theory of gearing : kinetics, geometry, and synthesis · gearing kinetics,geometry,andsynthesis...

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


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




9.4.5 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


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




11.3.5 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


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


20.1.1 Kinematics of the Interacting Tooth Flanks 569

20.1.2 Local Geometry of the Interacting Tooth Flanks 570


Intersected-Axis Gearing 574

20.2.1 Kinematics of Interaction of the Tooth Flanks 574


20.3 Local Geometry of the Interacting Tooth Flanks inCrossed-Axis Gearing ,

577

20.3.1 Kinematics of Interaction of the Tooth Flanks 577




20.4.1 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


22.2.2 Effective Length of Lines of Contact in


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


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

theory of gearing : kinetics, geometry, and synthesis · gearing kinetics,geometry,andsynthesis...

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