S e c o n d E d i t i o n

DUDLEY'S" H A N D B O O K OF

PRACTICAL GEAR DESIGN AND

MANUFACTURE

Stephen P. Radzevich

LßP) CRC Press VV J Taylors Francis Group

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

Contents

Preface xv Introduction xix Acknowledgments xxiii Author xxv

1 Gear Design Trends 1 1.1 Small, Low-Cost Gears for Toys, Gadgets, and Mechanisms 4 1.2 Appliance Gears 6 1.3 Machine Tools 7 1.4 Control Gears 8 1.5 Vehicle Gears 10 1.6 Transportation Gears 12 1.7 Marine Gears 13 1.8 Aerospace Gears 15 1.9 Industrial Gearing 16 1.10 Gears in the Oil and Gas Industry 18 1.11 Mill Gears 18 1.12 External Spur Gears 20 1.13 External Helical Gears 23 1.14 Internal Gears 24 1.15 Straight Bevel Gears 26 1.16 Zerol Bevel Gears 28 1.17 Spiral Bevel Gears 29 1.18 Hypoid Gears 30 1.19 Face Gears 31 1.20 Crossed-Helical Gears (Nonenveloping Worm Gears) 32 1.21 Single-Enveloping Worm Gears 34 1.22 Double-Enveloping Worm Gears 37 1.23 Spiroid Gears 39

2 Gear Types and Nomenclature 41 2.1 Types of Gears 41

2.1.1 Classifications 41 2.1.2 Parallel Axis Gears 44

2.1.2.1 Spur Gears 44 2.1.2.2 Helical Gears 45 2.1.2.3 Internal Gears 47

2.1.3 Nonparallel, Coplanar Gears (Intersecting Axes) 48 2.1.3.1 Bevel Gears 48 2.1.3.2 Face Gears (On-Center) 53 2.1.3.3 Conical Involute Gearing 54

2.1.4 Nonparallel, Noncoplanar Gears (Nonintersecting Axes) 55 2.1.4.1 Crossed-Axis Helical Gears 55 2.1.4.2 Cylindrical Worm Gearing 57

v

vi Contents

2.1.4.3 Single-Enveloping Worm Gearing 57 2.1.4.4 Double-Enveloping Worm Gearing 59 2.1.4.5 Hypoid Gears 60 2.1.4.6 Spiroid and Helicon Gearing 61 2.1.4.7 Face Gears (Off-Center) 63

2.1.5 Special Gear Types 63 2.1.5.1 Square or Rectangular Gears 63 2.1.5.2 Triangular Gears 64 2.1.5.3 Elliptical Gears 64 2.1.5.4 Scroll Gears 65 2.1.5.5 Multiple-Sector Gears 65

2.2 Nomenclature of Gears 66 2.2.1 Spur Gear Nomenclature and Basic Formulas 68 2.2.2 Helical Gear Nomenclature and Basic Formulas 68 2.2.3 Internal Gear Nomenclature and Formulas 70 2.2.4 Crossed Helical Gear Nomenclature and Formulas 70 2.2.5 Bevel Gear Nomenclature and Formulas 73

2.2.5.1 Straight 73 2.2.5.2 Spiral 74 2.2.5.3 Zerol 74

2.2.6 Worm Gear Nomenclature and Formulas 78 2.2.6.1 Cylindrical Worm Gears 80 2.2.6.2 Double-Enveloping Worm Gears 81

2.2.7 Face Gears 81 2.2.8 Spiroid Gear Nomenclature and Formulas 81 2.2.9 Beveloid Gears 83

3 Gear Tooth Design 85 3.1 Basic Requirements for Gear Tooth Design 85

3.1.1 Definition of Gear Tooth Elements 85 3.1.2 Basic Considerations for Gear Tooth Design 88

3.1.2.1 Continuity of Action 89 3.1.2.2 Conjugate Action 92 3.1.2.3 Pitch Diameter 93 3.1.2.4 Zones in Which Involute Gear Teeth Exist 94 3.1.2.5 Pointed Teeth 98 3.1.2.6 Undercut 100

3.1.3 Long- and Short-Addendum Gear Design 101 3.1.3.1 Addendum Modification for Gears Having a Few Teeth 102 3.1.3.2 Speed-Increasing Drives 103 3.1.3.3 Power Drives (Optimal Design) 103 3.1.3.4 Low-Friction Gearing 105

3.1.4 Special Design Considerations 107 3.1.4.1 Interchangeability 107 3.1.4.2 Tooth Thickness 107 3.1.4.3 Tooth Profile Modifications 109 3.1.4.4 Transverse of Profile Modification 109 3.1.4.5 Allowances for Errors of Gear Manufacture 109 3.1.4.6 Allowances for Deflection under Load 109

Contents vii

3.1.4.7 Axial Modifications 110 3.1.4.8 Root Fillets I l l 3.1.4.9 Effective Outside Diameter 113 3.1.4.10 Width of Tip of Tooth 113 3.1.4.11 Pointed Tooth Diameter 114 3.1.4.12 Purpose of Backlash 114 3.1.4.13 Backlash: Recommended Values 115

3.2 Standard Systems of Gear Tooth Proportions 116 3.2.1 Standard Systems for Spur Gears 116

3.2.1.1 Limitations in Use of Standard Tables 116 3.2.1.2 Standard Tooth Forms That Have Become Obsolete 118 3.2.1.3 Brown and Sharp System 119 3.2.1.4 AGMA 14.5° Composite System 119 3.2.1.5 Fellows 20° Stub Tooth System 120 3.2.1.6 AGMA 14.5° Full-Depth System 120 3.2.1.7 Cycloidal Tooth Profiles 121 3.2.1.8 Clockwork and Timer Tooth Profiles 121 3.2.1.9 Specific Spur Gear Calculation Procedure 122 3.2.1.10 Explanation and Discussion of Items in Table 3.5 126

3.2.2 System for Helical Gears 128 3.2.2.1 Selection of Tooth Form 130 3.2.2.2 Selection of Helix Angle 130 3.2.2.3 Face Width 131 3.2.2.4 Specific Calculation Procedure for Helical Gears 132

3.2.3 System for Internal Gears 133 3.2.3.1 Special Calculations 135 3.2.3.2 Specific Calculation Procedure for Internal Gears 136

3.2.4 Standard Systems for Bevel Gears 136 3.2.4.1 Discussion of 20° Straight Bevel Gear System 137 3.2.4.2 Discussion of Spiral Bevel Gear System 137 3.2.4.3 Discussion of the Zerol Bevel Gear System 138 3.2.4.4 Special Tooth Forms 138 3.2.4.5 Limitations in 20° Straight Bevel Gear System 139 3.2.4.6 Limitations in Spiral Bevel Gear System 141 3.2.4.7 Limitations in Zerol Bevel Gear System 141 3.2.4.8 General Comments 141

3.2.5 Standard Systems for Worm Gears 143 3.2.5.1 General Practice 143 3.2.5.2 Basic Tooth Forms for Worm Gearing 144 3.2.5.3 Specific Calculations for Worm Gears 145

3.2.6 Standard System for Face Gears 146 3.2.6.1 Pinion Design 148 3.2.6.2 Face Gear Design 148

3.2.7 System for Spiroid and Helicon Gears 149 3.2.7.1 Spiroid Gearing 151 3.2.7.2 Helicon Gearing 154 3.2.7.3 Detailed Calculations of Spiroid and Helicon Tooth Data 155

3.3 General Equations Relating to Center Distance 156 3.3.1 Center Distance Equations 156

viii Contents

3.3.2 Standard Center Distance 159 3.3.3 Standard Pitch Diameters 160 3.3.4 Operating Pitch Diameters 160 3.3.5 Operating Pressure Angle 161 3.3.6 Operating Center Distance 162 3.3.7 Center Distance for Gears Operating on Nonparallel

Nonintersecting Shafts 162 3.3.8 Center Distance for Worm Gearing 162 3.3.9 Reasons for Nonstandard Center Distances 163 3.3.10 Nonstandard Center Distances 163

3.4 Elements of Center Distance 165 3.4.1 Effects of Tolerances on Center Distance 165 3.4.2 Machine Elements that Require Consideration in Critical Center

Distance Applications 168 3.4.3 Control of Backlash 171 3.4.4 Effects of Temperature on Center Distance 172 3.4.5 Mounting Distance 174

4 Preliminary Design Considerations 177 4.1 Stress Formulas: Calculation 177

4.1.1 Calculated Stresses 177 4.1.2 Gear Design Limits 181 4.1.3 Gear Strength Calculations 181

4.1.3.1 Worst Load 184 4.1.3.2 Stress Concentration 185 4.1.3.3 Load Distribution 186 4.1.3.4 Dynamic Load 186 4.1.3.5 Finite Element 187

4.1.4 Gear Surface Durability Calculations 188 4.1.4.1 Hertz Derivations 189 4.1.4.2 K-Factor Derivations 192 4.1.4.3 Worst-Load Position 193 4.1.4.4 Endurance Limit 193 4.1.4.5 Lubrication Regimens 194

4.1.5 Gear Scoring 198 4.1.5.1 Hot and Cold Scoring 198 4.1.5.2 PVT Formula 199 4.1.5.3 Flash Temperature 202 4.1.5.4 Scoring Criterion 203

4.1.6 Thermal Limits 205 4.1.6.1 Thermal Limits at Regular Speed 205 4.1.6.2 Thermal Limits at High Speed 206

4.2 Stress Formulas: Special Considerations 206 4.2.1 Gear Specifications 207 4.2.2 Size of Spur and Helical Gears by Q-Factor Method 207

4.2.2.1 Face-Width Considerations 210 4.2.2.2 Weight from Volume 211

4.2.3 Indexes of Tooth Loading 213 4.2.4 Estimating Spur and Helical Gear Size by K-Factor 214

Contents ix

4.2.5 Estimating Bevel Gear Size 215 4.2.6 Estimating Worm Gear Size 220 4.2.7 Estimating Spiroid Gear Size 223

4.3 Data Needed for Gear Drawings 223 4.3.1 Gear Dimensional Data 223 4.3.2 Gear Tooth Tolerances 226 4.3.3 Gear Material and Heat Treatment Data 234 4.3.4 Enclosed Gear Unit Requirements 241

5 Design Formulas 243 5.1 Calculation of Gear Tooth Data 243

5.1.1 Number of Pinion Teeth 243 5.1.2 Hunting Teeth 248 5.1.3 Spur Gear Tooth Proportions 249 5.1.4 Root Fillet Radii of Curvature 251 5.1.5 Long-Addendum Pinions 253 5.1.6 Tooth Thickness 257

5.1.6.1 Backlash 257 5.1.6.2 Tolerances and Tooth Thickness 258

5.1.7 Chordal Dimensions 258 5.1.8 Degrees Roll and Limit Diameter 260 5.1.9 Form Diameter and Contact Ratio 264

5.1.9.1 Form Diameter 264 5.1.9.2 Contact Ratio 265

5.1.10 Spur Gear Dimension Sheet 266 5.1.11 Internal Gear Dimension Sheet 268 5.1.12 Helical Gear Tooth Proportions 272 5.1.13 Helical Gear Dimension Sheet 277 5.1.14 Bevel Gear Tooth Proportions 279 5.1.15 Straight Bevel Gear Dimension Sheet 283 5.1.16 Spiral Bevel Gear Dimension Sheet 285 5.1.17 Zerol Bevel Gear Dimension Sheet 288 5.1.18 Hypoid Gear Calculations 289 5.1.19 Face Gear Calculations 290 5.1.20 Crossed-Helical Gear Proportions 293 5.1.21 Single-Enveloping Worm Gear Proportions 296 5.1.22 Single-Enveloping Worm Gears 297 5.1.23 Double-Enveloping Worm Gears 299

5.2 Gear Rating Practice 303 5.2.1 General Considerations in Rating Calculations 303

5.2.1.1 Calculation Procedure 305 5.2.1.2 Grades of Material Quality 306 5.2.1.3 Reliability of Gears 307

5.2.2 General Formulas for Tooth Bending Strength and Tooth Surface Durability 307 5.2.2.1 Strength Formula 308 5.2.2.2 Durability Formula 308 5.2.2.3 Rating Curves of Stress versus Cycles 313 5.2.2.4 Rating Bevel Gears 315

X Contents

5.2.3 Geometry Factors for Strength 316 5.2.3.1 Lack of Load Sharing 317 5.2.3.2 Helical Gears with Narrow Face Width 317 5.2.3.3 Geometry Factors for Strength for Some Standard Designs.... 318

5.2.4 Overall De-Rating Factor for Strength 319 5.2.4.1 Application Factor Ka 322 5.2.4.2 Load Distribution Factor Km 322 5.2.4.3 Effect of Helix Error and Shaft Misalignment 324 5.2.4.4 Aspect Ratio Effects 327 5.2.4.5 Load Distribution Factor Km for Bevel Gears 328 5.2.4.6 Size Factor Ks 330 5.2.4.7 Dynamic Load Factors K, and Cv 330

5.2.5 Geometry Factors for Durability 333 5.2.6 Overall De-Rating Factor for Surface Durability 335

5.2.6.1 Size Factor Cs 336 5.2.6.2 Complementary Considerations 338

5.2.7 Load Rating of Worm Gearing 339 5.2.7.1 Crossed-Helical Gear Durability 339 5.2.7.2 Cylindrical Worm Gear Durability 341 5.2.7.3 Double-Enveloping Worm Gear Durability 346 5.2.7.4 Comparison of Double-Enveloping and Cylindrical

Worm Gear Rating Procedures 349 5.2.8 Design Formulas for Scoring 350

5.2.8.1 Hot Scoring 350 5.2.8.2 Cold Scoring 355 5.2.8.3 Design Practice to Handle Scoring 360

5.2.9 Trade Standards for Rating Gears 361 5.2.10 Vehicle Gear Rating Practice 363 5.2.11 Marine Gear Rating Practice 365 5.2.12 Oil and Gas Industry Gear Rating 367 5.2.13 Aerospace Gear Rating Practice ; 368

6 Gear Materials 371 6.1 Steels for Gears 371

6.1.1 Mechanical Properties 371 6.1.2 Heat-Treating Techniques 375 6.1.3 Heat-Treating Data 379 6.1.4 Hardness Tests 381

6.2 Localized Hardening of Gear Teeth 383 6.2.1 Carburizing 383 6.2.2 Nitriding 389

6.2.2.1 Features of Nitriding Process 389 6.2.2.2 Nitride Case Depth 392

6.2.3 Induction Hardening of Steel 393 6.2.3.1 Induction Hardening by Scanning 395 6.2.3.2 Load-Carrying Capacity of Induction-Hardened Gear Teeth 398

6.2.4 Flame Hardening of Steel 400 6.2.5 Combined Heat Treatments 401 6.2.6 Metallurgical Quality of Steel Gears 402

Contents xi

6.2.6.1 Quality Items for Carburized Steel Gears 402 6.2.6.2 Quality Items for Nitrided Gears 404 6.2.6.3 Procedure to Get Grade 2 Quality 405

6.3 Cast Irons for Gears 406 6.3.1 Gray Cast Iron 407 6.3.2 Ductile Iron 408 6.3.3 Sintered Iron 410

6.4 Nonferrous Gear Metals 411 6.4.1 Kinds of Bronze 411 6.4.2 Standard Gear Bronzes 413

6.5 Nonmetallic Gears 414 6.5.1 Thermosetting Laminates 414 6.5.2 Nylon Gears 416

7 Gear-Manufacturing Methods 419 7.1 Gear Tooth Cutting 419

7.1.1 GearHobbing 419 7.1.2 Shaping—Pinion Cutter 426 7.1.3 Shaping—Rack Cutter 431 7.1.4 Cutting Bevel Gears 435 7.1.5 Gear Milling 438 7.1.6 Broaching Gears 442 7.1.7 Punching Gears 444 7.1.8 G-TRAC Generating 446

7.2 Gear Grinding 448 7.2.1 Form Grinding 448

7.2.1.1 Ceramic Form Grinding 449 7.2.1.2 Borazon Form Grinding 452

7.2.2 Generating Grinding—Disk Wheel 453 7.2.3 Generating Grinding—Bevel Gears 456 7.2.4 Generating Grinding—Threaded Wheel 459 7.2.5 Thread Grinding 462

7.3 Gear Shaving, Rolling, and Honing 465 7.3.1 Rotary Shaving 465 7.3.2 Rack Shaving 470 7.3.3 Gear Rolling 470 7.3.4 Gear Honing 473

7.4 Gear Measurement 475 7.4.1 Gear Accuracy Limits 476 7.4.2 Machines to Measure Gears 481

7.5 Gear Casting and Forming 487 7.5.1 Cast and Molded Gears 487 7.5.2 Sintered Gears 488 7.5.3 Cold-Drawn Gears and Rolled Worm Threads 489

8 Design of Tools to Make Gear Teeth 491 8.1 Shaper Cutters 491 8.2 Gear Hobs 502 8.3 Spur Gear-Milling Cutters 515

Contents

8.4 Worm-Milling Cutters and Grinding Wheels 517 8.5 Gear-Shaving Cutters 522 8.6 Punching Tools 525 8.7 Sintering Tools 526

Kinds and Causes of Gear Failures 529 9.1 Analysis of Gear-System Problems 529

9.1.1 Determining the Problem 529 9.1.2 Possible Causes of Gear-System Failures 531 9.1.3 Incompatibility in Gear Systems 533 9.1.4 Investigation of Gear Systems 534

9.2 Analysis of Tooth Failures and Gear-Bearing Failures 535 9.2.1 Nomenclature of Gear Failure 536 9.2.2 Tooth Breakage 536 9.2.3 Pitting of Gear Teeth 540 9.2.4 Scoring Failures 542 9.2.5 Wear Failures 544 9.2.6 Gearbox Bearings 547 9.2.7 Rolling-Element Bearings 548 9.2.8 Sliding-Element Bearings 550

9.3 Some Causes of Gear Failure Other than Excess Transmission Load 552 9.3.1 Overload Gear Failures 553 9.3.2 Gear-Casing Problems 554 9.3.3 Lubrication Failures 555 9.3.4 Thermal Problems in Fast-Running Gears 561

Special Design Problems 565 10.1 Center Distance Problems 565 10.2 Profile Modification Problems 576 10.3 Load Rating Problem 585

Gear Reactions and Mountings 591 11.1 Mechanics of Gear Reactions 591

11.1.1 Summation of Forces and Moments 591 11.1.2 Application to Gearing 594

11.2 Basic Gear Reactions, Bearing Loads, and Mounting Types 595 11.2.1 Main Source of Load 595 11.2.2 Gear Reactions to Bearing 596 11.2.3 Directions of Loads 597 11.2.4 Additional Considerations 597 11.2.5 Types of Mountings 598 11.2.6 Efficiencies 600

11.3 Basic Mounting Arrangements and Recommendations 600 11.3.1 Bearing and Shaft Alignment 601 11.3.2 Bearings 601 11.3.3 Mounting Gears to the Shaft 602 11.3.4 Housing 603 11.3.5 Inspection Hole 603 11.3.6 Break-In 603

Contents xiii

11.4 Bearing Load Calculations for Spur Gears 604 11.4.1 Spur Gears 604 11.4.2 Helical Gears 606 11.4.3 Gears in Trains 606 11.4.4 Idlers 606 11.4.5 Intermediate Gears 607 11.4.6 Planetary Gears 609

11.4.6.1 One Planet 609 11.4.6.2 Several Planets 609

11.5 Bearing-Load Calculations for Helicals 610 11.5.1 Single Helical Gears 610 11.5.2 Double Helical Gears 612 11.5.3 Skewed- or Crossed-Helical Gears 613

11.6 Mounting Practice for Bevel and Hypoid Gears 615 11.6.1 Analysis of Forces 615 11.6.2 Rigid Mountings 616 11.6.3 Maximum Displacements 616 11.6.4 Rolling-Element Bearings 619 11.6.5 Straddle Mounting 619 11.6.6 Overhung Mounting 619 11.6.7 Gear Blank Design 622 11.6.8 Gear and Pinion Adjustments 625 11.6.9 Assembly Procedure 626

11.7 Calculation of Bevel and Hypoid Bearing Loads 627 11.7.1 Hand of Spiral 627 11.7.2 Spiral Angle 627 11.7.3 Tangential Load 628 11.7.4 Axial Thrust 629 11.7.5 Radial Load 629 11.7.6 Required Data for Bearing Load Calculations 637

11.8 Bearing Load Calculations for Worms 639 11.8.1 Calculation of Forces in Worm Gears 640 11.8.2 Mounting Tolerances 642 11.8.3 Worm Gear Blank Considerations 642 11.8.4 Run-in of Worm Gears 642

11.9 Bearing Load Calculations for Spiroid Gearing 644 11.10 Bearing Load Calculations for Other Gear Types 648 11.11 Design of the Body of the Gear 648

12 Gear Vibration 651 12.1 Fundamentals of Vibration 651 12.2 Measurement of Vibration 654

12.2.1 Examples of Sensing Devices 655 12.2.2 Practical Problems in Vibration Measurement 657

12.3 Some Examples of Vibration in Geared Units 657 12.4 Approximate Vibration Limits 660

12.4.1 Velocity Limits 661 12.4.2 Acceleration Limits 662 12.4.3 Proximity Probes 662

xiv Contents

12.4.4 Displacement Limits 662 12.4.5 General Vibration Tendencies 663 12.4.6 Trade Standard 663

12.5 Control of Vibration in Manufacturing Gears and in the Field 665 12.5.1 Testing of Gear Units at the Gear Factory 666 12.5.2 Tests of Assembled Power Package 666 12.5.3 Vibration Tests in the Field 667

12.6 Vibration Analysis Technique 668

Appendix A: Evolution of Gear Art 671

Appendix B: Complementary Material 717

Appendix C: Numerical Data Tables 771

Appendix D: On the Concept of Novikov Gearing and the Inadequacy of the

Terms "Wildhaber-Novikov Gearing" and "W-N Gearing" 803

References 835

Index 847