Rolling Contacts (Tribology in Practice Series)

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Rolling Contacts by T A Stolarski and S Tobe

Professional Engineering Publishing Limited London and Bury St Edmunds, UK

First published 2000 This publication is copyright under the Berne Convention and the International Copyright Convention. All rights reserved. Apart from any fair dealing for the purpose of private study, research, criticism, or review, as permitted under the Copyright Designs and Patents Act 1988, no part may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, electrical, chemical, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owners. Unlicensed multiple copying of this publication is illegal. Inquiries should be addressed to: The Publishing Editor, Professional Engineering Publishing Limited, Northgate Avenue, Bury St Edmunds, Suffolk IP32 6BW, UK.  T A Stolarski and S Tobe

ISBN 1 86058 296 6 ISSN 1470-9147

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

The publishers are not responsible for any statement made in this publication. Data, discussion, and conclusions developed by the authors are for information only and are not intended for use without independent substantiating investigation on the part of the potential users. Opinions expressed are those of the authors and are not necessarily those of the Institution of Mechanical Engineers or its publishers. Printed by J W Arrowsmith Ltd, UK.

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Professional Engineering Publishing (publishers to the Institution of Mechanical Engineers) offers a valuable and extensive range of engineering books, conference and seminar transactions, and prestigious learned journals, as well as a successful list of magazines. For the full range of titles published by Professional Engineering Publishing contact: Sales Department Professional Engineering Publishing Limited Northgate Avenue, Bury St Edmunds Suffolk, UK IP32 6BW Tel: C44 (0) 1284 724384 Fax: C44 (0) 1284 718692 E-mail: [email protected] www: pepublishing.com


Contents Series Editors’ Foreword

xiii

Preface

xv

Notation

xvii

Chapter 1 Introduction to Rolling Contacts 1.1 Historical perspective 1.2 Practical use of rolling contact 1.3 Need to lubricate the rolling contact 1.3.1 Lubrication in the contact area 1.3.2 Reasons for lubrication 1.4 References Chapter 2 Elements of Surface Contact of Solids 2.1 Introduction 2.2 Distribution of stresses within the contact zone 2.3 Deformations resulting from contact loading 2.4 Contact between bodies of revolution 2.4.1 Stress distribution within the contact 2.4.2 Contact with combined normal and tangential loads 2.4.3 Three-dimensional contact 2.5 Contact of real surfaces 2.6 Criterion for deformation mode 2.6.1 Surface plastic deformations 2.7 Thermal effects during rolling 2.7.1 Moving source of heat 2.8 Contact of bodies with interposing film 2.8.1 Background to the analysis 2.8.2 Case of contacting cylinders 2.8.3 Contacting spheres 2.9 Crack formation in contacting elastic bodies 2.9.1 Description of the contact

1 1 2 5 5 7 9 11 11 12 18 19 21 23 24 30 34 35 38 39 40 41 42 45 45 46

viii

Rolling Contacts

2.10 Contacts deviating from the Hertz theory 2.10.1 Friction at the contact interface 2.10.2 Adhesion at the contact interface 2.11 References

50 50 51 54

Chapter 3 Fundamentals of Rolling Motion 3.1 General features of rolling contact 3.2 Source of friction in rolling contact 3.3 Rolling friction force 3.4 Free rolling 3.4.1 Cylinder on a plane 3.4.2 Sphere on a plane 3.5 Material damping during rolling 3.6 Slip at the surface of contact 3.7 Internal friction 3.8 References

55 55 58 60 62 62 64 65 68 72 74

Chapter 4 Dynamic Characteristics of Rolling Motion 4.1 Introduction 4.2 Analytical evaluation of friction torque 4.2.1 Friction during rolling 4.2.2 Friction torque in the rolling contact 4.2.3 Total friction torque 4.2.4 Variable components of friction torque 4.3 Elastic and damping characteristics of the rolling contact 4.3.1 Static stiffness of the rolling contact 4.4 Dimensional accuracy and contact stiffness 4.4.1 Radial stiffness as a function of inaccuracies 4.4.2 Effect of variable dimensions and variable stiffness 4.4.3 Effect of waviness of raceways 4.5 Ball motion in a rolling contact bearing 4.5.1 Inertia forces and moments acting on the ball 4.5.2 Relative motions of the rolling elements 4.5.3 Friction at the contact interface 4.6 References

75 75 76 76 79 88 91 106 106 115 116

Chapter 5 Rolling Contact Bearings 5.1 Phenomenology of friction during rolling 5.2 Friction torque 5.2.1 Friction coefficient

145 145 150 152

121 122 125 125 129 135 143

Contents

5.3 Contact stresses and deformations 5.3.1 Contact between elastic bodies 5.3.2 Elastic deformations in bearings 5.3.3 Permanent deformations 5.4 Load distribution within bearings 5.4.1 Radial bearings 5.4.2 Thrust bearings 5.5 Kinematics of bearing elements 5.5.1 Rotational speed of the elements and the cage 5.5.2 Contact cycles due to rolling 5.6 Inertia forces 5.6.1 Centrifugal forces 5.6.2 Crankpin bearings 5.6.3 Forces of gyration 5.7 Load-carrying capacity 5.7.1 Dynamic capacity 5.7.2 Static capacity 5.7.3 Equivalent bearing loads 5.8 Lubrication of bearings 5.8.1 Elastohydrodynamic lubrication 5.9 References

ix

153 153 157 160 163 163 167 167 167 171 171 171 173 174 175 175 181 184 195 196 198

Chapter 6 Rolling Contacts in Land Locomotion 6.1 Rail–wheel systems 6.1.1 Traction at the rail–wheel interface 6.1.2 Braking process 6.1.3 Traction enhancing techniques 6.1.4 Consequences of wheel and rail wear 6.1.5 Ribbed tyre 6.2 Tyre–road interactions 6.2.1 Relationship between friction and traction 6.2.2 Characteristics of the traction 6.2.3 Analysis of dry road traction 6.2.4 Traction under wet conditions 6.2.5 Analysis of wet road traction 6.2.6 Practical approach to traction modelling 6.3 References

201 201 202 205 205 207 207 210 211 214 218 222 226 234 236

Chapter 7 Machine Elements in Rolling Contact 7.1 Contact of meshing gears 7.1.1 Peculiarities of contact between gear teeth 7.1.2 Geometry of contact between gear teeth

239 239 239 241

x

Rolling Contacts

7.2 Friction in meshing gears 7.2.1 Tooth losses 7.3 Outline of elastohydrodynamic theory 7.3.1 Estimates of film thickness 7.4 Application of elastohydrodynamic theory to gears 7.4.1 Film thickness between gear teeth 7.4.2 Operating temperature 7.4.3 Oil viscosity in relation to surface condition 7.5 Boundary contact in gear lubrication 7.5.1 Running-in process 7.6 Scuffing in meshing gears 7.6.1 Flash temperature as a criterion for scuffing 7.6.2 Phenomenon of scuffing 7.6.3 Probability of scuffing 7.7 Tooth face pitting 7.7.1 Fatigue fracture 7.7.2 Impact fracture 7.7.3 Tooth loading 7.8 Cam–follower system 7.8.1 Reciprocating engine cam 7.8.2 Analysis of the follower motion 7.8.3 Tangent cam with a roller follower 7.8.4 Camshaft torque 7.8.5 Convex cam with a roller follower 7.8.6 General case of a convex cam with a roller follower 7.8.7 Convex cam with a flat follower 7.8.8 Stresses within the cam–tappet contact 7.8.9 Lubrication of the cam–tappet contact 7.8.10 Design considerations 7.9 References Chapter 8 Non-metallic Rolling Contacts 8.1 General considerations 8.1.1 Approaches to polymer fatigue 8.1.2 Loading conditions in rolling contact 8.2 Phenomenology of polymer fatigue 8.2.1 Physical states of stressed polymers 8.2.2 Response to applied stress 8.2.3 Phenomenological description of fatigue

243 246 248 251 252 253 257 259 261 262 263 263 266 268 274 279 280 281 285 286 289 291 297 300 302 306 309 312 313 314 317 317 319 319 320 321 323 324

Contents

8.3 Behaviour of polymers in rolling contact 8.3.1 Characteristics of rolling contact conditions 8.3.2 Thermodynamic equilibrium in rolling contact 8.3.3 Mechanics of polymer rolling contact 8.3.4 Fatigue considerations 8.4 Model rolling polymer contact 8.4.1 Experimental setting 8.4.2 Kinematics of the model contact 8.4.3 Performance of some polymers in rolling contact 8.5 Technical ceramics in rolling contact 8.5.1 Ceramic materials 8.5.2 Ceramic bearings 8.5.3 Manufacture of silicon nitride balls 8.5.4 Dimensional quality of ceramic bearing components 8.5.5 Material quality 8.5.6 Surface quality 8.5.7 Failure modes of ceramics in rolling contact 8.6 References

xi

329 330 331 337 340 344 344 345 348 350 350 352 356 357 358 359 361 362

Chapter 9 Coated Surfaces in Rolling Contact 9.1 Introduction 9.2 Coating processes 9.2.1 Thermal spray 9.2.2 Electroplating 9.2.3 Physical vapour deposition 9.2.4 Chemical vapour deposition 9.3 Application of coatings to rolling contact elements 9.3.1 Rollers for steel forming 9.3.2 Rollers for papermaking and printing 9.3.3 Fracture of coatings during rolling 9.4 References

365 365 365 365 372 377 379 383 383 385 388 389

Chapter 10 Rolling in Metal Forming 10.1 Introduction 10.2 Forces acting in the contact region 10.2.1 Forces acting in the roll gap 10.2.2 Neutral point and no-slip angle 10.2.3 Expressions for the no-slip angle 10.2.4 Maximum value of the no-slip angle 10.2.5 Rolling when bar motion is impeded

391 391 392 392 395 397 398 399

xii

Rolling Contacts

10.3 Forward slip during rolling 10.3.1 Introduction 10.3.2 Expressions for forward slip 10.4 Friction between the rolls and the material 10.4.1 Role of friction in rolling 10.4.2 Friction in hot rolling 10.4.3 Friction in cold rolling 10.4.4 Evaluation of friction measuring methods 10.4.5 Coned compression tests 10.4.6 Friction coefficient variation along the arc of contact 10.5 Theories of metal rolling 10.5.1 Introduction 10.5.2 Equation describing the friction hill 10.5.3 Theory of von Karman 10.5.4 Simplification of von Karman’s equation 10.5.5 Modification of von Karman’s equation 10.5.6 Effect of front and back tension on the pressure distribution 10.5.7 Shortfalls of rolling theories 10.5.8 Theory of rolling by Orowan 10.5.9 Variations of the Orowan equation 10.6 Discussion of metal rolling theories 10.7 References Index

400 400 400 404 404 405 407 408 409 412 416 416 417 421 423 424 426 428 429 437 438 439 441


Series Editors’ Foreword Almost every aspect of industry and many fields of science and medicine require a practical understanding of tribology. The Tribology in Practice Series aims to be accessible and industry-relevant – bringing the latest developments in tribological research, as well as established ideas and techniques, to bear on real engineering problems. The Series Editors are pleased to introduce Rolling Contacts by T. A. Stolarski (Brunel University, UK) and S. Tobe (Ashikaga Institute of Technology, Japan) as the second title in this Series. This book is a rigorously researched monograph on rolling contacts covering all aspects of their use. It covers the fundamentals of rolling contacts of all types with particular attention paid to important engineering applications – including rolling bearings, gears, rail–wheel and road– tyre interactions, cam–tappet systems, as well as the roll-forming of materials. We are sure that this comprehensive treatment of the subject will be a valuable addition to the available literature and will prove useful to researchers, practising designers, and students. As a Series, Tribology in Practice is particularly concerned with design, failure investigation, and the application of tribological understanding to the products of various industries and to medicine. We are also including reviews of areas of research in the subject, which will be useful both for research workers and engineers in industry. The link with the IMechE through its publishing operation, Professional Engineering Publishing, is appropriate as tribology has its natural home within this branch of engineering. The scope of the series is as wide as the subject and applications of tribology. Wherever there is wear, rubbing, friction, or the need for lubrication, then there is scope for the introduction of practical, interpretative material. The Series Editors and the publishers would welcome suggestions and proposals for future titles in the Series. M J Neale Neale Consulting Engineers, UK T A Polak Neale Consulting Engineers, UK C M Taylor University of Leeds, UK

Preface Rolling friction is a very old problem in engineering and undoubtedly one of the most important from a practical point of view. According to estimates, the losses in the United Kingdom resulting from friction and wear related problems amount to £500 million annually. The availability of reliable, low-friction rolling contacts has become an important factor in the development of micro-machines and miniaturization in general. Many devices in high-precision engineering applications, such as magnetic storage and recording systems, miniature motors, laser scanners, machine tools for micro- and nano-level machining, and scanning microscope techniques, require bearings with extreme accuracy of motion. Undoubtedly, the modern ‘high-tech’ world depends upon and demands tribological systems of the highest quality. Despite this, many aspects of rolling friction are still not entirely understood, and research into mechanisms and processes governing the operation of rolling contacts at an atomic level is just starting to emerge. This book presents a general introduction to the fundamentals of rolling friction with the emphasis on important engineering applications of rolling contacts. Usually, a rolling contact is taken to be synonymous with a rolling contact bearing. This, however, is not necessarily true as there are a number of technologically important applications, such as gears, road–tyre and cam–tappet systems, and roll-forming of materials, where rolling contact configuration is at the heart of the matter. Analytical treatment of the topics discussed, wherever feasible, was considered to be of prime importance and, in the majority of cases, this was achieved. It is very much hoped that the procedures and techniques of analysis presented in this book will be found useful through improved understanding, selection, and design of rolling contacts for mechanical devices and systems. It is also hoped that the book will be seen as a comprehensive monograph on rolling contacts in all aspects of their utilization. Therefore, it should prove useful to practising designers, researchers, and postgraduate students. Students on engineering degree courses in universities should also benefit from this book, as it will give them an introduction to rolling contacts that are commonly used in engineering.

xvi

Rolling Contacts

Many years of research collaboration between the authors provided the inspiration to write this book. This was a natural progression, as the subject matter of the book is firmly rooted in their research interests. The material presented is grouped according to leading themes: sources of rolling friction, mechanics of contact between solid bodies, dynamics of rolling motion, land locomotion, rolling contact bearings, gears, the cam–follower system, non-metallic rolling contacts, coated surfaces in rolling contact, and rolling in the metal forming process. Chapter 1 presents a general introduction to the area of rolling contacts, with some information about the history of development of rolling contacts. Chapter 2 deals with elements of surface contact of solids. It is by no means a comprehensive treatment of the subject as there are specialist monographs that focus on contact mechanics. Nevertheless, the information contained in this chapter is considered to be adequate for proper understanding of the problems involved in contact between solids. Chapter 3 is an attempt to explain the origin and sources of friction during rolling motion. The dynamic characteristics of rolling motion, a topic frequently neglected in design practice, is discussed in Chapter 4. Rolling contact bearings are introduced in Chapter 5. As this is a topic dealt with in almost all books on tribology, it is treated with the assumption that a fairly good understanding of problems pertaining to rolling contact bearings already exists in the engineering community. Nevertheless, some topics such as inertia forces, although important in practice, are felt to be less publicized and therefore are presented more thoroughly. Chapter 6 is devoted to land locomotion where two applications of rolling contact are of prime importance, i.e. the rail–wheel system and tyre–road interaction. Both topics are treated with sufficient depth to allow understanding of the problems involved. Gears and cam–follower systems are commonly encountered machine elements where rolling motion plays an important role, and these are introduced in Chapter 7. Non-metallic rolling contacts, both polymeric and ceramic, are discussed in Chapter 8. In order to meet ever-increasing demands for better performance, surfaces in rolling contact are coated. Chapter 9 presents the various coating techniques available and investigates how coating can improve the performance of a rolling contact. Finally, Chapter 10 deals with rolling in metal forming – an important area of engineering. The authors would like to thank Ms Sheril Leich, Commissioning Editor, Professional Engineering Publishing, for facilitating the project and for having confidence in them.

Notation a A c E E′ Fe Fr G H l M n N p pm po R R′ t T Tf W V Y

width of contact area of contact radial clearance modulus of elasticity equivalent modulus of elasticity axial (thrust) load radial load shear modulus identation hardness contact length moment (torque) speed of rotation normal load contact pressure mean contact pressure maximum contact pressure radius of curvature equivalent radius of curvature traction tangential load flash temperature power velocity tensile yield strength

α β γ δ ε µ µr ν ρ

thermal conductivity asperity radius surface energy normal approach strain coefficient of friction coefficient of rolling friction Poisson’s ratio density

xviii

σ τ τ max φ (z) Ψ

Rolling Contacts

normal stress tangential stress maximum tangential stress distribution of peak heights plasticity index

Chapter 1 Introduction to Rolling Contacts

1.1 Historical perspective Technological progress usually leads to increasing demands on all fields of design and manufacture, and consequently on the design and application of contacts in relative motion supporting load. The solution of the diversified contact problems involved occasionally requires a detailed knowledge of friction and wear mechanisms and theories on the part of the designer. This knowledge is a part of an ever-growing area called tribology. The interdisciplinary nature of tribology, with knowledge drawn from different disciplines such as mechanical engineering, materials science, chemistry, and physics, leads to a general tendency for the chemist to describe in detail, for instance, lubricant additives, and for the mechanical engineer to discuss, for example, sliding journal bearings, with no overall guide to the subject. Also, it is difficult to find, in books dealing with tribology problems in general, a focused and advanced treatment of certain practically important topics in a comprehensive and thorough way. This is certainly true with rolling contacts – the subject of this book. It is quite probable that primitive man at a very early stage in the development of civilization discovered that it is far easier to move a heavy object over the ground by placing it on logs and rolling rather than sliding it. Though refuted by some scholars on the grounds of lack of positive archaeological evidence, it is difficult to see how the wheel with its axle and sliding bearings could have been developed other than from logs. It could thus be argued that experience of the rolling contact bearing predates that of the sliding bearing.

2

Rolling Contacts

The ancient Egyptians and Greeks are believed to have made effective use of the principle of the rolling contact, and a reference exists to a rolling bearing devised by the Greek Diades in 330 B.C. for a battering ram which incorporated the essential principles of a rolling bearing as made at the present time. Fragments of what appears to resemble a ball thrust bearing were found in Lake Nemi, Italy, in 1928 (1). It was speculated that it was used to support a rotatable statue and may have been made about 12 A.D. Leonardo da Vinci (2) studied, among other things, the differences between sliding and rolling, but this aspect of his work was not generally known until a publication that appeared in the late nineteenth century. On the British scene, an iron ball thrust bearing with many design characteristics of a modern bearing made its appearance about 1780 for use in a post mill in the Norwich area. A book published by Varlo in 1772 (3) describes a ball bearing he designed and fitted to his postchaise. British Patent 1580 was assigned to John Garnett of Gloucester in 1787 for interesting arrangements of various types of rolling element to form a bearing. In 1794, British Patent 2006 was granted to Philip Vaughan of Carmarthen for a radial ball bearing, the first of its kind on record. Important engineering developments of the rolling bearing continued in the early and middle part of the eighteenth century, but the main impetus that led to the foundation of the rolling bearing manufacturing industry came from the invention of the bicycle in Scotland in about 1840. In 1881, Heinrich Hertz (4) published in Germany his study on deformation of curved elastic bodies in contact, providing the growing industry with a mathematical theory that is used to the present time. Other than papers that have since been presented to learned bodies and other institutions, there appears to be very little literature in English that describes the development of the rolling bearing industry and provides information on the complex technology of design and production developed in the closing years of the nineteenth century and the first half of the twentieth century.

1.2 Practical use of rolling contact Though much study seems to have been devoted to rolling resistance, the basis of rolling itself seems to have been given comparatively little place in the literature. It may be helpful to understanding rolling motion if an ideal concept is visualized, a perfect cylinder on a perfect plane,

Introduction to Rolling Contacts

3

Fig. 1.1

both made of the same rigid, inelastic, frictionless material. If the cylinder is made to rotate about its own long axis, theoretically it might continue to rotate indefinitely. If pushed along the plane it might slide. If the coefficient of friction of the surface is then assumed to be raised by some means and the hypothetical experiment repeated, the cylinder should roll as soon as the value of static friction between roller and plane exceeds the value of the force previously applied to the cylinder, causing it to rotate or slide. The static friction may be regarded as a force acting in the opposite direction to the applied force, thus creating a couple. Since the same friction prevents it from sliding or rotating, it must roll. This is illustrated in Fig. 1.1. If the applied force is greater than the static friction the roller may rotate or slide as in the case of the locomotive wheel. The significance of the concept will not be altered if a sphere is substituted for the simpler case of the cylinder. If the case of a cylinder and plane both made of an elastic material such as steel is then considered, when the system is at rest, the metal in the contact area, as shown by Hertz, is deformed elastically under load (Fig. 1.2).

Fig. 1.2

4

Rolling Contacts

Application of a force tending to push the cylinder along the plane will cause it to start to roll owing to static friction. The displaced metal then forms waves preceding and following the rolling cylinder. Differences in the rate of recovery of the deformed area, because of elastic hysteresis, lead to an imbalance that produces a couple acting on the roller, causing it to continue to roll. This mechanism would theoretically apply even if the cylinder and plane were frictionless but elastic. In a rolling bearing, for example, the cylinder or sphere is not freely rolling as in the hypothetical case described but is in contact with two surfaces, one of which normally is stationary and the other, in motion, may be regarded as providing the force causing the rolling element to move. This, of course, is an oversimplification. However, what is often referred to as rolling friction is really rolling resistance, which will be considered later. The widespread use of the rolling contact principle in industry is, in large part, due to the lower power losses expected in rolling contacts compared with sliding contacts. However, in the practical situation, a significant amount of sliding motion occurs and it is important to consider this. The situation in the area of contact between rolling element and track is then examined first. Hertz was concerned with the deformation of curved elastic solids in dry contact and calculation of the stresses thereby created. Figure 1.2(a) shows the pattern of deformation and stress he proposed for a stationary cylinder resting on a flat surface, while Fig. 1.2(b) shows the change in the stress pattern created by rolling the cylinder under load. Some modification of the deformation pattern occurs when the surfaces are separated by a film of lubricant. Since steel is highly elastic, and provided the elastic deformation limit is not exceeded, recovery is almost instantaneous, but the rate varies to the extent that the elasticity is imperfect; other factors are also involved. The hysteresis effect produced by differences between deformation rate and recovery time accounts in large measure for resistance to rolling, while the repeated stress and relaxation cycles themselves in a rolling contact have the major influence on its fatigue life. A perfect cylinder stationary on a perfect plane has contact on a mathematical line if no load is applied. On the application of load, the projected area of contact is a rectangle increasing in size according to the load. Similarly, a perfect sphere would have point contact under noload conditions, but the projected deformation area would be a circle of size increasing with load. If a force acting parallel to the plane is applied to a cylinder or ball under loaded conditions, it will roll along the plane,


5

but the rolling cannot be perfect and a degree of slip must occur, producing some resistance to motion owing to increased friction. Osborne Reynolds (5) studied the nature of rolling even before he made his classic study of lubricated sliding. Using the simple system consisting of a cylindrical roller on a plane, Reynolds proposed a theory of rolling resistance due to microslip. In a more recent publication, Tabor (6) showed that the apparent slip observed by Reynolds was due to unequal stretching of the surfaces. Tabor also argued that resistance to pure rolling is due largely to the elastic hysteresis described earlier. Heathcote (7) demonstrated that any departure from mathematical straight line contact towards curvature introduces an element of slip arising from the variations in circumferential speed. In practice, this situation always applies even to a cylinder on a flat track, since neither is perfect in a mathematical sense. Any force operating to displace a ball or roller from its true rolling path will also cause slip. Surface finish inevitably affects the amount of friction when slip occurs and must therefore be included in the factors creating rolling resistance. Thus, it is possible to summarize the main components of resistance to rolling of the element on the track as: elastic hysteresis which will be affected by the properties of the materials; temperature; load and frequency of the stress relaxation cycle, that is, rolling speed; the shape and surface finish of the contacting surfaces; the effect of any deviation from the rolling path.

1.3 Need to lubricate the rolling contact 1.3.1 Lubrication in the contact area Taking the hypothetical case introduced earlier of an ideal system consisting of an inelastic true cylinder of perfect surface finish on a true plane of the same material, there would be mathematical line contact, no deformation, and rolling should take place without slip or wear. There must be sufficient static friction or interaction of surface force fields between the roller and plane to promote rolling. This might of course be inconsistent with perfect surface finish and the ideal system represents only a concept. In practice, all metal surfaces exhibit some surface roughness and, with exaggeration, the plane and roller may be conceived as resembling a crude rack and pinion (Fig. 1.3). If a lubricant is introduced at the contact and the roller is rotated on its own long axis, hydrodynamic lubrication may be established. For hydrodynamic conditions to be established, laminar flow must take

6

Rolling Contacts

Fig. 1.3

place within the lubricant film. Since the rolling element may be considered as being rotated by tangential forces from the two tracks transmitted through the lubricant film, slip must then occur in such a case, as a hydrodynamic film cannot transmit forces parallel to its motion without slip. Archbutt and Deeley (8) suggested that a hydrodynamic wedge is created between roller and track. Purday (9) made a mathematical study of hydrodynamic conditions which suggested they could exist between a roller and plane. Osterle (10), working with roller bearings and using Purday’s analysis, confirmed the existence of hydrodynamic conditions between roller and track. Smith (11), studying wear problems in roller bearings on the main shafts of aircraft gas turbines, demonstrated that rollers and cage could travel at different speeds. He found, by using a roller with magnetic inserts, that slip between the rotating inner race and the roller exceeded that between the roller and the fixed outer race. Sudden increase in radial load could cause breakdown of the hydrodynamic film with resultant wear if slip continued. It is possible that the somewhat unexpected experimental results reported by Fogg and Webber (12), working with cageless roller bearings at high speeds, may also be explained by postulating some form of hydrodynamic conditions both between rollers and tracks and between the rollers themselves. They found that the cageless bearings, which were made by removing the cage from a standard bearing and increasing the number of rollers, operated with lower torque and no increase in friction, compared with the standard caged bearing. Lubrication was by oil mist in both cases. The conclusions of Palmgren and Snare (13) about bearing behaviour in conditions of high speed and no load require some qualification in what must be taken as the normal case. They suggested that the regime changed from hydrodynamic to boundary lubrication when speed was reduced and load increased. However, since Grubin (14) published his work on the elastohydrodynamic theory of lubrication, the entire concept of lubricant behaviour in rolling contacts has changed.


7

1.3.2 Reasons for lubrication Before considering the significance of the elastohydrodynamic theory, it may be useful to review the reasons for lubricating rolling contacts at all. If, in the hypothetical simple case, rolling takes place in the dry state without deformation or sliding, and no wear occurs, lubrication is unnecessary. Even in a practical bearing, Goodman (15) claimed to have shown that introduction of lubricant into a well-designed rolling bearing increased friction. However, the potential advantage of lubricated rolling contacts over sliding contacts in respect of friction is illustrated by the following typical ranges of friction coefficient. Fully lubricated sliding bearing Lubricated rolling bearing

Static 0.1–0.3 0.002–0.005

Kinetic 0.001–0.005 0.0010–0.0018

Rosenfeld (16) stated that lubrication may greatly increase frictional resistance, but ball and roller bearings cannot operate for any considerable time without a lubricant. Rare exceptions may exist when it is expedient to operate a rolling bearing in a dry condition and the resulting wear is tolerable, but, in general, without lubrication the bearing life would be unacceptably short. The main reasons for lubricating rolling contact bearings can be summarized as follows: (a) to prevent metal–metal contact between races and rolling elements at points of sliding; (b) to eliminate any harmful effects of surface irregularities that cannot be completely removed even by the most careful polishing; (c) to support the sliding contact between the cage and the rolling elements and兾or race shoulder; (d) to carry away the heat developed in the bearing; (e) to protect the highly polished surfaces from corrosion. Thus, the main functions of a lubricant can be classified as lubrication, heat transfer, and protection. It has already been noted that some sliding takes place between the rolling element and the running track in commercial rolling bearings and, if diametral clearance and relaxation of load encourage it, a hydrodynamic regime is established. However, in normal loaded conditions the preponderant function of the lubricant in this area according to the elastohydrodynamic theory is the generation of a film between rolling element and track as a result of deformation. The shape of this is

8

Rolling Contacts

Fig. 1.4

shown schematically in Fig. 1.4 with the theoretical pressure distribution. In practice, as the pressure in this region is very high, possibly 1.5–3.0 GPa, the viscosity of this film is greatly increased. Therefore, to the extent that such a regime prevails in the contact area, it would seem likely that this film, momentarily of such high viscosity as to be comparable with dry static friction, transmits the elements of the couple which produces rolling. At the entry to the region the lubricant is under shear and may be regarded as behaving hydrodynamically, but within the contact zone negligible slip occurs. Clearly, the pressure–viscosity characteristics of lubricants are of importance in the behaviour of such elastohydrodynamic films. Usually, a close agreement between measured thickness of an oil film and predictions of the theory is found. Where anomalies are found, notably in the case of a silicone fluid and a solution of polymethylmethacrylate in oil, the possible effect of non-Newtonian behaviour is thought to be responsible. Where full hydrodynamic or elastohydrodynamic conditions can be maintained, the full fatigue life expectancy of the rolling contact may be achieved. If boundary conditions prevail even for part of the time, fatigue life may become unpredictable.

1.4 References (1) Cellini, B. (1949) Autobiography ‘The Life of Benevenuto Cellini’, (Phaidon Press, London).


9

(2) Da Vinci, L. Codie Atlanticus (British Museum Library, Milan). (3) Varlo, C. (1772) Reflections upon friction with a plan of the new machine for taking it off in wheel carriages, windlasses of ships etc., London. (4) Hertz, H. (1896) The Contact of Elastic Bodies (Macmillan, London). (5) Reynolds, O. (1875) On rolling friction. Phil. Trans. R. Soc., 166. (6) Tabor, D. (1954) The mechanism of rolling friction. Phil. Mag., 45. (7) Heathcote, H. L. (1920–21) The ball bearing in the making, under test and on service. Proc. Instn Auto. Engrs, 15. (8) Archbutt, L. and Deeley, R. M. (1927) Lubrication and Lubricants (Griffin, London). (9) Purday, H. F. P. (1949) Streamline Flow (Constable, London). (10) Osterle, J. F. (1959) On the hydrodynamic lubrication of roller bearings. Wear, 2. (11) Smith, C. F. (1962) Some aspects of the performance of high-speed lightly loaded cylindrical roller bearings. Proc. Instn Mech. Engrs, 176. (12) Fogg, A. R and Webber, J. S. (1955) The influence of some design factors on the characteristics of ball bearings and roller bearings at high speeds. Proc. Instn Mech. Engrs, 169. (13) Palmgren, A. and Snare, B. (1957) Influence of load and motion on the lubrication and wear of rolling bearings. In Proceedings of IMechE Conference on Lubrication and Wear, London. (14) Grubin, A. N. and Vinogradova, I. E. (1994) Book No. 30 (Central Scientific Research Institute for Technology and Mechanical Engineering, Moscow). (15) Goodman, J. (1912) Roller and ball bearings. Proc. Inst. Civ. Engrs, 189. (16) Rosenfeld, L. (1942) Friction of ball and roller bearings. Instn Auto. Engrs paper 1942兾6.


Chapter 2 Elements of Surface Contact of Solids

2.1 Introduction This chapter deals with the stresses and deformation resulting from the contact of the surfaces of two solid bodies. Usually, conforming and non-conforming contacts can be distinguished. A contact is defined as conforming when the surfaces of the two bodies fit exactly together without deformation. An example of a conforming contact is a journal bearing and a thrust bearing. A non-conforming contact is formed by bodies that have different profiles. Depending on the overall contact geometry, they will form a point contact or a line contact. A ball bearing represents the case of point contact because the ball makes point contact with both raceways. On the other hand, in a roller bearing, the roller makes line contact with raceways. In general, line contact is created when the profiles of the bodies are conforming in one direction and non-conforming in the perpendicular direction. The area of nonconforming contact is usually small compared with the dimensions of the bodies in contact. The stresses are highly concentrated in the region close to the contact zone and are not significantly influenced by the shape of the bodies at a distance from the contact area. In engineering applications, the points of surface contact are quite often executing complex motions and are required to transmit both forces and moments. For instance, the point of contact between a pair of gear teeth moves in space, while, at the same time, the two surfaces move relative to each other at that point and the motion combines both

12

Rolling Contacts

rolling and sliding. In this chapter a frame of reference will be defined in which the motions and forces that arise in any particular circumstances can be generalized. In this way, the problems of contact can be formulated and studied independently of any application context. This approach also facilitates the application of the results of such studies to the wide variety of engineering problems.

2.2 Distribution of stresses within the contact zone It should be stated right at the beginning that the stresses acting within the outermost layers of material (typically up to a depth of 1 mm or so) will mainly be considered here. The effects at several centimetres below the surface are of only secondary importance. For that reason, it is permissible to treat the surfaces as though they represent the surface of bodies of infinite depths; i.e. they may be considered as semi-infinite bodies. This approach facilitates the analysis of details of the surface contact of solids rather than considering their overall geometrical shape and thus leads to considerable mathematical simplification. Stresses produced by a single normal line load P per unit length in the plane xz and applied at a point O′ and defined by the coordinates ε , O on the surface (zG0) of a semi-infinite body and having the same value for all values of y are shown in Fig. 2.1(a). The elastic stress field in the plane xz is easily obtained. Considering a unit length in the y direction, the radial stress σ r is given by 2P σ r G− (2.1) cos θ πr as the tangential stress σ θ and the shearing stress τ rθ are equal to zero. The case depicted in Fig. 2.1(a) represents a state of simple radial compressive stress. The stress increases with decreasing radius r and decreasing angle θ . The use of the two-dimensional Mohr’s circle of stress, shown in Fig. 2.1(b), for these stresses gives the resulting Cartesian stresses with respect to O′ σr 2P zx2 σ x G (1Acos 2θ )Gσ r sin2 θ G− 2 π (x2Cz2)2

冤

冥

2P σr z3 (1Ccos 2θ )Gσ r cos2 θ G− 2 π (x2Cz2)2 2P σr z2x τ xz G sin(2θ )Gσ r sin θ cos θ G− 2 π (x2Cz2)2

σzG

冤冤

冥冥 (2.2)

Elements of Surface Contact of Solids

Fig. 2.1

13

14

Rolling Contacts

Equations (2.2) can be written with respect to the origin O as

σ X G−

2P

σ Z G−

2P

冦

Z(XAε )2

冦

Z3

冧

π [(XAε )2CZ 2]2

冧

π [(XAε )2CZ 2]2

τ XZ G−

Z 2(XAε ) 2P π [(XAε )2CZ 2]2

冦

冧

(2.3) In a similar way, the stresses due to a single tangential line T acting at O′ (Fig. 2.2) can be obtained 2T cos θ ′ σ r G− πr σ θ ′ Gτ rθ ′ G0 (2.4) and Z 2(XAε ) 2T π [(XAε )2CZ 2]2

冦冧 (XAε ) 2T G− 冦 π (XAε ) CZ ] 冧 Z(XAε ) 2T G− 冦 π [(XAε ) CZ ] 冧

σ X G−

3

σZ

2

2 2 2

τ XZ

2

2 2

(2.5)

Fig. 2.2


15

Taking into account the fact that TGµP, where µ is the appropriate friction coefficient, and adding the stress components due to P and T at any point (x, y), the stress distribution arising in a simple frictional contact can be obtained. Examination of equations (2.1) and (2.4) reveals that at O′ (rG0) the stresses are infinite which in practice is unacceptable. This is due to the assumption that the load acts at a single point, i.e. over zero contact area. In reality, there is always some finite area of contact associated and this changes the formulation of the initial problem. Figure 2.3 shows a uniformly distributed load producing a contact pressure p over the region O–a on the surface (zG0) of a semi-infinite body. Taking the length along the y direction to be equal to unity, the external load itself is given by a

冮 p dxGpa

PG

0

In the case of a very small load p dε at some point defined by the coordinates (ε , O), it is possible to find the stress at any point (X, Z) that is caused by this load using equations (2.3). In this case, P will be replaced by p dε . The total stress at a point X, Z that is due to the distributed load P is then obtained by the summation of the effects of all the p dε loads acting at different values of ε from O to a, or, in

Fig. 2.3

16

Rolling Contacts

mathematical terms

σ X G− σ Z G−

冮冦[(XAε ) CZ ] 冥 dε 2

2 2

0

a

冮冦[(XAε ) CZ ] 冧 dε

2p π

τ XZ G−

Z(XAε )2

a

2p π

Z3 2

2 2

0

2p π

Z 2(XAε )

a

冮冦[(XAε ) CZ ] 冧 dε 2

2 2

0

(2.6) If a tangential load TG µP acts over the region O–a (Fig. 2.4), then at every point it follows that t dxGµp dx, and thus

冮

a

TG

a

冮 µp dxGµP

t dxG

0

0

Using equations (2.5) for each elemental tangential load t dε acting on element dε (O, ε ), the stresses at any point (X, Z) that are due to the total distributed load T can be obtained

σ X G−

2t π

2t σ Z G− π

τ XZ G−

a

Z 2(XAε )

冮冦[(XAε ) CZ ] 冧 dε 2

2 2

0

a

(XAε )3

冮冦[(XAε ) CZ ] 冧 dε

2t π

2

2 2

0

a

冮冦 0

Z(XAε )2 dε [(XAε )2CZ 2]2

冧

(2.7)

Fig. 2.4


17

The total stresses for a sliding contact subjected to a normal load P uniformly distributed over the contact region O–a are given as the sum of the stresses defined by equations (2.6) and (2.7). It is obvious that the solutions for the normal and tangential point load may be used to obtain the resultant stress distribution for any type of load over the contact region. In the above solutions, the basic assumption is that of elastic behaviour of the bodies in contact. However, in reality the possibility of plastic effects must be taken into account. The simplest criterion defining the onset of plastic deformation assumes that this occurs when the maximum shear stress attains the critical level, k, for the material, where kGY兾2 (Y denotes the tensile yield stress). Under plane strain conditions, the maximum shear stresses always occur in the xz plane. The maximum shear stress in this plane is simply the radius of the Mohr’s circle of stress, which is shown in Fig. 2.5. Thus

τ max G

P σr G− cos θ 2 πr

By drawing a circle of diameter b in the way shown in Fig. 2.5(a), it can be found that rGb cos θ and

τ max G−

P πb

which means that the stress remains constant at all points on the circle. It is therefore beneficial to plot the stress distribution as isochromatics

Fig. 2.5

18

Rolling Contacts

or lines of constant τ max . It is then possible to determine the location at which τ max will reach its limiting value of k, i.e. the location of the onset of plastic deformation. These diagrams are useful since they also indicate the pattern of isochromatics obtained in photoelastic stress studies. In the case of a point normal load and a uniformly distributed normal load, analysis of τ max produces the pattern of isochromatics shown in Fig. 2.5(b) and (c). It is clear that in both cases the material will first reach a yield condition at the surface where increasing load gives τ max Gk, which denotes the yield strength of the material.

2.3 Deformations resulting from contact loading The logical step from assessing the contact stresses is to examine the displacements in a solid using the known relations between stress and strain. Thus, for a single normal load P acting at O′ [Fig. 2.1(a)], the horizontal and vertical displacements u and w respectively are ∂u

1 2P Ger G (σ rAνσ θ )G− cos θ ∂r E πrE

u ∂w 2P 1 C cos θ Geθ G (σ θ Aνσ r )Gν r r ∂θ E πrE r

∂u ∂w w 1 C A Gγ rθ G τ rθ G0 ∂θ ∂r r G

The solution of the above equations requires information on the boundary conditions. For this, it can be assumed that points on the z axis, i.e. at θ G0, have no lateral displacements and that at a point on the z axis at a distance b from the origin there is no vertical displacement. The displacements occurring at the boundary zG0 are of interest. Thus, by putting θ GJπ兾2 in the solution of the above equations it can be shown that the horizontal displacement is given by (u)z G0 G−

(1Aν)P 2E

(2.8)

This indicates that at all points on the boundary of the solid there is a constant displacement directed toward the origin. Also, it is possible to find the vertical displacement of a point on the boundary zG0 at a distance x from the origin (w)z G0 G

2P b (1Cν)P log A πE x πE

(2.9)


19

At the point of load application (xG0) the displacement in the vertical direction tends to infinity. This is the result of the assumption of a point load which, in reality, is not valid as the load is usually distributed over a small finite area. If the load is distributed over the region O–a, as shown in Fig. 2.3, giving rise to a constant pressure p, the vertical displacement at any point with coordinates X, O produced by an element of load p dε at a distance ε from point O is known from equation (2.9). It can be estimated by substituting p dε for P and (XAε ) for x so that the total displacement at point (X, O) is given by (w)z G0 G

2 πE

冮

a

0

p log

b XAε

dε A

(1Cν) πE

a

冮 p dε

(2.10)

0

All the above solutions are valid for two-dimensional problems only. The three-dimensional problems are far more complex and their detailed treatment can be found in standard books on the theory of elasticity.

2.4 Contact between bodies of revolution The contact between bodies whose geometry is defined by circular arcs is essential for rolling contacts. Hertz (1) was the first to solve this type of problem for elastic bodies and, for that reason, this contact is known as Hertzian contact. Figure 2.6 illustrates the contact of two identical cylinders under conditions of plane strain. Utilizing the symmetry of the contact, it can be

Fig. 2.6

20

Rolling Contacts

argued that the zone of contact is created by compression of the cylinders to generate a straight line, i.e. to produce a plane contact zone [Fig. 2.6(a)]. Although this is not strictly true for a cylinder in contact with a plane, the error is small and can be neglected. Thus, a plane contact zone may be assumed. When two identical elastic cylinders are in contact under a normal load P per unit axial length, the resulting plane contact zone has a width of 2a [Fig. 2.6(b)]. The normal deformation at the centre of the contact zone is greater than at the extremities, and the contact pressure distribution p is given by pG

2P πa

1冢

x2 1A 2 a

冣

(2.11)

Using a simple physical argument, it can be shown that Stress T(P兾a) Considering the deformation, it is justified to say that increasing the load increases a and, thereby, increases the strain. Thus Strain T(a兾R) where R is the radius of the cylinder. From the above two relations

冢冣

a P TE a R or a2T

PR E

The analytical solution for this case gives 2

a G

4PR(1Aν 2 ) πE

(2.12)

The solution defined by equations (2.11) and (2.12) is approximately true for other identical cylinders, i.e. plane contact geometries. Provided that the angle subtended by the contact width at the centre of the cylinder is less than 30°, the results may be used for other contact geometries


21

Fig. 2.7

shown in Fig. 2.7 by using the equivalent modulus of elasticity E′ and the equivalent radius of curvature R′. Thus 1 E′

G

1Aν 21 1Aν 22 C E1 E2

G

1 1 C R1 R2

and 1 R′

and finally a2 G

4PR′ πE′

In the case of contact between a cylinder and a plane, the radius of the plane is taken as infinity. Therefore, R′ becomes the radius of the cylinder only and for concave curvatures the radius is taken as negative. It is important to note that when E → S, the solids become rigid, resulting in a single point contact where a → 0.

2.4.1 Stress distribution within the contact It was argued earlier in this chapter that the onset of plastic deformation may be associated with the maximum shear stress reaching a critical value k. Therefore, it is of practical importance to examine the distribution of the maximum shear stress for a contact loaded by the pressure profile given by equation (2.11) and acting over the region from Aa to Ca. Using equations (2.6) for elemental loads p dε and integrating for the actual distribution of P will result in the Cartesian

22

Rolling Contacts

stress distribution within the body in contact. Thus 0

σ X G−A

冮冦 B

−a

Z(XCε )2 dε AA [(XCε )2CZ 2]2

冧

0

σ Z G−A

冮冦[(XCε ) CZ ] 冧 Z3

B

2

−a

0

τ XZ G−A

2 2

Z 2(XCε ) 2

−a

2 2

冮冦 B

0

a

dε AA

冮冦[(XCε ) CZ ] 冧 B

a

Z(XAε )2 dε [(XAε )2CZ 2]2

冧

冮冦[(XAε ) CZ ] 冧 dε B

Z3 2

2 2

0

a

dε AA

Z 2(XAε )

冮冦[(XAε ) CZ ] 冧 dε B

2

2 2

0

(2.13) where AG

4P π 2a

1冢

BG

ε2 a2

冣

1A

The maximum shear stress for plane strain conditions is given by the radius of Mohr’s stress circle, i.e.

1冤冢

τ max G

σ XAσ Z 2 Cτ 2XZ 2

冣

冥

(2.14)

where σ X, σ Z, and τ XZ are defined by equations (2.13). Therefore, equation (2.14) defines the values of τ max at all points. Equation (2.14) can be used to draw the isochromatics, from which it can be seen that the greatest value of τ max occurs below the surface at a distance of 0.67a. Besides, as the load is increased, τ max at this point also increases, attaining the value k when the maximum pressure at the centre of the contact zone po is 3.1k. This is because the surface elements are subjected to compressive stresses in all three orthogonal directions, allowing po to reach a value greater than 2k without producing yield. This is an important result since it means that contact pressures in excess of the yield value for the material do not result in plastic deformation. Thus, higher loads than might have been expected can be supported elastically within Hertzian contacts. In addition, even if yielding has taken place below the surface, very little plastic deformation takes place on the surface itself because the plastic zone is constrained by elastic material on all sides.


23

With the further increase in load, the plastic zone also increases in size and ultimately spreads to the surface of the body. Plastic flow may then occur quite readily and the cylinder will indent the surface of the body. This happens when the mean contact pressure pm is about 6k, i.e. more than twice the contact pressure at which initial yield occurred. The mean pressure under these conditions is essentially the indentation hardness value of the material, H, which is why for metals the following is applicable H⯝6k⯝3Y where Y is the material uniaxial tensile yield strength. Another case of contact loading that is important in practice concerns the combined action of a normal load, P, and a tangential load, µP. Furthermore, it is obvious that at all points within the contact zone the tangential traction is given by tGµp. Combining the stress distribution due to the normal and tangential loads and calculating the values of τ max leads to the isochromatics. When the pattern of isochromatics is plotted it will be seen that the location of the greatest value of the maximum shear stress is now much nearer the surface. Thus, plastic deformation can take place more readily than in the previous case. In practical terms it means that macroscopic plastic deformation is facilitated by the presence of friction traction.

2.4.2 Contact with combined normal and tangential loads Contact conditions under which bodies are subjected to tangential loads less than µP so that macroscopic sliding does not occur are quite frequently encountered in engineering applications. This takes place in situations where friction is used as the mechanism for preventing slip between mating components, for instance, nuts and bolts, interference fits, and friction devices such as clutches. This mechanism will be explained by considering a cylinder pressed against a plane and loaded by a tangential load less than µP. Under such loading conditions there is a central area within the contact zone in which no slip occurs, while at the two extremities a small degree of slip takes place as depicted in Fig. 2.8. The coexistence of a zone of sticking and zones of microslip is possible because of the deformable nature of the materials in contact and the deformation pattern being such as to allow slip at the extremities of the contact zone. As the value of T increases, the areas of microslip increase until, when TGµP, they meet at the centre of the contact and microslip occurs over the whole contact zone. It is possible for µ to have a constant value wherever slip takes place, that is, within the

24

Rolling Contacts

Fig. 2.8

slip regions tGµp while within the stick region tFµp. Since T is the integral of t over the contact zone, this can satisfy the requirement of the problem with µ always having a constant value. The case of TGµP is the next contact situation to be considered. Increasing the normal load induces equal compression strains ε x in both bodies so that no slip occurs owing to this effect. With the tangential load, on the other hand, slip must occur throughout the contact zone since the load must be acting in opposite directions on the two bodies in contact. It must be concluded that, even when no macroscopic motion takes place, some degree of microslip exists when TFµP and this gives rise to a phenomenon known as fretting. For more complicated contact geometries these arguments are still qualitatively valid and microslip will occur at the extremities of the contact zone.

2.4.3 Three-dimensional contact Many engineering applications of rolling contacts involve more complicated three-dimensional problems. In general, the patterns of behaviour are similar to two-dimensional contacts, but some of the previously introduced expressions must be modified. If two identical spheres are brought into contact under a normal load N (Fig. 2.9), the area of contact will be a plane circle of radius a and the pressure distribution will be of hemispherical form and is given by pG

3N 2πa2

1冢

冣

x2 z2 1A 2A 2 a a

(2.15)


25

Fig. 2.9

The magnitude of a is given by

1冢 2E′ 冣

aG

3NR

3

(2.16)

The contact of two dissimilar spheres does not result in a plane circular contact area, and the results given by equations (2.15) and (2.16) still hold with substantial accuracy. The contact area radius is defined by

1冢 4E′ 冣

aG

3NR′

3

(2.17)

where R′ is related to the radii of contacting spheres R1 and R2 by 1 R′

G

1

1 C R1 R2

In the case of contact between a sphere and a plane, R′ is equal to the radius of the sphere as the radius of the plane is taken to be infinity. Consider the contact of two bodies 1 and 2 whose geometry is defined by the principal radii of curvature of each body in two orthogonal

26

Rolling Contacts

Fig. 2.10

planes as shown in Fig. 2.10. The area of contact is now elliptical in shape and the contact pressure distribution is given by pG

3N 2πab

1冢

冣

x2 z2 1A 2A 2 a b

(2.18)

The size of the contact ellipse is defined by the semi-major and semiminor axes a and b as follows aGka

bGkb

1冢 1冢 3

3

冣

(2.19)

冣

(2.20)

3N 4E′(ACB) 3N 4E′(ACB)

where ka and kb are constants depending on the values of the principal curvatures of the contacting bodies and on the angle φ between the normal planes that contain these curvatures. If the principal radii of curvature of body 1 are denoted by R11 and R12 , the principal radii of curvature of body 2 are denoted by R21 and R22 , and then constants A and B are found from BAAG12 1(C 2CD 2C2CD cos 2φ ) ACBG

冢

1 1 1 1 1 C C C 2 R11 R12 R21 R22

冣

where CG1兾R11A1兾R12 and DG1兾R21A1兾R22 .

(2.21) (2.22)


27

In the above expressions a concave curvature is taken as negative. Coefficients ka and kb in equations (2.19) and (2.20) are numbers depending on the ratio (BAA)兾(ACB) and they can be found by introducing an auxiliary angle γ defined as cos γ G

BAA ACB

With the help of equations (2.21) and (2.22), the value of γ can be easily obtained. In order to determine the values of ka and kb corresponding to a certain value of γ , quite complicated numerical calculations involving elliptical integrals are required. Figure 2.11 shows typical results of such calculations. The assumption of a plane area of contact is no longer valid for complicated geometries. While the pressure distribution and the size of the contact as determined from the Hertz theory are generally correct, sometimes there is a need to know the actual shapes of such contacts. For materials having the same elastic properties it is sufficient to assume that the deformed surface, which has some common radius R c , is about mid-way between the two original surfaces as shown in Fig. 2.12. The value of the common radius of curvature is given by RcG

2R1R2

(2.23)

R1AR2

Fig. 2.11

28

Rolling Contacts

Fig. 2.12

Obviously, for two identical spheres in contact the above equation gives the expected result of a plane contact area. The radius is taken to be negative where concave curvatures occur. It is necessary for the analysis of contact to define the normal approach of a sphere owing to the application of normal load and the consequent deformation. Figure 2.13 depicts the contact of a sphere and a plane. It can be seen that the separation u of the surfaces at a distance r from the centre of the contact zone is given by uGRA1(R2Ar 2)GRAR

1冢

1A

冣

r2 r2 GRARC A· · · R2 2R

Fig. 2.13


29

If r is small compared with R, then uG

r2

(2.24)

2R

The normal approach is defined as the distance over which points on the two bodies remote from the deformation zone move together on application of a normal load. The reason for that is the flattening and general displacement of the surface within the deformation region. If a is the radius of the contact zone and w is the displacement of the sphere at the boundary of this zone, then the normal approach δ will be given by a2

δ GuCwG

Cw

2R

(2.25)

At the centre of the contact zone, δ is given by the degree of deformation and it is therefore justified to assume that the normal approach will be proportional to the flattening of the sphere. Thus

δT

a2 R

With the help of equation (2.17)

1冢 3

aT

冣

NR E′

so that

δT

1冢 3

N2

冣

E′2R

The exact solution gives

1冢

δG

3

冣

9N 2 16E′2R

or finally NG43 E′δ 1(Rδ )

(2.26)

Combining equations (2.17) and (2.26), the area of contact, A, is given by AGπa2 Gπ δ R

(2.27)

30

Rolling Contacts

Equation (2.27) indicates that the surface outside the contact region is displaced in such a way that the actual area of contact is only one-half of the geometrical area, which is equal to 2π δ R.

2.5 Contact of real surfaces All engineering surfaces are rough, and surface finishing processes available nowadays can substantially reduce the level of roughness but cannot eliminate it altogether. It is convenient to consider a simplified contact between a single rough surface with a perfectly smooth plane, as the result from such an approach is then reasonably indicative of the effects to be expected from the contact of real surfaces. Moreover, the problem will be simplified further by introducing a theoretical model for the rough surface in which the asperities are represented by spherical segments so that their elastic deformation characteristics may be described by the Hertz theory. Also, it is assumed that there is no interaction between individual asperities; i.e. the displacement due to a load on one asperity does not influence the heights of the neighbouring asperities. Figure 2.14 shows, schematically, a surface of unit nominal area composed of an array of identical spherical asperities all of the same height z with respect to some reference plane XX′ As the smooth plane moves towards the rough surface as a result of the application of load, the normal approach will be given by (zAd), where d is the current separation between the smooth surface and the reference plane. It is apparent that each asperity is deformed equally and carries the same load Ni so that for η asperities per unit area the total load will be equal to η Ni . For each asperity, the load Ni and the area of contact Ai are known from the Hertz theory [see equations (2.26) and (2.27)]. Thus, if β is the

Fig. 2.14


31

asperity radius, then Ni G43 E′β 0.5 (zAd)1.5 and Ai Gπ β (zAd) and the total load is given by 4

NG η E′β 3

0.5

冢π β 冣 Ai

1.5

Thus, the load is related to the total real area of contact AGη Ai by NG

4E′ 3π βη 0.5 1.5

A1.5

(2.28)

Equation (2.28) indicates that the real area of contact is related to the two-thirds power of the load when the deformation is elastic. In the case of a load causing plastic deformation of asperities characterized by a constant flow pressure H, which is closely related to the hardness, it is assumed that the displaced material moves vertically down and does not spread horizontally. In this way, the area of contact A′ will be equal to the geometrical area 2π βδ . The load on an individual asperity N′i is then given by N′i GHA′i G2πHβ (zAd) Therefore N′Gη N′i Gη HA′i GHA′G2HA

(2.29)

which means that the real area of contact is linearly related to the load. All engineering surfaces have asperity peak heights distributed in a probabilistic way. Therefore the surface model introduced earlier must be modified accordingly and the analysis of contact between real surfaces has to include a probability statement as to the number of asperities in contact. Assuming that the separation between the smooth surface and the reference rough plane is d, then there will be contact at any asperity whose height was originally greater than d as shown in

32

Rolling Contacts

Fig. 2.15

Fig. 2.15. If φ (z) is the probability density of the asperity peak height distribution, then the probability that a particular asperity has a height between z and zCdz above the reference plane will be φ (z) dz. Thus, the probability of contact for any asperity of height z is given by

冮

S

φ (z) dz

prob(zHd)G

d

Considering a unit nominal area of surfaces containing η asperities, it is not difficult to show that the number of contacts created, n, is equal to nGη

冮

S

φ (z) dz

(2.30)

d

The total area of contact is given by AGπ ηβ

冮

S

(zAd)φ (z) dz

(2.31)

d

and the expected load is defined by the following expression 4 NG ηβ 0.5E′ 3

冮

S

(zAd)1.5φ (z) dz

(2.32)

d

It is common practice to express the above equations in terms of standardized variables, i.e. hGd兾σ and sGz兾σ , where σ is the standard


33

deviation of the peak height distribution of the surface. Therefore nGη Fo (h) AGπ ηβσ F1(h) NG43 ηβ 0.5σ 1.5E′F3/2 (h) where F3兾2(h) is a statistical function which, in general terms, is given by

冮

S

Fm(h)G

(sAh)mφ *(s) ds

h

In the above expression, φ *(s) is the probability density function standardized by scaling it to give a unit standard deviation. In cases where the asperity obeys the plastic deformation mode, equations (2.31) and (2.32) are modified to assume the following forms A′G2π ηβ

冮

S

(zAd)φ (z) dz

(2.33)

d

N′G2π ηβ H

冮

S

(zAd)φ (z) dz

(2.34)

d

It is apparent that the load is linearly related to the real area of contact by N′GHA′ and this result is independent of the height distribution φ (z). The analysis presented above was based on a theoretical model of a rough surface. An alternative approach to the problem is to use actual surface roughness profilograms and obtain from them the surface bearing area curve. In the absence of asperity interaction, the bearing area curve provides a direct method for determining the area of contact at any given normal approach. Thus, if the bearing area curve is denoted by ψ (z) and the current separation between the smooth surface and the reference plane is d, then for a unit nominal surface area the real area of contact is given by

冮

S

AG

d

ψ (z) dz

(2.35)

34

Rolling Contacts

In the case of plastic surface deformation, the total load on the contact is NGH

冮

S

ψ (z) dz

(2.36)

d

It is apparent from the analyses presented above that the relationship between the real area of contact and the load will be dependent on both the mode of deformation and the distribution of the surface profile.

2.6 Criterion for deformation mode In most real contact situations the higher asperities are deformed plastically while the lower asperities are still elastic. Therefore, a mixed elastic–plastic system exists in most real contacts. Usually the greater the load, and hence the increase in the normal approach, the greater is the number of plastic asperity contacts. Thus, the normal approach can be regarded as an indicator of the extent of plastic deformation within the contact area. Taking into account equations (2.26) and (2.28), it is possible to define the mean pressure, pm, for the case of an elastic asperity contact as pm G

4E′δ 0.5 3π β 0.5

or

δ 0.5 G

3π β 0.5pm 4E′

(2.37)

The transition from a purely elastic contact to a completely plastic contact takes place over a range of loading for a contact between two spheres. Plastic deformation is initially located under the surface when the maximum contact pressure is 3.1k or the mean pressure is approximately equal to Y. The extent of plastic deformation becomes macroscopic when the mean contact pressure is about 3Y; i.e. it is equal to the hardness of the material. Thus, from equation (2.37) it can be seen that the transition from elastic to fully plastic behaviour occurs in a range of values of δ 0.5, and the initial deviation from elastic behaviour occurs when pm GH兾3, where

δ 0.5 G0.78

β 0.5H E′

冢

冣


35

The transition from elastic to fully plastic behaviour is gradual, and therefore a transition point can be assumed at

δ 0.5 ⯝

H 0.5 β E′

It is useful to normalize the above expression by dividing both its sides by σ 0.5

δ σ

冢冣

(δ *)0.5 G

0.5

G

H β

0.5

冢冣

E′ σ

where σ is the standard deviation of asperity heights. The left-hand side parameter decreases as the surface roughness, defined by σ , increases. It is common to name the inverse of (δ *)0.5 as the plasticity index

ΨG

E′ σ H β

0.5

冢冣

(2.38)

The plasticity index is indicative of the onset of plastic deformations within the contact zone. It is large when the contact is basically plastic, and small, less than unity, when the contact is essentially elastic.

2.6.1 Surface plastic deformations In this section, a heavy, rigid roller moving steadily over the surface of an elastic–plastic half-space is considered. Resulting deformations of the roller are assumed to be in plane strain. The elastic properties of the half-space are homogeneous and isotropic, and the plastic deformation is assumed to be in-plane shear on planes normal to the direction of rolling. The shear is taken to be of constant magnitude γ within a surface layer of thickness h, extending backwards through the rolled zone from the leading point of contact between the roller and the surface. In a Cartesian coordinate system (x1 , x2 , x3) the half-space is defined as x2‚0 and is occupied by an elastic–plastic body, isotropic and homogeneously elastic and subject to conditions of plane strain normal to the x3 axis. The surface of the half-space is loaded by a rigid roller assumed to move steadily in the positive direction of the x1 axis, leaving behind it a plastically deformed layer of material with thickness h. Owing to the assumed steadiness of the movement, the plastic deformation, when checked behind the roller, will be constant at a given depth under the surface.

36

Rolling Contacts

In front of the surface layer there will be a region in which plastic deformation takes place. It is assumed, tentatively, that this region is a narrow slip band emerging from the leading point of contact perpendicularly to the surface. Removal of the material in the slip band would leave the unloaded body with displacements v1 Gv 1s (x2)

(2.39)

where v2 G0 for ASFx1Fb and Ah‚x2 ‚0, and elsewhere v1 Gv2 G 0. Here, b is the coordinate in the x1 direction of the slip band. In particular, v1s (−h)G0, and so the body is left stress free. To calculate the deformation of the surface in the actual body that is due to plastic straining, let gλ α (x1At, x2 ) be the xα direction displacement of the point (x1 , x2 , x3 ) of the body when deforming a purely elastic material to a unit-concentrated line load acting in the xλ direction at (x1 , x2 )G(t, 0). The Greek subscripts range over the values 1 and 2. Also, let Gλ α β (x1At, x2 )G−

2 lλ lα lβ πr

(2.40)

be the in-plane stress of the body thus loaded. Additionally rG1[(x1At)2Cx22] and 1 lα G (x1At, x2 )G(cos θ , Asin θ ) r denote the distance between the points (x1 , x2 ) and (t, 0). In order to restore the body to its state prior to removal of the slip band, it is necessary to subject the flanks at x1 Gb− and x1 Gb+ to tractions T −α (x2 ) and T +α (x2 ) respectively, with T −α (x2)CT +α (x2 )G0

(2.41) − α

In this process, the flanks will experience displacements v (x2 ) and v +α (x2 ) respectively. The tractions are determined by the conditions v −1 (x2 )Cv 1s (x2)Gv +1 (x2 ) v −2 (x2 )Gv +2 (x2 ) This loading will give rise to fields of displacements vα (x1 , x2 ) and residual stresses σ α β (x1 , x2 ) satisfying the following conditions: σ α β ,β G0 for x2 ‚0 and σ α 2 G0 for x2 G0.


37

Utilizing the reciprocity relations

冮

−

S AC

s

Gγ α β (x1At, x2 )vα ,β (x1 , x2) dS

冮

G

S −AC s

gγ α ,β (x1At, x2 )σ α β (x1 , x2 ) dS

In the above expression, S − is the region x2 ‚0, taken up by the body, and C s denotes the slip band. Application of the divergence theorem is admissible for the region S −AC s. Thus

冮

0

Gλ 11 (bAt, x2 )v 1s (x2 ) dx2

vλ (t)G

(2.42)

−h

Here, vλ (t)Gvλ (t, 0) are the surface displacements. However, the derivative v′λ (t) rather than vλ itself is required. Therefore

冮

0

Gλ 11,1 (bAt, x2 )v 1s (x2 ) dx2

v′λ (t)GA

−h

冮

0

冮

0

Gλ 12,2 (bAt, x2 )v 1s (x2 ) dx2

G

−h

Gλ 12 (bAt, x2 )γ (x2 ) dx2

G

−h

where

γ (x2 )Gdv1s (x2 ) dx2

(2.43)

is the magnitude of slip. Finally, as l1 dx2 G−r dθ v′1 (t)G−

v′2 (t)G

2 π

2 π

冮

θo

γ cos θ sin θ dθ

0

冮

θo

γ sin2 θ dθ

0

where

θ Garctan

Ax2 ; bAt

θ o Garctan

h bAt

for tFb

38

Rolling Contacts

and v′1 (t)G− v′2 (t)G

2 π

2 π

冮

π

γ cos θ sin θ dθ

θo

冮

π

γ sin2 θ dθ

θo

where

θ GπAarctan

Ax2 tAb

;

θ o GπAarctan

h tAb

for tHb

If, for simplicity, it is assumed that γ is constant, then v′1 (t)G− v′2 (t)G

γ h2 π hC(bAt)2

(2.44)

γ h h(bAt) arctan A 2 π bAt h C(bAt)2

冤

冥

(2.45)

It should be noted that for γ H0 the surface has a wedge at tGb with a jump in inclination of magnitude γ . It can be assumed that the wedge is situated at the leading point of contact. There are, however, no shear stresses available to drive the plastic deformation in front of the contact zone, and the wedge would be flattened under the roll.

2.7 Thermal effects during rolling In order to create a qualitative physical picture of the problem, it is advisable to consider the situation shown in Fig. 2.16(a) where two

Fig. 2.16


39

contacting discs are assumed to be rolling with a very small amount of relative slip. It is clear that all particles on the surface of both discs pass through the contact zone where heat is being generated and afterwards undergo considerable periods of rest. Temperature rise for them will therefore be rather modest owing to the small magnitude of frictional work and low relative sliding velocity, and also because the generated heat will be readily dissipated during the rest periods. If one of the discs is stationary [Fig. 2.16(b)], then conditions of pure sliding will be created. In this case, surface particles on disc 2 will be subjected to relatively high temperatures when passing through the contact zone and have a considerable rest period outside the contact zone where cooling takes place. Surface particles on disc 1, however, never leave the contact zone and are subjected to a continuous build-up of temperature towards a steady state determined by the thermal properties of the whole system. Thus, the contact zone which is fixed in space can be regarded as a stationary heat source with respect to disc 1, while the contact zone with respect to disc 2 can be regarded as a moving heat source. It is obvious that for rolling contacts the relevant and important case is that of a moving heat source.

2.7.1 Moving source of heat The problem of a moving heat source traversing the surface of a semiinfinite body at a relatively high speed V can be somewhat simplified by neglecting the effects of the transverse flow of heat. Thus, the problem can be treated as one of linear heat flow. If the heat is supplied at a constant rate of q per unit area, then the mean temperature rise of a point on the surface of the body is given by

1冢π αρc冣 2qt

∆θ G

where t is the time during which heat is supplied, α is the thermal conductivity of the body, ρ is the density, and c is the specific heat of the body. If heat is supplied through a circular area of radius a, then, by assuming that qGQ兾πa2 and by considering the effective value of t for all points within this area, a mean surface temperature can be obtained. The time to traverse the contact area for any point defined by (x, y) is given by tG

2x V

G

2 V

1(a2Ay 2 )

40

Rolling Contacts

and therefore the mean effective time is tG

a

πa

1(a Ay ) dyG 2a 冮 (a Ay ) 4V 1

2

2

2

2

2

0

Thus

θmG

0.318Q 2Q1(πa) G 2 1 1 2πa (π αρcV ) a (αρacV )

(2.46)

It is convenient to introduce a normalized surface temperature θ * defined as

θ *G

ρcV θ πq

so that for a moving source of heat, utilizing equation (2.46), the result is

θm *G

0.3181(2acρV ) G0.438ξ 0.5 0.5 (2α )

(2.47)

where

ξG

ρac V 2α

2.8 Contact of bodies with interposing film The performance of rolling contact bearings coated with thin solid lubricant films, particularly those deposited by such processes as ion plating and sputtering, is of practical importance and interest, as in many cases the size of the contact zone is considerably influenced by elastic deformation of the film material as well as by that of the contacting bodies. In this section, expressions are provided from which the dimensions of all-elastic cylindrical and spherical contacts may be determined when the ratio of the contact dimensions to the film thickness is large and where the ratio of the modulus of elasticity, δ , of the film material to that of the contacting bodies is less than unity.


41

2.8.1 Background to the analysis It is known that a wide variety of contacts possess an approximately elliptic pressure distribution of the form qGqc

1冢

冣

r2 1A 2 a

(2.48)

where a is the contact half-width, q is the pressure at a distance r from the contact centre, and qc is the pressure at the centre. The above approximation is not justified for values of δ greater than unity. If the load on the contact is P′ per unit length, it is permissible to write a

P′G2

冮 q drG 2 πaq 1

(2.49)

c

0

and, making the reasonable assumption that the same pressure distribution in an axisymmetric form will apply in the case of spherical contact a

PG2π

冮 rq drG 3 πa q 2

2

c

(2.50)

0

in which P is the total contact load and a is now the contact radius. In the special case of Hertzian contact between equal cylinders or spheres of the same material (i) cylinder a2o G

4P′R(1Aν 2 ) πE

(2.51)

(ii) sphere a3o G

3PR(1Aν 2 ) 2E

(2.52)

where ao is the Hertzian half-width or radius for direct contact between surfaces of radius R with modulus of elasticity E and Poisson’s ratio ν.

42

Rolling Contacts

In the case of Hertzian contact, equation (2.48) becomes qGqh

1冢

1A

r2 a2o

冣

(2.53)

where qh is the pressure at the centre of the Hertzian contact. Fig. 2.17 shows the model of the contact under consideration. The contact consists of two equal cylinders or spheres (radius R and elastic constants E2 and ν2 ) pressed together with load P′ or P into a film of thickness h and elastic constants E1 and ν1 . As the contact is symmetrical, all the derivations are significantly simplified. However, the results of analysis may be translated into any conformal or counterformal cylindrical or spherical contact provided that RZaZh.

2.8.2 Case of contacting cylinders With reference to Fig. 2.17, the relative displacement normal to the contact surface between points at the contact edge (rGa) and the contact centre is given approximately by w(0)Aw(a)G

a2

(2.54)

2RA∆h

where ∆h is the central compression displacement of one-half of the film. Assuming the film to be sufficiently thin for the normal component of the pressure across it to be virtually constant, it is permissible to write ∆hG

hqc

(2.55)

2E1

Fig. 2.17


43

as it is assumed that there is no friction or bonding between the cylinders and film, i.e. that free compression conditions exist. At the other extreme, it is possible to assume that the film is bonded to both cylinders and, since the lateral strain is negligible at the centre, that the film in this region is subjected to bulk compression. Thus ∆hG

(1A2ν1)(1Cν1 )hqc

(2.56)

2(1Aν1)E1

It is convenient to write equations (2.55) and (2.56) as ∆hG

khqc

(2.57)

2E1

where k is a factor describing the nature of the bonding. Considering now an infinite line load of q∆r per unit length, acting on the surface of a semi-infinite plane, the relative perpendicular displacement between two points whose distances from the line load are A and B can readily be shown to be ∆w(A)A∆w(B)G

2(1Aν22 ) B ln q∆r πE2 A

冢冣

(2.58)

where E2 and ν2 are the elastic constants of the plane material. It is now possible to use equation (2.58) to define the relative displacement w(0)Aw(a). Figure 2.18 shows the contact with two line load elements at a distance r from the centre-line. The total relative displacement for

Fig. 2.18

44

Rolling Contacts

the whole contact is

冤冮冢冣 1冢

1Aν 22 πE2

w(0)Aw(a)G2qc

a

aAr r

ln

0

1A

冣

r2 dr a2

冮冢 r 冣 1冢 a

C

ln

aCr

1A

0

G2aqc

1Aν 22 πE2

r2

冣 dr冥

a2

冤冮 ln 冢1Aa 冣 1冢1A a 冣 d 冢a冣 1

r2

r2

2

2

r

0

冮冢a冣 1冢 1

A2

ln

r

1A

0

d a 冣冢a冣冥 r2

r

2

which can readily be evaluated by trigonometric substitution and integration by parts to give w(0)Aw(a)Gaqc

1Aν 22

(2.59)

E2

By combining equations (2.54), (2.57), and (2.59) aqc

1Aν 22 a2 khqc G A E2 2R 2E1

(2.60)

from which, after substituting for qc from equation (2.49), the following is obtained 1Aν 22 kh aC a G4P′R πE2 2γ (1Aν 22)

冤

3

冥

(2.61)

which reduces to the Hertz formula given by equation (2.51) when hG 0. By substituting from equation (2.51) and rearranging h ao

G2γ

1Aν 22 k

3

冤冢a 冣 Aa 冥 a

o

a

(2.62)

o

so that, provided that ao is known for Hertzian contact between cylinders of material 2, it is possible to define a for given values of h.


45

2.8.3 Contacting spheres In order to obtain w(0)Aw(a) for the case of contacting spheres, it is convenient to refer to some results of Timoshenko and Goodier (2), which may be presented in the form w(r)Gπqc

1Aν 22 (2a2Ar 2 ) 4aE2

(2.63)

where w(r) is the normal displacement at a point located at a distance r from the centre of an elliptical contact. Hence w(0)Aw(a)Gπaqc

1Aν 22 4E2

(2.64)

and, in a way analogous to that used in the case of contacting cylinders, the following is obtained 1Aν 22 h Gπ γ ao 2k

4

冤冢a 冣 Aa 冥 a

o

a

(2.65)

o

It is important for both cases considered to discuss the significance of factor k. It is clear that the extreme values of k are unity for no lateral constraint and (1A2ν1)(1Cν1)兾(1Aν1) for both surfaces rigidly bonded. By taking ν1 G0.25, these limits are 1 and 56 ; therefore it is justified to conclude that k is not expected to have a very pronounced effect for most materials. In friction situations, one surface of the film is bonded while the other is constrained to some extent by friction forces. Thus, it is reasonable to expect the lower value of k to be more applicable in this case.

2.9 Crack formation in contacting elastic bodies The problem of the development of a crack in contacting elastic bodies has some common elements with the contact problem of the theory of elasticity (3). This section analyses the problem of two elastic halfplanes pressed together, without any friction, by distributed loads of constant intensity q that are orthogonal to their common boundary, i.e. the x axis, and are applied far away from that axis.

46

Rolling Contacts

2.9.1 Description of the contact The model of the contact is shown in Fig. 2.19. It can be seen that two equal but opposite forces P and P′ tend to break the contact of these half-planes. Force P is oriented in the positive direction of the y axis and is the resultant of a uniformly distributed load p which is applied to the part of the boundary defined by −cFxF+c and yG0 of the upper half-plane. Force P′ has an analogous significance; the load p′, the resultant of which is P′, acts on the lower half-plane. It is obvious that 2pcGPGP′G2p′c Let it be assumed that, as a result of the action of forces P and P′, the contact of the elastic half-planes is broken along that part of the interface which is defined by −aFxF+a, where a gap, 2a, appears, the length of which has to be determined. In order to find the length of the gap, a plane problem of the theory of elasticity will be considered. In agreement with the earlier considerations, the following boundary conditions at the boundary of the upper half-plane are applicable

σ x → 0, σ y G0, σ y G−p,

τ xy → 0, τ xy G0 τ xy G0

σ y → Aq for y → S for AaFxF−c and for cFxFa for AcFxFc

Fig. 2.19


47

and, moreover, τ xy G0 and vG0 at all points of the part of the boundary of both half-planes at which they are in contact, i.e. for xHa, yG 0. Here, as is usual, σ x , σ y , and τ xy are the components of the stress tensor and u and v are the components in the x and y directions of the displacement of an arbitrary point on the elastic half-plane. By using well-known methods of solution of the plane problem of the theory of elasticity, the functions u and v of the variables x and y can be determined from the functions σ x , σ y , and τ xy which satisfy both equilibrium equations and the boundary conditions. It is found that, on the segment of the interface where there is no contact, i.e. 兩x兩Fa and yG0, the displacement of the points of the boundary of the upper elastic half-plane in the vertical direction is given by the following expression vGv(x, 0)G

c λ C2µ 2 p arcsin Aq 1(a2Ax2 ) 2µ (λ Cµ) π π

冢

冣

1(a2Ax2)C1(a2Ac2) p C c ln 1(a2Ax2)A1(a2Ac2) π

冤冨

Ax ln

冨

c 1(a2Ax2)Cx1(a2Ac2) c 1(a2Ax2)Ax 1(a2Ac2)

冨

冨冥

(2.66)

where λ and µ are Lame’s constants. The expression for the distributed load maintaining the contact between the two half-planes in the domain where there is contact has the following form

冢π p arcsin aAq冣 1(x Aa )

σyG

c

2

x

2

冤

2

a2Acx a2Ccx C arcsin Aarcsin π a(xAc) a(x¯c) p

冥

(2.67)

which is valid only for xHa. Let l denote the root in the interval (c, S) of the trigonometric equation c 2 p arcsin AqG0 π a

(2.68)

in which the quality a is unknown. Under the condition pHq, equation (2.68) will always have exactly one such root.

48

Rolling Contacts

It is not difficult to show that the half-length a of the gap formed as a result of forces P and P′ which tend to separate the half-planes is exactly equal to the root l of equation (2.68). Let it be assumed that aFl. In this case, the gap has the shape shown in Fig. 2.20. The tangent to the contour of the gap is vertical in its extreme points xGJa and yG0. In the immediate proximity of the ends of the gap, the stress σ y at the points of the x axis becomes positive. This means that the elastic half-planes must exert a tensile traction on each other in order to maintain contact. In the present case, as in the classic statement of the contact problem of the theory of elasticity, a tensile traction between contacting bodies is not admitted. Thus, the case aFl is impossible. Now, let it be assumed that aHl. Thus, everywhere in the contact area, i.e. for 兩x兩Ha and yG0, the stress σ y is negative and, consequently, the elastic half-planes are pressed against each other. In this case, however, the function vGv(x, 0), i.e. the displacement of the boundary of the upper elastic half-plane, is positive near the ends of the gap and negative in its central part. In consequence of this, the contour of the gap acquires the shape of a curve that intersects itself as shown in

Fig. 2.20


49

Fig. 2.21

Fig. 2.21. This means that the elastic half-planes penetrated one another which, of course, is impossible, and the case aHl has to be eliminated. It can then be concluded that the solution of the plane problem of the theory of elasticity leads, for a ≠ l, either to a physically impossible stress state or to an impossible displacement field. Thus, the only possible case left is lGa. In this case, the following is applicable 1(l 2Ax2)C1(l 2Ac2) λ C2µ p v(x, 0)G c ln 1(l 2Ax2)A1(l 2Ac2) 2µ (λ Cµ) π

冤冨

Ax ln

冨

冨

c1(l 2Ax2)Cx1(l 2Ac2) c1(l 2Ax2)Ax1(l 2Ac2)

冨冥

(2.69)

and

σ y (x, 0)G

冤

冥

p a2Acx a2Ccx arcsin Aarcsin π a(xAc) a(xCc)

(2.70)

Now the ends of the gap smoothly converge to each other. The tangent to the contour of the gap in the points xGJl and yG0 is directed

50

Rolling Contacts

along the x axis. The stress σ y and the distributed pressure of one elastic half-plane upon the other at the given points is equal to zero and increases gradually, tending to the negative value Aq as the distance from the gap increases. The length of the gap increases with growth of the cleaving forces P and P′. The equations of the theory of elasticity and the boundary conditions are satisfied.

2.10 Contacts deviating from the Hertz theory The theory of elastic contact put forward by Hertz has been proved to predict the area of contact and stresses remarkably well. It has been used by both the engineer and physicist to deal with problems arising from the contact between elastic bodies. In general, deviations from the Hertz theory are small; nevertheless in special circumstances, physical conditions that were omitted by Hertz can be important. These special cases are known as non-Hertzian contacts. It is useful to list the relevant restrictions in the Hertz theory: – the bodies in contact are homogeneous and isotropic; – their surfaces are smooth and continuous; – their profiles are represented by a second-order surface; – the stresses and displacements may be found from the small strain theory of elasticity applied to a linear elastic half-space; – there is no friction between surfaces in contact; – the surface tractions are the result of the contact forces only, i.e. adhesive forces are ignored. In this section, modifications to the theory that are necessary if some of the above restrictions are to be relaxed will be considered.

2.10.1 Friction at the contact interface No engineering surface is ever frictionless, so that an obvious practical extension of the Hertz theory is to remove that restriction and to consider the influence of friction at the contact interface. In the normal contact of spheres, friction introduces a first-order correction only when the two bodies have dissimilar elastic constants. Under the action of their mutual contact pressure, the two surfaces undergo radially inward tangential displacements whose magnitudes, in the absence of friction, would be proportional to the respective values of the parameter (1A2ν)兾G, where G is the shear modulus and ν is Poisson’s ratio. With dissimilar materials, the displacements will be different for the two surfaces and the resulting slip will be resisted by interfacial friction,


51

which will act radially outwards on the more compliant surface and inwards on the more rigid one. Theoretically, interfacial slip could be prevented by a sufficiently high coefficient of friction µ at the interface. This will constitute the fully adhesive or no-slip contact conditions. At the other extreme, if the coefficient of friction is very small, slip will take place over the whole contact area and the tangential traction will be radial and equal to Jµp everywhere ( p denotes normal contact pressure). This is the so-called complete slip contact condition. In reality, the contact comprises a central circular region of adhesion surrounded by an annulus of slip. This case is usually referred to as the partial slip contact condition. The contact behaviour is controlled by two non-dimensional parameters, namely

£G

[(1A2ν1)兾G1]A[(1A2ν2 )兾G2] [(1Cν1)兾G1]C[(1Cν2)兾G2]

and coefficient of friction µ . For practical values of the elasticity parameter £ (F0.4) and the coefficient of friction µ (F0.5) the influence of frictional traction upon the compliance and load-bearing capacity of most contacting bodies is small, so that the use of the Hertz theory is justifiable. An exception arises in the case of brittle solids which fracture as a result of the radial tensile stress surrounding the contact area. This component of stress in the absence of friction is given by

σ rr G

(1A2ν)P 2πr 2

(2.71)

and is valid for r„a. In the above expression, P is the load on the contact and r denotes the distance from the centre of the contact. It is particularly sensitive to frictional tractions at the interface.

2.10.2 Adhesion at the contact interface Owing to intermolecular forces, clean, smooth surfaces when pressed into intimate contact may in some circumstances strongly adhere together. The theory proposed by Hertz does not involve such forces and, as a matter of fact, measurable adhesion between non-conforming elastic bodies is rarely observed in practice. A notable exception to this common experience is the contact of an optically smooth rubber sphere

52

Rolling Contacts

of very low elastic modulus with a smooth glass surface. The combination of smooth surfaces and a compliant solid allows for intimate contact between the rubber and the glass surfaces to be realized throughout the whole of the apparent contact area. Under a compressive load, the sphere makes contact with the glass over a circular area appreciably greater than that predicted by Hertz theory and a measurable tensile force is required to separate the two surfaces. As expected, a thin layer of fluid with a wetting agent between the surfaces destroys the adhesion and the deformation of the sphere reverts to that predicted by Hertz theory. As it is not known what law governs the force acting agent against separation, so it is convenient to express the adhesive action in terms of a surface energy γ , defined as the work required to separate a unit area of the adhered surfaces. Then, if two elastic spheres of radii R1 and R2 are brought into contact under a force P1 , according to Hertz they make contact on a circle of radius a1 given by a31 G

RP1 M

(2.72)

where RG

R1R2 R1CR2

MG

4E′ 3

and 1 E′

G

1Aν 21 1Aν 22 C E1 E2

Reducing the load acting on the contact to Po , while maintaining the surfaces in contact by adhesion over the same contact radius a1 , allows the resulting distribution of surface traction to be found by subtracting the stress under a flat cylindrical punch given by Hertz theory, namely p(r)G

3P1 2πa21

1冢

1A

冣

1 r 2 P1APo A 2 2 a1 2πa1 1(1Ar 2兾a21)

(2.73)


53

Fig. 2.22

This traction is of tensile nature at the edge of the contact and compressive in the centre, as shown in Fig. 2.22 (line B). It is possible to estimate the elastic strain energy UE associated with the state of stress within the contact zone. Thus 1 UE G 15 M −2/3R 1/3 (P15/3C15P 2o P1−1/3C5Po P12/3)

(2.74)

where P1 is related to a1 through equation (2.72). The surface energy associated with a contact area of radius a1 is Us G− γ πa G−π γ 2 1

冢M冣 P1R

2/3

(2.75)

According to Griffith, the equilibrium value of a1 , for any given value of Po , occurs when the variation in elastic strain energy with a1 just

54

Rolling Contacts

matches the variation in surface energy, i.e. when d (EECVs)兾da1 G0. This condition results in Ma31 GP1 GPoC3π γ RC1[6π γ RPoC(3π γ R)2] R

(2.76)

If it is assumed that Po GP and a1 Ga, an expression is obtained for the equilibrium value of a under the action of load P in the presence of an adhesive force. Thus a3 G

R 1[PC3π γ RC36π γ RPC(3π γ R)2] M

(2.77)

When the surface energy is zero, equation (2.77) reduces to the simple Hertz equation. It is also apparent that an equilibrium contact area can be maintained even though the force is tensile up to a maximum value given by PG− 32 π γ R

(2.78)

This is also the force required to separate the contacting spheres.

2.11 References (1) Hertz, H. (1896) The contact of elastic bodies. Miscellaneous Papers (Macmillan, London). (2) Timoshenko, S. and Goodier, N. N. (1951) Theory of Elasticity (McGraw-Hill, New York). (3) Barenblatt, G. I. (1959) On the equilibrium cracks due to brittle fracture (in Russian). Prikl. Mat. i Mekh., 23.

Chapter 3 Fundamentals of Rolling Motion

Why is it that spherical and cylindrical forms are easier to move? Firstly because they have a very slight contact with the ground and secondly because there is no friction, for the angle is well away from the ground. Why is it that it is easier to convey heavy weights on rollers than on carts although the latter have large wheels and the former a small circumference? Is it because a weight placed upon rollers encounters no friction, whereas when placed upon a cart it has the axle at which it encounters friction?

This quotation from Mechanica, Section 8.11, by Aristotle adequately describes a popular understanding of the resistance to motion in rolling contacts. It also emphasizes the fact that there is a different way in which one surface can be moved relative to another and that is by rolling. It is undoubtedly much easier to roll surfaces than to slide them, an observation discussed above with characteristic clarity by Aristotle. With hard materials, the coefficient of rolling friction may be as little as 0.001.

3.1 General features of rolling contact It is a common experience of each of us that rolling contacts operate at a far lower resistance to motion than those involving sliding. However, in order to secure this low resistance to motion, the rolling component must be placed on a relatively hard and smooth substrate so that the stresses in both interacting surfaces are entirely elastic. For a given load and geometry, increasing the rigidity of the rolling components will result in further reduction in the resistance to motion. It is, however,

56

Rolling Contacts

not ‘free’ and there is a cost to pay in the form of increasing the surface and near-surface stresses at the point of contact. During the rolling motion, for instance, of a cylinder over a nominally flat surface under a load N, deformations of both the cylinder and the surface occur. As a consequence, the macroscopic contact between them takes place on a finite area ab (Fig. 3.1). Because of the roughness of both interacting surfaces, the real contact takes place at discrete locations within the nominal contact area which is analogous to a sliding contact. However, owing to deformations of interacting surfaces a special characteristic feature of the rolling contact is microslip. The first to point out the occurrence of microslip during rolling was Reynolds (1). Assuming that deformations at the contact between the cylinder and a flat surface are elastic, Reynolds pointed out that within the instantaneous contact area ab the cylinder is under compression and the flat surface under tension. Thus, as rolling progresses, points a and b, and for the matter other points located within the contact zone on the substrate surface, will have a tendency to get closer to each other, whereas corresponding points located on the cylinder surface will tend to move apart. This phenomenon is responsible for microslip taking place at discrete points within the contact zone where the energy of elastic deformation is greater than the cohesive energy of junctions formed as a result of intimate contact between the cylinder and the substrate. Thus, aa1 and bb1 are the regions where slip occurs and a1b1 is the region where there is no relative sliding between the

Fig. 3.1

Fundamentals of Rolling Motion

57

contacting bodies. Therefore, the resistance to rolling originates from elastic hysteresis losses within the material of the contacting bodies. Another particular feature of rolling motion is manifested by a continuously growing distance between surfaces at the rear of the contact and their steady approach in front of the contact. Thus, for the rolling to continue, surface forces and eventual adhesive junctions must be overcome at the rear of the contact. Some of the work expended on that can be retrieved by the action of surface forces in front of the contact. When a lubricant is applied to the substrate, the creation and destruction of discrete direct contacts between the interacting bodies must be preceded by action against lubricant viscosity. During rolling there is a continuous change in the nominal area of contact. Contact pressures within the nominal area of contact, as well as deformations caused by them, are uneven. Moreover, the relative motion of surface elements takes place along curvilinear trajectories. Therefore, it is justified to state that the rolling friction is a more complicated physical phenomenon than the sliding friction. For that reason it is more difficult to present rolling friction in analytical terms. Two familiar and practically important examples of rolling contacts are those of pneumatic tyres and steel-rimmed railway wheels. The loads on each are quite similar but the properties of the materials involved are very different. The rolling resistance of a tyre is very much greater than that of a steel wheel on a steel rail because of the very much lower value of the equivalent elastic modulus of a rubber tyre on a concrete road as compared with that of a steel wheel on a steel rail. On the other hand, the magnitude of the contact stresses, which to a large extent control the potential damage to both road and rail surfaces, is very much lower for a pneumatic tyre than for a steel-rimmed wheel. The smooth operation of a very large number of machines depends, at least to some extent, on the utilization in their design of rolling contacts. A typical practical example of this is a rolling contact bearing where a number of balls or rollers are placed between the two moving elements so that the sliding motion is transformed into a rolling motion. This consequently leads to a substantial reduction in the resistance to motion. Although friction in a rolling contact is generally much less than in a sliding contact, it is, nevertheless, finite and its origins have been the subject of much investigation. As a result of the difference in the elastic deformation of the contacting surfaces, there is always likely to be an element of microslip within the contact region, and this makes a contribution to the overall resistance to motion. However, in steel components rolling on one another, the magnitude of this element of slip and

58

Rolling Contacts

the friction associated with it are quite small and insufficient to account for the whole of the observed resistance to motion in rolling contacts. A second characteristic mechanism for rolling contacts is known as Heathcote slip. It contributes to the rolling resistance of ball bearings. If the geometry of the rolling components is conformal, exemplified by that of a sphere rolling in a groove, then within the contact zone there will be two lines along which there is zero surface slip. However, on either side of them there are regions in which the relative sliding between the sphere and the track is in opposite directions and therefore the rolling resistance is quite substantial. In addition, the small but finite hysteresis loss can also be an important factor. Finally, in real bearings the contact is usually lubricated and energy loss within the viscous lubricant film will also play a part.

3.2 Source of friction in rolling contact What is the ultimate source of rolling friction and why is it generally so small? Until recently, most engineers considered that rolling friction arises from minute slip between the ball and the surface over which it rolls. However, recent studies, both experimental and theoretical, have shown that, although such slip does occur, it generally contributes only a small part to the rolling resistance. This is supported by the observation that lubricants, which greatly reduce sliding friction, have very little effect on rolling friction (2). A clue as to the source of rolling friction can be obtained by considering what happens if a hard steel ball rolls over a softer metal such as lead or copper [Fig. 3.2(a)]. As it rolls along, the ball displaces metal plastically around and ahead of it and produces a permanent groove in the metal surface. It is easy to demonstrate that the force required to displace the metal is almost exactly equal to the observed rolling friction. Thus, the rolling friction is essentially a measure of the plastic work of deformation or grooving. It is now easy to see why lubricants have little effect on rolling friction. What happens, then, if rolling takes place over an elastic solid such as rubber? No permanent groove is formed. Does this mean that the rolling resistance is zero? The answer is no. As the ball rolls forward, it deforms the rubber ahead of it and in doing so does work on it. The rubber recovers elastically and does work on the rear portion of the ball pushing it forward [Fig. 3.2(b)]. If the recovered energy in the rear portion were equal to that expended on the front portion, the net work required to roll the ball would be zero. However, no material is ideally elastic. In the course of deforming and relaxing rubber, some energy is


59

Fig. 3.2

lost by elastic hysteresis. Hysteresis losses originate from the rubbing of rubber molecules over one another during the deformation process. Indeed, elastic hysteresis is sometimes described as internal friction. With a very bouncy rubber, the hysteresis losses are very small; only a few percent of the deformation energy may be lost in this way. On the other hand, with a very soggy rubber as much as 95 percent of the deformation energy may be lost. This lost energy appears as heat within the rubber. Ball bearings are made of hard steel and are so designed that the contact stresses are not high enough to produce plastic flow of the balls or races. Only elastic deformation occurs. For bearing steels, the hysteresis losses are extremely small (less than 0.5 percent) so that the rolling resistance is very small. In practice, the balls must be surrounded by a

60

Rolling Contacts

cage to separate them and prevent them rubbing on one another. The cage friction is often far greater than the rolling friction. Lubricants are used to reduce the sliding friction between balls and cage. As was said earlier, they play little part in rolling friction itself. With rubber-like materials, the rolling resistance can be very much larger. For example, with a ball loaded so heavily that it is half-buried in the rubber and using a soggy rubber of high hysteresis loss, a rolling resistance equivalent to a coefficient of 0.3 can be achieved. An interesting question in the context of rolling is about the role of adhesion at the regions of contact. The first point to observe is that the rolling process imposes much gentler deformation on the surface than the sliding process. Consequently, break-up of surface films is likely to be less marked so that strong adhesion is less likely but cannot be ruled out completely. Secondly, even if adhesion does occur, the junctions are peeled apart. This is a much easier process than shearing which actually takes place in sliding contact. The overcoming of interfacial adhesion consumes only a small part of the total energy expended during rolling. Thus, even if a lubricant reduces the adhesion, this has very little effect on the total rolling friction. This is the basic reason why lubricants play a small part in rolling friction.

3.3 Rolling friction force A fundamental principle describing rolling of a cylinder over a flat surface was formulated by Coulomb (3). It says that the rolling friction force is directly proportional to the applied load and inversely proportional to the radius of the cylinder TGf

N

(3.1)

r

where r denotes the radius of the cylinder and f is the coefficient of rolling friction, expressed in the same units as the radius. The rolling friction force acts in the opposite direction to the relative motion and its point of application is not precisely defined. The rolling friction coefficient depends on the type of material and surface finish but is independent of rolling velocity. According to Reynolds (1), the rolling friction coefficient is represented by tangent ac (Fig. 3.1). Figure 3.3 shows the forces acting on a cylinder rolling over a nominally flat substrate. Rolling motion is due to the application of force F. During steady state motion FGT,

NGZ,

NfGTr

(3.2)


61

Fig. 3.3

In Fig. 3.3, λ represents the shift of reaction force Z and is called ‘the arm of rolling friction’. The Coulomb law does not have a general character as, depending on the mechanical properties of the materials of the cylinder and the substrate, the relationship between friction force and load may be expressed by a different equation and, sometimes, the effect of rolling velocity may be included in it. Because of the practical importance of rolling contacts, a considerable amount of research effort has gone into studies of rolling resistance. As a result, new equations taking into account conditions under which rolling occurs have been derived. In Fig. 3.4 a schematic representation of forces acting on a driving wheel of a vehicle is shown. Shift a of a normal reaction depends on the deformation of the tyre, although the magnitude of a is influenced by the normal load on the wheel and the tractive force. The coefficient of resistance to rolling for a rigid wheel is given as the ratio of the force resisting rolling Pk and the normal reaction Zk

µrG

Pk

G

Pk

Zk Gk

(3.3)

Shift of the reaction force Zk is caused, in the case of a rigid wheel, by deformation of the substrate. The coefficient of rolling friction is given by a

µrG

r

(3.4)

62

Rolling Contacts

Fig. 3.4

In the case of a deformable wheel a Mk rkArd µrG A rd Zk rk rd

(3.5)

where rd denotes the ‘dynamic’ radius of the wheel, i.e. the distance from its centre to the substrate over which it is rolling, and rk is the ‘rolling’ radius of the wheel, i.e. the radius of a fictitious rigid wheel which, rolling without slip, has identical angular velocity and linear velocity to a real wheel.

3.4 Free rolling Free rolling is the most basic form of rolling motion. During free rolling, material ahead of the contact region is compressed, while material to the rear of the contact zone has this stress state relieved. For these conditions, the Hertz theory applies. Two cases will be analysed here: a cylinder on a plane and a sphere on a plane.

3.4.1 Cylinder on a plane The distribution of pressure p in the contact zone owing to an applied load N is equal to 2N

pG πal

1冢

1A

x2 a2

冣

where l denotes the length of the cylinder.

(3.6)


63

The above equation represents a semicircular pressure distribution with a maximum pressure at the centre of the contact zone of value 2N兾(πal ), as shown in Fig. 3.5. The value of the contact semi-width a is given by

1冤

4Nr 1Aν 21 1Aν 22 C πl E1 E2

aG

冢

冣冥

(3.7)

where ν is Poisson’s ratio, E is Young’s modulus, r is the radius of the cylinder, and suffixes 1 and 2 denote the cylinder and the plane respectively. During the forward compression of the material in contacting bodies, a resisting moment M about the centre of the contact region is developed a

MG

冮 plx dxG 3π

2Na

(3.8)

0

Under ideal conditions the cylinder would be subjected to an equal and opposite moment arising from the pressure distribution in the rear part of the contact zone, but the rearward recovery does not replace the

Fig. 3.5

64

Rolling Contacts

whole of the elastic work of forward compression, which is given by Mx

φG

2Nax

(3.9)

G

r

3πr

This imbalance may be explained by an energy dissipation due to the elastic hysteresis loss occurring in the complex straining of the material which must occur during the rolling process. If this loss is represented by a coefficient ε, then FxGεφGε

2Nax

(3.10)

3πr

where F is the force required to overcome the resistance to motion. The ratio F兾N is usually called the coefficient of rolling friction F 2εa µrG G N 3πr

(3.11)

It must be noted that µr depends on the geometry of the roller, the applied load, and the elastic constants of the two bodies in contact. Moreover, the hysteresis loss factor ε used in the expression for µr is not the same as that which would be measured in a simple tensile test. An approximate estimate of the hysteresis loss coefficient suggests that it should be almost 3 times the loss factor measured in a simple tension test.

3.4.2 Sphere on a plane The pressure distribution in the contact zone is given by 3N pG 2πa2

1冢

冣

r2 1A 2 a

(3.12)

where r denotes the radius of the sphere. The radius of the contact can be found from

冤4 冢

aG

3

Nr

1Aν 21 1Aν 22 C E1 E2

冣冥

1/3

(3.13)

The contact configuration of the case under consideration is shown in Fig. 3.6. The moment resisting the motion is 3N

MG 3 4a

a

冮 (a Ax )x dxG 16 2

0

2

3Na

(3.14)


65

Fig. 3.6

The work done in forward compression in rolling a distance x is Mx

φG

r

3Nax

G

16r

(3.15)

Introduction of a loss factor similar to the one discussed earlier gives FxGεφGε

3Nax 16r

(3.16)

Therefore, the coefficient of rolling friction is F 3εa µrG G N 16r

(3.17)

The hysteresis loss coefficient for this case is only about twice that which will be obtained in a simple tension test.

3.5 Material damping during rolling The rolling of a sphere on a nominally flat plate is a cyclic process. The stresses resulting from the contact of ball and plate are carried along with the moving contact. Elements of volume in the material are cyclically stressed as the stress field passes through them. This cyclic stressing

66

Rolling Contacts

is accompanied with a hysteresis loss that appears to be the primary source of energy dissipation during rolling. Figure 3.7 shows the ball and plate in the vicinity of the contact area subdivided into blocks of thickness ∆x, each bounded by the sample surface and by planes perpendicular to the direction of rolling. The strain energy stored in each of these blocks is a function of the contact pressure and the elastic constants of the material. The stored strain energy in the blocks in the ball V1 and in the blocks in the plate V2 is plotted as a function of the distance of each block from the centre of the contact area. The exact shape of these curves is not known, but the essential fact is that the strain energy is a maximum for the block at the centre of the contact area. As the ball rolls a distance ∆x, the energy dissipated is ∆x times the friction force F and the strain energy in the blocks is changed according to these curves. Effectively, this energy change between blocks is equivalent to one block in the ball and one block in the plane being subjected to the cycle of zero strain energy to maximum strain energy to zero strain energy. By definition, the specific damping capacity is the ratio of the energy dissipated during a cycle to the maximum strain energy stored during the cycle. Thus, the specific

Fig. 3.7


67

damping capacity for the rolling ball is F∆x ΩG V1 maxCV2 max

(3.18)

Since F is determined experimentally, all that is needed for calculation of the specific damping capacity is the value of Vmax for ball and plate. Choosing a coordinate system with the origin at the centre of the contact area, the x axis extending in the direction of rolling, and the z axis extending into the sample, the complete stress distribution in the y–z plane for a sphere contacting a plane is given by 1A2ν a2

σxGqo

Cqo

σyGAqo Aqo

σzGqo

y2

3

3

冤冢冣冥 1A

z 1u

1u z (1Aν)u a 2νC 2 Aqo (1Cν) tan−1 1u 1u a Cu a

冤

冥

1A2ν a2 y2

3

3

z 3 a2u 1u u2Ca2z2

冤冢冣冥冢冣 1A

z 1u

Aqo

1u z (1Aν)u a Aqo (1Cν) tan−1 A2 2 1u a Cu 1 a u

冤

冥

z 3 a2u 1u u2Ca2z2

冢冣

τyzGqo

yz2

(3.19)

a21u

u2Ca2z2 a2Cu

(3.20)

(3.21)

(3.22)

where u is defined by the equation y2

z2 C G1 a2Cu u

(3.23)

and 3 P qoG 2 πa2

(3.24)

68

Rolling Contacts

In the above expression, P is the force pressing the sphere against the plane, a is the radius of the circular contact area, and ν is Poisson’s ratio. The strain energy in an element of volume S

∆x

冢冮冮

S

冣

dy dz

−S

0

xG0

is Vmax and is given by ∆x

VmaxG

E

S

冮冮 0

S

[12 (σ 2xCσ 2yCσ 2z )

−S

Aν (σx σyCσy σzCσz σx )C(1Cν)τ 2yz] dy dz

(3.25)

where E is Young’s modulus. Assuming that νG0.3 and expressing relevant terms in equation (3.25) by parameters given by expressions introduced earlier, the resulting double integral can be evaluated numerically to give VmaxG0.1315

P2 a2E

∆x

(3.26)

Equation (3.18) for the specific damping capacity thus becomes

冤

Ω GF 0.1315

冢

P2 1 a2

冣冥

1 C E1 E2

−1

(3.27)

where E1 and E2 are the Young’s modulus for the ball and plate, respectively. Equation (3.27) is only applicable for values of the load that do not cause any plastic deformation. When plastic deformation does occur, the geometry is no longer a sphere contacting a flat plate and equations giving stresses within the contact zone are no longer valid.

3.6 Slip at the surface of contact If a perfectly rigid ball is rolling on a perfectly rigid flat substrate, a single contact point occurs at any moment in time, and the rolling resistance disappears. However, in practice, all solids have a finite elasticity, and the bodies will deform elastically or plastically if the local contact pressure reaches the yield stress of any of the solids. Thus, the actual contact area will be finite. It follows from the theory of elasticity that a tangential stress may develop at the interface between the ball and the substrate. If the local tangential stress becomes high enough, local slip will occur


69

between the surface of the ball and the substrate. During this local slip, external energy will be converted into heat motion; i.e. the slip will contribute to the rolling resistance. The slip velocities are generally small, usually less than 1 percent of the rolling velocity, but nevertheless produce in many cases a major part of the total resistance to rolling. The basic theory of slip contribution to rolling resistance is due to Carter (4). His main interest was in the action of a locomotive driving wheel, but he considered the simpler problem of two cylinders of radius R and length l of like materials pressed together and rolling on one another. One of the cylinders is subjected to a torque and the other to an equal countertorque (see Fig. 3.8). Thus, any state of stress or strain in one member, resulting from tangential tractive forces only, is matched by an equal and opposite state in the other, and the distribution of pressure between the members is unaffected by the traction, since the radial displacements of the surfaces in contact are complementary. Assuming that both bodies can be considered as half-spaces and that the theory of elasticity is applicable, the normal stress distribution in the contact area AaFxFa is given by the Hertz theory (5) 4Po

PG

π

1冢

x2

1A

a2

冣

(3.28)

where PoGL兾2al is the average pressure in the contact area. Carter used the Coulomb friction law but assumed that the static and kinematic friction coefficients are equal. Thus, in the contact area, if the tangential stress τ is smaller than µP(x), no slip occurs, but if the tangential stress reaches µP(x), local slip occurs.

Fig. 3.8

70

Rolling Contacts

(a)

P

leading a edge

x

c

-a slip zone

stick zone

(b) 1.0

F/F1= R/

A

0.5

0 0

1

2 v/v

3

4

Fig. 3.9

The solid line in Fig. 3.9(a) shows µP(x), which is equal to the maximum tangential stress possible in the contact area. The thicker line A shows the actual tangential stress distribution τ (x). No slip has occurred for cFxFa where τFµP(x). In the other part of the contact region, AaFxFc, slip occurs, and τGµP(x). It should be noted that, when the friction force F increases, so does the size of the slip region, until F reaches F1GµL, the maximum friction force, at which point slip occurs in the whole contact area. In general, the friction force is given by 2

冤冢 2a 冣冥

FGF1 1A

aAc

(3.29)

so that F→F1 as c → a (complete slip) and F → 0 as c →Aa (no slip). Thus, for a free-rolling cylinder (no applied driving torque), there is no rolling resistance due to local slip. When a torque is applied (with constant rolling velocity), the friction force F≠0, and the rolling speeds of the two cylinders differ. If Rω 1 (ω is the angular velocity) and Rω 2 are


71

the rolling speeds of the two cylinders, then the ratio ∆v兾v, where vG R(ω 1Cω 2)兾2 is the average velocity and ∆vGR(ω 1Aω 2 ) is the difference in the rotational velocities, increases in proportion to the fraction (aCc)兾2a of the contact area where slip occurs ∆v 4(1Aν)µPo (aCc) G v πGa

(3.30)

where G is the shear modulus. Using equations (3.29) and (3.30), it is possible to express the friction force as a function of ∆v兾v FGF1[1A(1Aκ ∆v兾v)2]

(3.31)

where

κG

πG 8(1Aν)µPo

The solid line in Fig. 3.9(b) shows F兾F1 as a function of κ ∆v兾v. It is obvious that FGF1 when 1 8(1Aν)µPo G G v κ πG

∆v

If only elastic deformations take place in the contact area, the pressure Po must be below the plastic yield stress σc of the solids in contact. Since G兾σc ≈100 for steel, and since typically 8(1Aν)µ兾π≈1, it follows that at the onset of complete slip ∆v兾vF0.01. In the wheel–rail contact area, the pressure Po is usually close to the plastic yield stress and in a typical case ∆v兾v≈0.005 at the onset of complete slip. Thus, if, for example, v≈20 m兾s, then ∆v≈0.1 m兾s. It is quite clear that the static friction coefficient is generally higher than the kinematic friction coefficient. Furthermore, the steady sliding friction coefficient may be velocity dependent and usually decreases with increasing sliding velocity. Taking into account these two effects, the relation between F and ∆v is of a qualitative nature, as indicated by the dashed line in Fig. 3.9(b). Another important effect may be the dependence of the static friction force on the time of stationary contact. If a train moves with the velocity v and if the contact area between the wheel and the rail has the diameter 2a, then the maximum time of stationary contact (for no slip) is τ *G2a兾v≈10−4 s. In the example, 2a≈0.01 m and v≈100 m兾s. This is a very short contact time and it is clear that the static friction coefficient will, in general, not have reached its maximum value before the contact is broken. The actual dependence

72

Rolling Contacts

of the static friction force on the time of stationary contact is known for the wheel–rail system and is probably not very well defined as it is sensitive to the nature of the contamination layer, which varies with the spatial and temporal location. It is well known, however, that the maximum traction of a locomotive decreases with increasing rolling speed, as is expected and in line with what has been said above. The curve denoted by the dashed line in Fig. 3.9(b) forms the basis for designing anti-skid braking systems. Since the friction is maximum at point A in Fig. 3.9(b) an optimally designed braking system should keep the motion of the wheel close to this point during emergency braking. However, even with a feedback system, it is difficult to stay continuously close to point A, since the dashed-line curve is a function of time, i.e. the coefficient of friction varies with changing road surface conditions. Most anti-skid systems for automobiles use a pulsed system where the wheel–road surface slip oscillates around the maximum denoted by A, continuously adjusting itself to the road conditions. In fact, the most important effect is not the increase in the sliding friction compared with gross slip (locked wheels), but rather the fact that rolling wheels allow steering or directional control which is otherwise lost.

3.7 Internal friction During rolling, different regions of the contact interface are first stressed, and then the stress is released as rolling continues. Each time a volume element in either body in contact is stressed, elastic energy is stored by it. Most of the elastic energy is later released as the stress is removed, but the cyclic deformations cannot occur entirely adiabatically and some energy will be converted into heat by the internal friction. This transfer of energy from the rotational and translational motion to heat will contribute to the rolling resistance. Numerous physical processes contribute to the internal friction of a solid and are usually described by a complex shear modulus GGG1CiG2 . For example, in metals, various modes of switching of segments of pinned dislocations can take place. In chain polymers the viscoelastic properties may be due to switching of chain segments between two configurations. In both cases, the stress, on its own, does not force the switch. Instead, its result is to bias the activation energies of the switch in the favourable and unfavourable directions, and the jump over the barriers occurs due to thermal excitation. For a weak applied external stress that varies very slowly with time, the deformations occur adiabatically and the deformation vanishes as the


73

external load is removed. However, at finite frequencies this is not the case and external energy is converted into heat in the solid. Any specific mechanism of internal friction will, in general, give rise to a characteristic peak in the frequency-dependent dissipative response G2(ω) of a solid at a frequency ω∼ 1兾τ, where τ is the relaxation time characterizing the average time a molecule (or a segment of a molecule) spends in a particular state before jumping to the next state. Most solids have many such peaks at different resonance frequencies. It needs to be noted that τ depends strongly (in fact, exponentially) on temperature, and the viscoelastic properties (at a fixed frequency) of a solid depend on the inverse temperature T much as they do on the frequency, often exhibiting several peaks as a function of T. The magnitude of internal friction varies considerably from material to material. Thus, for a steel ball rolling on a steel substrate, the contribution from the internal friction is usually rather small. Using the theory of viscoelasticity, it is possible to estimate that

µ∼ Po Im

冤G(ω)冥GP G CG G2

1

o

2 1

2 2

where Po is the average pressure in the contact area; G(ω) is evaluated at the frequency ω∼ v兾R, where v is the rolling velocity and R is the radius of the ball. For steel on steel, Po cannot exceed the plastic yield stress σc ∼ 0.01G1 so that Po兾G1F0.01. Furthermore, the ratio G2兾G1 for steel is typically smaller than 0.001 for the frequencies of interest, so that µF10−5. However, for a steel ball rolling on a polymer or a wheel rolling on a road, the internal friction may contribute a substantial part of the rolling resistance. It was stressed earlier that the internal friction of solids depends on the frequency ω of the external oscillating stress, and, in general, it will exhibit several peaks (resonances) at well-defined resonance frequencies. This fact is utilized in many practical applications. For example, in order to minimize the friction losses during steady driving of an automobile, the rubber used for tyres is designed in such a way that during rolling at normal speeds the deformations at the contact interface are mainly elastic, while during braking the tyres behave viscoelastically, producing high internal friction. This can be achieved since the typical deformation frequencies during rolling are different to those during sliding; in the latter case the relevant deformations result from surface asperities on the road sliding relative to the tyres. This generates pulsating forces acting on the tyres at characteristic frequencies ω S ∼ v兾r,

74

Rolling Contacts

Fig. 3.10

where r ∼ 0.1–1 mm is the typical extent of surface asperities on the road. With v ∼ 10–100 m兾s this gives ω S ∼ 104–106 s−1. During rolling, the characteristic frequency is instead the rolling frequency ω R ∼ v兾R, where R ∼ 0.1–1 m is the radius of the wheel, which gives ω R ∼ 10–103 s−1. Figure 3.10 shows schematically the complex shear modulus G(ω)G G1(ω)CiG2(ω) of a viscoelastic amorphous polymer. The shaded areas denote the frequency intervals typical for rolling and sliding friction. It should be noted that in the sliding regime G2HG1 , while in the rolling region G2FG1 . It is apparent that the friction coefficient will be much larger during sliding than during rolling.

3.8 References (1) Reynolds, O. (1876) On rolling friction. Phil. Trans. R. Soc., 166. (2) Tabor, D. (1955) The mechanism of rolling friction – Part II: The elastic range. Proc. R. Soc., A229. (3) Coulomb, C. A. (1809) Theorie des machines simples-en ayant en regard au frottement de leures parties et a la roideur des cordages (in French), Paris. (4) Carter, F. W. (1926) Proc. R. Soc., A112. (5) Hertz, H. (1896) The contact of elastic bodies. Miscellaneous Papers (Macmillan, London).

Chapter 4 Dynamic Characteristics of Rolling Motion

4.1 Introduction The dynamic characteristics of motion in rolling contact and their variation during rotation are of importance for the precision running of rotors supported by a rolling contact bearing. It is quite a difficult problem to study as the rolling contact bearing represents a complex mechanical system. A number of studies wholly or partly devoted to the problem of stiffness in rolling contacts have been undertaken. Novikov (1) derived expressions in a linearized form for the axial and radial stiffness of a radial thrust bearing, taking into account the non-linearity of elastic characteristics. Kharlamov (2) analysed the case of static equilibrium of a radial thrust bearing when the axial load appreciably exceeds the radial load. Simple expressions for the radial, axial, and transverse stiffness of the bearing were obtained for this case by linearizing the equations. The formulae obtained for radial and axial stiffness were compared with those obtained by Novikov. In the theoretical analysis carried out by Szucki (3) a procedure is given for computation of the elasticity of a ball bearing. The load is represented in the form of radial force, axial force, and the torque in the plane of these forces. The assumption was made that, in a mounted bearing, clearance and preload are both absent. In all the above studies the dependence of stiffness on angular rotation of the cage owing to changes in the position of rolling elements

76

Rolling Contacts

and errors of shape of the elements has not been considered. The importance of this problem was signalled in the theoretical work of Tamura and Shimizu (4) and Neubert (5), but the problem itself has not been solved. In this chapter, an advanced analysis of rolling motion dynamics will be undertaken. Also, the different types of resistance to rolling motion in bearings and their contribution to the total resistance will be considered.

4.2 Analytical evaluation of friction torque 4.2.1 Friction during rolling The resistance to relative motion in rolling contact bearings is due to many factors, the major one being rolling friction. This was long assumed to be the only resistance to motion in rolling contacts. As a result of studies into the nature of rolling motion it was established that the contribution of rolling friction to the overall resistance to motion is small, though its effect on wear and tear and operating temperature conditions is important. These factors are especially important for miniature instrument rolling contact bearings operating in very accurate mechanisms of servo-systems, magnetic recorder mechanisms, and other precision parts of instruments. The first research into rolling friction was carried out by Leonardo da Vinci in 1508. The concept of the coefficient of external friction arose as a result of these experiments. Leonardo da Vinci concluded that for identical surface roughness this coefficient is a constant for different bodies and equals 0.25. In early research the rolling friction was treated as a mechanical process; i.e. the interaction of rough surfaces of absolutely rigid solid bodies was considered. The concept of rolling friction as a mechanical process was introduced by Delagir and was further developed in studies by Parin, Euler, Lesley, and others (6, 7). Coulomb was the first to introduce the hypothesis of the dynamic nature of external friction. The experimentally obtained relationship of the friction force F due to rolling along a plane is given by the classic equation FGf

N GµN r

(4.1)

where f is the coefficient of rolling friction having the dimension of length, N is the normal load, r is the radius of the cylinder, and µ is a

Dynamic Characteristics of Rolling Motion

77

coefficient similar to the coefficient of sliding friction. This relationship, known as the Amonton–Coulomb law, relating frictional force and normal load, appeared to mark the end of a long period in the development of the science of rolling friction (7). In 1876, Osborne Reynolds (8) was the first to propose the hypothesis of relative slipping, according to which the frictional force due to rolling of a perfectly elastic body along a perfectly elastic substrate appears to result from relative slipping of contact surfaces on account of their deformation. This is also discussed by Solski and Ziemba (9). It can be seen from Fig. 4.1 that, in the region AC, surface layers of the cylinder are compressed (in a direction along the area of contact), and in the plane they are elongated. In the region CB of the contact area, these deformations take place in the opposite direction, as a result of which microslip occurs. Akhmatov, Tabor, and Tomlinson (10, 11, 12) have shown that slipping in the contact zone is not the only source of friction losses in rolling. Tomlinson and Tabor showed experimentally that, even during

Fig. 4.1

78

Rolling Contacts

the rolling of two cylinders or two balls of identical size made of identical materials, loss of energy occurs. According to Tomlinson, when two bodies approach one another, atomic and molecular interactions appear and energy is lost in overcoming them. This accounts for the additional energy loss. Taking into account the forces of molecular interactions during rolling of a cylinder over a plane, Tomlinson determined the frictional coefficient as follows KG

3 4

1冤

π

ef 2 1(Nrθ )

冥

(4.2)

where K is the coefficient of rolling friction, N is the load on the cylinder, r is the radius of the cylinder, e is a crystal lattice constant, f is the coefficient of sliding friction, and θ is a function of the elastic constants of the materials of the bodies in contact. Tomlinson (12) carried out experimental verification of the coefficientof rolling friction by introducing an additional coefficient λ related to the coefficient of rolling friction K (mm) by the expression KGλ l, where l is the moment arm applied to the rolling body. Coefficient λ was determined by the damping of the oscillations of a pendulum. The value of λ thus obtained was twice the theoretical value for amplitudes of pendulum oscillation ranging from 4 to 0.61°, which is explained by the fact that the relative displacement and molecular sizes are commensurable. Akhmatov (10) developed formulae for determination of the coefficient of rolling friction based on perfect contact between nominally flat surfaces. The works of Ishlinskii (13, 14), Drutowski (15), Palmgren and Snare (16), Tabor (11), and many others have great practical importance. They studied the origins of rolling friction during contact of real solids with imperfect elastic properties. Ishlinskii considered the resistance to motion in the case of the rolling of a perfectly rigid cylinder over a viscoelastic surface and along a plastically deforming plane. The contact stress distribution is shown in Fig. 4.2. The presence of frictional force is reflected by the asymmetrical contact stress distribution. Ishlinskii developed an expression for determination of the resistance force F (at low rolling velocities) in the case of the rolling of a perfectly rigid body along a plastically deforming plane FG

µv N Cr

(4.3)


79

Fig. 4.2

where N is the load on the cylinder, v is the rolling velocity, µ is the coefficient of internal friction of the substrate material, r is the radius of the cylinder, and C is the hardness of the substrate material. The coefficient of rolling friction is thus given by KG

µv C

(4.4)

The corresponding corrections, especially in the mechanical and molecular concepts of external friction, led to further development in this area. The large amount of research, basically of an applied nature, devoted to the mechanical concept has improved the understanding of the laws of elastoplastic deformation and failure.

4.2.2 Friction torque in the rolling contact In 1919, Palmgren and Snare (16) developed a theory of resistance to motion in roller bearings, assuming differential slipping as the main cause of rolling friction. According to Palmgren, the contact region is a point or a line in the case of an unloaded bearing, but in the case of a loaded bearing surface, contacts appear owing to elastic deformation of contacting bodies. When rolling motion commences there is, although small, relative local displacement and slipping. The energy spent on this slipping is considered as a loss. The hysteresis loss is added to the losses arising from elastic slipping. These losses depend on the load and the properties of the materials of the contacting bodies.

80

Rolling Contacts

Poritsky (17) extended the theory of friction in rolling contact by analysing the losses resulting from sliding friction caused by the gyroscopic effect. Kalker (18) analysed roller bearing friction under axial load as the friction due to differential slipping but he did not consider the loss arising from the motion of the rollers. Johnson (19) considered the movement of the rollers as a starting point to develop a theory of friction. However, he did not take into account all the factors acting on the roller, in particular the losses due to elastic hysteresis. The current tendency is to generalize the early theories into a unifying theory of total friction torque. When considering the total friction torque it is necessary to break down the torque into a number of its constituent components. It can be represented by the following equation (20) MG(MdsCMgyrCMhysCMdeCMcCMeCMT )K

(4.5)

where Mds is the friction torque due to differential slipping of rolling elements on a raceway, Mgyr is the friction torque arising from gyroscopic spin or deviation of the axis of rotation of a rolling element, Mhys is the friction torque arising from elastic hysteresis losses, Mde is the friction torque due to the deviation of rolling elements from the required geometric shape and surface roughness, Mc is the sliding friction torque along the edge of the raceways, Me is the friction torque due to shearing of the lubricant film, MT is the friction torque arising from temperature effects, and K is a correction coefficient taking into account any other factors contributing to the friction torque such as vibration. Friction torque due to differential sliding Consider the friction torque caused by differential sliding, Mds , for the case where the ball rolls along a groove with a radius of curvature R in a plane perpendicular to the direction of rolling. As shown in Fig. 4.3, pure rolling will occur along two lines located on an ellipse of contact at a distance 2ac . In other parts of the contact ellipse there will be sliding because of the unequal distance of contact points from the axis of rotation. Friction torque due to differential sliding of rollers can be expressed in terms of the work done, A, by the rolling contact bearing in unit time AG(Fi liCFo lo )z

(4.6)

where Fi and Fo denote sliding friction caused by differential sliding during rolling on a ball along the surface of the inner and outer ring


81

Fig. 4.3

respectively, li and lo are the distances travelled in unit time by a point located within the contact zone of the ball with the inner and outer ring respectively, and z represents the number of balls. The above equation is valid for the case where the inner ring rotates and the outer ring is stationary. The distances li and lo can be estimated from the following expressions li G2πRl i nb

(4.7)

lo G2πRlo (nAnb )

(4.8)

where n is the rotational speed of the inner ring of the bearing, nb is the rotational speed of the ball, and Rli and Rlo represent the radii of curvature of the groove on the inner and outer ring respectively. The friction torque resulting from differential sliding of the ball is Mds G

A 2πn

(4.9)

82

Rolling Contacts

or after expansion Mds G

PDo z 4db

冢

1A

d 2b D

2 o

冣

cos2 α (ro BoAri Bi )µ

(4.10)

where µ is the coefficient of sliding friction, Bi and Bo are the coefficients determined by the geometry of contact deformations of the inner and outer race respectively, and ri and ro are the respective radii of contact areas on the inner and outer ring. Friction torque due to gyroscopic spin Torque caused by gyroscopic spin, Mgyr , appears in the presence of contact angle α on the balls. In order to estimate this torque, it is necessary to determine the moment of inertia of the ball, I IG

π 60

d 2b

ρ g

(4.11)

where ρ is the density of the material of the ball. The friction torque due to gyroscopic spin then equals Mgyr GIω b z sin α

(4.12)

where ω b G(πnb )兾30 is the angular velocity of the ball and the rotational velocity of the ball is given by nb Gnshaft

D 2o d 2b cos2 α 2Do db

(4.13)

where nshaft is the rotational velocity of the shaft on which the bearing is mounted. Torque Mgyr increases with an increase in contact angle α and attains a maximum when α Gπ兾2. In the case of a radial bearing, α G0, so that Mgyr G0. In order to avoid gyroscopic spin of balls, it is necessary to satisfy the inequality MgyrFNdb µz

(4.14)

where N is the axial load. The effect of gyroscopic spin is especially noticeable in radial thrust bearings. Friction torque resulting from hysteresis losses Energy losses due to elastic hysteresis in the material of contacting bodies can be determined by assuming that during the loading cycle a specific quantity of energy is expended. Additional loading influencing hysteresis losses is created during high-speed rolling on account of centrifugal forces. Therefore, in all the expressions presented below, torque Mhys must be treated as a function of rotational speed.


83

Fig. 4.4

When a ball rolls along a raceway, Hertz’s contact ellipse is formed between them (Fig. 4.4). Energy developed during the rolling motion of a body about one point of the raceway is proportional to the load p and to the magnitude of deformation δ . The contact time t of the rolling body with the raceway is directly proportional to the length of the contact area in the direction of motion 2b and inversely proportional to the circumferential velocity rω (Fig. 4.5). Thus t⯝

2b

(4.15)

rω

Fig. 4.5

84

Rolling Contacts

Energy losses during contact of the rolling body with one point on the raceway are given by Mhys ω t⯝pδ

(4.16)

According to the Hertz theory, where there is point contact between two bodies the magnitude of deformation is expressed as 3 δ GK 1 ( p3 ∑ r)

(4.17)

where ∑ r is the sum of the reciprocals of the principal radii of curvature of the bodies in contact, given by (r1Cr2)I

(4.18)

(r1Cr2)II and K is a coefficient depending on the auxiliary function F(r)

冤

F(r)G

冥

(r1Ar2)IC(r1Ar2)II

∑r

(4.19)

Coefficient F(r) is usually determined from appropriate tables. It is possible to consider approximately Mhys G∑ p4兾3

(4.20)

The above expression is valid for a point contact when δ ⯝p2兾3 and b⯝p1兾3. A more exact equation for loss caused by elastic hysteresis is given in the following form Mhys G1.25B10−4Do db−2/3 ∑ pi4/3

(4.21)

where pi is the load on the ith ball. In practice hysteresis losses are independent of the shape of the raceway. Friction torque due to geometric errors The main cause of changes in friction torque are errors in the shape of rolling elements. Torque due to error of shape can be expressed as Mde G

dx N dk

(4.22)

where k is the angle of rotation of the moving ring relative to the stationary ring, x is the linear displacement of the moving ring in the direction of the load N acting on it and arising because of the deviation in the geometric shape and surface roughness of the contacting surfaces.


85

In a real radial bearing, the centre O of the circle of the moving inner ring does not coincide with the theoretical centre O1 (Fig. 4.6). Errors ∆z1 and ∆z2 arise owing to geometric deviations from the nominal dimensions. While determining the centre O1 , it is assumed that the bearing is loaded purely by a radial load R applied at point O1 . In the case analysed, the load is supported by two balls which roll without slip. Thus Mde GR Mde G

dx

(4.23)

dk R

sin(α π兾z)

冢 dk sin AA∆z 2D cos AC dk C∆z 2D cos α 冣

B

di

dz1

2

o

AG

di

dz2

1

1

o

2π Aα 1 z

Fig. 4.6

(4.24)

86

Rolling Contacts

where di is the diameter of the circle of the inner ring, ∆z1 and ∆z2 are the geometric errors, and Do is the diameter of the outer ring. Friction torque arising from raceway effects It it is assumed that the inner ring rotates about a vertical axis and touches, under the action of its own weight, the ball at one point only, then the friction torque due to the contact of rolling elements with the raceway is given by an expression of the form McI G

4 冢

Do

1A

d 2b D

2 o

冣冤

cos2 α sin α Ctan−1

冢

db sin α 2RI

冣冥 G µ c

(4.25)

where Gc is the raceway mass, RI is the radius of the inner ring, and db is the diameter of the ball. The bearing housing is in contact with the inner or outer ring and hence at the time of operation a friction force appears as a result of contact of the housing with the guiding edges. Consequently, the friction force is reduced or increased depending on the type of fit between the housing and guide rings. The expressions for the torque are as follows: housing fitted on to the outer ring M ocII G1.38B10−4Gc µn2Do ε

DoAdb cos α

冢

Do

冣

2

(4.26)

housing fitted on to the inner ring M icII G1.38B10−4Gc µn2Di ε

DoAdb cos α Do

冢

冣

2

(4.27)

where Do and Di denote the diameter of the outer and inner ring respectively, and ε is the eccentricity of the housing with respect to the bearing axis. The total torque is the sum Mc GMcICMcII

(4.28)

The housing is in its most suitable position when it rests on the outer ring as there is a reduction in the deformation of the housing owing to the centrifugal force. Friction torque due to shearing of the lubricant film The presence of a lubricant between contacting elements at the rolling contact leads to additional energy dissipation on account of the viscosity of the lubricant which, in turn, depends on the lubricant physical


87

characteristic, the contact pressure, the relative velocity of lubricant flow, and the temperature within the contact zone. The lubricant film formed in the contact zone prevents direct contact of rolling elements, with a subsequent reduction in wear and tear of the interacting surfaces. The cross-section of the contacting bodies is shown in Fig. 4.7. The energy lost on overcoming friction in the lubricant film per unit length of a cylindrical roller bearing is W oc G

6.5v2η 1(ho a)

(4.29)

where vG∂x兾∂t is the longitudinal velocity of the point of contact during rolling of the cylinder, η is the viscosity of the lubricant, ho is the minimum thickness of the lubricating film, and aG12 (1兾RxJ1兾R1 ). Here, Rx is the radius of curvature of the surface of the cylinder within the loaded contact zone, and R1 is the radius of curvature of the surface of the raceway within the loaded contact zone. The frictional loss for a ball has the following form W bo G

6πv2η 1a 4m2 ln (2b兾3a)1b ho

冢冣

(4.30)

where 2m is the maximum thickness of the lubricating film, and bG12 (1兾RyJ1兾R2 ), Ry and R2 are the radii of curvature perpendicular to the direction of rolling.

Fig. 4.7

88

Rolling Contacts

The energy expended on overcoming friction is sensitive to changes in the values of m and ho and is logarithmically dependent on the thickness of the lubricating film. The resistance moment due to shearing of the lubricating film can be approximately expressed in the following way Me G2zW bo

(4.31)

where z denotes the number of rolling elements in the bearing. Friction torque caused by elevated temperature Temperature does not uniformly affect friction torque. With an increase in temperature to around 100–120 °C, the friction torque actually decreases, which is explained by the decrease in the viscosity of the lubricant. An increase in temperature beyond 120 °C could cause an appreciable increase in the constant component of the friction torque as a result of changes in dimensions of individual elements of a rolling contact bearing.

4.2.3 Total friction torque So far the discussion has been confined to the existing state of the theory of friction torque in rolling contacts. However, the methods presented above are not very useful for practical applications. Quite often, it is sufficient to determine friction torque magnitude approximately, especially in the case of a bearing of secondary importance for the reliable operation of a whole system. In most practical applications, it is sufficiently accurate to assume that the coefficient of rolling friction, µ, is 0.0010 for a self-aligning ball bearing, 0.0011 for a cylindrical roller bearing, 0.0013 for a thrust ball bearing, 0.0015 for a deep groove ball bearing, 0.0018 for a tapered and bevelled roller bearing, and 0.0045 for a needle roller bearing. A simplified form of friction torque is MGµF

d 2

(4.32)

where F is the load on the contact and d is the diameter of a rolling element. The friction torque of an assembly of rolling contact bearings and also the torque due to individual bearings are determined in different ways by various researchers. According to Palmgren and Snare (16), the torque of resistance to rolling is expressed in the form MGM1(F )CMo (n)

(4.33)


89

where M1(F)Gf1 Fdb

冢C 冣 F

c

(4.34)

s

Mo (n)Gfo d 2b (γ n)2/3

(4.35)

where fo and f1 are factors depending on bearing design, load, and lubrication method, c is an exponent depending on the type of bearing, γ is the kinematic viscosity, Cs is the static carrying capacity of the bearing, db is the diameter of a rolling element, and n is the rotational velocity. The torque required to overcome the resistance to rolling motion can be approximately calculated using the following formula MG

Q µd 2

(4.36)

where Q is the radial load on the bearing and d is the diameter of the rolling body. The expression for the estimation of friction torque in a ball bearing is MG1.4Q

冢 d C1冣 µ Do

(4.37)

b

where Do is the diameter of the ball race of the outer ring and db is the diameter of the ball. When a radial load acts MGMoC1.25 µ

Do db

Pr

(4.38)

For a bearing supporting an axial load only MGMoC1.5 µ

Do Pa db

(4.39)

When both radial and axial loads act simultaneously MGMoC(1.5PaC1.25Pr )µ

Do db

(4.40)

All these expressions give an approximate value of friction torque since they do not take into account the influence of many factors present during the operation of a rolling contact bearing.

90

Rolling Contacts

The formula for the determination of the friction torque in a lubricated bearing is as follows MG− η πz

Do

LF

2

D1n 30db

[ACBCCDC16.5ao ]

(4.41)

and AG

4.5db ao E a2oCho db 9db ao E

BG

4a2oCho db

冤

CG

15F 3/2 9a20 1F A 21(2ho ) 2ho 1(2ho )

冤1(h d )Ctan ao

DGtan−1

o

EG FG

−1

b

冥冥

2ao 1(ho db)

db hoCa2o 2ho db 2

where η is the viscosity of the lubricant, n is the rotational velocity of the inner ring, z is the number of rolling elements, D1 is the diameter of the outer ring, LF is a coefficient depending on the ball race radius of curvature, ao is the length of the hydrodynamic lubricating film, and ho is the minimum thickness of the lubricant film. The above equation is not very handy for practical situations and an alternative to it has the form MG0.452

ζG

(1Aζ )(1Cζ )2 2 σ D o z 0.625B10−3 ζ λ K1r

db Do

λKG

1 1Cζ sin β

冢

6.5d −0.65 b

冣

(4.42)


91

where β is the contact angle, Do is the pitch circle diameter of the balls, σ is the allowable contact stress (usually 850–2800 MPa), and r is the radius of the ball race. When a bearing operates under vibration conditions and if its main axis is horizontal, then the friction torque can be computed using the formula MGµv Q

ds

(4.43)

2

where Q is the radial load on the bearing, ds is the diameter of the shaft, and µ v is given by

µ v Gk

2.8(1Cα ) Do ds g

d

V m V mv

where Vmv is the mean relative collision velocity of the inner ring and balls with the outer ring of the bearing, Vm is the mean frequency of impacts of the inner ring and balls against the outer ring, µ is the rolling friction coefficient, d is the ball diameter, g is the gravity constant, and α is the coefficient of speed restoration during impact. In the case of vertical vibration of a bearing with a vertical axis of rotation, the friction torque equation has the form MGfA Q

ds

(4.44)

2

where 2k(1Cα )

冢 d Ccos β 冣 f d g sin β V

fA G

Dri

mv

n

Vm

s

Dri is the diameter of the ball race on the inner ring, fn is the coefficient of non-uniformity of load distribution between balls (usually 0.9), and β is the angle at which the load acts on the ball. All these formulae give only an approximate value of friction torque or contain parameters that can only be determined experimentally.

4.2.4 Variable components of friction torque When a ball rolls along the ball race, the associated friction torque is then given by the formula MGW兾ω

(4.45)

92

Rolling Contacts

where W is the power expended during the rolling of the ball and ω is the angular velocity of the ball. Taking into account the contribution of the oil film to rolling losses, the power expended during rolling of the ball along the ball race can be expressed in the following form WG

17.3πv2η 1α (2β C3α )1β

(4.46)

Here the speed v is practically constant, the viscosity η can be considered independent of contact pressure, and α and β are parameters depending on the radii of curvature of the bodies in contact 1 1 αG J ; r Rin(out)

1 1 βG C r rz

For balls and ball races of ideal geometric form, the radii r, Rin(out) , and rz are constant. In actual bearings they are all variables. The geometric errors in ball races can be expressed in terms of a Fourier series. In the case of the outer ball race, the function for errors has the form m

φ 1(ϕ )G ∑ Ak cos(kϕCϕk )

(4.47)

k G1

and for the inner ball race m

φ 2 (ψ )G ∑ Bk cos(kψ Cψ k )

(4.48)

k G1

where m is the number of harmonics, Ak , and Bk are the amplitudes of the harmonic components, and ϕk and ψ k are the corresponding phase angles. The amplitude of the harmonic components is expressed in the form Ak G1(a2kCb2k)

(4.49)

where ak G

bk G

1 π 1 π

冮

2π

冮

2π

φ 1 cos kϕ dxG

0

0

φ 1 sin kϕ dxG

1 N ∑ ρoi cos ki∆ϕ∆x π i G1

1 N ∑ ρoi sin ki∆ϕ∆x π i G1


93

Here ∆ϕ is the interval for computing errors in a ball race. Usually, ∆ϕ ‚1°. Phase angles are expressed in the following way

ϕk Gtan−1

bk ak

Analyses carried out with the help of the expressions introduced above show that the magnitude of the variable component of friction torque is negligibly small compared with that obtained experimentally. The second approach to determination of the variable friction torque is as follows. Assume that the ball rolls without slip between two surfaces as shown in Fig. 4.8. The lower surface differs slightly from an ideal plane surface and is stationary. The upper surface is an ideal plane surface and moves with velocity v in such a way that at any moment in time no point on it changes its position along the height [Fig. 4.8(a)]. For the displacement of the upper plane it is necessary to overcome a certain rolling friction force Qc . Here, the ball will be either in the valley or on the peak. During rolling of the ball over the peak [Fig. 4.8(b)], it is necessary to do work proportional to the magnitude of elastic deformation and the height of the peak. The force Qc will also change. If the

Fig. 4.8

94

Rolling Contacts

height of the peak depends on its position along the length, then the force Qc will be a function of time. Furthermore, the model adopted also helps to explain the effect of lubricating oils of different viscosities on Qc (t). It is well known that the lubricant forms a film able to separate the contacting surfaces, and an increase in lubricant viscosity helps to achieve complete separation. Also, with increasing viscosity the amount of work done increases, since the deformation takes place within a larger nominal contact area. Force Qc changes proportionally to the work done. Thus, with an increase in the viscosity of the lubricant the amplitudes of the variable component of friction torque increase. Besides that, when the viscosity of the lubricant increases, the microroughness effects regarded as high-frequency components of the force are suppressed. These assumptions correspond well with the results of experiments which show that, with increase in viscosity, the highfrequency components of the variable friction torque are damped but the amplitudes of the lower components are increased. In the model outlined in Fig. 4.8 it can be seen that Qc (t) is possibly related to the force developed by the deformation of the projection and acting in the normal direction. The magnitude of the force when the ball approaches the uneven surface depends on the height of the bump and is described by the Hertz theory of elastic contacts. Thus, it is possible to write Qc GF(P). The friction torque developed when the ball rolls along the race can be expressed by

冦

MG 11.24

ηo 5 rrC0.5db ao bo ho rr db

冤

冥

2

e nσ o (nσ o )3/2

冧

B ω

(4.50)

where y 4 1 y 2 3 A C ao nσ o ao 2(nσ o )3/2

冢冣

BG

冢冣

In the above expressions, η o is the dynamic viscosity of the lubricant, ho is the thickness of the lubricating film, ao and bo are the major and minor axes of the ellipse of Hertzian contact respectively, rr and db are the radius of the ball race and the diameter of the ball respectively, n is the pressure–viscosity coefficient of the lubricant, σ o is the maximum compressive stress at the centre of the contact zone, and ω is the angular velocity of rolling of the ball. Equation (4.50) was obtained on the basis of the solution of a hydrodynamically lubricated contact. Since the major axis of the contact


95

ellipse is larger than the minor axis by one order of magnitude, it is assumed that the lubricant flows only in the direction of rolling (plane contact hydrodynamic problem). The resistance to differential slipping when the ball rolls along the ball race has also been taken into account. In addition, the hydrodynamic flow of lubricant, the deformation of surfaces, and the exponential dependence of viscosity on pressure are all included. Equation (4.50) is most suitable for heavily loaded contacts. An important element in the calculation of friction torque is the thickness of the lubricating film, ho . The following expression can be used to determine it hoin(out) G

3.17[η o (UaCUb )]0.75n0.6 K o0.15 ( ∑ ρ)0.45

(4.51)

where Ko G

3 31 3

8µ 4

3 1 [P 2E( ∑ ρ)2]

∑ ρG A db

Rin G

1

1 J Rin Rin(out)

0.5DinCrin

Rout G

cos β in 0.5DinArin cos β in A(rinA0.5db ) C0.5db cos β out cos β out

Ua ≈Ub ≈(ω inAω o )Din 兾2 In the above expressions, Ua and Ub are the relative speeds of the surfaces with respect to the contact point, µ is a coefficient that is a function of an elliptic integral, P is the normal load on the contact point, E is the elastic modulus, ω in is the angular velocity of the inner ring, and ω o is the angular velocity of the centre of the ball. Subscripts ‘in’ and ‘out’ represent the inner and outer rings.

96

Rolling Contacts

Equation (4.50) can be rewritten to facilitate its use at different loads. Thus

冤

MG

6.11η o s6µ5ν rinC0.5db 2 1.6 3/2tP 1/3 P e B ω kt 3/2 rin db

冢

冣

冥

(4.52)

where

冤冢 a 冣 ym

BG

o

tG

4

2 1 ym 2 1 A C0.407 3/2 1/2 1/3 3 tP ao t P

冢冣

冥

n πs2µv

sG1.08B10−3

kG

1/3

冢∑ ρ冣 1

3.48µ 0.15η o0.75 (UaCUb )0.75n0.6 E 0.05 ( ∑ ρ)0.45

ym 兾ao is the relative position of the trajectory of pure rolling, ao is determined from the Hertz contact theory, and v is a coefficient that is a function of an elliptical integral. Close estimation of the sum of two terms

冢a 冣 ym o

4

冢冣

2 1 ym A 3 tP 1/3 ao

2

shows that it does not exceed 5 percent of the term 0.407(t −3/2P −1/2). Hence, it is safe to neglect it in further analyses. In Fig. 4.9 the dependence of friction torque on lubricant viscosity is shown for two sliding velocities. From this graph it can be seen that the friction torque, even at low sliding velocity, sharply rises and can reach a very high value. This is contrary to experimental observations and is due to the close dependence of viscosity on pressure. A formula to determine shear stresses in an oil film that takes into account the dependence of the viscosity of the lubricant on pressure by the exponential law and on temperature by the polytropic law has the form

τG

21(427λ Ts η s) Z ho1γ 1[k2γ C2A1兾(a1C1)]

(4.53)


97

Fig. 4.9

where ZGln

kG

1(a1C1)(k 2γ C1C1{k 2γ [k 2γ C2A1兾(a1C1)]}) 1[k 2γ C(1Ca1)]

(UaAUb )1η s 21(427λ Ts )

In the above expressions, Ts is the temperature of the lubricating film, η s is the viscosity of the lubricant at the given pressure and temperature Ts , λ is the coefficient of heat conductivity of the lubricant, and γ and a1 are experimentally determined coefficients. The friction torque due to rolling is given by

冮冮 τ dF

MK G

(4.54)

F

The above equation is difficult to use because of its complexity. However, at low sliding speeds and small loads the shear stress can be

98

Rolling Contacts

computed using the classical formula for the isothermal domain

τ Gη o

v

(4.55)

ho

The following simplifying assumptions are introduced: (1) The contact radius of curvature ρm (Fig. 4.10) is determined, approximately, by the formula

ρm G

ri db

(4.56)

2riCdb

(2) The instantaneous rolling axis ω of the ball relative to the ball race passes through the trace of pure rolling (i.e. rolling without slipping)

Fig. 4.10


99

in the plane passing through the axis of the ring and the centre of the ball. (3) Since the stress in the contact zone is relatively small, it is possible to consider that viscosity does not depend on pressure. (4) The area of contact in the outer direction along the axis is assumed to be plane. According to Fig. 4.10, the distance from any point on the profile of the contact area to the axis of pure rolling ω is z≈

1 2ρm

y2

(4.57)

The sliding speed at the contact during rolling is VG

1 ( y 2mAy2 )ω 2ρm

(4.58)

The elemental resistance to sliding is equal to dTGη o

v ho

dx dyG

η o ( y2mAy2 ) dx dy 2ρm ho

(4.59)

The elemental friction torque in rolling with slipping is dMGdT(zmAz)G

ηo ω ( y2mAy2) dx dy ho 4ρ2m

The friction torque is ao

MG4

冮冮 0

bo

dMG

0

η o a5o bo 4 2 2 1 ymA ymC ω ho ρ2m 3 5

冢

冣

(4.60)

where ym Gym兾ao , i.e. the position of the trajectory of pure rolling. Detailed calculations show that y4mA23 y2m does not exceed 5 percent of the expression in brackets. Taking this into account, the expression for the torque, after transformation, is MG

0.8η o ω s6µ5v rinC0.5db k

冢

rin db

2

冣P

2.1

(4.61)

Equation (4.61) can be written in the following form MGCP 2.1

(4.62)

where C is a constant for the given type and identical geometric size of bearing.

100

Rolling Contacts

The power expended in rotation of the inner ring of the bearing is NGNinCNout

(4.63)

where Nin and Nout are the power spent during rotation of the ball along the raceway of the inner and outer rings respectively. In order to overcome the resistance to inner ring motion it is necessary to apply a torque given by MGMinC(MinCMout)

ωb ω

(4.64)

where ω b and ω are angular velocities of the ball and the inner ring of the bearing respectively. On the basis of equation (4.61), it is possible to write for one ball

冤

MG CinC(CinCCout )

ω b 2.1 Pi ω

冥

(4.65)

and for a bearing with s balls

冤

MG CinC(CinCCout )

ωb ω

冥∑P s

2.1 i

(4.66)

i G1

Here Cin and Cout are constants for the inner and outer rings respectively. Load acting on a single ball In the case of point contact, the load acting on the ith ball bearing is Pi GKδi3/2

(4.67)

where K is a coefficient depending on the curvature of the contact elements of the bearing at the contact points and on the elastic characteristics of the materials from which these elements are made, δ i is the elastic deformation of the contacting bodies. The deformation in the direction of the ith ball depends not only on the force due to preliminary tension and displacement of the rotor but also on the geometric errors in the rings. Taking the above into account, the deformation in the direction of the ith ball in the case of a radial bearing can be expressed in the form

δ i Gδ oCx cos ϕi cos α oCy sin ϕi cos α oCz sin α o Cρin (ψ i ) cos α oAρout(ϕi ) cos α 0


101

where x, y, and z are displacements of the rotor centre in directions X, Y, and Z. The static displacement of the rotor centre can be expressed by the equations xGF 1out(t)AF 1in (t) yGF 2out(t)AF 2in (t) zGF 3out(t)AF sin (t) (4.68) For a radial bearing, the radial load in the direction of the ball owing to axial preload has the following expression Pio GK{δ oC[F 1out(t)AF 1in (t)] cos ϕi cos α o C[F 2out(t)]AF 2in (t)] sin ϕi cos α oC[F 3out(t)AF 3in (t)] sin α o Cρin (ψ i ) cos α oAρout (ϕi ) cos α o }3/2 This formula is valid if the inequality δ oHδ x is satisfied, where δ x is the variable component of deformation. The load acting in the direction of the ith ball from the radial force is found using the formula Pi p G4.37

P s

(cos α i )2/3

(4.69)

where P is the radial load and α i is the angle between the direction of the radial load and the position of the ith ball. In this case, α i coincides with ϕi . Finally, the expression for the total load on the ith ball, taking into account the preload and radial load, takes the form Pi GPioCPi p

(4.70)

Friction torque resulting from the exponential dependence of viscosity on pressure The friction torque when a ball rolls along the ball race with an angular velocity ω is given by MG

W

ω

The power W used by friction while rolling in the lubricating film can be determined in the following way. Let an element of the surface of a

102

Rolling Contacts

Fig. 4.11

ball move with velocity V to the left (Fig. 4.11). Obviously, h at point xGx1 is a function of time t. On the basis of elementary considerations ∂h ∂t

G−V tan γ

(4.71)

where tan γ G∂h兾∂x, and hence ∂h ∂t

G−V

∂h

(4.72)

∂x

An element of the ball surface touching the film does an element of work on a unit distance −p dx dy∆h. The total work done is ∂h

冮冮 Ap dx dy ∂t dt

FG

S

Assuming the expression for the thickness distribution of a film hG hoCα x2Cβ y2, the power required is WG2α V

冮冮 px dx dy

(4.73)

S

where 12Vnη o1α 1 p′ pGA ln 1A n (2β C3α )1h3o

冤

冥

(4.74)


103

Here p′ in non-dimensional coordinates is expressed in the following way p′G

χ (1Cη C χ 2 )2 2

In the expression for pressure distribution [equation (4.74)] the following exponential dependence of viscosity on pressure is assumed: η Gη o e np . For further analysis, the following notations are introduced RG

2Vho3/2 n1β

LG

12Vη o n1α (2β C3α )1h3o (4.75)

After transformation WG−R

冮冮 x ln(1ALp′) dx dy

(4.76)

Ω

where p′G

x (1Cx Cy2 )2 2

The integration limits are determined by the following values

Ω G0FxFS Ω G兩y兩‚a From equations (4.74) and (4.75) it follows that the solution is meaningful in the interval 0FLp′F1 The function ln(1ALp′) is expanded into a Taylor series

冤

冥

(Lp′)2 (Lp′)3 (Lp′)n C C· · ·C ln(1ALp′)GA Lp′C 2 3 n (Lp′)k k k G1 S

G− ∑

(4.77)

104

Rolling Contacts

Taking into account equations (4.75) and (4.76), where kG1, 2, 3, . . . , the power loss is expressed by the expression

冮冮

R

S

∑

W(L)G

k G1

Ω

k

Lk

x kC1 (1Cx2Cy2 )2k

dx dy

(4.78)

It is not difficult to show that equation (4.78) converges uniformly relative to k. By putting 1Cy2 Gb2, the following is obtained 2Lk

S

W(L)GR ∑

k

k G1

a

冮冮 0

x kC1

S

dy

(b2Cx2)2k

0

dx

(4.79)

Equation (4.79) can be rearranged into the following form S

Lk

k G1

k

W(L)G2R ∑

冮

a

TkC1,2k dy

(4.80)

0

where

冮

x kC1

S

TkC1,2k G

[b2( y)Cx2]2k

0

dx

Assuming that a

TkC1,2k GAk ( y);

Ck G

冮 A ( y) dy k

0

the result is Ck k L k G1 k S

W(L)G2R ∑

(4.81)

In equation (4.81), Ck can be expressed as a

Ck GTkC1,2k

冮 1[(1Cy ) 1

2 3kA2

0

]

dy

The limits of integration are determined by the boundaries where the pressure is zero. Therefore, the limits of integration are taken as the mean value of pressure isobars in the direction of the y axis. A few isobars are taken for this purpose with different values of pressure.


105

Fig. 4.12

According to Fig. 4.12, aGa( p′), and therefore the following expression applies x

(4.82)

Ap′G0 (1Cx2Cy2)2 The value of a is found in the following way a( p′)G

1 x2Ax1

x2

冮

x1

1冤1冢

x

冣A(1Cx ) dx冥

p′

2

(4.83)

where x1 and x2 are points of intersection of isobars with the x axis. For final computation of power losses, the value of a is taken as the value corresponding to p′G0.0001. It is interesting that, in spite of the fact that the values of a are different for different p′, the integrals a

冮 1[(1Cy ) 1

2 3kA2

0

]

dy

(4.84)

differ very little. Thus, the final expression for determination of the power losses due to the rolling of a ball along the ball race, assuming an exponential

106

Rolling Contacts

dependence of viscosity on pressure, takes the form WG2R(2.4998LC0.03276L2C0.002619L3 C0.0005112L4C0.00009736L5C0.00002076L6 )

(4.85)

where R and L are determined from equation (4.75). It is possible to use two terms of the series to compute the power loss, and the error will not be more than 0.5 percent. The friction torque due to rolling of one ball is expressed in the following form MG

2R (2.4998LC0.03276L2) ω

(4.86)

where ω is the angular velocity of the ball. The friction torque of an assembly of balls when the inner ring rotates is given by MG

1

ω

s

∑ (WoutiCWini )

(4.87)

i G1

where s is the number of balls, and Wini and Wouti are the power losses due to rolling of the ball along the inner and outer rings respectively.

4.3 Elastic and damping characteristics of the rolling contact 4.3.1 Static stiffness of the rolling contact Stiffness of radial bearings The analysis of radial stiffness is based on the assumption that the bearing remains in static equilibrium throughout rotation. According to the Hertz theory, the total elastic force at the point of contact of the ith ball with the inner and outer rings is expressed in the following way Pi GKδi3/2

(4.88)

where δ i is the deformation of both rings in the direction of the ith ball and K is a proportionality coefficient. Equation (4.88) is also applicable to roller bearings, in which case the exponent is equal to 10兾9. Thus, the total elastic force at the contact points of rolling elements with the inner and outer rings can be


107

expressed as Pi GKδ ni

(4.89)

and the projection of this force on to the line of action of the applied radial load on the bearing is given by Pi GKδ ni cos ψ i

(4.90)

where n is the exponent mentioned above, ψ i GϕCiγ is the angle between lines of action of the radial load (direction of displacement of the moving ring) and the radius passing through the centre of the ith rolling element, ϕ represents the turning angle of the cage, γ G2π兾m is the angular distance between the rolling bodies, and m is the number of rolling elements in the bearing (iG0, 1, 2, . . . , mA1). It is assumed firstly that the bearing elements are ideal in shape, dimensions, and material properties. Then, the deformation in the direction of the ith rolling element is expressed by

δ i Gc cos(0.5θ max sin ψ i )Cδ 1Cδ 2 cos(θ max sin ψ i )

(4.91)

where c is the radial preload or clearance, and δ 1 and δ 2 are deformations caused by mutual displacement of rings at the contact points of the rolling element with the inner and outer rings respectively. With reference to Fig. 4.13, the magnitude of angle θ max is expressed by the formula

θ max G2 sin−1

z

(4.92)

d

Fig. 4.13

108

Rolling Contacts

where d is the diameter of the raceway of the bearing inner ring and z is the mutual displacement of the rings. Usually, θ max is small and may be taken as being equal to zero. Then equation (4.90) is transformed into Pi GK(cCx cos ψ iCy sin ψ i )n cos ψ i

(4.93)

where x is the displacement of the moving ring in the direction of the radial load, and y is the displacement of the moving ring in the direction perpendicular to the radial load. The reaction force of the complete bearing is equal to the sum of elastic forces of all rolling elements taking part in the transfer of load. Thus mA1

PG∑ Pi GK ∑ [cCx cos(ϕCiγ )Cy sin(ϕCiγ )]n cos(ϕCiγ ) i

(4.94)

i G0

In equation (4.94) only the terms with positive values in the square brackets are summed, since the rolling element does not participate in load transfer when it enters the unloaded zone. For the determination of y the condition of zero elastic force of the bearing along the y-axis will be used. Thus Py G0

(4.95)

since mA1

Py G∑ Pyi GK ∑ δ ni sin ψ i i

(4.96)

i G0

where Pyi is the projection on to the y axis of the elastic force of the ith rolling element. Equation (4.95) can be expressed in the following form mA1

K ∑ [cCx cos(ϕCiγ )Cy sin(ϕCiγ )]n sin(ϕCiγ )G0

(4.97)

i G0

Expanding equation (4.97) in a McLaurin series in powers of y and limiting it to the first two terms, the following is obtained mA1

0GK ∑ [cCx cos(ϕCiγ )]n sin(ϕCiγ ) i G0

mA1

CyKn ∑ [cCx cos(ϕCiγ )]nA1 sin2(ϕCiγ ) i G0

(4.98)


109

Hence, in a first approximation an expression for y is obtained

∑i G0 [cCx cos(ϕCiγ )] sin(ϕCiγ ) yG mA1 n ∑i G0 [cCx cos(ϕCiγ )]nA1 sin2(ϕCiγ ) mA1

n

(4.99)

The elastic properties of bearings can be conveniently studied by employing their stiffness, determined as the first derivative of the restoring force with respect to the displacement kG

∂P

(4.100)

∂x

Substituting equation (4.94) in equation (4.100) and taking into account equation (4.99), the following is obtained mA1

kGK ∑

i G0

冤

冤

cCx cos ψ iA

B cos ψ iA

A Bn

sin ψ i

CBnAAF(nA1) (Bn)2

冥

nA1

冥

sin ψ i cos ψ i

(4.101)

where mA1

AG ∑ (cCx cos ψ i )n sin ψ i i G0

mA1

BG ∑ (cCx cos ψ i )nA1 sin2 ψ i i G0

mA1

CG ∑ (cCx cos ψ i )nA1 sin ψ i cos ψ i i G0

mA1

FG ∑ (cCx cos ψ i )nA2 sin2 ψ i cos ψ i i G0

(4.102) Thus, the stiffness of each typical size of a bearing depends on preload or clearance c, the mutual displacement of the rings under the action of load x, and the angle of cage turn ϕ. However, equation (4.101) does not give a clear picture of how k depends on these values and is not useful for direct application. This equation can be used to obtain graphs of functions k(x) and k(ϕ ), employing a computer for further approximation by simple analytical expressions. In considering the stiffness of the bearings as a function of k(x), depending on c as a parameter, it is useful to take ϕ Gγ 兾2. For this

110

Rolling Contacts

position of the cage (radial load is exactly between two adjacent rolling elements) yG0, and equation (4.101) is appreciably simplified. The value yG0 also corresponds to the value ϕ G0, but this position is unstable in a bearing with initial clearance and can be impracticable. The derived expression for stiffness ought to be multiplied further by coefficient K, which takes into account the design and material of the bearing elements. Radial and torsional stiffness of radial thrust bearings In this section an analysis of the radial stiffness of a radial ball thrust bearing will be carried out. It is assumed that the contract angle of all balls is identical; i.e. it does not depend on radial displacement of the moving ring and the rotation of the cage remains constant. The centrifugal forces that appear during the rotation of a cage with balls affect the value of the contact angle. As a result, the angle between the balls and the outer ring somewhat decreases, and that between the balls and the inner ring increases. However, this phenomenon can be realized in practice only when the inner ring rotates at high speed (tens and hundreds of thousands of revolutions per minute), and is observed only in ball bearings. Let the outer ring of the bearing remain stationary while fixing the load, and let the inner ring and rolling elements be displaced to the right as shown in Fig. 4.14. The radial clearance of the free bearing co has an effect only on the value of contact angle β . When the ball is displaced from the initial position (the location of the centre of the ball is at O1 ) to a position with its centre at O3 , deformation takes place only in the region of displacement of the centre O2 , O3 and is equal to the axial preload ca . This load corresponds to radial preload c and gives the total given contact deformation of the ball resulting from preload

δgG

c cos β

(4.103)

The total contact deformation is equal to

δ 1 Gδ cCδ xiCδ yi

(4.104)

where δ xiCδ yi is the deformation of the ith ball caused by the radial displacement of the inner ring. Thus

δiG

c cos ψ sin ψ Cx Cy cos β cos β cos β

(4.105)


111

Fig. 4.14

Substituting equation (4.105) in equation (4.89) and adding the projections Pi on to the radial plane and on to the x axis, the following is obtained mA1

PGK ∑

i G0

冢cos β c

Cx

cos ψ i sin ψ i n Cy cos β cos ψ i cos β cos β

冣

(4.106)

Using the above equation, it is possible to determine the radial stiffness of a radial thrust bearing while neglecting the transverse displacements of the moving ring mA1

kGKn cos1Anβ ∑ [cCx cos(ϕCiγ )]nA1 cos2(ϕCiγ )

(4.107)

i G0

Now, let a torque act in the plane xOz (see Fig. 4.15) on a radial thrust ball bearing with a preload. As a result there appears a reaction torque of rings through an angle ξ. Let it be assumed that the twist appears

112

Rolling Contacts

Fig. 4.15

only in the plane xOz, and angular free transverse displacement is absent. Then the contact angle of each ball is changed by a value ε i , depending on ξ and ψ i (ε i Gε cos ψ i )

β i Gβ Cε cos(ϕCiγ )

(4.108)

The angle ε is very small when compared with angle β and hence it can be neglected. In such a case it is assumed that the contact angle of all balls remains unchanged and equal to β . Then the deformation of the ith ball caused by the twisting of rings will be expressed by

δ ξi G

ro sin ξ cos(ϕCiγ ) sin β

(4.109)

where rG(dkCdb )兾2, i.e. the radius of the circle passing through the centres of the balls, dk is the diameter of the raceway of the inner ring, and db is the ball diameter. The total contact deformation of the ith ball is determined by the expression

δiG

c ro sin ξ cos(ϕCiγ ) C cos β sin β

(4.110)


113

After summing i elastic moments from each rolling element, an expression for elastic torque about the y axis is obtained mA1

MGKro cos−n β ∑ [cCsin ξro cot β cos(ϕCiγ )]n cos(ϕCiγ )

(4.111)

i G0

Considering the magnitude of ξ, it may be presumed that sin ξ ≈ ξ. Then the contact stiffness of a radially supported bearing is determined from angle ξ as the first derivative of the elastic moment kξ GKnro sin−1 β cos1An β mA1

B ∑ [cCξro cot β cos(ϕCiγ )]nA1 cos2(ϕCiγ )

(4.112)

i G0

Consequently, the regularity of the variation in stiffness of a radial bearing is true for radial and torsional stiffness of a radial thrust bearing. Although equations (4.107) and (4.112) have been derived for ball bearings, they are also valid for roller bearings if nG10兾9 is assumed. Radial stiffness of thrust bearings For a thrust bearing (Fig. 4.16), deformation of the ith rolling element caused by the displacement of the inner ring in the axial direction with

Fig. 4.16

114

Rolling Contacts

axial preload and mutual twisting of rings is given by

δ i GcaCzCro sin ξ cos ψ i cos ξ

(4.113)

where ca is the axial preload and z is the displacement of the moving ring in the axial direction. Considering the low value of angle ξ and equation (4.89), the axial load transferred by the whole bearing is expressed by mA1

Pa GK ∑ [caCzCro ξ cosψ i ]n

(4.114)

i G0

and axial stiffness is given by ka G

∂P ∂z

mA1

GKn ∑ [caCzCξ ro cos(ϕCi γ )]nA1

(4.115)

i G0

In the above equation, δ iF0 means that the ith rolling element does not take part in the load transfer and hence the terms in the square brackets with a negative sign are not summed up. If the value of ca is negative (bearing with an initial radial clearance), then for zCξro ‚兩ca 兩 the axial rigidity of the bearing is equal to zero. For a radial thrust bearing the deformation of the ith rolling element is x cos ψ i

δ i GcaCzCro sin ξ cos ψ iC

(4.116)

cos β i

and the axial load transferred by the bearing is expressed in the form mA1

Pa GK ∑

i G0

x cos ψ i n caCzCro ξ cos ψ iC sin β cos β

冢

冣

(4.117)

In this case it is assumed that, for small values of ξ, sin ξ ≈ ξ. Differentiating equation (4.117) with respect to z, the following is obtained mA1

ka GK sin β ∑

i G0

冤

冢

冣

冥

x caCzC ro ξC cos(ϕCi γ ) cos β

nA1

(4.118)

In the case where caF0 the axial rigidity of a radial thrust bearing is equal to zero when

冢

冣

x zC ro ξC ‚兩ca 兩 cos β


115

From equations (4.115) and (4.118) it can be found that, when there is no radial preload or angle of mutual twisting of the rings, axial stiffness does not depend on the turn angle of the cage if the bearing is mounted ideally and accurately on a perfect rigid rotor without radial preload. If the shaft allows a constant deflection both in magnitude and direction, or if the moving ring is skewed or x≠0, the axial rigidity of the bearing varies periodically as it rotates. Hence, in actual conditions, axial stiffness of the bearing ka , like radial or torsional stiffness, depends on the turn angle of the cage ϕ. Another characteristic case will be considered now. Let the moving ring of the bearing have a twist with respect to the axis of the shaft. Then the deformation of the ith rolling element is expressed by x cos ψ i

冢

δ i GcaCzC ro ξC

cos β

冣 cos ψ Cr ξ cos qψ i

o

o

i

(4.119)

and the axial stiffness of the bearing is mA1

ka GKn sin β ∑

i G0

冤c CzC冢r ξCcos β 冣 cos(ϕCiγ ) x

a

o

Cro ξo cos q(ϕCiγ )

冥

nA1

(4.120)

where ξo is the angle of twist of the moving ring with respect to the axis of rotation and q is a coefficient depending on the relation of rolling element diameters to the circle of their centres. Since, in general, ϕq and ϕ are not repeated values, it follows that ka is not a periodic function if ϕ ≠0 and ξo ≠0 simultaneously.

4.4 Dimensional accuracy and contact stiffness Although components of precision rolling contact bearings can be manufactured with high accuracy, certain errors in the geometric shape and dimensions and deviations in the properties of materials are unavoidable. These errors affect only the periodic component of bearing stiffness. It is known that even in ideal bearings the amplitude of the periodic component of stiffness is considerable in the case of initial clearance. Hence the effect of errors is considered only for the case of preloaded bearings when the change in stiffness caused by imperfections of geometric shape and the characteristics and sizes of rolling elements outdo the changes in stiffness in an ideal bearing.

116

Rolling Contacts

4.4.1 Radial stiffness as a function of inaccuracies During the rotation of a bearing, the rolling elements roll along the raceways of the outer and inner rings and, hence, their contact points continuously change. In the case of an ideal bearing, this condition has no real significance. In an actual bearing, the rolling elements are either climbing over bumps or entering troughs while rolling along the raceway and the mutual position of the unevenness of the outer and inner rings changes continuously. At the initial moment let the ball or roller be in contact with point 1 (Fig. 4.17) of the stationary outer ring and with point 2 of the inner ring rotating in an anticlockwise direction. After a time t, point 2 moves away from stationary point 1 by an angle ω in t, where ω in is the angular velocity of the inner ring. The ball is displaced during this time in the same direction by an angle ω t, and ω tFω in t since the angular velocity of the cage, ω , is less than ω in . Then the rings make contact with the ball at points 1′ and 2′. This means that the point of contact of the ball with the outer ring moves along the profile of the ring in an anticlockwise direction through an angular distance ω t, and the point of contact of the ball with the inner ring moves in a clockwise direction through a distance ω in tAω tG(ω inAω )t, or qω t, if qG

ω in A1 ω

(4.121)

Fig. 4.17


117

Thus, during rotation of the cage through an angle ω tGϕ, the contact point between the stationary ring and the ball moves through an angular distance ϕ and the contact point of the inner ring with the ball moves through a distance qϕ. For the ith rolling element these angular distances can be written respectively in the form ϕCiγ and qϕCiγ , where γ G(2π)兾m is the angular distance between the rolling elements, m denotes the number of rolling elements in the bearing, and iG 0, 1, 2, . . . , mA1. It is possible to consider the raceway as a circle of variable radius. Since it is a closed curve, the function describing it is periodic and can be expanded in a Fourier series. Taking the above into account, the profile of the raceway of the outer ring can be represented by RoC∑ ∆Rl sin[l(ϕCiγ )Cα l ]

(4.122)

l

and the profile of the raceway of the inner ring is given by roC∑ ∆rp sin[ p(qϕCiγ )Cα p ]

(4.123)

p

where Ro and ro are the constant components of the radii of the raceways of the outer and inner rings respectively, l and p are the orders of the harmonic of unevenness in the raceways of the outer and inner rings respectively, ∆Rl and ∆rp are the amplitudes of these harmonics, and α l and α p are the phase angles. Consider the limit of summation of the series given by equations (4.122) and (4.123). Obviously, it is meaningful to take the upper limit as infinity since, for sufficiently large values of l and p, unevenness of the raceway is covered by the contact area and stops affecting the periodic component of stiffness. Besides, the upper limit of quantities l and p, which will be denoted by s, cannot be determined rigorously and uniquely since it depends on the value of the preload, the mutual displacement of the rings, the mechanical properties of the materials of the components, and other factors. The lower limit in equation (4.123) is equal to unity since, without exception, all lower harmonics of errors in raceways of the rotating ring affect the change in contact deformation during rotation under the condition xGconst. It is impossible to say the same about the stationary outer ring. The first harmonic of unevenness is caused by eccentricity which does not affect the change in contact deformation since the bearing is automatically centred in the inner ring. If, on account of design or some other factor, the centre of the outer ring still does not coincide with the centre of rotation of the inner ring, this misalignment is taken as the value of x in equation (4.124) given below. Thus, the lower limit of summation in equation

118

Rolling Contacts

(4.122) or the minimum value of l should be taken to be equal to 2, but the minimum value of p in equation (4.123) should be taken to be equal to unity. Let the value of the radial displacement x of the inner ring with respect to the outer ring remain constant during rotation of the bearing. Then the total contact deformation at places where the rings rub against the rolling elements is determined by the equation

冦

δ i G giC(x1Cx0 ) cos(ϕCiγ )C( y1Cy0 ) sin(ϕCiγ ) s

C ∑ ∆Rl sin[l(ϕCiγ )Cα l ] l G2

C ∑ ∆rp sin[ p(qϕCiγ )Cα p ] p G1

冧

(4.124)

where gi GgC∆Di and ∆Di is the deviation of the actual diameter of the ith rolling element from the nominal diameter, x1 is the mutual displacement of rings in the x direction under the action of radial load, x0 is the mutual displacement of rings in direction x for zero radial load, y1 denotes the mutual displacement of rings in direction y under the action of radial load in an ideal bearing, and y0 represents the mutual displacement of rings in direction y for zero radial load. Also, x0 GxAx1 . Then substitute the expression for total contact deformation [equation (4.124)] in the equation defining the applied radial load on the bearing Pi GKδ ni cos ψ i where n is the power index (for ball bearings nG3兾2 and for roller bearings nG10兾9) and ψ i is the angle between the line of action of the radial load (direction of displacement of the moving ring) and the radius passing through the centre of the ith rolling element. Summing with respect to i, an equation for the radial elastic force for the entire bearing is obtained mA1

冦

PG ∑ Ki giC(x1Cx0) cos(ϕCiγ )Cy0 sin(ϕCiγ ) l G0

s

C ∑ ∆Rl sin[l(ϕCiγ )Cα l ] l G2 s

n

冧

C ∑ ∆rp sin[ p(qϕCiγ )Cα p ] cos(ϕCiγ ) p G1

(4.125)


119

where Ki is the coefficient of proportionality. Its value, except for a number of geometric parameters, depends on the mechanical properties of the material of the rolling elements and the rings, i.e. the modulus of elasticity and Poisson’s ratio. It can therefore be said that the value of K for the contact of each rolling element with the rings depends on the mechanical properties of the material of a particular rolling element and can be used to describe the differences in the mechanical properties of the materials. In equation (4.125), there is no y1 , because here a bearing with a preload is considered for which the values of y1 are small and only slightly affect the stiffness. Values of x0 and y0 are determined from the condition that, in the absence of radial load, the projections of the elastic force of the complete bearing on to axes x and y are equal to zero, i.e. PG0 and Py G0. In this case, x1 G0 and y1 G0, and the condition written above takes the form mA1

冦

Py G ∑ Ki giCx0 cos(ϕCiγ )Cy0 sin(ϕCiγ ) l G0

s


冧

n

C ∑ ∆rp sin[ p(qϕCiγ )Cα p ] sin(ϕCiγ ) p G1

(4.126)

and mA1

冦

PG ∑ Ki giCx0 cos(ϕCiγ )Cy0 sin(ϕCiγ ) i G0

s


n

冧

C ∑ ∆rp sin[ p(qϕCiγ )Cα p ] cos(ϕCiγ ) p G1

(4.127)

By expanding equations (4.126) and (4.127) in a McLaurin series in powers of x0 and y0 and limiting the terms containing x0 and y0 to the zero and first order, the following is obtained ACBny0CCnx0 G0 BCCny0CEnx0 G0 (4.128)

120

Rolling Contacts

where mA1

n

冢

冣 sin ψ

AG ∑ Ki giC∑C∑ i G0

mA1

l

p

n

冢

冣 sin ψ

BG ∑ Ki giC∑C∑ i G0

mA1

l

冢

mA1

2

p

CG ∑ Ki giC∑C∑ i G0

l

p

冣

mA1

冢

l

冢

p

EG ∑ Ki giC∑C∑ i G0

i

nA1

sin ψ i cos ψ i n

DG ∑ Ki giC∑C∑ i G0

i

l

p

冣 cos ψ 2

冣

i

nA1

cos2 ψ i

s

∑ G ∑ ∆Rl sin[l(ϕCiγ )Cα l ] l

l G2 s

∑ G ∑ ∆rp sin[ p(qϕCiγ )Cα p ] p

p G1

Solving the system of equations given above with respect to x0 and y0 results in xG yG

AEACD n(C 2ABE ) BDAAC n(C 2ABE) (4.129)

After substituting these expressions in equation (4.125) and differentiating with respect to x1 , the formula for radial stiffness of a radial bearing incorporating the errors in the components is obtained mA1

冤

kGn ∑ Ki giCx1 cos(ϕCiγ )C i G0

C

BDAAC 2

n(C ABE )

AEACD cos(ϕCiγ ) n(C 2ABE )

sin(ϕCiγ )C∑C∑ l

p

冥

nA1

cos2 (ϕCiγ ) (4.130)


121

4.4.2 Effect of variable dimensions and variable stiffness In order to demonstrate the effect of variable dimensions and variable elasticity of rolling elements on the radial stiffness of the bearing it is assumed that: (i) there are no other errors, (ii) the bearing has an even number of rolling elements, and (iii) there are two identical defective rolling elements placed opposite each other. Under such conditions y1 Gy0 Gx0 G0 and equation (4.130) for the radial stiffness is considerably simplified mA1

kGn ∑ Ki [giCx cos(ϕCiγ )]nA1 cos2(ϕCiγ )

(4.131)

i G0

The maximum allowable value of variable dimensions of the rolling elements depends on the class of accuracy and the dimensions of the bearing. Coefficient Ki takes into account the difference in the mechanical properties of the rolling elements. Any difference in this coefficient, such as that between the rolling elements included in one assembly, is due to the prevailing manufacturing technology. The assembly of bearings does not guarantee that rolling elements with identical mechanical properties will be fitted into a particular bearing. Nor does it exclude the possibility of a difference in mechanical properties arising during heat treatment. The deviation in the value of Ki in equation (4.131) is equal to the deviation in the contact rigidity of the ball, since ki GnKi δ nA1 i

(4.132)

In Fig. 4.18 the function k(ϕ ) is graphically shown according to equation (4.131) for mG6 and nG3兾2. The graphs correspond to the following values of the parameters involved: (1) Ki G1 and gi G20 µm for all values of i (the case of an ideal bearing). (2) Ki G1 for all values of i, go Gg3 G19.7 µm, and g1 Gg2 Gg4 Gg5 G 20 µm (the stiffness of all balls is identical, but the deformation by preload of two balls is lower than the deformation of the remaining balls by 1.5 percent because of negative deviation in their diameters). (3) Ko GK3 G0.985, K1 GK2 GK4 GK5 G1, and gi G20 µm for all values of i (the stiffness of two balls is lower than the rigidity of the remaining balls by 1.5 percent; the diameter of all balls is identical).

122

Rolling Contacts

Fig. 4.18

A comparison of the graphs shows that classification of the rolling contact bearings, especially precision bearings with preload, on the basis solely of dimensional and geometric parameters is not enough. It is necessary to verify the rigidity of the rolling elements too. Besides, it is not rational to manufacture precision bearings with a small number of rolling elements since the change in their rigidity during rotation is considerable even in the case of absolute dimensional uniformity of the rolling elements.

4.4.3 Effect of waviness of raceways Assuming that the unevenness of the raceway of the outer and inner rings of the bearing comprises only harmonics whose order is equal to m, or less or more than that by a factor of 2, and that phase angles of all harmonics are equal, i.e. x0 Gy0 G0, it is possible to simplify equation (4.130) appreciably mA1

冦

s

kGn ∑ Ki giCx cos(ϕCiγ )C ∑ ∆Rl sin[l(ϕCiγ )Cα l ] i G0

l G2

s

C ∑ ∆rp sin[ p(qϕCiγ )Cα p ] p G1

nA1

冧

cos2(ϕCiγ )

(4.133)


123

The graphs shown in Fig. 4.19 were plotted using equation (4.133) for mG6, xG0, Ki GKG1, gi GgG5, qG1.64, and α l Gα p G0. All the parameters correspond to three cases: (a) the inner ring is ideal, and the raceway of the outer ring has three harmonics of unevenness whose amplitudes are ∆R3 G1, ∆R6 G0.5, and ∆R12 G0.3; (b) the outer ring is ideal but the inner ring has the same type of unevenness; (c) both rings simultaneously have the unevenness described above. As a result of superposing the periodic component owing to the unevenness of the raceways of the inner ring on the component arising from the unevenness of the outer ring, and in the general case also owing to variable dimensions, variable elasticity, and change in the position of rolling elements during rotation (x≠0), the function k(ϕ ) becomes almost periodic because the rotational speed of the moving ring and the cage are, in general, not repeated. The repetition of the rotational speeds is characterized by the quantity q, which is determined by the relation qG

ω in A1 ω

(4.134)

However nin

G

nc

2Do DoAdr cos β

(4.135)

where nin and nc are the rotational speeds of the inner ring and cage respectively. Also nin

G

nc

ω in ω

(4.136)

Do is the diameter of the circle passing through the centres of the rotating elements, and dr is the diameter of the rolling element. By substituting equation (4.135) in equation (4.134), the following is obtained qG

2Do A1 DoAdr cos β

(4.137)

124

Rolling Contacts

Fig. 4.19


125

Thus, the quantity q depends not only on the dimensional and geometrical parameters of the bearing, i.e. Do , dr , and β , but also on the value of the preload. With changes in the preload the effective values of diameters Do and dr and contact angle β also change. Consequently, the quantity q not only cannot be computed with sufficient accuracy but also does not remain a constant during the operation of the bearing.

4.5 Ball motion in a rolling contact bearing In modern high-speed ball bearings the pressure areas, which result from elastic deformations at the ball–ring contacts, are appreciably curved and interfacial slip can occur at most points within the pressure areas. These slippages give rise to friction forces acting on the balls which are held in equilibrium by reactions from the rings and the inertia effects of the motion of the balls. In this section, a method is derived for determining the motion of the ball and sliding friction in a high-speed, angular contact ball bearing under thrust load in terms of the inertia effects on the ball and the friction resistances resulting from interfacial slip at the contact areas. Possible elastic compliance at the interface, hysteresis, and dynamic perturbations of ball motion are neglected.

4.5.1 Inertia forces and moments acting on the ball The instantaneous position of an element of mass in the ball of a highspeed, angular contact bearing is shown in Fig. 4.20. The following notation is used (Fig. 4.20): x, y, zGfixed, right-handed coordinate system with x being the axis of the bearing about which the mass centre of the ball revolves; x′, y′, z′Gright-handed coordinate system with the origin at the mass centre of the ball and revolving about x at the radius e (x′ is parallel to x); U, V, WGright-handed coordinate system with the origin at the origin of x′, y′, z′ and moving with x′, y′, z′; U is directed along the axis of rotation of the ball about its own centre; W is in the plane of U and z′; U, r, θ Gcylindrical coordinates rotating with the ball; α Gangle between W and z′ axes; β Gangle between the trace of uz′ on x′y′ and the x′ axis; φ Gangle between the z′ and z axes. The ball motion is defined by θ˙ Gω B and φ˙ GΩ e . The position of the mass element dm in the ball at location U, r, θ is related to the fixed

126

Rolling Contacts

Fig. 4.20

coordinates x, y, z through UGU

(4.138)

VGr sin θ

(4.139)

WGr cos θ

(4.140)

x′GU cos α cos β AV sin β AW sin α cos β

(4.141)

y′GU cos α sin β CV cos β AW sin α sin β

(4.142)

z′GU sin α CW cos α

(4.143)

xGx′

(4.144)


127

yGe sin φ Cy′ cos φ Cz′ sin φ

(4.145)

zGe cos φ Ay′ sin φ Cz′ cos φ

(4.146)

From equations (4.138) to (4.146) xGU cos α cos β Ar(sin β sin θ Csin α cos β cos θ )

(4.147)

yGe sin φ CU(cos α sin β cos φ Csin α sin φ ) Cr(cos β sin θ cos φ Ccos α cos θ sin φ ) Ar sin α sin β cos θ cos φ

(4.148)

zGe cos φ CU(−cos α sin β sin φ Csin α cos φ ) Cr(−cos β sin θ sin φ Ccos α cos θ cos φ ) Cr sin α sin β cos θ sin φ

(4.149)

Treating α , β , ω B , and Ω e as constants, two differentiations of equations (4.147) to (4.149) with respect to time yield expressions for the instantaneous accelerations of a mass particle of the ball as x¨ Grω 2B (sin β sin θ Csin α cos β cos θ )

(4.150)

y¨ G2ω B Ω e r(−cos β cos θ sin φ Acos α sin θ cos φ Asin α sin β sin θ sin φ )CΩ 2e [−e sin φ CU(−sin α sin φ Acos α sin β cos φ )Cr(−cos β sin θ Acos α cos θ sin φ Csin α sin β cos θ cos φ )]Cω 2B r(−cos β sin θ cos φ Acos α cos θ sin φ Csin α sin β cos θ cos φ )

(4.151)

z¨ G2ω B Ω e r(−cos β cos θ cos φ Ccos α sin θ sin φ Asin α sin β sin θ cos φ )CΩ 2e [−e cos φ CU(cos α sin β sin φ Asin α cos φ )Cr(cos β sin θ sin φ Acos α cos θ cos φ Asin α sin β cos θ sin φ )]Cω 2Br(cos β sin θ sin φ Acos α cos θ cos φ Asin α sin β cos θ sin φ )

(4.152)

For the purposes of evaluating the forces and moments acting on the ball and referred to x′, y′, z′, the value of φ can be arbitrarily chosen.

128

Rolling Contacts

Setting φ G0 and with material density ρ r

Fx′ G− ρ

冮冮 −r

0

r

Fy′ G− ρ Fz′ G− ρ

√(r2AU 2)

冮冮

√(r2AU 2)

0

r

√(r2AU 2)

−r

x¨r dθ dr dU

(4.153)

y¨r dθ dr dU

(4.154)

z¨r dθ dr dU

(4.155)

0

−r

冮冮

冮

2π

冮

2π

冮

2π

0

0

0

The moments about the x′, y′, z′ axes are r

Mx′ G−ρ

冮冮 −r

√(r2AU 2)

0

冮

2π

{−y¨ [U sin α Cr cos α cos θ ]

0

Cz¨ [U cos α sin β Cr(cos β sin θ Asin α sin β cos θ )]}r dθ dr dU r

My′ G− ρ

冮冮 −r

√(r2AU 2)

0

冮

(4.156)

2π

{x¨ [U sin α Cr cos α cos θ ]

0

Az¨ [U cos α cos β Ar(sin β sin θ Csin α cos β cos θ )]}r dθ dr dU r

Mz′ G− ρ

冮冮 −r

√(r2AU 2)

0

冮

(4.157)

2π

{−x¨ [U cos α sin β Cr (cos β sin θ

0

Asin α sin β cos θ )]Cy¨ [U cos α cos β Ar(sin β sin θ Csin α cos β cos θ )]}r dθ dr dU

(4.158)

The integrations give Fx′ G0

(4.159)

Fy′ G0 Fz′ Gme Ω

(4.160) 2 e

(4.161)

Mx′ G0

(4.162)

My′ GIp ω B Ω e sin α

(4.163)

Mz′ G−Ip ω B Ω e cos α sin β

(4.164)

where m is the mass of the ball, and Ip is the polar moment of inertia of the ball.


129

4.5.2 Relative motions of the rolling elements The contact configuration between the ball and the outer race is shown in Fig. 4.21. The pressure surface is an ellipse of semi-axes ao and bo . The radius of curvature of the deformed pressure surface in the plane of the paper is Ro . Because the ratio bo 兾ao is small, the radius of curvature of the pressure surface in the rolling direction can be taken as infinity with little error. Assume that the ball centre is fixed in the plane of the paper. Let the outer race rotate with angular velocity ω o . The components of the ball rotational angular velocity that lie in the plane of the paper are ω x′ and ω z′ . There is some radius, as r′o , that suffices for determination of the relative translational velocities of ball and outer race. This radius is not

Fig. 4.21

130

Rolling Contacts

necessarily restricted to points that lie on the deformed pressure surface if gross slip between ball and race occurs. According to the Hertz theory Ro G

2fo d

(4.165)

2foC1

where fo is the ratio of outer race curvature radius to ball diameter, and d is the ball diameter. Through ω o cos β o , a point (xo , yo ) on the outer race has the linear velocity V1o V1o G− ω o cos β o B

冤cos β C1(R Ax )A1(R Aa )C1(r Aa )冥 e

2 o

2 o

2 o

2 o

2

2 o

(4.166)

o

V1o G−eω o A[1(R 2oAx2o)A1(R 2oAa2o)C1(r 2Aa2o)]ω o cos β o

(4.167)

Through ω x′ cos β o and ω z′ sin β o , a point (xo , yo ) on the ball has the linear velocity V2o V2o G−(ω x′ cos β oCω z′ sin β o ) B[1(R 2oAx2o)A1(R 2oAa2o)C1(r 2Aa2o)]

(4.168)

and the velocity with which the outer race slips on the ball in the Y direction is Vyo GV1oAV2o

(4.169)

Thus Vyo GAeω oC(ω x′ cos β oCω z′ sin β oAω o cos β o ) B[1(R 2oAx2o)A1(R 2oAa2o)C1(r 2Aa2o)]

(4.170)

Through ω y′ all points within the pressure area have a velocity of slip of race on ball, Vxo , in the direction of the major axis. Therefore, Vxo G− ω y′ [1(R 2oAx2o)A1(R 2oAa2o)C1(r 2Aa2o)]

(4.171)


131

Through the components of velocity that lie along the line defined by β o there is a spin of the race, ω so , with respect to the ball

ω so G− ω o sin β oCω x′ sin β oAω z′ cos β o

(4.172)

A similar condition exists at the inner race contact as shown in Fig. 4.22. The pressure surface is an ellipse of semi-axes ai and bi . The radius of curvature of the deformed surface in the plane of the paper is denoted by Ri . As with the outer race contact, the radius of curvature of the pressure surface in the rolling direction is taken as infinity. In Fig. 4.22, r′i is the effective rolling radius which determines the relative translational velocities of ball and inner race. As with r′o , r′i is not restricted to points within the inner race pressure area if gross slip occurs. The radius of curvature Ri is Ri G

2fi d

(4.173)

2fiC1

Fig. 4.22

132

Rolling Contacts

Through ω i cos β i a point (xi , yi ) on the inner race has the linear velocity V1i . Thus V1i Geω iC[1(R2iAx2i)A1(R2iAa2i)C1(r 2Aa2i)]ω i cos β i

(4.174)

Through ω x′ cos β i and ω z′ sin β i , a point (xi , yi ) on the ball has the linear velocity V2i V2i G(ω x′ cos β iCω z′ sin β i ) B[1(R2iAx2i)A1(R2iAa2i)C1(r 2Aa2i)]

(4.175)

The velocity with which the inner race slips on the ball in the Y direction is Vyi Vyi GV1iAV2i

(4.176)

Vyi GAeω iC(− ω x′ cos β iAω z′ sin β iCω i cos β i ) B[1(R 2iAx2i)A1(R 2iAa2i)C1(r 2Aa2i)]

(4.177)

Through ω y′ , all points within the area have a velocity of slip of race on ball in the direction of the major axis Vx′ G− ω y′ [1(R 2iAx2i)A1(R 2iAa2i)C1(r 2Aa2i)]

(4.178)

Through the components of velocity that lie along the line defined by β i there is a spin ω si of the race with respect to the ball

ω si Gω i sin β iAω x′ sin β iCω z′ cos β i

(4.179)

It is convenient to write the expressions for the linear velocities of slip and the angular velocities of spin in terms of the parameters α , β , and ωB. From Fig. 4.20

ω x′ Gω B cos α cos β

(4.180)

ω y′ Gω B cos α sin β

(4.181)

ω z′ Gω B sin α

(4.182)


133

Then, from equations (4.170) to (4.172) and (4.177) to (4.179) Vyo GAeω oC[1(R 2oAx2o)A1(R 2oAa2o)C1(r 2Aa2o)]

ωB

冢ω

B

cos α cos β cos β oC

o

ωB sin α sin β oAcos β o ω o ωo

冣

Vxo G−[1(R 2oAx2o)A1(R 2oAa2o)C1(r 2Aa2o)]ω o

(4.183)

ωB

冢ω 冣 cos α sin β o

(4.184)

ωB

ωB cos α cos β sin β oA sin α cos β oAsin β o ω o ωo o 2 2 Vyi G−eω iC[1(R i Axi )A1(R 2iAa2i)C1(r 2Aa2i)]

ω so G

冢ω

B

冣

ωB

(4.185)

ωB

冢 ω cos α cos β cos β C ω sin α sin β Ccos β 冣 ω i

i

i

i

i

(4.186)

i

Vxi G−[1(R 2iAx2i)A1(R 2iAa2i)C1(r 2Aa2i)]ω i

ωB

冢 ω 冣 cos α sin β i

(4.187)

冢

ω si G −

ωB ωB cos α cos β sin β iC sin α cos β iCsin β i ω i ωi ωi

冣

(4.188)

At the effective rolling radii r′o and r′i on the ball, the translational velocity of the ball and race is the same. From Fig. 4.21

冢cos β Cr′ 冣 ω

A

e

o

o

cos β o G−r′o (ω x′ cos β oCω z′ sin β o )

(4.189)

o

(eCr′o cos β o )ω o Gr′o (cos α cos β cos β oCsin α sin β o )ω B

(4.190)

ωB eCr′o cos β o G ω o r′o (cos α cos β cos β oCsin α sin β o )

(4.191)

From Fig. 4.22

冢cos β Ar′ 冣 ω cos β Gr′(ω

A

e

i

i

i

i

x′

cos β iCω z′ sin β i )

(4.192)

i

A(eAr′i cos β i )ω i Gr′i (cos α cos β cos β iCsin α sin β i )

(4.193)

ωB A(eAr′i cos β i ) G ω i r′i (cos α cos β cos β iCsin α sin β i )

(4.194)

134

Rolling Contacts

For the outer race to be stationary, the retainer must be given the absolute angular velocity Ω c , such that

Ω c G− ω o

(4.195)

Then the inner race rotates with the absolute angular velocity

Ω i Gω iCΩ c

(4.196)

From equations (4.191) and (4.194)

ω o G−

r′o (eAr′i cos β i ) cos α cos β cos β oCsin α sin β o ωi r′i (eCr′o cos β o cos α cos β cos β iCsin α sin β i

(4.197)

and from equations (4.194) to (4.197)

ωiG

Ωi 1CA1

(4.198)

and

ω o G− Ω c G

AΩ i 1CA2

(4.199)

where A1 G

r′o (eAr′i cos β i ) cos α cos β cos β oCsin α sin β o r′i (eCr′o cos β o) cos α cos β cos β iCsin α sin β i

A2 G

r′i (eCr′o cos β o ) cos α cos β cos β iCsin α sin β i r′o (eAr′i cos β i ) cos α cos β cos β oCsin α sin β o

Also

ωBG

AΩ i

(4.200)

B1CC1

where B1 G C1 G

r′i (cos α cos β cos β iCsin α sin β i ) eAr′i cos β i r′o (cos α cos β cos β oCsin α sin β o ) eCr′o cos β o


135

Likewise, for the inner race to be stationary the retainer must be given the absolute angular velocity Ω c such that

Ω c G− ω i

(4.201)

Then the outer race rotates with the absolute angular velocity Ω o

Ω o Gω oCΩ c

(4.202)

From equations (4.191), (4.197), (4.201), and (4.202)

Ωo 1CA3

ωoG

ω i G− Ω c G

(4.203) AΩ o 1CA4

(4.204)

where A3 G A4 G

r′i (eCr′o cos β o) cos α cos β cos β iCsin α sin β i r′o (eAr′i cos β i ) cos α cos β cos β oCsin α sin β o r′o (eAr′i cos β i ) cos α cos β cos β oCsin α sin β o r′i (eCr′o cos β o) cos α cos β cos β iCsin α sin β i

and

ωBG

Ωo B2CC2

(4.205)

where B2 G C2 G

r′o (cos α cos β cos β oCsin α sin β o) eCr′o cos β o r′i (cos α cos β cos β iCsin α sin β i ) eAr′i cos β i

4.5.3 Friction at the contact interface Figure 4.23 is an enlarged view of the pressure area at either race contact as viewed from outside the ball. Owing to the spin velocity ω s and the linear slip velocities Vy and Vx , an element of area dA at coordinates (x, y) has a resultant velocity of slip V of the race on the ball acting at an angle γ with respect to the Y direction.

136

Rolling Contacts

Fig. 4.23

The friction force with which the race acts on the ball is dF, taken over the area dA, and is in the same direction as V. The normal pressure is distributed over the elliptical pressure area in accordance with Sxy G

3P 2πab

1冢

冣

x2 y2 1A 2A 2 a b

(4.206)

With the coefficient of sliding friction µ, dF is dFG

3Pµ 2πab

1冢

冣

x2 y2 1A 2A 2 dx dy a b

(4.207)

Let xGaq

(4.208)

yGbt

(4.209)

thus dFG

3Pµ 1(1Aq2At 2) dq dt 2π

(4.210)


137

The component of dF parallel to the Y direction is cos γ dF and, for the whole ellipse Fy G

3Pµ

1

√(1Aq2)

−1

−√(1Aq )

冮冮

2π

2

1(1Aq2At 2 ) cos γ dq dt

(4.211)

The total force in the X direction is Fx G

3Pµ

1

2π 冮冮

√(1Aq2)

1(1Aq2At 2) sin γ dq dt

2

(4.212)

−√(1Aq )

−1

The moment of dF about the normal at the centre of the pressure ellipse is dQs Gρ cos(γ Aθ ) dF

(4.213)

With kGb兾a

(4.214)

dQs Ga1(q2Ck2t 2) cos(γ Aθ ) dq dt

(4.215)

Integration over the whole ellipse produces 3Pµ Qs G 2π

1

√(1Aq2)

−1

−√(1Aq )

冮冮

2

1(1Aq2At 2 ) 1(q2Ck2t 2 ) cos(γ Aθ ) dq dt (4.216)

where

θ Gtan−1

kt q

(4.217)

The moment of dF about the y′ axis is dQy′ G[1(R 2Ax2)A1(R 2Aa2)C1(r 2Aa2)] sin γ dF

(4.218)

Integration over the whole ellipse gives 3PµR Qy′ G 2π

1

冮冮

√(1Aq2)

−1 −√(1Aq2)

1(1Aq2At 2 ) sin γ [AABCC] dq dt (4.219)

138

Rolling Contacts

where

1冤冢冣冥 1冤冢冣冥 1冤冢冣冢冣冥

AG

1Aq2

BG

1A

CG

r

2

a

R

a

2

R

2

A

R

a

2

R

The moment about an axis through the centre of the ball, perpendicular to the line defining the contact angle and lying in the plane x′, z′, is dQR G[1(R 2Ax2)A1(R 2Aa2)C1(r 2Aa2)] cos γ dF

(4.220)

For the whole ellipse 3PµR QR G 2π

1

√(1Aq2)

−1

−√(1Aq2)

冮冮

1(1Aq2At 2 ) cos γ [AABCC] dq dt (4.221)

where A, B and C are defined above. The value of γ is found from Fig. 4.23 as tan γ G

ρω s sin θ AVx ρω s cos θ CVy

(4.222)

and tan γ G

ktAVx 兾(aω s ) qCVy 兾(aω s )

(4.223)

Figure 4.24 shows the moments acting on a ball. Figure 4.25 shows the forces acting on a ball. From Fig. 4.24 AQRo sin β oCQso cos β oCMz′CQRi sin β iAQsi cos β i G0

(4.224)

and AQRo cos β oAQso sin β oCQRi cos β iCQsi sin β i G0

(4.225)

My′AQyoAQyi G0

(4.226)


139

Fig. 4.24

From Fig. 4.25 APo cos β oAFxo sin β oCFz′CFxi sin β iCPi cos β i G0

(4.227)

Po sin β oAFxo cos β oCFxi cos β iAPi sin β i G0

(4.228)

FyoCFyi G0

(4.229)

In addition T GPo sin β oAFxo cos β o GPi sin β iAFxi cos β i n

(4.230)

There is also a constraint imposed by the physical dimensions of the bearing and the elastic deformations at the ball and race contacts.

140

Rolling Contacts

Fig. 4.25

Figure 4.26 shows the initial and final relative positions of the ball centre and the race curvature centres. In Fig. 4.26, β ′ is the initial contact angle of the bearing. The race centres are originally at (O) and (3) while the ball is at (1). When equilibrium is attained under high-speed conditions, the inner race curvature centre has moved axially from (3) to (4), by an amount δ H . The ball has moved from (1) to (2). These movements result from the elastic deformations δ o and δ i occurring at the outer and inner race contacts

δoG

δiG

Ko Po2/3 d 1/3

Ki Pi2/3 d 1/3

(4.231)

(4.232)


141

Fig. 4.26

Then

冤

冥

冤

冥

Ko Po2/3 Ki Pi2/3 ( foA0.5)dC 1/3 cos β oC ( fiA0.5)dC 1/3 cos β i d d G( foAfiA1)d cos β ′

(4.233)

The pitch radius of the bearing is also a function of the dynamic effects so that

冤

冥

Ko Po2/3 eGeoC ( foA0.5)dC 1/3 cos β oA( foA0.5)d cos β ′ d

(4.234)

A final condition is that the input and output torques on the races of the bearing are equal and opposite. The torque on the outer race is r o Gn Q

QRo (eCr′o cos β o ) CQso sin β o r′o

冤

冥

(4.235)

142

Rolling Contacts

and on the inner race r i Gn Q

冤

QRi (eAr′i cos β i ) r′i

AQsi sin β i

冥

(4.236)

Then QRi (eAr′i cos β i ) QRo (eCr′o cos β o ) CQso sin β oC AQsi sin β i G0 r′o r′i (4.237) Determination of the ball motion requires the evaluation of eight unknowns. These are r′o , r′i , α , β , Po , Pi , β o , and β i . The eight necessary simultaneous equations are as follows. From equations (4.224) and (4.225) 0GAQRo (sin β oCcos β o )CQso (cos β oAsin β o ) CQRi (sin β iCcos β i )AQsi (cos β iAsin β i )CMz

(4.238)

From equation (4.226) My′AQyoAQyi G0

(4.239)

From equation (4.227) APo cos β oAFxo sin β oCFz′CFxi sin β iCPi cos β i G0

(4.240)

From equation (4.229) FyoCFyi G0

(4.241)

From equation (4.230) T APo sin β oCFxo cos β o G0 n

(4.242)

Again, from equation (4.230) T APi sin β iCFxi cos β i G0 n

(4.243)

From equation (4.233)

冤

冥

Ko Po2/3 0G ( foA0.5)dC 1/3 cos β o d

冤

冥

Ki Pi2/3 C ( fiA0.5)dC 1/3 cos β iA( foAfiA1)d cos β ′ d

(4.244)


143

Finally, from equation (4.237) QRo (eCr′o cos β o ) QRi (eAr′i cos β i ) CQso sin β oC AQsi sin β i G0 r′o r′i (4.245) The solution of the eight simultaneous equations, equations (4.238) to (4.245), cannot be attained analytically since closed-form solution of the double integrations involved in some of them is not possible. However, iterative methods used in conjunction with a computer enable a numerical solution for any particular case, by means of which the ball motion is completely defined. Throughout the analysis it has been assumed that Coulomb friction law is applicable. Actually, the coefficient of friction is a complex function of a number of variables. Among these are: the unit contact pressure and sliding velocities at different points within the contact area, the nature of the contacting surfaces, the temperature, and the type of lubricant. The functional relationship between all factors is not known. However, measurements of friction torque in actual ball bearings have shown good agreement with calculated results using a coefficient of friction of 0.06–0.07.

4.6 References (1) Novikov, L. Z. (1964) On the elastic characteristics of radial-thrust ball bearings (in Russian). Izv. AN USSR, OTN, Mekhanika and Mashinostroenie, (3). (2) Kharlamov, S. A. (1962) Stiffness of radial-thrust ball bearings with axial loading (in Russian). Izv. AN USSR, OTN Mekhanika and Mashinostroenie, (5). (3) Szucki, T. (1955) Analysis of stiffness of radial ball bearing (in Polish). Z. Nauk. Warsaw Technical University. Mechanika, (27). (4) Tamura, H. and Shimizu, H. (1967) Vibration of rotor based on ball bearings. Bull. JSME, 10(41). (5) Neubert, G. (1968) Der Einfluss der Lager den Lauffehler einer Walzgelagerten Werkzeugmashinen-Hauptspindel (in German). Maschinenbautechnik, 17(6). (6) Kostetskii, B. I. (1970) Friction lubrication and wear in machines (in Russian) (Tekhnika, Kiev). (7) Kostetskii, B. I. and Edigoryan, F. C. (1964) Classification of the basic types of wear and elements of the theory of wear due to

144

(8) (9) (10)

(11) (12) (13) (14)

(15) (16)

(17) (18) (19)

(20)

Rolling Contacts

rolling friction (in Russian). Trudy Kievskogo Grazhdanskogo Vozdushnogo Flota, 4. Reynolds, O. (1876) On rolling friction. Phil. Trans. R. Soc., 166. Solski, P. and Ziemba, S. (1965) Problems of dry friction (in Polish) (Polish Scientific Publishers (PWN), Warsaw). Akhmatov, S. A. (1951) Influence of profile and physicochemical properties of rubbing surfaces on the type of dependence of friction resistance on lubricant. In Proceedings of 1st All-Union Conference on Friction and Wear, Moscow (USSR Acad. Sci.). Tabor, D. (1955) The mechanism of rolling friction. Proc. R. Soc., A229. Tomlinson, G. A. (1929) Molecular theory of friction. Phil. Mag., (7). Ishlinskii, A. Yu. (1938) Prikladnaya Matematika i Mekhanika (in Russian), 2. Ishlinskii, A. Yu. (1949) Friction and wear in machines (in Russian). In Proceedings of All-Union Conference, Moscow, Vol. 2 (USSR Acad. Sci.). Drutowski, R. C. (1959) Energy losses of balls rolling on plates. Trans. ASME, J. Basic Engng, June. Palmgren, A. and Snare, B. (1957) Influence of load and motion on the lubrication of and wear of rolling bearings. In Proceedings of IMechE Conference on Lubrication and Wear, London. Poritsky, K. E. (1950) Stresses and deflections of cylindrical bodies in contact. Trans. ASME, J. Appl. Mech., 17. Kalker, J. J. (1973) Simplified theory of rolling contact. Delft progress report, Series C, Vol. 1. Johnson, K. L. (1959) The influence of elastic deformation upon the motion of a ball rolling between two surfaces. Proc. Instn Mech. Engrs, 173. Sprishevskii, A. I. (1969) Rolling contact bearings (in Russian) (Moscow, Mashinostroenie).

Chapter 5 Rolling Contact Bearings

Rolling motion has been most extensively utilized in rolling contact bearings. Ball and roller bearings are typical representatives of that. They are considered to be machine elements that have found a use in almost all kinds of machinery and devices with rotating parts. Their properties have frequently contributed to technical and economic progress in different branches of engineering. Standardization of rolling contact bearings has made possible the assignment of their design and manufacture to specialists. Thus, the machine designer is not required to be expert on fundamental bearing problems. However, in order to utilize the properties of rolling contact bearings to the best advantage, engineers not only must be well acquainted with applied bearing engineering as used in their particular fields but also must have a general knowledge of the bearing elements themselves.

5.1 Phenomenology of friction during rolling Rolling contact bearings are commonly considered as offering lower resistance to motion than sliding bearings. Although rolling friction, or rather rolling resistance, is of small magnitude, it is a complicated phenomenon. There is no complete theory for predicting the magnitude of the rolling resistance under all possible conditions of bearing operation. Thus, only an outline of the principles involved will be given here, together with limited information concerning the frictional resistance in rolling contact bearings.

146

Rolling Contacts

Fig. 5.1

During rolling motion of a body along a substrate, certain forces resisting the motion develop within the contact zone. This can be illustrated by considering two bodies made of nearly perfectly elastic material and in contact with each other (Fig. 5.1). The load on the contact causes deformation of the contacting bodies so that a definite area of contact surface is produced. If one of the bodies begins to roll over the other one, the material in both bodies will be gradually compressed in the forward parts of the contact area and gradually relieved of the compressive stress in the rear parts. It is known that the relation between load and deformation is different for increasing load and decreasing load, although the limit of elasticity of the material is not exceeded. When the load is increasing, a given deformation corresponds to a higher stress than when the load is decreasing. Thus, an elastic hysteresis exists, which could be compared to the magnetic hysteresis. Rolling action, therefore, causes the surface pressure to be somewhat higher in the forward than in the rear part of the contact surface, which affects the resistance to the rolling motion. The difference in pressure for stresses within the limit of elasticity depends, to a certain degree, on the speed with which loading and unloading occurs. Therefore, the resistance to rolling must be influenced by the rolling speed. Temperature will also affect the rolling friction, as elastic properties are a function of temperature. It is known that elastic hysteresis is dependent not only upon the properties of the material, the rolling speed, and the temperature but also on the radii of curvature of the contacting surfaces and on the specific load acting on the contact. These last two factors determine the magnitude of deformation, and thus the magnitude of the hysteresis losses. When the loads are heavy and produce appreciable plastic deformation, the hysteresis losses increase considerably. When rolling occurs under load perpendicular to the contact, a forward displacement of the load resultant in the contact surface develops.

Rolling Contact Bearings

147

If an additional tangential force is also acting in the contact surface, this load displacement will considerably increase. Any tangential force resulting from the sliding friction at the interface develops elastic deformations both in a tangential and in a perpendicular direction. Perpendicular direction deformation causes the material to be pushed up into something resembling a bulge in front of the contact surface and in the direction of the tangential force, and also to be stretched into a cavity behind the contact surface (Fig. 5.2). The resultant effect may be different for different directions of the tangential force in relation to the direction of rolling and for different elastic properties of the material. When the rolling of the one body takes place in the same direction as the tangential force on the other body (the substrate), the rolling resistance will increase, particularly if the substrate is deformed more or is of more plastic material than the rolling body. However, the resistance decreases if rolling takes place in the opposite direction to the tangential force. In theory, there is one more kind of resistance to rolling in addition to the two described above. This is the resistance produced by damping of the elastic vibrations that occur under uneven contact pressure between two bodies in rolling contact. If the substrate is uneven or the rolling body unbalanced, vibrations develop that have their energy absorbed by the internal friction in the material, and thus contribute to the increase in the resistance to rolling. Apart from resistances due to the elastic properties of the material, resistance due to slippage between the surfaces, resulting from their shape (the original shape and the shape caused by deformation under load), should also be considered. Two perfect cylinders should roll on each other without slip when their axes of rotation are exactly parallel and their limited length and

Fig. 5.2

148

Rolling Contacts

Fig. 5.3

the length of the contact zone in the direction of rolling are neglected. However, if one or the other of the two surfaces has a curved generatrix, the actual contact surface between the two bodies will have a curved shape (Fig. 5.3). The various points in the contact area will thus be located at different distances from the centre of rotation of the roller. As the circumferential speed of each point relative to the centre of rotation is the product of the radius and the angular velocity of the whole body, the various points will move with different circumferential speed. This means that only a few points can be considered to have pure rolling motion on the substrate, while all the other points will slide with different velocities; some in a forward direction and others in a reverse direction. Those points which roll without slip form one or two lines, the neutral lines A (Fig. 5.3) being parallel to the direction of rolling. In the case depicted by Fig. 5.3, the points within area I on the roller slide backwards in relation to the direction of rolling, and those in area II slide in a forward direction. The location of the neutral lines is determined by the requirement of equilibrium, that the geometrical sum of the sliding friction forces in area I, the same forces in area II, and the outside tangential forces on the roller should be zero. This means that the distance between the neutral lines and their location relative to the centre of the contact surface is not always the same but varies with the distribution of the perpendicular forces and with the magnitude and direction of the outside tangential forces acting on the rolling element. Figure 5.4 shows the case where only one neutral line, A, develops. The forces in the contact surface form a moment that tends to turn the


149

Fig. 5.4

roller out of position. This is called pivoting. Only when the rolling element is spherical can pivoting take place continuously. For any other shape of the rolling element, the pivoting moment must be eliminated by the action of a counteracting outside moment in order for rolling to continue. The location of the neutral line depends, in this case, on the outside tangential forces which tend to displace it to one side or the other. Energy losses resulting from this type of sliding constitute a large proportion of the rolling resistance and are equal to the total of the product of sliding force and sliding distance, summed over all points of the contact surface. These losses are therefore influenced not only by the perpendicular load and the material properties but also by the properties of the lubricating film at the existing temperature, and finally by the deviation of the radii of the different points in the contact surface from the radius at the neutral line. This deviation depends on the original shape and the deformation of the two bodies and on the magnitude of the tangential forces, as the latter influence the location of the neutral lines, and the locations of these in turn influence the sliding distances of different points located on the contact surface. Microsliding within the contact surface also develops for other reasons than those discussed above. As a result of deformation caused by the perpendicular force, those parts of the contacting surfaces that are located within the area of contact expand or contract. The line acb on the periphery of the rolling element shown in Fig. 5.1 is shortened to the length aeb, while the section adb of the substrate is increased in length. A consequence of the differential deformation of contacting surfaces brings about microsliding which contributes to the overall resistance to rolling. Conditions favourable to microsliding may develop as a result of temperature at the interface. As a result of different types of energy losses at various points of the contact surface and the different shape and heat-dissipating ability of the two bodies in contact, the

150

Rolling Contacts

temperature increase may be higher in one part of the surface than in another; a net result of that is an unequal heat expansion leading to microsliding. A form of microsliding develops owing to the influence of outside tangential forces. If the rolling element is in contact with the substrate and a tangential force acts in a certain direction, elastic deformation in the tangential direction takes place. During rolling, microsliding occurs in the direction of the tangential force at the points located close to the periphery of the contact surface. The rolling element maintains a continuous motion in that direction. This special condition, which develops despite the fact that the points closer to the centre of the contact surface do not slide, has its roots in the known circumstance that the perpendicular load is not evenly distributed over the contact surface, but varies so that it is highest at the centre and decreases to zero at the periphery. As the coefficient of sliding friction is almost constant over the entire area, the tangential force acting on a surface element first attains the sliding friction force at or in the vicinity of the periphery and causes a microsliding there, while the friction force near the centre is still sufficient to prevent microsliding. A practical consequence of the phenomenon described above is that a ball never can remain rolling between surfaces that are inclined to each other, regardless of the magnitude of the angle of inclination. While rolling, the ball is always seeking surfaces that are exactly parallel. Macrosliding, which is the simultaneous sliding in a certain direction of all points within the contact surface, occurring coincidentally with rolling, should not be considered to be a part of the proper resistance to rolling. Such macrosliding between rolling surfaces develops in the case of non-parallel cylinders, or of cones with non-coinciding apexes. Finally, resistances that are caused by the bodies coming in contact with the lubricant and the atmosphere must be added to the overall rolling resistance. At the contact surface only a very small amount of lubricant participates in the rolling process as the excess lubricant is squeezed out. Within and immediately adjacent to the contact region, pressure and suction forces develop in the lubricant that oppose the movement of the surfaces towards or away from each other. This increases the hysteresis action during rolling. The resistance due to the motion of bodies through the surrounding atmosphere is less important but, nevertheless, must often be considered under practical conditions.

5.2 Friction torque Friction in ball and roller bearings results from several types of movement within the contact surface. In the contact surfaces between rolling


151

elements and raceways, rolling takes place. Because of the curved shape of the contact surfaces and other deviations from the requirement for pure rolling, microsliding develops at the interface. Moreover, certain losses due to sliding friction must always occur at the contacts between the rolling elements and the cage, between the rolling elements and the guide flanges in roller bearings, between the bearing parts and the lubricant, and within the lubricant itself. Even the atmosphere may be considered to be a source of losses as the bearing elements move in it. Apart from the material of the bearing and the finish of the contact surfaces, the following factors affect the total friction torque: (1) (2) (3) (4) (5) (6)

The bearing design. The bearing dimensions. The load and its distribution over the rolling elements. The speed of rotation. The quantity of lubricant. The lubricant characteristic and properties at the operating temperature, namely: (a) the viscosity, (b) the lubricity, (c) the strength of the surface film.

An equation to calculate the friction torque, encompassing all important parameters, would be rather complex. Besides, the influence of some individual factors on the friction torque is not known in an analytical form. Studies have usually been limited to examining the influence of different loads and speeds for a specific bearing size under constant conditions of lubrication for various types of bearing. It is, however, possible to put forward a simple formula for friction torque based on test results MF GM0CMGCMD

(5.1)

where M0 is the moment with zero loading. This is primarily due to friction with the lubricating film and, therefore, decreases with decreasing viscosity and increases with the speed of rotation and the amount of lubricant. In the loaded bearing, friction develops at the contact surfaces between rolling bodies and raceways. This friction is composed of a resistance to sliding motion, MG , and a damping resistance, MD . During rolling, microsliding takes place within a pressure region where a boundary film of lubricant is formed. Therefore, resistance to sliding

152

Rolling Contacts

motion, MG , does not follow the laws of hydrodynamic lubrication but is nearly independent of viscosity, speed of rotation, and amount of lubricant in the system. For a loaded ball bearing that has a constant contact angle, the moment due to microsliding usually varies according to the formula MG Gµf (const F 4/3Cconst F 5/3)

(5.2)

where µf is the coefficient of sliding friction in the pressure region and F is the load acting on the bearing. The constant depends on the bearing design and dimensions as well as on the load combination. The moment due to damping resistance, MD , is caused by the elastic hysteresis of the material. Experiments have shown that the damping losses increase with F 4兾3 for a bearing with point contact and a constant contact angle, while in the case of a deep groove ball bearing it is of the same order of magnitude as the loss due to microsliding. An approximate formula applicable to all types of bearing can be written as MF GM0CfF F c

(5.3)

Coefficient fF depends on the size of the bearing and its construction. It also depends on the bearing material and the lubricant characteristic. Load F is the bearing load or, more precisely, the imaginary radial or thrust load that would give the same friction moment as the actual combined load acting on the bearing. The magnitude of the exponent c is different for different bearing types. Generally, it is between 1.0 and 1.2 for roller bearings with good roller guiding and between 1.2 and 1.6 for various types of ball bearing and different load conditions; it is essentially dependent on the sliding loss caused by the curvature in the contact surfaces. During the start-up period of a ball or roller bearing, some dry friction must be overcome in the region where microsliding takes place. Therefore, the starting torque is slightly higher than the torque during steady state rotation.

5.2.1 Friction coefficient When carrying out approximate calculations, it is useful to use a constant value of the friction coefficient. In reality, the friction coefficient varies considerably, but a value can be chosen that can be applied to normal operating conditions and adequate lubrication without introducing any serious error. The friction coefficient values given below


153

represent the bearing load that will give a 1B109 revolutions life for the respective bearings (1). Self-aligning ball bearings

µ f G0.0010 Cylindrical roller bearings with flange guided short rollers

µ f G0.0011 Thrust ball bearings

µ f G0.0013 Single row deep groove ball bearings

µ f G0.0015 Tapered and spherical roller bearings with flange guided rollers

µ f G0.0018 Needle bearings

µ f G0.0045 All the above friction coefficients are referred to the bearing bore. Higher values apply for new bearings, particularly roller bearings that do not have run in guide flanges. This is also the case when starting bearings and when using an excessive amount of lubricant or a lubricant with high viscosity.

5.3 Contact stresses and deformations 5.3.1 Contact between elastic bodies During a loaded contact between two bodies with curved surfaces, an area of contact is created, the form of which depends on the curvatures of the surfaces at the point of contact, and the extent of which is a function of the force with which the bodies are pressed together. At zero load, the contact generally is a mathematical point and is commonly called point contact. However, when the two surfaces have the same curvature in any common plane, the contact is a mathematical line and is called line contact. This is the case, for example, in the contact between parallel cylinders. When one body is pressed against the other, an area of contact is formed; in the case of point contact it is an ellipse or a circle, whereas in the case of line contact it is a trapezoid or rectangle. The area of contact increases in size with

154

Rolling Contacts

increase in the load. Also, the bodies approach each other as a result of the flattening or deflection that takes place in and adjacent to the area of contact. The magnitude of deformation that develops is clearly dependent on the stresses in the material which are felt to some depth below the surface of the body. The stresses of a compressive nature are highest at the surface itself, but the shear stress, which is more serious because of the potential to cause surface fatigue or cracks, seems to develop inside the body at a certain small depth under the surface. The theory developed by Hertz (2) and its later modifications and amplifications are used to calculate stresses and deformations. The results are in accurate agreement with experiment regarding the deformations and the compressive stresses at the surface. It is thus possible to calculate the form and size of the contact area, the normal pressure on the contact area, its distribution within the area, and the compressive deformation of the body. However, as far as the static and dynamic bearing capacity is concerned, only a certain amount of guidance can be obtained from the present theory. Therefore, calculations of bearing capacity and endurance, of importance for practical application, ought to be supported by very comprehensive experimental data. It is mainly because of the friction between bodies in rolling contact that the shape and size of the contact area is of interest. The nearest approximation to pure rolling is obtained when the two surfaces in contact are parallel cylinders, or cones with a common apex. For any other form of the contact surface, more or less pronounced sliding motions develop. The theory of contact between elastic solids developed by Hertz assumes that the proportionality limit of the material is not exceeded and that the surfaces of the bodies are perfectly smooth, and thus no friction forces are allowed to exist in the area of contact. Half the major axis, half the minor axis, and the normal stress in the case of point contact can be calculated as a function of an auxiliary quantity F(r) (3). This is dependent on the principal radii of curvature which are located in perpendicular planes, and also on the angle between the planes of principal curvature of the two bodies. When these planes coincide F(r) G

(r1Ar2)IC(r1Ar2)II ∑r

(5.4)

where r1 and r2 are the reciprocals of the principal radii of curvature of the respective bodies at the point of contact, and ∑ rG(r1Cr2)I


155

C(r1Cr2)II is the sum of the reciprocals of all the principal radii of curvature. The radius of curvature is considered to be positive for a convex surface and negative for a concave surface. The half-axes of the contact ellipse are

1冢 1冢

aGλ a ΘE

3

冣

(5.5)

冣

(5.6)

W

∑r

bGλ b ΘE

3

W

∑r

where a is the major half-axis, b is the minor half-axis, W is the normal force acting on the contact, λ a and λ b are coefficients dependent on F(r) , and ΘE is a material constant. Constant ΘE can be calculated from the equation

ΘE G

1冤 3

115.5B109(E′ICE′II) E′I E′II

冥

(5.7)

where E′I G

EI 1A1兾v2I

E′II G

EII 1A1兾v2II

In the above expressions, EI and EII are the moduli of elasticity (N兾m2) and 1兾vI and 1兾vII are the Poisson’s ratios for bodies I and II respectively. The elastic compression, or the approach of distant points in the two bodies, is obtained from the equation

δ Gλ δ Θ 2E 31(W 2 ∑ r)

(5.8)

where λ δ is a coefficient whose magnitude is a function of F(r) . The mean normal pressure in the contact area is

σmG

W πab

(5.9)

and the maximum pressure at the centre of the area of contact is

σ max G1.5σ m G

1.5W πab

(5.10)

156

Rolling Contacts

From equations (5.2) and (5.3) it follows that

σ max G

1.5 π λ a λ b Θ 2E

1冤 3

冥

W( ∑ r)2

(5.11)

Assuming that the permissible load on the contact surface depends on a certain limiting value of σ max , and that a constant relationship between the radii of curvature in the contact exists (as in the case of bearings of the same type but of different size), i.e. if ∑ r is proportional to 1兾Dw , where Dw is the diameter of the rolling element, this limiting value for σ max is obtained when W兾D 2w reaches a certain value. The quantity W兾D 2w is known as the specific load and denoted by κ . For line contact between cylinders, the half-width of the contact area is bG0.0105B10−3ΘE3/2

1冢

W

冣

lw ∑ r

(5.12)

where lw is the length of contact (m). Deformation in line contact is rather difficult to express in a simple formula. When a cylinder of finite length is pressed between two plane surfaces of infinite size, and all three bodies are of the same material, then the approach between the axis of the cylinder and a distant point in either of the surfaces is approximately given by

δ G0.0003B10−3Θ 2.7 E

W 0.9 l 0.8 w

(5.13)

The above expression may also be used with a sufficient degree of accuracy in the case of ordinary roller bearings. It is quite clear that the diameter of the cylinder has no influence on the magnitude of the compression. The mean normal pressure in the contact area is

σmG

W 2blw

(5.14)

and the maximum normal pressure along the centre-line of the contact is 4 σ max G σ m π

(5.15)


157

By inserting equation (5.9), the following formula for the maximum normal pressure is obtained

σ max ≈

60B107

Θ

3/2 E

1冢

W lw

冣

∑r

(5.16)

When setting ∑ r ∼ 1兾Dw is permissible, as in the case of point contact, the specific load for line contact is

κG

W lw Dw

5.3.2 Elastic deformations in bearings When load is applied to ball and roller bearings, elastic deformations will be produced. They will develop at and adjacent to the point of contact and will have a significant influence on the degree of axial and radial rigidity with which the shaft is supported. Under a load, the bearing yields and the rings are displaced in relation to each other by a certain small amount. This elastic action can be calculated approximately in both magnitude and direction. Direction of elastic deformation When a bearing is under simultaneous action of a radial load Fr and a thrust load Fa , then the ratio of forces is Fa Fr

Gtan β

(5.17)

where β denotes the angle that the resultant load forms with the radial plane of the bearing (Fig. 5.5). Assuming that the outer ring of the bearing is stationary and that the load acts on the inner ring, the centre of the inner ring is displaced in a certain direction. This direction, however, does not always coincide with the direction of the resultant load. When the contact angle α of the bearing is constant and greater than 0, and the bearing has no internal clearance, the ratio of axial to radial displacements δ a 兾δ r can be calculated. Negative values of δ a 兾δ r indicate that the axial displacement takes place in such a direction that the bearing centre moves away from the plane in which the rolling elements are located. For a thrust load only, Fr G0 and δ r G0, so the following apply:

β G90°,

tan α 兾tan β G0,

δ r 兾δ a G0

158

Rolling Contacts

Fig. 5.5

When tan α 兾tan β G0.823 radial displacement takes place only in the case of point contact, and when tan α 兾tan β G0.785 radial displacement occurs in the case of line contact since, under these conditions, δ a 兾δ r G 0 and therefore δ a G0. It can be concluded that in all single-row angular contact bearings a pure radial displacement occurs when Fa ≈1.25 tan α Fr . A single-row bearing can support a pure radial load, i.e. β G0, only when the contact angle α G0°. In such a case, the displacement is also purely radial, i.e. δ a G0. Magnitude of elastic deformation When the maximum load W on one rolling element in the bearing is known, the compression in both contact areas of the most heavily loaded rolling element can be calculated provided that the material constants and the principal radii of curvature of the surfaces at the points of contact are given. The curvatures of the surfaces are similar for different bearing sizes and approximately proportional to the diameter Dw of the rolling elements for any ordinary standardized bearing type. Therefore, it is possible to derive approximate formulae of loading that give a value for the displacement from a zero location of the bearing centre in relation to the surrounding housing.


159

In the case of steel bearings under load conditions that give purely radial deflection (δ a G0), the following applies

δrG

1冢

0.0032B10−3

3

cos α

冣

W2 Dw

(5.18)

In the case of deep groove and angular contact ball bearings

δrG

0.002B10−3 cos α

1冢 3

冣

W2 Dw

(5.19)

In the case of roller bearings with point contact at one raceway and line contact at the other

δrG

4 0.001B10−3 1 W3 1lw cos α

(5.20)

In the case of roller bearings with line contact at both raceways

δrG

0.0006B10−3 W 0.9 cos α

(5.21)

l 0.8 w

In addition to elastic deformations, displacement resulting from possible bearing clearance should also be taken into account. Under practical conditions, particularly in the case of radial bearings, deformations in the bearing housing, which make the displacement of the shaft greater than the calculated value, should also be included. The axial deflection under pure axial load (δ r G0) is approximately as follows. Self-aligning ball bearings

δaG

0.0032B10−3 sin α

1冢 3

冣

W2 Dw

(5.22)

Angular contact ball bearings

δaG

0.002B10−3 sin α

1冢 3

冣

W2 Dw

(5.23)

Roller bearings with point contact at one raceway and line contact at the other

δaG

4 0.001B10−3 1 W3 1lw sin α

(5.24)

160

Rolling Contacts

Roller bearings with line contact at both raceways

δaG

0.0006B10−3 W 0.9 sin α l 0.8 w

(5.25)

Thrust ball bearings

δaG

0.0024B10−3 sin α

1冢 3

冣

W2 Dw

(5.26)

The case of deep groove ball bearings with α G0° is special, because rather large axial deflections develop under pure thrust load. Even for a very light thrust load an axial displacement of the magnitude 0.1 mm is possible. Any internal bearing clearance increases this displacement even further. Under heavier thrust loads, the displacement increases 4 Fa . The absolute value of the axial displacealmost in proportion to 1 ment mainly depends on the radius of curvature of the ball groove, the internal clearance, and the fits with which the bearing is mounted, and therefore no simple expression for calculation of the axial displacement can be derived. Occasionally, it is important to maintain as near constant deflection as is possible, even though the load may vary in both magnitude and direction. A practical way to achieve this is by avoiding internal clearance and clearance due to shaft and housing fits and by using a preload. The change in deflection caused by any given change in load is therefore reduced. In roller bearings this reduction is very small because the deflection is almost proportional to the load, and thus no advantage is usually gained by preloading. In the case of ball bearings, the gain due to preloading may be substantial.

5.3.3 Permanent deformations When a sufficiently high load is applied on the area of contact between two elastic bodies so that the elastic limit is exceeded at some point of the material, then a permanent deformation develops in the form of a local indentation at this point. A ball pressed against a flat surface with an appropriate force creates a bowl-shaped indentation in the flat piece; the ball is also flattened. In practical applications, permanent deformation occurs, even under relatively light loads, because the surfaces of the bodies are not perfect. Highest surface asperities are first subjected to pressure and are flattened by even a very small load because they represent an insignificant area of support. It has to be noted that this localized flattening affects the functioning of the bearing insignificantly.


161

Fig. 5.6

With increasing load, however, the permanent deformation gradually becomes more noticeable. Figure 5.6 is an illustration of that. At first, the total compression is almost entirely elastic and increases in proportion to W 2兾3. Gradually, the influence of permanent deformation becomes more important and a deviation from this proportionality takes place. This change is by no means abrupt. The increasing rate of permanent indentation is shown in Fig. 5.7, where the permanent deformation is a function of the load. It should be noted from this figure that surface imperfections cause permanent deformation even at very light loads. The Hertz theory cannot be used to estimate the magnitude of permanent deformation since it applies only to elastic contacts. However, relatively simple formulae can be derived, which give the magnitude as well as the variation of the permanent deformation within a limited range of the vicinity of the elastic limit. The point contact case can be represented by

δ p G1.25B10−8

W2 Dw

(r1ICr1II )(r2ICr2II )

(5.27)

where δ p is the total permanent deformation at a contact between a rolling element and its support, W is the load at the area of contact, Dw is the diameter of the rolling element, and r1I , r2I and r1II , r2II are

162

Rolling Contacts

Fig. 5.7

the reciprocals of the principal radii of curvature at the point of contact between body I and II. In equation (5.27), as in the equation given below, the radius of curvature is positive for a convex surface and negative for a concave surface. Permanent deformation for line contact is not uniform along the length of the roller if the roller generatrix has the same curvature along its entire length as the support. In such a case, the permanent deformation is considerably larger at the ends of the contact zone than it is in the middle, especially if the support has a greater length than the roller has in its axial direction. For this case, the following equation for the maximum total deflection can be used

δpG

2B10−11 W 1(rICrII ) 1Dw lw

冤

冥

3

(5.28)

where lw is the effective length of the roller, and rI and rII are the reciprocals of the radii of curvature of body I and II respectively in the radial plane of the rolling element. The distribution of permanent deformation between the rolling element and the bearing ring is such that one-third takes place in the rolling element and two-thirds in the ring. Before a fracture develops in either of the two bodies in contact, the permanent deformation may have assumed a considerable proportion. Therefore, in practice it is


163

necessary to take the risk of fracture into account only in exceptional cases.

5.4 Load distribution within bearings The transmission of external forces acting on the bearing occurs through the rolling elements from one bearing ring to the other. When two rolling elements are loaded simultaneously, a case of practical importance, the bearing is of statically indeterminate design. This means that the distribution of the forces among the individual rolling elements depends on the elastic deformation of the various bearing parts. The analysis of load distribution is important for static cases only, i.e. when the loaded bearing does not rotate. Endurance calculations for rotating bearings require not only the influence of the load distribution but also the number of stress cycles to which each point in the bearing material is subjected.

5.4.1 Radial bearings According to Stribeck (4), the maximum load per ball for a radially loaded radial contact bearing is Wmax G4.37

Fr n

(5.29)

where Fr is the radial load on the bearing. Equation (5.29) is quite accurate for any conventional number of balls n in a bearing, provided the internal clearance is zero. Stribeck found that these conditions are not often satisfied in practice, so that the calculated value for the maximum ball load has to be modified according to the expression Wmax G5

Fr n

(5.30)

A typical distribution of load over the balls in a ball bearing is shown in Fig. 5.8. The ordinate in the diagram gives the ball load as (nWmax)兾 Fr , so that it is directly comparable with the constants in equation (5.29) and (5.30). The abscissa ψ gives the angle between the annular location of the ball of interest and the direction of the load (Fig. 5.9). The dotted curve in Fig. 5.8 gives the calculated load distribution for the ideal case where there is no internal clearance and no out-of-roundness deformation of the rings.

164

Rolling Contacts

Fig. 5.8

Fig. 5.9

Elastic deformations in the areas of contact in a roller bearing with line contact between rollers and raceways are governed by different laws to those applicable to point contact. In theory, the maximum roller load under a radial force should be about 8 percent lower


165

than in the case of ball bearings. However, owing to the increased clearance required by the roller bearing, a greater deviation from the theoretical value is usually obtained. Nevertheless, practical experience supports the use of equation (5.30) for the ordinary types of roller bearing. In the general case of a bearing with an arbitrary but constant contact angle α , loaded by an external force F at an arbitrary angle β in relation to the radial plane (Fig. 5.10), the analysis is more involved and can be carried out only with the help of special tables. When the thrust load Fa G0, the inner ring moves away from the outer ring so that the bearing load becomes concentrated on a single rolling element. If, however, Fa increases above a certain critical value, the inner ring moves towards the outer ring, and more rolling elements are loaded. When Fa G Fr tan α , the inner ring does not move away from the outer ring, but only one rolling element is loaded by the force WGFr 兾cos α , while the other rolling elements remain unloaded. When Fa G1.25Fr tan α , the rolling elements located half-way around the bearing circumference are

Fig. 5.10

166

Rolling Contacts

loaded, and the load on the most heavily loaded rolling element is Wmax G4.37

Fr

(5.31)

n cos α

If Fa is further increased to about 1.7Fr tan α , all rolling elements are loaded, i.e. Wmax G3.93

Fr n cos α

G2.36

Fa n sin α

and Wmin G0. If β is then allowed to approach 90°, the bearing load gradually becomes more evenly distributed among the rolling elements and finally WG

Fa n sin α

In the case of double-row bearings with a constant contact angle, an internal thrust force develops between the two rows of rolling elements when there is a radial load on the bearing. Thus, even under pure radial load, the normal load distribution is obtained within the bearing, i.e. both rows of rollers are equally loaded around half the bearing circumference. Therefore Wmax G4.37

Fr 2n cos α

where n denotes the number of rolling elements in one row. Now, if additional thrust load is applied, the length of the loaded zone is increased in one row and reduced in the other. When Fa G 1.7Fr tan α , one row is loaded around its entire circumference and the other row is completely unloaded. The maximum ball load in one row is then WI max G7.86

Fr 2n cos α

G4.72

Fa 2n sin α

and in the other row, WII G0. In the case of a heavy thrust load, the loaded row acts in the same way as a single-row bearing. There are other more complicated cases that have not been discussed here. For example, any combination of two angular contact bearings, mounted opposed with separated centres and loaded with radial and axial forces as well as moments, can be analysed with the help of special


167

tables, provided the combination is symmetrical and the contact angle and distance between bearing centres remains constant.

5.4.2 Thrust bearings For a central force acting on a thrust bearing, the load on each of the rolling elements is WG

Fa n sin α

(5.32)

where Fa is the central thrust load and n is the number of rolling elements in the bearing. A thrust ball bearing able to support external load only in one direction with α G90° may be subjected to an eccentric thrust load (Fig. 5.11) making the load distribution non-uniform. If the eccentricity of the bearing load Fa is e and the pitch radius of the bearing is rm , one ball is unloaded when eG0.6rm . The load on the diametrically opposite ball is Wmax G2.36Fa 兾n. If e is increased to 0.8rm , only half the balls are loaded and Wmax G3.6Fa 兾n. When eGrm , only one ball is loaded and its load is WGFa .

Fig. 5.11

5.5 Kinematics of bearing elements 5.5.1 Rotational speed of the elements and the cage Firstly, the case where rings of the bearing rotate with different speeds is considered. In the analysis to follow, clockwise rotation is considered

168

Rolling Contacts

Fig. 5.12

as positive and counterclockwise as negative (Fig. 5.12). Assuming pure rolling conditions, the absolute speed of point A on the circumference of a rolling element, is equal to the circumferential speed vi at the instant A contacts the inner ring raceway. At the same time, point B on the rolling element contacts the outer ring raceway and has the absolute speed ve . The absolute speed of each point on the rolling element is a combination of the speed of the rolling element around its axis and the speed of the set of rolling elements around the bearing axis. The absolute velocities of points between A and B on the diameter of the rolling element are, at the instant shown in Fig. 5.12, parallel and vary linearly between the velocities of A and B. As the centre of the rolling element moves around the bearing axis only, its speed, which is the circumferential speed of the pitch circle, is given by viCve vm G (5.33) 2 Because πDi ni vi G 60 πde ne ve G 60


169

where ni and ne are the speeds of rotation of the inner ring and outer ring respectively, Di is the diameter of the inner ring, and de is the diameter of the outer ring. According to Fig. 5.12 Di GdmADw cos α de GdmCDw cos α where dm is the bearing pitch diameter, Dw is the diameter of a rolling element, and α is the contact angle. Finally, the circumferential speed of the rolling element centres and the cage can be expressed as vm G

π 120

冢

ni d m 1A

Dw dm

冣

cos α C

π 120

冢

Dw

ne d m 1C

dm

cos α

冣

(5.34)

Consequently, the speed of rotation of the set of rolling elements and the cage is

冢

nm G0.5ni 1A

Dw dm

冣

冢

Dw

cos α C0.5ne 1C

dm

cos α

冣

(5.35)

The difference between the absolute speed of rotation of the set of rolling elements and the speed of rotation of the inner ring constitutes the speed of rotation of the set of rolling elements relative to the inner ring. It is given by the following expression

冢

nmi GnmAni G0.5(neAni ) 1C

Dw dm

cos α

冣

(5.36)

In a similar way, the speed of rotation of the outer ring relative to the set of rolling elements can be found from

冢

Dw

nem GneAnm G0.5(neAni ) 1A

dm

cos α

冣

(5.37)

The speed of rotation of the rolling element around its own axis is given by nw G

Di nmi Dw

nw G

de nem Dw

or

170

Rolling Contacts

Using one of the above expressions, the following equation can be obtained nw G0.5

dm Dw

冢

(neAni ) 1A

Dw dm

冣冢

Dw

cos α 1C

dm

cos α

冣

(5.38)

It is rather rare that both rings of a bearing rotate simultaneously. Normally, one ring is stationary. If the outer ring is stationary (ne G0) and the inner ring rotates with the speed ni Gn, equations (5.35), (5.36), and (5.38) respectively reduce to

冢

nm G−nem G0.5n 1A

冢

Dw

dm

冢1Ad

nmi G−0.5n 1C

nw G−0.5n

Dw

dm

Dw

cos α

dm

cos α Dw

冣

冣

cos α

m

冣冢1Cd

Dw

cos α

m

冣

In the case of a stationary inner ring (ni G0) and the outer ring rotating with ne Gn, the following is obtained

冢

Dw

nm Gnmi G0.5n 1C

冢

nem G0.5n 1A

nw G0.5n

Dw dm

冢

dm

cos α

cos α

冣

冣

dm Dw 1A cos α Dw dm

冣冢1Cd

Dw m

cos α

冣

The case of a thrust bearing with α G90°, that is, cos α G0, when ni G n and the housing ring is stationary (ne G0) is described by the following equations nm G−nem G0.5n nmi G−0.5n nw G−0.5n

dm Dw


171

5.5.2 Contact cycles due to rolling Both the load-carrying capacity and the life of the bearing are of practical importance, and thus it is necessary to know the number of rolling elements, fi and fe respectively, that pass a given point on the inner ring or a given point on the outer ring, while one ring or the other makes one revolution. Each time the set of rolling elements has moved one full revolution in relation to one of the rings, the z rolling elements in a row have passed every point on the ring. The number of rolling elements passing per unit time is therefore znmi for the inner ring and znem for the outer ring. If one ring rotates with a speed of n revolutions per minute while the other ring is stationary, then fi Gz fe Gz

nmi n

nem n

冢

Dw

冢

Dw

G0.5z 1C

dm

G0.5z 1A

dm

cos α

冣

(5.39)

冣

(5.40)

cos α

In this analysis the speed of rotation of the set of rolling elements must always be taken as positive. Besides, the number of rolling elements passing a given point must not be confused with the number of stress cycles at a given point, because the point in question is subjected to load only when it passes or is passed by a rolling element that is in the loaded zone.

5.6 Inertia forces 5.6.1 Centrifugal forces Motion of a rolling element in a rotating bearing is associated with the creation of a centrifugal force acting on the rings. The magnitude of the centrifugal force is given by TG

G rm ω 2m g

(5.41)

where G is the weight of the rolling element, g is the gravity constant, rm is the pitch radius of the bearing, and ω m denotes the angular velocity of a set of rolling elements. In the case of a radial bearing, the force between the rolling element and the outer ring is increased by the given amount, while the force between the inner ring and the rolling element remains at the magnitude determined by the bearing load. The centrifugal force is usually relatively small and therefore does not have the effect of moving the most

172

Rolling Contacts

highly stressed point in the bearing from the inner ring to the outer ring in bearings in which the inner ring is the weakest element. Thus, in most types of radial bearing, the centrifugal force does not contribute to the reduction in the load-carrying capacity of the bearing. In bearings in which the outer ring is the weakest element, the overall loading is slightly increased by the centrifugal force. This increase, however, is usually so insignificant that it can be safely neglected. Only for very high speeds can the centrifugal force reach the magnitude of 10 percent of the outside load on the bearing. The centrifugal force in angular contact ball bearings causes a certain angle to exist between the directions of load at the two contact points of the ball with the rings. Usually, this does not have any effect of practical importance on the magnitude of the forces, but the areas of contact are displaced so that the microsliding within these areas increases. This phenomenon is most pronounced in thrust ball bearings where the balls at high speed must lean on the outer shoulders of the ball grooves (Fig. 5.13). The areas of contact, therefore, assume a position that deviates more from the theoretical cone of rolling than in the case when the centrifugal force is absent (Fig. 5.13). The balls also roll on a larger pitch diameter and may exert a pressure on the cage, thus facilitating the wear. A practical way to alleviate this effect is to make pockets oblong in a radial direction and keep a suitable thrust load on the bearing. When the bearing centre makes a circular motion, as in the case of an epicyclic gear, the rolling elements and cage are subjected to forces

Fig. 5.13


173

acting towards the centre of the circle. If the bearing has a cage, which is usually the case, it is centred on one of the rings and presses against this ring under pure sliding motion with a force equal to the centrifugal force of the cage. The forces created during acceleration owing to the eccentric motion and acting on the rolling elements are primarily evidenced as normal forces against the rings and as tangential friction forces on the rolling contact areas. The load on the rolling elements, which varies from one rolling element to another, is sufficient to develop a tangential friction force of the required magnitude in the contact areas for pressing the rolling elements against the side surface of the cage pocket under the condition of pure rolling. It is quite often advantageous to make bearings subjected to a crank motion cageless. The centrifugal forces then cause the rolling elements to press against each other within those parts of the circumference that are not located in the loaded zone. However, quite difficult lubrication problems are often encountered in such applications.

5.6.2 Crankpin bearings The kinematics of a crankpin bearing is quite complex, as it moves in several different ways. The inner ring rotates with the same speed of rotation as the crankshaft. Also, its centre moves with the crankpin, while the outer ring makes an oscillatory motion owing to the limited length of the connecting rod. As a result of this, the motions of the rolling elements are complicated, producing inertia forces within the bearing that may be of far greater importance for the proper functioning of the bearing than the external load. Especially important is the oscillatory motion of the outer raceway, which causes the speed of the rolling elements around the bearing centre to vary considerably. When the length of the connecting rod is 3 times the crack radius, the relative speed of rotation between the inner and outer raceway varies between 0.67 and 1.33 times the speed of rotation of the crankshaft. For example, at a crankshaft speed of 3000 r兾min the bearing speed changes from 2000 to 4000 r兾min and back during each revolution, that is, 50 times per second. For the rolling elements to maintain only rolling motion at their points of contact with the raceways during these rapid changes in speed, high tangential forces must exist in the area of contact; the heavier the rolling elements, the higher the forces must be. However, tangential forces can develop at the points of contact with the raceways only where the rolling elements are loaded, and these in turn must exert sufficient pressure on the unloaded elements so that the whole set takes part in the motion.

174

Rolling Contacts

Practice shows that cylindrical rollers are better for high-speed crankpin bearings than for ball bearings. This is due to the more favourable ratio of contact area to weight of the rolling element. Also, it is known that rollers of small diameter are preferred to rollers of large diameter.

5.6.3 Forces of gyration Any rotating body whose axis of rotation changes direction is subjected to gyratory forces. It is obvious that in ball and roller bearings the rolling elements change the direction of their axes of rotation if the contact angle α H0. A ball subjected to a gyratory moment has an axis of rotation that coincides with the tangent of the pitch circle. Thus Mg GJω o ω m sin α

(5.42)

where J is the mass moment of inertia of the ball, ω o is the angular velocity of the ball around its own axis, and ω m is the angular velocity of the ball centre around the bearing axis. According to equation (5.42), the gyratory moment increases with the contact angle α . It is highest in pure thrust bearings where α G90°. When the shaft ring rotates at n r兾min, the housing ring is stationary, the bearing pitch diameter is d m , and the ball diameter is Dw , then the angular velocity of the ball around its own axis of rotation is

ωoG

π

no G

30

π dm n 60 Dw

(5.43)

and the angular velocity of the ball around the bearing axis is

ωmG

π 30

nm G

π 60

n

(5.44)

The spinning of the ball in the direction of the gyratory moment is resisted by the friction moment caused by the ball load. Where sliding and rolling occur simultaneously, the coefficient of sliding friction is 0.07–0.08 for low rolling speeds. In the case of high speeds, however, it may assume considerably lower values owing to the formation of a lubricant film at the area of contact. An approximate formula for the friction moment in high-speed bearings is Mf G20

Dw Fa z

(5.45)


175

where Fa is the thrust load on the bearing and z denotes the number of balls in the bearing. The condition for sliding not occurring is that MfHMg , which means that FaH5.75B10−11zd mD 3w n2

(5.46)

Practical observations show that sliding between surfaces owing to spinning of the ball in the direction of the gyratory moment does not have an adverse effect on the bearing provided the specific load on the ball is less than 80 kN兾m2 and the lubrication is good. All radial bearings with a contact angle α H0 generate gyratory forces. However, they are small in these bearings since sin α is usually small and sliding can develop only when the ball load is low, so the bearing surfaces cannot be seriously damaged. In roller bearings with α H0, the gyratory moments have a potential to cause a slight change in the load distribution in areas of contact, but this is of no practical importance.

5.7 Load-carrying capacity 5.7.1 Dynamic capacity A rolling contact bearing can operate for a limited period of time only. If a ball or roller bearing is exposed to moisture or dirty surroundings, it may become unserviceable because of corrosion and wear after a period in service that cannot be predicted. On the other hand, if it is properly protected and well lubricated, all potential causes of damage are eliminated except one, i.e. the fatigue of the material owing to cyclic stresses produced by rotation. The manifestation of this fatigue process is flaking, which starts as a crack and develops into a spalled area on one or other of the load-bearing surfaces. Thus, ultimately, fatigue is unavoidable but the number of revolutions the bearing may make before flaking starts is a function of the bearing load. Concept of service life The concept of service life can be defined as a period of service that is limited by fatigue phenomena. The service life is measured in the number of revolutions of the bearing or in the number of hours of operation at a certain speed of rotation. Individual bearings that are identical and operate under identical conditions may, however, have different service

176

Rolling Contacts

Fig. 5.14

lives. The scatter of service life values is illustrated in Fig. 5.14. With a sufficiently large number of bearings tested, a minimum of 30, it is found that the longest life rarely exceeds 4 times the average life Lm , and that about 90 percent of the bearings have a longer life than onefifth of the average. On the basis of the relationship between life and bearing load, as well as between bearing load and material stress, it is possible to calculate that the material stress that 90 percent of the bearings can endure for a given length of time is only 16.5 percent smaller than that stress which can be endured for the same length of time by those bearings which have a life in excess of the average. The difference in material strength between individual bearings is therefore not as significant as might be thought when considering only the variation in service life. The scatter of results makes it necessary to formulate a precise definition of the concept of service life, which is taken to be the period of useful operation which can be anticipated with reasonable probability. Thus, the estimated service life means the number of bearing revolutions, or the number of working hours at a certain rotational speed,


177

that will be reached or exceeded by 90 percent of all bearings. This estimated service life is denoted by L in Fig. 5.14 and serves as a basis for bearing selections. The average service life Lm is 5 times as long as L. Analyses concerning the load capacity, equivalent load, and the like are carried out on the assumption that the most highly stressed individual point in the bearing material determines the capacity. The first fatigue cracks, however, do not always develop exactly at this point. Signs of fatigue could be observed first at certain other points, and their location depends on variations in strength inherent in the material. For that reason, the scatter of service life values is always observed. It is of practical importance to be able to estimate the probability of fatigue, permanent deformation, or fracture developing in various parts of a bearing. Link between load and service life In an imaginary test, if one group of bearings of a certain type and size is tested under the constant load F1 and another group under the load F2, but otherwise under identical operating conditions, the service life L1 and L2 respectively will be obtained for these two groups. It is found that a lower load always results in a longer service life. On the basis of a sufficiently large number of test results, the following has been established

1冢

F1 G F2

3

冣

L2

L1

or 3 3 F11 L1 GF2 1 L2 Gconst

(5.47)

Assuming that F is a constant pure radial load acting on a radial bearing whose inner ring rotates relative to the line of action of the load, and this bearing has a life of LN million revolutions, the constant in equation (5.47) is that load with which the bearing, under the given operating conditions, can attain one million revolutions. This load is called the dynamic specific load capacity or specific capacity C. In the case of thrust bearings, the specific capacity is expressed in terms of a pure thrust load. Because of their design, single-row angular contact bearings (α H0°) must always be subjected to a certain thrust load which is obtained by opposing the bearing to another bearing.

178

Rolling Contacts

Another well-known form of equation (5.47) is

冢P 冣

LN G

C

3

(5.48)

in which C is the specific capacity of the bearing, that is, the loadcarrying capacity for a service life of one million revolutions, P is the bearing load, and LN is the service life, in millions of revolutions, under the bearing load P. If the specific capacity of a bearing is known, then the permissible load for any service life can be calculated from equation (5.48) or the service life can be found for any known load. Estimation of specific dynamic capacity A number of factors influence the specific capacity of a bearing and, thus, its permissible load for a given life. Among the most important factors are: – the – the – the – the

properties of the material, degree of oscillation between rolling elements and raceways, dimensions of the rolling elements, number of rolling elements.

As theoretical estimation of the specific capacity is impractical, the only way to determine it is through time-consuming tests during which the influence of the different factors can be established. It has been found that the load-carrying capacity varies with the ball diameter Dw approximately as D 2w 1C0.02Dw It is known from the Hertz theory that the load-carrying capacity should be proportional to D2w . Practical experience shows that small balls have a relatively higher load-carrying capacity than large balls. This fact is accounted for by the correction factor 1兾(1C0.03Dw ). In roller bearings with short rollers of diameter Dw and length lw , the loadcarrying capacity varies with Dw lw . For rollers with greater lengths (lwH1.4Dw ) it is practically impossible to secure a uniform load distribution over the entire roller length owing to unavoidable skewing of the rollers, misalignment, and other conditions. Therefore, the loadcarrying capacity does not increase in proportion to the roller length. When the bearing has rolling elements made of a hard material, the fatigue phenomenon that determines the life usually develops on the raceway of one ring or the other. It is therefore obvious that the rolling


179

elements are not the weakest parts of the bearing, but their number influences the stress in the rings, partly because of the magnitude of load per roller and partly owing to the number of stress cycles per bearing revolution at the weakest point of the raceway. The magnitude of the force with which each rolling element acts on the weakest point of the raceway is inversely proportional to the number of rolling elements. Thus, for a constant number of stress cycles at the point of interest, the carrying capacity is proportional to the number of rolling elements. On the other hand, the number of stress cycles for one revolution of a bearing ring is proportional to the number of rolling elements z. Therefore, the same functional relationship between capacity and number of stress cycles exists as that between capacity and service life given by equation (5.48). In other words, for a constant ball or roller load the capacity is inversely proportional to the cube root of the number of rolling elements. Finally, the bearing capacity varies with z Gz2/3 3 1 z provided all other factors remain unchanged. The force of the rolling element acting against the raceway varies with 1兾cos α , so that the capacity is proportional to cos α . Finally, if the bearing has i rows of rolling elements, the capacity is i times higher than that of a single-row bearing, provided the load is distributed evenly among the several rows of rolling elements. This is, however, true only for double-row self-aligning bearings. For all other bearings, with two or more rows of rolling elements, it is necessary to allow for some reduction in bearing capacity owing to non-uniform load distribution. In summary, the specific load capacity of a ball bearing is CGfc

iD 2w z2/3 cos α 1G0.02Dw

(5.49)

and that of a roller bearing is CGfc iDw lw z2/3 cos α

(5.50)

where z is the number of rolling elements per row. A special explanation is necessary for coefficient fc . It depends, among other things, on the properties of the material, the degree of oscillation, and the reduction in the capacity on account of uneven load distribution within multiple row bearings and bearings with long rollers. The value of this coefficient can only be determined by comprehensive

180

Rolling Contacts

laboratory tests. It usually has one value for all sizes of a given bearing type. The values of coefficient fc are available in proprietary catalogues published by all leading manufacturers of rolling contact bearings. As far as the load-carrying capacity is concerned, there is no fundamental difference between radial bearings and thrust bearings. The term thrust bearing is used for those bearings whose contact angle is large, usually 45–90°, and whose rings take the form of washers. Thrust bearings are suitable for supporting loads that are primarily or exclusively thrust loads, and their specific capacity is therefore given in terms of pure thrust-carrying capacity. Thus, the loads on the individual rolling elements vary with 1兾sin α and the capacity with sin α . Other factors have the same effect as in radial bearings. The expression for calculating the specific capacity for single-row thrust ball bearings with α ≈ 90° is CGfc

D 2w z2/3 1C0.02Dw

(5.51)

and for single-row thrust roller bearings CGfc Dw lw z2/3 sin α

(5.52)

Effect of the speed of rotation on the load-carrying capacity With the help of equation (5.48) it is possible to find the required bearing size if the constant bearing load P and the desired life are both known. This can be carried out for service life expressed in number of revolutions or in number of hours at a certain constant speed of rotation. If Lh is the service life in hours and n is the speed of rotation, then LN G60B10−6nLh

(5.53)

The required specific capacity can be calculated from the equation describing service life. The bearing manufacturers provide tables showing the so-called load rating, i.e. the load-carrying capacity at different speeds of rotation based on a certain constant service life expressed in hours of operation, usually 500 h. Thus, for service life, equation (5.53) yields LN G0.03n According to equation (5.48), the load rating Cn at the speed of rotation n is C Cn G 3 Gfn C 10.03n

(5.54)


181

Coefficient fn is a function of rotational speed and can be easily found in catalogues published by bearing manufacturers. The specific capacity is equal to the load rating at 33.3 r兾min, as is apparent from equation (5.54) when Cn GC. Service life factors When all forces acting on a bearing are given and the equivalent load P is estimated on the basis of these forces, then the required specific capacity C can be calculated from the equation CGfN P

(5.55)

where fN is a factor that has a given relationship to the service life in million revolutions. From equation (5.48) the following is obtained 3 fN G1 LN

(5.56)

This service life factor can be used in cases where the service life cannot be easily expressed in hours as well as in the absolute number of revolutions. Examples of such instances are: automobiles, railroad boxes, and other vehicles where the service life is given in number of miles of operation. In the majority of applications, rolling contact bearings work with nearly constant speeds of rotation. Therefore, it is convenient to estimate the service life in number of operating hours. If the speed of rotation is constant, it is practical to start from the load rating Cn applicable at this speed. Therefore Cn Gfh P

(5.57)

where fh is the service life factor for the life given in hours. According to equation (5.47) 3 3 Cn 1 500GP 1 Lh

(5.58)

and therefore

1

fh G

3

Lh 500

(5.59)

5.7.2 Static capacity Quite often a bearing is under external load while it is not rotating. The service life equation (5.48) cannot be used in such a case because it gives PGS for LN G0. Understandably, there is a limit to the load that the bearing can carry under static conditions. However, this limit is not

182

Rolling Contacts

determined by the fatigue strength of the material but by the permanent deformations that are created in the load-supporting surfaces. It was mentioned earlier that permanent deformations could be produced even under relatively light loads and increase in size with increasing load. Consequently, there is no load limit below which the deformations are entirely elastic or at which plastic deformation suddenly begins to develop. It is practically impossible to avoid some permanent deformations, and the only question is what size of deformation can be safely tolerated if the bearing is to operate properly. Again, application experience demonstrates that permanent deformations have a negligible influence on the operation of the bearing provided that their combined magnitude at any one contact point is less than 0.0001 times the diameter of the rolling element. When the load is very high, permanent dents are formed in the raceways and consequently the bearing begins to run noisily and vibrate. Slight vibration and noise caused by the bearing when running at high speed have an important effect on certain applications. However, the static capacity implies the load to which a bearing can be subjected while stationary without deformations being produced, which would be noticeable when the bearing subsequently rotates under a lower load and normal requirements for smooth running. The pure radial load or pure thrust load that corresponds to this static load-carrying capacity of the bearing is called the specific static capacity and is denoted by Co . The above definition of static capacity implies that the bearing can safely be allowed to rotate under a load that is higher than the specific static capacity. If the maximum load is of limited duration and acts only while the bearing rotates, the permanent deformations that take place will be evenly distributed around the entire periphery of the raceways and cause no practical harm until the deformations become relatively large. The second implication of the definition of the specific static capacity is that the load may exceed the stated limit, in certain cases quite considerably, even while the bearing is stationary, provided that in subsequent running the need for smooth operation is not too important and the main requirement is that the bearing can still rotate. Magnitude of static capacity The maximum ball or roller load, Wmax, is used to determine the specific static capacity, Co . The magnitude of Wmax depends on the degree of oscillation in the weaker of the two ball or roller contacts as well as on the material characteristics. By setting a limit to the permissible permanent deformation of rolling element and bearing ring at a contact as


183

0.0001 times the diameter of the rolling element, the allowable static specific load po can be calculated for a given bearing from equations (5.27) and (5.28) respectively by substituting δ p 兾Dw G0.0001. Therefore, for ball bearings po G

2.8B107

(5.60)

Dw 1[(r1ICr1II )(r2ICr2II )]

and for roller bearings where the line contact at the inner ring is the location of interest po G

17B107

(5.61)

1[Dw(rICrII )]

In the above equations, the allowable static specific load, po , is given in N兾m2. Bearings of standard design have the relationship between diameter of rolling element and the radii of curvature of the ring surface fixed for different sized bearings of a given type. Thus, it is convenient to have average values for po for some bearing types prescribed in order to facilitate approximate analyses. Table 5.1 gives values of po . Since for ball bearings po G

Wmax D 2w

and po G

Wmax Dw lw

for roller bearings, the specific static capacity, Co , can be determined from the relations between bearing load and maximum ball or roller load which were discussed earlier. For radial ball bearings, the following Table 5.1 Bearing type

po (B107 N兾m2)

Ordinary ball bearings Self-aligning ball bearings Thrust ball bearings Deep groove ball bearings Roller bearings

1.5 1.7 5.0 6.2 11.0

184

Rolling Contacts

is obtained CoG15 Wmaxiz cos αG 15 po iz cos αD2w

(5.62)

and for radial roller bearings Co G15 Wmax iz cos α G15 po iz cos α Dw lw

(5.63)

The axial specific static capacity for thrust ball bearings is Co GWmax iz sin α Gpo iz sin α D 2w

(5.64)

and for thrust roller bearings Co GWmax iz sin α Gpo iz sin α Dw lw

(5.65)

Table 5.1 contains values of po that are applicable to bearings so designed and made that the load distribution within the bearings can be regarded as conforming to that calculated. In bearings where this is not the case, for example, in double-row radial contact ball bearings, roller bearings with long rollers, and the like, the static capacity is slightly reduced.

5.7.3 Equivalent bearing loads Equation (5.48) has to be used in all calculations of the service life of a bearing under given operating conditions. In this equation, LN is the service life in millions of revolutions, C is the specific bearing capacity, and P is the bearing load acting under operating conditions applicable to that specific capacity. It is assumed, therefore, that for a radial contact bearing the load P is constant and purely radial, and that the inner ring rotates in relation to the direction of the load P, while the outer ring is stationary in relation to the line of action of P. However, there are many practical applications in which the load is not constant and does not have a purely radial direction. Also, in these applications, the outer ring or both rings rotate in relation to the direction of load. In all these cases, the equation for the service life [equation (5.48)] can only be used if it is possible to calculate that constant pure radial load which, if only the inner ring were to rotate in relation to the direction of load, would have the same effect on the bearing as the actual load. In other words, it would give the same service life as that which the bearing will attain under the actual operating conditions. This imaginary load is called the equivalent load. The thrust bearing case requires calculation of the equivalent thrust load, because the specific capacity for this type of bearing refers to the capacity under constant centric pure thrust load. Calculation of the


185

capacity of a bearing that does not rotate under load also requires the use of an equivalent load, although this is related not to the fatigue process but to slight permanent deformations that limit the service potential of the bearing. It is known that the service life of a bearing, expressed in number of revolutions, is practically independent of the speed of rotation. Therefore, it is possible in the case of variable speeds to use the average speed, or to transform the product of time and number of revolutions per unit time during the different periods of operation into the total number of revolutions. Case of rotating bearings According to equation (5.48), the service life is inversely proportional to the third power of the load. Thus, a high load can exert a considerable influence on the life, even though it acts only during a relatively small part of the total service life. It is reasonable to assume that, if a load acts for a certain fraction of that number of revolutions during which the bearing can operate under this load before failing because of fatigue, the same fraction of the ability of the bearing to endure load under rotation is used up. Subsequently, if a load of another magnitude is applied, the bearing thus loaded can function for that number of revolutions for which this load alone would permit the bearing to operate before failing because of fatigue, minus the used up fraction of this last mentioned number of revolutions. Therefore, the relationship between the service life L1 and the radial load F1 is

冢F 冣

L1 G

C

3

1

If a1 denotes the fraction of the service life during which the load F1 acts, then a1L1 is the number of millions of revolutions during which the load F1 acts. This number is designated N1. Thus a1L1 GN1 Ga1

冢F 冣 C

1

or F 31N1 Ga1 C 3

3

186

Rolling Contacts

Using similar designations for other loads gives F 32 N1 Ga2 C 3 F 3n Nn Gan C 3 Adding the above equations 3 3 ∑ (F N)G∑ (a)C

According to the assumed principle for the consumption of the loadcarrying ability, it is apparent that ∑ (a)G1. Therefore 3 3 ∑ (F N)GC

or LN G

C 3LN 3 ∑ (F N)

By denoting Fm G

1冤 3

3 ∑ (F N )

LN

冥

(5.66)

it leads to 3

冢F 冣

LN G

C

m

Equation (5.66) shows that Fm represents the constant cubic mean load – the mean effective load which gives the same service life as the actual variable load acting on the bearing. Quite often it is possible to divide the bearing load history into a few periods with constant load within each period as shown in Fig. 5.15. By doing so it is easy to calculate Fm from equation (5.66). If a more accurate estimate of the load is required or the load history diagram has a complicated form, then it is necessary to divide the load diagram into an infinite number of small parts, i.e. replace ∑ (F 3N) by the integral

冮

LN

0

F 3 dN


187

Fig. 5.15

As a result, the following is obtained Fm G

1冢 3

L 兰0 F 3 dN N

LN

冣

(5.67)

If the load varies periodically, the cubic mean load for all periods is the same as the cubic mean load for one period. It is therefore possible in equation (5.67) to let LN represent the number of revolutions in one period. Figure 5.16 illustrates loading conditions in which F varies from a minimum Fmin to maximum Fmax . In such a case, the following approximate expression can be used Fm G13 FminC23 Fmax

(5.68)

All the equations for the cubic mean load apply to cases where the direction of the load is constant, that is, β Gconst. The cubic mean load can, however, be introduced in the service life calculations only after having been converted to pure radial load, acting on the bearing as a

188

Rolling Contacts

Fig. 5.16

rotating inner ring load in the case of radial bearings, or pure centric thrust load in the case of thrust bearings. It was concluded earlier that the maximum load on the ball or roller, as well as the length of the loaded zone and the load distribution within that zone, changes when the direction of the resultant load changes. For example, in a bearing whose weakest point is the inner ring ball or roller contact track and whose inner ring rotates relative to the direction of load, the pressure is exerted at a given point on the inner ring ball or roller contact path every time this point passes a rolling element located within the loaded zone. The magnitude of this pressure is dependent on the angular displacement ψ of the rolling element when it passes the point of interest, this displacement being measured from the action line of the bearing load (see Fig. 5.9). The relation between the magnitude of the force and its angular position under pure radial load is shown in Fig. 5.8. The potential fatigue effect that these cyclic loads have on the point under consideration is proportional to the number of times the point is loaded and to the cubic mean value of the different forces that act during each respective instant of load. Suppose that the bearing load acts at an angle β to the radial plane of the bearing (see Fig. 5.10). As a result of that, the loaded zone is longer and the load variation, from rolling element to rolling element within this zone, is not the same as that previously considered. Consequently, the fatigue effects are also different. In the meantime, other factors are introduced, such as the tangential stresses due to the tendency of various rolling elements in the bearing to have different peripheral speeds, and the increased probability of the occurrence of fatigue


189

owing to a greater number of points being stressed, and these all have the potential to influence the fatigue strength. Therefore, the load-carrying capacity of the bearing varies with the direction of load. This is not only because of change in the maximum ball or roller load but also for other reasons. The analysis is quite complicated and will be illustrated by an example of double-row and single-row angular contact bearings, both with a contact angle such that tan α G0.25 and in both of which the inner ring is the weakest element. A possible starting point for the analysis is that capacity which a bearing would have with various directions of load for a given life, provided that the maximum ball or roller load alone determined the capacity. Therefore, this bearing load is chosen as a unit load that theoretically could be applied to the bearing to give the same maximum ball or roller load with all directions of load. Figure 5.17 shows the conditions in the double-row bearing, where the curve i applies to the rotating inner ring load. Usually, the permissible load (i.e. the loadcarrying capacity of the bearing) is lower than the unit load mentioned above. Thus, in the case of a pure radial load, β G0, the load carrying capacity is only 0.875 of the theoretically calculated load, because the actual load distribution of the rolling elements is less favourable than the theoretical distribution owing to clearance within the bearing and between the outer ring and housing.

Fig. 5.17

190

Rolling Contacts

The increased number of stress cycles per revolution at the weakest point of the bearing must be taken into account when a thrust load is applied, that is, β H0°. As the angle β increases, the capacity gradually becomes a smaller fraction of that capacity which was determined by variations in the maximum ball or roller load only. When tan β G 1.7 tan α , that is, when β G23°, one row of rolling elements is loaded around the entire circumference and the other row is entirely unloaded. In order to obtain a certain life under a pure thrust load, β G90°, only half of the maximum ball or roller load can be allowed as that which theoretically would be permissible in the case of pure radial load. An analogous condition for a single-row bearing with the same contact angle is shown in Fig. 5.18. As argued earlier, β can never be smaller than α . Only when tan β G1.25 tan α will the bearing begin to operate normally and in the same way as each row in the double-row bearing when β G0. Thus, when β G17.5°, the carrying capacity is 87.5 percent of the theoretical. When tan β G1.7 tan α , that is, when β G23°, the row of balls is loaded around the circumference and then functions in exactly the same way as the row in the double-row bearing that carries the load. Therefore, the curves in Figs 5.17 and 5.18 coincide at β H 23°. When the inner ring rotates relative to the direction of the bearing load, a given point in the ball or roller contact path with the inner ring passes the different rolling elements. Within the unloaded zone, no

Fig. 5.18


191

pressure is exerted on the point in question. Within the loaded zone, a lightly loaded rolling element passes first, and thereafter other rolling elements that are more loaded pass successively until a maximum pressure is obtained when the point passes a rolling element that is located almost on the line of action of the bearing load. After this, less loaded rolling elements pass again. In a running bearing with the inner ring stationary in relation to the direction of load, the case of a fixed inner ring load, the point of the inner ring located on the line of action of the bearing load is always subjected to the maximum ball or roller load every time a rolling element passes this point. Thus, the load-carrying capacity of the bearing for a given service life is reduced. In the case of a pure radial load, it is usually about 70 percent of the capacity for a rotating inner ring load. As the bearing load gradually changes to a more axial direction, the load-carrying capacity changes according to the curves denoted by e in Figs 5.17 and 5.18. When there is a pure thrust load, it is unimportant which ring rotates so that curves i and e coincide at β G90°. In order to examine how the load-carrying capacity changes with the direction of load, it is instructive to consider as a unit the practical load-carrying capacity F1 of a single-row bearing with α G0°. Knowing the distribution of load over the rolling elements, it is possible to calculate theoretically the load-carrying capacity that the bearing should have for different directions of load, if the maximum ball or roller load Wmax were a limiting load. In the single-row bearing with α G0° and z denoting the number of rolling elements, the maximum load is Wmax G

5F1 z

In a double-row bearing under radial load F2 , the theoretical maximum load is Wmax G

4.37F2 2z cos α

and therefore F2 G2.29F1 cos α if Wmax is to be of the same magnitude in both cases.

192

Rolling Contacts

In theory, the double-row bearing should have 2.29 cos α times as high a capacity as the single-row bearing has in practical application. In the case of a pure thrust load F2 for the double-row bearing Wmax G

F2 z sin α

because only one row of rolling elements is loaded. Therefore, in this case the following is obtained 5F1 F2 G z sin α z or F2 G5F1 sin α In general, the following is applicable PGXFrCYFa

(5.69)

where P is the equivalent load, Fr is the radial component of the actual load, Fa is the axial component of the actual load, X is the bearing rotation factor, and Y is the bearing thrust factor. The rotation factor X is an expression for the effect on the bearing capacity of the conditions of rotation. The thrust factor Y is a conversion value for thrust loads. It has been found that equation (5.69) is applicable to most bearing types provided the load components Fr and Fa are constant during the entire bearing life. The values for X and Y are different for different bearing types and are available in catalogues published by rolling contact bearing manufacturers. Self-aligning ball bearings have the feature that the weakest point is located on the contact path of the outer ring. The factor Y for these bearings is smaller than that applicable to other double-row bearings. The number of stress cycles per revolution of the inner ring, produced by a thrust load, is the same at the most heavily loaded point on the outer ring contact path as under a radial load. When the inner ring load is fixed, the rings have comparable strength and, therefore, the factor X does not change its value in this case. The relationship between the load-carrying capacity and the direction of load for single-row bearings with a constant contact angle is exactly the same as in double-row bearings provided that tan β H 1.7 tan α .


193

The specific capacity and equivalent load P is only half of that for double-row bearings. Thus PG0.5FrC0.4 cot α Fa and for a fixed inner ring load PG0.7FrC0.4 cot α Fa It can be seen that the factors X and Y have only half the magnitude for a single-row bearing that they have for the corresponding double row. However, equation (5.69) is not entirely applicable to single-row bearings. In the case of a rotating inner ring load it must be remembered when analysing a single-row angular contact ball bearing that the thrust load should not be less than Fa G1.25 tan α Fr. For an even smaller relative thrust load, the load is concentrated on a reduced number of rolling elements, and when Fa G1.25 tan α Fr , only a single rolling element is loaded. Thus, the load-carrying capacity is reduced to a very small value for this direction of load. It was shown earlier that deep groove ball bearings do not have a constant contact angle. It is known, however, that, with the conventional radii of curvature of the ball grooves, which is approximately 4 percent larger than the ball radius, it is possible to use the factor YG 1.5–2, the lower value applying to relatively heavy loads. Equation (5.69) agrees well with test results under pure or predominantly radial loads. The pure thrust capacity of deep groove ball bearings can be raised by an increase in the radial internal bearing clearance to a maximum value of 0.005Dw. In this way, the thrust capacity is increased by a maximum of 25 percent, which corresponds to a doubling of the service life. There are many operating conditions under which the bearing loads are variable in magnitude as well as direction. If a radial load component continuously acts in the same axial plane in relation to the inner ring or outer ring but the load resultant is variable and changes direction in the plane of interest, then the following expression for the equivalent load is obtained as a result of insertion of the expression for P given by equation (5.69) instead of F resulting from equation (5.67) L 兰0 (XFrCYFa )3 dN

1冤

PG

3

N

LN

冥

(5.70)

194

Rolling Contacts

When the load changes in such a way that equation (5.68) can be applied, then the following approximation results PG13 (XFrCYFa )minC23 (XFrCYFa )max

(5.71)

If the load is not only variable but also has a radial component that changes direction in an irregular way, or if the load is periodic with the same frequency as the speed of rotation of the bearing, then a more sophisticated and advanced analysis is required. It is a well-known fact that pure thrust bearings, α G90°, never carry combinations of radial and thrust loads. In such a case, both bearing rings are usually equally strained and the equivalent thrust load may be calculated according to equations (5.67) and (5.68) if the load on the bearing is a variable centric axial load. With an eccentric thrust load, the equivalent load is increased in proportion to the maximum ball or roller load owing to the unchanged number of stress cycles per revolution. Nowadays, a special type of bearing is available which can be considered as a thrust bearing for which the specific capacity is given as pure thrust capacity even though the bearing can also support radial loads. One such type is the spherical roller thrust bearing which, as a matter of fact, may also be considered as a radial bearing with a very steep contact angle. The ratio of load-carrying capacity in a pure radial direction to that in a pure axial direction, as well as the ratio of the equivalent radial load P to the equivalent thrust load PA, is P

GY

PA Substituting P obtained from equation (5.69), the equivalent thrust load is given by X PA GFaC Fr Y or by denoting X Y

GYa

the result is PA GFaCYa Fr

(5.72)


195

Case of non-rotating bearings When a load of such magnitude as to damage a bearing is applied to a stationary bearing, then it is not subjected to fatigue comparable with that prevailing if the bearing rotates. The possibility of permanent deformation development in the material, which would affect the serviceability of the bearing, is quite real. In a case like that, the absolute maximum load must be taken into account, and the equivalent load must be calculated on the basis that it would cause the same deformation or give the same risk of fracture as the actual load combination. Thus, when calculating this static equivalent load Po for radial bearings, the following expression is recommended Po GXo FroCYFao

(5.73)

where Fro is the radial component of the maximum static load, Fao is the axial component of the maximum static load, and Xo and Yo are factors available from the bearing manufacturer catalogue. The eccentricity of the load in pure thrust bearings is the only factor that affects the equivalent thrust load. Thus, the static equivalent thrust load bears the same relation to the actual eccentric load as the maximum ball or roller load under eccentric load bears to the maximum ball or roller load under centric load. For a thrust bearing that can carry a radial load, the ratio of the static radial capacity to the static thrust capacity is 0.2 cot α . As the static equivalent radial load of the bearing is Po G0.5FroC0.2 cot α Fao the static equivalent thrust load is therefore PAo GFaoC2.5 tan α Fro

(5.74)

5.8 Lubrication of bearings As stated in Chapter 1, there are three basic functions performed by the lubricant, i.e. to lubricate, to remove heat, and to protect against the influence of environmental elements. It was also said that some sliding contact occurs between the rolling element and the running track in commercial bearings and, in some cases where diametral clearance and relaxation of load encourage it, a hydrodynamic lubrication regime is established. However, in normal loaded conditions the preponderant function of the lubricant in this area, according to the elastohydrodynamic theory, is the generation of a film between rolling elements and track as a result of deformation.

196

Rolling Contacts

5.8.1 Elastohydrodynamic lubrication The shape of the film of lubricant as predicted by Grubin and Vinogradova (5) and confirmed by Crook (6) is shown schematically in Fig. 5.19 with the theoretical pressure distribution. In practice, as the pressure in this region is very high, possibly 1.5–3.0 GPa, the viscosity of this film is greatly increased. Therefore, to the extent that such a regime prevails in the contact area, it would seem likely that this film, momentarily of such high viscosity as to be comparable to dry static friction, transmits the elements of the couple which produces rolling. At the entry to the region the lubricant is under shear and may be regarded as behaving hydrodynamically as suggested by Archbutt and Deeley (7), but within the zone, presumably, negligible slip occurs. Clearly, the pressure–viscosity characteristics of lubricants are of importance in the behaviour of such elastohydrodynamic films. Experimental studies into elastohydrodynamic lubrication confirmed a close agreement between film thickness measurements and values predicted by the theory. Where anomalies were found, notably in the cases of a

Fig. 5.19


197

silicone fluid and a solution of polymethyl methacrylate in oil, the possible effect of non-Newtonian behaviour was suspected. Tallian et al. (8) made a study of conditions in rolling contacts where load is shared between an elastohydrodynamic film and surface asperities. They called this partial elastohydrodynamic lubrication and considered that it could exist for a given system in speed conditions below those at which a full elastohydrodynamic film would be established. The critical nature of the conditions that determine whether a fully developed elastohydrodynamic film can be established in relation to the surface finish can be deduced from the following dimensions proposed by Dowson (9). The typical length of the Hertzian zone of contact (representing the region of effective pressure generation for elastohydrodynamic contact) is approximately 0.25 mm. The typical film thickness in the zone is approximately 0.00025 mm. The typical transit time for lubricant through the zone (according to rolling speed) is approximately 0.0001– 0.00001 s. Finally, the typical contact pressure on the lubricant is approximately 2 GPa. These quantities must be related to a typical surface finish for rolling contact bearing elements of about 12.5B10−6–2B10−4 mm. It is now commonly accepted that, when the film thickness exceeds the combined peak-to-valley heights of the rolling element surface and the raceway by about 3–4 factors, then a complete fluid film lubrication is established. Where full hydrodynamic or elastohydrodynamic conditions can be maintained, the full fatigue life expectancy of the bearing, insofar as it is determined by lubrication, may be achieved. If boundary conditions prevail even for part of the time, the fatigue life may become unpredictable. When considering the lubrication in the rolling element兾raceway contact, some reference must be made to the behaviour of a ball in conditions where it is forced to depart from relatively true rolling, as in an angular contact bearing. A study by Hirano and Tanoue (10) using a magnetized ball, which enabled tracking of its path, revealed that the motion of the ball is three-dimensional, including slipping and spinning. It was also found that slip of the ball occurs on the unloaded side, where its contact with the raceways is looser. Slip decreases with increase in speed and load. It increases with radial clearance. The instantaneous rolling axis relative to the ball changes regularly and this is closely related to the slip. The cause of the change in the rolling axis is spin during slip. The angular displacement of the spin is proportional to the mean slip. The spin increases with the asymmetry of the contact of the

198

Rolling Contacts

ball with the cage or with the grooves of the races. Further increase in asymmetry causes negative slip. Under pure thrust load, no slip is observed although the contact is still maintained. Friction resulting from the spin and slip described may be high in some cases and not only must it be included in the sum of lubricating requirements for the bearing but it may make severe demands necessitating special qualities in the lubricant. In general, lubrication of the rolling element sliding in the cage pocket presents no problems as the load at the contact point is negligible. Normal hydrodynamic lubrication would be expected to be maintained. Sliding of the roller end on the lip, particularly in taper roller bearings, can result in boundary conditions being produced under high loads, and heavy wear may ensue if special lubricants are not used. If the cage is centred on the inner or outer ring, lubrication is demanded between the bore or circumference of the cage, as the case may be, and the race. Again, the load may be negligible except where out-of-balance forces exist. Lubrication of this, in effect, plain bearing, is normally hydrodynamic, but in the case of the cage centred on the inner ring a speed limitation must be observed if the lubricant is a grease. If the limit is exceeded, the centrifugal force created by slight imbalance of the cage may cause the lubricant film to break and wear occurs, aggravating the tendency of the cage to precession. Finally, lubrication of the rubbing seal in sealed bearings must not be overlooked. Although strictly not a part of the bearing proper, like the inner centred cage this should be regarded as a plain bearing hydrodynamically lubricated and liable to wear if lubrication fails.

5.9 References (1) Harris, T. A. (1996) Rolling Bearing Analysis (John Wiley, New York). (2) Hertz, H. (1882) Uber die Beruhrung fester elastischer Korper (in German). J. Reine und Angewandte Mathematik, 92, 156–171. (3) Pinegin, S. W. (1965) Contact Strength in Machines (in Russian) (Mashinostroenie, Moscow). (4) Stribeck, H. R. (1901) Ball bearings for various loads (in German). Z. VDI, 45. (5) Grubin, A. N. and Vinogradova, I. E. (1949) Book No. 30 (in Russian) (Central Scientific Research Institute for Technology and Mechanical Engineering, Moscow).


199

(6) Crook, A. W. (1961) Elastohydrodynamic lubrication of rollers. Nature, 190. (7) Archbutt, L. and Deeley, R. M. (1927) Lubrication and Lubricants (Griffin, London). (8) Tallian, T. E., McCool, J. I., and Sibley, L. B. (1965) Partial elastohydrodynamic lubrication in rolling contact. In Proceedings of IMechE Symposium on Elastohydrodynamic Lubrication, London. (9) Dowson, D. (1965) Elastohydrodynamic lubrication: An introduction and a review of theoretical studies. In Proceedings of IMechE Symposium on Elastohydrodynamic Lubrication, London. (10) Hirano, F. and Tanoue, H. (1961) Motion of a ball in a ball bearing. Wear, 4.


Chapter 6 Rolling Contacts in Land Locomotion

The power–weight ratio of railway locomotives has been increasing consistently over the past two decades mainly owing to enlarged power plants, the extensive use of lightweight materials, and improved construction methods. The only limiting factor to a continued increase in this ratio seems to be the traction developed between the driving wheels and rails (1). This statement is supported by the pronounced tendency for wheel slip to occur, especially in starting from standstill and under wet conditions. Another major area of land locomotion and transportation involves the rolling performance of pneumatic tyres on road surfaces. The contact interface between tyre and road is determined by complex interaction events during free rolling, braking, driving, cornering, skidding, or any combination of these modes, and its properties reflect the result of these interactions (2). Nevertheless, in practice it is useful to separate the individual contributions of tyre and road in order to understand the fundamental events that subsequently determine the frictional coupling in the contact area.

6.1 Rail–wheel systems In order to gain a clear understanding of the fundamental mechanism governing the traction between wheel and rail, a memory effect also known as secondary conditioning has to be recalled. This phenomenon arises from the fact that, when certain substances, especially oils, are spread upon a rail, secondary effects take place which are manifested

202

Rolling Contacts

by the creation of a minute quantity of the substance so closely associated with the surface as virtually to form a part of the main material. Therefore, it is possible for two apparently identical steels to possess widely different coefficients of surface friction despite all the efforts to clean the track. This memory capability, which is generally detrimental for surfaces previously contaminated with oil, can have beneficial effects when the spark discharge method is employed to improve traction.

6.1.1 Traction at the rail–wheel interface Basically, there are only three methods for improving rail–wheel traction, namely: – employing additives on the rail surface (chemical), – scoring, abrading, and sanding of the railhead (mechanical), – plasma arc or spark discharge between an electrode and the rail (electrical). It appears that both the chemical and mechanical methods have either failed to provide satisfactory improvements in traction or they tend to introduce other unwanted effects. The list of undesirable effects given below is by no means comprehensive but illustrates well the problems involved: (1) The use of a colloidal dispersion of silica in water produces a considerable improvement in the traction of dry or wet rails having medium or low values of secondary conditioning. However, this treatment shows little improvement on rails covered by an oil. (2) Sodium hydroxide solutions are able to reduce the effects of oil contamination by attacking traces of oil which give low secondary conditioning, thus increasing adhesion. However, excess or longterm use of these compounds produces a sludge which may have an adverse lubricating effect. (3) Sanding of the rail surface provides a practical way of securing an instantaneous increase in rail–wheel traction. However, the sand must be absolutely dry, and there are practical difficulties with appropriate storage and particle size control. Besides, sanding is not permitted near switch gears and switching points because of the danger of clogging. Also, an increase in surface damage occurs owing to pitting. (4) Rails previously treated with silicone fluids tend to give high traction values even when subsequently treated with oil. On the other hand, initially clean rails give low traction when covered with oil. However, when the silicone-treated rails are covered with water, the

Rolling Contacts in Land Locomotion

203

formation of innumerable droplets brings about a large reduction in traction. The method of spark discharge, consisting of ionizing the air gap between an electrode placed ahead of each driving wheel and the rail, effectively removes the contaminants that normally appear on the rail, and thereby produces a marked improvement in traction under the most adverse conditions. Table 6.1 shows values of rail–wheel traction values under dry, wet, or greasy conditions (3). It is apparent that a large variation in the values of the traction coefficient exist, from a minimum value of 0.07 on damp rails to a maximum of 0.35 under dry, clean conditions. Lower traction values usually point to the presence of an oil contamination on the rail surface. Also, the slippery conditions are usually found on curves, near points, near stations, and at road crossings. Oil contamination from axles and lubricating pads flows on to the wheel rims and finds it way into the contact path. The traction coefficient values listed in Table 6.1 are approximate and most appropriate for low speeds. When speed is considered as a variable, then it can be seen that the traction systematically decreases with increasing locomotive speed. A number of empirical relationships have been put forward to approximate the adhesion versus speed. Usually, they take one of two forms k1 fT Gk1C VCk3

(6.1)

fT Gk4Ak5V n

(6.2)

Table 6.1 Examples of rail–wheel traction coefficients Condition of rail surface

Traction coefficient

Dry rail (clean) Dry rail (with sand) Wet rail (clean) Wet rail (with sand) Greasy rail Moisture on rail Sleet on rail Sleet on rail (with sand) Light snow on rail Light snow on rail (with sand) Wet leaves on rail

0.25–0.30 0.25–0.33 0.18–0.20 0.22–0.25 0.15–0.18 0.09–0.15 0.15 0.20 0.10 0.15 0.07

204

Rolling Contacts

where fT denotes the traction coefficient, ki represents positive constants (iG1–5), and n has the value 1 or 2. From the results of many investigations, the following approximate and simplified equations can be used for dry rail surfaces (a) 0FVF60 km兾h fT G0.25 (b) 60FVF225 km兾h fT G

30 VC75

In the case of a wet rail surface, the following is recommended: f ′T G0.6 fT . In general the traction coefficient decreases with increasing brake block pressure. This can be expressed by the following relationship fT ∼ p−0.38

(6.3)

where p is the pressure exerted by the brake block. The inverse dependence of fT on p is in accordance with the simple theory of adhesion for metals and is valid for pressures of up to about 2 MPa. It can be shown that at greater contact pressures the traction coefficient, which is directly related to the adhesion at the interface, begins to increase with increasing p and, at the same time, becomes independent of speed. The reason for this is probably the combination of high pressure and speed causing overheating of the brake block, softening it, and eventually bringing about more intimate contact with the rail. In order to secure an acceptable wear resistance, the brake blocks should have a hardness of 220–240 HB (Brinell hardness) and the wheel rims a hardness of 240–300 HB. There are a number of other variables affecting traction between rail and wheel, namely wheel load, wheel size, and braking or driving mode. The wheel load seems to have an insignificant effect on the traction coefficient fT. This is largely due to the fact that, for very high loading, the case of rail–wheel contact, the real area of contact approaches the apparent area size. The mean contact pressure is close to the yield limit in compression for steel, and, according to the simple theory of the adhesion component of friction, the traction coefficient remains invariant fT Gs兾p*

(6.4)


205

where s represents the shear strength of the weaker material in contact and p* denotes plastic flow or yield pressure. Increased wheel load will also increase the apparent area of contact, but this affects both the numerator and the denominator in equation (6.4) so that there is no overall change in fT. For clean surfaces, the value of s兾p* is about 0.8 for steel–steel contact. In most cases, however, the usual presence of oxides, rain, oil, or other contaminations reduces the effective shear strength of the interface, s, so that the adhesion values in Table 6.1 are obtained. With increasing wheel diameter, there is a small reduction in mean pressure and a corresponding increase in apparent contact area. Also, it appears that the traction coefficient is little different whether braking or driving conditions prevail at the wheel–rail interface.

6.1.2 Braking process It is instructive to consider the effect of braking a railway wheel from an initial travel speed of 100 km兾h. The time taken for wheellock to develop is approximately 1 s or less from the instant of brake application. During this period, both tractive effort and wheel angular velocity decrease progressively, whereas the velocity of slip between wheel and rail rises in a non-linear way to attain a final locked-wheel skidding value of 100 km兾h. The contact area between wheel and rail consists, during that time, of a region of adhesion and a slip zone. This is schematically shown in Fig. 6.1. Both theory and experiment indicate a longitudinal shear stress distribution within the area of contact, which takes the form of the upper curve in Fig. 6.1. It should be noted that the shear stress is confined to relatively low values within the adhesion zone and reaches its maximum value within the slip region. Thus, the well-known requirement that a certain relative slip velocity between surfaces is essential for a maximum friction is satisfied by the above observation. The distribution of longitudinal slip velocity in the contact zone tends to have a non-linear increase towards the rear of contact according to the lower curve in Fig. 6.1. The slip region is followed by a kind of overshoot as the band velocity attempts to return to its undeformed value just outside the contact area. Another interesting fact is that, although the apparent area of contact could be quite substantial, the real area of contact is much smaller. 6.1.3 Traction enhancing techniques One of the techniques used with considerable success in practical situations is the spark discharge method of volatilizing rail–wheel contaminations. It has shown a considerable improvement in traction values

206

Rolling Contacts

Fig. 6.1

after its application. In general, the technique consists of mounting a single electrode of mild steel ahead of the test wheel and at a height 15– 17 mm above the railhead. A similar electrode is placed to the right of the test wheel surface. The test wheel is rigidly attached to an axle, on the other end of which a similar wheel is free to roll. A hydraulically operated disc brake applies a braking torque to the experimental axle until the test wheel is on the point of slipping. This torque is proportional to the coefficient of traction between the wheel and rail and can be easily measured using a dynamometric arm. A high voltage circuit triggers and sustains a spark discharge simultaneously across the gap between the electrode and rail and across the gap between the electrode and wheel at prearranged time intervals.


207

6.1.4 Consequences of wheel and rail wear It is inevitable that, as a result of wear occurring at the interface formed by the contact between wheel and rail, the performance of the rolling contact is affected. For new wheel and rail profiles, the contact area is little more than 300 mm2, and the contact pressure approaches the yield limit in compression of the railhead material. Plastic flow of metal occurs both at the surface between wheel and rail and more readily at a small depth within the rail profile itself. This accounts for the smooth and shiny appearance of rails in service compared with the rusted appearance of those that have not been used for some time. As wear progresses in both the steel tyre of the wheel and the railhead, the contact area widens and becomes elliptical in a transverse direction. Figure 6.2 shows the relative profiles and contact conditions for three combinations of rail and wheel in both cornering and straight running. It can be seen that wear of the rail top causes a progressively increasing slope of its contact surface to the horizontal. Wear of both tyre and rail increases the slope of the contact interface from about 1 :20 for the new condition to 1 :6, while the centre of contact remains in about the same position. If the rail but not the wheel is reground to the original profile, this slope remains at a value of about 1:6, but the centre of contact moves inwards with respect to the vehicle or in the direction of the wheel flange. Meanwhile, the cornering condition produces only one contact spot for this combination compared with the normal two contact positions when the rail and tyre wear progressively together. 6.1.5 Ribbed tyre The wheel used in land locomotion may consist either of a flexible (pneumatic tyre) or rigid structure (locomotive wheel). The locomotion problem of a wheel on deformable terrain, such as soft soil or sand, finds particular application in farm tractors and off-road vehicles. Here, the properties of the soil are of primary importance in determining the tractive capability or flotation of the vehicle wheels. Soil, in general, has both plastic and frictional properties, and the shearing stress τ in soil depends on the coefficient of cohesion c and angle of friction ϕ. Thus τ GcCp¯ tan ϕ

(6.5)

where p¯ is the mean loading pressure acting at the wheel–soil interface. Plastic masses such as saturated clays or certain types of melting snow can be considered to have a zero angle of friction ϕ, and thus τ Gc. The more common granular type of soil exhibits no cohesion or internal

208

Rolling Contacts

Fig. 6.2

bonding (cG0), so that τ Gp¯ tan ϕ. One of the complications in attempting to apply equation (6.5) to an actual soil is that the relative proportion of c to p¯ tan ϕ is very sensitive to water content. Furthermore, soils generally lack homogeneity which is characteristic of other deformable materials. Figure 6.3 shows, schematically, the traction of a ribbed tyre in soft terrain. Both theory and practical experience indicate that the spuds or ribs of elastic tyre treads clog with soil once they are in contact with the ground. Therefore, they play an insignificant and secondary role in developing traction. This is in contrast to the effect of tread pattern on automobile tyres, as will be demonstrated later on in this chapter. The main effect of the ribs is to increase the effective wheel diameter from D to DC2t, where D is the undeflected smooth tyre diameter and t


209

Fig. 6.3

represents the rib depth as shown in Fig. 6.3. Let TD be the driving torque applied to the wheel and FD the average value of tractive force developed at distance h below the wheel centre. Then, taking moments about the centre O TD GFD hCWa

(6.6)

where a is the distance forward of the wheel centre at which the load reaction vector W is effective at the tyre–soil interface. Dividing both sides of the above equation by h gives TD GFDCFR h

(6.7)

where FR GW(a兾h) is the rolling resistance of the wheel. It should be noted that the ratio a兾h for locomotion in soft terrain is considerably larger in magnitude than that corresponding to the more usual tyre– road case. This is because the load is effectively supported on soft soil only from the point of initial contact to the maximum penetration position, i.e. along the arc bc in Fig. 6.3. If both sides of equation (6.7) are divided by W, the coefficients of friction rather than frictional forces are obtained on the right-hand side. Although the FD and FR terms in equation (6.7) appear to have the same sign, each is taken in a positive sense according to the directions indicated in Fig. 6.3. Thus, they are actually numerically subtractive. It is convenient, then, to divide each term in equation (6.7) by the contact area A, in order to obtain TD h A

Gτ Ap¯ fR Gp¯ (tan ϕAfR )

(6.8)

From equation (6.5), for granular soils, cG0. The shear strength τ of the soil at the shearing boundary is equal to FD兾A, p¯ GW兾A, and the

210

Rolling Contacts

coefficient of rolling resistance fR GFR 兾W. Equation (6.8) shows clearly that the application of driving torque TD to the wheel is a function of soil frictional properties and normal loading. It is important to assess the bearing capacity or flotation characteristics of soft soils and terrain, and this is expressed quantitatively by p¯ in equation (6.8).

6.2 Tyre–road interactions The most important aspect of tyre–road interactions seems to be traction at the interface of their mutual rolling contact. Traction is essentially a friction process where tangential forces are transmitted and controlled by varying the relative motion of rolling and sliding between the contacting bodies. Land transportation relies, to a large extent, upon tyre–road traction to achieve the expected high standard of mobility, efficiency of performance, and safety of road vehicles. The force of traction or the tractive force is defined as the resultant tangential force of friction in the plane of contact across which a normal force acts reciprocally between the tyre and the road. The traction forces transmitted by tyres to the vehicle are primarily responsible for the entire class of vehicle manoeuvres, i.e. acceleration, braking, cruising, cornering, steering, handling, and even parking. In order to achieve high performance and manoeuvrability of vehicles, the tyre and road should have the potential for generating large frictional forces in relation to the normal force being transmitted. Also, they should have the ability to control smoothly the magnitude and direction of traction forces as dictated by the dynamics of the intended manoeuvre. It is well known that pneumatic tyres perform remarkably satisfactorily on dry roads on account of some of their unique properties. The requirement of a large frictional force can be met easily because the friction of tyre tread rubber on dry surfaces can be, under suitable conditions, quite high. The smooth control of traction forces can be achieved mainly through the elastic compliance of the tyre. However, much of the benefits coming from the excellent frictional properties of rubber tyres may be lost when the tyre–road interface is wet or contaminated in any way. The frictional force resulting from shearing liquid films or contaminant layers is usually too small to generate adequate traction. Also, on wet roads, the tyre compliance may turn out to be a liability since it can promote elastohydrodynamic lubrication. In addition, the low elastic modulus of the tread rubber can induce microelastohydrodynamic or mixed lubrication depending upon the texture of the road surface. The final remaining problem pertaining to tyre–


211

road interaction is direct tyre–road contact across a boundary lubricated interface. The possibility of activating various kinds of lubrication mechanism on a wet road poses a serious problem for tyre and road designers.

6.2.1 Relationship between friction and traction Three distinct aspects are involved in the study of the process of traction between tyre and road. (a) the basic problem of generating friction through direct contact (dry or boundary lubricated) at the contact interface of the tyre and the road; (b) the mechanics of slipping of a deformable tyre under the combined action of rolling and sliding; (c) the mechanics of elastohydrodynamic lubrication on a wet road. In order to study the first aspect, there is a need to know how large the forces of static or sliding friction are, how they respond to the relative motion between contacting surfaces, the normal force, and other influential variables. The second aspect of traction is concerned with the mechanics of slipping, and thus the extent of sliding and rolling of the tyre is varied in order to control the force of traction. Owing to the elasticity of the tyre structure, it follows that the contact area under a normal force is finite and that distributions of normal force, relative motion, and other important variables that affect friction are not uniform within this area. These two aspects of traction are sufficient for analysing the traction process of tyres on dry roads. It is reasonable to assume that the same type of analysis also applies to traction on wet roads provided that water present at the interface gives rise to boundary lubrication only. The effect of boundary lubrication may be taken into account by appropriate modification of the basic laws of friction. The third aspect of traction becomes relevant on wet roads where the presence of water leads to some form of hydrodynamic lubrication of the contact interface. The region of direct tyre–road contact and the normal force on that region are both influenced by lubrication. How this affects the friction and traction between the tyre and the road will be discussed later in this section. The determination of available friction requires knowledge of the basic laws or constitutive relations of friction of an elastomer on hard solids. The frictional force is expressed in terms of a non-dimensional quantity, i.e. the coefficient of friction µ, which is the ratio of the tangential force and normal force transmitted through the contact area.

212

Rolling Contacts

From a fundamental point of view, µ is not strictly a basic property of the interface but an outcome of the friction process depending upon the adhesion of the surfaces. On a macroscopic scale, for µ to be viewed as a basic frictional property it is necessary for properties and the variables influencing friction to be prescribed uniformly over the entire contact area. In particular, the most important variable, i.e. the relative motion, should be uniform for all points at the contact interface. Taking into account the non-uniform distribution of governing variables within the contact of tyre with road, it is assumed that the basic laws of friction are only applicable locally. The local coefficient of friction µ at some point within the contact interface depends upon the relative motion of surface elements of tyre and road at that point. The local coefficient µ is defined as the ratio of the local tangential or shear stress and normal pressure. The resultant of the local friction forces of different magnitude and orientation is the tractive force, which, when divided by the normal force, is expressed as the non-dimensional coefficient of traction f. The second aspect of the traction process is concerned with the utilization of friction (represented by µ) by the slipping tyre to control the traction coefficient f. It should be noted that, unlike µ, coefficient f is not generally considered to be a basic tribological property. Under some optimum set of variables, the coefficient of friction attains the maximum value that can possibly be generated with a given combination of tyre, road, and interface. The maximum available friction sets an upper limit for the force of traction that can be generated by the tyre–road system. Typical values of the two key coefficients µ and f, representing the friction and the traction potential under different conditions, are shown in Fig. 6.4. It is interesting to note that the µ value of low-modulus rubber on smooth, clean, and dry surfaces can be very large under favourable conditions (low normal pressure and low sliding speed). The values of µ for dry friction of tread rubber compounds on road samples taken from typical roads are usually much lower. However, these values are of the same order of magnitude as the traction coefficients of tyres measured on dry roads. The traction coefficients of tyres on roads determined under different surface conditions also exhibit wide variations depending upon the variables affecting the traction process. On dry and reasonably clean roads, tractive forces of tyres are not very sensitive to small variations in road properties, speed, and load. The highest anticipated traction coefficients of tyres on dry roads are in the range 0.8–1.2; under conditions favourable to traction it is not


213

Fig. 6.4

unlikely to come across values of 1.5 on dry roads. In the case of racing tyres, it is possible to generate a traction coefficient on dry roads that exceeds a value of 2. Under laboratory conditions, the maximum friction coefficients for the various rubber compounds used in tyre treads range from 0.8 to 4 depending upon the type of counterface in sliding contact. These results show that the friction potential indicated by basic studies is not fully utilized for generating tyre-on-road traction. Some design and operational constraints are responsible for this but there is still scope for achieving further improvements. Figure 6.5 shows typical variations in µ with sliding speed (4). The data for the low speed part of the curve are based on laboratory tests with a tyre tread compound on a road surface of dry bituminous concrete. The part of the curve at high speeds is plotted from the results of tyre tests on roads made from materials similar to that used during laboratory tests. In the range of low sliding speeds, friction is almost an isothermal process, but for higher speeds, the temperature rises rapidly because of frictional heating. The non-stationary nature of heat transfer

214

Rolling Contacts

Fig. 6.5

strongly influences the transient value of µ. The problem of severe wear limits the duration of locked wheel sliding. It is possible to control the tyre–road traction by varying the extent of sliding motion of the rolling wheel. In order to define the kinematics of sliding and rolling more precisely, it is convenient to suppose that the wheel and the road are two rigid bodies linked together by an elastic body, i.e. the tyre. The translation of the wheel, together with rotation about its axis, implies that there is a relative motion between the wheel and the road in the plane of contact. The relative motion consists of translation in the contact plane and rotation about a normal to the plane. The two components of the relative motion are known as the slip and the spin. The elastic tyre has to accommodate this relative motion between the two rigid bodies if it is to roll without sliding. The ratio of the components of relative velocity between the wheel and the road, V r, referred to the interface, and the average rolling velocity of the wheel, V 0, is defined by a kinematic parameter termed the slip, ε . It may be generalized to include both the slip and the spin motions. The slip parameter may be regarded as an overall measure of relative motion of the wheel with respect to the road, but it does not contain any explicit information concerning the relative motion of sliding at the tyre–road interface. The kinematic description of rolling with slip is a relation involving three vector quantities, namely: the linear velocity of the rigid wheel, the angular velocity of the wheel about the axle, and the radius vector normal to the contact interface.

6.2.2 Characteristics of the traction It has been stated earlier that traction is strongly influenced by slip and it can be controlled by varying slip on the condition that it does not


215

lead to sliding of the entire contact area. The functional relation connecting the two non-dimensional quantities f and slip ε is known as the traction characteristics of the system consisting of the tyre, the road surface, and the interface. The traction characteristics are influenced jointly by the tyre and the road through an interaction of two kinds of properties, that is, the elastic properties of the tyre and the tribological properties of friction and lubrication of the tyre–road interface. Usually, the practical interest is in the effect of the resultant forces of traction on vehicle dynamics, and the usual approach is to treat traction as a black box system with ε and f taken as input and output quantities. Bearing in mind that the road surface and its contamination change continuously, the question of generality of the ε –f relation arises. Both the slip and traction, i.e. the force and moment, can be expressed as vector quantities with components in the longitudinal, lateral (slip angle), and spin (turnslip). The components of V r, expressed as three elements of slip ε (ε x , ε y , φ ), are the longitudinal, lateral slip (translation), and spin (rotation). The traction components are the two important force components in longitudinal and lateral directions Fx and Fy , the self-aligning torque Mz , the overturning couple Mx , and the wheel torque My . A set of ε –f functions for a given tyre–road interface system may be determined, corresponding to a set of time-invariant parameters such as the normal force, slip, and other vectors. They are known as the stationary traction characteristics. A family of such characteristics for various permutations and combinations of parameter values is useful for predicting the stationary type behaviour. In vehicle dynamic practice, non-stationary traction characteristics are also important because variables and parameters such as the slip, load, speed, and road inputs vary continuously with time and any attempt at analysis is an extremely difficult task. Even the simpler stationary problem, outlined later on in this section, poses a number of difficulties. The traction characteristics in the stationary state can be expressed in the form [Fx , Fy , Mx , My , Mz ]GΩ [ε x , ε y , φ ]

(6.9)

Keeping ε x , ε y , φ , V 0, and Fy constant and assuming constant properties of the tyre, the road surface, and the interface, only a short rolling distance equal to a few contact lengths is sufficient for generating a stationary response. It is convenient to study traction when only a single component of slip is considered at a time. This type of relation may be described as the partial traction characteristics. One of them is the lateral traction characteristic for stationary rolling. The variation in lateral

216

Rolling Contacts

traction and the self-aligning torque with lateral slip, with the other slip components equal to zero and the parameters constant, has the form [Fy , Mz ]GΩ1(ε y )

(6.10)

where function Ω1 defines the partial stationary characteristic of lateral traction. A typical stationary traction curve ( f as a function of ε ) may be described, qualitatively, without performing a detailed analysis. As shown in Fig. 6.6, the curve exhibits three broad regions. A region of small slip where the curve is almost linear, a region of finite slip where traction increases more slowly with increasing slip, and finally the region of large slip where substantial sliding takes place. In the last region, the traction curve can display a wide variety of shapes depending almost entirely on the process of friction. On dry roads and on wet roads where no hydrodynamic lubrication takes place, the initial part of the f–ε curve for very small ε has the same slope for all road surfaces. The reason for this lack of sensitivity of the initial slope to the tribological conditions of a dry and boundary lubricated contact interface is explained later in this section. In Fig. 6.6 the parameter λ M

Fig. 6.6


217

is defined as the ratio of the thickness of the undisturbed water layer T and D – a measure of the combined drainage of the road and the tyre surfaces; the parameter λ Ghm 兾σ , where hm is the minimum film thickness within the contact area, calculated on the basis of smooth profiles, and σ is the root mean square value of peak heights of the surface roughness of the road surface. Traction on a dry road From the practical point of view, the traction of a tyre on a dry road is of considerable importance as the driving takes place mainly on reasonably dry roads. The friction of representative tyres on a dry road is taken as the norm to design vehicles and to specify its performance. The performance estimates are based usually on the assumption that the coefficient of friction µ (kinematic) of a tyre on a dry road is roughly of the order of unity. The traction curve starts from vanishing slip corresponding to the conditions of free rolling where a small force of traction is needed to overcome the rolling resistance of the tyre. When a wheel rolls with small slip, the tangential forces are, on account of a finite static µ o , generated primarily by static friction. The slip is accommodated almost entirely by elastic deformation of the tyre. The sliding involved in this process is minimal and confined to a small region towards the trailing edge of the contact. The frictional properties such as the coefficients µ o and µ are not involved so long as the friction coefficients required to sustain the elastic deformation do not exceed µ o. Therefore, traction is insensitive to surface conditions. The action of static frictional force is passive because elastic deformation of the tyre is predetermined. This initial part of the f–ε curve can be described as the elastic regime. Because very little sliding is involved, the initial slope of the traction curve for a dry and a boundary lubricated road does not depend on values of µ o and µ. As slip increases, sliding friction becomes increasingly important and the traction depends essentially on the outcome of the detailed interaction between the friction and elastic forces acting on the tyre. This part of the traction curve could be named the triboelastic regime. Traction in this regime is a basic property neither of the tyre nor of the road, taken separately. For large magnitudes of slip, sliding takes place on a large scale, almost over the whole contact area. Then, traction is dictated mainly by the process of friction. Traction, under conditions of pure sliding, is of little use because the ability to direct traction forces and consequently the directional control of vehicle motion are completely lost. Besides, gross

218

Rolling Contacts

sliding at high speeds is clearly undesirable from the point of view of preventing damage by overheating and excessive wear. Practical interest is in modelling traction characteristics in the range of small to moderate slip. Occasionally, operation at higher slip is inevitable when, for instance, driving takes place on a low-friction icy road.

6.2.3 Analysis of dry road traction Triboelastic interactions of the tyre with the road govern the mechanics of how traction is generated and controlled by varying the slip of the wheel. It is concerned with the equilibrium of surface elements of the tyre as they pass through the contact interface under the action of the frictional and elastic forces. The kinematics of slipping influences the local sliding velocities and hence the frictional forces generated by the elements of the contact surface. The resultant of the local frictional forces within the contact is the tractive force generated by the tyre–road interface. When rolling with slip, the basic tribological variables, i.e. the speed of sliding and the local tangential stresses, vary from point to point within the contact. A proper analysis requires the independent specification of basic frictional and elastic properties, the formulation of the kinematics of rolling with slip, and solution of the rolling contact problem. Besides, heat dissipation caused by rolling and sliding of the tyre results in an increase in tyre temperature and consequently affects friction. Calculation of the temperature of the tyre surface under various operating conditions requires a thermal model of the contact between the tyre and the road. It is usual practice to obtain a rough estimate of the surface temperature of the tyre based on rolling and sliding speeds. This estimate utilizes simple semi-empirical equations without the use of a more elaborate thermal model. The basic relations of friction of elastomers on a dry and hard road surface are well established and can be found in published textbooks dealing with the friction of elastomeric materials. The elastic properties of interest here are those of the tyre because the road is relatively rigid. The specification of the elastic properties of the tyre required for a specific application is a difficult task. Modelling of the response of an elastic tyre Modelling of elastic deformation of a tyre is one of the most difficult tasks in traction research. The problems under consideration are mainly in the area of applied mechanics rather than pure tribology. Three major difficulties arise because the tyre is an air-inflated structure


219

which, when loaded, undergoes large displacements and deformation. Unlike Hertzian contacts, the standard analytical techniques cannot be applied to determine the shape and size of the contact area and the normal pressure distribution under a given normal load. Although finite element techniques may be used, nevertheless, it was found that their use is rather limited and it is common practice to determine these allimportant parameters by experimentally testing an existing tyre. A more serious problem of traction analysis is that the shape, the size, and the normal pressure distribution are affected significantly by tangential forces. Thus, the contact problem becomes strongly non-linear and extremely difficult to solve. However, if such effects are ignored in order to make the problem easier to analyse, the results, though approximate, are likely to be acceptable. Another task is to calculate the tangential deformations of the tyre on account of surface forces of friction. This is the most important part of any analysis of the triboelastic interaction. Assuming that the tyre has a linear elastic response in the tangential plane, the deformation due to the distributed contact force can be expressed by an integral of a Green-type function. However, the well-known influence functions of massive, semi-infinite bodies do not apply to tyres because of obvious structural differences. Since the analytical solutions for such complex structures are not known, it is necessary to determine the influence functions, with the help of finite elements, of empirical techniques. Assuming that the influence functions are somehow determined and provided that the tyre response is linear, the elastic displacement components u, v, w in the X, Y, Z coordinate system may be expressed by an integral of the influence functions. The X–Y plane is taken as the plane of contact (road surface) and Z is the normal to the plane from the centre of the wheel that intersects the contact plane at the origin of the coordinate system. The following influence, or Green, functions on the boundary surface are needed to specify the elastic response of the tyre to surface forces of traction, i.e. Gxx , Gyy , Gxy (GGyx ) for tangential displacements due to the tangential traction stresses and Gzz for the normal displacement due to the normal pressure on the contact. In order to describe interactions between the normal and the tangential stresses and displacements, additional functions are required. It is interesting to note that the elastic properties of the shell-like structure of the tyre are quite different from those of solid bodies. The boundary elastic response of a solid body is such that the influence of normal pressure on tangential displacement is more significant than that of tangential stresses on normal displacement. In the case of the shell-like

220

Rolling Contacts

structure of tyres, these dependencies are reversed. Radial tyres are especially sensitive in this respect as the lateral tangential components of surface tractions have a strong influence on the normal displacements to the tread surface. As a result, for a given normal load, the shape and size of the contact area and the distribution of normal pressure p are significantly influenced by lateral components of tangential stresses. The influence of longitudinal components is generally much smaller. The effect of the lateral traction stresses on the contact may be taken into account by introducing the influence function Gzy . Tangential displacement

冮冮 [τ (ξ, η)G

u(x, y)G

x

xx

(x, y; ξ, η )Cτ y (ξ, η )Gxy (x, y; ξ, η )] dξ dη

C

(6.11)

冮冮 [τ (ξ, η)G

v(x, y)G

x

yx

(x, y; ξ, η )Cτ y (ξ, η )Gyy (x, y; ξ, η )] dξ dη

C

(6.12) Normal displacement

冮冮 [ p(ξ, η)G (x, y; ξ, η)Cτ (ξ, η)G

w(x, y)G

n

y

zy

(x, y; ξ, η )] dξ dη

C

(6.13) The size and shape of the contact area C are determined by the compatibility of the deformed surface of the tyre with the road surface. Both the geometry of C and the distribution of normal pressure p are obtained for prescribed normal displacements within C. Taking into account the difficulties of solving normal contact problems, the modification of C on account of τ y has to be determined either by empirical means or by the use of the finite element method. Under a purely normal force, the contact geometry and normal pressure, denoted by CGC 0 and pGp0 respectively, have to be modified using certain empirical functions Ψ and ψ . Then, CGC 0 Ψ(w) and pGp0ψ (w). In addition, the following two conditions apply to points (x, y) on the tyre surface: pG0 outside C and pH0 inside C. Owing to the complexity of the elastic structural response of a pneumatic tyre, a number of simplified models have been proposed. Most of


221

them are well suited to represent, in an approximate way, the elastic deformation of a tyre for a single mode of slip. Rolling motion combined with slip The kinematics of rolling with slip is an important element of the analysis of the mechanics of interaction between pneumatic tyre and road surface. Consider a tyre rolling with a linear velocity of the wheel V 0. The slip components ε x and ε y and the spin φ resulting from relative motion between the wheel and the road surface, considered as rigid bodies, may or may not lead to relative motion of points within the contact zone. Whether the local sliding at certain points within the contact zone takes place depends upon the tangential displacements of the tyre. The kinematics of sliding is expressed by the relations defining the x and y components of the local sliding velocity, i.e. Vsx and Vsy

冢

∂u

V C ∂x冣 ∂t 0

冢

∂v

0

Vsx (x, y)G ε xAφ yA

Vsy (x, y)G ε yCφ xA

∂u

∂v

V C ∂y冣 ∂t

(6.14)

(6.15)

For stationary contact, the non-stationary terms disappear. Local sliding occurs when V 2sGV 2sxCV 2syH0

(6.16)

If Vs G0, then there is adhesion or static friction at that point. At any point given by (x, y) inside C, the tangential displacements of the tyre are caused by local stresses produced by frictional forces which, in turn, depend upon the local sliding velocity Vs and the normal pressure p at that location. The contact described by C can be split into two regions, namely the region C A of static friction (due to adhesion) and region C B representing kinematic friction (sliding). For (x, y) in C A, i.e. if Vs G0 1(τ 2xCτ 2y)‚ µ o ( p)p

(6.17)

and for (x, y) in C B, i.e. if Vs ≠ 0 1(τ 2xCτ 2y )Gµ (Vs , p, T)p

(6.18)

222

Rolling Contacts

where µ (Vs , p, T)Hµ o ( p). Also

τ x Gµp

冢V 冣

(6.19)

冢V 冣

(6.20)

Vsx s

τ y Gµp

Vsy s

The two non-linear functions µ o( p) and µ (Vs , p, T ) are the basic relations of dry friction. If a thermal model of the tyre is available, the calculations of T can be coupled to the present model to take into account the influence of µ on the surface temperature. In most applications it is sufficient to calculate the temperature T approximately using an elementary expression with heat input calculated from V 0 and ε , and with a reasonable estimate for µ. The contact problem, dominated by interactions between friction and elastic deformations, is completely formulated when elastic properties of the tyre given by the surface influence functions Gxx , Gyy , Gxy , Gzy , and Gn are specified, and the traction-free contact area C 0, the pressure p0, the functions Ψ and ψ for the traction influence, and the friction parameters are all known. The set of elastic response, kinematic, and friction law equations together form a mixed boundary problem where C A and C B are also unknown. The longitudinal and lateral components of the traction force, Fx and Fy , and the moments can simply be expressed as integrals over C, once the distributions of the tangential stresses have been determined. The solution of the equations requires a study of its own. An iterative approach is needed to determine C A and C B. In spite of the difficulties, which are mainly of a technical nature, the possibility of being able to predict tyre characteristics from the basic properties of the wheel–road system is of considerable practical interest.

6.2.4 Traction under wet conditions Undoubtedly, the friction on a wet road surface is a problem of considerable concern for safety of driving. The elastohydrodynamic lubrication (EHL) taking place at the interface is the main cause of the degradation of traction on wet roads. The main reason for that is the intrusion of water into the contact interface which is facilitated by the elastic deformation of the tyre. At some speed, a point is reached where the frictional interface created by the contact between the tyre and the road is separated by a film of water which, while transmitting the


223

normal force, ceases to transmit any sizeable tangential force. The viscous friction due to shearing in the water layer is negligible in relation to dry friction, and traction is neither sufficient nor controllable. Under wet conditions, three distinct mechanisms of film formation may be identified. The conditions under which one of these mechanisms is dominant depend upon the speed of rolling and the slip of the tyre. Also, the thickness of the water layer present on the road is of significance. In addition, surface features of tyre and road surfaces can play an important role in draining the water out of the contact zone. The geometrical features of the road surface are usually characterized by the average size of surface irregularities, i.e. a macroscale texture with a linear dimension of 1–10 mm and a microscale texture with features of the order of 0.1 mm. The depth of the tread profile of a tyre is of the same order of magnitude as the size of the macroscale texture of the road. The three EHL mechanisms are: the inertial or thick layer mechanism, the viscous or thin layer mechanism, and micro-EHL which occurs on tips of surface asperities. Inertial EHL most often takes place on roads flooded after heavy rain when thick layers of water may be present at various locations. If the combined drainage due to the road macrotexture and tread patterns is insufficient, the impact of water with the tyre causes large hydrodynamic pressure to build up ahead of the contact area. Clearly, the effect of fluid inertial forces acting on the tyre depends on the thickness of the water layer and the speed of rolling. Under the high pressure of water, the surface of the tyre is deformed inwards, which permits the water film to penetrate further into the contact zone. There is a critical speed at which the entire contact is water-borne. This inertial EHL where inertial forces are dominant is known as dynamic hydroplaning. The speed V D at which there is onset of dynamic hydroplaning is a function of the parameter λ M GT兾D, where D is a measure of the combined drainage of surfaces of the road and the tyre, and T is the thickness of the undisturbed water layer. The structural factors of the tyre are the membrane stiffness due to the inflation pressure pi and the flexural rigidity EI of the tread. The relation describing the propensity for hydroplaning has the following form: V兾V D Gfunction of [λ M , ρV 2兾( piCEI兾l 4 )]‚1, where ρ is the density of the fluid and l is the contact length. There are, basically, two ways to suppress dynamic hydroplaning. The first consists of reducing the cause, i.e. the force of impact of water. The strategy of a designer is to offer a part of the useful contact to ease the flow of fluid so that the remaining part is starved of lubricating

224

Rolling Contacts

fluid. This is usually done by providing grooves and macroroughness on the running surface of the tyre and the road. The second approach is to reduce the effect, i.e. the inward deformation of the tyre. This is accomplished by increasing the radial stiffness (the inflation pressure) of the tyre. Under flooded conditions, the combined drainage of the road and the tyre helps to reduce the hydrodynamic pressure due to the impact of water against the tyre surface. The importance of effective drainage is manifested at high speeds of rolling, especially when it takes place on a road with little or no macrotexture and with tyres whose tread grooves have been worn out. The use of open macrotexture and porous surfacing materials contribute to the effective drainage of water from the tyre–road surface contact zone. If the road macrotexture is inadequate, primary drainage can only take place through the grooves on the tyre surface. An inherently stiffer tread construction helps to reduce the deformation due to water pressure and also gives the tyre designer greater freedom in the geometrical layout of tread patterns for increasing the rate and dispersion of the flow of fluids. The danger of dynamic hydroplaning is very much reduced under normal driving conditions when tyres have an adequate groove depth. There is no significant benefit to be gained from the tread patterns on the tyre surface if the macrotexture of the road is sufficiently large to secure adequate drainage. On roads covered by a thin layer of water, full film lubrication can still occur owing to viscous effects. This can take place at relatively low rolling speeds of the tyre and is termed viscous aquaplaning or viscoplaning. Viscoplaning can occur, despite sufficient drainage, on smooth roads at vehicle speeds of only 40– 60 km兾h. Two different scales can be distinguished for the viscous EHL effect. Firstly when a thin but essentially continuous film is formed on a significant part of the contact, and secondly when very thin films are created locally at tips of asperities. The effectiveness of repelling water from the contact zone and preventing the build-up of EHL film by the first scale effect depends on the ratio λ m Ghm兾σ , where hm is the minimum film thickness calculated for smooth surfaces and σ is the root mean square value of peak heights of the surface roughness (ignoring the waviness) of the road surface. In order to prevent EHL film buildup, λ mF1. The parameters affecting viscous EHL and the structure of the equation describing isoviscous EHL of low-modulus materials have been thoroughly reviewed by specialists in the area of lubrication. However, utilization of the results for the case of tyre–road contact has not been fully implemented.


225

The second scale effect, taking place at the tips of asperities, may be viewed as a problem of point contact EHL applicable to soft materials. The way to prevent the formation of very thin films at the contact between individual asperities is to increase the intensity of local pressure so that the film breaks down. It appears that somewhat higher intensity of local pressure within the contact owing to tread patterns is not enough to destroy thin films. Thin micro-EHL films are formed readily on smooth tips of road surface asperities in contact with the low-modulus rubber tread of the tyre. The high pressure required to prevent the formation of thin EHL films can only be achieved on road surfaces having a sufficiently sharp microstructure. The microstructure of the road surface is usually described in qualitative terms; i.e. at one extreme the road surface may have asperities with smooth, polished and rounded tips while at the other extreme surfaces are described as harsh or sharp. A sharp microstructure is more effective in penetrating thin films and hence provides a significant reduction in the risk of viscoplaning. A rounded microstructure may actually help build up thin films. A change in microstructure from sharp to rounded may occur on roads where intense road traffic produces polishing of the tips of asperities. The choice of suitable road surfacing material as well as the geometry of surface asperities (average slope, tip curvature) are all important in controlling viscoplaning. The transition from micro-EHL to boundary lubrication may be defined by the ratio of the thickness of the boundary layer and the thickness of the microEHL film.

λµG

hb hµ

(6.21)

The significant contribution of tread patterns to wet road friction is generally recognized, but the precise function of the complex geometrical features is only vaguely understood. The distribution of pressure in the contact area depends on the structural behaviour of the tyre under normal load as well as on the geometry of tread patterns. Ideally, the pressure distribution should be such that more water is expelled much faster out of the contact through the grooves. Also, it is thought that the grooves serve as low-pressure reservoirs into which water may accumulate temporarily during the passage of a tread element through the contact zone. Several secondary features such as sipes or knife cuts are also provided, presumably for squeezing out the thin films. The narrow slits can collect water by a capillary action and act as tiny reservoirs. Experimental studies of friction on wet road surfaces do not report any

226

Rolling Contacts

significant effect of these features except on extremely smooth surfaces. There is no conclusive evidence on the benefits resulting from such elaborate and intricate designs. Undoubtedly, the deep grooves in the tread are clearly beneficial for the drainage of water. However, the influence of lowering the tread stiffness on traction under the action of inertial and viscous EHL should also be considered. The situation where the pressure in the central region is lower than at the boundary is regarded as unfavourable because water tends to be channelled towards the centre. This may arise when the tyre is underinflated or overloaded. There is also the accompanying deformation which tends to close the grooves and thereby increases the risk of dynamic hydroplaning at lower vehicle speeds. It is obvious that such a heuristic approach may not always lead to correct solutions of this complex problem. Nevertheless, the tyre designer must take care to verify that the tread patterns provided for drainage of the contact area do not help to enhance the build-up of hydrodynamic pressure. Only through a detailed analysis of surface deformation of the tyre, considered together with tread pattern and sipes, is it possible to assess whether the design is able to achieve the desired effect. It is very important to make sure that the tread surface does not act as a seal trapping pockets of water within the contact area.

6.2.5 Analysis of wet road traction Undoubtedly, the traction phenomenon on a wet road is much more complex than that on a dry road. The ingress of the film of water into the tyre–road interface influences the entire traction characteristic, including the initial slope of the traction curve in the elastic region. The extent to which the EHL film intrudes and how large the associated film pressure can become is determined by the competing action of hydrodynamic and elastic forces. As stated earlier, dynamic hydroplaning and viscoplaning are the two extreme conditions initiated by distinct EHL mechanisms operating during wet road traction. The first type is dominated by inertial forces and the second by viscous forces. It was emphasized earlier that the EHL due to viscous effects can occur at two levels. At the macrolevel, the formation of a continuous film is determined by the overall geometry and the elastic deformation of the tyre as a whole. At the microlevel, when the film is dispersed, the microEHL process occurs because a tread element has to go through the fluid film and squeeze the film away before it can establish a direct contact with the surface of the road.


227

There has been significant progress in understanding dynamic hydroplaning (5). A detailed analysis of the fluid flow between a flat road and a rigid surface having a shape that corresponds to the deformed shape taken by a hydroplaning tyre has been carried out. The importance of inertial forces in the phenomenon of dynamic hydroplaning of tyres has been stressed, while the role of viscous forces has been considered to be only marginal. The micro-EHL problem has also been studied (6). It has to be stated quite clearly that the conditions of contact between tyres and roads for ordinary driving are not such that tyres can encounter a thick layer of water at high speeds leading to dynamic hydroplaning. Full film dynamic hydroplaning occurs very rarely. Likewise, total viscoplaning due to the persistence of micro-EHL films on roads devoid of any texture is an extremely rare event. From the practical point of view, the regime of lubrication most representative of average driving conditions on wet roads is the partial EHL regime. This conjecture is based on the experience that, while the traction forces on wet roads are significantly smaller than those on dry roads, the forces are still quite substantial. This suggests that direct tyre–road contact takes place on a significant part of the contact interface. Partial EHL of contact interface The interaction between the fluid film and the elastic tyre forces is critically important for modelling EHL effects at the contact interface. The usual approach to the analysis of lubrication is to start from the assumption of a full film and then to calculate the minimum film thickness by solving the combined elastic and hydrodynamic equations. The two equations may be coupled together as integral or integral–differential equations which can be solved numerically. The more common techniques use either direct or inverse iteration and select a trial solution based on a physically justified simplification of the problem. Regardless of the method of solution, an implicit assumption made in the formulation of the problem is that the solids are separated totally by a continuous film of lubricant. The probability of solid asperities making contact is considered to be low if λ „2.5, where λ is the dimensionless ratio of the minimum film thickness, calculated assuming smooth surfaces, and the standard deviation of heights of asperities on interacting surfaces. In the outline of partial EHL presented here, the configuration is adopted a priori to ensure that direct solid–solid contact is established on a part of the interface bordering on the trailing edge of the contact area. Also, it is assumed that the pressure build-up in the inlet zone is

228

Rolling Contacts

Fig. 6.7

caused by inertial effects. The situation leading to partial EHL at the interface of the rolling tyre is depicted in Fig. 6.7. For simplicity, the region of interaction is rectangular and identical to the region of dry contact C. This assumption is probably justified if the water layer on the surface of the road is thin. It can be seen from Fig. 6.7 that the component of the fluid flow in the direction normal to boundary b, which separates zones II and III, is zero. The boundary is a plane curve b(x, y)G0, which cuts across the width of the contact and terminates at the side edges in the shoulder region of the tyre. The tangential component of the flow will decrease owing to the boundary layer along b. The interfacial area C is thus divided into two regions: region C f is the area of unbroken water film and region C b belongs to the third zone where direct tyre–road contact is established under the conditions of boundary lubrication. It is obvious that CGC fCC b. In an actual analysis, the curve b could be represented by a parabola y 2 Gα (xAx¯), with parameter α assumed to be a constant, and a variable parameter x¯ to locate the point of transition from zone II to zone III along the axis of symmetry, which is the x axis. In region C f, the fluid film thickness h(x, y) at any point is linked with the film pressure p f through the Reynolds equation of lubrication (7). The boundary conditions on b are defined by the requirement that there is no fluid flow normal to this boundary. In the following equations, superscript ‘f ’ is omitted for convenience and p denotes the film pressure p f 1 ∂p 3 1 Qx G− h C h(2V oAVsx ) (6.22) 12η ∂x 2


Qy G−

229

1 ∂p

1 h3C hVsy 12η ∂y 2

(6.23)

The scalar product of the flow and the local normal vectors vanishes at b; the expressions for Vsx and Vsy were introduced earlier in this section. Assuming isothermal conditions in the case of wet road traction, the behaviour of water may be taken as that of an isoviscous Newtonian fluid having dynamic viscosity η . Under conditions of stationary rolling with slip, the Reynolds equation for region C f expressed in a Eulerian frame has the following form ∂

∂p

∂

∂p

h C h ∂x 冢 ∂x 冣 ∂y 冢 ∂y 冣 3

3

G6η (2V oAVsx )

∂h ∂x

C6η Vsy

∂h ∂y

C6η hV o

∂2u

∂2v C ∂x2 ∂y2

冢

冣

(6.24)

In stationary rolling and slipping, the local surface velocity of the tyre varies both in the x and y directions depending on the elastic strains, which can be expressed in terms of the displacement components u and v occurring in these two directions. The variation in the tangential component of surface velocity resulting from elastic strain differentials of rolling bodies represents a stretch effect. The stretch effect can significantly affect the film pressure and thickness in the case of a slipping tyre. Under certain conditions, this effect could be utilized either to reduce or to increase the film pressure and thickness of the intruding fluid. The mechanics of fluid flow in inlet zone I is determined by the overall flow pattern around the tyre. If the water layer is thicker than the depth of the macrotexture on the road, inertial forces become significant and the theoretical framework has to start from the Navier–Stokes equations. In simplifying the complex problem, it is now propounded that the effect of the inlet zone is to boost the pressure pi at the inlet edge to some significant fraction of the stagnation pressure. If, however, the water layer is very thin, inlet pressure is boosted mainly by viscous forces which depend on the shape of the deformed surface of the tyre. The pressure at the inlet edge may be expressed as some fraction ζ of the stagnation pressure, although in the latter case this fraction may be quite small. The average pressure of the fluid will be reduced as a result of drainage through the channels on the surfaces of the road (macrotexture) and the tyre grooves. This reduction may be taken into account by decreasing further the value of factor ζ. The effective pressure at the

230

Rolling Contacts

inlet edge leads to the boundary condition pi Gζ1V o

(6.25)

where ζ F0.5. At all the other free boundaries, pG0. The finite pressure developed at the inlet edge will force the fluid to flow through the relatively narrow interfacial region bounded by the solid surfaces. This may be considered as a pressure-driven flow and it is governed mainly by viscous forces. In addition, there is the usual shear flow due to the velocity field imposed by the solid boundaries relative to the fluid film. The expressions relating the local elastic displacements of points on the surface of the tyre and the interfacial pressure and shear traction stresses are given by equations (6.11) to (6.13). These can be expressed explicitly once the influence functions of the tyre have been determined. The most characteristic feature of the EHL problem is the elastic response expressed by the normal surface displacement due to the film pressure in C f and the pressure exerted directly by the road in C b. Taking into account the highly compliant structure of the tyre and the road surface covered with a thin layer of water, the normal displacements of the tyre are expected to be much larger than the thickness of the fluid film. This implies that the pressure distribution in the contact region of the partial EHL interface may not be much different from that in the corresponding region of the dry interface. Using superscripts for the normal pressure and displacements (d for dry contact, f for the water film, and g for the boundary region), the EHL problem may be formulated in terms of the difference between the normal displacements in the dry and the partial EHL problems. The difference between normal displacements at any point (x, y) resulting from the difference in the normal pressures between the dry and partial EHL contacts consists of two main parts, i.e. the film thickness h(x, y) in C f and a small change in the rigid body displacement r o of points remote from the contact. The change in the normal pressure at any point from the dry contact situation to the partial EHL situation is denoted by ∆p(ξ, η ) which is in C f ∆p(ξ, η )Gp f (ξ, η )Ap d(ξ, η ) in C

(6.26)

g

∆p(ξ, η )Gp g(ξ, η )Ap d(ξ, η )

(6.27)


231

From equation (6.13) and ignoring the effect of shear traction in C f w f (x, y)Awd(x, y)Gh(x, y)Cr o

冮冮 ∆p(ξ, η)G (x, y; ξ, η) dξ dη

G

n

C

f


C

n

C

(6.28)

g

in C g wg(x, y)Awd(x, y)Gr o


G

n

C

f


C

n

C

(6.29)

g

The condition that the total normal load remains unchanged is

冮冮 ∆p(ξ, η) dξ dηC冮冮 ∆p(ξ, η) dξ dη G0 C

f

C

(6.30)

g

While the basic structure of the analysis is outlined here, it remains to investigate the detailed steps necessary to implement even an approximate numerical solution to this problem. The use of variational formulation may be required to solve the numerical problem. A simplified approach to the partial EHL problem characteristic for a tyre on a wet road is based on the assumption that the flow is onedimensional (longitudinal flow, the lateral flow is neglected). The analysis uses the choked film configuration as shown in Fig. 6.8, where the flowrate Qx G0. As an illustration of the approach, the pressure developed by the fluid for a slider with a parallel film profile will be considered. Since the film thickness is constant until the point of choking, it is observed that, if the inlet pressure is zero, the pressure profile increases linearly within the film. The solution can be obtained without making explicit use of the Reynolds equation as the condition that flow

232

Rolling Contacts

Fig. 6.8

vanishes, together with the inlet boundary condition, is sufficient to determine the pressure profile. Since there is also no pressure flow, the inlet pressure acts as hydrostatic pressure, and this will lead to a gross overestimation of the load supported by the film. This deficiency can be corrected if the decrease in the mean pressure owing to side leakage over the front part of the interface region can be estimated either empirically or from the approximate solution of two-dimensional flow. The decrease in the quasi-hydrostatic pressure along the film may be taken into account by letting ζ Gζ (x). The result of the parallel film problem may be extended to a more general film profile h(x). As an illustration, a normalized contact region (−1, 1) to represent the initial dry contact between a free rolling tyre and the road under a constant normal load Fz will be considered. The film of water initially enters the inlet region and the pressure is boosted at the inlet edge, that is, xG−1, just before entering the contact. The hydrodynamic pressure at the inlet edge is p(−1)Gpi Gζ (−1)1V o (6.31) It is assumed that the film intrudes into the interface up to the point xGx¯. Assuming that Qx G0 results in dp 12η V o G (6.32) dx h2 Also p(x)G12η V o

冮

x

dx 2 −1 h

(6.33)


233

Adding the quasi-hydrostatic component of p f produces the total fluid pressure pf in −1‚x‚x¯ in the form p f (x)G12η V o

冮

x

dx Cζ (x) 1V o 2 −1 h

(6.34)

The normal displacement w(x) at any point on the surface of the tyre can be expressed using the integrated one-dimensional version of the normal influence function Gn given by equation (6.13). Thus

冮

x

w(x)G

[ p(ξ )Gn (x; ξ )] dξ

(6.35)

−1

The formulation of the integral equation for the change in pressure ∆p(x) follows the same lines as that in the two-dimensional flow case discussed above. However, the numerical solution in the present case is likely to be easier to obtain. The tyre influence functions are based on a model of the tyre as a circular ring on an elastic foundation in both the radial and tangential directions. This simple model is able to represent and retain the most essential characteristics of the elastic response of a radial tyre. The simplicity of the model and the onedimensional approach make it easier to explore the influence of the tangential deformations of the tyre rolling with longitudinal slip. When steady state conditions prevail and only rolling with longitudinal slip is considered, the one-dimensional Reynolds equation is h G6η (2V AV dx 冢 dx冣

d

3

dp

o

sx

)

dh dx

C6η hV o

d 2u dx2

(6.36)

where η is the dynamic viscosity of water, V o is the linear velocity of the wheel, u is the longitudinal elastic displacement of the tyre, and Vsx is the longitudinal sliding velocity which was introduced earlier. The formulation of the partial EHL problem discussed above clearly indicates the nature of influence of the wet road conditions on the traction characteristics. The influence function of normal deformation has a role of some significance for wet road traction of the slipping tyre. It is not only limited to reduction in the maximum force of traction; the entire traction curve can be affected. The influence of the decrease in the region of direct contact on the initial slopes of the traction curves, at vanishingly small values of slip and spin, can have significant effects on vehicle handling and stability. The measures required to alleviate these undesirable effects are the responsibility of the tyre designer and

234

Rolling Contacts

tribologist, as this task is clearly beyond the scope of the vehicle engineer. However, the ability to estimate or predict the changes in initial slopes of the traction curves is one of the most basic requirements for counteracting and controlling the vehicle behaviour under these conditions.

6.2.6 Practical approach to traction modelling The specification, prediction, and control of the dynamic performance of vehicles require some sort of model able to predict the traction characteristics. In view of physical and mathematical complexities, only a few researchers have attempted to solve the triboelastic problem using the actual and independently determined properties of friction and elastic response of the tyre. Also, practising tyre and vehicle engineers have developed highly simplified models to deal with a wide range of traction problems. The main aim has been to use simple representations based on models or formulae, which, although relatively simple, exhibit qualitative features similar to results obtained from experimental work on traction. If the qualitative behaviour is similar, the quantitative behaviour may be matched empirically by using techniques of parameter identification. In this way, simple expressions to describe the dry traction characteristics of stationary rolling tyres when a single mode of slip is prescribed at a time have been constructed. They give a good fit, especially in the elastic regime. The empirical traction formulae are useful for the preliminary analysis of problems of vehicle dynamics. Two approaches have evolved for modelling the traction behaviour of the tyre–road interface system. The first one is a semi-empirical approach where the two basic properties of friction and elasticity are specified independently, although in a rudimentary form. Using these properties, the simplified mathematical problem of the triboelastic interaction of the slipping tyre is then solved. The parameters representing the fundamental properties are the best fits determined from suitable traction experiments in the laboratory or on the road. The experiments are performed with representative tyres and roads and with a dry interface. Usually, the elastic properties of the tyre are represented by a variety of simple mechanical elements such as a set of isolated springs or a beam on an elastic foundation. The frictional properties of the interface are modelled according to the Amonton–Coulomb law, sometimes including modifications inspired by experimentally observed trends of the traction characteristics. Taking into account the fact that the basic properties are independently specified but not determined in a rigorous way from fundamental


235

experiments, the results of analysis can serve only as qualitative guidelines. Although quantitative estimates may be made to match empirical results by providing a sufficiently large number of free parameters, it is doubtful whether such fine tuning of the model has any physical significance and meaning. Although the semi-empirical models are not entirely satisfactory, their main strength is that the formulae derived to represent the traction behaviour possess a basic structure that is qualitatively sound. It also implies some limited degree of generality for predicting the main trends of traction for small but finite deviation from the nominal operating conditions and parameters. The models are known to perform reasonably well when describing traction on dry roads and to a limited extent on wet roads under boundary lubrication conditions. The models do not strictly apply to traction under the more general conditions of wet roads because fundamental principles of lubrication are not included in such models. A more pragmatic approach to the problem is represented by the second type of model. Empirical models are derived directly by curvefitting of data obtained from indoor and outdoor traction measurements. One of the main goals is to find easy-to-use and time-efficient representation of the empirical relations. Among the various possible methods of function representations, those making use of special functions that match the required shape of the empirical curves are generally both accurate and economical. The value of such models can be further enhanced if the salient features of traction curves can be expressed directly and simply by the parameters of the special function. The main motivation for developing empirical traction expressions is twofold. Firstly, the vehicle dynamicist requires only the overall input– output relations of traction in order to connect the forces and moments acting on the wheel to the wheel slip, without going into details of interface tribology. Secondly, the modest traction requirements under average driving conditions correspond to operation of tyres mainly in the elastic regime, i.e. small slip, of the traction curve where traction is relatively insensitive to variations in the surface condition of the road. The empirical traction models make use of the special frictional behaviour of dry, high-quality road surfaces and, therefore, there are severe limitations to the range of road conditions and operational variables for which such models can provide useful predictions. Under more general surface conditions, the strong influence of friction is bound to cause large variations in traction in the triboelastic and frictional regime. Ideally, the traction versus slip curves should be determined

236

Rolling Contacts

from tests over the whole range of slip. Another problem with the accuracy of empirical models is the difficulty of specifying and controlling test procedures and subsequent evaluation and interpretation of test results. This is particularly true for the results pertaining to both the elastic (low slip) and the frictional (high slip) regimes. In the elastic regime, the difficulties arise partly on account of deviations and fluctuations in geometrical or physical properties which are inherent to the tyre and the road surfaces. Also, inaccuracies related to test equipment and procedures contribute to the difficulties mentioned above. The parameters of the traction model that are most difficult to determine are those representing the fundamental frictional property of the contact interface. The obvious option for determining the frictional parameters is to measure them under conditions of uniform sliding, i.e. when the wheel is locked. Unfortunately, the friction coefficient measured under locked-wheel sliding conditions is characterized by a large scatter as the value is sensitive to the duration of the test. The reason for this is the transient rise in the surface temperature which depends on the frictional heat dissipation. For the same reason, measurements carried out at different test speeds give different values of sliding friction coefficient. Locked-wheel testing at representative road speeds can be sustained only for a short period of time to avoid risk of damaging the tyre as a result of overheating and excessive wear. All the above difficulties of measurement and interpretation point to the complexity of the system and the futility of representing friction by a single constant µ coming from the Coulomb model. It is quite clear that the model parameters related to traction measurements in both regimes cannot be determined with a sufficiently high degree of accuracy. The data required to identify parameters are probably most accurate only for moderate slip values, i.e. in the triboelastic regime on condition that the test road surface is sufficiently isotropic. As it happens, traction in the triboelastic regime is governed by the combined properties of friction and elasticity, and therefore it is difficult to identify the separate parameters associated with the two basic properties. However, such a separation is required if the traction model is to apply for the same tyre rolling on different road surfaces.

6.3 References (1) Bekker, M. G. (1962) Theory of Land Locomotion (University of Michigan Press, Ann Arbor, Michigan).


237

(2) Hills, D. A., Nowell, D., and Sackfield, A. (1993) Mechanics of Elastic Contacts (Butterworth–Heinemann, Oxford). (3) Moore, D. F. (1975) Principles and Application of Tribology (Pergamon Press). (4) Grosch, K. A. (1963) The relation between the friction and viscoelastic properties of rubber. Proc. R. Soc., A274. (5) Browne, A. L. (1975) Mathematical analysis for pneumatic tyre hydroplaning. ASTM special technical publication, p. 583. (6) Moore, D. F. (1967) A theory of viscous hydroplaning. Int. J. Mech. Sci., 9. (7) Reynolds, O. (1886) On the theory of lubrication and its application to Mr Beauchamp Tower experiments including an experimental determination of the viscosity of olive oil. Phil. Trans. R. Soc., Lond., 177.


Chapter 7 Machine Elements in Rolling Contact

7.1 Contact of meshing gears 7.1.1 Peculiarities of contact between gear teeth If two parallel, curved surfaces, such as the profiles of meshing spur gear teeth made of a rigid material, were pressed together, they would make contact along a line, which implies that the area of contact would be zero, and the contact pressure infinite. However, there are no absolutely rigid materials, so deformation of an elastic nature occurs, and a finite, although small, area of contact supports the load. The case of two cylinders of radii R1 and R2 was solved by Hertz (1). If the case of two steel cylinders is considered, for which ν G0.286, then the maximum compressive stress is given by pmax G0.416

1冤

PE′

冢R CR 冣冥 1

1

1

(7.1)

2

where P is the compressive load per unit length of the cylinders and E′ is the equivalent Young’s modulus. If the radius of relative curvature R of the cylinders is defined as 1兾R1C1兾R2 , then pmax G0.416

1冢

PE′ R

冣

(7.2)

It should be noted that the stress given by equation (7.2) is one of the three compressive stresses and as such is unlikely to be an important factor in the failure of the material. The maximum shear stress occurs at a small depth inside the material and has a value of 0.3pmax. At the

240

Rolling Contacts

surface, the maximum shear stress is 0.25pmax. When sliding is introduced, however, a tangential stress field due to friction is added to the normal load. As the friction increases, the region of maximum shear stress, located at half the contact area radius beneath the surface, moves upwards while, simultaneously, a second region of high yield stress develops on the surface behind the region of contact. The shear stress at the surface is sufficient to cause plastic flow when the coefficient of friction reaches a value of about 0.27. These stresses are much more likely to be responsible for the failure of the gear teeth. The important point for the designer at this stage is that each of these stresses is proportional to pmax, and therefore for any given material they are proportional to 1(P兾R). For a number of reasons, however, this result cannot be directly applied to gear teeth. The analysis assumes two surfaces with constant radii of curvature and an elastic homogeneous isotropic material free of residual stresses. Firstly, a gear tooth profile has a continuously varying radius of curvature, and the importance of this departure from the assumption may be emphasized by considering the case of an involute tooth where the profile starts at the base circle. The radius of curvature, R1, is at all times the length of the generating tangent, so at this point it is, from a mathematical point of view, zero. However, it remains zero for no finite length of the involute curve, growing rapidly with the height of the tooth and having an unknown value within the base circle. If contact were to occur at this point, the stress would not be infinite, as an infinitely small distortion would cause the load to be shared by the adjoining part of the involute profile, so that there would be a finite area of contact. It is quite clear that Hertz analysis is rather inapplicable at this point; all that can be said is that the stresses are likely to be extremely high. In the regions where contact between well-designed gear teeth does occur, the rate of change in R1 is much less rapid, and it is not unreasonable to take a mean value at any instant for the short length which is of interest here. The assumption that the material is elastic will certainly break down if the resulting shear stress exceeds the shear yield strength of the material. The consequences are quite beyond the ability to predict them mathematically. An analysis might be possible if one load application at one instant were all to be dealt with; but the microscopic plastic flow that would then occur would completely upset the calculations for contact at the next point on the tooth profile, and so on. The situation when the original contact reoccurred would be quite different; and there is a need to deal with millions of load cycles as the gears revolve. All that

Machine Elements in Rolling Contact

241

can be said is that the repeated plastic flow is likely to lead to fatigue failure, but that it will not necessarily be so, since the material may perhaps build up a favourable system of residual stress, and will probably work harden to some extent. If such a process does go on, then there is no longer a homogeneous isotropic material free of residual stress. Gears transmitting more than a nominal power must be lubricated. The introduction of a lubricating film between the contacting teeth might be expected to alter the situation drastically, but it does not, simply because the lubricant assumes the form of an extremely thin film of almost constant thickness. In view of all these qualifying remarks, it is hardly to be expected that the gear teeth can be designed on the basis of the maximum shear stress being equal to 0.3pmax and having to equate to the shear strength of the material in fatigue. Nevertheless, the Hertz analysis is of vital qualitative value in indicating the parameter P兾R, which, for any given material, can be taken as a criterion of the maximum stress, the actual value to be allowed being determined experimentally.

7.1.2 Geometry of contact between gear teeth In order to apply fundamental equations given by the Hertz theory to the case of contact between a pair of gear teeth, it is necessary to define the reduced radius of curvature and the contact load for a specific gear. Figure 7.1 shows, schematically, two involute gears in mesh. The surface velocity at the point of contact C is expressed as UG

U1CU2 ω 1R1 sin φ Cω 2R2 sin φ G 2 2

(7.3)

where R1 and R2 are the pitch circle radii of the driving and driven gear respectively; φ is the angle of pressure defined as the acute angle between the contact normal and the common tangent to the pitch circles, and ω 1 and ω 2 are the angular velocities of the driving and driven gear respectively. Since R1 ω 2 G R2 ω 1 the contact surface velocity is therefore UGω 1R1 sin φ Gω 2R2 sin φ

(7.4)

242

Rolling Contacts

Fig. 7.1

Assuming that the total load is carried by one tooth only, then, referring to Fig. 7.1, the contact load in terms of the torque exerted is given by WG

T2 T2 G h2 R2 cos φ

(7.5)

where W is the total load on the tooth; h2 is the distance from the centre of the driven gear to the interception of the locus of the contact with its base circle (=R2 cos φ ), and T2 is the torque exerted on the driven gear. The torque exerted on the driving and driven gear, expressed in terms of the transmitted power, can be calculated from T1 G

T2 G

M

ω1

G9.55

M n1

M M G9.55 ω2 n2

(7.6)

(7.7)

where n1 and n2 are the rotational speeds of the driving and driven gear respectively (r兾s) and M is the transmitted power (kW).


243

Fig. 7.2

The line from C1 to C2 in Fig. 7.1 is the locus of the contact and it can be seen that the distance S between the actual contact point of the gear teeth and the pitch point P is continuously changing with the contact position during the meshing cycle of the gears. Thus, it is possible to model any specific contact position on the tooth flank of an involute gear by two rotating circular discs of radii R1 sin φ CS and R2 sin φ AS as shown in Fig. 7.1. This idea is applied in a testing apparatus generally known as a two-disc machine. Figure 7.2 shows, schematically, the principle of the two-disc machine.

7.2 Friction in meshing gears Attempts to measure the instantaneous coefficient of friction as a function of the meshing position have not been very successful, and at present it is necessary to rely mainly on disc tests for information on the relations between the coefficient of friction and other factors. Tests utilizing the two-disc machine are designed to reproduce the state of affairs at the point of contact between gear teeth, in a particular position of the mesh but under conditions permitting continuous observation. Two discs are run together in tangential contact with each other while a normal load calculated to reproduce the stress conditions under consideration is applied. The circumferential velocities ω 1r1 and ω 2 r2 correspond with the rolling velocities of the teeth under the conditions simulated (i.e. velocities of sweep of the point of contact), the sliding

244

Rolling Contacts

velocity Vs being given by Vs Gω 1r1Aω 2 r2

(7.8)

where the disc with the higher rolling velocity is identified as disc 1. The discs may be connected by gears or chain, the former reproducing the relative motion of the surfaces in actual gears; chain connection produces a condition that is abnormal in spur and helical gears, though it occurs on the inlet side of the teeth in worm gears. The practical conditions represented by a disc test depend not only on the surface velocities but also on the relative (reduced) radius of curvature. Denoting this by r, it is defined by 1 1 1 G C r r1 r2

(7.9)

and thus rG

r1r2

(7.10)

r1Cr2

It should be remembered that r1 and r2 are the radii of equivalent circles as shown in Fig. 7.1 and are, therefore, given by r1 GR1 sin φ CS r2 GR2 sin φ AS (7.11) The relative radius of curvature may be thought of as a measure of the degree to which the opposing contours conform to each other; the expression radius of conformity for this quantity is more informative, but relative radius of curvature is in general usage. In involute teeth the radii of curvature of course vary over the profiles. The relative radius of curvature also varies. At the pitch point it has a value given by 1 r

G

C sin φ 冢R R 冣

1

1

1

1

(7.12)

2

where φ is the pressure angle and R1 and R2 are the pitch circle radii. Qualitatively, there is a considerable measure of agreement between the results of various tests that have been carried out by different investigators with gear connected discs on the relations between the coefficient of friction, µ, and other variables. The discs that have been used in experiments have usually been given a finish similar to that typical


245

of high-quality gears, and the results thus obtained are consistent with the prevalence of quasi-hydrodynamic, or perhaps fully hydrodynamic, lubrication at the zone of conjunction. A comparative study leads to the conclusion that the principal correlatives of µ are the surface velocities, the relative radius of curvature, and the representative or effective viscosity of the oil, more specifically its viscosity at the surface temperature at the inlet to the lubricating film. To these may be added disc or tooth loadings, though its effect in this respect, like that of temperature, is secondary. In order to describe the influence of surface speeds on the coefficient of friction, it has been found expedient to accord primary importance to the entraining velocity, Ve , defined by Ve Gω 1r1Cω 2 r2

(7.13)

i.e. the sum of the velocities of sweep. The two velocities of sweep over mating profiles on involute teeth vary continuously along the path of contact. The sum of the two also varies, increasing in reduction gears from the point of first engagement to that of disengagement and similarly falling in speed-increasing gears. In a typical case the variation is of the order of 20–40 percent. At the pitch line the value is 2V sin φ , where V is the pitch line velocity and φ is the angle of pressure. The most noticeable of the relations between the coefficient of disc or tooth friction µ and other variables is an inverse dependence on Ve , attributable to its significance as a measure of the influence of surface velocity on the entrainment of oil into the load area, and to its effect on hydrodynamic relief of the contact load component, i.e. that portion of the load borne by areas in direct contact. The sliding velocity Vs has also been cited as another velocity factor governing the friction coefficient µ. At a given viscosity, the traction needed to shear the oil film will increase with the rate of shear, and the local rate of power loss will vary with the product of the sliding speed and the shear stress exerted locally on the rubbing surfaces. The result of an increase in sliding speeds is thus an increase in the temperature of the metal and hence a reduction in the viscosity of the oil beyond the film inlet, and possibly an increase in the bulk temperature, resulting in a reduction in the representative (inlet) viscosity itself. Recent work suggests that the net effect of Vs is comparatively limited and that the predominant velocity factor governing µ is Ve . There is also an inverse relation between µ and the relative radius of curvature r. It seems that this effect is again of the primary importance.

246

Rolling Contacts

The inverse relation between µ and the viscosity of the oil, η , is well established. The relevant viscosity, at the inlet to the zone of conjunction, is taken to be that of the bulk temperature of the discs or gear teeth. The magnitude of the variation in µ with η is the subject of some disagreement, but to a large extent the reason for this appears to be that in most of the earlier tests the oil viscosity was taken at the supply temperature and the bulk temperature of the metal is unknown. The variation in µ with tooth loading at constant temperature can be predicted theoretically, but its magnitude is relatively slight and some researchers have been unable to detect it. Tooth loading will, however, be reflected in the temperature at the metal surface, and hence in the effective viscosity of the oil at the inlet to the zone of contact, and will thereby indirectly influence µ to some extent. Hydrodynamic action in gears, discussed more fully later, is responsible for their successful operation, and it is interesting at this point to touch upon the reasons for the inverse relation between coefficient of friction and viscosity, an effect which is the opposite to that first expected from the theory of hydrodynamic lubrication as applied to bearings. Since viscosity is the coefficient of proportionality between the shear stress on the fluid and its rate of shear, if the latter is fixed it follows, of course, that the resistance to motion set up by the fluid, and hence the power it absorbs, will vary with viscosity. However, where lubrication in the zone of conjunction is only partially hydrodynamic, some of the load is still borne by direct contact. Under such conditions, the higher the inlet viscosity, the greater is the hydrodynamic relief of the contact load component, and accordingly the smaller is the coefficient of friction. Under fully hydrodynamic conditions, an increase in viscosity is accompanied in gears with a general increase in film thickness; where an oil film becomes thicker but the relative sliding velocity between its boundary surfaces remains the same, the rate of shear is thereby reduced. The effect of the increase in film thickness with increase in the viscosity of the oil would appear to predominate over the corresponding increase in resistance to shear. Thus, on both scores the coefficient of friction tends to vary inversely with viscosity.

7.2.1 Tooth losses For any particular gearset, µ′ (the overall mean value of friction coefficient µ) is essentially a measure of the transmission efficiency of the teeth. It can be derived by measuring tooth losses in the gearset concerned or in similar units.


247

In estimating tooth losses, or comparing different gearsets or operating conditions in this respect, empirical relations based on disc test data are helpful as a guide to the values of µ′ to be adopted. A number of empirical formulae have been put forward relating µ′ to other variables. To some extent they conflict with each other, and a good deal of scope remains for further work in this field. It would appear, however, that over the usual range of values Vs兾Ve the following relation gives a reasonable approximation of the mean coefficient of friction between the teeth of high-quality spur and helical gears

µ′G

308

(7.14)

0.25 0.5

v

r V 0.5 e

where v is the viscosity (cSt) at the bulk temperature of the gears, r is the relative radius of curvature at the pitch line (m), and Ve is the entraining velocity at the pitch line (m兾s). For spur gears, another form of the expression is 436

µ′G 0.25

v

sin φ

(7.15)

1冢

dD

冣V

0.5

dCD

where φ is the pressure angle, d and D are the pitch circle diameters of the gears (m), and V is the pitch line velocity (m兾s). For helical gears the comparable expression can be taken to be 436

µ′G 0.25

v

sin φ n

1冢

dD

冣V

cos0.5 β

(7.16)

0.5

dCD

where φ n is the normal pressure angle and β is the helix angle. The value of µ′ under any specific conditions of speed and geometry may, of course, be considerably increased because of inaccuracy, including misalignment, or wear of the gears. Tooth losses expressed as efficiency vary considerably with the size, proportions, and speed of the gears. In the largest and fastest sets, for example the reduction gears of marine turbine main engines, the tooth loss with modern units may be as low as 0.2 percent per train. At the other extreme, with spur and helical gears, the tooth loss in a small set with rather coarse teeth may be as high as 1.5 percent. A typical value in spur and helical gears of middling size is 0.7 percent per train. In

248

Rolling Contacts

other types of gear, notably worm gears, tooth losses may be very much higher.

7.3 Outline of elastohydrodynamic theory Under ideal conditions of hydrodynamic or fluid film lubrication the working surfaces are completely separated from each other by a continuous film of lubricant. A typical example of hydrodynamic lubrication in action is provided by plain journal or thrust bearings. Normally, in such bearings the thickness of the film at the loaded area is of a distinctly higher order of magnitude than that of the height of the asperities on the opposing surfaces. For many years it was commonly considered that in heavily loaded contacts, such as those in rolling bearings or between gear teeth, hydrodynamic lubrication was out of the question. This view rested on the conclusion drawn from classical hydrodynamic theory that the film thickness between gear teeth that could be attributed to viscous entrainment of oil was of a much lower order than that needed to separate the surfaces even under the most favourable conditions envisaged in that theory. Contact between gear teeth was therefore classed as a case of boundary lubrication, defined as a condition in which the load on the engaging surfaces is wholly, or nearly wholly, borne by direct contact. The view was commonly accepted that an attenuated film of oil remaining under such conditions owed its existence primarily to physical or physicochemical effects at the metal surface. The film was sometimes even regarded as monomolecular. Nevertheless, considerable practical evidence remained that in many gears the thickness of the fluid film is impossibly large for boundary conditions as thus defined, even where it does not separate the two surfaces completely. In particular, the long life with negligible wear often achieved with high-speed precision gears strongly suggested the existence between the teeth of lubricating films of substantial thickness by comparison with molecular dimensions. Classical hydrodynamic theory, as pioneered by Reynolds (2), rested on various assumptions, notably that the lubricant is of constant viscosity in the contact, and that bearing surfaces are perfectly smooth and rigid. From 1940 onwards a number of investigators attempted to modify the classical theory for application to heavily loaded conjunctions. A general agreement was reached that the thickness of the predicted film increased somewhat, by a factor of 2 or 3, when account was taken of the increase in the viscosity of the lubricant under pressure.


249

This theoretical refinement thus effected some improvement, but was still far from sufficient to account for the apparent existence of full, or nearly full, fluid films in practice. Attention turned also to the effect of elastic deformation of the contacting surfaces. Prediction of the operating film profile on this basis involves determination of the local deformation of the contours of the two rubbing surfaces, providing compatibility between the elastic effects in the metal and the hydrodynamic effects in the fluid. A theoretical solution that has proved to be essentially sound was published in 1949 by Grubin and Vinogradova (3) and by Petrusevich (4). According to their analyses, the surfaces confronting each other at the loaded area are almost parallel along the greater part of the contact zone, and the corresponding stress distribution is approximately Hertzian. At the outlet there is a pronounced constriction, which is often associated with a sharp and highly localized pressure peak. Figure 7.3 shows (a) the film profile and (b) the pressure distribution indicated in references (3) and (4). The analyses also predicted that the thickness of the film would be largely unaffected by load, given the effective oil viscosity. In 1958, Crook (5) reported the direct measurement of the oil film between the peripheries of the discs of a disc machine. He found that under heavy load the film thickness was of the same order as that predicted by Grubin and Vinogradova and was, in fact, highly insensitive to changes in load. The joint action of the elasticity of the surfaces and the increase in viscosity of the lubricant under pressure affords a striking example of synergy, i.e. the situation in which the whole is greater than the sum of its parts. The effect of the compressibility of the lubricant has also been taken into account by Dowson and Higginson (6). According to this study, the load capacity, or the film thickness at a given load, is not significantly affected by oil compressibility, but the shapes of the contact and of the pressure diagram are more rounded. The predicted shape of the contact zone is almost exactly as found experimentally by Crook. Because viscosity varies greatly with pressure and temperature, it has been a problem to decide on the value of the viscosity to be used in lubrication calculations. It is acceptable to take as the representative viscosity that existing at the conditions of pressure (i.e. atmospheric pressure) and temperature of the surfaces before they enter the zone of conjunction. The reason for this is twofold: in the first place the oil films on the contacting surfaces are only comparatively slowly displaced

250

Rolling Contacts

Fig. 7.3

by new oil coming from the bath or other source of supply; in the second place, because there is negligible side leakage from the zone of conjunction in the direction perpendicular to that of the peripheral velocities, the thickness of the film there is decided by the volume of oil


251

present and not directly by its viscosity, or variations in viscosity, whatever the case may be, as a result of the local increases in pressure and temperature.

7.3.1 Estimates of film thickness Grubin and Vinogradova (3) included the effects of elasticity and of variation in viscosity under pressure, although they introduced certain approximate assumptions. Their formula, arranged in non-dimensional groups, can be expressed as −0.091

冢冣

w h G1.17 r E′r

(α E′)0.73

η oVe E′r

0.73

冢冣

(7.17)

where h is the thickness of the nearly parallel part of the film, r is the relative radius of curvature of the contacting surfaces, given by 1兾r= 1兾r1C1兾r2 (r1 and r2 being the radii of curvature of two contacting surfaces), w is the load per unit length of contact, E′ is the reduced Young’s modulus for the materials of the contacting bodies, η o is the viscosity of the lubricant at atmospheric pressure and at the bulk temperature of the surfaces before contact, and α is the pressure coefficient of viscosity of the lubricant. The reduced Young’s modulus can be found from 1

G

E′

1 1Aν 21 1Aν 22 C 2 E1 E2

冢

冣

where ν is Poisson’s ratio. The pressure coefficient of viscosity is given by the formula

αG

冤冮

S

0

ηo dp ηp

冥

−1

where η p is the viscosity at pressure p. Under the elastohydrodynamic conditions for which the Grubin–Vinogradova formula applies, η p can be assumed to vary exponentially with p according to

η p Gη o eα p Also, Ve Gv1Cv2, where v1 and v2 are the velocities of the moving surfaces. Dowson–Higginson analysis of elastohydrodynamic lubrication produced a formula for minimum film thickness very similar to the

252

Rolling Contacts

Grubin–Vinogradova formula. Its form is as follows

冢冣

h min w G0.96 r E′r

−0.13

(α E′)

0.6

η oVe E′r

0.7

冢冣

(7.18)

The formula may be more conveniently written as IG0.96

G 0.6M 0.7 J 0.13

(7.19)

where I, J, G, and M represent the four non-dimensional groups respectively. The results from the Dowson–Higginson formula and the Grubin– Vinogradova formula are said never to differ by more than 20 percent. The formula can be further simplified, and quite drastically so, without serious error. Experience shows that, in practice, h min is virtually independent of E since the power to which it is raised is low and the range of variation for engineering materials such as steel and bronze is relatively limited. Furthermore, the value of α does not vary much for conventional lubricants, so that G 0.6 will remain fairly constant. Also, h min is only slightly dependent on the load parameter J. Dowson and Higginson therefore concluded that, to a reasonable degree of accuracy, G 0.6 and J −0.13 could be replaced by constants, so that the minimum film thickness h min could be simply calculated as follows h min G3.51(η oVe r)

(7.20)

where h min is in micrometres, η o in poises, Ve in centimetres per second, and r in centimetres. In the application of elastohydrodynamic theory to gears, the question arises as to the effects at the ends of the zone of contact. To the extent that elastic deformation under load might decrease somewhat at and near these positions, the oil film there might be slightly reduced in thickness.

7.4 Application of elastohydrodynamic theory to gears The above theories assume constant temperature conditions, but a useful guide to the interpretation of practical experience can be given in terms of elastohydrodynamic theory, together with the ‘flash temperature’ theory.


253

7.4.1 Film thickness between gear teeth Even in the many cases where complete fluid film lubrication is unattainable and some degree of actual contact has to be accepted, hydrodynamic relief of the contact load component remains important to the preservation of the gears. As indicated earlier, the film thickness is governed primarily by the viscosity of the oil at the temperature of inlet to the zone of conjunction, by the entraining velocity, and by the relative radius of curvature. The solution to the elastohydrodynamic problem achieved by Grubin and Vinogradova and Petrusevich and subsequently refined by others refers strictly to isothermal conditions. Such conditions characterize pure rolling, in which frictional losses and non-uniform film heating are quite slight, and low sliding speeds, when such losses and heating are still slight. When there is a considerable amount of sliding, the rate of generation of heat in the zone of conjunction will be much increased, but it does not appear that the film thickness will be directly affected, since the viscosity that is in this respect governing remains that at the inlet. It has been suggested that a forward flow of heat takes place from the conjunction, with the effect that the viscosity of the incoming oil is reduced. If this were so, the effect would be to reduce the film thickness to some extent. The reduction would be least at or near the pitch line and would rise to maxima at or near the points of first engagement and of disengagement. However, the grounds on which such a flow of heat has been inferred are not very convincing, and there is uncertainty as to its rate. According to flash temperature theory, which is discussed later, forward flow of heat from the conjunction would be negligible at any normal velocity of sweep. Moreover, the film thickness between rollers in sliding兾rolling contact has been found to be substantially as predicted by isothermal elastohydrodynamic theory over a wide range of conditions of speed, slide兾sweep ratio, and load. At the pitch line in spur gears, the entraining velocity Ve (m兾s) has a value of 0.08V sin φ , where V is the pitch line velocity (m兾s). For spur gears the radii of curvature r1 and r2 at the pitch line are r1 G r2 G

d sin φ 2 D sin φ 2 (7.21)

where d and D are the pitch circle diameters and φ is the pressure angle.

254

Rolling Contacts

The relative radius of curavature r at the pitch line is thus [see also equation (7.12)] rG

sin φ 2

冢dCD冣 dD

(7.22)

Assuming that the ratio of wheel to pinion diameters D兾dGρ results in rG

d sin φ 2

ρ

冢 ρC1冣

(7.23)

Comparing gears with the same centre distance but differing in ratio ρ, i.e. reduction ratio, with a given speed and viscosity, the film thickness at the pitch point will be less the higher the ratio. Beyond about 2 or 3, however, the effect of ρ will be limited. For comparative purposes, the film thickness in spur gears at the pitch line can thus be taken to be h min G0.7 sin φ

1冤

η o dV

ρ

冢 ρC1冣冥

(7.24)

where h min is in micrometres, η o is in poises, d is in metres, and V is in metres per second. The viscosity η o is that at atmospheric pressure and at the surface temperature of the teeth. For mineral oils, and for synthetic oils with specific gravities about equal to those of mineral oil, the formula becomes h min G0.065 sin φ

1冤

vo dV

ρ

冢 ρC1冣冥

(7.25)

where vo is the viscosity at atmospheric pressure and at the surface temperature of the teeth (cSt). It is important to note the effect of pressure angle φ on film thickness. In helical gears the relative radius of curvature appropriate to the calculation of Hertzian stress is that in the pitch ellipse of the virtual spur gear and has the value rG

ρ d sin φ n 2 2 cos β ρC1

冢

冣

(7.26)

where φ n is the normal pressure angle and β is the helix angle. At the pitch line, the corresponding value of the entraining velocity (m兾s) can


255

be taken to be Ve G0.08V sin φ n cos β On this basis, the foregoing relation for spur gears becomes h min G0.065

sin φ n 0.5

cos

β

1冤

vo dV

ρ

冢 ρC1冣冥

(7.27)

where h min is measured in the normal direction. The factor cos0.5 β has the value 0.931 for β G30° and 0.983 for β G15°. An approximate expression for h min in helical gears at the pitch line is thus h min G0.07 sin φ n

1冤

vo dV

ρ

冢 ρC1冣冥

(7.28)

As before, h min is in micrometres, vo is in centistokes, d is in metres, and V is in metres per second. In the transverse direction, the thickness of the film is, of course, h min sec β , while sin φ t Gsin φ n sec β where φ t is the transverse pressure angle. Assuming that isothermal elastohydrodynamic theory is valid for all positions of contact between the teeth, it is to be expected that the film thickness will vary over the active profiles of the teeth with the factor 0.43 V 0.7 , or approximately with 1(Ve r). e r The relative radius of curvature of involute profiles in mesh varies parabolically over the common tangent of the base circles. Figure 7.4 shows this effect. The interference points I1 and I2 are the points of tangency of the common tangent and the base circles, centres C1 and C2. At I1 and I2 the relative radius of curvature r is zero. At the point mid-way between I1 and I2 the relative radius r has its maximum value. The effect of r on the thickness of the oil film between the teeth is indicated by the lower curve, which on some scale represents r0.43. This curve has a comparatively flat top and falls off sharply near the interference points. The entraining velocity Ve , the sum of the velocities of sweep, is constant over the line I1I2 in gears of 1 :1 ratio, but in all other cases it varies from I1 to I2. The variation is linear, increasing from I1 to I2 in the simple relation that the ratio of Ve at I2 to Ve at I1 is equal to the gear ratio. Thus, in speed-reducing gears Ve increases continuously over the path of contact, and in speed-increasing gears it falls continuously. Figure 7.4 shows the variation in Ve with a gear ratio of 2 :1. If it is

256

Rolling Contacts

Fig. 7.4

taken that the thickness of the oil film will vary with V 0.7 e , the effect will be as indicated in the diagram. In a particular pair of meshing gears the magnitude of the variation in film thickness due to variation in r and Ve along the path of contact will depend on the position and extent of the section of the common tangent between I1 and I2 occupied by the path of contact. Figure 7.5 shows the combined effect of the variation over I1I2 of r and Ve on the foregoing basis, in terms of the ratio of the film thickness at any point to that at the pitch point. As the gear ratio increases, the pitch point, denoted by P in Fig. 7.4, moves towards I1. Except in gears of very low ratio, the effect insofar as Ve and r are concerned is that in general the film thickness will increase continuously over the active profile in speedreducing gears and will decrease continuously in speed-increasing gears. In general, the variation is greater the higher the gear ratio. Figure 7.5


257

Fig. 7.5

represents either a spur gear or a transverse section through a helical gear. In the latter case the base circles to be considered are those on which the involute helicoids are generated, and not the base circles of the virtual spur gears. The length of the path of contact in proportion to I1I2 will depend, at a given pressure angle, on the diametral pitch of the teeth. Figure 7.6 shows the approximate variation in the film thickness over the path of contact of standard full depth teeth with a 20° pressure angle, in terms of the percentage of the pitch point value at or near the point of first engagement and at or near that of disengagement. The film thickness is shown increasing from I1 to I2 , i.e. the condition of reduction gears. Increase in the pinion addendum, and decrease in that of the wheel, moves the point of first engagement nearer to the pitch point, and reduces the extent to which the film thickness at the point of first engagement falls short of that at the pitch line.

7.4.2 Operating temperature The significant operating temperature with respect to film thickness, i.e. that at which oil enters the zone of contact, is identified with the temperature of the metal surface prior to meshing. The viscosity– temperature dependence of lubricating oils is well known, but it is

258

Rolling Contacts

Fig. 7.6

relevant here to note its magnitude. Over the range of conditions encountered in gear lubrication, the viscosity of a mineral gear oil is halved by an increase in temperature of approximately 15 °C, the corresponding effect on film thickness being a reduction of about one-third. The temperature of the teeth is thus of primary importance: the lower the temperature, the thicker is the film of oil between the teeth. The viscosity grade of the oil is also important, but it should be noted that the use of a thicker oil, while reducing tooth friction, increases bearing friction and churning losses, and thus normally leads to a general increase in operating temperatures which partially offsets the viscosity increase at the surface of the teeth. Comparing geometrically similar gears of the same size, operating temperatures generally increase with speed and at any given speed they increase with the viscosity grade of the oil. With bath lubricated gears it is a reasonable assumption that the bulk temperature of the teeth will not differ much from the temperature of the oil charge. With spray lubricated gears, however, there is an element of uncertainty as to the temperature of the teeth. It is to be expected that it will vary in relation to the oil inlet temperature according to the amount of oil directed


259

on to the teeth and the effectiveness of application of the sprays. The temperature of the oil rebounding from the gears would give an indication of the surface temperature, but it is difficult to measure this reliably and in any case some doubt would remain as to the extent to which the temperature of the gears would exceed that of the oil. Some differential is inevitable since the period of contact of the bulk of the oil with the metal surfaces is very short indeed. Measurements of the operating temperature of the teeth in spray lubricated industrial gears led to the conclusion that with good design the margin over the temperature of the oil leaving the teeth is commonly in the range 5–10 °C and does not exceed 20 °C in the extreme cases. The tooth operating temperatures depend on the tooth losses in relation to the efficiency of cooling. Both of these factors may vary considerably. The tooth losses tend to increase with tooth loading and sliding speeds, but in practice designed loads are reduced and the teeth are made finer in pitch as peripheral speeds increase. Generally speaking, the operating temperature of the teeth does not show so much dependence on peripheral speed where lubrication is by spray. As a rule, the oil inlet temperature should not be more than about 10 °C below the estimated tooth temperature, so that the oil will not be of too high viscosity for effective spraying over the gears. In large units the inlet temperature is often about 32 °C. Typically, the temperature peak in the metal bounding the conjunction (the flash temperature) may, where sliding velocities are greatest (i.e. at the tips of the teeth), be perhaps 65–95 °C over the bulk temperature of the teeth outside the zone of contact. A temperature rise of 20 percent of the peak in the constriction has been suggested for the oil at entry, but the extent to which the oil temperature at entry is increased by forward flow of heat from the conjunction, if at all, is highly conjectural.

7.4.3 Oil viscosity in relation to surface condition The selection of the viscosity grade for gears is based on the general necessity for the viscosity grade of the oil, i.e. its viscosity at a reference temperature, to vary inversely with speed. If the viscosity grade typical of gears at a given speed is adopted for comparative purposes and the corresponding typical temperature of operation is taken into account, the characteristic value of film thickness in gears of given form shows the trend in relation to speed shown in Fig. 7.7. This has been drawn on the basis of pitch line conditions for a hypothetical spur gearset of standard design assuming a pinion diameter of 210 mm and a ratio of

260

Rolling Contacts

Fig. 7.7

diameters of 2.6. It is interesting, though perhaps not surprising, to note that over a wide range of speed the average film thickness is uniform. At low speeds it increases, and the reason for this could be that surface finish is not so good in slow running gears, with the effect that experience has led to the general use of oils of higher viscosity than would be expected by comparison with faster running gears. The values indicated in the diagram are those that would be obtained between geometrically true surfaces. In selecting the oil for a given gearset, distinction must be made between conditions during running-in and those when a smooth surface has been achieved. It would be reasonable to assume that for freedom from wear the minimum film thickness indicated by elastohydrodynamic theory should exceed the sum of the peak-to-valley heights of the asperities on the two surfaces, i.e. about 3–4 times the sum of the two centre-line average (c.l.a.) values. It appears, however, that freedom from wear is obtained with film thicknesses less than this, as though some depth of oil, in the valleys between asperities, acts as a foundation for the hydrodynamically induced film. Experience suggests that, as indicated by the theory, a film thickness of about twice the sum of the c.l.a. values will ensure freedom from wear. This condition may be achieved at the outset with some precision quality gears operating at high speed, but in general the initial roughness of the surfaces will exceed the value at which contact will occur, and the establishment of a predominantly hydrodynamic regime will depend on running-in. A practical question concerning film thickness is that of where in relation to the asperities it is presumed that the datum exists. Measurements of film thickness, using a voltage discharge method, indicate


261

variation in film thickness between pitch line and tips in gears of 1 :1 ratio, shown to be of the order of 5 :1. However, the base here is the average position of zero voltage, which will occur where contact takes place between the most prominent asperities. If the film thickness is assessed from the estimated position of the c.l.a. datum, the variation between pitch line and tips of the hydrodynamically induced film appears to be about 2 :1 or less. Variation in film thickness of this order exceeds that attributable to variation in the relative radius of curvature and has been cited as evidence of forward flow of heat from the constriction. If, however, the film thickness is defined as the effective clearance between the opposing asperities, it is arguable that the value of this at a particular position may, on average, vary with the rate at which opposing asperities are traversing that position; that is to say, the chances of an encounter or near-encounter between asperities over average size may increase with the local amount of sliding. Finely divided solid matter such as metallic debris of extremely small particle size in entrainment in the oil may similarly bridge the oil film with greater frequency the greater the sliding velocity. It is therefore possible that the elastohydrodynamic film thickness, assessed from, say, the c.l.a. datum, could be much the same at the tips of the teeth as at the pitch line, notwithstanding that the effective film thickness or clearance diminishes from the pitch line to the tips. If this is so, it is to be expected that the variation in film thickness will diminish with running-in, but the scales will always be weighted against the film thickness at the tips of the teeth.

7.5 Boundary contact in gear lubrication Full fluid-film lubrication implies that the surfaces are completely separated from each other, and partial fluid-film lubrication that the normal load is partly borne by areas of oil under pressure and for the remainder by limited areas of direct contact between opposing asperities separated only by adsorbed films of lubricant. The amount of such boundary contact can be referred to in terms of the proportion of the load so borne or in terms of the frequency of opposition or engagement of asperities. It is of note that, as sliding speeds vary, this amount will also vary with the amount of sliding, which is why sliding velocities are of more consequence under conditions of boundary lubrication than in fluid-film lubrication. For little or no contact between the surfaces, the calculated nominal film thickness must exceed some critical value in relation to the roughness of the asperities. In practice there are very large variations in

262

Rolling Contacts

surface roughness. Such variations are caused both by machining and wear, including running-in, and there is little doubt that many anomalies in gear performance can be traced to this source. Specifications for spur and helical gears do not usually include stipulations as to surface finish. Requirements as to surface accuracy imply particular conditions of surface finish, and in this respect there is less variation among gears of precision quality than among less accurate gears. Given that the oil film thickness indicated by elastohydrodynamic theory will have an order of magnitude of 0.75–1.25B10−3 mm, it is clear that in the large majority of gears much contact between asperities will be inevitable when the gears are first run. In gears of ordinary commercial quality, such contact will initially occur on a gross scale, while the load between the teeth is inevitably carried largely by opposing asperities.

7.5.1 Running-in process The process of running-in, particularly with comparatively ductile steels, consists primarily of the progressive smoothing of the surfaces through a combination of ploughing, abrasive wear, and plastic flow. With the more ductile steels the process is very rapid at the outset, with comparatively high rates of wear accompanied with rapid smoothing of the surface. As the surfaces become smoother, the rate of wear diminishes, falling swiftly to a low value in a matter of days. Subsequently, slow wear persists over a period of weeks or months, usually tailing off and in many cases falling to a negligibly low rate. This means that in general a state of near-hydrodynamic or marginal hydrodynamic lubrication will then have been reached. However, nothing much better than that marginal state can be achieved, even after prolonged running. Generally speaking, the harder the surface material of the gear teeth, the greater is their resistance to running-in. Ready running-in, leading to a highly polished surface condition, is characteristic of the steels of low to medium hardness widely used in industrial gears. Harder tooth surfaces may similarly undergo a progressive improvement in smoothness, though the process tends to take longer the harder the surface material and may occupy several years. Thus, with these harder gears the risks typical of a running-in period tend to cover a much longer period. Such teeth may well be rather sensitive towards the choice of lubricant. With hard teeth many anomalies have been noticed, and there is little doubt that to a considerable extent these are due to the influence of the material of the mating gear. If one gear is considerably harder than the other, the course of events tends to be governed by the surface condition of the harder of the two. A hard pinion with a surface produced by profile grinding may, for example, have a finish superior to


263

that of the mating wheel where the latter goes into service as hobbed, and the effect will be a smoother finish, or it will tend to plough and roughen the wheel. On the other hand, if the disparity in hardness is large, the pinion itself may not run in very well, or may hardly run in at all, with the result that real damage is done to the tooth faces of the wheel. As a rule, however, where both the gears have a surface hardness only slightly below the limit of machinability with normal cutting tools, running-in generally follows the same course as with the softer materials, though more slowly. Failure of the gears to run in well is associated mainly with the use of case carburizing or nitriding to obtain a fully hardened surface, particularly where the mating gear is not so hard. It is sometimes said that a certain degree of microroughness of the working surfaces of the teeth is advantageous, presumably because trapping of oil between asperities is supposed to help to maintain the oil film. There are indications that at increasing speeds the shear elasticity of the oil comes into play, i.e. that its behaviour begins to be to some extent that of an entrapped elastic solid. However, there is no evidence that there is an optimum degree of surface roughness within the range attainable in practice. The belief that the surface can have too high a polish is possibly due to the observation that breakdown of lubrication may be preceded by burnishing to a mirror-like finish. It may also be due to the fact that the harder teeth tend to receive the best initial finish, and that these have a prolonged and hazardous running-in period. During the early stages of running-in, gears are particularly vulnerable to scuffing. As running-in continues, the danger of scuffing becomes less acute, but it persists until the process has reached an advanced stage – say for 48 or 72 h running under load with comparatively soft gears, but for months or even years with hard gears. This is especially so if the gears are subjected to overload torque during these stages. Particularly with the harder tooth surfaces, it is often advantageous, and may be essential, to use an oil embodying an effective extreme-pressure (EP) additive during the running-in period. When the running-in process is completed it is usually not essential to retain an EP oil, but with hardened gears it may be 2 or 3 years before this stage is reached.

7.6 Scuffing in meshing gears 7.6.1 Flash temperature as a criterion for scuffing Scuffing is the severe surface damage attributed to local welding of microasperities on the mating surfaces followed by rupture of the welded microjunctions. It is apparently due to gross breakdown of

264

Rolling Contacts

lubrication over a distinctly defined area, which is likely to be badly damaged. Usually, scuffing is progressive and its development fairly slow, although it can develop at a catastrophic rate. It normally begins, or is more severe, when sliding speeds are highest, i.e. at the root and tip of the teeth, and will never occur at the pitch line, where rolling dominates and there is little or no sliding. For scuffing to occur, contact between the surfaces is essential. However, this is far from being the only requirement. Although it has been noted that a considerable amount of actual metal–metal contact takes place in practically all gears during the early stages of operation, nevertheless scuffing is normally avoided. It has been established that whether or not contact gives rise to scuffing can be related to the local increase in temperature in the adjacent metal as the zone of contact passes through the particular position considered. A reasonable estimate of this increase, or flash temperature, is given by the following equation derived by Blok (7). The transient contact temperature Tc is defined by Tc GTbCTf

(7.29)

where Tb is the bulk temperature of the teeth and Tf is the flash temperature superimposed on Tb. For spur gears, Tf is given by Tf G1.11

µ w 兩1U1A1U2 兩 b 1z

(7.30)

where µ is the instantaneous coefficient of friction for the contact area, bG(λρs)0.5 denotes the thermal contact coefficient for the tooth face material, in which λ is thermal conductivity, ρ is density, s is the specific heat per unit mass, w is the tooth normal load per unit length of contact in the meshing position considered (the actual tooth load should be used, allowing for tooth load sharing and the dynamic load increment), z is the Hertzian contact band width, and U1 and U2 denote tangential speeds of teeth 1 and 2 perpendicular to the line of action for the meshing position considered. The formula for Tf is valid for any consistent system of units. The meshing position considered has commonly been the lowest point of single-tooth contact. It is advisable, however, to calculate Tf for a number of positions on the profile to find the highest value. Only in a minority of gears is the lowest point of single-tooth contact likely to be the least favourable position. Unless there is some tip relief, either as manufactured or after running-in, it may be preferable for general


265

comparative purposes to take the value Tf at the position of maximum sliding velocity, i.e. the position of initial contact at the dedendum of the driving tooth and the addendum of the driven tooth. Practice has varied as to the load assumed. In the position of transition to singletooth contact, the whole tangential load, together with the dynamic load increment, is borne by the tooth; in positions where two teeth are in contact, some proportion of the tangential load upwards of one-half is assumed, depending on the accuracy and conditions of operation of the gears, but owing to dynamic loading this may exceed unity. Where more than two teeth can engage at once it is advisable to assume for this purpose that only two of them are load bearing at any instant. For helical gears, Tf is given by Tf G

µ w 兩1U1A1U2 兩 1(sin φ n csc φ t) b 1z 1.11

(7.31)

Since a number of teeth will be in contact at the same time, there is some doubt as to the value of w to be adopted. For comparative purposes it is assumed that the load is equitably distributed at any moment over the total length of line of contact. The appropriate value of µ in a particular case is likely to be difficult to decide. For the purpose of estimating the maximum flash temperature at which a thermally stable system can exist between the meshing gear teeth, the appropriate value is that under conditions approaching those of incipient scuffing, as might be indicated in a rig test by the beginning of a runaway increase in the bulk temperature of the gears. For some time a value of 0.06 for µ was thought appropriate in this context, and more recently a range of 0.07–0.09 has been put forward as more representative. It cannot be said with certainty that the scuffing temperature is a property of the oil, in the sense of it being a unique characteristic of the grade concerned, with the effect that in any situation where this value is attained the dynamic system in which the oil is taking part will undergo thermal collapse. There is room for speculation that the volatility of the oil, or perhaps other intrinsic properties, imposes such a limit, but there is not at present clear evidence of this. It is also possible that the limiting flash temperature with a particular oil may vary between gears differing in materials and surface finish. The value in a practical case may be to some extent higher or lower than that referring to laboratory gears used in scuffing tests. In estimating the flash temperature occurring in practice in a gear set, the relevant value of µ will be that under the least favourable operating

266

Rolling Contacts

conditions, including the effects of dynamic load increment. If, for example, the expected average value over the meshing cycle is 0.03, it may be appropriate to assume a value of 0.07–0.09 to take account of variation in the instantaneous value and dynamic loading effects.

7.6.2 Phenomenon of scuffing The term scuffing is used for the more rapidly damaging forms of wear due to oil-film breakdown under high load, characterized by welding and tearing and consequent roughening of the tooth surfaces. In many cases where gears are subjected to severe loads, the danger of scuffing can be avoided by the use of an extreme pressure additive. Scuffing varies in degree, being referred to as slight, moderate, heavy, and so on; the more severe cases show more obvious signs of metallic tearing. Severe scuffing, in which metal may be dragged radially over the tips of the teeth, is sometimes called galling. Scuffing results from asperity contact under conditions conducive to disruption of the oil film. Such conditions are characterized by the generation of heat at high rates at the points of contact and necessarily involve comparatively high sliding speeds, or a combination of high sliding speeds and loads, where contact occurs. The sliding speeds between gear teeth at a given surface temperature are seen to be rather immaterial to the thickness of the oil film at the point of contact, and the film thickness is also virtually independent of tooth loading. However, because of the influence of sliding speeds and load on surface temperatures, both of these quantities are subject to practical limitations for the avoidance of scuffing. The flash temperature theory discussed earlier is also applicable to full fluid-film lubrication, since the product µw may be replaced by fluid frictional force. The load w would still be influenced via the width z of the Hertzian area. Although the frictional force is still affected by sliding speed and load under elastohydrodynamic conditions, the influence of these factors increases comparatively greatly with the amount of direct contact, i.e. with the relative magnitude of the contact load component. Generation of heat thus increases with the frequency and severity of collisions between asperities. Scuffing will result from exceptionally violent engagement of asperities where the oil film is, or becomes, too thin in relation to their height. The inception of scuffing implies a situation both thermally and mechanically unstable, and its development tends to ‘snowball’ progressively. More often than not it spreads very rapidly. When scuffing occurs, it very often starts at the tips and roots of the teeth, where the


267

conditions of load and sliding speed are generally worst. Sometimes it starts at the points where tip relief ends or, in small spur gears, at the beginning or end of single-tooth contact. It tends to spread towards the pitch line, sometimes leaving a band there which remains unscuffed but is overstressed and consequently succumbs to progressive pitting. The designer should avoid unduly severe combinations of load and sliding in relation to pitch line speed. The pitch of the teeth and amount of addendum are important in this respect; large teeth, though stronger, have higher maximum sliding speeds, while increase in the addendum also increases sliding speeds. At a given peripheral velocity the tendency to scuff varies with the size of the teeth; gears with large numbers of small-pitch teeth are less likely to scuff than comparable gears with a small number of teeth of coarse pitch. This is because the smaller teeth are weaker in regard to bending fatigue and therefore the loads to which they are subjected are limited to lower values, and at the same time the maximum sliding velocities, at the tips and roots, are lower the smaller the teeth. Generally speaking, slow-running gears have little tendency to scuff, since the sliding speeds between the teeth are as a rule too slow; metal– metal contact tends to result in relatively slow abrasive wear rather than scuffing. As speeds increase, metallic contact is increasingly liable to result in scuffing, while running-in becomes more critical, and to counter this tendency increasingly high standards of finish and accuracy are required. At the same time, the hydrodynamic effect of increase in speed is to tend to increase the oil-film thickness. Thus, while metallic contact becomes more dangerous, the amount of such contact tends to diminish. In practice, the former effect appears to be predominant in mediumspeed gears and in some high-speed gears up to pitch line speeds of about 25 m兾s. At higher speeds, although torque or load must be reduced, the net effect with regard to power transmissible is a net increase in the margin of safety against scuffing. The choice of gear materials and their microstructure and hardness, as well as surface finish, are important to the scuffing resistance of the gears, some material combinations being more prone to scuff than others. It should be borne in mind, however, that what is being considered is the result of an interaction between the metal surfaces and the lubricant; in some cases the influence of the foregoing variables on scuffing resistance may be modified to some extent by changes in the lubricant. This applies particularly to changing from an inactive oil to an oil with extreme-pressure additives or vice versa. With spur and helical gears it is of note that a high content of nickel, or of nickel

268

Rolling Contacts

and chromium in combination, appears to reduce scuffing resistance, particularly where these materials are running together, while direct hardening steels free from nickel show very good scuffing resistance. Case-hardened and nitrided teeth are also highly resistant to scuffing. However, there are numerous anomalies in the relation between hardness and resistance to scuffing. The harder teeth are more resistant to running-in, and the gears are thereby exposed to a greater and more protracted danger of scuffing; extreme-pressure oils, or oils of higher viscosity grade, or both, may then be advisable. Any factor causing local or general overloading while sliding speeds are high enough may cause scuffing. Overloading may stem from many sources, including magnification of dynamic load increments as a result of abrasive wear, or reduction in the load-carrying area through pitting. Damage to the working surfaces of any kind will tend to increase local stresses and is capable of leading ultimately to scuffing. Discontinuities such as large pits also tend to cause an extended interruption of the oil film. Local overloading may be caused by inadequate tip relief, misalignment, or thermal distortion. Similarly, any factor causing a local or general reduction in the oil film thickness may lead to scuffing. This included, notably, increase in the operating temperature of gears. In gearsets lubricated by spray, inadequate supply of oil to the working face, or inefficient distribution of the oil, will result in local or general overheating and consequent reduction in the oil viscosity. Oil starvation of the tooth face will also be liable to lead to dry running. Conversely, an increase in the oil supply is generally advisable where scuffing threatens. In addition, the scuffing resistance of the oil is to be considered. The mechanism of oil-film breakdown may vary between different lubricants, and in some cases instability or dissociation at high temperatures may be involved. In mineral-based lubricants, resistance to scuffing depends on the viscosity grade and on the additives that may be contained in the oil. The hydrodynamically induced film thickness will increase with the effective viscosity of the oil. It should be noted, however, that, under conditions of partial hydrodynamic lubrication, a heavy oil showing a certain viscosity at a given operating temperature will give better protection against scuffing than a thinner oil at some lower temperature at which its service viscosity is the same.

7.6.3 Probability of scuffing It is permissible to say that scuffing is controlled by a number of factors such as load, lubrication, sliding speed between contacting bodies, surface topography, and temperature within the contact region. Although


269

a number of deterministic models and scuffing hypotheses have been put forward (8, 9), it seems that the only rational approach in predicting the potential for the onset of scuffing is in probabilistic terms. Lubrication regimes and thermal effect parameters As a first step in the analysis it is necessary to find out if the contact conditions are those defined by Martin (10). This is done by examining the values of the viscosity parameter gv and the elasticity parameter ge (11). gv G

αw Re

1冢

w ERe

冣

(7.32)

η oV

0.5

冢冣

ge G

w

1冢

w η oV

冣

(7.33)

where α is the pressure–viscosity coefficient, w is the normal load per unit length of contact, η o is the absolute viscosity, and 1

G

E

1Aν 21 1Aν 21 C E1 E2

1 1 1 G C Re R1 R2 VG

V1CV2 2

Hydrodynamic lubrication is possible when gvF1.5 and geF0.6. Operating conditions outside the limitations for gv and ge are usually defined as elastohydrodynamic (EHD) lubrication. Practical EHD lubrication is limited by the values of another pair of parameters: speed parameter gs Gα

E 3η oV Re

冢

冣

0.25

(7.34)

and load parameter gl Gα

1冢

wE

2πRe

冣

(7.35)

270

Rolling Contacts

For EHD lubrication 1.8FgsF100 and 1FglF100. Equations (7.32) to (7.35) help to establish whether or not the contact is in the hydrodynamic or EHD lubrication regime. Thickness of lubricant film For the hydrodynamic lubrication regime, the minimum film thickness, ho , in the case of smooth surfaces can be calculated (10) from ho G4.9

η oVRe w

(7.36)

where 4.9 is a constant for a rigid solid and an isoviscous lubricant. Moreover, equation (7.36) is valid for high velocities and low loads on the contact. In elastohydrodynamic contacts the oil-film thickness is nearly constant throughout the Hertzian zone. For line contacts of smooth surfaces, film thickness can be calculated (12) from ho G2.65

α 0.54 (η oV )0.7R e0.43 w0.13E 0.03

(7.37)

Film thickness at EHD point contacts is given (13) by ho G0.84(αη oV )0.74R e0.41

0.07

冢W冣 E

(7.38)

where W is the total load on the contact. The thickness of the film developed at the contact between smooth surfaces has to be related to the topography of real surfaces. Usually, the specific film thickness, defined as the ratio of the minimum film thickness for smooth surfaces, ho , to the peak-to-valley roughness of the contacting surfaces, is used

λG

2ho Rsk1CRsk2

(7.39)

where Rsk G1.11Ra is the root mean square height of the surface and Ra is the centre-line average height of surface asperities. The value of specific film thickness is an important indicator of the lubrication regime. If λ H3, the probability of metal–metal asperity contact is practically negligible and therefore no further analysis of the contact is required. However, if λ F1, the operating conditions are characteristic of boundary lubrication and the mode of imminent asperity contact should be examined. The parameter that can be used


271

to do this is the plasticity index. The range of λ values between 1 and 3 points to a mixed lubrication regime where the probability of asperity contact is much smaller than in the boundary lubrication regime. It is commonly accepted that most machined surfaces have asperity height distributions that are closely Gaussian. This is particularly true for the upper half of the surfaces, which includes the proportions of the surfaces in contact. This observation makes the mathematical characterization of surfaces much more tenable. It is therefore possible to determine a value for the standard deviation of the peaks, knowing the value of the standard deviation of the surfaces. Thus, the peak height r p1 GKm1 Rsk1 and distribution may be expressed approximately as X r Xp2 GKm2 Rsk2 and the standard deviation of the peaks is given by σ p1 G Kd1Rsk1 and σ p2 GKd2Rsk2 , where Km is the peak-to-surface mean proportionality factor determined for each type of surface manufacturing method and Kd is the surface-to-peak standard deviation factor. When λ F1, the mode of asperity contact has to be determined through examination of the plasticity index. The plasticity index is given by (14)

ψG

E Pm

1冢

σp r

冣

(7.40)

where Pm is the flow hardness of the softer material in contact, r is the average radius of curvature of an asperity peak, and E is the effective elastic modulus. When ψ F0.6 the contact between the asperities will be elastic under all practical loads, and when ψ H1 the contact will be partially plastic even at the lightest loads. The deformation mode is mixed when 0.6‚ ψ ‚1, and an increase in load can change the contact of some of the asperities from elastic to plastic. Temperature criterion of scuffing Scuffing is very unlikely when ψ F0.6, but when ψ H0.6 the probability of scuffing should be examined. This can be done by comparing the surface contact temperature Ts and the characteristic lubricant temperature. Special attention must be given to the relation between the plasticity index value and the surface topography as affected by the running-in process. It is commonly accepted that running-in improves the surface finish in the sense that the highest peaks are removed and the initial Gaussian height distribution is markedly skewed because of the flattening of the peaks. Taking into account the parameters involved in the plasticity index equation, it can be said that the elastic nature of

272

Rolling Contacts

contact will be a dominant feature of a well run-in surface. Entirely elastic contact requires ψ F0.6. If, however, the plasticity index ψ F1, it means that 98 percent of the contact area is composed of elastic asperity contacts. The remaining 2 percent will have yielded plastically. The surface contact temperature, Ts depends on many factors such as load, sliding velocity, the thermal and mechanical properties of contacting materials, and the coefficient of friction. An approximate estimate of the temperature rise due to frictional heating can be made using a simple theory proposed by Bowden and Tabor (15) Ts GToC∆T

(7.41)

where To is the bulk lubricant temperature entering the contact zone. The equation for estimating the temperature rise, ∆T, is ∆TG0.443

gµV1(WPm)

(7.42)

J(k1Ck2)

where µ is the friction coefficient, V is the relative sliding velocity of the contacting surfaces, W is the load on contact, J is the mechanical equivalent of heat, k1 and k2 are the thermal conductivities of the materials involved, and g is the gravitational constant. The fractional film defect strategy is most appropriate for examining the characteristic lubricant temperature Tcr . In physical terms, the fractional film defect, β, is a measure of the probability of two bare asperities coming into contact. It is given by the expression (16)

冦冤

β G1Aexp −

30.9B105 V

1冢

冣冥 exp 冢− RT 冣冧

Tm M

Ec

(7.43)

s

where V is the relative sliding velocity of the contacting surface, Tm is the melting temperature of the lubricant, M is the molecular weight of the lubricant, R is the universal gas constant, and Ec is the heat of adsorption of the lubricant on the surface being lubricated. Because of the double exponential dependence of the fractional film defect on the heat of adsorption, Ec , and the contact surface temperature Ts, a critical moment for the lubricating film durability is when β just starts to increase. This is defined by the point on the β versus Ts curve where the change in curvature first becomes a maximum. Mathematically, this is where the third derivative of the expression for β is zero. By calculating the third derivative and equating it to zero, an expression for the characteristic lubricant temperature, Tcr , can be


273

derived in the following form Tcr G

3[(Ec 兾RTcr)A2]A1{[5(Ec 兾RTcr)2A12(Ec 兾RTcr)C12]} [(30.9B105)兾V ]1(Tm兾V )(Ec 兾RT 2cr) exp(−Ec 兾RTcr) (7.44)

If the value of surface temperature, Ts , is greater than Tcr given by equation (7.44), the probability of scuffing is high, but when TsFTcr scuffing is not probable. Probability of asperity contact The fact that the thickness of the lubricant film is not sufficient to provide a complete separation of the contacting surfaces and asperity contacts take place does not necessarily mean the onset of scuffing. As long as the asperity contact is elastic the probability of scuffing is practically negligible. Plastic asperity contact is only considered as able to initiate scuffing. Thus, the probability of asperity contact in general can be estimated (17) from P(∆h‚0)G

1 1(2π)σ ∏

冮

S

exp

0

冤

−(hr A∆h)2 2σ 2∏

冥 d(−∆h)

(7.45)

where ∆h is the separation of the highest surface peaks and hr is the distance between the mean lines of the surface peaks. The probability of plastic contact is given by P(∆h‚ δ p )G

1 1(2π)σ ∏

冮

S

冤

exp

δp

−(hr A∆h)2 2σ 2∏

冥 d(−∆h)

(7.46)

Plastic asperity deformation, δ p , is given by

δ p Gr

π

2

冢2冣冢

0.6Pm E

冣

2

(7.47)

The composite standard deviation of surface peaks, σ∏, is estimated from

σ ∏G1(σ 2p1Cσ 2p2) If ∆h′G∆hCδ p , then the probability of asperity plastic contact is P(∆h′‚0)G

1 1(2π)σ ∏

冮

S

0

exp

−(hr Cδ pA∆h′)2

冤

2σ 2∏

冥 d(−∆h′)

(7.48)

274

Rolling Contacts

The probability P(∆h′‚0) of asperity plastic contact can be found from the normalized contact parameter ∆hr ′, where ∆hr ′G(hr Cδ p)兾σ ∏ is the number of standard deviations from the mean value of separation hr . Criterion for scuffing The probability of asperity contact given by equation (7.45) is related to the probability of scuffing, i.e. P(scuffing)FP(asperity contact) However, the probability of scuffing is mainly influenced by the plasticity index, ψ , and the temperature in the contact zone. The necessary condition for scuffing to be seriously considered is when ψ H0.6, which means plastic asperity contact. The sufficient condition is fulfilled when TsHTcr . When both conditions are simultaneously fulfilled, then P(plastic asperity contact)FP (scuffing)FP(asperity contact)

7.7 Tooth face pitting Elucidation of the origin of pitting has been quite an intractable problem. Areas undergoing pitting are found to have numerous minute pits visible only under the microscope, besides those, upwards of several micrometres across, visible to the naked eye. Although pitting generally involves a progression from such small beginnings, the term progressive is used only for the worst cases. Gears of the usual industrial quality very often develop pitting that does not proceed beyond an early stage in which the larger pits are no more than about 1 mm across. This socalled initial pitting is associated primarily with running-in, and commonly starts in the vicinity of the pitch line. It is attributable to local overstressing caused by the presence of asperities and minor departures from the geometrically correct profiles of the teeth; as running-in continues, and the asperities are reduced, the surface stresses become more uniform and the pitting is arrested. Sometimes, gears that have been in use for a considerable period develop pitting that similarly comes to a halt. For example, bearing wear may cause slight changes in alignment, accompanied with local overstressing of the teeth which is in due course reduced by corrective wear. Pitting beginning gradually and showing no tendency to increase rapidly is likely to be of this temporary kind. In a relatively few cases the pitting does not halt but continues to increase, the pits gaining in size or total area or both. This is the kind of pitting specificially known as progressive. It may not become manifest until the gears have been in service for some time, but if it occurs


275

at all it is most likely to do so within about 18 months of commissioning the gears. Unfortunately, in the initial stages of pitting it is generally hard to tell whether it will develop into the arrested or the progressive type. Progressive pitting is most often concentrated in the dedendum of the teeth, though it may extend to the pitch line area, and in some cases is concentrated there. The addendum is seldom much affected, until damaged by engagement with the opposing dedendum. If such pitting goes far enough, it will ultimately result in failure of the gears. It is well established that the pitting of gear teeth is a surface fatigue phenomenon, occurring where the material succumbs to repeated alternating shear and tensile stresses set up in rolling contact. Metals may be fractured by such stresses at values, considering the loaded region as a whole, well below their ultimate tensile stresses, which are determined under static conditions. Fatigue failure comes about as a result of local change in the material, or increase, brought about by stress alternation, in local stress concentrations in the vicinity of stress raisers such as inclusions or microcracks in the material or dislocations in the crystal structure. It is accepted that in many cases the alternating stress causes local gliding at the stress raisers, but with the effect that the material becomes work-hardened and eventually gliding is prevented. If in such a situation the applied stress exceeds a certain value, the metal will crack, and the crevice thus produced will in turn act as a stress raiser causing a stress concentration, enabling the process to continue as long as the local stress is sufficient for its propagation. Gear teeth commonly show surface cracks penetrating into the metal in a direction inclined with respect to the tooth face, this direction depending on that of sliding at each part of the tooth profile; metal tends to be displaced by frictional traction in the direction in which the opposing tooth slides over the surface, i.e. towards the root and tips of the driver and towards the pitch line in the follower (Fig. 7.8). More precisely, surface cracking tends to occur in lines parallel to the grain distortion near the surface, commonly at an angle to the tooth face in the range 5–20°. It has been observed that the general tendency of cracks to spread depends on the direction of sweep in relation to the orientation of the crack. The explanation may be that if, as in Fig. 7.9(a), a crevice is inclined in the direction of sweep, it will tend to be opened by the compressive stress immediately before the Hertzian band passes over it, with the result that, when the mouth of the crevice is in the loaded zone, oil will tend to be forced into it under pressure. Propagation of the crack will thus be furthered. On the other hand, it may be that, if a crack is inclined in the opposite direction, as in Fig. 7.9(b),

276

Rolling Contacts

Fig. 7.8

although the crack will be squeezed and some oil pressure may thereby be set up, the oil will tend to escape. Thus, although the crack may certainly spread, the oil will contribute little or nothing to its propagation. Conditions favouring crack propagation by oil pressure arise primarily where there is negative sliding, i.e. where sliding is opposite in direction to sweep. This is so in the dedendum of the teeth, both for the driver and the follower. In the addendum of the teeth there may be occasional random cracks with the inclination favouring propagation by oil pressure, but they will be exceptional. When progressive pitting reaches a certain stage, the metal between adjacent pits tends to be weakened and eventually breaks off. This effect, known as spalling, takes very different forms according to the scale and course of the pitting. It has been observed that in some cases the cracking causing the pitting tends to travel downwards into the metal to a certain depth, following a roughly uniform direction. In other


277

Fig. 7.9

cases the crack might well start in one direction only, but after some time a second crack may be initiated, so as to form a bifurcation. The new limb may spread more rapidly, with the effect that the main line of the crack changes course. There is a tendency for a crack to change course to a direction parallel with the surface when it reaches the plane of maximum shear stress. Where the cracks penetrate deeply into the metal, the result tends to be formation of deep, coarse pits at a relatively few localities. The collapse of intervening metal then produces gross enlargement of the surface craters through separation of quite large chips. If, however, the predominant tendency is for the cracks to travel laterally in a subsurface stratum, the result is the development of large numbers of pits of shallow and more or less uniform depth. As the intervening metal is undermined and flakes off, the surface is left rough but with a contour following the original fairly closely. Commonly, the process continues until the whole width of the tooth is affected. A certain additional period of operation is often required before further

278

Rolling Contacts

fatigue damage develops, and this may in turn take the form of fairly uniform spalling. In the meantime, the appearance of the gears may actually improve as the roughness remaining from the first stage of spalling is smoothed down by a renewed process of running-in. In practice, conditions intermediate between the two described are encountered, uniformly spalled areas being interspersed with some deeper pits. Any factor increasing the stresses at the surface of the teeth will be liable to promote pitting. Such an increase may result either from a general increase in the load on the gears or from variation in stress from point to point or between limited areas of the tooth face. In considering the loading of the gears, dynamic load increments must be taken into account. Experience suggests that detrimental changes in dynamic conditions, e.g. owing to vibration, are often the source of an increase in loading. Differences in performance between similar or identical gearsets may originate from differences in dynamic load increment. The time to fatigue pitting in gears is inversely proportional to load to some high power (probably 8–10). Hence, even a small increase in dynamic load may very greatly increase the tendency to pitting. Local variation in surface stress will vary from point to point with the roughness of the surface, and will vary between distinct areas with the accuracy of the gears. Usually, the better the surface finish and profile accuracy the greater the resistance to pitting. In this respect it is the initial state of the gears that matters. The attainment of a smooth surface through running-in does not substitute for initial smoothness and accuracy. Apart from the initiation of cracking during the relatively rapid deformation during running-in, the running-in process does not, in general, improve conformity of the surface with the geometrical prototype and may in fact reduce such conformity. Localized variation in loading, with a tendency to overstressing in places, will therefore tend to persist. Other sources of local overloading and consequent pitting include misalignment or thermal distortion, and also contamination of the oil by hard particles, sources of which include the pitting itself at earlier stages. In gears prone to destructive pitting, an increase in the viscosity grade of the oil used from the outset tends in some cases to defer the onset of the pitting and retard it subsequently. A change to a thicker oil for gears suffering destructive pitting similarly tends to retard its further development. These tendencies are attributable to the increase in oilfilm thickness obtained at the loaded area, and a reduction in the distortion of the surface layers of the tooth material. However, experimental results suggest that the time to pitting varies with viscosity to a low


279

power of the order of 0.2–0.3. By comparison with that of load, the influence of viscosity is thus slight. Other factors governing the film thickness, notably the temperature at inlet to the loaded area, will be capable of a similar effect. Improved cooling will have an effect similar to that of using a higher viscosity grade.

7.7.1 Fatigue fracture Gear teeth can break either through bending fatigue or through impact or overload. Fatigue failure results from the progressive extension of a crack, usually at or near the root of the tooth, until the remaining sound material is so attenuated as to give way under load. As a rule, the broken surface has a comparatively smooth, laminated texture, except for the part that finally snaps, which generally has a granulated appearance. Within the laminated area there is often a point from which the fault originated. When a tooth breaks off, the shock as the mating tooth jumps the gap in subsequent revolutions often carries away a number of other teeth. There is usually little doubt, however, which tooth failed first. If other teeth are not carried away at once, the gears may not break down immediately but may continue to run for some time; they are likely, however, to suffer scuffing or pitting, or accelerated development of scuffing or pitting, while running with a broken tooth. There is often evidence of a stress riser such as an inclusion or cavity at the initial point of fatigue failure. In other cases the failure must be attributed to the progressive extension of surface or subsurface cracks because of local overstressing, which have already resulted in heavy pitting. It is to be borne in mind that in some fields, e.g. the automotive field, the normal practice in the interests of reduction in size and weight is to require gears to have no more than a specific fatigue life beyond which ultimate fatigue fracture is practically certain; as a rule, however, this is not the case with industrial or marine gears, which are normally designed for a practically indefinite life. An important point is that fatigue endurance data usually assume dry conditions, by which is meant the absence of corrosive influences. If corrosion becomes involved, which usually means that moisture is present, the corrosion feeds on the fatigue but at the same time promotes it, e.g. work-hardened material tends to suffer preferential corrosive attack, probably through the development of corrosion cells owing to electrochemical differences between work-hardened and unaffected metal, while crevices tend to retain moisture and become the seat of corrosion cells through effects such as differential aeration. Under

280

Rolling Contacts

conditions of such corrosion fatigue there is no stress range below which stress alternation can go on indefinitely without progressive surface fatigue, which can therefore cause damage at stresses well below the limit under dry conditions. Commonly, small amounts of red stain can be seen on the laminated part of the broken surface of a tooth that has failed as a result of fatigue accelerated by corrosion. Occasionally, red staining is detected at the mouth of a crevice in which corrosion fatigue is going on. Unfortunately, the gears are liable to be in a dangerous state, if failure has not already occurred, before corrosion fatigue is diagnosed as such. The best preventive measure is to use an oil with good resistance to corrosion and as far as possible to keep the gears free from water, particularly salt water.

7.7.2 Impact fracture Failure through sheer overload does not in itself have anything to do with lubrication, particularly when a machine is suddenly stopped by unintentional jamming of one of its components. However, it may be the end-effect of progressive conditions in which lubrication was a factor. The surface of metal where impact or overload fracture has occurred is generally much coarser than that characteristic of fatigue and often includes areas where the material has been elongated and left with a fibrous appearance. Overload may be due simply to the application of excessive torque by the driving unit. It is, of course, unlikely that the steady load greatly exceeds the capacity of the gears unless they have been seriously weakened in some way. However, the dynamic load increments cannot always be predicted with total accuracy and are often liable to increase as a result of extraneous changes or deterioration of the gearset. Dynamic effects causing overload include transmitted shock, torsional vibration, lack of dynamic balance, misalignment of the gears, distortion of the casing or mounting, excessive shaft deflection, and bearing wear. Deterioration of the gears or bearing may also be due to inadequacy of lubrication. Except where wear has the effect of producing some beneficial tip relief, wear of the teeth usually increases the dynamic load increments as the teeth pass through the meshing cycle. Any departure of the teeth from the correct profile will have a similar effect, by causing either impact and shock, or excessive deflections, or both. A common result of severe overloading is plastic yielding of the surface material; this will in the same way increase the dynamic overloads and accelerate


281

deterioration of the gears. Frequently, overloading leads to cracking due to bending fatigue in the root fillet, where there is a tendency to stress concentration. Fracture will occur where the metal is weakest in relation to the loads on it. Commonly, this is at the root of the tooth, given that it is essentially a cantilever. Sometimes, however, progressive pitting at the pitch line or in the dedendum may initiate fracture in these positions. Casehardened teeth in which the case is too thick are left with too weak a core and are liable to break near the crest of the core, which may be as low as the pitch line. Fracture at the extreme ends of the teeth, rather than some distance towards the middle of the tooth face, may be an indication of misalignment or damage to the ends of the teeth during or prior to assembly, although a factor here is that the ends of the teeth are weakest since there is no material beyond them to act as a buttress. In other cases the pattern of fracture may be due to the haphazard position of stress raisers such as inclusions in the material.

7.7.3 Tooth loading The classic solution of the problem of stress distribution and deformation between elastic bodies in point or line contact was achieved by Hertz for cases where the dimensions of the actual contact area are small by comparison with the radii of curvature of the engaging bodies. In relation to spur and helical gears it is assumed that the relevant case is the comparatively simple one of line contact between cylinders with parallel axes. The contact zone between the cylinders takes the form of a narrow rectangular band. Across this band the pressure distribution is semielliptic, with a maximum at the mid-point. It follows that the maximum compressive stress qmax is given by qmax G

4w

(7.49)

πz

where w is the normal load per unit length of contact and z is the width of the contact band. The maximum shear stress is equal to 0.295qmax , and this occurs at a depth of 0.393z. The bandwidth has the value zG4

1冦

wr π

1Aτ 21

冤冢

E1

冣冢 C

1Aτ 22 E2

冣冥冧

(7.50)

282

Rolling Contacts

where r is the relative radius of curvature (radius of conformity) as defined below, τ 1 and τ 2 are the Poisson’s ratios for materials of cylinders 1 and 2 and E1 and E2 are the Young’s moduli for materials of cylinders 1 and 2. The relative radius of curvature r is defined by 1

1 1 G C r r1 r2

where r 1 and r 2 are the radii of cylinders 1 and 2. For steels in general, variation in Poisson’s ratio can be neglected. Assuming a value of 0.3 for τ 1 and τ 2 , the value of z becomes zG3.82

1冤

冢

冣冥

wr 1 1 C π E1 E2

(7.51)

A combination of materials having Young’s moduli E1 and E2 will produce the same stress conditions as if both have the harmonic mean value of Young’s modulus E′ prescribed by

冢

1 1 1 1 G C E′ 2 E1 E2

冣

Hence zG3.05

1冢

冣

wE′ r

(7.52)

and qmax G0.418

1冢

冣

wE′ r

(7.53)

In the United Kingdom, a criterion of surface stress known as the Sc value is defined by ScGw兾r

(7.54)

where w is the normal load per unit length of contact (N兾m) and r is the relative radius of curvature (m). The Sc value as defined above has the dimensions of stress but is not itself the actual maximum or mean stress at the loaded area. From the above qmax G0.4181(Sc E′)

(7.55)


283

Assuming for steel兾steel gears a value of 211 GPa for E′, this reduces to qmax G192B1031Sc

(7.56)

A somewhat more practical criterion of the stress conditions between gear teeth is that known both in the United Kingdom and the United States as the K factor. It is defined by KG

Ft ρC1 d

冢ρ冣

(7.57)

where Ft is the tangential load per metre of face width, d is the pinion pitch circle diameter, and ρ is the reduction ratio. In spur gears the relative radius of curvature r at the pitch line has the value rG

d sin φ 2

ρ

冢 ρC1冣

(7.58)

where φ is the pressure angle. Assuming the least favourable condition in which the whole of the load is borne by one pair of teeth, the maximum tangential load per unit face width, Ft, is Ft Gw cos φ

(7.59)

From qmax G192B1031(w兾r) qmax G192B103

1冤

2 Ft ρC1 sin φ cos φ d ρ

冢冣冥

or qmax G271.5B103

1冢

K sin φ cos φ

冣

(7.60)

With a pressure angle of 20° qmax G479B1031K

(7.61)

Thus, for spur gears with a pressure angle of 20° Sc G6.3K

(7.62)

284

Rolling Contacts

In helical gears the relative radius of curvature r at the pitch line has the value rG

ρ

sin φ n

2 冢 ρC1冣 cos β

d

(7.63)

2

where φ n is the normal pressure angle and β is the helix angle. The total load on the tooth normal to the tooth profiles and in the normal plane, say W, has the value WG

FtL

(7.64)

cos β cos φ n

where L is the face width. In accurate helical gears the load is distributed over a total length of contact, say ct, to which several pairs of meshing teeth contribute. This is the product of the full length of contact between one pair of meshing teeth, say c, and the number of axial pitches in the face width, say ln. Thus ct Gcln cG

nct pbt

(7.65)

sin β

where nct is the transverse contact ratio and pbt is the base pitch in the transverse plane ln G ct G

L tan β pbt

(7.66)

Lnct

(7.67)

cos β

Hence, the normal load per unit length of contact w has the value wG

Ft W G ct cos φ n nct

(7.68)

From qmax G192B1031(w兾r) qmax G192B103

1冤

2 cos2 β Ft ρC1 nct sin φ n cos φ n d ρ

冢

冣冥

(7.69)


285

or qmax G271.5B103

1冢

K cos2 β nct sin φ n cos φ n

冣

(7.70)

When the pressure angle is 20° and the helix angle is 30°, assuming a typical value of nct of 1.5 qmax G339B1031K

(7.71)

Sc G3.12K

(7.72)

or

7.8 Cam–follower system Generally speaking, a cam is understood to have rotary motion, either continuous or oscillatory, while the follower may have either a rotary or reciprocating motion. The common configuration of the cam–follower system is schematically shown in Fig. 7.10.

Fig. 7.10

286

Rolling Contacts

Cam motions have wide application in all classes of automatic machinery, since by varying the cam profile it is possible to impact any desired motion to the follower or driven element, within its range of action. The discussion presented here will, however, be limited to those types in which the follower moves in a plane perpendicular to the axis of rotation of the cam and which are generally referred to as radial or plate cams.

7.8.1 Reciprocating engine cam The cam plays the same part in an internal combustion engine as the eccentric of a steam engine. Cams operating the valves of internal combustion engines are of the type shown in Fig. 7.11. A tappet is usually placed between the cam and the valve and left with a small clearance to ensure that the valve closes properly when the tappet or push rod, which is the follower immediate to the cam, is in its lowest position. The cam profile will depend primarily upon the desired timing of the various operations in relation to the crankshaft angles of the engine, and upon the nature of the valve, i.e. whether inlet or exhaust. As an illustration of the principles involved, the valve settings on a four-stroke cycle engine, the indicator diagram of which is shown in Fig. 7.12(a), will be considered. At point 1, just before the completion of the exhaust stroke, the fuel and air inlet valve opens and remains open until point 2, just after the commencement of compression. Similarly, at point 4, just prior to the completion of the expansion stroke, the exhaust valve opens and remains open until point 5, just after the completion of the exhaust stroke. The corresponding angles of rotation of the crank are shown by 2α and 2β in Fig. 7.12(b). The object of the angle of overlap ψ between operations 1 and 5 is to improve scavenging effects on the exhaust gases by the incoming charge, and it is usually small except for high-speed engines. Typical average values for these two angles are 2α G196° and 2β G232°. Since the operations occur in two revolutions of the crank, it follows that the camshaft must rotate at half the speed of the crankshaft, so that average camshaft angles are α G98° and β G 116°. To take account of tappet clearance, these angles must be increased slightly in designing the cam profile. If the camshaft is gear driven it will rotate in the opposite direction to the crankshaft of the engine. The tangent or straight-sided cam was probably the first to be used in internal combustion engines and is shown in Fig. 7.11(a) and (b). Suppose the follower commences to lift when the roller makes contact at point A. Then, if clearance is neglected and the angle AOD represents


Fig. 7.11

287

288

Rolling Contacts

Fig. 7.12

the period of opening of the valve, the follower must reach its position of rest when the roller makes contact at D. Tangents AB and CD are drawn to the circle representing the least radius of the cam. This circle, of radius OA, will be referred to as the base circle, and AB and CD as the flanks of the cam. The profile of the nose BC of the cam is made up of one or more circular arcs, forming a continuous curve in a manner depending upon the total valve lift and the total angle of action during which valve lift occurs. Denoting the radius of the base circle by r1 and the total lift of the follower by L and considering the exhaust valve cam [Fig. 7.11(b)], the arc B′C′ of radius OB′GOC′Gr1CL subtends a small angle φ at the centre of rotation O of the camshaft, and extends equal distances on either side of the line of symmetry. The profile is completed by circular arcs BB′ and CC′ of equal radii connecting B′C′ with the straight portions. It will be shown in the next section that the velocity and acceleration of the follower progressively increase against the resistance of the spring control of the valve, while the contact point moves from A to B. When point B is reached, retardation of the follower commences, contact between the roller and the cam being maintained by the retarding force of the spring, until at B′ the velocity is reduced to zero.


289

From B′ to C′, represented by the angular displacement φ , the valve is temporarily at rest. This interval is termed a period of dwell and the valve is fully open. When point C′ is reached, the spring force accelerates the valve and follower, and the downward velocity reaches a maximum value at C. Thereafter, deceleration occurs and the velocity of the follower again falls to zero at D. The longer the period of dwell, the greater are the accelerating and decelerating forces to which the follower is subjected at points B and C. These forces are reduced to a minimum when there is no dwell at full lift, i.e. when the nose of the cam is circular, as shown for the inlet valve in Fig. 7.11(a). A more even distribution of acceleration and deceleration while the follower is passing from A to B and from C to D is obtained by using convex flanks [Fig. 7.11(c)]. In this case, a flat follower may be used. The greater the convexity, the smaller are the accelerating forces at B and C. The other extreme arises with concave flanks and a long period of dwell. This gives rapid opening and rapid closing of the valve, and the long period of maximum lift allows maximum piston speed. Stresses and associated wear of the valve gear is, however, excessive, and a cam of this nature results in noisy action as the valve comes abruptly into contact with the seating. Strong springs are needed to prevent the valve from bouncing. It ought to be noted that a flat follower cannot be used with cams having flat or concave surfaces.

7.8.2 Analysis of the follower motion Consider a radial cam with a roller follower in which the line of reciprocation of the follower is offset by a distance p from the centre of rotation O of the cam, as shown in Fig. 7.13. Suppose OX and OY are two mutually perpendicular axes of reference in the plane of the cam and rotating with it. The centre of the roller is at P, and at any instant (x, y) are the coordinates of P referred to the axes OX and OY respectively. Line OQ is fixed in space and perpendicular to the path of P shown by EP. Assuming the cam rotates with uniform angular velocity ω , then ωG

dθ dt

(7.73)

where θ is the inclination of OX to OQ at the instant considered. Furthermore, if h is the height of P above OQ at time t, then pGx cos θAy sin θ

(7.74)

290

Rolling Contacts

Fig. 7.13

and hGx sin θ Cy cos θ

(7.75)

Differentiating equation (7.74) with respect to t, p being constant, gives cos θ

dx dy Asin θ Gω (x sin θ Cy cos θ )Gω h dt dt

(7.76)

Similarly, from equation (7.75) VG

dh dt

Gω (x cos θ Ay sin θ )Csin θ

dx dt

Ccos θ

dy dt

and so VGω pCsin θ

dx dy Ccos θ dt dt

(7.77)

where V is the velocity of the follower along EP. Now, suppose that P moves on the curve yGF(x), where F denotes a function of x, referred to the moving axes. Then dy dt

G

dy dx dx dt

GF′(x)

dx dt

(7.78)


291

where dy兾dxGF′(x) is the slope of the path of P at (x, y). Substituting for dy兾dx, equations (7.76) and (7.77) become dx

VGω pC

dt

[sin θ CF′(x) cos θ ]

and

ω hG

dx dt

[cos θ AF′(x) sin θ ]

Eliminating dx兾dt, the velocity of the follower is given by VGω pCω h

sin θ CF′(x) cos θ

冤cos θ AF′(x) sin θ 冥

(7.79)

For the particular case where pG0 VGω h

sin θ CF′(x) cos θ

冤cos θ AF′(x) sin θ 冥

(7.80)

If F(x) is known and F′(x) is determined, the acceleration of the follower can be found by differentiation of the expression for V. The application of this result is illustrated by the special cases detailed below.

7.8.3 Tangent cam with a roller follower For simplicity, suppose the line of reciprocation passes through O, i.e. pG0. Referring to Fig. 7.14(a) and (b), let r1 be the radius of the base circle, r2 the radius of the roller, r3 the radius of the tip circle BB′, s the length of the straight flank AB, and d the distance of the centre O′ of arc BB′ from O. Roller is on the tangent This case is depicted by Fig. 7.14(a). Taking the axis of x parallel to AB, the point P will move along a line parallel to OX relative to the moving axes. Thus, yGk, where kGr1Cr2 Gconst, and so dy

GF′(x)G0

dx Hence the general equation for V reduces to VGω h tan θ

(7.81)

292

Rolling Contacts

Fig. 7.14


293

Fig. 7.14 Continued

From the figure, kGh cos θ or hGk sec θ so that VGk sec θ tan θ

(7.82)

Differentiating with respect to t dV Gω 2k sec θ (1C2 tan2 θ ) dt

(7.83)

Since s denotes the length of the straight flank AB, these results apply from θ G0 to θ Gtan−1 (s兾k). When point B is reached, i.e. the roller makes contact with the straight flank at B as shown in Fig. 7.14(b), then xGs,

yGk,

hG1(x2Cy2 )G1(s2Ck2 )

and thus VGω h tan θ G

ωs 1(s2Ck2 ) k

294

Rolling Contacts

and

冢

冣

2s2 dV Gω 2h(1C2 tan2 θ )Gω 21(s2Ck2) 1C 2 dt k

(7.84)

where kGr1Cr2 . Roller is on the circular arc In this case, illustrated in Fig. 7.14(c), suppose OX1 and OY1 to be the moving axes of reference, where OX1 passes through point O′. Further, let (x1 , y1) be the coordinates of the centre of the roller with reference to these axes, and θ 1 be the inclination of OX1 to fixed line OQ at the instant considered. The equation of the circular of P is then (x1Ad)2Cy 21 G(r2Cr3)2 where x1 GOG, y1 GPG, and OO′Gd. Upon differentiation with respect to x1, the following is obtained 2(x1Ad)C2y1

dy1

G0

dx1

or x1Ad dy1 GF′(x1)G− dx1 y1 Substituting in the general equation for V when pG0 gives VGω h

y1 sin θ 1Ax1 cos θ 1Cd cos θ 1 y1 cos θ 1Cx1 sin θ 1Ad sin θ 1

However, since pG0 x1 cos θ 1Ay1 sin θ 1 G0 x1 sin θ 1Cy1 cos θ 1 Gh The equation for V thus reduces to VGω h

d cos θ 1 hAd sin θ 1

Now hAd sin θ 1 GOPAONGPN d cos θ 1 GO′N

(7.85)


295

and thus VGω h

O′N PN

Gω h tan ψ

(7.86)

where ψ is the inclination of O′P to the line of reciprocation. For determination of ψ , use can be made of sin ψ G

O′N

G

d cos θ 1

O′P

(7.87)

r2Cr3

and for the height of P above OQ hGd sin θ 1C(r2Cr3) cos ψ

(7.88)

Alternatively, the expression for V can be determined by direct differentiation of the equation for h and sin ψ , thus VG

dh dt

Gω d cos θ 1A(r2Cr3) sin ψ

dψ dt

where cos ψ

dψ dt

G−ω

d r2Cr3

sin θ 1 G−ω

冢r Cr Acos ψ 冣 h

2

3

Eliminating θ 1 and dψ 兾dt, the velocity equation reduces to VGω h tan ψ

(7.89)

To determine the acceleration of the follower, the equation for V is differentiated, and thus dV dt

Gω V tan ψ Cω h sec2 ψ

dψ

(7.90)

dt

Eliminating V and dψ 兾dt, this becomes dV dt

Gω 2h tan2 ψ Aω 2h

d sec3 ψ sin θ 1 r2Cr3

or dV dt

G−ω 2h

冢r Cr sec ψ sin θ Atan ψ 冣 d

3

2

1

2

(7.91)

3

The two extreme positions B and B′ [Fig. 7.14(c)] are important, and it is necessary to examine them in detail.

296

Rolling Contacts

Contact point at B At this critical position, represented by Fig. 7.14(b), ψ Gθ , so that VGωh tan θ

(7.92)

and dV dt

G−ω 2h

冢r Cr sec θ sin θ Atan θ 冣 d

3

2

1

2

(7.93)

3

where tan θ Gs兾k and hG1(s2Ck2). The value of θ 1 in relation to θ will depend upon the cam profile. Thus, for the symmetrical cam of Fig. 7.14(b), if α is the total angle of rotation and φ is the angle of dwell, then

θ 1Aθ G 12 πA 12 (α Aφ )

(7.94)

and when φ G0

θ 1Aθ G 12 (πAα )

(7.95)

Comparing these results with those already obtained for the straight flank, it can be seen that the velocity when the contact point is at B is the same in each case. As the contact point passes on to the arc BB′, the positive acceleration of the follower suddenly changes to a retardation, the magnitude of which is given by equation (7.93). Contact point at B′ In this position, θ 1 G90° and ψ G0, so that VG0 and dV dt

G−ω 2h

d r2Cr3

Writing hGdCr2Cr3 , this becomes

冢

d G−ω 2d 1C dt r2Cr3

dV

冣

(7.96)

For the system to function properly, it is quite clear that external force is necessary to maintain contact between the follower and the cam. This force is supplied by a compression spring in the case of a cam-operated valve and tappet. Not only does the spring provide the necessary retarding force during the latter part of the upward movement of the valve, but in addition it produces a downward acceleration during the early part of the return stroke. During the latter part of the return stroke the motion of the follower is retarded by the cam profile.


297

The necessary spring stiffness and initial compression will depend upon the maximum value of the retardation during the upward movement of the valve, together with the valve lift, and for this reason the straight-sided cam is unsuited to high speed. For a cam with convex flanks, the change in curvature in passing from the flank to the nose arc is less marked and the instantaneous retardation at the point of discontinuity is therefore less than with the straight-sided cam. The effect of tappet clearance is shown in Fig. 7.15. Thus, if the clearance is denoted by c, the valve will commence to lift when hG r1Cr2CcGkCc, and if θ o is the corresponding value of θ , then hGk sec θ o and so cos θ o G

k

(7.97)

kCc

so that the effective angle of cam action is reduced to (α A2θ o ).

7.8.4 Camshaft torque Referring again to the case of a cam with tangent flanks and base circle radius r1, operating a follower through a roller of radius r2 (Fig. 7.14), suppose that the follower acts against a spring of stiffness S. Let x be the initial compression of the spring, M the equivalent mass effect of the valve, spring, and follower, assumed to be concentrated at the centre of the roller, l the valve lift at angle θ , and F the force exerted by the cam on the roller in a direction normal to the contact surfaces. Gravitational and frictional effects are neglected. Referring to Fig. 7.16, it can be seen that F cos ψ is the component of F in the direction of motion of the follower, and S(lCx) represents the spring force opposing the motion of the follower. Thus F cos ψ AS(lCx)GM

dV dt

or

冤

FG S(lCx)CM

dV dt

冥 sec ψ

(7.98)

Hence, the reaction torque on the camshaft is

冤

Rh sin ψ G S(lCx)CM

dV dt

冥 h sec ψ sin ψ

298

Rolling Contacts

Fig. 7.15

or

冤

Reaction torqueG S(lCx)CM where lGhA(r1Cr2).

dV dt

冥 h tan ψ

(7.99)


299

Fig. 7.16

Thus, when the contact point is on the straight flank, ψ Gθ and hG(r1Cr2) sec θ dV dt

Gω 2h(1C2 tan2 θ )

Again, if dV兾dt G−f, where f is the maximum retardation to which the follower is subjected, then, if contact is to be maintained S(lCx)AMfH0 that is SH

Mf lCx

(7.100)

300

Rolling Contacts

7.8.5 Convex cam with a roller follower Consider now the case of a cam with circular arcs for both flank and nose profiles. Let the centre of the flank arc be at O″ at a distance e from the centre of rotation O of the camshaft, so that nose and flank radii are r3 and r4 GeCr1 respectively. When the contact point is on the nose arc, the equations representing the motion of the follower are hGd sin θ 1C(r2Cr3) cos ψ

(7.101)

VGω h tan ψ

(7.102)

冢

dV d G−ω 2h sec3 ψ sin θ 1Atan2 ψ dt r2Cr3

冣

(7.103)

where sin ψ Gd cos θ 1兾(r2Cr3). When the contact point is on the flank arc, as shown in Fig. 7.17, the same expressions may be used by substituting r4 for r3, e for d, and (θ A12 π) for θ 1, so that hG(r2Cr4 ) cos ψ Ae cos θ

(7.104)

VGω h tan ψ

(7.105)

冢

dV e Gω 2h tan2 ψ C sec3 ψ cos θ dt r2Cr4

冣

(7.106)

where sin ψ Ge sin θ 兾(r2Cr4). The proportions of the cam will depend upon the desired ratio q, which is defined as the ratio of the duration of the acceleration period during lift to the duration of the deceleration period during lift. Thus, for a symmetrical cam with no dwell, the value of θ corresponding to the contact point at the junction of the nose and flank arcs is given by qG 1 2

θ α Aθ

and since θ 1Aθ G12 (πAα ), then θ 1 and θ are determined when the total angle of action α is known. Furthermore, suppose that the maximum lift L, the base circle radius r1 , and the radius of the roller r2 are given. Then LGdCr3Ar1

(7.107)

eGr4Ar1

(7.108)


Fig. 7.17

301

302

Rolling Contacts

and at the end of the acceleration period [Fig. 7.17(b)] sin ψ G

d cos θ 1 r2Cr3

G

e sin θ r2Cr4

(7.109)

Also, from the triangle O′OO″ [Fig. 7.17(b)], as cos(π A12 α)G−cos 12 α cos 12 α G

(r4Ar3)2Ad 2Ae2 2de

(7.110)

These four equations determine the nose and flank radii r3 and r4 , and the corresponding centre distance d and e respectively.

7.8.6 General case of a convex cam with a roller follower For the tangent cam and the convex cam with a circular nose and flank profile, described in the last section, the discontinuities in curvature result in a sudden change from acceleration to deceleration of the follower during lift. Consider now the general case in which the nose and flank profiles form a continuous curve such that the equation of the path of P referred to the moving axes OX1 and OY1 is given by y1 GF(x1) as illustrated in Fig. 7.18. At any instant, let ψ be the inclination to the line of reciprocation of the follower of the common normal at the point of contact of the roller and the cam profile. Then, for point P on the curve y1 GF (x1), the following applies dy1 dx1

Gtan[πA(θ 1Aψ )]G−tan(θ 1Aψ )

(7.111)

where θ 1 is the inclination of OX1 to fixed line OQ at the instant considered. The velocity of the follower is given by VGω h

sin θ 1CF′(x1) cos θ 1 cos θ 1AF′(x1) sin θ 1

so that VGω h

sin θ 1Atan(θ 1Aψ ) cos θ 1 tan θ 1Atan(θ 1Aψ ) Gω h cos θ 1Ctan(θ 1Aψ ) sin θ 1 1Ctan θ 1 tan(θ 1Aψ )


303

Fig. 7.18

or VGω h tan ψ This result is perfectly general and applies to any cam profile.

(7.112)

304

Rolling Contacts

Parabolic profile Suppose the profile is parabolic, and that A and D are the points of tangency where the parabola touches the base circle of radius r1. The cam is symmetrical and H is the value of h when θ 1 G12 π. The equation of the parabolic path of P may be written as y21 G2ρ(HAx1)

(7.113)

where ρ is the radius of curvature of the parabola at its vertex. Let r2 be the radius of the roller and α be the total angle of action. Then, writing kGr1Cr2, constants ρ and H are given by the condition y1 Gk sin 12 α ,

dy1兾dx1 G−cot 12 α ,

when x1 Gk cos 12 α

Thus 2y1 dy1 dx1

dy1 dx1 G

G−2ρ

ρ y1

and finally

ρ Gk cos 12 α

(7.114)

and HG12 ρ (sec2 12 α C1)

(7.115)

Now, for any position (x1 , y1), the equations y1 Gh cos θ 1 and dy1 dx1

G−tan (θ 1Aψ )G−

ρ h cos θ 1

are valid so that tan (θ 1Aψ )G

ρ h cos θ 1

(7.116)

Differentiating this expression, and recalling that dθ 1兾dtGω and dh兾dt GVGω h tan ψ , the following is obtained dψ

ρω

冢ω A dt 冣 sec (θ Aψ ) cos θ G h sin(θ Aψ ) sec ψ 2

2

1

1

1


305

However, from equation (7.106) the acceleration of the follower is given by dV dψ Gω 2h tan2 ψ Cω h sec2 ψ dt dt and eliminating dψ 兾dt

ρ sin(θ 1Aψ ) sec ψ Gω 2h tan2 ψ Csec2 ψ 1A dt h sec2(θ 1Aψ ) cos2 θ 1

dV

冦

冤

冥冧

Substituting cos2 θ 1 G

ρ2 2 cot (θ 1Aψ ) h2

then

冦

冤

h Gω 2h tan2 ψ Csec2 ψ 1A sec ψ sin3(θ 1Aψ ) dt ρ

dV

冥冧

(7.117)

Commencement of motion of the follower The following equations are applicable for this stage of cam–follower system operation

θ 1Aθ G12 (πAα ) so that, if θ G0, θ 1 G12 πA12 α , ψ G0, and hGk k α dV α Gω 2k 1A cos3 Gω 2k sin2 dt ρ 2 2

冢

冣

(7.118)

Maximum lift Assuming that, θ 1 G12 π, ψ G0, and hGH, it follows that H 1 dV α Gω 2H 1A GA ω 2H tan2 dt ρ 2 2

冢

冣

(7.119)

Consider now the case of a symmetrical cam with a roller follower, the cam having a parabolic tip with circular flanks. If the radius of the flank arc is made equal to the radius of curvature of the parabola at the junction of the two curves, then, by the crack and connecting rod analogy, sudden change in the magnitude of the acceleration of the follower as it leaves the circular flank will be avoided.

306

Rolling Contacts

As the follower passes over the circular flank, the acceleration will increase, reaching a maximum positive value when the contact point is at the end of the flank arc. Beyond this position the acceleration will diminish, reaching its maximum negative value at full lift. At the junction of the flank arc and the parabolic tip there will be a sudden change in the slope of the acceleration curve, resulting from the difference in the rate of change in curvature of the profile.

7.8.7 Convex cam with a flat follower Considering now the case of a radial cam operating a flat follower (Fig. 7.19), suppose that yGF(x) represents the profile of the cam referred to rectangular axes OX and OY in the plane of the cam and rotating with it. Point P is the point of contact of the profile with the surface of the follower, and at any instant (x, y) are the coordinates of P. The cam rotates with uniform angular velocity ω Gdθ 兾dt, where θ is the inclination of OX to fixed line OQ at the instant considered, and the contact surface of the follower remains parallel to OQ. Referring to Fig. 7.19(a), let h be the height of P above OQ at time t. Then hGx sin θ Cy cos θ so that dh dt

Gω (x cos θ Ay sin θ )Csin θ

dx dy Ccos θ dt dt

Now the slope of the profile at P referred to axes OX and OY is dy dx

Gtan(πAθ )

that is F′(x)G−tan θ Further dx dx dy GF′(x) G−tan θ dt dt dt so that dh dx Gω (x cos θ Ay sin θ )C (sin θ Acos θ tan θ ) dt dt


307

Fig. 7.19

The second term is zero and x cos θ Ay sin θ Gp

(7.120)

so that VGω p

(7.121)

308

Rolling Contacts

where V is the velocity of the follower in a direction perpendicular to OQ and p is the perpendicular distance from O to the normal at P at the instant considered. The acceleration of the follower is thus given by dV dt

Gω

dp dt

(7.122)

Equations (7.121) and (7.122) are valid for any convex cam with a flat follower. Applying the above results to a convex cam with circular nose and flank profiles [Fig. 7.19(b)], let OO′Gd and OO″Ge, where O′ and O″ are the centres of the nose and flank arcs respectively. The figure is drawn for the position where P is at the junction of the two arcs. When the contact point is on the circular flank arc, then pGe sin θ

(7.123)

VGω e sin θ

(7.124)

and dV Gω 2e cos θ dt

(7.125)

These equations apply from θ G0 to the position shown in the diagram, and during the corresponding period on the return stroke. When the contact point is on the circular nose arc, then pGd cos θ 1

(7.126)

VGω d cos θ 1

(7.127)

and dV dt

G− ω 2d sin θ 1

(7.128)

These equations apply from the position shown in the diagram to the position (πAθ 1), when the opposite extremity of the nose arc is in contact with the flat follower. Maximum lift occurs when θ 1 G12 π and the two sets of equations are connected by the relation

θ 1Aθ G 12 (πAα )

(7.129)

where α is the total angle of action of the cam. In order to determine the principal dimensions of the cam profile, suppose that θ and θ 1 relate to the position where P is at the junction


309

of the two arcs as shown in Fig. 7.19(b). Also, let L be the maximum lift of the follower, r1 the radius of the base circle, r3 the radius of the nose arc, and r4 the radius of the flank arc. Thus LGdCr3Ar1

(7.130)

r4 GcCr1

(7.131)

pGd cos θ 1 Ge sin θ 1

(7.132)

and from triangle OO′O″, since angle O′OO″ is equal to πA α 1 2

冢

冣

1 d 2Ce2A(r4Ar3)2 cos πA α G 2 2de

(7.133)

The above results show that, as point P passes the junction of the two arcs, the sudden change from acceleration to deceleration of the follower is accompanied with a sudden reversal in the sign of dp兾dt, i.e. in the lateral velocity of sliding at P. At this critical point pGd cos θ 1 G e sin θ has its maximum value. When P is on the flank arc, the maximum velocity of sliding occurs at the commencement of lift when θ G 0. Similarly, when P is on the nose arc, the maximum velocity of sliding occurs at full lift when θ 1 G12 π. The changes in the velocity of sliding are similar to the changes in acceleration of the follower.

7.8.8 Stresses within the cam–tappet contact The stresses within the contact between the cam and tappet are always an important aspect of the system operation, influencing, to a significant extent, its reliability and durability. Most tappets and cams can be classified into one of the forms shown in Fig. 7.20. Also, the Hertz theory can be used to evaluate the contact stresses. In the equations presented below, the following symbols are used: W is the load between the cam and tappet, b is the width of the cam, Rc is the cam radius of curvature at the point under consideration, Rt is the tappet radius of curvature, Rt1 is the tappet radius of curvature in the plane of the cam, and Rt2 is the tappet radius of curvature at rightangles to the plane of the cam. In the case of a flat tappet face on the cam [Fig. 7.20(a)], the maximum Hertzian stress at the point under consideration is pmax GK

1冢

冣

W Rc b

(7.134)

310

Rolling Contacts

Fig. 7.20

The centre-line of the tappet is often displaced slightly axially from the centre-line of the cam to promote rotation of the tappet about its axis. This improves scuffing resistance but is considered slightly to reduce the pitting resistance. Since the theoretical line contact of the flat tappet face on the cam is often not achieved, on account of dimensional inaccuracies including asymmetric deflection of the cam on its shaft, edge loading occurs. In order to avoid this, a large spherical radius is often used to the tappet face. Automotive engines use a spherical radius of 760–2540 mm. Usually, to promote tappet rotation, the tappet centre-line is displaced


311

slightly from the axial centre-line of the cam and the cam face is tapered. Alternatively, the longitudinal tappet axis is tilted by a suitable amount to the camshaft axis. The theoretical point contact extends into an elongated ellipse under load to give a better contact zone than with the nominally flat face. The maximum contact stress is given by pmax GXK

1冤冢

1 2 C W Rt Rc

3

2

冣冥

(7.135)

where K can be obtained from the diagram shown in Fig. 7.21 after evaluating (1C2Rc 兾Rt) and XG8380 for a steel on steel material combination, XG7223 for steel on cast iron, and XG6396 for cast iron on cast iron. The maximum contact stress for curved and roller tappets with a flat transverse face is given by the following expression pmax GY

1冤冢

冣冥

1 W C Rc Rt b

1

(7.136)

Fig. 7.21

312

Rolling Contacts

where YG1875 for a steel on steel material combination, YG1676 for steel on cast iron, and YG1526 for cast iron on cast iron. Finally, the contact pressure in the case of a curved tappet with large transverse curvature (crowning) can be obtained from pmax GXK

1冤冢

1 1 2 C C W Rt1 Rt2 Rc

3

1

冣冥

(7.137)

where the values of X for material combinations are as those listed above and K can be obtained from Fig. 7.21.

7.8.9 Lubrication of the cam–tappet contact The schematic geometry of the contact between a cam and flat follower is depicted in Fig. 7.22. The parameter λ relating the lubricating film thickness to the roughness of the contacting surfaces can be estimated from the following expression λ G10−34.35

1

σ

(bSn)0.74R0.26

(7.138)

Fig. 7.22


313

where n is the camshaft rotational speed (r兾min), SG(1011η o α p) is a lubricant parameter (η o is the lubricant viscosity at atmospheric pressure and α p is the pressure–viscosity coefficient), and bG兩2r1Al兩, where l is the distance from the nose tip to the shaft axis (Fig. 7.22) and r1 is the nose radius. The equivalent radius of the system is 1

1 1 G C R r1 r where r is the radius of the follower (in the cases of a flat follower rG S), σ G1(σ 21Cσ 22) is the composite roughness of the system (σ 1 and σ 2) denote the root mean square surface roughness (µm) of surface 1 and 2 respectively. If the arithmetical average Ra is available, it is necessary to multiply it by 1.3 to convert to the root mean square measure of surface roughness. In general, the value of λ in cam systems is well below unity. In this regime, elastohydrodynamic lubrication is not very effective and boundary lubrication is the last frontier in the battle against scuffing and excessive wear of contacting surfaces.

7.8.10 Design considerations Safe values of contact stress are dependent on a number of factors such as: the materials and the material combination in use; heat treatment of the materials, including the use of antiscuff treatments such as phosphating; surface finishes; the required fatigue life; the relative sliding velocity between the cam and tappet; the oil viscosity at operating temperature; the influence of any oil additives such as zinc dithiophosphates. Practical experience indicates that the maximum pitting tendency is usually at around half-speed in engines. Also, practical experience shows that the highest contact stresses can only be used with extremely good surface finishes. In internal combustion engines, which work to high contact stress levels, a cam surface finish of about 0.4 µm is used, and a tappet surface finish of about 0.15 µm. The required surface finish of the cam and tappet should be estimated in the light of the λ parameter given by equation (7.138). Experience shows that a high working oil viscosity is desirable, with a copious supply of lubricant for the best results. Studies on cams in relation to scuffing indicate that scuffing is likely to occur when the local metal temperature reaches about 200°C. For this reason, the oil should be supplied at the lowest practical temperature for the best

314

Rolling Contacts

results. In internal combustion engines, zinc dithiophosphates are added to the oil as an antioxidant and antiscuffing agent. However, there is evidence that this additive worsens the pitting tendency and that this effect rapidly increases when oil temperatures in excess of about 110 °C are reached, apparently with some corrosive influence.

7.9 References (1) Hertz, H. (1896) The contact of elastic bodies. Miscellaneous Papers (Macmillan, London). (2) Reynolds, O. (1886) On the theory of lubrication and its application to Mr Beauchamp Tower experiments including an experimental determination of the viscosity of olive oil. Phil. Trans. R. Soc., Lond., 177. (3) Grubin, A. N. and Vinogradova, I. E. (1949) Book No. 30 (Central Scientific Research Institute for Technology and Mechanical Engineering, Moscow); Trans. D.S.I.R., (337). (4) Petrusevich, A. (1951) Fundamental calculations from the contact– hydrodynamic theory of lubrication (in Russian). Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, 2. (5) Crook, A. W. (1961) Elastohydrodynamic lubrication of rollers. Nature, 190. (6) Dowson, D. and Higginson, G. R. (1963) The theory of roller bearing lubrication and deformation. In Proceedings of IMechE Convention on Lubrication and Wear, London. (7) Blok, H. (1937) Surface temperature under extreme pressure conditions. In Proceedings of Second World Petroleum Congress, Paris. (8) Blok, H. (1970) The postulate about the constancy of scoring temperature. In Interdisciplinary Approach to the Lubrication of Concentrated Contacts, SP-237 (NASA). (9) Snidle, R. W., Rossides, S. D., and Dyson, A. (1984) The failure of ehd lubrication. Proc. R. Soc., A395. (10) Martin, H. M. (1916) The lubrication of gear teeth. Engineering, August. (11) Johnson, K. L. (1970) Regimes of elastohydrodynamic lubrication. J. Mech. Engng Sci., 12. (12) Dowson, D. (1970) Elastohydrodynamic lubrication. In Interdisciplinary Approach to the Lubrication of Concentrated Contacts. SP-237 (NASA).


315

(13) Archard, J. F. and Kirk, M. I. (1961) Lubrication of point contacts. Proc. R. Soc., A261. (14) Greenwood, J. A. and Williamson, J. B. P. (1966) Contact of nominally flat surfaces. Proc. R. Soc., A295. (15) Bowden, F. P. and Tabor, D. (1954) Friction and Lubrication of Solids, Part I (Oxford University Press, London). (16) Stolarski, T. A. (1979) Adhesive wear of lubricated contacts. Tribology Int., 12. (17) Stolarski, T. A. (1989) Probability of scuffing in lubricated contacts. Proc. Instn Mech. Engrs, Part C, J. Mech. Engng Sci., 203, 361–369.


Chapter 8 Non-metallic Rolling Contacts

8.1 General considerations The issue of fatigue in engineering materials has been recognized as an important problem for the last 200 years. It came into prominence when the steam engine revolutionized transportation and fatigue in rolling stock axles became a serious problem. Thus, the stage was set for the scientific and applied study of fatigue in ferrous materials, which has been continued since then. A particularly strong incentive to research on fatigue in metals has been given by the advent of steel hulls, crude oil pipelines, and jet aircrafts. Fatigue can be defined as the loss of strength or other important properties by a material as a result of stressing fluctuating in magnitude and direction over a period of time. Alternatively, fatigue can be described as the ultimate failure of a material or component at the application of a varying load whose maximum amplitude is insufficient to cause failure. Various atomic or molecular processes may take place during the fatigue phenomenon. Some of them may be beneficial and some deleterious. The deterioration usually prevails over the strengthening and failure, generally as a result of stresses that are small in comparison with those required for failure resulting from a static stress to occur. In the case of certain materials, there may be a clearly defined so-called endurance limit. This is a limiting low stress below which failure does not occur within any practical time. Failure may also be defined in terms of the loss of functionality owing to the inability to meet certain design criteria, such as required strength, stiffness, or integrity of shape.

318

Rolling Contacts

Fatigue failure in polymers is controlled by a number of competing factors, such as loading conditions, material structure and morphology, composition, and time. It is therefore important that the engineer responsible for material selection knows the fatigue characteristics of polymers and understands the effects of loading and variables of materials. This is simply dictated by the nature of the fatigue process and the much more complex structure of polymers compared with metals. Cyclic loading is a characteristic feature of rolling contacts. For example, teeth in a polymer spur gear are subjected to loading–unloading every revolution, although the torque applied is constant. The same takes place in the case of a rolling contact bearing with inner and outer rings made of polymeric material. In material selection and component design for rolling contact applications, the empirical characterization of fatigue in polymers is useful but not sufficient. Growing sophistication in these functions requires a more fundamental understanding of the mechanisms underlying the overall fatigue process. Although curves of stress versus number of load cycles are helpful when selecting a material for a particular application, they usually tell nothing about the initiation and propagation of crazes or cracks, yielding, and drawing, or the storage, release, and dissipation of energy within the rolling contact. The state of the art in the fatigue of polymers is not as well advanced as is the case with ferrous materials. The same is true with fatigue of polymers resulting from rolling contact configuration. Another class of materials of interest to engineers and designers is that comprising technical ceramics. This is mainly due to demands on the load-bearing rolling contacts in all kinds of machinery to operate at high speeds, hostile environments, increased unit loads, and restricted lubrication. The design and manufacture of such contacts is at the limit of established technology. Ceramics as materials for rolling contacts show some considerable practical advantages over traditional bearing steels. The properties of ceramics, specifically low density and high stiffness, are of most interest to gas turbines and machine tool manufacturers. High hardness, a low coefficient of thermal expansion, and a high temperature capability are all properties most desired by rolling contact designers. Research over the past three decades on structure, quality control, and manufacturing technique of ceramics has produced materials of high quality and structural integrity. At present, however, there is rather little reliable information on the load-carrying capacity, wear, surface fatigue, and failure modes of ceramic materials when used in rolling contact applications.

Non-metallic Rolling Contacts

319

8.1.1 Approaches to polymer fatigue A number of major problems might be identified as being responsible for hindering the understanding of fatigue processes in polymers. The traditional continuum-mechanical approach to fracture developed by physicists and mechanical engineers excludes atomic and molecular processes taking place during polymer fatigue. The ultimate fracture is usually accompanied with large-scale, irreversible, and non-linear deformations. On the other hand, polymer scientists are accustomed to measure small-scale, reversible, and linear deformations in order to characterize the effects of the molecular properties and composition of a polymer. Besides, the exact nature of the competitive processes that are active during fatigue and are responsible for the damaging effect of cyclic loading is not sufficiently known. Thus, in order to progress, it seems necessary to accept the duality of continuum and molecular descriptions as representing two different aspects of the same reality and to investigate the limits of using a linear approximation to model non-linear systems such as polymers. 8.1.2 Loading conditions in rolling contact One of the promising applications of polymers is in rolling contact bearings. In practical circumstances, the contact between the rolling elements and the rings consists more of sliding than of actual rolling. The condition of no interfacial slip is seldom achieved because of material elasticity and geometric factors. Moreover, inherent to the state of loading on rolling elements and inner and outer rings is fluctuating load, although the external load applied to a bearing treated as a system is static. The fatigue life of a rolling contact bearing is a function of a number of factors which are interwoven in a highly complex manner. The effect of increasing speed on the fatigue life, for instance, is mainly manifested in the operating load due to the centrifugal force, with a corresponding reduction in the loading zone. The contact angles also change, that at the inner raceway increasing and that at the outer ring decreasing with rising speed. Another common example of a nominally rolling contact are gears. The fatigue failure of toothed gearing may occur in any one or a combination of three basic models, i.e. (i) tooth bending, which, if continued for a sufficient number of cycles, results in fracture at the root, (ii) surface distress as a result of mesh stresses exceeding the compressive fatigue limit in some localized areas of the face, and (iii) pitting resulting from rolling兾sliding action in the contact region of two meshing teeth.

320

Rolling Contacts

8.2 Phenomenology of polymer fatigue Polymeric materials subjected to strong mechanical and environmental excitation show, like many other materials, gradual deterioration in their performance including eventual failure. If the changes in properties are mostly due to chemical reactions, then the prevailing mode of deterioration is corrosion or radioactive degradation. When, however, the deterioration in material properties is caused by repeated cyclic or random application of mechanical stresses, then the resulting damage is of the fatigue type. Polymers and their composites exhibit a much more complex behaviour when subjected to fatigue loading than ferrous materials. The net effect of the several competing processes depends on a number of factors, which include the temperature, time, environment, and basic molecular properties of the polymer. The most important factors determining the fatigue behaviour of a polymer are: – thermal effects during the loading–unloading cycle, – morphological changes within the polymer, – transition phenomena, – molecular characteristics (molecular weight, thermodynamic state), – chemical changes (degradation of bonds), – homogeneous deformations, – inhomogenous deformations. It can be said that the deformation and flow in a polymeric material depend to a large extent on the characteristics listed above. However, the molecular structure, molecular weight, composition, and morphology are especially important in this respect. Polymers in a state of stress are, almost invariably, in a non-equilibrium state, an equilibrium state or steady state response being attained only under special test conditions. In fact, the mechanical properties, fatigue included, desired of polymeric materials in most engineering applications result in large measure from their non-equilibrium response to the applied stress. At elevated temperatures resulting, for instance, from frictional heating within a rolling contact zone, polymeric chains continuously undergo random changes in configuration. The rate of such diffusional processes, which depends strongly on temperature, is the dominant factor that affects the response to stress. Essentially, all available information on polymer fatigue is of an empirical nature. General data on the fatigue of engineering polymers

Non-metallic Rolling Contacts

321

Table 8.1 Fatigue strength (MPa) for common engineering polymers

>28 MPa at 104 cycles Polyimides Phenolics, moulded Polysulfones Acrylics Polycarbonates Nylons

Rolling Contacts (Tribology in Practice Series)

Engineering Tribology (Tribology Series)

Engineering Tribology (Tribology Series)

Tribology of Miniature Systems (Tribology Series)

Hydrostatic Lubrication (Tribology Series)

Mechanics and Chemistry in Lubrication (Tribology Series)

Multilevel Methods in Lubrication (Tribology Series, 37)

Tribology in Machine Design

Tribology in Machine Design

Tribology in Machine Design

Dissipative Processes in Tribology

Dissipative Processes in Tribology

Surfactants in Tribology

Rolling Stones

Rolling Stones

Rolling Thunder

Rolling Thunder

Lubricants and Special Fluids (Tribology Series)

Lubricants and Special Fluids (Tribology Series)

Mechanics of Coatings (Tribology Series 17)

Tribology in Total Hip Arthroplasty

The Third Body Concept: Interpretation of Tribological Phenomena : Proceedings of the 22nd Leeds-Lyon Symposium on Tribology Held in the Laboratorie De Mecanique Des contacts (Tribology Series)

Biobased Lubricants and Greases: Technology and Products (Tribology in Practice Series)

Coatings Tribology

Tribology In Chemical-Mechanical Planarization

Rolling Stones

Rolling Thunder

Rolling Thunder

Rolling Stones

Rolling Thunder

Rolling Contacts (Tribology in Practice Series)

Engineering Tribology (Tribology Series)

Engineering Tribology (Tribology Series)

Tribology of Miniature Systems (Tribology Series)

Hydrostatic Lubrication (Tribology Series)

Mechanics and Chemistry in Lubrication (Tribology Series)

Multilevel Methods in Lubrication (Tribology Series, 37)

Tribology in Machine Design

Tribology in Machine Design

Tribology in Machine Design

Dissipative Processes in Tribology

Dissipative Processes in Tribology

Surfactants in Tribology

Rolling Stones

Rolling Stones

Rolling Thunder

Rolling Thunder

Lubricants and Special Fluids (Tribology Series)

Lubricants and Special Fluids (Tribology Series)

Mechanics of Coatings (Tribology Series 17)

Tribology in Total Hip Arthroplasty

The Third Body Concept: Interpretation of Tribological Phenomena : Proceedings of the 22nd Leeds-Lyon Symposium on Tribology Held in the Laboratorie De Mecanique Des contacts (Tribology Series)

Biobased Lubricants and Greases: Technology and Products (Tribology in Practice Series)

Coatings Tribology

Tribology In Chemical-Mechanical Planarization

Rolling Stones

Rolling Thunder

Rolling Thunder

Rolling Stones

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