dooner kinematic geometry second edition red box rules are ... · kinematic geometry of gearing...

RED BOX RULES ARE FOR PROOF STAGE ONLY. DELETE BEFORE FINAL PRINTING.

Kinematic Geometry of Gearing

Kinem

atic Geom

etry of Gearing

Second Edition

David B. Dooner

Dooner

Kinematic Geometry of Gearing

Second Edition

SecondEdition

David B. Dooner, University of Puerto Rico-Mayagüez, Puerto Rico

Kinematic Geometry of Gearing Second Edition forms an indispensable tool of the trade forindustry engineers, academics and researchers working in kinematics and mechanics. It is alsoan invaluable reference for graduate and postgraduate students in mechanical engineering.

Cover design by Dan Jubb

The first edition of Kinematic Geometry of Gearing proposed a new system of curvilinearcoordinates specifically for describing general gear forms. This system of coordinates and newmathematical relations on the kinematics of ruled surfaces in mesh were new and believed to beinvaluable to the gearing world and the kinematics community. This development provided acompletely different perspective and approach to the state-of-the-art gearing. Since thepublication of the first edition in 1995, author David Dooner has continued the development ofthis dynamic new approach to the design, manufacture and evaluation of gears.

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

This new second edition has been extensively revised and updated with new and original material.

Key features include:• The methodology for general tooth forms, radius of torsure, cylinder of osculation,

and cylindroid of torsure.• A reworked ‘3 laws of gearing’.• A new chapter on gear vibration load factor and impact.• A companion website (www.wiley.com/go/dooner_2e) housing extensive software with GUI and development of all of the equations presented in the second edition.

www.wiley.com/go/dooner_2e

P1: TIX/XYZ P2: ABCJWST162-fm JWST162-Dooner February 16, 2012 20:40 Printer Name: Yet to Come Trim: 244mm×168mm


KINEMATIC GEOMETRYOF GEARING


KINEMATIC GEOMETRYOF GEARINGSECOND EDITION

David B. DoonerUniversity of Puerto Rico—Mayaguez

A John Wiley & Sons, Ltd., Publication


This edition first published 2012c© 2012 John Wiley & Sons, Ltd

Registered officeJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply forpermission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright,Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in anyform or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UKCopyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not beavailable in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names andproduct names used in this book are trade names, service marks, trademarks or registered trademarks of theirrespective owners. The publisher is not associated with any product or vendor mentioned in this book. Thispublication is designed to provide accurate and authoritative information in regard to the subject matter covered. It issold on the understanding that the publisher is not engaged in rendering professional services. If professional adviceor other expert assistance is required, the services of a competent professional should be sought.

Library of Congress Cataloging-in-Publication Data

Dooner, David B.Kinematic geometry of gearing / David Dooner. – 2nd ed.

p. cm.Includes bibliographical references and index.ISBN 978-1-119-95094-3 (hardback)

1. Gearing. 2. Machinery, Kinematics of. I. Title.TJ184.D66 2012621.8′33–dc23

2011050038

A catalogue record for this book is available from the British Library.

ISBN: 978-1-1199-5094-3

Typeset in 10/12pt Times by Aptara Inc., New Delhi, India

http://www.wiley.com


Contents

Preface xiii

Part I FUNDAMENTAL PRINCIPLES OF TOOTHED BODIES IN MESH

1 Introduction to the Kinematics of Gearing 31.1 Introduction 31.2 An Overview 31.3 Nomenclature and Terminology 51.4 Reference Systems 81.5 The Input/Output Relationship 91.6 Rigid Body Assumption 111.7 Mobility 111.8 Arhnold-Kennedy Instant Center Theorem 141.9 Euler-Savary Equation for Envelopes 181.10 Conjugate Motion Transmission 19

1.10.1 Spur Gears 201.10.2 Helical and Crossed Axis Gears 21

1.11 Contact Ratio 221.11.1 Transverse Contact Ratio 241.11.2 Axial Contact Ratio 25

1.12 Backlash 251.13 Special Toothed Bodies 26

1.13.1 Microgears 281.13.2 Nanogears 28

1.14 Noncylindrical Gearing 291.14.1 Hypoid Gear Pairs 291.14.2 Worm Gears 301.14.3 Bevel Gears 32

1.15 Noncircular Gears 331.15.1 Gear and Cam Nomenclature 381.15.2 Rotary/Translatory Motion Transmission 39

1.16 Schematic Illustration of Gear Types 40


vi Contents

1.17 Mechanism Trains 401.17.1 Compound Drive Trains 411.17.2 Epicyclic Gear Trains 431.17.3 Circulating Power 491.17.4 Harmonic Gear Drives 501.17.5 Noncircular Planetary Gear Trains 51

1.18 Summary 52

Part II THE KINEMATIC GEOMETRY OF CONJUGATE MOTIONIN SPACE

2 Kinematic Geometry of Planar Gear Tooth Profiles 552.1 Introduction 552.2 A Unified Approach to Tooth Profile Synthesis 552.3 Tooth Forms Used for Conjugate Motion Transmission 56

2.3.1 Cycloidal Tooth Profiles 562.3.2 Involute Tooth Profiles 592.3.3 Circular-arc Tooth Profiles 632.3.4 Comparative Evaluation of Tooth Profiles 64

2.4 Contact Ratio 652.5 Dimensionless Backlash 682.6 Rack Coordinates 69

2.6.1 The Basic Rack 712.6.2 The Specific Rack 762.6.3 The Modified Rack 772.6.4 The Final Rack 79

2.7 Planar Gear Tooth Profile 802.8 Summary 84

3 Generalized Reference Coordinates for Spatial Gearing—theCylindroidal Coordinates 85

3.1 Introduction 853.2 Cylindroidal Coordinates 85

3.2.1 History of Screw Theory 873.2.2 The Special Features of Cylindroidal Coordinates 87

3.3 Homogeneous Coordinates 893.3.1 Homogeneous Point Coordinates 913.3.2 Homogeneous Plane Coordinates 923.3.3 Homogeneous Line Coordinates 933.3.4 Homogeneous Screw Coordinates 96

3.4 Screw Operators 993.4.1 Screw Dot Product 993.4.2 Screw Reciprocal Product 993.4.3 Screw Cross Product 1013.4.4 Screw Intersection 1023.4.5 Screw Triangle 103


Contents vii

3.5 The Generalized Equivalence of the Pitch Point—the Screw Axis 1043.5.1 Theorem of Three Axes 1053.5.2 The Cylindroid 1073.5.3 Cylindroid Intersection 108

3.6 The Generalized Pitch Surface—Axodes 1103.6.1 The Theorem of Conjugate Pitch Surfaces 1153.6.2 The Striction Curve 116

3.7 The Generalized Transverse Surface 1213.8 The Generalized Axial Surface 1233.9 Summary 125

4 Differential Geometry 1274.1 Introduction 1274.2 The Curvature of a Spatial Curve 1274.3 The Torsion of a Spatial Curve 1294.4 The First Fundamental Form 1304.5 The Second Fundamental Form 1324.6 Principal Directions and Principal Curvatures 1354.7 Torsure of a Spatial Curve 1384.8 The Cylindroid of Torsure 1424.9 Ruled Surface Trihedrons 1484.10 Formulas of Fernet-Serret 1504.11 Coordinate Transformations 1514.12 Characteristic Lines and Points 1584.13 Summary 159

5 Analysis of Toothed Bodies for Motion Generation 1615.1 Introduction 1615.2 Spatial Mobility Criterion 1615.3 Reciprocity—the First Law of Gearing 1645.4 The Line Complex 1665.5 The Tooth Spiral 168

5.5.1 The Tooth Spiral Curvature 1705.5.2 The Tooth Spiral Torsion 173

5.6 Tooth Spiral Angle—the Second Law of Gearing 1745.6.1 The I/O Relationship 1795.6.2 The Phantom I/O Relationship 181

5.7 Reduced Tooth Curvature—the Third Law of Gearing 1835.7.1 Absolute Tooth Curvature 1875.7.2 Tooth Profile Modification 190

5.8 Classification of Gear Types 1925.9 Contact Ratio 194

5.9.1 Transverse Contact Ratio 1955.9.2 Axial Contact Ratio 196

5.10 Spatial Backlash 196


viii Contents

5.11 Relative Displacements 1975.11.1 The Sliding Velocity 1975.11.2 The Rolling Velocity 2005.11.3 The Pitch Line Velocity 202

5.12 Mesh Efficiency 2035.13 Summary 205

6 The Manufacture of Toothed Bodies 2076.1 Introduction 2076.2 Manufacturing Background 207

6.2.1 Form-Type Fabrication 2086.2.2 Generation-Type Fabrication 2086.2.3 Spiral Bevel/Hypoid Gear Fabrication 2126.2.4 Noncircular Gear Fabrication 215

6.3 Crossed Hyperboloidal Gears 2166.4 Fabrication of Cutters 220

6.4.1 The Hyperboloidal Cutter 2206.4.2 The Cutter Spiral Angle 2246.4.3 The Face Spiral Angle 2256.4.4 Cutter Constraints 2276.4.5 Speed Ratio 2286.4.6 Hyperboloidal Cutter Coordinates 231

6.5 Gear Cutting Machine Layout 2356.6 The Envelope of the Cutter 237

6.6.1 The Equation of Meshing 2396.6.2 Boolean Operations 241

6.7 Material Removal Rate 2426.8 Surface Cutting Speed 2426.9 Discretization Error 243

6.9.1 Scalloping 2436.9.2 Tessellation 245

6.10 Inspection 2466.11 Hyperboloidal Blank Dimensions 2476.12 Summary 248

7 Vibrations and Dynamic Loads in Gear Pairs 2497.1 Introduction 2497.2 Excitations 2497.3 Transmission Error 250

7.3.1 Static Transmission Error 2517.3.2 Loaded Transmission Error 2547.3.3 Dynamic Transmission Error 255

7.4 Fourier Transforms 2607.5 Impact Loading 2617.6 Mesh Stiffness 264


Contents ix

7.7 Inertial Properties 2657.7.1 Center of Mass 2657.7.2 Mass Moments of Inertia 267

7.8 Manufacturing Dynamics 2697.9 Summary 270

Part III THE INTEGRATED DESIGN AND MANUFACTURING PROCESS

8 Gear Design Rating 2758.1 Introduction 2758.2 Modes of Gear Failure 2758.3 Reaction Loads 2758.4 Gear Parameters for Specified Deflections 2808.5 The Fillet Stress 286

8.5.1 Discretization of Gear Tooth 2878.5.2 Element Stiffness Matrix 2898.5.3 Global Stiffness Matrix 2928.5.4 Boundary Conditions 2938.5.5 Nodal Strain 2948.5.6 Nodal Stress 294

8.6 Inertial Stress 2958.7 Contact Stress 2968.8 Minimum Film Thickness 2998.9 Wear 3018.10 Friction Coefficient 305

8.10.1 Sliding Friction 3058.10.2 Rolling Friction 309

8.11 Flash Temperature 3118.12 Thermal Stress 3138.13 Failure Analysis 314

8.13.1 Reliability Analysis 3148.13.2 Fatigue Analysis 3178.13.3 Cumulative Loading 320

8.14 Windage Losses 3218.15 Optimization 3258.16 Summary 326

9 The Integrated CAD–CAM Process 3279.1 Introduction 3279.2 Modular Components for Geometric Synthesis 327

9.2.1 The Motion Specification Module 3289.2.2 The Tooth Parameters Module 3289.2.3 The Gear Parameters Module 3319.2.4 The Cutter Parameters Module 332


x Contents

9.2.5 The Loading Parameters Module 3339.2.6 The Material Specifications Module 3339.2.7 The Lubricant Specifications Module 3349.2.8 The Dynamic Factors Module 3369.2.9 The Shaft Deflections Module 3379.2.10 The Manufacturing Specifications Module 337

9.3 The Integrated CAD–CAM Process 3389.4 Illustrative Example 3389.5 Summary 361

10 Case Illustrations of the Integrated CAD–CAM Process 36310.1 Introduction 36310.2 Case 1 36310.3 Case 2 36410.4 Case 3 36510.5 Case 4 36610.6 Case 5 36710.7 Case 6 36810.8 Case 7 36910.9 Case 8 37010.10 Case 9 37110.11 Case 10 37210.12 Case 11 37310.13 Case 12 37410.14 Case 13 37510.15 Case 14 37610.16 Case 15 37710.17 Case 16 37810.18 Case 17 37910.19 Case 18 38010.20 Case 19 38110.21 Case 20 38210.22 Case 21 38310.23 Case 22 38610.24 Summary 388

Appendix A Differential Expressions 389A.1 Derivatives of the Radius of the Axode 389A.2 Derivatives of the Included Angles 391A.3 Derivatives of the Generators 392A.4 Derivatives of the Pitch of the Instantaneous Twist 394A.5 Derivatives of the Parameter of Distribution 394A.6 Derivatives of the Striction Curve 394A.7 Manufacturing Expressions 396


Contents xi

A.8 Derivatives of the Transverse Curve 396A.9 Derivatives of the Angle Between the Generator and the Transverse Curve 397A.10 Derivatives of the Spiral Angle 398A.11 Derivatives of the Input Trihedron of Reference 399A.12 Derivatives of the Cutter Parameters 399

Appendix B On the Notation and Operations 401

Appendix C Noncircular Gears 409C.1 Torque and Speed Fluctuations in Rotating Shafts 409C.2 2-Dof Mechanical Function Generator 412C.3 Steering Mechanism 414C.4 Continuously Variable Transmission 416C.5 Geared Robotic Manipulators 418C.6 Spatial Mechanism for Body Guidance 420C.7 Nonworking Profile 421C.8 Multiple Reductions 422

Appendix D The Delgear© Software 425D.1 Installation 427

Appendix E Splines 429E.1 Cubic Splines 430E.2 Natural Splines 433

E.2.1 Derivatives 435E.3 NURBS 436

Appendix F Contact Stress 437F.1 Introduction 437F.2 Background 437F.3 Material Properties 438F.4 Surface Geometry 439F.5 Contact Deformations 442F.6 Contact Area 443F.7 Comparison 445

Appendix G Glossary of Terms 447

Appendix H Equilibrium and Diffusion Equations 455H.1 Equilbrium Equations 455H.2 Diffusion Equation Formulation 459H.3 Expressions 461

Appendix I On the Base Curve of Planar Noncircular Gears 465


xii Contents

Appendix J Spatial Euler-Savary Equations 471J.1 Planar Euler-Savary Equations 471J.2 Hyperboloid of Osculation 475J.3 Spatial Euler-Savary Equations 478

References 481

Index 489

P1: TIX/XYZ P2: ABCJWST162-preface JWST162-Dooner February 18, 2012 7:20 Printer Name: Yet to Come Trim: 244mm×168mm

Preface

This second edition is an expansion of the first edition of The Kinematic Geometry of Gear-ing; A Concurrent Engineering Approach, introducing a generalized integrated methodologyfor the design and manufacture of different types of toothed bodies. Several expressionsare modified from their original presentation along with a reorganization of the material.Included are changes in nomenclature to reduce subscripts, avoid conflicts with symbols,and aid in the implementation of computer software. The kinematic geometry of toothedbodies in mesh builds upon the original presentation and is supplemented with additionalfigures. The design and manufacturing sections are expanded to provide a more thoroughevaluation of the new geometric methodology. Biographical data on historical individuals areprovided in footnotes; much of this information is based on the work of J.J. O’Conner andE.F. Robertson.1 The ensuing presentation more thoroughly develops the single geometricmethodology for integrated design and manufacture of gear pairs. A computer simulation forthe integrated design and manufacture of generalized gear pairs has been completed (includ-ing a GUI or Graphical User Interface), showcasing the concurrent CAD–CAM of gear pairs.Prototype gear pairs have been fabricated and tested to illustrate the geometric methodologydeveloped.

Two bodies in direct contact where the position and orientation of the output element arespecified functions of a given input motion comprise gear pairs, threaded fasteners (i.e., boltsand nuts), as well as CAM systems. The overarching goal is a single geometric framework forthe generalized design and manufacture of gear pairs, with consideration to fasteners and CAMsystems. The importance of gearing continues in the twenty-first century. Gear elements rangein size from 4000 mm or 150 in. to 1 μm or 0.5 μin., where the speeds range from less than1 RPM to over 6 trillion RPM! The automotive industry is currently the biggest user of gearelements commanding over 60% of the world gear market. Such gears encompass spur andhelical gears used in transmission, worm and worm wheels used for window regulation,along with spiral bevel and hypoid used in a rear/front axle assembly. Twenty percent ofthe automotive gear market is targeted to right angle drive gear pairs. It is estimated that thereare over 800 million automobiles worldwide with 70 million produced annually. Gears arealso essential machine elements in industrial applications, as well as the aerospace and marineindustries.

Current gear practice for spatial gearing does not provide for bevel, hypoid, and wormgears to be treated with the same geometric considerations that are applicable to cylindrical

1 http://www-groups.dcs.stand.ac.uk/∼history/Mathematicians

http://www-groups.dcs.stand.ac.uk/%E2%88%BChistory/Mathematicians




xiv Preface

gearing (namely, spur and helical gears). These geometric considerations include generalformulations for the tooth profile, addendum and dedendum constants, profile modifications,crown, transverse and axial contact ratios, backlash, spiral angle variation, pressure anglevariation, inspection techniques, as well as manufacturing technology. The salient theme ofthis book is to present a single geometric theory for the concurrent CAD–CAM of toothedbodies in direct contact used to transmit power (motion and load) between two axes. The endresult is an axode-based theory analogous to that used to design and manufacture planar spurand helical gears. This unified approach is based on formulating a system of pitch, transverse,and axial surfaces, utilizing special curvilinear coordinates to parameterize the kinematicgeometry of motion transmission between skew axes. Screw theory or the theory of screwsis used as the basis for this geometric foundation. The same results can be obtained usingalternatives such as dual numbers, Lie algebras, geometric algebras, or vector algebra. Thepresented technique builds upon existing known relations and utilizes screw theory to establish

� cylindroidal coordinates,� theorem of conjugate pitch surfaces,� kinematic relations between generalized ruled surfaces,� three laws of gearing,� cylindroid of torsure, and� spatial analog of planar Euler-Savary equations.

This analytical foundation is further expanded by introducing a variable diameter cutter or hobcutter for gear manufacture and developing the accompanying kinematic relations necessary forgear fabrication using the variable diameter cutter. A subtle facet of the entire integrated processfor gear design and manufacture is the seamless integration of noncircular gear elements. Novelexamples of noncircular gear pairs include

� 2-dof mechanical function generator (variable NC gear pair),� spiral cylindrical and hypoid NC gear pairs,� variable face width NC gear pairs,� coiled NC gear pairs,� coordinated automotive steering with NC gears,� torque and speed balancing of rotating shafts, and� 1-dof mechanism for geared robotic manipulators.

AGMA published Gear Industry Vision in September 2004. The goal of this study was todefine a vision for the gear community over the next 20 years where gears remain fundamentaland the preferred solution in power transmission and control in the 2025 global marketplace.The final section of this study is Key Technological Challenges and Innovations. The top threeobjectives of this section are the following:

� To establish a single system of design and testing standards� To develop improved tribology modeling� To create predictive tools, virtual testing, and simulation tools


Preface xv

These objectives are central to this book. A single geometric system of design is presented byfocusing on noncircular hyperboloidal gears for motion transmission between nonorthogonalaxes. Although not immediately useful, such a system of design enables spur and helical,worm, and other forms to be readily obtained as a subset of noncircular hyperboloidal gears.About 15% ($5-8B) of the gear market targets rear axle assemblies consisting of spiral-bevel/hypoid gears. The basic differential gear train and spiral-bevel/hypoid gear set inherentin automotive axle assemblies has remained unchanged for over 75 years. Within the past100 years, spiral-bevel/hypoid gear machine tool manufacturers have focused on a specialfabricating process referred to as face milling and face hobbing. Inherent in this ‘‘face” cuttingprocess are limitations on the resulting end gear product. It is estimated that there exist over15,000 spiral-bevel/hypoid gear cutting machines with over 25 years of life where the processpresented in this book can eliminate some of these restrictions and lead to a new generationof gear fabrication.

As indicated, the central theme of this book is the presentation of a unified geometricmethodology for the parameterization of gear pairs in mesh. An overall evaluation of thekinematic geometry of the newly synthesized gear pair is taken into account by includingdesign rating formulas. These design rating formulas include fillet and inertial stress determi-nation using finite-element analysis, contact stress, dynamics loads, wear, flash temperature,contact and bending fatigue analysis, reliability analysis, minimum lubricant film thickness,and specific film thickness, in addition to mesh and windage losses. An evaluation of the man-ufacturing process is performed by providing the cutting time, material removal rate, cuttingpower, surface cutting speed, and relative position and orientation between cutter and gearblank. This concurrent CAD–CAM methodology enables the designer to synthesize gear pairswith increased efficiency, reduced noise, while improving strength and surface durability. Thisdevelopment differs from current gear design and manufacturing practice.

The book is split into three parts and addresses both theory and practice. The first partrevisits the concept of toothed bodies in mesh, their various forms for motion transmission,along with some terminology and nomenclature subsequently used to describe the concurrentdesign and manufacture of toothed bodies presented in Part Two. Part Two establishes themathematical model used for the integrated design/manufacturing methodology. It is thispart where contributions to the kinematic geometry of ruled surfaces in contact, differentialgeometry of surfaces in direct contact, along with toothed bodies in mesh are developed. PartThree includes design formulas to rate or evaluate gear pairs generated using the developedmethodology. Practicing gear engineers can bypass the analytical treatment of Parts One andTwo and focus on Part Three. Part Three discusses the design procedure based on the analyticaldevelopment and gives several examples to illustrate the capabilities of the new approach. Anoteworthy feature of the developed methodology is that the design and manufacturing datafor the toothed bodies that satisfy the stated requirements and the cutters used to produce themare synthesized concurrently and interactively in a PC environment. The synthesized shapesof the gear and cutter elements along with the surfaces of the teeth separately or in conjugateaction are displayed graphically. The designer can view and evaluate trial designs prior tofurther analysis or manufacturing. Sample displays are included as part of the final chapterto illustrate the process in a variety of nonconventional as well as conventional applications.Included are 10 appendices.

All the relations presented in this text have been coded and tested. Delgear©, a com-puter software package developed by the author, is included as part of this book. The


xvi Preface

requirements necessary to run Delgear© are standard with PCs and laptops today. This soft-ware enables the reader to specify motion (circular and noncircular gears), tooth type (involuteand cycloidal), gear type, cutter, and manufacturing parameters and view the results of theintegrated CAD–CAM process for generalized gear pairs. The included software is bundledinto an install package that prepares a windows-based environment to use the Delgear© pack-age. Installation instructions are provided in Appendix D. A user’s guide is included with theDelgear© software to assist its usage. Included are 22 illustrative examples of gear pairs, bothtraditional and nontraditional gear pairs, to illustrate features presented in this book.

Much of the mathematics used in this book is presented in existing textbooks and isnot summarized. However, the novel information of this book is preceded with a level ofbasic mathematics. Intermediate graphical displays of gear results and equations developed inChapters 2–6 are deferred to Chapter 9. The fundamental theory developed in Chapters 2–6 ispresented with figures and equations for individuals interested in the kinematic geometry ofgear elements. Gearing is not a field of study analogous to mathematics, vibrations, FEA, orfluid mechanics; and consequently, exercises at the end of each chapter are not included in atraditional textbook manner. The dedicated reader can use the Delgear© software package tocheck intermediate values at each stage of the presented methodology.

The webpage www.wiley.com/go/dooner_2e provides supplementary material to the Kine-matic Geometry of Gearing. This webpage provides a link that enables interested readers tofreely download and use software developed by the author. The developed software facilitatesthe geometric design and rating of various gear types including spur, helical, spiral bevel,straight bevel, spur and spiral non-circular gears, spur and spiral hypoid gears, non-orthogonalworm gears, along with nontraditional gear types.

Acknowledgments are of the order to express the author’s appreciation for facilitating thepresentation in this work. First, an acknowledgment is due to the late Prof. Ali Seireg for hiscollaboration and encouragement on the original work and sharing the importance of systemdesign. Behind-the-scenes facilitators include John Wiley & Sons, Ltd for their willingness tocontinue with this second edition along with IBM APL Product and Services for disentanglinga variety of programming woes; especially Nancy Wheeler and David Liebtag, formerly ofIBM APL Product and Services. An extended acknowledgement goes to Dr. Michael W.Griffis for his role in the presentation of this new gear approach by fielding many questionsand providing insight into the theory of screws. And finally, recognition to IMPO for thefortitude and patience to bear with me as I pieced together this manuscript and software.

David DoonerMayaguez, Puerto Rico

http://www.wiley.com/go/dooner_2e

P1: TIX/XYZ P2: ABCJWST162-c01 JWST162-Dooner February 20, 2012 9:41 Printer Name: Yet to Come Trim: 244mm×168mm

Part OneFundamentalPrinciples ofToothed Bodiesin Mesh

Before we can understand the future, we must learn about the past

—Anonymous


1Introduction to the Kinematicsof Gearing

1.1 Introduction

A brief history of gearing and some established gear concepts are presented in this chapteras an introduction to the development of a generalized kinematic theory for the design andmanufacture of gears. The primary objective is to familiarize the kinematician with gear termi-nology in a format that is familiar to them (compatible with established kinematic theory) aswell as to introduce the gear specialist to some of the relevant kinematic concepts that are usedin developing a generalized methodology for the concurrent design and manufacture of gearpairs. This approach includes the synthesis and analysis of the gear elements concurrently withthe design of the corresponding cutter elements used for their fabrication. These introductoryconcepts will be built upon throughout this book to develop a generalized methodology basedon kinematic geometry for the integrated design and manufacturing of appropriate toothedbody to transmit a specified speed and load between generally oriented axes and the constraintsthat may restrict implementation.

1.2 An Overview

An introduction to the complexities involved in the design and manufacture of toothed bodiesin mesh can be achieved by first examining the kinematic structure of conjugate motionbetween parallel axes. One purpose of this chapter is to introduce the concept of toothedwheels and demonstrate the basic kinematic geometry of toothed wheels in mesh as well astheir fabrication. This extended introduction is intended to establish a foundation that will beused as a corollary to exemplify the intricacies of spatial gearing (namely, worm and hypoidgearing). A similar introductory treatment on gears is presented in existing textbooks onkinematics and machine design (e.g., Spotts, 1964; Martin, 1969; Shigley and Uicker, 1980;Erdman and Sandor, 1997; Budynas and Nisbett, 2011). The elementary treatment providedin these textbooks on kinematics and machine design is essentially based on the books by

Kinematic Geometry of Gearing, Second Edition. David B. Dooner.© 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.

3


4 Kinematic Geometry of Gearing

Figure 1.1 South pointing chariot (reproduced by permission of Science Museum London/Science andSociety Picture Library)

Buckingham 1949 and Merritt (1971). Because of its practical importance, the design andmanufacture of toothed bodies continues to attract the attention of researchers in a varietyof fields (e.g., geometry, lubrication, dynamics, elasticity, material science, and computerscience). Dudley (1969) provides a brief account on the history of gears, and additionalinformation regarding the history of gears is provided by Cromwell (1884) and Grant (1899).An overview on the design and manufacture of gears is presented by Dudley (1984) and Drago(1988). Specialists in the gear industry have contributed to the second edition of Dudley’sGear Handbook edited by Townsend (1991). A more extensive and up-to-date analysis for thedesign and manufacture of gears is provided by the following organizations:

� American Gear Manufacturers Association (AGMA)� International Standards Organization (ISO)� Deutsches Institute fur Normung (DIN)� Japanese Gear Manufacturers Association (JGMA)� American National Standards Institute (ANSI)� British Gear Association (BGA)

One of the earliest documented geared devices is the South Pointing Chariot. A model ofa South Pointing Chariot is depicted in Figure 1.1. The function of this device is to serveas a mechanical compass in crossing the Gobi dessert. The statue atop of the wheeled cartmaintains a constant direction of pointing independent of the cart track. Various claims tothe date of the device range from 2700 bc to 300 ad. Heron of Alexandria devised manymechanical systems involving mechanisms (some geared). Example systems include specialtemple gates, mechanized plays, coin-operated water dispensers, and the aeolipile. Leonardoda Vinci is one of the most celebrated designers of all times. Da Vinci is credited with thevarious sketching of gears in Figure 1.2.

Norton (2001) credits James Watt as the “first” kinematician for documenting the couplermotion of a four-link mechanism. This documentation was part of his effort to achieve longstrokes on his steam engine. More noted is Euler (father of involute gearing) and his analyticaltreatment of mechanisms. Yet, Reuleaux is considered the “father” of modern kinematics forhis text Theoretical Kinematics. Reuleaux defined six basic mechanical components (namely,a link, wheel, cam, screw, ratchet, and belt). A gear can be considered a manifestation of thewheel, cam, and screw.


Introduction to the Kinematics of Gearing 5

Figure 1.2 Gear sketches by da Vinci (reproduced by permission of Biblioteca Nacional)

Geared devices remain vital components in many machine systems today. As a result, thefield of gearing endures an extensive pedigree and can require a devoted apprenticeship to mas-ter the subject. Due to the nature of the evolution of gearing, current research and practice, havefor the most part, built on concepts charted by nineteenth century geometricians. These contri-butions include modern concepts in kinematic synthesis and analysis, methods of manufacture,analysis of vibrations and noise, the development and integration of tribological behavior intothe field of gearing, and the widespread availability of digital computers. Improvements in thefield of gearing can be achieved by directing new energies toward these areas. In order to givethe field of gearing a new genesis, gears (special toothed bodies) are classified in general aselements of a mechanism that are used to control an input/output relationship between twoaxes via surfaces in direct contact. As this manuscript evolves the discrepancies, limitations,inconsistencies, different design philosophies, and the need for new technology within the gearcommunity will become more apparent and the concept of a gear will take on a new identity.The primary goal of this manuscript is to provide the gear designer with new technology andsimultaneously provide the gear designer with a practical and unified approach to design andmanufacture general toothed bodies. This unified approach provides the analytical foundationto better establish a correlation between theory and practice for generalized gear design andmanufacture. It is written with the assumption that the reader has access to the numerous textswhich illustrate traditional methods of gear design and manufacture.

1.3 Nomenclature and Terminology

An essential and important aspect of gear design and manufacture is to identify a nomenclaturethat distinguishes different phenomena with as few symbols as possible. Currently, each of the



Line of contact betweencylindrical pitch surfaces

Force

Axial plane

Input axisof rotation

Output axisof rotation

Force

Cylindrical

pitch surfa

ces

Transverse plane

Figure 1.3 Two cylindrical wheels (friction wheels) in line contact. An applied force F exists betweenthe two wheels in order to facilitate motion transmission

different gear types (planar, bevel, hypoid,1 worm, and worm gears) utilize a nomenclatureapplicable to the particular gear type. The vernacular of a gear specialist can be misleading andconfusing for the novice and may require clarification among gear specialists. Also, due to theinterdisciplinary nature of gear, design and manufacture some of the established nomenclaturewithin each discipline becomes nebulous. An attempt is made here to adhere to standard“gearing” nomenclature whenever possible.

The purpose of toothed wheels is to transmit uniform motion from one axis to anotherindependent of the coefficient of friction that exists between the teeth in mesh. Grant wasone of the first to document a treatise on toothed wheels in mesh (1899). He reveals thatat the close of the nineteenth century the design and manufacture of toothed bodies wasbecoming more analytical, and less of a craft. As the design and manufacture of toothedwheels became more analytical the nomenclature and terminology attained more significance.The following are some of the common terms presently used in the gear community, andadditional nomenclature and terminology will be established throughout this book as theanalysis of toothed bodies in mesh increases.

Pitch radius: When two cylindrical wheels (input and output wheel) are in linecontact as shown in Figure 1.3, the radii of the input and output cylinders arereferred to as the pitch radii upi and upo, respectively. Two cylinders are inline contact when the two axes of rotation are parallel. As the two cylindersrotate, there is no slippage at the line of contact. Motion transmission via two

1 The phrase Hypoid Gear Drive is a trademark of Gleason Works. Other forms of similar gear drives not fabricatedby Gleason are referred to as simply hypoid gears or skew axis gears. The term hyperboloidal gears is used frequentlyin this manuscript when referring the theoretical development of spatial gearing where current methods for the designand fabrication of hypoid gears are not applicable.



cylinders (friction wheels) in contact is limited by the applied radial force F andthe coefficient of friction that exists between the two cylinders.

Number of teeth: In order to maintain a desired speed ratio between two axes ofrotation, an integer number of teeth N must exist on each wheel. The combinationof the number of teeth on each wheel and the size of the two cylinders determinethe load-carrying capacity of the toothed wheels in mesh.

Transverse surface: For motion transmission between parallel axes, a transversesurface of any plane is perpendicular to the axis of rotation. The transverse surfaceis used to parameterize toothed wheels.

Pitch circle: The pitch circle is the intersection between a cylindrical wheel anda transverse surface. The pitch circle is used as a reference for which manycalculations are based. The radius of the input pitch circle is upi, and the radius ofthe output pitch circle is upo.

Diametral pitch: The diametral pitch Pd is a rational expression for the numberof teeth N divided by twice the pitch radius u: Pd = N/2u. The purpose forintroducing such an immeasurable quantity is to specify tooth sizes using integervalues. It is customary for SI designated standards to use the module m instead ofthe diametral pitch Pd to specify gear tooth sizes, where Pd = 1/m. The diametralpitch is always the same for two gears in mesh. Accordingly, Pd = Ni/2upi =No/2upo = (Ni + No)/2E, where the center distance E = upi + upo. The possibilityof specifying an irrational I/O relationship is alleviated by defining the pitch radiiin terms of the diametral pitch. Pd < 20 is considered coarse pitch; afterward finepitch (Pd ≤ 20).

Transverse pitch: The transverse or circular pitch pt is an irrational expressionfor the circumferential distance along the pitch circle between adjacent teeth:pti = 2πupi/Ni = pto = 2πupo/No = π/Pd.

Addendum circle: The addendum circle is a hypothetical circle in the transversesurface whose radius is the outermost element of any tooth. The addendum is theregion between the pitch circle and the addendum circle. The amount by which theradius of the addendum circle exceeds the radius of the pitch circle is expressedin terms of an addendum constant a: ua = up + a/Pd. The active region of the geartooth that lies in the addendum is referred to as the gear face.

Dedendum circle: The dedendum circle is a hypothetical circle in the transversesurface whose radius is the innermost element between adjacent teeth. The deden-dum is the region between the pitch circle and the dedendum circle.

Center line: The two points in the transverse plane where the two axes of rotationfor the input and the output wheel intersect, the transverse plane are instant centers.The line connecting these two instant centers is the center line. When the two axesof rotation are skew, the center line is the single line perpendicular to the two axesof rotation.

Center distance: The distance along the center line between the two axes ofrotation is the center distance. This length is sometimes referred to as the interaxialdistance.



Line of action: The line that passes through the point that is coincident with thetwo teeth in mesh and also perpendicular to the two teeth is the line of action.

Pitch point: The pitch point is the intersection between the center line and the lineof action.

Clearance: The distance along the center line between the dedendum of one gearand the addendum of its mating gear is the clearance. Like the dedendum andaddendum, the clearance is defined in terms of the clearance constant c and thediametral pitch Pd.

Tooth width: The distance along the pitch circle between adjacent profiles of asingle tooth is the tooth width tt.

Tooth space: The distance along the pitch circle between two adjacent teeth is thetooth space ts. The sum of the tooth width tt plus the tooth space ts must be equalto the transverse pitch pt (i.e., pt = tt + ts).

Backlash: The amount the tooth space of one gear exceeds the tooth width of itsmating gear. AGMA recommends that the face width b be proportional to toothsize. This is accomplished via the following AGMA recommendation:

9

Pd≤ b ≤ 14

Pd

Pressure angle: The included angle between the common tangent between the twopitch circles and the line of action.

IPS: A US customary system of measurements based on length, force, and timewhose units are inches, pounds, and seconds, respectively.

CGS: A SI system of measurements based on length, mass, and time whose unitsare centimeters, grams, and seconds, respectively.

1.4 Reference Systems

Three distinct coordinate systems are used to parameterize the geometry of a gear pair. Thethree distinct Cartesian coordinate systems are

1. (X, Y , Z) fixed to the ground,2. input (Xi, Y i, Zi) attached to the driving or input wheel, and3. output (Xo, Yo, Zo) attached to the driven or output wheel.

Each reference frame is a conventional right-handed Cartesian coordinate system as depictedin Figure 1.4. The zi-axis of the input reference frame (axis of rotation for the input body) iscollinear with the Z-axis of the fixed reference frame. The distance E between the two axesof rotation is a fixed distance directed along the positive X-axis of the stationary referenceframe. The zo-axis of the output reference frame (axis of rotation for the output body) isperpendicular to the X–Y plane of the fixed reference frame. Associated with each of thetwo Cartesian coordinate systems (xi, yi, zi) and (xo, yo, zo) are, respectively, two systems of



Figure 1.4 Three Cartesian coordinates systems (X, Y , Z), (xi, yi, zi), and (xo, yo, zo) are used toparameterize toothed wheels in mesh

curvilinear coordinates (ui, vi, wi) and (uo, vo, wo). The curvilinear coordinates (u, v, w) areintroduced to facilitate the parameterization of gear pairs and are indistinguishable from thecylindrical coordinates (r, θ , z) for motion transmission between parallel axes. The specialcurvilinear coordinates (u, v, w,) will be introduced in Chapter 3 where a single system ofcurvilinear coordinates can be used to analyze the general case of toothed bodies in mesh.

1.5 The Input/Output Relationship

The relationship between the fixed coordinate system (X, Y , Z) and the input coordinate sys-tem (xi, yi, zi) is defined by the net angular position vi about the input Z-axis as measuredfrom the fixed X-axis (see Figure 1.5). Similarly, the coordinate system (xo, yo, zo) is definedby the net angular position vo about a line parallel to the Z-axis and located at a distance Ealong the X-axis. The I/O relationship between the angular position vi of the input body tothat of the position (angular or linear) vo of the output body is defined as the transmissionfunction. The instantaneous gear ratio g is the ratio between the instantaneous angular displace-ment dvo of the output and the corresponding instantaneous angular displacement dvi of theinput; thus,

g ≡ dvo

dvi= Instantaneous angular displacement of the output body

Instantaneous angular displacement of the input body. (1.1)

Here, the differential displacements dvi and dvo refer to an instantaneous change in angularpositions vi and vo, respectively. The displacements dvi and dvo are angular displacementsabout the zi and zo axes, respectively. The angular speeds ωi and ωo are, respectively, theangular displacements dvi and dvo per unit time dt. For uniform motion transmission betweenfixed axes, the transmission function is linear and its slope is a constant equal to the gearratio. When this occurs the gear ratio is also defined by the ratio N i /No of gear teeth. Thisratio is defined to accommodate non-circular gears and is the reciprocal of the gear ratio usedby AGMA.



Figure 1.5 Basic terminology for toothed wheels in mesh

The zi-axis of the input moving reference frame and the zo-axis of the output moving frameare parallel for two external gears in mesh. The I/O relationship g is negative in this case fortwo external gears in mesh. Although the majority of gears are external gears, it is convenientto plot the I/O relationship g as positive for both two external gears and internal–external gearsin mesh with clarification on the gear type (namely, external–external or external–internal).The elements of a gear pair are usually identified as either the gear or pinion, where the pinionis the smaller of the two gears.2 It is possible in special circumstances regarding a hypoid gearpair that the pinion is physically larger than the gear and yet have fewer teeth! The reason forthis phenomenon will be presented in Chapter 5. Use of “gear” and “pinion” to identify twogears in mesh does not explicitly indicate if the gear pair is used for speed increasing or speeddecreasing. As a result, trailing subscripts “i” and “o” are added to identify the input and output

2 In Spanish, “pinion” translates to pinon and “gear” to Catalina or Catherine (literally, gear is engrenage whensimply referring to a generic “gear” element). Interestingly, St. Catherine of Alexandria has become emblematic forwheelwrights, machinist, and mechanical engineers. St. Catherine was condemned in 305 ad by the pagan emperorMaximian (305–313) for her confessed faith in Christianity. Accounts of this event vary, but one version is that aspecial machine consisting of wheels and axles was devised to shred Catherine. Another version is that a rack- andpinion-related device was used to stretch and torture Catherine. Both versions involve wheels and torture to discouragethe spread of Christianity. St. Catherine was invested by the Catholic Church and celebrated on November 25.



End radius

Edge radius

Top round

Fillet region

Figure 1.6 Edge radius, end radius, and top round reduce nicks and burrs encountered in shipping andhandling prior to assembly

respectively. Neither subscript is used in certain situations where a notation is applicable toboth the input and the output gears.

The simplest scenario of toothed wheels in mesh is motion transmission between parallelaxes. Depicted in Figures 1.5 and 1.6 is some terminology used to describe toothed wheels. Ingeneral, gear designers parameterize gear teeth in a plane. This same planar parameterizationis also applied to analyze bolts and nuts, presses, rotary compressors, and planar four-barlinkages. Since motion transmission between parallel axes can be adequately illustrated in aplane perpendicular to the axis of rotation it is commonly referred to as planar motion. Theease of visualizing planar motion attributes to its usage.

1.6 Rigid Body Assumption

Initially, when analyzing the kinematic geometry of toothed bodies in mesh, it is assumedthat the bodies in mesh are rigid although they will inevitably deform depending on thetransmitted load. These deformations are accounted for by the compliance of the housingused to support the bearings, the deflections in the bearing supports, the bending and torsionaldisplacements in the gear blanks and shafts, and the deflection of the teeth relative to the gearblank. The assumption of rigid bodies not only simplifies analysis but also necessary in orderto initially determine the geometry of the toothed bodies in mesh. The elastic deformations aresubsequently calculated and compensated for by profile modifications such as profile relief andcrowning of the teeth. Due to errors encountered in manufacturing, assembly, and operation ofa gear pair, the amount of profile modification varies for each gear type and is generally basedon experience. If the proper modifications are not incorporated then the smooth transmissionof motion from one axis to another can no longer be expected to occur, and the gear teeth willbe subjected to impulsive loading producing higher stresses and noise.

1.7 Mobility

Earlier in this chapter gears were described as elements of a mechanism. Reuleaux (1876)defines a mechanism as a closed kinematic chain where one of its links is held stationary.The stationary link or ground is usually indicated by feathered marks as shown in Figure1.7a. The mobility M or the degree of freedom (dof) of a mechanism refers to the numberof independent parameters that must be specified to uniquely determine the configuration or



Figure 1.7 Two mechanisms where (a) has mobility one and (b) has zero mobility

arrangement of the remaining links within the mechanism. One task of a kinematician is tospecify the configuration of a mechanism given the independent parameters. The mechanismshown in Figure 1.7a has no mobility (over constrained), whereas the special mechanismshown in Figure 1.7b has mobility one. The difference between the two mechanisms is thatthe links 2 and 3 in Figure 1.7a are connected by a pin, whereas the two links 2 and 3 inFigure 1.7b are tangent to one another at point c. A mechanism with zero or less mobility isa structure or truss. The concept of mobility is important and far reaching when consideringtoothed bodies in mesh. First, a brief discussion regarding planar three link 1-dof mechanismsis discussed; then, later in Chapter 5 the more general case of a five link 1-dof mechanismwill be discussed. More insight on mobility is found in many textbooks on mechanisms andkinematics (e.g., Hunt, 1978; Shigley and Uicker, 1980; Edman and Sandor, 1997).

The analysis of mechanisms involves identification of the types of motion that may existsbetween two objects. The displacement or change in position of a point relative to a fixedcoordinate system is defined as absolute displacement. The displacement or change in positionand orientation of an object relative to a fixed coordinate system is defined as vehiculardisplacement. The displacement of a point relative to another moving coordinate system isdefined as relative displacement.

Depicted in Figure 1.8 is a movable lamina 2 (planar body) relative to the fixed coordinatesystem (X, Y , Z). Three independent parameters X2, Y2, and θz2 are used to specify the position

Figure 1.8 Fixed coordinate moving coordinate systems

dooner kinematic geometry second edition red box rules are ... · kinematic geometry of gearing...

Documents