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Tecl^nical Drawing Series

Essentials of Gear

ANTHONY

TECHNICAL DRAWING SERIES

THE ESSENTIALS OF GEARING

A TEXT BOOK FOR TECHNICAL STUDENTS AND FOR SELF-INSTRUCTION,

CONTAINING NUMEROUS PROBLEMS AND

PRACTICAL FORMULAS

GARDNER C. ANTHONY, A.M.

Prokessor of Dravvint. in Tufts College; Dean of the Bromfield-Pearson School,

Author of "Elements of Mechanical Drawing," and " Machine Drawing;"

Member of American Society for the Promotion of Engineering Education;

Member of American Society of Mechanical Engineers

BOSTON, U.S.A.

D. C. HEATH & CO., PUBLISHERS

1897

By Gardner C. Anthony,

1897.

PEE FACE.

The most feasible method for the acquirement of a working knowledge of the theory of
gear-teeth curves is by a graphic solution of problems relating thereto. But it requires much
time on the part of an instructor, and is very difficult for the student, to devise suitable exam-
ples which, while fully illustrating the theory, shall involve the minimum amount of drawing.
It is tlie aim of the author to overcome these difficulties by the presentation of a series of pro-
gressive problems, designed to illustrate the principles set forth in the text, and also to encour-
age a thorough investigation of the subject by suggesting lines of thought and study beyond
the limits of this work.

In this as in the other books of the series the author would emphasize the fact that the
plates are not intended for copies, but as illustrations. A definite lay-out for each problem is
given, and the conditions for the same are clearly stated. This is accompanied by numerous
references to the text, so that a careful study of tlie subject is necessitated before performing
the problems.

Although specially addressed to students having no previous knowledge of the principles
of kinematics, it is also designed to serve as supplementary to treatises on this subject.

The methods and problems have already proved their usefulness in the instruction of stu-
dents of many grades : and it is hoped that tlieir publication may promote a wider interest in,

and more thorough studv of, the essentials of gearing.

GARDNER C. ANTHONY.
Tufts College, Sept. 24, 1897.

2065972

COISTTEI^TS.

CHAPTER I. PAOK

Introduction. — General Principles 1

I. Constant Velocity Ratio. 2. Positive Rotation. 3. (iearing.

CHAPTER II.
Odontoidal Curves • 4

4. Classes of Curves. 5. Cycloid. 6. Epicycloid. 7. Hypocycloid. 8. To Construct a Normal.
9. A Second Method for Describing the Cycloidal Cnrves. 10. Double Generation of the
Epicycloid aud Hypocycloid. 11. Epitrochoid. 12. luvolute.

CHAPTER III.

Spur Gears and the Cycloidal System 8

13. Theory of Cycloidal Action. 14. Law of Tooth Coutact. 15. Application. 16. Spur Gears.
17. Circular Pitch. 18. Diameter Pitch. 19. Face or Addendum. 20. Flank or De-
deudum. 21. Path of Contact. 22. Arc of Contact. 23. Arcs of Approach and Recess.
24. Angle of Obliquity or Pressure. 25. Rack. 26. Spur Gears Having Action on Both
Sides of the Pitch Point. 27. Clearance. 28. Curve of Least Clearance. 29. Backlash.
30. Conditions Governing the Practical Case. 31. Proportions of Standard Tooth. 32. In-
fluence of the Diameter of the Rolling Circle on the Shai>e and Efticiency of (iear Teeth.

VI rONTENTS.

PAGE

33. Intercliaii,n'eal)le (xears. 34. Practical Case of Cycloidal Gearing. 35. Face of (iear.
36. Comparison of Gears, illustrated in Plates 4, 5, and 6. 37. Conventional Representa-
tion of Spur Gears.

CHAPTER TV.

Involute System 26

38. Theory of Involute Action. 39. Character of the Curve. 40. Involute Limiting Case. 41. Epi-
cycloidal J^xtension of Involute Teeth. 42. Involute Practical Case. 43. Interference.
44. Influence of the Angle of Pressure. 45. Method for Determining the Least Angle of
Pressure for a Given Number of Teeth Having no Interference. 46. Defects of a System
of Involute Gearing. 47. Unsymmetrical Teeth.

CHAPTER V.

Annular Gearing 38

48. Cycloidal System of Annular Gearing. 49. Limiting Case. 50. Secondary Action in Annular
Gearing. 51. Limitations of the Intermediate Describing Curve. 52. Limitations of Exte-
rior and Interior Describing Curves. 53. The Limiting Values of the Exterior, Interior, and
Intermediate Describing Circles for Secondary Action. 54. Practical Case. 55. Summary
of Limitations and Practical Considerations. 56. Involute System of Annular Gearing.

CHAPTER VL

Bevel Gearing 45

57. Theory of Bevel Crearing. 58. Cliaracter of Curves Employed in Bevel Gearing. 59. Tredgold
Approximation. 60. Drafting the Bevel Gear. 61. Figuring the Bevel Gear with Axes at
90°. 62. Bevel (Jeai' Table for Shafts at 90°. 63- Bevel (iears with Axes at Any Angle.

CONTENTS. VU

CTIAPTKK VII.

I'AGK

Special Forms of Gears, Notation, Formulas, etc 57

64. Odoutograplis and Odoiitograph Tables. 65. Willis's Odoiitograph. 66. The Tliree-Poiiit
Odoutograph. 67. The (iraiit Involute Oduntograph. 68. The llobinsuu Odoutograph.
69. The Klein C'ooidinate Odoutograph. 70. Special Forms of Odontoids. and Their Lines
of Action. 71. Conjugate Curves. 72. \\'orni (Jeaiiug. "73. Literature. 74. Notation
and Formulas.

CHAPTER YIIL

Problems ...,.,.,, ,68

75. ^Method to l>e Observed in Performing the Problems —

PuoiiLK.M 1. Cycloidal Limiting Case. Face or Flanlc Only 69

'2. Cycloidal Limiting Case. Face and Flank 71

o. Cycloidal (iear. Practical Case 72

4. Involute Limiting Case , . . 74

5. Involute Practical Cases 75

6. Cycloidal Annular (iear 76

7. Involute Annular Gear 77

8. Cycloidal and Involute Bevel Gears. Shafts at !H)- 78

9. Cycloidal and Involute Bevel (iears. Shafts at Other than 90" 79

ERRATA FOR THE ESSENTIALS OF GEARING.

0.5
Page l(i, Hue 10, For o.oP read -—

Page 18, line 10. For one-sixth read one-seventh.

Page 18, line 11. For Fig. 6 read Fig. 9.

Page 34, line 4. For — read ^^

Page 69, problem 1, column (i. For A read a.

Plate 2. For A' on director circle read A.

Plate 10. Erase "Pitch line" near H K.

Plate 14. Fig. '2. The pinion is designed to have 5 teeth and the gear fi teeth.

THE ESSENTIALS OF GEARING.

CHAPTER I.

INTRODUCTION. — GENERAL PRINCIPLES.

1. Constant Velocity Ratio. Motion may be transmitted between lines of shafting by
means of friction surfaces ; and if there be no slipping of the contact surfaces, the circumference
of the one will have the same velocity as the circumference of the other. The number of revo-
lutions of the shafts will be inversely proportional to the diameter of the friction surfaces, and
this ratio will l)e maintained constant under the condition of no slip. Such friction surfaces
and shafts are said to have a co7istant velocity ratio.

2. Positive Rotation. In order to transmit force, as well as motion, and to insure its
being positive, it will be necessary to place cogs, or elevations, on one of the friction sur-
faces, and make suitable depressions in the other surface.

3. Gearing. The study of toothed gearing is a study of the shape of these cogs, teeth, or
odontoids, which are designed to produce a positire rotation Mhile preserving the condition of
constant velocity ratio.

GEARS CLASSIFIED.

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

Gears may l)e classified as fol-
lows : —

1. If the shafts are parallel,
the friction surfaces would be
cylinders (Fig. 1), and the gears
designed to produce the same
condition, as to the velocit}^ are
called Spur G-ears (Fig. 2).

2. If the shafts intersect, the
friction surfaces would he cones
(Fig. 3), and the gears called
Bevel Gears (Fig. 4).

3. If the shafts are neither in-
tersecting nor })arallel, the friction
surfaces will be hyperboloids of
revolution (Fig. 5), and the gears
called Ilt/perbolic, or Skeiv Grears
(Fig. 6).

In the preceding cases the ele-
ments of the teeth are rectilinear,
and the friction surfaces touch
each other along right lines.

4. If the elements of the teeth

GEARS CLASSIFIED.

in either of the first three cases
l)e made helical, an entirely dif-
ferent class of gearing will result.
The various forms are known as
Tivisted, Sph-(d, Worm^ and Screw
Crearing (Figs. 7 and 8). The
action of the latter is analogous to
that of a screw and nut.

One of these forms is generally
eni[)lo_yed as a siibstitnte for hy-
perbolic, or skew gears, by reason
of the ditficnlty experienced in cor-
rectly forming the teeth of snch
gears.

5. AnothtM-, allliough bnt little
used, form, is that known as Fare
(rf(irut(/. The teeth are iisnally
pins secnred to the face of cir-
cnlar disks having axes perpen-
dicnlar. The action takes place
at a point only.

None of the latter forms can
be represented by friction surfaces.

Fig. 7.

Fig. 8.

4 ODUNTOIDAL CUKVEtS.

CHAPTER II.

ODONTOIDAL CURVES.

4. The two classes of ciirves comnioiily employed in gear teeth are the cycloidal and the
involute. A knowledge of their characteristics and methods of generating is essential to an
understanding of their application in gearing.

5. Cycloid. Plate 1, Fig. 1. The cjcloid is a curve generated b}- a point in the
circumference of a circle A\hich rolls upon its tangent. The circle is called the describing,
or generating circle, and the point is known as the describing, or generating point. In
Fig. 1, Plate 1, B is the describing point, and B D C E the describing circle, which rolls on
its tangent E B'".

Assume a point, C, on the describing circle, and conceive the motion of the circle to be
from left to I'iglit. As it rolls upon its tangent, the arc E C will be measured off on E B'"
until point C becomes a point of tangency at C. The center of the describing circle will now
lie at A', in the perpendicular to E B"' at C\

From center A', with radius of describing circle, draw the ncAV position of describing circle.
The generating point must lie in this circle at a distance from C equal to the chord B C.

Therefore, with radius equal to this chord, from center C, describe an arc intersecting the
new position of the describing circle. The line B' C is called the instantaneous radius, or
normaU of the curve at B', it being a [)erpendicular to the tangent of the curve at this point.

CYCLOIDAL (UKVES. 5

The normal at B" would be B" D'. The radius A' B' is known as the dt'scrilniif/, or f/enerating
radius, and A' C is tlie contact radius, or the radius at the point of eontai-t. In like manner
other positions of the describing pt)int may be found, and the curve connecting them will l)e
the cycloid required.

6. Epicycloid. Plate 1, F'ig. 2. If the describing circle rolls upon the outside of an
arc, or circle, called the director circle, the curve generated will be an epicycloid, Fig. 2,
Plate 1. The method of descri1)ing this curve is similar to that for the cycloid, and the
lettering is the same. It must be observed, however, that any contact radius, as A' C, is a
radial line of the circle on which it rolls.

7. Hypocycloid. Plate 1, Fig. 3. If the describing circle rolls upon tlie inside of a cir-
cle, the curve generated will be an hypocycloid. P'ig. 3 illustrates this curve, the same letter-
ing Ijeing used as that of the preceding cases.

If it be re(j[uired to draw a normal at any point of this, or the two preceding curves, the fol-
lowing method may l)e employed : —

8. To Construct a Normal. From the given point on the curve, as a center, with radius of
generating circle, describe an arc cutting the path described by the center of the generating
circle. From this point draw the contact radius, thus obtaining the contact point. Connect
this with the given point, and tlie line will be the required normal.

9. A Second Method for Describing the Cycloidal Curves. Plate 2, Fig. 1. A B C is a
director circle, A D E, the generating circle for the epicycloid A A' A" H , and A K L the generating
circle for the hypoc3'cloid A L C.

b CYCLOIDAL ( UKVES.

To describe the epicycloid, assume any point, D, on tlie generating circle, and lay off the arc
A D' on the director circle, making it equal to arc A D. If A be the describing point, then A D
will be the normal when D shall have become a contact point, as at D'. With L as a center, de-
scribe the arc D A', llie describing point A must be in this arc when D shall be at D'. From D'
as a center, with radius equal to the chord A D , describe an arc intersecting A' D , and thus deter-
mine A', a point in the epicycloid. Similarly obtain other points, and draw the curve.

The hypocycloid may be constructed in like manner, as shown by the same figure. This
also illustrates a special case in which the hypocycloid is a radial line, A L C, and this is due
to the diameter of tlie describing circle being equal to the radius of the director circle.

The same method may also be employed in the construction of the cycloid.

10. Double Generation of the Epicycloid and Hypocycloid. Plate 2, Fig. 1. The epi-
cycloid may always be generated by either of two describing circles, which differ in diameter
by an amount equal to the diameter of the director circle. Thus in the case illustrated, the
epicycloid A A' A" H may be generated by the circle A D E , with A as a describing point, or by
the circle s T H , with H as a describing point. Similarly the hypocycloid is capable of being-
generated by either of two rolling circles, the sum of which diameters must equal that of the
director circle.*

11. Epitrochoid. Plate 2, Fig. 2, When the describing point does not lie on the cir-
cumference of the generating circle, a curve, connnonly called an epitrochoid, is described. If
the point lies without the circle, as at B , a looped curve, B B' B", called the curtate epitrochoid,

* For tlie geoiiietriiMl (Iriuuij.sUatiuu of this ])r'>l)lein see tUe aiipeiulix of Professor jNIacCord's ■" Kinematics," page old.

INVOLUTE CURVE. 7

is described: and if the point be within, as at D, the curve will be a prolate epitrot-hoid, as
D D' D".

To obtain a point in the former, assume any point, C, in the circumference of the describing
circle, and determine its position, C, when it shall have become a contact point. Draw the
contact radius A' C, and from C and A' as centei-s, with radii A B and C B, describe arcs intersect-
ing at B', a point in the curve. B' C is the normal at this point. In like manner obtain the
point D' in the })rolate epi trochoid.

12. Involute. Plate 1, Fig. 4. The involute is a curve generated by a point of a tan-
gent right line rolling upon a circle, known as the base circle, or the describing point may be
regarded as at the extremity of a fine wire which is unwound from a cylinder corresponding to
the base circle. In Fig. 4, A B C D is the arc of a base circle, and A the point from which the
involute is generated.

Layoff arcs A B, B C, C D, preferabh' equal to each other, and from points B, C. and D, draw
tangents equal in length to the arcs A B, A C, and A D. A line drawn through the extremity of
these tangents will be an involute of the base cii-cle A B C D.

b THE CYCLOIDAL SYSTEM.

CHAPTER III.

SPUR GEARS AND THE CYCLOIDAL SYSTEM.

13. Theory of Cycloidal Action. Plate 3, Fig. 1. ].iet H K and M L be the peripheries
of two disks, having centers G and F, and S the center of a third disk, also revolving in contact
with the arcs H K and M L. The largest disk will be known as disk 1, the second size as disk
2, and the smallest as disk 3, or the describing disk. Consider the ^peripheries of these disks in
contact at A, so that motion imparted to one will produce an equal motion in the circumference
of the other two, thus maintaining at all times an equal circumferential velocity, or constant
velocity ratio.

Imagine this to represent a model, disk 1 having a flange I extending below the other
disks, and the describing disk as being provided with a marking point at A, each of the disks
being free to revolve about their respective axes. Consider first the relation between the
describing disk and disk 1, the marking point being at A. Suppose motion to be given disk 3
in the direction indicated by the arrow, so that the describing point shall move from A to A'.
The point C of disk 1, which coincides with A when the describing disk is in the first position,
will now have moved on the circumference H K, to C, an arc equal to A A'. During this time,
the curve A' C will have been drawn upon the flange of disk 1 by the marking point. Next,
suppose the marking point to move from A' to A", then, since the circumferences of these disks
tra\erse equal spaces in equal times, C will have revolved to C", and the curve A' C will now
(ic'fupy the position E" C". But, since the niarking point has continued to describe a, curve upon

CYCLOIDAL ACTION. 9

the flange oi cli.sic 1. the curve E" C" will be extended to A". In like manner the marking- point
moves to A"', continuing to describe a curve, as C" A" revolves to C" A'". If now the describing
disk be freed from the axis on which we have supposed it to revolve, and Ije rolled on the cir-
cumference H K, the marking point would describe the same curve, h'" E'" C", as that already
drawn, which is an epicycloid.

In the same manner, we may imagine the marking point to describe a curve upon disk 2,
which curve, in its successive positions, would be shown by A' B', A" B", and A'" B'". For the
same reason, too, the arc A A' A" A'" will equal the arc B B' B" B"' ; and if, in a manner similar to
the preceding, we roll the describing disk on the inside of the arc M L, we shall describe the
same curve A'" D'" B'", and find it to be an hypocycloid.

Again, consider these curves, A'" C" and A'" B'", as l)eing traced at the same time l)y the
describing point A. If we now observe any special position of the point, as A", it will be seen
to be connnon to an epicycloidal, and a hypocycloidal curve, which have a common normal,
A" A, intersecting the line drawn through the centers, F and G, at the point of tangency of the
disks. This condition is true for all positions of the two curves.

If these curves A'" C"\ and A'" B'", be now used as the outlines for gear teeth, as in Fig. 2, G
and F being the centers and H K and M L the pitch lines, we shall have obtained a positive rota-
tion with a uniform velocity ratio, for it was under this latter condition that the curves were
generated, and the connnon normal to the curves at any point of contact will pass through the
point A (the pitch point). Such curves are said to be conjugate.

It is not necessary that the describing point be on the circumference of the circle, or that
the describing curve be a circle, in order to obtain two curves which, acting together, shall pro-
duce a constant velocity ratio.

10 LAW OF TOOTH CONTACT AM) .SPl'K (JEAHS.

14. Law of Tooth Contact. In order to preserve the eoiiditiou of coiistaiit velocity ratio,
the tootli outlines which act in contact must he such that the common normals at the point of
contact shall always cut the line of centers in the same point ; and in general, tiie curves must
1)6 such as may be simultaneously traced upon the planes of rotation of two disks, while re-
volving, by a marking point which is carried by a describing curve, moving in rolling contact
with both disks.

15. Application. Suppose action to take place between the odontoids, or gear teotli, shovvii
in Fig. 2, Plate 3. Let 1 l)e the driver, and suppose motion to begin from the position shown
in the figure, the contact being at A. As the motion takes place, points A', A", A'", will succes-
sively come into contact, their common normals passing through the PITCH POINT, A, at the
time of their contact, thus producing a constant velocity ratio, and the periphery, or pitch
LINE, of 1, will have the same velocity as the periphery, or pitch line, of 2. But this uniform
motion must cease when points k'" W" come into contact, and the velocity ratio will remain con-
stant no longer, unless a second pair of curves begin contact at this moment.

Plate 4 illustrates a pair of disks provided with a series of these curves arranged so as to
continue the motion indefinitely in eitlier direction.

16. Spur Gears. Plate 4. F is the center of a pinion having twelve teeth, and G the
center of a gear of eighteen teeth, only a segment of the latter being shown. A C K is the
describing circle, carrying the marking point C, which descril)ed the epicycloid C D, and
the hypocycloid C E. The depth of the pinion tooth must be made sufficient to admit the
addendum of the gear tooth, but only that portion of the curve between C and E will engage

CIRCULAR AND DIAMETER PITCH. 11

C D. The reinaiiider of the pinion Hank may l)e a continuation of tlio hypocN^chiicI, or any
other curve which may not interfere with the action of the gear tooth. The oppoSi.te sides of
the teeth are made alike in order that motion may take place in either direction. If the direc-
tion be that indicated by the arrows, the pinion bemg the driver, the sluuled side of the teeth
would have contact ; and if the direction be reversed the opposite faces would engage.

In order to accurately reproduce the dedenda of the pinion, a scroll may be used in the fol-
lowing manner : —

Having selected one to match the tooth curve, C E, continue the curve of the scroll by the
center F, from whicli a circle should l)e drawn tangent to the line of the scroll. jNlark that
point of the scroll in contact with the pitch circle. Having laid off the pitch, and thickness of
the teeth, place the marked point of the scroll to coincide with these points, and at the same
time tangent to the circle already drawn. Draw such part of the curve as lies between the
addendum and dedendum circles. Reverse the scroll for drawing the opposite side of the
teeth.

17. Circular Pitch. The distance A D, or A E, measured on the pitch line between cor-
responding points of consecutive teeth, is called the circular pitch, and is equal to the

circumference ot pitch circle
iiuiiiber ol teeth

Let P' denote the circular pitch, D' the diameter of the pitch circle, and N the numl)er of
teeth, then will P' = '^ (1), and, p- = ^ (2).

18. Diameter Pitch. In order to ex})ress in a more direct and simj)le manner the ratio
between the diameter of the pitch circle and the number of teeth, and to easily determine the

12 TOOTH PARTS.

proportions of the teeth, it hiis been found expedient to a})ply the term pitch, or more properly,
diameter pitch, to designate the ratio between the number of teeth and tlie diameter of pitch
circle. This is not an ahsolufe mrasvre^ hut a ratio ; and since it may usually be expressed by a
whole number, the proportions of the parts of a tooth, which are commonly dependent on
the pitch, may be more readily determined, and all the figuring of the gear simplified.

Designating the diameter pitch l)y P, P = ^, (3).

To obtain the relation between the diameter pitch and the circular pitch, compare formulas
2 and 3. -=— , — = ?; lience^, = P or P P' = 7r(4). 'J'liis last e(piation expresses the I'elation
between tlie two pitches in a simple form which may ])e easily remembered.

Illustration. — Tlie pinion represented in Plate 4 has 12 teeth, and is 3 inches in
diameter, k; = P? T "" ^- '^'^^^ pitch, therefore, is 4. The circular pitch, P' = □ = —7 — = .7854.
Having given any two of the terms, N, D', P, P', the other terms may l)e determined.

19. Face, or Addendum. Tliat portion of the tooth curve lying outside of the pitch circle
is called the face or adtlendum, as C D, Plate 4.

20. Flank, or Dedendum. That portion of the tooth curve lying inside of tht^ pitch circle
is called the flaidc or dedendum, as E H , Platp: 4.

21. Path of Contact. In Fig. 1, Plate 3, it will be observed that the contact between
the two curves takes place in the arc A A' k" A'". This is called the path of contact, or line
of action, and in the C3-cloidal system this line is an arc of the describing circle.

ARCS OF CONTArT. 13

22. Arc of Contact. The tire (leseril)e(l l)y a point on the pitch lin(; during' the time of con-
tact of two odontoids is called the arc of contact. It must not be less than the pitch. In this
case the arc of contact would be measured by the arcs A D or A E, and these arcs being equal to
the pitch, the case is called a limiting one. In ^)ractice it should be greater, wliicli w^ould be
accomplished by lengthening the addendum.

23. Arcs of Approach and Recess. There are four cases of contact that may take place
between the gear and pinion of Plate 4.

1. Gear as driver. Direction opposed to the arrows. Contact begins at A and ends at C.

2. Pinion as driver. Direction same as arrows. Contact begins at C and ends at A.

In each of these cases the action will take place between the shaded portions of the teeth.

3. Gear as driver. Direction same as arrows. Contact begins at A and ends at L.

4. Pinion as driver. Direction opposed to the arrows. Contact begins at L and ends at A.
In the last two cases there will be no contact between the shaded portions of the teeth.

In the lii'st and third cases the contact takes place from the pitch point, and is called an arc
of recess.

In the second and fourth cases the contact takes place tow^ard the pitch point, ending at A,
and is called an arc of approach.

It should also be observed that in the case illustrated the arc of contact must be either one
of approach or of recess; but had the teeth of each gear been provided with curves on both
sides of the pitch-line, as in Plate 5, the arc of contact would have consisted of an arc of
approach and of recess. (See Art. 30, page 10, for a further discussion of the relation between
these arcs.)

14 ANGLE OF PRIiSSURE. RACK.

24. Angle of Obliquity, or Pressure. Tlie angle which the conimoii normal to a pair of
conjugate teeth makes with the tangent at the pitch point, is called the angle of obliquity, or
angle of pressure. The angle CAP, Plate 4, is the angle of greatest obliquity. The greater
this angle, the greater the tendency to thrust the gears apart ; the friction will be increased
and the component of force tending to produce rotation will be decreased.

25. Rack. If the diameter of the gear be indefinitely increased, the pitch circle will finally
become a right line, and the gear will then be known as a rack.

The rack shown in Plate 4 has teeth only on one side of the pitch line, like the pinion and
gear, and the conditions of action are similar. The tooth-curve will be a cycloid, and the
rolling circle, M N 0, must be the same as that used for the engaging pinion, in order to fulfil
the general law for maintaining a constant velocity ratio (Art. 14, page 10).

26. Spur Gears having action on both sides of the Pitch Point, Plate 5. If we assume the
diameters of pitch and rolling circles to be the same as before, and the arc of action, C A, un-
changed, tlie addendum of gear and dedendum of pinion will be the same as those of Plate 4.
This case, however, differs from the preceding in that the number of teeth is but half as great,
and therefore the pitch will be doubled. This will require the arc of action to be doul)led, in
order that it shall equal the pitch (Art. 22, page 13). Such increase in the arc of action may
be made by continuing the path of contact to the other side of the pitch point, following the
circumference of a rolling circle which may or may not be equal to the other rolling circle.
Having laid off the arc A H equal to one-half the circular pitch, describe the curves H K and
H L , with H as the generating point of the new rolling circle. The former of these curves will

(UllVE OF LEAST CLEAIIAXCE. 15

beL-uiiiL' the addiMuliini of the pinion, ami the Litter the dedencliini of the gear tooth. The en-
gaging gears ^vill then liave both faces and flanks, the action ^vill l)egin at C and end at H , the
path of contact will be C A H, the arc C A being the path of approach, and A H the path of
recess, their snm being equal to the circular pitch.

In a similar manner the dedendum of the rack tooth may be described to engage the adden-
dum of the pinion tooth, and the contact begun at N Avill end at 0, N M being the path of
approach, and M the path of recess. That portion of the dedendum of rack tootli which
engages the addendum of the pinion is indicated b}- sectioning, but it is necessary to continue
the dedendum to a depth sufficient to allow the addendum of the engaging tooth to enter.

27. Clearance. The space between the addendum circle of one gear and the dedendum
circle of an engaging gear is called clearance. Fig. 9, page 17.

28. Curve of Least Clearance. If the pitch circle of the gear ])e rolled on that of the
pinion, and the epitrochoid of the highest point, C, of the gear tooth be determined, it will be
the curve of least clearance.

The successive positions of the tootli, when so revolved, are sliown by the dotted line in
Plate 5, and the line connecting these points wotdd ])e the desired curve. This nui}^ be
obtained as follows : Assume any point, R on the pitch circle of pinion, and la^• off ai'c A R' on
the pitch circle of gear, equal to arc A R. From R, with radius R C, equal to R' C, describe an
arc. Similarly describe other arcs, and draw a curve touching these arcs on the inside. This
curve will be tlie curve of least clearance.*

* See also the method of Akt. 71, page til.

16 CONDITIONS (;()A^Ei:\lN(; TIIK PRACTICAL CASE.

29. Backlash. In order to allow for unavoidable inaccuracies of workmanship and operat-
ing, it is customary to make the sum of the thickness of two conjugate teeth something less
than the circular pitch. This insures contact between the engaging faces only.

30. Conditions governing the practical case. From a consideration of the foregoing limiting
cases, the following principles are deduced, to which are also added the limitations and modifi-
cations established by practice.

1. The curves of gear teeth, which act to produce a constant velocity ratio, must be
described by the same circle rolling in contact with their respective pitch circles. (Akt. 14,
page 10.) Practical considerations limit the diameter of the describing circle to a maximum
of about — , or equal to the radius of the pitch circle, and a minimum of about 1^ P', or 6.5 P.

See also Art. 32, page 19.

2. The arc of contact must equal the circular pitch, and in practice exceed it as much as
possible.

8. The addendum of a gear tooth engages the dedendum of the pinion, and the action
between them either begins or ceases at the pitch point.

Since the addendum and dedendum of any tooth are independent curves, they may be
described by rolling circles differing in diameter.

4. In the limiting cases considered, the height of the tooth is dependent on the arc of con-
tact, but in practice, the arc of contact is made dependent on the height of the tooth.

While it is an almost universal custom to make the addenda of engaging teeth equal, there
are special cases, in which very smooth-running gears are required, where it would be advan-
tageous to make the addenda of the driver less than those of the driven gear, thus increasing
the arc of recess, or decreasing the arc of approach.

PROPORTIONS OF STANDARD TOOTH.

The approacliing uetion Ijeing- more (letrinientiil.
by reason of the friction induced, it is common to de-
sign clock gears so as to eliminate this by providing
the driver with faces only, and the driven with flanks
only. Or, if the gears are made with both faces and
flanks, to so round the faces of the driven gear tluit no
action may take place.

31. Proportions of Standard Tooth. The propor-
tions most connnonly accepted for cut gears are those
illustrated in Fig. !'. The dimensions are made depen-
dent on the pitch, as follows : —

Addendum, (S) = - ■. —

diam. -pitch
Dedendum, (s + f) =

~r. : — : + clearance = - + f .

diam pitch r

Thickness, (t) = ^ circular pitch = y = 7^ •
Clearance, (f) = ^ addendum = |= ^ or, f = ^^ thick-

t P' TT

ness of tooth = —=—: = „„ „ •
iO 20 20 P

In assuming this value for the thickness of the
tooth the backlash is taken as zero, but of course the

Fig. 9.

18 INFLUENCE OF THE ROLLING CIRCLE.

tooth must be slightly smaller than the space to permit of freedom in action. If there he
any backlash the value of t will be e.rcuiar pitch -backlash ^ j,^ rough cast gears the backlash may
be as great as ^Vth the circular pitch, but this amount is very excessive. It is, however, in-
consistent to base the values for backlash, or clearance, on the pitch, since an increase in the
size of the tooth, or pitch, does not necessarily mean a proportional increase in tlie allowance
to be made for the inaccuracies of workmanship. Indeed, l)()th these clearances nuist be left to
the judgment of the designer.

Fillets. The circular arc tangent to the flank and dedendum circle is called the fillet. It
is designed to strengthen the tooth by avoiding the sharp corner at the root of the tooth. A
good rule is that of making the radius of fillet equal to one-sixth of the space between the
teeth, measured on the addendum circle, as in Fig. 6. Tlie limit of size may be determined by
obtaining the curve of last clearance. Art. 28, page 15.

32. Influence of the Diameter of the Rolling Circle on the Shape and Efficiency of Gear
Teeth. If the height of the teeth be previously determined, any increase in the diameter of the
describing circle will increase the path of contact and decrease the angle of pressure. Bat
since an increase in the diameter of the describing circle produces a weaker tooth, by reason of
the undercutting of the flank, as shown in Fig. 12, page 21, the maximum limit of the diameter
is connnonly made equal to the radius of the pitch circle within which it rolls. As was shown
iji Art. 9, page 5, this will generate a radial flank. In the case of gears designed to trans-
mit a uniform force, and not subjected to sudden shocks, it is desirable that the teeth have
radial flanks, and consequently the diameters of the rolling circles will be equal to the radii
of the pitch circles within which they roll. If the force to be transmitted be irregulai-, and the

INFLUENCE OF THE KULLING (IKCLF.

teeth required to sustain sudden strains, it is better that
the flank be made wider at the dedendum circle, and a
describing- circle chosen of a diameter sufficiently small
to produce the desired result.

In general, the diameters of the descril)ing circles

D' 5

will lie between the values of -^ and - . The second

i. r

value was used for the describing circle of the gears in
Plate 5, and would describe radial flanks for a gear
having ten teeth.

Fig. 10 illustrates the effect of a change in the di-
ameter of the rolling circle on the path of contact and
angle of pressure. Two gears of equal diameter are
supposed to engage, and the teeth are described by roll-
ing circles of equal diameter.

K P is the addendum, and P L the dedendum of the
tooth described b}- the rolling circles C P, and P D,
which are of the same diameter, and equal to one-
quarter of the pitch diameter. A C being the ad-
dendum line of the engaging gear, C may be considered
as the first, and D as the last, point of contact. The
arcs C P and P D constitute the i)a-th of contact, and the
angle C P H is the angle of pressure.

Next consider the describing circle as increased, its

Fig. 10.

INTERCHANGEABLE GEARS.

Fig. 11.

diuineter being equal to one-half of the diameter of the pitch
circle. The form of the tooth will now be E P F, and the path
of contact A P B. In the latter case the arc of contact will
be greater, the maximum angle of pressure less, and the
tooth weaker than in the former.

The relation between the two cases may be more exactly
stated as follows : —

Diameter of describing curve, — — .

Arc of contact, i.08 P' 1.35 P'.

Maximum angle of pressure, 32.3° 20 3\

Again, the weakness of the tooth in the second case may
be partially overcome by reducing the height of the tooth,
and in general this would be advantageous, the so-called
standard tooth being too high for the best results.

33. Interchangeable Gears. Since the same diameter of
rolling circle must be used for the addendum of pinion tooth,
and the dedendum of engaging gear tooth, it follows that for
any system of interchangeable gears, the addenda and de-
denda of all teeth nuist be described by the same descril)ing
cur\^e. It is also necessary that the pitch, and proportion
of the teeth, be constant.

PRACTICAL CASE.

In pi-aetice, it is common to regard g'eai*s of twelve or t»f
fifteen teeth as the hase of the system, and the diameter of
the rolling circle is made equal to the radius of the corre-
sponding pitch circle, thus describing teeth with radial flanks
for the smallest gear of the set. If twelve be adopted as the
smallest number of teeth in the system, the diameter of the
pitch circle will be D' = - = — , and the diameter of the de-

scribino- circle will be t-t; = -•

& 2 P P

Again, if a fifteen-toothed gear be used as the base of

7 5

the system, the diameter of the describing circle will be — .

Figs. 11 and 12 illustrate a fifteen and a nine-toothed
gear engaging a rack. The diameter of the rolling circle by
which the teeth were described is — , which will equal 3.75
inches for a 2 pitch gear.

The fifteen-toothed gear Mill have radial flanks, but the
nine-toothed gear will have the flanks much undercut by
reason of the diameter of the rolling circle exceeding the ra-
dius of the pitt-h circle.

34. Practical Case of Cycloidal Gearing. Plate 6. Let
F and G be the centers of pinion and gear having twelve and
eighteen teeth respectively, and a diameter pitch of 4. The

Fig. 12.

22 SPUR C4EARS.

pitch diameters will equal 5=7"='" iiif'lit's, and p ^t =-l.V inches (Art. 18, page 12). If the

tooth be of standard dimensions, the addendum and dedendum lines may be determined and
drawn by Ai;t. 31, page IT. The diameter of the rolling circle is assumed to be 1^ inches for
the addendum and dedendum of both gears. Since the teeth should usually l^e shown in con-
tact at the })itch point, suppose the generating point of the describing curve to be at this point,
and describe the curves by rolling the circles from this position, first on the inside of one pitcli
circle, and then on the outside of the other pitch circle, thus obtaining the flank of one tooth,
and the engaging face of a tooth of the other gear.

An enlarged representation of these curves is shown in Plate (J. They may be di-awn by
the methods of Arts. 6 and 7, l^age 5, or by Art. 9, page 5, but care should be used to draw
them in their proper relation to each other, as sliown in the figure, so that it may not be neces-
sary to reverse the curves in order to incor})orate them into tooth forms. The order for the
drawing of the curves ma}* be A B, A T, A D, A S.

Instead of reproducing the tooth curves by means of scrolls, it is sufficiently accurate, and
much more rapid, to approximate them by circular arcs. Plate 2, Fig. 3, illustrates a simple
method which closely approximates the curves of this system, and suffices for the ordinary
drawing of a geai', but in no case should be used for descril)ing the curves for a templet. This
method consists, Jirsf, in the construction of a normal for a point of the curve at a radial
distance from the pitch line equal to two-thirds of the addendum or dedendum of the tooth;
8eco)id, in the finding of a center on this normal, such that an arc may Ije described through
the pitch point, and the point of the tooth already determined. A P is the height of the adden-
dum, and B a point radially distant from the pitch lint', equal to - A P, through which the arc

PRACTICAL CASE. 23

B E is drawn. When the point E of the descril)ing cnrve shall have become a point of contact,
as at E', the arc E' P being equal to E P, the })oint P will have moved to T, the chord T E' being
equal to the chord E P. T will be a point in the addendum, and T E' the normal for this point.
From a point, M , on this normal, and found l)y trial, describe the arc P T, limited by the ad-
dendum line. Similarly the curve of the dedendum may be determined.

Having determined such centers as may be required for describing the tooth curves, draw
circles through these centers, as indicated in Plate 6, to facilitate the drawing of other teeth.
The radius for the dedendum is often inconveniently great, and in such cases it is desirable to
use scrolls, employing the method of Art. 1(3, page 11. Next divide the pitch circle into as
many equal parts as there are teeth, beginning at the pitch point. From each of these divisions
lay off the thickness of the teeth. If there be no backlash, this thickness will equal one-half
the circular pitch ; but if an amount be determined for backlash, the thickness will equal

P' — backlash
2

The circle of centers having been drawn, the tooth curves sliould be described. These will
be limited l)y the addendum and dedendum circles already drawn. Finally draw the fillets.

The maximum angle of pressure between the pinion and gear will be 24°, the arc of
approach .52, the arc of recess .48, and tlieir sum, which is the arc of contact, 1 inch, or 1.27
times the circular pitch.

The rack teeth would be similarly described. The pitch line being a right line, the circular
pitch may be laid off directly by scale, or spaced from the pinion. The approximate method
may be used for the tooth curves, and lines drawn parallel to the pitch line, for the centers of
the arcs which ajjproximate the addenda and dedenda of the teeth.

GEAR FACE.

Fig. 14.

Fig. 15.

35. Face of Gear. In the previous consideration of
gear teeth no attention has been paid to the width of the
gear, or, as it is commonly termed, the face of the gear.
Tliis dimension is one of the factors to be considered in
determining the strength of the tooth, wliich is a subject
apart from the kinematics of gearing. It should be ob-
served, however, that the tooth having appreciable width,
nuist be generated by an element of a rolling cylinder in
place of the point of a rolling circle.

36. Comparison of Gears, illustrated in Plates 4, 5,
and 6. In tlie three cases previously considered, the di-
ameter of the pitch circles are equal, and only one diam-
eter of rolling circle has been used.

In Plates 4 and 5 the arc of contact is equal to the
circular pitch ; 1)ut the pitch of the latter is twice as great
as tlie former, hence there are but half as many teeth. In
Plate 6 the arc of contact is made dependent on the
height of the tooth, which is a standard so chosen as to
permit of an arc of contact sufficiently long for a practical
case. But in Plates 4 and 5 the height of the tooth is
dependent on the arc of contact, which latter is made the
least possible.

COXYENTIOXAL KKPIIE^ENTATION OF SPUR (iEAPvS. ZO

Till' number of teeili in the pinions of Plates 4 and 6 is the same; hut in tlie former the
action is only on one side of the pitch point, there being no a(hlenda to the teeth, hence the
limited arc of contact.

In Plates 4 and 5 there is contact between only one pair of conjugate teeth, save at the
instant of beginning and ending contact; while in the case of Plate 6, two pairs of conjugate
teeth may be in contact during a part of the arc of contact.

37. Conventional Representation of Spur Gears. In making drawings of gears, it is usually
best to represent them in section, as in Fig. 14. This enables one to give complete informa-
tion concerning all details of the gear, save the character of the teeth. If the latter be special,
an accurate drawing of at least two teeth and a space will be required. Should it be neces-
sary to represent the geai-s on the plane of their pitch circles, as in Plate 6, they may be shown
as in Fig. 13, thus avoiding the representation of the teeth. Again, if it be necessary to show
a full face view of the gears, the method illustrated in Fig. 15 may be employed to advantage.
This is simply a system of shading ; and no attempt is made to represent the proper number of
teeth, or to obtain their projection from another view.

26 INVOLUTE 8Y8TEM.

CHAPTER IV.

INVOLUTE SYSTEM.

38. Theory of Involute Action. If the describing curves l)e other than circles we shall obtain
odontoids differing in character from those already studied ; but so long as both pinion and
gear are described by the same rolling curve, the velocity ratio will remain constant. The
class of odontoids illustrated by Plate 7, Fig. 1, is known as the involute, or single-curve
tooth. This curve cannot be described by rolling circles, but may be generated by a special
curve rolling in contact with both pitch surfaces.* But as the curve may be described by a
much more simple process, the above statement is of interest only as showing the conformity
of the curve to the general law. (Art. 14, page 10.)

F and G, Plate 7, Fig. 2, are the centers of two disks designed to revolve about their
respective axes with a constant velocity ratio, which is maintained in the following manner: —
Suppose tlie disks to be connected by a perfectly flexible and inextensible band, D C B A, which
being wound on the surface of one, will be unwound from the other, after the manner of a
belt, producing an equal circumferential velocity in the disks. Conceive a marking point as
fixed to the band at A, so that during the motion from A to D, curves may be described on the
extensions of disks 1 and 2, in a manner similar to that described for the generating of the
cycloidal curves. When the point A, on tiie band, shall have moved to B, the curve Xj B will

* For descTlption of this method, .see MaeCord"s Kinematics, page Ui5, Airr. 279.

TIIK IX VOLUTE crRVE. 27

have been described on the exteii.siuii (if disk 2. and B Aj , on that of disk 1. When tlie motion of
the marking point shall have continned to C, Xg Yj C will have l)een described on the extension
of disk 2, and Ag B^ C. on that of disk 1. Finally, when the marking point shall have reached
D, the curve Xg ¥3 Z^ D ^\ill have lieen described on tlie extension of disk 2. and Ag B^ Cj D on the
extension of d'sk 1.

If these curves be made the outlines of ffear teeth, and the former act aoainst the latter
so as to produce motion opposed to that indicated by the arrows, a uniform velocity ratio will
be maintained between the disks. On investigation, these curves will be found to be involutes,
Ag D being an involute of the periphery of disk 1, and Xg D, an involute of disk 2. The curves
may. thei'efore, be descrilied liy tlie method for drawing an involute (Art. 12, page 7), the
path of contact, A D, being spaced off on the base circle from A to Ag, and the involute drawn
from Ag; or the line A D may be conceived as wrapped about the base circle l)eginning the curve
at D.

39. Character of the Curve. Plate 7, Fig. 1, represents the involute curve of Fig. 2
incorporated into gear teeth. It becomes necessary to continue the line of the tooth within the
periphery of the disk, which will now l>e designated as the base circle, so as to admit the
addenda of engaging teeth. This portion of the tooth is made a radial line.

The pitch point being at B, (the intersection of the line of centers and the line of action),
the pitch circles will be drawn through this point.

The circles from wliich tlie involute curves are described, are called base

Base Circle Defined. . , , ■ ^■ i i • iji i^it

cirdi's. J heu' diameters liear the same ratio to each otiier as do the (ham-
eters of the pitch circles.

28 CHARACTER OF THE INVOLUTE.

The Path of Contact '^'^ ^^'^^ "^^ actioii, OF patli of coiitact, is ii right lliie tangent to the' Ija.se cir-

a Right Line. (j[gy_ jj^ jg ^]jg jjj^g followetl b}' the marking point of the modeh Plate 7, Fig. 2.

Since the path of contact is a right hne, and as tlie common normals at the point of con-

constant Angle of ^'^•^^' i^^st alwajs pass through the pitch point (Art. 14, page 10), it fol-

pressure. lows tluit the line of pressure, or angle of the normals, is constant.

The action between the teeth of the gears in Fig. 1, begins at A, and ends at D, taking
Limit of Action placc Only bctwccn the points of tangency of the line of action and l)ase
circle. No involute action can take place within the, base circles.

If the distance between the centers of the gear be increased or decreased, the angle of pres-
sure, and length of the path of contact will be increased or decreased, but the involute curve,
which is dependent on the diameter of the base circle only, will remain unchanged. Hence, any
An Increase in the cen- c'liaugc iu the distauce bctwccn tliB ccuters of two involute gears will not
'^Affec\'^^hrve°ioci"°'^ chaugc tlic vclocity ratio, provided the arc of action is (Mpial to the circular
Ratio. pitch. The case illustrated by Fig. 1 is a limiting one ; and therefore an in-

crease in the center distance would mean an increase in the height of the tooth, in order that the
arc of action shouhl e(|ual tlie increased [)itch, an increase in the center distance necessitating
an increase in tlie diameters of the pitch circles, and therefore in the circular pitch. But
wliile the action between the teeth continued, the velocity ratio would remain constant.
Since the angle of pressure is constant, and the paths of the elements of a rack tooth are right
The Involute Rack b'lcs, it folh»ws tluit tlic tootli outliue of au iuvolute rack nnist be a right
tooth, a Right Line. ]j,j^,^ perpcudicular to the angle of pressure. Plate 8 illustrates a rack for
an inxohite gear, having an angle of pressure of about 30". (The section lined portions are
not involute.)

INVOLUTE LIMITIXO CASE. 29

40. Involute Limiting Case. Pi.ati-: S. Let the diameters of the pitch cireles, the angle
of pressure, and the nnniljer of teeth, l)e given. Having drawn the })iteh cireles ahout their
respective centers, F and G, ohtain the hase circles as follows : —

Through the pitch point, B, draw A D. making an angle with the tangent at the pitch point
equal to the angle of pressure, 'i'his will l)e the line of action ; and perpendiculars, F A and
D G, drawn to it from centers F and G. will determine the radii of the base circles, and the
limit of the action, or path of contact, at A and D. This is a limiting case, in that the path of
contact is a maximum, and the arc of contact equal to the circular jjitch. Next determine the
point, C, ])}• spacing the arc, D K C. e([ual to DA: A and C will he two points in the involute
curve of the base circle, D K c, from which other points may be obtained. Similarly describe
D P, the involute of the other l)ase circle, just beginning contact at D. The height of the teeth
will be limited by the addendum circles drawn through D and A, from centers, F and G. The
dedendum circles are made to admit the teeth without clearance. The pinion teeth are pointed,
and the gear teeth till the space, having no backlash. The circular pitch may be found by divid-
ing tlie circumference of the pitch circle into as many parts as there are teeth, or the teeth ma}'
be spaced on the base circle.*

The rack is made to engage the pinion in the following manner : — ■

being the pitch point of rack and pinion, the right line, R, drawn througli this point,
and tangent to tlie base circle, will be the path of contact for motion in the direction indicated
by the arrow. The contact will begin at R and end at S, the latter i)oint being that of the
intersection of the [)ath of contact and addendum circle. The rack tooth will be perpendicular
to the line of action, R S: and the thickness of tooth will equal that of the gear tooth, there
* For further details coneeniiiig the construction of this pinion anJ gear, see Problem 4, Page 74.

30 EPICYCLOIDAL EXTENSION OF INVOLUTE TEETH.

being no backlash in either case. The addendum of the rack tooth will be limited by the
parallel to the pitch line drawn through the first point of contact, R ; and the dedendum made
sufficiently great to admit the pinion tooth without clearance.

41. Epicycloidal Extension of Involute Teeth. The extent of the involute action between
the gear and the pinion of Plate 8 is limited to the path D A ; for while an increase in the
height of the gear tooth is possible, the limit of the engaging involute tooth is at A, since no
part of an involute curve can lie within its own base circle. It is, however, entirely feasible
to continue the contact by a cycloidal action, in the following manner : —

The angle FAB being a right angle, the circle described on F B as a diameter must pass
through A. This point may therefore be considered as a point in an epicycloid, described by
the rolling circle FAB, and having A B for its normal, which is also the normal for the involute.
I)ut this diameter of rolling circle being one-half the pitch circle within which it rolls, the
hypocycloid will be a radial line, and the dedenda of the teeth wall be radial within the base
circle. By rolling the same circle on the outside of the gear pitch circle, the addenda of the
gear teeth may be extended, and the path of contact continued to N, which is a limit in this
case, by reason of the gear tooth having become pointed.

Similarly, the addendum of the rack tooth may be extended l)v the same describing circle.
In the figure it is made sufficiently long to just clear the dedendum circle required for the
pointed gear tooth. The action will now begin at Q, follow the I'oUing circle to R, and then,
becoming involute, continue to S.

42. Involute Practical Case, Plates 9 and 10. Having given the number of teeth of
engaging gears, and the diameters of their pitch circles, it is required to determine the curves
for the involute teeth of a pinion, gear, and rack.

INVOLl'TE PHACTICAL CASE. 31

The diameters of the (lescrihing circles would be of lii-st consideration in cycloidal gearing ;
while in the involute system, the angle of pressure or line of action nnist first be established;
and tangent to this the base circles may be drawn. By reference to Plate 7, Fig. 1, it will
be seen that with a constant center distance, a decrease in the angle of pressure will necessi-
tate an increase in the diameter of the base circles, and a corresponding decrease in the path of
contact. That is to say, an increase in the possible length of the path of contact means an
increase in the angle of pressure. In Plate 7, Fig. 1, this angle is too great for actual prac-
tice, being about 30°; yet it cannot be lessened in this case, as the number of teeth is limited.
Practice lias limited this angle to 14^° or 15°, which is, unfortunately, too small; but as one
of these angles is generally adopted in the manufacture of gears, the latter will be used in the
follo\\ing problem : —

A pinion of 12 teeth is required to engage a gear of 30 teeth, and a rack, the diameter
pitch being 1. The former is illustrated by Plate 9, and the latter by Plate 10.

Pinion diameter = ci' = -=-=l2 inches.

Gear diameter = D' = ^ = ?^ = 30 inches. (Airr. 18, page 11.)

Since the teeth are to be of standaid dimensions (Art. 31, page 17), the addenda will Ije
1 inch, the dedenda li inches; and there being no backlash, the thickness of the teeth will be

half the circular pitch, or ^. The circular pitch, P', =p = 3.1416. Draw tlie pitch circles.

The line of action will pass through the pitch point, making tiie recpiired angle with the
common tangent at this point. Next draw the base circles tangent to tliis line, and determine
the points of tangeucy, D and A. Construct the involutes of these base circles in the manner

INTERFEREXCE.

Fig. 16,

indicated by Fig. l(i, and according to the nietliod for descril)-
ing an involnte, Ai;t. 12, page 7. It will now l)e seen that
the gear tooth will be limited by the arc drawn tlirougli D,
the point of tangency of liase circle and line of action. If,
however, the involnte cnrve be continued to the addendum
circle, as shown by the dotted line, C E, it will interfere with
the radial portion of the pinion flank, which lies within the
base circle. The pinion tooth will have no such limitation,
since the addendum circle intersects the line of action, D A,
at L, a considerable distance from the limit of involute action,
at the point A.

Similarly, the rack tooth will be found to interfere with
the pinion flank, if extended beyond the point C, Mhich
comes into contact at the point D, the limit of involute ac-
tion. But the pinion face may l)e extended indefinitely, so
far as involute action is concerned. The remedy for this
interference is treated of in the following article.

43. Interference. Since practical considerations demand
the maintenance of a standard proportion of tooth, two
schemes are adopted for avoiding or correcting this inter-
ference, observed in Plates 9 and 10.

The first is to hollow that j)art of the pinion flank lying

INFLUENCE OF THE ANGLE OF PRESSTRE. 33

within the base circle so as to clear the interfering part of the gear, or rack tooth. In this
case there will be no action beyond the point of tangency D. The second method consists in
making the interfering portion of the addendum an epicycloid descril)ed by a circle of a diam-
eter equal to the radius of the pinion pitch circle. Such a describing circle would generate a
radial flank for that part of the curve lying within the base circle. By this means, the action
will be continued and the velocity ratio maintained, although the action will cease to be
involute. AuT. 41, i)age 30.

44. Influence of the Angle of Pressure. The interference may be entirely obviated by
sufificiently increasing the angle of pressure ; but in the case cited (Plates 9 and 10) it would
necessitate an angle of 24.1°, which is too great for general use. Had the number of teeth in
the pinion been greater, the interference would have been less, and with 30 teeth in the pinion,
there would have l)een no interference. See Art. 45.

The angles of 14 i° and 15°, commonly adopted, are unfortunately small. There is, how-
ever, a tendency to increase this angle, and gears for special machines have been made with a
20° angle of pressure. This latter angle will permit gears having 18 teeth to engage without
interference, and the thrust due to this increase in the angle of pressure is an insignificant
amount. A system based on this angle of pressure would unquestionably be an improvement
over the present one.

45. Method for determining the Least Angle of Pressure for a Given Number of Teeth having
no Interference. Fig. 17.

Let A be the center of a gear having A B =^ R foi' the radius of pitch circle, and D B T the

LEAST ANGLE OF PRKSSURE WITHOUT INTERFERENCE.

Fig. 17.

angle of pressure to l)e (leterinined, the least
number of teeth l)eing N. Suppose the gear to
engage a rack having standard teeth, then will

C =

I D 2R

- = — - . D will be the last point of con-

P N N ^

tact, and A D = r, the radius of the base circle.

A C = A

C-R p-R N ~ N

A C:A D::A D:A B, hence, A D2 = r2 = A BxAC =
R'-2(N — 2) 1 ^ /N-2

The angle of pressure, D B T, is equal to an-
gle D A B = p , and

the cos. p = 1 =
R

V N

= y/NZ2

V N

Hence the

COS. of the angle of pressure

\/¥<''>-

By substituting in the above formula, it will
be seen that for a 12-toothed gear to engage a
rack without interference, the angle of action
must be 24.1°, and for 15 teeth the angle would
be 21.4°. Again, if the angle be 15°, the least

DEFE( T>s OF THE INVoLI'TE SY.STEM. OO

iiuinbei- of teeth that will engage without interference will be 80, while with a 20" aiigie of
pressure the least number would be 18.

46. Defects of a System of Involute Gearing. As in the case of the cycloidal system, it
is desirable to make all involute gears having the same pitch to engage correctly. In cycloidal
ge'dYs this was attained by the use of one diameter of rolling circle for all gears of the same
pitch (AiiT. 33, page 20). In the involute system we assume an angle of obliquity, or pressure,
which is constant for all geai-s ; but unless this angle be great, gears having so few as 12
teeth cannot be run together without interference. To obviate this difticulty we must adopt
one of the two methods already described (Art. 43, page 32) ; namely, the undercutting of the
interfering flanks, or the rounding of the interfering addenda. P^irst consider the hitter, which
is illustrated l)y Plates 9 and 10. We have seen how that portion of the gear tooth adden-
dum lying beyond the point C must be made epicycloidal in order to engage the radial part
of the pinion flank which lies within the base circle ; also that the pinion addenda might
be wholly involute since there would be nt) interference with the gear tooth flank, the action
between the latter taking place without the base circle. But if a 12-toothed gear be taken as
the base of the system, it will be necessary to round, or epicycloidally extend that portion of
the pinion addendum lying beyond the point K, since this would be the last point of involute
action between two 12-toothed geai-s. Tlierefore when the 12-toothed gear engages one having
a greater number of teeth, that part of the addendum lying beyond this point will no longer
engage the second gear, and the arc of contact will be greatly reduced. Again, suppose a pair
of 30-toothed gears to engage (each being designed to engage a 12-toothed pinion), the only
part of the tooth suitable for transmitting a uniform motion is that lying between the base

UNSYMMETRICAL TEETH.

Pig. 18.

circle and point C, Plate D, and the arc of contact would
be but 1.05 of the circular pitch. Now, one of the claims
made for the involute tooth is that the distance between
the centers of the gears may be changed without changing
the velocity ratio ; but in this latter case it cannot be done
without making the arc of contact less than the circular
pitch.

If the system of midercutting the flanks be adopted,
the addendum will be wholly involute; and in the case of
Plate 9 all of the })inion addendum would have been
available for action, Init the pinion flaidv, within the base
circle would have been cut away so that there would have
been no action of the gear addendum between C and E.
If, however, the engagement had been between two 30-
toothed gears, all of the tooth would have been available
for action, and the arc of contact would have been equal
to 1.91 of the circular pitch.

Thus it will be seen that involute gears should be de-
signed to engage the gears with which they are intended
to run, if the best results would be attained. This would,
of course, prevent the use of the ready-made gear or cut-
ter, but would insure a longer arc of action between con-
jugate teeth.

UNSYMMETinCAL TKETH. 37

47. Unsymmetrical Teeth. Fiy-. IS. A very dcsiialiK', altliou^li little used, form of tooth
is that known as the inisynimetrieal tooth, which usually combines the cycloidal and involute
systems. Fig. 18 illustrates a pinion and gear haying the same number of teeth as those illus-
trated l)y Plate 4, and the arc of contact is unchanged ; but the angle of pressure is much
reduced, and the strength of the tooth increased. As the involute face of the tooth is designed
to act only when it may be necessary to reverse the gears, and when less force would usually be
transmitted, the angle of pressure may be made greater than ordinary. In this case the angle
is 24.1°, which is sufficient to avoid interference in a standard 12-toothed gear (Art. 45,
page 33). But this angle is no greater than the maximum angle of pressure in Plate 4.
This reinforcement of the \ydck of the tooth makes it possible to use a much greater diameter
of rolling circle ; and in the case illustrated, the diameter is one-third greater than the radius
of the pitch circle. Tliis increase in the diameter of tlie rolling circle would have lengthened
the arc of contact, had not the height of the tooth been reduced to maintain the same ai'c as
that of Plate 4.

The cycloidal action begins at C and cuds at H, making a maxinnuu angle of pressure of
17". The same i-oUing circle has been ust'd for the face and flank of each gear; Imt the one
rolling within the pitch circle of the gear might have been nmch increased without materially
weakening the gear tooth.

The involute action would ))egin at D and end at B, making an arc of contact a little greater
than the pitch.

OO ANNULAR GP:AR1NG.

CHAPTER V.

ANNULAR GEARING.

48. Cycloidal System of Annular Gearing. If the center of the pinion lies within the pitch
circle of the gear, the latter is called an internal, or annular gear. The solution of problems
relating to this form of gearing differs in no wise from that of the ordinary external spur gear,
save in the consideration of certain limitations which will be treated of.

49. Limiting Case. Plate 11 illustrates a pinion engaging an internal and an external
spur geai'. The pinion has 6 teeth, and the gears have 13 teeth. The arc of contact is made
equal to the circular pitch, and equally divided between recess and approach. The pinion has
radial flanks, which therefore determines the diameter of the describing circle for the addenda
of the gears. The second describing circle, 2, is governed by conditions which will appear later.
It will be observed that the addenda of the annular gear teeth lie within, and the dedenda
without, the pitch circle. The height of the teeth is governed by the arcs of approach and
recess ; and the construction of the teeth does not differ from the limiting case considered in
Art. 26, page 14, and Plate 5. The action between the pinion and annular gear begins at
B, and ends at C, the pinion driving.

50. Secondary Action in Annular Gearing. We have already seen. Art. 10, page (>, that
every epicycloid may be generated by either of two rolling circles, which differ in diameter by

SECONDARY ACTION IN ANNULAR GEARING. 39

an amount equal to the diameter of the pitch circle. Also, that every hypocycloid may he
generated by either of two rolling circles, the sum of the diameters of which shall equal that
of the pitch circle within which they roll. Thus the addendum, C E, of the pinion, Plate 11,
may he described by the circle 2, or the intermediate circle 3. But in this case the circles 1
and 2 are so chosen that the intermediate circle 3 is the second describing circle for the hypo-
cycloid F G , as well as for the epicycloid C E ; consequently C E and F G will produce a uniform
velocity ratio, the contact taking place from A to D. The addendum C E has contact also with
the dedendum C F along the path A C ; hence, during a part of the arc of recess there must be
two points of each tooth in contact at the same time.

The plate illustrates the contact along the path A C as just completed ; but a second point
of contact will be seen on circle 3, between F and E, and action along this path will be con-
tinued to D. The case is therefore no longer a limiting one, inasmuch as the arc of contact is
greater than the circular pitch. 'J'he additional contact takes place during the arc of recess,
which is also advantageous.

In order to ol)tain this secondary action, the sum of the radii of the inner and outer rolling
circles must equal the distance hetween the centers of pinion and (/ear.*

For, letting r,, r.^, and r:; be the radii of the inner, outer, and intermediate rolling circles,
and Rp, Rg, the radii of pinion and gear, rg + r^ = Rg, (^6), and i-g — r.^ = Rp, (7), Art. 10, page (3.
Subtracting the second equation from the fii'st, rj + r., = Rg — Rp ='C = center distance (8).

Plate 12, Fig. 1, illustrates the same pinion and gear, the teeth having been described by
the intermediate circle only. In this case the action takes place wholly during recess, the arc

* The student is referred to Prof. MacCord's "Kinematics," images 104 to 10!) inclusive, for a very complete
demonstration of this law, together with other limitations of aniudar gears.

40 LIMITATION OF INTERMEDIATE DESCRIBING CURVE.

of recess being the same as before, about 1 j times the cireuhir pitch. Had the outer describing
circle been used to describe the dedenda of the gear teeth, as in the })reeeding cases, a secondary
action wouhl have taken place during the recess.

Special notice should be taken of the reduced angle of pressure in the secondarv action of
annular gearing, and of the possibility of obtaining a great arc of recess \vith little or no
approacliing action. These advantages are very apparent in Plate 11, in which the pinion
engages an external and an internal gear having an equal numl)er of teeth.

51. Limitations of the Intermediate Describing Circle. Plate 12, P'ig. 2, Suppose the
inner describing circle, 1, Plate 11, to l)e increased luitil it equals the diameter of the pinion
pitch circle, Of, the radius of the intermediate describing circle will then equal the center dis-
tance, 5}^, and the outer describing circle, 2, would be but \'^ radius. For by sul)stituting
Rp for rj in equations 6 and 8, Ai;t. 50, we shall obtain r, = Rg — Rp = c, and r, = C — Rp. Plate
12, Fig. 2, illustrates this case, the outer describing circle not being employed.

Since the pinion pitch circle has now become a describing curve, there will l)e an appi'oach-
ing action ; Ijut only one point of the pinion tooth will act, as the diameter of the describing
circle and pitch circle being equal reduces the pinion flank to a point. But if any further
increase be made in the diameter of the inner circle, whicli is equivalent to a decrease in the
intermediate describing curve, an interference will take place during approaching action ; since
the curves of gear and pinion teeth, generated by a circle greater than the pinion diameter, will
cross one another, which would make action impossible. Hence, the radius of the intermediate
describing circle ca)uiot be less than the line of centers.

52. Limitations of Exterior and Interior Describing Circles. Plate 12, Fig. 8. From

LIMITATION OF EXTEIJIOR AND INTERIOR DESCRIBING CURVES. 41

Art. 50, page 39, it was seen that the sum of the radii of tlie exterior aiul interior descril)ing
circles must equal the center distance if a secondary action be obtained. If either circle be
decreased without decreasing the other, the secondary action ceases ; but if either circle be
increased wdthout an equal decrease in the other, thus making the sum of their radii greater
than the center distance, the addenda will interfere. Thus, iii Plate 11, a decrease in describ-
ing circle 2 would produt-e a more rounding face, and C E would fail to engage F G; but had
this describing circle been increased in diameter without a corresponding decrease in 1, C E
would have interfered with F G . Hence, tlie limit of the sum of the radii of the exterior and
interior describing circles is the center distance,

Plate 12, Fig. •], illustrates a special case of the above condition, the interior describing
circle being reduced to zero, and the radius of tlie exterior circle made equal to the center dis-
tance, thus making the intermediate describing circle equal to the [)itch circle of the gear.
There will be double contact during a portion of the arc of recess, the contact l^eginning at A,
and following the outer describing circle to C, and the intermediate (or in this case the pitch
circle of the gear) to D. Tliis design is ol)jectionable in that the secondary action takes place
with only one point of the gear tooth.

53. The Limiting Values of the Exterior, Interior, and Intermediate Describing Circles for
Secondary Action. !Since r.^H- ri = C, either radius will equal C, when the other becomes zero;
l)ut if there l)e a secondary action, the minimum value of r., may not be zero, for r^ will be a
maximum when r., is a mininnim. as rj + rs = Rg, r- is a minimum when equal to C (Art. 52),
and substituting this value in the last equation, rj = Rg — C . Again suhstituting this value in
the equation, r._, + rj = C , r. = C — (Rg — C) = 2 C — Rg.

42 LIMITING VALUES OF DESCIUBING CIRCLES FOR SECONDARY ACTION.

Summary of the above limiting values and conditions governing secondary action: —

rj maximum = Rg — C ; r^ minimum = ; rs + rj = Rg . (G)

r.^ maximum = C ; r^ minimum = 2 C - Rg ; rs — r^ = Rp . (7)

rg maxinmm = Rg ; h mininuun = C ; Rg - Rp = C . (8)

54. Practical Case. If annular gears be made interchangeable with spur gears, it will be
necessary to have the number of teeth in the engaging gears differ b}- a certain number which
will depend on the base of the system. This is due to the limitation in the sum of the radii of
the describing circles, Art. 52, page 40. Thus, let 12 be the base of the system, and it is
required to find the least number of teeth in the annular gear that will engage the pinion. If
the pitch be 2, the diameter of the pinion will be 6, and that of the describing circles 8. But
since the center distance cannot be greater than the sum of the radii of the describing circles
(in this case 3), the diameter of the annular gear must be 12, and the least number of teeth in
the annular gear will be 24.

Using the notation of Plate 11, and Art. 17, page 11, n being the least number of teeth
in the gear, and n the least number in the' pinion, or the base of the system: —

n 1 „ „ r^ N n 1 n N n ^ .,

C = 2 r^ = — , also C=Rg-Rp= — - — , hence ^ = ^ - ^^ or 2 n = N .
The least number of teeth in the annular gear will be twice that of the base of the system.

55. Summary of Limitations and Practical Considerations, (a) The diameter of the inter-
mediate describing circle is equal to the diameter of the pinion, plus the diameter of exterior
describing circle, or diameter of gear minus interior describing circle. (Art. 10, page 6.)

SUMMARY OF LIMITATIONS AND PRACTICAL CONSIDERATIONS. 43

(^) There will be .secondaiy action only when the sum of the radii of the exterior and
interior deseril)ing cireles is e(|ual to tlie line of centers. (Art. 50, page 38.)

((?) The radius of the intermediate descril)ing circle cannot be less than the center distance.
(Art. 51, page 40.)

((?) The sum of the radii of exterior an.d interior describing circles cannot be greater than
the center distance. (Art. 52, page 40.)

(^) The nundjer of teeth in any pair of gears of an interchangeable system must differ by
an amount equal to the ])ase of the system. (Art. 54, page 42.)

(/) If the pinion drives, the exterior describing circle should be the greater in order tliat
the arc of contact may be chiefly one of recess.

(</) If the gear drives, the interior describing circle should be the greater, and the pinion
teeth may have flanks only, but in this case the teeth should be extended slightly beyond the
pitch circle in order to protect the last point of contact, which will be on the pitch circle.

56. Involute System of Annular Gearing. Fig. 11). The method of drawing the tooth
outlines for the involute annular gear does not differ from that of the spur gear. Pitch lines
having been determined, the base circles are drawn tangent to the line of action, and the invo-
lutes of those base circles will be the required curves. Care must be used in obtaining the
length of the teeth, in order to avoid a second engagement after the full action shall have
taken place. To determine if this interference takes place, it is necessary to construct the
epitrochoid of the point of the pinion tooth, or determine the path of least clearance, as in
Art. 28, page 15.

INVOLUTK 8Y8TKM OF ANNULAR (iEATvING.

A'V^

Fiy. 1*1 illustrates an aiiiinlar y;vin- of 20 teeth en-
gaging a })inion of 10 teeth, the angle of pressure
being 20°. The pinion driving in the direction indi-
cated will establish the first point of contact at A, and
the last point, B, will lie limited by the height of the

tooth, in this case

The limit of the Q-ear tooth will

be determined by the arc drawn from the center of
gear througli. the point A, Any extension of the in-
volute beyond this point will interfere with the pinion
flank. The stronger form of the annular gear tooth
permits of a greater clearance, whieh it is advantageous
to a(h)pt.

If the pillion and gear differ but little in diameter,
it is desirable to use the cycloidal system, in ^\ Inch case
the interference may be more easily avoided. It should
also be noted that the advantages to be derived from
an increase in the arc of contact and a decrease in tlie
angle of pressure are only to be obtained by the use of
the latter svstem.

Pig. 19.

TllKORV OF BEVEL (LEAKING. 45

CHAPTER VT.

BEVEL GEARING.

57. Theory of Bevel Gearing. In all cases previously considered, the elements of the teeth
were parallel, the surfaces having heen generated by a right line which was either an element
of a rolling cylinder, as in the cycloidal system, or hy an element of a flexible band parallel to
the axis of a cylinder from which it was unwrapped, as in the involute system. All sections
of the teeth made by planes perpendicular to the axis were alike, and therefore it was only
necessary to consider one. Under these conditions the pitch cyclinder became a pitch circle,
and the describing cylinder a describing circle. If we now consider the axes of the gears as
ijitersecting, the friction cylinders will become friction cones, the describing cylinder will be a
describing cone, and the elements of the teeth will converge to the point of intersection of the
axes, making all sections of the teeth to differ from one another.

Fig. 20, page 46, illustrates this case. A C B and BCD are two friction cones, or pitch cones,
having axes G C and H C. The outlines of the teeth are drawn on the s})herical l)ase of the
cone, that portion of the curve lying outside tiie pitch cone being a spherical epicycloid, and
that within, a s})herical hypocycloid. The dedendum, or surface of the tooth lying within the
])itcli cone A C B, was described by the element E F C of the describing cone, which is shown as
generating the acUlendum of the pinion tooth. Only that portion of the surface described by E F
would be used for the pinion tooth, the length of the gear tooth having been limited as shown.
The describing cone employed for generating the addendum of gear, and dedendum of pinion,

CHARACTER OF CURVES IN BEVEL GEARING.

is not shown ; Init the dia.meter of
its base would be governed by laws
similar to those already considered
for limiting the diameters of rolling
circles, Art. 32, page 18.

58. Character of Curves employed
in Bevel Gearing. The cycloidal
BEVEL TOOTH has already been con-
sidered in the previous article, and
the curve does not differ from that
employed in spur gearing, save that
it is described on the surface of a
sphere.

It is important to note that no
tooth can be made with a radial flank,
since no circular cone can be made
to generate a plane surface by roll-
ing within another cone, but the
flank may approximate closely to
such plane.

The INVOLUTE BEVEL TOOTH is

one havino- a srreat circle for its line

Pig. 20

TKP:I)G<)L1) AJ'lMtoXlMATlON.

of action. Fig. :21 illustrates a crowu gear of this type. A C
is a great circle of the sphere A D C E , and is tangent to the
circles A E and DC. If the circle A C he rolled on D C, so as
to continue tangent to D C and A E, the point B will descrihe
the spherical involute G B F. Conjugate teeth described by
this process maintain their velocity ratio constant, even while
undergoing a slight change in their shaft angles, thus conform-
ing to the general character of involute curves.

The OCTOID BEVEL TOOTH is one having a plane surface
for the addendum and dedendum, the plane being such as
Avould cut a great circle from the surface of the sphere. In
Fig. 22, G F is the plane which cuts the surface of the tooth
shown at B . The line of action, from which the tooth takes
its name, is indicated l)y the curve B C E B H K . This tooth
was the invention of Hugo lidgram, and is of interest in being
the only bevel tooth that can be formed in a practical manner
by the molding-planing process. The lUlgram machine, de-
signed to plane this tooth, is descril)ed in the Journal of the
Franklin Institute for August, 188G, and in the American Ma-
chinist for Mny 0, 1885.

59. Tredgold Approximation. I because of the diiticulty in-
volved in descril)ing the tooth form on the surface of a sphere.

Fig: 21.

Fig; 22.

DKAFTINC THE BEVEL GEAll.

it is custoniiiiy to draw' the outline on the developed surface of
a cone which is tangent to the sphere at the pitch circle.
Tliis cone is called the normal, or back cone. Plate 13 il-
lustrates a sphere A B D, from which the pitch cones A C B
and BCD have been cut. 'J'angent to the sphere at the
pitch circles, A B and B D, are the normal cones A G B
and B H D, the elements of which are perpendicular to
the intersecting elements ' of the pitch cones. The
error in the tooth curve due to this apjjroximation
is so small as to be inappreciable, save in exag-
gerated cases ; and the method is always em-
ployed for the drafting of bevel gears.

Fig. 23.

60. Drafting the Bevel Gear. Plate
13, and Fig. 23. The drawing usually
required is that illustrated by Fig. 23,
which is a section of a gear and pin-
ion, together with the development of
a portion of the outer and inner nor-
mal cones, only the tooth curves bt-ing
^o^ omitted.

The names of the parts of a bevel
gear are also given, and the lettering

DRA FATING THE BEVEL GEAR. 40

corresponds to that of Plate 1-5, Avliicli latter will be used to illustrate the method of
draAvi ng.

A B and B D, Plate 13, are the pitch diameters of a gear and pinion with axes at 90"^, and
liaving 15 and 12 teeth resi)ectively, the pitch being 3, when drawn to the scale indicated.
The pitch diameters being 5" and 4", lay off C K on the center line of gear, equal to one-half
the pitch diameter of pinion, and C L on the center line of pinion, equal to one-half the pitch
diameter of the gear. Through these points draw the pitch lines perpendicular to the axes of
the g-ears, and in this case perpendicular to each other. Draw the pitch cones A C B and BCD,
and perpendiculai' to these elements draw G A, G B H, and H D, elements of the normal cones.
Having figured the addendum and dedendum of the teeth, la}' off on the normal cone of pinion
B M and B N , D and D Q , and from these points draw lines converging to the apex of the pitch
cones. Similarly lay off addenda and dedenda of gear, limiting the length of the face at R by
drawing the elements of the inner normal cones at R S and R T. The face B R should not be
gi'eater than one-third B C, by reason of the objectionable reduction in small end of teeth.
Complete the gear blank, or outline, by drawing the lines limiting the thickness of the gear,
diameter and length of hub, diameter of shaft, etc., details which are matters of design.

The development of the normal cone of the gear, B G A, will be a circular segment described
with radius G B, and equal in length to the circumference of the pitch circle of the gear.
Since there are 15 teeth in the gear, the developed pitch circle will be divided into 15 parts,
as shown, and the circular 'pitch be thus determined. But it is unnecessary to obtain the
complete development as shown in the plate, since the shape of one tooth and space is alone
required. Therefore, space oft' on a portion of the arc of the developed 2')itch circle, the cir-
cular pitch, B V, A\hich is equal to p. Draw the addendum and dedendum circles with radii

50 DRAFTING THE BEVEL GEAR.

equal to distance of these circles from the a])ex of tlie normal cone, wliicli in the case of the
gear will be G E and G F.

Next determine the tooth cnrve as for spur gears, nsiiig tlie developed pitch circle instead
of the real pitch circle. In tlie case illnstrated, the cnrve is involute. B w is a part of the
line of action, making an angle of 75'^ with G hi, the line of centers. The hase circles drawn
tangent to this line will be the circles from which the involutes are described. Had the cycloidal
system been employed, the diameter of the rolling circle would have been made dependent on
the diameter of the developed pitcdi circle, instead of the pitch diameter A D.

In like manner obtain the development of the inner normal cones, having S R and T R for
elements, and describe the true curves of the small end of teeth. These pitch circles may be
drawn concentric with the developed pitch circles of the outer cones, or with S and T as centers,
the latter l)eing the method commonly adopted. I)otli methods have been employed in the
plate. If the development of the inner pitch cone of gear be di-awn fi'om the center G, the
reduced pitch, and thickness of tooth, may be obtained by drawing the radial lines from the
development of the outer cone as shown l)y the fine dotted lines. The addendum and deden-
dum circles will lie described with radii S Z and S Y, and the tooth curves may be drawn by
determining the leduced rolling circh% if the gear be cycloidal, or the reduced base circle if the
involute system be em])loyed.

A second method foi' describing the teeth on the inner normal cone would 1)0 to l)asc it directly
on the reduced ])itch, which may be determined by dividing the nundier of teeth by the diameter
of tlie base of the pitch cone at this point. In the plate, the value of P for small end of teeth

l5 12 . . .

is _— for gear, or r-~ for pinion -= 4 6 == P. TIk; addendum, dedendum, circular pitch, etc., may
now be obtained from this value of P, as was done in the case of the outer pitch cone. In like

FIGURING BEVEL GEARS.

manner we may obtain any otlier section
of the tooth, although a third section is
seldom required.

6i. Figuring the Bevel Gear with Axes
at 90°. Figs. 24 and 25. The dimensions
required for the figuring of a pair of bevel
gears will be : —

First : Those required for general refer-
ence, and consisting of pitch diameters,
number of teeth (or pitch), face (K), thick-
ness of geare (L and M) (U and V), diameter
and length of hubs.

Second : In addition to the above, the
pattern maker and machinist will require,
for the turning of the l)l;ink. the outside
diameter, backing, angle of edge, angle of
face.

Third : The cutting angle will be re-
quired for cutting the teeth.

The figures required for the fii*st set of
dimensions are all matters of design, but
the second and third dimensions must be

I BACKING

BEVEL GEARIN(;.

deteniiiiied from the data given in the first. To ol)tain these it is neeessar}- to figure the five
dimensions indicated in Fig. 25, three of which. A, B, and C, are angles, and two, E and F, are
necessary to determine the outside diameter and backing. Only one of tliese, A, is used
directly. B is called the angle increment, C the angle decrement, E is one-half the diameter
increment of the pinion, and F is equal to one-half the diameter increment of the gea,r.
In the similar right triangles a b t and t r m. Fig 25,

ta b=m t r = A.

ab = ^.

b t =

t m --

, d' n

tan B ^

2 sin A _ 2 sin A
P d' "^ n

E = - cos A

2 sm A

The angle decrement, C, is sometimes made equal to B, in which case the
dedendum of the tooth at the small end will he greater, as shown by the line
h u ; but if the bottom line of the tooth be made to converge to the apex of
the pitch cones, the angle t a h, or C, will be determined as follows:

F f P

tan L = —

sin

2.25

sin

Fig. 25.

Having determined tliese values, it is only necessary to combine them with those fixed by
the design to complete the figuring of the gear as shown in Fig. 24.

Tiie angles should be expressed in degrees and tenths, rather than in degrees and minutes.
It is also of importance that the outside diameter and backing be figured in decimals, to thou-
sandths, I'ather than in fractional e(|uivalents.

I5i:VEL GEAR TABLE. 53

62. Bevel Gear Table for Shafts at 9o\ In order to facilitate the figuring of bevel goal's,
tables or charts of the principal values are commonly employed. Such cliarty also make the
figuring possible to those unfamiliar with the solution of a right triangle. Some are designed
to solve the problems graphically, while others, like the following, pages 5-4 and 55, consist of
the trigonometrical functions for gears of the proportions commonly employed.

Desckiptiox of Table.*

CoLUMX 1. Ratio of Pinion to Gear. ,, ^

Column 2. Katio of Pinion to Gear expressed in decimals, or tang of center angle, tan ^ "" r, — 7,'

Column 3. Center angle of Pinion corresponding to tangent in column 2.

Column 4. Ten times the angle increment for a Pinion of 10 teetli. This increased value is employed to
simplify the figuring of gears having other than 10 teeth. Thus, the angle increment for miter gears
(1 to 1) having 10 teeth would be 8.2°, and for H teeth, {f of this value or J 7. There is, of course, a
slight error in deriving the angle increment for any number of teeth from these values, in that the
tangent and arc do not vary alike, but the error is inappreciable for small arcs.

Column 5. The diameter increment for a Pinion of one pitch, hence equal to '2 cos A.

Con'.MN 6. Center angle for Gear, or 90° — A.

Column 7. Ten times the angle increment for Gear of 10 teeth, whiclr of course equab that of the
engaging Pinion.

Column 8. Diameter increment for a Gear of one pitch, hence equal to 2 sin A.

Use of Table. — In columns 1 or 2 find the value corresponding to the ratio of given
gears. Against this value, in 3 and 6, the center angles for pinion and gear are given.

The angle increment may be found by dividing the value in 4 by the number of teeth in
tiie })inion, or l)y dividing the value in 7 by the number of teeth in the gear.

The diameter increment for the pinion is obtained by dividing the value in .) by P, and that
for the gear by dividing the value in 8 by P.

The value of the angle B may be determined with sufficient accuracy by making it | of B.

* The plan of this table is tliat adopted by Mr. George B. Grant. See "American Machinist/' Oct. 31, 1885, and
'•Odoiitics," page itO.

BEVEL GEAR TABLE.

BEVEL GEAR TABLE FOR SHAFTS AT 90°.

PROPORTION
OF

PINION.

QEAS.

2 E.

900 — A.

2 F.

Angle

Diameter

Angle

Diameter

PINION
TO GEAR.

(Center
Angle.

Increment.
Divide

Center
Angle.

Increment.
Divide

by n.

by P.

by N. ^

by P.

1.000

45.

80.5

1.414

45.

80.5

1.414

.900

41.98

76.0

1.486

48.02

84.5

1.337

.888

41.63

75.6

1.495

48.37

85.0

1.329

.875

41.18

75.0

1.504

48.82

85.5

1.317

.857

40.60

74.1

1.518

49.40

86.3

1.302

.833

39.80

73.0

1.536

50.20

87.4

1.280

.800

38.66

71.1

1.562

51.34

88.8

1.249

.777

37.85

70.0

1.579

52.15

89.6

1.228

.750

36.83

68.5

1.600

53.17

90.8

1.200

.714

35.53

66.2

1.628

54.47

92.5

1.162

.700

34.99

65.1

1.638

55.01

93.0

1.147

.666

33.68

63.2

1.664

56.32

94.5

1.109

.625

32.00

60.4

1.696

58.00

96.3

1 .060

\ 3

.600

30.96

58.7

1.715

59.04

97.3

1.029

^ 1

1 \'n

.571

29.75

o(}Aj

1.736

60.25

98.5

.992

\U^''>

.555

29.05

55.4

1.748

60.95

99.1

.971

— [il

.500

26.56

51.0

1.789

63.44

101.4

.894

--^ 4

.444

23.94

46.3

1.827

66.06

103.6

.812

BEVEL GEAR TABLE.

BEVEL GEAR TABLE FOR SHAFTS AT 90°.

PROPORTION

PINION

TO GEAR.

3: 7

.428

2: 5

.400

3: 8

.375

1: 3

.333

3:10

.300

2: 7

.285

1: 4

.250

2: 9

.222

5
11

G
13

; 7
;15

.200
.181
.166
.153
.143
.133

.125

.117

.111

.100

82.88 ! 112.2

83.30 1 112.3

83.67 112.4

84.30 1 112.6

.248
.233
.221
.200

n
^ d'

p-^.

d'
tan A = ^,

tan B

2 sin A

2.25 sin A

tan C =

2 E = p cos A ,

2 F = p sin A.

BEVEL GEARS WITH AXES AT ANY ANGLE.

63. Bevel Gears with Axes at any Angle.
If the axes of the gears intersect at angles other
than 90°, the drawing of the bhmks and devel-
o[)nient of the teeth do not differ from the cases
ah'eady descriljed. The figuring required is that
inthcated in Fig. 27, those in the heavy face
being used to determine the other vahies, and
not a[)pearing on the finished drawing.

tan A =

+ cos a

tan A' =

— + cos a

r, 2 sin A 2 sin A'
tan B = or

, r. 2.25 sin A 2.25 sin A'
tan C = or

E = - cos A ; E' = — cos A'.
P P

sin A

F' = — sin A'.
P

Fig. 27.

Or, the values for E, F, E', and F may be ob-
tained from the table for shafts at 90°, pages 54
and 55 by determining the center angles A and
A', and finding the values for 2 E and 2 F, corre-
sponding to each gear separately.

WILLIS S ODOXTOGllAril. 5/

CHAPTER VII.

SPECIAL FORMS OF ODONTOIDS, NOTATION, FORMULAS, ETC.

64. Odontographs and Odontograph Tables. If tooth curves are to be drawn according to
some estal)lislied system, in whieli the angle of pressure is constant, or but one diameter of
rolling circle be used, it may be desirable to employ some of the approximate methods for
shortening the operation. While it is unnecessary for the student to familiarize himself with
the theor3% or even the details, of operating the various systems of approximating these curves,
it is essential that a knowledge be had of the more useful tables and methods to which refer-
ence may be made when required.

Three methods are employed for approximating the odontoidal curves.

First, by circular arcs, the centers and radii of which are given in tables, or established by
instruments, designed for this purpose.

Second, by curved templets from \\liich the curves may be traced directly.
Third, by ordinates.

65. Willis's Odontograph. Among those of the first type, the oldest, best known, and least
accurate, are tlie odontographs designed by Professor Willis. When used for gears having a
large number of teeth, the error is very slight ; but in the case of involute teeth of small
number it is very noticeable. Fig. 28 illustrates the application of this instrument to the

THE GRANT ODONTOCJRAPIIS.

drawing of curves of the cycloidal system. The' centers for the circuhir arcs designed to
approximate the curves are found on tlie straight edge, A B, and at a distance from the zero
point of the scale to be found in tlie pul)lished table accompanying the instrument.

'Jlie theory and application of these odontographs
is clearly treated of in the instructions accompanying
these instruments, also in Stalil and Wood's " Elements
of Mechanism," pages 113 to 122, and more briefly in
MacConUs " Kinematics," pages 172 to 174.

66. The *' Three Point Odontograph," designed by

Mr. Geo. B. Grant, is a table for face and flank radii

and centers, figured for circular arcs passing through

j \ \ \ ^^^ three most important points of the tooth curves ;

/ \ \ ! viz., at addendum or dedenduni circles, pitch circle,

and a point midway between. This gives a very close
approximation to the true curve for the systeni which
has radial flanks for gears of twelve teeth. The tables
and instructions are pul)lished in Grant's " Odontics,"
pages 41 and 42, and in Stahl and Wood's '•* Elements of Mechanism," pages 124 and 125.

Fig. 28.

67. The Grant Involute Odontograph, designed by Geo. B. Grant, and puljlished in his
"Odontics," pages 29 and 80, gives a very close approximation to the involute for 15° angle
of pressure and epicycloidal extension, all gears being designed to engage a 12-tootlied gear
without interference.

THE ROBIXSOX AXD KLP:iN ODONTOCillAPHS

68. The Robinson Odontograph differs from
the preceding in that it is an instrument hav-
ing a curved edge which is used as a templet
to trace the tooth curve, tables being used to
determine the position of the instrument \^ith
relation to the pitch circle.

Fig. 29 illustrates the instrument in posi-
tion. The curve B C A is a logarithmic spiral,
and the curve B F H the evolute of the first, and
therefore a similar and equal spiral. By means
of this instrument, in connection with the pul>-
lished tables accompanying it, involute teeth may be drawn
as well as C3^cloidal, and a much larger range of the latter
is possiljle than is afforded by the Willis odontograi)h. Tlu^
theory of this instrument is best treated by Professor Rob-
inson in Van Nostrand's Eclectic Magazine for July, 18T(>, and
Van Nostrand's " Science Series," No. 24. Also see Stahl
and Wood's "Elements of Mechanism," pages 12(3 to 1-30.

69. The Klein Coordinate Odontograph. Fig. 30 is de-
signed to eliminate the labor of drawing pitch circles of large
radii by constructing the cuinc l)y ordinates from a radial
line. 'I'he tables and ex2)laiiatioii of tlie method may l)e

Fig. 30.

SPECIAL FORMS OF ODONTOIDS AND THEIR LINES OF ACTION.

Fig. 33.

found in Professor Klein's " Elements of ^Machine
Design," page 50.

70. Special Forms of Odontoids and their Lines
of Action. Gears maybe classified from the forms
of rack teetli, as follows :

System. Tooth Curve. Lixe of Action-.

Involute, Fig. 31, A right line, .V riglit line.

Cycloidal, Fig. o2, A cycloid, A circular arc.

Segmental, Fig. 00. A circular arc. Conchoid of Xicomedes.

In like manner other systems might be derived
from, and classified by, the forms of their rack
teeth.

It is of interest to note in connection with the
first two that any tooth of either system may be
derived from a right line. In the cycloidal system
the addendum of any gear tooth will properly
engage the radial flank of some gear. If. there
fore, the addenda of any gear tooth be made to fit
the dedenda of teeth consisting of radial flanks,
the resulting teeth must be cycdoidal. \ skilled
mechanic with file and straight-edge could in this

COXJUCIATE CURVES.

manner })roduce the templet for any de-
sired cyeloidal tooth without the aid of
other mechanism. Of course such a
method would require considerable skill
in producing a perfect tooth, and it is not
the best means to the end ; but it is of
much interest to the student as illustrat-
ing the relation between the mechanical
and graphic methods of attaining the same
end. In like manner we may produce templets for invo-
lute teeth from the right line rack tooth of the system.

71. Conjugate Curves. — The curves of any pair of
teeth being so related as to produce a uniform velocity
ratio are called conjugate, or odontoids, and if any tooth
curve of reasonal)le form be assumed, a second curve
may be obtained which shall be conjugate to the first.
By a reasonable form is meant the conformity to the
following principle : —

The normals to the curve must come into action
consecutively, as in Fig. 34, and not as in Fig. 35, in
which it will be seen that the normal E F will })ass
through the pitch point M, and the point E come into

WORM GEARING.

Elg. 37.

action before the point C, which is impossible. Let C, Fig. 36,
be any tooth form conforming to the above condition, and the
periphery of disk A its pitch line. Suppose it is required to
derive its conjugate having for its pitch circle the periphery of
disk B. This may be obtained by a graphic process, as in
Art. 28, page 15, or by the mechanical method known as the
molding process of Fig. 36. C is a templet of the given tooth
form, which is fastened to disk A , and revolving in contact with
disk B, the disks maintaining a constant velocity ratio. The
successive positions of C are then traced on the plane of disk B,
and the tangent curve will be that of the required conjugate
tooth.

The method is applicable to all forms of spur gear teeth,
but to only one form of bevel gear, the octoid.

72. Worm Gearing. A woi'm is a screw designed to oper-
ate a gear, called a worm wlieel or gear, the axis of the latter
being perpendicular to that of the worm. Art. 3, page 3.
The section of a worm and gear made by a plane perpendicular
to the axis of the gear, and including the axis of the worm, is
identical with that of a rack and gear of the same system and
pitch. The worm, or screw, may be single, double, etc. If
single, the circular pitch corresponds with the pitfli of the

LITERATURE. G3

thread : if double, the cireidai' iiitcli will he half the pitcli of the thread, etc. To avoid niis-
uiiderstandiug, it is customary to speak t)f the pitch of the tluead as the lead.

A drawing of the tooth foi-ni is required oiiiy in special cases of large cast gears, and the
usual representation is that shown by Fig. 37.

The diameter of the worm is connnonly made equal to foui' or live times the circular pitch,
and the angle A varies from (iU ' to \H) .

FoiiMULAS Full WoilISI AND (iEAll.

L = Lead of worm ;
m = Threads per iucli in worm;
d = Outside diameter of worm;
d' " Pitcli diameter of worm;
W = Wliole diameter of gear;
D =-- Tliroat diameter of gear;
D' = Pitcli diameter of gear;

L = — = P', for single tlireads,

2
L = — = 2 P'. for doiihie tlireads. etc.;
m

Tj and r., are dimensions reciiiircd for the liol), or

cutter, emploved in cutting the worm gear; >.; r^ , ^ / ^

' ' •' " • W = D + 2 rj — rj cos -

C = Center distance; \ ^

73. Literature. The following list of books and articles is published to assist the student
who may wish to puisue the subject beyond its elementary stage. Only those treatises have
been enumerated which are likely to be accessible and uscfid. The great works of Willis,

p, TT U

N + 2'

0..11-.^;

D=^ + ^

d 2
•■i - 2 P'

^2 = ^1 + ^ P;

r D + d
^ " 2

1
P'

W = D + 2 ( rj

— rj cos

(;4 LITEHATUHK..

Kaiikiiie, and Reuleux are omitted, as the student will derive more beuetit from the interpreta-
tion of these works by hiter authors tlian by a stndy of the original treatises.

'^The Mechanics of the Machinery of Transmission," revised by Professor Herrmann, is
Voh Iir., Part I., Sect. 1, of Weisbach's "Mechanics of Engineering." This work includes
one of the most valualde treatises on the subject of gearing, bnt it is somewhat dithcnlt.
Wiley, $')M.

"■ Kinematics," by Pi'ofessor MacCord, is chiefly devoted to the snbject of gearing. It con-
tains nuich original matter of importance. No student of the subject can afford to do witliout
this treatise. Wiley, -f!5.0().

" Elements of Machine Design," l)y Professor Klein, was published for the students of
Lehigh University. Several chapters are devoted to gearing, and include some excellent tables
and problems. The Klein coordinate odontograph is fully illustrated and explained. J. E.
Klein, Bethlehem, Pa., -16.00.

" Odontics," by Mr. Geo. B. Grant, is one of the most valuable modern treatises on gearing.
It is both theoretical and practical. It is concise, contains many useful tables, and is well
illustrated. The subject cannot be })ursued to advantage without its use. Lexington Gear
Works, Lexington, Mass., <fl.OO.

" Practical Treatise on Gearing," by Mr. O. J. Beale. An excellent practical treatment of
the design and construction of gears. It deals little with the theoiy, but that little is thor-
oughly and simply taught. Brown & Sliarpe Manufactin-ing Company, Providence, $L00.

" Eormidas in Geaiing." This is published by the Brown & Shari)e Manufacturing Com-
])any, and contains many useful formulas for the draftsman, and valuable hints for the cutting
of frears. .f2.00.

NOTATION AND FORMULAS. 65

" Elementary Alechainsin,'* by Professors Stahl and Wood, is a most comprehensive text
book on the subject of gearing. It is well classified, contains numerous examples, and is a
valuable reference book for the student. Van Nostrand, '^2.00.

The student is recommended to read the following articles published in the Ainei-ican
Machinist.

"Cutting- Bevel Gears in a Universal Milling Machine." by O. -I. Beale, June 20, 1895.
"Planed Bevel Gear Teeth," by George B. Grant, Dec. 9, 189G.
"Grant's Epicycloidal Bevel Gear Generator," June 7, 1894.
" Bilgrani Bevel Gear Cutting Machine," May 9, 1885.
"Bilgrani Gear Exhibit," Oct. 12, 1893.

" Bevel Gear Curves." Chart for plotting from (iranfs bevel gear chart, by H. AValden, Oct. 8 189(3
(chart corrected Xov. 5, 1896).

"The Strength of Gear Teeth," by Henry Hess, Feb. 18, 1897.
"Strength of Gear Teeth," by W. T. Sears, June 10. 1897.
"Gear Arm Proportions," by Henry Hess, April 29, 1897.

74. Notation and Formulas.

Spur Gears.

P' = Circular pitch. Art. 17, page 11; N = Number of teeth in gear;

P = Diameter pitch, Ar.T. 18, page 11; n = Number of teeth in pinion;

D' = Pitch diameter of gear; s = Addendum of tooth, Aiix. 31, page 17;

D = Whole, or addendum, diameter of gear; f = Clearance, Art. 27, page 15;

d' = Pitch diameter of pinion: t = Thickness, Art. 31, page 17;

d = Whole, or addendum, (liiimt'tei- of pinion; p ^ liCast angle of pressure, Aitr. 45, page 8;?;

NOTATIOX AND FORMrLAS.

TT -^ 3.1416;

N D' P'

TT D' N TT

P = 'TT- • ^, = ^, , Ai!T, 17, page 11;

N
P = j^,, P P' = TT, Akt. 18, page 12;

f = Q ~ o~B' '■^''T- 31, page 1/:

P' _ TT _ 1.57 .

t = ^ ^ ^ ^ . Art. 31, page 1 . ;

2 PD'+2 N4-2
D=D'+2S = D' + ^ = '^-y^-' = ^<

_ /N — 2
'^°^ P ~ V/ ~^' Ai!T. 45, page 33.

Annular Gears.

Rg ^ Radius of gear,

Rp = Radius of pinion,

fj = Inner describing circle,

r., = Outer describing circle,

r^ = Intermediate describing circle,

C = Center distance,

'Art. 50, page ;19;

Rg = r- + r, , Art. 50, page 30:

Rp = rg — r._, , Art. 50, page 3!)

C = Rg — Rp, Art. 50, page 39;
fj maximum = Rg — C ; rj minimum = C, Art. 52, page 40;
r.2 maximum = C ; r._> minimum = 2 C — Rg,

Ai;t. 52, page 40;
r3 maximum = Rg ; r^ minimum = C, Art. 51, page 40.

REV?:r. Gears, Shafts at 90°, Airr. 6i, Page 5G.

A = Tenter angle of piiiiuu;

B = Angle increment;

C = Angle decrement;

E = ()ne-half the (iianictci' inciemenl foi- pinion:

F ( )n('-lialf llic di;nii('tcr iiirrcinciit fur geai'.

tan A

d' _ n .
~ D' N'

tan B

2 sin A.

tan C

2.25 sin A

1 ,
= p cos A

sin A.

notation and formulas. 67

Bevel (iEaks, Shafts at Othei: than *J0^, Ai;t. 63, Page 5(3.

a = Aniile of .sliafl>; ^ , sm a

° tan A =

A = Center angle of pinion;

A' = Cenlei angle of gear; ^^p ^' —

B = Angle inerennMil :

„ . , , _, 2 sin A 2 sin A

C = Angle deeienienl : tan B =

—

cos

sin a

COS

n N

E = One-half the diameter inerenient for pinion; ^ 22.5 sin A 2 25 sm A'

tan C = =- ,

n N

E' = OiH'-half the diameter increment lor gear; ■, ,

E = _ cos A E' = p cos A';

F = Dimension i-e<|nireil for hacking of jiinion;

c 74- • • 1 .■ 1 1 ■ i- r = „ sm A F = — sm A ;

I- = Dimension re(iuued tor hacking ot gear. P P

\Voi:m (iEAEs, Ajrr. 72, I'AciE (32.

L = Lead of worm ; , 1 r^ . , , ,

L = — =^ P' tor .sinule threads;
m

nn = Threads per inch in worm; 2

L = — = 2 P' tor doiihle thread, etc.,
m
d = Outside diameter of worm; k, m i 9

D'= - • D = "*" :

P P

d = Pitch diameter of worm:

TT D ..,„,/ A'

P' -- • W = D + 2 1^ - ri COS

D = Thread diameter of gear; N + 2 y

C = ^-±- - -•
D' = Pitch diameter of gear; 2 P'

W = AVhole diameter of gear. ^^ ~ 7 P ' "^i ~ ''i """ 3 '^^

68 METHOD TO BE OBSERVED IX PEKEOKMJXC; THE PKOBLEMS.

CHAPTER VIII.

PROBLEMS.

75. Method to be Observed in Performing the Problems. No attempt slionld be made to
oraphically solve the following problems until the general principles involved are well under-
stood.

The first requisite to tliis is tlie mastery of C-hapter II., on Odontoidal Curves; and this
can be best acquired l)y the drawing of the various curves, together A\'ith a study of their
characteristics. No problems have been given on this topic, but the following course of
study would be desirable : —

Having prescribed diameters for rolling circles and director, or pitch circles, draw a cycloid,
epicycloid, and hypocycloid, as described in Arts. 5, G, and 7, page 5. Obtain a sufficient
number of points in each case to enable the curves to l)e drawn free-hand with considerable
accuracy, after wliich they may be corrected by the use of scrolls. Next prescribe a point on
each (not one already found), and draw^ normals to each by Akt. 8, page 5.

The second method, Art. 9, page 5, is the more practical, and shonld also l)e studied by
drawing a small part of each curve, beginning at a point on the director circle.

It is also desirable that one of the epitrochoidal forms be drawn, and a normal determined.
Art. 11, page 6.

The problems are designed to be solved on a sheet which shall measure 10" by 14" within

PKOBLKM 1. CYf'LOIDAL LIMITING CASE. 69

the margin line, and the hiy-out of these sheets is given on Plates 14 and 15, there being
four problems on each plate. ]\Ieasurements are from the margin line.

It is unnecessary to represent all the teeth in a gear, but such as are shown should be
drawn with the greatest accuracy attainable by the student. Without this care the study will
avail one little, and the time consumed in discovering errors will be great.

The inking of the curves may be omitted if time will not admit of its being w^ell done ;
but in either case it is desirable to emphasize the curves, and distinguish clearly between the
gears by making a very light wash of color on the inside of the curve, the width to be about
one-quarter of an inch. One color may be used for the pinion, and a second for the rack and

Cycloidal Limiting Case. Face or Flank only.

gear.

Problem i, Plate

14,

Fig.

Example.

121

lOL

7
8

12'

2'»

•^4

Statement of Problem. Having given the diameters of pitch circles, number of teeth,
and diameter of describing circle, it is re(]^uired to draw the teeth for pinion, gear, and rack,
liaving arcs of contact equal to the pitch, and contact on one side of pitch point only.

70 PROBLEM 1. CYCLOIDAL LIMITIXC; CASE.

Study Arts. 1 to 26 before performing this prol)lein.

Operations. 1. By Art. 18, page 11, determine the value of N, n, D' or d, one of whicli is
omitted from the table. Observe that - = -•

d n

2. Draw center and pitch lines and describing circle. Lay off the circular })itch on each
gear by spacing the circumferences into as many parts as there are teeth.

3. Obtain the first point of contact by laying off from the pitch point on the describing
circle an arc equal to the circular pitch, the directif)n being determined b}^ the rotation
required. Art. 16, page 10. Ar.t. 21, page 12. Arts. 22 and 23, page 13.

1. With the above describing point, generate the face and flank required. Arts. 14 and
15, page 10.

5. Draw the working faces of gear teeth, and assuming the gear teeth to l)e pointed, draw
opposite side of each. Art. 16, page 10.

6. Draw the working flanks of the pinion teeth, observing that the depth must be sufificient
to admit the gear teeth, but without clearance. Obtain the thickness, and draw the opposite
sides. Art. 16, page 10.

7. Draw the describing circle for rack. Obtain the first point of contact between pinion
and rack, and describe the cycloid for rack teeth. Construct rack teeth. Ar/r. 2o, i)age 14.
Note that thickness of rack tooth must equal space between pinion teeth, or thickness of
gear teeth, measured on the })itcli line.

8. To determine points of contact of conjugate teeth, assume any ])oint on face of gear
tooth, and determine, first, its position when in contact with the pinion ; second, the jjoint
of the pinion tooth engaging it. Since the contact must take place on the path of contact.
Art. 21, page 12, the assumed })oint will lie at the intersection of this arc and one described

PROBLEM 2. CYCLOIDAL LIMITING CASE. 71

through the given point from center of gear. To solve the second, describe an arc from the
center of the pinion through the point previously determined, and its intersection with the
pinion flank will be the engaging point required.

Next construct the normals for each of these points. Art. 8, page 5. They should he
equal to each other, and also to the distance from the pitch point to the point on the path of
contact in which they engage. Art. 14, page 10.

9. Obtain the maximum angle of obliquity, or pressure, between gear and pinion, pinion
and rack. Art. 24, page 14.

Problem 2, Plate 14, Fig. 2. Cycloidal Limiting Case. Face and Flank. Study Arts.
2(3 to 30.

Statement of Problem. The diameters of gears, number of teeth, and describing circles
being given, it is required to draw the teeth for pinion, gear, and rack, when the arc of
approach = the arc of recess = half the circular pitch, the flank of gear being radial.

Operations. 1. Draw center lines, pitch lines, and rolling circles, the second circle
being determined by Art. 9, page 0. Divide the pitch circle into the required parts to obtain
the circular pitch.

2. Lay off arcs equal to — on each of the rolling circles to obtain the first and last points
of contact, observing the direction of rotation prescriljed in Fig. 2.

3. With the point thus determined on small rolling circle, describe the addendum of gear
tooth and dedendum of pinion tooth. With the point on the second describing circle generate
the addendum of pinion tootli. The dedendum of gear tooth being radial may then be drawn.
Make the dedenda of pinion and gear deep enough to admit the engaging addenda, but allow
no clearance.

72 PROBLEM .?. CYCLOIDAL GEAR.

4. Draw the working faces of the pinion teeth and then the opposite faces to make the
teeth pointed. Simihirly draw the gear teeth, making them pointed also. The sum of the
thickness of tlie teeth cannot he greater that the circular pitch. Art. 29, page 16. In this
case it will he found to be about one-hundredth of an inch less, which will be the backlash.
An increase in the diameter of either rolling circle would make the solution impossible.

5. Draw the dedenda of pinion and gear teeth.

6. The describing circles for the rack teeth Avill be determined by Art. 14, page 10.
Draw the circles with their centers on the line of centers, and obtain the first and last points
of contact. These points should fall on the addendum and dedendum of pinion teeth already
drawn, as in Plate 5 at M and . From these points describe the addenda and dendenda of
the rack teeth. The thickness of these teeth must equal those of tlie gear.

7. Obtain the maximum angle of pressure for approach and recess between pinion and
gear and pinion and rack. It would also be desirable to obtain the curve of least clearance
in one case. Art. 28, page 15.

Problem 3, Plate 14, Fig. 3. Cycloidal Gear. Practical Case. ('om})lete Chapter III. be-
fore performing this problem.

N n A a C

15 12 4 5 31

21 12 4 4 3|
20 16 3i 4 3

22 12 3i 5 4

16 12 4" 5 3i
20 12 3 4 4^

^^X AMPLE

PROBLEM 3. CYC'LOIDAL GEAR. 73

Statement of Problem. Tlu' diameters of pitch circles and rolling circles being given,
and the number of teeth known, it is required to draw the teeth for gear, pinion, and rack,
to obtain the maximum angle of obliquity, and the arcs of approach and recess in each case.
The teeth will be standard with j^^" backlash. Art. 31, page 17. Art. 71, page 61.

Operations. 1. Figure the diameter of gear, circular, and diametral pitch. Arts. 17
and 18, page 11, and determine proportions of teeth. Art. 31, page 17.

2. Draw center lines, pitch lines, addendum, and dedendum circles, and rolling circles.
Divide the pitch circle into as many parts as there are teeth, beginning to space at the pitch
point.

3. Beginning at the pitch point, describe pinion flank, gear face, gear flank, and pinion
face, by Art. 9, page 5. See also Art. 34, page 21.

4. Lay off thickness of teeth. Art. 31, page 17, and describe addenda of pinion and gear
teeth by approximate method. Art. 34, page 22. Describe dedenda by Art. 10, page 11.
Draw fillets. Art. 31, page 18.

5. Describe rack teeth.

6. Determine the following for gear, pinion, and I'ack in tei-ms of P'. Arts. 21 to 24
inclusive, pages 12, 13, and 14, Art. 32, page 18.

Pinion and Gear. Pinion and Rack.
Arc of approach

Arc of recess

Arc of contact

Maxinunn angle of pressure ^

/4 PROBLEM 4. INVOLUTE LIMITING CASE.

Problem 4, Plate 14, Fig. 4. Involute Limiting Case. Study Arts. 38 to 42.

Statement of Problem. Number of teeth five and six. Pinion teetli pointed. No
Ijacklasli or clearance. Arc of contact equal to the circular pitch.

This problem being similar to that of Plate 8, reference will be made to that figure.

The case being a limiting one, the distance between the points of tangency of base circles
and line of pressure must equal one-sixth of the circumference of the gear base circle, or one-
tifth of the circumference of the pinion base circle. The tangent of the angle of pressure

will equal ^ = ^ = / ^ , but A D = D K C by construction, and D K G = tt. Also A F +

D C = 5i, hence, —= — — = — = tan. of the angle of pressure. The angle corresponding to

this tangent is 29° 44' 6". The distance between the centers will be ^ho'^ + A F + 'D~G^ =

The angle of pressure and distance between centers could have been determined graphi-
cally by laying off F A , in any direction, equal to the radius of pinion base circle, A D perpen-
dicular to FA, and equal to one-fifth of pinion base circle. Finally, D G perpendicular to A D,
and equal to the radius of gear l)ase circle.

Operations. 1. Draw the line of centers, base circles, and line of pressure. Deter-
mine the points of tangency, which limit the action in either direction, and through the pitch
point, determined by the intersection of the line of centers and line of pressure, draw the
pitch circles. It is desirable now to test A D by proving it equal to one-fifth of the pinion base
circle, or one-sixth of the gear l)ase circle.

2. Draw the involute A C, Plate 8, of the gear, and D p of the pinion. Airr. 12, page 7.
Art. 38, page 26. Determine the circular pitch, and lay off as many divisions as there are
teeth to be drawn. Copy the curves already drawn.

PROBLEM .-,. INVOLUTE PRACTICAL CASE. 75

3. Draw the opposite face of pinion tectli. making them pointed. To draw the opposite
faces of gear teeth proceed as follows : Since contact between the opposite faces must take
place along the line of action C E , Plate 8, the contact between the engaging teeth will be
at E. At E draw arc E 1 from center G . Bisect this arc, and lay off M and H from this radial
bisector equidistant with A and C. 'I'hrough these points describe the curve of opposite face,
and draw the remaining teeth.

That j)ortion of the teeth lying within tlie base circle will ])e radial, and extend sufficiently
to admit the engaging teeth, but without clearance.

4. Construct two rack teeth. Art. 40, page 29.

5. Epicycloidally extend the gear teeth so as to make them pointed. Similarly extend the
rack teeth, l)ut only as mucli as the clearance for tlie pointed gear tooth will permit. Art.
41, page 30.

Problem 5, Plate 15, Fig. i. Involute Practical Cases. Complete the study of Chapter IV.

State.mext of Pnop.LEMS. Several gears and racks are given to describe involute teeth
of standard dimensions. 'J'o determine the interference, if there be any, and to correct the
curves for the same.

QPERATroxs. 1. Draw three or four teeth of gear A, and two teeth of engaging pinion
B, the angle of pressure being 16°. Art. 42, page 32, Fig. 16. Make contact at pitch
point in all cases. Correct for interference by epicycloidal extension. Art. 31, page 17.
Art. 42, page 30. Art. 41, page 32.

2. Draw three or four teeth of o-ear A enofaofino- rack F .

3. Draw three teeth of pinion B engaging rack E, and correct rack teeth for interference.

PROBLEM 6. CYCLOIDAL ANNULAR GEAR.

4. Draw a portion of gear C and rack K , the angle of pressure being 20°. Test this for
interference by Art. 45, page 33, as well as by graphic method.

5. Draw a few teeth of gear D, the angle of pressure being 15°. Determine the least
number of teeth that will engage it without interference.

Problem 6, Plate 15, Fig. 2. Cycloidal Annular Gear. Study Arts. 48 to 56.

EXAMPLK.

191

171

17i

Statement of Problem. The number of teeth and diameters of pitch and describing
circles being given, it is required to draw the tooth outlines, and determine the increased arc
of contact due to secondary action. The arc of contact, not including that due to the
secondary action, is equal to the circular pitch, and the arc of approach equals the arc of
recess.

Operations. 1. Draw the center and pitch lines and describing circles.

2. Determine the circular pitch, and lay off half this amount from the pitch point on each
of the describing circles to determine the first and last points of contact.

3. Describe the curves of the teeth.

PROBLEM 7. INVOLUTE ANNULAR GEAR. i I

4. Determine the intermediate describing cnrve, and draw tlie same to obtain the limit of
secondary action.

5. Determine the maximnni angle of pressure for approach and recess. Also the angle
of pressure for the last point of secondary action, and the increase in the arc of contact.

Problem 7, Plate 15, Fig. 2. Involute Annular Gear. Complete Chapter V.

Example.

Angle

of Pressure.

-i

20°

15°

20°

15°

20°

Stateiment of Problem. The pitch diameters, number of teeth, and angle of pressure
being given, it is required to draw the tooth cui've, to determine if there Avill l)e any inter-
ference when the addenda of pinion teeth are made standard, and finally the length of the arc
of contact in terms of P'.

Operations. 1. Draw center and pitch lines, line of pressure, and base circles.

2. Make addenda of pinion standard if a second engagement does not take place. Art.
56, page 43, and limit addenda of gear by Art. 56, page 43.

3. Determine the arc of contact in terms of P'.

PROBLEM 8. CYCLOIDAL AND INVOLUTE BEVEL GEARS.

Problem 8, Plate 15, Fig. 3. Cycloidal and Involute Bevel Gears. Shafts at 90^. Study
Arts. 57 to 63.

IMPLE.

, P

•^8

3
4

"^8

3
4

If involute, make angle of pressure 15°.

If cycloidal, make diameter of rolling circles equal to the elements of normal cone of
pinion.

Statement of Problem. The proportions of the gear being given by the table, it is
required to draw the gear blanks, describe the development of the teetli on the normal cones,
and figure the gears.

Operations, 1. Having determined the pitch diameters, draw the gear blanks. Art.
60, page 48.

PROBLEM 9. CYCLOIDAL AM) INVOLUTE BEVEL (iEARS. 79

2. Describe two or three teeth of each gear on the developed surfaces of the outer and
inner normal cones. Akt. 60, page 48.

3. Figure the gears, Art. 61, i)age 51.

Problem 9, Plate 15, Fig. 4. Cycloidal and Involute Bevel Gears. Shafts at other than
90°. Study Art. 63.

EXAMT

'Lt;

40°

45°

50°

55°

60°

8t.

2.1

1^1
1*

1§

3^. i

If involute, make angle of pressure 15°.

If cycloidal, make diameter of rolling circles equal to the elements of normal cone of
pinion.

Statement of Problem. Tlie pro|)ortions of the gear being given by the table, it is
required to draw the gear ])lanks, describe the teeth on the development of the normal cones,
and figure the gear.

Operations. 1. Determine the ])it{h diameters from al)ove tal)le, and draw tiie gear
blanks.

2. Describe two or three teeth of each gear on tlie developed surfaces of the outer and
inner normal cones.

o. Figure the geai-s.

I ]^ D E X„

Addendum defined, 12: proportion for, 17.

Angle decrement, 52.

Angle increment, 52.

Angle of edge, 51 ; of face, 51.

Angle of obliquity, or pressure, 14 ; affected by rolling circle,
18; constant, 28; for involute, 31; influence of, 33;
method for determining, 33; reduced in annular gear-
ing, 40.

Annular gear, notation, and formulas, ()(>: epicycloidal prol)-
lem, 76; involute problem, 77.

Annular gearing, 38; secondary action in. ."W ; interchan-
geable with spur gearing, 42; involute system of. 43.

Approaching action detrimental, 17.

Approximate cycloidal curves. 22.

Approximation, Tredgold, 47 ; by circular arcs, 22

Arc of approach defined, 13.

Arc of contact defined, 13; relation to circular pitch, ](>.

Arc of recess defined, 13.

Backing, .52.

Back cone, 48.

Backlash defined, Hi; dimen.sions for, 18.

Base circle defined, 7, 27.

Base of system, 21 ; in annular gearing, 42.

Beale's "Practical Treatise on Gearing," 04.

Bevel gear defined, 2: Theory of, 45; character of curves
employed, 4(5; drafting the, 48; blank, 49; length of
face, 49; figuring the, 51 ; table for, 53, 54, 55; cliart for
plotting curves, 65 ; notation and formulas, 66 : prol>
lems, 78, 79.

Bevel gears with axes at any angle, 5(5.

Bilgram, Hiigo, inventor of octoid tooth, 47; machine for
cutting bevel gear teeth, 47, 65; exhibit, 65

Brown & Sharpe publications, (54.

Circular pitch defined, 11.

Character of curves in bevel gearing, 46.

Clearance defined, 15; proportion for, 17.

Clock gears, 17.

Conchoid of Nicomedes, (50.

Conditions governing the practical case, 16.

Conjugate curves defined, 9, 61.

Constant angle of pressure, 28.

Constant velocity ratio defined, 1.

Conventional representation of spur gears, 25.

Contact, point of, 5; radius, 5; path of, 12; arc of, 13.

Coordinate odontograph, .59.

Crown gear, 47.

Curtate epitrochoid, (i.

Curve of least clearance, 15.

INDEX.

Curves, odontoidal, 4.
Cutting bevel gear teetli, 65.
Cutting angle, 51.

Cycloid, defined, 4; problem relating to, 68.
Cycloidal action, Theory of, 8.

Cycloidal curves, second method for describing, 5 ; approxi-
mated, 22.
Cycloidal system of annular gearing, 38.
Cycloidal annular gear problem, 76.
Cycloidal bevel gear i^roblem, 78, 7fl.
Cycloidal limiting case problems, (59. 71.
Cycloidal practical case problem, 72.

Dedendum defined, 12; proportions for, 17.

Defects of involute system, 35.

Describing circle, defined, 4; a path of contact, 12; maxi-
mum and minimum, 16; influence on sliape and effi-
ciency of teeth, 18 ; relation to interchangeable gears, 20.

Describing disk, 8.

Describing point, 4.

Describing cone, 45.

Describing cylinder, 45.

Describing radius, 5.

Description of bevel gear table, 53.

Developed pitch circle, 50.

Development of normal cone, 49.

Diameter pitch, 11.

Director circle, 5.

Double contact in annular gearing, 3fl.

Double generation of epicycloid and hypocycloid, 6.

Drafting bevel gears, 48.

" Elements of Machine Design," (i4.

" Elementary Mechanism," 65.

Epicycloid, defined, 5; second method for describing, 5;
double generation, 6; spherical, 45; problem relating to,
68.

Epicycloidal extension, 30.

Epitrochoid, defined, 6; curtate, 6; prolate, 7: problem re-
lating to, 68.

Exterior (outer) describing circle, 39; limitations of, 40, 41.

Face gearing, 2.

Face of gear, 24.

Face of tooth, 12.

Flank of tooth, 12; radial, 18.

Figuring bevel gears, 51.

Fillet, 18 ; size of, 18.

" Formulas in Gearing," 64.

Formulas for worm and gear, 63.

Formulas, Notation and, 65.

Gearing, 1.

Gear arm proportions, 65.

Gears, interchangeable, 20: face of, 24; comparison of, 24.

Generating point, 4.

Generating radius, 5.

Grant, Geo. B., bevel gear chart, 53; three point odonto-
graph, 58; involute odontograph, 58; " Odontics," 58,
64; epicycloidal and bevel gear generator, 65.

Hyperboloid of revolution, 2.

Hyperbolic gears, 2.

Hypocycloid, defined, 5; second method for describing, 5; a
radial line, 6: double generation, (!; si)lierical, 45; prob-
lem relating to, 68.

INDEX,

Influence of the angle of pressure, 33.

Influence of the diameter of rolliiij; circle on shai>e and
efficiency of teeth, 18.

Inner desci'ibino; circle, 3i); limitations of, 40, 41.

Inner normal cone, 48, 50.

Instantaneous radius, 4.

Intermediate describing circle, 39; limitations of, 40.

Internal gear, see annular gear.

Interference, 32; in annular gearing, 43.

Interchangeable gears, 20.

Involute, 4 ; defined, 7 ; system, 26 ; curves, character of, 27 ;
rack, 28 ; system of annular gearing, 43 ; annular gear
l)roblem, 77; bevel gear tooth, 46; bevel gear problems,
78, 79 ; limiting case, 29 ; limiting case jiroblem, 74 ; prac-
tical case, 30; practical case problem, 75.

Involute action, Theory of, 26; limit of, 28.

Involute gearing, defects of system, 35.

Involute teeth, epicycloidal extension of, 30.

" Kinematics," MacCord's, (i4.
Klein's coordinate odontograph, 59;
Design," 64.

Elements of Machine

Law of tooth contact, 10.

Lead of screw, 04.

Least angle of i)ressure, method for determining, 33.

Least number of teeth in annular gears, 42.

Limit of involute action, 28.

Limiting ca.se, cycloidal, 10. 14; involute, 29; ainiular gear-
ing, 38.

Limitations of intermediate, exterior, and interior describing
circle, 40, 41.

Line of action a great circle, 47.

Literature, 63.
Logarithmic spiral, 59.

ISIacCord's "Kinematics," 64.

Method for determining least angle of pressure, 33.

Method to be observed in performing problems, 68.

"Mechanics of Engineering," 64.

" Mechanics of the Machinery of Transmission," 64.

Normal, defined, 4 ; to construct, 5 : law
Normal cone, 48 ; development of, 49.
Notation and formulas, 65.

governing, Gl.

Obliquity, angle of, 14.

Octoid bevel tooth, 47, 62.

" Odontics," Grant's, 64.

Odontoid defined, 1 ; special forms of, 60.

Odontoidal curves, 4 ; problems relating to, (i8.

Odontographs and odontograph tables, 57.

Odontograph, "Willis, 57: Grant involute, 58; Grant Three

point, 58; Robinson, 59; Klein, 59; coordinate, 59.
Outer describing circle, 39; limitations of, 40.
Outer normal cone, 48.

Path of contact defined, 12; affected by rolling circle, 18; a

right line, 28.
Path of approach defined, 15.
Path of recess defined, 15.
Pitch cone, 48.
Pitch line, 10.
Pitch point, 9, 10, 27.
Pitch, circular, 11 : diameter, 11.
Planed bevel gear teeth, ()5.

INDEX.

Positive rotation defined, 11.

Practical case, conditions governing the, 16; cycloidal, 21

involute, 30; annular, 42.
"Practical Treatise on Gearing," 04.
Pressure, angle of, 14.
Prolate epitrochoid, 7.
Proportions for standard tooth, 17.
Problems, method to be observed in performing, 68.

Rack, 14 ; involute, 28 ; gears classified by, 60.

Radial flank, 18 ; as base of system, 21, 60.

Radius, describing, 5 ; contact, 5.

Rankine, 64.

Reuleux, 64.

Robinson odontograph, 59.

Rolling circle, see describing circle.

Rotation, jiositive, 1.

Screw gearing defined, 3.

Scroll, use of, 11.

Second method for describing cycloidal curves, 5.

Secondary action in annular gearing, 38, 41.

Segmental system, 60.

Skew gear defined, 3.

Spiral gear defined, 3.

Special forms of odontoids, 45.

Spherical epicycloid, 45.

Spherical hypocycloid, 45.

Spur gear defined, 2; illustrated, 10; having action on one
side of pitch point, 10; having action on both sides of
pitch point, 14; conventional representation, 25 ; inter-
changeable with annular gears, 42 ; notation and formu-
las, 05.

Theory of cycloidal action, 8.
Theory of involute action. 20.
Thickness of tooth, 17.
Three point odontograpli, 58.
Tooth contact, law of, 10.
To construct a normal, 5.
Tredgold approximation, 47.

Unsymmetrical teeth, 37.
Use of bevel gear table, 53.

Velocity ratio constant, 1 ; not aeffcted by increase of center
distance in involute, 28.

Weisbacli's "Mechanics," 64.

Willis, odontograph of, 57 ; writings of, 64.

Worm gearing defined, 3, 62; notation and formulas for, 67.

Worm wheel, 62.

Plate I.

Cycloid, Epicycloid, Hypocycloid and Involute curves.

ItEFEIlENCES TO TEXT,

Art. 4, Page 4. Art. 8, Pao-e 5.

5, ^ 4. 9, 5.

G, 5. 12, 7.
7, 5.

Plate 1

i__^

kl?"

"7^

^C^^l^^x

/ V^

\\L^

^0.

\ 1

/ \

/ 2V^ \ k^^^'

\ \^ 5

' \

i >^

i- 'xL

g~ — \-

- Kr

'f- /

---At

^^-\ — ~~~-

|l\v^^

r /

" \

/ 1

\ /^^ \

' \

"/ J

\ \7c I 1/ A^,^

1 ^

V!><

J^V^/

3/v K

^ 'e

B'"

^'0

FIG

/ ^~~^^7<^:i

-^^^Z""i

^y^-y^

/O^

^\ /B,ic^

1 /

C 1 /' /—

""^/V^^^^ / /Fig

2 /^rVr

-^/A^\

1 >^

1 /^ \

/ A

/ \

^^^"--^

[

f^~~Jl

n \ a""^^'^^'"' /

-^A.

I^nM-^

/
/

. N

^^,^#^A^/V]\^-4?^ p/

/
/

"""^ Jf

~\//^'\

\
\
'\
\

1 /
i /
j /
i /
\ i /' /

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■/// K

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Iky'- §'

^XXA"" P

Fig. 3

Plate 2.

Epitrochoidal curves. Double generation of Epicycloid and
Hypocycloid. Approximate method.

REFERENCES TO TEXT.

Art. 10, Page 6.
11, 6.

34, 22.

Plate 2

Plate 3.

Mechanical method for describing Odontoidal curves.

REFERENCES TO TEXT.

Art. 13, Page 8.
15, 10.

21, 12.

Plate 3,

Plate 4.

Cycloidal Gear, Pinion and Rack having action on one side of
pitch point. Limiting case.

Airr,

REFERENCES TO TEXT.

IT). Page 10.

Art.

21,

Page 11.

10, ' 10.

25,"

11.

18, 12.

26,

14.

19, 12.

36,

24.

20, 12.

47,

37.

23, 13.

LATE 4.

Plate 5.

Cycloidal Gear, Pinion and Rack having action on both sides
of the pitch point. Limiting case.

ItEFEEENCES TO TEXT.

Art. 23, Pao-e 1-3. Art. 3(3, Page 24.

20, ' 14. 41), 38.

28, 15. PK015. 2, 72.

32, 10.

Plate 5.

Plate 6.

Cycloidal Gear, Pinion and Rack. Practical case.

REFERENCES TO TEXT.

Art. 34, Page 21.

36, 24.

37, 25.

Plate 6.

Plate 7.

Involute Gear and Pinion. Limiting case. Mechanical method
for describing the Involute.

REFEREXCES TO TEXT.

Art. 38, Page 26.
39, 27.

42, 31.

Plate 7.

Plate 8.

Involute Gear, Pinion and Rack. Limiting case.

REFERENCES TO TEXT.

Art. 39, Page 28. Art. 41, Pao-e 30.

40, 29. Pror. 4, 74.

PLATE 8.

Plate 9.

One Pitch Involute Gear and Pinion, showing Interference.

REFERENCES TO TEXT.

Art. 42, Page 30. Art. 44, Page 33.

43, 32. 46, 35.

Plate 9.

1 PITCH INVOLUTE GEAR & PINION
SHOWING INTERFERENCE

Plate 10,

One Pitch Involute Pinion and Rack, showing Interference.

REFERENCES TO TEXT.

Art. 42, Page 30. Art. 44, Page 33,

43, 32. 46, 35.

Plate to.

RACK

1 PITCH INVOLUTE PINION & RACK
SHOWING INTERFERENCE

Plate II.

Annular Gearing.

REFERENCES TO TEXT.

Art. 49, Page 38. Ai;t. 52, Page 41.

50, 39. 54, 42.

Plate 11

Plate 12.

Annular Gearing. Special cases.

REFERENCES TO TEXT.

Art. 50, Page 39.

51, 40.

52, 40.

Plate 12

12 3

116 1 I . I

ililinlilililililililililihlililililililililihlililihiilil '

Plate 13.

Bevel Gearing.

REFERENCES TO TEXT.

Art. 50, Page 47. Art. 60, Page 48.

Plate 13

Plate 14.

Problems i to 4 inclusive.

EEFEREXCES TO TEXT.

Art. 65, Page 69. Peob. 3, Page 72.

Peob. 1, 69. 4, 74.

2, 71.

Plate 14

I \ /<^ DRIVER

t /

/ \ /o-

__B ^ ^/ \

-9/ \

.Fig. 1

^i/

RADIAL FLANKS
ON GEAR.

■Fig. 2'

/.--~\.. //•'

> I \ / A /

/ I / >-a nJ;

/ ^/ \^ -■ --

-6.334

'^^■^ch'linS.

qt/-

PITCH LINE OF RACK

._. Y^

/ r\

//y \ / NX/ft

-H-^-^: — # -^

^V / "-^ /7V\ /I

^v/

PITCH UNtlOF^RACK

Fig. 3

Fig. 4

Plate 15.

Problems 5 to 9 inclusive.

REFERENCES TO TEXT.

Art. 65, Page 69. Prop.. 7, Page 77.

Prob. 5, 75. 8, 78.

6, 76. 9, 79.

Plate 15

o^^^^?-^e!^.o,.

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