theory of gearing : kinetics, geometry, and synthesis · gearing kinetics,geometry,andsynthesis...
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Theory of
GEARINGKinetics, Geometry, and Synthesis
Stephen P. Radzevich
/Ov CRC Press
yC*** J Taylor&Francis CroupBoca Raton London NewYork
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
Contents
Preface xix
Acknowledgments xxi
Author xxiii
Introduction xxv
PART I Synthesis
Chapter 1 Kinematics of a Gear Pair 3
1.1 Transmission of Motion through a Gear Pair 3
1.1.1 Transition from a Pair of Friction Disks to an
Equivalent Gear Pair 3
1.1.2 Meaning of the Term "Synthesis" in This Book 6
1.2 Vector Representation of Gear Pair Kinematics 7
1.2.1 Concept of Vector Representation of Gear Pair Kinematics 8
1.2.2 Three Different Vector Diagrams for Spatial Gear Pairs 10
1.2.2.1 Vector Diagrams of External Spatial Gear Pairs 11
1.2.2.2 Vector Diagrams of Internal Spatial Gear Pairs 15
1.2.2.3 Vector Diagrams of Generalized Rack-Type SpatialGear Pairs 17
1.2.2.4 Analytical Criterion of a Spatial Gear Pair 18
1.3 Classification of Possible Vector Diagrams of Gear Pairs 19
1.4 Complementary Vectors to Vector Diagrams of Gear Pairs 27
1.4.1 Centerline Vectors of a Gear Pair 28
1.4.2 Axial Vectors of a Gear Pair 28
1.4.3 Useful Kinematic and Geometric Formulas 30
1.5 Tooth Ratio of a Gear Pair 32
1.6 Example of the Application of Vector Diagrams of Gear Pairs 33
Endnotes 34
Chapter 2 Geometry of Gear Tooth Flanks: Preliminary Discussion 35
2.1 Pulley-and-Belt Transmission as an Analogy of a Gear Pair 35
2.2 Natural Form of a Gear Tooth Profile 37
2.3 Other Possible Forms of a Gear Tooth Profile 42
2.4 Possible Shapes of Gear Tooth Flanks 45
2.4.1 Spur Involute Gear Tooth Flank 45
2.4.2 Helical Involute Gear Tooth Flank 46
2.4.3 Bevel Gear with Straight Teeth Tooth Flank 52
2.4.4 Bevel Gear with Helical Teeth Tooth Flank 53
2.4.5 Gear for a Crossed-Axis Gear Pair Tooth Flank 55
2.4.6 Possible Form of a Gear Tooth in the Lengthwise Direction 58
2.5 Tooth Contact Ratio: General Considerations 60
Endnotes 62
viii Contents
Chapter 3 Geometry of Contact of Tooth Flanks ofTwo Gears in Mesh 63
3.1 Applied Reference Systems Associated with a Gear Pair 63
3.2 Possible Local Patches of a Gear Tooth Flank 65
3.2.1 Circular Diagrams of Local Patches of a
Smooth Regular Surface 66
3.2.2 Possible Classification of Local Patches of
Gear Tooth Flanks 70
3.3 Local Relative Orientation of Tooth Flanks at a Point of Contact 70
3.4 Second Order Analysis of the Geometry of Contact of the
Tooth Flanks of a Gear and of a Pinion 76
3.4.1 Preliminary Remarks: Dupin Indicatrix 76
3.4.2 Surface of Normal Relative Curvature 79
3.4.3 Dupin Indicatrix of the Surface of Relative Curvature 81
3.4.4 Matrix Representation of the Equation of the Dupin Indicatrix
of the Surface of Relative Curvature 82
3.4.5 Surface of Relative Normal Radii of Curvature 82
3.4.6 Normalized Relative Normal Curvature 82
3.4.7 Curvature Indicatrix 83
3.4.8 JrKCharacteristic Curve 85
3.5 Fourth Order Analysis of the Geometry of Contact of the
Tooth Flanks of a Gear and of a Pinion 87
3.5.1 Rate of Conformity of Two Smooth Regular Surfaces in the
First Order of Tangency 87
3.5.2 Indicatrix of Conformity of the Tooth Flanks $ and 0T. 89
3.5.3 Directions of the Extremum Rate of Conformity of the Tooth
Flanks & and 0s: 95
3.5.4 Asymptotes ofthe Indicatrix of Conformity 98
3.5.5 Comparison of Capabilities of the Indicatrix of Conformity
CnfR(^/^) and of the Dupin Indicatrix of the Surface of
Relative Curvature 99
3.5.6 Important Properties of the Indicatrix of
Conformity Caf^/ffp) 100
3.5.7 Converse Indicatrix of Conformity at a Point of Contact of the
Tooth Flanks $ and m. 101
3.6 Plucker's Conoid: More Characteristic Curves 102
3.6.1 Plucker's Conoid 102
3.6.1.1 Basics 102
3.6.1.2 Analytical Description 103
3.6.1.3 Local Properties 104
3.6.1.4 Auxiliary Formulae,
105
3.6.2 Analytical Description of the Local Topology of a Smooth
Regular Gear Tooth Flank & 106
3.6.2.1 Preliminary Remarks 106
3.6.2.2 Plucker's Conoid 107
3.6.2.3 Plucker's Curvature Indicatrix 108
3.6.2.4 8>fnR <j§) -Indicatrix of a Gear Tooth Surface $ 109
3.6.3 Relative Characteristic Curves Ill
3.6.3.1 Possibility of Implementation of Two Plucker's
Conoids Ill
3.6.3.2 ^S((g^)-Indicatrix of the Surfaces @ and m. 112
Contents ix
3.7 Possible Contacts of the Teeth Surfaces $ and 115
3.7.1 Possibility of Implementation of the Indicatrix of Conformityfor the Identification of Contacts of the Tooth Flanks § and 115
3.7.2 Impact of the Accuracy of the Computations on the Desirable
Parameters of the Indicatrices of Conformity Cnf(<^/^) 118
3.7.3 Classification of Contacts of the Tooth Flanks <f and 120
Endnotes 126
Chapter 4 Concept of Synthesis of a Gear Pair with Prescribed Performance 129
Endnote 132
PART II Ideal Gearing: Parallel-Axis Gearing
Chapter 5 Involute Gearing 135
5.1 Principal Features and Fundamental Theorems of
Parallel-Axis Gearing 135
5.1.1 Kinematics of Parallel-Axis Gearing 135
5.1.2 Willis Fundamental Law of Gearing 138
5.1.3 Euler-Savary Equation 140
5.2 Generation of an Involute Profile of a Gear Tooth 143
5.2.1 Geometry of the Tooth Flank of a Spur Gear 143
5.2.1.1 Generation of the Tooth Flank of a Spur Gear byMeans of a Rack 144
5.2.1.2 Addendum Modification (Profile Shift) 152
5.2.1.3 Determination of the Tooth Form Generated by a
Given Generating Rack Profile 155
5.2.1.4 Base Tangent Length 157
5.2.1.5 Tooth Thickness of a Gear 160
5.2.2 Geometry of the Tooth Flank of a Helical Gear 162
5.3 External Involute Gear Pair 172
5.3.1 Variation of the Tooth Flank Geometry 175
5.3.1.1 Normal Curvature of the Gear Tooth Flank 175
5.3.1.2 Variation of the Tooth Profile Angle and Helix Angle 180
5.3.2 Special Point of Meshing 180
5.3.3 Contact Ratio of an External Gear Pair 181
5.3.3.1 Transverse Contact Ratio 181
5.3.3.2 Face Contact Ratio 183
5.3.3.3 Total Contact Ratio 184
5.3.4 Contact Motion Characteristics 184
5.3.4.1 Sliding Conditions 184
5.3.4.2 Specific Sliding 186
5.3.5 Basic Equations for a Gear Pair with Addendum Modification 187
5.3.5.1 Principle of Addendum Modification 187
5.3.5.2 External Spur and Helical Gear Pairs 188
5.4 Internal Involute Gearing 190
5.4.1 Tooth Thickness Measurement of an Internal Gear 190
5.4.2 Contact Ratio in an Internal Gearing 191
5.4.3 Sliding Conditions in an Internal Gearing 194
X Contents
5.4.4 Mating Internal Gear Pair 196
5.4.5 Gear Coupling 197
5.5 Involute Gear-to-Rack Pair 198
5.6 Involute Gear Pairs with an Arbitrary Tooth Shape in
the Lengthwise Direction 200
5.7 Conditions to Be Fulfilled by Mating Gears 204
Endnotes 206
Chapter 6 Noninvolute Gearing 207
6.1 Spur Noninvolute Gear Pairs 207
6.1.1 Pin Gearing 207
6.1.2 Cycloidal Gearing 208
6.1.3 Root Blower 210
6.1.4 Spur Gear Pairs of an Oil Pump 212
6.2 Conditions for Smooth Rotation of a Noninvolute Gear Pair 214
6.2.1 Interaction of a Noninvolute Gear with a Rack 217
6.3 Helical Noninvolute Gear Pairs 221
6.3.1 Helical Gear Pair of a Root Blower 221
6.3.2 Infeasibility of Transmission of Rotation by a Noninvolute
Helical Gear Pair with a Positive Transverse Contact Ratio 222
6.3.3 Analysis of Wildhaber's Helical Gearing (US Patent
No. 1,601,750) as an Example of Noninvolute Helical Gearingwith a Positive Transverse Contact Ratio 224
6.4 Noncylindrical Gears in Designing Parallel-Axis Gearing 226
6.4.1 Conical Involute Gears 226
6.4.1.1 Kinematics of Conical Involute Gearing 226
6.4.1.2 Geometry of the Tooth Flanks of a Spur Conical
Involute Gear 226
6.4.1.3 Geometry of the Tooth Flanks of a Conical Involute
Gear with Helical Teeth 235
6.4.2 Toroidal Involute Gears 238
6.4.2.1 Spur Toroidal Involute Gearing 238
6.4.2.2 Toroidal Involute Gearing with Helical Teeth 246
Endnotes 250
Chapter 7 High-Conforming Parallel-Axis Gearing 251
7.1 Novikov Gearing: A Helical Noninvolute Gearing That Has a Zero
Transverse Contact Ratio 251
7.1.1 Essence of Novikov Gearing ; 254
7.1.2 Elements of Kinematics and the Geometry ofNovikov Gearing.... 258
7.1.3 Design Parameters of Novikov Gearing 261
7.2 High-Conforming Parallel-Axis Gearing 262
7.2.1 Fundamental Design Parameters of
High-Conforming Gearing 263
7.2.2 Boundary N-Circle in High-Conforming Gearing 264
7.2.3 Possible Tooth Geometries in High-Conforming Gearing 266
7.2.4 Permissible Location of the Culminating Point in
High-Conforming Gearing 273
Contents xi
7.2.5 Contact of Tooth Flanks in a High-Conforming Gear Pair 274
7.2.5.1 Configuration of Interacting Tooth Flanks at the
Culminating Point 274
7.2.5.2 Local and Global Geometry of Contact of InteractingTooth Flanks 276
7.2.5.3 Minimum Required Rate of Conformity between
Interacting Tooth Flanks 279
Endnotes 284
Chapter 8 Synthesis of Optimal Parallel-Axis Gearing 287
8.1 Geometrically Accurate Parallel-Axis Gearing 287
8.2 Peculiarities of the Problem of Synthesis of OptimalParallel-Axis Gears 291
8.2.1 Peculiarities of the Problem of Synthesis of
Optimal Involute Gears 291
8.2.2 Peculiarities of the Problem of Synthesis of
Optimal High-Conforming Gears 293
PART III Ideal Gearing: Intersected-Axis Gearing
Chapter 9 Geometrically Accurate Intersected-Axis Gear Pairs 299
9.1 Earliest Concepts of Intersected-Axis Gearing 299
9.2 Kinematics of Intersected-Axis Gearing 301
9.3 Base Cones in Intersected-Axis Gearing 305
9.4 Tooth Flanks of Geometrically Accurate (Ideal)Intersected-Axis Gear Pairs 307
9.4.1 Applied Coordinate Systems and Linear Transformations 307
9.4.1.1 Main Reference Systems 307
9.4.1.2 Operators of Rolling 308
9.4.1.3 Operators Associated with the
Gearing Housing 310
9.4.2 Tooth Flank of a Bevel Gear 312
9.4.3 Desired Tooth Proportions for Intersected-Axis Gears 317
9.4.3.1 Base Angular Pitch 318
9.4.3.2 Normal Pressure Angle 319
9.4.3.3 Angular Pitch 323
9.4.3.4 Angular Tooth Thickness and
Angular Space Width 326
9.4.3.5 Angular Addendum and Angular Dedendum 327
9.4.3.6 Specification of the Design Parameters in
Intersected-Axis Gearing 328
9.4.4 Contact Ratio in an Intersected-Axis Gearing 330
9.4.4.1 Transverse Contact Ratio 330
9.4.4.2 Face Contact Ratio 332
9.4.4.3 Total Contact Ratio 332
9.4.5 Tredgold's Approximation 333
Endnotes 334
x» Contents
Chapter 10 High-Conforming Intersected-Axis Gearing 335
10.1 Kinematics of the Instantaneous Motion in High-ConformingIntersected-Axis Gearing 335
10.2 Contact Line in High-Conforming Intersected-Axis Gearing 336
10.2.1 Bearing Capacity of High-Conforming Gearing 337
10.2.2 Sliding of Teeth Flanks in High-Conforming Gearing 338
10.2.3 Boundary N-Cone in Intersected-Axis
High-Conforming Gearing 339
10.3 Design Parameters of High-Conforming Intersected-Axis Gearing 340
Endnote 345
PART IV Ideal Gearing: Crossed-Axis Gearing
Chapter 11 Geometrically Accurate Crossed-Axis Gearing: ^-Gearing 349
11.1 Kinematics of Crossed-Axis Gearing 349
11.2 Base Cones in Crossed-Axis Gear Pairs 352
11.3 Tooth Flanks of Geometrically Accurate (Ideal)Crossed-Axis Gear Pairs 355
11.3.1 Applied Coordinate Systems and Linear Transformations 356
11.3.1.1 Main Reference Systems 356
11.3.1.2 Operators of Rolling/Sliding 357
11.3.1.3 Operators Associated with Gear Housing 359
11.3.2 Tooth Flank of a Crossed-Axis Gear 361
11.3.3 Desired Tooth Proportions in Crossed-Axis Gearing 369
11.3.3.1 Base Angular Pitch 369
11.3.3.2 Normal Pressure Angle 370
11.3.3.3 Angular Pitch 374
11.3.3.4 Angular Tooth Thickness and Angular Space Width
in the Round Basic Rack 376
11.3.3.5 Angular Addendum and Angular Dedendum of the
Round Basic Rack 376
11.3.3.6 Specification of the Design Parameters of
Crossed-Axis Gears..... 381
11.3.4 Contact Ratio in Crossed-Axis Gearing 382
11.3.4.1 Transverse Contact Ratio 383
11.3.4.2 Face Contact Ratio 384
11.3.4.3 Total Contact Ratio 384
11.3.5 Possible Analogy of Tredgold's Approximation for
Crossed-Axis Gearing 384
11.3.6 Peculiarities of Worm Gearing with Line Contact between the
Worm Threads and the Worm Gear Tooth Flanks 385
Endnote 387
Chapter 12 High-Conforming Crossed-Axis Gearing 389
12.1 Kinematics of the Instantaneous Relative Motion 389
12.2 Contact Line in High-Conforming Crossed-Axis Gearing 391
12.2.1 Bearing Capacity of Crossed-Axis
High-Conforming Gearing 391
Contents xiii
12.2.2 Sliding between Tooth Flanks of the Gear and of the Pinion in
Crossed-Axis High-Conforming Gearing 392
12.2.3 Boundary N-Cone in Crossed-Axis High-Conforming Gearing 393
12.3 Design Parameters of High-Conforming Crossed-Axis Gearing 395
PART V Ideal (Geometrically Accurate)Two-Degrees-of-Freedom Gearing
Chapter 13 Kinematics, Geometry, and Design Features of 2-DOF Gearing 403
13.1 Practical Examples of 2-DOF Gearing 403
13.2 Approach to Generate Tooth Flanks of the Gear and the Pinion in
2-DOF Gearing 405
13.3 Possible Auxiliary Generating Racks 406
13.4 Geometry of the Tooth Flanks of Geometrically Accurate 2-DOF
Crossed-Axis Gears 407
Endnote 411
PART VI Real Gears and Their Application: Real Gearing
Chapter 14 Desired Real Gearing: Spr-Gearing 415
14.1 Preliminary Considerations 415
14.1.1 Root Causes for Real Gears Differ from Ideal Gears 415
14.1.2 Applied Coordinate Systems 417
14.1.3 Displacements of a Gear Axis of Rotation from Its Desired
Configuration 418
14.1.4 Closest Distance of Approach between the Gear and the Pinion
Axes ofRotation 423
14.2 Tooth Flank Geometry of Desirable Real Gearing: 5pr-Gearing 427
14.2.1 Tooth Flank Geometry of Desirable Real Gearing 428
14.2.2 Possibility of Implementation of the Concept of Spr-Gearing in
the Design of Gear Coupling 436
14.2.3 Account for Normal Distribution of Manufacturing Errors onto
the Geometry of Base Lines 437
14.2.4 Preserving the Equality of Base Pitches at Different Values of
Axis Misalignment 438
14.2.5 Possible Simplifications 440
14.3 Possibility of Implementation of the Concept of 5pr-Gearing to Gear
Systems Featuring Point Contact of Tooth Flanks 441
14.4 Correlation among Gear Systems of Various Kinds 442
Endnotes 444
Chapter 15 Approximate Real Gearing 445
15.1 Approximate Real Parallel-Axis Gearing 445
15.2 Approximate Real Intersected-Axis Gearing 447
15.2.1 Root Causes for Referring to Real Intersected-Axis Gears as
Approximate Gears 448
xiv Contents
15.2.2 Approximate Real Intersected-Axis Gears 448
15.2.2.1 Straight Tooth Bevel Gears 448
15.2.2.2 Spiral Bevel Gears 450
15.2.2.3 Face Gears 451
15.2.3 Generation of Tooth Flanks of Intersected-Axis Gears 452
15.2.3.1 Generation ofTooth Flanks of Straight Bevel Gears 452
15.2.3.2 Generation of Tooth Flanks of Spiral Bevel Gears 455
15.2.3.3 Tooth Flanks of Bevel Gears Cut Using the
Continuously Indexing Method of Gear Machining 459
15.2.4 Examples of Approximate Real Intersected-Axis Gear Pairs 459
15.3 Approximate Real Crossed-Axis Gearing: Hypoid Gears 464
15.4 Worm Gearing 466
15.5 Tooth Flank Modification 471
15.5.1 Brief Historical Overview of Tooth Flank Modification 471
15.5.2 Requirements to Design Parameters ofModified Portions of
Tooth Flanks 472
Endnotes 473
Chapter 16 Generic Gear Shape 475
16.1 Origination of the Generic Gear Shape 475
16.2 Examples of Gear Pairs Comprising Gears with Various
Generic Shapes 476
16.3 Evaluation of the Total Number of Possible Generic Gear Shapes 478
16.3.1 Possible Profiles of the Generic Gear Shape Constructed in the
Axial Cross Section of the Gear 478
16.3.2 Profile of Generic Gear Surfaces Constructed in Cross Section
by a Plane at an Angle to the Gear Axis 486
16.4 Possibility of Classification of Possible Gear Pairs 490
16.5 Examples of Implementation of the Classification of
Possible Gear Pairs 491
Endnotes 495
Chapter 17 Gear Noise 497
17.1 Transmission Error 497
17.2 Base Pitch Variation 498
17.3 Influence of the Contact Ratio 499
17.4 Variation of the Load 501
17.5 Requirements to Design Parameters for Low
Noise/Noiseless Gear Drives 501
17.5.1 Ideal Gear Pairs 502
17.5.1.1 Ideal Parallel-Axis Gear Pairs 502
17.5.1.2 Ideal Intersected-Axis Gear Pairs 502
17.5.1.3 Ideal Crossed-Axis Gear Pairs 502
17.5.2 Desired Real Gear Pairs 502
17.5.2.1 Real (Approximate) Parallel-Axis Gear Pairs 503
17.5.2.2 Real (Approximate) Intersected-Axis Gear Pairs 503
17.5.2.3 Real (Approximate) Crossed-Axis Gear Pairs 503
17.5.3 Real (Approximate) Gear Pairs 504
Contents xv
PART VII Real Gears and Their Application: Gear Trains
Chapter 18 Gear Ratio of a Multistage Gear Drive 507
18.1 Principal Kinematic Relationships in a Multistage Gear Drive 507
18.1.1 Range Ratio of Speed Variation for a Gear Drive 509
18.1.2 Characteristic of a Transmission Group 509
18.2 Analytical Method for Determining Transmission Ratios 509
18.3 Rotational Speed Chart 510
18.4 Broken Geometrical Series 511
18.5 Minimum Number of Gear Pairs 512
18.6 Determining the Tooth Number of Gears of Group Transmissions 512
Endnote 513
Chapter 19 Split Gear Drives 515
19.1 Root Cause of Unequal Load Sharing in Multiflow Gear Drives 515
19.2 Mobility of Split Gear Drives 516
19.3 Epicyclic Gear Drives 517
19.4 Structural Formula for Planetary Gear Drives 519
19.5 Correspondence among Angular Velocities of All Members of a
Planetary Gear Drive 520
19.6 Problem of Equal Load Sharing in Planetary Gear Drives:
State of the Art 521
19.6.1 Planetary Gear Drives That Have Multiple Planet Pinions 521
19.6.2 Single-Row Planetary Gear Drives with Six Self-AlignedPlanet Pinions 528
19.6.3 Positive Planetary Gear Drives with LargeTransmission Ratios 530
19.6.4 Planar Planetary Gear Drives with Self-AlignedPlanet Pinions 531
19.6.5 Planetary Gear Drives with Free Carriers 533
19.6.6 Multiple and Closed Planetary Gear Drives 537
19.6.7 Method of Structural Groups for Investigating Self-Alignmentof Planetary Gearboxes 543
19.7 Alternative Approaches for Equal Torque Sharing in
Multiflow Gear Trains 553
19.7.1 Planetary Gear Drives with Flexible Pins 553
19.7.2 Load Equalizing in the Design of an Automotive
Differential 558
19.7.3 Elastic Absorbers of Manufacturing Errors 558
19.7.3.1 Elastic Properties of Elastic Absorbers of
Manufacturing Errors 559
19.7.3.2 Examples of Implementation of Preloaded Elastic
Absorber of Manufacturing Errors 560
19.7.4 Load Equalizing with the Elastic Absorber Common for all
Power Flows 563
19.7.5 Main Features of Multiflow Gear Trains with Preloaded Elastic
Absorbers of Manufacturing Errors 565
Endnotes 566
xvi Contents
PART VIII Real Gears and Their Application: PrincipalFeatures of Power Transmission and Loading of
the Gear Teeth
Chapter 20 Local Geometry of the Interacting Tooth Flanks 569
20.1 Local Geometry of the Interacting Tooth Flanks in
Parallel-Axis Gearing 569
20.1.1 Kinematics of the Interacting Tooth Flanks 569
20.1.2 Local Geometry of the Interacting Tooth Flanks 570
20.2 Local Geometry of the Interacting Tooth Flanks in
Intersected-Axis Gearing 574
20.2.1 Kinematics of Interaction of the Tooth Flanks 574
20.2.2 Local Geometry of the Interacting Tooth Flanks 575
20.3 Local Geometry of the Interacting Tooth Flanks inCrossed-Axis Gearing ,
577
20.3.1 Kinematics of Interaction of the Tooth Flanks 577
20.3.2 Local Geometry of the Interacting Tooth Flanks 579
20.4 Local Geometry of the Interacting Tooth Flanks in
High-Conforming Gearing 580
20.4.1 Kinematics of the Interacting Tooth Flanks 580
20.4.2 Geometry of the Interacting Tooth Flanks 582
Endnotes 585
Chapter 21 Contact Stresses in Low-Tooth-Count Gearing 587
21.1 Adopted Principal Assumptions 587
21.1.1 Comments on Analytical Description of the Local Geometryof Contacting Surfaces Loaded by a Normal Force: Hertz's
Proportional Assumption 587
21.1.2 Assumption of Equal Torque Sharing 590
21.2 Principal Features of Low-Tooth-Count Gears 591
21.3 Analytical Model for the Calculation of Contact Stresses 592
21.4 Combined Compressive and Shear Stresses in
Low-Tooth-Count Gearing 595
Endnotes 598
Chapter 22 Application of the Results Derived from Theory of Gearing 599
22.1 Bending Strength of a Gear Teeth: Comments on Lewis' Formula 599
22.1.1 Cantilever Beam of Equal Strength 599
22.1.2 Lewis' Formula for the Calculation of Gear Teeth Strength 601
22.2 Effective Length of the Line of Contact 604
22.2.1 Length of a Single Line of Contact in
Parallel-Axis Gearing 604
22.2.2 Effective Length of Lines of Contact in
Parallel-Axis Gearing 609
22.2.2.1 Effective Length of Lines of Contact in SpurParallel-Axis Gearing 609
22.2.2.2 Effective Length of Lines of Contact in Helical
Parallel-Axis Gearing 613
Contents xvii
22.3 Loading of Gear Teeth 617
22.4 Method for Simulating Interaction of the Gear and of the
Pinion Tooth Flanks 620
Endnotes 626
Conclusion 627
Appendix A: Elements of Coordinate Systems Transformations 631
Appendix B: Novikov's Gearing Invention Disclosure 643
Appendix C: Wildhaber's Gearing Invention Disclosure 651
Appendix D: Engineering Formulas for the Specification of Gear Tooth Flanks 659
Appendix E: Change of Surface Parameters 663
Appendix F: Notations 665
Appendix G: Glossary 669
References 675
Bibliography 681
Index 685