High Conformal Gearing

HigH-Conformal gearing K i n e m a t i c s a n d G e o m e t r y

HigH-Conformal gearing K i n e m a t i c s a n d G e o m e t r y

Stephen p. Radzevich

Boca Raton London New York

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

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20150518 International Standard Book Number-13: 978-1-4987-3918-4 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Radzevich, S. P. (Stepan Pavlovich) High‑conformal gearing : kinematics and geometry / author, Stephen P. Radzevich. pages cm Includes bibliographical references and index. ISBN 978‑1‑4987‑3918‑4 (alk. paper) 1. Gearing, Conical. 2. Convex geometry. I. Title. TJ193.R33 2016 621.8’33‑‑dc23 2015016072 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

This book is dedicated to my parents

Contents Preface.......................................................................................................................xi Acknowledgments............................................................................................... xiii Author......................................................................................................................xv Notations.............................................................................................................. xvii Introduction........................................................................................................ xxiii 1. A Brief Overview of Conformal Gearing: State of the Art.....................1 1.1 Criteria for Geometrically Accurate (Ideal) Gearing........................4 1.2 Ancient Designs of Conformal Gearings...........................................5 1.3 Improvements in and Relating to Gear Teeth....................................6 1.4 Toothed Gearing....................................................................................8 1.5 Wildhaber’s Helical Gearing................................................................... 11 1.6 Novikov Gearing..................................................................................... 18 1.7 Features of Tooth Flank Generation in Novikov Gearing.................22 2. Conditions for Transmitting a Rotation Smoothly................................ 27 2.1 Condition of Contact............................................................................ 27 2.2 Condition of Conjugacy...................................................................... 29 2.3 Equality of the Base Pitches................................................................ 35 2.4 Contact Ratio in a Gear Pair............................................................... 39 3. Conformal Gearing: Novikov Gearing.....................................................43 3.1 Novikov Gearing: A Helical Involute Gearing with a Zero Transverse Contact Ratio....................................................................43 3.1.1 Essence of Novikov Gearing....................................................44 3.1.2 Fundamental Design Parameters of Conformal Gearing................................................................. 52 3.2 Transition from Involute Gearing to Conformal (Novikov) Parallel-Axis Gearing..........................................................................54 3.3 Kinematics of Parallel-Axis Gearing................................................. 57 3.4 Plane of Action in Parallel-Axis Gearing.......................................... 59 3.5 Boundary N-Circle in Conformal Gearing......................................... 59 3.6 Possible Tooth Geometries in Conformal Gearing......................... 62 3.7 Tooth Profile Sliding in Conformal Gearing (in Novikov Gearing).............................................................................. 67 3.8 Elements of the Kinematics and the Geometry in Conformal Gearing (Novikov Gearing)............................................... 71 3.9 Designing a Conformal Gear Pair..................................................... 75 3.10 Conformal Gearing with Two Pseudo Paths of Contact................ 78 3.11 Tooth Flank Geometry in a Conformal Gear Pair........................... 78 vii

viii

Contents

3.12 Configuration of Interacting Tooth Flanks at the Culminating Point................................................................................ 82 3.13 Local and Global Contact Geometry of Interacting Tooth Flanks.........................................................................................84 4. High-Conformal Gearing............................................................................ 91 4.1 Contact Geometry in Conformal Parallel-Axis Gearing................ 91 4.2 High-Conformal Parallel-Axis Gearing........................................... 95 4.3 On the Accuracy Requirements for Conformal Parallel-Axis Gearing........................................................................ 100 5. Contact Geometry of the Gear and the Mating Pinion Tooth Flanks............................................................................................................. 107 5.1 Local Relative Orientation at a Point of Contact of Gear and Mating Pinion Tooth Flanks..................................................... 108 5.2 The Second-Order Analysis: Planar Characteristic Images........ 114 5.2.1 Preliminary Remarks: Dupin’s Indicatrix.......................... 114 5.2.2 Matrix Representation of Equation of Dupin’s Indicatrix of the Gear Tooth Flank..................................... 117 5.3 Degree of Conformity at a Point of Contact of Gear and Mating Pinion Tooth Flanks in the First Order of Tangency...... 118 5.3.1 Preliminary Remarks........................................................... 118 5.3.2 Indicatrix of Conformity at a Point of Contact of a Gear and a Mating Pinion Tooth Flank............................. 122 5.3.2.1 Conclusion.............................................................. 124 5.3.3 Directions of Extremum Degree of Conformity at Point of Contact of a Gear and a Mating Pinion Tooth Flanks.......................................................................... 130 5.3.4 Important Properties of Indicatrix of Conformity Cnf R(G/P) at a Point of Contact of the Gear and the Mating Pinion Tooth Flanks................................................ 133 5.3.5 Converse Indicatrix of Conformity at a Point of Contact of the Gear and the Mating Pinion Tooth Flanks in the First Order of Tangency............................... 134 6. On the Impossibility to Cut Gears for Conformal and HighConformal Gearing Using Generating (Continuously-Indexing) Machining Processes.................................................................................. 137 7. Kinematics of a Gear Pair.......................................................................... 141 7.1 Vector Representation of Gear Pair Kinematics............................ 141 7.1.1 Concept of Vector Representation of Gear Pair Kinematics..................................................................... 141 7.1.2 Three Different Types of Vector Diagrams for Spatial Gear Pairs.................................................................. 145

Contents

7.2 7.3

7.4

ix

7.1.2.1 Vector Diagrams of External Spatial Gear Pairs............................................................... 148 7.1.2.2 Vector Diagrams of Internal Spatial Gear Pairs............................................................... 153 7.1.2.3 Vector Diagrams of Generalized Rack-Type Spatial Gear Pairs.................................................. 156 7.1.2.4 Analytical Criterion of a Type of Spatial Gear Pairs............................................................... 157 Classification of Possible Types of Vector Diagrams of Gear Pairs............................................................................................ 159 Complementary Vectors to Vector Diagrams of Gear Pairs............................................................................................ 160 7.3.1 Centerline Vectors of a Gear Pair........................................ 162 7.3.2 Axial Vectors of a Gear Pair................................................ 163 7.3.3 Useful Kinematic and Geometric Formulas..................... 166 Tooth Ratio of a Gear Pair................................................................. 168

8. High-Conformal Intersected-Axis Gearing........................................... 171 8.1 Kinematics of the Instantaneous Motion in High-Conformal Intersected-Axis Gearing.................................................................. 171 8.2 Base Cones in Intersected-Axis Gearing........................................ 173 8.3 Path of Contact in High-Conformal Intersected-Axis Gearing.................................................................. 176 8.3.1 Bearing Capacity of High-Conformal Gearing................ 177 8.3.2 Sliding of Teeth Flanks in a High-Conformal Gearing................................................................................... 179 8.3.3 Boundary N-Cone in Intersected-Axis High-Conformal Gearing.................................................... 187 8.4 Design Parameters of High-Conformal Intersected-Axis Gearing................................................................................................ 189 9. High-Conformal Crossed-Axis Gearing................................................ 197 9.1 Kinematics of Crossed-Axis Gearing.............................................. 197 9.2 Base Cones in Crossed-Axis Gear Pairs.......................................... 201 9.3 Kinematics of the Instantaneous Relative Motion........................ 206 9.4 Path of Contact in High-Conformal Crossed-Axis Gearing........ 208 9.4.1 Bearing Capacity of Crossed-Axis High-Conformal Gearing................................................................................... 209 9.4.2 Sliding between Tooth Flanks of the Gear and the Pinion in Crossed-Axis High-Conformal Gearing.......... 210 9.4.3 Boundary N-Cone in Crossed-Axis High-Conformal Gearing................................................................................... 212 9.5 Design Parameters of High-Conformal Crossed-Axis Gearing....................................................................... 214

x

Contents

Conclusion............................................................................................................ 221 Glossary................................................................................................................223 References............................................................................................................ 233 Appendix A: On the Concept of “Novikov Gearing” and the Inadequacy of the Term “Wildhaber–Novikov Gearing” or “W–N Gearing”................................................................................................... 239 Appendix B: Elements of Vector Calculus..................................................... 277 Appendix C: Elements of Differential Geometry of Surfaces.................. 283 Appendix D: Elements of Coordinate Systems Transformations............. 303 Appendix E: Change of Surface Parameters................................................. 323 Index...................................................................................................................... 325

Preface The book deals with gears that feature convex-to-concave contact of the tooth flanks of the gear and the mating pinion. Gears of this type are commonly referred to as conformal gearings. Novikov gearing* and Wildhaber gearing are the most widely known examples of conformal gearing. The Bramley–Moore (otherwise known as the Vivkers, Bostock, and Bramley gearing, or just V.B.B.gearing) is another well-known example of conformal gearing. Conventional external involute gearing features convex-to-convex contact of the tooth flanks of the gear and the mating pinion. Because of this high contact, stresses are observed when involute gears operate. In conformal gearing, as well as in high-conformal gearing, the convexto-convex contact of the tooth flanks is substituted with convex-to-concave contact. Due to this, the gear teeth feature higher contact strength. The Novikov gear form is of the helical type but it only has face contact: There is no progressive profile contact as in the involute case. The profiles of the mating teeth at any section perpendicular to the wheel axis make contact with each other only for an instant and then separate. With the Novikov tooth, contact always occurs at a certain distance from the pitch point and therefore the sliding velocity is constant and unidirectional (for one particular direction of wheel rotation). A number of investigators have examined and manufactured gears of this type and load capacities of 3–6 times the corresponding involute tooth load have been reported. Conditions to transmit a rotation smoothly from the driving to the driven shaft are outlined in this book. Much of the discussion is based on the modern theory of gearing. In high-conformal gearing, the degree of conformity at point of contact of the tooth flanks of the gear and the mating pinion is greater than a prespecified critical value, the threshold. This is the main difference between conformal and high-conformal gearings. A set of conditions to meet when designing both conformal and high-conformal gears is specified. The principal differences between conformal gearing (as well as highconformal gearing) and Wildhaber helical gearing are outlined. It is shown that Wildhaber gearing on the one hand and Novikov gearing on the other hand are two completely different gear systems that cannot be combined into a common gear system. Therefore, the widely used terminologies like “Wildhaber–Novikov gearing,” “W–N gearing,” etc., are meaningless, and they need to be eliminated from use in the engineering and scientific community. * The Novikov gear form is named after Colonel M.L. Novikov, Dr. Eng. Sc., who was head of a department at the Zhuokovskii Military Aero Academy in Moscow. He developed this particular gear form but it was his colleagues who published his work under his name after his death in 1958 [19].

xi

xii

Preface

Those who want to demonstrate their unfamiliarity with gearing in a general sense, and with Novikov gearing in particular, loosely use the terms “Wildhaber–Novikov gearing” and/or “W–N gearing.” It is also shown that neither conformal gears (Novikov gears) nor high-conformal gears can be cut in the continuously-indexing (generating) process; that is, they cannot be hobbed, ground by worm grinding wheels, shaved, etc. Only form cutting tools can be used for machining conformal and highconformal gears: form milling cutters, form grinding wheels, etc. It should be noted in conclusion that the discussion in this book is limited only to the kinematics and the geometry of conformal and high-conformal gearing. Other important topics such as gear accuracy, gear loading, gear wear, gear lubricating, vibration generation, and noise excitation, etc., are not addressed in this book.

Acknowledgments I would like to share the credit for any research success with my numerous doctoral students with whom I have tested and applied the proposed ideas in industry. The contributions of many friends, colleagues, and students are overwhelming in number and cannot be acknowledged individually, and as much as my benefactors have contributed, their kindness and help must go unrecorded. My thanks also go to those at CRC Press who took over the final stages of preparing this book and coped with the marketing and sales of the fruit of my efforts.

xiii

Author Stephen P. Radzevich is a professor of mechanical engineering and a professor of manufacturing engineering at National Technical University of Ukraine “Kyiv Polytechnic Institute,” Kyiv, Ukraine. He earned his MSc in 1976, PhD in 1982, and Dr.(Eng)Sc in 1991, all in mechanical engineering. Dr. Radzevich has extensive industrial experience in gear design and manufacture. He has developed numerous software packages dealing with computeraided design (CAD) and computer-aided machining (CAM) of precise gear finishing for a variety of industrial sponsors. His main research interest is the kinematic geometry of part surface generation, with a particular focus on precision gear design, high-power-density gear trains, torque share in multiflow gear trains, design of special purpose gear cutting/finishing tools, and design and machine (finish) of precision gears for low-noise and noiseless transmissions of cars, light trucks, etc. Dr. Radzevich has spent over 40 years developing software, hardware, and other processes for gear design and optimization. In addition to his work in industry, he trains engineering students at universities and gear engineers in companies. He has authored and coauthored over 30 monographs, handbooks, and textbooks. The monographs Generation of Surfaces (RASTAN, 2001), Kinematic Geometry of Surface Machining (CRC Press, 2007; 2nd Edition 2014), CAD/ CAM of Sculptured Surfaces on Multi-Axis NC Machine: The DG/K-Based Approach (M&C Publishers, 2008), Gear Cutting Tools: Fundamentals of Design and Computation (CRC Press, 2010), Precision Gear Shaving (Nova Science Publishers, 2010), Dudley’s Handbook of Practical Gear Design and Manufacture (CRC Press, 2012), Geometry of Surfaces: A Practical Guide for Mechanical Engineers (Wiley, 2013), and Generation of Surfaces: Kinematic Geometry of Surface Machining (CRC Press, 2014) are among his recently published volumes. He also authored and coauthored about 300 scientific papers, and holds about 250 patents on inventions in the field (the United States, Japan, Russia, Europe, Canada, Soviet Union, South Korea, Mexico, and others).

xv

Notations Ag

apex of the gear in intersected-axis gearing and crossed-axis gearing Ap apex of the pinion in intersected-axis gearing and crossed-axis gearing Apa apex of the plane of action in intersected-axis gearing and crossed-axis gearing C center-distance ℄ center-line C1 ⋅ g, C2 ⋅ g first and second principal plane sections of the gear tooth flank, G C1 ⋅ p, C2 ⋅ p first and second principal plane sections of the pinion tooth flank, P Cnf R(G/P) indicatrix of conformity of the gear tooth flank, G, and the mating pinion tooth flank, P, at a current contact point, K Cnf k(G/P) converse indicatrix of conformity of the gear tooth flank, G, and the mating pinion tooth flank, P, at a current contact point, K Crv(G) curvature indicatrix at a point of the gear tooth flank G Crv(P) curvature indicatrix at a point of the pinion tooth flank P Dup(G) Dupin’s indicatrix at a point of the gear tooth flank G Dup(P) Dupin’s indicatrix at a point of the pinion tooth flank P E characteristic line Eg, Fg, Gg fundamental magnitudes of the first order of the gear tooth surface, G Ep, Fp, Gp fundamental magnitudes of the first order of the pinion tooth surface, P F face width Feff effective face width, or the face width of the active portion of the plane of action G tooth flank of the gear K point of contact of the tooth flanks, G and P (or a point within a line of contact of the surfaces, G and P) LA line of action LAinst instant line of action LC line of contact LAdes desirable line of contact Lg, Mg, Ng fundamental magnitudes of the second order of the gear tooth flank, G Lp, Mp, Np fundamental magnitudes of the second order of the pinion tooth flank, P xvii

xviii

Notations

length of a path of contact tooth number tooth count of the input member tooth count of the output member axis of rotation of the gear axis of rotation of the pinion tooth flank of the pinion plane of action path of contact current pitch point pseudo path of contact axis of instant rotation of the pinion in relation to the gear (pitch line) Pn normal pitch Pt transverse pitch Rc(PA ↦ G) operator of rolling/sliding (the operator of transition from the plane of action, PA, to the gear, G, in crossed-axis gearing) Rc(PA ↦ P) operator of rolling/sliding (the operator of transition from the plane of action, PA, to the pinion, P, in crossed-axis gearing) Rlx(φy, Y) operator of rolling over a plane (Y-axis is the axis of rotation, X-axis is the axis of translation) Rl z(φy, Y) operator of rolling over a plane (Y-axis is the axis of rotation, Z-axis is the axis of translation) Rly(φx, X) operator of rolling over a plane (X-axis is the axis of rotation, Y-axis is the axis of translation) Rlz(φx, X) operator of rolling over a plane (X-axis is the axis of rotation, Z-axis is the axis of translation) Rlx(φz, Z) operator of rolling over a plane (Z-axis is the axis of rotation, X-axis is the axis of translation) Rly(φz, Z) operator of rolling over a plane (Z-axis is the axis of rotation, Y-axis is the axis of translation) Rru(φ, Z) operator of rolling of two coordinate systems Rs(A ↦ B) operator of the resultant coordinate system transformation, say from a coordinate system A to a coordinate system B Rt(φx, X) operator of rotation through an angle φx about the X-axis Rt(φy, Y) operator of rotation through an angle φy about the Y-axis Rt(φz, Z) operator of rotation through an angle φz about the Z-axis R1 ⋅ g, R 2 ⋅ g first and second principal radii of curvature of the gear tooth flank, G R1 ⋅ p, R 2 ⋅ p first and second principal radii of curvature of the gear tooth flank, P Scx(φx, px) operator of screw motion about the X-axis Scy(φy, py) operator of screw motion about the Y-axis Scz(φz, pz) operator of screw motion about the Z-axis Tr(ax, X) operator of translation at a distance ax along the X-axis Lpc N Nin Nout Og Op P PA Pc Pi Ppc Pln

Notations

Tr(ay, Y) Tr(az, Z) Ug, Vg

xix

operator of translation at a distance ay along the Y-axis operator of translation at a distance az along the Z-axis curvilinear (Gaussian) coordinates of a point on the gear tooth flank, G Up, Vp curvilinear (Gaussian) coordinates of a point on the pinion tooth flank, P Ug, Vg tangent vectors to curvilinear coordinate lines on the gear tooth flank, G Up, Vp tangent vectors to curvilinear coordinate lines on the pinion tooth flank, P VΣ vector of the resultant motion of the pinion tooth flank, P, in relation to the gear tooth flank, G Zpa active length of the plane of action a tooth addendum b tooth dedendum d pitch diameter db ⋅ g base diameter of a gear db ⋅ p base diameter of a pinion do ⋅ g outer diameter of a gear do ⋅ p outer diameter of a pinion df root diameter dl start of active profile diameter do outside diameter ht total tooth height i current point of the path of contact k1 ⋅ g, k2 ⋅ g first- and second-principal curvatures of the gear tooth flank, G k1 ⋅ p, k2 ⋅ p first- and second-principal curvatures of the pinion tooth flank, P m module mp transverse (profile) contact ratio mF face contact ratio mt total contact ratio n unit normal vector of the common perpendicular at point of contact, K, of the gear tooth flank, G, and the pinion tooth flank, P ng unit normal vector to the gear tooth flank, G np unit normal vector to the pinion tooth flank, P pb base pitch pb ⋅ g linear base pitch of the gear pb ⋅ p linear base pitch of the pinion pb ⋅ op operating linear base pitch of the gear pair psc screw parameter (reduced pitch) of instant screw motion of the pinion in relation to the gear px axial pitch of the gear teeth rb ⋅ g radius of base circle/cylinder of a gear

xx

Notations

rg pitch radius of a gear rp pitch radius of a pinion rg position vector of a point of a gear tooth flank, G rp position vector of a point of a pinion tooth flank, P rN radius of the boundary N-circle in Novikov gearing and in parallel-axis high-conforming gearing rg pitch radius of a gear rp pitch radius of a pinion rcnf position vector of a point of the indicatrix of conformity, Cnf R(G/P) s space width sn normal space width st transverse space width t tooth thickness tn normal tooth thickness tt transverse tooth thickness t1 ⋅ g, t 2 ⋅ g unit tangent vectors of principal directions on the gear tooth flank, G t1 ⋅ p, t 2 ⋅ p unit tangent vectors of principal directions on the gear tooth flank, P ug, vg unit tangent vectors to curvilinear coordinate lines on the gear tooth flank, G up, vp unit tangent vectors to curvilinear coordinate lines on the gear tooth flank, G u gear ratio uω angular velocity ratio vΣ unit vector of the resultant motion of the pinion tooth flank, P, in relation to the gear tooth flank, G xg yg zg local Cartesian coordinate system having origin at a current point of contact of the teeth flanks, G and P

Greek Symbols Γl boundary N-cone angle (in intersected-axis as well as crossed-axis high-conforming gearing) crossed-axis angle (shaft angle) Σ Φ1 ⋅ g, Φ2 ⋅ g first- and second-fundamental forms of the gear tooth flank, G Φ1 ⋅ p, Φ2 ⋅ p first- and second-fundamental forms of the pinion tooth flank, P γ specific sliding γg slide/roll ratio for the gear tooth flank, G γp slide/roll ratio for the pinion tooth flank, P

xxi

Notations

μ ϕ ϕn ϕt ϕt⋅ω φb⋅op φb⋅g φb ⋅ p φb ⋅ op ψ ψb ωg ω rlg ω slg ωin ωout ωg ωp ωp

ω rlp ω slp

ωpl

angle of local relative orientation at a point of contact of the tooth flanks G and P profile (pressure) angle normal profile (pressure) angle in parallel-axis gearing transverse profile (pressure) angle in parallel-axis gearing transverse profile (pressure) angle in intersected-axis, and in crossed-axis gearing operating base angular pitch in intersected-axis, and in crossedaxis gearing angular base pitch of the gear angular base pitch of the pinion operating angular base pitch of the gear pair pitch helix angle base helix angle rotation vector of the gear rotation vector of pure rolling of the gear rotation vector of pure sliding of the gear rotation of the input member rotation of the output member angle that form the tangent vectors Ug and Vg on the gear tooth flank G angle that form the tangent vectors Up and Vp on the pinion tooth flank P rotation vector of the pinion rotation vector of pure rolling of the pinion rotation vector of pure sliding of the pinion vector of instant rotation of the pinion in relation to the gear

Subscripts a axial b base cnf conformity g gear max maximum min minimum n normal opt optimal p pinion gear t transverse

Introduction An important problem in transmitting a rotation from a driving shaft to a driven shaft is discussed in the book. The problem can be formulated in the following manner: How to transmit a rotation smoothly with the highest possible power density, or, in other words, how to transmit the largest possible amount of power through the smallest volume.

Historical Background Conformal and high-conformal gearings are discussed in the book. Gearings of these types feature convex-to-concave contact of the tooth flanks of the gear and the mating pinion. The concept of convex-to-concave contact of the tooth flanks of the gear and the mating pinion can be traced back to the fifteenth century (1493) when the famous book by Leonardo da Vinci was published [7]. Several examples of gearings with convex-to-concave contact of the tooth flanks are discussed in this book. Novikov gearing,* and Wildhaber gearing are the most widely known examples of conformal gearing. The Bramley–Moore (otherwise known as the Vivkers, Bostock, and Bramley gearing, or just V.B.B.-gearing) is another wellknown example of conformal gearing. It is critical to stress from the very beginning that only Novikov gearing is capable of transmitting a rotation smoothly. Gearings of all other types are not capable of transmitting a rotation smoothly as they are approximate gearings.

Importance of the Subject Today, industry is faced with the necessity of transmitting a rotation from a driving shaft to the driven shaft with the highest possible power density. This requires the necessity of gear boxes of the smallest possible size that are capable of transmitting the highest possible power. Conformal gearings * The Novikov gear form is named after Colonel M.L. Novikov, Dr. Eng. Sc., who was head of a department at the Zhuokovskii Military Aero Academy in Moscow. He developed this particular gear form but it was his colleagues who published his work under his name after his death in 1958 [19].

xxiii

xxiv

Introduction

and especially high-conformal gearings perfectly fit this requirement. In this book, conformal and high-conformal gearings are discussed.

Uniqueness of This Publication A first ever workable type of conformal gearing, namely, Novikov gearing, was proposed by Dr. M.L. Novikov in 1954 and was published in 1956 [23]. Since the time of Dr. M.L. Novikov, much effort was made with the aim of investigating and implementing this particular type of gearing in industry. Almost all of the efforts failed mostly because of poor understanding of the kinematics and the geometry of conformal gearing. For the first time ever, the kinematics and the geometry of conformal and high-conformal gearing is discussed in this book. This makes the book a unique source of information for those involved in design, production, and implementation of gears for high-power density gear transmissions in the aerospace industry, the automotive industry, electric wind power stations.

Intended Audience Mechanical and manufacturing engineers involved in continuous process improvement, research workers who are active or intending to become active in the field, and senior undergraduate and graduate university students in mechanical engineering and manufacturing will benefit from reading this book. This book is intended to be used as a reference book as well as a textbook.

Organization of This Book This book is written in strict accordance to the recently developed scientific theory of gearing by this author (2012) [38]. In the first chapter, a brief overview of conformal gearing is presented. The state-of-the-art achievements in the field are discussed here. The discussion begins with a brief consideration of the necessary and sufficient criteria that geometrically accurate (ideal) gearing needs to meet. This analysis is important as high-conformal gearings allow for transmitting a rotation smoothly from a driving shaft to the driven shaft. Ancient designs of

Introduction

xxv

conformal gearings, that is, pre-Leonardo da Vinci (1493) designs of gearing, are briefly discussed there. This discussion is followed by the analysis of several designs of conformal gearing proposed mostly at the beginning of the twentieth century. Weakness of these designs is mostly because they were developed based only on common sense and not on the scientific theory of gearing as the theory was not known at the time. Gearing in the twentieth century and in later times is sophisticated and was not developed based only on common sense: Special knowledge in the theory of gearing is a must. The absence of knowledge in the theory of gearing is the root cause of why all the proposed designs of conformal gearings, including those proposed by Wildhaber’s helical gearing, failed. Fortunately, in 1954 Novikov gearing was invented. Novikov gearing is a perfect example of conformal gearing. The author of this book does not believe that Novikov gearing was invented based on the scientific theory of gearing, definitely not. The theory of gearing was not available in the time of Dr. M.L. Novikov. No doubt, in his research, he was led only by intuition. This time the intuition resulted in success. Ultimately, a novel type of conformal gearing was invented. The first chapter ends with the consideration of the features of tooth flank generation in Novikov gearing. Conditions to transmit a rotation smoothly are discussed in Chapter 2 of the book. The set of three necessary and sufficient conditions to transmit a rotation smoothly is established. This set includes:

1. The condition of contact of the tooth flanks of the gear and the mating pinion 2. The condition of conjugacy of the interacting tooth flanks 3. The condition of equality of the base pitch of the gear, and the pinion to the operating base pitch of the gear pair

This set is complemented by the requirement according to which the total contact ratio of a gear pair must be greater than 1. Conformal gearing and Novikov gearing, in particular, are discussed in Chapter 3. It is shown that a zero transverse contact ratio is the principal essence of Novikov gearing. This consideration is followed by a detailed analysis of the fundamental design parameters of conformal gearing. It is proved that conformal (Novikov) parallel-axis gearing is a degenerate type of involute gearing when the gear and the pinion tooth flanks are truncated such that only one point on each tooth flank remains. This point is referred to as the involute tooth point. This discussion is followed by a consideration of the kinematics of parallel-axis gearing (in a general sense), and a configuration of the plane of action. This leads to an introduction to the consideration of the boundary Novikov circle (or just N-circle, for simplicity) in conformal (Novikov) gearing.

xxvi

Introduction

The introduced boundary N-circle imposes limitations on the possible tooth geometries in conformal gearing. In this section, the discussion of kinematics of parallel-axis gearing enables the performance of an investigation into the tooth profile sliding in conformal gearing (Novikov gearing), as well as proceeding with an in-depth investigation of the features of the kinematics and the geometry in conformal gearing (Novikov gearing). Designing of conformal gearing including conformal gearing with two paths of contact is mostly based on the discussed kinematics of the relative motion of the gear and the pinion when the gear pair operates. This also yields derivation of the analytical expressions for the tooth flanks of the gear and the mating pinion. Chapter 3 ends with a discussion of the configuration of interacting tooth flanks at the culminating point, and with the analysis of the local and global contact geometry of interacting tooth flanks. High-conformal gearing is discussed in Chapter 4. The contact geometry in conformal parallel-axis gearing and the accuracy requirements for conformal parallel-axis gearing are covered in the discussion. Contact geometry of the gear and the mating pinion tooth flanks is a corner stone in designing gearings with desirable performance. This topic is discussed in detail in Chapter 5. The discussion begins with the analytical description of local relative orientation of the gear and the pinion at a point of contact of a gear and a mating pinion tooth flanks on which the performed second-order analysis is based. Dupin’s indicatrix at a point of the gear tooth flank and especially its matrix representation is widely used for this purpose. The indicatrix of conformity Cnf R(G/P) at a point of contact of the gear and the mating pinion tooth flanks is used to quantify the degree of conformity of the interacting tooth surfaces in the first order of tangency. It is shown that the indicatrix of conformity Cnf R(G/P) is a powerful analytical tool in designing gearings with favorable design parameters. In Chapter 6, the infeasibility to cut gears for conformal and high- conformal gearing in generating (continuously-indexing) gear machining processes (by hobbing, shaping, shaving, worm grinding, etc.) is proved. This discussion is helpful in understanding the root cause of why almost all the efforts undertaken to design, to manufacture, and to implement conformal gearings have failed. Parallel-axis conformal and high-conformal gearing enables a generalization to conformal and high-conformal gearings of other types. This generalization is based on the fundamental analysis in the kinematics of gearing, which is discussed in Chapter 7. The analysis begins with the discussion of vector representation of gear pair kinematics. For this purpose, (a) the rotation of the gear, (b) the rotation of the mating pinion, as well as (c) their instant relative rotation are represented in the form of vectors. Regardless, rotations in nature are not vectors, yet they can be treated as vectors if certain care is undertaken. This approach yields a generalization in all types

Introduction

xxvii

of intersected-axis conformal and high-conformal gearings (Ia -gearing, for simplicity). A similar generalization is possible in the cases of all types of crossed-axis conformal and high-conformal gearings (Ca -gearing, for simplicity). All possible vector diagrams are constructed and are then classified. Use of the classification significantly simplifies designing conformal and highconformal gearings of any possible type. At the end of the chapter, complementary vectors to the vector diagrams of gear pairs are briefly discussed. Useful kinematic and geometric formulas along with the consideration of the tooth ratio of a gear pair are provided. Examples of the implementation of the classification of the vector diagrams of gear pairs can be found in Chapters 8 and 9. In Chapter 8, high-conformal intersected-axis gearing (Ia -gearing) is discussed. The discussion begins with the analysis of the kinematics of the instantaneous motion in high-conformal intersected-axis gearing. Then, two entities are introduced, namely, a plane of action together with the base cones. This makes the construction of the path of contact possible in a highconformal intersected-axis gearing. A configuration of the path of contact is a trade-off between the bearing capacity of the tooth flanks and the sliding between teeth flanks in a high-conformal Ia-gearing. The determined configuration of the path of contact permits a construction of the boundary Novikov cone (the boundary N-cone) in intersected-axis high-conformal gearing. The boundary N-cone is a generalization of the boundary N-circle in parallel-axis conformal and high-conformal gearing. This section of the book ends with the consideration of the design parameters of high-conformal intersectedaxis gearing. Another example of implementation of the classification of the vector diagrams of gear pairs relates to crossed-axis high-conformal gearing (Ca-gearing), and is discussed in Chapter 9. Again, the discussion begins with the analysis of the kinematics of the instantaneous motion in high-conformal crossed-axis gearing. Then, two entities are introduced, namely, a plane of action together with the base cones. This enables the construction of the path of contact in a high-conformal crossed-axis gearing. A configuration of the path of contact is a trade-off between the bearing capacity of the tooth flanks and the sliding between teeth flanks in a high-conformal Ca-gearing. The determined configuration of the path of contact permits a construction of the boundary Novikov cone (the boundary N-cone) in crossed-axis high-conformal gearing. (It should be noted here that Ca-gearings feature a boundary screw N-surface, and not an N-cone. N-cones could be a reasonable approximation to the boundary screw N-surface in cases when the tooth flank sliding is low). The boundary N-cone is a generalization of the boundary N-circle in parallel-axis conformal and high-conformal gearing, and of the boundary N-cone in crossed-axis gearing. This section of the book ends with the consideration of the design parameters of high-conformal crossed-axis gearing. The main obtained scientific results are briefly outlined in the book’s Conclusion.

xxviii

Introduction

For the reader’s convenience, a few appendices appear at the end of the book. The appendices aim to refresh the reader’s memory in the math that is used in the book. Taken as a whole, the topics covered will enable a reader to design optimal (most favorable) designs of conformal and/or high-conformal gear pairs for high power density gear transmissions for the needs of the aerospace industry, the automotive industry, electric wind power stations, etc. A book of this size is likely to contain omissions and errors. If you have any constructive suggestions, please communicate them to Dr. S. Radzevich ([email protected]).

1 A Brief Overview of Conformal Gearing: State of the Art It was the increased contact strength of gear teeth that initially was the main reason that initiated developments in the field of conformal gearing. Conventional (involute) gearing features a convex-to-convex contact of the tooth flanks, which is substituted with a convex-to-concave contact in conformal gearing. Due to such substitution, the contact strength of the gear teeth can be increased proportionally to the increase of the degree of conformity of the interacting gear teeth flanks. It is natural to assume that a decision for the replacement of a convex-to-convex contact with convex-to-concave contact of the gear teeth is based on observation. For instance, all joints in a human skeleton (Figure 1.1) feature a convex-to-concave contact of the bones and not a convex-to-convex contact. Later on, it became clear that the use of conformal gear pairs helps to increase the so-called power density being transmitted by the gear pair, that is, conformal gearing is capable of transmitting a higher power through a smaller volume occupied by the gear pair. Power density* is an important consideration in the gearing of many designs and applications. The higher the power density of a gear pair, the lighter the gear transmission and vice versa. Helicopter transmissions are a good example of gearing where the highest possible power density is vital. However, the aerospace industry is not the only industry that needs the application of small size and high capacity gear transmissions. Gearboxes for electric wind power stations also represent good examples of this specific type of gearing. Gas consumption of trucks, cars, tractors, and other vehicles, strongly depends on the weight of the vehicle, that is, the lighter the vehicle, the less gas consumption and vice versa. Therefore, the use of lighter gearboxes in the automotive industry is also desirable. More examples of the desirable application of lighter gearboxes with higher performance can be found in many industries. High-power density† gearing (HPD-gearing, for simplicity) is the global target for the future developments in the field of gearing. * Here, and in the following, the power density is understood as the ratio of the power being transmitted through a gear pair to the volume occupied by this gear pair. † It is instructive to note here that the broader application of high-power density gearing in the future will result in the broader use of gears with low tooth count (LTC-gearing, in other words), that is, of gears whose base diameter, db, is equal to or greater than the root diameter, df, and for which the inequality db ≥ df is observed. Commonly, LTC-gears are viewed in more narrow sense, that is, LTC-gears are the gears whose tooth count is equal to Ng = 12 or fewer.

1

2

High-Conformal Gearing

Spherical femoral head (ball)

Acetabulum (socket)

FIGURE 1.1 Convex-to-concave contact of the bones in a human skeleton.

The main goal for the development of conformal gearing and later on of high-conformal gearing (Hc -gearing, for simplicity) is to increase the power density transmitted through a gear pair. Two different ways to increase the power density through a gear pair are recognized. First, the power density through a gear pair can be increased by means of an increased rotation of the driving member (the pinion). Under such a scenario, the torque applied to the driving member remains the same. Second, the power density through a gear pair can be increased by means of an increased torque applied to the driving member (the pinion). In such a scenario, the rotation of the driving member remains the same. A combination of the first and the second approaches is possible as well. Conformal gearing features the convex-to-concave contact of the tooth flanks of the gear and the mating pinion. This particular type of contact of the tooth flanks enables an increase in the contact strength of the gear and the pinion teeth. It is known that the concave-to-convex contact of part elements is always stronger compared to their convex-to-convex contact. Because of this, numerous attempts were made in the past to design gears with convex-to-concave contact of the tooth flanks of the gear and the mating pinion. Generally speaking, conformal gearing can feature either parallel axes of rotation, Og and Op, of the gear and the pinion (i.e., parallel-axis gearing, or

A Brief Overview of Conformal Gearing

3

just Pa -gearing, for simplicity), or intersected axes of rotation, Og and Op, (i.e., intersected-axis gearing, or just Ia -gearing, for simplicity), or crossed-axes of rotation, Og and Op, (i.e., crossed-axis gearing, or just Ca -gearing, for simplicity). In this chapter, consideration is mostly focused on Pa -gearing with concaveto-convex contact of the tooth flanks. However, the discussed ideas can be enhanced to the cases of Ia -gearing as well as Ca -gearing. In most applications, especially when the input, ωin, and the output, ωout, rotations are high, use of gearings with a constant angular velocity ratio,* uω = ωin/ωout = Const, is preferred. Gearings with a constant angular velocity ratio are commonly referred to as geometrically accurate gearings or ideal gearings. Geometrically accurate (ideal) gearings of all types are capable of transmitting a rotation smoothly. Rest of the gearings, whose angular velocity ratio is not constant (i.e., uω = var, and u = Const), are not capable of transmitting a rotation smoothly. Gearings of these types are referred to as approximate gearings. Considered below, conformal gearing is capable of transmitting a rotation smoothly. Therefore, conformal gearing is a type of geometrically accurate gearing. This means that conformal gearing meets the following three requirements: 1. The condition of contact of the tooth flanks of the gear, G, and the mating pinion, P, is satisfied. It is a common practice to specify the condition of contact of two smooth regular surfaces by means of Shishkov’s equation of contact,† n ⋅ VΣ = 0, where n designates the unit normal vector of a common perpendicular through the point of contact, K, of the tooth flanks G and P (this vector is along the line of action, LA, of the gear pair), and VΣ designates the vector of the resultant relative motion of the tooth flanks G and P at K [38]. 2. At every instant of time, the line of action, LA, passes through a point within the instant axis of rotation of the pinion relative to the gear, or of the gear relative to the pinion. In the theory of gearing, the instant axis of rotation means the pitch line. In cases of parallel-axis gearing, this requirement is specified by Willis’ theorem.‡ For the

* It should be stressed here on the difference between the angular velocity ratio, uω, and between the gear ratio, u. The angular velocity ratio is specified as uω = ωin/ωout. The angular velocity ratio is of a constant value for geometrically accurate gear pairs, and is variable for approximate gear pairs. The gear ratio is specified in terms of the tooth count of the input, Nin, and the output, Nout, members of the gear pair u = Nin/Nout. The gear ratio is always of a constant value, for ideal as well as for approximate gearings (except of noncircular gearings). † This equation was proposed by Professor Shishkov as early as in 1948 (or even earlier) in his paper: Shishkov, V.A., Elements of kinematics of generating and conjugating in gearing, in: Theory and Calculation of Gears, Vol. 6, Leningrad: LONITOMASH, 1948. In detail, this equation is also discussed in the monograph: Shishkov, V.A., Generation of Surfaces in Continuouslyindexing Methods of Surface Machining, Moscow, Mashgiz, 1951, 152 pages. ‡ Willis, R., Principles of Mechanisms, Designed for the Use of Students in the Universities and for Engineering Students Generally, London, John W. Parker, West Stand, Cambridge: J. & J.J. Deighton, 1841, 446 pages.

4


cases of intersected-axis gearing and crossed-axis gearing, this condition is generalized by Dr. Radzevich in Reference 38. 3. At every instant of time, the base pitch of the gear is equal to the operating base pitch of the gear pair, and the base pitch of the mating pinion is also equal to the operating base pitch of the gear pair, that is, both the base pitches are equal to the operating base pitch of the gear pair [38]. In the author’s opinion, the efforts undertaken in the past to develop conformal gearing need to be discussed in detail in order to properly evaluate the results obtained in the field so far, and to separate potentially useful achievements of the research from those that are either useless or incorrect. Many ambiguities are observed in interpreting conformal gearing just because of the misunderstanding of the kinematics and geometry of this particular type of gearing. The following analysis is based on the fundamental results outlined in the scientific theory of gearing [38].

1.1 Criteria for Geometrically Accurate (Ideal) Gearing All known designs of gearings as well as all designs to be developed in the future fall into one of two groups: • Geometrically accurate (ideal) gearings comprise the first group of gearings. Gearings of this type are capable of transmitting a rotation smoothly from the input shaft to the output shaft. • Approximate gearings comprise the second group of gearings. Gearings of this type are not capable of transmitting a rotation smoothly from the input shaft to the output shaft, that is, when the input shaft is rotated steadily, the output shaft always rotates with oscillations. The consideration in this monograph is mostly focused on geometrically accurate (ideal) gearings. In order to be capable of transmitting an input rotation smoothly, that is, in order to be referred to a gearing as the geometrically accurate (ideal) gearing, the following three conditions need to be satisfied [38]: • The condition of contact of tooth flanks of the gear G and the mating pinion P that is specified by Shishkov’s equation of contact n ⋅ v = 0 [38,42,43]. • The condition of conjugacy of tooth flanks of the gear G and the mating pinion P. (In the case of Pa -gearings, this condition is specified by


5

Willis’ theorem [45]. In the cases of Ia-gearings and Ca-gearings, this concept is discussed in detail in Reference 38. • Base pitches of the driving and the driven gears must be equal to the operating base pitch of the gear pair* (i.e., φb.g ≡ φb.op and φb.p ≡ φb.op, or simply φb.g ≡ φb.p ≡ φb.op). To the best possible extent, this concept is discussed in Reference 38. Precision gears (geometrically accurate or ideal gears) of all types, that is, conformal and nonconformal gears need to fulfill all three conditions listed. In order to properly evaluate all known designs of conformal gearing (as well as gearings of other types), the earlier listed three conditions are of critical importance: any and all geometrically accurate (ideal) gearings need to fulfill all three conditions. Below, a few known designs of conformal gearings are discussed mostly following a chronological order.

1.2 Ancient Designs of Conformal Gearings The earliest (1493) design of a gearing with convex-to-concave tooth contact known to the author is discussed in the famous book by da Vinci (Figure 1.2) [42]. The addendum of the gear teeth features convex geometry, while the dedendum is concave (Figure 1.3). In the pinion, the addendum features a convex tooth profile, and the dedendum is also concave. It is assumed that convex-to-concave contact of the tooth flanks is observed in the gearing of this particular type. There are a few more illustrations of gearings with convexto-concave contact of the tooth flanks. It is not discussed in Reference 7 whether the gearing (Figure 1.3) is spur or helical. However, the gearing (Figure 1.3) is supposed to be a spur gearing as the invention of helical gearing is credited to Robert Hooke.† Because of very limited information on the design of the gearing (Figure 1.3), it is almost impossible to exactly evaluate the advantages and disadvantages of this particular ancient design of conformal gearing.

* It is known that in cases of Pa -gearings, base pitch of the gear and of the mating pinion must be equal to one another. However, nothing is said so far about the operating base pitch in ideal Pa -gearings (the concept of the operating base pitch in an ideal Pa -gear pair is introduced by Dr. S.P. Radzevich in Reference 38). Moreover, neither the concept of base pitch of the gear and the pinion in the cases of Ia-gearings, as well as in the cases of Ca-gearings is discussed so far in the public domain. † Robert Hooke (28 July [O.S. 18 July] 1635–3 March 1703)—was an English natural philosopher, architect, and polymath.

6


FIGURE 1.2 Title page of the book: da Vinci, Leonardo, The Madrid Codices, Vol. 1, 1493, Facsimile Edition of Codex Madrid 1, original Spanish title: Tratado de Estatica y Mechanica en Italiano, McGraw-Hill Book Company, 1974.

1.3 Improvements in and Relating to Gear Teeth The invention [20] relates to gear teeth. The objective of the invention is to provide gear teeth which possess the following characteristics: a. Easy to manufacture b. Small sliding action between gear teeth c. Tooth profiles envelope each other d. Section of gear tooth unusually strong e. Low obliquity on line of action


7

FIGURE 1.3 An example of gearing discussed in the book: da Vinci, Leonardo, The Madrid Codices, Vol. 1, 1493, Facsimile Edition of Codex Madrid 1, original Spanish title: Tratado de Estatica y Mechanica en Italiano, McGraw-Hill Book Company, 1974.

The tooth profiles of external involute gear teeth are always convex (except in the case of a rack which is straight sided). But two convex surfaces, in contact with each other, are not well adapted to carry heavy loads. It is often found that when involute gears are run under heavy load, a peculiar pitting of the teeth takes place near the pitch line. This trouble is caused by the rubbing action of two convex surfaces. In the invention, illustrated by Figure 1.4, the line of action is represented by MN. In the cycloidal system, the pivot arm revolves, whilst the pivot point remains stationary, whereas in the involute system, the pivot arm remains stationary, whilst the pivot point is moved tangentially to the gear blank. In the invention, providing the “enveloping” system of gear teeth, the line of action is traced out by the combined movements of the pivot arm and pivot point. As the pivot arm, AM is revolved, so the pivot point A is simultaneously moved, and if the gear blank be revolved at a suitable relative speed, then the end of the pivot arm traces out—relative to the blank—the enveloping tooth curve such as BN, as shown in Figure 1.4. The enveloping curves B1 and B2 correspond to intermediate positions of the pivot points A1 and A2. Figure 1.4a shows the pivot point A moving toward the gear blank, so that the position A1 corresponds to the enveloping tooth curve BN. Figure 1.4c shows the pivot point A moving in a curved path to the point A2 so that the end of the pivot arm traces out—relative to the blank—the enveloping tooth curves B1 and B2 corresponding to the pivot point positions A1 and A2. Figure 1.4d shows a few teeth, made according to the invention, in mesh with each other, with NMN being the line of action as before and M the pitch point of the gears.

8


(a)

(b) A3 A2 A1

N

(c)

A

A2

A1

N

A1

M

M

B

B1

B2

N

A

A M

B1

B2

B3 (d)

N

N

M FIGURE 1.4 Improvements in and relating to gear teeth. (After Bostock F.J. and Bramley-Moore, S., Great Britain. Improvements in and Relating to Gear Teeth. Pat. No. 186.436, October 2, 1922. Filed: July 2, 1921, No. 18,014/21.)

It is well known that in correctly designed gear teeth which provide uniform velocity transmission, lines drawn at right angles to the tooth contours from any points of contact between any two teeth always pass through the pitch point. The latter is, of course, the point where the two pitch circles touch each other. As illustrated in Figure 1.5, in the gearing (Figure 1.4), this requirement is not fulfilled. The instant lines of action, LAi, intersect the centerline, ℄, at different points, Pi, which in nature are the instant pitch points. This means that the gearing under consideration is not capable of transmitting a rotation smoothly, that is, the gearing (Figure 1.4) is a type of approximate gearing and not a geometrically accurate (ideal) gearing. The line of action, LA, and the path of contact, Pc, are congruent to each other only in cases of Eu-gearing (Euler gearing, that is, involute parallel-axis gearing). In the rest of the cases, the LA and Pc are the two different entities.

1.4 Toothed Gearing This invention [21] relates to the shape and arrangement of the working faces of intermeshing teeth for spur gears, bevel gears, spiral gears, and other forms of gearings for connecting rotating or oscillating bodies.

9


N

t1 t3

M LA3 Path of contact, Pc Instant lines of action, LAi

P2

LA2

N

t2

P3 LA1 P1

Instant pitch points, Pi

L

C

FIGURE 1.5 In the gearing shown in Figure 1.4, the instantaneous pitch point, reciprocates along the centerline, ℄, when the gears rotate.

The objects of the invention are to improve the working qualities and simplify the construction of gear teeth and gears, and to increase the strength thereof and the amount of power that can be transmitted thereby. Further objects and advantages of the invention appear below in connection with the following description of the embodiment of the invention shown in the drawings. In the drawings, which illustrate a preferred but not the only form of the invention, Figure 1.6a is a side elevation of two mating gears, showing the outline of the teeth on both sides and their relative positions; Figure 1.6b is a diagram illustrating the intermeshing of the teeth at various angular positions; and Figure 1.6c is an edge view of a portion of one of the gears shown in Figure 1.6a and b, supplementing the diagram. It is usual to make gears with teeth conformal in outline to an epicycloidal or involute curve. These gears are difficult to make and require special tools and cutters for shaping them. The invention substitutes simple forms of cutters having straight sides for the series of cutters of curved outline usually required for making gears earlier. Referring to Figure 1.6a of the drawings, gears A and B shown therein have 20 teeth each spaced uniformly around their circumferences. Gear B has a slightly smaller outside diameter than gear A and its teeth 11 have straight sides 12. The sides of teeth 13 of gear A are curved to arcs of circles having their centers half-way between the center lines of the successive pairs of teeth, as indicated by circle 14 in Figure 1.6a. The centers of the arcs are on or just inside of circle 15 through the base of teeth 13. Bottom spaces 16 between the adjacent teeth 11 and 13 are the same as the width of the extremities of teeth 11; and the width of the extremities of teeth 13 is slightly less. The width of the bottom spaces 16 is about one-quarter

10

(a)

(b)

20 teeth

17 16 12 11 13

12

B

11

B

16 13 11 13

16

11

13

13 11 13

A

16

16 14

14 16 15

16

(c)

B

20 teeth 11

11

FIGURE 1.6 Toothed gearing. (After Schmick, H.J., Toothed Gearing, US Patent 1,425,144, Patented: August 8, 1922, Filed: June 30, 1921, Serial No. 481.561.)


1 2 3 4 5 6 7

A


11

(or slightly less) of the circular pitch of the teeth, measured on the circle 17 through the bottom of teeth 11, as shown in Figure 1.6a. The teeth may be skewed or twisted as shown in Figure 1.6c, in which case the two opposite ends of each tooth are preferably spaced angularly one-half the center distance between adjacent teeth so that the end of every tooth on either side of the gear falls exactly opposite the space between it and the next tooth on the opposite side of the gear. In Figure 1.6b, the dotted lines indicate the intersections with the faces of the straight and curved teeth, respectively, of straight planes perpendicular to the axes of the gears on the dot and dash lines 1, 2, 3, 4, 5, 6, and 7 in Figure 1.6c. It is evident that the teeth may be cut by straight or circular sided cutters caused to cut in straight lines across the width of the gear. The number of teeth and dimensions of the gears may be varied considerably so long as the slope of the sides of the straight teeth and the radius of the circles forming the curved teeth are so chosen as to avoid interference between successive teeth. The invention is not restricted to the numbers of teeth and sizes and proportions shown in the drawings. The gearing shown in Figure 1.6 meets the condition of contact. However, the condition of conjugacy of the tooth surfaces and the condition of equality of the base pitches of the gear and the pinion to the operating base pitch are not fulfilled in this design of gearing.

1.5 Wildhaber’s Helical Gearing Helical gearing with a circular arc tooth profile [22] targets an improved power capacity of the gear pair. The invention relates to the tooth shape of gears, which run on parallel axes, and may be applied to helical gears, such as single helical gears and double helical gears or herringbone gears. The purpose of the invention is threefold:

1. To provide helical gearing with an improved tooth contact, so as to lessen surface stresses and wear 2. To provide helical gearing, which is capable of rapid and accurate production, and which may be ground without difficulty, if so desired 3. To provide accurate gearing of circular tooth profile The invention is illustratively exemplified in the accompanying drawings, in which Figure 1.7a is a side elevational view of the proposed gear showing parts thereof in section; Figure 1.7d is a normal sectional view of Figure 1.7a, taken on the lines 2–2 of the latter figure; Figure 1.7g is a

12


(a)

(b) 2

17

16

3

(c)

1

11 12

4

17

16

2

2 4

(d)

(e)

8′ 4 15

3′ 11 11′ 2′

7 12

2

10

8

15′ 14

6

3 13

(f )

9 14′ 1

(g)

(h)

23 21

24

20 25

26 27 27′

26′ 26′′ 27′′

23 22

22

35

37

39

31 36

31 40

30

32 33

30 34

38

FIGURE 1.7 Helical gearing. (After Wildhaber E., Helical Gearing, US Patent 1,601,750, Patented: October 5, 1926, Filed: November 2, 1923.) (Continued)

13


(j)

(k)

(l)

N

45

43

46 43

42

N′

(m)

(n)

41

42

44′

44

50

50′

52

52′

41′

(o)

(p)

60

61

(q)

62

(r) 65 65′ 65′′

64

66

63

FIGURE 1.7 (Continued) Helical gearing. (After Wildhaber E., Helical Gearing, US Patent 1,601,750, Patented: October 5, 1926, Filed: November 2, 1923.)

14


side elevational view of a pair of gears constructed in accordance with the invention; Figure 1.7h is a sectional view taken through a pair of gears; Figure 1.7b and c are sectional views of milling cutters used in the manufacture of gears of the proposed design; Figure 1.7e and f are elevational views of corresponding tools of rack shape, to be used in reciprocating machines for cutting helical gears in accordance with the invention; Figure 1.7k and l are side elevational views of the improved gear showing a pair of grinding wheels in different operating positions, the wheels being set to grind opposite tooth surfaces; Figure 1.7m is a view of a gear taken in normal section and showing the grinding wheels in operation position; Figure 1.7n is a view of a mate pinion sowing the grinding wheels in operating position; Figure 1.7o is a view of a modified form of gear made in accordance with the invention; Figure 1.7j is a sectional view taken through an internal gear and its pinion; Figure 1.7q is a normal section through helical teeth of composite outline, constructed from the invention; Figure 1.7p is a view of reciprocating tool of rack shape in operating position; and Figure 1.7r is a view of a modified type of reciprocating tools, in position to start a cut on a herringbone gear. Referring to the drawings, and particularly to Figure 1.7a and d, 1 denotes a helical gear having teeth 2 in contact with teeth 3 of a mating pinion 4. In order to clearly illustrate the degree of contact between the teeth of the gear and pinion, tooth 4 is shown in section in Figure 1.7a. It is customary to analyze helical gearing with reference to a normal section, that is, line 2–2 of Figure 1.7a, with line 2–2 being normal to the helix of the pitch circle. Figure 1.7d illustrates the said normal section 2–2 for both pinion 4 and gear 1. It has been assumed as an example that the tooth profiles 6 of gear 1 are circular arcs of radii 7 and centers 8, in the shown normal section. Centers 8 are situated close to the pitch circle 9 of the gear.* The corresponding teeth of pinion 4 are so shaped as to allow the rolling of the pitch circles 9 and 10 on each other, as well known to those skilled in the art. When the gear tooth 2 is in the position as shown in Figure 1.7a and d, and its center at 8, then it contacts with tooth 3 at point 11, which may be determined by a perpendicular to tooth 2 through point 12, with point 12 being the contact point between the two pitch circles 9 and 10. The said perpendicular in the present case is the connecting line between point 12 and center 8 of the tooth profile. Another position 2′ of the gear tooth and 3′ of the corresponding pinion tooth are shown with dotted lines in Figure 1.7d. The tooth profiles contact here at point 11′, which can be determined like point 11. It will be noted that the contact point has traveled from 11 to 11′ during a small angular motion

* Centers 8 need to be situated along the line of action, LA, that is a must. Otherwise, the condition of contact (n ⋅ VΣ = 0) is violated.


15

of the gears.* The contact point has passed practically over the whole active profile* during a turning angle 13 of the gear, which corresponds to a fraction only of the normal pitch14, 14′. The said normal pitch equals the circle pitch of the shown normal section. In gearing now in use, however, the tooth outline and the tooth proportions are so selected that the contact of corresponding normal profiles lasts for an angle, which, as a rule, corresponds to more than one full pitch. In gearing according to the invention, the contact point between two normal profiles passes over the whole active profile during* a turning angle, which corresponds to less than one-half the normal pitch and usually to much less than that. Gearing designed according to the invention allows the teeth to come into a better contact with each other inasmuch as the tooth surfaces remain much closer to each in a direction perpendicular to the contact line between two mating gears. This is illustrated by a section taken in the direction of lines 15, 15′ of Figure 1.7d. In Figure 1.7a, the lateral profile 16 of tooth 3 and profile 17 of tooth 2 of said section are shown to contact at point 11, and to remain close to each other on their whole length. The same holds true for other sections, taken parallel to section 15, 15′. Close contact between teeth is well known to reduce wear and to improve the efficiency of the gears. Although a circular arc is shown as the normal tooth profile of gear 1, in Figure 1.7d, it will be understood that this is not the only shape to affect the stated purpose of increasing the speed, at which the contact point travels over the tooth profile of a normal section. As a rule, however, the shape can be approximated by a circle whose center is close to the pitch center.* The gearing according to the present invention is strictly a gearing for helical teeth. It would not be advisable on straight teeth on account of the explained short duration of contact between tooth profiles. This would cause intermittent action, whereas on helical gears similar parts of the teeth are always in contact, on account of the twisted nature of the tooth surfaces. Figure 1.7g may be considered as a view taken in the direction of the axes of a pair of gears. The tooth profiles are the circles in a section, which is perpendicular to the axes. The gear is provided with helical teeth with working faces below the pitch circle 20, while the pinion teeth have working faces above the pitch circle 21 only. The working profiles 22 of the gear are concave and circular, and their centers are substantially situated on the pitch circle 20. The convex working profiles 23 of the pinion are also of circular shape. Their radii 24 are substantially the same as the radii 25 of the mate profiles. The centers 26, 26′, 26″ are similarly situated on the pitch circle 21. Profile centers 27, 27′, 27″ of the pitch circle 20 and profile centers 26 26′, 26″ of the pitch * Once the contact point has traveled from 11 to 11′ during a small angular motion of the gears, this immediately reveals that the transverse contact ratio, mp, in Wildhaber’s helical gearing is greater than 0, that is, mp > 0, which is not permissible.

16


circle 21 correspond to each other. They coincide during the mesh, which takes place on the whole tooth profile at once. Figure 1.7g may also be considered as a section perpendicular to the helical teeth and shows the normal tooth profiles. Figure 1.7h shows a refinement of the preferred embodiments of the invention. It is not only a normal section through the helical teeth, but can also be considered as a section perpendicular to the axes. Corresponding profiles 30 and 31 are circular, as in Figure 1.7g, but in this case the radius of the concave circular profile 30 is made a trifle larger than the radius of the convex circular profile 31. Consequently, the profile centers 32 and 33 do not exactly coincide during the mesh. The radii 34 and 35 of the circles 36 and 37, constituted by the profile centers 32 and 33, respectively, are not accurately identical with the pitch radii 38 and 39 of the two gears. The sum of the radii 34 and 35 is a trifle larger than the sum of the pitch radii. The radii 34 and 35 are so selected that the main tooth pressure runs about in a direction 33, 40. The slight difference of the radii of profiles 30 and 31 facilitates the tooth contact and allows for small errors in making and assembling. Figure 1.7b and c shows a pair of milling cutters for milling gear teeth. The cutters may be applied in the usual manner, their axis being inclined in correspondence with the tooth inclination, that is, with the helix angle of teeth. It will be found that the cutters to be inclined for an angle, which is a trifle smaller than the helix angle in the pitch circle, for producing the most accurate results. In Figure 1.7e and f, a pair of rack-shaped cutters is shown.* These cutters are for use in a reciprocating machine. The teeth of these tools are relieved inwardly, in the usual manner, as evident by the dotted lines. The convex grinding wheels shown in Figure 1.7k are illustrated in their operating positions, in a view which is taken perpendicular to the axis of the gear blank as well as to the axis of the grinding wheels, that is, in a view along the gear radii 41, 41′ of Figure 1.7m. The wheels which are to produce concave circular teeth profiles in a normal section are of a convex circular profile, with its radius 42 being the same as the radius of the concave circular profile. The grinding wheels are inclined for an angle 43, which equals the helix angle of the teeth in the pitch circle. The wheels grind along their profiles indicated with the dotted lines 44 and 44′, which are located in a normal section. As shown in Figure 1.7k, the two grinding wheels are coaxially arranged with respect to each other. The device shown in Figure 1.7l corresponds to that shown in Figure 1.7k, with the exception that the grinding wheels 45 and 46 are not coaxially arranged. Although the arrangement shown in Figure 1.7k imposes certain restrictions on the tooth design, it is frequently preferred. The arrangement of Figure 1.7l is advantageous when the grinding wheels are not free to run * As is shown later on in this book (see Chapter 6), conformal gears cannot be cut by rack cutters, and, more generally, they cannot be machined in any gear-generating process.


17

out for instance when they must clear against a shoulder or in the case of herringbone teeth. Referring particularly to Figure 1.7m, a normal section is illustrated and taken along lines N, N′ of Figure 1.7k. In this view, the axis of the coaxially arranged grinding wheels is situated in the said normal section. The wheels grind along the profiles 44 and 44′ of the shown normal section, while the blank performs a translator motion in the direction of its axis, and, in timed relation thereto, a turning motion about its axis. In other words, the blank is screwed past the grinding wheels. Figure 1.7n discloses a normal section through the teeth of the mating gear or pinion. Grinding wheels 50 and 50′ are provided with the concave circular profiles 52 and 52′ with which they grind the convex gear teeth. It will be understood that milling cutters might be used instead of the grinding wheels, as shown in Figure 1.7k through n, and also that grinding wheels of a shape shown in Figure 1.7b and c might be used, if so desired. The teeth ground according to Figure 1.7k, m and n are preferably so designed that the centers of opposite tooth arcs 44 and 44′, 52 and 52′, respectively, as shown in Figure 1.7n, coincide. In Figure 1.7m and n, the tooth arcs of every third tooth side have a common center. The tooth arcs of every fifth tooth side have a common center in the normal section, as shown in Figure 1.7o. In Figure 1.7m, the common center of opposite tooth arcs of alternate teeth is situated on the centerline of the intermediate tooth. The corresponding pinion shows convex circular profiles, of which opposite tooth sides of adjacent teeth have common centers in the middle of the intermediate tooth space. The normal section shown in Figure 1.7j shows an internal gear and its mate pinion, constructed in accordance with the concave tooth profiles. In external gears, similar preference is given to providing the larger gear with concave tooth profiles. The normal section through a pair of helical gears, as shown in Figure 1.7q, discloses opposite tooth profiles, with the addendum being convex and the dedendum concave. A rack-shaped planing tool* is illustrated in operating position in Figure 1.7p. Tools of this kind have been shown in another view in Figure 1.7e and f. The reciprocatory tool 60 moves in the direction 61, at an inclination, which equals the helix angle of the teeth. Gear 62, with its axis 63, is shown with the dotted and dash lines. In order to cut the proper tooth shape, gear blank 62 after every cut is slightly fed in a rolling generating motion with respect to a rack which is embodied by tool 60. Another reciprocatory tool 64 is shown in Figure 1.7s, the tool in this case being provided with stepped teeth 65, 65′, 65″ which allow it to clear

* See footnote * on page 16.

18


shoulders and herringbone teeth. The tool moves in the direction 66 of the helical teeth, which cuts. Other ways of producing gearing according to the invention,* that is, hobbing, planing with a pinion cutter, rolling and casting, may be contemplated, but it is not deemed necessary at this time to give a detailed explanation of the mechanism used in this connection. Briefly stated, the invention consists in providing helical gearing of such a profile, that the tooth contact passes rapidly over the normal profile of the teeth. This has been found to result in close contact between helical mate teeth. In a direction at right angles to the contact line, the mate teeth recede from each other only slightly, and, thus, provide a tooth contact, which is not very far from surface contact. Wildhaber’s gearing features a nonzero transverse contact ratio (mp > 0) and a nonzero face contact ratio mF > 0. The total contact ratio, mt, is greater than 1, that is (mt ≡ mp + mF > 1). Wildhaber’s gearing (see Figure 1.7) does not meet the condition of contact. The condition of conjugacy of the tooth surfaces and the condition of equality of the base pitches of the gear and the pinion to the operating base pitch are not fulfilled in this design of gearing. The invention by Wildhaber is explained to the best possible extent in Appendix A.

1.6 Novikov Gearing Novikov gearing [23] is an example of conformal helical gearing. For a long while, Novikov’s patent (S.U. Patent No. 109113 of 1956) was not available to most of the gear experts. Known designs of gearing, those featuring point system of meshing, feature low contact strength and are not widely used in practice. The contact strength of known designs of gearing with a line system of meshing, including the widely used involute gearing, is limited. The proposed gearing [23] features higher contact strength due to favorable curvatures of the interacting tooth flanks. Under equivalent contact stress, similar dimensions, and comparable remaining design parameters, greater circular forces are permissible by the proposed gearing. Lower sensitivity to manufacturing errors and to deflections under the load is the other advantage of the proposed gearing. The proposed gearing can be designed either with parallel, intersecting, or crossing axes of rotations of the gears. External gearing as well as internal gearing of the proposed system of meshing is possible. The tooth ratio of * See footnote * on page 16.

19


P

O1

B

B

O2 C

A

C

FIGURE 1.8 Helical gearing. (After Novikov, M.L. USSR, Gear Pairs and Cam Mechanisms Having Point System of Meshing. Patent 109,113, National Classification 47h, 6. Filed: April 19, 1956, published in Bulletin of Inventions No.10, 1957.)

the proposed gearing can be either of constant value or it can be variable, and time dependent. The proposed concept of gearing can be utilized in the design of cam mechanisms. In Figure 1.8, possible tooth profiles in the cross-section of tooth flanks by a plane that is perpendicular to the instant axis of relative rotation through the current point of contact are illustrated. Here, the point of intersection of the planar cross-section by the axis of instant relative rotation is denoted by P. O1 and O2 are the points of intersection of the planar cross-section by the axes of the gear and the pinion. A is the point of meshing (in its current location). PA denotes the line of action. ДAД is the circle centered at the point P that corresponds to the limit case of the tooth profiles (in the case the profiles are aligned to each other). Several curves, BAB, represent examples of the tooth profiles of one of the mating gears. The curves BAB are arbitrary smooth curves, which are located inside of the circular arc ДAД (i.e., the arcs are located within the bodily side of the limit tooth flank of one of the gears). The curves BAB are located close to the circular arc ДAД and they feature high degree of conformity to the circular arc.

20


Several curves, CAC, represent examples of the tooth profiles of the second of the mating gears. The curves CAC are arbitrary smooth curves, which are located outside the circular arc ДAД (i.e., the arcs are located within the bodily side of the limit tooth flank of another of two gears). The curves CAC are also located close to the circular arc ДAД and they feature a high degree of conformity to the circular arc. In the following, the entity of the invention is disclosed in detail. The location and orientation either of a straight path of contact or a smooth curved path of contact are specified in space in which the location and orientation of the axes of rotations of the gear and the pinion are given. The path of contact is located reasonably close to the axis of instant relative rotation of the gears. Either constant or time-dependent (smoothly varying in time) speed of motion of the contact point along the path of contact is assigned. A coordinate system is associated with the gear, and a corresponding coordinate system is associated with the pinion. In the coordinate systems, the moving contact point traces the so-called contact lines.* One of the paths of contact is associated with the gear and the other one is associated with the pinion. Certain smooth regular surfaces through the paths of contact can be used as the tooth flanks of the gear and the pinion. The following requirements should be fulfilled in order to use the surfaces as the tooth flanks: • At every location of the contact point, the tooth flanks should have a common perpendicular, and, thus, the requirements of the main theorem of meshing should be satisfied. • The curvatures of the tooth profiles should correspond to each other. • No tooth flanks interference occurs within the working portions of the surfaces. The proposed tooth flanks fulfill the earlier listed requirements and allow for high contact strength of the gear teeth. Consider a plane through the current contact point, which is perpendicular to the instant axis of relative rotation. Then, two circular arcs are constructed. The circular arcs are centered at points within the straight line through the pitch point and the contact point. The arc centers are located close to the pitch point. The constructed circular arcs can be considered an example of the tooth profiles of the gear and the pinion. The tooth flanks are generated as loci of the tooth profiles constructed for all possible locations of the contact point. The working portion of one of two tooth flanks is convex, while the working portion of the other tooth flank is concave (in the direction toward the axis of instant relative rotation). In a particular case, the radii of tooth profiles could be of the same magnitude and equal to the * The contact line is a term used by Dr. M.L. Novikov. Actually, the contact line is the path of contact.


21

distance from the contact point to the axis of instant relative rotation. The centers of both profiles in this particular case are located at the axis of instant relative rotation. Under such a scenario, point meshing reduces to a special line meshing. This would require an extremely high accuracy of the center distance and independence of it from operation conditions, which is impractical. Point meshing is preferred when designing tooth profiles. A small difference between the radii of curvature of the tooth profiles is desirable. It should be kept in mind that during the run-in period of time, point meshing of the gear teeth will transform to the earlier mentioned locally line meshing of the tooth profiles. However, the theoretical point contact of the tooth flanks will be retained. Tooth profiles can differ from the circular arcs. However, the tooth profiles of other geometries (those always passing through the contact point) should be located (for one gear) within the interior of the abovementioned circular arc profile that centers at the point within the axis of instant relative rotation as shown in Figure 1.8. For another gear, the tooth profile should be located outside the circular arc. The law of motion of the contact point (i.e., speed of the point and its trajectory) should be chosen so as to minimize the friction and wear loses. The friction and wear loses are proportional to the relative sliding velocity in the gear mesh. Therefore, it is desirable to reduce the sliding velocity as much as possible. For this purpose, the path of contact should be located not far from the axis of instant relative rotation. On the other hand, a too-close location of the path of contact to the axis of instant relative rotation is also not desirable as that reduces the contact strength of the gear tooth flanks. In addition, it is recommended to ensure favorable angles between the common perpendicular (along which the tooth flanks of one of the gears acts against the tooth flank of another gear) and the axes of rotations of the gears. Opposite sides of tooth profiles are designed in a similar manner to that just discussed. Tooth thicknesses and angular pitch are assigned to ensure the required bending tooth strength. The face width of the gear or the length of the gear teeth should correlate with their pitch to ensure the required value of the face contact ratio. Gear pairs can feature either one point of contact (when working portions of the tooth flank contact each other just in one point, excluding the phases of the teeth reengagement), or they can feature multiple contact points when the tooth flanks contact each other at several points simultaneously. For parallel axis gear pairs, it is preferred to use a straight line as the path of contact, which is parallel to the axes of rotations of the gear and the pinion. The speed of the contact point along the straight path of contact can be of constant value. In this particular case, the radii of curvature of the tooth profiles in all cross-sections by planes are equal to each other. Tooth flanks in this case are regular screw surfaces. Gears that feature tooth flanks of such a geometry are easy to manufacture, and they can be cut on machine tools available on the market.

22


An example of parallel axis gearing with the limit geometry of the tooth profiles is illustrated in Figure 1.8. The point contact of the tooth flanks in this particular case is transformed to almost the line contact of the tooth flanks. The curved contact line is located across the tooth profile. When axial thrust in the gear pair is strongly undesirable, herringbone gears can be used instead. Novikov gearing features a zero transverse contact ratio (mp = 0), while the face contact ratio mF is always equal to the total contact ratio, mt, and is greater than 1, that is (mt ≡ mF > 1). As was shown earlier in this chapter, for Wildhaber’s gearing, the following relations mp > 1, mF > 1, and mt = mp + mF > 1 are valid. Novikov gearing (see Figure 1.8) meets the condition of contact, as well as it meets the condition of conjugacy of the tooth surfaces, and the condition of equality of the base pitches of the gear and the pinion to the operating base pitch. Novikov gearing is a type of geometrically accurate gearing.* Novikov gearing is the only feasible type of gearing with the point contact of the tooth flanks that is capable of transmitting a rotation smoothly. As shown in the following, this is because Novikov gearing is a degenerate type of involute gearing [33]. The invention by Novikov is explained to the best possible extent in Appendix A. In Appendix A, the author’s comments on the concept of “Novikov gearing” and on the inadequacy of the terms “Wildhaber–Novikov gearing” and “W–N gearing” are presented.

1.7 Features of Tooth Flank Generation in Novikov Gearing In Novikov gearing, the transverse contact ratio is zero (mp = 0). Because of this, the contact point is stationary in the transverse cross-section when the gears rotate. The condition of conjugacy of the tooth profiles in Novikov gearing and the condition of equality of the base pitches of the gear and the pinion to the operating base pitch of the gear pair are fulfilled due to the contact point is motionless in all transverse cross-sections. The equality of the transverse contact ratio to zero (mp = 0) allows us to interpret Novikov gearing as a degenerated type of involute gearing, that is, * It needs to be noticed here that Dr. E. Wildhaber did not recognize the difference between Novikov gearing and the helical gearing proposed by him [22]. This conclusion immediately follows from Wildhaber’s statement: “I may say also that it gives me satisfaction to see the original concept vindicated through the Russian reinvention and effort and through subsequent efforts and articles.” This statement of Wildhaber’s can be found on page 949 in the paper by T. Allan [22] (see the Communications section). Other evidence in this regard is also known [29,38].


23

Novikov gearing is a type of involute gearing for which the involute tooth profile is truncated to a point [33]. The remaining portions of the gear teeth profiles are not active and do not come into contact with each other when the gears rotate. Therefore, the gear designer is free to modify the inactive portions of the gear teeth making them stronger in terms of contact strength. Once the entire profile of the gear teeth in Novikov gearing (including the inactive portions of the gear teeth) is noninvolute, this means that the profiles of this type cannot be machined in generating (continuously indexing) methods.* Therefore, neither gear hobs, nor shaper cutters, nor worm grinding wheels can be used for machining gears for Novikov gearing. Tooth profiles of the gear and the pinion in Novikov gearing cannot be generated by the so-called basic rack (Figure 1.9). The so-called basic rack does not exist in the case of Novikov gearing. Associating the so-called basic rack with gears for Novikov gearing is a huge mistake committed by many gear experts. Violation of both the condition of conjugacy of the tooth profiles and that of the equality of the base pitches in this case is the main reason for such infeasibility. The following conclusions can be drawn from the earlier statement:

1. Numerous inventions of different modifications of Novikov gearing based on the generation of tooth flanks by means of the so-called basic rack (German Patent No. 102,004,034,456, USSR Patents No. 95978, No. 800,472, No. 1,184,994, Russian Patents No. 2,426,023, No. 2,057,267, USA Patents No. 2,994,230, No. 3,180,172, No. 3,371,552, No. 3,533,300, No. 3,709,055, No. 3,855,874, No. 3,937,098, No. 3,982,445, No. 4,031,770, 4,051,745, 4,140,026, 5,022,280, No. 6,178,840, 6,205,879, No. 6,837,123, Japanese Patent No. 4,449,045, and numerous others) are wrong. 2. Numerous NASA contractor reports in which different modifications of Novikov gearing based on the generation of tooth flanks by means of the so-called basic rack are proposed (4771, AVSCOM Technical Report 87-C-18, 1987; 4089, Army Research Laboratory Contractor Report ARL-CR-339, 1997; 1406, Army Research Laboratory Contractor Report ARL-TR-1500, 1997; NASA/CR-2000209415, ARL-CR-428, 1997; NASA/CR-2000-209415, ARL-CR-428, 1997; and numerous others). All of them are wrong. 3. Dozens of scientific publications in the ASME Journal of Mechanical Design, ASME Journal of Manufacturing Science and Engineering, Mechanism and Machine Theory, along with publications in the * It should be mentioned here that the first pair of Novikov gears made out of aluminum alloy (a preprototype) had been cut on April 25, 1954 by means of a disk-type milling cutter [29,38]. Fifteen gear pairs for testing purposes had been machined in the summer of 1954 by means of a disk-type milling cutter [29,38].

24


(a)

p = πm

c ρa

ha

αn 2ha

αn ρf

c

(b) eu

ha

hk

ρa

αn

hja rj αn

ρf

hjf

hk

h

rg

ef FIGURE 1.9 The so-called basic racks in Novikov gearing according to (a) USSR standard and (b) Chinese standard. (Both the standards are wrong.)

proceedings of numerous International Gear Conferences, and so forth in which different modifications of Novikov gearing based on the generation of tooth flanks by means of the so-called basic rack are wrong. Efforts of hundreds of gear experts undertaken so far to improve Novikov gearing and tons of funds spent on these researches were just wasted because the fundamental principles of gearing were ignored when investigating Novikov gearing.


25

Again, gears for Novikov gearing cannot be cut in a continuously indexing (generating) process. For cutting the gears, disk-type milling cutters can be used. It is likely that the committed mistakes can be traced back to the publication by Prof. V. N. Kudriavtsev [15]. Prof. V. N. Kudriavtsev was the first (1959) who proposed cutting gears for Novikov gearing by means of specially designed gear hobs, the so-called Novikov hobs. The infeasibility of the correct generation of tooth profiles of gears for Novikov gearing in a continuously indexing (generating) process could be the root cause of the insufficient performance of Novikov gearing, especially in cases of case-hardened tooth flanks of the gears. The continuously indexing (generating) process is not capable of machining gears with a correct correspondence between the radii of curvature of the convex and concave tooth profiles of the gear and the pinion in a Novikov gear pair. Gears for Novikov gearing need to be cut by disk-type milling cutters, disktype grinding wheels, and so forth, that is, using form cutters and indexing processes for this purpose.

2 Conditions for Transmitting a Rotation Smoothly As discussed in this chapter, high-conforming gearing is a type of geometrically accurate gearing, that is, ideal gearing that is capable of transmitting a rotation smoothly from a driving shaft to a driven shaft. To transmit a rotation smoothly from a driving shaft to a driven shaft, a set of certain conditions need to be satisfied. In this section of the book, the set of necessary conditions to transmit a rotation smoothly is discussed with regard to parallel-axis gearing (Pa -gearing). In the subsequent chapters, the equivalent sets of conditions will be formulated for intersected-axis gearing (Ia-gearing) and for crossed-axis gearings (Ca-gearing). The first three conditions the geometrically accurate (ideal) gearing must obey are referred to as the fundamental conditions. These three conditions are discussed below.

2.1 Condition of Contact The first fundamental condition to be discussed in this section is the condition of contact of the tooth flanks of a gear and the mating pinion. The condition of contact of the tooth flanks is the first fundamental condition which all Pa -gearings (as well as Ia - and Ca-gearing) must obey. The condition of contact of the tooth flanks of the gear G and the mating pinion P is commonly expressed by Shishkov’s equation of contact* [38,42,43]:

n ⋅ vΣ = 0

(2.1)

where n is the unit vector of common perpendicular through the contact point of the tooth flanks G and P and vΣ is the unit vector of the speed of the resultant relative motion of the tooth flanks G and P at contact point K. * This equation was proposed by Professor Shishkov as early as in 1948 (or even earlier) in his paper: Shishkov, V.A., Elements of kinematics of generating and conjugating in gearing, in: Theory and Calculation of Gears, Vol. 6, Leningrad: LONITOMASH, 1948. In detail, this equation is also discussed in the monograph: Shishkov, V.A., Generation of Surfaces in Continuouslyindexing Methods of Surface Machining, Moscow, Mashgiz, 1951, 152 pages.

27

28


Equation 2.1 reveals that a component of the velocity vector vΣ along the common perpendicular n is equal to zero. Otherwise, either separation or interference of the tooth flanks G and P is observed. Neither separation nor interference of the tooth flanks G and P of the gear and the pinion is permissible. Therefore, the vector vΣ is either located in a common tangent plane or it is zero. Permissible instant relative motions of the tooth flanks G and P in a gear pair are illustrated in Figure 2.1. The relative motion of the tooth flanks G and P is not permissible along the common perpendicular, n, and the relative motion is allowed in any direction within the common tangent plane through the contact point K. It should be pointed out here that a swivel relative motion of the tooth flanks G and P around the axis along the common perpendicular, n, also meet the requirement specified by Equation 2.1. The swivel motion of the tooth flanks is not necessary to transmit a rotation from a driving shaft to a driven shaft. However, a motion of this nature can be observed in spatial gearing, that is, in Ca-gearing. It is necessary to stress here that regardless of the equation of contact (see Equation 2.1) that was proposed by Professor V.A. Shishkov in the mid of the twentieth century, the physics of the condition of contact was properly understood by the gear community in the time of da Vinci [7] and even in the earlier times. Other forms of analytical representation of the condition of contact of the tooth flanks G and P of the gear and the pinion are known as well. The fulfillment of the condition of contact is necessary but not sufficient to smoothly transmit a rotation from a driving shaft to a driven shaft.

n

vΣ

K

FIGURE 2.1 Permissible instant relative motions in geometrically accurate (ideal) gearing.

Conditions for Transmitting a Rotation Smoothly

29

2.2 Condition of Conjugacy The second fundamental condition to be discussed in this section is the condition of conjugacy of tooth flanks of the gear G and the mating pinion P. The condition of conjugacy of the tooth flanks G and P is the second fundamental condition which all Pa -gearings (as well as Ia - and Ca-gearing) must obey. For the first time ever, this condition was formulated by Dr. R. Willis as early as in 1841 (Figure 2.2). This condition is commonly referred to as the main theorem of gearing, or just as Willis’ theorem [45]. Nowadays, in the case of Pa -gearing,* Willis’ theorem is commonly formulated as follows: Willis’ Theorem 2.1 The common perpendicular to the conjugate tooth flanks of the gear and the pinion always passes through the pitch point of the parallel-axis gearing; the pitch point subdivides the center distance reciprocal to the angular velocities of the gear and the pinion. Willis’ theorem is commonly construed as the main theorem of parallelaxis gearing.†

FIGURE 2.2 The main theorem of gearing (Willis’ theorem) as was originally formulated by R. Willis on page 38 in his book: Willis, R., Principles of Mechanisms, Designed for the Use of Students in the Universities and for Engineering Students Generally, London, John W. Parker, West Stand, Cambridge: J. & J.J. Deighton, 1841, 446 pages. * In the cases of Ia - and Ca-gearings, this concept is discussed in detail in [38]. † It should be noted here that some authors claim that the main theorem of gearing was known to L. Euler (1707–1783) and to F. Savary (1779–1841).

30


In order to understand Willis’ theorem, the difference between the line of action, LA, and the path of contact, Pc, in Pa -gearing needs to be firmly made. A parallel-axis gearing is schematically shown in Figure 2.3. The gear and the pinion rotate about their axes of rotation, Og and Op, with angular velocities ωg and ωp, respectively. The axes Og and Op are at a center distance C form one another. The base diameter of the gear is designated as db.g, and the base diameter of the pinion is designated as db.p. Outside diameters of the gear and the pinion are designated as do.g and do.p, respectively. In accordance to the belt-and-pulley model of involute gearing, the line of action, LA, between the tooth flanks G and P of a gear and a mating pinion is tangential to both the base circle of the gear and the base circle of the pinion. Points of tangency are labeled as Ng and Np, respectively. The line of action, LA, intersects the centerline, ℄, at the pitch point, P. (The center distance, C, is a straight line segment of the centerline, ℄). Points of intersection Pg and Pp of the line of action, LA, by the circles of the diameters do.g and do.p are the extreme points of the active portion, Zpa, of the line of action, LA (that is, PgPp = Zpa). Op do.p

db.p ωp Np

Path of contact, Pc

ϕt Pg

P

VK Pp

Line of action, LA Ng C . sin ϕt

L

C

do.g db.g

ωg

C Og

FIGURE 2.3 In involute gearing, the line of action, LA, and the path of contact, Pc, align to each other.

31


A perpendicular through the pitch point, P, to the centerline, ℄, forms the transverse pressure angle, ϕt, with the line of action, LA. When the gears rotate, the transverse pressure angle, ϕt, is of a constant value at every instant of time, that is, for all configurations of the gear and the pinion in their relation to each other. Therefore, when the gears rotate, the contact point K between the tooth flanks G and P travels along the straight path of contact, Pc, and the path of contact is aligned with the line of action, LA. For Pa -gearing, the involute tooth profile is unique from this standpoint. Tooth profiles of other geometries are not capable of transmitting a rotation smoothly. Only involute gearing feature a straight path of contact that is aligned with the line of action. In gearings of other systems, a difference needs to be made between the line of action, LA, and the path of contact, Pc. To illustrate a difference between the line of action, LA, and the path of contact, Pc, cycloidal gearing is discussed in the following. The cycloid of a circle is used as a tooth profile in cycloidal gearing. A cycloidal curve is generated as the trajectory of a point of a circle rolling with no slippage over another circle (or over a straight line in a degenerate case). Henceforth, the difference between ordinary, extended, and shortened cycloids will be made. An example of cycloidal gearing is schematically shown in Figure 2.4a. In Figure 2.4a, the center of rotation of the gear, Og, and that of the pinion, Op, are at a certain center distance, C. The rotations of the gear and the pinion are denoted by ωg and ωp, respectively. The pitch radius of the gear is designated as rg and that of the pinion is designated as rp. The pitch point in the gear pair is denoted by P. (a) Rp

dp

(b) Op

op

ag

C

Pc

P rg

p Pc

ap

L

C

i

Pi

ωg Rg

ϕt.i rg

Og

Pc

P

g

og

rp

rpi

ωp

p

Pc

g

rp

ωp

ωg L

C

rgi

dg

FIGURE 2.4 Schematic of cycloid gearing: (a) path of contact and (b) instant line of action.

LAinst

32


Two auxiliary centrodes of radii rg and rp that have centers at og and op, respectively, are used to generate the addendum and dedendum of the tooth profile of the gear and the pinion. The generation of the gear tooth profile can be executed in two steps: first, to generate the gear tooth addendum, consider rolling with no slippage of an auxiliary axode (of a radius rp) over the gear pitch circle (of a radius Rg). The circles of radii rp and Rg are in external tangency in relation to one another. The pitch circle of the gear is considered as stationary. In such a relative motion, a point of the circle of radius rp traces the epicycloid, Pag , within the plane rigidly connected to the gear. A portion of the arc, Pag , is used as the profile of the addendum of the gear tooth. Second, to generate the gear tooth dedendum, consider rolling with no slippage of an auxiliary axode (of a radius rg) over the gear pitch circle (of a radius Rg). The circles of radii rg and Rg are in internal tangency in relation to one another. The pitch circle of the gear is considered as stationary. In such a relative motion, a point of the circle (of a radius rg) traces the hypocycloid, Pdg, within the plane rigidly connected to the gear. A portion of the arc, Pdg, is used as the profile of the dedendum of the gear tooth. Similar to the generation of the gear tooth profile, the generation of the pinion tooth profile can be executed in two steps as follows. First, to generate the pinion tooth addendum, consider rolling without slippage of the auxiliary axode of radius rg over the pinion pitch circle of radius Rp. The circles of radii rg and Rp are in external tangency in relation to one another. The pitch circle of the pinion is considered as stationary. In such a relative motion, a point of the circle of radius rg traces the epicycloid, Pap , within the plane rigidly connected to the pinion. A portion of the arc, Pap , is used as the profile of the addendum of the pinion tooth. Second, to generate the pinion tooth dedendum, consider rolling with no slippage of the auxiliary axode of radius rp over the gear pitch circle of radius Rp. The circles of radii rp and Rp are in internal tangency in relation to one another. The pitch circle of the pinion is considered as stationary. In such a relative motion, a point of the circle of radius rp traces the hypocycloid, Pdp , within the plane rigidly connected to the gear. A portion of the arc, Pdp , is used as the profile of the dedendum of the gear tooth. The path of contact, Pc, for a cycloidal gearing is a smooth, piecewise curve composed of two circular arcs of radii rg and rp. These two arcs, gP and pP, comprise the path of contact, gPp (Figure 2.4a). An enlarged view of two teeth in contact for cycloidal gearing is shown in Figure 2.4b. For the driving pinion and driven gear, the tooth flanks are engaged in contact at the starting point, p, of the path of contact, Pc. As the pinion rotates, ωp, the point of contact of the tooth flanks travels along the Pc from point p to point g. Point g is the end point of contact of the tooth flanks. While traveling along the path of contact, Pc, at a certain configuration of the gears, the contact point passes the pitch point, P. At every instant of time, the pinion tooth flank acts over the gear tooth flank along the common


33

perpendicular to the tooth flanks G and P, that is, in the direction tangential to the path of contact, Pc, at a current its point, i. As the path of contact, Pc, in cycloidal gearing is composed of two circular arcs, a straight line through point i tangential to Pc forms a different angle, ϕt.i, with the perpendicular to the centerline, ℄. Moreover, the location of the current pitch point, Pi, within the centerline, ℄, can be determined as the point of intersection of the centerline, ℄, by the line of action, LA, through the point of tangency, i. A straight line that is tangent at i to the path of contact, Pc, is referred to as the instant line of action. The instant line action is designated as LAinst. Because of the migration of the instant pitch point, Pi, back and forth along the center line, ℄, the current values of the pitch radii of the gear, Rgi , and the pinion, Rpi , differ from their nominal values (the inequalities Rgi ≠ Rg and Rpi ≠ Rp are observed). Under the uniform rotation of the driving pinion (when ωp = const) and constant center distance, C, the change to the pitch radii Rgi and Rpi of the gear and the pinion causes variation in the rotation, ωg, of the driven gear. Ultimately, the rotation, ωg, of the gear depends on the angle, φp, through which the pinion turns about its axis at a time, t, that is, a certain functionality ωg = ωg(φp) is observed for cycloidal gearing. Here, φp = ωp t. Therefore, cycloidal gearing is not capable of transmitting a rotation smoothly. Commonly, cycloid gears feature spur teeth. The conclusion just made states that spur gears which have noninvolute tooth profiles are not capable of transmitting a rotation smoothly. The geometry of helical gears with noninvolute tooth profiles is more complex than that of spur gears. Therefore, the capability of noninvolute gearing to transmit rotations smoothly also needs to be verified. Consider a helical gear pair composed of two gears that have teeth shaped in the form of smooth regular curves. The path of contact in the gear pair is also a planar smooth, regular curve. An example of the path of contact, Pc, of this gear pair is illustrated in Figure 2.5. As shown in Figure 2.5, the gear pair is composed of a helical gear and a helical pinion. Both members feature noninvolute tooth profiles. The gear and the pinion rotate about their axes of rotation, Og and Op, with angular velocities, ωg and ωp, respectively. The axes, Og and Op, are at a certain center distance, C, from each another. The location of the pitch point, P, is determined by the nominal value of the pitch radius of the gear, rg, and by the nominal value of the pitch radius of the pinion, rp. The transverse pressure angle at the pitch point, P, is denoted by ϕt. Let us assume that a rotation from the driving shaft to the driven shaft can be transmitted by means of the gear pair. It can be assumed then that both the gear and pinion are sliced by transverse planes perpendicular to the axes of rotation, Og and Op. The number of the slices, n, is reasonably large. Slices are numbered from 1 to n. Let us pick an arbitrary slice number, i (where i is an integer number within the interval 1 < i < n). For the ith slice, a corresponding point, i, within the path of contact, Pc, is constructed. The point of

34


Driving pinion

Driven gear rg

rpi–1

Pc

rgi+1

(i–1)

ωp

i (i+1) Op

rp

ωg

rgi

P i–1

P

P i–1 Pi

rpi+1 rpi

L

C

i–1 LAinst i LAinst i+1 LAinst

Og rgi–1

ϕt

C FIGURE 2.5 Schematic diagram of meshing of a helical noninvolute gear pair. i intersection of the centerline, ℄, by the instant line of action, LAinst , through i i point i is the instant pitch point, P . Two circles of the radii rg and rpi through point Pi are the pitch circles for the ith slice. Similarly, instant pitch points, Pi−1 and Pi+1, are constructed for the point (i − 1) preceding point i and point (i + 1) following point i. The corresponding instant i −1 i +1 lines of action, LAinst and LAinst , as well as the radii of instant pitch circles r gi−1 , r pi−1, and r gi+1, r pi+1, are shown in Figure 2.5. Instant transverse pressure angles ϕt.(i−1) , ϕt.i, and ϕ t.(i+1) are not shown in Figure 2.5 due to lack of space. If the pitch points for different slices of a gear pair are not coincident with one another, then the slices should rotate with different instant rotational speeds, which is physically impossible. A conclusion can be drawn from the analysis given in Figure 2.5:

Conclusion 2.1 Geometrically accurate helical gear pairs that have noninvolute tooth profiles and nonzero transverse contact ratios (mp > 0) are not feasible physically; the instant pitch point subdivides the center distance reciprocal to the instant angular velocities of the gear and the pinion. One can imagine a pinion of a noninvolute parallel-axis gearing being sliced into numerous slices by planes perpendicular to the axis of rotation of the pinion. If the tooth profiles are not involute, then each slice rotates


35

separately and independently from one another. However, the pinion rotates as a rigid body, and it cannot rotate with different angular velocities simultaneously. Therefore, tooth flanks in noninvolute parallel-axis gearing do not contact each other along a line; they contact each other at a distinct point instead. As the inequality of base pitches of the gear and the pinion to operating the base pitch of the gear pair is observed, the transmission of a uniform rotation from the driving shaft to the driven shaft by means of noninvolute parallel-axis gearing is impossible at all. This conclusion is in agreement with the main theorem of conjugate gear tooth surfaces (the theorem was formulated by R. Willis [44,45]), which can be expressed as follows (an alternative formulation of Willis’ theorem). Willis’ Theorem 2.2 To transmit a uniform rotary motion from a driving shaft to a driven shaft by means of gear teeth, perpendiculars to the tooth flanks of the interacting teeth at all points of their contact must pass through a stationary point within the centerline of the two shafts. When the condition of conjugacy is met, the condition of contact of tooth flanks G and P of the gear and the pinion is always satisfied. This means that the condition of conjugacy is more robust than the condition of contact of tooth flanks G and P. Fulfillment of the condition of conjugacy is necessary but not sufficient to transmit smoothly a rotation from the driving shaft to the driven shaft.

2.3 Equality of the Base Pitches The third fundamental condition to be discussed in this section is the condition that requires the equality of the base pitches, that is, the base pitches of the driving, φb.p, and the driven, φb.g, gears must be equal to the operating base pitch, φb.op, of the gear pair* (that is, the identities φb.g ≡ φb.op and φb.p ≡ φb. op or simply φb.g ≡ φb.p ≡ φb.op need to be satisfied). The condition that requires * It is known that in cases of Pa -gearings, the base pitch of the gear and the pinion must be equal to one another. However, nothing is said so far about the operating base pitch in ideal Pa -gearings (the concept of the operating base pitch of an ideal Pa -gear pair is introduced by Dr. S.P. Radzevich in [38]). Further, neither the concept of the base pitch of the gear and the pinion in the cases of Ia-gearings, as well as in the cases of Ca-gearings is discussed so far in the public domain. Moreover, the concept of the operating base pitch in the cases of Ia - and Ca-gearings is not discussed so far in the public domain. Again, the concept of the operating base pitch of Iaand Ca-gearings is introduced for the first time by Dr. S.P. Radzevich in [38].

36


the equality of the base pitches of the gear and the pinion to the operating base pitch of the gear pair is the third fundamental condition which all Pa -gearings (as well as Ia -, and Ca-gearing) must obey. To the best possible extent, this concept is discussed in [38]. It is convenient to begin the consideration of the base pitches with the beltand-pulley model of parallel-axis gearing. Shown in Figure 2.6 is a schematic of the belt-and-pulley model. The axis of rotation of the gear, Og, and that of the pinion, Op, are parallel to each other and are at a center distance, C, from one another. The base cylinder of a diameter, db.g, is associated with the gear. Similarly, the base cylinder of a diameter, db.p, is associated with the pinion. The plane of action, PA, in a parallel-axis gear pair is tangent to the base cylinders from the opposite sides. The gear and the pinion rotate, ωg and ωp, about their axes of rotation. The magnitudes ωg and ωp of the rotations ωg and ωp are synchronized with one another reciprocal to the tooth count, Ng and Np, of the gear and the pinion, that is, the ratio ωg Np = Ng ωp

(2.2)

is valid in ideal (geometrically accurate) Pa -gearing. ωp

db.p

i

LCdes Vlc

Op

i+1

LCdes

Plane of action pb.op C

db.g ωg Og

FIGURE 2.6 Definition of the operating base pitch, pb.op, in ideal (geometrically accurate) parallel-axis gearing.


37

When the gears rotate, the plane of action may be considered as a zero thickness film that is unwrapping from the base cylinder of the gear and wrapping on the base cylinder of the pinion. In such a motion, the plane of action travels with a linear velocity, Vlc. The magnitude of the linear velocity, Vlc, is timed with the rotations ωgand ωp so as to ensure rolling with no slippage of the plane of action, PA, over the base cylinders of the gear and the pinion. i A desirable line of contact, LCdes , of the ith pair of teeth of the gear and the pinion is drawn within the plane of action. As an example, the desirable line i of contact, LCdes , is shown for a case of helical involute gearing. In the case under consideration, this straight line segment forms the base helix angle, ψb, with the axes Og and Op (the base helix angle, ψb, is not shown in Figure 2.6). i+1 The desirable line of contact, LCdes , of the adjacent (i + 1)th pair of teeth of i i +1 i the gear and the pinion is parallel to the straight line, LCdes , (i.e., LCdes LCdes ), and is also located within the plane of action, PA. Measured in a common transverse cross-section of the gear pair, the disi+1 i tance, pb.op, between two adjacent desirable lines of contact, LCdes and LCdes , is * referred to as the operating base pitch of the gear pair. It is important to notice here that the operating base pitch, pb.op, of the gear pair is constructed before the interacting tooth flanks G and P of the gear and the mating pinion are generated. Use of such an approach gives an opportunity to design a gear with the correct geometry of the tooth flanks. For this purpose, the desirable line of contact, LCdes, between the tooth flanks G and P is used to generate the gear and the pinion tooth flanks G and P. When the gears rotate, the desirable line of contact, LCdes, travels together with the plane of action, PA, that is, the straight line segment LCdes is rigidly connected with the PA. When the plane of action rolls over the base pitch of the gear, the gear tooth flank G is generated as a family of consecutive positions of the desirable line of contact, LCdes, in its motion in relation to a reference system associated with the gear (Figure 2.7). Similarly, when the plane of action rolls over the base pitch of the pinion, the pinion tooth flank P is generated as a family of consecutive positions of the desirable line of contact, LCdes, in its motion in relation to a reference system associated with the gear as shown in Figure 2.7. Under such a scenario, the actual line of contact, LC, between the tooth flanks G and P is congruent to the desirable line of contact, LCdes, of these surfaces (LC ≡ LCdes). This enables the equality of the base pitches of the gear and the pinion to the operating base pitch of the gear pair (that is, pb.g ≡ pb.op and pb.p ≡ pb.op, or simply pb.g ≡ pb.p ≡ pb.op) as shown in Figure 2.8.

* It is also important to stress here that the operating base pitch of a gear pair, pb.op, is measured in linear units only in cases of ideal (geometrically accurate) parallel-axis gear pairs. In cases of intersected-axis and crossed-axis ideal (geometrically accurate) gearings, the operating base pitch of a gear pair, φb.op, is measured in angular units. Moreover, if the axis misalignment is taken into account, then the operating base pitch of a gear pair, φb.op, is measured in angular units in all cases, that is, in cases of Pa -, Ia -, as well as Ca-gearings.

38


db.p ωp Op Vlc

ψb a ψb

b

Plane of action P

LCdes

ωg Og

db.g

FIGURE 2.7 Generation of screw involute surfaces G and P of the tooth flanks of a pair of helical gears.

In cases of Pa-gearing, the identities pb.g ≡ pb.op and pb.p ≡ pb.op (or simply pb.g ≡ pb.p ≡ pb.op) are met only for geometrically accurate (ideal) involute gearing. For any and all types of noninvolute gearings, the identity cannot be satisfied, and, moreover, the base pitch of the gear and the pinion cannot be constructed, while the operating base pitch of the gear pair can be easily determined. pb.g pb.g

i–1

i+1 i

db.g

Og

FIGURE 2.8 Base pitch of a gear, pb.g, (of a pinion, pb.p) in an ideal (geometrically accurate) involute gear pair.


39

When base pitches of the gear and the mating pinion are equal to the operating base pitch of the gear pair, the condition of conjugacy as well as that of the contact of the tooth flanks G and P of the gear and the pinion is always satisfied. This means that Conclusion 2.2 The condition of conjugacy is the most robust condition compared to the condition of contact of tooth flanks G and P, and the condition of conjugacy of the tooth flanks G and P of the gear and the mating pinion. Generally speaking, the condition that requires the equality of the base pitches, that is, pb.g ≡ pb.op and pb.p ≡ pb.op (or simply pb.g ≡ pb.p ≡ pb.op), is the most general one. When this condition is met, no other conditions of gearing need to be verified.

2.4 Contact Ratio in a Gear Pair To transmit a continuous rotation from a driving shaft to a driven shaft, at least one pair of teeth must be engaged in mesh at every instant of time. The transmission of the rotation is interrupted when this condition is violated. The number of pairs of teeth simultaneously engaged in mesh is specified by the contact ratio of the gear pair. Three different types of contact ratio are distinguished, that is, transverse (or profile) contact ratio, mp; face contact ratio, mF; and total contact ratio, mt. The following expression is valid for the contact ratios:

mt = mp + mF

(2.3)

To transmit a continuous rotation from a driving shaft to a driven shaft, the following inequality needs to be observed:

mt ≥ 1

(2.4)

in addition to the three fundamental conditions just considered. In order to specify the transverse contact ratio, mp, refer to Figure 2.9 where an ideal (geometrically accurate) parallel-axis gear pair is schematically shown. The gear and the pinion rotate about their axes Og and Op. The axes are at a center distance C from one another. The gears rotate with the angular velocities, ωg and ωp, respectively, that are synchronized with each other. The transverse pressure angle in the gear pair is φt. The plane of action, PA, is tangent to the base cylinders of diameters db.g and db.p of the gear and the pinion.

40


The plane of action intersects the plane through the axes Og and Op along the pitch line P. The effective width of the plane of action is Feff. The outer diameters do.g and do.p of the gear and the pinion intersect the plane of action, PA, along two straight lines. These two lines form an effective portion of the plane of action in the form of a rectangle. The length of the effective portion of the plane of action is the so-called the length of action Zpa. Elementary analysis of the schematic depicted in Figure 2.9 reveals how the length of action, Zpa, can be expressed in terms of the design parameters of the gear pair. By definition, the face contact ratio, mp, in a parallel-axis gear pair is equal mp =

Zpa pb.op

(2.5)

In spur involute gearing, the equality mt = mp is always observed.

C db.p

do.g

Op

L

C

ωp

P

PA db.g

do.p Og

ϕt ωp

Vlc

PA

LCi

LCi+1

Feff

pb.op

FIGURE 2.9 Definition of the transverse contact ratio, mp, in the ideal (geometrically accurate) Pa -gearing.

41


px LCi+1 Zpa

LCi

pb.op

ψb PA

Feff

FIGURE 2.10 Definition of the face contact ratio, mF, in the ideal (geometrically accurate) Pa -gearing.

In the case of helical gearing, in addition to the transverse contact ratio, mp, the so-called face contact ratio, mF, needs to be taken into account when determining the total contact ratio of the gear pair. By definition, the face contact ratio can be calculated from the following formula (Figure 2.10): mF =

Feff px

(2.6)

where Feff is the effective face width of the gear pair and px is the axial pitch of the gear teeth. The total contact ratio of a gear pair, mt, is equal to the summa mt = mp + mF. In any and all gearings, the total contact ratio is always greater than 1 (i.e., the inequality mt ≥ 1 is always observed).

3 Conformal Gearing: Novikov Gearing In the previous section of the book, the following have been discussed: three fundamental conditions that all geometrically accurate (ideal) gearings need to satisfy; the consideration of the contact ratio in a gear pair (i.e., the total contact ratio mt must always be greater than 1, mt ≥ 1) that allows us to proceed with the analysis of the concept of conformal gearing (of Novikov gearing, or N-gearing for simplicity); and to then evolve the concept of conformal gearing to the concept of high-conformal gearing (further, Hc -gearing for simplicity). It should be noted from the very beginning that conformal gearing as well as Hc -gearing are the degenerate cases of involute gearing. Because of this, both conformal (Novikov) gearing and Hc -gearing meet the conditions one through three discussed in Chapter 2, as well as feature the total contact ratio mt ≥ 1.

3.1 Novikov Gearing: A Helical Involute Gearing with a Zero Transverse Contact Ratio Novikov gearing and, more generally, conformal gearing can be construed as a degenerate case of involute gearing when the involute tooth profile is truncated, and only one point of the involute tooth profile is left. This point is referred to as the involute tooth point (see Figure 3.7 later in the chapter). When Prof. M.L. Novikov carried out his research in the field of conformal gearing, he loosely assumed that to transmit a uniform, rotational motion, the gear teeth do not need to have special shapes, such as the involute of a circle. He meant that, if a gear is made helical then the helix itself can ensure uniform angular motion and tooth profiles can then be chosen with a view to minimizing contact stresses. This is incorrect: to transmit a rotation smoothly, the mating tooth profiles must be either involute or, in a degenerate case, they can feature the involute tooth point geometry. Novikov gearing is a type of helical gearing that has a zero transverse contact ratio. The equality of the base pitch of the gear and the pinion, to the operating base pitch of the gear pair is the principal feature of Novikov gearing that distinguishes it from helical noninvolute gearing of other types.

43

44


It is customary to associate Novikov gearing* with the patent Gear Pairs and Cam Mechanisms Having Point System of Meshing [23]. Evidence can be found out in the scientific literature revealing the unfamiliarity of the gear community around the world with this original publication [23] on Novikov gearing (see Appendix A for details). As early as 1955, before the invention application was filed, a doctoral thesis [18] on the subject had been defended by M.L. Novikov. The author’s familiarity with the practice of defending the doctoral thesis adopted in the former Soviet Union allows an assumption that the concept of Novikov gearing had been proposed in the late 1940s. After M.L. Novikov was granted with the patent [23], a monograph by him was published (Figure 3.1) [19]. The concept of Novikov gearing is discussed in detail in the two aforementioned valuable sources [18,19]. Unfortunately, none of them is quoted by gear experts in western countries or in the USA. This makes it possible to conclude that gear experts around the world are not familiar with these two valuable sources of information on Novikov gearing. 3.1.1 Essence of Novikov Gearing Novikov gearing was developed with the intent to increase the contact strength of the gear teeth. Gearing of this type features higher contact strength due to the favorable curvatures of the interacting tooth flanks. Under equivalent contact stress, similar dimensions, comparable values of the remaining design parameters, and greater circular forces are permitted by the proposed gearing. (The discussion in this section of the book mostly follows Dr. M.L. Novikov [23].) Another factor that contributes to the high load capacity of conformal gears is that they sustain a thicker film of lubricant, owing to the rapid rolling of the areas of contact along the helix, which provides a vigorous hydrodynamic action. The shape of the gear teeth designed to transmit power is traditionally based on the involute curve, and all gear tooth profiles in the past have been convex. However, if mating teeth are conformal, that is, one is convex and the other concave, stress for a given load can be reduced; alternatively, a heavier load can be carried for the same amount of stress. This point is made clear by the photographs of photoelastic models shown in Figure 3.2. Novikov gearing [23] is developed for, but not limited to, parallel-axis gear trains. However, gear pairs featuring intersected-axis and gear pairs that have crossed axes of the rotations of gears can be designed on the basis of the concept proposed by M.L. Novikov. External and internal gearing of the proposed system of meshing is possible. The tooth ratio of the proposed gearing can be of either constant or a variable value, as well as can be time dependent. * The first pair of Novikov gearing made of aluminum alloy (a preprototype) was cut on April 25, 1954, by a disk-type milling cutter. For testing, 15 gear pairs were machined in the summer of 1954 by the disk-type milling cutter. Hobs for cutting Novikov gears were proposed later on by Prof. V.N. Kudryavtsev (as early as 1956).

Conformal Gearing

45

FIGURE 3.1 Title page of the Novikov’s (1958) monograph, Gearing with a Novel Type of Meshing. (Adapted from Novikov, M.L., Gearing with a Novel Type of Teeth Meshing, Zhukovskii Aviation Engineering Academy, Moscow, 1958, 186pp.)

Possible geometries of tooth profiles of the Novikov gearing are schematically shown in Figure 3.3. In this figure, a section of the tooth flank intersected by a plane perpendicular to the instant axis of relative rotation is shown. The axis passes through the current point of contact of the tooth flanks. In Figure 3.3, the point of intersection of the planar section by the axis of instant relative rotation is denoted by P. The points of intersection of the

46


FIGURE 3.2 Comparison of distribution of contact stress: (a) an equivalent involute gearing and (b) Novikov gearing.

planar section by the axes of the gear and the pinion are designated as O1 and O2. A point, A, is the point of meshing (in its current location). The line of action is denoted by PA. Ultimately, ДAД (The notations in Figure 3.3 are mostly due to Dr. M.L. Novikov) is the circle* centering at the pitch point, P. The circle corresponds with the limiting case of the tooth profiles (in the case that the profiles are aligned to each other). Multiple curves denoted by BAB illustrate examples of possible tooth profiles of one of the mating gears. All the curves denoted by BAB are arbitrary smooth regular curves, which are located inside the limiting circular arc, ДAД (i.e., all the arcs BAB are situated within the bodily side of the limiting tooth flank of one of the gears). All the tooth profiles, BAB, feature a high degree of conformity to the limiting circular arc, ДAД. * The circle of a radius rN centered at the pitch point, P, was introduced in recent years by Prof. S.P. Radzevich. He proposed to refer to this circle as to the boundary Novikov circle or just as N-circle in honor of Prof. M.L. Novikov, the inventor of Novikov gearing. The concept of the boundary N-circle was not known to M.L. Novikov.

47

Conformal Gearing

ϕt The boundary Novikov circle

O1

P

rN

L

C

B

B

O2

C

A

C

LAinst C FIGURE 3.3 Concept of Novikov gearing. (After Novikov, M.L. USSR, Gear Pairs and Cam Mechanisms Having Point System of Meshing. Patent 109,113, National Classification 47h, 6. Filed: April 19, 1956, published in Bulletin of Inventions No. 10, 1957.; the boundary Novikov circle of a radius, rN, is introduced later on by Dr. S.P. Radzevich.)

Multiple curves denoted by CAC illustrate examples of possible tooth profiles of the second mating gears. All the curves denoted by CAC are arbitrary smooth regular curves, which are located outside the limiting circular arc, ДAД (i.e., all the arcs denoted by CAC are located within the bodily side of the limiting tooth flank of the second of two gears). All the curves CAC feature a high degree of conformity to the circular arc, ДAД. The location and orientation of either the straight path of contact or a smooth curved path of contact is specified in a space in which the location and orientation of the axes of rotations of the gear and the pinion are given. The path of contact is located reasonably close to the axis of instant relative rotation of the gears. Either constant or time-dependent (smoothly varying in time) speed of motion of the point of contact along the path of contact is assigned. A coordinate system is associated with the gear, and a corresponding coordinate system is associated with the pinion. In each of the coordinate systems, the moving contact point traces contact lines. One of the lines is associated with the gear and another is associated with the pinion. Certain smooth regular surfaces through these lines can be used as tooth flanks of the gear and the pinion. The following requirements should be fulfilled so that surfaces can be used as the tooth flanks of Novikov gearing: • At every location of the point of contact, the tooth flanks should have a common perpendicular and, thus, the requirements of conjugacy of meshing should be satisfied.

48


• The curvatures of the tooth profiles should correspond to each other. • No tooth flank interference is allowed within the working portions of the surfaces. If two surfaces are generated by the moving curves BAB and CAC, then the aforementioned requirements are fulfilled and the surfaces can be employed as tooth flanks for Novikov gearing. Consider a plane through the current contact point, which is perpendicular to the instant axis of relative rotation. Construct two circular arcs centered at points within the straight line through the pitch point and the contact point. The arc centers are located within the line of action and close to the pitch point. The constructed circular arcs can be considered as examples of the tooth profiles of the gear and the pinion. The tooth flanks are generated as the loci of tooth profiles constructed for all possible locations of the contact point. The working portion of one of two tooth flanks is convex, whereas that of the other tooth flank is concave (in the direction toward the axis of instant relative rotation). In a particular case, the radii of the tooth profiles can be of the same magnitude and equal to the distance from the contact point to the axis of instant relative rotation. The centers of both profiles in this particular case are situated at the axis of instant relative rotation. In such a scenario, the point contact is substituted by a special type of “line” contact. This requires the center distance to be extremely accurate and independent of the operation conditions, which is impractical. Point contact of the tooth flanks is preferred when designing tooth profiles. A small difference between the radii of curvature of the tooth profiles is desirable. It should be borne in mind that during the run-in period of time, point contact of gear teeth transforms to the aforementioned line contact of tooth profiles. However, the theoretical point contact of tooth flanks is retained. Generally speaking, it is not mandatory that tooth profiles have circular-arc shapes. Tooth profiles of another geometries (those always passing through the contact point) should be situated (for one gear) within the interior of the aforementioned circular-arc profile, ДAД, which centers at a point within the axis of instant relative rotation as shown in Figure 3.3. For another (mating) gear, the tooth profile should be located outside the circular arc, ДAД. Under all circumstances, the centers of curvature of both convex and concave tooth profile are located within the instant line of action, LAinst. The law of motion of the contact point (i.e., speed of the point and its trajectory) should be chosen so as to minimize losses caused by friction and wear. Friction and wear losses are proportional to the relative sliding velocity in the gear mesh. Therefore, it is desirable to reduce the sliding velocity as much as possible. For this purpose, the path of contact should not be too far from the axis of instant relative rotation. On the other hand, it is also not desirable that the path of contact be too close to the axis of instant relative rotation as this reduces the contact strength of the gear tooth flanks. In addition, it is

Conformal Gearing

49

recommended to ensure favorable angles between the common perpendicular (along which the tooth flanks of one of the gears act against the tooth flanks of the other gear) and the axes of rotations of the gears. The opposite sides of the tooth profiles are designed in a manner similar to that just discussed. Tooth thicknesses and tooth pitch are assigned so as to ensure the required bending strength of the teeth. The face width of the gear or length of the gear teeth should correlate with their pitch so as to ensure the required value of the face contact ratio, mF. Gear pairs can feature either one point of contact (when working portions of the tooth flank contact each other at just one point, excluding the phases of the teeth re-engagement) or multiple contact points (when tooth flanks contact each other at several points simultaneously). For parallel-axis gear pairs, it is preferable to use a straight line as the path of contact. The straight line is parallel to the axes of rotations of the gear and the pinion. The speed of the contact point as it moves along the straight path of contact can be constant. In this particular case, the radii of curvature of the tooth profiles in all sections of the tooth flank by planes are equal. The tooth flanks in this case are regular screw surfaces. Gears featuring tooth flanks of such geometry are easier to manufacture, and they can be cut on machine tools available in the market. An example of parallel-axis gearing with a limiting geometry of tooth profiles is illustrated in Figure 3.3. The point contact of the tooth flanks in this particular case is transformed to the line contact. The curved contact line is situated across the tooth profile. When axial thrust in the gear pair is strongly undesirable, herringbone gears can be used instead. A more detailed explanation of the early concept of Novikov gearing can be found in the book by Krasnoschokov et al. [14]. Tooth profiles contact each other only at an instant of time when the tooth profiles of both the gear, G, and the pinion, P, intersect the line of action, LAinst, in a common transverse section. At all instants of time before and after this instant, the tooth profiles, G and P, do not interact with one another* (Figure 3.4). To ensure continuous contact between the tooth flanks, G and P, the teeth of the gear and the pinion are of helical shape. The path of contact can be located either before or beyond the pitch point, P. Novikov gears of the first type are commonly referred to as “N by-gears,” whereas those of the second type are referred to as “N bf-gears.” The proposed method for generation of the tooth surfaces by Novikov is based on the trajectories of the points of contact between the tooth flanks. Novikov used to refer to these trajectories as to the contact lines. The method of generation of tooth flanks proposed by Novikov can be referred to as the contact lines method. The point of contact, K, of the tooth flanks, G and P, is often referred to as the point of culmination.

* Owing to this, M.J. French proposed [11] to refer to this point as culmination.

50


ωp OP RP

C L Dr

LAinst

rg rp = l

Or P K Rg

ϕt

ωg

P

K

FIGURE 3.4 Interaction between the tooth flanks of the gear, G, and the pinion, P, in a Novikov gear pair.

Both the pinion and gear are helical. The helices are of opposite hands, namely, one of them is right-handed, and the other is left handed. No spur Novikov gearing is feasible in nature. Because of the gears are helical and of opposite hands, the point of contact travels axially along the gears while remaining at the same radial position on both gear and pinion teeth, G and P. It is therefore fundamental to the operation of the gears that contact occurs nominally at a point and the point of contact travels axially across the full face width of the gears during a rotation. It should be stated as a condition of operation of Novikov gearing that for a given profile tooth surfaces should not interfere before or after culmination when rotated at angular speeds that are in the gear ratio. The transverses contact ratio, mp, of a Novikov gear pair is zero, (mp ≡ 0). The face contact ratio, mF, of the gear pair is equal to the total contact ratio (mF = mt), and is always greater than 1 (mF > 1). In every transverse section of the gear pair, the contact point, K, is motionless. This is because the length Lpc of the path of contact, Pc, in every transverse section of the gear pair is zero (i.e., Lpc ≡ 0). For parallel axes configuration, all paths of contact, Pc, comprise a pseudo path of contact, Ppc. The pseudo path of contact, Ppc, is a straight line through the contact point, K. The pseudo path of contact, Ppc, is parallel to the axes Og and Op as illustrated in Figure 3.5. When rotation is transmitted from a driving shaft to a driven shaft, the contact point, K, travels along the pseudo path of contact, Ppc (and it does not

51

Conformal Gearing

Og

ωg ωg

Pln

Ppc

LAinst K

ϕt

P

C

ωpl

ωp

ωp

L

C

Op

FIGURE 3.5 Configuration of the pseudo path of contact, Ppc, for a Novikov gear pair.

travel within the transverse cross-section of the gear pair), that is, parallel to the axes of rotation, Og and Op, of the gear and the pinion accordingly. This is because the transverse contact ratio is zero (mp ≡ 0) and the face contact ration is greater than 1 (mF > 1), as mentioned earlier in this section.* * Many gear engineers around the world loosely refer to Novikov gearing as Wildhaber–Novikov gearing or simply W–N gearing, which is incorrect. From Chapter 1, Figure 1.7, “Helical Gearing” in the patent by E. Wildhaber [22] should be referred to as Wildhaber gearing. “Gearing with Point System of Meshing” by M.L. Novikov [18,19,23] must be referred to as Novikov gearing. Finally, the terms “Wildhaber–Novikov gearing” or “W–N gearing” must be recognized as meaningless terms; and both terms need to be eliminated from the engineering literature. The comparison of Wildhaber gearing and Novikov gearing (see Appendix A for details) makes it possible to realize that the conclusion made by N. Chironis, “Novikov-type gears are similar to those developed by E. Wildhaber in the early 1920s (see p. 135,” Reference 6 is incorrect; further, that Wildhaber’s statement “all the characteristics of the Novikov gearing are anticipated by my patent. My gearing never had a real test here, although a pair of gears was made in the 1920s,” as quoted in the work by Chironis [6], is also incorrect. With the great respect to personality of Ernest Wildhaber, as well as to most of his contributions, let us assume that Wildhaber had correctly understood the benefits of his invention, “Helical Gearing” [22]. Then, being a smart gear expert, why did he not promote the invention in a practical application? Did he have no opportunities to do so? Definitely, he had!! According to the author’s personal opinion, the gear pair that was manufactured

52


ωp

do.p do.g

Op Pg

db.p

Vpa

Pp

P

Np

Ng

L

C

ϕt

db.g Og

Zpa ≡ 0

rN

pb.n

ωg

K1 c

Op

Ppc

VK

Feff

c

Og

K2 Pln ψK

pb.op

FIGURE 3.6 Base cylinders, db.g and db.p, normal base pitch, pb.n, and operating base pitch, Pb.op, in parallelaxis Novikov gearing.

3.1.2 Fundamental Design Parameters of Conformal Gearing The base diameters of the gear and the pinion, their base pitches, and the operating base pitch of the gear pair are referred to as the fundamental design parameters of conformal gearing. Because conformal gearing features a zero transverse contact ratio (mp ≡ 0), it is possible to interpret the kinematics of this gearing in the same way as those for parallel-axis involute gearing that have zero width of the field of action (Zpa = 0), schematically shown in Figure 3.6.

(as Wildhaber mentioned) never worked. The reason for this is clear to us now. Where had Wildhaber been for about 30 years, from 1926 till 1956? Why did he wait for the Novikov invention? It is likely that the unfamiliarity with the original publications by Novikov [18,19,23] of gear engineers in western Europe and the USA is the main reason for the incorrect reference to Novikov gearing. Much evidence to this end can be found in the literature on Novikov gearing; for example, Dyson et al. [8] referred to S.U. Pat. No. 109,750 as the patent on Novikov gearing. In reality, S.U. Pat. No. 109,750 was issued for a water sprayer, and not for Novikov gearing. Interested readers may wish to investigate this matter on their own.

53

Conformal Gearing

For a given center distance, C, and tooth ratio, u, the pitch diameter of the gear, dg, and the pinion, dp, are calculated by the following conventional formulas [38]: C u+1

(3.1)

C u+1

(3.2)

dg = 2u

dp = 2

Then, equations

db. g = dg cos φt db. p = dp cos φt

(3.3)

(3.4)

are used for the calculation of base diameters db.g and db.p of the gear and the pinion, respectively. In Equations 3.3 and 3.4, base diameters, db.g and db.p, are expressed in terms of the transverse pressure angle, φt. In parallel-axis gearing, the pressure angle, φt, is identical to the pressure angle in the involute parallel-axis gearing. Because conformal gears feature zero width of the field of action (Zpa = 0), the length of the line of contact of their tooth flanks shrinks to zero. Although the length of the line of contact is zero, the direction of the line of contact remains the same. Within the plane of action, the line of contact forms a base helix angle, ψb, with the axis of instant rotation, Pln, of the gear and the pinion. The base pitch helix angle, ψb, can be calculated from the following formula:

ψ b = tan −1(tan ψ cos φt )

(3.5)

In Equation 3.5, the pitch helix angle is denoted by ψ. Base pitch, pb, in the case under consideration is given by

pb = px sin ψ b

(3.6)

where px is the axial pitch of the teeth in a conformal gearing. Finally, the operating base pitch, pb.op, in a conformal gearing can be calculated from the following formula:

pb.op = px tan ψ b

(3.7)

54


The similarities between Equations 3.1 through 3.7 and the corresponding set of equations for parallel-axis involute gearing reveal that both gear systems originate from a common concept.

3.2 Transition from Involute Gearing to Conformal (Novikov) Parallel-Axis Gearing In parallel-axis geometrically accurate (ideal or perfect) involute gearing, the tooth flank of the gear, G, and the tooth flank of the pinion, P, make contact along the line of contact, LC. The line of contact is a planar curve of a reasonable geometry that is entirely located within the plane of action, PA. This could be either a straight line parallel to the axes of rotation of the gear and the pinion as in spur involute gearing, or a straight line at the base helix angle ψb to the axes Og and Op as in helical involute gearing. It can also be a circular arc or an arc of a spiral curve, or a curve of other reasonable geometry. The teeth flanks, G and P, of the gear and the pinion interact with each other only within the active portion of the plane of action. For the illustrative purposes, an example of the active portion of the plane of action is depicted in Figure 3.7. Referred to Figure 3.7a, NgNp is the total length of the plane of action. In reality, the active portion of the plane of action, PA, is of a smaller length Zpa (Figure 3.7b). This is the length of the straight-line segment PgPp. Points Pg and Pp are the points of intersection of the straight line NgNp by the outer circle of the pinion of a radius ro.p, and by the outer circle of the gear of a radius ro.g accordingly. Ultimately, the active portion of the plane of action is a rectangle of the size Zpa × Fpa. In involute helical Pa -gearing, the desirable line of contact, LC, between the tooth flank of the gear G and the pinion P (remember that the tooth flanks, G and P, are not constructed yet) is a straight-line segment that forms a base helix angle, ψb, with the axis of instant rotation, Pln (the desirable lines of contact of other geometries are not discussed here). As the contact ratio in a gear pair must be greater than 1, then there must be at least two portions of the desirable line of contact, LC, within the active portion of the plane of action. This makes it possible a definition for the axial pitch, px, of a tooth flank in a helical Pa -gearing: Definition 3.1 The axial pitch, px, in a parallel-axis gear pair is equal to the distance between points of intersection of two adjacent desirable lines of contact, LC, by a straight line parallel to the axis of instant rotation Pln.

55

Conformal Gearing

(a) ωp

rf.p

Op rb.p Np

C

rp

ϕt

ro.p Pg

K

Pp

P

PA

Ng rb.g

rf.g

rg

ωg

ro.g

(b)

pt PA Zpa

C sinϕt

Og b

LC

Feff

px

a ψ

ψb

b d

Zpa (c)

PA

Feff

LC

c

ψb

b K d

Zpa VK

(d) ψb

l

Feff

K Ppc

a

c

a

FIGURE 3.7 Elements of a parallel-axis gear pair featuring zero transverse contact ratio (mp = 0).

The following expression can be used for the calculation of the axial pitch, px:

px =

pt tan ψ b

(3.8)

Equation 3.8 is valid for all types of Pa -gearing capable of transmitting a rotation smoothly. When the base cylinders of diameters db.g and db.p rotate, the desirable line of contact, LC, travels (together with the plane of action, PA) in relation to two reference systems. One of the reference systems is associated with the

56


gear, and another one is associated with the pinion. In such a motion, the tooth flank of the gear, G (as well as the tooth flank of the pinion, P) can be construed as a family of consecutive positions of the desirable line of contact in the corresponding reference system. In the example illustrated in Figure 3.7b, the active portion ab of the involute tooth profile is specified by the radii of the outer cylinders of the gear and of the pinion, ro.g and ro.p, respectively. Point a corresponds to the start-ofactive-profile point (SAP-point), while point b corresponds to the end-of-activeprofile point (EAP-point). For both members of a gear pair, that is, for the gear and the pinion, the radius rEAP of the EAP-circle can be smaller than the outer radius of the gear ro.g (or than the outer radius of the pinion ro.p), for example, because of chamfering. In such a scenario (i.e., illustrated in Figure 3.7c; the radii rEAP for the gear and the pinion are not labeled there), the active portion of the plane of action becomes narrower. The SAP-point c, and the EAP-point d become closer to one another: the active portion cd of the involute tooth profile is shorter than that, ab, in the case illustrated in Figure 3.7b. This gives a certain freedom when selecting the geometry of nonactive portions ac and bd of the tooth profile. As these portions of the tooth profile do not interact with one another, the geometry of the segments ac and bd is not restricted by the conditions of meshing of the tooth profiles (which is the must for the active portion cd). All three conditions for geometrically accurate (ideal) gearing along with the requirement mt ≥ 1 can be ignored for the nonactive portions of the tooth profile of the gear, G, and of the pinion, of the gear, P. In the extreme case, the EAP-circles of the gear and the pinion can pass through a certain point K within the straight-line segment PgPp. Because of this, the length ZPA of the active portion of the plane of action becomes zero (ZPA = 0), and the active portion of the involute tooth profile shrinks to the point K (proper selection of the distance PK = l is considered below). The nonactive portions aK and bK of the tooth profile meet each other at the point K. These portions are not subjected to conditions of meshing of tooth profiles; thus this gives a certain freedom when selecting the geometry of nonactive portions aK and bK of the tooth profile. This particular case of Pa -gearing is illustrated in Figure 3.7d. As the width of the active portion of the plane of action is zero (ZPA = 0), and the involute tooth profile is shrunk to a point, the transverse contact ratio mp becomes zero. To meet the inequality mt ≥ 1, the following inequality must be satisfied:

mt = mp + mF = 0 + mF = mF > 0

(3.9)

The point system of Pa -gearing that is illustrated in Figure 3.7d gives much freedom when designing nonactive portions of tooth profiles of the gear and the pinion as the geometry of these portions is free of constraints imposed by conditions of meshing of two conjugate tooth profiles. In conformal

57

Conformal Gearing

gearing, this feature is used to increase the contact strength of the gear and the pinion. The involute gear/pinion tooth profile truncated to point K is a degenerate case of the involute tooth profile that conformal gearings feature. This degenerate tooth profile is referred to as the involute tooth point. The rest of the tooth profile is inactive and can be designed independently of conditions of interaction of tooth profiles. The above discussion reveals that no tooth flank modification is permissible in Novikov gearing. Neither profile modification nor crowning is allowed in Novikov gearing.

3.3 Kinematics of Parallel-Axis Gearing For a parallel-axis gearing, a vector diagram can be constructed. In the vector diagram (Figure 3.8), the rotation vector* of the gear ωg is along the axis of rotation Og of the gear. The magnitude ωg of the rotation vector ωg equals (a) rp

ωg

Op

ωp

Ap

Pln

–ωg

P C

rg

ωpl L

C

ωg

ωp

Ag

(b)

Og

db. p

ωp

ϕt

Op P

PA db.g

Og

ωg

FIGURE 3.8 Vector diagram of a parallel-axis gear pair (a), and a transverse cross-section (b). * It should be stressed here that a rotation in nature is not a vector at all. However, if special care is undertaken, then the rotations can be treated as vectors.

58


to ωg = |ωg|. The rotation vector of the mating pinion ωp is along the axis of rotation Op of the pinion. The magnitude ωp of the rotation vector ωp is ωp = |ωp|. The magnitudes, ωg and ωp, relate to one another as u=

ωp ωg

(3.10)

The axes, Og and Op, are apart from one another at a center distance C. The rotation vectors, ωg and ωp, form an angle Σ, that is

Σ = ∠(ω g ; ω p )

(3.11)

The angle, Σ, is referred to as crossed-axis angle. In a case of external Pa -gearing, the angle, Σ, is always equals Σ = 180° (and the angle, Σ, is always equals Σ = 0° in all cases of internal Pa -gearing). The principle of inversion of rotations can be implemented to a Pa -gearing. Let us assume that both the axes of the rotations, Og and Op, are rotated together with the rotation vector −ωg. Because the identity ωg + (−ωg) ≡ 0 is valid, the gear becomes stationary under the additional rotation, −ωg. The pinion is rotated with the rotation

ω pl = (ω p − ω g )

(3.12)

The vector of instant rotation, ωpl, of the pinion in relation to the gear is along the axis of instant rotation, that is, along the pitch line, Pln. Shown in Figure 3.8a is the vector diagram of the parallel-axis gear pair that is composed on the premises of the rotation vectors ωg, ωp, and ωpl. The axis Pln intersects the center-line ℄ at the pitch point P. The pitch point P is at a distance rg from the axis of rotation Og of the gear, and the pitch point P is at a distance rp from the axis of rotation Op of the pinion. The radii, rg and rp, are the pitch radii of the gear and the pinion, respectively. Summa of the radii rg and rp equals to the center distance, C

rg + rp = C

(3.13)

The radii rg and rp are signed values. They are positive in all cases of external gearing, and the radius rg is negative in a case of internal gearing. As the ratio u = ωp/ωg is valid in Pa -gearing, the ratio of the pitch radii is reciprocal to the ratio of the angular velocities in the gear pair, that is

rg ωp = rp ωg

(3.14)

The said allows proceeding to construction of the plane of action, PA, of the gear pair as illustrated in Figure 3.8b.

59

Conformal Gearing

3.4 Plane of Action in Parallel-Axis Gearing The plane of action, PA, in a parallel-axis gearing is a plane through the axis of instant rotation, Pln, as illustrated in Figure 3.8b. This plane forms a transverse pressure angle, ϕt, with the perpendicular to the plane through the axes of rotation, Og and Op, of the gear and the pinion, respectively. The base diameter of the gear, db.g, and that of the pinion, db.p, can be expressed in terms of the radii, rg and rp, of the pitch cylinders, and the transverse pressure angle, ϕt:

db. g = 2rg cos φt db. p = 2rp cos φt

(3.15)

(3.16)

In Pa-gearing, the plane of action is shaped in the form of a rectangle. In lengthwise direction, the rectangle is bounded by two straight lines of tangency of the plane of action with each of two base cylinders of diameters db.g and db.p. Width Feff of the rectangle equals to the length within which the face width of the gear, Fg, and the face width of the pinion, Fp, overlap each other. Once the base cylinders are determined, the transmission of a rotation from the driving member to the driven member of the gear pair can be interpreted with the help of the so-called belt-and-pulley analogy. Either of the Equations 3.15 and 3.16 can be used for the derivation of an expression for the calculation of the base pitch, pb, in a transverse cross-section of the gear pair: pb =

πdb. g πdb. p = Ng Np

(3.17)

Here, the tooth count of the gear and the pinion are designated as Ng and Np, respectively. Equation 3.17 is valid for all types of Pa -gearing capable of transmitting a rotation smoothly.

3.5 Boundary N-Circle in Conformal Gearing A performed analysis of the vector diagram for a Pa-gearing shown in Figure 3.8 clearly shows that a motion of the pinion in relation of the motionless gear is an instant rotation of the pinion about the pitch line Pln. Similarly, a motion of the gear in relation of the motionless pinion is an instant rotation of the gear about the pitch line Pln. It can therefore be drawn up that every point of contact, K, between the tooth flanks of the gear and the pinion, G and P,

60


ϕt LAinst

K

rg

ωp –l

ωg

L

C

Op

rN rp

P

+l

Og

Boundary N-circle K C FIGURE 3.9 A boundary N-circle in a conformal parallel-axis gear pair.

traces a circular arc that is centered at the pitch point, P. One of the circular arcs is associated with the gear, and another one is associated with the pinion. Based on this consideration, the concept of the so-called boundary Novikov circle was introduced by Dr. S.P. Radzevich [38]. The boundary Novikov circle is commonly referred to as the boundary N-circle. The procedure of constructing a boundary N-circle for a conformal gear pair is briefly outlined next. Consider two axes of rotations of the gear, Og, and the pinion, Op, in the design of a parallel-axis conformal gear pair, as schematically depicted in Figure 3.9. The axes of rotations, Og and Op, are at a certain center distance, C, from each other. The gear and the pinion are rotated about the axes Og and Op, and the rotations are labeled as ωg and ωp, respectively. The gear ratio in the conformal gear pair is equal to u = ωp/ωg. The center distance, C, is subdivided by point, P, into two segments, OgP and OpP. The ratio of the lengths of the straight line segments OgP and OpP is reciprocal to the gear ratio, u, of the conformal gear pair. If the straight line segments, OgP and OpP, are the pitch radii (OgP = rg and OpP = rp) of the conformal gear pair, then the equality rg/rp = u is observed. The point, P, is the pitch point of the conformal gear pair. The straight instant line of action,* LAinst, through the pitch point, P, is at the transverse pressure angle, ϕt, with respect to the perpendicular to the centerline, OgOp. Two points, both denoted by K, are within the straight line, LAinst, and are displaced at a certain distance, ±l, from the pitch point, P. The * The line of action, LA, in cases of conformal gearing (i.e., Novikov gearing) and high-conformal gearing (Hc -gearing) is referred to as the instant line of action, LAinst, because it travels in the axial direction together with the contact point, K, when the gears rotate. The projection of the instant line of action onto the transverse plane remains stationary as the transverse contact ratio in conformal gearing and Hc -gearing is zero.

Conformal Gearing

61

paths of contact are the two straight lines through the points K parallel to the axes, Og and Op, of the rotations of the gear and the pinion. This distance, that is, the displacement, l, of the paths of contact from the pitch point P, is one of the important geometrical parameters of conformal gearing. The strength of the gear teeth and the performance of the conformal gear pair strongly depend on the actual value of the displacement, l. The path of contact that is located beyond the pitch point, P (in the direction of rotation of the gears) features a positive displacement, that is, +l. A conformal gear mesh of this type is referred to as the Nby-mesh of conformal gear pair. The path of contact that is located before the pitch point (in the direction of rotation of the gears) features a negative displacement, that is, −l. A conformal gear mesh of this type is referred to as the Nbf-mesh of conformal gear pair. In order to avoid violation of the conditions of meshing, as well as to target wear reduction and reduction of friction losses, the paths of contact are displaced at a reasonably short distance from the axis of instant rotation, Pln. Let us imagine that the pinion is motionless; then, the contact point, K, traces a circle within the corresponding transverse section of the gear pair. The circle is centered at the pitch point, P. Similarly, the gear can be assumed stationary; then, the contact point, K, traces a circle within that same transverse section of the gear pair. This circle is also centered at the pitch point, P. It is clear from this consideration how the boundary N-circle of radius l can be constructed. A transverse section of a conformal gear pair is subdivided by a Novikov circle of radius rN = |l| into two areas.* The area within the interior of the boundary circle of a radius rN (including points within the boundary circle itself) represents the area of feasible shapes of the tooth profiles of one of the mating gears, and the area within the exterior of the boundary circle of the radius rN (including points within the circle itself) represents the area of feasible shapes of tooth profiles of the second mating gear. The boundary circle of a radius rN is referred to as a boundary Novikov-circle of a conformal gear pair or simply as N-circle. Definition 3.2 A boundary Novikov circle (or, a boundary N-circle, for simplicity) is a circle centered at the pitch point of a parallel-axis conformal gearing, the radius of which is equal to the distance of the point of contact of the tooth flanks of the gear and the pinion from the pitch point of the gear pair.

* Radius of the Novikov circle is designated as rN. The obsolete designation l was used by Dr. M.L. Novikov for the displacement of the contact point, K, from the pitch line, Pln. In reality, the equality rN = |l| is always observed.

62


It is the right point to stress here that the concept of the boundary N-circle is helpful for understanding the feasibility of conformal gearing that features locally line contact between the teeth flanks of the gear and the pinion. In an ideal case, when all the deviations are zero, the tooth flank of the gear, G, as well as the tooth flank of the mating pinion, P, can both be generated by that same arc of the boundary N-circle. In other words, an arc of the boundary N-circle can be used as the tooth profile of the gear, as well as the tooth profile of the pinion. In practice, a corresponding N-cylinder can be assigned to any and all parallel-axis conformal gear pair. The axis of rotation of the N-cylinder is aligned with the axis of instant rotation, Pln, of the gear and the pinion, G and P.

3.6 Possible Tooth Geometries in Conformal Gearing Designing of the mating tooth profiles for a conformal (Novikov) gear pair begins with the construction of the boundary N-circle. In Figure 3.10, the boundary N-circle of a radius rN is constructed for the pinion tooth profile (Figure 3.10a) and the mating gear tooth profile (Figure 3.10b) of a conformal gear pair. The displacement, l, is positive (l > 0) for the pinion addendum. The tooth profile* of the pinion addendum is a convex segment of a smooth regular curve, P ia (i = 1, 2,…), through the contact point, Ka. The radius of curvature, RP , of the addendum profile is equal to or less than the radius, rN, of the boundary N-circle (RP ≤ rN). The case of equality RP = rN is the limiting case, which is mostly of theoretical interest. Geometrically, the profile of the pinion addendum can be shaped in the form of a circular arc of radius rN. This case of the pinion addendum profile is the limiting one, which is of theoretical interest. It should be stressed here that none of the feasible profiles, P ia, of the pinion addendum intersects the boundary N-circle. The pinion addendum profile is entirely located within the interior of the boundary N-circle. Therefore, not any arc of a smooth regular curve can be used as tooth profile of the pinion addendum. The circular arc, an arc of an ellipse (at one of its apexes), and cycloidal profile containing its apex are examples of applicable curves for addendum tooth profiles. Spiral curves (involute of a circle, Archimedean * Recall that the active portion of the tooth profile in conformal gearing is limited to the so-called involute point. The rest of the gear and the pinion tooth profiles are not active, that is, they do not interact with each other. Because of this, there is a certain freedom for the gear designer in selecting the geometry of the inactive portions of the tooth flanks of the gear and the pinion.

63

Conformal Gearing

ϕt

(a)

ωp

b 2

b 1

LAinst

Kb

rg

b cr

–l Op

C

Og

+l

a 2

rp

L

a 1

P

rN

ωg

Ka

a cr

C

ϕt

(b)

a cr a 1

LAinst Ka

ωp

a 2

ωg

rg

–l

L

C

Op

P

rN rp

+l

b cr

b 2

Og b 1

Kb C

FIGURE 3.10 Examples of feasible tooth flank geometries of a conformal gear pair: feasible shapes of the tooth flank of (a) a pinion P and (b) the mating gear G.

spiral, logarithmic spiral, and so forth) are examples of smooth regular curves of which no arc can be used in designing a pinion tooth addendum. This is because the radius of curvature of a spiral curve (as well as of many others curves) changes uniformly when a point travels along the curve. This is schematically illustrated in Figure 3.11. In Figure 3.11a, an ellipse-arc, ab, is shown; the ellipse-arc is entirely located within the interior of the boundary N-circle. The ellipse-arc, ab, can be selected as the tooth addendum profile of a conformal gear pair. An ellipse-arc, cd (Figure 3.11a), is entirely located in the exterior of the boundary N-circle. The ellipse-arc, cd, can be selected as the tooth dedendum profile of a conformal gear pair. Finally, an ellipse-arc,

64


(a)

(b) 1

c

rN

a P

K b

e

rN

d

K P

f

2 Boundary N-circle

Boundary N-circle

3

FIGURE 3.11 Examples of ellipse-arc tooth profiles for conformal gears: (a) feasible and (b) not feasible.

ef (Figure 3.11b), intersects the boundary N-circle. The ellipse-arc, ef, cannot be used as the tooth profile in a conformal gear pair. The same is valid for all spiral curves. Therefore, at the point of tangency, K, spiral curves intersect the corresponding boundary N-circle, which is prohibited. Ultimately it should be clear that a variety of smooth regular curves can be used in the design of the tooth profile of a conformal gearing. The variety of curves is not limited only to circular arcs. The displacement, l, is a signed value. It is negative (l < 0) for the pinion dedendum (Figure 3.10a). The inactive tooth profile of the pinion dedendum is a concave segment of a smooth regular curve, P ib (i = 1, 2,…), through the contact point Kb. The radius of curvature, RP , of the dedendum profile is equal to or greater than the radius, rN, of the boundary N-circle (RP ≥ rN). The case of equality RP = rN is the limiting case, which is mostly of theoretical interest. Geometrically, the profile of the pinion addendum can be shaped in the form of a circular arc of the radius, rN. This case of the profile of the pinion addendum is the limiting one and is of theoretical importance. Constraints imposed on the geometry of the inactive tooth profile of the pinion dedendum are similar to those imposed on the geometry of the tooth profile of the pinion addendum. The dedendum profile is entirely located in the exterior of the boundary N-circle, shares a point with the N-circle (the contact point, Kb), and does not intersect the boundary N-circle. Not all types of smooth regular curves can be implemented in the design of the pinion tooth dedendum. An analysis that is similar to the one mentioned above regarding the pinion tooth profile can be performed for the gear tooth profile as well. a The analysis is illustrated in Figure 3.10b. The gear tooth addendum, G i , is entirely located within the interior of the boundary N-circle, whereas the gear tooth dedendum, G ib, is entirely located in the exterior of the boundary a N-circle. Both the profile of the gear tooth addendum, G i , and the profile b of the gear tooth dedendum, G i , share a common point with the boundary

65

Conformal Gearing

N-circle (point Ka in the first case and point Kb in the second). No intersection a of tooth profiles G i and G ib is permissible within tooth height of the gear and the pinion. The importance of the concept of the boundary N-circle for gear engineers is as follows: a boundary N-circle in conformal gear pairs is a constraint imposed on the geometry of the inactive tooth profiles of the gear and the pinion. The gear engineer is free to select an arc of any smooth regular curve to shape the tooth addendum profile if the arc is entirely located within the boundary N-circle. The gear engineer is also free to select an arc of any smooth regular curve to shape the tooth dedendum profile if the arc is entirely located outside the boundary N-circle.* The concept of the boundary N-circle has proved to be helpful in the theory of conformal gearing (Novikov gearing). As an example, Figure 3.12 illustrates a transverse cross-section of a conformal gear pair. In the case under consideration, the tooth flank of a gear, G, makes contact at a point, K, with the tooth flank of the mating pinion, P. The inactive circular-arc teeth profiles, G and P, are centered at the points og and op, respectively. The centers, og and op, are chosen so as to fulfill the necessary condition for the magnitudes ρg and ρp for the radii of curvature of the teeth profiles, G and P , at the point of tangency, K (ρg > ρp). However, as the circular arcs, G and P , intersect the boundary N-circle, the gearing of this particular type is not feasible. In the well-known helical gearing by Wildhaber [22] an unfavorable configuration of circular-arc teeth profiles is observed. This makes Wildhaber’s helical gearing not workable in nature. Another illustration of the infeasibility of helical gearing by Wildhaber (Figure 3.12) [22] is focused on an incorrect tooth profile orientation in relation to the boundary N-circle, as well as the instant line of action, LAinst. An

rN P

op

K ρp

og ρg

Boundary N-circle

FIGURE 3.12 Use of the concept of the boundary N-circle has proved to be helpful to distinguish whether a circular-arc profile is feasible for a conformal gearing or not.

* The concept of the boundary N-circle was introduced around 2008 by Dr. S.P. Radzevich; Dr. M.L. Novikov himself did not use the concept of the boundary circle.

66


Involute of a circle nbc tinv

pb

a

pb pb

Og

rb.g

FIGURE 3.13 An example of correct configuration of involute tooth profiles in relation to the base circle.

analogy of correct and incorrect tooth profile orientations in involute gearing can be used for the purpose of the illustration. Referring to Figure 3.13, consider a gear that has an involute tooth profile. Point a within the base circle is the starting point of the involute tooth profile. All the involutes are developed from the base circle of a radius rb.g. Hence, the unit normal vector, nbc, to the base circle at point a, and the unit tangent vector, tinv, to the involute curve at that same point a align with one another. As a result, the base pitch, pb, is of a constant value for any two adjacent tooth profiles and at any current point within an involute curve. In other words, the base pitch of the involute gear (Figure 3.13) is preserved as all the involutes are developed from a common base circle. Another example is shown in Figure 3.14. In this particular case, the gear teeth are shaped by means of the same involute curve as in the case shown in Figure 3.13. However, each involute curve is turned through an angle, ξ, about its corresponding starting point of the involute curve. All the shifted involutes are constructed from different base circles of a radius, rb.g, each. However, each base circle is centered at the point, Ogi , that does not coincide with the gear axis, Og. Hence, the unit normal vector, n bc, to the base circle of the true involute profile at point a and the unit tangent vector, t inv, to the shifted involute curve at the same point, a, make an angle, ξ. As a result, the base pitch, pb, cannot be specified in case of the involute gear

67

Conformal Gearing

ξ

nbc

tinv

Shifted involute “a”

a

rb.g

Oga

True involute of a circle at “a”

rb.g ξ

Ogi Og

i

nbc tinv

Shifted involute “i”

FIGURE 3.14 An example of the incorrect configuration of involute tooth profiles in relation to the base circle.

shown in Figure 3.14. Once the base pitch of the gear cannot be specified, this immediately results in the fact that the fundamental equality of the base pitch of the gear to the operating base pitch of the gear pair cannot be satisfied. The difference between the involute gear shown in Figure 3.13 and the gear* depicted in Figure 3.14 is of the same nature as the difference between Novikov gearing (see Figure 3.17 later in the chapter) [23] and helical gearing proposed by Wildhaber.

3.7 Tooth Profile Sliding in Conformal Gearing (in Novikov Gearing) The tooth profile sliding affects a permissible location of the culminating point in conformal gearing (Novikov gearing).

* It must be stressed here that involute gear is referred as such not only because its teeth are shaped in the form of an involute of a circle but also because the base circle of each involute is centered on the gear axis of rotation. Therefore, the gear with involute tooth profile as shown in Figure 3.14 is NOT an involute gear.

68


The culminating point in conformal parallel-axis gearing is located within the plane of action, PA. A portion of the plane of action, PA, within which the culminating point, K, is located is limited by the line of intersection of the plane of action by the outer diameter, do.g, of the gear and the outer diameter, do.p, of the pinion. The gear and pinion teeth must be designed so as to ensure the location of the culminating point within this interval. Geometrically, the culminating point, K, can be located between the points of tangency, Ng and Np, of the plane of action, PA, with two base cylinders of diameters, db.g and db.p, as illustrated in Figure 3.15. There is a trade-off between contact stress and the sliding of tooth flanks when determining the location of the culminating point. The smaller the radius of the boundary N-circle (i.e., when rN → 0), the smaller the sliding of tooth flank; however, contact stress in such a scenario increases as the allowed values for the radii of tooth profile curvature of the gear and the pinion decrease (ρg → 0, ρp → 0). The larger the radius of the boundary N-circle, the smaller the contact stress; however, the sliding of tooth flank is larger in this case. Theoretically, rN = 0 is the smallest possible radius of the boundary N-circle, and rN = NgP is the largest possible radius of the boundary N-circle. To make a correct decision regarding the appropriate value of radius, rN, of the boundary N-circle, both contact stress and the sliding of tooth flanks should be evaluated.

Y ωp φp

Vsl

Vg.K do. p do. g

Op

db. p

Vp.K

rp.K Np

K

Vpa

Pg

Pp rg.K

l = rN

Ng

P

ϕt

L

C

X db .g

φg

Og

ωg FIGURE 3.15 Tooth profile sliding in parallel-axis conformal gearing (Novikov gearing).

69

Conformal Gearing

For the calculation of contact stress, the radii of curvature of the gear and the pinion tooth profiles strongly correlate to the radius, rN, of the boundary circle. Some freedom is available for the gear designer in choosing the radius rN. The sliding of tooth profile depends on the distance of the culminating point, K, from the axis of rotation of the gear and the pinion. The radius, rp.K, at which the culminating point is located when rotating about the pinion axis of rotation, Op, can be calculated from the following expression:

rp.K =

0.5dp2 + 2rN2 + 2dp rN cos φt

rg .K =

0.5dg2 + 2rN2 + 2dg rN cos φt

(3.18)

A similar formula

(3.19)

is valid for the calculation of the diameter, rg.K, at which the culminating point is located when the gear pair is rotating about the gear axis of rotation, Og. For the further analysis, it is convenient to express the sliding velocity in a conformal gearing in terms of the radius rN of the boundary Novikov circle. Consider a triangle ΔKPOp in Figure 3.15. The angle φp in the triangle ΔKPOp can be determined using the cosine law

 rp2.K + rp2 − rN2  ϕ p = cos −1   2rp.K rp  

(3.20)

In Equation 3.20, pitch radius of the pinion is designated as rp. An expression similar to Equation 3.20 can be derived for the calculation of the angle φg in the gear. A simpler approach for calculation of the angle φg is based on the ratio φp/φg = u, where u is the gear ratio of the gear pair, that is, φg = φp/u. With that said, the vector Vp.K of the linear velocity of the point K in its rotation with the pinion is equal to

Vp.K = ω p rp.K [i ⋅ sin(φt − ϕ p ) + j ⋅ cos(φt − ϕ p )]

(3.21)

Similarly, the vector Vg.K of the linear velocity of the point K in its rotation with the gear can be represented in the form

Vg .K = u−1ω p r g .K [i ⋅ sin(φt + ϕ p /u) + j ⋅ cos(φt + ϕ p /u)]

(3.22)

70


The sliding velocity vector, Vsl, in conformal gearing can be calculated from the expression

Vsl = Vg .K − Vp.K

(3.23)

In parallel-axis gearing, the sliding velocity vector, Vsl, is always perpendicular to the plane of action, PA. For the specification of profile sliding of tooth flanks, G and P, of the gear and the pinion, a unitless parameter is used. This parameter is commonly referred to as specific sliding and is denoted by γ. Two different parameters, γ, are distinguished. First, the slide/roll ratio for the tooth flank, G, of the gear γg =

Vslm. g − Vslm. p Vslm. g

(3.24)

Second, the slide/roll ratio for the tooth flank, P, of the pinion γp =

Vslm. p − Vslm. g Vslm. p

(3.25)

The specific sliding, γ, is of positive value on the addendum portions of the tooth flanks. The parameter, γ, does not exceed 1. At the pitch point, P, it is equal to zero, and it is equal to 1 at the base circle of the mating gear. The specific sliding on the dedendum portion of the tooth flanks is of negative value. It is equal to zero at the pitch point, P, and it is approaching minus infinity at the base circle. The parameters Vslm. g and Vslm. p to be entered into Equations 3.24 and 3.25 are already available from Equations 3.21 and 3.22, respectively

Vslm. g = u−1ω p rg .K cos(φt + ϕ p /u) Vslm. p = ω p rp.K cos(φt − ϕ p )

(3.26) (3.27)

Commonly, the specific sliding, γ, is plotted along the line of action as depicted in Figure 3.16. Only the region, Zpa, within the path of contact comes into effect when investigating the engagement of the gear teeth. In the case of Pa-axis conformal (Novikov) gearing, the length of action is zero, that is, the equality Zpa = 0 is valid. The sliding is of a constant value in conformal as well as in high-conformal gearing; the specific sliding for the gear, γg, and the pinion, γp, are also of constant values [38]. Commonly, in involute gearing, the sliding is different

71

Conformal Gearing

Vm sl .g

Vm sl.p +γ 0

a

P

rN P

γp Zpa = 0

–2 –4

b

+γ 0 –2

γg

–4

–6

–6

–γ

–γ

FIGURE 3.16 Specific sliding, γ, in an Pa -axis conformal gear pair.

at different points of the tooth profile, and the specific sliding, γg and γp, are of a constant value. In involute gearing with a low tooth count, that is, in LTC-gearing, both the sliding and the specific sliding, γg and γp, are different at different points of the tooth profile.

3.8 Elements of the Kinematics and the Geometry in Conformal Gearing (Novikov Gearing) The kinematics and the geometry in conformal gearing (Novikov gearing) differ from that in involute gearing or gearing of other designs. From Figure 3.17, consider a conformal gear pair that is composed of a driving pinion and a driven gear. The gear is rotated about the axis, Og, and the pinion is rotated about the axis, Op. The axes of rotations, Og and Op, are at a certain center distance, C, from one another. The rotation of the gear, ωg, and the rotation of the pinion, ωp, are synchronized with each other in a timely manner. The pitch circle of the gear is of a radius Rg and the pitch circle of the pinion is of a radius Rp, respectively. The pitch circles of radii, Rg and Rp, are tangential to one another. The point of tangency of the pitch circles is the pitch point, P, of the gear pair. An instant line of action, LAinst, is a straight line through

72


ϕt

Ro. g

ωp Rp

Rg

cg P

Op Rf.p

rlim = rp b

ρp

ωg

ρf.g rg

cp

c

L

C

a

d

R*f .g

Og

Rf .g

K LAinst ag

Ro. p

ρg

C FIGURE 3.17 Kinematics and tooth flank geometry in a parallel-axis conformal gearing (Novikov gearing).

the pitch point, P, at a certain transverse pressure angle, ϕt, in relation to the perpendicular to the center line, ℄. The point of contact, K, of the tooth flanks of the gear, G, and the pinion, P, is a point within the straight line, LAinst. The further the contact point, K, is situated from the pitch point, P, the more freedom there is in selecting the radii of curvature of the tooth profiles. At that same time, the further the contact point, K, is situated from the pitch point, P, the higher the losses due to friction between the tooth flanks, G and P, and the higher the wear of the tooth flanks. Finally, the actual location of the contact point, K, is a trade-off between these two factors. Further, let us assume that the pinion is stationary and the gear performs the instant rotation in relation to the pinion. The axis, Pln, of the instant rotation, ωpl, is the straight line through the pitch point, P. The axis of the instant rotation, Pln, is parallel to the axes Og and Op of the rotations ωg and ωp. When the pinion is motionless, the contact point, K, traces a boundary circle of radius, rN, centered at P.

Conformal Gearing

73

The pinion tooth profile, P, can either align with an arc of the boundary circle, rN, or it can be relieved inside the bodily side of the pinion tooth. As a consequence, the location of the center of curvature, cp, of the convex pinion tooth profile, P, within the straight line, Linst, is limited to just the straight line segment PK. The pitch point is included in the interval, as shown in Figure 3.17, whereas the contact point, K, is not. On the other hand, the location of the center of curvature, cg, of the concave gear tooth profile, G, within the instant line of action, LAinst, is limited to the open interval P → ∞. Theoretically, the pitch point, P, can be included in that interval for K. However, this is completely impractical, and the center of curvature, cg, is situated beyond the pitch point, P. Hence, the radius of curvature, rp, of the convex pinion tooth profile, P, is smaller than the radius of curvature, rg, of the concave gear tooth profile, G (i.e., the inequality rp < rg is observed). It should be mentioned here that there are no physical constraints in designing a gear pair that has a convex gear tooth profile and concave pinion tooth profile. Both the pinion and gear are helical. The helices are of opposite hands, namely, one of them is right-handed, and the other is left-handed. No spur conformal gearing (Novikov gearing) is feasible at all. Because the gears are helical and of opposite hands, the point of contact travels axially along the gears while remaining at the same radial position on both gear and pinion teeth, G and P. It is therefore fundamental to the operation of the gears that contact occurs nominally at a point and the point of contact travels axially across the full face width of the gears during a rotation. It should be stated as a condition of operation of conformal gearing (Novikov gearing) that for a given profile, the tooth surfaces should not interact before or after culmination when rotated at angular speeds that are in the gear ratio. The transverses contact ratio, mp, of a conformal gear pair is zero (mp ≡ 0). The face contact ratio, mF, of the gear pair is always greater than 1 (mF > 1). The equality mF = mt is always observed in conformal gearing (here the total contact ratio is designated as mt). In the transverse section of the gear pair, the contact point, K, is motionless. For parallel axes configuration, the pseudo path of contact, Ppc, is a straight line through the culminating point, K. (Recall, that the length, Lpc, of path of contact, Pc, is zero). The pseudo path of contact, Ppc, is parallel to the axes Og and Op as illustrated in Figure 3.17. When rotation is transmitted from a driving shaft to a driven shaft, the contact point, K, travels along the pseudo path of contact, P pc (and it does not travel within the transverse cross-section of the gear pair), that is, parallel to the axes of rotation, Og and O p, of the gear and the pinion accordingly. This is because the transverse contact ratio is zero (mp ≡ 0) and the face contact ratio is greater than 1 (mF > 1), as mentioned earlier in this section.

74


FIGURE 3.18 As the gears rotate, contact patch in Pa —axis Novikov gearing: (a) travels in the direction of the axis of instant rotation, Pln, of the gear pair, and (b) is stationary in a transverse section of the gear pair.

The motion of the contact point, Ppc, along the axis of instant rotation, Pln, of the gear pair, and the motionless of that same point, K, in a transverse section of the gear pair is illustrated in Figure 3.18. These are the screen shots taken of an animation of a parallel-axis conformal gearing. A close-up of a conformal (Novikov) gear pair is illustrated in Figure 3.19 [8]. This is a conformal (Novikov) gear pair manufactured by Westland Helicopter Ltd.

75

Conformal Gearing

Pinion Gear

FIGURE 3.19 Close-up of a conformal (Novikov) gear pair manufactured by Westland Helicopter Ltd. (After Dyson, A., Evans, H.P., and Snidle, R.W., Proc. R. Soc. London, 1986, A403, 313–340.)

3.9 Designing a Conformal Gear Pair As an example, consider the calculation of design parameters of a Novikov gear pair that has a circular-arc tooth profile following the one proposed by Dr. M.L. Novikov [19]. The methodology disclosed in this section can be enhanced to Novikov gear pairs that have other geometries of the tooth profile in a transverse cross-section of the gear pair. For the calculation of the design parameters of a Novikov gear pair, the center distance, C, and the gear ratio, u = ωp/ωg, of the gear pair need to be given. The radii of the pitch circles of the gear, Rg, and the pinion, Rp, can be expressed in terms of the center distance, C, and the tooth ratio, u, as follows:

Rg = C ⋅

u 1+ u

(3.28)

Rp = C ⋅

1 1+ u

(3.29)

A distance, l, at which the path of contact, Pc, is away from the pitch point, P, must be known, as well as the transverse pressure angle, ϕt. The displacement,* l, is the principal design parameter in Novikov gearing. Many of the design parameters of a Novikov gear pair can be expressed in terms of the displacement, l = KP. * Recall here that the equality l = rN is observed.

76


For calculation of the radii of curvature of the tooth profile of the gear, rg, and the pinion, rp, the following formulas are used:

rg = l ⋅ (1 + k rg ) rp = l ⋅ (1 + k rp )

(3.30)

(3.31)

The actual value of the factor krp should fulfill the inequality krp ≥ 0. However, it is practical to set the factor krp equal to zero; then the equality rp = l is observed. The factor krg is within the range krg = 0.03, …, 0.10. The radius of the outside circle of the pinion, Ro.p, is calculated from the following formula:

Ro. p = Rp + (1 − k po ) ⋅ l

(3.32)

The addendum factor, kpo, of the pinion depends on the pressure angle, ϕt, absolute dimensions of the gear pair, accuracy of machining, and conditions of lubrication. It is common practice to set the pinion addendum factor, kpo, in the following range:

k po = 0.1 ÷ 0.2

(3.33)

The radius of the root circle of the pinion, Rf.p, can be calculated from the following equation:

R f . p = Rp − ag − δ

(3.34)

In Equation 3.34, the following are designated: ag is the dedendum of the mating gear [ag = (0.1,…,0.2) ⋅ l] δ is the radial clearance in the gear pair (δ = l ⋅ kpo) It is practical to set the fillet radius, ρp, in the range ρp = 0.3–l. The radius of the root circle of the gear, Rf.g, is given as follows:

R f . g = C − Ro. p

(3.35)

The radius of the outer circle of the gear, Ro.g, is calculated from the expression

Ro. g = Rg + ag

(3.36)

77

Conformal Gearing

The corner of the gear tooth addendum should be rounded with radius ρg, which is less than the fillet radius, ρp, of the pinion (ρg < ρp). The following relations among the design parameters of a Novikov gear pair are recommended by M.L. Novikov in Reference 19: rp = l, rg ≤ 1.10 ⋅ rp, ρp = 0.3 ⋅ l, mt/l = 0.8, tp/tg = 1.5, ϕt = 30°, λ = 60, …, 80° (ψ = 10°–30°), and circular pitch of the teeth p = tg + tp + B, where backlash B = 0.2, …,0.4 mm. The effective face width of the gear pair is given by

Feff = (1.1 ÷ 1.2) ⋅ p ⋅ tan λ

(3.37)

For a preliminary analysis of Novikov gearing, the following empirical expression returns a practical value for the displacement, l:

l = (0.05 ÷ 0.20) ⋅ Rp

(3.38)

The quality of parallel-axis Novikov gearing strongly depends on the following three design parameters: • The displacement, l • The transverse pressure angle, ϕt • The lead angle, λ It should be noticed here one more time that smooth rotation of the driven shaft under a uniform rotation of the driving shaft is possible only if the transverse contact ratio of the Novikov gear pair is always equal to zero (mp ≡ 0) and the face contact ratio is greater than 1 (mt = mF > 1). The application of Novikov gearing (N by -mesh of Novikov gearing in particular) featuring geometries of the tooth profiles known so far makes it possible to increase the contact strength of the gear teeth up to 2.0 ÷ 2.1 times and the bending strength up to 1.3 ÷ 1.5 times compared to involute helical gearing. Friction losses are up to 2.0 ÷ 2.5 less and tooth wear is 3 ÷ 4 times less in Novikov gearing [14]. All these application data are obtained for Novikov gearing that have hardness of tooth surfaces in the range up to HB 350. During the years when Novikov gearing was actively being investigated, Novikov gearing with harder tooth flanks was not investigated. The application of Novikov gearing makes the weight reduction of gear boxes possible (in average) 1.3 times. Uniform rotation of shafts in Novikov gearing is attained only due to face overlap of the gear teeth. Geometrically, meshing of gear teeth in a transverse cross-section is instant. Geometrically, the active portions of the tooth flanks of the gear and the pinion in Novikov gearing are represented by two conjugate helices, that is, by

78


two spatial curves. Under the applied load these portions spread over helical strips along the helices. The inactive portions of the tooth flanks are not conjugate to each other. Moreover, they are not envelopes to one another.

3.10 Conformal Gearing with Two Pseudo Paths of Contact Conformal gearing that features two pseudo paths of contact is feasible. Such a possibility immediately follows from the analysis of the schematic shown in Figure 3.10. The possibility of a conformal gear pair that has two contact points, K′ and K″, simultaneously, inspired R.V. Fed’akin to propose a conformal gearing that features not one pseudo path of contact, Ppc, as in the original Novikov gear system, but two pseudo paths of contact instead [10,25]. The invention by R.V. Fed’akin is schematically illustrated in Figure 3.20. Two pseudo paths of contact, Ppc.bf and Ppc.by, are straight lines parallel to the axis of instant rotation of the gears. The paths of contact, Ppc.bf and Ppc.by, pass through the points K′ and K″. They are at distances +l and −l from the pitch point, P, respectively. As conformal gears are helical, the contact points, K′ and K″, are displaced in axial direction in relation to one another at a distance, ΔZ. This distance can be calculated from the formula

∆Z = 2

l tan ψ

(3.39)

The axial displacement of the contact points results in a smoother rotation of the driven shaft of the conformal gear pair. The average number of contact points between the gear and pinion tooth flanks is doubled in a conformal gear pair of this design. When designing conformal gears, the gear designer is free to pick a favorable smooth curve to shape the inactive portions of tooth profile of the gear and of the pinion. An arc of the curve must be entirely located within the interior of the boundary N-circle for the tooth addendum, and a corresponding arc of the dedendum must be entirely located within the exterior of the boundary N-circle of radius rN.

3.11 Tooth Flank Geometry in a Conformal Gear Pair The radii of curvature of the interacting tooth flanks of the gear and the pinion in conformal gearing with two contact lines can be determined in the

79

Conformal Gearing

∆Z Ppc.bf K″

Zp

Zg

Z0

Ppc .by φp

φp

Xp

X0

Xp

P

φp

Op φp

Yp

Yg Y0 Y ′0

Xg Og

rg

Ppc .bf P

Ppc .by ϕt

X0

φg

φg

X ′0

ϕt

Op

ωg

K′

Yp

ωp

ωp

rp

Z0

ωg

Yg Y0

Og

φg

φg

Y ′0

Xg X ′0

FIGURE 3.20 Concept of a conformal gear system that has two paths of contact, as proposed by Fed’akin. (Adapted from Fed’akin, R.V., Investigation of Strength of Circular-Arc Gear Teeth, PhD thesis, Zhukovskiy Aviation Engineering Academy, Moscow, 1955.), and by Fed’akin and Chesnokov (Adapted from Pat. No. 182,462 (USSR), Gearing with Point System of Meshing and Having Multiple Paths of Contact. R.V. Fed’akin and V.A. Chesnokov, National Cl. 47 h, 6. Filed: November 20, 1963, Published in B.I. No. 7, 1966.)

following way: a boundary N-circle of a certain radius rN is centered at the pitch point, P, as illustrated in Figure 3.21. In a local reference system xNyN that has the pitch point, P, as the origin, the position vector, rN, of a point of the boundary N-circle can be expressed in matrix form as follows:

 rN cos ϕ N   r sin ϕ  N N  rN (ϕ N ) =    0   1  

(3.40)

where φN is the angular parameter of the boundary N-circle of the radius rN.

80


b

yN ρpb a

LAinst K1

ρga cbp

ρpa

rN

cap

xN

P cb g cag

ϕt K2

ρgb a

b

FIGURE 3.21 Tooth profiles G a -to-P b and G b -to-P a in a conformal gearing that has two pseudo paths of contact through K1 and K 2 correspondingly.

The instant line of action, LAinst, is a straight line through the pitch point, P. The instant line of action, LAinst, forms a transverse pressure angle, φt, with the pitch line through the pitch point, P. In the particular case under consideration, the addendum of the pinion, a P a, is shaped in the form of a circular arc of radius ρp . This circular arc is a centered at a point, cp , within the instant line of action, LAinst. The radius of curvature, ρap , is smaller than the radius, rN, of the boundary N-circle (i.e., the inequality ρap < rN1 is valid). In a local reference system xN yN, the position vector, r pa , of a point of the pinion addendum profile can be expressed in matrix form as follows:

ρap cos ϕ ap + (rN − ρap )cos φt   a  ρp sin ϕ ap − (rN − ρap )sin φt  rpa (ϕ ap ) =    0   1  

(3.41)

In Equation 3.41, the angular parameter of the pinion addendum profile is denoted by ϕ ap . In the particular case under consideration, the dedendum of the gear, G b, is also shaped in the form of a circular arc, the radius of which is ρbg . This circular arc is centered at a point, cbg , within the line of action, LA. The radius of curvature, ρbg , is larger compared to the radius, rN, of the boundary N-circle (i.e., the inequality ρbg > rN is observed). In a local reference system xNyN, the

81

Conformal Gearing

position vector, rgb , of a point of the gear dedendum profile can be expressed in matrix form as follows:

ρbg cos ϕ bg + (rN − ρbg )cos φt   b  ρ g sin ϕ bg − (rN − ρbg )sin φt  rgb (ϕ bg ) =    0   1  

(3.42)

In Equation 3.42, the angular parameter of the gear dedendum profile is denoted by ϕ bg . Similar to the way in which Equations 3.41 and 3.42 are derived, the corresponding expressions for the position vectors of a point of the pinion dedendum, rpb , and the gear addendum, rga , can be derived

(3.43)

ρbp cos ϕ bp − (rN + ρbp )cos φt   b  ρp sin ϕ bp + (rN + ρbp )sin φt  rpb (ϕ bp ) =    0   1  

(3.44)

ρag cos ϕ ag − (rN + ρag )cos φt   a  ρ g sin ϕ ag + (rN + ρag )sin φt  rga (ϕ ag ) =    0   1  

In Equations 3.43 and 3.44, the angular parameter of the pinion dedendum and the gear addendum are designated as ϕ bp and ϕ ag , respectively. Once the tooth profiles of the gear and the pinion addendum and dedendum are described analytically (see Equations 3.41 through 3.44), equations a b for the corresponding tooth flanks G , G , P a , and P b can be derived. For simplicity, but without loss of generality, Equations 3.36 through 3.39 are generalized as follows in the form of a single equation:

ρ cos ϕ + A   ρ sin ϕ + B   r(ϕ ) =    0   1  

(3.45)

where φ is the angular parameter of the circular-arc profile and the constants A and B are the values in terms of which coordinates of center of the corresponding point are expressed in a local reference system, xcrycr.

82


The operator, Rs(cr ↦ fl), of the screw motion of a circular-arc profile (see Equation 3.45) about the Z-axis can be represented in the form [38]

cos ϑ  sin ϑ Rs(cr fl) =   0   0

− sin ϑ cos ϑ 0 0

0 0 1 0

0  0  pϑ   1

(3.46)

where a b ϑ is the angular parameter of the helical tooth flank (either G , or G , or P a , or P b ) and p is the reduced pitch of the corresponding helical tooth flank. Equations 3.45 and 3.46 together make possible an expression for the posia b tion vector of a point, rfl, of the tooth flank (either G , or G , or P a,, or P b) in conformal gearing:

r fl (ϕ , ϑ ) = Rs(cr fl) ⋅ r(ϕ )

(3.47)

In expanded form, an expression for rfl becomes

(ρ cos ϕ + A)cos ϑ − (ρ sin ϕ + B)sin ϑ  (ρ cos ϕ + A)sin ϑ + (ρ sin ϕ + B) cos ϑ   r fl (ϕ , ϑ ) =    pϑ   1  

(3.48)

The derived equation (see Equation 3.48) for the position vector, rfl, makes it possible to further calculate the unit tangent vectors at a surface point, the unit normal vector to the tooth flank, and the first and second fundamental forms of the tooth flank. It should be stressed one more time that in conformal gearing, the tooth flanks of the gear, G, and the pinion, P, interact with one another in culminating point(s) only. The rest portions of the portions of the tooth profiles never interact with one another.

3.12 Configuration of Interacting Tooth Flanks at the Culminating Point Figure 3.22 shows a section in the transverse plane. The pinion, which has a left-hand helix, is rotating with an angular velocity, ωp, about its axis, Op,

83

Conformal Gearing

K

–l ϕt

ρp rN

ρg op Op

ωp

L

P C

C

og

ωg

Og

FIGURE 3.22 Design parameters of a high-conformal gear pair influence the geometry of contact of the teeth flanks G and P.

in a clockwise direction and is driving the gear. The gear is rotating with an angular velocity, ωg, about its axis, Og. The point of contact, K, moves in a direction at right angles to and into the plane of the paper in Figure 3.22. The pinion and the gear have working pitch radii of rp and rg = u ⋅ rp, respectively, where u is the gear ratio. The basic condition that the angular velocity ratio is equal to the gear ratio requires that the common normal at the point of contact between the teeth passes through the pitch point, P. The angle, ϕt, is the transverse pressure angle. With teeth of involute form, this condition is maintained as the gears rotate with the teeth in contact. With circular-arc teeth, however, the condition occurs at only one instant in any one transverse plane as the pitch circles roll together. Immediately before and immediately after the configuration shown in Figure 3.22, there is no contact in that particular plane between the teeth shown. French [11] referring to the instantaneous contact of profiles in a transverse section, proposed this as the culminating condition. When the gears are loaded, due to the elastic deformation of the gear materials, the contact point spreads over a certain area of contact, which results in a finite contact period. The contact lines on the gear tooth flank, G, and the pinion tooth flank, P , are helices of opposite hands. If the screw parameter, pp, of the pinion tooth flank (reduced pitch of the pinion), P, is given, then for the calculation of the screw parameter, pg, of the gear tooth flank G (reduced pitch of the gear) the expression pg = pp/u can be used. This means that high-conformal helical

84


gears, which are in point contact, will transform rotation with a constant gear ratio if their screw parameters pg and pp are related as follows: pg φg = pp φp

(3.49)

In Equation 3.49,

p g = rg tan λ g

(3.50)

where λg is the lead angle and rg is the pitch radius of the gear. Similarly,

pp = rp tan λ p

(3.51)

where λp is the lead angle and rp is the pitch radius of the pinion. Because conformal gears are helical and of opposite hands, the point of contact of the tooth flanks travels axially along the gears while remaining at the same radial position on both gear and pinion teeth. It is therefore fundamental to the operation of conformal gears that contact occurs nominally at a point and the point of contact moves axially across the full face width of the gears during a rotation. It is clearly a condition of operation that in a given profile, the tooth surfaces should not interfere before or after culmination when rotated and angular speeds are in the gear ratio.

3.13 Local and Global Contact Geometry of Interacting Tooth Flanks The tooth flanks of the gear and the pinion in a conformal gear pair are assumed to be smooth regular surfaces. The tooth flanks share a common point, which is in fact a point of culmination. Representation of two contacting tooth flanks, G and P, in the form of a surface of relative curvature is a practical and widely used kind of surfaces representation for the purpose of analytically describing the local geometry of contact of the tooth flanks. Approximation of this kind works perfectly in the differential vicinity of the point of contact. It also covers a greater area around the point of contact of the surfaces in cases when the radii of relative curvature are large enough and significantly exceed the size of the patch of contact. Under such conditions, the geometry of contact of the tooth flanks of the gear, G, and the pinion, P, can be perfectly described by the so-called

Conformal Gearing

85

ellipse of contact. Actually, the ellipse of contact is a three-dimensional (3D) curve whose projection onto the tangent plane through the point of contact of the surfaces resembles an ellipse. For a more accurate approximation of the geometry of contact of the tooth flanks of the gear, G, and of the pinion, P, of a conformal gear pair, the methods, discussed in Chapter 5 can be implemented. Studies of the area of contact and the shape of contact area are commonly based on the assumption that the difference between the profile radii of the tooth flanks, G and P, is equal to zero. In the differential vicinity of the point of contact of the tooth flanks, G and P, the patch of contact is bounded by an ellipse-like curve, that is, this curve can be expressed in terms of second order. However, the radii of relative curvature in the case under consideration are small enough. This is because a convex local patch of the tooth addendum is interacting with a saddle-like local patch of the tooth dedendum. The high degree of conformity of the contacting tooth flanks, G and P, results in small radii of relative curvature. A conclusion can be immediately entailed from the fact that the outside the differential vicinity of the point of contact boundary curve of the patch of contact between the tooth flanks, G and P, should differ from what is observed in the differential vicinity of the point of contact when the radii of relative curvature are small. This statement is proved analytically.* In a greater area around the point of contact of the tooth flanks of these high-conformal gears, the terms of the third and higher orders rapidly become important compared with the second-order terms, and they give rise to “banana-shaped” gap contours and to the region of potential interference. It is found that a third-order approximation is quite useful in that it gives an analytical expression for the gap, which remains a good approximation of the sufficient distance away from the point of contact so as to provide a good description of these unusual features. The qualitative results of the investigation of the contact area of conformal gears are illustrated in Figure 3.23. In this figure, the shape of the tooth profiles, shapes and configurations of the contact lines, shapes of the contact areas, and directions of their motion are illustrated for conformal gear pairs of various kinds. In Figure 3.23a, an example of an Nbf type of conformal gear pair is shown. This type of high-conformal gears features one contact line, Ppc.bf, which is a straight line parallel to the axis of instant rotation of the gears. The contact line, Ppc.bf, passes through the contact point, Kbf. The pinion features a concave tooth profile. The pinion is driving the gear, which has a convex tooth profile. The contact area between the tooth flanks, G and P, of the gear and the pinion is bounded by a banana-like contour. The wider side of the contact area faces toward the bottom of the gear tooth. * It should be pointed out here that because the teeth of gears of the type conform to each other so closely, the conventional Hertzian second-order equation may no longer be adequate.

86


ωp

(a)

Feff Ppc.bf

Kbf

Kbf

P

ωg ωp

(b)

Feff

P

Kby

Kby

Ppc.by

ωg

ωp

(c)

Kbf

Feff

Kbf

Ppc.bf

P Kby

Kby

Ppc.by

ωg

FIGURE 3.23 Contact patches between teeth flanks in conformal gear pairs: (a) meshing before the pitch point P; (b) meshing beyond the pitch point P; and (c) simultaneous meshing before and beyond the pitch point P.

An example of Nby -type of conformal gear pair is illustrated in Figure 3.23b. Conformal gears of this type also feature one contact line, Ppc.by, which is a straight line parallel to the axis of instant rotation of the gears. The contact line, Ppc.by, passes through the contact point, Kby. The pinion features a convex tooth profile. The pinion is driving the gear, which has a concave tooth profile. The contact area between tooth flanks, G and P, of the gear and

Conformal Gearing

87

the pinion is bounded by a banana-like contour. The wider side of the contact area faces toward the top-end of the gear tooth. The most widely used type of conformal gears features two contact lines, Ppc.bf and Ppc.by (Figure 3.23c). These contact lines are straight lines parallel to the axis of instant rotation of the gears. The contact line Ppc.bf passes through the contact point Kbf, and the contact line Ppc.by passes through the contact point Kby. The gear is driven by the pinion. The convex addendum of the gear tooth profile interacts with the concave dedendum of the pinion tooth profile, and the concave dedendum of the gear tooth profile interacts with the convex addendum of the pinion tooth profile. Two contact areas between the tooth flanks of the gear, G, and the pinion, P, are observed in this particular case. Both of them are bounded by banana-like contours. The wider sides of the contact areas face toward each other, and both face toward the axis of instant rotation of the gears. The shape and size of the contact area between the tooth flanks of the gear and the pinion are of importance in the stress analysis of conformal gears. As shown in Figure 3.23, the results of the analysis correlate with results of the corresponding experiments. Conformal gears that have various values of the design parameters, that is, various values of the profile angle, ϕt, pitch helix angle, ψg, displacement, l, and mismatch of the radii of profile curvature, Δr, were investigated [14,16]. For the experiments, an experimental rig with a closed load loop was used. Before beginning the experiments, every conformal gear pair underwent rotation for a run-in period of time. Then the gears were cleaned of the remains of the lubricant and were treated by a solution of copper sulfate. Finally, the tooth flanks were coated with a layer of silver just a few micrometers thick. Electrolytic technology was used for this purpose. After preparing them for testing, the gears were placed back in the rig in the same position with respect to each other. The experiments were carried out under light torque, which was applied to one gear of the gear pair. The other gear remained stationary. Angular vibrations were applied to one of the gears. The angular magnitude of the vibrations was in the range Δφ ≤ 15′. An increase in size of the contact area did not exceed 5%. Figure 3.24 is a reproduction of the photograph of the gear of a high- conformal gear pair that has one line of contact and a pitch helix angle ψg = 30°. The banana-shaped contact area is clearly seen from Figure 3.24. Reduction in the pitch helix angle results in a corresponding increase in the length of the contact area. Examples of various shapes of the contact area for high-conformal gear pairs that have different pitch helix angles are schematically depicted in Figure 3.25. The results of research studies similar to the ones discussed above align with those obtained by other researchers [2,46]. In comparison to conformal gears, a helical involute gear pair is schematically depicted in Figure 3.26. The active portion of the line of action, LA, in the transverse section of the gear pair is a straight line segment through the

88


Kby

FIGURE 3.24 An example of an experimentally obtained contact pattern between the teeth flanks of the gear, G, and the pinion, P, in a conformal gearing. (After Krasnoschokov, N.N., Fed’akin, R.V., and Chesnoschokov, V.A., Theory of Novikov Gearing, Nauka, Moscow, 1976, 173pp.)

(a)

Feff

(b)

(c)

FIGURE 3.25 Shape of the contact area between the teeth flanks in a conformal gear pair that has pitch helix angles: (a) ψ = 30°, (b) ψ = 20°, and (c) ψ = 10°.

89

Conformal Gearing

ωp

Feff

L P ωg

L

FIGURE 3.26 Contact area between the teeth flanks in a helical involute gear pair.

pitch point, P. The active portion of the line of action is terminated by points L and L. The line of contact, LC, is a straight line segment entirely located within tooth flank of the gear. Under the applied load, the straight line segment, LC, spreads over a narrow strip, which is the contact area between the interacting tooth flanks of the gear and the pinion. It should be stressed here that the conditions of contact of the involute tooth flanks are not favorable because both the contacting surfaces are convex, and the contact area is narrow and smaller compared to that for conformal gears. In addition to favorable conditions of contact, conformal gears enable better conditions for lubrication. When the gears rotate, the tooth flanks of the gear and the pinion roll over each other without sliding (or almost without sliding). The speed of the rolling contact point in the rolling motion significantly exceeds the linear speed of rotation of the gears. Hence, the oil film is thicker and the conditions of lubrication are significantly better.

4 High-Conformal Gearing The power density being transmitted by a gear pair is one of the most important criteria for the evaluation of how well or badly a particular gear pair has been designed and manufactured. An increase by all possible means of power density being transmitted through a gear pair is an important consideration in the future developments of the theory of gearing as well as in the manufacture and application of gears. The performance of conformal gear pairs is strongly correlated to the degree of conformity to each other of the gear tooth flank G and the pinion tooth flank P at every point of their contact. The more conformal the tooth flanks G and P at points of their contact, the better the performance of the conformal gear pair and vice versa.

4.1 Contact Geometry in Conformal Parallel-Axis Gearing General considerations of conformal gearing allow the conclusion that the substitution of convex-to-convex contact of the tooth flanks of the gear and the mating pinion (as it is observed in external involute gearing) by their convexto-concave contact (as it is observed in conformal gearing) allows an increase in contact strength in conformal gearing. Favorable conditions of contact of the tooth flanks of the gear and the mating pinion are the main anticipated advantage of a conformal gear pair. It can be assumed that the higher the degree of conformity the higher the load-carrying capacity of the contacting tooth flanks. This immediately entails a corresponding increase in power density through the gear pair, which is of critical importance for the user of the gears. Therefore, minimum possible mismatch in the curvature of the teeth of the gear and the pinion is desirable. In reality, the tooth flanks of the gear and the pinion in a conformal gear pair are displaced from their desirable positions. The undesirable displacements are mostly because of the manufacturing errors and mechanical deflections of the gear teeth, shafts, and housing that occur under the applied load because of the thermal extensions of the components and so on. Conformal gearing is sensitive toward tooth flanks displacements. To accommodate such displacements, some degree of mismatch in the curvature of gear and pinion teeth is necessary. Small mismatches are not 91

92


capable of accommodating the displacements. However, as the mismatch increases, the contact stresses also increase. A high contact stress may lead to various forms of surface failures such as heavy wear, pitting, or scuffing damage. Therefore, a minimum degree of mismatch in the curvature of the teeth of gear and pinion must be determined to make a workable conformal gear pair. Otherwise, one of the following two scenarios may be observed: • First, the gear pair is capable of absorbing the inevitable displacements of the tooth flanks, but the degree of conformity of the contacting tooth flanks is not sufficient for the high load-carrying capacity of the gear pair. • Second, the gear pair features sufficient degree of conformity of the tooth flanks, but is not capable of accommodating the tooth flanks displacements. In both cases, the gear pair has no chance of being successfully used in practice. For better understanding of the trade-off between the load-carrying capacity of conformal gearing and its capabilities being reasonably insensitive with respect to tooth flanks displacement, it is instructive to discuss the following simplified schematic. At every instance of time, the tooth flanks of a conformal gear pair contact each other at least at one point. When the gears rotate, the point of contact traces a line over each of the two tooth flanks. In reality, these lines are helices of opposite hands and equal axial pitch. As a result, at every contact point, K, the contact line of the gear, CLg, and the contact line of the pinion, CLp, share the common tangential straight line, tCL. Let us consider a section of the tooth flanks, G and P, that are intersected by a plane through the contact point, K. The plane is constructed so as to be perpendicular to the common tangential straight line, tCL. The constructed section of the tooth flanks is schematically shown in Figure 4.1. The section of the gear tooth flank is labeled G. Within the differential vicinity of the point of contact, the radius of curvature of the curve G is labeled Rg. The radius, Rg, is negative (Rg < 0), as the tooth profile is concave. The section of the pinion tooth flank before the load is applied is labeled P *. After the load is applied and the pinion tooth flank slightly penetrates the gear tooth flank, the same section, P *, is labeled P. It is assumed here that within the differential vicinity of the point of contact, the radii of curvature of the curves, P * and P, are of the same value, Rp. The radius of curvature is of positive value (Rp > 0), as the pinion tooth profile is convex. In the initial position of the tooth profiles, G and P, the contact point is labeled Kg. After the load is applied and the tooth flanks interfere with each other, the contact point is labeled Kp. The tooth profiles, G and P, intersect each other at two points, a and b. The distance, l, indicates the degree of conformity of the tooth profiles of radii Rg

93


cg ΔR

cp

Rg α

Rp

*

ng a

b

Kg Kp l

kRp

FIGURE 4.1 Section of the tooth flanks, G and P, of a conformal gear pair by a plane through a current point of contact: The plane is perpendicular to the trace of the contact point across the tooth flanks, G and P.

and Rp. The greater the distance, l, the higher the degree of conformity of the tooth flanks, and vice versa. The distance, l, between points a and b can be expressed in terms of the radii of curvature, Rg and Rp, and the displacement, k

l = 2 Rp sin α

(4.1)

For the calculation of the angle α(Rg, Rp, k), the following formula is derived:

 Rp2 − Rg2 + (Rp + Rg − kRp )2  α = cos −1   2Rp (Rp + Rg − kRp )  

(4.2)

Derivation of Equation 4.2 is based on the law of cosines. The angle α in Equation 4.1 depends on radii of curvature, Rg and Rp, as well as on the displacement k as follows from Equation 4.2. For convenience of further analysis of the plane section (Figure 4.1), all the design parameters in Equation 4.2 are normalized by the pinion radius Rp. The normalized design parameters are designated as follows:

Rp =1 Rp

(4.3)

94


Rg =K Rp

k Rp =k Rp

(4.4)

(4.5)

Angle α can be expressed in terms of the normalized design parameters in the following form:

 1 − K 2 + (1 + K − k )2  α = cos −1   2(1 + K − k ) 

(4.6)

The function l = l(k, K) is valid for both the convex-to-convex and convex-toconcave contacts of tooth flanks of the gear, G, and the pinion, P. For conformal gearing, only the case of convex-to-concave contacts of tooth flanks is of interest. Figure 4.2 shows a three-dimensional (3D) plot of the function l = l(k, K) that is constructed for the cases of convex-to-concave contacts of tooth flanks of the gear, G, and the pinion, P. Analysis of the 3D plots allows the following conclusions. Sections of the surface l = l(k, K) intersected by planes ki = Const. (Figure 4.2) are represented by curves that have asymptotes. For a particular curve,

l l = l(k, K)

0.5 0.4

Ki = Const.

0.3 0.2

ki = Const. k 0.01 0.008 0.006 0.004 0.002

0.1

–5

–4

–3

–2

–1

K

FIGURE 4.2 Three-dimensional plot of the function l = l(k , K ) constructed for convex-to-concave type of contact of the tooth flanks of the gear, G, and the pinion, P, in a conformal gear pair.


95

ki = Const., shown in Figure 4.2 in the bold line, the axis l and the straight line l = 1 are the asymptotes. The greatest possible degree of mismatch in the curvature of the teeth of gear and the pinion corresponds to the parameter K → −∞. An interval of changes in the parameter K starting from −∞ and going up to approximately K = −2 can conveniently accommodate any desirable displacement of the tooth flanks, G and P, from their correct location. However, within the interval −∞ < K < −2 of change of parameter K, the increase in the degree of conformity of the tooth profiles, G and P, is negligibly small. Within this interval of variation of the parameter K, the load-carrying capacity of a conformal gear pair remains approximately at the same range. Therefore, use of just the convex-to-concave contact of tooth flanks of the gear and the pinion gives almost no improvement in the load-carrying capacity of a gear pair. For the convex-to-concave contact, an additional requirement needs to be met to significantly improve the load-carrying capacity of a conformal gear pair. On the other hand, even a small change in the value of parameter K within the interval − 2 < K < −1 results in a significant increase in the degree rate of conformity of the teeth profiles, G and P. This immediately entails a corresponding increase in the load-carrying capacity of the gear pair.

4.2 High-Conformal Parallel-Axis Gearing High-conformal gears feature convex-to-concave contacts of the tooth flanks of the gear and the pinion similar to that of Novikov gearing* features. In addition, the degree of conformity at the point of contact of the tooth flanks of the

* The concept of Novikov gearing was not completely understood by the majority of gear experts in the years immediately following the Dr. M.L. Novikov disclosure. The lack of information on the new gear system was the root cause for this. Later on, after the concept of Novikov gearing was properly disclosed and made available for the use of western engineers, the principal differences between Novikov gearing [23] and Wildhaber gearing [22] became clear to most gear experts. The essence of Novikov gearing is disclosed in the S.U. patent [23], as well as in Novikov’s doctoral thesis [18] and monograph [19], whereas the essence of Wildhaber gearing is disclosed in the US patent [22]. A comparison of the principal features of the Novikov gear system claimed in Novikov’s patent [23] and shortly after discussed in Novikov’s doctoral thesis [18] and [19], and the principal features of Wildhaber’s gear system claimed in the patent [22] makes it easy to distinguish between the two. Unfortunately, beginners and less experienced gear specialists often make no difference between the Novikov gear system [23] and the gear system proposed by Wildhaber [22]. Many of them still loosely refer to Novikov gearing as to “W–N gearing.” This term is totally incorrect. It is instructive to point out here that to make the inconsistency of the term “W–N gearing” clear, one can provide a definition to the term “W–N gearing,” that is, to formulate what the term “W–N gearing” stands for. This definition can be compared with that of the Novikov gear system [18,19,23], as well as of the Wildhaber gear system.

96


Bearing capacity of the teeth flanks

gear and the pinion in high-conformal gearing exceeds a certain critical value, that is, the threshold. In the aforementioned example (Figure 4.2), the value of parameter K (i.e., the value of K ≈ −2) can be referred to as a critical value, that is, Kcr. This allows one to distinguish between conformal gearing (for which −∞ < K < Kcr) and high-conformal gearing (for which Kcr ≤ K < −1). Because of the favorable conditions of contact of the tooth flanks, high-conformal gears allow the transmission of higher power density. Without going into the details of this analysis, it is clear that high-conformal gears require tight tolerances for any possible displacements of the tooth flanks of the gear, G, and the pinion, P, from their desirable locations and orientations. This relates not just to the tolerances on the manufacturing errors, but to any and all possible displacements caused by thermal extension, elastic deflection, and so on. Otherwise, there could be no future for high-conformal gear systems. The areas of existence of conformal and high-conformal gearing are schematically illustrated in Figure 4.3. The performed analysis of the 3D plot shown in Figure 4.2 can be extended, although the extension is somewhat besides the main subject of this book. Consider sections of the surface l = l(k, K) intersected by planes Ki = Const. (Figure 4.2). An example of such sections is shown by the bold dashed line. For high-conformal gears, parameter Ki for these lines is within the interval Kcr ≤ Ki < −1. The degree of mismatch in the curvature of the teeth in highconformal gears is smaller compared to that in conformal gears. Without going into the details of this analysis, it is important to point out here that the teeth profiles of high-conformal gears feature convex-to-concave type of

C

High-conformal gearing

Conformal gearing B A a

dcnf

b

dcnf

c

dcnf

dcnf

[dcnf ] FIGURE 4.3 Impact of degree of conformity, dcnf, at a current point of contact of the gear tooth flank, [dcnf], a and the pinion tooth flank, dcnf , onto the bearing capacity of the teeth flanks.


97

contact, and the degree of mismatch in the curvature is small. The aforementioned features allow the conclusion that the Hertz formula is not applicable for the calculation of contact stress in high-conformal gears. First, the Hertz formula for the calculation of contact stress was derived [21] under the assumption that the dimensions of the contact patch between two contacting surfaces are significantly smaller in comparison to the corresponding radii of curvature of the surface of relative curvature. This requirement is violated by the aforementioned features: a small degree of mismatch in the curvature of the teeth profiles of high-conformal gears results in the sizes of the contact patch becoming comparable with the corresponding radii of curvature of the surface of relative curvature, which is not allowed. Second, the Hertz formula for the calculation of contact stress was derived [12] for cases of contact of two bodies of simple shape. Sphere-to-plane, sphere-to-sphere, and cylinder-to-plane are examples of shapes in relation to which the Hertz formula is valid. Generally speaking, to make the Hertz formula valid, the alignment of the principal directions of the contacting surface is a must. At a point of contact, the principal directions of the gear tooth flank are denoted by t1.g and t2.g. Similarly, at the same point the principal directions of the pinion tooth flank are denoted by t1.p and t 2.p. The Hertz formula is valid in either of the following cases: • t1.g is aligned with t1.p and t 2.g is aligned with t 2.p. • t1.g is aligned with t 2.p and t 2.g is aligned with t1.p. The greater the misalignment of the principal directions, the greater the deviation in the calculated values of contact stress from their actual values, and vice versa. The active portions of the tooth flanks in high-conformal gears are surfaces that have complex geometry. For these surfaces, the requirement of alignment of the principal directions t1.g, t 2.g, and t1.p, t 2.p is not fulfilled. This is the second reason the Hertz formula in not valid for the calculation of contact stress between the tooth flanks of high-conformal gears. Indicatrices of conformity of the kinds Cnf R(G/P) and Cnf k(G/P) are developed for the analytical description of the contact geometry of the interacting tooth flanks, G and P, of a gear pair (see Chapter 5). Characteristic curves of these kinds can be used to construct the contour of the contact patch between two high-conformal gears. This can be helpful when solving the contact stress problem for gearing of this system. Based on the aforementioned investigation, high-conformal gearing can be characterized by the following features, each of which is important and, moreover, all of which are sufficient to refer to this gearing as high-conformal gearing: • The transverse contact ratio is identical to zero (mp ≡ 0). • The total contact ratio, mt, is equal to face contact ratio, mF, and is greater than 1 (mt = mF > 1).

98


• The tooth profile of one member of the gear pair is convex, whereas that of the mating gear is concave. • The convex tooth profile of one member of the gear pair is entirely located within the interior of the boundary N-circle, whereas the concave tooth profile of the other member of the gear pair is entirely located within the exterior of the boundary N-circle. • The difference between the magnitudes of the radii of curvature of the concave tooth profile and the convex tooth profile in the gear pair is equal to or smaller than a given threshold beyond which the high conformity of the interacting tooth profiles contributes much to the bearing capacity of the gear pair. Novikov gearing and high-conformal gearing share the first four features. High-conformal gearing differs from Novikov gearing only in the last of the aforementioned features. The difference between the radii of curvature is required to make the gear pair capable of absorbing the tooth flank displacements due to manufacturing errors, deflections under the operating load, and deflections due to heat extension, as well as all other displacements. Comparing high-conformal gearing (as well as Novikov gearing) with involute gearing, the following should be noticed: • Proposed by L. Euler, spur involute gearing features a transverse contact ratio, mp, greater than 1 (mp > 1), a zero face contact ratio (mF = 0), and a total contact ratio, mt, equal to the transverse contact ratio (mt = mp > 1). Later on, the concept of spur involute gearing was enhanced to include the concept of helical involute gearing that has a face contact ratio greater than zero (mF > 0) and a total contact ratio mt = mp + mF > 1. [involute (Euler) gearing: mp > 1 and mF = 0, or just Eu-gearing for simplicity]. • Proposed by Dr. M.L. Novikov, so-called Novikov gearing (or just, N-gearing for simplicity) features a zero transverse contact ratio (mp ≡ 0), a face contact ratio greater than 1 (mF > 1), and a total contact ratio, mt, equal to the face contact ratio (mt = mF > 1). Later on, the concept of Novikov gearing was enhanced to include the concept of high-conformal gearing that has a degree of conformity at a point of contact of the interacting tooth flanks of the gear and the pinion equal to or smaller than a predetermined threshold [Novikov gearing: mp ≡ 0 and mF > 1]. As all feasible combinations of the values of transverse contact ratio, mp, and face contact ratio, mF, are covered by either involute (Euler) gearing or

99


Novikov gearing, it can be concluded that no new gear system can be developed based on the various combinations of contact ratios.* The term high-conformal gearing is broader than the term Novikov gearing. Novikov gearing features a convex-to-concave contact of the tooth flanks of the gear and the pinion, and a particular configuration of the tooth flanks in relation to the line of action under which the transverse contact ration of a gear pair is identical to zero (mp ≡ 0) and the face contact ratio is always greater than 1 (mF > 1). In addition, high-conformal gearing features a certain degree of conformity at a point of contact, K, of the tooth flanks G and P . The minimum diameter, dcnf, of the indicatrix of conformity, Cnf(G/P), at a current point of contact, K, of the tooth flanks, G and P, can be used as a quantitative measure of the degree of conformity of the interacting tooth flanks. The degree of conformity of the tooth flanks of the gear, G, and the pinion, P, exceeds a threshold beyond which a significant increase in the bearing capacity of the interacting tooth flanks is observed. Schematically, this property of high-conformal gearing is illustrated in Figure 4.4. a , at the point of contact of the tooth For a certain degree of conformity, dcnf flanks, G and P, the bearing capacity of the tooth surfaces can be evaluated by (a)

(b)

yg CnfR (

ng

t2.p

/ t2.g ϕ

Crv (

)

Crv (

t1.p

)

xg

t1.g

K

K

)

μ Crv (

)

np

FIGURE 4.4 An example of the indicatrix of conformity, Cnf R (G/P), at a point of contact of tooth flanks G and P of the gear and of the mating pinion: (a) local patches of the teeth flanks G and P in contact, and (b) the indicatrix of conformity, Cnf R (G/P), at point of contact, K, of the local patches of the surfaces G and P. * It is evident that the Helical Gearing patent proposed by E. Wildhaber does not meet the requirements of Euler gearing, nor does it meet the requirements of Novikov gearing. The widely adopted terminology “Wildhaber–Novikov gearing” clearly indicates poor understanding of the kinematics and geometry of both Novikov gearing and of helical gearing (proposed by E. Wildhaber). The incorrect terminology must be eliminated from use among proficient gear experts. The invention by Dr. M.L. Novikov must be referred to as Novikov gearing, and Wildhaber gearing must be referred to as Wildhaber gearing (or just the helical gearing patent as proposed by E. Wildhaber).

100


a number, A. If the degree of conformity of the tooth flanks of the gear, G, and b a , an insignificant increase the pinion, P, is increased from dcnf to a value of dcnf in the bearing capacity of the tooth flanks from number A to number B occurs. The increase in bearing capacity is insignificant in the case under consideration a b as both the degrees of conformity, dcnf and dcnf , are smaller than the threshold [dcnf] beyond which a significant increase in the bearing capacity of the tooth flanks, G and P, occurs. c Let us assume that the degree of conformity, dcnf , is greater than the threshc old [dcnf]. When the inequality dcnf > [dcnf ] is valid, the bearing capacity of the tooth flanks of the gear, G, and the pinion, P, grows fast. c ≥ [dcnf ] is always observed. For high-conformal gearing, the inequality dcnf

4.3 On the Accuracy Requirements for Conformal Parallel-Axis Gearing As it was discussed in the previous sections of this book, high-conformal gearing features a degree of conformity, dcnf, at the point of contact of the tooth flanks that exceeds a certain critical value of the degree of conformity [dcnf], that is, it exceeds a threshold for the degree of conformity (dcnf ≥ [dcnf]). This condition can be attained if tolerances for the accuracy of the gear and the pinion are tightened. In reality, none of the design parameters of a gear pair can be maintained with a zero deviation: all of the design parameters of a gear pair are within the corresponding tolerances for the accuracy. For practical needs, it is important to estimate the influence of the variation of the design parameters on the performance of a high-conformal gear pair. In a high-conformal gear pair, linear displacements of the gear and the pinion in relation to one another can be expressed in terms of the tree components along the axes of a Cartesian reference system. The reference system can be associated with the gear pair so as to minimize the total number of the components to be taken into account when calculating the design parameters of the gear and the pinion. If one of the axes is parallel to the pitch line, Pln, and another one is along the center line, ℄, then the total linear displacement of the gear and the pinion in relation to one another is equal to the actual value of variation, ΔC, of the center distance, C. In Figure 4.5 (see the upper portion of the figure), an ideal high-conformal gear pair is schematically shown. Here, the center distance is equal to a certain value, C. In reality, because of the manufacturing errors and for other reasons, the actual value of the center distance, C*, differs from C (see the upper portion of Figure 4.5). The alteration in the center distance inevitably entails corresponding changes to the diameters, dg.pc and dp.pc, of the gear and

101


ωp

C

dpc.p

Op db.p

Np

P

Vpa

dpc.g

Ppc

Ng

L

C

db.g

ϕt

Og rN

VK Opc

Pln

ωg

K1 O cg

Feff

Ppc K2

pb.op

ψb ωp

* dpc.p

C*

O*p

* dpc.g

Ppc

db.p Np* ΔCp

P

Vpa

Ng*

db.g

C*

L

ϕt*

Δϕt rN

Og

Og*

ωg ΔCg

FIGURE 4.5 Variation of the center distance, C, and the operating pressure angle, ϕt, in a real parallel-axis high-conformal gearing.

102


the pinion at which the pseudo path of contact is situated. The actual values * of the diameters, dg.pc and dp.pc, are labeled as dg . pc and dp* . pc, respectively. The operating pressure angle also changes from its nominal value, ϕt, to the actual value, φt* . The changes to the center distance and to the pressure angle need to be taken into account when designing the high-conformal gear pair. In the lower portion of Figure 4.5

POp* db. p = * POg db. g

(4.7)

and

POg* + POp* = C *

(4.8)

From these equations, the following expressions can be derived for calculating the parameters POg* and POp* : POg* = C * POp* = C *

db. g db. g + db. p db. p db. g + db. p

(4.9)

(4.10)

Commonly, the actual value of the center distance, C*, is not known. However, the tolerance, [ΔC], is known. Therefore, in the calculations, the actual center distance, C*, is substituted with the sum C* = C + [ΔC]. It should be noticed here that the tolerance [ΔC] is a signed value. With that said, Equations 4.9 and 4.10 cast into POg* = (C + [∆C]) ⋅ POp* = (C + [∆C]) ⋅

db. g db. g + db. p

db. p db. g + db. p

(4.11)

(4.12)

The calculated values of the parameters POg* and POp* yield the formula for calculating the operating pressure angle, φt* φt* = sin −1

db. g db. p = sin −1 * POg POp*

(4.13)

103


Equation 4.13 yields the calculation of the design parameters PN *g and PN *p PN *g = (C + ∆C ) ⋅ PN *p = (C + ∆C ) ⋅

db. g cos φt* db. g + db. p db. p cos φt* db. g + db. p

(4.14)

(4.15)

Once the operating pressure angle, and the design parameters, PN *g and * * PN *p , are known, then the diameters, dpc . g and dpc. p , can be calculated from the equations

* dpc .g =

db2. g + 4 ⋅ (PN *g − rN )2

* dpc .p =

db2. p + 4 ⋅ (PN *p + rN )2

(4.16) (4.17)

A change to the center distance, C, results in the tooth flanks G and P of the gear and the pinion interacting with one another not by involute tooth points within each of them; instead, they contact by points within the tooth profiles that were originally designed with the intent to not to be involved in the transmission of the rotation. The conditions of meshing are violated because of this. As an example, consider the influence of variation of the center distance alteration, ΔC, onto the required change in radii of curvature of the gear and the pinion tooth profiles. Figure 4.6 illustrates a portion of a concave gear tooth profile, G*, in a local vicinity of the involute tooth point. The involute tooth point coincides in Figure 4.6 with the projection of the pseudo path of contact, PPc, in the gear pair onto the plane of the drawing in Figure 4.6. The G* profile corresponds to a case when the actual center distance, C*, is greater than the desirable center distance, C, at a displacement, ΔC. For the comparison, the original concave gear tooth profile, G, constructed for zero displacement of the gears, is also shown in Figure 4.6. As the actual linear displacement is commonly unknown, the tolerance, [ΔC], for the displacement is considered instead. The tolerance [ΔC] is shared with the corresponding tolerances for the linear displacements [ΔCg] and [ΔCp] of the gear and the pinion in the following manner: [∆C g ] = [∆C] ⋅

db. g db. g + db. p

(4.18)

104


[∆C p ] = [∆C] ⋅

db. p db. g + db. p

(4.19)

The components [ΔCg] and [ΔCp] of the resultant tolerance [ΔC] are signed values. Evidently, the equality [ΔC] = [ΔCg] + [ΔCp] is observed. After the gears are displaced, the gear tooth profile, G*, must be entirely located outside the boundary N-circle of the radius rN. The original concave gear tooth profile, G, is centered at point cg. The radius, rg, of this profile is known (here and below only circular-arc gear tooth profiles are discussed as examples). The radius, rg, is greater than the radius, rN, (i.e., the inequality rg > rN is valid). The concave gear tooth profile, G*, is centered at point c*g . The center, c*g , is shifted parallel to the center line, ℄, at a distance [ΔCg]. As it follows from the analysis of Figure 4.6, the minimal permissible radius, r g* , of the concave gear tooth profile, G*, is greater compared to radius, rg. The expression can be used for the calculation of the radius r g*

r g* = r g + [∆C g ]2 + (c g P)2 − 2[∆C g ](c g P)sin φt

(4.20)

It is clear that the inequality r g* > r g is observed. ϕt υg L

C

rg

cg

rN PA

P

* Ppc

Ppc *

[ΔCg]

c*g

υg r*g

FIGURE 4.6 Configuration of the concave gear tooth profile, G*, in a high-conformal gear pair with the altered center distance (ΔC).

105


The original concave gear tooth profile, G, intersects the plane of action, PA, at a right angle. This is because the involute tooth point of the gear and the involute tooth point of the pinion make contact at culmination. Due to this, all three conditions of meshing are met when the displacement is zero ([ΔC] = 0). The concave gear tooth profile, G*, intersects the plane of action, PA, at a certain angle. The angle between the perpendicular to the plane of action, PA, and the tangent to the profile G* is designated as υg. The smaller the angle, υg, the better.* An equation similar to Equation 4.20 can be derived for the convex pinion tooth profile, P * (Figure 4.7)

r p* = r p − [∆C p ]2 + (cp P)2 − 2 [∆C p ](cp P) sin φt

(4.21)

ϕt

L

C

rp* rN

cp* P

[ΔCp] rp

cp

υp

*

* Ppc

Ppc

PA

υp

FIGURE 4.7 Configuration of the convex pinion tooth profile, P *, in a high-conformal gear pair with the altered center distance (ΔC). * In involute gearing, when the center distance changes, the contact point between the tooth flanks of the gear and the pinion travels along the involute tooth profiles. Because the tooth profiles are of involute geometry, this makes involute gearing insensitive to the variation of the center distance. In conformal and in high-conformal gearings, the involute tooth profiles are truncated to the involute point. When the center distance changes, the contact point between the tooth flanks of the gear and the pinion travels along the tooth profiles that are designed so as to not to be engaged in gear mesh. The last makes both conformal and high-conformal gearings sensitive to the variation of the center distance.

106


min The degree of conformity, dcnf , for the real high-conformal gearing need to be calculated taking into account Equations 4.20 and 4.21. The calculated min min value of the diameter dcnf is then compared with the threshold [dcnf ] . If the min min inequality dcnf < [dcnf ] is observed then the gearing can be referred to as min min high-conformal gearing. Otherwise, when the inequality dcnf > [dcnf ] is observed, the gearing represents an example of a conformal gear pair. In a case where the linear displacement, [ΔC], can take place in both the directions, that is, [ΔC+] and [ΔC−], the analysis needs to be performed for both the values of the displacements, [ΔC+] and [ΔC−]. In ideal high-conformal (as well as in conformal-Novikov) gearings, that is, when the displacement, [ΔC], is zero, then both the angles υg and υp are zero, and the resultant angle, υ, is also zero. In such a scenario, all three conditions for proper transmission of a rotation can be satisfied. All possible efforts need to be undertaken to minimize the angle υ = υg + υp. For example, gearings with a larger radius of the boundary circle, rN, are less sensitive to axes misalignment. Similarly to the linear displacements of the gears in high-conformal gearing, angular displacements can also be taken into account when designing a gear pair. For this analysis, the parallel-axis high-conformal gear needs to be considered as a crossed-axis gear pair having [ΔC] as the center distance, and having [ΔΣ] as the tolerance for the axis alignment [38]. Both [ΔC] and [ΔΣ] are signed values. This particular problem is more bulky compared to the case considered earlier. The discussion in this section of the book also indirectly confirms that neither profile nor longitudinal modifications are allowed for conformal and high-conformal gears.

5 Contact Geometry of the Gear and the Mating Pinion Tooth Flanks In the theory of gearing, the kinematics of gearing is considered as the prime element of the gear pair. Other important elements of gearing, namely (a) the shape and the geometry of the gear tooth flank, G, and (b) the shape and the geometry of the mating pinion tooth flank, P (as well as numerous others) are considered as the secondary elements of gearing. This does not mean that the importance of the secondary elements is lower than that of the primary elements. No, this is incorrect. This just means that the most favorable parameters of the secondary elements can be expressed in terms of the parameters of the prime element. Ultimately, the entire gear pair can be synthesized on the premise just of the prime element, that is, on the premise of the desirable kinematics of the gear pair. In other words, having just the desirable kinematics of the gear pair to be designed, it is possible to synthesize the rest of the design parameters of the gear pair. Only the kinematics of gearing is used for the purposes of synthesizing of the best possible gear pair for transmitting the input rotation and torque. The concept that establishes the priority of the kinematics of gearing over the remaining elements of the gear pair is the cornerstone concept of the developed scientific theory of gearing [38]. In order to solve the problem of synthesizing the most favorable gear pair, an appropriate analytical description of contact geometry of the gear tooth flank G and the mating pinion tooth flank P is required. The problem of analytical description of contact geometry between two smooth regular surfaces in the first order of tangency is a sophisticated one. Investigation of contact geometry of curves and surfaces can be traced back to the eighteenth century. The study of the contact of curves and surfaces was undertaken in considerable detail by J.L. Lagrange* in his “Theorié des Fonctions Analytiques” (1797) [17], and A.L. Cauchy† in his “Leçons sur les Applications du Calcul Infinitésimal á la Geometrie” (1826) [4]. Later on, in the twentieth century, an investigation in the realm of contact geometry of curves and surfaces was undertaken by J. Favard‡ in his “Course de Gèomètrie

* Joseph-Louis Lagrange (January 25, 1736–April 10, 1813)—an Italian born [born Giuseppe Lodovico (Luigi) Lagrangia] famous French mathematician and mechanician. † Augustin-Louis Cauchy (August 21, 1789–May 23, 1857)—a famous French mathematician. ‡ Jean Favard (August 28, 1902–January 21, 1965)—a French mathematician.

107

108


Diffèrentialle Locale” (1957) [9]. A few more names of the researchers in the field will be mentioned. The results of the research obtained in the field of contact geometry of two smooth regular surfaces are widely used in the theory of gearing. The problem of synthesizing of the most favorable gear pair can be solved on the premise of the analysis of topology of the contacting surfaces in differential vicinity of the point of their contact. Various methods for analytical description of contact geometry between two smooth regular surfaces have been developed. An overview of the methods can be found out in the monograph by Radzevich [31]. Latest achievements in the field are discussed in the papers [28,31,32], and in the monograph [41]. A detailed analysis of known methods of the analytical description of the geometry of contact between two smooth regular surfaces uncovered the poor capability of the known methods of solving problems in the field of designing efficient gear pairs. Therefore, an accurate method for analytical description of contact geometry between two smooth regular surfaces G and P in the first order of tangency, which fits the needs of the theory of gearing, is necessary. Such a method is worked out in this chapter. It is convenient to begin the discussion starting from the analytical description of the local relative orientation of the gear tooth flank G and the mating pinion tooth flank P. The proposed analytical description is relevant to the differential vicinity of the point of contact K of the tooth flanks G and P.

5.1 L ocal Relative Orientation at a Point of Contact of Gear and Mating Pinion Tooth Flanks When the gears rotate, the gear tooth flank G and the mating pinion tooth flank P are in permanent tangency with one another. Locally, the contacting surfaces G and P can be approximated by the corresponding quadrics as schematically illustrated in Figure 5.1. The requirement to be permanently in tangency to each other imposes a kind of restriction on the relative configuration (location and orientation) of the tooth flanks G and P and on their instant relative motions. In the theory of gearing, a quantitative measure of the relative orientation of the gear tooth flank G and of the mating pinion tooth flank P is established [28]. The relative orientation at a point of contact of the gear tooth flank G and of the mating pinion tooth flank P is specified by the angle μ local* relative * The surface orientation is local in nature because it relates only to differential vicinity of the point K of contact of the tooth flanks G and P.

109

Contact Geometry of the Gear and the Mating Pinion Tooth Flanks

zg

R1.p

C2.p

C1.p ng

R2.p

Tangent plane

yg

R2.g C2.p

t1.p

K

t2.g

yp

t2.p

m

t1.g

xp xg

np

R1.g zp

C1.g

FIGURE 5.1 Local configuration of two quadrics tangent to a gear tooth flank G and to a mating pinion tooth flank P at a point K of their contact. (Adapted from Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, CRC Press, Boca Raton, FL, 2012, 743pp.)

orientation of the surfaces. By definition, the angle μ is equal to the angle that the unit tangent vector t1.g of the first principal direction of the gear tooth flank G forms with the unit tangent vector t1.p of the first principal direction of the mating pinion tooth flank P. That same angle μ can also be determined as the angle that makes the unit tangent vectors t2.g and t2.p of the second principal directions of the surfaces G and P at a point K of their contact. This immediately yields equations for the calculation of the angle μ

sin µ = |t1. g × t1. p| = |t 2. g × t 2. p|

(5.1)

cos µ = t1. g ⋅ t1. p = t 2. g ⋅ t 2. p

(5.2)

tan µ =

|t1. g × t1. p| |t 2. g × t 2. p| ≡ t 2. g ⋅ t 2. p t1. g ⋅ t1. p

(5.3)

where t1.g, t2.g are the unit vectors of principal directions on the gear tooth flank G measured at a contact point K and t1.p, t2.p are the unit vectors of principal directions on the mating pinion tooth flank P at that same contact point K of the surfaces G and P.

110


Directions of the unit tangent vectors t1.g and t 2.g of the principal directions on the part surface G (as well as directions of the unit tangent vectors t1.p and t 2.p of the principal directions on the pinion tooth flank P) can be specified in terms of the ratio dUg/dVg (or in terms of the ratio dUp/dVp in case of the pinion tooth flank P). The corresponding values of the ratio dUg(p) /dVg(p) are calculated as roots of the quadratic equation

Eg ( p )

dU g ( p ) + Fg ( p ) dVg ( p )

Lg ( p )

dU g ( p ) + Mg( p) dVg ( p )

Fg ( p )

dU g ( p ) + Gg ( p ) dVg ( p )

Mg( p)

dU g ( p ) + N g( p) dVg ( p )

(5.4)

=0

In case of the point contact of the surfaces G and P, the actual value of the angle μ is calculated at the point K of contact of the surfaces. If the tooth flanks G and P are in line contact, then the actual value of the angle μ can be calculated at every point within the line of contact.* The line of contact of the tooth flanks G and P is commonly referred to as characteristic line E, or just as characteristic E. Determination of the angle μ of local relative orientation of the tooth flanks G and P of a gear and a mating pinion are illustrated in Figure 5.1. In order to calculate the actual value of the angle μ of local relative orientation of the tooth flanks G and P, unit vectors of the principal directions t1.g and t1.p are employed. Consider tooth flanks G and P in point contact, which are represented in a common reference system. The surfaces make a contact at a point K. For further analysis, an equation of the common tangent plane to the tooth flanks G and P at the contact point K is necessary (Figures 5.1 and 5.2)

(rtp − rK ) ⋅ u g ⋅ v g = 0

(5.5)

where rtp is the position vector of a point of the common tangent plane, rK is the position vector of the contact point K, ug, and vg are unit vectors that are tangent to Ug and Vg are coordinate lines on the gear tooth flank G at the contact point K. The angle ωg is the angle that is formed by the unit vectors ug and vg. The actual value of the angle ωg can be calculated from one of the following equations [38]:

sin ω g =

EgGg − Fg2 Eg G g

(5.6)

* It is worthy pointing out here that in a case of line contact, the relative orientation of the surfaces G and P is predetermined in a global sense. However, the actual value of the angle μ of the surfaces local relative orientation at different points of the characteristic E is different.


C2.p

111

C2.g ωg µ

ξp t2.g

vp

t2.p

up

C1.p

θ ug

K

ε

ξg

µ

t1.p t1.g

C1.g

vp

ωp

FIGURE 5.2 Local relative orientation at a point of contact of the tooth flanks of a gear G and a mating pinion P considered in a common tangent plane.

cos ω g = tan ω g =

Fg Eg G g

(5.7)

EgGg − Fg2 Fg

(5.8)

Equations similar to Equations 5.6 through 5.8 are also valid for the calculation of the angle ωp at a point on the pinion tooth flank P. Tangent directions ug and vg to the Ug and Vg coordinate lines at a point on the gear tooth flank G, and tangent directions up and vp to the Up and Vp coordinate lines at a point on the pinion tooth flank P are specified in terms of the angles θ and ε. For the calculation of the actual values of the angles θ and ε, the following equations can be used:

cos θ = u g ⋅ u p

(5.9)

cos ε = v g ⋅ v p

(5.10)

112


The angle ξg is the angle that the first principal direction t1.g on the gear tooth surface G forms with the unit tangent vector ug (see Figure 5.2). The equation for the calculation of the actual value of the angle ξg is derived by Radzevich [28,30,37] sin ξ g =

ηg 2 g

η − 2ηg cos ω g + 1

sin ω g

(5.11)

where ηg designates the ratio ηg = (∂Ug/∂Vg). In the event Fg = 0, the following equality tan ξg = ηg is observed. Here, the ratio ηg is equal to the root of the quadratic equation

( Fg Lg − Eg M g )η2g + (Gg Lg − Eg N g )ηg + (Gg M g − Fg N g ) = 0

(5.12)

which immediately follows from the equation Eg dU g + Fg dVg Lg dU g + M g dVg

Fg dU g + Gg dVg =0 M g dU g + N g dVg

(5.13)

The equation for the calculation of the actual value of the angle ξg allows for another representation. Following the chain rule, drg can be represented in the form drg = U g dU g + Vg dVg

(5.14)

By definition, tan ξg = (sin ξg/cos ξg). The functions sin ξg and cos ξg yield representation as sin ξ g =

|U g × drg| |U g| ⋅ |drg|

cos ξ P g =

(5.15)

U g ⋅ drg |U g| ⋅ |drg|

(5.16)

The last expressions yield tan ξ g =

sin ξ g |U g × drg| |U g × drg| = = cos ξ g U g ⋅ drg U g ⋅ (U g dU g + Vg dVg ) =

|U g × drg|⋅ dVg U g ⋅ U g dU g + U g ⋅ Vg dVg

(5.17)


113

By definition [38]

U g ⋅ U g = Eg U g ⋅ Vg = Fg U g × Vg =

(5.18)

(5.19)

EgGg − Fg2

(5.20)

Equations 5.14 through 5.20 yield the formula, tan ξ g =

EgGg − Fg2

(5.21)

ηg ⋅ Eg + Fg

for the calculation of the actual value of the angle ξg. Equations similar to those above Equations 5.11 and 5.21 are also valid for the calculation of the actual value of the angle ξp that the first principal direction t1.p at a point on the pinion tooth flank P forms with the unit tangent vector up. The performed analysis makes possible the following equations for the calculation of the unit vectors of principal directions t1.g and t 2.g:

t1. g = Rt(ξ g , n g ) ⋅ u g

  π t 2. g = Rt  ξ g +  , n g  ⋅ u g   2  

(5.22)

(5.23)

for the gear tooth flank G, and similar equations for the calculation of the unit vectors of principal directions t1.p and t 2.p,

t1. p = Rt(ξ p , n g ) ⋅ u p

  π t 2. p = Rt  ξ p +  , n g  ⋅ u p   2  

(5.24)

(5.25)

for the pinion tooth flank P. Equation 3.16 for the operator of rotation Rt(φA,A0) through an angle φA about an axis A0 is employed for the calculation of the operators of rotation in Equation 5.22 through 5.25.

114


5.2 The Second-Order Analysis: Planar Characteristic Images For a more accurate analytical description of contact geometry of the gear tooth flank G and the pinion tooth flank P, consideration of the second-order parameters is necessary. The second-order analysis incorporates elements of both of the first-order and of the second-order analysis. For performing the second-order analysis, familiarity with Dupin’s indicatrix is highly desirable.* Dupin’s indicatrix is a perfect startingpoint for consideration of the second-order analysis. 5.2.1 Preliminary Remarks: Dupin’s Indicatrix At any point of a smooth regular gear tooth flank G (as well as at any point of a smooth regular pinion tooth flank P) a corresponding Dupin’s indicatrix can be constructed. Dupin’s indicatrices Dup(G) at a point of a gear tooth flank G and Dup(P) at a point of the pinion tooth flank P are planar characteristic curves of the second order. They are used for graphical interpretation of the distribution of normal radii of curvature of a surface in the differential vicinity of a surface point. Dupin’s indicatrix at a point of the tooth flank G (and P the tooth flank) is of critical importance in the theory of gearing. Generation of this planar characteristic curve is illustrated with a diagram in Figure 5.3. A plane W through the unit normal vector ng to the part surface G at a point m rotates about ng. While rotating, the plane occupies consecutive positions

W1

W2

W3

ng

Q δ m

Dup (

)

FIGURE 5.3 Dupin’s indicatrix at a point of a smooth regular gear tooth flank G. * Fransua Pier Charles Dupin (October 6, 1784–January 18, 1873)—a French mathematician.

115


(a)

yg

R1.g

(b) Dup(

m

R2.g

)

Dup (

m

xg

)

yg

R1.g

Dup(

m

xg

)

xg

Rg

R2.g (d)

(c)

yg

Dup(

yg

Dup (

m

)

xg |R1.g |

(e)

R2.g

)

yg

Dup(

m

)

xg

|R1.g |

FIGURE 5.4 Five different types of Dupin’s indicatrices, Dup(G), at a point m of a smooth regular gear tooth flank, G. (a) Elliptic, (b) umbilic, (c) parabolic, (d) hyperbolic, and (e) minimal.

W1, W2, W3, and others. The radii of normal curvature of the line of intersection of the gear tooth flank G by normal planes W1, W2, W3 are equal to Rg,1, Rg,2, Rg,3, etc. The gear tooth flank G is intersected by a plane Q (see Figure 5.3). The plane Q is orthogonal to the unit normal vector ng. This plane is at a certain small distance δ from the point m. When the distance δ approaches zero (δ → 0) and when the scale of the line of intersection of the gear tooth flank G by the plane Q approaches infinity, then the line of intersection of the gear surface G by the plane Q approaches to the planar characteristic curve that is commonly referred to as Dupin’s indicatrix, Dup(G). In differential geometry of surfaces a surface is construed as a zero thickness film. Because of this, Dupin’s indicatrices of the five different types are distinguished in differential geometry of surfaces (Figure 5.4 a–e). Dupin’s indicatrix for a plane local surface patch does not exist. In the case of plane, all points of Dupin’s indicatrix are remote to infinity. For local surface patches having negative full curvature (Gg < 0), phantom branches (that is, the branches that are not intersected by a plane perpendicular to the unite normal vector, ng, to the gear tooth flank G at a point m) of the characteristic curve Dup(G) in Figures 5.4d and e are shown in dashed lines. An easy way to derive an equation of the characteristic curve Dup(G) is discussed immediately below.

116


Euler’s formula

k1. g cos 2 ϕ + k2. g sin 2 ϕ = k g

(5.26)

yields representation in the form

k1. g k 2. g cos 2 ϕ + sin 2 ϕ = 1 kg kg

(5.27)

Transition from polar coordinates to Cartesian coordinates can be performed using well-known formulas

x g = ρ cos ϕ y g = ρ sin ϕ

(5.28)

(5.29)

These formulas make the following expressions for cos 2 ϕ = x g2 /ρ2 and sin 2 ϕ = y g2 /ρ2 possible. After substituting the last formulas into Equation 5.27, one can come up with the equation

k1. g x g2 k 2. g y g2 ⋅ + ⋅ = 1. k g ρ2 k g ρ2

(5.30)

It is convenient to designate ρ = k g−1 . Principal curvatures k1.g and k2.g are the roots of the quadratic equation

L g − Eg k g M g − Fg k g

M g − Fg k g =0 N g − Gg k g

(5.31)

Substituting the calculated values of the principal curvatures k1.g and k2.g into Equation 5.30, and after performing necessary formulae transformations, an equation* for Dupin’s indicatrix Dup(G) can be represented in the form

k1. g x g2 + k2. g y g2 = 1

(5.32)

* The same equation of Dupin’s indicatrix could be derived in another way. Coxeter [13] considers a pair of conics obtained by expanding an equation in Monge’s form z = z(x, y) in a Maclauren series: 1 1 z = z(0, 0) + z1 x + z2 y + ( z11 x12 + 2z12 xy + z22 y 2 ) + = (b11 x 2 + 2b12 xy + b22 y 2 ) 2 2

This gives the equation (b11 x 2 + 2b12 xy + b22 y 2 ) = ±1 of Dupin’s indicatrix.


117

Equation 5.32 describes a particular case of Dupin’s indicatrix, which is represented in Darboux’ frame.* The general form of equation of Dupin’s indicatrix at a point m of a gear tooth flank, G, is often represented as [38] Dup( G ) ⇒

2Mg Ng 2 Lg 2 xg + xg y g + yg = 1 Eg Gg Eg G g

(5.33)

In Equation 5.33, the characteristic curve Dup(G) is expressed in terms of the fundamental magnitudes Eg, Fg, and Gg of the first, Φ1.g, and in terms of the fundamental magnitudes Lg, Mg, and Ng of the second order, Φ2.g at a point of the gear tooth flank, G. 5.2.2 Matrix Representation of Equation of Dupin’s Indicatrix of the Gear Tooth Flank Like any other quadratic form, equation of Dupin’s indicatrix of the gear tooth flank, G, can be represented in the matrix form as

Dup( G ) ⇒ [x g

yg

0

 Lg  E g   2Mg 0] ⋅   Eg G g  0   0

2Mg Eg G g

0

Ng Gg

0

0 0

∓1 0

 0   xg   yg  0  ⋅   = ±1  0   0  0   (5.34) 1 

In the Darboux frame this equation reduces to

Dup( G ) ⇒ [x g

yg

0

 Lg M g 0] ⋅   0   0

Mg Ng 0 0

0 0 ∓1 0

0  xg  0   y g  ⋅ = ±1 (5.35) 0  0     1  0 

It is convenient to implement matrix representation of equation of Dupin’s indicatrix (see above), for instance, when investigating spatial gearings, that is, crossed-axis gearings, when multiple coordinate system transformations are required. * Jean Gaston Darboux (August 13, 1842–February 23, 1917), a French mathematician.

118


The equation of Dupin’s indicatrix can be represented in the form

rDup (ϕ ) =

Rg (ϕ ) ⋅ sgn Φ2−.1g

(5.36)

The last equation reveals that the position vector of a point of Dupin’s indicatrix, Dup(G), in any direction is equal to the square root of the radius of curvature in that same direction.*

5.3 Degree of Conformity at a Point of Contact of Gear and Mating Pinion Tooth Flanks in the First Order of Tangency For an accurate analytical description of contact geometry of the gear and the mating pinion tooth flanks in the first order of tangency, a higher order analysis is needed. The method discussed below of a higher order analysis is targeting the development of an analytical description of the degree of conformity of the pinion tooth flank P to the gear tooth flank G at a current point K of their contact. The higher the degree of conformity of the tooth flanks G and P, the closer these surfaces are to each other in differential vicinity of the point K. This qualitative (intuitive) definition of degree of conformity of two smooth regular surfaces is needed in a corresponding quantitative measure. 5.3.1 Preliminary Remarks Implementation of the resultant deviation lcnf (Figure 5.5) [38] of two smooth regular surfaces in contact for the analytical description of contact geometry of two surfaces in contact is a type of straightforward solution to the problem under consideration. This approach is proven to be computationally ineffective. However, the approach gives an insight for how an effective method for solving the problem under consideration can be developed. As shown in Figure 5.5, three geometrical parameters are interrelated when a deviation of a surface from the tangent plane is considered in differential vicinity of a surface point. They are

a. the measure of the deviation, mg m*g , of a gear tooth flank G from the tangent plane, lcnf

* Similar to Dupin’s indicatrix Dup(G), a planar characteristic curve of another type can be introduced. Equation of this characteristic curve can be postulated in the form rDup.k (ϕ ) = |k g (ϕ )| ⋅ sgn Φ2−.1g . Application of the curvature indicatrix in the form rDup.k (ϕ ) makes avoiding uncertainty possible in cases of a plane surface. For a plane surface, the characteristic curve Dup(G) does not exist, while rDup.k (ϕ ) exists. It shrinks to the point m on the surface G.


119

Og

Rg

mg

tg Tangent plane

K

Km*g

0

mg m*g

lcnf

m*g

FIGURE 5.5 On transition from the resultant deviation, lcnf , of two tooth flanks to indicatrix of conformity Cnf(G/P) at a contact point K between a smooth regular tooth flanks G and P.

b. the distance, Km*g,, of a current point mg from the contact point K c. radius of normal curvature Rg of the gear tooth flank G at the contact point K

As a consequence from this relationship among the parameters mg m*g , Km*g, and Rg, any one of them can be used for the purposes of quantitative evaluation of the degree of conformity of the contacting tooth flanks G and P of the gear and the mating pinion. As it is following from Figure 5.5:

mg m*g = R g − Rg2 − (Km*g )2

mg → K

lcnf

(5.37)

Inversely, for the radius of normal curvature Rg at a point of the gear tooth flank, G, the following expression is valid: Rg =

(mg m*g )2 + (Km*g )2 2 ⋅ mg m*g

(5.38)

Ultimately, one may conclude that any legitimate analytical function of normal radii of curvature Rg and Rp at a point of contact of the gear tooth flank G and the pinion tooth flank P can be used for this particular purpose. Consider two smooth regular tooth flanks, G and P, in the first order of tangency that make a contact at a point K. The degree of conformity of the tooth flanks G and P can be construed as a function of the radii of normal curvature Rg and Rp of the contacting surfaces. The radii of normal curvature Rg and Rp of the surfaces G and P are taken in a common normal plane section through the point K. For a specified radius of normal curvature Rg of the part surface

120


G, the degree of conformity of the surfaces depends upon the corresponding value of radius of normal curvature Rp of the pinion tooth flank P. In most cases of gear meshing, the degree of conformity at a point of contact of the tooth flanks G and P is not constant and it changes as coordinates of the contact point change. The degree of the surfaces conformity to one another depends on the orientation of the normal plane section through the contact point K and changes as the normal plane section is turning about the common perpendicular ng. This statement immediately follows from the conclusion made above that the degree of conformity at a point of contact of the tooth flanks G and P yields interpretation in terms of the radii of normal curvature Rg and Rp. The change of degree of conformity of a gear tooth flank G and a mating pinion tooth flank P due to turning of the normal plane section about the common perpendicular ng is illustrated in Figure 5.6. In Figure 5.6, just twodimensional examples are shown, for which that same normal plane section of the gear tooth flank G makes contact with different plane sections Pi of the pinion tooth flank P. 1 In the example shown in Figure 5.6a, the radius of normal curvature Rp 1 1 of the convex plane section P of the gear tooth flank P is positive (Rp > 0). The convex normal plane section of the pinion tooth surface P makes contact with the convex normal plane section (Rg > 0) of a gear tooth surface G. (a)

R1p

(b)

ng

R2p

ng K

K 1

2

Rg (c)

R3p

Rg (d)

ng

ng

K

K 3

Rg

4

R4p

Rg

FIGURE 5.6 Sections of two smooth regular tooth flanks G and P in contact by a plane through the common perpendicular ng: contacts of (a) convex-to-convex, (b) convex-to-convex (Rp2 > Rp1 ), (c) convex-toplanar, and (d) convex-to-concave local patches of the tooth flanks G and P.


121

The degree of conformity of pinion tooth flank P to the gear tooth flank G in Figure 5.6a is relatively low as both the contacting surfaces are convex. Another example is shown in Figure 5.6b. The radius of normal curvature Rp2 of the convex plane section P2 of the gear tooth flank P is also positive 1 (Rp2 > 0). However, its value exceeds the value Rp of the radius of normal 2 1 curvature in the first example (Rp > Rp ). This results in the degree of conformity of the pinion tooth flank P to the gear tooth flank G (Figure 5.6b) being greater compared to that shown in Figure 5.6a. In the next example depicted in Figure 5.6c, the normal plane section P3 of the pinion tooth flank P is represented with the locally flatten section. The radius of normal curvature Rp3 of the flatten plane section P3 approaches 3 infinity ( Rp → ∞). Thus, the inequality Rp3 > Rp2 > Rp1 is valid. Therefore, the degree of conformity of the pinion tooth flank P to the gear tooth flank G in Figure 5.6c is also gets greater. Finally, for a concave normal plane section P4 of the pinion tooth flank P that is illustrated in Figure 5.6d, the radius of normal curvature Rp4 is of negative value (Rp4 < 0 ). In this case, the degree of conformity of the pinion tooth flank P to the gear tooth flank G is the greatest of all the four examples considered in Figure 5.6. The examples shown in Figure 5.6 qualitatively illustrate what is known intuitively regarding the different degree of conformity of two smooth regular surfaces in the first order of tangency. Intuitively one can realize that in the examples shown in Figure 5.6a–d, the degree of conformity at a point of contact of two tooth flanks G and P is gradually increased. A similar observation is made for a given pair of the tooth flanks G and P when different sections of the surfaces by a plane surface through the common perpendicular ng are considered (Figure 5.7a). When rotating the plane section about the common perpendicular, ng, it can be observed that the

(a)

ng

(b) Rp

K

K

Rg FIGURE 5.7 Analytical description of contact geometry of two smooth regular tooth flanks G and P of a gear and a mating pinion: (a) contacting convex local patches of the tooth flanks G and P, and (b) various sections of the tooth flanks G and P by planes through the common perpendicular, ng.

122


degree of conformity of the gear and the pinion tooth flanks G and P is different in different configurations of the cross-sectional plane (Figure 5.7b). The abovementioned examples provide an intuitive understanding of what the degree of conformity at a point of contact of two smooth regular tooth flanks G and P means. The examples cannot be employed directly for the purpose to evaluate in quantities the degree of conformity at a point of contact of two smooth regular tooth flanks G and P. The next necessary step to be made is to introduce an appropriate quantitative evaluation of the degree of conformity of two smooth regular surfaces in the first order of tangency. In other words, how can a certain degree of conformity of two smooth regular surfaces being described analytically? 5.3.2 Indicatrix of Conformity at a Point of Contact of a Gear and a Mating Pinion Tooth Flank This section aims to introduce a quantitative measure of degree of conformity at a point of contact between two smooth regular surfaces. The degree of conformity at a point of contact of two tooth flanks G and P indicates how the pinion tooth flank P is close to the gear tooth flank G in differential vicinity of a point K of their contact, say how much the surface P is congruent to the surface G in differential vicinity of the point K. This particular type of congruency between the contacting surfaces G and P can also be construed as the local congruency of the contacting surfaces. Quantitatively, the degree of conformity at a point of contact of a smooth regular surface P to another surface G can be expressed in terms of the difference between the corresponding radii of normal curvature of the contacting surfaces. In order to develop a quantitative measure of the degree of conformity of the tooth flanks G and P, it is convenient to implement Dupin’s indicatrices, Dup(G) and Dup(P), constructed at a point of contact of the gear tooth flank G and the pinion tooth flank P, respectively. It is natural to assume that the smaller difference between the normal curvatures of the surfaces G and P in a common cross-section by a plane through the common normal vector ng results in the greater degree of conformity at a point of contact of the tooth flanks G and P. Dupin’s indicatrix Dup(G) indicates the distribution of radii of normal curvature at a point of the gear tooth flank G as it had been shown, for example, for a concave elliptic patch of the surface G (Figure 5.8). For a gear tooth flank G, the equation of this characteristic curve in polar coordinates can be represented in the form

Dup( G ) ⇒ rg (ϕ g ) = |Rg (ϕ g )|

(5.39)

where rg is the position vector of a point of Dupin’s indicatrix Dup(G) at a point of the gear tooth flank G and φg is the polar angle of the indicatrix Dup(G).


yg

yp

xp t2.g

t2.p Dup (

φ

)

cφ bφ Dup (

)

123

K

μ

t1.p t 1.g

rcnf (φ, μ)

xg

aφ rp (φ) rg (φ)

FIGURE 5.8 Derivation of equation of indicatrix of conformity Cnf R(G/P) at a point of contact of a smooth regular gear tooth flank G and the mating pinion tooth flank P, which are in the first order of tangency.

The similar is true with respect to Dupin’s indicatrix Dup(P) at a point of the pinion tooth flank P as it had been shown, for instance, for a convex elliptical patch of the cutting tool surface P (Figure 5.8). The equation of this characteristic curve in polar coordinates can be represented in the form

Dup(P ) ⇒ rp (ϕ p ) = |Rp (ϕ p )|

(5.40)

where rp is the position vector of a point of Dupin’s indicatrix Dup(P) at a point of the pinion tooth flank P and φp is the polar angle of the indicatrix Dup(P). In the coordinate plane xg yg of the local reference system xg yg zg, the equalities φg = φ and φp = φ + μ are valid. Therefore, in the coordinate plane xg yg, Equations 5.39 and 5.40 cast into

Dup( G ) ⇒ rg (ϕ ) = |Rg (ϕ )|

Dup(P ) ⇒ rp (ϕ , µ) = |Rp (ϕ , µ)|

(5.41)

(5.42)

When degree of conformity at a point of contact of the pinion tooth flank P to the gear tooth flank G is greater, then the difference between the functions rg(φ) and rp(φ, μ) becomes smaller and vice versa. The last makes the following conclusion valid.

124


5.3.2.1 Conclusion The distance between the corresponding* points of Dupin’s indicatrices Dup(G) and Dup(P) constructed at a point of contact of a gear tooth flank G and a mating pinion tooth flank P can be employed for the purpose of indication of the degree of conformity at a point of contact of the gear surface G and of the pinion surface P at the contact point K. The equation of the indicatrix of conformity Cnf R(G/P) at a point of contact of a gear tooth flank G and the a mating pinion tooth flank P is postulated of the following structure: Cnf R ( G /P ) ⇒ rcnf (ϕ , µ ) = |Rg (ϕ )|sgn Rg (ϕ ) + |Rp (ϕ , µ )|sgn Rp (ϕ , µ )

= rg (ϕ )sgn Rg (ϕ ) + rp (ϕ , µ )sgn Rp (ϕ , µ)

(5.43)

Because of the location of a point aφ of Dupin’s indicatrix Dup(G) at a point of the gear tooth flank G is specified by the position vector rg(φ), and that of a point bφ of Dupin’s indicatrix Dup(P) at a point of the pinion tooth flank P is specified by the position vector rp(φ, μ), the location of a point cφ (see Figure 5.8) of the indicatrix of conformity Cnf R(G/P) at a point of contact K of the tooth flanks G and P is specified by the position vector rcnf (φ, μ). Therefore, the equality rcnf (φ, μ) = Kcφ is observed, and the length of the straight line segment Kcφ is equal to the distance aφ bφ. In Equation 5.43 rg = |R g| is the position vector of a point of Dupin’s indicatrix of the gear tooth flank G at a point K of contact with the pinion tooth flank P and rp = |R p| is the position vector of a corresponding point of Dupin’s indicatrix of the pinion tooth flank P. The multipliers sgn Rg(φ) and sgn Rp(φ, μ) are assigned to each of the functions rg (ϕ ) = |R g (ϕ )| and rp (ϕ , µ) = |Rp (ϕ , µ)| accordingly just for the purpose of remaining the corresponding signs of the functions, that is, to maintain that same sign that the radii of normal curvature Rg(φ) and Rp(φ, μ) have. Ultimately, one can conclude that position vector rcnf of a point of the indicatrix of conformity Cnf R(G/P) can be expressed in terms of position vectors rg and rp of Dupin’s indicatrices Dup(G) and Dup(P). For the calculation of the current value of the radius of normal curvature Rg(φ) at a point of the gear tooth flank G, the equality Rg (ϕ ) =

Φ1.g Φ2.g

(5.44)

can be used. * Corresponding points of Dupin’s indicatrices Dup(P) and Dup(T) share the same straight line through the contact point K of the surfaces G and P and are located at the same side of the point K.


125

Similarly, for the calculation of the current value of the radius of normal curvature Rp(φ, μ) at a point of pinion tooth flank P, the equality Rp (ϕ , µ ) =

Φ1.p Φ2.p

(5.45)

can be employed. Use of the angle μ of local relative orientation of the tooth flanks G and P indicates that the radii of normal curvature Rg(φ) and Rp(φ, μ) are taken in a common normal plane section through the contact point K. Further, it is well known that the inequalities Φ1.g ≥ 0 and Φ1.p ≥ 0 are always valid. Therefore, Equation 5.43 can be rewritten in the following form:

rcnf = rg (ϕ )sgn Φ2−.g1 + rp (ϕ , µ)sgn Φ 2−.p1

(5.46)

For the derivation of an equation of the indicatrix of conformity Cnf R(G/P), it is convenient to use Euler’s equation for normal radius of curvature Rg(φ) at a point of the part surface G [38]: R g (ϕ ) =

R1. g

R1. g ⋅ R2. g ⋅ sin 2 ϕ + R2. g ⋅ cos 2 ϕ

(5.47)

Here, the radii of principal curvature R1.g and R 2.g are the roots of the quadratic equation:

L g ⋅ R g − Eg M g ⋅ Rg − Fg

M g ⋅ Rg − Fg =0 N g ⋅ Rg − G g

(5.48)

Recall that the inequality R1.g < R 2.g is always observed. Equations 5.47 and 5.48 allow for the expression of the radius of normal curvature Rg(φ) at a point of the gear tooth flank G in terms of the fundamental magnitudes of the first order Eg, Fg, and Gg, and of the fundamental magnitudes of the second order Lg, Mg, and Ng. A similar consideration is applicable for the pinion tooth flank P. Omitting routing analysis, one can conclude that the radius of normal curvature Rp(φ, μ) at a point of the pinion tooth flank P can be expressed in terms of the fundamental magnitudes of the first order Ep, Fp, and Gp, and of the fundamental magnitudes of the second order Lp, Mp, and Np. Finally, on the premise of the above-performed analysis, the following equation for the indicatrix of conformity Cnf R(G/P) at a point of contact of the tooth flanks G and P can be derived

126


rcnf (ϕ , µ ) =

+

LgGg cos ϕ − M g 2

EgGg sgn Φ2−.g1 EgGg sin 2ϕ + N g Eg sin 2 ϕ

EpGp sgn Φ2−.p1 LpGp cos 2 (ϕ + µ) − Mp EpGp sin 2(ϕ + µ ) + N pEp sin 2 (ϕ + µ )

(5.49) Equation 5.49 of the characteristic curve* Cnf R(P/T) is known from the late 1970s and is published in Reference 27 and (in a hidden form) in Reference 26. Analysis of Equation 5.49 reveals that the indicatrix of conformity Cnf R(G/P) at a point of contact of a gear tooth flank G and the mating pinion tooth flank P is represented by a planar centro-symmetrical curve of the fourth order. In particular cases, this characteristic curve possess also a property of mirror symmetry. Mirror symmetry of the indicatrix of conformity is observed, for example, when the angle μ of local relative orientation of the tooth flanks G and G is equal to μ = ±π × n/2, where n designates an integer number. It is important to note here that even for the most general case of gearing, the position vector of a point rcnf (φ, μ) of the indicatrix of conformity Cnf R(G/P) is not dependent on the fundamental magnitudes Fg and Fp. Independence of the characteristic curve Cnf R(G/P) of the fundamental magnitudes Fg and Fp is because of the following. The coordinate angle ωg at a point of the gear tooth flank G can be calculated by the formula ω g = arc cos

Fg Eg G g

(5.50)

It is natural that the position vector rcnf (φ, μ) of a point of the indicatrix of conformity Cnf R(G/P) is not a function of the coordinate angle ωg. Although the position vector rcnf (φ, μ) depends on the fundamental magnitudes Eg, Gg and Ep, Gp, the abovementioned analysis makes it clear why the position vector rcnf (φ, μ) does not depend on the fundamental magnitudes Fg and Fp. Two illustrative examples of the indicatrix of conformity Cnf R(G/P) at a point of contact of a gear tooth flank G and a mating pinion tooth flank P are shown in Figure 5.9. The first example (Figure 5.9a) relates to the cases * Equation of this characteristic curve is known from 1. Pat. No. 1249787, USSR, A Method of Sculptured Part Surface Machining on a Multi-Axis NC Machine, S.P. Radzevich, B23C 3/16, Filed: December 27, 1984, [27], and (in hidden form) from: 2. Pat. No. 1185749, USSR, A Method of Sculptured Part Surface Machining on a Multi-Axis NC Machine, S.P. Radzevich, B23C 3/16, Filed: October 24, 1983 [26].

127


CnfR (

(a) ng

yg

)

dcnf

CnfR (

)

Crv (

φ

)

C1.g K

Crv (

C2.p )

xg

K Crv (

np

μ )

C2.g (b)

C1.p

yg

CnfR (

)

np Crv ( K

ϕ

)

C2.g

K

C1.p

Crv (

Crv (

)

dcnf

xg

) μ

np CnfR (

)

C1.g

C2.p

FIGURE 5.9 Examples of indicatrix of conformity Cnf R(G/P) at a point of contact K of a smooth regular gear tooth flank G and a mating pinion tooth flank P in the first order of tangency. Examples of contact of (a) an elliptic-to-hyperbolic, and (b) elliptic-to-parabolic local patches of the tooth flanks G and P, and the corresponding indicatrices of conformity, Cnf R(G/P), constructed at point of contact, K.

of contact of a saddle-like local patch of the tooth surface G and of a convex elliptic-like local patch of tooth surface P. The second one (Figure 5.9b) is for the case of contact of a convex parabolic-like local patch of the tooth surface G and of a convex elliptic-like local patch of tooth P. For both cases (see Figure 5.9), the corresponding curvature indicatrices* Crv(G) and Crv(P) at the point of contact of the tooth flanks G and P are depicted in Figure 5.9 as well. The imaginary (phantom) branches of Dupin’s indicatrix Dup(G) (not labeled in * See Reference 35 for details on the curvature indicatrices Crv(G) and Crv(P).

128


Figure 5.9a) for the saddle-like local patch of the gear tooth flank G are shown as a dashed line (see Figure 5.9a). A gear tooth flank G and the pinion tooth flank P can make a contact geometrically however physical conditions of their contact could be violated. Violation of the physical conditions of contact results in the bodies bounded by the contacting surfaces G and P interfering with one another. Implementation of the indicatrix of conformity Cnf R(G/P) immediately uncovers the surfaces interference if there are any. Three illustrative examples of the violation of physical condition of contact are illustrated in Figure 5.10. When correspondence between the radii of normal curvature of the contacting tooth flanks G and P is inappropriate, then the indicatrix of conformity Cnf R(G/P) either intersects itself (Figure 5.10a), or all of its diameters become negative (Figure 5.10b,c). Another interpretation of satisfaction and violation of physical conditions of contact of two smooth regular tooth flanks G and P is illustrated in Figure 5.11. The condition of physical contact is fulfilled when all diameters of the indicatrix of conformity Cnf R(G/P) are positive. In this case, the gear tooth flank G and the mating pinion tooth flank P may contact one another like two rigid bodies do. An example of indicatrix of conformity Cnf R(G/P) for such a case is depicted in Figure 5.11a. In cases when this planar characteristic curve has negative diameters as it is schematically shown in Figure 5.11b, the physical contact between the tooth flanks G and P is infeasible. The value of the current diameter* dcnf of the indicatrix of conformity Cnf R(G/P) indicates the degree of conformity to each other of the gear tooth flank G and the mating pinion tooth flank P in a corresponding cross-section of the surfaces by the normal plane through the common perpendicular. The orientation of the normal plane section with respect to the tooth flanks G and P is specified by the corresponding central angle φ. For the orthogonally parameterized gear tooth flank G and the mating pinion tooth flank P, equation of Dupin’s indicatrices Dup(G) and Dup(P) simplifies to

Lg x g2 + 2 M g x g y g + N g y g2 = ±1 Lp x p2 + 2 M p x p y p + N p y p2 = ±1

(5.51) (5.52)

After been represented in a common reference system, use of Equation 5.51 and 5.52 makes possible a simplified equation of the indicatrix of conformity Cnf R(G/P) at a point of contact of the tooth flanks G and P: * Diameter of a centro-symmetrical curve can be defined as a distance between two points of the curve, measured along the corresponding straight line through the center of symmetry of the curve.

129


(a)

CnfR ( ng

Crv (

C1.p

Crv (

CnfR (

C2.g C2.p

)

C1.g

K

yg

)

Crv (

) min < 0 dcnf

CnfR (

)

yg

ng K

CnfR (

C2.g

)

C1.p C2.p

C1.g Crv (

np

)

xg

K

np

(b)

)

Crv (

)

xg

K

Crv (

)

Crv (

)

)

min < 0 dcnf

(c)

Crv (

)

ng

yg C1.p C1.g

C2.g np K

K

C2.p min < 0 dcnf

Crv (

CnfR (

Crv (

) Crv (

)

)

xg

)

FIGURE 5.10 Examples of violation of physical condition of contact of a smooth regular gear tooth flank G and a mating pinion tooth flank P : cases of interference of (a) elliptic and hyperbolic, (b) hyperbolic and hyperbolic, and (c) parabolic and parabolic local patches of the tooth flanks G and P.

rcnf (ϕ , µ) = (Lg cos 2 ϕ − M g sin 2ϕ + N g sin 2 ϕ )−0.5 sgn Φ 2−.g1

−1 + [Lp cos 2 (ϕ + µ) − M p sin 2(ϕ + µ ) + N p sin 2 (ϕ + µ )]−0.5 sgn Φp.T

(5.53)

Equation 5.53 is valid for the orthogonally parameterized tooth flanks G and P.

130


(a)

(b) CnfR (

dcnf

yg

)

CnfR (

CnfR (

)

CnfR (

yg

)

C2.g C2.p

φ

C1.g

C2.p

)

K

C2.g

μ

xg

C1.g C1.p

xg

K

min dcnf < 0

C1.p

FIGURE 5.11 Another interpretation of satisfaction (a) and of violation (b) of condition of physical contact of a smooth regular gear tooth flank G and the mating pinion tooth flank P.

5.3.3 Directions of Extremum Degree of Conformity at Point of Contact of a Gear and a Mating Pinion Tooth Flanks The directions, those along which the degree of conformity at a point of contact of a gear tooth flank G and a mating pinion tooth flank P is extremum, that is, the degree of conformity reaches either maximal of its value or minimal of its value, are of prime importance for engineering applications. This issue is especially important for synthesizing a favorable gear pair. The directions of extremal degree of conformity of the contacting smooth regular tooth flanks G and P, that is, the directions pointed along the extremax min mal diameters dcnf and dcnf of the indicatrix of conformity Cnf R(G/P), can be found from the equation of the indicatrix of conformity Cnf R(G/P). For the reader’s convenience, the equation of this characteristic curve is transformed and is represented in the form rcnf (ϕ , µ ) = | r1. g cos 2 ϕ + r2. g sin 2 ϕ |sgn Φ −21. g

+ | r1. p cos 2 (ϕ + µ ) + r2. p sin 2 (ϕ + µ )|sgn Φ −2.1p

(5.54)

Two directions within the common tangent plane are specified by the angles φmin and φmax. These directions feature an extremum degree of conformity of the pinion tooth flank P to the gear tooth flank G. Actually, the angles are the roots of the equation

∂ rcnf (ϕ , µ ) = 0 ∂ϕ

(5.55)


131

It can be easily proved that in a general case of contact of two smooth regular tooth flanks G and P, the difference between the angles φmin and φmax is not equal to 0.5π. This means that the equality ϕ min − ϕ max = ±0.5πn

(5.56)

is not always observed, and in most cases the relationship ϕ min − ϕ max ≠ ±0.5πn

(5.57)

is valid (here n is an integer number). The condition [see Equation 5.56] φmin = φmax ±0.5πn is fulfilled only in cases when the angle μ of local relative orientation of the contacting surfaces G and P is equal to μ = ±0.5πn, and thus the principal directions t1.g and t2.g of the gear tooth flank G and the principal directions t1.p and t 2.p of the mating pinion tooth flank P are either aligned to each other or are directed oppositely. This enables one to make the following statement: Statement 5.1 In the general case of contact of two smooth regular tooth flanks, the directions along which the degree of conformity of the tooth flanks G and P is extremal are not orthogonal to one another. This statement is important for engineering applications. Solution to equation ∂rrel(φ)/∂φ = 0 returns two extremal angles φmin and φmax = φmin + 90° [where rrel(φ) denotes the position vector of a point of Dupin’s indicatrix at a point of the surface of relative curvature]. Equation 5.55 * * allows for two solutions ϕ min and ϕ max . Therefore, the extremal difference * ∆ϕ min = ϕ min − ϕ min , as well as the extremal difference ∆ϕ max = ϕ max − ϕ *max , can be easily calculated. Generally speaking, neither the extremal difference Δφmin, nor the extremal difference Δφmax is equal to zero. They are equal to zero only in particular cases, say when the angle μ of local relative orientation of the surfaces G and P fulfills the relationship μ = ±0.5πn. Let us consider an example that illustrates the statement 5.1. EXAMPLE 5.1 As an illustrative example, let us describe analytically the contact geometry of two convex parabolic patches of the contacting tooth flanks G and P (Figure 5.12). The example pertains to finishing a helical involute gear by a disk-type shaving cutter. In the example under consideration, the design parameters of the gear and of the shaving cutter, along with the specified the gear and the shaving cutter configuration, yield the following numerical data for the calculation. At the point K of the tooth flanks contact, principal curvatures of the gear tooth flank G are equal to

132


yg CnfRim ( Crv (

Crv (

Δφ2

)

π 2

)

ˆ2 φ

π 2

Δφ1

max tcnf

φ2

CnfR (

)

φ1 ˆ1 φ

K

)

xg

min

CnfR (

dcnf

)

Dup ( CnfRim(

) )

FIGURE 5.12 Example 5.1: Determination of the optimum instant kinematics for a gear shaving operation.

k1.g = 4 mm−1 and k2.g = 0. Principal curvatures of the mating pinion tooth flank P are equal to k1.p = 1 mm−1 and k2.p = 0. The angle μ of the local relative orientation of the tooth flanks G and P is equal to μ = 45°. Two approaches can be implemented for the analytical description of the contact geometry of the tooth flanks G and P. The first one is based on implementation of Dupin’s indicatrix of the surface of relative curvature. The second one is based on the application of the indicatrix of conformity Cnf R(G/P) constructed at a contact point K of the tooth flaks G and P. The First Approach In the case under consideration, normal curvature kR of the surface of relative curvature, R, can be analytically expressed as

k R = k 1. g cos 2 ϕ − k1. p cos 2 (ϕ + µ )

(5.58)

Therefore, the following equality

∂k R = −2k1. g sin ϕ cos ϕ + 2k 1. p sin(ϕ + µ )cos(ϕ + µ) = 0 ∂ϕ (5.59)

is valid for the directions of the extremum degree of conformity of the tooth flaks G and P at every point of their contact. For the directions of the extremal degree of conformity at the point of contact of the gear tooth flank G and the mating pinion tooth flank P,


133

Equation 5.59 yields the calculation of the extremal values φmin = 7° and φmax = φmin + 90° = 97° of the angles φmin and φmax. The direction that is specified by the angle φmin = 7° indicates the direction of the minimal diameter of Dupin’s indicatrix of the surface of relative curvature. That same direction corresponds to the maximal degree of conformity at the point of contact of the tooth flanks G and P. Another direction, that is specified by the angle φmax = 97°, indicates the direction of the minimum degree of conformity of the contacting tooth flanks G and P at that same contact point. The Second Approach For the case under consideration, use of Equation 5.49 of the indicatrix of conformity Cnf R(P/T) at a point of contact of the gear tooth flank G and the mating pinion tooth flank P makes the calculation of the extremal angles ϕ *min = 19 and ϕ *max = 118 possible. Imaginary branches of the indicatrix of conformity Cnf Rim ( G /P ) at the point of contact of the tooth flanks G and P in Figure 5.12 are depicted as a dashed line. It is important to stress the readers’ attention here to two issues. First, the extremal angles φmin and φmax that are calculated using the first approach are not equal to the corresponding extremal angles ϕ *min and ϕ *max that are calculated using the second approach. The relation* * ships ϕ min ≠ ϕ min and ϕ max ≠ ϕ max are generally observed. Second, the difference Δφ* between the extremal angles ϕ *min and ϕ *max * * is not equal to half of π. Therefore, the relationship ϕ max − ϕ min ≢ 90° * * between the extremal angles ϕ min and ϕ max is observed. In the general case of contact of two sculptured surfaces, the directions of the extremal degree of conformity of the gear tooth flank G and the mating pinion tooth flank P are not orthogonal to one another. The discussed example reveals that in the general cases of contact of two smooth regular tooth flanks, the indicatrix of conformity Cnf R(G/P) can be implemented for the purpose of accurate analytical description of the contact geometry of the surfaces. Dupin’s indicatrix of the surface of relative curvature can be implemented for this purpose only in particular cases of the surfaces G and P relative orientation. Application of Dupin’s indicatrix of the surface of relative curvature enables only approximate analytical description of the geometry of contact of the surfaces. The Dupin’s indicatrix of the surface of relative curvature could be equivalent to the indicatrix of conformity only in degenerate cases of contact of the surfaces. Advantages of the indicatrix of conformity over Dupin’s indicatrix of the surface of relative curvature are due to the fact that this characteristic curve, Cnf R(G/P), is a curve of the fourth order.

5.3.4 Important Properties of Indicatrix of Conformity CnfR (G/P ) at a Point of Contact of the Gear and the Mating Pinion Tooth Flanks The performed analysis of Equation 5.49 of the indicatrix of conformity Cnf R(G/P) at a point of contact of a gear tooth flank and a mating pinion

134


tooth flank reveals that this characteristic curve possesses the following important properties.

1. Indicatrix of conformity Cnf R(G/P) at a point of contact of the tooth flanks G and P is a planar characteristic curve of the fourth order. It possesses the property of central symmetry and, in particular cases, also the property of mirror symmetry. 2. Indicatrix of conformity Cnf R(G/P) is closely related to the surfaces G and P second fundamental forms Φ2.g and Φ2.p. This characteristic curve is invariant with respect to the kind of parameterization of the tooth flanks G and P, but its equation is. A change in the surfaces G and P parameterization leads to the fact that the equation of the indicatrix of conformity Cnf R(G/P) also changes, while the shape and parameters of this characteristic curve remain unchanged. 3. The characteristic curve Cnf R(G/P) is independent of the actual value of the coordinate angle ωg that forms the coordinate lines Ug and Vg on the gear tooth flank G. It is also independent of the actual value of the coordinate angle ωp that forms the coordinate lines Up and Vp on the mating pinion tooth flank P. However, parameters of the indicatrix of conformity Cnf R(G/P) depend upon the angle μ of local relative orientation of the tooth flanks G and P. Therefore, for a given pair of the tooth flanks G and P, the degree of conformity of the surface varies correspondingly to the variation of the angle μ while the pinion tooth flank P is spinning around the unit vector of the common perpendicular. 4. More properties of the indicatrix of conformity Cnf R(G/P) at a point of contact of a gear tooth flank G and a mating pinion tooth flank P can be outlined. 5.3.5 Converse Indicatrix of Conformity at a Point of Contact of the Gear and the Mating Pinion Tooth Flanks in the First Order of Tangency For Dupin’s indicatrix Dup(G/P) at a point of the surface of relative curvature, R, there exists a corresponding inverse Dupin’s indicatrix Dupk(G/P). Similarly, for the indicatrix of conformity Cnf R(G/P) at a point of contact of the tooth flanks G and P, there exists a corresponding converse indicatrix of conformity Cnf k(G/P). This characteristic curve can be expressed directly in terms of the surfaces G and P normal curvatures kg and kp:

cnv Cnf k ( G /P ) ⇒ rcnf (ϕ , µ ) = |k g (ϕ )| ⋅ sgn Φ2−.1g − |k p (ϕ , µ )| ⋅ sgn Φ2−.1p (5.60)

For derivation of an equation of the converse indicatrix of conformity Cnf k(G/P), Euler’s formula for a surface normal curvature is used in the following representation:

135


k g (ϕ ) = k1. g cos 2 ϕ + k 2. g sin 2 ϕ

(5.61)

k p (ϕ , µ ) = k1. p cos 2 (ϕ + µ ) + k 2. p sin 2 (ϕ + µ)

(5.62)

where in Equations 5.61 and 5.62, the principal curvatures of the gear tooth flank G are designated as k1.g and k2.g, while the principal curvatures of the mating pinion tooth flank P are designated as k1.p and k2.p. After substituting Equations 5.61 and 5.62 into Equation 5.60, one can come up with the equation cnv rcnf (ϕ , µ ) = |k1. g cos 2 ϕ + k 2. g sin 2 ϕ|sgn Φ2−.1g

− |k1. p cos 2 (ϕ + µ) + k 2. p sin 2 (ϕ + µ )|sgn Φ2−.1p

(5.63)

for the converse indicatrix of conformity Cnf k(G/P) at a point of contact of the tooth flanks G and P in the first order of tangency. Here, in Equation 5.63, principal curvatures k1.g, k2.g and k1.p, k2.p can be expressed in terms of the corresponding fundamental magnitudes Eg, Fg, and Gg of the first and Lg, Mg, and Ng of the second order of the gear tooth flank G, and in terms of the corresponding fundamental magnitudes Ep, Fp, and Gp of the first and Lp, Mp, and Np of the second order of the mating pinion tooth flank P. Following this way, Equation 5.63 of the inverse indicatrix of conformity Cnf k(G/P) can be cast to the form that is similar to Equation 5.49 of the ordinary indicatrix of conformity Cnf R(G/P) at a point of contact of the tooth flanks G and P. It can be shown that similar to the indicatrix of conformity Cnf R(G/P), the characteristic curve Cnf k(G/P) also possess the property of central symmetry. In particular cases of the surfaces contact, it also possesses the property of mirror symmetry. The directions of the extremal degree conformity of the gear tooth flank G and the mating pinion tooth flank P are orthogonal to one another only in degenerate cases of the surfaces contact. Equation 5.63 of the converse indicatrix of conformity Cnf k(G/P) is convenient for implementation when

a. either the gear tooth flank G, or b. mating pinion tooth flank P, or c. both of them

feature point(s) or line(s) of inflection. In the point(s) or (line(s)) of inflection, radii of normal curvature Rg(p) of the surface G(P) are equal to infinity. Points/lines of inflection cause indefiniteness when calculating the position vector rcnf (φ, μ) of a point of the characteristic curve Cnf R(G/P). Equation 5.63 of the converse indicatrix of conformity Cnf k(G/P) is free of the disadvantages of such kind and therefore it is recommended for practical applications.

6 On the Impossibility to Cut Gears for Conformal and High-Conformal Gearing Using Generating (Continuously-Indexing) Machining Processes In the discussion of machinability of gears for conformal and high-conformal gearing in a generating machining process, that is, in a gear hobbing operation, gear shaping process, worm grinding, shaving operation, etc, it is convenient to begin with the analysis of meshing of the gear to be machined and the gear cutting tool to be applied. The mesh of the gear to be machined with the gear cutting tool is commonly referred to as the gear machining mesh. It is shown earlier (see Figure 3.7) that both conformal and high-conformal gearings represent a degenerate case of a corresponding involute gearing. In involute gearing, the line of action, LA, is a straight line through the pitch point, P, of the gear pair. This line forms the transverse pressure angle, ϕt, with the perpendicular to the center-line, ℄. The path of contact, Pc, in involute gearing aligns with the line of action, LA, that is, the identity Pc ≡ LA is valid. It is instructive to note here that involute gearing is the only type of parallel-axis gearing for which the identity Pc ≡ LA is observed. Satisfaction of the condition Pc ≡ LA makes involute profiles conjugate to each other. No other shapes of the gear teeth allow for the path of contact, Pc, to be congruent to the line of action, LA. No other shapes of the gear teeth are conjugate to one another—gear teeth of other geometries can be enveloping to each other, but not conjugate to one another. Involute gearing is a unique gearing from this perspective. When either conformal or high-conformal gears rotate, the only point of the gear tooth profile, that is, the involute tooth point of the gear, interacts with the corresponding involute tooth point of the mating pinion. The rest of the tooth profiles of both the gear and the pinion are inactive and do not interact with each other. Moreover, the inactive portions of the gear and the pinion tooth profiles are not conjugate to one another. Because of this, a certain freedom is given to the gear designer when designing an inactive portion of the gear and the pinion tooth profiles. In contrast to the operating of conformal and high-conformal gearing, machining gears for conformal and high-conformal gear pairs, not point contact of the tooth flanks, is required in the gear machining mesh, as is observed in mesh of the tooth flanks G and P, of the gear and the mating 137

138


pinion. Instead, line contact between the tooth flanks of the gear, G, and of the gear cutting tool, T, is required in the gear machining mesh. This means that conjugacy of the tooth profiles G and T is a must when machining gears for conformal and high-conformal gearings. The same is valid with respect to the P to T machining mesh. The required line contact between the tooth flanks of the gear, G, and the gear cutting tool, T, in the gear machining mesh causes a huge difference in meshing compared to that when the high-conformal gears operate. When a high-conformal gear spins in a gear machining mesh (Figure 6.1), the gear tooth flank G and the machining surface T of the gear cutting tool contact each other along a line, that is, along the characteristic line E. The characteristic line E is a spatial 3D curve that intersects the plane of drawing in Figure 6.1 at a current contact point, Ki. The instant line of action, LAinst, at every contact point, Ki, is along the unit normal vector, ni (Figure 6.2), and therefore, it is tangent at Ki to the path of contact, Pc. The tangent to Pc at Ki is the instant line of action, LAinst. The instant line of action, LAinst, subdivides the center line ℄ at a current pitch point Pi. The instant pitch point, Pi, subdivides the center distance C onto two portions Og Pi and Op Pi (i.e., C = Og Pi + Op Pi). The instant ratio Og Pi/Op Pi is reciprocal to the instant gear ratio in the gear machining mesh, that is, Og Pi ωp = Op Pi ωg

(6.1)

Gear tooth profile

Common tangent

Ki+1

Common tangent L

C

Instant (i + 1)th line of action

Ki

Pi+1

Pi

P

Rack-cutter tooth profile

Path of contact

FIGURE 6.1 Instant lines of action, LAinst, and paths of contact, Pc, in generating (continuously indexing) machining processes of gears for conforming and high-conforming gearing.

139

To Cut Gears for Conformal and High-Conformal Gearing

where ωg and ωp are the rotations of the gear and the mating pinion correspondingly. When the contact point travels along the path of contact, Pc, the instant line of action intersects the center distance, ℄, at different points, Pi. This means that when the gear and the gear cutting tool rotate in a gear machining mesh, the rotation of the gear can be steady, while the rotation of the gear cutting tool must be unsteady. The last is impossible, for example, because there could be two contact points Ki and Ki+1 simultaneously, and, as a consequence, two instant pitch points, Pi and Pi+1. A rigid body, that is, the gear cutting tool cannot spin at two different angular velocities simultaneously. Another evidence of impossibility of machining gears for high-conformal gearing in generating (continuously-indexing) machining processes is as follows. Assume that a gear is sliced by planes perpendicular to the gear center line on an infinite number of slices. As gears for high-conformal gearing are always helical, each slice is turned about the center line through a certain angle. Because of this, the contact point within each slice is shifted in relation to each other along the path of contact. This immediately results in that there is an instant pitch point, Pi, for every slice. The orientation of unit normal vectors, ng and n*g, to the gear tooth flank, G, at different points, Ki and K* (Figure 6.2), within the instant line of contact, LC, in generating (continuously-indexing) machining processes of gears for conformal and highi conformal gearing is different. The instant lines of action, LAinst , are along the corresponding perpendicular to the gear tooth flank, G.

Og LAiinst

n*g

Line of contact

Ki

K∗

ng

FIGURE 6.2 Orientation of unit normal vectors, ng and n*g , to the gear tooth flank, G, at different points, Ki and K*, of the line of contact, LC, in generating (continuously indexing) machining processes of gears for conforming and high-conforming gearing.

140


Once there is a plurality of the pitch points in the gear machining mesh, machining of a gear for high-conformal gearing in a generating (continuously-indexing) gear machining process becomes impossible. It could be thought that in the cases when the active portion of the path of contact in the gear machining mesh is close enough to a straight line, the gear, as well as the pinion tooth flanks, G and P, can be generated within the tolerance band for the accuracy of the tooth profile. This is correct in general consideration, but is not practical because of the following. The geometry of the path of contact in the gear machining mesh is strongly correlated to the geometry of the tooth flank of the gear to be machined. The radii of curvature of the gear tooth flank of both the gear, G, and the pinion, P, is close to the radius, rN, of the boundary N-circle. The geometry of the boundary N-circle is different from that of the involute curve. As a result, the path of contact for a gear tooth profile with a limited mismatch from the boundary N-circle is different from that of a straight path of contact that occurs for involute tooth profiles. Therefore, only a wide tolerance band for the gear tooth profile accuracy makes an approximation of the actual path of contact in the gear machining by a straight line segment possible. Wide tolerances for the gear tooth profile accuracy are not applicable for conformal and, especially high-conformal gearings.

7 Kinematics of a Gear Pair The main purpose of a gear pair is to transmit and transform a motion from the input shaft to the output shaft. The kinematics of a gear pair includes rotations of the driving and driven gears about their axes, instant rotations of the driving and the driven gears in relation to each other, and axial and profile sliding of tooth flanks of the mating gears.

7.1 Vector Representation of Gear Pair Kinematics The kinematics of a gear pair comprises two rotations: (1) rotation of the gear with the rotation vector, ωg, about the gear axis of rotation, Og, and (2) rotation of the pinion with the rotation vector, ωp, about the pinion axis of rotation, Op. The instant screw motion of the gear in relation to the pinion, as well as the instant screw motion of the pinion in relation to the gear, can be determined based on the rotation vectors ωg and ωp, and the actual configuration of the axes Og and Op. Making use of the rotations ωg and ωp allows for the determination of axial and profile sliding of the tooth flanks of the mating gears. Ultimately, the kinematics of a gear pair can be entirely expressed in terms of the two rotation vectors* ωg and ωp [39,40]. Consider the most general case of gearing when the axes of rotation of the gear and pinion are skewed. In this general case, the configuration of the rotation vectors, ωg and ωp, can be expressed in terms of the center distance, C, and the crossed-axis angle Σ. 7.1.1 Concept of Vector Representation of Gear Pair Kinematics Referring to Figure 7.1, consider a crossed-axis gear pair (Ca-gearing) together with the associated rotation vectors ωg and ωp. A Cartesian coordinate system, XYZ, is associated with the Ca-gear pair. The rotation vectors ωg and ωp are at a center distance, C. The center-distance is the closest distance of approach

* Angular velocities are treated in this book as vectors directed along the corresponding axis of rotation in a direction defined by the right-hand screw rule. It is understood here and below that the rotations, ωg and ωp, are not vectors in nature. However, rotations can be treated as vectors with special care.

141

142


ωg L

C

ωp ωp Op

C

Z

ωg Og

X

Y

ωg L

C

ωp

Pln

Ag

Apa Ap

Op

C

ωpl

Og FIGURE 7.1 On the concept of vector representation of the kinematics of a gear pair with constant tooth ratio, u: the rotation vectors, ωg and ωp, of a crossed-axis gear pair are at a certain center- distance, C, from each other, and cross at a crossed-axis angle, Σ.

between the axes Og and Op. In a particular case under consideration, the crossed-axis angle, Σ, is equal to 90°. The rotation vectors of the gear, ωg, and the pinion, ωp, are in fact types of sliding vectors. They can be applied at any point within the gear axis, Og, and the pinion axis, Op, respectively. It is convenient to apply the rotation vectors, ωg and ωp, at points of intersection of the corresponding axes of rotation Og and Op by the centerline, ℄. In a case when the axes Op and Og intersect (i.e., when C = 0), it is convenient to apply the rotation vectors ωg and ωp at the point of intersection [39,40]. The last case is a degenerate case when the center distance, C, is zero (C = 0). The magnitude of rotation of the gear, ωg, is ωg = |ωg|, whereas the magnitude of rotation of the pinion, ωp, is ωp = |ωp|. The magnitudes of rotations ωg and ωp are synchronized with each other in a timely manner.

143

Kinematics of a Gear Pair

(a)

(b) L1

Σ

(c) L1

A1

A2 Σ*

L2

s1

Σ s2

A1

L2

L1

s2

A2

A1

A2

L2 s1

Σ

FIGURE 7.2 Definition of the crossed-axis angle for a gear pair. Parts (a)–(c) are discussed in the text.

The crossed-axis angle, Σ, is measured between the rotation vectors ωg and ωp, that is, the equality

Σ = ∠(ω g , ω p )

(7.1)

is observed for a gear pair. A more detailed explanation is required to make clear the concept of the crossed-axis angle, Σ. Consider two straight lines, L1 and L2, for which directions are not specified (Figure 7.2a). In the case under consideration, the straight line A1 A2 is the center-distance, C. Angular configuration of the straight lines L1 and L2 can be specified either by the acute angle, Σ, or by the obtuse angle, Σ*. The specifications of the crossed-axis angle of the straight lines L1 and L2 by means of the angles Σ and Σ* are equivalent to one another as long as the directions of the straight lines L1 and L2 are not specified. Once the directions of the straight lines L1 and L2 are specified (i.e., the directions are specified by unit vectors s1 and s2), it is easy to see when the crossed-axis angle Σ is acute (Figure 7.2b) and when it is obtuse (Figure 7.2c). Thus, no duality in specification of the crossed-axis angle, Σ, is observed for rotation vectors ωg and ωp of a gear pair. The use of rotation vectors, ωg and ωp, makes possible construction of the vector of instant rotation, ωpl, of the pinion in relation to the gear (or vice versa, the vector of instant rotation of the gear in relation to the pinion). Two options are available in this regard: First, the gear pair can be rotated about the pinion axis, Op, with the rotation vector, −ωp. Under this scenario, the pinion becomes stationary [ωp + (−ωp) = 0], and the resultant rotation of the gear is equal to the following

ω gp = (ω g − ω p )

(7.2)

Such a situation corresponds with the case of rotation of the gear in relation to the pinion, when the latter is considered motionless.

144


Second, the gear pair can be rotated about the gear axis, Og, with the rotation vector, −ωg. Under this scenario, the gear becomes motionless [ωg + (−ωg) = 0], and the resultant rotation of the pinion is equal to the following ω pg = (ω p − ω g )

(7.3)

Such a situation corresponds with the case of rotation of the pinion in relation to the gear, when the latter is considered stationary. Evidently, the rotation vectors, ωgp and ωpg, are opposite to each other (ωgp = −ωpg). In addition to the vector diagrams for rotation vectors ωg and ωp, corresponding vector diagrams can be constructed for torque vectors. Torque on the gear shaft is denoted by Tg, and torque on the pinion shaft is designated as Tp. One of the torques (commonly, the torque Tp) is the input torque, while another (usually, the torque Tg) is the output torque. An example of vector diagrams for the input and output torques Tg and Tp is schematically illustrated in Figure 7.3. In Figure 7.3a, a vector diagram for rotation vectors ωg and ωp is shown. Then a corresponding vector d iagram for the input and output torques Tg and Tp is constructed for the case in which the pinion is driving and the gear is driven (Figure 7.3b). This configuration corresponds to a case of reduction gears. In Figure 7.3c, a vector diagram for the input and output torques Tg and Tp, which is constructed for the case when the gear is driving and the pinion is driven, is shown. This configuration corresponds to a case of increasing gears.

(a)

(b)

ωg Ag

ωpl

Op

Ap

ωp Og

Og

Pln Apa

(c)

Ag Tp

Tg

C

Pln Ag

Op

Apa Ap C

C

Tg

Op

Pln

Apa Ap

Tp

Og

FIGURE 7.3 Vector diagram for the rotations, and two torque diagrams for a gear pair: (a), vector diagram for a crossed-axis gear pair, and two torque diagrams for the same gear pair corresponding to (b), the case of reduction gearing, when the pinion is driving and the gear is driven, and (c), the case of increasing gearing, when the gear is driving and the pinion is driven.


145

In both cases, the torque vectors Tg and Tp are pointed in the same direction, in contrast to the direction of the rotation vectors ωg and ωp. The actual direction of the torque vectors, Tg and Tp, depends on which of the two elements is the driving element, and which is the driven element (see Figure 7.3b,c). Torque diagrams can be constructed for all external and internal gearings, and gearing featuring crossing axes of rotation, as well as when the axes of rotation of the driving and driven shafts are parallel to one another. 7.1.2 Three Different Types of Vector Diagrams for Spatial Gear Pairs If two axes of rotations are positioned in space and the task is to transmit a motion and torque between them using gears of some type, then only three different types* of spatial (crossed-axis) gear pairs are distinguished. They are as follows: • Gear pairs for which the apex point Apa is located within the center-distance C. Commonly, gearings of this type are referred to as the external spatial gearing. However, in this case, gearings can be designed as the internal gearings as well [24,38]. • Gear pairs for which the apex point Apa is located outside the center-distance C. Commonly, gearings of this type are referred to as the internal spatial gearing. However, in this case, gearings can be designed as the external gearings as well [24,38]. • Rack-type spatial gear pairs for which the apex point Apa coincides with the gear apex Ag [38]. No other types of spatial gear pairs are feasible, and any known or newly designed gear pair falls into one of the three aforementioned types of spatial gear pairs. * It is preferable to represent these three possible types of spatial gearing in the following manner: • Negative-gearing: (external gearing and internal gearing), that is, gear pairs for which the apex point Apa is located within the center distance C [The term “negative” is related to the cases when the direction of the rotation changes by the gear pair, and therefore, the gear ratio, u, is of a negative value, (u < 0)]. • Positive-gearing: (internal gearing and internal gearing), that is, gear pairs for which the apex point Apa is located outside the center distance C [The term “positive” is related to the cases when the direction of the rotation is remained the same, and therefore, the gear ratio, u, is of a positive value, (u > 0)]. • Zero-gearing, that is, gear pairs, for which the apex point Apa coincides with the gear apex Ag [The term “zero” is related to the cases of rack-type gearing, when the gear ratio, u, is zero, (u = 0)].

146


Type of spatial gear pair, whether an external, internal, or rack-type spatial gear pair, depends on • The magnitudes of the rotations, ωg and ωp, of the gear and of the pinion, respectively • The crossed-axis angle, Σ, between the rotation vectors ωg and ωp • The center-distance, C Before proceeding with the analysis of vector diagrams, some new terminologies need to be introduced. Consider the vector diagram for an arbitrary gear pair given in Figure 7.4. The rotation vectors ωg and ωp of the gear and the pinion are at a certain center-distance, C, and they cross one another. Points Ag and Ap are points of intersection of the gear axis of rotation, Og, and the pinion axis of rotation, Op, respectively, with the centerline, ℄. The point Ag is referred to as the “gear apex,” and the point Ap is referred to as the “pinion apex.” The vector of instant rotation, ωpl, of the pinion in relation to the gear is a vector through the point Apa. This point is located within the centerline, ℄. The point Apa is referred to as the “plane of action apex.” The axis of instant rotation, Pln, is the straight line through the point Apa along the vector of instant rotation, ωpl. This straight line is also referred to as “pitch line.” Two straight lines through a common point uniquely specify a plane through these two lines. In the case under consideration, this is the plane Xln Zln

ωpl

Cln-plane Nln-plane

ωp

Ap

Op

Apa

Yln

Pln-plane

Ag

Pln

C Og

ωg

L

C

FIGURE 7.4 On the definition of the pitch-line plane (Pln-plane), centerline plane (Cln-plane), and normal plane (Nln-plane) in Ca-axes (spatial) gearing.


147

through the axis of instant rotation, Pln, and through the centerline, ℄. The plane is referred to as the pitch-line plane. Definition 7.1 The pitch-line plane (Pln-plane) in a crossed-axis gear pair is the reference plane through the centerline and the axis of instance rotation of the gear and the pinion. In a case of parallel-axis gearing as well as intersected-axis gearing, the pitch-line plane reduces to the plane through the axis of rotation of the gear and the pinion. For intersected-axis gearing, as well as for parallelaxis gearing, Pln-plane can be also defined as the plane through the axis of rotation of the gear, and the axis of rotation of the pinion. Two more important planes can be introduced here. They are so-called the centerline plane, (Cln-plane), and the normal plane, (Nln-plane), correspondingly: Definition 7.2 The centerline plane (Cln-plane) in a crossed-axis gear pair is the reference plane through the centerline perpendicular to the axis of instance rotation of the gear and the pinion. In a case of crossed-axis gearing, the centerline plane is perpendicular to the axis of instance rotation of the gear and the pinion. Definition 7.3 The normal plane (Nln-plane) in a crossed-axis gear pair is the referenced plane, the axis of instance rotation of the gear, and the pinion perpendicular to the centerline. The three planes, that is, Pln-plane, Cln-plane, and Nln-plane are of significant importance in the theory of gearing, and in the theory of high- conformal gearing in particular. They are referred to as the fundamental planes of a gear pair. A fundamental Cartesian reference system XlnYlnZln is associated with a gear pair. The axes of the reference system XlnYlnZln are along the lines of intersection of the fundamental planes as shown in Figure 7.4. With that said, let us consider vector diagrams for each type of spatial gear pairs in more detail [39,40].

148


7.1.2.1 Vector Diagrams of External Spatial Gear Pairs A vector diagram that is constructed for a certain combination of the rotation vectors ωg and ωp, crossed-axis angle, Σ, and center-distance, C, corresponds to an external spatial gear pair.* With two rotation vectors ωg and ωp, the corresponding vector of instant rotation, ωpl, of the pinion in relation to the gear can be constructed (ωpl ≡ ωpg = −ωgp). The instant rotation, ωpl, is performed about a straight line, Pln, which is the axis of instant rotation. The vector of instant rotation, ωpl, as well as other kinematical parameters of an external gear pair, can be determined graphically by implementing the methods developed in descriptive geometry. An example of such a construction is illustrated in Figure 7.5. For the purposes of construction of a vector diagram, a reference system π1π2 of two orthogonal planes of projection, π1 and π2, is implemented (Figure 7.5a). Following a convention adopted in descriptive geometry, the subscript “1” is assigned to projections onto plane π1 of all points, lines, and so on. Similarly, the subscript “2” is assigned to projections onto plane π2 of all points, lines, etc. The location and orientation of a pair of the rotation vectors ωg and ωp within the reference system π1π2 can be arbitrary. For convenience, the rotation vectors, ωg and ωp, are depicted in the reference system π1π2 parallel to the horizontal plane of projection π1. In this scenario, the crossed-axis angle, Σ, is projected onto plane π1 with no distortion. The centerline is projected onto plane π1 into a point. This point is denoted as C. Let us assume that a rotation, −ωg, is applied to a gear pair, that is, to the gear, pinion, and housing. The rotation −ωg does not affect the relative motion of the gear and the pinion. Under the additional rotation −ωg, the gear becomes stationary [ωg + (−ωg) = 0]. The rotation of the gear pair housing (denoted by rotation vector −ωg) is opposite to the rotation vector of the gear (denoted by ωg). Ultimately, the rotation of the pinion is the superposition of two rotation vectors, namely, rotation vectors ωp and −ωg. The resultant of two rotations ωp and −ωg is the instant rotation ωpl = (ωp − ωg) of the pinion about the pitch line Pln. Within the horizontal plane of projection π1, the vector of instant rotation, ωpl, can be determined as the vector difference of the rotation vectors ωg and ωp. Onto the plane of projection π1, the vector ωpl is projected with no distortion since the rotation vectors ωg and ωp are parallel to π1. The vector ωpl is applied at a certain point Apa within the centerline ℄. In a particular case, the plane of action apex, Apa, and the pitch point P can coincide with each other. Immediately after the rotation vector ωpl is determined, the axis of projections π1/π2 can be constructed so that it is parallel to the vector of instant rotation, ωpl. Such a configuration of the axis π1/π2 is not mandatory; the * See footnote * on page 145.

149


(a)

~ rp

Pln

Ap

ωprl

Op ωpl

π2 π3

ωpsl

Vgsl

Apa

r~g

π1

ωpl

Op

ωgrl

Ag

–ωg

Apa ωpsl

ωp

ωg

Σ > Σcr

(b)

Apa

C

Og

π2

ωpsl

Vpsl

ωpsl ωgsl Pln

Og

p

L

C

Op

ωpsl

ωg

Ap ωprl

Apa Ag

ωpl

ωp

Og

C g

Σ > Σcr

FIGURE 7.5 Vector diagram of an external crossed-axis (spatial) gear pair. Parts (a) and (b) are discussed in the text.

configuration can be arbitrary. Convenience is the only reason for selecting this particular orientation for the axis of projections π1/π2 in relation to the rotation vector, ωpl. Projections of the rotation vectors ωg and ωp onto the frontal plane of projections π2 are designated as ω rlg and ω rlp , respectively. The components ω rlg and ω rlp of the rotation vectors ωg and ωp are parallel to the axis of instant rotation, Pln. These components cause pure rolling of the a xodes of the gear and of the pinion. The following ratio [39,40]

rp rg C = = ω g ⋅ cos Σ g ω pl ω p ⋅ cos Σ p

(7.4)

150


is valid for magnitudes ω p, ωg, and ωpl of the rotation vectors ωg, ωp, and ωpl. In Equation 7.4, the distance between the apex, Apa, and the gear axis, Og, is designated as rg. The distances of the same point Apa from the pinion axis, Op, is designated as rp. The distances rg and rp are signed values. For an external gear pair, both of them are positive (i.e., rp > 0 and rg > 0). The angles Σg and Σp are specified by the following equalities: Σ g = ∠(ω g , ω pl )

Σ p = ∠(ω p , ω pl )

(7.5)

(7.6)

Evidently, the equality

rp + rg = C

(7.7)

is valid for an external spatial gear pair. The condition of pure rolling can be employed for the determination of the location of plane of the action apex, Apa, within the centerline, ℄. In compliance with the condition, the following ratio rg ω rlp = rl rp ωg

(7.8)

should be fulfilled. In Equation 7.8, the designations ω rlg =| ω rlg | and ω rlp =| ω rlp | are used. Generally speaking, magnitudes ω rlg and ω rlp of the vectors of pure rolling, rl ω g and ω rlp , are not equal to each other. The inequality, ω rlg < ω rlp , is commonly observed. The equality ω rlg = ω rlp is observed only in particular cases when the tooth number of the gear, Ng, and tooth number of the pinion, Np, are equal to each other (i.e., Ng = Np). From Equation 7.7, the distance rg can be expressed in terms of center- distance, C, and the distance rp

rg = C − rp

(7.9)

Substituting this expression for distance rg in Equation 7.8, a formula rp =

ω rlg ⋅C ω rlp + ω rlg

for calculating the distance rp can be derived.

(7.10)

151


Further, Equation 7.9 can be used for calculating the distance rg. After substituting Equation 7.10 in Equation 7.9, the equality rg = C − rp can be transformed as follows: rg =

ω rlp ⋅C ω + ω rlg rl p

(7.11)

For external spatial gear pairs, the plane of action apex, Apa, is located within centerline between the gear axis, Og, and the pinion axis, Op. Two others components, ω slg and ω slp , of the rotation vectors ωg and ωp are perpendicular to the axis of instant rotation, Pln. With no distortion these components are projected onto the frontal plane of projections, π3. The plane of projections, π3, is perpendicular to the axis of projections, π1/π2. The rotations ω slg and ω slp cause pure sliding of the axodes of an external spatial gear pair with respect to each other. Magnitudes ω slg =| ω slg | and ω slp =| ω slp | are equal (ω slg = ω slp ). The vectors ω slg and ω slp are pointed in opposite directions (i.e., ω slp = −ω slg ). Relative sliding of the axodes is created by both the gear and by the pinion. The vector of linear velocity of sliding that is created by the gear is equal to

Vgsl = rg ⋅ ω slg

(7.12)

Similarly, the vector of linear velocity of sliding that is created by the pinion is equal to

Vpsl = rp ⋅ ω slp

(7.13)

The expressions | ω slg |=| ω slp | and rg ≥ rp are valid for an external spatial gear pair; then, the component of sliding velocity, Vgsl, that is caused by the gear exceeds or is equal to the component of sliding velocity, Vpsl, that is caused by the pinion, that is, the inequality| Vgsl |≥| Vpsl |is always observed. The vectors of sliding velocities, Vgsl and Vpsl, are opposite to each other. The vector of the resultant velocity of sliding, Vgsl− p, of the gear in relation to the pinion is equal to the difference

Vgsl− p = Vgsl − Vpsl

(7.14)

The vector of the resultant velocity of sliding, Vpsl− g , of the pinion in relation to the gear is opposite to the vector Vgsl− p

Vpsl− g = − Vgsl− p = Vpsl − Vgsl

(7.15)

152


The magnitude of speed of the resultant sliding in an external spatial gear pair can be calculated from the following formula: Vsc = Vgsl + Vpsl

(7.16)

If the component vectors ω slg and ω slp are of the same magnitude and are opposite to each other, then they comprise a so-called pair of rotation. An equivalent velocity vector of the translation motion, Vsc, can be constructed for a given pair of rotations. The velocity vector, Vsc, is parallel to the vector of instant rotation, ωpl. The following formula Vsc = Vsc = C ⋅ ω p ⋅ sin Σ p = C ⋅ ω g ⋅ sin Σ g

(7.17)

can be used for calculating the magnitude of vector Vsc. Ultimately, the resultant instant relative motion of the gear and the pinion is composed of • An instant rotation, ωpl, about the pitch line, Pln • An instant translation, Vsc, along the pitch line Pln Superposition of the rotation, ωpl, and the translation, Vsc, results in a screw motion. The parameter of screw motion is designated as psc. The screw parameter, psc, is also often referred to as the reduced pitch. For the calculation of the reduced pitch, psc, the following formula is applied [39,40]: psc =

C ⋅ ω p ⋅ sin Σ p C ⋅ ω g ⋅ sin Σ g Vsc = = ω pl ω pl ω pl

(7.18)

An expression ω pl =

C ⋅ ω p ⋅ cos Σ p C ⋅ ω g ⋅ cos Σ g = rg rp

(7.19)

for the calculation of magnitude of instant rotation can be derived from Equation 7.4. Therefore, the parameter of a screw motion can be calculated from the following formula:

psc = rp ⋅ tan Σ g = rg ⋅ tan Σ p

(7.20)

This immediately returns the following proportion:

rp tan Σ g = rg tan Σ p

(7.21)


153

The resultant instant motion of the gear and the pinion can be construed as rolling with sliding of two hyperboloids of one sheet over each other. One of the hyperboloids, Ag, is associated with the gear, while the other hyperboloid, Ap, is associated with the pinion. In one particular case, the gear hyperboloid, Ag, can be considered stationary. In such a scenario, instant rotation is performed by the pinion hyperboloid, Ap. The hyperboloid Ag, which is associated with the gear, is generated by the axis of instant rotation, Pln, when the axis is rotated about the gear axis Og. Similarly, the hyperboloid Ap, which is associated with the pinion, is generated by the axis of instant rotation, Pln, when the axis is rotated about the p inion axis, Op. The instant rotation occurs about the pitch line, Pln. The instant translation is observed in the direction, parallel to the pitch line, Pln. As schematically shown in Figure 7.5b, two axodes, Ag and Ap, contact each other along the axis of instant rotation, Pln. The vectors, which are used for describing the kinematics of an external spatial gear pair are also depicted in Figure 7.5b. It should be mentioned here that the axodes Ag and Ap are shown just for illustrative purposes. The use of axodes for the analysis of the kinematics of gear pairs has been proved to be inconvenient because axodes cannot be drawn easily and are less informative compared to the vector diagrams. Because of this, axodes of the gear and the pinion have very limited use in this book. In all possible cases, axodes are replaced with the corresponding vector diagrams, which are more informative and can be drawn much more easily. 7.1.2.2 Vector Diagrams of Internal Spatial Gear Pairs A vector diagram for an internal spatial gear pair is constructed similar to that for an external spatial gear pair (see Figure 7.5). The similarity allows us to focus attention mostly on the peculiarities of vector diagrams for internal spatial gear pairs [39,40]. Consider an internal spatial gear pair* for which a set of parameters for which (ωg, ωp, Σ, and C) is given. An example of a vector diagram for an internal spatial gear pair is shown in Figure 7.6. The vector diagram (Figure 7.6) is referred to a system of two orthogonal planes of projections, π1 and π2. The vector of instant rotation, ωpl, is constructed as the difference of the rotation vectors ωg and ωp. In the case under consideration, the equality ωpl = ωp − ωg is valid. The vector of instant rotation, ωpl, is constructed so it is parallel to the plane of projections π1. Therefore, the rotation vector, ωpl is projected onto the reference plane π1 with no distortions. Similar to that above (see Figure 7.5), those components of the rotation vectors ωg and ωp that cause pure rolling of the axodes are designated as ω rlg and ω rlp , respectively. * See footnote * on page 145.

154


(a)

r~p

ωpl

Pln

π 2 π3 Apa

sl Ap ωprl Vg

Op r~g

ωpsl

π1

Op

ωgsl

–ωg

ωpl

Pln

Apa

ωg

ωp

ωpsl

ωgrl

Ag

π2

Apa

C

ωgsl

Og

(b)

Vpsl

ωpsl Σ < Σcr

Og C

L

C

Op

Apa p

ωg

ωpl ωp

Og

Σ < Σcr

g

FIGURE 7.6 Vector diagram of an internal spatial gear pair. Parts (a) and (b) are discussed in the text.

For an internal spatial gear pair, the plane of action apex, Apa, is located outside the center-distance, C. Instead, the pinion axis of rotation, Op, intersects the centerline, ℄, at a point located between the point Apa and the point of intersection of the centerline, ℄, by the gear axis of rotation, Og. Hence, the following equality

− rp + rg = C

(7.22)

is valid for an internal spatial gear pair. Equation 7.22 allows expression rg = C + rp. Making use of this equality and taking into account the conditions of pure rolling of the axodes, the following formulas

155


rg = rp =

ω rlp ⋅C ω rlp − ω rlg ω rlg ⋅C ω rlp − ω rlg

(7.23) (7.24)

for the calculation of distances rg and rp can be derived. Two others components, ω slg and ω slp , of the rotation vectors, ωg and ωp, cause pure sliding of the axodes of the gear and the pinion relative to each other. With no distortion, these components are projected onto the frontal plane of projections, π3. As it is already shown with respect to an external spatial gear pair, the sliding components, ω slg and ω slp , of the rotation vectors, ωg and ωp, are of equal magnitude and are opposite to each other (i.e., ω slg = −ω slp ). The vector of linear velocity of sliding that is created by the gear is equal to

Vgsl = rg ⋅ ω slg

(7.25)

Similarly, the vector of linear velocity of sliding that is created by the pinion is equal to

Vpsl = rp ⋅ ω slp

(7.26)

The following expressions: | ω slg |=| ω slp | and rg ≥ rp are valid for an internal spatial gear pair. Thus, the magnitude of the component Vgsl of sliding v elocity, caused by the gear, exceeds or is equal to the magnitude of the component, Vpsl, of sliding velocity caused by the pinion; that is, the inequality | Vgsl |≥| Vpsl | is always observed. The vectors of sliding velocities, Vgsl and Vpsl , are opposite to each other. The vector, Vgsl− p , of the resultant velocity of sliding of the gear in relation to the pinion is equal to the following difference:

Vpsl− g = − Vgsl− p = Vgsl − Vpsl

(7.27)

The vector, Vgsl− p , of the resultant velocity of sliding, of the pinion in relation to the gear is opposite to vector Vpsl− g :

Vgsl− p = − Vpsl− g = Vgsl − Vpsl

(7.28)

The magnitude of speed of the resultant sliding in an internal spatial gear pair can be calculated from the following formula:

Vsc = Vgsl + Vpsl

(7.29)

156


Similar to that of an external spatial gear pair, the components, ω slg and ω slp , of the rotation vectors ωg and ωp comprise a pair of rotation for an internal gear pair. The pair of rotation is equivalent to a straight motion. This allows for a formula for the calculation of the magnitude Vsc of speed of the resultant sliding similar to Equation 7.17. Two axodes, Ag and Ap, of a gear and a mating pinion, along with corresponding rotation vectors, ωg and ωp, are schematically illustrated in Figure 7.6. Again, the axodes, Ag and Ap, are significantly less informative in comparison with corresponding vector diagrams. It is inconvenient to draw the axodes for illustrative purposes. Therefore, in further discussions in this chapter, preference is given to vector diagrams rather than to axodes of a gear and of the mating pinion. 7.1.2.3 Vector Diagrams of Generalized Rack-Type Spatial Gear Pairs The performed analysis of external and internal spatial gear pairs makes it reasonable to assume that gear pairs with intermediate kinematics similar to that the gear-to-rack pair is for a cylindrical external and internal gear pairs are also feasible and they exist. Spatial gear pair of this nature is referred to as generalized rack-type spatial gear pairs. A generalized rack-type spatial gear pair* can be interpreted as the degenerate (critical) case of either external or internal spatial gear pairs when the tooth number of the gear (in external and internal spatial gearing) approaches infinity. In other words, there must exist a generalized rack-type gear pair as the limit case of either an external (Figure 7.5) or an internal (Figure 7.6) spatial gear pair. Without going into a detailed analysis of the vector diagrams depicted in Figure 7.4, it can be said that for an external spatial gear pair the gear angle, Σg, between the rotation vector, ωg, and the vector of instant rotation, ωpl,

Σ g = ∠(ω g , ω pl ) > 90

(7.30)

is an obtuse angle (see Figure 7.5). For an internal spatial gear pair, the gear angle, Σg, between the rotation vector, ωg, of the gear and the vector of instant rotation, ωpl

Σ g = ∠(ω g , ω pl ) < 90

(7.31)

is an acute angle (see Figure 7.6). It is reasonable to question the case when the gear angle, Σg, between the rotation vector, ωg, of the gear, and the vector of instant rotation, ωpl, is a right angle (i.e., ωg ⊥ ωpl)? * See footnote * on page 145.

157


The vector diagram of a spatial gear pair for which the equality

Σ g = ∠(ω g , ω pl ) = 90

(7.32)

is valid is shown in Figure 7.7. In the case under consideration, the axode of the gear, Ag, (a hyperboloid of one sheet), is reduced to a plane that is rotated about an axis perpendicular to the plane. The axode of the pinion, Ap, (a hyperboloid of one sheet) is reduced to a cone of revolution. The gear pair, for which the vector diagram is shown in Figure 7.7, can be interpreted as the case of rolling of the cone of revolution over the rotating plane. A spatial gear featuring this type of kinematics is referred to as generalized rack-type spatial gear pair. A critical value, Σcr, of the crossed-axis angle, Σ, corresponds to a generalized rack-type spatial gear pair. In other words, if the condition in Equation 7.32 is fulfilled then the equality Σ = Σcr is observed. Within the plane through the centerline, ℄, the linear speed, Vgsl , of the sliding of the axodes is due to the component ω slg of the rotation vector, ωg, of the gear. This component, ω slg , is produced by the rotating gear. Although the component ω slp of the rotation vector, ωp, is not equal to zero (i.e., ω slp ≠ 0), the linear velocity, Vpsl , is equal to zero (Vpsl = 0). The last equality is possible because the equality rg = C is valid for generalized rack-type spatial gear pairs. The equality rg = C entails the equality rp = 0. Ultimately, the resultant linear velocity, Vsc, of the sliding of the axodes in the case under consideration is equal to

Vsc = Vgsl

(7.33)

It must be stressed here that not every case of the rolling of a cone of revolution over the rotating plane corresponds with a generalized rack-type s patial gear pair. It is critical that the condition in Equation 7.32 is fulfilled in this regard. The vector diagrams of generalized rack-type spatial gear pairs is of particular interest in the designing of gear cutting tools for the machining of hypoid and Spiroid gears, and so forth [34]. 7.1.2.4 Analytical Criterion of a Type of Spatial Gear Pairs The angle formed by the rotation vector of a gear, ωg, with the vector, ωpl, of instant rotation of the pinion in relation to the gear, is the root cause for the principal difference between spatial gear pairs of different types, that is, between external, internal, and generalized rack-type gear pairs. These differences analytically are described by Equations 7.31 and 7.32. The equality

ω pl = ω p − ω g

is observed for a spatial gear pair.

(7.34)

158


(a)

π2 π3 Pln

Op

ωprl

ωpl

Vpsl

Apa ωpsl

Vgsl

Apa

ωpsl C ωgsl

Og

π2 π1

ωpl

Pln

Apa ωg

ωp

Op

ωgsl

–ωg

Osl

Σ = Σcr

Og

(b) L

C

Op Pln

Apa

–ωg

ωpsl

ωprl

ωgsl

ωp

ωg

ωpl

Og p

Σ = Σcr

C

g

FIGURE 7.7 Vector diagram of a generalized rack-type spatial gear pair. Parts (a) and (b) are discussed in the text.

159


TABLE 7.1 Analytical Criteria of a Type of Crossed-Axis Gear Pairs Type of Crossed-Axis Gear Pairs External spatial crossed-axis gear pair Generalized rack-type crossed-axis gear pairs Internal spatial crossed-axis pairs

Analytical Criterion ωg ⋅ (ωp − ωg) < 0 ωg ⋅ (ωp − ωg) = 0 ωg ⋅ (ωp − ωg) > 0

Equations 7.31, 7.32, and 7.34 make the representation of the analytical criteria of a type of a spatial gear pair as shown in Table 7.1 possible. Analytical expressions specifying the criteria for the spatial gear pair are composed of the premises of well-known properties of the dot product of two vectors.

7.2 Classification of Possible Types of Vector Diagrams of Gear Pairs Possible types of the vector diagrams of gear pairs can be classified based on the vector representation of gear pair kinematics. Such a classification is necessary for many purposes. The development of all possible types of gears, and then of all possible types of gear pairs, is one of the reasons for the development of the classification. Crossed-axis (spatial) gear pairs are considered in this book as the most general type of gear pairs. The remaining possible types of gear pairs can be interpreted as a reduction (simplification) of the corresponding type of the crossed-axis gear pairs. As stated at the beginning of this section of the book, there are only three different types of gear pairs those featuring crossed axes (see footnote * on page 145): • External crossed-axis gear pair • Generalized rack-type crossed-axis gear pair • Internal crossed-axis gear pair No other types of spatial gear pairs are feasible. External crossed-axis gear pairs are well-known and are widely used in the industry. For all types of external spatial gear pairs, the inequality ωg ⋅ (ωp − ωg) < 0 is observed (see Table 7.1). The component Σg of the shaft angle, Σ, exceeds right angle (i.e., Σg > 90°). An external crossed-axis gear pair can feature shaft angle of various values. In particular, the shaft angle, Σ, can be either acute (i.e., 0° < Σ < 90°), or it can be equal to Σ = 90°, or obtuse.

160


Three types of crossed-axis gear pairs comprise the first stratum of the classification of possible types of the vector diagrams of gear pairs as shown in Figure 7.8 (see footnote * on page 145): • External gear pairs • Generalized rack-type gear pairs • Internal spatial gear pairs Numbers 1.1, 1.2, and 1.3 are assigned to spatial gear pairs comprising the first stratum of the classification. High-conformal gear pairs can be designed according to each of the vector diagrams in the possible types of the vector diagrams of gear pairs as shown in Figure 7.8. Crossed-axis gear pairs can be reduced to gear pairs of a simpler design. There are two possible ways for the reduction: First, the center-distance, C, can be of zero value, and, second, the gear and the pinion axes of rotation, Og and Op, can be parallel to each other. In the second case, the crossed-axis angle, Σ, is equal to either Σ = 180° or Σ = 90°. Let us begin the consideration from the first case when the center-distance, C, of an external intersected-axis gear pair is reduced to zero. When the equality C = 0 is observed, the gear and the pinion axes of rotation, Og and Op, intersect each other at a point, Apa. The rotations vectors, ωg and ωp, are two vectors through the point Apa. They are along the axes Og and Op, respectively. For gear pairs of this type, it is convenient to investigate the engagement of the gear teeth on a sphere centering at the point Apa. Due to this, intersectedaxis gear pairs are loosely referred to as “spherical gear pairs.” The name “spherical” is because the tooth profiles of the gear and the pinion in this case are generated on spheres.*

7.3 Complementary Vectors to Vector Diagrams of Gear Pairs It is convenient to introduce a few more vectors for analytical description of a gear pair. Vectors along the centerline, as well as those along the gear and the pinion axes of rotations, are of particular importance.

* The term “spherical gear pair” is incorrect as gears of other types, for example, crossed-axis gear pairs, are also engaged in mesh on a sphere. Therefore, replacement of the obsolete and widely used term “conical gear pair” with the term “spherical gear pairs” is not valid. In order to avoid ambiguities in further discussions, gearing of this type is referred to as “intersected-axis gearing,” or just as “Ia -gearing” for simplicity.

161


Vector diagrams of gear pairs

External gear pairs (Σg > 90º)

ωpl

ωp

1.1

Ap

Σg

ωg

Op

Ag

Σp

Σ

Apa ω g

Ag

1.2.1

Apa

ωpl

C

Apa

Og

Ag

ωp

ωpl

Σp

ω Σ –ωg Apa g

ωg

1.3.1

Ap ~

ωpl 1.1.2

Op

C –ωg

Σ

Σ

Vg

Apa

Ap

Σp

Og

0.5 dp

ωp

Ag

Ap

Σg

ωp

ωpl

ωp

Op

1.3

Apa

ωg

ωp

–ωg

1.1.1

ωpl

Σg

Σg

Σp ωp

ωg

ωg

Σ

Og

–ωg

C –ωg

C

Σ

ωpl

Apa

Σp

Internal gear pairs (Σg < 90º)

1.2

ωpl

ωp

Apa Σp

Generalized rack-type gear pairs (Σg = 90º)

Vp

Σ

1.2.2

ωp

ωg

Ap

Ag

ωpl Apa

C

1.3.2 ωpl

ωp

ωpl ≡ 0

Vg

Ap

Apa

1.2.2.1

1.3.2.1

Ag ≡ Ap ≡ Apa

Vector diagrams of gear pairs

External gear pairs (Σg > 90º)

Generalized rack-type gear pairs (Σg = 90º)

Internal gear pairs (Σg < 90º)

1.1

1.2

1.3

1.1.1

1.1.2 1.1.2.1

1.2.1

1.2.2

1.3.2

1.3.1

1.2.2.1

1.3.2.1

1.3.1.1

FIGURE 7.8 Classification of possible types of vector diagrams of gear pairs.

Σg

162


7.3.1 Centerline Vectors of a Gear Pair Referring to Figure 7.9, consider the vector diagram of a gear pair.* The rotation vectors, ωg and ωp, are apart from each other by a center- distance C. A Cartesian coordinate system, XYZ, is associated with the rotation vectors, ωg and ωp, as depicted in Figure 7.9. The axis X is along the centerline of the rotation vectors ωg and ωp. This axis originates from the plane of action apex, Apa, and is pointed toward the pinion axis of rotation, Op. The Z-axis is along the axis Pln of the instant rotation, ωpl. Ultimately, the Y-axis c omplements the X and Z axes to a left-hand-oriented reference system, XYZ. Two vectors, Cg and Cp, are along the X-axis. These vectors specify the distances of the axes of the rotations of the gear, Og, and the pinion, Op, from the point Apa. The gear centerline vector, Cg, can be calculated from the following equation: C g = − rg ⋅ c

(7.35)

Another centerline vector, that is, the pinion centerline vector Cp, is specified as follows: C p = rp ⋅ c

(7.36)

In Equations 7.35 and 7.36 rg is the distance of the gear axis of rotation, Og, from the axis of instant rotation, Pln rp is the distance of the pinion axis of rotation, Op, from the axis of instant rotation, Pln c is the unit vector along the X-axis The magnitude of the centerline vector, Cg, is always greater in comparison with the magnitude of the centerline vector, Cp. Therefore, the inequality |Cg| ≥ |Cp| is observed.

* For gear pairs with varying tooth ratios, for example, for gear pairs composed of nonround gears, the parameters of the vector diagram, ωg, ωp, ωpl, C, C g, C p, Σ, Σg, Σp, and others should be considered as corresponding functions of time t, or (the same) as the corresponding functions of the angle of rotation either of the gear φg or of the pinion φp. Ultimately, these functions can be represented in a generalized way as ωg(t), ωp(t), ωpl(t), C(t), Cg(t), C p(t), Σ(t), Σg(t), Σp(t). All the parameters are synchronized with each other in a timely, proper manner.

163


Z X Op

Σp

ωp

ωprl

Ap

r~p

Apa

C Pln

r~g

ωpl

Vsc

Ap Cp

Cg

ωpsl Ag Y –ωg ωgsl

Ag ωgrl

ωg Σg

Og

Σ

L

C

FIGURE 7.9 Complementary vectors to the vector diagram of a gear pair.

7.3.2 Axial Vectors of a Gear Pair Three different locations of a gear in relation to the centerline are distinguished. First, a gear can be located such that the centerline, ℄, goes through the middle of the gear width, as schematically shown in Figure 7.10. Conventional helical gearing with skew axis of rotation features such a location for the gear and the pinion with respect to the centerline. In a more general case, a gear can be located at a certain distance from the centerline, ℄. These shifts in two opposite directions enable two more different locations of a gear in relation to the centerline, ℄. The second configuration features a positive shift of the gear in relation to the centerline, that is, to the position above the centerline, ℄, in Figure 7.10. The third configuration features a negative shift of the gear in relation to the centerline, that is, to the position below the centerline, ℄, in Figure 7.10. A hypoid gear pair is a perfect example of a gear pair with the gear and the pinion shifted in axial direction of the gear and the pinion correspondingly. The actual location of the gear in relation to the centerline is specified by the axial vector, Ag, of the gear (and by the corresponding axial vector, A p, of the pinion). The axial vector, Ag, associated with the gear is along the gear axis of rotation, Og. This vector is applied at the point of intersection of the gear axis, Og,

164


Zg ωg

~

Fg Fg Ag

L

C

C

Yg

Og

Xg

ωg

FIGURE 7.10 Possible configurations of the gear in relation to the centerline, ℄, in a spatial gear pair specified by the axial vector Ag.

and the centerline (Figure 7.10). The vector Ag can be expressed in terms of two parameters ag and Ag Ag = A g ⋅ ag

(7.37)

Here, the distance along the gear axis of rotation, Og, from the centerline, ℄, to the middle of the gear face width,* Fg , is denoted as Ag. The equality Ag = |Ag| is observed. The unit vector ag is the vector along the rotation vector, ωg, of the gear. The unit vector ag is dimensionless. It can be calculated from the following formula: ag =

ωg ⋅ sgn(ω g ⋅ ω pl ) |ωg |

(7.38)

The axial vector, Ap, associated with the pinion is along the pinion axis of rotation, Op. This vector is applied at the point of intersection of the pinion * The width of a gear, Fg, and the gear face width, Fg, are not identical. The width, Fg, of a cylindrical gear is equal to its face width, Fg, whereas the width of a conical gear, Fg, and its face width, Fg, correlate with each other as Fg = Fg ⋅ cos Γ. Here, the pitch angle of the conical gear is denoted by Γ.

165


axis, Op, by the centerline, ℄. The vector Ap can be expressed in terms of two parameters ap and Ap A p = Ap ⋅ a p

(7.39)

In Equation 7.39, the following are designated: Ap is the distance along the pinion axis of rotation, Op, from the centerline, ℄, to the middle of the face width Fp of the pinion ap is nondimensional unit vector along the rotation vector of the pinion, ωp; it can be calculated from the formula ag = ωg/|ωg|. The multiplier sgn(ωg ⋅ ωpl) in Equation 7.38 allows the accommodation of the unit vector ag for both kinds of gear pairs, that is, for external as well as for internal gear pairs. Once a gear is assumed stationary when determining the vector of instant rotation, ωpl, then the rotation vectors, ωp and ωpl, always form an acute angle. The multiplier sgn(ωp ⋅ ωpl) is always positive and, thus, it is not necessary to implement it in Equation 7.39. The angle between the vectors ωg and ωpl is obtuse for an external gear pair, and it is acute for an internal gear pair. Because of this, the gear and the pinion of a gear pair are located at the same side of the centerline, ℄, so the axial vectors, Ag and Ap, should always be acute. This is accounted for by the multiplier sgn(ωg ⋅ ωpl). If magnitude Ag of the vector Ag is known, then the following formula

rg =

rg2 + Ag2 ⋅ tan 2 Σ g

(7.40)

can be implemented for the calculation of pitch radius of the gear, rg. Conversely, if the pitch radius of the gear, rg, is given, then for the calculation of the axial shift of the gear the formula Ag =

rg2 − rg2 tan Σ g

(7.41)

can be used. Similar to Equations 7.40 and 7.41, the following expressions

rp =

rp2 + Ap2 ⋅ tan 2 Σ p

(7.42)

and Ap =

rp2 − rp2 tan Σ p

(7.43)

166


are valid for calculating the axial shift, Ap, and of pitch radius, rp, of a pinion. It can be easily shown that magnitude, Ap, of the axial vector, Ap, can be expressed in terms of magnitude Agof the axial vector Ag: A p = Ag

cos Σ p cos Σ g

(7.44)

Magnitudes, Ag and Ap, of the axial vectors, Ag and Ap, have the same sign. Both are positive (i.e., Ag > 0, Ap > 0), having zero value (i.e., Ag = 0, Ap = 0), or are negative (i.e., Ag < 0, Ap < 0). Consequently, three different locations of a gear in relation to the centerline can be distinguished. 7.3.3 Useful Kinematic and Geometric Formulas The proposed vector diagrams (rotations/torques) of gear pairs make it possible to derive numerous auxiliary formulas for calculating the kinematic and geometric parameters of gear pairs. For calculation of the distances, rg and rp, of the gear axis of rotation, Og, and the pinion axis of rotation, Op, from the axis of instant rotation, Pln, the following approach can be applied: Let us project the rotation vectors, ωg, ωp, and ωpl, onto a plane that is perpendicular to the centerline ℄ (Figure 7.9). The components ω rlg and ω slg of the rotation vector of the gear, ωg, and the components ω rlp and ω slp of the rotation vector of the pinion, ωp, are also depicted. The rolling components ω rlg and ω rlp of the rotation vectors, ωg and ωp, are within a plane through the centerline, ℄. The following expression can be derived on the premises of pure rolling in the gear pair:

ω rlg ⋅ rg = ω rlp ⋅ rp

(7.45)

For the distances, rg and rp , the following equality is valid:

rg + rp = C

(7.46)

If the distances, rg and rp , are considered signed values, then Equation 7.46 is valid for both, for external and internal gear pairs. The distance, rp , can be expressed in terms of the distance, rg , and the center-distance, C, as

rp = C − rg

(7.47)

167


This allows the representation of Equation 7.45 in the following form: ω rlg ⋅ rg = ω rlp ⋅ (C − rg )

(7.48)

This immediately returns a formula for the calculation of the distance, rg : rg =

1 + ωp − ωg ⋅C 1 + ωp

(7.49)

Once the distance rg is determined, for the calculation of the distance rp , Equation 7.47 can be implemented. In the case under consideration, Equation 7.47 allows the following formula: rp =

1 + ωg − ωp ⋅C 1 + ωg

(7.50)

It is right to discuss a few more formulas for the calculation of kinematic and geometric parameters of a gear pair here, which directly follow from analysis of Figure 7.9. The magnitude, ωpl, of a vector of instant rotation, ωpl, can be calculated from the following equation:

ω pl =

(ω rlg )2 + (ω rlp )2 − 2 ⋅ ω rlg ⋅ ω rlp ⋅ cos Σ

(7.51)

For calculation of the gear angle Σg between the vectors ω rlg and ωpl, the following equation can be used: Σg =

1 + ωp − ωg ⋅Σ 1 + ωp

(7.52)

Similarly, the pinion angle Σp between the vectors ω rlp and ω pl can be calculated from the equation: Σp =

1 + ωg − ωp ⋅Σ 1 + ωg

(7.53)

If the angle Σg = 90° is entered into Equation 7.52, then the expression Σ cr =

1 + ωg π ⋅ 1 + ωg − ωp 2

(7.54)

168


for the calculation of a critical value Σcr of the angle Σ between the gear axis of rotation, Og, and the pinion axis of rotation, Op, can be derived.

7.4 Tooth Ratio of a Gear Pair Gear pairs are designed and applied for two purposes:

1. Transmitting a rotation 2. Transforming a rotation

The tooth ratio of a gear pair is a design parameter, by means of which transformation of the rotation is specified. As shown in Figure 7.5 through 7.7, rotation vectors ωg and ωp can be represented as the summation of two components, ω rlg and ω slp − g , for a gear, and two components, ω rlp and ω slp , for a pinion. Transmission and transformation of rotation occurs due to the components ω rlg and ω rlp only. Components ω slg and ω slp neither transmit the rotation nor transform it. It would be natural to use the ratio of the components ω rlg and ω rlp for the evaluation of rotation transformation. Because it is not feasible to divide a vector by another vector, the ration of the components ω rlg and ω rlp is not used for specifying the tooth ratio u; instead, the ratio of magnitudes ω rlg and ω rlp is used for this purpose u=

ω rlp ω rlg

(7.55)

Both the magnitudes, ω rlg and ω rlp , in Equation 7.55 can be expressed in terms of the design parameters of the gear pair and magnitudes ωg and ωp the rotations ωg and ωp. For this purpose, the rotations ω rlg and ω rlp are expressed in terms of the magnitudes ωg and ωp and of the angles Σg and Σp. As it follows from the analysis of Figure 7.5, the following equalities are valid for the rotations ω rlg and ω rlp :

ω rlg = −ω g cos Σ g ω rlp = ω p cos Σ p

(7.56) (7.57)

The angles Σg and Σp can be calculated from Equations 7.52 and 7.53, respectively. Substituting the calculated values of angles Σg and Σp in

169


Equations 7.56 and 7.57 and then into Equation 7.55, the tooth ratio of a gear pair can be calculated based on the kinematics and design parameters of the gear pair. For an internal gear pair, the components ω rlg and ω rlp are pointed in the same direction. This means that these components are of the same sign. No change in the direction of rotation occurs in internal gearing. Therefore, the tooth ratio for an internal gear pair is a positive value (i.e., u > 0). In a particular case, the tooth ratio can be equal to one (i.e., u = 1). No rotation transformation is observed in this case. In gear couplings, for example, the rotation is just transmitted from the input shaft to the output shat, and is not transformed in this case. Because the pitch radius of an external gear is commonly considered as a positive value (i.e., rgrl > 0) and that of an internal gear is considered negative value (i.e., rgrl < 0), then the expression u=−

rgrl rprl

(7.58)

for a tooth ratio can be used instead of Equation 7.55. The negative sign allows us to avoid discrepancies when calculating the tooth ratio of a gear pair. The use of a signed value for tooth ration, u, for a gear pair has proved to be convenient in numerous applications. For an external gear pair, the components ω rlg and ω rlp are pointed in opposite directions. In external gearing, the direction of rotation of the output shaft is changed to the opposite of the input shaft. Therefore, these components have different signs. Thus, the tooth ratio for an external gear pair is of negative value (i.e., u < 0). The tooth ratio of a rack-type gear pair is equal to infinity (i.e., u = ∞). Rotation of the input shaft is transformed by the rack-type gear pair to a translation motion or vice versa. The tooth ratio u = ∞ is the maximum feasible tooth ratio of positive value. High-conformal gear pairs can be designed according to each of vector diagrams discussed in this section of the book.

8 High-Conformal Intersected-Axis Gearing High-conformal intersected-axis gearing can be used for transmitting a rotation from a driving shaft to a driven shaft that intersect one another. High-conformal gears are capable of transmitting a uniform rotation from the driving shaft to the driven shaft.

8.1 Kinematics of the Instantaneous Motion in High-Conformal Intersected-Axis Gearing For the investigation of the kinematics of instant rotation in high-conformal intersected-axis gearing, the use of a vector diagram is helpful. Referring to Figure 8.1, consider the rotation vector of the gear, ωg, and the rotation vector of the pinion, ωp. The rotation vectors, ωg and ωp, are the vectors through a common point, Apa. They form a shaft angle, Σ, with one another [i.e., Σ = ∠(ωg, ωp)]. Having constructed the rotation vectors, ωg and ωp, the rotation vector of instant relative rotation, ωpl, is constructed to fulfill the expression ωpl = ωg − ωp. Under such an assumption, the gear is considered motionless while the pinion performs an instant rotation in relation to the axis of instant rotation, Pln. Any and all intersected-axis gear pairs meet one of three expressions listed in Table 8.1. The angle between the vector of instant rotation, ωpl, and the rotation vector of the gear, ωg, is denoted by Σg. Accordingly, the angle between the vector of instant rotation, ωpl, and the rotation vector of the pinion, ωp, is designated as Σp. The angles Σg and Σp are signed values. Generally speaking, for an intersected-axis gear pair, the rotation vector of instant rotation, ωpl, does not align with the rotation vector of the gear, ωg, or with the rotation vector of the pinion, ωp. Thus, the rotation vector, ωpl, can be decomposed into two components, ω rlpl and ω slpl

ω pl = ω rlpl + ω slpl

(8.1)

The component ω rlpl of the vector of instant rotation, ωpl, is aligned with the axis of rotation of the gear, Og. This component causes pure rotation of the 171

172


ωpl Og rl ωpl

ωp

ωplsl

Σp

Apa Σg Op

ωg

Σ

Pln FIGURE 8.1 A vector diagram for a high-conformal intersected-axis gear pair.

TABLE 8.1 Analytical Criteria of the Type of Intersected-Axis Gearing Type of Crossed-Axis Gearinga

Analytical Criterion [C = 0 and Σ ≠ 0] ωg ⋅ (ωp − ωg) < 0 ωg ⋅ (ωp − ωg) = 0 ωg ⋅ (ωp − ωg) > 0

External intersected-axis gear pair Rack-type intersected-axis gear pair Internal intersected-axis gear pair a

See footnote * in Chapter 7 on page 145.

gear and the pinion. The magnitude, ω rlpl, of the rotation vector, ω rlpl, can be calculated from the formula ω rlpl = ω pl cos(180° − Σ p )

(8.2)

As the angle, Σp, can be expressed in terms of the rotations, ωg, ωp, and the shaft angle, Σ [38] Σp =

1 + ωg − ωp ⋅Σ 1 + ωg

(8.3)

Equation 8.2 casts into

 1 + ωg − ωp  ω rlpl = −ω pl cos  ⋅ Σ   1 + ωg

(8.4)

173

High-Conformal Intersected-Axis Gearing

The component ω slpl of the vector of instant rotation, ωpl, is perpendicular to the axis of rotation of the gear, Og. Therefore, the component ω slpl causes pure sliding (with no rolling) of the gear tooth flank and the pinion tooth flank. The magnitude, ω slpl, of the rotation vector, ω slpl, can be calculated from the formula

ω slpl = ω pl sin(180° − Σ p )

(8.5)

Substituting Equation 8.3 into Equation 8.5, we obtain the formula

 1 + ωg − ωp  ω slpl = ω pl sin  ⋅ Σ  1 + ωg 

(8.6)

for the calculation of the magnitude, ω slpl, of the rotation vector, ω slpl.

8.2 Base Cones in Intersected-Axis Gearing Geometrically accurate intersected-axis gear pairs (or, in other words, ideal intersected-axis gear pairs) are capable of transmitting a uniform rotation from the driving shaft to the driven shaft. From this perspective, geometrically accurate intersected-axis gear pairs resemble the previously discussed geometrically accurate parallel-axis gear pairs (Figure 2.7). The similarity between these two types of gears can be extended further [38]. Therefore, for convenience, it makes sense to consider geometrically accurate intersectedaxis gearing in comparison with geometrically accurate parallel-axis gearing, as parallel-axis gearing has been investigated much more deeply, and the meshing of the tooth flanks of the gear and of the pinion in parallel-axis gearing is better understood. Geometrically accurate parallel-axis gear pairs feature two base cylinders, as shown in Figure 2.7. Uniform rotation of the base cylinders allows for an interpretation of parallel-axis gearing as a corresponding belt-and-pulley analogy. This is also valid with respect to geometrically accurate intersectedaxis gear pairs. The base cones are associated with the gear and the pinion of any and all geometrically accurate intersected-axis gear pairs. This concept is schematically illustrated in Figure 8.2. An orthogonal intersected-axis gear pair is depicted here for illustrative purposes. Without going into details of the analysis, it should be stated here that that same approach is applicable with respect to angular bevel gears that have a shaft angle, Σ ≠ 90°. The schematic shown in Figure 8.2 is constructed based on the rotation vectors, ωg and ωp, of the gear and the pinion. The gear and the pinion rotate

174


PA

ωpa

Pln

Base cone (the pinion)

Σ

ωg Apa

ωp Γb

ωg

c1 e1

ωpl

a1

d1

Op

f1 lcp

PA

ωg , ωp Pln

ωpa PA

f5 e5

Base cone (the pinion) ωp

lcp c5

π4

Σp

d5

Feff

rl.pa

lcp

b1

Og

π1

Op

PA

lcg

Base cone (the gear)

γb

ωp

Σg

Pln

Og

ωg

ϕt.ω lcg

ωpa

Opa φpa

a5 ro.pa

lcg

b5

π5

π4


FIGURE 8.2 Base cones and the plane of action in orthogonal geometrically accurate intersected-axis gearing. (Adapted from Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, CRC Press, Boca Raton, FL, 2012, 743pp.)

about their axes, Og and Op, respectively. The rotation vectors, ωg and ωp, allow for the construction of the vector, ωpl, of instant relative rotation. The axis of instant rotation, Pln, is aligned with the rotation vector, ωpl. Based on the tooth ratio u = ωp/ωg, the corresponding ratio sin Σg/sin Σp of sines for the angles, Σg, of the gear and, Σp, of the pinion can be calculated [38] as

rp sin Σ g = r g sin Σ p

(8.7)

The plane of action, PA, is a plane through the axis, Pln, of instant rotation. The plane, PA, is in tangency with both base cones, namely with the base cone of the gear and with the base cone of the pinion. The plane of action, PA, is at a transverse pressure angle, ϕt.ω, in relation to a perpendicular to

175


the axis of instant rotation, Pln, within the plane through the rotation vectors, ωg and ωp. The pressure angle, ϕt.ω, is measured within a plane, which is perpendicular to the vector of instant rotation, ωpl. The left upper portion of the schematic in Figure 8.2 is plotted within the plane of projections, π1. Two others planes of projections, π2 and π3, of a standard set of planes of projections, π1, π2, and π3, are not used in this particular consideration. Instead, two auxiliary planes of projections, namely the planes, π4 and π5, are used. The axis of projections, π1/π4, is constructed so as to be perpendicular to the axis of instant rotation, Pln. The axis of projections, π4/π5, is constructed so as to be parallel to the trace of the plane of action, PA, within the plane of projections, π4. The plane of action can be imagined as a flexible zero thickness film that is free to wrap/unwrap from and onto the base cones. The plane of action is not allowed for any bending about an axis perpendicular to the plane, PA, itself. Under uniform rotation of the gears, the motion of the plane of action, PA, is a steady rotation about the axis Opa. The rotation vector, ωpa, is along the axis Opa. The vector ωpa is perpendicular to the plane of action. For intersected-axis gear pairs, the plane of action, PA, can be viewed as a round cone that has a 90° cone angle. As sin 90° = 1, the magnitude, ωpa, of the rotation vector, ωpa, can be calculated from the formula ω pa =

ωg ωp = sin Γ b sin γ b

(8.8)

For intersected-axis gear pairs, the base cone angles, Γb and γb, vary within the intervals 0° < Γb < 180° and 0° < γb < 180°, respectively. So, here and below all equations are valid for external, rack-type, and internal gear pairs.* Formally, the base cone angles, Γb and γb, can be considered in the narrower intervals, namely within the intervals 0° < Γb < 90° and 0° < γb < 90°, respectively. Under such a scenario, the following inequalities are valid for intersected-axis gear pairs of various types: 1. The base cone angles are of positive values (Γb > 0° and γb > 0°) for external gear pairs. 2. The base cone angle of the gear is equal to the right angle (Γb = 90° and γb > 0°) for rack-type gear pairs. 3. The base cone angles of the gear are of negative value (Γb < 0° and γb > 0°) for internal gear pairs. The face width of the plane of action, Fpa, or, in other words, the working portion of the plane of action, PA, is located between two circles of radii, ro.pa and rl.pa. The total portion of the plane of action spans within a central * See footnote * in Chapter 7 on page 145.

176


angle, ϕpa. The angle, ϕpa, is measured between the lines of contact, lcg and lcp, of the plane of action, PA, and each of the two base cones. Definition 8.1 Geometrically accurate intersected-axis gear pairs are those capable of t ransmitting a rotation smoothly. Intersected-axis gear pairs that do not allow for the construction of equivalent base cones and the plane of action, PA, are referred to as approximate intersected-axis gear pairs. The tooth flanks of approximate intersected axis gear pairs feature geometry for which no equivalent belt-and-pulley mechanism can be designed to replace the gear pair. Definition 8.2 Approximate intersected-axis gear pairs are those that are not capable of transmitting a rotation smoothly. Approximate intersected-axis gears are not discussed here as neither conformal nor high-conformal gear pairs can be designed on the premise of approximate intersected-axis gearings.

8.3 Path of Contact in High-Conformal Intersected-Axis Gearing The path of contact in a high-conformal intersected-axis gear pair is a trace of contact point when the gears rotate. Since the relative motion of the gear and the pinion is an instant r otation, ωpl, about the axis of instant rotation, Pln, the plane perpendicular to the vector of instant rotation, ωpl, at an arbitrary point, P, within the axis of instant rotation, Pln, can be constructed and the relative motion can be investigated within the normal plane (Figure 8.3). Within the normal plane, a boundary N-circle can be constructed. The center of the boundary N-circle is coincident with the point of intersection of the axis of instant rotation, Pln, by the normal plane. The radius, rN, of the boundary N-circle is equal to a desirable displacement, l, of the contact point, K [either in the positive direction to the position of the point, K+, or in the negative direction to the position of the point, K−], from the point, P (that is located within the pitch line, Pln), along the instant line of action, LAinst. The point P can be a pitch point in a particular gear pair. The desirable displacement, l (either of positive value, +l, or of negative value, −l), is a trade-off between the contact

177


Pln

Og

ωp

ϕt.ω

sl ωpr

sl ωlw

Op Σg

K–

LAinst

Apa

ωg

–l

Σ

ωplsl

rN

Σp

Pln

+l lla

P

K+

ωpl Boundary N-cone Boundary N-circle

Γl

FIGURE 8.3 Configuration of the boundary cone in a high-conformal intersected-axis gear pair.

strength of the gear teeth and the sliding of the teeth flanks of the gear and the pinion, G and P, in relation to one another. The larger the distance, l, the higher the contact strength of the gear teeth and the higher the sliding of the teeth flanks. The smaller the distance, l, the lower the contact strength of the gear teeth and the lower sliding of the teeth flanks. 8.3.1 Bearing Capacity of High-Conformal Gearing The influence of an increase in radius, rN, of the boundary N-circle onto a rise of contact strength in high-conformal gearing is schematically illustrated in Figure 8.4, where normal sections of the teeth flanks of a gear and a mating (a)

(b) r′p

r′g

r″p

r″g rrel rel

rrel

K l′cnf

δ

K l″cnf

δ

FIGURE 8.4 Impact of the magnitude of the radii of curvature of the gear tooth flank, G, and the pinion tooth flank, P, on the bearing capacity in gear pairs featuring an equal radius of relative curvature, rrel. Parts (a) and (b) are discussed in the text.

178


pinion for two high-conformal gear pairs are shown. Both normal sections feature equal radii of relative curvature, rrel. In the first case shown in Figure 8.4a, the radius of curvature of the gear tooth profile, G, is denoted by rg′ , while the radius of curvature of the pinion tooth profile, P, within that same plane is denoted by rp′ . The radius of relative curvature, rrel, of the interacting teeth flanks is equal to rrel = − rg′ − rp′ (as is adopted in this book, the radii of curvature are signed values: convex profiles feature radii of curvature of positive values while concave profiles feature radii of curvature of negative values). When a load is applied at a contact point, K, the teeth flanks, G and P, approach each other at a certain distance. This distance is designated as δ. Under the applied load, the contact point spreads over a certain area of contact. The width of the contact area within the normal plane section in this particular case is designated as l′cnf . In the second case, shown in Figure 8.4b, the radius of curvature of the gear tooth profile, G, is denoted by rg′′ while the radius of curvature of the pinion tooth profile, P, within that same plane is denoted by rp′′ . It should be stressed here that inequalities |rg′′|>|rg′| and rp′′ > rp′ take place in the consideration. The radius of relative curvature, rrel, of the interacting teeth flanks is equal to rrel = − rg′′ − rp′′. Let us assume that when a load is applied at contact point, K, the teeth flanks, G and P, approach each other at the same distance δ as in the abovementioned case (Figure 8.4a). Under the applied load, the contact point spreads over a certain area of contact. The width of the contact area within the normal plane section in this particular case is designated as lcnf ′′ . A detailed analysis is not necessary to make it evident that the arc, lcnf ′′ , is larger compared to the arc, lcnf ′ . As the inequality lcnf ′′ > lcnf ′ is valid, it becomes evident that the bearing capacity of high-conformal intersected-axis gearing depends not only on relative curvature, rrel, of the contacting tooth flanks, but also on the magnitudes of the radii of curvature of the tooth flanks, G and P, at a point of their contact. The larger the magnitudes of the radii, rg and rp, of normal curvature of the interacting teeth flanks, G and P, the greater the load capacity of high-conformal intersected-axis gearing and vice versa. Ultimately, this makes the following conclusion valid.

Statement 8.1 High-conformal gearing with the larger magnitudes of the radii of normal curvature of the tooth flanks feature higher load capacity. It is right to recall here that the magnitudes of the radii, rg and rp, of normal curvature of the interacting teeth flanks, G and P, strictly correlate to the radius rN of the boundary circle in the cross-section of the tooth flanks through the corresponding pitch point. Therefore, high-conformal gearing with the larger radius, rN, of the boundary N-circle features higher load capacity.

179


8.3.2 Sliding of Teeth Flanks in a High-Conformal Gearing The dependence of sliding of the teeth flanks, G and P, from the value of the displacement, l, is briefly discussed below. At a given point of contact of the gear tooth flank, G, and the pinion tooth flank, P, the linear velocity of sliding can be expressed in terms of the magnitude, ωpl, of the rotation vector, ωpl, and the distance of the contact point from the axis, Og. This is also true with respect to the pinion. The rotation vector, ω slpl, of sliding can be decomposed into two components (Figure 8.3) sl ω slpl = ω slpr + ω lw

(8.9)

One component, ω slpr, is along the axis of instant rotation, Pln. This component of the rotation vector of sliding, ω slpr, causes profile sliding of the tooth flank, G, of the gear and the tooth flank, P, of the pinion. The magnitude, ω slpr , of the rotation vector, ω slpr, can be calculated from the formula ω slpr = ω slpl sin(180 − Σ p )

(8.10)

Equation 8.10 casts into the formula  1 + ωg − ωp  ω slpr = ω slpl sin  ⋅ Σ + ω 1   g

(8.11)

or

 1 + ωg − ωp  ω slpr = ω pl sin 2  ⋅ Σ  1 + ωg 

(8.12)

for the calculation of the magnitude, ω slpr, of the rotation vector, ω slpr. sl Similarly, the component ω lw is perpendicular to the axis of instant rotation, Pln. The component of the rotation vector of sliding, ω slpl, causes sliding in the lengthwise direction of the tooth flank, G, of the gear and the tooth sl flank, P, of the pinion. The magnitude, ω sllw, of the rotation vector, ω lw , of sliding can be calculated from the formula

ω sllw = ω slpl cos(180 − Σ p )

(8.13)

Equation 8.13 casts into the formula

 1 + ωg − ωp  ω sllw = ω slpl cos  ⋅ Σ 1 + ω   g

(8.14)

180


or

 1 + ωg − ωp  ω sllw = ω pl sin 2  ⋅ Σ  1 + ωg 

(8.15)

sl for the calculation of the magnitude, ω sllw, of the rotation vector, ω lw . The unit vector, lla, is along the instant line of action, LAinst. The rotation vector, ω slpl, can be calculated from Equation 8.9. Then, the calculated value of ω slpl is used for the calculation of the vector of linear velocity of sliding, Vsl, in the following formula:

V sl = ω slpl × l la ⋅ l

(8.16)

The sliding velocity in high-conformal intersected-axis gearing can be determined using the methods developed in descriptive geometry of surfaces. Refer to Figure 8.5, consider a schematic of a high-conformal intersectedaxis gear pair. The linear velocity vector, Vgm, of a point m within the active portion (a helix line) in a gear tooth flank, G, is located in a plane, pl Vgm , through the point m. The plane, pl Vgm, is perpendicular to the axis of rotation, Og, of the gear (Figure 8.5a). Similarly, the linear velocity vector, Vgm, of a point m within the active portion (a helix line) in a pinion tooth flank, P, is located in a plane, pl Vgm, through the point m. The plane, pl Vgm, is perpendicular to the axis of rotation, Op, of the pinion (Figure 8.5a). m The velocity vector, Vrel , of the relative motion of the gear and the pinion m , through the point tooth flanks at a point m is located within a plane, pl Vrel m m. The plane, pl Vrel , is perpendicular to the axis of instant rotation, Pln, of the m gear and the pinion (Figure 8.5a). Later on, the velocity vector, Vrel , will be decomposed onto the velocity vector of pure rolling and the velocity vector of pure sliding of the tooth flanks, G and P. In the plane of projections, π6, the velocity vector Vgm is projected with no distortion. The magnitude of the velocity vector Vgm can be chosen arbitrarily. It is permissible to do because when calculating specific sliding of the gear, γg, and the pinion, γp, the ratio of the velocities is important (see Equations 3.24 and 3.25) and not the actual values of the velocities. The velocity vector Vgm is constructed in the plane of projections, π6, as shown in Figure 8.5b. The constructed velocity vector, Vgm, allows for a construction of projections of the m velocity vectors Vgm, Vpm, and Vrel onto the plane of projections π1. Therefore, m the corresponding projections Prπ1 Vgm, Prπ1 Vpm, and Prπ1 Vrel are considered as the known parameters. In the plane of projections, π7, the velocity vector Vpm is projected with no distortion (Figure 8.5c). The magnitude of the velocity vector Vpm can

181


(a)

Pln

ωg Apa

pl Vm g ωp

Og pl

(b)

Vm g

Opa

π6

Op

m

π1

ωpa

Vm rel

Pln PA

Pln Pc

m

Og

π4

ωpl Pc

pl Vm g

π1

Op

ϕt.ω

Vm rel

Pc

(c) ωg

ωp

Apa

pl Vm p

Op

ωpl

pl Vm g

pl

Prπ1 Vm p

Prπ1 Vm rel

Prπ1 Vm p

Op

Vm rel

m

Og

pl Vm rel m

π1 π7

pl Vm g Pln

Op

ωpl

Prπ1 Vm g

Og

π4

π5

ωp

Apa

pl Vm g

Og

m

ωg

Pln

m PA

m Vm p

Prπ1 Vm g Prπ1 Vm rel

(d)

m PrP Vm g ≡ PrP Vp ln

pl V m p

Pln Apa

φm

Op ωpl

Pc

pl Vm g

Og

ln

Pln

m Prπ1 Vm g

pl Vm rel

PrP Vm g

≠90°

ln

m PrP Vm p ln

Prπ1 Vm rel Prπ1 Vm p

(e)

Pln Apa

Op

φm ωpl

pl Vm g

m

Og

Prπ1 Vm g pl Vm g

π1

π4

pl Vm rel Prπ1 Vm g

Prπ1 Vm rel Pln Vsl = Prπ1 Vm rel

ωg , ωp ωpa m

Op

PA

ϕt.ω

Og

FIGURE 8.5 Sliding in a right-angle high-conformal gear pair: (a) rotations of the gear and of the pinion; (b) linear velocity of point m with the gear; (c) linear velocity of point m with the pinion; (d) velocity of the relative motion of the gear and the mating pinion at m; (e) sliding velocity.

182


be determined on the premise of the design parameters of the gear pair under consideration using for this purpose conventional rules developed in descriptive geometry of surfaces. Ultimately this allows to construct the projections, PrPln Vgm and PrPln Vpm, of the velocity vectors, Vgm and Vpm , onto the pitch line, Pln. These components of the velocity vectors cause rolling of the tooth flaks, G and P, over each other as illustrated in Figure 8.5d. m Finally, the sliding velocity vector, V sl = Prπ1 Vrel , is projected with no distortion onto the plane of projections π4 (Figure 8.5e). This due to the equality is valid

m |Vrel | = |V sl|2 +|Prπ1 Vgm|2 = |V sl|2 +|Prπ1 Vpm|2

(8.17)

The sliding velocity vector, Vsl, can be constructed at any point, m, within the active portion of the path of contact, Pc. In this way, the distribution of Vsl along the Pc can be plotted. However, to plot the distribution of the sliding velocity vector, Vsl, along the active portion of the path of contact, Pc, it is sufficient to construct the vector Vsl just for one point. The sliding velocity vector at other points can be determined as shown in Figure 8.6. The specific sliding of the gear, γg, and the pinion, γp, can be calculated from Equations 3.24 and 3.25 using for this purpose direct measurements of the velocity vectors taken from Figure 8.5. Otherwise, a set of equations can be derived for the calculation of the specific sliding of the gear, γg, and the pinion, γp, based on the graphical solution to the problem under consideration. The performed analysis (Figure 8.5) makes an analytical solution to the problem of sliding in high-conformal gearing possible. Consider a plane of action, PA, in an intersected-axis gear pair (Figure 8.7). The Cartesian reference system, XpaYpaZpa, is centered at the PA apex, Apa. The axis Ypa is along the pitch line, Pln. The axis Xpa is located within the plane of Vsl

Apa

Pc m

Opa FIGURE 8.6 The distribution of the sliding velocity vector, Vsl, along the active portion of the path of contact, Pc, in an intersected-axis high-conformal gear pair.

183


PA Pln

φm

Ypa Apa

m Rm

Xpa FIGURE 8.7 Plane of action, PA, in an intersected-axis gear pair.

action and is perpendicular to the axis Ypa. The axis Zpa is perpendicular to the coordinate plane XpaYpa (Figure 8.7). The coordinates of an arbitrary point m within the plane of action, PA, can be expressed in terms of the distance, Rm, of the point m from the origin of the coordinate system XpaYpaZpa, and the central angle, φm, that the position vector, rm, of the point m forms with the pitch line, Pln

 − Rm cos ϕ m   R sin ϕ  m m  rm (R m , ϕ m ) =    0   1  

(8.18)

The reference system, X0Y0Z0, is associated with the right-angle intersectedaxis gear pair as shown in Figure 8.8. The Cartesian reference system, X0Y0Z0, is centered at the PA apex, Apa. The axis X0 is along the gear axis of rotation, Og. The axis Y0 is along the pinion axis of rotation, Op. The axis Z0 is perpendicular to the coordinate plane X0Y0 (Figure 8.8). To represent the point m in the reference system X0Y0Z0, it is necessary to compose an operator of the resultant coordinate transformation, Rs ( pa 0), from the coordinate system XpaYpaZpa to the reference system X0Y0Z0. The operator, Rs ( pa 0), can be calculated as a dot product of two operators of rotations about the coordinate axis, that is,

Rs ( pa 0) = Rt (Σ p , Z0 ) ⋅ Rt (φt.ω , Ypa )

(8.19)

where the pinion angle is denoted by Σp and the transverse pressure angle is designated as ϕt.ω.

184


View A Z0 Og

ϕt.ω

XpaYpa

Zpa

Ypa

Zpa

X0Y0 Apa

Y0 Apa

Op

Σp

Σg

Xpa

Pln

A

X0 FIGURE 8.8 Reference system, X0Y0Z0, associated with a right-angle intersected-axis gear pair.

The operator of rotation, Rt(φt.ω Ypa), can be represented in the matrix form as

 cos(φ t.ω )  0 Rt(φt.ω , Ypa ) =   − sin(φ t.ω )  0 

sin(φ t.ω ) 0

0 1 0

cos(φt.ω ) 0

0

0 0  0  1

(8.20)

Similarly, the operator of rotation, Rt(Σp, Z0), can be represented in the matrix form as

 cos Σ p  − sin Σ p Rt(Σ p , Z0 ) =   0   0

sin Σ p cos Σ p 0

0 0 1

0

0

0 0  0  1

(8.21)

With that said, the operator of the resultant coordinate transformation, Rs ( pa 0), is considered below. In the coordinate system X0Y0Z0, the position vector, rm, of the point m can be analytically described as

rm0 (R m , ϕ m ) = Rs ( pa 0) ⋅ rm

(8.22)

185


Equation 8.22 can be rewritten in an expanded form

 − cos ϕ m cos φ t.ω cos Σ p + sin ϕ m sin Σ p   cos ϕ cos φ sin Σ + sin ϕ cos Σ  m t .ω p m p  rm0 (R m , ϕ m ) = Rm ⋅    cos ϕ m sin φ t.ω   1  

(8.23)

The point m within the tooth flank of the gear, G, is rotated about the gear axis of rotation, Og, as shown in Figure 8.9. In the plane

X0 = R m (− cos ϕ m cos φ t.ω cos Σ p + sin ϕ m sin Σ p )

(8.24)

it traces a circle of a radius rmyz . This radius is the distance, rmyz , of the point m from the axis Og, that is, from the axis X0 of the Cartesian coordinate system X0Y0Z0. In Figure 8.9, coordinates of the point m are as follows:

Ymyz = R m (cos ϕ m cos φ t.ω sin Σ p + sin ϕ m cos Σ p ) Zmyz = R m cos ϕ m sin φ t.ω

Z0

Og

(8.25) (8.26)

g Vm

m

yz rm

Y0

φg

φg = upa/g · φm

FIGURE 8.9 Linear velocity vector, Vmg, in the rotational motion of the gear in an intersected-axis gear pair.

186


The angle of rotation of the gear, φg, can be expressed in terms of the angle of rotation, φm, of the plane of action, PA (Figure 8.7), and the gear ratio, upa/g, of the pair gear-to-plane-of-action: ϕ g = upa/g ⋅ ϕ m

(8.27)

In the rotational motion of the point m, the linear velocity vector, Vmg , can be analytically described by a column matrix in the form

 Rm (cos ϕ m cos φ t.ω sin Σ p + sin ϕ m cos Σ p )   − rmyzω g cos ϕ g   g Vm =   rmyzω g sin ϕ g   1  

(8.28)

The angular velocity of the gear, ωg, is equal to ωg = upa/g ⋅ ωpa, where ωpa is the angular velocity of the plane of action, PA. Similar to that, the velocity vector, Vmg, is determined, and the velocity vector, Vmp, of the point m within the pinion tooth flank, P, can be determined. The point m within the tooth flank of the pinion, P, is rotated about the pinion axis of rotation, Op, as shown in Figure 8.10. In the plane

X0 = R m (cos ϕ m cos φ t.ω sin Σ p + sin ϕ m cos Σ p )

X0

(8.29)

Z0

p Vm

m

xz rm

φp

Op

φp = upa/g · φm FIGURE 8.10 Linear velocity vector, Vmp , in the rotational motion of the pinion in an intersected-axis gear pair.

187


it traces a circle of a radius rmxz. This radius is the distance, rmxz, of the point m from the axis Ogp, that is, from the axis Y0 of the Cartesian coordinate system X0Y0Z0. In Figure 8.10, coordinates of the point m are as follows:

Ymxz = R m (− cos ϕ m cos φ t.ω cos Σ p + sin ϕ m sin Σ p ) Zmxz = R m cos ϕ m sin φt.ω

(8.30) (8.31)

The angle of rotation of the pinion, φp, can be expressed in terms of the angle of rotation, φm, of the plane of action, PA (Figure 8.7), and the gear ratio, upa/p, of the pair pinion-to-plane-of-action

ϕ p = upa/p ⋅ ϕ m

(8.32)

In the rotational motion of the point m, the linear velocity vector, Vmp , can be analytically described by a column matrix in the form

  − rmxzω p cos ϕ p   Rm (cos ϕ m cos φt.ω sin Σ p + sin ϕ m cos Σ p ) g  Vm =   rmxzω p sin ϕ p   1  

(8.33)

The angular velocity of the pinion, ωg, is equal to ωp = upa/p ⋅ ωpa, where ωpa is the angular velocity of the plane of action, PA. The velocity vector, Vmrel , of the point m in the relative motion of the gear and the pinion tooth flanks, G and P, is equal to

Vmrel = Vmg − Vmp

(8.34)

The sliding velocity vector, Vmsl , is the projection of the vector Vmrel onto the direction of a perpendicular, npa, to the plane of action, PA

Vmsl = Prn. pa Vmrel

(8.35)

Other approaches can be used for the calculation of the sliding velocity vector, Vsl , in high-conformal intersected-axis gear pairs. 8.3.3 Boundary N-Cone in Intersected-Axis High-Conformal Gearing When gears rotate, the motion of the pinion in relation to the gear can be viewed as instant rotation about the axis of instant rotation, Pln. A boundary N-circle is traced up by the contact point, K, in such a relative motion.

188


In theory, the radius, rN, of the boundary N-circle is a trade-off between desirable high contact strength and low friction between the teeth flanks of the gear, G, and the pinion, P. In practice, run-out of the gear and the pinion, as well as displacements of other types of the tooth flanks, G and P, in relation to their desirable position should be taken into account. Under the assumptions that Δl = 0 and that manufacturing errors are zero, the point of contact, K, is located at the point of intersection of the boundary N-circle by the instant line of action, LAinst. At any point within the axis of instant rotation, Pln, a boundary N-circle of a certain radius, r Ni , can be constructed, and an instant line of action LAinst can be constructed as well. The pressure angle, φti.ω, is not mandatorily of the same value at all normal sections of the axis, Pln. The instant line of action, LAinst, is a line through the axis of instant rotation, Pln. No kinematical and/or geometrical constraints on the intersected-axis high-conformal gearing are violated in such a consideration. In practice, it is reasonable to keep the pressure angle, φti.ω, of a certain constant value, ϕt.ω, within the active face-width of the gear pair. Moreover, as a normal section through a point within the axis of instant rotation, Pln, approaches the apex, Pa, the radius r Ni of the boundary N-circle should be getting smaller accordingly. In this way, the path of contact, Pc, is the straight line through all the contact points, Ki. The path of contact passes through the apex, Apa, of the intersected-axis high-conformal gear pair. When the path of contact, Pc, is rotated about the axis of instant rotation, the boundary N-cone is generated as the loci of consecutive positions of the path of contact, Pc, in its rotation in relation to the axis Pln. Consider a straight line, Pc, through the point of contact, K, of the tooth flanks, G and P, of the gear and the pinion and through the common apex, Apa. When rotating about the axis of instant rotation, Pln, this line generates a cone of revolution. This cone of revolution is referred to as the boundary N-cone in intersected-axis high-conformal gearing. This makes the following definition possible: Definition 8.3 A boundary N-cone in intersected-axis high-conformal gearing is a cone of revolution that is generated by the rotation of the path of contact, Pc, about the axis of instant rotation, Pln. The convex tooth flank of one member of a gear pair (primarily of the inion, P) must be entirely located within the interior of the boundary N-cone. p The concave tooth flank of another member (primarily of the gear, G) of the gear pair must be entirely located outside the interior of the boundary N-cone.* * The concept of the boundary N-cone was not known in the times of Professor M.L. Novikov. This is newly introduced (circa 2008) concept to intersected-axis high-conformal gearing by Dr. S.P. Radzevich.

189


Pln

ωg Apa –ωg

rNi

Γi

ωpl ωp

Ai

FIGURE 8.11 Angle, Γi, of the boundary N-cone in high-conformal intersected-axis gearing.

The boundary cone angle, Γl, (Figure 8.11) can be expressed in terms (a) of the radius, r Ni , of the boundary N-circle at a current point within the axis of instant rotation, Pln, and (b) of the cone distance, Ai, of that point from the apex, Apa:

 ri  Γ l = tan −1  N   Ai 

(8.36)

In a more general case, not a boundary N-cone should be considered, but a boundary N-surface of revolution should be considered instead.

8.4 Design Parameters of High-Conformal Intersected-Axis Gearing The rotation vectors of the gear, ωg, and the pinion, ωp, should be given prior to beginning designing a high-conformal intersected-axis gear pair. Once the rotation vectors, ωg and ωp, are known, the vector, ωpl, of instant rotation, as well as the shaft angle, Σ, can be determined. The axes of rotations, Og, Op, and Pln, are the straight lines along the rotation vectors, ωg, ωp, and ωpl, respectively. The known configuration of the axes of rotations, Og, Op, and Pln, makes possible the determination of the tooth ratio, u, and the pitch cone angles of the gear, Γ, and the pinion, γ [23,38]:

  sin Σ Γ = − tan −1   + cos Σ ω / ω   p g   sin Σ γ = tan −1   Σ ω ω + cos /   g p

(8.37) (8.38)

190


The design parameters of a high-conformal intersected-axis gear pair can be specified based, to a great extent, on those of parallel-axis gearing. From this perspective, the vector of instant rotation, ωpl, and the axis of instant rotation, Pln, are of critical importance. As the instant motion of the pinion in relation to the mating gear is interpreted as instant rotation about the axis, Pln, the design parameters of a high-conformal intersected-axis gear pair can be specified within a reference plane through the pitch point, P. The pitch point, P, is at a cone distance, A, from the apex Apa. The reference plane is perpendicular to the axis of instant rotation, Pln, as depicted in Figure 8.12. The calculated values of the pitch cone angles, Γ and γ, along with the given cone distance, A, make it possible to calculate the pitch diameter of the gear, dg, and of the pinion, dp dg = 2 A cos Γ

dp = 2 A cos γ

(8.39)

(8.40)

The back cone distance of the gear, BCg, as well as the back cone distance of the pinion, BCp, can be calculated in a way similar to that above BC g = 2 A sin Γ

BC p = 2 A sin γ

(8.41)

(8.42)

dg Op

Σ

ωp ωpl

rN

ωg

–ωg

Pln

Apa

Γl

Og A

View A

P BCp op

dp

FIGURE 8.12 Configuration of a normal reference plane in relation to the axis of instant rotation, Pln, and to the pitch cones of the gear and the pinion in high-conformal intersected-axis gearing.

191


ϕt.ω BCo.g

LAinst

ρp

ρg

BCp cg op

P BCf.p

rN = rp b BCo.p

BCg

BCf.g

ρ f.g cp

rg

a

d

BC*f.g

og

c K

ag

cn = (BCg + BCp) FIGURE 8.13 Geometry of a high-conformal intersected-axis gear pair within the reference plane perpendicular to the pitch line, Pln.

Once the normal reference plane is constructed, the tooth profile parameters of the gear and the pinion can be specified. Referring to Figure 8.13, two points, namely og and op, are in nature the points of intersection of the axes, Og and Op, by the normal reference plane. The points, og and op, are at a distance cn = (BCg + BCp) from one another. Two circles of radii, BCg and BCp, that have the points og and op as the centers are constructed. The circles share a common point, which is the pitch point P. An instant line of action, LAinst, within the normal reference plane is the line through the pitch point, P. The line, LAinst, forms a certain transverse pressure angle, ϕt.ω, with the perpendicular to the center distance, cn. The point of contact, K, of the tooth flanks of the gear and the pinion is a point within the instant line of action, LAinst. The further the contact point, K, is from the pitch point, P, the more freedom in selecting the radii of curvature of the tooth profiles is observed. At the same time, the further the contact point, K, from the pitch point, P, the higher the losses on friction that occur between the teeth flanks and the higher wear of the teeth flanks of the gear and the pinion. Ultimately, the actual location of the contact point, K, is a kind of trade-off between the two aforementioned factors. Let us assume that the pinion is stationary and that the gear performs an instant rotation in relation to the pinion. The axis, Pln, of the instant rotation, ωpl, is the straight line through the pitch point, P. The axis of instant

192


rotation, Pln, is located within the plane through the axes, og and op, and it goes through the apex, Apa. When the pinion is motionless, the contact point, K, traces a boundary circle of radius, rN, centered at P. The pinion tooth profile, P, can either align with the circular arc of the boundary circle, rN, or it can be relieved in the bodily side of the pinion tooth. As a consequence, the location of the center of curvature, cp, of the convex pinion tooth profile, P, within the instant line of action, LAinst, is limited to the straight line segment, PK. The pitch point is included into the interval (P, K) as it is shown in Figure 8.13, while the contact point, K, is not. On the other hand, the location of the center of curvature, cg, of the concave gear tooth profile, G, within the instant line of action, LAinst, is limited to the open interval P → ∞. Theoretically, the pitch point, P, can be included in that interval for K. However, this is completely impractical, and the center of curvature cg is actually located beyond the pitch point P. Therefore, the radius of curvature, rp, of the convex the pinion tooth profile, P, is smaller than that, rg, of the concave the gear tooth profile, G, (the inequality rp < rg is observed). Both the pinion teeth and the gear teeth are helical and of opposite hand. Spur high-conformal gearing is not feasible in nature. Because both the gear and the pinion are helical with the helices of opposite hand, the point of contact will travel along the path of contact, Pc. It is therefore fundamental to the operating of the gears that contact occurs nominally at a point and that the point of contact travels across the full face width of the gears during the rotation. It is clearly a condition of operation that in a given profile the tooth surfaces should not interfere before or after culmination when rotated at angular speeds that are in the gear ratio. The transverses contact ratio, mp, in a high-conformal gear pair is zero (mp ≡ 0). The face contact ratio, mF, of the gear pair is always greater than one (mF > 1). The total contact ratio, mt, is equal to the face contact ratio, mF, that is, the identity mt ≡ mF is valid in intersected-axis high-conformal gearing. When a rotation is transmitted from the driving shaft to the driven shaft, the contact point, K, travels along the path of contact, Pc (and it does not travel within the transverse section of the gear pair), that is, within the normal reference plane. This is because mp ≡ 0 and mF > 1 as previously mentioned. For the calculation of the design parameters of a high-conformal gear pair, the center distance, cn, and the tooth ratio, u = ωp/ωg, of the gear pair should be given. The back cone distance of the gear, BCg, and the pinion, BCp, can be expressed in terms of the center-distance, cn, and the tooth ratio, u, as

BC g = cn ⋅

u 1+ u

(8.43)

BC p = cn ⋅

1 1+ u

(8.44)

193


A distance, l, at which the path of contact, Pc, is remote from the pitch point, P, must be known, as well as the transverse pressure angle, ϕt.ω. The displacement, l, is the principal design parameter of a high-conformal gear pair. In terms of the displacement, l, many of the design parameters of the high-conformal gear pair can be expressed (l = KP). For the calculation of radii of curvature, rg and rp, of the tooth profiles of the gear and the pinion respectively, the formulas

rg = l ⋅ (1 + k rg ) rp = l ⋅ (1 + k rp )

(8.45)

(8.46)

are used. The actual value of the factor, krp, should satisfy the inequality krp ≥ 0. However, as the factor, krp, is often set equal to zero, the equality rp = l is observed. The factor, krg, is within the range krg = 0.03 … 0.10. The radius of the outer back cone distance of the pinion, BCo.p, is calculated from the formula

BCo. p = BCp + (1 − k po ) ⋅ l

(8.47)

The addendum factor, kpo, of the pinion depends on the pressure angle, ϕt.ω, absolute dimensions of the gear pair, accuracy of machining, and conditions of lubrication. Commonly, the pinion addendum factor, kpo, is set in the range

k po = 0.1 ÷ 0.2

(8.48)

The root back cone distance of the pinion, BCf.p, is calculated from the equation

BC f . p = BC p − ag − δ

(8.49)

where ag is the dedendum of the mating gear [ag = (0.1 … 0.2) ⋅ l] and δ is the radial clearance in the gear pair (δ = l ⋅ kpo) It is practical to set the fillet radius, ρp, in the range of ρp = 0.3 ⋅ l. The root back cone distance of the gear, BCf.g, is equal to

BC f . g = cn − BCo. p

(8.50)

The radius of the outer back cone distance of the gear, BCo.g, is calculated from the expression

BCo. g = BC g + ag

(8.51)

194


TABLE 8.2 Design Parameters of High-Conformal Intersected-Axis Gearing Design Parameter Angular displacement Angular module Angular pitch Angular tooth thickness, gear Angular tooth thickness, pinion Angular space width, gear Angular space width, pinion Angular backlash Angular addendum, gear Angular addendum, pinion Angular dedendum, gear Angular dedendum, pinion

Symbol

Equation

φl φm.n pφ φt.g φt.p φw.g φw.p φB φa.g φa.p φd.g φd.p

φl = tan−1(l/A) φm.n = tan−1(mn/A) pφ = tan−1(p/A) φt.g = tan−1(tg/A) φt.p = tan−1(tp/A) φw.g = tan−1(wg/A) φw.p = tan−1(wp/A) φB = tan−1(B/A) φa.g = tan−1(ag/A) φa.p = tan−1(ap/A) φd.g = tan−1(bg/A) φd.p = tan−1(bp/A)

The designations: ag, bg, and ap, bp relate to the addendum and dedendum of the gear and the pinion, respectively. These design parameters are measured within the normal reference plane of the high-conformal intersected-axis gear pair.

The corner of the gear tooth addendum should be rounded with a radius, ρg, which is less than the fillet radius, ρp, of the pinion (ρg < ρp). The following relations among the design parameters of a high-conformal gear pair have been proved practical: rp = l, rg ≤ 1.10 ⋅ rp, ρp = 0.3 ⋅ l, mn/l = 0.8, tp/tg = 1.5, ϕt.ω =30°, λ = 60 … 80° (ψ = 10 … 30°), and circular pitch of teeth p = tg + tp + B, where backlash B = 0.2 … 0.4 mm. For the design parameters, l, p, tg, tp, mn, and B, corresponding angular values can be calculated (Table 8.2). The effective face width of the gear pair can be calculated as follows:

Feff = (1.1 ÷ 1.2) ⋅ p ⋅ tan λ

(8.52)

For a preliminary analysis of high-conformal gearing, an empirical expression

l = (0.05 ÷ 0.20) ⋅ BCp

(8.53)

returns a practical value for the displacement l. The functional face width and axial pitch of a high-conformal gear pair depend on each other. Consider a case when at a uniform rotation of the gear and the pinion, the contact point, K, travels along the contact line, CL, at a certain uniform liner speed. As the transverse contact ratio is zero (mp = 0), and the total contact


195

ratio, mt, is equal to the face contact ratio, mF, the axial pitch, pcl.g, of the helix on the gear tooth flank, G, can be calculated from the formula

pcl. g =

Feff ⋅ cos Γ mt

(8.54)

pcl. p =

Feff ⋅ cos γ mt

(8.55)

A similar expression

is valid with respect to the axial pitch, pcl.p, of the helix on the pinion tooth flank, P. The quality of high-conformal gearing strongly depends on the following design parameters: l, ϕt.ω, and λ. The tooth flanks, G and P, interact with one another only at a culminating point, K, that travels along the path of contact, Pc.

9 High-Conformal Crossed-Axis Gearing Crossed-axis gears (or just Ca-gearing, for simplicity) have wide applications in the industry. Early designs of crossed-axis gears can be found in Leonardo da Vinci’s famous book The Madrid Codices [7]. When motion is to be transmitted between two shafts whose axes cross, some form of bevel-like gear is applied. Although gears of this type are often made for a shaft angle of 90°, they can be produced for almost any shaft angle.

9.1 Kinematics of Crossed-Axis Gearing Transmission and transformation of rotation from a driving shaft to a driven shaft is the main purpose of implementation of crossed-axis gears. Both the input rotation and the output rotation can be easily represented by corresponding rotation vectors, ωg and ωp. The vectors, ωg and ωp, are along the straight lines, which cross one another. The closest distance of approach between the lines of action of the rotation vectors, ωg and ωp, is denoted by C. This distance is commonly referred to as the center-distance C. The variety of all possible types of crossed-axis gear pairs is limited to the total number of possible combinations of the rotation vectors, ωg and ωp, (a) of various magnitudes and (b) featuring different shaft angles Σ [remember that the shaft angle, Σ, is specified as the angle between the rotation vector, ωg, of the gear and the rotation vector, ωp, of the pinion, i.e., Σ = ∠(ωgωp)]. The total number of vector diagrams for different types of crossed-axis gear pairs is limited just to three diagrams when the actual configuration of the rotation vectors, ωg and ωp, of the gear and the pinion in relation to the vector of instant rotation, ωpl, is taken into account. These vector diagrams are depicted in Figure 9.1. Therefore, only three different types of intersectedaxis gear pairs are feasible.* The vector diagram shown in Figure 9.1a features an obtuse gear angle, Σg, between the rotation vector, ωg, of the gear and the vector of instant

* See footnote * in Chapter 7 on page 145.

197

198


(a) ωp

(b)

ω pl

ωp

Ap Apa

Σp

Σg

(c)

ωg

Pln Ag Σ

Og

Ap ≡ Apa

Op C

ω pl

ω pl ωp

Op

C

Σp Σ Og

ωg

Ag

Σp

Pln

–ω g

Σg Og

Apa Ap

Pln

C

Op

Σ ω g Ag –ω g

FIGURE 9.1 The total number of possible types of vector diagrams for crossed-axis gear pairs is limited just to three vector diagrams. Parts (a)–(c) are discussed in the text.

rotation, ωpl. The gear angle, Σg, can be expressed in terms of the shaft angle, Σ, and the magnitudes, ωg and ωp, of the rotation vectors, ωg and ωp

  sin Σ Σ g = tan −1   cos Σ ω / ω +  p g 

(9.1)

For a shaft angle of 90°, Equation 9.1 reduces to

 ωg  Σ g = tan −1    ωp 

(9.2)

The formulae for the calculation of the pinion angle, Σp, are similar to Equations 9.1 and 9.2

  sin Σ Σ p = tan −1    ω g /ω p + cos Σ 

(9.3)

and for a right shaft angle reduces to

 ωp  Σ p = tan −1    ωg 

(9.4)

For a gear pair of this particular type (Σ > 90°), the relation Σg = ∠(ωgωpl) > 90° is valid. This relation can be represented in an equivalent form

ω g ⋅ (ω p − ω g ) < 0

(9.5)

199

High-Conformal Crossed-Axis Gearing

or

ω g ⋅ (ω p − ω g ) = −1 |ω g| ⋅ |ω p − ω g|

(9.6)

The center-distance, C, can be interpreted as the summa of the pitch radii of the gear, rg, and the pinion, rp C = rg + rp

(9.7)

For external crossed-axis gearing of all types, both the pitch radii, rg and rp, are of positive values (rg > 0, rp > 0). The earlier-derived formulas [38] rg = rp =

1 + ωp − ωg ⋅C 1 + ωp 1 + ωg − ωp ⋅C 1 + ωg

(9.8)

(9.9)

can be used for the calculation of the pitch radii, rg and rp, of the gear and its pinion, respectively. The vector diagram (Figure 9.1a) corresponds to an external crossed-axis gearing. The configuration of the rotation vector of the gear, ωg, in relation to the vector of instant rotation, ωpl, is critical for the determination of whether or not a gear pair is external while the relative configuration of the rotation vectors, ωg and ωp, is of secondary importance in this consideration. In a particular case, the rotation vector of the gear, ωg, can be orthogonal to the vector of instant rotation, ωpl, that is, Σg = ∠(ωgωpl) = 90°. Two equivalent forms

ω g ⋅ (ω p − ω g ) = 0

(9.10)

and

ω g ⋅ (ω p − ω g ) =1 |ω g|⋅|ω p − ω g|

(9.11)

are valid for crossed-axis gearing that meet the condition Σg = ∠(ωgωpl) = 90°.

200


Crossed-axis gear pairs for which the condition ωg ⊥ ωpl is fulfilled feature pitch radii of the value: rg = 0 and rp = C accordingly (the condition C = rg + rp is still valid). The vector diagram for gear drives of this particular type is schematically depicted in Figure 9.1b. The vector diagram corresponds to a crossedaxis gear pair composed of a round rack (or face gear) and a conical pinion. Crossed-axis gearing of this type is analogous to the aforementioned pinionto-rack gearing in the case of the parallel axes of the gear and the pinion. Ultimately, a crossed-axis gear pair may feature acute angle, Σg, between the rotation vector, ωg, of the gear and the vector of instant rotation ωpl (Figure 9.1c). For a gear pair of this particular type, the relation Σg = ∠(ωgωpl) < 90° is valid. The last expression can be represented in two other forms: ω g ⋅ (ω p − ω g ) > 0

(9.12)

and ω g ⋅ (ω p − ω g ) = +1 |ω g| ⋅ |ω p − ω g|

(9.13)

Crossed-axis gear pairs for which the condition ωg ⊥ ωpl is fulfilled feature pitch radii of the value rg < 0 and rp > 0 (the condition C = rg + rp is still valid). A vector diagram of the type (Figure 9.1c) corresponds to an internal crossed-axis gearing. The analytically expressed conditions (see Equations 9.5 through 9.10) along with Equation 9.12 are summarized in Table 9.1. Any and all crossed-axis gear pairs meet one of three expressions those listed in Table 9.1. In particular cases, the centerlines of the driving shaft and the driven shaft cross each other at a right angle (Σ = 90°). This particular case is the most common in practice. Crossed-axis gear pairs of this particular type are referred to as orthogonal crossed-axis gear pairs. For gearing of this particular type, the cross product of the rotation vectors of the gear, ωg, and the pinion, ωp, is always equal to zero (ωg × ωp = 0). TABLE 9.1 Analytical Criteria of the Type of Crossed-Axis Gearing Type of Crossed-Axis Gearinga

Analytical Criterion [C ≠ 0 and Σ ≠ 0]

External crossed-axis gear pair Rack-type crossed-axis gear pair Internal crossed-axis gear pair

ωg ⋅ (ωp − ωg) < 0 ωg ⋅ (ωp − ωg) = 0 ωg ⋅ (ωp − ωg) > 0

a

See footnote * in Chapter 7 on page 145.


201

An orthogonal crossed-axis gear pair may feature equal tooth number of the gear, Ng, and the pinion, Np. Crossed-axis gearing of this particular type fulfills the requirement ωg × ωp = 0. It is evident that magnitudes, ωg and ωp, of the rotation vectors, ωg and ωp, in this case are equal (ωg = ωp). This gearing is often referred to as miter gears.

9.2 Base Cones in Crossed-Axis Gear Pairs Geometrically accurate crossed-axis gear pairs (or, in other words, ideal crossed-axis gear pairs) are capable of transmitting a rotation smoothly. From this perspective, geometrically accurate crossed-axis gear pairs resemble the earlier-discussed geometrically accurate parallel-axis gear pairs and intersected-axis gear pairs. This similarity can be extended further, namely crossed-axis gearing of a particular type can also transmit a uniform rotation from a driving shaft to a driven shaft. It should be noted here that in the case of crossed axes of rotation of the driving shaft and the driven shaft, there is no freedom in choosing a configuration of the axis of instant rotation, Pln, in relation to the rotation vectors ωg and ωp. Once the rotation vectors, ωg and ωp, and their relative location and orientation are specified, the configuration of the axis of instant rotation, Pln, can be expressed in terms of the rotations ωg and ωp, and the center distance, C. Recall that geometrically accurate parallel-axis gear pairs feature two base cylinders (see Figure 2.7). Smooth rotation of the base cylinders allows for an interpretation as a corresponding belt-and-pulley mechanism. Then, two base cones are associated with the gear and with the pinion in an intersected-axis gearing (see Figure 8.2). Smooth rotation of the base cones can be interpreted as a belt-and-pulley mechanism with the belt in the form of a round tape. The belt-and-pulley analogy is also valid with respect to geometrically accurate crossed-axis gearing. A base cone can be associated with the gear and another base cone can be associated with the pinion of any and all geometrically accurate crossedaxis gear pairs. This concept is schematically illustrated in Figure 9.2. The axis of rotation of the gear, Og, and the axis of rotation of the pinion, Op, cross each other at a shaft angle, Σ. The closest distance of approach of the axes of the rotations, Og and Op, is denoted by C. An orthogonal intersectedaxis gear pair is illustrated here for illustrative purposes only. Without going into details of the analysis, it should be stated here that the same approach is applicable with respect to angular bevel gears with a shaft angle Σ ≠ 90°, namely either an obtuse or acute shaft angle Σ. The schematic shown in Figure 9.2 is constructed starting from the rotation vectors, ωg and ωp, of the gear and the pinion. The gear and its pinion rotate

202


ωpa


Σg C Pln

Σ Ap

ωg

L

C

c1

a1

π1 Fp rl.pa

Ap

ωpl

Op

lcp

ωp Ap

π4

ωpa ωg

Pln

Ag PA

rp

d5 Feff

c5

Δφpa. p

f1

b1

Γb

ωp

d1

PA

Og


π4

ΣP

lcp

ωpl e 1 lcg

ωg

π1

Op

ωp

Ag A pa


γb

ωp

lcg

f5

Og

Pln

e5 φpa

rg ωg

PA


ωpa

Opa Ag

∆φpa.g L

C

a5

ro. pa

ϕt.ω

b5 Fg

FIGURE 9.2 Base cones and the plane of action, PA, in an orthogonal crossed-axis gear pair.

about their axes, Og and Op, respectively. The rotation vectors, ωg and ωp, allow for the construction of the vector, ωpl, of instant relative rotation. The rotation vector, ωpl, meets the requirement ωpl = ωp − ωg. The axis of instant rotation, Pln, is aligned with the vector of instant rotation, ωpl. The vector of instant rotation, ωpl, is the vector through a point, Apa, within the center-distance, C. The endpoints of the straight line segment, C, are labeled as Ag and Ap. Ag is the point of intersection of the center-line, ℄, and the gear axis of rotation, Og. Ap is the point of intersection of the center-line, ℄, and the pinion axis of rotation, Op. The point Apa is at a certain distance, rg,

203


from the axis of rotation, Og. At that same time, the point Apa is at a certain distance, rp, from the axis of rotation, Op. The following expression rg + rp = C

(9.14)

is valid. Here, in Equation 9.14, the distances rg and rp are signed values. The distances rg and rp are of positive values (rg > 0, rp > 0) when point, Apa, is located within the center-distance, C. When point, Apa, is located outside the center-distance, C, the distance, rg, is of negative value (rg < 0), while the distance, rp, remains of positive value (rp > 0). Equation rg ω rlp = rl rp ωg

(9.15)

makes it possible to calculate the distances, rg and rp [38]: rg =

1 + ωp − ωg ⋅C 1 + ωp

(9.16)

and rp =

1 + ωg − ωp ⋅C 1 + ωg

(9.17)

For a pair of rotation vectors, ωg and ωp, the ratio tan Σg/tan Σp can be calculated [38]

rp tan Σ g = rg tan Σ p

(9.18)

The plane of action, PA, is a plane through the axis of instant rotation, Pln. The plane of action, PA, is in tangency with both base cones, namely with the base cone of the gear and with the base cone of the pinion. Due to that, the plane of action, PA, makes a certain transverse pressure angle, ϕt.ω, in relation to a perpendicular to the plane associated with the axis of instant rotation, Pln. The perpendicular is constructed to the plane through the vector of instant rotation, ωpl, and through the center-line, ℄. The pressure angle, ϕt.ω, is measured within a plane that is perpendicular to the axis of instant rotation, Pln. The portion of the schematic plotted in the left upper corner in Figure 9.2 is constructed within the plane of projections, π1. Two others planes of

204


projections, π2 and π3, of the standard set of planes of projections, π1, π2, and π3, are not used in this particular consideration. Therefore, these planes, π2 and π3, are not shown in Figure 9.2. Instead, two auxiliary planes of projections, namely the plane of projections, π4 and π5, are used. The axis of projections, π1/π4, is constructed so as to be perpendicular to the axis of instant rotation, Pln. The axis of projections, π4/π5, is constructed so as to be parallel to the trace of the plane of action, PA, within the plane of projections, π4. The plane of action, PA, is projected with no distortions onto the plane of projections π4. The plane of action can be interpreted as a flexible zero thickness film. The film is free to wrap or unwrap from and onto the base cones of the gear and the pinion. The plane of action, PA, is not allowed to bend about an axis perpendicular to the plane, PA, itself. Under uniform rotation of the gears, the plane of action, PA, rotates about the axis, Opa. The rotation vector, ωpa, is along the axis, Opa. The rotation vector, ωpa, is perpendicular to the plane of action, PA. As the axis of instant rotation, Pln, and the axes of rotations of the gear, Og, and the pinion, Op, cross one another, the pure rolling of the base cones of the gear and of the pinion over the plane of action, PA, is not observed, but rolling together with sliding of the PA over the base cones is observed instead. The sliding of the plane of action, PA, is observed in the direction of the pitch line, Pln. For intersected-axis gearing, the plane of action, PA, can be understood as a round cone that has a cone angle of 90°. As sin 90° = 1, the magnitude, ωpa, of the rotation vector, ωpa, can be calculated from the formula ω pa =

ωg ωp = sin Γ b sin γ b

(9.19)

where ωg is the rotation of the gear, ωp is the rotation of the pinion, Γb is the base cone angle of the gear, and γb is the base cone angle of the pinion. For intersected-axis gear pairs, the base cone angles, Γb and γb, vary within the intervals 0° < Γb < 180° and 0° < γb < (180° − Γb), respectively. Thus, all the equations here and below are valid for (1) external crossed-axis gear pairs, (2) rack-type crossed-axis gear pairs, and (3) internal crossed-axis gear pairs. Formally, the base cone angles, Γb and γb, can be considered in the narrower intervals, namely within the intervals 0° < Γb < 90° and 0° < γb < 90°, respectively. Under such a scenario, the following three inequalities are valid for crossed-axis gear pairs of various types: 1. The base cone angles are of positive values (Γb > 0° and γb > 0°) for external gearing 2. The base cone angle of the gear is equal to the right angle (Γb = 90°, while γb > 0°) for rack-type gear pairs 3. The base cone angle of the gear is of negative value (Γb < 0°, while γb > 0°) for internal crossed-axis gearing


205

A desirable working portion, or, in other words, effective portion of the plane of action, PA, can be constructed in the following way. Consider a straight-line segment, ef, within the axis of instant rotation, Pln (Figure 9.2). When the gears rotate, the straight-line segment, ef, travels together with the plane of action, PA. The point f traces a circular arc of radius, ro.pa, while the point e traces a circular are of radius, rl.pa. The face width of the plane of action, Feff, or, in other words, working (functional) portion of the plane of action is located between two circles of radii, ro.pa and rl.pa. In order to get the desired face width of the plane of action, Feff, the face width of the gear, Fg, and the face width of the pinion, Fp, should be of values as shown in Figure 9.2. The appropriate radii of the outer circles, ro.g and ro.p, as well as of the inner circles, rl.g and rl.p, should be of values under which both the face width of the gear, Fg, and the face width of the pinion, Fp, overlap the face width, Feff. The radii ro.g and rl.g are centered at the gear apex Ag, while the radii ro.p and rl.p are centered at the pinion apex Ap. The inequalities, Fg > Feff and Fp > Feff, occur because the apexes, Ag and Ap, are not coincident to one another, and thus sliding in axial direction of the gear and the pinion is inevitable in crossed-axis gearing. The straight-line segments, lcg and lcp, are along the corresponding lines of contact of the plane of action, PA, with the base cones of the gear and the pinion. In angular directions, functional portion of the plane of action, PA, spans within the central angle

Φ pa = ϕ pa + ∆ϕ pa. g + ∆ϕ pa. p

(9.20)

The components Δφpa.g and Δφpa.p are due to the gear axis of rotation, Og, and the pinion axis of rotation, Op, are the straight lines, which do not pass through the apex, Apa, of the plane of action apex. In reality, crossed-axis gear pairs can be comprised of a gear and a pinion with tooth flank geometry for which base cones cannot be constructed. In such a case, the plane of action, PA, also cannot be constructed. Crossed-axis gear pairs of this type are referred to as approximate crossed-axis gear pairs. The tooth flanks of approximate crossed-axis gear pairs feature geometry for which no equivalent belt-pulley mechanism can be designed to replace the gear pair. The concept of high-conformal gearing is enhanced further and applied to transmit a rotation from a driving shaft to a driven shaft that has crossed axes of the gear rotation. Similar to parallel-axis high-conformal gearing and intersected-axis high-conformal gearing, this particular concept is applicable to crossed-axis gearing as well. It should be pointed out here that crossedaxis high-conformal gearing is capable of transmitting a rotation under uniform angular velocity of both a driving shaft and a driven shaft. The main features of crossed-axis high-conformal gearing are due to the kinematics of instantaneous relative motion of the gear and the pinion.

206


9.3 Kinematics of the Instantaneous Relative Motion Instant relative motion in a high-conformal gear pair is a kind of instant screw motion. The use of a vector diagram is helpful to investigate the instantaneous relative motion of the gear and of the pinion in crossed-axis high-conformal gearing. Referring to Figure 9.3, consider the rotation vector of the gear, ωg, and the rotation vector of the pinion, ωp. The rotation vectors, ωg and ωp, are at a certain distance, C, which is commonly referred to as the center-distance. The vector of instant rotation, ωpl, is the vector through a point, Apa. This vector is along the axis of instant rotation, Pln. The point Apa in nature is the point of intersection of the axis of instant rotation, Pln, by the centerline, ℄. The rotation vectors, ωg and ωp, form a shaft angle, Σ, with one another. Having constructed the rotation vectors, ωg and ωp, the vector of instant relative rotation, ωpl, is constructed so as to meet the requirement: ωpl = ωg − ωp. Under such an assumption, the gear is considered motionless while the pinion performs an instant rotation in relation to the gear about the axis of instant rotation, Pln. An angle between the vector of instant rotation, ωpl, and the rotation vector of the gear, ωg, is designated as Σg. Accordingly, an angle between the vector of instant rotation, ωpl, and the rotation vector of the pinion, ωp, is denoted by Σp. Both the angles, Σg and Σp, are measured so as to use the vector of instant rotation as the reference. Generally speaking, for crossed-axis gearing the vector of instant rotation, ωpl, does not align either with the rotation vector of the gear, ωg, or with the L

C

Op Ap Pln C

ωp

Apa ωg Ag

∑

ωpl

∑p Og

FIGURE 9.3 Vector diagram for a high-conformal crossed-axis gear pair.

∑g

207


rotation vector of the pinion, ωp. Due to this, the vector of instant rotation, ωpl, can be decomposed into two components ω rlpl and ω slpl :

ω pl = ω rlpl + ω slpl

(9.21)

The component, ω rlpl , of the vector of instant rotation, ωpl, is aligned with the axis of rotation of the gear, Og. This component causes pure rotation of the gear and the pinion. The magnitude, ω rlpl , of the rotation vector, ω rlpl , can be calculated from the formula

ω rlpl = ω pl cos(180 − Σ p )

(9.22)

As the pinion angle, Σp, can be expressed in terms of the rotations, ωg and ωp, and the shaft angle, Σ, [38] Σp =

1 + ωg − ωp ⋅Σ 1 + ωg

(9.23)

therefore Equation 9.22 can be cast into

 1 + ωg − ωp  ω rlpl = −ω pl cos  ⋅ Σ  1 + ωg 

(9.24)

The component, ω slpl, of the vector of instant rotation, ωpl, is perpendicular to the axis of rotation of the gear, Og. Because of this, pure sliding (with no rolling) of the gear tooth flank, G, and the pinion tooth flank, P, is observed. The magnitude, ω slpl, of the rotation vector, ω slpl, can be calculated from the formula

ω slpl = ω pl sin(180° − Σ p )

(9.25)

Substituting Equation 9.23 into Equation 9.25 yields a formula

 1 + ωg − ωp  ω slpl = ω pl sin  ⋅ Σ  1 + ωg 

(9.26)

for the calculation of the magnitude, ω slpl , of the rotation vector ω slpl. The center-distance, C, can be construed as the summa of the pitch radii of the gear, rg, and that of the pinion, rp

C = rg + rp

(9.27)

208


For external crossed-axis gearing of all kinds, both the pitch radii, rg and rp, are of positive values (rg > 0, rp > 0). The earlier derived formulas [38] rg = rp =

1 + ωp − ωg ⋅C 1 + ωp 1 + ωg − ωp ⋅C 1 + ωg

(9.28)

(9.29)

can be used for the calculation of the pitch radii, rg and rp.

9.4 Path of Contact in High-Conformal Crossed-Axis Gearing The path of contact in a high-conformal crossed-axis gear pair is a trace of contact point when the gears rotate. The path of contact, Pc, is commonly considered in a stationary reference system associated with the gear housing. As the relative motion of the gear and the pinion is an instant rotation, ωpl, about the axis of instant rotation, Pln, a plane perpendicular to ωpl at an arbitrary point P within Pln can be constructed. The relative motion of the gear and the pinion can be investigated within the normal plane (Figure 9.4).

Op

ϕt.ω

L

C

Ap Pln C

ωp

Apa ωg Ag ∑g

∑

K–

LAinst

Boundary N-cone

–l rN

ωpl

Pln Ila

∑p Og

+l

Γl

K+ Boundary N-circle

FIGURE 9.4 Configuration of the boundary N-cone in high-conformal crossed-axis gearing.


209

Within the normal plane, a boundary N-circle can be constructed. The center of the boundary N-circle is coincident with the point of intersection of the axis of instant rotation, Pln, by the normal plane. The radius, rN, of the boundary N-circle is equal to a desired displacement, l, of the contact point, K, from the pitch point along the instant line of action, LAinst. The displacements of both positive, +l, and negative, −l, are feasible. Therefore, two contact points, K+ and K−, are potentially possible. The magnitude of the desired displacement, l, is a trade-off between the contact strength of the gear teeth and between sliding between the teeth flanks, G and P, of the gear and the pinion in relation to one another. The larger the displacement, l, the higher the contact strength of the gear teeth, and the higher the sliding between the teeth flanks. The smaller the distance, l, the lower the contact strength of the gear teeth and the lower the sliding between the teeth flanks. 9.4.1 Bearing Capacity of Crossed-Axis High-Conformal Gearing The influence of an increase in radius, rN, of the boundary N-circle on the rise of contact strength in a crossed-axis high-conformal gear pair resembles much that of the cases of parallel-axis and intersected-axis high-conformal gearing. The aforementioned conclusion on the bearing capacity of intersected-axis high-conformal gearing is also valid with respect to the bearing capacity of crossed-axis high-conformal gearing. The bearing capacity of high-conformal crossed-axis gearing depends not only on the relative curvature, rrel, of the interacting tooth flanks, but also on the magnitudes of the radii of curvature of the teeth flanks, G and P, at a point of their contact. The larger the magnitudes of the radii, rg and rp, of normal curvature of the interacting teeth flanks, G and P, the larger the load capacity of high-conformal crossed-axis gearing and vice versa.* In other words: Statement 9.1 High-conformal crossed-axis gearing with larger magnitudes of radii of curvature of the tooth flanks feature higher load carrying capacity. As it can be concluded from Statement 9.1, high-conformal crossed-axis gearing with a larger radius, rN, of the boundary N-circle feature higher load carrying capacity. * It is right to recall here that the magnitudes of the radii, rg and rp, of curvature of the interacting teeth flanks, G and P, strictly correlate to the radius rN of the boundary circle in the crosssection of the tooth flanks through the corresponding pitch point.

210


9.4.2 Sliding between Tooth Flanks of the Gear and the Pinion in Crossed-Axis High-Conformal Gearing Sliding between the tooth flanks, G and P, of the gear and the pinion in crossed-axis high-conformal gearing depends on actual value of (1) the displacement, l, and (2) the crossed-axis angle, Σ. At a specified point of contact, K, of the gear tooth flank, G, and the pinion tooth flank, P, the linear velocity of sliding can be expressed in terms (1) of the magnitude, ωpl, of the vector ωpl of instant rotation, (2) of the crossed-axis angle, Σ, and (3) of the distance of the point, K, from the axis Og. This is also true with respect to the pinion. The rotation vector, ω slpl, of sliding can be decomposed into two components (Figure 9.5): sl ω slpl = ω slpr + ω lw

(9.30)

The component, ω slpr, of the rotation vector, ω slpl, of sliding is along the axis of instant rotation, Pln. This component of the rotation vector of sliding, ω slpl, causes profile sliding of the tooth flank, G, of the gear and of the tooth flank, P, of the pinion. The magnitude, ω slpr, of the rotation vector, ω slpr, can be calculated from the formula ω slpr = ω slpl sin (180° − Σ p )

(9.31)

As the pinion angle, Σp, can be expressed in terms of the rotations, ωg and ωp, and of the crossed-axis angle Σ, then Equation 9.31 can be represented in two equivalent forms Og

Apa

Pln

Apa

sl

ωpr

C

ωpl

ωg

sl ωlw

Op Σg

Ag Og

ωp

ωp

Ap

Op

Pln

ωpl

ωg

sl ωpl

Σ Σp

FIGURE 9.5 Determination of relative sliding of the tooth flanks, G and P, of the gear and the pinion in high-conformal crossed-axis gearing.

211


 1 + ωg − ωp  ω slpr = ω slpl sin  ⋅ Σ  1 + ωg 

(9.32)

or

 1 + ωg − ωp  ω slpr = ω pl sin 2  ⋅ Σ  1 + ωg 

(9.33)

for the calculation of the magnitude, ω slpr , of the rotation vector, ω slpr . Having calculated the angular velocity, ω slpr , the relative sliding of the tooth flanks, G and P, that is caused by this rotation is calculated by multiplying the magnitude, ω slpr, by the distance of the point of interest from the axis of instant rotation, Pln. sl Similarly, the component, ω lw , of the rotation vector, ω slpl, of sliding is perpendicular to the axis of instant rotation, Pln. This component of the rotation vector of sliding, ω slpl , causes the tooth flank, G, of the gear and the tooth flank, P, of the pinion to slide in a lengthwise direction. The magnitude, ω sllw , sl of the rotation vector, ω lw , of sliding can be calculated from the formula

sl ω lw = ω slpl cos(180 − Σ p )

(9.34)

As the pinion angle, Σp, can be expressed in terms of the rotations, ωg and ωp, and of the crossed-axis angle, Σ, then Equation 9.34 can be represented in two equivalent forms

 1 + ωg − ωp  ω sllw = ω slpl cos  ⋅ Σ  1 + ωg 

(9.35)

or

 1 + ωg − ωp  sl ω lw = ω pl sin 2  ⋅ Σ  1 + ωg 

(9.36)

sl for the calculation of the magnitude, ω sllw , of the rotation vector, ω lw . The unit vector, lla, is along the instant line of action, LAinst. The rotation vector, ω slpl, having been calculated (see Equation 9.30), the following formula can be used for the calculation of the vector of linear velocity of sliding, Vsl:

V sl = ω slpl × lla ⋅ l

(9.37)

212


Ultimately, the resultant sliding of the tooth flanks of the gear, G, and of the pinion, P, is the superposition of two components: (1) profile sliding due to the rotation, ω slpr and (2) sliding, Vsl, in the lengthwise direction. 9.4.3 Boundary N-Cone in Crossed-Axis High-Conformal Gearing A boundary N-cone in crossed-axis high-conformal gearing can be constructed in a similar manner to that of a boundary N-cone in intersected-axis high-conformal gearing (see Chapter 8). When the gears rotate, the motion of the pinion in relation to the gear can be construed as instant rotation about the axis of instant rotation, Pln. A boundary N-circle is traced up by the contact point, K, in such a relative motion. In theory, the radius, rN, of the boundary N-circle is a trade-off between the desired high contact strength of the interacting tooth flanks and low friction between the teeth flanks of the gear, G, and the pinion, P. In practice, run-out of the gear and of the pinion should be taken into account. Under the assumption that Δl = 0 and manufacturing errors are zero, the point of contact, K, is located at the point of intersection of the boundary N-circle by the instant line of action, LAinst. At any point within the axis of instant rotation, Pln, a boundary N-circle of a certain radius, r Ni , can be constructed, and a line of action, Liinst , can be constructed as well. The pressure angle, φ it.ω, is not mandatorily of the same value at all normal sections of the axis, Pln. The instant line of action, LAinst, is a line through a point within the pitch line, Pln. No kinematical and/or geometrical constraints are violated in such a consideration. In practice, it is reasonable to keep the pressure angle, φ it.ω , of a certain constant value, φ t.ω, within the active face width of the gear pair. Moreover, as a normal cross section through a point within the axis, Pln, approaches the apex, Apa, the radius r Ni of the boundary N-circle gets smaller. In this way, the path of contact, Pc, is the straight line through all the contact points, Ki. The path of contact passes through the apex, Apa. When the path of contact is rotated about the axis of instant rotation, the boundary N-cone is generated as loci of consecutive positions of the path of contact, Pc, in its rotation in relation to the axis, Pln. Definition 9.1 The boundary N-cone in crossed-axis high-conformal gearing is a cone of revolution that is generated by rotation of the path of contact, Pc, about the axis of instant rotation, Pln. The apex of the boundary N-cone is coincident with the plane of action apex, Apa.

213


It is naturally to assume that the concave tooth flank (primarily of the gear, G) is located outward from the boundary N-cone, while the convex tooth flank (primarily of the pinion, P) is located within the interior of the boundary N-cone. However, as the apex, Apa, is not coincident either with the gear base cone apex, Ag, or with the pinion base cone apex, Ap, the boundary N-cone is not the only constraint onto the geometry of the tooth flanks, G and P. The envelopes to consecutive positions of the boundary N-cone in their instant screw motion are used for this purpose instead. For the gear tooth flank, the constraint is generated when the boundary N-cone is rotated about the gear axis of rotation, Og. Similarly, for the pinion tooth flank, the constraint is generated when the boundary N-cone is rotated about the pinion axis of rotation, Op. Ultimately, the convex tooth flank of one member of the gear pair must be entirely located with the interior of the corresponding enveloping surface, while the concave tooth flank of another member of the gear pair must be entirely located outside the interior of the corresponding enveloping surface. The boundary cone angle, Γl, (Figure 9.6), can be specified in terms (1) of the radius, r Ni , of the boundary N-circle at a current point within the axis of instant rotation, Pln and (2) of the cone distance, Ai, of that point from the apex, Apa:  ri  Γ l = tan −1  N   Ai 

(9.38)

In general, a boundary N-surface should be considered. The boundary N-surface is a kind of Archimedean screw surface. This is possible geometrically under the assumption that manufacturing errors are zero. In practice, a boundary N-cone is a reasonable approximation to the screw boundary N-surface.

Ag

Ai

ωg

Pln

C

rNi

ωpl

Apa

Ap

Γl

ωp

FIGURE 9.6 The boundary N-cone angle and the cone angle, Γl, in high-conformal crossed-axis gearing.

214


9.5 Design Parameters of High-Conformal Crossed-Axis Gearing Designing of a high-conformal crossed-axis gear pair begins with the determination of the rotation vectors of the gear, ωg, and of the pinion, ωp, in a certain reference system. Once the rotation vectors, ωg and ωp, are known, the vector, ωpl, of instant rotation, as well as the shaft angle, Σ, can be determined. The axes of rotations, Og, Op, and Pln, are the straight lines along the rotation vectors ωg, ωp, and ωpl, respectively. Then, the known configuration of the axes of rotations, Og, Op, and Pln, makes it possible to determine the tooth ratio, u, and pitch cone angles, Γ, of the gear and the pinion, γ:

  sin Σ Γ = − tan −1    ω p /ω g + cos Σ    sin Σ γ = tan −1    ω g /ω p + cos Σ 

(9.39) (9.40)

The design parameters of a high-conformal crossed-axis gear pair can be specified based, to a great extent, on that for high-conformal intersected-axis gears. From this perspective, the vector of instant rotation, ωpl, and the axis of instant rotation, Pln, are of critical importance. As the instant motion of a pinion in relation to the mating gear is interpreted as instant rotation about the axis, Pln, the design parameters of a high-conformal crossed-axis gear pair can be specified within a reference plane through the pitch point, P. The pitch point, P, is at a cone distance, A, from the apex Apa. The reference plane is perpendicular to the axis of instant rotation, Pln, as schematically depicted in Figure 9.7. The calculated values of the pitch angles, Γ and γ, along with the given cone distance, A, make it possible to calculate the pitch diameter of the gear, dg, and of the pinion, dp

dg = 2 A cos Γ dp = 2 A cos γ

(9.41)

(9.42)

The back cone distance of the gear, BCg, as well as the back cone distance of the pinion, BCp, can be calculated in a similar manner to that mentioned above:

BC g = 2 A sin Γ

(9.43)

215


dg Σ Op ωp ωpl

rN

ωg

Pln

Apa

–ωg

Γl

BCp

Og

op

A

Ag

Og

View A

P

dp

C

Apa

Pln

View A

ωpl Ap

Op

FIGURE 9.7 Configuration of a normal reference plane in relation to the axis of instant rotation, Pln and the pitch cones of the gear and the pinion in a high-conforming crossed-axis gear pair.

BC p = 2 A sin γ

(9.44)

Once the normal reference plane is constructed, the tooth profile parameters of the gear and the pinion can be specified within the reference plane. Referring to Figure 9.8, two points, namely og and op, are in nature the points of intersection of the axes, Og and Op, by the normal reference plane. The points are at the distance cn = (BCg + BCp) from one another. Two circles of radii, BCg and BCp, that have the points og and op as the centers are constructed. The circles share a common point, which is the pitch point P.

216


ϕt.ω BC0.g

LAinst

ρp

BCp cg

op

BCf .g

ρf .g P

BCf .p

BCg

ρg

rN = rp

b

cp

c

rg

a

d

BC*f .g

og

K

BCo.p

ag

cn = (BCg + BCp) FIGURE 9.8 Geometry of a high-conforming crossed-axis gear pair within the normal reference plane.

A straight line of action, LAinst, within the normal reference plane is the line through the pitch point, P. The line LAinst forms a certain transverse pressure angle, ϕt.ω, with the perpendicular to the center-distance, cn. The point of contact, K, of the tooth flanks of the gear, G, and the pinion, P, is a point within the straight line LAinst. The further the contact point, K, from the pitch point, P, the more the freedom in selecting the radii of curvature of the tooth flanks. At that same time, the further the contact point, K, from the pitch point, P, the higher the losses on friction between the tooth flanks and wear of the tooth flanks of the gear and the pinion. Ultimately, the actual location of the contact point, K, is a trade-off between the two above-mentioned factors. Let us assume that the pinion is stationary, and the gear is performing the instant rotation in relation to the pinion. The axis, Pln, of the instant rotation, ωpl, is the straight line through the pitch point, P. The axis of instant rotation, Pln, is located within the plane through the axes, Og and Op, and it goes through the apex, Apa. When the pinion is motionless, the contact point, K, traces a boundary circle of a radius, rN, centered at P. In the normal cross section, the pinion tooth profile, P, can either align with a circular arc of the boundary circle, rN, or it can be relieved in the bodily side of the pinion tooth. As a consequence, the location of the center of curvature, cp, of the convex pinion tooth profile, P, within the straight line

217


of action, LAinst, is limited to the straight line segment, PK. The pitch point is included into the interval, (P, K) , as shown in Figure 9.8, while the contact point, K, is not. On the other hand, the location of the center of curvature, cg, of the concave gear tooth profile, G, within the straight line of action, LAinst, is limited to the open interval P → ∞. Theoretically, the pitch point, P, can be included in that interval for, K. However, this is completely impractical, and actually the center of curvature, cg, is located beyond the pitch point, P. Due to this, the radius of curvature, rp, of the convex pinion tooth profile, P, is smaller than that, rg, of the concave gear tooth profile, G, (the inequality rp < rg is always observed). Both the pinion teeth and the gear teeth are helical and of opposite hand. No spur high-conformal gearing is feasible in nature. Because both the gear and the pinion are helical and of opposite hand, the point of contact travels along the path of contact, Pc. It is therefore fundamental to the operating of the gears that contact occurs nominally at a point and that the point of contact travels across the full face width of the gears during rotation. It is clearly a condition of operation, that is, in a given profile the tooth surfaces should not interfere before or after culmination when rotated at angular speeds that are in the gear ratio. The transverses contact ratio, mp, of a high-conformal crossed-axis gear pair is zero (mp ≡ 0). The face contact ratio, mF, is equal to the total contact ratio (mt) of the gear pair and is always greater than one (mF = mt > 1). When rotation is transmitted from the driving shaft to the driven shaft, the contact point, K, travels along the path of contact, Pc (and it does not travel within the transverse cross section of the gear pair), that is, perpendicular to the normal reference plane. This is due to mp ≡ 0 and mF > 1 as mentioned above. To calculate the design parameters of a high-conformal crossed-axis gear pair, center-distance, cn, and the tooth ratio, u = ωp/ωg, of the gear pair should be specified. The back cone distance of the gear, BCg, and the pinion, BCp, can be expressed in terms of the center distance, cn, and of the tooth ratio, u

BC g = cn ⋅

u 1+ u

(9.45)

BC p = cn ⋅

1 1+ u

(9.46)

The displacement, l, at which the path of contact, Pc, is remote of the pitch point, P, must be known, as well as the transverse pressure angle, ϕt.ω. The displacement, l, is the principal design parameter of a high-conformal crossed-axis gear pair. Many of the design parameters of the high-conformal gear pair can be expressed in terms of the displacement (l = KP).

218


For the calculation of the radii of curvature, rg and rp, of tooth profiles of the gear and the pinion, respectively, the formulas

rg = l ⋅ (1 + k rg ) rp = l ⋅ (1 + k rp )

(9.47)

(9.48)

can be used. The actual value of the factor, krp, should satisfy the inequality krp ≥ 0. However, when the factor, krp, can be set equal to zero, the equality rp = l is observed. The factor, krg, is within the range krg = 0.03 … 0.10. The radius of the outer back cone distance of the pinion, BCo.p, is calculated from the formula

BCo. p = BCp + (1 − k po ) ⋅ l

(9.49)

The addendum factor, kpo, of the pinion depends on (1) the pressure angle, ϕt.ω, (2) absolute dimensions of the gear pair, (3) the accuracy of machining, and (4) the conditions of lubrication. The pinion addendum factor, kpo, can be set in the range

k po = 0.1 ÷ 0.2

(9.50)

The root back cone distance of the pinion, BCf.p, can be calculated from the equation

BC f . p = BC p − ag − δ

(9.51)

where ag is the dedendum of the mating gear [ag = (0.1 … 0.2) ⋅ l] and δ is the radial clearance in the gear pair (δ = l ⋅ kpo). The fillet radius, ρp, can be practically set in the range of ρp = 0.3 ⋅ l. The root back cone distance of the gear, BCf.g, is equal to

BC f . g = cn − BCo. p

(9.52)

The radius of the outer back cone distance of the gear, BCo.g, is calculated from the expression

BCo. g = BC g + ag

(9.53)

The corner of the gear tooth addendum should be rounded with the radius, ρg, which is less than the fillet radius, ρp, of the pinion (ρg < ρp).

219


The following relations among the design parameters in a high-conformal crossed-axis gear pair are anticipated to be practical (as the first approximation):

rp = l

(9.54)

rg ≤ 1.10 ⋅ rp ρp = 0.3 ⋅ l

(9.55)

(9.56)

mn = 0.8 l

(9.57)

tp = 1.5 tg

(9.58)

φ t.ω = 30°

(9.59)

λ = 60… 80°(ψ = 10… 30°)

(9.60)

circular pitch of teeth p = tg + tp + B, where backlash B = 0.2–0.4 mm. For the design parameters l, p, tg, tp, mn, and B corresponding angular values can be calculated (Table 9.2). The effective face width of the gear pair can be calculated as follows:

Feff = (1.1 ÷ 1.2) ⋅ p ⋅ tan λ

(9.61)

For preliminary analysis of high-conformal crossed-axis gearing, an empirical expression

l = (0.05 ÷ 0.20) ⋅ BCp

(9.62)

yields the value for the displacement, l, which could be practical. The functional face width and axial pitch of a high-conformal gear pair depend on each other. Consider a case when at a uniform rotation of the gear and the pinion, the contact point, K, travels along the path of contact, Pc, at a certain uniform liner velocity. Because the transverse contact ratio is zero (mp = 0) and the total contact ratio, mt, is equal to the face contact ratio, mF, the axial

220


TABLE 9.2 Design Parameters of High-Conformal Crossed-Axis Gearing The Design Parameter Angular displacement Angular module Angular pitch Angular tooth thickness (gear) Angular tooth thickness (pinion) Angular space width (gear) Angular space width (pinion) Angular backlash Angular addendum (gear) Angular addendum (pinion) Angular dedendum (gear) Angular dedendum (pinion)

Symbol

Equation

φl φm.n pφ φt.g φt.p φw.g φw.p φB φa.g φa.p φd.g φd.p

φl = tan−1(l/A) φm.n = tan−1(mn/A) pφ = tan−1(p/A) φt.g = tan−1(tg/A) φt.p = tan−1(tp/A) φw.g = tan−1(wg/A) φw.p = tan−1(wp/A) φB = tan−1(B/A) φa.g = tan−1(ag/A) φa.p = tan−1(ap/A) φd.g = tan−1(bg/A) φd.p = tan−1(bp/A)

Note: The following designations: ag, bg, and ap, bp relate to addendum and to dedendum of the gear and of the pinion respectively. These design parameters are measured within the normal reference plane of the high-conforming intersected-axis gear pair.

pitch pcl.g of the helix on the gear tooth flank, G, can be calculated from the formula p cl. g =

Feff ⋅ cos Γ mt

(9.63)

A similar expression p cl. p =

Feff ⋅ cos γ mt

(9.64)

is valid with respect to the axial pitch, pcl.p, of the helix on the pinion tooth flank, P. The quality of high-conformal gearing strongly depends first of all on the following design parameters: l, ϕt.ω, and λ. The tooth flanks of the gear, G, and the pinion, P, of high-conformal crossed-axis gearing are not conjugate surfaces; moreover, they are not enveloped to one another. High-conformal crossed-axis gearing has not been deeply investigated yet. This gearing has received only episodic application as no practical guide to design of this gearing has been developed.

Conclusion A brief overview of conformal gearing is performed from the perspective of how the known designs of conformal gearings meet the criteria for geometrically accurate (ideal) gearing. It is shown that none of the known designs of conformal gearings meet this criteria, that is, the known designs are either approximate gearings or they are not workable at all. Conditions to transmit a rotation smoothly, that is, the conditions ideal gearings need to satisfy, are briefly outlined. This set of conditions includes (a) the condition of contact of the tooth flanks, (b) the condition of conjugacy, (c) equality of the base pitches of the gear and the pinion to the operating base pitch of the gear pair, and (d) the requirement according to which the total contact ratio of the gear pair must be greater than one. It is shown that conformal gearing, and Novikov gearing in particular, is a degenerate case of parallel-axis involute gearing (Pa -gearing), for which the involute tooth profiles are truncated, and only one point of the involute tooth profiles remains. This point is referred to as the involute tooth point. The boundary Novikov circle (or just the boundary N-circle, for simplicity) in conformal gearing is introduced into consideration. This boundary circle establishes constraints onto the geometry of the tooth flanks of the gear and the pinion in conformal and high-conformal gearings. In later chapters of the book, the concept of the boundary Novikov circle is enhanced to the concept of the boundary Novikov cone (or just N-cone, for simplicity) in intersected-axis gearings (Ia -gearings). In a case of crossed-axis gearing (Ca -gearing), the corresponding screw boundary Novikov surface (or just N-surface, for simplicity) is introduced. A reasonable approximation to the boundary Novikov surface by a corresponding boundary Novikov cone is possible. In internal involute gearing, the degree of conformity at the point of contact of the gear and the pinion tooth flanks, G and P, is greater compared to that in external involute gearing. However, even in cases of internal gearing the concept of high-conformal gearing is useful as it allows achieving or even exceeding the desirable degree of conformity of the tooth flanks, G and P, that is, to proceed beyond the threshold that separates conformal gearing (Novikov gearing) from high-conformal gearing (Hc -gearing). Use of the concept of Hc -gearing makes it possible to take control over the contact geometry between the tooth flanks of the gear and the mating pinion. The load being transmitted by an Hc -gearing is applied at a point of contact of the tooth flanks of the gear, G, and the pinion, P. When the gears rotate, the contact point travels in the lengthwise direction, that is, along the path of contact, Pc. Therefore, even in cases of geometrically accurate Hc -gearing (i.e., ideal Hc -gearing), this could be an additional source of vibration and noise excitation from the gear pair. 221

222

Conclusion

Tooth profile sliding in conformal gearing (in Novikov gearing) and in highconformal gearing is investigated. The concept of high-conformal gearing is introduced. High-conformal gears feature a higher degree of conformity at the point of contact of the tooth flanks of the gear and the pinion. The accuracy requirements to conformal and to high-conformal parallelaxis gearing are investigated. It is proved that neither conformal nor high-conformal gears can be cut in a continuously-indexing process (i.e., in generating process). Gears can be cut by form milling cutters, form grinding wheels, etc. Based on the scientific classification of the vector diagrams for gearings of all possible types, intersected-axis as well as crossed-axis conformal and high-conformal gearings are considered. No profile modification is allowed to high-conformal gears, as well as to Novikov gears. This is, first of all, because Hc -gearing features the involute tooth flanks truncated to a point. Also, no lead modification is allowed to high-conformal and Novikov gears.

Glossary Here we list, alphabetically, the most commonly used terms in gearing. Most newly introduced terms are also listed below. Active profile: Part of the tooth profile that experiences contact during the mesh cycle. Addendum: In parallel-axis gearing, the addendum of a gear tooth is the radial distance from the nominal pitch circle to the top of the tooth (or, in other words, this is the portion of the gear tooth above the reference pitch surface). This concept can be applied to gearing of all other kinds. Addendum angle: In a bevel gear, this is the angle between elements of the pitch cone and the face cone. Addendum (chordal): The chordal addendum is used for the purpose of setting a gear tooth vernier to measure the tooth thickness, which is the chord length subtended by the two flanks of a gear tooth at the nominal pitch circle. This must be calculated from the known circular tooth thickness. The distance from the chord to the top of the tooth is known as the chordal addendum. Angular base pitch: This is an angle measured within the plane of action. The apex of the angle is coincident to the plane of action apex, Apa. The sides of the angle are through the corresponding points of intersection of two adjacent teeth flanks of the gear. Angular pitch: This is the angle subtended by the circular pitch, usually expressed in radians. Approximate gearing: A (a) parallel-axis gearing, (b) intersected-axis gearing, or (c) crossed-axis gearing, by means of which a rotation of the driving shaft at a uniform angular velocity is transmitted to the corresponding rotation of the driven shaft at a not uniform angular velocity. The tooth flanks of the gear and of the mating pinion in approximate gearing are shaped so that the base pitch of the gear is not equal to the base pitch of the pinion, and both of them are not equal to the operating base pitch of the gear pair (pb.g ≠ pb.op and/or pb.p ≠ pb.op; φb.g ≠ φb.op and or φb.p ≠ φb.op). Arc of approach: The arc on the pitch circle through which a tooth profile moves from its beginning of contact with a mating tooth profile until it reaches the pitch point. Arc of recess: The arc on the pitch circle through which a tooth profile moves from contact with a mating tooth profile at the pitch point until contact ends.

223

224

Glossary

Axial pitch: The pitch measured in the axial direction in helical gears. It is therefore normal circular pitch divided by the sine of the helix angle. Axial runout: Also known as “wobble,” this is the runout of the gear in the axial direction, measured at just below the root circle. It is expressed as a “total indicator reading.” Axis of instant rotation: The straight line through the plane of action apex along the vector of instant rotation; also referred to as the pitch line. Axodes: A pair of ruled surfaces that roll and slide upon one another in a special way such that there is no relative sliding perpendicular to the generators of the ruled surfaces; an obsolete term with limited usage (not recommended for use in the Theory of Gearing). Back angle: The elements of the back cone of a bevel gear extend from the outside diameter of the gear blank to its axis and are perpendicular to the elements of the pitch cone. It is usually equal to the pitch angle. Back cone distance: The distance along an element of the back cone from its apex to the pitch circle. Backlash: Amount the tooth space of one gear exceeds the tooth width of its mating gear. Base angular pitch: The angular distance between two adjacent lines of contact of the tooth flanks of the gear and the pinion in intersectedaxis and in crossed-axis gearing. This design parameter is measured within the plane of action of the gear pair. Base circle: In parallel-axis gearing, the base circle is the circle from which involute tooth profiles are developed. Base diameter: In parallel-axis gearing, this is the diameter of the base circle of an involute gear. Base helix angle: The angle of crossing of the straight generating line of a screw involute tooth surface with the gear axis of rotation. Base pitch: This is the pitch, in inches or millimeters, measured along the circumference of the base circle or along the line of action (or along the instant line of action). It is therefore the circumference of the base pitch divided by the number of teeth. It is the same if measured along the line of action because the corresponding profiles of involute gear teeth are parallel curves, and the base pitch is the constant distance between them along the common normal in the direction of rotation, which, by definition, is the line of action. Base spiral angle: The angle measured within the plane of action between tangent to the desirable line of contact and radial direction through the point of interest. Bottom land: This is the surface at the bottom of a tooth space which adjoins the fillets. For full fillet teeth, there is no bottom land as such. Boundary N-circle: In Novikov gearing and in parallel-axis high-conformal gearing, this is the circle that subdivides a transverse section of the

Glossary

225

gearing into two portions. The convex tooth profile of one member of the gearing must be entirely located within the interior of the boundary N-circle, while the concave tooth profile of another member of the gearing must be entirely located within the exterior of the boundary N-circle.* To be more precise, the concept of boundary N-circle should be referred to as the boundary N-cylinder. Boundary N-cone: In intersected-axis as well as in crossed-axis high-conformal gearing, this is the cone that subdivides the space onto two portions. The convex tooth flank of one member of the gear pair must be entirely located within the interior of the boundary N-cone, while the concave tooth flank of another member of the gear pair must be entirely located within the exterior of the boundary N-cone.† Boundary N-cylinder: See “Boundary N-circle.” Cartesian coordinate system: A reference system comprised of three mutually perpendicular straight axes through the common origin. Determination of the location of a point in the Cartesian coordinate system is based on the distances along the coordinate axes. Commonly, the axes are labeled as X, Y, and Z. Often, either a subscript or a superscript is added to designation of the reference system, XYZ. Center distance: In parallel-axis gearing, this is the distance between the axes of rotation of two gears in mesh with each other on parallel shafts. It is measured along the mutual perpendicular to the shafts, called the line of centers. In the case of crossed-axis gearing (skew axis helical gearing, worm gearing, hypoid gearing, etc.), the center distance is equal to the closest approach of two axes of rotation crossing in space. In particular case of hypoid gearing, the center distance is often referred to as the offset. Centerline: This is the straight line that is perpendicular to the two axes of rotation. Centerline plane: In a crossed-axis gear pair, the centerline plane is the reference plane through the centerline perpendicular to the axis of instance rotation of the gear and the pinion. Centrode: A line of intersection between axodes and their corresponding transverse planes. The term centrode is applicable to parallel axes gearing; an obsolete term with limited usage (not recommended for use in Theory of Gearing). Characteristic line: A limit configuration of the line of intersection of a moving surface that occupies two distinct positions when the distance between the surfaces in these positions approaches zero. In the limit case, a characteristic line aligns with the line of * The concept of the boundary N-circle was introduced at around 2008 by Dr. Radzevich, S.P. Dr. Novikov, M.L. himself did not use the concept of the boundary circle. † The concept of the boundary N-cone was introduced at around 2008 by Dr. Radzevich, S.P. This concept was not known to Dr. Novikov, M.L.

226

Glossary

tangency of the moving surface and with the envelope to consecutive positions of the moving surface. In other words, this is the curve of intersection of a single surface as defined by two distinct configurations. Circular pitch: The distance between the corresponding profiles of two adjacent teeth as measured along the pitch circle. It is therefore the circumference of the pitch circle divided by the number of teeth. Circular tooth thickness: The length of arc between the two flanks of a gear tooth, usually at the nominal pitch circle diameter. Clearance: A measure of the amount of space that exists between the tip of one gear tooth and the tooth-space bottom of the mating gear. Cone distance: In a bevel gear, the distance from the base of its cone to the cone’s apex is called the cone distance. Conformal gearing: This a gearing that features convex-to-concave contact between the teeth flanks of the gear and the mating pinion. Novikov gearing is a perfect example of conformal gearing. Conjugate: A term used to describe gear tooth forms which properly mate with each other. More generally, conjugate stands for reciprocally related and interchangeable as to properties, as two points, lines, etc. Reversibly enveloping form (i.e., Re -form) is another terminology for conjugate profiles/surfaces. Contact ratio: The measure of the average number of pairs of teeth in contact during the mesh cycle. In parallel-axis gearing, it is equal to the length of the arc of action divided by the base pitch. Culmination (culminating point, point of culmination): This is a point of intersection of the instant line of action and the boundary N-circle. In conformal and in high-conformal gearing the tooth flanks of the gear and the mating pinion interact with each other at culmination, and do not interact with one another before and after the culminating point. Darboux frame: In the differential geometry of surfaces, this is a local moving Cartesian reference system constructed on a surface. The origin of the Darboux frame is at a current point of interest on the surface. The axes of the Darboux frame are along three unit vectors, namely, along the unit normal vector to the surface and along two unit tangent vectors along the principal directions on the gear tooth flank. The Darboux frame is analogous to the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any nonumbilic point of a surface. It is named after a French mathematician Jean Gaston Darboux. Dedendum: The dedendum of a gear tooth is the radial distance from the nominal pitch circle to its root circle. In other words, this is a portion of the gear tooth below the reference pitch surface. Dedendum angle: In a bevel gear, this is the angle between elements of the root cone and pitch cone.

Glossary

227

Degree of conformity: A qualitative parameter to evaluate how close the tooth flank of one member of a gear pair is to the tooth flank of another member of the gear pair in the differential vicinity of a point of their contact (or at a point within the line of contact of the teeth flanks). Desirable line of contact: A planar curve that is entirely located within the plane of action of the gear pair. The interacting tooth flanks of a gear and a mating pinion in an ideal (geometrically accurate) gear pair are generated by means of the desirable line of contact. Diametral pitch: In spur gearing and transversal pitch-helical gearing, this is the number of teeth per inch of pitch diameter. EAP-point: (end-of-active-profile point); end-of-active-profile circle Effective length of the lines of contact: A portion of the total length of the line(s) of contact within which the driving gear is acting against the driven gear. Effective radius of curvature: A measure of the relative distance between two planar curves in tangency expressed in terms of the radius of curvature of each curve. Evolute: A locus of the centers of curvature for a planar curve. Face contact ratio: This is a design parameter equal to the face width divided by the axial pitch of the gear teeth. Face width: The length between two ends of a gear. It is equal to the teeth length of spur gearing. In helical or herringbone gearing, it is equal to the length of the teeth multiplied by cosine of the helix angle. Field of action: A portion of the plane of action bounded by two lines of intersection of outer surfaces of the gear and the pinion, and by two lines specified in terms of effective face width of the gear pair (it is also often called the zone of action). Fillet: A part of the tooth profile below the active region. Fillet radius: The radius of an arc approximating the root fillet curve. Gear apex: In a crossed-axis (spatial) gear pair, this is a point of intersection of the gear axis of rotation with the centerline. Gear machining mesh: This is the mesh between the gear to be machined and the imaginary (phantom) surface that is generated by the moving cutting edges of the gear cutting tool. This last surface is commonly referred to as the generating surface of the gear cutting tool. Gear ratio: The ratio between the instantaneous displacement of the output and the input. It is also known as the speed ratio, which is the number of teeth in the driven member (usually the larger, or the gear) divided by the number of teeth in the driver (usually the smaller, or the pinion). Generalized rack-type gear pair: This is a crossed-axis (spatial) gearing for which the vector of instant rotation is perpendicular to the gear axis of rotation. Geometrically accurate (ideal) gearing: A parallel-axis gearing, intersectedaxis gearing, or crossed-axis gearing that is capable of transmitting

228

Glossary

a uniform rotation of the driving shaft to a uniform rotation of the driven shaft. Heel: The thickest end of a bevel gear tooth. Helix angle: In helical and herringbone gears, this is the angle between the gear teeth and the axis of rotation of the gear. The orientation of the basic rack in relation to the gear axis of rotation is specified by the helix angle. Commonly, the helix angle is measured at the pitch circle. High-conformal gearing: This is a particular type of conformal gearing for which the degree of conformity at point of contact between the tooth flanks of the gear and the pinion exceeds a certain value, the so-called threshold. For simplicity, high-conformal gearing is often referred to as the Hc-gearing. Hunting ratio: A particular tooth ratio of a gear pair. This phenomenon used to describe two toothed bodies in mesh where the ratio of number of teeth cannot be reduced using a common integer. If the hunting ratio exists between two toothed bodies in mesh, each tooth of one body meshes with each tooth of the other body. Ideal gearing: A gearing with conjugate tooth flanks that has zero linear and angular displacements of the axes of rotations from their nominal configuration. Ideal gearing is commonly referred to as geometrically accurate gearing. Indicatrix of conformity: A planar centro-symmetrical characteristic curve of fourth order that is used for the analytical description of the geometry of contact of the gear tooth flank and the pinion tooth flank. In particular cases, the indicatrix of conformity also possesses the property of mirror symmetry. Instant line of action: A straight line that is tangent to the path of contact at an arbitrary point. Involute: A curve traced by a point on a flexible band at it is wrapped on/ unwrapped from another curve (the Evolute). Involute tooth point: The only point that remains from a truncated involute tooth profile. Lead: The axial advance of a helix for one complete turn, as in the treads of cylindrical worms and the teeth of helical gears. Lead angle: The angle between any helix and plane of rotation. It is the complement to the helix angle and used for convenience in worms and hobs. It is understood to be at the standard pitch diameter. Length of action: The distance along the line of action that a tooth moves from the beginning to the end of contact with its mating tooth. Length of path of contact (see Length of action): In conformal and highconformal gearings length of path of contact is zero. Line of action: The straight line that is tangential to the base circles of two mating gears. Its intersection with the centerline of the two base circles defines the pitch point. The line of action is the path the teeth follow while in contact.

Glossary

229

Line of contact: A line within which the gear tooth flank, G, and the pinion tooth flank, P, share common points. Low-tooth-count gear: A gear that has a base diameter, db.g, equal to or greater than the limit diameter, dl.g, of the gear, that is, the inequality db.g ≥ dl.g is valid with respect to low-tooth-count gearing. (LTC-gearing is the another term used for gears of this particular type). Master gear: Used in checking a production gear in a composite action inspection test, by rolling the master with the production gear in tight mesh (spring loaded) and measuring the variation in centers. Usually, the master gear is hardened, with ground teeth, and produced to a high degree of accuracy. Mesh cycle: The time length defined as the instance that two teeth come into contact until they get separated. The mesh cycle also yields interpretation in terms of angles of rotation or in terms of the length the contact point travels through. Module (of a gear): The design parameter used for specifying the size of gear teeth using ISO standards. The module is specified as the ratio of the pitch diameter in millimeters to the number of teeth. The module is reciprocal of diametral pitch. Mounting distance: In intersected-axis and in crossed-axis gearing, this is the distance from the crossing point of the axes of two bevel gears in mesh with each other to a locating surface of each. It is used in assembling bevel gearing. N bf -gears: These are conformal gears with the path of contact situated before the pitch point. N by -gears: These are conformal gears with the path of contact situated beyond the pitch point. Noninvolute gearing: In parallel-axis gearing, noninvolute gearing is comprised of gears that have noninvolute tooth profile. Noninvolute gearing is an example of approximate gearing. Normal plane: The normal plane (Nln-plane) in a crossed-axis gear pair is a reference plane through the axis of instance rotation of the gear and the pinion perpendicular to the centerline. Novikov gearing: (see conformal gearing). Number of teeth or threads: The number of teeth or threads contained in the whole circumference of the pitch circle. Operating base pitch: The angular distance between corresponding points within two desirable lines of contact of the tooth flanks of two neighboring teeth (measured within the plane of action, PA, in degrees/ radians). In ideal parallel-axis gearing, operating base pitch is a linear dimension. Pa -gearing: Gearing that features the axes of rotation of the gear and the pinion parallel to one another (parallel-axis gearing). Path of contact: Path of contact is a line segment within which the tooth flanks of the gear and the mating pinion interact with one another.

230

Glossary

The path of contact is measured within the plane perpendicular to the axis of instant rotation, Pln, of the gear pair. In conformal and high-conformal gearing, the length, Lpc, of the path of contact, Pc, is zero (i.e., Lpc = 0). Pinion: The smaller of two gears in mesh. Pinion apex: In a crossed-axis (spatial) gear pair, this is a point of intersection of the pinion axis of rotation with the centerline. Pitch: This is a measure of tooth spacing and size. Pitch circle: Circle through the pitch point that is centered on the gear/pinion axis of rotation (other definitions for the pitch circle are also known). Pitch diameter: Pitch nominal diameter is the diameter of the pitch circle of a gear. The nominal pitch circles of two gears in mesh at standard centers will be tangent to each other. The operating pitch circles of two gears when meshing at greater than standard centers, as when the pinion is oversize, will be greater than the nominal pitch circles and they will be tangential to each other. Pitch line: Straight line through the pitch point that is perpendicular to the centerline (other definitions for the pitch line are also known). Pitch plane: The pitch-line plane (Pln-plane) in a crossed-axis gear pair is the reference plane through the centerline and the axis of instance rotation of the gear and the pinion. Pitch point: Point of intersection of the centerline by the line of action in a parallel-axis gearing. Pitch point can also be specified as the point of tangency of two pitch circles, or of the pitch circle and the pitch line, and I on the line of centers (other definitions for the pitch point are also known). Pitch surfaces: A pair of ruled surfaces that roll and slide upon one another and are used as a reference when designing direct-contact mechanisms for spatial motion. In general, pitch surfaces are different from axodes. Pln -plane: In a crossed-axis (spatial) gear pair, this is the plane through the centerline and through the axis of instance rotation of the gear and the pinion. For intersected-axis gearing, as well as for parallel-axis gearing, the Pln-plane can be also defined as the plane through the axis of rotation of the gear and the pinion. Plane of action: The plane tangential to the base surfaces of two gears in mesh. It is perpendicular to the teeth flanks of two gears in mesh. Plane of action apex: In a crossed-axis (spatial) gear pair, this is a point of intersection of the axis of instant rotation with the centerline. Point of contact: Any point at which two tooth profiles/flanks touch each other. Power density: The amount of power transmitted per unit volume of the gearbox. Pressure angle: The included angle between the line of action and the plane tangential to two reference pitch surfaces where the line of action

Glossary

231

intersects. This angle is defined in terms of the transverse, axial, and normal pressure angles. Profile angle: The angle that makes a tangent to the gear tooth profile and centerline of the gear tooth. Profile contact ratio: This is the length of the path of contact in transverse cross section (either in millimeters or in degrees) divided by the base pitch also either in millimeters or in degrees (profile contact ratio is also referred to as Transverse contact ratio). Profile contact ratio is a unitless parameter. Pseudo path of contact: In conformal and high-conformal gearing, this is a line traced by the contact point in a stationary reference system, that is, in a reference system associated with the housing. Ppc is the commonly used designation for the path of contact. The pseudo path of contact equals to the effective face width of the gear pair. Rack: A toothed wheel whose pitch radius is infinite, pitch circle is a straight line, tooth number is infinite. Rotation vector: A vector along an axis of rotation that has a magnitude equal to the rotation about the axis. The direction of the rotation vector depends on the direction of the rotation. Commonly, the rotation vector is designated as ω. The magnitude of the rotation vector is commonly denoted by ω. Therefore, the equality ω = |ω| is valid. The rotation vector of a gear is designated as ωg, the rotation vector of the mating pinion is designated as ωp and the rotation vector of the plane of action is designated as ωpa. Rotation, in nature, is NOT a vector. Therefore, special care needs to be undertaken when treating rotations as vectors. Runout: Phenomenon describing the variation in pitch surface that results from nonzero eccentricity. There is a difference between the desired location of the axis of rotation and its actual location. It is measured in the radial direction and the amount of runout is the difference between the highest and the lowest reading in 360°. For gear teeth, runout is usually checked by placing a pin in tooth spaces and rolling past a dial indicator or by rolling with a master gear. SAP-point: (start-of-active-profile point); start-of-active-profile point (SAP-point). Shaft angle: The angle between the axes of two nonparallel gear shafts. Spacing: The term “spacing” is used as a general term to describe the accuracy with which teeth are spaced around the gear. Span measure: The measurement of the distance across several teeth of gears too large to use pin measurements. The measurement is made along a line tangent to the base circle. It is used to determine tooth thickness and tooth spacing accuracy. For such measurements a span measuring tool is used, set to touch the flanks of teeth at the ends of each span at or near the middle of the tooth height. A span measure tool will usually be set by a vernier and will be equipped

232

Glossary

with a dial indicator to indicate any deviation from the theoretical chord length. Toe: In a bevel gear or pinion, this is the thinnest end of a tooth. Tolerance: The amount by which a specific dimension is permitted to vary. It is usually expressed as the difference between the maximum and minimum limits allowed. Top land: The width, or thickness, of a gear tooth measured at its maximum (for external gears) or minimum (for internal gears) diameter. Total contact ratio: This is the summa of the transverse (profile) and the face contact ratios. Transmission function: The ratio between the instantaneous position of the output and the instantaneous position of the input. Transverse contact ratio: see Profile contact ratio. Transverse pressure angle: The pressure angle that is measured within a plane perpendicular to the axis of instant rotation of the gear and the pinion. Undercut: A condition during gear fabrication involving a generation process where auxiliary material is removed as a result of the relative motion between the cutter and the gear blank. For pinions with small numbers of teeth, the cutting tool will cut away that portion of the involute that is near and below the base circle. This cut away portion is called undercut, and it increases as the number of teeth becomes less. Vector of instant rotation: A vector along the axis of instant rotation either of a pinion in relation to the gear, or of a gear in relation to the pinion. The direction of the vector of instant rotation depends on the direction of rotation of the gear and the pinion. The rotation vector is commonly designated as ωpl. Whole depth: The distance from the top land of a gear tooth to its root. In parallel-axis gearing, full depth is the distance between the top land and the bottom land of a gear tooth. It is equal to half the difference between the outside diameter and the root diameter of a gear. Winding relationship: A manufacturing specification between the coordinates used to parameterize the cutter and those used to parameterize the desired gear. Working depth: The depth at which gear teeth are engaged. Zone of action: The portion of the plane of action within which the teeth flanks of the gear and of the pinion interact with one another (it is often called field of action).

References

1. Allan, T., Some aspects of the design and performance of Wildhaber–Novikov gearing, Proc. Inst. Mech. Engrs. Part I, 1964/1965, 179(30), 931–954. 2. An’ishchenko, V.V. and Koval’enko, G.D., Investigation of contact stress in Novikov gears using photo-elastic approach, in: Novikov Gearing, vol. 1, TsINTIAM, Moscow, 1964. 3. Bonnet, P.O., J. Ec. Polytech, 1867, 33, 31. 4. Cauchy, A.L., Leçons sur les Applications du Calcul Infinitésimal á la Geometrie, Imprimerie Royale, Paris, 1826. 5. Chironis, N., Design of Novikov gears, in: Gear Design and Application, Chironis, N.P. (ed.), McGraw-Hill Book Company, New York, 1967, pp. 124–135, 374. 6. Chironis, N., New tooth shape taking over? Design of Novikov gears, in: Gear Design and Application, Chironis, N. (ed.), McGraw-Hill Book Company, New York, 1967, pp. 124–135, 374. 7. da Vinci, L., The Madrid Codices, vol. 1, 1493, Facsimile Edition of Codex Madrid 1, original Spanish title: Tratado de Estatica y Mechanica en Italiano, McGraw Hill Book Company, 1974. 8. Dyson, A., Evans, H.P., and Snidle, R.W., Wildhaber-Novikov circular arc gears: Geometry and kinematics, Proc. R. Soc. London, 1986, A403, 313–340. 9. Favard, J., Course de Gèomètrie Diffèrentialle Locale, Gauthier-Villars, Paris, 1957, viii+553pp. 10. Fed’akin, R.V., Investigation of Strength of Circular-Arc Gear Teeth, PhD thesis, Zhukovskiy Aviation Engineering Academy, Moscow, 1955. 11. French, M.J., Conformity of circular-arc gears, J. Mech. Eng. Sci., 1965, 7(2), 220–223. 12. Hertz, H., Über die Berührung Fester Elastischer Körper (The contact of solid elastic bodies), Journal für die Reine und Angewandte Mathematik (J. Pure Appl. Math.), Berlin, 1981, pp. 156–171; Über die Berührung Fester Elastischer Körper und Über die Härte (The Contact of Solid Elastic Bodies and Their Harnesses), Berlin, 1882; Reprinted in: Hertz, H., Gesammelte Werke (Collected Works), 1, 155–173, 174–196, Leipzig, 1895, or the English translation: Miscellaneous Papers, translated by D.E. Jones and G.A. Schott), pp. 146–162, 163–183, Macmillan and Co., Ltd., London, 1896. 13. http://www.math.hmc.edu/faculty/gu/curves_and_surfaces/surfaces/ plucker.html. 14. Krasnoschokov, N.N., Fed’akin, R.V., and Chesnokov, V.A., Theory of Novikov Gearing, Nauka, Moscow, 1976, 173pp. 15. Kudriavtsev, V.N., Calculation and Design of Novikov Gearing, Leningrad Mozhaiskii Military Aviation Academy, Leningrad, 1959, 78pp. 16. Kul’ikov, G.V., Fed’yakin, R.V., and Chesnokov, V.A., An investigation of contact strength of Novikov gears, in: Novikov Gearing, vol. 2, Zhukovskiy Aviation Engineering Academy, Moscow, 1962. 17. Lagrange, J.L., Theorié des Fonctions Analytiques, Impr. De la République, prairial an V, Paris, 1797, 277pp.

233

234

References

18. Novikov, M.L., Fundamentals of Geometric Theory of Gearing with Point Meshing for High Power Density Transmissions, Doctoral thesis, Military Aviation Engineering Academy (MAEA), Moscow, 1955. 19. Novikov, M.L., Gearing with a Novel Type of Teeth Meshing, Zhukovskii Aviation Engineering Academy, Moscow, 1958, 186pp. 20. Pat. No. 186.436 (Great Britain). Improvements in and Relating to Gear Teeth. F.J. Bostock and S. Bramley-Moore, October 2, 1922. Filed: July 2, 1921, No. 18,014/21. 21. Pat. No. 1,425,144 (USA). Toothed Gearing. H.J. Schmick. Patented: August 8, 1922, Filed: June 30, 1921, Serial No. 481.561. 22. Pat. No. 1,601,750 (USA). Helical Gearing. E. Wildhaber. Patented: October 5, 1926, Filed: November 2, 1923. 23. Pat. No. 109,113 (USSR). Gear Pairs and Cam Mechanisms Having Point System of Meshing. M.L. Novikov, National Classification 47h, 6. Filed: April 19, 1956, published in Bulletin of Inventions No.10, 1957. 24. Pat. No. 163,857 (USSR), A Helical Gearing, B.V. Shitikov and N.A. Bayazitov, Int. Cl. F06h. Filed: November 25, 1963, Published: July 22, 1964. 25. Pat. No. 182,462 (USSR), Gearing with Point System of Meshing and Having Multiple Paths of Contact. R.V. Fed’akin and V.A. Chesnokov, National Cl. 47 h, 6. Filed: November 20, 1963, Published in B.I. No. 7, 1966. 26. Pat. No. 1.185.749 (USSR), A Method of Sculptured Surface Machining on a MultiAxis NC Machine. S.P. Radzevich, B23C 3/16. Filed: October 24, 1983. 27. Pat. No. 1.249.787 (USSR), A Method of Sculptured Surface Machining on a MultiAxis NC Machine. S.P. Radzevich, B23C 3/16. Filed: December 27, 1984. 28. Radzevich, S.P., Differential-Geometrical Method of Surface Generation, Doctoral thesis, Tula Polytechnic Institute, Tula, 1991, 300pp. 29. Radzevich, S.P., Dudley’s Handbook of Practical Gear Design and Manufacture, 2nd Edition, CRC Press, Boca Raton, FL, 2012, 896pp. 30. Radzevich, S.P., Fundamentals of Surface Generation, Monograph, Rastan, Kiev, 2001, 592pp. 31. Radzevich, S.P., Methods for Investigation of the Conditions of Contact of Surfaces, Monograph, UkrNIINTI, №759–Uk88, Kiev, 1987, 103pp. 32. Radzevich, S.P., On analytical description of contact geometry of part surfaces in highest kinematic pairs, Theory for Mechanisms and Machines, St. Petersburg Polytechnic Institute, 2005, 3(5), 3–14. 33. Radzevich, S.P., High-conformal gearing: A new look at the concept of Novikov gearing, in: Proceedings of International Conference on Gears, October 5–7, 2015, Technical University of Munich (TUM), Garching (Near Munich), Germany, 2015, pp. 1303–1314. 34. Radzevich, S.P., Gear Cutting Tools: Fundamentals of Design and Computation, CRC Press, Boca Raton, FL, 2010, 786pp. 35. Radzevich, S.P., Generation of Surfaces: Kinematics Geometry of Surface Machining, CRC Press, Boca Raton, FL, 2013, 738pp. 36. Radzevich, S.P., On the priority of Dr. M.L. Novikov in the development of Novikov gearing, Theory of Mechanisms and Machines, http://tmm.spbstu.ru (in press). 37. Radzevich, S.P., Sculptured Surface Machining on a Multi-Axis NC Machine, Monograph, Vishcha Schola, Kiev, 1991, 192pp. 38. Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, CRC Press, Boca Raton, FL, 2012, 743pp.

References

235

39. Shevel’ova, G.I., Theory of Surface Generation and of Contact of Moving Bodies, MosSTANKIN, Moscow, 1999, 494pp. 40. Radzevich, S.P., Vector representation of gear pairs. Part I, in: Theory of Mechanisms and Machines, 2008, 6(2), 74–81. www.http://tmm.spbstu.ru/journal.html. 41. Radzevich, S.P., Vector representation of gear pairs. Part II, in: Theory of Mechanisms and Machines, 2009, 7(1), 17–26. www.http://tmm.spbstu.ru/journal.html. 42. Shishkov, V.A., Elements of kinematics of generating and conjugating in gearing, in: Theory and Calculation of Gears, vol. 6, LONITOMASH, Leningrad, 1948. 43. Shishkov, V.A., Generation of Surfaces in Continuously-Indexing Methods of Surface Machining, Mashgiz, Moscow, 1951, 152pp. 44. Willis, R., On the teeth of wheels, Trans. Civ. Eng., 1838, 2. 45. Willis, R., Principles of Mechanisms, Designed for the Use of Students in the Universities and for Engineering Students Generally, London, John W. Parker, West Stand, J. & J.J. Deighton, Cambridge, 1841, 446pp. 46. Yakovl’ev, A.S. and Pecheniy, V.I., An experimental investigation of the load distribution within the patch of contact in Novikov gears, in: Reliability and Quality of Gearing, vol. 18–67–36, NIIINFORMT’AZhMASh, Moscow, 1967. 47. Wildhaber, E., Basic Relationship of Hypoid Gears, American Machinist, 1946, Vol. 90, No. 4, February 14. 48. Wildhaber, E., Basic Relationship of Hypoid Gears-II, American Machinist, 1946, Vol. 90, No. 5, February 28. 49. Wildhaber, E., Basic Relationship of Hypoid Gears-III, American Machinist, 1946, Vol. 90, No. 6, March 14. 50. Wildhaber, E., Surface Curvature, Product Engineering, 1956, May.

Bibliography Allan, T., Some aspects of the design and performance of Wildhaber–Novikov gearing, Proc. Inst. Mech. Eng., 1964, 179, 931–954. Astridg, D.G. et al., Tribology of high conformity gears, Proc. Inst. Mech. Eng., 1987, 179, 819–825. Chironis, N.P., Design of Novikov gears, Prod. Eng., 1962, Vol. 33(19), pp. 91–102. Chironis, N.P. (ed.), Gear Design and Application, McGraw-Hill, New York, 1967, 374pp. Davis, A.W., Marine reduction gearing, (28th Thomas Lowe Gray lecture), Proc. Inst. Mech. Eng., London, 1956, 170, 477. Davis, W.J., Novikov gearing, Machinery, London, 1960, 96(2461), 64–73. Discussion on Novikov gears, Power Transm., London, 1960, Vol. 29, p. 722. Discussion on Novikov gears, Power Transm., London, 1961, Vol. 30, pp. 32. Enveloping gear teeth, Engineer, London, 1924, 138, 257. French, M.J., Conformity of circular-arc gears, J. Mech. Eng. Sci., 1965, Vol. 7, pp. 220. Fed’akin, R.V., On Selection of Tooth Geometries with Circular-Arc Profile, Moscow Zhukouvskiy Military Aviation Academy, Moscow, 1957, pp. 63–94. Fed’akin, R.V., Chesnokov, V.A., Calculation of Cylindrical Novikov Gearings, and Friction Transmissions, Moscow Zhuokuovskiy Military Aviation Academy, Moscow, 1982, pp. 114.

236

References

Fed’akin, R.V., Chesnokov, V.A., Novikov Gearings, Vestnik Mashinostroyaniya, 1958, Vol. 4, pp. 3–11. Fletcher, H.A.G., Berry, G., Efficiency and noise of a Novikov gear pair, NEL Report No. 46. French, M.J., Gear conformity and load capacity, Proc. Inst. Mech. Eng., Part I, 1966, 180(43), 1013–1024. Gribanov, V.M., Fundamentals of Accuracy and Determination of Tolerances for Novikov Gearings, Doctoral Thesis, Lugansk, Lugansk Machin-Building Institute, 1990, pp. 410. Grishel, I.N., Effect of variation of centre distance on the load capacity of Novikov type gearing, Russ. Eng. J. (Translation of Vestnik Mashinostroyeniya), 1959, Vol. 39(5), pp. 11. Gunner, A.M., Comment on H. Walker’s. A critical look at the Novikov gears, Engineer, London, 1960, 209, 725. Gunner, A.M., First A.E.I. CirCarC gear for the science museum, South Kensington, Machinery, London, 1963, 102, 1051. Investigation and Implementation of Novikov Gearings, USSR Academy of Sciences Publishers, Moscow, 1960. Itkis, M.Ya., Geometrical Calculations of Cylindrical Novikov Gearings, Nishn’evolzhskoye Publishers, Volgograd, 1973, 312pp. Johnson, D.C., Gear teeth with circular arc profiles. Engineering, London, 1959, 188, 294. Johnson, D.C., Gear teeth with circular arc profiles: The novikov gearing system, Engineering, London, 1960. http://trove.nla.gov.au/work/18014966 Johnson, D.C., Novikov gear: Gear teeth with circular arc profiles, Engineering, 1959, Vol. 294. Klein, G.J., The wildhaber-Novikov system of gearing, DME/NAE Quarterly Bulletin, 1965 (1), National Research Council of Canada, April 1965. Korotkin, V.I., Kharitonov, Yu.D., Novikov Gearings, Rostov State University Publishers, Rostov-on-Don, 1991, 208pp. Korotkin, V.I., Onishkov, N.P., Kharitonov, Yu.D., Novikov Gearing: Achievements and Development, Mashinostroyeniye-1, Moscow, 2007, 384pp. Korotkin, V.I., Onishkov, N.P., Kharitonov, Yu.D., Novikov Gearing: Achievements and Development, Nova Science Publishers, Inc., New York, 2011, pp. 249. Krasnoshchokov N.N., Fed’akin, R.V., Chesnokov, V.A., Theory of Novikov Gearing, Nauka, Moscow, 1976, 174pp. Mack, J., Investigation of the Conformal Gear for Helicopter Power Transmission, Boeing Co, Morton PA, Vertol Div., PDF Url: AD0605844, June 1964, 136pp. More about Novikov, The Guardian, 20th October, 1959. Niemann, G., Novikov-gear system and other special gear systems for high load carrying capacity. Novikov gearing, The Guardian 18th November 1958. Parker, D.A., Gears that “conform”. New Scient., 1965, 25, 790. Progress report on wildhaber-Novikov gearing, Prod. Eng., 1962, Vol. 33(13), 88. Pavlenko, A.F., Fed’akin, R.V., Chesnokov, V.A., Novikov Gearing, Technika, Kiev, 1978, pp. 144. Roslivker, Ye G., Hyperboloid Novikov gearing, in: Novikov Gearings, Vol. 3, Moscow Zhuokuovskiy Military Aviation Academy, Moscow, 1964, pp. 87–112. Roslivker, Ye G., Investigation of hyperboloid cylindrical and bevel Novikov gearings, in: Novikov Gearings, Research Institute Publishers, Rostov-on-Don, 1964, pp. 5–85.

References

237

Russia adopts the Novikov system of gearing, The Guardian, 11th November 1958. Sevr’uk, V.N., Theory of Screw-Circular Surfaces in Design of Novikov Gearings, Kharkov University Publishers, Kharkov, 1972, 167pp. Shotter, B.A., The LYNX transmission and conformal gearing, SAE Paper 781041, Aerospace Meeting, Town & Country, San Diego, November 27–30, 1978, 7pp. Silich, A.A., Development of Geometrical Theory of Novikov Gearings, and Tooth Flanks Generation, Doctoral Thesis, Kurgan, 1999, 534pp. The Novikov Gearing System, Review of Soviet and Western literature 1957–60. A.I.D. Report 60–3. Thompson, G.B., The shape of gear teeth, New Scient., 1960, 7, 665. Tuplin, W.A., The Novikov system of gearing, Machinery, London, 1958, Vol. 93, pp. 1260. V.B.B. Enveloping tooth gearing, Mar. Eng., London, 1933, 56, 371. Walker, H., A critical look at the Novikov gear, The Engineer, London, 1960, Vol. 209, April 29, pp. 725–729. Wells, C.F. and Shotter, B.A., The development of a “CisCarC” gearing, A.E.I. Eng., 1962, Vol. 2, pp. 83. Wildhaber, E., Foundations of Meshing of Bevel and Hypoid Gearings, Translated and comments by A.V. Slepak, Moscow, Mashgiz, 1948, 176 pp. [In Russian]. Winter, H. and Looman, M.J., Tools for making helical circular arc spur gears, VDI Berichte, 1961, 47, 5, 13, 19, 25.

Appendix A: On the Concept of “Novikov Gearing” and the Inadequacy of the Term “Wildhaber–Novikov Gearing” or “W–N Gearing” The beginning of wisdom is to call things by their right names —An Old Chinese Proverb Novikov gearing was invented early in 1950 by Dr. M.L. Novikov. An increased power density through a gear train is the main advantage of Novikov gearing. The concept of Novikov gearing was disclosed by Dr. Novikov in three main sources. His doctoral dissertation [18] is the first source in which the novel concept of gearing is discussed. Second, the Soviet Union patent [23] had been granted to Dr. Novikov on his invention of the novel gear system. Third, a comprehensive discussion on the concept of Novikov gearing can be found out in the monograph by Dr. Novikov [19]. The kinematics and geometry of Novikov gearing is disclosed in detail in these three sources [18,19,23]. The discussion below is based mostly on the original works by Dr. Novikov.

A.1 Kinematics and Geometry of Novikov Gearing An attempt to demonstrate the principal features of the novel system of gearing, namely of Novikov gearing, had been undertaken by the author [36], where additional information on the issue can be found. A.1.1 Preamble For a long while, gear experts in western countries had very limited access to original sources in which the concept of Novikov gearing is discussed in detail. In western countries, gear experts who were deeply involved in research and development of Novikov gearing often had no access even to the S.U. Pat. No. 109113 [23] granted to Novikov gearing. For example, Dyson et al. [8] undertook an analysis of Novikov gearing in comparison to helical gearing earlier proposed by Ernest Wildhaber [22]. When making the

239

240

Appendix A

comparison, Dyson et al. mistakenly referred to the S.U. Pat. No. 109750* and not to the S.U. Pat. No. 109113 [23]—a very common mistake committed by many gear experts in western countries. The aforementioned, along with numerous other cases of using wrong sources of information for comparison, makes it clear that the authors [8], as well as many others, were not familiar with the main sources [18,19,23]; moreover, they were not familiar with the main patent [23]. This has resulted in a poorly made comparison of the two inventions [22,23]. Chironis [6] approached very close to the understanding and to the correct interpretation of the concept of Novikov gearing. But even he failed to make correct conclusions from the analysis that he undertook. It should be mentioned here that in his work [6], Chironis quoted to Wildhaber’s opinion: “All the characteristics of the Novikov gearing are completely anticipated by my patent. My gearing never had a real test here, although a pair of gears was made in the 1920s. It would be wrong to dismiss the Russian development as old and known.” The discussion below reveals that Dr. Wildhaber was mistaken when making the statement “All the characteristics of the Novikov gearing are completely anticipated by my patent.” Moreover, it is easy to show that helical gearing by Dr. Wildhaber (Pat. No. 1,601,750 [USA]) is NOT workable in nature. For this purpose, it is sufficient just to recall the result earlier obtained by Leonhard Euler in the eighteenth century on involute tooth profile for mating gears. A.1.2 Main Features of Novikov Gearing Novikov gearing had been developed with the intent to increase contact strength of the gear teeth. Gearing of this kind featureshigher contact strength due to favorable curvatures of interacting tooth flanks. Under equivalent contact stress, similar dimensions and comparable parameters for the rest of the design, greater circular forces are permissible by the proposed gearing. Novikov gearing was developed for but not limited to parallel axis gear trains. However, gear pairs featuring intersecting axis, as well as gear pairs having crossing axes of the rotations of the gears, can be designed on the basis of the concept proposed by Novikov. Possible geometries of tooth profiles of the Novikov gears are schematically shown in Figure A.1. Here, a section of the tooth flanks by a plane that is perpendicular to the instant axis of relative rotation is shown. The axis is through the current point of contact of the tooth flanks. In Figure A.1, the point of intersection of the planar section by the axis of instant relative rotation is denoted by P. The points of intersection of the * S.U. Pat. No. 109750, A Water Sprayer./P.F. Pisulin, National Cl. 36, d, 28b, 801. Filed: January 2, 1957.

241

Appendix A

O1

P

B

rN B C

C

O2

A

FIGURE A.1 The only schematic is used in the S.U. Pat. No. 109113 to explain the concept of Novikov gearing.

planar section by the axes of the gear and of the pinion are designated as O1 and O2, respectively. A point A is the point of meshing (in its current location). The line of action is denoted as PA. Ultimately, ДAД is the circle centering at the point P. The circle corresponds to the limit case of the tooth profiles (in the case that the profiles are aligned to each other). Multiple curves BAB illustrate examples of possible tooth profiles of one of the mating gears. All the curves BAB are arbitrary smooth regular curves, which are located inside the limit circular arc ДAД (i.e., the arcs BAB are situated within the bodily side of the limit tooth flank of one of the gears). All the tooth profiles BAB feature a high rate of conformity to the limit circular arc ДAД. Multiple curves CAC illustrate examples of possible tooth profiles of the second of the mating gears. All the curves CAC are arbitrary smooth regular curves, which are located outside the limit circular arc ДAД (i.e., the arcs CAC are located within the bodily side of the limit tooth flank of another of two gears). All the curves CAC feature a high rate of conformity to the circular arc ДAД. The location and orientation either of the straight line of meshing or of the smooth curved line of meshing is specified in a space in which the location and orientation of the axes of rotations of the gear and of the pinion are given. The line of meshing is located reasonably close to the axis of instant relative rotation of the gears. Either constant or time-dependent (smoothly varying in time) speed of motion of the point of contact along the line of meshing is assigned. A coordinate system is associated with the gear, and a corresponding coordinate system is associated with the pinion. In the coordinate systems, the moving meshing point traces contact lines. One of the contact lines is associated with the gear and the other one is associated with the pinion. Certain smooth regular surfaces through the meshing lines can

242

Appendix A

be employed as tooth flanks of the gear and of the pinion. The following requirements should be fulfilled so that the surfaces could be used as the tooth flanks of Novikov gearing: • At every location of the point of contact, the tooth flanks should have a common perpendicular and thus the requirements of the main theorem of meshing should be satisfied • Curvatures of tooth profiles should correspond to each other, and, finally • No tooth flanks interference is allowed within the working portions of the surfaces Once two surfaces are generated by one of the moving curves BAB and by one of the moving curves CAC, then the above listed requirements are fulfilled and the surfaces can be employed as the tooth flanks of Novikov gearing. Consider a plane through the current meshing point, which is perpendicular to the instant axis of relative rotation. Construct two circular arcs centering at points within the straight line through the pitch point and the meshing point. The arc centers are located within the line of action and close to the pitch point. The constructed circular arcs can be considered as an example of tooth profiles of the gear and of the pinion. Tooth flanks are generated as loci of tooth profiles constructed for all possible locations of the contact point. The working portion of one of two tooth flanks is convex, while the working portion of the other tooth flank is concave (in the direction toward the axis of instant relative rotation). In a particular case radii of tooth profiles could be of the same magnitude and equal to the distance from the meshing point to the axis of instant relative rotation. Centers of both the profiles in this particular case are located at the axis of instant relative rotation. In such a scenario, the point kind of contact reduces to a special kind of line contact. This would require an extremely high accuracy of the center distance and independence from operation conditions, which is impractical. Point contact is preferred when designing tooth profiles. A small difference between the radii of curvature of tooth profiles is necessary. It should be borne in mind that under the run-in period of time point meshing of the gear teeth will be transformed to the abovementioned line meshing of the tooth profiles. However, the theoretical point contact of the tooth flanks will be retained. Generally speaking, tooth profiles of circular arc shapes are not mandatory. Tooth profiles of another geometries (those always passing through the meshing point) should be located (for one gear) within the interior of the above mentioned circular arc profile that is centering at the point within the axis of instant relative rotation as shown in Figure A.1. For the other gear the tooth profile should be located outside the circular arc. The above discussion is easier to understand when a boundary circle of radius rN having the pitch point P as the center is drawn through the point

243

Appendix A

of contact point K (in Figure A.1, this circle was added by the author to the original drawing by Dr. Novikov). It is proposed to call this circle boundary Novikov circle or just boundary N-circle for simplicity. Novikov himself did not use the concept of the boundary circle (the concept of the boundary circle was introduced later by Dr. Radzevich, probably during 2008). This concept has proved convenient in the theory of high-conformity gearing, and in Novikov gearing in particular. As an example, Figure A.2 illustrates tooth flank of a gear G, which makes contact at the point K with tooth flank of the mating pinion P. Circular-arc teeth profiles G and P are centering at points Og and Op accordingly. The centers are chosen so as to fulfill the necessary condition for magnitudes ρg and ρp for radii of curvature of the teeth profiles G and P at the point of tangency K (ρg > ρp). However, as the circular arcs G and P intersect the boundary N-circle, gearing of this kind is not feasible. As shown below, helical gearing by Ernest Wildhaber has features which make the configuration of circular-arc teeth profiles unfavorable and not workable in nature [22]. The law of motion of the meshing point (i.e., the speed of the point and its trajectory) should be chosen to minimize friction and wear loss. Friction and wear loss are proportional to the relative sliding velocity in the gear mesh. Therefore, the sliding velocity should be reduced as much as possible. For this purpose, the line of meshing should not be too far from the axis of instant relative rotation. On the other hand, too close a location of the line of meshing to the axis of instant relative rotation is also not desirable as contact strength of the gear tooth flanks is thereby reduced. In addition, it is recommended to ensure favorable angles between the common perpendicular (along which tooth flanks of one of the gears act against the tooth flanks of the other gear) and between the axes of rotations of the gears. The pposite sides of tooth profiles are designed in a similar manner to that just discussed. Tooth thicknesses and the tooth pitch are assigned to ensure the required bending strength of the teeth.

rN P

op

K ρp

og ρg

Boundary N-circle

FIGURE A.2 Using the concept of the boundary N-circle is proven to be helpful to distinguish feasible and not feasible circular-arc tooth profiles for Novikov gearing.

244

Appendix A

The face width of the gear or length of the gear teeth should correlate to their pitch to ensure the required value of the face contact ratio, mF. Gear pairs can feature either one point of contact (when working portions of the tooth flank contact each other at just one point, excluding the phases of the teeth re-engagement), or they can feature multiple contact points when tooth flanks contact each other at several points simultaneously. For parallel axis gear pairs, it is preferable to employ a straight line as the line of meshing. The straight line is parallel to the axes of rotations of the gear and of the pinion. The speed of motion of the contact point along the straight line of meshing may be of constant value. In this particular case, the radii of curvature of tooth profiles in all sections by planes are equal to each other. Tooth flanks in this case are a type of regular screw surfaces. Gears that feature tooth flanks of such geometry are easy to manufacture, and they can be cut on machine tools available in the market. An example of parallel-axis gearing with limit geometry of tooth profiles is illustrated in Figure A.1. Point contact of the teeth flanks in this particular case is transformed to their line contact. The curved contact line is located across the tooth profile. When axial thrust in the gear pair is strongly undesirable herring-bone gears can be used instead. The kinematics and geometry of Novikov gearing is different from that for involute gearing as well as from gearing of other designs. Referring to Figure A.3, consider a parallel-axis Novikov gear pair that is comprised of a driving pinion and of a driven gear. The gear is rotated about the axis Og, while the pinion is rotated about the axis Op. The axes of rotations Og and Op are at a certain center-distance apart from each other. The rotation of the gear ωg and the rotation of the pinion ωp are synchronized with each other in a timely proper manner. The pitch circle of the gear is of radius Rg and the pitch circle of the pinion is of radius Rp accordingly. The pitch circles (Rg and Rp) are tangent to each other. The point of tangency of the pitch circles is the pitch point P of the gear pair. A line, Linst, is a straight line through the pitch point P at a certain transverse pressure angle, ϕt, in relation to the perpendicular to the centerline Og − Op. The point of contact K of tooth flanks of the gear and of the pinion is a point within the straight line Linst. The further the contact point K is situated from the pitch point P, the more freedom is there in selecting radii of curvature of the tooth profiles. At the same time, the further the contact point K is situated from the pitch point P, the higher the losses on friction between the tooth flanks and wear of the tooth flanks. Ultimately, actual location of the contact point K is a kind of trade-off between these two factors. Further, let us assume that the pinion is stationary and the gear is performing instant rotation in relation to the pinion. The axis, Pln, of the instant rotation, ωpl, is the straight line through the pitch point P. The axis of the instant rotation Pln is parallel to the axes Og and Op of the rotations ωg and ωp.

245

Appendix A

R2 R1

R2

H

ρ O1

R

1H

O2

r2H

R1r

R 2H

r1 r2

r 1H

r2

r

R 2r

A

αA

h′

FIGURE A.3 A schematic that illustrates the concept of Novikov gearing in more detail. (After Novikov, M.L., Gearing with a Novel Type of Teeth Meshing, Zhukovskii Aviation Engineering Academy, Moscow, 1958, 186pp.)

When the pinion is motionless, the contact point K traces a circle of limit radius rlim centering at P. The pinion tooth profile P can either align with a circular arc of the limit circle, rlim, or it can be relieved in bodily side of the pinion tooth. As a consequence, the location of the center of curvature, cp, of the convex pinion tooth profile P within the straight line Linst is limited just to the straight line segment PK. The pitch point is included into the interval as is shown in Figure A.3, while the contact point K is not. On the other hand, the location of the center of curvature, cg, of the concave gear tooth profile G within the straight line Linst is limited to the open interval P → ∞. Theoretically, the pitch point P can be included in that interval for K. However, this is completely impractical, and the center of curvature cg is situated beyond the pitch point P. Due to this, radius of curvature, rp, of the convex pinion tooth profile P is smaller than that, rg, of the concave gear tooth profile G (i.e., rp < rg). It should be mentioned here that there are no physical constraints to design a gear pair having a convex gear tooth profile, while the pinion tooth profile is concave.

246

Appendix A

Both the pinion and the gear are helical and of opposite hand. No spur Novikov gearing is feasible in nature. Because the gears are helical and of the opposite hand, the point of contact will move axially along the gears while remaining at the same radial position on both gear and pinion teeth. It is, therefore, fundamental to the operation of the gears that contact occurs nominally at a point and that the point of contact moves axially across the full face width of the gears during rotation. It is clearly a condition of operation that in a given profile the tooth surfaces should not interfere before or after culmination, when rotated at angular speeds that are in the gear ratio. Transverses contact ratio, mp, of the Novikov gear pair is zero (mp ≡ 0). Face contact ratio, mF, of the gear pair is always greater than 1 (mF > 1). In transverse section of the gear pair the contact point K is motionless. For parallel-axes configuration, instant line of action, Linst, is a straight line through the contact point K. Instant line of action, Linst, is parallel to the axes Og and Op. A.1.3 Construction of the Boundary N-Circle The boundary N-circle for Novikov gearing can be constructed in the way briefly outlined below. Consider two axes of rotations Og and Op of a parallel-axis Novikov gearing as it is schematically depicted in Figure A.4. The axes Og and Op are at a certain center-distance C. The gear and the pinion are rotating about the axes Og and Op, and the rotations are labeled as ωg and ωp correspondingly. The gear ratio of the Novikov gearing is equal to u = ωg/ωp. The center-distance C is subdivided by a point P on two segments OgP and OpP correspondingly. The ratio of lengths of the straight line segments OgP and OpP is reciprocal to the gear ratio u of Novikov gear pair. Once the straight line segments OgP and OpP are the pitch radii OgP = rg and OpP = rp of the Novikov gear pair, then the equality rg/rp = u is observed. The point P is the pitch point of the Novikov gearing. A straight line Linst through the pitch point P is at the transverse pressure angle ϕt.ω with respect to the perpendicular to the centerline OgOp. Two points K are within the straight line Linst and are displaced from the pitch point P a certain distance ±l from the pitch point P. Lines of action are the two straight lines through the points K parallel to the rotation axes Og and Op. This distance, namely, the displacement, l, of the line of action is one of important geometrical parameters of Novikov gearing. The strength of gear teeth and performance of a gear pair strongly depend on the value of the displacement, l. The line of action, which is located beyond the pitch point, P (in the direction of rotation of the gears) features positive displacement, ±l. A conformal gear mesh of this type is referred to as BY-mesh of Novikov gear pair. The line of action, which is located before the pitch point (in the direction of rotation

247

Appendix A

ϕt · ω

K

LAinst

rg

ωp

ωg

–l rN rp

P

Og

+l Boundary N-circle K C

FIGURE A.4 Construction of the boundary N-circle for Novikov gearing.

of the gears), features negative displacement, −l. A conformal gear mesh of this type is referred to as BF-mesh of Novikov gear pair. To avoid violation of the conditions of meshing, as well as targeting wear reduction and reduction in friction losses the contact lines are displaced at a reasonably short distance from the axis of instant rotation, Pln. Let us assume that the pinion is motionless, then the contact point, K, traces a circle within the corresponding transverse section of the gear pair. The circle is centering at P. Similarly, the gear can be assumed to be stationary, then the contact point, K, traces a circle within that same transverse section of the gear pair. This circle is also centering at P. It is clear from the above consideration how the boundary circle of radius l can be constructed. A transverse section of a Novikov gear pair is subdivided by a circle of radius rN = |l| onto two areas. The area within the boundary circle of radius rN (including points those within the circle itself) represents the area of possible shapes of tooth profiles of one of the mating gears, and the area outside the circle of radius rN (including points those within the circle itself) represents the area of possible shapes of tooth profiles of the other of the two mating gears. A.1.4 Possible Geometries of Teeth Flanks for Novikov Gearing Prior to design mating the tooth profiles of a Novikov gear pair, the N-circle should be drawn. Refer to Figure A.5, where N-circle of radius rN is constructed for the pinion tooth profile (Figure A.5a), and for the mating gear tooth profile (Figure A.5b) of a Novikov gear pair. The displacement l is of positive value (l > 0) for the pinion addendum. The tooth profile of the pinion addendum is a convex segment of a smooth

248

Appendix A

ϕt · ω

(a)

1

LAinst

ωp

Kb

b

rg

b cr

ωg

–l Op

1

P

rN rp

2

a

Og

+l

a

Ka

a cr

C ϕt

(b)

a cr a 1

LAinst Ka

ωp

a 2

Op

P

rN rp

rg

–l

+l

b cr

b 2

ωg

Og b 1

Kb C

FIGURE A.5 Examples of possible tooth flank geometries for Novikov gearing: Possible shapes of teeth flanks of a pinion (a) and of a mating gear (b).

regular curve, Pia(i = 1,2, …), through the contact point Ka. The radius of curvature, R P, of the addendum profile is equal to or less than the radius rN of the boundary N-circle (R P ≤ rN). The case of equality R P = rN is the limit case, which is mostly of theoretical interest. Geometrically, the profile of the pinion addendum can be shaped in the form of a circular arc of the radius rN. This case of profile of the pinion addendum is the limit one of theoretical importance. It should be stressed here that not one of the feasible profiles Pia of a pinion addendum intersects the N-circle. The pinion addendum profile is entirely located within the boundary N-circle. Therefore, not any arc of a smooth

249

Appendix A

regular curve can be used as tooth profile of the pinion addendum. Circular arc, arc of ellipse at one of its apexes, cycloidal profile containing an apex, etc., are examples of applicable kinds of curves for the addendum tooth profiles. Spiral curves (involute of a circle, Archimedean spiral, logarithmic spiral, etc.) are examples of smooth regular curves no arc of which can be used in design of a pinion tooth addendum. This, due to the radius of curvature of a spiral curve (as well as of many other curves), is changes steadily when a point travels along the curve. Schematically this is illustrated in Figure A.6. As shown in Figure A.6a, an ellipse-arc ab is entirely located within the boundary N-circle. The ellipse-arc, ab, can be selected as a tooth addendum profile of a high-conformity gear pair. An ellipse-arc cd (Figure A.6a) is entirely located outside the N-circle. The ellipse-arc, cd, can be selected as a tooth dedendum profile of a high-conformity gear pair. Ultimately, an ellipse-arc ef (Figure A.6b) intersects the boundary N-circle. The ellipse-arc, ef, cannot be used as a tooth profile of a high-conformity gear pair. The same is valid for most of spiral curves. Therefore, at the point of tangency, K, spiral curves intersect the corresponding N-circle, which is prohibited. Ultimately it should be clear that variety of smooth regular curves can be used in designing a tooth profile of Novikov gearing. The variety of curves is not limited to circular arc only. The displacement l is of negative value (l < 0) for the pinion dedendum (Figure A.5a). The tooth profile of the pinion dedendum is a concave segment of a smooth regular curve, Pib (i = 1,2,…), through the contact point Kb. Radius of curvature, R P, of the dedendum profile is equal to or exceeds the radius rN of the boundary N-circle (R P ≥ rN). The case of equality R P = rN is the limit case, which is mostly of theoretical interest. Geometrically, the profile of the pinion addendum can be shaped in the form of a circular arc of the radius rN. This case of profile of the pinion addendum is the limit one of theoretical importance. (a)

(b)

1

c

rN

a P

e

rN K

b

K P

d

f

2 Boundary N-circle

Boundary N-circle

3

FIGURE A.6 Examples of feasible (a) and not feasible (b) ellipse-arc tooth profiles for Novikov gearing.

250

Appendix A

Constraints imposed on tooth profile geometry of the pinion dedendum are similar to that imposed on tooth profile of the pinion addendum. The dedendum profile is entirely located outside of the N-circle, shares a point with N-circle (the contact point Kb), and does not intersect the N-circle. Not all smooth regular curves can be implemented in designing the pinion tooth dedendum. An analysis similar to that performed earlier with regard to the pinion tooth profile can be performed with regard to the gear tooth profile as well. The analysis is illustrated in Figure A.5b. The gear tooth addendum, Gia, is entirely located within the boundary N-circle, while the gear tooth dedendum, Gib, is entirely located outside the boundary N-circle. Both the profile of the gear tooth addendum Gia as well as the profile of the gear tooth dedendum Gib share a common point with the boundary N-circle (the point Ka in the first case and the point Kb in the second). No intersection of tooth profiles Gia and Gib is allowed within tooth height of the gear and of the pinion. The importance of the concept of the boundary N-circle for gear engineers is as follows. A boundary N-circle of a Novikov gear pair is a type of constraint imposed on the gear tooth profile and on the pinion tooth profile. The gear engineer is free to select an arc of any smooth curve to shape the tooth addendum profile if the arc is entirely located within the boundary N-circle. The gear engineer is also free to select an arc of any smooth curve to shape the tooth dedendum profile if the arc is entirely located outside the N-circle. A possibility of a Novikov gear pair having two contact points, K′ and K″ simultaneously, inspired R.V. Fed’akin to propose (1955) a kind of Novikov gear pair that features not one contact line CL, as Novikov gear system does, but two lines of action instead. The invention by R.V. Fed’akin is schematically illustrated in Figure A.7. Two paths of contact, namely Ppc .bf and Ppc .by, are straight lines parallel to the axis of instant rotation of the gears. The contact lines Ppc .bf and Ppc .by pass through the points K′ and K″. They are at distances +l and −l from the pitch point P, respectively. As Novikov gears are helical, the contact points K′ and K″ are displaced in axial direction in relation to one another at a distance ΔZ. This distance can be computed from the formula

∆Z = 2

l tan ψ

(A.1)

The axial displacement of contact points results in the smoother rotation of the driven shaft of the Novikov gear pair. The average number of contact points between the gear and the pinion teeth flanks is doubled in the Novikov gear pair of this design. When designing Novikov gears, the gear designer is free to pick a favorable smooth curve to shape the tooth profile of the gear and of the pinion. An arc

251

Appendix A

ΔZ Ppc.bf

Z0 P

φp ωp

Xp

X0′

ϕt · ω

Xg

φp Op φp

rg

Ppc.bf P

φg

ωg

Yg Y0 Y0′

Yg Y0

Og

Ppc.by

Yp ϕt

X0

φg

Og

Op

ωp rp

ωg

K′

Yp X0

Z0

Ppc.by

φp

Xp

Zg

K′′

Zp

φg

φg

Y0′

Xg X0′

FIGURE A.7 On the concept of Novikov gearing having two lines of contact that is proposed by R.V. Fed’akin.

of the curve must be entirely located inside the N-circle for the tooth addendum, and a corresponding arc of the dedendum must be entirely located outside the N-circle of radius rN. A.1.5 Contact of Teeth Flanks for Novikov Gearing The possibility to ensure favorable contact of teeth flanks of Novikov gearing is the major advantage of a gear system of this design. In order to systematically describe favorable conditions of contact of the teeth flanks design, parameters of a Novikov gear pair that influence geometry of contact of the teeth flanks G and P should be considered. Consider the configuration of the interacting teeth flanks at the point of culmination. Figure A.8 shows a section in the transverse plane. The pinion,

252

Appendix A

which has a left-hand helix, is rotating, ωp, about the axis Op in a clockwise direction and is driving the gear. The last is rotating, ωg, about the axis Og. The point of contact, K, moves in a direction at right angles to and into the plane of the paper in Figure A.8. The pinion and the gear have working pitch radii of rp and rg = u ⋅ rp, respectively, where u is the gear ratio. The basic condition that the angular velocity ratio is equal to the gear ratio requires that the common normal at the point of contact between the teeth passes through the pitch point P. The angle ϕt is the transverse pressure angle. With teeth of involute form, this condition is maintained as the gears rotate with the teeth in contact. With circular-arc teeth, however, the condition occurs at only one instance in any one transverse plane as the pitch circles roll together. Immediately before as well as after the configuration shown in Figure A.8, there is no contact in that particular plane between the teeth shown. French proposed (1965) to refer to the instantaneous contact of profiles in a transverse section as to the culminating condition. When the gears are loaded, then due to elastic deformation of the gear materials the contact point spreads over a certain area of contact, which results in a finite contact period. Contact lines on the gear tooth flank, G, and on the pinion tooth flank, P, are helices of the opposite hands. If the screw parameter, pp, of the pinion tooth flank (reduced pitch of the pinion), P, is given, then for the computation of the screw parameter, pg, of the gear tooth flank G (reduced pitch of the gear) the

–l

K ϕt

ρp rN

ρg op Op

ωp

P C

og

ωg

Og

FIGURE A.8 Design parameters of a Novikov gear pair that influence geometry of contact of the teeth flanks G and P.

253

Appendix A

expression pg = pp/u can be used. This means that Novikov helical gears, which are in point contact, will transform rotation with a constant gear ratio if their screw parameters pg and pp are related as follows:

pg φg = pp φp

(A.2)

Here is designated: pg = rg tan λg, and λg is the lead angle, and rg is the pitch radius of the gear. Similarly, pp = rp tan λp, and λp is the lead angle, and rp is the pitch radius of the pinion. Because Novikov gears are helical and of opposite hand, the point of teeth flanks contact will move axially along the gears while remaining at the same radial position on both gear and pinion teeth. It is, therefore, fundamental to the operation of Novikov gears that contact occurs nominally at a point and that the point of contact moves axially across the full face width of the gears during rotation. It is clearly a condition of operation that in a given profile the tooth surfaces should not interfere before or after culmination, when rotated at angular speeds that are in the gear ratio.

A.2 Main Features of Helical Gearing by Wildhaber In 1926, a famous American gear engineer Dr. Ernest Wildhaber was granted the U.S. Pat. No. 1,601,750 on helical gearing. The helical gearing by Dr. Wildhaber features a circular-arc tooth profile. The main features of gearing of this design are illustrated in Figure A.9. The invention relates to the tooth shape of the gears, which run on parallel axes, and may be applied to helical gears, such as single helical gears and double helical gears or herring-bone gears. To provide accurate gearing of a circular-arc profile is one of the purposes of helical gearing. No other tooth profiles except circular-arc profile are proposed in this invention. Referring to Figure A.9, 1 denotes a helical gear having teeth 2 in contact with the teeth 3 of a mating pinion 4. As is customary, the helical gearing is analyzed with reference to a normal section, that is, line 2–2 in the upper portion in Figure A.9 being normal to the helix of the pitch circle. The lower portion in Figure A.9 illustrates the said normal section 2–2 for both pinion 4 and gear 1. As an example, it has been assumed that the tooth profiles 6 of the gear 1 are circular arcs of radii 7 and centers 8, in the shown normal section. Centers 8 are situated close to the pitch circle 9 of the gear. Location of the centers 8 in relation to the line of action is not specified in the invention. The corresponding teeth of the pinion 4 are so shaped as to allow rolling of

254

Appendix A

Oct. 5, 1926.

E. Wildhaber

1,601,750

Helical gearing Filed Nov. 2, 1923 2

17

16

3

2 sheets–sheet 1

1

11 12

4

17

16

2

2 4

4 15

3′ 11 11′ 2′

7 12

8′ 8

10 15′ 14

6 2

3 13

9 14′

1

FIGURE A.9 Schematic of “helical gearing” by E. Wildhaber (U.S. Pat. No. 1,601,750, 1926).

the pitch circles 9 and 10 on each other, as well known to those skilled in the art. Therefore, no freedom in choosing the pinion tooth profile is allowed in the invention. When the gear tooth 2 is in position shown in Figure A.9, and its center at 8, then it contacts with tooth 3 at a point 11, which may be determined by a perpendicular to the tooth 2 through the point 12, the point 12 being the contact point between the two pitch circles 9 and 10. Commonly, the point 12 is referred to as pitch point. The said perpendicular is in the present case the connecting line between the pitch point 12 and the center 8 of the tooth profile. Another position 2′ of the gear tooth, and 3′ of the corresponding pinion tooth are shown in dotted lines in Figure A.9. The tooth profiles contact here at a point 11′, which can be determined like the point 11. It will be noted that the contact point has traveled from 11 to 11′ during the small angular motion

Appendix A

255

of the gears.* A certain line of action LA is passing through the points 11 and 11′. The contact point has passed practically over the whole active profile during a turning angle 13 of the gear, which angle corresponds to a fraction only of the normal pitch 14, 14′. The said normal pitch equals the circular pitch of the shown normal section. Omitting numerous inconsistencies and discrepancies between the design parameters of the gear pair, it is of critical importance to stress here that traveling of the contact point within a transverse section† of the gear pair indicates that transverse contact ratio mp of the “helical gearing” [22] is larger than zero (mp > 0). In the gearing according to the invention [22], the contact point between two normal profiles passes over the whole active profile during a turning angle, which corresponds to less than one half the normal pitch, and usually to much less than that. It is then claimed that helical gearing [22] is capable of ensuring better contact between the teeth of the gear and of the pinion in a direction perpendicular to the contact line between two mating teeth. Therefore, it is expected that the proposed helical gearing features line contact of tooth flanks of the gear and of the pinion. The gearing according to the invention [22] is strictly a gearing for helical teeth. It would not be advisable on straight teeth, on account of the explained short duration of contact between tooth profiles. It should be pointed out here that in the invention [22], a short duration of contact and not instant contact between tooth profiles is anticipated. The working profiles of the gear are concave and circular, and their centers are substantially situated on the pitch circle of the gear. The convex working profiles of the pinion are also of circular shape. Their radii are substantially the same as the radii of the mate tooth profiles. The centers of these profiles are similarly situated on pitch circle of the pinion. As the centers of the teeth profiles are situated within the corresponding pitch circles, the centers cannot be situated within the line of action. The performed analysis reveals that helical gearing [22] is a type of helical gearing having a noninvolute tooth profile, and featuring transverse contact ratio that exceeds zero (mp > 0). According to the results of research undertaken earlier by L. Euler (in the eighteenth century), gear pairs of this particular nature are not feasible physically.‡ In other words, the results of the research earlier obtained by L. Euler reveal that helical gearing by Dr. Wildhaber [22] is not workable in nature.

* The ability of contact point to travel over the tooth profile is mentioned several times in the patent description. † Once the contact point is traveling within the normal section 2–2, then the projection of the contact point onto transverse section is traveling within the transverse section. ‡ It should be stressed here that helical gearing [22] is a kind of mistake committed by Dr. Wildhaber. Unfortunately, this mistake became widespread within the gear engineering community.

256

Appendix A

The infeasibility of Wildhaber’s helical gearing [22] along with the principal features of Novikov gearing (to be considered below) make it possible to conclude that these two types of gearing cannot be combined into a common gearing. They must be considered individually and separately from one another.

A.3 Principal Differences between Novikov Gearing and Helical Gearing by Wildhaber The helical gearing by Dr. Wildhaber features a circular-arc tooth profile. This particular feature of Wildhaber’s invention was confusing for some gear engineers in western countries. Taking into account that both Novikov gearing as well as Wildhaber gearing are kinds of helical gears, less experienced gear engineers loosely decided to combine both gear systems into a common system, and to refer to Novikov gearing as to Wildhaber–Novikov gearing or just to W–N gearing for simplicity. This combination is incorrect, as outlined next. Therefore, the two gear systems cannot be combined in a common system, and they should be considered separately from each other. The above mentioned terms Wildhaber–Novikov gearing or and W–N gearing are meaningless. These should be eliminated from gear engineering vocabulary as is adopted by proficient gear experts. The main differences between two gear systems are as follows: • Transverse contact ratio mp for Novikov gearing always equals zero (mp = 0). This requirement is a must for Novikov gearing. Total contact ratio mt in this case is equal to face contact ratio mF, that is, the expression mt = mF > 1 is valid for Novikov gearing. • Transverse contact ratio mp for Wildhaber gearing always exceeds zero (mp > 0) as the contact point is traveling within the transverse section (“…The contact point has passed practically over the whole active profile during a turning angle 13 of the gear …”). Thus, total contact ratio mt in this case is equal to sum of transverse contact ratio mp, and of face contact ratio mF, that is, the expression mt = mp + mF > 1 is valid for Wildhaber gearing. It is clear that gearing of no kind can simultaneously feature two different values of transverse contact ratio, namely the transverse contact ratio mp = 0 and the transverse contact ratio mp > 0. This inconsistency makes it clear that combining Novikov gearing and Wildhaber gearing into a certain common gear system is not possible. As a consequence of the equality mp = 0, Novikov gearing can be designed so to have a reasonably small difference between the curvatures of the convex tooth profile of one member and the concave tooth profile of the mating member of a Novikov gear pair. Wildhaber gearing does not allow for that.

Appendix A

257

A.4 Possible Root Causes for the Loose Term Wildhaber–Novikov Gearing or W–N Gearing From the author’s standpoint, the unfamiliarity of western engineers with the original publications [18,19,23] is the root cause for incorrect interpretation of the concept of Novikov gearing. The absence of access to the original documents [18,19,23], some of which were classified for a long period of time, along with the so-called iron curtain created a significant barrier for western engineers interested in the novel system of gearing.* Because of that, in western countries the attention was wrongly focused on the similarity of the circular-arc tooth profiles (which is of secondary importance for Novikov gearing), and not on the zero transverse contact ratio (mp = 0), which is of critical importance. Once the equality mp = 0 is valid for a gear pair, then the gear engineer gets much freedom in designing teeth profiles of mating gears, making them maximum conformal to one another. It should be mentioned here that as early as in November 13–16, 1957, an All-Union Scientific Conference “Practice of Implementation of Novikov Gearing” was held in Moscow. A decision to call the novel system of gearing Novikov gearing had been adopted by the conference. This decision honors Dr. Novikov as the inventor of the novel system of gearing, and should be acknowledged by gear experts all around the world.

A.5 A Brief Biographical Sketch of Dr. Mikhail L. Novikov (1915–1957) Not too much is known about Dr. Mikhail L. Novikov. It is known that he was born on March 25, 1915 in the city of Ivanovo, Russia. He was born to a lower class family—his parents were workers. At the age of 15, he began working as an apprentice in a machine building factory. Beginning 1934, Novikov attended the Moscow Bauman Technical University (MBTU). After 2 years of study at MBTU, he switched to the Military Aircraft Engineering Academy (MAEA) that bears the name of the famous scientist Professor N.I. Zhukovsky. It is likely that this * No scientific publications by western gear engineers who quoted to original publications [18], and [19] by Novikov, M.L. are known to the author. The other original publication [23] by Dr. Novikov is often mistakenly quoted as: S.U. Pat. No. 109750, A Water Sprayer./P.F. Pisulin, National Cl. 36, d, 28b, 801. Filed: January 2, 1957.

258

Appendix A

transition was due to his significant achievements in education. After graduation from MAEA in 1940, he was offered work there in one of the special departments. Over a short period of time, he moved from the position of assistant professor to the chair of the Department with the military rank of colonel. Parallel to teaching engineering courses for academy students, which was a must for him, Mikhail L. Novikov was involved in intensive research. He had been granted numerous patents on inventions most of which were of critical importance for aviation. In particular, Dr. Novikov was concerned with the problem of increasing the bearing capacity of gear pairs. The research undertaken in this particular area of mechanical engineering ended with the development of a novel system of gearing later called Novikov gearing in his honor. Mikhail L. Novikov became very well known in the international engineering community for the invention of a novel system of gearing. His idea was that he could overcome the barrier caused by the relations between the curvatures of the contacting surfaces when the gear tooth surfaces are in line contact. For this purpose, he proposed to reduce transverse contact ratio, mp, of a helical gear pair to zero (mp = 0). Under such a scenario, total contact ratio, mt, of the gear pair equals the face contact ratio, mF, and, thus, the expressing mt = mF > 1 is valid. For gearing of this kind the gear designer has more freedom in setting the curvatures of the mating tooth flanks of a gear pair. Evolving this concept, Dr. Novikov proposed that arcs of smooth regular curves be chosen as the profiles of helical gears [23]. These arcs are constructed in a plane that is perpendicular to the axis of instant rotation of the gears (perpendicular to the pitch line of a gear pair). In particular (but NOT mandatorily), circular arcs could be used in design of the teeth profiles. The difference between the curvatures of the convex tooth profile of one of the gears and the concave tooth profile of the mate is chosen to be small to absorb manufacturing errors, as well as teeth flanks displacements under the load, etc. Researchers are not so proficient, having a wrong understanding of the kinematics and geometry of Novikov gearing, mistakenly identifying Novikov gearing with the helical gearing earlier proposed by E. Wildhaber [22]. For Wildhaber’s helical gearing the expression mt = mF > 1 is NOT valid, as for this gearing another relation among the design parameters observe (mp > 0, mF > 0, and mt = mp + mF > 1). They loosely call Novikov gearing, Wildhaber– Novikov gearing or, for simplicity, W–N gearing. Proficient gear engineers realize the inadequacy of these two last terms and avoid using them. Novikov’s invention became very popular in the former USSR and later in Russia, and he was awarded a prestigious national prize. Much research on Novikov gearing had been carried out in the United States as well as in other western countries. His invention inspired many researchers, which resulted in valuable contributions being made to the theory of gearing.

Appendix A

259

Through all of his life, Mikhail L. Novikov was a modest person regardless of the fame and power he had. Intensive research work ceased with his sudden death at a young age, a catastrophe for his family, colleagues, and students.

A.6 A Brief Biographical Sketch of Dr. Ernest Wildhaber (1893–1979) Dr. Ernest Wildhaber is one of the most famous inventors in the field of gear manufacture and design. He received 279 patents, some of which have a broad application in the gear industry because of his work as an engineering consultant for The Gleason Works. Dr. Wildhaber’s most famous inventions are (a) the hypoid gear drive, which is still used in cars and (b) the Revacycle method, a very productive way to generate straight bevel gears. Dr. Wildhaber graduated from the Technische Hochschule of Zurich University in Switzerland and then came to the United States in 1919. In 1924, he went to work for The Gleason Works where he began the most successful period of his career as a creative engineer and inventor. Some of Wildhaber’s former colleagues from The Gleason Works have recounted how deeply impressed they were with his creativity, imagination, almost legendary intuition, and alacrity. This last characteristic was even the source of a colleague’s complaint that Wildhaber was not as patient as university professors while giving explanations. Could it be that he expected his coworkers to comprehend at the same speed as his thoughts? They remembered that he even sped up the stairs, taking two steps at a time. Wildhaber’s inventions reveal signs of his originality. He proposed different pressure angles for the driving and coast tooth sides of a hypoid gear, which allowed him to provide constancy of the tooth top-land. Wildhaber’s milestone invention was the Revacycle and the unusual shape of the tool based on the location of blades on a spatial curve. His theoretical developments [47–50] also display his originality; for example, he found the solution to avoid singularities and undercutting in hypoid gear drives. Ernest Wildhaber’s talent was recognized not only in the United States but also by the world engineering community and particularly by his alma

260

Appendix A

mater, Zurich University, which awarded him an honorary doctorate in engineering in 1962 (Gear Pairs and Cam Mechanisms Having Point System of Meshing by M.L. Novikov [S.U. Pat. No. 109113]). Novikov’s patent (S.U. Pat. No. 109113 of 1956) is a rare publication which is not available to the most of gear experts. No translation of the patent from Russian to English is available to the public. This causes problems in the proper understanding and in the interpretation of the significance of this milestone invention. Because of this and for the readers’ convenience, an invention disclosure of Novikov’s gearing along with its translation from Russian to English is placed next for free discussion and for a comparison with the Wildhaber’s gearing.

Appendix A

261

262

Appendix A

Appendix A

263

264

Appendix A

O1

P

B C

B

O2 C

265

Appendix A

№109113

Classification 47n, 6 USSR INVENTION DISCLOSURE to the Certificate on Invention M.L. Novikov

GEAR PAIRS AND CAM MECHANISMS HAVING POINT SYSTEM OF MESHING Filed: April 19, 1956, application №550525 to Committee on Inventions and Discoveries at Council of Ministries of the USSR Known designs of gearing, featuring point system of meshing, feature low contact strength and are not widely used in practice. Contact strength of known designs of gearing having line system of meshing including the widely used involute gearing is limited as well. The proposed gearing features higher contact strength due to favorable curvatures of interacting tooth flanks. Under equivalent contact stress, similar dimensions and comparable parameters for the rest of the design, greater circular forces are permissible by the proposed gearing. Lower sensitivity to manufacturing errors and to deflections under the load is another advantage of the proposed gearing. The proposed gearing can be designed either with parallel or with intersecting, or with crossing axes of rotations of the gears. External gearing as well as internal gearing of the proposed system of meshing is possible. Tooth ratio of the proposed gearing can be either of constant value or it can be variable, and time-dependent. The proposed concept of gearing can be utilized in design of cam mechanisms. Possible tooth profiles in cross-section of tooth flanks by a plane that is perpendicular to the instant axis of relative rotation through the current point of contact is illustrated in Figure. Here, the point of intersection of the planar cross-section by the axis of instant relative rotation is denoted by P. O1 and O2 are the points of intersection of the planar cross-section by the axes of the gear and of the pinion. A is the point of meshing (current location). PA denotes the line of action. ДAД is the circle centering at the point P which corresponds to the limit case of the tooth profiles (in the case the profiles are aligned to each other). Several curves BAB represent examples of tooth profiles of one of the mating gears. The curves BAB are arbitrary smooth curves, which are located inside of the circular arc ДAД (i.e., the arcs are located within

266

Appendix A

the bodily side of the limit tooth flank of one of the gears). The curves BAB are located close to the circular arc ДAД and they feature a high rate of conformity to the circular arc. Several curves CAC represent examples of tooth profiles of the second of the mating gears. The curves CAC are arbitrary smooth curves, which are located outside the circular arc ДAД (i.e., the arcs are located within the bodily side of the limit tooth flank of another of two gears). The curves CAC are also located close to the circular arc ДAД and they feature a high rate of conformity to the circular arc. The entity of the invention is disclosed next in detail. Location and orientation of either straight line meshing or smooth curved line of meshing is specified in a space in which location and orientation of axes of rotations of the gear and of the pinion are given. The line of meshing is located reasonably close to the axis of instant relative rotation of the gears. Either constant or time-dependent (smoothly varying in time) speed of motion of the point of meshing along the line of meshing is assigned. A coordinate system is associated with the gear, and a corresponding coordinate system is associated with the pinion. In the coordinate systems, the moving meshing point traces contact lines. One of the contact lines is associated with the gear and the other one is associated with the pinion. Certain smooth regular surfaces through the meshing lines can be employed as tooth flanks of the gear and of the pinion. The following requirements should be fulfilled so that the surfaces could be used as the tooth flanks: • At every location of the point of meshing, the tooth flanks should have a common perpendicular and thus the requirements of the main theorem of meshing should be satisfied, • Curvatures of tooth profiles should correspond to each other, and, finally • No tooth flanks interference is allowed within the working portions of the surfaces. The proposed kinds of tooth flanks fulfill the above listed requirements and allow for high contact strength of the gear teeth. Consider a plane through the current meshing point, which is perpendicular to the instant axis of relative rotation. Construct two circular arcs centering at points within the straight line through the pitch point and the meshing point. The arc centers are located close to the pitch point. The constructed circular arcs can be considered as an example of tooth profiles of the gear and of the pinion. Tooth flanks are generated as loci of tooth profiles constructed for all possible locations of the meshing point. The working portion of one of two tooth flanks is convex, while

Appendix A

267

the working portion of the other tooth flank is concave (in the direction toward the axis of instant relative rotation). In a particular case radii of tooth profiles could be of the same magnitude and equal to the distance from the meshing point to the axis of instant relative rotation. Centers of both profiles in this particular case are located at the axis of instant relative rotation. Under such a scenario, point kind of meshing reduces to a special kind of line meshing. This would require an extremely high accuracy of the center-distance and independence from operation conditions, which is impractical. Point meshing is preferred when designing tooth profiles. A small difference between the radii of curvature of tooth profiles is necessary. It should be kept in mind that under the runin period of time point meshing of the gear teeth will be transforming to the abovementioned line meshing of the tooth profiles. However, the theoretical point contact of the tooth flanks will be retained. Tooth profiles can differ from the circular arcs. However, tooth profiles of another geometries (those always passing through the meshing point) should be located (for one gear) within the interior of the abovementioned circular-arc profile that is centering at the point within the axis of instant relative rotation as shown in Figure. For the other gear, the tooth profile should be located outside the circular arc. The law of motion of the meshing point (i.e., speed of the point and its trajectory) should be chosen to minimize friction and wear loss. Friction and wear loss are proportional to the relative sliding velocity in the gear mesh. Therefore, the sliding velocity should be reduced as much as possible. For this purpose, the line of meshing should not be too far from the axis of instant relative rotation. On the other hand, too close a location of the line of meshing to the axis of instant relative rotation is also not desirable as that reduces the contact strength of the gear tooth flanks. In addition, it is recommended to ensure favorable angles between the common perpendicular (along which tooth flanks of one of the gears acts against the tooth flank of another gear) and between the axes of rotations of the gears. The opposite sides of tooth profiles are designed in a way similar to that just discussed. Tooth thicknesses and pitch are assigned to ensure the required bending tooth strength. Face width of the gear or length of the gear teeth should correlate to their pitch to ensure the required value of the face contact ratio. Gear pairs can feature either one point of contact (when working portions of the tooth flank contact each other just in one point, excluding the phases of the teeth re-engagement) or they can feature multiple contact points when tooth flanks contact each other at several points simultaneously. For parallel-axis gear pairs it is preferred to employ a straight line as the line of meshing, which is parallel to axes of rotations of the gear

268

Appendix A

and of the pinion. Speed of the meshing point along the straight line of meshing can be of constant value. In this particular case, radii of curvature of tooth profiles in all cross-sections by planes are equal to each other. Tooth flanks in this case are a kind of regular screw surfaces. Gears that feature tooth flanks of such geometry are easy to manufacture, and they can be cut on machine tools available in the market. An example of parallel-axis gearing with limit geometry of tooth profiles is illustrated in Figure. Point contact of the tooth flanks in this particular case is transformed to their line contact. The curved contact line is located across the tooth profile. When axial thrust in the gear pair is strongly undesirable, herring-bone gears can be used instead. Subject of the Invention Gear pairs as well as cam mechanisms having a point system of engagement that differ from known designs in the following: (a) tooth profiles are created as the lines of intersection of the tooth flanks by planes, which are perpendicular to the axis of instant relative rotation and that is passing through the point of meshing in its current location (b), are circular arcs or other smooth regular curves, those conformal to radii of curvature of the circular arc centering at the point of intersection of the instant axis of rotation by the plane (c), while the line of action, that is, the loci of points of meshing in space (within which configuration of the axes of rotations of the gear and of the pinion are specified), is a straight line or a smooth regular curve.

A.6.1 Helical Gearing by Ernest Wildhaber (U.S. Pat. No. 1,601,750) With the greatest respect to Dr. Ernest Wildhaber and to his achievements in the field of gearing and gear machining, it should be mentioned here that his “helical gearing” (U.S. Pat. No. 1,601,760 of 1926) is a kind of mistake. This mistake could be forgiven of Dr. E. Wildhaber—we all are mistaken from time to time. Unfortunately, this mistake significantly affected further developments in the field of gearing, and ultimately it resulted in wide usage of the completely wrong term “Wildhaber–Novikov gearing” or simply “W–N gearing.” The combination of Wildhaber’s gearing with the gearing proposed by Dr. M.L. Novikov is incorrect, and, thus it should be eliminated from the scientific vocabulary. These two completely different type of gearings, namely the one proposed by Dr. E. Wildhaber and the other proposed by Dr. M.L. Novikov, must be considered individually, and can NOT be combined into the wrong term “Wildhaber–Novikov gearing.” For this purpose and for the readers’ convenience, an invention disclosure of Wildhaber’s gearing is placed below for free discussion and for a comparison with Novikov’s gearing.

Appendix A

269

270

Appendix A

Appendix A

271

272

Appendix A

Appendix A

273

274

Appendix A

Appendix A

275

Appendix B: Elements of Vector Calculus The vector, the key to all the theory of part surface generation, is a triple real number (in most computer languages these are usually called floating point numbers) and is noted in a bold typeface, for example, A or a. Care must be taken to differentiate between two types of vectors: • Position vector. A position vector runs from the origin of coordinate (0, 0, 0) to a point (X, Y, Z) and its length gives the distance of the point from the origin. Its components are given by (X, Y, Z). The essential concept to understand about a position vector is that it is anchored to specific coordinates (points in space). The set of points that are used to describe the shape of all part surfaces can be thought of as position vectors. • Direction vector. A direction vector differs from a position vector in that it is not anchored to specific coordinates. Frequently, direction vectors are used in a form where they have unit length; in this case they are said to be normalized. The most common application of a direction vector in the theory of part surface generation is to specify the orientation of a surface or ray direction. For this, we use a direction vector at right angles (normal) and pointing away from the part surface. Such normal vectors are also the key in many calculations in the theory of part surface generation. Vector calculus is a powerful tool for solving many geometrical and kinematical problems that pertain to the design and generation of part surfaces. In this book, vectors are understood as quantities that have magnitude and direction and obey the law of addition.

B.1 Fundamental Properties of Vectors The distance-and-direction interpretation suggests a powerful way to visualize a vector, and that is as a directed line segment or arrow. The length of the arrow (at some predetermined scale) represents the magnitude of the vector, and the orientation of the segment and placement of the arrowhead (at one end of the segment or the other) represents its direction. Vectors possess certain properties, the set of which is commonly interpreted as the set of fundamental properties of vectors.

277

278

Appendix B

Addition. Given two vectors a and b, their sum (a + b) is graphically defined by joining the tail of b to the head of a. Then, the line from the tail of a to the head of b is the sum c = (a + b). Equality. Two vectors are equal when they have the same magnitude and direction. Position of the vectors is unimportant for equality. Negation. The vector −a has the same magnitude as a but the opposite direction. Subtraction. From the properties of “addition” and “negation,” the following a − b = a + (−b) can be defined. Scalar multiplication. The vector ka has the same direction as a, with a magnitude k times that of a. Here, k is called a scalar as it changes the scale of the vector a.

B.2 Mathematical Operations over Vectors The following rules and mathematical operations can be determined from the above listed fundamental properties of vectors. Let us assume that a set of three vectors a, b, c and two scalars k and t are given. Then, vector addition and scalar multiplication have the following properties:

a + b = b+ a

a + (b + c) = (a + b) + c

k (t a) = kt a

(k + t ) a = k a + t a

(B.4)

k (a + b) = k a + k b

(B.5)

(B.1)

(B.2) (B.3)

Magnitude a of a vector a is

a = |a| =

ax2 + ay2 + az2

(B.6)

where ax, ay, and az are the scalar components of a. A unit vector a in the direction of a vector a is

a =

a a = |a| a

(B.7)

279

Appendix B

The components ax , ay , and az of a unit vector a are also the direction cosines of the vector a

cos α = ax

(B.8)

cosβ = ay

(B.9)

(B.10)

cos γ = az

It is a common practice to denote the components ax , ay , and az by l, m and n accordingly. Scalar product (or dot product) of vectors: The formula

a ⋅ b = ax bx + ay by + azbz = |a||b| cos ∠(a , b)

(B.11)

is commonly used for calculation of scalar product of two vectors a and b. Equation B.11 can also be represented in the form  bx    a ⋅ b = [a] ⋅ [b] = [ax ay az ] ⋅ by   bz  T

(B.12)

Angle ∠(a, b) between two vectors a and b is calculated from

 a⋅b ∠(a , b) = cos −1  |a||b|

(B.13)

Scalar product of two vectors a and b features the following properties:

a ⋅ a = |a|2

a⋅b = b⋅a

a ⋅ ( b + c) = b ⋅ a + b ⋅ c

(k a) ⋅ b = a ⋅ (k b) = k(a ⋅ b)

(B.14)

(B.15) (B.16)

(B.17)

If a is perpendicular to b, then

a⋅b = 0

(B.18)

280

Appendix B

Vector product (or cross product) of two vectors: Vector product of two vectors can be calculated from the formula

a × b = ( a y bz − a z b y ) i + ( a z b x − a x bz ) j + ( a x b y − a y bx ) k

(B.19)

Here, in Equation B.19, i, j, and k are unit vectors in the X, Y, and Z directions of the reference system XYZ, in which the vectors a and b are specified. Vector product possesses the following properties: in case a × b = c, then the vector c is perpendicular to a plane through the vectors a and b. Vector product of two vectors a and b features the following properties: i a × b = ax bx

j ay by

k az bz

(B.20)

a × b = |a||b| n sin ∠(a , b)

(B.21)

where unit normal vector to the plane through the vectors a and b is denoted by n.

|a × b| = |a||b| sin ∠(a , b)

(B.22)

Coordinates of the vector product a × b can also be expressed in the form

 0  |a × b| =  az  − ay 

− az 0 ax

a y   b x   − a z b y + a y bz       − a x  ⋅  b y  =  − a x bz + a z bx  0   bz   − ay bx + ax by 

(B.23)

a × b = −b × a

a × ( b + c) = a × b + a × c

(k a) × b = a × (k b) = k(a × b)

(B.26)

i × j = k , j × k = i, k × i = j

(B.27)

(B.24) (B.25)

If a is parallel to b, then

a × b = 0

(B.28)

281

Appendix B

Triple scalar product of three vectors. The product (a × b) ⋅ c is commonly referred to as triple scalar product of three vectors a, b, and c. Triple scalar product of three vectors a, b, and c features the following properties:

(a × b) ⋅ c = (b × c) ⋅ a = (c × a) ⋅ b

(B.29)

( b × c) ⋅ a = a ⋅ ( b × c)

(B.30)

(a × b) ⋅ c = a ⋅ (b × c)

(B.31)

az bz cz

(B.32)

ax a ⋅ ( b × c) = bx cx

ay by cy

Triple vector product of three vectors. The product (a × b) × c is commonly referred to as triple vector product of three vectors a, b, and c. The product (a × b) × c can be evaluated by two vector products. However, it can also be evaluated in a more simple way by use of the identity

(a × b) × c = (a ⋅ c)b − (b ⋅ c)a

(B.33)

It should be mentioned here that in general, the triple vector products (a × b) × c and a × (b × c) are not equal

(a × b) × c ≠ a × (b × c)

(B.34)

The analytical interpretation of many problems and results in the field of geometry of surfaces are simplified when vector calculus is used. Lagrange equation for vectors. For the purposes of calculation of mixed product of vectors a and b an equation

(a × b) ⋅ (a × b) = (a ⋅ a)(b ⋅ b) − (a ⋅ b)2

(B.35)

can be used. Equation B.35 is due to Lagrange.*

* Joseph-Louis Lagrange (January 25, 1736–April 10, 1813), a famous French mathematician, astronomer, and mechanician.

Appendix C: Elements of Differential Geometry of Surfaces The discussion in this book is primarily focused on the kinematics of conformal and high-conformal gearings and the geometry of tooth flanks of gears. The gear and pinion tooth flanks and their motion in space in relation with one another are analytically described in a reference system. An orthogonal Cartesian* reference system is a major kind of reference systems that is commonly used for this purpose. Mutually perpendicular coordinate axes of a Cartesian coordinate system are conventionally labeled as X, Y, and Z. In a Cartesian reference system, the axes can be oriented in either a left- or right-handed sense. A right-handed Cartesian reference system is preferred, and all algorithms and formulas used in this book assume a right-handed convention. A coordinate system provides a numerical frame of reference for the threedimensional space in which the theory is developed. Two coordinate systems are particularly useful to us: the ubiquitous Cartesian (XYZ) rectilinear system and the spherical polar (r, θ, φ) or angular system. Cartesian coordinate systems are the most commonly used, but angular coordinates are often helpful as well.

C.1 Specification of a Gear Tooth Flank A gear tooth flank could be uniquely determined by two independent variables. Therefore, we give a gear tooth flank G (Figure C.1), in most cases, by expressing its rectangular coordinated Xg, Yg, and Zg, as functions of two Gaussian† coordinates Ug and Vg in a certain closed interval‡

 X g (U g , Vg )  Y (U , V )  g g g  G ⇒ rg = r g (U g , Vg ) =   Zg (U g , Vg )    1  

(C.1)

* René Descartes (March 31, 1596–February 11, 1650), (Latinized form: Renatus Cartesius), a French mathematician, philosopher, and writer. † Johan Carl Friedrich Gauss (April 30, 1777–February 23, 1855)—a famous German mathematician and physical scientist. ‡ All the equations that are valid for the gear tooth flank, G, are also valid for the pinion tooth flank, P.

283

284

Appendix C

Ug – curve

ng

Vg – curve

Tangent plane vg

m

Zg

+Ug

ug

rg Xg

+Vg Yg FIGURE C.1 Principal parameters of local topology of a gear tooth flank, G.

U1. g ≤ U g ≤ U 2. g ; V1. g ≤ Vg ≤ V2. g

where rg is the position vector of a point of the gear tooth flank, G Ug and Vg are curvilinear (Gaussian) coordinates of the gear tooth flank, G Xg, Yg, Zg are Cartesian coordinates of the point of the gear tooth flank, G U1.g, U2.g are the boundary values of the closed interval of the Ug-parameter V1.g, V2.g are the boundary values of the closed interval of the Vg-parameter The parameters Ug and Vg must enter into Equation C.1 independently, which means that the matrix

 ∂X g  ∂U g M =  ∂X g   ∂Vg

∂Yg ∂U g ∂Yg ∂Vg

∂Z g  ∂U g   ∂Z g   ∂Vg 

(C.2)

has a rank 2. Positions, where the rank is 1 or 0 are singular points; when the rank at all points is 1, then Equation C.1 represents a curve.

285

Appendix C

Other methods of surfaces specification are known as well. Specification of a gear tooth flank by • An equation in explicit form • An equation in implicit form • A set of parametric equations are among the most frequently used in practice methods of surfaces specification. It is assumed here and below that any given kind of a gear tooth flank specification can be converted either into the vector form, or into the matrix form of its specification as it follows from Equation C.1.

C.2 Tangent Vectors and Tangent Plane: Unit Normal Vector The following notation is proven to be convenient in the consideration below. The first derivatives of rg with respect to Gaussian coordinates Ug and Vg are designated as

∂rg = Ug ∂U g

(C.3)

∂rP = VP ∂VP

(C.4)

Ug |U g|

(C.5)

and for the unit tangent vectors ug =

vg =

Vg |Vg|

(C.6)

Correspondingly,* the direction of the tangent line to the Ug-coordinate line through a given point m on the gear tooth flank, G, is specified by the unit tangent vector ug (as well as by the tangent vector Ug). Similarly, the direction * It is right to underscore here that the unit tangent vectors u P and vP are dimensionless values as it follows from Equations C.5 and C.6.

286

Appendix C

of the tangent line to the Vg-coordinate line through that same point m on a gear tooth flank G is specified by the unit tangent vector vg (as well as by the tangent vector Vg). The significance of the unit tangent vectors ug and vg becomes evident from the following considerations. First, unit tangent vectors ug and vg yield an equation of the tangent plane to a gear tooth flank G at a specified point m:

rt. p − rgm    ug  Tangent plane ⇒  =0  vg     1 

(C.7)

where, rt.p is the position vector of a point of the tangent plane to a gear tooth flank G at a specified point m and rgm is the position vector of the point m on a gear tooth flank G. Second, tangent vectors yield an equation of the perpendicular Ng, and of the unit normal vector ng to a gear tooth flank G at a given point m N g = U g × Vg

(C.8)

and ng =

Ng U g × Vg = = ug × v g |N g| |U g × Vg|

(C.9)

When the order of the multipliers in Equations C.8 and C.9 is chosen properly, then the unit normal vector ng (as well as the normal vector Ng) is pointed outward of the bodily side of the surface G.

C.3 A Local Frame Two unit tangent vectors ug and vg along with the unit normal vector ng comprise a local frame ug, vg, and ng having origin at a current point m on a gear tooth flank G. Unit tangent vector ug is perpendicular to the unit normal vector ng (i.e., ug ⊥ ng), as well as unit tangent vector vg is also perpendicular to the unit normal vector ng (i.e., vg ⊥ ng). Generally speaking, the unit tangent vectors ug and vg are not perpendicular to each other, they form a certain

287

Appendix C

angle ωg. In order to construct an orthogonal local frame, either the unit tangent vector ug in the local frame (ug, vg, ng) must be substituted with a unit tangent vector u*g, or the unit tangent vector vg in that same local frame (ug, vg, ng) must be substituted with a unit tangent vector v *g . For the calculation of the newly introduced unit tangent vectors u*g and v *g , the following equations can be used:

u*g = u g × n g v *g = v g × n g

(C.10)

(C.11)

It is convenient to choose that order of the multipliers in Equations C.10 and C.11, which preserves the orientation (the hand) of the original local frame (ug, vg, ng), namely, if the original local frame (ug, vg, ng) is right-hand oriented, then the newly constructed local frame [either the local frame (u*g , v g , n g ), or the local frame (u*g , v g , n g )] should also be a right-hand oriented local frame, and vice versa. It should be pointed out here that another possibility to construct an orthogonal local frame is also available. Local frame of this kind is commonly referred to as a Darboux* frame, and is briefly considered below in this section of the book. Unit tangent vectors ug and vg to a surface G at a point m are of critical importance when solving practical problems in the field of gearing. This statement is proven by numerous examples shown below.

C.4 Fundamental Forms of a Surface Consider two other important issues concerning the gear tooth flank geometry—both relate to intrinsic geometry in differential vicinity of a current surface point m. First fundamental form of a surface. The first issue is the so-called the first fundamental form Φ1.g of a gear tooth flank G. The metric properties of a gear tooth flank G are described by the first fundamental form, Φ1.g, of the surface. Usually, the first fundamental form, Φ1.g, is represented as the quadratic form

Φ1.g ⇒ dsg2 = Eg dU g2 + 2Fg dU g dVg + Gg dVg2

* Jean Gaston Darboux (August 14, 1842–February 23, 1917), a French mathematician.

(C.12)

288

Appendix C

Here, in Equation C.12 is designated: sg is the linear element on a gear tooth flank G (sg is equal to the length of a segment of a certain curve on a gear tooth flank G). Eg, Fg, Gg are fundamental magnitudes of the first order at a surface point. Equation C.12 for the first fundamental form, Φ1.g, is known from many advanced sources. In the theory of gearing, another form of analytical representation of the first fundamental form, Φ1.g, is proven to be useful

Φ1. g

 Eg F g ⇒ dsg2 = [dU g dVg 0 0] ⋅  0  0

Fg Gg 0 0

0 0 1 0

0   dU g  0   dVg  ⋅ 0  0     1  0 

(C.13)

This kind of analytical representation of the first fundamental form Φ1.P is proposed by Dr. Radzevich [28,38]. The practical advantage of Equation C.13 is that it can easily be incorporated into computer programs when multiple coordinate system transformations are used. The last is vital for the theory of gearing. Fundamental magnitudes of the first order Eg, Fg, and Gg, can be calculated from the set of the following equations: Eg = U g ⋅ U g

Fg = U g ⋅ Vg

Gg = Vg ⋅ Vg

(C.14)

(C.15)

(C.16)

Equations C.14 through C.16 can be represented in an expanded form as Eg =

∂rg ∂X g ∂X g ∂Yg ∂Yg ∂Z g ∂Z g ∂rg ⋅ = ⋅ + ⋅ + ⋅ ∂U g ∂U g ∂U g ∂U g ∂U g ∂U g ∂U g ∂U g

Fg = Gg =

∂rg ∂r g ∂X g ∂X g ∂Yg ∂Yg ∂Z g ∂Z g ⋅ = ⋅ + ⋅ + ⋅ ∂U g ∂Vg ∂U g ∂Vg ∂U g ∂Vg ∂U g ∂Vg ∂rg ∂rg ∂X g ∂X g ∂Yg ∂Yg ∂Z g ∂Z g ⋅ = ⋅ + ⋅ + ⋅ ∂Vg ∂Vg ∂Vg ∂Vg ∂Vg ∂Vg ∂Vg ∂Vg

(C.17)

(C.18)

(C.19)

289

Appendix C

Fundamental magnitudes of the first order, Eg, Fg, and Gg, are functions of the Ug- and Vg-coordinates of a point of a gear tooth flank G. In general form, these relationships can be represented in the form: Eg = Eg (U g , Vg )

(C.20)

Fg = Fg (U g , Vg )

(C.21)

Gg = Gg (U g , Vg )

(C.22)

It is important to point out here that fundamental magnitudes Eg and Gg are always positive (i.e., Eg > 0, Gg > 0), and the fundamental magnitude Fg can be equal to zero (Fg ≥ 0). This results in that the first fundamental form, Φ1.g at a point of a gear tooth flank G, is always positively defined (Φ1.g ≥ 0), and it cannot be of a negative value. By use of the first fundamental form, Φ1.g, the following major parameters of geometry of a gear tooth flank G can be calculated: a. Length of a curve-line segment on a gear tooth flank G b. Square of a gear tooth flank G portion that is bounded by a closed curve on the surface c. Angle between any two directions on a gear tooth flank G Length, sg, of a curve-line segment U g = U g (t)

Vg = Vg (t)

(C.23)

(C.24)

on a gear tooth flank G is given by the equation t

sg =

∫ t0

2

2

dU g dVg  dVg   dU g  Eg  + 2Fg + Gg  dt  dt   dt  dt dt

(C.25)

t0 ≤ t ≤ t1

For calculation of square, Sg, of a gear tooth flank G patch Σ, which is bounded by a closed curve on the surface G, the following equation can be used:

Sg =

∫∫ Σ

EgGg − Fg2 dU g dVg

(C.26)

290

Appendix C

Ultimately, the value of the angle, ωg, between two given directions through a certain point m on a gear tooth flank G can be calculated from one of the equations below cos ω g = sin ω g = tan ω g =

Fg Eg G g Hg Eg G g Hg Fg

(C.27) (C.28) (C.29)

For the calculation of the discriminant, Hg, of the first fundamental form, Φ1.g, the following equation can be used:

Hg =

EgGg − Fg2

(C.30)

It is assumed here that the discriminant, Hg, is always nonnegative—that is, H g = + EgGg − Fg2 . The first fundamental form, Φ1.g, represents the length of a curve-line segment, and thus it is always nonnegative—that is, the inequality Φ1.g ≥ 0 is always valid. The first fundamental form, Φ1.g, remains the same when the surface is banding. This is another important feature of the first fundamental form Φ1.g. Second fundamental form of a surface. The second fundamental form Φ2.g of a gear tooth flank G is another of the two above mentioned issues. The second fundamental form Φ2.g describes the curvature of a smooth regular surface G. Consider a point K on a smooth regular part surface G (Figure C.2). Location of the point K is specified by two coordinates Ug and Vg. A line through the point K is entirely located within the surface G. A nearby point m is located within the line through the point K. Location of the point m is specified by the coordinates Ug + dUg and Vg + dVg as it is infinitesimally close to the point K. The closest distance of approach of the point m to the tangent plane through the point K is expressed by the second fundamental form Φ2.g. Torsion of the curve Km is ignored. Therefore, the distance a is assumed to be equal to zero (a = 0). The second fundamental form, Φ2.g, describes the curvature of a smooth, regular part surface G. Usually, it is represented as the quadratic form (Figure C.2):

Φ 2.g ⇒ − drg ⋅ dn g = Lg dU g2 + 2 M g dU g dVg + N g dVg2

(C.31)

291

Appendix C

m m*

K m*

a=0

m U + dU , V + dV g g g g Tangent plane m*

K Ug , Vg

Zg

Yg

Xg

FIGURE C.2 On definition of second fundamental form, Φ2.g, at a point of a smooth gear tooth flank, G.

Equation C.31 is known from many advanced sources. In the theory of gearing, another analytical representation of the second fundamental form, Φ2.g, is proven to be useful:

Φ 2. g

 Lg M g ⇒ [dU g dVg 0 0] ⋅   0   0

Mg Lg 0 0

0 0 1 0

0   dU g  0   dVg  ⋅ 0  0     1  0 

(C.32)

This analytical representation of the second fundamental form, Φ2.P, is proposed by Dr. Radzevich [28,38]. Similar to Equation C.13, the practical advantage of Equation C.32 is that it can easily be incorporated into computer programs when multiple coordinate system transformations are used. The last is vital for both for the theory of gearing. In Equation C.32, the parameters Lg, Mg, Ng designate fundamental magnitudes of the second order. By definition, fundamental magnitudes of the second order are equal:

Lg = − U g ⋅

∂n g ∂U g = ng ⋅ ∂U g ∂U g

(C.33)

292

Appendix C

∂n g ∂n g  ∂U g ∂Vg 1 Mg = −  U g ⋅ + Vg = ng ⋅ = ng ⋅ ∂Vg ∂U g  ∂Vg ∂U g 2

N g = − Vg ⋅

∂n g ∂Vg = ng ⋅ ∂Vg ∂Vg

(C.34) (C.35)

For the calculation of the fundamental magnitudes of the second order of a smooth regular gear tooth flank G, the following equations can be used: Lg =

(∂U g /∂U g ) × U g ⋅ Vg EgGg − Fg2

Mg =

(∂U g /∂Vg ) × U g ⋅ Vg 2 g

Eg G g − F

Ng =

=

(C.36)

(∂Vg /∂U g ) × U g ⋅ Vg EgGg − Fg2

(∂Vg /∂Vg ) × U g ⋅ Vg EgGg − Fg2

(C.37) (C.38)

Equations C.36 through C.38 can be represented in an expanded form as

Lg =

∂ 2X g ∂U g2

∂ 2Yg ∂U g2

∂ 2Zg ∂U g2

∂X g ∂U g

∂Yg ∂U g

∂Z g ∂U g

∂X g ∂Vg

∂Yg ∂Vg

∂Z g ∂Vg

Mg =

EgGg − Fg2

(C.39)

∂ 2X g ∂U g ∂Vg

∂ 2Yg ∂U g ∂Vg

∂ 2Zg ∂U g ∂Vg

∂X g ∂U g

∂Yg ∂U g

∂Z g ∂U g

∂X g ∂Vg

∂Yg ∂Vg

∂Z g ∂Vg

EgGg − Fg2

(C.40)

293

Appendix C

NP =

∂ 2X g ∂Vg2

∂ 2Yg ∂Vg2

∂ 2Zg ∂Vg2

∂X g ∂U g

∂Yg ∂U g

∂Z g ∂U g

∂X g ∂Vg

∂Yg ∂Vg

∂Z g ∂Vg

EgGg − Fg2

(C.41)

Fundamental magnitudes of the second order, Lg, Mg, Ng, are also functions of the Ug- and Vg-coordinates of a point of a gear tooth flank G. In general form, these relationships can be represented in the form:

Lg = Lg (U g , Vg )

(C.42)

M g = M g (U g , Vg ) N g = N g (U g , Vg )

(C.43)

(C.44)

The discriminant, Tg, of the second fundamental form, Φ2.g, can be calculated from the following equation:

Tg =

Lg N g − M g2

(C.45)

We now come to the theorem, which is essential justification for considering the differential geometry of surfaces in connection with the six fundamental magnitudes. It has been proven (1867) first by Bonnet* [3], and may be enunciated as follows: Theorem C.1 When six fundamental magnitudes Eg, Fg, Gg and Lg, Mg, Ng are given, and when they satisfy the Gauss characteristic equation, and the two Mainardi† – Codazzi‡ relations, they determine a gear tooth flank G uniquely say as to its position and orientation in space.

* Pierre Ossian Bonnet (December 22, 1819–June 22, 1892), a French mathematician. † Gaspare Mainardi (June 27, 1800–March 9, 1879)—an Italian mathematician. ‡ Delfino Codazzi (March 7, 1824–July 21, 1873)—an Italian mathematician.

294

Appendix C

This theorem is commonly referred to as the main theorem in the theory of surface, or simply as the Bonnet theorem. According to the main theorem, two surfaces that have identical first and second fundamental forms must be either congruent or symmetrical to one another. By use of six fundamental magnitudes, all parameters of local geometry of a given part surface can be calculated.

C.5 Principal Directions on a Gear Tooth Flank Direction of vectors of principal directions, T1.g and T2.g at a point on a gear tooth flank G, can be specified in terms of the ratio dUg/dVg. For the vectors of the first, T1.g, and for the second, T2.g, principal directions at a point m of a smooth, regular part surface G, the corresponding values of the ratio dUg/dVg are calculated as roots of the quadratic equation

Eg dU g + Fg dVg Lg dU g + M g dVg

Fg dU g + Gg dVg =0 M g dU g + N g dVg

(C.46)

The first principal plane section, C1.g, is perpendicular to a gear tooth flank G at a current surface point m, and passes through the vector of the first principal direction T1.g. The second principal plane section, C2.g, is orthogonal to a gear tooth flank G at a current surface point m, and passes through the vector of the second principal direction T2.g. The principal directions T1.g and T2.g can be identified at any and all points of a smooth, regular gear tooth flank G except of umbilic points, and in flatten points of the surface. At umbilic points of a surface as well as at flatten points, principal directions cannot be identified. In the theory of gearing, it is often preferred to not use the vectors T1.g and T2.g of the principal directions, but, instead, to use the unit vectors t1.g and t2.g of the principal directions. The unit tangent vectors t1.g and t 2.g are calculated from the equations t1. g = t 2. g =

T1. g |T1. g|

(C.47)

T2. g | T2. g |

(C.48)

Correspondingly, the unit tangent vectors t1.g and t 2.g of principal directions at a point m on a gear tooth flank G along with unit normal vector ng at

295

Appendix C

that same point m comprise an orthogonal local frame (t1.g, t 2.g, ng). All three unit vectors t1.g, t 2.g, and ng are mutually perpendicular to one another. The local frame (t1.g, t 2.g, ng) is commonly referred to as a Darboux frame.

C.6 Curvatures at a Point of a Part Surface The first, R1.g, and the second, R 2.g, principal radii of curvature at a point of a gear tooth flank G are measured within the first and in the second principal plane sections, C1.g and C2.g, accordingly. For the calculation of values of the principal radii of curvature, the following equation is commonly used: Rg2 −

Eg N g − 2Fg M g + Gg Lg Hg =0 Rg + Tg Tg

(C.49)

Remember that algebraic values of the radii of principal curvature, R1.g and R 2.g, relate to one another as R2.g > R1.g. In particular cases, at umbilic points on a gear tooth flank G, no principal curvatures can be identified as all normal curvatures of the tooth surface G at an umbilic point are equal to one another. Another two important parameters of local topology of a gear tooth flank G are • Mean curvature, Mg • Intrinsic curvature (Gaussian or full curvature) curvature, Gg For the calculation of the curvatures Mg and Gg, the following equations are commonly used:

Mg =

k1. g + k 2. g Eg N g − 2Fg M g + G g Lg = 2 2 .(EgGg − Fg2 ) Gg = k1. g ⋅ k 2. g =

Lg N g − M g2 EgGg − Fg2

(C.50)

(C.51)

The expressions for the mean curvature Mg and for the Gaussian curvature Gg:

Mg =

k1. g + k 2. g 2

(C.52)

296

Appendix C

Gg = k1. g ⋅ k 2. g

(C.53)

considered together yield a quadratic equation with respect to principal curvatures k1.g and k2.g: k g2 − 2 Mg k g + Gg = 0

(C.54)

The following

k1. g = Mg +

Mg2 − Gg

k 2. g = Mg −

Mg2 − Gg

(C.55) (C.56)

are the solutions to Equation C.54. Here, in Equations C.55 and C.56, the first principal curvature of a gear tooth flank G at a current point m is designated as k1.g, and k2.g designates the second principal curvature of a gear tooth flank G at that same point m. The principal curvatures k1.g, and k2.g are the reciprocals to the corresponding principal radii of curvature R1.g, and R2.g k 1. g = k 2. g =

1 R1. g

1 R2. g

(C.57)

(C.58)

The first principal curvature, k1.g, is always larger than the second principal curvature, k2.g, of a gear tooth flank G at a current point m—that is the inequality

k1. g > k 2. g

(C.59)

is always valid. This brief consideration of the major elements of part surface geometry makes possible the introduction of two definitions that are of critical importance for further discussion. As it is already mentioned earlier in this section of the book, it has been proved by Bonnet [3] that the specification of the first and the second fundamental forms determines a unique surface if Gauss’ characteristic equation and Mainardi–Codazzi’s relations of compatibility are satisfied, and those two surfaces that have identical first and second fundamental forms

297

Appendix C

are congruent.* Six fundamental magnitudes determine a surface uniquely, except as to position and orientation in space. Specification of a surface in terms of the first and the second fundamental forms is usually called the natural kind of surfaces representation. In general form, this kind of part surfaces representation can be expressed by a set of two equations: The natural form of a ⇒ G = G (Φ1. g , Φ 2. g ) surface G representation

Φ1. g = Φ1. g (Eg , Fg , Gg )  Φ 2. g = Φ 2. g (Eg , Fg , Gg , Lg , M g , N g )

(C.60) Equation C.60 can be derived from Equation C.1. A given gear tooth flank G can be expressed in both forms, namely, either by Equation C.19, or by Equation C.1.

C.7 An Illustrative Example Consider an example of how an analytical representation of a surface in a Cartesian reference system can be converted into the natural representation of that same surface [28]. A Cartesian coordinate system Xg Yg Zg is associated with a gear tooth flank G as it is schematically shown in Figure C.3. The position vector of a point, rg, of the gear tooth flank G can be represented as summa of three vectors

rg = A + B + C

(C.61)

Each of the vectors A, B, and C can be expressed in terms of projections onto the axes of the reference system Xg Yg Zg. Then, Equation 1.61 casts into the equation

 rb. g cos Vg + U g cos τ b. g sin Vg   r sin V − U sin τ sin V  b. g g g b. g g  rg (U g , Vg ) =   rb. g tan τ b. g − U g sin τ b. g    1  

(C.62)

* Two surfaces with the identical first and second fundamental forms might also be symmetrical. Refer to the literature—Koenderink, J.J., Solid Shape, The MIT Press, Cambridge, MA, 1990, 699 pages.—on differential geometry of surfaces for details about this specific issue.

298

Appendix C

ψb.g

B Zg

Ug

rb.g

H

E

C

vg*

Base cylinder helix

m

rg λb.g

A

ng ug Xg F

Vg

М0

M Involute curve

Yg

FIGURE C.3 Derivation of the natural form of representation of a gear tooth flank, G.

This yields the calculation of two tangent vectors Ug (Ug, Vg) and Vg (Ug, Vg), which are correspondingly equal:

 cos τ b. g sin Vg   − cos τ cos V  b. g g  U g (U g , Vg ) =   − sin τ b. g    0  

 − rb. g sin Vg + U g cos τ b. g cos Vg   r cos V + U cos τ sin V  b. g g g b. g g  Vg (U g , Vg ) =    rb. g tan τ b. g   0  

(C.63)

(C.64)

Substituting the derived vectors Ug and Vg into Equation C.14, we can come up with formulas for the calculation of the fundamental magnitudes of the first order

Eg = 1 Fg = −

(C.65)

rb. g cos τ b. g

(C.66)

299

Appendix C

Gg =

U g2 cos 4 τ b. g + rb2. g cos 2 τ b. g

(C.67)

These expressions can be substituted directly to Equation C.12 for the first fundamental form Φ1.g of the gear tooth flank, G Φ1. g ⇒ dU g2 − 2

U g2 cos 4 τ b. g + rb2. g rb. g dU g dVg + dVg2 cos τ b. g cos 2 τ b. g

(C.68)

The derived expressions for the fundamental magnitudes Eg, Fg, and Gg (see Equations C.65 through C.67) can also be substituted to Equation C.13. In this way, a corresponding matrix representation of the first fundamental form Φ1.g of the gear tooth flank, G, can be calculated. The interested reader may wish to complete these formulas on his or her own. The discriminant, Hg, of the first fundamental form of the gear tooth flank, G, can be calculated from the expression:

H g = U g cos τ b. g

(C.69)

In order to derive an equation for the second fundamental form, Φ2⋅g, of the gear tooth flank, G, the second derivatives of the position vector of a point, rg(Ug, Vg), with respect to Ug- and Vg-parameters are necessary. The above derived equations for the tangent vectors Ug and Vg (see Equations C.63 and C.64) make it possible the following expressions for the derivatives under consideration:

(C.70)

0  0  ∂U g =  0  ∂U P   1 

(C.71)

cos τ b⋅ g cos Vg   cos τ sin V  ∂U g ∂Vg b⋅ g g  ≡ =   0 ∂Vg ∂U g   1  

(C.72)

 − rb⋅ g cos Vg − U g cos τ b⋅ g sin Vg   − r sin V + U cos τ cos V  ∂Vg b⋅ g g g b⋅ g g  =   0 ∂Vg   1  

300

Appendix C

Further, substitute these expressions (see Equations C.70 through C.72) into Equations C.36 through C.38. After the necessary formula transformations are complete, then Equations C.36 through C.38 cast into the set of formulas for the calculation of the fundamental magnitudes of the second order of the gear tooth flank, G. This set of formulas is as follows: Lg = 0

Mg = 0

(C.73)

(C.74)

N g = −U g sin τ b⋅ g cos τ b⋅ g

(C.75)

Further, after substituting Equations C.73 through C.75 into Equation C.31, an equation for the calculation of the second fundamental form of the gear tooth flank, G, can be represented in the form:

Φ 2⋅ g ⇒ − drg ⋅ dN g = −U g sin τ b⋅ g cos τ b⋅ g dVg2

(C.76)

Similar to Equation C.68, the derived expressions for the fundamental magnitudes Lg, Mg, and Ng of the second order can be substituted into Equation C.32 for the second fundamental form Φ2⋅g. In this way, a corresponding matrix representation of the second fundamental form, Φ2⋅g of the surface G can be derived. The interested reader may wish to complete this formula transformation on his or her own. The result of the formulas derivation are summarized in Table C.1. For the calculation of the discriminant, Tg, of the second fundamental form, Φ2.g, of the gear tooth flank, G, the following expression can be used: Tg = U g sin τ b ⋅ g cos τ b ⋅ g

(C.77)

The natural representation of the gear tooth flank, G, can be expressed in terms of the derived set of six equations for the calculation of the fundamental magnitudes of the first Eg, Fg, Gg and of the second Lg, Mg and Ng:

TABLE C.1 Fundamental Magnitudes of the First and the Second Order of the Gear Tooth Flank, G Eg = 1 Fg = (rb.g/cos τb.g)

Gg = (U g2 cos 4 τ b. g + rb2. g / cos 2 τ b. g )

Lg = 0 Mg = 0 Ng = −Ug sin τb.g cos τb.g

301

Appendix C

All major elements of local geometry of the gear tooth flank, G, can be calculated based on the fundamental magnitudes, Eg, Fg, Gg, of the first, Φ1.P, and Lg, Mg, Ng, of the second Φ2.g, fundamental forms. Location and orientation of the gear tooth flank, G, are the two parameters that remain indefinite. Once a part surface is represented in natural form—that is, it is expressed in terms of six fundamental magnitudes of the first and of the second order— then further calculation of parameters of a gear tooth flank G becomes much easier. In order to demonstrate significant simplification of the calculation of parameters of a gear tooth flank G, several useful equations are presented below as examples.

C.8 A Few More Useful Equations Many calculations of parameters of geometry can be significantly simplified by use of the first and of the second fundamental forms of a smooth, regular part surface G.

1. For the calculation of value of radius, Rg, of normal curvature within a normal plane section through a current point m on a gear tooth flank G and at a given direction the following equation can be used: Rg =

Φ1. g Φ 2. g

(C.78)

2. Euler’s formula for the calculation of normal curvature, kθ.g, at a point m in a direction that is specified by the angle, θ, can be represented as follows:

kθ. g = k1. g cos 2 θ + k 2. g sin 2 θ

(C.79)

Here, in Equation C.79, θ is the angle that the normal plane section, Cg, makes with the first principal plane section, C1.g. In other words, θ = ∠(tg, t1.g); here tg designates the unit tangent vector within the normal plane section Cg. Equation C.79 also is a good illustration of the significant simplification of the calculations when fundamental magnitudes, Eg, Fg, Gg, of the first and Lg, Mg, Ng of the second order are used. In order to get a profound understanding of differential geometry of surfaces, the interested reader may wish to go to advanced monographs in the field. Systematic discussion of the topic is available from many sources. The author would like to turn the reader’s attention to the books by doCarmo, Eisenhart, Stuik, and others

Appendix D: Elements of Coordinate Systems Transformations Coordinate system transformation is a powerful tool for solving many kinematical and geometrical problems that pertain to the theory of gearing. Consequent coordinate systems transformations can be easily described analytically with the implementation of matrices. The use of matrices for coordinate system transformation can be traced back to the late 1940s and early 1950s. The implementation of coordinate system transformation is necessary for representation of the pinion, and its motion relative to tooth flank of the gear in a common reference system. At every instant of time, the configuration (position and orientation) of the pinion relative to the gear can be analytically described by means of a homogeneous transformation matrix corresponding to the displacement of the pinion from its current location to its consecutive location.

D.1 Coordinate System Transformation In this section of the book, the coordinate system transformation is briefly discussed from the standpoint of its implementation for the purposes of the theory of gearing. D.1.1 Introduction Homogenous coordinates utilize a mathematical trick to embed threedimensional (3D) coordinates and transformations into a four-dimensional (4D) matrix format. As a result, inversions or combinations of linear transformations are simplified to inversion or multiplication of the corresponding matrices. D.1.1.1 Homogenous Coordinate Vectors Instead of representing each point r(x, y, z) in 3D space with a single 3D vector

x   r = y  z 

(D.1)

303

304

Appendix D

homogenous coordinates allow each point r(x, y, z) to be represented by any of an infinite number of 4D vectors T ⋅ x  T ⋅ y   r= T ⋅ z     T 

(D.2)

The 3D vector corresponding to any 4D vector can be calculated by dividing the first three elements by the fourth, and a 4D vector corresponding to any 3D vector can be created by simply adding a fourth element and setting it equal to 1. D.1.1.2 Homogenous Coordinate Transformation Matrices of the Dimension 4 × 4 Homogenous coordinate transformation matrices operate on 4D homogenous vector representations of traditional 3D coordinate locations. Any 3D linear transformation (translation, rotation, etc.) can be represented by a 4 × 4 homogenous coordinate transformation matrix. In fact, because of the redundant representation of three-space in a homogenous coordinate system, an infinite number of different 4 × 4 homogenous coordinate transformation matrices are available to perform any given linear transformation. This redundancy can be eliminated to provide a unique representation by dividing all the elements of a 4 × 4 homogenous transformation matrix by the last element (which will become equal to one). This means that 4 × 4 homogenous transformation matrix can incorporate as many as 15 independent parameters. The generic format representation of a homogenous transformation equation for mapping the 3D coordinate (x1, y1, z1) to the 3D coordinate (x2, y2, z2) is

 T * ⋅ x2   T * ⋅ a  *   * T ⋅ y 2  =  T ⋅ e  T * ⋅ z2   T * ⋅ i    * *  T  T ⋅ n

T* ⋅ b T* ⋅ f T* ⋅ j T* ⋅ p

T* ⋅ c T* ⋅ g T* ⋅ k T* ⋅ q

T * ⋅ d   T ⋅ x2     T * ⋅ h  T ⋅ y 2  ⋅ T * ⋅ m   T ⋅ z2     T *   T 

(D.3)

If any two matrices or vectors of this equation are known, the third matrix (or vector) can be calculated and then the redundant T element in the solution can be eliminated by dividing all elements of the matrix by the last element. Various transformation models can be used to constrain the form of the matrix to transformations with fewer degrees of freedom.

305

Appendix D

D.1.2 Translations along a Coordinate Axis The translation of a coordinate system is one of the major linear transformations used for the purposes of the theory of gearing. Translations of the coordinate system X2Y2Z2 along the axes of the coordinate system X1Y1Z1 are illustrated in Figure D.1. The translations can be analytically described by the homogenous transformation matrices of dimension 4 × 4. For an analytical description of translation along the coordinate axes, the operators of translation Tr(ax, X), Tr(ay, Y), and Tr(az, Z) are used. The operators yield matrix representation in the form

1 0 Tr( ax , X ) =  0  0

0

1 0 Tr( ay , Y ) =  0  0

0

0

0

1 0 Tr( az , Z) =  0  0

0 0 1 0

0 1

0

0

0 1

0

0

1 0

0 0 1

1 0 0

ax  0  0  1

(D.4)

0 ay  0  1

(D.5)

0 0  az   1

(D.6)

Here ax, ay, az are signed values that denote distances of translations along corresponding axes. (a)

(b) Z1

Z2

Z1

Y2

Z1

Z2

ax

X1

Y1

Z2

(c)

X2 Y2

X2

Y2

ay Y1

X2

az

X1

X1 Y1

FIGURE D.1 Analytical description of the operators of translation Tr(ax, X), Tr(ay, Y), Tr(az, Z) along the coordinate axes. Parts (a)–(c) are discussed in the text.

306

Appendix D

Consider two coordinate systems X1Y1Z1 and X2Y2Z2 shifted along the X1 axis on ax (Figure D.1a). A point m in the coordinate system X2Y2Z2 is given by the position vector r 2(m). In the coordinate system X1Y1Z1, that same point m can be specified by the position vector r 1(m). Then, the position vector r 1(m) can be expressed in terms of the position vector r 2(m) by the equation r1(m) = Tr( ax , X ) ⋅ r2 (m)

(D.7)

Equations similar to that above are valid for other operators Tr(ay, Y) and Tr(az, Z) of the coordinate system transformation (Figure D.1b and c). Any coordinate system transformation that does not change the orientation of a geometrical object is an orientation-preserving transformation, or a direct transformation. Therefore, the transformation of translation is an example of direct transformation. D.1.3 Rotation about a Coordinate Axis The rotation of a coordinate system about a coordinate axis is another major linear transformation used in the theory of gearing. The rotation of the coordinate system X2Y2Z2 about the axis of the coordinate system X1Y1Z1 is illustrated in Figure D.2. For an analytical description of rotation about coordinate axes, the operators of rotation Rt(φx, X), Rt(φy, Y), and Rt(φz, Z) are used. The operators yield representation in the form of homogenous matrices: 1 0 Rt(ϕ x , X ) =  0  0

(a)

(b)

Z1

Z2

X1

Y1 Y2

0 sin ϕ x cos ϕ x 0

Z1

Z2

0 0  0  1 (c)

X2

(D.8)

Z1

Z2 X2

φy

X2

φx

0 cos ϕ x − sin ϕ x 0

Y1

X1 Y2

φz

Y1

X1

Y2

FIGURE D.2 Analytical description of the operators of rotation Rt(φx, X), Rt(φy, Y), Rt(φz, Z) about the coordinate axes. Parts (a)–(c) are discussed in the text.

307

Appendix D

cos ϕ y  0 Rt(ϕ y , Y ) =   sin ϕ y   0

 cos ϕ z  − sin ϕ z Rt(ϕ z , Z) =   0   0

0 1 0 0

− sin ϕ y 0 cos ϕ y 0 sin ϕ z cos ϕ z 0 0

0 0 1 0

0 0  0  1 0 0  0  1

(D.9)

(D.10)

Here, φx, φy, and φz are signed values that denote angles of rotation about the corresponding axis: φx is the rotation around the X-axis (pitch); φy is the rotation around the Y-axis (roll), and φz is the rotation around the Z-axis (yaw). Consider two coordinate systems, X1Y1Z1 and X2Y2Z2, turned about the X1-axis through the angle φx (Figure D.2a). In the coordinate system X2Y2Z2, a certain point m is given by the position vector r 2(m). In the coordinate system X1Y1Z1, that same point m can be specified by the position vector r 1(m). Then the position vector r1(m) can be expressed in terms of the position vector r 2(m) by the equation

r1(m) = Rt(ϕ x , X ) ⋅ r2 (m)

(D.11)

Equations similar to that above are valid for other operators Rt(φy, Y) and Rt(φz, Z) of the coordinate system transformation. These elementary coordinate system transformations are schematically illustrated in Figure D.2b and c. D.1.4 Coupled Linear Transformation It is right to note here that a translation, Tr(ax, X), along X-axis of a Cartesian reference system, XYZ, and a rotation, Rt(φx, X), about the axis X of that same coordinate system, XYZ, obey the commutative law, that is, these two coordinate system transformations can be performed in different order equally. It makes no difference whether the translation, Tr(ax, X), is performed initially, followed by the rotation, Rt(φx, X), or the rotation, Rt(φx, X), is performed initially, followed by the translation, Tr(ax, X). This is because of the dot products Rt(φx, X) ⋅ Tr(ax, X) and Tr(ax, X) ⋅ Rt(φx, X) are identical to one another

Rt(ϕ x , X ) ⋅ Tr( ax , X ) ≡ Tr( ax , X ) ⋅ Rt(ϕ x , X )

(D.12)

308

Appendix D

(a)

Z1

Z2

Y1

Y*

Z1

Z*

ax

X1 ≡ X *

φx

(b)

Z*

Z2 ax

φx

X2

X1 ≡ X *

Y1 Y*

Y2

X2

Y2

FIGURE D.3 On the equivalency of the linear transformations, Rt(φx, X) ⋅ Tr(ax, X) and Tr(ax, X) ⋅ Rt(φx, X), in the operator, Cpx(ax, φx), of coupled linear transformation of a Cartesian reference system XYZ.

This means that the translation from the coordinate system X1Y1Z1 to the intermediate coordinate system X*Y*Z* followed by the rotation from the coordinate system X*Y*Z* to the finale coordinate system X2Y2Z2 produces that same reference X2Y2Z2 as in a case when the rotation from the coordinate system X1Y1Z1 to the intermediate coordinate system X*Y*Z* followed by the translation from the coordinate system X*Y*Z* to the finale coordinate system X2Y2Z2. The validity of Equation D.12 is illustrated in Figure D.3. The translation, Tr(ax, X), that is followed by the rotation, Rt(φx, X), as shown in Figure D.3a, is equivalent to the rotation, Rt(φx, X), that is followed by the translation, Tr(ax, X) as shown in Figure D.3b. Therefore, the two linear transformations, Tr(ax, X) and Rt(φx, X), can be coupled into a linear transformation

Cpx ( ax , ϕ x ) = Rt(ϕ x , X ) ⋅ Tr( ax , X ) ≡ Tr( ax , X ) ⋅ Rt(ϕ x , X )

(D.13)

The operator of linear transformation, Cpx(ax, φx), can be expressed in the matrix form

1 0 Cpx ( ax , ϕ x ) =  0  0

0

0

cos ϕ x − sin ϕ x 0

sin ϕ x cos ϕ x 0

ax  0  0  1

(D.14)

This expression is composed based on Equations D.4 through D.6 for the linear transformation Tr(ax, X), and on Equations D.8 through D.10 that describes the linear transformation Rt(φx, X). Two degenerate cases of operator of the linear transformation, Cpx(ax, φx), are distinguished.

309

Appendix D

First, it could happen that in a particular case the component, ax, of the translation is zero, that is, ax = 0. Under such a scenario, the operator of linear transformation, Cpx(ax, φx), reduces to the operator of rotation, Rt(φx, X), and the equality Cpx(ax, φx) = Rt(φx, X) is observed in the case under consideration. Second, it could happen that in a particular case the component, φx, of the rotation is zero, that is, φx = 0°. Under such a scenario, the operator of linear transformation, Cpx(ax, φx), reduces to the operator of translation, Tr(ax, X), and the equality Cpx(ax, φx) = Tr(ax, X) is observed in the case under consideration. The said is valid with respect to the translations and the rotations along and about the axes Y and Z of a Cartesian reference system XYZ. The corresponding coupled operators, Cpy(ay, φy) and Cpz(az, φz), for linear transformations of these kinds can also be composed 0 1 0

sin ϕ y 0 cos ϕ y

 cos ϕ y  0 Cp y ( ay , ϕ y ) =   − sin ϕ y   0

0

0

 cos ϕ z  − sin ϕ z Cpz ( az , ϕ z ) =   0   0

sin ϕ z cos ϕ z 0 0

0 0 1 0

0 ay  0  1

(D.15)

0 0  az   1

(D.16)

In the operators of linear transformations, Cpx(ax, φx), Cpy(ay, φy), and Cpz(az, φz), values of the translations ax, ay, and az, as well as values of the rotations φx, φy, and φz, are finite values (and not continuous). The linear and angular displacements do not correlate with one another in time, thus, they are not screws. They are just a kind of couples of a translation along, and a rotation about a coordinate axis of a Cartesian reference system. Introduction of the operators of linear transformation, Cpx(ax, φx), Cpy(ay, φy), and Cpz(az, φz), makes the linear transformations easier as all the operators of the linear transformations become uniform. The operators of linear transformations Cpx(ax, φx), Cpy(ay, φy), and Cpz(az, φz), do not obey the commutative law. This is because rotations are not vectors in nature. Therefore, special care should be undertaken when treating rotations as vectors—when implementing coupled operators of linear transformations in particular. D.1.5 Resultant Coordinate System Transformation The operators of translations Tr(ax, X), Tr(ay, Y), and Tr(az, Z) together with the operators of rotations Rt(φx, X), Rt(φy, Y), and Rt(φz, Z) are used for composing of the operator Rs(1 → 2) of the resultant coordinate system

310

Appendix D

transformation. The operator Rs(1 → 2) of the resultant coordinate system transformation analytically describes the transition from the initial coordinate system X1Y1Z1 to a certain coordinate system X2Y2Z2. Consider three consequent translations along the coordinate axes X1, Y1, and Z1. Suppose that a point m on a rigid body goes through a translation describing a straight path from m1 to m2 with a change of coordinates of (ax, ay, az). This motion can be described with an operator of the resultant coordinate system transformation, Rs(1 → 2). The operator Rs(1 → 2) can be expressed in terms of the operators Tr(ax, X), Tr(ay, Y), Tr(az, Z) of elementary coordinate system transformations. The operator Rs(1 → 2) is equal to

1 0 Rs(1 → 2) = Tr( az , Z) ⋅ Tr( ay , Y ) ⋅ Tr( ax , X ) =  0  0

0 1 0 0

ax  ay  (D.17) az   1

0 0 1 0

In this particular case, the operator of the resultant coordinate system transformation, Rs(1 → 2), can be interpreted as the operator Tr(a, A) of translation along an A-axis [Rs(1 → 2) = Tr(a, A)]. Evidently, the A-axis is always an axis through the origin. Similarly, three consequent rotations about coordinate axes can be described with another operator of the resultant coordinate system transformation Rs(1 → 2):

Rs(1 → 2) = Rt(ϕ z , Z) ⋅ Rt(ϕ y , Y ) ⋅ Rt(ϕ x , X )

(D.18)

In this particular case, the operator of the resultant coordinate system transformation Rs(1 → 2) can be interpreted as the operator Rt(φA) of rotation about an A-axis [Rs(1 → 2) = Rt(φA)]. Evidently, the axis A is always an axis through the origin. Practically, it is often necessary to perform coordinate system transformations that comprise translations along and rotations about the coordinate axes. For example, the expression

Rs(1 → 5) = Tr( ax , X ) ⋅ Rt(ϕ z , Z) ⋅ Rt(ϕ x , X ) ⋅ Tr( ay , Y )

(D.19)

indicates that the transition from the coordinates system X1Y1Z1 to the coordinate system X5Y5Z5 (Figure D.4) is performed in the following four steps: (1) translation Tr(ay, Y) (2) rotation Rt(φx, X), (3) second rotation Rt(φz, Z), and (4) the translation Tr(ax, X). Ultimately, the equality is valid.

r1(m) = Rt(5 → 1) ⋅ r5 (m)

(D.20)

311

Appendix D

Z1

ax1

Z5

Z4

Z2

X1, X2 Y5 Y1

Y4

Y2

⇒ X2

Y3

φz2 Y2

Z3

X3

Y3 Y4

ay4 ax1

X5

X4

⇓ Z3 Z2

Z5

Z1

ay4

m r1(m)

⇑

r5(m) X1

Z4 φy3

X5 X3

X4

Y5 Y1

FIGURE D.4 An example of the resultant coordinate system transformation.

When the operator Rs(1 → t) of a resultant coordinate system transformation is known, the transition in the opposite direction can be performed by means of the operator Rs(1 → t) of the inverse coordinate system transformation. The following equality

Rs(t → 1) = Rs −1(1 → t)

(D.21)

is valid for the operator Rs(1 → t) of the inverse coordinate system transformation. The operators of coupled linear transformations Cpx(ax, φx), Cpy(ay, φy), and Cpz(az, φz), (see Equations D.14 through D.16) (Figure D.5) can be used for the purpose of analytical description of a resultant coordinate system transformation. Under such the scenario, the operator, Rs(1 ↦ t), of a resultant coordinate system transformation can be expressed in terms of all the operators Cpx(ax, φx), Cpy(ay, φy), and Cpz(az, φz) by the following expression: t −1

Rs(1 t) =

∏ Cp (a , ϕ ) i j

i =1 j= x, y ,z

i j

i j

(D.22)

In Equation D.22, only operators of coupled linear transformations are used. D.1.6 Screw Motion about a Coordinate Axis Operators for the analytical description of screw motions about an axis of the Cartesian coordinate system are a particular case of the operators of the resultant coordinate system transformation.

312

Appendix D

(a)

Z1

Z2

ax

φx

X1

X2

Y1 Y2 (b)

Z1 ay

φy

Z2

X2 X1

Y1 Y2 (c)

Z2

X2

Z1 φz Y2 az

X1

Y1 FIGURE D.5 Analytical description of the operators (a) Cpx(ax, φx), (b) Cp y(ay, φy), and (c) Cpz(az, φz), of linear transformation of a Cartesian reference system XYZ.

313

Appendix D

ax = px . φx Z1

Z2

X1

Y1 Y2

X2

φx

FIGURE D.6 Analytical description of the operator of screw motion Sc x(φx, px).

By definition (Figure D.6), the operator Scx(φx, px) of a screw motion about X-axis of the Cartesian coordinate system XYZ is equal to

Sc x (ϕ x , px ) = Rt(ϕ x , X ) ⋅ Tr( ax , X )

(D.23)

After substituting of the operator of translation Tr(ax, X) (Equation D.4) and the operator of rotation Rt(φx, X) (Equation D.7), Equation D.14 casts into the expression:

1 0 Sc x (ϕ x , px ) =  0  0

0 cos ϕ x − sin ϕ x 0

0 sin ϕ x cos ϕ x 0

px ⋅ ϕ x  0  0   1 

(D.24)

for the computation of the operator of the screw motion Sc x(φx, px) about the X-axis. The operators of screw motions Scy(φy, py) and Scz(φz, pz) about the Y and Z-axes, respectively, are defined in a similar way to that above; the operator of the screw motion Scx(φx, px) is defined:

Sc y (ϕ y , py ) = Rt(ϕ y , Y ) ⋅ Tr( ay , Y ) Sc z (ϕ z , pz ) = Rt(ϕ z , Z) ⋅ Tr( az , Z)

(D.25)

(D.26)

Using Equations D.5 and D.6 together with Equations D.9 and D.10, we can come up with the expressions

cos ϕ y  0 Sc y (ϕ y , py ) =   sin ϕ y   0

0 1 0 0

− sin ϕ y 0 cos ϕ y 0

 py ⋅ ϕ y  0   1  0

(D.27)

314

Appendix D

 cos ϕ z  − sin ϕ z Sc z (ϕ z , pz ) =   0   0

sin ϕ z cos ϕ z 0 0

0 0 1 0

   pz ⋅ ϕ z   1  0 0

(D.28)

for the calculation of the operators of the screw motion, Scy(φy, py) and Scz(φz, pz), about the Y and Z-axes. Screw motions about a coordinate axis, as well as screw surfaces, are common in the theory of gearing. This makes practical use of the operators of the screw motion Scx(φx, px), Scy(φy, py), and Scz(φz, pz) when designing gears. If necessary, the operator of the screw motion about an arbitrary axis whether through the origin of the coordinate system can be derived in a similar manner to that used for the derivation of the operators Sc x(φx, px), Scy(φy, py), and Scz(φz, pz). D.1.7 Rolling Motion of a Coordinate System One more practical combination of a rotation and of a translation is often used when designing gears. Consider a Cartesian coordinate system X1Y1Z1 (Figure D.7). The coordinate system X1Y1Z1 travels in the direction of the X1-axis. The speed of the translation is denoted by V. The coordinate system X1Y1Z1 rotates about its Y1-axis simultaneously with the translation. The speed of the rotation is designated as ω. Assume that the ratio V/ω is constant. Under such a scenario, the resultant motion of the reference system X1Y1Z1 to its arbitrary position X2Y2Z2 allows interpreting it in the form of rolling with no sliding of a cylinder of radius Rw over the plane. The plane is parallel to the coordinate X1Y1-plane, and it is remote from it at the distance Rw. For the calculation of the radius of the rolling cylinder, the expression Rw = V/ω can be used. Because the rolling of the cylinder of radius Rw over the plane is performed with no sliding, a certain correspondence between the translation and the ax = Rw . φy Z1

Z2

Rw

V

ω

φy

X1 Y1

Y2

X2

FIGURE D.7 Illustration of the transformation of rolling, Rl x(φy, Y), of a coordinate system.

315

Appendix D

rotation of the coordinate system is established. When the coordinate system turns through a certain angle φy, the translation of origin of the coordinate system along the X1-axis is equal to ax = φr ⋅ Rw. The transition from the coordinate system X1Y1Z1 to the coordinate system X2Y2Z2 can be analytically described by the operator of the resultant coordinate system transformation Rs(1 ↦ 2). The operator Rs(1 ↦ 2) is equal

Rs(1 2) = Rt(ϕ y , Y1 ) ⋅ Tr( ax , X1 )

(D.29)

where Tr(ax, X1) designates the operator of the translation along the X1-axis, and Rt(φy, Y1) is the operator of the rotation about the Y1-axis. The operators of the resultant coordinate system transformation of the kind (see Equation D.29) are referred to as operators of rolling motion over a plane. When the translation is performed along the X1-axis, and the rotation is performed about the Y1-axis, the operator of rolling is denoted by Rlx(φy, Y). In this particular case, the equality Rlx(φy, Y) = Rs(1 ↦ 2) (see Equation D.29) is valid. Based on this equality, the operator of rolling over a plane Rl x(φy, Y) can be calculated from the equation

cos ϕ y  0 Rlx (ϕ y , Y ) =   sin ϕ y   0

0 1 0 0


ax ⋅ cos ϕ y   0  ax ⋅ sin ϕ y   1 

(D.30)

While rotation remains about the Y1-axis, the translation can be performed not along the X1-axis, but along the Z1-axis instead. For rolling of this kind, the operator of rolling is equal

cos ϕ y  0 Rlz (ϕ y , Y ) =   sin ϕ y   0

0 1 0 0


− az ⋅ sin ϕ y   0  az ⋅ cos ϕ y   1 

(D.31)

For cases when the rotation is performed about the X1-axis the corresponding operators of rolling are as follows:

1 0 Rl y (ϕ x , X ) =  0  0

0

0


sin ϕ x cos ϕ x 0

 ay ⋅ cos ϕ x  − ay ⋅ sin ϕ x   1  0

(D.32)

316

Appendix D

For the case of rolling along the Y1-axis and

1 0 Rlz (ϕ x , X ) =  0  0

0

0


sin ϕ x cos ϕ x 0

 az ⋅ sin ϕ x  az ⋅ cos ϕ x   1  0

(D.33)

For the case of rolling along the Z1-axis. Similar expressions can be derived for the case of rotation about the Z1-axis: sin ϕ z cos ϕ z 0

0 0

 cos ϕ z  − sin ϕ z Rlx (ϕ z , Z) =   0   0

0

0

sin ϕ z cos ϕ z 0

0 0

 cos ϕ z  − sin ϕ z Rl y (ϕ z , Z) =   0   0

0

0

1

1

ax ⋅ cos ϕ z  ax ⋅ sin ϕ z   0  1 

(D.34)

ay ⋅ sin ϕ z  ay ⋅ cos ϕ z   0  1 

(D.35)

Use of the operators of rolling (see Equations D.30 through D.35) significantly simplifies the analytical description of the coordinate system transformations. D.1.8 Rolling of Two Coordinate Systems When designing gears, combinations of two rotations about parallel axes are of particular interest. As an example, consider two Cartesian coordinate systems, X1Y1Z1 and X2Y2Z2, as shown in Figure D.8. The coordinate systems X1Y1Z1 and X2Y2Z2 are rotated about their axes Z1 and Z2. The axes of rotations are parallel to each other (Z1 ‖ Z2). Rotations ω1 and ω2 of the coordinate systems can be interpreted so that a circle of a certain radius, R1, which is associated with the coordinates system X1Y1Z1, rolls with no slippage over a circle of the corresponding radius, R 2, that is associated with the coordinate system X2Y2Z2. When the center distance, C, is known, then radii, R1 and R 2, of the circles can be expressed in terms of the center distance, C, and the given rotations, ω1 and ω2. For the calculations, the following formulas

R1 = C ⋅

1 1+ u

(D.36)

317

Appendix D

Y1

φ1

Y1*

Y2*

φ2

Y2

X1

R1

R2

X1* O2

P

O1

ω2

ω1

X2* φ2 X2

C FIGURE D.8 Derivation of the operator of rolling Rru(φ1, Z1) of two coordinate systems.

R2 = C ⋅

u 1+ u

(D.37)

can be used. Here, the ratio ω1/ω2 is designated as u. In the initial configuration, the X1 and X2 -axes align to each other. The Y1 and Y2 -axes are parallel to each other. In Figure D.8, the initial configuration of the coordinate systems X1Y1Z1 and X2Y2Z2 is labeled as X1*Y1* Z1* and X 2*Y2* Z2* . When the coordinate system X1Y1Z1 turns through a certain angle, φ1, the coordinate system X2Y2Z2 turns through the corresponding angle, φ2. When angle φ1 is known, the corresponding angle φ2 is equal to φ2 = φ1/u. The transition from the coordinate system X2Y2Z2 to the coordinate system X1Y1Z1 can be analytically described by the operator of the resultant coordinate system transformation Rs(1 ↦ 2). In the case under consideration, the operator Rs(1 ↦ 2) can be expressed in terms of the operators of the elementary coordinate system transformations:

 ϕ  Rs(1 2) = Rt(ϕ 1 , Z1 ) ⋅ Rt  1  ⋅ Tr(−C , X1 )  u, Z1 

(D.38)

Other equivalent combinations of the operators of elementary coordinate system transformations can result in that same operator Rs(1 ↦ 2) of the resultant coordinate system transformation. The interested reader may wish to conduct the exercise on his/her own deriving the equivalent expressions for the operator Rs(1 ↦ 2). The operator of the resultant coordinate system transformations of the kind (see Equation D.38) are referred to as operators of rolling motion over a cylinder.

318

Appendix D

When rotations are performed around the Z1 and Z2 -axis, the operator of rolling motion over a cylinder is denoted by Rru(φ1, Z1). In this particular case, the equality Rru(φ1, Z1) = Rs(1 ↦ 2) (see Equation D.38) is valid. Based on this equality, the operator of rolling Rru(φ1, Z1) over a cylinder can be calculated from the equation:

 u + 1   cos  ϕ 1 ⋅ u    u + 1  Rru (ϕ 1 , Z1 ) =  − sin  ϕ 1 ⋅   u    0  0 

u + 1  sin  ϕ 1 ⋅   u  u + 1  cos  ϕ 1 ⋅   u  0 0

 −C    0   0   1 

0 0 1 0

(D.39)

For the inverse transformation, the inverse operator of rolling of two coordinate systems Rru(φ2, Z2) can be used. It is equal to Rru (ϕ 2 , Z2 ) = Rru−1(ϕ 1 , Z1 ). In terms of the operators of the elementary coordinate system transformations, the operator Rru(φ2, Z2) can be expressed as follows:

 ϕ  Rru (ϕ 2 , Z2 ) = Rt  1  ⋅ Rt(ϕ 1 , Z2 ) ⋅ Tr(C , X1 )  u, Z2 

(D.40)

Other equivalent combinations of the operators of elementary coordinate system transformations can result in that same operator Rru(φ2, Z2) of the resultant coordinate system transformation. The interested reader may wish to conduct the exercise on his/her own deriving the equivalent expressions for the operator Rru(φ2, Z2). For the calculation of the operator of rolling of two coordinate systems, Rru(φ2, Z2), the equation can be used:

 u + 1  cos  ϕ 1 ⋅ u     u + 1 Rru (ϕ 2 , Z2 ) =  sin  ϕ 1 ⋅   u    0  0 

u + 1  − sin  ϕ 1 ⋅   u  u + 1  cos  ϕ 1 ⋅   u  0 0

0 0 1 0

 C   0  0  1

(D.41)

Similar to the expression (see Equation D.39) derived for the calculation of the operator of rolling Rru(φ1, Z1) around the Z1- and Z2 -axis, the corresponding formulas can be derived for the calculation of the operators of rolling Rru(φ1, X1) and Rru(φ1, Y1) about parallel axes X1 and X2, as well as about parallel axes Y1 and Y2.

319

Appendix D

Use of the operators of rolling about two axes Rru(φ1, X1), Rru(φ1, Y1), and Rru(φ1, Z1) substantially simplifies the analytical description of the coordinate system transformations.

D.2 Conversion of the Coordinate System Orientation The application of the matrix method of coordinate system transformation presumes that both of the coordinate systems, i and (i ± 1), are of the same hand. This means that it assumed from the very beginning that both of them are either right-hand-oriented or left-hand-oriented Cartesian reference systems. In the event the coordinate systems i and (i ± 1) are of opposite hand, for example, if one of them is a right-hand-oriented coordinate system while the other is left-hand oriented coordinate system, one of the coordinate systems will need to be converted into an oppositely oriented Cartesian coordinate system. For conversion of a left-hand-oriented Cartesian coordinate system into a right-hand-oriented coordinate system and/or vice versa, operators of reflection are used. In order to change the direction of the Xi-axis of the initial coordinate system, i, to the opposite direction (in this case, in the new coordinate system (i ± 1) the equalities Xi±1 = − Xi, Yi±1 ≡ Yi, and Zi±1 ≡ Zi are observed) the operator of reflection Rfx(YiZi) can be applied. The operator of reflection yields representation in the matrix form as

 −1 0 Rfx (Yi Zi ) =  0  0

0 1 0

0 0 1

0

0

0 0  0  1

(D.42)

Similarly, the implementation of the operators of reflections Rfy(XiZi) and Rfz(XiYi) results in reversal of the directions of Yi-axis and Zi-axes. The operators of reflections Rfy(XiZi) and Rfz(XiYi) in this case can be expressed analytically in the matrix form

1 0 Rfy (Xi Zi ) =  0  0

0 −1 0 0

0 0 1 0

0 0  0  1

(D.43)

320

Appendix D

1 0 Rfz (XiYi ) =  0  0

0 1 0 0

0 0 −1 0

0 0  0  1

(D.44)

A linear transformation that reverses the direction of the coordinate axis is an opposite transformation. Transformation of reflection is an example of orientation-reversing transformations.

D.3 Direct Transformation of Surfaces Fundamental Forms Every coordinate system transformation results in corresponding changes to the equation of the gear tooth surface, G. Because of this, it is necessary to recalculate the coefficients of the first Φ1.g and the second Φ2.g fundamental of the surfaces, G, as many times as the coordinate system transformation is performed. This routing and time-consuming operation can be eliminated if the operators of coordinate system transformations are used directly in the fundamental forms Φ1.g and Φ2.g. After calculation in an initial coordinate system, the fundamental magnitudes Eg, Fg, Gg, and Lg, Mg, Ng of the forms Φ1.g and Φ2.g can be determined in any new coordinate system using for this purpose, the operators of translation, rotation, and the resultant coordinate system transformation. Transformations of these fundamental magnitudes Φ1.g and Φ2.g become possible due to implementation of the formulas below. Consider a gear tooth surface, G, that is given by the equation rg = rg(Ug, Vg), where (Ug, Vg) ∈ G. For convenience, the first fundamental form, Φ1.g, of the gear tooth surface, G, is represented in the matrix form [28,38] [Φ1.g ] = [dU g

 Eg F g dVg 0 0] ⋅  0  0

Fg Gg 0 0

0   dU g  0   dVg  ⋅ 0  0     1  0 

0 0 1 0

(D.45)

Similarly, an equation of the second fundamental form, Φ2.g, of the surface, G, can be given by [Φ 2.g ] = [dU g

 Lg M g dVg 0 0] ⋅   0   0

Mg Ng 0 0

0 0 1 0

0   dU g  0   dVg  ⋅ 0  0     1  0 

(D.46)

321

Appendix D

The coordinate system transformation with the operator of the resultant linear transformation Rs(1 → 2) transfers the equation rg = rg(UgVg) of the gear tooth surface, G, that is initially given in X1Y1Z1, to the equation rg* = rg* (U g* , Vg* ) of that same surface G in a new coordinate system X2Y2Z2. It is clear that the position vector of a point of the tooth flank G in the first reference system X1Y1Z1 differs from the position vector of that same point in second references system X2Y2Z2 (that is, rg ≠ rg* ). The operator of the resultant linear transformation Rs(1 → 2) of the surface, G, that has the first, Φ1.g, and the second, Φ2.g, fundamental forms from the initial coordinate system, X1Y1Z1, to the new coordinate system, X2Y2Z2, results in that in the new coordinate system, the corresponding fundamental forms are expressed in the form [28,38]

[Φ1* .g ] = RsT (1 → 2) ⋅ [Φ1.g ] ⋅ Rs(1 → 2) [Φ *2.g ] = RsT (1 → 2) ⋅ [Φ 2.g ] ⋅ Rs(1 → 2)

(D.47) (D.48)

Equations D.47 and D.48 reveal that after the coordinate system transfor* mation is completed, the first, Φ1.g , and the second, Φ *2.g , fundamental forms of the surface, G, in the coordinate system X2Y2Z2 are expressed in terms of the first, Φ1.g, and the second, Φ2.g, fundamental forms, which initially are represented in the coordinate system X1Y1Z1. In order to convert the fundamental forms Φ1.g and Φ2.g to the new coordinate system, the corresponding fundamental form, either Φ1.g or Φ2.g, needs to be premultiplied by Rs(1 → 2) and after that, it needs to be postmultiplied by RsT(1 → 2). Implementation of Equations D.47 and D.48 significantly simplifies the transformations of formulas. Equations similar to Equations D.47 and D.48

[Φ1* .p ] = RsT (1 → 2) ⋅ [Φ1.p ] ⋅ Rs(1 → 2) [Φ *2.p ] = RsT (1 → 2) ⋅ [Φ 2.p ] ⋅ Rs(1 → 2)

(D.49) (D.50)

are valid for the pinion tooth flank, P. Implementation of the coupled linear transformations matrices for transforming the coordinate systems is a possible way to enhance the approach. In this case, just the operators of coupled linear transformations are necessary to be applied for all coordinate system transformation.

Appendix E: Change of Surface Parameters When designing a form cutting tool, it is often necessary to treat two or more surfaces simultaneously. For example, the cutting edge of the cutting tool can be considered as the line of intersection of the generating surface T of the form cutting tool by the rake surface Rs. Equation of the cutting edge cannot be derived on the premises of equations of the surfaces T and Rs as long as the initial parameterization of the surfaces is improper. When two surfaces, ri and rj, have necessarily been treated simultaneously, then it is required that they are not only represented in a common reference system, but the Ui- and Vi-parameters of one of the surfaces ri = ri(Ui, Vi) have to be synchronized with the corresponding Uj- and Vj-parameters of another surface rj = rj(Uj, Vj). The procedure of changing of surface parameters is used for this purpose. Use of the procedure allows representation of one of the surfaces, for example, the surface rj = rj(Uj, Vj) in the terms of Uiand Vi-parameters, say as rj = rj(Ui, Vi). If the parameterization of a surface is transformed by the equations U* = U*(U, V) and V* = V*(U, V), we obtain the new derivatives

∂r ∂ r ∂U ∂ r ∂V = ⋅ + ⋅ ∂U * ∂U ∂U * ∂V ∂U *

(E.1)

∂r ∂ r ∂U ∂ r ∂V = ⋅ + ⋅ ∂V * ∂U ∂V * ∂V ∂V *

(E.2)

 ∂r ∂r  A* =  = A⋅J * *   ∂U ∂V 

(E.3)

so that

where

 ∂U  ∂U * J=  ∂V  ∂U *

∂U  ∂V *   ∂V  *  ∂V 

(E.4)

This is called the Jacobian matrix of the transformation.

323

324

Appendix E

It can be shown that the new fundamental matrix G* is given by

G* = A *T A * = JT AT AJ = JT GJ

(E.5)

From this equation, we can see by the properties of determinants that |G*| = |J|2|G|. Using this result, and Equations E.2, we can show that the unit surface normal n is invariant under the transformation, as we could expect. The transformation of the second fundamental matrix can similarly by shown to be given by

D* = JT DJ

(E.6)

By differentiating Equation E.2 and using the invariance of n. From Equations E.5 and E.6, it can be shown that the principal curvatures and directions are invariant under the transformation. We conclude that the unit normal vector n and the principal directions and curvatures are independent of the parameters used, and are therefore geometric properties of the surface itself. They should be continuous if the surface is to be tangent and curvature continuous.

Index A Addition, 278 Angular velocities, 141 Approximate crossed-axis gear pairs, 205 Approximate gearings, 3, 4 Approximate intersected-axis gear pairs, 176 Arbitrary smooth curves, 19 Archimedean screw surface, 213 Axial pitch calculation, 55 Axial vectors, 163 gear pair, 163 hypoid gear pair, 163 magnitudes, 166 unit vector, 164, 165 B Base cones belt-and-pulley mechanism, 201, 205 in crossed-axis gear pairs, 201 in intersected-axis gearing, 173–176 intersected-axis gearing, 204 PA, 203, 204 and plane of action, 202 straight-line segments, 205 vector of instant rotation, 202–203 Base pitch helix angle, 53 Basic rack, 23, 24 Belt-and-pulley analogy, 59, 173 mechanism, 201, 205 model, 30, 36 BF-mesh of Novikov gear pair, 247 Bonnet theorem, 294 Boundary N-circle, 176, 209, 212 axis of rotation, 62 in conformal gearing, 59 conformal parallel-axis gear pair, 60 construction, 243, 246–247 instant line of action, 60 Nby -mesh of conformal gear pair, 61

Boundary N-cone in high-conformal crossed-axis gearing, 212–213 in intersected-axis high-conformal gearing, 187–189 C Cartesian coordinate system, 141–142, 162, 185, 187, 283, 311–312 Cartesian reference system, 100, 147, 182, 183, 283, 297 Center distance (C), 141–142, 150, 197, 202, 206 Centerline plane (Cln-plane), 147 Centerline vectors, 162–163 Characteristic line, 110 Condition of conjugacy, 29, 35 belt-and-pulley model of involute gearing, 30 cycloidal gearing, 31 gear and pinion tooth dedendum, 32 instant line of action, 33 instant pitch points, 34 main theorem of gearing, 29 meshing of helical noninvolute gear pair, 34 Condition of contact, 27–28 Conformal gearing, 1; see also Highconformal gearing ambiguities, 4 ancient designs, 5–6 boundary N-circle, 59–62, 62, 65 close-up of conformal gear pair, 74, 75 convex-to-concave contact of bones, 1, 2 dedendum profile, 64 elements of kinematics and geometry in, 71 ellipse-arc, 63, 64 features, 2, 22–25 fundamental design parameters, 52–54

325

326

Conformal gearing (Continued) geometrically accurate gearings, 3, 4–5 improvements in and relating to gear teeth, 6–8 incorrect configuration of involute tooth profiles, 67 involute tooth profiles in relation to base circle, 66 kinematics and tooth flank geometry, 72 Novikov gearing, 18–22 point of contact, 72 power density, 1 requirements, 3–4 toothed gearing, 8–11 tooth geometries in, 62 transverses contact ratio, 73 Wildhaber’s helical gearing, 11–18, 65 Conformal gear pair, 75 addendum factor, 76 inactive portions of tooth flanks, 78 quality of parallel-axis Novikov gearing, 77 transverse pressure angle, 75 Conformal parallel-axis gearing; see also Novikov gearing accuracy requirements for, 100 boundary N-circle, 104 Cartesian reference system, 100 concave gear tooth profile configuration, 104 convex pinion tooth profile configuration, 105 high-conformal gearing, 106 involute tooth point, 103, 105 parameters, 102 variation of center distance and operating pressure angle, 101 Conical gear pair, 160 Contact geometry; see also Gear tooth flank in conformal parallel-axis gearing, 91 considerations, 91 convex-to-concave contact, 95 of gear and mating pinion tooth flanks, 107 normalized design parameters, 93 relative orientation at point of contact of gear, 108–113

Index

scenarios, 92 second-order analysis, 114–118 three-dimensional plot of function, 94 tooth flanks, 93 tooth profiles, 92–93 Contact lines, 20, 49 Contact ratio in gear pair, 39 face contact ratio, 41 length of action, 40 transverse contact ratio, 39 Convenience, 149 Convex-to-concave contact of bones, 1, 2 Convex-to-convex contact of bones, 1 Coordinate axis rotation about, 306–307 screw motion about, 311–314 Coordinate system orientation conversion, 319–320 Coordinate system rolling motion, 314 analytical description, 316 Cartesian coordinate system, 314 rolling motion over plane, 315 Coordinate systems rolling, 316 Cartesian coordinate systems, 316 elementary coordinate system transformations operators, 317 for inverse transformation, 318 operators of rolling, 319 rolling motion over cylinder, 317 Coordinate systems transformations, 303 coupled linear transformation, 307–309 direct transformation of surfaces forms, 320–321 homogenous coordinate transformation matrices of dimension 4 × 4, 304 homogenous coordinate vectors, 303–304 resultant, 309–311 rotation about coordinate axis, 306–307 screw motion about coordinate axis, 311–314 translations along coordinate axis, 305–306 Coupled linear transformation, 307

327

Index

Cartesian reference system, 309 equivalency of linear transformations, 308 Crossed-axis angle, 58, 143 gearings, 27 gears, 197 Crossed-axis gear pairs, see Spatial gear pairs Cross product, see Vector product Culminating condition, 252 Cycloidal gearing, 31 D Darboux frame, 287, 295 Degree of conformity, 118; see also Contact geometry analytical description of contact geometry, 121 change of, 120 condition of physical contact, 128 converse indicatrix, 134–135 coordinate angle, 126 derivation of equation, 123 directions of extremum, 130–133 Dupin’s indicatrix, 122, 124 indicatrix, 122–130 indicatrix of conformity, 127 preliminary remarks, 118–122 properties, 133–134 quantitative evaluation, 122 radii of principal curvature, 125 tooth flanks, 119 on transition from resultant deviation, 119 violation of physical condition of contact, 129, 130 Differential geometry of surfaces elements, 283 curvatures at point of part surface, 295–297 equations, 301 example, 297–301 forms of surface, 287–294 gear tooth flank, 283–285 local frame, 286–287 principal directions on gear tooth flank, 294–295

tangent plane, 285–286 tangent vectors, 285–286 Direction vector, 277 Direct transformation, 306 of surfaces forms, 320–321 Dot product, see Scalar product Dupin’s indicatrix, 114, 122, 124 magnitudes, 117 matrix representation of equation of, 117–118 planar characteristic curve, 115 at point of smooth regular gear tooth flank, 114 principal curvatures, 116 E EAP-point, see End-of-active-profile point (EAP-point) Ellipse of contact, 85 End-of-active-profile point (EAP-point), 56 Equality, 278 of base pitches, 35, 39 belt-and-pulley model, 36 of gear, 38 generation of screw involute surfaces, 38 line of contact, 37 operating base pitch, 35, 36 External crossed-axis gear pairs, 159–160 External spatial gearing, 145 External spatial gear pairs vector diagrams, 148, 149; see also Conformal gearing center-distance, 150 horizontal plane of projection, 148 hyperboloid, 153 pair of rotation, 152 projections of rotation vectors, 149 vectors of sliding velocities, 151 F Face contact ratio, 39, 41 Floating point numbers, 277 Fundamental conditions, 27 Fundamental planes of gear pair, 147

328

G Gear-to-plane-of-action, 186 Gear apex, 146 Gear machining mesh, 137 Gear pair kinematics, 141; see also Spatial gear pairs axial vectors, 163–166 centerline vectors, 162–163 complementary vectors to vector diagrams, 160 kinematic and geometric formulas, 166–168 tooth ratio, 168–169 vector diagram for rotations, 144 vector diagrams types, 159–160 vector representation, 141–145 Gear tooth flank, 283; see also Conformal gearing; Teeth flanks principal directions on, 294–295 principal parameters of local topology, 284 surfaces specification, 285 Generalized rack-type spatial gear pairs vector diagrams, 156–157 Geometrically accurate gearing permissible instant relative motions in, 28 Geometrically accurate gearings, 3, 4–5 H Helical gearing by Wildhaber, see Wildhaber’s helical gearing Helical involute gearing with zero transverse contact ratio, 43 comparison of distribution of contact stress, 46 coordinate system, 47 essence of Novikov gearing, 44–52 fundamental design parameters, 52–54 geometries of tooth profiles, 45 interaction between tooth flanks, 50 involute tooth point, 43 law of motion of contact point, 48 multiple curves, 46 N by-gears, 49 pseudo path of contact configuration, 51

Index

similarities, 54 transverses contact ratio, 50 High-conformal crossed-axis gearing, 197; see also High-conformal intersected-axis gearing analytical criteria, 200 base cones in, 201–205 bearing capacity, 209 boundary N-circle, 209 boundary N-cone in, 212–213 configuration of boundary N-cone, 208 design parameters, 214–220 instantaneous relative motion kinematics, 206–208 kinematics, 197–201 path of contact in, 208 sliding between tooth flanks of gear and pinion, 210–212 vector diagram for, 206 High-conformal gearing, 27, 91, 106; see also Conformal gearing conformal parallel-axis gearing, accuracy requirements for, 100–106 contact geometry in conformal parallel-axis gearing, 91–95 parallel-axis gearing, 95–100 High-conformal intersected-axis gearing, 171; see also Highconformal crossed-axis gearing base cones in intersected-axis gearing, 173–176 bearing capacity, 177–178 boundary N-cone in, 187–189 configuration of boundary cone, 177 design parameters, 189–195 geometry, 191 kinematics of instantaneous motion in, 171–173 path of contact in, 176 teeth flanks sliding in, 179–187 vector diagram, 172 High-conformal parallel-axis gearing, 95 critical value, 96 degree of conformity, 100 Hertz formula, 97 impact of degree of conformity, 96

329

Index

indicatrices of conformity, 97, 99 Novikov gearing and high-conformal gearing, 98 High-power-density gearing (HPDgearing), 1 Homogenous coordinate transformation matrices, 304 Homogenous coordinate vectors, 303–304 HPD-gearing, see High-power-density gearing (HPD-gearing) Hypoid gear pair, 163 I Ideal gearings, see Geometrically accurate gearings Instantaneous relative motion kinematics, 206 crossed-axis gearing, 206–207 external crossed-axis gearing, 208 Instant line of action, 33, 60 Interacting tooth flanks configuration contact area between teeth flanks, 89 contact lines, 83 contact patches, 86 contact pattern, 88 at culminating point, 82, 83 design parameters, 87 design parameters of high-conformal gear pair, 83 ellipse of contact, 85 line of contact, 89 local and global contact geometry of, 84–89 point of contact, 84 shape of contact area between teeth flanks, 88 Internal spatial gearing, 145 Internal spatial gear pairs vector diagrams, 153, 154 axodes, 156 pinion axis of rotation, 154 vector of linear velocity of sliding, 155 Intersected-axis gearing, 27, 160 base cones in, 173–176 gear pairs, 171

Invention disclosure form, 261–268 Involute gearing, 137; see also Wildhaber’s helical gearing axial pitch calculation, 55 to conformal parallel-axis gearing transition, 54 elements of parallel-axis gear pair featuring zero transverse contact ratio, 55 involute tooth point, 57 line of contact, 54 tooth flank of gear, 56 Involute tooth point, 43, 57, 103, 105, 137 Iron curtain, 257 L Length of action, 40 Line of contact, 54, 89 Local frame, 286–287 Local relative orientation chain rule, 112 characteristic line, 110 local configuration of two quadrics, 109 at point of contact of gear and mating pinion tooth flanks, 108 tangent directions, 111 unit vectors of principal directions, 113 M Machining processes, 137 boundary N-circle, 140 instant lines of action and paths of contact, 138 unit normal vectors, 139 MAEA, see Military Aircraft Engineering Academy (MAEA) Main theorem of gearing, 29 MBTU, see Moscow Bauman Technical University (MBTU) Military Aircraft Engineering Academy (MAEA), 257 Miter gears, 201 Moscow Bauman Technical University (MBTU), 257

330

N Natural kind of surfaces representation, 297 N bf-gears, 49 Nbf -mesh of conformal gear pair, 61 N by-gears, 49 Nby -mesh of conformal gear pair, 61 Negation, 278 Negative-gearing, 145 Normal plane (Nln-plane), 147 Novikov gearing, 18, 239; see also Conformal parallel-axis gearing arbitrary smooth curves, 19 basic rack, 23, 24 boundary N-circle construction, 243, 246–247 contact lines, 20 contact of teeth flanks for, 251–253 contact strength of known designs of gearing, 18 essence of, 44–52 features, 22–25, 240–246 geometries of teeth flanks for, 247–251 helical gearing, 19, 256 kinematics and geometry, 239 preamble, 239–240 tooth profiles, 21 Novikov hobs, 25 Novikov, Mikhail L., 257 in international engineering community, 258 research work, 259 O Operating base pitch, 35, 36 Opposite transformation, 320 Orientation-preserving transformation, 306 Orientation-reversing transformations, 320 Orthogonal crossed-axis gear pairs, 200 P PA, see Plane of action (PA) Pair of rotation, 152

Index

Parallel-axis gearing, 2–3, 27 kinematics of, 57–58 plane of action in, 59 Paths of contact, 20 Pinion-to-plane-of-action, 187 Pinion-to-rack gearing, 200 Pinion apex, 146 Pitch-line plane, 147 Pitch line, 146 Pitch point, 217, 254 Plane of action (PA), 174, 183, 186, 203, 204, 205 Plane of action in parallel-axis gearing, 59 Point of culmination, 49 Position vector, 277 Positive-gearing, 145 Power density, 1 Profile contact ratio, see Transverse contact ratio R Rack-shaped planing tool, 17 Reduced pitch, 152 Reference plane, 190 Rolling motion over cylinder, 317 over plane, 315 S SAP-point, see Start-of-active-profile point (SAP-point) Scalar multiplication, 278 Scalar product, 279 Screw motion, 311 analytical description of operators, 312, 313 Cartesian coordinate system, 311–312, 313 about coordinate axis, 314 Second-order analysis, 114 Dupin’s Indicatrix, 114–117 matrix representation of Dupin’s indicatrix equation, 117–118 Shishkov’s equation of contact, 3, 27 Spatial gear pairs, 145, 159, 160 analytical criterion, 157–159 pitch-line plane, 147

331

Index

plane of action apex, 146 type, 146 vector diagrams of external, 148–153 vector diagrams of generalized rack- t ype, 156–157 vector diagrams of internal, 153–156 Specific sliding, 70, 71 Spherical gear pairs, 160 Start-of-active-profile point (SAP-point), 56 Subtraction, 278 Surface forms, 287 first fundamental form, 287 Gauss characteristic equation, 293–294 gear tooth flank, 289, 290 magnitudes of first order, 288 magnitudes of second order, 292, 293 second fundamental form, 290, 291 T Tangent plane, 285–286 Tangent vectors, 285–286 Teeth flanks; see also Gear tooth flank Cartesian coordinate system, 185, 187 Cartesian reference system, 182, 183 contact for Novikov gearing, 251–253 gear-to-plane-of-action, 186 linear velocity vector, 180, 185 for Novikov gearing, 247–251 operator of rotation, 184 plane of action, 183 sliding in high-conformal gearing, 179 sliding in right-angle high-conformal gear pair, 181 sliding velocity vector, 182 Toothed gearing, 8, 10 condition of contact, 11 objects of invention, 9 Tooth flank geometry in conformal gear pair, 78, 82 angular parameter, 81 boundary N-circle, 79 pinion addendum profile, 80 tooth profiles, 80

Tooth profile sliding in conformal gearing, 67 boundary N-circle, 68 parallel-axis conformal gearing, 68 sliding of tooth profile, 69 specific sliding, 70, 71 Tooth ratio, 168–169 Torque diagrams, 145 Total contact ratio, 39 Transmitting rotation smoothly, conditions for, 27 condition of conjugacy, 29–35 condition of contact, 27–28 contact ratio in gear pair, 39–41 equality of base pitches, 35–39 permissible instant relative motions in geometrically accurate gearing, 28 Transverse contact ratio, 39, 40 Transverse pressure angle, 75, 193 Triple scalar product, 281 Triple vector product, 281 Two pseudo paths of contact, conformal gearing with, 78 U United Stated patent office, 269–276 Unit normal vector, 285–286 V Vector(s) fundamental properties, 277–278 mathematical operations over, 278–281 product, 280 Vector calculus, 277 mathematical operations over vectors, 278–281 position vector, 277 vectors fundamental properties, 277–278 W Wildhaber–Novikov gearing (W–N gearing), 256 root causes for loose term, 257

332

Wildhaber, Dr. Ernest, 253, 259–260 United Stated patent office, 269–276 Wildhaber gearing, 256 Wildhaber’s helical gearing, 11, 12, 13, 65, 253, 254; see also Involute gearing contact point, 15 convex grinding wheels, 16 features, 18 infeasibility, 256

Index

pitch point, 254 purpose of invention, 11 rack-shaped planing tool, 17 tooth profiles, 14 working profiles, 255 Willis theorem, 3 Z Zero-gearing, 145

High Conformal Gearing

Recommend Documents