Tecl^nical Drawing Series 
 
 Essentials of gearing 
 
 ANTHONY 
 
 < ooG Looooo '^r-'^ ^
 
 S-^AT T.:/,Cf E3 s c L :<:e 
 SA..TA BARBARA. CALIFORNIA 
 ,^^9-?-^
 
 Technical Drawing Series 
 
 ANTHONY'S MECHANICAL DRAWING 
 
 ANTHONY'S MACHINE DRAWING 
 
 ANTHONY'S GEARING 
 
 ANTHONY AND ASHLEY'S DESCRIPTIVE GEOMETRY 
 
 DANIELS'S FREEHAND LETTERING 
 
 DANIELSS TOPOGRAPHICAL DRAWING 
 
 D. C HEATH & CO., PUBLISHERS
 
 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., Sc.D. 
 
 Professor of Drawing in Tufts College and Dean of the School of Engineering; 
 
 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 
 
 REVISED 
 
 D. C. HEATH & CO., PUBLISHERS 
 
 BOSTON NEW YOUK ClUCAGO
 
 Copyright, 
 
 Bt Gardner C. Anthony, 
 
 1897 AND 1311. 
 
 2 lO
 
 A 5 "'' lU^.1-0 
 
 Cjll PEEFAOE. 
 
 The most feasible method for the acquirement of a working knowledge of the theor}- 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 the 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 tliat the 
 plates are not intended for copies, but as illustrations. A definite la^'-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 the 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.
 
 iV PREFACE. 
 
 The methods and problems have already proved their usefulness in the instruction of stu- 
 dents of many grades ; and it is hoped that their publication may promote a wider interest in, 
 
 and more thorouerh study of, the essentials of gearing. 
 
 GARDNER C. ANTHONY. 
 Tufts College, Sept. 24, 1897. 
 
 PREFACE TO THE REVISED EDITION. 
 
 In revising The Essentials of Geaeing it seemed desirable to supplement the work by 
 
 an introduction which should better adapt it to tlie use of college students. In doing this the 
 
 chief aim of the author has been to present the general principles underlying the study of 
 
 velocity ratio, which should serve not only as an introduction to the study of Gearing, but as 
 
 supplementary to the many excellent treatises on Kinematics and Mechanism. It is in this 
 
 manner tliat the author makes use of the book in his classes. 
 
 GARDNER C. ANTHONY. 
 Tufts College, Jan. 2, 1911.
 
 co:n'te:nts. 
 
 Introduction i^ 
 
 I. Subject of Introduction. II. Velocity Ratio. III. Angular Velocity. IV. Instantaneous Center or 
 Centro. V. Determination and Proof of Instantaneous Center. VI. Centro of Rolling Curve. 
 VII. Velocity Ratio as Deterniined hy Intermediate Connectors. VIII. Velocity Ratio in Contact 
 Motion. IX. Conditionsof Constant Velocity Ratio. X. Flexible Connector.s. XI. Similarity 
 between the Three Modes of Transmission. XII. Sliding Action. XIII. Directional Rotation. 
 XIV. Positive Rotation. XV. Pure Rolling. XVI. The Three Modes of Transmission. 
 
 CHAPTER I. 
 
 General Principles 
 
 I. Constant Velocity Ratio. 2. Positive Rotation. 3. Gearing. 
 
 Odontoidal Curves 
 
 4. Classes of Curves 
 
 CHAPTER II. 
 
 5. Cycloid. 6. Epitycloid. 7. Hypocyclold. 8. To Construct a Normal. 
 9. A Second Method for Describing the Cycloidal Curves. 10. Double Generation of the 
 Epicycloid and Ilypocycloid. 11. Epitrochoid. 12. Involute. 
 
 CHAPTER 111. 
 
 Spur Gears and the Cycloidal System 
 
 13. Theory of Cycloidal Action. 14. Law of Tooth Contact. 15. Application. 16. Spur (Jears. 
 17. Circular Pitcli. 18. Diainotcr Pitch. 19. Face or Addeudinn. 20. Flank or Dedendum. 
 
 V
 
 VI CONTENTS. 
 
 PAGE 
 
 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 Govern- 
 ing the Practical Case. 31. Proportions of Standard Tooth. 32. Influence of the Diameter of 
 the Rolling Circle on the Shape and Efficiency of Gear Teeth. 33. Interchangeable Gears. 
 34. Practical Case of Cycloidal Gearing. 35. Face of Gear. 36. Comparison of Gears, illus- 
 trated in Plates 4, 5, and 6. 37. Conventional Representation of Sj^ur Gears. 
 
 CHAPTER IV. 
 
 Involute System 26 
 
 38. Theory of Involute Action. 39. Character of the Curve. 40. Involute Limiting Case. 41. Epi-- 
 cycloidal Extension 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 Exterior 
 and Interior Describing Curves. 53. The Limiting Values of the I^xterior, Interior, and Inter- 
 mediate Describing Circles for Secondary Action. 54. Practical Case. 55. Summary of 
 Limitations and Practical Considerations. 56. Involute System of Annular Gearing.
 
 CONTENTS. Vii 
 
 CHAPTER YL P^vGE 
 Bevel Gearing 45 
 
 57. Tlieory of Bevel Gearing. 58. Character of Curves Employed in Bevel Gearing. 59. Tredgold 
 Approximation. 60. Drafting the Bevel Gear. 61. Figiiring the Bevel Gear with Axes at 90°. 
 62. Bevel Gear Table for Shafts at 90°. 63. Bevel Gears with Axes at Any Angle. 
 
 CHAPTER vn. 
 
 Special Forms of Gears, Notation, Formulas, etc 57 
 
 64. Odontographs and Odontograph Tables. 65. The Three-Point Odontograph. 66. The Grant 
 Involute Odontograph. 67. AVillis's Odontograph. 68. The Robinson Odontograph. 69. TJie 
 Klein Coordinate Odontograph. 70. Special Forms of Odontoids, and Tiieir Lines of Action. 
 71. Conjugate Curves. 72. Worm Gearing. 73. Literature. 74. Notation and Formulas. 
 
 CHAPTER YITL 
 
 Problems 70 
 
 75. Method to be Observed in Performing the Pioblems — 
 
 PROBLK.-\r 1. Cycloidal f.imiting Case. Face or Flank Only 71 
 
 2. Cycloidal Limiting Case. Face and Flank 7^3 
 
 3. Cycloidal Gear. Practical Case 74 
 
 4. Involute Limiting Case 70 
 
 5. Involute Practical Cases 77 
 
 6. Cycloidal Annular Gear 7s 
 
 7. Involute Annidar Gear 79 
 
 8. Cycloidal and Involute Bevel Gears. Sliafts at 90° 80 
 
 9. Cycloidal and Involute Bevel Gears. Shafts at Other than 90° . . . .81
 
 THE ESSENTIALS OF GEARING. 
 
 INTRODUCTION. 
 
 I. This treatise deals with the principles governing the transmission of motion by gear- 
 ing. The introduction treats of the fundamental principles governing constrained motion in 
 one plane as it relates to the direction and relative velocities in link and contact mechanism, 
 and with special reference to gearing. 
 
 Those desiring a less mathematical treatment of the principles should omit the introduc- 
 tion, as the subsequent consideration of the subject is not necessarily dependent thereon. 
 
 The diagrams are lettered to correspond throughout and, in general, to accord with the 
 plates on gearing. 
 
 II. Velocity Ratio. In discussing the motion of the parts of a machine we are chiefly con- 
 cerned with their relative velocities, although the path described by a point, or points, may 
 in some cases be of equal or greater importance. 
 
 As the path of any point may be regarded as circular for any instant of time, we may com- 
 pare the velocities of connected parts by their respective revolutions, linear velocities of 
 peripheries, or the angles described in a unit of time. The I'atio between two such velocities 
 is called the velocity ratio, and the study of gearing relates chiefly to tlie methods used for 
 maintaining a constant ratio of velocities between two shafts while transmitting a positive 
 rotation.
 
 IXTRODUCTION", 
 
 III. Angular Velocity is the term used to exjiress the velocity of a rotating bod}'. It may 
 be expressed as follows : 
 
 1. Number of revolutions per unit of time; as 120 rev. per n 
 
 2. Number of degrees per unit of time ; as 720° per sec. 
 
 3. Number of radians per unit of time ; as 12.56 radians per sec. 
 These values are the same for all points in a revolving body. 
 The radian is a unit for angular measure. It is an arc of linear 
 
 measure equal to the radius, and subtends 57.3°, 
 
 hence there are 2 tt radians in 360°. A rotating 
 
 body making N revolutions per unit of time has 
 
 an angular velocity (co) of 2 7rN radians. This 
 
 , , linear velocitv 'IrrrH v v' 
 IS equal to 
 
 = - = - = «, 
 
 LINEAR VELOCITY OF A 
 
 Fig:. I. 
 
 radius r 
 
 V and I' ' being the linear velocities of points at ;• and r' radial distance from the center of rotation. 
 
 Since co is the same for all values of r. it is equal to - and, therefore, numerically equal to the 
 
 linear velocity of a point at unit distance from the center. 
 
 If the scale of velocities in Fig. I be 100 ft. per minute per inch, the number of radians 
 
 will be - = ^^ = 2G6.6. and the number of revolutions per minute would be — = " = 42.5. 
 r .<o '■ 2-77 6.28 
 
 Example. A disk 60" diameter makes 300 revolutions per minute, (a) What is its 
 
 angular velocity ? (J) What is the linear velocity of a point in its circumference ? 
 
 SoLUTiox. (a) tfj = 2 ttN = 6.28 x 300 = 188-4 radians per minute. (6) i' = &)r=1884 x 2| 
 
 = 4710 ft. per minute.
 
 IXTKODUCTIOX. 
 
 XI 
 
 IV. Instantaneous Center, or Centro. All bodies in motion may be considered as rotating 
 about some axis which can be determined for any instant of time. This axis is called the 
 instantaneous axis, or axode, and, being known, enables one to determine the direction of 
 motion and velocity of any point in a body, since the angular velocity is constant for all 
 points in the same body. If tlie motion considered is in a plane, the rotation takes place 
 about a point known as the instantaneous center, or centro. 
 
 EK, Fig. II, is a line having motion in the })lane 
 of the paper, point E moving in the direction ER with /'\ 
 
 a velocity equal to ER, and point K moving in the ^ z^-'"' \ ' 
 
 direction KL with a velocity to be determined. Point -{- — ^- — ^^'-\ ^"^ ' 
 
 E may be considered as rotating about some point in l\ j^^ J>^ I 
 
 EO which is perpendicular to ER at E, since the 
 direction of motion of any point in a rotating body 
 
 is perpendicular to the radius at tliat point. Simi- /^^T ^ ^ 
 
 larly K may be considered as rotating about any ^^' 
 
 point in KO perpendicular to KL at K. Their point of intersection will be common to both 
 radii, and therefore the center of rotation of EK for the instant ; hence, the instantaneous 
 center, or centro. According to Art. Ill, all points in the line will have an angular velocity 
 
 ^ V ER KL , 
 
 equal to - = — = — ; lience KL 
 ^ r OE OK 
 
 ER 
 
 OE 
 
 X KO. KL may be graphically obtained by laying off OE' 
 
 on OK, equal to OE, and drawing E'R' perpendicular to OK and equal to ER. The intersection 
 of the radial OR' and KL will determine the velocity of point K, which is KL. 
 
 Again, if we determine EH, the longitudinal component of ER, it must equal KM, the longitu- 
 dinal component of KL, otherwise the line EK would not maintain a constant length.
 
 Xll 
 
 INTRODUCTION. 
 
 V. Determination and Proof of the Instantaneous Center. Fig. III. EK is an 
 and inexteusiblc rod. KL indicates the motion of point K in direction and intensity. 
 ER indicates the motion of point E, the longitudinal components of l 
 
 ER and KL, which are EH and KM, being eqnal. 
 
 Draw IT, and Z will be the point about wliich the rod tends to 
 rotate by reason of the side components Kl and ET. YZ is the 
 horizontal component of point Z and, in common Avith all points 
 in the rod, is equal to EH and KM. B'rom K draw KG perpendicular 
 to KL, and from Z draw ZO perpen- 
 dicular to YZ. If the angle LOK 
 equals the angle YOZ, points Z and 
 K will have the same angular 
 velocity about (Art. Ill), and 
 hence will be the instantaneous 
 center. 
 
 inflexible 
 
 Similarly 
 
 = — , and 
 
 oz 
 
 Proof. In the similar triangles LKl and KOZ, — = — , but IL = MK = YZ, hence 
 therefore angle LOK is equal to angle YOZ. q.e.d. 
 
 Similarly angle ROE = angle YOZ = angle LOK, hence must be the center about which all 
 points in the rod EK revolve with equal angular velocity. 
 
 VI. If a circle, or other curve, rolls on a right line, or curve, its centre at any instant 
 will be the point of tangency of the rolling and fixed curve, since this will be the only point
 
 INTRODUCTION. 
 
 Xlll 
 
 of the circle whicli is at rest. In Fig. IV the rolling circle MAO has its centro at 0, its point 
 of tangency with the line on which it rolls. This may be determined by finding the direc- 
 tion of motion of any two points in the 
 curve, such as A and R, and drawing per- 
 pendiculars from these points. Their 
 intersection will be the instantaneous 
 center of rotation, or centro. If CD be the 
 velocity of translation, it will also be the 
 linear velocity of all points in the circum- 
 their rotation about the center C. Point A 
 will, therefore, have components of velocity AF and AE 
 equal to CD, and point R will have components RT and 
 RV also e(|ual to CD. Perpendiculars to the resultants 
 AB and RS from points A and R will intersect at 0, the 
 centro. 
 
 Again, if points C and M be taken, the centro will be 
 determined as follows : since both points move in horizontals, the point of intersection of the 
 perpendiculars from tliese points will be indeterminate, and other means must be emploj-ed 
 to determine the centro. The velocities of points C and M will be CD and MN. The latter is 
 the resultant of two liorizontal components equal to CD, one being the circumferential velocity, 
 and the otlier the velocity of translation common to all points of the circle. Therefore, 
 
 MN = 2CD; but = , hence must be the point about which rotation takes place. 
 CD CO 
 
 Fig-. IV.
 
 XIV 
 
 INTRODUCTION". 
 
 VII. Velocity Ratio as Determined by Intermediate Connectors. F and G, Fig. V, are the 
 fixed centers of the revolving arms 1 and 2, their free ends being connected by the link EK. 
 It is required to determine the velocity ratio between 1 and 2 for the instant shown. 
 
 Peoof. Let ER be the graphic rejDresentation of the linear velocity of the driving arm 1, 
 the longitudinal component of which is EH. Since the length of the link is constant, the 
 
 Fig. V. 
 
 longitudinal component of the motion of K in arm 2 must be KM = EH, and the linear velocity 
 
 ER 
 will be KL. Let co^ and co.^ designate the angular velocities of arms 1 and 2, then Wj = — and 
 
 cOr,= — . From tlie similar triansrles EHR and FCE, — = — Also from the similar triangfles 
 - GK ^ EF FC ^ 
 
 1 ^.,„ KL KM EH 1 CO, ER GK EH GN GN ^^ f .^ • -i 
 
 KML and GNK, — = — = — , hence -^ = — X — = — x — = — Also from the similar 
 GK GN GN &)o FF KL FC EH FC
 
 INTRODUCTION. 
 
 XV 
 
 triangles AFC and AGN, = — Hence the general law: The angular velocities of revolving 
 
 arms with an intermediate connector are inversely proportional to the segments into tvhich the line 
 of action divides the line of centers. 
 
 This law may be obtained also by using the centro in connection with centers F and G. 
 Let ©3 designate the angular velocity of E and K about the instantaneous center 0. Di'aw OZ 
 perpendicular to the line of the link EK and observe the similarity of triangles EHR and OZE, 
 
 also of KML and OZK. 
 
 ER 
 EF 
 
 KL 
 GK 
 
 ER KL 
 
 (Wo = — = 
 
 OE OK 
 
 "3 
 
 ER OE 
 
 - X — 
 
 EF ER 
 
 ^ — 
 
 KL OK 
 
 ^3 
 
 OK_OZ 
 <»3 GK KL GK~GN 
 
 If the center line of the link 
 intersects the line of centers 
 between the fixed centers, as 
 in Fig. VI, the only change in 
 the conditions will be in tlie 
 direction of rotation of arms 1 
 and 2. In Fig, V they are 
 alike and in Fig. VI they are 
 opposite. As tlie same letter- 
 ing is used in both figures, the 
 demonstration may be applied 
 to either. 
 
 <y, ft), 0). OZ GN GN 
 
 hence -^x— ^=— !^= — x — = 
 
 which is equal to 
 
 OE 
 EF ' 
 AG 
 
 OZ 
 FC 
 
 Also
 
 XVI 
 
 INTRODUCTION. 
 
 VIII. Velocity Ratio in Contact Motion. Fig. VII. F and G are the fixed centers of 
 revolving arms 1 and 2 having curved surfaces in contact. Motion is imparted from 1 to 2 
 by contact between the curved ends, the direction of rotation being indicated by arrows. 
 It is required to determine the velocity ratio at the instant shown, and to deduce tlie general 
 law. 
 
 Fi&. VII. 
 
 At the point of contact B draw the common tangent BT, and the normal BW, Let Bl, per- 
 pendicular to BF at B, represent the linear velocity of point B in arm 1, the angular velocity 
 
 being 
 
 Bl_ 
 
 bf" 
 
 Resolve Bl into the tangential component BV and normal component BW. The 
 
 direction of motion of point B in arm 2 will be perpendicular to BG, and the normal 
 component BW will be the same as for point B in arm 1. If this were not so, arm 1 would 
 recede from, or compress, arm 2. The tangential component must lie in BT and the value 
 
 BS 
 
 of the resultant motion will be BS, the angular velocity of arm 2 being 
 
 BG 
 
 The ratio between the angular velocities of arms 1 and 2 will be -^ = — x — 
 
 &j., BF BS 
 
 From the
 
 INTRODUCTION. 
 
 XVU 
 
 BW 
 
 BG GN 
 
 similar triangles IBW and BFC, —='f^; also from the similar triansrles SBW and BGN, — = 
 
 BF FC ^ BS BW 
 
 Substituting in the above, 
 
 BW GN GN , ^ ^ ^1^-1 1 GN AG 
 
 — X — = — , but from the triangles AGN and AFC, — = -^ 
 PC BW FC ^ FC AF 
 
 Hence the general law: The angular velocities of revolving arms in contact motion are in- 
 versely proportional to the segments into which the common normal divides the line of centers. 
 
 In Fig. VII the contact is such 
 that the arms rotate in the same 
 direction, but if the rotation be 
 opposite, the conditions may be 
 similar to Fig. VIII, which is let- 
 tered to correspond with Fig. 
 VII, and the same demonstration 
 may be employed. 
 
 IX. Condition of Constant Velocity Ratio. From V^ 
 the above law it will be seen that any change of point | 
 
 A will change the velocity ratio, hence : To preserve a constant velocity ratio the common normal^ 
 or line of action^ must always cut the line of centers in the same point. 
 
 Fig. VIII. 
 
 X. Flexible Connectors. If pulleys be substituted for the revolving arms of the link 
 mechanism in Fig. V, and if a flexible connector such as a belt, rope, or chain be used to re- 
 place the link, the velocity ratio may be maintained constant. And if the ratio of the pulley 
 
 diameters be that of — - or — , the velocity ratio will be the same as that of the link mecha- 
 FC AF
 
 XYUl 
 
 INTRODUCTION. 
 
 nism. Figure IX illustrates this case. FG is the fixed link, FC and GN (the radii at the point or 
 tangency with the flexible link) are the rotating arras, and CN the connector link. If 1 be the 
 driver, the driving side of the belt, or flexible connector, will be CN', but in other respects 
 the action of this mechanism, for the instant, does not differ from that of Fig. V and Fig. VII. 
 Figure X illustrates a case analogous to that represented by Fig. VI and Fig. VIII 
 
 C 
 
 STRIVING ^ 
 
 Tig. IX. 
 
 XI. Similarity between the Three Modes of Transmission. Figure XI illustrates a combina- 
 tion of Figs. V, VII, and IX. It will be observed that the center line of link EK, the normal 
 component BW, and the line of the belt CN all coincide, and that the common normal, or line 
 of action, intersects the line of centers AG in the point A, thus determining the common ve- 
 
 locity ratio, — ' = — 
 &)„ AF 
 
 Only one of these, the pulleys, will surely maintain a constant velocity 
 
 ratio, although the direct contact mechanism may be made to do this by so shaping the 
 contact curves that their common normals will continue to pass tlirough point A.
 
 INTRODUCTION. 
 
 XIX 
 
 ■Fig. XI.
 
 XX INTRODUCTION. 
 
 XII. Sliding Action. Figures XII, XIII, and XIV illustrate the sliding action taking place 
 between surfaces of contact. They are lettered to correspond with the preceding, and in 
 
 each case the velocity ratio of driver to follower is _. The velocity of the drivers at the point 
 
 AF 
 of contact is Bl and of the followers, BS ; the common normal component is BW, and the tan- 
 gential components of driver and follower, on which the sliding action depends, are BV and BT. 
 It will be observed that in Fig. XII the tangential components, BV and BT, act in opposite 
 directions, and the amount of sliding action will equal BV + BT. In Fig. XIII they act in the 
 same direction, and the total action is BV — BT. In Fig. XIV they are equal, and alike in di- 
 rection, and therefore no sliding action takes place. If rotation in one direction be considered 
 positive, and the opposite direction negative, the sliding action is equal to the algebraic difference 
 of the tangential components. 
 
 XIII. Directional Rotation. The relative direction of rotation is easily determined by 
 observation, but since all phases of the action can best be referred to the line of action, the 
 following law should be observed : The directional rotation of driver and follower are alike when 
 the fixed centers lie on the same side of the line of action^ and opposite if the centers lie on opposite 
 sides of the line of action. 
 
 XIV. Positive Rotation. If one cylinder transmits motion to a second by contact, we 
 know that motion is not positive, since it is dependent on the friction due to the imper- 
 fection of the contact surfaces. In the cases illustrated by Figs. XII, XIII, and XIV, a 
 positive rotation is apparent in the positions shown, but by continuing the rotation, a point 
 will be reached at which the driver will cease to move the follower. This will occur only
 
 INTRODUCTION. 
 
 ZXl 
 
 when the driving contact point has no com- 
 ponent in the direction of the normal, or line 
 of action. In this case Bl will coincide with 
 the tangent at the point of contact, and the 
 direction of the normal will be through the 
 center F. The dotted position of the contact 
 mechanism in Fig. XIII represents this phase, 
 and the normal B'F passes through the center 
 of the follower, since the normal component is 
 always perpendicular to the common tangent 
 of the contact surfaces. Whenever this con- 
 dition occurs, positive driving will cease and 
 the law may be expressed as follows: 
 
 Positive rotation can take place only when 
 the common normal^ or line of action^ does notjMSS 
 throiKjh the fixed center of driver, or follower. 
 
 XV. Pure Rolling. From the considera- 
 tion of Fig. XIV, in Art. XII, it was observed 
 that there was no sliding action when the tan- 
 gential components were alike in magnitude 
 and direction. Tliis can take place only when 
 
 Fig. XIV.
 
 XXU INTRODUCTION 
 
 the contact is on the line of centers. If such contact is maintained, the result will be pure 
 rolling such as exists between two cylinders, or circles. 
 
 Figures XV, XVI, XVII are examples of pure rolling. Fig. XV illustrates a pair of 
 logarithmic spirals rotating about centers F and G, with contact at B. The character of these 
 curves is such that the angle between the tangent and radius at any point is a constant ; the 
 locus of the points of contact will be FG, and there will be pure rolling with positive rotation, 
 but not a constant angular velocity ratio. As these curves are not closed, they cannot be 
 used to transmit continuous motion. 
 
 Figure XVI illustrates two elliptical curves revolving about their foci. The contact will 
 always be on the line of centers FG, and therefore pure rolling will take place. Tlie angular 
 velocity ratio will not be constant, but there will be positive rotation save when the major 
 axes coincide with the line of centers, at which time the common normal will pass through 
 the fixed centers. Art. XIV. 
 
 Figure XVII differs from the preceding in that pure rolling is accompanied with a constant 
 velocity ratio, but we have lost the positive rotation. 
 
 XVI. The Three Modes of Transmission. Constant velocity ratio, positive rotation, and 
 pure rolling cannot be maintained at the same time in any one mechanism. Only two of the 
 three conditions are possible. Figs. XV, XVII, XVIII illustrate these combinations. In 
 Fig. XV there is positive rotation and pure rolling, in Fig. XVII there is constant velocity 
 ratio and pure rolling, and in Fig. XVIII there is a constant velocity ratio and positive 
 rotation. The last condition is the one to be maintained in the design of gear teeth.
 
 INTRODUCTION. 
 
 XXlll 
 
 Fig:. XVIII. 
 
 Fig. XVII.
 
 THE ESSENTIALS OF GEARING. 
 
 CHAPTER I. 
 
 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 invei-sely 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 constant 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 suital»le 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 positive rotation while preserving the condition of 
 constant velocity ratio. 
 
 1
 
 GEARS CLASSIFIED. 
 
 Fig. 1. 
 
 Fig. 2. 
 
 Fig. 3. 
 
 Fig. 4. 
 
 Gears may he classified, as fol- 
 lows : — 
 
 1. If tlie shafts are parallel, 
 the friction surfaces would be 
 cylinders (Fig. 1), and the gears 
 designed to produce the same 
 condition, as to the velocity, are 
 called Sjjur Grears (Fig. 2). 
 
 2. If the shafts intersect, the 
 friction surfaces would be cones 
 (Fig. 3), and the gears called 
 Bevel Grears (Fig. 4). 
 
 3. If the shafts are .neither in- 
 tersecting nor parallel, the friction 
 surfaces will be hyperboloids of 
 revolution (Fig. 5), and the gears 
 called Hyperlolic, or Skew Crears 
 (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 eitlier of tlie first three cases 
 be made lielical, an entirely dif- 
 ferent class of gearing will result. 
 The various forms are known as 
 Twisted^ /Spiral, Worm, and Screw 
 G-earing (Figs. 7 and 8). The 
 action of the latter is analogous to 
 that of a screw and nut. 
 
 One of these forms is generally 
 employed as a substitute for hy- 
 perbolic, or skew geare, by reason 
 of the difficulty experienced in cor- 
 rectly forming the teeth of such 
 gears. 
 
 5. Another, although but little 
 used, form, is tliat known as Face 
 G-ear'uKj. The teeth are usually 
 pins secured to the face of cir- 
 cular disks having axes perpen- 
 dicular. The action takes place 
 at a point only. 
 
 None of the latter forms can 
 be represented by friction surfaces. 
 
 Fig. 7. 
 
 Fig. 8.
 
 4 ODONTOIDAL CURVES. 
 
 CHAPTER 11. 
 
 ODONTOIDAL CURVES. 
 
 4. The two classes of curves coninionly employed in gear teeth are the cycloidal and the 
 involute. A knowledge of their characteristics and methods of generating is essential to an 
 undei-standing of their application in gearing. 
 
 5. Cycloid. Plate 1, Fig. 1. The cycloid is a curve generated by a point in the 
 circumference of a circle which 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 right. 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 new position of describing circle. 
 The generating point nmst lie in this circle at a distance from C ec|ual 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 
 normal, of the curve at B', it being a perpendicular to the tangent of the curve at this point.
 
 CYCLOIDAL CURVES. 
 
 The normal at B" would be B" D'. The radius A' B' is known as the describing^ or generating 
 radius^ and A' C is the contact radius, or the radius at the point of contact. In like manner 
 other positions of the describing point may be found, and the curve connecting them will be 
 the cycloid required. 
 
 6. Epicycloid. Plate 1, Fig. 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 describing 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 the inside of a cir- 
 cle, the curve generated will be an hypocycloid. Fig. 3 illustrates this curve, the same letter- 
 ing being used as that of the preceding cases. 
 
 If it be required to draw a normal at any point of this, or the two preceding curves, the fol- 
 lowing method may be 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 the 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 hypocycloid A L C.
 
 6 CYCLOIDAL CURVES. 
 
 To describe the epic3'cloid, assume any point, D, on the generating circle, and la}- off the arc 
 A D' on llie director circle, making it equal to arc AD. 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'. The 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 intei-secting 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 A\"hich the hypocycloid is a radial line, A L C, and this is due 
 to the diameter of the 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, commonly 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 the geometrical demonstratiou of this problem see the appeudix of Professor MacCord's " Kinematics," page 319.
 
 INVOLUTE CURVE. 7 
 
 is described; and if the point be within, as at D, tlie curve will be a prolate epitrochoid, 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 intei-sect- 
 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 prolate epitrochoid. 
 
 12. Involute. Plate 1, Fig. 4. The involute is a curve generated bv 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, preferably 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 tano-ents will be an involute of the base circle A B C D.
 
 8 THE CYCLOIDAL SYSTEM. 
 
 CHAPTER III. 
 
 SPUR GEARS AND THE CYCLOIDAL SYSTEM. 
 
 13. Theory of Cycloidal Action. Plate 3, Fig. 1. Let 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 fii"st 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 wdth A wlien 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 
 traverse equal spaces in equal times, C will have revolved to C", and the curve A' C will now 
 occupy the position E" C". But, since the marking point has continued to describe a curve upon
 
 CYCLOIDAL ACTION. U 
 
 the flange of disK 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 be rolled on the cir- 
 cumference H K, the marking })()int would describe the same curve, A'" 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' k" A'" will equal the arc B B' B" B'" ; and if, in a manner similar to 
 the 2)i"eceding, 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 being traced at the same time b}- the 
 describing point A. If we now observe any special position of the point, as A", it will be seen 
 to be common 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 common normal to tlie curves at any point of contact will pass through the 
 point A (the pitch pointy. 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 AND SPUR GEARS. 
 
 14. Law of Tooth Contact. In order to preserve the condition of constant velocity ratio, 
 the tooth outlines which act in contact must be such that the common normals at the point of 
 contact shall always cut the line of centers in the same point; and in general, the curves must 
 be sucli 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 teeth, shown 
 in Fig. 2, Plate 3. Let 1 be the driver, and suppose motion to begin from the position shown 
 in the figure, the contact being at A, As the motion takes i)lace, 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 j)eriphery, or pitch line, of 2. But this uniform 
 motion must cease when points A'" A'" 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 either 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, wliich described 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 reiiiiiiiider of the pinion flank may be a continuation of the Iiypocych>id, or any 
 otlier curve which may not interfere with the action of the gear tooth. The opposite sides of 
 the teeth are unulc alike in order that motion may take pkice in cither direction. If tlie direc- 
 tion l)e that indicated hy the arrows, tiie pinion hcing the driver, the .shaded side of the teeth 
 would have contact ; and if the direction he 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 l>y the 
 center F, from which a circle should be drawn tangent to the line of the scroll. ]Mark 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 tangenl, 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 i)itch line between cor- 
 responding jjoints of consecutive teeth, is called the circular pitch, and is equal to the 
 
 circumference of pitch circle 
 number of teeth 
 
 Let P' denote the circular pitch, D' the diameter of the pitch circle, and N the number of 
 teeth, then will P' = '^ (1), and, ~, = ^, (2). 
 
 18. Diameter Pitch. Tn order to express in a more direct and simple 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 tee.th, it has been found expedient to apply 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 7iot an ahsoJufe measure, but 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 by P, P = q"/ (3). 
 
 To obtain the relation between the diameter pitch and the circular pitch, compare formulas 
 2 and 3. -, = —,, ^, = P I hence 5; = P or P P' = tt (4). This last equation expresses the relation 
 between the two pitches in a simple form which may be easily remembered. 
 
 Illustration. — The pinion represented in Plate 4 has 12 teeth, and is 3 inches in 
 diameter. 5; = P, ^ ^ '^' ^^^ pitch, therefore, is 4. The circular pitch, P' = ^ = — ^ = -7854. 
 Having given any two of the terms, N , D', P, P', the other terms may be determined. 
 
 19. Face, or Addendum. That portion of the tooth curve lying outside of the pitch circle 
 is called the face or addendum, as C D, Plate 4. 
 
 20. Flank, or Dedendum. That portion of the tooth curve lying inside of the pitch circle 
 is called the flank or dedendum, as E H , Plate 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' A" A'". This is called the path of contact, or line 
 of action, and in the c^xloidal system this line is an arc of the describing circle.
 
 ARCS OF CONTACT. lo 
 
 22. Arc of Contact. The arc described by a point on the pitch line 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 Ije 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 practice it should be greater, which would be 
 accomplished by lengthening the addendum. Plate 4. 
 
 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 first and third cases the contact takes place from the pitch point, and the arc 
 described by a point on the pitch line during tliis action is called the arc of recess. 
 
 In the second and fourth cases the contact takes place toward the pitch point, ending at A, 
 and the arc described by a point on the pitch line during this action is called the 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 PRESSURE. RACK. 
 
 24. Angle of Obliquity, or Pressure. The angle which the common 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 PLx^te 4 has teeth only on one side of the pitch line, like the pinion and 
 gear, and the conditions of action are simikir. 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, the addendum of gear and dedendum of pinion will be the same as those of Plate 4. 
 This case, liowever, differs from the preceding in tliat the number of teeth is but lialf as great, 
 and therefore the pitch will be doubled. This will require tlie arc of action to be doubled, in 
 order that it shall equal the pitch (Art. 22, page 13). Such increase in the arc of action may 
 be made b}' 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 circuhir 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
 
 CURVE OF LEAST CLEARANCE. 15 
 
 b«!coiiic the addiMidum of tin; piuiuii, and tliu latter the dedendum of tlie gear tooth. The en- 
 gaging gears Mill then lune l)oth faces and flanks, the action will begin at C and end at H, the 
 path of contact will be C A H, the arc C A being tlie path of approach, and A H the path of 
 recess, their snm being ecpial 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 will end at 0, N M being the path of 
 approach, and M the path of recess. That portion of the dedendum of rack tooth which 
 engages the addendum of the pinion is indicated by 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 tlie gear be rolled on that of the 
 pinion, and the epitrochoid of the liighest point, C, of the gear tooth l)e determined, it will l>e 
 the curve of least clearance. 
 
 The successive positions of the tooth, \\ hen so revoh cd, ai-e shown by the dotted line in 
 Plate 5, and tlie line connecting these points would be the desired curve. This may be 
 obtained as follows: .Vssume any point, R on the pitch circle of pinion, and lav off arc 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 deseri])e other ares, and draw a curve toui'hing these arcs on the inside. This 
 curve will be the curve of least clearance.* 
 
 * See also the method of Art. 71, page 03.
 
 16 CONDITIONS GOVERNING THE 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 curve rolling in contact with their respective pitch circles. (Art. 14, 
 page 10.) Practical considerations limit the diameter of the describing circle to a maximum 
 of about -z-5 or equal to the radius of the pitch circle, and a minimum of about 1| P', or -^. 
 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. 
 
 3. The addendum of a gear tooth engages the dedendum of the pinion, and the action 
 between them either l)egins 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. 
 
 17 
 
 The approaching" action being more detrimental, 
 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 that no 
 action may take place. 
 
 31. Proportions of Standard Tooth. The propor- 
 tions most commonly accepted for cut gears are those 
 illustrated in Fig. 9. The dimensions are made depen- 
 dent on the pitch, as follows : — 
 
 Addendum, (S) 
 Dedendum, (S + f) = 
 
 diam. pitch 
 I 
 
 diam pitch 
 
 -f- clearance 
 
 + f. 
 
 Thickness, (t) = ^ circular pitch = y = ^ 
 
 Clearance, (f) = -addendum = |^ 
 
 -^ or, f = - thick- 
 
 t P' 
 ness of tooth = ^ = ^ 
 
 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 CIPvCLE. 
 
 tooth must be slightly smaller than the space to permit of freedom hi action. If there be 
 any backlash the value of t will be '"'"'"^" '"^"^ ~ '^""'^'''^^ . In rough cast gears the backlash may 
 be as great as j^^jth the circular pitch, but this amount is very evcessive. It is, however, in- 
 consistent to base the values for backlash, or clearance, on the i)itch, since an increase in tlie 
 size of the tooth, or pitch, does not necessarily mean a proportional increase in the allowance 
 to be made for the inaccuracies of workmanship. Indeed, both these clearances must 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-seventh of the space between the 
 teeth, measured on the addendum circle, as in Fig. 9. The 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. But 
 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 commonly made equal to the radius of the pitch circle within which it rolls. As was shown 
 in Art. 9, page 5, this will generate a radial flank. In the case of geai"S 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 the}- roll. If the force to be transmitted be irregular, and the
 
 INFLUENCE OF THE ROLLING CIRCLE. 
 
 / 
 
 teeth required to sustain suddeu strains, it is better that 
 
 the flank be made wider at the dedenduni circle, and a 
 describiug circle chosen of a diameter sufficiently small 
 to produce the desired result. 
 
 In general, the diameters of the describing circles 
 
 D' 5 
 
 will lie between the values of -7^ and - . The second 
 
 value was used for the describing circle of the efears 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 
 su[)posed 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 by 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. Tlie 
 arcs C P and P D constitute tlie path of contact, and the 
 angle C P H is the angle of pi'cssure. 
 
 Next consider the describing circle as increased, its 
 
 Fig. 10.
 
 20 
 
 INTERCHAXGEABLE GEARS. 
 
 91 
 
 4 
 
 D' 
 2 
 
 38 P' 
 
 1.35 P'. 
 
 32.3° 
 
 20.3°. 
 
 Fig. 11. 
 
 diameter being equal to one-half of the diameter of the pitch 
 circle. The form of the tooth ^vill 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, 
 
 Maximum angle of pressure. 
 
 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 tlie dedendum of engaging gear tooth, it follows that for 
 any system of interchangeable gears, the addenda and de- 
 denda of all teeth must be described by the same describing 
 curve. It is also necessary that the pitch, and proportion 
 of the teeth, be constant.
 
 PRACTICAL CASE. 
 
 21 
 
 In practice, it is common to regard gears of twelve or of 
 fifteen teeth as tlie base 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 w ith 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 
 
 N 
 
 12 
 
 pitch circle will be D' = - = -— , and the diameter of the de- 
 scribing circle will be ^ = □• 
 
 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 will 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 pitch 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 GEARS. 
 
 pitcli diameters will equal p = t = ^ inches, and □ =-^ =4i^ inches (Art. 18, page 12). If the 
 
 tooth he of standard dimensions, the addendum and dedendnm lines may he determined and 
 drawn hj Art. 31, page 17. The diameter of the rolling circle is assumed to he 1^- inches for 
 the addendum and dedendum of hoth gears. Since the teeth should usually he shown in con- 
 tact at the pitch point, suppose the generating point of the describing curve to he at this point, 
 and describe the curves by rolling the circles from this position, first on the inside of one pitch 
 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 6. They may be drawn by 
 the methods of Arts. 6 and 7, page 5, or by Art. 9, page 5, but care should be used to draw 
 them in their proper relation to each other, as shown in the figure, so that it may not be neces- 
 sary to reverse the curves in order to incorporate them into tooth forms. The order for the 
 drawing of the curves may 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 simjjle 
 method which closely approximates the curves of this system, and suffices for the ordinary 
 drawing of a gear, but in no case should be used for describing the curves for a templet. This 
 method consists, frst, 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; 
 second, in the finding of a center on this normal, such that an arc ma}" be 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 line, equal to - A P, through which the arc
 
 PRACTICAL CASE. 23 
 
 B E is drawn. When the point E of the descn])ing curve sliall have become a point of contact, 
 as at E', the arc E' P being erjual to E P, the point P Avill have moved to T, the chord T E' being 
 equal to the chord E P. T will })e a point in the addendum, and T E' the normal for this point. 
 From a point, M, on this normal, and found by 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, di-aw 
 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(], 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 ; Ijut if an amount be determined for backlash, the thickness will equal 
 
 P^— backlash 
 2 
 
 The circle of centers liaving been drawn, the tootli curves should be described. These will 
 be limited l)y tlie addendum and dedendum circles already drawn. Finally draw the fillets. 
 
 The maxinuim angle of pressure between the pinion and gear will be 24°, the arc of 
 approach .52, the arc of recess .48, and their sum, Avliich is the arc of contact, 1 inch, or 1.27 
 times the circular ])itch. 
 
 The rack teeth would be similarly described. The })itch line l)einga 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 approximate the addenda and dedenda of the teeth.
 
 24 
 
 GEAR FACE. 
 
 ' 
 
 y/w, 
 
 P 
 
 m 
 
 m 
 
 m 
 
 ^////., 
 
 
 
 'mmm 
 
 ^ 
 
 ^ 
 
 
 ,_> I, 
 
 
 
 
 Fig. 14. 
 
 Fig. 15. 
 
 35. Face of Gear. In the previous consideration of 
 gear teetli no attention has been paid to the width of the 
 gear, or, as it is commonly termed, the face of the gear. 
 This dimension is one of the factors to be considered in 
 determining the strength of the tooth, which is a subject 
 apart from the kinematics of gearing. It should be ob- 
 served, however, that the tooth having appreciable width, 
 must 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 the 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 ; but the pitch of the latter is twice as great 
 as the 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.
 
 CONVENTIONAL REPRESENTATION OF SPUR GEARS. 25 
 
 The number of teeth in the pinions of Plates 4 and 6 is the same; but in the former the 
 action is only on one side of the pitch point, there being no addenda 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 enal)les 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 necessar}' to show 
 a full face view of the gears, the method illustrated in Fig. 15 may be employed to advantage. 
 This is simply a S3^stem of shading ; and no attempt is made to represent the proper number of 
 teeth, or to obtain their projection from another view.
 
 26 INVOLUTE SYSTEM. 
 
 CHAPTER IV. 
 
 INVOLUTE SYSTEM. 
 
 38. Theory of Involute Action. If the describing curves be 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 ilhistrated l)y 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 onl}- 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 the disks to be connected by a perfectly flexil)le and inextensible l)and, D C B A, wliich 
 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 the band, shall have moved to B, the curve X^ B will 
 
 * For description of this metliod, see MacCord's Kinematics, page 105, AuT. 279.
 
 THE INVOLUTE CURVE. 27 
 
 Juive Ik'C'H described on the extension of disk 2, and B Aj, on that of disk 1. Wlien the motion of 
 the nuuking- point sliall have continued to C, X2 Yj C will have been described on the extension 
 of disk 2, and A.^ B^ C, on that of disk 1. Finally, when the maiking point shall have reached 
 D, the curve X3 ¥3 Z^ D will ha\e lieen described on the extension of disk 2, and A3 B., C^ D on the 
 extension of d'sk 1. 
 
 If these curves be made the outlines of gear teeth, and the former act against 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, 
 A3 D being an involute of the periphery of disk 1, and Xg D, an involute of disk 2. The curves 
 may, therefore, be described by the method for drawing an involute (Airr. 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 A3; or the line A D may be conceived as wrapped about the base circle beginning the curve 
 at D. 
 
 39. Character of the Curve. Plate 7, Fig. 1, rejiresents 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 wmII now be 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 centere and the line of action), 
 the pitch circles will be drawn through this point. 
 
 The circles from which the invf>lute curves are described, are called base 
 
 Base Circle Defined. , , . .•,. , , . ,, iiT 
 
 nwr/es. 1 heir tliametei"s bear the same ratio to each other as do the diam- 
 eters of the pitch circles.
 
 28 CHARACTER OF THE INVOLUTE. 
 
 The Path ot Contact '^^^'^ ^"^® ^^ actioii, 01' ptitli of coiitact, is a right line tangent to the base cir- 
 
 a Right Line. qIq^^ j^ i,s the line followed by the marking point of the model. Plate 7, Fig. 2. 
 
 Since the path of contact is a right line, and as the common normals at the point of con- 
 
 constant Angle of ^'^^^ miist alwajs pass through the pitch point (Art. 14, page 10), it fol- 
 
 pressure. j^^yg that the line of pressure, or angle of the normals, is constant. 
 
 The action between the teeth of the gears in Fig. 1, begins at D, and ends at A, taking 
 Limit of Action placc oulj betwecu the points of tangency of the line of action and base 
 circle. No involute action can take place 2vithin 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- chaugc iu thc distaucc bctwccn the centers of two involute gears will not 
 ter Distance does not dianoe the velocitv ratio, if the arc of action is not less than the circular 
 
 Affect the Velocity & ^ ^ 
 
 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 tootli, in order that the 
 arc of action should equal the increased pitch, an increase in the center distance necessitating 
 an increase in the diameters of the pitch circles, and therefore in the circular pitch. But 
 while 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 li^GS, it follows that the tooth outline of an involute rack must be a right 
 tooth, a Right Line. |jj-jg^ perpcndicular to the angle of pressure. Plate 8 illustrates a rack for 
 an involute gear, having an angle of pressure of about 30°. (The section lined portions are 
 not involute.)
 
 INVOLUTE LIMITING CASE. 29 
 
 40. Involute Limiting Case. Plate 8. Let the cliametei's of the pitch circles, the angle 
 of pressure, and the niinihtT of teeth, be given. Having drawn the pitch circles about their 
 respective centers, F and G , obtain the base 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. This will be 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 pitch. Next determine the 
 point, C , by spacing the arc, D K C , equal to D A ; A and C will be 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 base 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 fill the space, having no backlash. The circular pitch may be found by divid- 
 ing the circumference of the pitch circle into as many parts as there are teeth, or the teeth may 
 be spaced on tlie base circle.* 
 
 The rack is made to engage tlie pinion in tlie following manner : — 
 
 ])eing the pitch point of rack and pinion, tlie riglit line, R. drawn through this point, 
 and tangent to the 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 point l)eing that of the 
 intersection of the path 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 concerning the construction of this pinion and gear, see Problem 4, Page 76.
 
 30 EPICYCLOIDAL EXTENSION OF INVOLUTE TEETH. 
 
 beino- no backlash in eitber case. The addendum of the rack tooth will be limited by the 
 parallel to the pitch line draAvu 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. Tlie 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 m an epicycloid, described by 
 the rolling circle FAB, and having A B for its normal, which is also the normal for the involute. 
 But this diameter of rolling circle being one-half the pitch circle within Avhicli it rolls, the 
 h}-pocycloid Avill be a radial line, and the dedenda of the teeth aaIII 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, Avliich is a limit in this 
 case, by reason of the gear tooth having become pointed. 
 
 Similarly, the addendum of the rack tooth ma}- be extended by 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 rolling circle to R, and then, 
 becoming involute, contintie to S. 
 
 42. Involute Practical Case, Plates 9 and 10. Having given the number of teeth of 
 engaging geai-s, and the diametei-s of their pitch circles, it is required to determine the curves 
 for the involute teeth of a pinion, gear, and rack.
 
 INVOLUTE PRACTICAL CASE. 31 
 
 The diameters of the describing circles would be of fii-st consideration in cycloidal gearing ; 
 while in the involute system, the angle of pressure or line of action must first be established ; 
 and tangent to this the base circles may be drawn. My reference to Platk 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 sa}', 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°; 3'et it cannot be lessened in this case, as the number of teeth is limited. 
 Practice has limited this angle to 14.V° 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 
 following problem : — 
 
 A pinion of 12 teeth is recpiired 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 = d' = -=-=12 inches. 
 
 Gear diameter ^ D' = ^ = I^ = 30 inches. (Art. 18, page 11.) 
 
 Since the teeth are to be of standard dimensions (Art. 31, page 17), the addenda will be 
 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 2>itch, p; =^=3.141G. Draw the pitcli cin-les. 
 
 The line of action will jkiss thi'ough tlie pitch point, making the i-iMpiired angle with the 
 common tangent at this point. Next draw the base circles tangent to this line, and determine 
 the points of tangency, D and A. Construct the involutes of these base circles in the manner
 
 32 
 
 INTERFERENCE. 
 
 Fig. 16. 
 
 indicated by Fig. 16, and according to the method for describ- 
 ing an involute, Art. 12, page 7. It will now be seen that 
 the gear tooth will be limited by the arc drawn through D, 
 the point of tangency of base circle and line of action. If, 
 however, the involute curve be continued to the addendum 
 circle, as shown by the dotted line, C E, it will interfere with 
 the radial portion of the j^inion 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 l^e found to interfere with 
 the pinion flank, if extended beyond the point C, which 
 comes into contact at the point D, the limit of involute ac- 
 tion. But the pinion face may be 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 fu-st is to hollow that part of the pinion flank lying
 
 INFLUENCE OF THE ANGLE OF PRESSURE. 33 
 
 within the l)ase circle so as to clear the interfering part of tlie 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 described 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. Art. 41, page 30. 
 
 44. Influence of the Angle of Pressure. The interference may be entirely olniated by 
 sufficiently 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 80 teeth in the pinion, 
 there would have been no interference. See Art. 45. 
 
 The angles of 14.1° and 15°, commonly adopted, are unfortunately small. There is. how- 
 ever, a tendency to increase this angle, and geai"S 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 when engaging a Rack. Pig. 17. 
 
 Let A be the center of a gear having A B = R for the radius of pitch circle, and D B T the
 
 34 
 
 LEAST ANGLE OF PRESSURE WITHOUT INTERFERENCE. 
 
 Pig. 17. 
 
 angle of pressure to be determined, the least 
 number of teeth Ijeing N. Suppose the gear to 
 engage a rack having standard teeth, then will 
 
 BC=- = -=-— . 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 B 
 
 C = R-p 
 
 2R_ R(N-2) 
 
 AC:AD::AD:AB, hcnce, A D2 = r2 = A BxAC = 
 R-^(N-2) 1 r, /N — 2 
 
 _^^andr=Ry/-^. 
 
 The angle of pressure, D B T, is equal to an- 
 gle D A B = p , and 
 
 the cos. 
 
 P=R = 
 
 V^ 
 
 N — 2 
 N 
 
 = jNzi2. Hence the 
 
 N 
 
 CDS. of the angle of pressure =1 
 
 N-2 
 
 N 
 
 (5). 
 
 By substituting in the above fornnila, it will 
 be seen that for a 12-tootlied gear to engage a 
 rack without interference, the -angle of action 
 must be 24.1°, and for 15 teeth the angle Avould 
 be '21 A°. Again, if the angle be 15°, the least
 
 DEFECTS OF THE INVOLUTE SYSTEM. 35 
 
 imiiiber of teetli that will engage -without interference will be 30, while with a 20° angle 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 
 gears this was attained 1)}- the use of one diameter of rolling circle for all geai-s of tlie same 
 pitch (Art. 33, page 20). In tlie involute system we assume an angle of oljliquity, or pressure, 
 which is constant for all geai-s ; but unless this angle be great, gears havings so few as 12 
 teeth cannot be run together without interference. To obviate this difficulty 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. Fii-st consider the latter, which 
 is illustrated by 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 i)art 
 of the pinion Hank which lies within the base circle; also that the j^inion addenda might 
 be wholly involute since there would be no 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. Therefore when the 12-tootlied gear engages one having 
 a greater numl)er 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 geai-s 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 Iving between the base
 
 36 
 
 UNSYMMETEICAL TEETH. 
 
 circle and point c, Plate 9, and the arc of contact ^yould 
 be but 1.05 of tlie circular pitch. Now, one of the claims 
 made for the involute tooth is that the distance between 
 the centers of the geare nmj be changed without changing 
 the velocity ratio ; but in tliis latter case it cannot be done 
 without making the arc of contact less than the circular 
 pitch. 
 
 If the system of undercutting the flanks be adopted, 
 the addendum will be wholly involute ; and in the case of 
 Plate 9 all of the pinion addendum would have been 
 available for action, but the pinion flank, 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 geai-s with which they are intended 
 to run, if the best results would be attained. This would, 
 of coui-se, prevent the use of the ready-made gear or cut- 
 ter, but would insure a longer arc of action between con- 
 jugate teeth.
 
 UNSYMMETRICAL TEETH. 87 
 
 47. Unsymmetrical Teeth. Fig. 18. A very desirable, although little used, form of tooth 
 is that knowu as the unsymmetrical tooth, Avhich usually comltines the cycloidal and involute 
 systems. Fig. 18 illustrates a pinion and gear liaving the same numl)er of teeth as those illus- 
 trated by 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 geai-s, and when less force would usually l)e 
 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 Ijack 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. This increase in the diameter of the rolling circle would have lengthened 
 the arc of contact, had not the height of the tooth been reduced to maintain the same arc as 
 that of Plate 4. 
 
 The cycloidal action begins at C and ends at H , making a maximum angle of pressure of 
 17°. The same rolling circle has been used for the face and flank of each gear; but the one 
 rolling within the pitch circle of the gear might have been much increased without materially 
 weakening th.e gear tooth. 
 
 The hivolute action would begin at D and end at B, making an arc of contact a little greater 
 than the pitch.
 
 38 ANNULAR GEARING. 
 
 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 hitter is called an iitternal, or annular gear. The solution of problems 
 relating to tliis form of gearing diffei-s in no wise from that of the ordinar}- 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 gear. 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 flanlcs, which therefore determines the diameter of the describing circle for the addenda 
 of the geare. The second describing circle. 2. is governed by conditions which will appear later. 
 It will be observed that the addenda of the auiuilar 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 6, 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 hy either of two rolling circles, the sum of the diametei-s of which sliall equal that 
 of the pitch circle within which they roll. Tlius the addendum, C E, of the pinion, Plate 11, 
 may be described by the circle 2, or tlie 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 ; consecjuently C E and F G will produce a uniform 
 velocity ratio, the contact taking place from A to D. Tlie addendum C E has contact also witli 
 the dedendum C F along the path A C ; hence, during a part of the arc of recess there nuist 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 ; bnt 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. The additional contact takes place during the arc of recess, 
 which is also advantageous. 
 
 In order to obtain this secondary action, the sum of the radii of the inner n)nl outer roUine/ 
 circles must equal the distance between the centers of pinion and gear.* 
 
 For, letting r, , r.,, and r;, l)e the radii of the inner, outer, and intermediate rolling circles, 
 and Rp, Rg, the I'adii of jjinion and gear, 1-3 + ri = Rg, (G), and r.j — r. = Rp, (7), AuT. 10, page (3. 
 Subtracting the second equation from the lii-st, ri + r^ = Rg — Rp = C = center distance (8). 
 
 Platp: 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 slntU'iit is roftMTod to Prof. MacCord's " Kinomatii's," pages 104 to 10!) inclusive, for a very complete 
 demonstration of tliis law, together with other limitations of annular gears.
 
 40 LIMITATION OF INTERMEDIATE DESCRIBING CURVE. 
 
 of recess being the same as before, about 1^ times the circular pitch. Had the outer describing 
 circle been used to describe the dedenda of the gear teeth, as in the preceding cases, a secondary 
 action would have taken place during the recess. 
 
 Special notice should be taken of the reduced angle of pressure in the secondary action of 
 annular gearing, and of the possibility of obtaining a great arc of recess with little or no 
 approaching action. These advantages are very apparent in Plate 11, in which the pinion 
 engages an external and an internal gear having an equal number of teeth. 
 
 51. Limitations of the Intermediate Describing Circle. Plate 12, Fig. 2. Suppose the 
 inner describing circle, 1, Plate 11, to be increased until it equals the diameter of the pinion 
 pitch circle, 9f, 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 substituting 
 Rp for rj in equations 6 and 8, Art. 50, we shall obtain rg = Rg — Rp = C, and ro = 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 be an approach- 
 ing action ; but 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, which 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 7'adius of the intet'mediate 
 describing circle cannot he less than the line of centers. 
 
 52. Limitations of Exterior and Interior Describing Circles. Plate 12, Fig. 3, From
 
 LIMITATION OF EXTERIOR AND INTERIOR DESCRIBING CURVES. 41 
 
 Art. 50, page 39, it was seen that the sum of the radii of tlie exterior and interior describing 
 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 without an equal decrease in the other, thus making the sum of their radii greater 
 than the center distance, the addenda will interfere. Thus, in Plate 11, a decrease in descril>- 
 ing circle 2 would produce 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, the limit of the sum of the radii of the exterior and 
 interior deserihitiy circles is the center distance. 
 
 Plate 12, Fig. 3, illustrates a special case of the above condition, the interior describing 
 circle being reduced to zero, and the radius of the exterior circle made equal to the center dis- 
 tance, thus making the intermediate describing circle equal to the pitch circle of the gear. 
 There will be dou])le contact during a portion of the arc of recess, the contact beginning at A, 
 and following the outer describing circle to C, and tlie intermediate (or in this case the pitch 
 circle of the gear) to D . This design is objectionable 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 ro-|-ri=C, either radius will equal C, when the other becomes zero; 
 but if there be a secondary action, the mininuun value of r^ may not be zero, for r^ will be a 
 maximum when r., is a minimum, as rj + rj = Rg, rs is a mininuim when ecjual to C (Art. 52), 
 and substituting this value in the last equation, r^ = Rg — C . Again substituting this value in 
 the equation, r., + rj = C , r.^ = C — (Rg — C) = 2 C — Rg.
 
 42 LIMITING VALUES OF DESCTJBIXG CIRCLES FOE SECONDARY ACTION. 
 
 Summarv of the above limiting values and conditions governing secondary action: — 
 
 ri maximum = Rg — C ; r^ minimum = : rg + r^ = Rg . (6) 
 
 r^ maximum = C ; r., minimum = 2 C - Rg ; rs — r^ = Rp . (7) 
 
 rg maximum = Rg ; rg minimum = C; Rg — Rp = C. (8) 
 
 54, Practical Case. If annular geai-s be made interchangeable with spur geare, it will be 
 necessarv to have the number of teeth in the engaging gears differ by 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 l)e 2, the diameter of the pinion will be 6, and that of the describing circles 3. 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 l)eing the least number of teeth 
 in the gear, and n the least number in the pinion, or the base of the system: — 
 
 ni~r,„ Nrii nNn ^., 
 
 C = 2r,= — , also C=Rg-Rp = — - — , hence ^ = ^-^. or 2n = N. 
 Tlie least number of teeth in the annuJar gear n'iU be twice that of the base of the si/stem. 
 
 55. Summary of Limitations and Practical Considerations. (</) 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 
 
 (6) There will be .secondiiiy action only when the sum of the radii of the exterior and 
 interior describing circles is equal to the line of centers. (Art. 50, page 38.) 
 
 (c) 'J'hc radius of the intermediate describing circle cannot l)e less than the cent^^r distance. 
 (Art. 51, page 40.) 
 
 (tZ) The sum of the radii of exterior and interior describing circles cannot be greater than 
 the center distance. (Art. 52, page 40.) 
 
 (ti) The number of teeth in any pair of gears of an interchangeable system must differ by 
 an amount equal to tlie base of the system. (Art. 54, page 42.) 
 
 (/) If the pinion drives, the exterior describing circle shouhl be the greater in order that 
 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 onl}-, 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 Ije on the pitch circle. 
 
 56. Involute System of Annular Gearing. Fig. 19. The method of drawing the tooth 
 outlines for the involute annular gear does not dift'er from that of the spur gear. Pitch lines 
 liaving been determined, the base circles are diawn 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.
 
 44 
 
 INVOLUTE SYSTEM OF ANNULAR GEARING. 
 
 Fig. 19 illustrates an annular gear of 20 teeth en- 
 gaging a pinion of 10 teeth, the angle of pressure 
 being 20°. The pinion driving in the direction indi- 
 cated will establish the first i)oint of contact at A, and 
 the last point, B, ^^•ill be limited by the height of the 
 
 tooth, in this case - . The limit of the gear tooth will 
 
 be determined by the arc drawn from the center of 
 gear through 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, which it is advantageous 
 to adopt. 
 
 If the pinion and gear differ but little in diameter, 
 it is desirable to use the cycloidal system, in which case 
 the interference may be more easily avoided. It sliould 
 also be noted that the advantages to be derived from 
 an increase in the arc of contact and a decrease in the 
 angle of pressure are to be obtained only by the use of 
 the latter system.
 
 THEORY OF BEVEL GEARING. 45 
 
 CHAPTER VI. 
 
 BEVEL GEARING. 
 
 57. Theory of Bevel Gearing. In the cases previously considered, the elements of the teeth 
 were parallel, the surfaces having been generated by a right line which was either an element 
 of a rolling cylinder, as in the cycloidal system, or by 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 
 intersecting, 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 spherical base of the 
 cone, that portion of the curve lying outside the pitch cone being a spherical epicycloid, and 
 that within, a spherical h>^ocycloid. The dedendum, or surface of the tooth lying within the 
 pitch cone A C B, was described by the element E F C of the describing cone, which is shown as 
 generating the addendum 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,
 
 46 
 
 CHARACTER OF CURYES IN BEYEL GEARING. 
 
 is not shown; but the diameter of 
 its base would be governed by laws 
 siniilar 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 
 BEYEL 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 Avithin another cone, but the 
 flank may approximate closely to 
 such plane. 
 
 The INYOLUTE BEYEL TOOTH is 
 
 one ha^dnof a grreat circle for its Hne 
 
 Fig. 20
 
 TREDGOLD APPROXIMATION. 
 
 47 
 
 of action. Fig. 21 illustrates a crown 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 be rolled on D C, so as 
 to continue tangent to D C and A E, the point B will describe 
 the spherical involute G B F. Conjugate teeth described by 
 this process maintain their velocity ratio constant, even while 
 undergonig a slight cliange in their shaft angles, tlius 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 
 would cut a great circle from the surface of the sphere. In 
 Fig. 22, G F is the plane Avhich 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 Bilgram, and is frequently confused 
 with the involute tooth. It can be formed in a practical manner 
 by the molding-planing process. The IJilgram machine, de- 
 signed to plane this toolli, is described m tlie Journal of the 
 Franklin Institute for August, 188G, and in the American Ma- 
 chinist for ]\Iay 9, 1885. 
 
 59. Tredgold Approximation. Because of the dilficulty in- 
 volved in describing the tooth form on the surface of a sphere, 
 
 Fig. 21.
 
 DRAFTING THE BETEL GEAR. 
 
 it is custoniar)- 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. Tangent 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 approximation 
 is so small as to be inappreciable, save in exag- 
 gerated cases : and the method is always em- 
 ployed for the drafting of bevel geai"S. 
 
 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 being 
 ^o^ omitted. 
 
 ^^ The names of the parts of a bevel 
 
 Q gear are also given, and the lettering 
 
 Fig. 23.
 
 DRAFTING THE BETEL GEAR. 49 
 
 corresponds to that of Plate 13, wliieli latter will he used to illustrate the method of 
 drawing. 
 
 A B and B D, Platk 13, aie the i)itch diameters of a gear and pinion with axes at 90", and 
 having 15 and lii teeth respectively, 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 tlie gear. Through these points draw the pitch lines perpendicular to the axes of 
 the gears, and in this case peipendicular to each other. Draw the pitch cones A C B and BCD, 
 and perpendicular 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, lay 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 
 greater 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 off on a portion of tlie arc of the developed pitch circle, the cir- 
 cular pitch, B V, whicli is equal to -. Draw the addendum and dedendum circles with radii
 
 60 DRAFTING THE BEVEL GEAR. 
 
 equal to distance of these circles from the apex of the normal cone, which in the case of the 
 gear will be G E and G F . 
 
 Next determine the tooth curve as for spur gears, using the developed pitcli circle instead 
 of the real pitch circle. In the case illustrated, the curve is involute. B W is a part of the 
 line of action, making an angle of 75° with G H, the line of centers. The base 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 pitch circle, instead of the pitch diameter A B . 
 
 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 being the method commonly adopted. Both methods have been employed in the 
 plate. If the development of the inner pitch cone of gear be drawn from 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 by the fine dotted lines. The addendum and deden- 
 dum circles will be described with radii S Z and S Y, and the tooth curves may be drawn by 
 determining the reduced rolling circle, if the gear be cycloidal, or the reduced base circle if the 
 involute system be employed. 
 
 A second method for describing the teeth on the inner normal cone would be to base it directly 
 on the reduced pitch, which may l)e determined by dividing the number of teeth by the diameter 
 of the base of the pitch cone at tliis point. In the plate, the value of P for small end of teeth 
 
 • 15 12 
 
 is T-TT for gear, or — r for pinion = 4.6 = P. The addendum, dedendum, circular pitch, etc., may 
 now be obtained from this value of P, as was done in the case of the outer pitcli cone. In like
 
 FIGURING BEVEL GEARS. 
 
 51 
 
 manner we may o])tain any other 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 re([uired for general refer- 
 ence, and consisting of pitch diametei-s, 
 number of teeth (or pitch), face (K), thick- 
 ness of gears (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 blank, 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 first set of 
 dimensions are all matters of design, but 
 the second and third dimensions must be 
 
 .H4-L-t£. 
 
 I BACKING 
 
 Tr — 
 
 Wm^^ 
 
 % 

 
 52 
 
 BEVEL GEARING. 
 
 determined from the data given in the first. To obtain these it is necessar}^ 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 these, 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 gear. 
 In the similar right triangles a b t and t r m, Fig 25, 
 
 tan A 
 
 
 t a b = m t r 
 
 tan B 
 
 a b = 
 
 D' 
 
 --f 
 
 2 sin A 2 sin A 
 
 d' 
 
 P d' 
 
 t m 
 
 cos A 
 
 F = - sin A 
 
 2 sin A 
 
 The angle decrement, C, is sometimes made equal to B, in which case the 
 dedendum of the tooth at the small end will be greater, as shown by the line 
 h k ; 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: 
 
 h t 8 P 9 ^ sin A _ 2.25 sin A 
 
 t a d' 4 n n ' 
 
 2 sin A 
 
 Having determined these 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. 
 
 The 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, rather than in fractional equivalents. 
 
 tan C 
 
 Fig. 25.
 
 BEVEL GEAR TABLE. 53 
 
 62. Bevel Gear Table for Shafts at 90". In order to facilitate the figuring of bevel geai-s, 
 tables or charts of the principal values are commonly employed. Such charts 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 54 and 55, consist of 
 the trigonometrical functions for geare of the proportions commonly employed. 
 
 Description of Table.* 
 
 Column 1. Ratio of Pinion to Gear. ,, 
 
 Column 2. Ratio of Pinion to Gear expressed in decimals, or tang of center angle. ^^" * ~ jy "= ju ' 
 
 Column 3. Center angle of Pinion corresponding to tangent in column 2. 
 
 Column 4. Ten times the angle increment lor a Pinion of 10 teeth. This increased value is employed to 
 simplify the figuring of gears having other tlian 10 teeth. Thus, the angle increment for miter geare 
 (1 to 1) having 10 teeth would be 8.2°, and for 14 teeth, \j of this value or J|. 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 inapjn-eciable for small arcs. 
 
 Column 5. The diameter increment for a Pinion of one pitch, hence equal to 2 cos A. 
 
 Column 6. Center angle for Gear, or 90° — A. 
 
 Column 7. Ten times the angle increment for Gear of 10 teeth, which, 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 
 tlie pinion, or by 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 5 by P , and tliat 
 for the gear by dividing the value in 8 by P. 
 
 The value of the angle C may be determined with sufficient accuracy by making it | of B . 
 
 * The plan of this table is that adopted by Mr. George B. Grant. See "American Machinist/' Oct. 31, 1885, and 
 "A Treatise ou Gear Wheels," page 90.
 
 54 
 
 BEVEL GEAR TABLE. 
 
 
 BEVEL GEAR TABLE FOR SHAFTS AT 90°. 
 
 d'+2E-; 
 
 Fig. 26. 
 
 
 PEOPORTION 
 
 OF 
 
 PINION 
 
 TO GEAR. 
 
 
 PINION. 
 
 
 
 GEAR. 
 
 
 
 A. 
 
 B. 
 
 2 E. 
 
 900 — A. 
 
 B. 
 
 2 F. 
 
 
 Center 
 Angle. 
 
 Angle 
 
 Increment. 
 
 Divide 
 
 by n. 
 
 Diameter 
 
 Increment, 
 
 Divide 
 
 by P. 
 
 Center 
 Angle. 
 
 Angle 
 
 Increment 
 
 Divide 
 
 by N. 
 
 Diameter 
 
 Increment 
 
 Divide 
 
 by P. 
 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 1 
 
 1 
 
 1.000 
 
 45. 
 
 80.5 
 
 1.414 
 
 45. 
 
 80.5 
 
 1.414 
 
 9 
 
 10 
 
 .900 
 
 41.98 
 
 76.0 
 
 1.486 
 
 48.02 
 
 84.5 
 
 1.337 
 
 8 
 
 9 
 
 .888 
 
 41.63 
 
 75.6 
 
 1.495 
 
 48.37 
 
 85.0 
 
 1.329 
 
 7 
 
 8 
 
 .875 
 
 41.18 
 
 75.0 
 
 1.504 
 
 48.82 
 
 85.5 
 
 1.317 
 
 6 
 
 7 
 
 .857 
 
 40.60 
 
 74.1 
 
 1.518 
 
 49.40 
 
 86.3 
 
 1.302 
 
 5 
 
 6 
 
 .833 
 
 39.80 
 
 73.0 
 
 1.536 
 
 50.20 
 
 87.4 
 
 1.280 
 
 4 
 
 5 
 
 .800 
 
 38.66 
 
 71.1 
 
 1.562 
 
 51.34 
 
 88.8 
 
 1.249 
 
 7 
 
 9 
 
 .777 
 
 37.85 
 
 70.0 
 
 1.579 
 
 52.15 
 
 89.6 
 
 1.228 
 
 3 
 
 4 
 
 .750 
 
 36.83 
 
 68.5 
 
 1.600 
 
 53.17 
 
 90.8 
 
 1.200 
 
 5 
 
 ( 
 
 .714 
 
 35.53 
 
 66.2 
 
 1.628 
 
 54.47 
 
 92.5 
 
 1.162 
 
 7 
 
 10 
 
 .700 
 
 34.99 
 
 65.1 
 
 1.638 
 
 55.01 
 
 93.0 
 
 1.147 
 
 o 
 
 3 
 
 .666 
 
 33.68 
 
 63.2 
 
 1.664 
 
 56.32 
 
 94.5 
 
 1.109 
 
 5 
 
 8 
 
 .625 
 
 32.00 
 
 60.4 
 
 1.696 
 
 58.00 
 
 96.3 
 
 1 .060 
 
 3 
 
 5 
 
 .600 
 
 30.96 
 
 58.7 
 
 1.715 
 
 59.04 
 
 97.3 
 
 1.029 
 
 ^ 4 
 
 7 
 
 .571 
 
 29.75 
 
 56.6 
 
 1.736 
 
 60.25 
 
 98.5 
 
 .992 
 
 \lu - 
 
 9 
 
 .555 
 
 29.05 
 
 55.4 
 
 1.748 
 
 60.95 
 
 99.1 
 
 .971 
 
 -4^1 
 
 2 
 
 .500 
 
 26.56 
 
 51.0 
 
 1.789 
 
 63.44 
 
 101.4 
 
 .894 
 
 -^ 4 
 
 9 
 
 .444 
 
 23.94 
 
 46.3 
 
 1.827 
 
 66.06 
 
 103.6 
 
 .812
 
 BEVEL GEAR TABLE. 
 
 55 
 
 
 BEVEL GEAR TABLE FOR SHAFTS 
 
 AT 90°. 
 
 
 
 >ORTION 
 OF 
 
 NION 
 GEAR. 
 
 PINION. 
 
 GEAR. 
 
 PRO] 
 
 A. 
 
 B. 
 
 2 E. 
 
 900 -A. 
 
 B. 
 
 2 F. 
 
 PI 
 TO 
 
 Center 
 A iigle. 
 
 Angle 
 
 Increment. 
 
 Idvide 
 
 Diameter 
 Increment. 
 
 Divide 
 
 Center 
 Angle. 
 
 Angle 
 
 Increment. 
 
 Divide 
 
 Diameter 
 
 Increment. 
 
 Divide 
 
 
 
 
 by n. 
 
 by P. 
 
 
 by N. 
 
 by P. 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 3: 7 
 
 .428 
 
 23.20 
 
 45. 
 
 1.838 
 
 66.80 
 
 104.1 
 
 .788 
 
 2: 5 
 
 .400 
 
 21.80 
 
 42.5 
 
 1.857 
 
 68.20 
 
 105.2 
 
 .743 
 
 3: 8 
 
 .375 
 
 20.55 
 
 40.3 
 
 1.873 
 
 69.45 
 
 106.0 
 
 .702 
 
 1: 3 
 
 .333 
 
 18.43 
 
 36.1 
 
 1.897 
 
 71.57 
 
 107.4 
 
 .632 
 
 3:10 
 
 .300 
 
 16.70 
 
 ■32.8 
 
 1.915 
 
 73.30 
 
 108.4 
 
 .575 
 
 2: 7 
 
 .285 
 
 15.95 
 
 31.4 
 
 1.923 
 
 74.05 
 
 108.8 
 
 .549 
 
 1: 4 
 
 .250 
 
 14.03 
 
 27.8 
 
 1.940 
 
 75.97 
 
 109.8 
 
 .485 
 
 2: 9 
 
 .222 
 
 12.53 
 
 24.6 
 
 1.952 
 
 77.47 
 
 110.5 
 
 .434 
 
 1: 5 
 
 .200 
 
 11.31 
 
 22.5 
 
 1.961 
 
 78.69 
 
 111.0 
 
 .392 
 
 2:11 
 
 .181 
 
 10.30 
 
 20.4 
 
 1.968 
 
 79.70 
 
 iii.a 
 
 .357 
 
 1: 6 
 
 .166 
 
 9.46 
 
 18.8 
 
 1.973 
 
 80.54 
 
 111.6 
 
 .329 
 
 2:13 
 
 .153 
 
 8.75 
 
 17.4 
 
 1.977 
 
 81.25 
 
 111.8 
 
 .304 
 
 1: 7 
 
 .143 
 
 8.13 
 
 16.4 
 
 1.980 
 
 81.87 
 
 112.0 
 
 .283 
 
 2:15 
 
 .133 
 
 7.60 
 
 15.0 
 
 1.982 
 
 82.40 
 
 112.1 
 
 .265 
 
 1: 8 
 
 .125 
 
 7.12 
 
 14.2 
 
 1.985 
 
 82.88 
 
 112.2 
 
 .248 
 
 2:17 
 
 .117 
 
 6.70 
 
 13.3 
 
 l.<)86 
 
 83.30 
 
 112.3 
 
 .233 
 
 1: 9 
 
 .111 
 
 6.33 
 
 12.6 
 
 1.988 
 
 83.67 
 
 112.4 
 
 .221 
 
 1:10 
 
 .100 
 
 5.70 
 
 11.3 
 
 1.990 
 
 84.30 
 
 112.6 
 
 lioo 
 
 P = - = " ; 
 
 P- = ^: 
 
 . d' n 
 tan A = ^, = ^ 
 
 tan B = 
 
 tan C = 
 
 2 sin A 
 n 
 
 2.25 sin A 
 
 2 E = p cos A ; 
 
 F = p sin A.
 
 56 
 
 BEVEL GEARS WITH AXES AT AXY ANGLE. 
 
 63. Bevel Gears with Axes at any Angle. 
 If the axes of the gears intersect at angles other 
 than 90°, tlie drawing of tlie Wanks and devel- 
 opment of the teeth do not differ from the cases 
 already described. The figuring required is that 
 indicated in Fig. 27, those in the heavy face 
 being used to determine the other values, and 
 not appearing on the finished drawing. 
 
 . ./ sin a 
 
 tan A = ■ ; 
 
 tan A = 
 
 sin a 
 
 N 
 
 - + cos a 
 
 
 !^ 
 
 tan B = 
 
 2 sin A 
 = or 
 
 
 n 
 
 tan C : 
 
 2.25 sin A 
 
 + COS a 
 
 2 sin A^ 
 N ' 
 
 E = — cos A ; 
 P 
 
 — sin A ; 
 P 
 
 2.25 sin A' 
 N 
 
 E' = — cos A'. 
 P 
 
 F' = — sin A'. 
 P 
 
 rig. 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 atid 
 A', and finding the values for 2 E and 2 F, corre- 
 sponding to each gear separately.
 
 ODONTOGRA.PHS. 57 
 
 CHAPTER VIT. 
 
 SPECIAL FORMS OF ODONTOIDS, NOTATION, FORMULAS. ETC. 
 
 64. Odontographs and Odontograph Tables. If tooth curves are to be drawn according to 
 some established system, as in the involute when the angle of pressure is constant, or in the 
 cycloidal when but one diameter of rolling circle be used, it may be desirable to employ some of 
 the approximate methods for shortening the operation of describing the outline of the teeth. 
 While it is unnecessary for the student to familiarize himself with the theory, or even the 
 details, of operating the various systems for approximating these curves, it is essential that a 
 knowledge be had of the more useful tables and methods to which reference may be made 
 when required. This is particularly true in the case of the involute tooth, which is the one 
 most used. 
 
 Three methods are employed for approximating tlie 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, l)y curved templets from which the curves may be traced directly. 
 
 Third, by ordinates.
 
 58 
 
 ODONTOGRAPHS. 
 
 _LINL_OF_FLANK 
 
 GRANT'S EPICYCLOIDAL ODONTOGRAPH.i 
 
 3 P. 21 T 
 
 65. The Three Point Odontograph, designed 
 by Mr. George B. Grant, is a table for the 
 face and flank radii, and location of centers 
 for circular arcs approximating the true curves 
 of epicycloidal teeth. It is designed for that 
 system which has for its base a twelve-toothed 
 gear with radial flanks. Art. 33, page 20. 
 
 In using this odontograph, proceed as fol- 
 lows : Draw the pitch circle, addendum, and 
 dedendum circles, and space the pitcli circle 
 lor the teeth. Obtain the radins of the cir- 
 cle for the flank centers by laying off out- 
 side the pitch circle the tabular distance for 
 flanks, as given in the sixth column, observing 
 that this value must be divided by the 
 
 
 Tfetii in 
 
 Fo 
 For a 
 
 R ONE Diametral Pi 
 
 rcii 
 
 nE BY 
 
 NUMUF.R 01 
 
 NY OTHER 
 
 Pitch divi 
 
 Ge 
 
 All 
 
 THAT 
 
 PiTl'II 
 
 
 Faces 
 
 Flank 
 
 Kxact 
 
 Intervals 
 
 Kad. 
 
 Dis. 
 
 Kad. 
 
 Uis. 
 
 12 
 
 12 
 
 2.01 
 
 .06 
 
 00 
 
 00 
 
 m 
 
 13-U 
 
 2.04 
 
 .07 
 
 15.10 
 
 9.43 
 
 lOi 
 
 15-lG 
 
 2.10 
 
 .09 
 
 7.80 
 
 3.46 
 
 1"' 
 
 17-18 
 
 2.14 
 
 .11 
 
 6.13 
 
 2.20 
 
 20 
 
 19-21 
 
 2.20 
 
 .13 
 
 5.12 
 
 1..57 
 
 23 
 
 22-24 
 
 2.2G 
 
 .15 
 
 4.50 
 
 1.13 
 
 27 
 
 25-29 
 
 2.33 
 
 .16 
 
 4.10 
 
 .96 
 
 3:3 
 
 30-36 
 
 2.40 
 
 .19 
 
 3.80 
 
 .72 
 
 42 
 
 37-48 
 
 2.48 
 
 .22 
 
 3.52 
 
 .63 
 
 58 
 
 49-72 
 
 2.60 
 
 .25 
 
 3.33 
 
 .54 
 
 97 
 
 73-144 
 
 2.83 
 
 .28 
 
 3.14 
 
 .44 
 
 290 
 
 145-300 
 
 2.92 
 
 .31 
 
 3.00 
 
 .38 
 
 00 
 
 Rack 
 
 2.96 
 
 .34 
 
 2.96 
 
 .34 
 
 1 By permission from Grant's "Treatise on Gear Wheels."
 
 ODONTOGRAPHS. 
 
 59 
 
 GKAXT'S INVOLUTE ODONTOGKAPH.i 
 
 diameter pitch. Similaiiy, ubtaiii the circle 
 for the face centers, which will be drawn 
 inside the pitch circle. Tlien, with the 
 corresponding radii for the given nunil)er of 
 teeth (divided by the pitch), describe the 
 required curves. 
 
 66. The Grant Involute Odontograph gives 
 a very close approximation to the involute 
 curve. It is designed for the system of 15° 
 angle of pressure with epicycloidal extension. 
 All gear teeth made by this odontograph will 
 engage a 12-toothed gear without interference. 
 
 Having drawn the pitch circle, addendum, 
 and dedendum circles, and spaced the teeth, 
 
 NfMHER 
 
 OP Teeth 
 
 1-2 
 
 1:5 
 U 
 1.-. 
 
 10 
 17 
 
 18 
 19 
 
 20 
 21 
 22 
 23 
 
 24 
 
 2.'> 
 2(3 
 27 
 
 28 
 
 Divide iiv the 
 Diametral 1'it<h 
 
 Face 
 Radius 
 
 2.51 
 2.02 
 
 2i)2 
 3.02 
 3.12 
 3.22 
 
 3.32 
 3.41 
 3.49 
 3.57 
 
 3.04 
 3.71 
 3.78 
 3.8.-) 
 3.92 
 
 Flank 
 Radius 
 
 .96 
 1.09 
 1.22 
 1.34 
 
 1.40 
 1..58 
 1.09 
 1.79 
 
 1.89 
 1.98 
 2.06 
 2.1.5 
 
 2.24 
 2.33 
 2.42 
 2.50 
 2.59 
 
 NiTMBER 
 
 OF Teetu 
 
 29 
 30 
 31 
 32 
 
 33 
 34 
 35 
 30 
 
 37-40 
 41-45 
 46-51 
 52-00 
 
 61-70 
 
 71-90 
 
 91-120 
 
 121-180 
 
 181-360 
 
 Divide by the 
 Diametral Pitch 
 
 Face 
 Radius 
 
 3.99 
 4.06 
 4.13 
 4.20 
 
 4.27 
 4.33 
 4.39 
 4.45 
 
 Flank 
 Radius 
 
 2.67 
 2.76 
 2.85 
 2.93 
 
 3.01 
 3.09 
 3.16 
 3.23 
 
 4.20 
 4.63 
 5.06 
 5.74 
 
 6..o2 
 
 7.72 
 
 9.78 
 
 13.38 
 
 21.02 
 
 > By permission from Grant's " Treatise on Gear Wheels.
 
 60 
 
 OD ONTO GRAPHS. 
 
 obtain the circle of centers for face and flank radii by describing a circle tangent to the 15° 
 line of pressure. This circle will be one-sixtieth of the pitch diameter inside the pitch circle. 
 Next, obtain the face and flank radii from the table, and, having divided the values by the 
 diameter pitch, describe the required arcs. Observe that the arc for the flank is drawn from 
 the pitch circle to the base circle, or circle of centers, the remainder of the flank being radial. 
 The method of drawing the rack tooth will be evident from the figure. 
 
 Fig. 28. 
 
 -"tH^^^-, 
 
 67. Willis's Odontograph. Among those of the 
 first tjpe, the oldest, best known, and least accurate, 
 are the 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. Figure 28 illus- 
 trates the application of this instrument to the drawing 
 of curves of the cycloidal system. The centers for the 
 circular arcs, designed to approximate tlie curves, are 
 / \ \ i found on the straight edge AB, and at a distance from 
 
 ' \ A I i]^Q 2ero point of the scale to be found in the publi-shed 
 
 table accompanying the instrument. 
 
 The theory and application of these odontographs 
 is clearly treated of in the instructions accompanying 
 these instruments, also in Stahl and Wood's "Elements 
 of Mechanism," pages 114-123, and more briefly in MacCord's " Kinematics," pages 172-174.
 
 THE ROBINSON AND KLEIN ODONTOGKAPHS. 
 
 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 with 
 relation to the i:)itch circle. 
 
 Fig. 20 illustrates the instrument in })osi- 
 tion. The curve B C A is a logarithmic spiral, 
 and the curve B F H the evolute of the fii-st, 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 cycloidal, and a much larger range of the latter 
 is i)ossible than is afforded by the Willis odontograph. The 
 theory of tliis instrument is best treated by Professor Rol> 
 inson in Xan Nosti-and's Edectie Magazine for July, 1876, and 
 Van Nostrand's "Science Series," No. 24. Also see Stahl 
 and Wood's " Elements of iNIechanism," pages 126 to 130. 
 
 69. The Klein Coordinate Odontograph. Fig. -'50 is de 
 signed to eliminate the labor of drawing pitch circles of large 
 radii by constructing the curve by oi-dinatcs from a radial 
 line. The tables and explanation of the method may be 
 
 Fig. 30.
 
 62 
 
 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 may be classified from the forms 
 of rack teeth, as follows : 
 
 System. Tooth Curve. Line of Action. 
 
 Involute, Fig. 31, A right line, A right line. 
 
 Cycloidal, Fig. 32, A cycloid, A circular arc. 
 
 Segmental, Fig. 33, 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 ma}^ be 
 derived from a right line. In the cycloidal system 
 the addendum of any gear ' tooth will properl}- 
 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 cycloidal. A skilled 
 mechanic with file and straight-edge could in this
 
 CONJUGATE CURVES. 
 
 manner produce the templet for any de- 
 sired cycloidal tooth without the aid of 
 other mechanism. Of course such a 
 method would i-equire 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 gra[)hic 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. 
 
 03 
 
 71. Conjugate Curves. — The curves of any [)air of 
 teeth being so related as to produce a uniform velocity 
 ratio are called conjugate, or odontoids, and if an}- tooth 
 curve of reasonable 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 
 foUowdng principle: — 
 
 The normals to the curve must come into actioi 
 consecutively, as in Fig. o4, and not as in Fig. 8"), in 
 which it will be seen that the normal E F will pass 
 through the pitch point M, and the point E come into 
 
 Fig. 36.
 
 64 
 
 WORM GEARING. 
 
 Eig. 37 
 
 action before the point C, which is impossible. Let C, Fig. 36, 
 be any tooth form conforming to the above condition, and the 
 peripliery 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 P'ig. 36. C is a templet of the given tooth 
 form, which is fastened to disk A , and revolving in contact Avith 
 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 worm is a screw designed to drive 
 a gear, called a worm wheel or gear, the axis of the latter being 
 perpendicular to that of the worm. AiiT. 3, page 3. The sec- 
 tion of a worm and gear made by a [)lane perpendicular to the 
 axis of the gear, and including the axis of the worm, is identical 
 witli that of a rack and gear of the same system and pitch. The 
 worm, or screw, may be single threaded, double threaded, etc. 
 If single threaded, the circular pitch corresponds with the pitch
 
 LITERATURE. 65 
 
 of the tliread ; if double, the circuhir pitch will be half the piteli of the thread, etc. To avoid 
 misunderstanding, it is customary to speak of the })itch of the thread as the lead. 
 
 A drawing of the tooth foi-m is required only in sijpcial cases of large cast geai-s, and the 
 usual representation is that shown by Fig. 37. 
 
 The diameter of the \\()rm is commonly made equal to four or five times the circular pitch, 
 and the angle A varies from 00° to 90°. 
 
 Formulas for Worm and Geai:. 
 
 L = Lead of worm; 
 m = Threads per inch in worm; 
 d = Outside diameter of worm ; 
 d' = Pitch diameter of worm; 
 W = Whole diameter of gear; 
 D = Throat diameter of gear; 
 D' = Pitch diameter of gear; 
 
 L = — = P', for single threads, 
 m 
 
 2 
 L = _ = 2 P', for douhle threads, etc. ; 
 m 
 
 fj and r., are dimensions required for the hob, or 
 
 cutter, employed in cutting the worm gear; 
 
 C = Center distance; 
 
 p. -n- D 
 N + 2' 
 
 
 O-.lf.-.^; 
 
 
 o. N + 2. 
 ° - P ' 
 
 
 d 2. 
 h - ^ P' 
 
 
 , 17 . 
 
 
 r D + d 
 ^ ~ 2 
 
 2. 
 P' 
 
 W = D + 2 ( r, 
 
 — 
 
 f ^1 - h cos - j 
 
 73. Literature. The following list of books and articles is published to assist the student 
 who may wish to jjursue the subject beyond its elementary stage. Only those treatises have 
 been enumerated which are likely to be accessible and useful. The great works of Willis,
 
 66 LITERATURE. 
 
 Rankiiie, and Reuleiix are omitted, as the student Avill derive more benefit from the interpreta< 
 tion of these worka by later authors than by a study of tlie original treatises. 
 
 "The Mechanics of the Machinery of Ti-ansmission," revised by Professor Herrmann, is 
 Vol. III., Part I., Sect. 1, of Weisbach's "Mechanics of Engineering." This work includes 
 one of the most valualjle treatises on the subject of gearing, but it is someAvhat difficult. 
 Wiley, .fS.OO. 
 
 " Kinematics," by Professor ^NlacCord, is chiefly devoted to the subject of gearing. It con- 
 tains much original matter of importance. No student of the ;sul)ject can afford to do witliout 
 this treatise. Wiley, Jif^oOO. 
 
 "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 odontogi'a[)h is fully illustrated and explained. J. F. 
 Klein, Bethlehem, Pa., -IG.OO. 
 
 " A Treatise on Gear Wheels," formerly called " 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 pursued 
 to advantage without its use. Philadelphia Gear Works, Philadelphia, $1.00. 
 
 "Practical Treatise on Gearing," by Mr. O. J. Beale. An excellent })ractical treatment of 
 the design and construction of gears. It deals little with the theor}^ but that little is thor- 
 oughly and simply taught. Brown & Sharpe Manufacturing Company, Providence, -fl.OO. 
 
 "Formulas in Gearing." This is published by the Brown & Sharpe ^bmufacturing Com- 
 pany, and contains many useful formulas for the draftsman, and valual)le hints for the cutting 
 of gears. l|2.00.
 
 NOTATION AND FORMULAS. 67 
 
 " Elementary Mechanism, " l)}- Professoi-s Stalil and Wood, is a most comprehensive text 
 book on tlie subject of oeaiino'. It is well classitied, contains numerous examples, and is a 
 valualjle reference book for the student. Van Xostrand, *2.00. 
 
 " Gear-cutting Machinery " by Ralph E. Flanders comprises a complete review of con- 
 tempoi^iry American and European practice for the forming and cutting of gear teeth. Also 
 an excellent logical classification and explanation of the principles involved in the several 
 machines described. Wiley, $3.00. 
 
 In addition to the above the student is referred to the files of the " American Machinist,'' 
 "Machinery," and the " ^Machinery Reference Series,"' for many valuable articles relating 
 to the subject. These articles are particularly intei'csting on account of the tests and novel 
 applications which they report. 
 
 The "Transactions of the American Society of Mechanical Engineers" contain many 
 most valuable and ini[)ortant papers and discussions on the latest theories and practice in 
 srearinof. 
 
 74. Notation and Formulas. 
 
 Spur Gears. 
 
 P' = Circular pitch, Art. 17, page 11; N = Number of teeth iu gear; 
 
 P = Diameter pitch, Art. 18, page 11; n = Number of teeth in pinion; 
 
 D' = Pitch diameter of gear; s = Addendum of tooth, Art. 31, page 17; 
 
 D = Whole, or addendum, diameter of gear; f = Clearance, Art. 27, page 1.); 
 
 d' = Pitch diameter of pinion; t = Thickness, Ai!T. 31, page IT: 
 
 d = Whole, or addendum, diameter of pinion; p = Least angle of pressure, ARr. 45, page 33;
 
 68 
 
 TT = 3.1416; 
 
 P = -Y' ' D' ^ P" ^^' ^^"^ ' 
 
 N 
 P = - , P P' = TT, Akt. i8, page 12; 
 
 NOTATION AND FORMULAS. 
 
 P^ _ TT _ 1.57 
 
 * ~ 2 2 P P 
 
 1_ 
 P' 
 
 f = I = ^, Akt. 31, page 17; 
 
 Art. 31, page IT; 
 
 ^, ,2 PD'+2 N + 2 
 D =D'+2S = D' + - = ^^— = — ^— ; 
 
 cos p = ^ / — ;-; — , Art. 45, page 33. 
 
 v'— 
 
 Rg = Radius of gear, 
 
 Rp = Radius of pinion, 
 
 Tj = Inner describing circle, 
 
 r.2 = Outer describing circle, 
 
 rs = Intermediate describing circle, 
 
 C = Center distance, 
 
 Annular Gears. 
 
 Rg = rs + Tj , Art. 50, page .39; 
 Rp = r.s — r.,, Art. 50, page 39; 
 C = Rg — Pp. Art. 50, page 39; 
 Art. 50, page 39; ^^ rnaximum = Rg - C ; r^ minimum = C, Art. 52, page 40; 
 
 r., maximum = C ; ro minimum = 2 C — Rg, 
 
 Art. 52, page 40; 
 
 fg maximum = Rg ; rs minimum = C, Art. 51, page 40. 
 
 Bevel Gears, Shafts at 90°, Art. 6i, Page 56. 
 
 A = Center angle of pinion; 
 
 B = Angle increment; 
 
 C = Angle decrement; 
 
 E = One-half the diameter increment for pinion: 
 
 F = One-half the diameter increment for gear. 
 
 d' n. 
 tan A = - - -, 
 
 tan B 
 
 tan C 
 
 2 sin A 
 
 2.25 sin A. 
 
 E = - cos A 
 
 F = p sin A.
 
 notation and formulas. 69 
 
 Bevel Gears, Shafts at Other than 90°, Art. 63, Page 5G. 
 
 a = Angle of shafts; 
 
 A = Center angle of pinion; 
 
 A' = Center angle of gear; 
 
 B = Angle inerenient; 
 
 C = Angle decrement ; 
 
 E = One-half the diameter increment for pinion; 
 
 E' = One-half the diameter increment for gear; 
 
 F = Dimension required for backing of pinion; 
 
 F' = Dimension required for backing of gear. 
 
 Worm Gears, Art. 72, Pac;e 62. 
 
 L = Lead of worm ; ■ 1 r,, , . , , , 
 
 L = — = P for single threads; 
 m 
 
 m = Threads per inch in worm; 2 
 
 L = — -= 2 P'for double thread, etc., 
 m 
 d = Outside diameter of worm; 
 
 d' — Pitch diameter of worm; 
 
 D = Thread diameter of gear; 
 
 D' = Pitch diameter of gear; 
 
 W = Whole diameter of gear. 
 
 
 
 N 
 
 — + cos a 
 
 n 
 
 tan A' 
 
 = 
 
 sin a 
 
 n 
 
 ^ -F cos a 
 
 tan B 
 
 = 
 
 2 sin A _ 2 sin A', 
 n N ' 
 
 tan C 
 
 = 
 
 2.25 sin A _ 2.25 sin A', 
 n N ' 
 
 E 
 
 = 
 
 1a C' 1 A' 
 
 p cos A E = p cos A ; 
 
 F 
 
 = 
 
 1 ■ A C. 1 ■ A' 
 
 sin A F = — sin A ; 
 
 °-^P • 
 
 
 D^''^^ 
 
 p-= '° 
 
 N + 2 
 
 • 
 
 W = D + 2 ^ 
 
 D + d 
 
 2 
 
 2. 
 P' 
 
 
 d 2 
 h - 2 p 
 
 • 
 
 •-2 = ^1 + 3p- 
 
 ^)
 
 70 METHOD TO BE OBSERVED IN PERFORMIXG THE PROBLEMS. 
 
 CHAPTER virr. 
 
 PROBLEMS. 
 
 75. Method to be Observed in Performing the Problems. Xo attempt should l)e made to 
 graphically solve the following problems until the general principles involved are ^vell under- 
 stood. 
 
 The first requisite to this is the mastery of Chapter II., on Odontoidal Curves ; and this 
 can be best acquired 1)y the drawing of the various curves, together with a study of their 
 characteristics. Xo problems have been given on this topic, ])ut the following couree 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 be drawn free-hand with considerable 
 accuracy, after whicli they may l)e corrected by the use of scrolls. Xext prescribe a i)oint on 
 each (not one already found), and draw normals to each by Art. 8, page 5. 
 
 The second method. Art. 9, page 5, is the more practical, and should also be 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
 
 PROBLE:\t 1. CYCLOIDAI. LIMITING CASE. I 1 
 
 tlie margin line, and the lay-out of these sheets is given on Plates 14 and lo. Measurements 
 are from the margin line. 
 
 It is unnecessary to represent all the teeth in a gear, but such as are shown should he 
 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 well done; 
 but in either case it is desiral)le to emphasize the curves, and distinguish clearly between the 
 gears by making a veri/ light wash of color on the inside of the curve, the width to be alx)ut 
 one-quarter of an inch. One color may be used for the pinion, and a second for the rack and 
 gear. 
 
 Problem i, Plate 14, 
 
 Fig. I. 
 
 Cycloidal Limiting 
 
 Case. 
 
 Face 
 
 or Flank only. 
 
 EXAMPLK. 
 
 D' 
 
 d' 
 
 N 
 
 n 
 
 
 a B 
 
 1 
 
 10 
 
 
 15 
 
 12 
 
 
 3 J 41 
 
 2 
 
 10 
 
 8 
 
 
 12 
 
 
 31 n 
 
 3 
 
 121 
 
 7 
 
 21 
 
 
 
 3 4 
 
 4 
 
 
 ^•V 
 
 15 
 
 10 
 
 
 3i 41 
 
 5 
 
 10 
 
 8 
 
 15 
 
 
 
 4| 4i 
 
 6 
 
 101 
 
 
 14 
 
 10 
 
 
 3i 4i 
 
 7 
 
 
 6 
 
 24 
 
 12 
 
 
 03 4. 
 
 8 
 
 12L 
 
 
 21 
 
 12 
 
 
 31 4.V 
 
 Statement of Pkoble.nl Having given the diameters of pitch circles, number of teeth, 
 and diameter of describing circle, it is required to draw the teeth for pinion, gear, and rack, 
 having arcs of contact equal to the pitch, and contact on one side of pitch point only.
 
 72 PROBLEM 1. CYCLOIDAL LIMITING CASE. 
 
 Study Arts. 1 to 26 before performing this problem. 
 
 Operations. 1. By Art. 18, page 11, determine the value of N, n, D' or d' one of which is 
 omitted from the table. Observe that - = -. 
 
 d n 
 
 2. Draw center and pitch lines and describing circle. Lay "off the circular pitch 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 direction being determined by the rotation 
 required. Art. 16, page 10. Art. 21. page 12. Arts. 22 and 23, page 13. 
 
 -1. AVith 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 be pointed, draw 
 opposite side of each. Art. 16. page 10. 
 
 0. Draw the working flanks of the pinion teeth, observing that the depth must be sufficient 
 to ailiait 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. Art. 25, page 14. 
 Note that thickness of rack tooth must equal space between pinion teeth, or thickness of 
 gear teeth, measured on the })itL'h line. 
 
 8. To determine points of contact of conjugate teeth, assume any point on face of gear 
 tooth, and determine, first, its position when in contact with the pinion ; second, the point 
 of the pinion tootb engaging it. Since the contact must take place on the path of contact. 
 Art. 21. page 12, the assuniid point will lie at the intei-section of this arc and one described
 
 PROBLEM 2. CYCLOIDAL LIMITING CASE. 73 
 
 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 it« intei-section with the 
 pinion Hank will be the engaging point required. 
 
 Next construct the normals for each of these points. Art. 8, page 5. They should be 
 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. 
 26 to 30. 
 
 Statp:ment of Problkm. Tlie dianietei"s of gears, nunil)er of teeth, and describing circles 
 being given, it is re(piired to diaw 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 l)y Art. 9, page 6. 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 prescribed in Fig. 2. 
 
 3. With the [)oint thus determined on small rolling circle, describe the addendum of gear 
 tooth and dedcnduni of pinion tooth. With the ])oiiit on the second describing circle generate 
 the addendum of pinion tooth. 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.
 
 74 PROBLEM 3. CYCLOIDAL GEAR. 
 
 4. Draw the working faces of the phiion teeth and then the opposite faces to make the 
 teeth pointed. Similarly draw the gear teeth, making them pointed also. The sum of the 
 thickness of the teeth cannot be greater that the circular pitch. Art. 29, page 16. In this 
 case it will be 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 will l)e determined by Art. 14, page 10. 
 Draw the circles with their centers on the line of centers, and obtain the firet and last points 
 of contact. These points should fall on the addendum and dedendum of pinion teeth already 
 drawn, as in Plate 5 at N and . From these points describe the addenda and dendenda of 
 the rack teeth. The thickness of these teeth must equal those of the 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. Complete Chapter III. be- 
 fore performing this problem. 
 
 X AMPLE 
 
 d' 
 
 N 
 
 n 
 
 a 
 
 A 
 
 c 
 
 1 
 
 9 
 
 18 
 
 12 
 
 4 
 
 5 
 
 31 
 
 2 
 
 8 
 
 21 
 
 12 
 
 4 
 
 4 
 
 ^ 
 
 3 
 
 10 
 
 20 
 
 16 
 
 3i 
 
 -4 
 
 3 
 
 4 
 
 8 
 
 22 
 
 12 
 
 3i 
 
 5 
 
 4 
 
 5 
 
 9 
 
 16 
 
 12 
 
 4 
 
 5 
 
 31 
 
 20 12 3 4 41
 
 PROBLEM 3. CYCLOIDAL GEAR. lO 
 
 Statement of Prorlem. Tlie diameters of pitch circles and rolling circles lieing given, 
 and the number of teeth known, it is required to draw the teeth for gear, pinion, and i-ack, 
 to obtain the maximum angle of obliiiuity, and the arcs of approach and recess in each case. 
 The teeth will be standard with .,\/' baeklasli. Art. 81, page 17. Art. 71, page 63. 
 
 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 j)inion flank, gear face, gear flank, aiul pinion 
 face, by Art. 0, page 5. See also Art. 34, page 21. 
 
 4. Lay off thickness of teeth, Airr. 31, page 17, and describe addenda of pinion and gear 
 teeth by approximate method. Art. 34, page 22. Describe dedenda l)y Art. 16, page 11. 
 Draw fillets. Art. 31, page 18. 
 
 5. Describe rack teeth. 
 
 6. Determine the following for gear, pinion, and rack in tei-nis 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 
 
 Maximum angle of pressure
 
 76 PROBLEM 4. INVOLUTE LIMITING CASE. 
 
 Problem 4, Plate 14, Fig. 4. Involute Limiting Case. Study Arts. 38 to 42. 
 
 StatExMENT of Problem. Number of teeth live and six. Pinion teeth pointed. No 
 backhisli or clearance. Arc of contact equal to the circular pitch. 
 
 This problem being similar to tliat 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- 
 fifth of the circumference of the pinion base circle. The tangent of the angle of pressure 
 
 will equal 7-3 = ^^-^ = •— = — -7;-::, but A D = D K C by construction, and D K C = tt. Also A F + 
 •^AFDGAF + DG ^ 
 
 D G = 5|, hence, = — = tan. of the angle of pressure. The angle corresponding to 
 
 this tangent is 29° 44' 6". The distance between the centers will be ^a~D''^ + A F -f- D~G^ = 
 
 V^r'-^ + 5:5^ = 6J. 
 
 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 base 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 base circle. 
 
 2. Draw the involute A c, Plate 8, of the gear, and D p of tlie pinion. Art. 12, page 7. 
 Art. 38, page 26. Determine the circular pitch, and lay off as many divisions as tliere are 
 teeth to be drawn. Copy the curves already drawn.
 
 PROBLEM 5. INVOLUTE PRACTICAL CASE. 77 
 
 3. Draw the opposite face of pinion teeth, 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. Through these points describe the curve of opposite face, 
 and draw the remaining tcctli. 
 
 That portion of the teeth lying Avithin the base circle will be radial, and extend sufficiently 
 to admit the engaging teeth, but without clearance. 
 
 4. Construct two rack teeth. Art. 40, page 29. 
 
 5. Epicj'cloidally extend the gear teeth so as to make them pointed. Similarly extend the 
 rack teeth, Init only as nuich as the clearance for the pointed gear tooth \\ ill pi-rmit. Aut. 
 41, page 30. 
 
 Problem 5, Plate 15, Fig. i. Involute Practical Cases. Complete the study of Chapter IV. 
 ■ Statement of Pmoblems. Several gears and racks are given to describe involute teeth 
 of standard dimensions. To determine the interference, if there be any, and to correct the 
 curves for the same. 
 
 Operations. 1. Draw three or four teeth of gear A, and t\\o teeth oi engaging pinion 
 B, the angle of pressure being 15°. , Art. 42, page 32, Fig. 16. Make contact at pitch 
 point in all cases. Correct for interference l)y epicycloidal extension. Art. 31, page 17. 
 Art. 42, page 30. Art. 41, page 32. 
 
 2. Draw three or four teetli of u'car A engasfino' lack F . 
 
 3. Draw three teeth of ijininn B cno-aoino- rack E, and correct rack teetli ft)r interference. 
 
 1 o o o ^
 
 78 
 
 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 Mill engage it without interference. 
 
 Problem 6, Plate 15, Fig. 2. Cycloidal Annular Gear. Study Arts. 48 to 5Q. 
 Example. D' d' N n A a B 
 
 1 
 
 191 
 
 9 
 
 13 
 
 6 
 
 7 
 
 3i 
 
 H 
 
 2 
 
 19^ 
 
 9 
 
 13 
 
 6 
 
 H 
 
 4 
 
 5h 
 
 3 
 
 191 
 
 9 
 
 13 
 
 6 
 
 6 
 
 H 
 
 5i 
 
 4 
 
 in 
 
 7 
 
 15 
 
 6 
 
 7 
 
 H 
 
 5h 
 
 5 
 
 ITi 
 
 7 
 
 15 
 
 6 
 
 7h 
 
 3 
 
 5i 
 
 Statement op Problem. The number of teeth and diametere 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. 79 
 
 4. Determine the intermediate describing curve, and draw the same to obtain the limit of 
 secondary action. 
 
 5. Determine the maximum angle of pressure for approacli 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. 
 
 D' 
 
 d' 
 
 N 
 
 n An 
 
 gle of Pressure. 
 
 B 
 
 1 
 
 15 
 
 'h 
 
 20 
 
 10 
 
 20° 
 
 H 
 
 2 
 
 15 
 
 6 
 
 30 
 
 12 
 
 15° 
 
 7 
 
 3 
 
 IG 
 
 8 
 
 16 
 
 8 
 
 20° 
 
 6 
 
 4 
 
 20 
 
 8 
 
 30 
 
 12 
 
 15° 
 
 7 
 
 5 
 
 24 
 
 18 
 
 24 
 
 18 
 
 20° 
 
 31 
 
 Statement of PK()nLE>L The pitch diameters, number of teeth, and angle of pressure 
 being given, it is required to draw the tooth curve, to determine if tliere will be 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. Determiiie the arc of contact in terms of P'.
 
 80 
 
 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. 
 
 Example. 
 
 P 
 
 N 
 
 n 
 
 Q 
 
 K 
 
 1 
 
 3 
 
 18 
 
 15 
 
 31 
 
 u 
 
 2 
 
 4 
 
 24 
 
 20 
 
 31 
 
 n 
 
 3 
 
 2 
 
 16 
 
 12 
 
 31 
 
 n 
 
 4 
 
 4 
 
 28 
 
 20 
 
 3 
 
 n 
 
 5 
 
 3 
 
 -21 
 
 15 
 
 3 
 
 n 
 
 6 
 
 2 
 
 14 
 
 12 
 
 31 
 
 n 
 
 7 
 
 3 
 
 21 
 
 18 
 
 H 
 
 u 
 
 8 
 
 2 
 
 18 
 
 14 
 
 4 
 
 n 
 
 9 
 
 4 
 
 20 
 
 16 
 
 3i 
 
 H 
 
 10 
 
 3 
 
 21 
 
 18 
 
 Si 
 
 u 
 
 w 
 
 13 
 
 2 
 13 
 
 li 
 
 2 
 
 H 
 U 
 
 3 
 
 21 
 
 4' 
 
 2i 
 2^ 
 31 
 3^ 
 4" 
 21 
 
 3i 
 
 1 
 1 
 1 
 
 H 
 
 n 
 If 
 n 
 n 
 
 H 
 
 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 Peoble^ni. The proportions of the gear l)eing given hy the tahle, it is 
 required to draw the gear blanks, describe the development of the teeth 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 AND INVOLUTE BEVEL GEARS. 81 
 
 2. Describe two or three teeth of each gear on the developed surfaces of the outer and 
 inner normal cones. Art. 60, page 48. 
 
 3. Figure the geai-s, Art. 61, i)age 51. 
 
 Problem 9, Plate 15, Fig. 4. Cycloidal and Involute Bevel Gears. Shafts at other than 
 90°. Study Art. 63. 
 
 EXAMPI.K 
 
 a 
 
 P 
 
 N 
 
 n 
 
 Q 
 
 J 
 
 K 
 
 L 
 
 M 
 
 H 
 
 w 
 
 X 
 
 U 
 
 V 
 
 Y 
 
 1 
 
 40° 
 
 3 
 
 24 
 
 15 
 
 9 
 
 SI 
 
 2i 
 
 i 
 
 n 
 
 g 
 
 lA 
 
 3 
 
 i 
 
 2i 
 
 li 
 
 2 
 
 45° 
 
 3 
 
 24 
 
 15 
 
 9 
 
 '^i 
 
 2i 
 
 i 
 
 If 
 
 § 
 
 n 
 
 3 
 
 i 
 
 2 
 
 u 
 
 3 
 
 50° 
 
 4 
 
 34 
 
 24 
 
 81 
 
 8 
 
 2 
 
 i 
 
 n 
 
 i 
 
 n 
 
 
 
 i 
 
 n 
 
 u 
 
 4 
 
 55° 
 
 3 
 
 27 
 
 21 
 
 9 
 
 8 
 
 21 
 
 3 
 
 n 
 
 h 
 
 n 
 
 3i 
 
 i 
 
 li 
 
 li 
 
 5 
 
 60° 
 
 2 
 
 20 
 
 12 
 
 8i 
 
 7i 
 
 2.1; 
 
 i 
 
 n 
 
 1 
 
 2 
 
 3i 
 
 i 
 
 2i 
 
 li 
 
 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 tlie gear blanks, describe tlie teeth on the development of the normal cones, 
 and figure the gear. 
 
 Operations. 1. Determine the pitch diameters from above table, and draw the gear 
 blanks. 
 
 2. Describe two or three teeth of each gear on the developed surfaces of the outer and 
 inner normal cones. 
 
 3. Fiourc the geai-s. _,,„ U-RA'^Y 
 
 8 AT: T'/CFERSC^L-EOE 
 SA.TA LAnD.;HA. CALIFORNIA 
 
 /•.(^.?...7-0.
 
 IIS^DEX. 
 
 Keferences ar; to pages. 
 
 Addendum dpfined, 12; proportion for, 17. 
 
 Anjjle decrement, 52. 
 
 Angle increment, 52. 
 
 Angle of edge, 51 ; of face, 51. 
 
 Angle of obliquity, or pressure, 14; affected by rolling circle, 
 18; con.stant, 28; for involute, 31; influence of, .'{.3; 
 method for determining, 3;}, reduced in annular gear- 
 ing, 40. 
 
 Annular gear, notation, and formulas, 08; epicycloidal prob- 
 lem, 78; involute problem, 79. 
 
 Annular gearing, 38; secondary action in, 38; interchangeable 
 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 contacrt defined, 13; relation to circular pitch, 16. 
 
 Arc of recess defined, 13. 
 
 Backing, 52. 
 
 Back cone, 48. 
 
 Backlash defined, !(!; dimensions for, 18. 
 
 Base circle defined, 7, 27. 
 
 Base of system, 21 ; in annular gearing, 42. 
 
 Beale's " Practical Treatise on Gearing," 66. 
 
 Bevel gear defined, 2; Theory of, 45; character of curves 
 employed, 4^!; drafting the, 48; blank, 49; length of 
 face, 49; figuring the, 51; table for, 53, 54, 55; chart for 
 plotting curves, (>7 ; notation and formulas, 68; problems, 
 80, 81. 
 
 Bevel gears with axes at any angle, 56. 
 
 Bilgram, Hugo, inventor of octoid tooth, 47; machine for cut- 
 ting bevel gear teeth, 47, 67 ; exhibit, 67. 
 
 Brown & Sharpe publications, (>6. 
 
 Circular pitch defined, 11. 
 
 Character of curves in bevel gearing, 46. 
 
 Clearance defined, 15; proportion for, 17. 
 
 Clock gears, 17. 
 
 Conchoid of Nicomedes, 62. 
 
 Conditions governing the practical case, 16. 
 
 Conjugate curves defined, 9, 63. 
 
 Constant angle of pressure, 28. 
 
 Constant velocity ratio defined, 1. 
 
 Conventional representation of spur gears, 25. 
 
 Contact, i)oint of, 5; radius, 5; path of, 12; arc of, 13. 
 
 Coiirdinate odontograph, 61. 
 
 Crown gear, 47. 
 
 8a
 
 84 
 
 INDEX. 
 
 Curtate epitrochoid, 6. 
 Curve of least clearance, 15. 
 Curves, odoutoidal, 4. 
 Cutting bevel gear teeth, 67. 
 Cutting angle, 51. 
 
 Cycloid defined, 4 ; problem relating to, 70. 
 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, 78. 
 Cycloidal bevel gear problem, 80, 81. 
 Cycloidal limiting case problems, 71, 73. 
 Cycloidal practical case problem, 74. 
 
 Dedendum defined, 12 ; proportions for, 17. 
 
 Defects of involute system, 35. 
 
 Describing circle defined, 4; a path of contact, 12; maximum 
 and minimum, 16; inHuenee on shape and efficiency of 
 teeth, 18; relation to interchangeable gears, 20. 
 
 Describing disk, 8. 
 
 Describing point, 4. 
 
 Describing cone, 45. 
 
 Describing cylinder, 45. 
 
 Describing i-adius, 5. 
 
 Description of bevel gear table, 53. 
 
 Developed pitch circle, 50. 
 
 Development of normal cone, 40. 
 
 Diameter pitch, 11. 
 
 Director circle, 5. 
 
 Double contact in annular gearing, .39. 
 
 Double generation of epicycloid and hypocycloid, G. 
 
 Drafting bevel gears, 48. 
 
 "Elements of Machine Design," 66. 
 
 "Elementary Mechanism," 67. 
 
 Epicycloid defined, 5; second method for describing, 5; 
 double generation, 6; spherical, 45; problem relating 
 to, 70. 
 
 Epicycloidal extension, .30. 
 
 Epitrochoid defined, 0; curtate, 6; prolate, 7; problem relat- 
 ing to, 70. 
 
 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," 66. 
 
 Formulas for worm and gear, 05. 
 
 Formulas, Notation and, 67. 
 
 Gearing, 1. 
 
 Gear arm iiroportions, 67. 
 
 Gears, interchangeable, 20 ; face of, 24 ; comparison of, 24. 
 Generating point, 4. 
 Generating radius, 5. 
 
 Grant, Geo. B., bevel gear chart, 53 ; three point odontograph 
 58; involute odoutograiih, 59; " Odontics," 66. 
 
 Hyperboloid of revolution, 2. 
 
 Hyperbolic gears, 2. 
 
 Hypocycloid defined, 5; second method for describing, 5; a 
 
 radial line, 6; double generation, 6; spherical, 45; pi'ob- 
 
 lem relating to, 70.
 
 INDEX. 
 
 86 
 
 Influence of tlie angle of pressure, 33. 
 
 Influence of the diameter of rolling circle on shape and effi- 
 ciency of teeth, 18. 
 
 Inner describing circle, 39; limitations of, 40, 41. 
 
 Inner normal cone, 48, 50. 
 
 Instantaneous radius, 4. 
 
 Intermediate describing circle, :i'.>, limitations of, 40. 
 
 Internal gear, see annular gear. 
 
 Interference, .32 ; in annular gearing, 43. 
 
 Interchangeable gears, 20. 
 
 Involute, 4; defined,?; system, 20; curves, character of, 27: 
 rack, 28; system of annular gearing, 43; annular gear 
 problem, 7!); bevel gear tooth, 4(;; bevel gear problems, 
 80,81; limiting case, 20; limiting case problem, 7(5; prac- 
 tical case, .'50; practical case problem, 77. 
 
 Involute action, Tlieory of, 2(5; limit of, 28. 
 
 Involute gearing, defects of system, 35. 
 
 Involute teeth, epicycloidal extension of, 30. 
 
 "Kinematics," MacCord's, (5(5. 
 Klein's cotlrdinate odontograph, (51; 
 Design," 06. 
 
 " Elements of Machine 
 
 Law of tooth contact, 10. 
 
 Lead of screw, (5(5. 
 
 Least angle of pressure, method for determining, 33. 
 
 Least number of teeth in annular gears, 42. 
 
 Limit of involute action, 28. 
 
 Limiting case, cydoidal, 10, 14; involute, 2!); annular gear- 
 ing, 38. 
 
 Limitations of intermediate, exterior, and interior describing 
 circle, 40, 41. 
 
 Line of action a great circle, 47. 
 
 Literature, 65. 
 Logarithmic spiral, 01. 
 
 MacCord's " Kinematics," 66. 
 
 Method for determining least angle of pressure, 33. 
 
 Method to be observed in jierforming problems, 70. 
 
 " Mechanics of Engineering," (5(5. 
 
 " Mechanics of the Machinery of 'I'lansmission," (50. 
 
 Normal defined, 4; to construct, 5; law governing, 03. 
 Normal cone, 48 : development of, 49. 
 Notation and formulas, (57. 
 
 Obliijuity, angle of, 14. 
 Octoid bevel tooth, 47, 04. 
 
 " Odontics," " A Treatise on Gear Wheels," Grant's, (50. 
 Odontoid defined, 1 ; special forms of, (52. 
 Odontoidal curves, 4: problems relating to, 70. 
 Odontographs and odontograph tables, .57. 
 Odoiitograi)h, Willis, 00; Grant involute, 59; Grant Three- 
 point, .58; Robinson, (il ; Klein, 01 ; coordinate, 61. 
 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, 13, 27. 
 Pitch circular, 11 ; diameter, 11. 
 Planed bevel gear teeth, 07.
 
 86 
 
 INDEX. 
 
 Positive rotation defined, 11. 
 
 Practical case, conditions governing the, 16; cycloidal, 21; 
 
 involute, 30; annular, 42. 
 " Practical Treatise on Gearing," G6. 
 Pressure, angle of, 14. 
 Prolate epitrochoid, 7. 
 Proportions for standard tooth, 17. 
 Problems, method to be observed in performing, 70. 
 
 Rack, 14; involute, 28; gears classified by, 62. 
 
 Radial flank, 18; as base of system, 21, 62. 
 
 Radius, describing, 5; ccntact, 5. 
 
 Rankine, 66. 
 
 Reuleux, 66. 
 
 Robinson odontograph, 61. 
 
 Rolling circle, see describing circle. 
 
 Rotation, positive, 1, 
 
 Screw gearing defined, 3. 
 
 Scroll, use of, 11. 
 
 Second method for describing cycloidal curves, 5. 
 
 Secondary action in annular gearing, 38, 41. 
 
 Seo;mental system, 62. 
 
 Skew gear defined, 3. 
 
 Spiral gear defined, 3. 
 
 Special forms of odontoids, 62. 
 
 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, 67. 
 
 Theory of cycloidal action, 8. 
 Theory of involute action, 26. 
 Thickness of tooth, 17. 
 Three-point odontograph, 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 affected by increase of center 
 distance in involute, 28. 
 
 Weisbach's " Mechanics," 66. 
 
 Willis, odontograph of, 60; writings of, 66. 
 
 Worm gearing defined, 3, 64; notation and formulas for, 69. 
 
 Worm wheel, 64.
 
 Plate I. 
 
 Cycloid, Epicycloid, Hypocycloid and Involute curves. 
 
 REFERENCES TO TEXT. 
 
 Art. 4, Page 4. Art. 8, Page 5. 
 
 5, 4. 9, 5. 
 
 6, 5. 12, 7. 
 
 7, 5.
 
 Plate l. 
 
 Fig. 3
 
 Plate 2 
 
 Plate 2. 
 
 Epitrochoidal curves. Double generation of Epicycloid and 
 Hypocycloid. Approximate method. 
 
 EEFEREXCES TO TEXT. 
 
 Abt. 10, Page 6. 
 11, 6. 
 
 34, 22. 
 
 -2l?l-E FOR EPltMOi'i'S
 
 Plate 3. 
 
 Mechanical method for describing Odontoidal curves. 
 
 REFERENCES TO TEXT. 
 
 Art. 13, Page 8. 
 15, 10. 
 
 21, 12.
 
 Plate 3,
 
 Plate 4. 
 
 Plate 4. 
 
 Cycloidal Gear, Pinion and Rack having action on one side of 
 pitch point. Limiting case. 
 
 TSF.li'F.KENCES To TEXT. 
 
 i'a.-f in. Ar.T. 24, 
 
 10. 2.-,, 
 
 18, 1:;. -M, 
 
 19, 1:;. 3ij, 
 
 14. 
 
 14. 
 14. 
 •24. 
 •37.
 
 Plate 5. 
 
 Cycloidal Gear, Pinion and Rack having action on both sides 
 of the pitch point. Limiting case. 
 
 KEFERESCES TO TEXT. 
 
 Art. 23, Page 13. 
 
 26, 14. 
 
 28, 15. 
 
 32, 19. 
 
 Art. 36, Page 24. 
 
 411, 38. 
 
 Pkob. 2, 74. 
 
 Plate 5. 
 
 
 
 
 
 u! r 
 
 ^\ 
 
 
 P^ 
 
 
 
 
 D 
 
 iiv/ 
 
 s 
 
 \ 
 
 ^^'^'^^^^~^^^^''^^--/ 
 
 /flT'^' \ -*- 
 
 
 
 
 / \\ '''' 1 
 
 ^\ 
 
 / 
 
 Y"'''/AvN' / ■''' "^ 
 
 
 
 \\i//' 
 
 \ 
 
 1 
 
 i \<^'rk^-'N^| ■■■ 1 ^\ 
 
 
 
 L-^!^-.4-.- 
 
 ] 
 
 F [ 
 
 i '■■^''^4 1 
 
 G 
 
 ^"^ii;^^ \ \ i 
 
 
 \ 
 
 ! *\ \l i /' 
 
 
 ^^*r'^^~"*^ V 
 
 
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 1 ,flV\ ]l / 
 
 
 y K'^i^' 
 
 PINION 
 
 
 
 
 .y^^i*' 
 
 
 
 /^^""^---...ilg)!^' '="''"^^™ 
 
 
 -^^^^T/x 
 
 
 
 \ *' /^ 
 
 
 f i/ \ 
 
 
 
 \ / 
 
 
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 \ / % 
 
 
 ^x 
 
 
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 ^" — M ' 
 
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 / 
 
 
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 Plate 6. 
 
 Plate 6. 
 
 Cycloidal Gear, Pinion and Rack. Practical case. 
 
 HKKKIIKNCKS TO TE.XT. 
 
 AiiT. 34, Page 21. 
 36, 24.
 
 Plate 7. 
 
 Involute Gear and Pinion. Limiting case. Mechanical method 
 for describing the Involute. 
 
 REFERENCES TO TEXT. 
 
 Art. 38, Page 26. 
 39, 27. 
 
 42. 31.
 
 Plate 7.
 
 TI 
 
 f 
 
 J
 
 PLATE 8. 
 
 Plate 8. 
 
 Involute Gear, Pinion and Rack. Limiting case. 
 
 EEFKKENCES T(J TEXT. 
 
 Art. 3fl, Page 28. 
 40, 29. 
 
 Art. 41, Page 30. 
 Peob. 4, 76.
 
 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. 
 
 GEAR 30 TEETH 
 
 1 PITCH INVOLUTE GEAR & PINION 
 SHOWING INTERFERENCE
 
 Plate lo. 
 
 Plate 10. 
 
 One Pitch Involute Pinion and Rack, showing Interference. 
 
 IlEFEKENCES TO TEXT. 
 
 Art. 42, Page 30. 
 43, 32 
 
 AuT. 44, Puge 33. 
 46, 35. 
 
 1 PITCH INVOLUTE PINION i RACK 
 SHOWING INTERFERENCE
 
 Plate II. 
 
 Annular Gearing. 
 
 REFERENCES TO TEXT. 
 
 Art. 49, Page 38. Art. 52, Page 41. 
 
 50, 39. 54, 42.
 
 Plate 11
 
 Plate 12. 
 
 Annular Gearing. Special cases. 
 
 KEFERENCES TO TEXT. 
 
 Art. 50, Page 39. 
 
 51, 40. 
 
 52, 40.
 
 Plate 12 
 
 x.Yv
 
 Plate 13. 
 
 Bevel Gearing. 
 
 KEFERENCES TO TEXT. 
 
 Art. 59, Page 47. Art. 60, Page 48.
 
 Plate 13
 
 NKS 
 
 i 
 
 ■Fig. 2 
 
 ^ / 
 
 Fig. 4
 
 Plate 14 
 
 Plate 14. 
 
 Problems i to 4 inclusive. 
 
 REFERENCES TO TEXT. 
 
 Art. 75, Page 70. 
 
 Prob. 1, U. 
 
 2, 73. 
 
 Prob. 3, Page 74. 
 4, 76. 
 
 I ^-i-rc^..^' 
 
 
 / j V yfty / 
 
 / I J// ./\;/i 6 TEETH 
 
 ;( DRIVER /■% / 
 
 ilv .^^' hi J 
 
 \< : 6.334 — ->-
 
 Plate 15. 
 
 Problems 5 to 9 inclusive. 
 
 REFERENCES TO TEXT. 
 
 Art. 75, Page 70. Prob. 7, Page 79. 
 
 Prob. 5, 77. 8, 80. 
 
 6, 78. 9, 81.
 
 Plate 15
 
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 STAMPED BELOW.
 
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