ the piteh and any multipies of
the same. This is sometimes used in saw mill feed motion,
where a fine feed is required with a coarse piteh ratchet.
Fig. 680.
A double ratchet is used in Weston’s Ratchet Brace, Fig. 680.
The pawls bx and b2 are placed one above and one below the
arm c, and act on the two parts of the double ratchet wheel
1
av a2. Another ratchet drill, also by Weston, with four pawls
is shown in Fig. 681. This has an internal ratchet wheel withTHE CONSTRUCTOR.
155
five teeth. Double ratcliets are also found in Uhlhorn’s coup-
ling, Fig. 454, and Pouyer’s coupling, Fig. 453.
If it is desired, the pitch may be halved, or divided into any
two chosen portions, in which case the pawls may be made in
One piece, Figs. 682, 683.
In each of these there is one pushing and one pulling pawl
upon the axis 3, the pitch being halved and the pawls acting
alternately. One form shows a spur wheel, the other an in-
ternal wheel. The form of the double pawl has caused this to
be called an “ anchor” ratchet.
If the wheel is a so-called “ face ” gear, that is, wdth the teeth
projecting from the face of a disc, two similar pawls may be
used, both pushing or both pulling, and forming the same
anchor, Figs. 684, 685.
If the teeth are set alternately in two concentric rings, the
two pawls may be merged into one, as in Fig. 686. This latter
form appears to be new.
i 243.
Step Ratchets.
A very instructive form of multiple ratchet gearing is obtained
by combining more than two pawls into one piece, and arranging
two such pawls to work together, and this form is capable of
a 1' is formed in the arc of a circle from the center 3, a farther
lifting of b will cause, without resistance, a fresh release of a,
again arrested at /? 2//, and a similar action again for the flank
y 2'" ; the points 2, c, /?, y ali lying on a circle struck from the
centre 1. Thus a continuous lifting of b will produce three suc-
cessive advanccs of a. The angle of each advance of a may be
called the angle of advance, and the corresponding angle of lift
of b the angle of release. In this case the angles of advance
are ali made equal to each other, as are also the angles of re-
lease. When the position in which 2 is arrested by the flank
y 2/f/ is reached, the angle of thrust o becomes so small that
further travel cannot well be obtained. If it is required to pro-
vide for stili further movement it can be done by making addi-
tional teeth behind 2, as II, II', III//, etc., which will engage
successively with b at 2///. The construction of “dead” form
of teeth is clearly shown in the diagram. As before, the angles
of advance and release are made uniform. The mechanism as
constructed will give nine successive engagements. The
ratchet surfaces on b are struck from 2, and the sliding surfaces
on a from 1 ; the flanks on a with a radius 3-2/// = 3 y, the flanks
on b w7ith a radius 1.2.
It is to be noted that the twro parts a and b are interchange-
able in their functions, so that when the extreme notch II> of
a has been reached, a may be reversed in movement and b
follow step by step to its former position.
Such step-ratchets are seldom used in practice, but many use-
ful applications are possible.
In Fig. 688 is given a form of step ratchet arranged to give a
uniform angle of advance together with uniform drop of the
pawl. The pawl a is acted upon by the force indicated by the
arrow, and teeth are upon a cam-shaped disc.
An arc with radius 1.2 passes through 3, the angles of release
on b are 30°, and the successive angles of drop of a are 50. This
form of ratchet is used in the striking mechanism of repeating
watches, and is known as a “snail” movement. The arm a in
this case is frequently made ot the form shown in dotted lines
at A. The construction of the snail is interesting. In order to
fulfill the given conditions the points 2.2/, 2"---must lie on
an abridged pericycloid ; in the given case, where 1.2 = 1.3 it is
the form known as a homccentric pericycloid.* The points of
the re-entering angles lie on a similar curve. The circles rolling
together to describe these curves are shown in the figure T a
rolling about 1, and T b about 3 ; their radii are inversely as the
angles of drop and advance. If the parts b and a move con-
tinuously, these circles roll on each other; for the actual rnove-
ments which take place, the drops of the pawl occur as the suc-
cessive ringed points coincide.
* See Reuleaux’s Thooretical Kinematics, § 24.THE CONSTRUCTOR.
156
In the preceding step ratchet (Fig. 687) the angle of drop and
of release were given the ratio 1:2. In this case the points of
the teeth were on cycloids, those on a being on a pericycloid,
those on b on a hypocycloid. The contact point 01 the gener-
ating circle falis without the figure on 3.1 prolonged. Since the
radii of the circles are as 1 : 2 with internal contact the hypo-
cycloid becomes an ellipse. A portion of the curve is given in
the figure ; 3 X----, and 3 V are the senii diameters. The sim-
plest form for the line of the teeth will be obtained by making
1.2 = 1.3, since for this case the ellipse for one diameter of the
base circle on b becomes the straight line 3 X.
1
Fig. 689.
If it is desired to combine in the same piece two step pawls,
Fig. 689, of which one set shall be in tension and the other in
compression, an anchor ratchet may be used. In this case a
back and forth motion of the anchor permits an intermittent
forward motion of the wheel. The anchor has ten steps and the
wheel four teeth. This may be considered the general case of
which Figs. 682 to 686 were special examples.
Numerous interesting problems may be solved by such de-
vices, such as the conversion of continuous rotation of one
piece into intermittent rotation of the second. Applications are
found in clock and w atch-making.
The various modifications which may be made in the relative
positions of the axes 2.1 and 2.3 permit a very great variety of
dtep ratchets to be made.
i 244-
Stationary Ratchets.
A stationary ratchet may be considered as a combination of a
pair of running ratchets with the teeth facing in opposite direc-
tions. The schetne of such a combination is showm in Fig. 690.
From the four possible positions of the parts 2.2', II and II' we
may make the following double combinations :
2 with II,	2' wdth IF,
2 with II',	2/ writh II.
The first two combinations are practically identical with the
stationary ratchet, Fig. 691. The flanks of the two wheels give
a notch for the space, while the teeth assume a dove-tail shape,
and this form of stationary ratchet may be called a notched
ratchet. The wheel will be firmly held by the so-called “ dead ”
tooth, or when (90° — c) < <p, $ 237. Many forms of this kind
are used in practice.
Fig. 692.	Fig. 693.
Figs. 692 and 693 show two modifications of the notched
ratchet. The distinction betwreen tension and compression
pawrls disappears, since the pawTl
is the same for either action.
If the distance between the axes
1 and 3 is made infinitely great,
the pawd becomes a sliding
bolt. Such a form is shown in
Fig. 694, which is for non-inter-
secting axes. The wrheel is a
crown wdieel, and the pawl may
have more than one notch.*
An other form of notched
ratchet with axes 1 and 3 infi-
nitely separated is shown in
Fig. 694.
Fig. 695, and is in-
tended to hold a shaft
from longitudinal	j
motion, being used	FiG. 695.
in connection wTith
the disengaging gear of hoisting machinery, lathes and other
similar machines.
In this case the radius a is infinitely great; the wheel a be-
comes a shaft.
The combination 2 wdth IF and 2' with II of Fig. 690, if we
make 3 . 2 = III . II, gives a stationary ratchet of the form
shown in Fig. 696.
The pawd becomes a segment of a cylinder and wTorks always
in compression, or in the modification given in Fig. 697, always
in tension. This form may be called a cylinder ratchet. The
form of Fig. 696 has many applications, as, for example, the
Thomas’ Calculating Machine and similar wrork.
* This form of ratchet will be recognized as similar to the common jaw
coupling. The shaft A carries the crown wheel a, the bolt corresponds to
the other half of the coupling b. The shaft B carries the part b, the fatter
sliding upon a feather.THE CONSTRUCTOR.
iS7
The cylinder b may be entirely cut through as in Fig. 698, so
that the segment sball fall entirely within the surrounding
circle. When it is placed op-
posite the teeth the wheel may
be revolved in either direction
as far as desired. If this move-
ment is to be limited, as, for
example, to a given pitch, it
can be aCcomplished by cut-
ting a corresponding space in
the cylinder, such as is shown
in Fig. 699 a.
It is not necessary that the
spaces in the wheel a should
conform to the circular profile
of the cylinder b (see $ 237);
the thrust is at two points on
the right and left of 1 . 3, and
it may be formed as at b, or
pin teeth used as at c. This
last figure showTs the modifi-
cation made in the notch of
Fig. 698 to reduce the back-
lash of the wheel a. In Fig. 699 a the pitch circle of the pin
gear a passes through the axis 3, and the gap in the cylinder is
increased propor iionally. When the wheel is impelled in the
direction of the arrow, the pin 2 slips into the space in the
Fig. 698.
Fig. 699.
cylinder as soon as the opening is tumed towards it far enough,
but cannot pass out until the cylinder has turned back the same
distance in the opposite direction, thus forming an intermittent
pitch movement.
This idea is more fully carried out in Fig. 699 e. In this case
the inner profile of the space is concentric with the outside of
the cylinder, as was also the case with the form shown in Fig.
697. In this case the tension and compression pawls are practi-
cally combined in one. When the opening moves into the
proper position, the pin 2 moves to the point 2and completes
the remainder of the pitch movement when the cylinder moves
to the left again. This form may be made free from backlash
by making the outside of the cylinder fili the space between two
teeth, as in Fig. 700. If it is required that the intermittent
movement should divide the pitch into two equal parts, the arc
of the pitch circle of a, which is the measure of the thickness
of the teeth, must be equal to the arc cut off by the space in the
cylinder. If backlash is permissible, the thickness of tooth
may be reduced.*
If we compare the various forms of cylinder ratehets with the
notched ratehets, as, for example, in Fig. 692, it will be seen
how the one may be derived from the other. If the pawl of
Fig. 692 is given a row of teeth similar to the tooth 2, placed in
a circle about a centre 3, and a space cut in a of the circular
profile indicated, we obtain the same general and important
form as is shown in Fig. 698.
In a similar manner the notched ratehet can be derived from
the cylinder ratehet, and also inverted by transposing the parts,
* If the preceding forms are compared with Fig. 682, a similarity will be
noticed. The dead” ratehet with pawls of circular profile, of Fig. 682, are
here, in Fig. 699, replaced by a gap of small angle ; the compression pawl is
at 2, the tension pawl at 2', the arc 2 — 2' is made very small, and the rela-
tive diameters very different.
and all the modified forms obtained. The interchangeability
of the two parts gives the midway form shown in Fig. 701, in
which both pieces are the same, each being wheel and pawl for
the other.*
For the varied positions which may be given to the axes, a
wide variety of cylinder ratehets can be made, many of these
possessing useful applications. If the axes are at right angles,
the cylinder may become a disc, as in Fig. 702 ; this form being
used in Thomas’ Calculating Machine, in which case the wheel
a is made with but a single tooth.
The form shown in Fig. 703 is derived from the globoid gear-
ing of Class III, \ 224, the ratehet being a cylindrical notched
ring. Fig. 704 showTs howr a pitch ratehet can be made on this
principle.
An examination of the preceding forms of stationary ratehets,
in which the pawl consists of a revolving member with a gap
cut in it, will show one common property in all of them. This
is the fact that an intermittent motion produced by successive
release and engagement may be made either by a continuous
rotation of the cylinder or by an oscillating movement. If,
therefore, we have a continuously revolving shaft to deal with,
or a vibrating member, the desired release or intermittent ac-
tion of the part to be acted upon may in either case be ob-
tained. Both forms are found successfully applied in actual
practice.
2 245.
Ratchets of Precision.
If we imagine the running ratehet of Fig. 682 so modified
that upon the release of the pawl 2 that at 2' shall enter at a
point nearer the tooth than the middle of the pitch, as there
shown, the principle will not
be changed. If this modification
is made to such an extent that
the angle 6 in both cases be-
comes zero, i. e.> the pawls so
made that one enters into e 1-
gagement at the instant of re-
lease of the other, we have the
form shown in Fig. 705.
In this case the wheel a, be-
ing impelled in the direction-------
of the arrow, can pass the
points of both pawls at once.
The slightest movement of the
member b in either direction,
however, will bring either 2 or
7/ into engagement and hold
the wheel. This form is called
a Ratehet of Precision, the
especial one given being a
running ratehet.
Fig. 705.
The principle is capable of various applications, and is also
suitable for stationary ratehets, two forms of which are showrn
* This form is similar to the running ratehet of Fig. 671.IS»
THE CONSTRUCTOR.
in Figs. 706 and 707. In the latter case the pawl assumes the
form of a bolt, shown in the illustration with several notches.
Fig. 706.
Fig. 707.
The practical applications of ratchets of precision are numer-
ous, and examples will be given hereafter.
I 247-
Dimensions oe Parts of Ratchet Gearing.
The great variety of ratchet gears in use makes it almost im-
practicable to prepare any compact rules for the determination
of the dimensions of the various parts. The general proportions
can be obtained for the various forms by comparison with simi-
lar preceding devices. For spur ratchet wheels similar propor-
tions may be used as for spur gears with thumb-shaped teeth.
$212. The action of the pawl tends to produce shocks and this
must not be overlooked in determining the thickness of the
teeth. It is generally most convenient to give the pawl a curved
profile, in which casd the discussion of combinedresistance, $ 18,
is to be considered. Pawls which are subject to frequent vibra-
tion are best made of Steel, as are also those in which the super-
ficial pressure is high.
?248.
Running Friction Ratchets.
I 246.
Generae Form of Toothed Ratchets.
We have already seeu that several forms of ratchet mechanism
which have been described possess numerous points of similar-
ity, and may be reversed and derived from each other, and
hence it is not unreasonable to expect that some general foim
may exist from which the various special modifications can be
derived, and in which the distinction between ratchet wheel
and pawl, or checked and checking member, shall not exist,
but each shall appear in both.
Fig. 708.
This general form is actually found in the combination of two
d^sc face wheels ($ 211), with their centres carried 011 the same
bar, Fig, 708, in such a manner that the teeth of both shall
engage and be engaged by the other. In the illustration is
shown such an arrangement made for a stationary notched
ratchet. The wheel b engages as a pawl with the wheel a at 2
and 2', and if it revolves a space of one-half a pitch, a is re-
leased. If a, however, revolves any given odd number of half-
pitch angles only, b will be checked, and a become the pawl.
In both cases we have a ratchet of precision of the same type
as in Fig. 706.
The pitch ratchet with anchor pawl may also be thus derived ;
it is true the anchor form cannot so readily be shown as a pair
of similar wheels, but it is clearly only another form of the
same problem. The zig-zag ratchet, notched ratchet, step
ratchet, or their combinations are all reducible to this general
form, the only condicion being that the direction of the force in
the position of engagement of the checking member shall be
such that the checked member cannot revolve. The intermedi-
ate forms show the “pawl lifting” action, $ 237. Itis evident
that in some cases the checked member may have a forward
movement, and in others a reverse movement. Since here, as
in $ 235, we may consider the link c as a checked memberwhcn
the wheel is held fast, we may, from the combination of these
parts, obtain four kinds of ratchets, viz. :
1.	c, stationary ; a, checked ; b, checking.
2.	c,	“	b,	“	a,
3.	a,	“	c,	“	b,
4.	b,	“	c,	“	a,
As a general statement of the fundamental principle we have :
A toothed ratchet consists of a combination of a pair of gear
wheels, or of portions of gear ivheels, in which the teeth are so
made that for certain positions of the wheels the resultant of the
pressures on the teeth of one of the ivheels either passes through
iis axis, or differs from such direction by less than the angle of
friction.
The mechanical devices which are constructed to modify the
relations between two moving bodies by means of friction, may
be called by the general term of friction clutches.* Such a de-
vice, when so arranged that one member opposes a positive
frictional resistance or check to the motion of the other in one
direction under the action of an impelling force, constitutes a
friction ratchet. Such de-
vices may be divided, as
before, into running and
stationary ratchets, $ 235,
and the first form will now
be considered.
In Fig. 709 is shown a
friction ratchet for parallel
axes. In this case the fric-
tion block b is carried by
the friction with the wheel \
a, when the latter begins \
to revolve in the direction \
of the arrow, that the pawl \
link c is crowded against \
the axis 4. Theradial com- \
ponent Qy in the direction	\
4.3, exerts a pressure upon	\
the brake block b. We
also have the tangential
component Sy which we
may consider as composed
of two forces and S2, act-
ing in the same direction,
which hold the friction at
1 and 2 in equilibrium. At 3 we have two opposite forces S3
and .S* which are capable of resisting the friction at 3 and 4 res-
pectively.
The moment M} of the four friction forces is : A/=	+ S2
— ^3 — >SV) (a + b). If we give the angles the symbols shown in
the illustration, and make 1 . 2 =a, 2 . 3== bf 3.4 — c, 4 . 1 -=dy
and call the radii of the several journals av bly and clf we have :
Fig. 709.
Qf±1
a + b
Qf ,

Qf*
a -f- b

s,--
R f c\
cos a	, 0
-----------, and 03 =
c cos v
= — (d a + d y)
d y
But we also have (a + b) sin a =
From this we get
= c sm y.
S,=
c cos a
Qf

(a + b) cos a
This gives for M:
cos g c cos o
(a	b) cos a
M — Qf (	—tfL-L— -ffr----+ —^ (a + b)
\ a + b	cos2 c c {a 4- b) cos a c J x
The force P which acts at 2, to revolve the wheel in the direc-
tion of the arrow, may be considered as a couple. We then
have for M— P a :
P_a_
a + b
-«/[■
a+ ax
i- (-
S2 G \C
bxd
a + b cos2 c \c (a 4- b) cos a
r) D
* This term only partially expresses the general scope of the German word
“ Bremswerke,” ior which there is no exact equivalent in English.—Trans.THE CONSTRUCTOR.
159
Pa
But Q is a function of P. and in fact we have ——- =<9tan <7.*
^	’	a-\- b
This gives:
sin cr cos <7 — /sin <7
a ~t~ a\
a + b
CL -f- CL\
a -f- b
(
______bj_d_________
c (a -f- b) cos a
and since the angles a and a are small, and become smaller
under the action of the pressure, a sufficiently close approxima-
tion will be obtained by putting :
The following conditions rnust be noted. If an independent
force outside of Q exerts a normal pressure N upon the circum-
ference of the wheel, the friction JV/will diminish the force
acting to turn the wheel backward. If this is to enter into the
resistance which is produced by Q, the magnitude of a as giveu
by equation (233) must be modified. If N becomes sufficiently
great, Q ruay become zero; in such a case we obtain a stationary
instead of a running ratchet.
The pressure R on the pawl may become very great. We
have R =	which may be made approximately:
cos a
R =
_Pa_______
(a + b) sin a
(234)
Example.—L,et a = 14.2", ax = 1.6", b = 2", b\ = 0.6", c = 11.8", C\ = 0.6", and
/“at ali four points = o.io,f we have d=a-\-b-{-c = 28" approximately, and
__	/15 8	0.6	28	0.6 \
sin 5 < 0.10 ( —----—5—.-------5 — —o )
—	\ 16.2	11.8 -f 15.8	11 8/
whence sin 0.0834, which gives <r = 40 47". To be 011 the safe side we will
make <r = 5U0, or sin <r = 0.0787, and then get R = P ——	— = 11.17 P.
10.2 -4- 0.0707
The exact length of d will be very slightly less than a + b -j- c.
As will be seen the ratio comes out unfavorably. The method
of remedying this will be discussed hereafter.
The pawl c may also extend within the circle of the wheel, as
in Fig. 710, in which a is an obtuse angle. The axis of the pawl
may be either at 4 or 4', on 3.4 prolonged; the pawl is in this
case a tension pawl. If a is made an internal wheel, we have
the arrangement shown in Fig. 711, the pawl being under com-
pression.
Especially noteworthy are those cases in which one or more
of the axes are infinitely distant. In Fig. 712 is shown the case
in which the length of pawl and also d and cx are of infinite
length. We have for the angle of thrust, from (233)
* The moment of the frictions produced at 2 and 1 by the force P is = P
{a + ax) — Pax = Pa.
t If various coefficients of friction are to be used we have for S\, S2, -S3 and
S4, corresponding values /\, /1, and /4.
° =f Q
{a -j- #i)
K
(a +' b)	(a + #i)
-]
When ax is very small, release is difficult, and the arrange-
ment does not appear to be very practical. If the arm a is made
infinitely long, so must also av and we get the case of Fig. 713.
The wheel becomes a sliding bar. The relations
.</[-* (A) -(4)]
give excellent action.
If with a and aY we make c infinitely long, we obtain the
construction of Fig. 714.
Fig. 716.
The conditions give : sin a fC /(2 — 1). The joint at 3 insures
full bearing for the surfaces at 1, 2 and 4. This is also the case
with the joint at 4 in Fig. 715, in which a, a1} blf c and d are in-
finitely great, while b is the difference between two infinitely
long but opposite distances, and hence is finite. We have the
relation <7 <C/ (2 — 1). By omitting the joint we obtain the
simple construction shown in Fig. 716. The friction block is
in the forni of a key or wedge, as in Fig. 715, and the number
of parts reduced to three (see also the following section).
The results as determined by calculatiou are not always prac-
ticable for the desired ratchet construction, which shows that
the selection of the relation between the parts must be made
with judgment, and care taken that those pieces which are sub-
jected to heavy pressure shall not readily be deformed. As the
preceding example indicates, the small size of the angle c ren-
ders it an important point for consideration. In this case the
actual length of d is only about greater than a + b -f- c. The
pressures Q and R act to lengthen d and shorten a, b and c, and
if P= 100 pounds the difference may readily = N' 1 so that
with only ■ additional wear, c becomes zero, the parts b, c
pass the centre, and the ratchet action ceases. This will indi-
cate the conditions under which the ratchet becomes a use fui
device.
The numerical magnitude of the parts alt c and d can be
chosen so as to render the unavoidable wear least hurtful. This
may be done with the sliding ratchet, Fig. 713, by making the
length c sufficiently great. It is also important to devise means
to prevent the block and pawl from being forced past the
centre. A method of accomplishing this is to substitute for the
pin joint between the block and pawl, a curve or cam bearing,
as in Fig. 717.
If the block is given a circular profile from the centre 1, and
the pawl an evolute outline developed from a circle about 4,
with radius d sin o, we shall have c nearly constant, notwith-
standing the elastic yielding of the parts and the unavoidable
wear. If the yielding between 1 . 4 is great, the radius of the
circle on which the evolute is generated may be made somewhat
greater than d sin a. This construction appears to be new. A
number of similar uiodifications may be made, for which see
the following section.i6o
THE CONSTRUCTOR.
It is desirable to examine the value of the coefficient of fric-
tiony, in order to increase it at the point 2. This cannot well
be done by choice of material, siuce wood can scarcely be used
in many cases, and lubrication of the rubbing surfaces is essen-
tial. The application of wedge profiles to
wheel and friction block enables greater fric-
tion to be obtained, Fig. 718, as in the case of
wedge friction wheels, § 196. Instead of the
coefficient /. we have the value	If
sin yz 0
the wedge angle 0 = 6o° this gives 2/; for 0
= 30°, nearly 4 f. By combining this pr-in-
ciple with the preceding forms, some very
useful devices may be made.
It is desirable to arrange the application of
the force R so as to exert as small a distorting
action upon the parts as possible. This may
sometimes be done by arranging two or more
friction pawls of similar kind to act upon one
wheel. Some examples of such devices will
be found in the following section. It must
not be forgotten that the conditions for <1 are
not changed by the repetition of parts, since
the numerical value of /Moes not enter into
its determination.
There is yet another form of friction ratchet
which is capable of being made very useful. By an examina-
tion of formula (223) it will be seen that the influence of the di-
mension ax is almost as great as that of a itself. If we increase
a1 to nearly the same magnitude as a, Fig. 719, we may approach
j
ii/
ii:
ii/
f
Fig. 718.
closely the minimum value of <x. This carries with it the dis-
advantage that the frictional resistance to the backward and
forward movement at 1, is greatly increased, but this effect may
be avoided by making a special bearing for the friction block
axis and rearranging the
parts somewhat as shown
m Fig. 720. The attempt
of a to move backward
causes the pieces b and d
to press upon the rim of
a from without and with-
in and grasp it firmly.
The angle c may now be
made twice as great as in
the previous forms with-
out danger, ali other
things remaining the
same. A practical form
of this device is shown in
Fig. 721, as applied to saw
mill feed motion. Here
the screw motion F G is
intended to permit of a
suitable degree of play
Fig. 721.	for the lever c.* If we
make ax > a} we have the
form shown in Fig. 722, which seems quite practical, and when
applied to a friction rack we obtain the form in Fig. 724. We
shall return to the consideration of these double friction ratchets
hereafter.
It must be remembered that these forms of friction ratchets
are also applicable to other positions of axes and some resulting
devices are in practical use.
&
Running Friction Ratchets.
If the force to be transmitted is not very great, the intermed-
iate friction block may be dispensed with and the curved con-
tact surface be made directly upon the pawl. This reduces the
mechanism to three parts: the wheel a, pawl b, and arm or con-
necting bar c, Fig. 724.
This form may be called a clarnp ratchet, or since the pawl
resembles the thumb-shaped teeth already described, the term
“ thumb-ratchet ” may be used. The determination of the angle
c may readily be determined by what has preceded, and the fol-
lowing relations established :
~ t) •;••••• *
A suitable profile for the thumb pawl may be obtained as in
Fig. 717, by using the evolute upon a base circle of radius c sin
a, rbout 3 as a centre. This may be approximated by a circular
arc struck from M> in which 3 M and 1 M are at right angles to
each other.
If a and c are made infinitely great we have a form similar to
Fig. 713, the straight profile 2 being an evolute of infinitely
long radius, and the profile 3 a portion of the circumference of
an infinitely great cylinder.
If the wheel be made a wedge friction wheel we have the
form shown in Fig. 725. The wheel may be made with several
grooves, by which means the pressure on each surface can be
reduced (see $ 196).
Fig. 725.
Fig. 726.
A variety of modifications can be made in the arrangement of
the pawls. A clatnp ratchet in which a repetition of pawls is
used to distribute the pressure, is Dobo’s ratchet, Fig. 726, which
is very effectively used by A. Clair in his indicator.f
If we adapt the idea of Fig. 717 to a revolving journal using
the “thumb ” pawl at 3, we obtain a very useful modification of
the clarnp ratchet. The curve which is applied between 2 and
3 may be variously arranged. A very simple form is obtained
* See Goodeve, Elements of Mechanism, Tondon, 1860, p. 49.
f See Morin, Notions geometriques sur les mouvements, Paris, 1861, p. 200-THE CONSTRUCTOR.
161
by making the curves at 2 and 3 portions of the same circle, and
the corresponding curve at 3 so found as to produce the required
clamping action.
The clamping piece
b becomes a cylin-
der, Fig. 727.
If we make the
angle O 2	3 = d,
prolong the radius
3 O to N, then will
3 O ^Vbe the normal
to the curve at the
point of contact
with b at 3, since
the angle 3 . 3 O =
d. The curve for c
is an arc of a circle
struck from a centre
My on 3 O N, found
by making 1 Mper-
pendicular to 3 O N. This curve is practically correct for a
smaller clamping cylinder as at O' 3/, since the angle of thrust
is very nearly the same as at O 1 . 2, or in other words the ef-
fectiveness of the clamping action is not impaired as the cylin-
der is reduced by wear.
Q	T
The pressures at 2 and 3, Fig. 728, are T —-”
Fig. 727.
R=-
= —, whence O
cos2 o
p Cc«+ai)/^)’
Fig. 728.
Fig. 729.
A practical application of the preceding form is shown in the
checking device for sewing machines, Fig. 729. In this case a
ball of rubber is substituted for the cylinder. Another similar
device is the ratchet check used on the old Eangen Gas Engine,
Fig- 730. In this case a number of roller checks are used in
order to distribute around the wheel a.
The whole forms a sort of continuous ratchet gearing in which
the backward and forward movement of c imparts a continuous
Fig. 73°-
Fig. 731.
forward‘movement to the wheel a. When c moves in the direc-
tion of the arrow II, a is clamped and driven, while the parts
are released when the motion is reversed in the direction I. The
action of the centrifugal force tends to keep the checking cylin-
ders in contact with the outer ring, and so insure prompt action
upon the reversal of motion. The piessure upon these roller
checks in the Langen Gas Engine was very great; wrought iron
rollers wore out rapidly and phosphor bronze was substituted,
although even these gradually altered their form under the
pressure.
Another ratchet check used by Langen for the same purpose,
is shown in Fig. 731. Here &gain we ha ve a repetition of the
parts, and also a return to the friction block, the rollers occu-
pying the place of pawls. Comparing this with Fig. 709, the
curved bearing surfaces correspond to the journals 3 and 4, and
the action is similar to Fig. 727. The block b is arranged so
that full clamping is obtained in a quarter turn. Friction ratch-
ets with double
clamps are also
used as in Fig. 721
and the same
principle appears
in Fig. 732, which
shows Saladin’s
‘‘friction pawl.”*
A similar de-
vice is shown in
Fig- 733, as ap-
plied to a rod
movement, and
upon inspection
the resemblance
to the action of
the “thumb”
pawl will be seen.f
As long ago as
Fig. 732.
1798 Hornblower applied this idea to a rotary engine as shown
in Fig. 7344
Fig. 733.	Fig. 734.
§ 250.
The Reeease oe Friction Pawes.
The release of a friction pawl under pressure requires a cer-
tain degree of force, since there is always a friction between the
rubbing surfaces which is at least equal to P\ which must be
overcome if the pawl is to be released under pressure. The
release is to be effected under
quite different conditions
from those which obtain with
toothed ratchets in which,
for example, with “dead”
engagement, only the “yth ”
part of P is exerted at the
pawl point. The force re-
quired for release may be
somewhat reduced by combi-
ning the action of two sets of
friction surfaces of opposite
direction of engagement,
Figs. 735 and 736. The mo-
tion in the direction of the
arrow tends to draw the pawl
2 into closer engagement,
Fig. 736.
Fig. 735-
and at the same time to release that at 2'. By altering the
relations of the distances 4-3 and 4~3/» etc., any proportion of
the moment may be used to hold the parts in gear. These
forms appear to be new, and may be called “ throttle ratchets.”
2 25*.
Stationary Friction Ratchets.
A stationary friction ratchet may be defined as one in which’
the clamping action is not dependent upon the direction of
* See Bulletin von Mulhausen,XII. 183S, p. 296, also Salzenburg’s Maschine-
details.
f A similar arrangement will be found to exist in the ring spinning frame.
Here the pawl b, Fig. 732, is made of wire and held at 4 by the thread pass-
ing through an eye. Since the angle a is made greater than we have taken
it above, a tension or brake is necessary.
| A similar device is used by Carter, both being found in Farey’s Steam
Engine, Pl. XV, Figs. 8 and 9, also in Severin’s Abhandlung, p. 141.
§It may be noted that friction pawl actions are found in nature. Some
fishes have such connections to certain bones or spines which thev can
thus elevate or depress. See O. Thiele, Die Sperrgelenke einiger Welse*
Dorpat, 1879.162
THE CONSTRUCTOR.
rotation of the wheel. Such a ratchet is shown in Fig. 737.
1 and 4 are parallel axes, the block acts with a radial pressure
Q, such that for the
Q*--------0
circumferential force
± i^the following con-
ditions may exist:
Qf(a -f a^> Pa, or :
e=7Gff5;)(’j6:i
Fig. 737.
ping
This
If Q is less than
the right haud expres-
sion, /* will only be par-
tially opposed, there
will be motion from a
toward d> with slip-
at 2, or in other words, we have a brake, see § 248.
fhis construction is frequently applied, although it requires a
relatively large force at Q', acting through the lever c c', giving
increased pressure on the axle and much wear on the block.
Various forms cf lever connection are used to modify the ratio
Qf : Q. By clearing the angle
which the axes 1 and 4 make
with each other, various con-
venient modifications may be
made. The general scheme
of such constructions is indi-
cated in Fig. 738, in which the
toggle connection gives a high
ratio of Q' to Q; the block
being guided in slides. By
making a au internal wheel,
a very practical arraugement
is obtained as shown in Fos-
sey’s coupling. Fig. 450.
Koechlin’s coupling, Fig. 449,
is also another form of fric-
tion ra tchet gearing, the pres-
sure in this case being applied
by the medium of a right and
left hand screw. The same is
true of other forms of friction
coupling, and the various ine-
thods of applying the pressure
and reducing the wear, given
in § 248, may also be applied
in the design of mechanism
for the purpose.
? 252.
REEEASING Ratchets.
Following the classification given in § 235, we have first dis-
cussed the various forms of ratchets for the general meaning of
the term, and the five special classes reinain to be considered,
the next being the so-called Releasing Ratchets. Such ratchets
must be considered primarily with regard to the question of re-
lease. When the release is to be effected by hand, various
forms of handles or other cotinections to the pawls are readily
<levised. In most cases, however, the release is automatically
effected, in which event, some mechanical tripping device is
required.
The resisting force in such gearing is practically the same as
the force required for release. It is applied usually by weights,
isprings, steam or air pressure, etc., and is variously intended to
tcause the released member to act with a predetermined velocity,
either slow or rapid, as may be required. Many millions of releas-
ing ratchetshave been made for gun locks, and the various forms
of releasing valve gears for steam engines, introduced by Corliss,
but first invented by Sickles,* are of this class. In designing
releasing valve gears, it is important that the valves should be
closed quickly yet without suddeu shock, and hence some form
of buffer is essential. It is in the various devices for applying the
force, for releasing, and for cushioning the released force that
the many gears differ from each other. The original form of
Corliss valve gear, and the modified form of Spencer & Inglis,
are but little used on the continent, but these are well known,
and hence examples will be given of some of the numerous
modified trip valve gears which have been put into practical use.
Example 1.—Valve gear by Cail & Co., Paris, Fig. 730. a is the driven
piece, a sector with one tooth, fast to the valve stem ; b is the pawl; c the
arm, loose on the hub of a; V is the pawl spring; d the releasing cam. The
* F. E. Sickles, of Providence, R. I., took out his^first patent for a “trip
cut-off” valve gear in 1842.
trigger is that part of the pawl upon which joumal 5 of the releasing cam
is carried. The tooth pronle at 2 should be “ dead,” but this is not the case,
as the curve is struck from the centre 3. The bearing points 011 pawl and
sector are of st£el, separately inserted. The force which closes the valve is
Fig. 739.
exerted by a spiral spring acting on the rod/, and the valve is opened by the
rod connecting the arm Ci with the engine motion. The cushion is effected
by an air dash pot, also acting through the rod ft and the instant of release
is determined by the governor.
Example 2.—Valve Gear by Wannich, of Briinn. In this case there are two
flat slide valves to be operated by the reciprocating movement of the piece c.
It will be seen that this is a form of ratchet rack gearing. The valves areTHE CONSTRUCTOR.
163
closed by steam pressure acting upon small auxiliary steam cylinderson the
rods a, the cushion being provided by air buffers as in the preceding exam-
ple. There are double pawls b, b, with “dead’’ tooth profiles, faced with
Steel; c is the pawl carrier, moved back and forth by the rod d; b' b' are the
triggers, and d d the releasing stops, the latter shown in three successive
positions; e is a guide rod. The rod d receives motion from an eccentric on
the engine shaft.
Example 3.—Valve Gear by Powel, of Rouen, Fig. 741.* This is a form of
rod ratchet gearing with bolt pawl. Here b is the driven piece in which the
bolt b and its spring are carried. The rod a is moved up and down by an
eccentric. The piece c is guided at c2. The trigger d acts sooner or later, as
the governor changes the position of the trip e. The force to close the valve
is steam pressure acting on the upper part of the rod Ci, which also carries
an air buflfer.
The use of releasing ratchets in valve gear of steam engines
is very old, being found in the old Newcomen pumping engines,
and in the Comish engine a similar gear is used to-day, while
in recent times trip valve gearing of various designs have come
into extended use, and some of the fornis are shown in Figs.
670 and 671, not only for closing the valves, but also for open-
ing them. These latter valve gears are intended to be operated
by direct connection with the piston movement, while those in
the preceding examples are operated from revolving crank
shafts.
Releasing gears which are to be operated by reciprocating
members, are sometimes constructed on quite a different prin-
ciple, viz. : that of a weighted lever in nearly unstable equi-
librium, so that it can be caused to fall to the right or left by
means of a slight thrust, and so operate a releasing member. A
form which was formerly much used, in which the lever is
carried on a horizontal axis, is shown in Fig. 742.
When the weight G is in the vertical position 1 . 2, the pres-
sure acts directly downward upon the axis, the journal friction
acting as a ratchet. The form is sometimes used on planing
machines, screw-cutting machines, etc.
Another form is shown in Fig. 743. Here the pressure is due
to a spring, acting through a link 3-2 upon 2.1.
A third form is that used in Shanks’ planing machine, Fig. 744.
In this case the lever, with
its axis a, is at right angles
to £, and the latter is pro-
vided with a roller. The
limit of measurement of a
is between 2/ and 2".
The forms of tumbling
ratchets described in § 239,
may be adapted as releas-
ing gears, but it must not be
forgotten that in such me-
chanisms provision must
be made for the middle
position of the ratchet.
A fourth form of tumbling gear, of which, indeed, there are
many varieties, is the so-called “loop” of Hofmann’s valve
gear, Fig. 745. The loop a is made in the arc of a circle from a
centre at 2, b is a heavy roller, whh additional wreight suspended
at d'. When the loop or curved link is in either of the positions,
30 or 3/, the weight acts to continue the motion in the direction
in which it started until the limit of travel is reached.
A swinging arm b may be substituted for the slot and roller,
Fig. 746, and it will be seen that during the movement from the
po*kion 20 30 to 2/ 3' the tumbling action will take place and the
arm a be carried over. The similarity to the previous tumbling
gear will be apparent. If 2.3 be made infinitely long, the loop will
become straight and the twro forms will coincide. Hofmann has
made the analogy to a ratchet train more complete by placing a
ratchet so as to engage with the point 3 in the positions 30 and 3',
the release being made at the proper time by means of a cataract.*
In some cases it is desirable to make a gearing which shall be
released by the action of a very small force. For this purpose
a second releasing gear may be introduced, itself being readily
released, and by its action permitting a blow to fall upon the
trigger of the main gear. Such a device forms a releasing gear
of the second order. Such an example is shown in the hair-
trigger of a rifle.f Releasing ratchet gearings of higher orders
are also found in textile machinery, as in the Jacquard loom,
also in the striking gear of tower clocks and of repeating watches.
Another example is found in the relay of the Morse telegraph, be-
sides many other applications which will be considered hereafter.
3 2S3-
Checking Ratchets.
Checking ratchets are used in a great variety of machines, but
their principal applications are found
in machinery for hoisting and lower-
ing heavy loads, as in mine lifts, ele-
vators, and the like, to guard against
accidents in case of the breakage of
the 2 ropes. In the opinion of the
writer these devices have not been as
yet regarded as they should be,
merely special caSes of ratchet con-
struction, and as such capable of util-
izing all the various principies here-
tofore considered. When examined
in this light their study will be greatly
facilitated.
As a scheme of a general system for
checking ratchets a rod friction ratchet
may serve, Fig. 747, in which the rod
a is held stationary, the loaded mem-
ber d carries the ratchet, and the pawd
c and friction block b are held out of
* Further details of this and the preceding gear will be found in the Aus-
trian Report on the Exposition of 1878. Section on Steam Engines by A.
Riedler, Vienna, 1879.
* See Zeitschrift des Vereius deutscher Ingenienre, 1860, Vol. IV., p. 209.
f Such hair-triggers were ingeniously applied in former times upon cross-
bows.164
THE CONSTRUCTOR.
engagement by the releasing lever /, and rod e as long as the
hoisting connections g and h, are under stress. If thetension is
released, the ratchet is thrown into gear and the parts clamped.
If a toothed ratchet is used instead of a friction device, the
block b is omitted. According to the manner in which the
various constructive details from a to h are arranged, we obtain
the various Systems of checking ratchets which have found
practical application.
A collection of such devices was exhibited by the Industrial
Association (Verein fur Gewerbfleiss) in 1879* More than 80
designs were shown, of which only a few can be described.
Mauy of the devices were rather designs for improved construc-
ti >u as regards strength and rigidity, rather than examples of
nn :hanical ingenuity.
} 1 most cases the clamping action takes place upon the up-
rignt timbers of the shaft; sometimes guide ropes are used.
The greater number of designs shown used friction clamps,
those of the type of Fig. 724 being shown, the thumb pawl
being roughened, however, or finely toothed. The one which
showed the most evidence of careful constructive design in
accordance with the principies previously laid down in % 248,
was that of Hoppe, shown, as attached to each side of the
hoisting car, in Fig. 748.
The form of friction pawl used is similar to that showrn in
Fig. 713, there being four pawls on each side of the car, or eight
in ali. The clamping action takes place upon the guide bars a,
made of T iron, as shown. At 1 are the guide rods between b
and a; at 2, the double clamp blocks of hardened Steel, which
are connected at 5 to the coupling rods e> e. The actuating
spring^* is a torsion spring (see Fig. VII, p. 19 ; also Fig. VIII,
p. 19), secured to the roof of the car at^, g, and operated by
the releasing gear f at 8, and transmitting action from 6 to 5 by
the rods e> e, the conuection being made by the links 9 to the
double chain in such a manner that the arm f cannot be drawn
too far out of position. The proper adjustment of the pawl
arms is obtain ed by the keys on the rods c, c. Hoppe has taken
into consideration the fact that the angle <j, see (233), must not
pass beyond certain limits, or too great pressure would be ex-
erted on the frame d, d, and hence has provided stops in the
frame for the tiavel of the pawls r, c. The parts are so propor-
tioned that a load of double that ever placed upon the car would
be supported by the friction clamps before there would be an
appreciable elastic yielding of the frame. The adjustments of
the rods c, c provide for the change of relations due to wear.
This apparatus does not bring the lowrering car to a sudden
standstill in case of breakage of the hoisting gear, but the shock
is avoided by the gradual action of the friction brakes.
By using the author’s device, shown in Fig. 717, at 3, the
value of o might be maintained constant, or by proper construe-
tion of the guides the wedge friction pawls, similar to Fig. 718,
may be used ; the blocks acting on both sides of the guide.
This would reduce the stress upon the frame very materially.
The System of brakes used upon railway trains are really
* Berliner Verhandlungen, 1879, p. 345. Prize essay by Dr. F. Nitzsch, on
Safety Checking Devices for Mining Apparatus. See also Mairs, Berg und
Huttenmann, Z., 1879, p. 361.
forms of friction checking ratchets. The shocks due to sudden
stoppage are also to be avoided, and if the wheels are braked
too firmly the sliding action is simply transferred to the rails.
? 254-
CONTINUOUS RUNNING RATCHETS.
Continuous ratchets (? 235, No. 4) consist of such combina-
tions of pawl mechanism as act to drive a member in a given
direction with practically a continuous determinate motion.
This may be effected by combining two single running ratchets
in such a manner that they both act upon the same wheel, one
pawl attached to the arm c, wiiich is stationary, the other swing-
ing about the axis 1, Fig. 749, this being a very common form.
a.	b.
Fig. 749-
In this case 3 . 2 is the checking pawl, and y . 2' the driving
pawl. A movement of the driving pawl, if a little more than
one tooth space, moves the wheel one tooth ; a little more
movement than twTo spaces moves it two teeth ; and a regular
back and forth motion gives a forward movement at intervals of
a single piteh space.
If this device is made with step ratchets, as in \ 243, the piteh
may be subdivided into 2, 3, 4 or more parts, and for some pur-
poses, such as saw-mill feed motions, this is very desirable.
If the arm which carries the feed pawl swings about an axis
4, removed from 1, Fig. 749 b, there will be a movement be-
tween the pawl and the point of application 2 on the wheel;
while in the arrangement shown at a the motion of the two is
identical, and hence no wear occurs.
The two pawrls may be connected so that both of them be-
come drivers. If they are arranged so that their movement is
alternate, as in Fig. 750 ay the wdieel will be moved forward for
the movement of the lever in each direction, giving a double-
acting ratchet motion, the so-called Lagarousse Ratchet.* This
may be also accomplished in various ways, as in Fig. 750 b.
For any movement of the arm which is less than 1 and more
than '/2 the piteh, the wheel will be moved 1 piteh for each
vibration, and hence for a half vibration a feed of a half tooth
may be obtained. Step pawls may also be used with these de-
vices to obtain further subdivisions.
If, in Fig. 750 a, we hold the lever cx c2 rigidly, and instead
permit the arm d to vibrate with the same angle about the axis
4, the wheel moving with it, we obtain the same relative feed
motion.f This has been used by Thomson in a telegraph
apparatus.
* Named from the inventor, M. de la Garousse, and used in 1737. Belidor,
Arch. Hydraulique.
f This is the ordinary kineraatic inversion.THE CONSTRUCTOR.
165
A continuous ratchet gearing may be so arranged that back-
ward movement of the wheel is utilized to compel a uniform
division of motion.
This is the case with the feed motion used by Gebriider
Mauser, of Oberndorf, in their revolvers, Fig. 751. In this case
a crown wheel is used (see Figs. 677 and 678). The wheel is at
a; bis the feed pawl, jointed at 3 to the slide c, the whole being
carried in the frame d. The zig-zag profile is formed in the rim
of the crown wheel, one portion being parallel to the axis, the
other spirally inclined, so that the angle of thrust is a 90° —
<p and > 0 (§ 237, cases 4 and 5). The movement of the pawl
produces a backward movement of the wheel. It should be
noted that at 2/ and 2" steps are made in ends of the tooth pro-
files in order to guide the pawl into the proper path and keep it
from reversing.
The anchor ratchet of Fig. 682 may be used for a feed motion,
as in Fig. 752, in which there is also the reverse action of the
wheel, in accordance with the notation of ? 237. Here the wheel
is at a and the anchor at b' b". When the latter is moved into
the position shown by the dotted lines, the wTheel is moved
backward ^ pitch, and the return vibration completes the pitch
movement. In order that the anchor shall enter the teeth pro-
perly, the movement should be quick, especially at the entrance
of the pawl into the space. This is well obtained by electro-
magnetic action.
8 255-
Continuous Ratchets with Focking Teeth.
If it is desired to use ratchets according to the method given
in Fig. 749, additional parts must be devised to move the pawl
in and out of gear. A simple method of accomplishing this re-
suit is to use a single tooth wheel for the driver, and operate
the pawl in the same manner as in Fig. 753.
Before the single tooth 5 begins to drive the wheel a, the arm
6 lifts the pawl b and lowers it into the next space just as the
tooth ceases to drive. In this case the usual gear tooth profiles
may be used. Stili better is the “ dead” tooth profile of Fig.
754, in which the entrance and withdrawal of the pin tooth
both lock the wheel while the pawl is being lowered.
This form may also be used for rack feed movement, Fig. 755.
In this case the profile of the pin tooth is formed in several
ares; 2' 2/// being struck from 3, and 2/r 2//r and 2/ 2^ being
the paths of the corners of the space (see § 203).
By using the cylinder ratchet, as shown in Fig. 696, the num-
ber of parts can be reduced, since the driving gear and check-
ing pawl may be combined in the same member. The resulting
forms, Figs. 756 to 758, are variously called : Maltese Cross;
Geneva Stop, used in Swiss watehes, in which case one of the
tooth sections is filled out; or after Redtenbacher we may call
them single tooth gears, although this is hardly correct, for the
general form of Fig. 758 may have several teeth, and a second
tooth is dotted in Fig. 756.
A great number of variations may be made of these cylinder
ratchet motions. An interesting form is the intermittent gear-
ing of Brauer, Fig. 759.*
Fig. 759.	Fig. 760.
The pinion a is the driver, and the wheel b is driven, and
between the passage of each tooth of the pinion the driven
gear remains stationary for a short space, about A of the pitch.
The points of the teeth of the driven wheel here act as ratchet
teeth, in a similar manner to the arc of repose of the single
ratchet gearing of Fig. 756.
The cylinder ratchet gearing of Fig. 760 is similar to that
shown in Fig. 700, and is used in the counting mechanism of
Fnglish gas meters. In Fig. 761 is a modified spiral ratchet of
Fig. 761.	Fig. 762.
the same general type as Fig. 702, with only a portion of the
path of b in a spiral, and a similar variation of Fig. 704 is shown
in Fig. 762.
FlG. 754.
Fig. 755-
* Royal German Patent, No. 5583, 1878.i66
THE CONSTRUCTOR.
i 256.
Locking Ratchets.
Locking ratchets include ali tke numerous devices by whicli
tbe parts of a mechanism are firmly held against the action of
external forces, and yet readily and definitely released when
desired (see # 235, No. 5) ; thus the varicus ciutch couplings are
included, also car-couplers and similar devices.
Locking ratchets occur frequently in the mechanism of fire-
arms, especially to prevent the danger of premature discharge,
etc. The great refinements which have been introduced in such
weapons during the last ten years include especially the appli-
cation of various forms of ratchets. The following single in-
stance will serve to illustrate :
The mechanism of the well-known Mauser revolver may be
divided into two series; one to effect the discharge and the
other to unload or remove the empty shell from the chamber.
The first may be called the discharging mechanism, the second
the unloading mechanism. We then have the following details:
A. Discharging Mechanism.
This includes the revolving chamber, barrel, hammer, spring
and accompanying smaller parts, giving as combinations :
1.	Hammer, spring-rod and trigger = ratchet rack, as Fig.
659-
2.	Spring-rod and trigger, acting as locking ratchet for the
above, as Fig. 664.
3.	Spring-rod, pawl and revolving chamber = continuous
ratchet with crown wheel and bolt pawl, as Fig. 751.
4.	Securing pawl and revolving chamber =r locking ratchet,
as Fig. 677.
5.	Revolving chamber and pawl, forming a ratchet gearing
with limited travel.
6.	Tumbling ratchet and securing pawl = ratchet gearing
for three positions, Fig 669.
7.	Catch on the axis of hammer = locking ratchet, as Fig.
695.
8.	Trigger guard and pin =■ locking ratchet and stationary
pawl.
9.	Checking-plug and trigger = locking ratchet with sta-
tionary pawl.
10.	Rifled barrel and bullet = screw and nut.
B. Unloading Mechanism.
This includes an axial slide which catches under the rim of
the empty cartridge shell to withdraw it, actuated by a toothed
sector and revolving clamp and axis called the ring clamp.
These include the following combinations :
11.	Unloading slide and sector = slide with rack and pin-
ion, Fig. 381.
12.	Axis of revolving chamber, with pawl to prevent end-
long motion, — locking ratchet gear, as Fig. 695.
13.	Ring clamp, barrel and chamber bearing == locking rat-
chet gear with stationary pawl, as Fig. 654.
14.	Ring clamp axis and axis of securing pawl =3 locking
ratchet, as Fig. 701, forming with (13) a locking ratchet
gear of the second order.
15.	Ring clamp axis upon'the reverse motion of the ring
clamp forms, with the axis of the securing pawl, a
locking ratchet gear, wThich combines with (4) to form
a similar gear of the second order.
16.	Securing pawl acts as a catch for the axis of the ring
clamp in the axial direction to forni a locking ratchet
gear, as Fig. 695, forming also with (4) a similar gear of
the second order.
17.	Ring clamp hub and axis of securing pawl = locking
ratchet, as Fig. 695, and with (4) gives one of the
second order.
This analysis shows that in the Mauser revolver there are 17
mechanical combinations; these are composed of 26 pieces.
Classified, these are as follows : 1 releasing ratchet, 1 continuous
ratchet, 2 driving ratchets, 11 locking ratchets, of which four
are of the second order, 1 screw motion and 1 slide motion.
A very important application of locking ratchet mechanism
is found in the signal apparatus of Saxby & Farmer for use on
railways, and made in Germany by Henning, Biising and others.
This includes many ratchets of higher orders, reaching to the
tenth, twelfth, or even higher. When this is used in combina-
tion with the electric Systems of Siemens & Halske, as in the
block System, we have the further combination of two systems
of the higher order with each other.
A branch of locking ratchets which exhibits a great variety
of applications is found in the different kinds of locks, such as
are used for securing doors, gates, chests, etc. These extend
from the most primitive forms, made of wood, to the most re-
fined productions of exact mechanism, and their study possesses
an historic and ethnographic interest in addition to their me-
chanical value.
A door forms itself a ratchet combination ; the door being
the part b, the strike the part c, and the bolt or other piece
which keeps it from being opened is the part a; doors with
latch bolts being running ratchets, and doors with dead bolts
being stationary ratch-
ets. A simple lift latch
and door, as the furnace
door shown in Fig. 763,
is really a section of
a Crown ratchet wheel
with running ratchet
gearing,
A door with sliding
dead bolt, as used on
common room doors, is
a similar section of rat-
chet gear with station-
ary ratchet.
In key locks, the key
is the releasing member
of the ratchet train, and also serves to actuate the bolt after it
is released. The key and ratchet mechanism are arranged in
most ingenious manners, so that numerous permutations can be
made to effect the release.
Some of the most important systems of lock construction are
given as examples:
Example /.—The common so-called French lock, Fig. 764, is similar to the
ratchet of Fig. 753. The bolt is a sliding rack, the “ tumbler ” b being ofteu,
as in this case, made in one piece with its spring. The case of the lock cor-
responds to the frame for the ratchet mechanism, and the key acts as the
releasing and actuating member.
Example 2.— The Chubb lock, Fig. 765, which is always made with a dead
bolt, forms with the door and door frame a ratchet gearing similar to Fig.
691.. Tne bolt is secured by means of several ratchets of precision, as in Fig.
706, and is moved by a ratchet as Fig. 755. The key, the axis 4, and the van-
ous bittings of the key form a system of pawls. The whole is a ratchet Sys-
tem of the second order with precision gear.
Example j.—The Bramah lock, Fig. 766« and Fig. 766 b, is differently con-
strucled. In this case the dead
___________________[Co________________ bolt is actuated through the
'■	.......—■* medium of a cylindrical driving
ratchet gear, which does not
contain the mechanism of se-
curi ty, the latter being in a
distinet portion of the lock,
Fig. 766 b. This consists of a
number of sliding precision
pawls, as Fig. 707, the number
being 6 to 8 (in the illustration
5). The member a of Fig. 707
is here made in the form of a
ring with internal teeth, se-
cured to the escutcheon a by
screws. The key is a prismatic
adj aster of the slides, and the
Fig. 766 a.	whole is a locking mechanism
'	of the third order with ratchets
of precision. The spiral spring around the pin restores the slides to their
extreme position when the key is withdrawn.
Example /.—The Yale lock, Fig. 767 a and b, is also a system in which the
mechanism of security is separated from the bolt mechanism. This is
again a system of the third order, with ratchets of precision. The key is a
flat prism (corrugated in recent locks) and serves to place precision bolts, or
llSL.
f 0 -
»0THE CONSTRUCTOR.
167
pin tumblers in proper line, and also operate the bolt. The figure shows
the method of connecting the cam b0 to the plug a.
The so called combination locks are locking ratchets with precision pawls,
operated without a key by being placed successively in the positions for
release in accordance with a previously selected series of numbers and dial
marks.
a.
The numereus systetns of Arnheim, Ade, Wertheim, Kleinert, Polysius,
Kromer, and others are mostly locking ratchet systems of the fourth order,
or combinations thereof. The American manufacturers, especially the
Yale and Towne Manufacturing Company of Stamford, Counecticut, ha ve
shown great ingenuity in this industry.*
3 257.
Escapements—Their Varieties.
Escapements may fairly be considered as among the most im-
portant mechanical devices, since it is by their means that the
elementary forces are used to regulate mechanical work. For
this purpose they are used in the greatest variety, ali forming
ratchet devices in which the driven member is altemately re-
leased and checked. The arc, angle or path through which the
driven member passes between the interval of release and check
is called the “range” of the escapement. During the passage
over this range there elapses a definite amount of time, which
may be called the “period” of movement of the escapement.
This is followed by an amount of time when the driven member
is stationary, called the period of rest. The sum of the two
forms the “ time of oscillation.” The range and the period of
oscillation may be (a) constant, (b) periodically variable, or (c)
variable at will.
We therefore have
a,	Uniform escapements,
b,	Periodical “
cy Variable “
and these will be briefly considered.
3 258.
Uniform Escapements.
If, in ordinary running ratchet, Fig. 76S, we have the wheel af
impelled by a weight or other force, and suppose the pawl b,
lifted and dropped quickly, as by the arm bv the wheel will'
move one space, and an escapement will have occurred. In
this case the range will be one pitch. If, after a definite time,
this operation is again and again repeated, we shall have a
uniform escapement. In mechanism the releasing and check-
ing action is produced mechanically and not by hand, the im-
pulse being obtained from the movement of the wheel.
* The ancientand modern Egyptian locks. also those of ancient Greece,
Rotne, India and China, oontain the principle of running ratchets with flat
pawls, actuated by a key pushed directly into the lock. The Egyptian lock,
with pin precision pawls, is quite similar to the Yale lock in principle, al-
though very different in construction. Ancient Roman locks, founct in
Pompeii, are similar in principle. Wooden locks are stili in use in China,
Persia, Bulgaria, Russia and Southern Italy, also in the Faroe Islands and
Iceland; At the suggestion of the author, Professor Wagner, of Tokio, suc-
ceeded in inducing some Japanese lockmakers to make a very complete and
intelligible collection of native locks for the kinematic cabinet of the Royal
Technical High School at Berlin.
The most general examples of uniform escapement are found
in watches. In these impulses are isochronous, and obtained
from the inertia of a vibrating body. The wheel a is called the
escape wheel. The vibrating member, or balance wheel, makes
its oscillations in nearly equal times for great or small vibra-
tions. If, therefore, in a watch escapement, the time of the fall
of the pawl is less than the time of oscillation, the most impor-
tant requirement is fulfilled, namely, that for uniform periods
of time the same number of teeth of the escape wheel shall pass,
and the corresponding angle may then be used as a measure of
time. A given amount of work may also be abstracted from the
motive power and used to produce the impulse. These impor-
tant points have been fulfilled in the design of escapements,
and it has been made possible to measure time with a great
degree of accuracy. When the highest accuracy is demanded
the greatest care must be given to the construction and execu-
tion, and to the reduction of friction and compensation of the
balance.
In the case of watches the duty of the impelling force is
simply that of overcoming the resistance of the mechanism,
the function of the escapement being to provide against any
acceleration of the rate motion, and the impulse which is re-
quired to operate the escapement may be considered as a por-
tion of the resistance of the mechanism.
A systematic discrimination between the various kinds of
watch escapements will show that they vary as to the checking
ratchet device, the impelling device, the release and the accel-
erating device. We may have Simple or Compound escape-
ments of the lower or higher orders. Some examples are here
given.
A. Simple Escapements.
Example 1.—The Free Chronometer Escapement (Jullien le Roy, Earn-
shaw, Arnold, Jurgensen), Fig. 769. The runuing ratchet gearing a, b, c, is
similar to Fig. 768. The pawl b is provided with a flat spring 3. The im-
pelling device is the balance wheel rf, which acts as a pendulum. The re-
leasing device is at 4.5, and is attached to d, and when it swings to the left,
impelled by the movement of the watch, it releases the pawl by means of a
second running ratchet at 5. At c' is a stop for the pawl b. At 5' is the ac-
celerator which, for each tooth of the escape wheel a, swings from 5' to 5".
As it returns, the pawl b engages with the tooth which has just left the point
5". The spring b' permits the releasing tooth sto pass back dnring the
return oscillation. The balance wheel can swing freely beyond 5" and back
without engagingwith the escape wheel, hence the name “ free ” escape-
ment.*
Example 2.—The Duplex escapement, Fig. 770, is derived from the ratchet
of Fig. 699. The escape wheel is upon the same axis as the checking pawl
* This beautiful movement is apparently the first form which was applied
as a pendulum escapement, having been used by Galileo in 1641.i68
THE CONSTRUCTOR.
2>; the accelerator is at 4, actiug upon the impelling pawl at every vibration
between 4 • 4'-
The so-called “verge ” escapement is similar in construction, except that
the arm H is longer and curved. The simplicity of this forni as compared
with the preceding is due to the fact that the impelling and checking pawls
are made in one member. It will be noticed that the entrance of the tooth
of the escape wheel into the space, causes a slight reverse movement at a,
due to the fact that b is really a tumbling ratchet gear. This escapement has
been called duplex by its English inventor, although some contend that it is
properly a double wheel escapement, although the two wheels are combined
m one.
Example3.—Another method by which the checking and impelling pawls
may be combined is shown in the Hipp escapement, Fig. 771. This consists
of a simple running ratchet a, b, c. The pawl & is a piate spring, which is
lifted and dropped by the passage of the teeth. The acceleration is given by
the deflection of the spring. If the impelling force upon the wheel a is
great, two teeth will pass, but this can be detected by the note emitted by
the spring, which will then be one octave higher than before.
B. Cornpound Escapements.
Example 4.—Lamb’s escapement. Those escapements which have two
escape wheels are properly classed as cornpound, and to this class belongs
I*amb’s escapement. This consists of a running ratchet gear, similar to
Example 1, and the same form of impelling device, but between these is an
internal wheel with pitch ratchet gearing, similar to Fig. 686, which is im-
pelled with each direction of vibration. Another double-wheel escapement
is Enderlein’s, based on Fig. 702, also one devised by the author, like Fig. 686.
Example5.—Mudge’s Escapement (also invented by Tiede), Fig. 772. This
is a double ratchet gear system, with one pawl in compression and one in
tension, b\ and <52. At 2' and 2" is a “deadM pawl action for checking, and
at IV and II" a running pawl action for impelling. (See Cases 5 and 7, § 237).
The pawls are lifted by the pendulum d. The releasing arms 3'. 5' and 3". 5"
are moved alternately by the pendulum ; for example, the arm &x, being
moved into the dotted position, lifts the pawl out of gear, and the weight of
the pawl and arm (sometimes assisted by a spring), gives an impulse to the
return vibration of the pendulum, the acceleration being provided by the
escape wheel acting on the portion II'. A similar action takes place on the
other side.
Example 6.—Bloxam’s or Dennison’s so called “ gravity’’ escapement, Fig.
773. The escapement is controlled by a pendulum suspended by a spring at
4. The escape wheel is made in two parts, as Fig. 686. The accelerating
surfaces IF and II" are much better arranged than in the preceding exam-
ple, the friction being reduced. A fan is used also, as shown at e, for the
purpose of preventing great acceleration of the escape wheel, which might
otherwise oecur in the large angle (6o°) of escape. The fan is not fast to the
axis of the escape wheel, but connected by running ratchet so that its mo-
mentum is not checked as the escape wheel is stopped.
Example 7.—Free Anchor Escapement, Fig. 774. The two pawls are com-
bined into one anchor, as in Fig. 682, and the action is much the same as
Fig. 772. The escape is controlled by a balance wheel at d. The pawls 2'
and 2" are operated through the armb3, and at the same time the impulses
are given by the action of the escape wheel upon the inclined surfaces IF
and IF'. The pawls are technicallv known as pallets. The tooth action at
5 is a continuous ratchet gear similar to Fig. 754. The arm b3 is limited in
travel by pins at 3' and 3", or in some forms by a fork at 4 Since there is a
ratchet at 5 and also at 2, this forms a system of the second order.*
* A watch escapement of the third order has recently been designed by A.
E. Mulier, of Passau. This is made with a cylindei ratchet, as Fig. 699 b%
between the arm and the escape wheel.THE CONSTRUCTOR.
169
Exampie 8.—Graham’s Escapement, Fig. 775. The construction is very
similar to the preceding. The connection 5 between the anchor-arra b% and
pendulum d, is different, and the arm b3 does not come to rest, but both it
and the pallets 2' and 2" slide upon the teeth while the escape wheel is
stopped. An earlier form of pallets for this escapement is shown at b\ and
bf2 (called Clemenfs Anchor, from Clement, 1680 ; but described by Dr. Hooke
in 1666). This form produces a brief reverse movement to the escape wheel
at each oscillation.
Exampie 9.—The form of ratchet of Fig. 684 is used in L,epaute’s escape-
ment, which was' really invented by the watchmaker Caron, afterwards
Marquis Beaumarchais.
Exampie 10.—Cylinder Escapement, Fig. 776. This is made from the
cylinder ratchet of Fig. 700, the impelling surfaces being divided between
the anchor and the teeth of the escape wheel. The cylinder b is attached to
the axis of the balance wheel, and the wide spacing of the teeth of the escape
wheel permits a correspondingly wide amplitude of oscillation. If we im-
agine the pallets of Graham’s anchor to be formed between two concentric
circles (as, indeed, most watch-nakers construet them), the “ cylinder" will
be seen to be a similar anchor.
Exampie //.—Crown Wheel Escapement, Fig. 777. Escapements con
structed with crown ratchet wheels (§ 241) are the oldest forms used in
Fig. 777-	Fig. 778.
ratehets.* The form of the pallets causes a reverse movement, and in the
old watehes using a balance with its centre of gravity in the axis of oscilla-
tion, without atiy assisting spring action, this reverse movement was a
necessity, which 'accounts for the long and extended use of this form of
escapement. Toward the end of the fifteenth century the hair spring was
introduced by Hele, in the form of a hog’s bristle, and in 1665 Hayghens
made the Steel hair spring, which made the construction of the modera
chronometer possible. The crown escapement is easily modified so as to
remove the reverse action, as was done by the author in 1864. We then have
a “dead ” tooth action, as Fig. 699. The modified escapement is shown in
Fig. 778; the pawls are praetically hyperboloidal in form.f
C. Power Escapements.
In the case of watch escapements the impelling force is only
used to overcome the resistance of the watch mechanism.
Escapements can also be used to regulate greater forces, such
as are intended to perform useful work, and these may be
C
called power escapements. Alarm and striking clocks are of
this class, and there are numerous other forms. The followdng
exampie will serve to illustrate :
* This has been used since the tenth century, having been invented by
Bishop Gerbert, afterwards Pope Sylvester II, about 990; also by Heinrich
von Wyck about 1370, and applied to a pendulum bv Huyghens. The oldest
tower clock in Nuremberg, built about 1400, has such an escapement.
f In the Kinematic cabinet of the Royal Technical High School there is a
schematic series of models of clock and watch escapements.
Exampie 12— Power Escapement for a Reciprocating Movement, Fig. 779.
At a bi ci and a b2 c2 are ordinary running ratehets, the pawls b\ and b2 of
which can be released and engaged by suitable auxiliary mechanism. This
mechanism is either a substitute for or identical with the iegulating device
(balance wheel, pendulun, etc.) of a watch escapement. The escapement is
intended to control the motion of the swinging arm Cby means of the lever
Ci and the descending arm Av This is accomplished by a double acting
ratchet system dx d? 5 (as Fig. 671), by means of the slide e, 'driven from 8 by
the arm cx.
The action is as follows: When the parts are in the position shown in the
figure, the motion of the wheel a to the right moves the arm C\ by means of
the pawl bi until the trigger 10" trips the pawl d2 and shifts the engagement
at 5 into the position 5' (in the small figure to the left). This action, by
means.of the trigger at 6", throws in the pawl b2 afid stops the wheel a. At
the same time bx is thrown out of gear by the connections dx, 6' and 7', and
the counterweight C\ returas the arm Ci to its original position. This brings
the trigger 10' against the lever dx, and again shifts the engagement at 5.
The pawl b\ falis into gear, and the pawl b2 is disengaged, leaving the wheel
a free for another forward movement.
The preceding escapement can be readily converted into a
double acting one by introducing a second ratchet wheel toothed
in the opposite direction, with proper pawl on cx and trigger
connections to d2; the other portions would remain the same.
This escapement appears to be new, and many important appli-
cations will suggest themselves.
£258.
Periodicae Escapements.
A great variety of periodical escapements are to be found in
the striking mechanism of clocks and repeating watehes. The
entire period is the revolution of the hour hand, and if the half
hours are struck the order will be
E b 2, I, 3, I, 4>	b 12,
making in all 90 strokes in the twelve hours. A fan regulator
is used to cause the strokes to follow each other uniformly.
There are two Systems of escapement in use for this purpose,
the German and the English, the latter also used for repeaters.
An essential piece of the latter, the so-called “snail,” has been
shown in Fig. 688 ; its function is to control the number of
strokes. Further subdivisions cannot be here discussed, but it
must be remembered that the striking arm is itself a ratchet
mechanism.*
Important applications of periodical escapements are found
in the self-acting spinning mule, and both these and the clock
striking mechanism are examples of powrer escapements.
The mechanism in Platt’s mule is here briefly shown. Fig.
780, a and 6. The shaft 1 is required to make rapid turns
b.
Fig. 780.
through 90° at intervals of different lengths of time. The wheel
a is an escape wheel with teeth in four concentric rings, I, II,
III, IV (compare Fig. 686), each ring having one tooth. The
other side of the wheel a is shown in Fig. b, where is the rat-
chet chain a d e. When a is released, the pressure of d at 5'
moves it slightly and brings the running friction wheel e into
contact, thus driving a through a quarter revolution, toward the
close of w7hich the pawl d again enters into engagement.
* See Ruhlmann R£dtenbacher, Denison.THE CONSTRUCTOR.
170
The recesses in a permit the friction wheel to run free when
a is at rest. This is evidently a form of ratchet gearing in it-
self. The order of escapements at 2 is as follows:
I II, II III, III IV, IV I.
This is controlled by a second escapement, shown in Fig. 781.
The pawl b of Fig. 780 is connected by the rod f to the beam a,
as shown. This mechanism is a step ratchet of four steps. The
steps are the pawls bv b2y b3y and the stop on the frame c; giving
the positions 21, 211, 2^1, 2iv. The action takes place in the four
following periods:
1.	Drawing and spinning—a checked at 21
2.	Stretching and twisting	“	“	211
3.	Holding and spun thread	“	“	2111
4.	Winding and returning	“	“	2iv
The succession of movements is as follows: At the termina-
tion of the first period a projection on the carriage strikes the
pawl bx at 5/. The step lever, which is heavier 011 the right end
than 011 the left, moves from position I to position II, in which
it is held by the pawl b2; this, by means of the rod fy places the
pawl b of Fig. 780 in the position 3 II, thus starting the second
period.
At the close of the second period the pawl b2 is released, the
lever falis to the position III, shiftiug the pawl b to 3 III, and is
held by the pawl b3 at 2///.
The third period, which is very brief, is terminated by the
wdnder striking 5///, releasing the pawl b3y and the lever as-
sumes the position IV, and the rod f moves the pawl b into the
position 3 IV, and the fourth period begins.
During this period the carriage returns, and just before the
close of its motion a roller acts upon the portion 50, bringing
the lever back into the first position. This returns the pawl b
to its original position 3 I, and the succession is repeated.
The entire mechanism fornis a periodical escapement of the
second order, or, when the connections are included, the third
order, and when taken together with the ratchet gearing, of the
fifth order ; while a sixth ratchet mechanism is used for the
primary control.
1259-
Adjustable Escapements.
An escapement can be so arranged that the checked member,
after the release, will again be checked by the impulse of its
fresh start, thus forming what may be called a self-acting
escapement. In a mechanism of this kind, the amplitude of
the escapement is dependent upon the amount of displacemeut
which is permitted to the releasing member. This may be
made greater or less, and hence sucli devices may be called
adjustable escapements. These devices are likely to play an
important part in modern machine design.
A simple forni of adjustable escapement is shown in Fig. 782.
This apparatus, designed by the author, is based upon that of
Fig. 674. The ratchet wheel a is stationary, being fastened to
the frame a' ; the pawl is at by and the link is in the form of a
disc c} driven by a force C, and checked by the escapement. At
3 . 5 is the guide for the pawl. This can be adjusted by the
wheel d, by turniug the latter more or less in the direction in
which c is impelled. If d is turned so far that the pawl b is
lifted out of gear, the force at C will set the disc c in motion.
This latter carries with it the axis 3 of the pawl, which, by the
action of the guide 5, draws the pawl into engagement again,
entering the space 2 and checking the disc. In order to avoid
an uncertain or irregular action, a brake may be used as at a".
If the w7heel d be moved forward regularly through two, three,
or four ares, the disc c will be released and checked successively
in similar manner.
It will be evident from the foregoing that the ratchet gearings
which form the foundation of the various kinds of adjustable
escapements are so varied that the different constructions which
may be used are very numerous. Among them may be men-
tioned those in which friction ratehets are used, these posses-
sing the advantage that the arc of motion of the escapement
may be varied from the smallest to the greatest without being
dependent upon any especial piteh.
We have already intimated that the various forms of coup-
lings may be considered as varieties of ratchet gearing. The
same is true of the present subject. If it is desired to use this
adjustable escapement as a disconnecting coupling, the follow-
ing arrangement may be adopted :
The disc c can be attached to the shaft which is to be set in
motion, and the wheel a to the driving shaft, wThich is supposed
to be in continuous revolution and is to be coupled to c. The
teeth are then to be so arranged that by the revolution of a the
pawl by disc c and wheel d will be carried around together.
When the disconnection is to be made, it is only necessary to
hold the wheel d from revolving. The pawl-axis 3 will then
move on and cause disengagement of the pawl at 2, and the
disc c will come to rest. If the wheel d is then turned a short
distance in the direction of rotation the pawl wTill again be
tbrown into gear and the parts once more connected. A coup-
ling thus formed from an adjustable escapement may be called
an adjustable coupling.
The suitability of the application of toothed ratchet gearing
for this purpose is open to question, and indeed toothed gearing
is only to be recommeuded for the lightest Service of this kind.
In most cases, if indeed not all, friction couplings are much
better. An adjustable friction coupling is to be seen by refer-
ence to Fig. 448, in which A is the friction wheel, B is the pawl,
disguised in the form of a cone, and b is the adjusting member.
If a combination is made of an adjustable friction coupling
with some form of transmission to a machine, such as a rope
or belt gearing, so that it is thrown into action when any re-
verse motion is attempted, we have what may be termed an
automatic friction brake.*
a'
*See German Patent, E. Langen, No. 21,922.THE CONSTRUCTOR.
171
Example.—Vig. 783 shows such an automatic brake device as applied to
the pontoon bridge at Cologne. At a is a friction cone combined with a spur
gear a', driven by the shaft and pinion a" in the direction to wiud up the
cord on the drum d. The drum is fast to the chaft c, but the cone a is loose
on the shaft. The wheel a is connected firmly to the shaft c, when the cone
6, which slides on a feather, is forced into engagement with it, and this en-
gagement is effected by the differential screw d and hand wheel d'. The use
of the differential screw enables the equisite pressure to be obtained, and
alse causes the motion of d' to be in the same direction as c' when lifting.
The friction of the cones binds the parts firmly together, so that a is practi-
cally secured to the shaft until d' is revolved backwards, when c' follows by
the action of the weight C, the cones slipping upon each other and the
pressure being. automaticallv regulated, and the motion at once checked
when d' is stopped.
Other and most important applications of adjustable escape-
ments will be giveu hereafter. It may, however, be here noted
that by means of such mechanism the most powerful combina-
tions may be controlled with the exercise of a minimum effort.
\ 260.
Generae Remarks upon Ratchet Mechanism.
Ratchet mechanism, as already discussed, is applicable to a
most extensive range of uses ; in this respect far excelling every
other form of mechanism. This is plainly due to the fact that
ratchets are suited either to produce the effect of relative motion
and relative rest. Considered in this light the six preceding
classes may be grouped as follows : Common ratchets, checking
ratchets, and locking ratchets are those which act to hinder
motion, while releasing and continuous‘ ratchets, as well as
escapements, act to produce definite motion. The motion pro-
duced by ratchets is intermittent while that produced by the
forms of mechanism previously considered, such as cranks,
friction, or toothed gearing, etc., is continuous. Mechanism for
continuous motion may be called “running gearing, ” * and
practically merges into ratchet gearing. The general province
of ratchet gearing has only been partially covered in the pre-
ceding pages, where such forms as may strictly be considered
machine elements have been included. An exception might be
made as to the allied forms of springs, some of which, indeed,
were referred to. There is, however, a large number of machine
elements of a different kind, which usually involve the continu-
ous action of the operative forces in one direction; these in-
clude tension organs, such as ropes, belts, chains, etc., compres-
sion organs, fluid connections, and many others, all of which
are considered in the following chapters. It will be seen that
these may all be so arranged as to be fairly considered ratchet
devices also ; as belts or chains may become friction or toothed
ratchet gears, and even the valves of fluid connections are really
pawls.f
The pawl mechanism must also be extended to include these
classes of machine elements, and their limits thus greatly
widened, especially in the case of pressure organs^ Bxamples
of this will be found in the pistons and valves of pumps, both
for liquids and gases, which may act as checking or locking
ratchets, or in hydraulic motors and steam engines as escape-
ments, and in gas engines, as escapements and continuous
ratchets combined. Similar comparisons may be made of the
ratchet principle in the use of accumulators for hydraulic cranes,
presses, riveting machines, and the like, and in the cataract for
single acting steam engines we find a complete analogy to the
ratchet. In these cases we have ratchet Systems of the higher
orders. The history of the development of these machines is
really that of their pawl membeis.
A very interesting example is that of Fig. 779, in which, if we
substitute a flow of steam for the ratchet wheel, we have the
arrangement of the single acting high pressure steam engine
with Farey’s valve gear. The numerous modifications of escape-
ment gear, which are included in the steam engine, have occu-
pied the activity of designers down to the present time. A
number of the more recent valve gears have been shown in § 252,
and similar devices are used on engines for steam steering gear,
called by the French “ moteurs asservis,’’ and such gear also
plays an important part in the mechanism of some of the so-
called ‘‘fish” torpedoes.
In this manner the applications of pawl ratchets may be ex-
tended before our eyes and yet the limitations are not reached,
and the further researches are carried the broader and more
general does the scope of this division of mechanism become.
Not only does it include fluid pressure organs, both liquid and
gaseous in a strictly mechanieal sense, as in the case of pumps,
etc., but also when these are considered in a physical sense with
regard to their internal stresses. This gives a branch which
may be called “physical” ratchet trains, of which the steam
boiler is the most important example. In this, when taken in
connection with a pipe full of steam, and suitable valves for
opening and closing, forming what has been termed a steam
* See the author's Theoretical Kinematics, p. 486, in which this classifica-
tion was originally made,
t See Theoretical Kinematics, p. 458 et seq.
column,* we have undoubtedly a physical ratchet train in which
the particles of vapor are considered as a physical aggregate,
which from the higher temperature, are under higher stress.
Another example of a physical ratchet train is the apparatus for
operation by liquid carbonic acid which has been recently used.
Electrical accumulators are also instances of physical ratchet
trains, as well as some applications of galvanic batteries, the
action taking place by make and break of electrical contact.
The dynamo-electric machine also becomes a physical running
ratchet and the electric motor a physical escapement, the whole
forming a physical running gear train.
Again we may consider a “chemical” ratchet train, such as
coal or any fuel, wrhich, during combustion, releases the energy
which is stored in it. This may be utilized in numerous ways,
but for our present considerations, mainly in the production of
motion. Chemical action is also included iu hot-air engines,
and in the operation of telegraph apparatus in a similar sense.
We may consider the principal factors in a steam motor piant
as portions of a ratchet chain, somewhat as followTs:
Chemical ratchet	= combustion of fuel,
Physical “	= steam generator, etc.,
Mechanieal escapement = steam cylinder and attachments,
Mechanieal running gear = crank shaft and wheel,
these four uniting to convert the released energy into mechani-
cal motion. If we consider a loeomotive engine, we have added
to this another running gear in the shape of the driving wheels
and rails, while the train and wTheels and journal bearings unite
to form a combination of the sixth order.
Another Chemical train may be formed by the use of explo-
sives, which are released either mechanically, as by percussion
or friction, or chemically, by combustion of some auxiliary
material. Again, we may have releasing gear of the first, second,
or higher orders.
In the case of most firearms the release is of the second order,
since the mechanism of the lock acts upon a fulminate by per-
cussion, and the heat of the latter releases the powder.
If wre examine and classify all mechanism of transmission in
the above manner, it wTill be apparent that ali forms are included
in one or the other of the followdng classes, viz.: mechanieal,
physical, or chemical ; these also entering into combinations of
the higher orders with each other.
The steam engine itself, as we have already seen, consists of a
driving train of the fourth order. Trains of stili higher orders
are of frequent occurrence.
In the recording telegraph, with relay, we have a physical
ratchet train of the second order, releasing a mechanieal run-
ning train and operating a recording train, both physical trains
actuated by chemical trains, the whole forming a combination
of the fifth order. The ordinary signal mechanism of a railway
station, when mechanically operated, is a system of the fourth
order.
The Westinghouse air brake, not considering the boiler, is a
train of the fifth order, consisting of an escapement (steam
cylinder), driving ratchet (air cylinder), intermittent ratchet (air
vessel), escapement (pistou and valve connections), friction
checking ratchet (brake gear). If we include furnace and
boiler, this becomes a train of the seventh order, and may be
stili further extended.
A stili more noteworthy example is found in the application
of compressed air for the purpose of operating pumping ma-
chinery at the bottom of deep mine shafts. In this case we
have:
1.
2.
3.
4.
5-
6.
7-
t».
Furnace	=
Boiler	—
Steam engine	—
Shafting and transmission
Air compressor,
Air chamber,
Air cylinder in mine,
Water cylinder in mine,
to
chemical ratchet train.
physical “	“
mechanieal escapement train.
“	running “
“	driving ratchet.
“	intermittent “
“	escapement train.
“	driv’g ratchet “
The preceding discussion and illustrations of the relationship
existing between mechanieal, physical and chemical trains show s
the necessity of combining mechanieal and'technical researcb,
and a complete mechanieal training therefore includes these three
branches, and also the later Science of electro-mechanics.
Modern methods of invention require research into all of these
lines of Science, and the constantly widening field of mechani-
cal engineering is thus extending its work, while at the same
time gathering into systematic form the many branches of
applied mechanieal Science.
* See Theoretical Kinematics, p. 493
f The system of clocks operated by pneumatic pressure from a central
station, designed by Mayrhofer, at Vienna, forms a combination of 33 dis-
tinet systems.THE CONSTRUCTOR.
172
CHAPTER XIX.
TENSION ORGANS CONSIDERED AS MACHINE ELEMENTS.
i 261
Various Kinds of Tension Organs.
The various fornis of machine elements which have already
been discussed, have been those which offered resistance to
forces acting in any given direction, forming more or less rigid
constructions. We now have a series of elements which are
only adapted to resist tension, and which are very yielding
under the action of bending, twisting or thrusting forces. These
include a great variety of rope, belt wire, chain belt and similar
transmission devices, all of which may be included under the
general terni of Tension Organs. Their usefulness is limited
by reason of the fact that they have only the single method of
resisting force, but at the same time the element of flexibility
permits the use of one and the same organ to transmit power in
changing directions, and hence gives rise to mauy useful com-
binations. A11 especially valuable feature of tension organs in
practice lies in the fact that mauy materials are excellently
adapted for such use, and cau be more economically applied.
Fig. 262.
Methods of Application.
A distinction is to be made between “ standing and running”
tension organs. The first are those used to suspend weights*
support bridges, also in the constructiou of mauy machine de
tails. Examples of such use are found in suspension bridges*
pontoon bridges, hawsers, guy ropes, standing tackle, etc*
Running tension organs are used in machine design in connec-
tion with other machine elements principally for the transmis-
sion of motion.
Running tension organs may again be divided into three
classes according to their action in connectiou with other
machine elements.
According as they are used :
1.	For guiding.
2.	For winding (hoisting or lowering).
3.	For driving, this also being possible by winding and un-
winding.
Combinations of these applications may be made, either wTith
or withouf the use of standing tension organs. In order to
understand the various applications it is desirable to consider
some of the most important combinations, hence these will be
briefly examined.
1. Guiding.—Fig. 784 shovrs s.veral combinations, adapted
solely for guiding. At a is the so-called stationary pulley, in
which a cord, led off at any angle, is used to raise and lower a
load Q. The dotted lines show the position of guides, or in the
absence of these the direction of motion is gcverned by the
action of gravity. At b we have the so-called movable pulley,
the pulley being combined with the moving piece; the weight Q
is here supported on two parts of rope. Form c is a combina-
tion of a and 6, and is the well known tackle block. Form d
consists of four sets of form a, and the action of the cords com-
pels the piece Q to maintain a parallel motion. This is practi-
cally applied in Bergner’s drawing board.
In like manner four pulleys of form b may be combined as in
form e. This is the old parallel motion for spinning mules, also
used as a squaring device for traveling cranes *
The use of pulleys and bearings is to reduce friction at the
point of bending, and roller bearings, as Fig. 566, are also used,
but when the bending surface is wTell rounded the pulleys may
be dispensed with. Fig. 785, at a, b, c> shows such arrangements,
the action being the same as before, but with greater friction.
The arrangement at d is a six-fold cord, aod in sail making eye-
lets are often used in similar manner, as at e. The friction is
great in all such devices, because the cord presses hard upon
the point of curvature ; its magnitude increases rapidly with the
* Form d is a kinematic inversion of the older form e.
arc of contact. This action, which here opposes the motion of
the cord, is in other instances made of great utility. Cord-
Fig. 785.
friction, which is to be considered as a particular case of sliding
friction, plays a very important part in constructions, involving
tension organs, and will be more fully considered hereafter.
In Fig. 786 is shown Riggenbach’s rope haulage system for
use on inclined trackways, or so-called “ramps.” In this
arrangement, the descending car is loaded at the top of the
ramp writh sufficient water to enable it to draw up the ascending
car by the power of its descent. The speed can be controlled
by the descending weight, and also a weight acting upon wheels
gearing into a rack z.f
2. Winding — The most important forms of winding gear are
Fig. 787.
showm in Fig. 787. At a is the common windlass, also known
as a winding barrel or drum, extensively used in many forms of
hoisting machinery ; b is a drum for spiral winding of a flat belt,
the belt being wTound upon itself, and side discs being provided
as guides for the belt; c is a spirally grooved drum for winding
chain ; d is a conical drum, with spiral groove, used in clocks
(there called a fusee), also for hoisting machinery with heavy
rope; and e is a rope “snail ” used on the self-acting mule, to
produce the varied speed of the carriage. Many combinations
of winding and guiding devices are made, also of winding de-
vices with each other.
In Fig. 788 are shown several lowering devices. At a is a
lowering drum for warehouse use; the unwinding coii at Wl
lowers the load Qt while the cord of the upward moving coun-
terweight Q2 is wTound on the drum at W2; a brake cau be ap-
plied at B, and when necessary, guide pulleys used as at LL.
Form b is a lowering apparatus for coal trucks, consisting of a
combination of two winding coils, with a brake at B. The
f Numerous illustrations are in use in Switzerland and elsewhere,with in-
clines varying from 25 to 57 per ceut.THE CONSTRUCTOR.
173
counterweight Q2 is in the form of Poncelefs chain, the action
being to vary the rate of descent of the load W2. This appa-
ratus, which is called a “Drop,” is much used in the coal
mining districts in England. Form c is Althan’s furnace hoist,
and consists of two drums with Steel bands. The load of water
at Qi, by its descent, raises the charge Q2 to the top of the fur-
nace, after Vvdiich the water is drawn off, and the empty car de-
scends and the water vessel is raised to the top again. The
speed is controlled by a brake at B.
Wrapping connections have been used from early times in
connection with beams and levers, as shown in Fig. 789 a, and
the form b is especially applicable to scroll-sawing machines.
Form fis a combination made with very fine Steel bands, and
used in the Emery wTeighing machine.
Combination windlasses are frequently used for lifting weights,
some forms being shown in Fig. 790, and other combinations
also in complete machines for hoisting, as in Fig. 791.
In Fig. 790, a is the so-called Chinese, or Differential Wind-
lass, consisting of two windlasses and one sustaining combina-
tion ; b is another differential combination used in a traveling
erane designed by Brown, of Winterthur, the arrangement
being intended to obviate the lateral motion of the load.
Another arrangement for the same purpose is shown at c (de-
vised by the author in 1862); it consists of two drums united in
one. The signal arms and automatic safety gates, now so much
used on railways, are operated by a combination of winding and
guiding members, chains being used on the winding barrels and
wire connections on the straight lines.
Winding and guiding members are much used in cranes and
hoisting machinery, several combinations being given in Fig.
791. A erane with boom of variable radius is shown at a \ b is
a pair of shears operated by three windlasses, W1 and IV2 for
moving and holding the shear legs, Wz for hoisting and lower-
ing the load; c is a form of bridge erane, nsing a trolley in
combination with two winches. If both winches are operated
in parallel direction and uniform speed, trolley travel is effected,
hoisting or lowering by unequal wind motion.
In Fig. 792 a, three drums and one guide sheave are used ; b is
made with four drums and two guide sheaves, a combination
used in steering machinery for operating the tiller; and c con-
sists of two drums and two guide sheaves so arranged that one
load is raised as tue other is lowered, this being used in mine
hoists. This is also used for inclines or “ramps.” When the load
is always to be lowered, the descending load doesaway with the
necessity of any motive power, and the speed is controlled by a
brake. Examples of this form are found in some mines and
stone quarries, and in apparatus for loading vessels, etc. (See
Chap. XXII.) Power-driven cable railways for passenger Ser-
vice on inclines are sometimes made with two cables, one for
driving, and a second for guiding and as an additional security,
an example being the old road up the Kahlenberg at Vienna.
When round ropes are used it is desirable to have the drums
made with spiral grooves, in order to reduce the wear on the
Fig. 793-
rope. The travel on the drum causes the angle of the rope be-
tween W and L to vary, and to prevent this the device shown
in Fig. 793 has been used by Riggenbach on the cable incline at
Eucerne ; two forms being given. The guide sheaves are trav-
ersed by screw motion, the rope being led off in a plane parallel
to the axis of the drum, and in the second form two guide
sheaves are used for a double cable.
3. Driving.—This application of tension organs is most ex-
tensive. The principal forms are given in Fig. 794. The cap-
Fig. 794.
stan a consists of a hollowed drum, the surface of which is
composed of numerous ribs and the rope is given several turns
about it. The axial travel produced by the spiral path causes
the rope to climb upon the larger diameter, from which it is
easily forced back to the middle from time to time by hand.
At b is a sprocket wheel with Y-shaped sprockets, much used in
many modifications; c is Fowler’s drum, a form of grip drum
which grasps the rope automatically, and which is discussed
more fully hereafter. At d is a simple rope pulley, partly en-
circled by a tension organ under such load as will produce suf-
ficient friction to prevent slippage; e is a chain wheel with
teeth to prevent the slipping of the links. In ali five cases the
wheel may drive or be driven by the tension organ.
By combination of driving and guiding devices many useful
transmissions are made.
Fig. 795.
Several forms are given in Fig. 795 : a is David’s Capstan,
with conical windlass, with a ring-shaped guide roller which
constantly leads the rope from its travel toward the base of the
cone. At b is a counter-sheave device, the main sheave T being
made with two grooves and the counter-sheave set at a corre-
sponding angle. This gives increased rope contact, which may
be multiplied stili more by increasing the number of grooves.
The counter-sheave may also form the second pulley of the
combination, as at c; this is used in rope transmission devices.
Driving tension devices are often capable of being used to174
THE CONSTRUCTOR.
greater advantage than winding devices, since the direction of
motion need not be changed and is not limited. For these
reasons driving combinations are frequently used instead of
drums, as in hoisting machinery. Chain sheaves with pockets
to receive the ordinary oval link chain are here applied (see
i 275)» or with flat link chain the sheave engages with the pins
of the chain.
Fig. 796.
Other driving Systems are shown in Fig. 796. At a is a double
lift with water counter-weight. T is a pulley for round or flat
belt; the weights Ql and Q.2 are nearly equal, so that a semi-
circle of contact is sufficient to prevent slipping at T, and the
friction of contact is sufficient.
A reference to the Riggenbach cable road gear, Fig. 786, will
show a similarity to this device, but in Fig. 786 a braking de-
vice is provided at Qx and Q2 to protect from accident in case of
breakage of the cable. A similar device, using strains at T} has
been applied by Green for operating the sluices of the Great
Western Canal. A* b is shown the grip-wheel, which has also
been used for cable driving. In this forni the loads may be
quite unequal without apprehension of a deep groove cutting
in the drum. Koppen’s System is shown at c; this uses a round
or flat belt with tightening pulleys L, Z, so that sufficient fric-
tion can be obtained for any given difference of loads; this
avoids the unequal action upon the heavily-loaded side of the
belt, by producing tension upon the otherwise slack side, and
might be applied with advantage to the driving system of Fig.
795 c} requiring but a single tightening pulley, and subjecting
the rope to only one kind of bending.
At d is shown a bucket gear, which combines driving and
guiding, and is much used fdr conveying in mills, grain eleva-
tors, etc. If the difference in weight between the sides is slight,
the tension organ may be a leather belt, but for heavy Service a
chain is used. This device has been in use from a very early
period for well buckets, and in modern times in mud dredging
machines. At e is the Weston differential pulley block. a modi-
fication of the Chinese windlass, Fig. 790 a. Tx and 7\are chain
sheaves fast to each other, producing a differential action due
to their difference in diameter, the whole forming a substitute
for the older tackle block gear, Fig. 784 c.
The forni shown at Fig. 796 d demands further consideration,
as it can be given a series of most important applications.
If the tension organ is made a band and placed in a horizon-
tal or nearly horizontal position, it can be used to convey finely
divided material simply poured upon its upper surface. Exam*
ples of this are found iti the transportation of grain, also in the
movement of paper pulp, and many other such purposes ; also
for conveying straw upon chain lattice conveyors, etc. In ali of
these cases the material is kept on the conveyor simply by
gravity. This condition may be avoided and the capacity ex-
tended by using a pair of belts, the material to be conveyed
being carried between thein. A very important application of this
principle is found in power printing presses, the delivery of the
sheets being effected by systems of tapes and bands with great
speed and accuracy. Band conveyors are also used in needle
machinery and in match tnaking machines, and many similar
situations.
An important application of driving gear is found in the con-
struction of inclined haulage systems for mine ramps.
In Fig. 797 is shown the inclined cable system of the Rhenish
Railway. The driving wind T Z, operated by a steam engine,
works the descending cable on one track and the ascending
cable on the other. At L' is a tension pulley to take up the
slack cable and maintain a proper tension. The trains Qx and
Q2, are connected to the brake cars Bx and B2y which are extra
heavy and control the rate of descent by proper brakes.
In the anthracite coal region of Pennsylvania haulage systems
are in extensive use for the transportation of coal, some being
constructed with iron bands, but most of them using ropes.
The arrangement will be understood from the diagrams in Fig.
798 and 799, which, with the accompanying data, have been ob-
tained by the author from their engineer and constructor, the
late Mr. W. Lorenz.
The car in which the coal is hauled is not attached directly to
the cable, bflt is driven by a dummy Z>, which is permanentiy
connected to the cable. This dummy runs on a narrow gauge
track, and at the foot of the incline the narrow track continues
on, so that the dummy D can go below the main track, as shown
in Fig. 799, and on the ascent it can thus be drawn up behind
Fig. 799.
the cars which have been placed by the shifting locomotive
The steam engine and drawing gear is placed at the head of the
incline, as shown in Fig. 798, and the cable is led, as shown by
the arrows, that it passes twice over the driving wheel Ty each
time covering about of its circumference. The dummy cars
Dx and D2 are connected by a secondary cable passing over the
tension sheave L'; this secondary cable maintains the proper
tension on the main cable, whether the load is at the head or
foot of the incline, or on the horizontal. The tension car is
given a play of 75 feet to provide for the necessary variation.
A different forni of cable haulage is found in the system in
use between Tiittich and Ans, and sketehed in Fig. 800 *
In this case the incline is divided into two sections, which
make an angle with each other as shown on the plan, and be-
tween which is a short level space. On this space is placed the
steam engine and driving wheels 7\t T2y T3, Z4, each wheel
having its own engine, two engines always diiving and two
being at rest; L' are the tension sheaves.
In this, as in the preceding case, it wiil be noticed that the
cable runs continuously in the same direction, differing in this
respect from the previously described winding and reversing
system. The cable is brought to rest in transferring the cars
from one plane to the other in order that this may be readily
and conveniently done, but should this be avoided by running
them over the connection, by momentum or otherwise, the ad-
vantage and usefulness of the system would be greatly increased.
This has been done in the cable tramways of Halliday and
Eppelsheimer, first used in San Francisco, and shown in dia-
* See \Veber\s “ Poitfolio John Cockerill.”THE CONSTRUCTOR.
175
gram in Fig. 801. This is most effectively applied on the trolley
streets of the city, for which it is admirably adapted.
The endless cable runs in an iron way between and beneath
the tracks, the power being at T and guide sheaves at L, L,
with suitable driving and tension mechanism. The cars grasp
the cable by a gripping device through a narrow slot in the
trackway. The guide sheaves at the bases of the inclines and
sides of the curves permit the grip to pass, and when the foot
of the hili at the end of the road is reached, the grip is released
and the car transferred to the other track as at and in sim-
ilar ruanner shifted at the other end, Wv The weight of the
cars on the down grades counterbalances those on the up
grades, and so the motive power has only to overcome the fric-
tional resistance.- The cable system of tramways has been ex-
tended to Chicago and many other American cities; also in
Tondon, and a cable system of canal towage has been projected
by Schmick for the proposed Strasburg-Germersheim Canal.
When it is practicable to propel the cars by a suspended cable
from overhead a different arrangement may be adopted.
Fig. 802.
Fig. 802 is a diagram of a system operated by a suspended
chain. The descending cars Q1 are loaded and the ascending
ones Q2 are empty, and the speed is coutrolled by a brake at B.
If the action is in the reverse direction, a driving engine must
be applied at T. A similar arrangement is much used in coal
mines which are entered by inclines. The chain is attached to
a fork on the cars.
The system of overhead cable tramway, which has been
brought to a high state of efficiency by Bleichert, is based on
the same principle as the preceding, but for much lighter loads.
The system consists of a cable tramway in which a stationary
cable is substituted for the trackway. The runuing cable is
commonly called the pulliug rope, and runs underneath the
stationary rope. The cars consist of a combination of grooved
sheaves, from which the bucket or other receptacle is suspended
by curved arms. The stationary cable is supported upon round
poles, and the arrangement of the stations is shown in the dia-
grams of Figs. 8030 and 803^.
counter-sheave, as in Fig. 795 b, to obtain increased tractive
power.
Fig. 804 shows a plan view of a double System.
At Ky is the motive power for systems / and //, and at K2 the
motor for system III. The driving sheaves are at 7, the coun-
ter-sheaves at Gy and the tension sheaves at Z/.
The supporting columns for the stationary cable must be
stiff, and often quite high.
a.	b.
___________Siandseil

Fig. 805.
Fig. 805 shows the forms used by Bleichert, a being used up
to 24 feet high, b for heights between 24 and 80 feet.*
In Fig. 806 is shown a combination of driving and guiding
systems in which the guiding and driving sheaves are combined
upon the car Q> and the tension organ is fastened at two points
So So on the path of the car Q.
Fig. 806.
The motive power is on the car and operates the shea\e 7.
In the form shown at a, a Fowler grip sheave is used at Ty this
form being suitable for a rope system, while the form shown ai
b is better adapted to be used with chain.
The system shown in Fig. 806 £ is also adapted for hauling
boats, and has been used by Harturch for operating the railway
ferry across the Rhine at Rhinehausen. The ferry boat in this
case is guided by a stationary cable securely anchored, as in
Fig. 807, the anchorage being up the stream, and the force of
Fig. 803*7.
The stationary cable connects with the suspended tramway at
SI SI1 and SnI S*K At So is the anchor of the stationary
o
n_
cable, with a tension weight at Z2. The driving sheave is at Tt
dnven by connections to the engine at and at L' is the ten-
sion device for the pulling cable. If the Service is heavy the
cable is carried twice around the driving sheave 7, using a
4--
Fig. 807.
the current keeping the cables taut. The equilibrium of these
forces enables this to act in the same manner as the stationary
cable of the Bleichert system, the difference only being that the
load, insteadof being suspended from the cable, exerts a lateral
stress. The driving cable is similar to Fig. 806 b> and is beneath
the surface of the water.
If we imagine, in the combination of Fig. 806, that the
traveling vehicle Q may be longer than the distance SQ SOJ
which is the full length of the tension organ, the principle will
not be altered, but the action will be modified, since the rela-
tions of the traveling vehicle and the tension organ are now
inverted. The ends of the tension organ can now be joined
* On the tramway at Iuker-Vashegy, poles of 140 feet high are used.176
THE CONSTRUCTOR.
together, or in other words itcau be made endless, and if heavy
enough, its weight can be caused to produce enough friction 011
the bed of the stream to furnish the necessary resistance. This
is the construction of Heuberger’s cliain propeller, Fig. 808, as
improved by Zede
Fig. 808.
T is the driving sheave for the chain, Z, Z, Z are guide
sheaves, Lx is a movable sheave to take up a portion of the
slack chain wlien passing into shallow water. The System is
made double, being placed 011 each side of the boat, and each
side is driven independently, so that sharp curves can be turned.*
If, in the case of a tension organ driven by a revolving pulley,
there is not sufficient tension given, the friction becomes insuffi-
cient to overcome the resistance of the load ; if the necessary
tension is externally supplied and removed periodically, a con-
tinuously revolving pulley can be caused to produce a lifting
and dropping action of a given load. This plan has been
adopted in some forms of drop-haminers, of which Fig. 809 is
the arrangement. Z is a pul-
ley running continuously in
the direction of the arrow, Q
is the drop weight, Ha handle
by which the operator applies
and releases the tension which
causes the pulley to drive or
slip.
The applications of running
tension organs which have been
thus far considered, are +nose
in^which the device has been
used either to lift weights or to
transport the same from place
to place. One of the most
important applications, how-
ever, is that of transmitting
rotative inotion from pulley to
pulley, an operation which can
be almost indefinitely repeated.
Fig. 809.
Fig. 810.
This combination includes ali numerous forms of belt, rope and
chain transmission, Fig. 810. The necessary tension for this
purpose is sustained by the journals and bearings of the pulleys,
also being modified by supporting or by tightening pulleys.
The two portions of the tension organ are distinguished as the
tight and slack sides respectively, and many modifications of
this forni of transmission are discussed more fully hereafter,
(see Chap. XX to XXII j.
There is one application, howrever, wThich is appropriately
discussed in this place, namelv, that in which rotative trans-
mission between pulleys upon stationary axes is combined writh
pulleys upon a movable member, thus enabling motion to be
transmitted from a stationary source to a moving body, Fig. 811.
a.	b.
In case a, one of the driven pulleys is mounted upon a car-
riage, saddle, trolley, or the like, and may be shifted in posi-
tion upon its ways or track; the tension is sustained by the
three guide sheaves. Applications of this forni, using belting,
are used upon planing machines by Sellers, Ducomtnun & Du-
bied and others. With rope driving gear it is used to operate
the spindles upon the carriage of the self-acting mule, also for
operating traveling cranes by Ramsbottom, by Tangye, and by
Towne ; being combined by the latter wdth the squaring device
as showTn in Fig. 784^, and effecting all the functions of the
erane, including bridge and trolley travel, as wTell as the hoist-
ing and lowTering of the load.
The forni of Fig. 811 b differs from a in that both sides of the
belt or rope are used to transmit powrer. The stationary pulleys
Tx and Z3 here drive the movable pulleys T2 and Z*. These
driven axes can be utilized in various manners, as, for example,
to operate a windlass device for the propulsion of the carriage
Q; an example of wdiich is found in Agudio’s cable locomotive.f
In this device the pulleys T2 and T4 drove a friction train which
operated a drum connected wTith a stationary cable as in Fig. 806.
A more recent device is shown in a modification of Fig. 811 a,
as shown in Fig. 8124
This construction, which is in use at the Soperga-Rampe at
Turin, consists of a double rack, placed between the rails as
showm at b, which also showrs the gearing by which car is
driven. The motive powrer is placed at the foot of the incline
at 7", G} the 500-horse powrer engine running continuously in
one direction. The cable is carried upon the overhead guide
sheaves Ll and passes around the pulley Z2, and through the
sheave system T T' of the locomotive, and is supported also 011
guide sheaves under the track, a tension pulley being placed at
Z7. The velocity of the driving cable is four times that of the
cars, and the descent is effected by gravity aione under control
of a brake. During the descent the bevel gears on the shaft of
the driving pulley are released by friction clutches at Zf, thus
rendering the car independent of the cable.
The foregoing condensed description is nevertheless fully
sufficient to indicate the extreme Service of which tension organs
arc capable in machine design. No less than seven Systems
have been shown for railway use, and four for boats. This is
the more significant since it will be remembered that cable pro-
pulsion had been abandoned for railway use, but yet appears to
now be revived with increasing success.
Our division into Guiding, Winding, and Driving systems
enables different devices to be placed in corresponding classes.
There yet remains to be considered the co-existing action of
many of the devices, such as pulleys, windlasses, cranes, etc.,
in which a negative motion may be given to the tension organ
by tlie descent of the load Q under the action of gravity. |
This action can be fully determined by reversing the previously
considered movement for the backw^ard motion. In the com-
mon belt transmission, Fig. 810, the action is reversible, as is
also the case writh the simple pulley, Fig. 794^-
The case is different, howTever, with the rope tackle Fig. 784^
and the differential block Fig. 796^, which are therefore here
considered in the more general form of Fig. 813.
«	If in these forms the
c	cord Z is pulled in either
direction the lower
sheave will be also
moved up or down pro-
portionally. At the pre-
sent time systems using
endless cords are under
j I	l(	0 consideration, but fre-
VJy g	^ S n	g quently choice is to be
yr\	made as to which por-
i I	i	[ l i\ tion is best used. It will
be seen that the system
of Fig. 806, which is
made with both ends of
the cable secured, can
also be considered as a
portion of an endless
system similar to Fig.
.	mi/	\»—jp	808, and other endless
\J^y	systems are found in
**	y Fig. 784 d and e\ also
Fig. 813 b, which differs
from a only in the run-
ning of the rope, the united ends being marked by a cross. If
i
Fig. 813.
* The following data of performance are given by Zede: Capacity, 500 tons;
length over all, 230 ft. ; breadth, ft.; depth, 6% ft.; midship draught, 31
in. The chains were of cast iron, weighing 275 pounds per yard- two en-
gines of 150 I. H. P. gave a speed of 3.72 miles (!) per hour.
!See Thomas Agudio. Memoire sur la Locomotive funiculaire, Tnrin, 1863.
See Bulletin de la Soc. d'Encouragement, Vol. XVI., 1869, p. 48.
Kinematic force closure. First discussed in the Author’s Theoretical
±4.mexnatics, p. 575.THE CONSTRUCTOR.
177
If we bring the applicatioris of Figs. 806 and 811 into a general
form in which the path of travel shall return upon itself, we
nave Fig. 814 a. If the guide sheaves are removed and the
c
Fig. 814.
cord crossed, the simpler forni of Fig. 814 b is obtained. The
rotation of the pulley Tx causes travel around the stationary
pulley Tr The old form of Agudio’s cable locomotive may be
represented by a similar diagram, Fig. 814 c. The shaded pulley
T2 is held stationary, while the concentric pulley Tz is assumed
to revolve ; this causes the System to revolve in a circular path,
the whole forming a differential or epicyclic System.
Finally it may be remarked that in electric transmission
Systems a similar analogy exists to the above combinations of
tension organs of wire and cable in various forms.
2 263.
Technological Applications of Tension Organs.
In addition to the preceding applications of tension organs,
they are also used in numerous forms of machine tools, i. e.> as
organs for the alteration of the form of bodies.
A straight blade of steel furnished with teeth forms the well-
known frame or gang saw used in numerous wood-working
machines. When made without teeth, and used wTith sand and
water, it becomes a stone-cutting saw, or in the form of a wire
charged with oil and emery or diamond dust, a saw for the
hardest materials, in which case a high tension must be given
to the wire to prevent lateral displacement. The saw blade may
be given a vibrating motion in a device such as Fig. 789 b for
use as a scroll saw. In all these cases a reciprocating motion
is used. Tension organs are also used as runniug members for
sawing, the form of Fig. 810 becoming the well-known band
saw. Very fine band saws have been made, and also saws of
wire, these having been used as long ago as 1877 by the writer,
suggested by the saws used for precious stones.
An ingenious form of wire saw has been made by Zervas for
cutting blocks of lava or stone from the original bed, as shown
in the diagram Fig. 815.
Fig. 815.
Two small shafts are sunk in the stone, and the guide pulleys
inserted as shown, the endless wfire being. fed down by the
screws. The cutting is effected by using water and sand, and
the cord is formed of three twisted wires, although more re-
cently a single smooth wire, with twisted one w^ound above it,
has been used, the outside diameters being to
A patent was taken out in Germany by Paulin Gay in 1882
for an apparatus for cutting a block of stone into slabs by the
use of a number of wire saws.
Polishing belts are another example of tension organs used as
tools, the flat side of the belt being used, impregnated with
polishing material. Such belts, used in the nickel-plating
establishment of Neumann, Schwartz & Weil, at Freiberg in
Breisgau, are operated at a speed of over 6500 feet per minute.
Tension organs are of frequent use in many details of spin-
ning machinery, acting both for guiding and winding; also in
numerous other forms of textile machinery.
Chains are especially useful for dredging machinery, working
in wet or dry material, also for handling coal.
In musical instruments we find tension organs of definite
dimension and stress, as sound producing machines.
? 264.
Cord Friction.
When a tension organ which is loaded at both ends is passed
over a curved surface, there is produced between the tension
organ and the surface a very consider;;ble sliding friction.
Since this friction will first be mathematically considered in
connection with the subject of cords, it will be given the gen-
eral name of cord friction. The curved surface over which the
cord is passed is the pulley, and the motion of the cord takes
place in the plane of the pulley. If the ^ension T on the
driving side of the cord is to overcome the cord friction Fy as
well as the tension t of the driven side, we have for the value
of the friction, F = T— t. It is dependent upon the magni-
tude of the angle of contact a and upon the coefficient of fric-
tion j\ but is independent of the radius R of the pulley ; it is
also dependent upon the iufluence of centrifugal force. For
these conditions we have :
T=tef°-{o — *)............................(237)
F— t (ef°- (1 — z) — 1)....................(238)
In these e is the base of the natural system of logarithms =■
2.71828, and z — 12
y v2
; v being the velocity of the tension
organ in feet per second, the stress in its cross section, y the
weight of a cubic inch of the material, and g the acceleration
of gravity = 32.2.
Example. - In the capstan shown in Fig. 794 a, let f — 0.21, a = 6 n = 3 con-
volutions, z — o. We then have f a = 0.21 X 6 X 3.14 = 3-958, say 4, and
F= t (2.7184 — 1) = t (54.6 — 1) = 53.6 t. This shows the friction upon ti.e
capstan drum to be nearly 54 times the pull upon the free end of theeord.
The influence of centrifugal force becomes important at high
speeds, and when the tension organ is under small stress. For
hemp or cotton rope, or for leather belting, we may take y =
0.035, and for wire rope about nine times so great.
The value of 6* in the formula z = 12
y ir
is properly con-
sidered a function of a, and wTe may therefore assume a con-
stant value for the arc a, and thus calculate the following table
for the values of 1 — z.
table.
s.	Value of Coefficient 1 — z for Centrifugal Force.					£
Hempen	Velocity of Rope in Feet per Second.					Wire
						
Rope.	20	40	60	80	100	Rope.
400 lbs.	0.987	O.948	0.882	0.791	0.674	3,600 lbs.
600 “	0.991	O.965	0.922	0.861	0.783	5,400 “
800 “	0.993	O.974	O.94I	0.896	0.837	7,200 “
IOOO “	0.995	O.980	0-953	0.916	0.870	9,000 “
1200 “	O.996	O.982	O.961	0.930	0.892	10,800 iL
1400 “	O.996	0.985	O.966	0.940	0.907	12,600 “
This table serves both for hemp and for wire rope by taking-
the ninefold value of S in the right hand column for wire rope.
It should be observed that the velocities are in feet per second.
It will be seen that for high speeds a high stress in the tension
organ is necessary in order to oppose the action of the centri-
fugal force.
In order to simplify practical calculations wTe may substitute
for the exponent /a (1 — z) in each case the form f' a ; that is,
instead of using the actual coefficient of friction f, taking an-
other one f, which is equal to (1 — z)f If it is a transmission
system, as Fig. 810, which is under consideration, the friction of
the cord, belt, chain, etc., must at least equal the transmitted
force P\ hence also must the stress be that of a cord friction
^ Pf which gives for a minimum value of T:
whence
(240)
Both 01* these values are absolute numbers.
The ratio p
indicates the amount of stress which must be given to the ten-j78
THE CONSTRUCTOR.
rsion organ, and hence may be called the stress modulus, and is
T
designated as r. The ratio —, we may, in like manner, call the
modulus of cord friction, and indicate as p. A series of values
for both are given in the following table.
Moduli for Cord Friction and Stress.
f'«	T P=T	k A II I-	S N a	T P=T	T 7 ~ P
0.1	I.I I	IO.4I	1.6	4-95	1.25
0.2	1.22	5.52	I*7	5.47	1.22
0.3	i-35	386	1.8	6.05	1.20
O.4	1.49	3-°3	1-9	6.69	1.18
0.5	1.65	2-54	2.0	7-39	1.16
0.6	1.82	2.22	2.2	903	"-J3
0.7	2.01	E99	2.4	11.02	1.10
0.8	2.23	1.86	2.6	13.46	1.08
0.9	2.46	1.69	2.8	16.44	1.07
1.0	2.72	1.58	3-o	20.09	105
I.I	3.00	1.50	3-2	24.53	1.04
1.2	3-32	M3	3-4	29.96	1.03
3-3	3.67	I*37	3.6	36.60	»03
M	4.06	r-33	3-8	44.70	1.02
15	4.48	1.29	4.0	54.60	1.02
Exatnple — Arc of contact = n ; coeffieient of friction f = o 16, velocity v =- 80
feet. The tension organ is a leatherbelt under stress of 400 lbs. persquareiuch.
We have from the first table 1 — z = 0.791, hencey' a = 0.791 X 0.16 rr = 3.976,
or n£arly o 4. From the second table this gives p = 1.49 and r = 3.03, that is,
over three tirr.es the above stress on the belt would be required to overcome
the frictional resistance. If v = 20 fit., the value 1 — z — 0.987, and f' a. —
0.496 or about 0 5, and the modulus of stress t = 2.54.
In order to make these relations more apparent, they are
shown graphically in the diagram, Fig. 816, in which the scale
upon the upper horizontal line gives the values for both moduli,
wliile the vertical scale on the left gives corresponding values
oi the product j7 a.
The superficial pressure p of the tension organ upon the cir-
cumference of the pulley mcreases as the belt or cord passes
from the slack to the tight side. It is equal to	in which
0' R d a
b' is the breadth of the surface of contact of the belt. Now for
any cross section q, the force Q — q S. Hence we have :
P_ = -q—
S b' K
(240
from which it will be seen that the pressure p can easily be kept
with moderate limits.
Special applications of this formula, and of the diagram, Fig.
816, will be given hereafter.
3265.
Ropes of Orgaxic Fibres.
Hemp Rope.—The form in most general use is a round hemp
rope twisted of three strands. This is twisted “loose” or
“tight,” accordiug as it is desired to be
more or less flexible. The cross section
of a threS-strand round rope, in wThich b
is the diameter of a single strand, is
3 — <J2, whence b bears the following
4
relation to the diameter d of the circum-
1'+-
>s 3oy
scribing circle: d = <5 ^1
= 2.15 d, Fig. 817. This gives for the
7T
cross section q — ——— d2. On account
6.16
of the spiral twisting of the strands, and
their compression upon each other, this
Fig. 817.
may be taken q — — d2, that is about 0.8 times the value of
the full cross section. Good hemp rope, wThen loosely twdsted,
wTill bear a stress of 1700 pounds, and when tightly twisted,
about 1 ^ times as much. For convcnience of calculation we
may assume the cross section to that of the full circle d, if, in-
stead of the full stress, wTe take only f as much, or 1400 lbs.,
and 2ico lbs. We then have for the force P, for:
loosely twisted rope d = 0.03	~P\ and P= mi d2 \
tightly “	“ d— 0.024 \/ P; “ P— 1677 d2 f
(242)
The radius R of the pulley should never be less than 3 to 4 d
for loosely twisted rope, and not less than 6 to 8 d for tightly
twisted rope, the diameter being measured to the centre of the
rope. For heavv Service, as for hoisting machines, R should be
not less than 25 d.
Flat hemp ropes are made bv sewing 4 to 6 round ropes to-
gether, each rope being then proportioned to bear J or J the
whole load.
The running weight GQ per foot is as foliows :
For loosely twisted rope, G0 — 0.325 d2 ]
For tightly “	“ G0 — 0.467 d2 >	.... (243)
and approximately for both P= 3400 GQ )
The latter assumption is based on the same number of fibres
in both cases. The following table gives values for three-straud
hemp rope.
Dia.	Loose Twist.		Hard Twist.	
d.	P.	S0.	P.	G0.
'A	276	O.081	397	O.Il6
%	621	O.183	893	0.263
7A	967	O.284	i."’89 1,588	0.408
1	1105	0.325		0.467
i*	1726	O.508	2,481	O.729
'A	2485	O.731	3.572	I.050
2	4420	I.3°0	6,351	1.868
*Az	6906	2.031	9.923	2.919
3	9945	2.925	14,290	4.203
According to (243) a rope L feet in length, hanging vertically.
is loaded —11— L of its working strength already by its own
weight. If L — 3400, the entire practical load would already
Fig. 816.THE CONSTRUCTOR.
U9
be applied, .and this may be considered practical working
length of the rope. We have for the available practical work-
ing load :	P' -\   L	P =	P or = P i — .
3400	V 34oo J
A vertically suspended rope will break by its own weight
when its length reaches about 2000 feet, since the modulus of
rupture is about 8500 lbs. for loosely twisted rope, and about
14,000 lbs. for tightly twisted rope. The above length (2000 ft.)
may be called the length of rupture. For a cord suspended
in the water, as for deep sea sounding, the length of rupture is
about twice as great. For very heavy stresses three simple
strands are iusufficient, and the strands themselves are each
made of smaller strands, as in cable construction. Very heavy
cables are also made of more than three strands.
Cotton Rope.—Cotton rope has been used of late for purposes
of transmission, and is usually made with three strands, very
loosely twisted. It opposes a resistance to rupture of about
75°° pounds, reckoning the full sectional area, and is operated
under stresses ranging from 1000 to 2000 pounds. It is used for
driving spindles in spinning frames and mules, and in the snail
drum movement, as in Fig. 787,* and is also used for operating
traveling cranes 011 the Ramsbottom System.
Driving ropes are usually operated over grooved pulleys, the
radius of the semicircular groove being slightly greater than
that of the rope. In machine construction the sheaves are
usually of cast iron, and in ship’s tackle they are made of
lignum vitae.
The sheaves revolve on cylindrical journals, and recently
jroller beariugs are being used, Fig. 8i8.f
When the pressure is moderate, the rollers may be made of
hard bronze, but for high pressures the rollers, ring and journal
■should all be made of hardened Steel. In case of extremely
high pressures bronze bearings with metaline may be used, the
metaline being a solid lubricant imbedded in recesses in the
box, Fig. 8194 Such bearings were used most successfully in
the construction of the East River Bridge at New York, oper-
ating for an entire year without requiring lubrication.
3 266.
Wire Rope.
Wire rope is usually round, and made with 36 wires, since six
strands are used, each containing six wires. Each strand con-
tains a small hemp core, and the strands are twisted about a
Central core of hemp. These hempen portions are of greatest
importance in the construction of wire rope for transmission
(see § 268), and should be made of the best material. For sta-
a.	b
f
i
tionary ropes the hempen strands may be replaced by wire,
giving 42 or 49 wires, and proportionally increasing the strength
* In a spinning mule of 844 spindles, by J. J. Rieter & Co., of Winter-
thure, a rope 22 mm. (0.866 in.) operates under a stress of 1.6 kg. (2275 lb.),
taking the full cross section.
f Martini’s design, used in the Italian navj*.
j John Wallace & Co., Eoudon ; Selig in Berlin.
of the rope. The strands of six wires may be combined to
make ropes of 48, 54, 60, 66, 72 wires, etc., and other combina-
tions are also used.
In Fig. 820 is shown at a a section of a rope of 36 wires, and
at b a diffeient form of 60 wires, both being made with cores of
hemp for the strands as well as for the ropes. For the external
diameter of the wire ropes of the preceding form, when the
wires lie in close contact, we have :
i = 36	48	54	60	66	72 'j
d	1
—^ = 8.00 10.25 11.33 12.80 13.28 14.20
in which i is the number of wires, d the diameter of a single
wire, and d the diameter of the rope.
In some later kinds of rope they do not lie in contact with
each other, but are separated slightly by the hemp, in which
case the diameter will exceed the previous figures by 10 to 15
per cent., but after a period of use the diameter becomes reduced
to the sizes given above. When the strands are made without
hemp cores they are arranged in the following manner: \
3	7	10	J4	16	19
wThile with hemp cores the numbers are
5	6	7	8	9	10.
The number of strands runs from 3 up to 4, 5, 6, which latter
is most used, and 011 up to 7, 8, 12, 14, 16, 19. For cables which
arc required to resist heavy stresses and also to possess great
flexibility, the same construction is employed as for hempen
cables, the strands themselves being composed of twisted ropes ;
the number of strands is 3, 4, 5 or 6. Flat cables are also made
of a number of parallel ropes. The number of ropes is 4, 6 or
8 ; the number of strands in each rope 4 to 6.
Example.—A heavy cable of Steel wire is made of 6 ropes, each rope of 19
strands, each strand containing 7 wires. The total number of wires —
6 X 19 X 7 — 798. Diameter of wire 6 — 0.055".
Well made rope is so wound that the load produces a uniform
stress upon all the wires, so that, when i — the number of wires,
Pthe load, S the stress 011 the wire, we have
P=Si—V........................................(2457
4
j . . . (244;
The diameter of wire varies from 0.04" to 0.14". If the ropt?
is required to be very flexible the wires should not be more than
o.i// in diameter.
I11 the passage of the rope over a sheave or pulley, of a tadius
R} the individual wires are subjected to bending, which, under
the action of tension and compression ^see \ 8), produces a
stress of a magnitude s = ~~j£t in which E is the modulus of
elasticity of the material. For steel or iron wire E may be
taken at = 28,440,000.
This gives 5 = 14,220,000	......................(246)
The stress s, which is produced on the tension side by bend-
ing, must be considered in connection with the stress N produced
by the load P} in order to arrive at the total stress. In order to
avoid a permanent set, it is necessary that the sum S-\- s should
not exceed the modulus of elasticity. The actual magnitude of
R becomes a minimum when s — 2 S\ that is, the stress due to
bending becomes double that due to the working tension.
Whatever may be the relation between the pulling stress S,
and the bending stress s, the total stress on the material will be
the sum S + *.
If it is desired to consider the security against rupture as well
as the possible overstepping of the elastic limit, the value of
S -|- s must be taken into account. The Prussian Government
rule places the modulus of rupture K if steel wire at 163000
pounds, or with a factor of safety of 6, the stress N =
163000
~6~~
= 27,166. If we take the case of a rope of 42 wires;, its diame-
ter d — iod, and making the pulley diameter === 75 d, we get
R — 37.5 d. This gives the bending stress, according to (246),
6*==
14,220,000 J
: 37,920.
37.5 X 10J
The sum S -{- s = 27,166 -f- 37,920 == 65,086. This gives an
163000
actual factor of safety of -7——- = 2.5.
65086
\ In American mining machinery, six strand ropes of 19 wires, with hemp
cores in the middle, are much used.i8o
THE CONSTRUCTOR.
The relation of the
stresses in the various
parts of the rope are
shown in Fig. 821.
On the right, the
tension side, there is
the tension stress ( -f- *S)
and the bending stress
(+ s)> giving a total _____
of 5* + s. On the left J
the tension stress (-{- 6")	\ s
is diminished by the
reverse bending stress
(— s). The neutral
axic is therefore shifted
from the middle at
N, to a point toward
the concave side of	Fig.	821.
the bent rope at 2V1.
Wire rope may be made either of iron or Steel wire, and its
fabrication has greatly advanced within recent years. The fol-
lowing data are applicable to the various grades : *
Material.	Elastic Limit.	Modulus of Rupture
Annealed Iron Wire .	42,000	56,000
Bright Iron Wire . .	56,000	80,000
Steel Wire			85,000
Steel Wire			142,000
Steel Wire			170,000
Steel Wire			213,000
Steel Wire			256,000
It will be evident that no general rule can be given as to
material, but that definite figures should be obtained for the
material to be used in each case. For high speed rope the wire
should be both smooth and strong, wTith a modulus of rupture
of about 170,000 lbs. If we then take a working stress 6* =
28,000 lbs., and a bending stress s = 28,000 lbs., we have S -j- «s
= 56,000 lbs., which gives about threefold security .f
14,220,000
For 5 = 28,000 we have R =------5------S = 500 6. If R is
28,000
made less, the security will be reduced ; if greater, it increases.J
The durability of the rope for mining servtce is increased by
galvanizing the wire.
For standing rigging of vessels galvanized annealed iron wire,
with a value K = 56,000 is used, while for running rigging Steel
wire rope (K = 170,000) is being more extensively used, this
also being galvanized. The latter rope is also suitable for
cables. Hawsers are frequently made from iron wire, with a
modulus of rupture K = 56,000 to 70,000. The cables for steam
plowing machinery should be made of the strongest Steel wire,
K — 256,000.
Wire Cables for power transmission are discussed in Chapter
XXI.
The cables for suspension bridges are not made from twisted
strands, but the wires are laid parallel and held in position by
bands of wire every two or three feet. \
* See the researches of J. W. Cloud on Steel wire in connection with the
Kmery Testing Machine at the Watertown Arsenal. Trans. Am. Soc. Mech.
Eng’rs, Vol. V.
f The Prussian rule requires S — —• K, which gives about 28,000, and R =
0
375 5» which gives s = 38,000, hence the security is only about 2^, or less than
given above.
Prscliibram has used with best results, .S = 23,000, j = 27,000; also 5 =
22,750, j = 36,000, but finds that a vali
preservation of the rope. (See § 268.)
diameter, the ratio to the diameter 5
to the diameter d of the rope.
t If -r- is made so small that S -f .
O
rope will receive a permanent set.
This, however, is not always dan-
gerous.
In Fig. 822 the curvature 1 . 1
may produce a stress upon the
concave side of the wires which.
when added to S, may not exceed
the elastic limit. If, however, a
reverse curvature be given, as at
3.3, there may resuit a set, as
3'. 3', and too Irequent repetition
of this reversal may become dan-
gerous. This is shown in the casa
of hoisting drums, such as Fig.
792 c, in which the rope	L2,
which is subjected to reverse bend-
ing, has been found to last only
about % as long as the rope Wi L\.
\ Among important suspension
bridges are those teuilt by Roebling
the East River bridges.
le s 27,000 to 28,0*0 lbs. is better for the
In considering the question of pulley
of the wire should be taken, not that
? is greater than the elastic limit, the
America, notably the Niagara, and
i 267.
Weight of Wire Rope and its Influence*
A rope of parallel iron or Steel wires, exclusi ve of any bands,
wdll weigh, per foot, 0.28
in which i is the num*
ber of wires and <5 the diameter of each wire. For twisted
rope, the twist and the hemp core increases this value from ij4
to 1% as much, or an average of 1 l/e times. This gives for the
running wTeight per foot
G0 = 3-92 — i = 3.07 i & <
4
(247)
This is also true for flat ropes, the value of the coefficient for
cable ropes being increased as above from \]/% to i#, usually
about il/e times. For deep mine hoists the wreight Go exercises
a marked influence upon the section of the rope. If L k is the
length in feet of the vertical hanging rope carrying a load P at
its end we have : P L Go = S
i <52, whence for ordinary
round wire rope :
(248)
Exampie 1.—Let the depth of shaft L = 1640 ft. Wire rope of Steel, K =
170,000, N = 28,000, P = 4400 lbs., and i = 36.
— X 4403
$2 = ------—-------------- = 0.0056
28,000 X 36 (1 — 0.229)
which gives S = 0.075. If L = O we get S~ = 0.0034, and^ = 0.058.
The above discussion enables us to determine the length Lt of
rope wrhich would produce by its own weight the stress .S in the
uppermost cross section :
= 0.25 5.......................................(249)
This may be called the load-length for the stress .S. Should
the shaft reach a depth equal to the load-length, no weight could
be suspended to the rope without exceeding the permissible
stress X If .S is equal to the modulus of rupture, the rope
wTould be broken by its own weight. This rupture-length may
be designated by Lz> and is
= 0.25 K.................................................(250
Exampie 2.—For round wire rope of uniform cross section the rupture-
length Lz is as below for the corresponding strength :
K	56,000	80,000	85,000	142,000
Lz	14,000	20,000	21,250	35i5°°
K	170,000	213,000	256000
L»	42,500	23,250	64,000
For very deep shafts it has been found advantageous to make
the rope a body of uniform resistance, which would make both
load-length and rupture-length unlimited. The formulae for
this purpose have been already given in \ 4. The taper to the
rope may be given in two different wTays. Fither a constant
diameter S of wire, and varying number 2, may be used ; or a
constant number i, and variable diameter 6. If the smaller
diameter of wire = 80, or the minimum number of wires = io,
we have for any depth x :
i	<52	x
log — or log = 0.4342945 y —.
Io	Oq
In this y is the coefficient of wreight wThich, for round rope,
wTe have found to be = 3.92. Substituting this value we get:
log L or log — i.68 --....................(251)
lo	00 1	S
Exampie 3 —If the value of .Sbe taken as 28,00®, we have for the following
depths:
x —	1000	1500	2000	2500	3000	3600
X ~S	0.036	0.054	0.0714	0.089	0.107	0.121
i io	1.115	1.123	1.3x8	i*4ii	1.512	x*597
S So	1.072	I.IIO	1.148	1.187	1.230	1.263
These values will serve to approximate the intermediate cases 5
H In the Prschibram mines taper ropes are in practical use. The rope in
the Adalbert shaft is as follows: P= 3850 lbs., 01 which 2200 is useful load,
R = 74.8", and the rope is mad«» in 7 sections of six part strands and eight
hemp strands.THE CONSTRUCTOR.
181
The great weight of the twisting rope has led to the use of a
double lift, each half of the rope assisti ug to counterbalance
the other half, or another plan is to use a conical drum, to
equalize the power.* The spiral winding of flat ropes also
serves to equalize the leverage of the drum, and by a judicious
selection of drum diameter, this may be very successfully done.
Flat ropes are little used in France, but are common in Bel-
gium, and their use is increasing in England and America.f
Ropes of copper wire are used for lightning conductors, and
these are also made of iron wire rope with a core of copper.
? 268.
Stiffness of Ropes.
The resistance of stiffness of ropes must be considered both
in hoisting and in driving ropes. The measure of this resistance
is the force required to move a rope hanging over a very easy
running pulley,. both ends of the rope bearing the given load Q.
It will be observed that the winding-up side of the rope does
not hangas closely to the pulley as does the other side, and that
the lever arm of the two sides is constantly changing. Eytel-
wein’s formula gives for the stiffness S of a hemp rope of diam-
ter d:
S = &j.Q.......................'..................(252)
in which, when R and d are given in inches, <5 = 0.463. Cou-
lomb gives the very inconvenient formula S —	^ ^
r + c2 a
Weisbach gives, from very limited data, for wire rope :
Q
R........................
.S = 1.078 -{- 0.093
(253)
Example /.—Given a hemp rope 1" diameter, with a load of 880 lbs., bent
over a pulley 4" radius, from EytelweiiFs formula we have :
880
S = 0.463	^	= 101.8 lbs.
which seems very high. Coulomb’s formula gives 66 lbs.
Example 2.—A wire rope, composed of 36 wires, each 0.039" diameter, with
a load of 550 lbs., is bent over a pulley 44 inches diameter. From Weisbach’s
formula we get:
5 = 1.078 + 0.093	= 3.403 lbs.
22
The utility of these formulas is doubtful, and a fuller investi-
gation of the subject is much to be desired. It will be seen
from formula (253) that for wire rope the value of R should be
taken stili greater than already considered for bending stresses
(formula 246) ; this subject is also discussed in Chapter XXI.
The above rui es are deficient in that they do not take into
account the kind of mechanieal work absorbed by the stiffness
of ropes. The angle embraced by the rope is, in the investiga-
tions of Amontons, Navier, Poncelet and Morris, assumed to be
constant, while in practice it is constantly changing, and exerts
a very material influence upon the resuit.
The author’s consideration of the subject is here given :
Referring to Fig. 823, it will be seen that the fibres or wires
on the concave side of
the rope which passes
over a pulley R, are
compressed, produc-
ing a reduction in the
form of the convex
side, the compression
originating w7ith the
load Qy being trans-
mitted along the
whole length of the
twisted strands. The
bent position of the
rope can no longer
retain its original sec-
tion, of diameter d,
but its volume must
be the same as that
of a corresponding
iength of the straight portion. The alteration in cross-section
The details are as follows :
:pth x.	Dia. of wire 6. Weight Go.		S.	J.	S -f- s.
3936	0.1043	1.52	23,210	27,260	50,470
3280	0.0984	1.36	22,910	25,710	48,620
2624	0.0925	1.19	22,820	24,170	46,990
1968	0.0866	i.oc6	22,860	22,640	45,5oo
1312	0.0807	0.912	23,*3°	21,090	44,220
656	0.0748	0.785	23,690	19,550	43,240
The twisting of the rope was commenced at the small end, and the diam-
eter of wires increased e very 5 meters (16.4 ft.) after the first 200 metres (656
fit). These ropes are very satisfactory, and last 3 to 4 years.
* Conical drums are used iu the American anthraci te coal mines,
f See Dwelsliauvers-Dery in Cuyper,’sRevue des Mines, 1874; also F. Krane
in Zeitschrift der Berg u. Hiittenwesen, 1864.
may be of two kinds ; first: uniform compression ; second,
when this has reached its limit, a flattening of cross section.
Both deformations are observed in practice. Ropes which are
very flexible are loosely twisted, and therefore readily com-
pressed as they pass over pulleys. The general compression
due to the tension of the load in the straight portion causes the
twisted strancsto pre^s firmly together towards the axis, so that
a heavily loaded rope is very hard. The compression is gener-
ally permanent, and not elastic, as may be deduced from the
permanent reduction in diameter of ropes after use, and is gen-
erally due, in the 1 ase of wire ropes, to the compression of the
hempen core ; as is shown by the observations of Eeloutre and
Zuber %
The preceding remarks have not considered those wire ropes
with metallic cores, used for running transmissions. Such ropes
are always very stiff, and permit little or no compression.
(According to Ziegler’s experiments, only 0.22 to 1.2 per
cent.)
It is really almost as important, so far as flexibility is con-
cerned, that a rope should have a suitable soft core as that it
should be made of the best and most elastic and flexible mate-
rial. This is shown by the fact that even with ropes made en-
tirely of hemp or of cotton, and used for transmission over
pulleys, the inner fibres, which are never in contact with the
pulleys, show great wear. This wear is evidently due to the
friction of the fibres against each other, due to the flattening
and changes of cross section. For this reason the desirability,
or rather necessity, of lubricating the wires or fibres is evident,
and this reduces the friction of the inner-lying portions of the
rope. Rieten & Co. state that in the case of cotton ropes, “the
rope always wears out by the internal friction of the strands
upon each other, and that a load-twisted rope becomes useless
in a shorter time than a soft, loosely twisted one, although the
actual strength of the latter is the smaller.”
In view of ali these conditions the insufficiency of the exist-
ing rules for stiffness will be evident. It is apparent that the
angle of contact must have a strong influence, and an entrance
is found in cable roads where, when the cable is deflected
through a small angle, small guide rollers are satisfactory, while
much larger ones are necessary for greater angles. At a certain
angle a, the deformation of the rope begins, and at another
angle a maximum is reached, beyond which the resistance of
stiffness is no longer dependent upon a. These pomts are of
greatest importance with wire rope. It must be expected that
the value of X will depend upon two functions of «, one for
compression, and one for flattening. The first may be unim-
portant with old and compressed ropes, the latter will be much
dependent upon the lubrication and upon the coefficient of
friction.
S 269.
Rope CONNECTIONS AND BUFFERS.
The connection of one rope with another, when a smooth
junction is required, must be effected by splicing. This may be
accomplished by the short or German splice; or by the long, or
1	2	3 M 4	5	G
6'	5'	4'	3'	2'	1'
Fig. 824.
Spanish splice. The latter is the form to be used for wire rope.
From the middle point M of the splice, Fig. 824, if, for exam-
ple, a six strand rope is in hand, strands 1, 2 and 3 on the left
are unwound, and strands 6', 5/, 4/, of the other rope wound in,
and the ends cut and worked iu. The same is done on the
other side, the whole length 1 — 6 being 30 to 50 feet.
To connect the end of a rope to another portion of the con-
struction, the so-called hangers are used; three fornis being
shown in Fig. 825. At a is the so-called “swan neck,” which
is secured to the rope by through rivets ; b is made with a coni-
cal socket, the wires being doubled up, and soft metal melted
and run in ; c is Kortunfs hanger, the rope being held by two
toothed wedges driven in, and secured by pins. Numerous
tests have shown this fastening to be as strong as the rope
itself.
In Fig. 826a is shown a buffer coupling used in the Zentrnm
mine at Eschweiler, designed by the superintendent, Oster-
t See Leloutre, Transmissions et courroies, cordes et cables. Paris, TignoI,
1884. Ziegler, Erfahrungs-resultant uber Betrieb and Instandhaltung der
Drahtseiltriebe, Winterthur, 1871.182
THE CONSTRUCTOR.
katnp. The wrought iron thimble in the bight of the rope	$ 270.
i? fitted with a wooden block. Fig. 826 b shovvs the so-called	0	~
s	Stationary Chains.
a.
b.
J
Chains may be considered as jointed rods. Running chains
are composed of very skort members, in order that they may
the easier pass over sheaves, while stationary chains, which are
used in bridge, and other numerous coustructions, are made
with quite long links.
Fig. 827 shows tlie'Admiralty form of stationary chain. Tha
links are made % fathom long, not including the thickness of
rnetal, and are divided into 10 fathom lengths, each length con-
Fig. S25.
“ friction hanger,” both this and the previous form being
arranged to be built into the upper part of the hoist cage.
Fig. 826.
I11 Osterkamp’s design the spring cage is built into the yoke of
the frame, thus economizing room.
sisting of 20 links. The lengths are jomed by a pin connec
tion, showu on the left, and the pin is made of steel, galvanized
Another form, known as the
Gemorsch chain, is shown in
Fig. 828, and is well known in
Germany.
Each long link is made 1.5
metres long, and these are co 1-
uected by short oval links. The
coupling link is secured by a
commoti, but heavy screw bolt.
The proportions in the illus-
trations are given in terms of
the diameter of the rod.
In order to enable such chains
to hang freely, the so-called
“swivel” is used. A heavy
swivel, for chains such as Fig.
827, is chosen in Fig. 829.
The swTivel bolt has a ring
attached which can be readily
opened, and is large enough
to receive two chain links,
while the upper ring can re-
ceive three. The limit of di-
mensions is the thickness of
metal of the chain of Fig. 827.
1	1
§ 271.
Running Chains.
a
The most important forms of running chains used in machine
construction are those shown in Fig. 830; a is an open link, and
b is a close link chain ; c is a stay link chain, and d a flat link
chain. This latter is especially suitable for a pitch chain, on
account of the parallel pins which are at uniform distance from
eacji other. The other three forms are made with a higher
order of linkage, viz.: the globoid form already discussed in
Fig. 224.
In the wide open link chain a the globoid action can readilyTHE CONSTRUCTOR.
183
be disarranged ; less so in the close links of b, and hardly at all
in the stay-link chain c, which latter closely resembles the
globoid link of Fig. a, p. 142.
The proportional dimensions of chain links are not very
closely determined. Those given in b and c are from the Ger-
man Admiralty. The British Admiralty gives both for open
and for stay-link chains, the pitch length 4 d, and width of link
3.6 d; in France, for open chains the length is made 3.25 d, and
width 3. 4 d, and for stay-link chains, 3.85 d and 3.75 d respec-
tively.*
In erane and hoisting machine construction, a very important
feature is the calibrating or adjusting of the links of chain.f
This is also a matter of much importance in connection with
the chain propulsion of boats nsed in France and Germany.
The chain used on the Sweetwater canal at Suez was made with
1
and a pitch of 3 d, and breadth 3 2 d. The Magde-
16
burg-Bodenbacher chain is very strong, d being given
15'
16
to
the links being proportioned as at b.
Flat link chains have been used by Neustadt, made of multi-
ple plates (see \ 94). The plates are made of the best quality
and the pins made to project a little, and riveted over cold.
Chains of this sort are also used for driving where heavy resis-
tances are overcome, as in wire drawing, and in some spinning
machinery.
I 272-
Cabcubations for Chains.
The chains which are made at the best 1 establishments are
always thoroughly tested, every link being subjected to a stress
closely within the limit of elasticity, or in some cases, slightly
exceeding the elastic limit. A few links, usually three, are
taken at frequent intervals every few weeks, and broken in the
testing machine. The usual proof-load is such as to give the
following stresses:
S = 20,000 lbs. per sq. in. for open link chain.
= 25,000 “	“	“	“ stay “
The tests of chain for the German navy give for 6*:
17.000	lbs. test of elasticity,
19.000	“ highest test,
25,600 “ proofload,
38,400 “ breaking load for
three links,
for open links;
for stayed links.
For hoisting chain the elongation should be considered, and
the metal should show an elongation before rupture of upwards
of 20 per cent.J
The permissible working stress per square inch section in
Germany § is:
For open link chain = 9,000 lbs.
For stay link chain = 13,000 lbs. ||
From these we get for the proper total load P:
For open links, P — 14,000 d1 >	, x
For stay links, P— 21,000 d2 j	^ ^4;
Flat link chains are subjected to the heaviest stress at the por-
tion which is in engagement with the toothed chain-wheel.
(See Fig. 837.) For this reason there should be not less than
five link pins in gear with the wheel at any time. If we assume
that the tooth pressure is in arithmetic progression as 1 :2 : 3 :4 :5
the pressure on the body of the last pin will be y P, and on
each journal also % Py they being impelled forward by y2 P.
If we put as a maximum stress in the bolts of 17,400 pounds,*
we have for the thickness of plates <?, pin diameter d, and num-
ber of piate i, for a given load Pt the following values:
* The length of flat links in Fig. 830 a is given as 5 + 2.8 a, and the projec-
tion of the ends as 2 4- 1.4 d. These are in millimetres, and for inches the
values o 1875 + 2-8 d, and 0.08 + *-4 d should be used.
t Kxcellent pitch chain is made at the Guttehoffnungshiitte at Oberhausen;
also by Schlieper at Iserlohn, and by Doremieux at St. Arnaud, and Plinchon
Havez at Guerigny, and by Hawkes Crawshay at Gateshead on Tyne.
X At the Guerigny Works the required elongatiou is:
For rods j%"	to	1"	 .18	per	cent.
For rods 1"	to	x/in..........................16 per cent.
For rods	.................................14	per	cent.
For rods X5S"..................................12 per cent.
For rods	.................................10	per	cent.
\ At the Gutenhoffnunghutte.
|j Henry R. Towne, Treatise on Cranes, Stamford, Conn., gives a permissi-
ble stress of 9,000 to 10,000 pounds.
or
^0.0107^
1 -i-1
d — 0.0063	up
-J- = °.58 (i + 2)
(255)
The thickness <1 is made the nearest convenient value, and %
must be a whole even number. For the latter we may take the
nearest wdiole number to the value given by the relation:
i=o.26yjP...................................(256)
The following table has been calculated from these formulae.
The metal for the plates should be especially tough. Neustadt’s
chains had an ultimate resistance of four to five times the work-
iug load.
Example i.—An open link chain of iv iron, according to (254) should have
a working load P — 14,000 lbs., while a stay link chain of the sanie iron
would permit a working load P — 21,000 lbs.
Example 2.—Required to proportion a flat link chain to carry 22,000 pounds,
We have from (256I i = 0.26 P= 0.26 •<§/ 22,000 = 7.32, say 8. Then in (255)-
.	0.0107 \/ 22000	.	.8 + 2	/--—	„
0 =----— ------= 0.176 ' also d = 0.0063 — —- \P 22000 = 1.04 The pitch
length l = 0.1875 + 2.8 X i-°4 = 3.1"; the.width of plates = 2.6 X 1.04 = 2.7";
the length of the body of the pins is 0.25 -f- 1.67 X 1.04 = 1.98, say 2^, the
diameter = 1.2 X 1 04 — 1.25" and the length of link beyoud the pin centre
is o 08 4- (0.9 X I.°4) = I-°2 say 1".
Table of Flat Link Chains.
Working I,oad P.	No. of Plates i	Thickness of Plates | 8	Breadth of Plates V.	Dia. of | Pin | d. {	Pitch /.
1,000	2 !	O.I25	0.625	0.25 ;	0.875
1,5°°	4	0.093	0.75	0.28	o-93
2,000 i	4	0.109	0.875	0.34	1.14
3,000	4	0.125	1.0625	0.40	I-375
4,000	4	O.I40	«•1875	0.46	i-45
6,000	6	0.I09	>,4375	0.56	H625
8,000	6	O.I40	1.6875	0.68	2.00
10,000	6	O.156	1875	o-75	2.3125
12,000	6	O.I7I	2.00	o-93	2-375
16,000	8	I 0-156	i 2.375	0-93	2.50
20,000	i 8	0.171	! 2.625	1.00	2.8125
1273.
Wfight of Chain.
The length of rod required to make a chain of a given
length L bears the sanie relation to L as the length y for a sin-
gle link does to the pitch l. We have for the chains a, b, c cfi
Fig. 830:
	Open	Close	Stay	Stay Pinks,
	Iyinks.*	Pinks.	Pinks.	including stay.
s 'J “	11.33	942	II.94	I3-25
T =	2.52	2.69	2.39	2.65
From these reiations the weight of iron rods required may be
determined (see \ 82). The greater the pitch of chain for a
given weight of iron, the more economical is the form of com
strue tion.lf
The load length and rupture length for chains (see $ 267) have
been extended sin ce that subject has been given practical con-
sideration, this being especially the case with anchor chains (see
next section). For this we may take the modulus of rupture
K at 37,000 lbs. for open links and 38,000 lbs. for stay links,
w?ith a modulus of safety T — 20,000 and 24,000 lbs. respectively.
We then have:
T
Lt ~ ------------and
6.29 X > X ^
l
K
6.29 X 7 X
I
cubic inch of wrought iron = 0.27 lb., and hence
, 7 being the weight of a
f The pitch for stay link chain in the German navy was formerly 3 d,1
but has recently been made 4 d.184
THE CONSTRUCTOR.
Open Iyinks. Close Links. Stay Links.
Lt =	4672	4377	5458
Lz =	8672	8127	8567
8 274.
Chain Couplings.
Chains which are used for transmission of niotion (so called
<l endless ” chains) require devices for coupling, as do also those
constructions with which chains are to be connected, and hence
we have a variety of eyes, rings, coupling links, swivels, and
the like.
a.
b.
FlG. 831.	FlG. 832.
A piece which is sometimes used with anchor chains is the
•so-called “twin” link, Fig. 831. This may be made of cast
steel, and because of limited space is formed with circular opeu-
;ngs. The ordinary coupling link is shown in Fig. 832 a. The
link is of wrought iron, the bolt and pin of Steel, both galvanized.
The pin is shorter than the diameter of the eye, and is secured
on both sides by a plug of !ead. The next link is made some-
what longer than the other links of the chain, so that the
coupling link may be more readily introduced. This. forni is
used for joining pieces of chain to form greater lengths. The
German Admiralty anchor chain is made with stay links, in
seven lengths of 25 metres (82 feet) each, joined with coupling
links, two of which are swivels. A bow anchor chain is given
two more lengths of chain and made of iron 3mm. (0.118")
thicker.*
The chains for the system of boat propulsion are fitted with a
coupling 1ink with rounded edges, and two are used together,
as in Fig. 832 b> which shows the chain used on the Elbe. This
coupling might also be suitable for power transmission chain.
The swivel is used to permit the chain to have a rotation
about its axis of length without twisting the links together.
Fig. 833.
The form of swivel used in the German Navy is shown in Fig.
833 and at Fig. 833 b is shown the English swivel.
* The lengths in the Hnglish Navy are 12^ fathoms.
Chains must also be provided with hooks for attachments to
the load to be raised.
A single hook is given in Fig. 834«, and a double hook at
Fig. 834 b. The construction of such hooks demands the great-
est care, and according to Glynn, more lives have been lost and
damage incurred by the breakage of hooks than by any other
part of a erane. The case is one of combined resistance and
leads to unexpectedly great dimensions.
The diameter dx of the shank of the hook may be obtained
from formula (72), so that we have for a load P:
dl = 0.02 P...........................(257)
This is based upon a stress of 3500 pounds, but an angular
pull may increase this five-fold. Taking dx as the unit, we may
obtain the proportions given in the illustrations in the following
manner. Let zv be the width of the opening of the hook, and
h the width of the body of the hook, the thickness at the same
point is made % h, and for a stress of 12,800 lbs. upon the metal
of the hook we have :
h	w . 5	h	_ ^ f w , 5
= I.30 V N- + ~~ OV —— = 0.026 V -7- +-f* • * •
dx
y/ P
(258)
The thickness at the point of the hook is made —, and hence
the outside of the hook is a circle of diameter D = w -f- 1.5 h.
We then have for :
70 £ T = c.6	0.7	0.8 0.9	z.o	1.1 1.2	i-3	1.4	i-5
h -T- = 1.77 di	1.82	1.86 1.91	4-95	1.99 2.03	2.08	2.12	2.16
h -7— = 0.035 >/ P	0.036	0.037 0.038	0.039	0 1 0 0 i	0.042 0.042		ro tT O d
w . — — 1.06 di	1.27	1.49 1.72	I-95	2.19 2.44	2.70	2.97	3-24
D ~T = 3-72 di	4.00	4.28 4.59	4.88	5.18 5-48	5.82	6.15	6.48
^ . . . w The most useful ratio is -7- = h			= 1.	In wharf cranes a w			eight
is often combined with the hook in order to facilitate the
lowering of the empty chain. This is shown in the dotted lines
in Fig. 834 A In the case of a double hook each portion is cal-
culated for its component Px of the entire load P. From this a
special unit d/ is obtained only for the dimensions w, hf and D.
Ejrantple^—Tfit the load upon a hook be 4400 lbs. We have from (257) dx
= 0.02 x/ P — 0.02 v' 4400 - 1.326". If we take w — h we get from the above
h = 1.99 X 1 326 = 2.638", and 7u is the same ; while D — 2.638 -f 3 957 — 6.6".
In the case of a double hook the angle between the coinponents is 600 ; we
then have P\ — -°-	= 2540, whence d\ = 0.02 \/ 2340 = 1.008
1 cos 300	0.866
say 1". If we make — = 0.9, h = 1.91 and w = 1.72. D = 1.92 4- 2.86 = 4.58.
For the upper portion we have as above dx — 1.326".THE CONSTRUCTOR.
185
? 275.
Chain Drums and Sheaves.
Chain drums and sheaves are usually made of a radius R =
10 to 12 d} measured to the middle of the chain. In some cases
a rim is made on the chain sheave, as in Fig. 835 a.
Fio. 835.
This form of sheave brings a bending action upon the linksas
shown in Fig. 835 b. Sometimes the flanges are omitted and the
edges of the sheaves bevelled as in the dotted lines, and in other
cases the links havea bearing as shown at Fig. 835 c, in which
the bending action is somewhat reduced. The bending is en-
tirely avoided, however, by the use of a pocketed sheave, as in
Fig. 836.
This form is useful both for chain Transmission, and as a sub-
stitute for wdnding drums in hoisting machinery, as it enables a
small pocketed sheave to serve instead of a large drum. When
such a sheave is made with only four pockets, they form a
square with a side D' = l + d 2(l—d) v/°.5~ 2.414 l—
0.414 d; while the side of the square of the alternate links is
D" = 1.414 l + 0.414 d. The first gives the minimum, and the
second the maximum, (double) lever arm with which the chain
acts upon the sheave. If the pockets, instead of 4 and 4 are :
6 and 6, we have D = 3.732 / — 0.264 d
8 and 8,	“	“ D — 5.026/—0.198^.
Chain sheaves of this form require accurately made pitch
chain.
When the load is heavy, the friction causes the chain to cling
to the sheave, and a stripper S, Fig. 836, is required to lead the
chain ofF in the proper directiou R, while the entrance is pro-
perly effected by a guide channel E.
For flat link chain, a toothed chain wheel is used, Fig. 837.
Fig. 837.
In this form a guide channel E, and stripper S, should alsobe
used. The tooth profile is a circular arc with its centre at the
link pin. If z, be the number of teeth, we have for the radius
of the pitch circle :
l
1800
2 sin —
z
(259)
whence we get, for
z =	8	9 10	12 14	16	18 20
II j-h	1.3066	1.4619 1.618	1.932 2.247	2563	2.879 3.106
The minimum number of teeth is 8.
Neustadt uses the following :
z — 8 for P — 500 to 6,000 pounds.
z =r 9 for P — 6000 to 50,000 pounds.
z =. 10 for P= over 50,000 pounds.
Guide sheaves for either kind of chain are made with 16 to
30 teeth.
For chain propelling cables ordinary smooth drums with
parallel axes are used, with a groove for the chain.
In Fig. 838 a is shown a section
b.
Fig. 838.
of the rim 01 the drum on the chain propelling gear on the
river Elbe. This is made with steel flanges and channels on a
wrought-iron rim. The last channel is made slightly larger ir
diameter in order to give a higher velocity to the driving sideoi
the chain. The wear upon the chain is an important item. Fig.
838 shows a link of a chain as worn after long Service. It
must not be overlooked that the winding around the drum pro-
duces a twist in the chain, giving as many half twists in the chain
as there are half convolutions about the drums. This twisting
is not injurious if the chain is bent as frequently in one direc-
tion as in the opposite. In fact, however, the chain is /isually
bent into more concave than convex bends. This causes a twist-
ing motion to the chain and as it drags upon the botfcom and
banks of the stream it produces much wear, and causes kinks to
be produced at the shallow places. The chain must therefore
frequently be raised at such points and a link open^d and the
twist taken out.
This twisting may be
prevented by using the
drum arrangement
shown in Fig. 839. This
consists of simple
drums ali lying in one
plane driven by gear-
ing so that the proper
relative motion is com-
pelled.
Fig. I39.
? 276.
Ratchet Tension Org^ns.
Tension organs may be combined with pawls, which in the
case of cords are friction pawls, ($ 248, 249), and for chains are
toothed pawls, acting upon the links iu the same manner as
upon ratchet wheels and ratchet racks.
The establishment of Felten & Guilieaume, at Miilheim a.
Rhein, have devised a grip pawl for boat-cable driving, in which
the rope is clamped to and released from a driving drum by an
evolute shaped thumb clamp, the shock being reduced by a
spring buffer.
Pawls for chains may be found used in connection with the
heavy bow an chors of large vessels ; Bernier, of Paris, has also
used such devices upon chain hoisting machinery.i86
THE CONSTRUCTOR.
CHAPTER XX.
BELTING.
i 2?6.
Self-Guiding Belting.
Belt pulleys are indirect acting friction wheels (£ 191) and the
belt itself is a tension orgau combining the functions of driving
and guiding (§ 261). Those belts which act without requiring
the use of special guiding devices may be called self-guiding
belts. This action is attained by the use of cylindrical pulleys
when the edge of the prismatic belt runs in a plane at right
angles to the axis of the pulley ; or in other words, when the
middle line of the advaucing side of the belt lies in the plane of
the middle of its pulleys.
When a belt runs upon a conical pulley in a direction normal
to its axis, its tendency will be to describe a conical spiral path
upon the pulley, as will readily be seeu upon the examination
of the development of the surface of the cone, Fig. 840.
If the pulley is made with a double cone face or a rounded
face, Fig. 841, the tendency will be for the belt to ruu at the
middle of the face even when the direction of the belt is not
exactly correct.
For leather belting, with a height of the crowning or curva-
ture of the face 5 = E of the width of face, the belt may devi-
ate from the plane of the pulley by 2)4° (tau = four per cent ),
while for cotton belting, ou account of the lesser elasticity of
the material, the crowning 5 should not exceed of the face,
thus reducing very materially the permissible deviation. I11
ordinary circumstances at least one of a pair of pulleys should
be made with rounded force.
• a.	b
The simplest arrangement of self guiding belting is that for
parallel axes, Fig. 842 a and b, a being for open belt and b for
crossed belt, ehher arrangement being suitable to run in either
direction.
For inclined and intersecting axes self-guiding belts are not
suitable, except in the case of inclined axes in which the trace
X S, Fig. 843, of the iutersection of the plaues of the two pulleys
passes through the points at which the belt leaves the pulleys.
The leading line then falis in the middle plane of each pulley,
but the following side of the belt does not, hence such svstems
can only be run in one direction. The leaving points in the
figures are at a and bv The arrangement gives an open belt
when the angle (3 between the planes of the pulleys — o°> and a
crossed belt when (3 = 1800. In the intermediate positions a
partial Crossing of the belt is produced. If /3 = 90°, the belt is
half crossed (or as commonly called, quarter twist); if fi = 450,
it is quarter crossed.*
*Theabove geometrical construction is only approximate; for an exact
solution see a paper by Prof. J. B. Webb, Trans. Am. Soc. Mech. Eng rs
Vol. IV., 1883, p. 165.
The leading off angle may be made as much as 250, which
occurswhen the distauce between the axes is equal to twice the
diameter of the largest pulley. Another rule for the minimum
distance between shafts for quarter-twdst belts is to make the
distauce never less than’v/ b D.
I 277.
Guide Pulleys for Belting.
When a belt transmission is arranged with guide pulleys, the
proper guiding action is obtained when each guide pulley is
placed at the point of departure of its plane with that of the
next following pulley .f
In Fig. S44 examples are given of guide pulleys for parallel
axes, ali three pulleys lying in the same plane.
At a is shown a belt transmission with tightening pulley, b is
a device for transmitting motiou when great difference of speed
is desired. In this case the guide pulley Cis as large as the
driver A, and if desired may also be arranged to act as a tight-
ener.J At c is Weaver’s device for similar uses.$ In this case
two belts are used, and the device has been used for driving
circular saws. The pulleys should be fitted to run very smoothly
in such devices.
The cases in Fig. 845-846 have parallel axes with two guide
pulleys. In the first case the guide pulleys are placed in planes
tangent to both operating pulleys, and hence driving may occur
in either direction. Usually, however, it is required to provide
f See also the paper of Prof. Webb, referred to in the preceding note,
t Eckert’s patent (German) for driving the drum of a threshing machine
\ See Cooper’s Use of Belting. Phila., 1878, p. 171.THE CONSTRUCTOR.
1Z7
Fig. 851 shows the general case for inclined axes. Two
points c and cx are chosen 011 the line of intersection S S the
planes of the two pulleys, and the tangents c a, c b, cx alt cx b±
for motion in but one direction, in which case the second forni
is used as being simpler of installation. The pulley B may be
used as one of the guide pulleys,	!■
in which case it may be placed
loose upon the same shaft as A,
and C or D be made drivers or
driven.
By placing the guide pulleys
between the axes of A and B,
instead of beyond them, they
will revolve in the same direc-
tion, and may be made fast
upon one shaft, as in Fig. S47;
this arrangement admitting of
motion in only one direction.
In Fig. 848 is an arrangement
for inclined axes, which is a
modification of Fig. 846, as will
be seen by the dotted lines.
The guide pulleys run in oppo-
site directions, but may con-
veniently be placed upon the
same shaft.
In Fig. 849 is shown an arrange-
ment of quarter-twist belts with
guide pulleys.. One side of the belt is placed in the intersec-
tion S S oi 'the planes of the two pulleys. From any point c
on .S S, the tangents c a and e b are drawn, and in the plane of
4 these the guide pulley C is placed. This arrangement permits
of rotation in either direction.
Another arrangement for the same purpose is shown in
Fig 850. The side of the belt leading off from A is inclined
towards B, the other side passing over the guide pulley C,
which is in the same plane as A and .S 5*. This arrangement is
well adapted for driving a number of vertical spindles from one
horizontal shaft.*
drawn, and in the planes of these tangents the guide pulleys
C and Cx are placed. Under these conditions the rotation may
be in either direction. The arrangement shown in Fig. 852
occurs when the line 6* .S passes through the middle of one of
the pulleys.
A simplification of the general case occurs when, as in Fig.
853, the guide pulleys fall upon one and the same geometrica!
axis which is parallel to the axes of both transmitting pulleys.
In this case the only inclination of the belt is that given to it
by the guide pulleys. The rotation can be in but one direction,
viz : that shown by the arrows; if the reverse is desired, the
guide pulleys must be placed as shown in the dotted lines.
If the inclination of the shafts is too great the belt will be
liable to drop off when the pulleys come to rest. The use of
guide pulleys involves special hangers, a practical forni for
which is shown in Fig. 854.1
* An example is Jacob’s grinding mill with 40 sets of stones; see UhlancTs f Patented in Germany by the Berlin-Anhaltischen Maschinenbau-Aktien'
praks. Masch. Konstr., 1868, p. 83, 1869, p. 242.	Gesellschaft.i88
THE CONSTRUCTOR.
The vertical axis is provided with an oil hole, and is fitted
foy a ball and socket bearing to tbe bracket D.' The flauge on
the lower edge of the pulley keeps the belt from falling off the
pulley when at rest. The form in Fig. 855 was designed by the
author for the arrangement of Fig. 848, both pulleys being loose
upon the wrought iron shaft.
If the position of the shafts can be so chosen that the line
.S .S* touches at least one of the pulleys, the very practical
arrangement shown in Fig. 856 can be applied. If the distance
^4 C is great in comparison with the width of beT., the pulleys
C and Cx can be placed side by side instead of over each other,
-Fig. 857, in which case round face pulleys should be used.
Fig. 858.
By the use of a fifth pulley the preceding arrangement may
be so modified that two pulleys, Bx and B2, can be driven from
one driver, A. This is shown in Fig. 858 as applied in a spin-
niug mill, in w7hich the pulleys Bx and Z?2are on different floors
of the building, and are also provided with loose pulleys.*
* See Fairba Mills and Millwork, II., London, 1863, p. 103. For the
*heoretical discussion of these various arrangements, see § 301.
parallel shafts, one of which intersects its axis at right angles,
the other passing beneath.
Another arrangement, devised by the author, is given in Fig.
860.	I11 this case the following side of the belt is passed over
an idler pulley, Cx or C2, and a second time around the driver
(see also Fig. 795) by w7hich the angle of contact a is doubled,
and the modulus of friction e/*	264) iucreased. This may be
called a double-acting transmission. The cross section of belt
may be made x\ of a single acting transmission, so that in spite
of the increase of length an economy of belting is obtained.
One of the guide pulleys may also be ased for a tighteuer.
These devices will also be considered in connection with rope
transmission (Chapter XXI.) to which they are especially appli-
cable.
§278.
Fast and Loose Pueeeys.
Fast and loose, or tight and loose pulleys, as they are some-
times called, are generally used in connection with another belt
transmission in order to throw the latter in and out of action,
the belt being guided by a belt shifter, which by the means of
forks or finger-bars, enables the moving belt to be shifted.
These shifting devices may properly be regarded as guide
pulleys, and are sometimes fitted w7ith rollers, as shown dotted
in Fig. 861, at c and cQ.f
It is preferable to have the loose pulley upon the driven shaft,
since the belt then can be shifted with a gradual spiral action
by the shifter Fig. 861. It is best for the driving pulley to
be made straight face, or if two fast pulleys are used side by
side on the driving shaft, these should have very slightly
rounded faces, if the belt is to be shifted promptly and readily,
and for the same object the shifter should be placed as close to
the driven pulleys as possible. The loose pulley should be kept
thoroughly lubricated, and for this purpose numerous oiling
devices have been made. The friction between the hub and
shaft acts as a driving force upon the loose pulley, and this has
been a source of numerous accidents. This action is avoided
in the arrangement in Fig. 862, in which the loose pulley is
carried 011 a consecutive and stationary sleeve D.%
A variety of mechanical belt shifting devices have been
made, § the desire being to prevent the action of the belt
from moving the shifter. A useful form is Zimmermann’s
Shifter, Fig. 863.
f Such rollers as especially necessary for shifting cotton belts, which are
liable to catch on the shifter'fingers, and even larger rollers are best in such
cases.
t See Berliner Verhandlungen, 1S69, p. 127. This has been used by the
Society for Prevention of Accidents, of Mulhouse.
g See Berliner Verhandlungen, 1868, p. 171, Rittershaus, Belt Shifters.THE CONSTRUCTOR.
189
The shifter bar F, to which the fork G can be clamped at any
desired point, is operated by the lever H, which turns upon an
axis at /, forming a ‘ ‘ dead ’ ’ ratchet mechanism. The similarity to
the ratchet devices of Figs. 754 and 755 will be observed. The
movement of the bar is effected by connection at K or Kv
Fig. 864 shows a shifter for quarter-twist belt. In this form,
devised by the author, the guide pulley, which is required to
support the belt, also serves as a shifter to move the belt to and
from the belt pulley B, and loose pulley BQ. If these pulleys
are given greater width than that of the belt, as shown on the
right, a vertical adjustment can be given to the upright shaft;
a condition sometimes required in grinding mills and similar
machines.
i 279*
Cone Pueeeys.
When a number of pulleys are placed side by side in order to
enable varied speeds to be obtained with belt transmission, and
are united together in one member, we obtain what is called a
cone pulley, such pulley being used in pairs. This construction
involves the problem of determining the proper radii for the
various pulleys, so that the same belt shall serve for all the
changes, i. e., so that the length of the belt shall be the same
for each pair of pulleys in the set. The problem may be solved
as follows :
a. Crossed Belts, Fig. 865. The belt makes the angle /3 with
the centre line of the pulleys R and R1; and the half length of
the belt, l—R (t+/0 +*1 (t+/3) + a cos /3, a being
the distance from centre to centre of the pulley. We then have :
/=(* + *,) (J-+/3) +a i1 ~• • (26°)
This value is constant when R -\- R1 is constant; that is,
when the increase to the radius of one pulley is equal to the
decrease in the radius of the other. Crossed belts are seldom
used for this Service, however, because of the injurious friction
between the rubbing parts of the belt.
d. Open Belts, Fig. 866. In this case we have :
/= (.R + Ri) -J- + (R-Ri) P + a cos p,
and also a sin p = R — Rly yrhich gives :
R =	~ (P sin P + cos P) -f- sin p
R1 = —-------— (P sin p + cos P)--------— sin P
7T	7T	2
(261)
This function is transcendental, but may be graphically repre-
sented in the following manner, Fig. 867. In the rectangle
A B B' A', with a radius A B — a, strike the quadrant B M C
about the centre A. Within this arc will fall all the values of
P which can occur. For any value of P = C A M} draw M N
perpendicular to M A and make M N = the arc M C — a p.
Drop the perpendicular M P to A C, and draw N O perpendicu-
lar to M P. NO will then = a p sin p. Through N drawr
Q N K parallel to A B, and we have A Q = P Q A P= a
{P sin p 4* cos P). By taking successively all the vacues of p
between o° and 90° in this manner, we can determine the path
of the point N, which will be the evolute of a circle, C N D
7r
B D being equal to the length of the arc B M C = — a. If we
now draw D E parallel to B A, and take its middle point Ft we
have D F == E F = —, and hence the proportion :
2
D F:D B= —
2
— a = a : 7r, and by similar triangles :
T K= — QA = —^ sin /3 + cos /3).
7T	7T
This value is dependent upon —. If we prolong B Nuntii it
7T
intersects A C prolonged, the resulting length A A' — B B'
will bear to Af B' the ratio By then working B G = /, and
l
drawing G H parallel to A' B\ we have G H = —. This
7r
l	a
length being transferred to IR gives IT—------------------(p sin p
7r	7T
cos P). We then have only to U9e ± ~ sin /3 to solve the
problem.
ci	a
Make A R — —, and we have the perpendicular R S = —
sin p. By laying this length off above and below T on Q Ky
we obtain the points U and Vy and this finally gives / U for the
radius R of the larger cone pulley and / V= Rv the radius of
the corresponding smaller cone pulley.
By Solutions for successive values of /3, we obtain the curve
D U X VE} which can be used for the determination of the
radii of any desired pair of pulleys, each pair of ordinates
measured from HI belonging to corresponding pulley on each
cone.
In practice it is usual to find one of the cone pulleys given
and the dimensions of the other required. In this case V U
may be taken as the difference R — Rl, between the radii, were
the steps uniform. By taking this difference R — RY in the
dividers, and finding the equivalent ordinate U V on the curve,
and then adding VI = Rlt the axis HI is found.
In order to use the curve conveniently, it may also be laid off
left-handed, as shown in the dotted lines D/ X E'.
The use of the diagram will be rendered stili more convenient
if we omit the unnecessary value L This enables us to distort
the curve in the direction of the abscissas to any desired extent.10o
THE CONSTRUCTOR,.
ofF toward Cf the corresponding radius X d and
prolong the axial line d d' to its intersection dr with
B E. Then lay off the given geometric ratio on C X,
considering Xd as i (shown in the diagram by the
small circles for the ratios J,	f, f, f), and draw rays
from df through the points of di vision, and these rays
will intersect the curve at the correspouding points
for the pulley radii Rv We then have for the radii:
a i and a i/ for the ratio i : 4
b 2 “ b 2f
c 3 “ c 3' ‘
d X “ d X/ ‘
e 5 “ cs' 1
c 0 ‘k c 0/ ‘
2	: 4
3	: 4
4	•* 4
5	: 4
6:4,
Cone pulleys may also be made continuous, thus
becoming conoids upon which the belt can be shifted
to any point by au adjustable guide or shifter. Such
conoids are used for driviug the rollers in spinniug
machinery. Such a pair of conoids are shown in
Fig. 869, the proportions having been determiued by
the graphical scale. The angular velocity varies in.
au arithmetical ratio as shown.
The curve E V A in the scale shows the limit to
which the axial line may approach A E; this dis-
tance must not be less than R Rx = a, from wThich
V Y= i (AB— VU).
? 280.
Cross Section and Capacity of Beets.
A belt of rectangular cross section of width b, and
thickuess 6t will be subjected to a tension T on the
tight side (see $ 264), which it must be proportioned
to sustain. If 6* is the permissible stress for the unit
of cross section, we have, therefore. T = b b S.
The minimum ratio w7hich T bears to the trans-
mitted force P is dependent upon the stress modulus
r, since T= r P (g 264). But r = ——, in which p
represents the modulus of friction efa. Hence, if X
is the horse power transmitted for a belt speed of v
C4-	-f	u	at	bbSv
feet per minute, we have: N =----------=----------.
33000	33000 r
This enables us to determine the cross section of the
belt, but in practice the width of the belt is the varia-
ble factor, the thickness usually being determined by
commercial considerations, and limited to few defi-
nite sizes.
If we let q represent the cross section of the belt
in square inches, we have:
33000r
This has been done in the proportional diagram for cone
pulleys, Fig. 868.
The method of using the diagram is as follows :
The sides A B and D E of the rectangle represent the dis-
tance a between the centres of the pulleys ; ali radii are given
in proportional parts of a, for which reason A B is sub-divided,
the size of the diagram being selected so that A B = 18 to 20
readily be done. If, for example,
inches. If, then, 1 «and i'
a are two given radii for a
pair of pulleys on a pair of
cones, we take the vertical
chord of the curve which
= 1 'a — 1 a, prolong the
chord downward until its
length = 1 a, and draw the
axis ab c d parallel to A E.
Then for the other pairs of
pulleys on the cones, we
have b 2 and b 2', 03 and
cy, etc., which can be
taken directly from the dia-
gram with the dividers. If
the given pair of radii to
which the cones are to be
made equal, the chord R —
R1 — o, and the axis will
pass through X at right
angles to C X.
If it is desired to construet
a pair of cone pulleys to any
given speed ratio, this can
ie given ratio is 1 : 1, we lay
This formula is very useful, since it may be used to determine
the capacity of a belt from its cross section and velocity. m If we
put N0 = — we have:
q v
L_.A
33000 T
(262)
The value depends upon the material and stress modulus, the
latter including the arc of contact a, and upon ft which itself
depends upon the material of both belt and pulley ; it may also
be considered as dependent upon c, independenf of the material,
in the same manner as was the subject of specific weight. The
author has called this value NOJ the specific capacity or a belt.
It will be seen that when this specific capacity is determined
for any kind of belt, the proper cross section for the transmis-
sion of a given horse power N can readily be found, since the
velocity v can be chosen, and we have at once
q
JV_
NQv
(263)
For the determination of the specific capacity of any kind of
belt it is necessary to find the constants S and r.
The materials used for belting are:
Tanned ieather,
Cotton, woven and treated wifh oil,
Rubber, interlaid with linen or cotton webbing.
In practice the value of 5“ to be used must depend much uponTHE CONSTRUCTOR.
judgment, the value being governed to a great extent by the
quality of the material. Customary values are for:
Leather ........ S — 4000 to 6000 lbs.
Cotton..........S = 3000 to 4000 lbs.
Rubber..........S — 3500 to 5000 lbs.
The thickness d for single leather belts varies from J3S// to ;
<louble, triple, quadruple, and eveu quintuple thicknesses being
sometimes used, the thicknesses being secured by cernent, gmd
sewed or rivetted together. Cotton belts are usually from -^I//
to thick, while rubber belts are made of any desired thick-
ness, a web of cauvas being interlaid between the successive
thicknesses of rubber.
The stress modulus r depends upon a and f, and the latter co-
cfficient varies with the age of the belt, being greater with belts
which have been used some time than with quite uew belts. It
is advisable, however, to make all calculations as for new belts,
in which case we have for smooth iron pulleys, for:
Leather and cotton, f = o. 16 to 0.25* *, p = 1.6 to 2.1
Rubber,	f— 0.20 to 0.25, /) = i.8to 2.1
These give as approximate values for .
T
Leather and cotton, — or r = 2.5 to 1.9
Rubber,	— or r = 2.2 to 1.9
By using these values together with those given for S, in (262)
we get for the specific capacity for belting :
Leather, N0 = 0.0062 to 0.0098 I
Cotton, N0 = 0.0036 to o 0068 V (265)
Rubber, N0 = 0.0050 to 0.0082 J
These are based upon low and moderate speeds ; say up to
3000 feet per minute, and the variations between the ltmits given
are those due to thedifferences in strength of various kinds of
leather and canvas used.
The resistance to bending or stiffness of a belt must be taken
into account, and the ratio of thickness d to pulley radius R,
must not be too great. Practical experience has shown that
-f- — — should not be exceeded to obtain best results.*
R 50
From the known stress and the thickness of the belt the
superficial pressure p, betw?een belt and pulley may be cal-
culated. We have only to substitute in (24r) for the width b'
of Jhe surface of contact, the width b of the belt itself, and
sin 'e q — b d, we get the simple relation :
= A
S R
(264)
Exantple 1.—Required a leather belt to transmit ioo H. P. and the speeds
of pulleys to be n =*= 80 n\ = 150 revolutions. Taking the specific capacity at
0.007 and the lineal velocity of belt at 3000 feet, we have q---------- = 4.8
3000 X 0.007
sq. in. cross section.
If we ust a double belt 0.4" thick, the width should be = 12 inches.
r n	04
For the driving pullev we have: 2 U--------- = v, and R = ^>° ^ T- = 7i~
& ^	12	2 tt n	'	‘
Bo y F z
say 72", or 154 inches. For the driven pulley we have R\ = —----------- = 38.4".
For the superficial pressure p, we have P =	IC° = J1»3 lbs. Also
T — 2.5 P= 2750, hence S1 —
- = 573* We have also t — 15 P = 1650,
0.4 X 72
which gives S2 — 343, or a mean of 458 lbs., which in (264) gives a mean value
458 X 0 4.
72
2.5 lbs. on the large pulley, and p = 458 ^ °~4 = 4.98, or
38.4
nearly 5 pounds. This is verified since, if f = 0.16:
72 X 3 T4 X 12 X 2.5 X 0.16 = 1100,
which is the value of Pas above.
Exatnple 2.—Whathorse power can be transmitted by a cotton belt 4 inches
wide and 0.25" thick, at a velocity of 2000 feet per minute ? Taking the speci-
fic capacity at 0.006, which has been found satisfactory in practice, we have
from (262) N — q v Nq = 4X 0.25 X 2000 X 0.006 == 12 H. P.
Exampte3.—A rubber belt is required to drive a centrifugal pump (rubber
"being especially adapted for damp locations). Ar=2o, the pump vane to
make 300 revolutions, and the driving shaft 80 revolutions per minute, and
the belt speed 2000 feet. Taking the specific capacity at 0.007, we have 20 =
q X 2000 X 0.007, hence q — 1 43 sq. in., and if we make 8 = 0.2 we have a
2000 V I2
width b = 7.1". For the driven pullev we have Ri = ------------- = i2."i, say
‘	"5> TT IOO	J
* For cotton the thinner belts from 0.25" to 0.4" are preferable.
191
12%", and for the driver R = - 2,-75-8^' - 3°- == 47.8". A mean value of S is 425
lbs., whence p = --5 *~g°‘2 = 1.78 on the large pulley and 425	= 6 7 on
47.0	12 71
the small pulley.
For extraordinary cases the fundamental formula should
always be applied. For double-acting belts, as in Fig. 860, in
which a = 2 tt instead of 7r, the value fa = 1, and the modulus
of stress is only 0 6 of the preceding value, hence q is reduced
in the same proportion. If the belt velocity v is very high, it
is 110 longer permissible to neglect the influence of centrifugal
force. For a speed v = 5000 feet and a stress .S* = 568 pounds
(see § 264) the exponent in the friction modulus becomes 0.84^0
instead of f a} which for f — 0.16 and a = 7r, gives f' a = 0.84
X 0.16 7r = 0.42. This gives r = 2.91 or about | of the normal
value, which requires one-sixth greater cross section q for the
belt. The highest limit of belt speed in ordinary practice
appears to be about 6000 feet per minute.f
l 2S1.
Exampfes of Beet Transmission.
The table of existing examples of belt transmission on next
page will serve to furnish data lor comparison with calculated
results.
The great variations in the values of .9 and N0i in the fol-
lowing table are not surprising when the differences in the
quality of material, and the various conditions are considered.
Many leather belts are working under high stresses which are
only practicable because of the excellence of the material.
Some such belting can be operated under stresses as high as
2000 pounds, which enables much lighter sections to be used.
Many belts which appear to have been excessively heavy have
simply been calculated to work at a moderate stress.
The plausible but erroneous idea that the pressure of the
atmosphere influences belt action cannot be admitted. It is
contradicted not only by the fact that the same coefficient of
friction exists for ropes as for belts, but also by the recent
and careful experiment made in a vacuum by Leloutre which
confirmed the theory of the modulus of friction.
i 282.
Beet Connections.
The various methods of connecting the ends of belts generally
give a greater stress at the point of connection than in the body
of the belt. The attempts to reduce this weakness and also
provide for the greatest facility in the making of the joint, has
caused a great variety of methods to be proposed ; some of the
best of these are here given :
In Fig. 870, a is a lap joint sewed wdch hempen thread ; b, a
lap joint secured with screw rivets \ c is a piate coupling, the
piate and prongs being made in one malleable casting and the
prongs bent over and clinched after insertion in the belt, several
clamps being used for belts more than 4 inches in width. At d
is shown belt lacings for use with single or double belts. The
upper one has the defect of giving intersections which make
the lacing cut itself, and the knot at the edge of the belt reduces
the strength of the joint.J These defects are both avoided in the
lower form, which is an American belt lacing.§
f I11 the construction of the Arlberg tunnel a hoisting machine was used
in which the belt had a velocity of 4700 feet per minute, which worked well
for fourteen months.
X Eeloutre has used the Kpper form of lacing for a belt of 26" wide, 0.66*
thick with excellent performance and durability.
I See Cooper, Use of Belting, p. 189.192
THE CONSTRUCTOR.THE CONSTRUCTOR.
193
Fig. 871 a, shows BSttei^s belt fastening. This is a forni of
belt hook which has been found very serviceable, reducing the
strength of the belt but little, and permitting easy renewal.
Another form is Moxon’s belt fastening,* shown at b, is a pin
a.	b,	c.	d.
point, the ends of the pin being riveted over, and from its con-
struction should be very strong. At c is a butt joint with a
reinforcement piece especially suited for cotton belts. When a
belt is made for special serviee it can be in several layers as at
d ; the joints overlapping, but thus giving no opportunity for
change of length.
The stretching and joining of heavy belts is a matter requir-
ing much care in order to secure the desired tension, = yz
( T + /) f Belts which are subjected only to light tensions may
be cemented by scarfing the ends and using a cernent composed
of common glue mixed with fish glue, or of rubber dissolved in
bisulphide of carbon,
§ 283.
The Proportions of Pueeeys.
Pulleys are usually made of cast iron and of single width, i. e.y
one set of arms. The arms, which formerly were made curved,
in order to resist the stresses due to contraction, are now made
straight, and for wide face oulleys two or even three parallel
sets of arms are used.
FiG. 872.
Fig. 872 shows both single and double arms. The dimensions
of arms and rim have been determined by experience, based
upon practical considerations. For the nutnber A of arms for
a single set, we get serviceable values from :
A = yz
(266)
which gives, for :
R
—- = i 2	3	4	5	6	7	8	9	10 ij 12	13
0
.4=34567	8	9.
The width h of the arm, if prolonged to the middle of the
hub, may be obtained from :
b 1 R
■ h — 0.25" + — + ^	...............(z67)
The width h1 of the arm at the rim is equal to 0.8 //, and the
corresponding thicknesses are e = y h, and ex — y hx.
Pulleys with two or three sets of arms may be considered as
* See Chronique industrielle, 1882, VoL 5, p. 97 ; also Mechanical World,
1882. Vol. 12, p. 56.
;Leloutre has used a dynamometric belt-stretcher for tensions of %
/) = 8800 pounds.
two or three separate pulleys combined in one, except that the
proportions of the arms should be 0.8 or 0.7 times that of single
arm pulleys, or in the proportion of yj ^ and yjH
The thickness of the rim may be made : k — i to y /2, this
being frequently turned much thinner. The width of face
should be from f to f the width of the belt.
The thickness of metal in the hub may be made W = h, to
y h. The length of hub may = b, for single arm pulleys and
2 b for double arm pulleys. Light pulleys are usually secured
to the shaft by means of set screws, as in Fig. 875 and 877 ;
heavier ones are keyed as in Fig. 191, either with or without
set screws.*
For many purposes pulleys are made in two parts, such being
commonly called “split pulleys. The forms of split pulleys
are shown in Figs. 873 to 875.
The arrangement of the two
halves is clearly shown, that
of Fig. 874 with hollow clamp-
ing section, being especially
good.f
The form in ' Fig. 875 is the
design of the Walker Mfg. Co.
ofCleveland, Ohio, the clamps
being made of malleable iron
or Steel. In all three cases
there is 110 especial method of
fastening to the shaft. I11
England and America pulleys
are frequently made with
wrought iron rims and cast
iron hubs. This construction
greatly simplifies the casting
of the arms, and at the same
time gives pulleys 25 to 60 per cent. lighter than those of cast
iron, which in large transmissions greatly reduces the friction
1
Fig. 874.
at the bearings of the shafting. Fig. 876 shows the Medart
pulley. The rim is curved in bendiug rolls, and also given a
rounding face, and is
countersunk for the
rivets at the attach-
mentof the arms. The
pads on the arms are
truly finished, as is
also the rim after it is
riveted on, thus giv-
ing an accurate and
useful pulley.J
A metal pnlley by
the Hartford Engin-
eering Company 6o7/
diameter and \W' face
weighed 320 pounds.
A cast iron pulley of
the same dimensions
madebytheBerlin-An
halt Works, weighed
700 pounds, and one
by Briegleb, Hansen & Co., a little narrower face weighed 528
pounds.
Fig. 873.
* In order to deterraine the necessary friction to secure a pulley to the
shaft, the force/> on the belt will serve. In ordinary cases, assuming a co-
efficient of friction on the key of one*half that on the belt, there should be
a pressure/»' on the key of about 4000 times that on the belt, which, accord.-
ing to l 20 will not give* more than 5000 to 7000 lbs. for p'.
j-This is the construction of the Berlin-Anhalt Machine Works,
t Made in England by George Richards & Co., Manchester.194
THE CONSTRUCTOR.
Fig. 877 shows Good-
win’s split pulley, with
wrought rim, the face of
the rim being rounded by
turning.
These constructions
naturally led to the use of
wrought iron arms also,
although these are some-
what difficult to make;
but for very large diame-
ters (say 16 to 25 feet)
they possess advantages.*
Pulleys made entirely of
Steel are used by J. B.
Sturtevant of Boston, in
connection with fan blowT-
ers, Fig. 878. The hub
■with web, is screwred on the steel shaft of the fan wheel, and
the rim, which has a groove turned in it, is expanded by warm-
ing, and shrinks into place,
the whole being finally
turned in position, and care-
fully balanced. Sturtevant
uses these pulleys up to 10
in. in diameter, and 7 in.
face, the thickness of rim
being from 0.08 to 0.16, and
the velocity at the rim reach-
ing 5°°° feet per minute.
By covering the rim with
leather the co-efficient of fric-
tion, y^andcan beincreased
betwreen the belt and pulley,
and the modulus of stress
r reduced, and the specific
capacity of the belt increased.
This is sometimes useful be*
cause a reduced moJulus of
stress r permits a smaller
cross section of belt and lighter pulley. I11 large transmissions
reduction of stress is important since it is accompanied with
reduced journal friction
and higher efficiency.
The observation of the
author leads him to be-
lieve the specific capa-
city of a belt is 11 ot
greater with leather
covered pulleys than
with uncovered ones,
and the cost of covering
is an important item.
The greater the angu-
lar velocity of a pulley
the more important it is
that its geometric axis
should be a so-called
“ free axis.” This re-
quires that the center of
gravity of the pulley
should be on the axis of rotation and also that the various por-
tions of the mass should be so distributed that the axis of
inertia should coincide with the axis of rotation and tbe centri-
Tugal moment equal zero.f This can be done empirically by so-
called balancing, the unequal distribution of material being
cqualized by attaching pieces of lead or other metal, or more
accurately by balancing when revolving, for which purpose a
beautiful appafatus has been inade by the Defiance Machine
Works, Defiance, Ohio. Careful balancing of pulleys is of great
importance at high speeds, the rapidly increasing vibrations
will soon limit the speed. This isto be considered in connection
Avith the advantages to be gained by the use of high speed
shaft as discussed in £ 146.
NoTE.—The recent investigations upon paper rim pulleys f
are instructive. This construction gives a very high modulus
of friction, the modulus of stress r being only 1.2. This gives
T — 1.2 P as agaitist 2.5 Py for iron pulleys. Hence follows a
great increase in the specific capacity of the belt, and. increased
efficiency with smaller and lighter pulleys. This leads the way
to further investigations which prove of material value in the
Science of belt transmission.
Fig. 877.
* Pulleys with wrought iron arms are made in Germany bv Starck & Co.,
Mainz ; in England by Hudswell, Clark & Co., Eeeds, these latter with
arms of round bar iron.
fSee an article by the writer, “ Ueber das Zentrifugal-Moment,” in Ber-
liner Verhandlung, 1876, p. 50.
X See Am. Machinist, May 23, 1885, p. 7.
? 284.
Efficiency of Beeting.
Three causes of loss exist in belt transmissions, viz.: journal
friction, belt stiffness, and belt creeping. For horizontal belt-
ing we have, according to formula (99) for the journal friction,
expressed at the circumference of the pulley a loss Ez when 7
=2.5 Pf t = 1.5 P:
(26S)
in which d and dx are the journal diameters, and f the coefficient
of journal friction. This loss is doubtless the greatest of the
three. For lack of better researches the loss of belt stiffness
may be deduced from Eytelwein’s formula for ropes. For the
coefficient of stiffness 5, for force S', which includes both pul-
leys ;
P
£.-sS±±
-s p
= 45
(269)
in which s = 0.009 — = 0.012.
7r
The loss from creep is due to the fact that the greater stress
on the driving pulley over that 011 the driven requires for a
given volume of belt, a longer arc of contact ; for the expendi-
ture of force G' for creep on both pulleys, we have for a stress
Si on the leading side of the belt:
__t
G' z, 1 ~f	0.4 S,
——E	- £ + St
1 + ~sT
■ ■ ■ (270)
In this E is the modulus of elasticity of the belt, which for
leather is 20,000 to 30,000 pounds. The losses from stiffness
and creep are small.
Example.—Eet d and d\ = 4"; R = R = 20", 5 = 0.2, f — 0.08, S = 0.012,
E = 28,440, Si = 425, we have E1 = P ^ ^	— X 0.4 —« 0.08 P\
also S- = P (0.048 X 2) . ?:2- = 0.0048 P,
20
and G1 = P -	* 425 - = 0.0059 P-
28,440 -f 425
The total loss is therefore : 0.08 + 0.0048 -f 0.0059 = 9.1 per cent.
CHAPTER XXI.
ROPE TRANSMISSION.
I 285.
Various Kinds of Rope Transmission.
If in the tension driving gear, shown in Fig. 810, the rope be
used only for the transmission of power wTe have what is called
a Rope Transmission. Since the details of construction must
vary, according as fibrous or wire rope is used, we may distin-
guish betweeu three kinds of rope transmission, viz. : those for
Hemp, Cotton or Wire Rope, and these wfill be considered in
this order. The oldest of ali these is hemp rope transmission,
but this was gradually being superseded by belting until
Combes, of Belfast, revived it, about 1860, since which time it
has been extensively used for heavy transmissions. The char-
acter of the material permits a wride variety of applications.
The same is true of cotton rope, which is extensively used for
driving spinning frames, travelling cranes and many other ma-
chines, the softness and flexibility of the material giving it ad-
vantages, but within limits. Wire rope transmissions, since its
introduction by the brothers Hirn, at Logeibach, in 1850, have
developed a high degree of efficiency and utility for long dis-
tance transmission. As will be seen hereafter, the applications
of rope transmission appear to be capable of stili further ex-
tensiou.THE CONSTRUCTOR.
195
A. HEMP ROPE TRANSMISSION.
\ 286.
Specific Capacity. Cross Section of Ropf.
It is important first to determine the specific capacity for
hemp rope (| 280). This is obtained from the general state-
ment according to (262) :
in which SY is the stress on the tight side of the rope, and r the
modulus of stress. The value for the co-efficient of friction fy
depends upon the form of the groove or channel in the sheave
over which the rope runs.
a.	f>	c.
Fig. 879.
If the groove is semicircular, as at b, Fig. 870, the friction is
but little greater than it is upon an ordinary cylindrical pulley,
as at a ; if, however, the groove is made wedge-shaped, as at c
(see wedge friction wheels § 196), the driving power is increased
although the surface of contact is reduced. In determining the
value of r, from formula (239) the influenee of the shape of the
groove can be included by using a corresponding co-efficient of
friction fK According to the recent investigations of Feloutre
and others, the value off for cylindrical pulleys and new hemp
rope is 0.075, for semicircular grooves, 0.088, and for wedge
grooves with an angle of 60°,/* — 0.15, which accords well with
the action of the wedge, doubling the pressure, see (185). For
Jl = 0.088 and a contact of a half circumference, we have /l a
= 0.3, and hence r = 3.86 ; with fl =0.15, fl a = 0.47, and r
= 2.67. The latter value, which is eveu reduced in actual
practice, may be adopted, since wedge grooves in general use.
The stress is usually taken while low, and may be put at 6* =
1	350
350 lbs., which, taking r = 2.67, gives N0 =---------. —— =
&	&	33000	2.67
0.0039 \ see (262). In practice N0 is found even one-half this
value, and we may take as a practical rule in hemp rope trans-
mission for the specific capacity, i. e, the horse power trans-
mitted per square inch of cross section, for each foot of linear
velocity per minute;
N0 — 0.004 to 0.002.........................(271)
•
the cross section being taken as in § 265, as that due to the full
outside diameter of the rope.
When great power is to be transmitted a number of ropes are
used side by side, the pulleys being made with a corresponding
number of grooves. For machine shop transmission such
ropes are conveniently made about two inches in diameter,
although they are used as small as 1%, and as thick as 2^
inches.
Example 1. A steam engine of 60 H. P. has its power transmitted through
five ropes of 2 inches, the pulley being 11.28 feet diameter, inaking 45 revo-
lutions per minute. This gives v = 1592 feet per minute. The cross section
of the rope 3.14 sq. inches. Hence N0 = E- --------------—---------- 0.0024. This
5	1592 X 3-14
is taken from an existing installatiou.*
Example 2. In the jute mills at Gera the fly wheel of the engine is grooved
for 30 ropes, o 1 2.36" diameter, each rope transmitting 25 H. P. ; the velocity
being 3000 feet per minute. This gives a specific capacity of NQ =
25
■------------= 0.0019.
3000 X 4-375
Example 3. The Berlin-Anhalt Machine Works has design rope transmis*
sions in which ropes of 1.18", 1.57", 1.97" diameter transmit forces, respec-
ti ve ly, of 92.4, 165 and 264 pounds. The respecti ve cross section s of the ropes
*	N	P
are 1.09, 1.93 and 3.04 square inches. Since---------= --------we have iVn =
7/	'lonnn	u
------which gives in each of the three cases NQ — 0.0026.
33000q
The cross section of the rim of a pulley for five ropes is
shown in Fig. 880. For large steam engines the grooves are
sometimes made on the
fly wheel, such con-
structions sometimes
being very large and
heavy.f
The application of
rope transmission in
manufacturing estab-
lishments simplifies the
mechanism very ma-
Fig. 880.	terially, since it enables
the jack shaft and gear-
ing to be dispensed with.
Such an arrangement is shown in Fig. 881, in which five
different lines of shafting are driven from one horizontal steam
engine, sixteen hemp ropes being used in all.
1287.
Sources of Loss in Hemp Rope Transmission.
The use of hemp rope transmission reduces many losses
which exist in other methods and which materially reduce the
efficiency ; the principal ones which need to be considered are
the resistances due to journal friction, stiffness of ropes, and
creep of ropes.
a. Journal Friction.—In rope transmissions from steam en-
gines the journal friction is usually great, because the large fly
wheel requires journals of large diameter. The usual calcula-
tions can only be given by indeterminate results, because the
tension of the ropes sometimes acts with the weight of the
other parts, and sometimes against it.
If we consider the rope tensions T and t by themselves, as
acting horizontally, we have from formula (100) the friction F
— —-f [T	t), which reduced to its corresponding resistance
to the rope, taking r = 2 —-, gives a loss due to one shaft —
3
f ^ 2 -j- + 1 (cJk~) *	^ we ta^e f' ~ °'°9 ^an(^
double the resuit for both shafts, calling this combined loss Ez
we have : Ez -
8
X 0.09 x 4-33

which reduces to;
(272)
Example i. In the first of the preceding examples we have also d =6.3
inches, and 2R = 135^ inches, hence — 0.046 or a little over 4 per
cent.
fSee Engineer, Jan., 1884, p. 38, for such a fly wheel 15 ft. face, 30 ft. dia.,
weighing 140 tons, to transmit 4000 H. P. by 60 ropes,
X See l 300.
♦ See Zeitschrift d. Verein deutscher Ingenieure, Vol. XXVIII, 1884, p. 640.196
THE CONSTRUCTOR.
b.	Stiffness of Ropes.—If we apply Eytelweir^s formula (252)
we liave Q = % (T-\- t) taking both pulleys iuto consideration,
and taking r = 2% and introducing T -f- t> gives Q = 4 J P.
It must be considered that the ropes are usually quite slack,
and that the co-efficient stiffness S, may be takeu somewhat
less than Eytelwein’s value. If we take % as a fair approxi-
mation. the ratio of loss is
S 2	. d2 1
- = _x 0.463 ^X4_
and calling this loss Es , we get:
Es — '.33 ..............................(273)
in which d is the diameter of the rope.
Ex ample 2. In the case of the preceding example, d =s 2", R =» 67.75".
This gives E» = 1.33 ■> 4 - = 0.078 or 7.8 per cent.
67-75
c.	Creep of Ropes.—The loss through creep is more important
in rope trausmission than with belting (see $ 284) and sliould
not be neglected, although it cannot be so readily determined,
owiug to the division of the power among a number of ropes.
It is practically impossible to insure a uniform tension upon a
number of adjacent ropes, or to have them of exactly the sanie
diameter, besides which the “ working ” diameters of the vari-
ous grooves differ slightly, so that additional slippage must oc-
cur * The resulting frictional loss is estimated by some at
as much as 10 per cent., \\Jien the number of ropes is 20 to 30,
and it is at all times important enough to be given considera-
tion. The losses from stiffness and creep should be investi
gated wkenever practicable, as the resulting information would
be of much technical value.
Assuming the loss from creep in the case previously consid-
ered to be 5 per cent., we have a total resistance of 4 + 7.8 -f 5
= 16.8 per cent.; which, siuce small values were taken in all
cases, is not to be considered higher than the actual loss.
This explains the numerous objectioris which have been raised
(as in England) against the use of hemp rope transmission for
very large powers (see \ 301).
2 288.
Pressure and Wear on Hemp Rope.
As already seen, the surface of contact of the rope and pulley
may be one of three kinds: upon a cylindrical pulley, in a
semicircular groove, or in a wedge-shaped groove (Fig. 879),
and to these formula (241) can be applied. In case a} we can
approximate b' as equal to \ the circumference of the rope.
This gives for the superficial pressure
P_
S
= S~
R
* whence:
-r"
(274)
For case b, we have b1 = — d, whence
2
p __ 1 d
~S~~2 R................... ’
(275)
In case c, the radial pressure Q} of the rope is divided iuto
two forces Q' acting normal to the wedge surfaces and equal
to —T*u whick 0 is the angle of the groove, and taking the
contact surface on each side as J the circumference of the rope,
we have
______1___d_
S sin 0	R
2
which, for d = 30°, gives approximately :
P d
(276)
Even under these uufavorable oonditions the superficial pres-
sure is not important, 011 account of the small value of S;
which, as already seen, is about 350 pounds.
Example.—If .S = 350 pounds, and ^	= —— we have for a cylindri-
R	25
cal pulley p = 350 X 2 X * =* 28 lbs. for semicircular grooves, p — lbs
and for wedge grooves, when 0 = 300, p = 56 lbs. per square inch.
These low pressures cause but little wear upon the rope,
hence the great durability of hemp transmission ropes, some-
times extendiug to two or three years of use.
2 289.
B. COTTON ROPE 7RANSMISSION.
Cotton rope is not so extensively used for purposes of trans-
mission as hemp rope, although it possesses the advantages of
great strength and flexibility ; the impediment to its use being
its higher price. The application of cotton rope for driving
spinning mule spindles, referred to ,in § 265, is shown in Fig.
882, in which Tx is the driving pulley and T% the driven pulley
on the carriage. This latter pulley is on the axis of a drum 7S
from which light cords drive the spindles 74. At L, Z, are
guide pulleys. The usual diameter of rope for 7 x T.2 is 0.86",
and for large machines with many spindles two such ropes are
used, the pulleys being made with double grooves, these always
being of semicircular section.
On the ring spinning frame cotton rope of 0.4" diameter is
used on cone pulleys of 12 steps, giving changes of speed from
3:1 to 2:3. The proportions of such pulley may be determined
as shown in § 279, the grooves being semicircular.
As already shown in § 265 cotton ropes have been used by
Ramsbottom for driving traveling cranes. For this purpose
ropes of J to £ inch diameter are used, running at speeds of
2500 to 3000 feet per minute, a weighted idler pulley keeping
the rope taut.
In view of the slow movement of the load, viz.: 20 to 40 feet
per minute, it is questionable whether cotton rope transmission
involving such a great transformation of speed, is advantageous.f
C. WIRE ROPE TRANSMISSION.
2 290.
Specific Capacity. Cross Section oe Rope.
In considering the transmission of power by means of wire
rope the points to be determined are the cross section of the
rope, and the deflection of the two portions of rope due to its
weight. The cross section will first be considered by determin-
ing the specific capacity (See \ 280). This we get from (262)
0	330°° * T
in which Sx is the stress in driving half of the rope, considered
either in connection with the driving or the driven pulley.
The modulus of friction p is taken somewhat higher than for
belting, since the angle of contact a is greater, and also because
the co-efficient of friction f \ for pulleys fitted with diagonal
leather strips (see below) is very high ; early and recent tests
giving^* = 0.22 to 0.25 and higher. The first value gives efa ~
2
2 (See Fig. 816), and also the stress modulus r = —-—-— = 2
(See 239). This gives, in (262) if we neglect centrifugal force :
iV0=-------.
33000
_A_
66000
(277)
* The variation in adjacent ropes may be shown by putting a little
coloring matter on the ropes and watchiug its distribution.
fln some instances leather transmission ropes are used, formed of
twisted thongs, these being used for light driving, as foot lathes, or
light spindles.THE CONSTRUCTOR.
197
This gives high numerical values, which is also borne out in
practice, since large powers are successfully transmitted with
wire ropes of small diameter. It is good practice to take S1
for iron wire as high as 8500 pounds, and for steel wire up to
20,000 ponnds and even higher. This gives for the specific
capacity, when:
Sx = 2000, 4000, 6000, 8000, 10,000, 12,000, 14,000, 16,000, 18,000,
20,000.
iVo 0.03, 0.06 0.09, 0.121, 0.151, 0.182, 0.212, o 242, 0.273, °-3°3
or approximately:	,
For Wrought Iron Wire N0 = 0.03 to 0.121.
For Steel Wire .... iV0 == 0.03 to 0.303.
The cross section q is readily obtained, since N — q v NQ
hence:
q = 66,000 ——.............................(278)
v
We then have, if i is the number of wires in the rope, a diam-
eter of work: S = i — d2. The speed v, of the rope may be as
4
high as 6000 feet, but should not exceed this velocity on ac-
count of the great stress upon the rim of the cast iron pulleys.
8 291.
Infeuence of Pueeey Diameter.
The bending of a rope about a pulley of a radius R produces
a stress in each wire equal to
Steee Wire.
5	S !	R 6
i 1422	49,770	286
2844	48,348	294
5688	45,504	313
8532	42,660	334
«i,376	39,816	357
14,220	36.972	385
17,064	34,128	417
19,908	31,284	455
22,732	28,440	500
25,596	25,596	55i
28,440	22,752	625
3«,284	«9,798	718
34,128	1 7,064	834
36,972	14,220	1000
39,8*6	11,376	1250
42,660	8532	1667
45,504	5688	2500
48,348	2844	5°°°
If a stili greater value of — is used for any given value of Sx
than in the above table, the durability of the cable will be in-
creased. The minimum pulley radius for any given sum of
<5
s
which, if we take both for iron and steel wire E =
gives:
6
s = 14,200,000 —
28,400,000,
. . . (279)
The driving half of the rope is therefore subject to a tension
stress, both at the point of advancing and departing contact
equal' to S1 + s in each wire. It is this sum which must be
considered in determining the stress upon the material, and it
must not be permitted to exceed the proper limits (See \ 266).
A practical upper limit for wrought iron wire is 25,000 pounds,
while for steel it may be taken much higher ; for hard drawn
steel wire of good quality as high as 50,000 or even 60,000
pounds.
If we take as upper limits 25,000 lbs. for wrought iron and
50,000 lbs. for steel wire, we have for the given values of S, the
R
following values of s and of
stresses + s is obtained when — = 2, which in the tables
gives for -j- = 833 and 417 respectively, as indicated by the
full-faced figures. Even in this advantageous proportion the
stress due to the bending of tbe wire around the pulley is double
that due to the tension of the driving force.
Example 1.— Uet N= 60 H. P. v — 2952. The material is iron wire, 5i =
8532 and the number of wires i = 36. We then have for the cross section of
rope:
q = 66,
60
2952 X 8532
= 0.16 sq. in
also a =
±
7T
= O.O76"
and the minimum pulley radius is R = 833 X 0.076 = 63.3" or appoximately
10 feet diameter. In order to obtain a velocity of 2952 feet per minute this
requires about 93 revolutions.
If we take Si = s = 12,798 we find:
q — 66,000
6d
2952 X 12,798
= 0.108"
Wrought Iron Wire.
5	5	R 6
711	24,885	571
1422	24,174	588
2844	22,752	625
4266	21,330	667
5688	19,908	714
7110	18,486	769
8532	17,064	833
9954	15,642	9°9
11,376	14,220	1000
12,798	12,798	IIII
14,220	i* ,376	1250
15,642	9954	1429
17,064	8532	1667
28,486	7110	2000
19,908	. 5688	2500
21,33°	4266	3333
22,752	2844	5000
24«74	1422	10,000
whence 5 =	- = 0.06"
26 JT
R = xni X 0.06 — 66.6" and n — 72.
The question here arises, to what extent should the effect of
centrifugal force be taken into account? If the velocity
v — 100 feet per second, with a stress S = 9000 lbs. we
have from the first table in \ 264 the value 1 — z = 0.87, so
that instead of fa we have fa' = 0.87 fa. Iffa= 0.22 we
have fa = 0.87 x 0.22 x n = 0.70, and if f = 0.22 we have
fa = 0.87 X o*22 X ^ —— 0.60. These give, byreference
to the second table, in \ 264, for the first value, the modulus of
T
friction p = — 2.01, and for the second, p = 1.82 and a modu-
lus of stress r = 2.22, which makes the specific capacity =
2	10
—— = — as great as previously obtained. This may be com-
pensated for by making the cross section of the rope 1.1 times
that obtained by the previous calculation. For lesser velocities
up to 20C0 to 3000 feet per minute the effect of centrifugal
force is much less and may safely be neglected, especially in
the case of steel cables, in which much greater stresses are per-
missible.
Example 2.—How many horse power can be transmitted by a cable of 36
wires, each 0.078" diameter; the velocity being 6500 feet per minute. We
have Na = —.	— = 0.117; also q — (0.078) —— X 36 = 0.172 sq. in.
0	11	66,000	'	4198
THE CONSTRUCTOR.
We then have N = q v NQ = 0.172 X 6500 X 0.117 = 130 H. P. For R we have
r r \	14,200,000 X o 078	,	„	- ,,
frotn (279) R — ----X1~t\------ ~ 64.9' say 65 .
Example 3.—What would the liorse power be if Steel wire were used ?
(See § 266). S\ = 17,000 Ibs. and A© will be twice = 0.234 whence N = 274
H. P. If we Ldesire durability of the cable we may make j only 28,440 in*
stead of 34,128, and thus obtain R — —°'°7- = 38.9 say only 40"
When the resistance P is directly given, which is rarely the
case, we have from the relation q Sl = t Pt taking r = 2.
P
Q = 2~S.....................................(280)
The maximum statical moment which may have to be over-
come upou the driven shaft is sometimes given, as in the case
of pumping machinery, etc. Dividing the precediug equation
by (279) we have
half, h2, and in the stationary rope ho. This gives for the tan-
gential force K at the point of suspension :
*“*(* + £)........................................(284)>
14,200,000 d Sx
PR
and since q — i — (i2, this reduces to :
4
<5 = 0.00564 V 4- V ~PR................
1 o1
(281)
and if we substitute for the moment P R the quotient of effect
JV
from formula (135) PR == 63,020 — we get
.	1 jf s N
1==a2S. VT S-s —
(282)
Example 4.—A pressure pump operated by a crank on a shaft driven by
wire rope, offers a resistance of 880 pounds, at a crank arm of 14.2 inches.
This gives a maximum statical moment of P R . — 14.2 X 880 = 12,496 inch
lbs. If we take i — 36 wires, and .Si = 8500 pounds, we have from (281):
*	3 / I	3f 17,00°	„
S = °.°°355 \J 3f;	V - s500' I2'«6 = °-°s"
This gives from the table R = 833 X 0.05 = 41.65, say 42".
§ 292.
Defeection of Wire Ropes.
In order that the desired tensions T and t shall be attained
in the two parts of a wire rope transmission, the deflections
must be of predetermined values. The centre line of the rope
will hang in a curve which lies between the catenary and the
elastic line and which approximates closely to a parabola.*
For the parameter c, of this parabola, we have for a deflection
h> in a horizontal rope, Fig. 883,
c=b........................................(2§3)-
in which a is the distance between two points of suspension ;
the deflection in the driving half being called k1} in the driven
All dimensions are to be taken in inches. For any cross sec-
tion q, wre have K — q S, and ^ 6 12 y, in which y is the
weight of the rope per cubic inch, and <p a coefficient, dependent
upon the twist of the rope and upon the hemp core. The weight
per cubic inch is y = 0.28 pounds, <j> is not always constant, but
may be taken = |. These values give p = \ x 0.28 x ^ “
0.3266^, and calling the coefficient 0.3266 = we have:
5= * (ji +-=0.3266	.... (287).
From this we get:
h
1
2
s r
V 4 Tp2
8
(288).
Since — = 3.061 we have, taking the negative sign.
h = 1.53 S— }[ ^ 1.53 5 ^	.........(2S9)-
If we neglect the first member in the parenthesis in (287) we
have for a close approximation :
h — 0.0408 —
(290).
Example. Eet a% which we may take as the distance from centre to centre
of pulley, be 262 feet, or 3144 inches; also let 5 in the driving side of the rope
be 8500 pounds, and 011 the driven side 4250 pounds. We have from (289)
h\ = 1.53 X 8500
^2 = i-53 X 4250 — 4/ (i.53 X 425°)* —	8
The approximate formula (290) gives:
y.-
-f
8
3i44z
hx
0.0408	= 47*45", and
* The equation of the catenary is as follows:
£
X= 2
+ *
in which the tangential, vertical,
and horizontal forces at a point
xy may be designated as pxt ps and
/c, p% being the weight for a unit of
length, and S = >/ x2 — c2. For
the point of suspension this gives:
K= p{h -f c), \A=/ \/A* T~2hc,
H — pc......................(286)
in which the parameter c is yet undetermined. In order to determine the
latter let the equation of the curve be developed into the following series:
x
+
1.2 c2
y3
1,2.3.C3
+1
y ,	y3
C	1.2 C2
y8
1.2.3.C3
Since the curve is always flat in rope transmission, the quotient is a
c
proper fraction, and both series are converging. Stopping at the third mem •
ber as giving sufficient accuracy we have :
=me+-irand
y2
— c = which is the equation of a parabola.
h2 = 0.0408	= 94.89".
4250
The following inethod may be used to show the deflection k,
graphically. The positive and negative signs before the radical
sign in (288) indicate two values for k, as will be seen in Fig. 884.
The greater value is not of practical use, as it gives unstable
{labii) equilibrium.
Between the two lies a value h =•
which is obtained when the quantity under the radical sign =
ip a
O, i.e. when 6* = — —. This we will call the “mean ” deflec-
y/ 2
tion and designate by hm. This deflection is important because
with it the absolute minimum stress exists in the rope (see note
at the end of this section) ; and this stress, which occurs with.
the deflection hm, will be designated Sm> and is:
Sm =
y/ 2
or for the preceding value ^ = 0.3266,
a
Sm ==----
4*33
(291).
(292).
and since hm= . —r, we have for the mean deflectionTHE CONSTRUCTOR.
199
hm = .......................... (293)-
F8
Dividing (288) by (293) we have, after some reductions :
h	S— v/5a —6V
hm	Sm
(294).
From this we obtain the following geometrical construction
of Fig. 885. With a diameter — describe the semi-circle
1.2.3, and join the point 3 of the quadrant 2.3 with 2, or 1;
a
Fig. 885.
Lay off this distance perpendicular to 1.2 at 2.4, and on any
scale (not too small) lay off from 2 to 5, the stress Sm, deter-
mined from (292). From 5 lay off, on the same scale, 5.6, equal
to the given stress S, and from 6 draw the arc 5.7. This gives
2.7	=6.5 — 6.2 = 6.5 — ✓ (6.5)2 — (5.2)2, which is = S —
>/S2— Sm2. If we now draw 4 • 8 parallel to 5.7 we have
2.8	2.7	A	, «	0	,
----=s------ — 7—, and hence 2.8 a= h.
2.4	2.5	Aw
The value h0 of the stationary rope is that of a parabola of a
length midway between those for hx and h2 and is equal to :
]h 2'4. jpL
ho — V ———- = o.6yh2 + 0.28^.................(295)
# It may readily be constructed graphically from the first expres-
sion. It is not essential that the driving part of the rope should be
the upper portion, as the lower part may drive, as in Fig. 886. The
ropes will not touch, when stationary, if h2 — hx < 2R. OwLng
to the fluctuations due to the action of wind, or of sudden
changes of load, the minimum distance should not be too small»
and is best kept greater than 20 to 24 inches.
NoTE.—We have from (287):
d S =	+ /4 ^0 — ^2 ^ which gives for the
mum of S:
mmi-
^ ~ ^ C 1 —	Wi *s the Parameter> or
hence 0 = 1-------- or Cm = hm = —7- and hence from (287):
hm ‘	v/g	v
In Fig. 884 is shown graphically how for each value of h, the
parameter c can be found, by constructing the proportion
a
— = —. In the figure, 2.5/ = 2.4' ■=. hf ; also 47.6' = —;
ac	2
2
and 6/ . 7' perpendicular to 6' . 5' gives at 4/ . 7' the parameter
c'. In like manner : 2 . 5" = 2.4" = h" ; also 4// . 67/ s
and 6" . 7f/ perpendicular to 6// . 5// gives the parameter
4// . 7// __
To determine the vertex 4// of the lower parabola we have :
h' + £• =h" + w whence h' ~h"=i Gf/ - f)
a2 h' — h"
“ T ~hrh77~'
— hm2.
This gives W h" — — which as shown above
o
If, as before, we make 2	8 = hm, and 2.5/ = hf and
draw through 8 a normal to 8.5' the normal will intersect
2.4" at 4// which is the desired apex. The lines 5.6, s/ •
5r/ . 6// intersect each other at the middle of the half-chord of
the parabola at 9. This may be used in the construction by
drawing from 9, the line 9.6, 9.6/, 9.6//, and the correspond-
ing normals give the parameter points 7, 7/, 7". The directrix
of the parabola lies at a point distant c from the vertex. For
the mean parabola the directrix is midway between 4 and 7,
and the focus Fm is at the middle of hm, and is also the centre
of the circle 5.6.7 . 1.
In the figure is also shown another curve which indicates the
values of S. The proportional value of h from formula (287)
taken from the line 2 . 11, shows that h is in inverse proportion
to the hyperbolic line io'. 10. io77. The ordinates of the hyper-
bola, taken from the axis of abscissas 2.7 gives the values of S
for the corresponding values of h. The ordinates 47 . io7 and
477 . io77 give the equal stresses S' and S77, and 4.10 the mini-
mum stress Sm The dotted hyperbola on the upper right, gives
the corresponding thrusts in a parabolic arch, and the curve in
an arch corresponding to the catenary is the line of thrust. In
this also we find the mean height the most economical, the
lower ones being stable, and the higher in an unstable equilib-
rium, dependent upon the thickness of arch ring and distribu-
tion of load for their stability.200
THE CONSTRUCTOR.
\ 293-
TIGHTENED driving ropes.
The defleetion of transmission cables often becomes incon-
veniently great. In many cases, however, it is possible to
reduce its amount by increasing the tension to a greater extent
than is necessary to prevent slippage. This requires the cable
to be made correspondingly stronger in order to resist the in-
creased tension. The modification in the preceding discussion
of forces and dimensions is here given, the various ternis being
given the subscript s to distinguish them, ( Ts, ts, Ss, $s, instead
of T} t, S, <1). The tension T, as shown in § 290, should not be
,.	.	„	14,200,000 x 0.026	T, .	.
12 feet. Accordmg to (279) R =---------n~376------- = 32'43’ sa^ inc^es.
which gives h2 — h\^y> 2 R and the driving part of the cable must be. above.
The above resuit shows that the centre of the pulleys must be more than
R -f h2 or 24ft + 2ft 8*4" above the ground in order to ciear. To reduce this
height we must tighten the cable. Suppose we made the diameter of the
wires = 0.04" instead of 0.024". This gives — 1.67, and from the table, coi*
o
umns 4 and 6, line u, 52« = 0.89, S = 12,650, and hence we have /12» —
0.040S	=162", and /rzs — h\ = 162 — 144 = iS".
12,650
We also have R =
“,376
X 0.04 = 50".
These values give his —

Fig. 887.
less than 2P, and if this is increased by a given factor m, we
have ts = Ts — P\ and also :
Ts — m T — 2 /// P,
ts = (27// —1) P}
ts _____ 277/ ---- I
Ts	277/
(296).
In order that the stress ^ in the driving part shall not be
changed we have for the stress in the driving part, instead of
•Si
-, the following :
& = si — —1,
277/
(297).
The diameter <1 of the wdre, if calculated from (280) is modi-
fied to
Ss = m................................(29^).
or if taken from (281) or (282), wre take
5, ^{l&m...............................(299).
from which the following table has been calculated. Tightened
cables are frequently applicable where moderate pow^ers are to
be transmitted.
TABLE FOR TIGHTENED CABLES.
JS Is \ ts
~T ~~P T
ts
~p
SoS S.2s
' S,
%
— ^ __v3/,
w 77/
1.6	3-2	2.2	0.69	) I 1.26	1 17
i.8	36	2.6	0.72	1.34	1.22
2.0	4.0	3-0	0.75 i	1.41	1.26
2.2	4-4	3-4	0.77	1.48	; 130
2.4	48	3-8	0.79	1-S5	i 1-34
2.6	5-2	4.2	0.81	1 61	! 1.38
2.8	5-6	46	0.82	1.67	1.41
3-0 j	6 0	5-0	0.83	1 73	1.44
3-2 !	6.4	5-4	0.84	1-79	1.47
3-4	6.8	5-8	0.85	1.84	
3-6	7.2	6.2	0.86	1.90	i-53
3-8	7.6	6.6	0.87	i-95	1.56
4.0 |	8.0	7.0	0.88	2.00 j	i-59
4.2	8.4	7-4	0.88	2.05	j 1.61
44 i	: 8.8	7.8	0.89	2.10	1.64
4-6 !	i 9 2	8.2	0.89	2.14	1 1.66
4.8 j	9.6	8.6	0.90	2I9	1.69
5.0	10.0	9°	0.90	2.24	I-71
2R and we may therefore place the driving side below without danger of
interference- The greatest defleetion occurs when the cable is at rest, and
from (295) we have hos = 149 inches, and the total height for the pulley cen-
tres is 149 -f 50 — 199" or 16« 7". This example is shown in Fig. 887, in which
the dimensions, however, are in the rnetric system.
$ 294.
SHORT SPAN CABLE TRANSMISSIONS.
When the distance between pulleys is short the defleetion
must not be too small if good results are to be expected. To
this end a small value should be taken for .Si, and hence the de-
flection is first to be chosen and the corresponding value deter-
mined from (287) wThich is readily done. For moderate powers
wire rope transmission may be used in this way for short spans
very successfully.
Example,—Let N= 5 horse power, to be transmitted over a span of 65.6 ft.,
or 787 4 inches; the number of revolutions to be 150, and the defleetion 40
) = 645 lbs. Taking iron
wire, and making 5 + j = 25,600 lbs. We have j = 25,600 — 645 = 24,955. If
inches. We have from (287) 51=0.3266 (40 -f
we make the number of wires i — 36 we have from (282)
3
5 = 0.251
4
24,957.
645
150
= 0.083"
We then have from (279)
R =
and v
14,2000.000 X o.083
25,600
150 X 2 7T X 46
= 45-9 say 46",
= 3600 feet,
all of which values are quite practicable.
\ 295-
TRANSMISSION WlTH INCLINED CABLE.
A transmission at which the pulleys are placed at different
heights is called an inclined transmission, and the curve in such
a case is unsymmetrical. For a given distance a, betwreen the
verticals through the ends of the curve, and for a difference in
height Hy we have for the deflections h/ = xl and h" =x2, Fig.
888, and for the ordinates yx and y2 of the two branches of the
curve:
xt — hf,=
a2	+	c	H2	H
Sc		2	~a2	2
a2	+	c	H2 ,	H
87C		2	a* +	2
(3°°)
and
yp
H
a
Example.—Given, N = 5 . 5, n — 100, a = 590.4 ft. = 7086 in. Itisrequired
to cover this distance with a single streteh of cable. If we take .Si = 14,220
lbs., and 4 = 11,376 lbs., we have —X	X —— = 0.044. If
’	^	S\	n	14,220	100
7 — 36 we have from (282) 5 = 0.251	0.044 = 0.024 inches. Wethen
-	-	-	_._o (7©86)2____„	t	t „	_
a , H
y2 —----J- c —
2	. a
in which the parameter c is yet unknown.*
(301)
* Deduced as follows: y- = 2 c xly y22 — 2 c x2, y\ -f y2—a. x2—jti==P whence:
>2* —	= 2c{x2 — xi) = 2cH i. e. {y2 +^1) (y2 —yi) =z cH and hencey2 —y\
H .THE CONSTRUCTOR.
201
For the parameter c, we have from (286) A^=^(^ + r) or
Sq = ip q {h -j- c) and if we consider the lower pulley as bearing
(302) C =
0.3266
+ O
^ ^0.3266^ —39^6“ __ 25)95llinches.

The defiection is:
39i62
12,962
8 X 12,962
whence
+ -—-— X 0.0025 —-98.5 = 67.1", and h2" — h2’ + 197—264.1,
yi = 1968 — 0.05 X 12,962 =
: 1319-9
The stress on the rope, iustead of being exactly 8500 and 4250 pounds, will
be, according to (304):
8500 + 0.3266 X 197 = 8564 lbs., and
4250 + 0.3266X 197 = 4314 lbs. respectively.
--4oom~
5/
the lighter load we have: S' = V' (f + x\) whence c = —----xv
Substituting the value of xv from (300) we obtain after reduc-
tion.
(3°2)
The plus sign before the radical indicates that we have chosen
the “stabil” parabola (see Fig. 884), and hence obtain the
greater of the two values for the parameter. The parameter
thus being determined, we have xx and yY from (300) and (301).
For the upper branch of the curve the stress S" is to be de-
termined at the upper ^pulley. We then have S" = {c + x2).
Subtracting from this S' =■ $ {c -f xx) we have
S"=zS' + $ (x2 — xl)=S'+tPJT...........(303)
and if ^=0.3266 we get:
S" = & + 0.3266 H.............(304)
Ex ample 1 —Let a = 328 felt = 3936", S' = 8500 lbs. If H= O, we have from
Fig. 889.
. The arrangement is shown in Fig. 889. the vertical diraensions being three
times the scale of the horizontal, and ali dimensions being in metres.
Example 2.—Suppose the distance a = 3936 inches, and Si = 8500, and S%
4250, as before, but the vertical distance //—1968", or ——. We then have
(a) For the Driving Side :
i
■ +
830°
'Z&H' + 984
2 4- 0.5-
hx — xi =
('Vyk' + 9SX)_______39361
V 2+0.5- J	8(1+0.1;
393^2
+ 0.25X
23361
125)
— 9S4 = 2019 inches.
23361, whence
8 X 23361
and y\ =* 1968 — 23361 X 0.5 = — 9712 inches,
the minus sign indicating th it the apex of the parabola lies without the
space between the pulley s.
(b) For the Driven Side:
o 3266
+ 984
- +


39 362
2 + 0.52
hn — xo = -
2 +0.5-
3Q362___
8 X 12271
and yi = 1968 — 12271 X 0.5 = 4167 inches,
and the apex again lies outside.
12271
-+0.25X —
8(1+0 125)
984 = 708 inches,
= 12271, whence
This value in (300) gives xx = x2 = hx = ~ = — ^ — = 74-62".
For the slack half of the rope we have S'2 = 4250, and
425°	f S 4250 X 2	rn
c « —— + y Q 03266^ _39|6“ = I2>862 Whence h2 = 8x^862
«= 150.5".
Suppose now that H— o 05 a = 197". We then have :
(a) For the Driving^Side:
8500
0.3266
+-98-5
\f	393^
* V 2 + 0.052 J 8 (1 + 0.00125)
2 + 0.05	T \ 2 + 0.052
which is slightly greater than when H=*0.
39362	,
- = 26,018 ;
We have also, from (300) h\ —
26,018
8 X 26,018
X 0.0025 -
197
= 8.45'',
and k{' = kxf + 197" = 205.45". *
The distance yx then becomes:
y\ —	— o 05 X 26,018 = 667.1.
(S) For the Driven Side:
.. ,
2 + 0.052
CU+98s)	393^	=;
V V 2 + 0.o52	/	8 C1 +O.OOI25)
The general arrangement is shown in Fig. 890 all d mensions being given
in the metric System, and the vertical and horizontal scales being the same.
The increase in the stresses is more marked than in the previous example,
on account of the increase in the value of H. We have S\ = 8500 + 0.3266 X
1968 = 9142 lbs., and Srf = 4250 + 0.3266 X 1968 = 4892 lbs.
VL/
Fig. 891.
The upper limit for this.form of ro
FROPERTY
DEPARTMENT
MACHINE DESTCN
SIBLEY SCHOOL
CORNELL UNiVERSITY
>e	jSwO31.	..202
THE CONSTRUCTOR.
which the parts of the rope are vertical, in which case the par-
ameter = oo. In this arrangement the necessary tension rnust
be obtained by the use of weights, spring, or the like. By using
guide pulleys, a combination of horizontal and vertical trans-
missions may be made, as in Fig. 891, and the tension obtained
by the deflection in the horizontal part.
i 296.
CONSTRUCTION OF THE ROPE CURVE.
We have considered the curve as an ordinary parabola.
When the apex C> Fig. 892, has been determined, bisect the
two parts Bx C and Dx C of the horizontal tangent Bx Dx, at Cx
and Cp join B Cx and D C2, and these two lines will give the
direction of tangents to the curve at the points of suspension B
and D. Then divide C Cx into equal parts C} 1, 2, 3-and
Cx B into the same number of equal parts Cx I, II, III-,
and by joining these points we obtain a number of tan-
gents which include the curve. The other portion C C2 D, of
the curve is constructed in a similar manner. When the apex
of the parabola falis beyond the lower pulley, only one portion
of the curve is used.
I 297.
Arrangement of Puueeys.
When the transmission pulleys are far apart, and not high
above ground, supporting pulleys must be used for the rope.
In some instances this is only necessary for the driven part of
the rope, the driving part being left unsupported, as in Fig. 893.
Each portion of rope between two pulleys may be called a
tl stretch ” of rope, so that in the above instance we have the
driving part in one stretch and the driven part in two stretches.
If it is necessary to support both parts it is often practicable to
use half as many supporting pulleys for the driving part of the
rope as for the driven part as in Fig. 894.
Fig. 894.
These pulleys are called guide pulleys to distinguish them
from the main transmitting pulleys and their supporting struc-
tures are called supporting stations.
An other arrangement has been used by Ziegler, as shown in
Fig. 895.
&	—	Sr	^		
		z!		 11 z	
t		
Fig. 895.
This consists of a number of shorter transmissions, using
double grooved pulleys, or two single grooved pulleys at each
station. In this arrangement it is advisable to make the
stretches of equal length so that a single reserve cable will
answer to replace anv one which may give out.
It is always desirable to run a transmission in a straight line,
and especial care must be taken to have the successive pulleys
all in the same vertical plane. If it is impracticable to run the
entire distance in a straight line it is necessary to introduce
angle stations. These may be constructed as in Fig. 896 a,
using vertical and horizontal guide pulleys, but this requires
six pulleys, three for each part of the rope. A simpler arrange-
ment is shown at Fig. 896 b, two pulleys and a pair of bevel
gears being used.
In many cases it is desirable to take off a portion of the power
at intermediate stations either by shafting or rope transmission,
and this may readily be done by a variety of arrangements of
gearing and shafting.
It is most important that the pulleys both for supporting and
transmission should be amply large in diameter. Many rope
transmissions have woru out rapidly, simply because the diam-
eter of the pulleys has been too small. The intermediate pulleys
for the driving side ought to be the same size as the main driv-
ing pulley in order that the total stress .S -f- s (see $ 291) shall
not be greater in the former case than in the latter. The sup-
porting pulley for the driven side may be smaller because the
stress S2 is smaller generally, being SXf or for tightened
transmissions (§ 289) being equal to (2m — 1) 2m Sx. The
smallest permissible pulleys may be determined from formula
(279) and the table of § 291.
Example 1.—In an ordinary wire rope transmission let Si = 8500, So —
4250, and the wire of wrougbtiron, 5 being = 0.06". From the table in § 291
we have for the minimum radius of pulley, R = 833 X 0.06 = 50'' or 8ft 4" dia.
and for the supporting pulleys: Ro — 667 X 0.06 = 40 or 6ft 8" dia.
Example 2.—Let 6= 0.04. .Si = 5688, S2 = 2844, and for iron wire we have
R = 28.56 say 30", R2 — 25".
Example 3.—In a tightened transmission let m = 3, and S = 0.06", Si =
S500. S2 =SX -(2 X ~~ - = Sx = 7080. R = 50" as before, and R2 = 769 X
2X3 t>
0.06 = 46", a difference which is hardly great enough to be of practical im-
portance.
*	? 298.
The Construction of Rope Pueeeys.
The low value of the coefficient of friction of iron on iron
makes it impracticable to run the wire cables directly upon the
bare metal rim of the pulley, and hence various atte’mpts were
early made to fit the groove of the pulley with some soft material.
After early experiments with wooden rims fitted with leather,
or rubber, it was practically shown that turned iron rims fitted
with leather filling placed edgewise in the bottom of the groove
gave the best results. *
a.	b.
!
Fig. 897.
In Fig. 897, is shown at a, a rim for a single pulley and at b,
for a double one, both being of cast iron. The proportions are
given in terms of the diameter d, of the cable, and in the illus-
trations the constants in the various proportions are in milli-
meters. The sides of the grooves are made at an angle of 30^
with the plane of the pulley in the case of the single groove
* See D. H. Ziegler, “ Erfahrungs resultate liber Betrieb und Instandhal-
tung des Drahtseiltriebs.” Winterthur, 1S71.
ITHE CONSTRUCTOR.
203
pulley, but this gives an excessively lieavy middle rib for the
double pulley, and hence tbe inner angles are made 150 as
shown. The smallest diameter of rope for practical use is
d — o.04//. The superficial pressure p, may be calculated from
d
(274). If, for example, i — 36, we have from (244) — — 8, and
if — — 1000 and .S = 8500 we have :
o
p = 2S : = 136 lbs. per sq. in.
a pressure readily borne by the leather filling.
The bottom grooves are made with a dovetail bevel in order
to keep the filling from being thrown out by centrifugal force.
The filling of leather may be made of pieces of old belting
placed on edge and forced by driving into the dovetail groove;
if new leather is used it should be softened by soaking in train
oil. Rope sheaves for hoisting machinery, which are only used
for guiding and supporting the rope, were formerly used with-
out any filling, the rope resting on the bare metal. It is be-
coming more and more the practice to use a filling in the bot-
tom of the grooves of such pulleys, vulcanized rubber giving
good resuits.
Fig. 899 a, in order to avoid injurious stresses from shriukage
in casting. The spaces are afterwards filled in with fitted pieces
of iron and a ring shrunk on each side to hold all together.
The proportions of hubs are the same as in \ 283.
a.	b
b.
The construction of the rim of Fowler’s “ Clamp Pulley,” re-
ferred to in Fig. 794 c} is shown in Fig. 898 a, the clamps being
pivoted to blocks by means of bolts with anchor-shaped heads.
The pressure upon the rope is the same as in the case of a wedge
groove of equal angle, and the pulley as made by Fowler,
has one clamp ring mounted upon a screw thread cut upon the
pulley, thus enabling adjustment to be made for wear upon the
clamps and for the reduction in the diameter of the rope. Fig.
898 b shows an American form of clamp pulley, somewhat sim-
pler in construction than Fowler’s. The clamps are pivoted on
half-journals (see \ 95) and the angle is not so small as in the
preceding form.
The arms of rope pulleys are usually made of cast iron as well
as the rim, although the intermediate supporting pulleys are
sometimes made with wrought iron arms, as in Fig. 901. Large
pulleys, when of cast iron, are usually made in halves, for con-
venience of transportation.
The number of arms A, may be obtained from :
A=A+ioRd........................(3°5>
Cast iron arms may be either oval or cruciform in cross sec-
tion, and the width of arm h} in the plane of the pulley, if pro-
longed to the centre is:
fi = 4<i+ X ^....(306)
The distance between jouruals for the intermediate and sup-
porting pulleys varies frotn \ R to J R. The load upon the
bearings consists of the sum of the weight of the pulley and the
vertical component of the various forces upon the rope, and
this can best be determined graphically as shown in Fig. 900.
Fig. 930.
b.
The weight G, of the pulley is so dependent upon slight
variations in the thickness and section of rim and arms that a
general formula of practical value cannot be given. The follow-
ing examples from practice are given :
Example 1.—In an executed transmission by Rieter & Co., at Oberursel,
near Frankfurt a. M., the pulleys are made with twelve straight arms of
oval section and are 12 ft. 3.6 in. diameter. The inain driving pulleys at the
end of the transmission, with single groove, each weigh 2525 lbs. and the
intermediate supporting pulleys, with double groove, each weigh 2780 lbs.
The rope is made of 36 wires, each being 0.07" diameter.
Example 2.—The Berlin-Anhalt Machine Works Company makes a line of
rope pulleys with wrought iron arms as in Fig. 901, the weigh ts being as
follows:
R = 20" 24"	28"	32''	36"	40"	50" 60" 70"
G = 176 211 248-30S 281-343 316-387 316-506 528-570 748 968
In these instances the weight upon the bearings is not great.
The journals for these pulleys should be made long, in order to
reduce the superficial pressure, and swivel bearings with cast
iron boxes ($ 116) can be used, which with self-oiling devices
will give good Service. Iu many cases the journals are made of
hardened steel in order to combine the greatest security with
the minimum size.
For arms of cruciform section, the thickness of the arms e
may be made J h, and the rib thickness e' — % e. Arms of oval
section may be made of the same proportions as for belt pulleys,
tbe thickness being made one-half h at all points and the width
at the rim being % h.
Arms of cruciform section are usually made straight as at a,
Fig. 899, but arms of oval section are frequently made curved
as at b.
To draw the curved arm make the circle O A of a radius =
y2 R and divide it into spaces for the desired number of arms.
Make A E — % A B, and draw O C normal to A O and C will
be the centre for half the arm, and the other centre will be at
D, the radius D E being equal to C E.
When straight arms are used the hub should be divided as in
Example 3.—The intermediate pulleys in Example 1, give a total pressure,
according to Fig. 900 b, upon the bearings, of 3036 pounds, or 1518 pounds on
each journal. If we make / = 1.5 d according to the table in § 91 we get for
d only in. In the actual case, however, the journals are 3% in. diam-
eter, giving a greatly reduced superficial pressure and thus insuring the
most complete lubrication. In this case we have the actual length l = 4.7,
whence p =
1518
3-75 X 4-7
= 86 lbs. per sq. in.
If, in order to use formula (89)
we take — = H and make 5 =
a
8500 as before we have:
d= s/ -^tVI5i8 = i-9"“-yj’
and =4d — 8''. This gives :
15*8
P = -2 x 8— = 95 lbs. per sq. in.204
THE CONSTRUCTOR.
which is such a low value tliat even half boxes, similar to those in Figs. 324-
325 could be used. By usmg hard Steel bearings even this small frictional
lesistance could be reduced to % the amount due to the above dimensions.
The pulleys for rope transmission should be most carefully
balanced, as any vibration causes serious oscillation of the rope
l 299-
CONSTRUCTION OF THE PUEEEY STATIONS.
The extraordinarily high specific capacity of wire rope trans-
mission has, as already said, caused it to be used especially for
1	1
Fig. 901.
the long-distance transmission of power. It has been found
particularly adapted for the transmission of the power of natural
falis of water to places where it can be utilized and has thus
materially advanced the use of natural sources of power. In
such transmissions one of the most important and difficult por-
tions of the work consists in the construction of the stations
upon which the pulleys are carried. * The following are exam-
ples of well designed and constructed stations.
Fig. 901 shows a design for an intermediate station of masonry.
The fouudation is of rough stone-work and the superstructure
of brick-work.
Stations similar to this are used in the transmission at Qber-
ursel, referred to in the preceding section, and erected in 1858.
This installation is used to transmit 104 horse power over a dis-
Fig. 903.
tance of 3168 feet (966 meters) divided into eight stretches, giv-
ing twro terminal and seven intermediate stations. Each
stretch —	= 396 feet long ; K = 74 inches, n = 114*5, v —
Fig. 904.
4400 ft, d — 0.07", i = 36. The difference in level betwTeen the
two terminal stations in this case is 145 feet.
The transmission of the wTater power from Schaffhausen, con-
* These have been fully discussed in a work by D. H. Ziegler treating of
the installations made by Joh. Jak. Rieter, Winterthur, 1876, and printed
privately.THE CONSTRUCTOR,
20$
structed byj. J. Rieter & Co. and in operation since 1866, is
used to transmit a total of 760 horse power developed by the
Falis of the Rhine. Of this 200 horse power is transmitted
direct to the left bank by means of shafting; 560 horse power
is carried across the Rhine in one stretch, the distanee a being
385 feet, using two similar ropes carrying 530 horse power.
(n =180, R = SS}4// v —4636ft.)andathird single rope carrying
30 horse power (n = 180, R = 35.4"). Of this powTer there is
about 480 horse power transmitted over three principal stretches
of 378, 332, and 455 feet. The number of wires in the heavier cables
is i — 80, the thickness of wire <5 = 01074", the rope being made in
8 strands of 10 wires each. One of the intermediate pulley sta-
tions is shown in Fig. 902, and this is an excellent example of
good style in construction. In this case there are twro pulleys,
side by side. There is a guard shown over the pulleys, to pre-
necessary„
Messrs. Rieter & Co. have also installed a System of turbines
and rope transmission at Freiburg, for the Societe generale
suisse des eaux forUs, of which 300 horse power is in a long-
distance transmission. The power is carried in five stretches of
502 feet each, to a saw mill, the difference in level being 268 feet.
One of the stations with two supporting pulleys is shown in
Fig. 903, this one being quite high ; a similar station, No. II, is
placed in a tunnel, through which the rope passes. The num-
ber of wires i = 90, the diameter of wire <5 = 0.072", the cable
being made in 10 strands of 9 wires each, R = 88.6", // = 81,
v = 3743 ft. From this point the power is divided by an angle
station and one part is delivered to the saw mill and balance
transmitted to a number of minor establishments.
A11 angle station is shown is Fig. 904, and this form is also
used when a portion of the power is to be taken off.
A fourth large installation of turbines and rope transmission has
been executed by the firm of Rieter & Co., for the Compagnie
generale de Bellegarde, at the latter place, for the utilization of
the well-known Perte du Rhone. The combined pow?er of the
Rhone and the Valserine is exerted upon five turbines of 630 horse
power each, giving a total of 3150 horse power w7hich is trans-
mitted by cable to the Plateau of Bellegarde.*
At Zurich, the city has utilized the power of the Limmat by
means of turbines and rope transmission built by the firm of
Escher, Wyss & Co. In this case the stations, which for various
reasons are quite high, are made of wrought iron, as shown in
Fig. 905. The entire installation develops 1150 horse power,
of which 750 horse powTer is used for the city water works.
At St. Petersburg a rope
transmission in ten stretches
is used to drive the Imperial
Powder Works, the power
being delivered into the
buildings by shafting from
each of the ten stations.
Amodification of Herland’s
device for putting on belts,
has been made by Ziegler for
the purpose of putting the
wire cables upon the pulleys.
As shown in Fig. 906, it
consists of a curved piece of
angle iron, clamped tempo-
rarily to the arm of the
pulley in such a manner as
to lead the rope into the
groove of the pulley. The	FiG. 906.
short radius to which the
rope is thus once bent does not
appear to have an injurious eflfect.
When a transmission rope is car-
ried over a public or private road
a guard should be used as a pro-
tection in case of breakage of the
rope. A simple forni used by Rieter
& Co. is shown in Fig. 907, and
consists of a sheet iron trough about
18 inches deep and ten feet wide,
carried by two stationary suspen-
sion cables is H H.
H
v.
'J
Fig. 907.
2 300.
Efficiency of Rope Transmission.
The injurious resistauces in wire rope transmission are mainly
those due to journal friction and stiffness of the rope ; the slip
and the atmospheric resistance of the pulley arms being in-
significant. t
a) Journal Friction.—We have from formula (100), F— —f Q>
7r
m which Q is the load upon the journal. For a circumferential
speed cf at the journal, we have a resistance in foot pounds :

or:
p = fndQ
3
(307)
Example /.—In the case of the transmission at Oberursel a number of ex-
perimental determinations were made. For a pair of journals Q — 2948 lbs.,
d = 3.75" and n = 114.6. For a coefficient ot friction/ — 0.09 (experimeutally
determined) we have :
Fc =
0.09 X 114-6 X 3-75 X 2948
3
= 37*658 foot lbs.
or	a— = 1.14 horse power.
33000
This gives for 8 stations a total loss of 8 X 1.14 = 9*1.2 horse power. The
maximum power transmitted is 104 H. P. and the minimum 40.3 H. P., so
that this gives a loss of about 9 per cent. of the maximum and 22 per eent.
of the minimum. This shows the objection to the use of too large journals.
* See Engineering, Vol. 37, 1874.
f See Iyeloutre.20 6
THE CONSTRUCTOR.
b) Stiffness o/ Robe.—Using Weisbach’s formula (253) given
in $ 268:
• S = 1.078 + 0.093 j?-
we have, calling T' the tension on the rope :
Sv = 0.093	v^11.6+ .(308)
for the resistance in foot pounds.
Example 2.— In the preceding case, v = 4400 ft., R — 73.8", and Tr = %
(jT+ /) = 0.5 X 202S = 1014 lbs., whence :
Sc = 0.093 X 4490 ^11.6 +	= io3^ ft* lbs*
This resistance comes twice at each station, and for eight stations we have a
total of 2 X 8 X 10,368 = t63,888 foot lbs., or nearly 5 horse power. Adding
to this the journal resistance we have a total of 9.12 -f 5 = 14.12 H. P. The
direct measurements of Ziegler gave T3-34i H. P , which is a reasonably close
verification of the calculatious. The total loss of efficiency is therefore :
—l* = 13.6 per cent. of the maximum,
104
and I^Q\' = 35 per cent. of the minimum,
the lesser of these being a very excellent resuit.
2 301.
Roteux’s System of Rope Transmission.
In the preceding sections the utility and importance of wire
rope transmission has been shown. The various applications of
the methods already discussed exhibit much ingenuity and abil-
ity 011 the part of the designers. At the saine time there ap-
pears to be a possibility of improvement, especially in the case
of the transmission of large powers over long distances involv-
ing a number of stretches.
The Ziegler system of intermediate pulleys has given excel-
lent results, but the following points may be enumerated as ob-
jectio 11 s :
a.	The great height of the supports usually necessary because
of the large size of the pulleys.
b.	The large base required for the supports, not only for ciear-
ance for the lower part of the rope, but also to resist the tension
of the rope.
c.	The necessity of making the supports of great strength
when gearing is to be carried.
These three points are ali well shown in the Zurich station,
Fig. 905.
d.	The resistance due to stiffness of the rope. This has
usually been considered unitnportant, uutil the recent investi-
gations have shown otherwise. (See the preceding section.)
e.	The loss of power when the rope becomes slack.
f.	The necessity of giving sufficient tension to the rope to in-
sure satisfactory actiou in warm weather and consequeut exces-
sive tension in winter.
g.	The unsightly soiling of the exterior of buildings caused
by the grease from the cable defacing the wall upou which the
receiving pulley is placed.
h.	The necessity of making the intermediate pulleys strong
enough to carry the heavy stress of the cable, thus iucreasing
the weight and consequently the journal friction.
It therefore appears advisable to devise a system which should
permit the supports to be made low and light, to use a light
cable under moderate tension, also to reduce the number of
splices, and to place the terminal pulleys inside of the building,
the pulleys being made as light as practicable.
Ali these points have been attained to a great extent in the
following system.
In the first place, the cable, whenever possible, is made in
one endless length from the driving to the driven pulley, thus
making the intermediate pulleys merely supports and permit-
ting them to be constructed very light. It is also desirable to
arrange the cable so that both parts shall be at the same height
from the ground and that this height should be as moderate as
possible.
In Fig. 908 is showru the arrangement of the power house, the
first driving pulley Tx being directly upon the motor shaft and
lying in a horizontal plane. The driving part of the rope then
passes around a sta ionary pulley Z, and is carried off in the
desired direction. The driven part of the rope passes around a
pulley L' mounted on a carriage runniug on a track parallel to
the direction of the line of transmission and by means of
weights a pull somewhat greater than 2/ is brought upon the
carriage. This tightener pulley L' is placed so as to bring the
driven part of the rope to the sanie height as the driving part.
The whole arrangement may be protected under roof as shown
and the rest of the building used for other purposes, but if
necessary the track and carriage may extend out of doors.
The intermediate stations may ali be supporting stations
meiely, unless power is to be taken off at an intermediate point.
If the transmission is a normal one, not using the method of in-
creased tension (see g 293) the same deflection will be obtained
in both portions of the rope by making the stretches for the
driven part half as long as those of the driving part, so that
every other station may be provided with a double-grooved
pulley, Fig. 909.
Fig. 909.
If no change in direction is necessary the cable is thus carried
to the driven pulley, the two parts being separated by a distance
equal to the diameter of the driving pulley Tx, and entering the
building where the power is to be received the cable passes over
guide pulleys Z6, Z-, and around the driven pulley T2.
When the load is reduced by throwing off machinery in the
manufactory,‘the tightener carriage is drawn tow^ard the turbine
(Fig. 908) by the driving part of the rope, since both parts give
a pull of % (T+ t). A spring buffer is provided to check the
motion of the carriage in that direction. A spring dynamometer
may be connected with the bearing of the other pulley Lx and
the tension thus measured experimentally. When the trans-
mission is set in motion
from a state of rest the
tightener pulley Z moves
slowly back until the
tension in the driven
part of the rope becomes
equal to t. Should the
rope have much stretch,
the carriage must have
sufficient travel pro-
vided, and when neces-
sary the rope must be
shortened. The stretch
of the cable is less in this
arrangement than with
intermediate driving
pulleys, because it is
bent less frequently
around the pulleys, and
the wear of the rope is
much reduced for the
same reason.
If angle stations are
needed the arrangement
of Fig. 910 is used ; this
requiring only two pulleys to each part of rope, instead of three,
as formerly, and the use of gear wlieels is avoided.
If the first driving pulley is in a vertical instead of a horizontal
plane, the arrangement shown in Fig. 911 a is used, this requiring
one more guide pulley than before. I11 this case the track for
the tightener carriage is inclined so that its weight is used to
produce the required tension. If it is desired to place the
tightener pulley horizontal the arrangement shown in Fig. 911 b is
used. In the cable of the Brooklyn bridge the tightener car-
riage is provided with a brake in order to check the suddenness
of motion due to variations of load. A friction device similar
Fig. 910.THE CONSTRUCTOR.
207
to c Fig. 709 will serve for this purpose if the angle d is made
somewhat greater than is given by formula (233).
If it is desired to place the driven pulley in the same plane
as ohe of the parts of the main line cable, the other part must
a.
be led over another angle pulley. If power is to be taken off at
intermediate stations these may be constructed as the angle sta-
tions of Fig. 910.
Various other forms of intermediate power stations may be
used without involving the use of gearing, as shown in Fig. 912,
in which a is for a shaft at right angles to the cable, and b and
c for inclined shafts for either direction of revolution.
The very moderate force which this system brings upon the
supporting pulleys permits them to be made very light. This
has been difficult of accomplishment with a cast iron rim. A
light wdieel can be made of wrought iron, using angle iron
riveted to a special shaped centre piece, as shown in Fig. 914.
Fig. 913.	Fig. 914.
These rims are bent by means of special rolls, and a tongue is
formed in the sides of the groove to hold the leather filling in
place. The arms are made of light flat iron and the hub of cast
iron ; the arms either beiug bolted fast or cast into the hub, the
latter being made in halves. Pulleys made in this manner are
very light.
The construction of the supports is also peculiar, as shown in
Fig. 913. The two posts are made of channel iron secured to a
block of stone in the ground by means of lead run in around
the holes in the stone. The whole is steadied by guy-rods, and
brackets are provided so that the bearings can be reached by a
ladder. In many cases these supports of iron are cheaper than
those built of stone.
b.
Fig. 915.
Pj
\
For the intermediate driving pulleys of cast iron, the form
shown in Fig. 915 is used. The hub is outside of both bearings,
but the plane of the pulley is midway between the journals.
The connection between the arms and the hub is made by
means of a hemispherical shaped device, somewhat resembling
the frame of an umbrella, and hence these have been called
“ umbrella ” pulleys. This construction enables the pulley to
be firmly secured and readily removed without disturbing either
bearing.
In Fig. 915 b, a modification of this form of pulley, the
umbrella-shaped hub being made separately, and a straight arm
pulley fitted upon it. This permits a single pattern to be used
for the centres of a number of sizes of pulleys, or wrought iron
pulleys may be used on cast iron hubs of this form. Instead of
two journals a single longer one may be used, two forms of
hangers being shown in dotted lines.
The use of the umbrella
pulley enables a very sim*	a.
ple form of support to be
used, either for single or
double stations.
Fig. 916 a, is a single
station composed of a
wooden post upon which
a projecting bearing is
bolted, and in which the
journal of the pulley runs.
At b, is a double station,
the post being made of
iron The dotted lines at
D indicate a small roof
to protect the bearings
from the weather. A
comparison of these forms
with the older style, as
for example, Fig. 903, will
show that merely the use	1 it)
of the continuous rope
and the umbrella pulley tjj 1LNI_*
will effect a great econ-
omy in construction. The
umbrella pulley is also
well adapted to be used
for rope sheaves for hoist-
ing machinery and for	Vig. gi£t
chain sheaves.*
* Various applications of the umbrella pulley will be shown hereafter.
The principle is also applicable to bell pulleys. At a, is a simple counter-208
THE CONSTRUCTOR.
A comparative example with that in § 300 will be a practical
illustration.
Example— The transmission at Oberursel is made in eight equal stretches
and seyen stations with two pulleys each, one driving pulley and oue driven.
This gives 16 semi-circular wraps of the rope about the pulleys, causing a
loss of 5.13 H. P. from stiffness. By the adoption of the new system there
would be three semicircular wraps at the power liouse (see Fig. 908), one 011
the driven pulley and two quarter wraps 011 the guide pulleys L$} L7 (see
Fig. 909) There are also 11 short ares of con tact, about ^ of a circle each, on the
supporting pulleys, which latter would be very light and on supports con-
structed as already described. The combined ares 01 contact rnake practically
about 5 semi-circular wraps or T5g of the resistance of the old arrangement, that
isT5B. 5.13 or about 1.6 H. P. This is nottoofavorableanestimate,as we havenot
included the effect of the excessive tension which often occurs by the con-
traction of the cable in cold weather, and which is entirely avoided by the
use of the tightener pulley and carriage.
The reduction of journai friction is also important, as the weight of the
pulleys and the effect of the rope tension are both rauch less. The total
weight of the pulleys will be only about % that of the old system, although
more pulleys would be used, and the journai diameter tnay be reduced to l/$
of the previous value. This gives a loss of ^ X § = i of the previous value of
9.36 H. P., which is 2.08 H. P. To this we must add a resistance of 0.40 H. P. for
the guide pulleys which have been addedin the new system, giving a total loss
of Noa —1.60 + 2.08 h 0.40= 4.08 H. P. The loss in the first instance is with the
new system 4 per cent. and in the second io per cent., as against 13 9 and 35.9
per cent. for the old system.
In this example there are tio intermediate power stations, the
entire amount of power less only the hurtful resistance. In
considering the question of the stress in the driving part of the
cable it is important to kuow whether the entire power is to be
transmitted to the end of the line or if a portion is to be taken
off at intermediate stations. If the initial forces at successive
intermediate power stations be indicated by Plf P2f P3i P±, etc.,
the successive tensions in the cable will be reduced, and hence
the deflection h should be determined for the stretches preced-
ing and following each station, and the tension in the cable will
vary according to the power taken off at intermediate poiuts.
The sum of ali the forces Py will in every case be determined by
taking the tension ty in the driven part at the first driven pulley,
from the initial tension Ty so that we have T — t —^LP. From
this equation we can deduce important results.
As an illustration we can assume the entire power transmitted
to be divided up among a number of intermediate stations, all
being operated by oue continuous cable, as shown in diagram
in Fig. 917.
Fig. 917.
In this case the rope passes the entire round of stations Plf
T2t T3, T4 to rn. returning to the main power house. The
rope returns to the power house at any angle with a tension /,
giving T= 2 P t. All stresses are regulated automatically
for each streteh of the rope, as the forces vary at each station.
If the work at any station is reduced or even becomes zero, the
tightener carriage responds and alters the deflection so that
T— t — 2 . P^ in which t remains constant. A transmission of
this kind, in which the cable makes a complete circuit of a num-
ber of stations, may be called a “ ring ” system. In Fig. 917, the
supporting stations are indicated by small rectangles or tri-
angles, according as the line is straight or makes an angle, and
shaft, at 3, a simple headstock for a small lathe, and at c, is a head for a
boring machine, the loose pulley runuing on a stationary sleeve, as already
shown ip Fig. 86*.
the power stations as shown are circles. At 7^ the rope passes
off into an auxiliary circuit, which may be called a “ring,>
transmission of the second order (see \ 260). The stations may
all be coustructed very simply. The supporting stations are
made with one pulley wrhen the line is straight, and wTth two
at the angle stations ; the power stations can generally be made
with only two pulleys, providing the necessary arc of contact at
is obtained, or three pulleys used if necessary, see Fig. 918.
In many cases it is desirable to use the system for under-
ground transmission, as in Fig. 919.*
Fig. 919.
In order to determine when an arc of contact o?, of the
proper magnitude has been obtained, we have, from (239), if P
is the greatest force to be transmitted by the pulley with a ten-
sion T':
P=- T'
e/a. — 1
efo-
jy
We will call the ratio which is the reciprocal of the modu-
lus of stress, the modulus of transmission, and let it be repre-
senbed by 0, whence :
1 ___ ef’°- — 1
t	e/'a
(309)
Neglecting the iufluence of centrifugal force, wTe have, from
§ 290, for f the values f — 0.22 and 0.25 to consider. Taking
these we get the following values for various angles :
Modulus of transmission 0.
Oc ==	15°	30°	45°	6o°	90°	120°	150°	1800	270°	360°	450°	540°
f— 0.22	O.06	O.II	0.16	0.21	0.29	O.38	0.44	0.50	0.65	0-75	0.86	0.88
= 0.25	O.07	O.I2	0.18	O.24	0.33	O.4I	O.48	0.54'	0.69	0.79	0.81	0.87
These values are shown graphically in the following diagram,
Fig. 920:
Fig. 920.
From this it wTill be seen that an arc of contact of 30° will per-
mit the transmission of ^ the power due to the tension T\ and
an arc of 90° gives about y3.
A convenient application of this principle is found in the
arrangement of a “ ring ” transmission when a large arc of con-
tact is obtained upon the first or main driving pulley by redu-
* This has been done in San Francisco by Boone, using a conduit for the
rope similar to a cable railway.THE	CONSTRU.
209
plication of the rope over a counter pulley, as in Fig. 795, and
also shown in the case of the double-acting belt transmission in
Fig. 860. By using a single-grooved counter pulley and double-
grooved driver we get oc ^ 360°, so that 0 is at least equal to 0.75.
In this way the specific capacity of the rope can be materially
increased, practically about 1%. times. If we give r = -i- the
V
value f in the first equation of \ 290, we have for the specific
capacity of a cable transmission with a counter pulley :
NQ = ——
33000
§1
A_
3
1
2475° *
•Si,
or say JVQ == -1—
25000
(3io)
The adaptation of the mechanism to receive the counter pulley
is usually not difficult.
The adaptability of the “ ring ” system of transmission to use
in distributing power in manufacturing establishments is appar-
ent, and for this purpose hemp rope is very suitable. This will
be shown by the following example :
Example 1.—The transmission shown in Fig. 881, § 286, contained 16 hemp
ropes 2 inches in diameter, each having a specific capacity N0 — 0.0021 and
v = 2360 feet per minute. The cross section of each rope is 3.14 sq. ins.
Hence N = NQ q v = 0.0021 X 3.14 X 2360 = 15.57 H. P. for each rope, or
249 H. P. for the 16 ropes.
Fig. 921.
Substituting the arrangement shown in Fig. 921, we take a single wire
cable composed of 60 Steel wires, and use a stress of 17,000 pounds in the
driving side of the cable and increase the speed to 3150 feet per minute. We
then have from (278):
q = 66,000
249
17,000 X 3I5°
= 0.307 sq. in.
Fig. 922.
If it is desired to use a counter pulley with theabove transmission, the ar-
rangement in Fig. 922 may be adopted. In this case the counter pulley G,
andtightened pulley L', are botli inclined so that the rope shall be properly
guided for the double grooves in the main driving pulley. The arc contact
a is in this case greater than 360°, and the specific capacity will be times
greater. This will enable the cross section of the rope to bereduced to § the
previous value, or q = §. 307 — 0.204 sq in. If we use wires instead of 60,
we have for the cross section of each wire
——— = 0.0056 sq. in., and 8 = 0.084 in.
30
The diameter of the rope will be from 8 to 9 6 or §" to the latter when
the rope is new.
The conditions of this example are hardly such as to demand
the introduction of the counter pulley, but when large powers
are to be transmitted its use is most advantageous. In some
instances the counter pulley may be arranged, as in Fig. 911,
so as to sustain a part of the weight of the fly wheel of the en-
gine, and hence materially reduce the journal friction.
In many instances the power in factories may be arranged so
as to use the “ring ” system of transmission, and dispense with
the use of the spur or bevel gearing, and some examples are
here given.
In Fig. 923 a is shown the usual arrangement of the trans-
mission of power in a weaving establishment.
	f 	- ’ . f >_ 		 ! \ ; .. !							
	* N:	.1				&-A		v.	.						
	r	2 i	i	ai ^ IF K. 1 » 1				
Fig. 923 a.
In this instance the two shafts which extend each way from
K, drive the line shafting by seven pairs of bevel gears, while
in some factories as many as 12 to 18 pairs are used.
and hence the area of each wire is
= 0.0051 sq. in., and the diameter 8 = 0.08".
Tn the original hemp rope transmission the main driving pulley had a
radius of 71 inches, and as we have increased the speed $ times, the driving
pulley must be proportionally increased, and hence the radius will be 95".
This gives a stress due to bending,
po.o8
j = 14,200,000 —7----— == 12,000 lbs. nearly; see formula (279),
this being not too great to 'give satisfactory results. We have, instead of a
wide face pulley made with 16 grooves, a single groove pulley made with
leather filling, as in Fig. 879 a, of 15 fit. 10" diameter. An important point
to be considered is the stress due to the bending of the rope over the pul-
leys T2, T3t etc. These pulleys were 36" radius for the hemp rope, and
hence §. 36 = 48" radius for the wire rope, or 8 feet diameter. We then have
from (279)
Fig- 923 £ shows bow a ring transmission can beused to drive
the same shafting, there being seven guide pulleys and one
tightener L', the guide pulleys being of the “ umbrella ” pat-
tern, as in Fig. 915. The tension weight for the tightener is
equal to 2 Tf.
I
Fig. 923 c.
S = 14,200,000
0.08
46
23,666, which added to the working stress of
17,000 lbs. gives a total.of 40,666 pounds, which is not too high for Steel wire,
according to § 266. The idler pulley Z, is made the same size as the driveu
pulleys T3, T4, etc., and the tightened pulley L’ can be made a little larger.
The loss of efficiency will be somewhat less than in the case of hemp rope,
since for wire rope there is a smaller modulus of stress r, (£. e. 2 instead of
2§, see § 287), and the initial force P, is smaller, because of the increase in
velocity and the loss from stiffness will be less. The loss from stoppage
and creep should also be considered as not unimportant (see § 287).
Another arrangement is shown in Fig. 923 r, this being used
when the alternate shafts are to revolve in opposite directions.
This permits the rope to be used double acting, as described in
§ 277 and shown in Fig. 860. Those portions of the rope
marked 1 in Fig. 923 c, are in one plane, and those marked 2
in a second plane, giving clearance to the parts of the rope,
and the rope is guided from one plane to the other by the
guide pulley Ll and tightener L/. Five of the seven driven
pulleys are double acting, and hence are made double grooved.210
THE CONSTRUCTOR.
Shafts which lie at right angles but in parallel planes, one
above the other, are also readily driven by use of a ring trans-
mission System.
Tn the preceding cases it is desired to obtain a double wrap
of the rope about the driving pulley K, the arrangement in
Fig. 924 may be adopted. In this case two idler pulleys Gl and
G2 are used to guide the rope from one plane to the other. The
rest of the rope, when either of the planes shown in Fig. 923 b
or c is used, is guided in a third plane by suitable pulleys. I11
Fig. 925 is shown an arrangement by means of which a series
of parallel vertical shafts, revolving alternately in opposite di-
rections, can be driven from a single horizontal shaft K.
The ring system is well adapted for driving a number of mill
stones, as arranged in Fig. 926, for example, in which ali the
mill spindles revolve in the satne direction. The direction of
the stones may be readily reversed by a corresponding change
in the cutting of the furrows, and lience the double-acting
arrangement as in Fig. 925 can be used if so desired.
The arrangement of the double-grooved pulley on the spin-
<Ile in this case is shown in Fig. 927. This is a modified form
of the umbrella pulley of Fig. 915, the hub being made in the
form of a hollow sleeve carrying a cone or other suitable clutch
Ky K', by which any pair of stones can be stopped without in-
terfering with the
others. An adjust-
able step for the
spindle is provided
at H.
In machine shop
transmissions it is
frequeutly required
to drive a series of
parallel c o u n t e r
shafts, which re-
volve in one direc-
tion, sometimes in
the opposite, and
any of wThich may
need to be stopped.
Such an arrange-
ment is shown in
Fig. 928. The rope
is carried over the
various pulleys of
one series around
the tightener pulley
L', and back over
the other series. At
Kx and K2 are fric-
tion clutches which
are throwu into en-
gagement on one
side or the other,
according to the di-
rection of revolution
required, or which may be left disengaged. If two adjacent
shafts are desired to revolve in the same direction, an inter-
Fig. 928.
mediate guide pulley is introduced, as shown at Lv The sub-
sequent belt trausmission from these counter shafts can be
greatly simplified by using the above system. In ali these ar-
rangements the modulus of transmission is determined as al-
ready discussed in formula (309) and the proper arc of contact
determined. For example, if in an arrangement similar to
Fig* 925> each part of the rope is in contact 30° on the pulley,
and the coefficient of friction fy is 0.22, we have from the pre-
ceding table for the modulus of transmission 0 = 0.11. If the
tension on the respective sides of cable be T' and t' y acting
upon the pulleys, we have for the maximum force transmitted
by the rope P' = o. 11 (T' + t'). In this case we have always
7y -f- t = T -f- t, and hence T = 2 2 P, t =2 P (see § 264).
Hence we have P' — 0.11 X 3 2 P, or about ^ 2 P. If there
were but three driven pulleys, each offering the same resist-
ance, the system would operate well, and stili better with a
greater number of driven pulleys. For mills of 20, 30 or more
pairs of stones, this arrangement is especially applicable, since
it furnishes a far simpler transmission system than heretofore.
This system, liowever, should not be carried beyond its proper
limits, and for small, light running mills, such as are used for
grinding paints, graphite, etc., belts are generally more advan-
tageous, being easier thrown in and out; the rope system
being better adapted for the trausmission of greater powers.
In all the various classes of heavier mills, such as are used
for grinding plaster, cernent, and the like, also for paper mill
machinery, the rope transmission is best adapted, replacing all
the heavy shafting, gearing and belting otherwise necessary.
An example will illustrate the method of applying the fore-
going principies.
Example 2.—I,et there be two sets of wood pulp mills each requiring 60
H. P. to be driven from a pair of turbines by a “ring transmission ” system,
the first moving shaft making 125 revolutions per minute. The main shaft
is driven bv the two turbines by means of spur gearing, and carries the
driving pulley of the rope system, making also 125 revolutions. We have
for the specific capacity of the rope, from (277) N0 =	and ifwe use steel
wire and take Si = 21,30x5 we get N0 = 0.323. Also we make the velocity v of
the rope = 3150 leet per minute, and we get for the cross section of the rope
from (278)
Q =
N
v N0
120
3150 X 0.323
= 0.118 sq. in.
If w^e make the cable of 36 wires wTe have for the cross section of one wire
o. 118"
—= 0.0033 and the diameter 5 = 0.073".
From the number of revolutions and the chosen speed of rope we have for
the pulley radius, R= 3^500^ = 48'', and using this in (279) we get for the
2 n- X 125
bending stress, f = 14,200,000 — = 21,500 lbs., which is satisfactory. The
A
entire 120 H. P. is carried by the rope to the first set, where 60 H. P. is used,
and the balance is transmitted to the second set, the necessary supporting
pulleys being introduced between the two points, and the required tension
t being given by the tightener carriage. Following the course of the rope,
we have at the driving pulley the tensions T and /, respectively equal to
2 v Pand 2 E, whence 2 P= 23000 * 12 ■ 1257 lbs. = t and T — 1257 X 2 =
25x4 lbs. The tension at the first set is reduced by F* = 628.5 lbs., whence
Tv = 2514 — 628.5 = 1885.5 lbs. At the second point again is taken off P' =
628.5 lbs., and the tension becomes 1885.5 — 628.5 = 1257 lbs., which is equal
to the above value of /, and which is obtained by loading the tightener car-
riage with 2514 lbs. or a little more.
This system requires the use of clutches for starting and stopping the
machines, and for this purpose AdjunaiPs coupling, (§ 307), is suitable.
In some instauces it is found practicable to drive two pulp
mills with one pulley, the pulley being between the machines
on an intermediate shaft with a fraction coupling at each end.
Another case may be given where a number of machines with
horizontal shafts each requiring the same amount of power, are
arranged in a row and drawn by a ring transmission system,
Fig. 929.
In this case friction clutches are placed at Kx Kx for stopping
and starting the machines, while the intermediate pulleys L Z,
which may be of the umbrella pattern, are carried *>n hangers
from the ceiling. The rope and the driving pulleys are covered
by guards S S to protect the workmen. This arrangement isTHE CONSTRUCTOR.
211
especially convenient if there is a second series of machines on
the floor above, when the pulleys L L become the driving
pulleys of the upper set, and no guide pulleys are required at
ali. It is sometimes desirable to make the driving pulleys of
umbrella form, supported on independent bearings, so that any
machine can be repaired or entirely removed without inter-
fering with the rest of the transmission.
It should not be forgotten that the ring system of rope trans-
mission generally involves an entire rearrangement of the
establishment, and that it can rarely be substituted for a shaft-
ing transmission to much advantage.
A comparison of the last example with the older system in
which a separate rope is used for each portion of the transmis-
sion will be of interest. In the previous method the pulleys
could not be brought close together because the tension would
require to be too great, and slight variations in temperature
would produce excessive variations in tension. These difficul-
ties are overcome in the ring system by the use of the tightener
carriage, which may also be used to much advantage in those
Systems of belt transmission which lie in one plane, such as
ha ve been shown in Fig. 844. The construction is similar to
that for rope transmission, and the umbrella hub may be used
to advantage. In many cases the specific capacity may be much
increased in this way.
The new system is also highly advantageous for long distance
transmission, especially where power is to be taken off at
several points, or it may be used in combination with the old
system, retaining the latter and using the new system for pur-
pose of distribution.
The difficulties of construction are much less for long dis-
tance transmission than with the old system, and the cost of
installation and supervision much smallei.
The application of the new system appears likely to increase
very greatly, since it involves less first cost than electrical
transmission piant, and also a higher efficiency when the losses
from transformation of electrical currents are considered.
This subject will be further considered in Chapter XXIV.
CHAPTER XXII.
CHAIN TRANSMISSION. S1RAP BRAKES.
I 302.
Specific Capacity of Driving Chains.
The use of chain for purposes of power transmission is neces-
sarily more restricted than the use of rope, but for single trans-
missions in special cases it is well adapted, and its applications
are increasing. Chain is especially capable of resisting varia-
tions of temperature and exposure to the weather and to dust,
and hence is well adapted for driving revolving drums in
mining machinery, washing machinery, the machines in bake-
ries, etc. In mining machinery chains are very extensively
tised, both above and below ground, not only for continuous
tramway driving, as in Fig. 802, but also for the transmission of
rotary motion over exteuded distances.
Chain sheaves are made either with smooth wedge shaped
grooves, or with pockets for the chain links as already indicated
in § 275. In the first case the driving is due to friction in the same
manner as with belting and ropes, while in the second case the
action is similar to that of toothed gearing.
The method of friction driving can be used with ordinary
link chain as at a, Fig. 930, and may also be used with the flat
link chain of Fig. 830 d, if so desired. The circumferential
friction F= T — t may be determined from the following rela-
tion (see \ 264):
T= t = Qi + 2/sin	...........(311)
in which T and t are the tensions in the driving and driven
sides of the chain respectively, and f is the coefficient of fric-
tion. The angle /? is that subtended by the pitch length of a
link of the chain at the centre of the chain sheave, and may be
obtained from r sin y fi = y l; the exponent m is the number
a
of link contacts, hence m = —. A sufficiently close approxi-
mation may be obtained by taking /3 = —, which gives for the
modulus of friction p :
p —7- = (y +/-0)“ ~.............(312)
In chain transmission the modulus of friction is not indepen-
dent of r, as with rope transmission, but varies somewhat with
y
the ratio —. This latter ratio in practice seldom goes below 5.
Taking this limit, and also putting/" =0.1, we have for practi-
cal values of p the following, in which u equals the number of
half wraps of the chain around the sheave :
1 v™
-f-0
whence:
The following table has been calculated for 1 to 8 half-wraps,
T
and gives the modulus of friction p = —, the modulus of stress
T
r — — and the modulus of transmission d (see (309)).
u —	1	2	3	4	5	6	7	8
p —	i-37	1.88	2.57	3-53	4-83	6.61	9.06	12.41
T =	3-69	2.13	1.64	i-39	1.26	1.18	1.12	1.09
6 =	0.27	0.47	0.61	0.72	°-79	0.85	0.89	0.92
These values for p and r are similar to those obtained for ten-
sion organs generally, as indicated in the diagram already given
in Fig. 816. It will be noted that the transmitting capacity of
chain even with a single half-wrap about a smooth sheave is
good.
Since the specific capacity of a driving tension organ (see
(262)) is equal to
Na =— — . — or — — S6
33000 T	33OOO
we have for ordinary open link chains the following values for
various stresses S:
5	u =	I	2	3	1 4	5	6	7	8
Sooo 7000 8500	JVo = A^0 = A^o =	0.042 O.O57 O.O7O	O.O7I O.O99 O.I2I	O.O92 O.I29 0.157	0.109 o.i53 0.185	0.120 0.168 0.203	0.129 0.180 0.219	0.135 0.189 0.230	0.140 0.197 0.237
The specific capacity is in all cases high, and for the generally
accepted stresses in the chain cross section it varies from
0.042 to 0.237. ‘Various applications permit variations in
the value of S, the value being taken lower when it is desired
that the wear through friction shall be reducedo The cross sec-
tion of chain is determined from the equation N = 2 q v
(see § 280) in which N is the horse power to be transmitted at
a velocity vy and q is the sectional area of the iron of which the
chain links are made. We have :
j_ _N_
2 v * N0
(314)
The value of v is always low, and hence the influence of cen-
trifugal force upon p may be neglected.
Example 1.—It is required to transmit 10 H. P., by means of a chain
making a half wrap about a smooth sheave, the velocity v being 1180 feet per
minute and S = 8500 lbs. We then have for the cross section q of metal:
2 X 1180	0.070
= 0.0609 sq. in.
which corresponds to a diameter of 0.3 in.212
THE CONSTRUCTOR.
Exatnple 2.—By using the counter sheave (Fig. 795) and thus obtaining
three half*wraps the value of 6 may be reduced to 5000 lbs., whence
? = ;
- = o 046 sq. in.
2 X 1180 * 0.092
or a diameter of 0.27 in.
This gives a lighter chain and at the same time a more durable one, as the
friction is materially reduced when entering and leaving the sheave (see
l 303)-
By using grooved and pocketed sheaves the specific capacity
may be greatly increased, the chain being held so securely that
Fig. 931.
as many as eigbt half-wraps may be used. Two very practical
arrangements for such sheaves are shown in illustrations, which
are from executed examples in the chain tramway of the Decido
iron mines in Spain, built by Brtill, of Paris. The dimensions
are given in millimetres, and the chain is operated under a
stress of about 5000 pounds per sq. in.
The sheave shown in Fig. 931 is for a 25 mm. (1" nearly)
chain, and is made with inserted teeth of steel, and the form of
Fig. 932 is similar, and is for an 18 mm. (0.7 in.) chain. In both
cases the teeth are radial, and formed to rec dve the chain links,
being secured by jam nuts in the second case, and by nuts
fitted with the Belleville elastic washers, which latter have
worked well in practice.
FiG. 933«THE CONSTRUCTOR.
213
In Fig. 933 is given an arrangement of chain sheave gearing,
including a solid massive form of bearing, as used in many
English collieries.* Here the sheave is made with eight semi-
circular ridges or ribs, similar to the old form of capstan shown
already in Fig. 794 a; and both parts of the chain are carried
on supporting pulleys. In many instances this arrangement is
used, by widening the face of the sheave, to receive several
wraps of chain, as shown in the upper right corner of Fig. 933.
If we may safely assume that the ridges increase the coefficient
of friction at least three times, in the preceding formulas (311)
and (312), we have for the corresponding modulus of friction p/:
p' = 2.$u..................(315)
which gives for
u		1	2	3	4
p' =	I-58	2.50	6.25	1563	39.06
r' —	2.72	1.67	I*I9	1 07	1.03
6' =	o-37	0.60	0.84	0.94	0.97
from which the security against zlippage and also the specific
transmitting capacity may be determined for any given case.
Within moderate limits chain transmission may be used as a
“ ring ” System, as for instance in driving the rollers of carding
machines, also in wood pulp grinding mills a ring chain trans-
mission is used for driving the feed rolls.
2 3°3-
Efficifncy of Chain Transmission.
The loss of efficiency in a chain transmission is due to jour-
nal friction, dependent upon the chain tensions Tand t; and
upon the friction of the links in entering and leaving the
sheaves. The journal friction is determined as already shown
in § 300, and for high values of 0, it is not great. The loss from
chain friction is due to the rotation of each liuk about its
adjoining link as an axis through an angle j3. This gives, with
a coefficient of friction fly a circumferentia! resisting force Tlt
due to chain friction (see formula 100)
____________*='■ (r+ 0 0) (4->
* The illustration is from Newcliurch colliery at Burnley.
In this case the pitch length l of the links is taken = 3.5 dy
and making r = 5 /, we get Fx = (T -f- /) 0.036 flt and if we put
for the loss at both sheaves:
we get:
Ek= 0072/, 4±-i.................(316)
Exantple 1.—Taking the coefficient of friction = 0.1 j on account of the
small bearing surface we have for a chain transmission on smooth sheaves
with half-wrap; p being = 1.37, as in the preceding section :
Ek = 0.072 X 0.15 -'37 = 0.0692
o-37
or say 7 per cent.
Exantple 2.—If the sheave is made with ridges, as in Fig. 933, we have fi
— 2.5, and hence
Ek = 0.072 x 0-15	~ 0,025
or only 2% per cent.	•
Exantple 3.—By using. carefully made pocketed teeth and making u — 8,
we have p = 12.41, whence
Ek = 0.072 X 0.25	= °*OI26
or only i}/x per cent., this reduction being due to the reduction in thetension
on the chain, showing the importance of considering the question of chain
tension in this connection.
In the preceding examples the friction of the links upon each
other has been considered, but not that of the links upon the
sheave. This latter is a very variable quantity, being unimpor-
tant with a smooth sheave, as Fig. 930 a, and sometimes
becoming excessive, as shown already in Fig. 838 b, § 275.
In every case all possible care should be taken to produce as
little rubbing contact as possible.
i 304.
Intfrmfdiatk Stations for Chain Transmission.
The most important applications of chain transmission are in
mining work, both above and below ground ; and especially in
coal mines. In this branch of work England takes the lead,
followed by America, where, however, wire rope is more exteu-
sively applied, while in Germany the most applications are
found in the Saarbruck district.
A very interesting application of endless long distance chain
transmission is shown in Fig. 934, which gives two views of the214
THE CONSTRUCTOR.
Gannow mine at Burnley in Lancashire. The driving pulley
is at T, and guide pulleys at Ly while at L' is a tightener pulle}'
hung between two idlers, a construction which is frequently used.
The rotation is modified in various ways in the English mines,
stations similar to those of rope transmission Systems being used.
Fio. 935.
In Fig. 935 is shown an intermediate station at 7\ T2> and
a^so on angle station at L. In many instances combinations of
bevel g*»aring and shafting are found in connection with chaiu
transmission, but the examples here given are confined to the
use of chain alone.
Fig. 936.
In Fig. 936 an intermediate station is shown at T3 T4, and a
change station at 7"i T2. At Tly Fig. 936, the chain makes an
entire wrap around the sheave, the latter being made with a
wide groove, and interference of the two parts of the chain pre-
vented by guide sheaves. The simple supporting stations are
made with small horizontal guide sheaves, with wide grooves.
The velocity of the chain varies froin 200 to 500 feet per minute.
2 305.
Strap Brakes.
Tf a driven pulley is embraced by a tension organ, either belt,
rope, strap or chain, the ends of which are subjected to tensious
T and and also keld from moving, the pulley is hindered from
moviug toward so long as the force acting to rotate it does
not exceed P — T — t. The tension organ then forms, with the
pulley and stationary frame work, a friction ratchet System in
which the tension organ forms the pawl. If the tension 7"be
reduced until T — t < Py the pulley will slip in the strap, over-
coming the frictional resistance due to T — t, and the motion
can be made slower, if T and t be made great enough, so long
as their difference is only slightly smaller than P. The tnechau-
ism then becomes a form of checking ratchet ($ 253) better
known as a friction brake, or simply as a brake. Such brakes,
when made with tension organs, are called strap brakes.
Strap brakes are made in various forms to suit the applica-
tion.
(a) Clamping Brakes.—When a strap brake is to be used to
act as a complete clamping brake, to check motion entirely, the
teusions T and must be determined. These are obtained from
formulas (239) and (240) or from the graphical diagram of Fig-..
816. Such strap brakes are frequently made with straps of iron
or steel. It is generally desirable to so arrange the parts that
the motion of the pulley acts to draw the strap into closer
engagement, which may be done in various ways. Fig. 937
shows several such arrangements.
The various parts are indicated as follows : 1 is the axis of the
pulley ; 2, the point of application of the brake ; 3, the attach-
ment of the tight side of the strap ; 4, the attachment for the
slack side ; 5, the axis for the brake lever. In Fig. 937 a, 3 and
5 are separate ; in Fig. 937 b they are combined in one, and in
Fig. 937 c both 3 and 5 are separate, but 3 and 5 are made rnova-
ble, and 3 and 5 are so nearly in line with T that a very slight
effect is produced on the lever by T.
In Fig. 938 a 3 and 4 are combined, and at the same time 3
and 5 are nearly in line with T% Fig. 938 £ is the so-called
b.
“ differential ” brake of Napier, in which 3 and 4 are so placed
that perpendiculars to the directions of T and t are inversely
proportional to those tensions, thus reducing the action of the
strap upon the brake lever to a small amount. Fig. 938 c shows
an arrangement adapted to permit the pulley to revolve in
either direction. The angle 3.5.4 can be so chosen that the
force upon the lever may be very small.
For keavy hoisting machinery, the braking power required
makes the arrangement shown in 939 suitable. In this case the
strap is filled with blocks of wood in order to obtain a bigher
coeflicient of friction and at 6 is shown an application of the
globoid worm and worm w7heel shown in Fig. 641.
Example.—Required a brake for a shaft driven bya force of 2200 poundsat
a lever arm of 7.875 inches. The forni chosen is that of Fig. 938 a. the arc of
contact cx of the strap being 0.7 of the circumference. The coefficient of
friction f— 0.1, the strap being lubricated. We then have f & — 0.1 X 14 tr,
= 0.14 7r = 0.43. We then have from the second table of § 264, the tension
T	T
modulus r = —p- = 2.88 nearly, and the friction modulus p = —y— = 1.5 (see
also the diagram, Fig. 816).
we make the brake pulley with a radius of 15.75 in., the braking force at the-
circumference of the pulley must be ~~~ . 2200 = 1100 lbs., and t — 1.02 X
15.75	*
1100 = 2112 pounds. and T — 2.88 X noo = 3168 pounds. If the brake is to be
operated by a hand lever wtth a force of 44 pounds, the ratio of the length of
the hand lever to lever arm 4 . 5 must be —----= 48. The strap is under a
tension of T = 3168 pounds. If we assume a permissible stress of S = 14,220
lbs. and a thickness of strap 8 — 0.08" the width will be:
3168
14,220 X 0.08
= 278",
which is quite practicable.THE CONSTRUCTOR.
215
The question of the pressure between the braking surfaces is of interest.
According to formula (241) -% = -J - we have for the tight end, where
o o K
S = 14,220.
o.oS
p = 14,220-----= 72 lbs.
15-75
and at the slack end, since	p = % . 72 = 48 lbs., both of #hich
such small values that the wear must be very slight.
This example showshow, in aproperly arranged construction,
a great ratio of force to resistance can be obtained. In large
winding engines the brake pulley can readily be cast in one with
the rim of the drum gear.
The method of securing the ends	a	b
of the metal strap is shown in Fig.	1	!
940. The form at a, is secured by .
countersunk rivets, and that at b,
by an anchor head and a single
small rivet to prevent lateral slip-	FiG. 940.
page.
(b) SlicLing Brakes.—In using clamp brakes operated by hand
for lowering heavy loads in hoisting machinery, great care must
be taken, since the throwing out of the checking pawls puts the
entire resistance on the brake. With this arrangement there is
always more or less insecurity, the safety depending upon the
handling of the lever, and serious accidents have frequently
occurred. This danger can be avoided by the use of automatic
sliding brakes, the following form being designed by the author,
and shown in two forms in Fig. 941. The brake pulley a, is
loose on the shaft, but engages with it by means of a ratchet
System a' b7 c'. The brake is subjected to a tension equal to
a.	b.	c.
Fig. 941.
the greatest braking force desired ; i. e. so that the weight K
must be raised in order to permit the load to run down. If the
lever is let go, for any reason, the descent is checked. I11 form
a, the pawls are attached to the pulley, and the ratchet wheel
a7 keyed to the shaft; in b, the pawl is on a disk c/. When the
load is raised the combination forms an ordinary ratchet train.
A silent ratchet, Figs. 673, 674 may be used for this device. At
c, is shown a pendulum couuterweight, which can be adjusted
so as to vary the braking power to suit various loads.
Another form of sliding brake, also designed by the author,
is shown in Fig. 942. In this design the strap b, is given such
tension /, by means of the screw e 7, and lever c, as to hold the
load from descending ; a rubber spring being introduced at 7.
If the load is to be lowered, the clamp e, is loosened, but is
again tightened on ceasing. When hoisting, the tension t at 1"'
is readily overcome. This is in realitv a form of running ratchet
gear, and as shown it is made with a strap of wedge section, the
angle 6 being 450. The wedge portion is made of wood on iron
at leasto.2ol increased by —~~p~	when used to multi -
sm —
2
ply the value of f cx , requires a very small force to overcome
the tension t.
§306.
Chain Brakes.
Chain may be
used as the ten-
sion organ in
brake construc-
tion, in which case
it is generally
lined with blockl
ofwood, as in Fig.
943. Thetensions
T and /, to be
given to the two
parts of the cha^ri.
are readily oV>-
tained from for»
mula (312). The
ratio of chain
pitch length /, to
the pulley radius
r, is increased be-
cause of the use of the wooden block. When l — r and the
arc of contact is less than 180°, we have :
'-f-0+f)’.................... •
For wood on iron we may take f = 0.3 (see section 193)»
This gives :
T
p = -j- = 1 . i9 — 2.35 ; also
and — = t — 1 = 0.74, or/ = 0.74 P.
These proportions should not be strictly foliowed for heavy
brakes such as in Fig. 939, as sueh should be determined for
each case.
2 307.
Internae Strap Brakes.
Strap brakes may be used in internal pulleys, in a manner
similar to the internal ratchet gear of Fig. 711, for example.
The outside of the strap then acts upon the inner surface of the
pulley, the strap being subjected to compression instead of ten-
sion,* thus becoming a pressure organ, a subject treated more
fully in the following chapter.
a..	"b.
The pressure of the internal strap brake is of the same mag-
nitude as with the external brake, but in the opposite direction,
so that the previously determined value of p from the forces T
and t, may be used. Fig. 944 shows three fornis of such brakes,
these being used for friction couplings, and hot in hoisting
machinery (see Fig. 449). Fig. 944 a, is Schurman’s friction
coupling. f The brake lever c, acts by means of a wedge 4,
upon one end of the strap. The other end of the strap is held
by a pin 3, to the member d, which is to be coupled to a by
means of the strap b. The lever c, is also pivoted to the mem-
ber d. For the forces T and /, wre may use formula (239), and
since o< is nearly = 2 tt, or say = 6, we have for f = o. 1 the
value/ ot = 0.6, which from the table of § 264 gives p — 1.82,
and r = 2.22, whence t — 1.22 P. The strap must be released
by the action of a spring.
Fig. 944 b, is Adyman’s coupling, J which is made with a
heavy cast iron riiig. The ring b, is made in halves, b/ and b",
fitted with projections 4' and 4" which eugage with an interme-
diate sheave keyed on the shaft.
* See Theoretical Kinematics, p. 167 ; p. 548.
f Zeitschrift des Vereins Deutscher Itigenuiere, Vol. V. p. 301.
X Made by Bagshaw & Sons, Batley, Yorkshire.2l6
THE CONSTRUCTOR.
The levers c' and c" have a common axis at 5, and when
separated by a wedge at 6, they press upon the ends of the ring
at y and 3/r. A pin at 7, keeps the levers from sliding in the
direction 7 . 1, as well as the ring b' b".
The coupling shown in Fig. 944 c} acts both ways, as an inter-
nal and external strap brake, and is used on a shaping machine
by Prentiss. * The Steel strap b, is covered with leather. When
the arms c' c" are drawn together it acts as an external strap
on the pulley a", and when they are forced apart it becomes an
internal strap in the pulley a'. The arms c' c" are carried on
sleeves and are rotated to or from each other by a screw action.
CHAPTER XXIII.
PRESSURE ORGANS CONSIDERED AS MA CHINE ELEMENTS.
I 3<>S.
Various Kinds of Pressure Organs.
In distinction from the various kinds of tension organs which
have been considered in the four preceding chapters, there
exists another group of machine elements of which the sole or
principal characteristic is that they are capable only of resist-
ing forces acting in compression. This group includes fluids,
both liquid and gaseous, whether limpid or viscous, such as :
Water, oil, air, steam, ali pasty substances, clay, molten metals;
also granular materials, ali kinds vf grain, meal, gravel, etc.
In all these materials the principal feature lies in the fact that
the particles are subdivided to such an extent that they can be
separated from each other by a very small force, while on the
other hand they are capable of opposing more or less resistance
to compression, this resistance in many instances, as, for exam-
ple, in the case of water, almost equalling that of metals. These
materials may be used as machine elements in a great variety of
ways, and in the following discussion they will be included
under the general title of Pressure Organs. Eike their coun-
terparts the tension organs already discussed, they are used
largely for the transmission of motion in various manners, but
are of stili greater importauce on account of the great variety
of physical conditions in which they appear.
I 3°9-
Methods of Using Pressure Organs.
The distinction which has been made between tension and
pressure organs enables various points of contrast and compari-
son to be made as regards the methods of ulilizing them, and
pressure organs may be divided in the same mauner as tension
organs (see § 262) into standing and running organs. These
divisions have but little practical application in this instance,
but the three followdng subdivisions in $ 262, viz.: Guiding,
Supporting (i. e, raising or lowering), and Driving are here
applicable also. We may therefore distinguish pressure organs,
when considered as machine elements, into the following
classes :
1.	For Guiding.
2.	For Supporting (including raising and lowering).
3.	For Driving.
These various methods of action may be used either separately
or in combination, and are found in most varied forms in many
machine constructions. The great variety of possible combiua-
tions makes it desirable for a general view of the entire subject
to be taken before discussing details.
? 310.
Guiding by Pressure Organs.
In order to use a pressure organ for guiding, i. e., to compel a
more or less determinate succession of motions, it is necessary
to use also two ether machine elements formed of rigid mate-
rials. These latter are for the purpose :
a} Of resisting the internal forces of the pressure organ
and keeping it within the desired limits.
b, Of connecting the pressure organ with the external forces
to be received and opposed.
Tubes, Conduits, Canals.—The tube a, Fig. 945, limits the
boundary of the particles of the pressure organ, and retains it
in the desired form and Controls its direction. A bend in a
tube corresponds to a pulley around which the pressure organ
is bent, and thus has its direction changed. Even when no
change of direction is made, the tube is necessary to oppose re-
sistance to the particle of the pressure organ, and hence at
every section it must offer resistance to tension as well as com-
pression. Conduits, or channels, as at b, are tubes with one
side left open, the force of gravity or the so-called “living
force” of the pressure organ serving to retain it within the
desired limits. Canals are merely conduits of larger dimen-
sions, as at e, and natural streams of water often serve the pur-
pose.
Driving Organs, Pistons and Cylinders.—The bodies by
means of which the pressure organ is connected with the exter-
nal forces and resistances with which it is intended to act
mechanically may be called generically, Driving Organs, and
are very varied in character. Among these are movable recep-
tacles, also moving surfaces or moving conduits (as in turbines),
and also moving pistons in tubes or cylinders. A piston serves
to oppose the stress in the pressure organ in the direction of its
motion, while the walls of the tube oppose their resistance at
right angles to the direction of motion. The inclosure in which
a piston acts is called, in general terms, the cylinder, and details
of construction will be given hereafter. The principal types
will here be considered briefly.
A complete working contact between piston and cylinder can
only be obtained when both surfaces are alike, and this is only
geometrically possible with three forms of bodies ; i. e., prisma-
tic bodies, bodies of rotation, and spirally formed bodies. Of
these the prismatics are most useful, and among the prismatic
bodies the form most extensively used is the cylinder.
The fit of a piston in its cylinder, entirely free from leakage,
is very difficult of attainment, and is rarely attempted in practice.
In steam indicators the piston is very accurately fitted directly
into the cylinder, but in most cases a practically satisfactory
resuit is obtained by the use of some intermediate packing
device.
a	b	c	d	e
In many cases a soft packing of hemp or leather is used, Fig.
946. At a is showm a piston with external packing, at b an
internal packing. In these cases one entire end of the cylinder
is open, the piston filling the entire cylinder and acting upon
the iuclosed pressure organ on one side, this constituting a
single-acting position. At c and d are similar double acting
pistons. Pistons of the forms showm in a and b are sometimes
called plungers, and the shorter inclosed pistons, as c ox d, are
also called piston-heads. At e is a double-acting piston used in
connection with a rod and stuffing box, the rod being connected
with external mechanism, and the stuffing box made either with
external or internal packing, as iudicated at 1 and i/. In many
instances pistons are made with openings which are fitted wdth
valves, and hence may be called “valved” pistons, while those
here showrn are termed closed or solid pistons.
The tightness of the packing is usually produced by the appli-
cation of some external force, but in the so-called forms of self-
acting packing the necessary pressure is supplied by the con-
fined fluid. This is shown in the following illustrations.
a	b	c
Fig. 497-
Fig. 947 a and b} Cup packing for piston or stuffing box ; meta!
♦See Mechanics Feb., 1884, p. 140.THE CONSTRUCTOR.
217
packing, usually for pistons, but also used in stuffing boxes.
The fluid in ali three cases enters behind the packing rings and
tightens the joint in proportion to the increased pressure.
In the class of self-acting packing may also be included the
various forms of liquid packing, some of which are given in
Fig. 948. The forms at a and b are practically plungers, while
in many cases an ordinary packing has its tightness increased
by a layer of water or oil upon the piston, as shown at c.
Another variety occurs when the connection between cylinder
and piston is made by means of a membrane or diaphragm, as
in Fig. 949.
These are among the oldest forms of transmission organs,
but are practically true pistons in principle and action. At a is
a single diaphragm, known as the monk’s pump : b is the so-
called ‘ * bello ws ’ ’ form ; c is a series of flexible metal diaphragms,
usually of Steel, brass or copper, used for pressure gauges or
other similar purposes involving but little movement. At d is
the so-called “bag” pump, in which the liquid does not come
in contact with either cylinder or piston, but is confined within
a flexible bag.
a	b	c	<1
i 3”-
Guide Mechanism for Pressure Organs.
The combination of a pressure organ and its accompanying
guide mechanism forms a pressure transmission System. Fx>
Fig. 951.
amples of such systems are given in outline in Fig. 951. At a
is an arrangement for raising the load Q vertically. The
plungers b and d are of the same diameter ; the pressure on b
must be the same as Q, neglecting friction. The column of
water is the same diameter as the plungers, and the direction is
changed an angle of 120°. It is desirable that distinguishing
names should be given to the various arrangements. If we
compare these with the corresponding parts in tension organs,
Fig. 784 and Fig. 785 a, we may properly call such an angle
transmission a hydraulic pulley, or water pulley, but a stili bet-
ter name is the “ hydraulic-lever ” or “ water-lever,” which will
be hereafter adopted.
At b is shown a free water-lever. The plungers b and d are
equal in diameter, the load Q is supported on two columns of
water, hence, if friction is neglected, the force on each plunger
will be yz Qy the angle of change of direction is 180°.
At c is a combination of case a with case b. The plungers
bXy b2y bSy are of the same diameter, and the load Q is supported
on these columns. These three cases correspond in principle
with the similar cases ab coi Fig. 784. Since the three plungers
bXi ^2> °f case c a11 exert the same force, they may also be
made to give the same resuit when made as shown at d, or if
the three plungers are combined in one, forni e is obtained.
The latter form is well known in practice as the hydraulic
press. The principle involved in ali these devices is the same
as appears in the various pulley systems of tension organs.
A comparison of Fig. 951 a with e shows that the same prin-
ciple exists in both. and case a may be considered as a water-
lever of equal arms, and case e as a lever of unequal arms.
Another class of pistons is that in which a tight packing is
not attempted, these usually being used only for air. Fig. 950 a
shows a deep piston with grooves formed in it, the fluid eudea-
voritig to pass the piston in the opposite direction to the motion
of the latter, becomes entrapped in the grooves, and before it
can pass, the direction of motion is changed and this action
reversed.* At b is a piston with a brush packing, used for a
blowing cylinder at Sydenham. In this class of pistons we may
also include floats which rise and fall with the motion of the
liquid. Such floats are shown at c and dy the former being open
and the latter closed. A solid block may also be used for this
purpose, if its weight is nearly counterbalanced by another
weight.
Details of piston and cylinder construction will be given in
Chapter XXVI. The corresponding machine elements to pis-
tons in tension organs will be found for ropes in Figs. S25-826,
and for chains in Figs. 831 to 834. The change of direction from
compression to tension dispenses with the necessity for a
cylinder.
Fig. 952.
The water-lever has been used in more or less complete de-
vices for balancing the weight of pump rods in deep mine
shafts. Fig. 952 shows Oeking’s water counterbalance.f The
* See Weisbach, Vol. III., Part 2, § 4:0.
f Zeitschrift Deutscher Ingenieure, 1885, p. 545. Oeking incorrectly call
the device a b an accumulator.218
THE CONSTRUCTOR.
pump rod is carried on the two plungers dx d2, and its weiglit
counterbalanced by the weighted plunger and cylinder a-b.
In the Eniery scales and testing machines water-levers of
uuequal arms are used in connection with metallic diaphragms.
Fig. 953 shows a combination of two hydraulic levers, each of
the form of Fig. 951 a. The weight Q travels in a straight line,
being kept parallel by the four equal plungers bxb2b3b4, and
crossed pipe connections. This construction is similar to the
cord parallel motion of Fig. 7S4 d.
In all of the devices described the rigid body is guided by the
motion of the pressure^organ. It must be remembered that
motion is merely a relative term, and the rigid body may move
through the fluid. An example of the latter is the rudder of a
vessel, which acts in one plane ; or in the case of the Whitehead
torpedo several rudders are used, guiding the torpedo in any
direction.
§ 312.
RESERVOIRS FOR PRESSURE ORGANS-
Reservoirs are used in connection with pressure organs in
order to enable a number of applicatious to be operated collec-
tively, and also to enable the pressure to be stored for subse-
quent Service, and in this respect they correspond to the various
forms of winding drums used with tension organs, and shown
in Fig. 787. The following illustrations will show the use of
such reservoirs.
Fig. 954 shows a tank for use with petroleum distribution, as
used in the American oil fields, and more recently in the oil
district of Baku. The oil wells are at ax, a2t a3> and the oil is
forced to the elevated reservoir at c by pumps. From the reser-
voir the oil flows to the point of shipment d, and the supply is
gauged by the fluctuations of level in the tank.*
The reservoirs used in connection with the water supply of
cities are similar in principle. Where the configuration of the
land demands it, the pipes are run in inverted siphons connect-
ing intermediate reservoirs. An illustration of this arrange-
ment is given in Fig. 955, whicii shows the waterworks system
of Frankffirt-am-Main designed by Schmick.
The highest spring is at ax, Vogelsberg, and the next at a2,
Spessart. These both deliver into the reservoir clt r2, at Aspen-
hainerkopf. The next reservoir is at c3, Abtshecke, from which
the water flows through b4 to the reservoir c4 and r5, from wdiich
the city is supplied. The elevations above sea level are given
* A system of this sort was built in 1887 from Baku to Batoum on the
Black Sea. The length of line.is 1005 kilometres (603 miles), 6 in. diameter,
and the reservoirs 3000 feet above sea level.
in metres. The flow between the various reservoirs is controlled
by suitable valves.f
Small tanks are in very general use at railway stations ; and
the various ponds and mill dams used in connection with water-
wheels are other examples. In many cases the water ways are
large enough to serve as reservoirs also, as in the case of canals.
Natural reservoirs are fouud in the case of many mountain
lakes, the Swiss lakes affording many numerous instances.J
Such basins are also formed artificially by constructing dams
across uarrow outlets, and so storing the water for use. Note-
worthy examples found in France, the basin at St. Etienne,
formed by damming the river Furens, being over 164 feet (50
metres) deep.£
Water may also be stored in accumulators at kigh pressures
from 20, 50, as higk as 200 atmospkeres, and can theu be used
for operating hydraulic cranes, sluice gates, drawbridges, etc.
These accumulators may be cousidered as a form of releasing
ratchet mechanism (see § 260). To this class of mechanical
action also belongs the system, used in the Black Forest, by
which the streams are temporarily damrned and then suddenly
released in order to float the logs down with the sudden rush of
the current.
I11 using high pressure water transmission it is sometimes
desirable to transform a portion to a lower pressure in order to
operate a lower pressure mechanism, or by a reversal of the
same principle, to convert a lower to a higher pressure. This
can be done by means of the apparatus devised by the author>
and shown in Fig. 956. ||
a	b	c
This is a form of hydraulic lever of unequal leverage, but
is different from those shown in Fig. 95Referring to Fig.
956 a, the high pressure water is delivered at a, and connected
with the lower pressure water ax by means of the plungers b, blt
the latter being in one piece of two different diameters. The
difference in pressure, neglecting friction, will be inversely as
the areas of the two plungers, or if they are of circular section,
inversely as the squares of their diameters. I11 this case the
lower pressure then acts in the cylinder cupon the plunger d.
The action of this arrangement may be cousidered as if the
plungers b and bx were upon the same axis and rigidly con-
nected, and the leverage compounded in a manner similar to
that of the rope erane of Fig. 792 a ; this comparison being
more clearly shown by referring to Fig. 956 b, This device may
also be used as a supporting hydraulic lever, similar to Fig. 951
If a communication is made between the two different water
columns, as shown in Fig. 956 c, the pressure will be equalized.
This gives a differential hydraulic lever similar in principle to
the Chinese windlass of Fig. 790 0, or the Weston Differential
Block of Fig. 796 e.
f A large inverted siphon is formed by the new Croton Aqueduct, which
passes under the Harlem Rxver at a depth of 150 feet below the surface of the
river, and a tunnel of 10^ feet in diameter driven through the solid rock.
See Mechanics, Nov., 1886, p. 241.
t This is examined in detail in a memorial on the better utilization of
water, published at Munich in 1883 by the German Society of Engineers and
Architects.
§ For further discussion of this subject the following references may be
consulted : Jaubert de Passa, Recherches sur les arrosages chez les peuples
anciens, Paris, 1846; Ditto, Memoire sur les cours d’eau et les canaux
d'arrosages des Pyrenees orientales; Nadault de Buffon, Cours d’agriculture
et d’hvdraulique agricole, Paris, 1853-1858; Ditto, Hydraulique agrieole. ap-
plication des canaux d’irrigation de 1’Italie septentrionale, Paris, 1861-1862 ;
Baird-Smyth, Irrigation in Southern India, London, 1856 ; Dupuit, Traite de
la conduite et de la distr. des eaux, Paris, 1865 ; Scott-MoncriefF, Irrigation in
Southern Europe, London, 1868; Linant de Bellefonds Bey, Memoire sur les
principaux travaux d’utilite publique en Egypte etc., Paris, 1873; Krantz,
Etude sur les murs de reservoirs, Paris, 1870; F. Kahn, TJeber die Thalsperre
der Gileppe bei Verviers, Civil ingenieur, 1879, P* 1> also an article by
Charles Grad in “ la Nature,” 1876, p. 55 ; also a brief article by the author
“ Ueber das Wasser,” Berlin, 1876.
B See Glaseris Anualen, 18S5, Vol. XVII., p. 234.THE CONSTRUCTOR.
The opposite extreme to a high pressure accumulator is found
in those pools or receptacles of water far below the natural sea
level, such as are found in mines, and in the polders or drainage
pools of Holland, Lombardy, and parts of Northern Germany.
Reservoirs are not confined to use with liquids. Examples of
Other fluids are found in the gasometers of gas works, in the
receivers for compressed air, so extensively used in mining and
tunneling, and in making the so-called pneumatic foundations.
Smaller reservoirs are found in the air-chambers on pumping
machinery, and the like.
The sewage System of Berlin, designed by‘von Hobrect, con-
sists of ten drainage pits, with the water level below the natural
level, arxanged on the so-called radial system. The sewage is
pumped from these pits and delivered by means of pipes to
sewage farms at a distance from the city.
Negative receivers, so-called, may be used for air, as in the
case of the coining presses of the English mint, where a vacuum
chamber is used to receive the air already used for driving the
machines, and kept pumped out by steam power. The venti-
lating apparatus for mines also often coutains such negative
reservoirs for air.
Reservoirs are also used for granular materials, such being
extensively used in connection with grain handling machinery.
A steam boiler may be considered as a physically supplied
reservoir, as well as a physical ratchet system (see § 260). A
combined physical and Chemical reservoir is found in the elec-
trical accumulator, which may properly be called a current*
reservoir. A combined physically and mechanically operated
negative reservoir is found in the various forms of refrigerating
machines.
A modern application of pressure organs, and one which is
rapidly extending in use, is that of the distribution of power in
cities. Following the impulse given by the introduction of the
high pressure water system of Armstrong, the use of gas in
motive power engines by Otto followed, and many other methods
of meeting the problem have been applied.
In long distance trausmissions of this sort, special reservoirs
are often used, in which force may be stored, so to speak, and
from thence distributed in a manner similar to the ring trans-
mission system for rope (see \ 301). In this method the pres-
sure organ after use is returned to the reservoir to be compressed
and used again, or it may be used as in the line transmission
and allowed to escape at the end of the line.*
The following cases are given as applications of pressure
organs in long distance transmission :
1.	The London Hydraulic Power Company distributes 300
H. P. by means of water at a pressure of 46 atmospheres (675
pounds). A similar and earlier instailation is in use at Hull.
2.	The General Compressed Air Company distributes power
by means of air at a pressure of 3 atmospheres (45 pounds) in
Leeds and Birmingham. The system is an open line, and 1000
H. P. are used in Leeds, and 6000 H. P. in Birmingham.f In
Paris the Compagnie Parisienne de l’air comprime, procedes
Victor Popp, distributes po\yer from three stations in quantities
varying from a few foot pounds up to 70 or 80 H. P., a total of
some 3000 H. P. The use of compressed air appears to be
destined to a widely extended use for this purpose.
3.	The distribution of power in New York by means of steam
maius is extensive and w^ell known.
4.	The vacuum system is used also in Paris by the Societe
anonyme de distribution de force a domicile. This is an open
line transmission, operating in 1885, about 200 H. P.
5.	Transmission by highly superheated water has been used
in Washington, by the National Superheated Water Co., dis-
tributing heated water at pressures from 26 to 33 atmospheres
(400 to 600 pounds), the water being converted into steam at the
point of utilization.
6.	The distribution of power by means of gas holders has
already been referred to, and the distribution by electric cur-
rents is rapidly being developed.
1313-
Motors for Pressure Organs.
The methods of applying pressure organs to the development
of motive powTer are even more varied as in the case of tension
organs. For this reason a general view of the subject will be
taken in order to obtain a classification which will simplify the
discussion. The main distinctions are those of the character of
the motion of the mechanism, and of the method of applying
the pressure organ to the motor.
The great difference in the character of the motion of the
* See a paper by the author in Glaser's Annalen, 1885, Vol. XVII., p. 226.
f See Lupton and Sturgeon, Compressed Air vs. Hydraulic Pressure, Leeds,
1886: Sturgeon, Compressed Air Power Schemes, London, 1886; also The
Birmingham Compressed Air Company, Birmingham, 1886.
219
mechanism lies in the fact that it may be either continuous or
intermittent, so that the motor may be either :
A running mechanism, or
A ratchet mechanism (compare § 260). The ratchet pawls for
pressure organs are the various forms of valves (see Chapter
XXVI).
The various fornis may also be classified according to the fol-
lowing importaut distinctions based on the method of driving.
The pressure organ may drive, or
It may be driven, or
The impelling mechanism may itself be propelled.
There is also a third distinctiou to be made, namely, whether
the pressure organ acts merely by its weight, or whether it acts
by its living force of impact. This last distinction cannot be
sharply observed in practice, but is especially to be considered
in discussing the theory of action of the various machines.
In the following pages the various applications will be shown
in a manner similar to that employed in \ 262 for tension organs,
following the system of classification outliued above, and be-
ginning with running mechanism as the simpler of the two
great divisions.
A. RUNNING MECHANISM FOR PRESSURE ORGANS.
\ 314-
Running Mechanism in which the Pressure Organ
Drives by its Weight.
With a few unimportant exceptions the motors of this class
are operated by liquids, which at moderate velocities practically
follow the laws of gravity.
In Fig. 957, a is an undershot water-wheel, and b is a half-
Fig. 957-
breast water. The water is guided in a curved channel and the
buckets are radial, or nearly so. The wheel is so placed that
the buckets pass with the least practicable amount of clearance
over the curved channel. At c is shown a high breast wheel,
and at d an overshot wheel (compare § 47). In these latter
wheels the buckets are so shaped that they retain the water in
the circular path, being closed at the sides also, while on
account of the moderate pressure they are left open above. At
e is shown the side-fed wheel of Zuppinger.
Fig- 958> a is an endless
chain of buckets, and b a
similar arrangement, using
disks running with slight
clearance in a vertical tube.
In the wheels shown in
Fig- 957 the water acts on
the wheel much in the same
manner as a rack acts wThen
driving a pinion, and in this
sense a water wheel may be
considered as a gear wheel.
When the water acts only
by gravity these construc-
tions are only practical when
the wheel can be made
larger in diameter than the
fall of water, and where
small diameters must be used the arrangements of Fig. 958 are
available. Very small wheels acting under high pressures may
be employed by making use of the so-called “chamber wheel
work,” X of which some examples are here given.
a	b
Fig. 959.
Fig- 959 # is the Pappenheim chamber wheel train. In this
the tooth contact is continuous, the teeth being so formed that
the continuous contact of the teeth at the pitch circle prevents
% See Berliner Verhandlungen, 1868, p. 42.220
THE CONSTRUCTOR.
the water frotn passing, while the points and sides of the teeth
make a close contact with the walls of the chamber. The
downward pressure of the water enters into the spaces between
the teeth and drives both wheels. The axes of the wheels are
also coupled by a pair of spur gear wheels outside the case,
thus insuring the smooth running of the inner wheels. This is
the oldest forni of chamber train mechanism known, and cau
also be used as a pump, operating equally well in either direc-
tion. Fig. 959 b is Payton’s Water Meter, with evolute teeth.
The flow is intermittent, but one contact begius before the
action of the previous one ceases.
Fig. 959 c is Eve’s chamber gear train. The ratio of teeth is
i to 3, and the flow is also intermittent. The theoretical volume
of delivery for ali fornis of chamber gear traius, whether con-
tinuous or intermittent in delivery, is practically equal to the
volume described by the cross section of a tooth of one of the
two wheels for each revolution.
Fig. 959 d is Behren’s chamber train. In this case each wheel
has but one tooth, as is also the case with Repsold’s train (de-
scribed hereafter), and the gears belong to tiie class of disc
wheels or so-called “ shield gears ” (see§2ii). This arrauge-
ment possesses the great advantage of offering an extended sur-
face of contact at the place between the two wheels where, in
the previous forins, there is but a line contact. This permits a
sufficient degree of tightness to be obtained without requiring
the parts to press against each other. Beliren’s chamber gear
makes an excellent water motor if the impurities of the water
are not sufficient to iujure the working parts.
The flow of water through chamber gear trains is not uni-
form, and the inequality of delivery increases as the uumber of
teeth in the wheels is diminished, hence they should be driven
only at moderate velocities when used as motors, in order to
avoid the shocks due to the impact of the water.
streams as a simple expedient, but of low efficiency; b is the
Borda turbine, consisting of a series of spiral buckets in a bar-
rel shaped vessel; c is tne so-called Danaide, the spiral buckets
being in a couical vessel, this form being mostly used in France.|
In the wheels which have been shown in the preceding illus-
trations from Fig. 958, the living force of the wrater acts by
direct impact through a single delivery pipe.
The following forms differ from the preceding, in that the
water acts simultaneously through a number of passages around
the entire circumference of the wheel. This form gives the so-
called hydraulic reaction in each of the inclosed channels, and
hence wheels of this class are commonly called reaction wheels,
or reaction turbines.!
Fig. 962 a is Segner’s wheel, the water entering the vertical
axis and dischargiug through the curved arms ; b is the screw-
turbine, eutirely filled with water ; c is Girard’s current turbine,
with horizoutal axis, and only partially submerged ; d is Cadiafs
turbine, with Central delivery, and e is Thomson’s turbine with
circum ferential delivery and horizontal axis, the discharge being
about axis at both sides. In all five of these examples the
column of water is received as a whole, and enters the wheel
undivided uutil it enters the wheel; in the following forms the
flow is divided into a number of separate streams.
1315-
Running Mechanism in which the Pressure Organ
Drives by Impact.
In driving running mechanism by impact, fluid pressure
organs, both liquid and gaseous, may be used, as will be seen
from the following examples.
Fig. 960.
Fig. 960 a is a current wheel, or common paddle wheel. The
paddles are straight, and either radial, or slightly inclined
toward the current, as in the illustratiou. The working contact
in this case is of a very low order.
Fig. 960 b is Poncelefs wheel. The buckets run in a grooved
channel, and are so curved that the water drives upwards and
then falis dowmwards, thus giving a inuch higher order of con-
tact. At c is shown an externally driven tangent wheel. The
buckets are similar to the Poncelet wheel, but with a sharper
curve inward. The discharge of the water is inwards, its living
force being expended. At d is an internally driven tangent
wheel, similar to the preceding, but with outward discharge.
The form shown at e is the so-called Hurdy-Gurdy wheel. The
water is delivered through curved spouts, and this form is prac-
tically an externally driven tangent wheel of larger diameter
and smaller number of buckets. This wheel, from a crude
makeshift, has become one of the most efficient of motors.*
Wheels with inclined delivery as made in the fornis shown in
Fig. 961.
Fig. 961. At a is shown a crude form, used 011 rapid mountain
* This is the Pelton Water Wheel, built in sizes as great as 300 H. P. See
Mining and Scientific Press, 1884, p. 246, and 1885, p. 21. This wheel is built
in Zurich, by Escher, Wyss & Co., with a casing, and used for driving
dynamos.
Fig. 963.
Fig. 963 a is the Fourneyron turbine, acting with Central
delivery ; the guide vanes are fixed and the discharge of the
water is at the circumference of the wheel ; b is a modification
of the Fourneyron turbine, the water being delivered upwards
from below, and sometimes called Nagel’s turbine ; c is the
Jonval or Henschel turbine, the guide vanes c being above the
wheel, which is entirely filled by the wrater column ; d is Fran-
cis’ turbine, with circumferential delivery through the guide
vanes c* ; e is the Schiele turbine, a double wheel with circum-
ferential delivery and axially directed discharge. I11 the latter
three forms a draft tube may be used below the wheel, to utilize
that portion of the fall, as indicated in forms c and d.
For gaseous pressure organs, of which wind is the principal
example, some forms are here given. Fig. 964 a is the German
windmill, with screw-shaped vanes ; b is the Greek and Anato-
lian windmill, with cup-shaped vanes. Both forms are similar
in action to the above described pressure wheels. At c is shown
the so-called Polish windmill, with stationary guide vanes ; ||
d is Halladav’s windmill, made with many small vanes, which
place themselves more and more nearly parallel with the axis
as the force of the wind increases, the rudder c} keeping the
wheel to the direction of the wind. The extreme position of the
vanes is shown at e. Anemometers and steam turbines are
examples of wheels in which other pressure organs than wind
are used.
f See Weisbach-Hernman, Mechanics of Engineering, Part II., Section 4,
p. 553.
X This use of the term reaction is hardlv desirable for this construction,
nor is the proposed name of “ action turbine,” and the name ‘‘pressure
turbines ” is to be preferred.
§ This form is well made by J. M. Voith.of Heidenheim, WUrtemberg.
| Recueil des Machines avantageuses, Vol. I., No. 31, 1699, also from thence
shown in Henning’s Sammlung von Machinen und Instrumenten, Niirn-
berg, 1740.THE CONSTRUCTOR.
221
. i 316.
RUNNING MECHANISM IN WHICH THE PRESSURE ORGAN IS
DrIVEN AGAINST THE ACTION OF GRAVITY.
Running mechanism for the purpose of elevating liquids, and
especially for lifting water, are of very early origin, and the
various machines for this purpose form the very oldest of
machine inventions.
Fig. 965-
Fig. 965 a is a bucket wheel, the vessels on the circumference
lifting the water ; this is driven by the power of men or animals,
or in many instances by a current wheel (as in Fig. 960«).* At
b is the Tynipanon of Archimedes, used down to modern times,
the sections deliver the water through openings into the axis;
c is a paddle wheel, only adapted to raise the water a small
height, much used in the polders of Germany, Holland and
Italy. The paddles are made either straight, or curved, or
sometimes slightly crooked at the end.f At d is the Archime-
dian screw, which, when placed at an angle as shown, is well
adapted to elevate water. The Archimedian screw is exten-
sivelv used in all positions for the granular and pulverized
materials, in which cases the outer cylinder is omitted and a
stationary channel substituted, as shown at c, in Fig. 965 e, and
if the transportation of material is in a vertical direction the
screw is eutirely surrounded by a stationary tube. A stili later
form is made with a wire spiral, by Kreiss of Hamburg.
Fig. 966 a is the spiral pump, in which the screw of Archi-
medes is replaced by a channel formed in a plane spiral. In
this form the inclosed air becomes compressed by the speed of
revolution of the mass, and the water can be forced quite a con-
siderable height.:}:	Fig. 966 b is a conical spiral pump called
after its inventor, Cagniard Latour, a Cagniardelle. The Cag-
niardelle is usually placed entirely in a trough, but the illustra-
tion shows how the end of the spiral may be modified so as to
require no enlargement of the delivery channel. The diameter
of the cone is adapted to the height to which the water is to be
lifted. The Cagniardelle may also be used as a blower, the in-
closed water driving the entrapped air before it.
The chain and bucket devices already shown in Fig. 958 as
motors are also well adapted to drive the pressure organ, and
are in practical use in numerous modificatious. Fig. 958 a is
extensively used in dredging machinery, grain elevators and
the like, and Fig. 958 b is much used for lifting water.
The various forms of chamber gear trains already described,
give by inversion corresponding forms of driving mechanism,
some examples of which are here given.
Fig. 967 a is Repsold’s pump ; each wheel has one tooth, the
profiles being formed as described in $ 207 ; b is Root’s blower,
the wheels having two teeth each, and the action being the
same as the Pappenheim machine, Fig. 9590. This device has
been very extensively used as a blowing machine. Since the
action of these machines in drawing air against pressure is simi-
lar to that of lifting water against the resistance of gravity,
* Earge wheels of this sort have been in use in Syria for many centuries,
as at Orontes, north of Damascus. The town of Hamath, of 40,000 inhabi-
tants, receives its water supply from twelve such wheels.
+ A recent installation ot such paddle wheels has been made at Atfeh. on
the Mahmudieh Canal, in Egypt. Eight wheels 32.8 feet diameter, each
driven by a separate steam engine lifting water from the Nile 8^ feet to the
canal. The eight wheels deliver 115,000,000 cubic feet in 24 hours. See
Engineer, 1887, p. 57.
t such pumps, made by Klein. Schanzlin & Becker, at Frankenthal, deli-
ver water from 2 to 30 feet, the revolutions being from 15 to 22 per minute,
and diameters from 20 to 70 inches.
there is no necessity for distinguishing in classification between
them as pumps for liquids or for gaseous fluids. Fig. 967 c is
Fig. 967.
Fabry’s ventilating machine for mine ventilation, consisting of
a double toothed combination chamber train, with unequal
duration of contact. Root has also used the form shown at d,
which has unequal contact duration, and which has since been
made by Greindl as a pump.g
c.
Fig. 968.
Greindl also makes the form shown in Fig. 968 a, w-ith gears
of one and two teeth, and rightly claims it to possess the advan-
tage of a greater freedom from leakage. The form showm at b
has been used by Evrard as a blower, but it does not differ in
principle from a. Baker’s blower, shown at c, is a triple cham-
ber train, also used by Noel as a pump.
It has already been stated that Behren’s pump, Fig. 959 d, has
also been used as a steam engine. As long ago as 1867 a steam
fire engine has been constructed by putting two of these
machines 011 the same axis, one being driven by steam. the other
forcing the water.
Chamber gear trains may also be used to be worked in con-
nection. Fig. 969 showTs an arrangement in which the chamber
Fig. 969.
train A delivers wrater to a distant one B, driving the latter and
receiving the discharge water from B through a return pipe to
be used again. The combination forms a transmission System
of the second order (see \ 26), and is similar to a belt or chain
transmission. The loss in efficiency in this device is not an un-
important consideration.
An important class of machines consists of those made with
tension organs for transporting granular materials. For this
purpose belts, chains, ete., are used, and wThen the transmission
is horizontal, or nearly so, grain is successfully transported on
wide belts.|| Another application is that of Marolles, using an
iron belt, 40 in. wide, 0.06 in. thick, for transporting mud.
Twelve such machines were used 011 the Panama Canal work,
the distance being 200 feet, and the speed of the band 12 to 40
feet, according to the nature of the material. Similar apparatus
at the Suez Canal handled material at a cost of 7.6 cents per
cubic yard.
8 317-
Running Mechanism in which the Pressure Organ is
Driven by Transfer of Living Force.
The method of driving pressure organs by a transfer of living
force is one which admits of numerous applications, as the fol-
lowing examples show.
Fig. 970 a is a centrifugal pump for moving liquids. The
driving mechanism consists of the curved blades, which in
§ The firm of Klein, Schanzlin & Becker. at Frankenthal, make a line of
pumps similar to Fig. 967 d, of a capacity of 1.77 to 177 cubic feet per minvite,
and discharge openings from 1.18 to 11.8 ins. diameter. These are driven
by belt and used beer-mash oil, acids, paper pulp, syrup, etc., as well as
water.
|| An excellent transmission is in use at Cologne. See also Trans. Am. Soc.
Mech. Engrs., Vol. VI., 1884-85, p. 400. At the Duluth elevator a rubber belt
50 inches wide, running 600 to 800 feet per minute, carries grain from 600 to
900 feet horizontallv. A 26" belt has carried 14,000 bushels per hour.222
THE CONSTRUCTOR.
many instances are made in one piece with the wheel itself, this
adding to the efficiency. These pumps have been most suc-
cessfully made by Gwynne, Schiele, Neut and Dumont among
Fig. 970.
others.* Centrifugal pumps have been successfully used as
dredging machines for lifting wet sand, gravel and mud, in-
stauces among others beingthe North Sea Canal at Amsterdam,
and the harbor at Oakland, California.
Fig. 970 b is the well known fan blower used everywhere for
producing a blast of air, and acting by centrifugal force. When
used as exhaust fan this is widely used in connection with
suitable exhaust pipes for removing foul air, sawdust, and other
impurities in workshops, as well as for the ventilation of mines, f
At c is shown a form of spiral ventilator, known as Steib’s ven-
tilator; it is similar to soine of the precediug forms, but is of
limited application, and is better adapted for lifting water, a
Service to which it has been applied in the polders of Holland.
At d is a centrifugal separator, a device of numerous applica-
tions for separating materials of different specific gravity by
centrifugal force. A notable example of this machine is the
centrifugal separator for removing cream from milk.
Auother variety of machines for driving pressure organs by a
transfer of living force, is that in which another pressure organ,
either liquid or gaseous, is used instead of a wheel as the im-
pelling meclianism. To this class beloug the various jet devices,
injectors, etc.
Fig. 971.
Fig. 972.
suction tube at b2, and the mixmg tube at b3; the regulation is
effected by a valve at the end of b3.
Steam jets are also used to produce a blast of air, or com-
pressed air may be used for the same purpose, as can also water
under pressure. A reversal of the last mentioued arrangement
occurs in Bunsen’s air pump, in which a jet of water is used to
produce a vacuum. Recent devices for utilizing jet action are
numerous. Among others, a jet of air has been used to feed
Petroleum into furnaces as fuel. Dr. W. Siemens proposed to
carry the petroleum in the hold of a vessel in bulk, and substi-
tute sea water, as it was consumed, in order to maintain the
ballasting of the ship undisturbed. Granular materials have
been handled by means of jet apparatus, usually impelled by
compressed air, sometimes by water jets.
An especial feature of jet pumps, and one which should not
be overlooked, is that they act either by guiding the pressure
organ stream, or that the driving action of the pressure organ
stream itself produces a guiding action, and that the existence
either of a reservoir or some exterual means
of driving must be presupposed The use of a
pressure organ in motion for driving mechan-
ism, is in this respect similar to the so-called
inductive action of an electric current.
An example of pure guiding action is
found in the “Geyser Pump” of Dr. W.
Siemens, Fig. 973. The water is to be raised
from a depth H, and the tube b is prolonged
downward to a depth Hx below the sump S.
The prolonged tube bx is open at the lower
end, and in the bottom opening T an air tube
c is introduced, and air is admitted at a pres-
sure slightly under that of a column of water
of height equal to Hx. The air mingles with
the w7ater and forms a mixture in ax which is
lighter than water, and the air pressure is then
capable of forcing the light mixture up to the
surface. The lifting action is assisted by the
expansion of the ascending air. Siemens
found that it was possible to produce this
action when //was equal to Hx, that is, the
specific gravity of the mixture of air and
water = x/2.

Fig. 973-
Fig. 971 a is Giffard’s injector in the improved and simplified
form made by the Delaware Steam Appliance Co. In this case
steam is used to drive a jet of water into a vessel already con-
taining water under pressure. The jet of steam rushing through
the nozzle bx draws the water in by the suction tube Z2, and both
pass through the mixing tube b3, and are discharged through
the outlet tube ; the outflow at b5 provides for the relief of
the discharge at starting, before the jet action is fully estab-
lished. The regulation of the flow of steam is effected by a
steam valve attached above bx. At b is Gresham’s automatic
injector, which is so made that should any interruption occur
in the supply of wrater at b2, the suction action is automatically
started, and the entering column of water is lifted again. This
is done by the introduction of a movable nozzle b6 between b3
and bA, which adjusts its position with regard to b3 according to
the variations in pressure above and below.
Fig. 972 is Friedmann’s jet pump. The mixing tube b3 is
divided into a number of sections, which permits a very free
entrance to the water, and gives an excellent action ; b is
Nagel’s jet pump, used for lifting water from foundations by
means of another jet of water. The entrance jet is at bl9 the
* A recent installation of magnitude is that of five centrifugal pumps built
by Farcot, of Paris, in 1887, for supplying the Katatbeh Canal in Egypt.
The wheels are 12 ft. 6" dia , and each deliver 17,660,000 cubic feet in 23 hours,
the lift varies from 1 to 12 feet.
t Fans for these purposes are made in great variety by J. B. Sturtevant.
Boston, Mass.
8 318.
Running Mechanism in which the Motor itseef is
.	Propeeeed.
The third division, in which the motor itself is propelled in
the liquid pressure organ, contains fewer varieties than the pre-
ceding oues but is of the greatest importance since to it belongs
the eutire subject of marine propulsion.
Fig. 974.
Fig- 974 0 is the so-called “ flying bridge,” the current flow-
ing in the direction of the arrow, causing the boats to swing
across the stream, describing an arc about the anchor to whichTHE CONSTRUCTOR.
223
they are held by a chain ; by is a sail-boat, the sail being the
driving organ transferring to the boat a portion of the living
force of the current of wind. At r, is a steamboat with side pad-
dle-wheels, and d, a stern-wheel boat; e> is a screw propeller.
A screw driven by a steam engine pressing the water backward
and the reaction of the water impelling the boat. At ff is a
so-called jet propeller, the reaction being produced by jets of
water forced through tubes at the side of the boat, the wrater
being driven by centrifugal putnps.* At g, is shown a current
wheel motor. The side paddle wheels are caused to revolve by
the action of the current, and by connection with a cable or
chain gearing (See Figs. 787 and 794) the boat is propelled up
the stream.
Direct acting reaction jets have been used for torpedo boats,
using carbonic acid gas, but this method has been superseded
by twin screw propellers driven by compressed air. Rockets
and rocket shells are examples of direct acting pressure organs.
B. RATCHET MECHANISM FOR PRESSURE ORGANS.
i 319-
FtUID Running Ratchet Trains.
The pawls in a fluid ratchet train are the valves. They may
be divided into two great classes,f similar to those existing in
ratchets of rigid materials, viz.
Running Ratchets, or Lift Valves, and
Stationary Ratchets, or Slide Valves.
In the first class we have flap valves, also conical and spheri-
cal valves, and in the second, the various lorms of cocks, cylin-
drical and disc valves and flat slide valves. In both kinds of
valves there exists an analogy to toothed and to friction rat-
chet gearing, since by use of contracted openings the effect of
friction is produced, and with full openings it is obviated.
This gives a division which does not exist in the case of friction
and toothed ratchet gearing.
Viewed according to the precediug classification, piston-
pumps, and piston tnachines are properly ratchet trains.J*
This idea does not seem to offer any practical difficulties, since it
can be made to include ali the numerous variations without crea-
ting more confusion than the former methods of classification.
It is not practicable to distinguish between the devices acting
by gravity and those acting by transfer of living force, since
both are frequently combined.
The oldest devices are those using air, and the oldest piston
is the membrane piston, (Fig. 949) in the form of a bag of skin
used as a bellowTs. In this primitive device the earliest valve
was the human thumb, and in the larger bellows the heel of
the operator, these being followed at a later date by valves of
leather.§ The working part of the bag was next strengthened
by a piate, (See Fig. 949«.) and developed into the common
bellows, next followed the disc piston, a very early improve-
ment || and later the plunger, from which the numerous modern
forms have grown. The followiug examples will illustrate.
Fig. 975 ay is the 'common lift and suction purup, a ratchet
train similar to Fig. 749 ; a, is the pressure organ stream (cor-
responding to the ratchet wheel a) b.2l the holding pawl in the
form of a valve, c2, is the receiver or cylinder for the water and
piston, cl9 is a pawl-carrier in the form of the piston, blf the
other pawl, or lift valve. The water here overflows at the tbp
* Used by Von Seydell in the Albert in 1856 ; by Ruthven in the Water-
witch, 1866, and recently in torpedo boats by Thorneycrofl.
t See the author’s Theoretical Kinematics, p. 459, et seq.
jThis treatment of the subject was first published by the author in Ber-
liner Verhandlungen, in 1874, p. 228 et seq., but had previously been used in
his lectures since 1866.
gContrary to Wilkinson and Ewbank, the bellows shown in the Egvptian
wall paintings have not flap valves, but the inlet opening is closed by the
heel of the workman, and the bellows used to-day in India use the heel or
thumb of the operator as an inlet valve.
fi See Belidor, Arch hydraulique, Paris 1739, H-» P-
of the cylinder, and if it is to be lifted to a greater height the
cylinder may be prolonged upward and the rod proportionately
lengthened. If the rod is to be kept short, the form shown at
bf is used. The top of the cylinder is closed and the rod
brought out through a stuffing box, and the discharge tube
only is prolonged. At c, is the so-called force pump with a
disc piston, and at d, the same form with plunger. In these
the discharge valve is in a separate chest. The water colunin a,
is divided into twTo divisions ax and a2} the low’er being impelled
in the up-stroke, and the latter on the down-stroke of the pis-
ton. A blow or shock is produced at each stoppage of the
motion of the water column and to reduce this action the speed
of flow must be kept down, and also the shock cushioned by
means of air vessels. At d, air vessels are shown both on the
suction and force pipes.
The precediug pumps are all single acting, discharging one
cylinder of water for each complete double stroke of the piston.
By cylinder of water is here meant the product of the piston
area by the length of stroke.Tf The space between valves and
piston is not included, this being merely clearance or water space.
The piston may be so constructed that it remains stationary
and the cylinder slides upon it, this forming an inversion of the
common form and possessing many applications.
Fig. 976 a is Muschenbrceck’s pump (1762) for moderate lifts,
b, is Donuadieu’s pump for deep wells, especially adapted for
Fig. 976.
artesian wells.** This latter form possesses the peculiaritj* that
cylinder and discharge pipe move, and the piston is stationary
while action is not chauged. (See Fig. 749 ) At c, is Althaus
so-called telescope pump, which does not differ from Fig. 975 af
except that the piston is longer and is operated by two side
rods instead of a single Central one.ft The form at d, is a mod-
ification of c, with external packing.
In the pumps shown in Fig. 975 a, b, and Fig. 976 a, the pis-
ton rod plunges into the wrater on the downward stroke and
hence acts as a piston, lifting water by its displacement. On
the upward stroke the water flows into the space again,
and so the volume of delivery is not altered but a slight portion
of the delivery takes place on the down stroke. This action
can be utilized, howrever, as was very early done in mine
pumps, by increasing the diameter of the rod, or forming it in-
to a plunger so as to cause the delivery to be divided equally
between the tw o parts of the stroke. This form may be called
a double delivery pump, or briefly a double pump, since it is
practically two pumps, using the same set of .valves. Some
examples follow\
Fig. 977 a, the plunger r2, is connected to the piston clt the
latter being twice the diameter of the former, this being the so-
called “ differential ” pump. In b, two plungers are used, both
valves being in separate chests JJ At r, two telescopic pistons
are used, this being by Rittinger, and well adapted for a mine
pump. The form shown at d, has an auxiliary piston and cyl-
inder parallel to the main cylinder, (designed by Trevethick in
l802.§§
1[In small and medium sized pumps the loss of cylinder capacity dimirn
ishes with the increase of speed. Experimental researches show
of the theoretical capacity (Konig, Pumps, Jena, 1869). In very large pumps
the momentum of the water shows an increase over the theoretical capac-
ity; the pump in the Beryberg mine, 1 metre diameter giving 4 per cent.
excess. See Portfenille John Cocquerill.
** See Poillon, Traite th. et prat. des pompes, Paris, 1885, Piate 27.
tt See the design of the Spaniards, Barnfet, Viciauain & Poillon, Plates 33
and 34, and p. 193.
11 See Poillon, Piate 7, Saigun Waterworks.
See Ewbank’s Hydraulics, New York, 1870, p 280.224
THE CONSTRUCTOR.
By making the suction valve also a moving piston, both the
water columns may be kept in motion for both movements of
the rod. This is a double acting ratchet mechanism (Fig. 750,)
and hence also a double acting pump.
Fig. 978 a, is a double acting pump with two opposing
valved pistons, described by Fourneyron, but much older; this
corresponds to the ratchet work of Fig. 7500.
The pumps shown in Fig. 978 b, and cf are similar, the first
by Stolz, the second by Amos & Smyth.*
Fig. 979 a, is Vose’s pump, in which the two pistons are
placed parallel to each other. This corresponds to the Laga-
rousse ratchet, Fig. 750 b. Similar double acting pumps may
be made with solid pistons, if it were
desirable ; the form of Fig. 979 b, de-
signed by the author, being an ex-
ample, and others might readily be
devised-t
Fig. 979c, is Downton’s pump. The
three pistons r3, c2) cZy keep the water
in constant flow, which is further
assisted by the air chamber. The foot
valve bAy may be omitted if desired.
The annexed sketch of a pump by
Lippold, (See Bach. Fire Engines,
Stuttgart, 1883, p. 41,) is not double
acting but contains practically one
piston split in two, and equivalent to
one of half the area and same stroke,
FiG. 980.	or two of the same area and half
stroke. This is also the case with
Franklin’s Double Pump, (See Konig, p. 55).
• See Theoretical Kinematics, p. 462.	f See Poillon, Piate 29.
By combining two complete fluid ratchet trains in such a
manner that they have a common cylinder and piston, a form
of pump is obtained which gives two full discharges for each
cycle, and which may hence properly be called a double acting
pump.
b	c
i
Fig. 981.
Fig. 981 a is a double-acting pump with disk piston, and Fig.
981 b, the same form with a plunger. In both cases the suction
pipe is at IV, and the discharge pipe at I. In double-acting
pumps it is usually not convenient to put a valve in the piston ;
this is, however, done in Fig. 981 c, in which we see two single-
acting pumps combined in one.
In Fig. 982 a, is shown Stone’s Pump, % which is much used
Fig. 982.
for ships, as is also Downton’s Pump. In this case there are
four pistons, operating in two cylinders, the latter being placed
one below the other on the same axis. The pistons cA and cz are
connected by one rod and connected by the same crank k 1.3,
and the other two pistons are, in like manner, connected and
operated by the crank k 2.4, which is set opposite the other
crank. The action may be more readily understood by examin-
ing Fig. 982 by which is similar to the preceding one, if we sup-
pose the pistons c2 and c4 to be held stationary and the other
pair clf c3 driven by a single crank of double the length of arm
of those shown. This will obviously not alter the volume of
delivery, and it will be evident that the lower pump is really a
double-acting force pump and the upper one a single-acting lift
pujnp, hence each revolution of the cranks will deliver three
cylinders of water, two on the up stroke and one on the down
stroke. In Stone’s pump the pistons r2 and cA are so disposed
that for each half revolution f cylinders of water are discharged,
and in other respects the pump is a double-ratchet tram. Fig.
982 c is Audemar’s Pump. In this form two double pumps
similar to Vose’s Pump (Fig. 979 a) are combined to make a
double-acting pump. §
| See Poillon, Piate 26.
\ See Poillon, Piate 6, p. 93.THE CONSTRUCTOR.
225
Fig. 983 is Norton’s so-called V shaped pump. In this device
the pistons c2 and c± form a single stationary piece, and the cyl-
inder and valves bx and b3 is
the moving part. It will
readily be seen how easily
the lift pump may be made
double-acting.
A double-acting lift pump
as used for a steam engine
air pump, by Watt, is shown
in Fig. 984. This is practi-
cally a combination of two
different pumps. It has tbree
valves, the foot valve b2, pis-
tou valve bl and upper valve
b3. On the downward stroke
the mixed air, water and va-
por passes tbrougb the piston
from the lower to the upper
part of the cylinder, and on
the up stroke this is dis-
charged through b3 and a
fresh cylinder full drawn in
through b2. This pump is
double acting, since the pic-
ton valve acts both in the up
and down stroke. This works
the same wbether pumping
liquid or gaseous fluids, the
action being the same as if
two valves only were used.
The upper valve is required
for other reasons, i. e. to con-
trol the discharge, as for
boiler feeding, etc.
Fig. 984.
The preceding examples will serve to illustrate the applica-
tion of fluid ratchet trains with running ratchets. It is impor-
tant in all cases, and especially with the higher velocities, that
provision should be made to have the valves close without
shock, or in other wrords, that the engagement of the pawls
should be quiet. This problem has already appeared in some
forms of ratchet mechanism (see § 240) and here offers stili
greater difficulties, especially when heavy moving masses are to
be controlled. The question is daily being considered in prac-
tical problems of construction * and a great variety of valves
has been designed. The present indications appear to be lead-
ing toward the use of valves operated mechanically by the
pump, instead of those operated by the fluid itself, but a final
solution of this problem has not yet been renched.
? 320.
Fluid Ratchet Trains with Stationary Ratchets.
As already shown in § 255, it is necessary, in ratchet trains
wTith locking teeth, to effect the engagement and disengagement
of the pawls by some additional mechanism. This is also the
case in those fluid ratchet trains which used stationary pawls,
i. e., sliding valves. An example is found in the case of the
simple single-acting air pump used in physical laboratories, which
since its invention by Otto von Gerike f has been made wTith
stationary pawls, and is shown in a crude form in Fig. 985.
The “receiver” d\
and its pipe connec-
tion forms a negative
reservoir, the pump
a c d bxb2 a ratchet
train for the propul-
sion of the column
of air a. The suction
valve is at b2, and the
discharge valve at b{, both being in the form of stop cocks.
The suction valve b2 is operated by hand when the piston is
drawn out, and when the end of the stroke is reached the valve
* See Fink, “ Kotistruction der Kolben-und Zentrifugalpumpen,” Berlin,
1872; also Bach, “ Konstruction der Feuerspritzen.”
t This name is spelled as given above in the earliest records, and not
“ Guericke,” as is often given.
6lt which had previously been closed, is opened, and the first
one closed, and the air expelled on the return stroke. A stop
cock, b3, is also placed close to the receiver.
There is but little difficulty in apply-
ing slide valves to single-acting pumps,
and they are also readily arranged
for double-acting cylinders. By exarn-
ining the arrangement of flap valves
in the compress double-acting pump,
Fig. 986, it will be seen that the valves
bl and b± open and close simultaneously,
and that the same is true of b2 and b3, |r j
and that the two actions alternate with
each other. The operation of the valves
is such that the four spaces / to IV are
connected alternately in the order /-//
and III-IV, and /-/// and II-IV.
From this it will be seen that if sliding
valves are used they may all be con-
nected together, or united in the same
construction. This may be done as
shown in Fig. 987 a, which represents
the so-called “ four-way ” cock. As here
shown, all four of the passages are
closed, this position corresponding to	pIG g^
the end of the piston stroke. When	' y
the plug is turned 450, as shown by the dotted lines, / and III
are connected, and also II and IV; and if it is turned the same
amouut in the other direction, / and //and III and IVare
Fig. 987.
connected. The portions b2 and b± may be omitted, as in Fig
987 b, and the passages //, IV and III brought closer together,
as shown at c. From this form it will readily be seen how the
passage / can be converted into a mere delivery pipi, and the
radius of curvature of the bearing surfaces, made of infinite
length, giving the well-known slide valve, Fig. 987 b. In like
manner other forms may be developed. It must not be forgot*
ten that this device really consists of four valves combined in
one, and in fact recent forms of steam engines contain the four
valves made separately, these often again being lift valves.
A noteworthy peculiarity in the forms shown in Fig. 987 a and
d must be considered. Iu both instances the valve overlaps
the port on both sides, this being technieally known as “lap.’*
It is also apparent that the lap on the two sides of one port may
differ, and that different laps may be used for different ports.
By use of this expedient the opening and closing of*the ports
need not be simultaneous, but may occur successively.
From the preceding considerations the following propo^itions
may be laid down ; the latter applying to all, and the former to
nearly all, lift valves :
The application of slide valves in all fluid ratchet trains de-
pends upon two principies:
1.	The combination of several valves into one piece.
2.	The control of the time of action of these valves by means
of the lap.
The application of a slide valve to a pump is shown in Fig.
988 a. In this case / is the discharge outlet, and IV the suction
connection. In such pumps it is necessary to provide some
mechanism to operate the valve, and such mechanism is termed
the “ valve gear.” This valve gear may be arranged in a great
variet)7 of ways.
A simple fform of gear is that shown in the figure, 98%a> in
which an arm 6, attacbed to the piston rod, moves the valve by
striking against tappets 5' and 5//f on the valve stem. This226
THE CONSTRUCTOR.
arrangement is similar to the locking ratcket of Fig. 753. It
lias the defect, however, of requiring the piston to move rapidly,
or else the valve will not be carried past the middle position,
and the pump will stop. This defect can be met by using a trip
gearing device such as shown in Figs. 742 and 743, to continue
the condition of the valve wThen started by the impulse of the
piston rod.
A somewhat simpler method is that in which the reciprocatiug
motion of the pump rod is used to revolve a shaft by means of
a crank, Fig. 988 b> from which the valve may be operated by
means of a return crank or eccentric. This arrangement is
often used, especially for blowing engines, etc.* It will be
apparent that a four-way cock device, Fig. 987, may be arranged
so as to be operated by contiuuous revolution, instead of a
reciprocating motion, and hence the eccentric may be omitted
and a rotary valve device substituted.
I11 Fig. 988 b the crank and crank shaft are used merely for
the purpose of actuating the valve gear. It is practicable,
however, when a crank is once admitted, to use it stili further
as one of the parts of the pump, such as in chamber trains.
Many such devices haae beeu proposed,f although but few of
these have been put to practical use. The three following de-
vices will illustrate.
l ig. 989« is Pattison’s pump, a forni of chamber-crank train.
The crank a here assumes the forni of an eccentric, the rod b
becomes a flat piston, the edges of wdiich form a tight joint
with the ends of the cylindrical chamber d. In the position
shown i% the illustration the spaces //and/and IIIand IV
are in communication. In the dotted positious III is connected
with /, followed again by II and I and III and IV. This trans
fer of communication is produced by the action of the crank,
and hence no other valves are necessary.
The forni shown at Fig. 989 b is made with an oscillating
cylinder. The piece c, which plays an inconspicuous part in
Fig. 989 a, is now used for the chamber, and .its oscillating
motion with regard to b supplies the necessary valve action.
Oscillating pumps are used in a variety of fornis.
Fig. 989 c is Beale’s gas exhauster, made with a so-called
“sliding crank ” c, which acts at the sanie time as crank and
piston. Without the use of special valves, the spaces II and
III interchange with / and IVby the revolution of d. Beale’s
exhauster is in successful and extensive operation in various
gas works.
In the examples cited and in the numerous modifications of
them, it will be noticed that the checking or ratchet action of
the liquid is invariably performed by slide valves.
One of the objections to the use of slide valves for ordinary
water pumps is the wear upon the surfaces due to impurities
in the water. When the water is free from such pbjectiouable
impurities, it is to be considered whether slide valves might not
be iniich more geuerally employed than has hitherto been the
case. If this forni of valve were given the benefit of practical
study and experience, it ought to be possible to avoid the
shocks due to concussion existitig in pumps made with lift
valves when operated at high speeds.J
A great number of valve fornis have been designed, \ using
combinations of single valves on the principle of the multiple
ratchet (see \ 242), the action of the valves being assisted by
weights, springs, etc., but these have not completely attained
the desired end. ||
* See Zeitschrift Deutscher Ingenieure, 1885, p. 929; also Herrraann’s
Weisbach’s Mechanics, Vol. III, Part 2, p. 1089.
f See the author’s “Theoretical Mechanics,” in which over 90 chamber
crank trains are described and analyzed.
X Poillon refers to the fact that the automatic action of mechanically oper-
ated slide valves enables high speeds to be obtained with less noise than
when lift valves are used, but also notes the wear of the slide valves as an
objection to their use.
I See Riedler, Zeitschrift d. Deutscher Ingenieure, 1885, p. 502 et seq.
|| See Bach, “ Konstruktion der Feuerspritzen, ” also in Zeitschrift d.
Deutscher Ing., 1886, p. 421.
When the pump is used for pure wTater, as for drinking supply,
the question of wear upon slide valves is not so important as
with pressure pumps. A fair comparison can hardly be made,
however, between pumps with slide valves and those with lift
valves, as the former have been but little used and also not
practically designed.
It is a matter of surprise that when occasional applications of
slide valves are made in pumping machinery, that such devices
should be considered as something new\ The difference be-
tween the action of water and air is well known, and yet even
wTith the slight weight of an air column the shock in blowdng
machines is most apparent. It can hardly be supposed that the
other forni wTould remain uninvestigated.
The pumps shown in Fig. 989 a and c are commonly known
as rotary pumps, wrhick title is manifestly incorrect, since in
forni a there is an oscillating piston wThich does not rotate,
while in form c, notwdthstanding the rotary motion the action
is similar to form a. Other so-called rotary pumps have been
devised with curved piston action, some of these being as early
as the I7th century. In some designs a radial slide acts in the
pump case as a ratchet, and is drawn in and out by a cam of
appropria ely curved profile. A large number of rotary pumps
have been made on this principle, many of which will be found
in Poillon’s treatise. These pumps are usually made wdth
metallic packing only, and are used in Italy and France for
pumping wdne and oli ve oil; they are also adapted for brewery
pumps.
The undeniable predilection in favor of rotary pumps on the
ratchet train principle is wrorthy of consideration. It is claimed
that they have a higher efficiency, but this remains to be estab-
lished ; also the rotary motion gives a contiuuous uniform motion
to the w7at:r column, but this is equally accomplished by the
fornis shown in Figs. 982 and 989. This uniform flow can only be
approximately attained, as must be the case from tbe nature of
the mechanism. The principle is that of a ratchet train which
is intermittent in principle, and hence differs from a contiuuous
running movement. The idea that such pumps give a coutinu-
ous aud uniform discharge is due to the fact that the column of
water is operated directly from the part wdiich is driven con-
tinuously, but this by no means follows. This combination of
a contiuuous running motion, with an intermittent ratchet
action wrhich is not apparent to the eye, will be showrn in other
cases hereafter.
1321-
Escapements for Pressure Organs.
Ratchet trains found wdth pressure organs also include
escapements as completely as is tbe case with the preceding
forms of rigid ratchet mechanism. The ratchet of $ 258, shown
again in Fig. 990 may be considered as an escapement if we
assume the checkiug of a by b to be uniformly opened aud
closed.
If now, in Fig. 991, the checked member a is made a pressure
organ, such as wTater, in communication at //with a pressure
reservoir, or w?ith a negative reservoir at T\ or both, the
regular lifting and closing of the valve b produces an escape-
ment acting in a similar manner to Fig. 990. By means of
such a device the pressure organ a can be coustrained in per-
forming mechanical work. The range of such an escapement
is not determined by the teeth of a wTheel, but 011 the contrary,
is similar to a friction ratchet, and can be varied at will.
The applications of escapements wi;h fluids are in principle
the sanie as those formed of rigid bodies, but in practice their
nature is very different. We have already distinguished between
watch escapements and power escapements, and in the present
instance the powrer escapements are by far the most important.
For this reason the latter will be considered first. Unperiodical
escapements are showm in the simple form of Fig. 991, in wrhich
the time of releasing and checking is regulated by hand ; a
form very seldom found in rigid escapements. Periodical
fornis, similar to watch escapements, are used with pressure
organs for measurement, but not for measurement of time, butTHE CONSTRUCTOR.
of volume. To these we may add the adjustable escapements
on the principle of those described in § 259, and we kave the
following classification :
a.	Unperiodical Power escapements.
b.	Periodical Power escapements.
c.	Adjustable Power escapements.
d.	Escapements for measurements of volume.
A. UNPERIODIC POWER ESCAPEMENTS FOR PRESSURE
ORGANS.
1322.
Feuid Escapements for Transportation.
One of the simplest practical applications of the principle of
Fig. 991 is Felbinger’s Postal Tube, shown in diagram in Fig.
992. The line tube d is connected with a reservoir of compressed
air at H, and at T with a similar negative reservoir. At b is a
sliding pawl, here shown open ; the piston, or carrier c, in the
form of a leather box containing letters, telegrams, etc., being
<iriven through the tube. A valve b' enables the end of the
tube to be thrown into communication with a second negative
reservoir, and this mechanism can be arranged at both ends of
the line so that the tube can be used for transmission in either
direction, Such postal pneumatic tubes are well knowm and
widely used.*
An atmospheric escapement operated by a negative reservoir
is found in the so-called “atmospheric railway,” invented by
Pinkus in 1834, and put into practical operation somewhat later
in England by Clegg and Samuda. This was operated on the
Kingston-Dalby road with a vacuum of l/e atmosphere in the
exhausted receiver, but it is no longer in operation.
When an escapement is intended to control the back and
forth movement of a piston in the same path, the single valve
shown in Fig. 991 is not sufficient, but at least a second must be
used, as is already indicated in Fig. 992. One of the most prac-
tical of ali fluid escapements is found in the lock used on canals
and shown in diagram in Fig. 993.
	—	c —-			
					
		u" ~ 	—			f** * *
		— -- - ~• -=r-'	%		
					
					
The canal is open on the upper side (see Fig. 945 b and c) ;
the valves bx and b2 are of the running ratchet form, and are in
reality double gates. Smaller by-pass valves bx' and b2/ are used
in order to enable the inlet and outlet of the water to be started
gradually. The boat c forms the piston, and when the motion is
upward, bx is the escapement valve, and when downward, b2 is
used.
The above canal lock device, while extremely useful, pos-
sesses a very low efficiency, since it not only uses a volume of
water equal to the displacement of the boat plus the necessary
clearance, but also discharges the whole lock chamber of water
each time it is used. Eater devices ha ve been made for the
same purpose, involving a less waste of water. If it is arranged
for the Service to be doubled by making two lifts adjacent to
each other, it is evident that the descending boat can counter-
balance an ascending one of the same weight, the only require-
ment being that there must be some connecting mechanism in-
volving the overcoming an additional resistance, and capable of a
* In 1887 the length of the postal pneumatic tubes in Berlin was over 26
miles.
227
reversal of 180°. This may be accomplished either by the use
of tension organs or pressure organs.
Fig. 994.
Fig. 994 shows a double canal lift constructed by Green tax
the Grand Western Canal in England in 1840, the connecting
mechanism being tension organs in the form of chains. The
boats are carried in tanks c1 c2, the ends of which are closed by
valves or gates bx and b2, and similar gates b/ and b/ also close
the ends of the canal sections. A small addition to the weight
on the descending side is sufficient to raise the other tank f
The substitution of a pressure organ for the chain wTas first
made by Mr. Edwin Clark on the Mersey Canal in 1875, in the
form of a hydraulic lever, as shown in Fig. 995. This shows
clearly the equivalence of the cord or chain and pulley and the
water lever, already referred to in g 311. The tanks cx and c2
are carried on plungers 3 feet in diameter, and are 75 feet long
and 15^ feet wide. A head of 6" of water is sufficient to over-
come the resistance of motion, and a lift of 50 feet is effected
in three minutes.J Smaller installations have been made by
Clark and by Stanfield, and other large ones at the Ea Louviere
Canal in Belgium, and the Neufosse Canal at Ees Fontinettes,
in France. The lifts are 43 ft. and 50 ft. respectively, and the
plunger diameters feet. The loss of water with these lifts is
only about of the quantity used by common locks of the
same capacity. §
The preceding escapement devices are made for open canals,
but escapements may also be constructed with closed tube con-
nections. This latter type includes numerous hydraulic eleva-
tors for lifting burdens of all kinds.
An example of a direct-acting hydraulic elevator is given in
Fig. 996. The two valves are combined in one cock. The water
under pressure enters at H, and the discharge against the
atmospheric pressure is at A. The weight of the plunger is
counterbalanced by two counterweights G with chains and
t See Weisbach-Hermann, Vol. III., Part 2, p. 633.
t See Duer Trans. Inst. C. E-, 1876; Colyer, Hydraulic Machinery, London,
Spon., 1881, p. 17 ; also Robinson, Hydraulic Machinery, Eondon Gnffin &
Co., 1887, p. 64.
I See Colyer, p. 29 ; Robinson, p. 69 : also Zentralbl. der pr. Bauverwaltung,
1882, p. 395 ; Hensch, Schiffshebung in Frankreich ; also Schemfil, Kanal
and Hafenwerkzeuge in Frankreich und England, Wien, Gerold, 1882, p. 15;
also Ernst, Hebezeuge, Berlin, Springer, 1883, p. 630. In Green’s lift the
loaded boats descended and the empty ones ascended, hence an excess of
water was raised, which was permitted to overflow. These lifts enable much
greater differences of level to be overcome than do the ordinary locks, and
make it practical to use long stretches of canal and ;;jake an entire lift at
one operation. It may be here noted that pneumatic lifts for canals were
designed in 1863 by the Swiss engineer. Seyler.228
THE CONSTRUCTOR.
pulleys, and the plunger operates the valve automatically by
means of the rod b'> when the highest
positiou is attaiued. This form of lift has
been much used, sometimes of very large
dimensions. The great passenger elevator
of the Hamilton St. Station of the Mersey
Tunnel has a plunger iS" in diameter,
with a lift of 87^3 feet, the car holding 50
passengers.*
A practical objection to direct-acting
lifts of this form lies in the heavy counter-
weights required, and also in the depth to
which the cylinder must be sunk. . A
different form has therefore been designed
in which a piston travel of moderate length
is multiplied by use of a tension organ
system, such devices being extensively
used for passenger elevators, notably by
the Otis Elevator Company.
Hydraulic cranes are also forms of high
pressure escapements, first designed by
Armstrong, and since used by many others,
especially in connection with Bessemer
Steel piant, in which hydraulic cranes
have proved most valuable.
Fig. 997 shows the mechanism of a
hydraulic erane by Armstrong. The piston
is double acting, and there are four valves
blt ^3> ^4» of the type shown in Fig. 986,
the external connections also being neces-
sary in order to complete the escapement.
The high pressure water enters at H} and
passes through the pipe /, and is discharged to the atmosphere
at IV. The rod c2 is made of half the area of the piston e1
T>1 l^tji
(compare Fig. 946 e). When bY and b3 are open, as in the illus-
tration, the forward stroke is made with one-half the full force ;
when bx and b± are open, the forward stroke is made with full
force. By opening b2 and b3f the return stroke is made by the
pull of the load upon the chain. At b' is a safety valve which
comes into action should the load descend too rapidly, by the
opening of b3 alone.*
§ 323-
Hydra uuc Tooes.
Hydraulic escapements, similar to those used for lifting loads
are also applicable to machine tools. Among these may be
noted the devices of Tweddell, for riveting, punching, bendiug,
etc. (see § 54).
Figs. 998 and 999 showT the arrangement of Tweddell’s rivet-
ing machine ; d is the piston, blt b2 the valves, one of which
connects with the pressure reservoir at H, and the other with
the atmosphere at A. When bx is opened by the lever e, the
hydraulic pressure enters above the piston d, and the stroke is
made. The return stroke is effected by means of the auxiliary
piston dlt which is fast to d, and under which the water pres-
sure is acting at all times. Closing blt and opening b2y enables
this to act and lift the main piston. This gives practically a
hydraulic Jever of unequal arms, the shorter arm always being
loaded wuth //, and the load on the longer arm varying between
H and A. The lever mechanism d', d"y dnfy Controls the
length of stroke of the die, by means of the tappets d'f and
d//;, which are connected with the lever e. This is also used
on the lift of Fig. 996, and shows the complete escapement.
The arrangement of valves is shown in detail in Fig. 9994
* See Robinson.
t See Weisbacli-Herrman, III., 2, p. 240 ; Colyer, p. 11; Robinson, p. 52.
t For fuller descriptions of Tweddell’s machine see: Proc. Inst. C. E.
I.XXIII., 1883, p. 64 ; Engineer, July, 1885, p. 88; August, p. 111; Revue Indus-
trielle, 1884, p. 5; 1885, p. 493; Mechanics, 1885, p. 272 j also Robinson, as
above, and Zeitschr. Deutscher Ing., 1886, p. 452.
The preceding apparatus resembles the hydraulic press. It is
in fact quite different, being a genuine ratehet train, capable of all
the modifications of such mechanisms as to speed, distance, and
arrangement. On account of these points the applications of pres-
sure organ escapements are becoming rapidly more important.
\ 324.
Pressure Escapements for Moving Eiquids.
The use of unperiodic pressure escapements for moving
liquids in machine coustruction has
been practiced from an early period,
and at the present time improved de-
vices for this purpose are much used.
An almost forgotten device of this
kind is Brindley’s boiler feeding ap-
paratus, Fig. 1000, this being based
upon the principies already given in
Fig. 991.
The necessary opening of the valvq b
is made by the float cy and the closing
by the counterweight cl (compare Fig.
950). This apparatus w7as first applied
to Watt’s boilers, the feeding of the
boilers of Newcomen’s engiues being
effected by a cock operated by the
attendant.
Fig. 1001 is Kirchweger’s st eam trap
for the removal of water of condensa-
tion. The escape valve b is opened by
the float cy which, in this instance, is
open at the top, so that the water flows
over the rim until it sinks, and thus
opens the valve, This valve motion is
in itself a ratehet train, checked and
released by the action of the float.
When the valve is opened the water in
the float is forced out by the pressure of
the steam. \
The slow moving float device, as in
Fig. 1000, has also been advantageously
used for operati ng steam traps, by
Tulpin, of Rouen ; Handrick, of
Buckau ; Puschel, of Dresden;
Dehne, of Halle, and others.
Similar escapements have been
designed to separate air from
steam, or air from water, as in
the devices of Andral, Kuhl-
mann, Klein and others ||
Other examples of escape-
ments of this kind are found in
the so-called Moutejus, used for
elevatingsyrup in sugar refitier-
ies, in the return traps of steam
heating systems, and in various
other forms of boiler feeders,
such as those of Cohnfeld. Rit-
ter & Mayhew, and others.
§ This form of trap is made in many varieties, the one shown being by
Eosenhausen, of Dusseldorf. A similar one by MacDougal is much used in
England, and a feed pump on this principle is made by Korting in Hanover ;
German Patent No. 36, 332.
P For illustratious of these devioes see ScholPs Fuhrer des Maschinisten,
10 Edit., p. 493.	^ See Scholl, p. 235.
Fig. iooi.
hiTHE CONSTRUCTOR.
229
B. PERIODICAL PRESSURE ESCAPEMENTS.
I 325-
PUMPING MACHINERY.
Periodical fluid escapement trains have a wider application
than unperiodical trains, since itis practicable, as already shown,
to use a fluid ratchet train to operate the valves in a simple
manner. This makes it possible to produce the opening and
closing of the valves in a periodical succession mechanically,
instead of by the fluid column. In this construction the fluid
column may therefore drive the piston, instead of being driven
by it. This idea seems very simple, and yet pumps had been
known for two thousand years, and had occupied the iuventive
energy of the preceding centuries before the simplest fornis of
the modern steam engiue were devised. It is therefore all the
more important in the study of machine design to investigate
the fundamental principies involved.
It is impossible, in the limited space which can here be given,
to go into this subject in its entirety ; the arrangement of the
valve gear of the Newcomen engine with tumbling bob gear, is
an instructive example.
In Fig. 1002 is shown Belidor’s single acting water pressure
engine.* *
\
f
Fig. 1002.
In the cylinder d is a piston ; ax is the entrance of the water,
a2 the discharge outlet. The valves bx and b2 are united in a
three-way cock (see Fig. 987). This valve is operated from the
piston rod c by a tumbling-bob gear (see Fig. 742). The tum-
bling lever E e1 e2> wTeighted at E, is connected writh the piston
rod at cly and moves about its axis independently of the lever f.
When the end of the piston stroke is nearly reached, the lever
E passes the middle point, and tips over, when the arm fx strikes
the lever f and carries it to the position V7, moving the lever of
the three-way cock from b to b'. The arm ex is behind E. The
return stroke of the piston moves the arm e2 of the tumbling
gear towards the right, and as the end of the stroke is reached,
the tumbling bob is again tripped, and the three-way cock
moved again into the position b. A cord secured at the ends to the
points e3 and e±, and fastened to E, limits the travel of the latter.
The piston rod is connected directly to the pump to be operated.f
It will be observed that this machine is a ratchet train of the
second order, the piston and valve forming an escapement, and
the valve gear a releasing ratchet train each operating the other.
Fig. 1003 is the single
acting water pressure en-
gine of Reichenbach. In-
stead of using a tumbling
bob gear to operate the
valve, Reichenbach uses
a second water escape-
ment, operating the valve
by a piston, the valve
being itself a piston valve.
The double piston valve
b3 bx of the second escape-
ment is operated by the
main piston rod, the tap-
pets 5 and 6 striking the
lever c1 as each end of the
stroke is reached. The
water under pressure en-
ters at a} and is discharged
at a2. The tappet 5 moves
the auxiliary valve into
the position b/ b/, which
places the space above b1
* Belidor, Architecture hydraulique, Paris, 1739, Vol. II., p. 238.
t The above described machine, designed by Relidor in 1737, for the water
works at the bridge of Notre Dame, does not appear to be altogether practi-
cable. It has been here given on account of the valve motion, which is his-
torically interesting and doubtless good, and has been re-invented several
times since. It was not new in 1737, having been in Newcomen’s steam
engine, as was already known to Belidor, since it is described by him in the
same volume of his treatise.
in communication with the discharge, and since b2 is larger than
blf the pressure between them moves them into the position
b/ b/. This puts the main cylinder in communication with the
discharge, and the piston sinks by the weight of the load upon
it. At the close of the stroke the tappet 6 moves the arm c/
into the position c1 again, and places the auxiliary valve in the
first position and a new stroke is made.J
This machine constitutes an escapement of the second order,
since the small and large escapements alternately release each
other; the lever device 5-6-^ lorms a third mechanism, so that
the machine, as a whole, is of the third order.
	|U|	|	: 1 ; i	1 d	§yl Hi	
te		i				
Fig.1004.
Fig. 1004 shows the double acting water pressure engine of Roux.§
The double action is obtained by combining the four valves in one,
and by communicating the admission and discharge alternately
with both sides of the piston. In this case the lever connection
c1 is replaced by an escapement. The small pistons b./ b/ are
acted on at the outer ends by the pressure water through the
small passages k./, k/. This gives an escapement of the third
order. The cup-shaped ends <r2, c3, of the main piston c form
the pump plungers. This machine should operate satisfactorily.
It is readily apparent that the piston steam engine may also
be considered as an escapement. The valve gears differ from
the preceding fornis only on account of the conditions of ex-
pansion and condensation. These are reducible to a limited
number of simple cases.
Fig. 1005.
Fig. 1005 is a single acting high pressure engine. The steam
t See Weisbach-Herrman, II., 2, p. 536 ; also Ruhlmann, allgem Masch-
inen Chre., I., p 348
l See Revue Iudustrielle, 1884, p, 114. Built by Crozet et Cie.230
THE CONSTRUCTOR,
enters at a}, and the discharge to the atmosphere is at a2. The
opening of the valve bl permits the steam to enter, forcing the
piston c down, and raising the weight G. The valves bx and b2
are operated by a ratchet train released by the tappets 5 and 6
on a rod moved by the main piston. The pawls are double-
acting, and are of the form shovvn in Fig. 671. When c reaches
the bottorn of the cylinder the tappet 5 releases the ratchet 7,
and closes the valve bx by means of the connections fx ex. The
release of 7 opens the valve b2 by means of the connections e2f2,
and permits the escape of the steam from below the piston.
This equalizes the pressure above and below the piston, from
which the valve b2 is called the equalizing valve. The upward
stroke of the piston causes the tappet 6 to reverse the ratchet 7
and operate the levers exfx, closing the equalizing valve and its
connections.
The device differs from the preceding in that the principal
escapement a bx b2 c d changes in character with the stroke.
The two ratchet trains can be seen in principle in the double
acting tumbling gear of Fig. 1002. The mechanism, when lift-
ing the valve, is of the third order, and when closing, of the
second order. The gear as shown is Farey’s; Fig. 779 shows
this principle in a rigid escapement train, the correspouding
form in single-acting train is the chronometer escapement, Fig.
769-
If the engine is a condensing one, a condenser valve b3 is
added, this being opened by the closing of b2, as is also a jet
valve in the condenser. When the steam is to be expanded, the
lever ex is so arranged, the closing of bx is produced earlier (see
the smaller diagram) by the position of the tappet 5, and the
correspouding counterweight lifted. This only operates the
ratchet 7, and f2 is released by a second train 8, which is
effected by the tappet rod or by the so-called cataract K,
released by a tappet 9, see $ 260.
The condenser is a negative reservoir, and was the principal in-
vention of Watt. It involves the use of two fluid ratchet trains ;
the air pump, and the cold water pump, and also usually in-
cludes a boiler feed pump. The entire engine is composed of a
collection of ratchet trains.
Steam pumping engines are by no means always made with
lift valves, and a great number of more recent designs are made
with slide valves (see Fig. 987). Rittenger has applied slide
valves successfully to single-acting engines, and they are espe-
cially applicable to double-acting non-rotative engines. In the
last decade especially have valve motion for steam pumps with
slide valves been multiplied, and some illustrations are here
given.
Fig. 1006 is Tangye’s direct-acting steam pump. The steam
entrance is at I, and the exhaust at IV. The slide valve b is
the so-called E form, combining the four valves of Fig. 986 in
one; b2 and b3 are the auxiliary pistons to move the valve, and
form part of an escapement ol w hich the valves b" and b"' are
operated by the mam piston c at each end of its stroke. The
latter valves communicate with the Cylinder posts II and III.
When b'" is lifted by the piston, the space R is in communica-
tion with the exhaust, and the pressure in L throws the valve
over, equilibrium being soon after established through the aper-
ture k2. The reverse action occurs 011 the return stroke. This
is a steam escapement of the second order, wTith an independent
starting lever, the whole forming a combination of the third
order. This has been much used by Tangye for steam pumps.
Fig. 1007 shows the valve motion of the Blake pump, which
is very extensively used in the United States. I11 this case there
is a movable seat bQ under the valve b} the opening through the
seat always being in communication with the posts II, III, IV,
although bQ is moved a short distance at each end of the stroke
by tappets on the piston rod. In the position of the parts
shown the steam entering at /will pass through ///and move
the main piston to *he left, as indicated by the arrow\ Just
before the end of the stroke is reached the seat bQ is moved as
much to the left of the centre as it now stands to the right. In
Fig. 1007.
the seat bQ, as show n in the figure to the right, there are addi-
tional valves formed, /32, /?3, /34, which act to operate the auxil-
iary pistons b2, b3, under which latter the small steam passages
can be partly seen. When bQ is moved to the left, a small post
is uncovered by /?.,, and live steam euters the cylinder L behind
b.L, while at the same time /?4 connects R with the exhaust. This
causes b2, b, b3, to move to the right and reverses the pump.
The reverse action takes place at the other end of the stroke,
the whole forming a combination of the third order.
Fig. 1008.
Fig. 1008 shows the valve gear of Deane’s steam pump, w7hich has
also been extensively used. The main valve is moved by means
of auxiliary pistons, as in the preceding instance. The auxil-
iary pistons are controlled by a separate valve b', which itself
is operated by lever connections writh the main piston rod.
This combination b', b2, b, forms again a mechanism of the third
order.*
If the last three devices described are compared with the
Reichenbach wTater-pressure engine, it will be seen that the
fundamental principle is the sanie in ali. The constructive
arrangements which may be adopted are clearly showm iu the
preceding examples, which may be modified in a variety of
ways. Among other widely used arrangements, that of Knowles
may also be mentioned ; in it the action of the auxiliary pistons
is controlled by a slight twisting motion given to the valve stem.
Fig. 1009.
Fig. 1009 shows Pickering’s steam pump.f Iu this design the
main piston c acts also as the valve for the auxiliary pistons
b2, b3, so that the spaces R and L are placed alternately in com-
* See Am. Machinist, Feb. 17, 18S3, p 4. For an excellent steam pump by
Dow, see Mining and Scientific Press, 1885, March, p. 169, and May, p. 313.
f See Poillon.THE CONSTRUCTOR.
231
munication with I and IV. The whole forms a steam escape-
ment of the second order.
Fig. 1010 shows Harlow’s valve gear, also used for pumping
machinery.* This is also a steam escapement of the second
order, similar to the preceding. The valve action for the
auxiliary pistons is formed in a prolongation of the piston rod,
the grooves cY and c2 placing the spaces R and L alternately in
communication with IV.
i	i
59*
Fig. ioio.
By comparing the preceding designs with the water pressure
eugine of Roux, Fig. 1004, the similarity will be apparent. Ali
the examples given show the fuudamental relation existing be-
tween these devices and the mechanical escapements of w7atch
movements. The escape wheel is replaced by the fluid column;
the anchor, by the valve ; the vibrating member, whether pen-
dulum or balance wheel has here not a free movement but a
determinate one against an external resistance. Similar
arrangements include steam hammers, also hammers and rock
drills, usually driven by compressed air, these latter consisting
of mechanism of the second, rather than the third order. A11
example will serve to illustrate the general arrangement of such
devices.
Fig. ioii shows the arraugement of Githen’s rock drill.f
The curved valve b, is operated by the action of the curved
outline formed in the piston c. The middle position of the
valve is a dead point, but this is overcome by the momentum
of the heavy piston.
* See Engineering and Mining Journal, Oct., 1884, p. 23
fSee Eng. and Mining Journal. March, 1887, p. 107. Also Halsey s rock
drill, Trans. A. S. M. E., 1884-5, p. 71.
The devices of the third order are capable of a very import-
ant modification which can be considered by examining for in-
stance the Deane gear, Fig. 1008, or either of the two preceding
it. An inspection will show that it is entirely practicable to
use the auxiliary piston to operate a pump cylinder, as indepen-
dently of that operated by the main piston d. It is only neces-
sary to make it larger in diameter and of proper length of
stroke ; and there is nothing to prevent making it of the same
diameter and stroke as the main piston.
The valve of each cylinder will then be operated by lever
mechanism connected to the rod of the other piston. This
arrangement involves the replacing of the E valve by the com-
moti D valve, which is not important, but is nevertheless an
advantage. The two escapements are conveniently placed side
by side for constructive reasons, and the double arrangement is
known as a “duplex” machine, this term being given to two
combined cylinders, of which the valve of each is operated by
the piston movement of the other. This type is now frequently
met, having been made for small apparatus very early, in
France by Mazellire and yet earlier, in 1859, in the United
States by Worthingtou.
Fig. 1012 shows a duplex pump by Mazelline.j The illustra-
tion shows one piston cly at mid-stroke with lts valve b1} at the*
end of its travel, and connected to the rod of the other cylinder
by the lever e2.
The wTork is divided into two portions which is provided for
by the doubling of the parts. If the two piston escapements
(cylinders, pistons, valves, steam, etc.) are indicated by [1] and
[3], and the valve movements by [2] and [4] the action will be
as shown in the following lines,
wrhence wre have
[1] m [3]
and	[3]	[4]	[*].
both being of the second order.
Fig. 1013.
Fig. 1013 shows a perspective view of Worthington’s Duplex
Pump, the arrangement of which is apparent from inspection.
The duplex steam cylinders are at the right, and the double
acting pump cylinders on the left.
The advantages obtained by using this form of pumping
machine practically outweigh the objections which rnight be
made against the duplication of p&rts. In double acting pumps
of the forms shown in Figs. 1006 to idio, the motion of the
water columns is interrupted, at low speeds, at each reversal of
|See Poillon, Piate IX232
THE CONSTRUCTOR.
the piston, while with the duplex putnp the discharge is practi-
cally continuous, because each cylinder begins its stroke just
before the other comes to rest.
An objection to ali the other forms of direct acting pumps
already described lies in the fact that to obtain uniform pump-
ing action it is necessary to carry the initial steam pressure for
the entire stroke of the piston, or in other words, the best action
of the water end is obtained by means of the least economical
action of the steam cylinders.
This defect was overcome in the earlier pumping engines,
such as the Cornish mine engines, by using the steam to lift
heavy weights, pump rods, etc., the living force of the mass
permitting an early cut-off and high expansion, and the uni-
form descent of the weight being used to force the water. By
this method the Cornish engines attained a high degree of
economy. This method being single acting, caused the entire
column of water to come to rest during the time required for
the up stroke of the pump rod, and hence the Cornish type of
pumping eugine gives a most economical action of the steam at
the expense of a defective action of the pumps.
In the larger sizes of Worthington pumping engines the ex-
pansion of the steam has been for a long time effected by using
compound cylinders, and excellent results attained in steam
economy. The efficiency, however, was by no meaus so high
as was desired. In 1886 the so-called Worthington equalizer
was introduced with a view of enabling the desired high duty
to be attained.
This device, shown in Fig. 1014, is a ratchet train of the tumb-
ling type, similarto that shown in Fig. 743, the springs being
replaced by wTater pressure from a high pressure air cham-
ber.* The air chamber forms a periodical storage reservoir.
The plungers/*, f, areattached to a cross-head connected to
the prolonged piston rod, and the cylinders are carried on
trunnions 7, 7. During the first half of the stroke the plungers
are forced into the cylinders the latter swinging about the cen-
tres 7, 7 ; and during the second half they are forced out by the
action of the stored euergy.f
The resistance and assistance which the pistons f give to the
steam piston is shown by a curve of the form of Fig. 1014 b, as
has also been obtained by the indicator.
If in Fig, 1015 a> we uiake P equal the component on each
portion of the pressure Q on the main piston rod, we have :
in which
This gives
Q — 2 Psin p
2 P tan P
v^i -f- tan P2
tan /3 = —
•T'
* An equalizer of this type was patented in Germany, by the Berlin
Anhalt Works in 1885.
fln kinematic notation this action is expressed by CP fCP-1-), as shown
by b. See Theoretical Kinematics, pp. 322, 325.
or if we make Q the ordinatey9 of the desired curve :
2 Px
y = -7———
v/*-2 -f b1
and substituting c for 2 P x
we have	y	x
c ~ ....................
(3*7)
which equation is readily expressed graphically.
If this curve is drawh upon the rectangle which represents
the resistance of the water, as in Fig. 1015 b, we get the actual
resistance curve fg h, and this resembles closely the expansion
line for a high degree of expansion, or in other words, the im-
pelling force and the resistance are practically made equal to
each other througho"t the stroke. The dotted curve a b c d e,
is that of an actual indicator diagram.j This shows that with
the Worthington high duty pumping engine the most efficient
action of the steam is obtained at the same time as the best
action of the water end. §
Fig. 1016.
Fig. 1016 shows a longitudinal section of a Worthington high
duty pumping engine. The equalizing cylinders and their air
chamber are se en on the right; the dotted lines e2 show the rod
of the second cylinder, which operates the valve bv
As it has already been seen that mauy forms of the third or-
der can be reduced to the second order, it may be inquired as to
the possibility of obtaining a pumping mechanism of the first
order. This has already been accomplished by uniting the
steam escapement with a water ratchet train. The device is the
Hali Pulsometer, shown in diagram in Fig. 1017.
The steam enters at a, at b, is the anchor shaped pawl, and d,
is the vessel corresponding to the frame-
work of a rigid escapement, (compare
Fig. 775). If the vessel d is closed as
shown by the dotted lines and a volume
of water c, iucluded, we obtain an action
of the first order. The efficiency is very
low ; about to ^ that of a piston
pump, but the simplicity and conveu-
ience is so great that this may often be
neglected.
Another escapement of the first order
is Montgolfier’s hydraulic ram, which is
a water checking-ratchet train, the effi-
ciency of which is low. A more recent
device is the application of a water
ratchet train to drive a pneumatic rat-
chet train, first used on a large scale by
Sommeillier in the construction of the
Mont. Cenis tunnel, and by means of
which the efficiency was brought up
nearly to 50 per cent.|| Pearsall has re-
cently improved the hydraulic ram and
raised its efficiency to nearly 80 per
cent., either for water or for air, but this
t See Mair, Experiments on a direct-acting steam pump. I^-oc. Inst. C. E.
London, 1886.
I The Worthington equalizer accomplishes an end sought by designersot
steam pumps for the past 200 years, for since Papin s first machines in Cas-
sel (1690) the desired aim has been to combine the action of a variable elastic
driving medium, and a uniform. non-elastic resistance.
Ii See the author's paper, Ueber die Durchbohrung des Mont-Cenis.
Schweiz. polyt. Zeitschr, 1857, p. 147.THE CONSTRUCTOR.
233
has been done by the introduction of a valve gear, making it a
device of the second order.*
I 326.
Feuid Transmission AT IyONG Distance.
When the motive power is intended to operate the piston of
a pump situated at a distance, some connecting mechanism
must be interposed between the two cylinders. Formerly this
was accomplished by using long rod connections; instead of
this a pressure-organ transmission may be employed. When
water is used as the medium for transmission, this may be
termed a “ water rod ” connection. This is used in connection
with water levers (see § 311).
Fig. 1018.
Fig. 1018 shows three devices for this purpose. At a is shown
a closed System with pistons of equal diameter; b is a similar
one with unequal pistons; and c is a form with combined
pistons. Such water-rod connections are adapted for use in
mines, and the following example will illustrate.
2 327-
Rotative Pressure Engines.
An effective method of obtaining an advantageous action of
the steam is to substitute for the reciprocating mass of the
Cornish engine a rotating mass. This is accomplished by
using the reciprocating motion of the piston to operate a crank
shaft upon which a fly-wheel is placed. Since it is practicable
to give the rim of the fly-wheel four to six times the velocity of
the crank piu, the magnitude of the moving mass can be much
smaller, and since the value varies as the square of the mean
velocity, the mass is reduced at least 16 times. It is therefore
possible by this means to give even small pumps an efficiency
equal to that of large pumping engines.:}:
It is not practicable to construet single-acting pumping
engines into fly-wTheels, because the piston speed v is too varia-
ble. If we draw a curve representing v, the ordinates being
the positions of the piston, we have for a connecting rod of
infinite length a circular curve, as in Fig. 1021 a. When the
a.	b
The arrangement of transmission in the Sulzbach-Altenwald
is shown in Fig. 1019, which represents the engine aboveground,
while Fig. 1020 shows the mechanism in the mine.
The arrangement is of the same form as Fig. 1018 b. The
steam piston c operates the twro plungers bx b2% which in turn
operate the plungers c1 c/, and c2 c2' in the mine, the pump
plungers ex e2 being placed in the tniddle f
*See Engineering, Vol. XLL, 1886, p. 47, also H D. Pearsall, Principle of
the hvdraulic ram applied to large machinery. Eondon. 1886.
f See Zeitschrift fur Berg, Hiitten und Salinenwesen, XXII. p. 179 ; XXIII.,
p. 6; XXIV., p. 35. The depth is 820 feet, the speed from 6 to 16 double
strokes per minute, with a pause of one second, giving about 420 feet piston
speed per minute. This engine, built by the Bayetithal Machine Works at
Cologne in 1858. has operated regularly for 29 years without any mterrup-
tiou worth mentiouing.
length of the rod is taken into account these curves are modi-
fied, as shown in Fig. 1021 £, which is drawn for a rod four
times the length of the crank. This curve also shows the ratio
of the tangential force on the crank pin to the pressure on the
piston. $
The variations in the value of v, which often differ widely
from the mean value Vnu must necessarily be communicated to
the mass of water, and hence great variations occur in the
stresses. For this reason the velocity of the column of water
must be kept within moderate limits, notwithstanding the use
of air vessels. These variations become much less serious
when two pumps are connected by cranks set at right angles
with each other. The corresponding velocity curve is shown in
Fig. 1021 c, and many pumping engines are now so made. More
recently triple cylinder engines are made with cranks 120°
apart. The velocity curves in this case are shown at d. It is
evident that both these fornis involve complicstions in con-
struction which compare unfavorably with the direct-acting
pump with equalizing cylinders (see § 325).
Instead of using a revolving fly-wheel, the mass of metal may
be arranged to swing in an arc of a circle of large radius. An
ingenious application of this principle has been made by Kley,
in his water works engine with auxiliary crank motion. The
proportion between the steam pressure and the vibrating mass
is so arranged that the auxiliary crank comes to rest either a
little before, or a little beyond the dead point, so that the re-
turn stroke in each case can be effected by the action of a
cataract. In the first case, the fly-wheel swings backward after
X The Gaskill pumping engine is a duplex pump with fly-wheel, and
cranks at right angles, and has given excellent results. See Porter’s “ Re-
port of the Gaskill Pumping Engine at Saratoga."
I Referring to the designations in Fig. 1022, we have y- = sin « -j- tan a234
THE CONSTRUCTOR.
the pause, and in the second case, forward.* The valve motion
of this form of engiue is considered in the following sectiom
3 328.
Valve Gears for Rotating Engines.
Rotative engines are distinguished frorn pure reciprocating
pressure organ escapements in that they deliver their effort in
the form of rotary motion adapted to be used for driving run-
ning machinery. Between the two forms there is also the
iutermediate kind, with merely auxiliary rotative mechanism,
such as have been already referred to. The translation of
reciprocating and rotary motion may be accomplished in a
variety of ways, but by far the most useful and best known is
that by which the rectilinear motion of a piston is transmitted
to the shaft by crank connection.
The variatious in the tangential component of the pressure
P' 011 the crauk pin, Fig. 1021, becomes stili greater when the
pressure P, on the piston also varies by reason of the expan-
sion of the steam. For this reason some form of equalizer is
required in the form of a fly wheel. This latter becomes a
reservoir for the storage of living force. Extreme examples of
this actioti are found in rolliug mill work in which withiu a
brief time a 1000 H. P. engine may be called upon to deliver
2000 H. P., a demonstration of action of the fly wheel as a
reservoir of power.
The valve geariug for rotative engines is an important and
extensive subject. In the preceding sections a series of valve
gears have already been described. These have all been based
upon the principle of operatiug the valves by a direct recipro-
cating motion, taken either from the piston or piston rod.
With rotative engines another method is used, the motion being
taken from the revolving portion of the machiue, and this
method may also be adopted for pumps with auxiliary crank
action. We may then distinguish between :
Reciprocating valve gears, and
Rotative valve gears.
Rotative valve gears are desirable even for direct acting
pumps, but in a stili greater degree are they desirable for rota-
tive engines. Watt’s rotative engine was made with a recipro-
cating valve gear,f and this form has oue advautage in that it
is adapted for rotation in either direction.
Hornblower, the inventor of the compound engine, also used
a reciprocating valve gear. The slide valve, invented by Mur-
dock, in 1799, led the way to the introduction of the rotative
valve gear in 1800, but the old reciprocating gear stili continued
to be used, and is even re-invented at the present time. The
later direct acting steam pumps with auxiliary rotative mechan-
ism are almost always made with rotative valve gear. Kley’s
pumping engine, referred to in the preceding section, is made
with reciprocating valve gear, since its motion is both before
and behind the dead points of the crank.
The use of the slide valve, combining four valves in one inem-
ber, enables a very simple valve gear to be made for the ordinary
double acting escapement, as the diagram of a plain slide valve
engine, Fig. 1023, clearly shows.
1
Fig. 1023.
The use of an eccentric rx and rod lx to operate the valve b>
is not the earliest form of gear, the first method being the use
of an irregularly shaped cani which brought the valve to rest
except at the time of opeuing or closing.J A feature of the
slide valve which was long overlooked was the fact that the
time of closing the steam ports II and III could be regulated
so as to effect the proper expansion of the steam. In order to
accomplish this resuit without impeding the exhaust of the
steam, the eccentric rx must be given the so-called angle of
advance 20 1 . 2' beyond the mid-position. The direction of rota-
* For a fuller account of this interesting engine (German Patent No. 2345),
Of which more thau fifty are in operation, see: Berg-u. Hiittenm. Zeitung
Gliickauf, 1877, No. 18, 1879, No. 98 ; Moniteur des int. materiels, 1877, No 20;
Compt. rend. de St. Etienue, 1877, June ; Berggeist, 1879, No. 85 ; Z. D. Ingen-
iure, 1879, p. 304, 1S81, p. 479, 18S3, p. 579. Dingler’s Journal 1881, p. 1, 18S2,
p. 349; Maschinenbauer 1881, p. 63; Oesterr. Ztg. f. Berg u. Hiittenwesen 1882 ;
Kohleninteressent (Teplitz) 1882, No. 34 ; Revista metalurgica (Madrid) 1883,
No. 968.
t See Farey, Treatise on the Steam Engine, London, 1827, p. 524. Engines
with slide valves were only made by Boulton and Watt, after James Watt
retired to private life.
X See Farey, p. 677.
tion of the crank is then governed by this angle, the arrange-
ment above giving rotation to the left, and the position 1 2"
for rlf giving right-hand rotation.
The action of the slide valve may readily be represented
graphically.§ The angle of advance and lap being given the
point of cut*off can be determined by the following method.
Fig. 1024. The circle 1 CQ represents the circle of the eccen-
tric and may also be taken as the crank circle on a reduced
scale. C" and C'ff are two symmetrically placed positions of
the piston at which it is desired that the cut-off shall take place.
Through these points with a radius 1.3 = /.describe ares from
centres 3// and 3/// ; their intersections E2 and E3 with the cir-
cle give the angles at wrhich the expansion CQ C" and C' C"'
occurs, in this iustance of the stroke. We now select the
point v2 of the crank circle at which the admission shall begin,
join V2 E2 and draw the equator 2 . 1 . 2' parallel to it, and the
angle 2.1 . O will be the angle of advance S, and the distance
of 2 . 1 from E2 F2, the outside lap e2 for the port //. The
width of port a must also be chosen, and must be so taken that
it is less thau rx— e2, and is represented by the parallel
When the crank reaches /2, in this iustance at of the stroke,
the exhaust begins, and the distance i2 i2 of the parallel /2 /2
from the equator is the inside lap.
The construction is similar for the other half of the stroke.
The angle S is already known, and hence the parallel Ez Vz
from Ez can be at once drawn, and the admission point V3 de-
termined. The outside lap es is somewhat less than e2, thus
giving a correspondingly wider port opening. The inside lap
/3 is made equal to /2, and the bridges b3 and b2 are made equal,
thus giving a symmetrical valve seat. A certain amount of dis-
cretion is permissible in the selection of b2— b3; care being
taken that there is sufficient bearing at the extreme valve stroke
to insure tightness. The points // and I/ are also of impor-
tauce, as they determine the closing of the exhaust. The cor-
responding piston positions Civ and Cvare not symmetrical,
because t3 — i2l but the inequality in the compression is not
serious.	%
The above method of considering the influence of the ratio
but the variation is slight. It must be noted that the distance
1 . 3 must be laid out to the actual scale of construction.
The application of
Zeuner’s diagram to
the same case is made
in the following man-
ner, Fig. 1025. The
circle 1 CQ represents
as before, the eccentric
circle and the crank
pin path. The angle
CQ. 1. 2 = C' . I . 2 =
90 — 6. With 1 as a
centre, describe circles
with radii ^and /, here
made alike for both
ends of the valve, also
one of radius e -f- a,
Fig. 1025.	Upon 1 . 2 and 1 . 2 as
diameters, describe circles, called the valve circles.
I Formerly the so-called “ valve ellipses ” were used : since 1860 Zeuner’s
diagram has superseded these, see Zeuner, Schiebersteuerungen, Freiburg,
Englehardt, first published in Civil Ingenieure, Vol. 2, 1856.THE CONSTRUCTOR.
235
The intersection of radii frorn 1, with these circles, give the
distance of the valve from its middle position for various crauk
positions. For the position 1 for instance, the admission
for the left stroke begins, at 1 E2 the expansion, at 1 / the ex-
haust, etc.*
The Zeuner diagram gives the valve position by means of
polar co-ordinates, while the writer’s diagram is based on par-
allel co-ordinates. To be strictly correct, the valve circles 1.2
and 1 . 2' of the Zeuner diagram should fall upon each other.
The arrangement shown has been adopted by Zeuner as more
convenient in practice.
It will be seen from the preceding that the rate of expansion
can be varied by altering the eccentricity and the angle of ad-
vance. This may be carried so far that the direction of rotation
is changed, giving what is termed a reversing motion. A variety
of reversing motions have been devised, which accomplish the
desired relation of parts by shifting a reversing lever. Ot these
the most practical are the so-called link motions, of which a
number will here be briefly shown.f
b.
Fig. 1026 a, is an outline diagram of Stephenson’s link
motion. The link 3' 3", of convex curvature towards the valve,
is giveri an oscillating motion by means of the two equal eccen
trics 1 . 2', 1 . 2//, and is suspended from its middle point 7,
from the bell crank lever 5 7'. The motion of the link is trans-
mitted to the valve by means of the sliding block 5, and rod 6.
Fig. 1026 b, is Gooch’s link motion. The link 4 is driven by
two eccentrics as before, but is curved in the opposite direction
with a radius 5.6, and is suspendfcd from its middle point 8 to
a fixed pivot 8', while the rod 5.6 is shified by means of the
lever connection 5* 10 . io'.
a.	b.
Fig. 1027 a> is the link motion of Pius Fink. In this form
the link is operated by a single eccentric instead of two, as in
the previous fornis. This simple mechanism is not as widely
used as its merits deserve.
Fig. 102j b, is the link motion of Allan, or Trick. In this
design the link 4, is straight, and both the link and th„ radius
rod are suspended and shifted by the lever conuections 8' . 8,
and g/ . 94
a.,	b.
Fig. 1028 af is Heusinger’s link motion. The link 4, vibrates
upon a fixed centre 9, and is operated by an eccentric 1 . 2. The
valve rodis moved from the main cross head by the connections
10 . 11 . 6 . 7, and also by the radius rod 5 . 6, which latter is
suspended from the bell crank .S. 12'.
Fig. 1028 b, is Klug’s valve gear, known in England as Mar-
shall’s. The curved link 4, is rigidly secured and does not
* It is usual to raake the valve symmetrical, i. e., e%= c2. which necessarily
causes the cut-off to take place at different poiuts for the back and forward
strokes.
f See also Zeuner, as above; Gustav Schmidt, Die Kulissensteurungen,
Zeitschr. d. dsten. Ing. u. Arch.-Vereins, 1866, Heft. II.; also Fliegner, Ueber
eine gelr. Eokomotiv-steuerungen, Schweiz. Bauzeitung, March, 1883, p. 75.
x See Reuleaux, Die Allanische Kulissen steuerung, Civ. Ing., 1857. p. 92.
move. The eccentric 1 . 2, moves the valve connection 6.7, by
means of the lever 2.3.6, which vibrates about the point 3, on
the end of the radius rod, the other end of the rod being held
by the link block 5. Instead of the link 4, a radius arm 40 5, is
often used, the centre 4? corresponding to the centre of curva-
ture of the link, the action being the same in both cases.$
Fig. 1029 is Brown’s valve gear, which differs from the pre-
ceding by the substitution of a straight link of adjustable angle,
for the curved guide link.
Fig. 1029 by is Angstrom’s valve gear. The point 3 of the pre-
ceding gear is guided by a parallel motion, and the point 6 is
between 2 and 3, instead of beyond.
The eight preceding valve gears operate the valve approxi-
mately in the same manner as if a single eccentric of variable
eccentricity and angular advance were used, the eccentric rod
being assumed of infinite length as compared with r. The patii
of the successive positions of the middle pomtof thisimaginary
ecentric is called the Central curve of the valve gear.
Fig. 1030 sbows the form of this curve for link motions in
general. Form a, is that for cases 1, 4 and 5 ; form b, for case
1, when the eccentric rods are crossed, and form c, in which
the curve becomes a straight Ime, is for cases 2, 3, and 6 to
8. In the latter instance, the lead, or opening for admission of
steam at the beginning of the stroke is constant, a point con-
sidered by many to be of much importance.
It is possible to arrange the mechanism in such a manner that
the centre of the valve motion may move directly in the desired
Central curve, as is shown in Fig. 1031.
This construction involves the rotation of the link about the
crank axis. The only point to be accomplished is to guide the
centre 2' in the path 2' . 2 . 2". Fig. 1031 c, is a direct guide
for the eccenttic with wedge adjustmeuts ; b, is SweePs valve
gear, in which the position of the eccentric is determined by a
centrifugal governor.|| This only uses the Central curve from
§ For a further account of this gear, see : Berliner Verhandl, 1877, p. 345,
1882, p. 52. Engineering, Aug. 13, Oct 1, Dec. 3, 1880; Nov. 4, 1881; June 23,
1882; Feb. 6 and 27, 1885; Jan. 12, 1886 ; Sept. 9, 1887. Engineer. May 26, 1887 ;
Feb. 23, Mar. 30, April 27, June 29, 1883 ; June 5, 1S85. Marine Engineer,
1885, No 1., Civ. Ing. Heft, 7 and 8, 1882 ; Zeitschr, D. Ing, 1885, p. 289, 1886,
PP 509-625; Revue universelle, 1882. p 421; Busley, Schiffsmaschine, I , p.
454; Hartraann, Schiffsmaschinendienst, Hamburg, 1884, p. 53; Blaha,
Steuerun^en der Dampfmasch, Berlin. 1885, p. 65.
|j See Rose, Mech. Drawing Self-taught, Philad. Baird; for sirailar gears.
see Am. Machinist, Grist, Oct. 5, 1883 ; Ball, ditto Aug. 18, 1883 ; Harmon,
Gibbs & Co., ditto Nov. 24, 1883. Also Sturtevant, The Engineer, New York,
Jan. 1888.2^6
THE CONSTRUCTOR.
2/ to 20, and the path is a curve produced by a radial arm as in
Klug’s valve gear. The valve is balanced, in order to reduce
friction to a minimum.
The last described valve gear possesses the advantage of great
simplicity but retains the disadvantage of all single valve gears
when used for a high expansion ratio, i. e., the admission and
exhaust of the steam do not remain uniform, and are often un-
satisfactory. For this reason many valve gears with indepen-
dent expansion valves have been designed.
Fig. 1032.
Three forms of gear with separate expansion valves are shown
in Fig. 1032. The forni a, is kuown as Gonzenbach’s ; that of b,
by various names; c} is the widely used Meyer valve gear.*
In France, Farcofs gear is used, having two loose cut-off
plates carried 011 the back of the main valve, and in America,
the excellent Porter-Allen engine with double valves operated
by two eccentrics, is much used. Rider’s valve gear is a modi-
fication of Meyer’s, Fig. 1032 c. The two cut-off plates forni re-
verse spirals, and slide in a concave seat on the back of the
main valve, the admission parts being also spiral shaped and
cut-off varied by twisting the cut-off valve axially.f
Instead of using eccentrics to operate the valves, cams of
irregular outline may be adopted, these permitting a rapid
opening and closing of the parts. Noteworthy examples of
cam valve movements are to be found in the steamboats of the
Western and Southern States in America. In its original forni
of a cock or cone a slide valve may be operated by an alternat-
ing motion as well as by continuous rotation. Such valves
have been used in steam engines by various builders, among
them the firm of Dingler in Zweibriicken, but the cost preveuts
wide use. A most extensive use of oscillating cylindrical
valves has been developed by Corliss and his followers.
The forms of oscillating and rotating chamber gear trains
already described involve other means of operating the valve
than are used for recip-
rocating engines, and
shown in Fig. 1023. As
an example, the water
pressure engine of
Schmid, of Zurich, Fig.
1033 is given. In this
instance the valve b, is
formed in the frame of
the machine, and is of
the type shown in Fig.
987 c. The regulation
of speed of rotative
water pressure engines
is a much more difficult
matter than is the case
with steam engines,
partly on account of
the lesser fluidity of
Fig. 1033.	the water and also be-
cause of its sligbt elas-
ticity. An air chamber in the admission pipe as shown in Fig.
1033 is therefore desirable, and when extreme changes of load
are anticipated the valve gear should be modified. If it is
desired to cut off the admission of water before the end of the
stroke is reached, it is necessary to arrange a special valve to
permit the discharge to continue. Excellent engines with this
arrangement have been made by Hoppe, of Berlin, for the
Mansfeld mines, and for the Frankfurt railway station.
Another method is applicable to power driven pumps, an
illustration of which may be found in the design of Franz Hel-
fenberger, of Rorschach.J This is made with a hydraulic
ratchet mechanism arranged in the crank disk in such a man-
ner as to move the crank pin to or from the centre, the ratchet
being operated by tappets which strike each time the crank
passes the dead centres. The throw of the crank is thus varied
to correct for variations of speed, the mechanism being con-
trolled by a regulator. The action is very satisfactory, giving
results varying from 90 to 82 per cent. for a change of power
*An excellent gear with two valves operated by a single eccentric, is
Bilgram’s. See Bilgram, Side Valve Gears. Philadelphia, 1878.
f This is an excellent problem in kinematics, the action of the valves
and spiral ports forming a kinematic Chain. See Theor. Kinematics, p. 333.
JGerman Patent, No. 12,01*, Jan. 27,1881.
from 1 to |, according to the investigations of Autenheimer,
Buss and Kuratli, in 1885. \
A later device is that of Rigg, which also acts by varying the
stroke. The machine is a so-called “chamber crank train ”
(described in Theoretical Kinematics, p. 359: English ed., p.
361), with four single acting cylinders carried on the revolving
wheel in the sanie manner as the machines of Ward, Schneider
and Moline. The length of stroke is controlled by a regulator,
similar to Sweet’s governor, Fig. 1031 b, which operates a hy-
draulic escapement and adjusts the radius. This device is used
by Rigg for steam and air engines to control the degree of ex-
pansiou. These latter machines are operated as high as 2000
revolutions without producing trerabling. ||
Besides the various forms of valve gear wThich have already
been described, there are also the numerous “ trip ” gears, of
which some examples have been given in $ 252. These gears
are made in many forms. The valve is made in four parts, as
indicated in Fig. 986, on account of the facility with which the
release can be controlled by the regulator.
The varieties of trip valve gears are most numerous, and
there can be little doubt that the subject has been overdone,
when it is considered that in many instances the entire mechan-
ism of the engine has no other aim than to determine the open-
ing and closing of the valves. In America, where this forni
originated, the reaction has already set in, and there is a dis-
position to return to the single slide valve, especial care being
taken, how^ever, to secure relief from pressure and to produce
correct motion.
There is to be found in some parts of Germany ‘a forin of
valve gear which may be called an “ inner ’ ’ and “ outer ” gear.
This form does not pos-
sess sufficient merit to
meet general application,
but may be briefly no-
ticed. The mechanical
action of parts is not
different, whether the
“ inner ” or “ outer ” con-
struction is used, and
either arrangement may
be adopted, at the dis-
cretion of the designer.
The following exam-
ples will make the ar-
rangement apparent, as
well as the illustration
already given of Schmid^
water pressure engine,
Fig. 1033. Fig. 1033 a is *
an “ outer ” valve gear |
for a blowing cylinder,
and Fig. 1034 b is a valve
for a vacuum pump.^f
Another example is Cu-
velier’s valve, wdiich is
placed entirely outside of
the steam cylinder, as is |
also the gear ofLeclerq.** f
The ordinary slide valve &
is partly without and	pig. 1034.
partly within the ma-
chine, being outside the cylinder and within the steam chest.
C. ADJUSTABLE POWER ESCAPEMENTS
l 329-
Adjustabee Pump Gears.
The principies of adjustable escapements have already been
discussed in g 259, and examples of rigid construction given.
§A third system is that devised by Hastie, of London (see Engineer,
August. 1878, and April, 1880. p. 304). ' Hastie Controls the position of the
crank pin by means of a pair of curved cams which act against increasing
external resistance, and opposed by a spiral spring for diminishing resis-
tance. This device does not give complete regulation for the following rea-
sons: 1. In order that the statical moments of the increasing resistance
and driving force on the crank pin shall be equal, the spring must act on
the curved cams in such a manner as to cause the crank pin to move out-
ward. This can be approximately accoinplished by a careful arrangement
of parts, but only approximately. Under this arrangement, however, there
is a tendency for the pin also to’move outward when the driving force in-
creases, insfead of moving promptly inward as should be the case. An
attempt to correct for this error, reverses it. 2. The angular velocitv of a
motor is not a function of the impelling force, i. <?.. the machine will run
fast or slow as the case may be, this fact also appearing in practice. This
is a common error into which inventors of “ dynamometric governors”
have fallen, even Poncelet, himself, having done so in his dynamometric
regulator.
|| See Rigg, Obscure Influences of Reciprocation in High Speed Engines,
Trans. Soc C. E., 1886; also Engineer, June 4, 1886. These engines are used
on the compressed air Systems of Birmingham and Leeds.
f See Oppermann. Portfeuille econ., Feb. 1888, p. 18.
**See Genie ind. 1864, also Schweiz. polyt. Zeitung, 1864, p. 83.THE CONSTRUCTOR.
237
Their action consists of the two following operations: 1. By
the adjustment of one part the release of another part to the
action of an impelling force is accomplished ; 2. By the attain-
ing of motion of the checked member, the checking is again
produced either directly or indirectly.
These principies also obtain for escapements for pressure
organs, and include a great number of important applications.
The general scheme of such an adjustable gear for a steam
pump cylinder is shown in Fig. 1035 a. The valve chest dx is
made separate from the cylinder d, and is capable of movement
parallel to it, the connections ax a2 being made flexible. The
slide valve b is operated from above by the lever b'. When the
lever b' is lifted the pressure is admitted under the piston c
through the port ///, while the space above the piston is in
communication with the exhaust IV. This causes the piston to
move upwards and hence the lever c' moves the steam chest dx
also upwards, thus closing the valve ports and checking the
movement of the piston. If the lever b' is again lifted this
action will again take place, and so on until the upper limit is
reached. A reverse motion of the valve lever produces a eor-
responding reverse motion of the piston * The same action
may be obtained by using the arrangeinent shown in Fig. 1035 b.
In this case the valve chest is fast to the steam cylinder, while
the valve is so arranged as to be moved both by the hand lever
b' and by the piston rod c. When the valve is moved by b', the
piston also moves and closes the valve by the lever c\ thus
bringing itself to rest again. The piston c follows the motion
of the lever b' in either direction ; starting when the lever is
started, and stopping when it is stopped. Auy resistance not
exceeding the force of the pressure at ax, can thus be overcome
while the resistance to the operator is only that due to the frac-
ture of the valve and connections. The practical value of this
device in many directions will be evident, and the examination
of the above simple forms will explain the action of the various
modifications.
Two constructions, designed by the author in 1866 for regula-
tors will be found described in the Civil Ingenieur.\ The
lever is connected with the valve by means of a double parallel
motion which is moved by the piston motion back into a posi-
tion parallel with the base line.
The operation was satisfactory but the apparatus was cum-
brous. In 1868 Farcot constructed a similar device, using an
approximate parallel motion, but the apparatus was too com-
plicated to be practically satisfactory.]; A somewhat simpler
construction was afterwards made by Farcot, but this was also
too complicated for practical use.£ Other designs have been
made by Farcot. Some recent constructions are here given.
Fig. 1036 shows the hydraulic steering gear of Bernier-Fon-
taine & \Vidmann,|| which is similar in principle to Fig. 1035 b.
In this case the controlling gear b\ consists of a small hy-
draulic piston. The water pressure is admitted to it through
the pipe a'. and is opposed by the spring a". The two plun-
gers Cx and C2, act as a double acting piston, the hydraulic
* The escapements shown in \ 259 are only single-acting, and do not admit
of a reverse motion.
f Civ Ingenieur, 1879-1880, Prof. Rittershaus, Ueber Kraftvermittler. Also
the models at the Roval Technical High School in*Berlin.
]See Civ. Ingenieur, 1879, 1880, Rittershaus, Ueber Kraftvermittler.
(Intermediate Power Mechanism.) A model of the design of 1866 is in the
cabinet of the Royal Technical High School at Berlin.
gSee Annales industiielles 1873, P- 5l8 ; also Oppermann, Portfeuille econ.
1874. p. 113.
flSee Revue Industrielle, i?S6, p. 373 5 l887> P »48.
pressure being supplied from an accumulator. The valve b is
operated by the plunger b' against the pressure of the springs
b'", and again reversed by the pistons Cx C2 and connection 5.
The piece at 6 is not a lever but is a cross head connected with
Fig. 1036,
the valve. The admission and release of water pressure through
a' forms a long distance transmission involving the use of an-
other escapement; the whole thus forming a.gear of the second
order.
Fig. 1037, is a neat regulator for steam engines by Guhrauer
& Wagner.^f In this, as in
Fig. 1035 a, the valve seat is
capable of movement in a
direction parallel to the pis-
ton e, and is made concen-
tric with the piston rod, the
valve b, being a piston valve
or rod moved by the gover-
nor. The piston c is subjec-
ted to full steam pressure
from ax on both sides through
the ports II and III, but as
soon as the valve b is moved
up or down, the holes bQ re-
lieve the pressure on one
side or the other, the equi-
librium is disturbed and the
wiredrawing of the steam
through the small ports II
or III prevents sudden action
and the piston moves until
the holes are closed. As
might be expected, this de-
vice is very satisfactory in
practice.
Devices of this type are
well adapted for steering
gears as well as for regula-
tors and a very delicate ap-
plication of the principle is
found in the Whitehead tor-
pedo, in which the steering
gear which determines the
depth of the torpedo beneath the water is thus controlled by a
barometric device.
I 33°-
Adjustabue Gears for Rotative Motors.
The principies of the gears described in the preceding section,
are also applicable to rotative motors although the arrangement
is not so simple as with direct reciprocating cylinders, since
the motion of the valve gear has also to be controlled. At the
same time it must be uoted that the application of adjustable
gears to direct acting reciprocating motors is the more recent
of the two. The earliest rotative gear of this sort, so far as the
author has been able to ascertain, is that designed by F. E.
Sickles, of Providence, R. I., in 1860 (See also g. 252).**
f Built by Ganz & Co , of Budapesth, with Meyer^s and with Rider’s valve
gears.
** According to the catalogue of the Centennial Exposition, in Philadel-
phia, in 1876, Vol. II. p. 52, Sickles made his first application in 1849 , his
patent was granted in 1860, and his first machine exhibited at the Eondon
Exhibition of 1862.
Fig. 1037.238
THE CONSTRUCTOR.
Sickles’ machine was made with two oscillating cylinders.
BotU eccentrics were fastened together and were loose on the
crauk shaft and operated by a hand wheel and spindle. The
steam chests oscillate with the cylinders. The crank «haft re-
volved in the same direction as the haud wheel is turned, but as
soon as the motion of the latter was stopped, the valve seat
moved under the valve and the ports were closed.
The more recent fornis of adjustable valve gears for rotative
engines are made after two distinet and important principies.
The first forni is that in which a double engine, without a fly
wheel and with ordinary slide valve gear without angular
advance is used, in order to permit rotation in either direction.
The ports I and IV are then made so as to be interchangeable
so that I can be connected either with the admission aY or ex-
haust a2; and IV with the exhaust a.2 or admission alf at will.
This change of connection is effected by means of an auxiliary
valve sometimes kuown as a “ hunting valve.” This hunting
valve can readily be controlled by hand for a steering en-
gine, for which it is well adapted, siuce the angular motion of
the rudder piu is limited, seldom exceediug 90°. The adjust-
ing valve can then be arranged according to either of the prin-
cipies of Fig. 1035 a or b. The following desigus will illustrate
the construction.
Fig. 1038 shows the steering gear of Dunning & Bossiere.*
The adjusting valve b rides upon a moveable valve seat bo. The
lower port A is always in communication with chests of the
two engine cylinders, while the upper port J is in communica-
tion with the Central port under the valves. The lever b' is
connected with the spindle b" by an iuternal gear. This spindle
Fig. 1039.	Fig. 1040.
has a screw thread of steep piteh, and is connected to the ad-
justing valve b. The moveable valve seat bo is connected to a
spindle bo"> which has on it a mucli slower screw thread, and
is also geared by bevel wheels to the axis cf of the drum of the
tiller chains. Whenever the engines are started by moving the
lever b\ the chain drum revolves and shifts the moveable seat
b0 until the ports are again closed. The parts are so propor-
tioned that the angle through which the rudder is moved is
equal to the angle through which the lever b' has also been
moved. This enables the position of the rudder to be deter-
mined at once by noticing the position of the adjusting lever.
The moveable valve seat bo will be recognized as the same in
priuciple as the moveable steam chest of Fig. 1035«. The
spindle bx' can be prolonged to operate an indicator on the
bridge for the inspection of the officer in charge of the ship.t
Fig. 1041.
Fig. 1039 shows Britton’s steeiing gear.:}:	The adjustment is
effected by a hand wheel and screw b' operating the lever b" at
6, and thus moving the valve b. At 7 this same lever is con-
nected to a nut 011 a screwT thread cut on the axis c' of the chain
drum, so that the motion of the latter closes the valve after it
has been opened by b'. This corresponds in principle to Fig.
1035 b.
Fig. 1040 shows the steering
gear of Douglas & Coulson.§
This is another application of the
same principle as the preceding
de vice. When the adjusting screw
b' moves the nut, lever and rod b
out of the mid-position, the re-
volving axis c of the chain drum
turns the nut bx' by means of the
spur gearing until the dead posi-
tion is again reached.
Fig. 1041 is a steering gear de-
signed by Davis & Co.|| This is a
simpie application of the principle
of Fig. 1035A The hand whetl
shaft bf has a screw thread at 6,
the nut being in the hub of the
worm wheel cfy the latter being
driven by a worm on the crank
shaft. Any adjustment of the
valve rod b by turning the hand
wheel results in a corresponding
readjustment by the motion cf
the worm wTheel and nut derived
from the engine.
The second kind of adjustable
gear for rotative engines is much
less frequenti y used than the first
form. In this arrangemeut the
adjustable valve is not connected
with the main valve gear, but is
operated independently, so that
the crank will make any desired
number of turns in either direc-
tion, according to the motion
which is given to the adjusting
valve.
Fig. 1042 shows Hastie’s steer-
ing gear.^f This is based 011 the
priuciple of Fig. 10350. The mov-
able valve seat bQ is operated by
the piston c, the eccentric cl being
so placed that bQ has a reduced motion coincident with that of
the piston c. The valve b is operated by the eccentric b//, which
f A steering- gear of sirailar design, with moveable valve seat bo and ad-
justing valve. is that of Hastie, English Patent No. 1742, 1875. Also that of
Holt; see Engineer, Sept , 1877, p. 221.
t See Revue Industrielle, 1884, p. 435.
\ See Engineering. April, 1882, p. 281.
I See Engineering, April, 1882, p. 398.
% See Rittershaus in Civil Ingenieur, already cited.
* See Revue Industrielle, 1866, p. 401.THE CONSTRUCTOR.
239
has the same throw as c±, and is moved by a hand wheel on b
The aetion which follows is that the crank shaft follows the move-
ment which is given to the hand wheel both in direction and
in revolutions. This aetion is similar to that of duplex pumps.
Any number of revolutions may be made in either directions,
and the device is a genuine rotati ve gear, as was also the ear-
liest type, i. e. Sickles’ gear, already described at the begin-
ning of this section. It would not be difficult to design a simi-
lar gear on the principle of Fig. 1035 b, but the author has no
knowledge that this has been attempted.
Adjustable valve gears for rotative engines have generally
been designed for steam steering engines, and in some of the
recent powerful marine engines they have also been used to
shift the link motion. There are many other applications
which might be made. The speed can also be controlled by the
adjusting wheel or lever b\ if desirable, by suitable connections
to the steam valve. As simplicity in construction is most im-
portant, steam economy is not considered in designing ma-
chines of this kind.
D.—ESCAPEMENTS FOR MEASUREMENT OF VOLUME.
?33«-
Running Measuring Devices.
In the classification of running mechanisms operated by
pressure organs, it was noted that these devices could be used
for measurement of volume. As already showm in g 321, fluid
escapements are better adapted for measurements of volume
than for measurement of time ; but there is a close resemblance
between the two operations, and many fluid meters might
properly be classed as time-pieces. When the fluid to be meas-
ured is a homogeneous liquid, the quantity and volume are in
direct proportion. With fluids wdiich are not homogeneous,
such as gases and the like, a knowledge of the density is neces-
sary in order to determine the quantity from the measured
volume; if the density is also to be determined at the same
time as the measurement, the problem becomes much more
complicated, as will hereafter be seen.
Liquids are frequently measured by means of continuous
running devices; but the choice of construction is very limited.
Among the open wheel devices there is available practically only
the form shown in Fig. 957 d, and that only when the liquid is
under moderate pressure. If, then, the liquid is slowly con-
veyed off below the horizontal plane of the axis so that the
acceleration of the wheel is uniform, then will the continuous
rotation of the wffieel be proportioned to the volume of the
liquid passing through it.
An instance of this construction is the measuring drum in
Siemen’s meter for spirits.* This is made with three buckets,
and has inward dei i very. Since the question of the density is
in this case important, Siemens has devised a very ingenious
float arrangement by which the couuting mechanism is regu-
lated to the volume of flow.
When liquids under high pressure are to be measured by
such a device, the wheel must be inclosed in a case in which
also a gaseous fluid must also be contained under a correspond-
ingly high pressure, which is usually inconvenient. For meas-
urements of high-pressure liquids, the chamber gear trains
already described are preferable, especially since it is practica-
ble to pack the working joints reasonably tight.f Wheels in
which the living force of the water acts are also adapted for
this Service, either as bucket wheels or in the form of reaction
wheels (see § 315). Siemens has constructed a meter of this
kind, in which a reaction wdieel is used, and the error of which
does not exceed 2 per cent., plus or minus.X
Another form, giving fair results for open channels, is based
on Woltmann’s fan.
Gaseous fluids of small and only slightly varying density
can be well measured by modifications of bucket water wdieels ;
the conditions being practically ieversed from those already
considered, and the water now being the surrounding medium,
and the gas the one to be measured.
One of the best known and most widely used devices for this
purpose is the ‘‘ wet” gas meter of Clegg and Crosley, shown
in Fig. 1043 a. The revolving drum is a wheel with four
buckets, which is driven by the passage through it of the gas.
The gas is introduced just above the horizontal plane through
*See Zeitschr. D. Ing., 1874, p. 108 This meter is in extensive use in
Russia, Sweden and Germany for measuring brandy and alcohol, and is
very satisfactory. In Sweden, in 1F83, a comparison showed that a most
careful hand measurement gave 15,365,931 litres of 50 per cent alcohol, and
the meter gave 15,450,775 litres, or an excess of onlv y2 of 1 per cent.
f See the Orown Water Meter in Schweizerische Bautzeitung, March, 1883,
p. 81; also Payton’s Water Meter.
X The older form (1857, Z. D. Ing., p. 164) was on the principle of Segner’s
wheel, Fig. 962 a ; the more recent design is made like the turbine of Fig
963 d. At the end of 1886 Siemens & Halske had made 88,500 meters, and
the English house of Siemens Bros., 130,000 of the old and new patterns.
the axis, and the liquid used is water; or, if there is danger of
freezing, glycerine may be substituted. If the level of the
water is lowered through evaporatiou or leakage, the volume
of gas passing through the meter at each revolution will be
increased, and to avoid this a float is so arranged that the gas
will be shut off if the water level falis too much.
b
For very accurate measurements of gas, Sanderson’s meter is
used. Fig. 1043 b i11 the water level remains unaltered so long
as the vessel is kept suppiied. The semi-cylindrical float is
pivoted on the axis C, and is so constructed that the centre of
gravity of all the sectors is the same as if the sheet metal body
were homogeneous. If the float moves through au angle a
with a sector A CB, the thrust of the sector A'C D of an angle
1800—2 a will pass through the axis, with force Pf for the sector
A'CB. The weights P of the two equal sectors A CB and
A'CB act downward through their centres of gravity, and are
also equal to P'. In order that there may be equilibrium, if
for any chosen value of a, P' shall equal 2 P\ the specific
gravity of the float (assumed to be homogeneous) must be
equal to one-half that of the liquid in the trough, i. e.> = x/z for
xvater, or = 0.63 for glycerine.
The preceding meters have heretofore been used only for
gases under low pressure, but are equally well adapted for
gases at high pressures, such as compressed air for power
transmission, simply by increasing the strength of the case.
This has been done at the author’s suggestion in connection
with the compressed air System at Birmingham, as has also
been the case with the “dry” meter described in the next
section.
Anemometers, used for measuring the flow of air, generally
belong to that class of running devices which are driven by the
living force of the pressure organ isee § 315). They are usually
screw turbines, or some modification of them. It is always
necessary to take into account the stress of the gaseous me-
dium, in order to obtaip the desired measurement, since the
apparatus only determines the volume.§
§ 332-
Escapements for Measurement of Fluids.
There exist certain defects in running devices wdien used as
fluid meters, such as the journal friction, and in chamber gear
trains the surface friction, which render the results inaccurate
for fluids of weak flow. For this reason piston meters have
been devised, these also utilizing the power of the fluid, and
for these the application of escapements is necessary. Meters
constructed on this principle have been used especially for
measuring water. Among these may be meutioned Kennedy’s
water meter, a form which has been extensively used.|| This
is usually made with a vertical cylinder, the valve being a
fourwav cock operated by a tumbling gear similar to that of
Belidor’s water-pressure engine (§ 325).
Jopling’s water meter is a piston escapement of the second
§ Running devices may also be used to measure time as well as volume,
and in fact the oldest constructions for this purpose, the clepsydra, the
sand glass, etc., belong to this class. Escapement clocks were introduced
only in the middle ages. There have been numerous recent attempts to
make running time-pieces. (See Redtenbacher, Bewegungs-mechanismen,
Heidelberg, 1861, p. 34, pl. 79 ; also Riihlmann, AUgemeine Maschinenlehre
I., Braunschweig, 1862, p. 62 ) The problem is a difficult one. since it in-
volves the construction of a running device which shall operate both with
a uniform and a determinate velocity. Examples are found in the driving
mechanism of astronomical Instruments, in which the motion is transmit-
ted by friction. controlled by revolving fly-wheel devices. With these may
be included the fan regulators for the striking mechanism of clocks, and
similar applications.
i See Revue Industrielle. 1881, p. 205.
f See Z. D. Ing., 1857, p. 164; Maschinenbauer, Vol. XVI., 1881, p. 324;
Technologiste, 1882, Vol. 42, p. 95.240
THE CONSTRUCTOR.
order. There are two parallel horizontal double-acting cylin-
ders, each operating the valve of the other.
Fig. 1044 shows a section of Schmid’s water meter. This is
made with two single-acting pistons, each also being the valve
of the other, and the whole fonning with the crank connection
an escapement of the third order.
Escapement meters are also used for gaseous fluids. A very
extensively used form is the so-called “dry” meter used for
measuring illuminating gas. These have, in many cases, super-
seded the “wet” meter, since the use of the liquid seal is
avoided. In order to prevent friction, these meters are con-
structed with flexiWe diaphragms iustead of pistons, much like
the diaphragm pumps shown in g 317. A good example is
Glover’s dry meter, which is an escapement of the second
order connected to a crank shaft which operates the counting
mechanism. The diaphragms are made of linen, made imper-
vious to gas by a gelatine sizing. These meters do not show a
higher degree of accuracy than the wet forms.
3 333-
Technological Applications of Pressure Organs.
The applications of pressure organs for technological uses
are not so numerous or important as those in which they act
in connection with the help of various machines. These appli-
cations are not dissimilar from those of tension organs, which
have already been discussed in \ 263. A general survey will
be of value for the better understanding of the whole sub-
ject.
The use of a pressure organ from a technological standpoint
consists in so using it that the resuit is a modification in form
or shape either of another body directly by the action of the
pressure organ or of the pressure organ itself by the other
body. This “forming” action adds a fourth manner of action
for pressure organs to those already classified in \ 309, so that
we now have:	#
1.	Guiding,
2.	Supporting,
3.	Driving,
4.	Forming,
as the four methods of action or application. Of these the
first three belong to all classes of machine elements used
in construction ; the fourth falis within the domain of tech-
nology.
I11 order to speak comprehensively of the action of pressure
organs, we will arrange them in live groups, according to the
rnethod of action, viz.: by Filling, Discharging, Internal Flow,
Jet Action and Inclosing or Covering.
a. Filling.
1.	The ease with which pressure organs can be led into de-
sired channels on account of their fluidity is applied in the
operations of casting. Metals which it is desired to make into
given forms are rendered fluid by heat, and thus converted
into pressure organs which can readily be run into moulds.
In similar manner wax, stearine, paraffin, etc., are cast, in
making candles and the like, the formed material resuming
its solid state on cooling. Plaster, cernent, magnesia or similar
materials may also be made fluid by mixing with water, and
then cast into forms which afterwards become hard by com-
bination with water and carbonic acid. Other and similar
methods are adopted for other materials.
2.	Glass, in a plastic state, is formed by pressure in moulds
or by passage between rollers.
3.	In cases where complete fluidity is unnecessary, the mate-
rial may be softened by heat, and then shaped in suitable
presses, the slight fluidity of the material being overcome by
the mechanical pi essure of the machines.
4.	Lead is sufficiently soft to be readily formed by the action
of a plunger press, and is thus formed into bullets in arsenals,
and also made into pipe.
5.	The forming of a pressure organ by cooling is showm in
the action of an ice machine, by means of which water may be
given the form of sheets, rods, blocks, etc.
6.	Copper, tin, zinc, etc., and also gold and silver are formed
under the drop press in dies. Steel and wrought iron are
heated for this purpose; but sheet Steel is stamped cold.
7.	Wire, already considered as a tension organ, may also be
treated as a pressure organ, having great similarity to a flowing
stream with its curves. Examples are found in the ingenious
machines for making hooks and eyes, and also wire chains,
made by William Prym, of Stolberg. Another illustration is
the machine of Hoff & Vogt for rolling spiral springs.
8.	Hydraulic or lever presses are used to press clay in a
plastic coudition into various dies to make bricks. Bricks are
also forms of compressed turf, culm, etc. Chocolate and cocoa
are also compressed in moulds.
9.	The so-called art work in pressed wood is composed mix-
ture of sawdust formed into a solid mass by great pressure in
suitable moulds.
10.	Papier mache is formed into shape from paper pulp re-
duced to a doughy consistency, and then subjected to heavy
pressure.
11.	In the use of moulding machines the pattern is first
pressed into the moist sand, this being a granular pressure
organ, this being followed by the casting of the liquid metal in
the mould thus formed. This gives two applications of form-
ing,—the first in moulding, the second in casting.
12.	Compresses, such as are used for packing merchandise of
powdered or fibrous nature, are also examples in point. These
are used for baling hay, cottou, wool and similar materials
under very great pressure.
b. Discharging; Formation of Jets.
When a pressure organ is contained in a guiding inclosure,
and by properly directed pressure is forced out through a suit-
able mouthpiece, the jet which is emitted is formed with a
cross section corresponding to that of the mouthpiece used.
Jets may be formed in this way not only from materials which
flow readily, but also from those which are of a tough or
doughy consistency, so that even moderately dense substances
may be thus formed:
1.	The clay presses made by Schlyckersen and others are
used to form tiles, drain pipes, etc., by this jet rnethod of form-
ing, the issuing stream being cut off at regular intervals by a
wire cutter. The clay in such machines is eflectively forced
through the discharge opening by an arrangement of screw-
propelling blades.
2.	In the model press the dough is forced by a piston up
through a die piate in which various shaped holes (such as
stars, circles and the like) are made, and the issuing streams
are sliced off by a wire cutter and dried.
3.	The so-called artificial silk of De Chardonne is a jet forma-
tion of nitro-cellulose. This is made into a semi-fluid mass
with iron or tin chloride and alcohol, and forced through a
tube of glass or platinum of about a sixteenth of an inch bore
drawn to a fine aperture, whence it issues in a hair-like thread.
It is then toughened by passing through acidulated water, after
which it is wound on a reel.
4.	In the manufacture of paper the liquid pulp is discharged
in a broad, flat sheet by its own pressure and the superfluous
water first removed by absorption, after which the paper is
dried and made into sheets.
5.	L-ead pipe is made by a process of jet formation in a pipe
press. The mass, which is only moderately heated, is forced
by pistou pressure through the die in a continuous stream.
6.	The insulating covering of gutta percha is formed upon
wires used for electrical conductors by a jet action.
7.	The common punching press, used for punching rivet holes
in plates, really works with a jet action, as has been shown by
the celebrated researches of Tresca upon the flow of metals.
8.	The drawing press for forming various cups, pans and
other household articles, also cartridge shells, from sheet metal,
operates by a kind of jet action, one part of the mouthpiece
being forced against the other. The powerful presses built by
Erdmann Kircheis at Aue, and by the Oberhagener Machine
Works, operate by means of cranks and cams, while those ofTHE CONSTRUCTOR.
241
Lorenz, of Carlsruhe, work. by hydraulic pressure. Drawing
presses are much used in the United States.*
9.	The drawing bench for the manufacture of wire as well as
rods is an example of jet action. The wire acts both as a ten-
sion and a pressure organ, since it is pulled through the die in
which it is formed. Drawn brass tubing is found in a similar
manner and of various shaped sections.
10.	The manufacture of shot is a variety of jet action, the
melted alloy of lead and arsenic being poured through a sieve
and permitted to fall in streams from the top of a shot tower,
the drops assuming a spherical shape during the fall.
11.	In gas lighting, the shape of the flame is formed by the
jet tip on the burner, the flat flame in oue form being made by
two round inclined jets impinging against each other.
c.	Internal Flouu.
There are a number of pressure organs which are not homo-
geneous, being composed of granular and fluid materials, or of
fluid materials of different density. It is a frequent problem in
technology to separate such substances so as to divide the
liquid from the solid, the large from the small, the light from
the heavy, etc. In general, this can only be done by some
application of the method of internal flow in the mass of the
pressure organ. The methods include the use of artificially
produced high pressures, the natural gravity of the material,
or in some instances by vibratory or other motion, i. e.y by the
action of the living force of the material, or rather by the un-
equal action of the various portions. The following examples
will illustrate the various methods :
1.	Presses used for the extraction of liquids (such as wine
presses), presses for seed oil, olive oil, also for oil cake, stear-
ine, beet root, yeast, etc., ali act to separate the liquid from the
solid portion by the acflon of internal flow.
2.	Filter presses act to separate the fluid from the more slug-
gish portion of the mass, the liquid passing through the minute
openings of the filter under the action of the high pressure,
while the slimy mass remains behind. Filter presses are used
for separation of colors, stearine, yeast, starch, sugAr, potters’
clay, etc.
3.	The purification of water under natural pressure is effected
by conducting it through settling and filtering tanks; also by
special devices (as that of G. Niemax, of Cologne, German
patent No. 30,032), by which the water is rendered harder or
softer, as may be required.t
4.	In mining and machine shop operations, the separation of
mingled pressure organs by difference of internal flow is effected
in various ways, showing very effective applications of the laws
of hydraulics.f
5.	Various applications of sieves are used to separate granular
materials of different sizes, as are also different devices which
act by shaking or jigging the material, the separation thus
being effected by differences of living force.
6.	Centrifugal machines are used for drying yarn, wet clothes,
etc., although the action in this case might be more properly
termed external, rather than internal flow.
7.	Another application of the centrifugal machine is for the
separation of materials by their difference in specific gravity,
as in the case of the machines for separating cream from
milk.
8.	In the Bessemer process the molten fluid mass of iron is
penetrated by a gaseous pressure organ, i. e., air, under high
pressure, producing a violent internal flow and agitatiou, and
burning out the carbon of the iron.
d.	Jet Action.
A considerable amount of living force may be stored up in a
fluid jet. This may be utilized in a limited number of ways, a
few of which are here given :
1.	The System of hydraulic gold mining used in California,
to a great extent, is an important application of the jet.§
2.	Tilghman’s sand blast acts by means of a jet of air which
sets a stream of sand particles in motion. This sand blast is
used to cut glass, surface metals, sharpen files, clean castings,
and has many other useful applications.
3.	Machines for cleaning grain are made to throw the grain
against frictional intercepting surfaces, thus removing dust and
other impurities.
*See address by Oberlin Smith (Jour. Frank. Inst., Nov., 1886, “Flow ol
Metals in the Drawing Press ”). The American presses are made for rapid
duplicate work, the German for more general Service with various special
attachments, which latter Mr. Smith commends. We may be permitted to
accept and return the compliment as regards the other side of the question.
f See Z. D. Ingenieure, April, 1888, p. 377-
JFor an.account of the separation of pulverized minerals by means of
currents of air into portions of various nneness, see Z. D. Ing., April, 1888,
p.381.
g See Appleton's Cyclopedia of Applied Mechanics, New York, 1880, Vol.
II., p. 434-
4.	The impinging of a rapidly issuing jet of steam against the
bell of a whistle causes a series of rapid vibrations, producing
sound.
5.	In the reed pipes of organs and similar musical instru-
ments, the notes are produced by the action of a jet of air
causing the reeds to vibrate.
6.	The “ Siren ” used for fog signals acts by setting a colurno
of air in vibration by rapidly succeeding jets of steam, causing
a shrill note to be emitted.
7.	In the simple organ pipe a column of air is set into musi-
ca! vibration by a jet of air. The church organ is probably the
oldest type of pressure organ escapement, the release being
effected by the hand of the performer. The modern church
organ consists of a series of the fifth order, namely: a water
motor (hydraulic escapement), bellows (escapement) and regu-
lator (checking escapement), stops (escapements), and key-
board with pneumatic action (escapement). In au organ man-
ual of 10 octaves there are 120 escapements, each with n stops,
with n different pipe connections arranged together. In this
connection also may be mentioned barrel organs, in which the
closing and releasing of the escapements is effected mechan-
ically.
e.	Inclosing or Co ver ing.
As a counterpart to the inclosing of a pressure organ in a
pipe or vessel, we have the inclosing or covering of a solid
body by a pressure organ. This occurs when a body is sub-
merged in a liquid, when its surface at least is covered with
the liquid. A partial covering may also take place, as, for
instance, one side of a flat piece, or by distribution after any
particular plan. These conditions appear in a number of tech-
nical operations, as will be seen:
1.	In the operations of dyeing, the articles are immersed in
a liquid containing the coloring matter, many machines having
been devised to assist in the operations.
2.	In the various operations of finishing fabrics, heavy flow-
ing liquids are used, distributed by various brush devices, this
forming at least a combination of the second order, since the
liquid must first be distributed to the brushes, and then to the
fabric.
3.	In coating paper with gum, the gum is distributed in the
form of a liquid solution.
4.	In the manufacture of colored papers and leathers, the
eoior is distributed in liquid form over the desired surfaces.
5.	The various operations of printing from surfaces of stone,
copper, zinc, Steel, etc., involve devices of the third and fourth
order for the distribution of the printing material before it is
finally transferred to the paper.
6.	In the operation of printing fabrics and paper hangings,
the printing surfaces are charged with color by a distributing
system usually of the third order, and then impressed upon the
fabric. The printed fabric is dried, usually, by a current of
warm air, which is merely a gaseous pressure organ. In some
instances the printed surfaces are dusted with felt dust while
yet sticky, and then finally dried.
7.	In printing woven fabrics, the processes of (5) and (6) are
used with a mordant liquid, and the material then immersed in
dye, and finally the color washed out of the unprinted portion
with water.
8.	The operations of electro plating surfaces with gold, silver,
copper, brass, zinc, nickel, etc., involve the use of a physical
apparatus, i. e., the galvanic battery. The disposition of the
covering may be modified by covering portions with a non-
conducting material. Another operation in electrotechnics con-
sists in the decomposition and deposition of minerals by means
of electric currents generated mechanically.
9.	In the use of illuminating gas, the method of making a
gas poor in carbon, and then enriching it either with a rich
hydrocarbon gas is a form of combination in the line of inclo-
sing. The incandescent gas lamps operate by the surrounding
of a network of magnesia or zircon with the flame of a weak
illuminating gas.
10.	The jet condenser acts by surrounding the discharge ot
exhaust steam with cold water.
11.	In the surface condenser the tubes are surrounded with
water and filled with the steam to be condensed, an arrange-
ment of the second order.
12.	The apparatus for cooling beer consists of an arrange-
ment of parallel surfaces of sheet metal between which the
cooling water flows, thus forming an apparatus of the second
order.
* * * * * * * *
The above outline of technological applications of pressure
organs is only an indication of the systematic treatment of
which the subject is capable, but cannot here be carried farther,
as it does not properly belong to the subject of machine de-
sign. The fifty examples given might each be made the sub-242
THE CONSTRUCTOR.
ject of a chapter, many of entire books. Even these can lay
no claim to be a complete survey of the subject; rather are
they merely a beginning. They will serve, however, to indi-
cate at least bow great a number of machines and mechanical
appliances are involved in the use of pressure organs, and how
these may ali be considered to rest upon the same founda-
tions.
CHAPTER XXIV.
CONDUCTORS FOR PRESSURE ORGANS.
I 334.
Empirical Formul.E for the Thickness of Cast Iron
Pipes.
Among the various forms of conductors for pressure organs
mentioned in \ 310, the most important are the various kinds
of pipes. These are made of a great variety of materials, such
as cast iron, Steel, copper, bronze, brass, lead, wood, clay,
paper, etc. For underground pipes for water, air and gas, cast
iron has been most extensively used, and it is yet a question
whether wrought iion or Steel will successfully displace it.
Cylindrical cast iron pipes, which will first be considered,
are subjected to so many varying conditions in the course of
manufacture that the determination of the proper thickness. to
resist moderate internal pressures caunot be made upon striet
theoretical principies, and recourse must be had to empirical
formulae. It is customary to subject cast iron pipes to a hy-
draulic test both at the foundry and also at the place where
they are to be used, to a pressure of 1 x/2 to 2 times the working
pressure to which they are afterwards to be subjected. As a
protection against rust, the pipes are also coated with asphal-
tum, applied at a high temperature, or in special cases where
the expense is permissible, they may be enameled.
Eet the internal diameter be D, and the thickness <J, we may
make:
For cast iron pipes for water, air or gas,
<5=0.315" + £	(318)
? 335-
Table of Weights of Cast Iron Pipe.
Diameter, D. Inches.	Weight per Foot, for Thickness S.						
	x	| H	A	H	*	A	1 1 !
2	5*53	9.69		i	| ■		
4	10.30	i 16.10					
O	15.40	i 23 50	32.0				
8	20.3	30.9	41.8				
10	25.1	38.3	51.7	65-3			
12	30.0	45-7	6I.5	77-7	1 94.I		
14	35.o	53-1	71.4	89.4	IO9	128	
16	39-1	60.4	81.2	102	124	145	
18	44-8	67.8	91.0	115	J39	163	! '187'
20		75-2	IOI	127	153	180	\ 207
22		82.6	III	139	168	I96	j 227
24	. . .	89.9	121	*52	183	214	246
26			t3I	164	198	231	266
28			I40	176	512	249	286
30		. . .	>5°	188	227	266	305
32			160	201	242	283	325
34			I70	213	257	300	345
36				225	27I	318	364
38				237	285	336	384
40				249	299	353	403
42				262	315	370	423
44				274	329	387	443
46				286	344	405 i	i 462
48				298	359	422	1 482 i
Sockets and flanges shouid be calculated separately.
2 336.
Pipes for High Pressures.
For cast iron steam pipes and air pump cylinders,
<5 = 0.472"+--	(319)
For bored cast iron steam cylinders and pump barrels,
<5 = 0.787 + —	. (320)
Example 1.—A pipe 12 inches bore, according to (318), shouid be of a
thickness ,J = 0.315 -f ** = 0.465", or in practice say % in., while for a steam
12
pipe, according to (319), we have: cJ = 0.472	— = 0.722", say ^ in.
Example 2.—The pipes used in sections	and F of the Frankfort water
system, already shown in Fig. 955, and subjected to a pressure of 270 pounds
(18 atmospheres) are 15% inches (400 mm.) and 20 inches (508 mm.) diam-
eter, and according to (318) the corresponding thicknesses shouid be:
d = 0.315 +	= o. 55 in., and J = 0.315 + g° = 0.565 in. The pipes for the
water systems of Salzburg, Bamberg, Carlsbad, Goslar and Iserlohn are ali
of thicknesses conforming to formula (31S).*
Example 3.—The pipes for conveying the compressed air in the construc-
tion of the Mont Cenis tunnel were 7% in. (200 mm.) bore, and subjected to
a pressure of about 75 pounds, and were exposed in lengths of over 2000 feet
long during winter and summer. The thickness of these pipes was 0.39 in.
(10 mm.), and by (318) the thickness would be 0.41 in.
Example 4.—The thickness of a locomotive cylinder 15^ in. diameter
would be, according to (320) :
15.75
J = 0.787 -i--= 0.944", say 1 inch.
100
The use of. cast iron pipes has greatly increased during the
past twenty years. Manufacturers have been disposed to make
them of excessive thickness, not only to obtain the increased
security, but to add to the cost,—a matter which public offi-
cials are sometimes not disposed to discourage, but which has
frequently caused such installations to be excessively expen-
sive. Hundreds of thousands of pounds of metal have thus
been uselessly buried in the earth,—a waste which the co-oper-
ation of hydraulic and gas engineers might join in reducing.
* The foundry of Roll, at Solothurn, using a high grade. make stili lighter
pipes, using the formula S 0.24"
In determining the thickness of pipes wdiich are to be sub-
jected to an unusually high pressure, tame’s formula (see \ 19)
may be applied to advantage, i. e. .*
.................<»■>
in which p is the internal pressure per unit of area, X the per-
missible stress in the material; the external pressure being so
small as to be neglected.
If in the preceding formula we substitute the external diam-
eter, D0 = D + 2<5, we get:
Do _ \f $ + P
D ~ 1 S — p
(322)
This shows that the internal pressure p shouid in no case ex-
ceed the permissible stress S of the material. If we make S
equal to the modulus of rupture for tension, and make p^_S>
the pipe will be burst according to either formula however
great the thickness 6 be made.
For given dimensions and pressures we have for the stress S
in the walls of a pipe:
•S	i + f
p DQ* — D2 1 - ^
(323)
in wThieh the ratio — is indicated by as already discussed in
§ 90. From this we have the following values:
4>	0.50	0.52	0.54	0.56	0.58	0.60	0.62	0.64	| 0.66	0.68	0.70
S	1.67		1.82								
p		i-74		1.91	2.01	2.13	2.25	2-39	2.54	2.72	2.92
*	0.72	0.74	0.76	0.77	co ts d	! 0.79	0.80	0.81	0.82	0.83	0.84
s	1						4o6	4.81			
p	3.15 ;	3-42 i	3-73	3-91	4.11	4-32			5.ii	5-43	5.79
*	0.85	0.86	0.87	0.88	0.89	0.90	0.91	0.92	0-93	0.94	0-95
s ~y~	6.26	6.68	7-23	7.86	8.62	9.52	10.63	12.01	13.80	16.17	19.51THE CONSTRUCTOR.
243
(324)
Ex ample 1.—In the case of the compressed air pipe of the Mont Cenis
tunnel, already mentioned, we have 8 =0.39, D= 7^75, Eo — 8.66, whence
ip —	= 0.91. For a pressure p = 72 pounds, we have from the above table
S — io%3 X 72 == 765 lbs., or, if tested at double the working pressure, the
metal would be under a stress of 1530 pounds per square inch.
When p is small, we may use instead of (321) for a sufficient
approximation (compare Case I., \ 19)
<5 _ , p , S D
D 2 5 aDd	p 26
Exampie 2.—Applying these formulae to the data of the preceding exam-
ples, we have S =	= 0.572— -7 875 = 710 lbs., as an approximate value.
S	0.39
Exampie 3 —A pipe 4 inches diameter is subjected to a water pressure of
1500 pounds. It is required that the stress S shall not exceed 3200 pounds.
S
This gives — = 2.13, which from the above table gives if/ = 0.60. From
P
this we have D0 = ----= —— ~ 6.66 in. In some instances the pressure
0.60	0.60
may reach as high as 2250 pounds, in which case the stress would reach
3200 X 1.5 = 4900 lbs. A 4 in. pipe at the Frankfurt Railway Station at 1500
lbs. has an outside diameter of 6.4 ins.
Exampie 4 —The Helfenberger Water Pressure Engine at Hersbrugg
near Rheineck, of 40 H. P., has a cast iron feed pipe under a head of 1312
feetgivinga pressure of 569 pounds. The pipe is 14,760 feet long, and 4.6
ins. diameter, and the thickness in the lower thirctof its length is 0.43 ins.
This gives DQ — 5.46 ins., D = 4.6, and from (333):
5 = 569 5.^63-4:S = 3357 lbs~
Exampie 5.—Using the empirical formula (318), we have for the following
sizes under the assuinptiou of 150 lbs. pressure :
D	= 4"	6"	12"	30"	48"
8	= 0.36	0.39	0.46	068	0 915
V	0.S9	0.88	0 92	o-95	0.96
S P S	8.62	7.86	12.01	190	24-51
	1293	1179	1800	2925	3176
The above values of 6” are taken as acting upon the longitu-
dinal section of the cylinder, which is the case when a pipe is
open at both ends. When the ends are closed, there is also to
be considered the stress on a section at right angles to the axis,
which is equal to x/z S. This, combined with the previous
value, gives for the inclined resultant, >/S'1 + (0.5 S)'1 = 1.12 .S
as the minimum. These conditions exist in the case of a
cylinder for a hydraulic press. These are usually made of cast
iron, and the iucreased thickness
adds greatly to the weight. It is
therefore important to use mate-
rial capable of withstanding a high
stress, and to take great care in
construction and in the disposition
of the material. Repeated melt-
ings of the iron give more homo-
geneous castings. Good results are
also obtained by adding wrought
iron in the cupola. By thus im-
proving the quality of the metal,
the permissible stress may be in-
creased. A stress as high as 10,000
lbs. may be permitted when the
casting is assuredly sound. Simi-
lar conditions obtain when bronze
is used. With good bronze, if no
alteration of form is to occur, the
stress should not be greater than
5000 lbs. If it i6 desired to go
higher, some harder composition,
such as manganese bronze, must
be used.
A few practical examples will be
given:
Exampie 6.—Iu raising the tubes of the
Conwav Bridge, a hydraulic press of the
following dimensions wTas used:* Diam-
eter of ram, K = 18 inches; bore of cylin-
der, D — 20 inches; thickness, 5 = %yx in.
The load was 650 tons = 1,456,000 pounds.
The pressure in the cylinder being 5900
lbs. per square inch, we get from (323)
the stress 5 = 10,500 lbs. The cylinder is
shown in Fig. 1045.
Exampie 7.—In the construction of the
Britannia Tubular Bridge several forms of hydraulic presses were used.
One of these was a double press with cylinders of the same dimensions
as in the preceding exampie. The load on each ram was only 460.5 tons,
and the cylinder pressure 4190 lbs., giving a stress on the metal S — 7460
pounds.
Exampie 8.—The press which sustained the heaviest load on this great
work was one which lifted 1144 tons, or 2,562,590 pounds. This was made
Fig. 1045.
with a single cylinder with a ram 20 inches diameter, cylinder 22 in. bore,
and 10 inches thick. The water pressure was 8400, and the stress in the
metal, according to (323), was 14,500 lbs. ! When the tube of the bridge had
been raised 24 feet, the cylinder gave way, and the load dropped upon the
safety suppoits, but was seriously damaged. The fracture was not longitu-
dinal, but in the cross section near the bottom of the cylinder, as shown in
Fig. 1046. The fracture was doubtless due to the sharp angle at the bottom.
A new cylinder was made and successfully used, the bottom being altered
in shape as indicated by the dotted lines. The first cylinder which was
cast for this press was moulded with the bottom up, but was rejected as
being porous; the second was cast bottom down, and gave way in use, as
above described ; the third, for which the iron was melted twice, was suc-
cessfully used to the end, while a fourth, which was maue as a reserve, was
not required.
Exampie 9.—A press designed for making compressed emery wheels has
the following dimensions: D = 28.35 in., D0 = 40.94 in. K = 27.56 in.
P= 2,640,000 lbs., from which p = 4425 pounds. We have 1h = z—~ = o.6g,
r 40.94
whence .S = 12,134 pounds, which must be considered a high stress.
More recently the cylinders for hydraulic presses have been
cast of Steel, permitting stresses as high as 20,000 to 28,000
pounds. Modifications in the method of construction may
also be made to enable cast
iron to stand higher pressures.
The danger due to casting the
bottom in one with the cylinder
may be avoided. The method
used by Hummel, of Berlin, is
to make the cylinder as a ring,
and the bottom as a separate
piate (Fig. 1047).
Lorenz, of Carlsruhe, makes
the bottom separately and screws
it in, as shown in Fig. 1048.
By increasing the diameter of
the ram to exert a given force,
the pressure of the water re-
quired will be reduced, and the
stress 6* will be less. This is not
attended with a proportional iu-
crease in the amount of metal
required, but on the contrary
with a reduction. If the cross
section of the cylinder be F\ we
(325)
Fig. 1046.
have F = n {D + d) 6. Substituting the value of 6 from (321),
7T F) ^ 2 fi
wre get F = X -------- and iutroducing Ky
o p
F=	JF.
W) s-P
which, for any chosen value of S, diminishes as / is reduced.
Exampie 10. —In a hydraulic press by Hummel, for making rollers of
compressed paper, there are two cylinders of the form shown in Fig. 1047,
placed side by side. The diameter Koi the ram is 23 inches, and the cylin-
der diameter D 24 inches, the thickness 8 being inches. The load P on
the ram is 2,200,000 pounds. The water pressure is 5174 pounds, and the
stress on the material about 10,000 pounds. If we increase K to 26 inches,
we have, since this is § the preceding value, the value of p reduced to (§)2 —
0.79 of the previous amount, or 4087 lbs. If we now make the relation be-
tween the inside and outside diameters of the cylinder the same as before,
we have the same relation between S and />, hence S = 10,000 X o 79 = 7,900
lbs., which is quite practicable. By leaving the relation ^ unchanged, we
find the relation between the cross sections of the two cylinders will be also
as 0.79 to 1. Hence this alteration in dimensions which reduces the pres-
sure in the cylinder also causes a reduction of about 20 per cent. in the
amount of material.
S 337-
Wrought Iron and Steed Pipes.
Wrought iron pipe is in very extensive use for conveying
gas, water, air, petroleum, as well as steam. These pipes are
made either by the process of welding during passage between
rollers or are riveted while cold. The former method produces
either a butt- or lap-welded joint, the seam being parallel to
the axis of the pipe, and more recently pipe has been made in
America with a spiral lap-welded seam.f After welding the
outside of wrought pipe is generally made smooth by passing
between another set of rolls after re-heating, whence it is
sometimes called “drawn” pipe. Pipe is also made of mild.
Steel in the same manner as if wrought iron. The Mannes-
mann system is also used for rolling tubing from the solid rod
of Steel, copper, delta metal, etc., the product being without
any seam.
Welded tubing possesses a great resistance to external pres-
sure and to tension, but a less resistance to internal pressure.
Butt welded pipe should not be subjected to a greater stress
than S — i 500 lbs.; but for lap-welded pipe 5* may reach 8000
* See Clark, The Britannia and Conway Tubular Bridges. Uondon, 1850.
t See Engineering and Mining Journal, April 7 and 14, 1888; also Scien-
tific American, June, 1888, p. 377.244
THE CONSTRUCTOR.
to 12,000 pounds. Spiral lap-welded tubing has been tested
to pressures corresponding to stresses from 30,000 to 40,000
pounds, according to the quality of the material used; but in
practical Service lower values are used. The Mannesmann
tubes have been used without deformation almost to the elastic
limit of the material, which, with cast Steel and with Siemen’s
open-hearth Steel, reaches 25,000 to 50,000 pounds, and there:
fore possess a utility to which welded tubes have not attained.
Fig. 1047.
As an example of the efficiency of this construction, Mr.
Hamilton Smyth cites an installation over two miles long and
under a head of 550 ft., the pipe lying on the surface of the
ground and only protected from changes of temperature by a
roof of roughly nailed boards, and in which the total loss by
leakage was only 3 to 4 cubic feet per minute.
As a consequence of the successful use of these pipes for
mining purposes, they were next used for more permanent
service as for water supply of cities, and with excellent results.
Two such pipes were put in for the supply of drinking water
for San Francisco, and a third pipe, many miles long, was sub-
sequently added. For large diameters in permanent installa-
tions the sections should be riveted together, while for smaller
diameters the joint may be made with lead, as hereafter de-
scribed. The following table will illustrate some important
constructions of this kind.
Name.	Date.	V	| Diameter.	Head of Water.	Stress S.	Description of Pipe.
Cherokee . . . Virginia City | Texas Brook . Humbug . . .	00 00 00 00 00	| ft. i 12,800 37.116 37.116 4,440 1,194	|in. 30 11 1 IO 17 26	ft. | 887 j 1722: 1722 781 120	lb. 17.500 14.000 14.000 17.000 11.500	Sheet Iron, double riveted. Lap welded, screw connections. Sheet Iron, double riveted. xy' sheet iron, single riveted.
Example 1 —In the oil pipe line shown in Fig. 954, 6 inch lap-welded pipe
is used, j5g in. thick, at a pressure of about 1000 pounds. We get from (324)
5 =
6 X 1000
2 X 0.3125
(6.625)2 + (6)2
= 9600 lbs. Fcotn (323) we have more accurately 5 = 1000
(6.625)2— (6)2
= 9887 lbs.
Example 2.—It a spiral lap welded pipe had been used for the preceding
example, the thickness 5 need only have been & in.
Example 3.-*-If a Mannesmann tube of Siemens Steel had been used for
the high pressure water service of Example 3, § 336, and the stress put at
the moderate limit of 15,000 pounds, we get from (324) D = 4",/ = 1500 lbs.,
1500.X 4
15,000 X 2
and from (323) we get more accurately
5= 1500
(4-4)2 + (4)2
(4.4)2-(4)2
15,750.
The Steel pipe would weigli only about \ that of the corresponding cast
iron pipe.}
Also may be noted the Kimberley water works in England,
.14 inches diameter, % in. thick and eighteen miles long. The
superior economy of wrought pipe over that of cast iron is
w?orthy of greater attention.
In order to illustrate the arrangement more fully of an in-
stallation of such pipe the inverted siphon in the valley of the
Texas Brook, constructed by Mr. Hamilton Smyth, is given,
Fig. 1049. The difference in the level is 303.6 feet, and the total
length 4438.7 ft. The pipe is in lengths of 20 feet and the
figures in the diagram indicate the gauge thickness of the sheet
iron in the various portions.
The average diameter of the pipe is 17 inches and the highest
value of the stres .S was calculated as equal to 16,500 pounds;
some of the plates were too thin and the stress in such
places reached 18,000 pounds. The inlet is so shapted that the
coefficient of contraction reaches 0.92. The pipe is bedded in
gravel 12 to 18 inches deep, and passes entirely under the bed
Fig. 1049.
Riveted pipe of wrought iron have been successfully used in
America for conducting over long distances, and valuable in-
formation has been furnished by Mr. Hamilton Smyth, Jr.,
upon this subject.*
Wrought iron riveted pipes were first used in California,
made Steel metal TV in. thick, to take the place of the canvas
hose then extensively used in the operations of hydraulic min-
ing. The pipes were made of ordinary sheet iron, there being a
single row of rivets, driven cold, and the joints made simply
by inserting the end of one section into the next, as in the case
of stove pipes. These first attempts succeeded beyond all ex-
pectations and were followed by numerous installations, in sizes
reaching as high as 22" to 30" diameter and sections 18 to 25
feet long. A satisfactory protection against rust was obtained
by immersing the finished pipes for a few minutes in a boiling
mixture of asphaltum and tar. If the fit of the ends was too
loose to make a good joint the smaller pipe was wrapped with
tarred cord, leaky places being stopped with wedges of wTood
and the small leaks being checked by sawdust admitted writh
the water.
* See Engineering and Mining Journal, May and June, 1884; also Journal
of the Iron and Steel Institute, 1886, No. I, p. 133.
of the stream. During a large part of the year the siphon is
not full of water and hence entraps much air. In order to per-
mit this to escape, air valves of the construction shown in Fig.
1050, are attached at suitable points, fcurteen in all being used.
Fig. 1050.
These are simply heavy cast iron flap valves with rubber ring-
packing. When the chamber is filled with air the valve falis
open by its weight but is closed by the action of the waterTHE CONSTRUCTOR.
245
when the air has escaped. In case of a rupture in the lower
portion of the pipe, the air valves in the upper portion prevent
the collapse of the pipe from atmospheric pressure.
338.
Steam Pipes.
When steam is to be conducted to considerable distances, the
condensation which is due to loss of heat through the walls of
the pipes becomes so great that it is necessary to surround the
pipes with a uon-conducting covering. Materials for covering
steam pipes play quite an important part in the Science of
steam economy and their manufacture constitutes an extensive
industry. The importance of this subject has long been appre-
ciated, having been considered, among others by the Industrial
Society of Mulhouse more than sixty years ago. In these in-
vestigations the measure of effect is the amount of water con-
densed by a unit of surface, as one square metre per second.
The following table will indicate some of the results obtained.*
Material of Covering.	Grammes condensed per sq metre per ! second.	Material of Covering.	Grammes condensed per sq. metre per second.
Uncovered Pipe . .	2.84 gr.	Clay Pipe		1.12 gr.
Pimonfs Mass.. . .	1.56 “	Cotton Waste . . .	1.39 “
Straw		O.98 “	Felt			I*35 “
The so-called PimonCs Mass, consists of loam and cows’ hair,
60 mm. (2in.) thick. The straw was first laid on longitudi-
nally 14 mm. (x\ in.) thick, and then wrapped with straw 15
mm. in.) thick. The cotton waste was 25 mm. (1 in.) thick
covered with canvas. The felt was saturated with rubber.
Straw shows the best results, the condensation being only one-
third that given by the un covered pipe.
These experiments have not great present value, partly be-
cause the comparison by condensation of water is not altogether
reliable, and partly because new material for covering pipes
have since come into use. The Society of German Fngineers
( Verein Dentscher Ingenieure) has undertaken a series of ex-
periments from which results of value are to be anticipaled.
In the United States, Prof. Ordway, of Boston, has made some
very beautiful investigations, the results being in two series,
the first by the method of measuring the condensed water, the
second by the calorimetric method.t The unsatisfactory char-
acter of the method of condensation is apparent, as it was
found, for example, that a portion of pipe 2 feet long condensed
328 grammes of water per square foot per hour, while 30 feet of
pipe gave only 140 grammes per square foot per hour. It is
also to be noted that Prof. Ordway’s first researches showed
much less condensation for the uncovered pipe than appeared
in the Mulhouse experiments, so that no definite conclusions
could be deduced. The calorimetric method appears to be
much more reliable, as the results appear to be more consistent.
From the great nutuber of experiments the two following tables
have been selected.
TabeE I.
Material.
1 Per Cent. i Kilo-Cent.
Solid Material. Heat Units.
Air Space		0.0	1302
Carded Cotton		1.0	310
Feathers		2-0	321
Wool			2.1	3QI
Calcined Magnesia		2.3	335
Cork Charcoal, coarse		3-i	343
Calcined Magnesia		4-9	340
Wool		5.6	220
Uampblack		5.6	266
Carbonate of Magnesia		6.0	37i
Fossil Meal		6.0	393
Wool		7.9	238
Asbestos	-	8.1	T329
Zinc, White		8.8	466
Fossil Meal 		11.2	426
Pine Charcoal		H.9	376
Carbonate of Magnesia		15.0	416
Hair Felt		18.5	177
Uampblack . . . . »		24.4	286
Chalk		25*3	56O
Graphite		26.1	1922
Calcined Magnesia		28.5	1156
Zinc, White		323	II64
Pumice Stone			845
* This table has been kept in the metric system, as it is only available for
comparison.— Trans.
fSee Trans. Am Soc. Mechanical Engineers, Vol V. p. 73; Vol. VI. p. i68„
Material.	Per Cent. Solid Material.	Kilo-Cent. Heat Units.
Plaster of Paris		36.8	839
Common Salt			48.0	1983
Anthracite Coal 			50.6	968
Fine Sand		 .	5M	1690
Coarse Sand		52.9	1684
Temperature of steam 1550 C. Ali coverings 1 inch thick
= 25.4 mm.
This table gives noteworthy, and in many cases unexpected
results. It is important to note that in all cases the trans-
mission of heat bears a definite relation to the percentage of
solid matter. For instance, calcined magnesia gives off 335 to
1156 he^t units when the percentage of solid matter ranges
from 2.3 to 28.5. Asbestos makes an unfavorable showing, and
lampblack gives good results but is inconvenient to use; wool,
is excellent. In practice the cost is of course au important
consideration.
TabeE II.
Temperature of steam 1550 C.
I Material.	Thickness. Milii- [ metres. 1	Per Cent. j Solid Matter. i	Kilo-Cent. Heat units.
Glazed Cotton Wadding . . . .	50	i-oj I	129.1
n n a .... i	40	I-3	193-4
(< a a	30	1-7	205.5
“ “ “ J	20	2.5	326.4
a tt it	15	3-4	424.2
u a n	IO	5-1	502.4
Wool Wadding		 .	25	5.6	219.8
Calcined Magnesia, loose . . .	25	2.3	335-2
“ •“ crowded . .	25	4.9	340.1
“ “ compressed	25	28.5	II55.9
Carbonate of Magnesia, loose .	25	6.0	370.9
‘‘ “ crowded	25	9.4	386.7
“ “ compressed	25	15	416.5
Fossil Meal, loose		25	6.0	393-4
” ” crowded		25	11.2	425-8
X Cork in Strips		15	?	87.1
\ Silicated Cork Chips		30	?	59-2
Paste of Fossil Meal and Hair .	9	1.0	69.4
Carded Cotton		50	?	157-7
Rice Chaff, straw board ....	12	?	7E9
This table gives a comparison of fibrous and granular ma*
terials. In the first cases the same material was successfully
compressed, reducing the thickness and increasing the density,
showing and increasing loss of heat. Ordway recommends
cork as the best material, especially in the form of cemented
chips, which may be formed into semi-cylindrical sections, as
has already been done in Germany.||
Ordway does not advise air space under the covering, but
rather recommends the filling such space with a light powder.
Of all the materials tried he recommends in the order given:
Hair Felt, Cork, Fossil Meal, Magnesia, Charcoal and Rice
Chaff.lf Prof. Ordway remarks that “ it is useless to make the
testing apparatus of cumbrous dimensions, for as in Chemical
analysis we use a gramme or less of the sample, instead of kilo-
grammes, so in physical experiments increase of size does not
necessarily enhance the accuracy of the results.”
In long stretches of steam pipe the expansion from the heat
demauds the use of some compensatory device or expansion
ioint**
Fig. 1051.
Some of the forms in general use are shown in Fig. 1051. As
a, is a packed expansion joint; b, is a bent copper pipe; c> a
drum with flexible steel diaphragms.
t The cork was put on like barrel staves, with a slight air space beneath.
| The cork was chopped into small chips and mixed with 1% tjmes ita
weight of water glass at 300 Beaurae.
|| See Z. D. Ingenieure, 1886, p. 38.
% A new, and efficient as well as cheap material made of coinmon flour
paste and saw dust, is described in the Revue Industrielle, Sept. 1888, p. 346.
** This “ compensation ” does not neutralize the expansion, as in a pendu-
lum, but only renders it harmlcss.240
THE CONSTRUCTOR.
Fig. 1052 shows a U joint with packed connections. The
forms given in Fig. 1051 generally require one position of the
pipe to be held fast; that in Fig. 1052 permits both lengths of
pipe to remain free.
Fig. 1052.
In calculations the actual amount of expansion due to any
2jiven temperature we may put the expansion, if ty be the dif-
ference of temperature in degrees for:
Fahrenheit.
_t__
162,180
Materia!.
Cast Iron
Centigrade.
t
90,100
Wrought Iron
_J__	___t__
84,600	155,280
Copper
___	____t__
58,200	104,760
Brass
_J__	___t__
53,500	92,300
Example. A cast iron pipe 98.4 ft. long, (= 1181.1 inches). At a tempera-
ture of 500 F. is filled with steam at 63 pounds pressure, =310° F. The ex-
pansion will then be
1181.1 X 26J
162,180
= i.Sg^in.
1339.
PlPES OF COPPER AND OTHER METAI,.
Brazed pipes of copper when used as conductors of steam,
should not be subjected to higher stresses than 1500 to 2000
pounds, since the brazed joint is not reliable and reduces the
strength of the cross section of the metal about one-third. The
heat due to the temperature of steam at pressures from 60 to
100 pounds also reduces the strength of the copper from 10 to
12 per cent.*
Seamless pipes made from the solid metal, or rolled by the
Mannesmann process, can stand stresses from 8,000 to 10,000
pounds, and when made by forcing in the hydraulic press (see
§ 333, b. 5) only a stress of about 700 to 800 pounds.
Wooden pipes for water conductors, made water-tight with
cernent, have been made by Herzog of Logelbach with excel-
lent results; the most recent being 71 in. diameter, and 5900
feet long.
Pipes made of paper coated with asphalt have been used to
a limited extent, but do not stand the heat of.the sun.
? 340.
Resistance to Feow in Pipes.
The resistances which oppose the motion of a liquid in a pipe
are due either to changes in the direction of motion, to changes
in the rate of motion, or to the resistance of friction. We can
only here consider a few cases, and those will be limited to the
flow through pipes.
Frictional Resistance.—When a flow of water takes place in a
vessel with flat walls, through a cylindrical tube, Fig. 1053, the
difference of level betweeu the surface of the water and the
*Investigations made after the explosions on the Elbe and the Lahn will
"be found in Engineering for August, 1888, pp. 113, 116, 125. These gave for
the modulus 01 rupture K for tension ; for hard brazed pipes K = 33,400 ;
for seamless electrically deposited pipes, 50,000. The reduction in
strength due to heat is given according to the old but reliabte expenment
of the Franklin Institute.
mouth of the discharge pipe being h, we have, according to
Weisbach:
*=(, + C0+f4)£...............(3*6)
in which l is the length and d the diameter of the tube in feet,
and v the velocity in feet per second. The volume of flow
will be:
Q — — d2v
4
(327)
per second.
In (326) C0 is the coefficient of friction for the orifice of influx,
and C the coefficient of friction for the rest of the tube.
The coefficient Co. when the entrance is a sharp angle, be-
comes considerable, having a mean value of 0.505, but when the
entrance is carefully rounded it may fall as low as 0.08. In the
latter case, for long tubes, CQ may be neglected.f
For the coefficient of friction C in the pipe various deductions
have been made. The conditions which affect the flow of water
in pipes are numerous and variable. In cylindrical pipes the
particles arrange themselves in such a manner that those in the
axis move with the greatest velocity, and each successive annu-
lar sheet moving slower, while the particles in contact with the
walls of the tube remain practically at rest, so that the velocity
of each annular film, from the wall to the axis is a function <r
the distance from the wall to the centre, increasing from zei
to the maximum.
In the case of gases the velocity of adjacent rings appro*.
mates much more closely than with liquids. In both instances
the resistance is the sum of the friction of the successive annu-
lar layers upon each other.
^ In practice, the variation of the section of the pipe from the
circular forni must be taken into accountf and also the rough-
ness of the walls. The mathematical expression of these rela-
tions cannot be a simple orre. In practice also many disturbing
influences exist, such, for instance, as ice, weeds, etc. In all
comparisons with calculated resistances it is therefore essential
that the walls of the pipe should be ascertained to be smootli
and clean.
The Society of German Architects and Engineers has in
progress modera investigations conducted by several of its
members with a view of determining the most useful formula
for finding the value of C for water. The value which such a
formula would possess is undoubted, but before it can be satis-
factorily determined the fundamental principies of the subject
must be determined.{ We are at present obliged to use for-
mulas previously determined. Among these the formulas of
Weisbach and of Darcy are most available. If we express the
loss of head in the height h by friction hv in feet, we have for
water, according to Weisbach :
h
l vi f	0.017155"
1 =	= (^°-oi439 + P-
[7i55\ /
T-Jy »•••<*«>
all dimensions being expressed in feet, and g being the accelera-
tion of gravity.g We have for:
V — 0.1	0.2	°*3	0.4	0.5
c = 0.0686	O.0527	0.0457	0.0415	0.0387
v — 0.6	0.7		0.8	0.9
C = 0.0365	O.O349		0.0336	0.0325
v — 1		''A	2	3
£ = 0.0315	c.0297	0.0284	0.0265	0.0243
v= 4	6	8	12	20
C = 0.0230	0.0214	0.0205	o.o193	0.0182
According to Darcy we have for water :
=	^=t°-OI989 +
: ^O.C
0.00166^ l vx
d J d 2g
• (329)
which gives somewhat greater values than does Weisbach'
formula for the higher velocities.||
f If the tube starts from another tube instead of from the side of a reser-
voir, the coefficient of resistance becomes much greater and much care must
be given to the shape of the entrance. See Hertel, Zeitschr. D. Ingenieure.
1885, p. 660, also W. Roux, Jenaische Zeitschr. fur Naturwisseuschaften,
Vol. XII., 1878.
t See a Memoir of the D. Arch- u Ing.-Vereine, edited by Otto Iben, pub-
lished by Meissner, Hamburg, 1880. This must be used with caution on
account of numerous typographical errors.
§ This and the two following formulas may also be used when in addition
to the height h\ a second height h\ is to be added due to the contraction ol
discharge. This is only of importance in the case of high pressure water
transmission, and experimental researches are to be dec^ed.
jj Recherches Experimentales, etc. Paris, Mallet-Bachelier, 1857.THE CONSTRUCTOR.
247
The formula of M. de Saint Venant, which gives lower results
than either of the above, is:
K = (°-03« v 7)~d2g..............................(33°)
If we insert in equation (327) the value for v from the equa-
l	v2
tion A, = f	p ■ — we get:
: 39-72541 di
By assuming C constant, as proposed by Dupuit, * we may
state the practical formula :
Dupuit makes C = 0.03025649, whence C becomes 1313, and
we have:
d'=i G&).............................<33,)
and hence for an approximate formula :
t(^)........................(332)
These formulas cau be applied so that first from the given
values of Q and l and the friction loss of head hv the diameter
D may be determined, and then by making Z?somewhat larger,
and applying the formula of Weisbach or Darcy, the excess of
head over friction determined. This will be illustrated by a few
examples :
Example 1 —The large inverted siphon described in § 337, Fig. 1049, gives
D = 17" = 1.417 ft., I = 4438 ft. and Q — 32 cu. ft. From th.s we get
— = 21.2 ft. per second.
0.7854 X (1.417)-
These give in Weisbach’s formula :
,	, 0.017155
<; = 0.01439 ----	= 0.0181.
V 2I-2
and from (328) h\ = 395.6 ft.
The actual difference of level is 303.6 feet, and hence the coefficient as de-
termined from Weisbach, is too high. The coefficient determined from the
given difference of level is £ = 0.0155, and as a flow takes place the actual
coefficient must be somewhat less. According to Saint Venant*s formula
(33°) ^1 = 293 feet, which is slightly under the actual difference of level.
Exatnple 2.—In the work of Iben, already referred to, is given a case in
Stuttgart in which l = 3614 ft., D = 0.33 ft , v = 2.063 ft. From these data,
Weisbach’s formula gives £ = 0.0263, and thence h\ — 19 ft. The actual value
is 23.2 ft., which corresponds to £ = 0.0332. The difference is probably due
to the construction, there being two stop valves and six elbows included in
the resistance.
Example 3.—Another instance in Stuttgart is as follows: l = 302 ft. ;
D = 0.0843 fit.; v = 5 897 ft.
The friction head as determined by observation for / = 328 fit., was 82.89 ft.
According to Darcy, the friction head would be 76.57, which is quite close to
the experimental results. In other instances, however, Darcy’s formula
has not agreed so well with experiment.
When air is used instead of water, Weisbach gives for the
height of a column of water equal to the frictional resistance :
, r / v1	l v2	.	,
~2ge ~ °'°25 ~d ' SgT-------------------(«3)
in which e is the ratio of the density of the air in the pipe to
that of the external atmosphere. Since is e always greater than
unity when the air in the pipe is under pressure, hx is smaller
than is the case for water, especially when the pressure of the
air is great. Valuable experiments upon the transmission of
compressed air have been made by Bngineer Stockalper at the
St. Gothard tunnel.f These showed that Darcy’s formula (329)
served well for air when the results are multiplied by the ratio
of the density of the air to water. Professor Unwin has given
some valuable researches upon the friction of air, in which he
shows the important influence which D exerts upon C-t
Example 4.—At the construction of the Hoosac Tunnel it was observed that
the pressure of compressed air feli from 821 pounds per square inch to 801
pounds in being transmitted a distance of about 118,000 feet.
Resistance in Angles and Bends.—The resistance due to an
angle, such as Fig. 1054 a is important, and is dependent upon
what Weisbach calls the semi-angle of deviation, /?, according
to the following formula :
—	= (0.9457 sin1 /3 + 2.047 ‘ p) . . . . (334)
*See Dupuit, Traite theoretique et pratique de la conduite et de la dis-
tribution des eaux. Paris, D.unod, ime ed. 1854; 2tue ed. 1865.
f Stockalper, Experiences, fates au Tunnel de Saint Gothard, sur Pecoule-
ment de l’air comprime. Geneve, 1879.
t The coefficient of friction of air flowdng in long pipes. Proc. Inst. C. E.
Eondon, 1880.
from which we get:
ft = 10	20	30	40	45	50	60	70
C2 = 0.046	0.139	0-364	0.740	0.985	1.260	1.861	2.431
Example 5.—In a right angle bend /3 = 450, the loss is practically equal
v-
to----.
Ih the case of bends, Fig. 1054 b, the resistance is not so great,
but is too large to be neglected since we have :
ft v1
= ^ 90 ' Tg...............................(335>
The ratio of the radius of the tube to the radius of the curva-
ture of the bend affects the coefficient as below:
0.5 D
	= O.I r	0.2	0.3	0.4	0.5
C2 = 0.131	0.138	0.158	0.206	0.294
= 0.6 r	0.7	0.8	0 9	1.0
C2 = 0.440	0.661	0.977	1.408	1.978
Example 6.—For a right angle bend in which r — D we have :
A2= 0.294 -4S-	= 0.147 ——
90 2g	2g
or only about} the resistance of a sharp bend with any curvature.
Resistances due to Sudden Changes of Cross-Seciion.—When
water which is moving at a velocity vl suddenly changes to
another velocity v, see Fig. 1055 a, it experiences a loss of
pressure which, according to Weisbach, is equivalent to a height:
v ^ — v2	f F \ 2 v2 v2
‘•- — -kr,-')	<33«>
F and Fx being the respective cross sections ; also Fv= Fxv
.	.	v2
Doubhng the cross section causes a loss of head equal to
Fig. 1055.
For gate valves, Fig. 1055 b, or cocks, Fig. 1055 c, there is a
loss due to the amount of contraction. For gate valves we
have from Weisbach :
Openings	= i	l4	Vi	y	%	X	7A
F1 F	= 0.159 0.	315	0.466	0.609	0.740	0.856	0.948
£3	= 97.8 17	.00	5-52	2.06	0.81	0.26	0.07
and for cocks:							
Angle =	lO° 20°	3o°	40°	50°	6o°	63°	82
F	O.85O O.692	o-535	i 0.385	0.250	0.137	0.091	0
C = 4.	O.29 I.56	5-47	17.3	52.6	206	486	00
From the above tables it will be seen how important an in-
fluence is exerted by valve chests, mud traps and the like upon
the flow of water. In all such cases it is important to modify
the suddenness of the change of velocity by rounding and curv-
ing all angles in the passages, and in this way a large pait of
the loss may be obviated. For gaseous fluids the resistance is
less, but is at the same time sufficiently important to be care-
fully considered. For a fuller discussion of the resistances
offered to water in canals and streams the reader must be re-
ferred to special treatises 011 the subject.24«
THE CONSTRUCTOR.
\ 34i.
Methods of Connecting Cast Iron Pipes.
One of tlie most frequently used methods of connecting cast
pipes is by means of the common flange joint, Fig. 1056.
Fig. 1056.
The proprortions are given in the illustration. Formerly it
was customary to raise a small bearing surface inside the bolt
circle, but this is generally omitted now, and the entire surface
of the flanges finished, making a much better joint, although
a trifle more expeusive. In many instances a ring of copper
wire, let into a groove, is used to make the joint. For pipes
which are uot subjected to very high pressures the number of
bolts A, is determined from the following :
^ = 2 + -7-..............................(337)
in which D is the diameter of the pipe inches. This would give
for a pipe 4 inches in diameter four bolts, and for one 3 inches
diameter 6 bolts. According to (337) an air pump cylinder 60
inches in diameter would have 2 -j- -\°- = 32 bolts.
When the pressure is known to be great, or for cylinder lids,
etc., the following formula is to be preferred:
a = (t).......................(338)
in which d is the diameter of the bolts, D the diameter of the
cylinder, and a the pressure in pounds per square inch. This
assumes the diameter of the bolt at the bottom of the thread to
be 0.8 d, and the stress in the bolts to be 3500 lbs. as in for-
mula (72).
Example.—A steam cylinder 40 inches in diameter, subjected toa pressure
of 60 pounds, would have according to (320) a thickness of 6 = 0.787 +	=
iT% in. This gives from Fig. 1056 for the bolts, d = $ X fi* = 1.58, say xx% in.,
and these values in (338) give for the number of bolts :
A =	• f	= 16 bolts.
2400	^1.58 J
(Compare close of Chapter XXVI).
!
1
Fig. 1057.
Flanges with ears, as shown in Fig. 1057, are frequently used,
the thickness being made 2 to 2.5 6, mstead of 1.6 d, on account
of the smaller flanges.
On the Prussian State railways flange joints are made with a
lenticular shaped ring inserted in the joint, as shown in Fig.
1058.
This permits a certain amount of motion and gives good re-
sults in practice. The following table of dimensions is based
on one used on the Prussian railways :
D	2	!			: 3 1	|	4	41/*	5	5 XA	6
A r s	zVs %	1	2^8 H	3K 2 H T9*	4 3 A	$ H |	5 3X X	sX 4 H	6% 4H X	6X 4X X	% ii
Fig. 1059 a shows a cast iron bend with flange. The bend
should not be too sharp, in order to avoid excessive resistance
to the flow of the water. (See Example 6, \ 340.) Bends of
this sort require a separate pattern to be made for every different
angle. Brown’s joint is more convenient in this respect, Fig.
1059 b. The bolt holes in this form should be drilled in only
one of the flanges first, and* the other flange marked off in
place. For any flange angle oc the pipes may be connected for
any angle betweeu 2 ot and 1800. In the illustration c* = 40°,
which answers for most practical purposes.
a	b
Fig. 1059.
Bell or Socket connections are much used for gas and water
pipe. The joint is caulked with lead, which may conveniently
be made in half rings and driven in, or run in in place, a pack-
ing of oakum being first driven in.
Fig. 1060.
The large end of the pipe is called the bell, the other the
spigot. The dimensions of the various parts in Fig. 1060 may
be taken as follows, the thickness S being determined from for-
mula (318), i.e., i = 0.315 +
Thickness of bell,
Thickness of bead,
Inside length of bell,
Length of bell reinforcement,
Outside length of bell,
Space for packing,
Depth of lead ring
Length of bead on spigot.
Thickness of bead,
Some makers put a bead around the inside edge of the bell
to assist in retaining the lead packing, but others consider this
but little use, owing to the softness of the metal. More recently
the bead has been altogether omitted from the spigot end, a
shoulder being cast on the inside of the bell instead.
In Belgium a joint is used in which a gum ring of globoid
form (see Fig. 637 a) is used instead of the lead packing, the
ring rolling in as the spigot is pushed into the bell.
Fig. 1061 is Petit’s pipe joint. A gum ring is inserted in the
short bell, and one clamp being connected the pipe is used as a
lever to compress the gum ring, the second clamp can be
secured. This coupling, which was used in the extensive water
system of the camp at Ch^lons, is cheap and can be rapidly
= °*375// + 0.0135 D.
k — 0.7" -f- 0.0025 D.
lx — 2.62$" -j- o-11 D.
1.2 = 2‘' -f 0.09 D.
I — 4.625// 4- 0.20 D.
b = 0.1875" 4- 0.007 D.
h = 1.125" 4- 0.07 D.
a — 1.2 6.
c — 6 4- b — 0.0625".THE CONSTRUCTOR.
249
connected, and possesses a certain flexibility which permits it
to be used in running a line of pipe over uneven ground.
FiG. 1061.
A form of screw connection for cast iron pipe is shown in
Fig. 1062. The screw thread is cast on the pipe and a leaden
FiG. ro62.
^asket is placed so as to pack the joint outside of the screw
connection.* This may be considered as a flange joint with a
single Central bolt, which latter is made large enough to permit
the pipe opening to pass through it (see $ 86). Since the pipe
must be revolved in tnaking the connection, it is necessary to
provide wrenches of suitable size for the purpose.
Fig. 1063.
Fig. 1063 shows Normandy’s pipe joint. The packing con-
sists of two rubber rings. This very simple joint is very useful
under certain circumstances, where the proper packing is avail-
able. It possesses the flexibility of Petit’s joint and is easily
connected and disconnected.
A similar form of joint has been made for water pipes, using
packing rings of lead. The sleeve may be considered as a
double bell and the pipes are perfectly straight without any
bead at either end. The distance from the centre of one joint
to that of the next constitutes a “length.” With cast iron
pipe this is made a minimum of about 4 to 7 feet, being made
as long as practicable for extensive lines of pipe. For gas and
water pipe with bell and spigot connections the following pro-
portions occur in practice :
D = 4 inches,	l	—	7	to	10	ft.
/? = 12	“	l	—	10	to	12	ft.
D = 24	“	and over, / = 12 ft.
A form of joint used by Riedler for high
pressure water connections is shown in
Fig. 1064. f The flanges are faced in the
lathe and bolted together without any
packing in the joint. A leather ring is
placed in a channel turned ir. the pipe and
held in place by a spring ring in two parts,
or this latter may sometimes be made in
one piece.
Joints with spherical contact surfaces
have been used by Hoppe for cast iron
high pressure pipes when they are to be
laid in yielding ground4
Three forms of construction are shown
in Fig. 1065. At a is a single ball joint.
* This joint is used by the Lauchhammer Works for pipe up to 2^ inches
diameter.
f See Zeitschr. D. Ing, Vol. XXXII., 1880, p. 481.
j German Patents, No. 42,126.
The bearing ring is held in position by a ring of bronze divided
at right angles to the axis ; this form permits a deflection of 50.
At b is shown a double joint constructed in a similar mannef
and permitting a deflection of io°. The third form, which is
the most recent, has 110 packing ring, and the bolts are made
with spherical heads to facilitate motion.
? 342.
Connections for Pipes of Wrought Iron and Steee.
Riveted pipes are often connected by means of wrought or
cast iron flanges, as shown in Fig. io6b a and b. When no
b
other data are at haud, the diameter and number of bolts may
be determined by assuming the pipe to be of cast iron, and
using the proportions given in the illustration. The actual
thickness 6 of the pipe may then be determined independently
according to the material, pressure, and other conditions.
Example.—Ps. wrought iron pipe 3 fit. 4 in. in diameter, for delivering water
to a turbine, is to be fitted with flanges of wrought iron. A cast iron pipe
of this diameter would have a thickness, according to (318)
8 = 0.315" +	= 0.815"
whence from Fig. 1056 d = § X 0.815 = 1.05", say iTV The number of bolts,
according to (337), will be 2 +	= 22. If the internal pressure is 30 pounds
per square in. we have, according 10,(324), taking S — 4200:
Fig. 1067.
For thin pipes a very practical form is that shown in Fig.
1067 a. The ends of the pipes are flanged over, and the turned-
over ends countersunk into the cast flange rings, the bolt heads
also being countersunk. A similar form with wrought iron
flange rings is shown at b.\ For the thin pipes described in
i 337, when subjected to a high internal pressure, the joint
shown in Fig. 1067 c is adapted. In this form a short sleeve is
riveted into one of the pipe ends and a loose ring slipped over
the outside of the joint, forming a space into which lead is run
and afterwards caulked. This also serves as a sort of expan-
sion joint (compare $ 338).
Many important constructions are made with wrought iron
pipe. The connections are usually made by screwing the parts
together, and for this purpose many special pieces are made,
known by the generic term of “pipe fittings.” For straight
connections the ordinary “socket” is used, while for angles
the so-called “elbows ” and “ tees ” are made.
I For description of a flange joint with welded conical rings, by De Naeyer,
see Zeitschr. D. Ing., Vol. XXX, 1866, p. 106.250
THE CONSTRUCTOR.
The American practice of making the thread tapering is much
to be recommended, since by means of a little cementing mate-
rial a tight joint may be made. The American Mechanical Kn-
gineers have given careful attention to the proportions of pipe
fittings, and since 1887 the fornis proposed by the late Mr.
Robert Briggs have been generally adopted.*
The system is as follows : The thread is of triangular section
with the angle 2 /3 = 6o°, as in Sellers’ system. The top and
bottom of the thread are flatteued of the theoretical depth
/0, so that the actual depth t — o.8tQ, and hence equal to 0.69 of
the pitch s, Fig. 1068«.
h
4>s—^ss-^k—
Fig. 1068.
»
The end of the pipe is given a taper of ^ on each side, the
length of the tapered part being T = (4.8 -f- 0.8 Z>) s, D being
the outside diameter of the pipe and s the pitch. Beyond the
taper portion comes a length Tx = 2 s, which threads are full at
the root but imperfect at the top, beyond which there is a
length T2= 45, consisting of imperfect threads blending into
the full outside diameter. The thickness d of the pipe is such
that the thickness of metal below the thread at the end of the
pipe is = 0.0175 D -f o.o25//. The pitch ^ is finer than for
bolts of the same diameter, there being only five different
pitches used, and the various dimensions are given in the
following table:
Tabus of Standard Pipe Threads.
Diameter of Pipe.			Thickness of Metal.	Screwed Ends.	
Nominal Inside.	Actual Inside. D.	Actual Outside. Dq.		Threads PerInch.	Length.
Inches.	Inches.	Inches.	Inch.	No.	Inch.
yi	0.270	0.405	O.068	27	O.19
X	O.364	0.540	O.088	18	O.29
X	O.494	0.675	0.091	18	O.30
X	0.623'	0.840	O.109	14	0-39
X	O.824	I.050	O.II3	14	O.40
I	I.048	I-3I5	O.134	11'A	O.51
	I 380	I.660	O.140	nH	0.54
	I.6IO	I.900	0.145	II %	0-55
2	2.067	2-375	0.154	11%	O.58
	2.468	2.875	0.204	8	0.89
3	3.067	3500	O.217	8	0.95
3 %	3-548	4.OOO	0.226	8	1.00
4	4.026	4.500	O.237	8	'-05
4/4	4.508	5.000	O.246	8	I.IO
5	5-045	5.563	O.259	8	1.16
6	6.065	6.625	O 280	8	1.26
7	7 023	7.625	O.30I	8	1.36
8	8.082	8.625	O.322	8	1.46
9	9.000	9.688	0.344	8	i*57
10	10.019	IO.750	0.366	8	1.68
Taper of conical portion of tube 1 in 32 to axis of tube.
It will be observed in the table that the thickness d agrees
very well with the formula 6 = 0.111 \/D0. This gives for the
diameters 0.405, 1.050, 4.000 and 10.750, the thicknesses 0.071,
0.114, 0.222, 0.364, which agree quite closely with the actual
values. The shape of the sockets is shown in Fig. 1069, the
thread being given a somewhat greater taper than 3^, so that
the greatest stress will come on the strongest part of the
Socket.
The increasing use of such pipe in Germany makes it most
desirable that a Standard of dimensions should be adopted.
The American system is manifestly unsuited for use with the
metric system. The general arrangement of the American
system may, however, be followed with some approximations
to adapt it to the metric measurements.
The angle of thread may be the same as in the American
System : 2 /? = 6o°. The depth of thread may also be abbre-
viated TV top and bottom, making t — o.8t0 = 0.685, and the
* See Trans. Am. Soc. Mech. Engineers, Vol. VII, pp. 311 and 414 ; also Vol.
VIII, pp. 29 and 347.
taper can also be made on a side. The length T of the
tapered portion may be made T = (5 -f ^ D0) s, which is
about the metrical equivaleut of the former expression, the
nearest even value being taken. The lengths Tx = 2 5 and Tx
= 45 may be retained.
For the thickness of pipe the American formula transformed
gives d —0.555 \/D0 in millimetres. Finally for the pitch we
may take
5 =	1	1.4	1.8	2.2	3.2	mm.
(0.94)	(1.41)	(1.81)	(2.21)	(3.17) in.
the values in parentheses being the corresponding equivalents
of the American pitches. The following table gives the values
from 10 to 325 mm. This system has been submitted by the
author to the manufacturers of the Mannesmann tubes in
Remschied, Saarbriick and Komotau, and by them adopted.
Metric Pipe Thread System.
Outside Diameter Bo-	Thickness 5.	Inside Diameter D.	Pitch	Length of Thread T.	Length Tx — 2 s.	Length Tg = 4 j.
IO	1*75	6-5	1.0	5*5	2.0	4
l5	2.00	II.O	1.4	7*5	2.8	5*6 ,
20	2.50	15.0	1.4	8	2.8	5*6
25	2-75	19*5	1.8	11	3.6	7*2
3°	3.00	24.	1.8	12	3.6	7.2
35	3*25	28.5	2.2	14	4*4	8.8
40	3.50	33*0	2.2	15	4.4	8.8
50	4.00	42.0	2.2	15	4*4	8.8
60	4.25	51.5	2.2	16	4.4	8.8
70	4 75	60.5	3-2	25	6.4	12.8
80	5.00	70.0	3*2	26	6.4	12*8
90	5.25	79.5	3-2	28	6.4	12.8
100	5*50	89.0	3.2	29	6.4	12.8
110	5-75	98.5	3*2	30	6.4	12.8
120	6.00	108.0	3*2	3i	6.4	12.8
130	6.25	II 7.5	3-2	33	6.4	12.8
140	6.50	127 O	3.2	34	6.4	12.8
150	6.75	136.5	3-2	36	6.4	12.8
175	7.25	160.5	3.2	38	64	12.8
200	7*75	184.5	3*2	42	6.4	12.8
225	8.25	208.5	3*2	45	64	12.8
250	8.75	232.5	3*2	48	6.4	12.8
275	9*25	256.5	3*2	5i	64	12.8
300	9.50	2§I.O	3*2	54	64	12.8
325	10.00	305.	3.2	58	64	12.8
In the preceding table the pipe is classified according to its
outside diameter DQ} but it is a question whether it would not
be better to follow the custom of designating the sizes by the
internal diameter D. The former, however, has an important
influence upon the dimensions of the fittings, which it is most
desirable to reduce to a Standard system. It will be seen by
reference to the table of American pipe dimensions that the
.actual internal diameter differs frequently from the nominal
size, the latter really being only a convenient name. By adopt-
ing a striet gradation for the sucessive sizes of DQ it would be
practicable to make the thickness 6 somewhat less than given
in the table, but in some cases it would be greater. When D0
is greater than 325 mm., <5 may in ordinary cases be made
— 10 mm.
The production of the screw threads both in pipe and fittings
must be carefully considered in order to insure the interchange-
ability which is necessary. Power fui and accurate machines
have been devised for cutting the threads, as well as devices for
producing the taps and dies, and also gauges to insure mainte-
nance of standards. This branch of the art has been carried to
a high degree of perfection in America.
Fittings for Wrought Pipe.
The simplest pipe fitting is the socket used for connecting
two pipes of equal diameter DQf and is made of wrought iron
FiG. 1069.
or of steel. It is made of sufficient length to give a thread in
each end of length equal to T\ as given in the preceding tables,
together with a slight clearance between the ends of the pipes,
Fig. 10693:. In many cases the socket must be made w7ith rightTHE CONSTRUCTOR.
25l
and left hand threads, as in Fig. 1069 by this being necessary to
connect two pipes which cannot be turned axially. For other
connections a variety of fittings are made, examples of which
are shown in Fig. 1070.
ab	c	de
Fic. 1070.
In Fig. 1070, a is a right angle; b an elbow (abbreviated in
practice to “ ell ”); c is a T ; d a cross ; and e a reducing socket.
These fittings are used as connections for a 11 sorts of gaseous
pressure organs. They may also be used for liquids, as water,
brine, oil, etc., when the velocity of flow is not great. For im-
portant installations it is becoming more and more the practice
to design the fittings in such forms as to produce a minimum of
resistance. By making the fittings of cast iron, as is done in
England and America, where pipe constructions are very ex-
tensively used, it is possible to adhere to accurately designed
Standard forms.
The most important fitting is the elbow, for the right angle
bend occasions far too much resistance to be used in important
cases. In Fig. 1071 three forms are shown, ali of which are
a	b	C
designed to be used with the thread already described. Of
these, form b is the most popular, although form a is frequently
used because of the smoothness and neatness of external ap-
pearance. Form c is here proposed as au additional design. A
comparison between the three forms will show a difference in
resistance which may be calculated as follows: The resistance
may be divided into two portions ; one due to the curvature, the
radius of curvature being made equal to DQ ; and one due to
the enlargement and consequent contraction of the passage.
Example 2.—In the three forms shown in Fig. 1071 let the radius of curva-
ture DQ = 1 inch, and let the velocity v be taken at 6.56 feet per second.
We then have from (335) for the resistance due to the curvature, h2 = &
= 0.334 C2, and £2 in the various forms
		a	b	c
	^1 to 6 i	= 0.66	0.54	0.39
	r			
	£2	= 0.573	0.352	0.201
whence	hft	= 0.191	0.118	0.067
We also have from (336) for the loss due to enlargement and contraction :
*» -2 [ * (%r) ] = '■***
when ce	a	b	c
F {	f 1-3125 v	/ 1.0625 \2	1.0
Fx 1	l 0.75 /	\ o-75 /	
& =	3-444	0.904	0 *
whence —	4.6	1.207	0
hence ho + h3 =	4-791	i*325	0 067
It will thus be seen that form a cannot be recommended, except for steam
for which the coefficient of loss is much less than for water; and that form
b occasions quite a perceptible loss. Form c is much to be preferred, both
because it offers the least resistance, and also because it is lighter, the pro-
portion of metal in the curved portion of the three forms being as 36 : 30 : 25.
The only dimension which is important in connection with a
Standard system of fittings is the distance DQ 4- T, which
should be taken from the preceding table. The thickness ^ is
mainly dependent upon matters of casting, and is here made
= & 4- 0.04" (S 1 mm.) the thickness of the collar being
= 2<5j.
An indispensable condition for any Standard system of fittings
is the constant length from end to centre for each size of elbow7,
cross, or T, so that at any time one fitting may be substituted
for another without affecting the length of the pipes. This
principle can also be observed when the fittings are used to
connect pipes of different diameters.f Such fittings are always
known by the name of the largest opening, whether T, elbow,
or cross, this dimension governing the proportions.
Fig. 1072 a shows a T, which is proportioned to permit one-
half the flow of water to pass off the side opening. This is
based on the form b, of the preceding illustration, and, as is
usual, the direct discharge opening is made the same size as the
entrance. D' is made equal to 0.7 D, thus giving one-half the
area, and making the velocity the same as in the entrance pipe ;
if the side opening had been kept full the velocity would have
been reduced one-half. The side outlet is shaped like an elbow,
with a sharp internal partition to direct the flow. According to
Roux, these partitions are of much importance, acting as wedges
to split the flow of the water. At b is shown another form, in
which both discharge openings are reduced, and every precau-
tion taken to give a smooth flow to the water. At c is a reduc-
ing fitting which will double the velocity of flow, the reduction
in diameter being made by gradual curves.
a	b	c
Fig. 1073 a shows a T with equal outlets, formed on the plan
of the elbow shown in Fig. 1071 b. This is made with a divid-
mg wedge, which is much better than the straight form shown
by the dotted lines. The latter form causes material loss by the
sudden reduction of velocity to one-half. The form shown at b
is intended stili further to reduce this loss. At c is shown a
cioss with three equal outlets designed on the same principle.
The previously described fittings have been given on the as*
sumption that the velocity of flow is to be kept uuiform from
the point of division both as regards the fittings and in the
pipes. In extensive installations, whether in residences, public
buildings or manufacturing establishments, this is not often the
case. Very often it is found that one portion of a system is
possessed of but little velocity of discharge, while a neighbor-
ing pipe has a flow of high velocity in it. The resistances in
such systems become quite material, but may be somewhat re-
duced by giving care to the shape of the fittings.
In adopting standaid dimensions for pipe fittings, which may
be based either upon form b or c, especial precautions must be
taken to insure interchangeability, this being the principal ad-
vantage to be obtained. This involves accurate tapping of the
threads both in the sockets and in the right-angle fittings,
which is accomplished by special devices which enable all these
operations to be performed without releasing the fitting, the
accuracy of angles and sizes then being readily controlled by
the machine. The sizes of the fittings are cast upon thein in
distinet figures, so that they may readily be determined.
I 343-
Connections for Pipes of Lead and other Metaes.
Lead pipes may be connected by means of separate flanges
of wrought iron which draw the expanded ends of the pipes
together.
* The small clearance for the screw thread may be neglected.
t See Trans. Am. Soc. Mech. Engrs., IV, p. 273.252
THE CONSTRUCTOR.
A good fiange connection for lead pipe is shown in Fig.
1074 ;* the pipes are expanded and a double cone socket of
brass inserted and drawn together by bolts. Fig. 1075 shows
anotber design, by Louck ; the pipes are drawn together by
means of screw flanges and a collar, the three pieces all being
made of cast iron.
Fig. 1076.
Fig. 1076 a shows a connection for joining lead to cast iron
pipe, and Fig. 1076 £ is for lead to wrought iron pipe ; the loose
collars in both forms are made hexagonal or octagonal exter-
nally, so as to be operated by wrenches.
8 344.
Feexibee Pipes.
For many purposes it is desirable to have a pipe which shall
be yielding or flexible, so tliat, for example, it may follow the
inequalities of the ground, or may accommodate itself to yield-
ing supports. In such cases the flauge connections may be
constructed to permit motion by meatis of ball and socket
bearings, as shown in Fig. 1065, such joints being especially
adapted for pipes to be laid under water. An example of such
constructiou is found in the water main built by G. Schmidt, of
Carouge, for the water supply of Geneva, laid on the bed of the
Lake of Geneva. The pipe is 47 X inches (1.2 metre) diameter,
and is made in lengths of 29^ feet of riveted wrought iron,
0.197 in. thick (5 mm.). The connections are ball and socket
flanges, riveted to the pipes.
Instead of making the pipe rigid and the joints flexible, the
joints may be made rigid and the pipe flexible. Familiar ex-
amples of flexible pipe are various kiuds of hose, made of
leather, canvas or rubber. Special fornis of couplings are made
for fire hose. If the hose is to be subjected to heavy pressure,
either internally or externally, special methods of increasing
its strength are used. This may be done by means of a spiral
of wire, or better by two separate spirals, one to resist internal
pressure and one to resist external pressure, as shown in Fig.
1077 a. The wire spirals furnish the strength and the hose the
h
Fig. 1077.
tightness. This idea may be stili further carried out by making
the material which makes the pipe tight, also in the spiral form.
This is shown in the flexible metallic tubing of Tevasseur, of
Paris, shown in Fig. 1077 £•+ This is composed of a spiral of
copper or similar metal, the section resembling somewhat the
figure 5. The spiral is wound upon a mandrel in a special
machine, a layer of rubber packing being wound in at the same
time, as shown in the illustration. This pipe has been found to
answer well for gas, water, steam, air, etc., and is adapted to
high internal or external pressures, being tested to 180 pounds
Flanges and other fittings are screwed ou to the spiral and
soldered carefully. This pipe is used, among other purposes,
for connections for air and vacuum brakes.
.? 345-
PlSTONS.
Next to the various kinds of pipes, as already discussed in
$ 310, the most important members in pressure organ mechan-
ism are the various forms of pistons, and with these the differ-
ent methods of packing wTill be considered. Pistons, properly
so called, are fitted with packing which presses outward against
the walls of the cylinder, while in the case of plungers the
packing presses inward. Both forms wdll be given considera-
tion.
The most important forms of pistons are those used in steam
engines. Some of the low-pressure engine pistons are yet
made with hemp packing; but for higher pressures, metallic
packing is used, this consisting of metal rings pressed against
the walls of the cylinder by spriugs and by the steam pressure.
In some instances a combination packing is used, the metal
rings having a backing of hemp instead of springs.
The unit upon which the dimensions of the following pistons
are based is determined from the formula:
s == 0.368 D — o 04 — o. 118	...........(339)
in which D is the piston diameter in inches.
The following table will aid by giving a series of values for
5 and D:
.9	D	5	D	s	D
0.4	4	0.65	20	0.90	58
0.45	57	0.7	24	0.95	70
0.5	8	o-75	30	1.00	85
0.55	11	0.8	40	1.05	100
0.6	14	0.85	48	I.IO	120
Fig. 1078.
Fig. 1078 shows a hemp packed piston by Pen*'. This is
made of a cored casting with a ring follower secured by bolts,
screwing into bronze nuts recessed into the piston. For pistons
of large diameter an increased depth is given at the centre ;
this increase may be made by making the depth in the middle
equal to 6s -j-	the depth at the edge being 7.8 s, and the
piston being made flat—when the latter value exceeds the
former.
Example.—Let D = 24 inches—for a hemp packed piston, as Fig. 107S, we
then have j = 0.7. This gives for the thickness of the packing 0.7 X|.8 =
1.26, say 1 yx in.; the depth of packing = 0.7X6 = 4.2 ln.; the depth of piston
at the edge = o 7 X 7.8 = 5.36 = say 5% in. The depth in the middle will be
equal to 6 X 0.7 X = 6.6, say 6Yz ins.
Fig. 1079 shows a good form of piston with metallic packing,
by Krauss. The packing consists of two steel rings, each cut
at an angle, a ring of white metal being cast on each steel ring.
If it is desired to make the cut in each ring tight, some one of
* German Patent, No. n,535.
f Made by the Metallic Tubing Co., Ld., Port Pool Eane, Gray’s Inn Road.
Eondon, N. C.THE CONSTRUCTOR.
253
the methods shown in Fig. 1080 may be used. In the first one
the overlap makes a tight joint, while in the others the inserted
piece is fitted steam tight. By filling the packing rings with
white metal the wear comes mainly upon the softer material
instead of on the cylinder, a most desirable feature, since the
rings are easily and cheaply renewed. For the same reason
Fig. 1079.
bronze rings are used, while iron or Steel are not to be recom-
mended, with the exception of soft cast iron, which works well,
the cylinder being made quite hard.
In Fig. 1081 is shown the so-called “Swedish” piston, as
used in a large blowing engine by EgestorfF. This piston is
a	b	c	d
Fig. 1080.
made with increased depth in the centre, similar to that in
Fig. 1078, and the holes shown in the sectional plan view are
for the purpose of removing the core from the casting. The
packing rings are made of cast iron, with the joint made as
1
Fig. 1081.
shown in Fig. 10S0#. The rings are kept in their proper posi-
tion by small pins. The method of securing the piston to the
rod is worthy of notice. The large key is secured and tightened
by a smaller key, the latter being held by a bolt, thus forming
a fastening of the third order.
Fig. 1082 shows a metallic piston in which the packing rings
are pressed out by an inner spring ring of Steel.*
The double cone shape of the inner ring enables the piston
to be closely fitted to the cylinder by tightening the bolts when
the engine is built. The nuts for the bolts are made of bronze,
as in Penn’s piston, the thread in this case being carried entirely
through the nut and the hole closed by a plug.
* E. Webers & Co., Machine Works, Rheine, Westphalia. This firm
makes a specialty of high class steam engines.
A piston for a single acting engine, with combination pack-
ing, is shown in Fig. 1083. The metallic packing rings are
backed with hemp, this combination presenting the advantage
Fig. 1082.
of elasticity together with durability. This style of packing is
well suited also for marine engines, as its elasticity renders it
less likely to be injured by the pitching and rolling of the vessel
than an entire metallic packing.
m OTlM
Fig. 1083.
Pistons for pump cylinders may be packed with leather so
long as the temperature of the liquid to be pumped does not
exceed88° F. (30° C.).
i
1	i	i
!	i	i
*.........rl*.......*
Fig. 1084.
A form of packing for this purpose is showm in Fig. 1084, the
principle being the same as the forms shown in the following
section. The units for the dimensions are the same as already
given.
3 346.
Peungers and Stuffing Boxes.
As already observed, the packing for plungers and rods acts
from the circumference inward, and such packings, in connec-
tion with the necessary parts, are known as stuffing boxes.
Two stuffing boxes for leather cup packing, especially adapted
for hydraulic presses and for pumps, are shown in Figs. 1085
and 1086, the former being for small and the latter for large
plungers. The double cup in Fig. 1085 is made with a spring
ring of iron between the cups to hold them in position before
the water pressure is applied. When the form shown in Fig.
1086 is used in the horizontal position, a ring of bronze made
in several parts is introduced below the packing, as shown in
dotted lines. This is intended to support the plunger and pre-
vent it from rubbing against the cast iron cylinder. The propor-
tions given in the illustrations are all based on the unit s, given
by formula (339).254
THE CONSTRUCTOR.
The friction existing between a plunger or piston rod in the
ordinar^ stuffing box in which the packing is tightened by
screws, cannot well be calculated, as it depends upon the pres-
sure which is put upon the packing. In those forms of stuffing
box in which the pressure in the cylinder tightens the packing
the friction may be calculated. According to the very elaborate
Fig. 1085.	Fig. 1086.
researches of Hick,* the friction of a well-lubricated cup leather
packing is independent of the depth of the packing, and is
directly proportioned to the water pressure and to the diameter
of the plunger. If P is the total pressure, D the diameter of
plunger, and ^the fractional resistance, we have :
F __ 0.04
~P ~~ ~~D~
(340)
For a new leather packing the friction is about 1 y2 times
greater. If instead of the total pressure P we use the pressure
p, in pounds per square inch we have :
F	7T
— = 0.0393 — ...............................(341)
P	4
Example.—For a piston rod 0.4 in. diameter, according to (340) the loss by
friction would be T\j, or 10 per cent., while for a plunger 24 in. diameter it
would be 0.0016, or % of 1 per cent. If, for example, the pressure is 4000
pounds per square inch, the* friction according to (341) would be
F — 4000 X 0.0393 X 0.7854 X 24 = 2963 pounds.
The total pressure on the plunger would be
F = 4000 X 0.7854 X 24- = 1,810,000 pounds.
Stuffing boxes for the piston rods of steam engines must be
capable of resisting the action of heat. Hemp packing is stili
much used for this purpose. The following illustrations show
two excellent forms of stuffing boxes to be used with hempen
packing.
Fig. 1087.	Fig. 10S8.
Fig. 1087 is intended to be used on the top of a cylinder;
Fig. 1088 is for an inverted cylinder. Both gland and box are
fitted with bronze rings, in order to reduce the wear upon the
rod. The wedge-shaped edge which is given to these rings was
introduced by Farcot, and is an improvement 011 the older style
of beveling the edge in one direction only, the latter method
often drawing the packing away from the sides of the box and
permitting leakage. In some designs the edge is left square, as
in Fig. 1090, or slightly rounded, as in Fig. 1089.
Fig. 1089.
Fig. 1090.
Fig. 1089 shows a form especially adapted to inverted cylin-
ders. The construction will be apparent on examination, and
it will be seen that the ordinary arrangement is reversed, and
the gland is cast upon the cylinder and the box containing the
packing is made separate. This prevents water from the cylin-
der from readily getting into the box.
In order to prevent the gland from binding on the rod it is
important that care should be taken to tighten both nuts
equally. In large marine engines, for example, the nuts are
made with worm wheels upon a common shaft. For small
stuffing boxes this is accomplished by having the screw thread
cut upon the outside of the box, as shown in Fig. 1090. This
box is intended to be made entirely of bronze. The nut is
made with six or eight notches in its circumference, to enable
it to be turned by a spanner wrench.
The dimensions of all the preceding figures are based upon
the unit s given by the empirical formula (339).
Example.—For a rod 3 ins. diameter, according to (339) we get 5 = 0.36.
The thickness of packing will then be 0.36 X 1.8 = 0.648, say % in. The
height of box for Fig. 1087 will be 0.36 X 12 = 4.32 ins., and for Fig. 1088 0.36
X 21 = 7 56 ins., and so for the other dimensions.
In horizontal stuffing boxes the length of the bronze collar9
should be made not less than 8 to 12 s, in order to reduce the
wear. The dimensions given in the illustrations may some-
times be modified in order to conform to the thickness of ad-
joining parts, so as to avoid difficulties in casting and shrinkage.
In some instances the stuffing boxes for valve rods for steam
engines are made in two parts, divided in a plane passing
through the axis of the rod. The flauge of the steam chest is
then made in the same plane, so that with this construction the
chest can be opened and valve and rod very conveniently re-
moved and replaced.
The large plungers for mine pumps are packed with hemp,
the stuffing boxes having 4 to 8 bolts.
More recently metallic packing has been introduced for
stuffing boxes of steam engines. An excellent example is
Fig. 1091.
Fig. 1092.
shown in Fig. 1091, which is made by Howaldt Brothers, of
Kiel.f The rings are made of white metal, in double cone
* See Verhandl. des Vereins F. Gewerbfleiss, 1866, p. 159.
f German Patent, No. 15.418. Over 9000 such boxes had been made up to
1888: one of these had been runniug eight years without opening.THE CONSTRUCTOR.
255
pairs as shown, thus causing the pressure to be exerted alter-
nately against the rod and the walls of the stuffing box. An
elastic washer is placed between the gland and the first ring to
equalize the pressure. Fig. 1092 shows the Standard metallic
packing introduced on the Prussian State Railways by Super-
intendent Neumann. This uses a single ring of white metal
made in two parts. The pressure is obtained from a steel spiral
spring placed in the bottom of the stuffing box, and acting
against a bronze pressure ring. The whole is enclosed in a
Steel cylinder which, together with its contents, can be drawn
out by inserting a hook into a T-shaped recess. The form
shown in the illustration is intended for a valve rod, but a
similar pattem is used for the piston rod.
? 347.
PiSTONS WITH VAI/VES.
Pistons with valves are used in lift pumps and in steam en-
gine air pumps. An example of such a pistoii, with leather
packing, intended for a mine pump, is shown in Fig. 1093.
\
Fig. 1093.
The packing is composed of conical rings of leather and
can vas, each three adjoining layers being sew^ed together. The
pressure of the water acts to tighten the packing. The acid
mine water often acts injuriously upon the leather packing of
the pump pistons, and in such cases metallic packing, with
rings of soft cast iron, is used. At Fahlun, in Sweden, after
many experiments the best material for packing was decided to
be birch wood. The proportions for Fig. 1093 are based upon
the unit 5. A valved piston for steam engine air pump is shown
in Fig. 984.
I 348.
Piston Rods.
Piston rods for steam engines are usually made of wrought
iron or steel, and recently compound rods of wrought iron sur-
rounded with hard Steel have been used. The rod ic either sub-
jected to tension only, as in single acting engines, or is alter-
nately subjected to tension and compression, in which case the
length and resistance to buckling must be taken into account.
For short rods the same results are obtained for both condi-
tions, but in no case should a rod subjected to alternate tension
and compression be made lighter than a rod under tension
only.
a. Dimensions of Piston Rods, Tension only.
D — diameter of cylinder in inches.
p = pressure in pounds per square inch.
The total pressure P on the piston will be P = — p D2. In
order that the stress on the rod should not exceed 8500 pounds
we have for the diameter d of the piston rod wheu made of
wrought iron, and is subjected to tension only :
■p- =00108 \f p
or for a close approximation :
d_ _ 57 + °-5
D	1000
(342)
(343)
Example.—If p = 60 pounds we have from (342),
for a 20 inch cylinder d — 20 X 0.0836 = 1.67 in.
pLy “ 0 0836, and hence
„	.	. The approximate formula
(343) gives	3- = 0.087, which for D = 20 gives d = 1.74 in.
Steel rods subjected to tension only may be made 0.8 times
the diameter of wrought iron rods.
If a piston rod is weakened by having a keyway cut through
it, or by a screw thread, the reduction in cross section should
be provided for by a proper increase in diameter. For this
reason the diameter of the rod is sometimes increased in the
cross head, an example of which will be seen in the locomotive
cross head, Fig. 539. This construction involves the necessity
of making the stuffing box gland in halves, as it could not be
slipped over the enlarged end of the rod.
b. Dimensions of Piston Rods for Buckling Stresses.
Using the preceding nomenclature and indicating the length
of stroke by A, we have :
• (344)
from which the following table has been calculated :
L	/ = 50 j = 60	= 70	= 80	= 9°	= 100	1 1 = 120	= 140	= 160	— 180
1.5	0.0967 0.100	0.104	0.108	O.III	0.114	OII9	0.124	0.129	0-133
2.0	O.III | 0.116	O.I2I	O.I25	0.128	0.132	0.138	0.143	00 6	0.153
2.5	0.124 1 0.130 1	0.135	O.I4O	0.144	0.148	0.154	0.161	0.166	0.171
These values will serve both for wrought iron and for steel
(compare \ 182, and table in \ 2).
Example.—For a steam cylinder 16 in. bore, 4 in. stroke, with a pressure
of 60 pounds, we have = 2.5, and d = 0.130 X 16 = 2.08, say 2 inches dia-
meter, either for steel or wrought iron.
The dimensions of steel keys to secure the piston to the rod
are so taken as to give shearing stresses from 5600 to 7500
pounds in the key. Care should be taken that the key be not
made too narrow, and the consequent superficial pressure be-
come too great. Pressures of 6000 to 7000 pounds per square
inch are found in stationary engines, and 10,000 to 15,00(7
pounds in locomotive engines.
d
D
77 = 0*0295 V 77

l 349*
Specific Capacity of Pressure Transmission Systems.
Having discussed the subject of conductors for pressure
organs, we return to the consideration of the various mechani-
cal devices which may be operated by pressure organs, although
these have already been described in Chapter XXIII. We are
now prepared to consider these in connection with the subject
of long-distance transmission of power, in a manner similar to
that in which tension organs are used in Chapter XXI. For
this purpose we may use to advantage the conception of specific
capacity. This method is especially desirable because its sim-
plicity and general character enables comparison to be made
between widely differing Systems.
The conception of specific capacity can be extended without
difficulty to motors operated by water, air, steam, etc., since for
all these we may put the general equation :
qv
deduced in \ 280. In this equation q represents the cross sec-
tion of the pipe or other conductor in square inches ; the mean
velocity in feet per minute — v, and iV being the horse power.
If, for example, in a water pressure engine, h is the available
head of water. Q the weight of water delivered per minute, and
h' the head equivalent to the resistance against which the water
leaves the engine, we have for the work delivered :
N-_ Q{h-h*)
33000
But Q == 0.0361 x 12 qv — 0.434 Q v, the coefficient 0.0361 being
the weight of a cubic inch of water, and the pressure p with
which the water acts = 0.434 hy whence h 2.3/. Substituting
these values we get:
N = P*3* ULX 2~3	STUzn = _JL_ qv (p_p,)
32,000	33000 *	r
and the specific capacity becomes :
N 1
qv 33000	r 1
a value of the same forni as that previously deduced in \ 280
[see formula (262)].
1060THE CONSTRUCTOR.
;;6
Example.—If the effective pressure p—p’ be 320 pounds, the specific ca-
paci ty will be N0 — 0.0097. If the pipe is 4.75 in. diameter, and the water has
a velocity of 236 feet per minute, we have:
N = 4.75 2 X —j— X 236 X 0.0097 = 40.56 H. P.
This is only the capacity of the pipe. The effective capacity will
be considered later.
Formula (345) can also be used for air pressure or for vacuum,
for steam or gas, by expressing the effective pressure in terms
of an equivalent head of water. For steam and air it may be
considered as an expression of the following form:
iron and steel, especially in the Mannesmann rolled tubes, per-
mit the use of high stresses; for wrought iron 6* = 17,000 lbs.
and for steel 35,000 to 40,000 lbs., or even higher, if necessary,
may be used. By neglecting the value of p in formula (347) we
have for:
Cast Iron, 5 =2 6,500, N0 = 0.197
Wrought Iron S = 17,000, NQ = 0.515
Steel	5	= 35,000, N0 — 1.060
This gives an indication of the efficiency of the pipe system of
power transmission and enables comparisons to be made with
other systems.
iVo =-----(p —p') u
33o°o
(346)
The coefficient n is very comprehensive; it increases with p
and with the rate of expansion e, and can be calculated from
these data, and also confirmed by observation. For e = 2, it
ranges from i}4 to 1^3, and increases to 3 to 4 for s = 20 to 30,
results whicli conform to the higher pressures and greater effi-
ciency of compound engines in which such high expansion
ratios are used.
With some transformations the equation for specific capacity
may also be used to solve another important problem, that is
the question of the best material to be used for the conducting
pipe.
If we assume the diameter of the pipe, the horse-power N
will be:
p = — D1
4	33°°°
For the thickness of pipe, we have from (321), for a stress S, in
the material:
*6+D = D)[-
And since 2 4 -f- D is the external diameter Doy we have for
the cross section qx of the pipe.
\ 35°-
The Ring System of Power Distribution with Pipe
Conductors.
Before proceeding with the further discussion of the preced-
ing equatious it is advisable to investigate further the subject of
power transmission by means of pipe conductors, as already in-
dicated in $ 312. It was there remarked that pressure organs
might be used in connection with pipe conductors so as to form
“ring” transmission systems in a manner similar to those
already described for rope. Taking into consideration first,
hydraulic systems, especially high pressure hydraulic systems,
we find two distinet kinds of “ring” systems which may be
used.
or
(D0 2 — ZF
)
— D2
4
fs + p
\S-P
X
;
7T
D2
_3_P_
S-p
Substituting the value of — D2 from its equation in the above
4
expression for N, we have

1	.S* — p
------qx---------- - p v =
33000	2 p
whence
1
33000
Q1 v
N0 = -A_ -= -1— (S — p ...................(347)
ipv	33000 V rJ	K
a form similar to the preceding expressions for JVQ.
This expression is very instructive. It is applicable to all
fornis of conducting pipes for power transmission. It shows
clearly the importance and value of a high value of S. A high
value of S reduces the proportional influence of py to a degree
which practically makes N0 dependent mainly upon .S. It fol-
lows that we may consider that the specific capacity of the pipe
in a pipe transmission system, is practically independent of the
pressure of the fluid used in it. Iu other words, the capacity of
a given pipe in horse-power is the same, whether the medium
be liquid or gaseous, high or low pressure, provided the stress
in the material of the cross section of the pipe is constant.
It is therefore desirable to use pipes of small diameter and
fluids at moderately high pressures. The friction in the pipe
need not prevent this, as care in avoiding sharp bends and
angles can be taken ; and as already shown in § 340 the friction
is independent of the pressure of the medium, at least so it
appears from such experimenta as have yet been made.
The value of the stress in the material of the pipe cannot be
taken very high ; ,S = 7000 lbs. being about the upper limit,
and S = 6500 lbs. appears to be quite high enough. Wrought
In the first method, Fig. 1094, the flow of water under pres-
sure starts from the plbwer station T°>. with a pressure p0, and
proceeds to the first station Tlt where it operates a water pres-
sure engine, and passes on with a reduced pressure pv It has
therefore operated at the station Tx with a pressure p0—pv
With the pressure px it passes on to the eecond, third, fourth —
— nth station Tn, each time losing pressure until it retums to
the power station with a final pressure p nt where it is again
raised to the initial pressure of pQ. This is practically a couu-
ter part of the rope transmission system of Fig. 917. It is
apparent that the water pressure engines (escapements) at Zj,
7V Z3,--------Z«, should all be of equal size in order to uti-
lize the entire flow without excessive resistance. Automatic
regulation, such as HelfenbergeFs, described in § 328, is also
desirable.*
The second system is showm in diagram in Fig. 1095. It will
be seen that at each station there is a branch or shunt tube,
leading through the motor (or escapement) Z2, and then re-
uniting with the main pipe. The main pipe Ay forks at the
station into the two branches B and Ct of which the first diverts
any required fraction of the power of the main flow, as	|,
as the case may be. At the fork is a swing valve C7, operated
by a speed governor B, driven by the motor. This governor
requires the assistance of some form of power reinforcement,
such, for example, as showTn in Fig. 1037. The discharge pipe
D of the motor unites with the by-pass Cy to form again the
main conductor E. At the entrance in the main pipe Ay we
have the pressure px of the original flow ; the motor Z2 is now
supposed to be stationary, the stop valve at B' having been
closed by hand. The flap valve C' which has been disconnected
* The Eondon Hydraulic Power Coinpany has instaUed separate ring sya-
tems, each with a single generator and motor.THE CONSTRUCTOR.
*S 7
from the regulator before stopping the motor, is also closed.
The flow of water then passes through C to E with the pres-
sure pv
When the motor T2 is to be started, the valve B' is opened
and the flap valve O gradually opened until the motor begins
to move, when it is connected to the governor, which regulates
it thereafter so as to keep the motor at its normal speed. When
a heavy load is thrown on, the valve is opened so that the pres-
sure p2 in B, becomes a greater fraction of plf and when the
work is less it is reduced. The pressure of discharge ps acts as
a back pressure so that the motor works with an effective pres-
sure p2—pz. The flow of water in the by-pass pipe C, also
passes the valve Cf with a pressure ps> and unites with the dis-
charge at E to be further utilized at subsequent stations until
it returns to the power station, where if it has reached the min-
imum pressure, it is permitted to flow into a tank, from which
it is again drawn by the pressure pumps. If the return water is
delivered under pressure it may be allowed to enter the suction
pipe of the pressure pumps direct and so form a closed ring
System to start anew on the Circuit.
This system has not yet to the Author’s knowledge been put
into practical operation *
The ring system of hydraulic power transmission is to be
recommended when the various stations are distributed over a
wide area and are readily connected by a continuous line of
pipe. The pipe can be kept from freezing in winter by occa
sional gas flames. as has already been demonstrated by exper-
ience with Armstrong’s hydraulic cranes. The ring system
should be carefully distinguished from those forms in which
the flow of water passes through the motor and is allowed to
flow off at lowest pressure of discharge. A corresponding dis-
tinction is to be made with other forms of power transmission.
The author distinguishes as “line ” transmissions, those forms
in which the transmitting medium does not return to itself in a
complete circuit, in contradistinction with the “ ring ” systems.
The older form of rope transmission ($ 297) is therefore a “ line”
system, while the system devised by the author and discussed
in \ 301 is a “ring” system. A hydraulic system in which
there is a free discharge of water from the motors is in like
manner a hydraulic “line” transmission system.
There is, however, an intermediate form possible, namely,
that in which water after passing through a series of motors as
in a ring system, is discharged freely from the last motor Tn.
A similar arrangement is possible with other systems of trans-
mission. We may therefore extend the definition of a “ring ”
system to include those forms in which the medium of trans-
mission returns to the place of starting. The distinction can
thenbe made between “open” and “closed” ring systems,
the latter being shown in diagram in Fig. 917.
High pressure hydraulic systems are well adapted for large
railway stations where numerous elevators as well as winding
hoists and other rotati ve machines are to be operated. For
suchinstallations a combination of “ring” and “line” systems
is best suited. The hydraulic elevators are more conveniently
arranged on a line system than in a fing circuit. An apparent
objection to the use of high pressure water to direct acting ele-
vators lies in the fact that the diameter of the plunger becomes
so small as to be hardly stiff enough to support the load on the
platform without buckling. This difficulty is readily overcome
by use of the hydraulic lever, as shown in Fig. 956 a, the con-
struction of which offers no difficulties, and it is unnecessary to
go into details.
Up to the present time air has only been used upon line sys-
tems. either with direct pressure or with vaacum. Gas engines
can only be operated on a line system since the gas is burned
intheengine. Steam has been used in a ring system in New
York for some time, on a long distance transmission, and short
ring systems exist in most cases of compound or triple expan-
sion steam engines as used in marine and stationary practice.
Steam at a high initial pressure is expanded successively in one
cylinder after anether, and between the last cylinder or station
Tn and the first, or boiler T0i is placed the surface condenser
* See the Author’s article in Glaser’s Annalen, Vol. XVII (1885), part 12.
from his paper to the Verein fur E)isenbahnkunde, Nov. 10,1885.
7 m, where the medium reaches the minimum pressure and is
converted into water to be returned to the boiler .and start anew
on the circuit. In order that the velocity of flow shall be uni-
form the successive passages for the expanding steam should
be made with continually increasing cross section as shown in
diagram in Fig. 1096. If a jet condenser is used instead of a
surface condenser the circuit becomes an open ring. The high
economy which has been attained by the application of the
“ ring ” system with steam in the form of multiple expansion
engines, points to the possibility of a similar economy in the
application of the ring system to wire rope transmission.
i,ehmann’s hot air engine, which is a true closed circuit, is
an example of the ring system confined within the limits of a
single machine.
? 35i.
Spfcific Capacity of Transmission by Shafting.
The subject of the specific capacity of shafting was not con“
sidered in Chapter IX, and it is introduced in this place in
order to obtain a basis for comparison with the other systems
of transmission.
If we have the moment PR and shaft diameter d, we have, if
S is the fibre stress at the circumference
PR=S~d*
16
(see l 144).
If we make the lever arm R = J d, we have P = the force at
the circumference of the shaft and hence P =
sta
Taking
v = the velocity at the circumference of the shaft and N the
number of horse-power transmitted, we have:

_Pv_
33,ooo
— S — d*v
2	4_______•
33,000
But — d2 = qy the cross sectional area of the shaft, whence
4
N-— ***
2 33OOO
(348)
and hence the specific capacity of the shaft is:
1	5
2	33^000 *
• • • (349>
This expression, which is of the same form as those already ob-
tained, does not give values numerically great, because S must
be taken low enough to avoid excessive torsion of the shaft. If
we require, as in $ 144, that the torsion shall not exceed 0.075°
per foot of length we must have S 630 d which gives for
shafting from 2 to 6 inches diameter 6* = about 1200 to 3700
pounds and the specific capacity
iV0 = 0.018 to 0.056................(35°)
In other words, such a shaft will transmit, at one foot per min-
ute circumferential velocity, 0.018 to 0.056 horse-power for each
square inch cross section.
In the application of shafting to long distance transmission
the friction of the journal bearings is a very important consid-
eration. The influence of friction may be determined in the
form of a general expression in a similar manner to that of the
friction of water in a pipe (§ 340). According to formula (100)
we have for the force F> exerted at the circumference to over-
come the journal friction F= — f times the weight of the
shaft, that is = — f
12 L X 0.28 in which L is
the
length of the shaft in feet, and 0.28 is the weight of a cubic
inch of wrought iron. It follows that the horse power Nx re-
quired to overcome the friction will be :
^1 =
F_v_
33000
X q X 12 L X 0.28
33000
and if we take the coefficient of friction f — 0.08 we have<58
THE CONSTRUCTOR.
N,
m 0.08 X 4 X 0-28 X 12 L q v L q v
33°oo tt
Ar	L
Nx = —--------q v
96,422
96,422
(351)
and if we wish the specific frictional resistance, we have:
,V. L
Cv‘
Jo~
qv
96,422
(352)
This resistance is by 110 means inconsiderable. Expressed as a
percentage it will be :
__Nx  L qv  L 1
N	96,422 * N	96,442 * AT0
• (353)
The value pr, it will be seen, is inversely proportional to the
specific capacity. If we apply this to (350) we have for a 2 inch
shaft
pr =
and for a 6 inch shaft
Pr
0.018 X 96,422
L
0.056 x 96,422
1735
_L_
5400
(354)
hence 1735 feet and 5400 feet are the limits of length respec-
ti vely for the two diameters given, at which the frictional resis-
tance will equal the total transmitting capacity. Much higher
efficiency is obtained by using hollow Steel shafting such as is
now produced by the Mannesmann process of rolling weldless
tubing. This furnishes a seamless tube, of sufficient truth as to
cylindrical shape, the journals of which may be made either
«ntirely of Steel or of so-called “compound steel.” *
*	d
If we take the ratio of outer to inner diameter ip = -~
dn
0.9
(compare ? 90) and the thickness of the journal d/ = 0.4 dQ we
have for JVQ :
at N	1 S f 1 r
N0= — = — .------( i-f ^
qv	2 33000 V
)
which for ip = 0.9 gives
== 1.81 x •-------.
33°00

96,422
or dividing again by N:
Nx___ L
^r	N 241,000
241,000
1
~N,
— q v
CAT\ 2
N0 = 0.0326
and for the 6 inch shaft:
and these give in (357) :
for the 2 inch shaft—
Na =-0.101
L
0.0326 x 241,000	7856
and for the 6 inch shaft—
.	L	L
pr = ;
0.101 X 241,000	24,341
(358)
(355)
(356)
which is decidedly higher than for the solid shaft. (The value
6* fC 630 dQ must be retained to avoid too great torsion). For
the frictional resistance at the circumference of the shaft we
have:
and if d' =- 0.4 dQ we have :
0.4 L qv __ L
(357)
With the values for N0 as given in the two preceding instances,
we have for the 2 inch shaft:
* The Mannesmann “compound” steel tubing is made with the interior
of soft wrought iron and the outside of bardened steel.
so that in both instances it is less than one-fourth the resistance
of the corresponding solid shafts, as given in (354). Hollow shaft-
ing thus greatly extends the capacity of shafting for long dis-
tance transmission and also permits an important economy in
material.
The subject of shafting made of steel tubing was not consid-
ered in Chapttr IX, and a brief discussion will therefore be
given here.
Let dQ be the ontside diameter, dx the inside diameter, let the
ratio = V?- Making V' = 0.9 as is usual in practice with such
“ o
tubing, the diameter for resistance to torsion, (compare formula
VI33) ) will be:
^0 = 0-394/ P r = 6.18 4/ —...............(359)
This requires that the number of revolutions be known or
assumed. If instead of nt the circumferential velocity v, be
given, wre have for the same shaft:
da — 7-25	....................(360)
v, being expressed in feet per minute at the circumference of
the shaft. The number of revolutions will be :
3-32	, . .
« = —v..........................(361)
“o
The diameterTor strength (compare (131)) wfill be :
=	=5-354/^..................(362)
If ip is not assumed as above, it may be taken at will and the
following formula used:
=	. . .063)
VI — Tp™ s Vi — yp*^ nS
in which, when :
a,
—i- = yp = 0.4 0.5 0.6 0.7 0.750.800.85 0.90
I
-------= I.oi 1.02 I.05 I.IO 1.14 1.19 I.242 I.427
V i—ip
The weights of tubular and solid shafting are to each other as
G9 (-*■)■
Exampie.—If iV= 60 horse-power, n — 120 revolutions per minute, we
have from (359)
^	, 0 4/IT
d° = 6-lSAlM=s-*
instead of
,	4/"6cT
d = *-7 V™=3-95m-
as would be the case for_ a solid 'shaft. The hollow shaft, however, weighs
only (£)■(- afc)" 0.33 times the weight of the solid shaft. The
circumferential velocity v — —° g^~~~ ~ i63 feet. *f a higher speed be
chosen, as may readily be done, on account of the small journaf diameter
dwe have from (360), making v = 300 ft., for exampie:
. 3/ 6o~
d0 = 725	= 4-24 in.
whence d* — 0.4 d0 = 1.7 in.
The number of revolutions will then be
n=3^0<3«>
The weight otfshafVwill be
times that of a solid shaft at 120 revolutions. The loss from friction will be
only 0.26 times that of the solid shaft.THE CONSTRUCTOR.
259
i 352.
Spfcific Vaeue of Long Distance Transmissions.
In the two preceding sections, equations have been given
showing comparative relations between various methods of
transmissions but at the «ime time the general equation by
which ali the various methods of long distance transmission
may at once be compared, has not yet been given. The point
which yet remains to be determined is the amount of material
which the tranimitted force carries in the shape of the trans-
mitting medium. Investigation reveals certain fundamental
points which may be applied either to a special case or to a
comparative judgment as to the value of different Systems.
The amount of material required for the principal transmit-
ting medium of a long distance transmission systemy may be
considered as a function of the number of horse-power required
to transport one pound of the material of which the conductor
is composed over the distance between the origin of power and
the point of application.
The name ‘ Specific Long Distance Value ” may properly be
given to this quantity. If it is high, the method is efficient, if
low it is less efficient for applications in which the distance
plays an important part.
In all the cases considered the medium of transmission may
be taken as a form of prism of constant cross section qy having
an endlong motion and the length of which is equal to the dis-
tance A from the point of origin to the point of application. A
chosen length A0 may be selected as a unit. The weight G of
such unit will then be :
G = 12 A0 q <r...................(364)
in which <x is the weight of a cubic inch of the material of which
the conductor is made. The work in horse power exerted dur-
ing the passage over the distance A0, is expressed by:
N — NQ q v
Dividing this value by the preceding we obtained the desired
resuit. It is desirable to select a Standard unit for A0 which
shall be generally applicable. Making A0 — 1 inch = TV ft,
jsr
and letting the quotient — be represented by A}, we have for
G
the general equation of “ specific loug distance value : ”
v	Sv
ATs = Ar0 — and also = C —..............(365)
These values hold good for all the systems which have been
considered with the exception of rope and belt transmissions.
In these the tension organ moves not only forward, but must
be returned back on the slack side, and hence for these cases
we must put:
N
Ao f
2 cr
(366)
When rope is used on the “ ring” transmission system, how-
ever, the preceding formula (365) is properly used, since a
single rope makes the entire circuit back to the origin of power
ana the last point of application usually lies near the starting
point
The formula for Ns is especially noteworthy because its
application reveals great and unexpected differences between
the various systems of long distance transmission.
When the conducting medium is operated at a high velocity
and also at a high working stress the specific value is very
high; when they are both small the specific value becomes
lower, since they are both multiplied together. The following
table shows both numerically and graphically the “ specific
long distance value ” for the several methods named :
The stresses in the table are xaRen at the maximum values
given in the preceding pages of this work, without approaching
too close to the upper limit. The high value of Steel cable trans-
mission is most noteworthy, and explains its frequent use.
The table does not even give as good a standing for wire rope
as might have been done, as when used with a tightening pul-
ley its specific capacity is increased 1J times, see formula (310).
Equally noteworthy with the value of wire rope is the poor
showing made by shafting, especially solid shafting ; as it occu-
pies the lowest position of all. The newer system of hollow
Steel shafting stands somewhat better, but stili very low.
Welded iron and Steel tubing when used for conductors give
fairly good results. To avoid misunderstanding, it must be
noted that where pipes are referred to in the table, v is the
velocity of the fluid passing through them. No distinction is
made between pipes for steam, compressed air or water, since it
is only the weight of the tube which is here considered. A cir-
cumstance worthy of note is that the reciprocal of Ns is pro-
portional to the weight of the transmitting material, omitting
N
connections, flanges, couplings, etc., so that Gs — —. If, for
As
example, 200 horse-power is to be transmitted by hydraulic
pressure over a distance A, of 984 feet — 11,808 inches ; the
weight of the bare Steel pipe will be
11,808 -J- 200
1448 ~
1631
pounds, the thickness of the pipe being so made that the stress
on the material shall be 34,000 pounds, and the diameter such
that the velocity shall be 787 feet per minute. Such calcula-
tions are very useful for a general and preliminary investiga-
tion. The designer must be careful not to lose sight of the fact
that the stress in the material, and the velocity bear a very im-
portant relation to each other.
The values of Ns> given above, are the gross values including
the entire work transmitted by the system. The net value
(A^j and its relation to the gross value, that is, the quotient
jyr
-A is the next question to be answered. This question is by no
means so simple as the preceding. The actual efficiency of a
long distance transmission depends so much upon the resis-
tauces of friction, stiffness, centrifugal force, heat, etc., all of
which differ for the different constructions, that only a very
general allowance can be made to include them. A brief glance
can only here be given to the method of determining this
point.
The greater the number of horse-power which can be trans-
mitted for each pound of material, the less, proportionally will
be the load upon the bearings and other points of loss, and
iience the smaller, proportionally, will be the loss of friction and
other hurtful resistances. In other words: The greater the
specific value of the system, the lessy in generaly will be the pro-
portion of hurtful resistance.
The values already given in the table for the gross specific
value, give also, therefore a measure of the net efficiency as
well.
While it can hardly be asserted that the above values for Ns
are inversely proportional to the losses from hurtful resistan-
ces, yet there is a relation existing between them, so that it
may be said that the net value (N2)s is in all cases higher than
the gross value Ns; higher in the sense, that, the greater gross
values are accompanied also with a higher net efficiency.
The difference will appear most distinctly by comparing wire
cable transmission with solid shafting. Such a comparison is
the more readily made because in both instances the resis-
tances can be closely determined.
SPECIFIC VALUE FOR LONG DISTANCE TRANSMISSIONS.
SYSTEM. J	I v	s i	cr	i #0	Ns	PERCENTAGE OF EFFICIENCY.	PER CENT.
Steel Cable, Ring System . . .	5900	21,000	O.32	0.318	5683		IOO.
Steel Cable, Line System . . .	59°°	21,000	O.32	0.318	2631		50.
Iron Cable, Ring System . . .	5900	8,500	O.32	O.129	2380	: ■■■■■■IHHi	40.6
Steel Conducting Pipe ....	787	34,000	0.28	0.515	1448	■■■■■■■	24.7
Iron Cable, Line System . . .	5900	8,500	O.32	O.I29	1190		20.3
Leather Belting, Line System .	5900	540	0.036	0.0068	1114	■■■■■	*9-
Wrought Iron Pipe ......	787	17,000	0.28	O.257	722		12.3
Hemp Rope, Line System . . .	5900	240	O.036	OOO36	295	HHH	5-02
Cast Iron Pipe		787	6,400	0.28	O.097	272	^■i	4.64
Hollow Steel Shafting ....	394	5,200	0.28	0.II4	160	■	2.73
Solid Shafting, Iron or Steel . !	l9 7	4,200	O.28	O.C63	45’	■	0.8THE CONSTRUCTOR.
260
The iron wire cable transmission at Oberursel, discussed in
? 300, showed a loss of about 14 per cent. in a transmission of
104 H. P., over a distance of 3168 feet. To transmit the same
power over this distance with solid shafting, we get from (353)
the frictional resistance:
.	8_ i_
Pr 95,422	* NQ
Taking R0 = 0.063, which is aniply high enough, we have pr —
0.52 or about J. The net specific value for long distance will
then be for iron cable, (1 — 0.14) 1190_= 1023 ; for solid shaft-
ing, (1—0.5) 45 = 22.5.
It will be seen that about 52 H. P. is absorbed in the friction
of the shaft; so that at periods of low water, when the turbine
yields only 40.3 H. P. it would not be able to overcome the
friction of the shaft alone.
The correctness of these considerations will be confirmed
when it is remembered that a wire cable runs at a very high
velocity and operates at a high stress which the journals of the
rope pulleys move at a very low velocity (scarcely ^ v) ; while
on the other hand the shaft can only be subjected to a low
stress, and the velocity at its circumference is not only low, but
it has to overcome the resistance of friction at the same veloc-
ity. This also explains clearly the reason why rope transmis-
sion has so frequently superseded shafting in actual practice.
CHAPTER XXV.
RESER VOIRS FOR PRESSURE ORGANS.
2 353-
Various Kinds of Reservoirs.
Reservoirs form a most important feature in connection with
the use of pressure organs, and are dividedinto tanks, receivers,
chambers of various kinds, in which the pressure organs may
be stored in greater or less quantity and drawu upon for use as
may be required. Such reservoirs may be used either for posi-
tive or negative pressure according to the system with which
they are used. Both kinds are shown in Fig. 993, in the case
of a canal lock. As already indicated in £ 312, the various
forms of reservoirs are very numerous. From the nature of the
subject we can only here discuss that branch of the subject
which relates to machine construction, including reservoirs of
cast and wrought iron, copper and Steel. These are applicable
both to gaseous and liquid organs and in most cases are of
special construction to meet the circumstances of use.
A reservoir when considered in connection wTith the appar-
atus for filling and emptying, as well as for controlling the
pressure, wThether positive or negative, forms a storage system
which may properly be considered as a ratchet train (see Chap.
XVIII).
For the present, however, it is here only the intention to
discuss the constructive features of the reservoir itself consid-
ered as a machine element.
2 354-
Cast Iron Tanks.
Cast iron tanks with flat sides are used only for very small
reservoirs and need not be discussed here ; for larger sizes the
walls are made cylindrical in order better to resist the internal
pressure. Cylindrical cast iron tanks can be advantageously
used for water up to 1000 cubic feet capacity. A good construc-
tion has already been shown in Chapter IV, as made by Lauch-
hammer’s Iron Works, of Groditz, and used in many places.
Fig. 1097 shows a tank of this sort. The water is delivered
at E ; A, is the discharge, and U the overflow. The thickness
of the walls is made about %-inch ; the flat bottom rests on a
strong floor of wood carried by heavy beams. The flange
joints are made as in Fig. 268, 269. If is the greates head of
water in the tank, the pressure per square inch on the bottom
will be p = 0.434 h, h being taken in feet, and we have for the
thickness w hen D is the inside diameter according to (324) :
<5 __ 1 p
~D~~2 ~S
0.434 h
2.S
h
°-2I7t
(367)
Example.
-If 5 = 0.25 in., D = 118 ins., h = 9.83 ft., we have S — o 217^-^?=
0
0.217
118 x 9-83
0.25
1009 lbs., which is such a moderate value that the tank is
amply secure. If we take the diameter of the bolts at %-in. for the joint 4.
inches deep at the bottom of the tank, and let n, be the number of bolts,
and further put the permissible load upon each bolt at 275 pounds, wc
have :
4 E> X 0.434 h = 2 n X 275
from which n =	^ 9-83 _ g which gives for the distance
2 X 275
from centre to centre of bolts, —- = 1.11 in. or about ins. For the joint
3.0
half-way between the top and bottom of the tank the pressure would be but
half that at the bottom and the bolts may be spaced proportionately wider,
say about 2 inches apart. The total contents of the tank will be = 742 cubic
feet = 5550 gallons.
In using cast iron tanks of this sort care must be taken to
avoid filling them with liauids which have an injurious action
upon the rubber packing of the joints.
\ 355.
Riveted Tanks.	x
When tanks of large capacity are required, wrought iron or
Steel must be used in their construction and these involve the
use of riveted joints. With tanks of large diameter construc-
tive difficulties arise in connection with the flat bottoms.
In the United States, oil tanks are made with flat bottoms,
carefully bedded in cernent, and similar tanks are used in Ger-
many for water. It is, however, found that greater facility of
construction, as well as economy of material, is obtained by
making the bottom convex, as will be shown.
A very frequent and useful form is that in which the bottom
b
is made in the shape of a spherical segment, Fig. 1098 a, the
tank being supported on a flanged ring riveted to its circumfer-
ence and the ring standing on a support of masonry.
The construction of the supporting ring is shown in Fig.
1098 b, from the design of Prof. Intze.
The tension in the inclined direction of the bottom of the
tank is carried by the lower half of the supporting ring, while
the upper portion is subjected to the pressure of the tank at
right angles to the vertical. This latter force is well resisted
by a ring of angle iron running entirely around the tank.
The calculation of the bottom of spherical segment shape is
as follows:
If R is the radius of the sphere of which the segment is a
part, we have from \ 19, Case II.:
A _p_
R ~ 2 5,
in which ^ is the thickness and S1 the stress therein due to the
pressure p. The pressure is the greatest at the lowest point of
the bottom where the height in feet of the column of liquid isTHE CONSTRUCTOR.
261
equal to hy so tbat if <x, is the weight of a cubic inch of the
iiquid p = 12 h <r. We then have:
R
which for water gives,
12na___n
2 S1	Sx
= 0.0361
§
R
0.217
A
Si
(368)
At each higher point of the bottom the pressure is less, until
at the edge of the bottom the height h, is diminished by the
depth fy of the bottom. For simplicity, however, it is custom*
ary to make the entire bottom of the same thickness which
is required for the lowest point.
For the thickness of the cylindrical walls of the tank at the
bottom we have the pressure p = 0.036 {h —f) both h and/,
being in inches, and from (367)
6
n
i_ ^434 0^—/)
2 •	5
= o.2.7---
this gives S in feet, hence we have for 6 in inches :
R	i i 0.5 O.55 0.60	0.625 °^5 j		1 !	1	1		
D ~				0.70,0.75 	1	!	o.8oj 0.85	0.90	0.95	1.0
/ D	05 0.32 0.27	0.25	0.23	i ! 0.2110.19	o.isj 0.16	0.15	0.14	o-i34
h D	1.0 j 0.71 0.68 1 j	0.67	0.66	0.70^ 0.76 1	o.88j 1.07	1-52	2.84	00
h — 0.5/ D j	! I o.75 o-55 0.54	0-54	0.56	0.59 0.67	o.79, °-99	11.45	2-77	00
-W=:	0.17; 0.07; 0.05 liti	0.04	0 b	j 0.03} 0.02 i i	! 0.02! 0.02 t i	i 0.02 1	i 0.01	O.OX
These relations are also shown graphically in Fig. 1099, and
the results are interesting. It will be seen that in order to have
= 6 when Sx = S we must always make R *< D. It also
appears that the best ratio of depth to diameter occurs when
— is about equal to 0.60, for then h — 0.5 f nearly approaches
0.5 D; this, however, is only approximate. It thus appears
that the two conditions of greatest economy of material and
equality of value ^ and cannot be attained at the same time
6 = 12 X 0.217 D —	— 2*6°4 D —. (369)
In order to obtain good proportions it should be considered
d
that as h diminishes, the ratio of — becomes smaller, while as
D increases the size and thickness of the bottom increases.
An approximate formula by which the minimum amount of
material will be required is :
£> — 1.366^1*2	.................(370)
tn which Q is the volume of the material in cubic feet to be
contained in the tank.
For the height H of the wetted portion of the surface we
have:
H + — = /. — -^ = —.......................(371)
2 22 )
if we assume, as we may with sufficiently close approximation,
the segment of the sphere to be practically that of a paraboloid.
The same remark about the most economical ratio of depth
to diameter applies here as in the note to ? 354.
Example /.—For Q = 47,000 cubic feet we have from (370),
D — 1.366	42,000 = 47.36.
A carefutly calculated tank at Halle, of this capacity (1200 cu. metres) was
made 51.88 feet diameter.
If O — 65,60-1 cu. ft. we have D = 1.366^70,000 = 56.3 ft., while a tank of
the same capacity at Esseu is 58 feet in diameter.
The water tower at Neustassfurt has a capacity Q = 21,160 cu. ft., and is
39.36 ft. diameter; according- to (370) it would be D — 1.366^21,160 = 37,79 ft
Ali three cases thus agree well with the formula.
For the depth/ of the concave bottom, we have for any given
radius R, the expression
2 Rf-P = i D\
from ■which we get
/ R
rv
U x/
(372)
Itisfound convenient, but not essential, to choose such a
value for Rt that 6X = when 5 = Sv To accomplish this re-
suit, the conditions which obtain for the equations both for
and d must be fuifilled. These are :
R h—f ^ h
-5 = —' whence-zr =
/
D__
~~ D
(373)
The following table gives a series of numerical values for
these relations:
exactly. The most useful ratio in practice will be obtained by
selecting a value for Dt according to (370).
The value R — 0.5 Dy which corresponds to a hemispherical
bottom, is useful to the extent that when the supporting ring
is placed at its upper edge there is no lateral pressure produced
tending to compress the ring, as there is in ali of the other
cases. The hemispherical bottom, however, offers too many
constructive difficulties to be much used.
Example 2.—Eet Q =* 53,000 cubic feet. We have from (370):
D = 1.336^ 53,000 = 50*28 feet and according to (371), h-'•$/— 0.5 D =*
25.14 fit., and combining these again we get: Q = 25.14 X 0.7854 (50.28)2
49.420 cu. ft., which is a little under the required content, but shows the cor-
rectness of the proportions.262
THE CONSTRUCTOR.
If we now make / = o 21 D = 0.21 X 50.28 = 10.56 we have from the above
table, R = 0.7 D = 0.7 X 50.28 = 35.2 ft. We have from (371) h = 0.5 D
0.5 f— 0.605 D — 30.42 it. The height of the wetted perimeter will be H —
h — /= (0.605 — 0.21) D — 0.395 D = 19.86 ft.
Taking for the stress in the metal at the lowest part of the walls of the
tank we have from (369):
8 = 2.604 d)
H
S
jyi
= 2.604 X 0.395 —pr — 0.372 in
For the bottom we have
81 2.604 R	- o,7X 2.604 X 0.605	= 0.4 in.
and 1.07 ; that is, the thickness of the bottom is 7 per cent. greater than
that of the lowest row of plates in the walls of the tank.
If we make the tank with six rings of 3 ft. width and one of 2 ft. we get for
the thicknesses:
Depth = 5	19.86	16.86	13.86	10.86	7.86	4.86	1.86
Calculated =	0.372	Q-3^	0.260	0.203	0.147	0.091	0.035
O In practice =	%"	tV*	W'	K'	M"		K"
The latter figures show an excess over the theoretical thickness, but the
excess is needed for stiffness and for constructive reasons. The thickness
of the bottom. as already calculated is 0.4 in., but in practice would prob-
ably be made
The riveting may be made the same as ordmary bofler riveting; and from
the table in | 59, we find for S = ^s", d = and for single riveting th®
modulus of efficiency is 0.47.
which seems rather too high.
This gives a stress of	= 15,000 pounds,
• 0.47
For this reason the two lower seams at least
lateral forces which act, each on one half the circumfe
the base ring of the tank :
a.	p.
5 =
G
2 cos a
producing a load sx per running foot:
should be made with double riveting: which gives a stress of = 11,800
o-59
pounds. The seams of the bottom should always be made double riveted.
Example 3.—L,et Q again be taken as 53,000 lbs. We will now proportion
the tank so that 61 = 8, and take D = 50 ft.
In order that 8X shall at least equal 8, we will take — 0.625 whence /
— 0.25 D = 12.5 ft. We then have h = 0.67 D — 33.5 ft., and h — —~ f =
(0.67 — 0.125) D — 0.545 D — 27.25 ft. We therefore have
5
2
Substituting for Gy its value
7 |-J1 (A _/) + JL.S 0.2? +/.) -]
in which y is the weight of a cubic foot of the liquid, we get:
Q = 0.7854 X 27.25 X (50)2 = 53,5oo cu. ft.
which agrees quite closely enough with the original assumed capacity.
H will be = to h —/=(0.67 — 0.25) D = 0.42 D — 21 ft.
We therefore have for the lowest cylindrical portion of the tank:
8 = 2.604 X 0.42 —— = 2.604 X 0.42 ------ = 0.3906"
o	7000
and for the bottom:
e *	0.625 X 0.67 IT-	^0.625 x 0.67 x (50)3	0	„
8l — 2.604 ------—^—-—— = 2.604*----------——— = 0.3894"
thus giving practically 8 = 82.
The tank will be heavier than the preceding proportions give, as might
be expected, but the excess weight will be only about 1 per cent.
2 35
Tanks with Concavf Bottoms.
The question of the action of the forces upon the bottom of
a tank as discussed in the preceding section, was first
thoroughly investigated by Prof. Intze, whose valuable re-
searches have practically revolutionized the construction of
riveted tanks.* The following discussion is based on Intze*s,
but the calculations are simplified and abridged.
Fig. 1100 shows two forms in which the spherical segment
may be used, a, with convex or hanging bottom, as already dis-
cussed, and b, with concave or reversed bottom. In both forms
the pressure of water on the bpttom produces a stress at the
base of the cylindrical portion of the tank in the direction of
the tangent to the curve of the bottom, the stress acting in-
wards in case a, and outward in case b. It is desirable to make
the construction such that this force is received by the base
ring and not by the shell of the tank. In every case, however,
an increase is required in the thickness of the bottom of the
tank.
There is also a force t, acting at right angles to the tangent
or normal to the curve of the bottom of the tank, and the deter-
mination of both of these forces is a matter of importance.
If G be the weight of the liquid, and a the angle which the
tangents make with the axis we have for case ay for the two
* See the article by Dr. Forchheimer: “ On the Construction of Iron
Tanks, for Water, Oil and Gas, according to the Calculations and System of
Prof. Intze, of Aacheu.', Schilling’s Journal tUr Gas-beleuchtung, 1884, p.
705-
-4[‘X + X(£)']
In this h is the distance from the level of the surface of the
liquid to the crown of the curve of the bottom, and for the case
b, we have :
The last member in the brackets is always very small in value
as will be seen by reference to the table in the preceding sec-
tion. It can therefore generally be neglected, when we have
for both cases:
i=7i(44) ......................(374)
The detailed determination of the forces tx and /2, need not
be gone into here, we have for both cases :
t = y R[h =F /) — -f = 7 — (h \ f) ■ ■ ■ (375>
There is also a third force u, acting upon the rim of the
spherical bottom in the direction of a great circle at right
angles to the plane of the drawing, for which we have per run-
ning foot:
u = y—(h =f/)...................(376)
and finally for the crown of the curve, where the force u0 in a
great circle is :
These formulas will be somewhat simplified if we take the
height Hy of the wetted portion of the cylinder, whence
h = H dz f. This gives:
= y—(	«H. II 1* ~\ * +1
2 V 2 J	2 V 2 J
R	„ R
“ ~yT	u0 = y~ {H dzf)
(378)THE CONSTRUCTOR.
263
These are the necessary formulae for the calculations of
spherical bottoms. The following points are to be noted :
i. For the convex bottom (Form. a) uQ has the greatest value,
that is, the stress must be calculated for the deepest point if «Jj,
is to remain constant; 2. For the concave bottom (Form. b) t
has the greatest value, and must be used to determine ; 3.
The sapporting rim should be capable of sustaining s, if the
shell is to be free from any stress due to the bottom of the
tank.
The determination of is the same as before.
If we divide the values for uQ and by 12, we get the stress
per running inch, and by using the weight 0 of a cubic inch of
the liquid and taking R in inches, we have for the convex
bottom:
12 hf (7
R ~	2 5i
H+f
12 o-e—
2
(379)
and for form d;
A
R
12 G
2S,
(383)
__ 12 G H
R ~~	2 s
(384)
The conical form of bottom, as will be found upon compari*
son, requires about 40 per cent. more material than the spheri-
cal, but as will be seen, its use under some circumstances is
advisable.
Instead of using a complete tone, the bottom may be made a
truncated cone, the tank being formed of two concentric cylin-
ders connected by a ring-shaped bottom, as in Fig. 1102.
and for the concave bottom :
12 hf a
6i
'R
2 Si
12 G

2 Si
(380)
[Fig. iioi.
Fig. 1102.
These may be made either projecting inward or outward.
Following the same line of investigation as in the previous
cases we have for case e:
G = y E(D0'-D*) H - 1 f ~r {D0 — D) H {D +
4	4
E(d0-D))
If the bottom is made conical, projecting either within or
without as in Fig. 1101, the height of the cone being/*, we have
for the weight of the body of liquid •
and taking the component as before in the direction of the
angle of the cone, we have:
and for case f:
G = yT-(D0*- D») H- r/^ {Do~D) (D + -f (R-D))
4	4	3
This gives for case e :
and for case f:
~iZ(3
0’+§-]}
(385)
5 =
2
_G__
2 cos a
whence:
2
But------is equal to the radius R of a sphere inscribed within
cos a
the cone ; whence we have :
s
(381)
D_R_
Dq 2

Do
D
(38h)
in which R is the radius of the sphere inscribed within the
truncated cone.* The forces t and u are obtained in a similar
manner as before.
The subject of. truncated conical bottoms will be discussed
again.
We have for the weight c, of a cubic inch of various liquids :
Water.................................0.0361	lbs.
Petroleum.............................0.0289	lbs.
Tinseed Oil, at 120 C. = 54° F........°-°339	lbs.
Bisulphide of Carbon, at o° C. = 320 F. 0.0459	lbs.
Glycerine, at o° C. = 320 F...........°-°455	lbs.
Beer, at o° C. = 320 F................0.0372	lbs.
Alcohol (absolute), at 20° C. = 68° F. . 0.0286 lbs.
We also have for t the same value as for uy and
t=u — y ~ H ..........................(382)
In the construction of tanks, it is necessary also to consider
the peculiar properties of the various liquids. For alcohol no
packing should be used in the joints, the tightness only being
secured by caulking the riveted seams
For the inverted hanging cone bottom, form c, the greatest
of the three forces is sf while for form d, in which the cone pro-
jects into the tank t — uy is the greatest, and we use in practice
for form c;
* If D0 be made o, the formulse will become those for complete eones, as
indicated in the dotted lines. The formulae for the weight might also be
symmetrically expressed: the form used has been selected because it makes
H the higher of the two walls, which is more convenient in numerical cal-
culation.264
THE CONSTRUCTOR.
i 357-
COMBINATION FORMS FOR TANKS.
In the forms of tanks already described the force s sin a
acts either to press the supporting ring inward or outward in a
direction radical to the axis, according as the forms at c, e> or
bf dy fy are used. This circumstance lends ltself very fortu*
nately to Prof. Intze’s method of construction, since by com-
bining both forms in one bottom the forces may be made to
equilibrate each other and thus relieve the supporting ring from
ali radial stresses.
Fig. 1103.
This idea may be carried out in many ways, as by combining
forms d and fy Fig. 1103 ay or forms e and b} Fig. 1103 b, or using
all three forms as in Fig. 1103 cy the inner vertical walls being,
in these combination forms omitted.*
The forms shown in the illustration also have the advantage
of reducing the diameter of the supporting ring and hence re-
quiring less extensive foundation walls.
In order that the supporting ring may be free from radial
stresses, the condition :
s' sin a'— s" sin a" = O...............(387)
must be satisfied. This simple equation cannot be briefly
solved numerically, hence au example is here giveu of its
application.
Example.—Given a water tank of the forni and dimensions of Fig. 1104, th
radius of curvature ofthe bottom being R". The first meniber of the equa
tion belongs to the outer, and the second to the inner portiou of the tank
For the first meraber we have for s\ from (385); DQ = 12, D = 4, H = 6,/=*
2.4, whence tan or = —=1.667 = tan 59°- This gives sin « ' = 0.8572, and
24
cos a *= 0.5150, and
R'
0.5 D
sin a’
—2— = 3.8?3 and
0515
4' sin a = 0.8572 y X 0.5	{	[ {^Y
T/[’ (1oY-{%)-'}} ” 0 55» (y X o-s D
(tt) (8X«-o.8Xh) =
* Combination forms of this sort have been patented by Prof. Intze, (Ger-
man Patents, No. 23,187, 24,951 and built for oil, water, gas, etc., at the works
of F. A. Neumann, at Aachen.
(y X 0.5 D) 0.8572 X 0.323 (48— xi.*, —
(y X 0.5 D) 0.2769 X 36.8 = 10.19 (y X 0.5 D).
For the second member we have from formula (378):
s” sin a' = sin a" y 0.5 R" {H— 0.5 f")
in which both R" and a" are unknown, hence we introduce /3" and have:
s" sin a" = y cos /3" R" (3 — 0.25 R" (1 — cos /3") ).
Introducing these into the equation of condition, we get:
10.19	X 0.5 Dy — cos fi" R" (3 — 0.25 R" (1 — cos /3") ) y = o
0.5 D
But rjpr sin £ whence:
tan 0" — 3 — °-°5 R? (1 co» P") =Q
10.19
We may obtain a first approximation for fi" by neglecting the second
member of the numerator. This gives tan fi" = —-— = 0.2054 = tan 160 25'.
10.19	y	*
The true value must be somewhat less. Assuming it to be fi" = z6° 20', the
tangent = 0.2930, the sine = 0.2S12, the cosine = 0.9596. We then have R" =■
sin = ~o~28i2 = 7-11 and 10,19 X °*293° — (3 — °-25 X 7-u X 0.0404) = o
nearly. Numerically this gives;
2.9S6 — 2.928 = 0.058
or, since the weight of a cubic foot of water = 62.4 lbs.,the unbalanced
radial force upon the ring is 62.4 X 0.058 = 3.62 lbs. per running foot, which
is so small as to be unimportant.
The question may properly be asked, as to the stresses upon the support-
ing ring when the tank is not fu 11, that is, when //varies. In answer, it is
true that the pressure on the ring necessarily changes. Suppose H = 3 ft.
We then have for the first member of the equation:
y X 0.5 D X 0.2769 (24 — 11.2) = y X 2 X 0.2769 X 12.8 = 7.088 y
and for the second member
y X 0.9596 X 7-11 ^.5 — °-25 X X 0.404) = 6.823 X 1.4*8 y = 9-743 y.
'■'his gives a pressure of
7.o88 y — 9.743 y = —2.655 X
or 2.655 X 62.4 = 165.67 lbs. per running foot acting from without inwards,
which is large enough to be worth considering. It istherefore important to
base the calculation upon a depth of water which will be usually main-
tained in the tank. The propoitions may also be so made that the forces
will be in equilibrium when the tank is half full, when a greater depth will
cause an outward pressure and a lesser depth au inward pressure.
Tanks constructed on the combination are well adapted for
use with gasholders, the level of the water remaining so nearly
uniform that the supporting ring may be kept free from any
lateral pressure.
3 358.
High Pressure Reservoirs or Accumueators.
The forms of tanks already described are intended to be
placed at such elevation either in buildings, or towers or on
natural elevations that the liquid is delivered through pipes at
the desired pressure.
In this way a water tank with a pump and the necessary pip-
ing forms a storage system, an overflow being provided as a
security against flooding the tank. Systems of oil storage are
constructed also in this manner ; and on a small scale the water
tank stations for railway Service come under the same classifi-
cation. These water stations are usually provided with steam
pumps, although windmills are often used, especially in the
United States.
It is a question whether the required pressure might not be
obtained by the use of compressed air, the tank being closed at
the top and the confined air exerting by its elasticity sufficient
pressure to obviate the necessity of elevating the tank upon a
tower to obtain the necessary pressure.
For high pressure water Systems for operating hydraulic
machinery the use of weighted devices, as suggested long since
by Armstrong, has superseded the open water column, such
devices being generally known as Accumulators.
The volume of such accumulators is generally quite small,
but the pumping mechanism is so efficiently devised as to ena-
ble them to possess a very extensive capacity. The pressure is
obtained by means of a weighted plunger, the overflow being
replaced by a safety valve.
Fig. 1105 shows an accumulator built by C. Hoppe, Berlin*
This is weighted to a pressure of 20 atmospheres, or nearly 300
pounds per square inch. The plunger is 17% in. diameter (450
mm.) weighted with shot which is enclosed in a cylinder. The
plunger is shown in the highest position. When it reaches the
position the lever and connections M M' act to shut off the
steam from the duplex pump, and at the same time the rod 5
* All dimensions in the illustration are in millimetres.THE CONSTRUCTOR.
relieves the safety valve. When the use of the water causes the
plunger to sink, the steam is turned on and the pump starts.
If the pressure should be suddenly released by the bursting of
Fig. 1105.
a pipe, the sudden drop is received by heavy beams, and at the
same time the stop Pf strikes the lever P and checks the water
flow in time to moderate the shock.
An accumulator for very high pressures is shown in Fig.
1106.* This is designed by Tweddell for use for operating riv-
eting machines, punches and similar tools. The plunger c> is
stationary; the cylinder d, sliding upon it, weighted with rings
dl of cast iron. In the lowest position the cylinder rests upon
vertical buffers of oak. The water is delivered under high pres-
sure at Ht while the water is taken off for use through suitable
valve gear at A; the safety valve is at b'. The plunger is of
the differentia! variety similar to thoseshown in Fig. 977 £, and
Fig. 981 b. The difference in diameter between the two por-
tions of the plunger is the space to be filled by the entering
water, the small annular area bearing the total weight, thus
giving a very high pressure per square inch. The pressure at-
tained when the cylinder is stationary is about 100 atmospheres
(1420 pounds), but experimental investigation has shown that
when the weighted cylinder is permitted to descend rapidly the
pressure reaches as high as 193 atmospheres, (2740 pounds), so
that it is worthy of note that the attainable water pressure in
such devices may reach double the statical pressure.
i 359-
Steam Boieers, Various Forms.
Steam boilers may properly bejconsidered as reservoirs for
vapor of water, while at the same time they serve as generatois
of force by the application of heat. The pressure is produced
by the heat, the feed is effected either by a pump, as Fig. 975 d,
or injector, Fig. 971. The overflow is represented by the safety
valve, and the observation of the water level is provided for in
a variety of ways.
The forms used for steam boilers are very numerous; the
great variations of size, the varying conditions of locality, and
* See Proc. Inst. C. E. Vol. EXXIII. 1883, p. 92.
265
the efforts to attain compactness, having led to a vast number
of modihcations of the original simple forms.
The various boilers used in Germany may be reduced to
Fig. 1106.
eight principal classes, examples of which will here be
given.
1. Plain cylinder boiler, Fig. 1107, usually placed in the hori-
zontal position, and now principally used in iron works whera
the waste gases from the furnaces are used.
2. Cylinder Boiler with Heater, Fig. 1108. The cylinder has266
THE CONSTRUCTOR.
added to it a “heater ” or auxiliary cylinder placed beneath and
forming a part of the boiler, beiug entirely filled with water
and surrouuded by heated gases. Besides the usual form, there
are HenschePs in which the heater is placed at right angles to
the main boiler, and the vertical form.
tnbe to produce a circulation by the difference of temperature
between the inner and outer tubes. Boilers with large and
small cross water tubes are shown at b and c.
A ninth group might be formed of special combinations of
the eight groups above shown.
Fig. 1109.
3.	Tubulous Boilers, Fig. 1109. This class includes boilers
made of tubes 6 inches in diameter and under. Of the arrange-
ment shown, a, is Belleville-s ; b, Root’s ; and c, Howard’s.
Fig. i i io.
4.	Flue Boilers, Fig. 1110. Flue boilers are constructed with
internal f-'es entirely surrounded by water and containing the
furnaces, fire and heated gases. The Cornish
boiler, a, is made with a single flue, and the
Lancashire boiler, b, with two flues.
5.	Flue Boilers with Cross Tubes, Fig. mi.
This form, also known as the Galloway boiler
is constructed with water tubes Crossing the
flue at various points.
6.	Plain Tubular Boilers, Fig. 1112. These
are made writh tubes of 6 inches or less in
diameter, through which the heated gases
pass. The tubes are lap-welded or seamless, and a distinction
is made between direct and return tubes.
Fig. ii ii.
Fig. i i 12.
7.	Fire Box Tubular Boilers, Fig. 1113. A fire box consists
of a box forming a part of the boiler containing the furnace and
surrounded with water. These boilers are also made with
direct and return tubes. These are made either with vertical
tubes as at a, or horizontal tubes b, and are much used for loco-
motives and portable boilers.
8.	Fire Box Boilers with Water Tubes. Fig. 1114. Of the
forms shown, a, is a boiler fitted with Field’s tubes. These
tubes are closed at the lower end, each containing a small inner
a
Fig. i i 14.
In England and America a different classification is made,
the boilers being divided into two great classes, those which
consist of a large shell, with the necessary auxiliary parts, and
those composed of numerous small elements, the number of
elements being governed by the size of the boiler. These two
classes are known as shell” boilers, and “ sectional ” boilers.
The third group shown above, consists of sectional boilers. A
popular form in many countries is the Harrison boiler, com-
posed of small spherical elements of cast iron. The relative
value between shell and sectional boilers is a question not yet
entirely settled. The latter form is incapable of destructive ex-
plosions, such as may occur with shell boilers containing large
volumes of water. Sectional boilers are also adapted for very
high steam pressures, but have the defect in many cases of pro-
ducing moist steam.
The latest police ordinances in Prussia, which are similar
to those of Austria, distinguish between “dwarf” boilers
and ordinary boilers. The former are boilers of small volume,
less than 18 cubic feet ()4 cubic meter) capacity, these together
with sectional boilers being permitted for small private indus-
tries.
i 360.
Boiler Detaies Subjected to Internae Pressure.
The walls of steam boilers are subjected to varied and some-
times complicated stresses greatly dependent upon the method
of construction. It will only be practicable here to discuss the
ordinary forms, first takiug the parts which have to resist the
internal pressure.
a. Cyli?idrical Details.
The Prussian ordinance relating to steam boilers, used the
formula of Brix, for cylinder boilers subjected to internal pres-
sure :
D /	000.3 a x
<*=—(*	— ij + o.i..............(388)
in which b and D are in inches and e is the logarithmic base =
2.71828, «being the pressure in atmospheres. This is closely
approximated by the simpler formula :
6 = 0.0015 a Z? 0.1..................(389)
The French formula is much the same but gives a slightly
greater thickness:
6 — 0.0017 a D 0.12..................(390)
only % °f this value being used in locomotive practice. On
account of the large constant added to provide for deterioration
all three formulae must be considered as empirical. At present
there are nearly everywhere government enactments which
prescribe the method of determining the thickness of steam
boilers and regulate by law the limits of construction.
In most cases the boilers must be subjected to a test pressure
which may reach double the working pressure.
The stress existing in the longitudinal seams of a cylindricalTHE CONSTRUCTOR.
267
boiler shell may be obtained witk sufficient accuracy from
(324), as:
2
D_
T
(390
p being the pressure in pounds per square inch, and D and <5
being in inches. If we calculate d from (389) and determine 6*
from (391) we have tlie following results
a =	4 = 60 lbs.		7 = 105 lbs.		10= 150 lbs.		113= J75 lbs.	
D	S	S i	6	S \	S	6* i	6	5
24	0.24	3000	°-35	3600	0.48	3750	00 d	3700
36	0.31	3500	0.48	3900	0.64	4000	0.80	4000
42	o-35	3600	o-54	4000	o.73	4300	0.92	4000
72	0.43	5000	IO 00 d	4400	1.18	4600	1.50 1 “	4200
This table shows that the formula gives for large diameters
and beavy pressures, thicknesses which are excessive, and with
quite moderate stresses. The stress at the riveted seam will be
greater, and from \ 59 we have for the stress in the perforated
piate :
for single riveting S/ —
S_
o'
for double riveting =
5
(392)
in which, if d, is the diameter of rivets and a, the pitch,
a
Even with this increase the stresses fall below the values
which good boiler piate should properly bear. In practice,
smaller values are often used for 6 than are given by (389)
especially since mild Steel came into use for boiler piate.
The stress which comes upon the rivets, according to § 59, is
greater than that upon the perforated portion of the piate.
This, however, should be considered in connection with the
fact that rivets are generally made of a stili better grade of iron
than the plates.
At the present time the disposition is apparent to break loose
from set rules for the thickness of boiler shells. Careful
designers aim more and more to investigate each case for itself
and endeavor to adapt both design and material so as to obtain
at the same time.the greatest strength and economy. The more
recently designed ocean steamers are fitted with boilers as
large as 16 feet. in diameter, operated at pressure from 160 to
250 pounds pressure. The older formulae canaot be used for
such extreme values, and every resource of the art must be used
to reinforce the streugth of the plates and the riveting. The
method of group riveting, (| 57) is here found of value and is
already used to some extent.
Longitudinal Seams.
For all large steam boilers the longitudinal seams are double
riveted. For plates T\// tkick and over a modulus <j>/ = 0.76 to
0.73 is obtained which corresponds to a ratio of S'2 : S of 1.32 to
1.37. It is more and more made a point of importance that
these joints shall not be exposed to the direct action of the
fi re.
A construction especially intended to meet this point is
shown in Fig. 1115 in which the entire shell is made of two
sheets, the lower sheet comprising obout f of the entire cir-
cumference.*
Another method which bids fair to become very important,
is to weld the longitudinal seams, this being more and more
and more used for large boilers. The welding is accomplished
* Boilers of this sort have been made at the Erie City Iron Works, Erie,
Pa. See Trans. Am. Soc. Mech. Engrs. Vol. VI., 1884-5, P- uo. Scheffler, A
New Method ofConstructing Horizontal Tubular Boilers. The first boiler
was 16 ft. by 60 in., thick, of mild Steel, 60,000 lbs. ultimate strength,
30,000 lbs. proof strength.
either by furnace beat or by water-gas burners, or aS more re-
cently by electric wrelding by the Bernados’ method.
Fig. 1115.
In Fig. 1116 is shown the cross section of a marine boiler,
constructed by H. C. Stiilken, of Hamburg, the two longitudi-
nal seams being welded.* Both seams are reinforced by double
riveted flaps the strength of the plates being reduced by the
rivet holes.
The joint, however, is preferable to a lap joint and needs no
strengthening. The pressure in this boiler is 180 pounds. The
strength may be calculated as follows ; the diameter being 76.5
ins. and the plates in. thick. From (391) we have : .S =
180 x 76.5
2 X 0.875
— 7868 pounds.
For the double riveting »in both
flaps the pitch a2 is 2.9 ins. and rivet diameter d — J4 in. This
gives for the modulus of efficiency b'2 = ——— c.7 atl(^
the stress in the perforated piate in the longitudinal seams is
7868
—— = 11,240 lbs. The thickness of plates according to for-
mula (389) would be 0.0015 X 12 x 76.5 + 0.1 = 1.5 ins., in-
stead of ins.
A third method of construction which may become impor-
tant is to construet the shell in a single piece of mild Steel by
the Mannesmann processof rolling. This method would be best
of all, since the question of the strength of the riveted seam
would be entirely eliminated, and tbe kigh elastic limit of the
material would permit correspondingly liigh working stresses.
At the present time, however, the Mannesmann rolling mills
cannot make tubes over 24 inches in diameter.
* See Zeitschr, D. Ing., 1886, p. 109.268
THE	CONSTRU.
Circumferential Seams.
The cross section of the boiler skell, when the head is fast to
it, is subjected to a force Z?2 p = S2tt D <5, in which S2 =
—pD
____; that is half as great as the stress S, in the longitudinal
d
seams. For this reason it is deemed necessary to use only
single riveting for the circumferential seams. It will also be
shown hereafter, that the cross section of the shell can be re-
lieved of this load.
Openings i?i the Shell.
soon abandoned on account of the demand for increased heat'
ing surface and small content. The spherical form is, however,
well adapted for units for sectional boilers.*
For spherical ends of cylinder boilers, as in Fig. 1107, and for
the heads of domes, and auxiliary drums, we have for the thick-
ness, R1 being the radius of the sphere :
<*! = ^	.......................(394)
which gives, when Sx = S the same value for the thickness d,
as in the shell when Rx = D. This latter condition cannot
always be fulfilled since the curvature of the boiler head is
usually controlled by the dies with which the press is provided.
The openings for the steam dome and manholes weaken the
boiler, and in some instances explosions have been caused by
cracks radiating from such openings. All such openings should
be carefully reinforced by riveting on rings of wrought irou or
preferably Steel, as shown hereafter in Fig. 1118. The size of a
manhole openiug should be about 12 by 18 inches, and when
practicable the short axis of the oval should be placed length-
wise of the boilers.
b. Spherical Details.
A sphere of the diameter Dx with an internal pressure /, will
be subjected to a force — Dx2 pf which is the same as already
4
found for the cross section of a cylinder, and one-half that on
the longitudinal seams. The thickness, therefore, need be only
half so great as that of a cylindrical shell of the same diameter,
i. e., D — D.x If, however, both vessels are to have the same
content we must have Dx D. If the cylindrical shell is made
7r	7r
with flat heads its content will be — D1 L — — Ds
4	4
and the spherical vessel will have a content = — Dx3; hence
6
we must have D\ = — Dz
1	2
For the thickness of metal we have :
T°P TA/ '
<5 = -3.-and <5,= -^- -
and for the respective surfaces :
F = 7r D L -|-	£>2, and Fx = 7r D2x.
Assuming the heads of the cylindrical vessel to be made the
same strength as the shell, we have for the material required
for each case:
Making ^ and putting for £>3X its value

we
get:
L_
5A _ 3 D
F6 4 L 1
R + 2
(393)
for the ratio between the amount of material required for
spherical and cylindrical vessels. We have for :
II oh	1 1	2 3
FUx Fi ~	O.50 O.56	0.60 0.64
showing that the spherical vessel
form.
The earliest boilers were made
456	00
0.67	0.68 0.70	0.75
is in all cases the lighter
in the spherical form, but
The head is usually joined to the shell by being flanged or
turned over around the edge in the flanging press, thus enab-
ling a joint to be made as at a, Fig. 1117 ; or it may be made
with a ring of angle iron, as at b. Here the circumferential
force, as considered in § 355, may be taken into consideration,
especially the radial component s sin «, since this acts to draw
the shell inward. It is, however, hardly necessary to take this
into account as theflauge of the head reinforces the shell amply
at this point.
c. Flat Surfaces.
Unstayed flat surfaces can only be used in boilers of small
dimensions, as already shown in § 19, and should only be used
for lieads of steam domes, auxiliary heaters, and the like.
Where extended flat surfaces are used, it is necessary to adopt
some method of staying; or in other words to subdivide the ex-
tended surface into supported portions small enough to be of
ample strength and at the same time of moderate thickness.
A number of methods of staying flat surfaces are in practical
use, those most generally employed being shown in Fig. 1118.
b
Stay bolts, such as shown in Fig. inSa, (see also \ 61) are
used for parallel surfaces which are near to each other. Those
shown at a are made with nuts instead of riveting the heads
as is sometimes done. Flat surfaces wThich are farther apart
are secured by anchor bolts, as shown at b ; these are practi-
* The Harrison Boiler, the pioneer of modera sectional boilers, is com-
posed of spherical units. Trans.THE CONSTRUCTOR.
269
cally long stay bolts. These are shown reinforced by large
riveted washers under the nuts.
Stay bars, as shown at c} are used for staying crown sheets of
fire boxes in marine and locomotive boilers. Stay tubes, such
as shown at dt are used to strengthen tube sheets. These are
heating tubes about X to T5S in. thick reinforced at the ends and
screwed into the tube sheets. Gusset plates e} Fig. b, are used
to stay flat heads to the shell, and are used both in land and
marine boilers.*
2 361.
Boieer Feues Subjected to Externae Pressure.
The stresses which appear in the case of a boiler flue subject-
ed to external pressure are similar to buckling stresses npon
eolumns, rods, etc., since beyond a certain increase in pressure
when a slight departure from the true cylindrical form occurs
a sudden collapse follows. The smaller sizes of flues used in
the ordinary tubular boiler possess ample strength against col-
lapsing, but for larger flues such as are used in Cornish and
Lancashire boilers the question of strength to resist collapsing
must be considered. The experiments of Fairbairn have demon-
strated that the length of the flue has an important influence
upon the resistance to collapsing, practically being inversely as
the length of the flue, or rather as the distance between the
points at which the flue is reinforced against external pressure.
stituting these in the formula it will be found if the flue is safe
against collapsing.
Example.'—In a Cornish boiler intended to work at 37pounds pressure,
the dimensions are / = 25 ft., D = 23 ins., 5 = 0.25 ins., the flue being made
with lap joints.
From (397) we have :
p — 368,000
0.25 3 /	0.25
23 V 25 X 23
= 303 lbs.
at whmh pressure the flue actually collapsed. It is evident that should the
thickness of the flue be only slightly reduced by corrosion, etc., an explo-
sion might readily follow.
A method of increasing the safety without using a greater
thickness of metal in the walls of the flue, is to reinforce it by
stiffening rings, thus practically reducing the length /, as noted
by Fairbairn.
Fig. i i 19.
Two forms of stiffening rings are shown in Fig. 1119, a being
Adamson’s and b, Hick’s. The first form is the more difiicult

Fig. 1120.
Fairbairn deduced from his experiments for the collapsing
pressure of such flues :
rf2.19
P' = 806,300-^—....................(395)
in which p' is the pressure in pounds per square inch, D and b
are in inches, and l is the length of the flue in feet.
If the dimensions are given in millimetres and p' is the pres-
sure in kilogrammes per square millimetres, this becomes :
d
100 p' = a' = 367,973 -j-g-...................(396)
Fairbairn’s experiments have been discussed more recently,
with a view of deducing a formula which should be more con-
venient to use.| The results of Dr. Wehage in connection with
later experiments, J give the following formula :
• (397)
in which the upper coefficient is to be used for flues made with
lap joints, riveted ; and the lower coefficient for flues in which
the joints are made with flap plates riveted on.
This formula gives results approximating very closely to
Fairbairn’s most important experiments. It is best used by
selecting the desired dimensions for D, l and b and then by sub-
* The question has been raised as to whether it is not best to stay only
one boiler head to the shell and then tie the other head tothe first by means
ot a number of parallel anchor bolts, thus closing one end of the shell in a
manner similar to the cylinder of a hydraulic press, and relieving the shell
of any stress due to the pressure on the heads, and permitting the use of
packing to make the joint tight. The author recollects such a construction
having been used in a portable engine and boiler but without knowledge of
any further attempts of the sort.
f See Grashof, Zeitschr. D. Ing. 1859. P- 234, Vol. III.; also Love, Civilin-
genieur, 1861, p. 238, Vol. VII., discussed by the author for the Hutte Society
in the Berliner Verhandlungen, 1870, p. 115.
X See Engineer, Vol. 51, 1881, p. 426, also Dingler’s Journal, Vol. 242,1S81,
p. 236.
of construction, but possesses the advantage of removing the
rivet heads entirely from the action of the fire. This form of
joint and stiffening piate is also frequently used in other parts
of boilers for the sole purpose of avoiding the action of the fire
on the heads of the rivets.
The use of corrugated iron for boiler flues enables great
strength against collapsing to be obtained. Fig. 1120 shows a
boiler writh corrugated flue, the lengths being welded together.
This boiler is made by Schulz, Knaudt & Co., of Essen, and is
86.6 inches in diameter (2.2 metre). Notwithstanding the con-
structive difficulties the use of the corrugated flues is constantly
increasing. In England corrugated flues are made by the in-
ventor, Sampson Fox & Co., of Eeeds. The depth of corruga-
tions is usually about 4 inches.
Corrugated fire boxes have been used in locomotive boilers,
Fig. 1121, showing Kaselowsky’s fire box. In this form the
1
f / g / ___ H 1 f > Ar	IX ' \ \y i 1	1—	i		NA/NA/WVi i		
\\ "v*		✓ II ] jjy			—1—	/	
Fig. i i 21.
stay bars to support the crown sheet, and the stay bolts at the
sides are entirely omitted. The cross section shows the method
of supporting the boiler by a cross beam below the grate bars.
The corrugated flue is attached to the boiler by a riveted joint,
either by flanging as in Fig. 1121, or by the use of angle iron,
as Fig. 1120.
Tubes of small diameter are treated practically as single I10I-
low rivets, the ends being inserted into holes in the tube sheets270
THE CONSTRUCTOR.
and expanded by an expanding tool, the ends being riveted over»
as shown in Fig. 1122 a.
In many establishments, as for example, the Esslingen loco-
motive works, the tubes are fitted with hard copper ferules
which stand the expanding and riveting better than tubes of
Steel or iron. The form of tube shown in Fig. 1122 b, is rein-
forced at the ends, and one end made conical, thus enabling
ol'd tubes to be more readily removed and replaced. This con-
struction is used by Pauksch & Freund, of Eandsberg, in Ger-
many, and by various French builders since 1867-
§ 362.
Future Possibieities in Steam Boieer Construction.
The discussion of the preceding sections has necessarily been
limited to a few constructive details, since a complete treatment
of such an extensive subject requires a special treatise. It is
proposed here to give only a broad general view of the subject
of boiler construction in its present and prospective condition.
The descriptions in the preceding sections and in the previous
chapter on riveting show that the art of boiler construction has
made little or no advance during the past twenty or thirty years,
although there is reason to believe that there is ample room for
improvement, especially in the matter of greater economy of
fuel. In the author’s opinion there are four points in construc-
tion which deserve the closest attention and to which efforts at
improvement should be directed, while in other directions also
senous wastes of force appear.
1.	Expenditur e of Materiat.—As already shown in § 359, the
expenditure of material is considerably greater in the present
forms of steam boilers than if the spherical form were more
generally used. It is questionable to what extent the spherical
form may be made practicable, but the possibilities in this direc-
tion have not been exhausted, at least for certain purposes, for
example, for boilers used solely for heating purposes. The
spherical vacuum pans only serve as reminders that this oldest
form of boiler (i. e.y that used with Newcomen’s engine), is no
longer used ; but it may be only a question of the increase in
the capacity of the flanging press ; or, in other words, of the
increased command over the working of iron and steel, when
the spherical form shall again be used.
Another point in the question of material, is the subject of
riveting. One of the greatest sources of weakness in steam
boilers is the reduction in strength due to the presence of riveted
seams. Even if the very best material obtainable is used for the
rivets, the reduction in strength for single riveting is about 40
per cent., and for double riveting, 25 per cent.* This weaken-
ing is unimportant so far as the circumferential seams of cylin-
drical shells are considered, but is well worthy of consideration
in connection with the longitudinal seams, especially since it
concems the largest and heaviest part of the boiler, i. e., the
main shell. It is for this reason that attempts have been made
to weld the longitudinal seams.
The meagre resuits which have been obtained for welded shells
subjected to internal pressure, as compared with welded flues
for external pressure, may be seen from the case shown in Fig.
1116. The welded seam is there reinforced by a riveted flap,
thus reducing the strength practically to that of an unwelded
seam. Experimental results with welded joints in the testing
machine, justify this distrust of welded seams, and do not war-
rant the idea that the weld is equal to the full strength of the
piate.
This leads to the remark that the coming boiler shell must be
without longitudinal seams of any kind, either riveted or welded.
Heating flues for external pressure are already made seamless,
and the Mannesmann process produces seamless tubes adapted
for internal pressure, and of a grade of material far superior to
that heretofore used, as experimental researches have demon-
strated. If this process can be so extended as to be made avail-
able for boiler shells, an economy of at least one-third of the
material can be obtained.
2.	Combustion.—The subject of economy of combustion of
the fuel is even more important than that of material. In the
♦When tbe rivets are made of no better material than the plates, the re-
duction for single riveting is about 53 per cent., and for double riveting
about 41 per cent. Triple riveting, as shown in Fig. 155, is too expensive to
come into general use.
general description given in the preceding sections it will be
seen that the present methods of firing are ali based upon the
principle of exposing portions of the boiler to the direct action
of the fire and of conducting the products of combustion into
contact with various portions of the boiler, arranged to act as
heating surface. This means that in nearly all cases boilers are
independently fired. For a long time the advantages of this
System have been doubted. It is manifestly impossible for a
complete combustion of the gases to be effected when they are
almost immediately brought into contact with surfaces which
have a temperature of 1200 to iSoodegrees lower than the flame.
The production of smoke and soot, that is, of unconsumed fuel,
is the necessarv resuit of these conditions, and hence a great re-
duction in efficiency. This subject has been actively worked
over, and an almost endless variety of furnaces and Systems
has been proposed. The true method of solving the problem
appears to have been first discovered by Frederick Siemens
(Dresden), and for a number of years he has been engaged in
developing the practical applications of his researches. f
The previous methods of firing were based upon the idea of
bringing the flame into direct contact with the surface to be
heated, but since about 1879 the method of construction, espe-
cially in glass furnaces, open hearth steel furnaces, smelting
furnaces, etc., has been to utilize the radiant heat from the
arched roof of the furnace, and to economize the heat of the
escaping gases in the regenerator. An economy in the use of
the heat of as much as 80 to 90 per cent. has resulted. This has
been followed by a stili more marked separation between the
two principal periods of combustion, and by the application to
steam generators where such a high economy cannot be expected,
although a saving of about 25 per cent. has been shown in actual
practice.J
It is therefore strongly recommended to use such furnace con-
structions as shall not bring the direct flame of the fire in con-
tact with the heating surface of the boiler, but to use radiating
surfaces and also to conduct the highly heated but fully bumed
gases through the flues, both of which can be accomplished in
various ways.§
The application of the principle to stationary boilers is not
difficult, and experiments have shown that it may also be suc-
cessfully applied both to marine and locomotive boilers. In all
cases it has been demonstrated that the fuel should be burned in
a combustion chamber lined with refractory material, and the
discharge of the heated gases retarded by a fire brick bridge or
screen before coming in contact wfith the boiler. It will be seen
from the preceding, that by using the Siemens’ method instead
of the older method of burning the fuel directly in the boiler,
an economy of about 25 per cent. can be obtained, and this fact
should always be kept in mind in future designs.
3.	Heating Surface.—The third point concems not so much
a variation in construction, as it does the lack of knowledge of
the fundamental principies, this subject having been much less
fully investigated than other portions. Recent investigations
show conclusively that the axiom that the heating surface is a
magnitude proportional to the desired efficiency of the boiler,
cannot be sustained. It is evident that there must be a very
considerable difference in the heating value of portions of the
surface which are at greatly different distances from the fire. A
very high temperature of the gases- at the beginning, and a
comparatively low temperature near the end, must mean a rapid
formation of steam near the fire and a weak production over
fThe following list will serve for those who aesire to refer to the original
and fundamental publications upon this subject:—Friedrich Siemens, Heiz-
ver fohren mit freier Flamment faltung, Berlin, Springer, 1882; Siemens’
Regenerativofen, Dresden, Raraming, 1854; Vortragvon Friedrich Siemens
uber Ofenbetrieb mit ausschliesslicher Benutzung der strahlenden Warme
der Flamine, Gesundheitsingenieur, 1884; Vortragvon demselben uber ein
neues Verbrennungs-und Heiz-system, Busch, Journ. f. Gasbeleuchtung,
etc., 1885; Vortrag von demselben in der Ges. Isis in Dresden uber die Dis-
sociation der Verbrennungsprodukte, Dresden, Blochmann, 18S6; Vortrag
von demselben im Sachs. Ing. u. Archit Verein uber die Verhiitung des
Schornsteinrauches, Civ. Ing. Bd., 32, Heft 5,1886; Vortrag vom demselben
im Bez-Ver. D. Ing. in Eeipzigam 8 Dez. 1886uber den Verbrennungsprozess,
2 Aufl , Berlin, Springer, 1887; Vortrag von demselben, gehalten in Ham-
burg im Ver. D. Gas-und Wasser fachmanner iiber Regenerativ—Gasbrenner,
etc., Dresden, Ramming, 1887; Ueber die Vortheile der Anwendung hocher-
hitzter Euft fur die Verbrennung, etc. 2 Aufl., Berlin, Springer, 1887.
X For example, a test by K. H. Kiihne & Co., of Dresden Eobtau on Feb.
16, 1884, showed a gain of 26 per cent. due to the substitution of a Siemens
furnace for one of the usual kind ; the conditions of drafit and cleanness of
flues being alike in both cases.
§ Two methods have been described by Dr. Siemens, both of which have
been applied by him to flue boilers. In the first, the combustion of the fuel
takes place upon a grate in a combustion chamber which is directly over
the grate. A bridge wall of fire brick is placed about half the length of the
grate further back, and beyond this are two ring shaped screens of fire brick,
which are so placed as to direct the products of combustion toward the axis
of the boiler flue; after passing through the flue the gases retura about the
outside of the shell and are then sufficieutly cooled to be permitted to pass
over the portions of the shell unprotected by water on the way to the chim-
ney. In the second method the fuel is burned to gas in a gas producer
separately constructed from the boiler, and tbe gas mixed with heated air
and thus delivered to the boiler flue, where it follows thesaure course as in
the first case.THE CONSTRUCTOR.
271
distant portions of the surface. It has been shown that in some
instances the heating surface of one and the same boiler may be
reduced one-half without causing any reduction in the steam
production. The usual method of proportioning the heating
surface in all kinds of boilers appears to be based upon previous
results with similar forms, and hence is often one-sided and
unsuited for systematic investigation. A new departure in the
discussion of this important subject has been made by the chief
director and engineer of the Swedish railways, Mr. F. Almgren.
He has made the subject of the proportioning of heating sur-
face the object of a series of experiments extending over a
number of years, and has placed the matter upon a much higher
plane of investigation than heretofore. The practical results
are of much importance, and in advance of the publication of
the whole the following general discussion has kindly been
placed in the author’s hands by Mr. Almgren, and is here given
in his own words.*
PRACTICAL RESEARCHES UPON LOCO MOTI VE BOILERS WITH
SMALL TUBES.
BY F. ALMGREN.
“According to the investigations of Geoffroy, as given by
Couche,f the amount of steam produced by tubular heating sur-
face depends upon the volume of heated gases passing through
the tubes per hour. The heating surface under experiment con-
sisted of portions 0.9 metre long of tubes, the total length of
which was 3.6 metres long each.
‘ ‘ I have found that the volume of gases may be considered as
a function of the length / of the tubes, the latter being con-
sidered as a variable, according to the following general expres-
sion ; in wThich i is the number of tubes, and L the number of
heat units given off by each tube per second.
a	,	b	,
~r = 1 + T........................(398)
‘ ‘ In this formula a and b are constants which depend upon
the mean temperature Te of the gases, upon the temperature <5
of the water, and upon the weight i G of the gases passing
through the tubes per second.
‘ ‘ As the resuit of a series of experiments I have found these
constants as follows:
a — 0.357 i G (Te — <5) \	,	^
b = 7.15 G°™7	/.............'399)
in which G is the mean weight of gases or products of combus-
tion for one tube. For the number of heat units L given off by
a single tube of a set, the following expression is given :
Z =	.....................(400)
i + lf-Gc«7
“In order to show the utility of these formulae, a table is
here given of the results of twenty-one experiments upon a
locomotive boiler, the walls of the fire box having been made
non-conducting by means of brick-work. A second table is also
given to show the great advantages resulting from these experi-
ments. The quantities given in the table are as follows :
i — the number of tubes.
G = the weight in kilogrammes of the products of com-
bustion passing through each tube per second.
Tr — b = the difference between the temperature of the smoke
and the water, the former being measured in the
smoke box.
Te — b = the difference between the mean temperature of the
gases in the tubes and the water = Tr — «5 -j-——
0.24 G.
Le — the mean value of L determined by experiment.
Lb =* the value of L determined by formula (400).
* It has been thought best to leave the formulae and tables in the metric
system, and temperatures in the Centigrade thermometer, also keeping the
French thermal units, and thus retaining the discussion in Mr. Almgren's
own figures, as the principies are equally well shown, and the unity of this
prelimmary presentation thereby retained.— Trans.
f Voie, materiei et exploitation des Chemins de fer, Tome III.
TABLE I.
Eocomotive boiler: pressure 4 atmospheres, tubes of brass, 2.934 metre»
long, 42 mm. diameter, somewhat scaled.
No.		i	G J	Tr — 4	Te —S	Le	Lh
1			0.00713 j	210° c !	901° C	I.184	1.248
2			O.OO60I j	185 “ i	916 “	I.035	1.090
3		110	0.00733	222 “ 1	969 “	I.304	1.370
4			0.00827	230 “ !	1009 “	I-53I	1-570
5 J			0.00900 :	235 “ j	1000 “	I.648	1.700
6 j			0.01795 ;	275 “ 1	1067 ‘ ‘	3.360	3.330
7			0.01871 i	285 “ !	1091 “	3.600	3-520
8			0.01832 :	278 “ |	1115 “	3.660	3.530
9		55	0.01479	290 ‘ ‘	1421 ?	4.OOO?	3.750
10			0.01514	240 “	1221° C	3-510	3.290
11			0.01303	255 “	1312 “	3.300	3.080
12			0.01091 j	235 “	1328 “	2.860	2.700
!3 1			0.00466 j	90 “	682 “	0.650	0.646
14 !■		88	0.00448	95 “	724 “	j O.67O	0.660
			0.00405	95 “	781 “	0.660	0.652
16 J			0.00360	95 “	709 “	0.530	0.530
17 "			0.00586	75 “	462 “	0.542	0-534
18			0.00529	70 “	368 “	O.376	0.388
19	-	110	0.00640	83 “	466 “	O.59I	0.586
20			0.00715	95 “	i 522 “	0-734	0.721
2!			0.00668	90 “	529 “ 1	O.695	0.686
“ Remarks.—Between each set of experiments the boiler was
blown off and both boiler and tubes cleaned. The 110 tubes of
the fourth set were only partially the same as those of the first
set. In the ninth experiment one of the cast iron plugs which
were used to close the tubes not in use was melted out.
“The correspondence between the experimental value Le and
the calculated value L b is very striking. A formula for special
practical cases has also been deduced, being adapted for the
special number of tubes as given in the preceding table, and
without the variation in G and Te which occur in single experi-
mental cases.
* ‘ Equation (400) shows that for a given length l for the tubes,
the production of steam is nearly proportional to the weight of
gases flowing through them, and that it also increases nearly in
direct proportion to the quantity of heat 0.24 G (Te — <$).
This indicat es that for a constant blast opening, the amount of
steam produced by the heating surface of the tubes will almost
exactly equal the amount of steam passing through the blast
nozzle, that is the amount of steam used by the engine. If it is
also remembered that an increase of draft also increases the
temperature of combustion, it will be seen that the tubes and
the blast nozzle of a locomotive boiler bear a most intimate
relation to each other, and that great and sudden variations in
the production of steam occur almost hourly.
“ Now the researches of Geoffroy show that the walls of the fire
box have a much less favorable action. In this portion of the
heating surface the production of steam responds much more
slowly to variations in the draft. The larger the fire box, the
more marked is its action in this respect, and consequently the
less effective will be the blast. Equation (400) shows that for a
given tube length, the production of steam of each tube in-
creases with the increase of the draft, and hence the number of
tubes and consequently the weight of the boiler may be kept at
a determinate minimum, which depends upon the permissible
force of blast and limit of size of grate. The formula also shows
that with a strong draft and high temperature even the latter
portion of long tubes is of excellent steaming value.
“ Since also a given amount of tube heating surface is lighter
and cheaper than the same amount of fire box surface, and since
by the reduction of the latter the products of combustion will
be cooled less and so en ter the tubes at a higher tempera-
ture, it will readily be seen that a material advantage can be
gained by removing that portion of the fire box surface which
is of the least value (that is, the side walls), and adding an
equivalent proportion by lengthening the tubes. As an exam-
ple may be cited the case of a locomotive boiler with 125 tubes,
3 metres long and 45 millimeters inside diameter in which a re-
duction of 7 square metres of fire box surface was made up by
an increase in length of tubes which gave 14 square metres of
surface, the force of draft being 40 millimetres water pressure.
This change removed the expensive stayed fire box walls, which
were replaced by a fire brick lining, and the reduction in weight
and cost amounted to about 700 kilogrammes and 1500 marks.272
THE CONSTRUCTOR.
1 * The latest boilers for the Swedish State Railways have been
constructed with the preceding principies in view as shown in
Fig. 1123. The fire brick lining of the fire box is shown at a> a,
while at b, by are openings for the admission of air, which can
be closed by sliding dampers c. A year’s experience with this
construction has given satisfaction, as the following table shows.
It will be seen that the new forin of boiler produced the sanie
amount of steam per unit of heating surface as the old forni, the
force of the draft and the temperature in the smoke box being
nearly the same in both instances.
“The external length of this fire box is 1.485 metres, and the
internal width is 1 metre. The diameter of the shell is 1.103
metres, with 144 tubes, 45 millimetres inside diameter, and the
fire brick lining is 74 millimetres thick.
“TABLE II.
A. Dimensions.
	Tubes.			Heating Surface.		
Boiler.	Length.	Diame- ter.	No.	Tubes.	Fire Box.	Ratio.
Old Style. New ‘ ‘	3.111 m. 3*305	46 mm. 46 mm.	184 102	77.28sq. m.1 50.83 “ “	7.82 sq. m. 2.19 “ “	9-9 23.1
B Performance.
1 Evapora tion per sq. meter per hour	Draft Pressure in millimetres of water.		Temperature in Smoke Box.	
	Old Style j	[ New style	Old Style.	New Style.
24 kg. 30 “	20 mm.	24 mm.	310 °c	315°^
	30 “	i 35 “	340°	340°
37-45 “	40-50 “	50-60 “	410°	395°
55 “	80 “	90 “ 1	470°	470°
“A patent has been applied for by Herm. Von Storckenfeldt
for the construction shown in Fig. 1123, and made from my cal-
culations and directions.”
The preceding brief description shows the nature and import-
ance of Almgren’s researches and appears to forni a starting
point for a change in methods of locomotive boiler construction.
Further investigations may develop a theoretical foundation
for this empirical formula. Especially interesting is the con-
formity of Almgren’s observations with the above described
results of Siemen’s. We also see the previous remarks upon
the subject of economy of material confirmed in the advantages
resulting from the replacing of flat stayed and riveted surfaces
by cylindrical welded tubes. A corresponding gain would be
attained were it possible to produce a shell free from riveted
«eams.
4. Artificial Draft.—The use of forced draft has been com-
mon for many years in locomotives and portable engines, and
by this means a much greater quantity of steam produced from
a unit of heating surface than with natural draft. More recent-
ly forced draft has been applied to marine boilers, the blast
generally being produced by fan blowers. Especially has this
been necessary in the case of torpedo boats, in which the high-
est speed is demanded. By the use of multiple expansion en-
gines operated by greatly increased steam pressure speeds of 18
to 20 knots are attained without an excessive increase in the con-
sumption of fuel. This, however, involves a much greater in-
crease in the steaming capacity of the boilers in proportion to
their weight, and this resuit is accomplished by the use of arti-
ficial draft.
This has been discussed very completely in a paper presented
before the Royal United Service Institution, by Naval Engineer
H. J. Oram, upon the subject of the moti ve power of modera
war ships. The large boilers of the English war ships ‘ ‘ Blen-
heim ” and “ Blake ’’ are 15 feet in diameter and 18 feet long,
with four furnaces at each end These are worked with closed
ash pit, and an air pressure of two inches of water, and at a
steam pressure of 150 to 165 pounds, the engines indicating
3,350 liorse power. The air pressure of two inches is ample for
the desired rate of combustion, and by reference to the prece-
ding table it will be seen that it is no greater than has long
been common in locomotive practice. The combustion is more
complete under this pressure than with natural draft, being
more unifomi and producing less smoke. It may be remarked
that the efficiency of the boilers may also be increased by pro-
per heating of the feed water and by use of the double distilling
apparatus. The use of forced draft also makes it practicable to
cool and ventilate the stokeholds.
The latest examples of construction, of American design, are
made to work at pressures as liigh as 250 pounds per square
inch, with boiler shells 16 to 17 feet in diameter. Mr. Oram
considers that there is a limit to increase in this respect due to
the increase in weight beyond practical limits, both of the boil-
ers and of the engines.
It is worthy of note that in the recent express steamers of the
French “Societ^ des Messageries Maritimes” the use of shell
boilers has been abandoned, and sectional boilers of the Belle-
ville type introduced. The increase in speed also appears to
have its limits, but the advantages of forced draft, however, as
regards the reduction in size and weight of the boiler, should at
least lead to its introduction in the future for stationary prac-
tice.
Taking into consideration all the points of the preceding dis-
cussion, it appears that an application of them to practical boil-
er construction should resuit in an economy both of construc-
tion and of operation of 25 to 33 per cent. with entire safeU%
\ 363-
Reservoirs for Air and Gas.
In the use of compressed air now so general in mining and
tunneling operations, cylindrical reservoirs similar to steam
boilers are used In tunnel construction, portable reservoirs are
sometimes found mounted upon tram locomotives, the engines
of which are operated by the compressed air instead of steam.
Compressed air locomotives have only been used to a small ex-
tent, however, for general tram Service. The so-called pneu-
matic method of sinking shafts and construction piers involves
the use of air reservoirs. In this case the air reservoir is the
caisson within which the work is canied on, the water being
kept out by the air pressure, and the workmen entering and
leaving by an air lock chamber with a double system of doors.
In the case of power transmission in cities by means of com-
pressed air, the entire System of piping is included in the reser-
voir capacity. Negative reservoirs for mingled air and steam
are found in the case of condensers for steam engines. These
are usually made of cast iron and are from one to two times the
capacity of the steam cylinder. The regular removal of the
contents by the air pump at each stroke of the engine renders a
larger capacity unnecessary. In some cases the flow of spring
has been increased by fitting a tight cover over the well above
the water level when the exhaustion of the air causes an in-
creased flow from the underground sources. The vacuum Sys-
tem of power distribution, as used in Paris and London, in-
volves the use of negative reservoirs similar to cylindrical
boilers. An important application of vacuum for air and vapor
of water is found in the vacuum pans used in sugar refineries.
These pans are made in the spherical form, already referred to
as most economical of material, the rnotive in this instance
being the high pnce of copper, of which they are constructed.
Gas holders for illuminating gas are reservoirs intended only
for very low pressures, the strength of the walls being mostTHE CONSTRUCTOR.
273
important in the matter of tightness against leakage. These
holders are composed of two principal parts, the holder proper,
or so-called “ bell,” often made telescopic, and the tank or res-
ervoir filled with water which acts as a liquid packing ; the bell
in this case acts as a piston (compare Fig. 948). Similar reser-
voirs are used in laboratories and Chemical works for many
kinds of gases. For very large gas holders, in which the inter-
nal pressure of the gas is insuflicient to sustain the weight, the
roof of the holder must be strengthened by intemal trussing.
Until now the gas holder has had no definite place in construc-
tion, but it will be seen from what has already been said, that
it, together with various other kinds of reservoirs, belong pro-
perly to machine construction, not only because of their char-
acter but also because of their intimate connection with the
entire subject of mechanical engineering.
2 364-
Other Forms of Storage Reservoirs.
The construction of reservoirs for water has been a most im-
portant subject from the earliest tiqies down to the present,
many of these being of great extent, although, as has already
been said, these have until now been considered rather as be-
longing to the domain of building construction than to machine
construction. To these must also be added the subterranean
reservoirs in mines, from the small pump to those of large
extent and capacity. Cther examples are found in the negative
reservoirs which exist in low-lying tracts of land, such as are
found in Northern Gennany and Holland, intersected by canals.
A notable example in Holland is the valley formed by the
drainage of the Harlern Lake, the water having been pumped
by steam engines out to the level of the sea and the latter kept
out by dykes.
Reservoirs for agricultural puqaoses are often formed by Sys-
tems of canals, as in Lombardv and in the south of France,
where this important subject of irrigation has proved of the
greatest benefit to the country. The nature of such Systems,
considered as rleservoirs, is more apparent when the magnitude
of the work involves the construction of artificial lakes for
water storage. Ancient examples of such storage reservoirs
are found in Lake Moeris, of ancient Egypt, and Lake Nitocris,
of Babylon, as well as the existing Lake Maineri, in Ceylon,
and many others. The mechanical nature of such constructions
is more apparent when the reservoir is made by building a dam
across a gorge or valley, with weirs to permit the periodical
release of the water, the analogy to ratchet action being quite
ciear.
Finally, another natural form of stored power may be men-
tioned, one which has not to the writer’s knowledge been con-
sidered in this light before, yet which possesses the greatest
significance in the climatic economy of nature. This is the
glacier. The vapor of water, raised from the level of the sea
by the heat of the sun, collects in the form of snow about the
highest mountain peaks. In the upper valleys the snow packs
together, and under gradual pressure forms the glacier ice, and
slowly the glacier flows down into the lower and warmer val-
leys and melts away. The mass of ice, consisting of hundreds
of millions of cubic feet, forms a reservoir of stored power,
flowing in an irresistible stream of almost uniform strength
from the highest snow field to the lower valley. Ali the actions
involved are of a nhysical and mechanical nature. Taken as a
whole the glacier forms a reservoir System of the fifth order :
evaporation of the water from the sea by the heat of the sun,
transformation of vapor into snow, fusion of the snow into a
mass, conversion by pressure into glacier ice, and melting of
the ice partly by the friction on its bed and partly by the heat
of the sun.
CHAPTER XXVI.
RATCHETS FOR PRESSURE ORGANS, OR VALVES.
2 365.
The Two Divisions of Vaeves.
The application of the ratchet principle to pressure organs,
that is, the periodical interruption of its motion, closely resem-
bles the same principle applied to constructions formed of rigid
elements ; the principal difference being that the pressure organ
is very easily separable into small portions. It might also be
remarked that the pressure organ is always confined in a con-
ductor of some kind, but this feature also belongs to some forms
of rigid constructions, such as bearings, guides and the like.
Ratchets for pressure organs may be divided into two princi-
pal classes, namely, those intended to check the motion in only
one direction, and those which check in both directions. The
name given to ratchets for pressure organs is valves.*
The difference between the two classes is shown in Fig. 1124.
In the form shown at a, the pressure organ is checked by the
flap valve b, from moving in the direction of the arrow at /,
a	b
but not against motion in the direction of the arrow at II. In
the form shown at by the flow is checked in both directions.
There is here a close analogy to the two kinds of rigid ratchets,
as will be seen in Fig. 1125, which is here reproduced from
2 235« The valve b, in Fig. 1124 a, corresponds to the pawl in a
Running Ratchet,
and the valve in case b, to a
Standing Ratchet
for the pressure organ a. The difference in construction will
also be seen to depend upon the fact that in form ay the valve
lifts from its seat during the passage of the pressure organ,
while in form b} the valve slides upon the seat. This permits
another classification into:
a, Lift valves;
b} Slide valves.
The variety of forms in which valves are constructed is fully
equal to that of rachets for rigid elements, as shown in Chapter
XVIII., and there is a close analogy existing between the two
groups, with one important exception, namely, that the form
of rigid rachet which has a tension pawl, has no counterpart
among the valves. This exception naturally follows from the
fact that the member to be checked is always subject only to
compression.
There is also an analogy between the numerous forms of
valves and the two classes of toothed and friction rachets, as
has already been mentioned in \ 319, valves which have but a
slight opening, acting like friction rachets (compare § 340), and
those with full opening and entire closing like toothed rachets.
This circumstance, however, reduces the number of sub-divi-
sions into which valves may be classified, so that the principal
basis upon which a classification is made depends upon the
character of the motion of the valve, and thence upon the
necessary variation in form. This basis of classification has not
been used in the case of rigid rachets, the divisions there
having been made upon the more practical idea of the variation
in form only. We have in rigid ratchets the two forms of
pawls, one of which moves about an axis 3 within a finite dis-
tance, as in Fig. 1124 ; and the other in which the axis is re-
inoved to an infinite distance. In the first case, every point of
the pawl (or valve) moves in circular arc about the axis, while
in the second, all points move in straight lines and equally far.
In rigid Systems these correspond to link pawls and bolt pawls.
*The author calls attention to the derivation of the German word “ ven-
tile,” from the mediseval name for valves used for checking wind in church
organs. The English word “valve’’ from the Eatin “vulva,” meaning
hinged doors, is therefore broader and more general — Trans.274
THE CONSTRUCTOR.
In addition to the circular and rectilinear motion of valves,
there is a third variety possible, although but little used in
practice, viz. : those having a spiral motion. We therefore
have three sub-divisions of the two main classes of valves,
according as the movernent is circular, rectilinear, or spiral.
Lift valves may be
1.	Hinged 01 Flap Valves.
2.	Disk, Cone, or Ball Valves.
3.	Spiral Lift Valves.
Slide valves may be
1.	Rotary Valves or Cocks.
2.	Rectilinear Slide Valves.
3.	Spiral Moving Slide Valves.
Although this sub-classification is not exhaustive, yet it gives
a convenient and practical arrangement, the few special fornis
being placed 111 the group they most nearly resemble.
A. LIFT VALVES.
§ 366.
HINGED OR FEAP VAI/VES.
Flap valves are most generally applicable to piston pumps,
which, as we have already seen, form fluid escapements, see
$319. Their tightness is often attained by the use of some
elastic material, such as leather, rubber, etc., but very generally
the joint is made between metallic surfaces, especially when no
small hard particles are likely to be found in the passing fluid.
It is always difflcult to keep the loss due to shock within small
limits, this loss being especially marked with flap valves, and
indeed in all liquid ratchet systems the loss frorn this cause is
by no means unimportant.
1
)
i
Fig, 1126.
The width of bearing 5 of the valve on its seat is given by the
following formula, in which D is the ciear opening through the
valve.
5 = \/ D -j- 0.16".....................(401)
For round valves D is the diameter of the opening , for rec-
tangular openings it is taken as the smaller side of the rec-
tangle.
The blow with which a valve strikes the seat increases in force
with the amount of lift (compare \ 368), and as the lift depends
upon the actual size of the valve, this objectionable feature is
reduced by using several valves pf smaller size instead of a
single large one.
Fig. 1128.
A flap valve with metal seat, which is so constructed as to
offer as little obstruction as possible to the flow of liquid, is
shown in Fig. 1126*. This is tapped out for the Standard pipe
thread system described in § 342, the cap gives access to the
valve, the screw plug limits the amount of lift, and a flexible
connection between the disk and the hinge enables the former
to obtain a fair bearing on its seat. The freedom frotn shock
would be somewhat less if the bottom of the case conformed to
the shape indicated by the dotted lines.
Fig. 1127.
Another form of straight-way flap valve is shown in Fig. 1127.
Both valve and seat are made of bronze, the seat being secured
in place by two wrought iron keys. The case is closed by a lid
shown reinoved in the illustration. The axis of the valve is
made to permit a slight degree of lateral play in order to permit
thebest bearing 011 the seat to be obtained. Valves of this sort
are used on air pumps for steam engines and for vacuum pans.
*Pratt's Straight*way Check Valve.
A double flap valve and valve chamber designed for a mine shaft
pump, is shown in Fig. 1128*7. The flaps are formed of pieces
of leather between plates of iron, secured either by screws or
by rivets. The door by which access is obtained is curved to
the shape of the valve chamber in order to avoid excessi ve dead
space, and so reduce the shock, and is supported upon hinges.
The stops are so placed that the valves open to an angle of
6o°.
Another design for a double flap valve is shown in Fig. 1128^,
this also being for a shaft pump.f In this^ instance the valves
are formed of three thicknesses of leather. At *; is shown a
quadruple valve. The proportions given are all based upon the
unit s, as given by formula (401).
■,	Fig. 1129 shows a circular valve of
rubber, this form being much used
for air pumps for steam engines. The
valve lifts approximately in a circular
path, forming a cup, the limit of
which is the shape of the guard. On
account of the flexibility of the rub-
ber, the bearing of the seat is rein-
forced by a grating, and the rubber is
from ^ to 1^ inch 111 thickness.
These values are now made also of
vulcanized fibre, in which case the thickness need be only about
one-third that of rubber disks of the same diameter, t
Quite similar in principle to the above disk valve, is the
leather rolling valve, Fig. 1130*7, used for water wheel gates, the
principal difference being that the bending of the valve takes
place at the edge of the valve, as shown in the illustration.
Fig. 1129.
fSee Riebler, Indikator versuche an Purapen und Wasserhaltungs ma-
schMien. p. 34. Munich, 1881.
X Made by the Vulcanized Fibre Company of New York.THE CONSTRUCTOR.
275
The same principle is ingeniously used in the hanging weir
of Camere,* Fig. 1130A The valve consists of a series of strips
of wood, each really forming a separate valve, these being con-
nected and operated by chain links of bronze as indicated in
the sketch.
b.
Fig. 1130.
An excellent installation is seen at the sluice gates at Geneva
(Passerelle de la machine), where forty such gates are used to
dam the right arm of the Rhone. The gates are rolled up by
the chains shown, these being connectedto suitable windlasses.
When a whole section is to be thrown entirely open the support-
ing posts are also tipped back into the horizontal position,
these being jointed at the bottom as shown, and this operation
being effected by another chain gearing. Bach gate is 3 ft. 8
inches wide ; the sets of connecting links are 27^ inches apart,
the number of strips is 39, each being about 3 inches wide, the
uppermost being 2^ inches thick, and bottom one 3^ inches.
The weir system at Geneva, of which the above forms only a
small portion of the entire work, was completed in 1889, as an
intercantonal system to control the level of the lake of Geneva
and maintain it between the limits of 1.30 and 1.90 metres (4 ft.
3^ in. and 6 ft. in.) of that of the Rhone. During the year
1888, when the system was not entirely completed, the differ-
ence feli to 1.95 metres (6 ft. 4^ in.) in the drought of June
of that year. Between October and May the entire series of
gates was kept closed.
2 367.
Round Seef acting Vaeves.
Rift valves for small openings are frequently made of con-
ical or spherical form, and in Fig. 1131 two forms are shown
which are intended for feed pumps.
a.	b.
i
Fig. 1131.
♦Chief Engineer of “ Ponts et Chausseesof France. The subject of
weirs and movable dams has been very skillfully worked out by French
engineer».
At a is shown a pair of conical valves. The upper valve and
seat are made of bronze to avoid rust. The lower one, which is
the suction valve, has an iron seat. If it is desired to provide a
bronze seat for both valves they may both be made the same
size and bevel. The width of bearing, s, may be made as in
formula (401). If the horizontal projection of the seat is made
sl = s — o.i6// — \/ D.................. (402)
the smaller valve will have a sharper bevel than the larger one.
In designing the valve chamber, it is important to proportion
the space over the valves so that the return flow of water shall
be high enough over the valves to insure their closing, as it is
possible for the return flow to get under the valves and hold
them up from closing.f The valves here shown are made with-
out any packing material.
At Fig. 1131^ is shown a ball valve. In this the width 5 of
the seat, and also its projection s1 are the same as in the pre-
ceding. The diameter of the ball is found by drawing lines at
right angles to the bevel of the seat from the middle of its
width, the intersection of the lines giving the centre of the
ball. i The high position of the outlet opening is necessary in
order to maintain a proper lift to the valve and keep the seat
in good condition.
In order that the opening through the valve shall be equal to
that of the pipe the lift, hy of the valve must equal % D. (See
§ 369)-
Disk valves are often made with soft packing upon the seat,
two examples being given in Fig. 1132. That shown at a is a
valve for a mine pump, packed with leather. The ribs are
shaped so as to form acylindrical guide for the valve, this con-
struction being also frequently adopted for conical valves. At
b is a disk valve with rubber packing, similar valves being used
on many of the Gaskill pumping engines; all the metallic parts
are made of bronze. § In many instances disk valves are made
in the form of a ring, the seat being in two positions, the bear-
ing being on both the inner and the outer edge of the ring.
b.
Fig. ii33*
Fig. 1133a shows the valve for the air pump of a Corliss
engine at Creuzot. In this case the valve is made of a hard
material instead of a soft one. The seat is made as usual, and
the valve is a ring of phosphor bronze, held down to the seat
by a strong flat helical spring. The form shown at b is another
style of ring valve much used in the air pumps of English
marine engines.
f See Zeitschr. Deutscher Ingenieure, 1886, p. 97.
i See Uhland, Prakt. Maschinen Konstrukteur, 1870, p, 83. Piate 24.
\ See Engineering and Mining Journal, April, 1886, p. 285.276
THE CONSTRUCTOR.
Fig. 1134 is a so-called
“bell” valve, used in
mine pumps. Here the
two seats for the ring of
the valve are in different
planes. The seats are
packed with oak with
the end grain up. The
outlet in this form is
around both the inner
and outer bearings, in
which respect it differs
from Fig. 1133^. The
lift p, which is required
7r
to give an area of — D2
is somewhat less than
before, being equal to
1	D1
Fig. 1134.
4	D1 +
The necessity for lim-
iting the lift of valves in
pumping machinery has
led to the use of a large
number of small valves in the same valve chamber in order to
obtain the required area with small lift.
A distinction may be made between two methods of arrang-
ing such valves. The first method consists in arrauging a
number of similar round disk valves each over its own opening
in a piate. An example of this is seen in Fig. 1016, in which
rubber valves similar to Fig. 1132^ are arranged in rows. The
phosphor bronze valve, Fig. 11330, is also used in this manner,
38 being placed on the suction side, and 27 on the discharge
side of the air pump.
In a round valve chamber the arrangement of the valves is
more difficult, both as to the placing of the valves and to pro-
vide guides to control their lifting and seating.
a.	b.
Fig. 1135.
Fig. 11350 shows a set of 19 valves as used in the Heidt shaft
at Hermsdorf, and Fig. 1135^ a set of 21 ball valves in the
Joseph’s shaft at Frohnsdorf* These are both shown inde-
pendently of the casing. This system has shown itself so
advantageous that it has been extended until sets of several
hundreds ofball valves, acting as a single valve, have been put
into use. Fig. 1135^ shows one feature which must always be
taken into account, namely, the relation which the size of the
valves and valve casing bearto the water pipe. In this instance
the diameters of the casing and pipe are 19^ in. and 7X in*,
and the areas as 7.4 to 1.
The secoud method of arranging a number of valves is sug-
gested by the bell shaped valve of Fig. 1134. In this case the
stream which flows toward the centre is above the one which
flows outward, thus providing sufficient room for the flow of
the upper stream. This idea is also used in the arrangement
* See Rieller, Indikator versuche, etc., p. 27, and piate 11.
shown in Fig. 1135^, the inner circle of balls being placed
higher than the outer circle. By extending this idea of super-
posing the discharge openings of a number of valves we obtain
a construction consisting of a number of ring valves, forming
what may be called a set or cone of valves,f of which three dif-
ferent forms are shown in Fig. 1136. The form shown at a is
used in the large pumping engine of the Scharley-Tiefban
mine, % the pumps being 1 metre diameter (39.37 in.). This
consists of a number of ring shaped valves of constantly dimin-
ishing diameter, constructed on the bell principle, the seat of
each valve being ou the one next below.
a
Fig. 1136.
The form at b is the design of Thometzek, § and is very prac-
tical. The ring valves are ali alike in size and form, each hav-
ing its own seat, these being built up as high as may be required
and held in place by a screw bolt through the lid of the valve
casing.
The design c is that of the Humboldt Machine Work at
Kalk. || The ring shaped valves of bronze are slipped over the
succession of seats which form a cone of stepped shape, also of
bronze. These seats, as in the system of Thometzek, are sep-
arate, and are held together by a screw bolt on top, with the
difference, however, that each valve in lifting strikes against
the next, the amount of lift increasing in an arithmetical ratio
from above downward, the uppermost valve being held down
by a spring. In this last construction the ratio to D is some-
what smaller than in form b. Ali of these designs are intended
f Gerraan “ StufenventilFrench “ Etagenventile."
t See Riedler, Indikator versuche, p. 21.
§ To the best of the author!s knowledge Director Thometzek, of Bonn,
was the first to use ring valves arranged in steps (1875), and his designs
have been widely and successfully used in practice.
J A very good summary of such valves is ibund in an article by Fngineer
Waldastel, entitled, “ Ueber Ringveutile fur Pumpen und Geblase,” in Z D.
Ingenieure, 1886, p'. 935.THE CONSTRUCTOR.
2 77
for water pumps, br: an excellent form is designed by the
Humboldt Machine Works for blowing engines also, the suc-
tion and discharge valves being concentrically arranged * * * §
§368.
Unbaeanced Pressure on Lift Vai/ves.
If we assume the joint of contact of a lift valve to be entireiy
tight and represent the projected area subjected to the pressure of
the discharge colutnn by FXf the area exposed on the under-
side being called F, we have at the instant of equilibrium of
the two columns as the valve is about to lift, p F = px Flt in
which p and px are the pressures per unit of area on each side,
and the weight of the valve is neglected or counterbalanced.
From this we have
P —A _ F\ — F
A “ F '
p'
or of the ratio p is put =	;
P—Px
Px
(4°3)
The pressure p — ptis the unbalanced pressure on the valve,
-p
Px
~ jy_
and the ratio - — ,	■— is the ratio of unbalanced pressure.
Upon this question of unbalanced pressure much depends,
and many calculations have been made for various sorts of
valves, the pressure tending to close the valve being much
reduced in bell shaped valves, such as shown in Fig. 1134.
Experimental researches, made upon pumps of various sizes,
however, have shown that only a small excess of pressure is
actually required. f At the same time the preceding formula
shows that the question of the unbalanced pressure is by no
means a subject to be neglected. t
As an instance of the effect of unbalanced pressure may be
cited a bell shaped valve, 1 metre ciear opening, in the shaft of
the Bleyberg mine, of which the seats could not be kept down
by their own weight, but would adhere to the valve, rising and
falling with it until secured by some other means.
Riedler has observed the fact that in arranging valves in a
series in a cone as in Fig. 1136^, the uppermost valve which is
subjected to the greatest excess of pressure according to (403),
lifts first, and is followed by the others, the lowest rising last.
It appears that a thin film of water is retained between the
bearing faces of valve and seat, which responds rapidly to the
pressure of the lower column px ,and thus tends to reduce the
value given by the above equation. If we first make the
assumption that such a film exists and acts in the manner indi-
cated, we have for two successive ring valves, arranged for
example as in Fig. 11364:, the following stresses in the liquid.
The weight of the valves, beginning from the top, is indicated
by Gx and G.l} and their projected areas by Fx and F2.
The experimental valve, shown in Fig. 1137, had an annular
seat of 6 in. outside and 2^ inside diameter, and was subjected
to a steam pressure px above,
and to the atmospheric pres-
sure p below. In the follow-
ing table p' indicates the
pressure per square inch
which would give the equiv-
alent of the actual pressure
P required to lift the valve,
while a is the area and d the
diameter of a circle for which
a (px — p) — P. This circle
Robinson calls the circle of
equilibrium, and it is always
smaller than the upper pro-
jection of the valve.
The valves under a and d
are taken approximately at
the nearest values. The un-
balanced pressure can readily
be determined from the table.	FlG. 1137.
fi\-p Pounds per SquareInch.	f Pounds per Square Inch.	a Square Inches.	d Inches.	d' Inches.
5	8	5-6	2 6	2.53
IO	17	5-8	2-7	2.85
15	26	6.0	2.8	2.92
20	36	6.2	2.8	3.02
25	46	6.4	2.9	3-°9
3°	57	6.6	2.9	3-i4
35	69	6.8	2.9	3-19
40	81	7.0	30	3.22
45	95	7*3	30	3-25
*5°	112	7-8	3-i	3-27
55	129	8.2	3-2	3-29
60	15°	8.7	3-3	3-3i
65	172	9i	3-4	3-33
70	198	9.8	35	334
75	230	10.5	3-7	3-35
If px — p = 45 lbs. we have, since d = 3 in. = yz 6 in. for the
excess pressure, one-fourth px — p ; for px — p = 75 lbs. it is
equal to 0.38 (px — p). The law of reduction of pressure
between the surfaces from px to p is not simple. The corres-
ponding curve is convex towards the axis of abscissas, as shown
in Fig. 1137. If it is desired to determine the mean pressure
pm we have from the table for px — p = 5 the value
Pm — {oTPi — P = 75 it is/„ = ——- —. For a rough
4-43	2-3°
(404)
P'=P, + and P" = A +	• • •
Now it appears by examination of the weights and areas that
G	G
under the circumstances - 2 is greater than which is then
A
also true for the entire second member of the value of p" §, so
that p' is the resistance which is overcome first. In the case of
the Bleyberg mine F2 is very much greater than Fx, and p//
becomes less than p' which explains tne action of the valve
seat.
The actual behavior of the film of liquid between the surfaces
of contact may not be so definite as indicated above, but it ap-
proaches to it as an approximation. This is shown by the
very valuable researches made by Prof. Robinson upon a valve
acting under steam pressure. || In two extensive series of ex-
periments he investigated the actual weight required to lift a
valve under pressure. The results showed that the unbalanced
pressure was much less than px —p.
* German Patent, No. 33,103
fReference is especiahy made to the numerous and valuable investiga-
tions of Prof. Ried’er.
t See the comprehensive papers of Prof. C. Bach, in Zeitschr. D. Ing. for
1886.	“ Versiiche zur Klarstellung der Bewegung Selbstthatiger Pumpen-
ventile.’’
§ In the case of the arrangement shown in Fig 1136«, the ratio of weight
and area for the three valves, proceeding from above downwards, is 50: 76:
85.
1 See Trans. Am. Soc. M. E , Vol. IV, 1882-1883, p. 350.
approximation we may put pm —	(px — p). Prof. Robinson
has deduced a theory from these experiments. He assumes
that between the surfaces there exists between the pressure px
at the outer circumference to the pressure p, at the inner cir-
cumference, a gradual increase of pressure from p to px. Under
the assumption that the fluid under consideration is incom-
pressible he obtained by pure analysis the following equation
for the value of d :
in which R and r are the inner and outer radii of the ring of
the seat. The values of d' as obtained from this equation are
given in the fifth column of the table. They increase nearly
as the experimental determinations of d, but with Robinson’s
assumption of an entireiy elastic fluid they are 10 to 15 per
cent. too great. Probably steam should be considered as mid-
way between an elastic and a non-elastic fluid.
The deductions from Robinson’s experiments are hardly ap-
plicable to pump valves because the lifting of the valve by the
action of the lower column is effected by a varying pressure,
while in the experiments p w*as uniform. If we accept Robin-
son’s theory we arrive in fact to what has been already stated,
namely, that when the value of p increases between the surfaces
until it reaches pXt the pressure p2 will be balanced, since in
equation (405) for p — px the value of d' = 2 r, that is, the
unbalanced pressure becomes zero. This also agrees with
Riedler’s indicator tests, since experiments with the indicator
failed to show appreciable unbalanced pressure.27$
THE CONSTRUCTOR.
These experiments appear to indicate that practically the
unbalanced pressure cannot be great, and in most cases for self*
acting valves it may be neglected. Prof. Robinson’s experi-
ments and theory may serve to determine with considerable
accuracy the pressures at which a safety valve begins to lift.
\ 369.
Ceosing Pressure oe Seef-acting Vaeves.
As already shown, a self-acting valve opens whenever the
pressure in the under column exceeds that above the valve. As
soon as the direction of pressure is reversed the valve should
close quickly. This is especially important, as Riedler has
shown in the case of suction valves, since when the closing is
delayed appreciably after the reversal of the pump piston, the
moving column of water is checked with a sudden shock. For
this reason the suction valves are given especial attention, as
shown in the example already cited from Creuzot, in which
there are 38 suction valves and only 27 discharge valves.
,	In order that the lift
,	shall not be too great and
to insure prompt closing,
the valve may be loaded
with a definite pressure,
K, obtaiued either from
the weight of the valve,
or by means of a spring,
or by both. This ques-
tion will here be exam-
ined. Referring to Fig.
1138, we have for the
lifting pressure due to
the under column :
		K Px	
i vmmmm			
i	LSL	ih	Wlj y
1	P		— n
Fig. 1138.
P = A +
R
K'
Pi + q
(406)
in which p —p\ = q the closing pressure per unit of area. For
a height h, and putting u = the circumference of the cylindri-
cal space inclosing the valve, we have :
wl h u = Fv
wx being the velocity of flow at the outer edge of the valve,
and v the velocity of flow in the under column, h being in feet.
Now if w is the velocity at the inner edge of the valve we have
that is:
Wi
w
But we also have
' =	2 gh' — ^ x 2.3 <
(since the pressure per square inch is equal to — ) and hence:
rh'\
2.3/ ‘
wx = J 2Z x 2-3?
^	a
Substituting, we get:
hu	yJHFML =F
a
Now it is desirable that w and wx should not be too great;
that is, the ratio of h u to Fshould be equal to, or less than,
unity. If we put h u — p F, we have :
a
and, putting for^* its value = 32.2, we get:
a v2	cl
148.12 f orsay=->-
from this formula we get for :
&
150
. . . . (407)
1
4
q — .006667 a v1 .01185 a v1 .02666 a v1 .1066 a v2
in which v is at its maximum value when it equals the velocity
of the pump piston. For purposes of numerical calculation we
stili require the value of a. Taking the width of bearing s, and
projection in the case of conical valves sx from (401) and (402)
we have :
Dia. D =	2 in.	4 in.	6 in.	8 in.	xo in.	12 in.	16 in.
Width of seat j =	0.44	0.56	0.64	0.72	0.80	0.84	0.96
Proiection jj = . .	0.28	0.40	0.48	0.56	0.64	0.68	0.80
Cone valve a = .	1.65	244	1.36		1.27	x.24	1.21
Flat Valve a = . .	2.17	1.64	i-44	2-39	i-35 •	1.30	2-25
An example will show how the pressure of closing can be
calculated :
Example x.—For a conical valve whose smallest diameter D = 4 inches,
and the greatest velocity v of the lower column is 6% feet per second the
area of inlet of valve hu = F\ and /3 = 1, we have a pressure of q = .006667 X
1.44 X (6.5)2 - 0.4 lbs. per sq. in. For the total pressure we have
K =
- (4 + 2 X 0.40)2 X 0.4 = 7.24 lbs.
Example 2.—For a flat valve of the same dimensions we have a = x.64 —
whence K — —7.24 = 8.24 lbs.
1.44
The method of calculation is simiiar for ring shaped valves
and can readily be applied. The formula (407) can only be
considered as an approximation as the variations in the jet of
water affect the pressure. It is evident, however, that K is
often quite large.
In the preceding calculation the momentum of the water
column has not been taken into account. In some cases this is
sufficient to hold the valve open until the piston has made a
great portion of its return stroke. This is well shown in the
case of the pump at the Bleyberg mine (§ 319, note) which ap-
parently showed a discharge of 104 per cent. If this action can
be made to exist during the entire stroke by giving the water
a sufficient velocity by contracting the tube that the discharge
valve does not close at ali, this valve may be entirely omitted.
This is the case with the single valved pump of Edmond
Henry,* which has only a suction valve and no discharge
valve. An analogy to this form of fluid ratchet is found in
L,angen’s fly wheel ratchet train, Fig. 730 and 731. In this case
the momentum of the fly wheel is sufficiently great for it to
suffer no perceptible loss of velocity during the return stroke
of the pawl.
2 370.
Mechanicaeey Actuated Pump Vaeves.
The numerous investigations of recent years have show 1
that by proper loading of the valves, combined with a reduc-
tion of lift, the shock of the water in a pump can be very rua-
terially reduced and kept within practical limits, even for high
piston speeds. The reduction of lift involves a great multipli-
cation in the number of the valves and a great increase in
dimensions. For this reason another solution of the problem
has been attempted, namely, that of abandoning the self-acting
feature, and actuating the valves by mechanical means. The
best arrangement seems to be that in which the valves are
opened by the action of the water, but closed by a positive
gear in advance of the shock. The application of this method
enables the size of the valves to be reduced, and as it is princi-
pally used for large pumping engines the valves can be oper-
ated by connection to the fly wheel shaft. Professor Riedler
has recently made very valuable investigations upon this
system.f
f
Fig. 1139.
Fig. 1139 shows the valve gear for the Riedler pumping
engine at the Wartinberg mine. The revolving cam d> closes
* See Revue Industrielle, p. 342, September, 1888, where the complete
theory of this form of pump is given.
fSee Riedler, Mine Pumps with Positive Valve Gear, Zeitschr. D. Ingen-
ieure, 1S88 p. 481.THE CONSTRUCTOR.
279
the valve b, just as the plunger is at the end of the stroke, and
permits it to open by the action of the water. The valve is
held to its seat by a
a	spiral spring. Pumps
of this construction
operate very smooth-
ly. Further details
of this construction
are given in the arti-
cles already cited. For
blowing engines, and
especially for air com-
pressors, positively
actuated valve gears
are much used. A
very simple action for
the ini et valves is
shown in Fig. 1140.
The piston rod c
moves the valve b,
by means of the fric-
tion of the rod in the
stuffing box, the ac-
tion taking place just at the reversal of the stroke. Examples
of this construction are to be found in the air pumps for use in
physical laboratories.
Fig. 1140.
t 371-
Vaeves with Spirae Movement.
It is not so convenient to construet a valve so that its motion
shall be both rotary and rectilinear axially, and this construction
is mainly limited to valves which are operated by hand.
I
Fig. 1141.
Fig. 11410 shows a conical valve with spiral motion, as used
on the Giffard injector. This arrangement enables a very fine
adjustment of the opening to be obtained ; a similar form is also
used in the so-called “ cataract ” for steam engines. The sharp
point of the cone has caused valves of this sort to be called
“needle” valves, and similar forms, without the spiral action,
are found in gas regulators. Stop valves for steam and for
water are frequently made writh spiral motion. An example is
showm in Fig. 1141&. When the valve is not in contact wfith its
seat it has b®th a vertical and a rotary motion. In the parti-
cular form shown the valve has a disk of asbestos tvhich forms
the surface of contact with the seat. This* general form is
known as a “globe ” valve on account of the form of the body,
and such valves are very extensively used for steam and water.
1372.
Baeanced Vaeves.
Valves which are to be operated by other means than by the
action of the fluid, are advantageously made so as to be relieved
from fluid pressure, and thus offer less resistance to operation.
Valves of the wing or flap construction are conveniently bal-
anced by combining twTo valves moving in opposite directions
into one valve of the form commonly called “throttle” valve,
Fig. 1142.
This is the counterpart of the throttle ratehet shown in \ 250,
and valves of this sort have been much used with throttling-
govemors for steam engines. The closing of such valves is im-
perfect, as the edge must be rounded near the hub of the valve,
thus giving only a line of contact.*
a.	b.
I	I
If it is desired to use throttle valves for regulation of water
pressure, as the case of turbines, etc., it must not be forgotten
that the resistance of the valve will materially affect the effi-
ciency.
For self-acting valves a variety of throttle valve may be used,
in which the area of one wing is only about % to that of the
other wring, thus partially balancing the valve. This form,
which is old, appears to be again coming into use. |
Lift valves which are situated in vessels which are not closed
at the top may be balanced in a simple manner by making -the
valve with a tubular continuation which extends above the sur-
face of the wTater. A balanced valve upon this principle, as
used for an outlet valve in a canal lock, as at bx' and b/,
♦The form shown at b is recomraended in Revue Industrielle, p. 205, May
26, 1888, as insuring a better balance, but from Robinson’s experiments,
already cited, this form would offer too much resistance to opening.
t See Belidor, Architecture Hydraulique, Paris, 1739, Vol. II. These
valves were of brass with metallic packing.28o
THE CONSTRUCTOR.
993» is shown in Fig. 1143. This valve, designed by
Constructor Cramer, is made with a cylindrical shell of sheet
iron extending to the surface of the water. The diameter of
this shell is the same as that of the yalve, and the weight of
the valve, which is by no means small, is partially counter-
balanced, leaving only sufficient to insure proper closing and
seating. * If it is desired to apply Cramer’s construction to
valves which are subjected to high pressure, this may be done
by using two stuffing boxes, one external and one intemal,
as shown in Fig. 1144, which, however, adds to the complica-
tion. For lift valves which are to act under high pressure a
better construction is the so-called “ double-beat ” valve,
which, like the throttle valve, consists of two similar valves in
which the pressures oppose and neutralize each other. Three
forms are shown in the accompanying illustrations. Fig. 11450
Fig. 11450.
being a double disk valve, and Fig. 11456 a tubular valve. Both
of these were invented by Hornblower in the latter part of the
last century. Fig. 1145c is a bell or Cornish valve. These
Fig. 11456.
valves each consist of a pair of conical lift valves, the varia-
tions appearing in the details of the connections and passages.
* See Annales des Ponts et Chaussees, 6me serie, Vol. XII, 1886, II Semes-
tre, p. 248, also Zeitschrift fur Bauwesn, 1880, p. 155.
When the projection of one seat falis within that of the other,
as in forms 6 and r, the unbalanced pressure is that due to the
projections of both seats. If so desired, however, these may be
Fig. 1145G
madeas Fig. 11450, with one seat directlv over the other, in
which case the pressure px — p need only be calculated for one
seat. For the preceding double seated valves we may make :
for the width of seat s —
and for the projection =
yi f 0.2	0.137
^ ( 0.2	J

• (408)
I11 form a the mean diameter D' of the valve is = 0.8 times
the diameter D of the pipe, while in forms b and c the diame-
ters of valve and pipe are the same. For the force required to
lift the valve, taking the projection sx into account and assum -
ing the pressure between the surfaces to be as in § 368, equal to
Yi (px—p), we have, neglecting the weight of the valve :
P> = V D' % (A — p).........................(409)
while for a single conical valve of the same diameter D it would
be:
P= Qj	W + % «■	{D + i,)] (A — /) •
(410)
P is proportionally very great, while H is not always unim-
portant.
Example.—For ZT = 12, we have for form a, Sx = % ^ 0.2 \/12 ^ =0.346".
If now P\ — p = 60 pounds per square inch we have :
P = it X 12 X 0.346 X % X 60 = 521 pounds.
jy
For a single valve the diameter would be D = —= 15 inches, and from
'402) sx = 0.2 \/ 15 = 0.^7, whence
P=	J52 + % X 0.77 ir ^ 15 + 0.77^ J 60 = 12,126 lbs.
so that P is nearly 24 times P.
It is very desirable for double seated valves which are to be
used for steam, that both valve and «eat be made of the same
material, in order to avoid unequal expansion.
Fig. 1146.
Double seated valves are also used for water. Fig. 1146
shows such a valve arranged for a sluice.THE CONSTRUCTOR.
281
This valve is made writh flat seats, the lower seat being faced
with rubber, and the upper one packed with leather secured to
the housing which is shown over the valve. The valve rod nris
through this housing and through a tube above the surface of
the water. The diameter D is 1400 mm. = 4 ft. 7 in. This is
practically a tubular valve, similar to Fig. 1145^, except that
the direction of flow is reversed; this arrangement has also
been used by Hornblower. The leather packing at 2" is made
flexible, since the projections of the valve seats lie one wdthin
the other so as to make a slight tendency for the valve to lift,
without entirely overcoming the weight of the valve. Balanced
valves of the kind described above are also adapted to large
steam engines. In some instances a small balanced valve is
arranged so that it is lifted first and admits steam under the
main valve before the latter is lifted.
Another device is that shown in Fig. 1147, known as Ait-
ken’s automatic steam stop. The main valve b, is closed by
being screwed up against its seat by the spindle and hand
wheel. Before opening, it is balanced by admitting steam
through the by-pass valve b'. The valve itself is loose on the
spindle, and if through any breakage in the pipe bevond the
valve a sudden or rapid flow of steam should take place, it will
be automatically closed by the force of the current.
1
Lift valves may also be balanced by making a balance piston
connected with the valve, the pressure of the steam acting upon
the piston in the opposite direction to the action on the valve.
This construction has also been applied to reducing valves in
the place of weighted levers or springs in various ways, but
space cannot here be given to the subject.
B.—SLIDING VALVES.
8 373-
Rotary Vai/ves and Cocks.
For rotary valves the bearing surfaces are conveniently made
conical, so that a simple endlong pressure on the valve will
hold it firmly to its seat. Valves of this construction are
known as cocks.
Fig. 1148 shows two forms of such cocks which are in general
use. The opening through the plug of the cock increased in
height in order to obtain a full area without requiring the
diameter of the plug to be too great; the area of the opening
through the plug being made equal to the area of the pipe, i. e.,
= — D\
4
According to the experiments of Edwards, a good taper for
the plug is -5- on each side. For the thickness 6 of the metal in
the body of the cock formula (319) may be used when the
material is of cast iron, which gives b = 0.472" -f- for bronze
the thickness may be made one-half to two-thirds this value.
The design shown in Fig. 1148^ has the plug entirely inclosed
in the body, and is made with two stufling boxes, one for the
plug and one for the spindle. The management of screw cap
and jam nut enables a fine adjustment to be obtained.*
Fig. 1148.
Fig. 1149 shows two forms with hollow plugs, these being
much used for injection cocks for jet steam condensers.
Fig. 1149.
When the angle of the apex of the cone becomes 180° the
plug becomes a flat disk, and this forni is often found in the
throttle valves of locomotives, and less frequently in the valve
gear of engines. True cylindrical plugs, i. e., those in which
the angle of taper is equal to zero, are rarely used, although
recommended by some. This form is better made in a portion of
a cylinder, and operated by an oscillating motion, as in the
Corliss and similar valves. A starting valve of this type, used
as the steam admission valve for a triple expansion engine is
shown in Fig. 1150.
%	b.	c.
1
Fig. 1150.
At a is a longitudinal section, b a cross section, and at c is
shown the seat looked at from above. In the one seat three
passages are controlled at Ff and ////. All three are closed
when the valve is in the position shown at b, but open at the
same time when the valve is moved to the left. The trapezoidal
opening in I' admits a small amount of steam to the high pres-
sure cylinder at the same time that a little live steam is admit-
ted through /// and /" to the intermediate and low pressure
cylinders, so that the engine is sure to start. The valve is then
thrown all the way over, closing I" and I'" and throwing I'
wride open.f
* Mosler’s German Patent, No. 33,912.
fSee Zeitschr. D. Ingenieure, p. 509, 1886, Meyer, Triple Fxpansion
Marine Engine.282
THE CONSTRUCTOR.
i 37 C
GATE VAI/VES FOR OPEN AND ClyOSFD CONDUCTORS.
A great variety of valves has been devised for open water con-
ductors in the form of gates by which the flow can be regulated.
Such gates have been preferably made of wood with the exeep-
tion of the operating mechanism. At the present time iron is be-
ginning also to be used for the gates, and as in the case of other
branches of work, wood is likely to be less and less used, being
limited to a few special cases. For very broad streams the con-
struction of such gates is now sometimes made upon the princi-
ple of subdivision. In such cases the breadth of the stream is
subdivided into a number of smaller streams, each with a sep-
arate gate, thus keeping the gates small enough to be movable
by hand.
A weir which is placed in a stream is both in principle and in
construction a valve. When the water in the stream is low the
flow is entirely checked ; for the mean flow the stream passes
through the reduced opening with a velocity due to the reduc-
tion in section, while for high wrater the enti re wddth of the
dam is overflowed. Movable weirs are plainly examples of
regulating valves. French engineers have given much atten-
tion to moveable danis with excellent results. A new design
for a moveable dam by Schmick is shown in Fig. 1151.*. This
dam consists of a number of pontoons, each three of which are
Fig. 1151.
secured together by a yoke and anchored by a chain to a point
up the stream. All three pontoons of each set are arranged
with variable water ballast in two or more compartments, 0/
and a2 . An adjustable valve bx enables communication to be
made with the upper water level, and the compartment 0/, and
a similar valve b2 connecting the compartment 0/ with the
lower level, while a third valve b2 enables communication to be
made between the two compartments. By varying the open-
FiG 11524.
* S. Schmick, Prahmwehr (Poutoon Dams), Zeitschrift fur Bankunde,
Munich, 1884, p. 502.
ings of the valves the pontoons can be caused to regulate the
difference of water level above and below the pontoons, while
if all three valves are closed the pontoons will rise and fall with
the variations in the level of the stream.
Gate valves are much used for water mains, and an example
of the many varietics used for the purpose is shown in Fig.
11520. The gate or disk of the valve is made of bronze, and is
wedge shaped, in order that it may be firmly pressed against its
seat when the screw is tightened (this fornis a pressure of the
second order) while the pressure is immediately relieved at the
commencement of opening. The screw is in this case made of
“ sterro-metal ” to avoid rusting.
1
Fig. i152A
Gate valves are also used for gas mains, and a valve for this
Service is shown in Fig. 1152A In this instance the valve is
operated by means of a rack and pinion. The motion is made
in the horizontal direction so that the valve will remain in any
position, the only resistance being that of friction.
2 375-
Si,ide Vaeves.
Slide valves are mainly used for the purpose of effecting the
distribution of steam in steam engines. This is such an im-
portant subject that all the forms in general use will here be
noticed.
a	b.
1. Plain D valve, Fig. 1153. This is the most important
forni of all. The action of this valve has already been discussed
in g 328, and hence the dimensions will only be considered
here. The width 0 of the steam ports is kept as small as is
practicable, while the length at right angi es to the plane of the
drawing is made quite large. When 0 is given, the dimensions
to be determined are the outside and inside lap e and i, the
bridges b, the width of face bo beyond the ports, the width a0 of
the exhaust port IVi the travel r, the length of the valve /, and
of the valve seat l0. The laps e and i, and under some circum-
stances two valves e2 and e3 for e are determined according to the
method given in Figs. 3024 and 1025. In the sanie manner also
is found the greatest distance s, Fig. 1153^, in which the edge
of the valves passes the edge of the port. This gives the width
of bearing t of the valve upon the bridge, since b — s + t. The
3
value of t varies greatly, the least permissible value is t = —,
and it is more frequently made to x/zn’. Approximately, for
we have, after assuming t as just given, a0 t — (* -f* 0 + i)
=- 0, in which e is taken as a mean between e2 and e3. We then
have:
Cio == 2 0 -|- l -f- i — t	I
whence r = a e s	S-...............(410
and l a 3 l i 2 s t J
The valve face must have an inner width of bearing t0 Fig. b
at least equal to /, whence for the total width of the valve face
we have the value
a0 -f 2 b -f 2 0 + 2 &>,or	\	' fA19x
l0 = Aa + 3e — i + 4s + t+2to(................{A'2)
The thickness of metal in the valve itself, when made of cast
iron should be about
200
-j- 0.4", which is about half theTHE CONSTRUCTOR.
283
thickness of the metal of the steam cylinder as given by formula
(3-0,. If the valve is faced with wliite metal the body of the
valve should be of bronze, the white metal itself not being strong
enough.
2. Double D valve, Fig.
1154. In this form the four
valves which in the plain
D valve are united in one
piece, are separated into
two portions, connected
by a rod. This construc-
tion is adopted to shorten
the steam passages / and
III, the width of each
Fig. 1154*	valve is = 3 a 4- 2 e 2 s
+ t + to or of both to-
gether = l= 6a-{-\e + ^sJ{-2t-\-2to.
3. Pipe Valve, Fig. 1155. This form is also intended to re-
duce the length of the ports II and ///, which is often an im-
j«^a-*d*~'-s+t+a+F--*u
-----hj------*
tejaXr-iitx;
+_a 4rj|ajgj
«• - ♦** ■»« —v---*.
uo.a; b2 i
... +’4-	M* - »•«* • *»
b\ : ao ; 0 ; a; b«;
Fig. 1155.
portant consideration in engines of long stroke. The total
length of valve bearing surface is / = 6 a + 5^ + 3J+/-f-
2 to.
Example 1.—If a — e — i — s = y&", t = tQ = T3S we have for a
plain D valve the width l.= 4 X 0.75 + 3 X °-75 + 0.6875 + 2 X 0.375 + 0.1875
™ 6.875".
For the double D valve we have l = 6 X 0.75 + 4 X 0.75 + 4 X 0.375 + 4 X
0.1875 = 8.75" and for a pipe slide valve as Fig. 1155, 1= 6 X 0.75 + 5 X
0.75 + 3X 0.375 + 0.6875 + 2 X 0.1875 = 10.4375". The work of friction in
moving the valves is directly in proportion to the above widths, since the
travel is the same in all three cases, being: 2r=2^ + 2«-(-2i = 2X 0.75
+ 2 X 0.75 + 2 X 0.375 = 3-75".
In order to reduce the work of friction in slide valves the
multiplication of valves has been resorted to, much as has
already been shown in the case of lift valves. A division of the
valve system into two parts has also been made for marine
engines with oscillating cylinders, the object being to place one
portion on each side of the cylinder and thus keep the entire
mass symmetrical with regard to the axis of oscillation. In
this arrangement the two slide valves correspond to eight sepa-
rate valves. In these as also in engines, with stationary cylin-
ders, the valves may be combined into one. This may be ac-
complished by using two or more sets of steam passages which
unite at one point and by making corresponding divisions in
valve and valve seat. The combination of several valves so as
to act as one is not limited to lift valves, as many useful fornis
of slide valves are made on this principle, some of the best
forms being here shown.
4. Penn’s Gridiron Valve, Fig. 1156. In this the steam port
a is divided into two ports, each having a width = —. To de-
Fig. 1156.
termine the total width of valve as in the previous cases, we
have : l = 5.5 a + 3.5 e + 3 s -f t 4- 2 ta + % i, and for the
travel: 2r=a-\-e-\-s, that is half as much as before. It is
C	'l
evident that the laps — and —- must bear the same relation to
2	2
— as the diagram gives for a: e: i, in the preceding forms.
2
Ex ample 2.—For —= ?-, — = 2_L
2	8*2	82
5" JL
16 *	2

3
= ~z6 WC haVC l== 5’5 * °'75	3‘5 ^ °’75 + 3 X 0.375 -f 0.1875 + 2 X 0.187S
+ o-3i25 = 8 75"
2 r = 0.75 + 0.75 + 0.375 = 1.875.
>f friction of
6-875 X 375
This gives for the work of friction of such a valve as compared with an
equivalent plain slide valve :
- = i-57-
8.75 X 1.875
which is an important gain.
5. Borsig’s Gridiron Valve, Fig. 1157. This is the same in
principle as the preceding, and differs only in construction, the
exhaust passages being carried on cach side of the valve instead
of above, as in Penn’s construction.
6. Hick’s Double
Valve, Fig. 1158. This
is intended for use with
compound engines
with parallel cylinders
(Hornblower and
Woolf), the ports II'
and III' are for the
high pressure cylinder,
and II" and III" for
the low pressure cylin-
der. The width / of
the valve is :
^=5^ + 3ai+6^-(-
4 s + ex -f fi X t0.
Usually ax is made

:	1 t !	•
• bo tt b b|
jAO i bj'ai'b a' bo'
Fig. 1158.
equal to a, which reduces the value of l somewhat.
7. Allan’s Double Valve, Fig. 1159, a valve for compound

a+e+i
bj a bo ' ao ' bo ’ a
Fig. 1159.
engines with tandem cylinders. The value of l is
l = 10 a + 7 e -f el + 6 J -f i + ix -f 3 tQ.
This construction not only economizes the work required to
operate the valve, but also gives a very simple arrangement of
steam passages.
8. The E Valve,
Fig. 1160, is used to
advantage in place of
the plain D valve»when
the use of a valve gear
actuated directly from
the piston rod requires
that the valve shall
move in the same di-
rection as the piston.
(See Fig. 1006 and
1008). This valve con-
sists of two D valves cast together, and the over travel beyond
the valve seat gives the admission.
We have as before : r — a -|- e + s, and
b = e + r= a + 2 e + s V...........................(413)
= a	JTHE CONSTRUCTOR.
284
This gives for the widtli of the valve;
l = 3a+2b+2b0 + 2toT:	\	(a,a
/ = 704-6^4-z4-4^+2^	J............^4 4;
which is considerably greater than for an ordinary D slide
valve.
Xlff
Ex ample 3.—If as in Exampie 1, we make a = e — 0.75", i = •—2J =
10
°-375">* = 0.1875", wethen have l = 13 'X 0.75 + 0.6875 + 4 X 0.375 + 2 X
o. 1875 = 12.3125" against 6.875" for the plain D slide valve. It will be evident
that the E valve is only available for small port widths and small laps, as
will also be seen in Figs. 1006 and 1008. The principal value of this valve
lies in the use of the outer edge of the valve seat as the edge of opening,
which principle also has a valuable application in the following valve.
1	9. Trick’s Valve,
Fig. 1161.* This is a
double valve and con-
sists of one D valve
over another, with a
steam passage be-
tween. As before,
we have r — a + e
-f- s, and also niake
b0 — 2 £ — ty i. e., the
inner edge of the
Fig. i i 61.	outer valve when the
valve is in mid-posi-
tion, is at a distance -= e from the edge of the valve seat. The
consequence is that when the valve is rnoved a distance equal to
ey say to the right, the passage through the valve opens to ad-
mit steam at the sanie instant as does the edge of the valve on
the left. This gives a steam admission twice as quickly and an
opening twice as great as would otherwise be the case. The
following positions from a to /, Fig. 1162, will show' the succes-
sive actions, the exhaust ports being omitted for simplicity.
a	e	f.
Fig. 1162.
Fig. 1163.
Fig. 1163, we must from the point Ay which indicates the port
opening, double the width given by the Zeuner circle until the
•This valve was invented and made by Trick at Esslingen in 1857, and by
Allan in England in 1858-1860; in the United States it is correctly kuown as
Trick’* valve.
1
entrance to the passage in the valve is wide open, as at b. By thus
doubling the opening in the diagram wre obtain the curve A Bv
b.	From this position on, the opening at the left continues to
grow wider, but that through the valve on the right does not,
hence on the Zeuner diagram from this point we return to the
opening which the regular valve circle gives, to which is added
the constant opening c — B Bx — C Cx indicated by the curve
Bx Cv This continues until the inner edge of the opening of
the valve passage on the left reaches the edge of the bridge as
at c.
c.	As the valve continues to move the passage through it is
gradually closed, but the steam port is opened to the same
amount, and hence the actual port opening remains constant.
Xhis continues until the position d is reached, when the passage
through the valve is entirely shut off. This is indicated in the
diagram by the arc Cx D, struck from the centre at 1.
d.	The valve continues to move to the right until it is en-
tirely upon the bridge, the corresponding portion of the dia-
gram being the arc D E of the valve circle.
e.	The valve from this position moves on the bridge beyond
the port until it has traveled a distance equal to s> as shown at
fy during which time the port opening remains constant, as in-
dicated in the diagram by the arc E E' struck from the centre
1. From this point the same actions take place successively in
the reversed order.
It will be seen that Tricas valve gives a much quicker opening
and also a much longer duration of the full opening than does
the plain slide valve. It remains to be seen how these features
can be used to the best advantage. According to Trick’s prac-
tice this is best done by making the value of s negative, and
also>/. This makes the port opening from Cx to C/ in the
diagram constant, as shown in the diagram.
In order that the apparent contraction of the ports by the
change in the sign of s shall not occur, the value of a is made
greater than would otherwise be the case. Under these condi-
tions we have for the exhaust port a0, the equation :
a,o -f- t — ex — a — i = a — s,
in which 5 is given the maguitude equal to the distance which
the edge of the valve is rnoved beyond the edge of the bridges.
(See Fig. 1162^). We then have :
For the exhaust port,	a0 = 2 a + ex 4- * — s—t
For the bridge,	b = e — ex s — t
For the passage through valve c = e — i — ex
For the total valve,	l=^a-\-^e—ex-\-i—35+/
Exampie 4.—Making j	^— and negative also t — —\— and the
16	10
other data the same as the plain slide valve of Exampie 1, and we have :
a — °-75 + 0.1875 = 0.9375"; e = 0.75; i = 0.6875 ex = 0.078", whenee :
r = a + e — s = 0.9375 + 0.75 — 0.1875 = 1.5"
ao = 2 a -f ex + i — $ — t — 2 X 0.9375 + 0.078 + 0.6875 — 0.1875 — 0.1875 =
2.2655"
b — e — ex + s — / = 0.75 — 0.078 -f 0.1875 — o 1875.
c = e — t — e1 = 0.4845"
and
l = 4a + 4e — ex + 1 — 3* + / — 3-75 + 3~ °-°78 + 0.6875 — 0.5625 + 0.1875
= 6.9835"
as against— 6.875" for the plain slide valve, which compares very favorably.
Fig. 1164.
Fig. 1164 shows the application of the author’s valve diagram,
already shown in Fig. 1024. The action of the inner portion of
the valve is the same as with the ordinary slide valve.
For comparison the following dimensions of an executed
valve by Trick, are given :
a — I.77'7 (45 mm.), ex = i = 0.078" (2 mm.), t = 0.216"
(5.5 mm.).THE CONSTRUCTOR.
285
b = 1" (25.5 mm.), e — 0.846" (21.5 mm.), c = 0.55"
(14 mm.).
s = — 0.374 (9.5 mm.), r = 2.24" (57 mm.), I = 5.27"
(134 mm.).
bo = M5" (37 mm.),	= 5.83" (148 mm.), <5 = 30°.
Trick’s valve is especially well adapted for use on compound
marine engines, and has recetitly been used in the forms of
double and gridiron valves, as in Nos. 4 to 7 preceding.*
i 376.
BAIyANCED SEIDE VAEVES.
The resistance to motion due to the pressure is not so great
with a slide valve as with a lift valve of the same area, because
in the former case it is only necessary to overcome the friction
between the valve and its seat. For large valves, however, it
becomes so great as to render some method of balancing neces-
sary. It is desirable that even small slide valves should be
balanced, as by this means the wear upon valve face anci seat
can be greatly reduced. Balancing is most important for steam
engiue valves, and the following examples belong to this class ;
for other kinds of Service it is unimportant.
But few researches have been made upon the subject of valve
friction, but from such as have been made, and for such good
bearing surfaces as are used, the coefficient of friction may be
taken at 0.05 to 0.04.
American engineers, as we have already seen, enter into very
practical investigations and prosecute them with patience and
success, and to one of these, Mr. C. M. Giddings, we owe the
following results.f
Balanced Valve.—Cylinder 6%" X 10".
Revolutions	Pressure	K. P. of	H. P. N7	Ratio of	Ratio
per Minute.	Pounds.	Engine—N.	for Valve.	preceding.	a F
125	IO	3	A	2 per ct. 1.2 “	48
175	30	9	i		6l
200	40	13-5	i	1.4 “	91
Unbalanced Valve.—Cylinder 9" x i2/;. n = 100.
H. P. by Brake.	Ratio v N	Ratio a F
4-5	4.5 per cent.	247
7.0	3.5	245
8.25	4.0	330
8.9	6.0	534
II.I	7-3	8lO
Balanced Valve.— Cylinder 9 7 x 14"- n = 100.
H. P. by Brake.	Ratio w N	Ratio a F
11-4	1.2 per cent.	*37
13-5	11 “	149
14.0	1.0 “	I40
15.6	1.0 “	
The last column in each of the three tables has been added
by the author of this work, and is obtained as follows :	If N
and N' are the values in horse power of the engine and of the
resistance of the valve, v and v*, the corresponding mean
velocities of pistou and valve, and P and P' the force upon
N'
each, the experiments give the relation — = ipor P'v' —tyPv.
Hencejit follows for the force required to move the valve:
rf) P V
V'
Now for a given engine v’ bears a constant re-
lation to the number of revolutions «, so that we may put
*See Zeitschr. D. Ing. 1888, p. 509, Triple Expansion Engine byJG. E. C.
Meyer, of Hamburg.
f See Trans. Ara. Soc. Mec. Engrs., Vol. VII, p. 631: C. M. Giddings, Descrip-
tion of a Valve Dynamometer for measuring the power required to move a
slide valve at different speeds and pressures.
tZ> 2V	t/> N
P/ = -r----whence a P' —--------. The value a P' shows the
a n	n
increase in power required to operate the valve. It is evident
that P' increases more slowly than the increase in steam pres-
sure, but the resistance becomes quite great for unbalanced
valves. The present difficulty lies in the limi' ed number of
engines upon which experiments have been made. Fig. 1165
shows the character
of diagram made by
Gidding’s apparatus,
and it will be seen
that the greatest re-
sistance occurs at the
beginning of the
stroke, diminishing j
toward the end to
nearly zero. The
inequality between
the resistance of the
back and forward
strokes is due to the action of the steam pressure upon ihe
area of the valve rod.
In considering the pressure upon unbalanced slide valves ihe
consideration mentioned already in connection with Robin-
son’s experiments on lift valves is that there exists a counter
pressure between the valve and seat which overcomes an im-
portant portion of the pressure on the valve. As a rough ap-
proximation we may take this pressure between the surfaces as
XA (P\ — P)- Gidding’s experiments show that the coefficient
of friction is not constant, but diminishes with increased speed.
More extensive experiments are much to be desired.
The method of balancing slide valves may be divided into
three classes :
a.	Removal of pressure from the back of the valve.
b.	Opposing the pressure on the valve by counter-pressure.
c.	Equalization of pressure on all sides.
Typical examples of these three Systems will here be given :
a. Removal of pressure frcm back of valve.
1.	The so-called long /2-valve, invented by Murray, and used
on engines built by Watt, the pressure was relieved from the
back by a form of stuffing box, which answered well, but was
not adapted for high pressures.
Fig. 1166.
2.	Fig. 1166a shows the balance ring of Boulton & Watt.
The under side of the steam chest lid is finished parallel to the
valve face. Against this surface a ring of soft cast iron or
bronze is fitted steam tight, this ring being fitted to the valve
by an elastic packing and moving back and forth with it. The
space within the ring is subject only to the exhaust pressure.
This form was used on the Great Eastern.
Fig. n66£ shows the balance ring of Kirchweger, much used
for locomotive engines. In this form the ring is pressed
against the lid by steam pressure instead of spring packing.
Both of these devices, as well as the similar ones of Penn,
Borsig and others, leave too great a portion of the steam pres-
sure unbalanced (at least 30 per cent. being left), and also pre-
vent the valve from leaving its seat should water be carried
into the cylinder J
b. Balancing by counter-pressure.
3.	Cave’s Valve with Balance Piston.
The valve in this form, Fig. 1167, is connected by a link to a
piston, which works in a cylinder formed in the steam chest
lid, and is subjected on the outside only to atmospheric
pressure. Bourne’s method of balancing is similar, except
that the other side of the balance piston is in communica-
tion with the exhaust.
4.	Valves with Rolling Support, Fig. 1168.
At a is shown Lindneris valve. The top of the valve
itself is formed into a piston, sliding up and down in the
valve and supported by two segmental rollers.
t An elegant construction for this form of balanced valve is that of Robio»
son. (See Trans. Ara. Soc. Mech. Eugrs., Vol. IV, p. 375.286
THE CONSTRUCTOR.
The degree of balancing is dependent upon the size of the
piston. At b is shown Armstrong’s roller supported valve. In
this case the valve is closed at top as usual, and the construc-
tion is very simple and practical. At c is BristoPs valve, in
which the valve is supported on a system of friction rollers.
This has been used by the works at Seraing for large marine
engines. To this class of methods of balancing belongs also
that used by Worthington, in which a cylinder is formed in the
top of the valve, as shown in Fig. 1016.
L
.....:
j ff*	
	.1 Filii»
	IU
Fig. i 168.
5.	Cuvelier’s Valve, with pressure beneath,* Fig. 11690. The
ordinary slide valve is here combined with another, both being
made in one piece, and the combined valve held down to its
seat by pressure rollers. Live steam is admitted through the
passage I into the space between the two valves. At b is
Fitch’s valve, also with pressure beneath. In this form the
a.	T).
pressure rollers are omitted, and the valve is held down by live
steam pressure in the steam chest. The steam is admitted
through very small holes B i?, and escapes to the exhaust
through similar holes B' B'f so that the supply is about equal
to the loss by condensation. An objection to the use of valves
with pressure beneath is the large area of valve seat which is
required.
6.	Double Seated Valves, Fig. 1170. At a is Brandau’s valve,
and at b is Schaltenbrand’s valve ; the former is analogous to
Hornblower’s lift valve, Fig. 11450, and the latter to the bell
valve of Gros, Fig. 1145^. In neither form is the degree of bal-
ancing so complete as is desirable.
c. Equalization of Pressure on all Sides.
Fig. 1171.
7.	A very complete equalization of pressure is obtained by
making the valve in the form of a piston. Fig. 1171 shows a
recently designed piston valve with its steam cylinder. The
flat valve seat here becomes a cylinder, and the valve a double
piston, the flat sides of the valve disappearing. The valve pis-
tons are each fitted with a single ring behind which steam is
admitted through a small hole, thus renderingspringsunneces-
sary. The principal defect in piston valves is the question of
wear. The best results appear to be obtained by making the
piston valve solid, and very accurately turned and polished, and
made about " smaller in diameter than the boreof the valve
cylinder, both valve and valve cylinder being made of the same
material. Piston valves fitted in this manner last a long time.
8.	Rotary Valves : For steam hammers, in which valve gear
operated by hand has been found preferable to automatic
movements, valves with rotary movement, formed like cocks,
are used to advantage. These have been well designed by
Wilson, the superintendent of Nasmyth’s works.
a.	b.
!
Fig. 1172.
Fig. 11720 shows an oscillating valve by Wilson. Opposite
to the ports //, ///, IV} are false ports or recesses of shallow
depth. The steam enters at the end of the valve into the sym-
metrical spaces /, /. The unbalanced area of the steam of the
valve causes a corresponding endlong pressure which is received
by a thrust bearing. If we neglect the slight pressure due to
the steam in the false ports when expansion takes place in II
and ///, the valve is balanced on all sides. Very large oscil-
lating valves of this sort are easily moved by hand.f
A modification of this valve enables it to be operated by rota-
tion instead of oscillation, as shown in Fig. 1172b. Here the
parts are symmetrically arranged, as was not the case with the
old four way cock of Fig. 987. The exhaust passages IV con-
nect with one end of the valve, and the admission /, with the
other end. There remains here also an unbalanced end-long
pressure which is received by a thrust bearing. With this ex-
ception the valve is entirely balanced, and when well made the
thrust bearing offers but little resistance. The construction of
such valves demands a high degree of accuracy, and a specialty
of this form is made by the establishments of Dingler of Zwei-
briicken and of PfafF in Vienna.
The brief examination which we have given to the preceding
methods of balancing does not include a method which, while
offering great difficulties of construction, appears to be gradu-
ally coming into use. This method consists in snrrounding the
ordinary flat side valve with equalizing pressure plates. Several
practical illustrations of this method will be given.
Compare Fig. i<“34-
fSee Zeitschr. D. Ing., 186S, Vol. II, p. 207.THE CONSTRUCTOR.
287
9. \Vilson’s Balanced Valvc. Fig. 1173. (First shown at the
London Exhibition of 1862.)
1	1
The valve is symmetrical, and slides between two parallel
and similar faces, the lower face having openmgs correspond-
ing to the ports, the upper face having similar false ports. The
close fitting and accurate parallelism of the surfaces was de
pended upon to obtain the balancing. In practice it was found
that the balance plates would spring under the pressure of the
steam unless made very stiff and strong, and that the weight of
the valve caused much friction and wear.
Both of these difficulties have been met in more recent
designs, as will be seen below.
10. Fig. 11740 shows the balancing, of the valves of the Por-
ter-Allen engine.* The pressure piate.is made very deep and
stiff and formed with inclined plane bearings and set screws, by
which the pressure can be very closely regulated.
Fig. 1174^ is Sweet’s balanced valve. The pressure piate is
here also made heavy and stiff, and is supported on longitu-
dinal wedge bearings on each side, adjustable at the ends by
screws. In both forms the pressure piate is fitted with springs
to allow the piate to yield in case of water getting in the cylin-
der. These forms of balanced valve have the objection that
the ignorant mechanic may render the balancing ineffective by
improper adjustment of the screws, permittiug the full pressure
of the live steam to act upon the valve.
§ 377-/
Fujid Valves.
Valves may be formed of fluids, or, more generally speaking,
may be constructed of pressure organs. Ratchets adapted to
pressure organs, as fluid valves are properly called, are in ex-
tensive use, but have not generally been recognized as valves.
They are all reducible to one of two principal forms, either the
direct or inverted siphon. Fig. 1175« and b (compare \ 312).
A direct siphon connects two quantities of the same fluid
above the level of both portions, these levels differing, for ex-
ample by a height h; an inverted siphon is similar, but con-
nects them below the surface levels. Let ax and a2 be liquids,
* $ee Trans. Am. Soc. Mech. Engrs , Vol. IV, p. 268. C. C. Collins, Bal-
anced Valves.
which do not combine with a.f If the pressures of ax and a2 are
equal, the fluid a will flow from the higher to the lower level
under a pressure due to the height h. In the inverted siphon
the flow is constant, but with the direct siphon the flow is
stopped, and the siphon empties as soon as the level falis below
the short end of the siphon. J If the upper vessel is again
filled, the flow will begin as soon as the fluid attains the height
h' of the bend in the siphon. The fluid in the siphon there-
fore forms a valve, which converts a continuous flow into the
upper vessel into a periodical flow into the lower one (see the
example in \ 324, where a similar action takes place with a
rigid valve). This action of the siphon has recently been ap-
plied to excellent advantage.
When the pressures in ax and a2 are different, as represented
by the heights hx and hv as is frequently the case, we have for
both forms for the height to which the flow is due, hx -j- h — h2
for the height in au equivalent column of the fluid #.
If this valve is positive, there will be an outflow, if it is zero,
the fluid will be stationary, and if it is negative, there will be a
reverse flow. In cases in which hx + h — h2 = O, h represents
the measure of the difference between h2 and hx. It therefore
follows that by means of fluid valves the relation between the
fluids ax and a2 can be checked or controlled as may be desired.
Applications of fluid valves are very numerous, as the fol-
lowing examples will indicate :
Fig. 1176« shows a
water trap in a pipe. This	a	b>
is a fluid valve (inverted
siphon) which checks a
gas a2 from mingling
with a gas ax so long as
1	a		
ini 1		I « 1 !h 1	
1 1 1 1 •		1 1	
1	 	 1 U-—■			Jr
			
S: 1			1
1 1 • .y.-.J	|		S

/Ov-


Fig. 1176.
h\ — h2 is less than twice
the height s of the
branches of the siphon.
If the pressure from above
upon a increases the over-
flow runs off through a2.
This latter pipe must not
be too srnall, however, or
a siphon action will oc-
cur, and all the water
will be drawn off. This
device is much used in
gas works, Chemical
works, laboratories, etc.
Fig. 1176^ shows the
same arrangement used
as a barometer, mano-
meter, vacuum gauge,
etc., the difference of
level indicating differ-
ences of pressure h2 —hx
for valves below 2 s. Applications of this principle are very
numerous, from the largest forms to the most delicate physical
instruments.
Fig. 1177« is an open stand-pipe, used on certain forms of
low pressure boilers. This is practically an inverted siphon, of
which the boiler shell forms one branch. The fluid valve
checks the steam a2 against the atmosphere ax. If the pressure
becomes so great that h2 > hx + h' the fluid valve will be
thrown out at the top of the pipe, the arrangement thus form-
ing a safety valve against an excess of pressure in a2. This
device was for a long time in use for low pressure boilers,
Brindley’s feeding device, Fig. 1000, being constructed on this
principle. Natural stand-pipes with periodical discharge exist
as geysers.
Fig. 1177^ is a closed stand pipe for steam boilers. The pipe
which has first.been filled wdth steam gradually filis with water
as the steam condenses. If the water level in a sinks below
the end of the pipe the wrater runs out and live steam filis the
pipe again. This action is utilized in safety devices by Black
and Warner, and by Schwartzkopf.
In the blast furnace the fluid iron with the slag floating upon
it forms an inverted siphon which checks the blast. In the
Bessemer converter the air pressure is so great that the iron is
kept in agitation by the air bubbling through it.
fln this statement is included such fluids as do not mingle by simple
coutact. In this sense steam and water will not mingle, and it they are not
of the same temperature the warmer will be transferred to the other. Air
and water will not mingle because the water has becotne saturated with air.
According to the researches of Colladon & Sturin (Memoire sur la compres-
sion des liquides, 1827, reprinted by Schuchart, Geneva, 1887), the saturation
of water with air appears topartake of the nature of an internal, Chemical
combination. As might be expected, water which is saturated with air
shows a smaller compressibility in the Piezometer than water wbich is free
from air, being 48.65 millionths to 49.65 millionths. The combination of air
with water ceases upon heating to the boiling point.
J Natural inverted siphons with branches of varying levels exist in the
case of artesian wells.288
THE CONSTRUCTOR.
In gas holders the water in the tank forming the seal is a
fluid valve of tiie inverted siphon type (compare Fig. 948^),
ft.	b.
man’s furnace, Fig. 1178, in which ax is air, and a2 smoke, the
the bell-shaped lid being sealed with an annular valve of sand.
Fig. i i;8.
Fig. 1179 is Wilson’s water gas furnace.* Inthis a mixture of
waste-slack and water forms a fluid valve. The mixture is
propelled by an endless screw and discharged at the end. The
atmosphere is at ax and the gas at a2, the latter being kept
under pressure by a steatn 1*et.
&2
Hero’s Fountain, Fig. 1180, consists of two inverted siphon
valves, in which ax and a3 have air at the atmospheric pressure,
a2 is air under pressure, and a is water (often Cologne water).
The action continues until the column h \ = h2.
A practical application of the principle of Hero’s fountain is
the water trap of Morrison, Ingram & Co., Fig. u8i.f In this
device there is a periodical action of fluid valves as follows : a
stream of water flows into the tank i^at E, gradually filling it,
Fig. 1181«. The inner tube C] and fixed bell Df form an in-
verted siphon, the shorter branch of which is the space between
C and D. As soon as the level of the water in the tank F rises
above the top of C an overflow begins, filling the cup B, at the
foot of the pipe Cf and forming there a second siphon and
making a seal between a3 and a2, Fig. 1181A The two siphons
now form a Hero’s fountain, in which the contiuuing flow at E
a,	*.
causes an outflow into the discharge pipe A. As the level con-
tinues to rise in F, the air in a2 becomes more and more com-
pressed, until finally the pressure column h becomes greater
than the diflerence in level of the lower siphon, causing its dis-
charge and consequent opening of the fluid valve into a3. This
relieves the pressure on the air in a2, thus permitting the upper
siphon to act, and causing an immediate and rapid discharge
of the contents of F. By adjusting the rate of flow at E this
action can be regulated so as to take place periodically at any
desired intervals of time.
Richard’s manometer, Fig. 1182, consists of alternate direct
and inverted siphons ; a is quicksilver. a, steam, a2 water and
a3 atmospheric air.
The spiral pump and the Cagniardelle shown in Fig. 9660
and b contain successive fluid valves in the same pipe, alter-
nately direct and inverted.
Langen’s device for discharging bone furnaces of the hot
granular burnt bone, is a ratchet system involving valves con-
* See A. Wilson,'Generation of Heating Gas, etc., Journal of Soc. of Chem-
ical Industries, Manchester, Nov., 1883.	fSee Revue Industrielle, June, 1888, p. 226.THE CONSTRUCTOR.	289
sisting of a granular pressure organ, Fig. 1183. The discharge
pipe d of the furnace is closed at the bottom by the sliding
piate c which is given a reciprocating movement (iu this in-
stance operated by a small hydraulic motor). This piate c is
made with a step as shown in the figure at a, which receives a
layer of the material, and on the return stroke, as shown at b,
this layer is discharged on the piate. This layer forms a sue-
tion valve when acting as at a, and a discharge valve, as at b,
while the piate c corresponds to a single acting piston, con-
sidering the whole as a pump. If the piate c is made writh
amiddlerib, as shown in Fig. 1190^, it works both ways and
becomes a double-acting pump. This is an illustration of the
fluid valve in its most general form as applied to a pump.
In many instances fluid valves are as good and sometimes
even better than valves composed of rigid materials. Especially
is this the case when they act continuously in one direction in
in a free, open pipe, for which purpose they excel ali other
forms of valves, as in jet pumps and the like (see Fig. 972).
?37S.
Stationary Vatves.
We have thus far considered valves as ratchets for pressure
organs, when they operate so as to check the motion of the
fluid at the intervals of time (see $ 365). If we consider this
definition in its most general sense we may take it to include
certain kinds of fastenings for closing apertures, and call these
also valves. These we may distinguish from ordinary valves
by the fact that they are not operated by the motion of the
machine, and hence to them may be given the name of
“stationary valves.”
Stationary lift valves are found in the lids of steam cylinders,
these belonging to the class of disk valves. These are required
to resist internal pressure, and must therefore be securely
bolted in place, the pressure being generally great, and resisted
by the bolts. Steam chest covers are generally rectangular,
flat, stationary valves, and an example of a stationary flap
valve is seen in the valve chest door shown in Fig. 1128, this
also being secured by means of bolts. Furnace doors, such as
shown in Fig. 763, also belong to this class. The more readily
such a valve is opened and closed the more nearly it approaches
in construction to the movable valves, and packing is sometimes
omitted in order to facilitate opening and closing. The valve
chest lids, shown in Fig. 1131, are readily recognized, these
being readily slipped into place and held by a yoke, or so-
called “gallows screw.” Numerous forms of stationary valves
are also found in various kinds of bottle stoppers, these being
effective substitutes for the older cork stoppers which often
were held in place only by friction. Stationary fluid valves are
also occasionally stili found in use for bottle stoppers in parts
of Italy and Greece.
In all the cases thus far mentioned the fastening by which
the stationary valve is held in
place must be at least slightly
stronger than the pressure beneath
the valve.
As a stationary valve in which
this is not the case, we have the
ordinary manhole piate as used
in steam boilers, Fig. 1184. In
this the pressure acts to hold the
piate to its seat. Other examples
are found in the spring valves
used in the so-called siphons of
soda water, and the particular
form of bottle stopper which con-
sists of a small ball valve held up
to the mouth of the bottle by the
pressure within. Stationary
slide valves are less frequently
used than lift valves, as the con-
ditions are less favorable for
proper packing, but examples are
to be found. It will be seen by
the instances already given how
far reachiug luto all bianches of
machine design the use of ratchets
for pressure organs extends.
2 379-
Stationary Machine EtEments in Generat*
It is not a peculiarity of valves alone to be used conveniently
in the “stationary ” form in the sense discussed in the preced-
ing section. Here, as we have arrived at the close of the book,
it is desirable to review the preceding pages in this respect. In
the first four chapters of Section III the subjects considered are
nearly always used as stationary elements.
Rivets do not differ in form from cylindrical journals, but
they are generally stationary because of two conditions; be-
cause of thefirm bindingof the surrounding metal, and because
there are generally two or more rivets placed side by side. If
only single rivet is used and no impediment to movement in-
troduced, the binding of the metal would soon give way to any
forces tending to cause rotation.
Forced connections resemble journals and their bearings in
form. The force by which the external piece grasps the inter-
nal one effectively resists all forces acting to produce rotation.
Keyed connections are especially adapted for stationary Service.
The particular examples showm in Figs. 618 and 619 are in fact
stationary keys in form, although really special cases of spiral
gear wheels. Screws, in by far the greater number of cases, are
used as stationary elements, probably in a greater variety of
applications, broadly considered, than any other machine ele-
ment. In § 86 a glance is given at the use of the screw as an
active machine element.
Journals are frequently conveniently used as stationary ele-
ments, as in the examples illustrated iu Figs. 251, 252, 253, 256,
257 and 258. In § 90 we have already distinguished between
“journals at rest” and “running journals,” the former corres-
ponding to the definition of stationary elements. Roller bear-
ings for bridge truss supports, \ 198, are also stationary ele-
ments.
Crank connections are found in the bottle stoppers already
mentioned, and in numerous other applications such connec-
tions are properly considered as stationary elements, Here
wheels are rarely used as stationary elements, but such applica-
tions are frequently found of ratchet wheels. Eongitudinal
keys used to secure hubs upon their axles are almostinvariably
stationary elements, practically corresponding to “stationary
ratchets,” as a comparison between Figs. 188 and 654 will show.
Ratchets also find numerous applications in stationary mechan-
ism for securing bolts, keys and the like. An examination of
Figs. 237 to 243 and 246 to 248 will illustrate this point. In the
couplings shown in Figs. 423 to 430 we also have a number of
stationary ratchets (see also Fig. 678).
In \ 309 I have referred to the possibility of using pressure
organs as standing or “ stationary ” elements, but these are as yet
unimportant. The pipes used as conductors for pressure organs,
however, furnish numerous instances of pressure organs.
The above distinctions are by no means merely theoretical,
but are of a highly practical nature. Every means which will
enable us to obtain a clearer and better comprehension of the
use of machine elements should be most welcome.
In the preceding arrangement the stationary elements have
therefore been grouped together for this end. It followTS that
those forms which as “stationary ” or “passive” e1ements are
extensively used ia building and civil works, as wrell as in ma-
chine design, formmg the connecting links between the works
of the civil and the mechanical engineer.
|
i
i
i
Fig. 1184.THE CONSTRUCTOR.
291
7
SECTION IV.
MATHEMATICAE TABLES.
3 380.
Tabees of Curves, Areas and Voeumes.
The following tables give in convenient form the most im-
portant geometrical and mechanical properties of the more
useful curves, areas and volumes. The significance of the let-
ters used in the formulae will be found indicated on the dia-
grams. The following remarks are also to be noted.
By the rectification of a curve is meant the length 5 of that
portion of the curve from the origin to the point x y, corres-
ponding to the angle $ ; and by 6* is meant the entire length of
the curve.
In the moment of inertia the mass of the body is assumed
= 1, in order to reduce the number of letters. In view of the
importance of this subject a few points are here given. The mo-
ments of inertia for surfaces are both equatorial and polar, each
referred an axis of moments. This latter is called an equatorial
axis when it lies in the plane of the surface, and a polar axis
when it is at right angles to the surface. Each equatorial
which passes through the centre of gravity is especially termed
an equator-axis, and a polar axis which passes through the cen-
tre of gravity is called a pole axis. Every surface, therefore,
has but one pole-axis, and an infinite number of equator axes.
The moment of inertia is called equatorial or polar, according
to the axis to which it is referred.
The moment of inertia Jp for any surface referred to the
polar axis is found by adding together the two equatorial mo-
ments of inertia Jg\ and Jq2, the axes of which intersect each
other at right angles in the polar axis :
Jp = Jq\ + Jq%................................(4*6)
The moment of inertia Jr of a surface, referred to any axis
sitnated at a distance a, from the centre of gravity S, is found
from the moment of inertia J referred to a parallel axis through
Sy by the following relation :
J' — J + a2 F................................(417)
in which i ? is the area of the surface. This relation also holds
good for solids, if the mass of the body is substituted for F.
For solids one of the preceding conditions does not hold.
For each different shape one of the axes which passes through
the centre of gravity, is taken as the pole-axis for ali sections
normal to it, and the section at right angles to this axis which
passes through the centre of gravity is called the Equatorial
Section, whence the equatorial and polar moments of inertia
are in these cases distinguished according to their position with
regard to this equatorial section. In all the examples of solids
here given, the actual equatorial and polar axes are meant.
For a right prism, of any given base having as the polar mo-
ment of inertia ip and the half-height = /, the polar moment
of inertia is:
Jp — 2. I ip.................................(418)
and the moment of inertia referred to an equatorial axis :
Jp = %flz + zliq.............................(419)
in which f is the area of the cross section, and iq the equatorial
moment of inertia of the cross section referred to the same
axis as Jq.
The centre of gravity and the moment of inertia for a surface
of irregular form is often readily obtained by grapho-static
methods, with sufficient numerical accuracy. For this purpose
the force and cord polygons are applicable according to the
methods already described in Section II.No.
I.
Circle.
ii.
iii.
Ellipse.
IV.
Curve.
Rectanguear Ec^uation.
From O: (x — a)2+(y — b)2 = r2.
From S: y2~2rx — x2.
From M: x1 -\-y2 = r1.
From S : y2 — 2 p x.
		y %	
* V. Catenary.	\ h i\ L «		JL-u	y
From M; y1 a2 + x% b2 = a2 b1.
y2 = C~	{2 ax — X1)
From S:
y2 = 2p X — —- X1.
From O: —y1 dl -f- x2 b2 = a2 b2.
From O: y = -^~ ( e6 + e~

Speciae Properties.	Poear Equation. ,	Radius oe Curvature.	Rectieication.
x Approximately, when — is small: 9' r y y 2 x	From O; fi2 -f* f1 — 2 />y«w (p — r2. From ^.* f)=:2r COS <f>.	P = r.	s = r<p S = 2 7r r
Semi-parameter =p A S— SF=P 2	From F: r=	\- x 2 r-_ . P .				+ l„e „„1 (Vy + >/ i+jJJ
L L — Directrix. F== Focus.	2 sin2J*_ 2		Approximately, when — is small: -[-KD-kf)']
Kccentricity — e = v' d2 — b2 e Numerically = e = Semi-parameter: b1	From F: a2 _ e'l rs=.p e x ~	 a — e cos (p 	P	 I — £ COS (p Radii: r=-a-j-ex; r'-=a — ex	a b For 5/ /> = — a _ A . d2 For A : p — - 0	<, = .(« + *) (1+ 4+”‘ + n6 \ . 25$ J a + l> 1
p^ccen. --= O F— a2 4- £2 e Numerically £ = —. Cl Axis b — e2 — 1 p Semi-param. — a (e2— 1) = —	From F: e-i_a2 r = p 4- £ x =	—— a — e cos <p =	p	 I — £ COS (p Radii: r — e x — a} r' = e x a.	(a4y2 -f- b4 x2) '2 ,, = (I. 6,-		Very complicated.
		*	
, / t \ Kl + 2 h c /*«*=_ V—^— L L = Directrix.		y2 pamT	-e ) — \/y2 — c2, / =\/h2-\-2 h c
VO
to
THE CONSTRUCTOR..Rectanguear Equation.
r	r' . V
x — r lu ——sin w \
A	r' \
y = rll----—COSi0 )
x= (R -f f) cos -jj<J—r'cos
R +r
— r* cos —— o)
R
r	. R+r
y —(R r) sin — « — r' sm —- — w
K	K	Jv
,, r , R—r
x=(R— r)cos -gU + r' cos
, x . r , . R—r
y-=z(R—r)sm—u — r' sin — — o
K	K	K
r	r—R
#== (r—R) cos — w — r' cos — — <.
K	K
r	. r—R
yz=(r—R)sin — o—r'sin— - <.
r	K	K
x = R' cos d -f- R&sin $
y = R/ sin & — R & cos &
r=Ra>
/
Rectification.
s = 4r

S=8r
s — 4r
R+r
Q-cost)
S=8r
R+r
R
R-r
s~4r-x
Q~cost)
5=8r
R-r
R
Radius of Curvature,
p = 4 r sm
R -f* r . w
P -- 4 r -------sm —
R + 2 r 2
p ~ 4 r
R — r . o)
R — 2 r 2
r—Rf	«\
^4r--^-cos-J
c e r—^
S=Sr
RV *
.£ =----
2
7?r
5=----[wv I -|- <
-f- log nat (w +s/ i + J2)]
v^i -|- (log nat a)2
log nat a
r — R . o)
P = 4r------------ sin —
2 r — R 2
Remarks.
^ — R \

(r* + ar)i
2 (r1 + 2 a1)
f> — r>/i + (log nat a)2
—	whence
sm a
cot a = log nat a.
rf is the radius to the describing point. When
r —t', as here assumed in the rectification and curva-
ture formulae the equations become those of the com-
moli orthocycloid, epicycloid, hypocycloid and peri-
cycloid. Also in the equation of the evolute R/ is
the distance of the describing point from 0 when
19 — 0; if R' = R the curve becomes the common
evolute. I lie length of the common cardoid is :
S= 8 X 2 R = 2.546 (2 n R)
Or approximately 0.81 ir (2 n R)
s, for 1 revolution = w (2 n R)
s, for 2 revolutions = 4 n (2 ir R)
s} for n revolutions = n2 ir (2 n R)
The Archimedian spiral is the prolonged evolute
of R.
s} for 1 revolution =■ 1.08 ir (2 n R)
s, for 2 revolutions = 4.09 t (2 n R)
s, for n revolutions s= n2 n (2 n R) approximately.
In the logarithmic spiral the tangent at any point
/^makes the angle a with O	whence cot a = log nat a.
to
vo
00
THE CONSTRUCTOR.294
THE CONSTRUCTOR.THE CONSTRUCTOR.
295
No.	Form.	SURFACE.	VOEUME.	Centre of Gravity. !	Moment of Inertia.
XXI. Trianolar Prisi.	ajLc	Sides: Fx = 2 / {a + b -f- c) One end : P2 = ^ 2	V=bhl	For Equator axis O : . r/2 , I /„ = »« ■ 1 =	1	
				Centre of Figure.	* L_ 3 i£>_J 3 01 For Pole axis PP: F b li3 h bh /p = m L~ g- + V («3 + **)-- (2 l
XXII. Rectanolar Prisi.		Sides: Fx — 4 / (b -f h) One end : F2 = b h	V= 2 bhl	Centre of Figure.	• For Equator axis O Q : r 12 hi \ J1~’“ (y + nr) For Pole axis PP: Wl JP=-- w + p)
XXIII. Rhoibic Prisi.	-fi-	1	b' Sides :i^=8/V/52+ - 4 One end : F2 = b h	r=2 bhi	Centre of Figure.	For Equator axis Q Q : For Pole axis P P: (?+£■)
XXIV. Heiagonal Prisi.		Sides : Fx = 12 l r One end : F2— ~p\/3— = 2.598 r2	K=3/^s/y= = 5.196 l r2	Centre of Figure.	For Equator axes Q Q and Qx Qx : ^=w/ (f+i'0 ! For Pole axis P/5: Jp = T2mr'
XXV. CyliMer.		Vertical surface: C = 4-/r One end : F2 = n r2	V — 2 7r l r1	Centre of Figure.	For Equator axis <2 <2 • J1 = "‘(f+t) For Pole axis PP: Jp-^ l2 m
XXVI. Hollow CyMer.		Vertical surface: /^=4^ l(rx-}-r2)=STTlt One end: F2 n (r,2 — r22) — 2 Krb	V=2ivl(r2—r2) — \r:rbl	Centre of Figure.	For Equator axis Q Q : r/2 , p , d2-] L3 2 8 J For Pole axis P P: Jp = ™ [n2 + = rn 3+
XXVII. ParaDoUcPrism. i		One end : P, = — x y 3 '	y= — i x y	5	! For Equator axis Q Q : For Pole axis PP:296
THE CONSTRUCTOR.
No.
Form.
XXVIII.
Glotoid Ring.
XXIX.
Rectangular
Pyramid.
XXX.
Risit Cone.
XXXI.
Trnncated Cone.
XXXII.
Spliere.

r	? 1 / .:;Bk jb	F\ = a	— 4
j.HHr.. j’	+ b	\J a2 ^ h2+ - 4
	Bottom :	0' Q II t*r
Sides :
=57 (''1 + rF(n—
— 2 7r r s
Ends:
F/ — rx2 F</' = r22 n
XXXIII.
SectorofSpbere,
XXXIV.
Sesment of
Spliere.
XXXV.
Spberoid.
XXXVI.
Paraboloid of
ReyolQtion.
SURFACE.
P=4 -2 R r
Inclined surface:
Fx = 7r fs/h2 -}- r2 — ir r s
Bottom : F* = ir r2
F=.4r*n
Conical surface:
Fx= a ~ r= 7r ;V2 r h — h'1
Curved surface :
Fx — 2 v r h = 7T (a2 -f- h2)
Bottom :
F2 = a2 7T, r =
2 h
Bottom : F2—rcy2
Voeume.
V=2rr2 Rr
v=Fi JL
3	3
V=
7r r2 h
V=-h\r2+r,r2+r2-}
Centre of Gravity Moment of Inertia.
Centre of Figure.
Jb =
r—
4"
For the surface onlv
*-±
3

For Equator axis <2 0 :
^ = w [?+ 8 'Q
For Pole axis /* P:
E^+v"2]
For Equator axis Q Q :
L80	^ 20 J
For Pole axis P P :
For Equator axis Q Q :
Ja = — ™
*	20
D-f]
For Pole axis P P :
3
10
Jp=-J-m ^
For Pole axis PP:
__________Pa_\
4' N+r^+r./ /	7» —
10 r,A — rJ
V=^r>
F= — Kph
3
Centre of Figure.
For Equator axis Q Q :
/=y»<!
For Pole axis P P:

-h2)
(—0
7-^(3«’ + A*)
3 (ar-A)»
For the surface only
2
F= — 7rabe
3
V=: — xy2
2
Centre of Figure.
.r
3
For Pole axis P P:
jp—m \_r'--\rh+
—FI ———
2° J 3#*—h
For Equator axis @ (?,
coincident with a :
For Equator axis Q Q :
For Pole axis P:
r m 2
JP = TyTHE CONSTRUCTOR,
29 7
§381.
Trigonometricae Tabide.
The following table contains, in convenient form, the sines, cosines, tangents and cotangents for angles from o° to 90°
lor every ten minutes, and also the corresponding ares to a radius of unity. At the foot of the table ares are also given for
small angles and also for some of the more frequently used angles greater than 90°.
ANGI, E.		are.	sine.	cosine.	tan. i	cot.	arc.	! angee.		ANGEE.		arc.	sine.	cosine.	tan.	cot.	arc. <	ANGEE.	
deg..	inin.							deg.	min.	deg.	min.							ieg.	min.
O	0	0.0000	0.0000	1.0000	0.0000	00	1.5708	90	0	10	0	0.1745	0.1736	0.9848	°.i763	5.6713	1.3963	80	0
	IO	0.0029	0.0029	1.0000	0.0029	343-77	1.5679		50		10	0.1774	0.1765	0.9843	0.1793	5.5764	1-3934		5°
	20	0.0058	0.0058	1.0000	0.0058	171.89	1.5650		40		20	0.1804	0.1794	0.9838	0.1823	5.4845	1.3904		40
	30	0.0087	0.0087	1.0000	0.0087	114.59	1.5621		30		30	0.1833	0.1822	0.9833	0.1853	5-3955	1.3875		3°
	40	0.0116	0.0116	0.9999	0.0116	85.940	1.5592		20		40	0.1862	0.1851	0.9827	0.1883	5-3°93	1.3846		20
	50	0.0145	0.0145	0.9999	0.0145	68.750	1.5563		10		50	0.1891	0.1880	0.9822	0.1914	5-2257	1.3817		10
I	0	0.0175	0.0175	0.9998	0.0175	57.290	1-5533	89	0	11	0	0.1920	0.1908	0.9816	0.1944	5.1446	1.3788	79	0
	IO	0.0204	0.0204	0.9998	0.0204	49.104	1.5504		50		10	0.1949	0.1937	0.9811	0.1974	5-0658	1-3759		50
	20	0.0233	0.0233	0.9997	0.0233	42.964	1-5475		40		20	0.1978	0.1965	0.9805	0.2004	4.9894	1.3730		‘40
	30	0.0262	0.0262	0.9997	0.0262	38.188	1.5446		30		30	0.2007	0.1994	o.9799	0.2035	4.9152	1.3701		3°
	40	0.0291	0.0291	0.9996	0.0291	34-368	I.54I7		20		40	0.2036	0.2022	0-9793	0.2065	4.8430	1.3672		20
	50	0.0320	0.0320	0.9995	0.0320	3X*242	1.5388		10		50	0.2065	0.2051	0.9787	0.2095	4.7729	1-3643		10
2	O	0.0349	0.0349	0.9994	0.0349	28.636	1-5359	88	0	j 12	0	0.2094	0.2079	0.9781	0.2126	4.7046	1.3614	78	0
	IO	0.0378	0.0378	0.9993	0.6378	26.432	1-5330		50		10	0.2123	0.2108	0.9775	0.2156	4-6382	1.3584		50
	20	0.0407	0.0407	,0.9992	0.0407	24.542	1-5301		40		20	0.2153	0.2136	0.9769	0.2186	4.5736	1.3555		40
	30	0.0436	0.0436	0.9990	0.0437	22.904	1.5271		30		30	0.2182	0.2164	0.9763	0.2217	4.5'o7	1.3526		3°
	40	0.0465	0.0465	0.9989	0.0466	21.470	1.5243		20		40	0.2211	0.2193	0.9757	0.2247	4.4494	1-3497		20
	50	0.0495	0.0494	O.9988	0.0495	20.206	1-5213		10 |		50	0.2240	0.2221	0.9750	0.2278	4.3897	1.3468		10
3	O	0.0524	0.0523	O.9986	0.0524	19.081	1.5184	87	0 1	S ! 1 13	0	0.2269	0.2250	0.9744	0.2309	4.3315	1-3439	77	0
	IO	0.0553	0.0552	0.9985	0.0553	18.075	I-5I55		50		10	0.2298	0.2278	0.9737	0.2339	4.2747	i.34io		5°
	20	0.0582	0.0581	0.9983	0.0582	17.169	1.5126		40.		2D	0.2327	0.2306	0.9730	0.2370	4.2193	i.338i		40
	30	0.0611	0.0610	O.998I	0.0612	16.350	1.5097		30		30	0.2356	0.2334	0.9724	0.2401	4.1653	1-3352		30
	40	0.0640	0.0640	O.998O	0.0641	15-605	1.5068		20		40	0.2385	0.2363	0.9717	0.2432	4.1126	1.3323		20
	50	0.6669	0.0669	0.9978	0.0670	14.924	1.5039		10		50	0.2414	0.2391	0.9710	0.2462	4.06ll	1.3294		10
4	O	0.0698	0.0698	0.9976	0.0699	14.301	1.5010	86	0	H	O	0.2443	0.2419	0.9703	0.2493	4.0108	1.3264	76	0
	IO	0.0727	0.0727	0.9974	0.0729	i3-727	1.4981		50		IO	0.2473	0.2447	0.9696	0.2524	3.9617	1.3235		5°
	20	0.0756	0.0756	0.9971	0.0758	13-197	I.495I		40		20	0.2502	0.2476	0.9689	0.2555	3-9136	1.3206		40
	30	0.0785 1	! 0.0785	O.9969	0.0787	12.706	1.4923		30		30	0.2531	0.2504	0.9681	0.2586	3.8667	1.3177		3°
	40	0.0814	| 0.0814	O.9967	0.0816	12.251	14893		20		40	0.2560	0.2532	0.9674	0.2617	3.8208	1.3148		20
	50	0.0844	0.0843	O.9964	0.0846	11.826	1.4864		10		50	0.2589	0.2560	0.9667	0.2648	3.7760	1.3119		10
5	O	0.0873	0.0872	O.9962	0.0875	11.430	14835	85	0	x5	O	0.2618	0.2588	0.9659	0.2679	3.7321	1.3090	75	0
	IO	0.0902	0.0901	0.9959	0.0904	n.059	1.4806		50		IO	0.2647	0.2616	0.9652	0.27II	3.6891	1.3061		50
	20	0.0931	0.0929	0.9957	0.0934	10.712	14777		40		20	0.2667	0.2644	0.9644	0.2742	3.6470	1.3032		40
	3°	0.0960	0.0958	0.9954	0.0963	10.385	14748		30		30	0.2705	0.2672	0.9636	0.2773	3.6059	1.3003		30
	40	0.0989	0.0987	0.9951	0.0992	10.078	14719		20		40	0.2734	0.2700	0.9628	0.2805	3.5656	1.2974		20
	50	0.1018	0.1016	O.9948	0.1022	9.7882	1.4690		10		50	0.2763	0.2728	0.9621	0.2836	3.5261	1.2945		10
6	0	0.1047	0.1045	0.9945	0.1051	9-5144	1.4661	84	0	16	O	0.2793	0.2756	0.9613	0.2867	3.4874	1.2915	74	0
	IO	0.1076	0.1074	0.9942	O.I080	9-2553	1.4632		50		IO	0.2822	0.2784	0.9605	0.2899	3.4495	1.2886		5°
	20	0.1105	0.1103	0.9939	O.IIIO	9.0098	1.4603		40		! 20	0.2851	0.2812	0.9596	0.2931	3.4124	1.2857		40
	30	0.1134	0.1132	0.9936	O.II39	8.7769	1-4573		30		30	0.2880	0.2840	0.9588	0.2962	3-3759	1.2828		30
	40	0.1164	0.1161	0.9932	O.H69	8.5555	1.4544		20		40	0.2909	0.2868	0.9580	0.2994	3.3402	1.2799		20
	50	0.1193	0.1190	0.9929	O.II98	8.3450	1-4515		10		50	0.2938	0.2896	0.9572	0.3026	3.3052	1.2770		10
7	O	0.1222	0.1219	0.9925.	0.1228	8.1443	1.4486	83	0	l7	0	0.2967	0.2924	0.9563	0.3057	3.2709	1.2741	73	0
	IO	0.1251	0.1248	0.9922	0.1257	7.9530	1-4457		50		10	0.2996	0.2952	0.9555	0.3089	3.2371	1.2712		5o
	20	0.1280	0.1276	O.99I8	0.1287	7.7704	1.4428		40		20	0.3025	0.2979	0.9546	' 0.3121	3.2041	1.2683		40
	30	0.1309	0.1305	°*99I4	0.1317	7-5958	1,4399		30		30	0.3054	0.3007	0.9537	0.3153	3.1716	1.2654		30
	40	0.1338	0.1334	0.9911	O.I346	7.4287	1.4370		20		40	0.3083	0.3035	0.9528	0.3185	3.1397	1.2625		20
	50	0.1367	0.1363	0.9907	O.I376	7.2687	I-434I		10		50	0.3113	0.3062	0.9520	0.3217	3.IO84	1.2595		10
8	O	0.1396	0.1392	0.9903	O.I405	7-H54	1.4312	82	0	18	0	0.3142	0.3090	0.9511	0.3249	3.0777	1.2566	72	0
	IO	0.1425	0.1421	0.9899	°-I435	6.9682	1.4283		50		10	0.3171	0.3118	0.9502	0.3281	3-0475	1.2537		50
	20	0.1454	0.1449	0.9894	0.1465	6.8269	1.4254		40		20	0.3200	o.3i45	0.9492	0-3314	3.0178	1.2508		40
	3°	0.1484	0.1478	0.9890	0.1495	6.6912	1.4224		30		30	0.3229	o.3i73	0.9483	0.3346	2.9887	1.2479		3°
	4°	0.1526	0.1507	0.9886	0.1524	6.5606	I.4I95		20		40	0.3258	0.3201	0.9474	0.3378	2.9600	1.2450		20
	5°	0.1542	0.1536	0.9881	0.1554	6.4348	1.4166		IO		50	0.3287	0.3228	0.9465	0.3411	2.9319	1.2421		10
9	0	0.1571	0.1564	0.9877	0.1584	6.3138	I.4I37	81	O	19	0	0.3316	0.3256	0.9455	'o.3443	2.9042	1.2392	7i	0
	10	0.1600	0.1593	0.9872	0.1614	6.1970	1.4^08		50		10	0.3345	0.3283	0.9446	0.3476	2.8770	1.2363		5°
	20	0.1629	0.1622	G.9868	0.1644	6.0844	1.4079		40		20	0.3374	°.33i1	0.9436	0.3508	2.8502	1.2334		40
	3°	0.1658	0.1650	0.9863	0.1673	5-9758	1.4050		30		30	0.3403	0.3338	0.9426	o.354i	2.8239	1.2305		3°
	40	0.1687	0.1679	0.9858	0.1703	5.8708	1.4021		20		40	0.3432	0.3365	0.9417	0.3574	2.7980	1.2275		20
	50	0.1716	0.1708	0.9853	0.1733	5.7694	1.3992		IO		50	0.3462	0.3393	0.9407	.0.3607	2.7725	1.2246		10
Angle.		are.	cosine.	sine.	cot.	i tan. i	arc.	Angle.		| Angle.		arc.	cosine	. sine.	cot.	tan.	arc.	Angle.	298
THE CONSTRUCTOR.
ANGEE.		arc.	sine.	cosine.	tan.	cot.	arc.	ANGEE.		ANGEE.		arc.	sine.	cosine.	tan.	cot.	arc. J	ANGEE*	
deg.	min.							deg.	min.	deg. j I	min.							deg-|	min.
20	0	0.349 <	0.3420	o.9397	0.3640	2.7475	1.2217	7°	0	31	0	0.5411	0.5150	0.8572	0.6009	1.6643	1.0297	59	0
	IO	0.3520	0.3448	0.9387	°-3673	2.7228	1.2188		50		10	0.5440	0.5175	0.8557	0.6048	1.6534	1.0268		5o
	20	0-3549	0.3475	0.9377	0.3706	2.6985	1.2159		40		20	0.5469	0.5200	0.8542	0.6088	1.6426	1.0239		40
	30	0.3578	0.3502	0.9367	0.3739	2.6746	1.2130		30		30	0.5498	0.5225	0.8526	0.6128	1.6319	1.0210		30
	40	0.3607	0.3529	0.93S6	0.3772	2.6511	1.2101		20		40	0.5527	0.5250	0.8511	0.6168	1.6212	1.0181		20
	50	0.3636	0*3557	0.9346	0.3805	2.6279	1.2072		10		50	o.5556	0.5275	0.8496	0.6208	1.6107 I i	1.0152		10
21	0	0.3665	0.3584	0.9336	o.3839	2.6051	1.2043	69	0	32	0	°.5585	0.5299	0.8480	0.6249	1.6003	i 1.0123	53	0
	IO	0.3694	0.3611	0.9325	0.3872	2.5826	1.2014		50		10	0.5614	0.5324	0.8465	0.6289	1.5900	1.0094		50
	20	0.3723	0.3638	0.93!5	O.39O6	2.5605	I-I985		40		20	0.5643	0.5348	0.8450	0.6330	1.5798	1.0065		40
	30	0.3752	0.3665	0.9304	0-3939	2.5386	i-i955		3°		30	O.5672	0.5373	0.8434	0.6371	1.5697	1.0036		30
	40	0.3782	0.3692	0.9293	0.3973	2.5172	1.1926		20		40	0.5701	0.5398	0.8418	0.6412	1.5597	1.0007		20
	50	0.3811	o.37i9	0.9283	O.4OO6	2.4960	1.1897		10		50	0.5730	0.5422	0.8403	0.6453	1.5497	0.9977		10
22	O	O.384O	0.3746	0.9272	O.4O4O	2.4751	I.I868	68	0	33	0	0.5760	0.5446	0.8387	0.6494	1-5399	0.9948	57	0
	IO	O.3869	0-3773	0.9261	0.4074	2.4545	1.1839		50		10	0.5787	o.547i	0.8371	0.6536	1.5301	0.9919		50
	20	0.3898	O.38OO	0.9250	0.4108	2.4342	I.I8IO		40		20	O.5818	o.5495	0.8355	0.6577	1.5204	0.9890		40
	30	0.3927	0.3827	0.9239	0.4142	2.4142	1.1781		3°		30	O.5847	o.55i9	0.8339	0.6619	1.5108	0.9861		30
	40	0.3956	0.3854	0.9228	O.4I76	2.3945	I.1752		20		40	O.5876	0.5544	0.8323	0.6661	1.5013	0.9832		20
	50	0.3985	O.388I	0.9216	0.4210	2.3750	1.1723		10		50	0.5905	0.5568	0.8307	0.6703	1.4919	0.9803		10
23	O	O.4OI4	0.3907	0.9205	0.4245	2.3559	I.I694	67	0	34	0	0.5934	o.5592	0.8290	0.6745	I .4826	0.9774	56	0
	IO	0.4043	0-3934	°*9I94	0.4279	2.3369	I.I664		50		10	0.5963	0.5616	0.8274	0.6787	1.4733	0.9745		50
	20	0.4072	0.3961	0.9182	0.43*4	2.3183	I.I636		40		20	0.5992	0.5640	0.8258	0.6830	I.464I	0.9716		40
	30	0.4102	0.3987	0.9171	0.4348	2.2QQ8	I.I606		30		30	0.6021	0.5664	0.8241	0.6873	I.455O	0.9687		30
	40	0.4131	O.4OI4	0.9159	0.4383	2.2817	I-I577		20		40	0.6050	0.5688	0.8225	0.6916	1.4460	0.9657		20
	50	0.4160	O.4O4I	0.9147	0.4417	2.2637	1.1548		10		50	0.6080	0.5712	0.8208	0.6959	1.4370	O.9628 ;		10
24	O	O.4I89	O.4O67	0.9135	0.4452	2.2460	I.I5I9	66	0	35	0	O.6IO9	0.5736	0.8192	0.7002	I.428l	0.9599	55 |	0
	10	0.4218	O.4O94	0.9124	0.4487	2.2286	1.1490		50		10	O.6I38	0.5760	0.8175	0.7046	I.4I73	0.9570		! 50
	20	0.4247	0.4120	0.9112	0.4522	2.2113	1.1461		40		20	O OI67	0.5783	0.8158	0.7089	1.4106	0.9541		I 40
	30	0.4276	O.4I47	0.9100	0.4557	2.1943	I.I432		3°		30	0.6 96	0.5807	0.8141	0.7133	1.4019	0.9512		i 30
	40	0.4305	0.4173	0.9088	0.4592	2.1775	I.IA03		20		40	0.0225	0.5831	0.8124	0.7177	1-3934	0.9483		j 20
	50	0-4334	0.4200	0.9075	U.4628	2.1609	I*I374		10		50	0.6254	0.5854	0.8107	0.7221	1.3848	0.9455 j	i	IO
25	O	0‘4363	0.4226	0.9063	O.4663	2.I445	I.I345	65	0	36	0	0.6283	0.5878	0.8090	0.7265	1-3764	0.9425	54	O
	10	0.4392	0.4253	0.9051	0.4699	2.1283	1.1310		50		10	0.6312	0.5901	0.8073	0.7310	I.368O	0.9306		50
	20	0.4421	0.4279	0.9038	0-4734	2.1123	1.1280		40		20	0.6341	0.5925	0.8056	o.7355	1.3597	0.9367		40
	30	0.445 1	0.4305	0.9026	0.4770	2.O965	1.1257		3°		30	0.6370	0.5948	0.8039	0.7400	I.35H	0.9338		30
	40	O.448O	o*4331	0.9013	O.4806	2.0809	1.1228		20		40	0.6400	0.5972	0.8021	o.7445	I*3432	0.9308		20
	50	0.4509	0.4358	0.9001	O.484I	2.0655	1.1199		10		5o	0.6429	o.5995	0.8004	0.7490	i.335i	0.9279		IO
26	! 0	0.4538	0.4384	0.89S8	0.4377	2.0503	I*II7°	64	0	37	0	0.6458	0.6018	0.7086	0.7536	1.3270	0.9250	53	O
	10	O.4567	0.4410	0.8975	0.4913	2.0353	1.1141		50		10	0.6487	0.6041	0.7069	0.7581	1.3190	0.9221		50
	20	0.4596	o.4436	0.8962	0.4950	2.0204	1.1112		40		20	0.6516	0.6065	o.795i	0.7627	1.3111	O.9I92		40
	30	0.4625	0.4462	0.8949	0.4986	2.0057	1.1082		3°		30	0.6545	0.6088	o.7934	0.7673	1.3032	O.9I63		30
	40	O.465A	0.4488	0.8936	0.5022	I.99I2	1.1054		20		40	0.6574	0.6111	0.7916	0.7720	1.2954	0.9134		20
	50	O.4683	0.4514	0.8923	0.5059	1.9768	1.1025		10		50	0.6603	0.6134	0.7898	0.7766	1.2876	0.9105		IO
27	0	0.4712	0.4540	0.8910	0.5095	1.9626	1.0996	63	0	38	0	0.6632	0.6157	0.7880	0.7813	1.2799	O.9O76	52	O
	10	0.4741	0.4566	0.8897	0.5132	1.9486	1.0966		50		10	0.6661	0.6180	0.7862	0.7860	1.2723	0.9947		50
	20	0.4771	0.4592	0.8884	0.5169	J-9347	1.0937		40		20	0.6690	0.6202	0.7844	0.7907	1.2647	O.9OI8		40
	3°	O.48OO	0.4617	0.8870	0.5206	1.9210	1.0908		30		30	0.6720	0.6225	0.7826	0.7954	1.2572	O.8988		30
	40	0.4829	0.4643	0.8857	0.5243	1.9074	1.0879		20		40	0.6749	0.6248	0.7808	0.8002	1.2497	O.8959		20
	50	O.4858	0.4669	0.8843	0.5280	1.8940	1.0850		10		5o	0.6778	0.6271	0.7790	0.8050	1.2423	O.893O		IO
28	0	O.4887	0.4695	0.S820	o.53i7	1.8807	1.0821	62	0	39	0	0.6807	0.6293	0.7771	0.8098	i.2349	O.89OI	51	O
	10	0.4916	0.4720	0.8816	0-5354	1.8676	1.0792		5°		10	0.6836	0.6316	o.7753	0.8146	1.2276	0.8872		50
	20	0.4945	0.4746	0.8802	o.5392	1.8546	1.0763		40		20	0.6865	0.6338	o.7735	0.8195	1.2203	O.8843		40
	30	0.4974	0.4772	0.8788	0.5430	1.8418	1*0734		3°		30	0.6894	0.6361	0.7716	0.8243	1.2131	0.8814		30
	40	0.5003	o.4797	0.8774	0.5467	1.8291	1.0705		20		40	0.6923	0.6383	0.7698	0.8292	1.2059	O.8785		20
	5°	0.5032	0.4823	0.8760	0.5505	1.8165	1.0676		10		50	0.6952	0.6406	0.7679	0.8342	1.1988	O.8756		IO
29	0	0.5061	! 0.4848	0.8746	0.5543	1.8040	1.0647	61	0	40	0	0.6981	0.6428	0.7660	0.8391	1.1918	0.8727	50	O
	10	O.5O9I	0.4874	0.8732	0.5581	1.7917	1.0617		50		10	0.7010	0.6450	0.7642	0:8441	1.1847	O.8698		50
	20	0.5120	0.4899	0.8718	0.5619	1.7796	1.0588		40		20	0.7039	0.6472	0.7623	0.8491	1.1778	0.8668		40
	30	0.5149	0.4924	0.8704	O.5658	1.7675	1.0559		30		30	0.7069	0.6494	0.7604	0.8541	1.1708	0.8639		30
	40	0.5178	0.4950	0.8689	O.6696	1.7556	1.0530		20		40	0.7098	0.6517	0.7585	0.8591	1.1640	0.8610		20
	5o	0.5207	0.4975	0.8675	0.5735	1-7437	1.0501		10		50	0.7127	0.6539	0.7566	0.8642	1.157i	0.8581		IO
3°	0	0.5236	0.5000	0.8660	0.5774	1.7321	1.0472	60	0	41	0	0.7156	0.6561	o.7547	0.8693	1.1504	0.8552	49	O
	10	0.5265	0.5025	0.8646	0.5812	1.7205	1.0443		50		10	0.7185	0.6583	0.7528	0.8744	1.1436	0.8523		50
	20	0.5294	0.5050	0.8631	0.5851	1.7090	1.0414		40		20	0.7214	0.6604	0.7509	0.8796	1.1369	0.8494		40
	30	0.5323	0.5075	0.8616	0.5890	1.6977	1.0385		30		30	0.7243	0.6626	0.7490	0.8847	1.1303	0.8465		30
	40	0-5352	0.5100	0.8601	0.593°	1.6864	i.°356		20		40	0.7272	0.6648	0.7470	0.8899	1.1237	0.8436		20
	5o	0.5381	0.5125	0.8587	0.5969	>•6753	l .0326		10		5°	0.7301	0.6670	0.7451	0.8952	1.1171	0.8407		IO
Angle.		arc.	cosine.	sine.	cot.	tan.	arc.	Angle.		Angle.		arc.	cosine.	sine.	cot.	tan.	arc.	Angle.	THE CONSTRUCTOR.
299
ANGlyE.		arc.	sine.	cosine.	tan. |	cot.	arC.	ANGITE.	
deg.	min.							deg.	min.
42	0	0-7330	0.6691	o.743i	0.9004	1.1106	0.8378	48	0
	10	0.7359	0.6713	0.7412	0.9057	1.1041	0.8348		50
	20	0.7389	0.6734	o.7392	0.9110	1.0977	0.8319		40
	30	0.7418	0.6756	0-7373	0.9163	1.0913	0.8290		30
	40	0.7447	0.6777	o.7353	0.9217	1.0850	0.8261		20
	50	0.7476	0.6799	o.7333	0.9271	1.0786	0.8232		10
43	0	0.7505	0.6820	o.73*4	0.9325	1.0724	0.8203	47	0
	10	0.7534	0.6841	0.7294	0.9380	1.0661	0.8174		5o
	20	0.7563	0.6862	0.7274	0.9435	I-°599	0.8145		40
	30	0.7592	0.6884	0.7254	0.9490	1-0538	0.8116		30
	40	0.7621	0.6905	0.7234	0.9545	1.0477	0.8087		20
	50	0.7650	0.6926	0.7214	0.9601	1.0416	0.8058		10
Angle.		arc.	cosine.	sine.	cot.	tan.	arc.	Angle.	
angee.		arc.		sine.		cosine.		tan.	cot.	arc.	ANGEE.	
deg.	min.										deg.	min.
44 45	0 10 20 30 40 50 0	0.7679 0.7709 0.7738 0.7767 0.7795 0.7824 0.7854		0.6947 0.6967 0.6988 0.7009 0.7030 0.7050 0.7071		0.7*93 0.7173 o.7i53 0.7*33 0.7112 0.7092 0.7071		0.9657 0.9713 0.9770 0.9827 0.9884 0.9942 1.0000	1.0355 1.0295 1.0235 1.0176 1.0117 1.0058 1.0000	0.8029 0.7999 0.7970 0.7941 0.7912 0.7883 0.7854	46	0 5o 40 30 20 10 0
Angle.		arc.		cosine.		sine.		cot.	tan.	arc.	Angle.	
ang.—o° 1' arc. = 0.0003			o° 5' 0.0015		1350 2.3562		180° 3-*4i6	2250 1 3-9270	270° 4.7124	3150 5-4978	360° 6.2832	
1.
2.
3-
4.
5.
6.
7-
8.
9-
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
22.
TRIGONOMETRICAL FORMUI^E.
sin (a dz fi) = sin a cos fi dz cos a sin fi
cos (a± fi) — cos a cos fi sin a sin fi
sin 2 a — 2 sin a cos a
sin 30 = 3 sin a — 4 sin a3 = sin a (4 cos a2 — 1)
cos 2 a = cos a2 — sin a1 — 2 cos a2 — 1 = 1 — 2 sin a3
cos 3 a = 4 cos a3 — 3 cos a = cos a (1 — 4 sin a2)
cos a -J- cos fi
fi — 2 sin	a + fi 2	a-fi COS	— 2
fi = 2 cos	±±.L 2	. a — fi stn	— 2
fi = 2 cos	a + fi 2	a — fi cos	 2
fi — 2 sin	a + $ 2	. fi — a stn	 2
sin a2 = X (1 — cos 2 a)
cos a2 == X (1 4- a?* 2 a)
Q3 = X (3 s*n a — sin 3 a)
cos a3 = X (3 60$ a + cos 3 a)
.	,	tang a dz tang fi
tang (a dz fi) ~ s	s
cotang (a dz fi) —
tang 2 a
1 -t- tang a tang fi
cotang a cotang fi -fi
— cotang a cotang fi
2 tang a
cotang 2 a
tang a — sj
cotang a ■■
1 — tang a2
cotang a2 — 1
2 cotang a
I — cos 2 a
stn 2 a
I + COS 2 a
I -f- 2 COS a
cos 2 a sin 2 a
1 — cos 2 a 1 — cos 2 a
sin (a dz fi)
cos a cos fi
_ sin (fi ± a)
tang a dz tang fi =
cotang a dz cotang fi---.	—. — --
*	*	stn a stn fi
sin a -4- sin fi  tang X (a 4- fi)
sin a — sin fi	tang X (a — P)
2 tang X a
1	— tang X a*
cotang X ft2 —1
2	cotang X a
23.300
THE CONSTRUCTOR.
TABUE OF NUMBERS.-I.									
	I P			1	1		1		1
n		«2	«8	Sn				v	 v n	
0.30	3333	0.090	0.027	0.548	1.826	0.669	*-495	0.740	i.35i
o-375	2.667	0.141	0.053	0.612	»•633	0.721	1-387	0.783	1.278
0.60	I.667	0.360	0.216	0.775	1.291	0.843	1.186	0.880	1.136
0.625	1.600	0.391	0.244	0.791	1.265	0.855	1.170	0.889	1.125
0.70	1.429	0.490	0.343	0-837	1.195	0.888	1.126	0.915	I-°93
0.75	1.333	0-563	0.422	0.866	1.155	0.909	I.IOO	0.931	1.075
o-875	I.I43	0.766	0.670	0-935	1.069	0.956	1.046	0.974	1.024
O.9O	I.III	0.810	0.729	0.949	1.054	0.965	1.036	0.987	1.013
I.io	o.qoq	I.2IO	1.331	1.049	0.953	1.032	0.969	1.024	0.976
1.2	0.833	1.440	1.728	1.095	0.913	1.063	0.941	1.047	0.955
1.25	0.800	1-563	1-953	I.Il8	0.894	1.077	0.928	1.057	0.946
1.50	0.667	2.250	3-375	1.225	0.816	1.145	0.874	1.107	0.904
1.75	o.57I	3-063	5-359	1.323	0.756	1.205	0.830	1.150	0.869
2.0	0.500	4.0	8.0	I.4I4	0.707	1.260	0.794	1.189	0.841
2.25	0.444	5-063	n.391	1.500	0.667	1.310	0.763	1.225	0.816
2.50	0.400	6.250	15.625	1.581	0.632	1.357	o-737	1.257	o.795
2.75	0.364	7-563	20.797	I.658	0.603	1.401	0.714	1.288	o.777
3-0	0-333	9.0	27.0	1.732	0.577	1.442	0.693	1.318	o.759
3-25	0.308	10.563	34.328	1.803	0.555	1.481	0.675	1.342	0-745
3-5°	0.286	12.250	42.875	I.87I	0.535	1-5*8	0.659	1.368	0.731
3.75	0.267	14.063	52.734	1-936	0.516	1-554	0.644	1.392	0.719
4.0	0.250	16.0	64.0	2.0	0.500	1.587	0.630 •	I.4I4	0.707
4.5	0.222	20.250	91.125	2.I2I	0.471	1.651	O.606	1.457	0.687
5.0	0.200	25.0	125.0	2.236	0447	I.7IO	0.585	1-495	0.669
5-5	0.182	30.250	166.375	2.345	0.426	1.765	0.567	i.53i	0.653
6.0	0.167	36.0	216.0	! 2.449	0.408	1.817	o.55o	i.565	0.639
6.5	0.154	42.25	274.625	i 2.550	0.392	1.866	0.536	1.597	0.626
7.0	0.143	! 49.0	243.0	2.646	0.378	1.913	0.523	1.627	Q.615
7.5	0133	56.250	421.875	!‘ 2.739	0.365	1-957	0.510	1.655	0.604
8.0	0.125	1 64.0	512.0	2.828	0.354	2.0	0.500	1.682	0595
8.5	0.118	72.250	614.125	2.915	0.343	2.041	0.490	1.707	0.586
9.0	O.III	81.0	729.0	3.000	0.333	2.080	O.48I	1.732	o.577
95	0.105	90.250	857.375	3.082	0.324	2.118	0.472	1.756	0.570
IC	0.100	100.0	1000.0	3.162	0.316	2.154	O.464	I.77S	0.562
11	0.091	121,0	1331.0	3-317	0302	2.224	0.450	1.821	o.549
12	0.083	144	1728	3-464	0.289	2.289	0.431	1.861	o.537
13	0.077	169	2197	3.606	0.277	2.351	0.425	1.899	0.527
H	0.071	196	2744	3-742	0.267	2.410	0.415	1.934	0.517
15	0.067	225	3375	3-873	0.258	2.466	0.405	1.968	0.508
16	0.063	256	4096	4.000	0.250	2.520	0.397	2.000	0.500
17	0.059	289	4913	4.123	0.243	2.571	O.389	2.031	0.492
18	0.056	324	5832	4.243	0.236	2.621	O.38I	2.060	0.485
19	0.053	361	6859	4.359	0.229	2.668	0.375	2.088	0.479
20	0.050	400	8000	4.472	0.224	2.714	O.368	2.115	0*473
50	0.020	2500	125000	7.071	0.141	3.684	0.271	2.659	0.376
100	0.010	10000	1000000	10.0	0.10	4.642	0.215	3.162	0.316
1000	0.001	I000000	1000000000	31.623	0.032	10.0	0.100	5.623	0.178
7T = 3.I42	0.318	9.870	31.006	1.772	0.564	1465	0.683	i.33i	0.751
2 ir = 6.283	0.159	39-478	248.050	2.507	0.399	1845	0.542	i.583	0.632
ir - = I.57I 2	0.637	2.467	3.87S	1.253	0.798	1.162	0.860	1.120	0.893
7T - 3 = 1-047	0.955	1.097	1.148	1.023	0.977	1.016	0.985	1.012	0.989
4 - 7T = 4.I89 3	0.239	17.546	73.496	2.047	0.489	1.612	0.622	1.431	0.699
ir = 0.785 4	1.274	i 0.617	0.484	0.886	1.128	0.923	1.084	0.941	1.062
i = °-S24	1.910	1 0.274	0.144	0.724	1.382	0.806	1.241	0.851	1.176
*r2 — 9.870	0.101	I 97-409	961.390	3.142	0.318	2.145	0.466	1.772	0.564
7r* =s 31.006	0.032	i ; 961.390	29809.910	5.568	1.796	3.142	0.318	2.360	0.424
— = 0.098 32	10.186	1 0.0095	0.001	0.313	3.192	0.461	2.168	0.560	1.782
2ir 76= °-589	1.698	0.347	0.204	0.768	1.303	0.838	1.194	0.876	1 1.142
i" = 32-2	0.031	1036.84	. 33386.24	5.674	. 0.176	3.181	0.314	2.381	0.419
2 j' = 64.4 .	0.015	4147.36	267090	8.025	0.125	4.007	0.249	2-833	0.337THE CONSTRUCTOR.
301
			TABLE OF NTJMBERS-					i H H			
n	n	& n	n	v/ n		n	>/ n		i n	\/ n	n
0.01	0.10000	0.21544	0.26	0.50990	0.63825	0.51	0.71414	O.79896	0.76	0.87178	0.91258
0.02	0.14132	0.27144	0.27	0.51962	0.64633	0.52	0.721 n	O.80415	0.77	°-8775o	0.9I657
0.03	0.17321	0.31072	0.28	0.52915	0.65421	o.53	0.72801	O.80927	0.78	0.88318	0.92052
0.04	0.20000	0.34200	0.29	0.53852	0.66191	0.54	0.73485	O.81433	0.79	0.88882	O.92443
0.05	0.22361	0.36840	0.30	0.54772	0.66943	o.55	0.74162	O.81932	O.SO	0.89443	0.92832
0.06	0.24495	0.39149	°*3I	0.55678	0.67679	0.56	0.74833	O.82426	0.8l	0.90000	0.93217
0.07	0.26458	041213	0.32	0*56569	0.68399	0.57	0.75498	O.829I3	0.82	0.90554	0-93599
0.08	0.28284	0.43089	°-33	0.57446	0.69104	0.58	0.76158	O.83396	O.83	0.91104	O.93978
0.09	0.30000	0.44814	0.34	0.58310	0.69795	0.59	0.76811	O.83872	O.84	0.91652	0.94354
0.10	0.31623	0.46416	o-35	0.59161	0.70473	0.60	0.77460	0.84343	0.85	0.92195	0.94727
O.II	0.33166	0.47914	0.36	0.60000	0.71138	0.61	0.78102	O.84809	0.86	0.92736	0.95097
O.I2	0.34641	o.49324	0-37	0.60828	0.71791	0.62	0.78740	O.8527O	0.87	0.93274	O.95464
0.13	0.36056	0.50658	0.38	0.61644	0.72432	0.63	0.79373	O.85726	.0.88	0.93808	O.95828
O.I4	0.37417	0.51925	o-39	0.62450	0.73061	0.64	0.80000	O.86177	0.89	0.94340	O.96 f 90
0.15	0.38730	0.53133	0.40	0.63246	0.73681	0.65	0.80623	O.86624	0.90	0.94868	O.96549
0.16	040000	0.54288	0.41	0.64031	0.74290	0.66	% 0.81240	O.87066	0.91	0*95394	O.969O5
0.17	041231	0-55397	0.42	0.64807	0.74889	0.67	0.81854	0.87503	0.92	0.95917	0.97259
0.18	042426	0.56462	043	0.65574	0.75478	0.68	0.82462	0.87937	o.93	0.96437	O.976IO
O.I9	043S89	0.57489	0.44	0.66332	0.76059	0.69	0.83066	O.88366	0.94	o-97954	0.97959
0.20	0.44721	0.58480	045	0.67082	0.76631	0.70	0.83666	O.8879O	o-95	0.97468	0.98305
0.21	045826	0.59439	0.46	0.67823	0.77194	0.71	0.84261	O.892 II	0.96	O.9798O	O.98648
0.22	046904	0.60368	0.47	0.68557	0.77750	0.72	0.84853	O.89628	0.97	O.98489	O.9899O
0.23	0.47958	0.61269	0.48	0.69282	0.78297	o.73	0.85440	O.9OO4I	0.98	0.98995	0.99329
0.24	048990	0.62145	0.49	0.70000	0.78837	0.74	0.86023	O.90450	0.99	0.99499	O.99666
0.25	0.50000	0.62996	0.50	0.70711	0.79370	0.75	0.86603	O.90856	1.00 .	I.OOOOO	I.OOOOO
		sin 30° = cos 6o° .== x/z ;			cos 30° = sin 6o° = y2			y/ 3 = 0.8660.			
		sin 75	0 = cos 150 = 0.9659; tan 30° = cot 6o° = % 0.5774;								
		cos 75	0 =• sin 150 = 0.2588 ; cotan 30° =				= tan 6o° =	v/T= 1.73**.			
		log 7T	= o.497H99.		log g = 1.507856.						
PROFUFZTY
DEPARTMENT
MACHTNE DESTGN
SIBLEY SCHOOL
CORNELL UNn/ERSITY
RECE1VED ...A T /PTT A TTFTFTC AX, USTDEX
ACCUMULATORS..................... 264
‘ ‘	. Hoppe’s.....264
“	Hydraulic..... 218
“	Tweddells'---- 265
Aetion of GearTeeth.............. 129
AdamsonJs Stiffening Ring........ 269
Addition and Subtraction of Forces 26
Addition, Graphical............... 26
Adjustable Escapements........... 170
“	Gears for Rotati ve.Mot-
ors. .....................	 237
“	Hangers............. 74
“	Power Escapements	..	236
“	Pump Gears......... 236
Admiralty Chain.................. 182
Adyman’s Coupling................ 215
Agudip’s Cable Locomotive........ 176
Air Compressors, Riedler*s....... 279
Air Pump, Bunsen’s............ 222
“	Von Gerike’s. .........225
“	Watt’s............. 225
Air, Reservoir for Compressed.... 272
Allan’s Link Motion.............. 235
Almgren’s Researches on Steam
Boilers. ....................  271
Althaus* Furnace Hoist........... 173
Althaus’ Pump...................  223
American Standard Car Bearing----	75
Amos & Smyth’s	Pump............. 224
Anehor Escapement,	Free.......... 168
“ Bolts........................ 56
“ Ratchet..................... 155
Anemometers...................... 239
Angle and T Iron Columns.......... 83
Angie of Torsion, Determination of 93
Angle of Rotation in Torsion.....	11
Angstr5m’s Valve Gear............ 233
Anti-Frietion Wheels.............  123
Anti-parallel..................... 22
Anti-projection................... 23
Application of Tension Organs.... 172
Archimedes, Tympanon of.......... 221
Archimedian Screw................ 221
Area of Polygons................. 23
“ Quadrilaterals.............. 24
Triangles................ 23
Arithmography..................... 22
Arm Sections, Table for Transform-
ing.....................      103
Armsof Gear Wheels............... 149
Armstrong Hydraulic Crane......... 228
Artificial Draft......... • .....272
Atmospheric Railway.............. 227
Attachment of Joumals............. 67
Audemar’s	Pump................. 224
Automatic	CoUpling............  101
“	Friction Brake........ 170
“	Steam Stop............ 281
Axis, Neutral...................... 3
Axle, American Railway Standard..	89
“ Prussian Railway Standard...	89
**	Simple Crank............... 106
Axles:... T....................... 85
“	for Water Wheels............ 91
“	Graphical Calculation of...	86
“ Loaded at Two Points......... 87
Axles 4.................   107
“ Non-Symmetrical.............. 86
“	Proof Diagrams of........... 87
“	Proportions of............. 86
*•	Railway..................... 88
“	Symmetrical................. 85
“	with Circular Section......	85
“	with Cruciform Section.....	90
M	with Inclined Loads........ 90
Axles with Three or more Bearings. 89
“ Wooden.......................... 92
BAG PUMP......................... 217
Baker*s Blower.................... 221
Balanced Valves.................. 279
Balance Wheel..................... 167
Balancing of Pulleys............. 194
Balanced Slide Valves............ 285
Balanced Valve, Cramer’s......... 280
Ball Bearings.................... 127
Ball Joints for Pipes............249
BandSaws.......................... 177
Base	Figures for Hyperboloidal
Wheels....................... 136
Bastard Gears..................... • 135
Beale’s Gas Exhauster.........'.. 226
Beams............................ 3
“	Double Trussed............ 35
“	Sections, Table of........ 5-7
“	Force Plans for Framed---- 38
*•	Multiple Trussed.......... 35
“	Scale..................... m
“	Simple Trussed............ 35
“	Triple Trussed............ 35
“	Walking................... 110
“	with Common Load.......... 11
Bearings, Ball.................   127
“ Design and Proportion of 68
“ Independent Step........... 75
“ Lateral.................... 68
“ Metaline.................. 179
“ Multiple Collar............ 77
“ Multiple Supports for.........	80
“ Pedestal................... 71
“ Roller.................... 126
Roller for Bridges...... 126
“ Simple Supports for...........	79
“ Standard American Car...	75
“ Standard Prussian Car---------	75
“ Special Forms of........... 74
„ V	Step..................... 75
Bearing Supports, General Principies 82
Bearings, Thrust...........65, 68, 75
“	Thrust with Wooden	Sur-
f ace................. 76
“	Supports for............. 79
Wall....................68, 71
“	Wall Step................ 75
“	with Three-part Boxes---	70
“	Yoke..................... 72
Becker’s Clutch.................. 101
Behren’s Chamber Gear Train...... 220
Belidor’s Water Pressure Engine	. 229
Bell Crank........................ no
Bellegarde, Rope Transmission at.. 205
Belleville Elastic Washers....... 212
Bellows Pump....................  217
Bell Pipe Connections............ 248
Bell Valve......................  276
Belt Connections................. 191
“ Fastening, Botter*s......... 193
“ Fastening, Moxon’s..........i.. 193
Belting.......................... 186
Cernent for.............. 193
Efficiency of............ 194
Specific Capacity of..... 190
Stress on................ 191
Various Examples of---.... 187
Table of Examples.-...... 192
Belt Lacing...................    191
Belts, Capacity of............... 190
“	Creep of................   194
“	Cross Section of.........  190
“	Path of.................  .186
Belts, Polishing.  ............... 177
“ QuarterTwist................. 186
Belt Shifters......................... 188
Belt Shifter, Zimmerman’s....... ... 189
Belts, Stiffness of................... 194
Belt Transmission, Examples of---- 191
Belts, Transporting................... 221
Bending, Bodies of Uniform Resist-
ance to............. 8
“	Load.................... 3
“	Moment.................. 3
“	Resistance to........... 2
Bergner’s Drawing Board............... 172
Berlin, Sewerage System of............ 219
Bevel Friction Wheels................. 124
“ Friction Wheels, Minotto’s... 125
**	Gears.................. 135
“	“	Construction Circles	for.	135
“	“	Spiral............ 141
“	Stepped........... 141
Beylich’s Universal Gears............. 136
Biquadratic Parabola................... 10
Blake’s Steam Pump.................... 230
Bleicherfs Cable Tramway System.. 175
Blower, Baker’s......................  221
Blower, Root’s........................ 221
Blowers, Fan.......................... 222
Bloxam’s Gravity Escapement....... 168
Boat, Sail............................ 223
Boat, Chain Propulsion of............. 183
Bodies of Uniform Strength.............. 2
Bogardus Mill......................... 126
Boiler Construction, Economy in
Combustion. 270
“	“	Economy of
Material in 270
“	Improvements
in Heating
Surface.. . 270
Boiler Details...... ..................266
“ Feeder, Brindley’s...............228
“ Feeders..........................228
“ Flues............................269
“ Flues, Corrugated................269
“ Riveting......................... 42
Boilers,	Almgren’s Researches	on..	271
“	Circumferential Seamsof..	268
“	Classified............265-266
“	Flat Surfaces of....... 268
‘*	for Swedish State Railway.	272
“	Longitudinal Seamsof... 267
‘ ‘	Thickness of..........	266
“	Spherical Details.......268
“ Steam........—..............265
Boiler Tubes........................ 270
Boiling Water, used for Shrinking..	47
Bolt Connections, Unloaded........... 60
“ Dead............................. 166
“ Gerber’s.......................... 57
“ Heads.........................	54
“ Latch............................ 166
Bolts, Anehor........................ 56
“ andNuts, Metric.................. 55
“ and Screws....................... 50
“ Maudslay’s Method of Secur-
ing........................... 58
" Parsons’.......................   57
“ Penn’s Method of Securing.. 57
“ Special Forms of................. 55
Borda Turbine....................... 220
Bored Guides........................ 122
Botter’s Belt Fastening............. 193
Boxes, Various Forms of Journal...	69
Brace, Weston’s Ratchet............. 154
Bracket Support for Bearings....	79304
ALPHABET1CAL INDEX.
Brackets, Wall.................... 72
Brake, Automatic Friction........ 170
Brake, Napier’s Differential....... 214
Brakes, Chain.......................215
“ Clamping..................... 214
“ Internal Strap............... 215
“ Sliding....................   215
“ Strap.....................211-214
Brake, Toggle Friction............. 162
Bramah Lock........................ 166
Brasses for Connecting Rod......... 112
Brauer’s Intermittent Gearing...... 165
Breaking Load........................ 1
Bridge Bolts........................ 59
“ Flying....................... 222
“ Roller Bearings ............. 126
44 St. Louis.................... 60
Briggs’ System of Pipe Threads----- 250
BrindLey’s Boiler Feeder........... 228
Britton’s Steering Gear............ 238
Brown’s Valve Gear................. 235
Brown*s Windlass................... 173
Buckling, Resistance to............. 13
Buckling Strains, Table of.......... 14
Buffer Couplings................... i3i
Built up Screw Propellers........... 57
Bunsen’s Air Pump.................. 222
Butler’s Coupling................... 96
CABLE, Ari^ngement of Pulleysfor 202
“	Drum, Fowler’s............ 173
“	Ferry System, Hartwich’s. 175
44	Grip Pawl................. 185
44	Haulaga Systems........... 174
44	Incline at Lucerne........ 173
44	Loeomotive, Agudio’s...... 176
44	Riiiways................   173
44	Rhenish Railway........... 174
“	Road, Kahlenberg.......... 173
Cables, Table for Tightened. ....... 200
Cable System for Canals, Schmick’s. 175
44 System, Riggenbaeh’s......... 174
44 Tramway, Chicago............. i75
“ Tramway, Overhead............. 175
44 Tram ways, San Francisco.... 174
44 Transmission, Ring System 208-211
44 Transmissions, Short Span... 200
44 Transmission with Inclined... 200
Cadiat Turbina.................... 220
Cagniardelle ..................... 221
Cail & Co., Valve Gear............ 162
Calculating Machine, Thomas’.. .153» 156
Calculation of Springs.............. 18
Calculations for Chains........... 183
Cambon’s Roller Bearing........... 127
Cam Valve gears..................  236
Canal Cable System, Schmick’s..... 175
44 Lift at Les Fontinettes..... 227
• 4	4 4 Green’s...,............ 227
“	44 La Louviere............ 227
4 4	4 4 Mersev................ 227
44 Locks......’................. 227
Cannon, Thickness of................ 15
Capacity of Belts.................. 19°
Capstan, David’s .................. *73
Car Baaring, American	Standard... 75
Car Baarings, Prussian	Standard...	75
Cardan’s Coupling................... 97
Cardioide.......................... 92
Casting........................... 240
Cast Iron Cranks................... 105
Central Curve of Valve Gear....... 235
Centre of Gravity GraphieallY De-
termined...................... 33
Centrifugal Force................. 177
44 Force of Wire Rope------- 197
44 Pumps.................... 222
Chain,	Admiralty ................ 182
44	Brakes.................... 215
44	Couplings................. 184
44	Drums..................... 185
44	Flat Link................. 183
44	Gemorsch.................. 182
44	Madgeburg-Bodenbacher.... 183
44	Open Link................. 182
44	Pawls for................. 185
44	Pitch..................... 183
Chain Propeller, Heuberger’s..... 176
44	Propelling Gear.......... 187
44	Proportions of........... 183
“	Propulsion of	Boats...... 183
Chains, Calculations for......... 183
Chain Sheaves................ 185, 211
Chain, Specific Capacity of......... 211
Chains, Running.................. 182
“	Stationary.............. 182
44	Tests for...............  183
Chain	Strippers................ 185
44	Swivels.................. 184
“	Transmission............. 211
“	Transmission, Efficiency of... 213
“	44	in Mines.... 213
4 4	4 4	Decido Mines. 212
44	Weight of................ 183
Chamber Gear Train, Behren’s..... 220
44	“	44 Eve’s.........220
“	“	“ Repsold*s. ... 220
44	Wheel Trains............  219
Channeled Connecting Rods........ 117
Checking Ratchets............150, 163
Check Valves..................... 274
Cheese Coupling.............. 99, 151
Chemical Ratchet Trains.......... 171
Cherefs Press, Friction Gear of-- 125
Chicago Cable Tramway............ 175
Chronometer Escapement........... 167
Chubb Lock....................... 166
Circular Piate, Deflection of..... 15
Circumference Scale.............. 128
Clamp Coupling.................... 95
Clamping Brakes.................. 214
Clamp Pulley, Fowler’s........... 203
Clamp Ratchet.................... 160
Clark’s Canal Lift............... 227
Clerk, Method of Shrinking Rings.	45
Clocks, Striking Mechanism for.... 169
Close Link Chain................. 182
Clutch, Becker’s...............   101
“	Cone...................... 99
“ Couplings...................95, 98
44	Coupling, Fossey’s....... 100
“	Dohmen-Leblanc’s......... 101
“	Forks..................... 99
Clutches, Friction................ 99
Clutch, Garand’s................. 101
“	Jaekson’s..............   101
44	Koechlin*s Friction...... 100
44	Napier’s................. 101
“	Reuleaux’s Friction...... 100
44	Sehurmann*s.............. 101
44	Toothed................... 98
44	Weston’s Friction........ 101
Coating Operations............... 241
Cock, Four Way................... 225
Cocks..........................   281
Coefficients of Resistance......... 1
Coefficients of Safety............. 1
Cold Forcing...................... 17
44 Forcing, Dimensions for....	47
44 Hooping..................    45
Collar Thrust Bearings............ 66
Columns, Calculations for Iron...	82
44	Fluted...................  83
44	Forms for Iron............ 84
44	Grouped................... 84
“	Hollow.................... 83
44	of Angle and T Iron......	83
“	of Uniform Resistance----	13
44	Strength of Cast Iron....	83
“	Stresses in............... 82
Combined Levers.................. 110
Compound Escapements............. 168
44	Link as Thrust Bearing. 67
4 4	Strains, Table of.......	15,
44	. Stresses. • ............ 13
Compressed Air for Power Distribu-
tion.......................  219
Compression, Resistance to......... 2
Condenser, Watt’s................ 230
Conditions of Equilibrium......... 29
Conductors for Pressure Organs... 242
Conduits for Pressure Organs.....216
Cone Clutch Coupling............   99
Cone Coupling,	Reuleaux’s........ 96
44 Pulleys..................... 189
4 4	4	4	Diagram for.......... 190
4 4	4	4	for Crossed Belts... 189
4 4	4	4	for Open Belts....... 1S9
Conical Gear Wheels.......*...... 135
Connecting Rod Brasses............. 112
44	End, Cast Iron-- 113
“	4* “ Krauss*..... 113.
‘ 4	4 4	4 4 Penn’s.... 113
“	44 “ Polonceau’s..	114
“	44 Porter- Allen... 117
“	Rods *......... 112
44	“‘Channeled......... 117
“	“ Forms of....... 118
“	“ Locomotive...... 116
“	“ Reetangular Section	117
“	Rod, Solid End for.. 113
“	Rod, Solid End for Lo-
motive. 113
“	Rods, Ribbed...... 117
44	Rods, Round...... 116
44	Red, Strap End for.. 112
44	Rod, Whip Aetion of....	116
Connections for Belting. 191
“	“ Cast Iron Pipes-- 248
44	“ Crank Axles.... 115
“	44 Lead Pipe...... 251
“	Neck Journals... 114
“	44 Wrought Iron Pipes	249
Construction Circles for Bevel Gears 135
Construction of Machine Elem’nts. 39-289
“	Pulley Stati ons.... 204
“	Rope Curve........... 202
44	Rope Pulleys........ 202
“	Screw Thread........	50
Continuous Ratchets.......•...... 150
“ Ratchets with Locking
Teeth.. 165
“	Running	Ratchets....... 164
Copper Pipes...................... 246
Cord Friction....................» 177
Cord Polygon ...................... 26
Corliss Valve..................... 236
Corliss Valve Gear................ 162
Cornish Valve..................... 280
Cornish Valve Gear............153, 163
Corrugated Boiler Flues........... 269
Corrugated Fire Box..............  269
Cotton Rope....................... 179
Cotton Rope Transmission.......... 196
Counterbalance, Oeking*s.......... 217
Countershaft Hanger, Sellers’....	74
Counting Gear for Gas Meter...... 165
Couples, Force..................... 2^
Coupling, Adyman*s................ 215
44	Butler’s.‘................ 96
44	Cardan’s.................  9?
44	Cheese...............99, 151
44	Clamp..................... 95
44	Cresson’s................. 96
44	Drag Link................. 97
44	Hooke’s................... 97
44	Muff...................... 95
44	01dham*s.................. 96
44	Piate...................   95
44	Pouyer’s................. 101
44	Prentiss................. 216
44	Ramsbottom*s Friction... 99
44	Reuleaux’s	Cone........... 96
Couplings........................   95
44	Automatic................ 101
44	Buffer................... 181
“	Schurman’s Friction..... 215
44	Clutch.................... 98
Coupling, Sellers*................. 96
Couplings, Flexible................. 96
44	for Chain................. 184
44	for Propeller	Shafts....	95
44	Sharp’s..........T......... 96
44	Link....................... 98
Coupling, Uhlhom*s................ 101
Cramer*s Balanced Valve............ 280^
Crane Hook, Proportions for. ..... 184
44 Pillars....................   89
44 Ramsbottom’s............... 176*ALPHABETICAL IXDEX.
305
Cranes, Cotton Rope Driven....... 196
“	Graphical Calculation.....	27
44	Hydraulic................. 228
*•	Squaring Device for....... 172
“	Varieties of.............. 173
Crane, Tangye’s.................. 176
Crane, Towne’s................... 176
Crank Axle, Graphostatic Calcula-
tion of........................... 106
44 Axles, Connections for...... 115
106
Calculation for
Calculation for
105
104
233
61
112
44 Axle, Simple.
44 Graphostatic
Retura----
44 Graphostatic
Single...................
44 Pin, Tangential Pressure on.
44 Pins........................
44 Pins, Connections for.......
44 Return...................... 105
Cranks, Cast Iron................ 105
Cranks, Classified............... 104
Crank Shaft, Graphostatic Calcula-
tion of.................... io7
Cranks, Hand..................... io9
Crank, Sliding................... 226
Cranks, Single Wrought Iron...... 104
Creep of Ropes................... I96
Creep of Belts................... J94
Cresson’s Coupling................ 96
Crossed Belts, Cone Pulleys for.. 189
Cross Heads.....................  118
44 for Guides............. **9
44 for Link Connections..	119
44 for Locomotives........ 121
44 for Marine Engines----- 120
Head, Slipper..............  121
Head, Superficial Pressure on. 120
Keyed Connections............ 48
Section of Belts............ I9°
44 Hemp Rope............ 195
44 Wire Rope............ 196
Crown of Pulleys................. 186
44 Ratchet..................... 154
44 Wheel Escapement............ 169
Cup Packing...................... 253
Current Motor.................... 223
Curve, Elastic..................... 3
Curves, Velocity................. 233
Cycloidal Curves................. I3°
44 Curves, Generation of....... 130
44 Sinoide...................... *3
. Cycloid, Spherical............... *35
Cylinder Escapement............... ^9
44 Ratchet................... J56
44 Ratchet Gearing........... 165
Cylinders........................ 216
Cylinders for Hydraulic Presses.. 243
Cylindrieal Spiral Gears......... J38
Cylindrical Vessels.............. 15
DANAIDE......................... 220
Darcy, Formula for Frietion of
Water........................ 246
David’s Capstan.................. i73
Davis & Co., Steering Gear....... 238
DeadBolt......................... 166
Dead Ratchet Tooth............... 152
Deane’s Steam Pump............... 230
Decido Mines, Chain Transmission of 212
Decomposition into Parallel Forces. 31
Deflection......................... 3
44 in Bodies of Uniform Re-
sistance................ 8
“	of	Circular Piate...... 15
44	of	Shafting............ 94
“	of	Shafting, Torsional-	92
44	of	Wire Ropes......... 198
Delisle's Screw Thread Systems--	53
Dennison’s Escapement..........  168
Design and Proportion of Bearings. 68
Diagram for Cone Pulleys........ 19°
Diametral Pitch................. 128
Diaphragm Pump.................. 217
Differentia! Brake, Napier*s.... 214
44 Hydraulic Lever........... 218
44 Pulley Block, Weston’s. 174
Differential Windlass............. 173
Dimensions of Gear Wheels......... 147
Disk Frietion Wheels...............124
44 Valves, Flat................. 275
44 Wheels with Pin Teeth........ 133
Distribution of Weight.............. 3
Division by Lines.................. 23
Division of Gear Wheels, Circumfer-
'ential....................... 128
Dobo’s Ratchet. . • .............. 160
Dohmen-Leblanc’s Clutch........... 101
Donnadieu’s Pump.................. 223
Door Locks........................ 166
Double Acting Pumps............... 224
44	Arm Pulleys................ 193
44	Beat Valve................. 280
44	Frietion Ratchets.......... 160
44	Spiral Gears............... 141
Douglas & Coulson’s Steering Gear. 238
Downton’s Pump.....................224
Draft, Artificial................. 272
Draft Keys......................... 48
Drag Link Coupling................. 97
Drawing Board, Bergner’s.......... 172
Driving by Tension Organs......... 173
Drop Hammer, Frietion............. 176
Drop Hammer, Merriirs............. 123
Drums for Chain................... 185
Dry Gas Meter..................... 240
Ducommun & Dubied’s Planing Ma-
chine........................ 176
Dunning & Boissiere’s Steering Gear 238
Duplex Escapement................. 167
“ Pump, Mazelline’s........... 231
“ Pump, Worthington’s.........231
ECCENTRICS........................ 109
Eccentric Straps.................. 115
Edge Keys......................... 49
Effieieney of Belting............... 194
44 of Chain Transmission.... 213
44 of Rope Transmission... 205
Equalizing Levers........... . .32, m
Equalizer, Worthington’s.......... 232
Equation of Elastic Curve......... 3
Equatorial Section Modulus........ 5
Equilibrium, Conditions of........... 29
44 of External Forces------	27
44 of Forces.................. 22
44 of Internal Forces. ...	28
“ of Three Parallel For-
ces ......................... 30
Elastic Curve........................  3
Elastic Curve, Equation of............ 3
Elasticity and Strength of Flexure.	2
Elasticity, Modulus of............1, 13
Elastic Limit..................... 1, 3
44 Limit in Beams............. 8
44 Line....................... 92
Washers, Belle ville.........212
Elbe, Chain Propelling Gear on.... 185
Elbow Fittings...................... 251
Elbow Fittings, Frietion in..........251
Electric Signals, Siemens & Halske. 166
Elements of Graphostatics.........22 38
Elevator, Hydraulic,................ 227
Elevator Safety Devices............. 164
Emery Weighing Machine.............. 173
Enderlein’s Escapement.............. 168
Engine, Porter-Allen............ . 236
Engines, Rotative Pressure........ 233
Engines, Valve Gear for Rotative..	234
Enlarged Screws.................... 58
Epicycloidal and Evolute Teeth
Compared....................... 135
Erhardt*s Flange Joint ............ 47
Escapement, BloxanFs Gravity.... 168
Chronometer............467
Crown Wheel........... 169
Cylinder.............. 169
Dennison’s............ 168
Duplex................ 167
Enderlein’s........... 168
Free Anchor........... 168
Hipp.................. 168
GrahanFs.............. 169
Escapement, Lamb’s.................. 168
44	Lepaute’s............... 169
44	Mudge’s.............. 168
44	Reuleaux’s.............. 168
Escapements..................150, 167
“	Adjustable.............. 170
“	Compound................ 168
“	for	Measurements	of
Fluids................ 239
“	for	Measurements	of
Volume................ 239
“	for	Moving Liquids,
Pressure.............. 228
44	for Pressure	Organs. .	226
44	Isochronous............. 167
“	Periodical................ .	169
44	Periodical Pressure. ..	229
44	Period of............... 167
“	Power................... 169
44	Power Adjustable..... 237
44	Range of................ 167
44	Simple.................. 167
44	Time of Oscillation....	167
44	Uniform................. 167
Escapement, Tiede’s................. 168
Eve’s Chamber Gear Train............ 220
Evolute Rack Teeth................ 132
Evolute Teeth for Interchangeable
Gears......................... 131
Examples of	Belting,	Table......... 192
44	of	Belt Transmission.. 191
“	of	Gearing............... 147
44	of	Journals............... 62
44	Thrust Bearings............ 78
Expansion Gear, Farcot’s............ 236
4 4	4* Gonzenbach’s........ 236
4 4	4 4 Meyer*s............. 236
“	Joints................ 245
44	Valves................ 236
External Forces, Equilibrium of--	27
Extraction of Roots................ 26
Eytelwein’s Formula for Stiffness of
Ropes.....................181, 196
FABRY’S VENTILATOR............... 221
Factor of Safety .................. 1
Fairbairn, Experiments on Boiler
Flues........................... 269
Fan Blowers....................... 222
Farcofs Stufl&ng Box................ 254
Farcot*s Valve Gear............... 236
Fast and Loose Pulleys............ 188
Felbinger’s Postal Tube............. 227
Ferules for Boiler Tubes ........... 270
Fink*s Link Motion................  235
Fire Box, Corrugated................ 269
Fire Box, Kaselowsky’s.............. 269
Fish Torpedo...................... 171
Flange Connections for Lead Pipe. 252
“	Joints....................... 58
“	Joint, Erhardt’s............. 47
“	Joints for Pipes............ 248
Flanges for Riveted Pipes........... 249
Flap Riveted Joints.................  40
Flap Valves........................  274
Flat Hemp Rope....................   178
4 4 Link Chain, Neustadt’s.... 1S3
“ Link Chain, Table of........... 183
4 4 Pivot Bearings................ 66
“ Ropes.......................... 181
Flexible Couplings.............95, 96
44 Pipes........................ 252
44 Rod Connection............... 114
Flexure, Elasticity and Strength of.	2
Flexure, Strains of................. 3
Flow of Metals...................  240
Fluid Escapements for Transporta-
tion ..........................  227
44 Running Ratchet Trains..... 223
44 Valves........................ 287
Fluted Columns..................... 83
Flying Bridge..................... 222
Fly Wheel, Oscillating............ 233
Fly Wheels........................ 233
Force, Centrifugal................ 177
Force Couples...................... 29306
ALPHABETICAL INDEX.
Forced Connections, Examples of...	46
Forced Draft...................... 272
Force Plans for Framed Structures. 34
‘ ‘ Plans for Roof Trusses.....	36
“ Polygon...................5	26
Forcing Fit....................... 17
Forces, Addition and Subtraction of 26
Equilibriuip of........... 22
44 Resultant of Several....... 26
Forcing........................... 45
Forks, Clutch..................... 99
Fork Journals...................... 63
Fork Journal, Stub End for........ 114
Forms for Iron Columns............ 84
Fossey’s Coupling................. 100
Foundation Bolts, Keying for......	48
Fourneyron Turbine................ 220
Four Way Cock..................... 225
Fowler’s Cable Drum................. 173
Fowler’s Clamp Pulley. -.............203
Friction Brake, Automatic........... 170
“ Brake, Toggle................. 162
‘ * Clutches..................... 99
“	Clutch,	Koechlin’s......... 100
44	44	Ramsbottom’s......	99
“	“	Reuleaux’s.......... 100
‘	“	Westons............. 101
“ Cord.......................... 177
“ Coupling, Schurman’s........ 215
“ Drop Hammer................... 176
“ Feed, Sellers*................ 126
“ Gear of Cherefs Press....... 125
“ Gear, Robertson*s............. 125
“	in Elbow Fittings........... 251
“	in Spur Gearing............. 134
‘ ‘ in Stuffing Box.... .........254
“	of Chain Transmission..... 213
“	of Journals.................. 64
“	of Pivot Bearings............ 66
‘ ‘	of Screw Thread.............. 50
“ of Spiral Gear Teeth........ 140
“ of Water in Pipes............. 246
Friedmann’s Jet Pump.............. 222
Friction Pawls.....................  1S9
44 Pawl, Saladin’s.............. 161
“ Pawls, Release of............. 161
“ Ratchets.....................  158
44 Double............... 160
“	44	Rod................. 163
44	“	Running............. 160
44	“	Stationary.......... 161
44 Rollers, Mechwarfs........... 127
44 Trains, Special....	  126
44 Wheels....................... 122
4 4	4	4	Bevel............... 124
14	“	Construction of.—	123
4 4	4	4	Disk...............  124
44	“	for Inclined Axes...	124
44	4	4	for Parallel Axes •	123
4 4	4	4	Material for........ 123
44	4	4	Minotto*s............125
44	44	Robertson’s......... 125
44	“	Sellers*............ 126
44	44	Two Applications of	123
44	“	Wedge............125,160
Francis* Turbine.................... 220
Framed Beams, Force PlansTor------	38
Framed Structures, Force Plans for 34
Frankfurt on Main, Water Supply of 218
Free Anchor Escapement.............. 168
Free Cross Heads...................  119
Freiburg, Rope Transmission at------ 205
French Lock......................... 166
Front Bearings....................... 68
Furnace Hoist, Althaus*............. 173
Furnace, Wilson’s Water Gas......... 288
Future Possibilities of Boiler Con-
struction ................... 270
GANNOW MINE..................... 214
Garand’s Clutch..................... 101
Gases, Reservoirs for............... 219
Gas Exhauster, Beale*s.... ..........226
“ Holders.......................  272
44 Meter, Counting Gear for...... 165
44	“ Dry....................... 240
44	“ Sanderson’s............... 239
Gas Meter, Wet................... 239
GateValves....................... 282
Gearing, Brauer’s Intermittent... 165
44 Calculation of Pitch and
Face.................... 144
Cylinder Ratchet.......... 165
Double Pin................ 132
Examples of...............^147
Friction in Spur Tooth... /134
Fundamental Formula for. 128
Globoid Worm ............. 143
Hawkin’s Worm:............ 143
Jensen’sWorm............ 143
Ratchet .................. 150
Shield.................. 133
Gearing, Step .......... 141
Toothed................. 127
Worm...................... 139
Gears,	Bastard..... .............135
“	Bevel...................... 135
44	Bevel Spiral............... 141
“	Beylich’s Universal.......... 136
“	Cylindrical Spiral........... 148
*•	Double Spiral.............. 141
“	Examples of Spiral..........  140
“	Globoid Spiral............... 142
44	Hoisting..................... 144
“	Parallel..................... 133
44	Precision.................... 139
44	Single Tooth...............   165
44	Spiral....................... 138
44	Stepped Bevel.............. 141
4*	Table of Cast Iron......... 144
“	Teeth for Hyperboloidal.... 138
44	Transmission...’............. 144
Gear Teeth,	Aetion of............. 129
“	44	Construction of Spur..	128
“	“	Epicycloidal and Evo-
lute Compared................. 135
4 4	4	4	Evolute Interchange-	131
able................ 131
4 4	4	4	Friction of Spiral..140
4 4	4	4	Interchangeable..... 130
4 4	4	4	Intemal............... 131
4 4	44	Line of Aetion of...129
4 4	44	Loss in Determined
Geometri cally.... 134
44	of Circular Ares.... 131
4 4	4 4	Pin................... 132
4 4	4 4	Pressure on.........	146
“	44	Section of............ 144
4 4	4	4	Stress in............. 145
4 4	4	4	Thumb Shaped.......... 134
4 4	4	4	Wearon..............   134
44	Tooth Outlines, General Solu-
tion................. 129
“	Wheel Arms, Table	of........ 149
4 4	4	4	Hubs.................. 150
4 4	4	4	Plane................. 136
“	Wheels, Arms of.............. 149
4 4	4 4	Circumferential Divis-
ion of.............. 128
4 4	4	4	Classified............ 127
“	44	Conical............... 135
“	“	Dimensions of......... 147
4 4	4 4	Diametral Pitch of... 128
4 4	44	Hyperboloidal......... 136
“	44	Interchangeable..... 128
“	44	Pitch of.............. 144
4 4	4	4	Pitch Radius of..... 128
4 4	4	4	Rim of................ 147
44	“	Weight of............. 150
Gemorsch Chain ................... 182
General Form of Toothed Ratchets. 158
General Remarks upon Ratchet Me-
chanism............................ 171
Generation of Cycloidal Curves___ 130
Geneva, Sluice Gates at........... 275
Geneva Stop....................... 165
Gerber*s Bolt ....................  57
Geyser Pump, Siemens*.............. 222
Gidding’s Slide Valve Experiments. 285
Giffard*s Injector ................ 222
Girard Turbine.................     220
Githen’s Rock Drill............... 231
Globe Valve........ .,........... 279
Globoids,.........................  142
Globoid Spiral Gears.............. 142
Globoid Worm Gearing.............. 143
Gooch’s Link Motion............... 235
Goodwin’s Split Pulley............ 194
Gonzenbach’s Expansion Gear...... 236
Graham’s Escapement............... 169
Graphical Addition................. 26
44 Calculation of	Axle....	86
44	44	of Powers ... 24
4 4	4 4	of Shafting... 95
of Crank Axle 106
of Multiple
Crank Shaft 107
44	4 4	Return	Crank	105
“	“	Single	Crank.	104
Graphostatics,	Elements of..22-38
Green*s Canal	Lift.......... 227
Greindl’s Pump .................. 221
Gresham‘s Injector............... 222
Grip Pawl for Cables............. 185
Grooves for Rope Transmission----	195
Grooved Fly Wheels............... 195
Grouped Columns.................   84
Group Riveting.................... 41
Guide Mechanism for Pressure Or-
gans........................ 217
Guide Pulleys for Belting........ 186
Guides and Guide Bars............ 121
44 Bored.....................  122
44 for Marine Engines..........122
Guide Sheaves.................... 185
Guides, Locomotive............... 122
Guiding by Pressure Organs....... 216
Guiding, Tension Organs for......	172
Gun Lock Mechanism............... 166
Gun Locks........................ 163
Guns, Hooping of.................. 16
HAIR SPRING...................... 169
Hair Trigger...................   168
Half Journals..................... 64
Hammer, MerrilPs Drop............ 123
Hand Cranks...................... 109
Hanger Boxes, Sturtevant’s........ 74
Hangers........................... 68
Hangers, Adjustable..............  74
Hanger, Sellers*.................. 74
Hanger, Sellers* Countershaft....	74
Hangers for Rope............... 181
“ Post...................... 73
4 4 Proportion of........... 73
44 Ribbed...... ............ 73
Harlow’s Valve Gear.............. 231
Hartwich’s Cable Ferry System.... 175
Hastie’s Steering Gear........... 238
Hauling System, Riggenbach’s.....	172
Haulage Systems, Pennsylvania----	174
Hawkin’s Worm Gearing............ 143
Helfenberger’s Regulator......... 236
Hemp Rope.......................  178
“	“	Transmission, Specific
Capacity of........ 195
«	“	Wear on.............. 196
“	“	Weight of............ 178
Hero’s Fountain.................. 288
Heusinger’s Link Motion.......... 235
Heuberger*s Chain Propeller...... 176
Hick, Experiments on Stuffing Box
Friction.....,.............. 254
Hick’s Stiffening Ring........... 269
High Duty Pumping Engine, Worfh-
ington...................... 232
Hipp Escapement...............  168
Hirn’s Experiments on Journals-	64
Hodgkinson*s Experiments........ 13
Hofmann’s Valve Gear........... 163
Hoist, Althaus*.	    173
Hoisting Devices............... 172
Hoisting Gears................. 144
Hollow Columns ................  83
Hollow Journals................. 62
Hooke*s Coupling..............   97
Hooks*........................  184
Hooping......................... 45
by Shrinkage............. 45
44 of Guns.................   16
Hoppe*s Accumulator.,.......... 264ALPHABETICAL INDEX.
307
Hose............................... 252
Howaldt’s Metallic Packing......... 254
Hubs for Rock Arms................  102
Hubs of Gear Wheels................ 150
Hunting Valve...................... 238
Hurdy Gurdy Wheel.................. 220
Hydraulic Accumulators........ 218,	264
44	Cranes................... 228
“	Elevator................. 227
“	Lever.................... 217
“	Lever, Differ en tial.. 218
“	Parallel Motion.......... 2r8
“	Power Distribution......	219
<4	Power Distribution, Ring
System...........• • • 256
44	Presses, Cylinders for...	243
44	Press, Thickness of Cyl-
inder.................. 16
“	Hydraulic Ram, Montgol-
fier’s...............•	232
44	Riveting Machine, Twed-
dell’s...............*	228
44	Steering Gear............ 237
“	Tools.................... 228
44	Transformer.............. 218
Hyperboloidal Gear Wheels........ 136
“	Gears, Teeth for... 138
44	Wheels, Base Figures
for............... 136
IDEAL BENDING MOMENT....	13
Ideal Twisting Moment............ 13
Impact Water Wheels..............220
Inclined Cable, Transmission with.. 200
Inde^endent Step Bearings........ -	76
Inertia, Moment of...............5~7
Influence of Pulley Diameter on
Wire Rope.................... J97
Influence of Weight of Wire Rope. 180
Injector, Giffard’s.............. 222
Injector, Gresham’s................ 222
Interchangeable Gear Teeth....... 130
Interchangeabe Gear Wheels....... 128
Intermittent Gearing, Brauer’s...165
Intemal Flow..................• • • * 240
44	Forces, Equilibrium	of.	28
“	Gear Teeth..................
44	Ratchet Wheels........... I51
44	Strap Brakes..............215
Intze*s Discussion of Tanks......261
Inverted Siphon..................244
Iron Columns, Caiculations for... 82
Iron, Weight of Round............ 55
Isochronous Escapements.......... 167
Isolated Forces in One Plane.....	26
Isolated Forces, Resultant of....	29
TACKSON’S CLUTCH................. 101
Jacquard Loom-----.... ............. ^3
James Watt & Co., Thrust Bearing 77
Jam Nut............................. 56
Jaw Clutch........................   98
Jensen’s Worm Gearing.............. 143
Jet Aetion..........................241
“ Mechanism......................240
44 Propeller.....................223
41 Pump, FriedmamPs........ - • 222
Joints, Expansi on................. 245
44 Flange.....................-	58
44 Strength of Riveted........-	40
Joint, Universal..................   97
Jopling*s Water Meter.............. 239
Journal Boxes, Various	forms----- -	69
44	Friction in Rope	Transmis-
sion	  195,205
“	“ of Chain Transmis-
sion	........-	213
Journals, Attachments of............ 67
44	Dimensions................ 61
44	Examples of............... 62
44	Fork...................... 63
44	for Levers............... 101
44 for Shafting.............-	94
44	Friction of..	   64
44	Half...................... 64
44	HbUow,,,,,,............... 62
Journals, Lateral................... 61
Lubrication of.... ........ 61
“ Multiple..................... 63
44 Neck........................ 62
“ Overhung .................... 61
44 Pressure on................. 61
44 Proportions of.............. 61
44 Stress on................... 61
“	—Various Kinds.60
KAHLENBERG CABLE ROAD . . 173
Kaselowsky’s Fire Box.............. 269
Kennedy’s Water Meter ..............239
Kernaul’s Key....................... 49
Keyed Connections................... 47
Keying ............................. 47
44	Foundation Bolts.............. 48
“	Peters* Method.............. 102
“	Screw Propellers..... ........ 49
Key, Kernaul’s...................... 49
Keys, Concave......................  48
“	Draft......................... 48
“	Edge.......................... 49
“	Flat.........................  48
“	for Rock Arms........ ........102
4 4	Longi tudinal................ 48
44	Methods of Securing........... 50
44	Recessed.....................  48
44	Stresses on................... 48
44	Taper of.....................  47
44	Unloaded...................... 49
Kirchweger’s Steam Trap............ 228
Kirkstall Forge Rolling Mill.... 126
Kirkstall Forge, The...............  94
Kley‘s Pumping Engine.............  233
Klug’s Valve Gear.................235
“Knot” in Cord Polygon.............. 27
Koeehlin’s Friction Clutch......... 100
Krauss* Connecting Rod	End..... 113
Krauss’ Piston....................  252
Kriiger, Investigation of Rivets ...	39
LACING, BELT....................... 191
Lagarousse Ratchet................  164
La Louviere Canal Lift............. 227
Lamb’s Escapement.................. 168
Langen Gas Engine.................. 161
Lap Joints, Riveted................. 40
Lap of Slide Valve..................225
Latch Bolt........................  166
Lateral Bearings.................... 68
Lateral Journals.................... 61
Lead Pipe Connections.............. 251
Lemielle’s Ventilator............... 82
Lepaute’s Escapement............ . 169
Levasseur's Metallic Tubing..... 252
Lever Arms, Calculation of......... 103
44 Arms of Combined Seetion... 103
44 Differential Hydraulic .......218
“ Hydraulic............... .... 217
Levers, Combined..................  110
Equalizing................  32
“ Journals for.................101
44 Simple....................  101
Lifting Frame for Screw Propeller. 151
Lift Valves.................. 223, 273
Liquids, Pressure Escapement for
Limitof Elasticity in Beams.
Locomotive Connecting Rod, Solid
End for............ 113
44	Cross Heads..........	120
“	Guides............... 122
Springs, Screws for__	58
Locks, Canal....................  227
Door..................... 166
“ Gun........................ 162
Lock, Yale.....................   166
Logarithmic Spiral...............  26
Long Distance Fluid Transmission. 233
Long Distance Power Transmission 259
Longitudinal Keys................. 48
Loom, Jacquard................... 163
Loss in Gear Teeth Det.ermined
Geometrically............... 135
Loss in Hemp Rope Transmission.. 195
Lubrication of Journals........... 61
Lucerne, Cable Incline at,. 173
MACHINE ELEMENTS, CON-
struction of.....39-289
Machine Riveting.................. 39
Mackay & McGeorge, Riveting Ma-
chine...........................m
Magdeburg-Bodenbaeher Chain______183
Maltese Cross Gear................165
Manholes......................... 289
Mannesmann Tubing.................243
Marine Cross Heads...............121
4 4 Engine Guides............122
44 Propulsi on............... 222
Marshairs Valve Gear............. 235
Materials—Strength of........... 1-21
Mathematical Tables.............. 291
Maudslay, Method of Securing Bolts 58
Maudslay, Thrust Bearing by......	77
Mauser’s Revolver................ 165
Mazelline’s Duplex Pump.......... 231
Measurement of Fluids, Escape-
ments for.................... 239
Measuring Devices, Running....... 239
Mechwarfs Friction Rollers....... 127
Medart Pulley...................  193
Merrill’s Drop Hammer............ 123
Metaline Bearings................ 179
Metallic Piston Packing.......... 253
Metallic Tubing.................. 252
Metals, Weight of Sheet.......... 43
Meter for Alcohol, Siemen’s...... 239
Methods of Securing Bolts......... 57
Methods of Securing Pawls........ 153
Metrical Screw Systems............ 52
Metric Bolts and Nuts............. 55
Metric Pipe Thread System........ 250
Meyer’s Valve Gear............... 236
Mill, Bogardus................... 126
Minotto’s Bevel Friction Wheels... 125
Mixed Tooth Outlines............. 133
Mines, Chain Transmission in...... 213
Modulus of Elasticity............1, 13
44 Resistance................ 1
“ Rupture..................1, 2
4 ‘ Transmission............208
Molinos & Pronnier, Speed of Rivet-
ing...:...................... 39
Moment of Inertia.............3, 5, 7
228	Moment of Inertia, Polar		
I	Mont Cenis Air Compressors		. 232
8	Montejus			
129	Montgolfier’s Hydraulic Ram		. 232
93	Morin’s Experiments on Journals..	. 64
98	Motors for Pressure Organs		
235	Moulding			
235	Moxon’s Belt Fastening		• 193
235	Mudge’s Escapement		
235	Muff Coupling		
234	Mule Post		
I	Mule, Spinning		
180 166	Multiple Collar Bearings		 44 Collar Thrust Bearings...	• 77 . 66
166	4 4 Crank Shafts		
166	44 Journals		• 63
150	44 Ratchets			
107	‘1 Supports for Bearings		
116	4 4 Trussed Beams	....	3°8
ALPHABETICAL IXDEX.
Multiple Valves.................. 276
Multiplication and Division Coni-
hili ed .......................... 23
Multiplication by Lines........... 22
Murdock’s Slide Valve............ 234
Muschenbroeck’s Pump............. 223
NAGEL TURBINE.................... 220
Napier’s Clutch.................. 101
Napier’s Differential Brake...... 214
Natural Reservoirs............... 218
Neck Journals........,........... 62
Neck Journals, Connections for... 114
Negative Reservoirs.............. 219
Neutral Axis....................... 3
Neutral Plane...................   10
Neustadfs Chain.................. 183
Newcomen Engine.................. 163
Normandy's Pipe Joint............. 249
Norton’s Pump.................... 225
Nut, Jam.......................... 59
Nut Locks......................... 56
Nuts, Washers and Bolt Heads.....	54
OBELISK, FORCES IN RAISING 28
Oberursel, Rope Transmission at.203,205
Oeking’s Water Counter-balance... 217
Oil Tanks...................... • 218
01dham’s Coupling................. 96
Open Belts, Cone Pulleys for..... 189
Open Link	Chain................ 182
Ordway, Experiments on Pipe Cov-
ering........................ 245
Oscillating	Fly Wheel............. 233
Oscillating	Pumps................. 226
Oscillation of Escapements, Time of 167
Osterkamp*s Rope Hanger.......... 181
Otis Elevator..................... 228
Overhead Cable Tramway........... 175
Overhung Journals................. 61
PACKING FOR HYDRAULIC
.Press................. 253
“	for Pump	Pistons.... 255
44	Howaldfs	Metallic.... 254
44	Piston.................. 216
“	Standard	Prussian Rail-
way. ................. 255
PagePs Elastic Washer.............. 57
Pallets........................; • 168
Pappenheim Chamber Wheel Train. 219
Parabola, Biquadratic ............. 10
Parallel Forces—Equilibrium of. .30, 31
“ Gears....................... 133
44 Motion, Hydraulic........... 218
44 Rods for * Locomotive En-
gines.................. 117
Parson’s Bolts..................... 57
*^Pattison’s Pump.................. 226
Pawl, Cable Gnp................... 185
Pawl. Saladin’s Friction.......... 161
Pawls for Chains. ................ 185
“ Friction.....................  159
“ Methodsof Securing........... 153
Pawl, Spring....................   153
Pawls, Release of Friction....... 161
Pawl, Thrust upon................. 152
Pawl, Thumb Shaped................ 160
Payton’s Water Meter.............. 220
Pedestal Bearings. ............... 71
Penn’s Connecting Rod End........ 113
“ Method of Securing Bolts..	57
“ Piston...................... 252
Pennsylvania Haulage Systems_____ 174
Periodical Escapements........... 169
Periodical Pressure Escapements... 229
Peters’ Method of Keying......... 102
Petit’s Pipe Joint..............  248
Pfalz-Saarbruck Screw Thread Sys-
tem................................ 53
Phoenix Column................ 59» 83
Physical Ratchet Tram............ 171
Pickering’s Steam Pump........... 230
Pillow Blocks..................... 68
44 Blocks, Adjustable.......69, 70
“ Block, Sellers*..	70
Pillow Blocks, Large.............
4 ‘ Blocks, Proportional Scale for
“ Block, Sturtevant’s...........
Pin Gearing, Double..............
“ GearTeeth.....................
*	‘ Ratchet Wheel...............
Pins, Crank......................
Pin, Split.......................
Pipe Connections, Socket.........
*	‘ Coverings...................
‘ ‘ Fittings....................
“ Joint, Normandy’s.............
“	“	Petit*s................
“	‘; Riedler’s...............
‘ ‘ Riveted.....................
Pipes, Ball Joints for...........
Pipes, Connections for Cast Iron...
‘4	Connections for Wrought Iron
“ Copper.......................
“ Flange Joints for............
“ Flexible.....................
“ for High Pressures...........
“ Resistance of Bends in.......
“ Resistance to Flow in.........
Pipe Sockets.....................
Pipes, Steam.....................
Pipe, Steel......................
Pipes, Thickness of Cast Iron....
Pipe Threads, Briggs’ System.....
|“ Thread System, Metric.........
“ Weight of Cast Iron...........
“ Wrought Iron..................
Piston, Krauss’..................
“ Packing..................216,
“ Packing, Metallic...........
“ Penn’s......................
‘ ‘	Pumps. ...................
44 Rods.......................
Pistons..................... 216,
Piston, Swedish..................
Pistons with Valves -............
Piston Valves....................
Pitch and Face of Gearing, Calcula-
tlon of........
“	44 Hoisting Gears..
“ Transmission
Gears.........
“ Chain........................
“ Circles, Table of............
“ of Gear Wheels..............
44 Radius of Gear Wheels.......
Pivot Bearings, Flat ............
Pivots, Formula for..............
44 Pressure on................
“ Proportions of...............
Plain Slide Valve................
Plain Slide Valve Gear...........
Plane Gear Wheel.................
Planing Machine, Ducommun & Du-
bied’s.........
4 4	4 4	Sellers..........
“	44	Shanks'..........
Piate Coupling...................
Plungers......................216,
Plunger Pumps....................
Pneumatic Power Distribution.....
Pneumatic Tube...............
Polar Moment of Inertia..........
Polishing Belts..................
Polonceau’s Connecting Rod End..
Polygons, Area of................
Poncelefs Chain..................
Poncelefs Water Wheel............
Porter-Allen Connecting Rod......
Porter-Allen Engine..............
Post Hangers.....................
Pouyer’s Coupling.........101, 152,
Powel Valve Gear.................
Power Distribution, Compressed Air
44	44	Hydraulic......
44	44	Hydraulic Ring
System ......
“	“	Pneumatic......
44	44	Steam..........
44	“	Systems........
44	44	Vacuum.. .....
69
68
70
132
132
152
61
56
248
245
249
249
248
249
244
249
248
249
246
248
252
242
247
246
250
245
243
242
250
250
242
243
252
253
253
252
223
255
252
286
144
144
144
183
128
144
128
66
65
65
65
282
234
136
176
176
163
95
253
223
257
227
11
177
114
24
173
220
117
236
73
153
163
219
219
256
257
219
219
219
Power Escapements................
Powers, Graphical Caleulation of.. .
Powers of Trigonometrical Func-
tions.........................
Power Transmission by Superheated
Water........................
Practical Resistance.............
Precision Gears..................
Precision Ratchets...............
Prentiss’ Coupling...............
Pressure Escapements for Moving
Liquids.............
44 on GearTeeth..............
44	Journals.............
“	Lift Valves..........
44	Pivots...............
44	Screw Threads........
“ Organs.....................
Conductors for....
Conduits for......
Escapements for...
Guiding by........
Guide Mechan i s m
for.............
Motors for........
Ratchet Mechanism
for..............
Reservoirs for. .218,
Running Mechanism
for.............
4 4	Technological Appli-
cations of...................
“ Superficial................
“ Transmission, Specific Ca-
pacity of.............
Proof Diagrams of Axles..........
Propeller Bearing, Ravenhill &
Hodgson’s.............
“ Jet........................
“ Liftmg Frame for............
“ Screw......................
“ Shafts, Couplings for.......
Propelling Chain............... :
Proportions of Axles.............
Chain..............
Flange Joints......
Hooks .............
Journals............
Pivots.............
Pulleys............
Propulsion, Marine...............
Prussian Standard Car Bearing....
Pulley Block, Weston’s Differential.
44 bjr Walker Mfg. Co..........
“ Diameter, Influence on Wire
Rope.....................
44 Fowler*s Clamp.............
“ Medart.......................
Pulleys, Balancing of...........
44 Cone.....................
44 Construction of Rope......
Crown of Face............
“ Double Arm.................
“ Guide.....................
44 Fast and Loose...........
44 for Cable, Arrangement of.
44 Proportions of...........
44 Split....................
Pulley Stations, Construction of ...
Pulleys, Tightening.............
Pulley, Sturtevanfs.............
PuUeys, Umbrella................
Pulleys, Vertical Supporting....
Pulsometer......................
Pump, Althaus’
“ Amos & Smyth’s.
“ Bag...............
44 Bellows..........
44 Diaphragm........
44 Donnadieu’s......
44 Downton^........
4 4 Friedmann’s Jet..
“ Gears, Adjustable
“ GreindTs.........
44 Engine, Kley’s....
169
24
25
219
139
157
216
228
146
61
277
65
58
216
242
216
226
216
217
219
223
260
219
250
255
87
74
223
151
223
95
185
85
183
59
184
61
65
193
222
75
174
193
197
203
193
194
189
202
186
193
186
188
202
193
193
204
186
194
207
188
232
223
224
217
217
217
223
224
222
236
221
223ALPHABETICAL INDEX.
309
Pumping Engine, Worthington’s
High Duty..................... 232
Pumping Machinery.................. 229
Pump, Mazelline’s Duplex............231
“	Muschenbroeck’s........... 233
“	Norton’s.................. 225
44	Pattison’s................ 226
“	Pistons..................  253
4 4	Pistons, Packing for...... 255
Regulator, Helfenberger’« . 236
Repsold’s.................. 221
“	Rittinger’s ............   223
Pumps, Centrifugal................. 222
4 * Considered as Ratchet
Trains.................. 223
44	Double Acting..............224
Pump, Siemens Geyser............... 222
Pumps, Oscillating................. 226
Pump, Spiral............... ••• • 221
Pumps,	Piston.................... 223
“	Plunger................... 223
44	Rotary...................  226
44	Stolz’s..................  224
Pump,	Stone’s................... 224
“	Valve Gear................ 225
“	Valves, Riedler’s......... 278
“	Vose's.................... 224
“	Worthington's Duplex...... 231
QUADRANTS........................ 153
Quadrilateral Figures, Area of..	25
Quarter Twist Belts.............. 186
Quarter Twist Belt, Shifter for. 189
RACK, RATCHET.................... 151
Rack Teeth, Evolute.............. 132
Railway Axles..................... 88
Ramsbottom*s Crane............... 176
Ramsbottom’s Friction Clutch,...	99
Ratchet, Anebor.................. 155
“	JBrace, Weston’s......... 154
“	Clamp.................... 160
“	Crown.................... 154
“	Cylinder............. 156
“	Dobo*s..................•	I6°
“	Gearing.................. 150
“	“	Cylinder......... 156
“	44	Dimensions of Parts
of............   i58
“	Gears, Toothed Running. 150
44	Lagarousse........... 164
“	Mechanism for Pressure
Organs.....223
General Re-
marks upon 171
Kinematically
Discussed.. 171
44	Rack..................... 151
“	Rod Friction............. 163
Ratchets, Checking......... • ----150
44	Continuous................ 150
“	Continuous Running....... 164
44	Double Friction........... 160
“	Friction.................. 158
“	General Form of Toothed 158
“	Locking...................150,	166
44	Multiple.................. 154
44	of Precision.............. 157
•4	Releasing.................150,	162
4 4	Running................... 150
44	Running Friction.......... 158
44	Silent.................... 153
44	Spring.................... 153
44	Stationary............150,	156
44	Stationary Friction...... 161
44	Step...................... 155
44	Step Anchor............... 157
44	Throttle.................. 161
44	with Locking Teeth, Con-
tinuous ...................   165
44	Teeth, Flanks of.......... 153
44	Teeth, Form of............ 150
•4	Tension Organs............ 185
44	Tooth, Dead............... 152
44	Train, Physical........... 171
Ratchets, Trains, Chemical........ 171
44	Wheels, Internal........ 151
44	Wheels, Special Forms... 154
4*	Wilber’s...............  153
RavenhUl & Hodgson, Propeller
Bearing......................... 74
Ravenhill & Hodgson, Thrust Bear-
ing by.......................... 77
Reciprocating Valve	Gears......... 234
Regulator, Guhrauer	&	Wagner’s.. 237
Regulator, Rigg*s.................. 236
Reichenbach’s Water Pressure En-
gine......................... 229
Releasing Ratchets...........150, 162
Releasing Valve Gears............. 162
Release of Friction Pawls......... 161
Rennie, Experiments on Joumals...	64
Repeating Watches................. 169
Repsold’s Chamber Gear Train..... 220
Repsold’s Pump..................   221
Reservoirs for Air and	Gas........ 272
44	for Gases............... 219
44	for Pressure Organs. 218, 260
4:	Natural................. 218
44	Negative................ 219
Resistance, Coefficients of......... 1
4 4	Modulus of................ 1
“	of Bends in Pipes...... 247
44	of Valves in Pipes..... 247
4 4	Practical................. 1
Theoretical............. 1
“ to Bending............ ...	2
44	Buckling............. 13
“	Flow in Pipes...... 246
44	Shearing.............. 2
“	Torsion ............. 11
Resultant of Isolated Forces.....	29
*“	Load on Water Wheel
determ’d Graphically.	34
“	Several Forces..... 26
Return Crank...................... 105
Retum Crank, Graphostatic Calcu-
lation for.................... 105
Reuleaux’s Coupling.............. 96
“	Escapement............. 168
44	Friction Clutch......... 100
44	System of Rope Trans-
mission............  206
44	Valve Diagram........... 234
*	44 Winding Drum............. 173
Reversing Gear, Globoid........... 143
Revolver, Mauser,-...........165, 166
Rhenish Railway Cable............. 174
Ribbed Axles....................... 91
Rib Profiles, Construction of....	91
Richard’s Manometer............... 288
Rider’s Valve Gear. .............. 236
Riedler’s Air Compressors......... 279
44 Pipe Joint................ 249
44 Valve Gear................ 278
Riggenbach’s Cable System......... 174
Riggenbach’s Hauling System...... 172
Rigg’s Regulator.................. 236
Rigid Couplings..................   95
Rim of Gear Wheel................. 149
Ring System of Cable Transmis-
sion......................208-211
Rittinger’s Pump.................. 223
Riveted Joints, Construction of An-
gles.............. 44
4 4	4	4	Junction	of	Plates.. 43
“	44	Proportional Scale
for............... 41
“	Reinforcement of
Plates............ 44
“	44	Special Forms...	43
“	44	Strength	of..... 40
4 4	4	4	Table of........ 40
44 Pipe...................... 244
44 Pipes, Flanges for........ 249
Rivet Heads, Proportions of........ 39
Riveting..........................  39
44	Boiler...................  42
44	Group...................  41
“ Machme...................... 39
4 4	4 4 Mackay & McGeorge m
Riveting Machine, Tweddell’s Hy-
draulic...................... 228
Riveting, Speed of................. 39
Rivets...........................    39
Robinson’s Experiments on Lift
Valves......................... 277
Robertson’s Friction Wheels........ 125
Rock Arms.......................... 162
Rock Drill, Githens...............  231
Rod Connection, Wiedenbruck’s....	50
Rod, Friction Ratchet.............. 163
Rods, Connecting................... 112
Rolled Shafting.................... 94
Roller Bearing, Cambon’s........... 127
44 Bearings..................... 126
44 Bearings for Sheaves......... 179
Roof Trusses, Force Plans for....	36
44 Truss, Polygonal.............. 37
4 4 with Simple Principals....... 36
44 with Trussed Principals....	36
Root’s Blower...................    221
Roots, Extraction of................ 26
Rope, Centrifugal Force of Wire.... 197
“ Connections................... 181
44 Cotton....................... 179
44 Cross Section of Wire-------- 196
44 Curve, Construction of..... 202
44 Hanger, Osterkamp’s.......... 181
44 Hangers...................... 181
44 Hemp......................... 178
44 Influence of Pulley Diameter
on......................... 197
Ropes of Organic Fibres........... 178
Rope Pulleys, Construction of.... 202
Ropes, Creep of. .................. 202
44 Deflection of Wire.......... 198
“ Flat....................178, 181
44 Loss from Stiffness......... 196
Rope, Specific Capacity of Wire.... 196
RopeSplice........................ 181
Ropes, Stiffness of................ 181
44 Tightened Driving........... 200
44 Ziegler’s Experiments on... 181
Rope Transmission.................. 194
44	“	at Bellegarde—	205
4 4	4	4	at Freiburg.... 205
“	“	at St. Petersburg	205
“	“	at Schaffhausen.	204
“	“	at Zurich........	205
44	“	Cotton........... 196
4 4	4	4	Cross Section for
Hemp.......... 195
“	44	Efficiency of....	205
“	44	Loss in Hemp. ..	195
44	44	Reuleaux’s Sys-
tem of......................... 206
“	“	Specific Capacity
of Hemp------- 195
“	44	Wire............. 196
“ Weight of Hemp................ 178
44 Wire......................... 179
Rotary Pumps..................... 226
44 Valves...................... 281
44 Valve, Wilson’s............. 285
Rotati ve	Motors, Adjustable Gears
for.................... 237
44	Pressure Engines......... 233
44	Valve Gears.............. 234
Round Connecting Rods............ 116
Round Valves..................... 275
Roux’s Water Pressure Engine... 229
Rubber Springs...................  21
Rubber Springs, Werder’s Experi-
ments............................. 21
Running	Chains................. 182
44	Friction Ratchets......158,160
44	Mechanism for Lifting
Water.................. 221
“	Mechanism for Pressure
Organs. ............... 219
“	Ratchet Gears, Toothed. .	150
“	Ratchets................  150
44	Ratchet Trains, Fluid___223
44	Tension Organs .... ..... 172
Rupp’s Variable Speed Gear..... 124
Rupture, Modulus of.............    13io
ALPHABETICAL INDEX.
SAFETY, COEFFICIENT OF.. .. i
Safety Devices for Elevators..... 164
Safety, Factor of................... 1
Sail Boat........................  223
St. Peter sburg, Rope Transmission at 205
St. Louis Bridge................... 60
Saint Venant, Frictionof Water___247
Saladin’s Friction Pawl........... 161
Sanderson’s Gas Meter............. 239
San Francisco Cable Tramways.... 174
Saxby & Farmer, Signal Apparatus. 166
Saw, Zervas’ Wire................. 177
Saws, Band.....................    177
Scale Beams.......................   m
Schaffhausen, Rope Transmission at 204
Schiele Turbine................... 220
Schmick*s Canal Cable System..... 175
Schmid’s Water Meter.............. 240
Schmid’s Water Pressure Engine . 236
Sehurman’s Clutch................ 101
Schurman’s Friction Coupling..... 215
Screw, Archimedian................ 221
44 Connections............... 58
“ Propeller................... 223
44 Propeller, Lifting Frame for 151
“ Propellers, Built up......... 57
“ Propellers, Method of Keying 49
Screws, Enlarged................. 58
Screw Thread, Construction of....	50
“	44 Dimensions of V-----	50
44	44 Frictionof...........	51
44	44 Pressure on......... 58
44 Threads, Special Forms of ..	57
44 Threads, Trapezoidal...... 58
Section of Gear Teeth............. 144
Section Modulus.............5, 7» n
Sections of Uniform Resistanee---	8
Secured Bolts...................... 57
Securing Keys, Methods of.......... 5°
Segner s Water Wheel.............. 220
Self Guiding Belting.............. 186
Seller’s Coupling.................. 96
44 Friction Feed.............  126
44 Hanger.....................  74
44 Pillow Block................ 70
44 Planing Machine............ 176
•4 Screw Thread System.......	52
44 Wall Bearing................ 71
Sewage System of Berlin........... 219
Sewing Machine Check.............. 151
Shafting..........................  92
44 Deflectionof................ 94
44 Dimensions of..... ......... 92
44 Examples of Torsion in....	94
44 Graphical Calculation of	94
44 Toumals for................. 94
44 Line.......................  93
44 Rolled...................... 94
44 Specific Capacity of....... 257
44 Torsional Deflection of...	92
44 Wooden...................... 94
44 Wrought Iron................ 93
Shank*s Planing Machine........... 163
Sharp’s Coupling................... 96
Sharp*s Strap End................. 112
Shearing, Resistanee to..........2, 10
Shearing Strain..................... 2
Sheaves, Chain..............185, 211
Sheaves, Roller Bearings for..... 179
Shield Gearing..................   133
Shifter for Quarter Twist Belt... 189
Shifters, Belt.... •• ............ 188
Shifting Eccentrics..............  235
Short Span Cable Transmissions---200
Shrinkage, Hooping by.............. 45
ShrinkingFit ...................... 17
44	Fits, made with Boiling
Water.................. 47
44	Rings, Clerk’s Method...	45
44	Temperatures............. 45
Sickles* Adjustable Valve Gear...237
Sickles* Valve Gear............... 162
Side Wheel Steam Boat............. 223
Siemens & Halske, Electric Signals. 166
Siemens* Alcohol Meter............ 239
Siemens’Geyser Pump............... 222
Signal Apparatus, Saxby & Farmer. 166
Silent Ratchets......................... 153
Simple Crank Axle......................  106
Simple Escapements...................... 167
Single Acting Steam Engine ............. 229
Single Tooth Gears...................... 165
Sinoide.................................. 91
Sinoide, Cycloidal....................... 13
Siphon, Direct.......................... 287
Siphon, Inverted........... ... .244, 287
Slide	Valve,	Common..................... 225
44	4	4	Gear, Plain................ 234
4 4	4	4	Lap of..................... 225
4 4	44	Murdock’s................... 234
44 Valves. .........223, 273, 2S1, 282
44 Valves, Balaneed.................. 285
Sliding Brakes.......................... 215
Sliding Crank........................... 226
Slipper Cross Head...................... 121
Sluice Gates at Geneva.....	275
Sluice Valve. .......................... 281
Snail...............................     169
Solid End for Connecting Rod . 113
Special Forms of Bearings ............... 74
“	44 of Bolts.......... •	55
“	44 of Ratchet	Wheels..	154
4 4	4 4 of Screw Threads.	...	58
Specific Capacity of Belting............ 190
“	“	of Driving Chains.	211
“	44	of Hemp Rope
Transmission.. .	195
“	44	of Pressure Trans-
missions...................-. ..	255
“	“	of Shafting....... 257
“	44	of Wire Rope...... 196
Speed Gear, Variable.................... 124
Spencer & Inglis Valve Gear....... 262
Spherical Cycloid....................... 135
44	Journal, Connection	for.. 115
44	Spiral....................    142
44	Valves....................... 275
Spiral Bevel Gears...................... 141
“	Gears................... 138
4 4	4 4 Double,................ 141
“	44 Examples of............... 140
4 4	4 4 Teeth, Friction of..... 140
“	Gears,	Globoid.. *. ... ....... 142
“	Pump...........................221
44 Spherical........................  142
“ Winding Drums...................... 181
4’	Wire Pipe................. 252
Spinning Mule.......................169,	196
Splice for Ropes..................... ;.	181
Split Pin..............................   56
44	Pulley, Goodwin’s............. 194
44	Pulleys................   193
Spring, Dudley’s........................  20
44	Pawl.......................... 153
44	Ratchets.................. 153
Springs, Best Material for............... 20
44	Calculation	of................. 18
44	Table of.......................18-19
44	Vulcanized	Rubber............. 21
Spur Gear Teeth, Construction of.. 128
Squaring De vice for Cranes............. 172
Square Thread............................ 40
Standing Tension Organs.....	172
Stand Pipes............................  287
Statical Moment........................... 3
Statical Moment, Graphically Con-
sidered............................. 33
Starting Valve.......................... 281
Star Pin................................ 153
Stationary Chains....................... 182
44	Friction Ratchets... 161
44	Machine Elements.....289
44	Ratchets..................150,156
44	Valves.............. 289
Steam Boat, Side Wheel...................223
44	Boilers........................  265
44	Distribution of Power..... 257
“	Engine, Single Acting Steam 229
44 Pipes...................... 245
“	Power Distribution.............. 219
44	Pump, Blakes.................... 240
4 4	4 4 Deane’s............ 230
Steam Pump, Pickering’s............ 230
e “ Tangye's................... 230
Steering Gear............... 238
“ Trap, Kirchweger’s........... 228
Steel Pipe......................... 243
Steering Gear.................... 171
“	44	Britton’s.......... 238
“	44	Davis & Co.’s. 238
“	44	Douglas &	Coulson’s.	238
“	44	Dunning &	Bossiere’s	238
4 4	4 4	Hastie’s...... 238
44	44	Hydraulic..... 237
“	“	Steam......... 238
Steib’s Ventilator................. 222
Step Anchor Ratchet................ 156
44 Bearings......................  75
4 4 Bearing, Support for ......,	80
44 Bearings, Wall................. 75
44 Gearing.....................   141
Stephenson’s Link Motion............235
Stepped Bevel Gears................ 141
Step Ratchets.... ................. 155
Step Valves................... . . 276
Stevart, Experiments on Springs.. .	21
Stiffness of Belts.' .............  194
“ of Ropes......................181
“	“	Eytelwein’s	For-
mula	181
“	44	Loss from....... 196
4 4	4 4	Weisbaeh’s	For-
mula	 181
44	“	Wire Rope.......206
Stolz*s Pump......................  224
Stone’sPump.......................  224
Stop, Geneva.........	  165
Storage Reservoirs, General........ 273
Strain, Shearing....................  2
Strains of Flexure................... 3
Strap Brakes.................211, 215
44 Brakes, Intemal............... 215
4 4 End for Connecting Rod..... 112
44* End, Sharp’s................. 112
Straps, Eccentric.................. 115
Strength of Cast Iron Columns....	83
“	of Materials...............1-21
“	of Wire Rope..............  179
“	Tensile................... 1
Stress Curve.......................  87
S tresses, Compound................. 13
44	in Columns...............   82
44	on Keys................... 48
Stress on Belting.................. 191
44	on Gear Teeth.............. 145
44	on Journals................ 61
44	S, Value of................. 8
Striking Mechanism for Clocks....169
Stub End, for Fork Journal......... 114
Stuffing Boxes..................... 253
“	Box, Farcot’s............ 254
44	Box, Friction	in..........254
Sturtevanfs Hanger Boxes............ 74
“ Pillow Block................ 70
44 Pulley.................... 194
Superficial Pressure................. 1
Superheated Water Transmission... 219
Supporting Pulleys, Vertical..... 188
Supports for Bearings............... 79
4 4	4 4	General Prin-
cipies......................	82
“	44	Simple.....	79
Supporting Power of Beams........	5
Swedish Piston......................253
Swedish Railway, Boilers for..... 272
Sweet*s Valve Gear...............235
Swivels..........................   182
Swivels fo* Chain.................  184
Symmetrical Simple Axles........	83
TABLE OF BEAM SECTIONS.5, 6, 7
Tables of Curves, Areas and Vol-
umes......................  291-296
Table of Numbers............... 300-301
Tackle Block....................... 172
Tangential Pressure on Crank Pin.. 233
Tangye’s Crane....................  176
Tangye’s Steam Pump................ 230ALPHABETICAL INDEX.
311
Tanks, Cast Iron................ 260
“	Combination Forms for..... 264
“	Intze’s Discussion of..260-264
“ Oil......................... 218
“	Witb Concave Bottoms...... 262
“	Wrought Iron............   260
Taper of Keys..................   47
Technological Applications of Pres-
sure Organs................. 24°
Technological Applications of Ten-
sion Organs................. *77
Tenacity.......................... 1
Tensile Strength.................  1
Tension Organs....... .......... *72
“	“	forDriving........ 173
44	“	for Guiding........i72
44	44	for Winding....... 172
44	“	Ratchet........... 185
14	“	Running..........  I72
14	44	. Technological Appli-
cations of.................	177
44 Resistance to..............  2
Testsfor Chain.................. 183
T Fittings...................... 251
Theoretical Resistance............ 1
Thickness of Cast Iron	Pipes.... 242
Thick Vessels, Walls of.......... 16
Thomas’ Calculating Machine. 153» x46
Thometzek’s Valve................276
Thomson^ Turbine................ 220
Three-part Bearings----*......... 7°
Throttle Valves...............   161
Throttle Vaves...................279
Thrust Bearing by James Watt & Co.	77
44	44	by Maudslay----....	77
“	44	by Penn............ 77
44	“	by Ravenhill &
Hodgson.......... 77
4 4	4	4	Compound Link as	67
44 Bearings..............65,68,75
“	“	Collar............. 66
44	“	Examples	of. 78
“	“	Multiple Collar...	66
“ with Wooden Sur-
f ace........... 76
“ upon the Pawl.............. 152
Thumb Shaped Pawl................ 160
Thumb Shaped Teeth..............  134
Tiede’s Escapement..............  168
Tightened Cables, Tablefor....... 200
Tightened Driving Ropes.......... 200
Tightening Pulleys............... 186
Tightening Pulley, Weaver’s......186
Toggle Friction Brake............ 162
Tools, Hydraulic................. 218
Toothed Gearing.................. 127
Tooth Friction in Spur Gearing--- 134
44 Outlines, General Solution of 129
4 4	4 4 Mixed............  133
4 4	4 4 of Circular Ares... 131
Torpedo, Fish.................    171
Torsional Deflection of Shafting...	92
Torsion,	Determination of	Angle of 93
“	Resistance to............ 11
“	Table.................... 12
4-	Uniform Resistance to___	13
Towne Crane...................... 176
Transformer, Hydraulic........... 218
Tranforming Arm Sections, Table for 103
Transmission at Long Distance,
Fluid..............233
Chain.............. 211
Gears.............. 144
Long Distance Power. 259
Modulus of......... 208
Rope............... 194
With Inclined Cable... 200
■Transportation, Fluid Escapement
ment for.................... 227
Transporting Belts .............  221
Trapezoidal Screw Threads........51, 58
Trap, Water.....................  287
Triangles, Area of................ 23
Trick’s Valve...................  284
Trigger, Hair.....................    153
Trigonometrical Formulae.....— 299
Trgionometrical Functions, Powers
of................. 25
“	Table.........297-299
Trussed Beams, Double....... ... 35
44	“ Simple................ 35
“	“ Triple................ 35
Tubing, Levasseur’s Metallic.....252
Tumbling Gears..................... 163
Turbine, Borda’s.................   220
44	Cadiat.....................220
“	Fourneyron.................220
44	Francis....................220
44	Girard.....................220
“	Nagel..................... 220
“	Schiele....................220
“	Thomson’s................. 220
TweddelFs Accumulator.............. 265
TweddelFs Hydraulic Riveter......228
Twin Link.........................  184
Twisting Moments, Graphically Con-
sidered......................   33
Tympanon of Archimedes............. 221
UHLHORN’S COUPLING... .101,153
Umbrella Pulley.................... 207
Uniform Escapements.............. 167
Uniformly Distributed Forces.....	32
Uniform Resistance, Columns of. ..	13
4 4	4	4	Sections of....	8
“	“	to Bending----	8
4 4	4 4	to Torsion....	13
“ Strength, Bodies of.......	2
Universal Gears, Beylich’s......... 136
Universal Joint..................... 97
Unloaded Bolt Connections........... 60
Unloaded Keys......................  49
Unperiodic Power Escapements for
Pressure organs............. 227
VALVE, ALLAN’S DOUBLE.... 283
44 Armstrong*s Supported.. 286
44 Bell................*.. 276
“	Boulton & Watt’s Bal-
anced.................. 285
44 Brandau’s Double Seated. 286
44 Cave’s Balanced............ 285
“ Corliss..................... 236
“	Cornish................... 280
4 4	Cramer’s Balanced....... 280
“ Cuvelier’s Underpressure 286
“ Double Beat................. 280
“ D....................283
4 4	4	4	Flap............. 274
“	Diagram, Reuleaux’s..... 234
“	Zeuner’s.................  234
“ Gear, Angstrom’s.............235
“	“	Brown*s.........235
4 4	4	4	Cail & Co......... 162
“	“	Cam............... 236
4 4	4	4	Corliss........... 162
44	“	Cornish..........  163
44	“	for Pumps......... 225
44	“	Harlow’s............231
“	“	Hofmann’s......... 163
“	Klug’s............ 235
“	“	MarshalFs......... 235
44	“	Plain Slide........ 234
44	44	Powel.............. 163
44	“	Rider’s............ 236
44 Gears for Rotati ve En-
gines............ 234
“	“	Reciprocating.___ 234
“	44	Releasing.......... 162
“	“	Rotative........... 234
“	Gear, Sweet*s............... 235
“	Gear, Wannich............... 162
44 Globe.................... .. 279
44 Gridiron................. 283
44	Hick’s Double........... 283
“	Injector.................  279
“	Kirchweger’s Balanced.	..	285
“	Lindner’s Balanced...... 285
44	Plain Slide.........225,282
“	Porter-Allen............ 287
“	Rubber Disk..............  274
Valves......................... 279
Valves, Balanced........	   279
Valves, Balanced Slide............... 285
Valve, Schaltenbrand’s Double Seat-
ed...........................  286
Valves, Check.......................  274
44 Closing Pressure of.............278
“ Conical......................... 275
44 Considered as Pawls... .223, 273
44 Flap........................... 274
“ Flat Disk....................... 275
44 Fluid.......................... 287
44 Gate..........................  282
44 Gidding’s Experiments	on.. 285
“ Lift....................223, 273
“ Mechanically Actuated....... 278
44 Multiple....................... 276
44 Piston.......................   286
“ Resistance of................... 247
“ Robinson’s Experiments on. 277
44 Rotary......................... 281
44 Round.......................... 275
44 Slide...............225, 273, 281
4 4 Spherical..................... 275
Valve, Starting.................. 282
Valves, Stationary....................289
44 Step........................    276
44 Throttle........................279
44 Unbalanced Pressure on Lift 277
Valve, Sweefs Balanced................287
Valves, Width of Seat.................274
Valve, Thometzek’s....................276
44 Trick’s........................ 274
44 Wilson*s Balanced.............. 287
44 Wilson’s Rotary................ 286
Valueof Stress S....................... 8
Vacuum Power Distribution............ 219
Variable Speed Gear.................  124
Variable Speed Gear, Rupp’s...... 124
Velocity Curves.....................  233
Ventilator, Fabry’s.............. 221
44 L6mielle’s............... 82
“ Steib’s....................... 222
Verge Escapement..................... 168
Volume, Escapements for Measure-
ment of.......................... 239
Von Gerike’s Air Pump................ 225
Vose's Pump.........................  224
V Screw Thread......................   50
WALKER MFG. CO., PULLEY
by..............................  194
Walking Beams........................ 110
Wall Bearings..................68,	71
44 Bearing, Support for............ 79
44 Bearing, Sellers’............... 71
“ Brackets......................... 72
Walls of Vessels, Resistance of........15
Wall Step Bearings.................... 75
Wannich Valve Gear................... 162
Washers.......................... 54
Watches, Repeating................... 169
Water Counterbalance, Oeking's. .. 217
“ Meter, Jopling’s................ 239
4 4	4	4	Kennedy’s.............. 239
44	“	Payton*s................220
“	44	Schmid’s............... 240
Pressure Engine, Belidor’s.. 229
“	44	“	Reichen-
bach’s. . 229
“	4	4	4	4	Roux’s ... 229
“	“	“	Schmid’s.. 236
“ Reservoir, of Frankfurt on
Main........................ 218
44 Rod Connection............ 233
“ Running Mechanism for Lift-
ing......................... 222
44 Trap.......................... 287
44 Trap, Morrison,Ingram & Co. 288
44 Wheel, Poncelefs.............. 220
44 Wheel, Resultant of Loadon 34
“ Wheels, Axles for............... 91
“ Wheel, Segner’s.................220
44 Wheels, Gravity............... 219
“ Wheels, Impact............ .	220
Watt’s Condenser...............,...	230
Wesar on Gear Teeth................ 134312
ALPHABETICAL INDEX.
Wear on Hfemp Rope.............. 196
Weaver’s Tightening Pulley...... 1S6
Wedge Friction Wheels.  .....125, 160
Weighing Machine, Emery’s....... 173
Weight of Cast Iron Pipe........ 242
of Chain............... 183
“	of	Gear Wheels......... 150
“	of	Hemp Rope........... 178
“	of	Round Iron........... 55
“	Sheet Metal.............. 43-
“	of	Wire Rope........... 1S0
Weir, Camere’s...................275
Weisbach, Formula for Friction of
Water....................... 246
Weisbach’s Formula for Stiffness of
Ropes.....................   181
Werder, Experiments on Springs...	21
Weston’s Differential Pulley Block. 173
“ Friction	Clutch__. _____101
“ Ratchet	Brace .......  154
Wet Gas Meter................... 239
Wheels, Classification of....... 122
Whip Aetion of Connecting Rod... 116
Whitehead Torpedo............... 237
Whitworth’s Screw System.......... 51
Whitworth’s Pipe Thread Scale___	51
Wiedenbruck’s Rod Connection----	50
Wilber’s Ratchet................  153
Wilson’s Rotary Valve............ 286
Wilson's Water Gas Furnace......288
Winding Drum, Reuleaux’s......... 173
“ Drums, Spiral.............. 181
“ Tension Organs for........ 172
Windlass.........................  172
“ Brown’s.................... 173
“ Differential............... 173
Windmills.......................  220
Wind Stresses, Graphieally Deter-
mrned........................  37
Wire Rope........................  179
“	“ Influence of Weight....	180
“	“ Load Length of..........	180
“	“	Strength of........... 179
“	“	Transmission.......... 196
“	“	Weight of............. i8c
“	Saw,	Zervas’.............  177
Wooden Axles, Proportions of....	92
Wooden Shafting................... 94
Worm and Worm Wheel............ jyg
“	Gearing,	Globoid......... 143
“	“	Hawkins......... 143
“	“	Jensen’s.........  ^3
Worthington High Duty Pumping
Engine..................... 232
Worthington’s Duplex Pump...... 231
Worthington’s Equalizer....... 232
Wrapping Connections.......... 173
Wrenches....................... 56
Wrought Iron	Cranks, Single... 104
“	“	Pipe............ 243
44	4	4	Shafting......... 93
“	“	Walking Beams.... m
YALE LOCK..................... 167
Yoke Bearings.................  72
ZERVAS’ WIRE SAW.............. 177
Zeun er’s Valve Diagram....... 234
Zimmermann’s Belt Shifter..... 189
Zuppinger’s Water Wheel....... 219
Zurich, Rope Transmission at... 205I