DAMIR JELASKA
`
`GEAR
`
`AND GEAI see
`
`LIBERTY EXHIBIT 1012, Page 1
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`

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`LIBERTY EXHIBIT 1012, Page 2
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`LIBERTY EXHIBIT 1012, Page 2
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`

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`GEARS AND GEAR
`DRIVES
`
`LIBERTY EXHIBIT 1012, Page 3
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`

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`LIBERTY EXHIBIT 1012, Page 4
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`LIBERTY EXHIBIT 1012, Page 4
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`

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`GEARS AND GEAR
`DRIVES
`
`Damir Jelaska
`
`University of Split, Croatia
`
`LIBERTY EXHIBIT 1012, Page 5
`
`

`

`This edition !rst published 2012
`© 2012 John Wiley & Sons Ltd
`
`Registered of!ce
`John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom
`
`For details of our global editorial of!ces, for customer services and for information about how to apply for permission to
`reuse the copyright material in this book please see our website at www.wiley.com.
`
`The right of the author to be identi!ed 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.
`
`Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in
`electronic books.
`
`Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product
`names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The
`publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accu-
`rate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is
`not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a
`competent professional should be sought.
`
`Library of Congress Cataloging-in-Publication Data
`
`Jelaska, Damir.
`Gears and gear drives / Damir Jelaska.
`p. cm.
`Includes bibliographical references and index.
`ISBN 978-1-119-94130-9 (cloth)
`1. Gearing.
`I. Title.
`TJ184.J415 2012
`621.8 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
`
`LIBERTY EXHIBIT 1012, Page 6
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`

`

`In memory of my mother
`
`LIBERTY EXHIBIT 1012, Page 7
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`

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`LIBERTY EXHIBIT 1012, Page 8
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`LIBERTY EXHIBIT 1012, Page 8
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`

`

`Contents
`
`Preface
`
`Acknowledgments
`
`1
`
`Introduction
`1.1 Power Transmissions and Mechanical Drives
`1.2 Classi!cation of Mechanical Drives
`1.3 Choosing a Mechanical Drive
`1.4 Multi-Step Drives
`1.5 Features and Classi!cation of Gear Drives
`1.5.1 Features of Gear Drives
`1.5.2 Classi!cation of Gear Drives
`1.6 List of Symbols
`1.6.1 Subscripts to Symbols
`
`2 Geometry of Cylindrical Gears
`2.1 Fundamentals of the Theory of Toothing
`2.1.1 Centrodes, Roulettes and Axodes
`2.1.2 Envelopes, Evolutes and Involutes
`2.1.3 Cycloid and Involute of a Circle
`2.1.3.1 Cycloid
`2.1.3.2
`Involute of Circle
`2.1.4 Main Rule of Toothing
`2.1.4.1
`Analytical Determining of Mated Pro!les
`2.1.4.2
`Radii of Curvature of Mated Pro!les
`2.2 Geometry of Pairs of Spur Gears
`2.2.1 Cycloid Toothing
`2.2.2 Involute Toothing
`Involute Teeth and Involute Gears
`2.3
`2.4 Basic Tooth Rack
`2.5 Fundamentals of Cylindrical Gears Manufacture
`2.5.1 Generating Methods
`2.5.2 Forming Methods
`
`xv
`
`xvii
`
`1
`1
`3
`7
`9
`12
`12
`12
`16
`16
`
`17
`17
`17
`18
`18
`18
`20
`21
`25
`27
`29
`29
`30
`33
`35
`38
`38
`43
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`LIBERTY EXHIBIT 1012, Page 9
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`

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`viii
`
`Contents
`
`45
`2.5.3 Gear Finishing
`48
`2.5.4 Basic Rack-Type and Pinion-Type Cutters
`49
`2.6 Cutting Process and Geometry of Gears Cut with Rack-Type Cutter
`49
`2.6.1 Pro!le Shift
`50
`2.6.2 Meshing of Rack Cutter with Work Piece, Basic Dimensions of Gear
`51
`2.6.3 Tooth Thickness at Arbitrary Circle
`52
`2.6.4 Tip Circle Diameter
`53
`2.6.5 Pro!le Boundary Point; Tooth Root Undercutting
`55
`2.6.6 Effect of Pro!le Shift on Tooth Geometry
`56
`2.6.7 Gear Control Measures
`56
`2.6.7.1 Chordal Tooth Thickness on the Arbitrary Circle
`57
`2.6.7.2 Constant Chord Tooth Thickness
`58
`2.6.7.3
`Span Measurement
`60
`2.6.7.4 Dimension Over Balls
`62
`2.7 Parameters of a Gear Pair
`62
`2.7.1 Working Pressure Angle of a Gear Pair
`63
`2.7.2 Centre Distance
`64
`2.7.3 Gear Pairs With and Without Pro!le Shift
`64
`2.7.3.1 Gear Pairs Without Pro!le Shift
`64
`2.7.3.2 Gear Pairs with Pro!le Shift
`66
`2.7.4 Contact Ratio
`70
`2.7.5 Distinctive Points of Tooth Pro!le
`71
`2.7.6 Kinematic Parameters of Toothing
`74
`2.8 Basic Parameters of Gears Generated by the Fellows Method
`74
`2.8.1 Pinion-Type Cutter
`75
`2.8.2 Dimensions of Gears Cut by Pinion-Type Cutter
`76
`2.8.3 Undercutting the Tooth Root
`77
`2.8.4 Geometry of Internal Gear Toothing
`78
`Interferences in Generating Processes and Involute Gear Meshing
`78
`2.9.1 Interferences in Tooth Cutting
`78
`2.9.1.1
`Tooth Root Undercutting
`79
`2.9.1.2 Overcutting the Tooth Addendum (First Order Interference)
`2.9.1.3 Overcutting the Tooth Tip Corner (Second Order Interference) 80
`80
`2.9.1.4
`Radial Interference (Third Order Interference)
`82
`2.9.1.5 Null Fillet
`83
`2.9.2 Interferences in Meshing the Gear Pair Teeth
`83
`2.9.2.1 Gear Root Interference
`84
`2.9.2.2
`Interferences of Tooth Addendum
`84
`2.9.2.3
`Radial Interference
`2.10 Choosing Pro!le Shift Coef!cients
`84
`2.10.1 Choosing Pro!le Shift Coef!cients by Means of Block-Contour
`Diagrams
`2.10.2 Choosing Pro!le Shift Coef!cients by Means of Lines of Gear Pairs
`2.11 Helical Gears
`2.11.1 Basic Considerations
`2.11.2 Helical Gear Dimensions and Parameters of a Gear Pair
`
`85
`88
`91
`91
`97
`
`2.9
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`LIBERTY EXHIBIT 1012, Page 10
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`

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`Contents
`
`2.11.3 Control Measures
`2.11.4 Helical Gear Overlaps
`2.11.4.1 Length of Contact Lines
`2.12 Tooth Flank Modi!cations
`2.12.1 Transverse Pro!le Modi!cations
`2.12.1.1 Pre-Finish Flank Undercut
`2.12.1.2 Tip Corner Chamfering and Tip Corner Rounding
`2.12.1.3 Tooth Tip Relief
`2.12.1.4 Tooth Root Relief
`2.12.1.5 Tooth Tip Relief of the Gear Generated by Pinion-Type
`Cutter
`2.12.1.6 Pro!le Crowning
`2.12.2 Flank Line Modi!cations
`2.12.2.1 Flank Line end Reliefs
`2.12.2.2 Flank Line Slope Modi!cation
`2.12.2.3 Flank Line Crowning
`2.12.3 Flank Twist
`2.13 Geometry of Fillet Curve
`2.13.1 Fillet Curve Equation
`2.13.2 Fillet Curve Radius of Curvature
`2.13.3 Geometry of Undercut Teeth
`2.13.3.1 Pro!le Boundary Point
`2.13.3.2 Contact Ratio of Gears with Undercut Teeth
`2.14 Tolerances of Pairs of Cylindrical Gears
`2.14.1 Control and Tolerances of Gear Body
`2.14.2 Control and Tolerances of Teeth
`2.14.2.1 Tooth Pro!le Control
`2.14.2.2 Helix Deviations
`2.14.2.3 Pitch Deviations
`2.14.2.4 Radial Runout of Teeth
`2.14.2.5 Tangential Composite Deviation
`2.14.2.6 Tooth Thickness Tolerances
`2.14.2.7 CNC Gear Measuring Centre
`2.14.3 Control of Gear Pair Measuring Values
`2.14.3.1 Systems of Gear Fits, Centre Distance Tolerances,
`Backlash
`2.14.3.2 Contact Pattern Control
`2.15 Gear Detail Drawing
`2.16 List of Symbols
`2.16.1 Subscripts to symbols
`2.16.2 Combined Symbols
`
`3
`
`Integrity of Gears
`3.1 Gear Loadings
`3.1.1 Forces Acting on the Gear Tooth
`3.1.2 Incremental Gear Loadings
`
`ix
`
`100
`102
`104
`106
`107
`107
`107
`108
`113
`
`114
`117
`117
`117
`117
`118
`119
`119
`120
`124
`125
`125
`126
`127
`128
`128
`130
`134
`135
`136
`136
`138
`143
`145
`
`145
`149
`151
`153
`154
`155
`
`157
`157
`157
`159
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`LIBERTY EXHIBIT 1012, Page 11
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`

`

`x
`
`Contents
`
`3.2 Causes of Gear Damage
`3.2.1 Gear Breakages
`3.2.2 Active Tooth Flank Damage
`3.3 Pitting Load Capacity
`3.3.1 Contact Stresses
`3.3.1.1 Nominal Value of Contact Stress
`3.3.1.2
`Real Value of Contact Stress
`3.3.2 Allowable Contact Stresses
`3.3.3 Dimensioning for Contact Stress
`3.3.4 List of Symbols for Sections 3.1, 3.2 and 3.3
`3.3.4.1
`Subscripts to Symbols
`3.3.4.2 Combined Symbols
`3.4 Tooth Root Load Capacity
`3.4.1 Tooth Root Stress
`3.4.2 Tooth Root Permitted Stress
`3.4.3 Dimensioning for Tooth Root Stress
`3.5 Gear Load Capacity at Variable Loading
`3.6 List of Symbols for Sections 3.4 and 3.5
`3.6.1 Subscripts to Symbols
`3.6.2 Combined Symbols
`3.7 Scuf!ng Load Capacity
`3.7.1 Safety Factor Against Scuf!ng for Flash Temperature Method
`3.7.2 Force Distribution Factor X
`3.7.3 Safety Factor Against Scuf!ng for Integral Temperature Method
`3.8 Micro-Pitting Load Capacity
`3.8.1 Elastohydrodynamic Lubricant Film Thickness
`3.8.1.1 Calculation of Material Parameter GM
`3.8.1.2 Calculation of Speed Parameter UY
`3.8.1.3
`Load Parameter WY
`3.8.1.4
`Sliding Parameter SGF
`3.8.2 Safety Factor Against Micro-pitting
`3.9 List of Symbols for Sections 3.6 and 3.7
`3.9.1 Subscripts to Symbols
`3.9.2 Combined Symbols
`
`4 Elements of Cylindrical Gear Drive Design
`4.1 Design Process
`4.1.1 Design Procedure for a Gear Pair
`4.1.2 Distribution of Gear Train Transmission Ratio
`4.1.3 Gear Materials and Heat Treatment
`4.1.3.1 Metallic Materials and their Heat Treatment
`4.1.3.2
`Sintered Materials
`4.1.3.3
`Polymer Materials
`4.1.4 Gear Drive Design
`4.1.4.1 Design of Housing
`4.1.4.2
`Vents
`
`164
`164
`166
`170
`170
`170
`175
`181
`189
`190
`191
`192
`193
`193
`200
`207
`208
`210
`211
`212
`213
`213
`217
`225
`229
`229
`230
`231
`232
`232
`232
`236
`237
`238
`
`241
`241
`241
`243
`244
`244
`248
`248
`249
`251
`255
`
`LIBERTY EXHIBIT 1012, Page 12
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`

`

`Contents
`
`Lubricant Drain
`4.1.4.3
`4.1.4.4 Design of Bearing Locations
`4.1.4.5 Design of Ribs
`4.1.5 Design of Gears
`4.2 Gear Drive Lubrication
`4.2.1 Selection of Lubricant
`4.2.2 Ways of Gear Lubrication
`4.2.2.1
`Bath Lubrication
`4.2.2.2
`Spray Lubrication
`4.3 Power Losses and Temperature of Lubricant
`4.3.1 Power Losses in Mesh
`4.3.1.1
`Power Losses in Mesh, Under Load, for a Single
`Gear Pair
`Power Losses in Idle Motion
`4.3.1.2
`4.3.2 Power Losses in Bearings
`4.3.2.1
`Rolling Bearings
`4.3.2.2
`Sliding Bearings
`4.3.3 Power Losses in Seals
`4.3.4 Power Ef!ciency of Gear Drive
`4.3.5 Temperature of Lubricant
`4.4 List of Symbols
`4.4.1 Subscripts to Symbols
`4.4.2 Combined Symbols
`
`5 Bevel Gears
`5.1 Geometry and Manufacture of Bevel Gears
`5.1.1 Theory of Bevel Gear Genesis
`5.1.2 Types and Features of Bevel Gears
`5.1.3 Application of Bevel Gears
`5.1.4 Geometry of Bevel Gears
`5.1.4.1
`Fundamentals of Geometry and Manufacture
`5.1.4.2
`Virtual Toothing and Virtual Gears
`5.1.4.3
`Basic Parameters of Straight Bevels
`5.1.4.4 Design of Bevel Teeth
`5.1.4.5 Undercut, Pro!le Shift
`5.1.4.6
`Sliding of Bevels
`5.1.4.7 Contact Ratio of Straight Bevels
`5.1.5 Geometry of Helical and Spiral Bevels
`5.1.6 Manufacturing Methods for Bevel Gears
`5.1.6.1
`Straight Bevels Working
`5.1.6.2
`Spiral and Helical Bevel Working
`5.2 Load Capacity of Bevels
`5.2.1 Forces in Mesh
`5.2.2 Pitting Load Capacity
`5.2.3 Tooth Root Load Capacity
`5.2.4 Scuf!ng and Micro-Pitting Load Capacities
`
`xi
`
`255
`257
`257
`258
`262
`262
`263
`263
`265
`266
`266
`
`266
`267
`268
`268
`269
`270
`270
`271
`275
`276
`276
`
`279
`279
`279
`280
`283
`284
`284
`287
`289
`291
`291
`292
`293
`293
`294
`294
`301
`306
`306
`307
`310
`311
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`LIBERTY EXHIBIT 1012, Page 13
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`

`

`xii
`
`Contents
`
`5.3 Elements of Bevel Design
`5.4 Control and Tolerances of Bevel Gears
`5.4.1 Pitch Control
`5.4.2 Radial Runout Control of Toothing
`5.4.3 Tangential Composite Deviation
`5.4.4 Tooth Thickness Control
`5.4.5 Bevel Gear Drawing
`5.5 Crossed Gear Drives
`5.5.1 Basic Geometry
`5.5.2 Speed of Sliding
`5.5.3 Loads and Load Capacity
`5.5.3.1
`Forces Acting on Crossed Gears
`5.5.3.2
`Ef!ciency Grade
`5.5.3.3
`Load Capacity of Crossed Gear Pair
`5.6 List of Symbols
`5.6.1 Subscripts to Symbols
`5.6.2 Combined Symbols
`
`6 Planetary Gear Trains
`6.1
`Introduction
`6.1.1 Fundamentals of Planetary Gear Trains
`6.1.2 Rotational Speeds and Transmission Ratio
`6.1.3 Features of Planetary Gear Trains
`6.1.4 Mating Conditions
`6.1.4.1 Condition of Coaxiality
`6.1.4.2 Condition of Neighbouring
`6.1.4.3
`Assembly Condition
`6.1.5 Diagrams of Peripheral and Rotational Speeds
`6.1.6 Wolf Symbolic
`6.1.7 Forces, Torques and Power of Planetary Gear Trains
`6.1.7.1
`Peripheral Forces and Torques
`6.1.7.2
`Power and Ef!ciency
`6.1.7.3
`Branching of Power
`6.1.7.4
`Self-Locking
`6.2 Special Layouts of Simple Planetary Gear Trains
`6.2.1 Bevel Differential Trains
`6.2.2 Planetary Gear Trains with Single Gear Pair
`6.2.3 Harmonic Drive
`6.2.4 Differential Planetary Gear Trains
`6.2.5 Planetary Gear Train of a Wankel Engine
`6.3 Composed Planetary Gear Trains
`6.3.1 Compound Planetary Gear Trains
`6.3.2 Parallel Composed Planetary Gear Trains
`6.3.3 Coupled Planetary Gear Trains
`6.3.4 Closed Planetary Gear Trains
`6.3.5 Reduced Coupled Planetary Gear Trains
`
`311
`316
`316
`318
`319
`319
`321
`321
`323
`324
`325
`325
`325
`326
`327
`328
`328
`
`331
`331
`331
`334
`341
`342
`342
`342
`343
`344
`347
`347
`347
`349
`352
`353
`356
`356
`358
`359
`361
`362
`364
`364
`364
`364
`366
`368
`
`LIBERTY EXHIBIT 1012, Page 14
`
`

`

`Contents
`
`6.3.6 Reverse Reducers
`6.3.7 Planetary Gear Boxes
`6.4 Elements of Planetary Gear Train Design
`6.4.1 Issues of Planetary Gear Train Design
`6.4.2 Calculations for Central Gears and Planets
`6.5 List of Symbols
`6.5.1 Subscripts to Symbols
`6.5.2 Combined Symbols
`
`7 Worm Gear Drives
`7.1 Concept, Features, Classi!cation
`7.2 Geometry and Working of Worm Gear Pair
`7.2.1 Geometry and Working of Worm
`7.2.1.1 Dimensions of Worm
`7.2.1.2 Worm Sections
`7.2.1.3 Worm Working and Shape of Flanks
`7.2.2 Geometry and Working of Wormwheels
`7.2.2.1 Wormwheel Geometry
`7.2.2.2 Wormwheel Working
`7.2.3 Calculation Values of Worm Gear Pair
`7.2.3.1 Centre Distance of Worm Gear Pair
`7.2.3.2
`Transmission Ratio and Gear Ratio
`7.2.3.3
`Tip Clearance of Worm Gear Pair
`7.2.3.4 Contact Ratio of Worm Gear Pair
`7.2.3.5 Worm Gear Pair Speeds
`7.3 Control Measures and Tolerances of Worm Gear Pair
`7.3.1 Control of Worm Measuring Values
`7.3.1.1
`Pitch Control
`7.3.1.2
`Thread Pro!le Control
`7.3.1.3
`Radial Runout Control
`7.3.2 Control of Wormwheel Measuring Values
`7.3.2.1
`Pitch Control
`7.3.2.2
`Tooth Pro!le Control
`7.3.2.3
`Radial Run-Out Control
`7.3.2.4
`Tooth Thickness Control
`7.3.2.5 Composite Deviation Control
`7.3.3 Measuring Values Control of Worm Gear Pair
`7.3.3.1 Centre Distance Control
`7.3.3.2
`Backlash Control
`7.4 Forces, Power Losses and Ef!ciency of Worm Gear Drives
`7.4.1 Forces Acting on Worm Gear Pair
`7.4.2 Power Losses and Ef!ciency of Worm Gear Pair
`7.5 Load Capacity of Worm Gear Pair
`7.5.1 Wear Load Capacity
`7.5.1.1 Calculation of Expected Wear
`7.5.1.2
`Permitted Wear
`
`xiii
`
`373
`374
`377
`377
`382
`384
`385
`386
`
`387
`387
`389
`389
`390
`390
`392
`392
`394
`397
`399
`399
`399
`399
`399
`400
`400
`401
`401
`401
`402
`402
`402
`402
`402
`403
`403
`403
`403
`404
`404
`404
`406
`409
`409
`410
`413
`
`LIBERTY EXHIBIT 1012, Page 15
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`

`

`xiv
`
`Contents
`
`7.5.2 Pitting Load Capacity
`7.5.3 Heating Load Capacity
`7.5.3.1 Heating Load Capacity at Bath Lubrication
`7.5.3.2 Heating Load Capacity at Spray Lubrication
`7.5.4 Wormwheel Bulk Temperature
`7.5.4.1 Wormwheel Bulk Temperature in Bath Lubrication
`7.5.4.2 Wormwheel Bulk Temperature in Spray Lubrication
`7.5.5 Wormwheel Tooth Root Load Capacity
`7.5.5.1
`Shear Stress in Wormwheel Tooth Root
`7.5.5.2
`Shear Fatigue Limit of Wormwheel Tooth
`7.5.6 Load Capacity for Worm Shaft De"ection
`7.6 Elements of Worm Gear Drive Design
`7.6.1 Design Procedure
`7.6.1.1
`Previous Choices
`7.6.1.2 Dimensioning the Worm Gear Pair
`7.6.2 Design Details of Worm Gear Drive
`7.7 List of Symbols
`7.7.1 Subscripts to Symbols
`7.7.2 Combined Symbols
`
`Further Reading
`
`Index
`
`414
`415
`416
`416
`417
`417
`417
`418
`418
`419
`420
`421
`421
`421
`422
`424
`427
`428
`429
`
`433
`
`437
`
`LIBERTY EXHIBIT 1012, Page 16
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`

`

`Preface
`
`Since gear drives operate with a power ef!ciency signi!cantly higher than any other mech-
`anical drive, or any electrical, hydraulical or pneumatical power transmission, they have the
`widest use in transforming rotary motion from the prime mover to the actuator, and their
`importance is growing day by day. Although ef!ciency is not the only criterion for choosing
`the type of transmission, the gear drive, due to its robustness and operational reliability, pres-
`ents an inevitable component of most mechanical engineering systems. Gear drives are known
`to be highly demanding in design, manufacture, control and maintenance.
`The entire !eld is well provided with standards, books and journal and conference papers.
`Thus, why a necessity for this book? There are three main reasons:
`
`1. Much knowledge has lost its validity through the statute of limitations, so it needs to be
`renewed. This book incorporates up-to-date knowledge.
`2. Despite the body of data available through the Internet, there is obviously still a lack of real
`knowledge. Namely, a basic knowledge is necessary for one to be able to apply the data.
`By collecting the data and by using the gear standards, a designer can get all the necessary
`information for gear drive design. Nevertheless, if someone wants to become a gear drive
`designer, he must primarily have basic knowledge. This book is conceived to enable both
`the basic knowledge and the data necessary to design, control, manufacture and maintain
`gear drives.
`3. There is no single book so far which incorporates almost all types of gears and gear drives:
`spur, helical, bevel and worm gear drives and planetary gear trains.
`
`This book is written with the presumption that the reader has a basic knowledge of mechan-
`ics and general mechanical engineering. It is primarily addressed to graduate and under-
`graduate students of mechanical engineering and to professionals dealing with the
`manufacturing of gears and gear drives. For all of these, it is supposed to be a primary text.
`Groups with an occasional need for this material are students of industrial engineering, tech-
`nology, automotive engineering, students of marine engineering, aviation engineering
`and space engineering and professionals in control and maintenance. The objective of this
`book is to provide all of these with everything they need regarding the subject matter in a
`single book: (i) a background for dealing with gears and gear trains (classi!cation, power,
`torque,
`transmission ratio distribution), (ii) a complete geometry and kinematics for
`almost any type of gears and gear drives, (iii) assessments of load capacities in accordance
`
`LIBERTY EXHIBIT 1012, Page 17
`
`

`

`xvi
`
`Preface
`
`with recent standards, including the calculation of micro-pitting load capacity, (iv) direc-
`tions and suggestions for the practical design of gears and gear drives, (v) detailed instruc-
`tions and formulae for determining the tolerances and procedures for measuring and
`controlling the accuracy of drives and their members in accordance with the latest standards.
`The reading matter is accompanied with a large number of !gures and every important for-
`mula is derived and discussed.
`This book consists of seven chapters. The !rst chapter introduces the reader to the funda-
`mental parameters of mechanical drives – transmission ratio, power, ef!ciency, torque and
`rotational speed – and explains the way for determining them. The classi!cation of mechani-
`cal drives and gear drives is also included. The second chapter explains in depth the geometry
`of cylindrical gear toothing as the basis of the entire !eld of gear drives, beginning with the
`idea of rolling, through the manufacturing of gears, the mesh and interference of teeth, tooth
`modi!cations, to the gear tolerances. The third chapter deals with the integrity of cylindrical
`gears, presenting the ways of calculating the load capacities for pitting, tooth root strength,
`scuf!ng and micro-pitting. In the fourth chapter the cylindrical gear drive design process is
`suggested and the selection of gear materials and their heat treatment are explained in depth,
`as well as gear drive lubrication and the ef!ciency and temperature of the lubricant. The !fth
`chapter deals with bevel gear drives: geometry, manufacturing, control, tolerances and load
`capacity checks. Crossed gear drives are also explained. In the sixth chapter simple plane-
`tary gear trains are !rst presented: transmission ratio, torques, ef!ciency of power and
`branching. Special trains, like harmonic and composed trains and also coupled, closed and
`reduced coupled trains are explained, as well as planetary reducers. The seventh chapter
`deals with worm gear drives: their geometry, manufacture, deviation control and load capac-
`ity assessments for the wear, pitting, heating, wormwheel tooth root and worm shaft
`de"ection.
`The book assumes that the reader is familiar with the metric (SI) system of units. However,
`some remarks are given herein: since standard modules are given in millimetres, all gear
`dimensions should be expressed in millimetres as well. Hence, in all equations where only
`length units appear, all physical quantities are to be substituted in millimetres. The exceptions
`are the allowance equations, where gear dimensions are to be substituted in millimetres to
`obtain the allowance in microns. In other equations, where the dimensions of physical quantit-
`ies are not only their lengths, the SI scale of units should be applied and gear dimensions
`should be substituted in metres, regardless of being marked in millimetres in the list of sym-
`bols at the end of each chapter. Relationship equations make an exception where the units of
`both sides of the equation are not the same. In each such equation the units of physical quanti-
`ties (those which are to be substituted) are speci!ed, as well as the unit of physical value
`obtained on the left side of the equation.
`
`LIBERTY EXHIBIT 1012, Page 18
`
`

`

`Acknowledgments
`
`Boris Obsieger, Technical Faculty, University of Rijeka, Croatia
`
`Irma R. Sharma, Hindustan Motors, Noida, India
`
`Josip Obsieger, Tehnical Faculty, University of Rijeka, Croatia
`
`Jozÿe Flasÿker, Faculty of Mechanical Engineering, University of Maribor, Slovenia
`
`Srdan Podrug, Faculty of Mechanical Engineering, Electrical Engineering and Naval Archi-
`tecture, University of Split, Croatia
`
`Srecÿko Glodezÿ, Faculty of Physical Sciences, University of Maribor, Slovenia
`
`Stanislav Pehan, Faculty of Mechanical Engineering, University of Maribor, Slovenia
`
`Zoran Ren, Faculty of Mechanical Engineering, University of Maribor, Slovenia
`
`LIBERTY EXHIBIT 1012, Page 19
`
`

`

`LIBERTY EXHIBIT 1012, Page 20
`
`LIBERTY EXHIBIT 1012, Page 20
`
`

`

`1 I
`
`ntroduction
`
`1.1 Power Transmissions and Mechanical Drives
`
`Mechanical power transmissions1 consist of units which, in distinction from electrical, pneu-
`matic and hydraulic ones, transfer power from the prime mover to the actuator (operational
`machine or operational member) with the assistance of rotary motion. These units are called
`mechanical drives and are situated between the prime mover and the actuator (Figure 1.1).
`The drive is connected with both the prime mover and the actuator by couplings or clutches
`forming an entirety whose function is de!ned by the purpose of the actuator.
`The embedding of a power transmission to link the prime mover and the machine operating
`member can be due to a number of reasons:
`
`The required speed of the machine operating member very often differs from the speeds of
`the standard prime movers.
`One prime mover has to drive several actuators.
`The driven side speed has to be frequently changed (regulated), whereas the prime mover
`cannot be used to full advantage for this purpose.
`Certain periods of the driven side operation may require torques far from those obtained on
`the motor shaft.
`As a rule, standard motors are designed for uniform rotary motion, while operating mem-
`bers have sometimes to move with varying speed or periodic halts.
`If a resonant vibration of some member in the chain of power transmission cannot be
`solved in any other way, the frequency of rotary motion can be changed by building-in a
`drive.
`Sometimes considerations of safety, convenience of maintenance or the dimensions of the
`machine, especially if the prime mover and operational machine shaft axes are not coaxial,
`do not allow the direct coupling of the prime mover shaft with operating member.
`
`1 The power is a feature of some machine or device and cannot be transmitted. Actually, only the energy is trans-
`mitted, but it is globally common to say that the power is transmitted.
`
`Gears and Gear Drives, First Edition. Damir Jelaska.
`© 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.
`
`LIBERTY EXHIBIT 1012, Page 21
`
`

`

`2
`
`Gears and Gear Drives
`
`Prime mover
`
`,
`
`Mechanical
`drive
`
`Actuator
`
`,
`
`Figure 1.1 Schematic account of a mechanical drive application
`
`The capital task of the designer is to select such an assembly ‘prime mover – transmission
`(drive)’ which should optimally meet the needs of the operational machine or member.
`This act of choosing is a complex task, whose solution depends on: (i) accessibility of the
`energy source and its price, (ii) ef!ciency of the entirety of prime mover – transmission –
`operational machine, (iii) investment costs, (iv) operational machine features, primarily the
`(v) variability of its speed of rotation, (vi) service conditions, (vii) drive maintainability and
`so on. Within the framework of this task, a particularly complex problem is de!ning the
`transmission: mechanical or some other? This question is beyond the scope of this book,
`but generally it may be af!rmed that the basic advantage of mechanical drives in relation to
`all the others is their very high ef!ciency, which is becoming more and more important day
`by day.
`The comparative advantages offered by possible transmissions and drives are outlined in
`Table 1.1 which gives only a general illustration. Recently, a prominent feature in power
`transfer has been the extensive employment of electric, hydraulic and pneumatic transmis-
`sions. Frequently, such transmissions together with mechanical drives are simultaneously
`used to actuate various mechanisms. The proper choice of a drive for each speci!c case can
`be made only by comparing the technical and economical features of several designs.
`The mechanical drive driving shaft receives power P1 at speed of rotation n1 from the
`prime mover driven shaft, and the mechanical drive driven shaft supplies power P2 P1 at
`
`Table 1.1 Advantages of transmissions and drives
`
`Advantage
`
`Transmission
`
`Mechanical drive
`
`Electric Hydraulic Pneumatic Friction Mesh
`
`Centralized power supply
`Simplicity of power transmission over large
`distances
`Easy accumulation of power
`Step by step speed change over a wide range
`Stepless change over a wide range
`Maintaining accurate transmission ratio
`High speed of rotation
`Simplicity of machine designed for rectilinear
`motion
`No effect of ambient temperature
`Comparatively high practically obtainable loads
`acting upon actuators of machine
`Easy control, automatic and remote
`
`LIBERTY EXHIBIT 1012, Page 22
`
`

`

`Introduction
`
`3
`
`speed of rotation n2 to the operational machine driving shaft. The difference P1 P2 PL is
`called power loss and the ratio:
`
`!
`
`P2
`P1
`
`P1
`
`PL
`
`P1
`
`1
`
`PL P1
`
`is called ef!ciency; it takes a special place amongst power transmission characteristics
`because it shows unproductive power expenditure and so indirectly characterizes the wear of
`the drive and its warming up – the capital problems in power transmissions. Warming up
`causes strength and lifetime decrease of drive parts. Their corrosion resistance and the func-
`tional ability of lubricant are also imperilled. The importance of ef!ciency is raised to a power
`by the global lack of increasingly expensive energy and its value also decisively affects the
`price of the drive.
`The power loss consists of constant losses which on the whole do not depend on load, and
`variable losses which on the whole are proportional to the load. The value of constant losses
`approximates the power of idle run, that is, the power needed to rotate the drive at P 0 on
`the driven shaft. It depends on the weight of the drive parts, the speed of rotation and the
`friction in the bearings and on other surfaces of contact.
`The second fundamental parameter of a mechanical drive is the transmission ratio i de!ned
`as the ratio of its driving n1 and driven n2 shaft speeds of rotation or angular speeds:
`
`1 1
`
`1 2
`
`i
`
`n1
`n2
`
`n2) the mechanical drive is called an underdrive and its member is called a
`1 (n1
`If i
`reducer. It reduces the speed of rotation and the transmission ratio is also called a speed
`reducing ratio. If i
`1 the mechanical drive is called an overdrive, its member is called a
`multiplicator and the transmission ratio is also called a speed increasing ratio. It multiplies
`the speed of rotation. An overdrive usually works less ef!ciently than an underdrive. This is
`especially true for a toothed wheel gearing.
`
`1.2 Classi!cation of Mechanical Drives
`
`The basic division of mechanical drives falls into:
`
`Drives with a constant transmission ratio.
`Drives with a variable transmission ratio.
`
`In constant transmission ratio drives, the constant speed of driving shaft rotation results
`in a constant speed of driven shaft rotation, n2
`n1/i. Their design should, as a rule, include at
`least the following data: (i) transmitted power of the driving (P1) or driven (P2) shaft or related
`torques, (ii) speed of rotation (rpm) of the driving (n1) and driven (n2) shaft, mutual location
`of the shafts and distance between them, (iii) overall dimensions and drive operating condi-
`tions, especially the dependence of driven shaft rpm or torque on time.
`In general, this design has several solutions, that is, given conditions can be used to develop
`drives of various types. All possible designs should be compared according to their
`ef!ciency, weight, size, original and operational costs in order to select the most advanta-
`geous one. Some general considerations, mainly the available experience of design,
`
`LIBERTY EXHIBIT 1012, Page 23
`
`

`

`4
`
`Gears and Gear Drives
`
`Table 1.2 Limit parameters of mechanical drives
`
`Drive type
`
`Transmission
`ratio
`
`Ef!ciency
`
`Power
`(MW)
`
`Rotational
`speed
`(min 1)
`
`Peripheral
`speed
`(m/s)
`
`Ratio
`mass/power
`(kg/kW)
`
`Cylindrical gears
`Planetary gear trains
`Bevel gears
`Hypoid gears
`Crossed gears
`Flat belt
`V-Belt
`Worm drives
`Friction drives
`Chain drives
`
`45
`1000
`8
`50
`100
`20
`15
`100
`8
`15
`
`0.99
`0.996
`0.98
`0.90
`0.95
`0.98
`0.94
`0.98a
`0.98b
`0.99
`
`35
`65
`4
`1
`0.08
`3.6
`4
`1.5
`0.25
`5
`
`100 000
`150 000
`50 000
`20 000
`20 000
`200 000
`8000
`50 000
`10 000
`30 000
`
`40
`200
`130
`50
`50
`120
`40
`70
`50
`40
`
`0.2 . . . 1.0
`0.4 . . . 1.8
`0.6 . . . 2.5
`0.7 . . . 3.0
`1.5 . . . 3.0
`1.5 . . . 6.0
`1.0 . . . 5.0
`0.2 . . . 4.5
`8 . . . 30
`6 . . . 10
`
`a Only for small transmission ratios.
`b Values are related to planetary friction drives.
`
`manufacture and operation of various drives enables us to outline generally the limits of prior-
`ity application of these drives.
`Therefore, the designer must take into account the limit values of main parameters
`which the mechanical drive can reach. These parameters are the maximum values of transmis-
`sion ratio, ef!ciency, power and speed of rotation. For underdrives, they are presented
`in Table 1.2.
`However, these limits are of a temporary nature: as new materials are produced and manu-
`facturing methods are improved, our knowledge of processes taking place in transmission
`becomes deeper and designs are perfected to suit broader !elds of application.
`For overdrives, these values are considerably smaller; such drives have a poor performance.
`It primary refers to ef!ciency and transmission ratio, but vibrations and noise levels
`are also much greater, especially in gear drives.
`It
`is due to the reasonable and
`experimentally approved fact that, with a similar error in the manufacture of the two meshing
`wheels, the driving wheel of greater diameter (in overdrive) causes larger angular accelera-
`tions (causing dynamic impacts) on the smaller driven wheel, while the reverse is applied to
`an underdrive.
`Mechanical drives with a constant transmission ratio are divided into: (i) drives with
`immovable axes – classical mechanical drives – and (ii) drives with movable axes – planetary
`mechanical trains.
`According to the mode in which they transmit motion from the driving wheel to the driven
`one, mechanical drives with immovable axes fall into the following types:
`
`Transmissions using friction: with direct contact of wheels (friction drives) or with a "exi-
`ble connection (belt drives).
`Transmissions using mesh: with direct contact (toothed and worm gears) or with "exible
`connection (chain and gear-belts drives).
`
`This classi!cation is clearly presented in Table 1.3.
`
`LIBERTY EXHIBIT 1012, Page 24
`
`

`

`Introduction
`
`5
`
`Table 1.3 Main classi!cation of classical mechanical drives with constant transmission ratio and
`immovable axes
`
`Mutual position of driving wheel and driven wheel
`Indirect
`Direct
`
`Friction
`
`Mesh
`
`Modeofmotiontransmittion
`
`Any individual drive is basically composed of a pair of wheels: gear wheels, rollers, belt
`pulleys and sprockets. It can operate on its own or can be built into mechanical trains of vari-
`ous machines and instruments and made in the form of an individual drive enclose