The Geometry of Involute Gears

The Geometry of Involute Gears

J.R. Colbourne The Geometry of Involute Gears With 217 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo

J.R. Colbourne Department of Mechanical Engineering The University of Alberta Edmonton, Alberta Canada T6G 2G8 Library of Congress Cataloging in publication Data Colbourne. J.R. The geometry of involute gears. Bibliography: p. Includes index. I. Gearing. Spur. I. Title. TJ 189.C65 1987 621.8'331 87-4843 1987 by Springer-Verlag New York Inc. Softcover reprint of the hardcover I st edition 1987 All rights reserved. This work may not be translated or copied ill whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue. New York, New York 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication. even if the former are not especially identified, is not to be taken as a sign that such names. as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. 987654321 [SBN-13: 978-1-4612-9146-6 e-[sbn-13: 978-1-4612-4764-7 00[: 10.1007/978-1-4612-4764-7

Table of Contents Introduction PART SPUR GEARS 1. The Law of Gearing 9 The Requirement for a Constant Angular Veloci ty Ratio 9 Rack and Pinion 13 Law of Gearing for Two Gears 19 Conjugate Profiles and the Basic Rack 23 2. Tooth Profile of an Involute Gear Basic Involute Rack Standard Pitch Circle The Involute Tooth Profile The Involute Function Pressure Angle of a Gear Tooth Thickness Specifying a Spur Gear 24 24 25 27 30 33 41 49 3. Gears in Mesh A Pinion Meshed wi th a Rack A Pair of Gears in Mesh Imaginary Rack Fundamental Circles of a Gear Advantages of the Involute Profile 53 53 65 72 78 79 4. Contact Ratio Interference and Backlash Contact Ratio Interference Backlash 83 83 91 97

vi Contents 5. Gear Cutting I Spur Gears Form Cutting Shaping with a Pinion Cutter Advantages of Generating Cutting Shaping with a Rack Cutter Hobbing Cutter Tooth Tip Geometry Undercutting 110 110 112 122 123 128 133 140 6. Profile Shift Definition of Profile Shift Geometric Design of a Spur Gear Pair Alternative Names for Profile Shift 148 151 155 168 7. Miscellaneous Circles 174 Highest and Lowest Points of Single-Tooth Contact 174 Form Diameter 177 Undercut Circle 179 8. Measurement of Tooth Thickness Gear-Tooth Vernier Caliper Span Measurement Measurement Over Pins 191 192 196 200 9. Geometry of Non-Involute Gears General Theory Fillet Shape Cut by a Rack Cutter Fillet Shape of an Undercut Gear Profile Modification Fillet Shape Cut by a Pinion Cutter 207 207 212 216 218 221 10. Curvature of Tooth Profiles Involute Radius of Curvature Euler-Savary Equation Gear Tooth Fillet Radius of Curvature 229 229 229 235 11. Tooth Stresses in Spur Gears Contact Force Intensi ty Contact Stress Fillet Stress 241 243 244 248

Contents vii 12. Internal Gears Tooth Profile of an Internal Gear Meshing Geometry of an Internal Gear Pair Tip Interference Axial and Radial Assembly Cutting Internal Gears Shape of the Fillet Undercutting Rubbing Geometric Design of an Internal Gear Pair Measurement of Tooth Thickness 259 259 266 272 275 279 283 288 292 294 297 PART 2 HELICAL GEARS 13. Tooth Surface of a Helical Involute Gear 305 The Basic Helical Rack 309 Standard pi tch Cylinder of a Helical Gear 313 The Helix and the Involute Helicoid 318 The Generator Through Point A 331 Direction of the Normal to the Tooth Surface at A 337 Tooth Thickness 350 Span Measurement 354 Tooth Profile in the Normal Section 359 Specifying a Helical Gear 362 14. Helical Gears in Mesh 366 A Pinion Meshed wi th a Rack 366 A Pair of Helical Gears in Mesh 375 Contact Ratio 382 Backlash 387 position and Orientation of the Contact Line 393 15. Crossed Helical Gears 405 Rack and Pinion 406 A Crossed Helical Gear Pair 414 Path of Contact 423 Contact Ratio 432 Backlash 437 Tooth Contact Force and Bearing Reactions 445

viii Contents 16. Gear Cutting II Helical Gears 451 Shaping wi th a Pinion Cutter 451 Shaping wi th a Rack Cutter 454 Hobbing 457 Hobbing Machine Gear Train Layout 469 Use of a Differential in the Hobbing Machine 472 Theoretically Correct Shape for the Hob Thread 479 Geometric Design of a Helical Gear Pair 483 17. Tooth Stresses in Helical Gears 489 Tooth Contact Force 489 Contact Length 491 Contact Stress 502 Fillet Stress, and the Equivalent Spur Gear 508 Bibliography 525 Index 527

Notation Lower Case Symbols a b c e f h k m p r s t u v w x,y,z Addendum Dedendum Clearance Profile shift Special function defined by Equation (13.89) Length on tooth of rack cutter Dimensionless factor Ratio pi tch Radius of circles not centered at the gear axis Coordinate, measured along path of contact Tooth thickness position of rack Velocity Space width; Tooth contact force intensity Coordinates Upper Case Symbols A B C D E F G H I,J Point on tooth profile Point where the involute meets the base circle; Backlash Gear center; Center distance Diameter Interference point; Young's modulus Face width Point where the generator meets the base cylinder End points of the contact path during cutting AGMA geometry factors

x Notation Upper Case Symbols (continued) K L M N P Q R S T W X,Y Dimensionless factor Lead Measurement over pins; Torque Number of teeth Pitch point; Diametral pitch End points of single-tooth contact Radius of circles centered at gear axis; Polar coordinate Span measurement End points of the contact path Tooth contact force Coordinates Ital ic Symbols m n p v Module Uni t vector Posi tion vector Veloci ty vector Greek Symbols a f p o T w General angle Angular position of gear Angle between involute tangent and tooth center-line Roll angle Coordinates Polar coordinate Lead angle Generator inclination angle; Poisson's ratio Radius of curvature Stress; Swivel angle Time Profile angle; Pressure angle Helix angle Angular velocity, measured in radians/sec Shaft angle

Notation xi Subscripts b c d e f g h i n p r s t u w x,y,z F L P R T L71, S >.. At the base cylinder Contact; Cutter At the defining circle of the equivalent spur gear Equivalent spur gear Fillet Gear Hob; At the highest point of the involute 1 or 2 Normal At the pitch cylinder Rack; Rack cutter Standard; At the standard pi tch cylinder Transverse; Tensile Undercut At the load point In the x,y,z direction Face Limit Profile At radius R At the tip cylinder In the L71,S directions In the direction of the helix binormal In the direction of the helix tangent Superscript c Cutting