GEARING. Theory of. Stephen. Kinetics, Geometry, and Synthesis. P. Radzevich. /Ov CRC Press yc*** J Taylor& Francis Croup Boca Raton

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1 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

2 Contents Preface Acknowledgments Author Introduction xix xxi xxiii xxv PART I Synthesis Chapter 1 Kinematics of a Gear Pair Transmission of Motion through a Gear Pair Transition from a Pair of Friction Disks to an Equivalent Gear Pair Meaning of the Term "Synthesis" in This Book Vector Representation of Gear Pair Kinematics Concept of Vector Representation of Gear Pair Kinematics Three Different Vector Diagrams for Spatial Gear Pairs Vector Diagrams of External Spatial Gear Pairs Vector Diagrams of Internal Spatial Gear Pairs Vector Diagrams of Generalized Rack-Type Spatial Gear Pairs Analytical Criterion of a Spatial Gear Pair Classification of Possible Vector Diagrams of Gear Pairs Complementary Vectors to Vector Diagrams of Gear Pairs Centerline Vectors of a Gear Pair Axial Vectors of a Gear Pair Useful Kinematic and Geometric Formulas Tooth Ratio of a Gear Pair Example of the Application of Vector Diagrams of Gear Pairs 33 Endnotes 34 Chapter 2 Geometry of Gear Tooth Flanks: Preliminary Discussion Pulley-and-Belt Transmission as an Analogy of a Gear Pair Natural Form of a Gear Tooth Profile Other Possible Forms of a Gear Tooth Profile Possible Shapes of Gear Tooth Flanks Spur Involute Gear Tooth Flank Helical Involute Gear Tooth Flank Bevel Gear with Straight Teeth Tooth Flank Bevel Gear with Helical Teeth Tooth Flank Gear for a Crossed-Axis Gear Pair Tooth Flank Possible Form of a Gear Tooth in the Lengthwise Direction Tooth Contact Ratio: General Considerations 60 Endnotes 62

3 viii Contents Chapter 3 Geometry of Contact of Tooth Flanks oftwo Gears in Mesh Applied Reference Systems Associated with a Gear Pair Possible Local Patches of a Gear Tooth Flank Circular Diagrams of Local Patches of a Smooth Regular Surface Possible Classification of Local Patches of Gear Tooth Flanks Local Relative Orientation of Tooth Flanks at a Point of Contact Second Order Analysis of the Geometry of Contact of the Tooth Flanks of a Gear and of a Pinion Preliminary Remarks: Dupin Indicatrix Surface of Normal Relative Curvature Dupin Indicatrix of the Surface of Relative Curvature Matrix Representation of the Equation of the Dupin Indicatrix of the Surface of Relative Curvature Surface of Relative Normal Radii of Curvature Normalized Relative Normal Curvature Curvature Indicatrix JrKCharacteristic Curve Fourth Order Analysis of the Geometry of Contact of the Tooth Flanks of a Gear and of a Pinion Rate of Conformity of Two Smooth Regular Surfaces in the First Order of Tangency Indicatrix of Conformity of the Tooth Flanks $ and 0T Directions of the Extremum Rate of Conformity of the Tooth Flanks & and 0s: Asymptotes of the Indicatrix of Conformity Comparison of Capabilities of the Indicatrix of Conformity CnfR(^/^) and of the Dupin Indicatrix of the Surface of Relative Curvature Important Properties of the Indicatrix of Conformity Caf^/ffp) Converse Indicatrix of Conformity at a Point of Contact of the Tooth Flanks $ and m Plucker's Conoid: More Characteristic Curves Plucker's Conoid Basics Analytical Description Local Properties Auxiliary Formulae, Analytical Description of the Local Topology of a Smooth Regular Gear Tooth Flank & Preliminary Remarks Plucker's Conoid Plucker's Curvature Indicatrix >fnR <j ) -Indicatrix of a Gear Tooth Surface $ Relative Characteristic Curves Ill Possibility of Implementation of Two Plucker's Conoids ^S((g^)-Indicatrix of the and m. 112 Ill

4 Contents ix 3.7 Possible Contacts of the Teeth Surfaces $ and Possibility of Implementation of the Indicatrix of Conformity for the Identification of Contacts of the Tooth Flanks and Impact of the Accuracy of the Computations on the Desirable Parameters of the Indicatrices of Conformity Cnf(<^/^) 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 Principal Features and Fundamental Theorems of Parallel-Axis Gearing Kinematics of Parallel-Axis Gearing Willis Fundamental Law of Gearing Euler-Savary Equation Generation of an Involute Profile of a Gear Tooth Geometry of the Tooth Flank of a Spur Gear Generation of the Tooth Flank of a Spur Gear by Means of a Rack Addendum Modification (Profile Shift) Determination of the Tooth Form Generated by a Given Generating Rack Profile Base Tangent Length Tooth Thickness of a Gear Geometry of the Tooth Flank of a Helical Gear External Involute Gear Pair Variation of the Tooth Flank Geometry Normal Curvature of the Gear Tooth Flank Variation of the Tooth Profile Angle and Helix Angle Special Point of Meshing Contact Ratio of an External Gear Pair Transverse Contact Ratio Face Contact Ratio Total Contact Ratio Contact Motion Characteristics Sliding Conditions Specific Sliding Basic Equations for a Gear Pair with Addendum Modification Principle of Addendum Modification External Spur and Helical Gear Pairs Internal Involute Gearing Tooth Thickness Measurement of an Internal Gear Contact Ratio in an Internal Gearing Sliding Conditions in an Internal Gearing 194

5 X Contents Mating Internal Gear Pair Gear Coupling Involute Gear-to-Rack Pair Involute Gear Pairs with an Arbitrary Tooth Shape in the Lengthwise Direction Conditions to Be Fulfilled by Mating Gears 204 Endnotes 206 Chapter 6 Noninvolute Gearing Spur Noninvolute Gear Pairs Pin Gearing Cycloidal Gearing Root Blower Spur Gear Pairs of an Oil Pump Conditions for Smooth Rotation of a Noninvolute Gear Pair Interaction of a Noninvolute Gear with a Rack Helical Noninvolute Gear Pairs Helical Gear Pair of a Root Blower Infeasibility of Transmission of Rotation by a Noninvolute Helical Gear Pair with a Positive Transverse Contact Ratio 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 Noncylindrical Gears in Designing Parallel-Axis Gearing Conical Involute Gears Kinematics of Conical Involute Gearing Geometry of the Tooth Flanks of a Spur Conical Involute Gear Geometry of the Tooth Flanks of a Conical Involute Gear with Helical Teeth Toroidal Involute Gears Spur Toroidal Involute Gearing Toroidal Involute Gearing with Helical Teeth 246 Endnotes 250 Chapter 7 High-Conforming Parallel-Axis Gearing Novikov Gearing: A Helical Noninvolute Gearing That Has a Zero Transverse Contact Ratio Essence of Novikov Gearing ; Elements of Kinematics and the Geometry ofnovikov Gearing Design Parameters of Novikov Gearing High-Conforming Parallel-Axis Gearing Fundamental Design Parameters of High-Conforming Gearing Boundary N-Circle in High-Conforming Gearing Possible Tooth Geometries in High-Conforming Gearing Permissible Location of the Culminating Point in High-Conforming Gearing 273

6 Contents xi Contact of Tooth Flanks in a High-Conforming Gear Pair Configuration of Interacting Tooth Flanks at the Culminating Point Local and Global Geometry of Contact of Interacting Tooth Flanks Minimum Required Rate of Conformity between Interacting Tooth Flanks 279 Endnotes 284 Chapter 8 Synthesis of Optimal Parallel-Axis Gearing Geometrically Accurate Parallel-Axis Gearing Peculiarities of the Problem of Synthesis of Optimal Parallel-Axis Gears Peculiarities of the Problem of Synthesis of Optimal Involute Gears 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 Earliest Concepts of Intersected-Axis Gearing Kinematics of Intersected-Axis Gearing Base Cones in Intersected-Axis Gearing Tooth Flanks of Geometrically Accurate (Ideal) Intersected-Axis Gear Pairs Applied Coordinate Systems and Linear Transformations Main Reference Systems Operators of Rolling Operators Associated with the Gearing Housing Tooth Flank of a Bevel Gear Desired Tooth Proportions for Intersected-Axis Gears Base Angular Pitch Normal Pressure Angle Angular Pitch Angular Tooth Thickness and Angular Space Width Angular Addendum and Angular Specification of the Design Parameters in Dedendum 327 Intersected-Axis Gearing Contact Ratio in an Intersected-Axis Gearing Transverse Contact Ratio Face Contact Ratio Total Contact Ratio Tredgold's Approximation 333 Endnotes 334

7 x» Contents Chapter 10 High-Conforming Intersected-Axis Gearing Kinematics of the Instantaneous Motion in High-Conforming Intersected-Axis Gearing Contact Line in High-Conforming Intersected-Axis Gearing Bearing Capacity of High-Conforming Gearing Sliding of Teeth Flanks in High-Conforming Gearing Boundary N-Cone in Intersected-Axis High-Conforming Gearing 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 Kinematics of Crossed-Axis Gearing Base Cones in Crossed-Axis Gear Pairs Tooth Flanks of Geometrically Accurate (Ideal) Crossed-Axis Gear Pairs Applied Coordinate Systems and Linear Transformations Main Reference Systems Operators of Rolling/Sliding Operators Associated with Gear Housing Tooth Flank of a Crossed-Axis Gear Desired Tooth Proportions in Crossed-Axis Gearing Base Angular Pitch Normal Pressure Angle Angular Pitch Angular Tooth Thickness and Angular Space Width in the Round Basic Rack Angular Addendum and Angular Dedendum of the Round Basic Rack Specification of the Design Parameters of Crossed-Axis Gears Contact Ratio in Crossed-Axis Gearing Transverse Contact Ratio Face Contact Ratio Total Contact Ratio Possible Analogy of Tredgold's Approximation for Crossed-Axis Gearing 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 Kinematics of the Instantaneous Relative Motion Contact Line in High-Conforming Crossed-Axis Gearing Bearing Capacity of Crossed-Axis High-Conforming Gearing 391

8 Contents xiii Sliding between Tooth Flanks of the Gear and of the Pinion in Crossed-Axis High-Conforming Gearing Boundary N-Cone in Crossed-Axis High-Conforming Gearing 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 Practical Examples of 2-DOF Gearing Approach to Generate Tooth Flanks of the Gear and the Pinion in 2-DOF Gearing Possible Auxiliary Generating Racks 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 Preliminary Considerations Root Causes for Real Gears Differ from Ideal Gears Applied Coordinate Systems Displacements of a Gear Axis of Rotation from Its Desired Configuration Closest Distance of Approach between the Gear and the Pinion Axes of Rotation Tooth Flank Geometry of Desirable Real Gearing: 5pr-Gearing Tooth Flank Geometry of Desirable Real Gearing Possibility of Implementation of the Concept of Spr-Gearing in the Design of Gear Coupling Account for Normal Distribution of Manufacturing Errors onto the Geometry of Base Lines Preserving the Equality of Base Pitches at Different Values of Axis Misalignment Possible Simplifications Possibility of Implementation of the Concept of to Gear 5pr-Gearing Systems Featuring Point Contact of Tooth Flanks Correlation among Gear Systems of Various Kinds 442 Endnotes 444 Chapter 15 Approximate Real Gearing Approximate Real Parallel-Axis Gearing Approximate Real Intersected-Axis Gearing Root Causes for Referring to Real Intersected-Axis Gears as Approximate Gears 448

9 xiv Contents Approximate Real Intersected-Axis Gears Straight Tooth Bevel Gears Spiral Bevel Gears Face Gears Generation of Tooth Flanks of Intersected-Axis Gears Generation oftooth Flanks of Straight Bevel Gears Generation of Tooth Flanks of Spiral Bevel Gears Tooth Flanks of Bevel Gears Cut Using the Continuously Indexing Method of Gear Machining Examples of Approximate Real Intersected-Axis Gear Pairs Approximate Real Crossed-Axis Gearing: Hypoid Gears Worm Gearing Tooth Flank Modification Brief Historical Overview of Tooth Flank Modification Requirements to Design Parameters ofmodified Portions of Tooth Flanks 472 Endnotes 473 Chapter 16 Generic Gear Shape Origination of the Generic Gear Shape Examples of Gear Pairs Comprising Gears with Various Generic Shapes Evaluation of the Total Number of Possible Generic Gear Shapes Possible Profiles of the Generic Gear Shape Constructed in the Axial Cross Section of the Gear Profile of Generic Gear Surfaces Constructed in Cross Section by a Plane at an Angle to the Gear Axis Possibility of Classification of Possible Gear Pairs Examples of Implementation of the Classification of Possible Gear Pairs 491 Endnotes 495 Chapter 17 Gear Noise Transmission Error Base Pitch Variation Influence of the Contact Ratio Variation of the Load Requirements to Design Parameters for Low Noise/Noiseless Gear Drives Ideal Gear Pairs Ideal Parallel-Axis Gear Pairs Ideal Intersected-Axis Gear Pairs Ideal Crossed-Axis Gear Pairs Desired Real Gear Pairs Real (Approximate) Parallel-Axis Gear Pairs Real (Approximate) Intersected-Axis Gear Pairs Real (Approximate) Crossed-Axis Gear Pairs Real (Approximate) Gear Pairs 504

10 Contents xv PART VII Real Gears and Their Application: Gear Trains Chapter 18 Gear Ratio of a Multistage Gear Drive Principal Kinematic Relationships in a Multistage Gear Drive Range Ratio of Speed Variation for a Gear Drive Characteristic of a Transmission Group Analytical Method for Determining Transmission Ratios Rotational Speed Chart Broken Geometrical Series Minimum Number of Gear Pairs Determining the Tooth Number of Gears of Group Transmissions 512 Endnote 513 Chapter 19 Split Gear Drives Root Cause of Unequal Load Sharing in Multiflow Gear Drives Mobility of Split Gear Drives Epicyclic Gear Drives Structural Formula for Planetary Gear Drives Correspondence among Angular Velocities of All Members of a Planetary Gear Drive Problem of Equal Load Sharing in Planetary Gear Drives: State of the Art Planetary Gear Drives That Have Multiple Planet Pinions Single-Row Planetary Gear Drives with Six Self-Aligned Planet Pinions Positive Planetary Gear Drives with Large Transmission Ratios Planar Planetary Gear Drives with Self-Aligned Planet Pinions Planetary Gear Drives with Free Carriers Multiple and Closed Planetary Gear Drives Method of Structural Groups for Investigating Self-Alignment of Planetary Gearboxes Alternative Approaches for Equal Torque Sharing in Multiflow Gear Trains Planetary Gear Drives with Flexible Pins Load Equalizing in the Design of an Automotive Differential Elastic Absorbers of Manufacturing Errors Elastic Properties of Elastic Absorbers of Manufacturing Errors Examples of Implementation of Preloaded Elastic Absorber of Manufacturing Errors Load Equalizing with the Elastic Absorber Common for all Power Flows Main Features of Multiflow Gear Trains with Preloaded Elastic Absorbers of Manufacturing Errors 565 Endnotes 566

11 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 Local Geometry of the Interacting Tooth Flanks in Parallel-Axis Gearing Kinematics of the Interacting Tooth Flanks Local Geometry of the Interacting Tooth Flanks Local Geometry of the Interacting Tooth Flanks in Intersected-Axis Gearing Kinematics of Interaction of the Tooth Flanks Local Geometry of the Interacting Tooth Flanks Local Geometry of the Interacting Tooth Flanks in 577, Crossed-Axis Gearing Kinematics of Interaction of the Tooth Flanks Local Geometry of the Interacting Tooth Flanks Local Geometry of the Interacting Tooth Flanks in High-Conforming Gearing Kinematics of the Interacting Tooth Flanks Geometry of the Interacting Tooth Flanks 582 Endnotes 585 Chapter 21 Contact Stresses in Low-Tooth-Count Gearing Adopted Principal Assumptions Comments on Analytical Description of the Local Geometry of Contacting Surfaces Loaded by a Normal Force: Hertz's Proportional Assumption Assumption of Equal Torque Sharing Principal Features of Low-Tooth-Count Gears Analytical Model for the Calculation of Contact Stresses Combined Compressive and Shear Stresses in Low-Tooth-Count Gearing 595 Endnotes 598 Chapter 22 Application of the Results Derived from Theory of Gearing Bending Strength of a Gear Teeth: Comments on Lewis' Formula Cantilever Beam of Equal Strength Lewis' Formula for the Calculation of Gear Teeth Strength Effective Length of the Line of Contact Length of a Single Line of Contact in Parallel-Axis Gearing Effective Length of Lines of Contact in Parallel-Axis Gearing Effective Length of Lines of Contact in Spur Parallel-Axis Gearing Effective Length of Lines of Contact in Helical Parallel-Axis Gearing 613

12 Contents xvii 22.3 Loading of Gear Teeth 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

DUDLEY'S" HANDBOOK OF PRACTICAL GEAR DESIGN AND MANUFACTURE. Stephen P. Radzevich

DUDLEY'S HANDBOOK OF PRACTICAL GEAR DESIGN AND MANUFACTURE. Stephen P. Radzevich Second Edition DUDLEY'S" HANDBOOK 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