Advances in Gear Design and Manufacture [1 ed.] 9781351049832, 9781351049818, 9781351049801, 9781351049825, 9781138484733

Table of contents :

1. Fundamentals of Transmission of Rotary Motion by Means of Perfect Gears


[Stephen P. Radzevich]


2. Optimization of Geometrical Engagement Parameters for Gear Honing


[Michael Storchak]


3. Design and Generation of Straight Bevel Gears


[Alfonso Fuentes-Aznar]


4. Interaction of Gear Teeth: Contact Geometry of Interacting Gear and Pinion Teeth Flanks


[Stephen P. Radzevich]


5. Elastohydrodynamic Lubrication of Conformal Gears


[R. W. Snidle and H. P. Evans]


6. Gear Drive Engineering


[Boris M. Klebanov]


7. Adaptive Gear Variators (CVTs)


[Konstantin S. Ivanov]


8. Kinematic and Power Analysis of Multi-Carrier Planetary Change-Gears through the Torque Method


[Dimitar Karaivanov and Kiril Arnaudov]


9. Powder Metal Gear Technology


[Anders Flodin]


10. Induction Heat Treatment of Gears and Gear-Like Components


[Valery Rudnev]


11. A Brief Overview on the Evolution of Gear Art: Design and Production of Gears, Gear Science


[Stephen P. Radzevich]

Citation preview

Advances in Gear Design and Manufacture

Advances in Gear Design and Manufacture

Edited by

Stephen P. Radzevich

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 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 International Standard Book Number-13: 978-1-138-48473-3 (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. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Contents Acknowledgments........................................................................................................................ vii Editor................................................................................................................................................ix Contributors.....................................................................................................................................xi Introduction.................................................................................................................................. xiii 1. Fundamentals of Transmission of Rotary Motion by Means of Perfect Gears.......... 1 Stephen P. Radzevich 2. Optimization of Geometrical Engagement Parameters for Gear Honing................. 35 Michael Storchak 3. Design and Generation of Straight Bevel Gears.............................................................65 Alfonso Fuentes-Aznar 4. Interaction of Gear Teeth: Contact Geometry of Interacting Gear and Pinion Teeth Flanks......................................................................................................................... 121 Stephen P. Radzevich 5. Elastohydrodynamic Lubrication of Conformal Gears............................................... 151 R. W. Snidle and H. P. Evans 6. Gear Drive Engineering..................................................................................................... 165 Boris M. Klebanov 7. Adaptive Gear Variators (CVTs)...................................................................................... 243 Konstantin S. Ivanov 8. Kinematic and Power Analysis of Multi-Carrier Planetary Change-Gears through the Torque Method.............................................................................................. 291 Dimitar Karaivanov and Kiril Arnaudov 9. Powder Metal Gear Technology....................................................................................... 329 Anders Flodin 10. Induction Heat Treatment of Gears and Gear-Like Components............................. 363 Valery Rudnev 11. A Brief Overview on the Evolution of Gear Art: Design and Production of Gears, Gear Science........................................................................................................ 417 Stephen P. Radzevich

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Contents

Appendix A: On the Inconsistency of the Term “Wildhaber-Novikov Gearing”: A New Look at the Concept of “Novikov Gearing”......................................... 487 Stephen P. Radzevich Appendix B: Applied Coordinate Systems and Linear Transformations....................... 503 Stephen P. Radzevich Index.............................................................................................................................................. 535

Acknowledgments The author would like to recognize Jonathan Plant, Executive Editor for Mechanical Engineering, and Ed Curtis, Project Editor, as well as the rest of the team at Taylor & Francis/CRC Press for their patience and efforts in publishing this book.

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Editor Stephen P. Radzevich is a professor of mechanical engineering and manufacturing engineering. He earned an MSc in 1976, a PhD in 1982, and a 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, particularly with a focus on precision gear design, high-powerdensity 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, and so on. Dr. Radzevich has spent over 40 years developing software, hardware, and other processes for gear design and optimization. Besides his work for the industry, he trains engineering students at universities and gear engineers in companies. He has authored and coauthored about 40 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), and Geometry of Surfaces: A Practical Guide for Mechanical Engineers (Wiley, 2013) are among his recently published books. He has also written or coauthored over 300 scientific papers and holds over 250 patents on inventions in the field (USA, Japan, Russia, Europe, Canada, Soviet Union, South Korea, Mexico, and others).

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Contributors Kiril Arnaudov Bulgaria

Boris M. Klebanov Israel, USA

H. P. Evans United Kingdom

Stephen P. Radzevich USA

Anders Flodin Sweden

Valery Rudnev USA

Alfonso Fuentes-Aznar USA

R. W. Snidle United Kingdom

Konstantin S. Ivanov Kazakhstan

Michael Storchak Germany

Dimitar Karaivanov Bulgaria

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Introduction There is nothing more practical than a good theory.

James C. Maxwell

Historical Background Gears and gear transmissions have been extensively used by human beings for centuries. Lots of practical experience in design, manufacture, and application of gears and gear drives has accumulated in the industry to this end. An enormous amount of the research in the field has been carried out, both from the theory side as well as from the experimental side. All the experience in the field of gearing is summarized in a few fundamental monographs. However, novel attempts have been undertaken in recent years, and important new results of the research have been obtained in the field of gearing. Nowadays, gear science is still extensively evolving.

Uniqueness of this Publication This book is based on the newest accomplishments in gear theory, gear design, gear production, and gear application. A team of world-leading experts in gear science have contributed their achievements in the field of gearing. The most important subjects of gear science are covered in the book. A gap between the current needs of advanced gear users and gear manufacturers is bridged by gear science. This makes the book unique.

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Introduction

Intended Audience This book is written by top level gear experts and for gear experts, that is, for the gear manufacturers, gear designers, and gear customers, and those used to advanced designs of gears and gear transmissions: first of all, of gear drives with a highest possible power density (or, in other words, “power-to-weight ratio”), and low noise (or almost “noiseless”) gear transmissions. Post gear engineers and gear researchers from the industry, as well as graduate students, will benefit from the book.

Organization of this Book The book contains eleven chapters and two appendix parts. The scientific theory of gearing, gear design, and gear production are covered by these chapters. The evolution of gear art (gear theory, gear design, and gear production) is briefly outlined in the ending chapter of the book. Fundamentals of transmission of a rotary motion by means of perfect gears are discussed in Chapter 1. This section of the book is contributed by Prof. Stephen P. Radzevich, an expert in the field of theory of gearing, gear design, gear production, gear inspection, and gear application. The discussion begins with introductory remarks and follows with the detailed analysis of three fundamental laws of gearing. Condition of contact between the interacting tooth flanks of a gear and a mating pinion is the first fundamental law of gearing. Condition of conjugacy of the interacting tooth flanks of a gear and a mating pinion is the second fundamental law of gearing. Ultimately, the third fundamental law of gearing requires equality of base pitches of interacting tooth flanks of a gear and a mating pinion to operating base pitch of the gear pair. Perfect gear pairs, that is, gear pairs that feature a constant angular velocity ratio, obey all three fundamental laws of gearing. Gear pairs that do not obey one or more fundamental laws of gearing are referred to as approximate gear pairs. Approximate gear pairs feature a variable in time angular velocity ratio. An illustrative example of perfect crossed-axes gearing with line contact between the tooth flanks of a gear and a mating pinion (“R-gear system”) is provided at the end of this section of the book. Optimization of geometrical engagement parameters for gear honing is considered in Chapter 2, contributed by Dr. Michael Storchak of Germany, a well-known expert in the field. Elaboration of principles for developing technological systems for the finishing of gears is discussed at the beginning. This discussion is followed by the consideration of a model of the machine engagement geometry. The development of the model of the machine engagement geometry along with the objective functions of the model are covered in this section of the chapter. The chapter ends with the discussion on the synthesis of the working layer forms of tools.

Introduction

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The design and generation of straight bevel gears are discussed in Chapter 3, written by Prof. Alfonso Fuentes-Aznar, a wellknown expert in the field of gearing and applications of gears. The chapter begins with an introduction that is followed by the determination of the pitch cone angles of bevel gears. Then, mathematical definition of the spherical involute profile is provided. Mathematical definition of the spherical involute profile and direct and indirect definitions are covered here. Spherical bevel gear tooth surfaces are defined in the next section of the chapter. Gear tooth thickness, the polar angle at the pitch cone, the base cone angle, the spherical involute bevel gear tooth surfaces, the face and root cone angles, along with the modified geometry for localization of contact are discussed. This discussion is followed by the analysis of geometry of the crown gear for generation of spherical involute straight bevel gears. The determination of the number of teeth, and of the base cone angle for the crown gear are disclosed. Then the tooth thickness and generating surfaces of the crown gear are specified. This analysis is followed by description of the geometry of the crown gear for generation of octoidal bevel gears, and by the geometry of spherical and octoidal bevel gears generated by a crown gear. Special attention is devoted to the generation of bevel gears by dual interlocking circular cutters. For this purpose, positioning the circular cutters is specified, definitions for the cutter swing angle and of the blade angle are provided, and localization of the cutting disks on the crown rack is discussed. The generating surface of the cutting disk and generation of straight bevel gears by dual interlocking circular cutters are investigated to the best possible extent. A numerical example is provided. The mechanical behavior of the spherical involute bevel gear drive is investigated, a comparison of spherical and octoidal geometries is performed, and mechanical behavior of bevel gears generated by dual interlocking circular cutters is explained. This section of the chapter ends with the discussion on the application of barrelshaped gear blanks to improve the maximum bending stresses of straight bevel gears. In Chapter 4, interaction of gear teeth from the perspective of the contact geometry of interacting gear and pinion teeth flanks is considered. This section of the book is contributed by Prof. Stephen P. Radzevich), an expert in the field of theory of gearing, gear design, gear production, gear inspection, and gear application. The discussion begins with the essence of the approach proposed by Heinrich Hertz and follows by the second-order analysis of the contact geometry. This includes, but is not limited to, the discussion of the local relative orientation of the tooth flanks of a gear and a mating pinion at a point of their contact along with planar characteristic images, that is, of “Dupin indicatrices” at a point of a gear, Dup (G ), of a mating pinion, Dup (P ) , and of the surface of relative curvature, Dup (G|P   ). The reader’s attention is focused on the advantages of the matrix representation of equation of the “Dupin indicatrix” at the point of a gear tooth flank. Then, the fourth-order analysis of the contact geometry of the tooth flanks is performed. A definition to the term “degree of conformity” of two tooth flanks at the point of their contact is introduced. This discussion is followed by the introduction of a characteristic curve of a novel type. This planar characteristic curve of the fourth order is referred to as the “indicatrix of conformity, Cnf R(G|P   )” at the point of contact of a gear and a mating pinion tooth flank. The directions of the extremum degree of conformity at the point of

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Introduction

contact of a gear and a mating pinion tooth flank are determined, and important properties of Cnf R(G|P   ) at the point of contact of a gear and a mating pinion tooth flank are outlined. This chapter of the book ends with the discussion of the converse indicatrix of conformity at the point of contact of a gear and a mating pinion tooth flank in the first order of tangency. Chapter 5, “Elastohydrodynamic Lubrication of Conformal Gears,” is contributed by Prof. R. W. Snidle and Prof. H. P. Evans of the UK, who are well-known experts in the area of EHD lubrication in mechanics in general, and in gearing in particular. At the beginning of the chapter, a brief introduction to elastohydrodynamic lubrication (further “EHD lubrication,” for simplicity) is outlined. This is followed by a detailed discussion of the conditions for EHL film-forming in conformal gears; detailed contact geometry, kinematics, load and Hertzian pressure; and ends with the analysis of the Chittenden et al. film thickness formulas. An example on the steps in the film thickness calculation for a typical gear set is provided. Operating conditions, principal radii of relative curvature at the tooth contact, normal load at the contact, Hertzian contact dimensions, maximum Hertzian pressure, entraining velocity, and predicted film thickness values are covered in this example. The chapter ends with the discussion section, notation section, and references. Chapter 6, a contribution in the field of gear drive engineering is written by Dr. Boris M. Klebanov, a well-known expert in the field. Disc type gears and design of large gears are covered in the “Gear Body Design” section of this chapter. This discussion is followed by a section called “Gear-Shaft Connections.” In this section, both fixed and movable connections are discussed. Then, compliance of shafts and bearings is discussed to the best possible extent. Bending of the shafts, and influence of the shaft’s bearings are considered. A separate section of the chapter is devoted to the analysis of the gear housing deformations. This analysis is followed by the discussion of planetary gear drives. Improvement of relative accuracy, flexible supports of planet gears, floating sun gears, along with floating ring gears are considered here. This chapter of the book ends with the detailed discussion of toothing improvements of cylindrical involute gears. Transverse contact ratio in spur gearings, tooth profile modification of spur gears, meshing geometry of helical gears, as well as tooth root design are covered in this section of the chapter. The chapter is complemented by the list of references. Chapter 7 is written by Dr. Konstantin S. Ivanov of Kazakhstan, a leading expert in the field of continuously variable transmissions (CVTs). In his chapter titled “Adaptive Gear Variators (CVTs),” a novel principle of creation of an adaptive gear variator is disclosed. The chapter begins with the analysis of the structure of the basic kinematic chain of gear variators, and kinematics of a basic initial kinematic chain. Special attention is given to the force analysis of an initial kinematic chain of a gear variator. This discussion includes, but is not limited to, the consideration of the effect of force adaptation (necessary condition of adaptation), energy circulation,

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condition of transfer of forces (sufficient condition of adaptation), method of the force analysis of an adaptive gear variator, examples of kinematical and force analysis of a gear variator, force interacting in the adaptive gear variator on start, as well as operation of the gear variator. This discussion is followed by the analysis of the dynamics of gear variators. The following problems are carried out in this section of the book: preconditions of dynamic research of a gear variator, description of a gear variator in dynamics, dynamics of transient of a gear variator in a starting regime, a numerical example of dynamic calculation for a starting regime, dynamics of transient of a gear variator in the stage of steady motion, a numerical example for the stage of steady motion, and a conclusion about the results of dynamic researches. Then, the designing of a gear variator is discussed. In this section, the engineering philosophy of a gear variator, description of design of a gear variator, operation of a gear variator, and design of a gear variator are considered. The experimental researches of an adaptive gear variator is the other topic to be discussed in this section of the book. This discussion includes the explanation of the experiment purpose, a testbed description, technique of tests of an adaptive gear variator, along with the efficiency of a gear variator. The chapter ends with the consideration of the problem of synthesis of an adaptive gear variator. For this purpose, the range of transmission ratios of an adaptive gear variator, the solution of problems of synthesis of an adaptive gear variator, and an example of a solution of the problem of synthesis are discussed. A conclusion and the list of references are provided at the very end of the chapter. Professor Dimitar Karaivanov and Professor Kiril Arnaudov, leaders in the field of design, kinematics, and application of planetary gear drives contributed Chapter 8 on the kinematic and power analysis of multicarrier planetary change gears through the torque method. This discussion begins with the simple AI-planetary gear train as a building element of planetary change gear trains, and is followed by a detailed analysis of the torque method—a simple and easy way for kinematic and power analysis of complex compound planetary gear trains. Essence of the method, kinematic analysis of a simple AI-planetary gear train, power analysis of AI-planetary gear train, and kinematic and power analysis of compound two-carrier planetary gear  trains are covered in this analysis. Then, the kinematic and power analysis of two-carrier and four-carrier change gear  is  performed. The discussion end with the consideration of the load spectra determination of particular elements of a change gear. Chapter 9 is on powder metal gear technology. This chapter is contributed by Dr. Anders Flodin, a well-known expert in the field. The chapter begins with a general introduction, selection of materials for gears, lubrication of materials, and alloying concepts. The alloying is discussed in more detail; mixing, pre-alloyed, organically bonded alloys, and diffusion alloyed are discussed to the best possible extent. The section on materials is followed by a section on manufacturing. Compaction, tooling, heat treatment, perfomance boosting processes, hard finishing, different process paths to make

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Introduction

gears, and tolerances are discussed here. This discussion is followed by consideration of design for powder metal, performance, and AM gears. Novel accomplishments in the area of induction heat treatment of gears and gear-like components are discussed in Chapter 10. This section of the book is contributed by Dr. Valery Rudnev, an expert in the field of heat treatment of gears and gear-like components. The chapter begins with an introduction that follows with the discussion of electromagnetic principles of induction heating. This includes, but is not limited to, commonly accepted definitions in skin effect, nonexponential distribution in skin effect, and eddy current cancellation. Then metallurgical subtleties of induction gear hardening are discussed. Material selection for induction gear hardening, impact of rapid heating and steel prior microstructure, super hardness phenomenon, along with specifics of induction hardening of powder metallurgy (P/M) gears are covered in this chapter of the book. Particular attention is paid to the technologies for induction gear hardening. This section of the book begins with general remarks, and is followed by an overview of tooth hardness patterns, and inductor designs and heating modes. Tooth-by-tooth hardening of gears, gear spin hardening (encircling inductors), quenching options, along with the heating modes for encircling inductors are covered when discussing this later topic. In the later sections of the chapter, the residual stresses at tooth working surface, hardening components containing teeth, as well as tempering of gears and gear-like components are discussed to the best possible extent. The chapter ends with the conclusion section, acknowledgement section, and references. In Chapter 11, a brief overview on the evolution of gear art is presented. Design and production of gears, as well as the accomplishments in gear science are covered in this discussion. This section of the book is contributed by Prof. Stephen P. Radzevich. The chapter begins with brief notes on the history of methods of machining gears and on the design of gear cutting tools. Early accomplishments in the design of toothed wheels and in methods for manufacture of gears, early designs of special purpose cutting tools to produce gear teeth, gear-cutting tools for the first production machines, and evolution of the gear-cutting tools for production machines are covered in this discussion. This discussion follows with a brief analysis of the development of the skiving internal gears process, and rotary gear shaving process, grinding hardened gears, along with the designs of gear-cutting tools for generating bevel gears. Later accomplishments in design of gear-cutting tools for the generating bevel gears are also considered. This section of the chapter ends with a brief discussion of the generating milling of bevel gears. In the next section of the chapter, the evolution of the scientific theory of gearing is briefly overviewed. The discussion is subdivided onto three periods of time that are labeled as (a) the pre-Eulerian period of gear art, (b) the origin of the scientific theory of gearing: the Eulerian period of gear art, and (c) the post-Eulerian period of the developments in the field of gearing. The reader’s attention is focused here mainly on the developments in the field of perfect gearings. These include but are not limited to G. Grant bevel gearing, contributions by Prof. N. I. Kolchin, Prof. M. L. Novikov, and by Prof. V. A. Gavrilenko. Then accomplishments in the investigation of the condition of conjugacy of the interacting tooth flanks of a gear and a mating pinion in crossed-axes gearing, and the equality of angular

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base pitches of a gear and a mating pinion to operating angular base pitch in intersectedaxes, and in crossed-axes gearing are outlined. This section of the chapter ends with a tentative chronology of the evolution of the theory of gearing, along with harmonic drive contributed by Walton Musser. In the third section of the chapter, the developments in the field of approximate gearings, Cone double-enveloping worm gearing, approximate bevel gearing, approximate crossedaxes gearing, and face gearing are discussed. The chapter ends with a brief summary of the principal accomplishments in the theory of gearing achieved by the beginning of the twenty-first century. These accomplishments form a set of three fundamental laws of gearing all gearings must obey to. All the discussions are summarized in the concluding remarks. Appendix A, titled as “On the Inconsistency of the Term ‘Wildhaber-Novikov Gearing’: A New Look at the Concept of ‘Novikov Gearing,’” is contributed by Prof. Stephen P. Radzevich. An advanced interpretation of the concept of “Novikov Gearing” is outlined in this appendix. Contributed by Prof. Stephen P. Radzevich, Appendix B is titled “Applied Coordinate Systems and Linear Transformations.” Novel accomplishments that pertain to the coordinate system transformations are briefly outlined in this appendix. It is likely this book is not free from omissions or mistakes; or that it is as clear and ambiguous as it should be. If you have any constructive suggestions, please communicate them to me via email: [email protected]. Stephen P. Radzevich Sterling Heights, MI September 22, 2018

1 Fundamentals of Transmission of Rotary Motion by Means of Perfect Gears Stephen P. Radzevich CONTENTS 1.1 Introductory Remarks............................................................................................................ 2 1.2 Three Fundamental Laws of Gearing..................................................................................4 1.2.1 Condition of Contact between Interacting Tooth Flanks: The First Fundamental Law of Gearing................................................................................... 4 1.2.2 Condition of Conjugacy of Interacting Tooth Flanks: The Second Fundamental Law of Gearing................................................................................... 8 1.2.2.1 Pulley-and-Belt: Analogy of a Gear Pair.................................................. 8 1.2.2.2 Camus-Euler-Savary Theorem................................................................. 10 1.2.2.3 Condition of Conjugacy of Interacting Tooth Flanks in Case of Crossed-Axes Gearing............................................................................... 14 1.2.3 Equality of Base Pitches of Interacting Tooth Flanks of a Gear and a Mating Pinion to Operating Base Pitch of the Gear Pair: The Third Fundamental Law of Gearing................................................................................. 18 1.3 Illustrative Example: Perfect Crossed-Axes Gearing with Line Contact between the Tooth Flanks of a Gear and a Mating Pinion............................................................. 20 1.3.1 Kinematics of Crossed-Axes Gearing.................................................................... 23 1.3.2 Base Cones in Perfect Crossed-Axes Gear Pairs................................................... 24 1.3.3 Tooth Flanks in Perfect Crossed-Axes Gears........................................................ 27 1.4 Conclusion............................................................................................................................. 32 References........................................................................................................................................ 32 Bibliography................................................................................................................................... 33 Three principal components are recognized in modern machinery. These are (a) a source of power, and (b) a working member, that are connected to each other by means of (c) a transmission. Transmitting and transforming an input motion is the main purpose of the transmission. Gears and gear transmissions are extensively used in today’s industry for transmitting and transforming an input motion. Principal features of perfect gear pairs are briefly discussed in this section of the book. A higher power, and a smaller size, is the main trend in the evolution of the power sources for the application in modern machinery. Therefore, along with a smaller size, the output rotation of the power sources (of an internal combustion engine, of an electric DC motor, as well as of other sources of motion) gets higher. For example, the output rotation of the electric DC motor spindle in the range of 30,000 to 40,000 rpm is common even in trivial engineering applications. For an electric motor of a specified power, the following 1

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Advances in Gear Design and Manufacture

correlation is observed: the smaller the size of the electric motor, the higher the rotation of the motor spindle, and vice versa. For many reasons, gear transmissions perfectly meet most of the requirements the transmissions have to fulfill, especially in cases when a gear transmission is capable of transmitting a uniform rotary motion with a highest attainable “power density.”1 Modern gear transmissions have to be as small in size as possible and be capable of transmitting as high amount of power as possible in order to meet these requirements.

1.1  Introductory Remarks Gear transmissions have been extensively used by human beings for a long time. The Antikythera2 mechanism is the oldest surviving relic containing gears [1]. It is named after the Greek island near where the mechanism was discovered in a sunken ship in 1900. Gears made of wood represent a perfect example of gears most extensively used in the past. Wooden gears were used in the design of windmills for pumping seawater, for producing flour, and so forth. An example of a gear transmission comprised of wooden gears is illustrated in Figure 1.1. Parallel-axes gear pairs (or just “Pa –gearing,” for simplicity), intersected-axes gear pairs (or just “Ia –gearing,” for simplicity), as well as crossed-axes gear pairs (or just “Ca –gearing,” for simplicity), were used at that time. Wooden gears were possible because the main sources of power—power of the wind, power of the water stream, and that of animals—produced a low torque, and a low rotation, thus, the gear

FIGURE 1.1 Use of wooden gears in the design of a gear transmission. “Power density” is understood here and below as a ratio of the amount of power being transmitted by means of a gear transmission to weight of the transmission. “Power-to-weight ratio” is another term used for this purpose. 2 The artefact was recovered between 1900 to 1901 from the Antikythera shipwreck off the Greek island of Antikythera. Its significance and complexity were not understood until decades later. Believed to have been designed and constructed by Greek scientists, the instrument has been dated either between 150 and 100 BCE, or, according to a more recent view, at 205 BCE. This precious example of antique genius complexity grade was so high that artefacts of a similar complexity and workmanship did not reappear for a millennium and a half, when mechanical astronomical clocks were built in Europe. 1

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Fundamentals of Transmission of Rotary Motion by Means of Perfect Gears

FIGURE 1.2 Areas of existence of parallel-axes gearings of different kinds.

transmissions featured a low power density. Actually, at that time, the power density of a gear transmission was not of importance to the gear users. Let’s consider only two critical parameters of a gear pair, namely, a torque (Trq) and a rotation (RPM) on the driving shaft of a gear pair. Then, in a Cartesian coordinate system, “Trq vs. RPM,” a point corresponds to an arbitrary gear pair. As an example, parallel-axes gear pairs3 are plotted in the reference system, “Trq vs. RPM,” in Figure 1.2. Two sets of parallel lines are plotted in Figure 1.2. The straight lines, RPM = RPMmin and RPM = RPMmax, correspond to physically permissible minimal and maximal rotations of the input shaft of a gear pair. Two other straight lines, Trq = Trqmin and Trq = Trqmax, correspond to physically permissible minimal and maximal torques of the input shaft of a gear pair. All the gear pairs can be mapped by a point that is located within the interior of the rectangle bounded by the lines RPM = RPMmin, RPM = RPMmax, and Trq = Trqmin, Trq = Trqmax, including the boundaries. No gearing is feasible if a gear pair is mapped by a point that is located in the exterior of the rectangle bounded by the just mentioned lines. The permissible area of existence of gear pairs can be narrowed based on the following analysis. A highest power, Pmax, that can be transmitted by a gear pair, equals

Pmax = RPMmax ⋅ Trqmax

(1.1)

Assuming Pmax = Const, a boundary hyperbola can be constructed in the reference system, “Trq vs. RPM.” 3

Similar diagrams can be constructed for intersected-axes gear pairs, as well as for crossed-axes gear pairs.

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Advances in Gear Design and Manufacture

Transmission of a rotation from a driving shaft to a driven shaft is possible only by means of gear pairs4 that are mapped inside the area shown in Figure 1.2. Points that correspond to wooden gear pairs are plotted in Figure 1.2 by points that are located close to the point of intersection of the straight lines, RPM = RPMmin and Trq = Trqmin. The areas of preferable applications of gear pairs of different types are also shown in Figure 1.2. As it follows from the analysis of Figure 1.2, future developments in the field of gearing have to be undertaken toward the gear systems that feature a highest permissible power density, that is, of the gear pairs that are mapped in Figure 1.2 closer either to the straight line RPM = RPMmax, or to the straight line Trq = Trqmax, or to the boundary hyperbola, Pmax = RPM max · Trqmax (see Equation 1.1). Gearings of these kinds obey to all three fundamental laws of gearing [6], and, thus, are referred to as “perfect gearings.”

1.2  Three Fundamental Laws of Gearing In the pre-Eulerian period of gear art, numerous intuitive attempts were undertaken to come up with the most favorable geometry of a gear tooth flank. However, it was Leonhard Euler who found out that the involute of a circle is a curve that perfectly fits the needs of gear design. This scientific accomplishment by Euler was far ahead of time, and got no proper understanding by the majority of the scientists and practitioners in the field of gear art. The considered below concept of “perfect gearings,” as well as “three fundamental laws of gearing,” can be traced back to two fundamental papers by Euler [2,3]. A uniform rotation from a driving shaft to a driven shaft can be transmitted by means of a gear pair, either with a constant angular velocity ratio, or with a time-dependent angular velocity ratio, uϕ. In cases when the angular velocity ratio, uϕ, is of a constant value (uϕ = ωinput/ωoutput = const), it is said that the rotation from a driving shaft to a driven shaft is transmitted smoothly. Here, rotation of the input shaft (i.e., rotation of the driving shaft) is designated as ωinput, and rotation of the output shaft (i.e., rotation of the driven shaft) is designated as ωoutput. Perfect gear pairs feature the angular velocity ratio, uϕ, of a constant value.5 Perfect gears rotate smoothly, producing almost no vibration and noise excitation. If the angular velocity ratio, uϕ, is time-dependent, that is, when uϕ = uϕ(t), it is said that the gear pair is approximate. Variation of the angular velocity ratio causes vibration generation, an excessive noise excitation, an excessive tooth flank wear, and so forth. It is important to identify a set of conditions under which the gear velocity ratio in a gear pair is of a constant value, that is, a set of conditions under which the equality uϕ = const is valid.  ondition of Contact between Interacting Tooth Flanks: 1.2.1  C The First Fundamental Law of Gearing The condition of contact between the tooth flanks of a gear and a mating pinion is reflected by the “first fundamental law of gearing” to be discussed below. The condition of contact 4 5

The schematic shown in Figure 1.2 is qualitative (not quantitative), and intuitive. Noncircular gears with a variable angular-velocity ratio are not considered here.

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between the tooth flanks is the first fundamental condition to be met in gearings of all types in parallel-axes gearing, in intersected-axes, as well as in crossed-axes gearing. Alignment of the linear velocity vector, VΣ, of the resultant relative motion to the common tangent at point of contact, K, of the interacting tooth flanks, G and P, of a gear and its mating pinion (VΣ || t, where t is a unit vector through contact point, K; for example, entirely located in the common tangent plane) is specified by the condition of contact. The necessity of alignment of the resultant linear velocity vector, VΣ, to the common tangent (or, the same, the necessity of perpendicularity, VΣ ⊥ ng, of the resultant velocity vector, VΣ, to the common perpendicular, ng) is illustrated by the following example. Consider a relative motion of the tooth flanks, G  and P, of a gear and a mating pinion as illustrated in Figure 1.3. For simplicity, but without loss of generality, normal sections through contact point of the gear and the mating pinion are depicted there. It is also assumed that the gear tooth flank, G, is motionless, and the pinion tooth flank, P, performs an arbitrary instantaneous motion, VΣ, in relation to the gear tooth surface, G. Three different scenarios can be distinguished when the pinion tooth flank, P, travels in relation to the gear tooth flank, G. First, an instantaneous motion of point, Ka, within the pinion tooth flank, P, is specified by an instant linear velocity vector, VΣa (see Figure 1.3a). Point, A, within the pinion tooth flank, P, is chosen so that the projection, Prn VΣa, of the linear velocity vector, VΣb, onto the unit normal vector, n ap , to the moving surface, P, at point, Ka, is pointed toward the interior of the motionless gear tooth flank, G, that is, the inequality, Prn VΣb > 0, is valid. In the differential vicinity of point, Ka, this results that the moving pinion tooth flank, P , penetrates into the motionless gear tooth flank, G. A relative motion of this kind is not permissible for the conjugate tooth flanks, G  and P, of a gear and a mating pinion. Second, an instantaneous motion of point, Kb, within the pinion tooth flank, P, is specified by an instant linear velocity vector, VΣb (see Figure 1.3b). Point, Kb, within the pinion tooth flank, P,  is chosen so that the linear velocity vector, VΣb, is perpendicular to the unit normal vector, nbp , and, thus, it is tangent to the gear tooth flank, G, at contact point Kb. The projection, Prn VΣb , of the linear velocity vector, VΣb, onto the unit normal vector, n ap , to the moving tooth flank, P, at point, Kb, is equal to zero (i.e., an equality Prn VΣb = 0 is observed). In the differential

FIGURE 1.3 On the necessity of alignment of the linear velocity vector of resultant relative motion, VΣ, to the common tangent to the contacting tooth flanks, G  and P , (a) penetration of the tooth flanks, G  and P, (b) proper contact tooth flanks, G  and P, (c) separation of the tooth flanks, G  and P, is observed.

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vicinity of point, Kb, this results that the moving pinion tooth flank, P, does not penetrate the motionless gear tooth flank, G. Instead, the pinion tooth flank, P, rolls and slides in relation to the gear tooth flank, G. In a particular case, either the rolling component, or the sliding component of the resultant relative motion of this kind can be zero. Relative motion of this particular kind is permissible for the conjugate tooth flanks, G  and P. Transmitting a motion from a driving shaft to a driven shaft is possible if, and only if, a relative motion of this particular kind is occurred. Third, an instantaneous motion of point, Kc, within the pinion tooth flank, P, is specified by an instantaneous linear velocity vector, VΣc (see Figure 1.3c). Point, Kc, within the pinion tooth flank, P, is chosen so that the projection, Prn VΣc , of the linear velocity vector, VΣc , onto the unit normal vector, ncp , to the moving pinion tooth flank, P, at Kc is pointed outward to the motionless gear tooth flank, G (i.e., the inequality Prn VΣc < 0 is valid). Therefore, in the differential vicinity of point, Kc, the moving pinion tooth flank, P, departs from the motionless gear tooth flank, G. No motion transmission is possible when a relative motion of this kind is occurred. The schematic depicted in Figure 1.3 shows the necessity of the proper alignment of the vector of the linear velocity, VΣ, of the resultant relative motion of the tooth flanks, G  and P, to the common tangent to the tooth flanks, G  and P, at every point of their contact. This condition is referred to as the “first fundamental law of gearing.” The first fundamental law of gearing is formulated as follows: “At every point of contact of tooth flanks of a gear and a mating pinion, a vector of their instant relative motion has to be perpendicular to the common perpendicular at every instant of time.” Various forms of analytical representation of the condition of contact of the tooth flanks, G  and P, of a gear and its mating pinion are known. To derive a convenient analytical representation for the condition of contact of the tooth flanks, G  and P, of a gear and a pinion, let’s turn our attention to the following. As shown in Figure 1.3, the dot product of vectors ng and VΣ • Is of a positive value (n ap ⋅ VΣa > 0) in the first case (see Figure 1.3a) • Equals zero (n ap ⋅ VΣa = 0) in the second case (see Figure 1.3b) • Is of a negative value (n ap ⋅ VΣa < 0) in the first case (see Figure 1.3c) The condition of contact between a gear tooth flank, G, and a mating pinion tooth flank, P, is violated in a misaligned parallel-axes gear pair as schematically illustrated in Figure 1.4. Edge contact between the tooth flanks, G  and P, is observed in the case under consideration. Therefore, no common perpendicular is observed, and the condition of contact of the tooth flanks, G  and P, is violated.

FIGURE 1.4 Example of violation of the condition of contact between a gear tooth flank, G, and a mating pinion tooth flank, P, in misaligned parallel-axes gear pair.

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The condition of contact of two conjugate tooth flanks of mating gear teeth can be expressed in the form of dot product, written as [10]

n g ⋅ VΣ = 0

(1.2)

where ng is the unit vector of common perpendicular through the contact point of the tooth flanks, G  and P, of a gear and its mating pinion, and VΣ is the linear velocity vector the resultant instantaneous relative motion of the tooth flanks, G  and P, at contact point K. Equation 1.2 reveals that a component of the linear velocity vector, VΣ, along the common perpendicular, ng, is equal to zero. Otherwise, either separation, or penetration of the tooth flanks, G  and P, in the gear pair occurs. Neither separation, nor interference of the tooth flanks, G  and P, of a gear and a mating pinion is permissible. Therefore, the linear velocity vector, VΣ, is either located in a common tangent plane or it is of a zero value. Equation 1.2 was proposed by Prof. V. A. Shishkov as early as 1948 (or even earlier [4,14]). As Prof. V. A. Shishkov is the first to propose to express the condition of contact in the form of dot product (see Equation 1.2), this equation is commonly referred to as the “Shishkov equation of contact, ng·VΣ = 0.” The interested reader may wish to discover why Prof. V.A. Shishkov is credited for the discovery of the equation of contact of two tooth flanks in the form ng·VΣ = 0 (see [6] for further reading). Equation of contact in the form ng·VΣ = 0 is practical in cases when the interacting surfaces feature simple geometry, and when the resultant relative motion is also of a simple nature. The first makes it possible to determine the unit normal vector, ng, without the derivation of the expressions for the derivatives of the equations of the contacting surface with respect to the surface parameters. The second allows the determination of the linear velocity vector, VΣ, without derivation of the equation of the moving surface with respect to the parameter of motion. Use of the “Shishkov equation of contact” in the form ng·VΣ = 0 simplifies the solution to the problem in this particular case. In cases when derivation of the equations of the derivatives for the purposes of determination of the vectors, ng and VΣ, cannot be avoided, use of the “Shishkov equation of contact” in the form ng·VΣ = 0 is less efficient. The above discussion can be summarized as follows. Permissible instant relative motions of the tooth flanks, G  and P, in a gear pair are illustrated in Figure 1.3. No relative motion of the tooth flanks, G  and P, is permissible along the common perpendicular, ng, and the relative motion is allowed in any direction within the common tangent plane through 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, ng, through an angle, ±ϕn, (Figure 1.5), also meets the requirement specified by Equation 1.2. The swivel motion, ±ϕn, of the tooth flanks is not necessary to transmit a rotation from a driving shaft to a driven shaft. However, motion of this nature can be observed in spatial gearing, that is, in Ca–gearing. It is necessary to stress the reader’s attention here on the following. Equation of contact in the form ng·VΣ = 0, was proposed by Prof. V. A. Shishkov in the mid-twentieth century (see Equation 1.2). However, the physics of the condition of contact (but not the equation of contact itself) was properly realized by gear people in the time of Leonardo da Vinci [7], and earlier. The condition of contact of a gear and a mating pinion tooth flanks, G  and P, is referred to as the “first fundamental law of gearing.” All perfect gearings have to fulfill this law of gearing.

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FIGURE 1.5 Permissible instant relative motions in “perfect gearing”.

Fulfillment of the condition of contact of a gear and mating pinion tooth flanks, G  and P, is a necessary but not sufficient condition to transmit a uniform rotation smoothly from a driving shaft to a driven shaft by means of a gear pair.  ondition of Conjugacy of Interacting Tooth Flanks: 1.2.2  C The Second Fundamental Law of Gearing The condition of conjugacy of tooth flanks of a gear and a mating pinion is the second fundamental condition to be fulfilled in perfect gearings of all kinds. Conjugacy is a specific property of a gear and mating pinion tooth flanks (tooth profiles) that roll over one another. This property pertains only to surfaces and curves that roll over one another. The condition of conjugacy of a gear and mating pinion tooth flanks is referred to as the “second fundamental law of gearing.” Below, the discussion on the second fundamental law of gearing begins with perfect parallel-axes gearing. Then, this discussion is evolved to the most general case of crossed-axes gearing. 1.2.2.1  Pulley-and-Belt: Analogy of a Gear Pair Let us begin the discussion with a trivial case of transmission of a rotary motion between two shafts that are parallel to one another. In the simplest case, a rotation from the driving

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FIGURE 1.6 Schematic of the transmission of a rotary motion by means of two pulleys connected by a belt.

shaft can be transmitted to the driven shaft by means of two disks (pulleys) connected by means of a belt, as schematically illustrated in Figure 1.6. Beginning the discussion of gearing with a pulley-and-belt analogy of a gear pair is convenient, especially for the readers less experienced with gears and gearings. The pulleys of diameters d1 (driving) and d2 (driven) are rotated about their axes, O1 and O2, respectively. The axes, O1 and O2, are at a certain center distance, C, from one another. The pulleys are connected to each other by a belt. The belt is tangent to the pulleys at points, a and b. The rotations, ω1 and ω2, of the driving and the driven pulleys are synchronized with each other so as to meet the following ratio



ω1 d 2 = ω2 d 1

(1.3)

The linear velocity of an arbitrary point, m, that travels with the belt, Vm, can be calculated from the formula

Vm = 0.5 ⋅ ω1 ⋅ d 1 ≡ 0.5 ⋅ ω 2 ⋅ d2

(1.4)

Shown in Figure 1.6, the pulley-and-belt mechanism is capable of transmitting a uniform rotation smoothly. Here is a good time to stress the reader’s attention on three important features of the pulley-and-belt analogy of a gear pair. First, when the pulleys rotate, arbitrary point, m, within the portion, ab, of the belt traces a “straight line” in a motionless reference system associated with the transmission housing. This straight line is the path of point m. Therefore, in a pulley-and-belt analogy of a gear pair, each point of the belt travels straightforward.6 6

This statement is valid only with respect to parallel-axes gearing, and it is not valid in cases of intersected-axes, as well as of crossed-axes gearings.

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Second, when a uniform rotation is transmitted from the driving pulley 1 to the driven pulley 2, the torque is transmitted by a force that acts along the belt, that is, along the straight line, ab, in a pulley-and-belt mechanism, a force can be transmitted only along the belt. A force acts only along a straight line, and it cannot be transmitted along a curve. Therefore, in a pulley-and-belt analogy of a gear pair, the straight-line segment, ab, is the “line of action” along which a force that is exerted in the driving pulley is transmitted to the driven pulley. Third, in a pulley-and-belt analogy of a parallel-axes gear pair, a straight-line path of a point, m, in the belt is aligned with the straight line of action along which a force that is exerted in the driving pulley is transmitted to the driven pulley. It is instructive to note here that the straight-line segment, ab, at that same time serves both, it serves as the path of point, m, and it also serves as the line of action in the pulley-and-belt mechanism (see Figure 1.6). Only in perfect parallel-axes gearing the path of contact, and the line of action, both, are straight lines that align to each other. It is critical to distinguish the “path of contact” from the “line of action,” and never combine these two different entities. 1.2.2.2  Camus-Euler-Savary Theorem The performed analysis of transmission of a rotary motion by means of a pulley-and-belt mechanism is helpful to understand the requirements the geometry of the tooth flanks in parallel-axes gearing must be aligned with. The condition of conjugacy of the interacting tooth flanks, G  and P, in cases of parallelaxes gearing is also referred to as the Camus-Euler-Savary fundamental theorem of gearing (or, in other words, as “CES–fundamental theorem of gearing,” for simplicity). This theorem received wide recognition mostly due to the book by Dr. R. Willis7 published as early as 1841 (see Figure 1.7). In Europe, the condition of conjugacy of the interacting tooth flanks is loosely referred to as “Willis’ theorem” [15]. The second fundamental requirement that governs the shapes that any pair of conjugate tooth profiles may have in parallel-axes gearing8 states: “In parallel-axes gearing, in order 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 located on the line of centers, that is, the pitch point P; the pitch point subdivides the center distance reciprocal to the angular velocities of the gear and the pinion.” In other words, two planar curves are said to be conjugate to one another9 if a contact perpendicular at point of their contact is along a straight line through the pitch point, P. The center of the instantaneous rotational motion is coincident with the pitch point, P. Tooth flanks of a gear, G, and a mating pinion, P, should be shaped so as to fulfill the requirements of the second fundamental law of gearing. This statement is also often called the “conjugate action law.” Robert Willis (1800–1875), a British engineer; a major contributor to the theory of gear teeth in the nineteenth century. 8 For the cases of intersected-axes gearing, and crossed-axes gearing, this concept is discussed in detail in R. Willis (1841) [15]. 9 It is wrong practice to define conjugate shapes in the commonly adopted manner: Definition: “A pair of transverse gear tooth profiles is said to be conjugate if a constant angular velocity of one profile produces a constant angular velocity in the meshing profile.” A constant output rotation is a “consequence” of conjugacy of the interacting tooth profiles. The property of “conjugacy” must to be expressed in terms of the kinematics and the geometry of the interacting tooth flanks, G and P , of a gear and a mating pinion. 7

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FIGURE 1.7 The main theorem of gearing (Willis’ theorem) as it was originally published 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 p.

The difference between the “line of action, LA,” and between the “path of contact, Pc” indirectly follows from the Camus-Euler-Savary fundamental theorem of gearing. One more consideration needs to be taken into account when discussing conjugate tooth profiles. In contact point, K, tooth flanks, G  and P, of a gear and a mating pinion slide over one another. Because of the sliding, the instantaneous rotation can be performed only about the pitch point, P, and not about any other point—otherwise the tooth flanks interfere into each other. When the condition of conjugacy of the tooth flanks, G  and P is violated, an interference of the gear and the pinion teeth is always observed. The far contact point, K, is located from the pitch point, P, the more sliding motion is observed, and vice versa. Refer to Figure 1.8 for more detailed analysis of the “conjugate action law.” In Figure 1.8, the directions of rotation of the driving and driven gears are reversed compared to that shown in Figure 1.6, as a driven pulley is “pulled” by the belt, while the driven gear is “pushed” by the driving gear. For properly designed tooth flanks of a gear and a mating pinion, the contact point of the tooth flanks, G  and P, traces a straight path of contact, Pc. When friction is not taken into consideration, a force is acting perpendicular to the common tangent plane, t − t, to the tooth flanks, G  and P. As long as the friction is not accounted, the acting force is always perpendicular to the tooth flanks, G  and P, at the current point of their contact. The straight line of action, LA, aligns with the straight path of contact, Pc, that is, in perfect parallel-axes gearing, the following identity LA ≡ Pc is observed at every instant of time when the gears rotate. It is critical to bear in mind that the “path of contact, Pc,” and the “line of action, LA,” are two completely different entities in the theory of gearing. Therefore, the difference between the line of action, LA, and the path of contact, Pc, in parallel-axes gearing has to be firmly realized. One more example is illustrated in Figure 1.9. Consider the two tooth profiles, G  and P, that contact one another at point, K, as shown in Figure 1.9. The tooth profiles, G  and P, are designed so as to smoothly transmit a rotary motion from the pinion axis of rotation, Op, to the gear axis of rotation, Og. The axes, Og and Op , are at a center distance, C. The common unit normal vector to the contacting profiles

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FIGURE 1.8 “Second fundamental law of gearing” (“Conjugate action law”): Generation of the natural form of a gear tooth profile in perfect parallel-axes gearing.

at the contact point, K, is designated as ng. A straight line that is aligned with the unit normal vector, ng, is referred to as the line of action, LA. The line of action, LA, intersects the center line, ℄, at the pitch point, P. According to “conjugate action law” (“CES–fundamental theorem of gearing”), the center distance, C, is divided by the pitch point, P, onto two segments, OgP = rg and OpP = rp, so that a proportion

Og P rg ω p = = =u Op P rp ω g

(1.5)

is observed. A point within the gear tooth profile, G, that contacts the pinion tooth profile, P, is denoted by Kg. Similarly, the point Kp within the pinion tooth profile, P, is specified. At the point of tangency of the tooth profiles the points, Kg and Kp, coincide with the contact point, K. The linear velocity vector, VKg, of the point, Kg, can be expressed in terms of the rotation vector, ωg, of the gear and the position vector, rKg, of the point Kg that is, the equality is valid.

VKg = ω g × rKg

(1.6)

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FIGURE 1.9 Conjugate tooth profiles of a gear, G, and its mating pinion, P, which fulfill the “CES−fundamental theorem of gearing”.

Similarly, the linear velocity vector, VKp, of the point, Kp, can be expressed in terms of the rotation vector, ωp, of the pinion and the position vector, rKp, of the point Kp, that is, the equality

VKp = ω p × rKp

(1.7)

is valid as well. The linear velocity vector, VΣ, of the resultant motion of tooth profiles, G  and P, in relation to each other must be aligned with a common tangent to the tooth profiles at K, or, in other words, it should be perpendicular to the unit normal vector, ng. Therefore, the radius of instant rotation, PK, is aligned with the normal vector, ng. Thus, to be conjugate, common perpendiculars at every point of the line of contact, LC, between the tooth flanks, G  and P, must pass through the axis of instant rotation of the surfaces at every instant of time, that is, for any and all possible configurations of the surfaces relative each other. In most cases, the theoretical analysis of gears and gear pairs is limited to fulfillment only of the “condition of contact” (or, in other words, of the “enveloping condition”) of the interacting tooth flanks of a gear, G, and its mating pinion, P. No analysis of the fulfillment/ violation of the condition of conjugacy of the tooth flanks, G  and P, is performed.

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The provided verbal description of the condition of conjugacy of the interacting tooth flanks, G  and P, of a gear, and a mating pinion can be complemented with an analytical description. 1.2.2.3 Condition of Conjugacy of Interacting Tooth Flanks in Case of Crossed-Axes Gearing In the second fundamental law of gearing, conjugacy is a specific property of the tooth flanks of a gear, G, and a mating pinion, P. Only surfaces that roll over one another can feature this unique property. Due to this property, in the rolling motion of a gear and a pinion over one another, the tooth flanks, G  and P, can be viewed as a kind of “reversiblyenveloping surface” (or just “Re–surface,” for simplicity) [6]. In crossed-axes gearing, when a gear and a mating pinion rotate steadily, the gear tooth flank, G, can be viewed as an envelope to consecutive positions of the mating pinion tooth flank, P. Generated in this way, the gear tooth flank, G, can be used to generate the mating pinion tooth flank, Pg. If the tooth flanks, G  and P, are conjugate, then the original pinion tooth flank, P, and the pinion tooth flank, Pg, generated by the gear tooth flank, G, are identical to one another (P g ≡ P ). Tooth flanks not of all geometries can be referred to as “conjugate surfaces,” or, the same, “reversibly-enveloping surfaces” [6]. In order to possess the property of conjugacy, a criterion to be fulfilled by two tooth flanks, G  and P, in a crossed-axes gear pair can be expressed analytically in terms of the design parameters of the gear, and the mating pinion. When the tooth flanks of a gear, G, and a mating pinion, P, interact with one another, straight lines that align to common perpendiculars, ng, through points within a current line of contact, LC, must always intersect the axis of instant rotation, Pln. The condition of conjugacy must be met at all points of a (desired) line of contact, LCdes, between the interacting tooth flanks, G  and P. This is the key requirement to be fulfilled by conjugate tooth flanks, G  and P, when the gears rotate. At arbitrary point, m, within a desired line of contact, LCdes, an instant line of action, LAinst, forms an angle with the axis of instant rotation, Pln. At every instant of time, every instant line of action, LAinst, intersects the axis of instant rotation, Pln, of the tooth flanks of a gear, and a mating pinion, G  and P —this is a must for conjugate tooth flanks. A component of the instant motion that is parallel to the axis of instant rotation, Pln, does not affect the conjugate action between the interacting tooth flanks, G  and P. Therefore, this component of the relative motion is not considered here. The other component of the relative motion is along a straight line through a point, P, that is located within the axis of instant rotation, Pln. The conjugate action between the interacting tooth flanks, G  and P, is considered for this component of the relative motion. Three vectors, pln, Vm, and ng, are constructed in Figure 1.10. By construction, two vectors, pln and Vm, are located within the plane of action, PA. The unit vector, pln, is along the axis of instant rotation, Pln. Therefore, in the Cartesian coordinate system, XpaYpaZpa, associated with the plane-of-action, it can be analytically represented as

pln = i (1.8)

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FIGURE 1.10 Condition of conjugacy of a gear, G, and a mating pinion, P, tooth flanks in crossed-axes gearing.

The linear velocity vector, Vm, of point of interest, m, is along an instant line of action, LAinst, through the point of interest, m. In the Cartesian coordinate system, XpaYpaZpa, associated with the plane-of-action, the linear velocity vector, Vm, can be analytically described as

Vm = i ⋅ Vm sin ϕ pa + j ⋅ Vm cos ϕ pa (1.9)

The magnitude, Vm, of the linear velocity vector, Vm, can be omitted from the further analysis as it does not affect the direction of the vector, Vm.

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The third unit vector, ng, is perpendicular to the gear tooth flank, G. Therefore, it can be calculated from the equation

ng =

∂rg ∂rg × ∂U g ∂Vg

(1.10)

∂rg ∂rg × ∂U g ∂Vg

where rg is the position vector of a point of a gear tooth flank, G; Ug and Vg are the curvilinear (the Gaussian) coordinates of a point of a gear tooth flank, G. In perfect crossed-axes gearing, the unit vector, ng, is entirely located within the plane of action, PA. Generally speaking, in real gearings, the unit vector, ng, deviates from the plane of action, PA, as it is calculated as a perpendicular to the real gear tooth flank. When the tooth flanks of a gear, G, and a mating pinion, P, are conjugate, then the three unit vectors, pln, Vm, and ng, are coplanar. The unit vectors, pln, Vm, and ng, are coplanar if, and only if, the triple scalar product, pln × Vm · ng, is zero, that is, if the equality [8]

pln × Vm ⋅ n g = 0

(1.11)

is valid. The triple scalar product, pln × Vm · ng, can be represented in a form of a determinant



 X pl  pln × Vm ⋅ n g = Vx.m   Xg

Ypl Vy . m Yg

Zpl   Vz.m  = 0  Zg 

(1.12)

In addition to Equation 1.12, the condition

n pl ⋅ n g ≠ 0



(1.13)

must also be fulfilled. The unit vector, npl, is entirely located within the plane of action, PA, and is perpendicular to the axis of instant rotation, Pln. Therefore, in the Cartesian coordinate system, XpaYpaZpa, associated with the plane-of-action, PA, it can be analytically represented in the form

n pl = j (1.14)

The parallelism of the directions specified by the unit normal vectors, npl and ng, is eliminated by this condition. No rotation can be transmitted by means of gears when the unit normal vector, ng, is parallel to the axis of instant rotation, Pln, (i.e., npl · ng ≠ 0). An infeasible case of gearing, for which the unit vectors, npl and ng, are parallel to one another is eliminated by this inequality. An expression

pln × n g ≠ 0



is an alternative representation of the condition specified by Equation 1.13.

(1.15)

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“Equation of conjugacy, pln × Vm · ng = 0,” can be rewritten in the form

ω pl × Vm ⋅ n g = 0

(1.16)

(ω p − ω g ) × Vm ⋅ n g = 0

(1.17)

or in the form

as the vectors, pln and ωpl, align to each other. The tooth flanks of a gear, G, and a mating pinion, P, are said to be conjugate if, and only if, the conditions, those specified by Equation 1.11 (or Equation 1.12) and Equation 1.13 (or Equation 1.15), are fulfilled for any and all points within the line of contact, LC, for any possible configurations of the gear and the pinion in relation to one another, written as [8] pln × Vm ⋅ n g = 0 (1.18)  pln × n g ≠ 0



The condition of conjugacy of the tooth flanks of a gear, G, and a mating pinion, P, is analytically described by Equation 1.18. The second fundamental law of gearing (general case): “In crossed-axes gearing, in order 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 intersect the axis of instant rotation.” This statement is also valid with respect to intersected-axes gearing (in which the center distance, C, is zero; for example the equality, C = 0, is observed). The second fundamental law of gearing is useful when an approximate gearing is analyzed. A point of a gear tooth flank, G, can be a contact point when the condition of conjugacy is met. For a corresponding angular configuration of a gear and a mating pinion, point of a pinion tooth flank, P, that is anticipated to be in contact with the gear tooth flank, G, can actually be a contact point when the condition of conjugacy is met. A possibility of contact of the tooth flanks, G  and P, in approximate gearing can be verified by means of comparison of coordinates of the “contact” point on the gear tooth flank, G, and that on the pinion tooth flank, P. If the coordinates of the points are identical to one another, then a point is a contact point. Otherwise, contact of the tooth flanks, G  and P, in these points is not possible at all. In modern practice, fulfillment of the condition of conjugacy of the interacting tooth flanks of a gear and its mating pinion is commonly limited to the analysis of the simplest case of perfect (with zero axes misalignment) parallel-axes gears, that is, to the case of perfect involute gearing. The condition of conjugacy of the tooth flanks for gears with another tooth flank geometry is not discussed at all.10 Examples can be readily found for gears that feature a cycloidal tooth profile, a circular-arc tooth profile, as well as tooth profiles of other tooth flank geometries, different from the involute of a circle. Moreover, the condition of conjugacy of the interacting tooth flanks of a gear and its mating pinion Poor practice is commonly adopted when analyzing whether the condition of conjugacy of the tooth flanks, G and P , is fulfilled, or not. The “Shishkov equation of contact,” ng · VΣ = 0, is loosely used for this purpose (here, in the equation, n is the unit normal vector to the tooth flanks, G and P, at point, K, of their contact, and VΣ is the velocity vector of the resultant relative motion of the surfaces, G and P , at the contact point, K). The condition of contact of the tooth flanks, G and P , and not the condition of conjugacy of the interacting tooth flanks of a gear and its mating pinion is analytically described by the “Shishkov equation of contact.” Unfortunately, this difference is often not recognized at all.

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is analyzed neither for gears that operate on intersected axes of rotation, nor for gears that operate on crossed axes of rotation. Fulfillment of the condition of conjugacy of the tooth flanks, G  and P, is necessary, but not sufficient for designing perfect gears. 1.2.3  E quality of Base Pitches of Interacting Tooth Flanks of a Gear and a Mating Pinion to Operating Base Pitch of the Gear Pair: The Third Fundamental Law of Gearing The third fundamental law of gearing is related to base pitches of a gear and its mating pinion. Tooth flanks in a gear for a crossed-axes gear pair can be viewed as a series of cam surfaces that act on similar surfaces of the mating gear to impart a driving motion. The third fundamental law of gearing requires the equality of angular base pitches of a gear, and that of a mating pinion, to operating the angular base pitch of the gear pair. As two (or even more) pairs of teeth make contact at the same time, the fulfillment of a condition caused by several pairs of interacting tooth surfaces in gears is a must in perfect gearings of all designs. The third fundamental law of gearing is discussed below with respect to the most general case of gearing, that is, with respect to perfect crossed-axes gearing. A plane of action, PA, in a crossed-axes gear pair is shown in Figure 1.11a. The operating base pitch, ϕb.op, of a crossed-axes gear pair is measured within the plane of action. The plane of action, PA, is intersected by a cylinder of revolution of an arbitrary radius, ry.pa, having the plane-of-action center line, Opa, as the axis of its rotation. As an angular operating base pitch, ϕb.op, of a gear pair is of a constant value, then the lengths

lb. pa = ry . pa ⋅ϕb.op

(1.19)



lb. g = ry . pa ⋅ϕb.op

(1.20)



lb. p = ry . pa ⋅ϕb.op

(1.21)

of the circular arcs also are of constant value, that is, in this particular analysis the angular base pitches ϕb.op, ϕb.g, and ϕb.p can be replaced with the lengths lb.pa, lb.g, and lb.p of the corresponding circular arcs. Once the lengths, lb.g = lb.pa and lb.p = lb.pa, are equal, then angular base pitches ϕb.g = ϕb.op, and ϕb.p = ϕb.op, are equal as well. The unfolded section, A − A, of the gear tooth flanks, G, by the cylinder of revolution is shown in Figure 1.11b; the unfolded section, A − A, of the pinion tooth flanks, P, by the cylinder of revolution is shown in Figure 1.11c, and the unfolded section, A − A, of the gear crossed-axes gear pair by the cylinder of revolution is shown in Figure 1.11d. Here and below, only local parches of the gear tooth flanks, G, are considered. The gear tooth flanks themselves are not yet defined. Therefore, only small portions of the tooth flanks, G, are depicted in Figure 1.11b. All these portions of the tooth flanks of a gear are located in the differential vicinity of points of intersection, Kg, of the tooth flanks, G, by the path of contact, Pc. Each tooth flank, G, is perpendicular to the path of contact, Pc, at points Kg. All the points, Kg, are evenly distributed along the path of contact. The distance between each pair of neighboring points, Kg, is equal to the angular base pitch, ϕb.g, of the gear. An analysis similar to that above can be performed with respect to the pinion tooth flanks, P, as shown in Figure 1.11b. All these portions of the tooth flanks of the pinion are located in the differential vicinity of points of the intersection, Kp, of the tooth flanks, P, by the path of contact, Pc. Each pinion tooth flank, P, is perpendicular to the path of contact, Pc, at points

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FIGURE 1.11 On the concept of equal base pitches in perfect crossed-axes gear pair: (a) plane of action, PA, and the unfolded section A – A, (b) of a gear, (c) of a mating pinion, and (d) of the gear pair.

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Kp. All the points Kp are evenly distributed along the path of contact. The distance between each pair of neighboring points, Kp, is equal to the angular base pitch, ϕb.g, of the pinion. When the equality of the angular base pitch of a gear, ϕb.g, and that of a mating pinion, ϕb.p, to operating angular base pitch of the gear pair, ϕb.op, is observed, that is, when the identities, ϕb.g ≡ ϕb.op and ϕb.p ≡ ϕb.op, are valid, then the gear and the pinion can be engaged in mesh as illustrated in Figure 1.11d. Each gear point, Kg, coincides with a corresponding pinion point, Kp. Because of this, the gear and the pinion points, Kg and Kp, are further designated as a contact point, K. When angular base pitches of a gear and a mating pinion are not equal to one another (i.e., ϕb.g ≠ ϕb.p), for example, the gear base pitch, ϕb.g, is smaller compared to the mating pinion base pitch, ϕb.p, and the inequality ϕb.g  ϕb.p is valid as shown in Figure 1.12b; again, only one pair of teeth is engaged in mesh. A gap between the other pairs of teeth of the gear, G, and the pinion, P, is observed. No gaps of this sort are permissible in perfect crossed-axes gearing. In this second example (see Figure 1.12b), the distribution of the gaps is inverse to that shown in Figure 1.12a. This is because the gear and the mating pinion are rigid bodies that physically cannot interfere with one another. Therefore, a uniform rotation can be smoothly transmitted by a crossed-axes gear pair if the following equalities are fulfilled at every instant of time

ϕb. g ≡ ϕb.op



ϕb. p ≡ ϕb.op (1.23)

(1.22)

With that said, the third condition to be fulfilled in a perfect crossed-axes gear pair can be formulated in the following manner, the third fundamental law of gearing: “In crossedaxes gearing, in order to transmit a uniform rotary motion from a driving shaft to a driven shaft by means of gear teeth, angular base pitch of a gear, and that of a mating pinion, must be equal to operating base pitch of the gear pair at every instant of time.” Only “conjugate” gear tooth flanks feature base pitch. Base pitch cannot be specified for nonconjugate gear tooth flanks. Therefore, only conjugate tooth flanks of a gear and a mating pinion can fulfill the third fundamental law of gearing. The equality of angular base pitches of a gear and a mating pinion to operating angular base pitch in a crossed-axes gear pair is the third fundamental law of gearing that all perfect crossed-axes gearings have to fulfill.

1.3 Illustrative Example: Perfect Crossed-Axes Gearing with Line Contact between the Tooth Flanks of a Gear and a Mating Pinion Various types of gearing that feature crossing axes of rotation of a gear and a mating pinion are used in the industry. Early designs of crossed-axes gears can be found in Leonardo da Vinci’s famous book The Madrid Codices (1493) [7]. Hypoid gearing, Spiroid gearing, Helicon

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FIGURE 1.12 Examples of violation of the condition of equal angular base pitches in crossed-axes gearing: (a) a case when the angular base pitch of a gear, ϕb.g, is smaller compared to the angular base pitch, ϕb.p, of a mating pinion (ϕb.g  ϕb.p); in both cases the gaps are observed.

gearing, along with numerous types of worm gearings, represent perfect examples of gearing with crossing axes of rotation of a gear and a mating pinion that are now used in the industry. In gearings of all of these types, it is vital to ensure line contact between the tooth flanks of the mating gears as this enables higher bearing capacity of the interacting tooth flanks. Maintaining line contact is critical to improve the power density of the gearbox. Perfect crossed-axes gearing with a line contact between the tooth flanks of the gear, G, and the mating pinion, P, is discussed below as an illustrative example of perfect crossed-axes gearing, that is, as an example of crossed-axes gearing that meets all three fundamental laws of gearing. The gearing of this type is the only type of Ca–gearing capable of transmitting an input rotation smoothly, with a constant angular velocity

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ratio uω = ωp/ωg = Const. Gearings for which the angular velocity ratio, uω is of a constant value (uω = Const) are commonly referred to as the “perfect gearings.” In order to better understand the term “perfect gearings,” it is instructive to note here that in a case of parallel-axes gearing, involute gearing11 is the only type of perfect gearing; similarly, in a case of intersected-axes gearing, gearing that features spherical involute geometry is the only type of perfect intersected-axes gearing12 [7]. Many past efforts aimed to investigate crossed-axes gearing. The most significant results of the research in the field obtained to this end are summarized and are generalized by Prof. J. Phillips [4]. It should be stressed here that, in terms of Prof. J. Phillips [9], “general spatial involute gearing” features point contact between the tooth flanks of a gear, G, and a mating pinion, P. The desirable line contact in spatial involute gearing is impossible, as in this particular type of gearing, the tooth flanks of both of a gear and a mating pinion are developed from the corresponding base cylinders. Because of this, the fundamental laws of gearing are fulfilled only at a current contact point, K, and are violated in the rest of the points of the tooth flanks, G  and P. Another approach to investigate crossed-axes gearings is implemented by Prof. H. Stachel [11–13], and others. Without going into the details of the performed analysis [11–13], it is important to point out the statement claimed in Corollary 2.2 (see page 38 in [12]): “If two helical involutes Φ1, Φ2 are placed such that they are in contact along a common generator and if their axis are kept fixed, then Φ1 and Φ2 serve as gear flanks for a spatial gearing with permanent straight line contact.” Two questions arise from the above cited statement [12]. First, yes, of course, two helical involutes Φ1 and Φ2 can be placed such that they are in contact along a common generator, which for helical involutes is a straight line. However, such contact means the contact along a “straight line,” and not along a line of contact in the form of a “planar curve” of an arbitrary reasonable geometry. Second, it is not evident that line contact along a common generator will be maintained when the gears rotate. Both of these questions are still not clarified so far. It can be shown that the statement [12] is valid only in a case of parallel-axes gearing, and is not valid for intersected-axes, as well as for crossed-axes gearings. The latter can be easily validated with fulfillment/violation of the fundamental laws of gearing in a case when two helical involutes, Φ1 and Φ2, make line contact in intersected-axes, or in crossedaxes gearings. In spatial involute gearing, line contact between the tooth flanks (i.e., contact along a straight line) is possible when crossed-axes gearing is reduced to a parallel-axes gearing. Only in this reduced case, the generating straight line of one of two involute helicoids can be aligned with the straight generating line of the other involute helicoid. To the best of the author’s knowledge, the considered below “R–gearing” (or “R–gear system,” in other words) is the only kind of perfect crossed-axes gearing with line contact between the tooth flanks of a gear and a mating pinion. Any planar curve of a favorable geometry (a straight line, a circular arc, a cycloidal curve, as well as numerous of others) can be used to generate tooth flanks of a gear, and a mating pinion in “R–gearing.” The gear As perfect parallel-axes gearing was proposed by Leonhard Euler (1760), this particular kind of gearing is referred to as “Euler gearing,” or just “Eu–gearing,” for simplicity. 12 As perfect intersected-axes gearing was proposed by George Grant (1887), this particular kind of gearing can be referred to as “Grant gearing,” or just “Gr–gearing,” for simplicity. 11

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tooth shape in the lengthwise direction is specified by the geometry of the planar curve, that is, by the desired line of contact. Perfect crossed-axes gearing (“R–gearing”) will feature a higher bearing capacity of the interacting tooth flanks, and, as a consequence, it will feature a higher power density that is vital in many applications. Helicopter transmissions and transmissions for rotorcraft are just a few to be mentioned. 1.3.1  Kinematics of Crossed-Axes Gearing Transmission and transformation of a rotary motion from a driving shaft to a driven shaft is the main purpose of implementation of crossed-axes gears. Both the input rotation and the output rotation can be easily represented by corresponding rotation vectors,13 ωg and ωp (see Figure 1.13). The rotation vectors, ωg and ωp, are along the gear and the pinion axes of rotation, Og and Op. The closest distance of approach between the lines of action of the rotation vectors, ωg and ωp, is denoted by C. This distance is along the center line, ℄, and is commonly referred to as the “center distance.” The instant rotation of the pinion in relation to the motionless gear is analytically described by a rotation vector ωpl = ωp − ωg. The “axis of instant rotation, Pln” (or the “pitch line, Pln,” in other words, as this line is a straight line through the pitch-point, P), is along the rotation vector, ωpl. It should be noted here that in the case of crossing 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 configuration are specified, the configuration of the axis of instant rotation, Pln, can be expressed in terms of the rotations, ωg and ωp, and of the center distance, C. In a particular case, the center lines 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-axes gear pairs of this particular kind are referred to as “orthogonal crossed-axes

FIGURE 1.13 Kinematics of crossed-axes gearing.

It is instructive to note here that rotations are not vectors in nature. Therefore, special care is required to be undertaken when treating rotations as vectors.

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gear pairs.” For gearing of this particular kind, the cross product of the rotation vectors of the gear, ωg, and the pinion, ωp, is always zero (ωg × ωp = 0). 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, written as



∑ g

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

(1.24)

For a shaft angle of 90°, Equation 1.24 reduces to Σg = tan−1(ωg/ωp). Similar equations are valid for the calculation of the pinion angle, Σp. The center distance, C, can be interpreted as the sum of the pitch radii of the gear, rg, and the pinion, rp, written as C = rg + rp (1.25)



This equation yields an expression for the calculation of the pitch radius, rg, as



rg =

1 + ωp − ωg ⋅C 1 + ωp

(1.26)

Then, the pinion pitch radius, rp, equals to: rp = C − rg . 1.3.2  Base Cones in Perfect Crossed-Axes Gear Pairs A belt-and-pulley analogy can be constructed for a crossed-axes gearing similar to that known for a parallel-axes gearing. For this purpose, two base cones are associated with the gear and the pinion. Smooth rotation of the base cones can be construed as a belt-andpulley mechanism with the belt in the form of a round tape. This concept is schematically illustrated in Figure 1.14. An orthogonal crossed-axes gear pair is shown here for illustrative purpose. That same approach is applicable with respect to angular bevel gears with a shaft angle Σ ≠ 90°, namely, with respect to either obtuse or acute shaft angle Σ. The schematic (see Figure 1.14) is constructed starting from the rotation vectors, ωg and ωp, of the gear and the pinion. The gear and its pinion rotate 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 an apex, Apa, within the center line, ℄. Point, Apa, is referred to as the “plane-of-action apex, Apa.” The endpoints of the straight-line segment, C, are labeled as Ag and Ap. Point, Ag, is the point of intersection of the center line, ℄, and the gear axis of rotation, Og. Point, Ag, is referred to as the “gear apex, Ag.” Point, Ap, is the point of intersection of the center line, ℄, and the pinion axis of rotation, Op. Point, Ap, is referred to as the “pinion apex, Ap.” For the convenience of the further analysis, three principal reference planes are introduced below. Two straight lines through a common point uniquely specify a plane through these two lines. In the case under consideration, this is the plane through the axis of instant rotation, Pln, and through the center line, ℄. The plane is referred to as the “pitch-line plane, Pln”: Two more important planes can be introduced here. They are the so-called the “center line plane,” (Cln–plane), and the “normal plane,” (Nln–plane), correspondingly.

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FIGURE 1.14 Base cones and the plane of action, PA, in an orthogonal crossed-axes gear pair.

The Cln–plane in a crossed-axes gear pair is the reference plane through the center line perpendicular to the axis of instance rotation of the gear and the pinion. The center line plane is perpendicular to the axis of instance rotation of the gear and the pinion. The Nln–plane in a crossed-axes gear pair is the referenced plane, the axis of instance rotation of the gear and the pinion perpendicular to the center line. All three reference planes, Pln–plane, Cln–plane, and N ln–plane, are of significant importance in the theory of gearing. They are referred to as the “main reference planes” of a gear pair. A main 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 1.15.

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FIGURE 1.15 Principal planes: the “pitch-line plane” (the Pln–plane), the “center line plane” (the Cln–plane), and the “normal plane” (the Nln–plane) associated with a gear pair.

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



rp tan Σ g = rg tan Σ p (1.27)

The plane of action, PA, is a plane through the axis of instant rotation, Pln. Let us assume that the plane of action rotates about the gear axis of rotation, Og. The base cone of the gear is an envelope to consecutive positions of the plane of action in such a rotation. Similarly, the base cone of the pinion is constructed. The plane of action, PA, is in tangency with the base cones of a gear and a mating pinion. Due to that, the plane of action, PA, makes a certain transverse pressure angle, ϕt.ω, in relation to the Pln − plane. The pressure angle, ϕt.ω, is measured within the Cln−plane. The portion of the schematic plotted in the upper left corner in Figure 1.14 is constructed within the plane of projections, π1. Two other planes of projections, π2 and π3, of the standard set of planes of projections, π1π2π3, are not used in this particular consideration. Therefore, these planes, π2 and π3, are not shown in Figure 1.14. Instead, two auxiliary planes of projections, 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 be bent about an axis perpendicular to the plane, PA. Under a uniform rotation of the gears, the plane of action, PA, rotates about the axis, Opa. The rotation vector, ω pa, 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

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the pitch plane, PA, is not observed, but rolling together with sliding of PA over the base cones is observed instead. For crossed-axes gearing, the plane of action, PA, can be viewed 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 Γb sin γ b

(1.28)

where Γb is the base cone angle of the gear, γ b and is the base cone angle of the pinion. In crossed-axes gearing, the base cone angles, Γb and γb, vary within the intervals 0°  0). However, its value exceeds the value, R1p , of the radius of normal curvature in the first example (Rp2 > R1p ). This results in that the degree of conformity of the pinion toot flank, P, to the gear tooth flank, G (Figure 4.9b), is greater compared to that shown in Figure 4.9a.

(a)

(b)

(c)

(d)

FIGURE 4.9 Sections of two smooth regular tooth flanks, G and P, in contact by a plane through the common perpendicular, ng.

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In the next example depicted in Figure 4.9c, the normal plane section, P  3, of the pinion tooth flank, P, is represented with a locally flattened section. The radius of normal curvature, Rp3, of the flattened plane section, P  3, approaches an infinity (Rp3 → ∞). Thus, the inequality, Rp3 > Rp2 > R1p , is valid. Therefore, the degree of conformity of the pinion tooth flank, P, to the gear tooth flank, G in Figure 4.9c, also gets greater. Finally, for a concave normal plane section, P  4, of the pinion tooth flank, P, that is illustrated in Figure 4.9d, the radius of normal curvature, Rp4, is of a 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 four examples considered in Figure 4.9. The examples shown in Figure 4.9, qualitatively illustrate what is already realized 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 4.9a through Figure 4.9d, 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 as illustrated in Figure 4.10a. When spinning the plane section about the common perpendicular, ng, it can be observed that the degree of conformity of the gear and the pinion tooth flanks, G and P, is different in different configurations of the crosssectional plane (see Figure 4.10b). The above 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, stands for. 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 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 be described, analytically?

(a)

(b)

FIGURE 4.10 On analytical description of contact geometry of two smooth regular tooth flanks, G and P, of a gear and a mating pinion.

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4.3.2 Indicatrix of Conformity at the Point of Contact of the Tooth Flanks of a Gear and a Mating Pinion This section aims to introduce a quantitative measure of the degree of conformity at a point of contact between two smooth regular tooth flanks, G and P. 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 the differential vicinity of a point, K, of their contact; for example, how much of the surface, P, is “congruent” to the surface, G, in differential vicinity of the contact 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 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 the difference between the normal curvatures of the tooth flanks, G and P, in a common section by a plane through the common normal vector, ng, results in the greater degree of conformity of the tooth flanks, G and P, at a point of their contact. Dup(G) indicates the distribution of the radii of normal curvature at a point of a gear tooth flank, G, as it had been shown, for example, for a concave elliptic patch of the surface, G (see Figure 4.11). For a gear tooth flank, G, equation of this characteristic curve in polar coordinates can be represented in the form

Dup(G ) ⇒ r g (ϕ g ) = |Rg (ϕ g )|

(4.40)

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

FIGURE 4.11 On derivation of equation of the indicatrix of conformity, CnfR(G /P ), at a point of contact of a smooth regular gear tooth flank, G, and a mating pinion tooth flank, P, which are in the first order of tangency.

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This is similarly true with respect to 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 pinion tooth flank, P (see Figure 4.11). Equation of this characteristic curve in polar coordinates can be represented in the form Dup (P ) ⇒ rp (ϕp ) = |Rp (ϕp )|



(4.41)

where rp is the position vector of a point of Dup(G ) at a point of the pinion tooth flank, P, and ϕp is the polar angle of Dup(P ). In the coordinate plane, xgyg, of the local reference system, xgygzg, the equalities, ϕg = ϕ and ϕp = ϕ + µ, are valid. Therefore, in the coordinate plane, xgyg, Equations 4.40 and 4.41 cast into

Dup(G ) ⇒ rg (ϕ ) = |Rg (ϕ )|

(4.42)



Dup(P ) ⇒ rp (ϕ , µ) = |Rp (ϕ , µ)|

(4.43)

When the degree of conformity at a point of contact of a gear tooth flank, G, and a mating pinion tooth flank, P, becomes greater, then the difference between the functions rg(ϕ) and rp(ϕ, µ) becomes smaller and vice versa. The last makes valid the following conclusion: Conclusion 4.1: The distance between the corresponding11 points of the “Dupin 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 tooth flank, G, and of the pinion tooth flank, P, at the contact point, K. The equation of the “indicatrix of conformity, CfnR(G/P )” at a point of contact of a gear tooth flank, G, and the a mating pinion tooth flank, P, is defined of the following structure: Cnf R (G /P ) ⇒ rcnf (ϕ , µ) = |Rg (ϕ )|sgn Rg (ϕ ) + |Rp (ϕ , µ)|sgn R p (ϕ , µ) = rg (ϕ )sgn Rg (ϕ ) + rp (ϕ , µ)sgn Rp (ϕ , µ)



(4.44)



As the location of a point, aϕ, of Dup(G) at a point of the gear tooth flank, G, is specified by the position vector, rg(ϕ), and the location of a point, bϕ, of Dup(P ) at a point of the pinion tooth flank, P, is specified by the position vector, rp(ϕ, µ), then the location of a point, cϕ (see Figure 4.11) of the 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. Here, in Equation 4.44, it is designated that rg = |R g | is the position vector of a point of the “Dupin indicatrix” of the gear tooth flank, G, at a point K of contact with pinion tooth flank, P, and rp = |R p | is the position vector of a corresponding point of the “Dupin indicatrix” of the pinion tooth flank, P. Here, in Equation 4.44, the multipliers, sgnRg(ϕ) and sgnRp(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 to remain as the corresponding sign of the functions, that is, to remain that same sign that the radii of normal curvature, Rg(ϕ) and Rp(ϕ, µ), have. Corresponding points of the “Dupin indicatrices,” Dup(P) and Dup(T), share the same straight line through the contact point, K, of the tooth flanks, G and P, and are located at the same side of the point, K.

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Ultimately, one can conclude that position vector, rcnf, of a point of Cnf R(G/P ) can be expressed in terms of position vectors, rg and rp, of the “Dupin indicatrices,” Dup(G ) and Dup(P ). For the calculation of a current value of the radius of normal curvature, Rg(ϕ), at the point of the gear tooth flank, G, the equality Rg (ϕ ) =



Φ1.g Φ 2.g

(4.45)

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



Φ1.p Φ 2.p

(4.46)

can be employed. Use of the angle, µ, of local relative orientation at a point of contact 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 4.44 can be rewritten in the following form:

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

(4.47)

For the derivation of the equation of Cnf R(G/P ), it is convenient to use the “Euler equation” for normal radius of curvature, Rg(ϕ), at a point of the gear tooth flank, G [12]:



Rg (ϕ ) =

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

(4.48)

Here, the radii of principal curvature, R1.g and R2.g, are the roots of the quadratic equation:



L g ⋅ R g − Eg M g ⋅ R g − Fg

M g ⋅ Rg − Fg =0 N g ⋅ Rg − G g

(4.49)

Recall, that the inequality, R1.g 1) is the ratio Rx/Ry, where R x = 1/(2A1) and Ry = 1/(2B1). It may be noted that the exponents of the two main nondimensional groups (expressing basic speed and load dependence) are the same for central and minimum film thickness, which implies that the ratio of central to minimum film thickness is constant for a given value of the radius ratio, k). However, in an earlier paper [12] on EHL of conformal gears, in which full numerical solutions were obtained for a range of heavy loads, it was found that although the Chittenden et  al. formula gave reasonable predictions of the central film thickness (which is virtually independent of load), the effect of increasing load on the minimum film 1

See Appendix A (by Prof. S.P. Radzevich) for details on the kinematics and geometry of conformal gearing (Novikov gearing).

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TABLE 5.1 Lubricant and Material Properties Temperature/°C Viscosity at ambient temperature/Pas

90 0.0195

120 0.0086

Pressure coefficient of viscosity, α/Pa−1

1.94 × 10−8 218

1.70 × 10−8 218

Reduced Young’s modulus, E′/GPa

Note: Constants in density pressure law, γ = 2.226 × 10−9 Pa−1; λ = 1.683 × 10−9 Pa−1.

meant that the Chittenden et al. formula significantly overestimated the minimum value at the heavy loads present in conformal gear contacts considered. Operating conditions assumed in [12] have been reworked and checked, using our up-to-date EHL solver, for the present contribution. Lubricant and material properties assumed are shown in Table 5.1. Two typical film thickness contour maps obtained from these analyses corresponding to relatively light and relatively heavy loads, respectively, at a temperature of 90°C are shown in Figures 5.6 and 5.7, respectively (where the x axis corresponds to the entrainment direction). Note that in these, and other contour maps, the boundary of the corresponding Hertzian elastic contact area is shown approximately by the 5 µm contour. A graph showing the ratio of minimum to central film thickness for the full range of six loads considered at this temperature is shown in Figure 5.8. It is of interest to note that the corresponding (constant) value of the ratio h min/h0, which would be obtained from the Chittenden et al. formula for the radius ratio considered (Rx/Ry = 8.55), is h min/h0 = 0.320. The marked sensitivity of minimum film thickness to load in elongated contacts of the type considered has also been noted by Venner and Lubrecht [13]. Therefore, it is concluded that the Chittenden et al. formula does not predict the severe thinning of the film that occurs at its transverse edges, particularly at heavy loads. Until a more useful minimum film thickness correlation based on analysis of a wide range of elongated contacts is available, it will, therefore, be

FIGURE 5.6 Lubricant film thickness contours (in µm) for load of 25 kN, R x = 0.663 m, Ry = 0.0775 m, (k = 8.55), temperature = 90°C, Ve = 28.5 m/s. Other operating conditions are given in Table 5.1.

FIGURE 5.7 Lubricant film thickness contours (in µm) for load of 120 kN, R x = 0.663 m, Ry = 0.0775 m, (k = 8.55), temperature = 90°C, Ve = 28.5 m/s. Other operating conditions are given in Table 5.1.

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FIGURE 5.8 Variation of the film thickness ratio h min/h0 with load.

necessary to carry out numerical solutions for specific conformal designs. An advantage of full numerical solutions is that they can be developed to take account of particular lubricant viscosity/pressure models and the inclusion of surface roughness effects.

5.5 Example: Steps in the Film Thickness Calculation for a Typical Gear Set 5.5.1  Operating Conditions The above equations from Dyson et al. [9,10] are applied to predict film thickness values in a typical conformal gear pair proposed for a multiple-pinion experimental aerospace reduction gearbox at a transmitted power of 385 kW per mesh. Physical properties of the materials and lubricant assumed are shown in Table 5.1, and salient details of the design are given in Table 5.2. The speed of the gears is taken to be constant. The pressure coefficient of viscosity is the quantity α in the following viscosity/pressure relation, which is appropriate for EHL film thickness calculations:

η = η0 exp(α p) (5.22)

Compressibility of the lubricant is considered in the numerical solutions according to the following empirical formula:



ρ = ρ0

(1 + γ p) (1 + λ p) (5.23)

The steps involved in obtaining the estimated film thickness values are as follows. 5.5.2  Principal Radii of Relative Curvature at the Tooth Contact The first step in the analysis is the fundamental center distance/pressure angle relation, which is simply obtained from the geometry of the transverse section of the teeth shown

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TABLE 5.2 Details of Conformal Gear Pair Design Tooth numbers, pinion/wheel Centres distance/mm Culminating pressure angle/degrees Pitch radius of generation of pinion teeth/mm Pitch radius of generation of wheel teeth/mm Profile radius of pinion teeth/mm Profile radius of wheel teeth/mm Helix angle/degrees Wheel speed/rev.min.−1

9/107 283.4 28.4 21.9 262.0 7.3 8.3 12.36 353

in Figure 5.5. With the teeth profile radii, pitch radii of generation, and the center distance given in Table 5.2 we obtain the working culmination pressure angle, ψ, of 28.4° and the distance x = 0.1805 mm. With the given helix angle, γ = 12.34°, the profile mismatch Δβ = 1.00 mm, and the gear ratio G = 107/9 = 11.89 we obtain from Equations 5.3 to 5.8 the sets of geometry coefficients B11, B22, B12 and b11, b22, b12. The coefficients C11, C12, C22 then follow from Equation 5.10 and the principal relative curvature coefficients at the contact point, A1 and B1, are given by Equation 5.16. The two principal radii of relative curvature at the point of contact are 1 = 965 mm and 2 A1 1 Ry = = 61.5 mm 2B1 (5.24) Rx =



5.5.3 Normal Load at the Contact, Hertzian Contact Dimensions, Maximum Hertzian Pressure Equation 5.19 is Pn =



H (1 + cos 2 ψ tan 2 γ )(1/2) (ωr cos ψ )

(5.25)

which gives, for H = 385 kW a nominal tooth load of Pn = 46 kN. With Pn = 46 kN; R x = 965 mm; Ry = 61.5 mm; E′ = 218 GPa we obtain, for these conditions, the elastic contact semi-axes a = 9.53 mm; b = 1.60 mm; the maximum Hertzian contact pressure as p0 = 1.44 GPa. 5.5.4  Entraining Velocity The entraining (rolling) velocity, which may be assumed to be directed along the major axis of the contact ellipse is, from Equation 5.18, (1/2 )



Ve = (u12 + u22 )

(5.26)

And with u1 and u2 from Equation 5.17, we obtain Ve = 45.3 m/s.

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TABLE 5.3 Predicted EHL Film Thickness Values Film Thicknesses from [5] Contact Load/kN 46 22.0

Film Thickness from Numerical Solution

h0/µm

h min /µm

h0/µm

h min /µm

1.03 1.08

0.31 0.33

2.26 2.13

0.16 0.21

FIGURE 5.9 Lubricant film thickness contours (in µm) for load of 46 kN, R x = 0.965 m, Ry = 0.0615 m, (k = 15.69), temperature = 120°C, Ve = 45.3 m/s. Other operating conditions are given in Table 5.1.

FIGURE 5.10 Lubricant film thickness contours (in µm) for load of 22 kN, R x = 0.965 m, Ry = 0.0615 m, (k = 15.69), temperature = 120°C, Ve = 45.3 m/s. Other operating conditions are given in Table 5.1.

5.5.5  Predicted Film Thickness Values A full numerical EHL analysis using the above values of load, entraining velocity, and principal radii of relative curvature were carried out using the given lubricant properties at a temperature of 120°C. An analysis was also performed at a lower contact load of 22 kN for which the contact semi-axes are a = 7.45 mm; b = 1.25 mm; and the maximum Hertzian contact pressure is p0 = 1.13 GPa. Table 5.3 shows predicted film thickness values from the two solutions together with those from the Chittenden et al. formula [5]. Corresponding film thickness contours are shown in Figures 5.9 and 5.10, respectively.

5.6 Discussion It is clear that the available formula [5] for minimum film thickness underestimates the severe load-dependent thinning that occurs under heavily loaded conditions typical of conformal contacts. At this stage of the development of EHL of elongated elliptical contacts it

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will, therefore, be necessary to carry out full numerical analyses of conformal gear contacts under the specific conditions of operation. The results of such analyses given here show that although significant films of micron order can be generated over much of the corresponding Hertzian contact area, the thinning occurring at the points of minimum clearance can lead to lubricant films of submicron order. Under such conditions, some degree of “mixed” or “micro-EHL” will occur in which the mechanism of lubrication depends on EHL at the asperity level. Considerable progress in understanding of micro-EHL is currently being made in this area [14], which helps to explain how gear tooth contacts can nevertheless benefit from effective EHL even under the most severe conditions of heavy load and, nominally, thin oil films.

Notation Hertzian contact major semi-dimension Hertzian contact minor semi-dimension profile radius of wheel tooth Young’s modulus 1 1− ν 2 E′ reduced elastic modulus, = E′ E G gear ratio (>1) hmin minimum EHL film thickness h0 central EHL film thickness H transmitted power p pressure p0 maximum Hertzian contact pressure Pn normal load on tooth r pitch radius of pinion Re entrainment radius Rx, Ry principal radii of relative curvature Ve entrainment velocity w load (point contact) w′ load per unit length (line contact) α pressure coefficient of viscosity β profile radius of pinion tooth γ helix angle Δβ B–β δ Hertzian elastic approach distance η viscosity ν Poisson’s ratio ψ pressure angle ω angular velocity of pinion a b B E

Reference frames (see Figure 5.5) y1, y2, y3 Y1, Y2, Y3

stationary frame with origin at center of rotation of pinion stationary frame with origin at center of rotation of wheel

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x1, x2, x3 frame embedded in pinion and rotating with it X1, X2, X3 frame embedded in wheel and rotating with it The x1, y1, X1, Y1 axes are at right angles to, and into the plane of Figure 5.5; they coincide with axes of rotation of the pinion (x1, y1) and of the wheel (X1, Y1) z1, z2, z3 stationary frame, origin at point of contact. The z1 and z2 axes are in the common tangent plane, the z3 axis lies along the common normal z1′ , z2′ principal axes of relative curvature between the teeth of the pinion and the wheel Other symbols are defined in the text.

References 1. Grubin, A. N. 1949. Fundamentals of the Hydrodynamic Theory of Lubrication of Heavily Loaded Cylindrical Surfaces, Book No. 30, Central Scientific Research Institute for Technology and Mechanical Engineering, Moscow (DSIR Translation No. 337). 2. Dowson, D. and Higginson, G. R. 1959. A numerical solution to the elastohydrodynamic lubrication problem. J. Mech. Eng. Sci., 1: 6–11. 3. Dowson, D. and Higginson, G. R. 1961. New roller-bearing lubrication formula. Engineering. 192: 158–159. 4. Evans, H. P. and Snidle, R. W. 1982. The elastohydrodynamic lubrication of point contacts at heavy loads. Proc R. Soc Lond. A382: 183–199. 5. Chittenden, R. J., Dowson, D., Dunn, J. F. and Taylor, C. M. 1985. A theoretical analysis of the isothermal elastohydrodynamic lubrication of concentrated contacts. Proc. R. Soc. Lond. A397: 271–294. 6. Shotter, B. A. 1977. Experiences with conformal/W-N gearing. Machinery. 131: 322–326. 7. Wells, C. F. and Shotter, B. A. 1962. The development of CIRCARC gearing. AEI Eng. March/ April: 83–88. 8. Radzevich, S. P. 2017. High-Conformal Gearing: Kinematics and Geometry, CRC Press. 9. Dyson, A., Evans, H. P. and Snidle, R. W. 1986. Wildhaber-Novikov circular arc gears: Geometry and kinematics. Proc. R. Soc. London. A403: 313–340. 10. Dyson, A., Evans, H. P. and Snidle, R. W. 1989. Wildhaber Novikov circular arc gears: Some properties of relevance to their design. Proc. R. Soc., London. A425: 341–363. 11. Dyson, A., Evans, H. P. and Snidle, R. W. 1992. A simple, accurate method for calculation of stresses and deformations in elliptical Hertzian contacts. Proc IMechE. J Mech Engng. 206C: 139–141. 12. Evans, H. P. and Snidle, R. W. 1993. Wildhaber-Novikov circular arc gears: Elastohydrodynamics. Trans. ASME J. Tribol. 115: 487–492. 13. Venner, C. H. and Lubrecht, A. A. 2010. Revisiting film thickness in slender elastohydrodynamically lubricated contacts. Proc. IMechE., J Mech Engng Sci. 225C: 2549–2558. 14. Evans, H. P., Sharif, K. J., Snidle R. W., Shaw, B. A. and Zhang, J. 2013. Analysis of microelastohydrodynamic lubrication and prediction of surface fatigue damage in micropitting tests on helical gears. ASME J. Tribol. 135(1). DOI: 10.1115/1.4007693.

6 Gear Drive Engineering Boris M. Klebanov CONTENTS 6.1 Gear Body Design............................................................................................................... 170 6.1.1 Disc-Type Gears...................................................................................................... 170 6.1.2 Design of Large Gears............................................................................................ 172 6.1.2.1 Welded Gear Body Design...................................................................... 173 6.1.2.2 Built-Up Gear Design.............................................................................. 174 6.2 Gear-Shaft Connections..................................................................................................... 178 6.2.1 Fixed Connections.................................................................................................. 178 6.2.1.1 Connections in Industrial Gear Drives................................................. 178 6.2.1.2 Connections in Lightweight Gear Drives............................................. 181 6.2.2 Movable Connections............................................................................................. 183 6.2.2.1 Gears of Variable-Speed Drives............................................................. 183 6.2.2.2 Idlers and Planet Gears........................................................................... 184 6.2.2.3 Double-Wheel Idlers and Planet Gears................................................. 186 6.3 Compliance of Shafts and Bearings................................................................................. 192 6.3.1 Bending of Shafts.................................................................................................... 192 6.3.2 Influence of Shafts’ Bearings................................................................................. 194 6.3.2.1 Displacement of Shafts in Rolling Supports........................................ 194 6.3.2.2 Types and Location of Supports............................................................ 198 6.4 Housing Deformations....................................................................................................... 202 6.5 Planetary Gear Drives........................................................................................................ 204 6.5.1 Improvement of Relative Accuracy...................................................................... 204 6.5.2 Flexible Supports of Planet Gears........................................................................ 205 6.5.3 Floating Sun Gears................................................................................................. 207 6.5.4 Floating Ring Gears................................................................................................ 207 6.5.5 Planet Carriers......................................................................................................... 211 6.6 Toothing Improvements of Cylindrical Involute Gears................................................ 213 6.6.1 Transverse Contact Ratio of Spur Gears.............................................................. 214 6.6.2 Tooth Profile Modification of Spur Gears........................................................... 218 6.6.2.1 Basics.......................................................................................................... 218 6.6.2.2 Parameters of Linear Tip Relief............................................................. 221 6.6.3 Meshing Geometry of Helical Gears................................................................... 232 6.6.4 Tooth Root Design.................................................................................................. 237 References...................................................................................................................................... 239

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The precision of gears achieved in production does not assure their precision in operation when the applied load and thermal strain change the geometry of the gears, shafts, and other parts of a drive. These deformations may cause undesirable deviations from the theoretical contact pattern between mating teeth of the gears and result in premature failure due to local overload. That is why the designer is to analyze the possible deformations of related drive elements in operation, even if they look sturdy in appearance. To understand the order of magnitude of the elastic deformations that may be of interest as applied to the gears, let us consider a simplified example of two misaligned teeth touching each other at angle γ (Figure 6.1a). Under load, the teeth become deformed, and their contact extends for some length L (Figure 6.1b). For the sake of simplicity, the load distribution is assumed to be linear as shown (this is not exact because of the end effect). In this case, length L and load concentration factor Kβ that allows for the nonuniformity of load distribution across the face width can be calculated from the following equations: L=



q 2Fn 2b 2 b 2 γ c′ mm ; K β = max = = . qav L γ c′ Fn

(6.1)

Here, Fn = normal force applied to the teeth (N); γ = angle of misalignment (rad); b = effective face width of the gears (mm); c′ = tooth pair stiffness (N/mm2, i.e., N per mm of face width per mm of tooth deformation); qmax and qav = maximal and average unit load on the tooth (N/mm). As the force grows, the length of contact L grows as well, and at L = b (Figure 6.1c), Kβ = 2. That means that the maximal unit load on the tooth is twice as big as the average value.

FIGURE 6.1 Tooth misalignment and load distribution along the contact line.

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Further increase of force Fn leads to more uniform load distribution (Figure 6.1d), and for this case, the equation for Kβ is Kβ = 1 +



γ b 2 c′ . 2Fn

(6.2)

From Equations 6.1 and 6.2, we can see that the greater the face width b and the smaller force Fn, the greater the nonuniformity of load distribution at the same angle of misalignment γ. Let us estimate the amount of misalignment—angular (γ) or linear (δ = γ ⋅ b)—that makes such a great nonuniformity of load distribution. Rearranging Equations 6.1 or 6.2 for Kβ = 2, we obtain δ=



2qav mm , c′

(6.3)

where, δ = γ ⋅ b (mm); qav = Fn/b – average unit load (N/mm); c′ = tooth pair stiffness (N/mm2). The average stiffness of the mesh (when the gears are made of steel) equals approximately 14 N/mm ⋅ µm = 14 ⋅ 103 N/mm2 (i.e., N per mm of face width per mm of mesh elastic deformation). The unit load qav admissible from the strength considerations depends on the mechanical properties of material and the tooth module m. For example, if a gear with tooth module m = 3 mm is made of tempered steel and loaded with a force of 50 N/mm, the linear misalignment δ obtained from Equation 6.3 equals 0.007 mm. If the teeth of the same gears are case hardened and the admissible load is 300 N/mm, then δ = 0.043 mm. One can see that even a tiny misalignment of teeth, just of hundredth parts of a millimeter or even of several microns, may double the unit load on one side of the teeth. The bigger the size of the gear drive, the more difficult to keep the misalignment of the teeth within such narrow limits. The tooth misalignment, besides the manufacturing errors, can also be caused by elastic and thermal deformations of the drive elements: pinions and wheels, shafts, bearings, housings, and so on. Figure 6.2 presents factors Kβ recommended as estimative for contact strength (K Hβ) and bending strength (K Fβ) calculations [1]. These diagrams give a pictorial presentation of the influence of the pinion proportion b/d1 (i.e., of the twisting and bending of the pinion) and the gear location relative to the supports (i.e., of the bending of the shaft and bearings compliance) on the load distribution across the face width. They also take into consideration the ability of nonhardened teeth to run-in and reduce the nonuniformity of load distribution due to plastic deformation and wear of the overloaded areas of tooth flanks. One can see that at b/d1 > 1.2 for nonhardened teeth and b/d1 > 0.8 for hardened teeth, flank line (helix) slope modification or crowning of the teeth may be required. The overhang-mounted gears are in an even worse situation. (1) For shaft B in gear drives III and IV (Figure 6.2), rigid rolling bearings with reduced clearance should be chosen to minimize the tilting of the shaft under oppositely directed tooth forces (see also Section 6.3). The same choice is preferred for the overhang mounting of the gear (I in Figure 6.2).

NO T E S :

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FIGURE 6.2 Diagrams of factor K Hβ and K Fβ. I to VI, examples of gear drive design; Ib and Ir , overhanging pinions on ball bearings and roller bearings, respectively; (a) and (b), teeth of at least one gear of the pair are not hardened (HV  350).

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(2) Pinions with low numbers of teeth are especially compliant because the diameter of their body is relatively small as compared with the tooth module. If the teeth are surface hardened, the load capacity is limited by the bending strength of the teeth and is nearly proportional to the tooth module. For example, for pinions with tooth numbers z = 15, x = 0.15, and z = 25, x = 0 the permissible tooth force is nearly the same, while the diameter of the smaller pinion equals only 61% of the larger pinion diameter. The stiffness is proportional to the 4th power of the diameter, both in bending and in twisting. Hence, at the same tooth load, the deformation of the smaller pinion is greater than the deformation of the bigger pinion by a factor of about 7 in bending and about 4.4 in twisting. (At twisting, the difference is less because the twisting moment reduces at smaller diameter.) Diagrams presented in Figure 6.2 are usable for strength calculations of gear drives with a relatively big safety margin. The engineering of high-power and low-weight gear drives requires detailed analysis of all factors influencing the load distribution across the gear face. These factors, besides the manufacturing errors, are as follows: • Bending and twisting deformations of the gear body caused by the tooth forces; • Uneven tooth stiffness across the gear face due to nonuniform gear rim thickness; • Nonuniform radial deformations of the toothed rim by centrifugal forces (essential for high-speed drives); • Deformations and clearances in hub-shaft connections; • Bending of shafts; • Compliance and clearances of bearings and adjoined parts of housing; • Deformation of the entire housing under loads applied by the mesh forces and other internal and external forces, including inertia forces, if the gear drive is mounted on a moving platform; • Nonuniform radial deformations of gear body due to nonuniform heating in operation. This enumeration can be continued. Many kinds of the elastic shape distortions listed above are controllable: their magnitude, and sometimes the direction of the deformations, can be changed with the aim of mutual neutralization of their adverse influence. For this purpose, the geometry of the gears and other parts can be varied, or additional flexible elements can be used, to obtain profitable deformations under load and compensate this way the influence of other, unavoidable, deformations. The attainment of uniform load distribution across the face width of the gears is an important part of the gear drive engineering. All the forces and their effects are computable, except the heating. Unlike the shape distortions caused by applied forces, the temperature of gears in different areas can hardly be calculated. It depends largely on the cooling quality: sort and temperature of cooling oil, degree of the oil jet fragmentation by the sprayer, distribution of the cooling oil over the gear face and, possibly, on the inner side of the gear rim, and so on. All this may also have a significant effect on the efficiency of the drive. The measurements of temperatures on the working drives and the collection of such measurements made on similar drives may be helpful in new design. The pinion is usually hotter than the wheel, and this may lead to angular misalignment of the helical teeth because the thermal elongation of a helical gear is accompanied by reduction of the helix angle. But not only that, the temperature is not uniform across the

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face width. For example, the difference in temperature between the ends of a pinion may be as high as 20–30°C and even higher (depending on the rim speed) at the face width of 300 mm [37]. As the thermal expansion coefficient of steel equals about 12 µm/m ⋅ °K, the difference of 20°C gives the difference in the gear diameter of 24 µm per 100 mm of the diameter. This factor should not be ignored. In this chapter, we discuss the design of gear drive elements, which influence the load distribution across the face width and between the meshing teeth.

6.1  Gear Body Design 6.1.1  Disc-Type Gears Like most of the pinions are designed as pinion-shafts, so most of the gear wheels have disc-type design (Figure 6.3a), that is, they consist of rim 1 and disc 2 (also called web) which connects the rim to a hub or shaft. The thickness of rim (SR) and disc (bs) are important in respect of their strength. Force Fn applied to the tooth bends both the tooth and the rim (Figure 6.3b, the initial shape is shown with dashed lines) and induces high local stress in the transition area between the rim and the disc (Figure 6.3c). If the disc is too thin, there can be fatigue failure and complete separation of the rim from the web because the stress wave runs around the circle each turn of the gear. It is recommended to make the rim thickness SR  ≥  1.6 m (mostly SR ≈ 2 m) because at thinner rim, the fatigue crack, starting from the tooth fillet, may propagate into the rim as shown in Figure 6.3d and result in an immediate and very grave failure. At thicker rim, the fatigue crack goes from the tension side to the compression side of the tooth root (Figure 6.3e), and if the tooth just breaks off and does not damage anything else, the drive mostly continues working until it is stopped. When the weight of the gear is critical, the thin rim can be provided with ring ribs as shown in Figure 6.4d, and large fillet radii strengthen the transition from the rim to the disc.

FIGURE 6.3 Bending deformation and failure of toothed rim.

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FIGURE 6.4 Load distribution across the face width.

However, not only the strength of the rim and web, but also their stiffness is important. As per ISO 6336-1, the tooth stiffness is proportional to factor CR determined as follows:



CR = 1 +

ln(bS /b) . 5e SR /( 5 mn ) (6.4)

For solid gear (i.e., at bs/b = 1) CR = 1. For SR = 1.6 m and bs/b = 0.2, we obtain CR = 0.766. (For bs/b  +1 TD min

(8.9)

because TDmax > TDmin (Figure 8.3). This innovation is especially important. It is connected to the lever analogy. 7. Ideal external torques are always in a certain ratio, expressed with the torque ratio t

TD min : TD max : TΣ = TD min : +t ⋅ TD min : −(1 + t)TD min = +1 : +t : −(1 + t),

(8.10)

in which always,

TD min < TD max < TΣ

(8.11)

irrespective of: • How many degrees of freedom F (F = 1 or F = 2) the gear train is running with; • Which shaft is fixed at F = 1 degree of freedom; • What is the direction of power flow; respectively, does the PGT works as a reducer or a multiplier at F = 1 degree of freedom; respectively, such as a summation or as a division differential at F = 2 degrees of freedom; • Does the PGT works as a single or as a component train along with others in a compound planetary gear train. 8. From those relationships, for the three ideal external torques of the PGT shafts, is not difficult to establish that there is an analogy of the gear train with a lever, loaded with three forces, that is, there is a lever analogy. In Figure 8.3, there is a comparison of the PGT with a straight lever, in which can be seen the complete analogy. This lever analogy with its visibility is very useful for ease of understanding and insight into the work of the planetary gear trains.

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8.3.3  Kinematic Analysis of a Simple AI-Planetary Gear Train From Figures 8.1 and 8.3, it can be seen that when applying the torque method on AI-PGT TDmin ≡ T1, TDmax ≡ T3, and TΣ ≡ TH. 8.3.3.1  Work with F = 1 Degree of Freedom For the possible six working modes with F = 1 degree of freedom (Figure 8.2) by Equation 8.6, the following dependencies can be derived. Work with fixed carrier (ωH = 0) is the pseudo-planetary train (Figure 8.5: arrows show the direction of power flow): As a reducer (|i13(H)| > 1)



i13( H ) =

ω1 ωA +t z T = =− B =− = −t = i0 = − 3 < −1, ω3 ωB +1 z1 TA (8.12)

As a multiplier (|i13(H)|  −1, ω1 ωB +t TA t i0 z3 (8.13)

In this case, both speed ratios are negative (i  1)



i1H ( 3 ) =

ω1 ω −(1 + t) z T = A =− B =− = 1 + t = 1 − i0 = 1 + 3 > 1; ωH ωB +1 z1 TA (8.14)

FIGURE 8.5 AI-planetary gear train working with fixed carrier H as a reducer (a) and as a multiplier (b). (From Arnaudov K. and D. Karaivanov. Sofia, 2017 (in Bulgarian). With permission.)

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FIGURE 8.6 AI-planetary gear train working with fixed ring gear 3 as a reducer (a) and as a multiplier (b). (From Arnaudov K. and D. Karaivanov. Sofia, 2017 (in Bulgarian). With permission.)

As a multiplier (i1H(3)  0). Work with fixed sun gear (ω1 = 0) (Figure 8.7): As a reducer (i3H(1) > 1)

i3 H (1) =

ω3 ω −(1 + t) 1 1 T z = A =− B =− = 1 + = 1 − = 1 + 1 > 1; ωH ωB +t TA t i0 z3 (8.16)

FIGURE 8.7 AI-planetary gear train working with fixed sun gear 1 as a reducer (a) and as a multiplier (b). (From Arnaudov K. and D. Karaivanov. Sofia, 2017 (in Bulgarian). With permission.)

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As a multiplier (i1H(3)  0). 8.3.3.2  Work with F = 2 Degrees of Freedom 8.3.3.2.1  Work of PGT as a Summation One At given angular velocities ωAI and ωAII of both input shafts, the angular velocity ωB of the output shaft is defined by the condition of the sum of the input PAI, PAII, and the output PB powers

∑P = P i

AI

+ PA II + PB = TA I ⋅ ωA I + TA II ⋅ ωA II + TB ⋅ ωB = 0. (8.18)

From three possible modes of work (Figure 8.2c), the output carrier is the most normally used. For this case, Equation 8.18 becomes

∑P = P

A1

i

+ PA 3 + PB = T1 ⋅ ω1 + T3 ⋅ ω3 + TH ⋅ ωH = 0.

(8.19)

From here, it can be defined as



ωH = ωB = −

T1 ⋅ ω1 + T3 ⋅ ω3 = iH 1( 3 ) ⋅ ω1 + iH 3(1) ⋅ ω3 . TH (8.20)

8.3.3.2.2  Work of PGT as a Division One In this case, the power equilibrium is as follows

∑P = P +P i

A

BI

+ PB II = TA ⋅ ωA + TB I ⋅ ωB II + TB II ⋅ ωB II = 0. (8.21)

At known input angular velocity ωA, output velocities depend on output torques (working resistances). Their ratio is constant (see Equation 8.10), but their value depends on operating conditions of a working machine. In the case of input sun gear (ωA = ω1), both output torques T3 and TH are in a definite relation through the torque ratio t (Equation 8.9) TH = −T3



1+ t t

(8.22)

and dependence Equation 8.21 becomes as follows

∑P = P +P i

A

BI

+ PB II = T1 ⋅ ω1 + T3 ⋅ ω3 + TH ⋅ ωH = 0.

(8.23)

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The work of PGT with F = 2 degrees of freedom is examined in more detail in the examples in [8]. Therefore, only the approach is given here. 8.3.4  Power Analysis of AI-Planetary Gear Train 8.3.4.1  Power Flows In examining the power flows in a PGT, it is convenient to assume that the input (transmitted, external, absolute, shaft) power PA in the planetary gear train is divided into two types of power (Figure 8.8): 1. Coupling power Pcoup , which transmits by coupling movement when PGT rotates as a coupling (as a whole) without relative rotation of gears to the carrier H. Consequently, this power is assumed to be transmitted without internal losses (the losses in the central element bearings are neglected). 2. Relative (rolling) power Prel , or the power in the mesh that is transmitted by relative movement of the gears with respect to the carrier. This is the power in PGT after inversion [8,44] (“pseudo-planetary” gear train with fixed carrier) that causes the meshing losses. These losses are considered by the basic loss factor ψ0, respectively by the basic efficiency η0. Of the above, follows PA = Pcoup + Prel . (8.24)



In the most common case, the sun gear 1 is input, the carrier H is output, and ωH is the coupling angular velocity. Then the following dependencies are in effect

PA = TA ⋅ ωA = T1 ⋅ ω1 ,

(8.25)



Pcoup = TA ⋅ ωH = T1 ⋅ ωH ,

(8.26)

Prel = T1(ω1 − ωH ) = T1 ⋅ ω1 rel = T1 ⋅ ω1( H )

  ω  1   < PA , = T1 ⋅ ω1 1 − H  = PA 1 −   ω1  i1H ( 3 ) 

FIGURE 8.8 Types of power and peripheral velocities in Sofia, 2017 (in Bulgarian). With permission.)

(8.27)

AI -planetary gear train. (From Arnaudov K. and D. Karaivanov.

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respectively, Prel 1 = 1− < 1. PA i1H ( 3 ) (8.28)



It can be seen that in the AI-PGT, the relative (rolling) power Prel is less than the input (absolute) power PA, which results in smaller losses and higher efficiency of this type of PGT compared to other PGTs (AA, II, etc.) [8]. This is shown in Figure 8.8, where the peripheral velocity v1 of sun gear 1 is represented as the sum of the coupling vcoup and the relative vrel velocity v1 = vcoup + vrel ,



(8.29)

and the coupling velocity is less than the relative velocity vcoup < vrel .



It should be kept in mind that in all PGTs operating with unfixed carrier PA ≠ Prel. In negative-basic-ratio PGTs (as AI-PGT) PA > Prel, but in positive-basic-ratio ones, it is vice versa. Therefore, the losses of the second ones are greater than when they are working with the fixed carrier [8]. 8.3.4.2  Real Torques When determining the torques, considering the losses (so-called real torques), the direction of relative (rolling) power must be considered [8,9]. It can either be from the sun gear 1 to the ring gear 3 (driving sun gear) or vice versa (driving ring gear). A driving element is one in which the torque and the relative angular velocity are unidirectional. The following rules may be used to determine the direction of relative (rolling) power in a simple PGT, regardless of whether it operates alone or as a part of a compound gear train: When one of the two elements (sun gear or ring gear) is input (or output) for the transmitted power, it is input (or output) for the relative power, too (see Figures 8.9, 8.10, and 8.13). At work of the simple PGT with F = 2 degrees of freedom, two cases are possible. If one of the two elements is an input and the other is an output for the transmitted power, this is the relative power direction from the input to the output (see Figures 8.15 and 8.16). If both elements are input (summation differential) or output (division differential), the direction of relative power is determined by the method of samples [34] (see Figure 8.29). Values of real torques depend on basic efficiency η0. At a certain torque on the driving element, the value of the real torque on the driven element is obtained as ideal but multiplied by η0, and at a certain torque on the driven element, after the division of η0: Driving sun gear 1:

1 T3 T3 ⋅ > = T1. t η0 t

(8.30)

1 T3 T3 < = T1 , resp. T3′ = ⋅ t ⋅ T1 > t ⋅ T1 = T3 . η0 t t

(8.31)

T3′ = η0 ⋅ t ⋅ T1 < t ⋅ T1 = T3 , resp. T1′ = Driving ring gear 3:



T1′ = η0

Real torques are used to determine the efficiency η of the gear train according Equation 8.8.

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FIGURE 8.9 Ideal and real torques determination in AI- PGT with fixed ring gear (ω3 = 0): (a) input sun gear 1; (b) input ring gear 3. (From Arnaudov K. and D. Karaivanov. Sofia, 2017 (in Bulgarian). With permission.)

FIGURE 8.10 Ideal and real torques determination in AI- PGT with fixed sun gear (ω1 = 0): (a) input ring gear 3; (b) input carrier H. (From Arnaudov K. and D. Karaivanov. Sofia, 2017 (in Bulgarian). With permission.)

8.3.4.3 Efficiency There are different ways of the theoretical determination of PGT efficiency η known from the literature [12,15,27,29,32]. The torque method is very appropriate because, unlike other methods, it is primarily very visual, also simple and easy to use [8,9]. To apply Equation 8.8, ideal and real torque should be determined. When PGT works with one degree of freedom, the following cases are possible (Figure 8.2): Work with fixed carrier (ωH = 0) – pseudo-planetary gear train: With sufficient accuracy for engineering practice, it can be assumed that the efficiency in both possible directions of power flow is the same [31,32] η13( H ) ≈ η31( H ) = η0 < 1. (8.32)



Work with fixed ring gear (ω3 = 0) (Figure 8.9): In case of input sun gear (Figure 8.9a) the formulae are



−(1 + t) TB z =− = 1+ 3 . +1 TA z1

(8.33)

 z  TB′ −(1 + η0 ⋅ t) = −1 + η0 3  . =  TA′ +1 z1 

(8.34)

ik = i1H ( 3 ) = − iT =

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 z  z −1 + η0 3  1 + η0 3  iT z1  z1 < 1. η1H(3) = − = − = z3 z3 ik 1+ 1+ z1 z1 (8.35)



Because TA′ = TA it is possible to use following formula [8,9] z 1 + η0 3 TB′ −(1 + η0 ⋅ t) z1 < 1. η1H ( 3 ) = = = z3 TB −(1 + t) 1+ z1 (8.36)



In case of input carrier (Figure 8.9b) the formulae are ik = iH 1( 3 ) = − iT =

TB′ = TA′



+1 1 TB = =− . −(1 + t) 1 + z3 TA z1

+1 1 =− .  z3  1 t   + ⋅ 1  −1 +  η0 z1 η0  

(8.37)

(8.38)

1 1 z3 z 1+ ⋅ 1+ 3 iT η0 z1 z1 < 1. = η H 1( 3 ) = − = − 1 1 z3 ik 1+ ⋅ z η0 z1 1+ 3 z1 −



(8.39)

Or (TB′ = TB )



z 1+ 3 TA −(1 + t) z1 < 1. η H 1( 3 ) = = = z3 1   TA′ t −1 +  1 + ⋅ η 0 z1 η0  

(8.40)

Work with fixed sun gear (ω1 = 0) (Figure 8.10): In case of input ring gear (Figure 8.10a) the formulae are





−(1 + t) 1 TB z =− = 1+ = 1+ 1 . +t TA t z3

(8.41)

 η  TB′ −(η0 + t) z  = = − 0 + 1 = −1 + η0 1 .  t   t z3  TA′

(8.42)

ik = i3 H (1) = −

iT =

Kinematic and Power Analysis of Multi-Carrier Planetary Change-Gears



 η  z −1 + 0  1 + η0 1  iT  t  z3 η3 H (1) = − = − = < 1. z 1 ik 1+ 1+ 1 t z3

305

(8.43)

Because TA′ = TA it is possible to use following formula [8,9]:



z1 η0 + 1 1 + η0 TB′ −(η0 + t) z 3 η3 H (1) = = = t = < 1. 1 z1 TB −(1 + t) 1+ +1 t z3 (8.44) In case of input carrier (Figure 8.10b) the formulae are ik = iH 3(1) = −

iT =

+t 1 1 TB = = =− . z 1 TA −(1 + t) 1 + 1+ 1 t z3

η0 .t TB′ 1 1 = =− =− . z 1 1 TA′ −(1 + η0 .t) 1+ 1+ ⋅ 1 η0 .t η0 z3 −

i η H 3(1) = − T = − ik

(8.45)

(8.46)

1 1+ 1 1+

1 η0 .t 1 t

z 1 1+ 1 z t = 3 = < 1. 1 1 z1 1+ 1+ ⋅ η0 .t η0 z3 1+

(8.47)

Because neither TA′ ≡ TA nor TB′ ≡ TB, the second way to define efficiency cannot be applied here [8,9]. However, this can happen if start from TB′ ≡ TB = +t instead TC′ ≡ TC = +1, where  1 1 the other torques will be TC′ = + and TA′ =  + t.  η0  η0 The above logic can also be applied in the case of the two degrees of freedom operation [8]. The method proposed herein for theoretical determination of the efficiency does not consider permanent, non-load-depending losses (in seals, from oil churning, etc.). For the more accurate determination of the efficiency, other factors have to be considered, some of which change during operating even with constant load and speed [8]. Acceptance of basic efficiency of the component PGTs for constant shows the influence of the structural scheme of the multi-carrier PGTs on the efficiency of the entire gear train and is very appropriate in their comparative analysis [19,20,22,35–37,39,40]. 8.3.5  Kinematic and Power Analysis of Compound Two-Carrier Planetary Gear Trains 8.3.5.1  Possible Ways of Connection and Work In this section, the most typical case of compound two-carrier PGT with two compound and three external shafts is considered (Figure 8.11a) [8,9]. Figure 8.12 shows various working modes of the PGT in question [8,9].

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FIGURE 8.11 Possible ways to connect two simple single-carrier planetary gear trains in a compound two-carrier planetary gear train: (a) with two compound and three external shafts; (b) with two compound and four external shafts; (c) with one compound and four external shafts. (From Arnaudov K. and D. Karaivanov. Sofia, 2017 (in Bulgarian). With permission.)

FIGURE 8.12 Various working modes of two-carrier compound PGT with two compound and three external shafts: (a) connected in series component PGTs; (b) closed power-loop differential PGT with internal division of power; (c) closed power-loop differential PGT with internal circulation of power. (From Arnaudov K. and D. Karaivanov. Sofia, 2017 (in Bulgarian). With permission.)

8.3.5.2  Analysis of PGT with Fixed External Compound Shaft Figure 8.13 shows the working as reducer PGT in question and its torques [8,9]. Torque ratios tI = 4 and tII = 3 as well as basic efficiencies η0I = η0II = 0.97 of both component PGTs are known. The sequence of determining the individual torques is indicated in the figure with numbers enclosed in circles.

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FIGURE 8.13 Connected in series two-carrier PGT and its torques. (From Arnaudov K. and D. Karaivanov. Sofia, 2017 (in Bulgarian). With permission.)

For convenience, not because it is compulsory, the set value for the smallest torque in the compound gear train is +1 [8,9]. In this case, it is the shaft of the sun gear 1 of the first stage. At the compound PGTs, the way in defining the shaft with the lowest torque does not represent any difficulty. Furthermore, it is not necessary to start with the shaft of some of the sun gears, neither with a numerical value +1. It is quite possible to start with the torque of any shaft, and with any arbitrary value without limitation in the algebraic sign “+” or “−” [8,9]. The three ideal external torques of each of the component PGTs, which are invariably in proportion, are determined by the relevant torque ratios tI and tII. As it is seen in Figure 8.13, both ends of the internal compound shaft operate equal in size, but different in direction torques (with different algebraic signs). The torque of the outer compound shaft is defined as an algebraic sum of both torques, acting on the coupling shafts (Figure 8.13). In the investigated compound PGT, it can be considered without any difficulty that the angular velocities of the sun gears 1 and 4 are larger than the angular velocity of the corresponding carriers HI and HII, so that in both cases, the relative power in the component PGTs is transmitted from the sun gears 1 and 4 to the ring gears 3 and 6. In this situation, the real torques of ring gears 3 and 6 are calculated using the following formulae

T3′ = η0 I ⋅ tI ⋅ T1′ and T6′ = η0 II ⋅ tII ⋅ T4′.

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Relative power direction is shown by dotted arrow. Ideal external torques

TA ≡ TD min = +1; TB ≡ TΣ = −20; TC ≡ TD max = +19

(8.48)

are in equilibrium Equation 8.3

∑T = T

D min

i

+ TD max + TΣ = +1 + 19 − 20 = 0,

(8.49)

as well as the real torques Equation 8.4

TA′ ≡ TD′ min = +1; TB′ ≡ TΣ′ = −19.08; TC′ ≡ TD′ max = +18.08

∑ T′= T′ i

D min

(8.50)

+ TD′ max + TΣ′ = +1 + 18.08 − 19.08 = 0. (8.51)

From speed ratio Equation 8.6 ik =



ωA −20 T =− B =− = +20. ωB +1 TA

(8.52)

and torque transformation Equation 8.8 iT =



TB′ −19.08 = −19.08. = +1 TA′

(8.53)

the efficiency is obtained



 iT  − = − −19.08   ik +20  T /T η = − B A =   = +0.954 = 95.4%.  ωA / ωB  TB′ −19.08  =   +20  TB 

(8.54)

As in this case the input torques, ideal TA and real TA′ are the same, that is, TA = TA′ = +1, coefficient of efficiency can be determined by the ratio of the two output torques, ideal TB and real TB′, as it is shown in Equation 8.54. 8.3.5.3  Internal Division or Circulation of Power This question is not only delicate, it is a real trap for the designer. Therefore, the following two examples are tasked to illustrate this problem and to clarify it [8,9]. They are more interesting than the hereinbefore described first simplest example, because they are closed power-loop differential gear trains, one of which is with internal power division (Figure 8.15) and the other one with internal power circulation (Figure 8.16), which is very unfavorable. In the literature, there are indeed ways to determine the presence of division or a circulation of power. They are not simple [25,26]. Much simpler is the following rule of algebraic signs [3,6,8,9,19], shown in Figure 8.14.

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FIGURE 8.14 Rule of algebraic signs in determining the internal power flows: (a) internal power division; (b) internal power circulation. (From Arnaudov K. and D. Karaivanov. Sofia, 2017 (in Bulgarian). With permission.)

Rule of algebraic signs for establishing the existence of internal power division or internal power circulation: When the algebraic signs of the external torques of the two shafts, forming the external compound shaft are the same, there is an internal power division. In different algebraic signs there is internal power circulation. Direction of the circulating power is determined by the following: Rule of algebraic signs to determine the direction of the circulating power: The direction of the circulating power Pcirc coincides with the direction of the input PA or output PB power at this one from both shafts, forming the external compound shaft, at which the algebraic sign of its torque coincides with the algebraic sign of the torque of the external compound shaft. Figure 8.15 shows the kinematic and structural scheme of a two-carrier closed powerloop differential PGT with internal division of power [8,9]. The same procedure is followed with the decision, as in the previous example (Figure 8.13). As in the previous example in both component PGTs, the relative power is transmitted from the sun gears 1 and 4 to the ring gears 3 and 6. It is easier to make calculations with concrete values of torque ratio and basic efficiency of component gears. Figure 8.16 shows the kinematic and structural scheme of a two-carrier closed-loop differential PGT with internal circulation of power [8,9]. Again, the procedure above is followed, starting from the shaft of the sun gear 4 of II-nd component gear train (T4 = +1). Different algebraic signs of the torques of the shafts, forming the external compound shaft, indicate the presence of internal power circulation, whose direction is determined by the rule of algebraic signs. The direction of relative power in both component PGTs is also obvious. Unlike the two previous examples, it is not possible here to use an alternative method for the coefficient of efficiency determination by the output torques, as the input torques, the ideal T4 = +0.25 and the real TA′ = +0.2943 , are not equal.

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FIGURE 8.15 Analysis of two-carrier closed-loop differential PGT with internal power division. (From Arnaudov K. and D. Karaivanov. Sofia, 2017 (in Bulgarian). With permission.)

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FIGURE 8.16 Analysis of two-carrier closed-loop differential PGT with internal power circulation. (From Arnaudov K. and D. Karaivanov. Sofia, 2017 (in Bulgarian). With permission.)

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In Figure 8.16, it is seen that the circulating power Pcirc is 2.4 times bigger than the input power:



T1′ −0.7057 ≈ 2.4. = TA′ +0.2943

Its size, moreover, because of losses, is changing along its path. Circulation of power determines significantly lower coefficient of efficiency of this gear train compared to the previous two (83% versus 95%), although their speed ratios ik are close (16, 17, 20) and it is assumed that the basic coefficients of efficiency of the component PGTs η0I and η0II are the same.

8.4  Kinematic and Power Analysis of a Two-Carrier Change-Gear Figure 8.17 shows the kinematic and structural scheme of a two-carrier compound changegear of a Mercedes-Benz® [30] which is relatively simple and very suitable for entering to the torque method. In Figure 8.17, there is also a table showing which of the gear train elements, the brakes Br.I and Br.II or the couplings Cp.I and Cp.II, are locked in the realization of the various speed ratios (gears)—three forward and one in reverse. In the forward direction, the input shaft is connected to the ring gear 3 by the coupling Cp.I. In the reverse direction, it is connected to the coupled sun gears 1 and 4 by the coupling Cp.II. The output shaft is always connected to the coupled carrier HI and ring gear 6. In various cases, the carrier HII or the coupled sun gears 1 and 4 are fixed. Figure 8.18 sets the speed ratios for the four operating cases (gears). Only the ratios of the planetary gear train are considered here. To be specific, the car can work with more gears (speeds) when changing the input torque (through a hydrotransformer, for example). Figure 8.19 shows the determination of the efficiency of the gear train at various gears as a function of torque ratios tI and tII and basic efficiencies η0I and η0II of component PGTs. At first gear, the I-st component PGT works as a power-division differential with the ring gear 3 as input. In this case, the relative (rolling) power PrelI is transmitted from the ring gear 3 to

FIGURE 8.17 Four speeds (gears) two-carrier change gear with two brakes (Br.I and Br.II) and two couplings (Cp.I and Cp.II).

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FIGURE 8.18 Determination of the speed ratios of the change-gear from Figure 8.17.

the sun gear 1. In the II-nd component PGT, things are even clearer, because the sun gear 4 is input for the transmitted power and the carrier HII is fixed, hence the sun gear 4 is also input for the relative (rolling) power PrelII. At second gear, only the I-st component PGT with ring gear as input and fixed sun gear 1 operates, hence the relative (rolling) power PrelI is transmitted from the ring gear 3 to the sun gear 1. At third gear, the compound PGT is blocked and rotates as a coupling. At fourth gear, all power is transmitted through the III-rd component PGT with a fixed carrier. After determining the real torques of all the shafts by Equations 8.30 and 8.31 and verification of the equilibrium condition of the three external torques by Equation 8.4, a torque transformation iT for each gear can be determined, and hence the corresponding efficiency by Equation 8.8. Table 8.1 shows the values of speed ratios and efficiency at various gears at specific values of torque ratio and basic efficiency of the component PGTs. High efficiencies at various gears are due to the fact that the kinematic scheme is simple and there is no circulation of power. At first gear, the power is divided into two flows through the component gears. Due to the fact that the carrier of the I-st component PGT

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FIGURE 8.19 Determination of the efficiency of change-gear from Figure 8.17.

TABLE 8.1 Speed Ratio and Efficiency of Planetary Change-Gear from Figure 8.17 at Various Gears, Considering tI = tII = 3 and η0I = η0II = 0.98 Gear i η

First (1)

Second (2)

+2.3333 0.980

+1.3333 0.995

Third (3) +1 1

Reverse (R) −3 0.980

is rotated and the relative (rolling) power PrelI (which creates the meshing loss) is smaller than the input power, the efficiency of the gearbox η1 is almost equal to the basic efficiency of the component PGTs. In the numerical example, η1 = 0.980185 is obtained. At second and reverse gears, the power passes only through one PGT, while the other is idling. The

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efficiency of the gearbox is equal to the efficiency of the working component PGT. It should be kept in mind that in these studies it is assumed that the basic efficiency accounts only for the meshing loss. Therefore, in the above formulae, permanent (non-load-depending) losses, such as in seals and oil churning, are not considered. In fact, the idling PGT will also generate losses, and the actual efficiency of the gearbox will be lower than the one calculated in Table 8.1. The torque method allows for the direct determination of the load on individual gears. If the operating time of a given gearbox load is known (for example, in the case of vehicles), it is possible to determine the load spectrum of the individual gears and, consequently, their equivalent loads and, ultimately, their safety factors SF and SH [8,9,17] (see Figure 8.31). This possibility is a very significant advantage of the torque method.

8.5  Kinematic and Power Analysis of a Four-Carrier Change-Gear The purpose of this chapter is not to promote new gearbox design solutions but to demonstrate the advantages of the torque method in their investigation. For this reason, a four-carrier change-gear from a Caterpillar heavy equipment vehicle from the middle of the twentieth century [43] is taken as the next example. In this case, the input torque is supplied by a hydrotransformer. Figure 8.20 shows the kinematic and structural schemes of the PGT (with four brakes) only, without the hydrotransformer before the PGT, and some gears of the transmission after the PGT. The first two stages (I and II) act as reverse. In this way, the gearbox performs two gears (speed ratios) in each direction. The figure also includes a table showing which of the brakes are locked at the various gears. Also, the two-carrier gear train in Section 8.4 is simple enough and in determination of the speed ratio ik = f(tI, tII) and efficiency η = f(tI, tII, η0I, η0II) the obtained formulas are not very long. In the analysis of four-carrier gearboxes

FIGURE 8.20 Kinematic and structural scheme of a four-carrier planetary change-gear with brakes (Br).

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ik = f(tI, tII, tIII, tIV) and η = f(tI, tII, tIII, tIV, η0I, η0II, η0III, η0IV) can also be determined [23,24]. These formulas are useful for the comparative analysis of different gear trains or for optimization of a given arrangement [22,39]. But the power (simplicity) of the torque method is manifested in the gear train analysis with specific parameter values. This is done in the kinematic analysis (Figures 8.21–8.24) and in the efficiency analysis (Figures 8.25–8.28) using the torque ratios t (basic speed ratios i0) of the actual arrangement.

tI =|i0 I |= 2.94



tII =|i0 II |= 2.8



tIII =|i0 III |= 3.04



tIV =|i0 IV |= 2.4



η0 I = η0 II = η0 III = η0 IV = 0.98

The direction of the relative (rolling) power in the component PGTs is determined by the rule set out in Section 8.3.4.2. Only in the III-rd PGT operating with unlocked brake Br.III, this direction is not obvious. The PGT works as a differential with a carrier as the input and both a sun gear and a ring gear as the output. The relative (rolling) power direction should be determined by calculations. The method of samples is very appropriate [34]. The procedure is

FIGURE 8.21 First gear—ideal torques and speed ratio.

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FIGURE 8.22 Second gear—ideal torques and speed ratio.

as follows: There are two possible directions of relative (rolling) power PrelIII, as shown in Figure 8.29. In both cases, for the III-rd component PGT, a lowered basic efficiency is assumed (e.g., η0III = 0.8), whereas for the I-st, II-nd and IV-th component PGTs, whose direction of relative power is clear, η0I = η0II = η0IV = 1 is taken. Then the real external torques are determined and it is found that in one of the cases (see Figure 8.29a for this case), for the efficiency of compound PGT, an absurd result is obtained (η > 1!) and in the other case (Figure 8.29b), a realistic one is obtained (η  1.  T     TA  Eq

(8.59)

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FIGURE 8.26 Second gear—real torques and efficiency.

Analogously, the load spectrum of each central element of the component PGTs can be determined. For determination of efficiency at all gears of this gearbox, see [23]. In general, the torque method allows direct determination of the load on all PGTs elements, without passing through the kinematic analysis of Willis’s or Kutzbach’s methods. This is undoubtedly a valuable asset of the torque method that is missing from other analysis methods.

8.7  Conclusion This chapter presents the basics of the torque method application for compound planetary gear trains analysis. Its advantages are particularly essential and dominant in the analysis of the complex compound multi-carrier planetary gear trains. The two classical methods of analysis used so far, Willis’s analytical method and the graphical method of Kutzbach, have actually proved their potential for many years, so they will continue to be used in the future. Nevertheless, when compared to them, the torque method used in the chapter has

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FIGURE 8.27 Reverse gear 1—real torques and efficiency.

both in general, and particularly in the analysis of the complex compound multi-carrier planetary gear trains, the following essential advantages: • It has more applications, not only the determination of speed ratio, as in Willis and Kutzbach, but determination of the internal power flows in size and direction and on this base defining the efficiency; also, determination of the load of the individual gear train elements. • The method is simple, particularly clear, and easy to use. • It combines the precision with the clarity that exist separately in the analytical and graphical methods. • It allows for the easy checking of the calculations by the sum of the external torques. • The designer’s dependence on literary sources is avoided so that one can act independently. Due to the clarity and accessibility, the method is suitable both for industrial practice (for designers), and for training (for students).

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FIGURE 8.28 Reverse gear 2—real torques and efficiency.

The chapter especially emphasizes on the application of the method in the kinematic and power analysis of multi-carrier planetary change-gears. Attention is drawn to the determination of the direction of relative (rolling) power in the component PGTs, which is crucial in determining the efficiency of the whole gear train.

Acknowledgments The authors would like to express their deepest gratitude to: • The editor of this book and the technical team of CRC Press for the patience and efforts made to improve the introduction; • Dr. Eng. Alexander Kapelevich for encouraging them to join this edition and for the invaluable help with terminology; • The translator Galina Koteva who helped to render the Bulgarian word order into English; • All readers who had the patience to reach these lines.

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FIGURE 8.29 Determination of relative (rolling) power direction through the method of samples: (a) with wrong direction of the relative power; (b) with right direction of the relative power.

FIGURE 8.30 Kinematic scheme of the main part (change-gear) of RENK® transmission for heavy vehicles. (From Arnaudov K. and D. Karaivanov. Sofia, 2017 (in Bulgarian). With permission.)

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FIGURE 8.31 Determination of load spectrum of the sun gear of the IV-th component PGT of the change-gear from Figure 8.30.

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References

1. Arnaudov, K. Engineering analysis of compound (two-carrier) planetary gear trains. Symposium “Heavy Machinery”. Varna [Bulgaria], 1984 [microfiche (48x)]. (in Bulgarian). 2. Arnaudov, K. and D. Karaivanov. Engineering analysis of the coupled two-carrier planetary gearing through the lever analogy. Proceedings of the Int. Conf. on Mechanical Transmissions. Chongqing [China]: China Machine Press, 5–9 April, 2001, pp. 44–49. 3. Arnaudow, K. und D. Karaivanov. Systematik, Eigenschaften und Möglichkeiten von zusammengesetzten Mehrsteg-Planetengetrieben. Antriebstechnik. 2005, Nr. 5, S. 58–65. 4. Arnaudov, K. and D. Karaivanov. Higher compound planetary gear trains. VDI-Berichte 1904-1, 2005, pp. 327–344. ISSN 0083-5560. 5. Arnaudov, K. and D. Karaivanov. The complex compound multi-carrier planetary gear trains – a simple study. VDI–Berichte 2108-2, 2010, pp. 673–684. ISSN 0083-5560. 6. Arnaudov, K. and D. Karaivanov. Alternative method for analysis of complex compound planetary gear trains: Essence and possibilities. Mechanisms and Machine Science, 13 (2013), Power Transmissions, Proceedings of the 4th International Conference, Sinaia, Romania, June 20–23, 2012, Editor Dobre G., Springer, Dordrecht Heidelberg New York London, pp. 3–20 (Plenary Paper). ISSN 2211-0984. 7. Arnaudov, K. and D. Karaivanov. The torque method used for studying coupled two-carrier planetary gear trains. Transactions of FAMENA, 2013, 37 (1), pp. 49–61. ISSN 1333-1124. 8. Arnaudov, K. and D. Karaivanov. Planetary gear trains. Sofia: Bulgarian Academy of Sciences Publ. “Prof. Marin Drinov,” 2017, 368 p., ISBN 978-954-322-885-0. (in Bulgarian). [This book has been translated and published as Arnaudov, K. and D. Karaivanov. Planetary gear trains. Boca Raton, FL: CRC Press, 2019, ISBN 978-1-138-31185-5.] 9. Arnaudov, K. and D. Karaivanov. Torque method for analysis of compound planetary gear trains. Beau Basin [Mauritius]: LAP LAMBERT Academic Publishing, 2017, 92 p., ISBN 978-3659806841. 10. Edwards, P. Tank steering systems (differentials, the theory and practice). Constructor Quarterly, September 1988, (1), pp. 47–48. 11. Esmail, E.L. and S.S. Hassan. An approach to power-flow and static force analysis in multiinput multi-output epicyclic-type transmission trains. ASME J. of Mechanical Design, 2010, 132 (1), 011099-011099-9. 12. Förster, H.J. Zur Berechnung des Wirkungsgrades von Planetengetrieben. Konstruktion, 1969, 21 (5), S. 165–178. 13. Giger, U. and K. Arnaudov. Redesign of a gearbox for 5MW wind turbines. Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE, Washington, DC [USA], 2011 August 28–31, 2011, DETC2011-47492. 14. Giger, U. and K. Arnaudov. New drive train design for ultra large 15 MW wind turbines. VDIBerichte 2199-1, 2013, pp. 101–112. ISSN 0083-5560. 15. Hardy, H.W. Planetary gearing: Design and efficiency. Andesite Press, 2017, 66 p. ISBN 978-13759190. 16. Henriot, G. Gear and planetary gear trains. Brevini, 1993. 17. Jelaska, D. Gears and gear drives. Chichester [UK]: John Wiley & Sons, 2012, 444 p. ISBN 9781119941309. 18. Kahraman, A., H. Ligata, K. Kienzle, and D.M. Zini. A kinematics and power flow analysis methodology for automatic transmission planetary gear trains. ASME J. of Mechanical Design, 2004, 126, pp. 1071–1081. 19. Karaivanov, D. Theoretical and experimental studies of influence of the structure of coupled two-carrier planetary gear trains on its basic parameters. Dissertation. Sofia: University of Chemical Technology and Metallurgy, 2000. (in Bulgarian). 20. Karaivanov, D. Structural analysis of the coupled planetary gears with considering the efficiency of the coupling gears. Proceedings of the 2th Int. Conf. on Manufacturing Engineering (ICMEN), Kallithea of Chalkidiki [Greece], October 5–7, 2005, pp. 381–387. ISBN 960-243-615-8.

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21. Karaivanov, D. and S. Troha. Examining the possibilities for using coupled two-carrier planetary gears in two-speed mechanical transmissions. Machinebuilding and Electrical Engineering, 2006, (5–6), pp. 124–127. ISSN 0025-455X. 22. Karaivanov, D. Structural analysis of compound planetary gear trains. Balkan Journal of Mechanical Transmissions, 2011, 1 (1), pp. 33–45. ISSN 2069-5497. 23. Karaivanov, D. et al. Analysis of complex planetary change-gears through the torque method. Machines Technologies Materials, 2016, 10 (6), pp. 38–42. ISSN 1313-0226. 24. Karaivanov, D., M. Velyanova and V. Bakov. Kinematic and power analysis of multi-stage planetary gearboxes through the torque method. Machines, Technologies, Materials, 2018, 12 (3), pp. 102–108. ISSN Print 1313-0226, ISSN WEB 1314-507X. 25. Kreynes, M.A. Efficiency and speed ratio of a gear mechanism. Proceedings of the Seminar on Theory of Machines and Mechanisms. Vol. 1. Moscow: Acad. of Sciences of USSR, 1941, pp. 21–48. (in Russian). 26. Kudryavtsev, V. and Y. Kirdyashev. Planetary trains. Handbook. Leningrad: Mashinostroenie, 1977. 27. Kurth, F. Efficiency determination and synthesis of complex-compound planetary gear transmissions. Dissertation. Munich: Technical University of Munich, 2012. 28. Kutzbach, K. Mehrgliedrige Radgetriebe und ihre Gesetze. Maschinenbau, 1927, Nr. 22, S. 1080. 29. Leistner, F., G. Lörsch und O. Wilhelm. Umlaufrädergetriebe. 3. Auflage. Berlin: VEB Verlag Technik, 1987. 30. Looman, J. Zahnradgetriebe – Grundlagen, Konstruktion, Anwendung in Fahrzeugen. 3. Auflage. Berlin: Springer-Verlag, 1996. 31. Müller, H.W. Epicyclic drive trains. Detroit: Wayne State University Press, 1982. ISBN 0-8143-1663-8. 32. Niemann, G. and H. Winter. Maschinenelemente. Band 2. Zahnradgetriebe – Grundlagen, Stirnradgetriebe. 2. Auflage. Berlin: Springer-Verlag, 1995. 33. Rajasri, I. Synthesis and analysis of epicyclic gear trains: Graph theory. Beau Basin [Mauritius]: LAP LAMBERT Academic Press, 2016, 252 p. ISBN 978-3659855405. 34. Seeliger, K. Das einfache Planetengetriebe. Antriebstechnik, 1964, S. 216–221. 35. Stefanović-Marinović, J. and M. Milovančević. The optimization possibilities at the planetary gear trains. Journal of Mechanics Engineering and Automation (David Publishing). 2012, 2 (6), pp. 365–373. ISSN 2159-5275 (print) 2159-5283 (online). 36. Stefanović-Marinović, J., S. Troha and M. Milovančević. An application of multicriteria optimization to the two-carrier two-speed planetary gear trains. FACTA Universitatis, Series: Mechanical Engineering, 2017, 15 (1), pp. 85–95. ISSN 0354-2025 (print) 2335-0164 (online). 37. Tenberge, P., D. Kupka and T. Panero. Influential criteria on the optimization of a gearbox, with application to an automatic transmission. VDI-Berichte 2294.1, 2017, pp. 253–265. ISSN 0083-5580. 38. Tkachenko, V.A. Design of multi-planet planetary gear trains. Kharkov: Kharkov National University Publ., 1961. (in Russian). 39. Troha, S., P. Petrov and D. Karaivanov. Regarding the optimization of coupled two-carrier planetary gears with two coupled and four external shafts. Machinebuilding and Electrical Engineering, 2009, 58 (1), pp. 49–55. ISSN 0025-455X. 40. Troha, S. Analysis of a planetary change gear train’s variants. Dissertation. [Croatia]: Engineering Faculty, University of Rijeka, 2011. (in Croatian). 41. Troha, S., R. Žigulić and D. Karaivanov. Kinematic operating modes of two-speed two-carrier planetary gear trains with four external shafts. Transactions of FAMENA, 2014, 38 (1), pp. 63–76. ISSN 1333-1124. 42. VDI-Richtlinie 2157 Planetengetriebe – Begriffe, Symbole, Berechnungsgrundlagen, Entwurf. 2010. 43. Volkov, D. and A. Kraynev. Transmissions of construction and road machinery. Moscow: Mashinostroenie, 1974. (in Russian). 44. Willis, R. Principles of mechanism. London: John W. Parker, 1841. 45. Wolf, A. Die Grundgesetze der Umlaufgetriebe. Braunschweig: Friedr. Vieweg und Sohn, 1958.

9 Powder Metal Gear Technology Anders Flodin CONTENTS 9.1 General Introduction.......................................................................................................... 330 9.1.1 Material.................................................................................................................... 330 9.1.2 Selection of Materials for Gears............................................................................ 330 9.1.3 Lubrication of Materials.........................................................................................334 9.1.4 Alloying Concepts.................................................................................................. 336 9.1.4.1 Mixing....................................................................................................... 336 9.1.4.2 Pre-Alloyed............................................................................................... 336 9.1.4.3 Organically Bonded Alloys.................................................................... 337 9.1.4.4 Diffusion Alloyed.................................................................................... 338 9.2 Manufacturing.................................................................................................................... 338 9.2.1 Compaction.............................................................................................................. 339 9.2.2 Tooling...................................................................................................................... 341 9.2.3 Heat Treatment........................................................................................................343 9.2.3.1 Sintering....................................................................................................344 9.2.3.2 Case Hardening.......................................................................................344 9.2.3.3 Induction Hardening...............................................................................345 9.2.3.4 Nitriding....................................................................................................346 9.2.4 Performance Boosting Processes..........................................................................346 9.2.4.1 Rolling Densification...............................................................................348 9.2.4.2 Forging....................................................................................................... 349 9.2.4.3 Peening...................................................................................................... 349 9.2.5 Hard Finishing........................................................................................................ 349 9.2.6 Different Process Paths to Make Gears............................................................... 350 9.2.7 Tolerances................................................................................................................. 351 9.3 Design for Powder Metal................................................................................................... 353 9.3.1 Light Weight Design............................................................................................... 353 9.3.2 Root Optimization.................................................................................................. 354 9.3.3 Asymmetric Design................................................................................................ 355 9.3.4 Microgeometry........................................................................................................ 355 9.4 Performance......................................................................................................................... 355 9.4.1 Allowable Tooth Root Bending Stress................................................................. 356 9.4.2 Allowable Contact Stress....................................................................................... 358 9.4.3 Impact Toughness................................................................................................... 359 9.4.4 Calculation Methods.............................................................................................. 359 9.4.5 Noise, Vibration, Harshness.................................................................................. 359

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9.5 AM Gears............................................................................................................................. 360 9.5.1 Introduction............................................................................................................. 360 9.5.2 Powder Steel for AM Gears................................................................................... 360 9.5.3 Manufacturing of AM Gears................................................................................. 360 9.5.4 Strength of Case-Hardened AM Steel Gears...................................................... 361 References...................................................................................................................................... 361

9.1  General Introduction Powder metal (PM) is commonly used for manufacturing different types of gears, sprockets, and pump gears as well as a multitude of other components. In order for the technology to be cost competitive, there are a few general guidelines that apply to powder metal parts in general, but also for powder metal gears. 1. A certain amount of geometrical complexity or tight tolerance settings so that machining becomes costly and stamping is not possible. 2. A certain number of parts have to be made to cover the tooling costs. Depending on the part and tool complexity, breaking even can be somewhere between 10,000 and 100,000 parts per year. 3. A certain durability or ultimate strength is required for the part to rule out plastics. So those criteria roughly set the boundaries for when compacted and sintered powder metal parts can be the better alternative compared to machined gears and parts. Bear in mind that powder metal is also used in many other types of applications such as synchronizer rings and hubs, gear pumps, the list goes on. In this chapter, the focus will be on compacted and sintered gears with optional post processes to improve durability and tolerances. Three-dimensional printed gears also use powder steel in a sintered condition and will be covered at the end of the chapter. 9.1.1 Material Thousands of different powder metal mixes and qualities exist. Figure 9.1 shows what the powder looks like when entering into the gear production process. Maybe the most important criteria for selecting a material for gears is that there is reliable fatigue data from gear testing available for that particular material and a well-known process that gave the material its fatigue properties, so that the fatigue performance is also valid for other PM gears. The material and the process for making the gears goes hand in hand as exemplified in Section 9.1.2. Figure 9.2 shows different types of powder particles that will naturally function differently in the part-making process and also give parts different properties. 9.1.2  Selection of Materials for Gears When it comes to the selection of material for a particular gear, it depends on the process path for the gear. The process path in its turn is set by the strength requirements that have been calculated in the design phase. Since the selection process starts with the output from the stress analysis, relevant data has to be available since it is relatively easy to calculate

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FIGURE 9.1 Powder metal.

FIGURE 9.2 Different types of powder particles. Left: gas atomized. Center: sponge process. Right: water atomized covered with graphite particles for alloying.

stress using gear software; however, to calculate a safety factor, the fatigue behavior of the material and process has to be known to ensure proper working of the gears. Since material cost is a factor that the designer or purchaser wants to keep as low as possible, it makes no sense to select a highly alloyed material for a low performance gear. Also, certain materials require specific furnaces and process equipment that is not available at every parts manufacturer. So, that also has to be kept in mind in the selection process.

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FIGURE 9.3 S-n diagram for tooth root bending fatigue for case hardened Astaloy 85 Mo 0.25%C.

EXAMPLE 9.1 5th drive gear in a manual transmission for a car. The drive cycle can be recalculated to stress at max torque for 5 million cycles. The contact stress for this torque is 1400 MPa. The bending stress is 560 MPa.

The bending fatigue S-n diagram can now tell which process is needed, since a safety factor against pitting and tooth root breakage has to be met; see Figure 9.3, where the diamond is below the runout level for bending, indicating a safe area to operate in. The same principle applies for pitting. The S-n diagram is different depending on the material and process route that has been used and can be provided by the supplier of the parts, the material manufacturer, or by certain gear calculation software. In this example, a process route that can sustain the loads will consist of:

1. Compaction 2. Sintering 3. Case carburizing 4. Grinding

The density should be 7.2 g/cc or higher. This means that a material that can be compacted to 7.2 g/cc and that can respond to a case carburizing and temper process has to be used. Grinding is there to meet automotive tolerance levels and setting the microgeometry. Lead modifications can’t be done in compaction and has to be ground, honed, shaved, or rolled into the gear flank.

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A typical material that will fulfill the requirements would be alloyed with 1.5% Molybdenum (Mo) and 0.25% Graphite (C), the remainder is iron (Fe). There are other alternatives but they might require different process equipment such as low-pressure furnaces. Another option could be a material alloyed with 1.8% Chromium (Cr) and 0.25% C. The 1.8% Cr material has certain advantages over a 1.5% Mo based material; apart from cost, it is more dimensionally stable in the thermal processes and it is better suited for densification process (rolling) that will be discussed later. However, Cr is prone to oxidation and requires a very well-controlled atmosphere during sintering and case carburizing, meaning that suitable processing equipment has to be in place. Another material for smaller modulus m 7.5 g/cc), the PM steel behaves very much like a regular steel during the heat treatment, but comes with the benefits of lower demands on process control and the material still being isotropic, unlike forged steels that often display a memory effect from previous process steps. Chapter 13 in the Steel Heat Treatment Handbook [3] has a good summary of powder metal heat treatment methods and their differences to the solid steel methods for more in-depth reading. 9.2.4  Performance Boosting Processes There are several ways of increasing the mechanical performance of powder metal gears. Heat treatment is one method covered in the previous paragraphs, and in the next coming paragraphs, methods of increasing density and removing porosity will be discussed. Common for them all is that the size and the number of pores will be reduced or eliminated. Pores act as stress concentration points from which cracks can grow and fail the gear in tooth root or surface fatigue-type failures. The smaller the pores, or the fewer they are, the better from a stress concentration or probability perspective. Single compaction and single sintering of gears will normally reach density levels of 7.1–7.25 g/cc depending on material, temperature, and lubricant. The first method to increase performance is to increase temperature in the sintering furnace. Normal

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sintering temperature is 1120°C but in the PM industry higher temperatures are used, up to 1280°C before the temperature is so high that the normal furnace materials have to be replaced with some more exotic and expensive materials and the melting point of the material itself is approaching (Fe 1560°C) and distortions increase. The higher temperature will make the pores rounder and slightly smaller which is beneficial for strength. Not all materials will behave like this but some PM gear materials will. The drawback is dimensional control that becomes more difficult, but if grinding or honing is used anyway as the final dimensional setting process it does not matter since distortions will be within the protuberance on the flanks. More information on the sintering process and the chemistry involved can be obtained from the Höganäs Sintering Handbook [5]. The next step-up would be to compact the gear twice with a short low temperature sintering (brown sintering) between compactions. The gear is then sintered under normal conditions. What this does is to increase the density to 7.4–7.5 g/cc. As mentioned previously, the porosity at these densities will not be interconnected anymore and heat treatment processes must be adjusted so they are similar or same as solid steel processes. The drawback is of course the extra cost for repressing and brown sintering, but this method is not uncommon for powder metal parts that needs to perform at solid steel density levels. Double compaction is discussed more further on. A different method is to selectively plastically deform the surface of a PM part, thereby removing the porosity, or at least drastically reducing the influence from it. There are different methods of doing this. The most common one is roll densification but there are others such as Densiform or Densgrad, which are trade names from two different providers of PM parts. Figure 9.17 shows a method for selectively densifying a gear bore. Ball bearings are pushed through in an automated press giving a fully dense layer of steel onto which, for instance, a needle bearing can be run.

FIGURE 9.17 Two variants of bore densification using ball bearings. The resulting surface will have improved wear and surface fatigue properties by removing surface porosity.

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FIGURE 9.18 Gear rolling for surface densification of gears.

9.2.4.1  Rolling Densification In this chapter, only the rolling densification will be discussed as a general and widely used technology by many powder metal parts suppliers. In the process chain, the rolling takes place between the sintering and the heat treatment when the gear is in a relatively soft state. Once hardened, it will be very difficult to introduce the required plastic deformation without breaking tools or parts, or both. In some cases, a grinding or power honing operation is used after heat treatment to reach ISO 6 tolerances for higher end transmissions such as in cars. The hardware required is a gear burnishing machine that is rigid enough and sufficiently controllable to create the sufficient force without machine deflection (see Figure 9.18). The most important know how for this process to be successful is the ability to design the tools and the gear preform. The gears have to have a modified involute that, after the repeated contact with the tool, takes the final shape optimized for the heat treatment distortions and the stress reliving that takes place at the elevated temperatures the gears will be subjected to during heat treatment. Normally it will take 2–3 iterations with small changes to the tools and the gears before a satisfactory result is obtained. A gear/tool grinder, and gear CMM is necessary to develop the process, together with a gear-burnishing machine. The result is a gear with similar performance characteristics as a solid steel gear with a tolerance class on par with a shaved gear. This process has been in use for many years and millions of cars are running with timing gears in their engines with this technology; also, recently, passenger car transmissions have been equipped with gears using this process. The best materials for this process are chromium alloyed powder metals such as Astaloy CrA followed by Astaloy CrM and Astaloy Mo. These materials exhibit the best plastic deformation properties under repetitive yield in compression. The carbon content should be kept low 0.15–0.20%C, since carbon reduces the ductility in the sintered state. Many gears are rolled using Astaloy 85 Mo, which is a leaner version of Astaloy Mo but in order to reach proper heat treatment response, higher graphite levels have to be used (0.3%C). This combination of graphite level and material alloying constituents is a tradeoff where hardenability and mechanical performance is sacrificed for a cheaper material. Figure 9.19 shows a cross section of a rolled gear where the porosity is reduced or eliminated in the near surface region.

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FIGURE 9.19 Cross sections of different parts of the gear. Left: whole tooth. Center: part of flank. Right: root.

9.2.4.2 Forging Powder forging of gears is a process where the gear is taken out of the sintering furnace at around 1100°C and put in a tool die where it is compacted one more time. This will eliminate any porosity in the gear and boost the mechanical properties above those of regular solid gear steels such as 20MnCr5. There are limitations as to the helix angle and tooth face width, but the process is used on industrial scale for transmission gears in cars as well as for differential gears and final drive gears. The teeth may be performed in the first press cycle or completely formed in the hot forging compaction step. The gears will need additional machining of the flanks to reach tolerances, normally a hard finishing process. 9.2.4.3 Peening Shot peening on sintered gears will have similar effects as it has on solid steel gears. The shot peening is done after heat treatment in order to increase the compressive stresses in the material. The benefit in bending is a 15–25% improvement of the runout level for a 7.2 g/cc material. Also, laser peening may be used. Laser peening has the advantage of creating a dense layer when applied in the sintered condition of the gear. This will improve fatigue properties without creating a detrimental surface roughness, which is the case when a sintered nonhardened surface is shot peened using steel or silica shots. The drawbacks of laser peening may still be the manual labor involved in preparing the gears as well as production speed. But the technology is relatively new and develops fast. 9.2.5  Hard Finishing Grinding and honing are two common methods to correct or improve the gear geometry after heat treatment. Care has to be taken to avoid grinding burns and compressive stress relief. Powder metal works well in both these processes, may it be profile grinding, threaded wheel grinding, or power honing. The thermal conductivity for a 7.2 g/cc powder metal steel is lower than for solid steels. So, this would point toward less sensitivity against grinding burns. Tool wear can be another concern but has not been noted at the gear manufacturer that has been used by the author. It has not been properly researched but the wear has been under observation for a few thousand gears without any anomalies in the wear of the grinding tools. The tolerance class achieved is not influenced by the material, only by the grinding process, so powder metal steel gears and solid steel gears will have the same accuracy after hard finishing in the same machine.

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9.2.6  Different Process Paths to Make Gears There are several different process paths to make powder metal gears. Combining the different process paths will give different durability, tolerance class, and cost. Generally, more processes are used to increase strength and tolerances, which also adds to the total manufacturing cost. The shortest methods have been discussed under the material heading. Here, only the sintered high-end gears suitable for car transmissions and the likes will be discussed and not every aspect will be covered since the combinations of processes can lead to a very high amount of process paths that will all produce a high-end gear. Often, it is the equipment at the manufacturer that sets the production steps involved. When making powder metal gears, single compaction is the most common process but double compaction is also an alternative method common in the industry. Double compaction means that the gear is compacted once, taken out of the die, and sintered, but only to what is known as a “brown” state. Brown state means sintering at around 800°C for around 15 minutes to get some stress relief but no carbon solution. The gear needs to be somewhat soft for the second press step. In the second press step, a different tool is used since there is normally some swelling of the part in the brown sintering step. The second compaction leads to an increase in density from, as an example 7.20–7.45 g/cc. Final tolerances are also improved, which is important for internal ring gears where hard finishing operations are uncommon but of course also for gears in general. The double compaction increases cost but it also increases performance relative to its density increase. It may also save money further down in the process chain due to a shorter cycle time in the grinding/honing machine and less scrapping due to tolerance problems. The sintering, irrespective of single or double compaction, may be a normal temperature (1120°C) or a high temperature (1280°C). Higher temperatures normally lead to better strength due to pore rounding. All pores, may it be in solid steel or powder metal steel, act as stress concentrations. The rounder the pores are, the less of a stress concentration they will be—that is the mechanism. For certain materials, the density is also slightly increased, meaning the pores will not only be rounder, but smaller in volume. The higher temperature may lead to higher costs for the sintering. The case carburizing process is favored by the low-pressure process or vacuum process. During the gas quenching, the temperature can be controlled much better as compared to oil quenching. A step quench process is introduced where the temperature is held constant to let the transformation take place. When carburizing using pulses of Methane  gas  into  the  chamber, the penetration depth can be controlled and over carburization causing brittle carbide networks can be avoided. The powder steel is isotropic and shows very little to no distortion during case carburizing, may it be a conventional Carburize, Quench, and Temper (CQT) process or a low-pressure process (LPC). The distortion comes in the sintering step where temperature is significantly higher than for case carburizing. A very attractive process combines the Sintering and LPC process in the same furnace. This has the advantage of not cooling down the parts to room temperature but instead cooling them to case carburizing temperature during boost-diffusion cycles, and then rapidly cooling the parts before tempering. This not only saves time but also total cost compared to dividing the process into one sintering step using a sintering furnace and then heating the parts up again for the CQT process in another furnace.

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The process would then consist of three steps: 1. Compaction 2. Combined sinter and case carburizing 3. Hard finishing This would give a gear that has 75% of the durability, in bending and pitting, of a casehardened and hard-finished gear made from 20MnCr5. If higher performance is needed, some of the techniques such as densification or double compaction has to be used. There is also the possibility of compacting a smart gear body without teeth but with features like splines, weight reduction holes, part number, and so on, and then machine the teeth when the same body is used for different variants of the same gear with different modulus and number of teeth. This creates greater flexibility since once the compaction die has a certain number of teeth it cannot be changed. 9.2.7 Tolerances The tolerances achievable using PM gears depends on the tooling and the processes. Since there has to be radial clearance between the different punches and rods, the more punches and rods, the wider the tolerance will have to be, since the worst-case scenario is that all the clearances between the tool parts adds up radially. This means that the core rod, that forms the bore of the gear, will move radially equal to the clearance times the number of punches and rod. Most presses cannot handle more than three upper and lower punches and one core rod meaning the runout of the bore is limited to four times the clearance. Typical clearance is 15 microns, which means that max runout will be around 60–65 microns. Less tool parts means less runout. The error induced by the radial movement from the outer tool parts (punches) forming the teeth will be less affected since the biggest radial movement of the outer punch is confined to the clearance between die and outer punch. The 60–65 microns in the example is only for the core rod. The other tool parts will exhibit less movement and therefore induce less errors to the geometry. This is a simplified explanation and there are more factors that influence the errors in tolerances that is generated in the compaction step. The runout can be effectively improved in the grinding process and be brought back to normal runout tolerances. This is especially important to consider when the bore has a spline or other geometric feature that has been created in the compaction and machining should be avoided. If the bore is just circular, it can be hard turned just like it would be for a hobbed gear before going into the grinding machine. In order for the grinding machine to be able to improve runout, flank deviations, spacing errors, and so on, flank protuberance is introduced. For typical automotive gears, the protuberance is 0.1 mm and this is also a suitable number for PM gears. The protuberance is built into the compaction tool and any PM gears that will be ground further down the process chain will have the protuberance directly from compaction. See Figure 9.20 that demonstrates the runout during the different process steps. In Figure 9.20, there are two measurements made on different coordinate measurement machines after the grinding. The gear that is measured in Figure 9.20 is depicted in Figure 9.21. It is a complicated gear and it is made with all the features such as holes, bore spline, helical gear teeth and different hub, rim and web thickness, in one compaction stroke.

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FIGURE 9.20 Runout during the process steps. The ground gear is measured on two different machines to compare results.

FIGURE 9.21 Net shape compacted PM gear made with three upper and three lower punches plus splined core rod.

The gear in Figures 9.20 and 9.21 is made according to the above described process using three upper and three lower punches in compaction and is, after that, sintered and case carburized in a combined LPC furnace and finally ground to tolerances without touching the spline. For a more in-depth examination of the compaction of this particular gear, see [6]. A more complete tolerance investigation where the tolerances of an automotive PM gear is investigated as it is subjected to the different manufacturing steps can be found in [7]. Typically, the flanks are ISO 6–7 after compaction but deteriorates in the sintering process to ISO 8–9. As mentioned previously, the following heat treatment, such as case carburizing, does not change the tolerance class very much and the gear flanks are still within ISO 8–9. The spacing error of a PM gear is usually very good. Normally ISO 5–6 and does not change very much during sintering and case hardening. The compaction die itself should be ISO 3–4, borderline to what is accurately measurable.

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SUMMARY Manufacturing of powder metal gears is quite different from conventional gear cutting. To a much larger extent, the performance of the gears will be connected to the processing. The density is the main focus for improving durability and this can be accomplished in a few different ways. Heat treatment requires no special hardware but different process settings and the process itself is more sensitive to time and temperature variations. A heat treatment process optimized for solid steel will work poorly with powder metal unless the density is 7.5 g/cc or higher. The first process to look at if there is a durability problem with a PM gear is the heat treatment. The material and compaction process is very stable just like the surface densification process or any machining processes but furnace processes are atmosphere dependent and the porosity makes the material susceptible to variations or perturbations in the atmosphere. The advantages lie in fewer process steps, less capital investment, and a larger freedom to optimize the gear body without wasting material or machining time, which, most of the time, is boiled down to cost.

9.3  Design for Powder Metal Due to the manufacturing method of powder metal gears and the difference in material characteristics compared to solid steel, some design modifications are necessary and some are optional. This is not a drawback, it is actually an advantage since lower weight, reduced noise, and less material can be used if thought of in the design phase. The mandatory design change is generated by the higher deflection in sinter gears. Simply copying a steel gear can lead to poor contact conditions or sub optimized transmission error in a PM gear. 9.3.1  Light Weight Design There are opportunities to reduce the mass in a PM gear. Opposite to conventional gear manufacturing where machining off steel in order to reduce the weight of gear is a cost, in powder metal gear manufacturing, material and cost is saved by optimizing the gear body weight. This is because in the PM gear manufacturing process, there is very little material waste. The wasted material is what is dusted off the die in the filling process, which is about 1%–2% and this can sometimes be reintroduced in the process or used in parts where quality demands are lower. As previously discussed, the shortest and most cost-effective path to 75% of the durability of case-hardened solid steel gear is a 7.2 g/cc single compaction case-hardened PM gear. The 7.2 g/cc density gives a weight reduction of 8% without changing anything else from the original solid steel gear design. This means that, within limits, the density can be tailored to meet a certain required durability and at the same time save weight and material cost. For most gears in car transmissions, the durability requirements are at 75% of the solid steel material strength or higher but there are gears that require significantly lower durability levels. Normally, these gears can be found in low power automatic transmissions and some of the CVT variants as well as in balancer shaft drive gears and engine timing gears. Lower weight and also the reduced Noise Vibration Harshness (NVH) these lower density gears can offer are benefits besides the lower inertia and cost.

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FIGURE 9.22 Left: original design. Right: PM weight reduced.

Since the manufacturing process is centered around the axial compaction, any feature that can save weight without compromising structural integrity of the gear or tool life can be introduced. This typically comprise a thinner web section of the gear or holes in the web or grooves that can be introduced by the punches. Typical weight and inertiasaving numbers for a manual transmission gear is 8–20%, depending on load and size of gear. Smaller gears, sub 50 mm in diameter, are typically difficult to tool up with three separate punches and a core rod since one of the punches typically becomes very thin, less than 3 mm, which is not good for tool life when compacting at 700 MPa for maximum density. Figure 9.22 shows how a gear in the Smart Fortwo AMT transmission was optimized for powder metal. The weight and inertia reduction was 25%. The gear in Figure 9.22 is an extreme example and, in reality, for the application under full torque, the tooth deflection is increased by 0.15 mm, which is too much but it serves as an illustration of what is possible in terms of gear body optimization. 9.3.2  Root Optimization When hobbing gears, the root is generated by the movement of the hob relative to the gear and the geometry of the hob tip. Tip wear of the hob has to be considered since tool life is important for the cost of the gear. Sending the hob for refurbishment and stopping production to change tools is nonproductive in time and cost. Gears in manual transmissions for cars are often designed as quite slender with a lower pressure angle (16–19°) since that gives quieter mesh engagement. However, the hobs wear faster and, therefore, short pitch hobs are used [8] giving a less favorable root geometry. The purpose of the root optimization is then to reduce the bending stress in the root but the drawback is that it becomes impossible to hob such a root unless a profile cutter is used, which hampers productivity in conventional gear manufacturing. Since the teeth of a PM gear is not generated by a hob or rack, but made from a die, the root shape can be optimized or at least improved. True optimization of the root requires higher level FEM knowledge using an iterative approach. The technique has been developed and explained by Kapelevich [9] and coworkers. An optimized root is slow to cut using conventional gear machining methods and is rarely used in mass production. In powder metal gear manufacturing, there is no penalty in cost or production speed when optimizing the root, only benefits in terms of higher safety factors against tooth root breakage. Some commercial gear software has the ability to create an elliptic root and thereby reduce the bending stress. The elliptic root is not true optimization but it can still reduce the stress significantly, and

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TABLE 9.2 Relative Stress Reduction Using Different Root Shapes Root Form Original Spline Elliptic

1st Input

1st Output

4th Input

4th Output

6th Input

6th Output

1 1 0.92

1 1 0.96

1 0.96 0.82

1 0.94 0.76

1 0.93 0.83

1 0.93 0.70

Maximum stress reduction is 30% as compared to a full root radius.

typical numbers vary from 5–20% reduction. With a proper optimization using FEM, a slightly lower but, more importantly, more accurate stress number may be calculated. Table 9.2 shows the result from a few different root modifications in a six-speed manual transmission. In Table 9.2, original design means the maximum constant radius possible. Spline means that a spline function has been used to straighten out the radius at the point of max stress and elliptic means that suitable major and minor axis for the elliptic function has been used in the root design. The elliptic and spline function parameters have been chosen by hand under iterations. There is no true optimization in Table 9.2, which would likely give even greater benefits. 9.3.3  Asymmetric Design Since the shape of the PM gear teeth is formed by a mold, asymmetric gears are simple to make with less limitations to the pressure angles that may be used. Also, grinding protuberance can be added, which is sometimes a problem in asymmetric designs when hobbing the gear teeth since too high/low pressure angles leads to interference between hob and protuberance on the low-pressure side of the adjacent flank in the gap being hobbed. Also, root optimization can be added to the asymmetric design. Kapelevich [10] has explained this in more detail. 9.3.4 Microgeometry Due to the lower density, Young’s modulus and Poisson’s number is lower for powder steel compared to solid steel. This means that tooth deflection and also the contact patch will be slightly bigger. The gears are more compliant, which might help to explain why PM gears are often perceived as quiet. However, if the micro design is copied from a solid steel gear and put onto a gear with lower Young’s modulus then care has to be taken to ensure that the meshing of the gears is still good. In many cases, the gears will still operate satisfactory but just as often they will not with premature failure or NVH problems as a consequence of the lack of mesh analysis. The optimized powder metal microgeometry parameters (i.e., Fα, Fβ, etc., according to DIN 3960 and DIN 3961) will be close to the optimal microgeometry parameters for solid steel and, depending on the  sensitivity of the chosen design, the difference can be more or less influential (see Henser [13]).

9.4 Performance The strength and durability of PM gears is set by a number of factors and since the normal density span varies between 6.9 and 7.8 g/cc, it is not possible to have every

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density, material, and heat treatment tested out. There has to be some assumptions as to how the S-n curve moves, as a function of density, alloys, and root radius, between two different known S-n curves. When studying the fatigue data for case-hardened gears, the actual hardness curve, as well as core hardness, plays an important role. The material composition is set to generate a certain heat treatment response and the alloying elements are better or worse at doing this at a given cost. However, different alloys also react with oxygen creating different types of oxides that are detrimental to the diffusion bonding of particles, achievable density, and tool wear. The oxidation is more pronounced in powder metals than in solid steels. For this reason, Manganese and Chromium is only used by those PM part manufacturers who have good process control and the proper equipment to handle material with these alloys, but when that is in place, the alloys perform very well. If performance is regarded as tooth root bending fatigue, pitting, and impact toughness then a rule of thumb for the process limited to compaction, sinter, CQT, and grinding/ honing renders 75% of the performance of 16MnCr5 gears in bending and pitting but only 50% of impact toughness. In the S-n graphs in Figures 9.23 and 9.24, hardness curves according to Figure 9.16 have been in place for the tested gears. This is important since the heat treatment result has a profound impact on the durability of the gears. 9.4.1  Allowable Tooth Root Bending Stress The tooth root bending data in Figure 9.23 is generated on spur gears in a pulsator rig and is a much faster process compared to generating the pitting data. The S-n curve can be used for calculating the safety factor for a gear provided that the specified material and density provided in Figure 9.23 is met and the case carburizing process generated hardness curves similar to Figure 9.16.

FIGURE 9.23 S-n curve for 7.2 g/cc PM gear in bending. Material is Astaloy 85 Mo 0.25%C. Failure probability 1% and 50%.

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FIGURE 9.24 Pitting fatigue curves for different failure probabilities using Astaloy 85 Mo 0.25%C case hardened and tempered. Curves based on more than 40 data points.

Tooth root breakage from repeated loading is also density dependent just like pitting. The fracture path is the same as for solid steel and the macro observation is that the fracture surface looks the same as for a solid steel gear. In a sweep electron microscope, there will be differences due to the porosity with the crack propagating both through particles and between particles. Ideally, the fracture should never be through the sintering necks, meaning between particles, since that would indicate poor diffusion in the sintering process, which could be caused by oxides, which was discussed earlier. In reality, it will be a mix of the two paths (see Figure 9.25).

FIGURE 9.25 Typical crack path after bending an FZG type-C pinon.

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9.4.2  Allowable Contact Stress The pitting phenomena has been extensively researched in solid steels and many models and hypothesis exist as to how the cracks grow under the influence of oil, sliding direction, friction, asperities, and so on. From extensive testing in back-to-back gear testers at Höganäs AB, while generating S-n curves for pitting of PM materials, it has been noted that the pits look the same in a light optical microscope for both solid steels and PM steels. There seems to be no apparent change in crack propagation directions even though there is probably a difference if studied on the microlevel since the pores are abundant and a crack can grow not only through powder steel particles but also between the particles through the sintering necks. The general shape of the pit and how it progresses from initiation through crack growth until a spall is removed from the surface still gives the surface the look of a traditionally pitted surface (see Figure 9.26). Also, the spread in the data is on the same order as for solid steels, with the exception of outliers that for solid steels are more common due to the forging process creating few long and narrow inclusions instead of many spherical inclusions that can be found in PM steels. Depending on the density, the pitting resistance will change. Higher density gives better pitting resistance, and higher surface hardness also improves pitting resistance. Hardness also changes the mechanism, both for PM-steel and solid steel. Given that the hardness profiles of the surface is correct, that the compressive residual stresses are in place, the pitting resistance will increase with density and will match that of solid steel as the density approaches solid steel density. The contact fatigue S-n curves presented in Figure 9.24 are tested out on an FZG backto-back gear tester for pitting using off-tool compacted FZG type-C gears. The number of data points is more than 40 and it takes more than 12 months of testing to generate one curve. The staircase method is used to hone in on the run-out level. The data is valid for the specified material, density, and heat treatment.

FIGURE 9.26 Pitted surfaces of PM gears in an FZG test. Load stage 7.5. All failed at 31–32 million cycles.

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9.4.3  Impact Toughness Impact toughness varies nonlinearly with the density. Surface densification or shot peening will not improve the toughness very much. For pitting and tooth root bending fatigue resistance, surface densification will make a big difference but not so much for impact toughness. The best method for improving the impact toughness is higher base density by double compaction, powder forging, or cold or hot isostatic pressing. For a transmissions gear in a manual car transmission, it can be argued if there really are impacts during abusive driving. There are a lot of mechanical elements between the friction in tire-asphalt contact and the crankshaft. Most of these machine elements will act as springs and prohibit the stress amplitude from reaching double the nominal stress. In manual transmissions, the synchronizer hubs are made exclusively from powder metal and they are, of course, also subjected to any stress waves from impact. But for other transmissions with more direct energy paths from impulse to gear, the impact toughness has to be considered. Tooth root optimization will, on a theoretical level, have a measurable positive influence on the stress wave during impact (see [8,11]). 9.4.4  Calculation Methods When calculating stress, deformation, strain, and so on, using powder metal, there are only two parameters that change in the equations as long as the gears are operating in the elastic regime: Young’s modulus and Poisson’s number. There are no special powder metal equations necessary, ISO 6336 and FEM can be used. The governing equations for Young’s Modulus and Poisson’s number as a function of density can be found below in Equations 9.2 and 9.3.  ρ 3.4 E = E0    ρ0 





(9.2)

 ρ 0.16 ν =   (1 + ν 0 ) − 1  ρ0  (9.3)

For all normal gear calculations, the material is linear elastic. In the plastic regime, which happens during roll-densification, there are some highly specialized material models that are applicable such as the Ponte Castaneda model. Angelopoulos [12] has made a good overview of models. 9.4.5  Noise, Vibration, Harshness Powder metal gears have an inherent damping characteristic that is dependent on density, carbon content, and structure [13]. The damping is clearly audible by tapping a gear with a finger or small mallet; however, the microgeometry of the gears has to be adopted to the lower Young’s modulus to avoid premature contact in the root of the pinion at the start of engagement. Powder metal may have attractive acoustic properties but it will not remedy a design that is optimized for solid steel with flank reliefs not designed for PM tooth deflections. There is also the possibility to design the density across the web and create density gradients that will dampen the excitation from the tooth engagement as it travels to shaft, bearings, and housing. Henser [13] has investigated this as well as [14–16].

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SUMMARY Powder metal gears can be tailored to the strength equivalent of solid steel gear steels. It is a matter of processing the gears to higher density toward full density. In many cases, the stresses can be reduced by designing the gear differently in the root or on the flanks. Root optimization does not only reduce the root stress, it also enables making gears with higher contact ratio, sacrificing some of the reduced root stress by introducing more teeth with a smaller modulus. Normal calculation standards may be used but with a reduced Young’s modulus and Poisson’s number according to Equations 9.2 and 9.3. Failure modes for case-hardened PM gears are similar to steel gears—the crack propagation paths follow the same paths as for a solid steel gear. The porous structure of the steel leads to a faster decay of the noise when excited by an impact, and this may be utilized in the design of the body of the gear using a varying density so augmenting the effect and reducing the transfer of vibrations from the teeth to the housing.

9.5  AM Gears 9.5.1 Introduction Using additive manufacturing in gear design and manufacturing opens up possibilities for the designer. For plastic gearing, a basic 3D printer using plastic filament such as ABS can be used to make functioning gears good enough for light loads and moderate temperatures. Even steel gears can be made with this method by using a filament that has powder metal particles admixed into the pellets that form the filament. The plastic is then burnt off in a desktop sintering furnace and the steel particles diffusion bond to one another just like in a metal injection molding (MIM) process. Predicting shrinkage is the key for this type of manufacturing and the surfaces and geometric quality is insufficient for industrial or automotive use but good enough for a basic prototype, power tool, or for further machining to tighter tolerances. The next level will be laser powder bed fusion technology (L-PFB) in terms of accuracy and strength but also cost. 9.5.2  Powder Steel for AM Gears The most common materials for L-PFB are stainless steels and tool steels; however, both of them are not suitable for making high-end steel gears. GKN developed a 20MnCr5 steel [17] for a high-end differential case with integrated ring gear suitable for race and high-end sports cars. The material is heat treatable and can be printed to full density and adequate strength compared to regular 20MnCr5 steel. The real advantage lies in the design and reduced weight and inertia of the gear. 9.5.3  Manufacturing of AM Gears In any AM method where a lot of heat is put into the powder there will be internal stresses leading to warping of the part or gear. There are no bulletproof methods to handle the warpage and care has to be taken as to the scanning path of the laser or electron

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beam. Also, the parts can be stress relieved in a furnace before cutting them from the build platform and even preheating the powder before printing since this reduces the thermal distortion. Because L-PFB is a layer-by-layer build, there will be anisotropy in the printed part meaning that the strength of the part depends on the direction of stress. The anisotropy will be reduced by the stress relieving but it will still be there. After stress relieving, the gears would have to be machined to remove any supports and the build plate. The next step would be heat treatment and subsequent hard machining of bore, face, and flanks since the tolerances are not within what is acceptable for automotive transmissions (ISO 6–7). 9.5.4  Strength of Case-Hardened AM Steel Gears Provided that the laser deposition, stress relieving, and heat treatment processes are all working properly, durability levels similar to 16MnCr5 or 20MnCr5 can be expected but there is a lot of work involved in fine tuning the processes. There is no public data available for case-hardened 3D printed gears since the normal steel materials for additive manufacturing does not work well when case hardening. Reference 17 describes the development of a automotive gear designed for AM using a material suitable for case carburizing and the L-PFB process.

References

1. ISO 6336-5. Calculation of load capacity of spur and helical gears Part 5: Strength and quality of materials, p. 21. 2. Rudnev, V., Loveless, D., Cook, D. Handbook of Induction Heating. Section 5.6. Marcel Dekker, Inc.: New york, ISBN 0-8247-0848-2. 3. Totten, G. Steel Heat Treatment Handbook. CRC Press: Boca Raton, ISBN 0-8493-8452-4. 4. Rodziňák, D. et al. Effects of Plasma Nitriding and Surface Densification on Contact Fatigue of Astaloy CrM Based PM Steel. Powder Metallurgy Progress, 2012, Vol. 12, No 1, pp. 17–26. 5. Production of Sintered Components. Höganäs Handbook of Sintered Components 2. Download from www.hoganas.com. 6. Larsson, M. et al. Compaction of a Helical Transmission Gear. Advances in Powder Metallurgy & Particulate Materials, 2014, Part 3, pp. 151–165. 7. To be published in gear technology October issue 2018 title not yet set. 8. Flodin, A. Tooth Root Optimization of Powder Metal Gears. Gear Technology., Randall Publication: Illinois, Andersson, June/July 2013, pp. 56–59. 9. Kapelevich, A., Shekhtman, Y. Tooth Fillet Profile Optimization for Gears with Symmetric and Asymmetric Teeth. Gear Technology, September/October 2009, pp. 73–79. 10. Asymmetric gearing ISBN 978-1-138-55444-3. 11. Andersson, M., deOro Calderon, C. Qualitative comparison of PM gear impact performance, equipment, methodology and results. Proceedings WorldPM 2016, Hamburg, Germany, 2016. 12. Angelopoulos, V. Improved PM gear rolling simulations using advanced material modelling. Proceedings from the JSME International Conference on Motion and Power Transmissions, Kyoto, Japan, 2017. 13. Henser, J., Micro geometry optimization of PM gears. M.Sc Thesis WZL-RWTH, Aachen, Germany, 2010. 14. Leupold, B. et  al. Validation of PM gears for eDrive applications. Conference proceedings 7th International Conference on Gears (2017). Munich, Germany, 2017.

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15. Sholzen, P. et  al. Optimization of NVH-behavior of gears by alternative gear materials. Conference proceedings 7th International Conference on Gears (2017). Munich, Germany, 2017. 16. Slavkovsky, E. et al. An analytical investigation of rattle characteristics of powder metal gears. Conference proceedings 2018 International Gear Conference. Lyon, France, 2018. 17. Kluge, M. et al. Design and production of innovative transmission components with additive manufacturing. Proceedings from 16th International CTI Symposium Automotive Transmissions, HEV and EV Drives, Berlin, Germany, 2017.

10 Induction Heat Treatment of Gears and Gear-Like Components Valery Rudnev CONTENTS 10.1 Introduction......................................................................................................................... 363 10.2 Electromagnetic Principles of Induction Heating.......................................................... 366 10.2.1 Skin Effect (Commonly Assumed Definition).................................................... 368 10.2.2 Skin Effect (Nonexponential Distribution)......................................................... 370 10.2.3 Eddy Current Cancellation.................................................................................... 371 10.3 Metallurgical Subtleties of Induction Gear Hardening................................................ 373 10.3.1 Material Selection for Induction Gear Hardening............................................. 373 10.3.2 Impact of Rapid Heating and Prior Microstructure.......................................... 374 10.3.3 Super-Hardness Phenomenon.............................................................................. 380 10.3.4 Specifics of Induction Hardening of Powder Metallurgy (PM) Gears............ 380 10.4 Technologies for Induction Gear Hardening..................................................................384 10.4.1 General Remarks.....................................................................................................384 10.4.2 Overview of Tooth Hardness Patterns................................................................ 385 10.4.3 Inductor Designs and Heating Modes................................................................. 387 10.4.3.1 Tooth-by-Tooth Hardening of Gears..................................................... 387 10.4.3.2 Gear Spin Hardening (Encircling Inductors)....................................... 393 10.4.3.3 Quenching Options................................................................................. 398 10.4.3.4 Heating Modes for Encircling Inductors.............................................. 401 10.5 Residual Stresses at Tooth Working Surface................................................................... 406 10.6 Hardening Components Containing Teeth.................................................................... 407 10.7 Tempering of Gears and Gear-Like Components..........................................................408 10.7.1 General Comments.................................................................................................408 10.7.2 Tempering Options................................................................................................. 409 10.7.3 Induction Tempering Subtleties........................................................................... 410 10.8 Conclusion........................................................................................................................... 414 Acknowledgments....................................................................................................................... 414 References...................................................................................................................................... 414

10.1 Introduction Over the years, gear manufacturers have increased their knowledge of the production of quality gears and gear-like components. This knowledge has led to many improvements including lower noise, lighter weight, and lower cost as well as increased load-carrying capacity to handle higher speeds and torque with a minimum amount of generated heat. 363

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Improvements in wear resistance, contact fatigue strength, endurance, and impact strength help to eliminate premature gearbox failure. Formation of considerable compressive residual stresses at the surface and in the subsurface help inhibit crack development and resist tensile bending fatigue. In recent years, heat treating by means of electromagnetic induction has become increasingly popular. The capability of in-depth heat generation in combination with high heat intensity (if required) quickly and at well-defined regions on the workpiece is a very attractive feature of this technology leading to low process cycle times (high productivity) with repeatable quality. Highly controllable heat intensities that range from moderate rates (e.g., as low as 2–3°C per second for tempering and stress-relieving applications) to high heat intensities (e.g., exceeding 800°C per second in gear surface hardening) allow the implementation of optimal process recipes/protocols [1]. Induction heating is also more energy efficient and inherently more environmentally friendly than most other heat sources including thermochemical diffusion processes such as carburizing or nitriding. Any smoke and fumes that may occur due to residual lubricants or other surface contaminants can be easily removed. A considerable reduction of heat exposure is another factor that contributes to the environmental friendliness and ergonomics of induction heaters. Advantages in safety (neither combustion nor environmental contaminants are used) in combination with low equipment maintenance cost, lean and green process with reduced labor, and improved efficiency make electromagnetic induction an attractive investment with great returns. Induction systems usually require far less start-up and shutdown time eliminating or dramatically reducing idle periods of unproductive heating. No energy is needed to build or to maintain the heat in nonoperative conditions. Other attractive features of induction systems are piece-by-piece processing capabilities with individual component traceability, high product quality, and repeatability, readiness for automation, advanced monitoring, and high dimensional stability of the heat-treated parts with low distortion. In contrast to carburizing and nitriding, induction hardening (IH) does not require heating the whole gear, pinion, or sprocket. Instead, the heating can be primarily localized to the areas where metallurgical changes are desired (e.g., the flank, root, and gear tip can be selectively hardened) [1–4]. Selective hardening of specific areas of gear teeth producing a fine-grain martensitic layer helps not only to optimize critical mechanical properties and gear performance but also to minimize distortion. In some countries, especially in Europe, induction hardening (IH) is gaining popularity off a wave of environmental mindedness. The current (2018) political climate in Europe is motivating many heat treaters to consider shifting their production processes from vacuum carburizing to IH, which is a more environmentally friendly process. Some gear processing companies don’t even feel there’s room left for consideration, and that carburizing isn’t an option anymore. However, IH isn’t a guaranteed solution for all gear heat treatment applications. The process has its own challenges and limitations, for example, with complexly shaped gears such as double helical or herringbone gears. The more complex a gear’s shape, the more difficult it is to develop a uniform case-hardened pattern, and eventually, the complexity can reach a point where IH simply cannot evenly austenitize a tooth surface, leading to varying case depths on different areas of the gear and potentially producing some undesirable metallurgical results. In these cases, heat treaters applied vacuum carburizing or nitriding for their gears instead, whichever is more suitable. Although not all gears and pinions are well suited for IH, external spur and helical gears, worm gears, internal gears, racks, and sprockets are among those that are typically induction hardened (Figure 10.1).

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FIGURE 10.1 Spur and helical gears, worm gears, internal gears, racks, and sprockets are among the parts that are typically induction surface hardened. (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

Similarly, sharp corners on gear teeth can exhibit certain undesirable metallurgical structures. Sharp corners sometimes get excessively heated compared to the rest of the teeth surface, which leads to grain coarsening and, in some instances, even grain boundary liquation (incipient melting) making them to be susceptible to crack development. Thus, appropriate chamfering and rounding of sharp edges is always welcomed by induction heat treaters . Required gear performance characteristics (including load condition and operating environment) dictate the needed grade of steel, cast iron, or powder metallurgy (PM) material, their prior microstructure, surface and core hardness, hardness profile, and desirable residual stress distribution. IH also often requires using different steel grades to start with. Thermochemical diffusion processes (e.g., carburizing or nitriding) alter the actual chemical composition of the tooth surface layer, thus hardening it. But one critical difference between these processes and IH is that the latter does not alter the chemical composition of the material, which in turn means a stronger base material is required with sufficient carbon content to form the required amount of martensite at a tooth’s working surface in order to provide adequate strength and wear properties. It is not always possible to obtain a fully martensitic case depth. Depending on the steel chemical composition, the presence of a certain amount of retained austenite (RA) within the case depth might be unavoidable (unless cryogenic treatment is used). Up to a certain point, some amount of RA does not noticeably reduce the surface hardness. However, it might bring some ductility and provide better absorption of impact energy, which could be imperative for heavily loaded gears. In addition, because of its unstable nature, RA may, with time, transform into martensite, introducing additional compressive residual stresses and increasing the surface hardness. From this perspective, a small amount of RA may not only be harmless but might even be considered beneficial in some cases. However, in the great majority of applications, an excessive amount of RA is undesirable and can even be detrimental because it may noticeably reduce the surface hardness, weaken bending fatigue properties, decrease wear resistance and the magnitude of useful compressive residual stresses within the case depth, promote shape distortion, and can result in the appearance of a crucial amount of brittle untampered martensite during the gear’s service life.

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Thanks to surface hardening capabilities of electromagnetic induction and depending upon tooth geometry, a gear tooth’s core could remain relatively cool, which allows it to act as a shape stabilizer for the heated surface and, in turn, reduce the distortion caused by the heat-treating process (e.g., eliminating sagging) while also making that distortion more predictable and repeatable. A typical gear-hardening procedure for steels and cast irons involves heating the alloy to the austenitizing temperature range, holding it (if necessary) at temperature for a period long enough for completion of the formation of fully or predominantly austenitic structure, then rapidly cooling/quenching it below the Ms critical temperature where martensite starts to form. Rapid cooling allows replacement of the diffusion-dependent transformation of austenite by diffusion-less shear-type transformation producing a much harder constituent of martensite.

10.2  Electromagnetic Principles of Induction Heating Induction heating is a multifaceted, multidimensional phenomenon comprised of complex interactions involving electromagnetics, heat transfer, materials science, metallurgy, and circuit analysis. These physical phenomena are tightly interrelated, and highly nonlinear. Application of alternative heating methods (e.g., gas furnaces, infrared ovens, or fluidized baths) is practically immune to a variation of electromagnetic properties of heated material (including electrical resistivity ρ and relative magnetic permeability µr). In contrast, implications of using IH are noticeably different and associated with deviations in not only thermal physical properties but also electromagnetic properties. The main components of an induction heating system are a heating inductor, power supply, load-matching station, water cooling and quenching (for hardening applications) systems, controls, monitoring, bus network, and the workpiece itself (e.g., a gear, pinion, sprocket, rack with teeth, and others). Heating inductor, inductor, induction coil, and coil are all terms used interchangeably for the electrical apparatus that provides the contactless heating effect in the workpiece. In everyday practice, an inductor is often simply called a “coil,” but its geometry does not always resemble, by any stretch of the imagination, the shape of classic circular coil. As an example, Figure 10.2 shows an array of a variety of geometries of inductors for hardening various types of gears and gear-like components. Depending upon application specifics, an inductor may encircle a gear (its outside or inside diameter) or may be positioned in close proximity to gear teeth. The basic electromagnetic phenomena of IH have been discussed in several textbooks including college physics. An alternating voltage applied to an induction coil will result in an alternating electric current (AC) flow in the coil circuit. An alternating coil current produces in its surroundings a time-variable magnetic field that has the same frequency as the coil current. This magnetic field generates eddy currents flowing in the electrically conductive (e.g., metallic) workpiece. These induced currents have the same frequency as the coil current; however, their direction is opposite. There are two mechanisms of heat generation in the induction heating of electrically conductive materials [1]: • The primary mechanism of heat generation is associated with heat produced by the inherent resistance of electrical conductors to induced electrical current

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FIGURE 10.2 An array of a variety of geometries of inductors for hardening various types of gears and gear-like components. (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

flow (Joule’s effect). Joule heat generation is often referred to as I2 R heating as the magnitude of heat generation is proportional to the material’s electrical resistance R and the square of the electrical current I flowing within it. • The second mechanism of heat generation occurs when heating ferromagnetic materials (e.g., plain carbon steels) and is associated with magnetic hysteresis (magnetization–demagnetization cycles). Thermal energy is dissipated during the reversal of magnetic domains due to internal friction between molecules. Magnetic hysteresis heat generation is proportional to the applied frequency and the area of the hysteresis loop, which is a complex function of chemical composition, prior structure, grain size, temperature, magnetic field intensity, and frequency. Heating rates are typically controlled by the coil voltage/current. A key factor is the degree of electromagnetic coupling between the workpiece and the magnetic field of the coil. Coupling is determined by the number of imaginary magnetic flux lines that enter the workpiece. This flux density is roughly proportional to the coil current, and the amount of energy transferred is proportional to the square of the number of imaginary flux lines intercepted by the workpiece. The electrical frequency of coil current also influences the pattern of induced eddy currents. Because of several electromagnetic phenomena, the distribution of electrical currents within an inductor and workpiece is not uniform [1]. This includes but not limited to;

1. Skin effect, 2. Proximity effect, 3. Ring effect, 4. Slot effect, 5. End and edge effects, and some others.

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10.2.1  Skin Effect (Commonly Assumed Definition) Recognizing the importance of all electromagnetic phenomena, the skin effect represents a fundamental property of induction heating. It is convenient to illustrate the skin effect by positioning the electrically conductive workpiece inside of a solenoid-style induction coil. As one may know from the basics of electricity, when a direct current flows through an electrical conductor that stands alone, the electrical current distribution within the conductor’s cross section is uniform. However, when an AC current flows through the same conductor, the current distribution is not uniform. The maximum value of the current density will be located on the surface of the conductor with homogeneous electromagnetic properties; the current density will decrease from the surface of the conductor toward its center. According to the commonly accepted theory of electromagnetic induction, eddy currents primarily flow near the surface layer (or “skin”). This “skin effect” must be clearly understood since it is associated with the surface region where a great majority of heat sources will be generated. Because of the skin effect, approximately 86% of the power generation will be concentrated in the surface layer of the conductor that is called the current penetration depth (or reference depth) δ. The degree of skin effect depends on the frequency, electromagnetic properties (ρ and µr) of the conductor, and the geometry of the workpiece. Figure 10.3 shows a commonly accepted appearance of the skin effect when heating solid cylinders showing distribution of current density and power density from the workpiece surface toward the core. As one can see from the figure, at one penetration depth from the surface (y = δ), the current density will equal approximately 37% of its surface value. However, the power

FIGURE 10.3 Commonly accepted appearance of the skin effect when heating solid cylinders showing distribution of current density and power density from the workpiece surface toward the core. (From V. Rudnev et al., Handbook of Induction Heating, 2nd edition, CRC Press, New York, 2017.)

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density will equal 14% of its surface value [1]. From this, we can conclude that 63% of the current and 86% of the power will be concentrated within a surface layer of thickness δ. Mathematically speaking, penetration depth is described in meters as δ = 503



ρ µr F

(10.1)

where, F = frequency, Hz (cycle/sec) ρ = electrical resistivity of the electrically conductive material Ω*m µr = relative magnetic permeability or in inches,



δ = 3160

ρ µr F

(10.2)

where electrical resistivity ρ is in Ω* in. As one can see, the value of δ varies with the square root of electrical resistivity and inversely with the square root of frequency and relative magnetic permeability; µr indicates the ability of a material (e.g., carbon steel) to conduct the magnetic flux better than a vacuum or air. During the heating cycle for hardening, the ρ of most carbon steels can increase to four to six times its value at room temperature. The ferromagnetic property of the material µr is a complex function of structure, chemical composition, prior treatment, grain size, frequency, magnetic field intensity, and temperature. As one can see from Figure 10.4, left, the same carbon steel grade at the same temperature and frequency can have a noticeably different value of µr due to differences in the intensity of the magnetic field (coil power). For example, even at room temperature,

FIGURE 10.4 Effect of temperature and field intensity on relative magnetic permeability µr (left) and the Curie temperature of plain carbon steel versus carbon content (right). (From V. Rudnev et al., Handbook of Induction Heating, 2nd edition, CRC Press, New York, 2017.)

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TABLE 10.1 Current Penetration Depth (in mm) versus Frequency and Heating Stage for Typical Conditions of Induction Hardening of Medium Carbon Steel Frequency (kHz) Heating Stage Initial heating stage (room temperature) At temperature above Ac3

0.5

3

10

30

70

200

3.6–3.9 17.5

1.4–1.6 10

0.7–0.85 5.6

0.42–0.5 3.2

0.3–0.38 2.1

0.15–0.22 1.2

Source: V. Rudnev et al., Handbook of Induction Heating, 2nd edition, CRC Press, New York, 2017 [1].

the µr of magnetic steels commonly used in induction heat treating can vary from small values (e.g., µr = 5 or 8) for hardening to very high values (exceeding µr = 150) for induction tempering and stress relieving, depending on the magnetic field intensity H. It would be beneficial at this point to review a case study. Consider that two identical gears fabricated from the same grade of steel, (for example, SAE 1055) having same prior microstructure being placed into corresponding identical coils that are powered by AC power generators of the same electrical frequency. Magnitude of power applied to the first coil corresponds to powers commonly used for induction tempering. However, power applied to the second coil is noticeably greater and corresponds to levels typical for hardening applications. Regardless of the fact that same steel grade has been used in both cases, that steel will respond to induction heating differently and would appear as two different materials exhibiting different 3D heat generation (including different heat generation depth), coil electrical efficiency, and so on. The temperature at which a ferromagnetic body loses its magnetic properties becoming nonmagnetic is called the Curie temperature (Curie point) and also often referred to as A2 critical temperature. Figure 10.4, right, shows a portion of the Fe–Fe3C phase transformation diagram that illustrates the A2 critical temperature being a function of carbon content for plain carbon steels. The value of current penetration depth δ varies at different heating stages. Table 10.1 shows the δ for medium carbon steel versus frequency and heating stage (ferritic-pearlitic vs. austenitic microstructure). Therefore, one of the most distinguished features of IH compared to alternative heattreating methods (for example, thermo-chemical diffusion processes or flame hardening) is an impact of electromagnetic physical properties on practically all important parameters of an induction system and its performance. 10.2.2  Skin Effect (Nonexponential Distribution) It is important to keep in mind that a commonly accepted introduction of the skin effect phenomenon provided in Section 10.2.1 is only correct for a homogeneous solid body with constant distribution of ρ and µr. Therefore, realistically speaking, this classical assumption can be made only for some unique cases because for the great majority of IH applications, including surface gear hardening, there are always thermal gradients within the heated workpiece. These thermal gradients result in nonconstant distribution of ρ and µr. Thus, the main postulation of exponential heat source distribution (as shown on Figure 10.3) does not “fit” its principle assumption due to the presence of the nonlinearities of electromagnetic properties of heated metallic workpiece. In reality, at different stages of IH, the power density (heat source) distribution along the radius/thickness of the workpiece

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may have a unique wave shape [1], which differs significantly from the commonly assumed exponential distribution. There might be a maximum of power density/heat sources at the surface. Then power density starts to decline from the surface toward the subsurface. However, at a certain distance it might suddenly begin to rise again, reaching its second maximum before its final falloff. At the initial stage of heating when surface temperature is below Ac1 critical temperature, the steel is magnetic; thus, the δ is quite small and the skin effect is highly pronounced. With time, temperatures of the gear surface and near-surface exceed both Ac1 and A2 critical temperatures. Since the surface layer became nonmagnetic, there is a noticeable reduction of the heat intensity there. Decrease of µr and an increase of ρ cause a corresponding increase in δ compared to its values during the initial heating stage. Although the surface layer is nonmagnetic, the internal region, having temperatures below the Curie point, retains its ferromagnetic properties. If the thickness of a nonmagnetic surface layer is sufficiently small compared to current penetration depth in nonmagnetic steel, there will be substantial magnetic field penetration beyond a surface layer heated above A2 critical temperature. Two peaks of an induced power density (heat sources) may occur at this point. The first maximum of power density would be located at the surface and then the power density decreases toward the core. However, once it reaches a certain distance below the surface, the power density might start to rise again, and after reaching the second maximum, it starts its final decline. In such case, the power density distribution would have a unique nonexponential distribution (“wave-shaped” profile), which is very different from the commonly assumed exponential distribution. The first maximum of power density (heat sources) would be located at the tooth surface. The second peak would occur where there is a “below Curie– to–above Curie” border. If with time, the surface layer heated above the Curie point is expanded so far that its thickness become greater than the δ in hot steel for a given frequency, then the “waveshaped” distribution disappears and a classical exponential power density distribution will then take place. A nonexponential (wave-shape) heat source distribution has a noticeable impact on the selection of process parameters, transient and final temperature distribution, as well as hardness pattern. This is so because if for surface hardening the frequency has been chosen correctly, the thickness of the austenitized layer (the nonmagnetic surface layer) is less than δ in austenitized steel and the wave-shape heat source distribution takes place during the majority of the heating cycle for induction surface hardening. By comparison, through hardening applications or when heating magnetic steels before hot working, the impact of this nonexponential (wave-shape) heat source distribution phenomenon is less significant, because the duration of the “hot” stage (when the entire cross section is heated above the Curie point) occupies a much greater portion of the heat cycle diminishing the impact of the nonexponential heat source distribution. Numerical computer modeling allows to accurately take this phenomenon into consideration. 10.2.3  Eddy Current Cancellation Under certain conditions, a phenomenon of partial or complete eddy current cancellation may occur. According to this phenomenon, certain regions of the gear teeth (more typically tooth tip and addendum region) may start absorbing only a negligible amount of energy regardless of applied coil power and being practically transparent to the electromagnetic field. As a result, there will be only small amounts of the heat generation appearing in the

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area that is commonly referred to as an electromagnetically thin region. The phenomenon of eddy current cancellation is not only a function of the physical size of the gear teeth but also the ratio of tooth thickness-to-δ. If the electrical frequency is too low producing too large δ compared to tooth thickness, then eddy currents circulating in opposite sides of the tooth profile and being oriented in opposite directions might start canceling each other. In order to avoid eddy current cancellation, δ should be no more than one-fourth of the thickness of the tooth. It should be noted that in some cases certain degree of a localized eddy current cancellation might be beneficial assisting to obtain preferable hardness pattern. Numerical modeling provides more accurate assessment of a cancellation phenomenon. If eddy current cancellation of appreciable degree occurs, then due to a lack of heat generation it may manifest itself in insufficient austenitization of an entire tooth or its certain regions (e.g., tooth tip or addendum region). As an example, Figure 10.5 shows selective hardening patterns on the carbon steel shaft with teeth using different combinations of power, frequency, and heat time. This case study illustrates a wide diversity of induction hardening patterns obtained thanks to proper control of an eddy current cancellation phenomenon. White color areas indicate etched regions where martensite was formed. Note: in all cases, hardening inductor has encircled the shaft during austenitization having noticeably smaller “tooth tip diameter-to-inductor” gap compared to “tooth root diameterto-inductor” gap. Figure 10.6 illustrates the eddy current flow in two extreme cases: application of high frequency (left) and low frequency (right). When high frequency is applied, δ is relatively small and the induced eddy currents generally follow the contour of the gear teeth (Figure 10.6, left). This leads to a heat source surplus in the tip of the tooth compared to that in the root. Besides that, the tip of the tooth has a substantially smaller mass of metal to be heated, compared with the dedendum and root area, where a much larger thermal heat sink is located. These two major factors result in greater heat intensity at the tip, with a corresponding temperature rise and deeper case depth upon quenching. This also explains conditions for selective hardening of tip and addendum region without hardening tooth root. Application of low frequency (Figure 10.6, right) is associated with a dramatic increase in δ, potentially leading to eddy current cancellation at the tooth tip (top land) and possibly at entire addendum area. This makes it much easier for induced current to take a shorter path, following the root circle of the gear instead of following along the tooth profile, leading to more intensive heat generation in the tooth root area compared with its tip.

FIGURE 10.5 Selective hardening patterns on the carbon steel shaft with teeth using different combinations of power, frequency, and heat time. (From V. Rudnev et al., Handbook of Induction Heating, 2nd edition, CRC Press, New York, 2017.)

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FIGURE 10.6 Illustration of an eddy current flow in two extreme cases: application of high frequency (left) and low frequency (right). (From V. Rudnev et al., Gear heat treating by induction, Gear Technology, March 2000.)

As expected, the terms high frequency and low frequency have relative meanings. For example, depending on the tooth geometry, a frequency of 3 kHz may act as a high frequency for coarse teeth and 300 kHz may act as a low frequency for splines, threads, fine teeth, or skinny teeth. In-depth discussion on this subject will be provided later in Section 10.4.3.2. However, at this point, it is beneficial to stress the importance of taking into consideration not only a skin effect but an eddy current cancellation as well when developing recipes for gear-hardening applications. In some cases, both phenomena might be “friends,” in others they could be “foes” preventing obtaining the required hardness pattern.

10.3  Metallurgical Subtleties of Induction Gear Hardening There are tremendous amounts of publications devoted to metallurgical subtleties associated with rapid induction hardening [1–8]. This includes an impact of steel’s chemical composition and prior microstructure, homogeneity/heterogeneity of formed austenite phase, influence of alloying, and residual elements among critical factors. Due to space limitation, only a brief overview regarding a few selected points will be provided in this section. 10.3.1  Material Selection for Induction Gear Hardening Several guidelines, recommendations, and standards exist with helping to select an appropriate material for gears. This includes but is not limited to the following: • ANSI/AGMA 2004-C08 standard: “Gear Materials, Heat Treatment and Processing Manual” • AGMA 923-B05 standard: “Metallurgical Specifications for Steel Gearing” • ASTM A536-84 standard: “Specification for Ductile Iron Castings” • ASTM A48/A48M-03 standard: “Specification for Gray Iron Castings”

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• ANSI/AGMA 6008-A98 standard: “Specifications for Powder Metallurgy Gears” • Heat Treater’s Guide: Practices and Procedures for Irons and Steels, ASM Int’l. As sated earlier, thermo-chemical diffusion hardening processes, such as carburizing and nitriding, modify the actual chemical composition of the steel. The major difference between these processes and induction hardening is that the latter does not alter the chemical composition of the parent material; instead it modifies its crystal structure. Therefore, since induction hardening does not change the chemical composition of steel (rear cases of forming surface decarb are outside of present discussion), the steel grade must have sufficient carbon and alloy content and be capable of achieving a required surface hardness, case depth, and core strength to accommodate needed engineering properties (for example, sufficient strength, wear, and fatigue properties, to name just a few). An exception from this rule is induction hardening of cast irons. In steels, the carbon content is fixed by chemistry and, upon austenitization, cannot exceed this fixed value (rare attempts to use a carbon-enriched environment are excluded from consideration). In contrast, in cast irons, there is a “reserve” of carbon in the primary (eutectic) graphite particles [1,9]. At temperatures of austenite phase, some of the graphite particles can go into solution by diffusing (fully or partially) into the matrix in proximity to nodules (for ductile cast irons) or flakes (for gray cast irons). This can locally deviate chemical composition, increase the carbon level in the austenite, shift continuous cooling transformation (CCT) curves, affect martensitic formation and Ms temperatures, as well as the amount of retained austenite (RA). This tendency is an important metallurgical factor representing one of the major differences between hardening cast irons versus steels and potentially resulting in a variable amount of carbon in as-quenched structure. Taking into consideration that the great majority of gears and gear-like components are made of steels or PM materials, there will not be a discussion on metallurgical subtleties of induction hardening of cast irons here. Interested readers should review publications [1,9] on this subject. Low-alloy and medium-carbon steels with 0.4–0.55% C (e.g., SAE 1040, 15B41, 4140, 4340, 1045, 4150, 1552, and 5150) are commonly used in induction gear hardening [1]. In some cases, high-carbon steels (SAE 5160, 1065, 1080, and 52100) are used, as well as martensitic stainless steels, hardenable PM materials and proprietary micro alloy steels. It is important to have a sufficiently “friendly” prior microstructure (structure of parent material) when induction hardening gears. It is highly desirable to use sufficiently homogeneous (both chemically and structurally) prior structures without excessive segregation and severe banding. Quenched & Tempered (Q&T) microstructure with a hardness range of 30–36 HRC leads to a fast and consistent steel response to IH. In contrast, steels with large carbides (i.e., spheroidized steels) have poor response to rapid hardening, requiring prolonged heating and higher temperatures for austenitization. A combination of high temperatures with longer heat times may lead to grain growth, data scatter, and excessive gear distortion. Coarse martensite has a negative effect on tooth toughness as well on impact strength and fatigue life. These are the reasons why steels in normalized and Q&T conditions are frequently used in rapid induction gear hardening. 10.3.2  Impact of Rapid Heating and Prior Microstructure In case of hardening plain carbon steels, when iron is alloyed with different percentages of carbon, the critical temperatures are sometimes determined by the iron-iron carbide phase transformation diagram (Fe–Fe3C diagram) or, in case of low-alloy steels, by correspondent

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more complex diagrams (e.g., ternary diagrams) or mathematical correlations (i.e., formulas or equations) that indicate an effect of certain chemical elements on the positioning of critical temperatures (e.g., A1, A3, Acm). There have been several attempts to develop convenient mathematical expressions that would allow the heat treater to calculate the critical temperatures of transformation for different phases. Some of the formulas that can be used for a rough estimation of A1, A3, Bs, and Ms temperatures of industrial carbon steels are provided in [5]. Unfortunately, some heat treaters are unaware that those classical phase transformation diagrams and correlations might be misleading in the majority of induction-hardening applications, because they are valid only for the equilibrium conditions. Transformations taking place at A1, A2, A3, and Acm critical temperatures are diffusion-driven transformations and rapid heating permits less time for diffusion. The intensity of heating (heating rate) in induction gear-hardening applications often exceeds 200°C/s and, in some cases, reaches 800°C/s and even higher (e.g., in simultaneous dual-frequency gear hardening). Therefore, such a process cannot, by any means, be considered as equilibrium. Actually, with induction hardening, we always deal with a certain degree of nonequilibrium even with seemingly slow heating [1]. In case of ferritic-pearlitic initial prior microstructures, practically all of the carbon is contained in the pearlite. Therefore, regardless of relatively high solubility of carbon in austenite, some minimum time is required during solid-state transformation (such as austenitization) for carbon to be diffused from pearlitic regions into areas occupied by ferrite. Therefore, it is required to create conditions conducive to the needed diffusiondriven processes in order to develop an essentially homogeneous austenitic structure with sufficiently uniform carbon distribution before quenching that is desirable for the great majority of induction gear hardening. If an austenite exhibits a nonuniform distribution of carbon, then, upon quenching, a decomposition of heterogeneous austenite begins in the lower-carbon regions shifting the CCT curves there to the “left” compared to steels with nominal carbon content and resulting in greater probability of forming upper transformation products. The CCT curves for regions having excessive amounts of carbon will be shifted in the opposite direction with a corresponding reduction of Ms temperatures and greater probability of having a larger amount of localized RA. Despite the fact that austenitic transformation takes place, its markedly heterogeneous nature might potentially lead to an unacceptably heterogeneous as-quenched microstructure. It is important to be aware that in contrast to conventional thermochemical heat treatment processes that apply relatively long heat times, as a result of rapid induction heating, a certain amount of undissolved carbides and carbonitrides as well as various concentration gradients of not only carbon but also alloying elements may be present in austenite. Keep in mind that complex carbides of some alloy metals often dissolve noticeably slower than Fe3C. Rapid heating causes a measurable increase of critical temperatures (e.g., Ac1, and Ac3) compared to the corresponding temperatures A1 and A3 indicated by the equilibrium phase transformation diagram. The nucleation and growth of austenite in the rapid heating of measurably heterogeneous initial structures are more concerning than in the case of thermo-chemical diffusion processes, which apply significantly longer process times. Selection of hardening temperatures based on the equilibrium Fe–Fe3C diagram and failure to choose the proper hardening temperatures taking into consideration the appreciably nonequilibrium nature of induction heating may result in incomplete transformation and mixed as-quenched structures. Metallographic evaluation helps reveal the presence of

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FIGURE 10.7 Failure to choose proper hardening temperatures based on the nonequilibrium nature of induction heating can produce mixed structures after quenching of heterogeneous austenite. 4% Nital etched, ×100. Two images (left) and (right) of heterogeneous as-quenched structures. Light areas represent martensitic regions and dark areas are pearlitic regions. (From V. Rudnev et al., Handbook of Induction Heating, 2nd edition, CRC Press, New York, 2017.)

“ghost pearlite” and other upper transformation products in the specimens (Figure 10.7) that are often associated with not fully transformed structures or with the presence of severely heterogeneous austenite before quenching. The degree of heterogeneity in the as-quenched microstructure can be reduced by increasing the hardening temperatures and/or by lengthening the time at the austenite phase temperature range. A tremendous amount of metallurgical research has been directed at the determination of the effect of heat intensities and prior microstructures on steel’s response to IH. Rapid heating considerably affects the kinetics of austenite formation, shifting it toward higher temperatures according to the continuous heating transformation (CHT) diagrams. As an example, Figure 10.8 shows the effect of the heating rate on the Ac3 critical temperature of medium-carbon steel having different prior microstructures [7,8]. The summaries of the results of some of those studies can be found in [1,5,6].

FIGURE 10.8 Effect of initial microstructure and heating rate on Ac3 critical temperature for SAE 1024 steel. (From S.L. Semiatin, D.E. Stutz, Induction Heat Treatment of Steel, ASM International, Materials Park, OH, 1986, p. 88; W. Feuerstein, W. Smith, Trans. ASM, 46: 1270, 1954.)

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A comprehensive study regarding CHT diagrams in steels and the correlation of heat intensity versus positioning of Ac1, Ac2, Ac3, and Acm critical temperatures, as well as the ability to obtain homogeneous austenite was conducted by J. Orlich, A. Rose, and colleagues at the Max-Planck-Institut fur Eisenforschung GmbH, Dusseldorf, Germany [10,11]. Atlases for a variety of steels were developed as a result of their study, taking into consideration a wide range of heating rates (from 0.05°C/s to 2400°C/s). These atlases consist of numerous diagrams indicating conditions for formation of homogeneous and heterogeneous austenite, as well as positioning of critical temperatures that were obtained by conducting laboratory experiments. The results presented in the CHT diagrams are more appropriate for rapid induction hardening in making decisions regarding the appropriate hardening temperatures. The study shows that differences in the Ac3 critical temperatures (rapid heating vs. equilibrium conditions) could be quite high; exceeding 150°C. An interesting observation [11] can be made from the data experimentally obtained by J. Orlich, A. Rose, and colleagues [10,11] that is particularly applicable to IH. When heat intensities exceed approximately 20°–30°C/s (which is very typical for the great majority of induction gear-hardening applications), instead of the normal order of critical temperatures, Ac1, Ac2, Ac3, rapid heating can switch the order to Ac2, Ac1, Ac3. This might be essential knowledge in some induction heating applications, turning an easy job into an almost impossible job. On the basis of the information provided by Orlich et al., there are many steels that exhibit such behavior. As can be concluded from Figure 10.8 and the discussion above, in addition to heat intensity, the prior microstructure also makes a measurable impact on the selection of hardening temperatures, and its effect can be summarized in the following main points [1]: • Homogeneous fine-grain Q&T initial microstructures with a hardness range commonly being approximately 28–36 HRC are the most favorable for rapid induction hardening, ensuring fast transformation, which allows a reduction in the hardening temperatures compared to alternative initial structures. The nucleation and growth of austenite in the rapid heating of measurably heterogeneous initial structures are more concerning than in case of conventional austenitization, which applies longer process times. • Q&T structures result in a consistent response to IH with least amounts of grain growth, shape/size distortion, and minimum required heating energy. These prior structures are associated with a well-defined (crisp) pattern having short transition zone and may also produce slightly higher than expected hardness levels (when hardening medium-carbon steels) and deeper case depths, and form greater compressive residual surface stresses compared with other types of initial structures. As an example, Figure 10.9 shows the effect of three different initial microstructures on an SAE 1070 carbon steel bar in response to surface hardening [7]. • Normalized structures consisting of a uniformly distributed fine-grain mixture of ferrite and pearlite also provide a rapid response to IH, allowing one to reduce the required hardening temperatures and heat times almost as much as Q&T prior structures and resulting in fast and consistent transformations. • If the steel’s prior structure has a significant amount of coarse ferrites, clusters, or thick bands of ferrites, then the structure cannot be considered “favorable.” Ferrite does not contain a sufficient amount of carbon required for martensitic transformation suitable for common industrial applications (practically speaking). During austenitization, large areas (clusters or bands) of ferrite require noticeably

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FIGURE 10.9 Effect of initial microstructure in SAE 1070 steel bars in response to surface hardening using a 450-kHz induction generator operated at a power density of approximately 2.5 kW/cm2 (15.9 kW/in.2). (From S. L. Semiatin, D. E. Stutz, Induction Heat Treatment of Steel, ASM International, Materials Park, OH, 1986; T. Spencer et al., Induction Hardening and Tempering, ASM International, Materials Park, OH, 1964.)

longer times for carbon to diffuse into carbon-depleted areas. These ferrite clusters and bands could be retained in the austenite upon rapid heating, and after quenching, mixed structures can be formed. These mixed structures (Figure 10.7) are not typically permitted by customer specifications, because scattered soft and hard spots and poor engineering properties (e.g., low fatigue resistance and poor wear resistance, just to name a few) are associated with such structures. Gears having such prior microstructures can still be induction hardened but appreciably higher temperatures and longer heating/holding times are needed, as an attempt to avoid the formation of undesirable as-hardened mixed structures. Therefore, severely banded initial microstructures of “green” parts exhibiting severe chemical and microstructural segregation should be avoided in short-time IH; otherwise, higher temperatures and/or longer times at the austenite phase temperature range are needed, resulting in a lower production rate, reduced useful compressive surface residual stresses, exhibiting potential grain coarsening, and may also negatively affect distortion characteristics. It is certainly understandable that, practically speaking, modern steels almost always consist of some degree of heterogeneity, segregation, and banding. However, severely segregated and banded initial microstructures of “green” parts should be avoided in short-time induction gear hardening. • Fully annealed and spheroidized prior structures (Figure 10.10, left) and other initial structures that consist of coarse stable carbides also have a poor response to shorttime hardening, requiring prolonged heat cycles, noticeably higher temperatures, and extended times at elevated temperature to complete austenitization. Otherwise, data scatter, a combination of soft-and-hard regions, may occur. It should be noted that regardless of achieving seemingly high hardening temperatures, some stable complex carbides (e.g., titanium carbides TiC) and nitrides might not dissolve and remain within the austenite and respectively will be present in the as-quenched structure of the martensite matrix. As expected, the necessity of using higher

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FIGURE 10.10 (Left) Spheroidized structures and other initial structures that consist of coarse stable carbides also have poor response to rapid induction hardening requiring higher temperatures and extended times at elevated temperature to complete austenitization. (Right) Decarburized layer present in the initial microstructure remained in as-quenched sample (thin white color region indicates a free-ferrite region at ×400). (From V. Rudnev et al., Handbook of Induction Heating, 2nd edition, CRC Press, New York, 2017.)

temperatures (compared to those that would be conventionally employed in IH of friendlier initial structures) and longer duration at those temperatures needed to dissolve the coarse spheroidal carbides could potentially provoke an excessive grain coarsening, the formation of brittle martensite, grain boundary liquation (incipient melting), extended heat effected zone (HAZ), greater amounts of RA, and worsening the shape distortion. Factors like these can have a detrimental impact on toughness, impact strength, bending fatigue strength, wear resistance, and other critical properties of gears. • Recommendations suggested in the literature (including [5–12]) for a selection of IH temperatures as functions of heat intensities should still be used only as basic guidelines. This is particularly true for modern plain carbon steels, which are often melted using a large percentage of scrap that may contain microalloyed highstrength low-alloy steels. Thus, the steel may still be considered to be nominally of the plain carbon steel composition but may also contain trace amounts of niobium, vanadium, titanium, boron, and other elements whose presence may noticeably affect the response of steel to IH. For this reason, it is beneficial to check the complete chemistry on each lot of steel and to determine the proper austenitizing temperatures experimentally. It is important to remember about the effect of prior microstructure not only on the positioning of critical temperatures during rapid heating but also on the thermal and, in particular, electromagnetic properties of a steel of a particular grade. This factor alone can cause the need to adjust the process recipe/protocol. • The presence of decarburized (carbon depleted) layers at the surfaces of “green” parts is highly undesirable and even prohibited in majority of IH applications. A substantial amount of decarb is commonly present when dealing with hot rolled bar stock, as well as as-forged, annealed, and spheroidized structures. Because of the short process time, IH itself does not typically produce any measurable amount of decarb. Incoming decarb cannot be repaired during rapid hardening under normal processing conditions. Since the carbon content determines the achievable steel’s hardness level, surface decarburization causes a “soft” surface that can lead to localized severe hardness and strength reduction in as-quenched steel and it may even reverse the residual stress distribution, forming undesirable localized peak of tensile surface stresses, which in turn can cause premature failure of gears during

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their service life. Decarburized layers that are present in the initial microstructure will remain within the as-quenched structure, making a considerable negative impact on other mechanical properties, including, wear resistance, fatigue strength, impact and bending properties, and so on. Thus, if the prior microstructure of green gear exhibits the presence of decarb, then the weakened near-surface material should be machined off. Metallographic examination is the most reliable way to determine the presence and extent of decarburization (Figure 10.10, right). 10.3.3  Super-Hardness Phenomenon When surface hardening steels, a combination of using “friendly” prior microstructures, fast heating rates, and intense quenching could result in the so-called super-hardness phenomenon or super hardening [1,6]. This phenomenon refers to obtaining greater hardness levels in the case of surface hardening compared to through hardening or hardness levels that would normally be expected [14]. Due to this phenomenon, for identical steel composition, the surface hardness of an induction case hardened part could be 2–4 HRC higher (1–3HRC being more typical) than normally expected for a given carbon content and steel chemical composition. The super-hardness phenomenon is not clearly understood and its origin has not been widely accepted by metallurgists worldwide. However, it has been obtained experimentally on numerous occasions and several interpretations have been offered. Review of potential causes of the super-hardness phenomenon is provided in the 2nd Edition of the Handbook of Induction Heating (CRC Press, 2017 [1]). 10.3.4  Specifics of Induction Hardening of Powder Metallurgy (PM) Gears During the last several decades, the use of PM materials in several industries has been expanded at an impressive pace; however, the biggest market for iron-based PM components remains to be the transportation industry, particularly the automotive and agricultural sectors [1,15–17]. An ability to manufacture net-shape complex geometry components offering competitive performance at an economical cost is an attractive feature of PM parts, including gears, timing sprockets, cams, splined hubs, shock absorbers, and so on. Figure 10.11 shows a selection of PM gears that regularly undergo induction surface hardening.

FIGURE 10.11 Selection of PM gears suitable for induction surface hardening. (From V. Rudnev, Intricacies of Induction Hardening PM Parts, Professor Induction Series, in: Heat Treating Progress, ASM Int’l, Materials Park, OH, November/December 2003, pp. 23–24.)

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Density and porosity in the sintered compact are major factors affecting strength and hardenability of iron-based PM materials; both properties have been noticeably improved in the last decade, allowing closely approaching respected properties of fully dense wrought steels. Carbon content is another major factor that affects achievable hardness and strength. Similar to steels, alloying helps enhance a desirable combination of strength, load-bearing capacity, ductility, and fracture resistance of PM materials. Copper, nickel, molybdenum, and, to a lesser extent, chromium, phosphorus, and silicon are the most commonly used alloying elements in iron-based hardenable PM materials. Modern PM materials are also noticeably more homogeneous with minimized chemical segregation. Still, strength, fatigue and wear resistance, and fracture toughness of PM materials remained somewhat lower compared to the respective properties of fully dense wrought materials. Induction hardening of PM gear-like components has several subtleties compared to gears made from wrought steels. These features primarily deal with the marked difference in physical properties noticeably affecting the PM’s response to induction heating and quenching. Sintering specifics, sintering temperature, and secondary processing are other factors that affect the response of a PM material to IH. Challenges associated with differences in thermal properties (e.g., thermal conduction, specific heat, etc.) of PM materials are common for any type of heat treatment. However, differences in electromagnetic properties are specific for processing those materials applying electromagnetic induction. Table 10.2 shows the effect of a mass density reduction and porosity increase on selected material properties and their influence on induction hardening of PM gears. The µr of PM materials is significantly lower than the µr of the corresponding wrought steels, while their ρ being greater to some extent. Both factors lead to noticeably larger δ during the heating cycle, affecting not only heat generation and temperature profiles but also the magnitude and distribution of transient and residual stresses as well as coil electrical parameters (including coil impedance, load-matching characteristics, etc.). These features make hardening recipes for PM materials noticeably different compared to hardening wrought equivalents. When induction hardening PM gears, it is good practice to have a minimum density of at least 7.0 g/cm3 (0.25 lb/in.3). This will help obtain consistent heat treat results. When hardening surfaces that have teeth, holes, splines, undercuts, and TABLE 10.2 How Density Reduction (Porosity Increase) Affects Some PM Part Properties and Induction Hardening Parameters Property

Change

Influence on Induction Process

Thermal conductivity

Decrease

Electrical resistivity Magnetic permeability Hardenability Structural homogeneity

Increase Decrease

Less soaking action from high-temperature to low-temperature regions. Larger temperature gradients and thermal stresses during heating. Slower cooling during quenching Larger current penetration depth Larger penetration depth and lower coil electrical efficiency

Decrease Worse

More severe quench is required to provide the same case depth Inconsistency of hardening; variations in surface hardness, case depth, hardness scatter, and residual stress data. Increased tendency for cracking during hardening.

Source: V. Rudnev, Intricacies of induction hardening powder metallurgy parts, Professor Induction Series, in: Heat Treating Progress, ASM Int’l, Materials Park, OH, November/December, 2003, pp. 23–24.

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other geometrical discontinuities and stress risers, it is preferable to have a minimum density of 7.2 g/cm3 (0.26 lb/in.3). One reason for low-density PM gears to be prone to cracking during rapid heating is associated with a penetration of the gases into the subsurface areas of the part through the interconnected pores. Interconnected pores contribute to decreased part strength and rigidity compared with wrought materials. In addition, the poor thermal conductivity of porous PM parts encourages the development of localized hot spots and excessive thermal gradients and also requires the use of quenchants with intensified cooling rates to obtain the required hardness and case depths, because an increase in pore fraction and a reduction in density negatively affect the hardenability of PM materials compared to their wrought equivalents [1]. Aqueous polymer solutions of various concentrations (with 2–8% being the most typical) with some type of rust inhibitors and water (containing appropriate additives) are often used in surface hardening PM materials. Rust inhibitors help prevent internal corrosion. Oil quenching is sometimes specified when hardening PM gears, particularly for those with stringent dimensional stability requirements and those having a pronounced tendency to cracking. Concern about fires and environmental restrictions are obvious drawbacks to using oils and oil-based quenchants. Note that quench oils may require higher temperatures than polymer quenchants and water. Highly porous PM parts have a greater tendency to corrode and to take on an unsightly appearance. The causes of both can be traced to residual water-based quenchant trapped in subsurface pores. It is quite common for PM materials to absorb approximately 2% oil by weight. Therefore, intensive ventilation must be incorporated into machine design when heat treating PM gear-like components. It is also imperative to make sure that steps are taken to ensure that the reusable quenchants remain sufficiently clean and have appropriate characteristics when running high production. Obviously, density and porosity are not the only factors that affect the process of heat treatment of PM materials and probability of cracking. Other factors include the material composition, homogeneity of the microstructure (the degree of segregation), surface conditions (including, surface roughness), and the process recipe, as well as specifics of prior operations such as sintering of the green compact. In the case of sintering, factors include the process sequence, atmosphere used, pressure, temperature, degree of sintering, and segregation. High-temperature sintering is preferred because it improves microstructural homogeneity and ensures good diffusion. However, decarburization of the surface before IH should be avoided [1]. The alloying method used to produce the powder can also have a marked effect on heat treat results. Among alloying techniques are admixing, diffusion alloying, pre-alloying, hybrid alloying, and the metal injection molding (MIM) method. The technique used can affect material segregation and chemical and microstructural heterogeneity, because of different areas of the component undergoing abnormal phase transformations during cooling. For example, large inclusions may form, serving as stress risers, increasing the potential for cracking of the part, and causing inconsistent hardness readings. Under certain conditions, structural heterogeneity and porosities may even affect an eddy current flow. Depending on the specific composition, some PM gears may have a greater tendency for cracking during heat treatment. For example, special attention must be paid when developing IH recipes for copper alloyed iron-based PM. Although iron-based PM materials with a high copper content provide a good blend of properties at a competitive cost, they

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are often more prone to cracking. Besides, there is always a legitimate concern in regard to incipient melting when using copper alloyed iron-based PM utilizing excessive hardening temperatures. This makes it imperative to have sufficiently accurate process control and monitoring when hardening gears fabricated from such materials. It would be appropriate to review a couple of case studies at this point [1]. In order to reduce the crack sensitivity of a gear-like component with holes made of DIN Sint D11 (ISO P2045), it was specified having gentle preheating in the furnace (600°C for 40–45 min) and induction surface hardening using dunk quench in oil or comparable high concentration polymer quenchant followed by tempering. Application called for the following composition of this iron-based PM alloy: 0.6–0.8% C and 1.6–2% Cu. Preheating and closely controlled process recipe are needed to avoid crack developing during or after IH. As-quenched hardness was within the 71–73 HRA range with noticeable amount of RA. After tempering, hardness was increased to 76–79 HRA accompanied by some reduction in RA (it was allowed to have 16–18% of RA within Q&T structure). Besides preheating and in order to avoid delayed brittleness, it was also important to minimize the HAZ as well as peak temperature and to apply the proprietary tempering recipe immediately after quenching. Another case study discussed in [1] reveals that the microstructural heterogeneity in a copper alloyed iron-based PM gear-liked component may be so severe that, when examined under a microscope; it almost looked like copper had been electroplated on the surface of the part. This was the result of incomplete copper diffusion during alloying. Such heterogeneity can easily result in the redirection of the localized eddy current flow if high frequencies are applied potentially affecting hardness pattern. Iron-based PM components are often more prone to inconsistencies to IH because of variations in prior microstructures, sintering conditions, composition deviations, and segregation caused by “lot-to-lot” processing differences (including sintered density and composition) compared to wrought steels. Thus, greater inconsistency compared to hardening wrought steels may be observed. It should also be mentioned that although it is strongly recommended that PM inductionhardened gears should have a density of not less than 7.0 g/cm3 (0.25 lb/in.3), there is a number of successful applications where a hardening has been done on PM components with a density as low as 6.8 g/cm3. In some cases, it might be considered beneficial for induction surface hardening if PM parts have a variable “surface-to-subsurface” density. For example, the density at the surface to be hardened might be as high as 7.5 g/cm3 or even higher, and it gradually decreases to a base density of 7.0–7.2 g/cm3 below the surface. This helps maximize beneficial compressive residual stresses at the surface and increase the strength. It is important to avoid having sharp corners/edges on PM gears within regions required to be induction hardened. Sufficient chamfering and radii should be applied. When determining process parameters for hardening of PM gears, energies and frequencies higher than those used for wrought equivalents are often needed. Closer process control is also required. Preheating or pulse heating might be beneficial for obtaining the required hardness patterns and avoiding excessive heat surplus and crack developing. The PM industry continues to improve its technology. PM gears were sometimes tagged “low strength.” The low strength and high porosity of PM parts held back the widespread adoption of induction hardening. Not anymore. Improvements in PM gear manufacturing and a greater awareness of IH specifics have emerged in recent years. A number of different tools (including numerical computer modeling) are now available to develop intelligent process recipes that ensure success in induction hardening of PM gears.

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10.4  Technologies for Induction Gear Hardening 10.4.1  General Remarks Because gears provide transmission of motion and force, they belong to a group of the most geometrically accurate power transmission components. A gear’s geometrical accuracy and ability to provide a required fit to its mate greatly affect gear performance characteristics. Typical required gear tolerances are measured in microns; therefore, the ability to control such undesirable phenomena as gear warpage, ovality, conicality, out-of-flatness, tooth crowning, bending, growth, shrinkage, and the like plays a dominant role in providing quality gears [1]. This is why hardness pattern consistency, minimum shape/size distortion, and its repeatability are among the most critical parameters that should be satisfied. Gears are often manufactured with lightning holes to reduce weight (Figure 10.12). In induction hardening of gears with internal lightening holes, including hubless spur gears and sprockets, cracks can develop appreciably below the case depth in the inner hole areas. This crack initiation is caused an unfavorable stress distribution during or after quenching. Proper material selection, improved hardening process protocol (including quenching technique), and modification in gear design or the required hardness pattern can prevent crack development in the lightening hole areas. The first step in designing a gear-hardening machine is to specify the required surface hardness and hardness profile. Insufficient hardness as well as an interrupted (broken) hardness pattern at the tooth contact areas will shorten gear life due to poor load-carrying capacity, premature wear, tooth bending fatigue, rolling contact fatigue, pitting, and spalling, and can even result in some plastic deformation of the teeth. Among other factors, operating load condition (whether there are occasional, intermittent, or continuous loads) has a pronounced effect on the tooth geometry and needed hardness pattern. A through-hardened gear tooth with a hardness reading exceeding 60 HRC may sometimes be too brittle, exhibiting lack of toughness and potentially causing a premature brittle fracture. Hardened case depth should be adequate (not too large and not too small) to provide the required gear tooth properties without excessive distortion. There is a common misconception that a uniform contour profile is always the best pattern for all gear-hardening applications. It is not. In some cases, a certain hardness gradient can provide a gear with superior performance although, in many applications, a true contour hardening pattern is highly desirable yielding the best gear characteristics, maximizing the beneficial compressive residual stresses within the case depth, and minimizing tooth distortion.

FIGURE 10.12 Gears are often manufactured with lightning holes to reduce weight.

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10.4.2  Overview of Tooth Hardness Patterns A variety of hardening patterns achievable with induction hardening is shown in Figure 10.13 [1]. Pattern “A” is a flank hardening pattern that has been used since the late 1940s for hardening large sprockets and gears (with teeth modulus of eight and larger). The hardness region occupies the tooth flank area and ends before the tooth fillet. This pattern might provide the required wear resistance, but bending fatigue fracture is the typical failure mode. In the hardened-to-nonhardened transition region of tooth fillet or root land, the residual stresses change from compression in the hardened area to tensile in the nonhardened area. The maximum tensile residual stresses are located just below the end of the hardened pattern. A combination of applied tensile stresses with tensile residual stresses creates a favorable condition for early crack development in the root/fillet area, particularly for moderate and heavily loaded gears. Therefore, a mechanical hardening (i.e., roll or ball hardening) of the roots is required, developing the needed compressive residual stresses that will resist bending fatigue. However, mechanical hardening dramatically increases the overall cost, and it is normally preferable to use alternative hardness profiles such as Patterns “G” or “H”. Pattern “B” is a flank and tooth hardening pattern. This pattern has a shortcoming similar to the previous one, featuring poor load-carrying capacity, yet might be used in cases where wear resistance is of primary concern. Patterns “E,” “F,” and “G” provide better results when a combination of wear, tear, and bending fatigue resistance is required. Pattern “C” is a tooth tip hardening pattern. In this case, the gear has minimum shape distortion. However, the application of gears with this pattern is extremely limited because

FIGURE 10.13 Variety of hardening patterns achievable with induction hardening. (A) Flank hardening; (B) Flank and tooth hardening; (C) Hardening of tooth tip; (D) Root hardening; (E) Hardening of entire tooth and root; (F) Profile hardening. Nonuniform pattern; (G) Profile hardening. Uniform pattern; (H) Flank and root hardening. (From V. Rudnev et al., Handbook of Induction Heating, Marcel Dekker, 2003.)

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the two most important tooth areas (flank and root) are not hardened. As a matter of fact, because of the unfavorable residual stress distribution, the bending fatigue strength of a gear with this pattern, as well as Patterns ”A” and “B,” can even be lower compared to the “green” gear. In most cases, Patterns “F” and “G” would be better choices. Pattern “D” is a root hardening pattern. The maximum bending stresses are located in the tooth fillet area; therefore, this pattern provides good fillet/root strengthening. Since the root is reinforced, the maximum tensile residual stresses are shifted far away from the surface to a depth where tensile residual stresses will not complement the tensile applied stresses during service, reducing the probability of bending fatigue fracture. However, application of this pattern is quite limited. Since the tooth flank is not hardened, this pattern provides poor wear resistance that may result in removal or displacement of particles from the gear surface and tooth deformation. Theoretically, it is possible to imagine the necessity of using this pattern; however, it is more practical to use another pattern, such as Pattern “H.” Pattern “E” is one of the most typical patterns when hardening gears with small teeth as well as sprockets and splines. Because the tooth is through hardened, it might exhibit some brittleness unless fine-grain and ultrafine-grain martensite is formed. There is a danger of having a brittle fracture in gears with through-hardened teeth, particularly those subjected to shock/impact loads. The core of the tooth that exhibits sufficient toughness, ductility, and strength should be able to withstand impact loads and prevent plastic deformation of the gear teeth. Low-temperature tempering helps provide a combination of those properties. Low-temperature tempering lowers the final hardness down to 54–58 HRC (depending on application specifics). It is very important to avoid forming a microstructure of coarse martensite at tooth tip, addendum, and dedendum regions, including tooth core. Coarse martensite negatively affects impact strength, fatigue and bending strength making tooth brittle. Through-hardened teeth are commonly associated with lower compressive residual stresses at tooth surface or, in some extreme cases, even with localized reversal of compressive residual stresses to tensile residual stresses at working surface of tooth. Compressive stresses help inhibit crack development and resist bending fatigue. Among other factors, tooth fatigue properties are directly related to hardness (strength) and magnitude of compressive residual surface stresses. Induction surface-hardened gears are typically having impressive magnitude of compressive residual stresses at tooth surface and in its root area. Gears hardened using dual frequency and producing true contour profile or contour-like hardness pattern may have similar residual compression as case carburized gears or, even higher. However, lower magnitude of compressive residual surface stresses in the case of through-hardened teeth might be associated with lower fatigue life. This is the reason why the closer your hardness profile would be to a contour hardness profile, the better it will typically be for gear fatigue life. Therefore, it would be beneficial making attempts to achieve as close hardness profile to contour-like pattern as possible. This will increase a magnitude of useful compression at the surface. Patterns “F” and “G” are popular patterns for medium-size gears and referred to as profiled or contour-like hardness pattern and true contour (uniform profile) hardness pattern, correspondently. According to Pattern “F,” a case depth in the tooth root area is typically 30–40% of the depth in the tooth tip. Slightly larger hardness depth at the tooth pitch line compared to the root may be beneficial in some cases as a preventive action against spalling and pitting. It is very important to properly harden the entire gear perimeter, including the flank and root area. An uninterrupted hardened pattern of all contact areas of the tooth indicates good wear properties of the gear and it ensures the

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existence of an uninterrupted distribution of desirable compressive stresses at the tooth working surface. Because gear teeth are not hardened through, a relatively ductile tooth core (30–44 HRC) and a hard surface (56–62 HRC) provide a good combination of such important gear properties as wear resistance, toughness, bending strength, and the needed gear durability (assuming an appropriate gear design). Pattern “H” is very typical for large gears and pinions having weight that exceeds several tons with coarse teeth. This pattern provides an exceptional combination of fatigue strength and wear properties as well as resistance to shock loading and scuffing, which is very important for heavily loaded gears and pinions experiencing loads of appreciable magnitude. It is recommended that for these applications, surface hardness should not be too high, typically in the range of 54–59 HRC. If surface hardness exceeds 61–62 HRC, the gear might be too brittle. 10.4.3  Inductor Designs and Heating Modes Depending on the size of the gear, the required hardness pattern, and tooth geometry, gears are induction hardened by encircling the whole gear (external or internal) with an induction coil (the so-called “spin hardening”) or, for larger gears, hardening them toothby-tooth with either gap-by-gap or tip-by-tip heat-treating techniques [1]. Spin hardening provides higher production rates but requires a considerably greater amount of power and capital equipment investment because of the necessity of austenitizing the entire perimeter of the gear (particularly when hardening large gears). In contrast, the power demand of tooth-by-tooth inductors is substantially lower; however, the production rate is also noticeably reduced. 10.4.3.1  Tooth-by-Tooth Hardening of Gears As the name implies, each tooth is heated individually. The tooth-by-tooth method comprises two noticeably different techniques: tip-by-tip or gap-by-gap hardening. The tip-by-tip method can apply a static heating mode or scanning mode (although static is more typical). A gap-by-gap technique exclusively applies a scan hardening mode. Both techniques (tip-by-tip and gap-by-gap hardening) are not suitable for small and fine-pitch gears (modules smaller than six). In tip-by-tip hardening, an inductor encircles the body of a single tooth. Inductor geometry depends on the shape of the teeth and the required hardness pattern. In the case of static hardening of large sprocket teeth, a solenoid coil or a split-return inductor can be used. In other cases, a hairpin inductor can be applied to scan harden the tooth working surfaces. The use of this method produces hardening patterns “A,” “B,” and “C” (Figure 10.13). One of the typical concerns associated with tip-by-tip hardening is the problem of undesirable heating and softening (tempering back) of the areas adjacent to the hardened area, because of the external magnetic field of the inductor. This effect can be minimized to some extent using flux concentrators as shields. However, in cases when the allocation of concentrators can be difficult because of space limitations, the undesirable heating of the adjacent teeth can be reduced by applying thin copper shields (copper caps). Presently, tip-by-tip hardening is very rarely used because the hardening patterns usually do not provide the needed fatigue and impact strength of the root. This is the reason why the term tooth-by-tooth hardening is often exclusively associated with the gap-by-gap hardening method and, below, both tooth-by-tooth and gap-by-gap hardening will be used interchangeably.

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FIGURE 10.14 Inductors for tooth-by-tooth (also referred to as gap-by-gap) surface hardening of large gears. (Left) Close-up view of a single-butterfly style inductor and integral spray quench; (middle) Single-butterfly style inductor; (right) Double butterfly style inductor. (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

The tooth-by-tooth hardening concept can be applied to external and internal gears and pinions, and it requires the inductor to be symmetrically located between two flanks of adjacent teeth (Figure 10.14, middle). It can be designed to heat only the root and flanks of the tooth, leaving the tip and tooth core tough and ductile. This is one of the oldest IH techniques; however, recent innovations continue to improve the quality of gears that are heat treated using this method. Induction-hardened gears can be fairly large, with outside diameters easily exceeding 3 m and can weigh several tons. Gears used in wind turbines are typical examples where tooth-by-tooth induction hardening is effectively used [1,2]. There is a limitation to applying this method for hardening internal gears. Typically, it is required that the internal diameter of an internal gear exceed 200 mm (8 in.) and, in some cases, 250 mm (10 in.) or more. Since most wind turbines are constructed on remote sites, the size and weight of turbines in combination with the expenses associated with their repair demand superior strength and higher quality of wind energy generator components. Therefore, the quality of surface-hardened large gears directly affects the longevity of wind turbines and their competitiveness. This emphasizes the importance of taking steps to ensure the reliability and repeatability of hardness patterns. Gap-by-gap technique applies the scanning mode, requiring a high level of skill, knowledge, and experience. Scanning rates can reach 10 mm/s and even higher (though 6–8 mm/s is more typical). It is a time-consuming process with low production rates. Power requirements for these techniques are usually quite low. This can be considered a significant advantage, because if spin hardening is used, a large gear would require an enormous amount of power, which could diminish the cost-effectiveness of the heat treating or make it not even feasible. There is a variety of tooth-by-tooth inductor designs to accommodate the vast variety of gear types, tooth profiles, and sizes. Some of the most popular inductor designs are shown in Figure 10.14. Originally, the tooth-by-tooth inductor was developed in the 1950s by the British firm Delapena. As one can see in Figure 10.15, the path of the induced eddy current has a butterfly-shaped loop. The maximum current density is located in the root area (the body of the butterfly). This is the reason why this inductor style is also called a butterfly inductor. In order to further increase the power density induced in the root, a magnetic flux concentrator is applied. A stack of laminations is typically used as flux concentrators.

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FIGURE 10.15 Path of the induced eddy current resembles a butterfly-shaped loop. (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

Laminations are oriented across the gap. In some cases, it is advantageous to use a so-called double-butterfly inductor (Figure 10.14, right). Figure 10.16 shows the hardness profiles that are typical for gap-by-gap hardening. Although the eddy current path appears to be of a butterfly-like path, when applied with a scanning mode, the temperature is distributed within the gear roots and flanks quite uniformly. At the same time, since the eddy current makes a return path through the flank and, particularly through the tooth tip, proper care should be taken to prevent overheating the tooth tip. Overheating of the tip can substantially weaken the tooth. Applied frequencies are usually in the range of 3–30 kHz. At the same time, there are cases when a frequency of 70 kHz or higher has been used [1]. Pattern uniformity is quite sensitive to coil positioning. Asymmetrical positioning of an inductor results in a nonuniform hardness pattern due to electromagnetic proximity effect. For example, an increase in the air gap between the coil copper and the flank surface on one side will result in a reduction of hardness and shallower case depth there, altering the mechanical properties. Because of relatively small inductor-to-tooth air gaps (with 0.8–2 mm being typical) and harsh working conditions, these coils require intensive maintenance and have a relatively short life compared to inductors that encircle the gear. Too small an air gap can result in local overheating or even melting of the gear surface. Some arcing can occur between the inductor and the gear surface if the air gap is too small [1,18,19].

FIGURE 10.16 Etched hardness profiles that are typical for gap-by-gap hardening (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

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Precise inductor fabrication techniques, inductor rigidity, and superior alignment techniques are essential. Special locators or electronic tracking systems are used to ensure proper inductor positioning in the tooth space. Thermal expansion of metal during heating should also be taken into consideration when determining the proper inductor-to-tooth gap. After loading and initial coil positioning, the process runs automatically based on the application process protocol. When developing tooth-by-tooth gear hardening, particular attention should be paid to electromagnetic end/edge effects and the ability to provide the required pattern in the gear end areas, as well as along the tooth perimeter. To obtain the required temperature uniformity, it is necessary to use a complex control algorithm: Inductor Power versus Scan Rate versus Inductor Position. A short dwell at the initial and final stages of inductor travel is often used. Thanks to preheating attributed to thermal conduction, the dwell at the end of coil travel is usually shorter compared to the dwell at the beginning of travel or is not applied at all. With the scanning mode for hardening gears with wide teeth, two techniques can be used: a design concept where the inductor is stationary and the gear is moveable, and a concept that assumes the gear is stationary and the inductor is moveable. There can be noticeable shape/size distortion when applying tooth-by-tooth hardening. Shape distortion is particularly noticeable in the last heating position where the last tooth can be pushed out 0.1–0.8 mm. Hardening every second tooth can minimize distortion. Obviously, this will require two revolutions to harden the entire gear. Still, final grinding may be required. There is a linear relationship between the volume of required metal removal and the grinding time. Thus, excessive distortion leads to a prolonged grinding operation and increases the cost. Repeatable distortion may be compensated for during gear fabrication. Even though there might be an appreciable distortion when hardening large gears and pinions (e.g., mill, marine, and large transportation gears, etc.), its magnitude is not typically as significant when compared to carburized gears. Carburizing requires soaking of gears for many hours (in some cases, up to 30 h or longer) at temperatures of 850°C (1562°F) to 950°C (1742°F). At these temperatures, the large masses of metal expand to a much greater extent compared to the case when only the surface layer is heated by electromagnetic induction. The expansion of a large mass during prolonged heating during carburizing and the contraction during cooling/quenching “move” the metal to a much greater degree, resulting in larger gear distortion. Besides that, large gears being held at temperatures of 850°C (1562°F) to 950°C (1742°F) for many hours have little rigidity; therefore, they can sag and tend to follow the movement of their supporting structures during soaking and handling. With induction hardening, areas unaffected by heat as well as areas with temperatures corresponding to the elastic deformation range serve as shape stabilizers and lead to not only lower but more predictable distortion. One typical concern when applying tooth-by-tooth hardening is the challenge related to undesirable heating and softening of the areas adjacent to the hardened region (tempering back). The main cause of undesirable softening/temper back is associated with the thermal conductivity phenomenon. Heat is transferred by thermal conduction from a high temperature region of the tooth toward a lower-temperature region and is a function of the temperature difference, distance, and the value of thermal conductivity. Most metals are relatively good thermal conductors. During steel hardening, the surface temperature exceeds the critical temperature Ac3. Therefore, when austenitizing one side of the tooth, there is a danger that the opposite side of the tooth will be heated by thermal conduction to

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FIGURE 10.17 Undesirable softening of tooth tip area of previously hardened areas. (From V. Rudnev et al., Handbook of Induction Heating, 2nd edition, CRC Press, New York, 2017.)

an inappropriately high temperature. This can result in undesirable softening of previously hardened areas (Figure 10.17). Whether a hardened side of the tooth will be unacceptably softened depends on several factors, including the applied frequency, gear module, tooth shape, heat time, case depth, and others. In the case of shallow case depths and thick teeth, the root of the tooth, its fillet, and the dedendum area are typically not unduly softened because of thermal conduction. The massive area below the tooth root serves as a heat sink, which helps protect the hardened side of the tooth from tempering back. Conversely, the addendum and, in particular, tooth tip region can be considered as areas of concern because there is a relatively small mass of metal. In addition, heat has a short distance to travel from one (austenitizing) side to the other (already hardened) side of the tooth (Figure 10.17). To overcome the problem of tempering back, special cooling spray blocks are applied. Additional cooling protects already hardened areas while austenitizing unhardened areas of the tooth. Even though external cooling is applied, there still may be some unavoidable softening depending on the tooth shape and process parameters. This tempering back is typically insignificant, well controlled, and acceptable (Figure 10.16). Tooth-by-tooth hardening can be applied for gears submerged in a temperature-controlled quench tank. This technique was applied in the original Delapena induction hardening process. In this case, quenching is practically instantaneous. However, noticeably higher power is needed to compensate for the cooling effect of the quenchant during heating. The fact that a gear is submerged in quenchant also helps prevent the tempering back problem. In addition, the quenchant serves as a coolant to the inductor. Therefore, in submerged hardening, an inductor might not have to be water cooled. On the other hand, there may be some obvious challenges to set up this system because of the quenchant obstruction issues. Figure 10.18 shows two standard tooth-by-tooth induction hardening machines built by Inductoheat Inc. (left) for hardening large gears and by Inductoheat-Europe (right) for heat treating large bearing rings for a wind energy turbine with teeth located on the inside diameter of the ring. A bearing ring O. D. can be as large as 140 in. (3556 mm), and the maximum weight exceeds 11,000 lb (5000 kg). The required case depth is 2.5–3.5 mm. The z-axis scan height (tooth width) is 13.75 in. (350 mm). As expected, in any induction surface hardening, applied frequency has the greatest effect on depth of heat generation and on an appearance of electromagnetic end and edge effects. It is advantageous to apply various combinations of frequency, power density, and scan rates at various stages of the gear scan hardening cycle. This would allow improving the metallurgical quality of teeth, effectively addressing the presence of end zones (sides

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FIGURE 10.18 Two tooth-by-tooth induction machines built by Inductoheat Inc. (Left) for hardening large gears and by Inductoheat-Europe (right) for heat treating large bearing rings for a wind energy turbine with teeth located on inside diameter of the ring. (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

Frequency, Power

of gear) as well as specifics of gear tooth nomenclature. Unfortunately, the majority of inverters do not have such capability. A new generation IGBT-type inverter (Statipower-IFP) recently developed by Inductoheat Inc. (Figure 10.19, left) eliminates this limitation and simplifies achieving the required hardness pattern [20]. The patented technology is specifically developed for induction scan hardening needs. It enables instant and independent adjustment of frequency (from 5 to 60 kHz) and power (Figure 10.19, right) in a preprogrammed manner during the heating cycle, optimizing electromagnetic, thermal, and metallurgical conditions. The capability of IFP inverters to instantly and independently change power and frequency during scan hardening is essential for better control of end/edge effects, help to avoid edge overheating and cracking, and it is particularly beneficial when hardening a variety of gears with appreciably different tooth geometry and case depth requirements.

Frequency Power

Process Time FIGURE 10.19 New generation of transistorized inverters. The Statipower-IFP (left) provides Independent Frequency and Power control allowing instantly and independently controlled frequency and power during induction scan hardening (right). (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

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This innovative technology effectively addresses industry needs for cost-effectiveness and enhanced process flexibility, greatly expanding induction equipment capabilities and further improving metallurgical quality of induction hardening. 10.4.3.2  Gear Spin Hardening (Encircling Inductors) Spin hardening of gears utilizes a single or multiturn inductor that encircles the gear (Figure 10.20). It is typically used for small- and medium-sized gears. Unfortunately, spin hardening sometimes cannot be easily used for certain gears with complex geometries because of the difficulty in obtaining a contour-like (uniform) austenitized surface layer before quenching. Besides, in case of appreciable size gears, it might also require an excessively large amount of power owing to the necessity of short heat times suppressing thermal conduction in order to obtain the desired hardness pattern uniformity. Still, gear spin hardening is the most popular technology for hardening a variety of gears of small and moderate size. As always in life, there are some exceptions, and encircling coils have also been used for hardening large gears and gear-like components. For example, Figure 10.21, left, shows an induction machine for hardening large sprockets. A multiturn encircling inductor is used for hardening sprockets with a major diameter of 27.6 in. (701 mm), root diameter 24.3 in.

FIGURE 10.20 Examples of spin hardening of gears. (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

FIGURE 10.21 Two examples of induction hardening machine applying encircling coils. (Left) A multiturn encircling inductor is used for hardening sprockets with a major diameter of 27.6 in. (701 mm), root diameter 24.3 in. (617 mm), and face width 3.125 in. (79 mm). (Right) Induction hardening of output gear 24 in. (610 mm) outside diameter with teeth located on inside diameter. Solenoid-style inductor encircles inside diameter. (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

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(617 mm), and face width 3.125 in. (79 mm). Figure 10.21, right, shows induction hardening of output gear 24 in. (610 mm) outside diameter with teeth located on inside diameter. Solenoid-style inductor encircles inside diameter. Gears are usually rotated during heating to ensure even distribution of energy compensating for the field fringing (fish-tail) effect [1]. At the same time, there are applications where workpiece rotation is not required (Figure 10.20, right). Conventionally designed single-turn inductors and some multiturn coils have areas where there is an inevitable distortion of the magnetic field, which could lead to the appearance of regions with a heat deficit and where undesirable microstructures can be formed. One region is related to an area where copper buses that transmit electrical current from a power source are connected to an induction coil. These connectors sometimes have the shape of a fish tail, and the region is often called a fish-tail region (although the proper term is field fringing region) of the inductor. In case of using a single-turn solenoid coil, the positioning of the field-fringing is associated with an area of incoming and outgoing electrical currents oriented in opposite directions at the regions where the connection buses (coil terminals) join the inductor copper. This results in magnetic field distortion that leads to a deficit of the heat generation within the corresponding region. One of the typical solutions to compensate for this fishtail effect is to rotate the part during heating, ensuring that all regions of the workpiece absorb the same energy during the entire process cycle. When applying encircling coils, there are five parameters that play a dominant role in obtaining the required hardness pattern: frequency, power density, heat time, quenching conditions, and coil geometry. Certain combinations of these parameters can result in different hardness patterns. As it has been discussed earlier, Figure 10.5 shows a diversity of hardening patterns that were obtained on the same carbon steel shaft with teeth using different combinations of heat time, frequency, and power [1,3]. Review of an eddy current flow in two extreme cases discussed in Section 10.2.3 (Figure 10.6): application of high frequency (left) and low frequency (right) helps to better understand conditions for determining process recipes for obtained hardness patterns. Finite element analysis (FEA) illustrates the effect of frequency on temperature distribution and hardness pattern while applying different frequencies. Figures 10.22 through 10.25 show the dynamics of temperature distribution during heating and quenching of a fine pitch gear using various frequencies: RF (300 kHz), moderate frequency (30 kHz), and lower frequency (10 kHz) [21]. Advantage was taken with respect to periodicity and symmetry of the gear teeth, allowing the modeling of only half of the tooth (right side). As expected, when 300 kHz is applied, the eddy current induced in the tooth follows its contour (Figure 10.22). Because the highest concentration of current density will be in the tip of the tooth, there will be a heat source surplus there compared to the root. Also, taking into account that the tip of the tooth has a smaller mass of metal to be heated compared with the root, the tip will experience the most intensive temperature rise during the heating cycle. In addition, from a thermal perspective, the amount of metal beneath the root circle represents a much larger cold sink compared with the addendum area and particularly in the tooth tip region. Another factor that contributes to more intensive heating of the tip is better electromagnetic proximity between the inductor and tooth tip versus its root. Higher frequency has the tendency to make the proximity effect more pronounced. These factors provide rapid austenitization of the tooth tip (Figure 10.22, top images). Temperatures of two critical nodes (N1, tip; N2, root) are in degrees Celsius and X–Y dimensions are in meters. Spray quenching results in a martensitic layer at the tooth tip only (Figure 10.22, bottomright). Note that the dedendum and the root area have not been hardened, because insufficient heat was generated there for austenitization.

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FIGURE 10.22 Temperature distribution during heating and quenching of a fine-pitch gear using RF (300 kHz); heat time is 1.2 s. (From V. Rudnev, Spin Hardening of Gears Revisited, Professor Induction Series, in: Heat Treating Progress, ASM Int’l, Materials Park, OH, March/April, 2004, pp. 17–20.)

The next simulation was conducted using a moderate frequency, which is 10-fold lower (using 30 kHz, Figure 10.23). At the end of the heat cycle, the whole tooth was heated quite uniformly, achieving the austenite phase temperature and resulting in a through-hardened tooth upon quenching (Figure 10.23, bottom-right). The study was continued by further reduction of frequency, using 10 kHz, representing a 30-fold reduction compared to the case study shown on Figure 10.22. With 10 kHz, the eddy current flow and temperature distribution in the gear tooth are quite different (Figure 10.24). Keep in mind that the heat time in all three cases discussed so far was kept the same (1.2 s). A frequency reduction from 300 to 10 kHz noticeably increases the eddy current penetration depth in the hot steel, from 1 to 5.4 mm. In a fine-toothed gear under consideration, such an increase in current penetration depth results in eddy current cancellation in the tooth tip and addendum (pitch circle) area heated above A2 critical temperature. This makes it easier for eddy current to take a short path, following the root circle of the gear instead

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FIGURE 10.23 Temperature distribution during heating and quenching of a fine-pitch gear using moderate frequency (30 kHz); heat time is 1.2 s. (From V. Rudnev, Spin Hardening of Gears Revisited, Professor Induction Series, in: Heat Treating Progress, ASM Int’l, Materials Park, OH, March/April 2004, pp. 17–20.)

of following the tooth contour. The result is more intensive heating of the tooth root area compared with its tip and the development of martensite upon quenching. In contrast to using 300 kHz, with 10 kHz upon reaching the Curie point, the temperature rise in the tip and entire addendum region was stopped; thus, proper austenization does not occur and hardening did not take place there. Notice the effect of a slight increase in heat time from 1.2 to 1.5 s when using 10 kHz (compare Figure 10.24 with Figure 10.25), leading to hardening not only the root but also the dedendum region. As a rule, when it is necessary to harden only the tooth tips, a higher frequency and highto-moderate power densities are applied (Figure 10.6, left). When hardening the tooth root, a lower frequency in combination with short time and high-power density is used (Figure 10.20, left; Figure 10.6, right). Low power density and extended heat time produces a deep pattern with an enlarged transition zone. It is imperative to keep in mind that depending on gear geometry, besides frequency, the variation of applied power density can shift the heat intensity within the gear tooth. For

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FIGURE 10.24 Temperature distribution during heating and quenching of a fine-pitch gear using lower frequency (10 kHz); heat time is 1.2 s. (From V. Rudnev, Spin Hardening of Gears Revisited, Professor Induction Series, in: Heat Treating Progress, ASM Int’l, Materials Park, OH, March/April 2004, pp. 17–20.)

example, the application of sufficiently high frequency in combination with a relatively low power density could result in tip hardening or hardening only the tip and addendum area. However, if the power density is substantially increased (keeping the same frequency), it could result in more intense heating of the root area instead of the tooth tip. Higher power density reduces µr by saturating the steel and leading to a considerable increase of δ, thus modifying the eddy current flow within the gear tooth. Figure 10.26 shows an example of applying excessive power density and the use of lower than-optimal frequency when heating a gear in an encircling coil. Gear teeth being approximately 8 mm tall have some undercuts. Regardless of the fact that the tips of the teeth were located much closer to the current-carrying face of the coil copper compared to the roots, the great majority of eddy currents have completed their loop following the root circle, resulting in severe overheating there. As expected, the terms high frequency and low frequency have relative meanings. For example, depending on the tooth geometry, a frequency of 3 kHz may act as a high

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FIGURE 10.25 Temperature distribution during heating and quenching of a fine-pitch gear using lower frequency (10 kHz); heat time is 1.5 s. (From V. Rudnev, Spin Hardening of Gears Revisited, Professor Induction Series, in: Heat Treating Progress, ASM Int’l, Materials Park, OH, March/April 2004, pp. 17–20.)

frequency for coarse teeth and 300 kHz may act as a low frequency for splines, threads, fine teeth, or skinny teeth, for example. In addition to the factors discussed above, the hardness pattern and its repeatability depend strongly on the relative positioning of the gear with respect to the induction coil and the ability to maintain gear concentricity during the heat cycle. 10.4.3.3  Quenching Options The primary objective of quenching is to provide the required rate of heat removal to arrive at the desired microstructure, hardness level, and pattern, which produce needed industrial characteristics of the gears while optimizing the residual stress distribution. There are several ways to accomplish quenching for gear spin hardening [1]. One technique is to submerge the gear in a quench tank upon completion of austenization (Figure 10.21, right). This technique (also referred to as dunk or immersion quenching) is

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FIGURE 10.26 Example of applying an excessive power density and the use of very low frequency when hardening gears using an encircling coil. (From V. Rudnev et al., Handbook of Induction Heating, 2nd edition, CRC Press, New York, 2017.)

commonly used for moderate and relatively large-sized gears where gear spin hardening is applicable. Small- and medium-sized gears are usually spray-quenched in place, using an integrated quench. The quench barrel and quench holes can be integrated into a hardening inductor. This design is commonly referred to as a machined integral quench (MIQ) inductor. The third technique requires the use of a separate concentric spray quench (ring, barrel, follower, or block may be used), which is separate from the induction coil and is typically located below the inductor or in its close proximity or austenitized gear may be sprayquenched off place. A variety of different quench media may be used, beginning with water and progressing through various aqueous solutions including water-based polymer solutions, petroleum oil–based quenchants, and vegetable oils. Quench oils raise concerns not only with respect to their disposal but also with regard to combustion, flammability, and fire hazards. In induction gear hardening, quench oils are typically used with dunk/immersion quenching only. In some cases, a flame will be seen and quickly extinguished during gear immersion. Smoke is frequently produced at that time, requiring an appropriate exhaust ventilation system. Oil-based quenchants also require a closed system to contain the oil and special techniques to clean the parts before their transfer to the next operation. In a high-production environment, there is the danger of quench oils accumulating the heat from hot gears and exceeding the temperature of the flash point. Therefore, means must be adapted for their adequate cooling. Concerns about fires, the necessity of removing oil drag out, and certain environmental restrictions are obvious drawbacks to using oils and oil-based quenchants. The use of vegetable oils (including canola, soybean, corn, cottonseed, and sunflower oils) is associated with lesser environmental restrictions compared to petroleum oil–based quenchants. Still, their application in IH is limited. Quench oils are not recommended with spray quenching because of the fire hazard. These are some of the reasons why aqueous polymer solutions are the most popular quenchants for induction gear hardening. There are a variety of complex thermo-hydro-dynamic processes involved in spray quenching, with the cooling intensity being a function of several factors including the temperature of the austenitized phase, gear rotation, type and purity of quenchant, pressure/flow, design of the quench assembly, number and distribution of quench holes (orifices), size of orifices and angle of drilled holes (impingement angle), quenchant temperature, and others. Usually, spray quench devices are directed at an austenitized

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surface of the gear with the intent to completely cover the austenitized area with quench fluid providing sufficient uniformity. Quench orifices are placed facing the surface of the gear at approximately 4- to 6-mm intervals in a staggered pattern. The orifice diameter is related to the specifics of the quenching requirements, including the coil and workpiece geometries, the air gap between the quench device and the gear, the type of quenchant, its concentration, required flow, and others. The quench assembly may be positioned quite closely to the inductor. In order to minimize unnecessary eddy current losses within the quench assembly, it is usually made of nonmetallic materials. Dirt and foreign deposits (e.g., hard water deposits) can clog the orifices, which may build up slowly in quench devices, reducing their cooling capacity and introducing cooling nonuniformity and may necessitate extensive cleaning or replacement. There is a common misunderstanding regarding the ability to apply the widely published, classical cooling curves for immersion quenching in spray hardening applications. Classical cooling curves representing three stages of quenching—vapor blanket (Stage A), nucleate boiling (Stage B), and convective cooling (Stage C)—cannot be applied directly to spray quenching. The differences are both quantitative and qualitative, and include, but are not limited to [1]: • Specifics of film formation and heat transfer through the vapor blanket during the initial stage of quenching. The thickness of the vapor blanket film is typically much thinner during spray quenching than when the part is submerged in a quench tank, and depends on flow rate, impingement angle, part rotation, and other characteristics of the quenching system. This vapor film is unstable and could be frequently ruptured. • The kinetics of formation, growth, and removal of bubbles from the surface of the heated component during nucleate boiling (Stage B) is noticeably different. During nucleate boiling, bubbles are smaller because they have less time to grow. And much larger numbers of bubbles form during spray quenching and the intensity with which they remove heat from the surface of the component is substantially greater compared with immersion quenching. • Due to the nature of spray quenching, Stages A and B are greatly suppressed in time, while cooling during the convection heat transfer stage (Stage C, where the cooling is accomplished by heat convection from the gear surface to the quench medium) is noticeably more intense, compared with the process represented by classical cooling curves. • In addition, the transition between Stages A and B is smoother with spray quenching than that shown by classical cooling curves for immersion quenching. • Another factor that has a considerable effect on quench severity in induction surface hardening is the existence of the “cold sink” effect. In the majority of gear surface hardening applications, the temperature of subsurface regions and, in particularly below the root circle, does not rise significantly, due to a skin effect, high power density, and short heating time. Colder regions of a gear complements spray quenching by further increasing the cooling intensity of austenitized surface. Press quenching or die quenching are used in special cases where distortion must be minimized while heat treating relatively thin gear-like components, (for example, tooth plates or flywheels with teeth, as well as some spiral bevel gears). A component may be held in position by a press or die that is water- or oil-cooled in order to reduce the temperature of

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FIGURE 10.27 In many applications, obtaining a true contour hardening pattern or contour-like pattern are highly desirable, yielding the best gear characteristics, maximizing beneficial compressive residual stresses within the case depth, and minimizing distortion. (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

the part while maintaining its geometry as near to its original shape as possible. If required, preferential cooling rates at different regions of the workpiece can be achieved. Since parts must be handled individually, this process can be quite expensive and time-consuming. More discussion on a subject of quenching in induction hardening can be found in [1,26]. 10.4.3.4  Heating Modes for Encircling Inductors In many applications, obtaining a true contour hardening pattern or contour-like pattern (Figure 10.13 “F” and “G”; Figure 10.27) are highly desirable, yielding the best gear characteristics, maximizing beneficial compressive residual stresses within the case depth, and minimizing distortion. Unfortunately, obtaining such patterns can be a challenging task due to unfavorable eddy current flow for some gear geometries. Both vertical and horizontal designs for induction gear hardening have been used by different manufacturers, although horizontal gear processing is more popular. Depending upon the process requirements, size, and geometry of the gear, hardening equipment can be designed as a relatively simple apparatus with manual loading/unloading or can involve sophisticated fully automated high-production machinery. Different heating modes have been applied as an attempt of achieving required gear hardness patterns. The five most popular induction-heating concepts that employ encircling-type coils are as follows [1]:

1. Conventional single-frequency concept. 2. Pulsing single-frequency concept. 3. Pulsing dual-frequency concept.

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4. IFP technology. 5. Single-coil, dual-frequency concept (also referred to as simultaneous dualfrequency concept). 10.4.3.4.1  Conventional Single-Frequency Concept (CSFC) CSFC [1] is commonly used for hardening splines and gears with medium and small teeth. In these applications, it is often acceptable to through harden teeth (Figure 10.13, Patterns “B” and “E”), and CSFC can be a cost-effective approach to do so. Quite frequently, CSFC can also be successfully used for medium-size gears without through hardening the entire tooth, but having deeper case depth at the tip and shallower case depth at the root with a pattern somewhat similar to that shown in Figure 10.13, “F”. Although CSFC is the most cost-effective approach for small- and medium-sized gears, there are cases where this concept has been successfully used for heat treating larger gears, pinions, and sprockets as well (Figure 10.21). Figure 10.28, left, shows another example of a two-station induction gear-hardening machine that applies CSFC. The gear-like component being heat treated in this application is an automotive transmission part for a parking brake having helical fine teeth on the inside diameter (I.D.) and coarse teeth on the outside diameter (O.D.). Both the inside and outside diameters require hardening (Figure 10.28, middle). Because of the difference in size of teeth, the hardening of the fine teeth requires a noticeably higher frequency than the outside diameter. Frequency of 200 kHz was chosen for I.D. heating and 10 kHz was chosen for the O.D. Regardless of using 200 kHz being RF, the fine teeth are through hardened, but the coarse teeth reveal some contouring of the hardness pattern despite of using a much lower frequency of 10 kHz (Figure 10.28, right). Therefore, electromagnetically speaking, a frequency of 200 kHz “acts” as low frequency and, in contrast, 10 kHz “acts” as a high frequency in this application. At the end of the austenization, the quenchant is applied in place; that is, no repositioning is required. Practically instantaneous quenching provides a consistent metallurgical response and precise hardness pattern control. There is a sufficient gear rotation during heating, but during quenching, there is reduced gear rotation to ensure that the polymerbased aqueous quenchant penetrates all critical areas of the teeth. Quenching reduces the gear temperature to the level suitable for safe handling.

FIGURE 10.28 Automotive transmission component with helical fine teeth on the inside diameter (I.D.) and coarse teeth on the outside diameter (O.D.) for a parking brake being induction hardened (left). Both the inside and outside diameters require hardening (middle and right). (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

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Signature process monitoring is applied at each station. It verifies critical machine settings and provides confidence in the quality of processing for each individual gear. This includes energy input, quench flow rate, temperature, quench pressure, heat time, and others. Parts are accepted or rejected based on all the major factors that affect the hardening process. Precise control of the hardening operations and appropriate coil design minimize the distortion, forming the desirable magnitude and distribution of residual stresses. The hardened gear is then inspected and moved to the next operation. Quite often, in order to optimize gear performance and reduce distortion, it is desirable to make a hardness distribution as close to contour hardening as possible. Unfortunately, CSFC does not always allow doing so. 10.4.3.4.2  Pulsing Single-Frequency Concept (PSFC) In order to overcome the drawbacks of CSFC and to improve the hardness distribution, PSFC was developed. In many cases, PSFC allows the user to noticeably improve the shortcomings of CSFC and obtain a hardening pattern that is more contoured than patterns obtained using the CFSC concept. Pulsing assists in providing the desirable heat flow toward the root without overheating of the tooth tip. A typical PSFC protocol consists of a moderate-power or low-power preheat, soaking stage, short high-power final heat, and quenching. This approach is also known as dual pulse hardening. Preheating improves the heated depth at the root, enabling an enhancement of the metallurgical result. Preheat times typically range from several seconds to a several dozen seconds, depending on the size and gear geometry. Depending on the application specifics, the preheat stages can consist of several heating pulses alternated with powerless dwell/soak stages. Soak times might be ranging from less than 1–10 s to achieve more suitable thermal conditions within the tooth prior to a final heating. Final heat times can range from less than 1 s to a few seconds. As expected, preheating reduces the amount of energy required in the final heat. 10.4.3.4.3  Pulsing Dual-Frequency Concept (PDFC) The idea of using two different frequencies to produce the desired contour hardness patterns has been around since the late 1950s. This concept was primarily developed to obtain a contour hardening profile for helical and straight spur gears. Obviously, since that time, the process has been refined and several innovations have been developed. According to PDFC, the gear is preheated using lower frequency to a temperature that is usually 350–100°C below the critical temperature Ac1. As expected, the higher the preheat temperature is associated with lower the power required for the final heat. However, excessively high preheat temperatures can result in increased distortion. As with PSFC, PDFC can be accomplished using a single-shot mode or scanning mode. The scanning mode might be applied for hardening shafts with teeth or gears having wide faces. Preheating is usually accomplished by using a medium frequency (3–10 kHz). Depending on the type of gear, its size, and material, higher frequency (30–450 kHz) in combination with high power density is applied during the final heat, where the selected frequency allows eddy currents to penetrate only to the desired depth, helping to obtain a contour-like hardness pattern. Originally, a two-coil arrangement was used. In the case of a single-shot heating mode, a two-step index-type approach is used. One coil provides preheating, and another coil provides final heating. Both coils work simultaneously if the scanning mode is applied. In some cases, dual-frequency machines produce components with lower distortion having a more favorable distribution of residual stresses compared to IH techniques

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relying upon single-frequencies. However, depending on the tooth profile and presence of undercuts, the time delay between low-frequency preheating and high-frequency final heating could have a detrimental negative effect for obtaining truly contour hardened patterns. Attempts have been made to minimize the gear indexing time, keeping it within the 0.25–0.60 s range [4]. However, further reduction is limited by inertia and the mechanical capability of the transferring/indexing mechanisms. Even such a seemingly short transfer time delay between preheating and final heating may have a sufficient negative impact that discourages the achievement of optimal gear heat treat properties. 10.4.3.4.4  Independent Frequency and Power Control Concept (Statipower-IFP Technology) One of the impressive technologies that can significantly improve the encircling gearhardening process is IFP Technology. This technology has been reviewed in Section 10.4.3.1 for tooth-by-tooth hardening (Figure 10.19). IFP Technology allows independent and instantaneous changing of the power and frequency (within the 5–60 kHz range) during heating cycle. Therefore, this technology allows eliminating some shortcomings associated with PSFC. The Statipower-IFP inverter’s (Figure 10.19, right) ability to independently and instantly (like a CNC machine) change both frequency and power during the heating cycle allows heat treaters to use a lower frequency for preheating the root areas, while an instant change to higher frequency helps ensure sufficient heating of tooth flanks and tips when hardening moderate-sized gears. This technology eliminates the need for using expensive high-speed transferring/indexing mechanisms and noticeably improves the hardness pattern and its repeatability. 10.4.3.4.5 Single-Coil, Dual-Frequency Concept (Also Referred to as Simultaneous Dual-Frequency Concept) Limitations of the PDFC concept have initiated the development of an alternative technology called a single coil simultaneous dual-frequency gear hardening as a way to further improve the quality of induction-hardened gears [1,4,22]. The core of simultaneous dual-frequency gear-hardening technology is associated with the development of solid-state inverters capable of producing two substantially different frequencies simultaneously. Figure 10.29 shows some examples of the waveforms generated by an Inductoheat’s simultaneous dual-frequency inverter. Waveforms representing coil voltage (top) and coil current (bottom) comprise two appreciably different frequencies that can be applied at the same time to a single inductor [22]. The lower-frequency output of the power supply helps austenitize the roots of the teeth and the high frequency helps austenitize the flanks and tips. Thus, two appreciably different frequencies can be applied to a hardening inductor simultaneously, making it much easier to obtain a true contour hardening of the gear teeth (examples of some of true contour hardening patterns are shown in Figure 10.27). Experience reveals that it is not always advantageous to have two different frequencies working simultaneously. Many times, depending on the gear geometry, it is preferable to apply lower frequency at the beginning of the heating cycle, and after achieving the desirable root heating, the higher frequency can complement the initially applied lower frequency, completing the job by working together. Figure 10.30, left, shows Inductoheat’s single-coil, dual-frequency induction gearhardening system utilizing a total output power of 1.4 MW. As expected, for different applications and gear sizes, the output power might not need to be so high. Figure 10.30, right, shows a sketch of one possible circuitry of this technology. In this scheme, two single-frequency modules generate substantially different frequencies. Specially designed

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FIGURE 10.29 Examples of the waveforms generated by a simultaneous dual-frequency inverter. Waveforms representing coil voltage (top) and coil current (bottom) comprise two appreciably different frequencies that can be applied at the same time to a single inductor. (From V. Rudnev, Single-Coil Dual-Frequency Induction Hardening of Gears, in: Heat Treating Progress, ASM Int’l, October 2009, pp. 9–11.)

FIGURE 10.30 Inductoheat’s single-coil dual-frequency induction gear-hardening system utilizing a total output power of 1.4 MW (a) and a sketch of one possible circuitry of this technology (b). (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

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FIGURE 10.31 Effect of different combinations of time–power–frequency of hardening patterns of spur gears. (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

filters prevent undesirable interactions between modules. Those modules can work not just simultaneously but in any sequence desirable to optimize the desired properties of the gear-hardening process [22]. The relatively high cost is the main shortcoming of power supplies that produce two different frequencies simultaneously. Nevertheless, single-coil simultaneous dual-frequency technology has a number of obvious advantages over conventional single-frequency hardening. Higher cost can be justified using this technology when it is appropriate. As an example, Figure 10.31 illustrates the effect of different combinations of time, power, and frequency on hardening patterns of the spur gears.

10.5  Residual Stresses at Tooth Working Surface Compressive residual stresses after induction hardening of gears and gear-like components applying single frequencies are typically within −400 to −550 MPa. The magnitude of residual stresses depends greatly upon material, its prior microstructure, hardness pattern profile, and the process recipe/protocol. Shorter heat time normally produces higher compressive residual surface stresses. Simultaneous dual frequencies commonly allow one to minimize the heat time, producing true contour or closer-to contour hardness patterns, and thus allowing increase compressive residual stresses that could reach −600 to −700 MPa [3]. However, some gears with “skinny” teeth having a narrow tooth face/thickness might not allow for a true contour hardened pattern almost regardless of the applied frequencies or heating modes reducing a magnitude of compressive residual stresses at the tooth surface. It is also critical to make sure that in through-hardened regions of tips or addendum, there is fine martensitic structure, not a coarse martensitic structure. This will help to improve toughness, ductility, and impact strength regardless of the through hardening tip area [1,4]. Some gear manufacturers applied shot peening after induction surface hardening of gears, further increasing compressive residual stresses at the surface and subsurface and improving fatigue strength, bending strength, and obtaining needed pitting prevention characteristics [3]. Sometimes, double shot-peening is applied. During the first stage of peening, the larger particles are used, allowing one to increase the depth of compression. Then, for the second stage, they apply high-intensity shot-peening utilizing smaller size particles (so-called conditioned cut wire particles), further increasing the magnitude of residual compressive stresses at tooth surface and making sure that hard-to-reach regions are properly treated. In such cases, residual surface compression may exceed −900 MPa. It is important to apply

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so-called conditioned cut wire particles; as such particles have a special finish with sharp edges removed without damaging the tooth surface and improving pitting characteristics. Large shot-peening particles are typically within 0.6–0.8 mm dia., and small particles are within 0.1–0.25 mm dia. Again, not all induction surface-hardened gears and gear-like components undergo a single or double shot peening, but some of them may apply it.

10.6  Hardening Components Containing Teeth Besides gears, pinions, and sprockets, induction hardening is successfully used for hardening various components containing teeth. Steering racks are typical representatives of such components [1]. Steering racks (Figure 10.32, left) are an essential component of automotive power-steering systems. Size and geometry specifics of the toothed steering racks as well as their heat treat specification depend on the design of the particular steering system. Steering racks consist of toothed and ball-screw sections needed to be surface hardened. Minimization of shape distortion after hardening without compromising the metallurgical characteristics is particularly important, because any elongated products have a tendency to grow, bow, and contract during the heating and quenching cycle, particularly in cases when only the single-sided toothed area of the rack is specified to be heat treated. In some cases certain distortion of as-quench parts is desirable because during sequential tempering operation metal may move in opposite direction relieving some stresses and producing sufficiently straight as-tempered parts. The necessity to minimize the weight of vehicles has initiated the use of smaller diameter solid/hollow steering racks in combination with improved hardness patterns to provide the needed strength. There are two most frequently used processes for induction hardening of toothed steering racks. The first approach is scan hardening using horizontal or vertical (more typical) arrangements with single or multiple spindles/axes. The second approach is static hardening. Machines can be built with single or multiple stations for high production rates. Specially designed scan hardening inductors (including but not limited to conventional single-turn and multiturn coils, as well as so-called “D”-shaped or “Z”-shaped inductors and some others) and support fixtures are used for scan hardening of steering racks. Certain inductor design configurations may necessitate part rotation while others may

FIGURE 10.32 Steering racks are an essential part of automotive power-steering systems (left) being heat treated using medium frequency (right). (Courtesy of Inductoheat Inc., an Inductotherm Group company.)

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process without rotation, requiring a particular orientation of the inductor with respect to the toothed area of the rack and its back region. Various sensing devices may be used as an attempt to measure and minimize distortion during scan hardening, although they noticeably complicate the system making it more expensive. With single-shot processing, a modified split-return, butterfly-style, or modified hairpin inductor can be used. In other applications, AC current electrical resistance heaters can apply medium-frequency or high-frequency systems (Figure 10.32, right). A number proprietary techniques has been developed over the years in the industry to minimize the distortion while ensuring sufficient strength of toothed steering racks.

10.7  Tempering of Gears and Gear-Like Components 10.7.1  General Comments Heat treaters are often faced with the necessity of balancing required hardness and strength with sufficient ductility and toughness in order to enhance specific engineering characteristics. Tempering is a form of subcritical heat treatment producing an attractive blend of microstructures and properties [1,5,6,23]. In some cases, induction-hardened steels are also tempered to obtain specific distribution and magnitude of residual stresses and to ensure dimensional stability. As-quenched martensitic structures are frequently associated with high hardness and strength but low ductility and toughness. There is a belief that untampered martensite is too brittle for commercial applications and tempering of as-quenched components is always required. Depending on the application specifics, chemical composition of material, hardness levels, morphology of martensite, geometry of the component, and its function, this may or may not be true. Applications of hardening low-carbon steels and cases where improvements in wear properties and maximization of compressive surface residual stresses are the primary goals may serve as examples where tempering might not be specified. Nevertheless, the requirements of having untampered martensitic structures are more the exception than the rule, because the as-quenched martensite is typically too brittle for the majority of commercial applications since it promotes notch sensitivity and crack development. Untampered martensite is also characterized by a high level of internal residual stresses, which may be relieved during a gear service life, resulting in shape distortion and potentially affecting certain engineering characteristics. When making a decision as to whether tempering is required, consideration must be given to more than just the hardness levels and strength [23]. If some hardened components are left in the untampered condition, delayed cracking attributed to residual stress can occur. Therefore, it is important to minimize the time between the quench and temper operations (also referred as “transient time”). Among other factors, the probability of delayed cracking is dependent on the processing temperature, hardness level, case pattern, morphology of martensite, chemical composition of the alloy (e.g., carbon content in steels or carbon equivalent CE in cast irons), as well as the gear geometry. Reheating of steel for tempering after hardening relaxes internal stresses forming a tempered martensite microstructure. Tempering temperatures for steels are always below the lower transformation temperature Ac1.

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In induction hardening of some medium-carbon steels, the majority of high-carbon steels, cast irons, and some alloy steels, there is a certain amount of retained austenite (RA) present in the as-quenched structure (unless cryogenic treatment is used). There is always a concern that RA might transform to untampered martensite during the component’s service life introducing brittleness and potential shape distortion. Tempering helps decompose the RA. Depending on the application specifics, tempering temperatures are commonly within the range of 110°C (230°F) to 650°C (1200°F) enhancing ductility and toughness to a considerable extent [1,5,8]. If the steel is heated to less than 110°C (230°F), then (practically speaking) there is no change in microstructure that occurs as the result of short-time induction tempering. Lowtemperature tempering of carbon steels typically occurs at 120°C (248°F) to 250°C (482°F). The hardness reduction in this case typically does not exceed 2–4 points HRC. As stated earlier, surface compressive residual stresses are usually considered to be beneficial. They provide protection against the initiation and propagation of cracks caused by microscopic scratches and geometrical stress risers, delay fatigue cracking, and improve the performance of parts that experience bending and torsional stresses during service. It is important to note that the residual stress system is self-equilibrating; that is, there is always a balance of stresses within the workpiece owing to mechanical equilibrium. If certain regions have compressive residual stresses then there must be offsetting tensile stresses somewhere else. If the stresses were not balanced, “movement” would then result. In induction surface-hardened components, the maximum tensile residual stress is commonly located just beneath the hardened case or within the hardness transition zone. This is a zone of potential subsurface crack initiation. The maximum of applied tensile stresses during service is often located at the part surface or near surface and then drops off (as it is in the case of a combined effect of rolling and sliding in spur gears). Therefore, one of the important “duties” of tempering is not only the reduction of tensile residual stresses but also the shift of the maximum of these stresses from the surface further away from maximum of the applied tensile stresses. Regardless of the fact that in the majority of gear-hardening applications, an increase in tempering temperature results in a monotonic reduction of the hardness and strength, a change in impact toughness may not be monotonic with an increase of tempering temperature. Embrittlement phenomena can occur after tempering at certain temperature ranges, leading to a drop-in impact toughness. It is important to keep in mind that there are several types of embrittlement that can be associated with temper recipes and as-tempered structures. A detailed discussion on these and other embrittlement phenomena can be found in a classical book authored by Professor George Krauss [5] and work of his colleagues at Colorado School of Mines. It should be also noted that the magnitude of both phenomena, tempered martensite embrittlement and temper embrittlement, is noticeably less pronounced (in some cases even does not exist practically speaking) when short-time induction tempering is used compared to a conventional long-time oven tempering [27]. 10.7.2  Tempering Options Tempering can be done as either a batch or a continuous process [23]. The batch process requires that gears be accumulated after hardening and then moved to the tempering operation. In a continuous process, an in-line system moves as-quenched gears from the hardening stage into the tempering operation. Continuous processing can be very beneficial

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in eliminating or dramatically reducing the probability of delayed cracking, because it minimizes the time between hardening and tempering. Furnace or oven tempering is a well-proven, robust process typically specifying to soak the gear at the tempering temperature for approximately 1–2 hours. Most of the reduction in hardness of plain and majority of low-alloy carbon steels occurs in a time that is shorter than this, so the process becomes very stable. Tempering of alloy steels may require longer times and multiple tempering cycles. Induction tempering is also a proven process for various workpieces (for example, shafts, rods, crankshafts, etc.) exhibiting several operational advantages, such as single part processing capability, reduced tempering time, and a smaller footprint (floor space) for equipment, to name just a few. However, an applicability of induction tempering is noticeably limited when dealing with gears and gear-like components. 10.7.3  Induction Tempering Subtleties With induction tempering, the results are achieved in a matter of seconds or dozens of seconds rather than hours; therefore, the heating cycle must be controlled more precisely. There are several ways to determine the time–temperature correlation between conventional longertime lower-temperature furnace/oven tempering and shorter-time higher-temperature induction tempering. A considerable amount of work has gone into establishing the time– temperature relationships that result in identical hardness values for a variety of steels. Most of these correlations show that hardness is a logarithmic function in the form of a Larsen–Miller parameter, which suggests that it is the product of the absolute tempering temperature times the sum of a constant and the logarithm of the tempering time, written as

Hardness is a function of: T × [C + log 10(t)]

(10.3)

where T is the temperature, t is time, and C is a constant that depends on the alloy composition. According to Hollomon and Jaffe’s work [24], it has been established that the value of the constant C is generally between 10 and 18 for steels. Another parametric method for correlating equivalent tempering time–temperature conditions, which is very similar to that of Hollomon and Jaffe [13,24], was established by Grange and Baughman [25]. Unlike the Hollomon–Jaffe correlation, however, Grange and Baughman revealed that the parametric constant (C) can have a constant value of C = 18 regardless of the alloy content for a variety of steels. Although the establishment of induction tempering parameters might look straightforward, it should be understood that the abovementioned correlations can serve only as a rough estimation technique. Some of these factors are the chemical composition of the steel, the microstructural subtleties before tempering, the morphology of martensite, grain size, the presence of residual heat, hardness profile, and thermal history (heating rate, temperature gradients, and the cooling rate after induction tempering). The last factor (thermal history) is often the most neglected when determining short-time induction tempering parameters. Induction tempering is a continuous process of heating and subsequent cooling (soaking) for safe handling. Therefore, tempering conditions are affected by both heating and cooling stages. Although the maximum tempering temperature is achieved at the heating stage, the cooling/soaking stage is typically much longer (unless water cooling is used immediately after heating). Because tempering is a function of time and temperature, both factors will affect the as-tempered conditions.

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One of the most extensive investigations of the effects of the thermal history (heating rate, cooling rate, peak temperature, etc.) and hardenability on tempering response was that conducted by Semiatin and colleagues [8]. It has been demonstrated that the specifics of thermal history are critical in defining the deviations in the effective tempering parameter of induction-tempered parts. Several case studies of using subroutines for predicting the results of induction tempering developed by Semiatin and colleagues and comparison with experimental data are provided in [8]. The decision as to which tempering process to use should be carefully weighed. With furnace/oven tempering, the gear geometry and uniformity of heating is not normally an issue, although it is important to consider furnace loading conditions and to make certain all gears within a load reach the proper temperature for the specified amount of time. In induction tempering of gears, sprockets, and splines with coarse teeth, the challenge is to generate a sufficient amount of energy into the root of the tooth without overheating its tip and addendum region. The root is the critical area where the maximum concentrations of applied stresses are typically located. As a result, fatigue cracks frequently occur in the root area. However, several factors complicate this task of uniform heating in the case of induction tempering using encircling coils. This includes the following: • The first factor is related to a poor electromagnetic coupling between the encirclingtype coil and the tooth root and dedendum area compared to tooth tip and addendum region. An unequal electromagnetic coupling poses some challenges in generating sufficiently uniform heat within the gear teeth. • The second factor deals with the existence of a significantly larger cold sink that is located under the pitch circle and in particular under the tooth root compared to its tip. The substantial mass of metal located just below the root results in an intense heat sink that complements the reduced electromagnetic coupling and complicates the task of obtaining uniform heating of entire tooth. • The third factor derives from the fact that the tempering temperatures are always below the Curie point. Therefore, the steel is always exhibiting ferromagnetic properties during the temper cycle (Figure 10.4, left) and, generally, the skin effect is always pronounced (similar to the case study shown on Figure 10.6, left), resulting in a power surplus in the tip of the tooth and its addendum compared to that in the root. Higher electrical frequencies worsen this imbalance resulting in more heat generation in the tip compared to the root. Also, for certain types of gears it might be challenging to provide heat uniformity within the face of the gear (e.g., active profile area, top land, and root land) due to a three-dimensional eddy current distribution and a tendency to have a heat surplus at edges and sharp corners. In order to overcome these difficulties, low frequency, loose coil coupling, and low power density are more suitable for induction tempering in combination of using special temper inductor designs. These are some of the reasons why the majority of gears are tempered in furnaces/ovens. At the same time, some gears and gear-like components are successfully tempered/stress relieved by electromagnetic induction using conventional low- and medium-frequency solenoid-style coils (in particularly when dealing with fine teeth gears and splines) or C-core tempering inductors (if there is a hole/bore of sufficient size) [1,23]. A C-core tempering inductor represents a typical transformer-type design with one or several C-shaped magnetic cores (Figure 10.33, two images on the left). Laminated lowcarbon steel thin sheets are used for core fabrication. One or several multiturn coils are

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wound around the C-core to create the common magnetic flux. The intensity of heating of the C-core inductors depends on the strength of that magnetic flux. As the gear is ferromagnetic and is not laminated, the induced eddy currents will heat it quite effectively thanks to the Joule effect and the heat produced by magnetic hysteresis without generating any appreciable heat in the laminated magnetic core (assuming its appropriate design). There are a number of design approaches applying this technique [1], but the main principle is very simple. The gear with a sufficiently large inside hole/bore, (for example, 3 in. or 75 mm diameter) is placed around a certain portion of the magnetic core. In this case, the heated gear represents the electrically short-circuited secondary winding of a transformer. Several C-shaped magnetic cores (e.g., two or four cores) may also be used to improve the heat uniformity for components with larger bores (Figure 10.33, bottom-right). The most commonly applied electrical frequencies are in the range of 50–300 Hz. Laminations of the C-core should be electromagnetically thin enough. For low-frequency applications, individual lamination is typically 0.1–0.5 mm thick. Because laminations are electromagnetically thin, there will be a current cancellation of eddy currents induced in each lamination and, therefore, the Joule losses will be small enough and will not generate

FIGURE 10.33 Sketches of design options of C-core inductor designs for heating hollow gear and a thermal image of temperature distribution of ring level gear. (From V. Rudnev et al., Handbook of Induction Heating, 2nd edition, CRC Press, New York, 2017.)

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a large amount of heat. More discussion on a selection of laminations as well as other magnetic flux concentrators is provided in [1]. Laminations conduct the majority of the magnetic flux with minor heat generation there and having minimum magnetic flux leakage. Because of Joule losses and magnetic hysteresis losses within the gear, there will be very efficient and uniform heating with electrical efficiencies often exceeding 90% (Figure 10.33, top-right). An application of medium and high frequencies leads to a reduction of the electrical efficiency due to the higher kW losses in the laminated C-core and can also lead to localized heat surpluses in the heated gear. The use of line or low frequencies results in larger eddy current penetration depth into the heated material and, therefore, produces a smaller temperature difference compared to using medium or high frequencies. More desirable eddy current flow while using C-core inductors also complements more uniform heating compared to applying solenoid-style coils. C-core tempering inductors have several beneficial features, including but not limited to the following: • The ability to use line and low frequencies leads to system simplicity and low capital cost investment. • Radial, axial, and circumferential temperature gradients are typically quite small, producing fairly uniform temperatures even when heating such complex geometries as ring gears (Figure 10.33, top-right). • The heated gear does not need to be rotated to obtain circumferential heat uniformity. In some applications of heating workpieces with sizable diameters instead of C-core, using an E-core inductor or double or triple E-core inductors might be beneficial. The number of poles might be further increased, further improving heat uniformity of larger diameter gears. • Design simplicity and process robustness. • High electrical efficiency (typically 85–92%) and inductor power factor (COSφ 0.75–0.85). However, the wide utilization of C-core inductors for gear tempering is limited due to several challenges [1]: • Complications in handling certain geometries for in-line processing and necessity of having a hole/bore of sufficient size. • Ineffective for heating asymmetrical complex-shaped workpieces. • Relatively low production rate unless multi-post design is used. • Equipment might be quite noisy (exceeding 80 db) at higher power levels. • Does not always suit well for tempering selective areas and for certain geometries (e.g., shafts with teeth). • If an inverter is not used, there could be challenges associated with power factor correction and phase balancing (when a single-phase inductor is used). • Parts can be magnetized and the value of retained magnetism could exceed the permissible maximum level requiring post-degaussing. Although, as an option, it is possible to incorporate special power control for C-core inductors providing a degaussing effect and minimizing retained magnetism.

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10.8 Conclusion Due to space limitation, only the most essential aspects of gear heat treatment using electromagnetic induction have been reviewed above. More information on subtleties of using induction heating for a wide range of processing metallic materials can be found in [1,26].

Acknowledgments Materials for this article are chiefly adapted from the 2nd edition of the Handbook of Induction Heating, by V. Rudnev, D. Loveless, and R. Cook, CRC Press, 2017. CRC Press has granted permission to publish these materials.

References 1. V. Rudnev, D. Loveless, R. Cook, Handbook of Induction Heating, 2nd edition, CRC Press, New York, 2017, 750p. 2. V. Rudnev, Induction Gear Hardening: Part 1, Gear Solutions, February 2018, pp. 40–45. 3. V. Rudnev, Induction Gear Hardening: Part 2, Gear Solutions, March 2018, pp. 46–51. 4. V. Rudnev, J. Storm, Induction Hardening of Gears and Gear-like Components, in: ASM Handbook, Vol. 4C: Induction Heating and Heat Treatment, V. Rudnev and G. Totten, (eds.), ASM International, 2014, pp. 187–210. 5. G. Krauss, Steels: Processing, Structure and Performance, 2nd edition, ASM International, 2015, 680 p. 6. D. Matlock, Metallurgy of Induction Hardening of Steel, in: ASM Handbook, Vol. 4C: Induction Heating and Heat Treatment, V. Rudnev and G. Totten, (eds.), ASM International, 2014, pp. 187–210. 7. W.J. Feuerstein, W.K. Smith, Elevation of critical temperature in steel by high heating rate, Trans. ASM, 46: 1954, pp. 1270–1281. 8. S.L. Semiatin, D.E. Stutz, Induction Heat Treatment of Steel, ASM International, 1986. 9. V. Rudnev, Induction hardening of cast irons, Thermal Processing, July, 2018, pp. 40–45. 10. J. Orlich, A. Rose, P. Wiest, Atlas zur Warmebehandling der Stahle, Vol. 3, Zeit-TemperaturAustenitisierung-Schaubilder, Verlag Stahleisen M. B. H., Düsseldorf, Germany, 1976, 282 p. 11. J. Orlich, A. Rose, P. Wiest, Atlas zur Warmebehandling der Stahle, Vol. 4, Zeit-TemperaturAustenitisierung-Schaubilder, Verlag Stahleisen M. B. H., Düsseldorf, Germany, 1973, 264 p. 12. V. Rudnev, Technology Innovations and Challenges When Induction Heating and Heat Treating Long Steel Products. Proceedings of Int’l Conference on Advances in Metallurgy of Long and Forged Products, Vail, Colorado, 12–15 July, 2015, pp. 188–199. 13. J. Hollomon, L. Jaffe, Ferrous Metallurgical Design, Wiley, New York, 1945. 14. Heat Treater’s Guide. Practices and Procedures for Irons and Steels, ASM International, 1996, 670 p. 15. ASM Handbook, Vol. 7: Powder Metallurgy, ASM International, Materials Park, OH, 2015. 16. V. Rudnev, Specifics of Induction Hardening of Powder Metallurgy (P/M) Components, Induction Thoughts Series, in: Heat Processing, Vulkan-Verlag GmbH, Vol. 4, 2018, pp. 67–69. 17. J. Newkirk, Heat Treatment of Powder Metallurgy Steels, in: ASM Handbook, Vol. 4D: Heat Treating Irons and Steels, J. Dosset and G. Totten (eds.), ASM International, 2014, pp. 253–273.

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18. V. Rudnev, Systematic Analysis of Induction Coil Failures and Prevention, in: ASM Handbook, Vol. 4C: Induction Heating and Heat Treating, V. Rudnev and G. Totten (eds.), ASM International, Materials Park, OH, 2014, pp. 646–672. 19. V. Rudnev, Systematic Analysis of Induction Coil Failures, Part 8: “Gap-by-Gap” Gear Hardening Coils, Professor Induction Series, in: Heat Treating Progress, ASM International, Materials Park, OH, November/December, 2006, pp. 19–20. 20. G. Doyon, V. Rudnev, C. Russell, J. Maher, Revolution – not evolution – necessary to advance induction heat treating. Advance Materials & Processes, September 2017, pp. 72–80. 21. V. Rudnev, Spin hardening of gears revisited, Professor Induction Series, in: Heat Treating Progress, ASM International, Materials Park, OH, March/April 2004, pp. 17–20. 22. V. Rudnev, Single-Coil Dual-Frequency Induction Hardening of Gears, in: Heat Treating Progress, ASM International, Materials Park, OH, October 2009, pp. 9–11. 23. V. Rudnev, G. Fett, S.L. Semiatin, Tempering of Induction Hardened Steels, in: ASM Handbook, Vol. 4C: Induction Heating and Heat Treatment, V. Rudnev and G. Totten, (eds.), ASM International, 2014, pp. 130–159. 24. J. Hollomon, L. Jaffe, Time–temperature relations in tempering steel, Trans. AIME, 1945, p. 162. 25. R.A. Grange, R.W. Baughman, Hardness of tempered martensite in carbon and low alloy steels, Trans. ASM, 48: 1956, pp. 165–197. 26. V. Rudnev and G. Totten, (eds.), ASM Handbook, Vol. 4C: Induction Heating and Heat Treatment, ASM International, Materials Park, OH, 2014. 27. V.K. Judge, J.G. Speer, K.D. Clarke, K.O. Findley, A.J. Clarke, Rapid Thermal Processing to Enhance Steel Toughness, Scientific Reports, January 11, 2018, pp. 1–6. DOI:10.1038/s41598-017-18917-3.

11 A Brief Overview on the Evolution of Gear Art: Design and Production of Gears, Gear Science Stephen P. Radzevich CONTENTS 11.1 Brief Notes on the History of Methods of Machining Gears and on Design of Gear-Cutting Tools......................................................................................................... 418 11.1.1 Early Accomplishments in the Design of Toothed Wheels and in Methods for Manufacture of Gears����������������������������������������������������������������� 419 11.1.2 Early Designs of Special Purpose Cutting Tools to Produce Gear Teeth.......422 11.1.3 Gear-Cutting Tools for the First Production Machines................................. 424 11.1.4 Evolution of the Gear-Cutting Tools for Production Machines....................425 11.1.5 Development of the Skiving Internal Gears Process...................................... 427 11.1.6 Development of the Rotary Gear Shaving Process......................................... 427 11.1.7 Grinding Hardened Gears................................................................................. 427 11.1.8 Gear-Cutting Tools for Generating Bevel Gears............................................. 428 11.1.9 Later Accomplishments in Design of Gear-Cutting Tools for Generating Bevel Gears�������������������������������������������������������������������������������������430 11.1.10 Generating Milling of Bevel Gears................................................................... 435 11.2 A Brief Overview on the Evolution of the Scientific Theory of Gearing.................... 439 11.2.1 Preliminary Remarks..........................................................................................440 11.2.2 Pre-Eulerian Period of Gear Art........................................................................445 11.2.3 The Origin of the Scientific Theory of Gearing: Eulerian Period of Gear Art................................................................................................................ 450 11.2.4 Post-Eulerian Period of the Developments in the Field of Gearing.............454 11.2.5 Developments in the Field of Perfect Gearings............................................... 461 11.2.5.1 Grant Bevel Gearing........................................................................... 461 11.2.5.2 Contribution by Professor N. I. Kolchin.......................................... 462 11.2.5.3 Novikov Conformal Gearing............................................................463 11.2.5.4 Contribution by Professor V. A. Gavrilenko................................... 466 11.2.5.5 Condition of Conjugacy of the Interacting Tooth Flanks of a Gear and a Mating Pinion in Crossed-Axes Gearing������������ 466 11.2.5.6 Equality of Angular Base Pitches of a Gear and a Mating Pinion to Operating Angular Base Pitch in Intersected-Axes, and in Crossed-Axes Gearing���������������������������������������������������������� 469 11.2.6 Contribution by Walton Musser........................................................................ 471 11.2.7 Tentative Chronology of the Evolution of the Theory of Gearing................ 472 11.3 Developments in the Field of Approximate Gearings................................................... 474 11.3.1 Cone Double-Enveloping Worm Gearing........................................................ 475 11.3.2 Approximate Bevel Gearing............................................................................... 475 417

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11.3.3 Approximate Crossed-Axes Gearing................................................................ 476 11.3.4 Face Gearing......................................................................................................... 476 11.4 A Brief Summary of the Principal Accomplishments in the Theory of Gearing Achieved by the Beginning of the Twenty-First Century............................. 477 11.4.1 Condition of Contact of the Interacting Tooth Flanks of a Gear and Pinion............................................................................................................. 477 11.4.2 Condition of Conjugacy of the Interacting Tooth Flanks of a Gear and Pinion............................................................................................................. 478 11.4.3 Condition of Equal of Base Pitches of the Interacting Tooth Flanks of a Gear and Pinion������������������������������������������������������������������������������������������480 11.4.4 On the So-Called “Russian School of Theory of Gearing”........................... 481 11.5 Concluding Remarks.......................................................................................................... 482 References...................................................................................................................................... 482 Bibliography.................................................................................................................................484 Gear art has a long history of evolution as gears have been used by human beings for many centuries. At the beginning, gears were produced by smart handicrafts without any theoretical analysis of the gear design. Therefore, the origin of gear art is associated with the manufacture of gears. Later on, after certain experience in gear production and gear operating had been accumulated, and it was necessary to transmit more power at a higher rotation, rudiments of the theory of gearing began to appear. This sequence, that is, gear production that follows by gear design is adopted in the discussion below. It should be mentioned here that both gear manufacture, and gear theory partially overlap each other, and stimulated the evolution of one another, that is, accomplishments in gear manufacture stimulated evolution of gear theory and, vice versa, achievements in the theory of gearing stimulated methods of gear production.

11.1 Brief Notes on the History of Methods of Machining Gears and on Design of Gear-Cutting Tools Gears are the means by which power is transferred from source to application. A sizeable section of industry and commerce in today’s world depends on gearing for its economy, production, and livelihood. Today, as it has always been in the industry in the past, the design and manufacture of gears is an area of extremely specialized knowledge. Gear manufacturing practice is in a constant state of change. This results from the continuing search for novel and lower-cost methods of producing key components that are among the most difficult and expensive to produce. Practical men were able by various empirical means to get gears adequate for their needs, at least until the early nineteenth century, when the mathematician’s work was translated into practical language. Purely empirical solutions for the form of gear teeth can only be accounted for by the fact that gears operated at “low speeds” and under “small loads.” The developments in the field of novel methods of gear cutting, novel designs of machines for cutting gears, and novel designs of gear-cutting tools are tightly connected to each other, that is, it is common practice that a novel design of a machine for cutting gears, as well as a novel design of the gear-cutting tool, are commonly initiated by an invented novel method of cutting gear teeth. The gear broaching process is a perfect illustration of this: the

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design of gear broaching machines, as well as the design of broaches for machining gears, are the consequences of the proposed method of gear broaching. The main goal of this chapter is to briefly outline most of the fundamental accomplishments in the methods of cutting gear teeth, and the novel designs of special purpose gear-cutting tools. In the best-case scenario, the following principle should be adopted when writing this chapter: • The principal contributions to the methods and mean for cutting gear teeth should be identified, and a list of the principal accomplishments in the field should be composed • The names of the gear experts who are credited with the corresponding accomplishments should be named For example, • The importance of the involute tooth profile to the gear-cutting tool designer is widely recognized. Academician Leonhard Euler is credited with this accomplishment • Analytical representation of the condition of contact of the tooth flank of the workgear and the gear-cutting tool tooth flank in the form of dot product n · VΣ = 0 is of critical importance for the gear-cutting tool designer. Prof. V.A. Shishkov is credited with this accomplishment. Commonly, the equation n · VΣ = 0 is referred to as the “Shishkov equation of contact, n · VΣ = 0” • The condition of conjugacy of the work-gear tooth profile and the gear-cutting tool tooth profile is vital when designing cutting tools for cutting/finishing precision gears. This theorem is commonly referred to as the “main theorem” in the theory of gearing. However, Charles Camus, Leonhard Euler, and Felix Savary are credited with this accomplishment. Therefore, it makes sense to refer to this theorem as to “Camus-Euler-Savary theorem,” or just to “CES − theorem,” for simplicity In this manner, a corresponding name can be assigned to all principal accomplishments in the field of gear-cutting tool design. Following this rule, not many gear experts are credited with the principle accomplishments in the field of gear machining—only those who were the first to discover a fundamental result of the research in the field. Unfortunately, in the meantime, this approach is not applicable as not all the methods of gear cutting, as well as not all the designs of gear-cutting tools, can be associated with the corresponding names of the gear experts who are the first to propose a solution to a particular engineering problem that pertains to the methods and means of gear cutting. Therefore, the consideration below in this chapter are limited just to “brief notes” on the subject. A full and complete history of evolution of methods of cutting gear teeth, and of the corresponding designs of gear-cutting tools, is to be developed in the future. 11.1.1 Early Accomplishments in the Design of Toothed Wheels and in Methods for Manufacture of Gears The art and science of gearing have their roots before the Common Era. The earliest account of gears comes from ancient Chinese and Greek literature. Because of force-multiplying properties of gears, early engineers used them for hoisting heavy loads such as building

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materials. The mechanical advantage of gears was also used for ship anchor hoists and catapult pretensioning. At the beginning, transmitting and transformation of a rotation was the main purpose of gearing. The quality of rotation of the output shaft, that is, smoothness of its rotation, was out of importance in the earliest designs of gears. As a result, the gear tooth profile geometry was not considered at all, and pin gears successfully met all the requirements of that time. The earliest written descriptions of gears are said to have been made by Aristotle [3] in the fourth century BCE. It has been pointed out that the passage attributed by some to Aristotle, in Mechanical Problems of Aristotle (c. 280 BCE), was actually from the writings of his school. In the passage in question, there was no mention of gear teeth on the parallel wheels, and they may just as well have been smooth wheels in frictional contact. Therefore, the attribution of gearing to Aristotle is most likely incorrect. The real beginning of gearing was “probably” with Archimedes who, in about 250 BCE, invented the endless screw turning a toothed wheel, which was used in engines of war. Archimedes also used gears to simulate astronomical ratios. The Archimedean were early forms of the wagon mileage indicator (odometer) and the surveying instrument. These devices were “probably” based on “thought” experiments of Heron of Alexandria (c. 60 CE), who wrote on theoretical mechanics and the basic elements of mechanisms. Judging from the history books is one thing. Finding hard evidence of actual gears is another. The biggest problem in finding archeological evidence of gears is that early gear materials were not built to last. Gears made during the classical at the time of ancient Greek were probably made of bronze. When bronze tools and mechanical pieces broke, they were simply melted down and refashioned into something else. The original gear teeth were wooden pegs driven into the periphery of wooden wheels and driven by other wooden wheels of similar construction. An example of wooden gear transmission is illustrated in Figure 11.1. Neither special methods of cutting the gear teeth, nor the gear-cutting tools of a special design were necessary to manufacture wooden gears used in ancient gear transmissions. At that time, smart craftsmen were governed solely by common sense and by previously accumulated experience when designing and making wooden gears and gear trains. Primitive common purpose cutting tools were used in gear production. The oldest surviving relic containing gears is the Antikythera mechanism, named after the Greek island near which the mechanism was discovered in a sunken ship in 1900. The mechanism is not only the earliest relic of gearing but is also an extremely complex arrangement of epicyclic differential gearing. The mechanism is identified as a calendrical Sun and Moon computing mechanism and is dated to about 87 BCE. The Antikythera mechanism (see Figure 11.2) is the oldest artefact consisting gears known so far. An image of the original Antikythera mechanism is shown in Figure 11.2a. Figure 11.2b illustrates the Antikythera mechanism overlapped with the image of the corresponding replica. The device has more than 30 gears, although some scientists suggested as many as 72 gears, with teeth formed through equilateral triangles. To be reproduced in metal, the simple geometry of the gear teeth used in the design of Antikythera mechanism did not require any special methods for cutting gear teeth, as well as any gear-cutting tool of special design. This job had been done by smart craftsmen governed solely by common sense and by previously gained experience. Primitive common purpose cutting tools were used for cutting the gear teeth. The history of gear-cutting mechanisms, as well as of gear-cutting tools, begins in the early sixteenth century, and may be even associated with some of Leonardo da Vinci’s

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FIGURE 11.1 Old-style gear transmission comprised of wooden gears. (a)

(b)

FIGURE 11.2 The Antikythera mechanism, (205 BCE to 100 BCE).

drawings [7]. The first machines to cut gears were designed and used for machining gears, that is, at the end of the seventeenth century and at the beginning of the eighteenth century. Robert Hooke’s wheel cutting engine (c. 1672), Christopher Polhem’s machine to cut clockwork gears (c. 1729), and a few others to be mentioned, are perfect examples of early designs of machines to cut gear teeth.

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Only common purpose cutting tools of simple designs were used for cutting the gear teeth on gear-cutting machines of these designs. Accomplishments in the field of gearing and gear art since the earliest times are briefly summarized immediately below: • Various primitive designs of wooden gears were developed with the purpose to transmit a rotation between two shafts; • Gears that operate on: (a) parallel shafts—that is, parallel-axes gearing, (b) intersected shafts—that is, intersected-axes gearing, and (c) crossed shaft—that is, crossed-axes gearing, are known; • All the gearings operate at low rotations, and transmit a low torque; • No tooth flank geometry was taken into account, first of all because of the absence of necessity of doing that; low power density wooden gears that operate at low rotations met the current customer requirements of that time. Old-style gearings are far from to be referred to as “perfect gearings” as they are not capable of transmitting a rotation smoothly. Geometrically inaccurate gears (those that feature variable angular velocity ratio) are still used even in today’s industry in cases when the rotations are low, and the transmitted power is also low. 11.1.2  Early Designs of Special Purpose Cutting Tools to Produce Gear Teeth As our progress in the use of mechanical devices increased so did our use of gears, and the form of the gear teeth changed to suit the application. The contacting sides or profiles of the teeth changed in shape until eventually they became parts of regular curves. The changes to the gear tooth profile entailed the corresponding changes to the design of cutting tools for cutting the gear teeth. The first known gear-cutting machine is described by Morales [6]. Juanelo Turriano (see Figure 11.3) is credited with the invention of this early design of a gear-cutting machine. An Italian mechanic and clockmaker, Turriano1 was the inventor of the first known gear-cutting machine. He spent 20 years on this project and paid a high price for his intensive work. He was sick twice and almost died before he successfully created his manually operated machine. Maybe his reward was self-satisfaction. As is shown by Robert. S. Woodbury [35], Turriano had a rotary file cutter not unlike those of two hundred years later, but since it cuts iron, it must have been hardened. No details on the rotary file cutter tooth profile geometry are discussed by Woodbury [35]. The “oldest” known gear cutter (rotary file) is shown in Figure 11.4. This gear-finish tool was made in France, circa 1782. Jacques de Vaucanson (Figure 11.5), a famous French inventor, is credited for the invention of the form file for finishing wooden models of cylindrical gears. The gear models made of wood were used in finishing gears made of metal. By the beginning of the nineteenth century, several other designs of gear-cutting tools (form gear milling cutters) were proposed. The concept of the design of form gear milling cutters can be traced back to the Vaucanson’s form rotary file (see Figure 11.4). Relieving of the clearance surfaces of teeth of form milling cutters became practical later on. 1

Juanelo Turriano (Italian: Gianello Torriano; born Giovanni Torriani, c. 1500–1585) was an Italo-Spanish clockmaker, engineer, and mathematician. He was born in Cremona. He died at Toledo in 1585.

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FIGURE 11.3 Juanelo Turriano (c. 1500–1585).

FIGURE 11.4 The “oldest” known gear cutter (rotary file), circa 1782.

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FIGURE 11.5 Jacques de Vaucanson (1709–1782).

Originally, gear teeth were produced with form mill cutters on milling machines equipped with an index head. Nowadays, designs of disc-type form gear mill cutters that are originated from the Vaucanson’s form rotary file are used in low-volume production of gears. 11.1.3  Gear-Cutting Tools for the First Production Machines Robert S. Woodbury in his book on the history of gear-cutting machines [35] noticed that the first production machines to cut gears appeared in the first half of the nineteenth century (i.e., 1800 to 1855). Approximately at this period of time, the first gear-cutting tools for the production machines appeared on the market. These mechanisms and gear-cutting tools developed actively until 1930, when most of the primary types seem to have become stabilized. The entire sequence, thus, covers over a century. In his patent of 1839, Bodmer describes gear-cutting tools of two types. Those for cutting wooden patterns for cast gears are of the fly-cutter type. Those for cutting actual metallic gears are of the milling type, segmented for convenience in hardening. All are formed cutters. The problem of sharpening them gave great trouble and expense. As cast gears came more and more into use in the last half of the eighteenth century, there were two trends: toward increased accuracy of gears, and toward increased size of gears. It is likely that the gear-generating principle was utilized for the first time in the Saxton’s gear-generating machine (1842) [35]. The gear cutting of a simple geometry was used to cut gears.

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The molding-generating principle had already appeared in Witworth’s patent of 1835 [35]. In this machine, the gear blank is geared by means of a train of gears with a worm-wheel hob in exact gear ratio without slippage. This was the first machine to generate involute teeth. In 1856, Schile took out a patent for a machine designed to cut spur gears with a hob, also having gearing to rotate the blank and the hob at the correct relative speed. It seems clear that almost all types of gear-cutting tools for production type gear-cutting machines embodying all the basic methods of forming the teeth had been developed by 1850. 11.1.4  Evolution of the Gear-Cutting Tools for Production Machines Although in 1765 Leonhard Euler had already discovered in the involute curve a suitable tooth form for the kinematically correct transmission of rotary motion, there was still a long way to go before the design and making of gear-cutting machines that could generate an involute profile. It had been recognized that the worm is a form of continuous rack and all that was necessary to cut gears with it was to provide cutting edges on it—to make a hob shown in Figure 11.6. According to [35], teeth had been cut by this method probably for the first time by Ramsden in 1768. But this was only instrument work. The first application of the hobbing principle to a production machine was made in 1835 by Joseph Whitehead, but his machine would only cut spiral gears. In 1839, Pfaff used a hob to cut worm gears, as had Whitehead in 1853. In 1856, Christian Schiele obtained a patent for a screw-shaped

FIGURE 11.6 A screw-shaped cutter to manufacture cylindrical gears by Christian Schiele.

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cutter to manufacture cylindrical gears, the forerunner of the modern hob. Schiele’s patent specifications provided for gearing the blank to the hob in the proper ratio. This was the first hobbing machine to do this. Schiele used his hob 90° to cut spur and helical gears, and also cut worm gears on his machine. Although Schiele’s machine is clearly the beginning of the gear-hobbing type, it is not known if it ever appeared in practice. The backed-off formed type of milling cutters has been invented by Joseph R. Brown as early as in 1854. Backed-off formed mill cutters appeared on the market in 1867. The second half of the nineteenth and the beginning of the twentieth century was the “golden” era of pioneering developments in gear manufacturing. At that time, a special reliefturned milling cutter was used for each cylindrical gear to cut the slots between the teeth [1]. Three inventors most significantly improved the gear-cutting process. Their proposals were pioneer in creating gear-producing automotive machines based on the continuous relation of motions between the generating tool and the gear being generated. Such machines provide continuous indexing and generate the gear tooth surface as the envelope to the family of tool surfaces. These inventions are as follows. First, in 1897, Robert Hermann Pfauter invented the process (and machine) for generating spur and helical gears by hobs. The process of invention is a mystery and its crowning moment like a lightning bolt. For many years, Pfauter noted the disadvantages of the existing manufacturing process for spur and helical gears: • The necessity of indexing the gear for the generation of each tooth space. • The generation of the tooth space only as a copy of the tool surface. His patent for generating spur and helical gears was free of these disadvantages and although it claimed to be limited to the generation of helical gears, it could actually be applied to the generation of spur and worm gears by hobs. In this respect, Pfauter was the precursor of a group of inventors who later proposed other gear generation processes based on principles similar to those proposed in his patent. With the widespread use of the hobbing process, the gear machining processes, those based on application of the templates, is rapidly going out of use. Erwin R. Fellows is the second name to be mentioned. Fellows’ invention was based on the application of shapers. Fellows invented a revolutionary method to generate gears by shaping. His method was based on the simulation of two gears meshing during gear rotation about two parallel axes. The manufacture of spur gears by shaping originated at the Fellows Gear Shaper Company in Springfield, Vermont, in 1896. Fellows’ invention was a significant contribution because it enabled the automation of the internal and external gear manufacturing process. The automotive industry’s broad application of the Fellows gear-shaper process confirmed the importance of the invention. Continued development produced machines capable of generating helical and face gears. In 1897, the Fellows Company designed and built a machine for grinding involute profiles on gear-shaper cutters. This process was later recognized for its application in grinding the various tooth profile cutters that were used to shape spur, helical, and special tooth profile gears. One more significant accomplishment in the field of cutting-tool design and manufacture is due to Dr. Max Maag. In 1908, Dr. Maag developed the geometry of nonstandard involute spur and helical gears, which enabled designers to avoid undercutting and to increase the tooth thickness by just modifying the installation of the tool (the rack-cutter) with respect to the gear being generated. His research led to the development of the Maag system of

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generating nonstandard involute gears. In this system, he applied the method and machine invented by Samuel Sunderland of Keighley, Great Britain, and, in 1909, received the first patent for cutting nonstandard involute gears based on the application of Sunderland’s machine. The advantage in generating gears by a rack-cutter was that it was a simpler and more precise tool than a hob and a shaper. A significant achievement of the Maag Company was the development of the method to grind spur and helical involute gears. The grinding tool was a saucer-shaped wheel with a 15° profile angle, a prototype of which was built in 1913. This invention was important because it provided the first automatic compensation for the wear of the grinding wheel, making it the first and most famous automatic control system in machine tool history. In 1950, the method of grinding spur and helical gears by a plane was invented. The originality of the idea lay in the application of two grinding planes with a zero-pressure angle, each plane simulating the surface of a rack tooth with a zero-pressure angle. 11.1.5  Development of the Skiving Internal Gears Process On March 1, 1910, Wilhelm von Pittler registered a patent that aimed to significantly increase the productivity of the manufacture of internal gears. His patent for skiving, with the title, “Procedure for the Cutting of Internal Gears by Means of an Internal Gear-Like Cutting Tool on which the Faces of the Teeth are Provided with Cutting Edges,” was initially reminiscent of the method for shaping internal gears that was already known at that time. In order to produce rings with internal gearing, one is still forced to rely on broaching, form cutting, or shaping. It is precisely at this point that skiving opens up new opportunities. An internal gear with cutting on the abutting face is used as the tool. In contrast with shaping, however, the cutting movement is not generated by an oscillating stroke movement. It is far more the case that the intersecting axis alignment of the tool and workpiece creates an axial relative speed that makes the cutting movement possible. During the rotation of the tool, each cutting edge cuts through different tooth depths on the workpiece—a movement that is required for the cutting process. The intersection of the axes accounts for the fact that the helix angle of the tool and the helix angle of the internal gear to be manufactured differ by the amount of the axis intersection angle as illustrated in Figure 11.7. Von Pittler thus developed a continuously revolving gear tooth generating process, which drastically increases productivity for internal gearing. As a rule, skiving makes it possible to generate any kind of periodic structures on axially symmetrical lateral areas. 11.1.6  Development of the Rotary Gear Shaving Process The development of the rotary gear shaving process can be traced back to the early 1930s when Robert S. Drummond was granted with the U.S. Patent on a method of lapping gears (see Figure 11.8). The gear finishing tool is engaged in tight mesh with a gear to be finished at crossed axes of rotation of the gear finishing tool and the gear. When rotated, the gear finishing tool is reciprocating in the axial direction of the gear to be finished. Later on, about four dozen U.S. Patents on inventions were granted to Drummond on the methods of and means for rotary shaving of gears. 11.1.7  Grinding Hardened Gears The first machine for grinding the hardened gears for automobiles was the work of J. E. Reinecker of Germany.

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FIGURE 11.7 Wilhelm von Pittler’s 1910 sketch of the skiving process.

These methods of grinding high-precision involute spur and helical gears are based on the application of two types of grinding worms: • A cylindrical worm whose design parameters depend mainly on the normal pressure angle and the normal module of the gear to be ground • A globoidal worm that is designed as the conjugate2 to the gear The cylindrical worm and the gear being ground are in instant point contact whereas the globoidal worm and the gear are in line contact. Methods for dressing the grinding worms have also been developed. A device by which the wear on the formed grinding wheel was corrected by two diamond points to maintain the correct form “automatically”; and the resulting wear on the grinding wheel was compensated for in the gear-grinding machine, also “automatically” was proposed by Ward and Taylor of the Gear Grinding Machine Company, of Detroit [35]. 11.1.8  Gear-Cutting Tools for Generating Bevel Gears For generating straight bevel gears, a design of a gear planning machine was proposed by James E. Gleason (1874). The principle of a bevel gear generating method in a Gleason bevel gear planning machine is illustrated in Figure 11.9. Approximate (not perfect) straight bevel gears can be cut using the proposed method of generating a bevel gear tooth flank. 2

It is important to stress the readers’ attention here on the loosely used term “conjugate”. Actually, in this particular case, a globoid worm is not conjugate to the gear (for details, see [29]).

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FIGURE 11.8 A few of the illustrations to Drummond’s U.S. Patent No. 1,989,650 on the method of lapping gears.

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FIGURE 11.9 Principle of “Gleason” bevel gear planning machine (1874) [7].

The principle behind all the molding-generating machines grows out of Sang’s theoretical analysis,3 which showed how a rack-tooth-shaped cutter could be used in an intermeshing method to produce interchangeable gears. The first machine of this type was the invention of Hugo Bilgram in 1884. This method of bevel gear generating was the first to be utilized. As early as in 1889, a gear generator using the describing generating principle was patented by George Grant. A single-point gear-cutting tool is used in this machine to cut straight bevel gears. Use of the Grant’s approach makes possible the cutting of perfect interchangeable bevel gears. The first spiral bevel gear generator, which operated using the single indexing method, was developed in the United States in 1913. The tool was a face cutter-head for a 5-cut method [1]. 11.1.9 Later Accomplishments in Design of Gear-Cutting Tools for Generating Bevel Gears Until the automobile, there was little demand for new types of gears or gear trains. The appearance of the automobile produced a much greater demand for types of gears seldom used before 1900, and this was reflected in gear-cutting machines designed to cut helical gears, both spur and bevel, as well as skew-bevel gears. Machines utilizing the generating principle were produced to meet this demand, but the hobbing method was soon to dominate this field. In the 1920s, automotive industry designers needed a gear drive to transform motions and power between crossed axes, and a lower location for the driving shaft. In 1898, Gleason invented a bevel-gear generating machine. He also invented a two-tool bevel generator that enabled manufacturers to reduce the production time of straight bevel gears twice: two reciprocating blades could generate the opposite sides of the tooth space (1905), and the “Formate” cut method for spiral bevel gears and hypoid gears (1938). The need for more accurate and quieter running gears became obvious with the advent of the automobile. Although the hypoid gear was within our manufacturing capabilities 3

Note that bevel gears only for “approximate gear pairs” can be cut in this method.

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FIGURE 11.10 Nikola John Trbojevich, also known as Nicholas J. Terbo (1886–1973).

by 1916, it was not used practically until 1926, when it was used in the Packard automobile. The hypoid gear made it possible to lower the drive shaft and gain more usable floor space. By 1937, almost all cars used hypoid-geared rear axles. The success with the design, manufacture, and application of the contemporary crossedaxes gearing is credited to two famous gear experts, Nikola Trbojevich (also known as Nicholas Terbo) and Ernest Wildhaber. Nikola Trbojevich (Figure 11.10), a world-known research engineer, mathematician, and inventor, was a nephew and friend of Nikola Tesla. Mr. Trbojevich is the first key developer of the methods of and the means for machining hypoid gearing. He held nearly 200 U.S. and foreign patents, principally in the field of gear design. Mr. Trbojevich’s most notable work that brought him international recognition was the invention of the “hypoid gear” along with a method and design of a cutting tool to cut gears of this novel design (Figure 11.11). First published in 1923, it was a new type of spiral bevel gear employing previously unexploited mathematical techniques. The “hypoid gear” is used in the majority of all cars, trucks, and military vehicles today. Together with his invention of the tools and machines necessary for its manufacture, the “hypoid gear” became an integral part of the final drive mechanism of automobiles by 1931. Its effect was immediately apparent in that the overall height of rear-drive passenger automobiles was reduced by at least four inches. The accomplishments of Trbojevich in the design of hypoid gears and the methods of and means for their production are granted with numerous U.S. and international patents on inventions. It should be noted here that the “hypoid gears” produced by the method and means of those proposed by Trbojevich are a kind of approximate gear, that is, they are not capable of transmitting a rotary motion smoothly. Trbojevich is also credited with numerous others accomplishments in the field of methods for cutting gears, as well as of novel designs of gear-cutting tools for these purposes.

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FIGURE 11.11 A few of the illustrations to Trbojevich’s U.S. Patent No. 1,465,149 on the design of and a method for cutting hypoid gears.

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FIGURE 11.12 Heinrich Schicht (1886–1962).

The first continuous indexing generator for spiral bevel gears was presented in Germany in 1923 using a conical hob. Heinrich Schicht (Figure 11.12) is the second key developer of the methods of and the means for machining hypoid gears. Schicht took up the idea of hobbing cylindrical gears and transferred it to bevel gears using a conical hob instead of a cylindrical hob to manufacture spiral bevel gears. Schicht and Preis applied for a patent with this idea in 1921. Schicht is credited first of all with the invention of “Palloid gearing.” In the “Palloid” gearcutting method, two hobs with opposite threads are required to produce a pinion and wheel. Teeth flanks in “Palloid gears” are generated using specially designed conical hobs as shown in Figure 11.13. Several Great Britain Patents [No. 230885 (1925), and No. 234131 (1926)], as well as U.S. Patents [No. 2,037,930 (1936), and No. 2,146,232 (1939)] are granted to Schicht on the Palloid gearing, the methods of and means for their production. The first machine that used the Palloid method for generating spiral bevel gears and hypoid gears with a conical hob was built in 1923. This method was invented in the 1920s by engineers Schicht and Preis and was later replaced by the face-milled and face-hobbed methods. It is often loosely claimed that the “Palloid gears,” which are cut by hobs, feature a tooth form that follows a true involute curve. This is incorrect. It should be stressed here that the “Palloid gears” produced by the method and means those proposed by Schicht also are a kind of approximate gears, that is, they are not capable of transmitting a rotation smoothly. Dr. Ernest Wildhaber (Figure 11.14) is the third key developer of the methods of and the means for machining hypoid gears. Wildhaber is one of the most famous inventors in the field of gear manufacture and design. He was granted with as many as 279 patents on inventions, some of which have a broad application in the gear industry. The hypoid gear drive is of the most famous invention by Dr. Wildhaber. 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. The proposed designs of crossed-axes gears in today’s industry are examples of approximate gearing as they are developed and manufactured based on application of

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FIGURE 11.13 A few of the illustrations to Schicht’s G.B. Patent No. 230,884 on the improvements in the method of milling helicoidal bevel gears, and appliances therefor.

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435

FIGURE 11.14 Dr. Ernest Wildhaber (1893–1979).

the imaginary crown gear with straight-sided profile (basic crown rack). Because of this, today’s crossed-axes gears of all types, both, face-milled and face-hobbed, do not meet all the requirements that perfect gears needs to meet. The U.S. Patent No. 1,622,014 (1927) is the first patent granted to Dr. Wildhaber on the design of hypoid gears, as well as in the field of methods for cutting gears, as well as of novel designs of gear-cutting tools for these purposes. Later on, dozens of the U.S. Patents were granted to Dr. Wildhaber in this particular area of gear design and production. Teeth flanks of hypoid gears (see Figure 11.15) are generated by a cutter-head. The gear generators of this type can produce high volumes of gears very efficiently. Due to economic factors, the majority of spiral and hypoid gear sets manufactured today are of the Wildhaber design. This is the main reason why Wildhaber’s methods of and means for production of hypoid gears are used most widely in today’s industry. It should be stressed here that the “hypoid gears” produced by the method and means those proposed by Wildhaber are a kind of approximate gear, that is, they are not capable of transmitting a rotation smoothly. Dr. Wildhaber is also credited with one more milestone accomplishment in production of bevel gears, that is, he invented the so-called “Revacycle” method of cutting straight bevel pinions, and a design of a rotary broach for this purpose. The concept of a “Revacycle” method is illustrated in Figure 11.16. A few more U.S. Patents on inventions [No. 2,271,753 (1942), No. 2,278,576 (1942), No. 2,288,058 (1942), No. 2,294,014 (1942), No. 2,315,147 (1943), No. 2,327,296 (1943), No. 2,357,153 (1944), No. 2,376,465 (1945), No. 2,392,278 (1946)] are granted for the “Revacycle” method of cutting straight bevel pinions. In 1946, a face-hobbing machine with a cutter-head for the continuous-indexing method of gear machining was developed for the first time in Switzerland. Tooth depth was constant and the tooth lengthwise shape was an elongated epicycloid. 11.1.10  Generating Milling of Bevel Gears A different course was followed by Oscar Beale who, around 1900, developed a generating method for the production of bevel gears using two disc-shaped cutters, which could machine both flanks at the same time. Paul Böttcher improved this concept and, in 1910, presented a face mill cutter system, which produced spiral-shaped teeth on bevel gears.

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Advances in Gear Design and Manufacture

FIGURE 11.15 A few of the illustrations to Wildhaber’s U.S. Patent No. 1,622,014 on the design of and a method for cutting hypoid gears.

A Brief Overview on the Evolution of Gear Art

437

FIGURE 11.16 A few of the illustrations to Wildhaber’s U.S. Patent No. 2,267,182 on the design of a cutter for cutting gears, splined shafts, and the like.

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Advances in Gear Design and Manufacture

The molding-generating machine with the greatest commercial importance was the Brown & Sharpe machine, designed in 1900 by O. Beale. In this machine [35], both sides of a space are finished at the same time. This is done by using two toothed discs mounted on inclined axes with the teeth of one occupying the spaces of the other. The outer cutting faces of the discs act as the two sides of an involute worm-gear tooth. There is no motion of the axes of the cutters; all other motions—rotation and translation—are given to the blank. Provision is made for taking a finishing cut as well as to form bevel gears of various diameters, pitch, and pressure angle (Figure 11.17).

FIGURE 11.17 Generating milling of a straight bevel gear.

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439

Common sense and the gained practical experience (empiricism), along with a limited use of scientific approach are the main features of designing gear-cutting tools and methods for cutting gears that were developed in the past.

11.2  A Brief Overview on the Evolution of the Scientific Theory of Gearing The art and science of gearing have their roots before the Common Era. Yet many engineers and researchers continue to delve into the areas where improvements are necessary, seeking to quantify, establish, and codify methods to make gears meet the ever-widening needs of advancing technology. It should be stressed here that the scientific theory of gearing is a foundation to design, production, and application of perfect gears and geared mechanisms. It should be realized here that there are two different considerations when state-of-the-art gearing is discussed. Gear designs and gear manufacture, both based on common sense of smart handicrafts is one of them. An engineering approach that is based on scientific accomplishments in the theory of gearing is the other one. Considering that gears and gear transmissions have been investigated for a long time, today’s knowledge of gear theory is poor and is completely insufficient. Not much has been contributed to the theory of gearing since the time of Leonhard Euler (mid-eighteenth century) who is recognized as the founder of the scientific theory of gearing. It is of importance to make a brief overview on the evolution of the scientific theory of gearing. This will help us to identify what is already done in the field to this end, where we are now, and what to do in the future. Such an analysis needs to be carried out to credit the correct gear researchers with correct accomplishments in the scientific theory of gearing. Unfortunately, in the meantime, numerous achievements in the field of gearing cannot be attributed to a specific gear researcher. A gear researcher who has contributed significantly to the theory of gearing deserves to be credited with a corresponding scientific result. In the meantime, several fundamental achievements in the theory of gearing cannot be credited to the correct person, as this information is lost (lost, but maybe not forever). The names of the contributors are missing for: • Condition of contact of tooth flanks, G and P, of a gear and a mating pinion (it could happen that this accomplishment is NOT exactly from gearing, but it is from another area of the theory of machines and mechanisms) • Equal base pitches of a gear teeth and mating pinion teeth (in a case of parallelaxes gearing) A few more achievements and names to mention. These accomplishments, as well as numbers of others, are vital to the scientific theory of gearing. It is desired to know who should be credited with these meaningful results, and in what way these results were achieved. The main goal of this chapter of the book is to briefly outline “all” known “fundamental accomplishments” in the scientific theory of gearing, and to credit the correct gear researchers with the corresponding scientific achievements in the field, that is, an effort is undertaken in order to associate each of the fundamental accomplishments in the scientific theory of gearing with the corresponding name of the gear researcher who contributed a particular accomplishment. In order to mention all the key researchers in the field and to miss none of them, the following approach is adopted below.

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First, all (with no exclusions) the fundamental accomplishments in the scientific theory of gearing are listed in a chronological order. Second, a correct gear researcher name is assigned to each of the accomplishments, that is, the name of a researcher who was the first to discover, or who contributed the most, to a particular accomplishment in the scientific theory of gearing. For example, Leonhard Euler (and none other) is credited for the application of the involute of a circle to the gear tooth profile, as he was the first to prove that the involute tooth profile fits the best needs of the gear tooth geometry, regardless that the involute of a circle was known long before Euler made his discovery in 1760. In addition to that, a few huge mistakes committed by the gear researchers when investigating gears are also mentioned in order to better understand the theory, and to properly value those scientists who contributed much to the scientific theory of gearing. Mistakes of this type can be referred to as the “tremendous mistakes” in the theory of gearing. Following such an approach, it is helpful to separate the names of the key contributors to the scientific theory of gearing from those who contributed less to the subject, and, moreover, from those who committed mistakes that significantly affected the evolution of the theory of gearing. The results of the research carried out by the author, and a few papers earlier published by the author [21,22,28] are extensively used in this section of the book. Other sources are extensively used as well [3,35]. The consideration begins with ancient gear designs that are created only due to common sense and ends with the modern scientific theory of gearing. Tons of the sources (including sources like [10] on gear cutting tool design) were investigated prior to making a possible representation of the principal accomplishments in the scientific theory of gearing in a chronological order. A limited number of the sources were selected for a more detailed analysis. These sources are summarized in Table 11.1. The reported analysis is mostly based on the results of the research listed in Table 11.1 (the rest of the sources that are less informative are not included here). 11.2.1  Preliminary Remarks Gears are used to transmit and transform a rotation from an input shaft to an output shaft. Depending on a particular application, gearings have to meet certain additional requirements, that is, high accuracy of the transmission of rotation, high power density, and so forth. The development and investigation of gearings with a constant angular velocity ratio (i.e., gearings for which the equality ωp/ωg = const is valid) is one of the main goals of the scientific theory of gearing [29]. Gearings with a constant angular velocity ratio (or gearings with a prespecified function of the angular velocity ratio) are commonly called “perfect gearings,” or “geometrically accurate gearings,” or just “ideal gearings,” for simplicity. More generally, design of gearings with a prescribed function of the angular velocity ratio (that is, (a) noncircular gearings with a constant center distance, (b) noncircular gearings with a variable center distance, (c) gearings with a variable shaft angle, and (d) gearings with a variable center distance, and a variable shaft angle simultaneously) are also covered by the scientific theory of gearing. In the past, many efforts were undertaken by hundreds of researchers aiming to develop the “theory of gearing” that fits the main laws of mechanisms and machines science. However, not many of them have contributed to the theory. In this section of the book, the evolution of gearing from the earliest times to the nowadays is concisely discussed with the emphasis on the “theory of gearing.” The

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TABLE 11.1 Main Contributions to the Theory of Gearing No.

Year

The Name of the Key Contributor

1.

1493

da Vinci, Leonardo

2.

1754

Euler, L.

3.

1841

Willis, R.

4.

1842

Olivier, T.

5.

1861

Reuleaux, F.

6.

1863

Gibbs, J.W.

7.

1886

Gochman, H.I.

8.

1887

Grant, G.B.

9.

1912

Flanders, R.E.

10.

1917

Oberg, E.

11.

1935

Dicker, Ya.I.

12.

1936

Cormac, P.

13.

1948

Frifeldt, I.A.

14.

1948

Shishkov, V.A.

15.

1948

Wildhaber, E.

16.

1948

Fraifeld, I.A.

Source of Information da Vinci, Leonardo, The Madrid Codices, Volume 1, 1493, Facsimile Edition of “Codex Madrid 1,” original Spanish title: Tratado de Estatica y Mechanica en Italiano, McGraw Hill Book Company, 1974. Euler, L., De Optissima Figura Rotatum Dentibus Tribuenda. Suplementum de Figura Dentium Rotatum, Novi Commentarii Academial Petropolitanae, 1754/55, 1765. 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 p. Willis, R., “On the Teeth of Wheels,” Trans. Civ. Eng., Vol. II, 1838. Olivier, T., Théorie Géométrique des Engrenages destinés, Bachelier, Paris 1842, 132 p. [in French]. Reuleaux, F., The Constructor, a Hand-Book of Machine Design by F. Reuleaux, Authorized translation, complete and unabridged from the 4th enl. German ed. By Henry Harrison Suplee. Philadelphia, H.H. Suplee, 1893. -312 p. [First German edition published in 1861 with title: Der Constructeur. Ein Handbuch zum Gebrauch beim Maschinen-Entwerfen.] Gibbs, J.W., On the Form of the Teeth of Wheels in Spur Gearing, Doctoral Dissertation, Yale University, New Haven, Conn., 1863. Gochman, H.I., Theory of Gearing Generalized and Developed Analytically, Odessa, 1886, 229 p. [In Russian]. U.S. Pat. No. 407.437. Machine for Planing Gear Teeth. /G.B. Grant, Filed: January 14, 1887 (serial No. 224,382), Patent issued: July 23, 1889. Grant, G.B., A Treatise on Gear Wheels, 11th edition, Philadelphia Gear Works, Inc., Philadelphia, 1906, 105 p. Flanders, R.E., Bevel Gearing, 4th Edition, Machinery’s Reference Series, Number 37, The Industrial Press, 1912, 48 p. Oberg, E., Spur and Bevel Gearing, The Industrial Press, New York, 1917, 274 p. Dicker, Ya.I., Involute Gearing, Moscow, Orgametal, 1935, 220 p. [In Russian]. Dicker, Ya.I., Internal Gearing: Spur and Helical, Moscow, Orgametal, 1938, 138 p. [In Russian]. Cormac, P., A Treatise on Screws and Worm Gear, Their Mills and Hobs, London, Chapman & Hall, Ltd., 1936, 138 p. Frifeldt, I.A., Continuously Indexing Cutting Tools, Moscow, Mashgiz, 1948, 252 p. [In Russian]. Shishkov, V.A., “Elements of Kinematics of Generating and Conjugating in Gearing,” in: Theory and Calculation of Gears, Vol. 6, Leningrad: LONITOMASH, 1948. [In Russian]. Shishkov, V.A., Generation of Surfaces in Continuously Indexing Methods of Machining, Moscow, Mashgiz, 1951, 152 p. [In Russian]. Wildhaber, E., Foundations of Meshing of Bevel and Hypoid Gearings, Translated and comments by A.V. Slepak, Moscow, Mashgiz, 1948, 176 p. [In Russian]. – This book is not available in other languages. Fraifeld, I.A., Cutting Tools that Work on Generating Principle, Moscow, Mashgiz, 1948, 252 p. (Continued)

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TABLE 11.1 (Continued) Main Contributions to the Theory of Gearing The Name of the Key Contributor

No.

Year

17.

1949

Kolchin, N.I.

18.

1949

Buckingham, E.

19.

1950

Davidov, Ya.S.

20.

1953

Nikolayev, A.F.

21.

1955

Novikov, M.L.

22.

1959

Kudr’avtsev, V.N.

23.

1960

Litvin, F.L.

24.

1961

Saari, O.E.

25.

1961

Baxter, M. L., Jr.

26.

1964

Korostel’ev, L.V.

27.

1968

Dus’ev, I.I. and Vasilyev, V.M.

28.

1968

Lashn’ev, S.I.

Source of Information Kolchin, N.I., Analytical Calculation of Planar and Spatial Gearings, Moscow-Leningrad, Mashgiz, 1949, 210 p. [In Russian]. Buckingham, E., Analytical Mechanics of Gears, Dover Publications, Inc., New York, 1988, 546 p. (The 1st print – 1949). Davidov, Ya.S., Non-Involute Gearing, Moscow, Mashgiz, 1950, 179 p. [In Russian]. Nikolayev, A.F., Kinematical Foundations of the Theory of Spatial Gearing, Doctoral Thesis, Moscow, STANKIN, 1953. [In Russian]. Novikov, M.L., Fundamental Issues of Geometrical Theory of Point Gear Meshing for Application in Powerful Gear Transmissions, Doctoral Thesis, Moscow, Zhukovsky Air Force Engineering Academy, 1955. [In Russian]. Novikov, M.L., Gearing and Cam Mechanisms with Point System of Meshing, Pat. No. 10913 (USSR), Cl. 47, 6. Filed:April 19, 1956. [In Russian]. Novikov, M.L., Gearing with New Type of Meshing, Moscow, Zhukovsky Air Force Engineering Academy, 1958, 186 p. [In Russian]. Kudr’avtsev, V.N., Calculation and Design of Novikov Gearing, Leningrad, Zhukovsky Air Force Engineering Academy, 1959, 78 p. [In Russian]. Litvin, F.L., Non-Circular Gears, 2nd edition, Moscow-Leningrad, Mashgiz, 1956, 311 p. [In Russian]. (1st edition: 1950, 218 p). [In Russian]. Litvin, F.L., Theory of Gearing, 2nd edition, Moscow, Nauka, 1968, 584 p. (1st edition: 1960). [In Russian]. Litvin, F.L., Theory of Gearing, NASA Reference Publication 1212, AVSCOM Technical Report, 88-C-035, 1989, 470 p. Litvin, F.L., Gear Geometry and Applied Theory, Prentice Hall, Englewood Cliffs, NJ, 1994, 724 p. Litvin, F.L., Fuentes, A., Gear Geometry and Applied Theory, 2nd Edition, Cambridge University Press, Cambridge, UK, 2004, 800 p. Saari, O.E., Analytical Theory of Gear Tooth Surfaces, Illinois Institute of Technology, 1961, 114 p. Baxter, M.L., Jr., “Basic Geometry and Tooth Contact of Hypoid Gears,” Industrial Mathematic, Vol. 11, No. 2, pp. 19–42, 1961. Baxter, Meriwether L. Jr., “Basic Theory of Gear-Tooth Action and Generation”. This is the opening chapter of Gear Handbook, 1st Edition, Editor Darle Dudley, McGraw Hill, New York 1962. Korostel’ev, L.V., Geometrical and Kinematical Indicators of Bearing Capacity of Spatial Gearing, Doctoral Thrsis, Moscow, Stankin, 1964, 48 p. [In Russian]. Dus’ev, I.I. and Vasilyev, V.M., Analytical Theory of Spatial Gearing and its Implementation to Hypoid Gearing, Rostov-on-Don, Book Publishers, 1968, 148 p. [In Russian]. Dus’ev, I.I., Analytical Theory of Spatial Gearing and its Implementation to Hypoid Gearing, Rostov-on-Don, Doctoral Thrsis, Novocherkassk, Novocherkassk Polytechnic Institute, 1970. [In Russian]. Lashn’ev, S.I., Fundamentals of the Theory of Surface Generation by Means of Disk-Type, Rack-Type and Worm-Type Cutting Tools, Doctoral Thesis, Tula, Tula Polytechnic Institute, 1968, 268 p. [In Russian]. Lashn’ev, S.I., Generation of Gear Tooth Flanks by Means of Rack-Type and Worm-Type Cutting Tools, Moscow, Mashinostroyeniye, 1971, 216 p. [In Russian]. (Continued)

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TABLE 11.1 (Continued) Main Contributions to the Theory of Gearing The Name of the Key Contributor

No.

Year

29.

1969

Gavrilenko, V.A.

30.

1969

Chasovnikov, L.D.

31.

1969

Dyson, A.

32.

1969

Timofeyev, B.P.

33.

1971

Merritt, H.E.

34.

1972

Yerikhov, M.L.

35.

1972

Pismanik, K.M.

36.

1974

Sakharov, G.N.

37.

1976

Gul’ayev, K.I.

38.

1977

39.

1978

Lopato, G.A., Segal’, M.G. Shtipelman, B.A.

40.

1979

Henriot, G.

41.

1980

Schulz, V.V.

42.

1985

Goldfarb, V.I.

43.

1987

Colbourne, J. R.

44.

1989

Sizrantsev, V.N.

45.

1989

Xuezhu Dong

46.

1992

Wu, D.R. and Luo, J.S.

Source of Information Gavrilenko, V.A., Fundamentals of Theory of Involute Gearing, Moscow, Mashinostroyeniye, 1969, 432 p. [In Russian]. Chasovnikov, L.D., Gearing, Moscow, Mashinostroyeniye, 1969, 486 p. [In Russian]. Dyson, A., A General Theory of the Kinematics and Geometry of Gears in Three Dimensions, Clarendon Press, Oxford, 1969, 141 p. Timofeyev, B.P., Synthesis and Analysis of Spiral Bevel Gearing, Doctoral Thesis, Leningrad, Leningrad Polytechnic Institute, 1969. [In Russian]. Merritt, H.E., Gear Engineering, Putman Publishing, London, New York, Toronto, 1971, 489 p. Yerichov, M.L., Principles of Systematization, Methods of Analysis, and Problems of Synthesis of Schematics of Gearing, Doctoral Thesis, Khabarovsk, Khabarovsk Polytechnic Institute, 1972, 373 p. [In Russian]. Pismanik, K.M., Theoretical Foundations of Tooth Flank Generation in Bevel and Hypoid Gearing, Doctoral Thesis, Saratov, Saratov Polytechnic Institute, 1972, 408 p. [In Russian]. Pismanik, K.M., Hypoid Gearing, Moscow, Mashinostroyeniye, 1964, 27 p. [In Russian]. Sakharov, G.N., Theoretical Issues of Continuously Indexing Cutting Tools, Doctoral Thesis, Moscow, STANKIN, 1974, 320 p. [In Russian]. Sakharov, G.N., Continuously Indexing Cutting Tools, Moscow, Mashinostroyeniye, 1983, 232 p. [In Russian]. Gul’ayev, K.I., Theoretical Foundations of Synthesis and Finishing of Bevel Gearings, Doctoral Thesis, Leningrad, 1976. [In Russian]. Lopato, G.A., Kabatov, N.F., Segal’, M.G., Spiral Bevel and Hypoid Gearing, 2nd Edition, Moscow, Mashinostroyeniye, 1977, 423 p. [In Russian]. Shtipelman, B.A., Design and Manufacture of Hypoid Gears, John Wiley & Sons, New York, 1978, 394 p. Henriot, G., Traité Théorique et Pratique des Engrenages: Théorie et Technologie, Vol. 1, 6th Edition, Bordas, Paris, 1979, 662 p. Schulz, V.V., Geometrical Optimization of Worn Kinematical Pairs, Doctoral Thesis, Kiev, 1980, 32 p. [In Russian]. Goldfarb, V.I., Fundamentals of the Theory of Automated Geometrical Analysis and Synthesis of Generalized Type of Worm Gearing, Doctoral Thesis, Izhevsk, Izhevsk Mechanical Institute, 1985, 432 p. [In Russian]. Colbourne, J. R., The geometry of involute gears, New York, Springer-Verlag, 1987, 532 p. Sizrantsev, V.N., Synthesis of Cylindrical Gearing with Localized Contact, Doctoral Thesis, Kurgan, Kurgan Polytechnic Institute, 1989, 428 p. [In Russian]. Xuezhu Dong, Theoretical Foundation of Gear Meshing. China Machine Press, Beijing, 1989. [In Chinese]. Xuezhu Dong, Design and Modification of Hourglass Worm Drives, China Machine Press, Beijing, 2004. [In Chinese]. Wu, D.R., Luo, J.S., A Geometric Theory of Conjugate Tooth Surfaces, World Scientific Publishing, River Edge, NJ, 1992, 192 p. (Continued)

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TABLE 11.1 (Continued) Main Contributions to the Theory of Gearing The Name of the Key Contributor

No.

Year

47.

1994

Shishov, V.P.

48.

1994

Wang, X.C.

49.

1995

Vulgakov, E.B.

50.

1995

Dooner, D.B.

51.

1999

Sheveleva, G.I.

52.

1999

Silich, A.A.

53.

2000

Dudas, I.

54.

2003

Phillips, J.

55.

2005

Pavlov, A.I.

56. 57.

2007 2008

Krenzer, T. Radzevich, S.P.

58.

2009

Xuntang, W.

59.

2011

Tsukanov, O.N.

60.

2013

Stadtfeld, H.J.

Source of Information Shishov, V.P., Theory, Mathematical Foundations, and Synthesis of High Power Density Gearings for Industrial Transportation, Doctoral Thesis, East Ukrainian State University, Lugansk, 1994, 580 p. [In Russian]. Wang, X.C., Ghosh, S.K., Advanced Theories of Hypoid Gears, Studies in Applied Mechanics, Vol. 36, Elsevier, Amsterdam, 1994, 341 p. Vulgakov, E.B., Theory of Involute Gearing, Moscow, Mashinostroyeniye, 1995, 320 p. [In Russian]. Dooner, D.B., Kinematic Geometry of Gearing, 2nd Ed., John Wiley & Sons, Inc., New York, 2012, 512 p. (1st edition: Dooner, D.B., Seireg, A.A., The Kinematic Geometry of Gearing. A Concurrent Engineering Approach, John Wiley & Sons, Inc., NY,1995, 450 p.). Sheveleva, G.I., Theory of Surface Generation and of Contact of Moving Bodies, Moscow, STANKIN, 1999, 494 p. [In Russian]. Silich, A.A., Development of Geometrical Theory of Novikov Gearing and Generation of their Tooth Flanks, Doctoral Thesis, Kurgan, Kurgan Polytechnic Institute, 1999, 534 p. [In Russian]. Dudas, I., The Theory and Practice of Worm Gear Drives, London, Penton Press, 2000, 314 p. Phillips, J., General Spatial Involute Gearing, Springer, Berlin Heidelberg, 2003, 498 p. Pavlov, A.I., Modern Theory of Gearing, Kharkov, Kharkov National Automotive State University, 2005, 100 p. [In Russian]. Pavlov, A.I., Synthesis of High Loaded Gearing on the Basis of Linear Gear Meshes with Convex-to-Concave Contact of Tooth Flanks, Doctoral Thesis, East Ukrainian State University, Lugansk, 2009, 42 p. [In Russian]. Krenzer, T., The Bevel Gear, Published October 1, 2007, 252 p. Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, CRC Press, Boca Raton, Florida, 2012, 743 p. Radzevich, S.P., Gear Cutting Tools: Fundamentals of Design and Computation, Boca Raton Florida, 2010, 754 p. Radzevich, S.P., Geometry of Surfaces: A Practical Guide for Mechanical Engineers, Wiley, 2013, 264 p. Radzevich, S.P., Generation of Surfaces: Kinematic Geometry of Surface Machining, CRC Press, Boca Raton, Florida, 2014, 738 p. Radzevich, S.P., High-Conformal Gearing: Kinematics and Geometry, CRC Press, Boca Raton, Florida, 2015, 368 p. Radzevich, S.P., Fundamentals of Surface Generation, Monograph, Kiev, Rastan, 2001, 592 p. (In Russian). Radzevich, S.P., Differential-Geometric Method of Surface Generation, Dr. Sci. Thesis, Tula, Tula Polytechnic Institute, 1991, 300 p. Xuntang, W., Principle of Gearing, Xi’an Jiaotong University, Xi’an, 2009. [In Chinese]. Tsukanov, O.N., Fundamentals of Synthesis of Non-Involute Gearings in Generalized Parameters, Chel’abinsk, South-Ural State University Publishers, 2011, 140 p. [In Russian]. Stadtfeld, H.J., Gleason Kegelradtechnologie: Ingenieurwissenschaftliche Grundlagen und modernste Herstellunsverfahren für Winkelgetribe, Renningen, Expert-Verlag, 2013, 491 p. Stadtfeld, H.J., Gleason Bevel Gear Technology: The Science of Gear Engineering and Modern Manufacturing Methods for Angular Transmissions, The GLEASON Works, Rochester, New York, 2014, 491 p.

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consideration is mainly focused on the kinematics of gear pairs, the geometry of the interacting tooth surfaces, as well as on some other kinematical and geometrical aspects of gears and gearing pairs. 11.2.2  Pre-Eulerian Period of Gear Art The art of gearing was carried through the European Dark Ages, appearing in Islamic instruments such as the geared astrolabes that were used to calculate the positions of the celestial bodies. Perhaps the art was relearned by the clock- and instrument-making artisans of fourteenth-century Europe, or perhaps some crystallizing ideas and mechanisms were imported from the East after the crusades of the eleventh through the thirteenth centuries. It appears that an English abbot of St. Alban’s monastery, born Richard of Wallingford in 1330, reinvented the epicyclic gearing concept. He applied it to an astronomical clock that he began to build, which was completed after his death. A mechanical clock of a slightly later period was conceived by Giovanni de Dondi (1348–1364). Diagrams of this clock, which did not use differential gearing, appear in the sketchbooks of Leonardo da Vinci (Figure 11.18), who designed geared mechanisms himself [4]. Numerous designs of gearings are discussed in a famous book by Leonardo da Vinci [4]. In 1967, two of Leonardo da Vinci’s manuscripts, lost in the National Library in Madrid since 1830, were rediscovered [4]. One of the manuscripts, written between 1493 and 1497 and known as Codex Madrid I (Figure 11.19) [4], contains 382 pages with some 1600 sketches. Included among this display of Leonardo’s artistic skill and engineering ability are his studies of gearing. Among these are tooth profile designs and gearing arrangements that were centuries ahead of their “invention.”

FIGURE 11.18 Leonardo di ser Piero da Vinci (1452–1519).

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FIGURE 11.19 Title page of the book The Madrid Codices by Leonardo da Vinci, 1493 [4].

For a long while, the most accurate gears were produced by clockmakers and instrument makers. Questions of exact tooth form, pressure angle, and strength did not enter into the designs of the clockmakers and instrument makers. Contemporary gears for the uniform transmission of power and rotation are based much on the application of mathematical curves discovered by scientists in the sixteenth and seventeenth centuries, in the design of teeth flanks. In the period 1450–1750, the mathematics of gear-tooth profiles and theories of geared mechanisms became established.

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FIGURE 11.20 Girard Desargues (1591–1661).

Albrecht Dürer is credited with discovering the epicycloid shape (c. 1525). The first record on the use of cyclic curves as the tooth profile of a gear is related to Gerard Desargues (Figure 11.20). Desargues’ work on gearing is mostly known from the records made by his student Philippe de La Hire. A treaty on epicycloids and their usage in mechanics is discussed in his book [5]. Philippe de La Hire (Figure 11.21) was the first to describe the

FIGURE 11.21 Philippe de La Hire (1640–1718).

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FIGURE 11.22 Charles Camus (1699–1768)

use of epicycloids for gears that ensured (as he loosely meant) a uniform transmission of rotation. De La Hire considered the involute as the best among exterior cycloids, since he recognized that it is the special case in which the generating circle’s radius is infinite. He also noted that the involute tooth gives the teeth of the corresponding rack as having straight sides. It took 150 years before this principle found practical application. The first mathematician to work the theory of gear teeth into a systematic and general theory of mechanism was Charles Etienne Louis Camus. Camus (Figure 11.22) was the first [2] to formulate the condition that, in his opinion, has to be fulfilled for a pair of gears to be able to transmit rotation smoothly. According to Camus, this condition can be formulated as follows: If, in a uniform rotation, power is to be transmitted via a pair of teeth, then the normal to the teeth flanks at the contact point (within the path of contact) must pass through the pitch point.

Another formulation of that same condition by other researchers is represented in the form: If an auxiliary curve is rolling on the pitch circles of circular gears, any point attached to this curve traces conjugate profiles.

This sounds similar to the fundamental theorem of gearing known nowadays (see Chapter 1). The Camus principle of gearing is illustrated with a 1733 schematic (Figure 11.23). In this schematic, the path of contact, that is, a curved line segment KBC, and the line of action at different angular configurations of the mating gears, that is, BC, MP, and RQ, do not align to one another. The schematic (Figure 11.23) reveals that Camus did not correctly understand the difference between the “path of contact” and between the “line of action.” Therefore, Camus was close to discovering the fundamental theorem of gearing, but he did

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FIGURE 11.23 Illustration of Camus’ gearing principle (1733).

not succeed in doing that. As shown in Chapter 1 for gears that operate on parallel shafts, the only feasible case is when both the “path of contact” and the “line of action” are straight lines that align with one another as it is observed in involute gearing. According to some authors, the fundamental theorem of gearing was known to Leonhard Euler and to Felix Savary. This statement needs to be verified for correctness and completeness. Savary is also credited with the so-called “Euler-Savary equation.” A correlation between radii of curvature of the centrodes in a gear pair, and corresponding radii of curvature of the interacting tooth flanks, is established by this equation. In detail, the “Euler-Savary equation” is considered below. Camus repeated much of de La Hire’s work, although he added many important elements of his own. He gives a detailed analysis of the teeth desirable for the combination of a spur and lantern gear. Camus did, however, correct de La Hire in that he recognized the fact of sliding of even the epicycloid teeth one upon the other and said that this phenomenon is one of the principal sources of friction and wear in gearing. The action of engaged teeth relative to the line of centers is discussed, and he points out that the action is best when engagement takes place after the working face of the driving tooth has passed the line of centers, that is, during the receding action. Camus goes on to consider the problem of the minimum number of teeth and that of the proper form for the ends of the teeth. He deals with true bevel gears and uses the rollingcone principle for their analysis. But he considers only the case of interaction of a crown and a bevel gear.

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Camus does not consider the involute tooth at all. Although he analyzes trains of gears, he says nothing of the form of teeth required in a series of three or more gears. This can probably be accounted for by the fact that he had only clockwork in mind. The mills of this era seldom had trains of more than two gears engaged. Clearly, Camus had the basis for a theory of mechanism of gear teeth, but it was not systematically and completely worked out, as in Willis [34]. The main accomplishments in the field of gearing in the pre-Eulerian period of time are summarized by Desargues, de La Hire, Camus, and Savary. These accomplishments are briefly outlined below: • It became clear that performance of a gear pair depends on a specific tooth profile of the mating gears, that is, teeth wear in gearing depends on the actual shape of teeth of a gear and a mating pinion • Mathematicians indicated an interest to a special tooth profile of a gear and a mating pinion that allows the lowest tooth flank wear • Epicycloid is investigated as a potential candidate that can be used to shape the gear teeth, and epicycloid tooth flank geometry was proposed for gearings that operate on parallel shafts • It was realized in the pre-Eulerian period of time that a rotation cannot be transmitted smoothly, that is, with a constant angular velocity ratio, if gearings with epicycloid teeth are used • Involute of a circle was known at that time. However, it was not realized that this curve best meets the needs of gearing Desargues, de La Hire, and Camus are the main contributors to gear art in the preEulerian period of time. Even though the mathematicians began investigating some curves aiming their application for the purpose of gearing, no foundations in the theory of gearing was made at that time. 11.2.3  The Origin of the Scientific Theory of Gearing: Eulerian Period of Gear Art The interest of mathematicians (at the beginning such as Desargues, de La Hire, and Camus, and later on of Euler) seems to have come from a desire to increase efficiency and reduce wear in mills of various types where, although the speeds were low, the load was substantial. Indirectly, these problems were associated with the quality of the transmitted rotation, that is, with the smoothness of rotation of the output shaft. The beginning of the scientific theory of gearing can be traced back to the middle of the eighteenth century when Leonhard Euler (Figure 11.24) published his famous paper on the geometry of the gear tooth profile [8]. The mathematics of the involute curve and its application to gear teeth had been worked out by Euler, a great Swiss mathematician. In this work (Figure 11.25), Euler proved the usefulness of the involute of a circle to be used as the shape of gear teeth flanks. In this paper [8], along with his next paper on gearing [9] (Figure 11.26), Euler already shows the grasp and precision of his great mathematical mind. He specifically states the conditions: • Uniform rotary motion of both gears • In the mutual action of the teeth “nullus atritus oriatur” (no interference between the mating teeth flanks) [however, a gap between the mating teeth flanks, that is, equality of base pitches of the mating gears is not considered yet]

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FIGURE 11.24 Leonhard Euler (1707–1783).

The proposed Euler parallel-axes involute gearing with zero axes misalignment/ displacement deserves to be referred to as “Euler gearing,” or simply as “Eu– gearing”:Definition 11.1Euler gearing is a kind of parallel-axes involute gearing that features zero axes misalignment/displacement. Such a terminology can be used at least of science purposes, similar to that the terms “Newtonian fluid,” “absolutely rigid body,” “absolutely dark body,” and so forth, are extensively used in science publications. Besides the invention of the involute tooth profile being of critical importance, at the time of Euler, the difference between the line of action and the path of contact in a gear pair had not been understood in detail. This is, mostly, because in the case of parallel-axes gearing, both the lines, that is, the line of action, LA, and the path of contact, Pc, are straight line segments that align to one another. Later on, this inconsistence in interpretation of involute gearing was the root cause of many mistakes when gearings of other designs where proposed and investigated. This is because of the following. For gearings that operate on parallel shafts, an involute tooth profile is the only tooth geometry under which the tooth flanks: (a) are enveloping to one another, and (b) are conjugate to each other (or, in other words, they are reversibly-enveloping surfaces, i.e., they are a kind of Re–surfaces, for simplicity [26]). Epicycloid tooth flanks of the mating gears are enveloping to each other, but they are not conjugate to one another—they are not a type of Re–surfaces. Euler details the principle of common tangent. He specifically points out the need for the proper design of gear teeth to avoid friction and wear and indicates this application for clocks. Most clockmakers, however, ignored this, if they ever heard of it. Euler’s treatment of gear teeth was very general and was carried out by the application of principles of analytic geometry using both differential calculus and integral calculus. He set up mathematical expressions for gears to move without friction between their teeth (actually for a minimum

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FIGURE 11.25 Title page of the paper by Euler, L., (1754–1755), “De Aptissima Figure Rotarum Dentibus Tribuenda” (“On Finding the Best Shape for Gear Teeth”), in: Academiae Scientiarum Imperiales Petropolitae, Novi Commentarii, t. V, pp. 299–316.

value of friction). Then, he set up expressions for gears to move with uniform motion. Then he showed in his famous paper (1754–1755) [8,9], that the developed equations can be satisfied only by involute or “epicycloid” teeth. Euler and Savary devised together an analytical method for determining the curvature centers of gear teeth flanks. The importance of the “law of conjugate action” worked out by Euler (gears designed according to this law have a steady angular velocity ratio ratio), became correctly realized much later.

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FIGURE 11.26 Title page of the paper by Euler, L., “Supplementum. De figura dentium rotarum,” Novi Comm. Acad. Sc. Petropol, 1767. (Originally published in Novi Commentarii academiae scientiarum Petropolitanae 11, 1767, pp. 207–231).

For over a century, the invention of an involute tooth profile was not used in practice. The industrial revolution in Great Britain in the eighteenth century saw an explosion in the use of metal gearing. A science of gear design and manufacture rapidly developed through the nineteenth century. The invention and the beginning of application of steam and gas

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turbines that operate at high rotations and produce lots of power immediately turns the attention of engineers to involute gearings. The contribution by Euler is incomplete, as he proposed the involute tooth profile for parallel-axes gear pairs; however, the concept of the “gear/pinion base pitch” (linear base pitch), as well as the concept of the “operating base pitch” (linear operating base pitch) in a parallel-axes gear pair was not known to Euler. Accomplishments in the field of gearing in the Eulerian time can be briefly summarized as follows: • It is proven by Euler in the mid-eighteenth century that an involute tooth profile meets the best requirements of parallel-axes gearing • It is likely the fundamental theorem of parallel-axes gearing was already known due to Camus, Euler, and Savary. This makes it reasonable to refer to this theorem as to “Camus-Euler-Savary fundamental theory of parallel-axes gearing,” or just as to “CES–fundamental theory of parallel-axes gearing” • There is no evidence that a difference between the line of action, LA, and the path of contact, Pc, was recognized at this time, as in cases of parallel-axes gearings these two lines align to each another • No significant accomplishments at that time are done in the area of intersected-axes as well as crossed-axes gearings The invention of an involute tooth profile for parallel-axes gearings is one of the cornerstone accomplishments in the scientific theory of gearing. It is likely this achievement can be referred to as the beginning of the scientific theory of gearing. 11.2.4  Post-Eulerian Period of the Developments in the Field of Gearing In the nineteenth century, a profound investigation of mechanisms in the general sense was undertaken by Robert Willis (Figure 11.27). In his 1841 book [34] titled Principles of Mechanisms (Figure 11.28), Willis compiled the lectures for his students and knowledge about gears that could be used in practice. In the book [34], gearings were discussed by the author to the best extent possible in his time. The “fundamental theorem of gearing” is known now mostly due to the book by Willis [34] (Figure 11.29): Fundamental theorem of parallel-axes gearing (according to Willis): The angular velocities of the two pieces are to each other inversely as the segments into which the “line of action” divides the line of centers, or inversely as the perpendiculars from centers of motion upon the line of action. This is the reason why in Eastern Europe, the fundamental theorem of gearing is commonly referred to as “Willis’s theorem.” However, as this theorem was already known to Camus, Euler, and Savary long before Willis, it makes sense to refer to the “fundamental theorem of gearing” as the “Camus-Euler-Savary fundamental theorem of gearing” (or as to “CES–theorem of gearing,” for simplicity). A contribution by Camus is also covered by the term “CES–theorem of gearing.” As early as 1842, a monograph by Theodore Olivier (Figure 11.30) on the theory of gearing [17] was published. This monograph (Figure 11.31) is the first monograph ever to be titled “Geometric Theory of Gearing” (“Théorie Géométrique des Engrenages”). In the monograph, the first and the second principles of generating of enveloping surfaces are

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FIGURE 11.27 Reverend Robert Willis (1800–1875).

proposed. Later on, both these principles got extensive usage by gear scientists. Graphical methods developed in descriptive geometry were used by Olivier in his book [17]. In the general case of gear meshing, both the principles proposed by Olivier (1842) are incorrect as the condition of conjugacy of the interacting surfaces is ignored when performing this analysis. Both the principles are valid just in degenerate cases, when moving surfaces allow for sliding over themselves in the direction of the enveloping motion. In these reduced cases, the principles by Olivier become useless. Therefore, there is no reason in applying “Olivier principles” for the purpose of generation of conjugate tooth flanks in a gear pair. Olivier cannot be considered as a contributor to the scientific theory of gearing as his accomplishments are a kind of mistake that has significantly affected further development of gear science. In 1848, the curved tooth configuration was proposed by A. C. Semple. Proposed in the first half of the nineteenth century, the curved tooth configuration captured the interest of many mechanical engineers and inventors. The second known monograph on the theory of gearing was published in 1852 by E. Sang [30]. This book, titled A New General Theory of the Teeth of Wheels, is nothing more than a collection of the known achievements in the field of gearing. No contribution to the theory of gearing has been made by E. Sang. In 1886, a new effort to evolve the theory of gearing was undertaken by Chaim Gochman. In his master’s thesis (Figure 11.32), Gochman converted the results earlier obtained by Olivier (who used graphical methods for solving problems in the field of gearing) into that same results obtained using the methods developed in analytical geometry [12]. As it is claimed on page 7 in the research by Gochman [12], no new scientific results are

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FIGURE 11.28 Title page of the 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 p.

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FIGURE 11.29 The fundamental theorem of gearing as formulated in: 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 p.

FIGURE 11.30 Théodore Olivier (1793–1853).

contributed by Gochman to those already obtained by Olivier [17]. The interested reader is referred to [28] for details on this research. Taking into account this later conclusion, the accomplishments by Olivier and those by Gochman are considered together. Olivier in his book [17] (and, later on, Gochman in his master’s thesis [12]) loosely considered the tooth flanks of a gear and a mating pinion only as surfaces enveloping to one another, that is, the requirement of conjugacy of the mating tooth flanks was ignored, which is a huge mistake. Fulfillment of the condition of contact is sufficient only in the cases when “no” rolling motion is observed. Otherwise, this condition needs to be complemented with (a) the condition of conjugacy, and (b) the equality of a gear base pitch and its mating pinion base pitch to the operating base pitches of a gear pair [29].

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FIGURE 11.31 Title page of the first ever monograph on gearing published by Olivier, T., Théorie Géométrique des Engrenages destinés, (Geometric Theory of Geairng), Bachelier, Paris, 1842, 118 p.

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FIGURE 11.32 Title page of Master Thesis by Ch. Gochman: Theory of Gear Teeth Engagement, Generalized and Developed by Implementation of Mathematical Analysis, Odessa (Ukraine), 1886, 229 p.

It must be clearly realized that the terms “conjugate surfaces” and “enveloping surfaces” are not equivalent to one another: all conjugate surfaces are enveloping to each other but NOT vice versa, that is, not all enveloping surfaces are conjugate to one another. In detail, the committed mistake is discussed by Prof. S. P. Radzevich in [28]. The condition of proper meshing in perfect gearing was correctly specified by Euler, Savary, and later on, by Willis. Camus was also close to the correct understanding of the condition of proper meshing in perfect gearing.

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Then, the condition of conjugacy has been ignored in the research undertaken by Olivier, and by all of the followers of this approach in the theory of gearing (Gochman, Litvin, and so forth). The developments in the theory of gearing were significantly affected by the mistake committed by Olivier, and later on repeated by Gochman. The names of Prof. V. A. Shishkov, Prof. F. L. Litvin, Prof. G. I. Sheveleva, and many others are among those who loosely followed Olivier’s approach. There is no chance to develop a scientific theory of gearing based only on the condition of contact, and ignoring:

a. the condition of conjugacy of the interacting tooth flanks b. equality of base pitches of a gear and a mating pinion to operating base pitch of the gear pair, and so forth

The direction of evolution of the gear theory that strictly follows the Olivier–Gochman approach represents the dead end in the evolution of the theory of gearing. Among the experts in the field of gearing of that period of time, the name of Thomas Tredgold (Figure 11.33) should be mentioned as well. As a gear person, he is mostly known for proposing the approximation of bevel gears, (i.e., of intersected-axes gears) by appropriate cylindrical gears, (i.e., by parallel-axes gears). The proposed approximation, that is, the so-called “Tredgold approximation,” significantly simplifies the calculation of bevel gearings in engineering practice. Accomplishments in the field of gearing in post-Eulerian time can be briefly summarized in the following manner: The fundamental theorem of parallel-axes gearing, (i.e., the “Camus-Euler-Savary fundamental theorem of gearing”) is formulated. Later on, this theorem was published in the book by Willis [34], and sometimes is loosely referred to as “Willis’s fundamental theorem of gearing”

FIGURE 11.33 Thomas Tredgold (1788–1829).

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• The importance of the “condition of contact” between two interacting tooth flanks, (i.e., the “enveloping condition”) is realized; various forms of verbal, as well as analytical representation of this important condition are known at that time • Investigation into intersected-axes and crossed-axes gearings started at this time • A huge mistake in the interpretation of the interaction between the tooth flanks of mating gears has been committed by Olivier [17] (1842) and repeated by Gochman [12] (1886). All the research in the field of gearing in the years since 1842 through recent years is significantly affected by this mistake. The “fundamental theorem of parallel-axes gearing,” and the “contact condition,” (i.e., the “enveloping condition”) can be considered as the main contribution to the scientific theory of gearing attained at this time. In the period until the end of the nineteenth century, the development of the tooth flank profile shape was more or less completed for the case of parallel-axes gearing. Since that time, involute gearing prevailed as the most advantageous shape of the gear teeth flanks. 11.2.5  Developments in the Field of Perfect Gearings Regardless of unavailability of the scientific theory of gearing till the beginning of the twentyfirst century, gear practitioners on their own have proposed designs of “perfect” gearings. 11.2.5.1  Grant Bevel Gearing In this concern, the invention [33] by George Grant (Figure 11.34) should be mentioned first of all. Use of the invention [33] allows generating “perfect” bevel gears. This is due to that in one of the possible applications of the invention, “… the rolling cone is increased in

FIGURE 11.34 George Barnard Grant (1849–1917).

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FIGURE 11.35 The essentials of Grant’s invention [U.S. Pat. No. 407.437. Machine for Planing Gear Teeth. /G.B. Grant (1887)].

size until its center angle is ninety degrees, and it becomes a plane circle. Its element will form an epicycloidal surface as before, but it is now called an “involute” surface.” (Figure 11.35). Therefore, the bevel gear tooth flanks are generated by the describing method adopted to the case of intersected-axes gearing, that is, bevel gearing. This is a significant scientific achievement by Grant in the field of scientific theory of gearing. Figure 11.36 is good evidence of perfect tooth flank geometry in a bevel gear, correctly realized by Grant at the end of nineteenth century. An elementary device (Figure 11.37) was used in the past to demonstrate the principal features of meshing in a perfect bevel gear pair. The proposed gearing design by Grant [33] is a kind of perfect gearing that operates on intersecting axes of rotation of a gear and a mating pinion. This invention by Grant deserves to be referred to as “Grant gearing,” or just “Gr–gearing” in his honor. The contribution by Grant is incomplete, as he proposed only a method of generation of tooth flanks of a gear for intersected-axes gear pairs. The concept of the “gear/pinion angular base pitch,” as well as the concept of the “operating angular base pitch” of a bevel gear pair was not known to Grant. However, Grant was a gear practitioner, and not a researcher, and (per the author’s personal opinion) he did not properly value this as his accomplishment, which is of significant importance to the scientific theory of gearing. In addition, in the time of Grant, there was no necessity in more accurate bevel gears compared to those produced by the gear generating method. Because of this, the invention by Grant was forgotten for over a century. 11.2.5.2  Contribution by Professor N. I. Kolchin In the mid-twentieth century, an important analytical research in gearing (in bevel gearing in  particular) was undertaken by Professor A. I. Kolchin of the USSR (Figure 11.38) [13].

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FIGURE 11.36 The involute tooth flank in a bevel gear according to Grant [see Figure 143 in: Grant, G.B., A Treatise on Gear Wheels, 6th edition, Philadelphia Gear Works, Inc., Philadelphia, 1893, 105 p.].

Professor A. I. Kolchin analytically described the results discovered and known in the public domain before his book was published. However, his contribution to the theory of gearing was important as a profound mathematical analysis of gears has been started from his research [13]. 11.2.5.3  Novikov Conformal Gearing In the late 1940s and at the beginning of the 1950s, extensive research work in the field of gearing was carried out by Dr. M. L. Novikov (Figure 11.39) in Moscow, at Zhukovsky Air Force Engineering Academy. Ultimately, a novel design of high-performance gearing was proposed [14,18]. Later on, the results of the research were summarized in the doctoral thesis [15], and in the monograph [16] by Novikov (Figure 11.40). The proposed design of gearing features “concave-to-convex” contact between the interacting tooth flanks of a gear and a mating pinion. The gear designer is free to design the rest of the gear and the pinion tooth profiles. When Novikov carried out his research in the field of conformal gearing, he loosely assumed that in order 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 a bit confusing: in order 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 length of the field of action, that is, the equality Zpa = 0 is valid in “Novikov gearing” (this entails a zero transverse contact ratio, mp = 0, in “Novikov gearing”). The equality of base pitches of a gear and a

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FIGURE 11.37 Demonstration of principal features of meshing in a bevel gear pair.

mating pinion, to operating base pitch of the gear pair is the principal feature of “Novikov gearing” that distinguishes it from helical non-involute gearing of other types. It is customary to associate “Novikov gearing” with the patent “Gear Pairs and Cam Mechanisms Having Point System of Meshing” [18]. Evidence can be found in scientific literature revealing the unfamiliarity of the gear community around the world with this original publication [18] on “Novikov gearing.” As early as 1955, before the invention application was filed, a doctoral thesis [15] on the subject had been defended by 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 Novikov was granted with the patent [18], a monograph was published by him (Figure 11.40) [16]. The concept of “Novikov gearing” is discussed in detail in the two aforementioned valuable sources [15] and [16]. Unfortunately, none of them is quoted by gear experts in Western countries or in the United States. 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.” Formally, the tooth flanks have a circular-arc profile. Actually, as shown later by Prof. S. P. Radzevich [27], “Novikov conformal gearing” is a reduced case of involute gearing in which the involute tooth profile is shrunk to a point, and the rest of the tooth profiles are shaped in the form of a circular arc.4 Because of this, “Novikov conformal gearing” is a 4

For details, see: Radzevich, S.P., “An Examination of High-Conformal Gearing,” Gear Solutions, February, 2018, pp. 31–39.

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FIGURE 11.38 Title page of a monograph by Dr. Kolchin, N.I., Analytical Calculation of Planar and Spatial Gearing, Moscow, Mashgiz, 1949, 210 p.

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FIGURE 11.39 Dr. Mikhail L. Novikov (1915–1957).

type of perfect gearing (a reduced type of involute gearing) that is capable of transmitting a rotation smoothly. 11.2.5.4  Contribution by Professor V. A. Gavrilenko An extensive research in the field of gearing in the 1930s through the 1960s was carried out by Prof. V. A. Gavrilenko (Figure 11.41). He spent decades running extensive research in the field of gearing, particularly in the geometrical theory of involute gearing. In the author’s opinion, the most systematic discussion on involute gearing ever can be found in the monograph by V. Gavrilenko [11]. Unfortunately, the fundamental monographs by Gavrilenko are not known by most gear experts in Europe or in the United States. 11.2.5.5 Condition of Conjugacy of the Interacting Tooth Flanks of a Gear and a Mating Pinion in Crossed-Axes Gearing Conjugacy is a specific property of the tooth flanks of a gear, G, and a mating pinion, P. Only surfaces that roll over one another can feature this unique property. Due to this property, in rolling motion of a gear and a pinion over one another, the tooth flanks, G and P, can be viewed as a kind of “reversibly-enveloping surfaces” (or just “Re–surfaces,” for simplicity) [26]. Tooth flanks not of all geometries can be referred to as the “conjugate surfaces,” or, the same, “reversibly-enveloping surfaces” [26]. In order to possess the property of conjugacy, a criterion to be fulfilled by two teeth flanks, G and P, in a crossed-axes gear pair can be expressed analytically in terms of the design parameters of the gear, and the mating pinion.

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FIGURE 11.40 Title page of Novikov’s (1958) monograph, Gearing with a Novel Kind of Meshing, 1958 [16].

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FIGURE 11.41 Professor Vladimir A. Gavrilenko (1899–1977).

When the tooth flanks of a gear, G, and a mating pinion, P, interact with one another, straight lines that align to common perpendiculars, ng, through points within a current line of contact, LC, must always intersect the axis of instant rotation, Pln. The condition of conjugacy must be met at all points of a (desired) line of contact, LCdes, between the interacting tooth flanks, G and P. This is the key requirement to be fulfilled by conjugate tooth flanks, G and P, when the gears rotate. Three vectors, pln, Vm, and ng, are constructed in Figure 11.42. When a gear, G, and a mating pinion, P, tooth flanks are conjugate, then the three unit vectors, pln, Vm, and ng, are coplanar. The unit vectors, pln, Vm, and ng, are coplanar if and only if the triple scalar product, pln × Vm · ng, is zero, that is, if the equality ([29])

pln × Vm ⋅ n g = 0

(11.1)

is valid. In addition to Equation 1.11, the condition ([29])

n pl ⋅ n g ≠ 0

11.2)

must also be fulfilled. The tooth flanks of a gear, G, and a mating pinion, P, are said to be conjugate if, and only if, the conditions ([29])



pln × Vm ⋅ n g = 0  pln × n g ≠ 0

(11.3)

are fulfilled for any and all points within the line of contact, LC, for any possible configurations of the gear and the pinion in relation to one another.

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FIGURE 11.42 Condition of conjugacy of the tooth flanks of a gear, G, and a mating pinion, P, in crossed-axes gearing.

Another form of representation of the condition of conjugacy of the tooth flanks of a gear and a mating pinions crossed-axes gearings are also known. Equation 11.3 yields the formulation of the second fundamental law of gearing in crossedaxes gearing ([29]): In crossed-axes gearing, in order 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 intersect the axis of instant rotation. (Radzevich, 2008)

This statement is also valid with respect to intersected-axes gearing (in which the center distance, C, is zero, i.e., the equality, C = 0, is observed). Fulfillment of the condition of conjugacy of the tooth flanks, G and P, is necessary, but not sufficient for designing perfect gears. 11.2.5.6 Equality of Angular Base Pitches of a Gear and a Mating Pinion to Operating Angular Base Pitch in Intersected-Axes, and in Crossed-Axes Gearing Tooth flanks in a gear for a crossed-axes gear pair can be viewed as a series of cam surfaces that act on similar surfaces of the mating gear to impart a driving motion. Discussed below, the third fundamental law of gearing requires the equality of angular base pitches of a gear, and that of a mating pinion, to operating angular base pitch of the gear pair. As two (or even more) pairs of teeth make contact at the same time, the fulfillment of a condition caused by several pairs of interacting tooth surfaces in gears is a must in perfect gearings of all designs. A plane of action, PA, in a crossed-axes gear pair is shown in Figure 11.43a. Operating base pitch, ϕb·op, of a crossed-axes gear pair is measured within the plane of action.

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(a)

(b)

FIGURE 11.43 On the concept of equal base pitches in perfect crossed-axes gear pair: (a) plane of action, PA, and (b) the unfolded section A − A of the gear pair.

The plane of action, PA, is intersected by a cylinder of revolution of an arbitrary radius, ry .pa, having the plane-of-action centerline, Opa, as the axis of its rotation. As angular operating base pitch, ϕb·op, of a gear pair is of a constant value, then the lengths: lb. pa = ry . pa ⋅ϕb.op

(11.4)

  lb. g = ry . pa ⋅ϕb.op

(11.5)

  lb. p = ry . pa ⋅ϕb.op

(11.6)



of the circular arcs also are of constant value, that is, in this particular analysis the angular base pitches ϕb.op, ϕb.g, and ϕb.p can be replaced with the lengths lb.pa, lb.g, and lb.p of the corresponding circular arcs. Once the lengths, lb.g = lb.pa and lb.p = lb.pa, are equal, then angular base pitches ϕb.g = ϕb.op and ϕb.p = ϕb.op, are equal as well. The unfolded section, A − A, of the gear teeth flanks, G, by the cylinder of revolution is shown in Figure 11.43b. When the equality of the angular base pitch of a gear, ϕb.g, and that of a mating pinion, ϕb.p, to operating angular base pitch of the gear pair, ϕb.op, is observed, that is, when the identities, ϕb.g ≡ ϕb.op and ϕb.p ≡ ϕb.op, are valid, then the gear and the pinion can be engaged in mesh as illustrated in Figure 11.43b. Each gear point, Kg, coincides with corresponding pinion point, Kp. Because of this, the gear and the pinion points, Kg and Kp, further are designated as contact point, K. Therefore, a uniform rotation can be smoothly transmitted by a crossed-axes gear pair if the following equalities are fulfilled at every instant of time ([29]):



ϕb. g ≡ ϕb.op  ϕb. p ≡ ϕb.op (11.7)

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With that said, the third condition to be fulfilled in a perfect crossed-axes gear pair can be formulated in the following manner ([29]): In crossed-axes gearing, in order to transmit a uniform rotary motion from a driving shaft to a driven shaft by means of gear teeth, angular base pitch of a gear, and that of a mating pinion must be equal to operating base pitch of the gear pair at every instant of time. Radzevich, 2017

Only “conjugate” gear tooth flanks feature base pitch. Base pitch cannot be specified for nonconjugate gear tooth flanks. Therefore, only conjugate tooth flanks of a gear and a mating pinion can fulfill the “third fundamental law of gearing.” The equality of angular base pitches of a gear and a mating pinion to operating angular base pitch in a crossed-axes gear pair is the third fundamental law of gearing that all perfect crossed-axes gearings have to fulfill. 11.2.6  Contribution by Walton Musser In the late 1950s, Walton Musser (Figure 11.44) proposed a novel type of transmission, the so-called, “harmonic drive.” Although this invention revolutionized the theory of “machines and mechanisms,” harmonic drives are not gear drives in the sense considered in this monograph. This kind of transmission is out of the scope of the book. Accomplishments in the field of gearing attained in that period of time can be briefly summarized as follows: • A breakthrough invention in the field of intersected-axes gearing has been made by Grant. He proposed a “Machine for Planing Gear Teeth” (U.S. Pat. No. 407.437, [33]) that is capable machining perfect straight bevel gears. The geometry of a

FIGURE 11.44 C. Walton Musser (1909–1998).

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straight bevel gear tooth flank, (i.e., equivalent to the involute of a circle in cases of parallel-axes gearing) is proposed by Grant for the case of intersected-axes gearing • A novel design of conformal gearing was proposed by Novikov [18]. Grant’s invention [33] is an important contribution to the theory of gearing. Novikov’s invention completely aligns with the well-developed theory of parallel-axes involute gearing, as “Novikov conformal gearing” is a reduced case of involute gearing. 11.2.7  Tentative Chronology of the Evolution of the Theory of Gearing Summarizing the above discussion, the benchmarking achievements in the theory of gearing are schematically illustrated in Figure 11.45. Contributions to the field of gearing by Desargues, de La Hire, and Camus comprise the pre-Eulerian period of evolution of the theory of gearing. In the schematic (see Figure 11.45), number “0” is assigned to the pre-Eulerian period evolution of the theory of gearing. Invention of involute gearing by Euler (1760) is a benchmarking achievement in the theory of gearing. Per the author’s opinion, the origin of the “scientific theory of gearing” has to be associated with this accomplishment. In the schematic (see Figure 11.45), number “1” is assigned to the invention of involute gearing by Euler. The next step in the development of the theory of gearing was made by Euler and Savary, who are granted with the “fundamental theorem of gearing” (along with Camus). Later on, in 1841, this theorem was published in the book by Willis [34]. Number “2” is assigned in the schematic (see Figure 11.45) to the achievement in the scientific theory of gearing. The “CamusEuler-Savary fundamental theorem of gearing” is valid only for parallel-axes gearings. In 1842, a huge mistake had been committed by Olivier, who proposed his version of the theory of gearing based just on the enveloping condition of conjugacy of the interacting tooth flanks of a gear and a matting pinion. The condition of conjugacy of the tooth flanks was not taken into account by Olivier. This event is labeled as “3” in the schematic (see Figure 11.45). The mistake committed by Olivier [17] (1842) and repeated by Gochman [12] (1886), significantly influenced further fundamental developments in the theory of gearing (“4,” “8,” and others in Figure 11.45). The accomplishments in the theory of gearing labeled as “5” through “7” are not associated with the necessity to meet the condition of conjugacy of the interacting tooth flanks, and are applicable in both branches, that is: (a) in the “approximate gearings” of the theory (“4,” “8,” and others in Figure 11.45), as well as (b) in the way that leads to the selfconsistent scientific theory of gearing (“5” through “14,” and others in Figure 11.45) [29]. No perfect intersected-axes and crossed-axes gearings can be designed following the first way. No correct tooth flank modification in parallel-axes gearing is possible—only a trial and error method can be used to determine the parameters of the tooth flank modification if the first way is chosen. The condition that requires equal base pitches of a gear and its mating pinion (only in cases of parallel-axes gearings) has been known for a long time (note, the “operating base pitch” of a gear pair is not known yet). Per the author’s estimate, this requirement, that is, item “5” in Figure 11.45, has been known since the mid nineteenth century. Unfortunately, in the meantime, it is not possible to identify the name of the gear scientist who should be credited with this significant accomplishment in the scientific theory of gearing.

FIGURE 11.45 Principal accomplishments in the scientific theory of gearing.

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Spherical involute in perfect bevel gearing (item “6” in Figure 11.45) has been known since 1887. The “Shishkov equation of contact” (item “7” in Figure 11.45) deserves to be mentioned here, as the use of this equation makes possible significant simplifications of the kinematic method of surface generation, especially in cases when both the contact perpendicular, n, and the instant linear velocity vector, VΣ, can be determined with no derivatives of equations of the tooth flanks, G and P, as well as the parameters of the kinematics of a gear pair. Conditions of contact (item “9” in Figure 11.45) of the interacting tooth flanks, G and P, of a gear and a mating pinion in a higher order are described by means of the indicatrix of conformity, CnfRG/P, at point of contact, K, of the interacting tooth flanks of a gear and a mating pinion (Radzevich, 1983) [19,20,23–25,29], and others. Then, the listed below accomplishments: • Condition of conjugacy of the tooth flanks for gear pairs of all types (item “10” in Figure 11.45), including intersected-axes gear pairs, and crossed-axes gear pairs, • The concepts of (a) “angular base pitch” in intersected-axes gear pairs, and crossedaxes gear pairs, and (b) the “operating angular base pitch” in gear pairs of all types (item “11” in Figure 11.45), • The equality of base pitches of a gear and its mating pinion to the “operating base pitch” in gear pairs of all types (item “12” in Figure 11.45), • Design of perfect crossed-axes gearing with line contact between the tooth flanks, G and P, that is, R–gearing (item “13” in Figure 11.45), • A scientific classification of vector diagrams of gear pairs of all types (item “14” in Figure 11.45), and • Design of perfect (crossed-axes) gearing insensitive to the axes’ misalignment, that is, Spr–gearing (item “15” in Figure 11.45), were contributed by Prof. S. P. Radzevich circa 2008. Equation of conjugacy, pln × Vm · ng = 0, of the interacting tooth flanks, G and P, of a gear and a mating pinion is derived by Prof. S.P. Radzevich, in 2017 “16” in Figure 11.45). The development of a theory of favorable approximate gearings (item “?” in Figure 11.45) is a challenging problem for the future developments in the field of theory of gearing. It should be realized that the diagram shown in Figure 11.45 is tentative. More accomplishments in the scientific theory of gearing and the corresponding gear researcher’s names can be added in Figure 11.45 if a more detailed investigation into the evolution of the scientific theory of gearing is undertaken. Only the key (the fundamental) achievements in the scientific theory of gearing are included in the diagram (Figure 11.45) in its current stage. Generally speaking, perfect gear pairs of any type can be designed based on the scientific theory of gearing.

11.3  Developments in the Field of Approximate Gearings To meet the current needs of the industry, practical gear engineers proposed numerous approximate designs of gearings. Initially, when the designs were proposed, it was loosely assumed that each of them is capable of transmitting a rotation smoothly. Unfortunately, it was shown later on that they do not meet all the requirements perfect gears needs to meet.

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11.3.1  Cone Double-Enveloping Worm Gearing First rudimentary “double-enveloping” worm gear drive has been known since the times of Leonardo da Vinci [4]. In more recent times, double-enveloping worm gearing was proposed as early as 1891 by Dr. Friedrich Wilhelm Lorenz of Germany. In his invention Dr. Lorenz proposed methods to generate the worm and the gear of the double-enveloping worm-gear drive, and he received two patents for these accomplishments. A bit later (c. 1920), a similar double-enveloping worm gearing was independently proposed by Mr. Samuel Cone of the United States. Wilhelm Lorenz and Samuel Cone understood very well the advantages of the drives they had invented, particularly, the increased load capacity due to the higher contact ratio in comparison with that of conventional worm-gear drives. Although the geometry of the Lorenz and Cone drives differs, both types offer this advantage. Double-enveloping worm gearing is an example of approximate gearing as it does not meet all the requirements the perfect gears needs to meet. 11.3.2  Approximate Bevel Gearing Early accomplishments in the field of bevel gearing are tightly connected with the name of William Gleason (Figure 11.46). In 1874, his invention of the straight bevel gear planer for the production of bevel gears with straight teeth substantially advanced the progress of gear making. The early part of the twentieth century was the beginning of the automotive industry, which required a broader application of bevel gears to transform rotation and power between intersected axes. In the 1920s, automotive industry designers also needed (a) a gear drive to transform motions and power between crossed axes and (b) a lower location for the driving shaft. The Gleason Works engineers met these needs with pioneering

FIGURE 11.46 William Gleason (1836–1922).

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developments directed at designing new types of gear drives and the equipment and tools to generate the gears for these drives. The proposed designs of bevel gears in today’s industry are examples of approximate gearing as they are developed and manufactured based on the application of the imaginary straight-sided crown gear (basic crown rack). Because of this, today’s bevel gears of all types, that is, straight bevel gears, skew bevel gears, spiral bevel gears, and others, both, face-milled and face-hobbed, do not meet all the requirements perfect gears needs to meet. 11.3.3  Approximate Crossed-Axes Gearing The concept of the gearing that operates on crossing shafts can be traced back to the times of Leonardo da Vinci (see Figure 11.18) [4]. The need for more accurate and quieter running gears became obvious with the advent of the automobile. Although the hypoid gear was within our manufacturing capabilities by 1916, it was not used practically until 1926, when it was used in the Packard automobile. The hypoid gear made it possible to lower the drive shaft and gain more usable floor space. By 1937, almost all cars used hypoid-geared rear axles. The success with the design, manufacture, and application of the contemporary crossedaxes gearing is credited to two famous gear experts, Nikola Trbojevich (also known as Nicholas Terbo), and Ernest Wildhaber. Nikola Trbojevich (Figure 11.10), a world-known research engineer, mathematician, and inventor, was a nephew and friend of Nikola Tesla. Mr. Trbojevich held nearly 200 U.S. and foreign patents, principally in the field of gear design. Mr. Trbojevich’s most notable work that brought him international recognition was the invention of the “hypoid gear.” First published in 1923, it was a new type of spiral bevel gear employing previously unexploited mathematical techniques. The “hypoid gear” is used in a majority of all cars, trucks, and military vehicles today. Together with his invention of the tools and machines necessary for its manufacture, the “hypoid gear” became an integral part of the final drive mechanism of automobiles by 1931. Its effect was immediately apparent in that the overall height of rear-drive passenger automobiles was reduced by at least four inches. Ernest Wildhaber (Figure 11.14) is one of the most famous inventors in the field of gear manufacture and design. He was granted with 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. The hypoid gear drive is one of the most famous inventions by Dr. Wildhaber. 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. The proposed designs of crossed-axes gears in today’s industry are examples of approximate gearing as they are developed and manufactured based on application of the imaginary crown gear with straight-sided profile (basic crown rack). Because of this, today’s crossed-axes gears of all types, both, face-milled and face-hobbed, do not meet all the requirements perfect gears needs to meet. 11.3.4  Face Gearing Face gearing can be viewed as a reduced case either of intersected-axes gearing, or of crossed-axes gearing when the pitch cone angle increases to the right angle. All known designs of face gearings, both, intersected-axes gearings, and crossed-axes gearings, are approximate gearings as they do not meet all the requirements perfect gears need to meet. The face cutting technique used to produce crossed-axes gears is supplied by these three

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companies: The Gleason Works, Klingelnberg-Oerlikon, and Yutaka Seimitsu Kogyo, LTD, and is based upon an empirical and manufacturing technology that predates World War II. Accomplishments in the field of gearing in that period of time can be briefly summarized as follows: • Double-enveloping (approximate) gearing was proposed by Wilhelm Lorenz of Germany (1874), and a bit later (c. 1920) by Samuel Cone of the United States; • Design of and methods for machining of approximate hypoid gearing were proposed by Nikola Trbojevich, and later on improved by Ernest Wildhaber, both of the United States; • Face gearing are widely used in the design of Fellow’s gear-shaping machines. The most significant contributions to the field of gearing at that time are made to approximate gearing.

11.4  A Brief Summary of the Principal Accomplishments in the Theory of Gearing Achieved by the Beginning of the Twenty-First Century It should be stated here, from the very beginning, that no self-consistent (or potentially self-consistent) scientific theory of gearing had been developed by the beginning of the twenty-first century (c. 2010). Among others, a self-consistent scientific theory of gearing must possess two important properties: first, a self-consistent scientific theory of gearing must cover all known designs of gears and gearings with no exclusion and second, a self-consistent scientific theory of gearing must cover all (with no exclusion) unknown yet designs of gears and gearings, that is, the theory must possess the property to predict novel designs of gears and gearings. All the books published so far under the subject “theory of gearing” (starting from the first 1841 book by Théodore Olivier [17], and ending with the latest publications in the field c. 2010) consist of no scientific theory of gearing. These books cannot be referred to as a “theory of gearing,” rather they are collections of known achievements in the field of gearing, having no ability to predict novel unknown designs of gears and gearings. No doubt, a scientific theory of gearing is necessary to the gear researchers and practical engineers as it is a powerful tool for the development of novel designs of gears and gearings with a prescribed performance. Such a scientific theory of gearing can be developed now. With that said, it is important to revise the earlier obtained accomplishments in the field of gearing and select those of them that can be useful in the development of the fundamental scientific theory of gearing. 11.4.1  Condition of Contact of the Interacting Tooth Flanks of a Gear and Pinion The “condition of contact” of the interacting tooth flanks of a gear, G, and a mating pinion, P, is the first scientific result of fundamental importance that can be used in the foundation of the scientific theory of gearing. The “condition of contact” is also known as the “enveloping condition.” The contact condition states that: At every point of contact of the tooth flanks

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of a gear, G, and a mating pinion, P, the projection of the relative velocity vector onto the common perpendicular to the interacting tooth flanks is zero. The condition of contact of two interacting tooth flanks in a gear pair has been known for centuries. Per the author’s opinion, this important condition was already known to Camus (1733) [2] or even to Desargues. Since the time when gear scientists started realizing the importance of the “condition of contact,” the forms of its representation were different. Without going into details of the analysis of this particular problem, it should be stressed here that the condition of contact is finally represented in the form of equal to zero of the dot product of the unit vector of the common perpendicular, n, at point of contact of the tooth flanks G and G, and the instant velocity vector, VΣ, of the resultant relative motion of the tooth flanks, G and P, that is

n ⋅ VΣ = 0 (11.8)

In the form of dot product [see Equation 11.8], the condition of contact was proposed by Professor V. A. Shishkov in his paper [31]. This equation can be found out in his monograph [32] (Figure 11.47), as well as in his later publications. As it is shown from the research undertaken by Prof. Radzevich [22] that Dr. Shishkov is the first (no later than 1948) to represent the condition of contact of two smooth regular surfaces in the form of dot product n · VΣ = 0 of the unit vector of a common perpendicular, n, by the vector of the velocity of the relative motion of the interacting surfaces at a point of their contact. The equation of contact in the form n · VΣ = 0 is known as “Shishkov equation of contact” [22,29], and others. The “Shishkov equation of contact, n · VΣ = 0” is a valuable contribution to the scientific theory of gearing. 11.4.2  Condition of Conjugacy of the Interacting Tooth Flanks of a Gear and Pinion The condition of conjugacy of two interacting tooth profiles of a gear and a mating pinion is a bit tricky. Informally, the condition of conjugacy can be interpreted in the following manner. Assume that a profile of one member of a gear pair is given. Then, the tooth profile of the mating member of the gear pair can be generated as an envelope to consecutive positions of the first member in its motion in relation to the second member. Then assume that the tooth profile of the second member of a gear pair is known, and the tooth profile of the first member of the gear pair is generated as an envelope to consecutive positions of the second member in its motion in relation to the first member. Then, compare the obtained tooth profiles of the first member of the gear pair with its original profile. If they are identical to one another then the interacting tooth flanks are conjugate to one another. Otherwise, the interacting tooth flanks are not conjugate to one another. Conjugate tooth profiles/surfaces are also known as reversibly-enveloping profiles/ surfaces (or just Re– profiles/surfaces for simplicity) [26]. For the cases of Pa–gearings, the problem of conjugacy of the tooth profiles/flanks was solved by Euler in the eighteenth century (1760). In his famous work [8], Euler proposed an equation for the involute tooth profiles of a gear and a mating pinion that are conjugate to one another. No special attention is made neither on the fulfillment of the condition of contact (n · VΣ = 0), nor on that the interacting tooth flanks are conjugate to one another. There is no evidence that Euler himself realized the importance of the condition

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FIGURE 11.47 Title page of the monograph (1951) by Prof. V. A. Shishkov [32].

of conjugacy for the interacting tooth flanks of a gear and a mating pinion. Moreover, amazingly, the solution to the problem of conjugacy of the tooth flanks is not understood in all detail by most of gear community all around the world. Per the author’s opinion, the root cause for the poor understanding of necessity of the condition of conjugacy is because of the following.

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The condition of conjugacy of interacting surfaces is more robust rather than the enveloping condition. All conjugate surfaces are enveloping to one another, but not vice versa—not all enveloping surfaces are conjugate. In involute gearing (Figure 11.48), the line of action, LA, and the path of contact, Pc, align to each other at every point of contact, K, of the tooth flanks G and P, of the gear and the pinion, correspondingly. This is possible as both the line of action LA and the path of contact Pc are straight lines through the pitch point, P, at transverse pressure angle, φt, to a perpendicular to the center line. This feature of involute gearing is the root cause of confusion as the line of contact and the path of contact are commonly not distinguished from one another in the cases of Ia–gearing, as well as in the cases of Ca–gearing. To correct the mistake committed by Olivier, and later on followed by Gochman, and many other followers, it is necessary to make the difference between the line of action and the path of contact in a gear pair. The “Camus-Euler-Savary fundamental theorem of gearing” (see Figure 11.29), has to be met at every instant of meshing of a gear and a mating pinion. The “CES–fundamental theorem of gearing,” is a valuable contribution to the scientific theory of gearing. For the first time ever, the condition of conjugacy of the tooth flanks of a gear, G, and a mating pinion, P, is described analytically by Prof. S. Radzevich (2017) in the form of a triple scalar product pln × Vm · ng = 0. 11.4.3 Condition of Equal of Base Pitches of the Interacting Tooth Flanks of a Gear and Pinion In order to transmit a rotation between two shafts, at certain periods of time, more than one pair of teeth needs to be engaged in mesh simultaneously. To meet this requirement,

FIGURE 11.48 The line of action, LA, and the path of contact, Pc, in parallel-axes involute gearing.

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base pitches of the mating gears must be equal to one another. This fundamental requirement is only known for the cases of perfect parallel-axes gearing with zero axis misalignment. The “condition of equality of base pitches” of two mating gears is a valuable contribution to the scientific theory of gearing. Accomplishments in the field of gearing in that period of time can be briefly summarized as follows: • In the earlier known condition of contact of the interacting tooth flanks of a gear and a mating pinion, Prof. Shishkov proposed to represent in the form of dot product n · VΣ = 0. This equation of contact is a key equation in the kinematic method of surface generation. Commonly, this equation is referred to as “Shishkov equation of contact, n · VΣ = 0” • The condition of conjugacy of the interacting tooth flanks of a gear and a mating pinion is not understood and, in most cases, this important condition is ignored. This is a consequence of the mistake committed by Olivier in the nineteenth century. • The requirement according to which the base pitches of the mating gear and pinion is construed is only in part, and only for the case of perfect parallel-axes gearing. The concept of the operating base pitch of a gear pair is not realized at all. It is likely that only a contribution by Prof. Shishkov, (i.e., the “Shishkov equation of contact, n · VΣ = 0”) can be referred to as a significant contribution to the theory of gearing at that period of time. 11.4.4  On the So-Called “Russian School of Theory of Gearing” In recent years, numerous papers authored/coauthored by Russians have been published (both, in English, and in Russian languages), in which a claim on the so-called “Russian school of theory of gearing” has been made. Some of these publications are listed in the “Bibliography” section of this chapter, and a few more can be found in the public domain. The discussion in this chapter of the book along with the results of the earlier performed retrospective analysis on the history of evolution of the scientific theory of gearing [21] and [29], reveal that this aggressive claim has been made with no sufficient validity (see Figure 11.45, for details). Is there a fundamental accomplishment of the “Russian school of theory of gearing” to the scientific theory of gearing that is not taken into account (and not indicated in Figure 11.45)? Feel free to name it, if any! This could be a good contribution helpful for the enhancement of understanding of the evolution of the scientific theory of gearing (see the chart in Figure 11.45). In the published papers and monographs authored by the leading Soviet/Russian gear researchers, there is no evidence of understanding of the kinematics and geometry of:

a. b. c. d.

“Novikov gearing” Spiroid gearing (and perfect worm gearing in a more general sense) Perfect intersected-axes, and crossed-axes gearing (more generally) Perfect gears with the axes’ misalignment.

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What can be expected from the less-experienced researchers? One of the Soviet coryphaeus in the field of gearing, Prof. Ya. S. Davidov, in his “Memories…”5 correctly compared all the Russian gear theoreticians with the “swamp.” A following dialog took place between Prof. F. L. Litvin and Prof. Ya. S. Davidov, when they were discussing the features of “Novikov gearing”: “In one of the conversations with me, F. L. Litvin very correctly compared the work of Novikov to the stone thrown into the swamp and caused a stirring of water.”6 Can someone ignore this opinion of two well-known Soviet/Russian gear researchers (of Prof. F. L. Litvin, and Prof. Ya. S. Davidov), when they have compared all the Russian gear theoreticians with the “swamp”? This comparison is one more evidence of that the claim on the so-called “Russian school of theory of gearing” is at least doubtful. The just made statement has to be taken into account when the readers meet the meaningless term “Russian school of theory of gearing” (as well as similar terms introduced by Russians in the recent years: “classical school of theory of gearing” and “the gold age of theory of gearing”). In the meantime, experienced readers are skeptical with that and are commonly having a laugh when they are reading about the so-called “Russian school of theory of gearing.”

11.5  Concluding Remarks A brief overview on gear art and on the evolution of the scientific theory of gearing was discussed in this chapter. Among others, the discussion is aimed to initiate an in-depth investigation into the field of the origins of the scientific theory of gearing. More names of gear researchers deserve to be mentioned. However, consideration in this chapter is limited to the evolution only of the theory of gearing. Therefore, the number of names of the researchers is limited only to those who contributed to the kinematics and the geometry of gearing. A comprehensive research on the evolution of the theory of gearing is necessary to be undertaken in the nearest future. The research needs to be based on an in-depth study of the original scientific works of all principal investigators of the topic. The history of engineering is not less important than the engineering itself. The better we know the past, the better we can predict the future.

References 1. Bevel Gear: Fundamentals and Applications, by Klingelnberg, J., (Editor): Springer-Verlag GmbH, Berlin Heidelberg, 2016, 328 p. 2. Camus, C.-É.-L., “Sur la figure des dents des rouës, et des ai les des pignons, pour rendre les horloges plus parfaites,” 1733. 3. Coy, J.J., Townsend, D.P., Zaretsky, E.V., “Gearing,” NASA Reference Publication 1152, AVSCOM, Technical Report 84-C-15, 1985, 76 p. 4. 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. 5 6

http://referat.znate.ru/text/index-8600.html. It is likely the comparison of the gear community in the Soviet Union/Russia with a “swamp” is correct.

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5. de la Hire, P., Mémoires de Mathématique et de Physique, Impr. Royale, Paris, 1694. 6. de Morales, A., Las Antigüedades de las ciudades de Espana, Alcalá, 1575, pp. 91–93. 7. Dudley, D.W., The Evolution of the Gear Art, AGMA, Washington DC, 1969, 93 p. 8. Euler, L., “De Aptissima Figure Rotarum Dentibus Tribuenda” (“On Finding the Best Shape for Gear Teeth”), in: Academiae Scientiarum Imperiales Petropolitae, Novi Commentarii, 1754–55, t. V, pp. 299–316. 9. Euler, L., “Supplementum. De Figura Dentium Rotarum,” Novi Commentarii Adacemiae Petropolitanae, 1767, 11, pp. 207–231. 10. Fraifeld, I.A., Cutting Tools that Work on Generating Principle, Mashgiz, Moscow, 1948, 252 p. 11. Gavrilenko, V.A., Fundamentals of the Theory of Involute Gearing, Mashinostroyeniye, Moscow, 1969, 432 p. 12. Gochman, Ch.I., Theory of Gear Teeth Engagement, Generalized and Developed by Implementation of Mathematical Analysis, Odessa (Ukraine), 1886, 229 p. 13. Kolchin, N.I., Analytical Calculation of Planar and Spatial Gearing, Mashgiz, Moscow, 1949, 210 p. 14. Nieman, G., “Novikov Gear System and other Special Gear Systems for High Load Carrying Capacity,” VDI, Berichte, 1961, p. 47. 15. Novikov, M.L., Fundamentals of Geometric Theory of Gearing with Point Meshing for High Power Density Transmissions, Doctoral Thesis, Zhukovsky Air Force Engineering Academy (AFEA), Moscow, 1955. 16. Novikov, M.L., Gearing of Gears with a Novel Type of Teeth Meshing, Published by Zhukovsky Air Force Engineering Academy, Moscow, 1958, 186 p. 17. Olivier, T., Théorie Géométrique des Engrenages destinés à transmettre le movement de rotation entre deux axes ou non situés dans un měme plan, (Geometric Theory of Gearing), Bachelier, Paris, 1842, 118 p. 18. Pat. No. 109,113, USSR, Gear Pairs and Cam Mechanisms Having Point System of Meshing./M.L. Novikov, National Classification 47 h, 6; Filed: April 19, 1956, published in Bull. of Inventions No.10, 1957. 19. Pat. No. 1,185,749, USSR, A Method of Sculptured Surface Machining on Multi-Axis NC Machine./S.P. Radzevich, Int. Cl. B23c 3/16, Filed: October 24, 1983. 20. Pat. No. 1,249,787, USSR, A Method of Sculptured Surface Machining on Multi-Axis NC Machine./S.P. Radzevich, Int. Cl. B23c 3/16, Filed: December 27, 1984. 21. Radzevich, S.P., “A Brief Overview on the Evolution of the Scientific Theory of Gearing: A Preliminary Discussion,” in: Proceedings of International Conference on Gears 2015, October 5–7, 2015, Technische Universität München (TUM), Garching (near Munich), Germany, 2015, pp. 1035–1046. 22. Radzevich, S.P., “Concisely on Kinematic Method and about History of the Equation of Contact in the Form n · VΣ = 0,” Theory of Mechanisms and Machines, 2010, 1(15), 8, pp. 42–51. http://tmm. spbstu.ru. 23. Radzevich, S.P., Differential-Geometric Method of Surface Generation, Dr.Sci. Thesis, Tula Polytechnic Institute, Tula, 1991, 300 p. 24. Radzevich, S.P., Fundamentals of Surface Generation, Monograph, Rastan, Kiev, 2001, 592 p. 25. Radzevich, S.P., Generation of Surfaces: Kinematic Geometry of Surface Machining, CRC Press, Boca Raton, Florida, 2014, 747 p. 26. Radzevich, S.P., Geometry of Surfaces: A Practical Guide for Mechanical Engineers, Wiley, Chichester, 2013, 264 p. 27. Radzevich, S.P., “High-Conformal Gearing: A New Look at the Concept of Novikov Gearing,” in: Proceedings of International Conference on Gears 2015, October 5–7, 2015, Technische Universität München (TUM), Garching (near Munich), Germany, 2015, pp. 457–470. 28. Radzevich, S.P., “On Master Thesis: Gochman, Ch.I., Theory of Gear Teeth Engagement, Generalized and Developed by Implementation of Mathematical Analysis,” Theory of Mechanisms and Machines, 2011, 17(1), pp. 33–43. http://tmm.spbstu.ru/01_2011.html. 29. Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd edition, revised and expanded, CRC Press, Boca Raton, Florida, 2018, 898 p. [First edition: Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, CRC Press, Boca Raton, Florida, 2012, 760 p.].

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30. Sang, E., A New General Theory of the Teeth of Wheels, A&C Black, North Bridge, Edinburgh, 1852, 257 p. 31. Shishkov, V.A., “Elements of the Kinematics of Generating, and the Conjugation in Gearing,” in: Theory and Computation of Gears, Vol. 6, LONITOMASH, Leningrad, 1948. 32. Shishkov, V.A., Generation of Surfaces in Continuous-Indexing Methods of Surface Machining, Mashgiz, Moscow, 1951, 152 p. 33. U.S. Pat. No. 407.437. Machine for Planing Gear Teeth./G.B. Grant, Filed: January 14, 1887 (serial No. 224,382), Patent issued: July 23, 1889. 34. 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 p. 35. Woodbury, R.S., History of the Gear-Cutting. A Historical Study in Geometry and Machine, The M.I.T. Press, 1958, 135 p.

Bibliography Babichev, D.T., Lagutin, S.A., Barmina, N.A., “Overview of the Works of the Russian School of Theory of and the Geometry of Gearing. Part 1. Origins of the Theory of Gearing, and its Heyday Time in 1935–1975,” Theory of Mechanisms and Machines, Vol. 14, 2016, 3(31), pp. 101–134. http://tmm. spbstu.ru/31/Babichev_Lagutin_Barmina_31.pdf. Babichev, D.T., Lagutin, S.A., Barmina, N.A., “Overview of the Works of the Russian School of Theory of and the Geometry of Gearing. Part 2. Development of the Classical Theory of Gearing and Establishment of the Theory of Real Gearing in 1976–2000,” Theory of Mechanisms and Machines, Vol. 15, 2017, 3(35), pp. 86–119. http://tmm.spbstu.ru/35/Babichev_Lagutin_Barmina_35.pdf. Babichev, D.T., Volkov, A.E., “History of Evolution of the Theory of Gearing,” Journal of Scientific and Technological Development, 2015, 5(93), с. 25–42. www.vntr.ru. Bayazitov, N., Helical Gears with a New Type of Gearing, Ph.D. Thesis, Kazan’ Technological & Chemical Institute, Kazan’, 1964. Bobil’ov, D.K., “On a Motion of a Surface, that Retains in Tangency to another Surface, Motionless,” Notes of Imperial Academy of Sciences, 1887. Crosher, W.P., A Gear Chronology: Significant Events and Dates Affecting Gear Development, Xlibris Corporation, 2014, 260 p. ISBN 1499071191, 9781499071191. Davidov, Ya.S., Non-Involute Gearing, Mashgiz, Moscow, 1950, 179 p. Dürer, A., Underweysung der Messung mit dem Zirckel und Richtscheyt, 1525, pp. 6–17. Field, J.V., Wright, M.T., “The Early History of Mathematical Gearing,” Endeavour, 1985, 9(4), pp. 198–203. Gol’dfarb, V.I., Fundamentals of Theory of Computer Aided Geometric Analysis and Synthesis of General Kind of Worm Gearing, Doctoral Thesis, Moscow Aviation Institute, 1986, 48 p. Goldfarb, V.I., “What We Know About Spiroid Gearing,” in: Proceedings of the International Conference on Mechanical Transmissions, China, Vol. 1, Science Press, 2006, рp. 19–26. Goldfarb, V.I., Glavatskikh, D.V., Trubachev, E.S., Kuznetsov, A.S., Lukin, E.V., Ivanov, E.V., Puzanov, V.Yu., Spiroid Gearboxes for Pipeline Valves, Moscow, Veche, 2011, 222 p. Golovin, A.A., Tarabarin, V.B., “Russian Models from the Mechanisms Collection of Bauman University,” History of Mechanism and Machine Science, 2008, 5(Springer), 238 p. Hooke, R., Lectiones Cutlerianae, London, No.2, “Animadiversions on Helvius ‘Machina Coeledits,’” 1679, pp. 70–72 and Figs. 20 and 21 (the date of 1666 is Hooke’s). https://en.wikipedia.org/ wiki/Juanelo_Turriano.

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Kaestner, A.G., “De Dentibus Rotarum,” 1781, in: Commentationes Societatis Regiae Scientiarum Gottingensis, Classis Mathematicae, t. IV, pp. 3–25, and idem., “De Dentibus Rotarum,” t. 5, 1782, p. 3. Lagutin, S.A., Barmina, N.A., “Prof. F.L. Litvin: Contribution to the Formation of the Russian School of the Theory of Gearing,” in: Goldfarb, V., and Barmina, (Editors): Theory and Practice of Gearing and Transmissions, Mechanisms and Machine Science 34, Springer International Publishing, Switzerland, 2016, pp. 19–36. Lewis, W., “Interchangeable Involute Gearing,” Journal A.S.M.E., October 1910, p. 1631, and: Am. Mach., 1909, pp. 307–314. Prudhomme, R., and Lemasson, G., Cinématique, École Nationale Supérienre d’Arts et Métiers, École d’Engenieurs, Donod, Paris, 1906, 1955. Radzevich, S.P., “An Examination of High-Conformal Gearing,” Gear Solutions, February 2018, pp. 31–39. Somov, I.I., Rational Mechanics: Part I. Kinematics, Saint Petersburg, 1872. Verkhovskii, A.V., Geometric Modeling in the Analysis and Synthesis of Worm Gearing of General Type, Doctoral Thesis, Specialty 05.02.08 – Theory of Machines and Mechanisms, Moscow, Moscow State University of Electronics and Mathematics, 2000, 254 p. Volkov, A.E., Babichev, D.T., “History of Gearing Theory Development,” 25th Working Meeting of IFToMM Permanent Commission for Standardization of Terminology on MMS, Saint-Petersburg, Russia, June 23–29, 2014, pp. 71–102.

Appendix A: On the Inconsistency of the Term “Wildhaber-Novikov Gearing”: A New Look at the Concept of “Novikov Gearing” Stephen P. Radzevich CONTENTS A.1 Introduction......................................................................................................................... 487 A.2 A Brief Historical Overview.............................................................................................. 488 A.3 Principal Design Features of “Novikov Gearing”......................................................... 489 A.3.1 Vector Diagram of a Gear Pair.............................................................................. 490 A.3.2 Plane of Action in Parallel-Axes Gearing............................................................ 490 A.3.3 A Desired Line of Contact in a Parallel-Axes Gearing...................................... 491 A.3.4 Design Features of “Novikov Gearing”.............................................................. 495 A.3.5 Principal Design Parameters of “Novikov Gearing”........................................ 496 A.4 High-Conformal Gearing.................................................................................................. 496 A.4.1 Critical Degree of Conformity in “Novikov Gearing”...................................... 497 A.4.2 A Minimum Required Degree of Conformity at Point of Contact of the Interacting Tooth Flanks........................................................................................ 497 A.4.3 Conclusion...............................................................................................................500 References...................................................................................................................................... 501 A brief historical overview on “Novikov gearing” is presented below. It is shown that “Novikov gearing” is a reduced case of involute parallel-axes gearing. The concept, earlier introduced by the author, of the “boundary Novikov circle” (or just “N–circle,” for simplicity) enables a clear analysis of the kinematics, and geometry of “Novikov gearing.” Necessary and sufficient conditions for perfect operating of conformal gearing, (i.e., “Novikov gearing”), and “high-conformal gearing” are outlined. Use of the concept of “reversibly-enveloping surfaces” (or just “Re –surfaces,” for simplicity) makes possible a statement that neither gears “Novikov gearing,” nor (more generally) for “high-conformal gearing” can be machine-finished in a continuously-indexing machining process, that is, the gears cannot be finish-cut by hobs, by shapers and rack-cutters, by worm grinding wheel, by gear shavers, and so forth.

A.1 Introduction This article is about “Novikov gearing” and (more generally) “high-conformal gearing” are discussed below aiming to reveal that the term “Wildhaber-Novikov gearing” (or “WN– gearing,” for simplicity) is inconsistent, and, thus, it must be eliminated from scientific literature on gearing. After being invented at the beginning of 1950, “Novikov gearing” 487

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FIGURE A.1 Close-up of a conformal gear pair (“Novikov gear pair”) manufactured by Westland Helicopter, Ltd. (After Dyson, A., Evans, H.P., and Snidle, R.W., 1986, “Wildhaber-Novikov Circular Arc Gears: Geometry and Kinematics.” Proceedings of the Royal Society London A, v. 403, pp. 313–40.).

is extensively investigated by many scientists and engineers. There is a large body of published scientific work on “Novikov gearing” available in the public domain. “Novikov gearing” is successfully used in helicopter transmissions1 (Figure A.1) [1,2], as well as in other applications. Numerous state standards on “Novikov gearing,” as well as on hobs for cutting gears for “Novikov gearing” are active in the USSR/Russia, China, and other countries. Unfortunately, even to date, the kinematics and geometry of “Novikov gearing” is not understood by the vast majority of the gear experts all around the world. Many scientists and researchers in the field of gearing still loosely refer to “Novikov gearing” as to “Wildhaber-Novikov gearing,” or just to “WN–gearing.” This indicates an insufficient training in the theory of gearing, especially in the absence of understanding of the kinematics and geometry of both “Novikov gearing,” and “Wildhaber gearing.” Below, the difference between two gear systems, that is, between “Novikov gearing,” and “Wildhaber gearing” is clearly outlined. It is shown that “Novikov gearing” and “Wildhaber gearing” must be considered only separately, and it is a huge mistake combining two gear systems in a common gear system, (i.e., to “WN–gearing”). Potential improvements to “Novikov gearing” are also disclosed.

A.2  A Brief Historical Overview Transmission and transformation of a motion is the main purpose of gearing of all kinds. Generally speaking, axes of rotation of the driving and the driven gears can be arbitrarily

1

These gears in Westland Helicopters, Ltd. were finish-cut by a disc-type grinding wheel.

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oriented in relation to one another. From this perspective, three kinds of gearing are commonly distinguished:

a. Parallel-axes gearing (or “Pa–gearing,” for simplicity), b. Intersected-axes gearing (or “Ia–gearing”), and c. Crossed-axes gearing (or “Ca –gearing”).

For simplicity, but without loss of generality, the consideration below is limited only to parallel-axes gearing. In the past, pin gearing, as well as other primitive kinds of gearing, were in use. None of them is capable of transmitting a rotation smoothly. It was the famous Swiss mathematician and mechanician Leonhard Euler who proposed involute gearing—the only kind of parallel-axes gearing that is capable of smoothly transmitting a steady rotation from a driving shaft to a driven shaft. Gearing of no other kind is capable of doing that. The equality of the base pitches can be observed only in involute gearing. In external involute parallel-axes gearing, a “convex” involute tooth profile of the driving member contacts a “convex” involute tooth profile of the driven member. In other words, external involute parallel-axes gearing features “convex-to-convex contact” of the mating tooth profiles. As the contacting tooth profiles are convex, this imposes a strong limitation on the bearing capacity of the involute gearing because of high contact stress. It is highly desired to replace two convex contacting tooth profiles of the gear teeth with their “convexto-concave contact.” In conventional involute gearing, this is not feasible as this inevitably entails the violation of three fundamental laws of gearing [3]. The breakthrough invention in the realm of gearing was made in the 1950s. As early as 1956,2 a novel gearing was proposed by Dr. M. Novikov [4]. The concept of the proposed gear system is illustrated in Figure A.2 [4]. Later on, “Novikov gearing” was investigated in his doctoral thesis [5] and this is summarized in the monograph [6]. Below, we do not follow the approach used by Dr. M. Novikov to design a gear pair of the novel design. Instead, the concept of “Novikov gearing” is derived on the basis of conventional external parallel-axes involute gearing. Gearing of this kind is chosen for the derivation, as “Novikov gearing” is a reduced case of involute gearing.3

A.3  Principal Design Features of “Novikov Gearing” For the designing of a pair of “Novikov gearing,” let’s assume that a location and orientation of the axes of rotation of the driving, and driven members of the gear pair to be designed is specified, and the gear ratio is given. The desired value of the transverse pressure angle is also known. With that said, a pair of “Novikov gears” can be designed following the routine briefly outlined below. It should be mentioned here that the first pair of “Novikov gears” made out of an aluminum alloy (a preprototype) had been cut on April 25, 1954, by means of a disc-type mill cutter. Fifteen gear pairs for testing purposes had been machined in the summer of 1954 by means of a disc-type mill cutter. 3 The proposed by Euler involute gearing deserves to be called “Euler gearing,” or just as “Eu–gearing,” for simplicity. 2

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FIGURE A.2 On the concept of “Novikov gearing.” (After Dr. M. Novikov; USSR Pat. No. 109,113, 1956 [the “boundary Novikov circle” of a radius, rN, is introduced later on by Dr. S.P. Radzevich].)

A.3.1  Vector Diagram of a Gear Pair Designing of a pair of “Novikov gears” begins with the construction of the vector diagram of the gear pair to be designed. The rotation vector4 of the gear, ωg, is along the axis of rotation, Og, of the gear. The magnitude, ωg, of the rotation vector, ωg, equals 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. The axes of rotation, Og and Op, are at a certain “center distance, C.” The rotation vectors, ωg and ωp, form a “crossed-axes angle, Σ,” that is: Σ = ∠(ωg; ωp). In a case of external parallelaxes gearing, the angle, Σ, is always equals Σ = 180°. The principle of inversion of rotations can be implemented to the gear pair to be designed. Let’s 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: ωpl = (ωp − ωg). The vector of instant rotation, ωpl, of the pinion in relation to the gear is along the axis of instant rotation, Pln. More details about the vector diagrams are discussed in [3]. A.3.2  Plane of Action in Parallel-Axes Gearing. The plane of action, PA, in parallel-axes gearing is a plane through the axis of instant rotation, Pln. This plane forms a transverse pressure angle, φt, with the perpendicular to 4

It should be stressed here that a rotation in nature is not a vector at all. However, if special care is undertaken, rotations can be treated as vectors.

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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 pitch radii, rg and rp, of the pitch cylinders, and the transverse pressure angle, φt:

db. g = 2rg cos φt

(A.1)



db. p = 2rp cos φt

(A.2)

Once the base cylinders are determined, then 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 “pulley-and-belt analogy” (Figure A.3). Either of the Equations A.1 and A.2 can be used for the derivation of an expression for the calculation of the base pitch, pb, in a transverse section of the gear pair:



pb =

π db. g π db. p = Ng N p (A.3)

Equation A.3 is valid for parallel-axes gearings that are capable of transmitting a rotation smoothly, that is, this equation is valid for all perfect parallel-axes gearings. A.3.3  A Desired Line of Contact in a Parallel-Axes Gearing The tooth flank of the gear, G, and that of the mating pinion, P, make contact along a desired line of contact, LCdes (see Figure A.3), or just a line of contact, LC. The line of contact, LC, is a planar curve of a favorable geometry that is entirely located within the plane of action, PA. The tooth flanks, G and P, of the gear and the mating pinion interact with one another only within the active portion of the plane of action shown in Figure A.4.

FIGURE A.3 The kinematics of perfect parallel-axes gearing.

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(a)

(b)

(c)

(d)

FIGURE A.4 Elements of a parallel-axes gear pair that features a zero-transverse contact ratio (mp = 0).

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Referred to Figure A.4a, the straight-line segment, 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 (see Figure A.4b). [In a case of “Novikov gearing,” the equality, Zpa = 0, is observed]. In involute helical parallel-axes gearing, the desired line of contact, LC, between the tooth flanks 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 total contact ratio, mt, can be expressed in terms of the transverse contact ratio, mp, and the face contact ratio, mF, written as

mt = mp + mF

(A.4)

where mp = Zpa/pb and mF = Fpa tanψb/pb. The inequality, mt ≥ 0, must be observed for any and all parallel-axes gear pairs. When the base cylinders of diameters, db.g and db.p, rotate, the desired line of contact, LC, travels (together with the plane of action, PA) in relation to the reference systems, one of which is associated with the gear, and another one is associated with the pinion. In such a motion, the tooth flank, G, of the gear (as well as the tooth flank, P, of the pinion) can be interpreted as a family of consecutive positions of the desired line of contact in the corresponding reference system. It is of importance for further analysis to distinguish an active gear tooth profile from a full gear tooth profile. In an involute gear, the active tooth profile is shaped in the form of an involute of a circle. The full tooth profile in an involute gear tooth profile is comprised of the involute of a circle, circular-arc top-land (with or without chamfer and/or roundness), and circular-art bottom land with two fillets. An involute gear is referred to as the “involute gear” only due to the specific geometry of the active portion of the tooth profile, and not because of the geometry of the rest of the gear tooth profile. In the example shown in Figure A.4b, the active portion, ab, of the gear tooth profile5 is shaped in the form of an involute of a circle. The profile, ab, is specified by the radii of the outer cylinders of the gear and of the pinion, ro.g and ro.p, correspondingly. Point a corresponds to the “start-of-active-profile” point (SAP − point), while point b corresponds to the “end-of-active-profile” 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, due to chamfering. Under such a scenario (see Figure A.4c), the active portion of the plane of action gets 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, illustrated in Figure A.4b. This gives a certain freedom to the gear designer when selecting a desired 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. In the extreme case, the EAP − circles of the gear and of 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 point, K. This point is referred to as the “involute tooth 5

The tooth involute profile is called “involute” because the active portion of the tooth profile is shaped in a form of an involute of a circle, regardless if the rest of the gear tooth profile (fillets, bottom-lands) are not involute.

494

Appendix A

point.” The nonactive portions, aK and bK, of the tooth profile meet each other at point, K. These portions are not subject to conditions of meshing of tooth profiles, thus this gives a certain freedom to the gear designer when selecting the geometry of nonactive portions, aK and bK, of the tooth profile (see Figure A.4d). More detail on the transformation of an involute gear tooth profile into the “involute tooth point” is illustrated in Figure A.5. As depicted in Figure A.5, the original involute gear tooth profile in Figure A.5a is reduced to a truncated involute gear tooth profile as illustrated in Figure A.5b. The later can be reduced to the “involute tooth point” shown in Figure A.5c that features a zero-transverse contact ratio (mp = 0). Ultimately, the gear tooth profile truncated to an “involute tooth point” can allow a locally circular-arc tooth profile, for which only an “involute tooth point” is active, while the rest of the tooth profile is not engaged in mesh with the teeth of a mating pinion. As the width of the active portion of the plane of action is zero, (i.e., Zpa = 0), and the involute tooth profile is shrunk to point, the transverse contact ratio, mp, gets a zero value. In order to meet the inequality, mt ≥ 0, the following inequality must be met: mt = mp + mF = 0 + mF = mF > 0 (A.5)

(a)

(b)

(c)

(d)

FIGURE A.5 Gradual transformation of a true involute tooth profile (a) to a truncated gear tooth profile (b) and, finally, to the “involute tooth point” (c) that allows for a locally circular-arc tooth profile (d) with a zero-transverse contact ratio (mp = 0).

Appendix A

495

The point system of meshing in parallel-axes gearing (see Figure A.4d) 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. A.3.4  Design Features of “Novikov Gearing” The concept of “Novikov gearing” is based on the schematic depicted in Figure A.4d. For this case (see Figure A.4d), Dr. M. Novikov proposed to replace “convex-to-convex contact” of the teeth profiles with their “convex-to-concave contact.” So, the replacement only becomes possible in a case when the active portion of the involute tooth profile is shrunk to point (and it is infeasible in cases when the active portion of the involute tooth profile is of a certain length [3]). Figure A.2 shows the first (in time) schematic that illustrates that the concept of “Novikov gearing” [4] is far from the best and consistent. Point of contact, K, of the tooth flanks, G and P, is located within the straight line of action, LA. The larger the distance of the contact point, K, from the pitch point, P, the more freedom there is for the gear designer in selecting the radii of curvature of the interacting tooth profiles. At the same time, the larger the distance of the contact point, K, from the pitch point, P, the higher losses on friction between the tooth flanks, G and P, and the higher the tooth flanks wear (see Figure A.6). Ultimately, the actual location of the contact point, K, is a tradeoff between the two just mentioned factors. Further, let’s assume that the pinion is stationary, and the gear performs an instant rotation in relation to the pinion. The axis, Pln, of instant rotation, ωpl, is a straight line through the pitch point, P. When the pinion is motionless, the contact point, K, traces a “boundary Novikov circle”6 of a radius, rN, that is centered at P, as illustrated in Figure A.2. The pinion tooth profile, P, can either align with a circular-arc of the “boundary Novikov circle” of a radius, rN, or it can be relieved in the bodily side of the pinion tooth. The pitch point is included into the interval, while the contact point, K, is not. From the other hand, location of the center of curvature of the concave gear tooth profile, G, within the straight line, LA, is limited to the open an interval P → ∞. Theoretically, the pitch point, P, can be included in this interval for K. 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, thus, the inequality rp  1). A.3.5  Principal Design Parameters of “Novikov Gearing” From a historical perspective, it is interesting to consider the calculation of the principal design parameters of a Novikov gear pair following the approach proposed by Dr. M. Novikov [6]. In today’s terminology and designations, the calculation of the principal design parameters of a Novikov gear pair is considered in [3].

A.4  High-Conformal Gearing Conditions, under which “convex-to-concave contact” between the tooth flanks of a gear and a mating pinion becomes feasible is the fundamental achievement by Dr. M. Novikov. Once the active tooth profiles in a parallel-axes involute gearing shrunk to a point, (i.e., to the “involute tooth point”), then a favorable contact between the tooth flanks, G and P, can be attained. A capability to accommodate for the manufacturing errors and for the displacements under an operating load is the only consideration when determining the geometry of the interacting tooth profiles, G  and P, by Dr. M. Novikov.

Appendix A

497

A.4.1  Critical Degree of Conformity in “Novikov Gearing” It was assumed from the very beginning that “convex-to-concave contact” between the tooth flanks of a gear and a mating pinion is sufficient for a significant increase of bearing capacity of the contact area between the tooth flanks, G  and P. The analysis reveals that in the case of “Novikov gearing,” “convex-to-concave contact” is necessary but not sufficient for a significant increase of power capacity of Pa –gearing. A certain critical degree of conformity at point of contact, K, of the tooth flanks, G  and P, must be attained in order to make the “convex-to-concave contact” beneficial. An increase in the degree of conformity a b under its critical value, that is, from δcnf to δcnf , makes possible a limited increase of the bearing capacity of the gear pair. A low increase of the bearing capacity is because if both a b the values of the degree of conformity, (that is, δcnf and δcnf ) are smaller, the more critical a b its value [δcnf ], and thus, the inequalities δcnf < [δcnf ] and δcnf < [δcnf ] are observed. However, c c > [δcnf ]), then when the actual degree of conformity, δcnf , gets larger than the critical value (δcnf even a small increase in the degree of conformity at point of contact of the tooth flanks, G  and P, causes a significant increase in bearing capacity of the tooth flanks in “Novikov gearing.” Therefore, a following intermediated conclusion can be formulated: Intermediate conclusion: The substitution of “convex-to-convex contact” between the tooth flanks in Pa –gearing with their “convex-to-concave contact” (as in “Novikov gearing”) is necessary, but not sufficient for a significant increase in power capacity of a parallel-axes gear pair. In addition to that, a certain critical degree of conformity, [δcnf], at point of contact between the tooth flanks of the gear, G  and of the pinion P, must be exceeded.

Gearing for which degree of conformity, δcnf, at point of contact of the tooth flanks, G and P, is larger than its critical value, [δcnf ], that is, the gearing for which the inequality, δcnf > [δcnf ], is observed, is referred to as the “high-conformal gearing.” The intuitively understood qualitative term “degree of conformity” can be quantified. For this purpose, a characteristic curve called the “indicatrix of conformity, Cnf R(G/P ),” at point of contact of the tooth flanks of a gear, G, and a mating pinion, P, is commonly used [3,7]. The Cnf R(G/P ) is a planar centro-symmetrical curve of the fourth order. Position vector of a point, rcnf, of the indicatrix of conformity corresponds to degree of conformity of the tooth flanks, G and P, in a corresponding direction through the contact point K [3,7]. The smaller the radius, rcnf, the larger the degree of conformity at point of contact of the surfaces, and vice versa. A.4.2 A Minimum Required Degree of Conformity at Point of Contact of the Interacting Tooth Flanks Favorable conditions of contact of the tooth flanks of the gear and the pinion is the main anticipated advantage of a high-conformal gear pair. The higher the degree of conformity, the higher the load-carrying capacity of the contacting tooth flanks. Therefore, a minimum possible mismatch in the curvature of the teeth of gear and pinion is desired. In reality, tooth flanks of a gear and a mating pinion in a high-conformal gear pair are displaced from their desired position. The undesired displacements are mostly because of: (a) the manufacturing errors, and (b) the elastic deflections of the gear teeth, of the gear shafts, of the housing that is occurred under the applied load, thermal expansions of the components, and so forth. High-conformal gearing is sensitive toward the tooth flanks displacements.

498

Appendix A

To accommodate for the inevitable displacements, a certain degree of mismatch in the curvature of the teeth of a gear and a pinion is required. A small mismatch cannot be capable of accommodating for the displacements. However, as the mismatch increases, the contact stresses increase as well. High contact stress may lead to various forms of surface failure such as heavy wear, pitting, or scuffing damage. Therefore, a minimum required degree of mismatch in the curvature of the teeth of a gear and a pinion is necessary to be determined. Otherwise, one of two scenarios could be observed. First, the gear pair is capable of absorbing the inevitable displacements of the tooth flanks, but the degree of conforming 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 for the tooth flanks displacements. In both cases, the gear pair has no chance of being successfully used in practice. Shown in Figure A.7, is a three-dimensional plot of the function δcnf = δcnf (k , K ). Figure A.7 relates to the cases of “convex-to-concave contact” of tooth flanks of the gear, G, and the pinion, P . The performed analysis of the 3D-plots allows for the following conclusions. The sections of the surface, δcnf = δcnf (k , K ), by planes ki = Const (see Figure A.7) are represented with curves that have asymptotes. For a particular curve, ki = Const, shown in Figure A.6 in bold line, the axis, δcnf, and the straight line, δcnf = 1, are the asymptotes. The greatest possible degree of mismatch in the curvature of the teeth of gear and pinion corresponds to the parameter, K → −∞. The interval of alteration to the parameter, K, starting from − ∞ and going up to approximately, K = −2, is convenient to accommodate for any desired displacement of the tooth flanks, G  and P, from their correct configuration. However, within the interval (−∞ < K < −2) of alteration in K −parameter, an increase of the degree of conforming of the tooth profiles, G  and P, is negligibly small. Within this interval of K − parameter, the load-carrying capacity of a conformal gear pair is remained approximately at the same range. Therefore, just “convex-to-concave contact” between the tooth flanks of a

FIGURE A.7 Three-dimensional plot of the function, δcnf = δcnf (k , K ), constructed for “convex –to–concave” kind of contact between the tooth flanks of a gear, G, and a mating pinion, P, in a “conformal” gear pair.

Appendix A

499

gear and a mating pinion gives a limited improvement in the load-carrying capacity of a gear pair. Being “convex-to-concave,” an additional requirement needs to be satisfied in order to get not just “conformal gearing,” but, instead, to get “high-conformal gearing.” From another hand, even a small change to the value of the K −parameter within the interval −2 < K < −1 results in significant increase of the degree of conformity of the tooth flanks, G and P. This immediately entails a corresponding increase in load-carrying capacity of the gear pair. In the above considered example, the value of the K −parameter, (i.e., the value of K ≈ −2) can be referred to as its critical value, K cr. This allows for distinguishing between just “conformal gearing” (for which −∞ < K < K cr ) from “high-conformal gearing” (for which K cr ≤ K < −1). Without going into details of the analysis, it is clear that gears for “high-conformal gearing” require tighter tolerances for any possible displacements of the tooth flanks, G and P, from their desired location and orientation. Otherwise, there could be no future for the application of a high-conformal gear system. Based on the results of the performed analysis, the following statement is valid: “conformal gearing” and “high-conformal gearing” meet all three fundamental laws of gearing [3,7,8]. They are capable of transmitting an input steady rotation smoothly. As a consequence, “conformal gearing,” as well as “high-conformal gearing,” feature: • The transverse contact ratio is identical to zero (mp ≡ 0) • The total contact ratio, mt, is equal to the face contact ratio, mF, and is greater than one (mt = mF > 1) • The tooth profile of one member of the gear pair is convex, while the tooth profile of the mating gear is concave • The convex tooth profile of one member of the gear pair is entirely located within the interior the boundary N − circle, while the concave tooth profile of another member of the gear pair is entirely located within the exterior of the boundary N − circle • The difference between the magnitudes of radii of curvature of the concave tooth profile and the convex tooth profile in the gear pair is equal to, or smaller, than the given threshold beyond which higher conformity of the interacting tooth profiles contributes much to the bearing capacity of the gear pair. The principal difference between “Novikov gearing,” and between “Wildhaber gearing” is perfectly illustrated in Figure A.8. In “Novikov gearing,” the common unit perpendicular, ng, is always aligned with the instant line of action, LAinst, as illustrated in Figure A.8a. The gear tooth flank, G, is properly configured in relation to the boundary N − circle of a radius, rN. In “Wildhaber gearing,” the common unit perpendicular, ng, is not aligned with the instant line of action, LAinst, as illustrated in Figure A.8b. The gear tooth flank, G *, is improperly configured in relation to the boundary N − circle in a gear pair. Neither “transverse contact ratio, mp,” nor “face contact ratio, mF” can be defined for “Wildhaber gearing.” Helical gearing with a circular-arc tooth profile proposed by Dr. E. Wildhaber [9] does not meet the three fundamental laws of gearing. Insufficient understanding of gearing of this particular kind clearly follows from the paper by T. Allen [13]. The comparison of the “Novikov gearing” [4] and the “Wildhaber gearing” [9] started decades ago by N. Chironis

500

(a)

Appendix A

(b)

FIGURE A.8 Correct, G (a), and incorrect, G* (b) configurations of the circular-arc tooth profile of a gear in “Novikov gearing” (the common perpendicular, ng, is aligned with the instant line of action, LAinst), and in “Wildhaber gearing” (the common perpendicular, ng, does not align with the instant line of action, LAinst).

[10], is incomplete. To the best possible extent, the comparison was later on accomplished by Dr. S. Radzevich, and is summarized in [3,8,11]. Two points need to be mentioned here. First, neither gears for “Novikov gearing,” nor gears for “high-conformal gearing” can be finish-cut in a continuously-indexing method of gear machining. That is, the gears cannot be hobbed, shaped, shaved, or ground by the worm grinding wheel. This is because the three fundamental laws of gearing are not fulfilled in the gear machining mesh [7]. Only Re–surfaces can be accurately generated in a continuously-indexing method of gear machining. Cutting tools for machining gears for both, for “Novikov gearing” as well as for “high-conformal gearing” are considered in [8,12]. Second, neither tooth profile modification, nor longitudinal modification of the tooth flanks of the gear and the pinion is applicable to “Novikov gearing,” as well as to “highconformal gearing” in a more general case. It is clear now from the above discussion that “Novikov gearing” and “Wildhaber gearing” cannot be combined in a common gear system that is called “Wildhaber-Novikov gearing.” These two gear systems must only be considered separately. A.4.3 Conclusion A brief historical overview of “Novikov gearing” is presented in the paper. Principal features of the kinematics, and geometry of “Novikov gearing” (and, more generally, of “high-conformal gearing”) are discussed. It is stressed in the paper that poor understanding of the kinematics and geometry of the novel kind of gearing proposed by Dr. M. Novikov is the root cause for the loose combination of “Novikov gearing” with “Wildhaber gearing.” These two types of gearing must only be considered separately from one another. The terms “Wildhaber-Novikov gearing,” as well as “W − N gearing,” are incorrect, and thus they must be eliminated from scientific communication on gearing. It is a mistake to refer to the “Novikov gear system” as to gears with a “circular-arc tooth profile.” “Novikov gearing” is NOT a kind of gearing with a “circular-arc tooth profile”

Appendix A

501

like gears in the “Wildhaber gear system” are. “Novikov gearing” is a reduced kind of “involute gearing.” In “Novikov gearing,” a working involute tooth profile is shrunk to a point. This point is referred to as the “involute tooth point.” The rest of the tooth profile is inactive, and, thus, can be shaped with no constraints imposed by the three fundamental laws of gearing. The acting standards on “Novikov gearing” (in USSR/Russia, China, and other countries), and on gear cutting tools for cutting gears for “Novikov gearing,” are out of date due to their incorrectness. Gears for “Novikov gearing” cannot be cut (finish-cut) by hobs, shaper cutters, shavers, worm grinding wheels, and others. No generating machining of gear tooth flanks are allowed. Only disc-type mill cutters, disc-form grinding wheels, and so forth, can be used for this purpose. High-conformal gearing has a huge potential for application in high-power-density transmissions, and in low-noise gear transmissions.

References 1. Shotter, B.A., “Experiences with Conformal/W-N Gearing.” Proceedings of World Congress on Gearing, Paris, France, June 22 – June 24, 1977, pp. 527–540. 2. Shotter, B.A., “The Lynx Transmission and Conformal Gearing.” SAE Technical Paper 781041, 1978. 3. Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd edition, CRC Press, Boca Raton Florida, 2018, 898 pages. [First edition: Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, CRC Press, Boca Raton Florida, 2012, 743 pages.]. 4. Pat. No. 109,113, (USSR). Gearing and Cam Mechanisms Having Point System of Meshing./M.L. Novikov, National Classification 47 h, 6; Filed: April 19, 1956, published in Bull. of Inventions No.10, 1957. 5. Novikov, M.L., The Principles of the Geometric Theory of Point Meshing of Gearing for the Purpose of Transmitting of High Power, Doctoral Thesis, Zhukovsky Air Force Engineering Academy, Moscow, 1955. 6. Novikov, M.L., Gearing that is Featuring a Novel Kind of Meshing, Published by Zhukovsky Air Force Engineering Academy, Moscow, 1958, 186 pages. 7. Radzevich, S.P., Geometry of Surfaces: A Practical Guide for Mechanical Engineers, Wiley, 2013, 264 pages. 8. Radzevich, S.P., High-Conformal Gearing: Kinematics and Geometry, CRC Press, Boca Raton, Florida, 2015, 352 pages. 9. Pat. No. 1,601,750, (USA). Helical Gearing. /E. Wildhaber, Filed: November 2, 1923, published October 5, 1926. 10. 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 pages. 11. Radzevich, S.P., Dudley’s Handbook of Practical Gear Design and Manufacture, 3rd edition, CRC Press, Boca Raton Florida, 2017, 606 pages. [Second edition: Radzevich, S.P., Dudley’s Handbook of Practical Gear Design and Manufacture, 2nd edition, CRC Press, Boca Raton Florida, 2012, 880 pages.]. 12. Radzevich, S.P., Gear Cutting Tools: Science and Engineering, 2nd edition, CRC Press, Boca Raton Florida, 2017, 606 pages. [First edition: Radzevich, S.P., Gear Cutting Tools: Fundamentals of Design and Computation, CRC Press, Boca Raton Florida, 2010, 786 pages.]. 13. Allan, T., “Some Aspects of the Design and Performance of Wildhaber-Novikov Gearing.” Proc. Inst. Mech. Engrs., Part I, v. 179, n. 30, 1964/1965, pp. 931–954.

Appendix B: Applied Coordinate Systems and Linear Transformations

Stephen P. Radzevich CONTENTS B.1 Coordinate System Transformation.................................................................................504 B.1.1 Homogeneous Coordinate Vectors......................................................................504 B.1.2 Homogeneous Coordinate Transformation Matrices of the Dimension 4 × 4...............................................................................................505 B.1.3 Translations.............................................................................................................505 B.1.4 Rotation about a Coordinate Axis........................................................................ 507 B.1.5 Rotation about an Arbitrary Axis through the Origin...................................... 509 B.1.5.1 Conventional Approach.......................................................................... 509 B.1.5.2 “Eulerian Transformation”..................................................................... 510 B.1.6 Rotation about an Arbitrary Axis not through the Origin............................... 511 B.1.7 Resultant Coordinate System Transformation................................................... 512 B.2 Complex Coordinate System Transformation................................................................ 514 B.2.1 Linear Transformation Describing a Screw Motion about a Coordinate Axis................................................................................................... 514 B.2.2 Linear Transformation Describing Rolling Motion of a Coordinate System...... 515 B.2.3 Linear Transformation Describing Rolling of Two Coordinate Systems....... 517 B.2.4 Coupled Linear Transformation........................................................................... 520 B.2.5 An Example of Nonorthogonal Linear Transformation................................... 522 B.2.6 Conversion of a Coordinate System Hand.......................................................... 522 B.3 Useful Equations................................................................................................................. 523 B.3.1 RPY-Transformation............................................................................................... 524 B.3.2 Operator of Rotation about an Axis in Space..................................................... 524 B.3.3 Combined Linear Transformation........................................................................ 524 B.4 Chains of Consequent Linear Transformations and a Closed Loop of Consequent Coordinate System Transformations......................................................... 525 B.5 Impact of the Coordinate System Transformations on Fundamental Forms of the Surface............................................................................................................................ 531

Consequent coordinate systems transformations can be easily analytically described with implementation of matrices. The use of matrices for the coordinate system transformation1

1

Matrices were introduced into mathematics by A. Cayley in 1857. They provide a compact and flexible notation particularly useful in dealing with linear transformations, and they presented an organized method for the solution of systems of linear differential equations.

503

504

Appendix B

can be traced back to the mid-1940s2 when Dr. S. S. Mozhayev3 began describing coordinate system transformation by means of matrices. Below, coordinate system transformation is briefly discussed from the standpoint of its implementation in the theory of gearing.

B.1  Coordinate System Transformation Homogeneous coordinates utilize a mathematical trick to embed three-dimensional coordinates and transformations into a four-dimensional matrix format. As a result, inversions or combinations of linear transformations are simplified to inversions or multiplication of the corresponding matrices. B.1.1  Homogeneous Coordinate Vectors Instead of representing each point r(x, y, z) in three-dimensional space with a single threedimensional vector,



x   r =  y  z  

(B.1)

homogeneous coordinates allow each point r(x, y, z) to be represented by any of an infinite number of four-dimensional vectors:



T ⋅x    T ⋅ y    r=  ⋅ T z    T   

(B.2)

The three-dimensional vector corresponding to any four-dimensional vector can be calculated by dividing the first three elements by the fourth, and a four-dimensional vector corresponding to any three-dimensional vector can be created by simply adding a fourth element and setting it equal to one.

Application of matrices for the purposes of analytical representation of coordinate system transformation should be credited to Dr. S. S. Mozhayev. [Mozhayev, S. S., General Theory of Cutting Tools, Doctoral Thesis, Leningrad, Leningrad Polytechnic Institute, 1951, 295 p.] Dr. S. S. Mozhayev began using matrices for this purpose in the mid-1940s. Later on, matrix approach for coordinate system transformation has been used by Denavit & Hartenberg [Denavit, J. and Hartenberg, R.S., (1955), “A Kinematics Notation for Lower-Pair Mechanisms Based on Matrices,” ASME Journal of Applied Mechanics, Vol. 77, pp. 215–221. (Manuscript received by ASME Applied Mechanics Division, December 14, 1953, paper №54-A-34)], as well as by many other researchers. 3 S. S. Mozhayev is a soviet scientist mostly known for his accomplishments in the theory of cutting tool design. 2

505

Appendix B

B.1.2  Homogeneous Coordinate Transformation Matrices of the Dimension 4 × 4 Homogeneous coordinate transformation matrices operate on four-dimensional homogeneous vector representations of traditional three-dimensional coordinate locations. Any threedimensional linear transformation (translation, rotation, and so forth) can be represented by a 4 × 4 homogeneous coordinate transformation matrix. In fact, because of the redundant representation of three-space in a homogeneous coordinate system, an infinite number of different 4 × 4 homogeneous coordinate transformation matrices are available to perform any given linear transformation. This redundancy can be eliminated to provide a unique representation by dividing all elements of a 4 × 4 homogeneous transformation matrix by the last element (which will become equal to one). This means that a 4 × 4 homogeneous transformation matrix can incorporate as many as 15 independent parameters. The generic format representation of a homogeneous transformation equation for mapping the threedimensional coordinate (x1, y1, z1) to the three-dimensional 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 

(B.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 constraint the form of the matrix to transformations with fewer degrees of freedom. B.1.3 Translations Translation of a coordinate system is one of the major linear transformations used in the theory of part surface generation. Translations of the coordinate system X2Y2Z2 along axes of the coordinate system X1Y1Z1 are depicted in Figure B.1. Translations can be analytically described by the homogeneous transformation matrix of dimension 4 × 4.

FIGURE B.1 Analytical description of the operators of translations Tr(ax, X), Tr(ay, Y), Tr(az, Z) along the coordinate axes of a “Cartesian” reference system XYZ.

506

Appendix B

For an analytical description of translation along coordinate axes, the operators of translation Tr(ax, X), Tr(ay, Y), and Tr(az, Z) are used. These operators yield matrix representation in the form 0

0



1  0 Tr( ax , X ) =  0 0 

1 0 0

0 1 0

0

0



1  0 Tr( ay , Y ) =  0 0 

1 0 0

0 1 0

0

0



1  0 Tr( az , Z) =  0 0 

1 0 0

0 1 0

ax   0  0  1 

(B.4)

0  ay   0  1 

(B.5)

0  0  az  1 

(B.6)

Here, in Equations B.4 through B.6, the parameters ax, ay, and az are signed values that denote the distance of translation along the corresponding axis. Consider two coordinate systems, X1Y1Z1 and X2Y2Z2, displaced along the X1-axis at a distance ax as schematically depicted in Figure B.1a. A point m in the reference system X2Y2Z2 is given by the position vector r2(m). In the coordinate system, X1Y1Z1, that same point m can be specified by the position vector r1(m). Then the position vector r1(m) can be expressed in terms of the position vector r2(m) by the equation

r1 (m) = Tr(ax , X ) ⋅ r2 (m)

(B.7)

Equations similar to Equation B.7 are valid for the operators Tr ( ay , Y ), and Tr ( az , Z) of the coordinate system transformation. The latter is schematically illustrated in Figure B.1b and c. Use of the operators of translation Tr(ax, X), Tr(ay, Y), and Tr(az, Z) makes it possible for an introduction of an operator Tr(a, A) of a combined transformation. Suppose that point, p, on a rigid body goes through a translation describing a straight line from a point p1 to a point p2 with a change of coordinates of (ax, ay, az). This motion of the point, p, can be analytically described with a resultant translation operator Tr(a, A):



1  0 Tr( a, A) =  0 0 

0

0

1 0 0

0 1 0

ax   ay   az  1 

(B.8)

The operator Tr(a, A) of the resultant coordinate system transformation can be interpreted as the operator of translation along an arbitrary axis having the vector A as the direct vector. An analytical description of translation of the coordinate system X1Y1Z1 in direction of an arbitrary vector A to the position of X2Y2Z2 can be composed from Figure B.2. The operator

507

Appendix B

FIGURE B.2 Analytical description of an operator, Tr(a, A), of translation along an arbitrary axis (vector A is the direct vector of the axis).

of translation Tr(a, A) of that particular kind can be expressed in terms of the operators Tr(ax, X), Tr(ay, Y), and Tr(az, Z) of elementary translations:

Tr( a, A) = Tr( az , Z) ⋅ Tr( ay , Y ) ⋅ Tr( ax , X )

(B.9)

Evidently, the axis along vector A is always the axis through the origins of both the reference systems X1Y1Z1 and X2Y2Z2. Any and all coordinate system transformations that does not change the orientation of a geometrical object are referred to as “orientation-preserving transformation,” or “direct transformation.” Therefore, transformation of translation is an example of a direct transformation. B.1.4  Rotation about a Coordinate Axis Rotation of a coordinate system about a coordinate axis is another major linear transformation used in the theory of part surface generation. A rotation is specified by an axis of rotation and the angle of the rotation. It is a fairly simple trigonometric calculation to obtain a transformation matrix for a rotation about one of the coordinate axes. Possible rotations of the coordinate system X2Y2Z2 about the axis of the coordinate system X1Y1Z1 are illustrated in Figure B.3. For analytical description of rotation about a coordinate axis, the operators of rotation Rt(ϕx, X1), Rt(ϕy, Y1), and Rt(ϕz, Z1) are used. These operators of linear transformations yield representation in the form of homogeneous matrices:



1  0 Rt(ϕx , X1 ) =  0 0 

0 cos ϕx − sin ϕx 0

0 sin ϕx cos ϕx 0

0  0 0 1

(B.10)

508

Appendix B

FIGURE B.3 Analytical description of the operators of rotation Rt(ϕx, X), Rt(ϕy, Y), and Rt(ϕz, Z) about a coordinate axis of a reference system X1Y1Z1.



 cos ϕy   0 Rt(ϕy , Y1 ) =  − sin ϕy  0 

0 1 0 0



 cos ϕz  − sin ϕz Rt(ϕz , Z1 ) =   0  0 

sin ϕz cos ϕz 0 0

sin ϕy 0 cos ϕy 0 0 0 1 0

0  0  0 1

(B.11)

0  0 0 1

(B.12)

Here ϕx, ϕy, and ϕz are signed values that denote the corresponding angles of rotations about a corresponding coordinate axis: ϕx is the angle of rotation around the X1-axis (pitch) of the “Cartesian” coordinate system X1Y1Z1; ϕy is the angle of rotation around the Y1-axis (roll), and ϕz is the angle of rotation around the Z1-axis (yaw) of that same “Cartesian” reference system X1Y1Z1. Rotation about a coordinate axis is illustrated in Figure B.3. Consider two coordinate systems X1Y1Z1 and X2Y2Z2, which are turned about the X1-axis through an angle ϕx as shown in Figure B.3a. In the reference system X2Y2Z2, a point m is given by a position vector r2(m). In the coordinate system X1Y1Z1, that same point m can be specified by the position vector r1(m). Then, the position vector r1(m) can be expressed in terms of the position vector r2(m) by the equation

r1 (m) = Rt(ϕx , X ) ⋅ r2 (m)

(B.13)

Equations those similar to that above Equation B.13, are also 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 B.3b and c accordingly.

Appendix B

509

B.1.5  Rotation about an Arbitrary Axis through the Origin When a rotation is to be performed around an arbitrary vector based at the origin, the transformation matrix must be assembled from a combination of rotations about the “Cartesian” coordinate. Two different approaches for analytical description of a rotation about an arbitrary axis through the origin are discussed below. B.1.5.1  Conventional Approach Analytical description of rotation of the coordinate system X1Y1Z1 about an arbitrary axis through the origin to the position of a reference system X2Y2Z2 is illustrated in Figure B.4. It is assumed here that the rotation is performed about the axis having a vector A0 as the

FIGURE B.4 Analytical description of the operator Rt(ϕA, A) of rotation about an arbitrary axis through the origin of a “Cartesian” coordinate system X1Y1Z1 (the vector A is the directing vector of the axis of rotation).

510

Appendix B

direction vector. The operator Rt(ϕA, A0) of rotation of that kind can be expressed in terms of the operators Rt(ϕx, X), Rt(ϕy, Y), and Rt(ϕz, Z) of elementary rotations: Rt(ϕ A , A 0 ) = Rt(ϕz , Z ) ⋅ Rt(ϕ y ,Y ) ⋅ Rt(ϕx , X )



(B.14)

Evidently, the axis of rotation (a straight line along the vector A0) is always an axis through the origin. The operators of translation and of rotation also yield linear transformations of other kinds. B.1.5.2  “Eulerian Transformation” The “Eulerian transformation” is a well-known kind of linear transformation widely used in mechanical engineering. This kind of linear transformation is analytically described by the operator Eu(ψ, θ, ϕ) of the “Eulerian4 transformation.” The operator Eu(ψ, θ, ϕ) is expressed in terms of three “Euler angles” (or “Eulerian angles”) ψ, θ, and ϕ. Configuration of an orthogonal “Cartesian” coordinate system X1Y1Z1 in relation to another orthogonal “Cartesian” coordinate system X2Y2Z2 is defined by the “Euler angles” ψ, θ, and ϕ. These angles are shown in Figure B.5. The line of intersection of the coordinate plane X1Y1 of the first reference system by the coordinate plane X2Y2 of the second reference system is commonly referred to as “line of nodes.” In Figure B.5, the line OK is the line of nodes. It is assumed here and below that the line of nodes, OK, and the axes Z1 and Z2, form a frame of that same orientation as the reference systems X1Y1Z1 and X2Y2Z2 do. The “Euler angle, θ” is referred to as the “angle of nutation.” The angle of nutation, θ, is measured between the axes Z1 and Z2. The actual value of this angle never exceeds 180°.

FIGURE B.5 “Euler angles.”

4

Leonhard Euler (1707–1783): a famous Swiss mathematician and physicist who spent most of his life in Russia and Germany.

511

Appendix B

The “Euler angle, ψ” is referred to as the “angle of precession.” The angle of precession, ψ, is measured in the coordinate plane X2Y2. This the angle between the line of nodes, OK, and the X2-axis. Direction of the shortest rotation from the axis X2 to the axis Y2 is the direction in which the angle of precession is measured. When the angle of nutation is equal either θ = 0° or θ = 180°, then the “Euler angles” are not defined. Operator Eu(ψ, θ, ϕ) of “Eulerian transformation” allows for the following matrix representation: Eu(ψ , θ , ϕ ) − sin ψ cos θ sin ϕ + cos ψ cos ϕ  − sin ψ cos θ cos ϕ − cos ψ sin ϕ =  sin θ sin ϕ   0 

cos ψ cos θ sin ϕ + sin ψ cos ϕ cos ψ cos θ cos ϕ − sin ψ cos ϕ − cos ψ cos θ 0

sin θ sin ϕ sin θ cos ϕ cos θ 0

0  0 0 1

(B.15)

It is important to stress here on the difference between the operator Eu(ψ, θ, ϕ) of the “Eulerian transformation,” and between the operator Rt(ψA, A0) of rotation about an arbitrary axis through the origin. The operator Rt(ψA, A) of rotation about an arbitrary axis through the origin can result in that same final orientation of the coordinate system X2Y2Z2 in relation to the coordinate system X1Y1Z1 as the operator Eu(ψ, θ, ϕ) of the “Eulerian transformation” does. However, the operators of linear transformations Rt(ψA, A0) and Eu(ψ, θ, ϕ) are the operators of completely different nature. They can result in identical coordinate system transformation, but they are not equal to one another. B.1.6  Rotation about an Arbitrary Axis not through the Origin The transformation corresponding to rotation of an angle ϕ around an arbitrary vector not through the origin cannot readily be written in a form similar to the rotation matrices about the coordinate axes. The desired transformation matrix is obtained by combining a sequence of elementary translation and rotation matrices. (Once a single 4 × 4 matrix has been obtained representing the composite transformations it can be used in the same way as any other transformation matrix). Rotation of the coordinate system X1Y1Z1 to a configuration, which the coordinate system X2Y2Z2 possesses, can be performed about a corresponding axis that features an arbitrary configuration in space (see Figure B.6). The vector A is the direction vector of the axis of the rotation. The axis of the rotation is not a line through the origin. The operator of linear transformation of this particular kind Rt(ψA, A) can be expressed in terms of the operator Tr(a, A) of translation along, and of the operator Rt(ψA, A0) of rotation about an arbitrary axis through the origin:

Rt(ϕA , A) = Tr(−b, B*) ⋅ Rt(ϕA , A 0 ) ⋅ Tr(b, B)

(B.16)

Here, in Equation B.16 is designated: Tr(b, B) – is the operator of translation along the shortest distance of approach of the axis of rotation and origin of the coordinate system

512

Appendix B

FIGURE B.6 Analytical description of the operator, Rt(ϕA, A), of rotation about an arbitrary axis not through the origin (vector A is the direct vector of the axis of the rotation).

Tr(− b, B*) – is the operator of translation in the direction opposite to the translation Tr(b, B) after the rotation Rt(ψA, A) is completed. In order to determine the shortest distance of approach, B, of the axis of rotation (that is, the axis along the directing vector B) and origin of the coordinate system, consider the axis (B) through two given points rB.1 and rB.2. The shortest distance between a certain point r0, and the straight line through the points rB.1 and rB.2 can be calculated from the following formula:



B=

|(r2 − r1 )×(r1 − r0 ) | | r2 − r1 |

(B.17)

For the origin of the coordinate system, the equality r0 = 0 is observed. Then,

B = | r1 | ⋅sin ∠[r1 ,(r2 − r1 )]

(B.18)

Matrix representation of the operators of translation Tr(ax, X), Tr(ay, Y), Tr(az, Z) along the coordinate axes, together with the operators of rotation Rt(ϕx, X), Rt(ϕy, Y), Rt(ϕz, Z) about the coordinate axes is convenient for implementation in the theory of part surface generation. Moreover, use of the operators is the simplest possible way to analytically describe the linear transformations. B.1.7  Resultant Coordinate System Transformation The operators of translation Tr(ax, X), Tr(ay, Y), and Tr(az, Z) together with the operators of rotation Rt(ϕx, X), Rt(ϕy, Y), and Rt(ϕz, Z) are used for the purpose of composing the

513

Appendix B

operator Rs(1  2) of the resultant coordinate system transformation. The transition from the initial “Cartesian” reference system X1Y1Z1 to the other “Cartesian” reference system X2Y2Z2 is analytically described by the operator Rs(1  2) of the resultant coordinate system transformation. For example, the expression: Rs(1  5) = Tr( ax , X ) ⋅ Rt(ϕz , Z) ⋅ Rt(ϕx , X ) ⋅ Tr( ay , Y )



(B.19)

indicates, that the transition from the coordinate system X1Y1Z1 to the coordinate system X5Y5Z5 is executed in the following four steps (see Figure B.7): • • • •

Translation Tr(ay, Y) followed by Rotation Rt(ϕx, X), followed by Second rotation Rt(ϕz, Z), and finally followed by the Translation Tr(ax, X).

Ultimately, the equality

r1(m) = Rs(1  5) ⋅ r5 (m)

(B.20)

is valid. When the operator Rs(1  t) of the resultant coordinate system transformation is specified, then the transition in the opposite direction can be performed by means of the operator Rs(t  1) of the inverse coordinate system transformation. The following equality can be easily proven:

FIGURE B.7 An example of the resultant coordinate system transformation, analytically expressed by the operator Rs(1  5).

514



Appendix B

Rs(t  1) = Rs−1(1  t)

(B.21)

In the above example illustrated in Figure B.7, the operator Rs(5  1) of the inverse resultant coordinate system transformation can be expressed in terms of the operator Rs(1  5) of the direct resultant coordinate system transformation. Following Equation B.21, one can come up with the equation

Rs(5  1) = Rs−1(1  5)

(B.22)

It is easy to show that the operator Rs(1  t) of the resultant coordinate system transformation allows for representation in the following form:

Rs(1  t) = Tr( a, A) ⋅ Eu(ψ , θ , ϕ )

(B.23)

The linear transformation Rs(1  t) [see Equation B.23] can also be expressed in terms of rotation about an axis Rt(ϕA, A), not through the origin [see Equation B.16].

B.2  Complex Coordinate System Transformation In particular cases of complex coordinate system transformations that are repeatedly used in practice, special purpose operators of coordinate system transformations can be composed of elementary operators of translation, and operators of rotation. B.2.1  Linear Transformation Describing a Screw Motion about a Coordinate Axis Operators for 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. By definition (see Figure B.8), the operator Scx(ϕx, px) of a screw motion about X-axis of the “Cartesian” coordinate system XYZ is equal to:

Sc x (ϕx , px ) = Rt(ϕx , X ) ⋅ Tr( ax , X )

(B.24)

After substituting of the operator of translation Tr ( ax , X ) [see Equations 3.4 through 3.6], and the operator of rotation Rt(ϕx, X) [see Equation B.10], Equation B.24 casts into the expression:



1  0 Sc x (ϕx , px ) =  0 0 

0 cos ϕx − sin ϕx 0

0 sin ϕx cos ϕx 0

px ⋅ ϕx   0  0  1 

(B.25)

for the calculation of the operator of the screw motion Scx(ϕx, px) about the X-axis. The operators of screw motions Scy(ϕy, py) and Scz(ϕz, pz) about the Y- and Z-axis correspondingly are defined in a way similar to that of the operator of the screw motion Scx(ϕx, px) is defined:

515

Appendix B

FIGURE B.8 On analytical description of the operator of screw motion, Scx(ϕx, px).



Sc y (ϕ y , py ) = Rt(ϕ y , Y ) ⋅ Tr( ay , Y )

(B.26)



Sc z (ϕz , pz ) = Rt(ϕz , Z) ⋅ Tr( az , Z)

(B.27)

Using Equations B.5 and B.6 together with Equations B.11 and B.12, one can come up with the expressions



 cos ϕy   0 Sc y (ϕy , py ) =   sin ϕy  0 



 cos ϕz  − sin ϕz Sc z (ϕz , pz ) =   0  0 

0 1 0 0

− sin ϕy 0 cos ϕy 0 sin ϕz cos ϕz 0 0

0 0 1 0

0   py ⋅ ϕy   0  1 

(B.28)

0   0  pz ⋅ ϕz  1 

(B.29)

for the calculation of the operators of the screw motion Scy(ϕy, py) and Scz(ϕz, pz) about the Y- and Z-axis. Screw motions about a coordinate axis, as well as screw surfaces, are common in the theory of part surface generation. This makes it practical to use the operators of the screw motion Scx(ϕx, px), Scy(ϕy, py), and Scz(ϕz, pz) in the theory of part surface generation. In case of necessity, an operator of the screw motion about an arbitrary axis either through the origin of the coordinate system or not through the origin of the coordinate system can be derived following the method similar to that used for the derivation of the operators Scx(ϕx, px), Scy(ϕy, py), and Scz(ϕz, pz). B.2.2  Linear Transformation Describing Rolling Motion of a Coordinate System One more practical combination of a rotation and of a translation is often used in the theory of part surface generation.

516

Appendix B

FIGURE B.9 Illustration of the transformation of rolling, Rlx(ϕy, Y), of a coordinate system.

Consider a “Cartesian” coordinate system X1Y1Z1 (see Figure B.9). The coordinate system X1Y1Z1 is traveling in the direction of the X1-axis. Velocity of the translation is denoted by V. The coordinate system X1Y1Z1 is rotating about its Y1-axis simultaneously with the translation. Speed of the rotation is denoted 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 interpretation 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 radius of the rolling cylinder, the expression Rw = V/ω can be used. Due to rolling of the cylinder of a radius, Rw, over the plane is performed with no sliding, a certain correspondence between the translation and the rotation of the coordinate system is established. When the coordinate system turns through a certain angle ϕy, then the translation of origin of the coordinate system along the X1-axis is equal to ax = ϕr · Rw. 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 Rs(1  2) is equal:

Rs(1  2) = Rt(ϕ y , Y1 ) ⋅ Tr( ax , X1 )

(B.30)

Here, 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. Operator of the resultant coordinate system transformation of the kind [see Equation B.30] is referred to as the “operator 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 as Rlx(ϕy, Y). In this particular case, the equality Rlx (ϕ y , Y ) = Rs(1  2) [see Equation B.30] is valid. Based on this equality, the operator of rolling over a plane Rlx(ϕy, Y) can be calculated from the equation



 cos ϕy   0 Rlx (ϕy , Y ) =   sin ϕy   0

0 1 0 0

− sin ϕy 0 cos ϕy 0

ax ⋅ cos ϕy    0  ax ⋅ sin ϕy   1 

(B.31)

517

Appendix B

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 ϕy   0 Rlz (ϕy , Y ) =   sin ϕy   0

− sin ϕy 0 cos ϕy 0

0 1 0 0

−az ⋅ sin ϕy    0  az ⋅ cos ϕy   1 

(B.32)

For the cases when the rotation is performed about the X1-axis, the corresponding operators of rolling are as follows:



1  0 Rl y (ϕx , X ) =  0   0

0 cos ϕx − sin ϕx

0 sin ϕx cos ϕx

0

0

 0   ay ⋅ cos ϕx  −ay ⋅ sin ϕx   1 

(B.33)

 0  az ⋅ sin ϕx   az ⋅ cos ϕx   1 

(B.34)

for the case of rolling along the Y1-axis, and



1  0 Rlz (ϕx , X ) =  0 0 

0 cos ϕx − sin ϕx

0 sin ϕx cos ϕx

0

0

for the case of rolling along the Z1-axis. Similar expressions can be derived for the case of rotation about the Z1-axis: sin ϕz cos ϕz 0 0

0 0 1 0

ax ⋅ cos ϕz   ax ⋅ sin ϕz    0   1 

(B.35)



 cos ϕz  − sin ϕz Rlx (ϕz , Z) =   0   0

sin ϕz cos ϕz 0 0

0 0 1 0

ay ⋅ sin ϕz   ay ⋅ cos ϕz   0   1 

(B.36)



 cos ϕz  − sin ϕz Rl y (ϕz , Z) =   0   0

Use of the operators of rolling Equations B.31 through B.36 significantly simplifies analytical description of the coordinate system transformations. B.2.3  Linear Transformation Describing Rolling of Two Coordinate Systems In the theory of part surface generation, combinations of two rotations about parallel axes are of particular interest.

518

Appendix B

As an example, consider two “Cartesian” coordinate systems X1Y1Z1 and X2Y2Z2 shown in Figure B.10. The coordinate systems X1Y1Z1 and X2Y2Z2 are rotated about their axes Z1 and Z2. The axes of the rotations are parallel to each other (Z1||Z2). The rotations ω1 and ω2 of the coordinate systems can be interpreted so that a circle of a certain radius R1 that is associated with the coordinates system X1Y1Z1, is rolling with no sliding over a circle of the corresponding radius R2 that is associated with the coordinate system X2Y2Z2. When the center distance C is known, then radii, R1 and R2, of the circles, (i.e., of centrodes) can be expressed in terms of the center distance, C, and of the given rotations, ω1 and ω2. For the calculations, the following formulae

R1 = C ⋅

1 1+ u

(B.37)

R2 = C ⋅

u 1+ u

(B.38)

can be used. Here, the ratio ω1/ω2 is denoted by u. In the initial configuration, the X1- and X2-axes align to each other. The Y1- and Y2-axes are parallel to each other. As shown in Figure B.10, 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, then the coordinate system X2Y2Z2 turns through the corresponding angle ϕ2. When the angle ϕ1 is known then the corresponding angle ϕ2 is equal to ϕ2 = ϕ1/u. Transition from the coordinate system X 2Y2Z 2 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:

Rs(1  2) = Rt(ϕ1 , Z1 ) ⋅ Rt(ϕ1 / u, Z1 ) ⋅ Tr(−C , X1 )

FIGURE B.10 On derivation of the operator of rolling, Rru(ϕ1, Z1), of two coordinate systems.

(B.39)

519

Appendix B

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 exercise on their own deriving the equivalent expressions for the operator Rs(1  2). The operator of the resultant coordinate system transformations of the kind [see Equation B.39] are referred to as the “operators of rolling motion over a cylinder.” When rotations are performed around the Z1- and the Z2-axis, the operator of rolling motion over a cylinder is designated as Rru(ϕ1, Z1). In this particular case, the equality Rru (ϕ1 , Z1 ) = Rs(1  2) [see Equation B.39] is valid. Based on this equality, the operator of rolling Rru(ϕ1, Z1) over a cylinder can be calculated from the equation



    cos ϕ1 ⋅ u + 1    u     u + 1  Rru (ϕ1 , Z1 ) = − sin ϕ1 ⋅   u    0   0 

 u + 1  sin ϕ1 ⋅   u   u + 1  cos ϕ1 ⋅   u  0 0

0 0 1 0

 −C     0   0  1 

(B.40)

For the inverse transformation, the inverse operator of rolling of two coordinate systems Rru(ϕ2, Z2) can be used. It is equal to Rru (ϕ2 , Z2 ) = Rru−1(ϕ1 , Z1 ). In terms of the operators of the elementary coordinate system transformations, the operator Rru(ϕ2, Z2) can be expressed as follows:

Rru (ϕ2 , Z2 ) = Rt(ϕ1 / u, Z2 ) ⋅ Rt(ϕ1 , Z2 ) ⋅ Tr(C , X1 )

(B.41)

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 exercise on their 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



    cos ϕ1 ⋅ u + 1    u     u + 1  Rru (ϕ2 , Z2 ) =  sin ϕ1 ⋅  u     0   0 

 u + 1  − sin ϕ1 ⋅   u   u + 1  cos ϕ1 ⋅   u  0 0

0 0 1 0

 C    0   0  1 

(B.42)

can be used. Similar to how the expression [see Equation B.40] is derived for the calculation of the operator of rolling Rru(ϕ1, Z1) around the Z1- and Z2-axis, corresponding formulae 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.

520

Appendix B

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. B.2.4  Coupled Linear Transformation It should be noted here that a translation, Tr(ax, X), along the X-axis of a “Cartesian” reference system, XYZ, and a rotation, Rt(ϕx, X), about the X-axis 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 initially performed, which is followed by the rotation, Rt(ϕx, X), or the rotation, Rt(ϕx, X), is initially performed, which is 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(ϕx , X ) ⋅ Tr( ax , X ) ≡ Tr( ax , X ) ⋅ Rt(ϕx , X )

(B.43)

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 the 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 B.43 is illustrated in Figure B.11. The translation, Tr(ax, X), that is followed by the rotation, Rt(ϕx, X), as shown in Figure B.11a, is equivalent to the rotation, Rt(ϕx, X), that is followed by the translation, Tr(ax, X) as shown in Figure B.11b. Therefore, the two linear transformations, Tr(ax, X) and Rt(ϕx, X), can be coupled into the linear transformation

Cpx ( ax , ϕx ) = Rt(ϕx , X ) ⋅ Tr( ax , X ) ≡ Tr( ax , X ) ⋅ Rt(ϕx , X )

(B.44)

The operator of the linear transformation, Cpx(ax, ϕx), can be expressed in matrix form (see Figure B.12a) as

FIGURE B.11 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.

521

Appendix B

FIGURE B.12 Analytical description of the operators Cpx(ax, ϕx), Cpy(ay, ϕy), and Cpz(az, ϕz), of linear transformation of a “Cartesian” reference system XYZ.



1  0 Cpx ( ax , ϕx ) =  0 0 

0 cos ϕx − sin ϕx 0

0 sin ϕx cos ϕx 0

ax   0  0  1 

(B.45)

This expression is composed based on Equation B.4 for the linear transformation Tr(ax, X), and on Equation B.10 that describes the linear transformation Rt(ϕx, X). Two reduced cases of operator of the linear transformation, Cpx(ax, ϕx), are distinguished. 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. This 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 (see Figure B.12b,c):



 cos ϕy   0 Cp y ( ay , ϕy ) =  − sin ϕy  0 

0 1 0 0



 cos ϕz  − sin ϕz Cpz ( az , ϕz ) =   0  0 

sin ϕz cos ϕz 0 0

sin ϕy 0 cos ϕy 0 0 0 1 0

0  ay   0  1 

(B.46)

0  0  az  1 

(B.47)

522

Appendix B

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 to one another in time, thus, they are not screws. They are just a couple of translations along, and a rotation about, a coordinate axis of a “Cartesian” reference system. Introduction of the operators of linear transformation, Cpx(ax, ϕx), Cp y(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 transformation Cpx(ax, ϕx), Cpy(ay, ϕy), and Cpz(az, ϕz), do not obey the commutative law. This is because rotations are not the vectors in nature. Therefore, special care should be undertaken when treating rotations as vectors—when implementing coupled operators of linear transformations in particular. The operators of coupled linear transformations Cpx(ax, ϕx), Cpy(ay, ϕy), and Cpz(az, ϕz), [see Equations B.45 through B.47] 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

(B.48)

In Equation B.48, only operators of coupled linear transformations are used. B.2.5  An Example of Nonorthogonal Linear Transformation Consider a nonorthogonal reference system X1Y1Z1 having certain angle ω1 between the axes X1 and Y1. Axis Z1 is perpendicular to the coordinate plane X1Y1. Another reference system X 2Y2Z2 is identical to the first coordinate system X1Y1Z1, and is turned about the Z1-axis through a certain angle ϕ. Transition from the reference system X1Y1Z1 to the reference system X 2Y2Z2 can be analytically described by the operator of linear transformation



 sin(ω1 + ϕ )   sin ω1   sin ϕ Rt ω (1 → 2) =  −  sin ω1   0   0

sin ϕ sin ω1 sin(ω1 − ϕ ) sin ω1 0 0

0 0 1 0

 0    0   0  1

(B.49)

In order to distinguish the operator of rotation in the orthogonal linear transformation Rt (1 → 2) from the similar operator of rotation in a nonorthogonal linear transformation Rtω (1 → 2), the subscript “ω” is assigned to the last. When ω = 90°, Equation B.49 casts into Equation B.12. B.2.6  Conversion of a Coordinate System Hand The application of the matrix method of coordinate system transformation presumes that both of the reference systems “i” and “(i ± 1)” are of the same hand. This means

523

Appendix B

that it assumed from the very beginning that both of them are either right-hand- or lefthand-oriented “Cartesian” coordinate systems. In the event the coordinate systems i and (i ± 1) are of a opposite hand, say one of them is the right-hand-oriented coordinate system while another one is the left-hand-oriented coordinate system, then one of the coordinate systems must be converted into the oppositely oriented “Cartesian” coordinate system. For the conversion of a left-hand-oriented “Cartesian” coordinate system into a righthand-oriented coordinate system or vice versa, the operators of reflection are commonly used. In order to change the direction of 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 matrix form, written as



−1  0 Rfx (Yi Zi ) =  0   0

0

0

1 0 0

0 1 0

0  0  0 1

(B.50)

Similarly, implementation of the operators of reflections Rf y(XiZi) and Rfz(XiYi) change the directions of the Yi axis and Zi axis onto opposite directions. The operators of reflections Rf y(XiZi) and Rfz(XiYi) can be expressed analytically in the form



1  0 Rfy (Xi Zi ) =  0 0 



1  0 Rfz (XiYi ) =  0 0 

0

0

−1

0 1 0

0 0 0

0

1 0 0

0 −1 0

0  0  0 1

(B.51)

0  0 0 1

(B.52)

A linear transformation that reverses direction of the coordinate axis is an “opposite transformation.” Transformation of reflection is an example of an “orientation-reversing transformation.”

B.3  Useful Equations The sequence of the successive rotations can vary depending on the intention of the researcher. Several special types of successive rotations are known, including “Eulerian transformation,” “Cardanian transformation,” two kinds of “Euler–Krylov transformations,”

524

Appendix B

and so forth. The sequence of the successive rotations can be chosen from a total of 12 different combinations. Even though the “Cardanian transformation” is different from the “Eulerian transformation” in terms of the combination of the rotations, they both use a similar approach to calculate the orientation angles. B.3.1  RPY-Transformation A series of rotations can be performed in the order “roll matrix, (R),” by “pitch matrix (P),” and finally by “yaw matrix, (Y).” The linear transformation of this kind is commonly referred to as “RPY–transformation.” The resultant transformation of this kind can be represented by the homogenous coordinate transformation matrix: RPY(ϕx , ϕy , ϕz ) =  cos ϕy cos ϕz + sin ϕx sin ϕy sin ϕz cos ϕy sin ϕz − sin ϕx sin ϕy cos ϕz   − cos ϕx sin ϕz cos ϕx cos ϕz   sin ϕx cos ϕy sin ϕz − sin ϕy cos ϕz − sin ϕx cos ϕy cos ϕz − cos ϕy sin ϕz   0 0 

cos ϕx sin ϕy sin ϕx cos ϕx cos ϕy 0

0  0 0 1

(B.53) The “RPY–transformation” can be used for solving problems in the field of part surface generation. B.3.2  Operator of Rotation about an Axis in Space A spatial rotation operator for the rotational transformation of a point about a unit axis a0(cosα, cosβ, cosγ) passing through the origin of the coordinate system can be described as follows, with a0 = A0/|A0| designating the unit vector along the axis of rotation A0. Suppose the angle of rotation of the point about a0 is θ, the “rotation operator,” is expressed by:  (1 − cos θ)cos α cos β − sin θ cos γ (1 − cos θ)cos 2 α + cos θ  (1 − cos θ)cos α cos β + sin θ cos γ (1 − cos θ)cos 2 β + cos θ Rt(θ , a 0 ) =   (1 − cos θ)cos α cos γ − sin θ cos β (1 − cos θ)cos β cos γ + sin θ cos α  0 0  (1 − cos θ)cos α cos γ + sin θ cos β 0  (1 − cos θ)cos β cos γ − sin θ cos α 0  0 (1 − cos θ)cos 2 γ + cos θ 0 1  

(B.54)

The solution to a problem in the field of part surface generation can be significantly simplified by implementation of the rotational operator Rt(θ, a0) [see Equation B.54]. B.3.3  Combined Linear Transformation Suppose a point, p, on a rigid body rotates with an angular displacement, θ, about an axis along a unit vector, a0, passing through the origin of the coordinate system at first, and then followed by a translation at a distance, B, in the direction of a unit vector, b. The linear transformation of this kind can be analytically described by the homogenous matrix

Appendix B

 (1 − cos θ)cos α cos β − sin θ cos γ (1 − cos θ)cos 2 α + cos θ  (1 − cos θ)cos α cos β + sin θ cos γ (1 − cos θ)cos 2 β + cos θ Rt(θa 0 , Bb ) =   (1 − cos θ)cos α cos γ − sin θ cos β (1 − cos θ)cos β cos γ + sin θ cos α  0 0  (1 − cos θ)cos α cos γ + sin θ cos β B cos α  (1 − cos θ)cos β cos γ − sin θ cos α B cos β   B cos γ  (1 − cos θ)cos 2 γ + cos θ 0 1 

525

(B.55)

More operators of particular linear transformations can be found in the literature.

B.4 Chains of Consequent Linear Transformations and a Closed Loop of Consequent Coordinate System Transformations Consequent coordinate system transformations form chains (circuits) of linear transformations. The elementary chain of coordinate system transformation is composed of two consequent transformations. Chains of linear transformations play an important role in the theory of part surface generation. Two different kinds of chains of consequent coordinate system transformations are distinguished. First, transition from the coordinate system XgYgZg associated with the gear tooth flank, G , to the local “Cartesian” coordinate system xgygzg having the origin at a point, K, of contact of the gear tooth flank, G , and of the pinion tooth flank, P . This linear transformation is also made up of numerous operators of intermediate coordinate system transformations (XinYinZin). It forms a chain of direct consequent coordinate system transformations illustrated in Figure B.13a. The local coordinate system, xg ygzg, is associated with the gear tooth flank, G . The operator Rs (G → K g ) of the resultant coordinate system transformation for a direct chain of the linear transformations can be composed using for this purpose a certain number of the operators of translations [see Equations B.4 through B.6], and a corresponding number of the operators of rotations [see Equations B.10 through B.12]. Second, transition from the coordinate system, XgYgZg, to the local “Cartesian” coordinate system, x py pzp, with the origin at a point K of contact of the tooth flanks, G and P . The local coordinate system, xpypzp, is associated to pinion tooth flank, P . This linear transformation is also made up of numerous intermediate coordinate system transformations (XjYjZj), for example, transitions from the coordinate system X hY hZh associated with gear housing, to numerous intermediate coordinate system XiYiZi. The linear transformation of this kind forms a chain of inverse consequent coordinate system transformations shown in Figure B.13b. The operator Rs (G → K p ) of the resultant coordinate system transformations for the inverse chain of transformations can be composed using for this purpose a certain number of the operators of translations [see Equations B.4 through B.6], and a corresponding number of the operators of rotations [see Equations B.10 through B.12].

526

Appendix B

FIGURE B.13 An example of direct chain (a), of reverse chain (b), and a closed loop (c) of consequent coordinate system transformations.

Chains of the direct and of the reverse consequent coordinate system transformations together with the operator of transition from the local coordinate system, xpypzp, to the local coordinate system, xgygzg, form a closed loop (a closed circuit) of the consequent coordinate system transformations depicted in Figure B.13c. If a closed loop of the consequent coordinate system transformations is complete, implementation of a certain number of the operators of translations [see Equations B.4 through B.6], and a corresponding number of the operators of rotations [see Equations B.10 through B.12] returns a result that is identical to the input data. This means that the analytical description of a meshing process specified in the original coordinate system remains the same after implementation of the operator of the resultant coordinate system transformations. This condition is the necessary and sufficient condition for existence of a closed loop of consequent coordinate system transformations. Implementation of the chains, as well as of the closed loops of consequent coordinate system transformations, makes it possible consideration of the meshing process of the tooth flanks, G and P , in any and all of reference systems that make up the loop. Therefore, for consideration of a particular problem of part surface generation, the most convenient reference system can be chosen. In order to complete the construction of a closed loop of a consequent coordinate system transformation, an operator of transformation from the local coordinate system, xpypzp, to the local coordinate system, xgygzg, must be composed. Usually, the local reference systems, xgygzg and xpypzp, are the kind of semi-orthogonal coordinate systems. This means that the

527

Appendix B

FIGURE B.14 Local coordinate systems, xgygzg and xpypzp, with the origin at contact point, K.

axis, zP, is always orthogonal to the coordinate plane, xgyg, while the axes, xg and yg, can be either orthogonal or not orthogonal to each other. The same is valid with respect the local coordinate system, xpypzp. Two possible ways for performing the required transformation of the local reference systems, xgygzg and xpypzp, are considered below. Following the first way, the operator Rtω(p → g) of the linear transformation of semiorthogonal coordinate systems (see Figure B.14) must be composed. The operator Rtω(p → g) can be represented in the form of the homogenous matrix



 sin(ωp + α)   sin ωp   sin α Rt ω ( p → g ) =  − sin ωp   0   0 

−

sin(ω g − ωp − α) sin ωp

0

sin(ω g − α) sin ωp

0

0 0

−1 0

 0    0  0  1

(B.56)

Here, it is designated: ωg – the angle that make Ug and Vg coordinate lines on the gear tooth flank, G ωp – the angle that make Up and Vp coordinate lines on the pinion tooth flank, P α – the angle that make the axes, xg and xp, of the local coordinate systems xgygzg and xpypzp The auxiliary angle β in Figure B.14 is equal to β = ωT + α. The inverse coordinate system transformation, that is, the transformation from the local coordinate system, xgygzg, to the local coordinate system, xpypzp, can be analytically described

528

Appendix B

by the operator Rtω(g → p) of the inverse coordinate system transformation. The operator Rtω(g → p) can be represented in the form of the homogenous matrix



 sin(ω g − α)   sin ω g   sin α Rt ω ( g → p) =   sin ω g  0   0 

sin(ω g − ωp − α) sin ω g

0

sin(ωp + α) sin ω g

0

0 0

−1 0

 0    0  0  1

(B.57)

Following the second way of transformation of the local coordinate systems, the auxiliary orthogonal local coordinate system must be constructed. Let’s consider an approach, according to which a closed loop (a closed circuit) of the consequent coordinate system transformations can be composed. In order to construct an orthogonal normalized basis of the coordinate system xgygzg, an intermediate coordinate system x1y1z1 is used. Axis x1 of the coordinate system x1y1z1 is pointed out along the unit vector ug that is tangent to the Ug-coordinate curve (see Figure B.15). Axis y1 is directed along vector vg that is tangent to the Vg-coordinate line on the gear tooth flank, G . The axis, z1, aligns with unit normal vector, ng, and is pointed outward from the gear tooth body. For a gear tooth flank, G , having orthogonal parameterization (for which Fg = 0, and therefore ωg = π/2), analytical description of coordinate system transformations is significantly simpler. Further simplification of the coordinate system transformation is possible when the coordinate Ug- and Vg-lines are congruent to the lines of curvature on the part surface G . Under such a scenario, the local coordinate system is represented by a “Darboux frame.” In order to construct a “Darboux frame,” the unit tangent vectors, t1.g and t2.g, of the principal directions at a point on the gear tooth flank, G , must be calculated. In the common tangent plane, orientation of the unit vector, t1.g, of the first principal direction on the gear tooth flank, G , can be uniquely specified by the included angle, ξ1.g, that the unit vector, t1.g, forms with the Ug-coordinate curve. This angle depends on both, on the gear tooth flank, G , geometry, and on the gear tooth flank, G , parameterization. Depicted in Figure B.16 is the relationship between the tangent vectors Ug and Vg, and the included angle ξ1.g. From the law of sine’s, Gg sin ξ1. g



(

=

Eg sin[π − ξ1. g − (π − ω g )]

=

Fg sin(ω g − ξ1. g )

(B.58)

)

Here, ω g = cos−1 Fg / EgGg . Solving the expression above for the included angle, ξ1.g, results in:



ξ1. g = tan−1

EgGg − Fg2 Eg + Fg



(B.59)

Another possible way of constructing of orthogonal local basis of the local “Cartesian” coordinate system, xPyPzP, is based on the following consideration. Consider an arbitrary nonorthogonal and not normalized basis, x1x2x3 (see Figure B.17a). Let’s construct an orthogonal and normalized basis based on the initial given basis, x1x2x3.

529

Appendix B

FIGURE B.15 Local coordinate system, xgygzg, associated with the gear tooth flank, G .

FIGURE B.16 Differential relationships between the tangent vectors, Ug and Vg, the fundamental magnitudes of the first order, the included the angle, ξ1.g, and the direction of the unit tangent vector, t1.g.

The cross product of any two of three vectors, x1, x2, x3, for example, the cross product of the vectors x1 × x2, determines a new vector, x4 (see Figure B.17b). Evidently, vector, x4, is orthogonal to the coordinate plane, x1x2. Then, use the calculated vector, x4, and one of two original vectors, x1 or x2, for instance, use the vector, x2. This yields calculation of a new vector, x5 = x4 × x2 (see Figure B.17c). The calculated basis, x1x4x5, is orthogonal. In order to convert it into a normalized basis, each of the vectors, x1, x4, and x5 must be divided by its magnitude:



e1 =

x1 |x1 |

(B.60)

530

Appendix B

FIGURE B.17 A normalized and orthogonally parameterized basis, e1e4e5, that is constructed from an arbitrary basis, x1x2x3.



e4 =

x4 |x 4 |

(B.61)

e5 =

x5 |x 5 |

(B.62)

The resultant basis e1e4e5 (see Figure B.17d) is always orthogonal, as well as it is always normalized. In order to complete the analytical description of a closed loop of consequent coordinate system transformations it is necessary to compose the operator Rs(Kp → Kg) of transformation from the local reference system, xpypzp, to the local reference system, xgygzg (see Figure B.13c).

531

Appendix B

In the case under consideration, the axes, zg and zp, align with the common unit normal vector, ng. The axis, zg, is pointed out from the bodily side to the void side of the gear tooth flank, G . The axis, zg, is pointed oppositely. Due to that, the following equality is observed:

Rs(K p → K g ) = Rt(ϕz , z p )

(B.63)

The inverse coordinate system transformation can be analytically described by the operator

Rs(K g → K p ) = Rs−1(K p → K g ) = Rt(−ϕz , z p )

(B.64)

Implementation of the discussed results allows for: a. Representation of the gear tooth flank, G , and the pinion tooth flank, P , of the form cutting tool, as well as their relative motion in a common coordinate system, and b. Consideration of meshing of the gear tooth flank, G , in any desired coordinate system that is a component of the chain and/or the closed loop of consequent coordinate system transformations. Transition from one coordinate system to another coordinate system can be performed in both of two feasible directions, for example, in direct as well as in inverse directions.

B.5 Impact of the Coordinate System Transformations on Fundamental Forms of the Surface Every coordinate system transformation results in a corresponding change to equation of the gear tooth flank, G , and/or pinion tooth flank, P . Because of this, it is often necessary to recalculate coefficients of the first Φ1.g and of the second Φ2.g fundamental forms of the gear tooth flank, 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 to the fundamental forms Φ1.g and Φ2.g. After being calculated in an initial reference system, the fundamental magnitudes Eg, Fg, and Gg of the first, Φ1.g, and the fundamental magnitudes Lg, Mg, and Ng of the second, Φ2.g, fundamental forms can be determined in any new coordinate system using, for this purpose, the operators of translation, of rotation, and of the resultant coordinate system transformation. Transformation of such kind of the fundamental magnitudes, Φ1.g and Φ2.g, becomes possible due to implementation of a formula, that can be found out immediately below. Let’s consider a gear tooth flank, G , that is given by equation rg = rg(Ug, Vg), where (Ug, Vg) ∈ G. For the analysis below, it is convenient to use the equation of the first fundamental form, Φ1.g, of the gear tooth flank, G , represented in matrix form:

532

Appendix B

Φ1.g  =  dU g   

dVg

0



 Eg   Fg 0 ⋅  0 0 

Fg Gg 0 0

0 0 1 0

0  dU g     0  dVg  ⋅  0  0  1  0 

(B.65)

Similarly, equation of the second fundamental form Φ 2.g of the gear tooth flank, G , can be given by:

Φ 2.g  =  dU g   

dVg



0

 Lg   Mg 0 ⋅   0  0 

Mg Ng 0 0

0 0 1 0

0  dU g     0  dVg  ⋅  0  0  1  0 

(B.66)

The coordinate system transformation that is performed by the operator of linear transformation Rs(1 → 2) transfers the equation rg = rg(Ug, Vg) of the gear tooth flank, G , initially given in X1Y1Z1, to the equation rg* = rg* (U g* , Vg* ) of that same gear tooth flank, G , in a new coordinate system X2Y2Z2. It is clear that rg ≠ rg* . In the new coordinate system, the gear tooth flank, G , is analytically described by the following expression: rg* (U g* , Vg* ) = Rs (1 → 2) ⋅ rg (U g , Vg )



(B.67)

The operator of resultant coordinate system transformation Rs(1 → 2) casts the column matrices of variables in Equations B.65 and B.66 to the form: [dU g*



dVg*

0]T = Rs(1 → 2) ⋅ [dU g

0

dVg

0]T .

0

(B.68)

Substitution of Equation B.68 into Equations B.65 and B.66 makes it possible for the * * expressions for the fundamental forms, Φ1.g and Φ2.g , in the new coordinate system to be Φ1* .g  = [Rs(1 → 2) ⋅ [dU g  

dVg

0

0]T ]T ⋅ [Φ1.g ] ⋅ Rs(1 → 2) ⋅ [dU g

dVg

0

0]T (B.69)

Φ*2.g  = [Rs(1 → 2) ⋅ [dU g  

dVg

0

0]T ]T ⋅ [Φ 2.g ] ⋅ Rs(1 → 2) ⋅ [dU g

dVg

0

0]T (B.70)

The following equation is valid for multiplication:

[Rs(1 → 2) ⋅ [dU g

dVg

0

0]T ]T = RsT (1 → 2) ⋅ [dU g

dVg

0

0]

(B.71)

Therefore, Φ1* .g  = [dU g  

dVg

0

0]T ⋅ {RsT (1 → 2) ⋅ [Φ1.g ] ⋅ Rs(1 → 2)} ⋅ [dU g

dVg

0

0]

(B.72)

Φ*2.g  = [dU g  

dVg

0

0]T ⋅ {RsT (1 → 2) ⋅ [Φ 2.g ] ⋅ Rs(1 → 2)} ⋅ [dU g

dVg

0

0]

(B.73)

*   *  It can be easily shown that the matrices Φ1.g  and Φ 2.g  in Equations B.72 and B.73 represent quadratic forms with respect to dUg and dVg.

533

Appendix B

The operator of transformation Rs(1 → 2) of the gear tooth flank, G , having 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:

Φ1* .g  = RsT (1 → 2) ⋅ [Φ1.g ] ⋅ Rs(1 → 2)  

(B.74)



Φ*2.g  = RsT (1 → 2) ⋅ [Φ 2.g ] ⋅ Rs(1 → 2)  

(B.75)

Equations B.74 and B.75 reveal that after the coordinate system transformation is * completed, the first Φ1.g and the second Φ*2.g fundamental forms of the gear tooth flank, G , in the coordinate system X2Y2Z2 are expressed in terms of the first, Φ1.g, and the second, Φ2.g, fundamental forms initially represented in the coordinate system X1Y1Z1. In order to do that, the corresponding fundamental form (either the form, Φ1.g, or the form, Φ2.g) must be pre-multiplied by Rs(1 → 2) and after that it been post-multiplied by RsT(1 → 2). Implementation of Equations B.74 and B.75 significantly simplifies formulae transformations. Equations similar to those above Equations B.74 and B.75 are valid with respect to pinion tooth flank, P . In case of use of the third, Φ 3.g , and of the fourth, Φ4.g , fundamental forms, their coefficients can be expressed in terms of the fundamental magnitudes of the first and of the second order.

Index Note: Page numbers followed by “n” with numbers indicate footnotes. A ABS, 360 Absolutely dark body, 451 Absolutely rigid body, 451 Absorption of impact energy, 365 AC, see Alternating electric current Accuracy, 292 A2 critical temperature, see Curie temperature Active drive system, 341–342 Adaptation, 248 Adaptive gear variator, 244 advantages, 247 brake, engine, and variator parameters, 278 design of units, 276 efficiency, 280–282 example of solution, 286–287 experimental researches, 276–282 experimental traction characteristics, 279 experiment purpose, 276 kinematics of basic initial kinematic chain, 249–252 lever system for measurement of start moment, 279 parameters of intermediate mode of motion, 278 principle of creation, 248 in projections, 265 solution of problems, 285–286 synthesis, 282–287 technique of tests, 277–280 test-bed description, 276 test rig with, 277 Adaptive mechanism theory, 244 Adaptive variator, 246 Agglomeration, 335 Algebraic signs, 309 Allowable contact stress, 358 Allowable tooth root bending stress, 356–357 Alloying, 336, 382 diffusion alloyed method, 338 mixing, 336 organically bonded alloys, 337 pre-alloyed method, 336

Alternating electric current (AC), 366 electrical resistance heaters, 408 AM gears, 360 manufacturing, 360–361 powder steel for, 360 strength of case-hardened AM steel gears, 361 “Angle of nutation,” 510 “Angle of precession,” 511 Angular base pitch, 474 contact bearings, 203 displacement, 205 operating base pitch, 470 speeds of rotation and spinning, 235 Angular velocity ratio, 4 of satellites, 251 of variator, 250 Antikythera mechanism, 2, 420–421 Applied frequencies, 389, 391–392 Approximate gearings, 474; see also Bevel gears approximate bevel gearing, 475–476 approximate crossed-axes gearing, 476 cone double-enveloping worm gearing, 475 face gearing, 476–477 Aqueous polymer solutions of concentrations, 382 As-quenched martensitic structures, 408 Astaloy CrA, 348 CrM, 348 material, 336 Mo, 348 Asymmetric design for PM, 355 Atomized material, 336 Austenite, 375 Automatic control system, 427 gearboxes, 245, 247–248 Automaton, 244 Automobiles, 427 Axial force, 213

535

536

B Barrel-shaped gear blank application, 114–118 Base cone angle determination, 77 for crown gear, 82 Base pitches equality of gear and mating pinion, 18–20 Basic speed ratio, 293 Batch process, 409–410 Bearing contact, 124 Bearing stress, 180 Belt-and-pulley analogy, 24 Bending of shafts, 192–194 stresses, 121 Bevel gears, 66; see also Approximate gearings blade angle, 90 cutter swing angle, 89–90 cutting disk localization on crown rack, 90–92 by dual interlocking circular cutters, 87 generating surface of cutting disk, 92–93 mechanical behavior, 101 positioning circular cutters, 87–89 straight bevel gear generation by dual interlocking circular cutters, 93–94 Big-sized gear drives, 234 Big-size welded gears, 173 Blade angle, 90–91 Boundary Novikov circle (Boundary N–circle), 487 Brown state, 350 Built-up gear design, 174–178 Butterfly inductor, 388 C Ca–gearing, see Crossed-axes gearing Calculation methods, 359 Camus-Euler-Savary theorem of gearing (CES−theorem of gearing), 419, 454 fundamental theorem of gearing, 10–14, 472, 480 fundamental theory of parallel-axes gearing, 454 Camus, Charles, 448 Carbon content, 381 Carburize, Quench, and Temper process (CQT process), 350 Carburizing, 364, 374 Cardanian transformation, 523–524 Cartesian coordinate system, 3, 132, 508 Case carburizing, 344, 350

Index

Case hardening, 344–345 Casting, 66 C-core tempering inductors, 411–413 CCT curves, see Continuous cooling transformation curves Center line plane (Cln–plane), 24–25 “Central” film thickness, 153 Centro-symmetrical curve, 142n13 CES−theorem of gearing, see Camus-EulerSavary theorem of gearing Change-gears, 295 Chromium, 356 CHT diagrams, see Continuous heating transformation diagrams Circular-arc conformal gears, 155 Classical cooling curves, 400 Clockwise direction (CW direction), 73 Closed contour, 247–248 kinematics, 252 Closed loop of consequent coordinate system transformations, 525–531 “Cold sink” effect, 400 Combined linear transformation, 524–525 Compaction, 339–340, 342 Complex coordinate system transformation, 514 conversion of coordinate system hand, 522–523 coupled linear transformation, 520–522 linear transformation describing rolling motion of coordinate system, 515–517 linear transformation describing rolling of two coordinate systems, 517–520 linear transformation describing screw motion, 514–515 nonorthogonal linear transformation example, 522 Compliance of shafts and bearings, 192 bending of shafts, 192–194 influence of shafts’ bearings, 194–202 Compound two-carrier planetary gear trains; see also Planetary gear trains (PGT) analysis of PGT with fixed external compound shaft, 306–308 connected in series two-carrier PGT and torques, 307 internal division or circulation of power, 308–312 kinematic and power analysis of, 305–312 ways of connection and work, 305–306 Compressibility of lubricant, 160 Computer programs, 232 “Concave-to-convex” contact, 463 Concave elliptic patch, 138

Index

Concave normal plane section, 137 Concentric spray quench, 399 Conditioned cut wire particles, 406–407 Condition of conjugacy of interacting tooth flanks of gear and pinion, 478–480 Condition of contact, 461 between interacting tooth flanks, 4–8 of interacting tooth flanks of gear and pinion, 477–478 Condition of equality of base pitches, 481 of interacting tooth flanks of gear and pinion, 480–481 Cone-shaped discs, 174 Cone double-enveloping worm gearing, 475 Conformal-helical gears, 153 Conformal gearing, 499 Conformal gears, EHL film-forming in, 154–155 Conjugacy, 10n9, 14 of interacting tooth flanks of gear and mating pinion, 466, 468–469 Conjugacy condition of interacting tooth flanks, 8–18 in case of crossed-axes gearing, 14–18 CES theorem of gearing, 10–14 pulley-and-belt analogy, 8–10 Conjugate action law, see Second fundamental law of gearing Conjugate surfaces, 459, 466 Conjugate tooth profiles/surfaces, see Reversiblyenveloping surfaces (Re–surfaces) Consequent coordinate system transformations, 525 closed loop of, 525–531 Consequent linear transformation, chains of, 525–531 Contact condition between interacting tooth flanks, 4–8 geometry, 121–122, 134 mechanics field of materials, 123 modified geometry for contact localization, 79–80 patch, 124 stresses, 121 of surfaces of simple geometries, 123 Continuous cooling transformation curves (CCT curves), 374–375 Continuous heating transformation diagrams (CHT diagrams), 376 Continuously revolving gear tooth generating process, 427 Continuously variable transmission (CVT), 244 Continuous process, 409–410

537

Contour-like hardness pattern, 386 Contour-like pattern, 401 Conventional approach, 509–510 Conventional austenitization, 377 Conventional single-frequency concept (CSFC), 402–403 Converse indicatrix of conformity at point of contact, 148–149 Convex elliptic-like local patch, 141 Convex elliptical patch, 139 Convex parabolic-like local patch, 141 Coordinate system transformations, 503–504 conventional approach, 509–510 Eulerian transformation, 510–511 homogeneous coordinate transformation matrices, 505 homogeneous coordinate vectors, 504 impact on fundamental forms of surface, 531–533 resultant, 512–514 rotation on arbitrary axis, 509, 511–512 rotation on coordinate axis, 507–508 translations, 505–506 Coordinate transformation method, 72–73 Coupled linear transformation, 520–522 Coupling, 367 power, 301 Course de Gèomètrie Diffèrentialle Locale (1957), 122 CQT process, see Carburize, Quench, and Temper process Critical degree of conformity, 497 Crossed-axes gearing (Ca–gearing), 14–18, 30n15, 31, 466, 468–469, 489 Crown gear base cone angle determination for, 82 determination of number of teeth, 80–82 generating surfaces of, 82–84 for octoidal bevel gear generation, 84–87 spherical bevel gear generation by, 86–87 for spherical involute straight bevel gears generation, 80 tooth thickness of, 82 CSFC, see Conventional single-frequency concept Curie temperature, 370 Current penetration depth, 368, 370 Cutter swing angle, 89–90 Cutting disk generating surface of, 92–93 localization on crown rack, 90–92 Cutting tool design theory, 504n3 CVT, see Continuously variable transmission

538

CW direction, see Clockwise direction Cylindrical helix, 213 Cylindrical involute gears, toothing improvements of, 213 meshing geometry of helical gears, 232–237 tooth profile modification of spur gears, 218–232 tooth root design, 237–239 transverse contact ratio of spur gears, 214–218 Cylindrical roller bearings, 200 D “Darboux frame,” 133 da Vinci, Leonardo di ser Piero, 445–446 Decarburized layers, 379–380 Decomposition, 36 Deformations, 166 Degree of conformity, 497 Degree of freedom, 298–301 de La Hire, Philippe, 447 Delapena induction hardening process, 391 Densgrad, 347 Densiform, 347 Desargues, Girard, 447 Design object, 36 Design solution Synthesis, 36 Detrimental surface roughness, 349 de Vaucanson, Jacques, 424 DHG, see Double-helical gears DICC bevel gear set, 104, 109–113 Die quenching, 400–401 Diffusion alloyed method, 338 zone, 346 DIN Sint D11 (ISO P2045), 383 Direct transformation, 507 Disc-type gears, 170–172 Disc stiffness, 171 Displacement of shafts in rolling supports, 194–198 Distaloys, 338 Double-butterfly inductor, 389 Double compaction, 350 Double-disc planet carrier, 211–212 Double-enveloping (approximate) gearing, 477 Double-helical gears (DHG), 174, 234, 236 Double-helical mesh, 200, 210 Double-tooth-contact (DTC), 216 Double-tooth zones, 213

Index

Double-wheel idlers and planet gears, 186–192 planet gears, 205 Dowson-Higginson equation, 154 DTC, see Double-tooth-contact Dual interlocking circular cutters bevel gears generation by, 87–94 mechanical behavior of bevel gears generation, 101–114 straight bevel gear generation by, 93–94 Dual pulse hardening, 403 Dunk quenching, 398 Dupin indicatrix, 130–133 at point of gear tooth flank, 133–134 Dynamic model, 215 Dynamics of gear variator, 264 description, 265–266 dynamics of transient of gear variator, 267–270 numerical example for stage of steady motion, 271 numerical example of dynamic calculation for starting regime, 269–270 preconditions, 264–265 results of dynamic researches, 271–272 tractive characteristic of adaptive gear variator, 269 E EAP−point, see End-of-active-profile point Eddy current cancellation, 371–373 path, 389 Edge impact, 218–219 Efficiency of adaptive gear variator, 280–282 EHL, see Elastohydrodynamic lubrication Elastic deformation, 151–152, 166, 195 Elastic sphere, 123 Elastohydrodynamic lubrication (EHL), 151 Chittenden et al. film thickness formulas, 158–160 conditions for EHL film-forming in conformal gears, 154–155 film thickness calculation for typical gear set, 160–162 Electrical presses, 339 Electromagnetically thin region, 372 Electromagnetic principles of induction heating, 366 eddy current cancellation, 371–373 mechanisms of heat generation, 366–367 skin effect, 368–371

539

Index

Elliptic function, 355 Elliptic local surface patch, 131 Encircling inductors, 393–398 End-of-active-profile point (EAP−point), 493 Energy circulation, 248, 256–257 Entraining velocity, 161 Enveloping condition, see Condition of contact Enveloping surfaces, 459 Epicycloidal surface, 462 Epicycloid teeth, 452 flanks, 451 Equality of angular base pitches of gear and mating pinion, 469–471 Etched hardness profiles, 389 Eu–gearing, see Euler gearing Euler-Savary equation, 449 Euler gearing (Eu–gearing), 22n11, 451, 489n3 Eulerian period of gear art, 450–454 Eulerian transformation, 510–511, 523–524 Euler–Krylov transformations, 523 Euler, Leonhard, 451, 453 External involute parallel-axes gearing, 489 Extremum degree directions of conformity at point of contact, 144–148 F Face gearing, 476–477 Face-hobbing machine, 435 FEA, see Finite element analysis Fe–Fe3C diagram, see Iron-iron carbide phase transformation diagram Fellows Gear Shaper Company in Springfield, Vermont (1896), 426 FEM, see Finite element method Ferrite, 377–378 Ferromagnetic property of material, 369 Filling process, 353 Film thickness calculation for typical gear set, 160 entraining velocity, 161 normal load at contact, 161 operating conditions, 160 predicted film thickness values, 162 principal radii of relative curvature at tooth contact, 160–161 Film thickness formulas, 158–160 Finite element analysis (FEA), 394 Finite element method (FEM), 94, 172 First fundamental law of gearing, 4–8 First order of tangency, 122, 134–135, 137–138, 148–149

First production machines, gear-cutting tools for, 424–425 First spiral bevel gear generator, 430 Fish-tail region, 394 Fixed carrier, 298, 303 Fixed connections, 178 connections in industrial gear drives, 178–180 connections in lightweight gear drives, 181–183 Fixed ring gear, 298, 303 Fixed sun gear, 299 Flank hardening pattern, 385 “Flash” temperature, 152 Flexible supports of planet gears, 205–206 Floating ring gears, 207–211 Floating sun gears, 207 Force adaptation, 246 effect, 252–256 Force analysis of initial kinematic chain of gear variator, 252, 259; see also Gear variator condition of transfer of forces, 257–259 energy circulation, 256–257 examples, 260–263 force adaptation effect, 252–256 force interacting in adaptive gear variator, 263–264 operation, 264 Force-multiplying properties of gears, 419–420 Forging, 349 “Formate” cut method, 430–431 Four-carrier change-gear, kinematic and power analysis of, 315–317 Fourth-order analysis, 134 directions of extremum degree of conformity at point of contact, 144–148 indicatrix of conformity at point of contact of tooth flanks, 138–144 preliminary remarks, 134–137 properties of Indicatrix of conformity, 148 Fracture path, 357 Frequency reduction, 395 “Full complement” design, 186 “Fundamental theorem of gearing,” 454, 472 of parallel-axes gearing, 454 Furnace or oven tempering, 410 G Gap-by-gap technique, 387–388 Gavrilenko, V. A., 466, 468

540

Gear and pinion angular base pitch, 462 base pitch, 454 condition of conjugacy of interacting tooth flanks of, 478–480 condition of contact of interacting tooth flanks of, 477–478 condition of equal of base pitches of interacting tooth flanks of, 480–481 Gear art evolution developments in field of approximate gearings, 474–477 evolution of scientific theory of gearing, 439–474 machining gears methods and on design of gear-cutting tools, 418–439 principal accomplishments in theory of gearing, 477–482 Gear body design, 170 design of large gears, 172–178 disc-type gears, 170–172 Gear-cutting tools, machining gears methods and on design of, 418–439 design for generating bevel gears, 430–435, 436–437 design of toothed wheels and in methods for manufacture of gears, 419–422 development of rotary gear shaving process, 427, 429 development of skiving internal gears process, 427 for first production machines, 424–425 for generating bevel gears, 428, 430 generating milling of bevel gears, 435, 438–439 grinding hardened gears, 427–428 for production machines, 425–427 special purpose cutting tools to produce gear teeth, 422–424 Gear drive engineering compliance of shafts and bearings, 192–202 gear-shaft connections, 178–192 gear body design, 170–178 gear drive design, 168 housing deformations, 202–204 planetary gear drives, 204–213 toothing improvements of cylindrical involute gears, 213–239 tooth misalignment and load distribution, 166 Gear finishing fragment of design scheme structure, 45 graph of technological system for gear finishing, 41

Index

LSD TS, 44–45 metasystem, 36–37 multiplicity X formation on corresponding decomposition level, 39 parametric model of functional process, 38 parametric model of machining process, 40, 42 principle elaboration for developing technological systems, 35 second level of decomposition, 43–44 structural-hierarchical model of technological system, 40 synthesis of technological systems, 46 Gear honing, 58 machine engagement geometry model, 46–61 principle elaboration for developing technological systems, 35–46 synthesis of working layer forms of tools, 61–63 Gearing, 489; see also Novikov gearing base pitches equality of gear and mating pinion, 18–20 condition of conjugacy of interacting tooth flanks, 8–18 condition of contact between interacting tooth flanks, 4–8 fundamental laws of, 4 Gear/pinion base pitch, see Linear base pitch Gear(s), 384 apex, 24 broaching process, 418–419 gear-generating principle, 424 gear-grinding machine, 220 gear-like components, 363 machining processes, 426 manufacture, 419–422 manufacturers, 363 misalignment angle, 198 pair analogy, 8–10 ratio, 67 spin hardening, 393–398 system, 489 theory, 439 transmissions, 1–2 Gear-shaft connections, 178 fixed connections, 178–183 movable connections, 183–192 Gear teeth interaction converse indicatrix of conformity at point of contact, 148–149 fourth-order analysis, 134–148

Index

Heinrich Hertz approach, 122–125 second-order analysis, 125–134 Gear tooth flanks, 130–131, 133–134, 138–149 thickness, 75–76 Gear variator, 246, 248; see also Adaptive gear variator description of design of, 272–274 design, 274 dynamics, 264–272 engineering philosophy, 272 kinematic chain structure, 249 knots, 275 operation, 274 preconditions, 244–248 Generating bevel gears, 430–435 gear-cutting tools for, 428, 430 Generating milling of bevel gears, 435, 438–439 “Geometrically accurate gearings,” 440 “Geometric Theory of Gearing,” 454–455 “Ghost pearlite,” 376 Gleason Coniflex® method, 66 Gleason Revacycle®, 66 Gleason, William, 475 bevel gear planning machine, 430 “Global” sense, 127n6 Glue, 337 Gochman, Chaim, 455, 459 Grant bevel gearing, 461–462 involute tooth flank, 463 meshing in bevel gear pair, 464 Grant gearing (Gr–gearing), 22n12, 462 Grant, George Barnard, 461–462 Grinding, 349 hardened gears, 427–428 H Hardening components containing teeth, 407–408 Hard finishing, 349 Hardness, 410 Harmonic drive, 471 Harshness, 359 HAZ, see Heat effected zone HCR, see High contact ratio Heat effected zone (HAZ), 379 Heating intensity of, 375 rates, 367 Heating modes, 387–406 CSFC, 402–403

541

for encircling inductors, 401 IFP technology, 404 PDFC, 403–404 PSFC, 403 single-coil, dual-frequency concept, 404–406 Heat sink, 391 Heat treaters, 375 Heat treatment, 343 case hardening, 344–345 hardness curves for Chromium alloyed PM gear, 345 IH, 345 nitriding, 346 sintering, 344 Heavy-loaded planet gear, 186 Heinrich Hertz approach, 122–125 Helical gear, 169–170, 180, 499 meshing geometry of, 232–237 Helical mesh, 173 Hertz, 123 theory, 124 Hertzian contact, 153 dimensions, 161 High-conformal gearing, 496–500 critical degree of conformity in “Novikov gearing,” 497 minimum required degree of conformity at PC, 497–500 High contact ratio (HCR), 217, 219 meshing, 224 Higher-order analysis, 134 High frequency, 397 High-power gear drives, 169 planetary drives, 210 High-pressure region, 152 High-speed gears, 174 pinion, 202 High-speed steel (HSS), 341 High-temperature sintering, 382 Hobbing gears, 354 Hollomon–Jaffe correlation, 410 Homogeneous coordinates, 504 transformation matrices, 505, 524 vectors, 504 Homogeneous fine-grain Q&T initial microstructures, 377 Homogeneous matrices, 507, 527–528 Honing, 349 Housing deformations, 202–204 HSS, see High-speed steel Hydraulic CNC-presses, 339–340

542

Hydrodynamic entrainment velocity, 157 film-forming potential, 151–152 Hyperbolic local surface patch, 131 Hypoid gear, 431–432, 476 Hypoids, 153 Hysteresis loop, 367 I Ia–gearing, see Intersected-axes gearing Ideal gearings, 440 Idlers gears, 184–186 IFP technology, see Independent frequency and power control concept IH, see Induction hardening Immersion quenching, 398 Impact toughness, 359 In crossed-axes gearing, 469–471 Independent frequency and power control concept (IFP technology), 404 Indicatrix of conformity (CnfR(G/P)), 148, 497 at point of contact of tooth flanks of gear and mating pinion, 138–144 Induction gear hardening inductor designs and heating modes, 387–406 metallurgical subtleties of, 373–383 technologies for, 384 tooth hardness patterns, 385–387 Induction hardening (IH), 345, 364–365 vacuum carburizing to, 364 Induction heating, 364 electromagnetic principles, 366–373 Induction systems, 364 Induction tempering, 410 subtleties, 410–413 Inductoheat’s simultaneous dual-frequency inverter, 404–405 Inductor-to-tooth air gaps, 389 Industrial gear drives, connections in, 178–180 In methods for manufacture of gears, 419–422 Intermediate gears, 183 Intermediate relief, 223 Internal meshing, 293 Internal moments, 257 Internal power division, 308–312 Intersected-axes gearing (Ia–gearing), 30n15, 460, 489 Intralube HD, 335 Invention process, 426 Inverse coordinate system transformation, 531 Involute

Index

gear, 493 polar angle, 68 profile, 68 roll angle, 68 surface, 462 “Involute tooth point” geometry, 463–494, 496 Iron, 374–375 Iron-based PM materials, 382–383 Iron-iron carbide phase transformation diagram (Fe–Fe3C diagram), 374–375 Isothermal assumptions, 152 J Joule heat generation, 367 K Kinematic analysis of compound two-carrier planetary gear trains, 305–312 of four-carrier change-gear, 315–317 of simple AI-planetary gear train, 298–301 of two-carrier change-gear, 312–315 Kinematic(s) of basic initial kinematic chain, 249–252 chain structure of gear variator, 249 Kolchin, A. I., 462–463, 465 L LA, see Line of action Laminations, 389, 412–413 Large gears design, 172 built-up gear design, 174–178 welded gear body design, 173–174 Larsen–Miller parameter, 410 Laser peening, 349 Laser powder bed fusion technology (L-PFB technology), 360–361 Law of conjugate action, 452 of sine, 528 LC, see Line of Contact LCR gears, see Low Contact Ratio gears Leçons sur les Applications du Calcul Infinitésimal á la Geometrie (1826), 121–122 Legitimate analytical function, 135 Length of action, 214–215 Lever analogy, 297 Light weight

543

Index

connections in lightweight gear drives, 181–183 design, 353–354 Limit load, 320 Linear base pitch, 454 Linear speeds, 252 Linear tip relief parameters, 221 calculation of angle between lines normal to main involute profile, 230 example, 225 extension of linear tip relief, 222 extension of tip relief, 223 HCR meshing, 224 relief involutes, 229 TE, 232 tooth half-thickness, 227 tooth tip thickness, 231 Linear transformation combined, 524–525 describing rolling motion of coordinate system, 515–517 describing rolling of two coordinate systems, 517–520 describing screw motion on coordinate axis, 514–515 Line of action (LA), 11, 214–215, 448–449, 480 Line of Contact (LC), 491 desired LC in parallel-axes gearing, 491–495 Line of nodes, 510 Link, 35 Load concentration factor, 166 Load distribution, 167 Load spectra determination of particular elements of change-gear, 317–321, 325 Local congruency, 138 Local coordinate system, 528–529 Local relative orientation of tooth flanks of gear, 126–130 Logical scheme for design of technological systems (LSD TS), 42, 44–45 Loss factor in stop regime, 282 Low-alloy steels, 374 Low Contact Ratio gears (LCR gears), 232 Low frequency, 397 Low-pressure carburizing process, 343 Low-pressure process (LPC), 350 Low-temperature tempering, 386 Low-weight gear drives, 169 LPC, see Low-pressure process LSD TS, see Logical scheme for design of technological systems L-PFB technology, see Laser powder bed fusion technology

Lubricants, 335 Lubricated Hertzian contact, 152 Lubrication, 151 of materials, 334–335 M Machined integral quench inductor (MIQ inductor), 399 Machine engagement geometry model, 46 algorithm for decoding geometrical parameters of gears, 49 as apparatus of gear theory, 46–47 developed model algorithm, 48 development, 47 existence area of, 51–52 independent parameters, 50–51 limits of change in angle of engagement and radius-vector, 55 objective functions of model, 55–61 options for existence of submultiplicity, 53 relative position of disk tool and machined gear, 50 tooth profile, 53–54 “Machine for Planing Gear Teeth,” 471–472 Machining gear methods design of gear-cutting tools for generating bevel gears, 430–435, 436–437 design of toothed wheels and in methods for manufacture of gears, 419–422 development of rotary gear shaving process, 427, 429 development of skiving internal gears process, 427 gear-cutting tools for first production machines, 424–425 gear-cutting tools for generating bevel gears, 428, 430 gear-cutting tools for production machines, 425–427 generating milling of bevel gears, 435, 438–439 grinding hardened gears, 427–428 and on design of gear-cutting tools, 418 special purpose cutting tools to produce gear teeth, 422–424 Madrid Codices, The (Leonardo da Vinci), 446 Magnetic hysteresis heat generation, 367 “Main theorem,” 11, 419 Manganese, 356 Mating pinion, 138–149 crossed-axes gearings, 469 at point of contact, 126

544

Matrices, 503n1, 504n2 method of coordinate system transformation, 522–523 representation of equation of Dupin indicatrix, 133–134 Maximal unit load, 171 Maximum Hertzian pressure, 161 Mechanical presses, 339 Medium-carbon steels, 374 Medium-sized gears, 399 Meshing geometry of helical gears, 232–237 Metal injection molding process (MIM process), 360, 382 Metallographic examination, 380 Metallurgical subtleties of induction gear hardening, 373 density reduction, 381 material selection, 373–374 impact of rapid heating and prior microstructure, 374–380 specifics of IH of PM gears, 380–383 super-hardness phenomenon, 380 Metasystem (S), 36–37 Microgeometry, 355 Microslip, 177 Mid-impact, 218 MIM process, see Metal injection molding process Minimal interference, 178 Minimal local surface patch, 131 Minimum film thickness, 153, 158 MIQ inductor, see Machined integral quench inductor Mixing, 336 20MnCr5 steel, 349, 351, 360–361 Mobile closed contour, 264 circulation of energy in, 274 theorem on, 253–256 Modern gear transmissions, 2 Modern plain carbon steels, 379 Modified geometry for contact localization, 79–80 Molding-generating machine, 438 Moment lever, 248, 259 adaptive gear variator with, 258 Movable connections, 183 double-wheel idlers and planet gears, 186–192 gears of variable-speed drives, 183–184 idlers and planet gears, 184–186 Multiple-pinion experimental aerospace reduction gearbox, 160 Multitronic frictional variator, 245

Index

Multiturn coils, 394 Multiturn encircling inductor, 393 Musser, Walton, 471–472 N Natural frequency, 208 N–circle, see Novikov circle NCR gears, 232 Nelder–Mead method, 63 Newtonian fluid, 451 Newton rings, 123 NF, see Nonfixing Nicholas J. Terbo, see Trbojevich, Nikola John Nitriding, 346, 364, 374 Noise, 359 Noise vibration harshness (NVH), 353 Nominal position, 204 Nonfixing (NF), 198–199 Nonorthogonal linear transformation example, 522 Nonuniformity of load distribution, 167 Normal curvatures, 125, 135–136 Normalized structures, 377 Normal load at contact, 161 Normal plane (Nln–plane), 24–25 Novikov circle (N–circle), 487 Novikov conformal gearing, 463–464, 466–467, 472 Novikov gearing, 463–464, 487–488, 490 axes of rotation, 488–489 close-up of conformal gear pair, 488 design features of, 495–496 desired LC in parallel-axes gearing, 491–495 high-conformal gearing, 496–500 PA in parallel-axes gearing, 490–491 principal design features of, 489 principal design parameters, 496 vector diagram of gear pair, 490 Novikov, Mikhail L., 466 Numerical modeling, 372 NVH, see Noise vibration harshness O Object, 35 Octoidal bevel gear generation, 98–101 comparison of spherical geometries and, 98–101 crown gear for, 84–86 Oil-based quenchants, 399 Oil quenching, 382 Old-style gear transmission, 421

Index

Olivier principles, 455 Olivier, Théodore, 457–458 One-rim planet, 293 Operating angular base pitch, 462 in intersected-axes, 469–471 Operating base pitch, 454 Operational motion regime, 272 Operator of linear transformation, 522 of rolling motion over cylinder, 519 of rolling motion over plane, 516 Organically bonded alloys, 337 Orientation-preserving transformation, 507 Orientation-reversing transformation, 523 Orthogonal crossed-axes gear pairs, 23–24 Orthogonally parameterized gear tooth flank, 144 Orthogonal planes, 194 Oscillating process, 216 Overhang-mounted gears, 167 P PA, see Plane of action Pa–gearing, see Parallel-axes gearing “Palloid” gear-cutting method, 433 Parabolic local surface patch, 131 Paraboloid of revolution, 125 Parallel-axes gearing (Pa–gearing), 489 PA in, 490–491 Parallel-axes gearings, 3, 10, 460 Parametric synthesis, 36 Path of contact (Pc), 448–449, 480 Pattern uniformity, 389 PDFC, see Pulsing dual-frequency concept Peening, 349 Penetration depth, 369 Perfect crossed-axes gearing with line contact, 20 base cones in, 24–27 gearings, 21–22 kinematics of crossed-axes gearing, 23–24 R–gearing, 22–23 tooth flanks in, 27–32 violation of condition of equal angular base pitches, 21 Perfect gearings, 4, 422, 440, 461–471 A. I. Kolchin, 462–463, 465 conjugacy of interacting tooth flanks of gear and mating pinion, 466, 468–469 equality of angular base pitches of gear and mating pinion, 469–471 grant bevel gearing, 461–462, 463, 464 Novikov conformal gearing, 463–464, 466–467

545

permissible instant relative motions in, 8 V. A. Gavrilenko, 466, 468 Performance boosting processes, 346 forging, 349 peening, 349 rolling densification, 348 variants of bore densification using ball bearings, 347 Peripheral velocity of sun gear, 302 Permissible axial displacement, 200 PGT, see Planetary gear trains Pinion apex, 24 Pitch-line plane (Pln), 24, 26 Pitch cone angle determination, 67–68 polar angle determination at, 76 Pitting phenomena, 358 Plain bearings, 194 Planar characteristic images, 130 “Dupin indicatrix,” 130–133 matrix representation of equation of Dupin indicatrix, 133–134 Planar involute of circle, 68–69 Plane of action (PA), 31, 470, 490 minimum required degree of conformity at, 497–500 in parallel-axes gearing, 490–491 Planetary gear drives, 204 flexible supports of planet gears, 205–206 floating ring gears, 207–211 floating sun gears, 207 planet carriers, 211–213 relative accuracy, 204–205 Planetary gear trains (PGT), 291–292 analysis with fixed external compound shaft, 306–308 as division one, 300–301 with external shafts, torques, and lever analogy, 295 kinematic and efficiency analysis of, 291–292 simple AI-PGT as building element of planetary change-gear trains, 292–294 as summation one, 300 Planet carriers, 211–213 Planet gears, 183–187 elastically deform, 208 Plasma-assisted nitriding methods, 346 Plastic deformation, 167, 182 Plastic regime, 359 Plus elastic deformation, 187 PM, see Powder metal; Powder metallurgy Poisson’s number, 355 Poisson’s ratio, 158

546

Polar angle determination at pitch cone, 76 Ponte Castaneda model, 359 Porosity, 382 Post-Eulerian period of developments in field of gearing, 454–461 Powder circulation, 308–312 forging of gears, 349 source of, 1 steel for AM gears, 360 Powder metal (PM), 330–331 alloying concepts, 336–338 AM gears, 360–361 design for, 353–355 elevated temperature compaction temperatures and density, 335 examples, 333–334 gear for car seat adjustment, 333 lubrication of materials, 334–335 manufacturing, 338–353 material, 330 particles, 335 performance, 355–359 powder particles, 331 S-n diagram for tooth root bending fatigue, 332 selection of materials for gears, 330–334 Powder metallurgy (PM), 365 specifics of IH of, 380–383 Power density, 2 equilibrium, 300 flows, 301–302 power-to-weight ratio, 2 Power analysis of AI-planetary gear train, 301–305 of compound two-carrier planetary gear trains, 305–312 efficiency, 303–305 of four-carrier change-gear, 315–317 power flows, 301–302 real torques, 302–303 of two-carrier change-gear, 312–315 Pre-alloyed method, 336 Precision of gears, 166 Predicted film thickness values, 162 Pre-Eulerian period of gear art, 445–450 Press quenching, 400–401 Principal radii of relative curvature at tooth contact, 160–161 Process paths to making gears, 350–351 Production machines, gear-cutting tools for, 425–427

Index

Profiled hardness pattern, 386 Protuberance, 238–239 Pseudo-path of contact (Ppc), 495 Pseudo-straight tooth flanks, 31 PSFC, see Pulsing single-frequency concept Pulley-and-belt analogy, 8–10, 491 Pulsing dual-frequency concept (PDFC), 403–404 Pulsing single-frequency concept (PSFC), 403 Q Quality gears, 363 Quantitative measure, 126 Quenched & tempered microstructure (Q&T microstructure), 374, 377 Quenching options, 398–401 Quench media, 399 Quench oils, 399 Quench orifices, 400 R RA, see Retained austenite Rapid heating, impact of, 374–380 Real torques, 302–303 Reference depth, 368 Regime of motion, 281 Relative accuracy, 204–205 Relative orientation, 147 Relative power, 301, 324 Relief involutes, 229 RENK® change-gear for heavy vehicles, 318, 324 Repeatable distortion, 390 Re–profiles/surfaces, see Conjugate tooth profiles/surfaces Residual stresses at tooth working surface, 406–407 Resonance vibrations, 214 Re–surfaces, see Reversibly-enveloping surfaces Retained austenite (RA), 365, 374, 409 Revacycle method, 435 Revacycle® broaching process, 66 Reversibly-enveloping surfaces (Re–surfaces), 27n14, 466, 478, 487 Reynolds equation, 151–152 R–gearing, 22–23, 32 Rolling densification, 348 Roll matrix, pitch matrix and yaw matrix transformation (RPY-transformation), 524 Root hardening pattern, 386

Index

Root optimization, 354–355 Rotary gear shaving process, 427 Rotation, 3, 438 on coordinate axis, 507–508 operator on axis in space, 524 RPY-transformation, see Roll matrix, pitch matrix and yaw matrix transformation Runout, 351–352 “Russian School of Theory of Gearing,” 481–482 Rust inhibitors, 382 S Saddle-like local patch, 141 Schicht, Heinrich, 433–434 Schiele, Christian, 425–426 Scientific theory of gearing, 439 development and investigation, 440 development in field of perfect gearings, 461–471 Eulerian period of gear art, 450–454 post-Eulerian period of developments, 454–461 pre-Eulerian period of gear art, 445–450 tentative chronology of evolution of theory of gearing, 472–474 theory of gearing, 441–445 Walton Musser, 471–472 Scuffing, 387 Second-order analysis, 125 local relative orientation of tooth flanks of gear, 126–130 planar characteristic images, 130–134 Second fundamental law of gearing, 8–18 Self-consistent scientific theory of gearing, 477 Shaft bending deformation, 198 Shaft deformation, 192 Shafts bearings, 194 displacement of shafts in rolling supports, 194–198 types and location of supports, 198–202 Shape distortion, 390 Sharp corners, 365 Shishkov equation of contact, 17n10, 474, 478, 481 Shock loading, 387 Shot peening, 349 Shrink-fitted rim, 177–178 Shrinkage prediction, 360 Shutdown regime, 264 Simple AI-PGT as building element of planetary change-gear trains, 292–294 kinematic analysis, 298–301

547

Simultaneous dual-frequency concept, see Single-coil, dual-frequency concept Single-coil, dual-frequency concept, 404–406 Single compaction, 350 Single-disc planet carriers, 212 Single indexing method, 430 Single-point gear-cutting tool, 430 Single-tooth-contact (STC), 216 Single-tooth zones, 213 Single-turn inductors, 394 Sintering, 344, 350, 357 Skin effect commonly assumed definition, 368–370 nonexponential distribution, 370–371 Skiving process, 428 internal gears process, 427 “Slender” contacts, 153 Sliding bearings, 186 Slope angles manually calculation, 192–194 Small-sized gears, 399 S-n curves, 356 Speedup, 267–268 “Sphere-to-plane” contact, 124 Spherical bevel gear generation comparison of octoidal geometries and, 98–101 crown gear for, 86–87 Spherical gearings, 30n15 Spherical involute bevel gear mechanical behavior of, 94–98 tooth surfaces, 78–79 Spherical involute profile, 68–69 direct definition, 68–72 indirect definition, 72–74 Spherical involute straight bevel gears generation, 80–84 Spherical trigonometry, 70 Spin hardening, 387 Spline function, 355 Spray quenching, 394 devices, 399–400 Spur gears tooth profile modification, 218–220 transverse contact ratio of, 214–218 Stainless steels, 360 Starmix, 337 Statipower-IFP technology, 392, 404 STC, see Single-tooth-contact Steady motion dynamics of transient of gear variator in stage, 270 numerical example for stage, 271 Steering racks, 407

548

Step quench process, 350 Stop regime, 281–282 Straight bevel gears, 66 application of barrel-shaped gear blank, 114–118 base cone angle determination, 77 bevel gears generation by dual interlocking circular cutters, 87–94 comparison of spherical and octoidal geometries, 98–101 crown gear for octoidal bevel gear generation, 84–86 crown gear for of spherical involute straight bevel gear generation, 80–84 face and root cone angles, 79 gear tooth thickness, 75–76 mathematical definition of spherical involute profile, 68–76 mechanical behavior of bevel gears generation, 101–114 mechanical behavior of spherical involute bevel gear drive, 94–98 modified geometry for contact localization, 79–80 numerical example, 94 pitch cone angle determination, 67–68 polar angle determination at pitch cone, 76 spherical and octoidal bevel gears generated by crown gear, 86–87 spherical involute bevel gear tooth surfaces, 78–79 straight-tooth bevel gears, 66 tooth surfaces, 75 Stress concentration effect, 237 Structural synthesis, 36 Super-hardness phenomenon, 380 Super hardening, see Super-hardness phenomenon Support on frame, 248, 258 T TCA, see Tooth Contact Analysis TE, see Transmission errors Technological system, 35 Tempering, 345 of gears and gear-like components, 408 induction tempering subtleties, 410–413 options, 409–410 Tensile surface stresses, 379–380 Tentative chronology of evolution of theory of gearing, 472–474 Theorié des Fonctions Analytiques (1797), 121–122

Index

Theory of gearing, 440–445, 472–474, 477 condition of conjugacy of interacting tooth flanks of gear and pinion, 478–480 condition of contact of interacting tooth flanks of gear and pinion, 477–478 condition of equal of base pitches of interacting tooth flanks of gear and pinion, 480–481 “Russian School of Theory of Gearing,” 481–482 Thermo-chemical diffusion processes, 364–365 hardening processes, 374–375 Third fundamental law of gearing, 18–20, 471 Three-dimensional linear transformation, 505 Three-dimensional printed gears, 330 Through-hardened gear tooth, 384 TiC, see Titanium carbides Tip-by-tip method, 387 Tip relief, 219 Titanium carbides (TiC), 378 Tolerances, 351–352 Tooling, 341–343 Tool steels, 360 Tooth-by-tooth hardening of gears, 387–393 Tooth Contact Analysis (TCA), 94, 122 Toothed adaptive variators, circuit designs and advantages of, 247 Toothed variator, 259 Toothed wheels design and in methods for manufacture of gears, 419–422 Tooth flanks, 419, 457 in perfect crossed-axes gears, 27–32 Tooth flanks of gear and mating pinion base pitches equality of interacting, 18–20 perfect crossed-axes gearing with line contact, 20–32 at point of contact, 126 Tooth half-thickness, 227 Tooth hardness patterns, 385–387 Toothing improvements of cylindrical involute gears, 213–239 Tooth involute profile, 493n5 Tooth misalignment, 167 Tooth profile modification of spur gears, 218 basics, 218–220 parameters of linear tip relief, 221–232 Tooth root breakage, 357 Tooth root design, 237–239 Tooth tip region, 391 Torque (Trq), 3 ratios, 297, 318 transformation, 296 transmission, 183

549

Index

transmit ratio, 296 Torque method, 294 essence of method, 295–297 kinematic analysis of simple AI-planetary gear train, 298–301 kinematic and power analysis of compound two-carrier planetary gear trains, 305–312 modified symbol of Wolf with external shafts and torques, 297 power analysis of AI-planetary gear train, 301–305 Transformations, 31 matrix, 91–92 of motion, 488–489 of reflection, 523 Transform rotary motion, 121 Transient time, 408 Transitive motion, 267 Translation, 438 of coordinate system, 505–507 Transmission, 1 of force in planetary train, 248 of motion, 488–489 ratio range of adaptive gear variator, 282–285 transmitted torque, 214 Transmission errors (TE), 216, 222–223, 232 Transmission of rotary motion areas of existence of parallel-axes gearings, 3–4 fundamental laws of gearing, 4–20 perfect crossed-axes gearing with line contact, 20–32 wooden gears in gear transmission design, 2 Transverse contact ratio, 214–215 of spur gears, 214–218 Transverse edges, 53 Transverse overlap, 214 Transverse section, 158 Trbojevich, Nikola John, 431, 476 Tredgold approximation, 460 Tredgold, Thomas, 460 True contour hardness pattern, 386, 401 Tungsten carbide, 341 Turbine reduction gears, 190 Turriano, Juanelo, 423 Two-carrier change-gear, kinematic and power analysis of, 312–315 Typical conformal gear pair, 160

Unit tangent vector, 74, 127 Unit vector, 16 Untampered martensite, 408 U.S. Patent on method of lapping gears, 427, 429 V Variable load, 245 Variable output force of resistance, 259 Variable-speed drives gears, 183–184 Variable “surface-to-subsurface” density, 383 Variator, 244 angular velocities of, 250 assembly drawing, 275 three-dimensional image of details, 275 Vector diagram of gear pair, 490 Vibration, 214, 359 Virtually independent of load, 158–159 Virtual work principle, 259 Viscosity, 151 Viscosity/pressure relation, 160 Viscous friction, 152 von Pittler, Wilhelm, 428 W Warm die compaction, 335 “Wave-shaped” profile, 371 Wedging, 273–274, 279 Welded gear body design, 173–174 Wheel cutting engine, 421 Wildhaber-Novikov gearing (WN–gearing), 487–488, 500 Wildhaber, Ernest, 435–437 Wildhaber gearing, 499 Willis, Reverend Robert, 455–457 “Willis’ theorem,” 10–11, 454 Willis’s fundamental theorem of gearing, 460 Wind turbines, 388 WN–gearing, see Wildhaber-Novikov gearing Wöhler’s curve of gear material, 320 Wooden gears in gear transmission design, 2 Working member, 1 Wright “Cyclone,” 205 Y Young’s modulus, 158, 355

U Umbilic local surface patch, 131 Uniform load distribution, 167

Z Zero load, 157