Table Of ContentRolling Contacts
by
T A Stolarski
and
S Tobe
Professional
Englneerlng
Publishlng
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, with-
out 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.
0 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 infor-
mation 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.
Notation
a width of contact
A area of contact
radial clearance
C
E modulus of elasticity
E’ equivalent modulus of elasticity
Fe axial (thrust) load
F* radial load
G shear modulus
H identation hardness
1 contact length
A4 moment (torque)
n speed of rotation
N normal load
P contact pressure
Pm mean contact pressure
maximum contact pressure
Po
R radius of curvature
R’ equivalent radius of curvature
t traction
T tangential load
Tf flash temperature
W power
V velocity
Y tensile yield strength
a thermal conductivity
P asperity radius
Y surface energy
6 normal approach
E strain
Y coefficient of friction
Yr coefficient of rolling friction
V Poisson’s ratio
P density
xviii Rolling Contacts
0 normal stress
z tangential stress
,,,z maximum tangential stress
@(z) distribution of peak heights
Y plasticity index
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 avail-
ability 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 scan-
ning 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 con-
tacts 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. Ana-
lytical treatment of the topics discussed, wherever feasible, was con-
sidered 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 engineer-
ing 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 infor-
mation 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 appli-
cations 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 cer-
amic, 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 investi-
gates 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.
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 11
2.1 Introduction 11
2.2 Distribution of stresses within the contact zone 12
2.3 Deformations resulting from contact loading 18
2.4 Contact between bodies of revolution 19
2.4.1 Stress distribution within the contact 21
2.4.2 Contact with combined normal and tangential
loads 23
2.4.3 Three-dimensional contact 24
2.5 Contact of real surfaces 30
2.6 Criterion for deformation mode 34
2.6.1 Surface plastic deformations 35
2.7 Thermal effects during rolling 38
2.7.1 Moving source of heat 39
2.8 Contact of bodies with interposing film 40
2.8.1 Background to the analysis 41
2.8.2 Case of contacting cylinders 42
2.8.3 Contacting spheres 45
2.9 Crack formation in contacting elastic bodies 45
2.9.1 Description of the contact 46
viii Rolling Contacts
2.10 Contacts deviating from the Hertz theory 50
2.10.1 Friction at the contact interface 50
2.10.2 Adhesion at the contact interface 51
2.1 1 References 54
Chapter 3 Fundamentals of Rolling Motion 55
3.1 General features of rolling contact 55
3.2 Source of friction in rolling contact 58
3.3 Rolling friction force 60
3.4 Free rolling 62
3.4.1 Cylinder on a plane 62
3.4.2 Sphere on a plane 64
3.5 Material damping during rolling 65
3.6 Slip at the surface of contact 68
3.7 Internal friction 72
3.8 References 74
Chapter 4 Dynamic Characteristics of Rolling Motion 75
4.1 Introduction 75
4.2 Analytical evaluation of friction torque 76
4.2.1 Friction during rolling 76
4.2.2 Friction torque in the rolling contact 79
4.2.3 Total friction torque 88
4.2.4 Variable components of friction torque 91
4.3 Elastic and damping characteristics of the rolling
contact 106
4.3.1 Static stiffness of the rolling contact 106
4.4 Dimensional accuracy and contact stiffness 115
4.4.1 Radial stiffness as a function of inaccuracies 116
4.4.2 Effect of variable dimensions and variable
stiffness 121
4.4.3 Effect of waviness of raceways 122
4.5 Ball motion in a rolling contact bearing 125
4.5.1 Inertia forces and moments acting on the ball 125
4.5.2 Relative motions of the rolling elements 129
4.5.3 Friction at the contact interface 135
4.6 References 143
Chapter 5 Rolling Contact Bearings 145
5.1 Phenomenology of friction during rolling 145
5.2 Friction torque 150
5.2.1 Friction coefficient 152
Contents IX
5.3 Contact stresses and deformations 153
5.3.1 Contact between elastic bodies 153
5.3.2 Elastic deformations in bearings 157
5.3.3 Permanent deformations 160
5.4 Load distribution within bearings 163
5.4.1 Radial bearings 163
5.4.2 Thrust bearings 167
5.5 Kinematics of bearing elements 167
5.5.1 Rotational speed of the elements and the cage 167
5.5.2 Contact cycles due to rolling 171
5.6 Inertia forces 171
5.6.1 Centrifugal forces 171
5.6.2 Crankpin bearings 173
5.6.3 Forces of gyration 174
5.7 Load-carrying capacity 175
5.7.1 Dynamic capacity 175
5.7.2 Static capacity 181
5.7.3 Equivalent bearing loads 184
5.8 Lubrication of bearings 195
5.8.1 Elastohydrodynamic lubrication 196
5.9 References 198
Chapter 6 Rolling Contacts in Land Locomotion 201
6.1 Rail-wheel systems 20 1
6.1.1 Traction at the rail-wheel interface 202
6.1.2 Braking process 205
6.1.3 Traction enhancing techniques 205
6.1.4 Consequences of wheel and rail wear 207
6.1.5 Ribbed tyre 207
6.2 Tyre-road interactions 210
6.2.1 Relationship between friction and traction 21 1
6.2.2 Characteristics of the traction 214
6.2.3 Analysis of dry road traction 21 8
6.2.4 Traction under wet conditions 222
6.2.5 Analysis of wet road traction 226
6.2.6 Practical approach to traction modelling 234
6.3 References 236
Chapter 7 Machine Elements in Rolling Contact 239
7.1 Contact of meshing gears 239
7.1.1 Peculiarities of contact between gear teeth 239
7.1.2 Geometry of contact between gear teeth 24 1
X Rolling Contacts
7.2 Friction in meshing gears 243
7.2.1 Tooth losses 246
7.3 Outline of elastohydrodynamic theory 248
7.3.1 Estimates of film thickness 25 1
7.4 Application of elastohydrodynamic theory to gears 252
7.4.1 Film thickness between gear teeth 253
7.4.2 Operating temperature 257
7.4.3 Oil viscosity in relation to surface condition 259
7.5 Boundary contact in gear lubrication 26 1
7.5.1 Running-in process 262
7.6 Scuffing in meshing,gears 263
7.6.1 Flash temperature as a criterion for
scuffing 263
7.6.2 Phenomenon of scuffing 266
7.6.3 Probability of scuffing 268
7.7 Tooth face pitting 274
7.7.1 Fatigue fracture 279
7.7.2 Impact fracture 280
7.7.3 Tooth loading 28 1
7.8 Cam-follower system 285
7.8.1 Reciprocating engine cam 286
7.8.2 Analysis of the follower motion 289
7.8.3 Tangent cam with a roller follower 29 1
7.8.4 Camshaft torque 297
7.8.5 Convex cam with a roller follower 300
7.8.6 General case of a convex cam with a roller
follower 302
7.8.7 Convex cam with a flat follower 306
7.8.8 Stresses within the cam-tappet contact 309
7.8.9 Lubrication of the cam-tappet contact 312
7.8.10 Design considerations 313
7.9 References 314
Chapter 8 Non-metallic Rolling Contacts 317
8.1 General considerations 317
8.1.1 Approaches to polymer fatigue 319
8.1.2 Loading conditions in rolling contact 319
8.2 Phenomenology of polymer fatigue 320
8.2.1 Physical states of stressed polymers 321
8.2.2 Response to applied stress 323
8.2.3 Phenomenological description of fatigue 324
