Rough Surfaces

Rough Surfaces Second Edition

Rough Surfaces Second Edition

Tom R.Thomas Production Engineering Department, Chalmers University of Technology, Sweden

Imperial College Press

Published by

Imperial College Press 203 Electrical Engineering Building Imperial College London SW7 2BT Distributed by

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British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library

First edition published in 1982 by Longman Group UK Limited

ROUGH SURFACES, Second Edition Copyright 0 1999 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 1-86094-100-1

Printed in Singapore by Eurasia Press Pte Ltd

For Ann

CONTENTS

xi

PREFACE

...

Xlll

ACKNOWLEDGEMENTS

1. INTRODUCTION 1.1. Surface Roughness 1.1.1. What Causes Roughness? 1.1.2. Why Is Roughness Important? 1.2. Principles of Roughness Measurement 1.2.1. Range and Resolution 1.3. References

2. STYLUS INSTRUMENTS 2.1. Mechanical Instruments 2.2. Electrical Instruments 2.2.1. Stylus and Skid 2.2.2. Transducers 2.2.3. Pickup 2.2.4. Output Recording 2.3. Sources of Error 2.3. I . Effect of Stylus Size 2.3.2. Effect of Stylus Load 2.3.3. Other Sources of Error 2.4. Calibration and Standards 2.5. References

11 11 13 15 16 18 19 20 20 23 25 28 29

3.

35 36 36 37 44 46 47 49

OPTICAL INSTRUMENTS 3.1. Profiling Techniques 3.1.1. Optical Sections 3.1.2. Optical Probes 3.1.3. Interferometers 3.2. Parametric Techniques 3.2.1. Specular Reflectance 3.2.2. Total Integrated Scatter vii

Rough Surfaces

viii

3.3.

3.2.3. Angular Distributions 3.2.4. Direct Fourier Transformation 3.2.5. Ellipsorvietry 3.2.6. Speckle References

50

52 52 54 56

4. OTHER MEASUREMENT TECHNIQUES 4.1. Profiling Methods 4.1.1. Taper Sectioning 4.1.2. Electron Microscopy 4.1.3. Capacitance 4.1.4. Scanning Microscopies 4.2. Parametric Methods 4.2.1. Mechanical Methods 4.2.2. Electrical Methods 4.2.3. Fluid Methods 4.2.4. Acoustic Methods 4.3. References

63 63 63 64 66 68 71 71 77 80 83 84

5.

91 91 95 97 100 102 106

OTHER MEASUREMENT TOPICS 5.1. 3D Measurement 5.2. Relocation 5.3. Replication 5.4. In-Process Measurement 5.4.1. Optical Techniques 5.5. References

6. DATA ACQUISITION AND FILTERING 6.1. Data Acquisition 6.2. Filtering 6.2.1. Envelope Filters 6.3. References

113 113 115 125 130

7. AMPLITUDE PARAMETERS 7.1. Extreme-Value Parameters 7.2. Average Parameters 7.3. The Height Distribution

133 134 138 139

Contents

7.4, 7.5,

Bearing Area References

8. TEXTURE PARAMETERS 8.1, Random Processes 8.2, The Profile as a Random Process 8.3, Practical Computation 8.4. Fractal Roughness 8.5. References 9.

SURFACES IN THREE DIMENSIONS 9.1, Filtering 9.2, Parameters 9.3, Random Processes in Three Dimensions 9.4, The Surface as a Random Process 9.5, Practical Computation 9.6. Anisotropy 9.7. References

ix

144 147 151 152 157 159 162 168 171 172 173 177 180 185 188 195

10. APPLICATIONS: CONTACT MECHANICS 10.1. The Contact of Rough Surfaces 10.2. Rough Contact Mechanics 10.2.1. Contact of Curved Sulfaces 10.2.2. Joint Stiffness 10.3. The Plasticity Index 10.4. References

199 201 205 212 214 2 15 220

11. TRIBOLOGY 11.1. Friction 11.2. Lubrication 11.3. Wear 11.4. Seals 11.5. References

225 225 227 229 235 238

12. SOME OTHER APPLICATIONS 12.1. Contact Resistance 12.2. Noise and Vibration

247 247 250

Rough Su$aces

X

12.3. 12.4. 12.5. 12.6. 12.7. 12.8.

INDEX

Fluid Flow Dimension and Tolerance Abrasive Machining Bioengineering Geomorphometry References

25 1 255 257 259 26 1 263

269

PREFACE

This book is intended for graduate scientists and engineers who need to know somethmg more about roughness, how to measure and describe it and what practical problems it might cause them. It assumes a general familiarity with scientific and engineering terms and concepts and a mathematical level nowhere above that of a final-year engineering course. For a non-mathematical introduction to the subject at an undergraduate level, the reader is commended to the book by Mummery cited extensively in the text. A more comprehensive and rigorously mathematical account, to which the reader will also often be referred in these pages, is that of Professor Whitehouse. The first edition of Rough Surfaces, the first comprehensive monograph on the subject in English, was published in 1982 as a multi-author work. Several of the original authors have since enjoyed professional careers of some distinction, reflecting the increased importance of the subject since that time. Advances in the intervening period have required the addition of much new material and the updating of most of the original work, so that the present book is almost entirely a new production. Some of the new material is based on lectures which I have given at Chalmers University over the last few years. The book treats roughness primarily as an engineering phenomenon, reflecting its author’s interests and background in tribology and production engineering. I am very conscious, however, of the scientific and technical communities, of hydrodynamicists, geographers, optical engineers and many others, for whom roughness is equally important, and I have tried to keep the discussion as general as possible to reflect their needs and preoccupations. This is considerably helped by the conceptual unity of the subject; techniques of characterisation used successfully by the atomic-force microscopist can often be applied virtually without change to the problems of the geomorphologist, and vice versa. For those with more specialist interests, there are monographs (on scattering, for instance, by Ogilvy, and Bennett and Mattsson) which will be referred to where appropriate in the text. The subject naturally divides into three parts: measurement, characterisation and applications, and this division will be followed in the structure of the book. A large number of people have contributed to this book in various ways and I thank them all for their help and for their encouragement over what must have Xi

xii

Rough Surfaces

seemed a very long period. I am grateful to my colleagues at Chalmers University for many valuable discussions and other lundnesses, and particularly to BengtGoran Rosen and Robert Ohlsson for their fruitful collaboration, and to Professor Ralph Crafoord for extending to me the hospitality and facilities of the Production Engineering Department. I am also grateful to the University itself for the award of a Jubilee Professorship during 1997/98 which considerably aided the writing of this book.

ACKNOWLEDGEMENTS

I am grateful for permission to use copyright material from the following copyright holders: American Institute of Aeronautics and Astronautics for my Figs. 8.8 and 12.2 from Thomas & Sayles 1975, Prog. Astronaut. Aeronaut. 29, 3-20 Figs. 5 & 8; American Physical Society for my Fig. 4.5 from Hansma & Tersoff 1987, J. Appl. Phys. 61, Rl-R23 Fig. 1, my Fig. 4.6 from Alexander et al. 1989, J. Appl. Phys. 65, 164 Fig. 1, my Fig. 10.6 from Archard 1961, J. Appl. Phys., 32, 1420-1425 Fig. 2; American Society of Mechanical Engineers for my Fig. 6.14 from Olsen 1963, Proc. Int. Prod. Eng. Res .Con$, Pittsburgh, 8, 655-658 Fig. 3, my Figs. 7.11, 9.10 and 9.11 from Sayles & Thomas 1979, J Lubr Techno1 101, 409-417 Figs. 2, 11, 12, my Fig. 7.16 from Williamson et al. 1969 in: Surface mechanics. Ling, F.F. (ed.), 24-35 Fig. 2, my Figs. 9.8 and 9.9 from Nayak 1971, J. Lubr. Tech., 93, 398-407 Figs. 4-6, 10, my Fig. 10.3 from Majumdar & Bhushan 1990, Journal of Tribology 112, 205-216 Fig. 11, my Fig. 10.10 from Greenwood & Tripp 1967, J. Appl. Mech. 34, 153-159 Fig. 5, my Fig. 12.9 from Thomas et al. 1980, J. Biomech. Eng., 102, 50-57 Fig. 3; American Society for Testing Materials for my Fig. 4.10 from Doty 1975, 42-61 Fig. 1, and my Fig. 4.13 from Henry & Hegmon 1975, 3-17 Fig. 1, both in Surface Texture versus Skidding: Measurements, Frictional Aspects and Safety Features of Tire-pavement Interactions, STP 583; ARRB Transport Research Ltd. for my Fig. 12.10 from Potter et al. 1992, Road & Transport Research 1, 6-27 Fig. 2; British Hydromechanic Research Association for my Fig. 11.9 from Thomas et al. 1975, Proc. 7th. Int. Con$ on Fluid Sealing, Paper J32, my Fig. 12.3 from Thomas & Olszowski 1974, Proc. 6th. Int Gas Bearing Symp. D6, 73-92 Fig. 5; British Standards Institution for my Fig. 6.6 from BS1134 Part 1 1988 Fig. 20, my Fig. 6.9 from BS1134 Part 2 1972 Fig. 4; Cassell plc, London, for my Fig. 2.1 from Galyer & Shotbolt 1990, Metrologyfor engineers 5e Fig. 9.3; Chalmers University, Goteborg, for my Figs. 2.6 and 5.2 from Desages & Michel 1993 Figs. 2.9, 3.5, 3.6, 3.8; Elsevier Science, Oxford, for my Fig. 1.5 from Stedman 1987, Prec. Engng., 9, 149-152 Fig. 2, my Fig. 3.16 from Vorburger & Teague 1981, Precis. Eng. 3,61-83 Fig. 19, my Fig. 7.1 from Thomas & Charlton 1981, Precis. Eng. 3, 91-96 Fig. 3, my Fig. 10.2 from Sayles & Thomas 1976, Appl. Energy, 2 , 249-267 Fig. 1, my Figs. 2.7 and 2.8 from Radhakrishnan 1970, Wear, 16, 325-335 Figs. 1 & 9, my Figs. 5.4 & xiii

xiv

Rough Surfaces

11.6 from Thomas 1972, Wear, 22, 83-90 Figs. 2 & 4, my Fig. 5.6 from George 1979, Wear, 57, 51-61 Fig. 4, my Figs. 5.8 & 5.9 from Clarke & Thomas 1979, Wear, 57, 107-116 Figs. 2, 4, 5, my Fig. 6.4(b) from Thomas 1975, Wear, 3 3 , 205-233 Fig. 1, my Fig. 6.16 from Fahl 1982, Wear 83, 165-179 Fig. 3, my Fig. 6.17 from Shunmugam 1987, Wear 117, 335-345 Fig. 3c, my Fig. 8.6b from Thomas & Sayles 1978, Tribology International, 11, 163-168 Fig. 2, my Fig. 8.7 from Thwaite 1978, Wear, 51, 253-267 Fig. 3, my Fig. 10.8 from So & Liu 1991, Wear 146, 201-218 Fig. 8, my Fig. 10.9 from Woo & Thomas 1979, Wear, 5 8 , 331-340 Figs. 1 & 2, my Fig. 10.11 from Wu & Zheng 1988, Wear, 121, 161-172 Fig.1, my Fig. 11.1 from Koura & Omar 1981, Wear, 73, 235-246 Fig. 11, my Fig. 11.2 from Ogilvy 1993, Wear 160, 171-180 Fig. 6, my Fig. 11.8 from Golden 1976, Wear, 42, 157-162 Fig. 3, my Fig. 3.6 from Brown 1995, Int. J. Mach, Tool Manufact. 35, 135-139 Fig. 1, my Fig. 4.15 from Wager 1967, Int. J. Mach. Tool Des. Res., 7, 1-14 Fig. 5, my Fig. 12.7 from Sayles & Thomas 1976, Int. J. Prod. Res 14, 641-655 Figs. 3 & 6, my Figs. 1.3 & 1.6 from Thomas 1998, Int. J. Mach. Tool Manufact. 38, 405-41 1 Figs. 1 & 2, my Fig. 9.16 from Zahouani 1998, Int. J. Mach. Tool Manufact. 38, Fig. 11, my Figs. 8.13 & 10.13 from Rostn et al. 1998, Int. J. Mach. Tool Manufact. 38, Figs. 2 & 3, reprinted with permission; the European Commission for my Table 9.1 from Stout et al. 1993, The development of methods f o r the characterisation of roughness in 3 dimensions, EUR 15178 EN Fig. 12.22; Feinpriif Perthen GmbH, Gottingen, for my Figs. 7.2, 7.3 and 8.2 from Sander 1991, A practical guide to the assessment of surfice texture Figs. 12a, 13, 27; Hallwag Verlags GmbH, Ostfildern, for my Fig. 3.15 from Lonardo 1978, Ann. CIRP27, 531-533 Fig. 5, my Fig. 4.7 from Goch & Volk 1994, CIRP Ann. 43, 487-490 Fig. 6; Hommelwerke GmbH, Schwenningen, for my Figs. 6.11, 6.12, 6.15, 7.4, 7.7, 7.8, 7.12-7.15 from Mummery 1990, Sugace texture analysis: the handbook, Figs. 2.10, 2.1 1, 3.2, 3.3, 3.4, 3.6, 3.13, 3.17, 3.18, 3.19, 3.25, 3.26, 3.29, 3.30; I. F. S. (Publications) Ltd., Bedford, for my Fig. 5.7 from Dutschke & Eissler 1978, Proc. 3rd. Con$ on Automated Inspection & Product Control 19-30 Fig. 1; Indian Institute of Technology, Madras, for my Fig. 4.14(b) from Radhakrishnan & Sagar 1970, Proc. 4th. All-India Machine Tool Design & Research Con$ Fig. 1; Industrial Press Inc., New York, for my Figs. 2.4, 3.2 and 4.14 from Farago 1982, Handbook of dimensional measurement 2e, Fig. 6.1, Tables 15.2 & 15.4, used with permission; IOP Publishing Ltd., Bristol and the authors for my Figs. 3.3, 3.8 and 3.9 from Whitehouse 1994, Handbook of surfme metrology, Figs. 4.112 & 4.1 17, my Fig. 4.3 from Bugg & King 1988, J. Phys. E: Sci. Instrum., 21, 147-151 Fig. 2, my Fig. 4.9 from Powell 1957, J . S c i . Instrum., 34, 485-492 Fig. 1; Institution of

Acknowledgements

xv

Electrical Engineers for my Figs. 2.9 and 2.10 from Reason 1944, J. Inst. P r d . Engrs., 23, 347-372 Figs. 4 & 16; Institution of Mechanical Engineers for my Fig. 2.2 from Lackenby 1962, Proc. I. Mech. E., 176, 981-1014 Fig. 1, my Fig. 3.1 from Keller 1967/8, Proc. I. Mech. E., 182, Part 3K, 360-367 Fig. 3.5, my Fig. 3.1 1 from Westberg 1967/68, Proc .I .Mech .E., 182, Part 3K, 260-273 Fig. 25.1, my Fig. 7.5 from Hydell 1967/8, Proc. I. Mech. E., 182, Part 3K, 127-134 Fig. 15.7, my Fig. 8.4 from Peklenik 1967/68, Proc. I. Mech. E., 182, Part 3K, 108126 Fig. 24.18, my Fig. 11.5 from Leaver et al. 1974, Proc. 1. Mech. E., 1 8 8 , 461-469 Fig. 1, my Fig. 11.7 from Thomas 1978, Proc. 4th Leeds-Lyon S y m p . , 99-108 Fig. 5, by permission of the Council of the Institution; International Business Machines for my Fig. 4.4 from Binnig & Rohrer 1986, ZBM Journal of Research and Development 30; 355-369 Fig.1; Japanese Society of Precision Engineers for my Fig. 2.12 from Nara 1966, Bull. Jap. SOC. Precision Engng., 1, 263-273 Figs. 5 & 8; Kluwer Academic Publishers, Dordrecht, with kind permission for my Fig. 2.14 from Song 1988, Sulface Topography, 1, 29-40 Fig. 2, my Fig. 3.4 from Bristow 1988, Sulface Topography, 1, 281-285 Fig. 1, my Fig. 3.5 from Sayles et al. 1988, Sulface Topography, 1, 219-227 Fig. 1, my Figs. 4.1 and 4.2 from Garbini et al. 1988, Surface Topography, 1, 131-142 Figs. 1 & 6, my Figs. 4.1 1 and 4.12 from Lieberman et al. 1988, Sulface Topography, 1, 115-130 Figs. 3 & 6 , my Fig. 11.4 from Chandrasekaran 1993, J. Mat. Sci. Lett. 12, 952-954 Fig. 5; Lasertec Corporation for my Fig. 3.7; Macmillan Magazines Ltd., Basingstoke, and the authors, for my Fig. 8.9 from Sayles & Thomas 1978, Nature 271, 431-434 Fig. 2, reprinted with permission; Macmillan Press, Basingstoke, and the authors, for my Fig. 2.11 from Agullo & Pages-Fita 1974, Proc. 15th. Int. Machine Tool Des. & Res. Conf., 349-362 Fig. 9, my Fig. 12.6 from Thomas 1973, Proc. 13th Int. Machine Tool Des. & Res. Conf., 303308 Fig. 3b, reprinted with permission; the McGraw-Hill Companies, New York, for my Fig. 5.1 from Terman 1937, Radio Engineering 2e, Fig. 430, my Fig. 12.4 from Hunsaker & Rightmire 1947, Engineering applications offluid mechanics Fig. 43; McGraw-Hill Publishing Co., Maidenhead, for my Fig. 6.5 from Golten 1997, Understanding signals and systems Fig. 7.8; Optical Society of America for my Fig. 3.10 from Creath 1987, Applied Optics 26, 2810-2815 Fig. 6, my Fig. 3.12 from Birkebak 1971, Appl. Opt. 10, 1970-1979 Fig. 1, my Fig. 3.17 from Fujii & Lit 1978, Appl. Opt. 17, 2690-2705 Fig. 1, my Fig. 3.13 from Bennett & Mattsson 1989, Introduction to sulface roughness and scattering Fig. 17; Plenum Publishing Corporation, New York, and the authors, for my Fig. 10.4 from Russ 1994, Fractal surfaces, p. 67 Fig. 9, my Fig. 12.8 from Longfield et al. 1969, Biomed. Engng. 4, 517-522 Fig. 1; Random House (UK) Ltd. for my Figs. 1.2 and 6.4a from Brooker

XVi

Rough &$aces

1984 ed., Manual of British standari in engineering metrology (Hutchinson), pp. 184 Fig. 1 and 191 Fig. 10.10; the Royal Society and the authors for my Fig. 9.12 from Greenwood 1984, Proc. Roy. SOC. Lond. A393, 133-157 Fig. 7, my Fig. 10.5 from Pullen & Williamson 1972, Proc. R. SOC.Lond. A327, 159-173 Fig. 3; Science History Publications Ltd. for my Fig. 1.1 from Thom & Thom 1972, J. Hist. Astron. 3, 11-26 Fig. 5; SociCtC Belge des Mecaniciens for my Fig. 3.3 from Whitehouse 1975, Rev. M. Mec. 21, 19-28 Fig. 11; Society of Manufacturing Engineers for my Fig. 2.13 and Table 11.1 from Thomas et al. 1975, S.M.E. Paper IQ75-128 Fig. 2; Society of Photo-optical Instrumentation Engineers and the authors for my Fig. 3.14 from Church et al. 1977, Opt. Eng. 16, 360-374 Fig. 11, my Fig. 9.15 from Thomas 1991, Proc. SPZE 1573, 188-200 Figs. 5 & 6 ; Society of Tribologists and Lubrication Engineers for my Fig. 10.7 from Lee & Ren 1996 Trib. Trans. 39, 67-74 Fig. 13, my Fig. 11.3 from Akamatsu et al. 1992, Trib. Trans. 35, 745-750 Fig. 8; Taylor Hobson Pneumo, Leicester, for my Fig. 6.7 from Whitehouse & Reason 1965, The equation of the mean line of surface texture found by an electric wavefilter, Figs. 11-14, my Figs. 7.6 and 9.14 from Dagnall 1980, Exploring surjiie texture, Figs. 17 & 46; VCH Verlagsgesellschaft mbH, Weinheim, and the author for my Table 4.1 from DiNardo 1994, Narwscale characterisation of surfaces and inte$aces, p. 145 Table 2. I have attempted to trace the copyright holder of all the material reproduced in this book and apologize to copyright holders if permission to publish in this form has not been obtained.

CHAPTER 1

INTRODUCTION

1.1. Surface Roughness

We are used to the idea that materials have intrinsic properties such as density, conductivity and elastic modulus. Surfaces, representing material boundaries, have perhaps rather more insubstantial properties, but we still think of some of these properties as intrinsic, like colour. There are other properties, however, which are easy to define but whose value seems to depend on the technique or scale of measurement: hardness, for instance. Roughness seems to be such a property, with the added difficulty that it is not always so easy to define as a concept. "I can't define roughness, but I know it when I see it". When we speak of "rough country", "a rough road", "a rough fabric", we imply very different scales of feature in each case, but we understand well enough what sort of surface is meant. The Concise OED has more than a column of meanings, starting with "of uneven or irregular surface, not smooth or level or polished, diversified or broken by prominences.. . .. . .. .coarse in texture.. . To develop this idea a little, it seems to have something to do with our scale of view. If the countryside, or the stretch of road, or the patch of fabric which we observed, simply rose steadily in our field of view, or contained a single prominence, we would not refer to it as "rough"; if it contained, say, ten prominences, we probably would so categorise it; if it contained a hundred such, we certainly would. So already, it seems, the ideas of sampling interval and sample size which we will introduce later are emerging and are bound up with the very concept of roughness at quite a fundamental level. .'I.

I . 1.1. What Causes Roughness? A geographer might find this a strange question. To him or her, accustomed to the scale of features of the natural world where roughness is the norm, a better question would be, "What causes flatness (or straightness)?". Generations of students have been accustomed by their education to the normality of ideal straight lines and flat surfaces. This has been exacerbated by the wide use of computer1

2

Rough Surfaces

aided drafting software, where straightness and flatness are the default assumptions. This is a conceptual problem not only for scientists and engineers, but for everyone with a Eurocentric education. The idea of straightness is built in to all Indo-European languages as a concept linked with good, power and approval, where "right", "rule", "regal" all have the same root. This is so deeply embedded in our mental software that we are disturbed by the alignements of Brittany, whose megalithic architects spent thousands of person-years constructing adjoining lines of huge boulders, kdometres long, equidistant but nowhere straight (Fig. 1.1). Clearly they understood the concept of straightness (otherwise how could they have made the lines equidistant?), but culturally it held no importance for them.

-.,.. ...... .................... ... .. ... .... ............... ......... .... .. ......... .- ._. ..... . ..._ * . ............ *.*. .... -.----....-. . . ....... ." .......... , .... . ..... ........ .... ... --. . . ..... .... .... ...........----........ .. .*.. . . .. ,...............-L.(..* . . . ........... * ' ....-.* ........ ...- . ...... . _.,.--- ....... ... .. *....... -. ., .,.. -.. .............. , . .*-........ ................ . ...... ........ -.. . ...... ..... 0

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1

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. . . . . . . . . U . " . *

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-

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50

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Figure 1 . 1 . Part of the Camac alignments, adapted tiom Thom & Thorn (1 972). Each mark represents a menhir; a typical menhir is 2m high and lm in diameter.

The fact is that roughness is the natural state of surfaces, and left to its own devices Nature will make sure they are rough. The roughness of a surface is a measure of its lack of order. Disorder is entropy under another name, and if we consider a solid surface as a closed system then the Second Law of Thermodynamics predicts that its entropy will tend to a maximum. To reduce its roughness we must reduce its entropy, and the Second Law tells us that we can only do this by doing work. Thus if we transpose the axes of the well-known figure which relates machining time to roughness, we see that it is nothing but an entropy diagram (Fig. 1.2). Many, perhaps most, natural surfaces are fractal (see Chapter 9), and it is characteristic of fractal surfaces that their roughness increases without limit. So it is that in a universe of fractal surfaces man's attempts to reduce entropy by imposing straightness and smoothness extend over only a very small range of dimensions (Russ 1994).

Introduction

3

work Figure 1.2. (a) Relationship of surface texture to production time (Brooker 1984); (b) The same figure replotted as work reducing entropy

1.1.2. Why is Roughness Important?

When I was starting research in the field of surface roughness years ago I was advised against it by a distinguished academic engineer on the grounds that roughness is essentially a second-order effect in physical systems, and would therefore never assume an important place in engineering science. Time has, I think, vindicated my judgement rather than hs, for two rather interesting reasons (Thomas 1988). The first is that while it is certainly true that roughness is a second-order effect, it is a second-order effect across a very wide spectrum of technical activity; not just tribologists and production engineers, but cartographers, radar engineers, highway and aircraft engineers, hydrodynamicists and even bioengineers find increasingly that from time to time it obtrudes into their particular specialty (Fig. 1.3). The second is that all the easy first-order problems have been solved: whatever happened, for instance, to heavy electrical engineering as an academic discipline? We live increasingly in a world where second order effects present the major remaining challenge; any fool can make an internal

Rough &$aces

4

combustion engine that works, the trick is to make one that will run for a million miles at 100 miles to the gallon.

'

100 1970

1975

1980

1985

1990

1995

Figure 1.3. Cumulative number ofpublications on surface roughness (Thomas 1997)

1.2. Principles of Roughness Measurement

By "measurement" we mean something more than mere inspxtion. We will define measurement in the present context as a process which gives, or is capable of giving, quantitative information about individual or average surface heights. Thus we exclude many forms of optical examination. These may give information about the existence and direction of the lay on machined surfaces, or about the presence and spacing of feed and chatter marks or other individual defects, but this does not fall within our definition. There are some general considerations in choosing any measuring instrument: cost, ease of operation, size and robustness. There is also the issue of whether a measurement is comparative or absolute. In addition, for roughness measuring instruments, it is necessary to decide whether or not the instrument should make physical contact with the surface, and whethei it needs to be able to measure an area of a surface or only a section or profile through it. Most important of all are the horizontal and vertical range and resolution. Some of these criteria are self-explanatory, but the issue of comparative versus absolute measurement is worth a few moments' digression. Many roughness measuring instruments, for instance stylus instruments, give absolute measurements of local heights. Thus they can be calibrated against secondary length standards such as slip gauges and so in principle at least are traceable to primary standards. Other instruments, for instance glossmeters, give average values of some surface parameter, which may depend on material properties and may vary from one finishing process to the next. Such instruments must be

Introduction

5

calibrated against an absolute instrument used under the same conditions. Under these conditions they may still be traceable, but in a much more tightly restricted way. This is likely to be of some practical importance in a manufacturing environment where the roughness instrument is part of a quality system under I S 0 9000. Vorburger and Teague (1981) classify these two kinds of instruments as "profiling" and "parametric" techtuques. Sectional measurement is usually quicker, simpler and easier to interpret than areal measurement, and all current roughness standards, as we shall see later, are written in terms of sectional measurements. For many practical purposes sectional measurements are adequate, and sectional techruques should be preferred unless there is some good reason to the contrary. However, most engineering interactions of surfaces, including all contact phenomena, are areal in nature, and the information necessary to describe their function must similarly be areal. Often this information can be inferred mathematically from sectional information, as discussed in later chapters. The problem arises when the events about which information is sought are comparatively rare. For instance, under most engineering loads the area of real contact between two surfaces is likely to be less than 1% of their nominal area. Thus a sectional measurement may underestimate or miss altogether the features of practical interest (Fig. 1.4). The special problems of areal, or three-dimensional (3D), measurement will be discussed separately below.

Figure 1.4. A sectional measurement may underestimate the number and size them altogether

.If

important features or miss

Rough Sur$aces

6

I.2.1. Range and Resolution We can say crudely (Thomas 1978) that roughness exists in two principal planes: at right angles to the surface, when it may be characterised by some kind of height, and in the plane of the surface, identified as "texture" by Reason (195 1). There are thus two sets of limitations which we need to discuss with reference to each roughness measuring instrument or technique: the largest and smallest differences of height which it will resolve, and the longest and shortest surface wavelengths with which it can cope. It is important to remember that every instrument or technique is subject to these limitations of resolution, and that the actual figures involved will vary from instrument to instrument. A very useful way of defining and comparing instrument performance at a glance is due to Stedman (1987), who suggested plotting the horizontal range and resolution of an instrument as an envelope in two-dimensional space (Fig. 1.5). If z,, and z,,, are the maximum and minimum heights that can be measured by the instrument, and similarly A,, and are the longest and shortest wavelengths, these will define a rectangle in z-/2 space outside which the instrument will not measure. But the practical operating envelope is subject to further restrictions, for instance the steepest slope em, which the instrument will measure, and the sharpest curvature C,, which it will follow. These may also be displayed in z-/2 space if some assumption is made about the form of the surface. For mathematical convenience Stedman assumes a sinusoidal surface

z

=

Rpsin(2xx//2)

where Rp is the amplitude. Slopes and curvatures are then the first and second differentials respectively: B

C

=

=

( 2 xRp/A) cos (2 n x / R )

- (4 ?Rp/A2) sin ( 2 x x / A )

The maxima of these functions occur when the trig functions are unity, so taking logs, logRp logRp

=

=

10g(8,,/2x)

+ log1

log(Cm,/4?)

+ 210gA

Introduction

7

On a logarithmic plot these are lines of slope 1 and 2 respectively which further restrict the operating envelope (Fig. 1S ) .

Figure 1.5. Envelope of instrument pe.rfomance in amplitude-wavelengthspace (after Stedman 1987)

Other restrictions on the operating envelope could be devised; Stedman includes, for instance, the effect of angular uncertainties in the instrument reference plane. The assumption of sinusoidal surfaces is also somewhat of a simplification. Nevertheless, Stedman diagrams are such a straightforward and convenient way of defining and comparing instrument performance that they will be used throughout to illustrate our discussion of measurement techmques, omitting for the sake of simplicity the restrictions due to slopes and curvatures. Readers should be aware, however, that these Stedman diagrams represent a maximum working envelope, not all of which may be available to the same instrument at the same time. Also, the examples given will usually refer to a particular instrument, that is to an exemplification of the technique rather than as a general statement about the technique. For instance, it is possible to build stylus instruments on quite different scales of size, and it would be misleading to try to present a generic Stedman diagram which would cover all possible realisations of the stylus principle. It is interesting to construct a Stedman dagram to examine summarily the global coverage of topographical measurement techniques (Fig. 1.6). A single envelope covers the current range of roughness measuring instruments, stylus, optical and AFM. Electron microscopy extends this coverage to shorter

Rough Surfaces

8

wavelengths and larger amplitudes, though it is difficult (though not impossible; for a discussion see Whitehouse 1994) to extract quantitative height information. At the upper end of the roughness range, coordinate measuring machines take over. A final envelope of surveying techniques covers everything from autocollimators at the low-range end to satellite ranging techniques at the upper end.

I

Inm

1w

1mm

lm

Figure 1.6. Global Stedman diagram (Thomas 1997).

Fig. 1.6 is not intended to be comprehensive or definitive, and for the sake of simplicity many useful measurement techniques have been omitted. It does make the point, however, that there are large areas in range-resolution space which are not accessible to any current technique. Does this matter? The lower right area of Fig. 1.6 represents small amplitudes at long wavelengths. At the moment there does not seem to be any technological requirement in this area, but it is worth noting that the error in the Hubble telescope mirror (Parks 1991) would only just come withm a shaded area; a future generation of Hubbles might well fall outside. The upper left area perhaps represents a more pressing practical problem, that of the unavailability of techruques for measuring large amplitudes at short wavelengths. It is certainly possible to think of existing artefacts whose topography falls in this area, for instance a hairbrush! More importantly, there are very many biological structures with large vertical but small horizontal dimensions, starting on a cellular scale

Introduction

9

(Boyan et a1 1996), continuing up through growing crops (Gilley & Kottwitz 1994) and ending with forest canopies (Gallagher et al. 1992). Many such structures are of great economic importance, and at the moment we have no way of describing their topography comprehensively. This task would demand a much greater ratio of vertical range to resolution than is available from current instruments. But when we reflect on the improvement in this ratio in recent years, the implications for the future are promising (Thomas 1997). At the start of the 1980s few roughness instruments offered a ratio of better than 103:1 (Farago 1982). The current state of the art provides examples both of stylus instruments (Garratt 1982) and optical instruments (Caber et al. 1993) with ratios of better than 105:1. There does not seem to be any fundamental law of instrument design preventing further improvements, if not in these techniques then perhaps in newer ones; chemical balances, for instance, have for many years been constructed with a range 108 times their resolution (Cook & Rabinowicz 1963). Bearing the above principles in mind, we will begin in the next chapter by discussing the stylus instrument, still the most popular and widely used method of measuring surface roughness. We will go on to consider the increasing use of optical instruments, both profiling and parametric. Many other techniques of measuring roughness have been developed, and some of the more popular of these are highlighted and their use in scanning mode for 3D measurement in various microscopy systems is discussed. Finally in this part of the book we examine some associated measurement questions, such as replication and in-process measurement.

1.3. References

Boyan, B. D., Hummert, T. W., Dean, D. D., Schwarz, Z., "Role of material surfaces in regulating bone and cartilage cell response", Biomaterials 17, 137-146 (1996) Brooker, K. ed., Manual of British standards in engineering metrology (Hutchmson, London, 1984) Caber, P. J., Martinek, S. J., Niemann, R. J., "A new interferometric profiler for smooth and rough surfaces", Proc. SPIE 2088 (1993) Cook, N. H., and Rabinowicz, E., Physical measurement & analysis (Addson-Wesley, Palo Alto, 1963)

10

Rough Surfaces

Farago, F. T., Handbook of dimensional measurement 2e (Industrial Press, New York, 1982) Gallagher, M. W.; Beswick, K. M.; Choularton, T. W., "Measurement and modelling of cloudwater deposition to a snow-covered forest canopy", Atmospheric Environment 26A, 2893-2903 (1992) Garratt, J.D., "Applications for a wide range stylus instrument in surface metrology", Wear 83, 13-23 (1982). Gilley, J.E.; Kottwitz, E.R., "Darcy-Weisbach roughness coefficients for selected crops", Trans. ASAE 37, 467-471 (1994) Parks, R. E., "The Hubble space telescope investigation", Optics & Photonic News 2,28 (1991) Reason, R. E., "Surface finish", Australasian Engr., 44, 48-64 (1951). Russ, J. C., Fractal surfaces (Plenum Press, New York, 1994). Stedman, M., "Basis for comparing the performance of surface-measuring machines", Prec. Engng., 9, 149-152 (1987) Thom, A,, and Thom, A. S., "The Carnac alignments", J. Hist. Astron. 3, 1126 (1972) Thomas, T. R., "Surface roughness: the next ten years", Surface Topography 1, 3-9 (1988) Thomas, T. R., "Trends in surface roughness", Trans. 7th. Int. Con$ on Metrology & Properties of Engng. Surfaces (Goteborg, 1997) Thomas, T.R., "Surface roughness measurement: alternatives to the stylus", Proc. 19th. Machine Tool Design & Research ConJ, 383-390 (UMIST, Manchester, 1978) Vorburger, T.V., Teague, E.C., "Optical techniques for on-line measurement of surface topography", Precis .Engng. 3, 61-83 (1981). Whitehouse, D. J., Handbook of surface metrology (Institute of Physics, Bristol, 1994)

CHAPTER 2 STYLUS INSTRUMENTS

We shall commence in this chapter by dealing with the techniques most commonly used for roughness measurement: those based on the use of the stylus instrument. The two most natural ways to establish the roughness of a surface are to look at it and to run a finger over it. Whitehouse (1994) has pointed out that all roughness measurement techniques can be classified as analogues of one or the other of these elementary methods. The stylus instrument is the embodiment of this second way; there were estimated to be 25,000 stylus roughness measuring instruments in the U.S.A.alone (Young & Bryan 1974).

2.1. Mechanical Instruments

The principle of the phonograph or gramophone, where a sharp probe traverses a surface and transforms its minute irregularities into another form of energy, seems ideally suited in retrospect to apply to the measurement of surfaces. Strangely enough it was a generation after the invention of the phonograph before surfaces were first measured with a stylus instrument. One early instrument used an optical lever to magnify the stylus movement (Schmaltz 1936). Another amplified the vertical movement of the stylus mechanically by a system of levers until it sufficed to cause visible fluctuations in a continuous scratch on a smokedglass plate (Reason 1944) (Fig. 2.1). This had the advantage that the stylus need not move at constant speed. In another version (Abbott et al. 1938) the vertical movements of the probe were transmitted to a mirror forming part of an optical lever, and the deflections of a light beam thus amplified were recorded on a moving photographic film. An interesting variant of the mechanical stylus instrument is the Flemming integrator (Way 1969). As the stylus moves over the rough surface an ingenious mechanical arrangement sums its displacements in a downward direction only. If this sum is divided by the length of traverse it is easy to show that the result is half the mean absolute slope. The actual figure measured will depend on the stylus 11

12

Rough Surfaces

dimensions; F l e m i n g proposed the use of two styli of Merent radii to distinguish between the slopes associated with roughness and those associated with waviness, probably the first recorded recognition that mean slope is not an intrinsic property of a surface.

FI RO

HORIZONTAL MOTION O F INSTRUMENT BODY 6

L

E

A

F SPRING SMOKED GLASS

Figure 2.1. Tomlinson roughness meter (Galyer & Shotbolt 1990)

Before we leave mechanical stylus instruments mention must be made of the wall gauge (Lackenby 1962). This instrument was developed by the British Ship Research Association (BSRA) to measure the roughness of :hips' hulls. These are very much rougher than most machined surfaces and a larger instrument is consequently appropriate. The BSRA wall gauge consists of a railway about 76 cm long carrying a trolley on which the stylus and recording gear is mounted (Fig. 2.2). The stylus is a steel ball, and through an arrangement of levers its vertical motion is amplified and recorded on a moving smoked-glass microscope slide. T h s arrangement does not require the trolley to be moved at constant speed, which is just as well because it is advanced along the track manually by the operator. The slide is subsequently mounted in a projector with a cylindncal lens to increase its

13

Stylus Instruments

relative vertical magnification and measurements are made of the projected image by hand. As gauge moves 16 cm holder moves 7.6 cm

Holder carrying standard 7 . 6 m x 2 . S m

giving horizontal reduction of 10 :I

glass slide coated with colloidal

- -

\

Pair of feet

Probe with 1.5 mm dia. ball point constrained 10move

normal to track

10 cm

Direction of travel of gauge

Figure 2.2. BSRA wall gauge (Lackenby 1962)

The device is thus an ingenious compromise. The measurement and analysis are split into two separate tasks. The former can be carried out by relatively unskilled personnel and the instrument is robust enough to survive the rigorous environmental conditions of a dry dock. The latter needs skilled technical assistance but can be performed entirely in the laboratory at leisure. These are important considerations when it is borne in mind that up to 80 measurements may be needed on a single hull. An electrically recording version of this instrument has since been developed (Chuah et al. 1990).

2.2. Electrical Instruments Finally, however, the obvious step was taken and the stylus was given a transducer to convert its vertical movement into electrical oscillations. The Abbott profilometer (Abbott & Firestone 1933) ushered in a new era in surface measurement. (We shall refrain from using the word 'profilometer' hereafter, firstly because it is apparently a regstered trade-mark in the United States and thus

Rough Surfaces

14

may not be used in a generic sense, and secondly because it is bad practice to mix two classical languages.) In its original form it had all the important components which stylus instruments have embodied ever since: a pickup, driven by a gearbox, which draws the stylus over the surface at a constant speed; an electronic amplifier to boost the signal from the stylus transducer to a useful level; and a device, also driven at constant speed, for recording the amplified signal (Fig. 2.3). Pickup

Gear-box

Datum Stylus

Transducer

Amplifier

11111

1nm

1Iy”

1mm l r n

Data logger

Chart recorder

Figure 2.3. Schematic stylus instrument

The vertical range of a stylus instrument depends on the dynamic range of the transducer and can be as much as 1 mm or more. The vertical resolution is ultimately limited by background mechanical vibrations and thermal noise in the electronics; a commercial instrument is available with a claimed resolution of 2 nm (Moody 1968). At this level of sensitivity the instrument is quite an efficient seismograph and the most stringent precautions must be taken to ensure a stable thermal and mechanical environment. The story is told that during its development a mysterious transient, thought at first to be an electronic malfunction, was finally traced, after the entire electronics had been unsuccessfully rebuilt, by observing that its appearance coincided with the departure of the milk train for London from a station several miles away! The horizontal range of stylus instruments is set by the length of pickup traverse; horizontal resolution depends on stylus dimensions. The disadvantages of the stylus instrument are manifest: its bulk, its complexity, its relative fragility, its

Stylus Instruments

15

high initial cost, its limitation to a section of a surface, the necessity of a skilled operator for any measurement out of the ordinary. The single advantage which outweighs all these is the availability of an electrical signal, which can be subjected to all the conditioning processes of modem electronics to yield any desired roughness parameter, or can be recorded for display or subsequent analysis. The stylus instrument is thus by far the most popular method of surface measurement and outnumbers all other instruments combined. It is also the instrument in terms of which all national roughness standards are defined. It is therefore appropriate to discuss its component parts in more detail.

2.2.1. Stylus and Skid

In early instruments the stylus was often a phonograph needle (Abbott & Goldschmidt 1937; Barash 1963) and more recently a sewinq needle has been used as a stylus (Gray & Johnson 1972). However, phonograph needles were found to be too large and too heavily loaded, and caused unacceptable surface damage. Diamond styli are now universally employed. In many instruments they are cones of 90" included angle and tip radius 4-12 pm (Williamson 1947). However, in one of the most popular stylus instruments the stylus is a truncated pyramid. The angle between the faces is 90" and the dimensions of the rectangular flat at the tip vary; a small flat is classed as a high-resolution stylus, while an average one is about 3 pm x 8 pm (Jungles & Whitehouse 1970). The short edge is parallel to the direction of motion. Thus the stylus cannot resolve a wavelength shorter than 6 pm (see later chapters), and integrates over a narrow strip of surface 8 pm wide. The slopes of most real surfaces are so gentle that penetration of valleys is not usually a problem. However, this is not always the case. It is sometimes necessary to measure surfaces which consist effectively of more or less smooth planes containing relatively steep-sided and deep craters, such as those of wood (Elmendorf & Vaughan 1958) or machining tools (Tsao et al. 1968). More common still is the requirement to measure surfaces of abrasive composites, either coated abrasives or grinding wheels P a u l et al. 1972; Deutsch et al. 1973; Friedman et al. 1974, Fugelso & Wu 1977), where no ordinary stylus will penetrate the gaps between the individual grits. All the above authors have described instruments designed to overcome this problem by the employment of a stylus vibrated by an electrical transducer in a vertical plane with an amplitude much greater than the anticipated variation of height on the surface. It is clearly necessary that the frequency of vibration should be different from the expected

16

Rough Su$aces

frequencies of variation of height so that it can be removed from the signal by electronic filtering. The quantity actually measured by the stylus transducer is the change in the vertical separation of the stylus and the transducer. If the pickup is constrained to move in a horizontal plane or in a curve of fixed radius then the transducer will give the instantaneous height difference between the stylus and this independent datum. Such a system involves a tedious levelling procedure, particularly so at high vertical magnifications, and for many purposes it is more convenient to use a skid. The transducer senses the difference in level between the stylus and the skid and no skilled setting up is required. The skid is attached to the pickup and rests on the surface either beside or in line with the stylus (Figure 2.4). In-line skids, either in front of or behind the stylus, are the usual arrangement, and are preferred for tracing inside small bores. Straddling skids, either as buttons or as a V-shaped support member, provide guidance parallel to the axial plane of parts with curved surfaces.. Farago (1982) describes a number of detailed skid designs for various specialised inspection applications.

Figure 2.4. (a) Aligned and (b) straddling skids (Farago 1982)

2.2.2. Transducers As with the cartridge of a high-fidelity phonograph, the performance of a stylus instrument is only as good as its transducer. Some of the less expensive stylus instruments used a piezoelectric crystal as the transducer element. Changes in pressure due to stylus movement cause a small change in charge, which can then be amplified. In practice the charge leaks steadily away from the crystal at such a rate that the transducer will not correctly transmit low frequencies, resulting in a loss of accuracy at long surface wavelengths (Reason 1956).

Stylus Instruments

17

In another system, the stylus moves the anode of a triode through a flexible diaphragm, thus causing a large change in the effective electrical resistance for a comparatively small stylus displacement (Underwood & Bidwell 1953; Chinick 1968). Because of the fragility of the vacuum tube it is essential that excessive anode movement be avoided. This is achieved by coupling the stylus to the anode extension through a viscous liquid. For high-frequency small-amplitude displacements the coupling behaves as if it were rigid, but permits increasing shear as the frequency decreases, thus acting as a high-pass filter; unfortunately its transmission characteristic, being viscosity-dependent, is a function of temperature. Moving-coil transducers are sometimes used, but their output depends on the velocity of the stylus rather than its position and must be integrated to give an amplitude. As the integrating circuits are inefficient at low frequencies this also has a built-in high-pass filter (Reason 1956; Chinick 1968). A capacitance transducer has also been described (Miyazaki 1965) where the stylus vertically displaces, through a lever arm, one plate of a capacitor whose other plate is a conducting liquid. Difficulties are reported, not surprisingly, in levelling the system.

\ & Knife edge

Figure 2 . 5 . Schematic LVDT transducer

Many modern stylus instruments use a linear variable differential transformer (LVDT). In a typical modified LVDT two coils are wound on opposite arms of an E-shaped core (Fig. 2.5). An armature attached to the stylus and pivoted about the centre arm increases the air gap of one coil as it decreases that of the other, causing a differential change in inductance. This alters the output of a bridge circuit excited by an oscillator at a frequency much higher than the maximum anticipated frequency of stylus displacement. Thls camer frequency must subsequently be removed from the signal by demodulation. The advantage of this system is that it

18

Rough Surfaces

will measure low frequencies right down to a static stylus displacement (Reason 1956; Chinick 1968). Garratt (1982) describes an interferometric transducer. The stylus arm is a pivoted lever, at one end of whtch is the stylus and at the other end of which is a reflector which acts as the measurement arm of a Michelson laser interferometer. The vertical resolution is 5 nm and the range is 2 mm, giving a ratio of range to resolution of 5 x lo5.

2.2.3. Pickup

One instrument manufacturer has used an endless rubber belt driving a trolley carrying the stylus and transducer along an optically flat guideway. Another has used a linkage which in fact drives the pickup in a pair of arcs forming a very shallow letter W. This is adequate when the stylus is used with a skid, but causes some difficulty if a smooth surface is measured relative to an absolute datum. It is sometimes inconvenient to move the pickup over the surface and instead the pickup is held motionless and the test piece is moved below the stylus. Stages are commercially available for this purpose (see the discussion of 3D measurement.

Figure 2.6. Pickup speed measured as a hnction oftime for a commercial stylus instrument (Desages & Michel 1993). AC: Traverse length; AB:run-up length; BC: evaluation length

It is essential that the pickup be driven at constant horizontal speed while measuring, as the overall design of many instruments assumes that equal intervals of time correspond to equal intervals of horizontal distance. The actual length over which measurements are made, the evaluation length (BS1134, 1981), is shorter

Stylus Instruments

19

than the total length of pickup travel, the traverse length, to allow an initial run-up for pickup acceleration and a final run-out for the pickup to slow down and stop (Fig. 2.6). The commercial instrument tested in Fig. 2.6 in fact consistently slowed down by about 3% after the first quarter of the evaluation length. Very often it is necessary to measure workpieces whose surfaces are not flat. If they have a section of constant curvature this can be achieved by constraining the stylus to move in an arc of the same curvature, and a number of ingenious mechanical devices are available commercially for this purpose. If the test piece is circular or cylindrical, matters are much easier. Instruments are commercially available which will either rotate the component by rollers against a fixed stylus or will move the stylus in an arc of variable radius around the workpiece. The slower the stylus moves, the finer the detail that can be resolved (subject to the limitations of stylus size) and accessories for commercial instruments have been available which will reduce the speed to less than 5 p d s . At the lowest speeds it is apparently difficult to guarantee constant speed because of stick-slip in the translation mechanism. At the other end of the scale, the stylus cannot be traversed too fast or it will lose contact where the surface falls away steeply. The effect of this will be to skew the slope distribution, that is to record negative slopes as being more gentle than they really are. The exact proportion of slopes misrecorded depends on the interaction of the pickup dynamics and stylus load with the geometry of the particular surface being measured (discussed in detail by Whitehouse 1994). It is dlficult to avoid thls effect completely if measurements are to be made in a reasonable time, and the design of most commercial instruments is therefore a compromise. The fastest speed of traverse normally employed is about 1 mm/s; a stylus instrument which has been reported to traverse at 5 mm/s will not follow slopes steeper than about 6 degrees (Morrison 1995).

2.2.4. Output Recording

After removal of the carrier, if any, and amplification, it is necessary to transform the signal again into some form which the operator can understand. In some early instruments the amplified signal drove a loudspeaker (Harrison 1931) or an oscilloscope (Abbott & Firestone 1933). Neither of these could provide a permanent record, and it quickly became and has remained the practice to use a chart recorder. Clearly it is essential that the recorder should run at constant speed to avoid distortion of the signal, but this constant speed may be increased to increase the horizontal magnification of the recorded signal. (The same effect may

Rough Sudaces

20

of course be achieved by slowing the pickup traverse speed relative to that of the recorder.) Chart recordings of surface profiles are clear, unambiguous and selfexplanatory; the only difficulty in interpretation arises from the distorted ratio of vertical to horizontal magnification, a problem which we discuss below. Surface parameters of various kinds can be measured from the chart recordings, but this is rather slow, and would be unsuitable for a quality-control application where a large number of workpieces must be measured in a short time. Most stylus instruments, therefore, either have additional averaging circuitry of some kind which displays a selected parameter directly on a meter, or are connected to a microcomputer which performs the same function. The definitions of these parameters and the details of the procedures for calculating them are discussed in a later chapter.

2.3. Sources of Error

2.3.1. Effect of Stylus Size The stylus is not a mathematical point but an artefact of finite dimensions. This implies that the stylus must fail to follow peaks and valleys faithfully and hence must produce a distorted record of the surface. How serious is this distortion?

/--\

/

Figure 2.7. Distortion of measured profile due to f ~ t dimensions e of stylus tip (exaggerated) (Radhakrishnan 1970)

The so-called "traced profile" (IS0 3274, 1996) recorded by the stylus instrument is the locus of the centre of the stylus. If the contacting portion of the

21

Stylus Instruments

stylus is assumed to be spherical in section the effective profile will correspond to the contacting envelope (Hullproflo of the E or envelope system (von Weingraber 1957) where the radius of the rolling circle is that of the spherical portion of the stylus tip. The radius of curvature of a peak may be exaggerated, while a valley may be represented as a cusp (Fig. 2.7). To appreciate the likely magnitude of this effect it is necessary to consider more closely the geometry of the stylus. According to I S 0 3274 a stylus may have an included angle of 60' or 90' and a tip radius of curvature of 2, 5 or 10 pm. The measurement of the actual tip dimensions is extremely difficult as they approach the limit of resolution of optical techniques (Williamson 1947; Jungles & Whltehouse 1970); Williamson succeeded in measuring the tips of four conical styli and reported average radii of curvature of from 2.5 pm to 53 pm. It would seem, therefore, that a profile containing many peaks and valleys of ra&us of curvature 10 pm or less, or many slopes steeper than 45O, would be likely to be more or less badly misrepresented by a stylus instrument. Much play has been made with the problem of measuring the Caliblock and similar specimens, roughness standards on which a very nearly rectangular one-dimensional profile is etched (Reason 1951; Peres 1953), and clearly indeed such a profile can never be followed very closely by a standard stylus.

1

0.5 2.5

I

5

1

10

1

20

1

50

I

I

la, 200 Tracing stylus radius &m

Figure 2.8. Effect of stylus tip radius on measured roughness for various machined surfaces (Radhakrishnan 1970): (1) planed (2) electro-eroded (3) milled (4) ground; (5) electrochemically sunk, ( 6 )honed

22

Rough Surfaces

In measurements of real surfaces, however, this difficulty is largely illusory, as their slopes are for the most part very gentle at the scale on which the stylus measures them. This is confirmed by the experimental findings of a number of workers (Williamson 1947; Radhakrishnan 1970) that styli of standard dimensions do not significantly misrepresent the average roughness of the surface (Fig. 2.8). Whitehouse (1974) has reached the same conclusion by a different stochastic argument. Arguments which rely on a representation of the surface as a single pure sinusoid (Nakamura 1966), though they may reach similar conclusions, are to be avoided. The misconception has arisen from the way in which profiles are commonly presented. A slope of 1" is almost imperceptible to the human eye, and the instrument manufacturers have therefore found it convenient to exaggerate the vertical magnification over the horizontal on their chart recorders. This has given generations of metrologists a totally false impression of surface microgeometry. The contact of two rough surfaces, far from resembling Bowden's famous analogy of "Austria turned upside down on top of Switzerland" (Bowden & Tabor 1950) more closely resembles Iowa on top of the Netherlands (Thomas 1973). To emphasize this point Reason (1944) showed a conventional chart recording of a ground surface with the vertical scale exaggerated 35:l over the horizontal; the stylus, distorted correspondingly (Fig. 2.9) appears now as a mere sliver. On the same figure he drew the same recording at a 1: 1 ratio. As he himself remarks, it is difficult to reconcile the two representations even with the aid of fiducial marks. There is no room here to reproduce the whole of Reasonk figure, which unfolded from the original paper to a length of a metre or so!

Y

25:1

3 -

X Y

Figure 2.9. Effect of horizontal compression of chart recording on profile presentation (adapted from Reason 1944):(a) true appearance of section XY'; (b) representation on chart recording

Stylus Instruments

23

2.3.2. Effect of Stylus Load As the dimensions of the stylus are finite, so also is the load on it. Although the load is small, 0.75 mN according to I S 0 3274, the area of contact is also so small

that the local pressure may be sufficiently high to cause significant local elastic downward deformation of the surface being measured. In some cases the local pressure may exceed the flow pressure of the material and plastic deformation, i.e. irreversible damage, of the surface may result. It is fairly easy to calculate the elastic behaviour of a surface under a chiselshaped stylus. The average vertical deflection 6 of a homogeneous isotropic elastic half-space of Hertzian modulus E' and Poisson's ratio v by a rigid indenter of rectangular cross-section ab under load W is (Timoshenko & Goodier 1951): S

=

mW(l-9)/Erd(ab)

where m is a dimensionless constant whose numerical value is a function of ah. Taking m = 0.9, W = 1 mN, v = 0.33, E' = 2 x lo5 N/mm2 for steel gives 6 = 0.83 nm. Clearly there is no danger here of appreciable error in measurements on any metal. A calculation for the spherical stylus tip specified in I S 0 3274 (1996) would gwe a similar result. The problem with metals at least, then, if any, is not elastic but plastic deformation. This has been investigated by Quiney et al. (1967), Tucker and Meyerhoff (1969) and Guerrero and Black (1972). Quiney et al. were able to make a scratch about 1.7 pm deep on an aluminium surface by using a stylus force about 10 times higher than the recommended standard. Tucker and Meyerhoff presented scanning electron micrographs of a lead surface damaged by a stylus, but admitted that no scratches could be found on harder materials. Guerrero and Black presented a number of scanning electron micrographs of so-called stylus damage, and calculated that stylus scratches on a steel surface were as much as 50 nm deep; however, their assumptions concerning stylus tip geometry are rather questionable. The trouble seems to be that the stylus generally does make a visible scratch, and the existence of this scratch is held to be prima facie evidence of unacceptable damage. Here the key word is "unacceptable". If we can show, for instance, that the surface is everywhere deformed by the same amount, the output of the instrument will then be a true profile displaced downward by a constant distance. Reason et al. (1944) traversed a profile with a stylus load of 0.06 mN, repeated the measurement with an increased load of 0.8 mN and then returned to 0.06 mN for a third traverse; the profiles were nearly identical (Fig. 2.10). Schwartz and Brown

24

Rough Surfaces

(1966) found that the stylus measurements of a step in a silver film on a glass substrate agreed with interferometric measurements to within 24 nm, while Estill and Moody (1966) found no more than 43 nm deformation even on a soft gold film. Williamson (1968), in a series of careful experiments, could find no evidence of the information from stylus measurements being affected by plastic deformation. As he remarked elsewhere, a bulldozer traversing a range of hills would leave a scar visible from many miles up, but a recording barometer carried on the vehicle would return a profile of the topography accurate enough for most practical purposes.

Figure 2.10. Effect of stylus load (Reason 1944): (a) profile measured at 6 rng load; (b) relocated profile at 80 mg load (c) relocated profile at 6 mg load again

2.3.3. Other Sources of Error

A question was raised above concerning the dynamic response of the stylus: whether there is any possibility of it losing contact with steep reverse slopes of the surface as a consequence of the speed of traverse. Again this is to overestimate the average steepness of real surfaces. Nakamura (1966) and Damir (1973) have considered the effect of stylus geometry on its dynamic response, but the input surface models in both cases are somewhat unrealistic. Nakamura's results, however, indicate that with typical surface conditions, and with styli dimensions as quoted in the international standards, the errors incurred are negligible. Funck et al. (1992) found that speed of traverse significantly affected stylus measurements of roughness on wood surfaces, but it is not clear whether these effects were due to the timedependent mechanical properties of wood. Another possibility of error lies in the lateral deflection of the stylus by asperities. Verkerk et al. (1978) recommend that the ratio of axial to lateral stiffness should be less than 0.00 I . AguIIo and Pages-Fita (I 974) have shown (Fig. 2.11) that lateral deflection can amount to as much as 1 pm between extremes

Stylus Instruments

25

when traversing a rough surface. However, the RMS excursion is more like 0.3 pm, and in any case, as we have already seen, a typical stylus is in effect integrating over a strip 8 pm wide.

Figure 2.1 1. Effect of lateral deflection of stylus (Agullo & Pages-Fita 1974): (a) artificial two-dimensional square-wave surface traversed at an angle to the lay; (bJ and (c) typical manufactured surfaces of different roughness

Rough Su$aces

26

It has been claimed that the skid itself can cause damage to the surface. Tucker and Meyerhoff (1969) observed deformation of soft surfaces such as lead and niobium by a skid. Here the same argument applies as was used in the case of stylus deformation: will it actually affect the measurements? According to Reason et al. (1944), the difference in the deformation of a peak on a steel surface relative to that of a hollow amounts to about 40 nm. It seems unlikely that this will be an important source of error on hard surfaces.

“1. b

a a

Reciprocalwavelength

Figure 2.12. Effect of skid (Nara 1966):(a) profile as seen by (top to bottom) stylus with absolute datum; stylus with skid; skid with absolute datum: (b) power spectra of (a): solid line. stylus with absolute datum; broken line: stylus with skid

Stylus Instruments

27

More importantly, the skid acts as a mechanical high-pass filter. This has two consequences. Firstly, information about longer wavelengths is lost; this is irremediable and if these wavelengths are deemed to be relevant to the problem under investigation then a skid must not be used. Secondly, the filter introduces a phase lag meason et al. 1944) which might be supposed to distort the appearance of the surface. Experiment suggests, however, that this distortion is not obtrusive (Nara 1966) (Fig. 2.12a). It appears also that the mechanical filter embodied by the skid has quite a sharp cut-off (Fig. 2.12b) and that the power spectrum, which of course contains no phase information, is relatively unaffected at shorter wavelengths. Ishigaki & Kawaguchi (198 1) conclude that varying the separation &stance of skid and stylus has little effect. The question of damage to soft surfaces has been discussed above, but what of measurements on surfaces whch easily yield elastically? Elastomers are widely used in engineering, particularly as elements in static or dynamic seals. The surface finish of the metal elements is known to have an important effect on sealing properties, and it seems reasonable to assume that the finish of the elastomer will also play a part. To assume that all the asperities on the elastomer are somehow squashed flat is to beg the question; no matter how compliant the elastomer or heavy the load, a surface wavelength will exist below which the elastomer will not conform. Calculations such as those described above suggest that a stylus under its standard load will deform the surface of an elastomer by more than 100 pm. Of course it does not follow that measurements are therefore impossible; if every element of the surface is displaced vertically by exactly the same amount then we will still record a true profile. However, as most commercial elastomers are composite materials this may be rather a severe requirement.

I mm

Figure 2.13. Stylus measurement of a compliant surface (Thomas et al. 1975): (a) profile of elastomer cooled below its glass transitiontemperature; (b) a subsequent relocated measurement at room temperature

28

Rough &$aces

To investigate compliant yielding under the stylus, measurements were made on an elastomer at two different temperatures using a relocation table (Thomas et al. 1975). The elastomer was first frozen below its glass transition temperature by a stream of evaporating liquid nitrogen. In t h s condition it is no more compliant than steel. When it had warmed up to room temperature it was measured again (Fig. 2.13), and no significant differences in the two profiles were apparent. The explanation of this remarkable and useful result is not clear, but there is some evidence that a thin surface layer of the elastomer may be much less compliant than the bulk material.

2.4. Calibration and Standards

Because the stylus instrument was the earliest roughness mzasuring instrument to achieve general acceptance, and because it is still so widely used, roughness standards are still written largely with stylus instruments in mind. Production instruments are usually calibrated from secondary calibration specimens, of which IS0 5436 (1985) distinguishes four main types. Type A, used for checking vertical magnification, has wide grooves of a known depth. Type B, for checking the condition of the stylus tip, has a series of narrow grooves of various depths and widths. Type C, for checking parameter meters, has repetitive grooves of sinusoidal or triangular section. Type D has a pseudo-random profile which extends the whole width of the standard to provide a more realistic, but less accurate, overall system check. Type C specimens of triangular section were prodused in the U.S.A. by General Motors under the trade name Cali-Block (Young & Scire 1972) with an included angle of 150 degrees. The production of sinusoidal Type C specimens has been described by Sharman (1967/8) and by Teague et al. (1982), who found that the rigorous NIST specification could best be satisfied by diamond turning. Square-wave sections have been proposed for Type C specimens (Berger 1988) but interaction with stylus geometry makes these prone to error (Peres 1953). The best-known Type D specimens are those produced at the PhysikalischeTechnische Bundesanstalt by Hasing (1965). These have a measuring area of random profiles, obtained by grinding, in the direction of traverse, which repeat every 4 mm, and cover a range of roughnesses from 1.5 pm down to 0.15 pm.. Song (1988) has improved on the PTB design by adding a smooth reference surface at each end of the traverse to provide a datum for the skid, and has extended the range of roughness down to 12 nm (Fig. 2.14).

Stylus Instruments

29

Figure 2.14. Type D modified PTB roughness calibration specimen with 8 consecutive identical profiles (Song1988)

But how are these secondaq calibration specimens to be calibrated themselves? Small height increments can be measured directly by interferometry (Spragg 1967/8, IS0 5436). In most instances, however, the most suitable instrument for the purpose is another stylus instrument, which itself must be traceably calibrated, that is there must be an unbroken and documented chain of calibration from a primary standard of length, as required by IS0 9000 (1987). For the static deflection of the stylus and transducer, traceability may be provided by a series of gauge blocks arranged in steps ( I S 0 5436), which serves to check the linearity of the transducer as well as its sensitivity. If the steps are too coarse, they can be scaled down by a reducing lever (Spragg 1967/8). Dynamic calibration of the stylus instrument is often effected by a vibrator which can be driven at variable frequency (Van Hasselt & de Bruin 1962/3, Parkes 1969, Bendeli et al. 1974); the traceability of such vibrators is discussed by Barash & Reznikov (1983).

2.5. References

Abbott, E. J. and Firestone, F. A,, "Specifying surface quality", Mech. Engng., 55, 569-572 ( 1 933).

30

Rough Sudaces

Abbott, E. J., Bousky, S. and Williamson, D. E., "The profilometer", Mech. Engng. 60, 205-216 (1938). Abbott, E. J. and Goldschmidt, E.," Surface quality", Mech. Engng., 59, pp. 8 13-25. ( 1937). Agullo, J. B. and Pages-Fita, J., "Performance analysis of the stylus technique of surface roughness assessment: a random field approach", Proc. 15th Int. Machine Tool Des. & Res. Con$, 349-362 (Birmingham University, 1974). Barash, M. M., "Measuring the finish of rough surfaces", Int. J. Mach. Tool Des. Res., 3,97-I00 (1963). Barash, V. Y. and A. L. Reznikov, "Standard vibrator in metrological certification of contact methods of roughness measurement", Measurement Technology, 26, 658-661 (1983) Baul, R. M., Graham, D. and Scott, W., "Characterization of the working surface of abrasive wheels", Tribology, 2, 169-176 (1972). Bendeli, A,, Duruz, J. and Thwaite, E.G., "A surface simulator for the precise calibration of surface roughness measuring equipment", Metrologia, 10, 137-143 (1974). Berger, J., "A new surface roughness standard fabricated using silicon technology", Surface Topography, 1, 4 1-47 (1988) Bowden, F. P. and Tabor, D., The friction and lubrication of solids Part 1, (Oxford University Press, 1950). BS 1134, "Assessment of surface texture Part 1. Methods and instrumentation" (British Standards Institution, London, 1988) Chinick H. P., "LVDT puts precision in surface texture measurement", Cutting Tool Engng., 20, 13 (1968). Chuah, K. B., Dey, S. K., Thomas, T. R., Townsin, R. L., "A digital hull roughness analyser", Int. Workshop on Marine Roughness & Drag (RINA, London, 1990) Damir, M. N. H., "Error in measurement due to stylus kinematics", Wear, 26, 219-227 (1973). Desages, F. and Michel, O., "Calibration of a 3-D surface roughness measuring device", Production Engng. Dept Report (Chalmers University, Goteborg, 1993) Deutsch, S. J., Wu, S. M. and Straklowski, C. M., "A new irregular surface measuring system", Int. .J. Mach. Tool Des. & Res., 13,29-42 (1973). Elmendorf, A. and Vaughan, T. W., "A survey of methods of measuring smoothness of wood", Forest Products J., 8, 275-82 (1958).

Stylus Instruments

31

Estill, W. B. and Moody, J. C . , "Deformation caused by stylus tracking on thin gold film", Z.S.A. Trans., 5, 373-378 (1966). Farago, F. T., Handbook of dimensional measurement 2e, (Industrial Press, New York, 1982) Friedman, M. Y., Wu, S. M., and Suratkar, P. T., "Determination of geometric properties of coated abrasive cutting edges", Trans. A.S.hLE.: J .Engng .Ind., 96B, 1239-1244 (1974). Fugelso, M., Wu, S. M., "Digital oscillating stylus profile measuring device", Irzt. J. Mach. Tool Des Res 17,191-195 (1977) Funck, J. W.; Forrer, J. B.; Butler, D. A,; Brunner, C. C.; Maristany, A. G., "Measuring surface roughness on wood: a comparison of laser-scatter and stylustracing approaches", Proc. SPIE 1821, 173-184 (1993) Galyer, J. F. W., and Shotbolt, C . R., Metrologyfor engineers 5e (Cassell, London, 1990) Garratt, J. D., "A new stylus instruent with a wide dynamic range for use in surface metrology", Prec. Engng. 4, 145-151 (1982) Gray, G. G. and Johnson, K. L., "The dynamic response of elastic bodies in rolling contact to random roughness of their surfaces", J. Sound & Vibration, 22, 323-342 (1972). Guerrero, J. L. and Black, J. T., "Stylus tracer resolution and surface damage as determined by scanning electron microscopy", Trans. A.S.M.E. J . Eng. Ind., 94B, 1087-1093 (1972). Hamson, R. E. W., "A survey of surface quality standards and tolerance costs based on 1929-1930 Precision-Grinding practice", Trans. ASME 53, 11-25 (193 1). Hasing, J., "Herstellung und Eigenschaften von Referenznormalen fur das Einstellen von Oberflachenmefigeraten",Werkstattstechnik 55, 380-382 (1965) Ishigaki, H. and I. Kawaguchi, "Effect of a skid on the accuracy of measuring surface roughness", Wear, 68, 203-21 1 (1981). I S 0 3274, "Geometric product specifications - surface texture: profile method nominal characteristics of contact (stylus) instruments" (International Organisation for Standardization, Geneva, 1996) I S 0 5436, "Calibration of stylus instruments" (International Organisation for Standardization, Geneva, 1985) IS0 9000, "Quality management and quality assurance standards" (International Organisation for Standardization, Geneva, 1987) Jungles, J. and Whitehouse, D. J., "An investigation of the shape and dimensions of some diamond styli", J. Phys: Sci. Instrum., 3E,437-440 (1970).

32

Rough Suvfaces

Lackenby, H., "The resistance of ships, with special reference to skin friction and surface condition", Proc. I. Mech. E., 176, 981-1014 (1962). Miyazaki, K., "Electronic method, based on the surface of a liquid, for measuring flatness", Microtechnic, 19,74-76 (1965). Moody, J. C., "Measurement of ultrafine surface finishes", I.S.A. Trans. 7, 67-7 1 (1968). Morrison, E., "A prototype scanning stylus profilometer for rapid measurement of small surface areas", Int. J. Mach. Tools Manufact. 35, 325-331 (1995) Nakamura, T., "On deformation of surface roughness curves caused by finite radius of stylus and tilting of stylus holder arm", Bull. Jap. Soc. Precision Engng., 1, 240-248 (1966). Nara, J., "On CLA value obtained with direct reading surface roughness testers - effects of skid and high pass filter", Bull. Jap. Soc. Precision Engng., 1, 263-273 (1966). Parkes, D. H., "Calibration, certification and traceability of surface roughness measuring equipment", A.S. T.M.E. Tech. Paper IQ69-505 (1969). Peres, N. J. C., "Geometrical considerations arising from the use of square wave calibration standards of surface finish", Aust. J. Appl. Phys., 4, 380-388 (1953). Quiney, R. G., Austin, F. R. and Sargent, L. B., "The neasurement of surface rougbness and profiles on metals", A.S.L.E. Trans. 10, 193-202 (1967). Radhakrishnan, V., "Effect of stylus radius on the roughness values measured with tracing stylus instruments", Wear, 16, 325-335 (1970). Reason, R. E., "Surface finish and its measurement", J. Inst. Prod. Engrs ., 23, 347-372 (1944). Reason, R. E., "Surface finish", Australasian Engr., 44, 48-64 (195 1). Reason, R. E., Hopkins, M. R. and Garrod, R. I., Report on the measurement of surface finish by stylus methods", (Taylor Hobson, Leicester, 1944) Reason, R. E., "Significance and measurement of surface finish part 2: how transducers affect instrument performance; how to select proper cutoff values", Grinding &Finishing, 2, 32-36 and 41 (1956). Sayles, R. S., Thomas, T. R., Anderson, J., Haslock, I. and Unsworth, A,, "Measurement of the surface microgeometry of articular cartilage", J. Biomechanics, 12, 257-267 (1979) Schmaltz, G., Technische Oberfliichenkunde (Springer-Verlag, Berlin, 1936)

Stylus Instruments

33

Schwartz, N. and Brown, R., "A stylus method for evaluating the thickness of thin films and substrate surface roughness", Trans. of the 8th National Vacuum Symp., 836-845 (1966). Sharman, H. B., "Calibration of surface texture measuring instruments", Proc. I. Mech. E., 182, Part 3K, 319-326 (1967/68). Song, J. F., "Random profile precision roughness calibration specimens", Surface Topography, 1, 29-40 (1988) Spragg, R. C., "Accurate calibration of surface texture and roundness measuring instruments", Proc. I. Mech. E., 182, Part 3K, 397-405 (1967/68). Teague, E. C., Scire, F. E., Vorburger, T. V., "Sinusoidal profile precision roughness specimens", Wear 83, 61-73 (1982) Thomas, T. R., "Influence of roughness on the deformation of metal surfaces in static contact", Proc. 6th. Int .Conf on Fluid Sealing, B3, 33-48 (BHRA Fluid Engineering, Cranfield, 1973). Thomas, T. R., Holmes, C. F., McAdams, H. T. and Bernard, J. C., "Surface features influencing the effectiveness of lip seals: a pattern - recognition approach", S.M.E. Paper IQ75-128, (1975). Timoshenko, S., and Goodier, J. N., Theory of elasticity (McGraw-Hill, New York, 1951) Tsao, K. C., Husein, A. B. and Wu, S. M., "Cutting tool crater wear measurement by the lapping-comparator technique", Znt .J. Mach. Tool Des .Res., 8, 15-26 (1968). Tucker, R. C. and Meyerhoff, R. W., "An SEM study of surface roughness measurement", Proc. 2nd Annual Scanning Electron Microscopy Symp., 389-396 (Illinois Inst. of Technol., Chicago, 1969). Underwood, A. F. and Bidwell, J.B., "New instrument for roughness measurement", Mach. & Tool Blue Book, 49, 202-215 (1953). Van Hasselt, R., and de Bruin, W., "Comparative investigation of industrial surface-roughness measuring instruments", Ann. CIRP 11, 193 (1962/3) Verkerk, J.; Orelio, J. M. B.; Willemse, H. R., "Ratio of axial to lateral stiffness, a quality parameter for stylus surface profile traciqg instruments", Int J Mach Tool DesRes 18, 107-116 (1978) Von Weingraber, H., "Suitability of the envelope line as a reference standard for measuring roughness", Microtecnic, 11,6-17 (1957) . Way, S., "Description and observation of metal surfaces", Proc. Con$ on Friction & Surface Finish, 2e, 44-75 (MIT, Cambridge, 1969). Whitehouse, D. J., Handbook of surface metrology (Institute of Physics, Bristol, 1994)

34

Rough Surfaces

Whitehouse, D. J., "Theoretical analysis of stylus integration", Ann. C.I.R.P. 23, 81-82 (1974). Williamson, D. E., "Tracer-point sharpness as affecting roughness measurements", Trans. A.S.M.E., 69, 3 19-323 (1947). Williamson, J. B. P., "Topography of solid surfaces", in Ku P. M. ed., Interdisciplinary approach to friction and wear, SP-181, 85-142 (NASA, Washington, 1968). Young, R. D. and Bryan, J. B., "The role of NE3P in the US National Measurement System for surface finish", Ann. C.I.R.P., 23, 183-184, (1974) Young, R. D. and Scire, F. E., "Precision reference specimens of surface roughness: Some characteristics of the Cali-Block", J .Res .Nut .Bur .Stand., C76C, 2 1-23 (1 972).

CHAPTER 3

OPTICAL INSTRUMENTS

When electromagnetic radiation is incident on a rough sudace a proportion of its energy, depending on the local physical properties of the surface, will be reflected. The reflected beam will carry information about the roughness on which the design of an instrument may be based. This information may appear in several different ways. The radiation may be reflected either specularly or diffusely or both (Fig. 3.1). Reflection is totally specular when the whole energy in the incident beam obeys Snell's law, that is, the angle of reflection is equal to the angle of incidence, and a surface which reflects radiation in this manner is said to be smooth. Reflection is totally hffuse when the energy in the incident beam is distributed as the cosine of the angle of reflection (Lambert's law). In practice, matters are not as simple as this. Reflections from most real surfaces are neither completely specular nor completely diffuse. Clearly the relationship between the wavelength of radiation and the texture of the surface will affect the physics of reflection; thus a surface which is smooth to radiation of one wavelength may behave as if it were rough to radiation of a different wavelength (Ogilvy 1991).

Figure 3.1. Modes of reflection of electromagnetic radiation from a solid surface (Keller 1967/8). (a) Combined specular and diffuse; @) specular only; (c) diffuse only.

35

36

Rough Surfaces

The angular arc through which reflected energy is scattered, and the proportion of specular to diffuse reflection, both depend on the surface roughness. Instruments which measure these angles and ratios directly are glossmeters or scatterometers. Other instruments may extract more detailed roughness information by further optical processing. A general account of optical roughness measuring techniques is given by Bennett & Mattsson (1989). Reviews and comparisons of optical roughness measurement techniques have been made by Vorburger and his co-workers at NIST (Young et al. 1980, Teague et al. 1981, Vorburger 1992). Vorburger & Teague (1981) give more than 200 references to optical work. There is also a lengthy discussion of optical techniques in Whitehouse (1994). The following summary relies heavily on the above accounts. We will follow Vorburger & Teague (1981) in dividing optical techniques into profiling and parametric. Profiling techniques are associated with specular reflection, parametric techniques mainly with diffuse reflection.

3.1. Profiling Techniques

3.1.1. Optical Sections

In the light-section microscope the image of a slit is thrown on to the surface at an incident angle of 45" and viewed by a microscope objective at a reflected angle of 45" (Fig. 3.2). The reflected image will appear as a straight line if the surface is smooth, and as an undulating line if the surface is rough. The relative vertical magnification of the profile is the cosecant of the angle of incidence, in this case 1.4. Resolution is about 0.5 pm and it is quite easy to measure peak-to-valley roughness. Light-section microscopes have been commercially available. The image need not be of a slit; a straight-edge, such as a razor blade, will suffice as object (Kayser 1943; Way 1969). Shaw and Peklenik (1963) used a variation of this technique to measure the roughness of a razor-blade edge by projecting its image on to an inclined screen. They reported discrepancies in the apparent magnification which they attributed to the finite thickness of the razor blades; this agrees with the suggestions made above concerning the integrating effect of profile width. In a later development, Howes (1974) modified the lightsection technique to observe the virtual image of the slit, from which local slopes

37

Optical Instruments

‘f. I

lnm

lprn

Imm l m

Figure 3.2. Principle of light-section microscope (Farago 1982).

and curvatures can be measured. The technique can be applied to surfaces too rough for the standard light-section microscope. Johnson et al. (1993) measured terrain roughness by projecting a bar of light from a flash gun onto a rough surface and photographing its image at an oblique angle. The photographic negative was digitized back in the laboratory. Horizontal and vertical resolutions were said to be 1 mm and 2 mm respectively; horizontal and vertical ranges were 0.5 m and 1 m respectively. A similar system, but using a video camera for data acquisition, has been described by Davies et al. (1994).

3.1.2. Optical Probes

One possible method of optical measurement is simply to use the light beam as a non-contacting stylus for profile measurement. The most straightforward method is to detect the change in the angle of specular reflection as the surface is translated under an incident beam (Ramgulam et al. 1993). While adequate for its intended purpose, the measurement of textile roughness, the lateral and vertical resolutions of 25 pm and 10 pm respectively make such a system unsuitable for finer surfaces. A number of more sophisticated variations exist. In constructing their Stedman dagrams below, it will be assumed unless otherwise stated that the range in the plane of the surface is 100 mm for a generic translation stage. Whitehouse (1994) summarises the designs of several other optical profilers in addition to the ones described below. The first method employs the so-called Foucault knife-edge test (Dupuy 1967/68). An image of a spot is formed on the surface (Fig. 3.3). By using a halfsilvered mirror the image of the spot can be imaged itself to the knife-edge. A field lens is placed here to image the objective lens on to a screen, I. If the conjugates of the lens, 0, are at the knife-edge and surface respectively, a uniform disc appears

Rough Sur&aces

38

on the screen. If the surface is moved away the knife-edge intercepts the rays of light in a different way resulting in a non-uniformity of light on the screen. This non-uniformity is a measure of the distance moved by the surface. Electrically maintaining the focus by means of cells A and B (which produce signals to move the objective or knife-edge) and monitoring the movement gives the required transducer effect. Vertical and horizontal resolutions of 0.01 pm and 0.5 pm respectively, and a vertical range of 60 pm, are claimed for this instrument, a specification which compares favourably with that of many stylus instruments. Whitehouse (1975) has pointed out, however, that the system has a poor frequency response and is unduly sensitive to tilt. A development of this system described by Thwaite (1977) is claimed to have vertical and horizontal resolutions of 1 nm and 1 pm respectively

1-1

mechanism

paition I

Polirioa 2

P d o n3

Figure 3.3. Dupuy optical probe (Whitehouse 1975): (a)Schematic layout; (b, simplified view; (c) view of objective.

Optical Instruments

39

Another optical probe utilizing a somewhat similar principle was described by Keller (1967/68). A spot object is imaged on to the surface and the diffuse reflection at some arbitrary angle, which can be varied to increase sensitivity, impinges on a pair of photoresistors differentially connected to give a null signal when the spot is in focus. The out-of-balance signal drives a servo system which alters the height of the probe until focus is again achieved. The vertical sensitivity and range are respectively 2.5 pm and several millimemes. The horizontal resolution, however, is only about 5 mm, as the probe is designed primarily for the measurement of errors of form. Also intended for form measurement is the system described by Ennos & Virdee (1986). Autocollimation of a reflected laser beam measures the local slopes, which must then be integrated to obtain a height profile. The lateral range is 15 mm but the lateral resolution is only 0.1 mm, and a single slope measurement takes 3 s. The vertical range is 0.8 pm and the vertical resolution is 0.1 nm. A system based on Nomarski microscopy is described by Bristow (1988). A laser beam passes to a translation stage which holds two mirrors (Fig.3.4). The mirrors are in a pentaprism arrangement so that small mechanical motions of the stage will not affect the 90' turning angle of the beam. The beam then passes

I

Figure 3.4. Long-pathlength optical profiler (Bristow 1988)

through a Nomarski (modified Wollaston) prism which shears it into two orthogonally polarized components. The objective focusses both beams onto the

40

Rough Surfaces

surface, the two beams separated by about a quarter of the focal spot diameter. After reflection at the surface the beams spatially recombine at the Nomarski prism retaining their polarization identities as they pass to the turning mirrors and the non-polarizing beamsplitter. The beams are finally split info their respective components by the polarizing beamsplitter and directed to either of two detectors. The surface height difference is related to the phase difference between the two beams focussed on the surface and is proportional to the voltage difference between the two detectors. The spot diameter on the surface ranges between 1 and 1.8 pm in diameter depending on the choice of the objective. Translation of the turning mirrors causes the focal spots to scan across the surface, with a maximum scan length of 100 mm. The surface slope is calculated at each point, sampled 1 pm apart, and the profile is calculated by integrating the slope data. A vertical resolution of 0.025 nm and range of 2 pm is claimed. W t R DIODE

lrn,

, ,

,

,

, ,

,

,

T L C O U l MATOR

FOCUS1NC ENS CLASS C U T E

SRCIMEN

Figure 3.5. CD player as an optical profiler (Sayles et al. 1988).

The sensor of a compact disc player is a kind of optical stylus whose operating envelope overlaps with the characteristics of machined surfaces, and it is not surprising that a number of systems have been based on its focus-detection principle. Wehbi & Roques-Carmes (1986) reported a system using both white light and laser sources which split the reflected beam so that it fell on two detectors, the ratio of their outputs being proportional to the vertical distance by

41

Optical Instruments

which the local surface was out of focus. The principle of operation was demonstrated successfully but the performance of the actual system was too poor for a practical instrument. All focus-detection systems seem to suffer from a problem in coping with sudden sharp surface discontinuities. Sayles et al. (1988) adapted an actual CD reader, in which the out-of-balance signal from the two detectors servos the moveable objective back into focus (Fig. 3.5). They extended the vertical range to 30 mm by axially displacing the light source with a stepping motor. Vertical and horizontal resolutions of 0.1 pm and 1 pm were reported. The system worked well enough at long surface wavelengths, but spurious short wavelengths were generated by inappropriate damping of the servo. In a commercial realisation of the focus-detection principle (Brown 1995), the vertical displacement of the objective is measured independently (Fig. 3.6). The light source is an infrared laser diode, and a separate optical system permits simultaneous viewing of the workpiece for setting-up purposes. Vertical range and resolution are said to be 0.1 mm and 6 nm respectively.

1 analogue output

7,

UBC14 Controller

1

(1

F------

-'

2 ! ! 3

1: Laser diode 2: Prismwithbeam splitter 3: IRmirror 4: Window 5: Photodiodes 6: Leafspring 7: Coil 8: Magnet 9: Collimator lens 10: Objective 11: Tube 12: Light barrier measurement system 13: Test surface 14: PC control card 15: Microscopewith illumination 16: CCD camera

'4 -I 1nm

13

Figure 3.6. UBM optical profiler (Brown 1995).

u inm Im lpm

Imm

42

Rough S u ~ a c e s

The confocal microscope (Hamilton & Wilson 1982) is a focus detector of sorts. A pinhole is interposed in the detector path so that the axial position of sharpest focus is also the position of maximum intensity (Fig. 3.7). The distance between the workpiece and objective is varied incrementally and the workpiece is scanned at each axial position. By recordmg the axial position of greatest intensity for each pixel, a composite picture of great depth of field can be built up. In one realisation, the vertical range and resolution are 600 pm and 10 nm respectively, and the horizontal range and resolution are 0.4 mm and 0.25 pm respectively. Confocal microscopy was not developed primarily for roughness measurement but has been adapted for this purpose (Lange et al. 1993, Sandoz et al. 1996). REAL TIME IMAGE

FOCUS PLANE SAMPLE

[b)

\ PIN HOLE DETECTOR

Figure 3.7. Confocal microscope: (a) schematic @) construction of image by stacking sections

In the polarizing interferometer of Downs et a1 (1985, 1989), which needs no independent reference, a birefringent lens preceding the microscope objective splits the light into two polarisation components (Fig. 3.8). One component, the probe beam, is focussed on the surface, and the other polarisation component, the reference beam, is unfocussed. The phase difference between these two components, the probe beam and the area-averaging reference beam, yields the surface profile. The bandwidth is limited between the focussed spot size of 1 pm

Optical Instruments

43

Beam exoonder

Polorizer

L-c

Non-polar~zmg

beam solltter

Objective I m c I.zl.1

1 ,.

'\

/ I

2

Error

:eh:s,ve

slgna'

detector

Lens vibrator

Figure 3.8. Downs polarizing interferometer (Whitehouse 1994)

and the reference spot size of 10 pm, but the vertical resolution is an impressive 0.05 nm. Another polarising instrument has been described by Sommargren (198 la, b). A heterodyne laser beam is divided by a Wollaston prism into its two polarisation components which are focussed at different places on the workpiece. (Fig. 3.9). The reflected beams are then recombined at the prism. The reference beam is coincident with the axis of rotation of the workpiece, and the measuring beam traces a circular path on the surface as the workpiece is rotated. The resulting surface profiles of about 1 mm circumference are defined by the phase difference between the probe beam at the traversing point and the stationary reference beam. Vertical range and resolution are 0.5 pm and 0.1 nm respectively. Olsen and Adams (1970) described an instrument based on a rather different principle for measuring the profiles of ocean waves from a low-flying aircraft. An amplitude-modulated laser beam is reflected from the surface at normal incidence. Changes in height will produce a path difference causing a phase shift between the transmitted and reflected signals. The vertical resolution and range are 15 mm and 3 m respectively, and the horizontal resolution is 18 mm, but the effective signal-

44

Rough Surfaces

i -

Ywoble

-

- 7

:..-.-J

neutral fblter

+ .c-

I

I

.

_]

/9

//

Rotoiable A 1 2 Dlate Fmed M i plote

Reterenre poth

-

Laser

to-noise ratio is only about 20: 1. The reference datum is the time-averaged height of the aircraft, in effect a high-pass filter. An elegant application of the phase-measuring principle has been reported by Pettigrew and Hancock (1978). A laser resonating simultaneously in several different modes is employed as the light source for a Michelson interferometer. A phase detector is used which is sensitive to the beat frequency between any two of these modes. The effective wavelength is then very much longer than either of the two beating wavelengths, giving a vertical range of up to 24 +m for a phase change of 2n. The spot dameter is only 2.5 pm. The vertical resolution is not quoted but appears to be of the order of 10 nm.

3.1.3. Interferometers

The distinction between profilers and interferometers is to some extent arbitrary. A number of the instruments in the preceding section are based on an interference

45

Optical Instruments

principle, while several of the interferometer systems described below are referred to by their designers as profilers. Many beautiful interferograms of rough surfaces were produced by Tolansky (1960, 1970a, b), but the usefulness of interferometry for roughness measurement was originally limited by several factors. In the absence of coherent light sources it was difficult to obtain fringes of sufficient sharpness and contrast. If the amplitudes of surface roughness were greater than the wavelength of the incident light, there was no easy way of distinguishing between fringes of different order. Finally, there was no convenient way of converting the surface map of the interferogram into spot heights suitable for quantitative analysis. These difficulties are confronted in a phase-shifting interferometer originally conceived by Wyant and his co-workers (1986). Fringes from a Michelson, Linnik or Mireau interferometer, depending on the required magnification, are imaged onto a charge-coupled diode and the resulting intensity variations are stored in a computer (Fig. 3.10). A slightly different design (Biegen & Smythe 1988) uses a Fizeau interferometer. The entire interferometer optics are shifted axially relative to the workpiece by a piezoelectric transducer to change the optical path, giving a different set of fringes. Three such sets of measurements give enough recorded information to solve the intensity equations point by point for local heights.

Q Reid

+piece

SbD

PZT transducer (microprocessorcontrolled

Mhu mi

chrance wrtace Minu hterfemmete amspliier plate

...._...:.

::

.'..I

;

fbrtsurtsce

Figure 3.10. Phase-shft interferometer (Creath 1987).

46

Rough SurJaces

The vertical resolution is about 0.1 nm, but early realisations of this principle were limited in vertical range to half the wavelength of the illuminating light, say about 0.3 pm (Bennett & Mattsson 1989). Combining measurements made at different wavelengths extended the vertical range to 15 pm (Creath 1987). In a more recent development, the vertical range is extended still further by combining phase-shift interferometry with what is described iis vertical scanning interferometry (Caber et al. 1993). The visibility of fringes from a white-light source drops off rapidly from its maximum value at minimum optical path difference. If the interferometer is translated axially, these maxima, and hence the local height, can be extracted on a point-by-point basis by signal processing. A typical commercial realisation of this technique claims horizontal range and resolution of 0.2 mm and 0.4 pm respectively, vertical range and resolution of 150 pm and 1 nm. The maximum measurable slope at this magnification is stated to be 14". A somewhat similar system, rather misleadingly described as "coherence radar", has been described by Hausler & Neumann (1992).

3.2. Parametric Techniques Scattering is a subject of interest to optical engineers, radar engineers and many others, and has an extensive literature of its own. The present account will only attempt to cover it in sufficient depth to treat its application to roughness measurement. The mathematical foundations of scattering theory are set out in Beckmann & Spizzichino (1963) and more recently by Ogilvy (1991). Experimental techniques are discussed at length by Bennett & Mattsson (1989). The following account follows very closely the review of Vorburger & Teague (1981). When a beam of light is reflected by a rough surface, the intensity and pattern of the scattered radiation depend on the roughness heights, the spatial wavelengths and the wavelength of the light. In general, short and long surface wavelengths diffract the light into large and small angles respectively relative to the specular direction. For most surfaces, there is a broad spectrum of surface wavelengths, and the light is therefore diffracted into a range of angles. Five main mechanisms of interaction may be distinguished between rough surfaces and electromagnetic radiation (Vorburger & Teague 1981). For surfaces whose roughness is much less than the wavelength of the incident radiation, most of the reflected light propagates in the specular direction. As the roughness increases, the intensity of the specular beam decreases while the diffracted

Optical Instruments

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radiation increases in intensity and becomes more diffuse. I a addition, the angular distribution of diffuse radiation consists of a fine grainy structure called speckle, whxh shows up as intensity contrast between neighbouring points in the scattered field. Finally, the light wave may undergo a change in its polarization state upon reflection from the surface. All of these phenomena, the relative intensity or reflectance in the specular direction, the total intensity of the scattered light, the diffuseness of the angular scattering pattern, the speckle contrast, and the polarization, depend on the surface roughness, and all five have served as the bases for potential surface measuring instruments.

3.2.1. Specular Reflectance

One way of assessing specular reflectance is to measure what is called image clarity: the ability of the surface to reflect a clear image of a row of posts (Elmendorf & Vaughan 1958) or of a grid (Westberg 1967168). To quantify this, Halling (1954) designed an instrument in which the reflection of a series of vertical bars in the surface is observed at the specular angle. This angle is gradually decreased until the bars can no longer be distinguished. He found that the roughness was inversely proportional to the cosine of the angle of extinction.

Figure 3.11. Dflerent designs of glossmeter and definitions of their associated measurements (Westberg 1967168).

48

Rough Sufaces

A simpler approach is to measure the intensity of the specular beam. Commercial instruments following this general approach are sometimes called glossmeters, though gloss is a tenn not easily defined; Westberg (1967/68) quotes a review of 44 different definitions of gloss. He himself has reviewed the design, construction and operation of a number of designs of glossmeter at some length, and distinguishes six basic combinations of specular and diffuse measurement (Fig. 3.11). The measurement of roughness relies on the inverse correlation which exists between the specular reflectance p and the roughness Rq. For rougher surfaces (Rq > ;iy / 10 ) the true specular beam effectively disappears so p is no longer measurable. However, even in these circumstances there are a number of empirical studies that show an inverse correlation between the light intensity scattered in the specular direction and the surface roughness (Tanner & Fahoum 1976, Murray 1973, Spurgeon & Slater 1974). The main advantage of this method is speed. If the instrument is properly calibrated, a single measurement immediately yields a value for Rq. Therefore, the specular reflectance method is ideal for routine comparisons of similar surfaces. The technique is capable of studying isotropic or anisotropic surfaces and the ultimate vertical resolution is about 1 nm (Cunningham et al. 1976).

0

Figure 3.12. Roughness measured by specular reflectance and stylus methoas by six different groups of workers (adapted kom Birkebak 1971).

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There are a number of disadvantages to techniques based on specular reflection. Agreement with stylus measurements is not very good (Fig. 3.12). The measured intensities must be normalized for each material under inspection. Measurements are affected by the direction of the lay (Vashisht & Radhakrishnan 1974), though this dependence dlsappears at normal incidence (Ollard 1949). Also, even for fairly small apertures the diffuse component will be significant unless p/pois fairly close to unity. This implies that Rq must be much less than the optical wavelength 4, which means that visible light is suitable only for measuring the smoothest surfaces; for most engineering surfaces infrared radiation must be used. A final drawback of this technique is that it is primarily a function of surface amplitude and is not suitable for measuring surface wavelengths.

3.2.2. Total integrated scatter

The total integrated scatter (TIS) method is complementary to specular reflectance. Instead of measuring the intensity of the specularly reflected light, one measures the total intensity of the diffusely scattered light (Bennett & Mattsson 1989). TIS has much the same strengths and weaknesses as specular reflectance, i. e. the technique is fast, is based on an approximate theory, is practical only for surfaces smoother than the wavelength of the incident radiation, and has similar bandwidth limitations. The fundamental difference between the two is that the TIS is more dlrectly related to Rq. TIS is a fundamental quantity for the functioning and testing of optical components (Bennett et al. 1979). In many optical applications, if the TIS can be measured accurately and routinely, the roughness need not be. Using the technique for practical roughness measurements presents several problems, however. To begin with, the experimental set-up is more elaborate than that required for specular reflectance. Moreover, the bandwidth limitation arises from the fact that the collecting optics normally contain some sort of aperture or stop to prevent the specular beam from being detected along with the scattered radiation. Such an aperture or stop inevitably also blocks a portion of the scattered light and hence limits the surface wavelengths that can be detected (Bennett et al. 1979). The net result is that the technique does not seem to have been successfully applied to surfaces with Rq greater than about 10 nm. Even in the regime of very smooth surfaces, the roughness results obtained from TIS, although self-consistent, do not seem to agree well with measurements made by non-scattering techniques (Fig. 3.13). It appears then that TIS is an important technique for rapid

Rough SurJaces

50

measurements of the optical scattering characteristics of very smooth surfaces with short wavelengths. Like specular reflectance, TIS is not suitable for measuring surface wavelength parameters.

=m 200

0

8

2t

0 NWC TIS 0 BALZERS TIS A LIVERMORE OHP

0 N W C WYKO TOPO-20 0 NWC TALYSTEP

A I

20

I

I

I

I

I

I

I

I

50 100 AVERAGE IVIS ROUGHNESS. TIS d l

200

Figure 3.13. Summary of measurements on five roughness standards made by: 0 0 , TIS instruments; AO, optical profilers; 0,stylus instrument (Bennett & Mattsson 1989).

3.2.3. Angular Distributions

In principle, the entire angular distribution of the scattered radiation contains a great deal of information about the surface topography. In addition to rms roughness, measurements of the angular distributions (Tanner & Fahoum 1976, Stover 1976, Thwaite 1979) can yield other surface parameters such as the average wavelength or the average slope. The angle of incidence is normally held constant and the angular distribution is measured by an array of detectors or by a movable detector and is stored as a function I(@ where is the angle of scattering. A good example of the resulting angular distributions for a diamond-turned specimen illuminated by coherent light is shown in Fig. 3.14 (Church et al. 1977). The upper curve shows the angular distribution in the plzne of incidence and perpendicular to the predominant lay of the surface. It contains an intense specular beam at 8, =0, a broad scattering distribution due to the random component of the roughness, and a series of dwrete lines due to a periodic component of the roughness caused by the feed rate of the diamond tool. The lower curve is an

51

Optical Instruments

angular distribution measured parallel to the lay direction and it shows another broad distribution characteristic of the random roughness pattern in this direction. In principle then, one can distinguish between effects due to periodic and random roughness components and can detect the directional properties of surfaces.

6'

.-

lo-'

e lo-'

sB ,o-Q lo-'

lo-* lo-' 10.6

- 4 o ~ w - z I p o0

w

Scattering angle

200

30"

SOD

Figure 3.14. Angular distribution of light scattered &om a diamond-turnedsurface (adapted from Church et al. 1977). Upper curve, across lay; lower curve, along lay.

The kind of surface information that may be obtained from angular Qstributions depends on the roughness regime. For Rq >>% and surface spatial wavelengths il >> A,. one is working in the geometrical optics regime where the scattering may be described as purely specular scattering from a series of facets (Church 1979). The angular distribution is therefore related to the surface slope distribution, and its width is a measure of the characteristic slope of the surface. As Rq and il decrease, the distribution becomes a much more complicated function of both surface slopes and heights and is difficult to interpret. Some work has been done by Leader (1979), Smith & Hering (1970) and Chandley (1976) for Leader's comparisons between calculated and measured surfaces with Rq A., angular distributions yield interesting, qualitative idormation about the topography of painted and dielectric surfaces. The most interesting regime is where Rq << because then it can be shown theoretically that the angular distribution should directly map the power spectral density G (2x14 of the surface topography. The bandwidth is limited at the low end by the maximum scattering angle of 90" and at the high end by the minimum scattering angle which can be measured with respect to the specular beam, about 0.5" in practice (Church 1979), thus the maximum surface wavelength is only

-

52

Rough Surfaces

about 100 times the minimum wavelength. The angular distribution technique may be better suited to measuring wavelength parameters than roughness amplitude parameters. In summary, the scattering methods described so far are generally limited by available theories to studies of surfaces whose roughness is much less than the wavelength of the incident radiation. There are a few studies, mostly empirical, which have pushed beyond this limit. With a HeNe laser as the light source, the above constraint means that these techniques have been used mainly on optical quality surfaces where Rq < 0.1 pm. Within that limited regime, they can provide high speed quantitative measurements of the RMS roughqess of both isotropic surfaces and those with a pronounced lay. With rougher surfaces, angular distribution may be useful as a comparator for monitoring both amplitude and wavelength surface properties.

3.2.4. Direct Fourier Transformation

Ribbens and Lazik (1968) have described a technique in which a transparent replica of a rough surface is made. A photographic film is illuminated through the replica and developed; the local transmittance of the developed film is a function of the thickness at the corresponding position on the replica The power spectral density of the film transmittance can be measured directly and hence the power spectrum of the original surface can be determined. Anderson (1969) proposed a modification of this technique in which the replica itself is used as the object of the Fourier transforming lens. He pointed out that the relationships assumed by Ribbens and Lazik are correct only for fairly smooth surfaces. However, he suggested that rougher surfaces could be accommodated by reducing the effective variations of optical path length by interposing a liquid of appropriate refractive index, a device also proposed by Nagata et al. (1973). Thwaite (1979) used a direct optical transform in reflection on a number of periodic surfaces and reported good agreement between the power spectra measured in this way and by stylus instruments.

3.2.5. Ellipsometry

Ellipsometry measures the change in the polarization state of a beam of light when it is reflected from a surface. For an extensive review, see Azzam & Bashara

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(1977). In traditional null ellipsometry the important quantities are the angle of incidence of the beam of light and the rotational positions of the polarizing and analysing elements in the light path that produce a null in the detector. These

measurements allow one to determine the ratio of the complex reflection coefficients for the p- and s-components of the electromagnetic field. Ellipsometry measurements of the index of refraction are sensitive to a number of surface properties including composition; surface structurt such as damage, defects or surface crystal faces; temperature; strain state ; and surface roughness. Since the roughening of surfaces can significantly change the results from the ellipsometer, thereby obscuring the other surface changes to be observed, the question arises whether ellipsometry can be used to measure the surface roughness of engineering surfaces directly. Investigations so far have been largely empirical and they are in disagreement.

Figure 3.15. Variation of Lonardo's(1978) parameter a with roughness for various angles of incidence

Lonardo (1978) studied ellipsometry as a potential tool for in-process detection of surface roughness during manufacture. He worked with ground and polished steel surfaces of roughness, measured by stylus techniques, ranging approximately from 0.01 pm to 1.1 pm, and a derived ellipsometry parameter was found to vary almost linearly with Ra (Fig. 3.15). Overall results show that ellipsometry parameters cannot be related simply to roughness and slope. Vorburger and Ludema (1980) and Williams et al. (1988) reached a similar conclusion. If the key problem of quantlfication could be solved, ellipsometry would have several strengths as a surface measurement technique. The sensitive roughness range of the technique seems to be approximately 0-1 pm, which is in the range of

54

Rough Sudaces

interest for machined surfaces. Also, ellipsometry measurements should not be sensitive to fluctuations in the scattered-light intensity due to surface vibration since the technique measures the polarization state rather than the intensity of the scattered light.

3.2.6. Speckle

When a rough surface is illuminated with partially coherent light, the reflected beam consists in part of random patterns of bright and dark regions known as speckle. These patterns can be interpreted in terms of Huygen's principle whereby the intensity at a field point is caused by the interference of wavelets, scattered from different points within the illuminated area, with their phases randomized by height variations of the surface. The spatial pattern and contrast of the speckle depend on the optical system used for observation, the coherence condition of the illumination, and the surface roughness of the scatterer. A review (Briers 1993) cites more than 100 references on roughness-related aspects of speckle. Two broad classes of measurement methods for determining the roughness properties of a surface from speckle patterns can be discussed: speckle contrast and speckle pattern decorrelation. In both methods the roughness properties are obtained from the speckle patterns by empirically relating either the contrast or the degree of pattern correlation to the roughness of the surface under study. The empirical relationship is then interpreted with first order theories of speckle pattern formation, which generally assume that only single scattering of the electromagnetic wave takes place and that the scattering surfaces can be characterized by a Gaussian distribution of heights with a correlation length much less than the dimensions of the scattering region. It is doubtful that these assumptions are fully satisfied in most practical scattering problems. The results given below, however, demonstrate that withm these two classes of measurement techniques are methods to determine the roughness of a surface over the large range of 10 nm to 30 pm with a reasonable degree of confidence. In speckle contrast measurements the intensity variations are quantified in terns of an average contrast defined as the normalized standard deviation of intensity variations at the observation plane. The intensity variations are determined by either moving the specimen and thereby the speckle pattern past a fixed detector with an aperture smaller than the speckle size, or by moving the detector through the speckle field. High-contrast speckle patterns are produced when all the interfering wavelets have sufficient phase difference ( > 271 ) to give

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complete destructive interference at some points in the pattern and the illumination has a h g h degree of spatial and temporal coherence. Spatial coherence means that the phases of the electromagnetic field at two points spaced across the propagating wavefront are highly correlated, and temporal coherence means that the phases of the field at two points spaced along the direction of light propagation are highly correlated. Speckle contrast is unity for fully coherent monochromatic light illuminating a surface whose roughness is much larger than 4 so that the wavelet phases are uniformly distributed over the interval from 0 to 27c. Correspondingly, for coherent monochromatic illumination, as the reflecting surface becomes smoother and less complete destructive interference occurs, the contrast V decreases toward zero. Experiments to relate surface roughness to the contrast of speckle patterns produced by coherent monochromatic illumination (Fig 3.16) showed that a strong linear correlation exists between V and Ra determined by stylus profilometry for Ra values up to 0.13 pm.

0

m o D a J 3 a a Q D Q y I

2Ra (microns) Figure 3.16. Maximum average contrast of speckle intensity variations as a hnction of roughness for surfaces of various metals and finishes (Vorburger & Teague 1981).

The second broad class of techniques for relating surface roughness and speckle is speckle pattern decorrelation measurement. Here two speckle patterns are obtained from the test surface by illuminating it with different angles of incidence or different wavelengths of light. Correlation properties of the speckle patterns are then studied by recordmg the patterns on the same photographic plate by double exposure or by photoelectric detection of the two patterns. The primary attribute of this type of speckle measurement is that Rq values as large as 30 to 50 pm can be measured. Fujii & Lit( 1978) applied a speckle decorrelation technique

56

Rough Sur$aces

to the measurement of a range of ground glass and ground metal surfaces. They found good correlation between roughness deduced from correlation measurements and roughness measured by stylus instruments over a range of roughness from 0.13 pm to 6 pm (Fig. 3.17).

Figure 3.17. Roughness deduced &om speckle decorrelation measurementscompared with stylus roughness measurements for glass (circles) and metal (triangles) surfaces (Fujii & Lit 1978).

As the examples of this subsection have demonstrated, speckle patterns are rich in information about the microtopography of a test surface, though the field has not yet yielded techniques for obtaining characterizations of roughness other than the Rq value. The range of Rq values measurable with speckle techniques and their apparent insensitivity to the type of material and type of surface forming process indicate that these techniques have a high potential for roughness measurements. However, Briers (1993) has noted that speckle techniques have not in general been converted into practical instruments; he describes them as a "solution in search of a problem".

3.3. References

Anderson, W. L., "Surface roughness studies by optical processing methods", Proc. I.E.E.E., (Letters), 57, 95 (1969).

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Azzam, R. M. A,, and Bashara, N. M., Ellipsometry and polarised light (North Holland, Amsterdam, 1977) Bailey, W., "Optical inspection of cylinder bores", Trib. Int. 10, 319-322 ( 1977) Beckmann, P. and Spizzichino, A,, The scattering of dectromagnetic waves porn rough surfaces (Pergamon Press, Oxford, 1963). Bennett, J. M., Burge, D. K., Rahn, J. P., Bennett, H. E., "Standards for optical surface quality using total integrated scattering", Proc. SPIE 181, 124-128 (1979) Bennett, J. M., and Mattsson, L., Introduction to surface roughness and scattering (Opt. SOC.Am., Washington, 1989) Biegen, J. F. and Smythe, R. A,, "High-resolution phase-measuring laser interferometric microscope for engineering surface metrology", Surface Topography, 1, 287-299 (1988) Birkebak, R. C., "Optical and mechanical RMS surface roughness comparison", Appl. Opt. 10, 1970-1979 (1971) Briers, J. D., "Surface roughness evaluation", in Speckle metrology, R. J. Sirohi ed., (Marcel Dekker, New York, 1993). Bristow, T. C., "Surface roughness measurements over long scan lengths", Surface Topography, 1, 281-285 (1988) Brown, A. J. C., "Rapid optical measurement of surfaces", Int. J. Mach. Tool Munufact. 35, 135-139 (1995) Caber, P. J., Martinek, S. J., Niemann, R. J., "A new interferometric profiler for smooth and rough surfaces", Proc. SPIE 2088 (1993) Chandley, P. J., "Determination of the autocorrelation function of height on a rough surface from coherent light scattering", Opt. Quantum Electron. 8, 329-333 (1976). Church, E. L., "The measurement of surface texture and topography by differential light scattering", Wear, 57, 93-105 (1979). Church, E. L.; Jenkinson, H. A.; Zavada, J. M., "Measurement of the finish of diamond-turned metal surfaces by differential light scattering", Opt. Eng. 16, 360-374 (1977). Church, E. L.; Jenkinson, H. A.; Zavada, J. M., "Relationship between surface scattering and micro-topographic features", Opt. Eng. 18, 125-131(1979). Creath, K., "Step height measurement using two-wavelength phase-shifting interferometry", Applied Optics 26, 2810-2815 (1987)

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Cunningham, L. J., and Braundmeier, A. J., "Measurement of the correlation between the specular reflectance and surface roughness of Ag films", Phys. Rev. 14B, 479-488 (1976) Dainty, J. C . , "The statistics of speckle patterns", in E. Wolfed., Progress in Optics 14 (North Holland, Amsterdam, 1976) Davies, T.; Kun, X; Luxmoore, A. R., "Digital measurement of surface profiles by automated optical sectioning", Measurement Science & Technology 5, 710-715 (1994) Downs, M. J., Mason, N. M., Nelson, J. C. C., "Measurement of the profiles of super smooth surfaces using optical interferometry", Proc. SPIE 1009, 14-17 (1989) Downs, M. J., McGivern, W. H., Ferguson, H. J., "Optical system for measuring the profiles of super-smooth surfaces", Prec. Engng. 7, 2 11-2 15 (1985) Dupuy, O., "High-precision optical profilometer for the study of microgeometrical surface defects",. Proc .I .Mech .E., 182, Part 3K, 255-259 (1967/68). Edwin, R. P., "Light scattering as a technique for measuring the roughness of optical surfaces", J. Phys. E6, 55-59 (1973) Elmendorf, A. and Vaughan, T. W., "A survey of methods of measuring smoothness of wood", Forest Products J.,8, 275-282 (1958). Elson, J. M., and Bennett, J. M., "Relation between the angular dependence of scattering and the statistical properties of optical surfaces", J. Opt. SOC.Am. 69, 31-39 (1979) Elson, J. M., Rahn, J. P., Bennett, J. M., "Light scattering from multilayer optics: comparison of theory and experiment",Appl. Opt. 19, 669-675 (1980) Ennos, A. E. and M. S. Virdee, "High accuracy profile measurement of quasiconical mirror surfaces by laser autocollimation",Precis. Engng. 1, 5-8 ( I 982) Farago, F. T., Handbook of dimensional measurement 2e (Industrial Press, New York, 1982) Fujii, H., and Lit, J. W. Y., "Surface roughness measurement using dichromatic speckle pattern: an experimental study", Appl. Opt. 17, 2690-2705 (1978) Halling, J., "A reflectometer for the assessment of surface texture", J. Sci. Instrum., 31, 3 18-320 (1954). Hamilton, D. K.; Wilson, T., "Three-dimensional surface measurement using the confocal scanning microscope", Appl Phys 27,2 1 1-2 13 (1982). Hard, S., and Nilsson, O., "Laser heterodyne apparatus for roughness measurements of polished surfaces", Appl. Opt. 17, 3827-383 1 (1978)

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Hausler, G., and Neumann, J., "Coherence radar - an accurate 3D sensor for rough surfaces", Proc. SPIE 1822, 200-205 (1992) Howes, V. R., "An angle profile technique for surface studies", Metallography 7, 43 1-440 (1974). Johnson, F.; Brisco, B.; Brown, R. J., "Evaluation of limits to the performance of the surface roughness meter", Canadian Journal of Remote Sensing 19,140-145 (1993) Kayser, J. F., "Optical cut method for the determination of surface roughness", Foundv Trade J., 70, 137-138 (1943) Keller, B. E., "Non-contact surface contour analyser", Proc. 1. Mech. E., 182, Part 3K, 360-367 (1967/68). Lange, D. A.; Jennings, H. M.; Shah, S. P., "Analysis of surface roughness using confocal microscopy", Journal of Materials Science 28,3879-3884 (1993) Leader, J. C., "Analysis and prediction of laser scattering from rough-surface materials", J. Opt. SOC.Am. 69, 610-619 (1979) Lonardo, P. M., "Testing a new optical sensor for in-process detection of surface roughness", Ann. CZRP 27, 53 1-533 (1978) Murray, H., "Exploratoly investigation of laser methods for grinding research", Ann. CIRP 22, 137-139 (1973) Nagata, K., Umehara, T. and Nishiwaki, J., "The determination of RMS roughness and correlation length of rough surface by measuring spatial coherence function", Japan. J. Appl. Phys., 12, 1693-1698 (1973). Ogilvy, J. A,, Theory of wave scatteringfrom random rough surfaces (Adam Hilger, Bristol, 1991) Ollard, E. A., "Surface reflectometer for evaluating polished surfaces", J. Electrodepos. Tech. SOC.,24, 1-8 (1949) Olsen, W. S. and Adams, R. M., "A laser profilometer", J. Geophysical Res., 75,2185-2187 (1970). Parry, G., "Some effects of surface roughness on the appearance of speckle in polychromatic light", Opt. Comm. 12, 75-81 (1974) Pettigrew, R. M., and Hancock, F. J., "An optical profilometer", Proc. NELEX Conf: (Nat. Engng. Lab., Glasgow, 1978) Ramgulam, R. B.; Amirbayat, J.; Porat, I., "Measurement of fabric roughness by a noncontact method", Journal of the Textile Institute 84,99-106 (1993) Ribbens, W. B. and Lazik, G. L., "Use of optical data processing techniques for surface roughness stuhes", Proc. I.E.E.E. ('Letters) .56, 1637-1638 (1968).

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Sandoz, P., Tribillon, G., Gharbi, T., Devillers, R., "Ruughness measurement by confocal microscopy for brightness characterisation and surface waviness visibility evaluation", Wear 201, 186-192 (1996) Sayles, R. S., Wayte, R. C., Tweedale, P. J. and Briscoe, B. J., "The design, construction and commissioning of an inexpensive prototype laser optical profilometer", Surface Topography, 1 , 219-227 (1988) Shaw, M. C. and Peklenik, J., "A light projection technique for studying surface topology", Ann. C.Z.R.P.,12, 93-7 (1963). Smith, T., "Effect of surface roughness on ellipsometry of aluminium", Surf: Sci. 56, 252-259 (1976) Smith, T. F. and Hering, R. G., Tomparison of bidirectional reflectance measurements and model for rough metallic surfaces", Proc. 5th Symp. Thermophys. Properties, 429-435 (ASME, New York, 1970). Sommargren, G . E., "Optical measurement of surface profile", Precis. Eng. 3, 131-136 (1981a) Sommargren, G . E., "Optical heterodyne profilometry", Appl. Opt. 20, 610618 (1981b) Spurgeon, D. and Slater, R. A. C., "In-process indication of surface roughness using a fibre-optics transducer", Proc. of the 15th Int. Machine Tool Des. & Res. Conf.',Birmingham, 339-347 (1974). Stover, J. C . , "Spectral-density function gives surface roughness", Laser FOCUS12,83-85 (1976). Tanner, L. H. and Fahoum, M., "A study of the surfacc parameters of ground and lapped metal surfaces, using specular and diffuse reflection of laser light", Wear, 36,299-3 16 (1976) Teague, E. C., Vorburger, T. V., Maystre, D., Young, R. D., "Light scattering from manufactured surfaces", Ann. C I W 30,563-569 (1981). Thwaite, E. G., "The direct measurement of the power spectrum of rough surfaces by optical Fourier transformation", Wear, 57, 71-80 (1979). Thwaite, E. G., "The roughness of surfaces",Australian Physicist (November 1977) Tolansky, S . , Multiple-beam interference microscopy of metals (Academic Press, London, 1970a). Tolansky, S . , Multiple-beam interferometry of surfazes and films (Dover Publications, Inc., New York, 1970b). Tolansky, S., Surface microtopography (Longmans, London, 1960). Vashisht, S. K. and Radhaknshnan, V., "Surface studies with a gloss meter", Tribology Znt., 7, 70-76 (1974).

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Vorburger, T. V., and Ludema, K. C., "Ellipsometry of rough surfaces", Appl. Opt. 19, 561-569 (1980) Vorburger, T. V., "Methods for characterising surface topography", in Tutorials in Optics, 137-151 (Opt. Soc. Am., Washington, 1992) Vorburger, T. V.; Teague, E. C., "Optical techniques for on-line measurement of surface topography", Precis. Eng. 3,61-83 (1981). Way, S., "Description and observation of metal surfaces", Proc. Con$ on Friction & Surface Finish, 2e, 44-75 (MIT, Cambridge, 1969). Wehbi, D. and C. Roques-Carmes, "Physical limitations of optical defocussing technique," Wear 109, 287-295 (1986) West, R. N., and Stocker, W. J., "Automatic inspection of cylinder bores", Metrology &Inspection 9, 9-10 (1977) Westberg, J., "Development of objective methods for judging the quality of ground and polished surfaces in production", Proc .I .Mech .E., 182, Part 3K, 260273. (1967/68). Whitehouse, D. J., "Modern trends in the measurement of surfaces", Rev. M. Mec., 21, 19-28 (1975). Whitehouse, D. J., Handbook of surface metrology (Institute of Physics, Bristol, 1994) Williams, M. W., Ludema, K. C. and Hildreth, D. M., "Mueller matrix ellipsometry of practical surfaces", Surface Topography, 1, 357-372 (1988) Wyant, J. C., Koliopoulos, C. L., Bhushan, B., Basila, D., "Development of a three-dimensional noncontact dlgital optical profiler", Trans. ASME: J. Trib. 108, 1-8 (1986) Young, R. D.; Vorburger, T. V.; Teague, E. C., "In-process and on-line measurement of surface finish", Ann. CIRP 29,435-440 (1980).

CHAPTER 4

OTHER MEASUREMENT TECHNIQUES

We have considered stylus and optical methods in some detail because equipment using these methods comprises the major part of the installed base of roughness measuring instruments. However, a large number of other techniques have been used for the measurement of surface roughness. Some are mainly of historic interest, but may still be worth studying because the principles involved might suggest an application to some measurement problem insoluble by other means. Others are novel and still under development, and cover ranges or offer other special features which complement stylus and optical methods. We will continue to make the convenient distinction between profiling techniques, which can yield point-by-point information about the surface topography, and parametric techniques, which give directly some average measure of the surface roughness.

4.1. Profiling Methods

4.1.1. Taper Sectioning

In taper sectioning, as its name implies, a section is cut through the surface to be examined at a shallow angle, thus effectively magnifying height variations by the cotangent of the angle, and subsequently examined by optical microscopy. The technique was first described by Nelson (1969) and has since been employed by a number of other workers (Broadston 1944; Tarasov 1945; Darmody 1946; Shaw and Peklenik 1963; Dorinson 1965). Practical details are dlscussed at length by Nelson and by Rabinowicz (1950). It is necessaly to support the surface to be sectioned with an adherent coating which will prevent smearing of the contour during the sectioning operation. This coating must adhere firmly to the surface; must have a similar hardness; should not dlfise into the surface; and should not be affected by any subsequent etching. For steel these requirements are met by electroplating with nickel to a thickness of 0.5 mm. The specimen is then ground on a surface grinder at an angle of between 1 63

64

Rough Surfaces

and 6 degrees, depending on the required magnification, till the interface between coating and substrate has advanced halfway along the specimen. The taper section so produced is lapped, polished and possibly finally lightly etched or heat tinted to provide good contrast for the optical examination. The main advantage claimed for taper sectioning is its accuracy; indeed Shaw and Peklenik (1963) have gone so far as to describe it as 'probably the most accurate method that has ever been devised for studying the profile of a surface'. This seems rather an excessive claim for a technique whose vertical resolution is admitted by the Same authors to be only 0.25 pm. Great play is made, however, by Tarasov (1945) among others, of the ability of the method to show deep scratches which a stylus will not penetrate. The only measurement which can conveniently be made from a micrograph of a taper section is the peak-to-valley height, and Tarasov compares this measurement with the RMS roughness found by a stylus instrument for a number of surfaces. According to other results quoted by Shaw and Peklenik, a comparison of taper-section profiles with those of a stylus instrument revealed larger peak-to-valley roughness in every case, up to a maximum of 100 per cent discrepancy. This is not surprising when the integrating effect of taper sectioning is taken into account. Tarasov quotes peak-to-valley roughness of between 1 pm and 5 pm at a relative vertical magnification of 25. His sections must therefore have represented profiles of effective width from 25 pm to 125 pm. This compares with 8 pm for the width of a typical stylus (Jungles and Whitehouse 1970), which according to Guerrero and Black (1972) normally 'sees' an even narrower strip as it tends to ride on one edge only. Taper sectioning is thus equivalent, at a conservative estimate, to measuring 3-5 profile lengths from a stylus instrument. As the greater the profile length the higher the probability of encountering a high peak or deep valley, it is small wonder that the taper section gives a larger measurement. The other disadvantages of the technique are too obvious to need comment; we have discussed it at this length mainly because of the inflated claims made for its accuracy.

4.1.2. Electron Microscopy

Electron microscopy is thought of as primarily a technique for visualisation, but the very short wavelengths of electron beams offer the possibility of very high resolutions for quantitative work. Transmission electron microscopy (TEM), dealing as it does with specimens thin enough to pass an electron beam, is of little

Other Measurement Techniques

65

interest from the point of view of roughness measurement. Scanning electron microscopy (SEM) is at first sight more promising. An electron beam incident on the surface excites the emission of secondary electrons which are then detected. As with the STM, specimens must be electrically conducting and measured under vacuum. However, SEM has some vely attractive potential advantages. The depth offield is unusually large: features a few Angstroms high stand out plainly, and the vertical range is of the order of mm. The horizontal resolution is limited only by the beam diameter to a few nm. The beam is deflected virtually instantaneously by field coils, obviating the need for mechanical translation. The steepest slope which can be measured is about 15 degrees (Whitehouse 1994). For a general review of electron microscopy see Grundy & Jones (1976). The resulting intensity signal bears a striking resemblance to a profile measured by a stylus instrument. Unfortunately this resemblance is more apparent than real. Secondary electrons originate from regions 100 nm or so below the specimen surface, and local convexities increase the local electron flux, exaggerating topographic features. Point-by point comparisons of secondary electron images with stylus measurements (Samuels et al. 1974) show rather poor agreement. It is possible to extract true height information by analysing stereo pairs (Howell & Bayda 1972, Lee & Russ 1989), but this is laborious and may lose much of the potential high resolution. A number of workers have tried to get around this difficulty in other ways. One technique of SEM analysis employed a modified detector which followed the line-of-sight properties of back-scattered electrons (McAdams 1974). These electrons, which travel in straight-line trajectories, are thresholded according to direction and produce an electron optical sectioning of the surface along the critical trajectory direction. Surface elevation is therefore recorded as variations in detector position, as opposed to signal intensity in the standard SEM. A multipledetector technique (Lebiedzik & White 1975) produced reasonable agreement between measurements of average roughness. Holburn & Smith (1982) employed an autofocussing technique with a claimed vertical resolution of 1 pm. Myshkin and his co-workers (Kholodilov et al. 1985, Grigor'ev et al. 1988) have obtained roughness parameters from SEM measurements of secondary electron emission, but it is not clear whether they were able to reconstruct a true profile. Rasigni et al. (1981) have studied transmission electron micrographs of carbon replicas of calcium fluoride films and other surfaces, using a microdensitometer. They show that the micrograph transmittance is approximately proportional to the slope of the surface elements, which enables determination of the surface profile by integration of the microdensitometer data. This technique is

Rough Sugaces

66

able to measure roughnesses of 2 nm with a lateral resolution of better than 1 nm over an area of about 1 pm x 1 pm.

4. I.3. Capacitance

Capacitance techniques are discussed in more detail in the section on parametric techniques, but there are two designs of capacitance-based profilers which will be discussed here. Garbini et al. (1988) use a so-called fringe-field technique. A thin electrode held normal to the specimen is translated in a direction parallel to the plane of the electrode (Fig. 4.1). Lateral range is 6 mm; vertical range is not stated but appears to be about 10 pm.

Figure 4.1. Fringe-field capacitance probe (Garbini et al. 1988)

Although the electrode is only 0.3 pm wide, the capacitance between it and the specimen is influenced by regions of the specimen adjacent to the electrode in a manner analogous to the weighting function of a low-pass filter, so it is rather difficult to determine the lateral resolution. The equivalent stylus width is the length w of the electrode (about a millimetre in the practical realisation), so "profile" measurements are only meaningful on a surface produced by a process such as shaping or turning which is basically two-dimensional. In spite of these limitations, the instrument agrees with stylus measurements over a restricted range of roughness (Fig. 4.2) and appears to work well at relatively high (25 m d s )

67

Other Measurement Techniques

translation speeds. The fringe-field capacitance technique has also been used for parametric measurement (Nowicki & Jarkiewicz 1997). 4

-

3

3

'I 0

I

2

3

Surface roughness. RB

4

5

Cm)

Figure 4.2. Maximum, average and minimum roughnesses for various test samples (Garbini et al. 1988). (0)Stylus instrument; (x) fringe-field profilometer.

An instrument of much higher resolution is described by Bugg & King (1988). In the scanning capacitance microscope the electrode is a fine vertical wire. Most of the capacitance between the electrode and the specimen is due, as in the fringe-field device, to the surrounding field, but the effect of this field is ingeniously removed by vibrating the wire in a vertical plme and measuring the differential capacitance. In practice a servo arrangement is used to keep the separation of transducer and specimen constant during translation. The height variation is given by the servo signal, thus obviating the inherent non-linearity of the transducer (Fig. 4.3). In a commercial realisation of this instrument (Bugg 1991), lateral range and resolution are given as 26 mm and 10 pm respectively. The vertical resolution is O.lpm, and the vertical range is said to be 5 mm, presumably with the aid of some vertical translation device.

Rough Surfaces

68

Figure 4.3. Scanning capacitance microscope ( B u g & King 1988)

4.1.4. Scanning Microscopies

Although the techniques described below are all basically profiling techniques, they are generally used in a raster scan mode to make area measurements. The lateral displacements required are too small for conventional mechanisms requiring relative motion between their components, so piezo drives are used for translations in all three axes. Piezo drives are well suited for this, but are limited in lateral range to some fraction of a millimetre. They are also susceptible to hysteresis, which can then appear as an apparent form error. T h s wide variety of techniques shares many common elements (Teague 1988): servo control of tip-specimen spacing to maintain constant reaction; precise mechanical scanning of the tip with respect to the specimen; high sensitivity of the output to tip-specimen spacing, requiring stiff microscope structures and isolation from mechanical noise; lateral resolution determined by tip dimensions, with a resultant emphasis on the problem of probe formation. If a conducting probe is placed very close to a conducting surface a small potential difference across the gap will encourage electrons to cross the gap by quantum tunnelling. The resulting current is highly sensitive to the width of the gap. As the probe is translated across the rough surface the width of the gap, and thus the tunnelling current, changes. This is the principal of the scanning tunnelling microscope (STM) (Binnig & Rohrer 1986) (Fig. 4.4).

Other Measurement Techniques

69

Figure 4.4. Principle of STM (Binnig & Rohrer 1986)

There are two possible modes of operation (Fig. 4.5). The probe can be servoed to maintain a constant gap as it is translated, in which case the restoring servo voltage is a measure of the local height. This is relatively slow but can more easily follow the rougher surfaces. Alternatively, the probe can be maintained at a constant height and the change in tunnelling current can be measured. This is quicker but works best on smooth surfaces. CONSTANT CURRENT Y M E

I

CONSTANT NIGHT YXIE

Figure 4.5. Alternative modes of operation of STM (adapted from Hansma & Tersoff 1987)

The vertical resolution is a few Angstroms, and if a sufficiently fine probe is used the horizontal resolution is in principle atomic. The vertical range is limited in practice by the exponential fall-off in the tunnelling current as the gap increases. Also, the specimen must be in a vacuum chamber and must be of a conducting material. Within these limitations the S T M is a very powerful and flexible

70

Rough Surfaces

instrument and many commercial versions are available. Fu et al. (1992) have described an STM with a lateral range of 0.5 mm and lateral resolution of 1 nm. A recent review (DiNardo 1994) lists 400 or so references to STM and related techniques. In the atomic force microscope (AFM) (Binnig & @ate 1986), a probe mounted on a cantilever is repelled by the van der Waal's forces as it travels over the rough surface. The force deflects the cantilever and the deflection is sensed, either by an STM (Binnig & Quate 1986) or by the angular displacement of a reflected laser beam (Fig. 4.6), giving the added amplification of an optical lever (Alexander et al. 1989). Either repulsive forces or attractive electrostatic forces can be used, sometimes in the same instrument. Again the measured force may either be recorded directly or used as the control parameter for a feedback circuit which maintains the force at a constant value (McClelland et al. 1987).

Figure 4.6. AFM with optical lever mounted on the cantilever (Alexander et al. 1989)

The AFM avoids two disadvantages of the STM, its restriction to conducting specimens and the requirement for a hard vacuum. Note that although the AFM appears at first sight to be a kind of small-scale stylus instrument, the probe does not actually make contact with the surface. Translation arrangements, ranges and resolutions are similar to those of the STM, and the AFM is also commercially available in a number of models. In the scanning near-field acoustic microscope (SNAM), the friction of the air and other damping effects in the small gap between a vibrating tip and the measured surface change the frequency of vibration (Goch & Volk 1994). A standard diamond tip fixed to the constantly excited tuning fork from a wristwatch

Other Measurement Techniques

71

is guided along the surface at a separation of about 100 nm. A servo moves the fork up and down to maintain constant frequency, hence constant separation (Fig. 4.7), so the servo signal gives the varying surface height. Vertical resolution is about 1 nm, but lateral resolution is limited to about 0.5 l m by the radius of curvature of the diamond tip.

,nm

I F

4-

tm

inspected surface

Figure 4.7. Schematic of an acoustic microscope (Goch & \ olk 1994)

DiNardo (1994) describes a number of other techniques operating on a similar scale to STM and AFM and using similar translation systems (Table 4.1). These all yield topographic information of some kind so could in principle be used to measure roughness.

4.2. Parametric Methods

4.2.1. Mechanical Methods

The basic idea of a tactile test is that a probe of some kind is run across the surface to be measured and the friction between the surface and the probe is compared with that from a similarly machined surface of known roughness. The simplest and cheapest probe is the human fingernail, and it is surprisingly effective. Indeed one American engineer (Broadston 1947) waxed lyrical at the thought that every machinist carried $100,000 worth of surface-measuring equipment about his person, i.e. 10 fingers at $10,000, the cost of a good stylus instrument, each. The human fingernail is more sensitive to some frequencies than to others (Abbott & Goldschmidt 1937), so there is presumably an optimum speed with which it should be drawn along the surface. Schlesinger (19 12) performed some

Rough Surfaces

72

Table 4.1. Near-field surface characterisationprobes (adapted from DiNardo 1994).

Measurement principle

Lateral resolution

Reference

Scanning thermal profiler (STP)

surface temperature

< 30 nm

Williams & Wickramasinghe (1986)

Scanning chemical potential microscope (SCPM)

thermoelectric voltage

< 1 nm

Williams & Wickramasinghe (1991)

Optical absorption microscope (OAM)

effects of light absorption

< 1 nm

Weaver et al. (1989)

Scanning ion conductance microscope (SICM)

ion current

< 0.1 nm

Hansma et al. (1989)

Laser force microscope (LFM) Attractive force microscope

probe vibration amplitude van der Waals force

N.A.

Whitehouse (1994)

< 100 nm

DiNardo (1994)

Charge force microscope ( C M , EFM)

electrostatic force

< 200 nm

Whitehouse (1994)

Magnetic force microscope @EM)

magnetic force

< 100 nm

Whitehouse (1994)

Scanning near-field optical microscope (SNOM or NSOM)

near-field optical reflection

< 25 nm

Pohl et al. (1984)

Photon scanning tunnelling microscope

near-field optical transmission

50 nm

Reddick et al. (1990)

Instrument ......

73

Other Measurement Techniques

careful tests in which subjects were asked to differentiate between pairs of test pieces of increasingly different roughness. He found that for some finishes differences in roughness of as little as 20 per cent could be detected by the majority of his test panel (Fig. 4.8). A similar experiment by Haesing (1961) found a correlation between the subjects' assessments and stylus readings which, not unexpectedly, was stronger for peak height than for average roughness.

30 20

0-025 0.05

0.1

0.2

0.4

0.8

1.6

3.2

6.4

I3

25

Ra (microns) Figure 4.8. Tactile comparison (Schlesinger 1942). Percentage difference in roughness which could be assessed by 9 out of 10 testers: A, lapped, honed and ground B, milled C, turned and shaped.

This technique is of some scientific interest as it appem to be sensitive only to a narrow band of wavelengths, namely to those corresponding to the thickness of the probe. The maximum height difference detectable is presumably set by the protrusion of the nail beyond the fingertips while the minimum is set by the sensitivity of the human nervous system. It is quicker, cheaper and simpler than any other method providing a suitable range of reference specimens is available, but to give reliable results a fair amount of experience is probably necessary on the part of the tester. Watanabe & Fukui (1995, 1996) have shown that the subjective sensation of roughness is affected by vibrating the measured surface at ultrasonic frequencies. A development is the Mecrin tester, where a thin flexible steel blade is pushed along the surface at a gradually increasing angle till it buckles (Rubert 1967/68). As the device presumably relies on just failing to overcome static friction it must be sensitive to the mean slope. It is not clear what is gained in performance over the fingernail for the extra complexity.

74

Rough Su8aces

Another friction method is based on the retardaticn of the swing of a pendulum due to friction between the tested surface and a smooth shoe attached to the pendulum. The pendulum is released from a position 30 degrees from the vertical and the apparatus simultaneously begins to eject a paper tape at constant speed. The length of tape ejected before the pendulum comes to rest is taken as a measure of the finish of the test piece (Jost 1944). Dynamic friction is influenced by at least two surface parameters, RMS roughness and mean slope (Hirst & Hollander 1974), so the instrument must be used with reservations even as a comparator. The long-wavelength cut-off will depend on the nominal contact area, which in turn will depend on the load and the geometry of the contacting surfaces. The vertical resolution will depend ultimately on the finish of the shoe. The apparatus is easy and quick to operate but rather expensive to build; its measurements will also be affected by the cleanliness of the test surface. In a method employing spherical contact a ball of radius r rolls down an inclined plane as soon as the angle a of tilt exceeds a value which increases with the roughness of the planes (Bikerman 1970). If Rp is the peak-to-valley roughness, Rp

=

r(1 - cos a)

This relation is stated to hold approximately for surfaces rougher than 1 pm. Although this method makes use of the phenomenon of static friction the equation shows that it is independent of material parameters. Assuming contact is elastic, the area measured is probably not larger than 10 pm in diameter. The method gives an absolute measurement. It is very cheap, robust and simple to use, and might be suitable for production-line gauging, though it might be easier to set up in the form of a rolling cylinder. The thetameter is a device which presses a smooth steel sphere into the test surface under a known load. The increase in load required to increase its penetration by a fixed amount is measured (Tornebohm 1936). The 'theta' of its title is the effective change in the Hertzian elastic modulus of the test piece due to its roughness. A rigorous theory of the elastic contact of a sphere with a rough plane was not developed until many years later by Greenwood and Tripp (1 967), apparently in ignorance of the existence of an instrument bssed on this principle. The effect of roughness prevails only at light loads, and then not in a simple relationship with load. Asperity density and curvature are also involved. As the instrument reading is the result of a combination of at least two surface parameters

Other Measurement Techniques

75

it is probably not suitable even for a comparison without exhaustive calibration against a less ambiguous instrument using the range of surfaces to be tested. On the other hand the thetameter is reasonably cheap, robust, quick and easy to use and reproducible. Its sensitivity will decrease with decreasing roughness and the limit of vertical resolution will be set by the load-measuring device. The long-wavelength cutoff will depend on the Hertzian contact area. No dimensions are given in the reference, but an estimate of 10 pm diameter for the contact area seems plausible. In another technique involving contact with a sphere, two 16 mm diameter metal balls are mounted in a block of thermally insulating material (Fig. 4.9) (Powell 1957). One is completely recessed while the other protrudes slightly. They are connected by a differential thermocouple. The apparatus is placed in an oven till it attains constant temperature and then removed and placed, under a load of about 1 N, on the surface whose roughness is to be measured. The protruding ball will cool faster because it is in contact with the test surface, and this is quantified as a voltage reading from the differential thermocouple after a certain time has elapsed from contact.

Phosphor-bronze balls Balsa wood

m' I

I

,--

Figure 4.9. Thermal comparator (Powell 1957).

Clearly, this technique relies on far too many parameters, most of them difficult to quantifj, to be suitable for an absolute determination of roughness. The rate of cooling of the ball not in contact will depend on its initial temperature and on various material and atmospheric properties. The increased rate of cooling of the ball in contact will depend on the thermal conductance of the contact, which in turn will depend on the thermal conductivities of the contacting materials and their elastic moduli or relative hardness, depending on whether the contact is elastic or

Rough Surfaces

76

plastic, in addition to the surface properties and the load (Thomas & Probert 1972). The relevant surface properties are probably the RMS roughness and the mean slope (Thomas & Probert 1970). The sensitivity increases as the test surfaces become smoother. The useful upper roughness limit is probably about 4 pm RMS. The lower limit would probably be set by the roughness of the balls themselves. The range of wavelengths measured depends on whether contact is elastic or plastic, but the long-wavelength cut-off for this particular instrument is probably about 10 pm. An instrument based on this principle might be useful on a production line as a golno-go gauge, possibly with a built-in heater; it would be relatively cheap and robust and very simple to operate. Thermal comparators for general-purpose use have been commercially available. A scraping technique uses molten asphalt poured on to the surface whose roughness is to be measured (Blkerman 1970). After cooling and solidlfication, the excess is scraped off with a razor blade, leaving only the asphalt below the highest peaks over the area A measured. Its volume V is determined and the ratio v/! taken as RpI2. The ratio of VIA to the RMS roughness measured with a stylus instrument was reported as between 2.4 and 1.9. The height resolution is presumably set by the straightness of the edge of the razor blade. The method is simple and cheap but rather tedious.

I 1

g

1.15

0.10

I

II

-

I

I I

I

10

I

20

I

30

I

40

Test number r Figure 4.10. Scatter in sand-patch measurements of roughness (Doty 1975)

On somewhat similar lines, an open-bottomed vessel is placed on the surface to be measured and filled with fine sand to a predetermined level. The volume of sand required to fill the interstices of the surface is deduced by comparing the amount of sand needed with that needed on a perfectly flat smooth surface, and

Other Measurement Techniques

77

hence the average depth of the surface roughness is calculated @oty 1975). This test is intended for road surfaces, but could be used for finer surfaces if a finer particle size were employed. The longest and shortest wavelengths are clearly set by the diameter of the vessel and the diameter of a sand grain respectively. The latter dimension also limits the height resolution in theory, though in practice the mass or volume measurement of the differential quantity of sand would probably be the limiting factor. The upper height limit is set only by the height of the vessel. The method is simple, cheap and fairly quick, but is unduly sensitive to long wavelengths, as confirmed by the very large scatter in reported results (Fig. 4.10).

4.2.2. Electrical Methods

Capacitance profilers have been dealt with above, but the capacitance principle, being an areal rather than a sectional phenomenon, is better suited to parametric applications. The capacitance between two conducting elements is directly proportional to their area and the dielectric constant of the medium between them and inversely proportional to their separation. If a rough surface is regarded as the sum of a number of small elemental areas at different heights it is fairly easy to work out the effective capacitance between it and a smooth plate for various deterministic surface models (Sherwood & Crookall 1967/68; Ten Nape1 & Bosma 1970/71). Unfortunately, real surfaces are rarely deterministic. The capacitance of a condenser, one of whose plates is rough with a probability distribution of surface heights p(z), is proportional to

-m

where h is the separation of the mean planes. The difficulty is at once apparent: unless we have some grounds to truncate the height distribution at a height less than the mean plane separation we will always end up with infinite capacitance. Some numerical solutions for a Gaussian height distribution truncated at various arbitrary heights indicate that the capacitance is very sensitive to the mean plane separation and to the height of the highest point on the surface (Thomas 1978). Instruments for measuring surface

78

Rough Su$aces

roughness based on a capacitance principle have been commercially available (Fromson et al. 1976, Brecker et al. 1977, Risko 1981). The readings of Fromson's instrument correlated well with stylus measurements over a rather restricted range of roughness, but as the roughness increased the relationship became non-linear. A more versatile capacitance-based instrument was described by Lieberman et al. (1988). The electrode is a flexible diaphragm which when pressed against the rough surface conforms to more and more peaks as the load P increases (Fig. 4.11). Modelling the diaphragm as a two-dimensional elastic beam, the degree of conformity is calculated iteratively from stylus profile traces by assuming the maximum load is that which gives stable capacitance for an electrode in contact with a sinusoidal surface of wavelength 0.8 mm. The effective plate separation can then be calculated and agrees reasonably well with measured values. Vertical resolution is stated to be 25 nm. P =0 (Rigid Beam)

Figure 4.1 1. Progressive stages of the sagging-beam calculation for modelling the compliance of the capacitance probe (Liebeman et al. 1988).

When these values are compared with stylus measurements of a wide range of roughnesses and finishing processes (Fig. 4.12), although the surface model is twodimensional, it seems to work just as well for finishes without a lay. However, while the overall trend is clear, the scatter is such that many individual measurements disagree with stylus values by 50% or more. This is a pity, as the method, being fast and non-destructive, is potentially well-suited to production applications. The inductance between two magnetic surfaces will also be a hnction of their roughness, again because inductance falls off with increasing separation. Radhakrishnan ( 1977a) has measured the inductance between a magnetic recording head and a number of rough surfaces, and compared his results with

Other Measurement Techniques

79

../ '

Figure 4.12. Capacitance versus stylus roughness far 41 surfaces, with best fit straight line (Lieberman et al. 1988)

stylus measurements. Useful correlation with average roughness was obtained only when the comparison was restricted to a particular machining process. A stronger correlation was reported with peak density, again indicating a sensitivity to local maxima. As the measurement is quick and cheap this technique also has possibilities for quality control, though of course it is restricted to magnetic materials. Skin resistance is a also a phenomenon affected by roughness. Alternating electric currents of high frequencies are shifted from the central to the peripheral annuli of a wire; thus, the major part of the current flows in a surface layer, which, for copper, would be about 0.4 pm thick at 25 GHz (Bikerman 1970). Thus, at high frequencies, the thickness of the actively conducting region is of the order of magnitude of the height of surface hills. Consequently, the experimental resistivity of a wire deviates from that calculated under the assumption of no rugosity, and the degree of roughness can be deduced from this deviation. The largest height difference which can be detected decreases as the fiequency increases. The smallest detectable height difference will depend on the resolution of the resistance change. The long-wavelength cut-of€ will be set by the wavelength of the a.c. current and the velocity of its propagation in the metal; for the above frequency it would be about 5 mm for a steel wire. The method is suitable only for measurement of wire specimens, but for these it might well be the only practicable technique.

Rough Su#aces

80 4.2.3. Fluid Methods

In the outflow meter, an open-bottomed vessel with a compliant annulus at its lower end is placed in contact with the surface to be measured and filled with water to a predetermined level. The time taken for a given volume of water to escape through the gap between the compliant seal and the rough surface is measured. Moore (1965) has defined the ratio of the total effective cross-sectional area of the gap to the perimeter as the mean hydraulic radius (MHR). According to his analysis, which assumes laminar flow, the time of escape is inversely proportional to the fourth power of the MHR and should therefore be very sensitive to changes in surface texture. This device was intended to measure road surfaces, though there seems no reason why the principle should not be applied to much smoother surfaces. To work out the vertical and horizontal ranges is rather difficult, as the roughness measured is actually that of an annular section. The compliant seal acts as a highpass filter whose cut-off depends in a complicated way on its elastic properties and on the roughness itself. A low-pass cut-off of sorts is set by the width of the annulus. The maximum height range is governed only by the rate of discharge of water through an open pipe, and the minimum height resolution by the operator's patience!

Timer contacts Ground contact

Figure 4.13. Sectional view of an outflow meter (Henry & Hegmon 1975).

Other Measurement Techniques

81

A more sophisticated version has been described (Henry & Hegmon 1975) (Fig. 4.13) in which a marked temperature dependence has been found for smooth but not for coarse surfaces. This leads the authors to suggest that flow is turbulent for coarse surfaces and that Moore's analysis is therefore invalid. The device is fairly simple and fairly cheap and should give reproducible results, though no one seems to have compared its measurements with those of more orthodox equipment on the same surfaces. If the outflow meter employs a compressible rather than an incompressible fluid its theory becomes a little more complicated, but the basic strategy remains the same: to measure the effective cross-sectional area of outflow. However, as the viscosity of compressible fluids is so much lower an instrument employing one is more suited to engineering surfaces. Pneumatic gauges are used extensively in manufacturing industry; the general principles of their design, and various practical realisations, are discussed at length by Farago (1982). They were first employed to measured surface roughness by Nicolau (1937), but the theory was not worked out till rather later (Graneek & Wunsch 1952).

Figure 4.14. (a) Principal elements of a pneumatic gauging system (Farago 1982): (1) continuous supply of pressurised air; (2) pressure reducing valve; (3) metering device; (4)pressure indicator; (5) gauge head (6) specimen surface: (b) various nozzle cross-sectionsused for roughness measurement, outer diameter 25-30 mm (Radhakrishnan & Saga 1970)

The essential elements are a gauging nozzle in proximity to the test surface connected through a metering device to a source of air at constant pressure (Fig.

82

Rough Su$aces

4.14a). Air escapes between the gauging nozzle and the rough surface, thus lowering the pressure downstream of the metering device. A circular nozzle was orignally used, but various other shapes have been tried including slits and cruciform sections (Fig. 4.14b). A sufficiently fine slit should provide something nearly like a profile measurement, with the high-pass cutoff set by its length. The horizontal and vertical resolutions presumably depend in principle on the gas laws, but in practice the useful horizontal range of measurement is set by the range of linearity of the relationship between pressure and roughness. Measurements have been reported (Graneek & Wunsch 1952; Wager 1967) which correlate very well with roughness readings from a stylus instrument over a range from 0.1 pm to 5 pm (Fig. 4.15). Tanner (1979, 1980, 1981, 1982) has described a pneumatic Wheatstone bridge for roughness measurement.

Figure 4.15. Variation ofpneumatic gauge reading with roughness (Wager 1967)

This pneumatic technique seems to have possibilities which would repay development. With a circular nozzle it is unduly sensitive to long wavelengths, but offers the compensating advantage of giving a reading independent of orientation of a surface with a lay. Its reading depends on height parameters only. Up till now it has only been used as a comparator, but a more rigorous theory could probably be devised to give an absolute interpretation. It is cheap, robust, simple, quick and non-contacting, in fact well suited to a production line. When the sensitivity is improved by electrical pressure sensing and the nozzle is miniaturised, it can even be used as a profiler, as described in the section on in-process measurement.

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83

In the oil-droplet method, a droplet of oil of volume V is placed on to a rough solid and squeezed with an optical flat (Bikerman 1970). If the greatest area of the oil patch which can be achieved is A , then V/A is the average thickness of the patch. This is claimed to be approximately equal to half Rp, though this relationship almost certainly depends both on the finish of the surface and on the size of the oil droplet. The vertical resolution will depend on the waviness of the optical flat. The method is simple and cheap, but slow and not very reliable. In the stagnant-layer method, a plate is covered with a non-volatile liquid and suspended vertically (Bikerman 1970). The mass and thus also the volume Vof the liquid remaining on the plate after time t is determined from time to time. The experimental function V = f (t) deviates from that predicted by hydrodynamics in such a manner as if a stagnant layer were present. The deduced thickness of the layer is from one to two times the RMS roughness. The height resolution will depend on the resolution of the weighing arrangements; a difference in roughness of 0.1 pm on an area of 100 cm2 would cause a change of mass of only about 1 mg in a total mass of perhaps 100 g. The method is not very simple and requires quite expensive equipment; it is slow and not very reliable; and it is only suitable for workpieces of the appropriate geometry. Using a somewhat similar principle, an oil drop is timed as it flows down a fixed length of an inclined test piece (Kamnev 1966). The theory is not discussed, but one might expect both the mean slope and the RMS roughness to affect the performance. This is confirmed by a difference in behaviour between surfaces finished in various ways. Roughness was found to be proportional to the 3.7 power of the time for filed and milled surfaces but the 1.74 power for etched surfaces. Sensitivity falls off rapidly as the surface gets rougher. The long-wavelength cutoff must be set by the diameter of the drop. The method is simple, cheap and surprisingly reproducible, but rather slow.

4.2.4.Acoustic Methods

Acoustic radiation interacts with rough surfaces in ways analogous to the various interactions of electromagnetic rahation, for instance by backscattering (Ogilvy 1988, Blakemore 1993). In addition acoustic waves can be transmitted through a rough interface, and the transmission may yield information about the roughness itself (Nagy & Adler 1987, Pecorari et al. 1992, 1995b) and the real area of contact (Krolikowski & Szczepek 1992, Polijaniuk & Kaczmarek 1993, Pecorari et al. 1995a).

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If the acoustic wavelengths are short compared to the surface wavelengths then specular reflection may occur, and the topography of underwater surfaces is investigated in this way by sonar, giving vertical resolutions of better than a meter and lateral resolutions of a few meters (Russ 1994). On a smaller scale, Blessing & Eitzen (1988) have obtained profiles of machined surfaces acoustically, but with rather poor resolution. Acoustics show more promise for parametric measurement of roughness. De Billy et al. (1976), using backscattering techniques, have measured roughnesses in the range from 3 pm to 100 pm, and Stor-Pellinen & Luukkala (1995) have investigated the possibility of measuring the roughness of paper using ultrasound. Chiang et al. (1994) have suggested that ultrasound could measure the roughness of articular cartilage in vivo. Gezanhes et al. (1982) measured the roughness of cmcrete surfaces using acoustical speckle correlation. The surface to be measured was illuminated by two coherent ultrasonic waves at different angles of incidence. The correlation between the two scattered waves depends on the roughness, the angles of incidence and the acoustic wavelength. With a so-called "acoustic interferometer" constructed on this principle they were able to measure roughnesses of between 25 pm and 2.4 mm.

4.3. References

Abbott, E. J. and Goldschmidt, E., "Surface quality", Afech. Engng., 59, 813825 (1937). Alexander, S., Hellemans, L., Marti, O., Schneir, J., Elings, V., Hansma, P. K., "An atomic-resolution atomic-force microscope implemented using an optical lever", J. Appl. Phys. 65, 164 (1989). Bikerman, J.J., Physical surfaces, (Academic Press, New York, 1970). Binnig, G., and Quate, C. F., "Atomic force microscope", Phys. Rev. Lett. 56, 930-933 (1986). Binnig, G.; Rohrer, H., "Scanning tunnel microscopy", IBM Journal of Research and Development 30, 355-369 (1986) Blakemore, M., "Scattering of acoustic waves by the rough surface of an elastic solid", Ultrasonics 31, 161-174 (1993) Blessing, G. V. and Eitzen, D. G., "Surface roughness sensed by ultrasound", Surface Topography, 1, 143-158 (1988) Brecker, J. N., Fromson, R. E.; Shum, L. Y., "Capacitance-based surface texture measuring system", Ann. C I W 26,375-377 (1977).

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Broadston, J. A., "Control and measurement of surface finishes"' Steel, 120, No. 2, pp.82-3, 116, 118 and 121 (1947). Broadston, J. A., "Measuring methods described for surface roughness specification", Prod. Engng. 15, 806-810 (1944). Bugg, C. D., "Noncontact surface profiling using a novel capacitive technique: scanning capacitance microscopy", Proc. SPIE 1573, 2 16-224 (1 99 1) Bugg, C.D. and King, P.J., "Scanning capacitance microscopy", J. Phys. E: Sci. Instrum., 21, 147-151 (1988) Chiang, E. H.; Adler, R. S.; Meyer, C. R.; Rubin, J. M.; Debrick, D. K.; Laing, T. J., "Quantitative assessment of surface roughness using backscattered ultrasound: the effects of finite surface curvature", Ultrasound in Medicine and Biology 20, 123-135 (1994) Darmody, W, J., "Tapering for surface inspection", Am. Mach., 90, 134-135 (1946). de Billy, M., Cohen-Tenoudji, F., Jungman, A,, Quentin, G. J., "Possibility of assigning a signature to rough surfaces using ultrasonic backscattering diagrams", IEEE Trans Sonics Ultrason SU-23,356-363 (1976). DiNardo, N. J., Nanoscale characterisation of surfaces and interfaces, (VCH, Weinheim, 1994) Dorinson, A., "Microtopography of finely ground steel surfaces in relation to contact and wear", A.S.L.E. Trans., 8, 100-108 (1965). Doty, R. N., "Study of the sand patch and outflow meter methods of pavement surface texture measurement", in Rose, J. G. ed., Surface Texture versus Skidding: Measurements, Frictional Aspects and Safety Features of Tire-pavement Interactzons, STP 583, 42-61 (ASTM, 1975) Farago, F. T., Handbook of dimensional measurement 2e (Industrial Press, New York, 1982) Fromson, R. E.; Shum, L. Y.; Brecker, J. N., "Universal surface texture measuring system", SME IQ76-597, (1976) Fu, J., Young, R. D., Vorburger, T. V., "Long-range scanning for scanning tunnelling microscopy", Rev. Sci. Instrum. 63, 2200-2205 (1992) Garbini, J. L., Jorgensen, J. E., Downs, R. A. and Kow, S. P., "Fringe-field capacitive profilometry", Surface Topography, 1, 13 1-142 ( 1 388) Gezanhes, C ; Calaora, A.; Condat, R., "Acoustical measurement of surface roughness by speckle correlation", Signal Process n 2-3 (Apr 1982) Goch, G., Volk, R., Tontactless surface measurement with a new acoustic sensor", CIRP Ann. 43,487-490 (1994)

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Graneek, M. and Wunsch, H. L., "Application of pneumatic gauging to the measurement of surface finish", Machinery, 81, (1952). Greenwood, J. A. and Tripp, J. H., "The elastic contact of rough spheres", Trans. A.S.M.E: J.Appl. Mech., 34E, 153-159 (1967). Grigor'ev, A. Ya., Myshkin, N. K., Semenyuk, N. F. and Kholodilov, O.V., "Evaluating specific surface area by the secondary electron emission method", Trenie i Iznos, 9, 793-798 (1988) Grundy, P. J., and Jones, G. A,, Electron microscopy in the study of materials, (Edward Arnold, London, 1976) Guerrero, J. L. and Black, J. T., "Stylus tracer resolution and surface damage as determined by scanning electron microscopy", Trans. A.S.M.E: J. Eng. Ind., 94B, 1087-1093 (1972). Haesing, J., "Determining surface finish of workpieces by means of surface standards", Microtecnic, 15, 24-28 (1961). Hansma, P. K., and Tershoff, J., 5canning tunnelling microscopy", J. Appl. Phys. 61, Rl-R23 (1987). Hansma, P. K., Drake, B., Marti, O., Goud, S. A. C., Prater, C. B., Science 243, 641-643 (1989) Henry, J. J. and Hegmon, R. R., "Pavement texture measurement and evaluation", in Rose, J. G. ed., Surface Texture versus Skidding: Measurements, Frictional Aspects and Safety Features of Tire-pavement Interactions, STP 583, 317 (ASTM, 1975) Hirst, W. and Hollander, A. E., '*Surfacefinish and damage in sliding", Proc. R. SOC.Lond. A337,379-394 (1974). Holburn, D. M. and Smith, K. C. A., "Topographical analysis in the SEM using an automatic focusing technique", J. Microscopy, 127, 93-103 (1982) Howell, P. G. T. and Bayda, A., Proc. 5th. SEMSymp., Chicago, 1972 Jost, H . P., "A case for the qualitative inspection of surface finish", Machy. 65,483-486 (1944) Jungles, J. and Whitehouse, D. J., "An investigation of the shape and dimensions of some diamond styli", J. Phys. E: Sci. Instrum., 3,437-440 (1970). Kamnev, V. V., "Integral evaluation of surface roughness", Meas. Tech., 2, 261-263 (1966). Kholodilov, 0. V., N. K. Myshkin and A. Y. Grigor'ev, "Microtopography evaluation with scanning electron microscope", Soviet Journal of Friction Wear, 6, 133-136, (1985) Krolikowski, J.; Szczepek, J., "Phase shift of the reflection coefficient of ultrasonic waves in the study of the contact interface", Wear 157, 51-64 (1992)

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Lebiedzik, J. and White, E. W., "Multiple detector method for quantitative determination of microtopography in the SEM", Proc. 8th Annual Symp. Scanning Electron Microscopy, 101-188 (Illinois Inst. of Technol., Chicago, 1975). Lee, J. H., and Russ, J. C., "Metrology of microelectronic devices by stereo SEM", J. Computer-assisted Microscopy 1,79-90 (1989) Lieberman, A. G., Vorburger, T. V., Giaque, C. H. W., Risko, D. G. and Rathbun, K. R., "Comparison of capacitance and stylus measurements of surface roughness", Surface Topography, 1,115-130 ( 1988) McAdams, H. T., "Scanning electron microscope and the computer: new tools for surface metrology", Modem Machine Shop, 82-9 1 (1974). McClelland, G. M., Erlandsson, R., Chiang, S., "Atomic force microscopy: general principles and a new implementation" in Review of progress in quantitative nondestructive evaluation, 6B, D. 0. Thompson and D. E. Chimenti eds., 1307-1314, (Plenum, New York, 1987). Moore, D. F., "Drainage criteria for runway surface roughness", J. Roy. Aeronaut. SOC.,69, 337-342 (1965). Nagy, P. B. and L. Adler, "Surface roughness induced attenuation of reflected and transmitted ultrasonic waves", Journal of the Acoustical Society of America, 82, 193-197 (1987) Nelson, H. R.,"Taper sectioning as a means of describing the surface contour of metals", Proc. ConJ on Friction & Surface Finish, 2e, 217-237 (MIT, Cambridge, 1969). Nicolau, M. P., "Application du micrometre solex a la mesure de l'etat des surfaces", Mecanique, 80-83 (Mar-Apr 1937) Nowicki, B., and Jarkiewicz, A,, "The in-process surface roughness measurement using fringe field capacitive method", Trans. 7th Int. Con$ on Metrology & Properties of Engng Surfaces, 325-332 (Gothenburg, 1997) Ogilvy, J. A., "Computer simulation of acoustic wave scattering from rough surfaces", J. Phys. D: Appl. Phys., 21,260-277 (1988) Pecorari, C., Mendelsohn, D. A,; Adler, L., "Ultrasonis wave scattering from rough, imperfect interfaces. Part I. Stochastic interface models", Journal of Nondestructive Evaluation 14, 109-116 (1995a) Pecorari, C., Mendelsohn, D. A,; Adler, L., "Ultrasonic wave scattering from rough, imperfect interfaces. Part 11. Incoherent and coherent scattered fields", Journal of Nondestructive Evaluation 14, 117-126 (1995b) Pecorari, C.; Mendelsohn, D. A.; Blaho, G.; Adler, L., "Investigation of ultrasonic wave scattering by a randomly rough solid-solid interface", Journal of Nondestructive Evaluation 11,211-220 (1992)

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Pohl, D. W., Denk, W., Lam, M., Appl. Phys. Lett. 44,651-653 (1984) Polijaniuk, A,, Kaczmarek, J., "Novel stage for ultrasonic measurement of real contact area between rough and flat parts under quasi-static load", Journal of Testing &Evaluation 21, 174-177 (1993) Powell, R. W., "Experiments using simple thermal comparator for measurement of thermal conductivity, surface roughness and thickness of foils or of surface deposits", J.Sci. Instrum., 34, 485-492 (1957). Rabinowicz, E., "Taper sectioning, A method for the examination of metal surfaces",Metal Industry, 76, 83-86 (1950). Radhakrishnan, V. and Sagar, V., "Surface roughness assessment by means of pneumatic measurement", Proc. 4th. All-India Machine Tool Design & Research Con$, (Indian Inst. Tech., Madras, 1970) Radhakrishnan, V., "Application of inductive heads for non-contact measurement of surface finish", Proc. Int. ConJ Prod. Eng. 2 (Inst. of Eng., Calcutta, 1977). Rasigni, M., Rasigni, G . , Palmari, J.-P., Llebaria, A,, "Study of surface roughness using a microdensitometer analysis of electron micrographs of surface replicas:- 1. surface profiles", J. Opt. SOC.Am. 71, 1124-1133 (1981). Reddick, R. C. R., Warmack, R. J., Chilcott, D. W., Sharp, S. L., Ferrell, T. L., Rev. Sci. Instrum. 61, 3669-3677 (1990) Risko, D. G., "Quick, non-destructive method for measuring surface finish using capacitance", Carbide Tool J 13,26-29 (1981) Rubert, M. P., "Functional assessment of surface roughness", Proc. I. Mech. E., 182, Part 3K, 350-359 (1 967/68). Russ, J. C., Fractal surfaces, (Plenum Press, New York, 1994). Samuels, J. M., Hoover, M. R., Tarhay, L., Johnson, G . C . , White, E. W., "Quantitative SEM and raster profilometer analysis of fracture surfaces", in Bradt R. C. et al. Eds., Fracture mechanics of ceramics, (Plenum Press, New York, 1974) Schlesinger, G., Surface finish, (Inst. of Prod. Engrs., London, 1942). Shaw, M. C. and Pekleruk, J., "A light projection technique for studying surface topology", Ann. C.I.R.P., 12, 93-97 (1963). Shenvood, K. F. and Crookall, J. R., "Surface finish assessment by an electrical capacitance technique", Proc. I. Mech. E., 182, Part 3K, 344-349 (1967/68). Stor-Pellinen, J., Luukkala, M., "Paper roughness measurement using airborne ultrasound", Sensors and Actuators, A: Physical 49, 37-40 (1995)

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Tanner, L. H., "A self-balancing pneumatic Wheatstone bridge for surface roughness measurement", Wear 83, 37-47 (1982) Tanner, L. H., "An improved pneumatic Wheatstone bridge for roughness measurement", J. Phys: Sci. Instrum. 13E., 593-594 (1980) Tanner, L. H., "Pneumatic Wheatstone bridge for surface roughness measurement", J. Phys: Sci. Instrum. 12E, 957-960 (1979) Tanner, L. H.,"A self-balancing pneumatic potentiometer and Wheatstone bridge with electrical readout. Applications to surface roughness measurement, pneumatic gauging and to measurement of pressure difference ratios", Precis. Eng. 3,201-207 (1981). Tarasov, L. P., "Relation of surface roughness readings to actual surface profile", Trans. A.S.M.E, 67, 189-194, (1945). Teague, E. C . , "Scanning tip microscopies: an overview and some history", in G. W. Bailey ed., Proc. 46th. Annual Meeting of the Electron Microscopy Society ofAmerica, 1004-1005 (San Francisco Press, San Francisco, 1988) Ten Napel, W. E. and Bosma, R., "The influence of surface roughness on the capacitive measurement of film thickness in elastohydrodynamic contacts", Proc. I. Mech. E., 185,635-639 (1970/71). Thomas, T. R. and Probert, S. D., "Correlations for thermal contact conductance in vacuo", Trans. Am. SOC.Mech. Engrs., 94C, 176-180 (1972) Thomas, T. R. and Probert, S. D., "Thermal contact resistance: The directional effect and other problems", Int. J. Heat Mass Transfer, 13, 789-807 (1970). Thomas, T. R., "Surface roughness measurement: alternatives to the stylus", Proc. 19th. MTDR ConJ, 383-390 (UMIST, Manchester, 1978) Tornebohm, H., "Modern tolerance requirements and their scientific determination", Mech. Engng., 58, 41 1-417 (1936). Wager, J. G., "Surface effects in pneumatic gauging", Int. J. Mach. Tool Des. Res., 7, 1-14 (1967). Watanabe, T.; and Fukui, S., "Control of tactile surface-roughness sensation using ultrasonic vibration", Trans. JSME C62, 1329-1334 (1996) Watanabe, T.; and Fukui, S., "Method for controlling tactile sensation of surface roughness using ultrasonic vibration", Proc. IEEE Int. Con$ on Robotics andAutomation 1, 1134-1139 (IEEE, Piscataway, NJ, 1995) Weaver, J. M. R., Walpita, L. M., Wickramasinghe, H. K., Nature 342, 783785 (1989) m t e h o u s e , D. J., Handbook of surface metrology, (Institute of Physics, Bristol, 1994)

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Williams, C. C. and Wickramasinghe, H. J., J. Vac. Sci. Technol. B9, 537540 (1991) Williams, C. C. and Wickramasinghe, H. J., Proc. Ultrasonics Symp. 393397 (IEEE, Piscatawy, NJ, 1986)

CHAPTER 5

OTHER MEASUREMENT TOPICS

There are a number of important topics connected with the measurement of roughness which cannot readily be considered under the various headings of instrument design. Before we leave the subject of measurement and move on to characterisation, it is convenient to collect these topics here for discussion. We begin with the special problems of 3D measurement (3D characterisation will be discussd at length later). We go on to consider the Miculties of repeatedly finding a small area on a surface in a sequence of measurements, and the techniques necessarily employed if for some reason it is not possible or convenient to bring the instrument to the surface under investigation. Finally we discuss the measurement or inspection of roughness during manufacture.

5.1. 3D Measurement

By 3D measurement we mean more than a measurement of average area properties such as that given by glossmeters. We mean an actual determination of surface relief over an area, so that at least in principle a topographic map can be constructed. On a surveying scale this was traditionally done by triangulation; that is, starting from an original baseline, the position of individual points on a landscape was determined by measuring their angular displacement from two fixed points and solving the resulting triangle. Interestingly, this is basically a digital method, though not of course based on a rectangular grid. More recently, height information has been extracted from pairs of stereo photographs of the same area taken at slightly different angles by stereophotogrammetry, a technique which has been applied to surface roughness (Unsworth & Hepworth 1971). Contour maps on a smaller scale may be obtained directly by interferometry (Tolansky 1960, 1970a,b) but do not by themselves give quantitative height information. The volume of data from 3D measurements is so large and its processing so tedious that in practice, whatever the original measuring technique, relief information is at some stage extracted in the form of individual heights scanned on a rectangular grid. 91

92

Rough Sudaces

There is a paradox inherent in 3D measurement in the sense in which we have defined it above. To approximate as closely as possible to spot height measurement the footprint of the probe or sensor on the surface must be as small as possible. How can this small window on the surface map an area? The traditional answer is a raster scan, used for many years by television engineers to build up pictures on cathode ray tubes. The raster moves the transducer over the surface in a number of closely spaced parallel lines (Fig. 5.1). If the output of the transducer is processed and displayed according to an appropriate protocol, a picture of the surface being scanned will be built up line by line. 1Aperture

%nes indicafing path and direction foJlowedby aperiwe

Figure 5.1. Raster scanning of an area (Terman 1937). The "aperture"correspondsto the footprint of a roughness measuring probe.

In a television the scanning device is an electron beam, easy to move, and the display is analogue. For roughness measurement, movement is not usually so easy and acquisition and processing of data are digital. The aim of the measurement procedure is to build up a matrix of individual height readings in the computer whch can then be processed numerically, either to create graphical representations of the surface or to extract quantitative information about the relief. In principle such a scan need not be Cartesian; polar scans (Edmonds et al. 1977, Newman et al. 1989) and even spiral scans (Mollenhauer 1973) have been employed, but the non-Cartesian spacing is inconvenient for subsequent analysis. Thus there are no 3D measuring instruments as such: all so-called 3D instruments are basically profilers, and any of the profiling instruments described in previous chapters can in principle be used for 3D measurement using raster

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scanning. Individual scanning devices have individual strengths and weaknesses which have already been discussed. But there are some general problems, of which the first is translation in the plane of the measured surface. In electron microscopy this is achieved by deflecting the electron beam. In scanning interferometers the fringes from an area are imaged onto an array of detectors which is simply read out sequentially by the computer. In STM's and similar instruments a deflecting voltage is applied to a piezo element. In general, however, it is more convenient to keep the measuring probe stationary and move the workpiece. This is partly due to the ready availability of precision x-y translation tables already developed for optical engineering. The requirements for such tables are formidable, as they must not only offer precisely controlled movement in very small increments but also give accurate positional information and provide an absolute reference datum of height. Desages & Michel (1993) measured two kinds of error on a proprietary translation table used for roughness measurements. A laser interferometer measured the errors in x and y separately at various table positions (Fig. 5.2a, b). Positional errors of more than 60 pm were detected in a 280 x 280 mm area.

Figure 5.2. Positional errors (a) in x, (b) standard deviation of (a), (c) in z,of a translation table (Desages & Michel 1993).

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Rough Surfaces

They went on to measure the error in z by mounting an optical flat on the table and translating it in the x-direction (Fig. 5 . 2 ~ ) .Height errors of more than a micron were detected, with a marked periodicity. Both positional and height errors occur at wavelengths of the order of millimetres, and their effect would probably be filtered out in many roughness calculations. Note also that there are considerable differences in the time taken by different instruments to map the same area; while data acquisition by a scanning interferometer is virtually instantaneous, a scanning stylus profiler may take several hours. This is a period of time long enough for changes in the environment, for instance temperature changes, to affect the measuring system. Such changes usually appear as spurious errors of form and again will probably be filtered out in roughness calculations. In Fig. 5.1 the probe is moved in a true raster, that is with each measuring scan always in the same direction and followed by a return passage when no measurements are taken. This presents no difficulties when the probe is an electron beam whose movement is effectively instantaneous, but with a mechanically translated probe the return passage may represent a significant time overhead. Instrument designers often attempt to reduce this overhead by employing boustrophedon scanning (pouozpocpqFov = "as the ox turns", i.e. in ploughing a field), where measurements are taken on the return stroke also (Teague 1988). With many probes this causes no particular problem, but users should be aware that styli are usually designed to be dragged, and the dynamic effects of pushing them instead can introduce serious measuring errors (Sayles & Thomas 1976). Analysis of 3D data will be dealt with in a later chapter. Before analysis can begin, however, there is a problem of visualising the data. The amount of data collected in a single area measurement is so large that it is difficult to make a preliminary appreciation unless the data is processed by software to appear in some visual form. 3D visualisation software is now widely available in mathematical packages such as MathematicaTMand MatlabTMand even in general-purpose spreadsheets like ExcelTM,and most 3D roughness measuring systems are bundled with their own proprietary software. At the very least a customer should expect to be able to see his data as a contour map or as an isometric wire-frame view, often with perspective and rendering available as well (Fig. 5.3). There should be facilities to zoom in on areas of arbitrary size, to change the number and heights of contours, to reject obviously defective data points, and to export the data in a portable format. The facility to reject data points, though desirable to clean up measurements which may be impossible to repeat, should be treated with care. Problem data

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Other Measurement Topics

points usually represent regions where the local surface is too high, or too low, or too steep, for the sensor to follow. If the user is only interested in average or typical features, this may not matter, but extreme-value roughness parameters may be sigrufcantly affected by missing data. Any area measurement which contains more than say 10% of rejected heights should be discarded unless there are pressing reasons to the contrary. If changing the instrument settings does not improve the rejection rate, the wrong instrument is being used. Particular care should be taken to read the small print of the program manual; the default settings of some software packages simply interpolate over the top of bad data without flagging it, a recipe for drawing wrong conclusions.

a

b

r

I

Y

d Figure 5.3. (a) 0.12 mm x 0.12 mm of a plateau-honed surface mapped with an AFM, height contours at 1 p intervals; @) isometric view of (a); (c) 40 pn x 20 p &om (a), contours at 0.2 pn intervals; (d) isometric view of @)

5.2. Relocation

There are many situations, particularly in research, where it would be very useful to look at a single profile through a surface before and after some experiment such

96

Rough Sui$aces

as running-in to see what changes have occurred to the surface geometry. It is clearly essential that exactly the same section is traversed each time, otherwise the changes observed could be attributed to the displacement of the profile and small but significant changes might not be observed at all. This requirement is very difficult to achieve in practice; the action of returning the profiling transducer to the start of its traverse is often enough to displace it laterally by a few microns, and as a typical stylus is only 8 pm wide, and other sensors may well be smaller, this is sufficient to invalidate the result. The problem was overcome by Williamson and Hunt (1968) who designed what they called a relocation table. The table is bolted to the bed of the stylus instrument, and the specimen stage is kinematically located against it at three points and held in position pneumatically. The stage can be lowered and removed, an experiment of some kind performed on the specimen, and the stage replaced on the table. Relocation of the stylus then occurs to within the width of the original profile. It is necessary to raise the stylus during the return stroke of the pickup, and of course the specimen may not be removed from the stage during the course of the experiment.. Fig. 5.4 shows a succession of relocated profiles of a surface which was initially rough turned (Thomas 1972). After each measurement the stage was transferred to the table of a surface grinder and a slightly deeper cut taken. The progressive dtsappearance of the peaks and the persistence of the valleys can be followed in great detail.

Figure 5.4. Relocated profiles of an initially turned surface at progressive stages of grinding (Thomas 1972)

Other Measurement Topics

97

The original device has been widely copied and used to study wear in rolling and sliding contact (Grieve et al. 1970; Efeoglu et al. 1993a, b), contact mechanics (OCallaghan & Probert 1973), rolling of sheet metal (Atala & Rowe 1975), running-in (Stout et al. 1977), effect of successive coats of paint (King & Thomas 1978), and impact wear (Engel & Millis 1982). Detailed designs of relocation table have been described by Edmonds et al. (1977) and Sherrington & Smith (1993). When we move on from profile measurement to area measurement there is a tendency to assume that relocation is no longer so important because of the increased statistical reliability of the much larger data set. For measurement of average roughness parameters this is probably true in many cases, but there are other situations where a more detailed examination of the relocated surface is necessary. Several workers have found it necessary to use relocation for 3D work (Bengtsson & Ronnberg 1984, 1986, Jeng & Lalonde 1992) The very homogeneity of a well-machined surface can make it difficult to return to exactly the same area on an apparently featureless plain. A 99% match in x and y implies a mismatch of 2% in area, quite enough to interfere with extreme-value calculations; in fact Newman et al. (1989), scanning wear scars with a stylus instrument, found linear relocation within 0.1% necessary to avoid sigdicant error in scar volumes. Rather than rely on mechanical relocation, Newman et al. marked the workpiece with a pattern of indentations which could be realigned visually, a practice also followed by Blunt et al.( 1994).

5.3. Replication

Replication of the original surface to be measured is needed with some optical methods to provide a transparent specimen (Anderson 1969, Dyson 1955, Herschman 1945, Lech et al. 1984). It is also needed in electron microscopy (Andersson 1974, Butler 1973, Chan et al. 1976) to provide a conducting specimen from a non-conducting workpiece. Its use with stylus instruments is generally to obtain measurements on parts which are not easily accessible, such as internal surfaces (Timms & Scoles 1948) or underwater surfaces (Sawyer 1953), or which cannot conveniently be brought to the instrument, such as gear teeth (Timms & Scoles 1948, Young & Clegg 1959), crankshafts (Davis 1979), the rollers from steel mills (Pearson & Hopkins 1948), ships' hulls (Karlsson 1978, King et al. 1981, King 1982), large optical components (Gourley et al. 1985) and human teeth

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Rough Sudaces

(Mathia et al. 1989a,b). It has also been used with compliant surfaces in the belief that direct measurement would damage or misrepresent the surface (Dawson et al. 1967/68). The principle is usually to place the surface to be measured in contact with a liquid which will subsequently set to a solid, hopefully faithfully reproducing the detail of the original as a mirror image, what might be termed a negative. Materials such as plaster of Paris and dental cement have been employed, but it is now customary to use a polymerizing liquid. The vital question is how closely the replica reproduces the features of the original. Lack of fidelity may arise from various causes. The liquid may not wet the surface completely; usually it will first be necessary to degrease the surface carefully. If the surface is itself already wet, as may be the case for biological specimens, there may be problems of diffusion or even of chemical reaction during setting. Portions of the replica may adhere to the surface as they are parted unless a release agent is used. In any case the replica is a negative and a stylus instrument does not respond to a valley bottom in the same way as to a peak, so a further positive replica may need to be made (Fig. 5.5). In the case of transparent replicas, optical techniques often rely on detecting an optical path difference which is a function of refractive index. Misinterpretation can occur here due to inhomogeneity of the replica or to changes in refractive index due to temperature. A rigid replica may not reproduce short wavelengths faithfully, while a flexible replica may not be faithful to long wavelengths.

Figure 5 . 5 . Plateau-honed cylinder liner and positive replica (Ohlsson & Rosen 1993)

One series of careful comparisons made (Sayles et al. 1979) has found replicas of an optical flat of negligible measured roughness to show roughnesses of between 0.03 pm and 0.13 pm, and replicas of machined surfaces to disagree in

Other Measurement Topics

99

roughness with the originals by up to 17 per cent. Shunmugam & Fbdhakrishnan (1976), Narayanaswamy et al. (1979) and George (1979) have attempted to compare the power spectra of replica and original. This approach can show dramatically the range of wavelengths over which the replication material is effective (Fig. 5.6). Wavelength in p m

Wavelength in p m

Strand

dais

Warwick chemicals plymaster

resin B

-I 10

in I00 Spatial frequency in cycled mm-'

I@l

Spatid frequency in cycledmm

-1

Wavcleneth in am

Wavelength In p m

I

I

-I I

Parent wrface hefore and after acrulile replica

Strand glass rcsin C

in

I

In1 Spatial frequency in cycler/mm-'

,

-

I

-

IIY)

Spatial frcqucncy in cyclcrimm-1

Figure 5.6. Comparison of power spectra of original and replica for three diEerent replicating materials (George 1979). Effect of the act of replication on the original surface is also shown. Bands are 50% confidence limits

Replicas of calibration specimens have been made by electroforming (Song et al. 1988). Again this is a two-stage process ending up with a positive replica. The originals were essentially two-dimensional, that is to say the same "random" profile was reproduced across the whole width of the specimen, so the issue of relocation

100

Rough Su$aces

did not arise in comparing replica with original. Flatness did not reproduce well, and the nickel replicas were not as hard as the originals, but comparison of a number of different roughness parameters showed agreement usually within 2%, which is of the order of the traceable uncertainty of roughness measurements. This technique is probably too specialized and laborious for routine use in quality control.

5.4. In-process Measurement

A method of measuring the surface roughness of a component during its machining would clearly be valuable, either to terminate the machining process as soon as the required finish was obtained and thus increase throughput per machine, or to take part in some adaptive-control loop. Slow attainment of the required finish, for instance, might signal the need for tool replacement. The requirements for in-process measurement are fairly stringent. Measurement must be continuous and rapid and should preferably provide an electrical signal of some kind. The sensor must be robust and relatively insensitive to environmental changes such as temperature and the intermittent presence of films or sprays of lubricating or cutting fluids. It should be small and adaptable to as wide a range as possible of workpiece shapes and sizes. For obvious reasons the sensor should preferably not contact the workpiece. Young et al. (1980) conclude that only area (i.e. parametric) measurements made by optical instruments will satisfy a similar list of requirements, but in fact, as we will see below, a variety of methods have been used with varying degrees of success. A similar problem applies to inspection (Thomas 1997). Of course the majority of roughness instruments sold are used for inspection purposes, usually off-line and on a statistical sampling basis. But modem trends of quality control increasingly require 100% inspection (Kennedy et al. 1987), and this is now the norm for many other product parameters. Most existing roughness instruments are too slow for 100% inspection of mass-produced components. For a component produced at a rate of 3000 an hour, not excessive for many production lines, the total time available for setting up, data acquisition and processing is not much more than a second. Although stylus instruments have made impressive advances in speed (Morrison 1995), the only current techniques which can approach the speeds required for inspection are capacitance (Garbini et al. 1988, Table 5.1) or optical. For many comparative purposes these work well enough; difficulties arise when it is necessary to validate their performance in terms of legal standards,

Other Measurement Topics

101

which are still exclusively written in terms of stylus instruments and their limitations. Pneumatic gauging looks promising for in-process work, and it has been established (Wager 1967) that the dynamic effects of the moving workpiece are not serious. This is rather surprising, as the theory of the pneumatic gauge (Graneek & Wunsch 1952) assumes isothermal conditions whereas fluctuations at, say, turning speeds are more likely to be adiabatic. The pneumatic gauge is robust and the air jet will help to clear unwanted surface fluid. It also measures a parametric roughness integrated over the entire path of movement of the surface. Its dlsadvantages are that a nozzle fixed relative to the workpiece will be unduly affected by waviness, and that it is too insensitive to measure fine finishes. By miniaturising the nozzle and pressure transducers, Woolley (199 1) has succeeded in making a pneumatic profiler whch will resolve height differences of 12 nm at very high rates of translation (Table 5 . l), though the lateral resolution is no better than 75 pm.

I.Casing

2. Wheel flanges 3. Workpiccc 4. Bearing 5. Leaf-springjoint 6 . Leaf springs 7. Guide

8. Tracer pin

13

4

14

9. Leaf springs 10. Ferrite I I . Coil

12. Shock absorber 13. Inductive signal 14. Mount

Figure 5.7. Sectional schematic ofrotating stylus device for in-process roughness measurement (Dutschke & Eissler 1978).

The stylus instrument, which might be thought a priori unsuitable, has proved itself a serious contender in the measurement of turned surfaces (Dutschke & Eissler 1978). The measuring device is a steel cylinder which rotates in contact with the workpiece being machined. At every revolution of the cylinder a stylus, piercing a hole in its circumference, is deflected, producing an electrical signal

Rough Su$aces

102

which is read out through telemetry (Fig. 5.7). Surprisingly good correlations with orthodox roughness measurements were reported. A development of this design by Zhao & Webster ( 1 989) achieved translation speeds of 1 . 1 m/s (Table 5.1). Acoustic techniques have also been applied to in-process measurement of roughness. Blessing & Eitzen (1988) measured the amplitude of ultrasonic backscattering from stationary and moving surfaces. Roughnesses of between 1 pm and 40 pm were successfully determined at speeds of up to 5 m/s (Table 5.1). This work has since been extended by Coker and Shin (Shin et al. 1995, Coker & Shin 1996). The issue of speed would certainly seem to favour parametric methods, most of which are effectively instantaneous by comparison with machining speeds. However, for in-process measurement the critical speed is not the cutting speed but the speed of translation of the workpiece, which is generally much lower. Profiling techniques are steadily increasing in speed (Table 5.1) and are now well within the range of translation speeds for many machining techniques. Table 5.1. Profilers for in-process measuremenf in order oftranslation speed (adapted &om Thomas 1997).

Technique

Speed ( d s >

Reference

Capacitance

0.025

Garbini et al. 1988

Stylus

1.1

Zhao & Webster 1989

Optical

1.7

Mitsui 1986

Ultrasound

5

Blessing & Eitzen 1988

Pneumatic

52

Woolley 199 1

5.4.I . Optical Techniques

Optical techmques for in-process measurement have been reviewed extensively by workers at NIST (Young et al. 1980, Vorburger & Teague 1981). More recently, Mitsui (1986) has described a number of optical instruments designed for in-

Other Measurement Topics

103

process use. Instrumentation can be removed to a safe and convenient place by using fibre-optic techniques to interpose a flexible conduit (Adkins 1969; Spurgeon & Slater 1974, Takeyama et al. 1976, Mitsui 1986). Then there are the problems of variation in reflectivity and optical path length due to the intervention of fluid spray or films or particles of swarf, or an actual alteration in colour due to thermal changes during machining. Some of these variations might be removed by signal processing, for instance by taking the first or second differential of the reflected intensity of a continuously chopped signal (Tipton & Roberts 1967/68). A more promising technique might be polarization of coherent light; here the only measured quantity is the ratio of polarization of incident and reflected light, which is insensitive to any amplitude changes in the signal (Gee et al. 1975). The advent of coherent light sources has led to their application to in-process measurement of roughness (Fad1 & Parsons 1978; West & West 1978, Shiraishi 1980, Yanagi et al. 1986, Persson 1992). Takaya and Miyoshi (Takaya et al. 1995, Miyoshi et al. 1995) have used Fraunhofer diffraction to measure the roughness of polished silicon wafers in-process to better than 1 nm, calibrating their optical system against a stylus instrument. Optical stylus methods have also been used. Mitsui et al. (1985) built a focus-detection system based on astigmatic focussing with a vertical range and resolution of 1 pm and 1 nm respectively. Bodschwinna & Bohlmann (1991) measured the finish of milled crankshaft housings with a proprietary optical stylus; although this system was part of the production line it does not actually seem to have been used in-process. Kiyono et al. (1994) describe a system using two optical styli simultaneously, one of which has a greater vertical range but poorer lateral resolution. The coarse stylus thus acts as an optical skid for the fine stylus, filtering out waviness and also cancelling out extraneous sources of noise. In the laser scanning analyser (Clarke & Thomas 1979) a laser beam is reflected from a polygonal mirror rotating at high speed, down on the workpiece surface, where it is reflected into a fixed photodetector receiver with wide aperture (Fig. 5.8). The detector output is amplified and applied to the vertical deflection coils of an oscilloscope whose time base is provided by the rotation of the mirror. Localized variations in the reflectance of the surface thus appear as changes in signal strength whose position on the workpiece can be established from the time elapsed since start of the current scan. The spot diameter can be set from 200 prn upwards. When used to measure roughness the receiver aperture is masked to a narrow slit and the angular reflectance function is produced as the spot scans a strip whose width is dependent on the range from the surface and the angular width required.

104

Rough Sur$aces

At a given moment in any scan the fixed detector is receiving light scattered from the single point on the strip which happens at that instant to be illuminated by the deflected beam. The picture &splayed on the oscilloscope screen is therefore a symmetrical curve whose height at any point is proportional to the intensity of light scattered into the corresponding angle, and whose maximum corresponds to the reflection received at the specular angle. If the curve is characterized by its width at half the maximum amplitude the reflectance is effectively normalized. When this half-width is plotted against measurements by a stylus instrument for a range of surfaces correlation is better with slope than with roughness (Fig. 5.9).

Figure 5.8. Schematic of a laser scanning analyser (Clarke & Thomas 1979).

Another optical technique intended for in-process roughness measurement, this time by ellipsometry, has been described by Lonardo (1978). The quantity used to characterize the roughness is the deviation between the value of one of the ellipsometric angles for a smooth reference surface and its value for the rough surface measured. He found good agreement, approximately linear in some cases, between this deviation and the average roughness as measured by stylus. He also investigated the effect of contaminants on in-process ellipsometric measurements

Other Measurement Topics

105

of roughness during grinding, and found the error to be generally less than 10 per cent.

Figure 5.9. Variation of half-width with (a) roughness (b) mean absolute slope (Clarke & Thomas 1979). A: milled B: turned; C: spark eroded; D: shaped E: ground;; F: criss-cross lapped G: parallel lapped.

Several instruments measure the ratio of the specular intensity to the intensity at an off-specular angle. Since this ratio generally dec:eases with increasing surface roughness, it could provide a measure of the roughness itself. Peters ( 1 965) used this technique with the detector held 40 degrees off specular to determine the roughness of cylindrical parts while they were being ground. His results show good correlation between the diffuseness and roughness over a range of roughness up to 0.3 pm. Even under different lubrication conditions (oil, water, dry) the results are well fitted bv a single curve. A similar instrument was developed by Corey (1978) to measure roughness in the range 0.2 pm to 2 pm for high-speed quality control of the surface finish of machined hemispherical parts. Essential features of the instrument are its nondestructive capability and its ability to scan the entire surface of the part. The instrument uses the ratio of the intensity measured 15 degr2es off-specular to the specular intensity to yield a value for roughness. In order to make meaningful roughness measurements for a particular type of surface, a set of roughness specimens that have been manufactured in a way similar to the test specimens with known roughness values are required. Another system, developed by Takeyama et al. (1976), measures the ratio of the specular intensity at the surface-normal to the back-scattered intensity 30 degrees off-normal. This system is designed to be quite insensitive to surface vibration with its use of fibre optics bundles to transmit the incident and reflected

106

Rough Surfaces

light. Indeed, the measured signals from spinning parts are fairly stable with time. Takeyama et al's ratio measurements were performed on machined surfaces with high peak-to-valley roughness ranging from 5 to 80 pm. They found that the experimental curves of intensity ratio against roughness were a function of the tool radius used to machine the surfaces. This dependence on the manufacturing process again implies a need for a set of calibration specimens. Takeyama et a1 claimed that the curves of intensity ratio versus roughness were independent of the surface material, but this claim does not seem to be supported by all of their data.

5.5. References

Adkins, H., "A look at surface finish", Am. Mach., 113, 111-116 (1969). Anderson, W. L., "Surface roughness studies by optical processing methods", Proc. I.E.E.E., (Letters), 57, 95 (1969). Anderson, S., "Plastic replicas for optical and scanning electron microscopy", Wear, 29, 271-274 (1974) Atala, H. F. and Rowe, G. W., "Surface roughness changes during rolling", Wear, 32, 249-268 (1975). Bengtsson, A. and A. Ronnberg, "Absolute measurement of running-in.", Wear, 109, 329-342, (1986) Bengtsson, A. and A. Ronnberg, "Wide range three-dimensional roughness measuring system", Precision Engineering, 6 , 141-147, (1984) Blessing, G. V. and Eitzen, D. G., "Surface roughness sensed by ultrasound", Surface Topography, 1, 143-158 (1988) Blunt, L., Ohlsson, R., Rosen, B.-G., "A comprehensive comparative study of 3D surface topography measuring instruments", in P. Hedenqvist, S. Hogmark and S. Jacobson eds., Proc. 6th. Nordic Symp. On Tribology, Uppsala (1994). Bodschwinna, H., and Bohlmann, H., "Online surface roughness measurement in production lines for process control", 12th. IMEKO World Congress (Beijing, 1991) Butler, D. W., "A stereo electron microscope technique for microtopographic measurements", Micron, 4, 410-424 (1973) Chan, E. C.; Marton, J. P.; Brown, J. D., "Evaluation of surface roughness of metal films by transmission electron microscopy and ellipsometry", J. Vac. Sci. Technol. 13,981-984 (1976). Clarke, G. M., T. R. Thomas, "Roughness measurement with a laser scanning analyser", Wear, 57, 107-116 (1979).

Other Measurement Topics

107

Coker, S. A., and Shin, Y. C., "In-process control of surface roughness due to tool wear using a new ultrasonic system", Int. J. Machine Tools & Manufacture 36, 411-422 (1996) Corey, H. S., "Surface finish from reflected laser light", Proc. SPIE 153, 27 (1978) Davis, F. A,, "Replica techniques in the study of crankshaft journal topography", Automotive Engr. 46-47 (ApriWMay 1979) Desages, F. and Michel, O., Calibration of a 3 - 0 surface roughness measuring device, (Prodn. Engng. Dept., Chalmers University, Gothenburg, 1993) Dutschke, W. and Eissler, W., "A new sensor for measuring the surface roughness in-process on a grinding machine", Proc. 3rd. Con. on Automated Inspection & Product Control, 19-30 (Nottingham University, 1978) Dyson, J., "Examining machined surfaces by interferometry", Engineering, 179, 274-276 (1955). Edmonds, M. J., A. M. Jones, P. W. O'Callaghan and S. D. Probert, "A threedimensional relocation profilometer stage", Wear, 43, 329-340 (1977) Efeoglu, I.; Amell, R.D.; Tinston, S.F.; Teer, D.G., "Mechanical and tribological properties of titanium nitride coatings formed in a four magnetron closed-field sputtering system", Surface & Coatings Technology 57, 61-69 (1993a) Efeoglu, I.; Arnell, R. D.; Tinston, S. F.; Teer, D. G., "Mechanical and tribological properties of titanium aluminum nitride coatings formed in a four magnetron closed-field sputtering system", Surface & Coatings Technology 57, 117-121 (1993b) Engel, P.A. and H. B. Millis, "Study of surface topography in impact wear", Wear, 75,423-442 (1982). Fadl, M. F. A,, and Parsons, F. G., "Electro-optical flaw detection", Proc. 3rd. C o n . on Automated Inspection & Product Control, 111- 118 (Nottingham University, 1978) Garbini, J. L., Jorgensen, J. E., Downs, R. A. and Kow, S. P., "Fringe-field capacitive profilometry", Surface Topography, 1, 131-142 (1988) Gee, S., King, W. L., and Hegmon, R. R., "Pavement texture measurement by laser: a feasible study", in Surface texture versus skidding: measurements, frictional aspects and safety features of tire-pavement interactions, STP 583, 2941 (ASTM, 1975). George, A. F., "A comparative study of surface replicas", Wear, 57, 51-61 (1 979).

108

Rough Su6ace.s

Gourley; D., H. E. Gourley and J. M. Bennett, "Evaluation of the microroughness of large flat or curved optics by replication." Thin Solids Films, 124, 277-282, (1985) Graneek, M. and Wunsch, H. L., "Application of pneumatic gauging to the measurement of surface finish", Machinery, 81, 701-707 (1952). Grieve, D. J., Kaliszer, H., and Rowe, G. W., "A normal wear process examined by measurements of surface topography", Ann. C.I.R.P., 18, 585-592 (1970). Herschman, H. K., "Replica method for evaluating finish of a metal surface", Mech. Eng. 67, 119-122 (1945). Jeng, Y.-R., Lalonde, G. A., "3-D surface topography measurement system and its applications", in Special Publications 936, 175-182 (SAE,Warrendale, PA, 1992) Karlsson, R.I., "Effect of irregular surface roughness on the frictional resistance of ships", Proc. Int. Symp. on Ship Viscous Resist. 9, 1-20 (Gothenburg, 1978) Kennedy, C. W., Hoffman, E. G., Bond, S. D., Inspection and gaging 6e (Industrial Press, New York, 1987). King, M. J. and Thomas, T. R., "Stylus measurement of the microgeometry of a coated surface", J. Coatings Tech., 50, 56-61 (1978) King, M. J., "The measurement of ship hull roughness", Wear 83, 385-397 (1982). King, M. J., Chuah, K. B., Olszowski, S. T. and Thomas, T. R., "Roughness characteristics of plane surfaces based on velocity similarity laws", ASME Paper 81-FE-34 (1981) Kiyono, S., Yamatani, M., Ohe, A., Huang, P., Suzuki, H., "Critical angle type optical stylus with optical skid (2nd report) - construction by two optical sources and one receiving optical system", Seimitsu Kogaku Kaishi/Journal of the Japan Society for Precision Engineering 60, 114-118 (1994) Lech, M., I. Mruk and J. Stupnicki, "Comparison of tribological parameters of surfaces determined by the stylus method and by the immersion method of holographic interferometry." Wear, 93, 167-179 (1984) Lonardo, P. M., "Testing a new optical sensor for in-process detection of surface roughness", Ann CIRP 27, 531-534 (1978) Mathia, T. G., Brugirard, J. L., Duarte, J. and Maurin-Perrier, B., "Enamel and hydrocolloide dental replica surfaces: Part 1. Statistical characterisation of enamel topography" Surface Topography, 2, 157-172 (1989a)

Other Measurement Topics

109

Mathia, T. G., Brugirard, J. L., Balleydier, M., Duarte, J. and MaurinPerrier, B., "Enamel and hydrocolloide dental replica surfaces: Part 2. New statistical criteria for evaluation of replica fidelity", Surface Topography, 2, 173191 (1989b) Mitsui, K., "In-process sensors for surface roughness and their applications.", Precision Engineering, 8, 212-220, (1986) Mitsui; K., N. Ozawa and T. Kohno, "Development of a high resolution inprocess sensor for surface roughness by laser beam.", Bulletin of the Japan Society of Precision Engineering, 19, 142-143, (1985) Miyoshi, T., Takaya, Y., Saito, K., "Nanometer measurement of silicon wafer surface texture based on Fraunhofer diffraction pattern", CIRP Ann. 44, 489-492 (1995) Mollenhauer, C., "Surface topography measurement techniques", Proc. Int. Con$ on Surface Technol., Pittsburgh, 173-186 (SME, 1973). Morrison, E., "A prototype scanning stylus profilometer for rapid measurement of small surface area", Int. J. Mach. Tools Manufact. 35, 325-33 1 (1995) Narayanasamy, K., V. Radhakrishnan and R. G. Narayanamurthi, "Analysis of surface reproduction characteristics of different replica materials", Wear, 57, 6369 (1979) Newman, P. T., Radcliffe, S . J. and Skinner, J., "The accuracy of profilometric wear volume measurement on the rough LClB coated surfaces of an articulating joint", Surface Topography, 2, 59-77 (1989) O'Callaghan, P. W. and Probert, S. D., "Effects of static loading on surface parameters", Wear, 24, 133-145 (1973). Ohlsson, R., and Rosen, B.-G., "On replication and 3D stylus profilometry techniques for measurement of plateau-honed cylinder liner surfaces", in R. J. Hocken ed., Proc. ASPEAnnual Meeting, Seattle, 146-149 (1993) Pearson, J. and Hopkins, M. R., "Plastic replicas for surface-finish measurement",J. Iron & Steel Inst. 67-70 (May, 1948). Person, U., "Real time measurement of surface roughness on ground surfaces using speckle-contrast technique", Optics and Lasers in Engineering 17, 6 1-67 ( 1992) Peters, J., "Messung des Mitterauswertes Zylindrischer Teile Wahrend des Schleifens", VDI-Berichte 90, 27 (1965) Sawyer, J. W.,"Method for recording roughness of submerged surfaces", Am. SOC.Nav. Eng. J., 65, 816-821 (1953).

110

Rough &$aces

Sayles, R. S., Thomas, T. R., "Mapping a small area of a surface", J. Phys. E: Sci. Instrum. 9,855-861 (1976). Sherrington, I.; Smith, E. H., "Design and performance assessment of a Kelvin clamp for use in relocation analysis of surface topography", Precision Engineering 15, 77-85 (1993) Shin, Y. C.; Oh, S. J.; Coker, S. A., "Surface roughness measurement by ultrasonic sensing for in-process monitoring", Trans. ASME: J. Eng. Ind. 117, 439-447 (1995) Shiraishi, M., "In-process measurement of surface roughness in turning by laser beams", ASME Paper 80-WAPROD- 17 (1980) Shunmugam, M. S. and Radhaknshnan, V., "An analysis of the reference lines of the surface profile and its true replica", Wear, 40, 155-163 (1976) Spurgeon, D. and Slater, R. A. C., "In-process indication of surface roughness using a fibre-optics transducer", Proc. 15th Int. Machine Tool Des. & Rex Con$, Birmingham, 339-347 (1974). Song, J. F., Vorburger, T. V. and Rubert, P., "Comparison between precision roughness master specimens and their electroformed replicas. ", Precision Engineering 14, 84-90 (1992) Stout, K. J., King, T. G. and Whitehouse, D. J., "Analytical techniques in surface topography and their application to a running-in experiment", Wear, 13, 99-1 15 (1977) Takaya, Y., Miyoshi, T., Arai, M., Hayashi, K., Setaka, M., "Development of random micro-roughness measuring apparatus based on Fraunhofer diffraction subnanometer measurements by the new error calibration method", Seimifsu Kogaku Kaishi/Journal of the Japan Society for Precision Engineering 61, 377381 (1995) Takeyama, H., Sekiguchi, H., Murata, R., Matsuzaki, H., "In-process detection of surface roughness in machining", Ann CIRP 25,467-471 (1976) Teague, E. C., "Scanning tip microscopies: an overview and some history", in G. W. Bailey ed., Proc. 46th. Annual Meeting of the Electron Microscopy Society ofAmerica, 1004-1005 (San Francisco Press, San Francisco, 1988) Terman, F. E., Radio Engineering 2e (McGraw-Hill, New York, 1937) Thomas, T. R., "Trends in surface roughness", Trans. 7th Int. Con$ on Metrology & Properties of Engineering Surfaces (Chalmers University, Gothenburg, 1997) Thomas, T. R., "Computer simulation of wear", Wear, 22, 83-90 (1972). Timms, C. and Scoles, C. A., "Some applications of the plastic replica process to surface finish measurement", Machinery, 73, 871-875 (1948)

Other Measurement Topics

111

Tipton, H. and Roberts, J. I., "New optical method of assessing surface quality", Proc. I. Mech. E., 182, Part 3K, 274-278 (1967/68). Tolansky, S., Multiple-beam interferometry of surfaces and f h s , (Dover Publications, Inc., New York, 1970a). Tolansky, S., Multiple-beam interference microscopy of metals, (Academic Press, London, 1970b). Tolansky, S., Surface microtopography, (Longmans, London, 1960). Unsworth, A., and Hepworth, A,, "A new stereo-adapter for use with the scanning electron microscope", J. Microscopy 94, 252 (1971) Vorburger, T. V.; Teague, E. C., "Optical techniques for on-line measurement of surface topography", Precis. Eng. 3,61-83 (1981). Wager, J.G., "Surface effects in pneumatic gauging", Znt. J. Mach. Tool Des. Rex, 7, 1-14 (1967). West, R. N., and West, P., "New applications of laser scanners for on-line product inspection", Proc. 3rd. Conf on Automated Inspection & Product Control, 133-138 (Nottingham University, 1978) Williamson, J. B. P. and Hunt, R. T., "Relocation profilometry", J. Phys .E: Sci .Instrum., 1, 749-752 (1968). Woolley, R. W., "Pneumatic method for making fast, high-resolution. noncontacting measurement of surface topography", Proc. SPIE 1573, 205-2 15 (1991) Yanagi; K., T. Maeda and T. Tsukada, "Practical method of optical measurement for the minute surface roughness of cylindrical machined parts.", Wear, 109, 57-67, (1986) Young, A. P. and Clegg, B. H., "Replica method for examining surface profiles" Rev. Sci. Instrum., 30, 444-446 (1959). Young, R. D.; Vorburger, T. V.; Teague, E. C., "In-process and on-line measurement of surface finish", Ann. CIRP 29,435-440 (1980). Zhao, Y. W. and Webster, J., "An in-process roughness measuring system for adaptive control of plunge grinding", Surface Topography, 2, 247-26 1 ( 1989)

CHAPTER 6

DATA ACQUISITION AND FILTERING

6.1. Data Acquisition

The use of digital techniques is now so widespread that one is unlikely to find any new roughness measurement instrument which relies solely on analogue methods. Even if the transducer itself is non-electrical, processing and presentation are likely to involve digital electronics. The process of analogue-to-digital conversion (ADC) amounts to the representation of the continuous analogue signal by a series of discrete numbers. This discretisation occurs in two ways (Fig. 6.1). In the amplitude domain, the signal is split into a number of levels parallel to the plane of the surface. This process is called quantisation. In the frequency or wavelength domain, the instantaneous value of the signal is recorded at equal intervals in the plane of the surface. This process is called sampling. Amplitude domain: quantisation

V

Frequency domain: sampling

Figure 6.1. Analogue-to-digital conversion: quantisation and sampling

113

114

Rough Suflaces

The number of quantisation levels is determined by the resolution of the ADC hardware, usually expressed as a number of bits (powers of 2). Thus an 8-bit converter will quantize to 256 levels and so on, that is it will resolve amplitude variations to 1 part in 256. There are several points to watch here. One is that quantisation is likely to represent the limiting resolution of the overall measurement system; there is no point spending money on a transducer capable of resolving to one part in a thousand if its output is to be processed by an 8-bit ADC. The second is that the computing arrangements may themselves embody several components, and the limiting quantisation of the system will be the lowest of any of the components. A 16-bit ADC followed by an 8-bit processor, or a processor running software which will only represent numbers to 8 bits, is still an 8-bit system.

u ”

ul

L

Figure 6.2. Too few quantisation levels cause loss of detail

In deciding on an appropriate quantisation level it must be remembered that the quoted figure represents the entire range of the transducer, what for analogue instruments would be called the full-scale deflection (FSD). In practice, an instrument is usually set up so that the hghest peak in the entire signal is safely below FSD and the lowest valley is similarly above zero. Thus it can easily happen that for a nominal quantisation of 8 bits, for two-thirds of the length of a signal with a Gaussian amplitude distribution the variation in height is actually represented by only about 50 levels. This can lead to a sih:ation (Fig. 6.2) where small peaks which may be functionally significant are missed altogether. The remedy is obviously to increase the quantisation, but remembering that there will be a corresponding price to pay in longer conversion times and larger data storage requirements. Most commercial systems use at least 16 bits, and quantisation of less than 10 or 12 bits is not recommended. Similarly if the signal is sampled too often it will lead to data storage and processing difficulties. But if it is sampled too seldom, then any wavelengths

Data Acquisition & Filtering

115

present in the signal which are shorter than the sampling interval may be misinterpreted as longer wavelengths of the same amplitude (Fig. 6.3a). This effect is known as aliasing. According to the Nyquist sampling theorem (Wade 1994), the shortest measurable wavelength AN is twice the sampling interval. The effect of aliasing is to mirror the power spectrum of the aliased frequencies about (Fig. 6.3b), so that a real frequency oN + o appears as the Nyquist frequency oN, an aliased frequency of wN - w.

'Apparent waveform Sample points

Figure 6 . 3 . Aliasing: (a) short wavelengths misinterpreted as longer wavelengths by sampling too seldom; (b) aliased frequencies reflected about Nyquist frequency falsify power spectrum.

The sampling interval is usually matched to the dimensions of the probe. The simplest way to sample is "on-the-fly", that is to keep the translation mechanism in constant motion and sample the resultant signal at equal intervals of time, assuming that these represent equal intervals of horizontal distance. This will only be true if the translation is at constant speed, which is not always safe to assume (see Section 2.2.3). It must also be remembered that the actual ADC process takes a finite time, and that this time represents an integration of fine surface detail over a finite length of the signal, though this length is usually small compared with the sampling interval. The alternative to on-the-fly sampling is the added complication of an independent measurement of horizontal displacement.

6.2. Filtering

It very rarely happens that the output of any instrument is perfectly matched to the application for whch it is intended. Sometimes the instrument produces too little

116

Rough Surfaces

information for the intended purpose. More often it produces too much information, and the useful information has to be extracted or the extraneous information suppressed. In electrical engineering terms, theuseful information is the signal and the extraneous information is noise, and separating the noise from the signal is an essential preliminary to characterisation. This process of separation is calledjltering. The concept of filtering is borrowed from electronic engineeering, where the signal and the noise are both treated as essentially sinusoidal, continuous (i.e. indefinitely long signals are available for measurement and analysis) and stationary (i.e. increasing the length of the signal does not change the information present in it), and the problem therefore resolves into one of sorting groups of sinusoids. This approach is ill-suited to describing surfaces for several reasons: common experience tells us that surfaces are not sinusoids; the instruments which measure them do not produce continuous signals; and many real surfaces are not stationary. Nevertheless, because the properties of signals have been widely worked out in sinusoidal terms, it will be convenient for the time being to use this terminology. Restricting ourselves to two dimensions for the time being, we may think of a generalised measured profile consisting of a continuous spectrum of surface wavelengths. The width of the spectrum, i.e. the range of wavelengths, is fixed by the measuring instrument itself. The instrument will be unable to ”see” any wavelengths longer than its traverse length, and in fact the longest wavelength reliably represented in the spectrum will be only a fraction of the traverse length (see the later discussion on measurement of power spectra). At the other end of the spectrum, the instrument will be unable to see any profile wavelengths smaller than the dimensions of its own sensor. But there is no a priori reason why the spectrum of wavelengths produced by the instrument, designed as a general purpose device to suit a range of applications, should coincide with the spectrum of wavelengths associated with any particular application. In engineering metrology, for instance, the spectrum was once divided into six (DIN 4760, 1982), of which only the first three classifications need detain us here. The longest wavelengths are associated with errors of form; shorter wavelengths constitute waviness; and the shortest wavelengths are called roughness (Fig. 6.4a). DIN 4760 prescribes that the length of an error of form should be at least 1000 times its amplitude, and the ratio of wavelength to amplitude of the waviness should be between 1OO:l and 1OOO:l. This division is quite arbitrary; wavelengths associated with errors of form on a machined surface would be described as roughness on a ship’s hull, for instance. In manufacturing engineering, the problem is usually to remove the waviness and errors of form from

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the signal so that the remaining property of the surface wtiich is assessed is the roughness. Note again that there is no generally agreed wavelength which divides roughness from waviness, it is a matter for subjective assessment.

of danimni

Catternl Error of

1 I Wavelength

Figure 6.4. (a) Surface characteristics(Brooker 1984) and (b) their associated power spectrum (Thomas 1975)

One important reason for this removal is that for almost all real surfaces, the longer the wavelength, the larger the amplitude (Fig. 6.4b). This is a typical property of self-af€ine fractals, as we shall see in a later chapter, and it has the consequence that the numerical value of any parameter which depends on the amplitude properties of a profile will be dominated by the longest wavelengths present. For many practical purposes, then, filtering means hzgh-puss filtering. In the language of communication engineering, a high-pass filter stops long wavelengths (low frequencies) but passes short wavelengths (high frequencies). Using this terminology permits us access to the large body of existing work on digital filter design (see for example Wade 1994, Rorebaugh 1997, Golten 1997). Although this body of work is mainly concerned with low-pass filters, which stop short wavelengths but pass long wavelengths, the conceptual arguments are similar, as a high-pass filter is equivalent to subtracting the output of a lowpass filter from the original signal. Consider the internal computer representation of a profile, after analogue-todigital conversion, as a series of discrete heights. Conceptually a filter may be thought of as a sequence of weighting terms (the

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118

impulse response) which is moved along the profile z(i), multiplying it term by term and thus smoothing it as it goes (Fig. 6.5).

WelgM1ng function

m

Figure 6.5. Convolution as a low-pass filter (Golten 1997)

The sequence itself may be chosen arbitrarily, say of length 2m + 1 terms, and the output of the filter z’(i) is the convolution of the impulse response with the original signal :

The transmission coeflcient at a given wavelength is the ratio of the amplitude of the filtered signal at that wavelength to that of the unfiltered signal. The cutoffof the filter is the wavelength at which the transmission coefficient is some specified fraction of the input amplitude (Fig. 6 . 6 ) . The roll-off characteristic of the filter describes how sharply the transmission coefficient varies on either side of the cutoff. Clearly for roughness work as sharp a roll-off as possible is desirable. Eqn. 6.1 implies that an infinitely sharp cutoff, where transmission is 100% right up to the cutoff and zero immediately after it, is not realisable. Thus all real high-pass filters let some long-wmelength power bleed through and simultaneously stop some short-wavelength power which should have been admitted. All filter design is therefore a compromise. The large number of hfferent designs of filter which have been proposed all have particular advantages and disadvantages.

119

Data Acquisition & Filtering

025

06

25

80

Wnvelerqth in mm

Figure 6 . 6 . Transmission curves showing roll-off charactenstics at four standard cutoff lengths for a 2CR high-pass analogue filter (BS 1134, 1988)

Causal filters can only operate with historic data, that is for values of z (x + i) for which i < 0. All physically realisable filters are causal, because a filter operating in real time cannot "see" future data. But this is not a limitation for many roughness measurements, where pre-recorded data are being filtered, and non-causal filters may operate on values of z(x + i) wit]? positive values of i . Recursive filters are modified by a feedback from the output. Recursive filters are also called inJnite impulse response (IIR) filters because their weighting sequence cannot be represented by a fixed number of terms, whereas the weighting sequence of non-recursive filters is finite (FIR). IIR filters may potentially be unstable, whereas FIR filters are always stable. Because of the feedback mechanism implicit in IIR filters, fewer terms are needed to achieve a similar roll-off performance and so IIR filters are more efficient. On the other hand, linear phase response characteristics are impossible to achieve with IIR filters, whereas non-causal FIR filters do not introduce any phase distortion. The relative advantages of IIR and FIR filters for roughness assessment are discussed at length by Medhurst (1989). To determine the frequency response of the filtx, that is how the transmission coefficient varies with wavelength, it is necessary to take the Fourier transforms of the weighting sequence and the input signal and multiply them together. Their product is the Fourier transform of the frequency response, which may be recovered by inversion. This multiplication in the frequency domain is formally equivalent to convolution in the distance domain. So it is possible to proceed in the reverse direction, to decide on a required frequency response and hence to deduce the appropriate weighting sequence.

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Rough Su$aces

To begin with filters were analogue, operating on the actual instrument output in real time with physical components. The first internationally agreed filter (IS0 3274, 1975) was a cascade of two resistor-capacitor circuits, the socalled 2CR filter (Fig. 6.6), with a cutoff of 75%. The mean line with respect to which roughness parameters were calculated was taken as the zero voltage level of the output (the so-called M-system). This filter had an asymmetrical weighting function, the practical effect of which was to cause phase distortion of the output signal (Fig. 6.7).

Figure 6.7. Phase distortion caused by 2CR filter (Whitehouse & Reason 1965)

To avoid this distortion, Whitehouse (1967/68) proposed the use of a phasecorrected (PC) filter. He noted that a suitable weighting function would need to be concentrated in a central lobe and die away quickly on either side, though not so quickly as to produce undesirable harmonics. The effects on various machined surfaces of phase-corrected filters were computed by Shunmugam & Radhakrishnan (1976a). Whitehouse considered various weighting functions, including a Gaussian weighting function which satisfied the above criteria, but practical realisation was limited by the existing state of technology. With the advent of fast cheap digital processing, Gaussian PC filters are now practicable and have become the standard (DIN 4777, 1990). I S 0 11562 (1996) prescribes the weighting function as

where a = 0.4697 is a numerical constant. The cutoff is now defined as 50% transmission so that the mean line of the profile may be extracted by subtracting

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the high-pass filtered profile from the original, and the transmission coefficient at wavelength A, that is the ratio of the unfiltered amplitude a. to the filtered amplitude, is an/ao=

1 - exp f

-71

(&/A)

}

As well as desirable phase characteristics, the PC filter also has a sharper roll-off than the 2CR filter (Fig. 6.8).

Figure 6.8. Frequency responses of 2CR and PC filters compared for a cutoff of 0.8 mm

There are many other possible filtering techniques, some of which are in common use. Most are not capable of representation in terms of Eqn. 6.1, so although it is usually possible to define their cutoff, it is rarely possible to specie their frequency response. Earlier standards described the decomposition of the profile into a number of consecutive equal sample lengths (BS1134, 1972). To each sample length a separate straight mean line is fitted, by least squares or an equivalent method, then all the mean lines are joined in a single straight line (Fig. 6.9). This is an effective high-pass filter, but its operation cannot be specified in terms of Fourier transforms. Also, unless a separate algorithm is used to match the ends of the profile segments, the resulting discontinuities will give rise to spurious short wavelengths in the output spectrum.

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Figure 6.9. High-pass filtering by fitting straight lines to consecutive sample lengths (BSll34, 1972)

A polynomial filter (Fig. 6.10) can use any one of a number of well-known numerical techniques to fit a least-squares polynomial to the input profile. This and the previous filter are interesting as examples of techniques which will work adequately for roughness measurements with their short signal lengths, but would be unsuitable for the continuous signals of communications engineering.

Figure 6.10. Profile (a) fitted with a 12" order polynomial (b) after subtractingthe polynomial

Another filtering technique whose behaviour is difficult to quantify is the valley suppression filter (Schneider et al. 1988), prescribed by DIN 4776 (1990) and I S 0 13565 (1996) for use with surfaces containing deep valleys. Filtering takes place in three stages (Fig. 6.11). First, a high-pass PC filter is applied to determine the mean line. Next, all valleys below the mean line are removed and the profile is filtered again. Finally, the original profile is referred to the mean line from the second filter. The practical implementation of a filter algorithm is fraught with problems. How wide should the impulse response be? If it is too narrow, it will be fast but the filter will ring. If it is too wide, it will be slow, it may not have enough profile to process in order to give a stable result, or alternatively if it overlaps the ends of the profile it may create end effects. Unfortunately the width of the weighting function

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Step 1

step 3

Figure 6.1 1 . The three stages of the valley suppression filter (Mummery 1990)

is not specified in the standards, and manufacturers tend to make their own decisions on commercial grounds, decisions whch they are not likely to share with their customers. This is called "method divergence" and is just one reason why software from two suppliers, both conforming to the letter of the standards, may give quite different roughness values for the same profile (Fig. 6.12). However, from Eqn. 6.2, the standard deviation of a Gaussian filter 0Y 0.19Ac. If the width is taken as ?C 3 s equivalent to 99.5% of the possible area of the weighting function, then 6 0 = 1.I&; that is, conveniently, the width of the window is about the length of the cutoff. Low-pass filters, which let through long wavelengths and stop short wavelengths, are not so much used in roughness work &cause the instrument sensor, as already pointed out, is its own low-pass filter, and because in any case short wavelengths, with their small amplitudes, have little effect on amplitudebased parameters. If for whatever reason a low-pass filter is required, a simple running average is often enough. If something more elaborate is required, many mathematical software packages offer spline filters. There is a good deal to be said for a first difference check on new profile data in any case. If this throws up occasional slopes of say 30 degrees or greater, when the steepest slope which the sensor will measure is 10 degrees, then there is a problem which needs dealing with before any further processing. International standards prescribe a range of preferred cutoff wavelengths, roughly in the ratio of 3 : l . The cutoffs most used on measuring instruments for

124

Rough Sur$aces

Unfiltered profile

2 Rc fitter

Phase Corrected filter (DIN 4777)

ValleySup ression

mer (DIN8776)

Figure 6.12. The effect of three different standard filters on the same profile (Mummery 1990)

. .:

. . . . .

. a

.

. .. . ... .

- ...

..

..

.. .. .

. _.

. *

.I

1.

2

4

6

U

10

12

14

16

18

28

Rq (microns) at 2.5 mm cutoff

Figure 6.13. Roughness of 200 profiles measured on the same surface at high-pass cutoffs of 2.5 mm and 50 mm (Medhurst 1989)

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production engineering are 2.5 mm, 0.8 mm, and 0.25 mm. As has already been mentioned, amplitude-dependent roughness parameters increase with high-pass cutoff wavelength (often approximately as its square root, see later), so when a roughness parameter is quoted, on a drawing or elsewhere, it is necessary also to spec@ the cutoff. If no cutoff is specified, a default value of 0.8 mm is assumed. This curious distance apparently arises because it was the width of the field of view in the earliest light-section microscopes, though I cannot trace a reliable source for this story. It is sometimes suggested that choice of cutoff is not particularly important if measurements are being used for comparative purposes only. It is argued that if roughness values are ranked in a particular order when measured at one cutoff, then they will probably be ranked in the same order when measured at any other cutoff. This amounts to claiming that there is a strong correlation between measurements made at different cutoffs. Fig. 6.13 should be enough to disabuse most readers of this misapprehension.

6.2.1. Envelope Filters

The M- or mean-line system, now undisputed, was once in contention with the socalled E- or envelope system. In the E-system (von Weingraber 1957), the reference lines are defined by the loci of centres of circles of different radii rolled along the profile. The locus of the centre of the larger circle gives the curve of form (Formprojl) while that of the smaller circle gives the contacting profile (Hullprojil) (Fig. 6.14). The geometrical projle is now drawn; this is defined as a profile of the surface determined by the design, neglecting errors of form and surface roughness. The area between the geometrical profile and the curve of form represents the errors of form; the area between the curve of form and the contacting profile represents the secondary texture or waviness; and the area between the contacting envelope and the effective profile (defined as the nearest instrumental approximation to the true profile) represents the primary texture or roughness. The E-system mean line is defined as the contacting envelope displaced downwards by a distance such that the areas enclosed by the effective profile above and below it are equal. The advantages of the E-system over the M-system are claimed to be that the E-system is physically more significant in that many engineering properties of a surface are determined by its peaks, and that the mean line is easier and quicker to construct graphically. The contacting profile and curve of form can be obtained

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126

Geometrical profile I

Locus of centre for rolling circle

w w -

-

Curve of form (Formprofil)

Errors of form

Contacting envelope (Hiillprofil)

Waves

Effective profile

Peak to valley

Figure 6.14. Terminology ofthe E- (envelope) system of reference lines in which the filters are two circles of radius r and R rolling along the profile (Olsen 1963)

mechanically by using skids of appropriate shape, and instruments so designed were apparently commercially available at one time. Algorithms for computing the various reference lines have also been published (Shunmugam & Radhakrishnan 1976b). The skids were in fact spheres rather than circles, causing an integrating effect which can alter the form of the various reference lines (Shunmugam & Radhakrishnan 1974). Standard radii were 25 mm for roughness and 250 mm for waviness, though other radii have been proposed (Radhakrishnan 1972). The system does, however, exhibit certain disadvantages. On a visual display, where the vertical magnification is exaggerated, the rolling circle becomes a "rolling ellipse". The mean line, composed as it is of a succession of intersecting arcs, is mathematically discontinuous. Attempts to reconcile the two systems (Ishlgaki & Kawaguchi 1981) have not been successful as it is difficult to represent the envelope action as a weighting function. A successor to the E-system has gained wider acceptance. This is the French system of so-called motif analysis, which is now the subject of an international standard (IS012085, 1996). Motif analysis is more than just a filtering technique, it is a whole new method for classifying surfaces, but it will be convenient to deal with it here. It is basically a set of algorithms which purports to embody the collective expertise, based on subjective visual judgement of profiles, of inspectors

Data Acquisition & Filtering

Figure 6.15. Motifanalysis (Mummery 1990). For explanation see text.

127

128

Rough Surfaces

in the French automobile industry. The conceptual foundation of the method is described by Fahl (1982). An experienced inspector, scanning a profile chart recording by eye, picks out a number of characteristic featuies or motifs which he knows are typical of the surface and on which the fitness of the surface for its purpose may be assessed. This is essentially a pattern-recognition process at which the human brain is known to excel. Motif analysis is an attempt to reduce this complex and almost instinctive process to a list of instructions.

T?<0,6 TR

Figure 6.16. The four conditions for motif combination (Fahl 1982). For explanation see text

The process consists of a sequence of steps (Fig. 6.15). (1) The profile is divided into a number of windows, each normally 250 pm wide, The peak-to valley height of the profile within each window is calculated and averaged over all the windows. (2) Peaks less than 5% of the average peak-to-valley height are eliminated. (3) The profile is divided into stylized patterns, each of which must

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contain a valley bounded by two peaks. (4) The patterns are combined according to a complex and arbitrary set of rules, given below, so as tr, minimise their total number. (5) The profile is now represented by a series of linked patterns. (6) The peak-to valley height of the profile within each window is again calculated and averaged over all the windows. Any peaks or valleys which are more than 1.65 standard deviations from the mean are reduced to this value, and the average peakto-valley height is once more calculated. (7) The upper envelope line is determined by joining the tops of the roughness patterns. (8) Waviness patterns are determined from the upper envelope line, using the same criteria for motif combination, but this time starting with windows 2.5 mm wide. In arriving at numerical values the rules for combination of motifs are clearly crucial. There are four conditions (Fig. 6.16): two adjacent motifs cannot be combined if their common middle peak is larger than thc two outer peaks; (b) combined motifs cannot be wider than 500 pm; (c) adjacent motifs may not be combined if the characteristic depth (the smaller of the two individual peak-tovalley heights within each motif) of the result is less than the characteristic depth of each of the original motifs; (d) adjacent motifs may not be combined if at least one has a characteristic depth less than 60% of the combined characteristic depth. These conditions are to be applied iteratively until no more motifs can be combined. I1101

Wavelength, l l f k jm

Figure 6.17. Comparison of the power spectra of various attempts to filter a profile of a ground surface (Shunmugam 1987). Continuous line, unfiltered profile; chained line, 2CR filter mean line; dashed line, rolling circle envelope; dotted line, motif upper envelope

130

Rough Su$aces

Considerable claims have been made for motif analysis. Boulanger (1992) describes the method as a filter for separating waviness from roughness with "an absolutely sharp cutoff'. Dietzsch et al. (1997) compare an unfiltered profile assessed by motif analysis with a profile high-pass filtered according to the Msystem, and not surprisingly find that the M-system has removed the waviness. Clearly the process described above is not readily represented in algebraic terms, though Scott (1992) has developed a formally rigorous definition of a motif. However, Shunmugam (1987) has tried to compare the motif method's performance with that of other filters for specific surfaces. In Fig. 6.17, where the power spectrum of the upper envelope line from a m o analysis ~ is compared, for the same profile, with the envelope of a rolling circle of 3.2 mm radius and the mean line of a 2CR high-pass filter with 250 pm cutoff, the roll-off of the motif envelope does not seem to be particularly sharp.

6.3. References Boulanger, J., "'Motifs' method. Interesting complement to I S 0 parameters for some functional problems", International Journal of Machine Tools & Manufacture 32,203-209 (1992) Brooker, K. ed., Manual of British standards in engineering metrology (British Standards Institution, London, 1984) BS 1134 Part 1, Assessment of surface texture: methods and instrumentation (British Standards Institution, London, 1988) BS 1134 Part 2, Method for the assessment of surface texture: general information and guidance (British Standards Institution, London, 1972) Dietzsch, M., Papenfuss, K., Hartmann, T., "The motif-method - a suitable description for functional, manufactural and metrological requirements", Trans. 71h. Int Con$ On Metrology & Properties of Engng Surfaces, 231-238 (Gothenburg, 1997) DIN 4760, Surface irregularities - terms and definitions - classification system (Deutsches Institut f i r Normung, Berlin, 1982) DIN 4776, Measurement of surface roughness: parameters Rk, Rpk, Rvk, Mrl, Mr2 for describing the material portion (projile bearivg length ratio) in the roughness profile: measuring conditions and evaluation procedures (Deutsches Institut f i r Normung, Berlin, 1990) DIN 4777, Profile filters for electric stylus instruments - phase-corrected filters (Deutsches Institut f i r Normung, Berlin, 1990)

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Fahl, D., "Motifcombination", Wear 83, 165-179 (1982) Golten, J., Understandingsignals and systems (McGraw-Hill, London, 1997) Ishigaki,H. and I. Kawaguchi, "Analysis of the initial contact height when a curved body approaches a rough flat surface", Wear, 54,273-289 (1979) I S 0 3214, Instruments for the measurement of surface roughness by the profile method - contact (stylus) instruments of consecutive projle transformation - contact profile meters, system M (ISO, Geneva, 1975) IS0 11562, Geometricproduct specijications (GPS) surface texture: projle method - metrological characteristics of phase correctfilters (ISO, Geneva, 1996) I S 0 12085, Geometric product specification (GPS) surface texture: profile method motifparameters (ISO, Geneva, 1996) I S 0 13565-1, Geometric product specifcations - surface texture: profile method surfaces having strati$ed functional properties: Part 1. Filtering and general measuring conditions (ISO, Geneva, 1996) Medhurst, J. S., The systematic measurement and correlation of the frictional resistance and topography of ship hull coatings, with particular reference to ablative antifoulings, PhD thesis, Newcastle University (1989) Mummery, L., Surface texture analysis: the handbook, (Hommelwerke, Muhlhausen, 1990). Olsen, K.V., "The standardization of surface roughness", Proc. Znt. Prod. Eng. Res .Con$, Pittsburgh, 8,655-658 (1963) Radhakrishnan, V., "Selection of an enveloping circle radius for E-system roughness measurement", Int. J. Mach. Tool Des. & Res. 12, 151-159 (1972). Rorabaugh, C. B., Digital filter designer's handbook 2e (McGraw-Hill, New York, 1997) Scott, P. J., "Mathematics of motif combination and their use for functional simulation", International Journal of Machine Tools & Manufacture 32, 69-73 ( 1992) Shunmugam, M. S., "Comparison of motif combination with mean line and envelope systems used for surface profile analysis", Wear 117,335-345 (1987) Shunmugam, M. S., Radhaknshnan, V., Tomparison of the mean lines of surface profiles obtained from different electrical filters", Zsr. J. Technol. 14, 253260 (1976) Shunmugam, M. S. and Radhakrishnan, V., "Two- and three-dimensional analyses of surfaces according to the E-system", Proc. I. Mech. E., 188, 691-699 (1974). ~

~

~

~

132

Rough Surfaces

Shunmugam, M. S.; Radhakrishnan, V., "Comparison of different methods for computing the two-dimensional envelope for surface finish measurements", Comput Aided Des. 8,89-93 (1976). Thomas, T. R., "Recent advances in the measurement and analysis of surface microgeometry", Wear, 33, 205-233 (1975). Von Weingraber, H., "Suitability of the envelope line as a reference standard for measuring roughness", Microtecnic, 11, 6-17 (1957) . Wade, G., Signal coding and processing 2e (Cambridge University Press, 1994) Whitehouse, D. J. and Reason, R. E. The equation of the mean line of surface texture found by an electric wave filter", (Rank Precision Industries, Leicester, 1965). Whitehouse, D. J., "An improved type of wavefilter for use in surface finish measurement", Proc. I. Mech. E., 182, Part 3K, 306-3 18 (1967/68).

CHAPTER 7

AMPLITUDE PARAMETERS

In presenting roughness parameters we will proceed in more or less historical order. We will begin with amplitude parameters, starting with extreme-value parameters and going on to average parameters and properties of the height distribution. We will then introduce the height distribution and its moments, which will lead us to a discussion of multiprocess surfaces and their description. Turning to texture parameters, we will discuss autocorrelation and the correlation length, leading to the power spectrum and its properties. This will enable us to deal with surfaces in three dimensions and to touch on the problems of nonstationarity. Fractals will be suggested as the logical treatment of nonstationary surfaces, and finally we will consider the difficulties presented by anisotropy. First a note of caution. Because roughness is implicated in so many technological and scientlfic problems, many workers without a background in metrology have invented new roughness parameters to describe their own application, not realising that conventions for the description of roughness already exist. The consequence of this is what Whitehouse (1982) has termed the "parameter rash, a proliferation of parameters, possibly running into hundreds, many of which are redundant in the sense that they are only slightly different ways of describing the same property, or that they may be derived from each other. No completely comprehensive account of these parameters exists, and the following account tries to concentrate on the main parameters of theoretical or historical importance or which are defined in standards. Hopefully this will help the reader, on coming across a new parameter, at least to place it into context. One problem in specifying numerical values of roughness parameters is their large uncertainty compared to other common engineering length measurements, because of their inherent statistical nature. In the United Kingdom calibration system the smallest uncertainty permitted to be claimed for a measurement of Ra was f 4% at one time (my own calibration laboratory was permitted *7% by the accrediting body). This may be compared with a typical calibration uncertainty of better than *0.001% for the length of a gauge block. A common measure of uncertainty is the coefficient of variation, that is the ratio of the standard deviation 133

Rough Sudaces

134

to the mean value of a set of measurements. Fig. 7.1 compares the coefficients of variation of a number of commonly used roughness parameters from 10 parallel profiles measured on a ground surface. Few of these are within sight of 4%: this is the reality of roughness measurement. Readers who study this figure will perhaps come to share a certain impatience with standards committees which argue for years about minuscule points of parameter definition. Skewness

0 5 0 4 0 3 0 2 01

0

01 0 2 0 3 0 4 0 5

50 40

0

10

30 20

10

20 30 40 50

Coefficient of variation. %

Figure 7.1. Coeficients of variation of a number of roughness parameters measured on a ground surface across (left) and along (right) the lay, showing also the effect of high-pass filtering (Thomas & Charlton 1981)

7.1. Extreme-value Parameters

The development of parameters to describe roughness followed closely on the development of instruments to measure roughness. The first reliable measuring instrument was the light-section microscope,and the only parameter which this could conveniently measure was the vertical separation of the highest peak and lowest valley of the unfiltered profile, Pt (DIN 4771, 1977) (Fig. 7.2). (Roughness parameters are normally denoted by a capital and some following lower-case letters. Sometimes the following letters are written as subscripts and sometimes on the line, by analogy with the abbreviations for dimensionless numbers in fluid

135

Amplitude Parameters

dynamics. We will follow the I S 0 convention of writing all amplitude-dependent parameters with following letters on the line.) Referenca line

I

Evaluation length ,I

m

Figure 7.2. Pt is the vertical distance between two parallel straight lines enveloping the unfiltered profile within the evaluation length Zm (Sander 1991)

When electrical filters became available it was possible to define the maximum roughness depth Rt (DIN 4762, 1960) in similar terms for the filtered profile (Fig. 7.3). When Rt is evaluated over only a single sample length it is known as Ry. Rt does not provide much useful informaticn by itself, so is often split into Rp, the height of the highest peak above the mean line, and Rm, the depth of the lowest valley below the mean line. Upper enveloping line

Evaluation length ,I

Figure 7.3. Rt is the analogue ofPt for a filtered profile (Sander 1991)

Extreme-value parameters are what the statisticians call inefficient parameters, that is they are unrepresentative of the surface because their numerical values vary so much from sample to sample, as we can see from Fig. 7.1. Averaging over several consecutive sampling lengths reduces this variation but it is still too large to be practical for most purposes. Rt has largely been replaced by the

136

Rough Surfaces

ten-point height Rz, defined as the vertical separation of the average of the 5 highest peaks, and the average of the 5 lowest valleys. Confusingly, there are two versions of Rz in common use (Fig. 7.4). Rz(IS0) (IS0 4287, 1984) is the average distance between the 5 highest peaks and the 5 deepest valleys within the assessment length. Rz(DZN) (DIN 4762, 1989) is the mean of 5 individual values of Ry for 5 consecutive sample lengths. Needless to say, these two definitions in general give two different numerical values for the same profile, though Sander (1991) claims that "for economic reasons" some instruments which pretend to measure the IS0 parameter actually measure the DIN parameter. A variation of &(DIN) is R3z, based on the third highest peaks and third lowest valleys. This is unfortunately far from an exhaustive account of extreme-value parameters; for a more extensive list see the useful monographs by Mummery (1990) and Sander (1991).

(a

1

Figure 7.4. Definitions of& according to (a) I S 0 (b) DIN (Mummery 1990)

Even more elaborate peak-to-valley definitions have been devised, notably the Allison system (Hydell 1967/8) developed by General Motors, and the Swedish Hsystem (SMS 47, 1994). Each system positions the peak and valley reference lines by truncating the higher peaks and lower valleys. The Allison or "general surface texture" system (Fig. 7.5) truncates the 10% highest peaks and lowest valleys, similarly the Swedish technique involves a 5% truncation for the upper reference line and a 10% truncation for the lower line. In the Allison system the reference lines are termed the upper and lower GST lines and enclose the general surface texture which is considered as the workable surface between initial wear-in and a severe wear condition. The difference between the reference lines is used as the roughness parameter (GST). Outside the GST reference lines the system is

Amplitude Parameters

137

reported to offer a control on the highest peaks and lowest valleys by specifying a permissible peak height and a a permissible valley depth.

Figure 7.5. The Allison system ofprofile characterisation (Hydell 1967/8)

In discussing extreme-value parameters there is a general problem of the definition of a peak. Consider Fig. 7.6. Most readers would agree that the large features A and B are peaks, and probably the smaller feature f, but what about points a-e and g? Clearly the numerical values of peak-rzlated parameters will depend on the answer to this question, which does not seem to be generally agreed. The simplest mathematical definition of a peak is a local maximum in z(x), but this has two undesirable consequences. Firstly, a peak so defined can occur below the mean line ( e g point g in Fig. 7 . 6 ) . Secondly and more seriously, the profile as we deal with it is not a continuous function, and the discrete form of our peak definition is a point higher than its nearest neighbours. This implies that the number of peaks depends on the sampling interval( point e is a peak at sampling interval A, but not at A*), and also on the quantisation level (Whitehouse 1978). Nevertheless we shall use this definition of a peak from here on unless noted to the contrary.

Figure 7.6. Peak definition: point e is a peak at sampling interval A1 but not at A1 (adapted from Dagnall 1980)

Rough Surfaces

138

7.2. Average Parameters

A parameter which represented the average properties of a profile would clearly be an improvement, and the extraction of such parameters became practical as soon as electrical instruments became available. The Americans passed the electrical signal through an AC voltmeter which responded to the root mean square average of the signal, so they defined the RMS roughness:

where L is the evaluation length (in practice usually several cutoff lengths for a filtered profile). The Europeans preferred to pass the AC signal through a rectifier so that it could be used to charge up a capacitor, so they defined the centre-line average (CLA) roughness:

The total area of the material-filled profile above the mean line should be equal to the total area of voids below the mean line (Fig. 7.7). The CLA roughness was known for a whle in the USA as the arithmetic average (AA) roughness, but finally, as Rq ceased to be used, both terms dropped out of use and Ra is now known simply as the average roughness.

R

.-

Figure 7.7. Derivation ofthe average roughness Ra (Mummery 1990)

The victory of Ra is rather a pity from a scientific point of view, as it has no theoretical application, whereas Rq, as we shall see below, has quite a fundamental

Amplitude Parameters

139

mathematical significance. It is easy to show that R a B q = 23f2n = 0.9 for the special case of a sinusoid (and RdRq = (2/7r)'/* = 0.8 for a Gaussian height distribution, see below). In practice, for many surfaces it is possible to ignore the difference and use Ra and Rq interchangeably, as the inherent statistical variation in all roughness measurements, to which we referred earlier, is of the same order. This is not necessarily true for surfaces with more valleys than peaks (or vice versa, whch is much rarer), as the squared term in Rq makes it more sensitive to outliers, and the ratio Ra'Rq can then differ significantly from unity (King & Spedding 1982). Unfortunately Ra does not discriminate very well between profiles of quite different characters (Fig. 7.8).

Figure 7.8. Three profiles with similar Ra values (Mummery 1990)

7.3. The Height Distribution

Unlike the extreme-value parameters, the average parameters contain no information about the spatial or textural variation of the profile, that is the variation in height from point to point of the surface. Logically, therefore, one could go a step further and discuss the variation of surface heights in terms of a statistical distribution, completely free of textural considerations. Theoretical representation of surface relief in statistical terms was well-established by the end of the Fifties (e.g. Longuet-Higgins 1957, Iwaki & Mori 1958, Bennett & Porteus 1961). The drudgery of mechanical calculation required by a statistical approach postponed its more general application until the advent of digital techniques, when

140

Rough Surfaces

a number of workers took the method up independently at about the same time (e.g. Kubo 1965, Whitehouse & Reason 1965, Greenwood & Williamson 1966). Instead of a height z(x) at a point x, we now consider the probability density p(z) of a distribution of heights (Fig. 7.9). The probability of a height lying in the between z and z+dz is p(z) dz,and the cumulative probability that a height will be below some level h is just

P(h) = J -m h p(z)dz

p(zj

0

50%

loo%

Figure 7.9. Profile height distribution p(z) and cumulative height distribution P(z)

The distribution p(z), whose mean we assume is at the height of the mean line of the profile, may be characterised by its central moments

The second moment ,uz is the variance and describes the excursions of the distribution from its mean, in our terms the roughness. Because the variance is in units of (height) squared it is customary to use dp2 instead; this is the standard deviation of the distribution, usually denoted by o, and is formally identical to the RMS roughness Rq. Many surfaces have more or less symmetrical height distributions, and many workers have found it convenient to assume that such surfaces can be represented by the well-known Gaussian distribution (Fig. 7.10):

141

Amplitude Parameters

p (z)dz=P(z
Hatched area =P(hS

z

I

Figure 7.10. Gaussian probability density and cumulative probability functions, with abscissa normalised by

Rq

If this is so, then it gives us access to the very large existing body of statistical results using the Gaussian distribution, whose properties are extensively tabulated. Certainly many surface height distributions look very closely Gaussian (Fig. 7.1l), and Williamson et a1.(1969) have argued that a process of surface formation consisting of a large number of random independent events, such as shot-blasting, will produce a Gaussian height distribution by the action of the central limit theorem. But are matters quite so straightfonvard?

Rough Surfaces

142

Surface height ($)

Figure 7.1 1. Distribution of 403,200 heights c om 5 nun x 7.5 mm of a ground surface, Rq compared to a Gaussian distribution (Sayles & Thomas 1979)

=

2.4 pm,

The usual way of establishing whether data fit a particular distribution is to apply a goodness-of-fit test, such as the test, which calculates the probability that a particular sample comes from a given parent population. The larger the sample, the smaller the differences between sample and population must be to satisfy the criterion. When a test is applied to the very large sample of Fig. 7.11, it fails; the sample is not from a Gaussian population. Indeed it has been shown that the technically important process of grinding should not produce a Gaussian height distribution (Sayles & Thomas 1976). There is a further complication: some authors confuse distributions of peaks on a profile, or summits on a surface, with distributions of all surface heights, and assume that all these distributions are Gaussian. In fact they are quite separate distributions, and it has been shown (Wlutehouse & Archard 1970, Nayak 1971) that if the distribution of all surface heights is Gaussian, then the distributions of peaks and summits must be nonGaussian. Should we then abandon the convenience of the Gaussian assumption? Probably not. Goodness-of-fit tests should be made on statistically independent observations, and surface heights, as we shall see in a later section, are internally correlated and hence not independent in the sense required. The effective sample size must therefore be much less, and the criterion for agreement correspondingly less rigorous, though it is difficult to quantlfy this. In any case the Gaussian assumption is rather robust, and even for undeniably non-Gaussian surfaces it yields statistical predictions which are in good agreement with experiment (Sayles

2

2

Amplitude Parameters

143

& Thomas 1979). We will continue therefore with due caution to assume Gaussian properties where it is mathematically convenient. We should bear in mind that this assumption is most questionable in the tails of a dlstribution, that is for the highest peaks and lowest valleys. Unfortunately the highest peaks are the regions most critical in such applications as contact mechanics. Attempts have been made to fit other distributions to surface height data, including beta ( e g Tzeng & Saibel 1967, Dakshina Murthy & Raghavan 1972, Whitehouse 1978, Spedding et al. 1980) and log-normal, gamma and Rayleigh distributions (Izmailov & Kuriya 1983), but they do not appear to have attracted much subsequent imitation.

Figure 7.12. (a) Skewness and (b) kurtosis (Mummery 1990)

The third central moment p3 is the skewness, usually normalised as Sk = p3 / Rq3. Because it is an odd moment, it is a sensitive measure of the degree of asymmetry of the distribution, as its name suggests (Fig. 7.12a). Most machined surfaces tend to be at least slightly negatively skewed, because peaks are more easily removed than valleys. The fourth moment is the kurtosis, normalised as K = / Rq4, and describes the overall shape of the distribution (Fig. 7.12b). A Gaussian distribution has a kurtosis of 3. Distributions which have a sharper central peak and longer tails are called leptokurtic. Distributions which are flatter are platykurtic. Although many software packages for roughness analysis offer kurtosis as well as skewness, not much practical use has been found for it. If

Rough Surfaces

144

numerical values are available for skewness and kurtosis, the ratio Ra/Rq can be calculated (King & Spedding 1982).

7.4. Bearing Area

In 1933 Abbott & Firestone defined the bearing area fraction at a given height above the mean line as the proportional length of all the plateaux which would result if the surface were abraded away down to a level plane at that height (Fig. 7.13). The sum of the lengths of individual plateaux at a particular height, normalised by the total assessment length, is the bearing ratio tp (DIN 4762 Part 1, 1960). If the bearing area fraction defined in this way is measured over a range of heights and plotted against height, the result is the bearing area or material ratio cuwe. Values of tp are sometimes specified on drawings, but this can lead to large uncertainties if the bearing area curve is referred to the highest and lowest points on the profile.

Figure 7.13. Derivation ofthe material ratio curve and the parameter tp (Mummery 1990)

We can now see that the bearing area at height h is simply the complement of the cumulative probability bstribution function:

t* = [;p(z)dz = 1-

Amplitude Parameters

145

For a Gaussian height distribution the function P(h) is tabulated in terms of Rq, hence if Rq is known the numerical value of tp at any height may be found immediately by inspection. The bearing area is a useful tool in characterising a large group of surfaces of some practical importance. Many technical surfaces, particularly those employed in machine components involving tribological interactions, are not produced in a single operation but in a sequence of machining operations. Typically the initial operation establishes the general shape of the surface with a more or less coarse finish, and subsequent operations refine this finish to give the final properties required by the design. Such a sequence of operations may remove the peaks of the original process and superimpose a finer texture on the resulting plateaux, but leaving the deep valleys of the initial process untouched. Such processes were termed interrupted $finishes (Martz 1949, Williamson et al. 1969) and more recently multiprocess (Whitehouse 1985) or stratijed (IS0 13565, 1996) surfaces. Characteristically their height distributions are negatively skewed, making it difficult for a single average parameter such as Ra to represent them effectively for speclfication or quality control purposes.

Figure 7.14. Derivation of (a) the core roughness Rk and (b) the peak height Rpk and valley dzpth Rvk (Mummery 1990)

146

Rough &$aces

The bearing area curve should contain all the information needed to categorise these surfaces. One way of extracting this information has been proposed by Schneider et al. (1988) (Fig. 7.14). A straight template covering 40% of the total is offered to the central and flattest portion of the bearing area curve and moved until its slope is a minimum. This straight line is projected through the axes, and the height which separates the two intercepts is defined as the core roughness Rk (Fig. 7.14a). The area above the intercept is now considered, and a right-angled triangle of the same area is constructed. The height of this triangle is the peak height Rpk. A similar construction at the other intercept finds the valley depth Rvk (Fig. 7.14b). This technique has been incorporated into a German standard (DIN 4776, 1990) and is now an international standard (IS0 13565, 1996). This technique certainly yields numerical values of the various parameters which discriminate successfully between two different surfaces with the same Fb (Fig. 7.15). It has been criticized by Zipin (1990) on the grounds that the upper and lower curved portions of the bearing area have nothing to do with peaks and valleys or stratified textures but are simply mathematical properties of the Gaussian distribution. A further and more serious criticism might be that the construction is quite arbitrary and lacks any theoretical basis.

Figure 7.15. Comparison ofRk, Rpk, Rvk for different profiles with the same Ra value (Mummery 1990)

Zipin himself proposes a more soundly based procedure originally due to Williamson (1967/68), who drew attention to &he possible application for roughness characterisation of the cumulative probability scale, the vertical axis of which is deliberately distorted so that a Gaussian distribution plots as a straight line (Fig. 7.16b). The slope of the line is the standard deviation of the distribution.

147

Amplitude Parameters

or in our terms Rq. If the height distribution of a multi-process surface is plotted on such a scale, it appears as two straight lines of different slopes, intersecting at a height which corresponds to the depth to which the final finishing process has superimposed itself (Fig. 7.16a, c). The two slopes are proportional to the two Rq values which may thus be determined immediately (Malburg & Raja 1993). If the straight lines are not very well-conditioned, their slopes and point of intersection may be found by fitting a hyperbola to the probability plot (ISODIS 13565-3, 1995). In the fitting process it should be borne in mind that not all points are of equal weight; the points in the central region of the distribution represent many times more data values than the points in the tails. v v

99 i

"i -K"

v v

w

A A

w

A

A

0 0

. ?

w

I'

Figure 7.16. Cumulative distributions of heights from three surfaces produced by two-stage processes, plotted on a scale distorted so that a Gaussian distribution appears as a straight line (Williamson et al. 1969): the second process makes the upper region of the surface (a)rougher (b) ihe same (c) smoother

7.5. References

Abbott, E. J. and Firestone, F. A,, "Specifying surface quality", Mech. Engng., 55, 569-572 (1933). Bennett, H. E. and Porteus, J. O., "Relation between surface roughness and specular reflectance at normal incidence", J. Opt. Soc. Am., 51, 123-129 (1961).

148

Rough Sudaces

Dagnall, H., Exploring surface texture (Rank Taylor Hobson, Leicester, 1980). Dakshina Murthy, H. B. and Raghavan, M. R., "Compliance of rough cylinders in compression", Wear, 20, 353-369 (1972). DIN 4762 Part 1, Assessment of 2nd to -fth order irregularities of surface conjguration by means of sections of surfaces: definitions relating to reference systems and dimensions (Deutsches Institut fur Normung, Berlin, 1960) DIN 4762 Part 2, Assessment of 2nd to Jib order irregularities of surface configuration by means of sections of surfaces: evaluation of profile sections based on geometrically ideal reference projile (Deutsches Institut f i r Normung, Berlin, 1960) DIN 4762, Surface roughness - terms and dejnitions (Deutsches Institut fiir Normung, Berlin, 1989) DIN 4771, Messung von der Profiltiefe Pt von Oberfachen (Deutsches Institut f i r Normung, Berlin, 1977) DIN 4776, Measurement of surface roughness: parameters Rk, Rpk, Rvk, M r l , Mr2 for describing the material portion (Projle bearing length ratio) in the roughness profile: measuring conditions and evaluation procedures (Deutsches Institut fir Normung, Berlin, 1990) Greenwood, J. A. and Williamson, J. B. P., "Contact of nominally flat surfaces", Proc. Royal SOC.A295, 300-319 (1966). Hydell, R. R., "Today's need - functional surface roughness control", Proc. I. Mech. E., 182, Part 3K, 127-134 (1967/68). I S 0 13565, Geometric product specijkations - surface texture: profile method - surfaces having stratij2ed functional properties (ISO, Geneva, 1996) IS0 4287, Surface roughness - terminologv - part 1: surface and its parameters (ISO, Geneva, 1984) ISO/DIS 13565-3, Surface texture (projle method) - characterisation of surfaces having stratified functional properties - part 3: height characterisation using the material probability curve of surfaces consisting of two vertical random components (ISO, Geneva, 1995) Iwaki, A. and Mori, M., "On the distrihution of surface roughness when two surfaces are pressed together", Bull. J.S.M.E., 1, 329-337 (1958). Izmailov, V. V. and M. S. Kuriva, "Use of the Beta-distribution for calculating the contact characteristic of rough-bodies", Trenie i Iznos, 4, 983-990, (1983) King, T. C. and T. A. Spedding, "On the relationships between surface profile height parameters", Wear 83, 91-108 (1982)

Amplitude Parameters

149

Kubo, M., "Instrument for the measurement of slope and height distribution of surface roughness", Rev. Sci. Instrum., 36, 236-237 (1965). Longuet-Higgins, M. S., "The statistical analysis of a random, moving surface", Phil. Trans. Royal SOC.,A249,321-387 (1957). Malburg, M. C., and Raja, J., "Characterisation of surface texture generated by plateau honing process", Ann. CIRP 42, 637-639 (1993). Martz, L. S., "Preliminary report of developments in interrupted surface finishes", Proc. I . Mech. E ., 161, 1-9 (1949). Mummery, L., Surface texture analysis: the handbook (Hommelwerke, Muhlhausen, 1990). Nayak, P. R., Tandom process model of rough surfaces", Trans. A.S.M.E. Ser. F. J. Lubr. Tech., 93, 398-407 (1971). Sander, M., A practical guide to the assessment of surface texture (Feinpruf Perthen, Gottingen, 1991). Sayles, R. S., Thomas, T. R., "Stochastic explanation of some structural properties of a ground surface", Int J Prod Res 14,641-655 (1976). Sayles, R. S.; Thomas, T. R., "Measurements of the statistical microgeometry of engineering surfaces", J Lubr Technol Trans ASME 101,409-417 (1979). Schneider, U., Steckroth, A,, Fbu, N. and Hiibner, G., "An approach to the evaluation of surface profiles by separating them into functionally different parts", Surface Topography, 1, 343-355 (1988) SMS Rapport 47, Ytjamnhet: nya parametrar, angivning pa ritnning och projildjupet H (Sveriges Mekanstandardiering, Stockholm, 1994) Spedding, T. A.; King, T. G.; Watson, W.; Stout, K. J., "Pearson system of distributions: Its application to non-gaussian surface metrology and a simple wear model", J Lubr Technol Trans ASME 102,495-500 (1980). Thomas, T. R.; Charlton, G., "Variation of roughness parameters on some typical manufactured surfaces", Precis. Eng. 3 , 9 1-96 (198 1). Tzeng, S. T. and Saibel, E., "Surface roughness effect on slider bearing lubrication", A.S.L.E. Trans., 10, 334-338 (1967) Whitehouse, D. J. and Archard, J. F., "The properties of random surfaces of significance in their contact", Proc. R. SOC.Lond., A316, 97-121 (1970). Whitehouse, D. J. and Reason, R. E., The equation of the mean line of surface texture found by an electric wave j l t e r (Rank Precision Industries, Leicester, 1965). Whitehouse, D. J., "Beta Functions for surface typologie?", Ann. CIRP, 27, 491-497 (1978)

150

Rough Su$aces

Whitehouse, D. J., "The digital measurement of peak parameters on surface profiles", J. Mech. Eng. Sci.,20, 221-227 (1978) Whitehouse, D. J., "The parameter rash - is there a cure?", Wear 83, 75-78 (1982) Whitehouse, D. J., "Assessment of surface finish profiles produced by multiprocess manufacture", Proc. Instn. Mech. Engrs., 199, B4, 203-270 (1985) Williamson, J. B. P., "The microtopography of surfaces", Proc. I. Mech. E., 182, Part 3K, 21-30 (1967/68). Williamson, J. B. P., Pullen, J. and Hunt, R. T., "The shape of solid surfaces", in: Surface mechanics. Ling, F.F. (ed.). 24-35 (A.S.M.E., New York, 1969). Zipin, R. B., "Analysis of the Rk surface roughness parameter proposals", Prec. Eng., 12, 106-108 (1990)

CHAPTER 8

TEXTURE PARAMETERS

From the preceding chapter it is plain that a description of a surface restricted to its variations in amplitude will be incomplete for practical purposes. If stranded on a mountain top at nightfall, it is useful to know that the average height of the surrounding terrain is 1000 metres, but it is more importan: to know how quickly the height changes with position. In terms of roughness, it is necessary to find some convenient means of describing the variation of relief in the plane of the surface. How, for instance, are we to distinguish between the two profiles of Fig. 8.1, which are visibly different but for which all the amplitude parameters are the same?

T

T

Figure 8.1. Two surfaces with the same amplitude parameters but different ''textures", and their respective autocorrelationfunctions

Historically various measures were proposed which could be measured readily from the output of a stylus instrument. The high-spot count HSC is the number of peaks per unit length of a profile, and its reciprocal, the high-spot spacing or mean spacing of profile irregularities Sm, is the mean distance between peaks (Sander 1991). This immediately returns us to the problem of peak definition; too large a peak will result in a parameter too sensitive to waviness, too small a peak will 151

152

Rough Sur$aces

result in a parameter swamped by the fine detail. In ISODIS 4287 (1984) a peak is defined, for this purpose only, as the distance between two mean line crossings (Fig. 8.2).

Mean line

s,

=

+ % + Snu + S, mm I

Figure 8.2. Mean spacing ofprofile irregularities (Sander 1991)

An alternative is the profile length ratio, fr (DIN 4762, 1989), also known as the profile roughness parameter RL (Gokhale & Drury 1993), defined as the ratio of the developed length of the profile to its nominal length. This is rather hard to measure and if profile slopes are small it is rather insensitive to changes in slope. It is also instrument-dependent; the finer the detail revealed by the sensor, the longer the real length. Sensor dependence is also a problem in measuring profile slopes, and for the same reason. Using the mean slope is no use, as its value will tend to zero unless the surface has teeth pointing one way; the mean of the moduli of the slopes da, or alternatively the FWS slope, must be used instead (IS0 DIS 4287, 1984), but the sampling interval, on which the slope depends, is not defined in standards. Spragg & Whitehouse (1974) proposed a parameter 2&a/Aa which they called the average wavelength h.By analogy there is a corresponding RMS wavelength Aq (IS0 DIS 4287, 1984), but neither of these parameters seems to have won wide acceptance.

8.1. Random Processes

In 1945 Womersley & Hopkins suggested that differences in surface texture might be more systematically described by a correlogram, that is by investigating the correlation between pairs of points on a profile as the separation of the pairs is vaned. The technology to implement their suggestion did not then exist, but by the late sixties Peklenik (1967, 1967/8) was computing various random-process

153

Texture Parameters

functions for machined surfaces. These are functions developed to describe time series and used in communications engineering for signal processing, principally the autocovariance function (ACVF) and its Fourier transform, the power spectral density function (PSDF). For a profile of length L the ACVF is

R(Z) =

1 ~

L-Z

L-S

Iz(x)z(x

+ 7)dx

0

where z(x) and z(x+Q are pairs of heights separated by a delay ACVF has dimensions of height squared.

7 (Fig.

8.3). The

Figure 8.3. Construction ofthe autocovariance function

This is often normalised as the autocorrelation function (ACF)

where R(0) is the variance Rqz of the height distribution. The ACF is dimensionless with an initial value of unity. The ACVF and ACF are very sensitive to periodic components of the surface and will detect these even when they are obscured by random components (Fig. 8.4). Lf the profile is entirely random, the autocorrelation function will decay asymptotically to zero at a rate which depends on the open-ness of the texture, and will thus distinguish between the two profiles of Fig. 8.1. Like the bearing area in the last chapter, the ACF is a function, not a number, and needs to be characterised numerically in some way if it is to be of practical use. The correlation length is defined as the length over which it decays to some fraction of its initial value, sometimes taken as a tenth, sometimes as lle. Pairs of points separated by distances greater than the correlation length are statistically independent. Note that random and Gaussian are not synonyms, though often used

154

Rough Sur$aces

as if they were. Random simply means uncorrelated; a table of random numbers would be expected to have a rectangular distribution (otherwise some lottery numbers would be luckier than others!), and it is perfectly possible to imagine a periodic profile with a Gaussian height distribution.

0.05

'0

2

4

6

8

10 pm

Figure 8.4. (a) Profile, @) height distribution, (c) ACF, (d) PSDF for a profile of a fine turned surface (Peklenk 1967/68). Note the periodicity due to the turning feed detected by the ACF.

The corresponding function in the frequency domain is the power spectrum or power spectral densify function (PSDF). The Wiener-Khinchine relation defines the PSDF (Fig. 8.4) as the Fourier transform of the ACVF (Bendat & Piersol 1966): (8.3)

Texture Parameters

155

where w = 2x / /z is an angular frequency with dimensions of reciprocal length. The PSDF is a probability density function like the height distribution, and its dimensions are frequently a source of confusion. A single value of the PSDF has dimensions of height squared per unit frequency, and like the height distribution it has moments m,: m, =

ia, a2

w"G(w)do

The zeroth moment mo is the variance of the height Istribution. Thus if the power spectrum is integrated over a pass-band of surface frequencies between q and m,the area under the function is the square of the RMS roughness, Rg2. The second and fourth moments are the variances of the distributions of slopes and curvatures respectively. Sayles & Thomas (1977) proposed the use of the structure fimction L-7

1

1 S( 2) = - {z(x) - z(x + 2)12rn L-2

(8.5)

0

as an alternative to correlation for surface roughness investigations. The structure function is related to the autocorrelation function by

It contains the same information as the ACF and the PSD, biit offers some practical advantages: it is stable and easy to compute, it does not impose a periodogram model on the surface, and it does not require prior hgh-pass filtering. Another useful property is that for a random stationary profile it tends asymptotically to a value of 2Rq2 as 2 + 00. The structure function can show functional changes more clearly than the ACF (Fig. 8.5). Smith & Walmsley (1979, Yolles et al. 1982) have proposed the use of Walsh functions as an alternative to Fourier analysis for surface profiles. Based on square waves, Walsh functions are binary hence lend themselves to fast computation. Whitehouse (1994) suggests that they are particularly suitable for representing surfaces with sharp discontinuities,but points out that they require many terms to represent periodic surfaces. Another alternative is the use of autoregressive

156

Rough Sur$aces

moving-average ( M A ) techniques, which treat the surface as the output of a system (the manufacturing process) which modifies white noise (DeVor & Wu 1971). ARMA models have been used to describe surfaces prduced by abrasive machining (Pandit et a1 1976), but Whitehouse again points out that they do not represent periodic processes well, and accurate models make heavy computational demands (Kovacevic & Zhang 1992).

--

c 1.0 9

0.0

Figure 8.5. (a) Autocorrelation functions and (b) structure functions ofthe same pair ofworn and unworn profiles

157

Texture Parameters

8.2. The Profile as a Random Process

Whitehouse and Archard (1970) investigated a model of a random profile with a Gaussian distribution of heights and an exponentially decaying autocorrelation function,

where p* is the correlation length. Defining a peak as a point higher than its two nearest neighbours, they obtained results for a number of statistical properties of the profile’s geometly. These are worth quoting at length, as they are among the more important results in the literature: Ratio of the number of peaks to the number of all heights:

Ratio of mean peak height to Rq:

Ratio of variance of peak heights to Rq2 (adapted from Whitehouse 1978: the equation in Whitehouse & Archard 1970 appears to be misprinted)

Mean peak curvature x

2 I Rq:

Variance of profile curvature (i.e. second differential of profile) x 6 - 8p + 2p2

2 / Rg:

Rough Sut$aces

158

Mean absolute slope:

It is instructive to present these results in terms of the delay t-, which for this purpose we may treat as the sampling interval. As the sampling interval decreases, the number of peaks decreases from a third to a quarter of the total number of heights, the mean peak height asymptotically approaches the profile mean line, and the peak standard deviation tends to Rq (Fig. 8.6a). The variation of slopes and curvatures is rather more dramatic (Fig. 8.6b); as the sampling interval gets smaller, the slope increases as its square root, while the pe& radius of curvature decreases as the 3/2 power. From these figures it is clear that the numerical result obtained for any texture parameter depends on the sampling interval at which the data is measured. The shorter the interval, the steeper the slopes and the more and sharper the peaks. It is hard to overemphasize the importance of this result, which implies that none of these texture parameters is an intrinsic property of the profile. It follows that any measurement of texture parameters must quote the sampling interval in order to be meaningful, and that it is impossible to compare results from Merent sets of measurements unless the respective sampling intervals are known.

0.8

0.4

I

/

0.2

0.01

0.10

1.oo

10.00

Sampling interval I correlation length Figure 8.6a. The Whitehouse-Archardrelations: Ratio of peaks to all heights, and mean peak height and standard deviation normalised by Rq, as a function of normalised sampling interval

Texture Parameters

159

I . pm

Figure 8.6b. The Whitehouse-Archard relations: variation of mean absolute slope and peak radius of curvature with sampling interval, for a profile of roughness 1 pm and correlation length 50 pm (Thomas & Sayles 1978)

8.3. Practical Computation

The practical problems of measuring distributions of slopes and curvatures are now seen to be considerable. The steepest slope and sharpest curvature which can be measured depend on the construction of the sensor; if a contact probe has an included angle of 90 degrees then it will record any slope steeper than 45 degrees as a 45 degree slope, unless an error trap is built into the acquisition software. The numerical values obtained will of course depend on the sampling interval as discussed above, and also on the numerical formula used for their calculation. Seven-point, five-point and three-point central difference formulae will all give different answers. The more points, the lower the uncertainty, but the higher the overhead of computation time; increasing the number of points used to calculate a slope or curvature amounts to low-pass filtering (Thwaite 1978). The effecl of different numerical techniques is discussed by Whitehouse (1978). Direct measurement of the mean peak radius of curvature (MPRC) presents a special problem, in that a plateau will return an infinitely large radius of curvature, usually

160

Rough &$aces

resulting in the computer refusing to divide by zero. The solution is to measure its reciprocal, the curvature, remembering of course that the numerical value of the mean reciprocal peak curvature is not in general the same as that of the MPRC. Direct measurement of power spectra from Eqn. 8.3 is time-consuming, and instead the fast Fourier transform (FFT) is universally employed (e.g. Wade 1994). This makes best use of the computer’s facility for modulo 2 arithmetic to speed up the computation of a spectrum of n data points by a factor n / log2n, = 100 for n = 1024. It has the consequent disadvantage that n must be a power of 2, so if, say, 2000 data points have been laboriously collected the FFT can only use 1024 of them. This is one reason why so many modern instruments are designed to acquire an integral power of 2 data points.

Figure 8.7. Ensemble averaging of power spectral estimates from profiles on a ground glass surface improves their stability (Thwaite 1978)

Computing power spectra has other problems. As the spectral wavelength approaches the length of the record, fewer and fewer data points are available for the estimate of power. Bendat (1958) has shown that this results in a systematic underestimate of power at long wavelengths, resulting in a spurious peak in the spectrum. When such long-wavelength peaks are seen in the literature they should always be suspected unless some special physical argument can be advanced to explain them. Bendat also showed that random errors in the spectral estimate increase as the record length decreases; for record lengths typical of roughness

Texture Parameters

161

measurement, these errors may be of the order of the estimates themselves. Raw estimates of spectral power may be "smoothed" by convolution with various windows (see for example Bendat & Piersol 1966), but this requires a prior assumption that no spectral peaks exist which the smoothing will obscure. Thwaite (1978) recommends the technique of ensemble averaging over a number of profiles (Fig. 8.7). The autocorrelation function can be computed directly, but in practice is usually computed by an inverse Fourier transform from the PSDF. Again delays of the order of the record length should be avoided, as the estimates of correlation will be based on very few data points and the function may go unstable. The correlation length turns out, disappointingly, to be rather sensitive to high-pass filtering (Fig. 8.8), and so in spite of its fundamental theoretical significance has not been used much for practical characterisation.

Figure 8.8. Autocorrelation functions of a profile of a shotblasted surface after high-pass filtering at cutoff wavelengths of (a) 40 mm, (b) 20 mm, (c) 2 nun, (d) 0.6 mm (Thomas & Sayles 1975)

So far we have drawn attention to the difficulties of obtaining unambiguous numerical values for texture parameters, and have tacitly assumed that amplitude parameters are well-defined by comparison. Unfortunately amplitude parameters too have their problems of definition. An assumption usually made in dealing with time series is that they are stationary, that is that there exists some finite length of record beyond which further measurement will reveal no new information. This appears not to be the case for a wide range of natural and man-made surfaces (Sayles & Thomas 1978), for the spectra of which power appears to increase

162

Rough Surfaces

indefinitely as approximately the square of the wavelength (Fig. 8.9). But, as we have already seen, the variance of the height distribution, and hence Rq, is a moment of the power spectrum. If the PSDF increases indefinitely, so must Rq and other amplitude parameters. Thus we have arrived at a situation where many, if not most, real surfaces, including the majority of technical surfaces, appear to have no intrinsic geometrical properties whatever, and any numerical values we obtain will depend entirely on the scale of measurement. This is a uniquely unsatisfactory state of affairs in metrology; indeed elsewhere in engineering only fluid dynamics suffers a similar indignity. Is there no way out of this impasse?

8.4. Fractal Roughness

The spectra of the surfaces of Fig. 8.9 can all be represented by a relation of the type

where p i s a dimensionless constant, and B is a constant with dimensions of length which Sayles & Thomas (1978) called the topothesy. These constants appear to be intrinsic properties of the surface; if they define the power spectrum, it should be possible to express most of the parameters which we have discussed in the last two chapters in terms of them. Mandelbrot (1978, private communication) pointed out that such surfaces are examples of fractals. Berry (1979) redefined the topothesy as the horizontal separation of pairs of points on a surface corresponding to an average slope of one radian. Fractals are functions which are continuous but not differentiable (Mandelbrot 1983). They possess the property of self-szmzfarity,that is they appear the same at any scale of magnlfication (Fig. 8.10). Self-similar fractals can be completely characterised by a single parameter, the fractal dimension D. Examples of self-similar fractals in nature include fracture surfaces (Baran et al. 1992) and natural terrain (Snow & Mayer 1992). However, man-made surfaces in general are subject to the further restriction that they appear to measuring instruments to be single-valued. T h s implies that smaller features must always have steeper slopes than larger features, as Whltehouse and Archard (1970) concluded on quite different grounds. Thus when the scale of observation is changed, a scaling factor with dimensions of length must be introduced to restore the appearance of self-

Texture Parameters

163

Figure 8.9. Variation of power spectral density with wavelength for 23 natural and man-made surfaces (Sayles & Thomas 1978). Solid line is best fit of slope 2.

164

Rough Su$aces

similarity. This scaling factor turns out to be the topothesy, and single-valued fractals of this kind are described as self-afine.

Figure 8.10. Self-similarityof surface profiles (Thomas & Thomas 1986)

Most methods of calculating the fractal dimension were developed for selfsimilar fractals, like coastlines, cracks in rocks and particles of powder (Russ 1994). They do not work very well for self-sine surfaces with gentle slopes. An effective way to calculate the fractal parameters of a profile is to compute the structure function. It can be shown that for a fractal profile (Russ 1994):

In other words, the structure function of a fractal profile obeys a power law, so it plots as a straight line on a log-log scale (Fig. 8.11). This is an easy way of establishing fractal behaviour, and from the slope and intercept of this straight line both the fractal dimension D and the topothesy A can easily be calculated. This is the fractal equivalent of Eqn. 8.8, and the respective slopes and intercepts are related by (Russ 1994):

p= B

=

2 0 - 4

( 2 ~ 2) r (~- p ) cos (

where T i s a Gamma function.

(8.10)

- p ~ / 2A3ffl )

(8.11)

Texture Parameters

165

1 log

T

Figure 8.11.Fractal parameters and the structure function

Fractal descriptions of engineering surfaces have been given by a number of workers (Wehbi et al. 1992, Stupak et al. 1991, Vandenberg & Osborne 1992, Yordanov & Ivanova 1995, Brown et al. 1997). However, some of this work should be treated with caution, as not all authors have distinguished between selfsimilar and self-fine fractals, and different methods of calculation can yield significantly M e r e n t numerical values of the fractal parameters (Russ 1994). Many machined surfaces seem to be fractal at least at shorter wavelengths (Fig. 8.12). Fractal dimensions may vary between the theoretical limits of 1 (a straight line) and 2 (a space-filling curve), whle topothesies may be represented by very short lengths (Table S.l), the physical significance of which is not immediately obvious (Russ 1994).

Figure 8.12. Structure functions of (a) spark-eroded (b) ground surfaces showing fiactal behaviour at short wavelengths, measured with standard (open circles) and fine (filled circles) stylus instruments. Circles are ensemble averages, error bars are standard deviations (Thomas& Thomas 1986).

166

Rough Surfaces

1-

bl-

-t ++'

0 0 1-

Figure 8.12 (continued). Structure functions of (c) bead-blasted (d) turned surfaces showing fractal behaviour at short wavelengths, measuredwith standard (open circles) and fme (filled circles) stylus instruments. Circles are ensemble averages, error bars are standard deviations (Thomas & Thomas 1986). Table 8.1. Fractal parameters for some machined surfaces (Thomas & Thomas 1986)

Machining process

D

A (nm)

Grinding Turning Bead-blasting

1.17 1.18 1.14

0.338 0.274 0.427

; In practice no real surface can be fractal over an infinite range of wavelengths, because no natural or man-made process can operate over an infinite range of wavelengths. A real surface will be formed by several different processes each with its characteristic features. For instance, a mountain landscape may be formed by erosion on a scale from kilometres down to meters. Below this scale the landscape may be covered with vegetation, which may also be fractal down to a scale of millimetres, but with completely different values of fractal properties. Such surfaces are called multij?uctul (for an extensive discussion, see Russ 1994) and typically will present a structure function as two or more straight lines of different slope meeting at a more or less sharp discontinuity (Fig. 8.13). Such a structure function is evidence of multifractal behaviour, and the wavelength corresponding to the discontinuity will mark the transition between two different

I67

Texture Parameters

mechanisms of surface formation. This transition point has been termed a corner frequency (Majumdar & Tien 1990).

0)

0,001

0,Ol

I

10

100

MID, unworn (7.4)

lo00

10000

Figure 8.13. Structure functions ofregions ofthe same cylinder liner in worn and unworn conditions showing multifi-actal behaviour with comer frequencies, measured with AFM and stylus instrument ( R o s h et al. 1997).

Machined surfaces are likely to be multifractal because they are usually produced by several processes. The approximate shape of the surface is produced by casting or rough preliminary machining. This will produce the form and waviness of the surface, possibly depending on the dynamics of the machine tool (chatter, spindle runout and so on). One or more finishing processes will produce the final roughness of the surface. One would expect corner frequencies to depend on the dlmensions of the actual cutting element, which may produce surface features far smaller than itself but cannot produce features any larger. Fractal parameters may conveniently be extracted from bifractals by adapting a method originally recommended for a similar problem, the extraction of Rq values from bearing area curves of stratified profiles plotted as two intersecting straight lines on a probability scale (IS0 DIS 13565-3, 1995). A pair of best hyperbolae are fitted to the structure function by least squares (Fig. 8.14). The intersection of the asymptotes gives the corner frequency, and the slopes, and hence the fractal parameters, are found by differentiating the lower hyperbola at each end of the structure function.

168

Rough Sur$aces

5.41 '

1

5.61.

Figure 8.14. Extraction of fractal parameters from a bifractal structure function by hyperbola fitling ( h i n i et al. 1998)

8.5. References

Baran, G.R.; Roques-Carmes, C.; Wehbi, D.; Degrange, M., "Fractal characteristics of fracture surfaces",Journal of the American Ceramic Society 75, 2687-2691 (1992) Bendat, J. S., Principles and applications of random noise theory (Wiley, New York, 1958) Bendat, J. S., Piersol, A. G., Measurement and analysis of random data (Wiley, New York, 1966) Berry, M. V., "Diffractals",J. Phys. A12, 781-797 (1979) Brown, C., Johnsen, W. and Hult, K., "Scale-sensitivity, fractal analysis and simulations",Trans. 7th. Int. Con$ On Metrology & Properties of Engng Surfaces, pp. 239-243 (Goteborg, 1997) DPJ 4762, Surface roughness - terms and dejnitions (Deutsches Institut fir Normung, Berlin, 1989)

169

Texture Parameters

DeVor, R. E. and Wu, S. M., "Surface profile characterisation by autoregressive-moving average models", Paper 7 1-WA/Prod-26 (ASME, New York, 1971). Gokhale, A,; Drury, W. J., "Surface roughness of anisotropic fracture surfaces", Materials Characterization 30,279-286 (1993) ISODIS 428711.2, Surface roughness terminology Part I : surface and its parameters (ISO, Geneva, 1984) ISODIS 13565-3, Surface texture (projle method) characterisation of surfaces having stratrfied functional properties - Part 3: Height characterisation using the material probability curve of surfaces consisting of two vertical random components (ISO, Geneva, 1995) Kovacevic, R.; Zhang, Y. M., "Identification of surface characteristics from large samples", Proc. I. Mech. E: Mech. Eng. Sci. 206C, 275-284 (1992) Majumdar, A., and C. L. Tien, "Fractal characterisation and simulation of rough surfaces", Wear 136, 313-327 (1990) Mandelbrot, B. B., The fractal geometry ofnature 3e (Freeman, New York, 1983) Panht, S. M., Suratkar, P. T. and Wu, S. M., "Mathematical model of a ground surface profile with the grinding process as a feedback system", Wear, 39, . 205-217 (1976) Peklenik, J., "Investigation of surface typology", Ann. C.I.R.P. 15, 381-385 (1967). Peklenik, J., "New developments in surface characterization and measurements by means of random process analysis", Proc. I. Mech. E., 182, Part 3K, 108-126 (1967/68). Rosen, B.-G., R. Ohlsson and T. R. Thomas, "Nano metrology of cylinder bore wear", Trans. 7th. Int. Conf: On Metrology & Properties of Engng Surfaces, 102-1 10 (Goteborg, 1997) Russ, J. C., Fractal surfaces (Plenum Press, New York, 1994). Sander, M., 4 practical guide to the assessment of surface texture (Feinpruf Perthen, Gottingen, 1991). Sayles, R. S. and T. R. Thomas, "The spatial representation of surface roughness by means of the structure function: a practical alternative to correlation", Wear 42,263-276 (1977). Sayles, R. S. and T. R. Thomas, "Surface topography as a non-stationary random process", Nature 271,43 1-434 (1978) Smith, E. H. and W. M. Walmsley, "Walsh functions and their use in the assessment of surface texture", Wear, 57, 157-166 (1979) ~

~

~

170

Rough Surfaces

Snow, R. S., and Mayer, L., "Introduction to special issue-Fractals in geomorphology", Geomorphology 5, 1-4 (1992). Spragg, R. C. and Whitehouse, D. J., "An average wavelength parameter for surface metrology", Rev. M. Mec. 20, 293-300 (1974). Stupak, P. R., C. Y. Syu and J. A. Donovan, "The effect of filtering profilometer data on fractal parameters", Wear 154, 109-114 (1992) Thomas, T. R. and Sayles, R. S., "Random-process analysis of the effect of waviness on thermal contact resistance", Prog. Astronaut. Aeronaut. 29, 3-20 (1975) Thomas, A. and Thomas, T. R., "Experimental measurement of surface roughness on a sub-micron scale", Proc. Int. Con$ Modern Prod. & Prod. Metrology, Tech. Univ., Vienna, 97.3-1 11.3 (1986) Thomas, T. R. and Sayles, R. S., "Some problems in the tribology of rough surfaces", Tribology International, 11, 163-168 (1978) Thwaite, E. G., "The numerical interpretation of topography", Wear, 51, 253267 (1978). Vandenberg, S., and C. F. Osborne, "Digital image processing techniques, fractal dimensionality and scale-space applied to surface roughness", Wear 159, 17-30 (1992). Wade, G., Signal coding and processing 2e (Cambridge University Press, 1994) Wehbi, D.; Roques-Carmes, C.; Tricot, C., "Perturbation dimension for describing rough surfaces", International Journal of Machine Tools & Manufacture 32, 211-216 (1992). Whitehouse, D. J., "The digital measurement of peak parameters on surface profiles", J. Mech. Eng. Sci. 20,221-227 (1978) Whitehouse, D. J., Handbook of surface metrology (Institute of Physics, Bristol, 1994) Whitehouse, D. J. and J. F. Archard, "The properties of random surfaces of significance in their contact", Proc. R. Sac., A316, 97-121 (1970). Womersley, J. R., & M. R. Hopkins, "Suggestions concerning the use of the correlogram for the interpretation of measurements of surface finish", Journees des etats de surface, 135-139 (1945) Yolles, M. I., Smith, E. H., Walmsley, W. M., "Walsh theory and spectral analysis of engineering surfaces", Wear 83, 151-164 (1982) Yordanov, 0. I., and K. Ivanova, "Description of surface roughness as an approximate self-sine random structure", Surface Science 331-333, 1043-1049 (1995).

CHAPTER 9

SURFACES IN THREE DIMENSIONS

The characterisation of surfaces in three dimensions of course increases the time and effort of computation, and introduces some new topics such as anisotropy which we have been able to disregard in the case of profiles. But much of the analysis is a reasonably straightforward extension of the previous two-dimensional discussion, and it will be convenient to deal with it in a similar order. First we should consider any special problems of three-dimensional filtering, then we can go on to discuss height and texture parameters as before, and introduce random process theory and fractals in three dimensions. Practical computation will be treated at some length as it poses some special problems in three dimensions. Finally we need to look briefly at the problem of anisotropy which is unique to three dimensions. We begin by making an important distinction, following Williamson and Hunt (1967/8). A local maximum on a profile will be called a peak, and a local maximum on a surface will be called a summit. A profile will more often than not pass over the shoulder of a surface feature rather than its summit. The shoulder will, nevertheless, appear as a peak on the profile (Fig. 9.1). Thus, as we shall see below, the dm-ibutions of peak and summit heights and curvatures may be quite Merent.

Figure 9.1. Peaks on profilesare not in general summits on surfaces 171

172

Rough Surfaces

9.1. Filtering

The special problems of 3D filtering are discussed at length by Stout et al. (1993). The first stage in processing raw data is to remove the DC level (often called the piston term in the optical literature), the trend and any errors of form. In two dimensions this is generally done by fitting a polynomial of appropriate order. In 3D it is more convenient to do this in stages. First the DC level and trend in x and y are removed by fitting a least-squares mean plane z * (x, y)

=

a + bx + cy

(9.1)

and subtracting it from the data. From here on values of z will be assumed to be referred to the mean plane unless stated to the contrary. The coefficients a, b, c are found in the usual way by minimising the sum of the squares of the differences between the mean plane and the data. The discrete solutions are given in the section on computation. Form errors, if any, may then be removed by fitting polynomials. Stout et al. give a general expression for a least squares polynomial surface, but point out that in practice higher order polynomials are rarely required. Often on inspection it suffkes to fit separate polynomials serially in x andy (Fig. 9.2).

Figure 9.2. 3D filtering of a ground surface (x-direction is into the paper): (top left) raw data; (top right) 3'd order polynomial in x fitted to remove form error, then (bottom left) aftex high-pass and (bottom right) lowpass Gaussian filtering at 0.08 mm cutoff.

Surfaces in 3 0

173

Stout et al. identify general requirements for a 3D filter: it should be zero phase so as to preserve the shape of surface features, and it should be separable, so profiles in the x- and y-directions can be filtered separately, thus saving computation time. They recommend the use of a Gaussian filter for general purposes, and a modified zonal filter if sharp cutoffs are particularly needed, both of which filters satisfy these requirements, and they recommend implementation by FFT for h g h computational efficiency (Fig. 9.2). Envelope filters also exist in three dimensions. The 3D analogue of a rolling circle is a rolling ball. Algorithms for a rolling sphere were developed by Shunmugam (Shunmugam & Radhakrishnan 1974a, b). Little difference was found between the power spectra of rolling circle and rolling sphere on the same profile (Shunmugam 1977). An extra difficulty presents itself in three dimensions. If the rigorous geometric condition is imposed that the sphere is always supported by the three highest points as it rolls in, say, the x-direction, then the locus of its centre can no longer be constrained within the x-z plane but must wobble in the ydirection. Also, the locus of the centre of the ball, which ir the reference surface, will be unduly sensitive to high summits. For this reason ball filters have been developed (Jonasson et al. 1997; Schmoeckel & Staeves 1997) where the virtual ball is allowed to penetrate the surface to varying degrees. A 3D motif filter has also been developed (Zahouani 1997), where the individual motif is a pit surrounded by summits.

9.2. Parameters Stout et al. (1993) have proposed a set of 14 parameters for characterising a surface in 3D. As the so-called "Birmingham 14" are at present the only serious proposals they are worth examining here in some detail. The set comprises 4 amplitude parameters, 4 texture parameters, 3 hybrid parameters, and 3 so-called "functional" parameters (Table 9.1). They are presented below in continuous form, adapting the original discrete forms where necessary; some of the discrete forms will be discussed later. The amplitude parameters are RMS deviation Sq, ten-point height Sz, skewness Ssk and kurtosis Sku, all defined by analogy with the two-dimensional forms, for instance

d:s

sq= - zZ(x,y)dxdy A

(9.1)

174

Rough Sursaces

where the domain of integration is the area of measurement A . The ten-point height is the difference in height between the mean of the 5 highest summits and the 5 lowest valleys.

Table 9.1. The Birmingham 14 (Stout et al. 1993)

Amplitude Parameters

Spatial Parameters

Hybrid Parameters

Functional Parameters

RMS Deviation sq

Density of Summits Sds

RMS Slope

s4

Surface Bearing Index Sb i

Ten Point Height

Texture Aspect Ratio Str

Mean Summit Curvature ssc

Core Fluid Retention Index Sci

Skewness Ssk

Texture Direction Std

Developed Area Ratio Sdr

Valley Fluid Retention Index Svi

Kurtosis Sku

Fastest Decay Autocorrelation Length Sal

sz

The texture parameters are the summit density Sds, the texture aspect ratio Str, the texture direction Std and the shortest correlation length Sul. The summit density is the number of local maxima of z (x, y) per unit area of the surface; thus a local maximum at the bottom of a large valley will be included by the definition. The texture aspect ratio is the ratio of the longest correlation length to the shortest correlation length (Tsukada 62 Sasajima 1983) (Fig.9.3). Correlation length is defined as the radial distance required for the area ACF to decay to 0.2. The texture direction is the angle relative to the polar axis normal to the lay, for which the angular variation of the area PSDF at some unspecified power is a maximum.

Su$aces in 3 0

175

1

b 0 3

0 2

0 1

001

'

'

0.4

'

'

0 3 0'4 0 6 0 mn.210 p / m

0 2

0 5

C

m

Figure 9.3. Contour of p (",y) = 0.2 for the surface of Fig. 9.2. The texture aspect ratio is the ratio of aa' to bb', about 2.5. Note that the scales of this figure and Fig. 9.4 are Cartesian, not polar, so radial distances must be scaled accordingly

The hybrid parameters are the RMS slope Sdq, the mean summit curvature Ssc and the developed area ratio Sdr. The developed area ratio is the analogue of the profile length ratio. The slope at any point x, y is 2

z'(x,y) =

/(E)2+(g) a &

so

The curvature at any summit is

Remembering that the sum of the curvatures of a surface at a point is equal to the sum of the principal curvatures,

176

Rough SurJaces

where there are N summits. The functional parameters are the surface bearing index Sbi, the core fluid retention index Sci and the valley fluid retention index Svi. The surface bearing index is the ratio of Sq to the height above the mean plane at 5% bearing area; for a Gaussian height distribution Sbi = 0.61. The core and valley fluid retention indices are respectively

and

(9.7)

where z* = z / Sq and the suffices refer to bearing area fractions. The statistical approach to quantifying roughness makes it clear that the bearing length fraction of a profile through a random surface must be identical to the bearing area fraction of the whole surface, because the height distribution of any profile must be the same as the surface height distribution. This may seem obvious but at one time was disputed by a school of thought which claimed that the profile bearing length fraction should be squared to obtain the bearing area fraction. It is of course possible to construct a surface for which this is the case, for instance a rectangular array of rectangular towers, but such a surface does not satisfy the condition that all profiles through it have the same statistical properties. The computation of some of these parameters will be discussed later. Some general comments may be made here. As in two dimensions, the amplitude parameters, and those depending on the autocorrelation function ,are sensitive to long wavelengths, and the other texture and hybrid parameters are sensitive to sampling interval. Stout et al. make no specific proposals for filtering on the grounds that this should be carried out on a functional basis. In addition, the summit density and mean summit curvature are sensitive to the definition of a summit. The definition of texture direction is ambiguous; if a fine textured finish were superimposed on a coarser finish, the texture direction as defined could vary with the level of power (Fig. 9.4). It seems likely that the so-called "functional" parameters are redundant; as they depend only on the amplitude distribution, enough information to reconstruct them is probably combined in the RMS

177

Surfaces in 3 0

deviation, skewness and kurtosis. So far few instrument manufacturers seem to offer the Birmingham 14; ironically, the 3D parameter most widely available on proprietary systems is Sa, the 3D analogue of Ra, which is not even on the list.

i

I

3 9 rnrn.64

P/mrn

t

.

1

99

rnrn - 6 4 P-’rnm

Figure 9.4. Contours of equal power for a plateau honed surface at two diferent levels of power. The texture direction is easier to discern ifthe level of power is defined.

9.3. Random Processes in Three Dimensions

Analogues of the autocovariance and autocorrelation functions, the power spectrum and the structure function all exist in three dimensions. The 3D ACVF is

and the 3D ACF

where the normalising factor R(0, 0) is the variance Sq2 of the surface height distribution. For an anisotropic surface the ACVF has the serious practical drawback of being multi-valued at the origin. The ACF, being normalised, does not suffer from thus drawback, but must be interpreted cautiously as it scales differently in different angular directions (Fig. 9.5). Presentations like Fig. 9.5 make use of the symmetry properties of the ACF, i.e.

178

Rough Surfaces

A

m

v

Fig. 9.5. (a) Gritblasted, (b) ground and (c) plateau honed surfaces with their 3D autocorrelationfunctions and power spectra (Amini et al. 1998)

Sufaces in 3 0

P (-L-rJ

=

p (rx,5)

179

(9. 10)

to plot the origin at the centre of the figure, but the axes are Cartesian, not polar. The 3D PSDF (Fig. 9.5) is

The 3D structure function is (Fig. 9.6)

Figure 9.6. 3D structure functions ofthe surfaces of Fig. 9.5, in the same order. Vertical scales are in pn2, horizontal scales are in mm (Amini et al. 1998)

zJ is the expected value of the squares of the differences in height between all the pairs of points on the surface whch are separated by d(r? + q2)(Fig. 9.7). If the ratio between ry and r, is maintained at some constant value a, then a straight line through (rx= 0, ry = 0) at an angle a to the r, axis is the ensemble average structure function for all profiles which could be drawn at an angle tan-' a to the xaxis. S(r,, 0 ) and S(0, 5) are the ensemble average SFs for all profiles parallel to the x and y axes respectively. S( r- c) where c is a constant is the ensemble average

S(z,

Rough Sufaces

180

of all pairs of heights separated by a distance r., on pairs of profiles themselves separated by a distance c; a kind of cross-correlation.

Y

Figure 9.7. 3D structure knction (Amini et al. 1998). Pairs of points z (1. j ) in the x,y plane separated by d(h2 + zi2)

9.4. The Surface as a Random Process

The description of surfaces in three dimensions requires the study of functions of several random variables. Nayak (1971), followed by Whitehouse & Phillips (1978, 1982) and Greenwood (1984), pointed out the engineering significance of the work of Longuet-Higgins (1957a, b, 1962). The description of ocean surFaces led Longuet-Higgins to develop in a classical series of papers the theory of statistical geometry. The full power of statistical geometry applies to situations in which the surface height gradient and curvatures are random and furthermore the surface is in motion. Herein we consider only cases in which the surface is static. Nayak assumed the surface to be stationary and random, with a Gaussian distribution of heights. Starting with profile statistics, he showed that a number of properties of the profile statistical geometry could be expressed in terms of the first three even moments of the profile power spectrum. The density of zero crossings, that is the number of times per unit length that the profile crosses its own mean line, which is equivalent to some definitions of the high-spot density, is given by

Surfaces in 3D

181

and the density of extrema, that is the number of local maxima and minima per unit length, is similarly given by

The mean absolute profile slope is (9.15) He goes on to define a bandwidth parameter

a

=

rngn4/rn;

(9.16)

=

The bandwidth parameter is related to the breadth of the surface PSD. As a + 1.5, its limiting value, the spectrum gets narrower, and as a + 00, it gets broader. The probability density of peak heights is a function of a; as a + co, the distribution approximates to the distribution of all surface heights, while at the other extreme as a + 1.5 the distribution is clearly non-Gaussian and the peak height lies more than one standard deviation above the profile mean line (Fig. 9.8). The mean curvature of peaks is also a function of a (Fig. 9.9). As a + co, the curvature becomes independent of peak height, while as a -+ 1.5, the higher peaks become sharper than the lower peaks.

-2.0

-1.0

0

1.0

in

2~

-29

-19

0

1.0

3.0

2.0

ZlRq

ZIRs

Figure 9.8. Probability densities of heights of (left) peaks (right) summits for various values of cz (Nayak 1971)

182

Rough Surfaces

-

e' z

$ 3

:

$ -3' ,z

0

ij' 9 a c

g

6 E

E l

l

0

0 1.0

I .o

0

1.0

2.0

z/Rq

3.0

z/Rq

Figure 9.9. Dimensionlessmean curvatures of (left) peaks (right) summils for various values of a (Nayak 1971)

Extending the analysis to an isotropic surface, he showed that the first three even moments of the surface power spectrum are identical to the corresponding profile moments, and hence that the density of summits is given by Sds = (1 I 67143) (m4/ m2)

(9.17)

and the absolute mean surface slope is A,

=

47cm2i2)

(9.18)

(n/ 2) Ap

(9.19)

so A,

=

The probability density of summit heights and the mean summit curvatures follow trends similar to those of the peaks (Figs. 9.8, 9.9), and as a + 00, the peak and summit distributions both tend to the distribution c;f all surface heights. Othenvise, the distributions show distinct differences; the mean summit height is much higher than the mean peak height as CL + 1.5. As a --+ co, peak and summit curvatures tend to a constant value, the summit curvature a little larger than the peak. But as a -+ 1.5, curvatures become sharper with increasing height, and the peaks become a little sharper than the summits. These are clearly important results. They imply that for a Gaussian isotropic surface, much of the information needed to predict the practical behaviour of the surface in, for instance, contact mechanics, may be obtained simply by measuring the power spectrum of any profile. The theory is sufficiently robust that its

183

Sur$aces in 3 0

predictions are substantially in agreement with experiment even for visibly nonGaussian surfaces (Fig. 9.10)

z/Rq

Figure 9. D. Ground surface 5 x 7 mm, sampled on a 7.8 x 12 pxn grid (Sayles & Thomas 979). (a) 389803 heights; smooth line is Gaussian distribution with the same standard deviation: @) 34997 summit heights; smooth line is Nayak's prediction

It is of some interest to see how the moments and their associated geometrical parameters behave for the regions of real profile spectra which we can measure. For power-law spectra of the form of Eqn. 8.8 the moment equations 8.4 become

(9.20) OH

integrating over a pass-band of profile frequencies between a high-pass cutoff and a low-pass cutoff m. If m is set by the sampling interval so that a = 251 / Ax and a N m, and if -3 < p < -1, as is usually the case, then the first three even moments are approximately

184

Rough Sur$aces

(9.21)

(9.22)

(9.23)

In other words, the roughness is independent of the sampling interval but depends on the high-pass cutoff, while the slopes and curvatures are independent of the high-pass cutoff but depend on the sampling interval. Furthermore, the summit density and the bandwidth parameter become

(9.24)

and

(9.25)

So the bandwidth parameter depends on the ratio of cutoffs, as one might expect, and the summit density decreases as the square of the sampling interval. Note that the summit density is a function only of the sampling interval and the rate of decay of the power spectrum, and so at any particular scale of measurement it is more or less independent of the process by which the surface is produced (Sayles & Thomas 1979). Eqns. 9.21-9.23 imply that the numerical values of the moments, and hence the geometrical properties of the surface which depend on them, are not an intrinsic property of the surface but are dependent on the scale of measurement (Fig. 9.1l),as a number of workers have pointed out (Nayak 197 1, Greenwood 1984, Majumdar & Bhushan 1990, Kant 1996). None of these expressions converges, so in the absence of any natural short-wavelength limit to the scale of surface features, it seems that as the scale of measurement gets smaller, so the number of summits increases and they become steeper and sharper indefinitely.

Surjaces in 3 0

185

Figure 9.1 1. Variation of spectralmoments with sampling &al on a grttblasted surface (Sayles & Thomas 1979). Open Circles: 6om Da 0.; closed circles: from correlation; continuous lines: from distributions

Power-law spectra are characteristic of fractal surfaces, and the moments can be related to the fractal dimension and the topothesy through Eqns. 8.10 and 8.11, hence the statistical geometry of the surface can be quantified from fractal parameters. Note that from Eqns. 8.10, 9.24 and 9.25, the summit density and bandwidth parameter are independent of the scaling factor and depend only on the cutoffs and fractal dimension.

9.5. Practical Computation The coefficients of the least-squares mean plane sampled at intervals Ax, Ay are

a=

(7mn +m+n - 5)w - 6(n + 1)n - 6(m + l)v mn(m + l)(n + 1) b = -6

2u-(m-l)~

Ax mn(m - l)(m + 1)

c=-

6 2v-(n-l)~ Ay mn(n - l)(n + 1)

(9.26)

(9.27)

(9.28)

186

Rough Surfaces

where (9.29) i=l J = I n

v=

m

XT(Z- l)Zi,

(9.30)

? = I j=1

n r n

w

zi,j

=

(9.3 1)

i=l j = 1

For practical implementation of filtering algorithms the reader is referred to specialist texts, e.g. Wade 1994, Golten 1997, Rorabaugh 1937. Turning to the Birmingham 14 parameters, the discrete form of Eqn. 9.1 is

(9.32)

Other amplitude parameters such as Su may be evaluated similarly, e.g. the discrete skewness is

(9.33)

Measurement of the ten-point height presents the same problem as measurement of the summit density: the definition of a summit. The first difficulty is that the number of summits counted per unit area depends strongly on the sampling interval, as discussed above. The second, and associated, difficulty is that a local maximum of a discrete process can only be defined in terms of its neighbours, and a summit higher than, say, its 4 nearest neighbours may not be higher than its 8 nearest neighbours (Fig. 9.12). Greenwood (1984) has calculated that for Nayak's surface model, 22% of 5-point summits are not 9-point summits. The dependence of summit properties on summit definition is discussed exhaustively by Whitehouse (Whitehouse & Phillips 1978, 1982, Whitehouse 1994).

187

Surfaces in 3 0

Ridge

SUMMIT

NO SUMMIT

Saddle point

SUMMIT

N O SUMMI'I

Figure 9.12. Discrepancies in summit definition: A and B are higher than their 4 nearest neighbours, but not lugher than their 8 nearest neighbours (Thomas 1997, after Greenwood 1984).

The numerical values of slopes and curvatures also depend on sampling interval, as discussed in the previous section. Stout et al. recommend a two-point formula to compute the slopes and a three-point central difference formula to compute summit curvatures:

i

z(i

+ 1, j ) + z(i - 1, j ) - 2x(i, j ) + z(i,j (W2

+ 1)

- 1) + ~ ( ji ,

-

2z(i, j )

(AYI2

(9.35) The discrete forms of the 3D ACVF and structure function (Eqns. 9.8 and 9.12) are

Rough Surfaces

188

and

The nearer the area ~5 approaches to the area of measurement A, the fewer data points are available for the computation of R(rmrJ and the less reliable its estimates will be, the limiting case being 42;5 < A. In practice the ACVF is often obtained by Fourier transforming the power spectrum. The PSDF is best computed by FFT, and again the reader is referred to texts on signal processing. Nayak's moments can of course in principle be computed directly from the power spectrum, but in practice it is rarely convenient to do this. There are three other approaches which may be used instead (Sayles & Thomas 1979). After first measuring Sq = dmo,the first and simplest method to find the higher moments is from Fqns. 9.13 and 9.14 by Counting the number of peaks, valleys and mean line crossings in some length of the profile. Unfortunately this turns out to be the least accurate. The second method is to measure the variance of profile slope and curvature distributions. The third method is to merentiatr: the profile ACVF at the origin, as m2 and m4 are respectively its second and fourth differentials. In discrete form, the first three points of the ACVF are needed:

(9.38)

m, =

2@R(0)- 4R(Ax) +R(2&))

(W4

(9.39)

9.6. Anisotropy

So far in this chapter the topics which we have treated have all had their 2D counterparts. We now come to a topic which has no counterpart in 2D because it deals with the directional properties of surfaces. Up to this point in the book we

Surfaces in 3 0

Figure 9.13. Results of some common machining processes

I89

190

Rough &$aces

have usually assumed, explicitly or implicitly, that roughness is an isotropic phenomenon, that is to say that surfaces will have the same microgeometric properties no matter what direction they are measured in. In fact this is not at all the case; most common machining processes produce surfaces with highly dmctional properties (Fig. 9.13). These anisotropic surfaces are said in traditional engineering parlance to possess a lay. Lay is rather Micult to define mathematically but rather easy to recognize visually; it is the principal direction or directions in which machining marks seem to the eye to line up with each other. Aniscltropy is not a feature unique to man-made surfaces; anyone who has tried to cross the South Wales mining valleys from east to west instead of north to south will agree that the local topography possesses a distinct lay. Processes like surface grinding and shaping can produce a surface which is almost two-dimensional, where a profile parallel to the lay will appear to be nearly smooth. Face turning produces a surface with a circular lay, while some milling and honing processes produce surfaces with a complex pattern which although clearly visible is difficult to describe (Fig. 9.14). Furthermore, the initial and final machining operations on a stratified surface may create lays in two distinct directions; the upper load-bearing region of the surface may have ridges running in one direction, while the deep valleys may tend to line up in quite a different direction. Clearly these directional properties are likely to affect the functional behaviour of the surface. Summits which are long and narrow are likely to have different load-bearing properties from more symmetrical summits, for instance, and similarly the passage of fluid between contacting surfaces is likely to be influenced by the lay. It becomes important, therefore, to find if possible some quantitative way of characterising anisotropy. There are really two different problems: to find the direction of the lay, and to quantify the degree of anisotropy. Dealing with the first problem, Boudreau & Raja (1992) observed that in 3D data from a raster scan, the lay will show up as a feature repeating between pairs of parallel profiles. The delay in the repetition will depend in a simple geometric way on the angle between the lay and the profiles; if the lay is circular, of course, this angle will change from profile to profile. The delay, which they call the relative shift, may be found by cross-correlating pairs of parallel profiles z(x,), z(x3 and measuring the displacement of the peak of the cross-covariance function (Peklenik & Kubo 1968, Kubo & Peklenik 1968) (9.40)

Sur$aces in 3 0

TYPE

Parallel

LAY

191

SYMBOL

/c_I

Perpendicular

Crossed

d.

Multi -directional

Particulate

Circular

Radial

Figure 9.14. Different kinds of lay and associated drawing symbols (Dagnall 1980; see also BS 1134:1988)

192

Rough Sudaces

from the origin. They succeeded in determining the angle of the lay for several surfaces with parallel lays, and the characteristic radius of machining for several surfaces with circular lays. For surfaces whose lay is not curved, the polar or quasi-polar presentation of the power spectrum recommended by Stout et al. (1993) can highlight the principal directions of the lay effectively (Fig. 9.4), and the texture aspect ratio provides a measure of the degree of anisctropy (Fig. 9.3). Longuet-Higgins' (1962) analysis of a surface as a random process included a discussion of anisotropy. He found that in general 9 moments of the surface PSDF, including 3 odd moments, were necessary to characterise an anisotropic surface, but that these could be combined in only 7 independent combinations. A ratio of these combinations defined a measure of anisotropy which he called the longcrestedness lly. For an isotropic surface y = 1 and for a "two-dimensional" surface on whch all the waves have infinitely long crests y = 0. McCool (1984) pointed out that if profile slopes are measured at various angles to some arbitrary reference on the surface, yis just the ratio of the maximum and minimum RMS slopes.

Figure 9.15. Isotropy and anisotropy (Thomas199 1). (a) isotropy: 1 mm x 1 mm of a shotblasted surface; (b) weak anisotropy: outlined area of (a) stretched 7.5 times in the y-direction, (c) strong anisotropy: 1.2 x 1.3 mm of a ground surface

193

Surfaces in 3 0

Longuet-Higgins' anisotropic analysis was subsequently extended by Nayak (1973) and Semenyuk (1986a, b). Nayak showed that the 7 invariants of LonguetHiggins could be derived from only 5 nonparallel profiles, which is still rather discouraging for practical purposes. Bush et al. (1979) made further progress by drawing an important distinction between strong and weak anisotropy. In the general case of weak anisotropy, indwidual surface features are exaggerated in one direction; a weakly anisotropic surface may be thought of as an isotropic surface which has been stretched out in one direction (Fig. 9.15). In the special case of strong anisotropy, on the other hand, the major axes of the surface features are aligned. Bush et al. were able to show that a strongly anisotropic surface can be characterised by only 5 independent parameters which may be obtained from two profiles only, one parallel and one at right angles to the lay (though Rudzit 1984 recommends a 45 degree profile as easier to measure). Fortunately, many machining processes are likely to produce strongly anisotropic surfaces.

-16.6

8% 1'z 25 J%

->

42.h

r4

-24.8

-)

1%

w x

-28.-

98

8% 1% g%

-28.9

->

z

-24.8~1~

Figure 9.16. Anisotropy of a milled surface at various height levels (Zahouani 1997)

Approaches based on, or conceptually similar to, moM analysis have been used to characterise anisotropy. Grigoriev et al. (1997) used an approach based on pattern-recognition algorithms adapted from image processing and succeeded in classlfj.ing more than 100 AFM measurements of surfaces into 4 texture groups. Zahouani (1997) developed 3D motifs and by selecting the characteristic height of the motif was able to detect different directions of anisotropy at different levels of

1 94

Rough Su$aces

the surface (Fig. 9.16). Barre et al. (1997) adapted techniques from geomorphology to detect analogues of catchment areas and watersheds on machined surfaces. To give meaninml results it was necessary to increase the size of motif so as to lose the finer detail. The process looks very much like low-pass filtering, and appears to give results rather similar to the PSDF method of Stout et al. discussed earlier.

18

0 ow1

17

5

-E 1 E-05

L

P

'I 1.6

6

15

5

a

5 14

X'

1E-06

1E-07

0 c

1 E-08

Y

13 1 E-09 1 .2

':I

1E-10

,

o

10

, M

,

,

,

,

,

,

,

30

40

50

60

70

a0

90

1E-11

0

20

10

0.01

30

40

50

60

70

80

90

Angle to x-axis (degrees)

Angle to x-axis (degrees)

I , 0

10

20

30

40

I

I

50

60

70

,

,

80

90

A"@s to x-axis (degrees)

Figure 9.17. Variation of fi-actaldimension and topothesy with angle to the x-axis for the surfaces of Fig. 9.5 (Amini et al. 1998). Circles: gritblasted; lozenges: ground; triangles and crosses: plateau honed, short and long wavelengths respectively

The fractal dimension of an isotropic surface is well established to be 1 + the fractal dimension of any profile through the surface ( e g Russ 1994), but what is the fractal dimension of an anisotropic surface? Russ discusses this question at length, and argues that the answers are different for weak and strong anisotropy. For a weakly anisotropic surface, the fractal dimension of a profile will be independent of the angle of measurement, but the topothesy will change. For a

Su$aces in 3 0

195

strongly anisotropic surface, both fractal dimension and topothesy will change; the fractal dimension along the lay will be less than that across the lay (Thomas & Thomas 1988). Davies & Hall (1998) predict that for a strongly anisotropic surface the fractal dimension will be the same in all angular directions except along the lay, where it will decrease. Rather confusingly, in the fractal literature "isotropic" and ''anisotropic'' are sometimes used as synonyms for self-similar and self-afhe respectively (e.g. Arvia & Salvarezza 1994). In an attempt to investigate the effect of anisotropy on fractal parameters, Amini et al. (1998) measured profile fractal dimension and topothesy as a function of angle for the three surfaces of Fig. 9.5 by taking sections through the 3D structure function. The gritblasted surface was isotropic, the ground surface was strongly anisotropic and the plateau-honed surface was an example of a stratified surface with a more complex anisotropy. For the gritblasted surface, both the fractal dimension and the topothesy were found to be independent of the angle of measurement (Fig. 9.17). For the ground surface, the fractal dimension rose from a minimum across the lay to a more or less constant value, then fell to a sharp minimum along the lay. The topothesy also showed a sharp minimum along the lay, where it fell by a dramatic 6 orders of magnitude. Fractal parameters for the plateau-honed surface showed minima parallel to the direction of the honing scratches.

9.7. References

Amini, N., B.-G. Rosen and T. R. Thomas, "Fractal surfaces characterised by a 3D structure function", Proc. Fractal 98 (in the press) Arvia, A. J., and Salvarezza, R. C., "Basic aspects regarding irregular metal surfaces and their application in electrochemistry", J. de Physique 4, C1, 39-53 (1994) Barre, F., Lopez, J., Mathia, T. G., "New method for characterising the anisotropy of engineering surfaces", Trans. 7'h. Int. Con$ On Metrology 8 Properties of Engng Surfaces, 479-486 (Goteborg, 1997) Boudreau, B. D.; Raja, J., "Analysis of lay characteristics of threedimensional surface maps", International Journal of Machine Tools & Manufacture 32, 171-177 (1992) BS 1134 Part 1, Assessment of surface texture: methods and instrumentation (British Standards Institution, London, 1988)

196

Rough Sugaces

Bush, A. W., Gibson, R. D. and Keogh, G. P., "Stroqgly anisotropic rough surfaces", Trans. ASME: J. Lub. Tech., 101F, 15-20 (1979) Dagnall, H., Exploring surface texture, (Rank Taylor Hobson, Leicester, 1980). Davies, S., and Hall, P., "Fractal analysis of surface roughness using spatial data", J. Roy. Statist. SOC.Ser. B 61 (in the press) Golten, J., Understanding signals and systems (McGraw-Hill, London, 1997) Greenwood, J. A., "Unified theory of surface roughness", Proc. Roy. SOC. Lond. A393, 133-157 (1984) Grigoriev, A. Y., Chizhik, S. A,, Myshkin, N. K., "Texture classification of engineering surfaces with nanoscale roughness", Trans. 7th. Int. Conf: On Metrology & Properties of Engng Surfaces, 3 19-324 (Gotebcrg, 1997) Jonasson, M., Wihlborg, A., Gunnarsson, L., "Analysis of surface topography changes in steel sheet strips during bending under tension friction test", Trans. 71h. Int. Con$ On Metrology & Properties of Engng Surfaces, 38-46 (Goteborg, 1997) Kant, R.,"Statistics of approximately self-afline fractals: random corrugated surfaces and time series", Phys. Rev. E53, 5749-5763 (1996) Kubo, M., and Peklenik, J., "An analysis of micro-geometrical isotropy for random surface structures", Ann. CIRP 16,235-242 (1968) Longuet-Higgins, M. S., "The statistical analysis of a random, moving surface", Phil, Trans. Royal SOC.,A249,32 1-387 (1957a). Longuet-Higgins, M. S., "Statistical properties of an isotropic random surface", Phil. Trans. Royal Soc., A250, 157-174 (1957b). Longuet-Higgins, M. S., "The statistical geometry of random surfaces", Proc. 13th Symp. on Appl. Maths, 105-143 (Amer. Maths. SOC.,1962). Majumdar, A,; Bhushan, B., "Role of fractal geometry in roughness characterization and contact mechanics of surfaces", Trans. ASME. Journal of Tribology 112, 205-216 (1990) McCool, J. I., "Characterisation of surface anisotropy", Wear 49, 19-31 (1978) Nayak, P. R., 'IRandom process model of rough surfaces", Trans. A.S.M.E: J. Lubr. Tech., 93F, 398-407 (1971). Nayak, P. R., "Some aspects of surface roughness measurement", Wear, 26, 165-174 (1973). Peklenik, J. and Kubo, M., "A basic study of a three-dimensional assessment of the surface generated in a manufacturing process", Ann. C.I.R.P., 16, 257-265 (1968).

Sur$aces in 3 0

197

Rorabaugh, C. B., Digital filter designer's handbook 2e (McGraw-Hill, New York, 1997) Rudzit, Y. A. ,"Methodology of analyticexperimental- determination of friction surface microtopographic parameters", Soviet Journal of Friction Wear, 5 , 57-63, (1984) Russ, J. C., Fractal surfaces, (Plenum Press, New York, 1994). Sayles, R. S.; Thomas, T. R., "Measurements of the statistical microgeometry of engineering surfaces", JLubr Techno1 Trans ASME 101,409-417 (1979). Schmoeckel, D., and Staeves, J., "Function-oriented 3D filtering for tribological assessment of sheet metal surfaces in deep-drawing", Trans. 7fh.Int. Con$ On Metrology & Properties of Engng Surfaces, 438-4A4 (Goteborg, 1997) Semenyuk, N. F., "Average values of total and mean summit curvatures and heights of asperities of an anisotropic rough surface." Soviet Journal of Friction and Wear, 7,47-56, (1986) Semenyuk, N. F., "Summit height probability density and anisotropic rough surface summit characteristics", Trenie i Iznos, 7, 1017-1024 (1986) Shunmugam, M. S. and Radhakrishnan, V., "Computation of the threedimensional envelope for roughness measurement", Int. J. Mach. Tool Des. Res., 14,211-216 (1974). Shunmugam, M. S. and Radhakrishnan, V., "Two- and three-dimensional analyses of surfaces according to the E-system", Proc. I. Mech. E., 188, 691-699 ( 1974). Shunmugam, M. S., "Effectiveness of the E-system in three-dimensional roughness measurement", Proc. Int. Con?Prodn. Engng. 2, (Inst. Engrs. (India), Calcutta, 1977) Stout, K. J., Sullivan, P. J., Dong, W. P., Mainsah, E., Luo, N., Mathia, T. and Zahouani, H., The development of methods for the characterisation of roughness in 3 dimensions, EC Contract No.3374/1/01170/90/2, Phase I1 Report, Vol.1 (March 1993) Thomas, T. R., "Trends in surface roughness", Trans. 7". Int. Con$ On Metrology & Properties of Engng Surfaces, (Goteborg, 1997) Thomas, T. R. and Thomas, A. P., "Fractals and engineering surface roughness", Surface Topography, 1, 1-10 (1988) Thomas, T. R., "Some problems in the characterisation of surface microtopography", Proc. SPIE 1573, 188-200 (1992) Tsukada, T. and Sasajima, K., "An assessment of anisotropic properties in three-dimensional asperities", Bull. Japan SOC.of Prec. Eng., 17, 26 1-262 (1983)

198

Rough Sudaces

Wade, G., Signal coding and processing 2e (Cambridge University Press, 1994) Whitehouse, D. J., Handbook of surface metrology, (Institute of Physics, Bristol, 1994) Whitehouse, D. J. and Phillips, M. J., "Discrete properties of random surfaces", Phil. Trans. R. SOC.Lond., A290, 267-298 (1978) Whitehouse, D. J. and Phllips, M. J., "Two-dimensional discrete properties of random processes", Phil. Trans. R. Soc. Lond., A305, 441-468 (1982) Williamson, J. B. P. and Hunt, R. T., "Relocation profilometly", J. Phys. E; Sci. Instrum., 1, 749-752 (1968). Zahouani, H., "Spectral and 3D motifs identification of anisotropic topographical components. Analysis and filtering of anisotropic patterns by morphological rose approach", Trans. 7th.Int. Con$ On Metrology & Properties of Engng Surfaces, 222-230 (Goteborg, 1997)

CHAPTER 10 APPLICATIONS: CONTACT MECHANICS

Although it is clear that the existence of surface irregularities has profound effects in numerous engineering situations, knowledge of these effects has until comparatively recently been largely qualitative. Examples of the successful quantitative relation of specific surface parameters to engineering function are rather rare. The reasons for this are to be found in earlier chapters of this book. To restate them briefly here, none of the conventional surface parameters is an intrinsic property of a surface, hence its correspondence with any particular surface phenomenon will be at best accidental. All surface parameters vary with the scale over which they are measured. To apply a surface measureIlient to an engineering problem it is essential that the scale of the problem and the scale of the measurement be related. As an illustration (Thomas & King 1977), imagine taking a 1:50 000 geographic map and progressively enlarging it by linear factors of 10. The smallest feature we could resolve initially would be about l00m across. After only one enlargement the topography starts to have an engineering effect; height variations with a wavelength of 10 m will cause vibrations in the suspension of an aircraft as it lands which will have to be allowed for in the design. After another enlargement, to 1 m, a similar effect will be produced on the suspension of road and rail vehicles. Amplitudes on this scale may vary from 10 to 100 mm. At 10 cm we are down to the scale of surface features which influence tyreroad interactions such as skidding. We are also for the first time within the range of machined surfaces; the performance of a machine tool, for instance, will be influenced by features on this scale which transmit vibrations through its joints. Features of this size may also slow down ships by increasing hull friction (amplitude 0.1 - 1 mm). Below this we are firmly in the region of machined surfaces. Undulations of wavelengths from 1 mm down to 1 pm may increase friction and wear and promote noise, rough running and finally failure in bearings of all kinds. They may also affect the performance of nuclear power stations and space satellites by increasing their thermal resistance, or the functioning of telephone exchanges and other switchgear by affecting electrical resistance (amplitude 0.0 1 -- 10pm). 199

200

Rough Surfaces

At 1 pm and below another set of properties becomes of engineering importance, namely the reflection or diffraction of electromagnetic radiation. When surface irregularities are present at wavelengths comparable with those of visible light the appearance of the surface will alter, e.g. a painted surface such as a car body may appear dull instead of glossy (amplitude 0-10 nm). Thus the essential problem of the quantitative application of surface measurements is the choice of scale. In a previous chapter it was shown that the numerical values of heightdependent parameters depend on the longest wavelengths measured, while those of texture-dependent parameters such as slopes and curvatures depend on the shortest wavelengths measured. If one attempts to include the effect of all the wavelengths in the continuous spectrum present on most real surfaces one will obtain the trivial result that all roughness parameters tend to infinity or zero. To obtain finite numerical values for surface parameters it is necessary to reject certain portions of the spectrum at both its short-wavelength and its longwavelength ends. A measuring instrument does this as a matter of course by virtue of its design and construction; the finite dimensions of the probe remove certain short wavelengths (the action of the so-called "footprint" described by Newland 1986) and the filter circuits remove certain long wavelengths. Unfortunately the portions of the spectrum thus rejected are chosen quite arbitrarily, and there is no a priori reason why the roughness values thus obtained should have any functional significance.

L

Wavelengths tw long ffect interacbon

0

a

UL

OH

1 I Wavelength

Figure 10.1. Functional filtering:only the pass-band of surface wavelengths between high-pass cutoff oHand low-pass cutoff wL take part in the surface interaction

The obvious solution is to confine measurement to the portion of the spectrum of wavelengths which actually take part in the phenomenon under investigation (Fig. 10.1). The definition of this band of wavelengths is equivalent, in the

Contact Mechanics

20 1

terminology of communications engineering, to the application of a band-pass filter, and we have termed the process "functional filtering" (Thomas & Sayles 1973). The pass-band is specific not to the surface but to the particular interaction involved; the roughness and asperity density "seen" by reflected electromagnetic radiation, for instance, will be quite different from those "seen" by the contact of, say, a journal bearing on the Same surface. The problem then reduces to the choice of the cut-offs which define the pass-band. The cutoff which rejects the long wavelengths - the high-pass cutoff, in electronics terminology - is the easier one to select, as it fairly clearly must be related to the largest horizontal dimension of the surface interaction. In many contact problems, for instance, the high-pass cutoff is set by the dimensions of the nominal contact area. This is simply to say that wavelengths much longer than the nominal contact area will not affect what goes on inside it. The dimensions of the nominal contact area are not always the obvious ones; for instance, in a constrained reciprocating contact the critical dimension may be the stroke. Whether the numerical value of the cutoff and contact size should be the same is not entirely clear. It has been suggested that a cutoff of twice the contact size is more realistic (Leaver et al. 1974) because it allows for the lack of sharpness (the roll-off) characteristic of all real filters. However, as the roughness often increases only as the square root of the high-pass cutoff, the exact value chosen is probably not critical. In the following pages we will discuss some practical problems in which surface roughness is involved, in terms of functional filtering.

10.1. The Contact of Rough Surfaces

There are many situations, in engineering and other disciplines, where rough surfaces are brought together and it is important to know the topography of the contacting area. We will discuss the actual mechanics of contact a little later; for the time being we can start from the observation of Bowden and Tabor (1950) that the real area of contact is independent of the nominal area and is in most practical situations only a tiny fraction of the nominal contact area. We will also simplify the model to the contact of a rough surface with a smooth flat plane. Greenwood & Tripp (1971) have discussed the contact of two rough surfaces and have shown that this can be reduced to the contact of a single equivalent rough surface with a smooth flat plane. In practice one contacting surface is often in any case so much smoother than the other that no important information is lost by considering it as perfectly smooth. What we are interested in for the moment is the relationship

202

Rough Surfaces

between the approach of the surfaces and the real area of contact, and the way in which this real contact area is distributed over the nominal area of contact. The simplest model is the intersection a of rough surface with a plane parallel to the rough surface’s mean plane. In physical terms this is equivalent to contact with complete loss of the displaced material, or an abrasive process which removes all material above a given height without disturbing the material below. This model may seem somewhat unrealistic, but in fact the resulting errors only become appreciable when the surfaces approach each other very closely (Pullen & Williamson 1972). The fractional area of contact at any height h from the mean plane of the rough surface is then by definition the bearing area or bearing length introduced in Section 7.4. If we normalize this height as a dimensionless separation t = h / CT = h / Rq, then for a Gaussian random surface, in terms of the dimensionless height s = z / 4

and the fractional real area of contact becomes (10.2)

where @ (t) is the cumulative Gaussian probability function. Note that by this definition of separation, the fractional area of contact is still only % at zero separation; negative separations are possible; and complete contact is never attained. The next question to investigate is how the real area of contact is made up. How many discrete areas of contact, or contact spots, are there at any separation, and how big are they? By extending Nayak’s (1971) theory it can be shown (Sayles & Thomas 1976, 1978) that the density (i.e. the number per unit nominal area) of closed contours at any separation is

D, = ( 2 7 ~ ) ~ / ~ ( m 2 / m o ) q ( t )

(10.3)

and an upper bound for the mean contact spot radius is

(10.4)

Contact Mechanics

203

In other words, as the surfaces get closer, the numLer of closed contours increases but their average size stays more or less the same (Fig. 10.2), so that the increase in the real area of contact is almost entirely due to the increase in the number of contacts. Of course a closed contour may be a hole inside a region of contact, but at large separations the probability of this is small enough to be ignored (Sayles & Thomas 1978). Note that up to this point the discussion has been purely geometrical; we have made no assumptions about the actual mechanism of contact or about the relationship between separation and load.

7/

lo-.t

I /

/ //

/

C’

ii*

I I

l0-Y

W* Figure 10.2. Dimensionless mean contact spot radius, contact density and thermal conductance as a function of separation and dimensionless load (Sayles & Thomas 1976)

Sayles & Thomas (1978) suggested that the distribution of contact spot sizes

was log-normal. Majumdar, however, (Majumdar & Tien 1990, Majumdar & Bhushan 1990) observed that for fractal surfaces at any separation the number of contact spots Nw larger than a given area % follows a power law: (10.5)

where !RiL is the area of the largest spot at that separation (Fig. 10.3). He pointed out that although the number of infinitesimally small spots is infinitely large, their

Rough Surfaces

204

contribution to the total area of contact is negligible, an observation confirmed by Klimczak (1992).

-MAGNETIC TAPE 0

10’

38MR

MAGNETIC RIGID DISK

. .

too

Id

101

AREA.

Id

Id

.[lUn’l

Figure 10.3. Size distribution of contact spots for a magnetic tape surface under two different loads, and a magnetic disk (Majumdar & Bhushan 1990)

Such fractal spots are not generally circular; there is a power-law relationship between their perimeter and their area, but the exponent is significantly larger than % (Russ 1994) (Fig. 10.4). This has obvious implications for the calculation of conductance or contact stress.

Log area Figure 10.4. Total perimeter length as a function of total area of closed contours on a simulated isotropic fractal surface (Russ 1994)

Contact Mechanics

205

10.2. Rough Contact Mechanics

Now that we have some idea how the real area of contact varies with separation, it is time to consider how the separation might vary with the load. The following discussion concentrates on those aspects of contact mechanics which depend on the general topography of the surface; for a more general account of the principles of contact mechanics and their application to rough surfaces, the reader is referred to the books by Johnson (1985) and Hills et al. (1993). Rough contact is also reviewed by Greenwood (1992). Up to now we have considered rough surfaces as a continuum, but to calculate their behaviour during contact it is necessary to model them as an array of discrete physical objects distributed in some way both in the plane of the surface and perpendicular to it. These objects are often termed asperities in the literature, and we may think of them as related to the summits of the Nayak theory (though remembering that Nayak’s summits are local maxima and so may occur at any height, whereas contact must take place at least initially only at the highest parts of the surface). The local load on an individual asperity will cause it to deform, at first elastically. If the load continues to increase beyond its capacity to recover elastically it will suffer irreversible plastic deformation. We will consider later how to determine the mode of deformation. In general some of the asperities may deform elastically and others plastically, depending on their original geometry, the load and the physical properties of the bulk material. The easiest cases to consider are those where the mode of deformation is everywhere the same, that is where all the asperities deform plastically or where they all deform elastically. We will deal with the plastic case first as the load P per unit nominal contact area can be related directly to the separation through the real area of contact and the hardness H by means of the bearing area: A,

=

P/H

=

1

~

@(t)

(10.6)

Knowing the separation, the variation of number of contacts and mean contact spot size can be obtained from Eqns. 10.3 and 10.4. The number of contacts increases almost linearly with load while the mean contact size remains almost constant, a result confirmed by many experiments (e.g. Thomas & Probert 1970, Uppal et al. 1972, though not universally accepted: see Tian & Bhushan 1996). Note that these results are obtained without assuming any particular geometrical model for an individual asperity. Care should be taken in selecting a value for the hardness; the indentation hardness, commonly taken as 3 x the yield

206

Rough Surfaces

strength of the bulk material, is not appropriate for predicting the behaviour of an unsupported asperity, which will deform under a normal pressure much closer to the yield strength itself (Thomas et al. 1971). At very high loads, as in drawing operations, one would expect behaviour to change, as some of the simplifying assumptions made above no longer hold. Also as the mean separation of individual contact spots approaches their mean size, the asperities will begin to lend one another mutual support and the effective hardness will increase. Pullen & Williamson (1972) maintained that as the load increased the non-contacting parts of the surface moved uniformly upward, though they were never able to suggest a mechanism for this arresting phenomenon. The wellknown figure which purports to show this effect (Fig. 10.5) in fact refers all heights to their respective profile mean lines as separately calculated. But there was no independent height datum for the series of experiments, hence no evidence that the successive mean lines remained at the same height relative to the bulk material. The load progressively flattening the tips of the asperities has the effect of censoring the upper tails of the successive height distributions and so pushing their recalculated means closer to the lower tails; if the successive means are then constrained to line up, the rest of the distribution will of course appear to move upwards.

Load (kN)

Figure 10.5. Spurious upward movement ofthe non-contacting surface under increasing load (Pullen & Williamson 1972)

Contact Mechanics

207

Elastic contact is not quite so straightforward. To apply the theory of elasticity it is necessary to have some geometrical model of an individual asperity or at least of its tip, usually the model of an elastic sphere originally due to Hertz (Timoshenko & Goodier 1951). For an elastic sphere of radius R in contact with a semi-infinite elastic half-space, Hertz obtained the area of contact A and the load W in terms of the compliance w (the distance by which points outside the deforming zone approach): A W

=

=

nRw

(10.7)

(413) E'd@w2)

(10.8)

where the harmonic elastic modulus E' is given by 1 - 1-v,

E'

E,

2

+-1-v,

E2

2

(10.9)

and v is Poisson's ratio. Archard (1957) modelled a rough surface as an array of hemispherical asperities covered with smaller asperities, each of which is covered by even smaller asperities, and so on (Fig. 10.6). By applying Hertz's relations successively he showed that for such a surface the relationship between load and real area of contact becomes closer and closer to proportionality as the number of layers of asperities is increased. He explained that "the essential part of the argument was not the choice of asperity model: it was whether an increase in load creates new contact areas or increases the size of existing ones" (Greenwood & Williamson 1966).

Figure 10.6. Multiple-scale asperity models for elastic contact (Archard 196 1): real area of contact is proportional to the load raised to the power (a) 2/3 (b) 415 ( c ) 819 (d) 14/15 (e) 26/27 (t)44/45

208

Rough Sut$aces

Greenwood & Williamson combined Archard's model of a surface covered with hemispherical elastic asperities with a statistical distribution of asperity heights each of the same radlus, obtaining expressions for the density of contact spts, real contact area and load per unit nominal area:

0,= Sds F o ( t ) A, P

=

=

ZSdsR o F I ( t )

(10.10) (10.1 1)

d/2F3,2 ( t )

(10.12)

F,(t) = j ( ~ - t ) ~ p ( s ) d s

(10.13)

( 413 ) Sds E' R'"

where m

t

This is a general expression for any probability distribution; if p(s) is Gaussian, the area of contact is found proportional to the load. The GW theory is in good qualitative agreement with experiment (HandzelPowierza et al. 1992), and many attempts have been made to obtain quantitative agreement. They have mostly foundered on the usual reef, the difficulty of obtaining unique values of summit density and curvature. In addition the assumption that all summits have the same curvature and a Gaussian height distribution has been superseded by the work of Nayak discussed previously. For this reason Bush et al. (1975) attempted an "asperity-free" model of elastic contact based on Nayak's approach and showed that at large separations,

A, =

e x p - t 2 I 2)

(10.14)

2 t G

(10.15) Eliminating the separation gives strict proportionality between area and load:

(10.16)

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209

The real area of elastic contact is just half the bearing area, that is half the real area of plastic contact. Unfortunately the theory does not distinguish between number and size of contact spots, and again suffers from the difficulty of defining a unique second moment. McCool(1986) makes a detailed numerical comparison of the predictions of Bush et al’s theory and Greenwood & Williamson’s theory with those of an asperity simulation model and finds that they are in broad agreement. Chang et al. (1987) modified the GW theory to allow for the plastic deformation of the most highly loaded asperities, using a model based on volume conservation of the plastically deformed region. Their model predicts significantly higher separations at high loads but otherwise agrees with the GW model within a few percent. The Chang-Etsion-Bogy (CEB) model is now widely used, though there seems little practical gain to justify the extra complexity. Bhushan and his co-workers have carried out extensive investigations of rough contact (Oden et al. 1992, Ganti & Bhushan 1995, Tian & Bhushan 1996, Poon & Bhushan 1996a, b, Yu & Bhushan 1996). As we saw in Section 9.4, for fractal surfaces, all the moments of the power spectrum which define roughness, slopes and curvatures can be expressed in terms of fractal parameters. It should be possible, then, to recast theories of elastic contact in fractal terms. They began (Majumdar & Bhushan 1990, 1991) by modelling a self-affine fractal surface as a Weierstrass-Mandelbrot function, but it proved rather difficult to extract the characteristic parameters from a real surface for purposes of experimental comparison. However, they were able to show that the real area of contact, and the relative proportions of it in elastic and plastic contact, are quite sensitive to the fractal parameters. In an extension of this work to bifractal surfaces (Bhushan & Majumdar 1992), the relationships could be quantified by invoking one extra parameter, the size of the largest contact spot. Fractal parameters were found from a profile structure function, and the largest contact spot was presumably measured directly, though the paper does not make this clear. In any event the theory underestimated the real area of contact by an order of magnitude, possibly due to the uncertainty of statistical inference from an extreme value. A model of elasticplastic contact of self-af€ine fractal surfaces based on the Cantor set has been proposed by Warren & Krajcinovic (1999, but only relates load to displacement. Lee & Ren (1996) generated computer models of Gaussian rough surfaces with varying degrees of weak anisotropy and simulated elastic-plastic contact numerically. They obtained relationships between separation and real area of contact in terms of dimensionless hardness and dimensionless load normalised by the correlation length. They found that at high loads the variation of contact area with load was quite non-linear (Fig. 10.7).

Rough Sudaces

210

E

.-0 Y

u

k 1 .o

1

."

0.5

.o

0.0

m b

Figure 10.7. Real contact area as a function of dimensionless hardness and dimensionless load (Lee & Ren 1996)

All the theories we have considered so far have assumed isotropy, but as we saw earlier, most machined surfaces are anisotropic. Bush et al. (1978) obtained a solution for elastic contact of strongly anisotropic rough surfaces in terms of moments of the surface, rather than the profile, power spectrum. At low loads the area of elastic contact was proportional to the load. So & Liu (1991) extended this to include plastic contact. They found that the elastic portion of the contact area was almost independent of anisotropy, but the proportion of plastic contact varied significantly with the degree of anisotropy (Fig. 10.8). Lee & Ren, however, concluded that for weak anisotropy, contact could be treated as two-dimensional for asperity aspect ratios greater than 6. McCool (1986) compared the predictions of the Bush et al. model with his own numerical model for an isotropic surface and a surface with 10:1 anisotropy. The models agreed that at a given separation the real area of contact in the anisotropic case was the same as in the isotropic case, but real pressures were much lower.

Contact Mechanics

21 1

U

a \ a a

-2

'Bp;

16'

Figure 10.8. Proportion of plastic contact as a function of dimensionless load for various degrees of anisotropy (So & Liu 1991)

As Chang et al. (1987) observed, theories of rough contact are extraordinarily hard to verify experimentally. There are various reasons for this: it is difficult to measure the area of elastic contact as the asperities will recover their original geometry; displacements normal to the surface are very small, and the stiffness of the asperities may be of the order of the stiffness of the whole experimental rig; the height of the first point of contact and the absolute height of the rough surface mean plane are difficult to determine; small contact spots may be below the limit of resolution of the measuring system. Woo & Thomas (1979) reviewed the published experimental results for plastic contact and found real area of contact proportional to the 0.83 power of load up to dimensionless loads of 0.1, and separation inversely proportional to log (load ) (Fig. 10.9). No correlation was found between size or number of contact spots and load, as insufficient information was available in the published literature to normalise the data by any of the methods dscussed above.

Rough Surfaces

212

I

5

1

I

Bearing AIM

6

it2

Ralio

163

la4

1

IbC

113)

1

16

I

16'

I

.l

I

I0

Oimcnsionlcrs Lood

Figure 10.9. Collation of measurements of variation of rough plastic contact with dimensionless load from 7 sources (Woo & Thomas 1979): (above) real area of contact, solid line is best power law fit; (below) separation, solid line is best log-linear fit

10.2.1. Contact of Curved Surfaces

This is a case of some practical importance as many, perhaps the majority, of engineering contacts are curved. The problem here is complicated by the difficulty

Contact Mechanics

213

that because of the gross curvature the pressure distribution varies over the nominal area of contact in an unknown way. An iterative approach to this problem was suggested by Greenwood and Tripp (1967) for spherical contacts and investigations were extended to cylindrical contacts by Lo (1969), Dobychin (1988) and Merriman & Kannel (1989). Rather than a rough sphere on a flat plane, Poon & Sayles (1994) modelled a smooth sphere on a rough plane. Greenwood and Tripp found that the maximum pressure, and the area over which the load was distributed, could differ significantly from the classical Hertzian solution for smooth surfaces (Fig. 10.10).

Radial distance Figure 10.10. Dimensionless pressure as a function of dimensionlessradial distance from the centre of an elastic contact between a rough sphere and a plane (Greenwood & Tripp 1967). Broken line: classical Hertz solution for a smooth surface; solid line: rough-surface solution

Again, to obtain numerical solutions a pass-band of surface wavelengths must be defined. Such a calculation has been carried out for two practical cases (Thomas 1979). The first case was the contact of a pushrod of cup radius 10 mm with a rocker arm of ball radius 5 mm under a mean working load of 1 kN, taking their combined roughness as 0.5 pm. The low-pass cut-off was then estimated as 33 pm from the plasticity approach, giving 1,100 asperities available for contact per unit area with a mean tip radius of curvature of 310 pm. After iteration the final rough contact area was found to be 20% greater in diameter than the corresponding smooth contact area, and the maximum stress was 95 per cent of

214

Rough Su$aces

that for a smooth contact. It appears, therefore, that in this practical case the roughness does not have a significant effect. In the second case a journal bearing with a bronze liner was considered, having a nominal radius of 25 mm, a clearance ratio of 1 in 1,000, an axial length of 50 mm, a working load of 8kN and a combined roughness of 1 pm. Because of the relative softness of the bronze liner the low-pass cut-off was as high as 0.6 mm, i.e. all surface wavelengths shorter than this are immediately flattened plastically. The density of asperities was correspondingly low at 3.7 mm-2and their radius of curvature was no less than 11.3 mm. Convergence after three iterations yielded an effective roughness of 6 pm. The nominal contact area was 2.2 times wider than that for a smooth contact and the maximum stress was only 64 per cent that of the smooth case. The implication would seem to be that it is worth making a journal bearing as rough as is consistent with adequate lubrication.

10.2.2 Joint Stiffness Machine-tools are not generally manufactured as a continuous casting or fabrication; the reasons for this are functional, such as the necessity to incorporate guideways, and also difficulties in manufacture and transportation. Most practical designs for machine-tool structures, therefore, incorporate some form of connection between the basic elements (Back et al. 1973). The stiffness or lack thereof of such a joint is likely to affect the dynamic properties of the machine. It is known that the dynamic stiffness of machine-tool joints is proportional to their preload, and this result can be derived by considering the joint as an assembly of Gaussian elastic asperities after Greenwood and Williamson's model (Thomas & Sayles 1977). The constant of proportionality was calculated for some experimental stiffness measurements of Thornley and Lees (1971) on joints of various planforms, and reasonable agreement was found between theory and experiment. To find the numerical value of the constant it is necessary to know the density and mean curvature of the asperities, both of which, as noted above, depend on the pass-band of surface wavelengths chosen. A detailed calculation was carried out from surface measurements of the contact between the bed of a lathe and its saddle (Thomas & Sayles 1977). Here the high-pass cut-off was set at 61 cm by the nominal length of the contact. The roughness measured at this cut-off was 3.3 pm. The low-pass cut-off was found from the plasticity argument given above to be 19 pm. The mean plane separation could then be calculated by the method of

Contact Mechanics

215

Greenwood and Williamson to be 17 pm under a load of 1 kN supported by only 16 asperities, and the stiffness was calculated as 1.4 MN/mm.

10.3. The Plasticity Index

Greenwood & Williamson (1966) looked for a criterion which would determine the mode of deformation of an array of asperities of varying heights. They found that the mode of deformation of the highest asperities was almost independent of load. Sharp asperities would deform plastically even under the lightest loads, while blunt asperities would deform elastically even under the heaviest loads. The criterion of sharpness was the ratio of the standard deviation of the height distribution, o,,to the asperity radius of curvature R, and their result applied to any exponential distribution of asperity heights. For the special case of a Gaussian distribution, they showed that the top 2% of asperities would deform plastically under any load for ry > 1, and elastically under any load for ly < 0.6, where y~

=

(E'/H)t/(o,/R)

(10.17)

In the region 0.6 < ly < 1 the mode of deformation is dependent on the load. The plasticity index ry is thus a dimensionless figure of merit which can predict the dominant mode of deformation. Many workers have used the plasticity index, with some success, as a qualitative index of comparison witlun a series of tests, i.e. the higher the plasticity index, the more likely a surface is to suffer wear or similar problems. There are difficulties in comparing results between laboratories or in using it as an absolute index, however, because of the difficulties in quantifying the surface parameters R and o,. Because these are both properties of the peak distribution they depend on the definition of a peak, and raise the problems which we have encountered previously. For this reason various other formulations of the plasticity index have been proposed, for instance one due to Mikic (1974) which replaces peak parameters by the mean slope. Bush et al. (1978) developed an expression for the plasticity index of a strongly anisotropic surface in terms of Nayak's moments: (10.18)

216

Rough Sudaces

where moo is the variance of surface heights, moa m2a mO4and m40 are the secondand fourth moments of the power spectra of profiles parallel to and across the lay, respectively, and (10.19)

moo 'mqo .

where

a,=

- moo

2 , a 2 17220

Om04 2 m02

(10.20)

According to Bush et al., deformation will be plastic for y > 0.7, elastic for ty < 0.5, and load-dependent in the intermedate region. How anisotropic must a surface be to require the rather laborious anisotropic treatment? According to Wu & Zheng (1988) the correction for anisotropy increases very slowly with the degree of anisotropy y (Fig. 10.11). Their paper does not appear to distinguish between strong and weak anisotropy, but this does not affect the present argument. The degree of anisotropy is the ratio of the major and the minor asperity radii of curvature, which is approximately the square of the ratio of the major and minor axes of the ellipse projected when the asperity intersects a plane parallel to the surface mean plane (Bush et al. 1978).

2.0

1.8 1.6

1.4

1.2 1 .o 0.8

1

1 . 5 2

3

4

5

Figure 10.11. Anisotropy correction factor for plasticity index as a function of degree of anisotropy y(Wu & Zheng 1988)

Contact Mechanics

217

Remembering that the plasticity index only deals with the highest regions of the surface, one may observe that the higher one looks on any surface, however anisotropic, the less elliptical the tips of asperities appear (Fig. 10.12). If one attempts to fit a best ellipse to some of these irregular shapes, the ratio of major to minor axes is not more than about 3, corresponding to y = 0.1, in which case, from Fig. 10.11, the anisotropy correction would be only about 10%. Bearing in mind the large statistical uncertainties in measurement of somc of the other surface parameters, it seems likely for many surfaces that isotropic calculations will suffice, in which case Eqn. 10.18 simplifies to

(10.21)

Figure 10.12. "And we in dreams behold the Hebrides": contours more than 1 prn above the mean plane on a plateau-honed surface do not look very elliptical

218

Rough Surfaces

Recalling that the second moment of the profile power spectrum is related to the mean slope Bby Eqn. 9.15:

so (10.21)

or (10.22)

This formulation of the plasticity index simplifies the measurement problem considerably. Instead of measuring asperity radii of curvature or power spectra directly, we can replace these by the much easier measurement of the profile slope, which merely requires computation of the standard deviation of the first differential of the profile. But there are still some remaining practical obstacles. The slope is a function of the sampling interval and increases as the sampling interval decreases. This simply indicates that as asperities get smaller they also get sharper. It follows that, by varying the sampling interval, we can obtain any desired value of the slope, and hence of the plasticity index. To put this another way, there always will be features on the surface so small and sharp that they will deform plastically on contact. In obtaining numerical solutions in all the above cases the basic problem is the same: the choice of the low-pass cutoff. This amounts to asking the question: What is the smallest surface feature which will affect contact? There does not seem to be any general answer to this question. Many workers have implicitly, and Ganti & Bhushan (1995) have explicitly, assumed that no features smaller than the resolution of the particular instrument are important, but it is difficult to find physical grounds to justify this. One possible approach is to work back from the plasticity index itself (Thomas & Sayles 1977). As asperities get smaller and sharper, a size will be reached below which they will deform plastically during the very first cycle of contact and so dlsappear (Archard 1961). During the subsequent lifetime of the component, it will behave elastically as if the corresponding range of surface wavelengths did not exist; in other words, the initial encounter of the surface will act as a natural low-pass filter. The critical wavelength at which this filtering

219

Contact Mechanics

occurs may be found from the relationship between the second moment and the plasticity index. If vCis the critical value of the plasticity index above which deformation will be plastic at any load, then the critical second moment r n Z c = (71( 2

- nJ2 )I-”

wc2 ( H I E ’ )

(10.23)

The exact form of Eqn. 9.22 is m2 = B (3

+ p).’ ( 271)-p

(

e3+p - m3+p)

(10.24)

If p> 1 and the bandwidth of surface wavelengths is reasonably wide, i.e. oL)) oH, then Eqn. 10.24 reduces to

m2 E B The critical wavelength dc= 271/

Q , ~ +1~( 3

+ p ) ( 271)p

CL)LC, so combining Eqns.

(10.25) 10.23 and 10.25,

(10.26)

From Bush et a1 1978, IY, , = 0.7. Combining this with the numerical constant, replacing B and p by the corresponding fractal parameters from Eqns. 8.10 and 8.11 and rearranging, we have finally 20-1

(10.27)

where f ( D ) = (-)@z)iO-LT(I1175 1-20

2D)cos- 2 - 0

(10.28)

2

This is a relationship between three dimensionless numbers: the critical wavelength normalized by the topothesy, the fractal dimension and the material property ratio. The dimensionless wavelength is highly sensitive to the other parameters (Fig. 10.13), and for a given fractal dimension and material property

220

Rough Su?$aces

ratio the critical wavelength increases as the topothesy. Thus we can in principle now find a unique short-wavelength cutoff, depending only on material properties and intrinsic topography parameters, which we can use to determine the elastic behaviour of the contact.

90 0

E-IH

Figure 10.13.

Dimensionless critical wavelength as a function of material ratio and fiactal dimension ( R o s h et al. 1997)

10.4. References Archard, J. F., "Elastic deformation and the laws of friction", Proc. Royal SOC.,A243, 190-205 (1957). Archard, J. F., "Single contacts and multiple encounters", J. Appl. Phys., 32, 1420-1425 (1961). Back, N., Burdekin, M, and Cowley, A., "Review of the research on fixed and sliding joints", Proc. 13th Int. Machine Tool Des. & Rex Con$, 87-97 (1973). Bhushan, B.; Majumdar, A., "Elastic-plastic contact model for bifractal surfaces", Wear 153, 53-64 (1992) Bush, A. W., Gibson, R. D., Keogh, G. P., "Strongly anisotropic rough surfaces", Trans. ASME: J. Lub. Tech. 101, 15-20 (1979)

Contact Mechanics

22 1

Bush, A. W., Gibson, R. D. and Thomas, T. R., "The elastic contact of a rough surface", Wear, 35, 87-111 (1975). Dobychin, M. N., "Elastic contact of rough cylindrical bodies", Soviet Journal ofFriction and Wear 9, 1-5 (1988) Ganti, S., Bhushan, B., "Generalized fractal analysis and its applications to engineering surfaces", Wear 180, 17-34 (1995) Greenwood, J. A., "Contact of rough surfaces", 37-56 in I. L. Singer & H. M. Pollock eds., Fundamentals of piction: macroscopic & microscopic processes, (Kluwer, Dordrecht, 1992). Greenwood, J. A. and Tripp, J. H., "The contact of two nominally flat rough surfaces", Proc. 1. Mech. E., 186,625-633 (1970/71). Greenwood, J. A. and Williamson, J. B. P., "Contact of nominally flat surfaces", Proc. Royal SOC.A295, 300-319 (1966). Handzel-Powiem, Z., Klimczak, T. and Polijaniuk, A., "On the experimental verification of the Greenwood-Williamson model for the contact of rough surfaces", Wear, 154, 115-124 (1992) Hills, D. A., D. Nowell and A. Sacwield, Mechanics of elastic contacts (Butterworth-Heineman, Oxford, 1993). Johnson, K. L., Contact mechanics (Cambridge University Press, London, 1985). Klimczak, T., "Predicting microcontact spots size distribution in contact problems", Ann. CIRP 41,609-612 (1992) Lee, S. C., and Ren, N., "Behavior of elastic-plastic rough surface contacts as affected by surface topography, load, and material hardness", Trib. Trans. 39, 6774 (1996) Lo, C. C., "Elastic contact of rough cylinders", Int. J. Mech. Sci., 11, 105-115 (1969). Majumdar, A.; Bhushan, B., "Role of fractal geometry in roughness characterization and contact mechanics of surfaces", Trans. ASME. Journal of Tribology 112,205-216 (1990) Majumdar, A.; Bhushan, B., "Fractal model of elastic-plastic contact between rough surfaces", Trans. ASME. Journal ofTribology 113, 1-11 (1991) Majumdar, A.; Tien, C. L., "Fractal characterization and simulation of rough surfaces", Wear 136, 313-327 (1990) McCool, J. J., "Comparison of models for the contact of rough surfaces", Wear 107,3760 (1986)

222

Rough Surfaces

Merriman, T. and J. Kannel, "Analysis of the role of surface roughness on contact stresses between elastic cylinders with and without soft surface coating." Journal of Tribology, 111,87-94, (1989) Mikic, B. B., "Thermal contact conductance: theoretical considerations", Int. J. Heat Mass Transfer, 17,205-224 (1974). Moalic; H., J. A. Fitzpatrick and A. A. Torrance, "A spectral approach to the analysis of rough surfaces", Journal of Tribology, 111, 359-363, (1989) Nayak, P. R., "Random process model of rough surfaces", Trans. A.S.M.E. Ser. F. J. Lubr. Tech., 93, 398-407 (1971). Newland, D. E., "The effect of a footprint on perceived surface roughness", Proc. Roy. SOC.Lond. A405,303-327 (1986) Men, P. I.; Majumdar, A.; Bhushan, B.; Padmanabhan, A,; Graham, J. J., "AFh4 imaging, roughness analysis and contact mechanics of magnetic tape and head surfaces", Journal of Tribology, Transactions of the ASME 114, 666-674 (1992) Poon, C. Y., and Bhushan, B., "Nano-asperity contact analysis and surface optimisation for magnetic head slider/disk contact", Wear 202,83-98 (1996) Poon, C. Y., and Bhushan, B., "Numerical contact and stiction analyses of Gaussian isotropic surfaces for magnetic head slideddisk contact", Wear 202, 6882 (1996) Poon, C. Y.; Sayles, R. S., "Numerical contact model of a smooth ball on an anisotropic rough surface", Journal of Tribology, Transactions of the ASME 116, 194-200 (1994) Pullen, J . and Williamson, J. B. P., "On the plastic contact of rough surfaces", Proc. R. SOC.Lond. A327, 159-173 (1972). Ro&, B.-G., R. Ohlsson and T. R. Thomas, "Nano metrology of cylinder bore wear", Trans. 7th. Int. Con$ On Metrology h Properties of Engng Surfaces, pp. 102-110 (Giiteborg, 1997) Russ, J. C., Fractal surfaces (Plenum Press, New York, 1994). Sayles, R. S. and T. R. Thomas, "Computer simulation of the contact of rough surfaces", Wear, 49, 273-296 (1978). Sayles, R. S. and Thomas, T. R, "Thermal conductance of a rough elastic contact", Appl. Energy, 2, 249-267 (1976) So, H.; Liu, D. C., "An elastic-plastic model for the contact of anisotropic rough surfaces", Wear 146,201-218 (1991) Thomas, T. R., "Calculation of elastic contact stresses for rough-curved surfaces", ASLE Trans., 22, 184-189 (1979)

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223

Thomas, T. R., and King, M. J., Surface topography in engineering: a state of the art review and bibliography (BHRA, Cranfeld, 1977) Thomas, T. R. and Probert, S. D., "Establishment of contact parameters from surface profiles", J. Phys. D: Appl. Phys., 3,277-289 (1970). Thomas, T. R. and Sayles, R. S., discussion to Radhakrishnan, V., "Analysis of some of the reference lines used for measuring surface roughness", Proc. 1. Mech. E., 187, 575-582 (1973). Thomas, T. R. and Sayles, R. S., "Random-process approach to the prediction of joint stiffness", Trans. ASME: J. Eng. Znd. 99B, 250-256 (1977) Thomas, T. R., Uppal, A. H. and Probert, S. D., "Hardness of rough surfaces", Nature Physical Sci., 229, 86-87 (1971). Thornley, R. H. and Lees, K., "The effect of planforni shape on the normal dynamic characteristics of metal to metal joints", I. Mech. E., Paper C62/71, (1972). Tian, X., and Bhushan, B., "A numerical three-dimensional model for the contact of rough surfaces by variational principle", Trans. ASME: J. Trib. 118, 3342 (1996) Timoshenko, S., and Goodier, J. N., Theory of elasticity (McGraw-Hill, New York, 1951) Uppal, A. H., Probert, S. D. and Thomas, T. R., "The real area of contact between a rough and a flat surface", Wear, 22, 163-183 (1972). Warren, T. L.; Krajcinovic, D., "Fractal models of elastic-perfectly plastic contact of rough surfaces based on the Cantor set", Internatanal Journal ofSolids and Structures 32,2907- 2922 (1995) Whitehouse, D. J. and Archard, J. F., "The properties of random surfaces in contact", In: Surface Mechanics. Ling, F.F. (Ed.) 36-57 (A.S.M.E., New York, 1969). Williamson, J. B. P., Pullen, J. and Hunt, R. T., "The shape of solid surfaces", In: Surface Mechanics. Ling, F.F. (Ed.). 24-35 (A.S.M.E., New York. 1969). Woo, K. L. and T. R. Thomas, "Contact of rough surfaces: A review of experimental work", Wear, 58, 33 1-340 (1980). Wu, C. and Zheng Linqing, "General expression for plasticity index." JVear, 121, 161-172, (1988) Yu, M. M., and Bhushan, B., "Contact analysis of three-dimensional rough surfaces under frictionless and frictional contact", Wear 200, 265-280 (1996)

CHAPTER 11

TRIBOLOGY

Discussion of the effect of roughness on contact mechanics leads naturally to a discussion of the effect of roughness on friction, lubrication and wear. As we would argue that all real contact is rough contact, it follows that roughness is a complication in all tribological situations. There is no space here to do justice to the full scale of the tribological implications of roughness, and we will confine ourselves to discussing a few of the more interesting examples which illustrate some of the previous topics. No comprehensive account of rough surface tribology appears to exist, though Bhushan (1990) has a useful review.

11.1. Friction

It was recognised very early that friction was associated with surface roughness, to the point where Coulomb suggested that friction was due to the effort required for one surface to climb up the asperities of the other during translation (Bikerman 1944, Bowden & Tabor 1950). When it was pointed out that the energy thus dissipated would be largely recovered when the surface slid down the reverse slopes of the asperities, the Coulombic theory lost some of its popularity, but microgeometry remains a important factor in friction. Manj workers have found a correlation between roughness parameters and friction; friction increases with average roughness (Furey 1963, Koura 1980) and also with mean profile slope (Myers 1962, Ghabrial & Zaghlool 1974, Eiss & Warren 1975, Koura & Omar 1981, Moalic et al. 1987) (Fig. 11.1). Road roughness is llkely to influence friction between a wet road and a tyre as asperities pierce the intervening film of water (Taneerananon & Yandell 1981). An opposite effect of roughness on friction has also been reported. Ogilvy (1991, 1993) developed a statistical roughness model for adhesive friction which predicted that friction would decrease with increasing roughness, tending to a constant value independent of roughness. A wholly elastic and a mixed elasticplastic versionof the model were developed, dependin3 on two roughness parameters, the RMS roughness height and the correlation length. In tests with a 225

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226

thin molybdenum disulphide film on a rough steel subtrate, the theory gave good qualitative agreement, and reasonable quantitative agreement, with experiment (Fig. 11.2).

Figure 11.1. Variation of dynamic friction coefficient with mean profile slope (Koura & Omar 198 1)

-

dosli-plastic model w b l l y elastic model

I ’ 0.00

0-10

cipwimenlol volucs

0.20 0-30 0.40 rms heipht (microns)

0 50

Figure 11.2. Variation of fiction coefficient with roughness for steel coated with MoSz (Ogilvy 1993)

Chapman & Rizkallah-Ellis (1979) provide an interesting example of the effect of scale of roughness on tribological interactions. They found a pronounced directional effect in the coeficient of friction of automobile brake linings which, after eliminating other possible causes, they concluded was due to the surface topography. Reflectance measurements showed directional differences which could

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not always be detected by a stylus instrument. The implication is that at least some of the surface features responsible for friction in this casc were smaller than a stylus can resolve. Proctor & Coleman (1988) and Harris and Shaw (1988) showed that the roughness of floor surfaces was an important parameter in determining pedestrian slip-resistance, at levels of floor roughness much lower than had been reported by other researchers. It was well known that values of peak-to-trough surface roughnesses (Rtm) of 15-600 pm had an effect (Jung and Riediger, 1982). However, this level of roughness is many times higher than the Rtm values found by Harris and Shaw to have an important influence on the safety of water wetted floors. The significance of this finding was that it opened up new possibilities for ensuring the safety of floors exposed to contamination by water. In essence, all that is required is to incorporate a degree of roughness into the floor surface, that is small enough not to detract from the aesthetic appearance or create problems for cleaning. Since water is the most common floor contaminant, especially in public areas exposed to wet footwear, this is a significant development. Other researchers have confirmed the importance of surface roughness for assessing both floor surfaces and footwear (Manning et al., 1991, Stevenson, 1989. Gronqvist et al., 1990). Manning found that footwear soles of a granular construction that maintain surface roughness during wear, retain their slip resistant properties, whereas footwear soles that wear smooth, do not. He found a correlation between sole roughness Rtm and coefficients of friction measured during walking. He also found that the footwear ranked in approximately the same order on seven floor/contaminant combinations; this suggest; that surface structure determines the footwear ranking (Proctor 1993). Lloyd & Stevenson (1992) obtained a correlation between slip resistance and a combination of Rq, skewness and average wavelength.

11.2. Lubrication A large number of papers have appeared describing or modelling the effect of roughness on various lubrication regimes. Rough boundary lubrication has been investigated by Hisakado (1978), Nivatvongs et al. (1991) and Denape et al. (1992, 1995). Christensen (1965/6, 1971, 1972), Rao & Mohanran: (1993), Wakuri et al. (1995) and Liu et al. (1996) have reported work on rough mixed lubrication. Rough hydrodynamic lubrication has been treated by Bush & Gibson (1980), Kumar (1980), Nakahara et a1.(1984), Bayada & Chambat (1988), Cheng & Xie

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(1992), Sugimara & Yamamoto (1995) and Tonder and his co-workers (Christensen & Tonder 1971, Tonder & Christensen 1972, Prakash et al. 1979, Tonder 1980, 1987). Rough elastohydrodynamic lubrication (EHL) has been investigated by Tallian et al. (1965/6),Coy & Sidik (1979), Johnson et al. (1972), Kaneta & Cameron (1980), Karami et al. ( 1987), Lubrecht et al. (1988), Sadeghi & Sui (1989), Sinha et al. (1987), Fan & Zheng (1991), Kaneta (1992), Chang et al. (1993, 1994) and Ishibashi & Sonoda (1994). Chang (1995) has also reported on rough partial EHL. Micro-EHL, which by definition deals with rough surfaces, has been investigated by Baglin (1986), Kweh et al. (1989, 1992), Huang & Wen (1993), and Chang & Zhao (1995); the topic has been reviewed by Chang (1995). The specific application of lubrication in rough sliding has been dealt with by Tzeng & Saibel (1967), Patir & Cheng (1979), Shukla & Kumar (1979), Hughes (1981) and Anderson & Salas-Russo 1994. Other practical applications of rough lubrication include compliant surfaces (Darbey et al. 1979), improvement of roller bearing fatigue life (Akamatsu et al. 1991. 1992) (Fig. 11.3) and gear contacts (Peeken et al. 1990). Unfortunately some of these researches have confined themselves to simple deterministic roughness models and are thus of limited applicability.

Skewness Figure 11.3. Influence of skewness on relative fatigue life of rolling bearings (Akamatsu et al. 1991)

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An early application of functional filtering was to the case of a roller-bearing operating in a regime of mixed lubrication (Leaver et a1 1974). The surface finish of the bearings and races was measured before and after running-in, and a pronounced difference was noted, though no scuffing or other damage was apparent on the run-in surfaces. Measurements of the film thickness, however, showed it to be much smaller than the combined roughnesses of the contacting surfaces, and it was difficult to reconcile this with the absence of damage. Application of the principle of functional filtering suggested that all wavelengths longer than the Hertzian contact width should be ignored, as they took no part in the interaction with the lubricant film. The effect of this was to reduce the effective roughnesses to about a tenth of their measured valces, thus obtaining at a stroke a far more plausible set of film-thickness ratios and allowing the no-contact times to be predicted in fair agreement with experiment. One suspects that if this principle were more generally employed it would result in a substantial increase in the numerical values quoted in the literature for the so-called lambda ratio, the ratio of oil film thickness to roughness. Cann et al. (1994), in a review of the literature on lambda ratios, conclude that the behaviour inside the Hertzian contact zone depends on the mean slope and the asperity density as well as the roughness. They admit that many systems are known to run successfully at low lambda ratios. They point out that as soon as the lubricated surfaces are rough and the roughness heights are not negligible compared with the mean oil film thickness, the local pressure fluctuations cause3 by the asperities will have an influence on the elastic deformations of the surfaces.

11.3. Wear

The accommodation of two sliding surfaces over a period of time, variously known as running-in, brealung-in or shakedown, causes changes in their initial topography, and itself depends on that topography (Kapoor et al. 1994, Anderson et al. 1996). Kang & Ludema (1986) reported an optimum initial roughness of about 0.1 pm; smoother and rougher surfaces failed more quickly. Chandrasekaran (1993) found a proportionality between reniprocal wear rate and the reciprocal of initial roughness (Fig. 11.4). Summers-Smith (1969) dstinguishes two basic types of running-in mechanism. On the one hand he describes what he calls a “plastic squeezing” of the surface, that is a change in its shape by redistribution of material due to plastic flow without net loss. On the other hand there are the various wear mechanisms, adhesive or abrasive, all of

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which do involve net material loss. These two types of running-in are associated with quite different geometrical effects.

Figure 11.4. Wear rate of steel sliding against steel as a function of initial roughness (Chandrasekaran 1993)

The plastic redistribution of material during running-in is related to the finishing process known as roller burnishing (Black & Kalen 1973). The unit event is the compression of a single asperity by the roller until the plastic limit is exceeded, when the asperity will deform plastically so as to redistribute its material into the adjoining valleys. In general it will change its shape only until the new contact area is large enough to support the stresses elastically. The zone of affected asperities will therefore approximate to the nominal area of Hertzian contact between the roller and the rough sphere. Surface features significantly larger than this zone wilt not be affected by the burnishing process. If the length of the nominal contact is small compared with that of the cut-off of the measuring filter, very little change in roughness will be apparent from this type of running-in, because the roughness measurement is weighted heavily by the greater power associated with longer wavelengths. Whitehouse and Archard (1969), for instance, found a maximum change in R a of less than 20% in these

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23 1

conditions, although the changes in the actual profile were clearly visible to the naked eye. One way of overcoming this difficulty is to compare worn and unworn values of roughness as the filter cut-off is progressively shortened (Fig. 11.5a). If burnishing has occurred then power will have been lost by some band of wavelengths and hence some cut-off will be reached where the roughnesses diverge. The power spectrum is a more rational way of presenting the same information (Fig. 11Sb). (a)

I

i

2 6.(m

Cut-off wavelength (urn)

10 -6

f

I

I

I

1

1

2

3 4

6

810

1

1

1

1

I

l

l

I

20

3040 6 0 8 0 1 0 0 200 Frequency cyclelmrn :lmm

Figure 11.5. Topography of worn (circles) and unworn (crosses) inner races of a taper roller bearing characterised as (a) roughness v. high-pass filter cutoff (b) power spectra (Laver et al. 1974)

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Rough Surfaces

The simplest case of abrasive wear is a clean removal of the tops of the asperities without smearing or tearing. This may be performed experimentally under rather artificial conditions (Thomas 1972), but probably does not often occur in practice. However, it is convenient to investigate because it is easily simulated by computer (Thomas 1972; Willn 1972). As successive layers of the surface are abraded, parameters such as roughness, mean slope and mean peak curvature decrease in a systematic manner (Fig. 11.6).

A

I

P

I T'i

-0 ,

IS

I

I

I

10

5

0

-5

- 10

Height of worn surface above original mean line (rm) Figure 11.6. Effect of pure abrasion on (circles) RMS roughness (triangles) mean peak curvature (squares) profile curvature standard deviation (lozenges) mean absolute slope. Open symbols are simulated results, filled symbols are experimental. P = highest peak, V = lowest valley on unworn profile. Solid line is Gaussian bearing-area curve (Thomas 1972)

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These parameters are secondary rather than primary, as they can all be represented by joint probability distributions of heights. A more fundamental representation, then, is the height distribution. Hence it is important to describe correctly what happens to the distribution. Suppose we have a distribution, originally p{z), abraded until there are no heights higher than h above the mean line. This is sometimes described as being truncated at h, so that the new distribution

p{z)

p{z) for z 2 h Oforz>h

=

=

In fact the change is correctly described as being censored at h (Marcus 1967): p(z) =

= p(z) for z < h p(h)forz>h

Figure 11.7. Proposed typology of running-in (Thomas 1978). From top to bottom: unworn profile with power spectrum and height distribution; censoring without filtering; low-pass filtering without censoring; high-pass filtering without censoring; high-pass filtering with censoring

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That is, for a truncated distribution the fraction of values of z greater than h vanishes, but for a censored distribution it becomes equal to h, causing a Dirac spike at h (Fig. 11.7). The distinction makes a considerable difference to the moments of the distribution. It is more characteristic of abrasive wear in general, and also of adhesive wear, that changes in topography are due to the progressive removal of many small particles over long periods of time. Golden (1976) made a mathematical investigation of the effect on an initially Gaussian height distribution of a wear mechanism such that the rate of loss in height of an individual asperity with time is proportional to the depth to which the asperity has penetrated the opposing surface. He concluded that the resulting topography is that of the original surface up to some height representing the mean separation of the contacting surfaces. Above this height is superimposed another distribution, also Gaussian but smoother; as time progresses it will remain Gaussian but become smoother still (Fig. 11.8). Topographies of this type have been found experimentally by Ostvik and Christensen (1968/69) and also by Williamson et al. (1969) (see Section 7.4).

10-5 10-6

-

-

lo-'10-8 10-91

I

I

I

I

-10.00-R.00-6.00 -4.00 -2.00

I

I

0.00 2.00

I

I

4.00

6.00

Height, units of standard deviation Figure 11.8. Transitional double Gaussian height distribution of a surface at progressive stages of wear (Golden 1976)

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Censoring a random process reduces the amplitude of the autocovariance h c t i o n but has little effect on its form. When the autocovariance function is normalized to the autocorrelation function by the profile variance, the change due to censoring is barely perceptible even when abrasion has progressed as far as the mean line, as shown theoretically by Marcus (1967) and King et at. (1978) and confirmed by experiment (Willn 1972). Radhakrishnan (1977) divided the profile along the mean line and cross-correlated worn peaks and unworn valleys, but without any greater success. It follows that no change will be seen in the normalized power spectrum as this merely presents the same information in a m e r e n t form. A tentative classfication of run-in topographies was proposed by Thomas (1978) (Fig. 11.7). The original surface may be censored without filtering, or it may be high-pass filtered or low-pass filtered or both with or without censoring. In principle, then, it should be possible to define a run-in topography fairly closely by three or four parameters: the cut-off wavelength or wavelengths, the censoring level of the height distribution and, if a transitional topography, the roughness of the superimposed finish.

11.4. Seals

Roughness is a factor in determining the rate of leakage through seals, both static w t c h e l l & Rowe 1967/8, 1969, Wallach et al. 1968, Chivers & Hunt 1978, Hehn 1970, Kazamaki 1974, Otto 1974, Warren et al. 1988, Matsuzaki et al. 1992, 1993, Etsion & Front 1994) and dynamic (Lucas et al. 1994, 1995). Vacuum seals are a special case of static seals where the performance criterion in terms of leakage rate is particularly rigorous (Roth 1966, 1971, Yanagisawa et al. 1991). The performance of radial lip seals is known to depend on roughness, but the actual mechanism is not clear (Horve 1991, van Bavel et al. 1995). In the absence of a satisfactory physical model, a discriminant analysis approach has been applied (Thomas et al. 1975a, b) to distinguish between a set of rubber lip seals, individual members of which had either sealed or leaked. Worn and unworn profiles on the seals were measured and first 9 and then 11 parameters were computed. Two basic and relatively straightforward procedures were implemented in the analysis of the data. In the first, each of the nine features was examined individually. The procedure of evaluation was based on simple analysis of variance. The seals were first sorted into sealed or leaked categories. Then one of the surface measurements (e.g. zero-crossing density) was examined. An average value for this parameter

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Rough Su$aces

was computed for each performance category, and the overall or grand average value for all data was calculated. Then the pooled sum of squares was computed for the difference between each individual measurement and its corresponding group mean. This quantity is called the within sum of squares. Next, a sum of squares was computed by observing the difference between each group mean and the grand mean. This sum of squares, expressed on a per-observation basis, is called the between sum of squares. The ratio of the latter to the former sum provides a measure of the discriminating information available from the measurement in question. It is reasonable to consider that the greater this ratio the better the ability to discriminate between good and bad seals on the basis of a single parameter. In particular, if the ratio is large, it connotes a wide separation between the two groups of measurements but a relatively close clustering of individual measurements within each group. Accordingly, the several measurements can be ranked according to their (individual) ability to differentiate between effective and ineffective seals. It turns out that this ranking is different for the worn and unworn cases. Table 1 1 . 1 . Discrimination between sealed and leaked in terms of 1 1 surface parameters measured from worn and unworn profiles of 15 lip seals (Thomas et al. 1975a)

Discriminants

Rq Ra

Eigenvector x 10 Unworn -6.9 7.2 0.8

WOA.n

-0.5 0.4 -0 6

DO Peak height 5.8 -2.6 Valley depth 3.4 -0 1 Peak curvature -0.8 10.0 Valley curvature 1.4 -3.0 Profile curvature 0.1 -7 1 Slope -6.4 3.9 Sk 1 .o 07 3 ........ ................. ............K .*/................. .................. 0.5 ................................_.. .-1 ...............__ Rather than attempt a classification of seals on the basis of a single 'best' parameter, one may make much more effective use of the infomation contained in

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the data by means of linear discriminant functions. In this approach, the various parameters are combined as a weighted sum and the weights are adjusted so as to maximize the ratio of between to within sum of squares. The construction of the transformations appropriate to multiple linear discriminant functions is straightfonvard and well documented. In essence, the approach defines a sum of squares ratio in terms of the coefficients of a linear expansion. The ratio is then differentiated with respect to the coefficients and solved for the values which maximize the ratio. The problem reduces to finding the eigenvectors of a nonsymmetric matrix, and can be solved in a straightforward manner by any of a number of standard eigenvectodeigenvalue routines. The eigenvector is composed of the nine coefficients in the linear expansions. The associated eigenvalue provides a measure of the amount of variance accounted for by the discriminant function. Separate analyses for worn and unworn surfaces ezch separated the two categories and showed that the order of importance in which the parameters were ranked for the worn surfaces was quite different from that of the unworn (Table 11.1). The hypothesis advanced to explain this was that the geometry of the worn surfaces directly affects the sealing process, whereas that of the unworn surfaces affects it indirectly only in so far as it influences the production of the final geometry of the worn surfaces. As a result of the analysis it is possible to reconstruct ideal models of successfully and unsuccessfully sealing surfaces (Fig. 11.9).

Figure 11.9. Reconstructions from pattem-recOgnition analysis of profiles of the contacting surfaces of (a) an ideally good @) an ideally bad lip seal (discussion to Thomas et al. 1975b)

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Rough Surfaces

Although this form of analysis cannot replace a proper understanding of a particular problem it can aid it in several ways. By its assignation of degrees of importance to particular parameters it can offer the theoretician a useful initial guide for the formulation of ideas. In some practical cases it may be necessary to go no further, and the discriminant functions themselves may serve as part of the manufacturer's armoury of quality controls. The technique is also well suited to interactive computing work, where successive parameters can be dropped from the analysis until the engineer's subjective judgement decides that separation is no longer adequate.

11.5. References

Akamatsu, Y., Tsushima, N., Goto, T. and Hibi, K., "Influence of surface roughness skewness on rolling contact fatigue life", Trib. Trans. 35, 745-750 (1992) Akamatsu, Y., Tsushima, N., Goto, T., Hibi, K., Itoh, K., "Improvement of roller bearing fatigue life by surface roughness modification", SAE Trans. 100, 4449 (1991) Anderson, P., Juhanko, J., Nikkila, A.-P., Lintula, P., "Influence of topography on the running-in of water-lubricated silicon carbide journal bearings", Wear 201, 1-9 (1996) Anderson, S., Salas-Russo, E., "Influence of surface roughness and oil viscosity on the transition in mixed lubricated sliding stee: contacts", Wear 174, 71-79 (1994) Archard, J. F., "Elastic deformation and the laws of friction", Proc. Royal SOC.,A243, 190-205 (1957). Archard, J. F., "Single contacts and multiple encounters", J. Appl. Phys., 32, 1420-1425 (1961). Baglin, K. P., "Micro-elastohydrodynamic lubrication and its relationship with running-in", Proc. I. Mech. E, 200, 415-424 (1986) Bayada, G. and M. Chambat, "New models In the theory of the hydrodynamic lubrication of rough surfaces."Journal of Tribology, 110, 402-407, (1988) Bhushan, B., Tribology and mechanics of magnetic storage devices (Springer-Verlag, New York, 1990) Bikerman, J. J., "Surface roughness and sliding friction", Rev. Mod. Phys., 16, 53-68 (1944).

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Black, J. T. and Kalen, S. E., "The anatomy of a rolier burnished surface", Proc. Int. Conf: on Surface Technol., 507-526 (SME, Dearborn, 1973) Bowden, F. P. and Tabor, D., The friction and lubrication of solids Part 1 (Oxford University Press, 1950). Bush, A. W.; Gibson, R. D., "Effect of surface roughness and elastic deformation in hydrodynamic lubrication - A perturbation approach", in Surface roughness effects in hydrodynamic and mixed lubrication, 173-191 (ASME, New York, 1980). Cann, P.; Ioannides, E.; Jacobson, E.; Lubrecht, A. A,, "Lambda ratio - a critical re-examination", Wear 175, 177-188 (1994) Chandrasekaran, T., "On the roughness dependence of wear of steels: a new approach", J. Mat. Sci. Lett. 12,952-954 (1993) Chang, L., "Deterministic model for line-contact partial elasto-hydrodynamic lubrication", Tribology International 28,75-84 (1995) Chang, L., "Deterministic modeling and numerical simulation of lubrication between rough surfaces - a review of recent developments", Wear 184, 155-160 (1995) Chang, L.; Jackson, A,; Webster, M. N., "Effects of 3-D surface topography on the EHL film thickness and film breakdown", Tribology Transactions 37, 435444 (1994) Chang, L.; Webster, M. N.; Jackson, A,, "On the pressure rippling and roughness deformation in elastohydrodynamic lubrication of rough surfaces", Journal of Tribology, Transactions of the ASME 115,439-444 (1993) Chang, L. ; Zhao, W., "Fundamental differences between Newtonian and nonNewtonian micro-EHL results", Journal of Tribology, Transactions of the ASME 117,29-35 (1995) Chapman, R. J. and A. A. Rizkallah-Ellis, "Effect of the surface finish of brake rotors on the performance of brakes", Wear, 57, 345-356 (1979). Cheng, Y., Xie, Y., "Study of squeeze film between non-normal rough surfaces under partial hydrodynamic lubrication", Journal of Xi 'an Jiaotong University 26,59-65 (1992) Chivers, R. C. and Hunt, R. P., "The achievement of minimum leakage from elastomeric seals", 8th Znt. ConJ on Fluid Sealing, Paper 24 (BHRA, Cranfeld, 1978) Christensen, H. and Tonder, K., "The hydrodynamic lubrication of rough bearing surfaces of finite width", Trans. A.S.M.E. Ser.F. J.Lubr.Tech., 93, 324-330 (1971).

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Christensen, H., "A theory of mixed lubrication", Proc. I. Mech. E., 186, 421-430 (1972).

Christensen, H., "Nature of metallic contact in mixed lubrication", Proc. Instn.Mech. Engrs., 180, Pt.3B (1965/66) Christensen, H., Yome aspects of the functional influence of surface roughness in lubrication", Wear, 17, 149-162 (1971). Coy, J. J., S. M. Sidik, "Two-dimensional random surface model for asperity contact in elastohydrodynamic lubrication", Wear, 57, 293-3 11 (1979). Darbey, P. L.; Higginson, G. R.; Townend, D. J., "Lubrication of rough compliant solids", , Proc. 5th. Leeds-Lyon Trib. Symp. 398-403 (MEP. London, 1979) Denape, J.; Marzinotto, A,; Petit, J. A., "Roughness effect of silicon nitride slidmg on steel under boundary lubrication", Wear 159, 173-184 (1992) Denape, J.; Masri, T.; Petit, J.-A,, "Influence of surface roughness and oil ageing on various ceramic-steel contacts under boundary lubrication", Proc. I. Mech. E: J. Eng. Trib. 2095, 173-182 (1995) Eiss, N. S. and Warren, J. H., "The effect of surface finish on the friction and wear of PCTFE plastic on mild steel", S.M.E. Paper IQ75-125 (1975). Etsion, I.; Front, I., "Model for static sealing performance of end face seals", Tribology Transactions 37, 1 1 1-1 19 (1994) Fan, Y., Zheng, L., "A study on the limit criterion of full and partial lubrication", Wear 143,22 1-229 (199 1) Furey, M. J., "Surface roughness effects on metallic contact and friction", A.S.L.E.Trans., 6,49-59 (1963). Ghabrial, S. R. and Zaghlool, S. A,, "The effect of surface roughness on static friction", Int. J. Mach. Tool Des. Res., 14, 299-309 (1974). Golden, J. H., "The actual contact area of moving surfaces", Wear, 42, 157162 (1977). Gronqvist, R., Roine, J., Korhonen, E., Rahikainen, A,, "Slip resistance versus surface roughness of deck and other underfoot surfaces on ships", J. Occup. Accid. 13, 291-302 (1990) Harris, G. W. and Shaw, S. R., "Slip resistance of floors: users' opinions, Tortus instrument readings and roughness measurements", J. Occup. Accid. 9, 287-298 (1988) Hehn, A. H., "Effects of friction and wear on a sealing interface", Lubr. Engng. 26, 206-212 (1970). Hisakado, T., "Influence of surface roughness on friction and wear in boundary lubrication", J. Mech. Eng. Sci. 20,247-254 (1978).

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Tribology

Home, L., "Correlation of rotary shaft radial lip seal service reliability and pumping ability to wear track roughness and microasperity formation", SAE Trans. 100,620-627 (1991) Huang, P.; Wen, S. Z., "Sectional microelastohydrodynamic lubrication", Journal ofTribology, Transactions of the ASME 115, 148-151 (1993) Hughes, W. F., "A cell theory of rough surface lubrication", Wear, 67, 31-53 (1981). Ishibashi, A., Sonoda, K., "Mirrorlike finishing of precision rollers and changes on the roller surfaces caused by loaded running", JSME International Journal, Series 3 35,286-293 (1992) Johnson, K. L., Contact mechanics (Cambridge University Press, London, 1985). Johnson, K. L., Greenwood, J. A. and Poon, S. Y., "A simple theory of asperity contact in elastohydrodynamiclubrication", Wear, 19, 91-108 (1972). Jung, K. and Riediger, G., "Recent developments regarding the inspection of non-slip floor coverings", Die Berufgenossenschaft 6,l-7 (1982) Kaneta, M.; Cameron, A., "Effects of asperities in elastohydrodynamic lubrication", ASME Papern 79-Lub-6 (1979). Kaneta, M., "Effects of surface roughness in elastohydrodynamic lubrication", JSME International Journal, Series 3: 35,535-546 (1992) Kang, S. C., and Ludema, K. C., "The breaking-in of lubricated surfaces", Wear 108,375-384 (1986) Kapoor, A.; Williams, J. A.; Johnson, K. L., Steady state sliding of rough surfaces", Wear 175,81-92 (1994) Karami; G., H. P. Evans and R. W. Snide, "Elastohydrodynamic lubrication of circumferentially finished rollers having sinusoidal roughness." Proc. I. Mech. E. Part C, Mechanical Engineering Science, 201,29-36, (1987) Kazamaki, T., "An investigation of air leakage between contact surfaces. (3rd report, in which iron and brass were used as specimen)", Bull. J.S.M.E., 17, 1321-1331 (1974). King, T. G., Watson, W., Stout, K. J., "Modelling the microgeometry of lubricated wear", Proc. 4th. Leeds-Lyon Symp., 333-343 (MEP, London, 1978) Koura, M. M., "The effect of surface texture on friction mechanisms", Wear, 63, 1-12 (1980). Koura, M. M. and M. A. Omar, "The effect of surface parameters on friction", Wear, 73, 235-246 (1981) Kumar, S., "Stochastic models with variable viscosity for hydrodynamic lubrication of rough surfaces", Wear, 62, 329-336 (1980). 'I

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Kweh; C. C., H. P. Evans and R. W. Snidle, "Micro-elastohydrodynamlc lubrication of an elliptical contact with transverse and three-dimensional sinusoidal roughness." Journal of Tribology, 111,577-584, (1989) Kweh, C. C., Patching, M. J., Evans, H. P. and Snidle, R. W., "Simulation of elastohydrodynamic contacts between rough surfaces", J. Trib., ASME, 114, 412419 (1992) Leaver, R. H., Sayles, R. S. and Thomas, T. R., "Mixed lubrication and surface topography of rolling contacts", Proc. I. Mech. E., 188,461-469 (1974). Lee, S. C., and Ren, N., "Behavior of elastic-plastic rough surface contacts as affected by surface topography, load, and material hardness", Trib. Trans. 39, 6774 (1996) Liu, K., Liu, X. J., Xie, Y. B., "Definition of the transition from fluid lubrication to mixed lubrication for rough surface", Lubrication Science 8, 287-295 (1996) Lloyd, D. G., and Stevenson, M. G., "An investigation of floor surface profile characteristics that will reduce the incidence of slips and falls", Mech. Eng. Trans: Inst. Engrs. Australia ME17, 99-105 (1992) Lubrecht; A. A., W. E. ten Nape1 and R. Bosma, "The influence of longitudinal and transverse roughness on the elastohydrodynamic lubrication of circular contacts."Journal of Tribology, 110, 421-426, (1988) Lucas, V., Bonneau, O., Frene, J., "Roughness influence on turbulent flow through annular seals including inertia effects", ASME Paper 95-TRIB-11 (1995) Lucas, V., Danaila, S., BOMeaU, O., Frene, J., "Roughness influence on turbulent flow through annular seals", Journal of Tribology, Transactions of the ASME 116,321-329 (1994) Manning, D. P., Jones, C., Bruce, M., "A method of ranking the grip of industrial footwear on water wet, oily and icy surfaces", Safety Science 14, 1-12 (1991) Marcus, A. H., "Statistical model of a flooded random surface and applications to lunar terrain", J. Geophysical Rex, 72, 1721-1726 (1967). Matsuzaki, Y., Funabashi, K., Hosokawa, K., "Effect of surface roughness on contact pressure of static seals. (Effect of tangential force on conical inside-seal surface)", JSME International Journal, Series 3: 36, 119-124 (1993) Matsuzaki, Y., Hosokawa, K., Funabashi, K., "Effect of surface roughness on contact pressure of static seals", JSME International Journal, Series 3: 35,470-476 (1992)

Tribologv

243

Merriman, T. and J. Kannel, "Analysis of the role of surface roughness on contact stresses between elastic cylinders with and without soft surface coating. " Journal ofTribology, 111, 87-94, (1989) Mitchell, L. A. and Rowe, M. D., "An assessment of face seal performance based on the parameters of a statistical representation of surface roughness", Proc. I. Mech. E., 182, Part 3K, 101-107 (1967/68). Moalic; H., J. A. Fitzpatrick and A. A. Torrance, "A spectral approach to the analysis of rough surfaces", Journal of Tribology, 111, 359-363 (1989) Myers, N. O., "Characterization of surface roughness". W7ear, 5 , 182-189 (1962). Nakahara, T., M. Takesue and H. Aoki, "Effects of surface roughness and bearing slenderness ratio on hydrodynamic lubrication." Journal JSLE Int Ed, No.5, 65-70, (1984) Nivatvongs, K., Cheng, H. S., Ovaert, T. C. and Wilson, W. R. D., "Influence of surface topography on low-speed asperity lubrication breakdown and scuffing", Wear, 143, 137-148 (1991) Ogilvy, J. A,, "Predicting the friction and durability of MoS2 coatings using a numerical contact model", Wear 160,171-180 (1993) Ogilvy, J. A,, "Numerical simulation of friction between contacting rough surfaces", Journal of Physics D (Applied Physics) 24, 2098-2 109 (1991) Ostvik, R. and Christensen, H., "Changes in surface topography with running-in", Proc. I. Mech. E., 183, part 3P, 57-65 (1968/69). Otto, D. L., "Triangular asperities control seal leakage and lubrication", S.A.E. Paper No. 740201, (1974). Patir, M., Cheng, H. S., "Application of average flow model to lubrication between rough sliding surfaces", J. Lubr. Technol. Trans. ASUE 101, 220-230 (1979). Peeken, H.; Ayanoglu, P.; Knoll, G.; Welsch, G., "Measurement of lubricating film thickness, temperature and pressure in gear contacts with surface topography as a parameter", Lubrication Science 3, 33-42 (1990) Poon, C. Y., and Bhushan, B., "Numerical contact and stiction analyses of Gaussian isotropic surfaces for magnetic head slideddisk contact", Wear 202, 6882 (1996) Poon, C. Y., and Bhushan, B., "Nano-asperity contact analysis and surface optimisation for magnetic head slideddisk contact", Wear 202, 83-98 (1996) Prakash, J.; Tonder, K.; Christensen, H., "Micropolarity roughness interaction in hydrodynamic lubrication", ASME Paper 79-Lub-8 (1979).

244

Rough SurJaces

Proctor, T. D., "Slipping accidents in Great Britain - an update", Safety Science 16, 367-377 (1993) Proctor, T. D., and Coleman, V., "Slipping, tripping and falling accidents in Great Britain - present and future", J. Occup. Accid. 9, 269-285 (1988) Fbdhakrishnan, V., "The application of correlation functions in wear measurements", Wear 41, 169-177 (1977). Rao, A. Ramamohana; Mohanram, P. V., "Study of mixed lubrication parameters of journal bearings", Wear 160, 111-118 (1993) Rosen, B.-G., R. Ohlsson and T. R. Thomas, "Nano metrology of cylinder bore wear", Trans. 7th. Int. ConJ On Metrology & Properties of Engng Surfaces, 102-110 (Goteborg, 1997) Roth, A., "The interface-contact vacuum sealing processes", J. Vacuum Sci. & Technol. 9, 14-23 (1971). Roth, A,, Vacuum sealing techniques (Pergamon, Oxford, 1966) Sadeghi, F. and P. C. Sui, "Compressible elastohydrodynamlc lubrication of rough surfaces." Journal of Tribology, 111, 56-62, (1989) Shukla, J. B. and S. Kumar, "Effects of viscosity variation and surface roughness in the lubrication of a slider bearing", Wear, 52, 235-242 (1979) Sinha; P., J. S. Kennedy and C. M. Rodkiewicz, "Effects of surface roughness- and additives in lubrication: generalised Reynolds equation and its application to elastohydrodynamic film." Proc. I. Mech. E. Part C, Mech. Eng. Sci., 201, 1-9, (1987) Sugimara, J., and Yamamoto, Y., "Hydrodynamic lubrication of self-affhe fractal surfaces", Trans. JSME 61C, 475 1-4756 (1 995) Summers-Smith, D., An introduction to tribology in industry (Machinery Publishing Co., London, 1969) Tallian, T. E., McCool, J. I. and Sibley, L. B., "Partial elastohydrodynamic lubrication in rolling contact", Proc. I. Mech. E., 180, Part 3 8 , 169-84 (1965/66). Taneeranonon, P. and W. 0. Yandell, "Microtexture roughness effect on predicted road-tyre friction in wet conditions", Wear, 69, 321-337( 1981). Thomas, T. R., "Computer simulation of wear", Wear, 22, 83-90 (1972). Thomas, T. R., "The characterisation of changes in surface topography during running-in", Proc. 4th Leeds-Lyon Symp., 99-108 (MEP, London, 1978) Thomas, T. R., Holmes, C. F., McAdams, H. T. and Bernard, J. C., "Surface features influencing the effectiveness of lip seals: a pattern - recognition approach", S.M.E. Paper 1475-128 (1975a).

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245

Thomas, T. R., Holmes, C. F., McAdams, H. T. andBernard, J. C., "Surface microgeometry of lip seals related to their performance", Proc. 7th. Int. Con$ on Fluid Sealing, Paper J32 (BHRA,Cranfield, 1975b) Tonder, K. and Christensen, H., "Waviness and roughness in hydrodynamic lubrication", Proc. I. Mech. E., 186, 807-812 (1972). Tonder, K., "Effects of skew unidirectional striated roughness on hydrodynamic lubrication. Part 2- Moving roughness." Journal of Tribology, 109, 671-678, (1987) Tonder, K., "Numerical investigation of the lubrication of doubly periodic unit roughness", Wear, 64, 1-14 (1 980). Tzeng, S. T. and Saibel, E., "Surface roughness effect on slider bearing lubrication", A.S.L.E. Trans., 10,334-338 (1967) Van Bavel, P. G. M.; Ruijl, T. A. M.; Van Leeuwen, H. J.; Muijdennan, E. A., "Upstream pumping of radial lip seals by tangentially deforming, rough seal surfaces", ASME Paper 95-TRIB-13 (1995) Wakuri, Y., Hamatake, T., Soejima, M., Kitahara, T., "Study on the mixed lubrication of piston rings in internal combustion engine", Nippon Kikai Gakkai Ronbunshu, 61C, 1123-1128(1995) Wallach, J., Hawley, J. K., Moore, H. B., Rathben, F. V. and Gitzendanner, L. G., "Calculation of leakage between metallic sealing surfaces", A.S.M.E. Paper 68-LUB-15, (1968). Warren; W. E., J. G. Curro and D. E. Amos, "On the nature of O-rings in contact with rough surfaces. Journal ofTribology, 110,632-637, (1988) Willn, J. E., "Characterisation of cylinder bore surface finish - A review of profile analysis", Wear, 19, 143-62 (1972). Yanagisawa, T., Sanada, M., Koga, T., Hirabayashi, H., "Influence of designing factors on the sealing performance of C-seal", ME Trans. 100, 651-657 (1991) Yu, M. M., and Bhushan, B., "Contact analysis of threedmensional rough surfaces under frictionless and frictional contact", Wear 200,265-280 (1996) Is

CHAPTER 12

SOME OTHER APPLICATIONS

So far we have had the opportunity to discuss the application of surface roughness studies to contact mechanics in some detail, and we have also been able to outline briefly some of the more important areas of application in tribology. Regrettably, it is not possible within the scope of tlus book to cover the whole range of the other engineering and scientific effects of roughness. In this concluding chapter we will select a few of these applications, chosen at least partly to illustrate our earlier arguments about functional filtering. Contact resistance is interesting to electrical engineers and also important in heat transfer studies. The effect of roughness on fluid flow is appreciated in many technical fields, from aero- and hydrodynamics to chemical engineering, and also to workers in the earth sciences. Machining problems are the domain of manufacturing and production engineers, and Qmension and tolerance are also important to quality control personnel and inspectors. Finally we conclude by touching briefly on two topics of interest to their eponymous communities, bioengineering and geomorphology.

12.1. Contact Resistance

In many engineering applications involving contact it is sufTicient to know the real contact area. However, an important practical class of problems where a more detailed knowledge is necessary is that of thermal or electrical contact resistance. When two surfaces are in contact and electricity or conductive heat passes between them, it does so through the discrete areas of contact. The lines of equal potential, parallel at a distance from the interface, become increasingly distorted as the contact zone is approached, and the flow lines bunch together to pass through the individual contacts (Fig. 12.1). Holm (1958) has shown that the conductance of a single contact spot is proportional not to its area but to its radius. The conductance per unit nominal area of the interface is

c, = 2Kn t/a 247

(12.1)

248

Rough Sursaces

where a is the mean contact spot area, n is their number per unit area and K is the thermal or electrical conductivity. It is therefore necessary to determine the variation with load both of the number of contact spots and of their individual size.

Figurel2.1. Contact resistance due to constriction of flow lines

Electrical conductivity is of practical importance in the electronics and electrical power industries, and the effect of roughness has received some attention (Barkan & Tuohy 1965, Harada & Mano 1968, Hisakado 1977, Lanchon et al. 1986, Bryant 1994, Clausen & Leistiko 1995). The economic importance of thermal contact resistance is also considerable (Thomas & Probert 1972). In the form of the resistance between the fuel element and its container, it affects the economics of nuclear power (Boeschoten & van der Held 1$57, Tillack & Abelson 1995). In space technology also, the electronic equipment inside a satellite or space vehicle generates heat which can only be dissipated by solid conduction to the outside (Chung et al. 1993, Chung 1995, Chung & Shefield 1995). The property of the contiguous solids most difficult to define quantitatively is the topography of their contacting surfaces (Yip & Venart 1967/68, Cooper et al. 1969, Mikic 1974, Thomas 1982, Majumdar & Tien 1991, McWaid & Marschal 1992). Not only the surface roughness is involved but its waviness also, and a number of workers have considered the combined resistances of both (OCallaghan et al. 1989, Lambert & Fletcher 1995, Torii & Nishino 1995). It seems generally to be felt that the resistances are additive, so that the total resistance of the interface is the sum of a macroscopic resistance due to wavkiess and a microscopic resistance due to roughness. The microscopic resistance is due to the convergence of lines of flow through individual microscopic contact areas which are due to the

Other Applications

249

surface roughness. Clusters of these spots are held to be contained within larger macroscopic contour areas due to waviness. The effective of waviness on thermal contact resistance was investigated (Thomas & Sayles 1975) by considering the behaviour of the interface at very light loads W when it could be treated as a three-point Hertzian elastic contact. Here the high-pass cut-off was again a hnction of the diameter of the bar, and a low-pass cut-off was sought such that the enclosed bandwidth would contain only three asperities. This was found by substituting the appropriate asperity density into the integrated moment equation and solving the resulting transcendental equation iteratively. The strategy was as follows. One starts by assuming a form for the spectrum of wavelengths and goes on to define a pass-band representing waviness. The long-wavelength limit of this band must be set by the size A of the nominal contact area; the low-pass cut-off is found in terms of the contact size from the initial contour area density. Knowledge of the bandwidth permits calculation of the mean height and radius of curvature of the contacting asperities. The usual Hertzian formula will then give the area of real contact in terms of the nominal contact size and the total roughness at that size. This total roughness may be found from the measured roughness, leading to a figure of merit, sensitive to the effect of waviness on resistance, which is a function of readily measurable parameters.

Figure 12.2. Thermal contact resistance:experimental results of Fried (1965) replotted as a fhnction of waviness number <(Thomas & Sayles 1975)

The waviness number < = W/E’cr& so defined has a number of properties whlch make it suitable as a figure of merit for determining contact mechanism. Although it cannot at this stage be used to describe the lower limit of microscopic

250

Rough Sudaces

resistance, it has been argued, with some experimental sup?ort, that the effect of waviness can be neglected for 6 > 1. Its predictions are qualitatively reasonable; for a high load on a small smooth contact of low Hertzian modulus, jwill be large and the waviness will flatten to let the microscopic resistance predominate; for a low load on a large rough contact of high elastic modulus, cwill be small and the macroscopic resistance will predominate. One set of experimental results replotted in terms of waviness number can be seen in Fig. 12.2. The sharp dog-leg indicating a change of regime occurs near <= 1 as predicted. It turns out to be very difficult to find experimental data in the literature for which this condition is satisfied. The implication is that for the range of conditions of engineering interest the effect of waviness can never be neglected. There is some support for thw conclusion. When resistance is entitely due to waviness it can be inferred that the total resistance of the interface is proportional to W o 3 3 . From results on surfaces specially prepared with roughness but no waviness, it is known that when resistance is entirely due to roughness the exponent of W is between -0.95 and -0.99 (Thomas & Probert 1970). However, for the vast majority of results in the literature the exponent lies between these extreme values at about 0.73 (Thomas & Probert 1972). It seems likely, therefore, that in most practical situations thermal contact resistance is due to the combined effects of waviness and roughness. The case where microscopic resistance only is present is rather more difficult to tackle if plastic contact is assumed. The reason is that there are then no grounds for truncating the power spectrum at a short-wavelength limit and consequently the higher moments are infinite. Lf elastic contact is assumed then we can again use the argument of repeated contact to impose a low-pass filter based on the plasticityindex criterion. This allows us to find an upper-bound solution for the average contact spot size and an exact solution for the number of contacts, and hence an upper-bound solution for the conductance (Sayles & Thomas 1976a). All these are in terms of the separation of the contacting surfaces, but as we also know the variation of load with separation we can plot all three in terms of load (Fig. 10.2).

12.2. Noise and Vibration

Small surface irregularities can give rise to noise and vibration in rolling contact (Nayak 1972, Ananthapadmanaban & Radhakrishnan 1982, Hess & Soom 1991). This causes problems in rolling bearings (Yhland 1967/8, Kanai et al. 1987), railroad operation (Gray & Johnson 1971, Thompson 1993a, b) and transmission

Other Applicatiom

25 I

design (Mengen & Weck 1992, Aziz & Seireg 1994). The dynamic characteristics of rolling element bearings (Poon & Wardle 1978) are such that an extremely high number of resonances exist just in the bearing itself. When housing alignment, geometry and the general bearing environment are considered the problem becomes more complex, and a bearing design which would avoid or attenuate vibration seems very unlikely. The problem is best solved by removing or reducing the source of vibration, namely the wavelengths of the specific surface features which are responsible for the input of vibration energy. In achieving this it is first necessaq to identi@ these wavelengths and under what circumstances they can produce vibration (Su et al. 1993), and secondly to establish how they can be effectively monitored and removed in a production environmat. The input of vibration energy is due to random surface interaction occurring predominantly at the rolling-element interfaces. In some cases a significant energy input can be supplied by ball-cage interaction, macroslip and spin, but at conventional speeds and reasonable loads these effects are of secondary importance. By considering the possible partial and full-film lubrication conditions, and the physical interpretation of these mechanisms in terms of surface contact and the interchange of energy, the surface interaction can be split into three distinct mechanisms: rolling response due to form and waviness effects, shock noise due to local elastic deformation, and impulsive shocks due to asperity collisions and debris. Having established the size of asperities which can Aastically conform, it becomes possible to determine their density and therefore frequency of interaction (Sayles & Poon 1981). This can be easily accomplished through Eqn. 9.17, modfied to take account of anstropy. It is achieved by a circumferential and an across-lay measurement from the raceway components. From such measurements on all the conventional finishing processes it appears that this form of vibration generation predominates over the others and can affect all frequencies, although the most severe effects are apparent in the range of about 300-10,000 Hz.

12.3. Fluid Flow

Roughness is of interest and importance in a number of applications involving fluid flow. Flow through a rough gap or between rough planes, for instance, is a phenomenon affecting the performance of air bearings (Lau & Harman 1975) and of static seals such as vacuum seals (Thomas 1973a). The elemental mass flow

252

Rough Sui$aces

rate dQ through a length dx of a onedimensional rough gap of mean width h is given by (Thomas & Olszowski 1974) dQ

K

(h -z)"dx

(12.2)

where z is the instantaneous height of the rough surface above its mean plane and n depends on the flow regime. As z is not an analytic hnction of x we cannot integrate this directly over the length of the gap. However, we can transform the integration into one over the height probability distribution p(z). The fraction of the gap length at height z is p(z)dz, hence the mass flow per unit length is dQ

K

(h - z)" p(z)dz

(12.3)

and the total mass flow rate will be

(12.4)

I

-2

-1

0

I

2

t

3

Figure 12.3. Variation ofdimensionless effective gap width with dimensionless mean plane separation, accordingto Eqn. 12.5:(1) laminar, n = 3; (2) smooth turbulent, n = 12/7; (3) rough turbulent, n = 3/2 (Thomas& Olszowski 1974)

253

Other Applications

The values of n for laminar and turbulent flow are 3 and 3/2 respectively. If the height distribution is Gaussian, the integral is closed-form for n integer and can be evaluated numerically for other n. Jt is convenient to present the result as an equivalent gap 6 defined such that

(12.5)

The results indicate (Fig. 12.3) that the mean gap "seen" by the fluid diverges markedly from the real gap as the surfaces approach. This has implications for, for instance, the design of pneumatic gauges (see Section 4.2.3).

-

2-

1.1

1.0

v

-

=n.Y

-2 n . u 0.7

0.6

0.5 0.4

0.3 0.2 2.6 2.X

3.1) 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4 6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

1% 10 Re Figure 12.4. Skin fiction as a function of Reynolds number in pipes coated with increasinglylarge grains of sand; each curve represents grains twice as large as those for the curve below (Hunsaker & Rightmire 1947 after Nikuradse)

An application of even greater economic importance involves the effect of roughness on skin friction. This has been well known since Nikuradse's classic experiments on flow in rough pipes (see, for instance, Hunsaker & Rightmire 1947). He roughened a pipe artificialiyby cementing sand grains of similar size on its interior surface and repeated this with a number of pipes for various sizes of sand grain. For a smooth pipe the friction factor .ro/pu2,where is the wall shear and p and tl are the fluid density and velocity respectively, is inversely proportional

254

Rough Surfaces

to the logarithm of the Reynolds number Re, the constant of proportionality depending on whether flow is laminar or turbulent. For a rough pipe, however, the friction factor departs from its logarithmic dependence at some value of Re and tends to a constant value; that is, the shear stress increases as the square of the velocity. This constant friction factor increases, and the critical Reynolds number decreases, as the roughness is increased (Fig. 12.4). This has a profound effect on the performance of ships, where skin friction can amount to 80-90% of the total resistance to motion (Lackenby 1962). Ships tend to become rougher in service due to paint breakdown, corrosion and fouling, and this can necessitate an increase of power of as much as 40 per cent to maintain the same speed. An increase in roughness of 25 pm increases resistance by about 2%%; to put this in perspective, the roughness of a new hull is on average about 175 pm, at a high-pass cut-off of 50 mm (Lackenby 1962). Measurements on replicas of ships’ hulls (King et al. 1981) have shown that their height distributions are reasonably close to Gaussian and their power spectra are of the familiar inverse-square form; in other words they are mathematically very similar, on a larger scale, to the machined surfaces discussed throughout this book. Surprisingly little fundamental research has been carried out on this phenomenon since Nikuradse’s original work. The qllalitative explanation generally offered for the roughness resistance is that extra turbulence is caused by the projection of the roughness elements through the laminar sublayer immediately adjacent to the solid surface; if the height of the roughness elements is small relative to the thickness of the sublayer then the surface will behave as though it were hydraulically smooth, and the transition to rough behaviour is due to the decrease of thickness of the laminar sublayer with increasing Reynolds number. It is usual to express the results of any fluid measurement involving a rough surface in terms of an equivalent sand-grain number, a descriptive approach which is not altogether satisfactory in modem terms. Two points are apparent in the light of today‘s knowledge of surfaces. The first is that most surfaces generally have much gentler slopes than abrasive surfaces, and the abrupt changes in height between adjacent grains may themselves be likely to induce a special sort of turbulence not present for surfaces of a different texture. The second is that real surfaces are not composed of roughness elements of the same geometric form and at the same height. A Gaussian height distribution would imply a gradual increase in the number of summits penetrating the laminar sublayer as its thickness decreased, resulting in a transition region between “fully smooth and “fully rough regimes, and indeed such a transition region is apparent (Fig. 12.4). This transition regon is much wider for “natural” surfaces than for

Other Applications

255

Nikuradse’s pipes where no doubt an attempt was made to level the grains to the same height. The actual process of momentum transfer from a fluid to a rough surface does not seem to have been at all widely studied. A theoretical treatment in terms of the power spectrum of surface wavelengths (Singh & Lumley 1971) yielded predictions at variance with experiment. One might expect a priori that both the roughness and the slope would exert an important effect, and also thaf as usual it should be possible to isolate a pass-band of surface wavelengths which affect skin friction. What is the shortest wavelength, for instance, to which the laminar sublayer will conform? Clearly, wavelengths longer than thw will play no part in roughness effects. A correlation is reported between increased power requirement and mean apparent amplitude measured at 50 mm cut-off, but this cut-off is admittedly arbitrary (Lackenby 1962). An apparent peak in the surface power spectrum around 50 mm wavelength (Chaplin 1967) is probably an artefact of computation.

12.4. Dimension and Tolerance

A relationship between surface finish and dimensional tolerance has long been suggested (Schlesinger 1944, Ber & Yarnitzky 1968, Bryan & Lindberg 1973, Osanna 1979). It has also been well known for many years that the tolerance of machned components must be increased with component size. The reasons for k s are not clearly defined, although in BS 4500: 1969 it is stated that to some extent this can be explained by an increased difficulty in manufacture. Thermal expansion effects must also be considered, as these increase with component size but are generally small in relation to tolerance with components below 500 mm diameter. BS 4500 gives empirical rules which relate tolerance to diameter which are supposedly derived from “extensive practical investigations”, but the fundamental reasons why this should be so are still somewhat obscure. As an example, a 3 cm diameter shaft, produced on say a lathe, would be associated with tolerance under the IT10 grade of 40 pm, whereas the same machine and tolerance grade allows a 25 cm shaft 84 pm. Section 1.6 of BS 4500 suggests that geometry, form and surface texture must also be considered in some circumstances. In fact this is the principal reason why tolerance is necessary. The increase in Wiculties in manufacture with component size stated as a reason by BS 4500 is simply another way of stating the arguments leading up to the non-stationarity result of Section 8.4, in that as the size increases the number of potential surface geometric errors also increases.

256

Rough Surfaces

Fig. 12.5 shows the fractal topography-length law (Sayles & Thomas 1978) plotted against the empirical equations governing tolerance grades IT6-16 of BS 4500 up to 500 mm. The parameters of the fractal equation are chosen so that the curves overlap; however, the values are typical of an average ground surface. The fact that the curves do overlap is unimportant other than it provides the best way of comparing the trends. The agreement in trend is surprisingly good, even to the extent of anticipating tolerance requirements above 500 mm, and given by a separate equation in BS 4500 which is shown by the dashed line. The figure demonstrates that for a given tolerance grade, to maintain a constant grade of fit we are increasing the tolerance in the same way as the surface topography is changing. In other words we are maintaining the same relative surface interface conditions.

Figure 12.5. Tolerance Ias a function ofworkpiece diameter D. Dashed line:I = 0.4501'3+ 0.0010 (D 5 500 m)(BS 4500: 1969); dotted line: I = 0.0040 + 2.1 (D > 500 m)(BS 4500: 1969); solid line: I = 0.4 + $10" x D ) (Sayles &Thomas 1978)

From a production engineering point of view the fractal relationship can be considered at two different levels. Firstly it gives us an insight into the way in which the increase in tolerance is linked to size; an empirical fact whch has long been established and taken for granted. Secondly, and on a more practical level, it can provide the production engineer with a means of monitoring the condition of a machine. We know that machines producing the same components can generate differing values of fractal parameters, a good measure of machine condition and environment. We have shown how the fractal parameters can be related to a class of tolerance; thus it seems possible to classlfy a machme quality in terms of its

257

Other Applications

ability to produce components within a given tolerance grade. Periodic checks on the fractal parameters of surfaces being produced would also act as a good indication of the potential useful life of the machine.

12.5. Abrasive Machining

Ground surfaces have a welldefined lay owing to the parallel orientation of grinding scratches, which are much longer in the direction of rotation of the grinding wheel than they are wide. It is of interest in the study of grinding processes to determine the average dimensions of a grinding scratch. Thts is not readily done by direct observation, as the scratches are superimposed on each other to a confusing degree. A possible solution is to examine profiles taken at various angles to the lay; it can plausibly be shown that the correlation length should represent the average dimension of a scratch in that direction (Fig. 12.6). There can be seen a marked peak in the direction of the lay; the average length and breadth of a grinding scratch in this case were deduced a$ 252 pm and 34 pm respectively (Thomas 1973b).

E

0

0

120 '

0

0

3 IOU

'B 000

0

I

I

80

I

I

7 0 6 0

I

1

5 0 4 0

I

M

I 20

I 10

I

I

0 - 1 0

Angle to lay (degrees) Figure 12.6.Correlation length as a function of angle of stylus traverse kom the lay of a ground surface (Thomas1973b)

The height distribution of a ground surface is often cited as Gaussian and a practical example of the central limit theorem. However, Sayles & Thomas (1979) found a highly sigruficant negative skewness in samples of several hundred thousand height readings from a number of ground surfaces (Fig. 7.11). The

258

Rough Sur$aces

distributions were truncated at their upper ends, in effect possessing fewer high peaks than a normal distribution. A study of the literature revealed that a similar skewness is in fact almost always present in height distributions from simple abrasive processes, and is often present to a lesser extent in ground surfaces. Kapteyn (as quoted in Hald 1960) developed a statistical theory of skew distributions in terms of a function which can be interpreted as the effect of the abrasion or grinding mechanism at the interface. This mechanism has been the subject of much dxussion, but it has generally been established that a combination of conventional cutting, ploughing and plastic deformation exists. Such a mechanism suggests that the geometry of the abrasive surface imposes itself on the machined surface, irrespective of the metal removal or deformation mechanism involved at each individual grain. If this is so, then where light cuts are involved and several passes are made without increasing the feed, Kapteyn's theory would predict a Gaussian height distribution; conversely, with heavy cuts and single passes a truncated distribution would result. Distributions of both types are reported in the litenture; it is suggested, however, that the strict Gaussian distribution, heretofore accepted as the norm, is an artefact of the care taken to obtain a specimen, and that on the production line, where single-pass and plunge grinding are extensively employed, the height distributions created on many engineering components are negatively skewed. If so, this may have important practical implications: for instance, the degree of truncation of the height distribution has been found to influence the running-in of cylinder bores (Campbell 1972), and it might also be supposed that in a seal with a ground surface the sealing action would be affected by the number of high peaks on the surface. A detailed theoretical consideration of the grinding process (Sayles & Thomas 1976b) enabled a prediction of an effective height distribution for a grinding wheel which showed a strong positive skewness. This will produce its mirror image, a negatively skewed height dwtribution, on the ground surface. The exact form of the distribution depends on the number of effective profiles n on the grinding wheel which intersect with the surface; for small n the distribution is closely Gaussian, but becomes increasing skewed as n increases (Fig. 12.7a). The parameter n is a function of wheel and work speed, number of passes and wheel specification; the ratio of workpiece roughness to grit size can thus be calculated explicitly (Fig. 12.7b). Roughness is approximately inversely proportional to the square of the logarithm of n, in accordance with everyday experience that changing to a finer wheel is an easier way to improve the finish than prolonged machining with a coarser wheel.

Other Applications

N

d

c,.

259

1.0 0.8

0.6 0.4

0.2 0.0

-3.0

-2.0

0.0

-1.0

1.0

2.0

3.0

0.4

5.0

z/R9

0.04

: : :

0.m 0.0010'

102

103

104

105

106

Figure 12.7.(above) Height distributionsofthe envelopes of n effective abrasive profiles, assuming a normal distribution for a single profile; (below) variation of workpiece roughness, normalised with respect to grain size r, with number of effective abrasiveprofiles: circles are experimental, solid line is theory (Sayles & Thomas 1976b)

12.6. Bioengineering

In human joints the bone ends are separated by a soft porous tissue known as articular cartilage. The entire joint is enclosed in a sealed capsule containing a lubricating medium called synovial fluid (Fig. 12.8). Considered as a bearing, then, the joint behaves as two porous compliant surfaces backed by rigid solids and separated by a fluid. Roughness of artificial hip joints is known to be associate with increased wear perbyshire et al. 1994). The mechanism of lubrication of this bearing is far from clear, but its roughness is believed to be an important factor (Tandon & Rakesh 1981).

Rough Sur$aces

260

Hydrodynamic, elastohydrodynamic and boundary regimes have all been postulated, but there are experimental objections to all of them. One theory of "boosted lubrication" (Longfield et al. 1969) relied on the observation that articular cartilage is rough. It was suggested that lubrication under light or moderate loads is normally hydrodynamic. Under high-impact loads, as when walking or jumping, the sudden approach of the surfaces squeezes the watery components of the synovial fluid into the pores of the cartilage. The larger molecules which are left are trapped in the pools formed by the interlocking asperities and act as a boundary lubricant

Spmd membne Artmrlar cartilage

Bone

Figure 12.8. Schematic of a human joint (Longtield et al. 1969)

The problem of direct measurement of a yielding surface with a stylus instrument was discussed in Section 2.3.3. It turns out that the surface is highly irregular and that the distribution of heights is quite closely Gaussian (Fig. 12.9). A hip joint is basically a ball-and-socket joint in engineering terms, and calculations can be carried out on it in exactly the same way as for other engineering contacts (Thomas et al. 1980). Assuming a 40 mm diameter for the femoral head the high-pass cut-off becomes 20 mm and from the cartilage measurements the effective roughness was 20 pm. The plasticity approach gave a low-pass cut-off of 4.1 pm. It is of some medical interest to know the configuration of the joint under a high transient load, for instance at heel-strike in the normal walking cycle. This can be deduced in some detail from the surface measurements and material properties, using now the elastic theory of Eqns. 10.1410.16 which requires only the first two even moments of the profile power spectrum-

Other Applications

26 1

It turns out that the real area of elastic contact calculated under these conditions is 104 mm2, the mean plane separation is 58 pm, the real contact pressure is 24 N/mm2 and the stlffness is 185 kN/mm, more than two orders of magmtude less than that of the machine-tool interface quoted previously. None of these predictions is in serious conflict with existing experimental measurements. The volume enclosed between the cartilage surfaces at heel strike can be calculated to be about two-thirds of the volume when standing still, thus lending support to the theory of boosted lubrication of human joints.

lo p m

I

H I mm

Height above mean line ( g m ) Figure 12.9. (a) Profile of surface of human articularcartilage: (b) distribution of profile heights; broken line is Gaussian distribution with the same standard deviation (Thomas et al. 1980)

Roughness also affects the assimilation to the body of surgical implants and prostheses of various kinds. Where these are inserted into bone, it is essential that the bond between the implant and the living tissue should possess mechanical strength. To ensure this, individual cells must adhere to the surface of the implant itself or to its fasteners, and the local roughness is an important determinant of this adhesion. Wennerberg (1996) reviews the extensive literature on the effects of implant roughness with more than 200 references.

12.7. Geomorphometry

Geomorphometry has been defined as the science which treats of the geometry of landscape, and plays a vital role in both military and civil engineering (Bekker

262

Rough Su$aces

1969, Mitchell 1991). Geomorphometry is a recognized sub-field within geology, geography, geomorphology, hydrology and digital cartography (Thorn 1988, Richards 1990, Clarke 1990). Its specific applications range from measuring highway roughness (Hegmon 1979, Xu et al. 1992, al-Mansour et al. 1994), mapping sea-floor terrain (Hennings et al. 1994), and assessing soil erosion (Hagen & Armbrust 1992, 1994, Govindaraju & Kawas 1994) to meteorology (Wieringa 1992, Roberts et al. 1994, Hignett & Hopwood 1994), and analysing wildfire propagation (Kasischke et al. 1994). Recent technical advances have presented information on geographical relief as Cartesian data sets similar to those acquired by scanning microscopes and similar instruments, though of course on a much larger scale. Such data sets are referred to in the literature as digital elevation models OEMs). The field is reviewed by Plke (1995a), who has also published a bibliography of the geomorphometric literature (Pike 1993) which with supplements (Pike 1995b, 1996) runs to more than 3000 entries.

PUR 10 1

0

I 0

50

100 150 Roughness (dkm)

200

Figurel2.10. Ride quality (Pavement User Rating) as a function ofzero crossing density (Potter et al. 1992)

The measurement of terrain roughness is an essential preliminary to the study of vehicle dynamics (Van Deusen 1967). Levels of vibration in truck shipments (Marcondes & Singh 1992), dynamic pavement loadings (Gyenes et al. 1992, Lin et al. 1994) and bridge loadings (Hung et a1 1992) can be predicted from road roughness. Subjective perception of ride quality also appears to depend on pavement roughness (Potter et al. 1992, Gerardi & Schmerl 1995); comfort decreases as the average wavelength shortens (Fig. 12.10). Wambold & Henry (1982) review techniques for measuring road roughness with nearly 40 references.

Other Applications

263

12.8. References Ananthapadmanaban, T. and Radhakrishnan, V., "An investigation on the role of surface irregularities in the noise spectrum of rolling and sliding contacts", Wear 83, 399-409 (1982) Aziz, S. M. A,; Seireg, A,, "Parametric study of frictional noise in gears", Wear 176,2528 (1994) Barkan, P. and Tuohy, E. J., "A contact resistance theory for rough hemispherical silver contacts in air and in vacuum", Trans. I.E.E.E. PAS-84, 1132-1143 (1965). Bauer, T. W., Taylor, S. K., Jiang, M., Medendorp, S. V., "An indirect comparison of third-body wear in retrieved hydroxyapatite-coated, porous and cemented femoral components", Clinical Orthopaedics & Related Research 298, 11-18 (1994) Bekker, M. O., Introduction to Terrain-Vehicle Systems (Univ. Michigan Press, Ann Arbor. 1969) Ber, A. and Yarnitzky, Y., "Functional relationship between tolerances and surface finish", Microtecnic, 22, 449-45 1 (1968). Boeschoten, F. and Van der Held, E. F. M., "The thermal conductance of contacts between aluminium and other metals", Physica 23, 37-44 (1957) Bryan, J. and Lindberg, E., "Relationship of surface finish to dimensional tolerance", Proc. Int. Con$ on Surface Technol., Pittsburgh, 117-130, (S.M.E., 1973). Bryant, M. D., "Resistance buildup in electrical connectors due to fretting corrosion of rough surfaces", IEEE Transactions on Components, Packaging, and Manufacturing Technology 17A, 86-95 (1994) Campbell, J. C., "Cylinder bore surface roughness in internal combustion engines: its appreciation and control", Wear, 19, 163-168 (1972). Chaplin, P. D., "The analysis of hull surface roughness records", European Shipbuilding, 16, 40-47 (1967). Chung, K. C.; SheEeld, J. W.; Sauer, H. J. Jr.,; O'Keefe, T. J.; Williams, A,, "Thermal contact conductance of a phase-mixed coating layer by transitional buffering interface", Journal of Thermophysics and Hear Transfer 7, 326-333 (1993) Chung, K. -C., "Experimental study on the effect of metallic-coated junctions on thermal contact conductance", JSME International Journal, Series B: Fluids and Thermal Engineering 38, 100-107 (1995)

264

Rough Sursaces

Chung, K.-C., Sheffield, J. W., "Enhancement of thermal contact conductance of coated junctions", Journal of Thermophysics and Heat Transfer 9, 329-334 (1995) Clausen, T., Leistiko, O., "Surface roughness and specific contact resistance of AuGeNflnP ohmic contacts", Materials Research Society Symposium Proceedings 355,389-394 (Materials Research Society, Pittsburgh, 1995) Cooper, M. G., Mikic, B. B., and Yovanovich, M. H., "Thermal contact conductance", Int. J. Heat Mass Transfer 12, 279-300 (1969). Derbyshire, B.; Fisher, J.; Dowson, D.; Hardaker, C.; Brummitt, K., "Comparative study of the wear of UHMWPE with zirconia ceramic and stainless steel femoral heads in artificial hip joints", Medical Engineering & Physics 16, 229-236 (1994) Dowson, D., El-Hady Diab, M M., Gillis, B. J., Atkinson, J. R., %fluence of counterface topography on the wear of ultra high molecular weight polyethylene under wet or dry conditions", in Lee ed., Polymer wear and its control 171-187 (Amer. Chem. Soc.,St. Louis, 1985) Fried, E., "Study of interface thermal contact conductance", General Electric Co. Report 65SD4395 (1965) Gerardi, T., Schmerl, H., "Ride quality as a part of airport pavement management systems", Transportation Congress, Proceedings 1, 588-599 (ASCE, New York, 1995) Gray, G. G. and Johnson, K. L., "The dynamic response of elastic bodies in rolling contact to random roughness of their surfaces", J. Sound & Vibration, 22, 323-42 (1972). Gyenes, L.; Mitchell, C. G. B.; Phlipps, S. D., "Dynamic pavement loads and tests of road-friendliness for heavy vehicle suspensions", SAE Technical Paper 922464 (1992) Hald, A., Statistical theory with engineering applications (Wiley, New York, 1960) Harada, S. and Mano, K., "The effects of surface roughness on contact resistance of sphere-plane contact", Proc. 4th Int. Res. Svmp. Electric Contact Phenomena, Chicago, 25-28 (Illinois Inst. of Technol., Chicago, 1968). Hess, D. P.; Soom, A,, "Normal vibrations and friction under harmonic loads. 11. Rough planar contacts", Transactions of the ASME. Journal of Tribology 113,87-92 (1991) Hisakado, T., "Effects of surface roughness and surface films on contact resistance", Wear, 4, 345-359 (1977). Holm, R., Electric contacts handbook 3e (Springer-Verlag, Berlin, 1958)

Other Applications

265

Huang, D., Wang, T.-L., Shahawy, M., "Impact analysis of continuous multigirder bridges due to moving vehicles", Journal of Structural Engineering 118,3427-3443 (1992) Hunsaker, J. C. and Rightmire, B. G., Engineering applications of fluid mechanics (McGraw-Hill, New York, 1974) Kanai; H., M. Abe and K. Kido, "Estimation of the surface roughness on the race of bails of ball bearings of vibration analysis." Journal of Vibration, Acoustics, Stress, and Reliability in Design, 109,60-68 (1987) King, M. J., Chuah, K. B., Olszowski, S. T. and Thomas, T. R., "Roughness characteristics of plane surfaces based on velocity similarity laws", ASME Paper 81-FE-34 (1981) Lackenby, H., "The resistance of ships, with special reference to skin friction and surface condition", Proc. I. Mech. E., 176, 981-1014 (1962). Lambert, M.A.; Fletcher, L.S., Thermal contact conductance of nonflat, rough metals in vacuum", ASMEIJSME Thermal Engineering Joint Conference Proceedings 1 , 3 1-42 (ASME, New York, 1995) Lanchon; H., B. Makaya; J. Saint Jean Paulin; E. Kroener and A. Mirgaux, "Mathematical study to obtain qualitative effects of roughness in technical problems." Wear, 109, 99-111, (1986) Lau, H. and Harman, C.M., "Externally pressurized compliant air bearing operating on a rough moving surface", Trans. A.S.M.E.,Ser. F., J. Lubr. Tech., 97, 63-68 (1975). Lin, W.-K., Chen, Y.-C., Kulakowski, B. T.; Streit, D. A., "Dynamic wheeVpavement force sensitivity to variations in heavy vehicle parameters, speed and road roughness", Heavy Vehicle Systems 1, 139-155 (1994) Lon@ield, M. D., Dowson, D., Walker, P. S., Wright, V., "Boosted lubrication of human joints by fluid enrichment and entrapment", Biomed. Engng. 4 , 5 17-522 (1969) Majumdar, A.; Tien, C. L., "Fractal network model for contact conductance", Transactions of the ASME. Journal ofHeat Transfer 113, 516-525 (1991) Marcondes, J., Singh, S. P., "Use of road roughness to predict vertical acceleration in truck shipments", Advances in Electronic Packaging 2, 999-1004 (ASME, New York, 1992) McWaid, T. H.; Marschall, E., "Application of the modified Greenwood and Williamson contact model for the prediction of thermal contact resistance", Wear 152,263-277 (1992)

266

Rough Surfaces

Mengen, D.; Weck, M., "How to ensure precise analysis of gear surfaces and diagnosis of changes during operation", Proc 92 Int Power Transm Gearing Conf DE43,605-612( ASME, New York, 1992) Mikic, B. B., "Thermal contact conductance: theoretical considerations", Int. J. HeatMass Transfer, 17,205-214 (1974). Nayak, P. R., "Contact vibrations", J. Sound & Vibration, 22, 297-322 (1972). O'Callaghan, P., Babus'Haq, R. and Probert, S., "Predictions of contact parameters for thermally-distorted pressed joints", Am. Inst. Aeron. & Astron., 24th Thermophysics Con$, Paper AIAA-89-1659 (1989) Osanna, H. P., "Surface roughness and size tolerance", Wear, 57, 227-236 (1979). Plke, R. J., A bibliography of geomorphometry, with a topical key to the literature and an introduction to the numerical characterization of topographic form. - U.S. Geol. Survey Open-file Rept. 93-262-A (1993) Pike, R. J., "Geomorphometry - progress, practice and prospect", in Pike & Dikau eds., Advances in geomorphometry, 2. Geomorph. Supplementband 101, 221-238 (1995a) Pike, R. J., A bibliography ofgeomorphometry, Supplement 1.0. - U.S. Geol. Survey Open-file Rept. 95-046 (1995b). Pike, R. J., A bibliography ofgeomorphometry, Supplement 2.0. - U.S. Geol. Survey Open-file Rept. 96-726 (1996). Poon, S. Y. and Wardle, F. P., "Running quality of rolling bearings assessed", Chartered Mechanical Engineer (April 1978) Potter, D., Hannay, R., Cairney, P., Makarov, A,, "Investigation of car users' perceptions of the ride quality of roads", Road and Transport Research 1, 6-27 (1992) Sayles, R. S. & S. Y. Poon, "Surface topography and rolling element vibration", Precis. Eng 3, 137-144 (1981). Sayles, R. S. and Thomas, T. R., "Thermal conductance of a rough elastic contact", Appl. Energy, 2, 249-267 (1976a) Sayles, R. S., Thomas, T. R., "Stochastic explanation of some structural properties of a ground surface", Int J Prod Res 14,64 1-655 (1976b) Sayles, R. S. and Thomas, T. R., "Surface topography as a nonstationary random process", Nature, 271,43 1-434 (1978) Sayles, R. S.; Thomas, T. R., "Measurements of the statistical microgeometry of engineering surfaces", J Lubr Techno1 TransASME 101,409-417 (1979).

Other Applications

267

Schlesinger, G. "Surface finish and the function of parts", Proc. I. Mech. E., 151, 153-158 (1944) Singh, K. and Lumley, J. L., "Effect of roughness on the velocity profile of a laminar boundary layer", Appl. Sci. Res. 24, 168-186 (1971). Strahler, A. N., "Quantitative geomorphology of drainage basins and channel networks", in: Chow, V. (ed.)Handbook of applied hydrology 4, 39-76 (McGrawHill, New York, 1964). Su, Y.-T.; Lin, M.-H.; Lee, M.-S., "Effects of surface irregularities on roller bearing vibrations", Journal of Sound and Ebration 165,455-466 (1993) Tandon, P. N. and L. Rakesh, "Effects of cartilage roughness on the lubrication of human joints", Wear, 70, 29-36 (1981). Thomas, T. R., "Influence of roughness on the deformation of metal surfaces in static contact", Proc. 6th Int. ConJ on Fluid Sealing, B3, 33-48 (BHRA, Cranfeld, 1973a). Thomas, T. R., "Correlation analysis of the structure of a ground surface", Proc. 13th Int. Machine Tool Des. & R e x Con$, Manchester, 303-308 (1973b). Thomas, T. R., "Defining the microtopography of surfaces in thermal contact", Wear 79, 73-82 (1982). Thomas, T. R. and Olszowski, S. T., "Theory, design and performance of a porous-diaphragm hoverpallet", Proc. 6th. Int Gas Bearing Symp. D6, 73-92 (BHRA, Cranfeld, 1974) Thomas, T. R. and Probert, S. D., "Thermal contact resistance: The directional effect and other problems", Int. J. Heat Mass Tran.sfir, 13, 789-807 (1970). Thomas, T. R. and Probert, S. D., "Correlations for thermal contact conductance in vacuo", Trans. Am. Soc. Mech. Engrs., 94C, 176-180 (1972) Thomas, T. R. and Sayles, R. S., "Random-process analysis of the effect of waviness on thermal contact resistance", A.I.A.A. Paper No. 74-691, (1974). Thomas, T. R., Sayles, R. S. and Haslock, I., "Human joint performance and the roughness of articular cartilage", Trans. Am. SOC.Mech. Engrs: J. Biomech. Eng., 1026,50-57 (1980) Thompson, D. J., "Wheel-rail noise generation, Part I: Introduction and interaction model", Journal of Sound and Vibration 161,387-400 (1993) Thompson, D. J., "Wheel-rail noise generation, Part V: Inclusion of wheel rotation", Jourvlal of Sound and Vibration 161,467-482 (1993) Tillack, M. S.; Abelson, R. D., "Interface conductance between roughened Be and steel under thermal deformation", Fusion Engineering and Design 27,232-239 (1995)

268

Rough Surfaces

Toni, K., Nishino, K., "Thermal contact resistance of wavy surfaces", Revista Brasileira de Ciencias Mecanicas 17,56-76 (1995) Van Deusen, B. D., "A statistical technique for the dynamic analysis of vehicles traversing rough yielding and non-yielding surfaces", NASA Report CR659 (1967). Wambold, J. C., and Henry, J. J., "Evaluation of pavement surface texture significance and measurement techniques", Wear 83, 351-368 (1982) Wennerberg, A., On surface roughness and implant incorporation, PhD thesis, Goteborg University (1996) Yhland, E., "Waviness measurement - an instrument for quality control in rolling bearing industry", Proc. I. Mech. E., 182, Part 3K, 438-445 (1967/68). Yip, F. C. and Venart, 3. E. S., "Surface topography effects in the estimation of thermal and electrical contact resistance", Proc. I. Mech. E., 182, Part 3K, 81-91 (1967/68).

INDEX

Abrasive composites, 15 Abrasive machining, 156 ACF. See autocorrelation function Acoustic interferometer, 84 speckle, 84 waves, 83,84 ACVF. See autocovariance fimction ADC.See analogue-todigital conversion Adhesive Giction, 225 AFM. See atomic force microscope Air bearings, 251 gap, 17 jet, 101 Aircraft, 1,43,44 engineers, 3 Aliasing, 115 Alignements, 2 Allison system, 136, 137 Amplifier, 14 Amplitude parameters, 1 33, 151, 161, 173,186 Analogue methods, 113 Analogue-todigital conversion, 1 13, 117 Angle of extinction, 47 Angular distribution, 47, 50, 5 1, 52 reflectance function, 103 uncertainties, 7 Anisotropy,48, 133, 169, 171, 177, 190, 1 92- 1 98 correction factor, 2 16 Anode, 17 Area of contact, 23 Area of real contact, 5 ARMA. See autoregressive movingaverage Armature, 17 Articular cartilage, 84,259-261,267

Asperities, 225,229,230,232,241,243. 251,260 collisions, 25 1 density, 74,229,249 Astigmatic focussing, 103 Asymmetry, 143 Atomic force microscope, 70 Autocollimation, 8, 39 Autocorrelation, 133, 151, 153, 155, 157, 161, 176-178 Autocovariance fimction, 153,235 Autofocussing, 65 Automobile brake linings, 226 Autoregressive moving-average, 156 Average slope, 50, 162 wavelength, 152, 170,227,262 Averaging circuitry, 20

Back-scattered electrons, 65 Backscattering, 83, 84, 85 Bad data, 95 Ball filter, 173 Ball-cage interaction, 25 1 Bandwidth, 181, 184,185,249 Beamsplitter, 40 Bearing area, 144, 145, 146, 153, 167, 176 Gaction, 144, 176 ratio, 144 Beta distribution, 143 Bifractal, 167, 168 Birmingham 14, 173, 174,177, 186 Boosted lubrication, 260,261 Boundary lubrication, 227,240 Boustrophedon scanning, 94 Breaking-in, 229,241 Bridge circuit, 17

269

270

Rough Surfaces

Bridge loadings, 262 Bulldozer, 24 Burnishing, 230,23 1

Calcium fluoride films, 65 Cali-block, 2 1,28, 34 Calibration, 28,29, 30, 32, 33, 133 specimens, 28,29, 33, 99, 106 Cantilever, 70 Cantor set, 11,25 Capacitance, 17, 66-68, 77,78, 85, 87, 88,100 Carbon replicas, 65 Carnac, 2 Carrier, 17, 19 Cartographers, 3 Cartridge, 16 Casting, 16, 167 Catchment areas, 194 Cathode ray tubes, 92 Causal filters, 119 Censoring, 233-235 Central limit theorem, 141 Characteristic depth, 129 Charge force microscope, 72 Charge-coupled diode, 45 Chart recorder, 19,22 Chatter, 167 Chemical balances, 9 Chisel-shaped stylus, 23 Circular lay, 190, 192 profiler, 44 Closed contours, 4, 5,6 Clusters of contacts, 249 Coefficient of variation, 133 Coherence radar, 46 Coherent light, 45, 50, 54, 57, 103 Compact disc player, 40 Compliant seal, 80 surfaces, 27,98,228,259 Compressible fluids, 8 1 Concrete surfaces, 84

Conductance, 247,250,263-267 Conducting probe, 68 Confocal microscope, 42 Constriction of flow lines, 248 Contact spots, 248 Contacting envelope, 2 1 Convolution, 118, 119, 161 Coordinate measuring machines, 8 Core fluid retention index, 176 Comer frequency, 167 Correlation length, 11, 133, 153, 159, 161, 174,257 Correlogram, 152, 170 Coulombic theory of friction, 225 Crankshaft, 97, 103, 107 Crops, 9, 10 Cross-correlation, 180,235 Cross-covariance function, 190 Crystal faces, 53 Cutoff, 74, 76, 79, 80, 83, 118-125, 130, 138,161, 183,184,231 cutting fluids, 100 speed, 102 Cylinder bores, 258

Damping effects, 70 Data acquisition, 94, 100 Datum, 16, 18,26,28 Default settings, 95 DEM. See digital elevation models Demodulation, 17 Density of extrema, 181 Dental cement, 98 Diamond styli, 15 Diamond-turned, 50,51, 57 Dielectric surfaces, 5 1 Differential thermocouple, 75 Diffracted radiation, 47 Diffuse reflection, 36, 39, 60 Digital cartography, 262 elevation models, 262 filter, 117

Index Dimensional tolerance, 255,263 Directional properties, 5 1, 188, 190 Discretisation, 113 Discriminant analysis, 235 Dry dock, 13 Dynamic range, 14,31 response, 24,31 stiffness, 16

Effective hardness, 8 profile, 2 1 Elastic contact, 250,261 Elastohydrodynamiclubrication, 228, 238-244 Elastomers, 27 Electrical contact resistance, 1,247,268 Electroforming, 99 Electromagnetic radiation, 35,46 Electron beam, 92-94 microscopy, 7,64, 86, 93, 97, 106 Electrostatic forces, 70 Ellipsometry, 53,60,61, 104, 106 Ensemble averaging, 161, 179 Entropy, 2 , 3 Error trap, 159 Errors of form, 39,94, 116, 125 E-system, 21, 125, 126, 131 Evaluation length, 18, 135, 138 Excel, 94 Extreme-value parameters, 133, 136, 137,139

Face turning, 190 Facets, 51 Fast Fourier transform, 160 Fatigue life, 228,238 Feed rate, 50 Feedback circuit, 70 Femoral head, 260,264 Fibre optics, 103, 105

27 1 Field coils, 65 Film-thwkness ratio, 229 Filtering, 116, 117, 121, 122, 126, 134, 159, 161, 170, 172, 176, 186, 197, 198,229,233 Finger, 11 Fingemail, 71, 73 Finite impulse response, 119 FIR. See finite impulse response Flagging, 95 Flash gun, 37 Flatness, 1, 100 F l e m i n g integrator, 11 Flexible diaphragm, 78 Floor surfaces, 227 Flow pressure, 23 Fluid dynamics, 162 Fluid flow, 247,25 1 Focus-detection, 40,41, 103 Footprint, 2,24, 92 Footwear, 227,242 Forest canopies, 9 Form error, 68 Formprofil, 125 Foucault knife-edge, 37 Fouling, 254 Fourier transform, 119, 121, 154, 160, 161 Fourier transforming lens, 52 Fractal, 162, 164, 165, 166, 167, 168, 169, 170, 185, 194, 195, 196,244, 256 dimension, 165 Fracture surfaces, 162, 168, 169 Fraunhofer diffraction, 103, 109, 110 Frequency response, 119, 121 Friction, 225-22'7,238-245,253,254, 264,265 Fringe-field technique, 66 Fringes, 45,46, 93 FSD. See full-scale deflection Fuel element, 248 Full-scale deflection, 114 Functional filtering, 3, 229 parameters, 176

272

Rough Surfaces

Gamma function, 164 Gauge block, 29, 133 Gauging nozzle, 8 1 Gaussian distribution, 114, 139-147, 157, 180, 183,234,254,261, filter, 123, 173 weighting function, 120 Gear contacts, 228,243 teeth, 97 General surface texture, 136 Geography, 262 Geology, 262 Geomorphology, 194,247,262,267 Geomorphometry, 261,266 Glass transition, 27,28 Glossmeter, 36,47,48, 91 Goodness-of-fit test, 142 Gramophone, 11 Grinding, 134, 142, 149, 169, 172, 195, 257,258 scratch, 257 wheel, 257,258 Gritblasting, 185, 194, 195 Ground glass, 56, 160 GST. See general surface texture

Hairbrush, 8 Half-width, 104, 105 Heavy electrical engineering, 3 Heel-strike, 260 Height distribution, 10, 133, 140-149, 155, 176,234,235,253,254,257, 258 Hemispherical parts, 105 Hertzian elastic modulus, 74 Heterodyne laser, 43 High-fidelity, 16 High-pass filter, 117, 118, 121, 130, 155,161 High-spot count, 151 spacing, 151

Highway roughness, 262 Hip joints, 259,264 Honing scratches, 195 Horizontal compression, 22 magnification, 19 range, 6 resolution, 14,65,69 Housing aligumnt, 251 HSC. See high-spot count H-system, 136 Hubble, 8, 10 Hull fnction, 1 Hiillprofl, 2 1, 125 Human teeth, 97 Huygen’s principle, 54 Hybrid parameters, 173, 175, 176 Hydrodynamic lubrication, 227,238-245 Hydrodynamicists, 3 Hydrology, 262,267 Hyperbola, 147, 167, 168 Hysteresis, 68

W. See infinite impulse response Image clarity, 47 Impact wear, 97,107 Implant, 261,268 Impulse response, 1 18, 1 19, 122 Inclined plane, 74 Indenter, 23 Index of refraction, 53 Inductance, 17,78 Infrared laser diode, 4 1 In-process measurement, 100, 102, 103 Inspection, 16, 91, 100, 111, 172,241 Instrument performance, 6 , 7 reference plane, 7 Intensity equations, 45 signal, 65 Interferometry, 42-46, 91, 93, 94, 107, 108,111 Internal combustion engine, 4

Index

Interrupted finishes, 145 Inverse Fourier transform, 161 Isometric display, 94, 95

Kurtosis, 143, 173

Lambda ratio, 229 Lambert's law, 35 Laminar flow, 80 sublayer, 254,255 Landscape, 9 I , 166 Laser force microscope, 72 interferometer, 18 scanning analyser, 103, 104, 106 Lateral deflection, 24,25 resolution, 66,68,70,71, 84, 101 stiffness, 24, 33 Lathe, 255 bed, 16 saddle, 16 Leakage rate, 235 Length of traverse, 11 Leptokurtic, 143 Levelling, 16, 17 Light-section microscope, 36, 37, 125, 134 Linear discriminant functions, 237 Lip seals, 235,236,244,245 Log-normal distribution, 143 Long-crestedness, 192 Loudspeaker, 19 Low-pass filter, 66, 117, 118, 123, 194 LVDT. See linear variable differential transformer

Macroscopic resistance, 248,250 Magnetic force microscope, 72 Material ratio curve, 144

273 Mathernatica, 94

Matlab, 94 Mean absolute slope, 105, 158, 159 contact spot radius, 4, 5 hydraulic radius, 80 line crossing, 152 peak curvatwe, 157 peak height, 157,158,182 peak radius of curvature, 159 plane, 172, 176, 185,252,261 Mechanical noise, 68 vibrations, 14 Mecrin tester, 73 Megalithic, 2 Menhir, 2 Meteorology, 262 Method divergence, 123 MHR. See mean hydraulic radius Michelson interferometer, 44 Microdensitometer, 65, 88 Microscopic resistance, 248,250 Milling, 190, 193 Mireau interferometer, 45 Mixed lubrication, 227,229,239-245 Molten asphalt, 76 Molybdenum disulphide film, 226 Moment of distribution, 133, 140, 143, 155,162,183,249 Monochromatic illumination, 55 Motif analysis, 126, 130, 193 combination, 128, 129, 131 filter, 173 Moving-coil transducer, 17 MPRC. See mean peak radius of curvature M-system, 120, 125, 130 Multifractal, 166, 167 Multiprocess surface, 133, 145, 147

Nearest neighbows, 186, 187 Negative slopes, 19

274

Rough Surfaces

Nickel replicas, 100 Noise, 116, 156, 168,250,251,263,267 Nomarslu microscopy, 39 Nominal contact area, 74 Non-causal filters, 1 19 Non-contacting stylus, 37 Non-recursive filters, 119 Nonstationarity, 133 Nozzle cross-sections, 81 Nuclear power, 248 power stations, 1 Nyquist frequency, 1 15 sampling theorem, 1 15

Ocean surfaces, 180 waves, 43 Oil-droplet method, 83 On-the-fly sampling, 1 15 Operating envelope, 6, 7 Optical absorption microscope, 72 flat, 83, 94, 98 instruments, 9 lever, 11 path, 45,46,52,98,103 probe, 37-39 profilers, 37, 50 sections, 36 stylus, 40, 103, 108 Oscillator, 17 Oscilloscope, 19, 103, 104 Outflow meter, 80, 81, 85 Out-of-balance signal, 39,41

Paper tape, 74 Parameter rash, 133, 150 Partial EHL, 228 Pass-band, 155, 183 Pattern recognition, 128, 193 Pavement loadings, 262

roughness, 262 user rating, 262 PC. See phase-corrected filter Peak height, 73, 137,146,157,158,181 182 Peak-to-trough roughness, 227 Peak-to-valley height, 64, 74, 128, 129 Pendulum, 74 Perspective view, 94 Phase diirerence, 40,42,43, 54 distortion, 119, 120 lag, 27 shift, 43 Phase-corrected filter, 120, 130 Phonograph, 11, 15, 16 Photodetector, 103 Photographic film, 52 Photographic negative, 37 Photon scanning tunnelling microscope, 72 Photoresistors, 39 Pickup, 14, 16, 18-20 Piezo drives, 68 Piezoelectric crystal, 16 transducer, 45 Pmhole, 42 Piston term, 172 Plaster of paris, 98 Plastic deformation, 258 Plateau honed surface, 95, 177, 178 Platykurtic, 143 Pneumatic gauging, 81,86, 89, 101, 108,253 profiler, 101 Poisson’s ratio, 9 Polar scans, 92 Polarization, 40,42, 43,47, 52, 54, 103 Polarizing interferometer, 42,43 Polygonal mirror, 103 Polynomial filter, 122 fitting, 172 Portable data format, 94 Positional error?, 93

Index

Positive replica, 98, 99 Power spectra, 11, 18,20,27, 51, 52, 99, 115, 117, 129, 130, 133, 153-155, 160, 163, 173, 177, 178, 182, 192, 231,233,235,250,254,255 Pressure transducers, 101 Pnmary standards, 4 Probability density, 140, 141, 155, 181, 182, 197 Production engineers, 3 Production time, 3 Profile curvature, 157,232 length ratio, 152 Profilometer, 13, 30, 32 Projector, 12 Prostheses, 261 PSDF. See power spectrum Pt, 134, 135, 148 PTB, 28,29 Pushrod, 15 F'yramid, 15

Quality control, 20, 100, 105, 145,238, 247, 268 system, 5 Quantisation, 113, 114, 137 Quantum tunnelling, 68

R3z, 136 Ru, 133, 138, 139, 143-149,230 Rru'Rq, 139 Radar engineers, 3 Railroad operation, 250 Raster scan, 68, 92, 93 Rayleigh distnbution, 143 Razor blade, 36,76 Real contact area, 4, 10 Reciprocating contact, 3 Recording barometer, 24 Recursive filters, 119

275 Reflected beam, 35,40,43, 54 Refractive index, 52, 98 Relocation, 24,27, 28, 96, 97 Rendering of image, 94 Replica, 52,98,99, 100, 108, 109, 110, 254 Repulsive forces, 70 Resolution, 4, 6, 8, 9 Resonances, 25 1 Reverse slopes, 24 Reynolds number, 253,254 &de quality, 262,264 Rk,145,146, 148,150 RI, 152 Rm,135 Road roughness, 225,262,265 Road surfaces, 77, 80 Rocker arm, 15 Roller burnishing, 230 Rolling ball, 173 bearings, 229,250,266 circle, 2 1 contact, 250,264 ellipse, 126 Rolling of sheet metal, 97 Roll-off, 118, 119, 121, 130 Rough pipe, 253 sliding, 228, 243 Roughness of floor surfa.:es, 227 regime, 51 standards, 5, 15,21, 28 Rp, 74,76,83, 135, 145, 146, 148 Rq, 138, 139, 142, 144, 145, 147, 158, 162,167,236 Running average, 123 Running-in, 96, 97, 106, 110,229, 230, 233,238,243,244 Rm-out, 19 Rvk, 145, 146, 148 Ry, 135,136 Rz, 136

276

Rough Surfaces

SAq, 174,175

Safety of floors, 227 Sample length, 121,122,135 Sampling, 113, 115, 135, 137, 152, 158, 159, 176, 183-187 interval, 1, 137, 158, 159, 183-185 Sand grain roughness, 253 Sand-patch, 76 Satellite, 1, 248 ranging techniques, 8 Sbi, 174, 176 scanning chemical potential microscope, 72 ion conductance microscope, 72 near-field acoustic microscope, 70 near-field optical microscope, 72 thermal profiler, 72 tunnelling microscope, 68, 72 Scattering, 46,84 Scattering, 84 angle, 51 theory, 46 Scatterometers, 36 Sci, 174, 176, 198 Scraping technique, 76 Scuffing, 229,243 Sds, 174, 182 Sea-floorterrain, 262 Seals, 235-251 Second law of thermodynamics, 2 Secondary electrons, 65 length standards, 4 Second-order effect, 3 Sectional measurement, 5 Seismograph, 14 Self-afine fractals, 117, 164, 165, 170 Self-similarity, 162 SEM. See scanning electron microscopy Separable filter, 173 Servo control, 68 Sewing needle, 15 Shakedown, 229 Shaping, 66, 190 Sharpest curvature, 6 Shps' hulls, 12, 97,254

Shock noise, 251 Shortest correlation length, 174 Shot-blasting, 141 Silicon wafers, 103 Sinusoidal surface, 6 , 7 Skewness, 143, 173, 177, 186,227,228, 238,257 Skid, 126 Skid, 16, 18,26-32, 126 Skin friction, 253 resistance, 79 Sliding contact, 97 Slip gauges, 4 Slope, 6,7,20,65,73,74,76,83, 104, 105, 123, 146, 149, 152, 158, 159, 162,164,166,175, 181, 182, 188, 225,226,229,232,255 distribution, 19 measurement, 39 Sm,151,155,169,170 Smoked-glass plate, 11 Smoothing, 118, 161 SNAM. See scanning near-field acoustic microscope Snell's law, 35 Soil erosion, 262 Sonar, 84 Space vehcle, 248 Spark erosion, 166 Spatial coherence, 55 Speckle, 47,54-59, 84, 85 contrast, 47, 54, 55 pattern decorrelation, 54,55 Specular reflection, 36, 37,47-50, 58, 84, 147 Speed of traverse, 19,24 Spherical contacts, 15 Spindle runout, 167 Spiral scans, 92 Spline filters, 123 Spot diameter, 103 Sq, 173,174,176,177,188 Stagnant-layer method, 83 Static friction, 73,74

Index

seals, 235,242,251 Stationarity, 116, 155, 161, 169, 180, 255 Std, 174 Stedman diagram, 7, 8, 37 Steepest slope, 6,65, 123, 159 Stereo pairs, 65 Stereophotogrammetry, 9 1 Stick-slip, 19 Stm,68 Str, 174, 196 Straddling skids, 16 Straightness, 1,2 Strain state, 53 Stratified surface, 145, 146, 148, 190, 195 Strong anisotropy, 192, 193,194 Structure h c t i o n , 155, 156, 164-169, 177,179,180 Stylus dimensions, 12, 14 geometry, 24,28 instrument, 4, 7, 9-21,28-32, 96-104, 167,227,260 load, 19,23,24 tip, 20,21,23,28 transducer, 14, 16 width, 66 S h t , 10, 142, 171-190, 197 definition, 186 Surface bearing index, 176 damage, 15,23,3 1 discontinuities, 41 slope, 182 vibration, 54, 105 Surgical implants, 26 1 Surveying, 91 Svi, 174, 176 Swad, 103 Switchgear, 1 Synovial fluid, 259,260

Tactile test, 71

277 Taper roller bearing, 23 1 sectioning, 63,64 Telemetry, 102 Television, 92 TEM. See transmission electron microscopy Temporal coherence, 55 Ten-point height, 136, 173, 174, 186 Terrain roughness, 262 Textile roughness, 37 Texture, 1, 3,6 aspect ratio, 174, 175, 192 direction, 174, 176, 177 Thermal comparator, 75, 76 conductance, 75 expansion effects, 255 noise, 14 resistance, 1 Thermoelectric, 72 Thetameter, 74, 75 Three-dimensional filtering, 171 Time series, 153, 161 Tip radius, 15,21 TIS. See total integrated scatter Tolerance, 255,256 Tool radius, 106 replacement, 100 Topothesy, 162, 164, 185, 194, 195 Total integrated scatter, 49, 57 T,’ 139, 144, 145 Traceability, 29, 32 Traceable uncertainty, 100 Traced profile, 20 Transducer, 113, 114 Transducer, 13-18,29, 32, 113, 114 Transition region, 254 Transitional topography, 235 Translation, 115, 125 speeds, 67,102 stage, 37,39 tables, 93 Transmission characteristic, 17

278

Rough Surfaces

coefficient, 1 18, I 19, 121 design, 251 Transmittance, 52,65 Transparent replica, 52 Traverse length, 19 Trend removal, 172, 182 Tribology, 225,244 Triode, 17 Truck shipments, 262,265 Tuning fork, 70 Tunnelling current, 68,69 Turbulent flow, 253 Tyre-road interactions, 1

Ultrasonic back-scattering, 102 Ultrasound, 84, 85,88 Uncertainty, 11, 133 Underwater surfaces, 84,97

Vacuum chamber, 69 seals, 235 Valley depth, 137,145,146 fluid retention index, 176 suppression filter, 122, 123 Van der waal's forces, 70 Vehicle dynamics, 262 Vertical range, 4, 9, 14, 37-46,65-69, 103 resolution, 14, 18, 3748, 64-69,74, 75, 82-84 scanning interferometry, 46 Vibration, 250,251,262-267 Vibrator, 29, 30 Video camera, 37 Viscosity, 81,238,241,244 Visibility of fringes, 46

Wall gauge, 12 Walsh functions, 155, 169

Watersheds, 194 Waviness, 12, 83, 103, 116, 125, 126, 130, 167, 170,248-251,267 number, 250 Weak anisotropy, 192, 193 Wear, 227-244,259,263,264 scars, 97 Weighting function, 66, 120-126 sequence, 1 19 terms, 117 Wheatstone bridge, 82, 89 White light, 40 Wiener-Khinclune relation, 154 Wildfire propagation, 262 Wire-frame view, 94 Wollaston prism, 39,43 Wood, 15, 17,24, 30, 31, 33 Wristwatch. 70

Zero crossing density, 262 Zonal filter, 173 Zoom, 94