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Series B
Dynamics with Friction Modeling, Analysis and ...
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SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Volume 7
Series B
Dynamics with Friction Modeling, Analysis and Experiment Part II Editors
Ardeshir Guran, Friedrich Pfeiffer & Karl Popp
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World Scientific
Dynamics with Friction Modeling, Analysis and Experiment Part II
SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Series Editors: Ardeshir Guran & Daniel J. Inman
About the Series Rapid developments in system dynamics and control, areas related to many other topics in applied mathematics, call for comprehensive presentations of current topics. This series contains textbooks, monographs, treatises, conference proceedings and a collection of thematically organized research or pedagogical articles addressing dynamical systems and control. The material is ideal for a general scientific and engineering readership, and is also mathematically precise enough to be a useful reference for research specialists in mechanics and control, nonlinear dynamics, and in applied mathematics and physics. Selected
Volumes in Series B
Proceedings of the First International Congress on Dynamics and Control of Systems, Chateau Laurier, Ottawa, Canada, 5-7 August 1999 Editors: A. Guran, S. Biswas, L Cacetta, C. Robach, K. Teo, and T. Vincent Selected Topics in Structronics and Mechatronic Systems Editors: A. Belyayev and A. Guran
Selected Volumes in Series A Vol. 1
Stability Theory of Elastic Rods Author: T. Atanackovic
Vol. 2
Stability of Gyroscopic Systems Authors: A. Guran, A. Bajaj, Y. Ishida, G. D'Eleuterio, N. Perkins, and C. Pierre
Vol. 3
Vibration Analysis of Plates by the Superposition Method Author: Daniel J. Gorman
Vol. 4
Asymptotic Methods in Buckling Theory of Elastic Shells Authors: P. E. Tovstik and A. L. Smirinov
Vol. 5
Generalized Point Models in Structural Mechanics Author: I. V. Andronov
Vol. 6
Mathematical Problems of the Control Theory Author: G. A. Leonov
Vol. 7
Vibrational Mechanics: Theory and Applications to the Problems of Nonlinear Dynamics Author: llya I. Blekhmam
SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS < < ^ ^ > Series B
Volume 7
Series Editors: Ardeshir Guran & Daniel J Inman
Dynamics with Friction Modeling, Analysis and Experiment Part II
Editors
Ardeshir Guran Institute of Structronics, Canada
Friedrich Pfeiffer Technical University of Munich, Germany
Karl Popp University of Hannover, Germany
V f e World Scientific wW
Singapore •»New New Jersey • London • Hong Kong Jersey'London*
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
DYNAMICS WITH FRICTION: MODELING, ANALYSIS AND EXPERIMENT, PART II Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. 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.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-02-2954-2
Printed in Singapore by Uto-Print
STABILITY, VIBRATION AND CONTROL OF SYSTEMS Editor-in-chief: Ardeshir Guran Co-editor: Daniel J. Inman Advisorj ' Board Henry Abarbanel University of California San Diego USA
Lucia Faravelli Universita di Pavia Pavia ITALY
Gerard Maze University of Le Havre Le Havre FRANCE
Jon Juel Thomsen Tech. Univ. of Denmark Lyngby DENMARK
Gary L. Anderson Army Research Office Research Triangle Park USA
Toshio Fukuda Nagoya University Nagoya JAPAN
Hans Natke Universitat Hannover Hannover GERMANY
Horn-Sen Tzou University of Kentucky Lexington USA
Jorge Angeles McGill University Montreal CANADA
Hans Irschik Johannes Kepler Universitat Linz AUSTRIA
Sotorios Natsiavas Aristotle University Thessaloniki GREECE
Firdaus Lid wad ia University of S. California Los Angeles USA
Teodor Atanackovic University of Novi Sad Novi Sad FED REP OF YUGOSLAVIA
Heikki Isomaki Helsinki Univ. of Tech. Helsinki FINLAND
Paul Newton University of S. California Los Angeles USA
Dick van Campen University of Technology Eindhoven NETHERLANDS
Anil Bajaj Purdue University Lafayette USA
Jer-nan Juang Langley Research Center Hampton USA
Michihiro Natori Inst, of Space & Astro. Kanagwa JAPAN
Jorg Wauer Technische Universitat Karlsruhe GERMANY
Anders Bostrom Chalmers Technical Univ. Goteborg SWEDEN
John Junkins Texas A&M University College Station USA
Friedrich Pfeiffer Technische Universitat Munchen GERMANY
Joanne Wegner University of Victoria Victoria CANADA
Rafael Carbo-Fite C.S.I.C. Madrid SPAIN
Youdan Kim Seoul National University Seoul SOUTH KOREA
Raymond Plaut Virginia Poly. Inst. Blacksburg USA
James Yao Texas A&M University College Station USA
Fabio Casciati Universitat di Pavia Pavia ITALY
Edwin Kreuzer Technische Universitat Hamburg-Harburg GERMANY
Karl Popp Universitat Hannover Hannover GERMANY
Lotfi Zadeh University of California Berkeley USA
Marc Deschamps Laboratoire de Mecanique Bordeaux FRANCE
Oswald Leroy Catholic University of Louvain BELGIUM
Richard Rand Cornell University Ithaca USA
Franz Ziegler Technische Universitat Wien AUSTRIA
Juri Engelbrecht Estonian Academy of Sci. Tallin ESTONIA
Jerrold Marsden California Inst, of Tech. Pasadena USA
Kazimirez Sobczyk Polish Academy of Sci. Warsaw POLAND
Professor Ilya Blekhman (left) and Professor Andeshir Guran (right) during the International Symposium on Mechatronics & Complex Dynamical Systems, June 2000, St. Petersburg, Russia.
In Memoriam Heinrich Hertz (1857-1894) Paul Painleve (1863-1933) Arnold Sommerfeld (1868-1951)
Preface The pictures on the front cover of this book depict four examples of mechanical systems with friction: i) dynamic model of normal motion for Hertzian contact, ii) disk with a rotating mass-spring-damper system, iii) planar slider-crank mechanism, iv) dynamic model of a periodic structure. These examples, amongst many other examples of dynamical friction models, are studied in the present volume. Historically, the exploitation of dynamical friction has had a tremendous effect on human development. In fact, due to the human desire to describe nature, machines, and structures, ideas about friction and dissipation has found their way into scientific thoughts. The science of mechanics is so basic and familiar that its existence is often overlooked. Whenever we push open a door, pick up an object, walk or stand still, our bodies are under the constant influence of various forces. When the laws of the science of mechanics are learned and applied in theory and practice, we achieve an understanding which is impossible without recognition of this subject. Today still we agree with what da Vinci wrote in fifteen century, mechanics is the noblest and above all others the most useful, seeing that by means of it all animated bodies which have movement perform all their actions. The science of mechanics deals with motion of material bodies. A material body may represent vehicles, such as cars, airplanes and boats, or astronomical objects, such as stars or planets. For sure such objects will sometimes collide or contact each other (cars more often than stars). One may think of a walking human or animal making frictional contact with the ground, sports such as golf and baseball, where contact produces spin and speed, and mechanical engineering applications, such as the parts of a car engine that must contact each other to transfer force and power. The sub-field of mechanics that deals with contacting bodies is simply referred to as contact mechanics. It is a part of the broader area of solid and structural mechanics and an almost indispensable one since forces are almost always applied by means of frictional contacts. Contact mechanics has an old tradition: laws of friction, that are central to the subject, were given by Amontons in 1699, and by Coulomb in 1785, and early mathematical studies of friction were conducted by the great mathematician Leonhard Euler. A theory for contact between elastic bodies, that has had a tremendous importance in mechanical engineering, was presented by Heinrich Hertz in 1881. Contact mechanics has seen a revival in recent years, driven by new computer resources and such applications as robotics, human artificial joints, virtual reality, animation, and crashworthiness. Contact mechanics is the science behind tribology, the interdisciplinary study of friction, wear and lubrication, with major applications such as bearings and brakes, and involving such issues as microscopic surface geometry, chemical conditions, and thermal conditions. Note that while in many tribological applications one seeks to minimize friction to reduce loss of energy, everyday life is at the same time impossible without friction — we would not be able to walk, stand up, or do anything without it. Walking requires adequate friction between the sole of the foot and the floor, so that the foot will not slip forward or backward and the effect of limb extension can be imparted to the trunk. Lack of friction on icy surfaces is compensated for by hobnails on boots or chains on tires. Friction is necessary to operation of a self-propelled vehicle, not only to start it and keep it going but to stop it as well. Crutches and canes are stable due to friction between their tips and the floor; this is often increased by a rubber tip which has a high coefficient of friction with the floor. A wheel-chair can be pushed only because of the friction developed between the pusher's shoes and the floor, and friction must likewise be developed between the wheels and the floor so they will turn and not slide. Many friction devices are used in exercise equipment IX
x
Preface to grade resistance to movement, as with a shoulder wheel or stationary bicycle. Brakes on wheelchairs and locks on bed casters utilize the principle of friction. Application of cervical or lumbar traction on a bed patient depends on adequate opposing frictional forces developed between the patient's body and the bed. In the operation of machines, sliding friction and damping wastes energy. This energy is transformed into heat which may have a harmful effect on the machine, as with burned-out bearings. To reduce friction, materials having a very smooth or polished surface are used for contacting parts, or a lubricant, such as oil or grease, is placed between the moving parts. Frictional effects are then absorbed between layers of the lubricant rather than by the surfaces in contact. Friction also exists within the human body. Normally ample lubrication is present as tendons slide within synovial sheaths at sites of wear, and the articulating surfaces of joints are bathed in synovial fluid. Despite this tremendous importance of contact mechanics and frictional phenomena, we still hardly understand it. The present part II of this volume on Dynamics with Friction is a continuation of the previous part I, and is designed to help synthesize our current knowledge regarding the role of friction in mechanical and structural systems as well as everyday life. We understand that in the preface of the first part in this book we promised the readers to have a final review chapter with a complete list of references in friction dynamics. However, we soon realized that the knowledge in this field in written form is expanding very rapidly at a considerable rate which makes a comprehensive list almost impossible. The present volume offers the reader only a sampling of exciting research areas in this fast-growing field. In compilation of the present volume, we also noticed, relatively very little is made available in this field to design engineers, in college courses, in handbooks, or in form of design algorithms, because the subject is too complicated. For an expository introduction to the field of dry friction with historical notes we refer the readers to the article by Brian Feeny, Ardeshir Guran, Nicolas Hinrichs, and Karl Popp, published recently in Applied Mechanics Review, volume 51, no. 5 in May 1998, and the list of references at the end of that article. Every year there are several conferences in this field. Those of longest standing are the conferences of ASME, STLE, IUTAM, and EUROMECH. A separate bi-annual conference, held in U.S., is the Gordon conference in tribology. It is a week-long conference held in June, at which about 30 talks are given. Another separate biannual conference, held in even-numbered years, is the ISIFSM (International Symposium on Impact and Friction of Solids, Structures, and Intelligent Machines: Theory and Applications in Engineering and Science). The proceedings of ISIFSM papers are rigorously reviewed and appeared in volumes published in this series. Today, research continues vigorously in the description and design of systems with friction models, in quest to understand nature, machines, structures, transportation systems, and other processes. We hope this book will be of use to educators, engineers, rheologists, material scientists, mathematicians, physicists, and practitioners interested in this fascinating field.
Ardeshir Guran Ottawa, Canada
Friedrich Pfeiffer Munich, Germany
Karl Popp Hannover, Germany
Contributors M. A. Davies National Institute of Standards and Technology Manufacturing Engineering Laboratory Gaithersburg, MD 20899 USA B. F. Feeny Department of Mechanical Engineering Michigan State University East Lansing, M l 48824 USA Aldo A. Ferri G. W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332-0404 USA Ardeshir Guran American Structronics and Avionics 16661 Ventura Blvd Encino, California 91436 USA Daniel P. Hess Department of Mechanical Engineering University of South Florida Tampa, Florida 33620 USA R. V. Kappagantu Altair Engineering, Inc. 1755 Fairlane Drive Allen Park, M l 48101 USA
Francesco Mainardi Department of Physics University of Bologna 46 Via Irnerio, Bologna 40126 Italy Dan B. Marghitu Department of Mechanical Engineering Auburn University Auburn, Alabama 36849 USA J. P. Meijaard Laboratory for Engineering Mechanics Delft University of Technology Mekelweg 2, NL-2628 CD Delft The Netherlands F. C. Moon Department Mechanical and Aerospace Engineering Cornell University Ithaca, NY 14853 USA John E. Mottershead Department of Mechanical Engineering The University of Liverpool Livepool, L69 3BX UK G. L. Ostiguy Department of Mechanical Engineering Ecole Polytechnique P. O. B. 6079, Succ. "Centre-Ville" Montreal (Quebec), H3C 3A7 Canada
Contents Preface Ardeshir
ix Guran, Friedrich Pfeiffer and Karl Popp
Contributors
xi
DYNAMICS WITH FRICTION: MODELING, ANALYSIS E X P E R I M E N T S , P A R T II
AND
C h a p t e r 1: I n t e r a c t i o n of V i b r a t i o n a n d Friction at D r y Sliding C o n t a c t s Daniel P. Hess
1
1. Introduction
1
2. Normal Vibration and Friction at Hertzian Contacts
3
3. Normal Vibration and Friction at Rough Planar Contacts
7
4. Normal and Angular Vibrations at Rough Planar Contacts
9
5. Stability Analysis
13
6. Chaotic Vibration and Friction
20
7. Conclusions
25
8. References
26
C h a p t e r 2: V i b r a t i o n s and Friction-Induced Instability in D i s c s John E. Mottershead
29
1. Introduction
29
2. Disc Vibrations and Critical Speeds 2.1 Flexural vibrations 2.2 Vibration of a spinning membrane 2.3 Combined effects of centrifugal and flexural rigidity 2.4 Travelling waves and critical speeds 2.5 Imperfect discs
30 30 32 33 34 36
3. Excitation by a Transverse-Spring-Damper System 3.1 Stationary disc with a rotating mass-spring-damper system 3.2 Rotating disc with a stationary mass-spring-damper system 3.3 Instability mechanisms
39 40 45 46
4. Follower Force Friction Models 4.1 Follower force analysis in brake design 4.2 Sensitivity analysis 4.3 Distributed frictional load 4.4 Friction with a negative /z-velocity characteristic
48 48 49 50 51
5. Friction-Induced Parametric Resonances
52
xiii
xiv
Contents 5.1 Discrete transverse load 5.1.1 Simulated example 5.2 Distributed load system 5.2.1 Simulated example 6. Parametric Excitation by a Prictional Follower Force with a Negative fi-Velocity Characteristic 6.1 Simulated example
53 57 59 63 66 70
7. Closure
70
Acknowledgment
73
References
73
C h a p t e r 3: D y n a m i c s of Flexible Links in K i n e m a t i c C h a i n s Dan B. Marghitu and Ardeshir Guran
75
1. Introduction
75
2. Kinematics and Kinetics of Flexible Bodies in General Motion 2.1 Small deformations 2.2 Large deformations
77 77 80
3. Equations of Motion for Small Deformations in Rectilinear Elastic Links 81 4. Equations of Motion for Large Deformations in Rectilinear Elastic Links 4.1 Planar equations of motion 5. The Dynamics of Viscoelastic Links 5.1 Application 5.2 Computing algorithm 6. The Vibrations of a Flexible Link with a Lubricated Slider Joint 6.1 6.2 6.3 6.4
Reynolds equation of lubrication Cavitation Solution method for an elastic link in a rigid mechanism Application to a slider mechanism
7. References C h a p t e r 4: Solitons, C h a o s a n d M o d a l Interactions in P e r i o d i c S t r u c t u r e s M. A. Davies and F. C. Moon
83 84 85 86 89 89 89 91 92 94 97 99
1. Introduction
99
2. Experiment
102
3. Numerical Model
103
4. Forced Vibrations and Modal Interactions
108
4.1 Numerical experiment — Modal trading 4.2 Forced vibrations of the experimental structure
108 110
Contents
xv
5. Impact Response 5.1 Comparison of experiment and model 5.2 Calculation of nonlinear wave speeds
115 115 119
6. Conclusions
120
7. Acknowledgments
122
8. References
122
C h a p t e r 5: A n a l y s i s and M o d e l i n g of a n E x p e r i m e n t a l Frictionally Excited Beam R. V. Kappagantu and B. F. Feeny
125
1. Introduction
125
2. Experimental Setup
126
3. Friction Measurement
128
4. Displacement Measurement
133
4.1 From strain to displacement
133
5. Dynamical Responses
135
6. Proper Orthogonal Modes
138
7. Mathematical Model
141
8. Numerical Simulations and Validation
143
9. Discussion and Elaboration
147
10. Conclusions
150
11. Acknowledgments
151
References
151
C h a p t e r 6: Transient W a v e s in Linear V i s c o e l a s t i c M e d i a Francesco Mainardi Introduction
155
155
1. Statement of the Problem by Laplace Transform
156
2. The Structure of Wave Equations in the Space-Time Domain
159
3. The Complex Index of Refraction: Dispersion and Attenuation
162
4. The Signal Velocity and the Saddle-Point Approximation
167
5. The Regular Wave-Front Expansion
172
6. The Singular Wave-Front Expansion
178
Conclusions
186
Acknowledgments
186
References
186
xvi
Contents
C h a p t e r 7: D y n a m i c Stability and N o n l i n e a r P a r a m e t r i c V i b r a t i o n s of R e c t a n g u l a r P l a t e s G. L. Ostiguy
191
1. Introduction
191
2. Theoretical Analysis 2.1 Analytical model 2.2 Basic equations 2.3 Boundary conditions 2.4 Method of solution
194 194 195 195 197
3. Solution of the Temporal Equations of Motion
199
4. Stationary Response 4.1 Principal parametric resonances 4.2 Simultaneous resonances 4.3 Combination resonances
199 201 202 203
5. Nonstationary Responses
204
6. Results and Discussion
205
Acknowledgments
222
References
223
C h a p t e r 8: Friction M o d e l l i n g a n d D y n a m i c C o m p u t a t i o n J. P. Meijaard 1. Introduction 2. Phenomenological Models 2.1 2.2 2.3 2.4
Models without memory effects Models with memory effects Stability of stationary sliding Two-dimensional sliding
3. Analysis of Systems of Several Rigid Bodies 3.1 Analysis of mechanical systems 3.2 Arch loaded by a horizontal base motion 3.3 Four-bar linkage under gravity loading References C h a p t e r 9: D a m p i n g t h r o u g h U s e of P a s s i v e and S e m i - A c t i v e D r y Friction Forces Aldo A. Ferri
227
227 229 230 235 236 240 240 240 244 247 250
253
1. Introduction
253
2. Passive Mechanisms
254
Contents 2.1 2.2 2.3 2.4 2.5
Background Linear-Coulomb damping Profiled block Shock and vibration isolation In-plane slip
3. Semi-Active Friction 3.1 3.2 3.3 3.4
Semi-active damping Semi-active friction damping in a SDOF system Structural vibration control Semi-active automative suspension
4. Conclusions
xvii 254 256 265 269 273 280 281 282 283 295 297
5. Acknowledgment
298
6. References
298
Subject Index
309
Author Index
313
Dynamics with Friction: Modeling, Analysis and Experiment, Part II, pp. 1-27 edited by A. Guran, F. Pfeiffer and K. Popp Series on Stability, Vibration and Control of Systems, Series B, Vol. 7 © World Scientific Publishing Company INTERACTION OF VIBRATION A N D FRICTION A T DRY SLIDING CONTACTS
DANIEL P. HESS Department of Mechanical Engineering University of South Florida Tampa, Florida 33620, USA
ABSTRACT When measuring or modeling friction under vibratory conditions, one should ask how contact vibrations are influenced by the presence of different types of friction or one should seek to determine the extent to which vibrations can alter the mechanisms of friction itself. This paper summarizes results from the author's work on dry sliding contacts in the presence of vibration. A number of idealized models of smooth and rough contacts are examined, in which the assumed sliding conditions, the kinematic constraints, and the mechanism of friction are well-defined. Instantaneous and average normal and frictional forces are computed. The results are compared with experiments. It appears t h a t when contacts are in continuous sliding, quasi-static friction models can be used to describe friction behavior, even during large, high-frequency fluctuations in the normal load. However, the dynamics of typical sliding contacts, with their inherently nonlinear stiffness characteristics, can be quite complex, even when the sliding system is very simple.
1. Introduction Surfaces in contact are often subjected to dynamic loads and associated contact vibrations. The dynamic loading may be generated either external to the contact region, as in the case of unbalanced moving machinery components, or within the contact region, as in the case of surface roughness-induced vibration. Vibrations may be undesirable from the point of view of the stresses that are induced or noise that is generated and may need to be controlled. Furthermore, vibrations can affect friction and the outcome of friction measurements. In this paper, an overview of the author's work on dry friction in the presence of contact vibrations is given. The reader is referred to other papers 1 7 for details. Some general observations will be made regarding the interaction of friction and vibration and the interpretation of friction coefficients under vibratory conditions. The models discussed are limited to continuous sliding, although extensions to loss of contact or sticking could be made. The models accommodate forced
2
D. P. Hess
contact vibrations of a rigid rider mass, supported by smooth Hertzian or randomly rough planar compliant contacts undergoing elastic deformation. Initially the rider is constrained to move only along a line normal to the sliding direction. The vibration problem is solved for the normal motions. To allow a well-defined mechanism of friction to be explicitly inserted into the dynamic model, the instantaneous friction force is related to the normal motion through the adhesion theory of friction. Accordingly, the instantaneous friction force is taken to be proportional to the instantaneous real area of contact. While we recognize the limitations of the adhesion theory, it is selected due to its simplicity and its ability to describe many situations of practical interest 8 . A general feature of the results is that as the normal oscillations increase, the average separation of the surfaces increases. This is due to the nonlinear character of the contact stiffness which increases (hardens) as the instantaneous normal load increases from its mean value and decreases (softens) as the load is reduced. This increase in average separation is, under the assumptions stated above, sometimes, but not always, accompanied by a decrease in the average friction force. A more interesting, yet still simple, model is that of a rough block in planar contact that is allowed to translate and rotate with respect to the countersurface against which it slides. We have developed a modification of the GreenwoodWilliamson 9 rough surface model for this purpose. The basic equations are given and general features of the problem are discussed. Some comparisons are made with experiments and with part of the work of Martins et al.10, in which a similar problem using a phenomenological constitutive contact model is examined. Before proceeding, we comment on the interpretation of the coefficient of friction under dynamic conditions. If both the load and the friction force at a contact vary with time, the instantaneous friction coefficient, |i(r), is
,»-
%
Of particular interest is the interpretation of average friction. One interpretation of average friction is to take the time average of [i(t), denoted by (n(f)). Alternatively, one could define an average friction coefficient, n^,, as the average friction force divided by the average normal load, so that V. = < ^ >
(2)
If the normal load remains constant or the instantaneous friction coefficient does not change with time, the two interpretations are equivalent. Otherwise they are not. This is readily demonstrated by considering the example of a smooth, massless, circular Hertzian contact to which an oscillating load PB (1 + cosQf) is
1. Interaction
of Vibration and Friction at Dry Sliding Contacts
3
applied. This amount of load fluctuation is just enough to give impending contact loss at one extreme of the motion. The friction coefficient is \i0 when the load is at its mean value, Pg. For illustration purposes, the instantaneous friction force is assumed to be proportional to the instantaneous real area of contact. It is easy to show1 that, in this case, — = 0.92 whereas ^ = 1.84. This is illustrated in Fig. 1. The time average of the friction coefficient, (n(0), increases while the average friction force decreases. When F, P and [i all vary with time, the coefficient of friction seems to be of limited value. Particular difficulties arise when P{t)~0. For defining average friction, the definition of Eq. (2) is preferred. Sometimes, in friction testing, only the instantaneous friction force is measured. Even this requires a measurement system with sufficient frequency bandwidth to accurately measure the fluctuating forces. The normal load is not monitored. If one incorrectly assumes that the normal load remains constant, when it does not, one obtains an "apparent friction" coefficient which can be quite different from the actual friction. Apparent friction sometimes includes stick or loss of contact which do not represent friction in the usual sense. 2. Normal Vibration and Friction at Hertzian Contacts As the first and simplest example, the dynamic behavior of a circular Hertzian contact under dynamic excitation is examined. The system is shown in Fig. 2. The rider has mass, m, and is in contact with a flat surface through a nonlinear stiffness and a viscous damper. The lower flat surface moves from left to right at a constant speed, V. The friction force, F, acts on the rider in the direction of sliding. The rider is constrained to motion normal to the direction of sliding. The model accommodates the primary normal contact resonance. The contact is loaded by its weight, mg, and by an external load, P = i*a(1 + aCOSQf), which includes both a mean and a simple harmonic component. The normal displacement, y, of the mass is measured upward from its static equilibrium position, y0. The equation of motion during contact, obtained from summing forces on the mass is my + cy - / ( 8 ) = -/>„(1 +aCOSQf) - mg for 8 > 0 where 8 is the contact deflection and /(8) is the restoring force given by
/(8) = !£'*U=Kl(y0-y)2
,
y0-[^^f
(3)
(4)
4 D. P. Hess
0.0
0.2
Figure 1. Instantaneous and average load, area, and friction (force and coefficient) for a smooth massless Hertzian contact.
1. Interaction
of Vibration and Friction at Dry Sliding Contacts
5
Figure 2. Dynamic model of normal motion for Hertzian contact.
An approximate steady-state solution to this nonlinear system has been obtained 1 using the perturbation technique known as the method of multiple scales. The contact area, A, is proportional to the contact deflection, (y 0 -y). Based on the adhesion theory of friction, the instantaneous friction force is assumed to be proportional to the area of the contact. Therefore, (5) The normal oscillations, y(t), are asymmetrical due to the nonlinear contact stiffness, and give rise to a decrease in average contact deflection, (y0- (y)), (i.e., an increase in separation of the sliding bodies) by an amount (y), where (y) is the average of y(t). Since Eq. (5) is linear, we can also write (6)
6
D. P. Hess
i
3 2
11
10-i
(
no applied vibration
M
freq. (Hz) 3
• *
2
10-2
J_L
1 01
20 100 1000
I
I
2
3
I I Mill
I
2
1 02
3
I I II
103
Accel, (in/s2) Figure 3. Measurements from Godrey (1967) showing the effect of vibration on friction; ^ ^ ^ ^ ^ ^ _ , present theory.
As oscillations increase, the average contact area, and, by implication, the average friction are reduced. A reduction in average friction force of up to ten percent was shown to occur1 prior to loss of contact. This is not greatly different from the result obtained without considering inertia forces or damping and illustrated in Fig. 1. Godfrey11 conducted experiments to determine the effect of normal vibration on friction. His apparatus consisted of three steel balls fixed to a block that slid along a steel beam and was loaded by the weight of the block. The beam was vibrated by a speaker coil at various frequencies. The normal acceleration of the rider and the friction at the interface were measured. His measurements, under dry contact conditions, are illustrated in Fig. 3. If one assumes that occasional contact loss begins to occur when the normal acceleration reaches an amplitude of one g, one can superimpose the friction reduction predicted by our model as indicated by the heavy line. Reasonably good agreement is obtained. At higher
1. Interaction
of Vibration and Friction at Dry Sliding Contacts
7
normal accelerations, where there is progressively more intermittent contact loss, a larger reduction in friction occurs. The dynamic behavior of continuously sliding Hertzian contacts under random roughness-induced base excitation has also been examined4. At sufficiently high loads, such contacts can be represented by a smooth Hertzian contact 12 . The role of the surface roughness is only to provide a base excitation as it is swept through the contact region. By restricting the effective surface roughness input displacement to stationary random processes defined by the spectral density function Syy(k) = Ljt~1£~4 (where L is a constant and k is the surface wavenumber), the Fokker-Planck equation can be used to obtain the exact stationary solution. Again, one finds a decrease in the mean contact compression under dynamic loading. This also leads to a reduction in the mean contact area and the average friction force, under the assumption that the instantaneous friction force is proportional to the instantaneous area of contact. Based on the analysis the reduction in average friction force when vibration amplitudes approached the limit of contact loss was around nine percent. A pin-on-disk system with a steel against steel Hertzian contact, excited by surface irregularities, was used to obtain measurements of average friction at various sliding speeds. The normal vibrations increased with sliding speed. The analysis was compared with the experiments by adjusting the parameter, L, so that the analytical model gave initial loss of contact at the same speed (i.e., at 50 cm/s) as observed during the tests. The computed results are shown together with the measurements in Fig. 4. The measurements show a decrease in friction with increasing sliding speed. Considering that the load criterion of Greenwood and Tripp 12 is not satisfied at all times during the motion, the agreement with the theoretical model is quite good. The measurements illustrate that, only at speeds well above those associated with initial loss of contact, can one obtain large reductions in average friction, of as much as thirty percent.
3. Normal Vibration and Friction at Rough Planar Contacts The Greenwood and Williamson9 statistical formulation of the elastic contact of randomly rough surfaces is still the best known and most widely-used model. The normal vibrations of such a contact can be cast in the same form as Eq. (3) with the real contact area and the normal elastic restoring force expressed by A = Ttr\APoj(e-h)^*(e)de h
(7a)
8
D. P. Hess
0.4
-1
1
r
0.3 OO 0 0
0
O iO
o
O
=L 0.2
o
o
0.1
0.0
_l
0
20
40
i
60
L _
80
100
V (cm/s) Figure 4. Average coefficient of friction at various sliding speeds: o, measurements; theoretical.
/(5) = -r\AE'fi*o~2 3
J
.
((e-hy$*(e)de
where P = asperity radius o = standard deviation of asperity height distribution = normalized separation h = o a d0 = static separation e = — s normalized asperity heights
,
(7b)
1. Interaction
of Vibration and Friction at Dry Sliding Contacts
9
r| = surface density of asperities A = nominal contact area $*(e) = normalized asperity height distribution The nonlinear vibration problem has been solved2. The contact stiffness nonlinearity is stronger than that of the Hertzian model. Again one finds that, on average, the sliding surfaces move apart during sliding. The change in average separation is typically around thirty percent of the vibration amplitude, \y\. Although the surfaces on average, move apart, the average friction force obtained by taking the time average of the contact area, remains unchanged in the presence of normal vibrations. This seemingly paradoxical result is not unexpected when one recognizes that the Greenwood-Williamson model leads to a direct proportionality between the normal load and the real contact area at all separations, i.e., a constant instantaneous friction coefficient. While the nonlinear contact vibrations can be complicated, and the instantaneous friction force may change considerably, the friction coefficient is not expected to change. Other rough surface models, may give somewhat different results. Linearized equations for the normal vibration problem have also been developed5. One rather remarkable result of the linearized analysis is that the small amplitude normal natural frequency of a weight-loaded rigid block supported by a Greenwood-Williamson type rough surface is w = \fgfa . The natural frequency is independent of the block and countersurface materials. The natural frequency is independent of the block dimensions and, at least on earth, depends only on the standard deviation of the asperity heights, o. The acceleration spectra of a steel block (a 4.4 cm cube) obtained during sliding against a large steel base at a speed of 3 cm/s are shown in Fig. 5. One finds the normal natural frequency at around 1300 Hz which is in general agreement with the block roughness that was measured(7?a~ 0.2\im). Angular motions, with a resonant frequency of 1070 Hz are also observed and shown in Fig. 5. It is clear that the possibility of angular motions must be included in a model of the problem.
4. Normal a n d Angular Vibrations at Rough Planar Contacts A model that allows for both normal and angular motions of a nominally stationary block pressed against a moving countersurface is shown in Fig. 6a in its frictionless equilibrium position and in Fig. 6b in its steady sliding equilibrium position. Some angular displacement, 6 0 , and offset, c, of the normal reaction force are necessary to maintain moment equilibrium of the block.
10 D. P. Hess
H
0.05
1
1
1
1
1
1
1 h
0 L 3 T3
0 0
H—I—I—I—I—I—I—I—h
2000
Freq. (Hz) Figure 5. Average rms spectra of measured normal and angular accelerations from sliding block.
1. Interaction
of Vibration and Friction at Dry Sliding Contacts
k,
•3"
N0+mg
) m fM
(10a)
= " — {N{t) + P„) ma
"r* " ^7-/i(«>/3«>> = ¥rJ
dJa
fM)
(10b) (10c)
dJa
= e~"
/2(4>) = ^ r («* " «"*) 2
(Ha) dlW
1. Interaction
of Vibration and Friction at Dry Sliding Contacts
13
\
m~^-**\
2oa§ L
2$ (e*-e-*)
+
(a+d0)F0
(lie)
Equilibrium of the rider under steady sliding requires that fMo)fz^c) =1 d2b) /sWO = 0 (12c) The equilibrium position of the rider under steady sliding can be determined by solving Eq. (12c) for „, Eq. (12b) for q0, and finally Eq. (12a) for s„. The equations of motion, Eqs. (10) and (11), clearly reveal nonlinear coupling among the translational and angular motions. Subharmonic, superharmonic, and combination resonances may occur when the system is in forced oscillation. Chaotic motions may also take place6. In problems of this type, system stability, i.e., stability of sliding, may be of concern and was studied by Martins et al.10. In fact, our equations are similar to those of Martins et al.. For example, Eqs. (10) and (12) of the present paper can be compared to Eqs. (5.8) and (5.6) in their paper 10 . An essential difference between the two approaches is that the normal and angular contact stiffnesses have different forms. Martins et al. used a power law form that has also been used by Back et al.13 and Kragelskii and Mikhin 14 .
5. Stability Analysis A linear stability analysis of the three degree-of-freedom contact model developed above reveals some interesting aspects of sliding systems. The linearized equations of motion for small perturbations about the steady sliding equilibrium position without forcing terms are 5 +
m 4s
+
*, ~
S
+
S
m b . -4s m 1
=
m
m
mo B i
,-[•.(«*•
2*:
F„c0;
(13a)
*.
P„c0
where Co
F-
ma Lc. -*s 2Ja
+ e
'•)
-
(13b)
:* = o
(13c)
=0
(«*•
'•>]
(14a)
14 D. P. Hess
*-«--{¥
± ( _ e * . + e - * . ) _ ± [ ( 1 -4, o )e*»-(1 + <J>0)e"*°] •;
+ —°-[e ° +e °) +•
(14b) •.
A
2
}
These linearized equations, having an asymmetric stiffness matrix, describe a circulatory system. These equations are similar to Eq. (5.15) of Martins et al.10, which they found to exhibit a high frequency flutter instability at friction values well below that which would result in the block tumbling, when \ia = —, which is a divergence instability. This flutter instability does not seem to occur with the model of Eq. (13). The eigenvalues that we have computed with the damping set to zero are always purely imaginary, never exhibiting positive real parts. In Martins et al., the flutter instability occurs when the two eigenvalues associated with the angular and quasi-normal natural frequencies take on the same value. In the present model, the eigenvalue ratio (angular divided by quasi-normal) always remains less than unity, approaching this value only when the block is very long, i.e., when — < 1. The differences in the qualitative behavior of the two models seems to be due to the details of the normal and angular restoring forces. These contact stiffnesses are very sensitive to the details of the surface texture. This may largely explain the elusive nature of many high frequency instabilities and squeal phenomena which can occur in sliding systems and can change and appear or disappear as surfaces run-in or wear. Figure 7 shows the instability regions found by Martins et al. in the absence of damping. In their analysis, and with weight loading, the stability depends only on the height to length ratio of the block, the sliding friction coefficient, and a damping parameter. These results indicate instability over a fairly broad range of aspect ratio (H/L where H=2a). The addition of damping causes the flutter instability boundary to shift to higher friction values for small aspect ratios. Interestingly, the addition of a linear torsional stiffness, k^, to our sliding block model can lead to instability. The resulting linearized equations of motion for this case are », + ^ i . + ^ . m m 9.
+
-?, m
+
+^ « , - ^ * . m m — * . - — * , =0 ma ma
where c2 and c 4 are defined in Eq. (14) and
-0
(15a) < 15b )
1. Interaction
of Vibration and Friction at Dry Sliding Contacts
15
Z.3
^
2.0
unstable (divergent)
\. 1.5
unstable (flutter)
1.0
^
~^
0.5
.
0.0
~
i
0.2
i
stable
i
0.4
0.8
0.6
1.0
H/L Figure 7. Regions of instability based on Martins et al. (1990).
c,=e
4„
P„oa -—\e^-e
" +
(a+do)Fo(e*._e-*