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Kinematic Geometry of Gearing, 2nd Edition

ISBN: 978-1-118-38555-5
512 pages
April 2012
Kinematic Geometry of Gearing, 2nd Edition (1118385551) cover image
Building on the first edition published in 1995 this new edition of Kinematic Geometry of Gearing has been extensively revised and updated with new and original material. This includes the methodology for general tooth forms, radius of torsure’, cylinder of osculation, and cylindroid of torsure; the author has also completely reworked the ‘3 laws of gearing’, the first law re-written to better parallel the existing ‘Law of Gearing” as pioneered by Leonard Euler, expanded from Euler’s original law to encompass non-circular gears and hypoid gears, the 2nd law of gearing describing a unique relation between gear sizes, and the 3rd law completely reworked from its original form to uniquely describe a limiting condition on curvature between gear teeth, with new relations for gear efficiency are presented based on the kinematics of general toothed wheels in mesh.  There is also a completely new chapter on gear vibration load factor and impact.

Progressing from the fundamentals of geometry to construction of gear geometry and application, Kinematic Geometry of Gearing presents a generalized approach for the integrated design and manufacture of gear pairs, cams and all other types of toothed/motion/force transmission mechanisms using computer implementation based on algebraic geometry.

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Preface xiii

Part I FUNDAMENTAL PRINCIPLES OF TOOTHED BODIES IN MESH

1 Introduction to the Kinematics of Gearing 3

1.1 Introduction 3

1.2 An Overview 3

1.3 Nomenclature and Terminology 5

1.4 Reference Systems 8

1.5 The Input/Output Relationship 9

1.6 Rigid Body Assumption 11

1.7 Mobility 11

1.8 Arhnold-Kennedy Instant Center Theorem 14

1.9 Euler-Savary Equation for Envelopes 18

1.10 Conjugate Motion Transmission 19

1.10.1 Spur Gears 20

1.10.2 Helical and Crossed Axis Gears 21

1.11 Contact Ratio 22

1.11.1 Transverse Contact Ratio 24

1.11.2 Axial Contact Ratio 25

1.12 Backlash 25

1.13 Special Toothed Bodies 26

1.13.1 Microgears 28

1.13.2 Nanogears 28

1.14 Noncylindrical Gearing 29

1.14.1 Hypoid Gear Pairs 29

1.14.2 Worm Gears 30

1.14.3 Bevel Gears 32

1.15 Noncircular Gears 33

1.15.1 Gear and Cam Nomenclature 38

1.15.2 Rotary/Translatory Motion Transmission 39

1.16 Schematic Illustration of Gear Types 40

1.17 Mechanism Trains 40

1.17.1 Compound Drive Trains 41

1.17.2 Epicyclic Gear Trains 43

1.17.3 Circulating Power 49

1.17.4 Harmonic Gear Drives 50

1.17.5 Noncircular Planetary Gear Trains 51

1.18 Summary 52

Part II THE KINEMATIC GEOMETRY OF CONJUGATE MOTION IN SPACE

2 Kinematic Geometry of Planar Gear Tooth Profiles 55

2.1 Introduction 55

2.2 A Unified Approach to Tooth Profile Synthesis 55

2.3 Tooth Forms Used for Conjugate Motion Transmission 56

2.3.1 Cycloidal Tooth Profiles 56

2.3.2 Involute Tooth Profiles 59

2.3.3 Circular-arc Tooth Profiles 63

2.3.4 Comparative Evaluation of Tooth Profiles 64

2.4 Contact Ratio 65

2.5 Dimensionless Backlash 68

2.6 Rack Coordinates 69

2.6.1 The Basic Rack 71

2.6.2 The Specific Rack 76

2.6.3 The Modified Rack 77

2.6.4 The Final Rack 79

2.7 Planar Gear Tooth Profile 80

2.8 Summary 84

3 Generalized Reference Coordinates for Spatial Gearing—the Cylindroidal Coordinates 85

3.1 Introduction 85

3.2 Cylindroidal Coordinates 85

3.2.1 History of Screw Theory 87

3.2.2 The Special Features of Cylindroidal Coordinates 87

3.3 Homogeneous Coordinates 89

3.3.1 Homogeneous Point Coordinates 91

3.3.2 Homogeneous Plane Coordinates 92

3.3.3 Homogeneous Line Coordinates 93

3.3.4 Homogeneous Screw Coordinates 96

3.4 Screw Operators 99

3.4.1 Screw Dot Product 99

3.4.2 Screw Reciprocal Product 99

3.4.3 Screw Cross Product 101

3.4.4 Screw Intersection 102

3.4.5 Screw Triangle 103

3.5 The Generalized Equivalence of the Pitch Point—the Screw Axis 104

3.5.1 Theorem of Three Axes 105

3.5.2 The Cylindroid 107

3.5.3 Cylindroid Intersection 108

3.6 The Generalized Pitch Surface—Axodes 110

3.6.1 The Theorem of Conjugate Pitch Surfaces 115

3.6.2 The Striction Curve 116

3.7 The Generalized Transverse Surface 121

3.8 The Generalized Axial Surface 123

3.9 Summary 125

4 Differential Geometry 127

4.1 Introduction 127

4.2 The Curvature of a Spatial Curve 127

4.3 The Torsion of a Spatial Curve 129

4.4 The First Fundamental Form 130

4.5 The Second Fundamental Form 132

4.6 Principal Directions and Principal Curvatures 135

4.7 Torsure of a Spatial Curve 138

4.8 The Cylindroid of Torsure 142

4.9 Ruled Surface Trihedrons 148

4.10 Formulas of Fernet-Serret 150

4.11 Coordinate Transformations 151

4.12 Characteristic Lines and Points 158

4.13 Summary 159

5 Analysis of Toothed Bodies for Motion Generation 161

5.1 Introduction 161

5.2 Spatial Mobility Criterion 161

5.3 Reciprocity—the First Law of Gearing 164

5.4 The Line Complex 166

5.5 The Tooth Spiral 168

5.5.1 The Tooth Spiral Curvature 170

5.5.2 The Tooth Spiral Torsion 173

5.6 Tooth Spiral Angle—the Second Law of Gearing 174

5.6.1 The I/O Relationship 179

5.6.2 The Phantom I/O Relationship 181

5.7 Reduced Tooth Curvature—the Third Law of Gearing 183

5.7.1 Absolute Tooth Curvature 187

5.7.2 Tooth Profile Modification 190

5.8 Classification of Gear Types 192

5.9 Contact Ratio 194

5.9.1 Transverse Contact Ratio 195

5.9.2 Axial Contact Ratio 196

5.10 Spatial Backlash 196

5.11 Relative Displacements 197

5.11.1 The Sliding Velocity 197

5.11.2 The Rolling Velocity 200

5.11.3 The Pitch Line Velocity 202

5.12 Mesh Efficiency 203

5.13 Summary 205

6 The Manufacture of Toothed Bodies 207

6.1 Introduction 207

6.2 Manufacturing Background 207

6.2.1 Form-Type Fabrication 208

6.2.2 Generation-Type Fabrication 208

6.2.3 Spiral Bevel/Hypoid Gear Fabrication 212

6.2.4 Noncircular Gear Fabrication 215

6.3 Crossed Hyperboloidal Gears 216

6.4 Fabrication of Cutters 220

6.4.1 The Hyperboloidal Cutter 220

6.4.2 The Cutter Spiral Angle 224

6.4.3 The Face Spiral Angle 225

6.4.4 Cutter Constraints 227

6.4.5 Speed Ratio 228

6.4.6 Hyperboloidal Cutter Coordinates 231

6.5 Gear Cutting Machine Layout 235

6.6 The Envelope of the Cutter 237

6.6.1 The Equation of Meshing 239

6.6.2 Boolean Operations 241

6.7 Material Removal Rate 242

6.8 Surface Cutting Speed 242

6.9 Discretization Error 243

6.9.1 Scalloping 243

6.9.2 Tessellation 245

6.10 Inspection 246

6.11 Hyperboloidal Blank Dimensions 247

6.12 Summary 248

7 Vibrations and Dynamic Loads in Gear Pairs 249

7.1 Introduction 249

7.2 Excitations 249

7.3 Transmission Error 250

7.3.1 Static Transmission Error 251

7.3.2 Loaded Transmission Error 254

7.3.3 Dynamic Transmission Error 255

7.4 Fourier Transforms 260

7.5 Impact Loading 261

7.6 Mesh Stiffness 264

7.7 Inertial Properties 265

7.7.1 Center of Mass 265

7.7.2 Mass Moments of Inertia 267

7.8 Manufacturing Dynamics 269

7.9 Summary 270

Part III THE INTEGRATED DESIGN AND MANUFACTURING PROCESS

8 Gear Design Rating 275

8.1 Introduction 275

8.2 Modes of Gear Failure 275

8.3 Reaction Loads 275

8.4 Gear Parameters for Specified Deflections 280

8.5 The Fillet Stress 286

8.5.1 Discretization of Gear Tooth 287

8.5.2 Element Stiffness Matrix 289

8.5.3 Global Stiffness Matrix 292

8.5.4 Boundary Conditions 293

8.5.5 Nodal Strain 294

8.5.6 Nodal Stress 294

8.6 Inertial Stress 295

8.7 Contact Stress 296

8.8 Minimum Film Thickness 299

8.9 Wear 301

8.10 Friction Coefficient 305

8.10.1 Sliding Friction 305

8.10.2 Rolling Friction 309

8.11 Flash Temperature 311

8.12 Thermal Stress 313

8.13 Failure Analysis 314

8.13.1 Reliability Analysis 314

8.13.2 Fatigue Analysis 317

8.13.3 Cumulative Loading 320

8.14 Windage Losses 321

8.15 Optimization 325

8.16 Summary 326

9 The Integrated CAD–CAM Process 327

9.1 Introduction 327

9.2 Modular Components for Geometric Synthesis 327

9.2.1 The Motion Specification Module 328

9.2.2 The Tooth Parameters Module 328

9.2.3 The Gear Parameters Module 331

9.2.4 The Cutter Parameters Module 332

9.2.5 The Loading Parameters Module 333

9.2.6 The Material Specifications Module 333

9.2.7 The Lubricant Specifications Module 334

9.2.8 The Dynamic Factors Module 336

9.2.9 The Shaft Deflections Module 337

9.2.10 The Manufacturing Specifications Module 337

9.3 The Integrated CAD–CAM Process 338

9.4 Illustrative Example 338

9.5 Summary 361

10 Case Illustrations of the Integrated CAD–CAM Process 363

10.1 Introduction 363

10.2 Case 1 363

10.3 Case 2 364

10.4 Case 3 365

10.5 Case 4 366

10.6 Case 5 367

10.7 Case 6 368

10.8 Case 7 369

10.9 Case 8 370

10.10 Case 9 371

10.11 Case 10 372

10.12 Case 11 373

10.13 Case 12 374

10.14 Case 13 375

10.15 Case 14 376

10.16 Case 15 377

10.17 Case 16 378

10.18 Case 17 379

10.19 Case 18 380

10.20 Case 19 381

10.21 Case 20 382

10.22 Case 21 383

10.23 Case 22 386

10.24 Summary 388

AppendixA Differential Expressions 389

A.1 Derivatives of the Radius of the Axode 389

A.2 Derivatives of the Included Angles 391

A.3 Derivatives of the Generators 392

A.4 Derivatives of the Pitch of the Instantaneous Twist 394

A.5 Derivatives of the Parameter of Distribution 394

A.6 Derivatives of the Striction Curve 394

A.7 Manufacturing Expressions 396

A.8 Derivatives of the Transverse Curve 396

A.9 Derivatives of the Angle Between the Generator and the Transverse Curve 397

A.10 Derivatives of the Spiral Angle 398

A.11 Derivatives of the Input Trihedron of Reference 399

A.12 Derivatives of the Cutter Parameters 399

AppendixB On the Notation and Operations 401

AppendixC Noncircular Gears 409

C.1 Torque and Speed Fluctuations in Rotating Shafts 409

C.2 2-Dof Mechanical Function Generator 412

C.3 Steering Mechanism 414

C.4 Continuously Variable Transmission 416

C.5 Geared Robotic Manipulators 418

C.6 Spatial Mechanism for Body Guidance 420

C.7 Nonworking Profile 421

C.8 Multiple Reductions 422

AppendixD The Delgear Software 425

D.1 Installation 427

AppendixE Splines 429

E.1 Cubic Splines 430

E.2 Natural Splines 433

E.2.1 Derivatives 435

E.3 NURBS 436

AppendixF Contact Stress 437

F.1 Introduction 437

F.2 Background 437

F.3 Material Properties 438

F.4 Surface Geometry 439

F.5 Contact Deformations 442

F.6 Contact Area 443

F.7 Comparison 445

AppendixG Glossary of Terms 447

AppendixH Equilibrium and Diffusion Equations 455

H.1 Equilbrium Equations 455

H.2 Diffusion Equation Formulation 459

H.3 Expressions 461

Appendix I On the Base Curve of Planar Noncircular Gears 465

Appendix J Spatial Euler-Savary Equations 471

J.1 Planar Euler-Savary Equations 471

J.2 Hyperboloid of Osculation 475

J.3 Spatial Euler-Savary Equations 478

References 481

Index 489

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David B Dooner, University of Puerto Rico-Mayagüez, Puerto Rico and Ali A Seireg, University of Wisconsin at Madison and University of Florida at Gainesville, USA
David B Dooner is a Professor in the Department of Mechanical Engineering at the University of Puerto Rico-Mayagüez. He received his doctorate from the University of Florida at Gainesville in 1991 where he remained as a Post-Doctoral Fellow from 1991-1994. He worked at the General Motors Gear Center in 1989 and was a visiting scientist at the Mechanical Sciences Research Institute of the Russian Academy of Sciences in Moscow in 1992.
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Kinematic Geometry of GearingVisit the companion website to access the Delgear© software.
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