GEARS AND GEAR DRIVES GEARS AND GEAR DRIVES Damir Jelaska University of Split, Croatia This edition first published 2012 # 2012 John Wiley & Sons Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permis- sion to reuse the copyright material in this book please see our website at www. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.
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Library of Congress Cataloging-in-Publication Data Jelaska, Damir. Gears and gear drives / Damir Jelaska. Includes bibliographical references and index.80 33–dc23 2012014164 A catalogue record for this book is available from the British Library. Print ISBN: 9781119941309 Set in 10/12 pt Times by Thomson Digital, Noida, India In memory of my mother Contents Preface xv Acknowledgments xvii 1 Introduction 1 1.1 Power Transmissions and Mechanical Drives 1 1.2 Classification of Mechanical Drives 3 1.3 Choosing a Mechanical Drive 7 1.4 Multi-Step Drives 9 1.5 Features and Classification of Gear Drives 12 1.1 Features of Gear Drives 12 1.2 Classification of Gear Drives 12 1.6 List of Symbols 16 1.1 Subscripts to Symbols 16 2 Geometry of Cylindrical Gears 17 2.1 Fundamentals of the Theory of Toothing 17 2.1 Centrodes, Roulettes and Axodes 17 2.2 Envelopes, Evolutes and Involutes 18 2.3 Cycloid and Involute of a Circle 18 2.2 Involute of Circle 20 2.4 Main Rule of Toothing 21 2.1 Analytical Determining of Mated Profiles 25 2.2 Radii of Curvature of Mated Profiles 27 2.2 Geometry of Pairs of Spur Gears 29 2.3 Involute Teeth and Involute Gears 33 2.4 Basic Tooth Rack 35 2.5 Fundamentals of Cylindrical Gears Manufacture 38 2.2 Forming Methods 43 viii Contents 2.4 Basic Rack-Type and Pinion-Type Cutters 48 2.6 Cutting Process and Geometry of Gears Cut with Rack-Type Cutter 49 2.1 Profile Shift 49 2.2 Meshing of Rack Cutter with Work Piece, Basic Dimensions of Gear 50 2.3 Tooth Thickness at Arbitrary Circle 51 2.4 Tip Circle Diameter 52 2.5 Profile Boundary Point; Tooth Root Undercutting 53 2.6 Effect of Profile Shift on Tooth Geometry 55 2.7 Gear Control Measures 56 2.1 Chordal Tooth Thickness on the Arbitrary Circle 56 2.2 Constant Chord Tooth Thickness 57 2.4 Dimension Over Balls 60 2.7 Parameters of a Gear Pair 62 2.1 Working Pressure Angle of a Gear Pair 62 2.3 Gear Pairs With and Without Profile Shift 64 2.1 Gear Pairs Without Profile Shift 64 2.2 Gear Pairs with Profile Shift 64 2.5 Distinctive Points of Tooth Profile 70 2.6 Kinematic Parameters of Toothing 71 2.8 Basic Parameters of Gears Generated by the Fellows Method 74 2.1 Pinion-Type Cutter 74 2.2 Dimensions of Gears Cut by Pinion-Type Cutter 75 2.3 Undercutting the Tooth Root 76 2.4 Geometry of Internal Gear Toothing 77 2.9 Interferences in Generating Processes and Involute Gear Meshing 78 2.1 Interferences in Tooth Cutting 78 2.1 Tooth Root Undercutting 78 2.2 Overcutting the Tooth Addendum (First Order Interference) 79 2.3 Overcutting the Tooth Tip Corner (Second Order Interference) 80 2.2 Interferences in Meshing the Gear Pair Teeth 83 2.1 Gear Root Interference 83 2.2 Interferences of Tooth Addendum 84 2.10 Choosing Profile Shift Coefficients 84 2.1 Choosing Profile Shift Coefficients by Means of Block-Contour Diagrams 85 2.2 Choosing Profile Shift Coefficients by Means of Lines of Gear Pairs 88 2.2 Helical Gear Dimensions and Parameters of a Gear Pair 97 Contents ix 2.4 Helical Gear Overlaps 102 2.1 Length of Contact Lines 104 2.12 Tooth Flank Modifications 106 2.1 Transverse Profile Modifications 107 2.1 Pre-Finish Flank Undercut 107 2.2 Tip Corner Chamfering and Tip Corner Rounding 107 2.3 Tooth Tip Relief 108 2.4 Tooth Root Relief 113 2.5 Tooth Tip Relief of the Gear Generated by Pinion-Type Cutter 114 2.6 Profile Crowning 117 2.2 Flank Line Modifications 117 2.1 Flank Line end Reliefs 117 2.2 Flank Line Slope Modification 117 2.3 Flank Line Crowning 118 2.13 Geometry of Fillet Curve 119 2.1 Fillet Curve Equation 120 2.2 Fillet Curve Radius of Curvature 124 2.3 Geometry of Undercut Teeth 125 2.1 Profile Boundary Point 125 2.2 Contact Ratio of Gears with Undercut Teeth 126 2.14 Tolerances of Pairs of Cylindrical Gears 127 2.1 Control and Tolerances of Gear Body 128 2.2 Control and Tolerances of Teeth 128 2.1 Tooth Profile Control 130 2.4 Radial Runout of Teeth 136 2.5 Tangential Composite Deviation 136 2.6 Tooth Thickness Tolerances 138 2.7 CNC Gear Measuring Centre 143 2.3 Control of Gear Pair Measuring Values 145 2.1 Systems of Gear Fits, Centre Distance Tolerances, Backlash 145 2.2 Contact Pattern Control 149 2.15 Gear Detail Drawing 151 2.16 List of Symbols 153 2.1 Subscripts to symbols 154 2.2 Combined Symbols 155 3 Integrity of Gears 157 3.1 Forces Acting on the Gear Tooth 157 3.2 Incremental Gear Loadings 159 x Contents 3.2 Causes of Gear Damage 164 3.2 Active Tooth Flank Damage 166 3.3 Pitting Load Capacity 170 3.1 Nominal Value of Contact Stress 170 3.2 Real Value of Contact Stress 175 3.2 Allowable Contact Stresses 181 3.3 Dimensioning for Contact Stress 189 3.4 List of Symbols for Sections 3.1 Subscripts to Symbols 191 3.4 Tooth Root Load Capacity 193 3.1 Tooth Root Stress 193 3.2 Tooth Root Permitted Stress 200 3.3 Dimensioning for Tooth Root Stress 207 3.5 Gear Load Capacity at Variable Loading 208 3.6 List of Symbols for Sections 3.1 Subscripts to Symbols 211 3.7 Scuffing Load Capacity 213 3.1 Safety Factor Against Scuffing for Flash Temperature Method 213 3.2 Force Distribution Factor XG 217 3.3 Safety Factor Against Scuffing for Integral Temperature Method 225 3.8 Micro-Pitting Load Capacity 229 3.1 Elastohydrodynamic Lubricant Film Thickness 229 3.1 Calculation of Material Parameter GM 230 3.2 Calculation Speed Parameter UY 231 3.3 Load Parameter WY 232 3.4 Sliding Parameter SGF 232 3.2 Safety Factor Against Micro-pitting 232 3.9 List of Symbols for Sections 3.1 Subscripts to Symbols 237 3.2 Combined Symbols 238 4 Elements of Cylindrical Gear Drive Design 241 4.1 Design Procedure for a Gear Pair 241 4.2 Distribution of Gear Train Transmission Ratio 243 4.3 Gear Materials and Heat Treatment 244 4.1 Metallic Materials and their Heat Treatment 244 4.4 Gear Drive Design 249 4.1 Design of Housing 251 4.2 Vents 255 Contents xi 4.4 Design of Bearing Locations 257 4.5 Design of Ribs 257 4.5 Design of Gears 258 4.2 Gear Drive Lubrication 262 4.1 Selection of Lubricant 262 4.2 Ways of Gear Lubrication 263 4.3 Power Losses and Temperature of Lubricant 266 4.1 Power Losses in Mesh 266 4.1 Power Losses in Mesh, Under Load, for a Single Gear Pair 266 4.2 Power Losses in Idle Motion 267 4.2 Power Losses in Bearings 268 4.3 Power Losses in Seals 270 4.4 Power Efficiency of Gear Drive 270 4.5 Temperature of Lubricant 271 4.4 List of Symbols 275 4.1 Subscripts to Symbols 276 4.2 Combined Symbols 276 5 Bevel Gears 279 5.1 Geometry and Manufacture of Bevel Gears 279 5.1 Theory of Bevel Gear Genesis 279 5.2 Types and Features of Bevel Gears 280 5.3 Application of Bevel Gears 283 5.4 Geometry of Bevel Gears 284 5.1 Fundamentals of Geometry and Manufacture 284 5.2 Virtual Toothing and Virtual Gears 287 5.3 Basic Parameters of Straight Bevels 289 5.4 Design of Bevel Teeth 291 5.5 Undercut, Profile Shift 291 5.6 Sliding of Bevels 292 5.7 Contact Ratio of Straight Bevels 293 5.5 Geometry of Helical and Spiral Bevels 293 5.6 Manufacturing Methods for Bevel Gears 294 5.1 Straight Bevels Working 294 5.2 Spiral and Helical Bevel Working 301 5.2 Load Capacity of Bevels 306 5.1 Forces in Mesh 306 5.2 Pitting Load Capacity 307 5.3 Tooth Root Load Capacity 310 5.1 Scuffing and Micro-Pitting Load Capacities 311 xii Contents 5.3 Elements of Bevel Design 311 5.4 Control and Tolerances of Bevel Gears 316 5.2 Radial Runout Control of Toothing 318 5.3 Tangential Composite Deviation 319 5.4 Tooth Thickness Control 319 5.5 Bevel Gear Drawing 321 5.5 Crossed Gear Drives 321 5.2 Speed of Sliding 324 5.3 Loads and Load Capacity 325 5.1 Forces Acting on Crossed Gears 325 5.2 Efficiency Grade 325 5.3 Load Capacity of Crossed Gear Pair 326 5.6 List of Symbols 327 5.1 Subscripts to Symbols 328 5.2 Combined Symbols 328 6 Planetary Gear Trains 331 6.1 Fundamentals of Planetary Gear Trains 331 6.2 Rotational Speeds and Transmission Ratio 334 6.3 Features of Planetary Gear Trains 341 6.1 Condition of Coaxiality 342 6.2 Condition of Neighbouring 342 6.5 Diagrams of Peripheral and Rotational Speeds 344 6.7 Forces, Torques and Power of Planetary Gear Trains 347 6.1 Peripheral Forces and Torques 347 6.2 Power and Efficiency 349 6.3 Branching of Power 352 6.2 Special Layouts of Simple Planetary Gear Trains 356 6.1 Bevel Differential Trains 356 6.2 Planetary Gear Trains with Single Gear Pair 358 6.4 Differential Planetary Gear Trains 361 6.5 Planetary Gear Train of a Wankel Engine 362 6.3 Composed Planetary Gear Trains 364 6.1 Compound Planetary Gear Trains 364 6.2 Parallel Composed Planetary Gear Trains 364 6.3 Coupled Planetary Gear Trains 364 6.4 Closed Planetary Gear Trains 366 6.5 Reduced Coupled Planetary Gear Trains 368 Contents xiii 6.7 Planetary Gear Boxes 374 6.4 Elements of Planetary Gear Train Design 377 6.1 Issues of Planetary Gear Train Design 377 6.2 Calculations for Central Gears and Planets 382 6.5 List of Symbols 384 6.1 Subscripts to Symbols 385 6.2 Combined Symbols 386 7 Worm Gear Drives 387 7.1 Concept, Features, Classification 387 7.2 Geometry and Working of Worm Gear Pair 389 7.1 Geometry and Working of Worm 389 7.1 Dimensions of Worm 390 7.3 Worm Working and Shape of Flanks 392 7.2 Geometry and Working of Wormwheels 392 7.3 Calculation Values of Worm Gear Pair 399 7.1 Centre Distance of Worm Gear Pair 399 7.2 Transmission Ratio and Gear Ratio 399 7.3 Tip Clearance of Worm Gear Pair 399 7.4 Contact Ratio of Worm Gear Pair 399 7.5 Worm Gear Pair Speeds 400 7.3 Control Measures and Tolerances of Worm Gear Pair 400 7.1 Control of Worm Measuring Values 401 7.2 Thread Profile Control 401 7.3 Radial Runout Control 402 7.2 Control of Wormwheel Measuring Values 402 7.2 Tooth Profile Control 402 7.3 Radial Run-Out Control 402 7.4 Tooth Thickness Control 403 7.5 Composite Deviation Control 403 7.3 Measuring Values Control of Worm Gear Pair 403 7.1 Centre Distance Control 403 7.4 Forces, Power Losses and Efficiency of Worm Gear Drives 404 7.1 Forces Acting on Worm Gear Pair 404 7.2 Power Losses and Efficiency of Worm Gear Pair 406 7.5 Load Capacity of Worm Gear Pair 409 7.1 Wear Load Capacity 409 7.1 Calculation of Expected Wear 410 7.2 Permitted Wear 413 xiv Contents 7.2 Pitting Load Capacity 414 7.3 Heating Load Capacity 415 7.1 Heating Load Capacity at Bath Lubrication 416 7.2 Heating Load Capacity at Spray Lubrication 416 7.