UNIVERSITY OF CALIFORNIA
LOS ANGELES
IA.
WORKS OF J. H. CROMWELL
PUBLISHED BY
JOHN WILEY & SONS.
A Treatise on Toothed Gearing.
izrno, cloth, $1.50.
A Treatise on Belts and Pulleys.
i2ino, cloth, $1.50.
A TREATISE
BELTS AND PULLEYS.
FULL EXPLANATIONS OF FUNDAMENTAL PRINCIPLES; PROPER
DISPOSITION OF PULLEYS ; RULES, FORMULAS, AND TABLES
FOR DETERMINING WIDTHS OF LEATHER AND VUL-
CANIZED-RUBBER BELTS AND BELTS RUNNING
OVER COVERED PULLEYS; STRENGTH AND
PROPORTIONS OF PULLEYS, DRUMS, ETC.
TOGETHER
THE PRINCIPLES OF AND NECESSARY RULES FOR ROPE-GEARING
AND TRANSMISSION OF POWER BY MEANS OF
METALLIC CABLES.
BY
J. HOWARD CROMWELL, Pn.B.,
AUTHOR OF A TREATISE ON TOOTHED GEARING.
FIRST EDITION.
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THIRD THO USA ND.
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NEW YORK:
JOHN WILEY & SONS.
1903.
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Copyright, 1885,
BY JOHN WILEY & SONS
PRESS OP
BRAUNWORTH A 'CO.
BOOKBINDERS AND PRINTERS
BROOKLYN. N. V.
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PREFACE.
IN the manufacture of modern machinery, which in the
great majority of cases embodies a vast deal of careful study
and precise calculation, there is probably no one element which
enters as largely into the calculations and forms as important
a part in the daily operations in the machine-shop as the end-
less belt for the transmission of power. The lathe, the drill,
the planer, the shaping-machine in short, almost without ex-
ception, all machine-tools are commonly driven from the
shop-shaft by means of belts and pulleys ; and we can scarcely
glance into a shop or factory of any description without en-
countering a mass of belts which seem at first sight to mo-
nopolize every nook in the building and leave little or no room
for anything else.
Notwithstanding the countless thousands of belts for trans-
mission in use and constantly being replaced in the shops and
factories of America; notwithstanding the fact that many
thousands of dollars are consumed every year by the rapid
wear and destruction of our machine-belts, and the immense
field thereby opened for the practical study and application of
the principles of economy in this connection there is no
branch of machine-construction which is to day in as crude
and unsatisfactory a state of development as this all-important
transmission by belt and pulley. Strange as it may seem, it is
iv PREFACE.
nevertheless true, that there is scarcely a machine-shop in
America which can definitely and correctly calculate the proper
width of a leather belt which will safely transmit a given horse-
power. Nor are the manufactures of machine-belting in any
degree in advance of the shops, for I have never yet seen the
manufacturer who has any better solution for this apparently
simple problem than his own " judgment."
Having taken the pains to write to a large number of the
best-known machine shops and belt manufacturers throughout
the country, asking for information concerning belting, and in
every case having received an answer to the communication, I
am compelled to assert that among all the letters received not
a single one contained any definite information on the subject.
As specimen answers to these letters I may quote the following
extracts :
" We have no particular method of calculating widths of
belts aside from tables found in books of reference." "There
is no rule for the width of belting that we know of: it is always
determined by the width of the pulley upon which it is to run."
" We determine the width of belts more by experience than by
any fixed rule." " We always try and make the strain as light,
in pounds per inch of width, as possible, and when we are
limited for room we use double belts. 100 pounds per inch
of width is about the ultimate strength of transmission, and if
you can reduce the working strain to 50 pounds, it means long
life to the belt." " It is difficult to give any positive rule about
belting that would apply to all cases." From one of the largest
and best-known belt manufacturing concerns in the country
comes the following : " We have no rules or formulas for esti-
mating the power of belts other than those given in works on
mechanical engineering, nor do we apply these strictly. It is
PREFACE. V
very much a question of judgment. . . . You will consider this
letter very indefinite, but we do not know how to make it less
so." Here are extracts from a letter received from another
well-known belt-maker : " We wish to express the fear that
what we have to say will be disappointing to you, to say the
least. ... As to the horse-power, we have no rule. . . . We
have made no tests of the tensile strength of leather, for the
reason that we do not consider it a matter of any importance.
. . . We have made no efforts to obtain the coefficient of fric-
tion. . . . When we can obtain a homogeneous material which
will be easily workable and a perfect substitute for leather, the
manufacture, sale, use, and study of belting may begin to be a
matter of satisfaction ; in the meanwhile they are puzzling, if
not indeed exasperating."
These extracts (many more of similar nature might be given)
show almost no knowledge at all, on the part of our great belt-
manufacturers and machine-shops, concerning the subject; and
worse still in some cases, that little or no effort has been made
to obtain any knowledge other than that of rough guesswork
and rule of thumb. Small wonder is it, then, that the ordinary
mechanic's practical knowledge of the subject is infinitely small.
Several of the parties above referred to state that they use the
rules found in the various books of reference ; let us look over
some of these works and endeavor to reach fair conclusions
concerning the rules and formulas in common use to-day.
Arnold, in his " Mechanical Principia," gives the rule for belt-
widths : "Multiply 36000 by the number of horse - powers ;
divide the amount by the number of feet the belt travels per
minute ; divide this quotient by the number of feet in length of
belt contact with the smaller drum or pulley, and divide this
by 6 : the result is the required width of belt in inches."
Vi PREFACE.
Professor Reuleaux offers the formula b 18 ^~P~, b represent-
ing the width of the belt in millimetres and P the force in
kilograms transmitted by the belt.
Unwin, in " Elements of Machine Design," gives the formula
2 P
ft = in which ft is the belt-width in inches, P the force
transmitted in pounds, and /the safe working tension per inch
of width, which he takes at 70 pounds for a belt -fa of an inch
thick. The formula is to be used only when the belt embraces
about 0.4 of the smaller pulley-circumference.
In Nystrom's Mechanics we find b = ^ , b denoting the
belt width in inches, H the horse-power transmitted, d the
diameter of the smaller pulley in inches, and a the number of
degrees occupied by the belt on the circumference of the
smaller pulley.
Let us now assume an example which will serve to determine
the variations in the results of calculations from the above
rules and formulas. Suppose we wish to determine the proper
width for a belt which will transmit a force of 25 horse-power;
the smaller pulley having a diameter of 5 feet = 60 inches, and
the velocity being 10 feet per second = 600 feet per minute.
The belt embraces 0.4 of the pulley-circumference = 0.4 x 15.7
= 6.28 feet = 360 x 0.4 = 144 degrees. For the force trans-
mitted, in pounds, we have P = - = 1375 pounds.
With these quantities as data, Arnold's rule, given above, gives
us for our required belt-width -^ ^ - = 39.8 inches.
OOO X O.2o X O
If we divide the force 1375 pounds by 2.2, we obtain 625 kilo-
grams, and Reuleaux's formula gives b 18 4/625 = 450 milli-
metres = 450 x 0.04 = 18 inches. From Unwin's formula we
PREFACE. Vll
obtain ft = -- = 39-3 inches, and from the formula of
Nystrom b = = 21.7 inches. Haswell in his " Engi-
60 x 144
neer's and Mechanic's Pocket-book," gives a rule by which our
belt-width would be 42 inches. Summing up our results will
show that, for the same belt, under the same circumstances,
the width is according to the authorities named as follows :
Haswell 42 inches.
Arnold 39.8
Unwin 39.3
Nystrom 21.7 "
Reuleaux 18
Of these different values the greatest is 2\ times the least.
Probably Arnold, Haswell, and Nystrom are in use in our shops
more than the others, and these give results, for the belt-width
in question, differing from each other by more than 20 inches.
According to a list of prices for double, white-oak tanned belt-
ing, which is before me, the difference in cost for the above-cal-
culated belt, supposed to be double and 100 feet long, between
Nystrom and Haswell would be six hundred and sixteen dol-
lars, to say nothing of the difference in the cost of the pulleys,
shafts, etc.
These great differences between the results from the rules of
different authors are apparently due to the difference of opinion
concerning the value of the coefficient of friction, which is
taken all the way from 0.22 to 0.40, and to the fact that each
writer on the subject has striven to obtain simple rather than
accurate rules. At best we are dealing with an uncertain ma-
terial when we attempt to deduce rules for the strength of
leather belts, and if the elements of belt-thickness, method of
Vlil PREFACE.
lacing or fastening, etc., are entirely or partially neglected, the
uncertainty of accurate results must be very greatly increased.
In the matter of joint-fastening alone, a glance at the table on
page no will show that a 2o-inch leather belt ^ inch thick run-
ning over two equal cast-iron pulleys will transmit a force of over
1800 pounds with a riveted joint or 1250 pounds when fastened
with a double raw-hide lacing, while with a single leather-lac-
ing the same belt will transmit but 976 pounds. In other
words, to transmit a force of 1000 pounds over two equal cast-
iron pulleys by means of a leather belt -fa inch thick we will
need a belt-width of 12 inches for riveted joint, 16 inches for
double raw-hide lacing, and 22 inches for single leather lacing.
I believe that it is utterly impossible for any man to write an
entirely simple work on the subject of belting, which will be of
any practical use to the mechanical world. The subject is
complicated by difficulties far greater than are ordinarily met
with in dealing with mechanical questions, and to attempt to
simplify it beyond a reasonable limit is simply to omit certain
necessary considerations, and thereby render the investigation
worthless. My object in writing this work on belts and pulleys
is, therefore, to present to the mechanical public a small yet
comprehensive and, above all, an accurate book on the subject.
I have constantly endeavored to have due regard for simplicity,
yet when I have found it necessary to sacrifice either simplicity
or accuracy, I have invariably chosen the former. All measure-
ments and dimensions are given in English units in order to
avoid the confusion sometimes resulting from the use of the
Metric System, and I have endeavored by numerous simple ex-
amples throughout the book to fully illustrate the use of the
various rules and formulas. In translating the part devoted to
metallic cables from Reuleaux, the formulas and tables have
PREFACE. IX
been transformed from the metric system into English measures,
which will, I trust, satisfactorily explain the unusual numbers
which have resulted in a few instances.
In the hope that my humble endeavors to furnish accurate
information on the subject of belt-transmission to those whom
it may concern may be in a measure, if not entirely, successful,
and trusting that in the present instance I may receive from
mechanical men the same generous support and encourage-
ment that have attended my previous efforts in the field of
mechanical literature, I present to the public my " Treatise
on Belts and Pulleys."
J. H. C.
NEW YORK, May i, 1885.
TABLE OF CONTENTS.
SECTION I.
PACK
Introduction Absence of early Mechanical Records Uncertain
Origin of the Belt and Pulley Probable Origin I
SECTION II.
Fundamental Principles Direction of Rotation Relations be-
tween Circumference, Diameter, and Radius Velocity Revo-
lutions Power Horse power 9
SECTION III.
Rules for the Proper Disposition of Pulleys Axes which coin-
cide geometrically Parallel Axes Axes which intersect each
other Axes which cross without intersecting 28
SECTION IV.
Transmissions by Belts without Guides Half-crossed Belt
Conditions necessary for maintaining the Belt on the Pul-
leys Distance between Pulleys 29
SECTION V.
Transmissions by Belts with Pulley-guides Half-crossed Belt
with Pulley guide Half-crossed Belt with Movable Pulley-
guide General Case of Crossed Arbors Arbors at Right An-
gles ; 33
xii TABLE OF CONTENTS.
SECTION VI.
PAGE
Length of Belts Open Belt Open Belt, approximate Formula
Crossed Belt Belts with Guides and intricately arranged 45
SECTION VII.
Speed cones Stepped Cones Open Belt Crossed Belt Graph-
ical Method Continuous-speed Cones 51
SECTION VIII.
Materials used for Belting Leather Vulcanized Rubber Intes-
tines of Animals Rawhide Hemp and Flax Leather and
Metallic Wire 65
SECTION IX.
Lacing and other Modes of Fastening Shortening Single and
Double Lacing Belt-hooks Cleat-fastening 68
SECTION X.
Strength of Leather Belts Resistance to Slipping Coefficient of
Friction Tensions on Belts Breaking-strength Width for
Different Kinds of Fastening Width necessary to transmit
certain Powers 75
SECTION XI.
Leather Belts over Leather-covered Pulleys Coefficient of Fric-
tion Tensions Width for Different Kinds of Fastening-
Width necessary to transmit certain Powers 115
SECTION XII.
Vulcanized Rubber Belts Number of Layers of Duck Thick-
ness Breaking-strength Coefficient of Friction Width for
Different Kinds of Fastening Width necessary to transmit
certain Powers Rubber Belts over Leather- and Rubber-cov-
ered Pulleys 140
TABLE OF CONTENTS. Xlii
SECTION XIII.
PAGE
Rim, Nave, and Fixing-keys for Pulleys Rounding of the Rim
Flanged Rim Rim of Pulley for Belt with Circular Cross-
section Split Pulleys Approximate Weight of Pulleys 159
SECTION XIV.
Arms of Pulleys Oval Cross sections Number of Arms
Strength of Arm s Straight Arms Single and Double Curved
Arms 166
SECTION XV.
Shafts Safe Shearing Stress Steel Wrought iron Cast-iron
Diameter necessary to transmit certain Powers 171
SECTION XVI.
The Tightening-pulley Fast and Loose Pulleys Reversing by
means of Fast and Loose Pulleys Fast and Loose Pulleys for
Belts with Circular Cross sections 179
SECTION XVII.
Rope-belts Tension almost entirely due to the Weight Pulley
for several Rope belts Proper diameters for Rope-belts Di-
ameters of Pulleys for Rope-belts 185
SECTION XVIII.
Jointed Chain-belts Rouiller's Chain-belt Metallic Belt of Go-
din Jointed Chain-belt of Clissold Coefficient of Friction
Dimensions 192
SECTION XIX.
Tensions of Metallic Cables Number of Strands and Wires Co-
efficient of Friction 196
xiv TABLE OF CONTENTS.
SECTION XX.
PAGE
Calculation of Diameters of Cables Formulas and Tables of Di-
ameters of Cables for Different Numbers of Wires 200
SECTION XXI.
Deflections in the Cable of a Horizontal Transmission Deflec-
tion of Cable in Motion Deflection in a State of Repose De-
flection in the Driving and Driven Parts 207
SECTION XXII.
Transmission by Cable with Increased Tension Increased Di-
ameters of Cable and Wires 212
SECTION XXIII.
Transmission by Inclined Cable Tensions in Inclined Cables
Deflections Height above the Ground 217
SECTION XXIV.
Method of Tracing the Curves of Cables Approximately Para-
bolic Curves 221
SECTION XXV.
Transmission by Cable with Pulleys near together Small Value
of Si 222
SECTION XXVI.
Rim of Cable pulleys Single Cable Several Cables upon one
Pulley 223
SECTION XXVII.
Arms and Nave of Cable-pulleys Number of Arms Oval Cross-
sections Flanged Cross-sections Straight Arms Curved
Arms Reserve Cables 226
TABLE OF CONTENTS. XV
SECTION XXVIII.
1AGR
Pulley-supports and Intermediate Pulleys Stations at the Ex-
tremities Intermediate Stations Changing the Direction of
the Cable 230
SECTION XXIX.
Dimensions of Pulley supports Ratio between the Radius of
the Pulley-support and Diameter of the Wires 234
SECTION XXX.
Pressure upon Axes of Pulley supports Weight of Large Pulleys 235
SECTION XXXI.
Station Pillars Brick and Stone Piers Pedestals Two Pulleys
side by side 238
APPENDIX I.
Experiments for determining various Coefficients of Friction
Leather over Cast-iron Pulleys Leather over Leather-covered
Pulleys Vulcanized-rubber Belts over Cast-iron and Covered
Pulleys 243
APPENDIX II.
Special Applications of Principles of Belts and Pulleys Devices
for changing Motion and Direction of Rotation Increasing
and Decreasing Speeds Intermittent Motion Different Meth-
ods of arranging Principal Pulley and Shop Shafts in Mills. . . 252
\A U FO&/
BELTS AND PULLEYS.
i. Introduction.
Says Thomas Ewbank in his famous " Hydraulics and
Mechanics :" " Tradition has scarcely preserved a single
anecdote or circumstance relating to those meritorious
men with whom any of the useful arts originated ; and
when in process of time History took her station in
the temple of Science, her professors deemed it beneath
her dignity to record the actions and lives of men who
were merely inventors of machines or improvers of the
useful arts; thus nearly all knowledge of those to
whom the world is under the highest obligations has
perished forever. ... A description of the foundries
and forges of India and of Egypt, of Babylon and
Byzantium, of Sidon and Carthage and Tyre, would
have imparted to us a more accurate and extensive
knowledge of the ancients, of their manners and cus-
toms, their intelligence and progress in science, than
all the works of their historians extant, and would
have been of infinitely greater service to mankind.
" Had a narrative been preserved of all the circum-
stances which led to the invention and early applica-
tions of the lever, the screw, the wedge, pulley, wheel
2 BELTS AND PULLEYS.
and axle, etc., and of those which contributed to the
discovery and working of metals, the use and manage-
ment of fire, agriculture, spinning of thread, matting of
felt, weaving of cloth, etc., it would have been the
most perfect history of our species the most valuable
of earthly legacies. Though such a work might have
been deemed of trifling import by philosophers of old,
with what intense interest would it have been perused
by scientific men of modern times, and what pure de-
light its examination would have imparted to every
inquisitive and intelligent mind !"
Rollin, writing of " The Arts and Sciences of the
Ancients" many years ago, finds fault with the world
for neglecting the great inventors and admiring the
military heroes of antiquity. " Of what utility to us
at this day," he asks, " is either Nimrod, Cyrus, or
Alexander, or their successors, who have astonished
mankind from time to time? With all their magnifi-
cence and vast designs they are returned into nothing
with regard to us. They are dispersed like vapors and
have vanished like phantoms. But the inventors of the
arts and sciences labored for all ages. We still enjoy
the fruits of their application and industry ; they have
procured for us all the conveniences of life ; they have
converted all nature to our uses. Yet all our admira-
tion turns generally on the side of those heroes in
blood, while we scarce take any notice of what we
owe to the inventors of the arts"
In like manner, Robertson, in his work on India,
laments the loss of, or rather absence of, early records
concerning the useful arts and sciences. He says : " It
is a cruel mortification, in searching for what is in-
INTRODUCTION. 3
structive in the history of past times, to find the ex-
ploits of conquerors who have desolated the earth, and
the freaks of tyrants who have rendered nations un-
happy, are recorded with minute and often disgusting
accuracy; while the discovery of useful arts and the
progress of the most beneficial branches of commerce
are passed over in silence and suffered to sink into ob-
livion."
The origin, age, first application, and use of the
mechanism known to us as the " endless belt and
pulley" are entirely unknown ; as far back into the
history of the ancients as we can see by means of the
earliest mechanical records, we find the endless belt
running continuously around the pulley precisely as it
does to-day. We may theorize, and assume a probable
origin ; we may bring up, in support of our assumption,
all the reason and logical conclusions at man's dispo-
sal ; we may even convince mankind that we have cor-
rectly traced and explained the path over which the
mechanism has come down to us from the dim ages of
the past. But here we must stop; we can go no fa--
ther: and the fact will yet remain that the real age and
origin for which we are searching are still undiscovered
and unknown. If, however, we cannot know with
certainty the real age and origin of belts and pulleys,
it is nevertheless a satisfaction to us to be able to trace
out, by analogy, by reason, and by the known existence
of things which must have necessitated the use of pul-
leys, what seems to us to have been the origin, the
successive modifications, and the line of improvement
by which this most useful contrivance has been handed
down to us.
4 BELTS AND PULLEYS.
In searching for an uncertain origin or beginning of
anything, we most naturally start by determining up-
on the very simplest and most rudimentary form
(knowing that simplicity almost always precedes com-
plexity, and that a thing must of necessity have a skele-
ton before it can have a form), and then strive to fix
upon its exodus from the conception to the tangible
thing itself. In order then to trace the growth of the
thing in question from its origin to its present much
altered and improved form, we strive to imagine the
slightest possible change, in the right direction, which
can be given to the original. Having successfully
achieved the first transformation or alteration, we con-
tinue to pick out each slight alteration and improve-
ment in proper order, until we have reached the present
most improved form.
If we assume, as is claimed by some writers, that the
mechanism of the belt and pulley was among the first
mechanical contrivances of primitive man, we must
search for its origin among what we judge to be the
first necessities of the human race and the modes of
obtaining these necessities. Although many claim that
the human race, in the beginning, passed through a
fireless period, that men lived without the use of fire
or artificial heat, we must nevertheless conclude that
this element was one of the first necessities of human
life, and that the first effort made by prehistoric man
in the line of invention was for the purpose of produc-
ing fire. It is very generally admitted that the first
" fire-machine" (Reuleaux concludes that this was the
first machine of any description. See Kencdcy's Eng-
lish translation " Kinematics of Machinery," London,
INTRODUCTION. 5
1876, p. 204) consisted of an upright piece of wood,
having one end pointed. This, fitted into a hollow in
another piece of wood and being twirled rapidly back-
wards and forwards with the hands, generated sufficient
heat to set fire to some small fragments of dry wood
or other combustible material (Fig. i). Here we
have the first belt and pulley hardly recognizable, it
is true, but none the less the probable origin. The
upright piece of wood here constitutes the pulley and
FIG. 2.
the human hands the belt. The first transformation
seems to have been the substitution of a cord wound
several times around the upright piece (as shown in
Fig. 2) in place of the direct application of the hands.
This rude contrivance, though it produced only an
oscillating motion, was used for other purposes than
that of producing fire; the primitive drill, lathe, pot-
ter's wheel, etc., were driven to and fro in this manner,
the work being done only on the forward turn, and the
O BELTS AND PULLEYS.
backward turn serving only to place the work in such
a position that the operation of cutting could be again
continued. The change from this contrivance to the
rope and pulley used for drawing water from deep
wells, and for lifting the vast blocks of stone, columns,
etc., used by the ancients in building, was indeed
slight, and may reasonably have taken place not long
after the first introduction of the improved form of
" fire-machine."
For how long a period this oscillating motion suf-
ficed for the rough manufacturing purposes of the age,
or at just what era in the life of man the change was
made to the endless belt, which transformed the oscil-
lating into a continuous rotary motion, is indeed a
mystery. Whole generations even centuries may
have been needed to impress upon the primitive mind
the advantages of continuous rotation and to accom-
plish the necessary change in the mechanism. It
seems most probable to us that the loss of time in-
curred by the useless backward motion in lathes, drills,
etc., and the natural desire on the part of these an-
cient artisans to accomplish more and more work in
less and less time, must have led to the adoption of the
two pulleys and the endless belt. Gradually, very
gradually, the slight but all-important change was
made. Some early thinker now unknown even in
the uncertain histories of the past ages connected the
loose and separated ends of the single cord, passed the
now endless cord over two cylindrical sticks, fitted
roughly into a frame to hold them apart, and caused
both to rotate by turning one with a crank. Next
some primitive inventor obtained the friction neces-
INTRODUCTION. ^
sary for the transmission of considerable forces by
winding the cord several times around each pulley ;
and so in process of time, in his attempts to obtain
and transmit greater powers, the man of the ages long
since forgotten at last discarded the round cord for the
broad flat band or belt of the present era. Reuleaux
says "the crossed belt appears to be the older;" but to
us it seems most probable that the flat band was first
used in its simplest form, i.e., open, and that the
crossed belt was afterwards introduced in order to pre-
vent (by its additional embracing of the pulleys) slip-
ping, and to produce a rotation of the driven pulley in
a direction contrary to that of the driver.
As to the material of the primitive cord and belt, we
can prove nothing : it is, however, reasonable to sup-
pose, since the skin of wild animals was the easiest
material to obtain, and since, from the earliest records
of history, skins have been used for clothing, bow-
strings, etc., that the material of the primitive belt
differed from the leather of to-day only in that it was
untanned and unfinished, and perhaps taken from a
different animal. Doubtless the fixing together or
lacing of the ends of belts was the source of considera-
ble difficulty to the ancients, for in all cases where
such a belt could be made to perform the necessary
work, round cords tied together at the ends seem to
have been used.
It is supposed, and very reasonably, from certain
known circumstances, that the first idea of continuous
rotary motion which was developed in the mind of
man took the form of an undershot water-wheel,
driven by the current of a stream or river. The Chi-
8 BELTS AND PULLEYS.
nese have doubtless used these water-wheels, for pur-
poses of irrigation and drawing water, for many centu-
ries, and, according to tradition, they Avere also used at
an early date in ancient Assyria, Mesopotamia, and
other countries of Asia Minor. These pristine water-
wheels consisted of a rough axle and two or more long
blades, usually built up of sticks and bamboo, some-
times with rough buckets formed out of mud or clay.
It is not at all unlikely that the first attempts to con-
F.G. 3.
struct large pulleys were founded upon the principles
of construction seen in the water-wheels, and that the
pulleys were used without rims, as shown in Fig. 3.
From the fact that wagon-wheels with entire rims and
fellies are known to have been in existence in the
earliest Greek and Egyptian times, we may very fairly
conclude that the use of the complete wooden, if not
also iron, pulley reaches far back into antiquity, and
that its advent into the world probably took place not
long after the discovery of the endless belt.
History informs us that the ancient city of Nineveh
FUNDAMENTAL PRINCIPLES. 9
was surrounded by a massive stone wall over 100 feet
high, and that the city was fortified with 1500 towers,
each 200 feet in height. Babylon, "the noblest city
ever built by man," had a fortified wall which reached
to the incredible height of 360 feet, and her famous
hanging-gardens were built of " flat stones of amazing
size." The Tower of Babel is said to have been "40
rods square at the bottom, and upwards of 600 feet
high." These gigantic structures supposed to have
been built about the year 2200 B.C. could not have
been erected without the aid of strong ropes and pul-
leys, or similar contrivance. Thus for over four thou-
sand years have been known and used successfully the
cord and pulley which we use to-day. For how many
centuries in the unknown ages of the prehistoric period
men toiled and labored with their crude " fire-ma-
chines," perhaps even lived and died without reaching
that much of " the machine," we must leave for future
investigation and development to decide.
2. Fundamental Principles.
The mechanism known in modern mechanics as the
" endless belt and pulleys" is, primarily, a device, the
object of which is to transmit a continuous rotary mo-
tion from one shaft or arbor to another parallel shaft,
and the first fundamental principle of the mechanism
may be clearly expressed as follows: If two drums or
pulleys be placed in certain positions relative to each
other, each being allowed the motion of rotation about
its fixed axis, and no other, and if an endless band be
passed tightly over the circumferences of the pulleys
IO BELTS AND PULLEYS.
as represented in Fig. 4 ; then, if a continuous rotary
motion be given to one of the pulleys, the friction be-
tween it and the band will cause the latter to move
around the circumference, and the second pulley will
FIG.
(because of the friction between it and the band)
therefore be caused to rotate continuously about its
fixed axis that is, the continuous rotary motion of
the driving-pulley will be directly transmitted through
FIG. 5.
the endless band to the second pulley. In this defini-
tion it is presupposed that the friction between the
band and pulleys is sufficiently great to overcome
the resistance of the pulleys ; otherwise the driving-
pulley will simply slide around upon the band without
FUNDAMENTAL PRINCIPLES. \\
causing it to move, and consequently the second or
driven pulley will remain motionless.
(a) Direction of Rotation. Belts may be either open,
as shown in Fig. 4, or crossed, as in Fig. 5 : in the
former case the two pulleys rotate in the same direc-
tion, while in the latter case the driven pulley rotates
in a direction contrary to that of the driver.
(b) Relations between Circumference, Diameter, and
Radius. The circumference C of a circle, the diameter
of which is represented by D, is given by the expres-
(i)
in which n represents the constant quantity 3.14159.
RULE. To determine the circumference of a circle
in inches or feet, multiply the diameter in inches or
feet by the constant 3.14159.
Since the radius of a circle is equal to one half its
diameter, if we denote the radius by R, we shall have
R = , or D = 2R, and formula (i) becomes by sub-
stitution
C = 27tR ....... (2)
RULE. To determine the circumference of a circle
in inches or feet, multiply the radius in inches or feet
by the constant 2rc = 6.28318.
From formula (i), by transposing the quantities, we
may write
-.' .'.... (3)
12 BELTS AND PULLEYS.
RULE. To determine the diameter of a circle in
inches or feet, divide the circumference in inches or
feet by the constant 3.14159.
In a similar manner from formula (2) we may obtain
RULE. To determine the radius of a circle in in-
ches or feet, divide the circumference in inches or feet
by the constant 27t = 6.28318.
If we let C and C denote the circumference of two
circles, D and D ', R and R 1 ', the respective diameters
and radii, we shall have, from formulas (i) and (2),
C = TtD = 2nR, and C = nD' = 2-nR' :
and we may write the proportions
C : C :: nD : nD' :: 2nR : 2nR.
in the form of an equation,
C 7tD
which, by cancelling the equal constants in numerator
and denominator, becomes
-
D' ~ R
(0
RULE. The ratio of the circumferences of any two
circles is equal to the direct ratio of their diameters or
radii.
FUNDAMENTAL PRINCIPLES. 13
(c) Velocity, The circumferential velocities of two
pulleys which are connected by one and the same belt
(supposing there is no slipping of the belt on either
pulley) must obviously be the same, each being equal
to the velocity of the belt. For the belt must unroll
from the driving-pulley just as fast as it is developed
from the pulley-circumference; it must also roll upon
the circumference of the driven pulley with the same
velocity, else the belt would constantly tend to become
tighter on one side and looser on the other, and sliding
or rupture would necessarily ensue.*
The circumferential velocity of the driven pulley and
the velocity of the belt are entirely independent of the
pulley-diameters, and depend solely upon the circum-
ferential velocity of the driving-pulley. Thus, if the
circumferential velocity of the driver is 10 feet per
second, 10 feet of circumference, and no more or less,
can be developed per second upon the belt, be the
driver ever so large or ever so small. In the same
manner, just 10 feet of belt can roll per second upon
* The tensions on the two sides (or parts) of the belt are not the
same (as will be seen farther on); consequently the circumferential
velocities of the two pulleys are not absolutely the same. According
to Professor Reuleaux, if v and v denote the circumferential velo-
cities of the two pulleys, t and T the tensions on the two parts of the
belt, E the coefficient of elasticity of the belt, and S the strain on the
driving part of the belt, the true velocities will be given by the ex-
t
v 1 -v i 7"
pression = . Reuleaux says, " The loss of velocity due
~S
to the sliding has for a mean value about \ percent; it is accompanied
by a loss of work, which is transformed into heat and produces wear
of the belt and pulleys."
H BELTS AND PULLEYS.
the circumference of the driven pulley, without refer-
ence to its size or diameter.
(d} Revolutions. Since the circumferential velocities
of any two pulleys, which are connected by one and
the same belt, are the same without regard to the
diameters of the pulleys, and since the circumferences
of the two pulleys are directly proportional to their
diameters (formula 5) ; if one of the pulleys has a
diameter equal to twice that of the other, the circum-
ference of the former will also be equal to twice that
of the latter, and the former will need just twice as
much time in which to perform one entire revolution
as the latter. In other words, the larger pulley will
make just one half as many revolutions in a given time
as the smaller. In a similar manner, if the diameter
of the larger pulley is three or four times that of the
smaller, the former will need three or four times as
much time for each revolution as will the latter, or the
larger pulley will make only one third or one fourth
the number of revolutions in a given time as the
smaller. In formula, denoting by ;/ and ri the num-
bers of revolution of the two pulleys, and by C and C ',
D and D', and R and R the respective circumferences,
diameters, and radii, we shall have
n C D' R'
RULE. The ratio of the numbers of revolutions of
two pulleys, which are connected by one and the same
belt, is equal to the inverse ratio of their circumfer-
ences, diameters, or radii.
'FUNDAMENTAL PRINCIPLES. 15
If we represent by n the number of revolutions per
minute, by v m the velocity in feet per minute, and by
R f and C f , respectively, the radius and circumference
of the pulley in feet, we shall have for the velocity the
expression
v m = 27tR f n =Cjn (7)
RULE. To determine the velocity, in feet per min-
ute, with which a pulley rotates, multiply the circum-
ference of the pulley, in feet, by the number of revolu-
tions per minute. ,
If R and C denote respectively the radius and cir-
cumference of the pulley in inches, we shall have, be-
D
tween R and R f , C and C f the relations R f = and
C f z= . These values, substituted in formula (7), give
27t -Rn Cn
=~- 0.5236^. ... (8)
RULE. To determine the velocity of a pulley in feet
per minute, multiply the circumference of the pulley in
inches by the number of revolutions per minute, and
divide the product by 12, or multiply 0.5236 times the
radius in inches by the number of revolutions per
minute.
Let v represent the velocity of the pulley in feet per
second; we shall then have the expression
1 6 BELTS AND PULLEYS.
and formula (8) becomes, by substitution,
27i Rn Cn
6ov= ~iT = ^>
which reduces to
RULE. To determine the velocity of a pulley in
feet per second, multiply the circumference of the pul-
ley in inches by the number of revolutions per minute,
and divide the product by 720, or multiply 0.00873
times the radius in inches by the number of revolutions
per minute.
If we substitute the value v m = 6ov in formula (7),
we shall obtain the expression
6ov = 27t R f n = C f n,
which reduces to
v = -JJT-- = o.iotfRjn. . . . (10)
oo
RULE. To determine the velocity of a pulley in
feet per second, multiply by the circumference of the
pulley in feet by the number of revolutions per
minute, and divide the product by 60; or multiply
0.1047 times the radius in feet by the number of re-
volutions per minute.
FUNDAMENTAL PRINCIPLES. IJ
By transposing formula (7), we may obtain, for the
number of revolutions per minute, the formula
i> m i> m
RULE. To determine the number of revolutions per
minute with which a pulley turns, divide the velocity
of the pulley in feet per minute by the pulley-circum-
ference in feet.
In a similar manner, by transposing formulas (8), (9),
and (10) we may obtain the following formulas for the
number of revolutions per minute :
n ^ = m ^ ,-, ..... (12)
C 0.5236^
RULE. To determine the number of revolutions per
minute, divide 12 times the velocity in feet per minute
by the circumference of the pulley in inches, or divide
the velocity in feet per minute by 0.5236 times the
radius of the pulley in inches.
,
RULE. To determine the number of revolutions per
minute, divide 720 times the velocity in feet per second
by the circumference of the pulley in inches, or divide
the velocity in feet per second by 0.00873 times the
radius of the pulley in inches.
1 8 BELTS AND PULLEYS.
RULE. To determine the number of revolutions per
minute, divide 60 times the velocity in feet per second
by the circumference of the pulley in feet, or divide
the velocity in feet per second by 0.1047 times the
radius of the pulley in feet.
The numbers of revolutions per minute of two or
more pulleys, which are fixed upon one and the same
shaft, must plainly be the same, for the shaft at each
revolution will carry each and all of the pulleys just
once around without reference to the diameters of the
pulleys. If, therefore, we denote by n the common
number of revolutions, and by v and v' the circum-
ferential velocities of two pulleys, which are fixed upon
one and the same shaft, we shall have, from formula
(9), the equations
Cn
v = = 0.008 7 3^,
720
and v' = = o.oo873^',
C, R, C, and R' denoting respectively the circumfer-
ences and radii of the two pulleys. From these two
equations we may write the proportion
v : v' :: : :: 0.00873^ : 0.0087 $R'n.
By cancelling out the equivalent quantities, and writ-
ing the proportion in the form of an equation, we have
v C R D
FUNDAMENTAL PRINCIPLES. IQ
RULE. The ratio of the velocities of two pulleys
which are fixed upon one and the same shaft is equal
to the direct ratio of the pulley circumferences, radii,
or diameters.
(e] Power. By the power of a pulley we mean the
force with which the circumference of the pulley turns:
it is equal to that force which, if applied to the pulley-
circumference in a direction opposite to that in which
the pulley rotates, would be just sufficient to stop the
c
FIG. 6.
motion of the pulley. The powers of two pulleys
which are connected by one and the same belt are
equal ; for the driving-pulley transmits all its circum-
ferential force to the belt, and the belt in turn trans
mits the same force to the driven pulley (less a very
slight amount which is consumed in the stretching of
the belt).
Let the circles of which the radii are R, R', r, and A
(Fig. 6) represent four pulleys, connected by belts as
shown in the figure, A being the driving-pulley and R
20
BELTS AND PULLEYS.
and r being fixed upon one and the same shaft. The
power Pot the driving-pulley is transmitted directly to
the pulley R through the belt xy. We may consider
the imaginary line abc as a simple lever, the fulcrum of
which is at the point a, and the arms of which are ac
and ab. If now we let P represent the power of the
pulley r, which is transmitted directly to the pulley R
through the belt x'y ', we shall have, from the principles
of the simple lever, the relation
PR = P'r,
P _ r
p ~ 7e
(16)
RULE. The ratio of the powers of two pulleys
which are fixed upon one and the same shaft is equal
FIG. 7.
to the inverse ratio of the pulley-radii (diameters or
circumferences).
Let the circles of Fig. 7 represent a number of
pulleys, connected by belts as shown in the figure, and
together constituting a " pulley-train." Let A be the
driving-pulley, and let r" be arranged to lift the weight
FUNDAMENTAL PRINCIPLES. 21
P" by means of a cord wound around its circumfer-
ence, as shown in the figure. From formula (16) we
shall have the expression
P r PR
P = K or p = T--
Also, we shall have
P r' PR'
-P> = W> r P = -r '
Substituting, in the last-found equation, the value of
P determined above, gives
p , f _
- rr > -
From formula (16) again we may write the equation
P" r" P'R"
= or p = -->
and by substituting in this the last-found value of P' ,
we shall finally obtain the formula
(I?)
rr'r"
Then, inversely, P = vv 1 &"
RULE. To determine the power of an increasing
pulley-train (one in which the powers of the pulleys
22 BELTS AND PULLEYS.
constantly increase from the driver), multiply the
power of the driver by the continued product of all
the larger pulley-radii (diameters or circumferences)
except that of the driver, and divide the result by
the continued product of all the smaller pulley-radii
(diameters or circumferences) except that of the
driver. To determine the power of a decreasing pulley-
train (one in which the powers of the pulleys con-
stantly decrease from the driver),* multiply the power
of the driver by the continued product of all the
smaller pulley-radii (diameters or circumferences) ex-
cept that of the driver, and divide the result by the
continued product of all the larger pulley-radii (diame-
ters or circumferences) except that of the driver.
From formula (15) we know that the circumferential
velocities of two pulleys which are fixed upon one and
the same shaft vary directly as the pulley radii,
diameters, or circumferences. We may therefore ob-
tain, by combining formulas (15) and (16) and denoting
the circumferential velocities of the pulleys R and r
(Fig. 6) by Fand v respectively,
P v
P=V 09)
RULE. The ratio of the powers of two pulleys
which are fixed upon one and the same shaft is equal
* If the pulley-train represented in Fig. 7 were a decreasing in-
stead of an increasing train, the " direction" of the train would be
reversed. That is, the pulley R" would be the driver and the pulley
A the one which lifts the weight
FUNDAMENTAL PRINCIPLES. 2 3
to the inverse ratio of the circumferential velocities of
the pulleys.
A glance at formula (19) will show that the increased
power which we obtain by means of an increasing pul-
ley-train necessitates a loss of time corresponding to
the gain in power. For since the power varies in-
versely as the velocity, if we increase the power two,
three, or four fold we necessarily decrease the velocity
two, three, or four fold also. Thus, if by means of the
train represented in Fig. 7 we can lift a weight of 1000
pounds with a circumferential force on the driving-
pulley amounting to say 200 pounds only, we will need
just ^y> = 5 times as much time as if we apply the
force of 1000 pounds directly to the pulley which lifts
the weight. Nevertheless there is a real gain repre-
sented in the increasing pulley-train ; because, without
it or a similar contrivance, we might tug, with our 200
pounds of power, for a lifetime, and still be unable to
lift the 1000 pound weight one inch from its resting-
place.
(/) Horse-power. The term " horse-power," as com-
monly used, is equivalent to 33,000 foot-pounds: it is
that amount of force or power which will lift a weight
of 33,000 pounds one foot high in one minute, or a
weight of one pound 33,000 feet high in one minute.
If we represent the horse-power of a pulley by H, and
the circumferential force or power in pounds byP, then
H X 33,000 pounds lifted one foot high per minute
will represent the power of the pulley. If therefore
we denote by v m the circumferential velocity of the
pulley in feet per mimite, we shall have, for the power
in pounds, the expression
24 BELTS AND PULLEYS,
33000#
(20)
Pv
And inversely, H = -- ....... (21)
RULE. To determine the power of a pulley in
pounds, divide 33000 times the horse-power by the cir-
cumferential velocity of the pulley in feet per minute :
to determine the horse-power, multiply the power of
the pulley in pounds by the circumferential velocity in
feet/^r minute and divide the product by 33000.
If v denote the circumferential velocity of the pul-
ley in feet per second, we shall have the relation
v m = 6ov, and formula (20) becomes, by substitution,
33000/f
6ov '
(22)
Pv
Inversely, H = - ........ (23)
RULE. To determine the power of a pulley in
pounds, divide 550 times the horse-power by the cir-
cumferential velocity in feet per second ; to determine
the horse-power, multiply the power of the pulley in
pounds by the circumferential velocity in feet per
second, and divide the product by 550.
The size of a pulley is usually given in terms of its
diameter: thus a "36-inch pulley" is a pulley the
FUNDAMENTAL PRINCIPLES. 2$
diameter of which is 36 inches; a "4-foot pulley" is
one the diameter of which is 4 feet.
Example I. The diameter of a pulley is 10 inches;
it is required to find the circumference. From formula
(i) we have C = TtD = 3.14159 X 10 or C= 31.4159".
Also we have R = = 5", and formula (2) gives
C = 27tR 6.28318 X 5 or C = 31.4159".
Example 2. The circumference of a pulley is C =
314.159"; it is required to find the diameter. We have,
from formula (3), D = - = - 3 ~ - = ioo".
* 3I4I59
Example 3. The diameters of two pulleys, which
are connected by one and the same belt, are D = 30"
and D' = 10" ; the larger pulley makes n = 120 revo-
lutions per minute. It is required to determine the
number of revolutions per minute of the smaller pul-
n D' 120 10
ley. rrom formula (6) we have = -~ or -,- = .
n D n 30
, 30 X 120
rrom this, n = = 360.
Example 4. A pulley, the radius of which is 2 feet,
makes 100 revolutions per minute; it is required to de-
termine the circumferential velocity in feet/^r minute.
We have, from formula (7), v m = 2rtR f n, or v m =
6.28318 X 2 X ioo = 1256.6.
Example 5. The radius of a pulley is 24 inches, and
the number of revolutions per minute ioo; it is re-
quired to determine the circumferential velocity of the
pulley in feet per minute. From formula (8) we have
27iRn 6.28318 X 24 X ioo
z> - or v- = = 1256.6.
12 12
26 BELTS AND PULLEYS.
Example 6. The radius of a pulley is 24 inches and
the number of revolutions per minute loo; it is re-
quired to determine the circumferential velocity of the
pulley in feet per second. Formula (9) becomes, by sub-
stituting the numerical data, v = 0.00873 X 24 X IOO,
or v = 20.95.
Example 7. The circumferential velocity of a pul-
ley is 1256.6 feet per minute, and the radius 2 feet ; it
is required to find the number of revolutions per
v
minute. From formula (u) we have // - =- =
2:TtJ\.f
1256.6
=
6.28318 X 2
Example 8. It is required to determine the number
of revolutions per minute of a pulley of which the
radius is 24", and the circumferential velocity, in
feet per second. 20.95. From formula (13) we have
v 20.95
~ 0.00873^ ~ 0.00873 X 24 "
Example 9. A shaft which makes 100 revolutions
per minute bears two pulleys of which the radii are
R = 36 inches and R' 24 inches; it is required to
determine the circumferential velocities of the two
pulleys in feet/^r second. From formula (9) we have,
for the circumferential velocity of the pulley R', v' =
0.00873 X 24 X IOO = 20.95 feet per second, and from
v 36 20.95 X 36
formula (15) we have = , or v --- =
20.95 24' 24
31.425 feet per second.
Example 10. In an increasing pulley-train we have
the following data: Power of the driving-pulley / ) = IOO
pounds, radii of the pulleys (of which there are six
FUNDAMENTAL PRINCIPLES. 27
besides the driver, and arranged as shown in Fig. 7),
R = R f = R" $6" and r = r' = r" 12" ; it is
required to determine the power of the pulley-train.
By substituting the above values in formula (17) we
Example u. Suppose the circumferential velocity
of the driving-pulley in Example 10 is 1200 feet per
minute ; it is required to determine the circumferential
velocity of the pulley r". From formula (19) we
P v m ' ff ioo v m m ,,,
have-^rz-pj-or = . From this, v m =
I2OO X IOO
-- = 44 44 feet per minute.
2700
Example 12. Required to determine the power of a
pulley which transmits 60 horse-power at a circumfer-
ential velocity of 10 test per second. From formula (22)
550 X 60
we have P = - or P = -- = 3300 pounds.
Example 13. The circumferential force or power of
a pulley is 3300 pounds, and the velocity 10 feet per
second ; it is required to determine the horse-power
transmitted by the pulley. Formula (23) gives // =
Pv 3300 X 10
550 550
= 60.
28 BELTS AND PULLEYS.
3. Rules for the Proper Disposition of Pulleys.*
The axes of two pulleys which are connected by one
and the same belt may bear to each other the follow-
ing relations :
1. They may coincide geometrically.
2. They may be parallel.
3. They may intersect each other.
4. They may cross, without being in the same plane.
In these different cases the belt passes from the
driving to the driven pulley, either directly or by
means of intermediate pulleys or pulley-guides. It is,
first of all, indispensable that the pulleys be placed in
such a manner that the belt shall maintain its proper
position upon both pulleys without running off or
compelling recourse to special guides. The geometric
disposition of the pulleys by which this condition may
be fulfilled is called the "arrangement" of the belt.
The preceding condition will be satisfied if the pul-
leys are so placed with reference to each other that, for
each of them, the median line of that portion of the belt
ivJiich runs toward the pulley is in the middle plane of
the pulley.
In pulleys which have rounded fellies (see I3)slight
variations from this rule (from |- to f) may be ad-
missible.
* 3. 4, and 5 from Reuleaux.
T&ANSMlSSfONS BY BELTS WITHOUT GUIDES.
4. Transmissions by Belts without Guides.
The simplest and most common arrangements of
pulleys are those in which the belt passes directly from
one pulley to the other without guides of any kind ;
the simplest of these dispositions, which corresponds
to the case in which the axes cf the pulleys are parallel,
is represented in Fig. 8. In the left-hand figure the
belt is open, and the pulleys rotate in the same direc-
FIG. 8.
tion : in the figure on the right the belt is crossed and
the pulleys rotate in opposite directions. In these two
arrangements the belt may run in either direction, the
condition which prevents its running off the pulleys
being fulfilled for either direction of rotation.
For pulleys the axes of which coincide geometrically,
as for those in which the axes intersect, it is evidently
impossible to establish transmission without guides.
BELTS AND PULLEYS.
For the case, however, in which the axes cross without
being in the same plane, belts without guides may be
used with the arrangement of pulleys represented in
Fig. 9, which is very frequently seen in practice.
This disposition allows us to dispense with all ex-
terior guides, if we are careful to place the pulleys in
such a manner that the line of intersection of their
middle planes shall be tangent to the circles contained in
TRANSMISSIONS BY BELIES WITHOUT GUIDES. 31
these planes at the points in which the belt leaves the
pulleys. In Fig. 9, in which a and b l are these points,
the belt must run in the direction indicated by the
arrows. If we wish to run the belt in a contrary direc-
tion it is necessary to move the pulleys upon their
arbors until the line of intersection of their middle-
planes becomes tangent to the circles at the points a t
and b. This condition is fulfilled when, with reference
to the crossing K of the pulley-axes, the new positions
occupied by the pulleys are found to be symmetrical
with the positions of the pulleys before the change.
The transmission represented in Fig. 9 may be con-
sidered as the general solution of transmissions by belts
without guides. It gives, in fact, the transmission by
open belt, when the angle ft included between the
middle planes of the pulleys is equal to O, and the
transmission by crossed belt when this angle is equal
to 180. In all intermediate positions the belt is only
partially crossed : for ft = 90, we have a half-crossed
belt, for ft = 45 a crossing of one fourth, etc.
In short, partially crossed belts, the tendency to run
off the pulleys is very great. According to Redten-
bacher, in order that this accident may be avoided,
the distance between the centres of the pulleys should
not be less than twice the diameter of the largest
pulley ; that is, the angle of deviation of the belt should
not exceed 25. Moreover, in order that the wear of
the belt may not be excessive, the distance between
the centres of the pulleys should not be less than
IO VbD, b representing the width of the belt and D
the diameter of the driving-pulley. It is evident that,
in each particular case, it is advantageous to take, for
3 2 BELTS AND PULLEYS.
the separation of the pulleys, the greater of these two
values.
5. Transmissions by Belts with Pulley-Guides.
RULE. In a transmission by belt with pulley-guides,
in order that the belt may run properly upon the
pulleys and pulley-guides, the point in which the belt
leaves each pulley must be the point of tangency be-
FIG. 10. FIG. n.
tween the pulley and the line of intersection of its
middle plane with that of the following pulley.
Figs. 10 and II represent transmissions of this kind
for pulleys with parallel axes. In Fig. 10 the middle
planes of the two pulley-guides are tangent to the two
pulleys of transmission A and B, and their common
diameter is equal to the distance between the middle
TRANSMISSIONS BY BELTS WITH GUIDES. 33
planes of these pulleys. This disposition of pulleys
permits of the movement of the belt in either direction.
When, as is most commonly the case, a movement of
the belt in one direction is sufficient, we may make use
of the simpler disposition of pulleys represented in
Fig. II, in which the axes of the pulley-guides coincide
geometrically. A and B are the pulleys of transmis-
sion ; the middle planes of the pulley-guides are par-
allel, and are tangent respectively to the pulleys A and
B at the points in which the belt leaves the latter
pulleys. The common diameter of the pulley-guides
is equal to the distance between the middle planes of
the pulleys of transmission. As indicated in the figure,
the pulleys of transmission A and B rotate in opposite
directions.
3
34
BELTS AND PULLEYS.
If we consider B as a pulley-guide (in which case it
may run loose upon the arbor of A), the two pulleys
C and D may be taken as pulleys of transmission, and
fixed upon two separate arbors, the directions of which
are the same.
If the pulley-guides C and D are placed between the
arbors of A and B, as is indicated in Fig. 12, they will
rotate in the same direction, and may consequently be
FIG. 13.
fixed upon one and the same arbor. The pulleys of
transmission A and B will also rotate in the same direc-
tion. In this case the belt can move in one direction
only, and remain properly upon the pulleys and guides.
The two pulley-guides C and D may be replaced by a
single pulley, provided it is placed obliquely so as to
run on both sides of the belt without causing displace-
ment.
TANSMISSIOA 7 S BY BELTS WITH GUIDES- 35
Fig. 13 represents a transmission by belt for two
pulleys, the axes of which intersect each other. In
this disposition, which differs from that of Fig. n only
in the inclination of the axis of the pulley B, the
movement of the belt can take place only in one direc-
tion. To obtain a movement in the other direction, it
is necessary to move the pulley-guides along their
common axis until the condition necessary for main-
taining the belt in position is fulfilled for this particu-
lar case. It must be remembered that the two pul-
ley-guides rotate in contrary directions, and therefore
cannot be fixed to the arbor upon which they run.
From the arrangement shown in Fig. 12, that of
Fig. 14 maybe devised; this disposition corresponds
to the case in which there is a very slight angle between
the arbors, and the pulley-guides rotate in the same
direction.
) BELTS AND PULLEYS.
The disposition represented in Fig. 15 is still more
simple, and may be used for a
greater angle between the axes
as great as 25.
FIG. 1 6. Half- crossed belt with
pulley-guide. In this case the re-
lative positions of the pulleys of
transmission are such that the dis-
position represented in Fig. 9
could be used, except that the
separation of the pulleys is too
slight, and the belt would there-
fore tend to run off. To deter-
mine the arrangement of the belt,
we begin by giving to the part 55
the direction of the line of intersection of the middle
planes of the pulleys A and B\ then from the point c,
chosen arbitrarily upon the line 55, we draw, to the
circumferences of the pulleys, the tangent lines ca and
FIG.
cb. The plane of these tangents determines the middle
plane of the pulley-guide O; to which the lines are also
tangents. Rotation may take place equally well in
either direction. Because of the cramped position of
TRANSMISSIONS BY BELTS WITH GUIDES. 3/
the pulleys and the consequent difficulty in placing
the arbor of the pulley-guide in proper position, this
arrangement is very rarely seen in practice.
FIG. 17. Another disposition for transmission by half-
crossed belt with pulley-guide. In this figure the pulleys
of transmission are so placed that the line of intersec-
tion SS of their middle planes is the common tangent
to the circles contained in the planes, and the middle
plane of the pulley-guide C coincides with that of the
FIG. 17.
pulley of transmission A. The portion of belt which
leaves the pulley A is inclined (as shown in the figure)
as in the crossed belt in order that it may properly roll
upon the pulley B, while the portion which leaves the
pulley B is guided by the pulley-guide C. The pulley-
guide is in contact with the line of intersection SS, and
with a tangent to the circle A drawn from an arbitrary
point upon the line SS. In this disposition the direc-
tion of rotation must be as indicated in the figure.
463^59
35 BELTS AND PULLEYS.
This mode of transmission is very convenient when we
wish to drive a series of vertical arbors from one hori-
zontal shaft ; it also finds frequent employment in mills
for grinding various materials, and when the separa-
tion of the pulleys of transmission is necessarily slight.
FlG. 1 8. Half-crossed belt with mozable pulley-guide.
In this disposition, which is used for a greater separa-
tion of the pulleys of transmission than in that of Fig.
17, we may, by moving the pulley-guide from the posi-
tion C to the position C (shown by the dotted lines),
cause the belt to pass from the fixed pulley B to the
idle pulley B : in a similar manner, the pulley-guide
may be used for running the belt off the pulleys en-
tirely. The position C a should be so chosen that the
tensions upon the belt for the two positions will be
the same or slightly less for C than for C.
General case of crossed arbors. When the pulleys of
transmission cannot be so placed that the line of inter-
TRANSMISSIONS BY BELTS WITH GUIDES. 39
section of their middle planes is a common tangent to
the circles contained in the planes, it becomes necessary
to make use of two pulley-guides. Fig. 19 represents
an arrangement which may be adopted in such cases,
and which may be regarded as the general solution of
the problem of transmission by belts with pulley-
guides. Fig. 20 represents a special application for
the case in which the line of intersection SS of the
middle planes passes through the centre of the middle
circle of one of the pulleys of transmission ; in this
figure the axis of the pulley B is supposed to be situ-
ated in a plane parallel to the pulley A. After having
obtained the line of intersection 55, we choose upon
4 o
BELTS AND PULLEYS.
it two arbitrary points c and , through which we draw,
to the middle circles of the pulleys of transmission, the
tangent lines ca, cb, c^^ and cjb^ The planes cab and
c 1 a 1 d l which are thus determined are those of the two
pulley-guides, which should be placed respectively in
contact with the above-named tangent lines. With
T? ..
FIG..2I.
this disposition, rotation may take place equally well
in either direction.
The mode of transmission represented in Fig. 19
may be simplified by giving to the axes of the two
pulley-guides a common direction mm parallel to the
two pulleys of transmission (Fig. 21). In this figure 55
represents the intersection of the middle planes of the
TRANSMISSIONS BY BELTS WITH GUIDES. 4!
two pulleys of transmission, ac and b l c l the intersec-
tions of planes perpendicular to S5 with the middle
planes of the pulleys of transmission A and B respec-
tively. In the perpendicular planes, tangentially to the
right lines ac and bj r we place the two pulley-guides
C and C r The arrows indicate the directions of rota-
tion ; to obtain a movement of the belt in a direction
contrary to the one indicated, it is necessary to give to
FIG. 22.
the pulley-guides C and C l the positions indicated at
C and C\ by the dotted lines.
It may be remarked here that the belt, instead of
passing from c to a and from c l to a r may be made to
pass from c to a^ and from c l to a, which causes a
change in the direction of rotation. The pulley guides,
instead of being horizontal, as in the figure, may be
placed vertically that is, respectively in the planes of
the pulleys of transmission A and B ; in this case, how-
4 2
BELTS AND PULLEYS.
ever, it becomes necessary to take account of the angle
of deviation (see 4).
When the pulleys of transmission can be so placed
that the intersection 55 of their middle planes is tan-
gent to one of the pulleys, and the distance between
the parallel planes containing the axes of the pulleys
A and B is sufficient, we may substitute, for the dis-
position shown in Fig. 20, the one represented in Fig.
FIG. 23.
FIG. 24.
22. This arrangement is often seen in practice ; the
axes of the pulley-guides are parallel to that of the
pulley of transmission A. The middle planes of the
pulleys A and B may make any desired angle with
each other.
If the distance AC is great compared with the width
of the belt, the pulley-guides, instead of being the one
above the other, may be placed upon the same axis, as
shown in Fig. 23. If the distance between B and C is
sufficiently great, the arbor B may be provided with
two pulleys, one fixed and the other idle.
TRANSMISSION'S BY BELTS WITH GUIDES. 43
When, on account of lack of space, it is impossible to
make use of one of the dispositions which we have de-
scribed above, we ought to seek at least to place the
axes of the pulley-guides in the middle plane of one of
the principal pulleys and the pulley-guides themselves
parallel to each other, as, for example, in Fig. 24. In
this case we first draw the tangent line ab; then in a
plane drawn through this line normally to the plane of
FIG. 25.
the figure we place the pulley-guide C in such a man-
ner that it is tangent at the point a to the line of in-
tersection of the middle planes of the pulleys A and C.
Through the point a l we then draw the line a^ paral-
lel to ab, and in a plane drawn through this line parallel
to the plane of the pulley-guide C, we place the second
pulley-guide tangent to the intersection of the middle
planes of the pulleys A and C l and to the middle plane
of the pulley .Z?. In this manner the axes mm and m l m l
44 BELTS AND PULLEYS.
of the pulley-guides are found parallel to each other,
and also situated in a plane parallel to that of the
pulley B.
By making the belt of Fig. 23 pass over a fourth
pulley we may obtain an arrangement by which we
may drive two pulleys B and C by means of a single
driving-pulley A.
Fig. 25 represents a disposition of this kind much
used in spinning-mills. The arbors B and 7 arc in dif-
ferent stories of the building, and each bears two pul-
leys, one fixed and the other loose; we use, in this
case, the permissible deviation of the belt from its
exact position mentioned in 3.
Fig. 26 represents another mode of transmission by
belt, in which the two parallel arbors B and C are
driven by a single pulley A. The axes of these arbors
are both perpendicular to that of the arbor A ; the first
intersects it, while the second crosses it without inter-
secting. In the machinery of spinning-mills a great
number of transmissions are found in which three, four,
or even a greater number of pulleys are driven by
means of a single driver. It may be remarked here,
that in all cases of transmission by leather belt in
which pulley-guides are used which are in contact with
LENGTH OF BELTS.
45
the upper surface of the belt, it is advantageous to
place the belt so that the contact of the pulleys is
always upon the same surface the flesh or wrinkled
side.
6. Length of Belts.
It is often necessary in practice to calculate the
proper length of a belt for a given separation of the
axes of the pulleys upon which the belt is to run and
for known pulley radii or diameters. Thus when we
have two pulleys, the bearings and positions of which
are already fixed, if we can determine the proper length
for the belt, we can save time and prevent waste of
belt in cutting too long or too short.
Open Belt. Let us denote by L the total length of
the required belt; by Z, the distance between the
centres of the two pulleys upon which the belt is to
run ; by R the radius of the larger pulley, and by r that
of the smaller. Let Fig. 27 represent the pulleys con-
46 BELTS AND PULLEYS.
nected by an open belt. In the figure the lines ob and
o'c are parallel, because each is perpendicular to the
line bc\ hence the angles xob and yo'c are equal. Let
us denote each of these angles by tp. It is evident
from the figure that the total length of the belt must
be
L = 2(bc -(- arc ab -f- arc cd).
Draw the line ck parallel to oo : we shall have ck = L v
because ob and o'c are parallel. In the triangle bkc, in
which the angle kbc is a right angle, we shall have
be = ^ cli bk or be = * L* bk*
But ok = o'c - r and bk = ob ok = R r; hence
bc =
The arc ab is equal to the arc ax -\- the arc xb\ arc
271 R 7lR(p
ax = = I.57& and arc xb =
Therefore
arc ab = l.tfR -f 0.017 $Rtp = (1.57 -f o.
Also the arc cd is equal to the arc dy the arc yc ;
2nr
arc dy =-. - = 1.57^ and arc ^ = -^ = 0.0175^.
Hence we shall have
arc cd i-5/r 0.0175^ = (1.57 o.oi75<p)r.
LENGTH OF BELTS. 47
Add together these values of be, arc ab, and arc cd, and
we shall have for the total length of the belt
-] (24)
In the right-angled triangle kbc we have, from trigo-
nometry, sin angle bck = 7- = j . But since the
sides of the triangles kbc and xbo are respectively per-
pendicular to each other, the triangles are similar, and
the angle bck = (p. Hence we shall have
sin (p = J^-* ( 2 5)
Crossed Belt. In crossed belts the lengths differ con-
siderably from those of open belts under the same cir-
* In open belts the angle q> is generally quite small, and we may
without serious error take the sine of th" angle equal to the angle it-
self expressed in circular measure. Thus we shall have, from for-
R-r R r
mula (25), (p c = sin <p = j 0.0175^, or cp = Q , tp e repre-
senting the angle in circular measure. This value substituted in for-
mula (24) gives for the total length of the belt
Representing R -\- r by 2 and R r by A, the above expression be-
comes
This formula is simpler than but not so accurate as formula (24).
4 8
BELTS AND PULLEYS.
cumstances of separation of pulleys and pulley-radii.
Let Fig. 28 represent two pulleys connected by a
crossed belt. As in open belts, we shall have
L = 2(bc -f- arc ab -\- arc cd).
Draw the line ck parallel to oo' and produce bo as far
as its intersection with ck. As before, we shall have
ok = o'c = r. Hence bk~ R-\-r and ck = L,.
FIG.
In the right-angled triangle kbc we shall have
be V~^k* - bk 1 = VL? (R + r)\
The triangles obx and kbc are similar, and the angle
bck = <p. We shall therefore have
sin angle bck = T-,
or
sin <p =
(26)
LENGTH OF BELTS. 49
From the figure we have arc ab = arc ax -j- arc xb =
(- 0.0175 A* <p. Also arc cd = arc yd -j- arc j/ =
27T7-
- + 0.01757-9.
Hence, by adding together these values of be, arc ab,
and arc cd, we shall obtain for the total length
Example I. Suppose we have two pulleys having
radii of R = 20" and r = 10", and a distance between
the axes of Z, = 10' = 120". Required the length for
an open belt which will properly connect the two pul-
leys. From formula (25) we shall have
20 10 47
Sm * = ^o~~ = ' 0833 ' r * = 4 to~*
Formula (24) therefore becomes, on substituting the
abo-ve data,
r~ / '
L = 2\ V~Mf> - (20 - Mp-H I.S7 + o OI 75 X 4 ^
+ M-57 0.0175 X4^j
or
L = 2(119.58 + 33.07 + 14.86) = 335.02" = 27' ii".
If we wish to use formula (2 5 A) instead of formula
(24), we proceed as follows : 2 = 20 -f- 10 = 30, A =;
4
5O BELTS AND PULLEYS.
20 10 = 10. Hence, from the formula, we shall have
L = 2y - o + 1.57 X 30 +
= 2(1 19.58 + 47.10 + 0.833) = 335-03".
Thus the difference in the results from formulas (24)
and (25A) is in this case practically o.
Example 2. Taking the data of Example I, it is
required to calculate the proper length for a crossed
belt which runs on the above pulleys. From formula
(26) we shall have
20 + 10 20
sm 9 = ~i2o~~ = - 25 ' or <p = l4 iz = I4 ' 5
Formula (27) therefore gives for the proper length of
the belt
L= 2[yi2o (20+ io) 2 + 1.57(20+ io) + 0.0175
X 145 ( 20 + I0 )l = 2(116.19 + 47.10 + 7.61),
or L 341.80" = 28^ feet.
In transmissions by belts with pulley-guides, and in
all cases where the intricate arrangement of the belt
renders arithmetical calculation long and tedious, the
proper length of the belt may be determined more
easily and with sufficient exactness graphically, by
measurement with the rule. To illustrate: Suppose we
have an arrangement of pulleys such as is represented
in Fig. 12, which figure is a sketch (containing two
projections) of the transmission, drawn to a scale of -fa.
SPEED-CO.VES. 51
The separation of the pulleys A and B is 5 feet = 60",
and the diameters respectively 21 "and 13". From the
figure it is evident that the total length of the belt is
L = arc xy + yD -f Dx' + arc x'y' + y'C + Cx. By
measuring with the compasses the above arcs, we find
xy = 34^" and x'y' = 20". The line yD in the left-
hand projection is given in its true length by the line
ND in the right-hand figure ; hence, by measuring
ND, we obtain for the true length yD = 30". The
distance Dx' is given in its true length in the left-hand
figure, and therefore, by direct measurement, we ob-
tain Dx' = 34". In a similar manner we obtain by
measuring KC the true length y' C = 36'', and by di-
rect measurement, Cx = 26". We have consequently
L = 34i + 30 + 34 + 20 + 36 + 26 = 180*" = is' i".
In a similar manner in Fig. 18, by measuring the arcs
xy and x'y' , and the length, NX, Ky', and yz, we may
obtain the length necessary for a belt which will
properly run on the pulleys represented in the figure.
7. Speed-cones.
The contrivance known to mechanics as "speed-
cones" consists of two stepped pulleys arranged as
shown in Fig. 29. The object of speed-cones is to ob-
tain different speeds for the driven arbor from the con-
stant speed of the driving-shaft. To illustrate : Suppose
in Fig. 29 we assume between the radii of the pulleys
the relations R = $r, R' = r' , and R" = $r". We have
seen from formula (6) that the ratio of the revolutions
of two pulleys which are connected by one and the
same belt is equal to the inverse ratio of the pulley-
52 BELTS AND PULLEYS.
radii. Hence, if we assume that the driving-shaft xy
makes IOO revolutions per minute (N= 100) when the
belt is on the pulleys R and r, we shall have for the
revolutions of r (and consequently of the shaft x'y'}
r>
n N = IOO X 3 = 300. When the belt is on the
pulleys R' and r' we shall have for the revolutions of
r>/
r' (and consequently of the shaft x'y'} n' = N -, = 100
X I = IOO. Similarly, when the belt is on the pulleys
x-
-y'
FIG 29.
R" and r" we have for the revolutions of R" and the
shaft x'y' N" = N -^77 = IOO X | 33i- Such differ-
ent speeds for the driven arbor are necessary in many
machine-tools, as the lathe, drill, etc., because the
speed of the mandrel and spindle must vary with the
SPEED-CONES, ^C^ii^-^ 53
material to be worked and with character of the work
to be done.
Open Belt. From formula (25A) we have for the
length of the belt
L =
in which 2 = R -f- r and A = R r (see Fig. 29).
Since now the length of the belt must be the same for
each pair of pulleys in the cone, we shall have
2\ j-*\ i 1.^1 j-> | j-
jLj
or
VL; - j- + 1.572 + - ^A 2 - ^ 2 + 1-57^'+
in which 2' = R' + r' and A' = R' - r'.
By means of the binomial formula we may extract
the square roots of the quantities under the radical
signs as follows :
and
A n A'* A' 6
But since L, is usually very large compared with A,
A* A*
^y-j and 6 are very small quantities, and may with-
54 BELTS AND PULLEYS.
out serious error be neglected. Similarly, we may
j/4 j/e
neglect the quantities ^y- 3 and >. ,. 6 . Hence we shall
have
which reduces to
2' = 2 + (28)
3-I4A
If we represent by N the constant number of revolu-
tions per minute of the driving-shaft (corresponding to
R), and by n the number of revolutions per minute of
the driven shaft when the belt is on the pulley r, we
shall have, from formula (6),
R n n
R n' n' . n'
Also r -TF7 = T7 or R = T ~T7,
r N N N'
in which ri represents the revolutions per minute of
the driven shaft when the belt is on the pulleys R and
r', and N' the revolutions per minute of the driving-
shaft, which being constant is equal to N. Hence we
shall have
SPEED-CONES. 55
which substituted in formula (28) gives
=. . (29)
3- MA
We shall also have (as above for the quantity 2')
Example i. Suppose we have two shafts, the dis-
tance between which is L l = 100": the revolutions
per minute of the driving-shaft is N= 100, and we
wish to construct a pair of speed-cones such that the
revolutions per minute of the driven shaft correspond-
ing to the pulleys r, r f , r", and R" shall be n = 300,
ri 200, n" = 100, and N'" = 50. From formula (6)
we shall have
R n 300
7- = N = 1^ = 3 ' r R =
We may choose any convenient value for r, and find
from the above expression the corresponding value of
R. Suppose we take r = 4" ; hence R = y = 3 X 4
= I2 /7 . Then 2 = 12 -f- 4 = 16 and J = 124
= 8. From formula (30) we shall have
f r'( 2 I } = r'
\IOO I
56 BELTS AND PULLEYS.
and formula (29) becomes
,/2OO \ 64 r' 2 64 r' 2 .
r'J -- h i = i6H --- , or ir = 16 -4- -
\ioo ~ 3-HX i oo' 314
From this by reducing we shall have
r" + 942^' = 5024 + 64.
Adding [~~ J = 471* to each side of this equation gives
r' 8 + 942r' -f- 221841 = 5024 + 64+ 221841 = 226929.
Extracting the square root of this expression gives
r' -\- 471 = ^226929.
From this r' = ^226929 471,
or r' = 476.38 471 = 5.38".
TU K n> 20 D/ /
Then r = -== = - , or R = 2r = 10.76 .
r' N loo
In the same manner for the pulleys R" and r" we
shall have from formula (30)
An /
A" = r'\ -^ i I = r \ --- I ] = o ;
\N I \ioo /
and formula (29) becomes
,,f lo , \ ,64
^Viw+V* =16 + =16.204,
or r" 8.iO2 x/ .
Also 7-, = -- , or R" = r" 8.IO2'',
r 100
SPEED-CONES.
For the pulleys r" and K" we shall have
57
Hence formula (29) gives
y" = 16 + ^{ > 942r" = 5024 + 64 - r"'\
Hence , r"" + ypr"' = 5088.
As before, adding - to each side, and extracting
roots, we shall have
r'" = ^5088 + 221841 - 47 1 = S-SS".
Then
R" N 100
' = 2r'" = 10.76".
7 = ~ = r = 2, or
Crossed Belt. The calculation of the radii of the
speed-cone pulleys becomes very much simpler when
58 BELTS AND PULLEYS.
crossed belts are used. If, in Fig. 30, we assume the
relations 2 = R -(- r = R' -f- r' = R" + r", etc., we
shall have for the corresponding angles, (p, <p', <p", etc.;
R + r 2 R'+r' 2
sin (p = j = y-, sm <p = j = y~, sin cp =
L/i i -^i i
Dtl I // -^
j = j , and consequently (p = (p' = q>", etc.
A A
The conditions that the length of the belt must be the
same for each pair of pulleys, and that the belt must
bear the same tension for each pair of pulleys, will
therefore be fulfilled if we take the sums of the radii
of each pair of pulleys equal to each other. Or, which
is the same thing, we shall have
R' = Z-r' (31)
Letting R' -\- r' = 2', we shall have from above
From formula (6) we may write
R n
R' n' n' ,ri
and 7= W = -N r R=r N'
Hence 2 = R + r = r~ + i),
SPEED-CONES. 59
Example 2. Taking the data of Example i, it is
required to calculate the radii of the speed-cone pul-
leys for crossed belt. We obtain, as in Example i,
R = 12", r = 4", and 2 = 16". From formula (32)
we shall then have
r > -. .(300 + ioo\ ._ 4
- 4 V^o~T~W ~ 4 3 "
Formula (31) then gives
R = 16 5.33 = 10.67".
For the third pair of pulleys formula (32) gives
and from formula (31) we shall have
R" = 2 - r" = 16 - 8 = 8".
For the fourth pair of pulleys from formula (32) we
shall have
4
= 4 x = IO - 6 7 "
Formula (31) then gives
r'" =2- R" = 16 - 10.67 = 5.33".
6O BELTS AND PULLEYS.
Suppose now that we wish to add to the speed-cones
another pair of .pulleys (R iv and r lv ] having such radii
that the number of revolutions per minute of the
driven shaft, when they are in use, shall be JV iv = 33^.
We shall have from formula (32)
I n -\- N \ /3<x> 4- ioo\
R = '^ = 4 " =4x3= 12",
and from formula (31)
r iv = 2 - R" = 16 - 12 = 4".
We have now two speed-cones, which are made up
of pulleys as follows :
First Cone. Second Cone.
R = 12" r = 4"
R' = 10.67" r' = 5.33"
R" = 8" r" = 8"
r'" = 5.33" R'" = 10.67"
r" = 4" R iv = 12"
A glance at this table will show that the two cones
are similar and equal, but so placed on their shafts
that they taper in opposite directions. We may there-
fore write the following:
Rule for Speed cones. Crossed Belt. Use two equal
and similar stepped cones tapering in opposite direc-
tions.
Mr. C. A. Smith, in the American Machinist, Feb-
ruary 25, 1882, gives a very neat graphical method for
determining the radii of speed-cone pulleys for open
belt, as follows: Lay off (Fig. 31) AB equal to the
SPEED-CONES.
61
given distance between the two shafts (AB = Z.,),
drawn to any convenient scale. Strike the circles repre-
senting the pulleys R and r (the radii of which are deter-
mined, as in Examples I and 2 of this section, from the
given revolution-ratio -TV], and draw the portion of belt
ab. Lay off (from the smaller pulley centre) BC =
AB X 0.496 = O-496Z-,, and erect the perpendicular
CD = ,. Then from D as a centre strike the cir-
3.1416
cle x tangent to ab. Divide AB = Z, into as many
FIG. 31.
equal parts as the shaft B is to revolve, less one, while
the shaft A makes one revolution, when the belt is on
the required pulleys R' and r'. Lay off, from the cen-
tre of the smaller pulley, BO equal to one of these
parts (BO = L l -r- ^ i), and from o draw the line oa!
tangent to the circle x. The circles drawn from B and
A as centres and tangent to oa' give the required
radii r' and R. When we wish to have the revolutions
62
BELTS AND PULLEYS.
of the driven shaft B less than those of the driving-
shaft A, or when the smaller pulley is to be on the
shaft A, we lay off (for r'" and R'"] the distance Ao' =
N
L l -~ T~77 I, draw o'b' tangent to the circle x, and
the circles r'" and R" give the required radii.
Crossed belts are not so often used for speed-cones
as open belts, and the speed-cones for the former are
so easily calculated from formula (32), that it is un-
necessary to give graphical
methods for determining
the radii.
Continuous Speed-cones.
Sometimes (especially in
cotton machinery and in
machines requiring gradu-
ally increasing or decreas-
ing speeds for the driven
arbors) continuous speed-
cones are used instead of
the stepped speed-cones
already described. It may,
however, be remarked that
in ordinary shop machin-
ery, such as lathes, planers,
drills, etc., etc., continuous
speed cones are very rarely
FIG. 33. seen.
To construct a pair of continuous speed-cones for
open belt we may proceed as follows : Having given
several of the different numbers of revolutions re-
quired of the driven shaft (for example, n = 300, n'=,
SPEED-CONES.
n" = 100, N'" = , N' 1V = 50, and the revolutions of
the driving shaft being N= 100), lay off (Fig. 32)
ab =. a'b' = the width of the belt -(- the proper clear-
ance X the number of changes in the speed of the
driven shaft : in this case there are five changes.
Then calculate, from formulas (29) and (30), the radii
R, r, R", r" , r lv , and R' v , corresponding to the known
z'
_a_f!
FIG. 33. FIG. 34.
numbers of revolutions, and draw the pulleys of which
R, r, etc., are the radii, and which are represented by
the dotted rectangles in the figure. Through the cen-
tres of the step-widths (x, y, z, x' , etc.) draw the Curves
xyz, x'y'z', and the outlines of the cones are complete.
Rankine gives for continuous speed-cones for open
belt the rule, " Use two equal and similar conoids taper-
6 4
BELTS AND PULLEYS.
ing in opposite ways and bulging in the middle, accord-
ing to the formula r =
r _i_ r ( r _ r \ 2
- 4 ~~"
r a is the radius in the middle, r, and r^ the radii of the
larger and smaller ends respectively, and c the distance
between the centres of the shafts. Fig. 33 represents
a pair of continuous speed-cones, open belt, calculated
from this rule, taking r, = 10", r^ = 4", c = loo",
r = I0+4
2
= 7.057", and ab = a'b' 14".
628
To construct a pair of continuous speed-cones for
FIG. 35.
crossed belt, calculate from formula (32) the radii R, r,
R", r", r iv , R {v (Fig. 34), and connect the centres of
the step-widths by the curves xyz, x'y'z', in the same
manner as in Fig. 32. Or we may use two equal and
similar cones tapering in opposite directions (Fig. 35).
An example will best explain the mode of calcula-
tion for a pair of continuous speed-cones by which we
wish to obtain a given gradual change in the speed
of the driven arbor. Suppose our driver makes 100
revolutions per minute, and that we wish, by slowly
MATERIALS USED FOR BELTING. 65
sliding the belt along the cones, to obtain for the
driven arbor a speed varying from 100 to 10 revolu-
tions per minute. According to the rapidity with
which we wish the changes to take place we choose
the number of changes let us say in this instance
10. Of these changes, the number of revolutions per
minute of the first is 100. With the 9 remaining
changes we must therefore gain 100 10 = 90 revolu-
tions per minute, or 10 each. The revolutions of the
changes are therefore as follows: 1st, 100 ; 2d, 90;
3d, 80; 4th, 70; 5th, 60; 6th, 50; 7th, 40; 8th, 30; 9th,
20 ; loth, 10. We may now calculate the diameters as
for stepped cones, and by drawing curves through their
face-centres obtain the outlines for the required con-
tinuous cones.*
8. Materials used for Belting.
Belts are most commonly made of leather, cut into
strips of the required width, and riveted together at
their ends to make up the required length. Strips
taken from the back part of the hide, and oak or hem-
lock tanned, are generally considered the best, although
some kinds of patent-tanned leather are said to have
greater adhesive power. Cow's hide is almost invari-
ably used for the leather of belts ; the skins of horses,
elephants, and other animals have, however, been util-
* In designing continuous speed-cones it is always best to make the
curves as gradual in taper as possible for the given changes, in order
to avoid the excessive stretching and wear of the belt which would
otherwise occur.
66 BELTS AXD PULLEYS.
ized for this purpose, in some cases with very good
results. For very heavy work, belts made of two or
more thicknesses of leather are used, in which case the
strips are fastened together with cement or rivets, and
the joints carefully " broken." In order to gain strength
and prevent stretching, leather belts are sometimes
edged on the upper side with narrow strips of leather,
which are riveted, laced, or cemented fast to the belts.
It has also been proposed (and to our knowledge in
one case at least tried) to strengthen belts by riveting
along their edges thin strips of brass, steel, or other
metals.
Of late years vulcanized- rubber belts have been very
successfully introduced in this country. They are
usually made continuous, thus avoiding the use of
rivets, and consist of one or more layers of cotton-duck
placed between layers of vulcanized rubber, the rubber
covering the edges in order to protect the seams from
injury. Rubber belts are now made in widths about
the same as leather ; they weigh nearly the same, and
are said to be equally strong and pliable.
The intestines of sheep, cats, and other animals have
been used to a considerable extent for belts ; they are
exceedingly strong and tough, and can be obtained, it
is said, thirty or forty feet in length. Gut belts are
either round, to run in grooved pulleys, or woven into
flat bands for use on ordinary flat-faced pulleys. Raw-
hide possesses, it is claimed, fifty per cent more
strength than tanned leather ; but belts of this material,
unless constantly oiled, soon become stiff and ungov-
ernable, and are not to be depended upon for general
purposes of transmission. Belts of hemp, flax, canvas,
MATERIALS USED FOR BELTING. 6?
sheet-iron and steel, and several combinations of leather
and metallic wire, have been proposed, and in some
cases used ; but these at present offer no practical
advantages over leather and vulcanized rubber.
For all practical purposes, then, we have two kinds
of belting leather and rubber, between which we may
offer the following comparison : Those who favor
leather belts claim that they are in the main stronger
than rubber, and that they will wear much longer,
especially when used for cross or half-cross pulleys ;
that leather belts cease to stretch after once or twice
shortened and relaced, while those of rubber do not ;
and that leather will bear contact with oil and grease
without harm, while rubber thus exposed will soften,
and stretch out of shape. Wide leather belts can be
cut up into narrow ones, while rubber belts cannot be
cut without injuring the finished edges ; also, leather
can be more easily repaired when injured than rubber.
On the other hand, rubber belts do not need to be
riveted, but are made continuous ; they do not slip so
easily on the pulley-faces as leather, and are cheaper
at first cost for the same sizes. It is also claimed that
rubber belts endure exposure to cold and wet much
better than leather, retain their flexibility better, and
do not lose strength so rapidly from wear. Leather
and vulcanized belts both are good. Thousands of
each perform well their arduous duties all over the
civilized world. Each has hundreds of admirers and
champions. We therefore deem it best to express no
preference on our own part, preferring rather to have
each purchaser choose for himself, assuring him that
either good leather or good vulcanized rubber will do
68 BELTS AND PULLEYS.
his work as faithfully and well as any reasonable man
should desire.
9. Lacing and other Modes of Fastening.
Endless belts, of whatever material they are made,
when subjected to a considerable strain for any length
of time become lengthened or stretched. As a result
of this lengthening, the belts hang loosely upon their
pulleys, and consequently slip and slide. It is there-
fore necessary to have some ready means of shortening
belts to their proper lengths, and thus make them again
fit tight upon the pulley-faces. This is very generally
done by leaving the belt with two ends (i.e., not end-
less), and then lacing together the free ends with leather
thongs or cords. When a laced belt becomes stretched,
it is unlaced, cut off to the proper length, and laced up
again, new holes having been punched at the cut end.*
Lacing-thongs are commonly made of leather or
good clean rawhide, softened and stretched somewhat
to render it firm and even ; they vary in width from
one quarter to three quarters of an inch, and in thick-
ness from one sixty-fourth to nearly one eighth of an
inch, according to the width. We may say very simply,
in lacing belts, punch the holes just large enough to
easily admit the lacing-thong inch to I inch from the
ends of the belt (no more material than is necessary
* Sometimes belts of considerable length are shortened to take up
the stretch by simply running off one pulley and twisting the belts
until the proper lengths are obtained. This practice is, however, a
very bad one, because the twists cause the belts to become cracked
and to wear out rapidly, and should never be indulged in except in
cases of immediate necessity.
LACING AND OTHER MODES OF FASTENING- 69
should be cut out, because this tends to weaken the
belt) ; use for small belts a -J-inch thong ; for belts from
4 inches to 8 inches wide, a f-inch thong ; for belts from
8 inches to 15 inches wide, a ^-inch thong; and for belts
over 15 inches in width, a f-inch thong. The first requi-
site in lacing together the free belt-ends is to have the
ends square -that is, at right angles with the sides of
the belt ; if the ends are not square the belt will not lie
straight on the pulleys, and will tend, consequently, to
FIG. 36. FIG. 37.
run off the pulleys, and otherwise interfere with the
proper motion of the machine.
The simplest mode of lacing belts, which is repre-
sented in Fig. 36, consists in starting at one side, and
lacing over and over through all the holes until the
other side of the belt is reached. This does well
enough for small belts not to be subjected to any
severe strain, although even they will do more satis-
BELTS AND PULLEYS.
factory work if laced differently ; but for larger belts
better and safer methods must be used.
Fig. 37 shows a style of lacing quite common among
machinists, and which combines quickness of operation
with strength about as well as any of the simpler
methods. Begin at the side a in the figure, and lace
with both ends of the thong, as shown, fastening the
ends at b in a knot or other convenient manner.
A still better lacing is represented in Fig. 38. The
thong is here crossed on one side of the belt only
the upper side, and care should be taken not to cross
unevenly the double parts on the pulley-side.
In heavy-driving belts, and in all belts where the
strain is severe, double rows of holes should be punched,
and the joining thus rendered
doubly secure against breakage.
Messrs. J. B. Hoyt & Co., manu-
facturers of leather belting, New
York, inform me that all their
belts are laced according to the
double method represented in
Fig. 39, in which a is the side
to be placed next the pulley.
This lacing has the advantage
that all its parts on the outside
of the belt are parallel to the di,
FIG. 38- rection of motion, and the ten-
dency is therefore to keep the ends of the belt at all
times in their proper positions. The above-mentioned
gentlemen, after many years of experience with leather
belting, have come to believe this method the best in
prdinary use,
LACING AND OTHER MODES OF FASTENING-
An excellent style of lacing for large belts is given
by Mr. John W. Cooper in his " Use of Belting," which
FIG. 39.
we represent in Fig. 40. Begin with one end of the
lacing-thong at the point a, and lace successively
through the holes i, 2, 3, 4, 5, and
so on, all around the rows of holes
until the point a is again reached,
where the thong is fastened off as
shown in the figure. Although in
this case the parts of the thong are
not parallel to the direction of mo-
tion, yet they are so slanted on
the pulley-side in one direction and
on the outside equally in the other
that the result is practically the
same, and the lacing is, beyond
doubt, one of the best in existence.
Several kinds of metallic belt- FIG. 4 o.
hooks or fasteners have been from time to time con-
BELTS AND PULLEYS.
trived and introduced never, however, to our knowl-
edge, with any great degree of success. For small belts
the best of these hooks do well enough, and lessen the
work of relacing and shortening; but large driving-
belts, and those used to transmit large powers, must,
for good results, be strongly laced by one of the methods
already given, or an equally good one. Among the
(MM>
Frc. 41
various metallic belt-hooks we may give the following
as probably the best in use: Fig. 41 represents a kind
of belt-hook which is quite extensively used for light
belts. Figure a is the hook itself. To fasten, proceed
as follows : Cut slits in the belt-ends parallel in length
to the length of the belt ; place the ends as shown in
Fig. b ; force through the slits the belt-hooks as in the
figure, turn them, and flatten out the belt as in figure c.
LACING AND OTHER MODES OF FASTENING. 73
Figure d represents the pulley-side of the belt and
figure c the outside.
In Fig. 42 the hook (figure a) has a double hold on
the belt through the two rows of holes, and is there-
fore a stronger fastener than the preceding hook.
Figure b represents the outside of a belt fastened with
i
5
6
n n n
1
n n n
d\
FIG. 42.
these hooks, figure c the pulley-side, and figure d a
section through the two ends of the belt showing one
hook.
An ingenious buckle for fastening together the belt-
ends is given in Mr. Cooper's " Use of Belting," and
credited to a Canadian inventor, The fastener consists
74
BELTS AND PULLEYS.
of two separate parts, one containing a series of parallel
metallic tongues (represented by the dotted lines in
figure 43 a) which are inserted through holes in the
belt-ends, and the other a rectangular cover which is
slipped over the projecting ends of the tongues after
they have been forced through the belt. Figure 43 a
represents the outside and figure b the pulley-side of
the belt. Figure c is a sectional drawing showing a
pair of tongues and the cover.
FIG. 43
All belt-hooks and metallic fasteners used for belts
to be run over pulleys should be of brass, copper, or
other soft metal, in order to prevent scratching the
surface of the pulley, and the consequent additional
wear of the whole belt.
A very simple, if not very firm and secure, method
of fastening, without the use of lacing thongs or hooks
of any kind, is shown in Fig. 44. One end of the belt
STRENGTH OF LEATHER BELTS.
is cut into cleat-shaped pieces, shown in figure b at
y> y> y> an d the other punched with oblong slots, figure
a, x, x, x. The cleats are forced through the slots, the
belt-ends hammered out flat, and the joining is complete.
Figure c shows a section through the ends of the belt,
88
I
with the cleat and slot fastening. Such a fastening as
this is at best weak and uncertain, and should not be
used at all in practice, except for some exceptionally
light work, where lacing-thongs or belt-hooks are not
easily to be obtained.
10. Strength of Leather Belts Resistance to Slipping.
The discussion of the strength and resistance to
slipping of leather belts is attended with well-nigh in-
7 BELTS AND PULLEYS.
surmountable difficulties, from the fact that the sub-
stance with which we have to deal is almost wanting
in homogeneity. We are able by means of standard
rules and formulas to calculate closely the strength of
a cast-iron column or wrought girder, because within
reasonable limits cast-iron and wrought-iron are homo-
geneous; in other words, if we know the breaking
strength and safe-working strength of a certain kind of
iron, we know these strengths of other iron of the same
kind : they are approximately the same. Other metals
also are even in texture and homogeneous in nature ;
many kinds of wood possess this valuable homo-
geneity to a remarkable extent. But this is by no
means true of leather. Few substances, if any, with
which mechanical men have to deal show such widely
varying results under apparently similar circumstances
as the leather which furnishes for us the countless
number of transmission-belts seen in nearly every shop
and factory in the land. In a series of tests made by
a prominent firm of leather-belt manufacturers in New
York City, strips of leather two inches wide were cut
from one of the ordinary sides used for belting, and
carefully tested in the same testing-machine and under
precisely similar circumstances. These strips were
broken at strains varying all the way from 1400 pounds
to 3475 pounds ; which result elicits the strange fact,
that one strip of leather may be nearly two and a half
times as strong as another strip equal in width and
thickness, and taken from the same side of leather.
The strips in question when in their original positions
in the skin were but 15 inches apart at their nearest
points. Nor is this all : in two strips which, in the
STRENGTH OF LEATHER ELTS. 77
side of leather, joined each other, lay immediately side
by side, the difference in breaking strength was 675
pounds, or 337^ pounds per inch of width ; a variation
of 32 per cent of the greater strength and of nearly 47
per cent of the smaller.
A gentleman for many years engaged in the manu-
facture of leather belting has informed the author that
he once cut off twelve inches of solid part (i.e., without
rivets or splicing) from a roll of two-inch belting ; cut
the piece longitudinally into two parts ; tested them
in a correct machine ; and found that one part with-
stood 400 pounds greater tensional strain than the
other. The gentleman also said that he had tested
with a good dynamometer two eight-inch belts, made
from similar leather in his own factory, running over
pulleys equal in size, doing the same kind of work, and
carefully stretched over their pulleys with as nearly as
possible the same tensions, and found that one would
transmit nearly a horse-power more work without slip-
ping than the other. Many other similar examples
from practice might be cited to show with how much
of uncertainty and variation from averages the investi-
gator of belt-transmissions is compelled to deal. Let
the examples already given, however, suffice for this
purpose ; and let us, keeping always well on the safe
side, endeavor to calculate, as simply as the compli-
cated nature of the subject will allow, the proper
strengths and sizes for the various transmission-belts in
use in practice.
The strain brought to bear upon an ordinary endless
belt running continuously over its pulleys, leaving out
of the question considerations due to centrifugal force,
78 BELTS AND PULLEYS.
etc., etc., is one of simple tension ; and were it not for
other complicating elements which enter into the cal-
culations, the proper strength for a belt to withstand
a certain strain could be quite easily calculated. For
example, if we represent by Pthe actual strain on the
belt in pounds, by A the cross-section of the belt in
square inches, and by f the safe working tensional
stress in pounds per square inch for the material of the
belt, we can write the formula
P= Af,
and, by transposing, A = -^.
From this simple formula, were the tensional strain
all which we must take into account, we could easily
calculate our belt widths and thicknesses. But, un-
fortunately for the simpli-
city of our calculations,
other considerations must
be looked into before we
can correctly obtain the
necessary rules and formu-
las. In the first place,
probably nine belts out of
ten in ordinary use will slip
around on their pulleys
before they will break ; that
is, the resistance of the
belt to slipping is not equal to its strength. It there-
fore becomes necessary to embody in our calculations
STRENGTH OF LEATHER BELTS. J<)
for strength considerations which will prevent slipping
of the belt upon its pulleys.
Let ACB (Fig. 45) represent a band or cord drawn
over an angle of a solid, as shown in the figure. Let
forces, represented by T and /, act at the ends of
the cord in the directions shown, and let a represent
the angle DCB. In drawing the cord over the angle
or corner the friction between the block and cord
must be overcome. By the principles of the parallelo-
gram of forces, the resultant normal pressure R of the
forces T and t is given by the expression
R= Vr + f-2Ttcos<x; .... (33)
and if we represent by F the friction and by cp the
coefficient of friction, we shall have
F cpR = (p V~T* -\-f-2Tt cos a.
In order to move the cord over the angle in the direc-
tion of the force 7", this force must be able to over-
come the force t acting in an opposite direction, and
also the friction ; that is, we must have
From this, by squaring,
Substituting this value of T* in the above equation for
the friction, and neglecting the quantity F", gives us the
equation
F= <p Vf -\- 2tF-\- f 2? cos a 2.Ft cos a
80 BELTS AND PULLEYS.
and by factoring we obtain
F q> V2(i cos a) (f -\- tF\
From trigonometry we find
V (i cos a) = sin \a,
which, multiplied by 4/4, becomes
cos a) = sin a
or V2(i cos a) = 2 sin \a.
Consequently
F 2(p sin ^ Vf -f- tF.
From the binomial formula, neglecting the small
terms after the second, we may extract the square
root of the quantity under the radical sign, and write
Hence F = 2g> sin -\t -| -- J,
F = 2<pt sin \- <pF sin -,
r* _, . or .or
F tpFsm - = 2(pt sin -,
STRENGTH OF LEATHER BELTS. 8 1
. fx
2q>t sin
and finally F = ..... (34}
i (p sin
The force, then, which is required to draw the cord
over the angle in the direction of T is
. a
2cpt sm
T=t+F=t+
I cp sm -
- ..... (35)
When the angle a is very small we may say correctly
enough
Of
i (p sin = i,
and formula (35) becomes
\
.... (36)
Suppose now instead of one angle over which to
draw the cord we have several, as shown in Fig. 46, the
angles being equal each to each. Let / be the tension
at one end of the cord, ^ that at the first angle, / 2 that
at the second angle, etc., to the tension T= t n at the
other end. From what precedes, we shall have for the
force necessary to draw the cord over the first angle
82
BELTS AND PULLEYS.
f,= t(i+2<psm-.
For the force necessary to draw the cord over the
second angle we shall have
Hence
or
* = t (\ + 2cp sin ^(i + 2g> sin |),
FIG. 46.
In a similar manner
2<p sn - i 2cp sn -= i 2g> sn -
STRENGTH OF LEATHER BELTS.
And finally
/ = T = t
i -f- 2(p sin
(37)
By means of this formula we are able to calculate
the forces which tend to cause
an endless belt to slip upon its
pulley, the tensions in the belt
necessary to prevent slipping,
and consequently the strength
and width of the belt itself.
Let K, Fig. 47, be a pulley,
over which, embracing a centre
angle BCA = EDB = a, a belt
tABT passes as shown in the
figure. We can assume the arc
AB to be composed of an infinite number () of in-
finitely small sides ; each will then be expressed by .
From formula (37) we have for the force T the ex-
pression
T= t
a \n
2^ sin-),
and since for the infinitely small arcs their sines are
(X Cf
equal to the arcs themselves, we may write, sin = ,
7 2n 2n
and therefore
This expression we may develop by means of the bi-
nomial theorem into
84 BELTS AND PULLEYS.
IX 2# a
I X 2 X 3
v
5
and since we have assumed n to be infinitely great, we
may write n I = n 2 = n 3 = n. Our last
equation therefore becomes
This is in the form of the series
in which e represents the base of the Naperian or hyper-
bolic system of logarithms (e = 2.71828), and the above
equation reduces to
T=U* ....... (38)
From this we have
hyp. log T hyp. log / = tpa,
and * hyp. log <pa ..... (39)
* This maybe very neatly demonstrated by means of the integral cal-
culus as follows: Let a represent the entire arc embraced by the belt,
and d. a one of the small portions which we represented above by.
The tension at the point A is t, that at the next portion of arc t -\-d.t;
STRENGTH OF LEATHER BELTS. 8$
Common logarithms are better known and more
easily handled than the hyperbolic. To reduce for-
mula (39) to common logarithms it is necessary only to
multiply by 0.434. Thus
T
log-- = o.434<7>, (40)
where a is expressed in circular measure, i.e., parts of n.
(X.71
If a is taken in degrees, substitute a = -_ , and we
obtain
T
log - = 0.007578^ (41)
If a is taken in fraction of the circumference, sub-
stitute a = 2ita. We obtain thus
log = 2.7297** (42)
the increase is J.t, and this is due to the friction in the unit of arc.
This friction is d.F= zcpt sin - ; or, since d.a is very small,
d.a
d.F = 2<pt <ptd.a.
Hence we have d.t = cptd.a or '- = <pd.a.
Integrating between the limits T and /, a. and o, gives us
r T d. t ('" T
I - : = <p I d. a, or hyp. log = (pa.
U t t t/o '
86 BELTS AND PULLEYS.
The best and most recent experiments made use of
for determining the coefficient of friction permit us to
use, for ordinary belt-leather over cast-iron pulleys, the
value
* 9 = -4 (43)
This value substituted in formula (40) gives
T
log -- = 0.434 X 0.4*,
or, when a is in circular measure,
T
log- = 0.1736* (44)
Substituting q> = 0.4 in formula (41) gives
log - = 0.007578 X 0.4*,
or, when a is in degrees,
log - = 0.00303* (45)
>See Appendix I.
STRENGTH OF LEATHER BELTS. 87
Similarly, by substituting in formula (42),
log -- = 2.729 X 0.4*,
or, where a is a fraction cf the circumference,
T
log - = l.OQlfa (46)
The following table, calculated from formulas (44),
T
(45), and (46), gives values of - for different values of
the arc a from 30 to 300 corresponding to from 0.524
to 5.236 in circular measure, and from -^ = 0.083 to
I = 0.833 in fractions of the circumference.
To illustrate the application of the table, suppose
we have a pair of cast-iron pulleys over which we pro-
pose to run a leather belt. Suppose the arc embraced
by the belt, upon the pulley over which it is most
likely to slip (the pulley having the smaller amount of
contact with the belt, or the smaller pulley), is 75 =
1.309 in circular measure = -/^ = 0.208 in fraction of
the circumference.
We look along the column of degrees until we find
the value 75, along the column of circular measures
until we find 1.309, or along the column of fractions
of the circumference until we find / = 0.208, and, op-
posite to these values we find the required value for
the ratio of the tensions, - = 1.689.
88
BELTS AND PULLEYS.
TABLE OF TENSIONS FOR LEATHER BELTS OVER CAST-IRON
PULLEYS.
a =
T
t
In degrees.
In circular
measure.
In fractions of the
circumference.
30
0.524
T V = 0.083
1-233
45
60
75
0.785
1.047
1.309
J= 0.125
= 0.167
=0.208
1.369
I.52I
1.689
90
I-57I
i = 0.250
1.874
105
I-833
T&- = 0.292
2.082
120
2.094
i = 0.333
2.312
135
2.356
1 = 0.375
2.565
ISO
2.618
A = 0.417
2.849
165
2.880
H = 0.458
3.163
180
3-I42
i = o 500
3-514
195
3-403
M = 0.541
3.901
2IO
3-665
A = 0-583
4-333
24O
4.189
\ = 0.667
5-340
270
300
4.712
5-236
= 0.750
I = 0.833
6.589
8.117
The greatest strain brought to bear upon an endless
belt, or the strain tending in the greatest degree to
cause breakage, is the tension in the driving part of
the belt, that is T. This tension acts in one direction
and the lesser tension / in a contrary direction. Con-
sequently it is the excess of the greater over the lesser
tension which overcomes the resistance of the pulley
and causes rotation. If we represent the force of re-
sistance in pounds at the circumference of the pulley
(which is the force transmitted by the pulley) by P, we
shall have the expression
P = T - t.
(47)
STRENGTH OF LEATHER BELTS. 89
Hence T = P + t,
which may be put in the form
By substituting for P within the parenthesis its value
from formula (47), we obtain
T=f ( l+ _L r \
But
Hence
(48)
90 BELTS AND PULLEYS.
by means of which and the preceding table the ten-
sion T for different values of a may be determined.
The following table, calculated from formula (48),
L
gives values of for different values of the arc a.
_ !
TABLE OF GREATEST TENSION FOR LEATHER BELTS OVER CAST-IRON
PULLEYS.
In degrees.
In circular
measure.
In fractions of the
circumference.
J - r X
30
45
0.524
0.785
T V = 0.083
i= 0.125
5-29
3-71
60
1.047
i = 0.167
.92
75
1.309
^=0.208
-45
90
I-57I
i = 0.250
.14
105
1.833
1& = 0.292
93
120
2.0Q4
4 = 0.333
77
135
2.356
1 = 0.375
.64
150
2.6l8
fV = 0-417
54
165
2.880
ii =0.458
47
1 80
3-I42
i = 0.500
.40
195
2IO
3-403
3-665
H = 0.541
A = 0.583
35
30
240
4.189
t = 0.667
23
270
4.712
= 0.750
.18
300
5.236
1 = 0.833
.14
To illustrate the use of the table: Suppose the force
transmitted by a pulley is P = 500 pounds and the angle
embraced by the belt a = 105. In the table opposite
to the value a = 105 we find the value 1.93. Hence
T = P X 1.93 = 500 X 1.93 or T = 965 pounds.
STRENGTH OF LEATHER BELTS. 9!
We have now developed rules by which the actual
strain upon the belt may be determined : we have still
to determine the strength of the belt, or, in other
words, the amount of material necessary in the belt to
safely sustain the given strain. We have said that the
strain T upon an endless belt is a tensional strain. If,
therefore, we represent by b the breadth of the belt in
inches, by 8 its thickness, also in inches, and by f the
greatest safe-working stress in pounds per square inch,
we shall have, for the relation between the strain and
the strength, the expression
T=bSf t ...... (49)
and consequently bd = - ........ (50)
Because of the great variations in the strength of
leather the quantity /can be only approximately de-
termined. Experiments and tests upon the strength
of leather, be they ever so numerous and carefully
made, serve only to impress more strongly upon the
mind of the experimenter this unfortunate lack of ho-
mogeneity in the substance with which he is dealing.
In this predicament he who would investigate the sub-
ject of leather belts must be satisfied with an average
value taken from a great many widely differing values
for his coefficient of strength ; and until our manufac-
turers are able to produce leather which shall be to a
reasonable extent uniform, the subject of strength of
belting must remain as it is now the most uncertain
and indefinite one with which mechanical men have
to deal.
92 8ELTS AND PULLEYS.
The weakest part of an endless belt is obviously at
the joint : the value of the safe-working stress /"must
therefore be taken for this part. The author has dur-
ing the last three years tried a great many experiments
with the view of obtaining the average strength of
laced and riveted joints. These average breaking
strengths he has found to be about as follows :
For ordinary single leather-lacing,
950 pounds per square inch ;
For ordinary single rawhide-lacing,'
1000 pounds per square inch ;
For good double leather-lacing,
1 200 pounds per square inch ;
For good double rawhide-lacing,
1400 pounds per square inch ;
For ordinary riveted joints,
1750 pounds per square inch.
We may therefore take for our safe-working stress
in pounds per square inch the following values :
Single leather-lacing, f = 325 ;
Single rawhide-lacing, ^"=350;
Double leather-lacing, f = 375 ;
Double rawhide-lacing, f = 400 ;
Riveted joints, / = 575.
By substituting these values successively in formula
(50), we obtain the following formulas :
STRENGTH OF LEATHER BELTS. 93
For single leather-lacing, bS = ; .... (51)
For single rawhide-lacing, bd = ; . . . . (52)
T
For double leather-lacing, bd =. ; . . . . (53)
For double rawhide-lacing, bd = ; . . . . (54)
T
For a riveted joints, bd = (55)
Example. Required the width of a leather belt \
inch thick, which will safely transmit a force of P = 600
pounds when laced according to each of the above-men-
tioned methods, the pulleys over which the belt is to
run being of the same diameter that is, the angle em-
braced by the belt being a 180.
From the table on page 90 we have, T = P X 1.40
= 600 X 1.40 = 840 pounds. From formula (51),
therefore, we have
v 1 - 8 -^ A _ 4 X 840
X 4 - 325' ~ 325 '
or, for single leather-lacing,
b = 10.34" =
94 BELTS AND PULLEYS.
From formula (52),
i 840 4 X 840
X 4 = &' ~35^~'
or, for single rawhide-lacing,
b = 9-6" - 9ft".
From formula (53\
A v 1 = ?40 A _ 4 X 840
4 ~ 375' 375
or, for double leather-lacing,
b = 8.96" = 8|i".
From formula (54),
x L = 8 1 j _ 4 X 840
' 4 ~ 400' 400
or, for double rawhide-lacing,
b = 8./.0 " =: 8f".
From formula (55),
i = 840 4 X 840
x 4 575' 575
or, for a riveted joint,
b = 5-84" - 5tt"-
STRENGTH OF LEATHER BELTS. 95
The following tables of formulas have been calculated
from the table on page 90 and formulas (51), (52), (53),
(54), and (55), respectively. The above example may
be calculated from these tables as follows : We have
for our data, P = 600 pounds, a = 180, and d = %'.
From formula (66), for single leather-lacing,
bS = 0.0043 1 X 600 ;
b 0.00431 X 600 X 4 = 10.34".
From formula (82), for single rawhide-lacing,
bS = 0.004 X 600 ;
b 0.004 X 600 X 4 = 9.60".
From formula (98), for double leather-lacing,
bS = 0.00373 X 600 ;
b 0.00373 X 600 X 4 = 8.952".
From formula (114), for double rawhide-lacing,
bd = 0.0035 X 600 ;
b = 0.0035 X 600 X 4 = 8.40".
From formula (130), for a riveted joint,
bd = 0.00243 X 600 ;
b = 0.00243 X 600 X 4 = 5-832".
9 6
BELTS AND PULLEYS.
TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST-IRON PULLEYS.
Single Leather Lacing.
a. in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
TV = 0.083
bS = o.oi62&P
56
45
0.785
i = 0.125
68 = O.OU42P
57
60
1.047
\ = 0.167
l>8 = o.ooSgS/'
58
75
1.309
^ = 0.208
5 = o. 007547*
59
90
I-57I
J = 0.250
68 = 0.006587'
60
105
1 20
I-833
2.094
A = 0.292
t = 0.333
35 = O.OO594/'
68 = o.oo545/>
61
62
135
2.356
I = 0.375
68 = 0.005057'
63
150
2.618
T = -4I7
35 = 0.004747'
64
165
2.880
ft= 0.458
t> = 0.004527*
65
1 80
3-I42
-J- = O.5OO
35 = o. 004317*
66
195
3-403
if = 0.541
35 = 0.004157*
67
2IO
3-665
5 = 0.583
35 = 0.004007*
68
240
4.189
| = 0.667
5 = 0.00378/ 1
69
270
4.712
4 = 0.750
M = O.OO363/'
70
300
5-236
1 = 0.833
35 = O.OO35I/*
71
TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST-IRON PULLEYS,
Single Rawhide- Lacing.
a. in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
45
0.524
0.785
rV = 0-083
i = 0.125
68 = o.oisii/ 1
5 = O.OIO6O/ 3
72
73
60
1.047
% = 0.167
33 = o.oo834/>
74
75
90
1.309
I-57I
A = 0.208
i = 0.250
35 = o. 00700 P
68 = o.oobiiP
a
105
I-833
& = 0.292
68 = o. 00551^
77
1 20
2.094
i = 0.333
68 = o. 00506 P
78
135
2.356
f = 0.375
68 = 0.004697'
79
150
165
180
2.618
2.880
3-I42
A = 0.417
H = 0.458
i = 0.500
68 = O.00440/'
65 = O.OO42O/'
35 = 0.004007'
80
81
82
195
3-403
ti = 0.541
35 = 0.003867'
83
210
3-665
A = -583
35 = O.OO37I/*
84
240
4.189
f = 0.667
35 = 0.0035I/'
85
270
300
4.712
5-236
t = 0.750
1 = 0.833
35 = 0.00337/ 1
35 = o.oo326/ >
86
87
STRENGTH OF LEATHER BELTS.
97
TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST-IRON PULLEYS.
Double Leather-Lacing.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
T V = 0.083
bS 0.014117*
88
45
60
0.785
1.047
i = 0.125
i = 0.167
bS o. 009897*
b8 = 0.007797*
89
90
75
1.309
-ff = 0.208
l>d = 0.006537*
Qi
90
I-57I
i = 0.250
68 = 0.005717*
92
105
1.833
A = 0.292
l>d o. 005147*
93
120
2.094
i = 0.333
l>8 = 0.004727*
94
135
2.356
1 = 0.375
b8 = 0.004377*
95
150
165
2.618
2.880
A = 0.417
tt = 0.458
d8 = 0.0041 \P
b8 = o. 00392 P
96
97
1 80
3-142
| = 0.500
b8 0.003737*
98
195
3.403
if = 0.541
bS = o. 00360 P
99
210
3.665
^ = 0.583
bS = 0.003477*
100
240
4.189
f = 0.667
l>8 = 0.003287*
101
270
4.712
f = 0.750
l>8 0.003157*
102
300
5-236
1 = 0.833
b8 = 0.003047*
103
TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST-IRON PULLEYS.
Dotible Rawhide- Lacing.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
T V = 0.083
l>8 = o. 013237*
104
45
0.785
i = 0.125
M = o. 009287*
^05
60
1.047
| = 0.167
5 = 0.007307*
106
75
1.309
-ff = 0.208
l>8 = 0.006137*
107
90
I-57I
i = 0.250
b8 = 0.005357*
1 08
105
r-833
ft = 0.292
l>8 0.004837*
109
120
2.094
i = 0.333
b8 =: 0.004437*
IIO
135
2.356
1 = 0.375
b8 0.004107*
in
150
165
2.618
2.880
fV = 0-417
H = 0.458
bS = 0.003857*
b8 = 0.003687*
112
H3
1 80
3.T42
i - 0.500
b8 = 0.003507*
"4
195
3-403
M = 0.541
^5 =: 0.003387*
115
2IO
3-665
A = 0-583
b8 0.003257*
116
240
4.189
| = 0.667
bS = 0.003087*
117
270
4.712
f = 0.750
l>8 = 0.002957*
118
300
5-236
1 = 0.833
bS = 0.002857*
119
9 8
BELTS AND PULLEYS.
TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST- IRON PULLEYS.
Riveted Joint.
a in
degrees.
o in circular
measure.
o in fractions of
circumference.
Formula.
No.
30
0.524
oV = 0.083
bS = 0.009207'
120
45
0.785
i = 0.125
lid = 0.00645/*
121
60
1.047
= 0.167
bd = o.oosoS/"
122
75
1.309
fs = 0.208
5 = o.oo426/ >
123
90
I.57I
i = 0.250
5 = O.OO372/"
124
105
1.833
A = o- 2 92
bS = O.OO336/ 1
125
1 20
2.094
* = 0-333
^5 = o. 00308 P
126
135
2.356
1 = 0.375
bS = O.OO285/*
127
150
2.618
A = 0.417
bS = 0.00268P
128
165
2.880
H = 0.458
I'd = 0.002567'
129
180
3-142
i = 0.500
^5 = O.OO243/ 5
130
195
3-403
M = 0.541
l>d = O.OO235/'
131
2IO
3.665
A = 0.583
^5 = 0.00226P
132
240
4.189
$ = 0.667
^5 = 0.002I4/ 5
133
270
4.712
4 = 0.750
^5 = O.O02O5/'
134
300
5-236
f = 0.833
M = o.ooigS/*
135
STRENGTH OF LEATHER BELTS. 99
Often, when we know the horse-power to be trans-
mitted, it is convenient to calculate belt-widths from
this, without finding the circumferential force. From
formula (20) we have, when v m represents the velocity
in feet per minute, and //the horse-power,
(.36)
and from formula (22), when v represents the velocity
in feet per second,
By substituting this last value of P in formulas (56)
to (135), and reducing, we may obtain the following
tables of formulas for calculating belt-widths from the
horse-power transmitted and the velocity in feet per
second :*
* By substituting the value of P given in formula (136) in formulas
(56) to (135), we may obtain formulas for belt- widths in terms of the
horse-power and velocity in feet/^r minute. For example, formula
(68) gives b8 =. 0.004 = 132 . Such formulas are, how-
ever, seldom needed in practice, the velocity being almost always
taken in feet per second.
100
BELTS AND PULLEYS.
TABLE OF FORMULAS FOR LEATHER-BELTS OVER CAST-IRON
PULLEYS.
Single Leather-Lacing.
a in
degrees.
a in circular
measure.
o in fractions of
circumference.
Formula.
No.
30
0.524
T V = 0.083
bd = 8. 954 f
138
45
0.785
i = 0.125
bd = 6.281-
V
139
60
1.047
= 0.167
If
bo = 4.939-
"V
140
75
1.309
f = 0.208
bd H
V
141
90
I-57I
i = 0.250
bd = 3.610-
V
142
105
1.833
fa = 0.292
bd = 3.267-
V
M3
1 20
2.094
* = 0.333
If
V
144
135
2-356
1 = 0.375
bd = 2.778-
Z'
145
150
2.618
T <V = 0.417
. H
bo = 2.607
V
146
165
2.880
H = 0.458
bd = 2.486^
V
147
180
3-142
\ = 0.500
bd = 2.371^
148
195
3-403
H = 0.541
l>8 = 2.283^
z>
149
210
3.665
A = 0.583
tjr H
DO = 2.2OO
V
150
240
4.189
I = 0.667
bd = 2.079
i5i
270
4.712
i = 0.750
b8 = 1-997^
152
300
5.236
I = 0.833
H
bd = I-93I-
153
STRENGTH OF LEATHER BELTS.
101
TABLE OF FORMULAS FOR LEATHER-BELTS OVER CAST-IRON
PULLEYS.
Single Haw hide- Lacing.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
A = - o8 3
bS = 8
3 "f
154
H
45
0.785
i = 0.125
bS = 5.
155
H
60
1.047
$ = 0.167
68 = 4
156
75
1.309
-% = 0.208
68 = 3.
H
t57
90
I-57I
J = 0.250
bS = 3.
3 6l f^
158
H
105
I-833
^ = 0.292
b8 - 3.
031
159
1 20
2.094
= 0.333
bS = 2.
783 f-
160
135
2.356
1 = 0.375
68 = 2.
58of
161
jcr
150
2.618
A = 0-417
bS = 2.
420^
162
165
2.880
M = 0.458
bS = 2.
3iof
163
180
3.142
| = 0.500
6d = 2.
200^
164
z/
195
3-403
f = 0.541
bd = 2.
123-^
165
V
2IO
3-665
A = 0-583
45 = 2.
O4.f
166
240
4.187
f = 0.667
tt = x.
93lf
167
270
4.712
4 = 0.750
M = l.
854^
168
z/
H
300
5.236
1 = 0.833
5 = i.
793-
169
102
BELTS AND PULLEYS.
TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST-IRON
PULLEYS.
Double Leather-Lacing.
a in
degrees.
measure.
circumference.
Formula.
No.
30
0.524
-fa = 0.083
bS - 7.761"^
170
45
0.785
J- = O.I25
H
05 = 5-440
171
60
1.047
t = 0.167
05 = 4.285"^
172
75
1.309
ff = 0.208
05 = 3-592^
173
90
I-57I
i = 0.250
0S = 3-i4i-
174
H
105
1.833
^ = 0.292
0S = 2.827-
175
1 20
2.094
i = 0.333
05 = 2.596^
176
135
2.356
$ = 0.375
05 = 2.404^
177
150
2.618
T <V = 0.417
05 = 2.26i^
178
//
165
2.880
tt = 0.458
05 = 2.156^-
179
//
180
3.142
i = 0.500
05 = 2.052-
1 80
H
195
3-403
if = 0.541
05 = 1.980-
181
210
3.665
rV = 0-583
05 = 1.909
182
//
240
4.189
f = 0.667
05 = 1.804-
p
183
270
4.712
4 = 0.750
5 _ !. 733 ^
184
9
300
5 .*36
1 = 0.833
0S = 1.672^
^
185
STRENGTH OF LEATHER BELTS.
103
TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST-
PULLEYS.
Double Rawhide-Lacing.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
A = 0.083
W = 7. a77
1 86
45
0.785
i = O.I25
TT
bS = 5.104
187
60
1.047
$ = 0.167
H
bo = 4.015
1 88
75
1.309
^f = 0.208
H
bS = 3-372-
189
90
I-57I
i = 0.250
bS = 2.943-
190
V
105
I.833
A = 0.292
. H
bo = 2 .657
191
V
I2O
2.094
t = o 333
bS = 2.437^
192
V
135
2.356
$ = 0.375
bO =2.22$-
193
V
150
2.618
A = 0.417
bo = 2.118-
V
194
?r
165
2.880
H = o 458
bo = 2.024-
195
V
1 80
3-142
i = 0.500
bo = i.g2S-
196
V
195
3-403
= 0.541
bS = 1.8^9^
197
V
2IO
3.665
& = 0.583
bo = 1.788^
V
198
2 4
4.189
t = 0.667
bS = 1.694^
199
V
if
270
4.712
f = 0.750
bo = 1.623^
200
H
300
5.236
1 = 0.833
bo = 1.568-
2OI
104
BELTS AND PULLEYS.
TABLE OF FORMULAS FOR LEATHER
PULLEYS.
Riveted Joint.
BELTS OVER CAST-IRON
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
g = 0.083
68 = 5.060^
2O2
45
0.785
i = 0.125
68 = 3.54*"
203
60
1.047
i = 0.167
b8 = 2.794-
204
75
1.309
-fa = 0.208
68 = 2.343^
205
90
I-57I
J = O.25O
fj
bS = 2.046
206
105
1.833
-JX = 0.292
//
68 = 1.848
207
1 20
2.094
t = 0-333
H
68 = 1.694-
208
135
2.356
f = o.375
H
bS = 1.568-
209
150
2.618
ft - 0.417
ff
bd = 1-474
210
165
2.880
tt ~ 0.458
H
b8 =1.408 -
211
1 80
3.142
i = 0.500
^5 = 1.337^
212
195
3.403
M = 0.541
H
bo = 1.293
213
2IO
3-665
h = 0.583
TT
68 = i . 243
214
240
4.189
= 0.667
-!.!*
215
270
4.712
= 0.750
W = 1-128^
216
300
5-236
1 = 0.833
H
bd = 1.089-
217
STRENGTH OF LEATHER BELTS. 1 05
Example. Required the width for a leather belt -^
inch thick which will transmit a force of 15 horse-
power at a velocity of 10 feet per second, the angle a
being 105. In this case = - = 1.5 and d = -=L
v 10 16
Hence from formula (143) we have
b X f6 = 3.267 ^ = 3-267 X 1.5-
Therefore b = 3.267 X 1.5 X ,
or, for single leather-lacing,
b 15.682" i$! x/ nearly.
From formula (159),
X-j= 3-031 X 1.5,
16
= 3.031 X 1-5 X y,
or, for single rawhide-lacing,
b = 14-549" = I4tt"-
From formula (175),
b X " = 2 ' 827 X I>5>
16
= 2.827 X 1-5 X ,
106 BELTS AND PULLEYS.
or, for double leather-lacing,
b= 13-570" =
From formula (191),
b = 2.657 X 1-5 X y,
or, for double rawhide-lacing,
b = 12.754" = i2f".
From formula (207),
b X j| = 1-848 X 1.5,
16
= 1.848 X 1.5 X y,
or, for a riveted joint,
b = 8.870" = 8$".
In the majority of cases leather belts (single) are ap-
proximately -fa inch thick. Very often the arc em-
braced by the belt is 180 that is, the pulleys are
equal ; and, perhaps, more often the arc a is about
135 = | the circumference. For these cases, then, we
may obtain formulas which will prove very useful in
practice.
STRENGTH OF LEATHER BELTS.
IO7
By substituting 8 ^ inch in formulas (66), (82),
(98), (114), (130), (148), (164), (i 80), (196), and (212),
successively, and reducing, we obtain the following
formulas :
When a = 180 and = ^ inch,
Single leather-lacing, & = o.oiyfP; . . . . (218)
Single rawhide-lacing, b = O.OiS^P; .... (219)
Double leather-lacing, b = o.O(7\P; .... (220)
Double rawhide-lacing, b = o.o 1 6oP; .... (221)
Riveted joint, b = o.omP. .... (222)
TT
Single leather-lacing, =10.839 J
TT
Single rawhide-lacing, =10.057 ;
Double leather-lacing, b= 9.381
TT
T T
Double rawhide-lacing, b = 8.800 ;
TT
Riveted joint b 6.112.
(223)
(224)
. . (225)
. . (226)
. . (227)
By substituting 3 = -^ inch in formulas (63), (79).
(95), (ill), (127), (145), (161), (177), (193), and (209),
successively, we obtain the following formulas :
When a = 135 and 8 ^ inch,
Single leather-lacing, b = 0.023 iP; .... (228)
Single rawhide-lacing, = o.O2i4/ > ; .... (229)
Double leather-lacing, b O.O2OOP; .... (230)
Double ra whide-lacing, b = o.oiSfP; . . . . (231)
Riveted joint, b = o.oi3oP. .... (232)
IO8 BELTS AND PULLEYS.
TT
Single leather-lacing, b = 12.699 ; . . . (233)
TT
Single rawhide-lacing, =11.794; . . . (234)
TT
Double leather-lacing, b = 10.990 ; ... (235)
TT
Double rawhide-lacing, b= 10.171 ; ... (236)
TT
Riveted joint, b 7.168. . . . (237)
Example. Required the width of a -^-inch leather
belt, single leather-lacing, to transmit a force of 600
pounds, the pulleys being equal.
From formula (218) we have
b 0.0197 X 600,
b= 11.82"= ii|".
Example. Required the width of a ^yinch leather
belt, single rawhide-lacing, to transmit a force of 15
horse-power at a velocity of 10 feet per second, the
pulleys being equal.
From formula (224) we have
b = 10.057 X ^,
*= 15.085" = isA".
Example. Required the width of a ^-inch leather
STRENGTH OF LEATHER BELTS. IOQ
belt, double rawhide-lacing, to transmit a force of 600
pounds, the arc embraced by the belt being about
I35-
From formula (231) we have
b = 0.0187 X 600,
b 11.22" = n^V".
Example, Required the width of a -j^-inch leather
belt, riveted joint, to transmit a force of 15 horse-power
at a velocity of 10 feet per second, the arc embraced
by the belt being about 135.
From formula (237) we have
b= 7.168 X |^,
b 10.75" = iof ".
The following tables give the forces in pounds (P\
and the values of the horse -power divided by the
/ ff\
velocity in feet per second ( j, corresponding to dif-
ferent widths of g^-inch leather belts from I inch to 30
inches for a = 180 and en = 135 for each of the five
methods of joint-fastening given above. For a great
many cases which arise in practice the tables will prove
convenient and labor-saving.
no
BELTS AND PULLEYS.
TABLE OF WIDTHS OF LEATHER BELTS OVER CAST-IRON PULLEYS,
WHEN a = 180 AND <5 = ^j". From Formulas (2i8)-(222).
Width
in
inches.
/>, single
leather-
lacing.
P, single
rawhide-
lacing.
P, double
leather-
lacing.
/>, double
rawhide-
lacing.
P, riveted
joints.
No.
I
50.76
54.64
58.48
62.50
90.09
I
Ii
76.14
81.97
87.72
93-75
135.14
2
2
101.52
109.29
116.96
125.00
180.18
3
2i
I 26 . 90
136.61
146.20
156.25
225.23
4
3
152.28
I63-93
175-44
187.50
270.27
5
3i
177-66
191 .26
204.68
218.75
3I5.32
6
4
203.05
218.58
233.92
250.00
360.36
7
4i
228.43
245.90
263. 16
28l 25
405-4I
8
5
252.71
273-22
292 40
312.50
450-45
9
5*
279.19
3co- 46
321 64
343-75
495-50
10
6
304-57
327-87
350-88
375-00
540.54
ii
7
355-33
382.51
409.36
437-50
630.63
12
8
406.09
437-16
467.84
500.00
720.72
13
9
456.85
491.80
526.32
562.50
810.81
14
10
507.61
546.45
584-80
625.00
900.90
15
ii
558.38
601.09
643-27
687.50
990.99
16
12
608.57
655-74
701.75
750.00
1081.08
17
14
710.00
765-03
818.71
875-00
1261.26
18
16
812.18
874-32
935.67
IOOO.OO
1441.44
19
18
9I3-7I
983 61
1052.63
1125.00
1621 .62
20
20
1015.22
1092.90
1169.59
1250.00
i 801.80
21
22
1116.75
1202.19
1286.55
1375.00
1981.98
22
24
1218.27
1311-48
1403-51
1500.00
2162.16
23
26
I3I9-79
1400.77
1520.47
1625.00
2342.34
24
28
1421.31
1530.05
1637.43
1750.00
2522.52
25
30
1522.84
1639-34
1754-44
1875.00
2702.70
26
STRENGTH OF LEATHER BELTS.
Ill
TABLE OF WIDTHS OF LEATHER BELTS OVER CAST IRON PULLEYS,
WHEN a = 180 AND S = -fa". From Formulas (223)-(227).
//
H
H
H
Width
-, single
~, single
, double
, double
~, riveted
No.
in
leather-
rawhide-
'leather-
rawhide-
iiuhes.
lacing.
lacing.
lacing.
lacing.
joints.
I
0.0923
0.0994
0.1066
0.1136
0.1636
I
I*
0.1384
0.1491
0.1599
0.1705
0.2454
2
2
0.1845
O.lgSg
0.2132
0.2273
0.3272
3
2*
0.2307
0.2486
0.2665
0.2841
0.4090
4
3
0.2768
0.2983
0.3198
0.3409
0.4908
5
3*
0.3229
0.3480
0-3731
0-3977
0.5726
6
4
o . 3690
0-3977
0.4264
0.4546
0.6544
7
4*
4152
0.4474
o 4797
0.5H4
0.7362
8
5
4613
0.4972
0.5330
.5682
0.8181
9
5*
5074
0.5469
0.5863
.6250
0.8999
10
6
553 6
0.5966
0.6396
.6818
0.9817
ii
7
-6458
o . 6960
0.7462
7955
I-I453
12
8
-7331
0-7954
.8528
9091
I 3089
13
9
-8303
0.8949
9594
.0228
I-4725
14
10
.9226
0-9943
.0659
.1364
1.6361
15
ii
.0149
1.0937
.1726
.2500
I 7997
16
12
.1071
1.1932
.2792
3637
I-9633
17
14
.2916
1.3920
.4924
5910
2.2905
18
16
.4762
i . 5909
.7056
.8182
2.6178
19
18
.6607
1.7897
.9188
0455
2-9450
20
20
.8452
1.9886
.1318
.2728
3-2722
21
22
.0297
2.1875
3452
.5001
3-599-1
22
24
.2142
2.3863
.5584
7274
3.9266
23
26
.3988
2.5852
.7716
9546
4-2539
24
28
.5333
2.7840
.9848
3.1819
4.5811
25
30
.7678
2.9829
3-1977
3.4092
4-9083
26
112
BELTS AND PULLEYS.
TABLE OF WIDTHS OF LEATHER BELTS OVER CAST-IRON PULLEYS,
WHEN a = 135 AND S = -fa". From Formulas (228)-(232).
Width
in
inches.
/*, single
leather-
lacing.
P, single
rawhide-
lacing.
P, double
leather-
lacing.
P, double
rawhide-
lacing.
P, riveted
joints.
No.
I
43-29
46.73
50.00
53.48
76 9 2
I
Ii
64.94
70.09
75.00
80.21
"5.38
2
2
86.58
93.46
JOO.OO
106.95
I53.85
3
*i
108.23
116.82
125.00
I33-69
192.31
4
3
129.87
140.19
150.00
160.43
230.77
5
3i
I5I-5I
163.55
175.00
187.17
269.23
6
4
173.16
186.92
200.00
213.90
307.69
7
4i
194.81
2IO.28
225.00
240.64
346.15
8
5
216.45
233-65
250.00
267.38
384-62
9
51
238.10
257.01
275.00
294.12
423-08
10
6
259-74
280.37
300.00
320.86
461.54
ii
7
303.03
327-IO
350.00
374-33
538.46
12
8
346.32
373-83
400 . oo
427.81
615.38
13
9
389-61
420.56
450.00
481.28
692.31
14
10
432.90
467.29
500.00
534.76
769-23
15
ji
476.19
514.02
550.00
588.24
846.15
16
12
519.48
560.75
600.00
641.71
923.08
17
14
606.06
654.21
700.00
748.66
10/6.92
18
16
692.64
747-66
800.00
845.62
1230.77
19
18
779.22
841.12
900.00
962.57
1384.61
20
20
865.80
934-58
IOOO.OO
1069.52
1538.46
21
22
952.38
1028.04
IIOO.OO
1176.47
1692.31
22
24
1038.96
1121.50
I 200.00
1283.42
1846.15
23
26
1125.54
1214.95
i 300 . oo
1380.38
20OO.OO
24
28
1212.12
1308.41
1400.00
T497-33
2I53-84
25
30
1298.70
1401.87
i 500 . oo
1604.28
2307.69
26
STRENGTH OF LEATHER BELTS.
TABLE OF WIDTHS OF LEATHER BELTS OVER CAST-IRON PULLEYS,
WHEN a = 135 AND d = &". From Formulas (233X237).
Width
inches.
f, single
leather-
lacing.
? Si <* 1C
rawhide-
lacing.
-, double
V
leather-
lacing.
, double
V
rawhide-
lacing.
, riveted
joints.
No.
I
0.0787
0.0879
0.0910
0.0983
0.1395 I
I*
0.1181
0.1272
0.1365
0.1475
0.2093 2
2
0.1575
0.1696
0.1820
0.1966
0.2790 3
2k
0.1969
0.2120
0.2275
0.2458
0.3488 4
3
0.2362
0.2544
0.2730
0.2950
0.4185 5
3^
0.2756
0.2968
0.3185
0-3441
0.4883
6
4
0.3150
0.3392
0.3640
0-3933
0.5580
7
4*
0-3544
0.3816
0.4095
0.4424
0.6278
8
5
0.3937
0.4239
0.4550
0.4916
0.6976
9
5*
0.433C
0-4663
0.5004
0.5108
0.7673
10
6
0-4725
0.5087
0-5459
0.5899
0.8371
ii
7
0.5512
0-5935
0.6396
0.6882
0.9766
12
8
0.6300
0.6783
0.7279
0.7866
1.1161
13
9
0.7087
o 7631
0.8189
0.8849
1-2556
!4
10
0.7875
0.8479
0.9099
0.9832
I-395I
15
ii
0.8662
0.9327
1.0009
1.0815
1.5346
16
12
0.9450
I.OI75
1.0919
1.1799
1.6741
17
.14
1.1024
I.I87O
1-2739
1.3765
1-9532
18
16
1.2600
1-3566
1-4558
I-573I
2.2322
19
18
1.4174
1.5262
1.6378
!.76 9 8
2.5112
20
20
1-5749
1.6958
1.8198
1.9664
2 . 7902
21
22
1.7324
1.8654
2.0018
2 . 1630
3.0692
22
24
1.8900
2.0349
2.1838
2-3597
3-3482
23
26
2-0475
2.2045
2.3657
2.5563
3.6273
24
28
2 . 2048
2-3741
2-5477
2-7530
3.9063
2 5
30
2.3623
2.9676
2.7297
2.9496
4-I853
20
Example. Required the force in pounds which can
be safely transmitted by a leather belt 20 inches wide
and ^ inch thick, running over two pulleys of equal
diameters (a = 180), the joint being fastened by a
double rawhide-lacing.
In the table on page 1 10, column of belt-widths, line
21, we find our width of 20 inches, and corresponding
114 BELTS AND PULLEYS.
to this, in the column for double rawhide-lacing, we
find the required force P = 1250 pounds.
Example. Required the width of a leather belt -^
inch thick, which will safely transmit a force of 1000
pounds running over two pulleys of equal diameters,
the fastening being a riveted joint.
In the table on page no, column for riveted joints,
line 17, we find P 1081.08 pounds, the nearest
value, not less than 1000 pounds, and, in the column
for belt-widths, we find the value corresponding to
P= 1081.08, b = 12 inches.
Example. Required the horse-power which can be
safely transmitted by a leather belt 12 inches wide and
-y 7 ^ inch thick, running over two pulleys of equal diame-
ters at a velocity of 15 feet per second, the fastening
being a single rawhide-lacing.
In the table on page in, column of belt-widths, line
17, we have b = 12 inches, and, in the column for single
rawhide-lacing, the corresponding value
H
TT
Hence = 1.1932, H = 15 X 1.1932,
or H 17.90.
Example. Required the velocity at which a leather
belt 12 inches wide and ^ inch thick can be driven
over two pulleys of equal diameters, in order to transmit
a force of 17.90 horse-power, the fastening being a sin-
gle rawhide-lacing.
LEATHER-COVERED PULLEYS. 115
In the table on page ill, column for single rawhide-
lacing, we find, corresponding to a belt-width of 12
inches,
H
= 1.1032.
v
Consequently ^ = 1.1932, v
or v = 15 feet per second.
Example. Given the data a = 135, $ = jfa inch,
H = 30, v 15, double leather-lacing, required the
belt-width. In this case
*=3?=2.
V IS
The table on page 113, column for double leather-lac-
ing, line 22, gives
= 2.0018,
and a corresponding belt-width of b = 22 inches.
II. Leather Belts over Leather-covered Pulleys.
As we have demonstrated in the foregoing pages,
the average leather belt will not transmit a force equal
to its strength, for the reason that it will slip upon its
pulley before it will break. If we can conveniently in-
crease the adhesion between the belt and pulley, i.e.,
increase the coefficient of friction, and in this way pre-
1 1 6 BEL TS A ND P ULLE YS.
vent slipping, the belt can be made to do more work
without increasing its size. Various methods have
been from time to time proposed for obtaining a greater
coefficient of friction, such as coating the pulley-faces
with gum, rosin, etc.; but these methods have more
often than otherwise proved useless, from the fact that
the belt is soon rendered stiff and clumsy by the sub-
stance placed upon the face of the pulley. Probably
the best of all contrivances in use for this purpose is
the pulley with a leather-covered face. The leather is
easily fastened securely upon the pulley, and we have
then practically a leather belt running over a leather
pulley. A series of carefully tried experiments has
given the coefficient of friction for leather belts over
leather-covered pulleys equal to 0.45-0.05 greater than
that for leather belts over cast-iron pulleys.*
If we substitute (p = 0.45 successively in formulas
(40), (41), and (42), and reduce, we shall obtain for
leather belts over leather-covered pulleys the follow-
ing expressions :
logy = 0.1953*; .... (238)
when a is expressed in circular measure,
logy =0.00341*; .... (239)
* Reuleaux says : " For a covering entirely new the value of is
between 6 and 7; after some service this value decreases, but still does
not become less than 4 to 5 ; the arc embraced by the belt being equal
to it. The smaller value, i.e., = 4, corresponds to 0.44 for the
coefficient of friction." See also Appendix I.
LEA THER- CO VERED PULLE YS.
when a is expressed in degrees,
log = l.22$a;
117
(240)
when a is expressed in fractions of the circumference.
The following table, calculated from the above
T
formulas, gives values of for different values of a
from 30 to 300. The arrangement is similar to that
of the table on page 88.
TABLE OF TENSIONS FOR LEATHER BELTS OVER LEATHER-COVERED
PULLEYS.
a =
T
In degree^.
I:i ircular
in asure.
In fractions of
circumference.
t
50
524
^=0.083
1.266
45
-735
i = 0.125
1.424
60
.047
= 0.167
1.601
75
309
^ = 0.208
1.802
90
571
4- = 0.250
2.027
105
833
& = 0.292
2.281
1 20
.004
i = 0.333
2.566
135
.356
1 = 0.375
2.886
150
165
.618
.880
A = 0.417
H = 0.458
3-247
3-653
1 80
3.142
4 = 0.500
4.110
195
3-403
M = 0.541
4-623-
2IO
2 4
3-665
4.189
-h = 0.583
t = 0.667
5-201
6.583
270
4.712
* = 0.750
8.331
300
5-236
1 = 0.833
12.655
By substituting the successive values of from the
above table in formula (48), we obtain the following
table, similar to the one on page 90 :
US
BELTS AND PULLEYS.
TABLE OF GREATEST TENSION FOR LEATHER BELTS OVER LEATHER-
COVERED PULLEYS.
a =
In degrees.
In circular
measure.
In fractions of
circumference.
T P X
30
0.524
A = - 8 3
4.76
45
0.785
0.125
3.36
60
1.047
i = 0.167
2.66
75
1.309
^r = 0.208
2.25
90
I-57I
= 0.250
97
105
1 20
1.833
2.094
A = 0-292
* = 0.333
79
.64
135
2-356
1 = 0.375
53
150
2.6l8
^ = 0.417
44
165
2.880
tt = 0.458
38
1 80
3.142
| = 0.500
32
195
3-403
if = 0.541
.28
2IO
3-665
A = 0.583
.24
240
4.189
t = 0.667
.18
270
4.712
f = 0.750
14
300
I = 0.833
.09
Example. A leather belt running over a leather-
covered pulley transmits a force of 500 pounds. It is
required to determine the greatest tension on the belt,
assuming that the belt embraces | the circumference
of the pulley. From the table we find, corresponding
to a = | the circumference,
or
T= P X 1.18 = 500 X 1.18,
T = 590 pounds.
Example. The greatest tension on a leather belt,
running over a leather-covered pulley and embracing
the circumference, is T = 792 pounds. Required the
force in pounds which it can transmit. The table gives
T= PX 1-32,
LEA THER- CO VERED PULLE YS.
as the greatest tension corresponding to a =
cumference. Hence
the cir-
792 = P X 1.32,
P = 600 pounds.
P =
792.
1.32'
By substituting the values of Ttrom the above table
successively in formulas (51), (52), (53), (54), and (55),
the following tables of formulas have been obtained.
The application of ;hese formulas will be easily un-
derstood from the explanation of the similar tables on
pages 96-98.
TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER COVERED
PULLEYS.
Single Leaf her- Lacing.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
-h = - 8 3
bS = 0.014647'
241
45
0.785
| = 0.125
bS = 0.010347*
242
60
1.047
i = 0.167
b*> = o.oo8i87 >
2-13
75
1.309
.^j = 0.208
68 = 0.006927'
244
90
I-57I
J = 0.250
bS = o. 006067*
245
105
I-833
A = o- 2 92
bS = 0.005517*
246
120
2.094
& = 0.333
bS = 0.005037'
247
135
2.356
f = o.375
5 = 0.004717*
248
ISO
2.618
2.880
S= 0.417
= 0.458
5 = 0.004437*
l>8 = 0.004247*
249
250
1 80
3.143
i = 0.500
^5 = 0.004067*
251
195
3-403
? = 0.541
/>8 0.003947*
252
210
3-665
= 0.583
/55 = 0.003827*
253
240
4.189
0.667
68 = 0.003637*
254
2 7
4.712
f = 0.750
M = O.0035I/'
255
300
5-236
1 = 0.833
(55 = 0.003357*
256
120
BELTS AND PULLEYS.
TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-COVERED
PULLEYS.
Single Rawhide-Lacing.
degrees.
measure.
circumference.
Formula.
No.
30
0.524
T V = 0.083
l>8 = 0.013607*
257
45
0.785
i = 0.125
68 = 0.009607*
258
60
1.047
= 0.167
l>d = 0.007607*
259
75
1.309
& = .208
b8 0.006437*
260
90
I-57I
i : .250
/><5 = 0.005637*
261
105
1-833
A - 292
l>8 = 0.00511 7*
262
120
2.094
*= -333
l>8 = 0.004697'
263
135
2.356
f= -375
l>8 0.004377*
264
150
2.618
T5 = -417
b8 = 0.004117*
265
165
2.880
li= -458
b8 = 0.003947*
266
i So
3-142
i = 0.500
/>5 = 0.003777*
267
195
3-403
M = 0.541
5 = 0.003667'
268
2IO
3-665
A = 0-583
M = 0.003547*
269
240
4.189
1 = 0.667
5 = 0.003377*
270
270
4.712
i = 0.750
b8 = 0.003267*
271
300
1 = 0.833
M = 0.003117*
272
TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-COVERED
PULLEYS.
Double Leather-Letting.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
45
0.524
0.785
T* = 0.083
i = 0.125
b8 = 0.012697*
b8 = 0.008967*
2/3
274
60
1.047
i = 0.167
b8 = 0.007097*
275
75
1.309
-ff = O.2O8
M = o oo6oo7*
276
I-57I
J = 0.250
6d = 0.005257*
277
105
1-833
& = 0.292
bS = 0.004777*
278
120
2.094
i = 0.333
l>8 = 0.004377*
279
135
2.356
I = 0.375
b8 0.004087*
280
150
2.618
! = 0.417
b8 = 0.003847*
281
165
2.880
= 0.458
bS = 0.003687*
282
180
3-142
= 0.500
b8 = 0.003527*
283
195
3-403
= 0.541
bS = 0.003417*
284
2IO
3-665
= 0.583
bS = 0.003317*
285
240
4.189
= 0.667
68 .= 0.003157*
286
270
4.712
f = 0.750
/;5 = 0.003047*
287
300
5.236
1 = 0.833
b8 = 0.002917*
288
LEA THER- CO VERED P ULLE YS.
121
TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-COVERED
PULLEYS.
Double Raw hide -Lacing.
a in
degrees.
a circular
easure.
a in fractions of
circumference.
Formula.
No.
30
.524
T L = 0.083
35 = o.ongoT*
289
45
-785
i = 0.125
35 = 0.008407*
2 go
60
047
i = 0.167
35 = 0.006657*
2gi
75
.309
ff = 0.208
35 = 0.005637*
292
90
571
i = 0.250
35 = 0.004937*
293
105
-833
^ = 0.292
35 = 0.004487*
294
120
.094
* = 0.333
35 = 0.004107*
295
135
-356
1 = 0.375
35 = 0.003837*
296
ISO
.618
35 = 0.003607*
297
165
.880
fl = o.458
35 = 0.003457*
298
1 80
3.142
i = 0.500
35 = 0.003307'
299
195
3-403
M = 0.541
35 = 0.003207*
300
2IO
3-665
A = 0-583
35 = 0.003107*
301
24O
4.189
t = 0.667
35 = 0.002957*
302
270
4.712
t = 0.750
35 = 0.092857*
303
300
I = 0.833
35 = 0.002737*
304
TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-COVERED
PULLEYS.
Riveted Joint.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
A = 0.083
35 = 0.008287'
305
45
0.785
i = 0.125
35 = 0.005847'
306
60
1.047
i- = 0.167
35 = 0.004637*
307
75
1.309
-f = 0.208
35 = 0.003917*
308
go
I-57I
i = 0.250
35 = 0.003437*
309
105
1.833
^ = 0.291
35 = 0.003117*
120
2.0 9 4
1 = 0.333
35 = 0.002857*
311
135
2.356
1 = 0.375
35 = 0.002667'
312
165
2.618
2.880
A = 0.417
ti = 0.458
35 = 0.002507*
35 = 0.002407*
313
314
180
3.142
i = 0.500
35 = 0.002297*
315
195
3-403
M = 0.541
35 = O.OO2227*
2IO
3-665
A = 0-583
35 = 0.002167*
317
240
4.189
1 = 0.667
35 = 0.002057'
270
4.712
t = 0.750
35 = o.ooigST*
319
300
5-236
I = 0.833
35 = o.ooigoT*
320
122 BELTS AND PULLEYS.
Example. A leather belt ^ inch thick, running over a
leather-covered pulley, transmits a force of 500 pounds.
Required the width of the belt for single leather-lacing
and single rawhide-lacing, taking a = 45. From
formula (242) we have
b X i = 0.01034 X 500, b = 0.01034 X 500 X 4,
or, for single leather-lacing,
b = 20.68" = 20fJ-" nearly.
From formula (258) we have
b X i = 0.00960 X 500, b = 0.00960 X 500 X 4,
or, for single rawhide-lacing,
b = 19.20" = I9ff.*
Example. With the data a = 1.833, circular meas-
ure, d = ^ inch, and b = 20 inches, required the forces
in pounds which the belt can transmit for each of the
* If we take the above data, P = 500, a = 45, <5 = J inch, and cal-
culate the width of a leather belt running over a cast-iron pulley, we
shall have, from formula (57), for single leather-lacing, l> = 0.01142
X 500 X 4 = 22.84 inches. The difference between the widths of the
belt necessary for transmission over cast-iron and leather-covered pul-
leys is therefore 22. 84 20.68 = 2.16 inches, which shows a gain for the
leather-covered pulley of nearly 10 per cent over the cast-iron pulley.
LEATHER-COVERED PULLEYS. 12$
above methods of joint-fastening, supposing the belt
to run over a leather covered pulley.
From formula (246) we have
or, for single leather-lacing,
P= 907.44 pounds.
From formula (262),
20 X 0.25
20 X = 0.005 I IP, P =
or, for single rawhide-lacing,
P= 978.47 pounds.
From formula (278),
or, for double leather-lacing,
P= 1048.22 pounds.
124 &EL7"S AND PULLEYS.
From formula (294),
20 x i = 0.00448^, P =
or, for double rawhide-lacing,
P= 1116.07 pounds.
From formula (310),
or, for a riveted joint,
P= 1607.71 pounds.
The formulas of the following tables, obtained by
TT
substituting P= 550-- in formulas (241) to (320), and
similar to the formulas on pages 100-104, will prove
convenient in calculating widths of leather belts over
leather-covered pulleys from the horse-power trans-
mitted and the velocity in feet per second :
LEA THEK- CO VERED P ULLE VS.
125
TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-COVERED
PULLEYS.
Sinyle Leather Lacing.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
A = 0.083
b8 = 8.052^
321
V
if
45
0.785
i = 0.125
** = 5.687-
322
60
1.047
| = 0.167
bS = 4-499-
323
H
75
1.309
ff = O.2O8
b8 = 3.806-
324
H
90
I-57I
i = 0.250
45 = 3.333-
325
H
105
1-833
-jlf O.292
b8 = 3.031
326
H
I2O
2.094
i = 0.333
b8 = 2.767^
327
135
2-356
t = 0.375
bS = 2.591
328
H
150
2.618
T <V = 0.417
bS = 2.437-
329
165
2.880
ii- = 0.458
b8 = 2.332^
330
180
3.I42
1 = 0.500
//
M = 2.233-
331
2/
195
3.403
tt = 0.541
M = 2.167-
332
2IO
3-665
A = 0-583
ff
bS 2.101
Z/
333
240
4.189
t = 0.667
35 = 1.997^
334
H
270
4.712
f = 0.750
b8 = 1.931-
335
z/
300
5.236
f = 0.833
77
35 = 1.843-
336
126
BELTS AND PULLEYS.
TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-COVERED
PULLEYS.
Single Ra'ahide Lacing.
a in
a in circular
a in fractions of
Formula.
No.
ff
30
0.524
T V = 0.083
b8 = 7.480^-
337
ff
45
0.785
i = 0.125
l>8 = 5.280-
338
H
60
1.047
= 0.167
339
75
1.309
^ = 0.208
35 = 3.537^
340
H
90
I.57I
i = 0.250
^ = 3-097-
34i
105
I-833
-t = 0.292
//
b = 2.811
342
120
2.094
* = 0.333
ff
bS = 2.580
343
H
135
2.356
1 = 0.375
^ = 2.404-
344
150
2.618
1% = 0.417
TT
bS 2.261
345
I6 5
2.880
tt= 0-458
. H
oo = 2.167
346
H
1 80
3-I42
I = 0.500
bS = 2.074-
347
195
3.403
if = 0.541
H
bo = 2.013
348
2IO
3-665
A = 0.583
ff
bS = 1.947-
349
240
4.189
= 0.667
TT
bS = 1.854
350
H
270
4.712
f = 0.750
bS = i . 793 -
35^
V
ff
300
5.236
1 = 0.833
bS = 1.701-
V
352
LEA THER- CO VERED P U&IE YS.
TABLE OF FORMULAS
FOR LEATHER BEITS OVER LEATHER-
PULLEYS.
Double Leather- Lacing.
a in
degrees.
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
rV 0.083
35 = 6.980-
353
H
45
0.785
i = 0.125
354
60
1.047
i = 0.167
H
35 = 3-900-
355
H
75
1.309
2 5 = O.2O8
35 = 3.300-
356
90
I-57I
= 0.250
35 = 2.888-
357
H
105
I-833
g f = o 292
35 = 2.624
358
I2O
2.094
i = 0.333
35 = 2.404-
359
//
135
2.356
1 = 0.375
35 = 2.244-
360
150
2.618
^ = 0.417
35 = 2.112
V
361
165
2.880
ft = 0.458
H
3d = 2.024
362
180
3-142
= 0.500
H
36 = 1.936
363
195
3-403
if = 0.541
H
35 = 1.876-.
364
H
210
3-665
A = 0.583
35 = i. 821-
V
365
240
4.189
1 = 0.667
H
^5 = 1.733-
366
270
4.712
I = 0.750
35 = 1.672-
367
300
5-236
I = 0.833
35 = 1.601^
368
128
BELTS AND PULLEYS.
TABLE OF FORMULAS FOR LEATHER-BELTS OVER LEATHER-COVERED
PULLEYS.
Double Raivhidc- Lacing.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
rV = 083
b8 =6.545^
369
45
. 0.785
^ = O.I25
bS = 4.620-
370
60
1.047
i = 0.167
b8 = 3-658^
371
75
1.309
-fa = 0.208
bS -. 3.097^
372
9
I-57I
i = 0.250
a
bo = 2.712
373
If
105
I-833
A = 0.292
bS = 2.464-
374
H
120
2.094
i = 0.333
bo = 2.255-
375
H
135
2.356
1 = 0.375
^5 = 2.107-
376
150
2.618
rV = o.4i7
H
35 = i. 9 8o-
377
165
2.880
M = 0-458
If
bo = i . 898 -
378
V
1 80
3-142
- = 0.500
If
35 = z.8 I5 -
379
H
195
3-403
H = 0.541
bd = 1.760
' z/
380
H
2IO
3-665
A = 0.583
bo = 1.705-
381
240
4.189
I = 0.667
TT
b8 = 1.623
382
H
270
4.712
f = 0.750
bo - 1.568-
383
H
300
5.236
* = 0.833
bo = 1.502
384
LEA THER- CO VERED P ULLE YS.
129
TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-COVERED
PULLEYS.
Riveted Joints.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
ff
30
0.524
A = 0.083
bS = 4.554-
385
V
ff
45
0.785
i = 0.125
bS = 3-212-
386
60
I '.047
i = 0.167
68 = 2.547*
387
75
1.309
^ = 0.208
bS = 2.151-
388
V
H
90
I-57I
i = 0.250
bS = 1-887^
389
105
1-833
& = 0.292
bS = 1-7"^
390
I2O
2.094
* = 0.333
ff
bS = 1.568-
391
V
ff
135
2.356
t = 0.375
bS = 1.463-
V
392
H
150
2.618
TS = 0.417
5 = 1.375-
393
H
165
2.880
H = 0.458
bS = 1.320
394
180
3 142
-| = O.5OO
bS = 1.260^
V
395
195
3-403
H = 0.541
bS = 1.221^
v
396
ff
2IO
3-665
A = 0.583
bS = 1.188-
V
397
ff
2 4
4.189
f = 0.667
bS = 1.128-
V
398
270
4.712
4 = 0.750
if
68 = 1.089-
399
300
5-236
1 = 0.833
bS = 1.045^
400
I3O BELTS AXD PULLEYS.
Example. A leather belt \ inch thick, running over
a leather-covered pulley, transmits a force of 20 horse-
power at a velocity of 15 feet per second. Required
the width of the belt for single and double rawhide-
lacing, assuming that the belt embraces an arc of the
pulley equal to 2.880 circular measure. From formula
(346) we have
b X i = 2.167 X ~, b = 2.167 X ^y X 4,
or, for single rawhide-lacing,
b= n.557" = HyV-
Formula (378) gives
b X i = 1.898 X ~ b= 1.898 X ~ X 4,
or, for double rawhide-lacing,
b 10.123" = 10$*.
Example. A leather belt running over leather-cov-
ered pulleys is % inch thick and 12 inches wide.
Required the velocity at which the belt can transmit a
force of 10 horse-power, assuming a = 45, and that
the belt has a double leather-lacing. We have from
formula (354)
10 4.928 X 10
.2 x A = 4.928 x -, 1 8 x A '
or v = 18.77 ft- P er second.
LEATHER-COVERED PULLEYS. 13!
Example. A leather belt (with a riveted joint)
running over leather-covered pulleys is 16 inches wide
and T 3 inch thick ; the arc embraced by the belt on
the smaller pulley is 150, and the velocity of the belt
IO feet per second. It is required to determine the
horse-power which can be transmitted by the belt.
From formula (393) we have
l6xA=I . 375X f, /r.-i
or H = 21.82.
By substituting d = ^\ in formulas (251), (267),
(283), (299), (315), (331), (347), (363), (379), and (395),
successively, we obtain the following formulas :
When a = 180 and d = J/',
Single leather-lacing, b = o.oi86/>. .... (401)
Single rawhide-lacing, b = o.oi72/ > . .... (402)
Double leather-lacing, b = o.oi6iP. . . . . (403)
Double rawhide-lacing, b = o.oi$iP. .... (404)
Riveted joint, b = o.oio$P. .... (405)
T_r
Single leather-lacing, b = 10.208 ..... (406)
TT
Single rawhide-lacing, b 9.481 (47)
132 BELTS AND PULLEYS.
IT
Double leather-lacing, b 8.850 (408)
IT
Double rawhide-lacing, b = 8.297 (409)
IT
Riveted joint, b = 5.760 (4 IQ )
By substituting 6 = ^ in formulas (248), (264), (280),
(296), (3 1 2), (328), (344), (360), (376), and (392), succes-
sively, the following formulas may be obtained
When a = 135 and 3 = y,
Single leather-lacing, b = 0.0215/1 .... (411)
Single rawhide-lacing, b = O.O2OOP. . . . . (412)
Double leather-lacing, b 0.0187/1 . . . . (413)
Double rawhide-lacing, b = 0.0175/1 .... (414)
Riveted joint, b = O.OI22/*. .... (415)
TT
Single leather-lacing, b= 11.845 (416)
TT
Single rawhide-lacing, b = 10.990 (417)
TT
Double leather-lacing, b = 10.258 (418)
LEATHER-COVERED PULLEYS. 133
TT
Double rawhide-lacing, b =. 9.632 . . . . (419)
TT
Riveted joint, b 6.6887-. . . . (420)
Example. A leather belt, running over leather-cov-
ered pulleys, transmits a force of 600 pounds. The
pulleys are of equal diameters (a = 180) and the thick-
ness of the belt is -^ inch. Required the width of
the belt for double leather-lacing. We have from
formula (403)
b 0.0161 X 600,
b = 9.66" = 9fl*.
Example. A -^-inch leather belt, running over two
equal leather-covered pulleys, transmits a force of 15
horse-power at a velocity of 10 feet per second.
Required the width of the belt for a riveted joint.
Formula (410) gives
b = 5-760 x
b = 8.64^ = 8|".
Example. A 7 -inch leather belt (double rawhide-
lacing), running over leather-covered pulleys, transmits
a force of 600 pounds. The arc embraced by the belt
on the smaller pulley is 135. Required the width of
the belt. From formula (414) we have
b = 0.0175 X 600,
b = 10.50" = loj".
134 BELTS AND PULLEYS.
Example. A leather belt -% inch thick, running over
leather-covered pulleys, transmits a force of 15 horse-
power at a velocity of 10 feet per second. It is re-
quired to determine the width of the belt, for single
leather-lacing, taking a = 135. Formula (416) gives
b= 11.845 X jo'
b = 17-77" = i7tf"
Example. A leather belt ^ inch thick and 20 inches
wide, running over leather-covered pulleys, transmits a
force of 20 horse-power. The arc embraced by the
belt on the smaller pulley is 135. It is required to de-
termine the velocity at which the belt can be driven for
double rawhide-lacing. We have from formula (419)
20 9-632 X 20
20 = 9.632 X -. = *-*^ -- .
or v = 9.632 = 9^ feet per second.
The following tables, calculated from formulas (401)
to (420), give the forces in pounds (P)and the values of
the horse-power divided by the velocity in feet per
(ff\
J corresponding to different widths (from i
inch to 30 inches) of ^V-inch leather belts running over
leather-covered pulleys for a = 180 and a 135 for
each of the five methods of joint-fastening given above:
LEA THER-CO VERED PULLE YS.
135
TABLE OF WIDTHS OF LEATHER BELTS OVER LEATHER-COVERED
PULLEYS, WHEN a 180 AND S = ^". From Formulas (401)-
(405)-
Width
inches.
P, single
leather-
lacing.
P. single
rawhide-
lacing.
P, double
leather-
lacing.
P, double
rawhide-
lacing.
P, riveted
joint.
No.
I
53-38
58.04
62.15
66.27
95-51
I
I*
80.82
87.06
93-23
99-40
143 27
2
2
107.76
116.08
124.30
132.54
igl.02
3
2j
I34-70
145-10
155.38
165.67
238.78
4
3
161.64
174.11
186.45
198.81
286.53
5
3*
188.58
203.13
217-53
23I-94
334-29
6
4
215-52
232.15
248.60
265.08
382.04
7
4i
242.46
261.17
279.68
298.21
429.80
8
5
269.40
290.19
310.75
331-35
477.56
9
5i
296.34
3i9- 21
34I-83
364.48
525-3I
10
6
323-28
348.23
372.90
397-61
573-07
ii
7
377-16
406.27
435-05
463.88
668.58
12
8
431-03
464-31
497 - 20
530.15
764.09
13
9
484.91
522.34
559-35
596.42
859.60
14
10
538.79
580.38
621.50
662.69
955-H
15
ii
592.67
638.42
683.65
728.96
1050.62
16
12
646-55
696.46
745.80
795-23
1146.13
17
14
754-31
812.54
870.10
927.77
I337.I5
18
16
862.07
928.61
994.40
1060.31
1528.18
19
18
969-83
1044.69
1118.71
1192.84
1719.20
20
20
1077.59
1160.77
1243.01
I325-38
1910.22
21
22
H85-34
1276.84
1367.31
1457.92
2101.24
22
24
1293.10
1392-92
1491.61
1590.46
2292.26
23
26
1400.86
1509.00
1615.91
1723.00
2483.29
24
28
1508.62
1625.07
1740 21
1825.53
2674.31
25
30
1616.38
I74LI5
1864.51
1988.07
2865.33
26
136
BELTS AND PULLEYS.
TABLE OF WIDTHS OF LEATHER BELTS OVER LEATHER-COVERED
PULLEYS, WHEN a = 180 and 3 = ^". From Formulas (406)-
(410).
Width
in
inches.
" single
leather-
lacing.
f , single
rawhide-
lacing.
-, double
v
leather-
lacing.
" t double
rawhide-
lacing.
, riveted
joint.
No.
I
o . 0980
0.1055
0.1130
0.1205
0.1736
I
ii
.1469
0.1582
0.1695
0.1808
0.2604
2
2
1959
0.2109
O.226O
0.2410
0.3472
3
2i
.2449
0.2637
0.2825
0.3013
0.4340
4
3
-2939
0.3164
339
0.3616
0.5208
5
3*
3429
0.3692
3955
0.4218
0.6076
6
4
.3918
0.4219
.4520
0.4821
0.6944
7
4k
.4408
0.4746
5085
0.5-424
0.7812
8
5
.4898
0.5274
5650
0.6026
.8681
9
5i
-5388
0.5801
.6214
0.6629
-9549
10
6
.5878
0.6328
.6779
0.7231
.0417
ii
7
.6857
0.7383
.7909
0.8437
2153
12
8
7837
0.8438
9039
0.9642
.3889
13
9
.8817
0.9493
.0169
1.0847
.5625
14
10
.9796
1-0547
.1299
1.2052
.7361
15
ii
.0776
I . l6O2
2429
1-3258
.9097
16
12
1755
1.2657
3559
1.4463
-0833
17
14
3715
1.4766
.5819
1.6873
.4306
18
16
5674
1.6876
.8078
1.9284
7778
19
18
.7633
1.8985
0338
2.1694
3.1250
20
20
9592
2.1095
.2598
2.4105
3-4722
21
22
-1552
2.3204
4858
2.6515
3-8194
22
24
35"
2.5314
.7118
2.8926
4.1667
23
26
5470
2.7423
9377
3.I336
4-5I39
24
28
.7429
2-9532
3-1637
3-3747
4.8611
25
30
2.9389
3-1642
3.3897
3-6I57
5-2083
26
LEA THER- CO VERED P ULLE YS.
137
TABLE OF WIDTHS OF LEATHER BELTS OVER LEATHER-COVERED
PULLEYS, WHEN a = 135 AND 5 = -fa". From Formulas (411)-
(415).
Width
in
inches.
.P, single
leather-
lacing.
P, single
rawhide-
lacing.
P, double
leather-
lacing.
P, double
rawhide-
lacing.
P, riveted
joint.
No.
I
46.45
50.05
53.62
57."
82.24
!
I*
69.67
75.08
80.43
85.67
123.36
2
2
92.89
100.10
107.24
114.22
164.47
3
2*
116.12
125.13
I34.05
142.78
205.59
4
3
139-34
150.15
160.86
I7I-33
246.71
5
31
162.56
175.18
187.67
199.89
287.83
6
4
185.79
200.20
214.48
228.44
32895
7
4t
209.01
225.23
241 . 29
257-00
370.07
8
5
232.23
250.25
268.10
285.55
411.18
9
5*
255.46
275.28
294.91
314.11
452.30
10
6
278.68
300.30
321.72
342.66
493.42
ii
7
325.13
350.35
375-34
399-77
575-66
12
8
371-57
400.40
428.95
456.88
657.89
13
9
418.02
450.45
482.57
513.99
740.13
14
10
464.47
500.50
536.19
571.10
822.37
15
ii
510.91
550.55
589-81
628.21
904.60
16
12
557.36
600.60
643.43
685.32
986.84
17
14
650.26
700.70
750.67
799-54
1151-32
18
16
743-15
800.80
857.91
913.76
I3I5.79
19
18
836.04
900.90
965-15
1027.98
1480.26
20
20
928.94
IOOI.OO
1072.39
1142.20
1644.74
21
22
1021.83
IIOI.IO
1179.62
1256.42
1809.21
22
24
1114.72
I2OI.2O
1286.86
1370.64
1973.68
23
26
1207.62
1301.30
1394.10
1484.87
2138.16
24
28
1300.51
I4OI . 40
1501.34
1599.09
2302.63
25
30
1393.40
1501.50
1608.58
I7I3-3I
2467 . 10
26
138
BELTS AND PULLEYS.
TABLE OF WIDTHS OF LEATHER BELTS OVER LEATHER COVERED
PULLEYS, WHEN a = 135 AND d = -fa". From Formulas (416)-
(420).
Width
inches.
f, single
leather-
lacing.
p single
rawhide-
lacing.
, double
leather-
lacing.
, double
rawhide-
lacing.
, riveted
V
joint.
No.
I
0.0844
0.0910
0.0975
0.1038
0.1495
I
Ii
0.1266
0.1365
0.1462
0-1557
0.2243
2
2
0.1689
0.1820
0.1950
0.2076
0.2990
3
a*
O.2III
0.2275
0-2437
0.2596
0.3738
4
3
0.2533
0.2730
0.2924
0.3"5
0.4486
5
3i
0.2955
0.3185
0.3412
0.3634
0.5233
6
4
0.3377
0.3640
0.3899
0.4153
0.5981
7
4*
0.3799
0.4095
0.4387
0.4672
0.6728
8
5
0.4221
0.4550
0.4874
0.5191
7476
9
5i
0.4643
0.5005
0.5362
0-5710
.8224
10
6
0.5066
0.5460
0.5849
0.6229
.8971
ii
7
0.5910
0.6370
0.6824
0.7267
.0466
12
8
0.6754
0.7280
0-7799
0.8306
.1962
13
9
0.7598
0.8189
0.8773
0.9344
3457
14
1C
0.8443
0.9099
.9748
.0382
4952
15
ii
0.9287
.0009
.0723
.1420
.6447
16
12
I.OI3I
.0919
.1698
.2458
.7942
17
14
I.I820
2739
-3f>47
4535
0933
18
16
1.3508
4559
5597
.6611
3923
19
18
I-5I97
.6380
7547
.8688
.6914
20
20
1.6885
.8199
.9496
.0764
.9904
21
22
1-8574
.0019
.1446
.2840
3.2894
22
24
2.0262
.1839
-3396
.4917
3-5885
23
26
2.195[
3658
5345
6993
3-8875
24
28
2 . 3640
5478
7295
.9070
4.1866
25
30
2.5328
.7298
9245
3-1146
4-4856
26
Example. Required the force in pounds which can
be transmitted by a ^inch leather belt, 20 inches wide,
running over two equal leather-covered pulleys, the
belt-joint being riveted. From the table on page 135,
column for riveted joint, line 21, we have
P = 1910,22 pounds,
LEATHER-COVERED PULLEYS. 139
Example. A^V-inch leather belt running over leather-
covered pulleys, and embracing an arc of 135 on the
smaller pulley, transmits a force of 1000 pounds. It is
required to determine the proper width for the belt for
single rawhide-lacing. The table on page 137, column
for single rawhide-lacing, line 21, gives, corresponding
to P = 1001.10 pounds, a width of
Example. A 7 ^-inch leather belt, running over two
equal leather-covered pulleys at a velocity of 10 feet
per second, transmits a force of 22 horse-power. Re-
quired the width for the belt for a double leather-lac-
H 22
ing. We have in this case = 2.2, and the
v 10
table on page 136, column for double leather-lacing, line
J_T
21, gives for 2.2598 a belt-width of
v
b = 20"
Example. A leather belt ^ inch thick and 28 inches
wide, running over leather-covered pulleys and embrac-
ing an arc of 135 on the smaller pulley, transmits a
force of 25 horse-power. It is required to determine
the velocity at which the belt can be driven for a
double rawhide-lacing. From the table on page 138,
column for double rawhide-lacing, line 25, we have
H 25 25
= - = 2.9070, v = - ,
v v 2.9070
I4O BELTS AND PULLEYS.
or v =. 8.60 feet per second.
Example. A leather belt -^ inch thick and 28 inches
wide, running over leather-covered pulleys and embrac-
ing an arc of 135 on the smaller pulley, is driven at a
velocity of 8.60 feet per second. It is required to de-
termine the horse-power which can be transmitted by
the belt, the joint-fastening being a double rawhide-lac-
ing. From the table on page 138, column for double
rawhide-lacing, line 25, we have
TT TT
H = 8>6 X 2 ' 9 70 '
or H 25.
12. Vulcanized-rubber Belts.
Vulcan ized-rubber belts are usually made, as ex-
plained in 8, by placing one or more layers of cotton
duck between layers of vulcanized rubber. The num-
ber of these layers is indicated by the term ply : thus
a one-ply belt contains one layer of duck, a three-ply
belt contains three layers, etc. The thickness of each
layer of duck varies more or less according to the
amount of material and the force with which the lay-
ers are pressed together in the manufacture. We may,
however, with sufficient correctness for ordinary pur-
poses, take for the average thickness of a ply -^ inch.
A three-ply belt is therefore approximately ^ inch thick,
a four-ply belt f inch thick, etc.
VULCANIZED-RUBBER BELTS. \\\
The strength of vulcanized-rubber belting seems to
be about that of leather of the same thickness. A
series of tests made for the author by Messrs. Fair-
banks & Co., on their standard testing-machine, gave
for superior new vulcanized-rubber belting an average
strength of nearly 4000 pounds per square inch of sec-
tion. A great number of other tests made by the
author on ordinary vulcanized-rubber belts which had
been in practical use for a short time gave results es-
sentially the same as for leather.
We shall therefore use for the safe-working stress in
pounds per square inch for vulcanized-rubber belting
the following values, given in 10:
Single leather-lacing, / = 325 ;
Single rawhide-lacing, f=^o;
Double leather-lacing, / = 375 ;
Double rawhide-lacing, f = 400 ;
Riveted joint, / = 575.
The coefficient of friction for vulcanized rubber over
cast-iron seems to be slightly greater than for leather
over leather-covered pulleys.* Since, however, rubber
belts are very seriously injured by slipping about their
pulleys, and for this reason greater care should be taken
to prevent slipping, we propose to neglect the ap-
parently small difference and take the coefficient equal
* See Appendix I.
142
BELTS AND PULLEYS.
to that for leather over leather-covered pulleys. We
have then
cp = 0.45.
The formulas for widths of vulcanized-rubber belts
over cast-iron pulleys may be copied directly from
those for leather belts over leather-covered pulley^
without the trouble of copying the preliminary tables
and formulas.
TABLE OF FORMULAS FOR VULCANIZKD-RUBBER BELTS OVER CAST-
IRON PULLEYS.
Single Leather-Lacing.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
^ = 0.083
5 = 0.014647*
421
45
0.785
i = 0.125
68 = 0.010347*
422
60
1.047
= 0.167
68 = o.ooSiST*
423
75
90
1.309
I-57I
ff = O.2O8
i = 0.250
68 = 0.006927*
68 = 0.006067*
424
425
105
120
135
1-833
2.094
2.356
A = 0-292
t = 0.333
1 = 0.375
68 = 0.005037*
68 = 0.004717*
426
427
428
150
2.618
4 = 0.417
68 = 0.004437*
429
165
2.880
= 0.458
68 = 0.004247*
430
ISO
3-I42
= o 500
68 = 0.004067*
431
195
3-403
if = 0.541
68 = 0.003947*
432
2IO
240
3-665
4.189
A = 0-583
t = 0.667
68 = 0.003827*
68 = 0.003637*
433
434
2 7
300
4.712
5-236
f = 0.750
1 = 0.833
68 = 0.003517*
5 = 0.003357*
435
436
VUL CANIZED-R UBBER BEL TS.
143
TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER CAST-
IRON PULLEYS.
Single Rawhide Lacing.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
A =0.083
68 = 0.013607*
437
45
0.785
i = 0.125
68 = 0.009607*
438
60
.047
\ = 0.167
68 = o. 00760 P
439
75
309
^- = 0.208
68 = o 006437*
440
90
571
i = 0.250
bS 0.005637'
441
105
120
.833
.094
A = 0.292
4 = 0.333
68 = 0.005 1 1/ 5
6$ o. 004697*
442
443
135
.356
1 = 0.375
68 = o. 004377*
444
150
165
1 80
.618
.880
3-142
4= 0.417
= 0.458
= 0.500
68 = 0.004117*
63 = 0.003947*
65 = 0.003777"
445
446
447
195
210
3-403
3-665
M = 0.541
A = 0.583
bS = o 003667*
65 = 0.003547*
448
449
240
4.189
f = 0.667
b8 o. 003377*
450
270
4.712
f = 0.750
68 = 0.003267'
451
300
5-236
I = 0.833
b8 = O.OO3H7 1
452
TABLE OF FORMULAS FOR VULCANIZED RUBBER BELTS OVER CAST-
IRON PULLEYS.
Dottble Leather- Lacing.
a. in
degrees.
a in circular
measure.
o in fractions of
circumference.
Formula.
No.
30
0.524
T V = 0.083
l>d = 0.012697*
453
45
0.785
i = 0.125
68 = o. oo8 9 67*
454
60
1.047
= 0.167
b8 = 0.007097*
455
75
1.309
ff = 0.208
b8 = o.oo6oo7*
456
90
I-57I
J = 0.250
bS = 0.005257*
457
105
1-833
& = 0.292
bS = 0.004777*
458
120
2.094
i = 0.333
5 = 0.004377*
459
135
2.356
1 = 0.375
68 = 0.004087*
460
150
2 618
fV = 0.417
bS = 0.003847*
461
165
2.880
tt= 0.458
68 = 0.003687*
462
1 80
3-I42
i = 0.500
bS = O.O03527"
463
195
210
3.403
3-665
H = 0.541
A = 0.583
68 = 0.003417*
b8 = 0.0033 1 P
464
465
240
4.189
* = 0.667
b8 = 0.003157*
466
270
4.712
= 0.750
b8 = 0.003047*
467
300
5-236
1 = 0.833
b8 = 0.002917*
468
144
BELTS AND PULLEYS.
TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER CAST-
IRON PULLEYS.
Double Rawhide-Latin?.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
fa = 0.083
bd = o.ongoP
469
45
0.785
i = 0.125
bd = o.oofyoP
470
60
1.047
= 0.167
68 = 0.00665^
4/1
75
1.309
ff = 0.208
bd = o. 005637*
4/2
90
I-57I
J = 0.250
bd = O.O0493/*
473
105
1.833
v\ = O.292
bd = 0.0044&P
474
120
2.094
"i = 0.333
bd = o.oojioP
475
135
2.356
f = 0.375
bd = o.oo383/>
476
150
2.618
^ = 0.417
bd = o. 003607'
477
2.880
M = 0-458
bd = 0.00345/ 5
478
180
3.142
4 = 0.500
35 = O.O0330/'
479
T 95
3.403
it = 0.541
35 = O.O032O/'
480
2IO
3-665
& = 0.583
bd = o.oo^ioP
481
240
4.189
3- = 0.667
bd = O.OO295/*
482
270
4.712
f = 0.750
*5 = 0.00285/ 1
483
300
5.236
1 = 0.833
35 = O.OO273/ 1
484
TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER CAST-
IRON PULLEYS.
Riveted Joints.
a in
degrees.
o in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
45
0.524
0.785
TV = 0.083
i = 0.125
35 = o.Oo828/>
35 = 0.00584/*
485
486
60
75
90
1.047
1.309
I-57I
i = 0.167
^ = 0.208
i = 0.250
35 = O.O0463/ 7
35 = O.O039I/*
35 = o. 00343^
487
488
489
105
1.833
A = 0.291
35 = O.OO3H/"
490
1 20
2.094
i = 0.333
35 = O.OO285/'
491
135
2.356
$ = 0.375
35 = O.OO266/'
492
150
165
2.618
2.880
A = 0.417
tt = -458
35 = O.OO25O/ 1
35 = O.OO24O/ 3
493
494
180
3-I42
i = 0.500
35 = 0.00229/ 3
495
195
3-403
if = 0.541
35 = O.OO222P
496
2IO
3-665
A = 0-583
bd = o.oo2i6P
497
24O
4.189
f = 0.667
bd = O.OO2O5-P
498
270
300
4-712
5-236
J = 0.750
f = 0.833
35 = o.ooigSP
bd = o.ooigoP
499
500
V UL CA NIZED-R UBBER BEL TS.
TABLE OF FORMULAS
FOR VULCANIZED-RUBBER BELTS OVER
IRON PULLEYS.
Single Leather-Lacing.
CAST-
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
& = 0.083
b8 = 8.052-
501
45
0.785
1 = 0.125
bd = 5.687-
502
H
60
1.047
^ = 0.167
bS = 4.499-
503
75
1.309
ft = 0.208
bS = 3.806^
504
H
90
I-57I
i = 0.250
3-333^
505
H
105
1-833
g f = 0.292
l>8 = 3.031
506
H
120
2.094
i = 0.333
dS = 2.767
507
135
2.356
I = 0.375
bS = 2.591
508
150
2.618
& = 0.417
H
bS = 2.437-
509
165
2.880
ii = 0.458
bS = 2.332-
510
V
180
3.142
i = 0.500
1>S = 2.233^
511
V
195
3-403
if = 0.541
bS = 2.167^
512
2IO
3-665
A = 0.583
H
1/8 = 2.101
513
240
4.189
3- = 0.667
M = i.99 7 f
514
270
4.712
i = 0.750
TT
6S = 1.931
515
300
5-236
1 = 0.833
H
bS = 1-843-
516
10
146
BELTS AND PULLEYS.
TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER CAST-
IRON PULLEYS.
Single Rawhide-Lacing.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
h = 0-083
68 = 7.480"
517
45
0.785
i = 0.125
68 = 5.280"
518
60
1.047
J = 0.167
If
bS = 4.180-
^|
519
75
1.309
1 = 0.208
W = 3 .53 7 f
520
90
I-57I
i = 0.250
M = 3-097^
521
105
1.833
^ = 0.292
//
M = 2.811-
522
1 20
2.094
t = o 333
//
M = 2.580-
p
523
135
2.356
1 = 0.375
i* 7/
b6 = 2.404-
524
150
2.618
A = 0.417
//
^5 = 2.261
p
525
165
2.880
M = o 458
<55 2.167-
V
526
1 80
3.142
| = O.i JO
bS = 2.074-
7'
527
195
3-403
if = 0.541
5 = 2.013-
V
528
2IO
3.665
A = 0-583
W = i.947"
529
240
4.189
t = 0.667
68 = i . 854
530
270
4.712
f = 0.750
M = 1.793"
531
300
5-236
1 = 0.833
35 = 1.701
532
VULCANIZED-R UBBER BEL TS.
147
TABLE OF FORMULAS
FOR VULCANIZED-RUBBER BELTS OVER
IRON PULLEYS.
Double Leather-Lacing.
CAST
degrees.
a in circular
measure.
a in fractions of
circumference.
Formula.
No.
30
0.524
=0.083
35 = 6.980
z/
533
45
0.785
i = 0.125
**= 4.928-
534
60
1.047
i = 0.167
35 = 3-900-
535
//
75
1.309
/j = 0.208
35 = 3-300-
536
00
I-57I
i = 0.250
//
35 = 2.888 -
537
105
1.833
g f = o 292
35 = 2.624^
538
If
1 20
2.09;
\ 0.333 ^5 = 2.404
539
135
2.356
1 = 0-375
35 = 2.244^
540
150
2.618
T 5 = 0.4:7
35 = 2.II2-
541
165
2.880
4i = -458
//
35 = 2.024
p
542
1 80
3.M2
J = 0.500
35 = 1.936^
543
195
3-403
^ = 0.541
//
35 = 1.876-
544
o
210
3-665
A = 0-583
35 = 1.821^
545
s?
240
4.189
= 0.667
//
546
270
4.712
t = 0.750
TT
35 = 1.672-
547
300
5.236
1 = 0.833
//
35 = 1.601
548
p
148
BELTS AND PULLEYS.
TABLE OF FORMULAS FOR VULCANIZED RUBBER BELTS OVER CAST-
IRON PULLEYS.
Double Rawhide- Lacing.
a in
degrees.
a in circular
measure.
a in fractions of
circumference.
P a,
No.
30
0.524
T V = 0.083
II
l>8 = 6.545-
549
45
0.785
i = 0.125
l>8 = 4.620
550
60
1.047
i = 0.167
1,8 = 3.658-
551
z>
75
1.309
-f = . 208
bS = 3 .c v 7-
552
V
U
90
I-57I
i = 0.250
b8 = 2.712-
V
553
[
105
I-833
T^f = O.2Q2
b8 = 2.464
554
120
2.094
i = 0.333
W--a.?25^
555
r/
135
2.356
I = 0.375
W = 2 . 107 -
556
Z
ISO
2.618
A = 0.417
1* H
bo = i . 980
&
557
165
2.880
H = 0.458
^5 = 1.898
558
180
3-I42
i = 0.500
= x.8i S f
559
195
3-403
M = 0.541
IT
68 = i . 760 -
560
p
210
3.665
A = 0.583
^5 = 1.705
56i
240
4.189
1 = o.66 7
<55 = 1.623-
V
562
270
4.712
f = 0.750
b8 = 1-568^
563
300
5.236
f = 0.833
//
M = 1.502-
564
VUL CANIZED-R UBBER BEL TS.
149
TABLE OF FORMULAS
FOR VULCANIZED-RUBBER BELTS OVER
IRON PULLEYS.
Riveted Joints.
CAST-
a in
degrees.
a in circular
measure.
o in fractions of
circumference.
Formula.
No.
30
0.524
A = 0.083
H
35 = 4-554-
565
H
45
0.785
i = 0.125
35 = 3-212-
566
ff
60
1.047
i = 0.167
35 = 2.547-
567
75
1.309
f = O.2O8
35 = 2.151^
568
90
I-57I
i = 0.250
tr
35 = 1.887-
569
105
1.833
^=0.292
-,.
570
V
120
2.094
i = 0.333
If
35= 1.568-
571
V
135
2.356
1 = 0.375
TT
35 = 1.463
572
150
2.618
A = 0-417
35= 1.375^
573
I6 5
2.880
H = 0-458
H
3d = 1.320
574
V
1 80
3-142
i = 0.500
35 = i . 260^
V
575
H
195
3.403
M = 0.541
35 = I.22I
V
576
2IO
3-665
A = 0.583
35 = 1.188^
577
V
240
4 189
1 = 0.667
35 = 1.128^
V
578
270
4.712
= 0.750
35 = i.oSg-
V
579
300
5-236
f = 0.833
35 = 1-045^
58o
ISO BELTS AND PULLEYS.
The formulas for vulcanized-rubber belts -^ inch
thick (say three-ply) over cast-iron pulleys are as
follows :
When a = 180,
Single leather-lacing, b = O.OI86P; .... (581)
Single rawhide-lacing, b o.O]J2P; . . .
(582)
Double leather-lacing, b O.oi6iP; . . .
(583)
Double rawhide-lacing, b 0.015 iP; . . .
(584)
Riveted joint, b o.oio$P. . . .
(585)
TT
Single leather-lacing, b = 10.208 ; . .
. (586)
TT
Single rawhide-lacing, b= 9.481-7-; . .
(587)
TT
Double leather-lacing, b = 8.850 ; . .
V
. ( 5 88)
TT
Double rawhide-lacing, b = 8.297 ; . .
(589)
TT
Riveted joint, b = 5.760 . . .
V
(590)
When a = 135,
Single leather-lacing, b = 0.02 i$P', . . .
(590
Single rawhide-lacing, b O.O2OOP; . . .
(592)
Double leather-lacing, = o.oi87/ 3 ; . . .
(593)
Double rawhide-lacing, b = O.OI75/*; . . .
(594)
Riveted joint, b O.O122P. . . .
(595)
TT
Single leather-lacing, =11.845 ; . . . (596)
TT
Single rawhide-lacing, b 10.990 ; . . . (597)
VULCANIZED-RUBBER BELTS.
TT
Double leather-lacing, b 10.258 ;
TT
b= 9- 6 32 ;
6= 6.668.
v
Double rawhide-lacing,
Riveted joint,
(598)
(599)
(600)
TABLE OF WIDTHS OF VULCANIZED-RUBBER BELTS OVER CAST-IRON
PULLEYS, WHEN a = 180 AND S = 5 y. From Formulas
(585).
Width
/*, single
leather-
P, single
rawhide-
P, double
leather-
P, double
rawhide-
P, riveted
No.
inches.
lacing.
lacing.
lacing.
lacing.
joints.
I
53.88
58.04
62.15
66.27
95-51
I
It
80.82
87.06
93-23
99.40
143-27
2
2
107.76
116.08
124.30
132.54
191.02
3
2*
134-70
145.10
155.38
165.67
238.78
4
3
161.64
174.11
186.45
198.81
286.53
5
3i
188.58
203.13
217-53
23I-94
334-29
6
4
215-52
232.15
248.60
265.08
382.04
7
4*
242.46
261.17
279.68
298.21
429.80
8
5
269.40
290.19
310-75
331-35
477.56
9
5*
296.34
319.21
341.83
364-48
525-3I
10
6
323.28
348.23
372.90
397-61
573-07
ii
7
377-i6
406.27
435-05
463.88
668.58
12
8
43I-03
464.31
497.20
530.15
764.09
13
9
484.91
522.34
559-35
596-42
859.60
14
10
538.79
580.38
621.50
662.69
955-n
15
ii
592.67
638 42
683.65
728.96
1050.62
16
12
646.55
696.46
745-80
795-23
1146-13
17
14
754-31
812.54
870.10
927.77
i337- I 5
18
16
862.07
928.61
994-40
1060.31
1528.18
T 9
18
969-83
1044.69
1118.71
1192.84
1719.20
20
20
1077-59
1160.77
1243.01
I325-38
1910.22
21
22
"85.34
1276.84
1367-31
1457.92
2101.24
22
24
1293.10
1392.92
1491.61
1590.46
2292.26
23
26
1400.86
1509.00
1615.91
1723.00
2483.29
24
28
1508.62
1625.07
1740.21
1825.53
2674.31
25
30
1616.38
1741 15
1864.51
1988.07
2865.33
26
152
BELTS AND PULLEYS,
TABLE OF WIDTHS OF VULCANIZED-RUBBER BELTS OVER CAST-IRON
PULLEYS, WHEN a 180 AND 8 = -fa". From Formulas (586)-
(590).
Width
in
inches.
H
-, single
leather-
lacing.
H
-, single
rawhide-
lacing.
H
, double
leather-
lacing.
, double
rawhide-
lacing.
, riveted
joint.
No.
!
o . 0980
0.1055
O.II3O
.I2O5
0.1736
I
I*
0.1469
0.1582
0.1695
.1808
0.2604
2
2
0.1959
0.2109
0.2260
.2410
0.3472
3
a*
0.2449
0-2637
0.2825
.3013
0.4340
4
3
0.2939
0.3164
0.3390
.3616
0.5208
5
J*
0.3429
0.3692
0-3955
.4218
0.6076
6
4
0.3918
0.4219
0-4520
.4821
0.6944
7
4i
0.4408
0.4746
5085
-5424
0.7812
8
5
o 4898
0-5275
5650
.6026
0.8681
9
5*
0.5388
0.5801
.6214
.6629
0-9549
10
6
0.5878
.6328
6779
-7231
1.0417
ii
7
0.6857
.7383
.7909
-8437
1-2153
12
8
0.7837
.8438
9039
.9642
1.3889
13
9
0.8817
-9493
.0169
-0847
1-5625
14
10
0.9796
0547
.1299
.2052
1.7361
15
ii
1.0776
.1602
.2429
.3258
1.9097
16
12
I-J755
.2657
3559
4463
2.0833
17
14
I-37J5
.4766
.5819
-6873
2.4306
18
16
I-5674
.6876
.8078
.9284
2.7778
19
18
I-7633
-8985
0338
.1694
3-1250
20
20
!-959 2
.1095
.2598
.4105
3-4722
21
22
2.1552
3204
.4858
.6515
3-8I94
22
24
2.35H
5314
.7118
.8926
4.1667
23
26
2.5470
7423
9377
3-I336
4-5I39
24
28
2.7429
2-9532
3-1637
3-3747
4.8611
25
30
2.9389
3-1642
3.3897
3.6157
5 - 2083
26
VULCANIZED-RUBBER BELTS.
153
TABLE OF WIDTHS OF VULCANIZED-RUBBER BELTS OVER CAST-IRON
PULLEYS, WHEN a = 135 AND d = $". From Formulas (591)-
(595)-
Width
inches.
/', single
leather-
lacing.
t>, single
rawhide-
lacing.
P, double
leather-
lacing.
/>, double
rawhide-
lacing.
P, riveted
joint.
No.
I
46.45
50.05
53-62
57-n
82 24
I
H
69.67
75-08
80.43
85.67
123.36
2
2
92.89
IOO.IO
107.24
114.22
164.47
3
2*
II6.I2
125.13
I34-05
142.78
205.59
4
3
139-34
150.15
160.86
171 33
246.71
5
3*
162.56
175.18
187.67
199.89
287.83
6
4
185.79
200.20
214.48
228.44
328.95
7
4i
209.01
225.23
241.29
257-00
37 -07
8
5
232.23
250.25
268.10
285.55
411.18
9
5^
255.46
275.28
294.91
3!4-ii
452.30
10
6
278.68
300.30
321.72
342.66
493-42
ii
7
325-13
350.35
375-34
399-77
575-66
12
8
371-57
400.40
428.95
456.88
657-89
13
9
418.02
450.45
482.57
5'3-99
740.13
14
10
464.47
500.50
536.19
571.10
822.37
15
ii
510.9!
550.55
589-81
628.21
904 . 60
16
12
557.36
600.60
643-43
685.32
986.84
17
M
650.26
700.70
750.67
799-54
1151-32
18
16
743-15
800.80
857-91
913.76
I3I5.79
19
18
836.04
900.90
965-15
1027.98
1480.26
20
ao
928.94
IOOI .OO
1072.39
1142.20
1644.74
21
22
1021.83
IIOI. 10
1179 62
1256.42
1809.21
22
24
1114.72
I2OI.20
1286.86
1370-64
1973.68
23
26
1207.62
1301.30
1394.10
1484.87
2138.16
24
28
1300.51
1401.40
1501.34
1599.09
2302.63
25
30
1391-40
1501.50
1608.58
I7I3-3I
2467.10
26
154
BELTS AND PULLEYS.
TABLE OF WIDTHS OF VULCANIZED-RUBBER BELTS OVER CAST-IRON
PULLEYS, WHEN a = 135 AND 5 = ^". From Formulas (596)-
(600).
Width
in
i iches.
7' sin & le
leather-
lacing.
f , sing,e
rawhide-
lacing.
, double
V
leather-
lacing.
-, double
9
rawhide-
lacing.
", riveted
joint.
No.
I
0.0844
O.OglO
0.0975
0.1-038
0.1495
I
I*
0.1266
0.1365
0.1462
0-1557
0.2243
2
2
0.1689
O.I82O
0.1950
0.2076
o . 2990
3
si
0.2III
0.2275
0.2437
0.2596
0.3738
4
3
0.2533
0.2730
0.2924
0.3115
0.4486
5
3*
0.2955
0.3185
0.3412
0.3634
0.5233
6
4
0-3377
o . 3640
0.3899
0.4153
.5981
7
4i
0-3799
o 4095
0.4387
0.4672
.6728
8
5
O.422I
0.4550
0.4874
0.5I9I
.7476
9
5*
0.4643
0.5005
0.5362
0.5710
.8224
10
6
0.5066
0.5460
0.5849
0.6229
.8971
ii
7
O.59IO
.6370
0.6824
0.7267
.0466
12
8
0.6754
.7280
7799
0.8306
.1962
13
9
.7598
.8189
8773
0-9344
3457
14
10
.8443
.9099
.9748
1.0382
4952
15
ii
.9287
.0009
.0723
I . 1420
.6447
16
12
.0131
.0919
.1698
1.2458
.7942
17
14
.1820
2739
.3647
1-4535
-0933
18
16
.3508
4559
5597
1.6611
3923
19
18
.5197
.6380
7547
I.S6S8
.6914
20
20
.6885
.8199
.9496
2.0764
.9904
21
22
8574
.0019
.1446
2.2840
3.2894
22
24
.0262
.1839
3396
2.4917
3-5885
23
26
1951
3658
5345
2.6993
3-8875
24
2S
.3640
5478
7295
2.9070
4.1866
25
30
2.5328
.7298
9 2 45
3.1146
4.4856
26
Example. Required the width for a vulcanized-rub-
ber belt inch thick which will transmit a force of 1200
pounds, the fastening being a single rawhide-lacing and
the arc embraced by the belt on the smaller pulley
being a =. 90.
VULCANIZ'ED-RUBBER BELTS. 155
Formula (441) gives
b X - = 0.00563 X 1200.
4
Hence b 0.00563 X 1200 X ,
or b = 9".
Example. Required the width for the above belt
with riveted joint instead of single rawhide-lacing.
We have from formula (489)
b X - 0.00343 X 1 200,
4
b = 0.00343 X 1200 X -,
or b = 5-489" = Sir-
Example. A vulcanized-rubber belt -J-inch thick
embraces an arc equal to \ the circumference of its
smaller pulley, and transmits a force of 20 horse-power
at a velocity of 10 feet per second. Required the
proper width for double leather-lacing.
Formula (539) gives
b X \ = 2.404 X ^
b = 2.404 X 2 X 4,
or b= 19.232" = i9$|/'.
J$6 BELTS AND PULLEYS.
Example. A three-ply vulcanized-rubber belt run-
ning over two equal pulleys transmits a force of 1275
pounds. Required the proper width for single raw-
hide-lacing. The table on page 151, column for single
rawhide-lacing, line 22, gives, corresponding to P =
1276.84 pounds,
b = 22".
Example. Given the data H = 20, v = 20, a =
135, d = -% inch. Required the proper width for the
belt, for single rawhide-lacing.
TT
The table on page 1 54 gives, corresponding to = I ,
a belt-width of II inches. (Column for single rawhide-
lacing, line 16.)
Vulcanized-rubber belts are very rarely seen running
over leather or rubber covered pulleys. We may, how-
ever, take for the coefficients of friction of rubber on
leather and rubber on rubber, respectively,
and ^ = 0.55.
The general formula (50) for the cross-section of any
belt for a given tension is
- T f
* Obviously this coefficient may be used for leather belts over
rubber-covered pulleys. See Appendix I.
VULCANIZED-RUBBER BELTS.
This may be put in the form
157
(601)
and the value of x for each special case determined
from the tensions T and /, as in 10 and 1 1. The fol-
lowing table gives values of x for all cases likely to
occur in practice :
TABLE OF GREATEST TENSION FOR VULCANIZED RUBBER BELTS OVER
LEATHER AND RUBBER-COVERED PULLEYS.
x, leather-
j-, rubber-
jf, leather-
x, rubber-
a in
degrees.
covered
pulleys.
covered
pulleys.
a in
degrees.
covered
pulleys.
covered
pulleys.
30
4-35
3-99
150
37
31
45
3-08
2.8 5
165
31
.26
60
45
2.28
1 80
.26
.21
75
.08
95
195
.22
.18
90
85
73
210
.I 9
15
105
.67
57
240
.14
.11
120
54
.46
270
.IO
.08
^35
44
38
300
.08
.06
Example. Required the proper width for a vulcan-
ized-rubber belt \ inch thick, and transmitting a force
of 8.OO pounds over leather-covered pulleys, taking the
angle a = 120, and the fastening a single rawhide-
lacing.
The table gives for the value of the variable coefficient
and the value of the safe-working stress for single raw-
hide-lacing is
158 BELTS AND PULLEYS.
Hence formula (601) becomes
, ^ I _ 800 X 1.54
* 4 ~ 350
or b 14.0$" =
Example. A vulcanized-rubber belt \ inch thick,
running over rubber-covered pulleys, transmits a force
of 25 horse-power at a velocity of 10 feet per second.
Required the proper width for double rawhide-lacing,
the arc embraced by the belt on the smaller pulley
being 135.
From the table, x 1.38,
and from page 141, /= 400.
TT
We also have P = 550-.
Substituting these values in formula (601) gives
b *^ = 550 x ^-x 1.38 ^400.
1Ience
or b = 18.98" ;= i|".
Example. A (\ inch thick) vulcanized-rubber belt
12 inches wide runs over leather-covered pulleys, and
embraces an angle of 90 upon the smaller pulley.
Required the force in pounds which may be safely
transmitted by the belt with a double rawhide-lacing.
RIM, NAVE, AND FIXING-KEYS FOR PULLEYS. I $9
The table gives x 1.85,
and we have also f = 400.
Hence formula (601) gives
I _ P x *- 8 5 p - 12 X 0.25 X4QQ
X 4 ~ 400 ~ ' 1.85
or P= 648.65.
13. Rim, Nave, and Fixing-keys for Pulleys.*
The rim of a pulley intended to carry a flat belt is
generally slightly rounded (Figs. 48 and 49), in order
that the belt may remain in the centre of the pulley-
face, instead of working to one side, as is the case with
flat-faced pulleys. The amount of this rounding (s)
may be taken equal to ^ the width of the belt.
For isolated pulleys the face-width B is taken some-
what greater than the width of the belt (b] ; often we
take
(602)
When, however, several pulleys are placed side by
side in order to receive alternately the same belt the
face -width B should be taken only very slightly
greater than the belt-width b.
The thickness k of the edge of the rim, or the
* From " Reuleaux."
i<5o
BELTS AND PULLEYS.
thickness at the ends of the face-width, may be easily
calculated from the formula
k = 0.08 + .
r 100
(603)
High-speed pulleys and those subjected to consider-
able shock and vibration are often provided with late-
ral flanges cast on the rims, as shown in Fig. 49, or are
replaced by grooved pulleys carrying belts with circu-
lar cross-section (Fig. 50).
Example. Required the rim dimensions for an iso-
FlG. 48.
FIG. 49.
lated pulley which is to carry a belt 12 inches wide.
From formula (602) we have for the face-width
12 =
and from formula (603), for the thickness of the rim at
the edges,
= 0.23".
RIM, NAVE, AND FIXING-KEYS FOR PULLEYS. l6l
For the amount of rounding of the pulley-face, s =
b = - X 12 = o.6 7/ . The thickness of the rim at
20 20
the centre is, therefore,
2k -f s = 2 X 0.23 + 0.6 = 1.06".
If we wish to provide .the pulley with rim-flanges, as
in Fig. 49, we have for the height of the flanges 8 =
8 X 0.23 = 1.84", and take the thickness of the flanges
equal to k. _..
Nave. The thickness (w, Fig. -^4) of a pulley-nave
is given by the formula
w = 0.4 + g + --, . . . . (604)
in which d represents the diameter of the shaft upon
which the pulley is keyed, and R the radius of the
pulley.
The length of the nave should not be taken less than
L 2.50^ ....... (605)
Often (in idle pulleys, for example) the length L is
taken equal to the face-width B of the pulley.
Example. A pulley of 36 inches diameter is keyed
upon a shaft of 4 inches diameter ; required the nave
dimensions. From formula (604) the thickness is
= 0.4 + '+~ = I- 427
1 62
BELTS AND PULLEYS.
and from formula (605) we have for the length of the
nave
= 2.5ox 1.427 = 3:5675".
In idle pulleys the interior diameter of the nave, or
the eye of the pulley, is taken slightly greater than the
FIG. 51.
diameter of the shaft upon which the pulley is to run ;
often the eye of an idle pulley is furnished with a coat-
ing of bronze or white metal, in order to diminish the
friction.
Keys. There are three kinds of keys which are used
to fix pulleys upon their arbors : the hollow key (Fig.
FIG. 5*.
FIG. 53.
FIG. 54.
54), used for light pulleys : the flat key (Fig. 52), used
for pulleys of medium size ; and the countersunk key
(Fig. 53), used for very large and heavy pulleys.
RIM, NAVE, AND FIXING-KEYS FOR PULLEYS. 163
The width s and thickness s, of the -fixing- key are
given by the expressions .
(6o6)
s. = o. 16 -I- ,
~ 10'
. (607)
and the inclination varies from T ^ to 2-J . .
Example. Required the width and thickness of the
fixing-key for the pulley of the preceding example,
in which the diameter of the shaft is d = 4". For-
mulas (606) and (607) give for the required width and
thickness, respectively,
and
s = 0.16 + - = o.(
s = O .i
^- = 0.56".
10
FIG. 55.
Split pulleys (Fig. 55) are often used for light work.
They offer the advantage of being easily put up and
taken down without interfering with the shaft-hang-
ings. With pulleys of this kind fixing - keys may be
'J
i6 4
BELTS AND PULLEYS.
dispensed with, the two parts of the pulley being
pressed upon the shaft by means of the nuts a, a, with
sufficient force to prevent slipping. For this purpose
the eye of the pulley is made slightly less than the
diameter of the shaft upon which the pulley is to be
fastened. When the division passes through a pair of
FIG. 56.
arms, as in the figure, each half of the split arm must
be as strong as an entire undivided arm, and conse-
quently of the same dimensions as the entire arms.*
Weight of Pulleys. The weights of pulleys can evi*
dently be calculated from one formula only approxi-
mately, since the arms, nave, etc., vary considerably in
* A better and stronger form of split pulley is represented in Fig.
56. In th's case all the arms are entire, and the pulley presents a
better appearance, as well as a simpler form. According to Un\yin
(see "Elements of Machine Design," 168), the net section of the
bolt at the rim should be one quarter the section of the rim plus
square inch, and that of the bolt at the nave one quarter the section
of the nave plus \ square inch.
RIM, NAVE, AND FIXING-KEYS FOR PULLEYS. 1
different pulleys. We may. however, calculate the
weights of pulleys with sufficient accuracy for ordinary
purposes from the formula
r> / r?\ 2 / rx 3\
G= (0.163^ + 0.0 15^] + 0.00309^) j 3 , .(608)
in which G is the weight of the pulley in pounds, R
and ^respectively the radius of the pulley and width
of the belt.
The following table gives values of -^ for different
values of - r :
o
TABLE OF WEIGHTS OF PULLEYS.
R
b
G
*
R
J
G
t s -
R
b
G
b*
R
7
G
6*
.O
0.181
2-5"
0-550
S-o
1-579
8.25
4.III
.1
O.202
2.6
0.580
5-2
1.691
8.50
4-378
.2
0.223
2.7
O.6I2
5-4
1.807
8-75
4.657
3
0.244
2.8
0.642
5-6
1.929
9.00
4-947
4
0.266
2.9
0.675
5-8
2.057
9-25
5-250
5
0.289
3-0
0.708
6.0
2.190
9-50
5.567
.6
0/312
3-2
0-777.
6.2
2.329
9-75
5-895
7
0-335
3-4
0.850
6.4
2-473
IO.OO
6.237
.8
0.360
3-6
0.926
6.6
2.623
10.25
6.592
9
0.385
3-8
1.007
6.8
2.780
10.50
6 961
.0
O.4II
4.0
1.089
7.0
2-943
11.00
7-742
.1
0-437
4.2 V
I.lSo
7-25
3-155
11.50
8.581
.2
0.464
4 4
1-273
7-50
3-378
12. OO
9.482
3
0.492
4.6
1.370
7-75
3.611
12.50
10.446
4
0:520
4-8
1.472
8.00
3-856
13.00
n-475
Example. The radius of a pulley is 16 inches, and
the width of the belt which runs upon the pulley 4
"inches ; required the approximate weight of the pulley.
Here , = 4. From formula (608),
l66 BELTS AND PULLEYS.
G = (0.163 X 4 + 0-015 X 1 6 + 0.00309 X 64)64,
G = (0.652 -f- 0.240 + 0.19776)64 = 1.08976 x 64;
or, G = 69.74 pounds.
Example. Required the approximate weight of a
pulley for the data R = $6", b = 4^". In this case
T- = ^r =8, and & 91.125. From the table we find
v 4?
V = 3-856.
Hence G = 3.856 X 91.125 = 351. 378' pounds.
14. Arms of Pulleys.*
Ordinarily the arms of pulleys have oval cross-sec-
tions, the diameter in the plane of the pulley being
twice the smaller diameter. The profile of such a cross-
section may be drawn by circle-
arcs as shown in Fig. 57. The
dotted circle is drawn on the
greater diameter //, of the pul-
ley-arm, and the arcs ab and
i v > wf / a'b' have their centres respec-
\ \ \X /' tively in the points c and c'.
Jt I^rSl The arcs ab and a'b' are con-
nected at their ends by small cir-
cle-arcs as shown in the figure.
The axes of pulley-arms may be straight as in Fig.
*From "Reuleaux."
ARMS OF PULLEYS.
I6 7
58, curved as in Fig. 59, or double curved in the form
of a letter 5". Single-curved arms may be drawn in the
following manner: Take (Fig. 59) the arc AE equal to
-$ the arc EF, determined by the centre s of the arms at
the rim of the pulley, and draw A^O perpendicular to
AO. From the centre D draw CD perpendicular to
FIG. 58.
FIG. 59.
OE, and the point C of intersection of DC and OC is
the centre for the curved axis of the arm.
The number of arms (TV) necessary for pulleys of
different sizes may be determined by means of the
formula
(609)
or the following table calculated from it :
^-=1234567
9 10 ii 12 13
789
1 68 BELTS AND PULLEYS.
The formula
r r>
= 0.24 + + - (610)
gives the greater diameter for the pulley-arms. The
diameter or width h is taken at the nave as shown in
Fig. 58, and the width 7i 1 at the rim may be conven-
iently taken equal to \h. These expressions have been
determined, with a certain approximation from the
most accurate formulas; for large and medium sized
pulleys they are especially applicable, but for small
light pulleys the dimensions should be slightly in-
creased in order that the pulleys may be easily cast
without taking special precautions.
Example. Required the number of arms and the
arm dimensions for a pulley having a radius of 18 inches,
the belt for the pulley being 6 inches wide. Here
R 18
From the above table we find the number of arms to
be A' 4, and formula (610) gives for the width of the
arms in the plane of the pulley
6 l8
The width at right angles with the plane of the pulley
is therefore
^, = f X 2.19= 1.46*.
ARMS OF PULLEYS.
[69
To trace the profiles of the arms proceed as follows:
Straight arms (Fig. 60). Having drawn the diameter
EOC, take ab = cC = Cd = f h, and draw the lines ac and
bd, which give the limits of the profile. Connect ac and
bd with the rim and nave by small circle arcs, and the
profile is complete. Curved arms (Fig. 61.) The centre
C for the axis having been determined, draw the straight
line ad, then take aE = Eb = - and Cc = Cd ^ ; the
points c and d thus determined are the centres for the
arcs which limit the profile, and cb and da are the radii.
Double-curved arms.* Fig. 62 shows a simple
method for drawing double-curved arms. Draw the
radial line oA, making 30 with the horizontal. Take
oc = \oA, and through the point c draw the line pD,
making 60 with the horizontal. Intersect the line
* From the author's "Treatise on Toothed Gearing."
l^O BELTS AND PULLEYS.
pD by a horizontal line through the point A : the points
D and/ are respectively the centres for the arcs oc and
cA, which together form the axis of the arm. Lay off
the arm-widths as shown in the figure. From the
point p as a centre strike the arcs ab and ef, and find
upon the line oD the centres for the remaining arcs bd
and/?.
Another very similar method for drawing double-
curved arms is shown in Fig. 63. Draw the radial line
oAi making 45 with the horizontal. Take oc = -^oA,
and through the point c draw the vertical line pD.
SHAFTS.
171
Intersect the line pD by the horizontal line Ap. The
points/ and D are the centres for the arcs of the axis.
Lay off // and h v as shown in the figure, and proceed,
as in Fig. 62, to strike the arcs ab, ef, bd, and fk'.
15. Shafts*
When a shaft is so supported by its bearings as to
be subjected to a torsional strain only, as is almost in-
variably the case in pulley-shafts (the bending strain
due to the weight of the pulley and the force trans-
mitted by the belt being ordinarily slight enough to be
safely neglected), the calculation of the proper strength
for the shaft may be made as follows :
FIG. 64.
In Fig. 64, P represents the total force tending to
twist the shaft, i.e., the total force transmitted by the
belt ; R the distance from the centre of the shaft to
the point at which the force acts, i.e., the radius of the
pulley ; and d the diameter of the pulley-shaft. The
*From the author's " Treatise on Toothed Gearing."
1 72 BELTS AMD PULLEYS.
greatest safe torsional strain which can be sustained by
the shaft is given by the expression
= * 19635
in which f is the greatest safe shearing stress in
pounds per square inch for the material of the shaft.
From this,
PR
Q-I9535/ 7 '
/>>r)
or, d= ijioj/ -yr (611)
RULE. To determine the diameter of a pulley shaft
of any material multiply the total force transmitted by
the belt by the radius of the pulley, divide this pro-
duct by the greatest safe shearing stress in pounds per
square inch for the material of the shaft, extract the
cube root of the quotient thus obtained, and multiply
the result by 1.720.
Example. Required the diameter for an oak shaft
upon which is a 6o-inch pulley transmitting a force of
1000 pounds, taking/"' = 500 pounds. From formula
(6 1 1) we have
=,. 720 y
IOOOX3O
-= 1.720x3-915 = 6.734"=
We propose to take for steel f = 12000 pounds;
for wrought-iron/"' = 8000 pounds; and for cast-iron
f = 4000 pounds. These values of f are nearly
SHAFTS. 173
mean between those used by Stoney, Hasvvell, and
Unwin, which differ far more than is conducive to any
degree of accuracy. Substituting the above values of
f successively in formula (611) and reducing, we
obtain,
For steel, d = 0.075 V/Y?. .... (612)
For wrought-iron, d 0.086 *tf~PR (613)
For cast-iron, d = o.\o*V~PR. .... (614)
RULE. To determine the diameter for a pulley-
shaft of steel, wrought or cast iron, multiply the total
force transmitted by the radius of the pulley, extract
the cube root of the product, and multiply the result
by 0.075 f r steel, o.c86 for wrought-iron, and 0.108
for cast-iron.
Example. A 48 inch pulley transmits a force of
IOOO pounds. Required the diameter for a steel shaft.
From formula (612) we have
d = 0.075 V 1000 X 24 = 0.075 X 28.84,
or, d = 2.163" 2^" nearly.
Example. Taking the data of the preceding ex-
ample, required the diameter for a shaft of cast-iron.
Formula (614) gives
d 0.108 Viooo X 24 = 0.108 X 28.84.
or, d = 3.115" = 3$" nearly.
Formulas for the diameters of pulley-shafts in terms
174 BELTS AND PULLEYS.
of the horse-power transmitted and the revolutions pel
minute may be obtained as follows:
As before explained, we have the expression
/>= 63000 .
//"representing the horse-power, ^ the radius of the
pulley, and n the number of revolutions per minute.
Substituting this value in formulas (611), (612), (613),
and (614), and reducing, we obtain the following:
r~ff
General formula, d = 68 44 /t / ^ (615)
For steel, d = 2.984 A/^- .... (616)
/ H
For wrought-iron, d= 3.422 A / (617)
fff
For cast-iron, a? = 4.297 /J/ (618)
RULE. To determine the diameter for a pulley-
shaft of any material from the horse-power and num-
ber of revolutions per minute, divide the horse-power
by the product of the number of revolutions into the
greatest safe shearing stress in pounds per square inch
for the material of the shaft, extract the cube root of
the quotient thus obtained, and multiply the result by
6844.
SHAFTS. 175
To determine the diameter for a pulley-shaft of
steel, wrought or cast iron, from the horse-power and
number of revolutions per minute, divide the horse-
power by the number of revolutions, extract the cube
root of the quotient, and multiply the result by 2.984
for steel, 3.422 for wrought-iron, and 4.297 for cast-iron.
Example. Required the diameter for an oak pulley-
shaft which transmits a force of 10 horse-power and
makes 40 revolutions per minute. If we take for the
greatest safe shearing stress for oak f = 500 pounds
per square inch, we shall have, from formula (615),
\ / = 68.44 \ = 68.44 X-
Y 40x500 ^Y 200 *
12.60'
or, d = $.432" = 5 T y nearly.
Example. Taking the data of the preceding ex-
ample, required the diameters for shafts of steel and
wrought-iron.
From formula (616).
d = 2.984 / = 2.984 Vo25 = 2.984 X 0.62996,
or, for steel, d = 1.88" = i|".
From formula (617),
3 /IO
d = 3.422 Y 40 = 3422 x ' 62 99 6 >
or, for wrought-iron,
</= 2.1557" =2^".
BELTS AND PULLEYS.
Pulley-shafts are most commonly of wrought-iron;
when, however, wrought-iron shafts, in order to give
the necessary strength, become so large as to be incon-
venient, steel shafts are used. Cast-iron shafts are. as
a rule, unreliable and treacherous ; they are therefore
seldom used except for the transmission of slight
powers and in cheap, inferior machinery. The follow-
ing tables, calculated from formulas (612), (613), (616),
and (617) to the nearest -fa inch, will be found very
convenient in designing pulley-shafts of steel and
wrought-iron :
TABLE OF SHAFT-DIAMETERS.
p*
d for steel.
rf for
wrought-iron.
I'R
d for steel.
fl'for
wrought-iron.
250
*
.
l|*
6OOOO
2 !f
3-ir
500
ill
|
700OO
3sV
3f
IOOO
j.
1
8OOOO
3 if
3sf
1500
2OOO
If
g
QOOOO
100000
3ff
3fJ
4
25OO
i*
-
i
I IOOOO
3j|
4i
3OOO
3500
il
'
s
120000
130000
M
i
4000
4500
5000
6OOO
ia
1
i
ft
140000
150000
175000
20000O
1
4g
4F
i
7OOO
8000
1
i
2500OO
5000OO
4-
5]
IOOOO
III
1
750000
6 T
?j
12500
I*
1 000000
7i
8|
15000
111
1.
1500000
8|J
9i
2OOOO
2 sV
20OOCOO
9 ff |.
loj
25COO
8 ft
2500000
iH
n|
30000
2 fr
3000000
IM
' i 2 !^
35000
2f|
3500000
"ft
i3rV
4OOOO
2 ^
4000000
lift
!3l5
45000
2 IT
3'
4500000
I2f
I4lf
5OOOO
2 lf
,,
J
i
5000000
tail
I4
SHAFTS.
TABLE OF SHAFT-DIAMETERS.
177
wrouglit-iro
(I'for
wrought-iron
0.025
0.050
0.075
O.IOO
0.150
O.200
O.250
O.30O
0-350
O.40O
o 500
0.600
0.700
0.800
0.900
I.
1.25
1.50
1-75
2.
2.25
2-50
2-75
3-
3-25
3-50
3-75
4
4-25
4-50
4-75
5
5-50
6
6.50
7
f
Jft
efl
Example. Required the diameter for a wrought-
iron shaft for a 4O-inch pulley which transmits a force
of 1000 pounds. In this case
PR = 1000 X 20 = 20000,
and from the table on page 176, the value of d for
wrought-iron corresponding to PR 20000 is d 2||
inches.
Example. The diameter of a wrought-iron pulley-
178 BELTS AND PULLEYS.
shaft is 4|- inches. Required the force which the shaft
can safely transmit by means of a 24-inch pulley. From
the table on page 176 the value of PR corresponding
to d = 4^- inches for wrought-iron is 110,000; hence
we have
IIOOOO IIOOOO
P- B = 9107 pounds nearly.
Example. A pulley transmitting a force of 20 horse-
power makes 200 revolutions per minute. Required
the diameter for a shaft of steel. We have
H 20 I
and from the table on page 177 the value of d for
rr
steel corresponding to = o.ioo is d = i|-- inches.
Example. A 2-inch steel shaft transmits a force of
25 horse-power. It is required to determine the
proper number of revolutions per minute. From the
table on page 177 the value of which corresponds to
rr
d = 2 inches for steel is = 0.300 ; hence we have
-= - = o. 3 oo,
or n = 83^ revolutions per minute.
THE TIGHTEN1NG-PULLEY.
179
16. The Tightening-Pulley.Fast and Loose Pulleys.
Tightening-pulleys are used to tighten loose belts,
or, in other words, to increase the tension, and thus
prevent slipping upon the principal pulleys. Fig. 65
represents a tightening-pulley as commonly used in
the shops. A and B are the principal pulleys, and C
the tightening pulley, which is pressed against the belt
or raised off the belt by means of the lever d. Often
the weight of the tightening-pulley is sufficient to pro-
duce the required tension ; if not, extra weights are
hung to the pulley, or the lever fastened up in its proper
position. When the pulley C is lifted off the belt en-
tirely, the belt relieved of its tension no longer runs
upon the driver ; the driven pulley is then at rest, or
the belt is disengaged. The tightening-pulley obviates
the necessity of taking up the slack caused by the
stretching of the belt, for as the belt becomes longer
180 BELTS AND PULLEYS.
and consequently looser upon the principal pulleys by
stretching, it may be tightened by simply lowering the
tightening-pulley. A glance at the figure will show
that by means of this pulley the arcs of contact be-
tween the belt and the principal pulleys are increased
to a considerable extent which in itself is an impor-
tant consideration. The tightening-pulley is also a
valuable means of increasing the duration of a belt ;
for since the wear upon the belt increases with the ten-
sion to which it is subjected, it is important that the
tension be no greater than is sufficient to prevent slip-
ping, and this may be easily regulated by lifting or
lowering the lever which controls the position of the
pulley. By placing the tightening-pulley below the
belt the contrivance may also be made to take the
place of a pulley support. With high-speed belts con-
siderable care is necessary to keep the tightening-pulley
in its proper position.
Fast and loose pulleys are used as a means of en-
gaging and disengaging the belt, and thus starting or
stopping the driven pulley without interfering in any
way with the driver. This is a very necessary con-
sideration in cases where several machines are driven
by a single driving-pulley, as is almost always the case
in practice. Many contrivances have been from time
to time devised for this purpose, but few if any have
proved as simple and sure as the fast and loose pulleys
seen in nearly every shop and factory in the land.
Fig. 66 represents a pair of such pulleys. A is keyed
fast to the shaft C C, while the pulley B runs loose
upon the shaft. The belt is made to pass from one
pulley to the other by means of a lever, or similar device.
"HE TIGHTENING-P ULLE Y.
181
When the belt is on the fast pulley A, the motion
of the driving-pulley is transmitted to the shaft C C.
When the belt is on the loose or idle pulley B, this
pulley simply rotates upon the shaft without giving to
it any motion. In many cases the loose pulley is placed
upon the driven shaft ; the belt then continues its mo-
tion when upon the loose pulley. It is preferable,
however, to have the loose pulley on the driving-shaft,
because when the belt is out of gear it remains mo-
tionless, thus saving it from unnecessary wear. Often
there are two loose pulleys, one on each shaft ; the
driving fast pulleys are then of the same face-width,
while with one loose pulley the driving-pulley on the
other shaft must have a face-width equal to those of the
loose pulley and its neighboring fast pulley together.
A device introduced some years ago for the pur-
pose of diminishing the tension upon a belt while
182
BELTS AND PULLEYS.
upon a loose pulley is shown in Fig. 67. A is an or-
dinary pulley keyed fast to the shaft D D, and B a
loose pulley, which is somewhat smaller than the fast
pulley, and which carries a conical flange C C, the out-
side diameter of which is equal to that of the fast pul-
ley A. When the belt passes from A to B, the tension
FIG. 67.
upon it is diminished, the belt slackens, and while out
of work is not subjected to any considerable strain.
In ordinary fast and loose pulleys the tension upon the
belt is constant, whether the belt is at work or at rest.
Fig. 68 represents a common application of the
principle of fast and loose pulleys, by which an alter-
nate rotating motion in both directions is obtained
from the continuous rotary motion of the driving-shaft.
The pulleys A, C, A', and C' are fast, while B and B'
run loosely upon their shafts. Two belts, one open
and the other crossed, are placed side by side in such
a manner that one rests upon the loose pulleys, while
the other runs upon one or the other pair of fast pul-
THE TIGHTENING-PULLEY.
183
leys. When the belts are in the positions shown in
the figure, the crossed belt is the driver, and the open
belt remains motionless. By sliding the two belts
over the pulley-faces the open belt is placed upon the
fast pulleys A and A', and the crossed belt upon the
two loose pulleys. This reverses the direction of
FIG. 68.
rotation of the driven shaft, and by sliding the belts
back into their first positions the motion of the driven
shaft is again reversed.
The most familiar example of this reversing gear is
seen in the planing-machine, where the forward and
backward motion of the table which carries the work
is thus accomplished.
Belts with circular cross-sections, such as round
leather-belts, rope-belts, etc., generally have pulleys
with grooved faces. The ordinary fast and loose pul-
1 84
BELTS AND PULLEYS.
leys obviously cannot be used in such cases. Fig. 69
shows a fast and loose pulley for round belts, which
seems to answer the purpose very well. The part B
of the fast pulley is keyed fast to the shaft dd, and
the part A may be moved away from the part B by
means of the lever f. In the figure the parts of the
FIG. 69.
fast pulley are together, and the belt gg therefore
drives the shaft. When the parts are separated the
belt slides from the part B to the inside loose pulley
C, which then rotates about the shaft without trans-
mitting to it its motion. Upon sliding the part A
ROPE-BELTS.
185
again into the position shown in the figure the thin
rim slides under the belt and
lifts it into the groove, in "*
position for work.
Another fast and loose
pulley for round belts is
represented in Fig. 70. In
this case the pulley runs
loosely upon the shaft dd
when in the position shown
in the figure. The conical
key B is fast to the shaft,
and, when forced into the
hub of the pulley by means
of the lever f, bites with suf-
ficient force to secure the
pulley. A collar kk fast to
the shaft prevents the pul-
ley from sliding away from FIG. 70.
the key. For light transmissions this pulley may work
satisfactorily, but for heavy or unsteady work it can
hardly be spoken of as reliable.
17. Rope-Belts.
Hemp and cotton ropes are sometimes used for
transmission-belts, the principal pulleys being placed
from 25 to 60 feet apart. Three-strand ropes, such as
is represented in section in Fig. 71, are most commonly
used ; the diameters vary from inch to 2\ inches,
and by placing several ropes side by side upon the
principal pulleys, large powers may be transmitted.
1 86 BELTS AND PULLEYS.
As a general rule, rope-belts work almost entirely by
means of their weights, being hung loosely upon the
pulleys instead of tightly stretched over the pulleys as
in leather and vulcanized-rubber belts. When only
small powers are to be transmitted, the pulleys may
have semicircular grooves upon their faces, and the
rope-belts may run in the bottoms of the grooves.
The weight of the belt in such cases furnishes sufficient
friction to prevent slipping. Fig. 72 represents a pul-
ley-rim of this kind for a single rope-belt. When,
FIG. 72.
however, large powers are to be transmitted, the grooves
in the pulley-faces should be V-shaped, so that the
ropes may be wedged between the sides, and thus fur-
nish the friction lost by diminishing the initial tension.
A pulley-rim for four rope-belts transmitting large
power is represented in Fig. 73. The dimensions of
the pulley may be calculated as for ordinary pulleys,
and the sides of the grooves commonly make angles of
45 with each other, as shown in the figure.
The coefficient of friction for rope-belts running in
the bottoms of semicircular grooves without biting
ROPE-BELTS.
I8 7
against the sides (Fig. 72) may be taken, for cast-iron
pulleys,
y 0.30.
For V grooves, of which the sides are inclined at
angles of 45, as in Fig. 73,
FIG. 73.
By substituting this value in formula (41), we obtain for
the ratio of the tensions
log -- = 0.0053**,
in which a is expressed in degrees.
(620)
If in formula (48) we make the quantity yr-=
t i
we have
T=Px (621)
188
BELTS AND PULLEYS.
The following table gives values for and x for all
values of a. likely to be needed in practice :
degrees.
a in circular
measure.
a in fractions of
circumference.
T
t
X.
45
0.785
i = 0.125
1.732
2-37
60
75
90
1.047
1.309
I-57I
\ = 0.167
-% = 0.208
i = 0.250
2.O8O
2.498
3.OOI
93
67
50
105
120
1.833
2.094
-fa = 0.292
t = 0.333
3.602
4.325
-38
30
135
2.356
1 = 0.375
5.I94
.24
1 80
3.142
i = 0.500
8-995
.12
2IO
3-665
A = 0.583
12.97
.08
240
4.188
t = 0.667
18.71
.06
If we represent the diameter of the rope by <?', we
have for the area of cross-section
and this substituted for H in formula (50) gives
0.785^" - ~
The safe working stress in pounds per square inch may
be taken
/= 1200,
but for greatest durability and best performance of
ROPE-BELTS. 189
work this is in practice about T V the above value.
Hence we use
/= 120,
which substituted in the above formula gives
or tf = o.i03*',
and consequently
tf' = 0.103 V7 ..... (622)
As before explained, we have the expression
in which H represents the horse-power transmitted and
v the velocity in feet per second. By substituting this
in formula (622) we obtain
or 6' =
/If
' = 2.416 A/ X (623)
Example. Required the proper diameter for a rope-
belt which will transmit a force of 1000 pounds over
IQO BELTS AND PULLEYS.
two equal V-grooved pulleys. In this case a = 180,
and the table gives
x = 1. 12.
Hence, from formula (622),
6' = 0.103 l/iooo X 1. 12 = 0.103 X 3347,
or 6' = 3.447" = 3 ||..
Rope-belts as large as this are seldom used in prac-
tice. In the above example, therefore, we should use
two ropes instead of one. Each rope would then
transmit a force of = 500 pounds, and we should
have d f 0.103 I 7 5OO X 1.12,
or S f = 2.437" = 2 T V".
Example. A rope-belt embracing an angle of 135
upon its smaller principal pulley transmits a force of
15 horse-power at a velocity of 30 feet per second. It
is required to determine the proper diameter for the
belt. From the table we have
x = 1.24,
and from formula (623)
6' = 2.416 ^ = 2
or 6'= i .90 1' 7 = iff
ROPE-BELTS. igl
Example, It is required to transmit a force of 800
horse-power at a velocity of 80 feet per second by
means of 15 rope-belts. The arc embraced by each
belt upon the smaller principal pulley is equal to f the
circumference. Required the diameters for the rope-
belts.
It is evident that each belt must transmit a force of
= 50 horse-power at a velocity of 80 feet per
second.
The table gives, for a = f the circumference,
x = 1.24,
and from formula (623) we have
*' = 2. 4 i6 A/ -^^- = 2.416 X 0.88,
or 6' 2.126" = 2\".
Because of the circular cross-sections of rope-belts
and the character of the material generally used, it is
necessary that the wear due to the bending of the
ropes about the pulleys be reduced as low as possible.
To this end very large principal pulleys are used from
about 7 to 15 feet in diameter commonly. It is a safe
rule, that the diameter of the smaller principal pulley
should not be less than thirty times the diameter of
the rope, and when small ropes are used we may con-
veniently increase the durability by taking the diameter
of the smaller pulley equal to 45 to 60 times the diame-
ter of the rope. Thus in the three examples given
I Q2 BELTS AND PULLEYS.
above we may have for the diameters of the smaller
principal pulleys 30 X 2.44 = 73.20" = 6\ feet, 30 X
1.9 = 57 V = 5 feet nearly, and 30 X 2.13 = 63.9" =
5i feet.
The ends of rope-belts are usually spliced together
by pressing them firmly together and winding about
with stout small rope. The spliced part should be as
long as possible in order to bend properly over the
pulleys and give the necessary strength. The weight
per foot of length of rope-belts is approximately given
by the formula
= 0.3*" (624)
1 8. Jointed Chain-Belts*
Of late years numerous attempts have been made
to replace ordinary leather belts by traction-bands.
Among the various systems proposed we mention in
the first place the chain-belt (leather) of Rouiller : this
contrivance, which at first appeared destined to do
good service, has not justified this hope, but has fallen
into disuse because of its want of durability. Belts
formed of twisted metallic wires (Godin) have produced
results scarcely more satisfactory. As for leather belts
covered with gutta-percha, they cannot, in reality, com-
pete with ordinary leather belts ; and at the present
time there is scarcely a transmission-band, with the ex-
ception of rubber-belts with layers of hemp or cotton,
which seems to be as advantageous in practice as or-
dinary leather belts, especially when used for the trans-
mission of considerable forces.
* From Reuleaux.
JOINTED CHAIN-BELTS.
J 93
in certain special cases, for the transmission of large
forces, and for unsteady work, such as in agricultural
machines, the ordinary leather belt may be successfully
replaced by the jointed chain-belt of Clissold (Fig. 74).
In this chain the joints are bound together, two by
two, by leather bands wound several times around, and
bevelled at the edges to fit properly in the trapezoidal
groove which forms the face of the pulley. Angstrom
has used instead of the leather bands pieces of wood
trimmed with iron.
In calculating the tensions for jointed chain-belts it
is necessary to introduce the friction of the joints in-
stead' of the rigidity which figures in formulas for
leather-belts. The formulas for tensions of leather-
belts may be used in the present case by putting
cpa = -PTy representing the angle of the bevelled
edges of the chain-belt.
For (p = 0.24, a 0.8 TT, = 30, we obtain
IQ4 BELTS AND PULLEYS.
p=0.20, -p 1.23, p =1.43, ^=0.163; (625)
and for (p 0.28, a 0.95^, = 30,
^, = 0.12, ^=1.15, -i- = 1.27, ^ = 0.105. (626)
By making use of these values we may obtain for
the diameter of the joint pivots (d, Fig. 74) the ex-
d = 0.0146 VP= 3-656 A/-^; .... (627)
d = 0.0414 /~(/^ = i. 6 44 / .. (628)
We should take for jointed chain-belts the following
proportions (see Fig. 74) :
/ b c I e i //
rf = 3' 5 = 2 *' rf = ? d = F 1 d = 2 *'
For small pulleys it is convenient to take
/ **
In practice ^/ should not be taken less than 0.32 inch,
even when a smaller diameter would be sufficient for
strength. In jointed chain-belts the limit of the force
* P = force in pounds transmitted, //= horse-power, n = revolu-
tions pe; - minute.
JOINTED CHAIN-BELTS. 195
P which may be transmitted (supposed to be applied
at the circumference of the pulley) is about 500 pounds,
which would require a width of about 1 1 inches in a
simple leather belt.
Example. Given the data H=2O,n = 50, n l = 100.
Required the dimensions for a jointed chain-belt, sup-
posing the radius of the smaller pulley to be R, = 5/.
Formula (628) gives
2O
From formula (629) then we obtain / = 3 X 0.5624 =
1.6872", b = 2f X 0.5624 = 1.5466", c 0.2", e
0.12", h 21 X 0.5624 = 1.2185", ^i = 5^= 8.436",
R = 2 X -, = 16.872".
Clissold has also invented a transmission by means
of a thick belt with trapezoidal section. This, how-
ever, has proved poorly because of its want of dura-
bility.*
* The experiments of Wedding of Berlin have shown that in an
angular groove, the angle being 30 (Fig. 50), the force necessary to
produce slipping of the cable is twice that coi responding to a cable
lying in a round groove. This confirms the preceding expressions,
i i
since - 5 = -.
sm 30 2
TRANSMISSION BY METALLIC CABLE.
19. Tensions of Cables.
Transmission of forces by means of metallic cables
was first introduced about the year 1850, by the Hirn
brothers.f The use of metallic cables, by means of
which we are able to transmit great forces at distances
as great as several thousand feet without notable loss,
depends essentially upon the principles of transmission
by belt, the principal difference being that with a
metallic cable the tension is due to its own weight.
The two principal pulleys of a transmission by cable,
as a general thing, have their axes parallel ; also the
pulleys are in the same plane, so that the cable may
be driven without guides. Moreover, the axes of the
principal pulleys are ordinarily in the same horizontal
plane, forming what is termed a horizontal transmis-
sion. An inclination of the plane of the axes to the
* From Reuleaux.
f In this first application the axes of the pulleys were about 280 feet
apart; the force transmitted was 42 horse-power, at 60 revolutions per
minute.
TENSIONS OF CABLES. 197
surface of the ground constitutes an oblique transmis-
sion. Vertical transmissions by metallic cable are very
rarely used. When the driven pulley transmits to a
third pulley the force which it receives from the driver
the transmission is said to be compound. In a simple
transmission by cable the two pulleys are ordinarily of
the same diameter.
In order to prevent the cable from touching the
ground, when the height of the pulleys above the
ground is insufficient and the separation of the axes
great, intermediate rollers are used to support the
cable. By inclining the rollers more or less they may
be used for guides when the axes of the pulleys cross
or intersect each other. We meet, however, very few
examples of transmission by cable in which the axes
of the pulleys are not parallel. When it becomes
necessary to give to the cable a considerable deviation,
we can place between two vertical rollers a horizontal
guide ; but it is preferable in such cases to rely upon a
compound transmission, with pulleys placed obliquely
to each other.
The inferior limit for the separation of the pulley-
axes in transmissions by metallic cable should be about
50 feet.
The distances between the rollers which support the
cable are determined by the flexibility of the cable
and its position above the ground.
The transmission-cables ordinarily used are com-
posed of 36 iron wires divided into six twists, each
containing six wires twisted around a central core of
hemp ; the six twists are likewise twisted around a
larger core, also of hemp (Fig. 75). When it is neces-
198 BELTS AND PULLEYS.
sary to strengthen the cable, we may, without serious
disadvantage, replace the central hempen core by a twist
of iron wire similar to the six others. It has also been
proposed to replace by an iron wire the smaller
hempen cores of the separate twists, in order to over-
come the looseness of the cable, which may tend to
produce a rapid wear. The value of such an arrange-
ment yet remains to be established. It has the dis-
FIG. 75.
advantage of destroying the elasticity of the cable.
When the cores are of hemp, it is of first importance
that first quality hemp be used in their manufacture,
instead of the inferior qualities which have been hither-
to extensively used for this purpose. The wires com-
posing the cable should be forced firmly together, so
that the diameter of the cable is not more than eight
times that of the wire.
In cables having more than 36 wires the number of
twists is generally six, and the large and small cores of
hemp.
TENSIONS OF CABLES. 199
While there is no absolute necessity of limiting the
number of twists to six, this number is almost always
used: in the different cables in use the total number
of wires is therefore 36, 48, 54, 60, 66, 72, etc.
Fig. 76 represents a cross-section of a cable of 60
wires. In these different cables the relations between
the external diameter d and the diameter 8 of the
wires are as follows :
For the number of wires
i = 36 48 54 60 66 72,
d
^=8.00 10.25 11.33 12.80 13.25 14.20.
In order to obtain the tensions T and t in metallic
cables we make use of the formulas determined for
tensions in ordinary belts. By substituting in these
formulas a coefficient of friction q> = 0.24, and an arc
of the pulleys equal to |- the circumstance, a = iSo
= TT, we may obtain the relations
t T T4-t t
-p = 0-97, p = 2.02, -^- = 2.99, y.= 0.48 ; (630)
or, in round numbers,
~
. (631)
*.The loss of velocity due to the shipping of ihe cable does not
ordinarily exceed ^ per cent; it may therefore be neglected alto-
gether in our calculations,
2OO BELTS AND PULLEYS.
20. Calculation of Diameters of Cables.
In a transmission by metallic cable composed of i
wires the tension T in the cable corresponds to a ten-
sion S in the wires ; this tension should not exceed
25601.4 pounds per square inch of section.* To de-
termine the diameter 6 of the wires the following for-
mulas may be used :
For a resistance of P pounds acting at the circum-
ference of the pulley,
<5 = i .6
For a force of H horse-power, with a velocity of v
feet per second at the circumference of the pulley,
8 = 37-867^%.. (633)
in which v should not materially exceed 100 feet per
second.
For a force of H horse-power at n revolutions of the
pulleys per minute,
(634)
If we represent by s = 25601.4 5 the tension pro-
duced in the wires by the bending of the cable around
the pulleys, and by (PR) the statical moment of rota-
tion of the driven pulley, we shall have
(. (635)
* 18 kilograms per square millimetre.
CALCULATION OF DIAMETERS OF CABLES. 2OI
Finally, if in place of the moment (PR) we have the
horse-power and revolutions per minute,
H
(636)
It is, moreover, important that the ratio of the radius
of the pulleys to the diameter of the wires be taken
not less than the limit,
14223000
s
(637)
This relation serves to calculate the following table :
s
J
R
S
s
s
R
~S
711-15
24890.25
571
12800.70
12800.70
mi
1422.30
24179.10
588
14223.00
11378.40
1250
2844.60
22756.80
625
15645.30
9956-IO
1429
4266 . 90
21334-50
667
17067.60
8533.80
1667
5689.20
19912.20
714
18489.90
7III-50
2OOO
7111-50
18489.90
769
19912.20
5689.20
2500
8533.80
17067.60
833
21334-50
4266 . 90
3333
9956. to
15645-30
909
22756.80
2844.60
5000
11378:40
14223.00
IOOO
24179.10
1422.30
ICOOO
For a constant value of S -{- s the minimum value of
the radius of the pulleys is given by the table by
making = 2* This minimum value corresponds to
* We may obtain from formulas (636) and (637) R = Ky -j , .
The sum s -\- S being constant, the maximum valu.s of the product
s*S is obtained by making = 2.
202
BELTS AND PULLEYS.
T~>
5=8533.8,*= 17067.6,^ = 833.
1T>
For values of -j
o
nearly equal to 833 the numerical value of R differs
very little from the minimum value; we may there-
fore safely give somewhat greater values to R when,
by so doing, we can make use of patterns and models
already on hand.
The two tables which follow have been calculated
from formulas (632)-(634), and (635) and (636) respec-
TT
tively. In the first table we have given 1000 -~-- in
order to avoid the small numbers which result from
H
SRn
D
i =
p
fj
Pf
36
42
48
60
72
~s
Sv
1000 ITn
O.O2O
O.OI84
o 0172
0.0156
o . 0140
0.0054
O.OOOOIO
0.000088
O.O24
O.022O
0.0208
0.0184
0.0168
0.0078
0.000015
0.000123
O.O28
O.O26O
0.0244
0.0216
0.0196
0.0107
O.OOOO2O
0.000175
0.032
0.0296
0.0276
0.0248
o 0228
0.0139
o . 000026
0.000229
0.036
0.0332
0.0312
0.0280
0.0256
o 0176
0.000033
0.000281
o 040
0.0368
0.0348
0.0308
0.0284
0.0218
o . 000040
0.000352
0.048
0.0444
o . 04 i 6
0.0372
o . 0340
0.0313
o . 000060
0.000492
0.056
O.05I6
0.0484
0.0432
0.0396
0.0426
0.000079
0.000686
o 064
0.0592
0.0556
0.0484
0.0452
0.0557
0.000103
0.001072
0.072
0.0664
o . 0624
0.0556
0.0508
0.0705
0.000131
0.001125
0.080
o . 0740
0.0692
o . 0620
0.0564
0.0870
0.000160
0.001389
o 088
0.0812
0.0764
0.0680
0.0624
0.1331
0.000195
0.001688
0.096
0.0888
0.0832
0.0744
0.0680
0.1408
0.000232
0.002004
0.104
0.0960
o . 0900
0.0804
0.0736
0.1471
0.000272
0.002356
O.II2
0.1036
o . 0968
0.0868
0.0792
0.1586
0.000306
0.002725
0.120
0.1108
0.1040
0.0928
0.0848
0.1958
0.000364
0.003329
In metallic transmission-cables, wires of less than 0.02
inch or more than 0.08 inch diameter are very seldom
CALCULATION OF DIAMETERS OF CABLES. 2O3
used. The values of d given in these two tables, in
the second to the fifth columns, are taken from values
contained in the first column, and should in practice be
taken in round numbers. The quality of the metal
used for transmission-cables is of first importance, from
the fact that only superior qualities can withstand for
any length of time the rapid wear to which the cables
are subjected. Swedish iron, which possesses at the
same time a remarkable tenacity and great strength, is
especially adapted for the wires of transmission-cables.
In order to reduce as much as possible the number of
joints, only long wires should be used. Experience has
shown that for transmission-cables wires of steel offer
no advantages over those of good iron.
Diameter of wire S for number of wires / =
j (PX)
s H
S n
36
4 2
48
60
72
O.O2O
0.0188
0.0180
0.0168
0.0160
1554
0.025
O.O24
0.0228
0.0220
O.O204
0.0192
2685
0.043
0.028
o . 0264
O.0256
0.0236
o 0224
4264
0.068
O.O32
o . 0304
o . 0292
O.0268
0.0252
6365
O.IOI
0.036
o 0340
O.O328
o . 0304
0.0284
0062
0.144
0.04O
0.0380
o . 0364
0.0336
0.0316
12431
0.197
0.048
0.0456
0.0436
o . 0404
0.0380
21481
0.341
0.056
0.0532
0.0508
0.0472
0.0444
34112
0-542
0.064
0.0608
0.0580
0.0540
o 0508
50919
0.894
O.O72
0.0684
0.0656
o 0608
0.0572
72499
I.I52
O.080
0.0764
0.0728
0.0676
0.0636
99451
1.580
0.088
0.0836
O.OSOO
0.0744
0.0700
132369
2.103
0.096
0.0912
0.0872
0.0808
0.0760
171851
2.730
O.IO4
0.0988
0.0944
0.0876
0.0824
218493
3-471
O.II2
0.1064
O.IO16
0.0944
0.0888
272892
4-335
O.I2O
0.1140
0.1092
O.IOI2
0.0952
335646
5-332
In the formulas (632)-(634) the radius R of the pul-
leys is supposed to be known ; the values of S given
204 BELTS AND PULLEYS.
r>
by them are admissible only when the ratio -^ gives
for the tension s a value which, added to S, does not
exceed 25601.4 pounds. In the case where s-{- S ex-
ceeds this limit, it is convenient to begin the calculation
by giving to R a greater value. To make use of the
preceding formulas and tables, we must begin by fixing
upon a value for the tension S. This may easily be
done with the aid of the considerations contained in
the following paragraph, and in the examples which
we now give we shall suppose this preliminary opera-
tion already accomplished.
Example It is proposed to transmit, by means of a
metallic cable running over pulleys 9.84 feet in diame-
ter, a force of 550 pounds : required the proper diameter
for the wires of the cable, supposing the number to be
i = 36.
If we take 5 = 9956.1, we shall have -= = =-- =
0.0552, which in the first table (column 6, line 9) cor-
responds to a diameter of 6 0.064 inch. From
this we obtain -^ = ' = 922, which, in the table on
S 0.064
page 201, corresponds nearly to .S = 9956.10, and is
therefore admissible. If we had taken R = 48 inches,
n .o
we should have had -= = -z~ 75O a value less
o 0.064
than the limit mentioned above, and it would therefore
be necessary to increase the value of R.
Example. The force transmitted by a metallic cable
is 300 horse-power, and the velocity v = 82 feet per
CALCULATION OF DIAMETERS OF CABLES. 2O5
second; taking 5 = 11378.4, and consequently s =
TT
25601.4 11378.4 = 14223, we shall have -~r- =
- = 0.000322. In the first table the near-
1 1370-4 X 02
TT
est value of -^- is 0.000306 (column 7, line 15). The
377
diameter for the wires is therefore d = 0.112 inch for
i = 36, S = 0.0868 inch for i = 60. For the value
s = 14223, we have, for the radius of the pulleys, R =
14223000 X 0.0848
= 84.8 inches. The expression v =
^- gives for the number of revolutions per minute
_ 82 X 12 X 60 _
~~ ~6^8~X8i8~
Example. It is required to calculate the horse-power
which may be transmitted by a cable of thirty-six wires,
the diameter of the wires being 0.08 inch, the diame-
ter of the pulleys 9.84 feet, and the number of revolu-
tions per minute 90. In this case we have -~ =
= 738, which, from formula (637), gives s = - ~
= 19272.3 and 5 = 6329.1. For d = 0.08 and i = 36,
j_ r
the first table furnishes the value 1000-7^5-- =
SRn
0.001389; hence
0.001389 0.001389 X 6329.1 X 59-04 X 90
H = ~1^~ SRn = - looo
= 46.71 horse-power. With a pulley of 8 feet diameter
2O6 BELTS AND PULLEYS.
R 48 14223000
we would have ^ = ^ = 600, s= ^~ = 23705,
S= 1896.4. Consequently
0.001389 X 1896.4 X 48. X 90
H= - - = 1 1. 40 horse-power.
Example. Upon the driven arbor of a transmission
by cable a resistance of no pounds acts continuously
with a lever arm of 40 inches. Required the proper
diameter for the 36 wires of the cable, supposing we
give to the pulleys the smallest admissible radius. In
order to satisfy this last condition, we ought to take
(from what precedes) s 17067.60 and 5 = 8533.80,
which gives -^ (PR) 2 X 1 10 X 40 = 8800. In the
second table (column 6, line 5) we find, for the nearest
value oi-~(PR), S = 0.036 inch. From the table on
r>
page 201, therefore, we obtain -~ = 833, R = 833 X
0.036 = 30 inches.
Example. A cable of 42 wires transmits a force of
30 horse-power at a velocity of 100 revolutions per
minute. Required the proper diameter for the wires
of the cable, taking 5 = 8533.80. In this case s =
s H 17067.60 30
,7067.60, and 5 - = -- X = 0.6. The sec-
ond table gives, for the nearest value of -~ to 0.6,
6 = 0.056 inch. From formula (637), then, we have
142^^000
for the radius of the pulleys R = 0.056 X " =
833 X 0.056 = 46.65 inches.
DEFLECTIONS IN A CABLE. 2O/
21. Deflections in the Cable of a Horizontal Transmission.
In order that, in the two parts of a transmission-
cable, the tensions Tandt have proper values (not too
small, for then the cable will slip on its pulleys ; nor
too great, because the wear is then great), the deflec-
tion which we give to each part, in a state of repose,
must be a determined quantity. It is equally necessary
that we know the deflections which are produced dur-
ing the motion of the cable, in order to leave sufficient
room for the passage of the cable. The deflection of
a cable depends upon the tension of its wires.
Let us represent by
A the separation of the pulleys of a horizontal trans-
mission in feet; h the deflection of the cable in feet
(Ji l for the driving part, h^ for the driven part, and /i a
for the state of repose) ; S the tension per square inch
in the wires (S l for the driving part, S^ for the driven
part, and S for the state of repose).
For a metallic cable of any number of wires we have
the relations
A
and
'' = 0.3535 [0.369^ - y (-3<59;|)' - i] (638)
- A = 3.8029^- +17 J. . . . (639)
By means of these formulas the following table has
2O8 BELTS AND PULLEYS.
been calculated. As a first approximation we may
take simply
h A
- = 0.475 5 - (640)
In order to make use of the table, we begin by
A
determining from the given quantities the ratio -=.- of
the separation of the pulleys to the tension developed
in the wires, and then find in the table the number
nearest to this ratio. From this we obtain the value
of-,-, which gives the amount of deflection h. The
A
tension S of the cable in a state of repose is not the
arithmetical mean between S t and S z ; we may, by a
more complicated calculation, however, determine it
from the length of the two cable parts. The value
which we need to know is the deflection h in the two
parts of the cable for a state of repose, and we have
approximately
h o = * + *'* = .6;// 2 -f 0.28*.. . (641)
This expression gives for k a a value slightly too
great, but which approaches more nearly the true value
as the tensions 5, and S z become less. The error may
be still farther decreased by using, instead of exact
values of h, and // those furnished by formula (640).
The driving part of the cable does not necessarily
DEFLECTIONS IN A CABLE.
209
occupy the higher position, as is the case in Fig. 77 :
it may be placed in the lower position, as in Fig. 78.
In the latter, the space required by the deflection of
FIG. 77.
the cable is considerably less than in the former. The
two parts of the cable do not intersect each other as
long as h^ //, < 2R. With a cable in motion, we
may place, at the lowest point of the curve, a gradu-
ated rule, by means of which we may observe at any
instant the tensions. The graduation of the rule may,
moreover, be such as to give directly the tension 5.
14
2IO
BELTS AND PULLEYS.
TABLE OF DEFLECTIONS IN METALLIC CABLES.
h
~A
5
A
A
5"
A
.9
O.O03
0.006
0-033
0.069
0.063
0.128
0.093
0.183
0.004
0.008
0.034
0.071
0.064
0.130
0.094
0.185
O.OO5
O.OII
0-035
0.073
9.065
I 0.132
0.095
0.186
O.OO6
0.013
0.036
0-075
0.066
I 0.134
0.096
0.188
O.007
0.015
0.037
o 077
0.067
0.136
o 097
O.lgO
0.008
0.017
o . 038
0.079
0.068
0.138
0.098
O.lgl
o.oog
0.019
0.039
O.08 1
O.Ofg
0.140
0.099
0.193
O.OIO
O.O2I
O.O40
O.OS3
O.O7O
0.142
O.IOO
0.195
o.on
0.023
O.O4I
0.085
O.OJI
0.144
O.IOI
0.196
O.OI2
O.025
O.O42
0.087
O.072
0.145
O. IO2
O.igS
0.013
O.O27
0.043
0.089
0.073
0.147
0.105
0.203
0.014
O.O29
0.044
O.Ogi
o 074
0.149
O.IIO
O.2II
0.015
O.O3I
0.045
0.093
o 075
0.151
o. 115
O.2I9
0.016
0.034
0.046
0.095
o 076
^153
O.I2O
O.226
0.017
0.036
o 047
o 097
0.077
155
O.125
0.234
0.018
0.038
0.048
0.099
0.078
0.156
0.130
0.241
0.019
O.O40
0.049
O.70I
0.079
o. 158
0.135
0.248
O.O2O
O.O42
0.050
0.103
0.080
o.i 60 '
0.140
0-255
O.O2I
0.044
0.051
O.I05
o.oSr
0.162
0-145
0.261
O.O22
0.046
0.052
o. 107
O.Of2
0.164
O.I5O
0.267
O.O23
0.048
0-053
0.109
0.083
0.165
0.155
0.274
O.O24
O.050
0.054
O. 1 11
0.084
0.167
o . 1 60
0.279
O.O25
O.052
0.055
o. 113
0.085
0.169
0.165
0.285
O.026
0.054
0.056
0.115
o.o6
0.171
o. 170
0.291
0.027
0.056
0.057
0.117
0.087
0.173 j
0.175
0.296
O.O28
O.O59
0.058
0.119
0.088
0.174
0.180
0.301
O.O29
O.O6 1
0.059 1
O.I2I
0.089
0.176
0.185
0.305
O.O3O
0.063
0.060 1
0.123
0.090
0.178
0.190
O.3IO
0.031
0.065
0.061
0.125
0.091
0.179
0.195
0.315
0.032
0.067
0.062
0.127
0.092
0.181
O.2OO
0.319
Example. In the last example of 20 the separation
A of the pulleys is 360.8 feet, and we take the tension
5, = 8533.8 pounds per square inch. Required the
deflections in the parts of the cable. For the driving
pare of the cable the relation = =
360.8
z = 0.0423
DEFLECTIONS IN A CABLE.
211
corresponds in the table (column 2, line 18) to the value
- =z 0.02. Hence we have //, = 360.8 X 0.02 = 7.216
A
feet. For the driven part of the cable we have from
formula (631) 5 = -
8533.'
= 4266.9, and consequently
->4 360 8 A
-7, = ^ = 0.0845. For this value of -~ the table
S 4266.9 5
gives (column 4, line 9) --T- = 0.041, and we have // =
A
360.8 X 0.041 = 14.79 f eet - From formula (641) the
deflection of the cable in a state of repose is k =
FIG. 78.
0.67 X 14-79 + - 28 X 7.216 = 11.93 feet. We have
also /i. 2 //, = 14.79 7-2 1 6 = 7.574 and 2R =
2 X 3.8875 = 7.7750 feet. Since 2A* > A t //,, we may
if necessary make use of the disposition of Fig. 78.
(See first example of 22.)
Example. In the third example of 20 the distance
below the line of centres of the pulleys is 9.84 feet ; it
is required to determine the proper distance between
the pulley-centres. Assuming that we can make use
of the disposition represented in Fig. 78, the greatest
admissible value for the distance of separation of the
pulleys may be calculated from the deflection of the
cable while in a state of repose. Making use of the
approximate formula (640), and remembering the value
212 BELTS AND PULLEYS.
Sj = 6642.141 pounds per square inch, we shall have
/p
and h, = 2/1,. Formula (641)
then gives
(0.67 X 2 + o.28>4 8 X 0.4755
feet.
22. Transmission by Cable with Increased Tension.
When the pulleys of transmission are very distant
from each other the deflections given by the preced-
ing formulas become so great that it is often necessary
to place the pulleys at a great elevation, or to provide
a deep trench for the cable when we wish to dispense
with intermediate pulleys and guides (see 28). In a
great many cases we may arrive at the same result by
simply giving to the cable a greater tension than is
necessary to prevent slipping, and taking care to give
to the cable a diameter sufficient to withstand the ad-
ditional strain. This artifice may be employed all the
more easily when the transmission is to be used for
moderate forces, and consequently a small diameter oi
the cable is sufficient. It is only necessary to examine
carefully the rules which follow, to be convinced that
a rational use of this method presents in reality little
or no difficulty.
A transmission by cable, established under the
above conditions, constitutes, by way of distinction
from ordinary cable-transmission, what we term a
TRANSMISSION WITH INCREASED TENSION. 21$
transmission with increased tension. We may distin-
guish it from ordinary transmissions by giving the sign
s to the forces and dimensions connected with it
(T s , f s , S s , <$ s instead of T, t, S, and tf). The tension
T, in the ordinary mode of transmission, ought not to
be less than 2P; in a transmission with increased
tension the tension ought to be increased by a certain
factor which we shall designate by m. We shall there-
fore have
T. = mT, *.= (2*-l)/, -f s = 2 ^-- (642)
The tension 5, in the driving part of the cable is not
changed, but in the driven part the tension 5 M is no
longer equal to '. We take instead
, . . . .
The diameter S a of the wire is deduced from the
diameter d given by one of the formulas (632) to (634),
by means of the relation
6 S d Vm. ..... (644)
If, however, d is calculated from formula (636) or (638),
we must take
6 S = dVm. ..... (645)
From these formulas the following table has been
calculated. It is important to remark, that in cables
with increased tension the strain in the wires is no
2I 4
BELTS AND PULLEYS.
greater than in ordinary cables, because they have a
proportionately greater diameter. The cable is heavier
in the former than in the latter case, and should there-
fore be strained more firmly over the pulleys in order
to reduce the deflection in the driven part.
Ts
Ts
ts is Szs
528 ts
s_ / _
Ss 3 ,
m =T
p
7 ~ j> ~ ~s*
Si ~ rl
y = ^<
1.2
2.4
1.4
0.58
.10
.06
1-4
2.8
1.8
0.64
.18
.12
1.6
3-2
2.2
0.69
.26
17
1.8
3.6
2.6
0.72
34
.22
2.O
4.0
3-0
0.75
.41
.26
2.2
4-4
3-4
0.77
.48
3
2.4
4.8
3-8
0.79
55
34
2.6
5-2
4-2
0.81
.61
38
2.8
5-6
4.6
0.82
67
.41
3-o
6.0
5-0
0.83
73
44
3-2
6.4
5-4
0.84
79
47
3-4
6.8
5-8
0.85
.84
50
3-6
7-2
6.2
0.86
.90
53
3-8
7-6
6.6
0.87
95
56
4.0
8.0
7.0
0.88
.00
59
4.2
8.4
7-4
0.88
05
.61
4-4
8.8
7-8
0.89
.10
.64
4.6
9.2
8.2
0.89
.14
.66
4.8
9.6
8.6
0.90
.19
.69
5-0
IO.O
9.0
0.90
.24
7i
5-5
II. O
IO.O
0.91
.36
75
6.0
12.
II. O
0.92
45
.82
6-5
13-0
12.0
0.92
55
.87
7.0
I4.O
13-0
0-93
65
.91
7-5
15-0
14.0
0-93
74
.96
8.0
16.0
15.0
0.94
83
.00
Example. In the first example of 21 the driven
part of the cable has a deflection of 7/ 2 = 11.76 feet,
and the diameter of the wire is 0.056 inch. If we wish
to diminish the value of h^ by using a cable with
increased tension, the value of 8 must be increased ac-
cordingly. If we take m = 2, the table gives (col-
TRANSMISSION WITH INCREASED TENSION. 215
umn 4, line S)~f = O-7S, $** = O./5 X 8533.8 = 6400.35
A 360.8
pounds. Consequently -~- = ^ = 0.056, which,
in the table of 21, corresponds to -j- = 0.027 or
h 0.027 X 360.8 = 9.74 feet. The tension of de-
flection s has the same value as if for an ordinary ca-
ble ; the quotient -~- does not change its value, and
Oj 11
consequently d may be determined by means of formu-
la (636). The preceding table gives, then, d a 1.266
= 1.26 X 0.56 = 0.07 inch.
When, in calculating the diameter S for an ordinary
cable of 36 wires, we obtain a very small value, the
cable itself may have such a small diameter that its
manufacture involves as great an expense as for a cable
of larger diameter. In such a case we cannot recom-
mend too highly the use of a transmission by cable
with increased tension, which has the advantage of re-
ducing the deflection in the driven part of the cable
without appreciably increasing the expense of manu-
facture. As a general rule, we should never make use
of wires of a less diameter than 0.04 inch, so that the
minimum diameter of cable may be 0.32 inch.
Example. For a transmission by cable, we have
given H 5-5, n 100, and A = 590.4. If we as-
sume 5, = 14223 and s = 11378.4, we have -~- - :
o, n
= 1Z.JZ x JLi = 0.044, which, for i 36 (table on
14223 100
page 203) gives, for the diameter of the wire S = 0.024
2l6 BELTS AND PULLEYS.
A 590.4 A
inch. We have also -^r- = " 0.0415, -~- =
590.4
- = 0.0830, and consequently, from the table of
page 210, h, = 0.0198 X 590.4 = 11.69 f eet > ^ =
0.04 X 590.4 = 23.616 feet, //, //, = 23.616 11.69
I A?'?
= 11.926 feet. But since R = d = 1250
1 1378.4
X 0.024 = 30 inches, h. t //, is greater than 2R. In
this case, therefore, we cannot place the driven part of
the cable above the driving part, and the axes of the
pulleys must have a height above the ground at least
equal to R -\- // = 2.5 -f 23.62 = 26.12 feet. Sup-
pose now we take for the cable diameter 0.32 inch,
instead of 8 X 0.024 = 0.192 inch; that is, we take
0.04 inch for the diameter of the wires. We have then
-TT = '- - = 1.67, and the preceding table gives (col-
umns 6 and 4, line 18) S ys == 14223 X 0.89 = 13058.47.
A 590.4
Consequently -~- = = 0.0452 and k at = 0.0228
^ 2S I 35-47
X 590.4 = 13.46 feet, // M /*, = 13.46 11.69 = 1-77
feet. As before, R = 1250 X <$ s 50 inches and 2R
= 8.33 feet: the inequality A M //, < 2R is now sat-
isfied, and we may give to the cable the desired ar-
rangement. The maximum deflection in this case cor-
responds to the state of repose, for which we have, from
formula (641), // M = 12.28 feet. The height of the pul-
ley-axes above the ground must be at least h M -j- R
= 12.28 -f~ 4-i6$ = T 6-445 feet; that is, less by nearly
10 feet than for the first calculated cable.
TRANSMISSION BY INCLINED CABLE,
23. Transmission by Inclined Cable.
Of the various transmissions by metallic cable, the
one which has met with the greatest development cor-
responds to the case in which the pulleys are not on
the same level, one being higher than the other, and
constitutes, therefore, what we call an inclined trans-
mission. We give here the rules necessary for such
transmissions. In the cable BCD, Fig. 79, which rep-
resents a part of an inclined transmission, the summit
of the curved axis is not in the middle of the distance
between the points of suspension, and the deflections
are therefore different from those in the cable of a
horizontal transmission. The deflections may, how-
ever, be easily determined in functions of the elements
of a horizontal transmission, having the same separa-
tion of pulleys and sensibly the same tensions.
Let us represent by
h and A, respectively, the deflection of the cable
and the separation of the pulleys of a horizontal trans-
mission;
2l8 BELTS AND PULLEYS.
S the tension corresponding to the point of sus-
pension of the part of the cable under consideration ;
// and h", respectively, the smallest and greatest de-
flsction (FC and EC} in an inclined transmission, in
which the separation of the pulleys measured horizon-
tally is equal to A ;
a' and a", respectively, the distances CB l and CD l of
the summit of the curve from verticals through the
points of suspension ;
5' and S", the tensions (at B and D] at the lower
and higher points of suspension respectively ;
H the difference between the levels (EF) of the
points of suspension.
The values of h and S may be determined by means
of the rules already given. We have then
; (646)
H\
a"=A-a-, (647)
S' = S- 3.8o4(//-/0, S" = S-f 3.8o4(/T - /*),
S" - S' = 3.804^7. . . (648)
In certain cases the value of a' may be negative ; the
summit of the curve of the cable prolonged is then sit-
uated beyond the lower pulley. The tension of flec-
tion s, and consequently the diameter of the pulleys,
are determined when we have obtained the value of the
tension S", which very often does not differ materially
from 5. The difference between the two tensions be-
TRANSMISSION BY INCLINED CABLE. 2 1C)
comes important only in cases where several inclined
transmissions are taken from a single higher pulley.
Example. A transmission by cable, the data of
which are the same as in the fifth example of 20, has
its pulleys placed at different heights; taking for the
difference in the levels of the pulleys H = 16.4 feet, it
is required to determine the deflections and the posi-
tions of the cu^ve-summits.
For the driving part of the cable we have
5, = 8533.8, 7/i = 7.216 feet, // = 16.4 feet, A = 360.8 feet.
Stating at the lower pulley, we have, from formula
(646),
/ I 16.4' 1.4
//, = 7.216(1 + iB -j^j - -^ ;= ,.35 feet,
h'\ = 7.216 -f- 1.35 = 8.566 feet ;
. .
a\ = *-f-(i - - ^~D= 180.4x0.432 = 77-93 feet,
a'\ = 360.8 + 77-93 = 382.87 feet.
For the driven part of the cable,
S 2 = 4266.9, // 2 = 14.79 f eet I
consequently
/ i 16.4" \ 16.4
*, = .4.79(1 + ,- 6 = - ~ = 773 feet,
A", = 16.4 + 7.73 = 24.13 feet.
220 BELTS AND PULLEYS.
For the state of repose,
h, = 0.67 X 14-79 + - 28 X 7.216 = 12.05 feet ;
hence
/ i i6.4 2 \ 16.4
h . - 12.05^1 + Y 6 ^J - 2 ~ = 5-24 feet,
h'\ 16.4 -j- 5.24 = 21.64 feet;
i i6
4 -i
a" = 360.8 119.06 = 241.74 feet.
The tensions in the driving part of the cable are as
follows: S', = 8533.8 - (7.216- 1.35)3.804 = 8511.5,
S", = 8533.8 + (8.566 - 7.216)3.804 = 8538.94; the
values of S\ and 5", differing so slightly from S 1 that
we may neglect the difference.
The heights which the calculations furnish for the
deflections of an inclined transmission should be laid out
in the drawing to a scale three or five times that of the
horizontal lines ; we then trace the curve of the cable
as an arc of a parabola (see the following paragraph),
and try if the conditions of the ground will permit us
to use the curve obtained. If this prove not the case,
we must recommence the calculation by adopting new
values for the tension until we have obtained a curve
which will satisfy the conditions. With a little prac-
tice, it is easy to determine by the eye the proper val-
ues to be adopted, and the calculation may then be
made without difficulty.
METHOD OF TRACING THE CURVES OF CABLES. 221
24. Method of Tracing the Curves of Cables.
The curve of a cable may be drawn with sufficient
accuracy for ordinary purposes by assuming it to be an
arc of a parabola. After having determined the summit
C of the part of the cable BCD, Fig. 80, as explained
in the preceding section, divide into two equal parts,
at the points C l and C v the two distances B.^C and
D (B l D l being tangent to the curve of its summit),
and through the points C, and , draw the lines BC^
and DC^ which give the directions in which the cable
leaves the pulleys. Divide the distances CC t and C,B
into a certain number of small equal parts at the
points i, 2, 3, etc., and I, II, III, etc.; by joining il,
2ll, 3!!!, etc., we obtain a series of lines tangent to
the required parabola. By a similar method with CC^D
we obtain the other part of the curve. When the sum-
mit C of the curve falls outside of the pulleys, on the
side of the pulley which occupies the lower level, a part
of the parabola near the summit cannot be made use
of. but the construction is still the same.
222 BELTS AND PULLEYS.
25. Transmission by Cable with Pulleys near together.
When the distance between the pulleys of a trans-
mission by cable is small, it is necessary, first of all,
that the deflections have not too small values, in order
that the cable may run properly upon the pulleys, and
also that we may be able to shorten the cable without
seriously increasing the tension. We adopt then for
S t a very small value, and thus determine upon a value
for the deflection ; then, by means of formula (638)
and the table calculated from it, obtain 5, ; / and R
are then calculated as we have already indicated. For
a small tangential resistance and a small separation of
the pulleys, transmissions by cable may still be used
with satisfactory results.
Example. A metallic cable transmits a force of 6
horse-power at 150 revolutions per minute ; the separa-
tion of the pulleys is 65.6 feet and the deflection in the
driven part of the cable 3.28 feet. We have then r
A
= 0.05, which, from 21, corresponds to -~ = 0.103,
and we obtain 5, = = 637. In order to find the
value of tf, we must know that of s. Assuming that
s -f- 5, is still equal to 25601.4. we have s = 25601.4
s H 24964.4 6
- 637 - 24961.4, which g,ves - s - - = ^- - =
1.57. The second table of 20 gives (column 7, line
n), therefore, S = 0.08 inch, for i = 36. From for-
mula (637) we have for the radius of the pulleys R =
RIM OF CABLE-PULLEYS. 22$
0.08 ^ = 45.6 inches. From what precedes, we
find that these values of tf and R are perfectly admis-
sible. If we wish to take for the diameter of the ca-
ble, d = 8# = 0.48 inch, that is, <$ is reduced to 0.06
inch, it is only necessary to give to R a smaller value.
In this case the table of 20 gives (column 7, lines 8
and 9) ~- = 0.718, hence s = 0.7 185, -73. = 0.718 X
O, 11 fl
637-!- = 11434.15, and formula (637) gives R
0.06 = 74 inches. In some cases pulleys of
large radii cannot be conveniently used, and we are
obliged to use pulleys of different radii in order to
make the deflections great enough. For the transmis-
sion of considerable forces, we obtain good results only
on the condition of giving to the pulleys a certain ve-
locity of rotation, the limits for which are indicated at
the end of the following paragraph.
26. Rim of Cable-pulleys.
When first used, the rims of cable-pulleys were made
of wood covered with leather, but practice soon de-
monstrated the fact that rims of metal are preferable,
and at the present time the latter are used almost ex-
clusively in all cases where durability forms an impor-
tant factor. Figs. 81 and 82 represent two cast-iron
rims, single and double. The sides of the groove in
the single rim are inclined at an angle of 30 with the
middle plane of the pulley. In the double rim such
224
BELTS AND PULLEYS.
an inclination would necessitate too great a weight
for the projection between the two grooves ; the in-
clination of the sides of this projection is therefore less
than 30. In Fig. 82 (which represents a portion of a
large pulley) this inclination is 15. All the dimensions
indicated in the figures are in terms of the diameter d
FIG.
of the cable. Since cables of less than 0.4 inch diameter
are seldom used, we may consider the value of d = 0.4
inch as the inferior limit of the unit for the construction
of cable-pulleys. The grooves in the faces of the pul-
leys are bottomed with gutta-percha driven into the
dovetails, as shown in the figures ; or small pieces of
wood, which are introduced into the dovetails through
openings in the side of the rim. Fig. 82 shows two
RIM OF CABLE-PULLEYS. 22$
openings of this kind covered up by pieces which are
bolted in after the insertion of the wooden pieces. Of
late years grooves with leather bottoms have come in-
to use for very heavy cables ; to this end old belts cut
into strips and wedged into the dovetails may be ad-
vantageously used. Professor Fink has successfully
employed bottoms formed by winding twine tightly
around in the dovetails ; bottoms thus made give great
resistance to slipping. Bottoms of cork have also been
used, but while they offer the advantage of being in-
expensive, they have not been tested sufficiently in prac-
tice to determine their utility for transmission by cables
where there is danger of slipping. When we wish to
make use of bottoms of twine, the depth of the dove-
tails need not be so great as that indicated in the fig-
ures. In the first three modes of furnishing the grooves
with bottoms which present more resistance to slip-
ping than cast-iron (gutta-percha, wood, and leather),
the profile of the groove upon which the cable rests
may be hollowed out after the introduction of the ma-
terial into the dovetails. Pulleys of 12 to 15 feet in
diameter are ordinarily cast in two pieces, which makes
them easier to handle and transport ; projections are
cast upon the inside of the rim by means of which the
two parts may be bolted together.
In order that no harm may come to the rim through
excessive centrifugal force, the velocity of rotation of
the rim should not exceed 100 feet per second. The
velocity of about 90 feet per second, which is now com-
monly given to metallic cables, may be considered as
without disadvantages in ordinary practice
226 BELTS AND PULLEYS.
27. Arms and Nave of Cable-pulleys.
The body and rim of a cable-pulley are ordinarily of
cast-iron, as is often the case with the entire pulley.
We however sometimes find arms of wrought-iron set
into cast-iron rims (see Fig. 96). In any case the num-
ber of arms A is determined from the expression
T /?
The cross-sections of cast-iron arms arc oval or flanged ;
in either case the width in the plane of the pulley is
given by the formula
..... (650)
4 A
In a flanged cross section the thickness of the prin-
cipal flange (in the plane of the pulley) is e = , and
that >f the secondary flange e' = \e. In an oval cross-
section the thickness is one half the width, as in pul-
leys for transmission by belt. The width at the rim
may be taken equal to -f the width at the nave.
Arms with flanged cross-sections are generally
straight (Fig. 83), and eight in number, while those
having oval cross-sections are curved, either single, as
explained in 14, or double, as in Fig. 84.
To draw double-curved arms for cable-pulleys, we
p
begin by striking a circle with a radius OA -, then
ARMS AND NAVE OF CABLE-PULLEYS.
227
take upon the circle the lengths AB and BC, correspond-
ing to the division by the arms. Draw the arc OE
representing one portion of the double curve, in the
same manner as for single-curved arms. Through the
centre of curvature C of this arc (which, for eight
arms, is on the circumference ABC) draw the line CED,
FIG. 83.
FIG. 84.
and taking ED EC, obtain the radius of curvature
corresponding to the part EF o( the arm. To draw
the curves which limit the profile, it is necessary only to
follow the method of 14, remarking that the centres
for the arcs are found upon the line CD.
When straight arms are used the nave is sometimes
cast with grooves, into which iron rings are afterwards
placed ; by putting on the rings hot, and allowing them
to cool, they are very firmly fixed, and add greatly to
228 BELTS AND PULLEYS.
the strength of the pulley. The dimensions of the
nave are determined, as already explained for pulleys
for transmission by belt, in 13.
Example. -In a transmission by cable the radius of
the pulleys is 50 inches, the diameter of the arbor is
4.8 inches, and that of the cable 0.48 inch ; it is re-
quired to determine dimensions of the pulley. From
formula (649) the number of arms is A = 4 -)
40 0.40
= 7. The width of the arms at the nave is, from
formula (650), h = 4 X 0.48 -| = 1.92 -(- 1.8 =
4 7
3.72 inches. Formula (604), in which d represents the
diameter of the arbor, gives for the thickness of the
4.8 50
nave w = 0.4 -j- 7 ^- + v~ = 0.4 + 0.8+1 = 2.2 inches.
The length of the nave (Z.) ought to be at least equal
to 2$- X 2.2 = 5.5 inches.
For very important transmissions it is prudent to
have a reserve cable ; that is, to divide the force to be
transmitted between two cables, each having sufficient
strength to transmit the whole force. An arrangement
of this kind is in use at Schaffhouse, in a transmission
by metallic cable of 600 horse-power, of which we shall
have occasion to speak farther on. In this transmis-
sion the two pulleys are placed upon one driving arbor,
as shown in Fig. 85. The pulleys which run loosely
upon the arbor are fixed to the two gear-wheels B and
D, which engage with the intermediate gears A and C.
The latter gears run loosely upon their journals, which
form a part of and rotate with the driving arbor. By
means of this arrangement each cable is made to trans-
ARMS AND NAVE OF CABLE-PULLEYS.
229
mit an equal share of the total force. If one of the
cables breaks, the pulley over which it ran is free to
rotate in the opposite direction, and the gears are thus
put in motion. In order to prevent too rapid motion
in the pulley, which by the breaking of a cable may be-
come loose upon the arbor, the transmission at Schaff-
house is provided with a powerful brake, by means
of which the motion of the motive turbine-wheel may
FIG. 85.
be almost instantaneously arrested. Instead of the
intermediate gears A and C, simple sectors, such as are
represented in the figure on the right, might be used in
this trrnsmission. In this case as soon as a break in
one of the cables occurred, the sectors would be put in
motion, and when the toothless parts came opposite
the gears D and B the motion of the pulleys would be
stopped, and danger of further accident avoided.
230
BELTS AND PULLEYS.
28. Pulley-Supports and Intermediate Pulleys
When the principal pulleys of a transmission by cable
are placed far apart, and especially when they are not
high above the ground, it is often necessary to support
the cable by other pulleys. In certain cases it is suffi-
cient to support at a single point the driven part of
the cable while the driving part is left free, as shown in
Fig. 86. When several pulley-supports are necessary,
the driving part is also supplied with at least one, as
shown in Fig. 87. In other cases the number of pul-
ley-supports is the same for both parts of the cable ; it
FIG. 87.
is then best to place the pulleys of the driving part
directly over those of the driven part, instead of juxta-
positing them, as has been several times attempted, and
which causes rapid wear of the cable, consequently
produces a wearing friction upon the pulley-grooves,
and also tends to make the cable run off the pulleys.
P ULLE Y-SUPPOR T& INTERMEDIA TE P ULLE YS. 2 3 1
In the arrangement represented in Fig. 88 the pulley-
supports of the driving part are placed under those of
the driven part in order to gain space above the ground.
In most cases when the distance between the princi-
pal pulleys makes a great number of pulley-supports
necessary, this arrangement may be advantageously
replaced by a series of successive transmissions (Ziegler),
Fig. 89. The pulley-supports of Fig. 88 are then re-
placed by intermediate double-grooved pulleys placed
at as near the same distances apart as possible, so that
in case of breakage in any of the cables a single reserve
cable may be used to replace it."*
*This has been done by Ziegler at Frankfort-on-the-Main, where a
force of 100 horse power is transmitted at a distance, of 984 metres
nearly f of a mile.
2 3 2
BELTS AND PULLEYS.
The different points at which a cable is supported
are called stations ; those which correspond to the
principal pulleys of the transmission are called the sta-
tions at the extremities and the others intermediate
\
FIG. 90.
stations. Sometimes it is necessary to change the
directions of the cable at an intermediate station ; Hirn
has proposed to accomplish this change of direction by
means of a horizontal pulley, Fig. 90, while it has also
been suggested to use a pair of bevel gears, Fig. 91.
The use of transmissions by cable is very convenient
when we wish to divide between several establishments,
belonging to different proprietors, the force derived
from a single motor: to do this we have simply to
..
OF
P ULLE Y-SUPPOR TS INTERMEDIA T.
233
make the intermediate stations the starting-points or
stations at one extremity of supplementary transmis-
sions. Stations of this kind are called division-stations.
Pulley-supports are also used in the special case in
which the driven arbor is placed almost vertically above
or below the driving-arbor. There would be serious
difficulty in making use of an inclined cable, connect-
ing directly the two pulleys A and B, Figs. 92 and 93 ;
it is preferable by far to use the pulley-supports T, T,
placed in such a manner that one part of the cable, TA
or TB, may be horizontal. It is then sufficient to
FIG. 92.
FIG. 93.
determine, by means of the preceding rules, the proper
tensions to give to the horizontal part of the transmis-
sion without reference to the inclined part.
The use of cables for the transmission of forces to
great depths into the shafts of mines, for example is
still in a period of development. We may say, how-
ever, from attempts already made in this direction, that
satisfactory results have been obtained.*
* Review of Society of German Engineers. 1866, p. 371. Werner,
" Use of transmissions by metallic cables for the shafts of mines,"
234
BELTS AND PULLEYS.
We meet with a remarkable example of this mode of
transmission in the arrangement at Schaffhouse, where
a force of about 600 horse-power, taken from the current
of the Rhine, is received by turbines at the left bank,
and is intended to be transmitted across the river to the
right bank, there to be divided among several factories.
This important application, credit for which is due to
the Society of Hydraulic Engineers of Schaffhouse, is
very nearly completed, and affords, in all its details,
information of the greatest interest to engineers.
29. Dimensions of Pulley-supports.
The pulleys intended to support the driving part of
the cable ought properly to have the same diameter as
the pulleys of transmission ; those supporting the
driven part may, in normal transmissions, have smaller
dimensions. The following table indicates the limits
below which we should not take the radius R of the
pulley-supports.
The numbers contained in the table have been cal-
culated by means of the formula
R _ 28446000
d ~ 51202.8 - S; ' '
(650
Si
$
T 2
*i
-
^?o
711.15
24890.25
563
12800.70
12800.70
741
1422.30
24179.10
57i
14223.00
11378.40
769
2844.60
22756.80
588
15645-30
9956.10
800
4266 . 90
21334-50
606
i 7067 . 60
8533-80
833
5689.20
19912.20
625
18489.90
7111.50
870
7111-50
18489.90
645
19912.20
5689.20
909
8533-80
17067.60
667
21334-50
4266.90
952
9956.10
15645.30
690
22756.80
2844.60
IOOO
11378.40
14223.00
7H
24197.10
1422.30
1053
PKESSUKE ON PULLEY-SUPPORT AXES. 2$$
The values contained in the table furnish excellent
dimensions for jR principally for large values of S t . In
transmissions with increased tension (see 21) the
difference between R and R is so small that we may
take, without disadvantage, R = R . In compound
transmssions (see 28) there is no difference in size be-
tween the principal pulleys at the extremities and the
intermediate pulleys.
30. Pressure upon the Axes of Pulley-supports.
In a transmission by cables, which we have taken
care to calculate for its entire length, we should know
the tensions at each station, and (from the curves of
FIG. 94.
FIG. 95.
the cables traced according to 23) the directions of
the different parts which are to be supported by in-
termediate pulleys. For example, in Fig. 94, for an
intermediate pulley we should know the values of T,
236 BELTS AND PULLEYS.
t, TV and /j, and their directions. We can then deter-
mine by means of formulas already given the approxi-
mate weight of the pulley, which allows us to trace
graphically (Fig. 95) the resultant Q of the different
forces. To accomplish this we draw the lines A B,
B C, CD, D E, and EF respectively equal and parallel
to T, T lt f, / and G. The line A F, which completes
the polygon, represents in amount and direction the re-
sultant Q.
Pulley-supports are ordinarily in construction iden-
tical with the principal pulleys for the same diameter
of cable. By virtue of the rules of 26 and 27, the
following formulas may be obtained for the approxi-
mate weights of the pulleys :
For single-grooved pulleys,
G rf , 145-6 , ii5.52\/y?\ , /
-^ = 0.034375 1^45 + ~^~ + -#r-)\ril + l-33 +
For double-grooved pulleys,
G rYo , 26 5- 6 , 2i2.S\/R\ , f
f = 0.034375^84 + -~ + -f- }\- d } + (0.33 +
o.oo28
Example. In the fourth example of 20 for a radius
of 30 inches the diameter of the wires (of which there
are 36) is 0.036 inch. The diameter of the cable
itself is therefore d = 8 X 0.036 = 0.288 inch, which
PRESSURE ON PULLEY-SUPPORT AXES. 237
gives , = - r Q = 104. The weight of the pulley for
Cl O.2oo
a single groove is, from formula (652), G 0.024 X
FIG. 96.
O.O028\
OOO5 -f- go~j II24804 = 2O4
pounds.
238 BELTS AND PULLEYS.
Example. For the transmission of 300 horse-powei
of the second example of 20 we have d = 0.087,
which for a cable of 60 wires gives d = 12.8 X 0.087 =
yT> O -
1. 1 1 inches, R = 85 inches. Consequently -3 = =
77. The weight of the pulley for double grooves is
therefore G = 1.37 X 0.034375 [(84 + ^- + ^~)
, / , 0.464 , o.ii5\ , / 0.0028
77 -t- ^0.33 + ~ l - rr + ; J 59^9 + (,0.005 + -7^
45^533 = 2I 93 pounds.
For very large transmission-pulleys the weights be-
come important considerations, as may be seen by the
last example. For this reason engineers have sought,
by modifying the system of construction, to reduce the
weights. By adopting the arrangement represented in
Fig. 96, in which the arms are formed by two series of
inclined rods meeting in pairs at the rim of the pulley,
the weights may be reduced to about three quarters
those given by the preceding formulas. In Sweden,
where transmission by cable has already been firmly
established, pulleys constructed of sheet- iron have
been successfully employed.*
31. Station Pillars.
Fig. 96 represents the arrangement of a station fof
the intermediate pulleys of a compound transmission.
To support the pedestals for the axle of a pulley of
this kind we may with propriety build up a frame-work
*See Annals of the Society of German Engineers, 1868, p. SQI.
STA TION PILLARS.
239
of wood ; it is, however, preferable to use a solid pier
of brick or stone, upon which are fixed either low
pedestals, as in the figure, or high pedestals, such as
FIG. 97.
Figs. 97 or 98, which are especially advantageous when
the height of the pulley-axes above the ground is great.
The pedestal-plates are fastened to the pier by means
..j 1...
STA TION PILLARS.
2 4 I
of four strong anchor-bolts passing through the pier
and into the foundation. The length of the axle be-
tween the centres of the journals is generally taken
equal to the radius of the pulley. In stations for two
pulleys the pier is divided to a greater depth, and the
axle of the upper pulley is supported by high pedes-
tals. In certain cases the two pulleys are placed side
by side, as indicated by the dotted lines ir Fig. 96
FIG. 99.
an arrangement especially convenient for putting on
the cable. Because of the weight of metallic cables
this operation is by no means simple ; to accomplish it
Ziegler has employed an arrangement similar to Her-
land's tool for putting on belts. Fig. 99 represents
the arrangement, which consists of a curved piece of
angle-iron, fixed in the groove of one of the pulleys by
means of hooked bolts (see figure in centre). In the
left-hand figure the cable is at the side of the pulley;
242 BELTS AND PULLEYS.
in the right-hand figure it rests in the groove of the
pulley.
Although throughout this entire chapter we have
assumed that the two pulleys of transmission have the
same diameter, it does not follow that we may not use
transmission-pulleys of different diameters. Indeed it
may sometimes be necessary to have such an inequal-
ity of pulleys. In all cases of this kind it is best to
confine ourselves to the determination of the dimen-
sions of the smaller pulley and the corresponding
diameter of the cable ; taking care, however, not to
lose sight of the fact that, in order to obtain the best
results from our transmission, it is essential, first of all,
that the diameters of our pulleys be no smaller than
the limits indicated in the preceding pages.
APPENDIX.
i.
ACTUATED by a desire to obtain, by experiment with
the belts and pulleys in ordinary practical use, the co-
efficient of friction which should be used in belt-calcu-
lations, the author provided himself with apparatus,
and, before making use of the coefficient value (p = 0.40
in this work, very carefully proved this value as the
mean of a number of trials. The apparatus consisted
of the following arrangement :
Fig. 100. A pulley A securely fastened by the pins
x, x, so that it could not move in any direction ; a belt
B, B passed around the pulley, and its ends attached
to the levers abc and a'b'c ; two weights w = 20
pounds and W 40 pounds, the latter being arranged
so that it could be moved along the lever-arm be at
will. Belts and pulleys which had been used for some
time not, however, badly worn or injured were pur-
posely chosen in order to obtain more practical results.
The fulcrums b and V were metallic knife-edges, and
the friction between them and their levers therefore
practically nothing. The weight w = 20 pounds was
244
BELTS AND PULLEYS.
fixed upon the lever a'b'c' ', the arms being a'b' = 4
inches and b'c = 12 inches. The tension / was there-
fore, from the principles of the simple lever, 20 X
4
= 60 pounds. In each experiment the arm ab was 4
inches long and the weight W was moved along the
lever-arm be until the tension T was such that the belt
was just on the point of slipping ; the corresponding
w
arm was then carefully measured with an accurate
hundredth rule, and the tension T calculated as above
for/.
Experiment i. The angle embraced by the belt was
a = 1 80 degrees, the tension / = 60, and the lever- arm
be = 22.50 inches. The tension T was therefore T =
22.50
X 40 = 225 pounds.
T 225
Hence - = -=
APPENDIX.
245
log - = log 3.75 = 0.57403. From formula (41), by
transposing, we obtain for the coefficient of friction
the expression (p = log -f- 0.0075780', which in the
0.57403 0.57403
present case becomes <p =
or cp = 0.42083.
In this experiment, although tried with five different
pulleys and as many different belts, the greatest value
obtained for the coefficient was <p = 0.4248, and the
smallest tp = 0.41997. The value determined above
was nearest to the average of the different experiments.
Experiment 2. The angle embraced by the belt was
a = 90. The arrangement of apparatus for this ex-
periment is shown in Fig. 101, being the same as in
Experiment I, except that the belt was held away from
246 BELTS AND PULLEYS.
the pulley by means of two small rollers y, y, in order
to obtain the required value for the angle a. The
friction of the rollers was so small that when used for
an angle a = 180, with the belt and pulley which
gave the mean result in Experiment i, and with the
same angle of deviation (kyz, Fig. 101), made a differ-
ence of only 0.00197 in the results practically none at
all.
The tension / was as before / 60 pounds, and the
lever-arm be 11.64 inches. The greater tension was
therefore T =- X 40 116.4 pounds. Hence
1164 T
= -ftp= ! -94> log - = log 1.94 = 0.28780, <p =
0.28780 0.28780
~
95 -538555 ~
average of different experiments.
Experiment 3. The angle embraced by the belt was
a = 45, tension t = 60 pounds, lever-arm be = 8.4
8.4 T
inches. Consequently T = X 40 = 84 pounds, =
4 t
84 T 0.14613
=!* I< = log M = 0.14613. * =
O.I46I3
= - -,or cp = 0.42852 nearest average value.
In this experiment the angle of deviation (kyz, Fig.
101) was so great that the friction of the rollers y,y
must have had an appreciable effect, which probably
accounts for the increased value of (p.
Experiment 4. The angle embraced by the belt was
210; the arrangement of apparatus in order to obtain
APPENDIX.
247
this angle is shown in Fig. 102. Tension t = 60 pounds,
and the lever-arm be = 28.4 inches. Therefore the
greater tension was T X 40 = 284 pounds, and
= -^ = 4.733. The logarithm of this ratio is
/ DO
0.67514, and we have, for the coefficient,
0.67514 0.67514
<z> = T/ = ^ = 0.42425 nearest
0.007578 X 210 1.59138
average value.
Example 5. The angle embraced by the belt was
a b
FIG. 102.
nr = 250, tension t = 60 pounds, and the lever-arm
#39.15 inches long. The greater tension was there-
fore T =
pounds> T = 3911 =
248
BELTS AND PULLEYS.
6.525,
= 0.81458.
0.81458
^- - = 0.42997 nearest average value.
1.0945
In each of these experiments five trials were made
with different belts and pul-
leys, the values worked out
above being about mean for
each separate experiment.
A mean between the five
values given above is there-
fore a mean value deter-
mined by twenty-five very
careful experiments, and
may be relied upon for
practical calculations. This
FlG - I0 3- T gives for us our coefficient of
friction between leather belts and iron pulleys* 9? =
0.42507. Since in this coefficient there is not the same
need of a factor of safety as in calculations with proof-
strengths and to prevent breakage, we may take very
nearly the full value without running risk of any serious
accident. We have taken, and shall use throughout
this work, the value
(p = 0.40.
All the belts with which the above experiments were
made had been oiled to a moderate extent with castor-
oil.
*This value practically agrees with the results of the experiments
of Messrs. Briggs and Towne, as given \n Journal of the Franklin In-
stitute, January, 1868.
APPENDIX. 249
A series of 18 experiments with new dry leather belts
hung over a fixed pulley and weighted at each end
(see Fig. 103) gave an average value of
cp = 0.304.
The angle embraced by the belt in each case was
1 80, the weights on the ends varying from 10 and 25^
pounds to 90 and 229 pounds.
The author also tried 21 experiments with some old,
gummy leather belting which had lain in a dry room
for nearly two years, and to his astonishment found an
average value of q> 0.61 for the coefficient of fric-
tion. These belts, which were 2 inches wide and T 3 7
inch thick, broke through the solid parts when tested
for strength, at an average strain of 1088 pounds. This
o
would give for the ultimate strength 1088 X , or about
2900 pounds per square inch very little if any below
that of ordinary belt-leather.
Leather over Leather-covered Pulleys. Using the
belting and pulleys of the first five experiments men-
tioned in this Appendix, the author tried the following
experiments with leather-covered pulleys :
Experiment I. With apparatus of Fig. 100. a =
1 80, t = 60 pounds. The lever-arm be was 26.15
inches long when the belt began to slip. Hence T =
26.15 T 261.5
- X 40 = 261.5 pounds, log - log ^- = log
4-358 - 0.63929. Consequently <p =
= 0-4687.
1.36404
2$0 BELTS AND PULLEYS.
Experiment 2. With apparatus of Fig. 101. a =
go, t = 60 pounds. The lever-arm be was 12.61 inches
long. Hence T = X 40 = 126.1, log - = log
126.1 0.32263
? = log 2.102 = 0.32263, and (p = - ^^--- =
60 0.007578 X 90
0.32263
Experiment 3. With apparatus of Fig. 101. a =
45, t 60 pounds. The lever-arm be was 8.75 inches
o * y
long. Hence T X 40 = 87.5 pounds, log - = log
4 i
87.5 0.16376
-~ log 1.458 = 0.16376, and cp =
60 ~ ~ 0.007578 X 45 ~
0.16376
^- = 0.4802.
0.34101
Experiment 4. With apparatus of Fig. 102. a =
210, t =. 60* pounds. The lever-arm be was 34.99
inches long. Hence T = - X 40 = 349.9 pounds,
4
log - = log /* = log 5.832 = 0.76582, and cp =
t oo
0.76582 _ 0.76582 _
'6 ^x ~~,~^ -~VTo 0.4012.
0.007578 X 210
Experiment 5. With apparatus of Fig. 102. a =
250, t = 60 pounds. The lever-arm be was 49.77
inches long. Consequently T = - X 40 = 497-7
pounds, log - = log I\ = log 8.295 = 0.91882, and
0.91882 0.91882 _
9 = 0.007578^X250 ~~ ''
APPENDIX. 251
Each of the above experiments is the one giving the
nearest average value of q> out of five tests. The aver-
age of these five experiments is therefore the average
of twenty-five carefully-made trials. This average
value is <p = 0.4776, which permits us, after making
fair allowances for the friction of the rollers y, y used
in the apparatus, to use, for the coefficient of friction
of leather over leather-covered pulleys, the value
cp = 0.45.
Experiments tried with new dry leather over pulleys
covered with the same gave for the coefficient of
friction
(p = 0.348.
Vulcanized-rubber Belts. Th2 author has made
nearly sixty different trials with vulcanized-rubber
belts over cast-iron pulleys, the belts and pulleys hav-
ing been used to a moderate extent for practical pur-
poses ; the mean value for the ratio of the tensions for
a = 1 80 was found to be
7 = 4-55-
From this we have
log- = log 4.55 = .0658011,
and consequently
0.658011 _ 0.658011 _
~~ ~
0.007578 X 1 80 ~~ 1.36404
2$2 BELTS AND PULLEYS.
This value is slightly greater than that obtained for
oiled leather belts over leather-covered pulleys ; for
reasons given in 12, however, we take the coefficient
of friction for vulcanized-rubber belts over cast-iron
pulleys^ the same as for leather over leather-covered
pulleys, i.e.,
cp = 0.45.
II.
THE principles of the endless belt and pulleys, as ap-
plied for numerous different purposes of the shops and
factories, have developed from time to time a vast
number of ingenious contrivances by means of which
many motions other than that of simple, continuous
rotation may be easily obtained. Thus, while the
fundamental mechanism itself has remained essentially
unchanged from the forgotten ages which gave it birth
to the present'time, instead of the simple band trans-
mitting the motion of one pulley to another parallel and
equal pulley, we have now devices by means of which
we may transmit the motion of the driver to one or more
shafts oblique in almost any direction to that of the
motive pulley ; to increase and decrease the speeds of
the various pulleys at will ; to reverse and change
the directions of rotation of any or all of the driven
pulleys ; to almost instantly impart or arrest their
motions; and to transform the continuous, rotary
motion of the driver into rectilinear, reciprocating, in-
termittent, and intricate compound motions. Indeed
it is not so far from the truth as is sometimes sus-
APPENDIX.
253
pected, that "almost anything can be accomplished
with a room full of pulleys and belts."
Some of the most common of the many devices
made use of to obtain the different motions necessary
for the various kinds of work known to the artisan
have been gathered together and explained, as briefly
as is consistent with a clear understanding of the
mechanisms, in the following pages.
Fig. 104 represents an arrangement of pulleys by
means of which the motion of the driving-pulley A
FIG. 104.
may be transmitted to the pulley B, at right angles
with the driver. The guides C, C hold the band in
such positions that it runs easily upon the driven
pulley, about which it makes one entire turn.
Fig. 105 represents a device for transforming a rec-
tilinear into a reciprocating rotary or oscillating mo-
tion. A rocking motion is given to the arm by
means of the connecting-rod A and the motive-piece
254
BELTS AND PULLEYS.
F. The required motion is then transmitted to the
two pulleys C and D by means of the half-pulley B and
the belt gg. Inversely, by giving an oscillating motion
to one of the pulleys C and D a reciprocating recti-
linear motion may be given to the piece F.
In the combination of pulleys represented in Fig.
106, the uniform rotary motion of the driver A gives
to the pulleys B, C, and D uniform rotary motions at
different speeds, according to their diameters, as ex-
FIG. 105. FIG. 106.
plained in 2. The driven pulleys are in different
planes, and the face-width of the driver is great enough
to carry the several belts without interfering with each
other.
The contrivance shown in Fig. 107 is intended to
give two rotaiy motions in contrary directions to two
coincident shafts. The pulley A is the driver, and
carries a crossed belt running to the pulley B, and also
an open belt running to the pulley C. The latter pul-
leys consequently rotate in opposite directions at the
APPENDIX.
255
same or at different speeds, according as their diameters
are equal or different.
Fig. 108 represents an arrangement by means of
which a great increase of friction between the band
and principal pulleys may be obtained. The band
passes several times around the principal pulleys, A and
B, and is properly guided by means of the small rollers
b and c, as shown in the figure. Let us suppose that
FIG. 107.
the band passes four times around the pulleys : the
arc embraced is then a = 360 X 4 = 1440. If we
take for the coefficient of friction cp = 0.30, we shall
have for the ratio of the tensions [see formula (41)],
T
log - = 0.007578 X 0.30 X 1440 = 3-2737,
or
- = 1878.
2 5 6
BELTS AND PULLEYS.
If the band passed only \ times around the pulleys,
we should have a = 180, and consequently
log - = 0.007578 X 0.30 X 1 80 = 0.4092,
or - = 2.566.
In other words, by means of the above arrangement
the ratio of the greater to the smaller tension is in-
creased over seven hundred-fold.
w
FIG.
An ingenious device for transmitting a rotary mo-
tion to a movable pulley by means of an ordinary belt
is represented in Fig. 109. A is the driver and B the
movable driven pulley. The intermediate pulley C is
suspended by means of a cord passing over the roller
D and the weight W\ it is thus free to move up and
down in the frame F, and the pulley Z? may be moved
through considerable distances without interfering with
APPENDIX.
257
F C'
the motion of the belt, as is indicated by the dotted
lines.
Fig. no represents an arrangement of pulleys for
transmitting two different speeds from the driving-
shaft AA to the driven
shaft BB. Each shaft car-
ries two fast pulleys (C, C,
C', and E) and two loose
pulleys (F, F, F, and G),
and the two belts xx and
yy are moved together,
backward and forward,
across the pulley-faces.
When the belts are in the
positions shown in the fig- FIG. no.
ure, the belt xx is at work and yy at rest. When the
belts are moved to the pulleys F, F and C', E, the belt
xx is at rest and the belt yy transmits to the driven
shaft a fast motion because of the small diameters of
the pulleys G and E.
G E
A mode of transforming a reciprocating rectilinear
motion into an alternating rotary motion is shown in
Fig. in. To the piece A A a cord, which also passes
2 5 8
BELTS AND PULLEYS.
around the pulley C, is attached, and by moving the
piece AA backward and forward, the required motion
of C is obtained. Inversely, by giving an oscillating
motion to the pulley C, a reciprocating rectilinear mo-
tion may be given to the piece A A.
FIG. 112.
Fig. 112 represents an arrangement by means of
which the continuous rotary motion of the driver C
may be transmitted to two pulleys A, A at right an-
gles with the motive-pulley. A portion of the band is
horizontal, and runs without difficulty upon the pulleys
A, A ; the other part is guided by the intermediate
pulleys B, B, as shown in the figure.
W
FIG. 113.
Fig. 1 1 3 represents a device by means of which a
variable motion for the driven shaft may be obtained
from the uniform motion of the driver. The pulley A
is the driver, and bears upon its face a deep groove, as
shown by the dotted circle. The pulley B is mutilated,
one part having a greater radius than the other, and
APPENDIX.
259
also has a deeply grooved face. When the pulleys are
in the positions shown in the figure, the belt is tight,
and drives uniformly the pulley B. When, however,
the smaller part of the pulley B comes opposite the
belt, the tension is greatly lessened and the weight W
jerks the pulley quickly around until the belt again be-
comes tight, when the operation is repeated.
In Fig. 1 14, the arm C bearing the pulley B, when
moved up and down in treadle
motion, transmits a rotary mo-
tion to the shaft 5 by means
of the belt and eccentric pul-
ley A. By making the pulley
A the driver, the pulley B may
be given a rotary motion, and
at the same time an oscillating
motion about the point D.
Fig. 1 1 5 represents a device FlG< II4>
for transmitting a continuous rotary motion to a mov-
able shaft. The pulley A is the driver, and the driven
pulley B may be moved
about in the frame C, as
shown by the dotted lines,
without interfering with the
motion of the belt. The
radius of curvature of the
axis of the frame is equal
to the distance between the
centres of the pulleys.
Another mode of trans-
FIG. 115. mitting a rotary motion to
a movable pulley is shown in Fig. 1 16. A is the driv-
260
BELTS AND PULLEYS.
FIG. 116.
ing-pulley and carries an elastic belt of india-rubber.
By stretching the belt, the driven
pulley B may be moved about in al-
most any direction, as indicated by
the dotted lines. This device is used
extensively in hair-cutting and clip-
ping machines, and dental apparatus
for boring and drilling.
Fig. 117 represents a device known
to artisans as the " frictionless bear-
ing," or " anti-friction bearing. The shaft b of the pul-
ley-^, instead of turning in an ordinary pedestal or
hanger, rests upon the circum-
ferences of six small rollers, c,
c, etc. The friction due to the
weight of the pulley and shaft
is thus distributed among the
six rollers, and, since the shaft
rolls upon the rollers instead of
sliding around in the pedestal
as with common bearings, the
friction of sliding is eliminated. FIG. n 7 .
Considerable difference of opinion exists among me-
chanical men as to the best method of connecting the
various shafts in shops and mills with the driving
drum or pulley. Some engineers claim that but one
belt should be used to drive all the shafts in the mill ;
that this method is the most advantageous, because of
the great duration of the driving-belt and because of
the simplicity of the arrangement. Others suggest two
belts one connecting the driving-pulley with the first
shop-shaft, and the other passing from the first shop-
APPENDIX.
26l
shaft to all the other shafts ; while by many it is claimed
that each principal shaft should have its own belt
connecting it, cither directly or indirectly, with the
driving-pulley.
FIG.
Fig. 118 represents a section of a three-story mill,
the shafts of which are driven by means of a single
belt. The arrangement of the various pulleys is surft-
262 BELTS AND PULLEYS.
ciently clear in the figure without further explanation.
The objections offered to this method of transmitting
to the different shafts the power of the motor are the
following : the belt must necessarily be very long
often nearly or quite 500 feet ; it must be very strong,
and consequently wide and heavy, since it must trans-
mit the entire power of the mill ; the expense is there-
fore great, and the tendency to stretch greater than in
a short belt ; because of the weight and length the op-
eration of shortening and tightening the belt is much
more difficult than in ordinary cases; since the belt
cannot be easily slipped from one pulley to another,
the use of fast and loose pulleys for engaging and dis-
engaging the shafts is extremely difficult. The advan-
tages are simplicity, supposed long wear (we however
doubt very much the truth of this, since the belt is con-
stantly bent in both directions and run on both sides
upon the various pulleys), the fact that the driving-
pulley need be no wider than is necessary to carry the
one belt, and economy of pulleys, the number of which
is less than if the power of each shaft was obtained by
means of a second pulley from its nearest neighbor.
Fig. 119 represents a three-story mill, in which each
principal shaft is connected by its own belt directly
with the driving-pulley. Disadvantages : the shop is
so cut up by the many belts that valuable space is sacri-
ficed ; the driving-pulley must be wide enough upon
its face to carry all the belts in the figure there are
seven belts ; if they average six inches wide and we al-
low one quarter of an inch between each two belts, the
face of the driver must be over three and one half feet
wide ; the use of fast and loose pulleys for the shafts
APPENDIX.
263
is rendered difficult. Advantages : each belt transmits
the force of one shaft only, and the belts may therefore
be light ; if any one belt breaks it may be removed and
FIG. 119.
the remaining shafts driven as if no accident had oc-
curred ; each belt may be made large or small, accord-
ing as it has heavy or light work to perform.
26 4
BELTS AND PULLEYS.
In Fig. 1 20, we show a section of a three-story mill
driven in the manner most common at the present time
throughout this country. A main driving-belt, heavy
enough to transmit the entire work of the mill, runs
from the motive-pulley A to the nearest shop-shaft B.
From the latter shaft to the third-story main shaft runs
a belt sufficiently strong to transmit the work of the
APPENDIX. 265
third story. The other shafts on each story are con-
nected by separate belts each with its nearest neigh-
bor, as shown in the figure. In this arrangement the
belts are all overhead and out of the way, except two
which run close to the ends of the building. Thus no
valuable space is used up by the belts. Fast and loose
pulleys may easily be used, because none of the belts
(except the driver) pass over the driving-pulley. This
mode of transmitting power is open to the objection
that the breakage of one of the principal belts causes
a stoppage of several shafts, for instance if the hori-
zontal belt from the pulley B breaks, the entire second
story is thrown out of gear ; but its other advantages
more than compensate for this risk, and it has there-
fore come to be the favorite in most of our shops and
factories.
INDEX.
A
PAGE
Belts, open
PAGE
116
66
6j
Angle between middle p'anes .
, s'-eet-iron
67
between shafts
Anti-friction bearing
3 1
... 260
. - - - 83
, vulcanized-rubber
without guides
.... 66
29
. .. if;
3
, examples of
, formulas for
, method of drawing
, of cable- pulleys
. ... 168
.... 167
.... 169
. . . 226
169
Binomial formula
Briggs and Towne
Breaking strength of leather. .
of rawhide
.... 80
.... 248
.... 92
. .. 66
188
, of pulleys
. . . 166
167
of vulcanized rubber
141
66
Arnold's rule
V
" Arts and Sciences of the Ancie
Axes
B
Babylon
US" 2
28
C
Cables, deflections of
, diameter of
, examples of
.... 207
200
204
84
196
66
Clissold's
93
double
66
196
66
Cast-iron shafts
. ... I 7 6
,gut
, half -crossed
, hemp
leather
.... 66
... 31
66
65
Circumference
, examples of
, formulas for
rules for . . .
25
II
, metallic
.... 192
Circumferential velocity
3
268
INDEX.
PAGE
PJ
Double lacing
\r,E
71
77
198
I So
viii
X
168
*S
164
W9
5
C*
*7
9
7
**9
4
*94
163
49
161
7
rf
5
5
160
18,
55
6
4j
3
93
i i H
>54
S7
Hi
?43
49
5*
Clissold's belt
Coefficient of friction of cables
of jointed chain-belts
of leather over cast-iron
of leather over leather
199
"93
86
n6
187
Dynamometer
E
Elasticity of cables
Engaging and disengaging
of rubber over leather
of rubber over rubber
Coinciding axes
Common logarithms
Comparison of formulas
of leather and rubber
Conditions necessary for maintain-
ing belt on pulleys
Continuous motion
speed cones
Cooper, J. W
Cores of metallic cables
Cork
156
156
28
85
vti
67
28
7
62
198
225
116
28
7
167
22
207
254
257
2S5
2 S 8
255
259
241
25
12
202
I 7 6
7 6
234
31
66
169
Enbank, Thomas
Examples of arms
of circumference
Examples of continuous speed cone
of deflection
of diameter of cables
of greaiest tensions
of horse-power
of inclined transmission
of increased tension
of jointed chain-belt
-of keys
Crossed axes
Crossed belt
of length of belts
of power
of pulley-train
of radius ...
of revolutions
of rope-belts
of speed-cones
of transmission with pulleys near
D
Decreasing pulley-train
Deflections in cables
Device for changing motion
for increasing speed
or increasing tension
- or obtaining intermittent motion
or obtaining opposite motions..
or obtaining variable motion. .
or putting on cables
Diameter, examples of
, formulas for
of cables
of shafts
Difficulties found in belting
Dimensions of pulley-supports
Direction of rotation
Distance between pulleys
between pulley-supports
Double belts
curved arms
Examples of velocit ies
of weight of pulleys
of weight of principal pulleys. . .
of width of leather belts over cast-
iron pulleys
Examples of width of leather belts
over leather-covered pulleys
Examples of width of rubber belts
over cast-iron pulleys
Examples of width of rubber belts
over leather-covered pulleys
Experiments with leather over iron
with leather over leather
with rubber over iron
INDEX.
269
Experiments with rubber ov
PAGE
IT
. 252
. iv
. 179
. 68
4
PAGE
Formulas for width of leather
belts 93-114
for width of rubber belts 150
G
Godin's belt 192
Graphical method ... 60
Extracts from letters
F
Fast and loose pulleys
116
60
834
3*
vii
31*
06
241
196
OS
03
03
217
aa
197
84
030
ay)
:>S
66
IQ2
2 4 6
4
x6a
4
First human necessity
Gutta ercha
machine
- transformation
Fixing-keys
4
S
. 162
. 66
23
. 167
. 226
. 207
. 12
3'
'59
. 163
2 4
. 219
. 213
'93
47
. 161
H
FUx belts
Haswell's rule
Height of cable above ground
Hemp belts
Formulas for arms
for arms of cable-pulltys ....
for cable diameters
for circumference
for deflections
for diameter
for distance between pulleys..
for face-width
for fixing-keys
for horse-power
for inclined transmission
for increased tension
for jointed chain-belts
for length of belts
Holes for lacing
Horizontal transmissions
Horse-power
" Hydraulics and Mechanics"
Hyperbolic logarithms.
I
Increased tension
Inferior limit of separation of pul-
leys
Integral calculus
Intermediate pulleys
stations
for power
for pressure on axes
for pulley -supports
for radius
for ratio of powers
for ratio of revolutions
for ratio of velocities
for revolutions
for rim
. 20
- 236
234
. 12
. 22
M
M
.. 160
.. 187
. 172
54
. 84
'99
>5
165
. 236
J
Jointed chain-belts
" Journal of Franklin Institute" . . .
K
Kenedy's translation
Keys
for shafts
for speed-cones
for tensions
for tensions in cables
for velocities
for weight of pulleys
for weight of principal pulleys.
Kilogram
" Kinematics of Machinery"
270
INDEX.
Lacing 68
Lack of knowledge of belting iv
of space 43
Leather belts 65
, examples of 93
, formulas for 93-114
, tables of 88-138
Leather-covered pulleys 115
Logarithms, common 85
, Naperian 84
Long belts 262
M
Material of belting 65
Median line 28
Metallic belts 192
cables 196
Middle plane 28
Methods of arranging pulleys 260
Method of tracing cable-curves ... 221
Middle plane 28
Mill-shafts 260
Mutilated pulley 258
N
Naperian logarithms. ..
Nave of cable-pulleys. .
of pulleys
Nineveh
Nystrom's formula
Open belt
Origin of belt and pulley .
Oscillating moijon
Parallel axes
Pedestals
Permissible deviation ...
Power, examples of
, formulas for
, ratio of
Pressure on axes
Primitive lathe, drill, etc.
water-wheel
Probable origin of pulley
Profiles of arms
PAGE
Proper disposition of pulleys 28
Pulley arms 166
, cable 226
, flanged 160
nave 161
--"* i59
, rounding of 159
-.split 163
supports 230
train 21
with light arms 238
R
Radius, examples of 25
, formulas for J2
Ratio of circumferences . 12
of power 22
of revolutions 14
of velocities 19
Rawhide belts 66
Reserve cables 228
Resistance to slipping . 73
Reuleaux, Prof , v i, 28
Reversing. . . 182
Revolutions 14
Rim of cable-pulleys 203
of pulleys , 59
Robertson 2
Rollin 2
Rope-belts 185
Rosin 1 1 6
Rotation it
Rouiller's belt. 192
Rounded fillies 28
Rules for arms 167
or belts with pulley-gufdes 32
or circumference.. . it
or diameter 1 1
or distance between pulleys. . . 31
or horse-power 24
r power 20
ir proper disposition of pulleys. 28
for radius . . 12
for ratio of circumferences
for ratio of powers
for ratio of velocities
for revolutions
for shaft-diameters.. . .
INDEX.
PAGE ;
271
PAGE
Table, metallic cables aio
Safe shearing stress 142
, number of arms
707
working stress, leather .. 92
, shaft-diameters
I 7 6
working stress, rubber 141
, tensions for leather 88,
1*7
Scale for cable-curves 220
. tensions for me
allic cables
201
p-hplts
188
Shafts 171
of mines 233
, widths of leather belts no,
\j 5
Sheet-iron pulleys 238
, widths of rubber belts 151
Shop-shafts 260
Tensions in cable-wires 200
Shortening 68
in belts
79
Single lacing 69
in inclined transmissions
220
Size of pulleys 24
Thickness of rubber belts 140
Slipping 45
Tightening- pulley.
I
Slow growth of belting 6
Torsional strain . . .
I?1
Smith, C. A 60
Tower of Babel...
9
Speed-cones 51
Transmission by cable with pulleys
Spinning-mills 44
near together . . .
222
Split pulleys ... 163
Transmission with
in^lin^H /-aH1*
Stations 232
" Treatise on Toothed Gearing"..
- 1 /
Station pillars 238
Steel cables 203
u
-^shafts 176
Uncertain origin of belt and pulley
3
Stepped cones 65
Unwin's formula.
vi
Strength of gut 66
of leather 75
V
Velocities
~~ of vulcanized rubber
Vulcanized-rubber belts
06
Swedish iron 203
w
T
Weakest part of belt. 92
Fable, deflections 210
Weight of pulleys
104
, dimensions of pulley-supports.. 234
Wrought-iron shafts 176
, formulas for leather belts 96-129
formulas for rubber belts
z
, greatest tensions 90, 118, 157
i Zeigler's machine
for putting on
, increased tension 214
metallic cables.
341
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Ingalls's Handbook of Problems in Direct Fire . . . : 8vo,
* Ballistic Tables 8vo,
* Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II . . 8vo. each,
* Mahan's Permanent Fortifications. (Mercur.) 8vo, half morocco,
Manual for Courts-martial i6mo. morocco,
* Mercur's Attack of Fortified Places. . . ... I2mo,
Elements of the Art of War. . .
Metcalf's Cost of Manufactures And the Administrationlof. Workshops, Public
and Private 8vo, 5 oo
* Ordnance and Gunnery I2mo, 5 oo
Murray's Infantry Drill Regulations iSmo/paper, 10
* Phelps's Practical Marine Surveying 8vo, 2 50
Powell's Army Officer's Examiner I2mo, 4 oo
Sharoe's Art of Subsisting Armies in/War i8mo, morocco, i 50
2
* Walke's Lectures on Explosives 8vo,
* Wh'eeler's Siege Operations and Military Mining 8vo,
Winthrop's Abridgment of Military Law I2mo.
Woodhull's Notes on Military Hygiene i6mo, 5
Young's Simple Elements of Navigation i6mo morocco, oo
Second Edition, Enlarged and Revised i6mo, morocco.
ASSAYING.
Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe.
1 2 mo, morocco, I 50
Furman's Manual of Practical Assaying 8vo, 3 oo
Miller's Manual of Assaying 12010, I oo
O'Driscoll's Notes on the Treatment of Gold Ores 8vo, a oo
Ricketts and Miller's Notes on Assaying 8vo, 3 oo
Hike's Modern Electrolytic Copper Refining 8vo, 3 oo
Wilson's Cyanide Processes i2mo, i 50
Chlorination Process i2mo, i 50
ASTRONOMY.
Comstock's Field Astronomy for Engineers 8vo,
Craig's Azimuth 4to,
Doolittle's Treatise on Practical Astronomy 8vo,
Gore's Elements of Geodesy 8vo,
Hayf ord's Text-hook of Geodetic Astronomy 8vo,
Merriman's Elements of Precise Surveying and Geodesy 8vo,
* Michie and Harlow's Practical Astronomy 8vo, 3 o
* White's Elements of Theoretical and Descriptive Astronomy tamo, a oo
BOTANY.
Davenport's Statistical Methods, with Special Reference to Biological Variation.
i6mo, morocco, i 35
Thome and Bennett's Structural and Physiological Botany i6mo, a as
Westermaier's Compendium of General Botany. (Schneider.) 8vo, a oo
CHEMISTRY.
(idriance's Laboratory Calculations and Specific Gravity Tables iamo, t 2$
Allen's Tables for Iron Analysis 8vo, 3 oo
Arnold's Compendium of Chemistry. (Mandel.) (In preparation.')
Austen's Notes for Chemical Students lamo, i 50
Bernadou's Smokeless Powder. Nitro-cellulose, and Theory of the Cellulose
Molecule zamo, a 5*
Bolton's Quantitative Analysis 8vo, i 50
* Browning's Introduction to the Rarer Elements STO, I 30
Brush and Penfield's Manual of Determinative Mineralogy 8vo, 4 oo
Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.) . . . 8vo, 300
Cohn's Indicators and Test-papers lamo, a oo
Tests and Reagents 8vo, 3 oo
Copeland's Manual of Bacteriology. (In preparation.)
Craft's Short Course in Qualitative Chemical Analysis. (Schaeffer.) iamo, i 50
Drechsel's Chemical Reactions. (Merrill.) 12 mo, i 25
Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 4 eo
's Modern High Explosives 8vo, 4 oo
Effront's Enzymes and their Applications. (Prescott.) 8vo, 3 oo
Erdmann's Introduction to Chemical Preparations. (Dunlap.) izmo, i 25
Fletcher's Practical Instructions in_Quantitative Assaying with the Blowpipe.
I2mo, morocco, 1.50
Fowler's Sewage Works Analyses I2mo, 2 oo
Fresenius's Manual of Qualitative Chemical Analysis. (Wells.) 8vo, 5 oo
Manual of Qualitative Chemical Analysis. Part I. Descriptive. ( Wells. )
Svo, 3 oo
System of Instruction in Quantitative Chemical Analysis. (Cohn.)
2 vols. (Shortly.)
Fuertes's Water and Public Health i2mo, i 50
Furman's Manual of Practical Assaying Svo, 3 oo
Gill's Gas and Fuel Analysis for Engineers xamo, i 25
Grotenfelt's Principles of Modern Dairy Practice. ( Woll.) I2mo, 2 oo
Hammarsten's Text-book of Physiological Chemistry. (Mandel.) Svo, 4 oo
Helm's Principles of Mathematical_Chemistry. (Morgan.) I2mo, i 50
Hinds's Inorganic Chemistry Svo, 3 oo
* Laboratory Manual for Students i2mo, 75
Holleman's Text-book of Inorganic Chemistry. (Cooper.) Svo, 2 50
Text-book of Organic Chemistry. (Walker and Mott.) Svo, 2 50
Hopkins's Oil-chemists' Handbook Svo, 3 oo
Jackson's Directions for Laboratory Work in Physiological Chemistry. .Svo, i oo
Keep's Cast Iron Svo, 2 50
Ladd's Manual of Quantitative Chemical Analysis I2mo i oo
Landauer's Spectrum Analysis. (Tingle.) Svo, 3 oo
Lassar-Cohn's Practical Urinary Analysis. (Lorenz.) I2mo, i oo
Leach's The Inspection and Analysis of Food with Special Reference to State
Control. (In preparation.)
Lib's Electrolysis and Electrosynthesis of Organic Compounds. (Lorenz.) I2mo, i oo
Mandel's Handbook for Bio-cherrical Laboratory izmo, i so
* Martin's Laboratory Guide to Qualitative Analysis with the Blowpipe . . I2mo, 60
Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.)
3d Edition, Rewritten Svo, 4 oo
Examination of Water. (Chemical and Bacteriological.) 12010, i 25
Meyer's Determination of Radicles'in Carbon Compounds. (Tingle.). . i2mo, i oo
Miller's Manual of Assaying I2mo, i oo
Mixter's Elementary Text-book'of Chemistry ...... i2mo, i 50
Morgan's Outline of Theory of Solution and its Results * 1 2mo, i oo
Elements of Physical Chemistry I2mo, 2 oo
Nichols's Water-supply. (Considered mainly from a Chemical and Sanitary
Standpoint, 1883.) Svo, 2 50
O'Brine's Laboratory Guide'in Chemical Analysis Svo, 2 oo
O'Driscoll's Notes on the Treatment of Gold Ores Svo, 2 oo
Ost and Kolbeck's Text-book of Chemical Technology. (Lorenz Bozart.)
(In preparation.)
* PenfieW's Notes on Determinative Mineralogy and Record of Mineral Tests.
Svo, paper, 50
Pjctefs The Alkaloids 'and their Chemical Constitution. (Biddle.) (In
preparation.)
Pinner's Introduction to Organic Chemistry. (Austen.) I2mo, i 50
Poole's Calorific Power of Fuels Svo, 3 oo
* Reisig's Guide to Piece-dyeing Svo, 25 oo
Richards and Woodman's Air .Water, and Food from a Sanitary Standpoint . Svo, 2 oo
Richards's Cost of Living as Modified by Sanitary Science I2mo, i oo
Cost of Food a Study in Dietaries ". I2mo, i oo
* Richards and Williams's The Dietary Computer Svo, i 50
Ricketts and Russell's Skeleton Notes upon Inorganic Chemistry. (Part I.
Non-metallic Elements.) Svo, morocco, 75
4
Ricketts and Miller's Notes on Assaying 8vo, 3 oo
Rideal's Sewage and the Bacterial Purification of Sewage 8vo, 3 30
Ruddiman's Incompatibilities in Prescriptions 8vo, 2 oo
Salkowski's Physiological and Pathological Chemistry. (Orndorff.)
(Shortly.)
Schimpf s Text-book of Volumetric Analysis 12010, 2 50
" Essentials of Volumetric Analysis 12010, I 25
Spencer's Handbook for Chemists of Beet-sugar Houses i6mo, morocc'o, 3 oo
Handbook for Sugar Manufacturers and their Chemists. . i6mo, morocco, 2 oo
Stockbndge's Rocks and Soils 8vo, 2 30
* Tillman's Elementary Lessons in Heat 8vo, I 50
* Descriptive General Chemistry 8vo 3 oo
Treadwell's Qualitative Analysis. (HalL) 8vo, 3 oo
Turneaure and Russell's Public Water-supplies 8vo, 5 oo
Van Deventer's Physical Chemistry for Beginners. (Boltwood.) I2mo, i 30
* Walke's Lectures on Explosives 8vo, 4 oo
Wells's Laboratory Guide in Qualitative Chemical Analysis 8vo, i 50
Short Course in Inorganic Qualitative Chemical Analysis for Engineering
Students I2mo, i 50
Whipple's Microscopy of Drinking-water 8vo, 3 50
Wiechmann's Sugar Analysis Small 8vo. 2 30
Wilson's Cyanide Processes 12010, i 50
Chlorination Process I2mo i 50
Wulling's Elementary Course in Inorganic Pharmaceutical and Medical Chem-
istry izmo, 2 oo
CIVIL ENGINEERING.
BRIDGES AND ROOFS. HYDRAULICS MATERIALS OF ENGINEERING.
RAILWAY ENGINEERING.
Baker's Engineers' Surveying Instruments 12010, 3 oo
Bixby's Graphical Computing Table Paper 19^X24$ inches. 25
* Burr's Ancient and Modern Engineering and the Isthmian GmaL (Postage ,
27 cents additional.) 8vo, net,
Comstock's Field Astronomy for Engineers 8vo,
Davis's Elevation and Stadia Tables 8vo,
Elliott's Engineering for Land Drainage 1 21110,
Practical Farm Drainage 1 2010,
Folwell's Sewerage. (Designing and Maintenance.) 8vo,
Freitag's Architectural Engineering. 2d Edition, Rewritten 8vo,
French and Ives's Stereotomy 8vo,
Goodhue's Municipal Improvements 12010,
Goodrich's Economic Disposal of Towns' Refuse 8vo,
Gore's Elements of Geodesy 8vo,
Hayford's Text-book of Geodetic Astronomy 8vo,
Howe's Retaining Walls for Earth izmo,
Johnson's Theory and Practice of Surveying Small 8vo.
Statics by Algebraic and Graphic Methods 8vo,
Kiersted's Sewage Disposal 12010,
Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) 12010,
Mahan's Treatise on Civil Engineering. (1873) (Wood.) 8vo,
* Descriptive Geometry 8vo,
Merriman's Elements of Precise Surveying and Geodesy 8vo,
Elements of Sanitary Engineering .
Merriman and Brooks's Handbook for Surveyors i6mo, morocco,
Nugent's Plane Surveying 8vo,
Ogden's Sewer Design 12010,
Patton's Treatise on Civil Engineering 8vo half leather,
5
23
oo
00
25
Reed's Topographical Drawing and Sketching 4to, 5 oo
Rideal's Sewage and the Bacterial Purification of Sewage 8vo, 3 50
Siebert and Biggin's Modern Stone-cutting and Masonry 8vo, I 50
Smith's Manual of Topographical Drawing. (McMillan.) 8vo, 2 50
Sondericker's Graphic Statics, witn Applications to Trusses. Beams, and
Arches 8vo, 2 o
* Trautwine's Civil Engineer's Pocket-book i6mo, morocco, 5 oo
Wait's Engineering and Architectural Jurisprudence 8vo, 6 oo
Sheep, 6 50
Law of Operations Preliminary to Construction in Engineering and Archi-
tecture 8vo. 5 oo
Sheep. 5 SO
Law of Contracts * 8vo, 3 oo
Warren's Stereotomy Problems in Stone-cutting 8vo, 2 50
Webb's Problems in the U=e and Adjustment of Engineering Instruments.
i6mo, morocco, r 25
* Wheeler's Elementary Course of Civil Engineering 8vo, 4 oo
Wilson's Topographic Surveying 8vo, 3 50
BRIDGES AND ROOFS.
Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .8vo, 2 oo
* Thames River Bridge 4to, paper, 5 oo
Burr's Course on the Stresses in Bridges and Roof Trusses, Arched Ribs, and
Suspension Bridges 8vo, 3 50
Du Bois's Mechanics of Engineering. Vol. II Small 4to, 10 oo
Foster's Treatise on Wooden Trestle Bridges 4to, 5 oo
Fowler's Coffer-dam Process for Piers 8vo, 2 50
Greene's Roof Trusses 8vo, i 25
Bridge Trusses 8vo, 2 50
Arches in Wood, Iron, and Stone 8vo, 2 50
Howe's Treatise on Arches 8vo 4 oo
Design of Simple Roof-trusses in Wood and Steel 8vo, 2 oo
Johnson, Bryan, and Turneaure's Theory and Practice in the Designing of
Modern Framed Structures Small 4to, 10 oo
Merriman and Jacoby's Text-book on Roofs and Bridges:
Part I. Stresses in Simple Trusses 8vo, 2 50
Part II. Graphic Statics 8vo, 2 50
Part HI. Bridge Design. 4th Edition, Rewritten 8vo, 2 50
Part IV. Higher Structures 8vo, 2 50
Morison's Memphis Bridge 4to, ro oo
Waddell's De Pontibus, a Pocket-book for Bridge Engineers. . . i6mo, morocco, 3 oo
Specifications for Steel Bridges v I2mo, I 25
Wood's Treatise on the Theory of the Construction of Bridges and Roofs. 8vo, 2 oo
Wright's Designing of Draw-spans:
Part I. Plate-girder Draws 8vo, 2 50
Part II. Riveted-truss and Pin-connected Long-span Draws 8vo, 2 50
Two parts in one volume 8vo, 3 50
HYDRAULICS.
Bazin's Experiments upon the Contraction of the Liquid Vein Issuing from an
Orifice. (Trautwine.) 8vo, 2 oo
Bovey's Treatise on Hydraulics 8vo, 5 oo
Church's Mechanics of Engineering 8vo, 6 oo
Diagrams of Mean Velocity of Water in Open Channels paper, i 50
6
Coffin's Graphical Solution of Hydraulic"'Problems i6mo, morocco, 2 50
Flather's Dynamometers, and the Measurement of .Power i2mo, 3 oo
Folwell's Water-supply Engineering 8vo, oo
Frizell's Water-power 8vo, oo
Fuertes's Water and Public Health i2mo, 50
Water-filtration Works iamo, 50
Ganguillet and Kutter's General Formula for the Uniform Flow of Water in
Rivers and Other Channels. (Hering and Trautwine.) 8vo, oo
Hazen's Filtration of Public Water-supply 8vo, oo
Hazlehurst's Towers and Tanks for Water- works 8vo, 50
Herschel's 115 Experiments on the Carrying Capacity of Large, Riveted, Metal
Conduits 8vo, 2 oo
Mason's Water-supply. (Considered Principally from a Sanitary Stand-
point.) 3d Edition, Rewritten 8vo, 4 oo
Merriman's Treatise on Hydraulics, oth Edition, Rewritten 8vo, 5 oo
* Michie's Elements of Analytical Mechanics 8vo,/ 4 oo
Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water-
supply Large 8vo, 5 oo
** Thomas and Watt's Improvement of Riyers. (Post., 44 c. additional), 4to, 6 oo
Turneaure and Russell's Public Water-supplies 8vo, 5 oo
Wegmann's Desicrn and Construction of Dams 4to, 5 oo
Water-supply of the City of New York from 1658 to 1893 4to, 10 oo
Weisbach's Hydraulics and Hydraulic Motors. (Du Bois.) 8vo, 5 oo
Wilson's Manual of Irrigation Engineering Small 8vo, 4 oo
Wolff's Windmill as a Prime Mover 8vo, 3 oo
Wood's Turbines 8vo, a 50
Elements of Analytical Mechanics 8vo, 3 oo
MATERIALS OF ENGINEERING.
Baker's Treatise on Masonry Construction 8vo, 5 oo
Roads and Pavements 8vo, 5 oo
Black's United States Public Works Oblong 4to, 5 oo
Bovev's Strength of Materials and Theory of Structures 8vo, 7 50
Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edi-
tion, Rewritten 8vo, 7 50
Byrne's Highway Construction % 8vo, 5 oo
Inspection of the Materials and Workmanship Employed in Construction.
i6mo, 3 oo
Church's Mechanics of Engineering 8vo, 6 oo
Du Bois's Mechanics of Engineering. VoL I Small 4to, 7 50
Johnson's Materials of Construction Large 8vo, 6 oo
Keep's Cast Iron 8vo, 2 50
Lanza's Applied Mechanics . . . .' 8vo, 7 50
Martens's Handbook on Testing Materials. (Henning.) a>ols 8vo, 750
Merrill's Stones for Building and Decoration 8vo, 5 oo
Merriman's Text-book on the Mechanics of Materials. ,. f 8vo, 4 oo
Strength of Materials I2mo, i oo
Metcalf's Steel. A Manual for Steel-users I2mo, 2 oo
Patton's Practical Treatise on Foundations 8vo, 5 oo
Rockwell's Roads and Pavements in France 1 2010, i 25
Smith's Wire: Its Use and Manufacture Small 4to, 3 oo
Materials of Machines , I2mo, i oo
Snow's Principal Species of Wood 8vo, .j 50
Spalding's Hydraulic Cement I2mo, 2 oo
Text-book on Roads and Pavements i amo , a oo
7
Thurston's Materials of Engineering. 3 Parts 8vo, 8 oo
Pan I. Non-metallic Materials of Engineering and Metallurgy 8vo, 2 oo
Part II. Iron and Steel 8vo, 3 SO
Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents 8vo, 2 50
fhurston's Text-book of the Materials of Construction 8vo, 5 oo
Pillson's Street Pavements and Paving Materials 8vo, 4 oo
Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.) . . i6mo, mor , 3 oo
Specifications for Steel Bridges I2mo, i 25
Wood's Treatise on the Resistance of Materials, and an Appendix on the Pres-
ervation of Timber 8vo, 2 oo
Elements of Analytical Mechanics 8vo, 3 oo
Wood's Rustless Coatings. (Shortly.)
RAILWAY ENGINEERING.
Andrews's Handbook for Street Railway Engineers. 3X5 inches, morocco, i 25
Berg's Buildings and Structures of American Railroads 4to, 5 oo
Brooks's Handbook of Street Railroad Location i6mo. morocco, i 50
Butts's Civil Engineer's Field-book i6mo, morocco, 2 50
Crandall's Transition Curve i6mo, morocco, i 50
Railway and Other Earthwork Tables 8vo, i 50
Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 5 oo
Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 oo
* Drinker's Tunneling, Explosive Compounds, and Rock Drills, 410, half mor., 25 oo
Fisher's Table of Cubic Yards * Cardboard 25
Godwin's Railroad Engineers' Field-book and Explorers' Guide i6mo, mor., 2 50
Howard's Transition Curve Field-book i6mo, morocco, i 50
Hudson's Tables for Calculating the Cubic Contents of Excavations and Em-
bankments 8vo, i oo
Molitor and Beard's Manual for Resident Engineers i6mo, i oo
Nagle's Field Manual for Railroad Engineers i6mo, morocco. 3 oo
Philbrick's Field Manual for Engineers i6mo, morocco, 3 oo
Pratt and Alden's Street-railway Road-bed 8vo, 2 oo
Searles's Field Engineering i6mo, morocco, 3 oo
Railroad Spiral l6mo, morocco, i 50
Taylor's Prismoidal Formulae and Earthwork 8vo, i 50
* Trautwine's Method of Calculating the Cubic Contents of Excavations arid
Embankments by the Aid of Diagrams '. 8vo, 2 oo
The Field Practice of jLaying Out Circular Curves for Railroads.
1 2 mo, morocco, 2 50
* Cross-section Sheet Paper, 25
Webb's Railroad Construction. 2d Edition, Rewritten i6rnn. morocco, 5 oo
Wellington's Economic Theory of the Location of Railways Small 8vo, 5 oo
DRAWING.
Barr's Kinematics of Machinery 8vo, 2 50
* Bartlett's Mechanical Drawing 8vo, 3 oo
* " ' " Abridged Ed 8vo, i 50
Coolidge's Manual of Drawing 8vo, paper, i oo
Durlev's Kinematics of Machines 8vo, 4 oo
Hill's Text-book on Shades and Shadows, and Perspective . 8vo, 2 oo
Jones's Machine Design:
Part I. Kinematics of Machinery 8vo, i 50
Part II. Form, Strength, and Proportions of Parts 8vo, 3 oo
8
MacCord's Elements of Descriptive Geometry 8vo, 3 oo
Kinematics ; or, Practical Mechanism 8vo, 5 oo
Mechanical Drawing 4*o, 4 oo
Velocity Diagrams 8vo, I 50
* Mahan's Descriptive Geometry and Stone-cutting 8vo, I 50
Industrial Drawing. (Thompson.) 8vo, 3 50
Reed's Topographical Drawing and Sketching 4to, 5 oo
Reid's Course in Mechanical Drawing 8vo, 2 oo
Text-book of Mechanical Drawing and Elementary Machine Design . . 8vo. 3 oo
Robinson's Principles of Mechanism 8vo, 3 oo
Smith's Manual of Topographical Drawing. (McMillan.) 8vo. 2 50
Warren's Elements of Plane and Solid Free-hand Geometrical Drawing . . I2mo, I oo
Drafting Instruments and Operations I2mo, I 25
Manual of Elementary Projection Drawing 1 21110, I 50
Manual of Elementary P-roblems in the Linear Perspective of Form and
Shadow I2mo, i oo
Plane Problems in Elementary Geometry I2mo, i 25
Primary Geometry I2mo, 75
Elements of Descriptive Geometry, Shadows, and Perspective 8vo, 3 50
General Problems of Shades and Shadows 8vo, 3 oo
Elements of Machine Construction and Drawing 8vo, 7 50
Problems. Theorems, and Examples in Descriptive Geometry 8vo, 2 50
Weisbach's Kinematics and the Power of Transmission. vHermann ard
Klein.) 8vo, 5 oo
Whelpley's Practical Instruction in the Art of Letter Engraving I2mo, 2 oo
Wilson's Topographic Surveying 8vo, 3 So
Free-hand Perspective , 8vo, 2 50
Free-hand Lettering 8vo, i oo
Woo If "s Elementary Course in Descriptive Geometry Large 8vo, 3 oo
ELECTRICITY AND PHYSICS.
Anthony and Brackett's Text-book of Physics. (Magic.) , . . .Small 8vo, 3 oo
Anthony's Lecture-notes on the Theory of Electrical Measurements lamo, i oo
Benjamin's History of Electricity 8vo, 3 oo
Voltaic Cell 8vo, 3 oo
Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.). .8vo, 3 oo
Crehore and Sauier's Polarizing Photo-chronograph 8vo > 3 oo
Dawson's "Engineering" and Electric Traction Pocket-book. . i6mo, morocco, 5 oo
Dolezalek's Theory of the Lead Accumulator. (Storage Battery.)
(Shortly.) (Von Ende.)
Dtihem's Thermodynamics and Chemistry. (Burgess.) 8vo, 4 oo
Flather's Dvnamometers, and the Measurement of Power I2mo, 3 oo
Giioert's De Magnete. (Mottelay.) 8vo, a 50
Hanchett's Alternating Currents Explained. (Shortly.)
Holman's Precision of Measurements 8vo, 2 oo
Telescopic Mirror-scale Method, Adjustments, and Tests Large JJvo, 75
Lanaauer's Spectrum Analysis. (Tingle.) 8vo, 3 oo
Le Chatelier's High-temperature Measurements. (Boudouard Burgess. )i2mo, 3 oo
Lob's Electrolysis and Electrosynthesis of Organic Compounds. (Lorenz.) 12010, i oo
* Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II. 8vo, each, 6 oo
* Michie. Elements of Wave Motion Relating to Sound and Light 8vo. 4 oo
Niaudet's Elementary Treatise on Electric Batteries. (Fishoack. ) izmo, 2 50
* Parshall and Hobart's Electric Generators Small 4to. half morocco, 10 oo
* Rosenberg's Electrical Engineering. (Haldane Gee Kinzbrunner.). . . .8vo, i 50
Ryan, Norris, and Hoxie's Electrical Machinery. Vol. 1 8vo, 2 50
Thurston's Stationary Steam-engines 8vo, 2 50
* Tillman's Elementary Lessons in Heat 8vo, i 50
9
Tory and Pitcher's Manual of Laboratory Physics Small 8vo, 2 oo
Dike's Modern Electrolytic Copper Refining 8vo, 3 oo
LAW.
* Davis's Elements of Law 8vo. 2 50
* Treatise on the Military Law of United States 8vo, 7 oo
Sheep, 7 so
Manual for Courts-martial i6mo, morocco, i so
Wait's Engineering and Architectural Jurisprudence 8vo, 6 oo
Sheep, 6 50
Law of Operations Preliminary to Construction in Engineering and Archi-
tecture 8vo, 5 oo
Sheep, 5 So
Law of Contracts 8vo, 3 oo
Winthrop's Abridgment of Military Law 1 21110, a 50
MANUFACTURES.
Bernadou's Smokeless Powder Nitro-cellulose and Theory of the Cellulose
Molecule I2mo, 2 50
Holland's Iron Founder izmo, 2 50
" The Iron Founder," Supplement I2mo, a so
Encyclopedia of Founding and Dictionary of Foundry Terms Used in the
Practice of Moulding 1 2 mo , 3 oo
Eissler's Modern High Explosives 8vo, 4 oo
E ffront's Enzymes and their Applications. (Prescott.) 8vo, 3 oo
Fitzgerald's Boston Machinist i8mo, i oo
Ford's Boiler Making for Boiler Makers 1 8mo , i oo
Hopkins's Oil-chemists' Handbook 8vo, 3 oo
Keep's Cast Iron 8vo, a 50
Leach's The Inspection and Analysis of Food with Special Reference to State
Control. (In preparation.)
Metcalf's Steel A Manual for Steel-users tamo, a o
Metcalfe's Cost of Manufactures And the Administration of Workshops,
Public and Private 8vo. 5 oo
Meyer's Modern Locomotive Construction 4to, 10 oo
* Reisig's Guide to Piece-dyeing '. 8vo, 25 oo
Smith's Press-working of Metals 8vo, 3 ob
Wire: Its Use and Manufacture Small 4to, 3 oo
Spalding's Hydraulic Cement I2mo, 2 oo
Spencer's Handbook for Chemiits of Beet-sugar Houses i6mo, morocco, 3 oo
andboox tor sugar Manufacturers and their Chemists.. . i6mo, morocco, a oo
Thurston's Manual of Steam-boilers, their Designs, Construction and Opera-
tion 8vo, 5 o
* Walke's Lectures on Explosives 8vo, 4 oo
West's American Foundry Practice ^ I2mo, 2 50
Moulder's Text-book iamo. 2 so
Wiechmann's Sugar Analysis Small 8vo. 2 50
Wolff's Windmill as a Prime Mover 8vo, 3 oo
Woodbury's Fire Protection of Mills : 8vo, x 50
MATHEMATICS.
Baker's Elliptic Functions 8vo, i 50
* Bass's Elements of Differential Calculus i zmo, 4 oo
B7iggs's.Elements>f Plane Analytic Geometry I2mo, i oo
10
Compton's Manual of Logarithmic Computations 12010,
Davis's Introduction to the Logic of Algebra 8vo,
* Dickson's CoUege Algebra Large I2mo,
* Introduction to the Theory of Algebraic Equations Large izmo,
Halsted's Elements of Geometry 8vo,
Elementary Synthetic Geometry 8vo.
Rational Geometry. (Shortly.)
* Johnson's Three-place Logarithmic Tables: Vest-pocket size paper, 15
100 copies for 5 oo
* Mounted on heavy cardboard, 8X10 inches, 25
10 copies for 2 oo
Elementary Treatise on the Integral Calculus Small 8vo, i 50
Curve Tracing in Cartesian Co-ordinates 1 2mo, I oo
Treatise on Ordinary and Partial Differential Equations Small 8vo, 3 50
Theory of Errors and the Method of Least Squares I2mo, i 50
* Theoretical Mechanics I2mo, 3 oo
Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) i2mo, 200
* Ludlow and Bass. Elements of Trigonometry and Logarithmic and Other
Tables 8vo, 3 oo
Trigonometry and Tables published separately Each, 2 oo
Maurer's Technical Mechanics 8vo, 4 o
Merriman and Woodward's Higher Mathematics 8vo, 5 oo
Merriman's Method of Least Squares 8vo, 2 OO
Rice and Johnson's Elementary Treatise on the Differential Calculus. Sm., 8vo, 3 oo
Differential and Integral Calculus. 2 vols. in one Small 8vo, 2 50
Wood's Elements of Co-ordinate Geometry 8vo, 2 oo
Trigonometry: Analytical, Plane, and Spherical izmo, I oo
MECHANICAL ENGINEERING.
MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS.
Baldwin's Steam Heating for Buildings i zmo, 2 50
Barr's Kinematics of Machinery 8vo, 2 50
Bartlett's Mechanical Drawing 8vo, 3 oo
Abridged Ed 8vo, i 5*
Benjamin's Wrinkles and Recipes 121110, 2 oo
Carpenter's Experimental Engineering 8vo, 6 oo
Heating and Ventilating Buildings 8vo, 4 oo
Gary's Smoke Suppression in Plants using Bituminous Coal. (In prep-
aration.")
Clerk's Gas and Oil Engine Small 8vo, 4 oo
Coolidge's Manual of Drawing 8vo, paper, i oo
Cromwell's Treatise on Toothed Gearing I2mo, i 50
Treatise on Belts and Pul.eys i amo, I 50
Durley's Kinematics of Machines 8vo, 4 oo
Flather's Dynamometers and the Measurement of Power I2mo, 3 oo
Rope Driving 1 2 mo , 2 oo
Gill's Gas and Fuel Analysis for Engineers i2mo, i 25
Hall's Car Lubrication . . i imo, i oo
Button's The Gas Engine 8vo, 5 o
Jones's Machine Design:
Part I. Kinematics of Machinery 8vo, i 50
Part II. Form, Strength, and Proportions of Parts 8vo, 3 oo
Kent's Mechanical Engineer's Pocket-book i6mo, morocco, 5 oo
Kerr's Power and Power Transmission 8vo, 2 oo
MacCord's Kinematics; or, Practical Mechanism 8vo, 5 oo
Mechanical Drawing 4to, 4 oo
Velocity Diagrams 8vo, i 50
11
Mahau's Industrial Drawing. (Thompson.) 8vo, 3 50
Poole's Calorific Power of Fuels 8vo, 3 oo
Reid's Course in Mechanical Drawing 8vo. 2 oo
Text-book of Mechanical Drawing and Elementary Machine Design. . Svo, 3 oo
Richards's Compressed Air 1 21110, i 50
Robinson's Principles of Mechanism 8vo, 3 oo
Smith's Press-working of Metals 8v<\ 3 oo
Thurston's Treatise on Friction and Lost Work in Machinery and Mul
Work 8vo, 3 oo
Animal as a Machine and Prime Motor, and the Laws of Energetics, i arno, i oo
Warren's Elements of Machine Construction and Drawing 8vo, 7 50
Weisbach's Kinematics and the Power of Transmission. Herrmann
Klein.) . . . . 8vo, 5 oo
Machinery of Transmission and Governors. (Herrmann Klein.). .8vo, 500
Hydraul.cs and Hydraulic Motors. (Du Bois.) 8vo, 5 oo
Wolff's Windmill as a Prime Mover 8vo, 3 oo
Wood's Turbines 8vo, 2 50
MATERIALS OF ENGINEERING.
Bovey's Strength of Materials and Theory of Structures 8vo, 7 50
Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edition,
Reset 8vo. 7 50
Church's Mechanics of Engineering 8vo, 6 oo
Johnson'o Materials of Construction Large 8vo, 6 oo
Keep's Cast Iron 8vo, 2 50
Lanza's Applied Mechanics ? 8vo, 7 50
Martens's Handbook on Testing Materials. (Henning.) 8vo, 7 50
Merriman's Text-book on the Mechanics of Materials 8vo, 4 OO
Strength of Materals 1 2 nio , i oo
Metcalf's SteeL A Manual for Steel-users I2mo 2 oo
Smith's Wire: Its Use and Manufacture Small 4to, 3 oo
Materials of Machines I2mo i oo
Thurston's Materials of Engineering 3 vols , Svo, 8 oo
Part H. Iron and Steel Svo, 3 50
Part HI. A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents Svo 2 50
Text-book of the Materials of Construction Svo, 5 oo
Wood's Treatise on the Resistance of Materials and an Appendix on the
Preservation of Timber Svo, a oo
Elements of Analytical Mechanics Svo, 3 oo
Wood's Rustless Coatings. (Shortly.)
STEAM-ENGINES AND BOILERS.
Carnot's Reflections on the Motive Power of Heat. (Thurston.) I2mo, i 50
Dawson's "Engineering" and Electric Traction Pocket-book. . i6mo, mor., 5 oo
Ford's Boiler Making for Boiler Makers i8mo, i oo
Goss's Locomot' ve Sparks .* Svo, 2 oo
Hem<-nway's Indicator Practice and Steam-engne Economy I2mo. a oo
Hutton'* Mechanical Engineering of Power Plants Svo, 5 oo
Heat and Heat-engines Svo, 5 oo
Kent's Steam-bo ; ler Economy Svo, 4 oo
Kneass's Practice and Theory of the Injector Svo i 50
HacCord's Slide-valves Svo, 2 oo
Meyer's Modern Locomotive Construction 4to, 10 oo
Peabody's Manua, of the Steam-engine Indicator 121110, I 50
Tables of the Properties of Saturated Steam and Other Vapors 8vo,' I oo
Thermodynamics of the Steam-engine and Other Heat-engines 8vo, 5 oo
Valve-gears for Steam-engines 8vo, a 50
Peabody and Miller's Steam-boilers 8vo, 4 oo
Fray's Twenty Years with the Indicator Large 8vo, 2 50
Pupln's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors.
(Osterberg.) izmo, i 25
Reagan's Locomotives : Simple, Compound, and Electric izmo, 2 50
Rontgen's Principles of Thermodynamics. (Du Bois.) 8vo, 5 oo
Sinclair's Locomotive Engine Running and Management i2mo, 2 oo
Smart's Handbook of Engineering Laboratory Practice I2mo, 2 50
Snow's Steam-boiler Practice 8vo, 3 oo
Spangler's Valve-gears 8vo, 2 50
Notes on Thermodynamics I2mo, i oo
Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo, 3 oo
Thurston's Handy Tables 8vo. i 50
Manual of the Steam-engine 2 vols. 8vo, 10 oo
Part I. History. Structuce, and Theory 8vo, 6 oo
Part II. Design, Construction, and Operation 8vo, 6 oo
Handbook of Engine and Boiler Trials, and the Dse of the Indicator and
the Prony Brake 8vo, 5 oo
Stationary Steam-engines 8vo, 2 50
Steam-boiler Explosions in Theory and in Practice . . I2mo i 50
Manual of Steam-boiler? , Their Designs, Construction, and Operation . 8vo, 5 oo
Weisbach's Heat, Steam, a 1 Steam-engines. (Du Bois.) 8vo, 500
Whitham's Steam-engine 1 isign 8vo, 5 oo
Wilson's Treatise on Steam- boilers. (Flather.) i6mo, 250
Wood's Thermodynamics. Heat Motors, and Refrigerating Machines. . . ,8vo. 4 oo
MECHANICS AND MACHINERY.
Barr's Kinematics ot machinery 8vo, 2 50
Bovey's Strength of Materials and Theory of Structures 8vo, 7 50
Chase's The Art of Pattern-making I2mo, 2 50
Chordal. Extracts from Letters I2mo, 2 oo
Church's Mechanics of Engineering 8vo , 6 oo
Notes and Examples in Mechanics 8vo, 2 oo
Compton's First Lessons in Metal-working I2mo, i 50
Compton and De Groodt's The Speed Lathe I2mo, i 50
Cromwell's Treatise on Toothed Gearing I2mo, i 50
Treatise on Belts and Pulleys I2mo, i 50
Dana's Text-book of Elementary Mechanics for the Use of Colleges and
Schools I2mo, i 50
Dingey's Machinery Pattern Making i2mo, 2 oo
Dredge's Record of the Transportation Exhibits Building of the World's .
Columbian Exposition of 1893 4to, half morocco, 5 oo
Du Bois's Elementary Principles of Mechanics:
VoL I. Kinematics 8vo, 3 SO
Vol II. Statics 8vo, 4 oo
Vol. III. Kinetics 8vo, 3 50
Mechanics of Engineering. Vol. I Small 4to, 7 50
Vol. II Small 4to, 10 oo
Durley's Kinematics of Machines 8vo, 4 oo
Fitzgerald's Boston Machinist i6mo, i oo
Flather's Dynamometers, and the Measurement of Power I2mo, 3 oo
Rope Driving I2mo, 2 oo
Goss's Locomotive Sparks 8vo, 2 oo
13
Hall's Car Lubrication I2mo, i oo
Holly's Art of Saw Filing i8mo 75
* Johnson's Theoretical Mechanics I2mo, 3 oo
Statics by Graphic and Algebraic Methods 8vo, 2 oo
Jones's Machine Design:
Part I. Kinematics of Machinery 8vo, i 50
Part H. Form, Strength, and Proportions of Parts 8vo, 3 oo
Kerr's Power and Power Transmission 8vo, 2 oo
Lanza's Applied Mechanics 8vo, 7 50
MacCord's Kinematics; or, Practical Mechanism 8vo, 5 oo
Velocity Diagrams 8vo, i 50
Maurer's Technical Mechanics 8vo, 4 oo
Merriman's Text-book on the Mechanics of Materials 8vo, 4 oo
* Michie's Elements of Analytical Mechanics 8vo, 4 oo
Reagan's Locomotives: Simple, Compound, and Electric I2mo, 2 50
Reid's Course in Mechanical Drawing 8vo, 2 oo
Text-book of Mechanical Drawing and Elementary Machine Design . . 8vo , 3 oo
Richards's Compressed Air I2mo, i 50
Robinson's Principles of Mechanism 8vo , 3 oo
Ryan, Norris, and Hoxie's Electrical Machinery 8vo, 2 50
Sinclair's Locomotive-engine Running and Management I2mo. 2 oo
Smith's Press-working of Metals 8vo, 3 oo
Materials of Machines i2mo, i oo
Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo, 3 oo
Thurston's Treatise on Friction and Lost Work in Machinery and Mill
Work. 8vo, 3 oo
Animal as a Machine and Prime Motor, and the Laws of Energetics. i2mo, i oo
Warren's Elements of Machine Construction and Drawing 8vo, 7 50
Weisbach's Kinematics and the Power of Transmission. (Herrmann
Klein.) 8vo, 5 oo
Machinery of Transmission and Governors. (Herrmann Klein.). 8vo, 5 oo
Wood's Elements of Analytical Mechanics 8vo, 3 oo
Principles of Elementary Mechanics I2mo, i 25
Turbines 8vo, 2 50
The World's Columbian Exposition of 1893 -4to, i oo
METALLURGY.
Egleston's Metallurgy of Silver, Gold, and Mercury:
VoL I. Silver 8vo,
VoL II. Gold and Mercury 8vo,
** Iles's Lead-smelting. (Postage 9 cents additional.) I2mo,
Keep's Cast Iron 8vo,
Kunhardt's Practice of Ore Dressing in Europe 8vo,
Le Chatelier's High-temperature Measurements. (Boudouard Burgess.) . 1 2 mo
Metcalf's Steel. A Manual for Steel-users I2mo,
oo
Smith's Materials of Machines I2m
Thurston's Materials of Engineering. In Three Parts 8vo, 8 oo
Part II. Iron and Steel 8vo, 3 50
Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents 8vo, 2 50
Hike's Modern Electrolytic Copper Refining 8vo, 3 oo
MINERALOGY.
Barringer's Description of Minerals of Commercial Value. Oblong, morocco, 2 50
Boyd's Resources of Southwest Virginia 8vo, 3 oo
Map of Southwest Virginia Pocket-book form, 2 oo
14
Brush's Manual of Determinative Mineralogy. (.Penfield.) 8vo, 4 oo
Chester's Catalogue of Minerals 8vo, paper, i oo
Cloth, i 25
Dictionary of the Names of Minerals 8vo, 3 50
Dana's System of Mineralogy Large 8vo, half leather, 12 50
First Appendix to Dana's New "System of Mineralogy." Large 8 vo, i oo
Text-book of Mineralogy 8vo, 4 oo
Minerals and How to Study Them . . . : xamo, i 50
Catalogue of American Localities of Minerals Large 8vo , i oo
Manual of Mineralogy and Petrography i2tno, 2 oo
Bakle's Mineral Tables. (Shortly.)
Egleston's Catalogue of Minerals and Synonyms 8vo, 2 50
Hussak's The Determination of Rock-forming Minerals. (Smith.) Small 8vo, 2 oo
Merrill's Non-Metallic Minerals. (Shortly.)
* Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests.
8vo, paper, o 50
Rosenbusch's Microscopical Physiography of the Rock-making Minerals.
(Iddings.) 8vo, 5 oo
* Tillman's Text-book of Important Minerals and Docks 8vo, 2 oo
Williams's Manual of Lithology 8vo, 3 oo
MINING.
Beard's Ventilation of Mines I2mo, 2 50
Boyd's Resources of Southwest Virginia ." 8vo, 3 oo
Map of Southwest Virginia Pocket-book form, 2 oo
* Drinker's Tunneling, Explosive Compounds, and Rock Drills.
4to, half morocco, 25 oo
Eissler's Modern High Explosives 8vo, 4 oo
Fowler's Sewage Works Analyses ' I2mo, 2 oo
Goodyear's Coal-mines of the Western Coast of the United States 12 mo, 2 50
Ihlseng's Manual of Mining 8vo, 4 oo
** Iles's Lead-smelting. (Postage QC. additional.) I2mo, 2 50
Kunhardt's Practice of Ore Dressing in Europe 8vo, i 50
O'Driscoll's Notes on the Treatment of Gold Ores 8vo, 2 oo
* Walke's Lectures on Explosives 8vo, 4 oo
Wilson's Cyanide Processes I2mo, i 50
Chlorination Process I2mo, i 50
Hydraulic and Placer Mining 1 2 mo, 2 oo
Treatise on Practical and Theoretical Mine Ventilation X2mo i 25
SANITARY SCIENCE.
Copeland's Manual of Bacteriology. (In preparation.)
Folwell's Sewerage. (Designing, Construction and Maintenance.) 8vo, 300
Water-supply Engineering 8vo, 4 oo
Fuertes's Water and Public Health 12010, i 50
Water-filtration Works . . 12010, 2 50
Gerhard's Guide to Sanitary House-inspection i6mo, i oo
Goodrich's Economical Disposal of Town's Refuse Demy 8vo, 3 50
Hazen's Filtration of Public Water-supplies 8vo, 3 oo
Kiersted's Sewage Disposal 12010, i 25
Leach's The Inspection and Analysis of Food with Special Reference to State
Control. (In preparation.)
Mason's Water-supply. (Considered Principally from a Sanitary Stand-
point.) 3d Edition, Rewritten : . . 8vo, 4 oo
Examination of Water. (Chemical and Bacteriological.) 12010, i 25
15
Merriman's Elements of Sanitary Engineering 8vo, 2 oo
Nichols's Water-supply. (Considered Mainly from a Chemical and Sanitary
Standpoint.) (1883.) 8vo, 2 50
Ogden's Sewer Design 1 2mo, 2 oo
* Price's Handbook on Sanitation I2mo, I 50
Richards's Cost of Food. A Study in Dietaries I2mo, I oo
Cost of Living as Modified by Sanitary Science 12 mo, r oo
Richards and Woodman's Air, Water, and Food from a Sanitary Stand-
point 8vo, 2 oo
* Richards and Williams's The Dietary Computer 8vo, i 50
Rideal's Sewage and Bacterial Purification of Sewage 8vo, 3 50
Turneaure and Russell's Public Water-supplies 8vo, 5 oo
Whipple's Microscopy of Drinking-water 8vo, 3 50
Woodhull's Notes and Military Hygiene i6mo, i 50
MISCELLANEOUS.
Barker's Deep-sea Soundings 8vo, 2 oo
Emmons's Geological Guide-book of the Rocky Mountain Excursion of the
International Congress of Geologists Large 8vo, i 50
Fen-el's Popular Treatise on the Winds 8vo, 4 oo
Haines's American Railway Management I2mo, 2 50
Mott's Composition, Digestibility, and Nutritive Value of Food. Mounted chart, i 25
Fallacy of the Present Theory of Sound i6mo, i oo
Ricketts's History of Rensselaer Polytechnic Institute, 1824-1894. Small 8vo, 3 oo
Rotherham's Emphasized New Testament Large 8vo, 2 oo
Steel's Treatise on the Diseases of the Dog 8vo, 3 50
Totten's Important Question in Metrology 8vo, 2 50
The World's Columbian Exposition ot 1893 4to, i oo
Worcester and Atkinson. Small Hospitals, Establishment and Maintenance,
and Suggestions for Hospital Architecture, with Plans for a Small
Hospital I2mo, i 25
HEBREW AND CHALDEE TEXT-BOOKS.
Green's Grammar of the Hebrew Language 8vo, 3 oo
Elementary Hebrew Grammar i2mo, i 25
Hebrew Chrestomathy 8vo, 2 oo
Gesenius's Hebrew and Cbaldee Lexicon to the Old Testament Scriptures.
(Tregelles.) Small 4to, half morocco, 5 joo
Letteris's Hebrew Bible 8vo, 2 25
16
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