Theory of GEARING Kinetics, Geometry, and Synthesis Stephen P. Radzevich /Ov CRC Press yc*** J Taylor& Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business
Contents Preface Acknowledgments Author Introduction xix xxi xxiii 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 Spatial Gear 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 of the 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 3.6.3.2 ^S((g^)-Indicatrix of the Surfaces @ and m. 112 Ill
Contents ix 3.7 Possible Contacts of the Teeth Surfaces $ and 115 3.7.1 Possibility of Implementation of the Indicatrix of Conformity for 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 by Means 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 Gearing with 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 Interacting Tooth 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 Optimal Parallel-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 9.4.3.6 Specification of the Design Parameters in Dedendum 327 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-Conforming Intersected-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 of Rotation 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 to Gear 5pr-Gearing 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-Aligned Planet Pinions 528 19.6.3 Positive Planetary Gear Drives with Large Transmission Ratios 530 19.6.4 Planar Planetary Gear Drives with Self-Aligned Planet 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-Alignment of 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: Principal Features 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 in 577, Crossed-Axis Gearing 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 Geometry of 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 Spur Parallel-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