dynamics of contacting thermoelastic bodies
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Dynamics of Contacting Thermoelastic BodiesTRANSCRIPT
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NONSMOOTH DYNAMICS OF CONTACTING
THERMOELASTIC BODIES
Advances in Mechanics and Mathematics VOLUME 16 Series Editors David Y. Gao (Virginia Polytechnic Institute and State University) Ray W. Ogden (University of Glasgow)
Advisory Board Ivar Ekeland (University of British Columbia, Vancouver) Tim Healey (Cornell University, USA) Kumbakonam Rajagopal (Texas A&M University, USA) Tudor Ratiu (École Polytechnique Fédérale, Lausanne)
David J. Steigmann (University of California, Berkeley)
Aims and Scope Mechanics and mathematics have been complementary partners since Newton’s time, and the history of science shows much evidence of the beneficial influence of these disciplines on each other. The discipline of mechanics, for this series, includes relevant physical and biological phenomena such as: electromagnetic, thermal, quantum effects, biomechanics, nanomechanics, multiscale modeling, dynamical systems, optimization and control, and computational methods. Driven by increasingly elaborate modern technological applications, the symbiotic relationship between mathematics and mechanics is continually growing. The increasingly large number of specialist journals has generated a complementarity gap between the partners, and this gap continues to widen. Advances in Mechanics
and Mathematics is a series dedicated to the publication of the latest developments in the interaction between mechanics and mathematics and intends to bridge the gap by providing interdisciplinary publications in the form of monographs, graduate texts, edited volumes, and a special annual book consisting of invited survey articles.
J. Awrejcewicz
NONSMOOTH DYNAMICS OF CONTACTING
THERMOELASTIC BODIES
Yu. Pyryev
Jan Awrejcewicz
Technical University of Lodz Lodz, Poland [email protected]
Series Editors:
David Y. Gao Ray W. Ogden Department of Mathematics Department of Mathematics Virginia Tech University of Glasgow Blacksburg, VA 24061 Glasgow, Scotland, UK [email protected] [email protected]
ISBN: 978-0-387-09652-0 e-ISBN: 978-0-387-09653-7 DOI: 10.1007/978-0-387-09653-7
Mathematics Subject Classification (2000): 70-xx, 34-xx, 35-xx, 37-xx, 65-xx © 2009 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
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The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are
Library of Congress Control Number: 2008939859
Technical University of Lodz Lodz, Poland
Department of Automatics Yu. Pyryev
and Biomechanics and Biomechanics
Department of Automatics
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Object of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Equations of motion of bodies in contact . . . . . . . . . . . . . . . . 11.1.2 Boundary and contact conditions . . . . . . . . . . . . . . . . . . . . . . . 91.1.3 Dynamics of contacting bodies . . . . . . . . . . . . . . . . . . . . . . . . 171.1.4 Contact thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.1.5 On some equations governing discontinuous systems
dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.2 Aim and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Thermoelastic Contact of Shaft and Bush
in Wear Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.1 Analysed system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Mathematical formulation of the problem . . . . . . . . . . . . . . . . . . . . . . 33
2.2.1 Equations for rotational movement of absolutely rigid bush 332.2.2 Thermoelastic shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.2.3 Rotational motion of the shaft . . . . . . . . . . . . . . . . . . . . . . . . . 392.2.4 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Kinematic external shaft excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.1 Block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.2 Stationary process associated with a constant shaft
velocity rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.2.1 Block diagram of the linearized system . . . . . . . . . 432.3.2.2 Stationary process without wear . . . . . . . . . . . . . . . 432.3.2.3 Analysis of steady-state solution in the presence
of wear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.3.3 Numerical analysis of the transient solution . . . . . . . . . . . . . 48
v
vi Contents
2.3.4 Chaotic motion of the shaft/bush with kinematic externalexcitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.3.4.2 Melnikov function . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.3.4.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.3.4.4 Calculation of Lyapunov exponents . . . . . . . . . . . . . 642.3.4.5 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3.5 Chaotic motion of the bush subject to mechanicalexternal excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.3.5.1 Melnikov’s method . . . . . . . . . . . . . . . . . . . . . . . . . . 712.3.5.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.3.6 Analysis of the bush motion with wear and cylinderkinematic excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.3.7 Dynamics with external temperature perturbation . . . . . . . . . 782.4 External shaft mechanical excitations . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.4.1 Shaft inertial motion with tribological processes . . . . . . . . . 812.4.1.1 Solution properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.4.1.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.4.2 Inertialess shaft and bush dynamics and frictional heatgeneration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.4.2.1 Application of the Laplace transform . . . . . . . . . . . 892.4.2.2 Stationary dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 902.4.2.3 Stick-slip process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.4.2.4 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.4.2.5 Acceleration process . . . . . . . . . . . . . . . . . . . . . . . . . 962.4.2.6 Braking process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.4.3 Inertial damped dynamics of cylinder and bush andtribological processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.4.3.1 Stick-slip process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.4.3.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.4.3.3 Cylinder and bush dynamics during acceleration . 1012.4.3.4 Cylinder and bush dynamics during braking . . . . . 104
2.5 Dynamics of contacting bodies with impacts . . . . . . . . . . . . . . . . . . . 1052.5.1 Mathematical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.5.1.1 Equations for shaft rotational movement of anabsolutely rigid bush . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.5.1.2 Thermoelastic shaft . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.5.2 Algorithm of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.5.3 On the periodic motion with impacts . . . . . . . . . . . . . . . . . . . 1102.5.4 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.62.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1262.6.2 One degree-of-freedom model . . . . . . . . . . . . . . . . . . . . . . . . . 1272.6.3 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1292.6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Stick-slip vibrations (continuous friction model) . . . . . . . . . . . . . . . . 126
Contents vii
2.6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3 Thermoelastic Contact of Parallelepiped Moving Along Walls . . . . . . 1353.1 Kinematically driven parallelepiped-type rigid plate . . . . . . . . . . . . . 136
3.1.1 Mathematical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.1.2 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.1.3 Stationary process subject to kinematic external excitation . 1413.1.4 Algorithm and solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.2 Rigid plate dynamics subject to temperature perturbation . . . . . . . . 1453.2.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.2.2 Application of the Laplace transform . . . . . . . . . . . . . . . . . . . 1493.2.3 Stationary process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.2.4 Constant friction coefficient and nonstationary process . . . . 1533.2.5 Variable friction coefficient and nonstationary process . . . . 155
3.3 Dynamics of a two degrees-of-freedom system with friction andheat generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1603.3.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613.3.2 Mathematical problem formulation . . . . . . . . . . . . . . . . . . . . . 1623.3.3 Solution of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1643.3.4 Steady-state solution analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.3.5 Numerical analysis of transient solution . . . . . . . . . . . . . . . . . 169
3.4 Tribological dynamical damper of vibrations with thermoelasticcontact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1703.4.1 Mathematical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713.4.2 On the heat transfer influence on dynamical damper of
self-vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1783.4.3 Nonlinear dynamics of a dynamical damper with wear
processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4 Contact Characteristics During
Braking Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1874.1 Contact characteristics of three-layer brake models . . . . . . . . . . . . . . 188
4.1.1 Mathematical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894.1.2 Contact characteristics of the metallic–ceramic frictional
strap and the metal disk during braking . . . . . . . . . . . . . . . . . 1944.2 Computation of the contact characteristics of the two-layer brake
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1984.3 Computation of the contact characteristics of the two semi-space
brake models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1994.3.1 Contact temperature and wear during braking . . . . . . . . . . . . 2004.3.2 Contact temperature and wear during braking and
harmonic load excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
viii Contents
5 Thermoelastic Contact of Two Moving Layers with Friction and
Wear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2075.1 Analysed system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2075.2 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2115.3 Algorithm of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2135.4 Solution analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2145.5 Frictional thermoelastic instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Preface
In this work, methods of analysis and models of contacting systems dynamics,including heat generation and wear exhibited by such systems, are presented. Itshould be emphasised that the methods and mathematical models of contacting
far applied separately. Tribological processes occurring on a contact surface werenot taken into consideration in the analysis of the dynamic rigid or elastic body
took tribological effects into consideration did not allow for their inertia analysis.This study contributes to the development of this field, as the models presentedhere yield prediction of the behaviour of contacting systems taking into accountboth mentioned aspects simultaneously.
When considered from the mathematical point of view, the method of analysisis reduced to the solution of the system of differential equations describing thevelocities of contacting bodies and Volterra integral equation modelling contactpressure. The latter equation is obtained with the use of the Laplace integraltransform.
Many aspects of the problem are considered in our analysis: the influence ofvarious types of nonlinearities, damping, and excitations on the system behaviourare studied. Apart from Duffing elastic nonlinearity, the nonlinearity of frictionalenergy flux, the nonlinear relation between the friction factor and the velocity ofslipping and contact temperature, and the nonlinearity of a wear factor are takeninto consideration. The problem defined and discussed is solved with the useof perturbation theory, Melnikov’s method, Laplace transforms, Volterra integralequation theory, and numerical analysis.
Numerical solutions of the problems are obtained, and the influence of heatgeneration and frictional wear on the motion of the bodies in contact is shownin many diagrams, because the authors intend to present the obtained results in aform useful for practical engineering.
The aim of this work is to define the dynamics of bodies in contact, the thermalfield, the tensor of quasi-static stresses, displacements, and wear amount. Themethods presented here apply to the problems defined using one or two coordinates
ix
models. On the other hand, most of the introduced models of bodies in contact that
systems exhibited by rigid/elastic bodies and heat/wear processes have been so
x Preface
of space and time coordinates. First, one-dimensional nonstationary contact modelsare considered, which enable an analytical solution. These simple models providedefinitions of the basic dynamic characteristics of the analysed systems.
The subjects of this research are nonstationary dynamic phenomena and thermalconduction, as well as thermal stresses and wear in the bodies being in contactand subject to external load action.
For the solution of the problems concerning movable couplings, strains of theelements of kinematic pairs are taken into account, unlike in the research of manyother authors. Energy dissipation is a result of the friction force that acts on thesurfaces of bodies slipping against each other. In the case of movable couplings,self-excited vibrations may occur, depending on the properties of frictional contact.The self-excited vibrations may sometimes cause fast wear and destruction of theanalysed kinematic pair.
The problem is formulated with the use of a classical linear theory of ther-moelasticity. Special emphasis is put on the assumption that the material ishomogeneous and isotropic as far as thermal and mechanical constants are con-cerned, and that its physical features are independent of temperature. Unlike otherworks, this study takes into account the inertial effects and the influence of ther-moelastic coupling in the models of bodies in contact.
In Chapter 1 both the subject and aim of the study, as well as the currentstate-of-the-art of knowledge in the field of the dynamics of contacting bodies arepresented. General rules of modelling contact problems are also outlined.
Chapter 2 is devoted to mathematical modelling of vibrations of a friction pairthat consists of a rigid body (a bush) connected with a casing by means of springsand dampers and a rotating thermoelastic shaft.
Frictional heat generation, wear of a bush, and thermal expansion of a cylinder(shaft) are taken into account. The model of a rigid body and a cylinder is a systemwith two degrees-of-freedom. The model of a thermoelastic cylinder is describedwith the use of the theory of thermal stresses. The contact of a bush placed on acylinder is governed by the Stribeck model of friction. In order to obtain the valueof a contact pressure of friction, the Laplace method of transformation is applied.Eventually, the analysed problem is cast (Section 2.2) into the system of nonlineardifferential equations and an integral equation governing the angular velocities ofa cylinder and of a bush and contact pressure. The cases of kinematic (Section 2.3)and mechanical (Section 2.4) external excitations of a cylinder for various initialconditions of the bodies in contact are considered.
A new model is analysed, in which the changes of rotational speed of a bushand a cylinder, the changes of contact pressure, contact temperature and wearare interrelated. The analysis of the stability of solutions is also conducted. Theanalytical prediction of the occurrence of periodic and stick-slip motions is carriedout, and then it is verified via numerical calculations. Possibilities of the occurrenceof self-excited vibrations and chaos are analyzed in such real-like systems. Thefollowing have been shown, among others: (i) how stick-slip dynamics decreasesand vanishes due to time and wear; and (ii) how stable regimes are extended dueto shaft heating and wear.
Preface xi
It has been shown that in the considered system where an inertia of the movingbody is taken into account, frictional stability loss cannot be achieved and thesystem cannot be overheated.
In the case when the motion of one of the contacting bodies has a constantvelocity, contact pressure increases, yielding an increase either of frictional forceor frictional torque. Frictional heat generation increases, and the analyzed systemis overheated. Energy is supplied to the system owing to an increase of force ortorque required for keeping constant velocity of the system.
In the case when the body inertia is taken into account, in spite of that thesystem heat extension is bounded and contact pressure may increase by increasingfrictional force or torque, overheating cannot appear, because the system starts itsbraking process. Owing to heat exchange, a cooling process occurs; both contactpressure and friction force (torque) decrease which again causes an increase of thebodies’ velocity and frictional heat generation. The mentioned process is repeated.In other words, the system controls its heating energy input to resist overheating.
A rigid body vibration model (bush–shaft) inserted into a shaft moving atnonconstant velocity is analysed. Melnikov’s method is used to study chaoticdynamics of the externally driven bush. An analytical form of Melnikov’s functionis formulated and various nonlinear phenomena are analysed, including chaoticbehaviour. The following conclusions are drawn: (i) Melnikov’s function can bedefined analytically, and hence one may control the system behaviour by realisingeither regular or chaotic dynamics; and (ii) there are zones of parameters whereMelnikov’s function can be significantly simplified to give an explicit formula forchaos occurrence.
Chaos thresholds, phase portraits, Poincare cross-sections, Lapunov exponents,and the frequency spectra of vibrations are determined. Numerical calculationslead to many conclusions concerning the influence of frictional heat generationand wear on the character of system vibrations. Contact pressure and temperatureare monitored numerically, and analysis of wear kinetics is conducted. In thiswork conditions of occurrence of periodic motion in systems with dry frictionand impacts are investigated. We are aimed at estimation of parameters of theinvestigated system associated with occurrence of periodic motion. In particular,both the restitution coefficient and period of periodic dynamics are defined. For anarbitrary restitution coefficient two periodic orbits (stable and unstable) presentedon the phase plane are studied.
In Chapter 3, vibrations of a cubicoid-shaped body and the phenomenon offrictional heat generation during contact with a surrounding medium are anal-ysed. Two cases of thermoelastic contacts between two bodies have been studied.The bodies move vertically and they are situated between two walls. In the firstconsidered model (Section 3.1) a rigid cubicoid (plate) is coupled with a masslessspring and it can vibrate. Walls compress the body (plate) and move down at aconstant speed. In the second model (Section 3.2) a contact between the plateand unmovable walls occurs owing to wall heating and thermal plate extension.The novelty of the proposed model includes inertia of the bodies, dependence offriction on the relative velocity, and the wear phenomenon. The proposed model
xii Preface
consists of the system of both differential and integral equations that govern theplate velocity and contact pressure behaviours. Stability of stationary solutionshas been analysed and conditions for stick-slip vibrations have been derived. It isexpected that the proposed model may serve for estimation of the contacting char-acteristics of moving engineering tribological systems. A novel thermomechanicalmodel of frictional self-excited stick-slip vibrations is proposed. A mechanical sys-tem consisting of two masses that are coupled by an elastic spring and movingvertically between two walls is considered. It is assumed that between masses andwalls a Coulomb friction occurs, and stick-slip motion of the system is studied.The applied friction force depends on a relative velocity of the sliding bodies.Stability of stationary solutions is considered. A computation of contact param-eters during heating of the bodies is performed. The possibility of existence offrictional autovibrations is illustrated and discussed. One degree-of-freedom sys-tem driven by either a force or a kinematic excitation is studied. An additionalmass is added to the mentioned mechanical system via a special pressing de-vice initiating dry friction occurrence on the contacting surfaces. We are focusedon solution to the nonlinear problem of thermal stresses and strongly nonlinearequations governing dynamics of the investigated system. The carried-out analysisyields directions for a proper construction of the mechanical vibration dampers. Itshould be emphasised that the dynamic damper with dry friction may not achievethe expected properties. As we have shown, heat generation on the contactingsurface between the damper and the oscillating body as well as heat expansioneliminate resonance phenomena. The real system is self-regulating, that is, itcontrols achievement of an optimal contacting pressure. The thermoelastic paral-lelepiped extends itself according to the conditions of both sliding velocity and heattransfer.
In Chapter 4 contact characteristics are studied (velocity of slipping, contacttemperature, contact pressure, and wear) during the process of braking. A solutionof the system of equations of heat dynamics of friction and wear is analysed(HDFW) for a tribomechanical system consisting of three different layers. Such amodel gives not only the possibility of taking the thickness of frictional elementsinto account, but also enables determination of thermal regime and wear of atribomechanical system consisting of the system of metal–ceramic weld and a metalplate. A solution of the thermal problem for the nonlinear relation between frictionand wear coefficients and contact temperature is found. Contact characteristics ofthe relation of friction coefficients and wear on contact temperature are analysednumerically, and contact characteristics of a two-layer model of a brake duringthe braking process (the change of dimensionless velocity of slipping, contacttemperature, and the kinetics of wear) are determined. Moreover, the maximumcontact temperature, as well as braking time and the so-called effective thickness ofa body are calculated. A particular case of a linear relation yielding the analyticalform of wear is also studied.
The obtained solution can be used for monitoring heat regimes and wear of fric-tional surfaces of brakes working in the regime of a linear dependence of frictioncoefficient on temperature. Additionally, the case of harmonic variation of external
Preface xiii
load has been analysed. A numerical analysis has shown that the vibration ofcontact pressure causes a minor vibration of contact temperature, and both ofthem do not influence velocity and wear during the braking process.
The following conclusions have been obtained owing to the analysis of thesteel–ceramic layer inserted on the steel basis of a frictional pad during braking:(i) for a constant thickness of the plate cover and disc an increase of the steel–ceramic layer induces an increase of contact temperature and decrease of wear;and (ii) there is thickness of the plate cover and disc such that a change of thesteel–ceramic layer thickness does not influence contact characteristics.
Chapter 5 concerns analysis of the one-dimensional model of thermoelasticcontact of two layers. The upper layer slips on the surface of the other oneunder the action of a force. Friction forces generate heat on the contact surfaceof friction pairs and result in wear. Heat exchange between external surfacesand their surroundings takes place, owing to Newton’s law. Special emphasisis put on the research concerning frictional contact of two layers with differentthicknesses.
The main novelty of the presented model lies in taking into account the inertiaof one of the layers in contact, as well as the relation of the friction coefficientand wear on contact temperature, allowing us to provide the analysis for the caseof close proximity of the layers. In the latter case, contact pressure is an unknownvalue and depends on heat expansion of the layers in contact.
It is illustrated how Frictional ThermoElastic Instability (FTEI) occurs in thecase of constant velocity of motion of one layer. The effect of nonstationary heatgeneration and wear on the conditions of FTEI occurrence is also studied. FTEI ischaracterized by an exponential increase in time of contact characteristics, that is,temperature, pressure, and wear, when the relative velocity of slipping is higherthan a certain critical velocity.
In the case of constant relative velocity of moving bodies, the mechanism ofoccurrence of FTEI of the bodies in contact modelling dynamics of seals, brakeblocks, and other tribological systems is studied. An equation for critical speeddetermination is derived. It is illustrated how the inclusion of wear results in ahigher critical speed, which means that FTEI occurs. In the case when wear ishigher than heat expansion, FTEI does not occur. The obtained results can beapplied in resistance calculations for movable elements of machines and mecha-nisms, as well as in introducing recommendations concerning improvement of thereliability and durability of various mechanisms and machines with friction, espe-cially in formulating the criteria of an optimal choice of materials of rubbing pairsand in the analysis of the processes of heat generation and wear while grinding orbraking of transport mechanisms.
The proposed theory and modelling can be applied in a numerical study ofsystems with many degrees-of-freedom or other more advanced models for wearand friction.
Owing to the introduction of a moment applied to a shaft (characteristic ofthe energy source) dependent on rotational velocity one may also obtain various
xiv Preface
models of self-excited systems with a nonideal energy source and with varioustribological processes.
Finally, we wish to acknowledge the financial support by the Ministry ofEducation and Science of Poland for years 2005–2008 (Grant No. 4TO7C04429and grant No. 4TO7A03128).
Lodz Jan AwrejcewiczJuly 2008 Yuriy Pyryev
Nomenclature
al thermal diffusivity (l = 1, 2) [m2 s−1]
Bi Biot number
Bl moment of inertia of the shaft (l = 1) and the bush (l = 2) for alength unit [kg m]
c coefficient of viscosity resistance (for a length unit) [N s m−2]
El Young’s modulus [N m−2]
f(Vr) friction coefficient
Ffr dry friction for a contact surface unit [N m−2]
fs coefficient in a function F (ω1 − ϕ)
F kinematic friction coefficient depending on relative velocity
h dimensionless coefficient of viscosity resistance
hM (t) prescribed dimensionless moment acting on the shaft
hT (t) environmental temperature for Newton’s cooling law of Equa-tions (2.10) and (3.32)
H(τ) unit jump function (H(τ) = 1, τ > 0, H(τ) = 0, τ < 0)
k coefficient of restitution
k1 rigidity coefficient of a radial spring (for a length unit) [N m−2]
k2 rigidity coefficient of a spring, tangentially directed (for a lengthunit) [N m−2]
kw dimensionless wear coefficient
Kw wear coefficient [m2 N−1]
M moment acting on the shaft per length unit [N]
xv
xvi Nomenclature
Mfr moment of friction force (per length unit) [N]
Me moment of elasticity force (per length unit) [N]
M(τ0) Melnikov function
N(t) normal component of the reaction (normal load per length unit)[N m−1]
P (t) contact pressure [N m−2]
P∗ characteristic contact pressure value [N m−2]
p dimensionless contact pressure
p∗ perturbation of contact pressure
pst dimensionless steady-state contact pressure
r dimensionless radius
R1 radius of a cylinder [m]
R2 external radius of the bush [m]
s Laplace transform parameter
sm roots of a characteristic equation ∆(s) = 0
t time [s]
tD characteristic system time related to the bush oscillations period [s]
tT characteristic time related to heat transfer [s]
t∗ characteristic time of a system [s]
tc period of time in which the bodies stay in contact [s]
T temperature of a contact surface [◦C]
Tsm temperature of the surrounding medium [◦C]
Tl body temperature (l = 1, 2) [◦C]
Tp period [s]
u dimensionless displacement
uw dimensionless wear
U0hU (t) compression [m]
Uw wear (displacement of the working surfaces along Nc due towear) [m]
U displacement component along radial direction in the shaft [m]
Vr relative velocity of the contact bodies [m s−1]
Nomenclature xvii
αl coefficient of thermal expansion of the bodies (l = 1, 2) [◦C−1]
αT heat transfer coefficient [W m−2 ◦C−1]
γ dimensionless thermomechanical parameter
∆(s) characteristic function
ζ dimensionless amplitude of external force excitation
ζk dimensionless amplitude of external kinematic excitation
θ dimensionless temperature
θ∗ temperature perturbation
θst dimensionless steady-state contact temperature
λl, μl Lame constants (l = 1, 2) [N m−2]
λl thermal conductivity (l = 1, 2) [W m−1 ◦C−1]
μm roots of a characteristic equation
νl Poisson’s ratios (l = 1, 2)
σR stress component along radial direction in the shaft [N m−2]
τ dimensionless time
φ dimensionless shaft position angle
φ = ω1(τ) dimensionless angular velocity of a cylinder
ϕ(τ) dimensionless bush position angle
ϕ∗ perturbation of the angle of bush rotation
ϕst dimensionless static angle of bush rotation
ϕl(t) angle of the shaft (l = 1) and the bush (l = 2) position
ϕl(t) angular velocities of a cylinder (l = 1) and the bush (l = 2) [s−1]
ωr(τ) dimensionless relative velocity
ω relation of time characterizing the vibrations of a material systemto the time characterizing the process of temperature equalizing
ω′ frequency of external excitation [s−1]
ω0 dimensionless frequency of external perturbation
ω Fourier transformation parameter
Ω angular velocity [s−1]
Chapter 1
Introduction
1.1. Object of the study
Friction, wear, heat generation accompanying friction, and heat expansion (orcontraction) are all very complex phenomena that interact and form one complexmultidimensional dynamic system analysed together with friction. For the nonsta-tionary friction process, all of its time-dependant parameters are interrelated.
This study deals both with the phenomena occurring in material systems andwith methods of analysing them. The problems discussed here concern the analysisof dynamic phenomena occurring in machine systems and are closely related totheoretical mechanics, friction theory, mechanics of materials, methods of theanalysis of differential equations, analysis of thermal stresses, tribology (friction,wear), and to dynamic process control.
The authors are interested mainly in the search for approximate solutions of theobtained differential and integral equations and the analysis of their properties. Forthis purpose approximate analytical methods and numerical methods are applied.What is very important from the practical point of view is finding periodic stick-slipsolutions. So-called deterministic chaos [Awrejcewicz (1989), (1996)] may occurin the studied systems, in which case the minimal change of initial conditions maylead to the exponential divergence of the neighbouring trajectories of the phaseflow.
1.1.1 Equations of motion of bodies in contact
Let us consider the system of two bodies taking in the Euclidean space ℜmdim ,(mdim = 1, 2, 3) occupying volumes Ω1, Ω2 bounded by the surfaces Γ 1 = ∂Ω1,Γ 2 = ∂Ω2 (Fig. 1.1).
For a general case Γ l = Γ lu ∪ Γ l
σ ∪ Γ luσ ∪ Γ l
c or Γ l = Γ lT ∪ Γ l
q ∪ Γ lTq ∪ Γ l
c ,
(l = 1, 2), where Γ lc = Γ l
s ∪ Γ lsl ∪ Γst, Γ l
u, Γ lσ , Γ l
uσ denote the boundaries for
1DOI: 10.1007/978-0-387-09653-7_1, © Springer Science+Business Media, LLC 2009 J. Awrejcewicz and Yu. Pyryev, Nonsmooth Dynamics of Contacting Thermoelastic Bodies,
2 1 Introduction
Fig. 1.1: Modelling the contact of two bodies.
each point, where displacements, stresses, or Winkler conditions are given. Γ lT ,
Γ lq , Γ l
Tq , Γ lc denote the boundaries where temperature together with its gradient,
and Newton condition or the condition of thermal contact, are determined. Γ ls,
Γst, Γ lsl are the boundaries where clearance, stick, or slip occurs.
Let Ω denote the sum of the volumes Ω1 and Ω2 (Ω = Ω1 ∪ Ω2), whereasΓ = Γ 1 ∪ Γ 2. The contact in the system under the influence of loads (bothmechanical and nonmechanical) is accompanied by friction. The models (systemsof equations) describing both mechanical and nonmechanical processes can be castin the form:
ℑl(∂/∂x, ∂/∂t)U l(x, t) = −Φl(x, t), l = 1, 2, (x, t) ∈ T∞, (1.1)
where ℑl(∂/∂x, ∂/∂t) is the operational matrix of the form
ℑl(∂/∂x, ∂/∂t) = ‖ℑlnm(∂/∂x, ∂/∂t)‖N×N . (1.2)
In the above U l = ‖ul1, u
l2, . . . , u
ln‖T is an N -dimensional single-column matrix
that governs the processes taking place in the bodies; the symbol ‖ · ‖T denotes acolumn matrix; the operators ℑl
nm(∂/∂x, ∂/∂t) for homogeneous bodies are thepolynomials obtained from the partial differentials ∂/∂xm (m = 1, 2, 3), ∂/∂twith constant coefficients; T∞{(x, t) : x ∈ Ω, t ∈ [0,∞)}; T∞ ∈ ℜmdim+1;x ∈ (x1, . . . , xmdim
) is the point of Euclidean space ℜmdim (mdim = 1, 2, 3); andt denotes time.
1.1 Object of the study 3
Nonstationary thermal and thermoelastic processes in the bodies are analysed inreferences [Parkus (1959), Carslaw, Jaeger (1959), Melan, Parkus (1959), Boley,Weiner (1960), Nowacki (1962), Kovalenko (1975), Awrejcewicz, Krysko (2003)].Various models of the bodies are applied for computations, and they can be found,for instance, in the monographs [Nowacki (1970), Nowacki, Olesiak (1991), Pyryev(1999), Jaeger (2005), Awrejcewicz et al. (2004)], where the models of thermo-elastic bodies are also studied.
Let us assume that the loads (mechanical and thermal) generate small strainsand stresses, and that for the system under analysis the mechanical and thermalinteraction of the bodies in contact occurs. The system of the equations of thermo-elasticity, expressed with the use of the vector of displacement ul = (ul
1, ul2, u
l3)
(l = 1, 2) and temperature gain Tl(x, t) (l = 1, 2) measured in relation to thesurrounding temperature, can be presented in the form (1.1), where N = 4, U l =‖ul
1, ul2, u
l3, Tl‖T . The following are the components of the matrix differential
operator (1.2):
ℑlnm(∂/∂x, ∂/∂t) = μlδnm(∇2 − c−2
l2 ∂2/∂t2) + (λl + μl)∂2/∂xn∂xm, (1.3)
ℑln4(∂/∂x, ∂/∂t) = −γl∂/∂xn, (1.4)
ℑl4m(∂/∂x, ∂/∂t) = −ηl∂
2/∂t∂xm, n, m = 1, 2, 3, (1.5)
ℑl44(∂/∂x, ∂/∂t) = ∇2 − a−1
l ∂/∂t, l = 1, 2, (1.6)
where
γl = αl(3λl + 2μl), cl2 =√
μl/ρl, l = 1, 2,
δnm = 1, n = m; δnm = 0, n = m.
In the above ∇2 = ∂2/∂x21 + ∂2/∂x2
2 + ∂2/∂x23 is a Laplace operator, λl, μl are
Lame coefficients, ρl is the density of the body material, αl is a thermal expansioncoefficient, al is thermal diffusivity, and ηl is a thermodynamic constant. Thesystem of differential Equations (1.1)–(1.6) is coupled by the element (1.5) andtherefore these equations are called coupled equations of thermoelasticity. Thetheoretical studies [Nowacki (1962), Pyryev (1999)] and experiments show thatthe coupling usually has a small effect on the system behaviour. If one neglectsthe element ηl∂
2/∂t∂xm in (1.5), Equations (1.1)–(1.6) create the system ofdifferential equations of Lame–Neumann quasistatic uncoupled thermoelasticity.
Position of a point of the body l (l = 1, 2), occupying space Ωl and boundedby the surface Γ l in its unloaded state, is characterized by vector rl
1 = xlj ij =
xl1i1 + xl
2i2 + xl3i3. For the power load of the system, bodies can (the main vector
of external powers and its moment are not equal to zero) exhibit a motion.As mentioned before, the mechanical uΓl, tΓl, and thermal loads T Γl, qΓl
(l = 1, 2) influence the bodies in the initial moment. We assume that the mainvector and the main moment of external loads bl, tΓl are different from zero,
4 1 Introduction
Fig. 1.2: Distribution of general displacement ul1
(l = 1, 2).
which causes the motion of bodies. At some point in time t > 0 the bodies willhave a certain new configuration in space, close to the configuration of that ofperfectly rigid bodies, respectively, and their geometry and density correspond tothe characteristics of unloaded elastic bodies. The point rl
1 takes the place of rl:rl = rl
1 + ul1, where ul
1(xl1, x
l2, x
l3, t) is the vector of displacement of the point
whose coordinates are ul11, u
l12, u
l13 (Fig. 1.2).
We assume that unit elongation (and shortening) and angular strains are smallerthan one. The shift of the bodies and their displacements related to their rotation(we treat them as rigid solids) are not small. In what follows we apply a Greenstrain tensor el = el
ijii ⊗ ij of the form
elij =
1
2
(∂ul
1i
∂xj+
∂ul1j
∂xi+
∂ul1m
∂xj
∂ul1m
∂xi
), l = 1, 2, i, j = 1, 2, 3 (1.7)
and nonlinear equations of the motion of elastic bodies according to Lagrange[Kappus (1939), Novogilov (1958), Ulitko (1990)]
divP l + bl = ρl ∂2ul
1
∂t2,
(∂P l
ij
∂xi+ bl
j = ρl∂2ul
1j
∂t2
), (1.8)
P lij =
(δij +
∂ul1i
∂xk
)σ∗l
kj , (1.9)
where P l = P lijii ⊗ ij is the Piola pseudostress tensor; σ∗l = σ∗l
ij ii ⊗ ij is the
Kirchhoff pseudostress tensor, defined via stress tensor σl = σlij ii ⊗ ij in the
Lagrange variables σ∗lij =
√1/(1 + 2el
jj)(da∗li /dal
i)σlij . Observe that dal
i, da∗li
1.1 Object of the study 5
denote surfaces of a body deformed element situated vertically to the axis xli,
whereas bl is the vector of the volume forces expressed in Lagrange’s variables.Note that for small deformations the mentioned tensors can be treated as approxi-mately equalones.
Kinematic Equations (1.7) and dynamic Equations (1.8) are supplemented bythe physical Duhamel–Neumann equations of the form
σl = 2μlel + (λlθl − γlTl)I , (σlij = (λlθl − γlTl)δij + 2μlel
ij), (1.10)
where θl = el11 + el
22 + el33, (l = 1, 2).
The mechanical boundary conditions of the uncoupled bodies (without acontact) can be formulated in the following form,
Nl · P l = tΓl, (N liP
lij = tΓl
j ), x ∈ Γ lσ, l = 1, 2, (1.11)
where N li are directional cosines of an undeformed surface Γ l
σ (Nl = N ljij),
and tΓlj are the initial loading components regarding the undeformed surface
(tΓl = tΓlj ij).
It should be emphasised that displacements ul1 are not small. What is more, the
nonlinear problem formulated like this includes the problem related to the dynamicsof thermoelastic bodies. There is no need to add the equations of motion and to for-mulate boundary conditions by introducing the additional equations. For instance,the theorems of the motion of a centroid and of the shift of angular momentumresult [Ulitko (1990)] directly from the motion equations and boundary conditions.
By integrating the equations of motion along the initial volume and by changingtriple integrals into surface integrals (Gauss–Ostrogradski theorem) and taking intoaccount boundary conditions (1.11), we obtain [Ulitko (1990)] the following vectorequations,
∫∫∫
Ωl
ρl ∂2ul
1
∂t2dv =
∫∫©Γ l
σ
tΓldS +
∫∫©Γ l
c
pcldS +
∫∫∫
Ωl
bldv. (1.12)
The position of the centres of elastic and homogeneous masses is determinedon the basis of equation
Ul =1
Ωl
∫∫∫
Ωl
rldv =1
Ωl
∫∫∫
Ωl
(rl1 + ul
1)dv. (1.13)
The eventual equation describing the motion of centroids will have the followingform.
M l d2Ul
dt2= Rl, Rl =
∫∫©Γ l
σ
tΓldS +
∫∫©Γ l
c
pcldS +
∫∫∫
Ωl
bldv, (1.14)
where M l = ρlΩl, (Ωl = const).
6 1 Introduction
The equation governing angular momentum variation possesses the followingform.
ρl d
dt
∫∫∫
Ωl
(rl × ∂ul
1
∂t
)dv
=
∫∫©Γ l
σ
(rl × tΓl)dS +
∫∫©Γ l
c
(rl × pcl)dS +
∫∫∫
Ωl
(rl × bl)dv. (1.15)
In accordance with the study [Ulitko (1990)] we assume that the motion of anelastic body ul
1 consists of (as for the perfectly rigid body) the motion of the cen-troid and the rotation around it and of the relative motion. After the introduction ofthe system of movable coordinates X, Y, Z (Fig. 1.2), next to the initial immovablesystem of coordinates, the general vector of displacement has the following form,
ul1 = Ul + rl + ul − rl
1, (1.16)
where Ul = O1O is the displacement of centroids (l = 1, 2); rl is the vector ofthe point in the system of movable coordinates that had the same coordinates inthe initial immovable system (rl = xl
jelj , el
1 = il, el2 = jl, el
3 = kl); and ul isthe vector of displacement that takes the deviation of the points of elastic bodiesfrom the perfectly rigid bodies into account.
Orientation of the movable coordinates X, Y, Z regarding the fixed onesX1, Y1, Z1 is defined by the following equations,
il = i1cosαl1 + i2cosβl
1 + i3cos γl1,
jl = i1cosαl2 + i2cosβl
2 + i3cos γl2,
kl = i1cosαl3 + i2cosβl
3 + i3cos γl3.
Taking into account vector (1.16) and the Duhamel–Neuman relations, the equa-tions of motion have the following form [Ulitko (1990)],
21 − νl
1 − 2νlgrad divul − rot rotul +
1
μlbl =
γl
μlgradTl +
ρl
μl
∂2ul1
∂t2, (1.17)
where the differentiation in the left-hand side of the equation of the vector ofrelative displacement ul = ul
1il +ul
2jl +ul
3kl takes place in the system of movable
coordinates (il, jl,kl). The right-hand side of this equation includes the total dis-placement expressed by the vector of general displacement ul
1. While introducingthe vector ul to the right-hand side of the equation we assume that the massmoments of inertia for elastic bodies are equal to the mass moments of inertia ofperfectly rigid bodies. In addition, we assume that in initial state the fixed Cartesiancoordinates coincide with the central axes of an ellipsoid of inertia. Eventually,we obtain:
1.1 Object of the study 7
21 − νl
1 − 2νlgrad divul − rot rotul +
1
μl
(bl − γlgradTl −
ρl
MlRl
)
=ρl
μl[ul + (ωωωl × rl) + ωωωl × (ωωωl × rl) + 2(ωωωl × ul)], (1.18)
∇2Tl =1
al
(∂Tl
∂t+ UlgradTl
), l = 1, 2, (1.19)
where ωωωl = plil + qljl + rlkl is the vector of instantaneous angular velocitydescribed by the following Euler equations,
(Al ˙pl + (Cl − Bl)qlrl, Bl ˙ql + (Al − Cl)rlpl, Cl ˙rl + (Bl − Al)plql)
+ 2
∫∫∫
Ωl
ρl(rl × (ωωωl × ul))dv +
∫∫∫
Ωl
ρl(rl × ul)dv
=
∫∫©Γ l
σ
(rl × tΓl)dS +
∫∫©Γ l
c
(rl × pcl)dS +
∫∫∫
Ωl
(rl × bl)dv, (1.20)
whereas Al, Bl, Cl are mass moments of inertia of the bodies (l = 1, 2)
Al =
∫∫∫
Ωl
ρl((xl2)
2 + (xl3)
2)dv,
Bl =
∫∫∫
Ωl
ρl((xl3)
2 + (xl1)
2)dv,
Cl =
∫∫∫
Ωl
ρl((xl1)
2 + (xl2)
2)dv.
It is clear that Equations (1.20) are related to Equations (1.18) by the momentsof Coriolis forces and inertial forces of relative motion.
Analysis of a stress–strain body state and its 3D motion is reduced to integrationof Equation (1.18) and Euler’s Equations (1.20) including the following relations,
pl = jl · kl = −kl · jl
= −αl2sin αl
2cosαl3 − βl
2sin βl2cosβl
3 − γl2sin γl
2cos γl3
= +αl3sin αl
3cosαl2 + βl
3sin βl3cosβl
2 + γl3sin γl
3cos γl2,
ql = kl · il = −il · kl
8 1 Introduction
= −αl3sin αl
3cosαl1 − βl
3sin βl3cosβl
1 − γl3sin γl
3cos γl1
= +αl1sin αl
1cosαl3 + βl
1sin βl1cosβl
3 + γl1sin γl
1cos γl3,
rl = il · jl = −jl · il
= −αl1sin αl
1cosαl2 − βl
1sin βl1cosβl
2 − γl1sin γl
1cos γl2
= +αl2sin αl
2cosαl1 + βl
2sin βl2cosβl
1 + γl2sin γl
2cos γl1.
In order to determine stresses in a body (movable) coordinates, the followingDuhamel–Neumann equations are used
σl = 2μlel + (λlθl − γlTl)I , (σlij = (λlθl − γlTl)δij + 2μlel
ij), (1.21)
where θl = el11 +el
22 +el33, (l = 1, 2); σl = σl
jkelj ⊗el
k, el1 = il, el
2 = jl, el3 = kl.
The symmetrical tensor (el = elT , eljk = el
kj ) of the strain el = eljke
lj ⊗ el
k,(l = 1, 2) in the body coordinates has the following components
eljk =
1
2
(∂ul
j
∂xk+
∂ulk
∂xj
), l = 1, 2, i, j = 1, 2, 3. (1.22)
Taking mechanical uΓl, tΓl, bl and thermal loads T Γl, qΓl, Ql (l = 1, 2)changing very (sufficiently) slowly in time (time of the change of the load islonger than time bl/c12, where bl denotes the size of the body l) allows us to usethe quasi-static equations of thermoelasticity (the inertial elements in the equationsof motion or the operator μlc−2
l2 ∂2/∂t2 in (1.3) can be neglected). The disturbingpulse for which a quasi-static solution can be applied has been determined in thework [Awrejcewicz, Pyryev (2003b)] on the basis of a classical Lamb problem[Awrejcewicz, Pyryev (2003a)].
It follows that the theoretical analysis of contact dynamics can be fully imple-mented in some simple cases. It takes place when displacement of the contactingbodies can be reduced either to rotation around an axis (Chapter 2) or to themotion along one axis (Chapters 3–5) and assuming that the studied system char-acteristics do not depend on the mentioned axes. For the material systems analyzedin this study, the mathematical models with one and two degrees-of-freedom areconstructed. Equations (1.14) and (1.20) are expressed in a dimensionless form
ϕ + 2hϕ + Φ(ϕ) = ε[ζ cos(ω0τ) + F (vr)p(θ, uw, τ)], (1.23)
φ = aM [M(τ) − F (vr)p(θ, uw, τ)], (1.24)
Φ(ϕ) = −aϕ + bϕ3, vr = φ − ϕ − ζksin(ω0τ), a ∈ ℜ1, (1.25)
where φ, ϕ are dimensionless displacements of each body; M(τ) is a mechani-cal load that drives one body; ζ and ζk are the amplitudes of both mechanical
1.1 Object of the study 9
and kinematic perturbations of the other body; F (vr) is a coefficient of kineticfriction; vr is the velocity of slipping of the bodies in contact; Φ(ϕ) is a nonlinearconservative force; and p(θ, uw, τ) is contact pressure that depends on contacttemperature, wear, and time. In the case φ = const = V we deal with the motionof one body with the given velocity (Equation (1.23)). In this case, the analysedsystems are self-excited ones, which means that in some areas of the plane (ϕ, ϕ)the following condition will be satisfied: εF (vr)pvr − 2hv2
r < 0.The numerical analysis of the problem has been carried out using the Runge–
Kutta method for Equations (1.23) and (1.24) and the quadrature method forintegral equations regarding contact pressure computations as well as the use ofiterative methods. The numerical computations support in full the conclusionsobtained on the basis of our theoretical study.
For each of the analysed bodies (during the analysis of classic problems), firstwe solve the equations of thermal conduction, which allows us to determine therange of temperature. Next, differential equations are solved for the known loadof body forces, γl∂/∂xn being its elements. Furthermore, it has to be empha-sised that during the analysis of contact problems, contact conditions must bemodelled.
1.1.2 Boundary and contact conditions
The stress vector pNl operating on the surface element dΓ l with a normalNl(N l
1, Nl2, N
l3) is determined in the following way: pNl = Nl · σl (pNl
j ≡ N liσ
lij )
(Fig. 1.1), assuming that Nc ≈ N1c ≈ −N2
c leads to a relation N1c · σ1 =
−N2c · σ2 = −pc. Furthermore, we also introduce the vector of a relative dis-
placement of bodies in the form of w = u2 − u1.Let us resolve the stress factor pc into the component −pNc in a normal
direction and the tangent component pτ in a plane dΓc (Fig. 1.3a): p = −pc ·Nc,pτ = pc+pNc = (I−Nc⊗Nc)p
c, pτ =√
(pc)2 − (p)2 (p, normal pressure; pτ ,tangent stress). Let us resolve the vector of displacement ul into the componentul
NNc in a normal direction and the tangent component ulτ in a plane dΓc: ul
N =
ul · Nc, ulτ = ul − ul
NNc = (I − Nc ⊗ Nc)ul. Furthermore, let us resolve
the vector of relative displacement w into the component wNNc in a normaldirection and a tangent component wτ (tangent slip) in a plane dΓc: wN = w ·Nc,wτ = w − wNNc = (I − Nc ⊗ Nc)w.
Classical mechanical boundary conditions have the following form.
ul = uΓl, (ulj = uΓl
j ), x ∈ Γ lu, (1.26)
pNl ≡ Nl · σl = tΓl, (pNlj ≡ N l
iσij = tΓlj ), x ∈ Γ l
σ, l = 1, 2, (1.27)
whereas classical temperature boundary conditions are governed by the followingequations.
10 1 Introduction
Fig. 1.3: The distribution of a stress vector pc into components (a), distance definition g (b),
and the definition of proximity of the bodies in contact gp (c).
Tl = T Γl, x ∈ Γ lT , (1.28)
λl∂Tl/∂N l = qΓl, x ∈ Γq, (1.29)
λl∂Tl/∂N l = αlT (Tsm − Tl), x ∈ ΓTq, l = 1, 2, (1.30)
where Tsm is the temperature of the medium contacting the body with the sur-face Γ l
Tq; ulj are the components of the vector of displacement; uΓl is a given
displacement for Γ lu; N l
j are the components of a unit vector to ∂Ωl; and tΓlj are
the components of the external load given for Γ lσ.
Let us introduce the notion of a perfect contact of two bodies 1 and 2 byassuming that walls of the bodies in contact have the same temperatures anddensities of the heat flow that crosses those walls in a given point of their contact,and that the equality of displacements and stresses is binding. It is expressed bythe following relations
T1 = T2, x ∈ Γc, q1 − q2 = 0 or λ1∂T1
∂Nc= λ2
∂T2
∂Nc, x ∈ Γc,
(1.31)
wN = 0, wτ = 0, x ∈ Γc, (1.32)
p1N = p2
N , p1τ = p2
τ , x ∈ Γc, (1.33)
where ql = λl∂Tl/∂Nc.
1.1 Object of the study 11
In fact, a set of phenomena called the external friction occurs between thebodies in contact.
Friction processes occurring in the joints of machine elements have great effecton the operating costs and durability of machines. Although a lot of effort hasalready been put into the research of friction (see, for instance, the review [Oden,Martins (1985), Ibrahim (1994)] and a monograph [Kragelsky, Shchedrov (1956)]),and a lot of practical knowledge on this topic has been gained, a homogeneous andgeneral theory of friction so far has not been formulated. We lack proper physicalmodels sufficiently explaining complex processes of friction, and we do not havemathematical relations useful in solving complex problems of modern engineeringin machine construction and operation.
Another separate and equally important phenomenon is the interaction of thefrictional surfaces with their surroundings. For high velocities, oxidation of the sub-surface layer takes place as a result of diffusion of atmospheric oxygen to metals,and then the compounds of oxygen and metal are formed in this layer. The intensityof the ionic diffusion of the molecules of the environment to the frictional bodydepends mainly on the temperature and strains. Various processes of mechanicthermodiffusion were described in the monographs [Nowacki, Olesiak (1991)] and[Pyryev (1999)]. The axiosymmetrical problems for the cylinder with its generalmechanical, temperature, and diffusional load were studied in the work [Olesiak,Pyryev (1997)], and, with the processes of ion exchange taken into account, in thework [Mokrik, Pyryev (1993)].
The surfaces of real bodies always have the microgeometrical condition ofirregularity (roughness, corrugation, inexact shape), which results in the geometri-cal surface of contact of such bodies decomposed into many small areas of contactand clearances. The notions of geometrical, real, and elementary contact surfaceare introduced (see, for instance [Solski, Ziemba (1965), Chichinadze et al. (1979),Lawrowski (1993)]). The systems with frictional centres with the corrugated con-tact surface were analysed in the monographs [Shtaerman (1949), Jonson (1985)]and in the works [Varadi et al. (2000), Pauk, Woźniak (1999)].
In order to analyse the characteristics of real surfaces of the bodies in con-tact, we apply a phenomenological approach that assumes the introduction ofthe so-called ‘third body’ [Aleksandrov, Annakulova (1992), Ganghoffer, Schultz(1995), Zmitrowicz (2001), Dragon-Louiset (2001)]. The macroscopic propertiesof this body include all possible microscopic phenomena that occur during the con-tact of a surface with microirregularities. The ‘third body’ is treated as a whole,without analysing its internal structure. Therefore, the mathematical descriptionis simplified and the status of the system can be determined with relatively fewparameters.
Friction, as a resisting force, has a great influence on the dynamics of feedmotion and on the precision of the positioning of many operating units responsiblefor the accuracy of shape, size, and the quality of obtained surfaces.
Model of a friction force allows us to determine this force during the macro-scopic slip of one of the analysed surfaces along the other one. In this caseFfr ≡ pτ = fp, where Ffr, p denote, respectively, a tangent and a normal
12 1 Introduction
Fig. 1.4: Models of friction forces used in the literature for modelling the stick-slip phenomenon:1, [Blok (1940)]; 2, [Bell, Burdekin (1969)]; 3, [Chin, Chen (1993)]; 4, [Awrejcewicz, Pyryev(2002)]; 5, [Brockley et al. (1967)]; 6, [Bell, Burdekin (1969)]; 7, [Banerjee (1968)]; 8, [Panovko(1980)]; 9, [Capone et al. (1993)]; 10, [Derjagin et al. (1957)] (numbers relate to the referencesgiven in the text).
component of the contact stress, and f is the coefficient of kinetic friction. In ageneral case [e.g., Chichinadze et al. (1979), Kragelsky, Gitis (1987), Lawrowski(1993)] f depends on the contact pressure and the slip velocity (but it does notdepend on the velocity when the frictional surfaces are already sufficiently groundin) and on the temperature of a contact surface; that is, it is the function:
f = f(p, wτ , T ). (1.34)
In the monographs [Blau (1996), Kragelsky et al. (1982), Kragelsky, Gitis(1987), Chichinadze et al. (1979)] and in the works [Martins et al. (1990)], manyphenomena of dry friction are studied, and the computation methods are presentedfor the estimation of such friction. The results of experimental research of therelation between the friction coefficient and the slip velocity are also presentedthere. It was discovered that this relation can have various patterns, that is, withits minima and maxima, and with a monotonic drop of value. Also, the patternswith a constant value of a frictional coefficient for a changing slip velocity areprovided (Fig. 1.4).
In the monograph [Blau (1996)] a multidisciplinary approach to static andkinetic friction, both with and without lubrication, and reviews of the conven-tional and novel methods used to measure friction are addressed. The elementaryproblems found in the mechanics of sliding objects and machine components, andthe effects of contact pressure, sliding speed, surface roughness, humidity, andtemperature on friction are discussed.
The survey [Awrejcewicz, Olejnik (2005)] is devoted to a significant role of var-ious dry friction laws in engineering sciences. Both advantages and disadvantages
1.1 Object of the study 13
of a frictional process are illustrated and discussed, but excluding the nature offriction. It is shown how the classic friction laws and modern friction theoriesnowadays exist in pure and applied sciences.
In the works [Blok (1940), Brockley et al. (1967), Halling (1975), Cockerham,Cole (1976), You, Hsia (1995)] the linear models of friction were used (curves 1in Fig. 1.4); in the works [Banerjee (1968), Kauderer (1958)] the parabolic modelwas applied (curve 7 in Fig. 1.4).
The pattern of friction behaviour in the case of oiled surfaces in the function ofslipping velocity and in stationary conditions is best described by the friction char-acteristic called the Stribeck curve. For small slipping velocities, the friction forceis determined mainly by mechanical, structural, and physical–chemical propertiesof the material of frictional surfaces (dry friction). For larger slipping velocities,grease microclines occur, and the resistance of combined hybrid friction decreases.For the further increase of slipping velocity, the total separation of rubbing sur-faces takes place. Only the liquid friction remains, and its value increases togetherwith the increase of velocity. In this study, we consider only dry or semi-dry fric-tion. The Stribeck curve (curve 4 in Fig. 1.4) has already been used in the works[Awrejcewicz, Pyryev (2002) (2003c) (2004a), Pyryev (2004)] to which the au-thors contributed. The simplified Stribeck curves (curves 3, 9 in Fig. 1.4) are usedin the works [Capone et al. (1993), Chin, Chen (1993)]. The model of frictionforce of the curve 6 presented in Fig. 1.2 is analysed in the work [Van De Velde,De Beats (1998), Cockerham, Symmons (1976), Bo, Pavelesku (1976)]. In thestudy [Andrzejewski, Awrejcewicz (2005)], the “Magnum” model (based on thephenomenon of friction between the road surface and a tyre) is applied for thesimulation of the dynamics of wheeled vehicles. This model embraces interactionsbetween the forces of adhesive, Coloumb, and viscotic friction (adhesive force),and the velocity of a wheel slip. They are presented graphically as a so-calledStribeck curve (curve 4 in Fig. 1.4). For some road surfaces friction modellingwith the use of curve 7 (Fig. 1.4) is suggested.
The above-mentioned relation, in accordance with [Moore (1975), Kragelsky,Shchedrov (1956)], has the following form,
f = sgn(wτ )((a + b|wτ |)exp(−c|wτ |) + d). (1.35)
The relation approximating curve 7, according to [Kauderer (1958)], has theform of
f = sgn(wτ )
(fs − fmin
1 + c|wτ |+ fmin
), (1.36)
where a, b, c, d, fs, fmin are the coefficients with constant values.In order to emphasise the nonreflectivity of friction forces [e.g., Stefański et al.
(2003)], the model of dry friction force that depends on acceleration wτ is studied.The relation of the function f = f(T ) to the contact temperature is used inthe works [Chichinadze et al. (1979), Chichinadze (1995)]. The kinetic frictioncoefficient can also be presented [Sadowski (1999)] as the sum of coefficients oftemperature friction and mechanical friction.
14 1 Introduction
Model of contact rigidity. This generally characterizes the contact of tworubbing surfaces (interference, negative allowance, tension, clamp, and normalcontact strains) gp of the bodies in contact (Fig. 1.3c) in relation to the contactload and contact pressure, contact temperature, and time; that is,
gp = εN (pτ , p, T, t). (1.37)
For instance, in the work [Shtaerman (1949)], a Winkler model is appliedgp = kNp, where a constant kN is a parameter determined experimentally. In thework [Pyryev, Mandzyk (1996)], during the analysis of the problem of a frictionalcontact of two moving thermoelastic cylinders, this model is applied in order todescribe their contact with their containment. Sometimes, the models gp = kNpα
[Kragelsky et al. (1982), Hess, Soom (1991)] are also used. In the work [Zajtsev,Shchavelin (1989)], in order to take roughness into account, contact rigidity ismodelled by the equation gp = α(pτ + β)−1 − γ. In the study [Martins et al.(1990)] p = cngmn
p + bnglnp gp is used, where cn, bn, mn, ln, α, β, γ are con-
stant parameters dependent on the type of materials and on processing of rubbingsurfaces. It can be assumed on the basis of the works [Kudinov (1967)], and alsoon the basis of the studies of many other researchers [e.g., Hebda, Wachal (1980),Martins et al. (1990)], that in order to understand and describe the behaviour ofcombined friction, the model of a rubbing pair should be applied, with at leasttwo degrees-of-freedom, and with contact strains in a normal direction in rela-tion to the friction surface taken into consideration. In the case of the systems offeed motion of machine tools, the mathematical model of friction is applicable fordetermining the friction forces while positioning and starting, and for the studyof the conditions of the loss of motion stability and the occurrence of a stick-slipphenomenon.
Model of initial displacement. Initial displacements take place before the pro-cess of slipping with the increase of the contact load from zero to the staticfriction force. If a contact force pτ is smaller than the developed static frictionforce pτs = fsp between the bodies in contact under the contact load, the slipwill not occur. However, certain slight displacement of the bodies in the directionof the load operation will take place, and the system will take a different balanceposition. This phenomenon is a subject of research [Kragelsky et al. (1982)]. Thiswork does not take this phenomenon into consideration. Generally, initial displace-ments depend on the contact load, contact pressure, contact temperature, and time;that is,
ετ = ετ (pτ , p, T, t). (1.38)
Model of wear. Friction occurs on the nuclear level, but its effects are observedon the macroscopic one, as the changes of surface and the occurrence of separateparticles called the products of wear. The wear process is the process of a gradualchange in the body dimensions caused by the process of friction and manifestedby the separation of the material from the rubbing surface. Wear is the result ofthe wear process. While studying the process of attrition of machinery elementson the basis of the changes in the values of attrition occurring in time, the notions
1.1 Object of the study 15
of the measure of attrition and the intensity of this process are required. Thechange of weight, volume, or linear dimensions of the body is usually taken asthe measure of attrition (after some period of time of friction). In this work, thethickness Uw of the rubbed layer of material is the measure of attrition.
In the research of tribological wear of machinery, total wear Uw is the basis forestablishing the intensity of attrition Uw as the quantity that better characterizesthe process of occurrence of the products of wear.
The basic methods of research and problems of the theory of wear are describedin the works [Solski, Ziemba (1965), Chichinadze et al. (1979), (1995), Lawrowski(1993), Sadowski (1999)]. Friction and wear cause huge energy and material losses.Nevertheless, the general relation between the value of friction force and wear hasnot been determined thus far.
Empirical models leading to the fuller understanding of the processes are pre-sented in many of the works mentioned above; for example,
Uw = gw(p, wτ , T, H, t), (1.39)
where H is the hardness of the softer body. According to Archard [Archard (1953)],the model of wear has the following form,
Uw = Kwp|wτ |. (1.40)
The models of wear (1.40) are applied in several works [e.g., Stromberg et al.(1996)]. The model (1.40) is a special case of the model of wear
Uw = Kwpα|wτ |β . (1.41)
For the changing external load of a tribomechanical system delay may occur[Hebda, Chichinadze (1989)]. In works [Hebda, Chichinadze (1989), Yevtushenko,Pyryev (1998), (1999b)], models that account for the processes of inheriting andageing are applied:
Uw(t) = Kw|wτ |t∫
0
K1(τ)K2(t − τ)p(τ)dτ , (1.42)
where K1(t) = 1 + c exp(−γt), K2(t − τ) = 1 − exp(−γ′(t − τ)) are thenuclei of inheriting and memory. The case of the following model of wear(K1(t) = 1) is analysed in the work [Kuzmenko (1981)] and in the study[Yevtushenko, Pyryev (1999a)], where for the model of contact of the thermoelas-tic layer with a thermally insulated plate, the obtained solution allowed for the heatgeneration and wear (1.42). The evolution of contact pressure, contact tempera-ture, and wear is presented there. Also, the conditions of the areas of applicabilityof the analysed model are determined as well as conditions of the occurrence ofthe so-called frictional thermoelastic instability (FTEI).
16 1 Introduction
Using the models analysed above, we denote the following general contactconditions,
(i) Kinematic contact conditions on the surface:
g = u2N − u1
N + g0 + Uw > 0, pN1 = 0, pN2 = 0, x ∈ Γs,(1.43)
gp = εN (pτ , p, T, t), p ≥ 0, x ∈ Γsl ∪ Γst, (1.44)
(ii) Forcing contact conditions on the surface:
|pτ | < fs(p, T, t)p, wτ = u2τ − u1
τ = ετ (pτ , p, T, t), x ∈ Γst,(1.45)
|pτ | = f(p, wτ , T )p, wτ = −λpτ/|pτ |, x ∈ Γsl, (1.46)
(iii) Thermal contact conditions:
q1 + q2 = (T2 − T1)/R(p), q1 − q2 = (1 − η)f(p, wτ , T )wτp,
x ∈ Γsl, (1.47)
q1 + q2 = (T2 − T1)/R(p), q1 − q2 = 0, x ∈ Γst, (1.48)
q1 + q2 = (T2 − T1)/R(u2N − u1
N + g0 + Uw), q1 − q2 = 0, x ∈ Γs,(1.49)
where T = (T1 + T2)/2 denotes contact temperature; g = u2N − u1
N + g0 is aclearance (a gap, e.g., an air-gap), if g > 0; gp is the initial interference (Fig. 1.3c)when g = 0; g0 is the initial clearance, when g0 > 0 and it is the initial interfer-ence when g0 < 0; R(·) is heat resistance that depends on the contact pressure(x ∈ Γst) or on the clearance (x ∈ Γs); ul
N , ulτ denote the normal and the tan-
gent component of the displacements of the point Γ , respectively; wτ is a tangentslip; εN(pτ , p, T, t) is contact rigidity; f(p, wτ , T ) is a kinetic coefficient of fric-tion that generally depends on the contact pressure, relative velocity, and contacttemperature. Moreover, we assume that f(p,−wτ , T ) = −f(p, wτ , T ) and thenthe density of generated heat is described by the inequality f(p, wτ , T )wτ > 0for the arbitrary slipping direction. The density of the frictional energy stream isq(t) = f(p, wτ , T )wτp. Mathematical generalization of the function g is given inthe works [Wriggers (1995), Agelet de Saracibar et al. (1999)].
Let us derive the second Equation (1.47). We use the equations of energybalance formulated in relation to the area of the elementary infinitesimal surface ofmacroscopic contact dΓ of solid bodies and infinitesimal time dt of the followingform,
q + q0 = q1 − q2 + qw, (1.50)
1.1 Object of the study 17
where q = pτ wτ is the density of friction power [W · m−2]; ql = λl∂Tl/∂Nc;q = −λlgradTl is the density of the heat stream (l = 1, 2); qw = ηpτ wτ is thedensity of mechanical dissipation; η is equal to the relation q/qw; q0 is the densityof the power of the elastic energy gain of the system (for the mechanical andthermal loads changing sufficiently slowly we assume [Aleksandrov, Annakulova(1990)] that q0 = 0). Equation (1.47) is now easily obtained. Let us notice that theparameter η for the boundary condition has a different sign than in the case of theconditions obtained in the works [Stromberg et al. (1996)] (the case of q1 = 0) andit has the same sign as in the works [Aleksandrov, Annakulova (1992), Sadowski(1999)].
1.1.3 Dynamics of contacting bodies
The obtained dimensionless Equations (1.23), (1.24) are nonlinear models thatcan be encountered in virtually all fields of science, for example, in techno-logy [Kononienko (1964), Moon (1987), Schuster (1995)], biology [Riznichenko(2002)], sociology, and economy [Kapica et al. (2003)], or in the problems ofenvironmental protection of humans (the effect of vibrations and noise on a humanbeing) [Engel (1993)].
For the purpose of the analysis of systems with one or more degrees-of-freedom,asymptotical methods can be applied (method of a small parameter, the Krylov–Bogolubov–Mitropolsky method, equivalent linearization, and others), that arediscussed in monographs [Andronov et al. (1966), Osiński (1979), Nayfeh (1981),Awrejcewicz (1989), (1996)]. It turns out, however, that those methods cannot beapplied directly in solving the obtained Equations (1.23), (1.24).
As has already been mentioned, the material systems of bodies in con-tact can show self-excited vibrations. This type of vibration is discussed inmany monographs devoted to nonlinear vibrations [e.g., Schmidt, Tondl (1986),Kragelsky, Gitis (1987), Kurnik (1997), Giergiel (1990), Nayfeh, Balachandran(1995), Awrejcewicz (1996)] and in many other articles [e.g., Oden, Martins(1985), Awrejcewicz, Pyryev (2005), (2007)]. Self-excited vibrations belong tononextinguishing vibrations that are supported by external energy sources in anonlinear conservative system.
Self-excited stick-slip vibrations occur in many mass-elastic systems with slipfriction, both in technology and in everyday life. In the case of machines, self-excited vibrations usually have detrimental effects, as they can lead to the damageof the vibrating object. Here, the following phenomena can be enumerated: thevibrations resulting from the air flow around vibrating strands, rods, and layers(e.g., the wing flutter in a plane or the vibrations of conductors). Self-excitedvibrations caused by the frictional contact of the bodies moving in relation toeach other (e.g., in a slip bearing with no lubrication) occur frequently, as well asself-excited relaxational vibrations.
18 1 Introduction
The basic methods and problems of the analysis of self-excited vibrations arepresented in the monographs [Osiński (1979), Andronov et al. (1966), Kragelsky,Gitis (1987), Awrejcewicz (1996), Awrejcewicz et al. (1998), and others]. Thestudy [Tondl (1970), (1978)] shows that the introduction of additional dry frictionto the system produces a stable balance point. The larger is the area of attractionof this solution the greater is the input of dry friction, up to the moment whenself-excited vibrations extinguish.
For relative motion of rubbing bodies, the real velocity of slipping can changein steps within the scope of small velocities. This type of stroke–slip motion iscalled stick-slip. In the case of the tribomechanical systems characterized by thistype of motion, mechanical relaxational vibrations that can disturb machine workare observed. For instance, in the system cutter–holder–slide, or in the system:processed element–machine, those vibrations may significantly worsen the qualityof the processed surfaces. Such vibrations also disturb the work of a driver whileengaging the clutch on a car, as well as the work of slip units in measuring instru-ments, the vibrations of turbine blades, and the vibrations of suspension bridges.Depending on the characteristics of tribomechanical systems and on operatingconditions, relaxational vibrations may have various features, namely they mayoccur as sudden leaps with the frequency of 8–10 Hz (e.g., in a car clutch) up to4000–5000 Hz (e.g., in a braking system of a vehicle).
The models discussed can be useful in the analysis of the stick-slip phenomenonthat occurs in the power–feed systems of machine tools [Marchelek (1991)].
Stepwise movement is observed with friction in the case of relative slip andrelative rest, and it occurs as a result of self-excited vibrations with the reductionof the friction coefficient and the increase of slip velocity. The main source ofself-excited vibrations is the positive difference between the rest forces pτ0 (staticfriction) and slip (kinetic friction), and the value of friction force decreases whenrelative velocity increases (see Fig. 1.4).
Most of the theoretical works concerning this field deal with pure self-excitedvibrations caused by friction [Awrejcewicz et al. (1998), Awrejcewicz (1996),Kragelsky, Gitis (1987), Martins et al. (1990), Ibrahim (1994), Andronov et al.(1981)].
From the many studies are devoted to stepwise movement, let us enumeratehere only the surveys [Kragelsky, Shchedrov (1956), Ibrahim (1966)].
For small slip velocities, nonuniformity of motion is observed, namely periodi-cally changeable motion and rest (interrupts).
Motion of a system with two degrees-of-freedom that is in contact with amovable plane is analysed in the work [Richard, Detournay (2000)]. The mass ofa system may vibrate on the vertical spring fixed in the centre of the body thatslips horizontally. The friction coefficient is considered a constant quantity. In thiscase, stick-slip motion may occur.
Unstable feed motion of working units (stepwise movement occurring with lowrate of feed and fluctuations of slip velocity) is very troublesome for machinetool constructors, producers, and users. It is still much of a problem, because therequirements related to the precision of machines and their work are very strict.
1.1 Object of the study 19
Self-excited vibrations of an oscillator caused by dry friction are analysedin many works [e.g., Panovko (1980), Balandin (1993)]. In the work [Balandin(1993)], periodic self-excited vibrations of a body moving in the area of the clear-ance in fitting with perfectly rigid walls are studied (curve 2 in Fig. 1.4 denotesthe model of friction force). It is stated in this work that there are four typesof periodic motion possible: with one stroke without tacking during a period ofvibrations; with one stroke and tacking during a period; with two strokes withouttacking during a period; and with two strokes and tacking during a period.
The obtained Equations (1.23), (1.24) describe the models of material systemsin which the external mechanical and kinematic excitations are presented as aperiodic function whose amplitudes ζ and ζk are on the right-hand side of differ-ential equations (case φ = 0). There is, however, no influence of the system onthe source of vibrations. Such a model is called a system with a perfect sourceof energy (a perfect system). When there is an interaction in the system betweenthe source of energy and the vibrating object, it is called a system with nonper-
fect source of energy [Kononienko (1964)]. If we assume for the analysed modelsthat one of the bodies in contact is a source of vibrations for the other one, thenthe obtained system of Equations (1.23), (1.24) is the system with nonperfectsources of energy. Similar equations for vibrations of a self-excited system withdry friction are studied in the works mentioned above, with various characteristicsof the source of the system energy assumed (various M(φ)). Analytical researchwas conducted with the use of the method based on the asymptotical Krylov–Mitropolski–Bogolubov method [Bogolubov, Mitropolski (1961)]. The amplitude,phase, and frequency of vibrations are determined in a stationary state and a non-stationary one at the moment of resonance. The influence of self-excited vibrationsgenerated by dry friction and having a nonperfect source of energy is presented inthe work [Giergiel (1990)]. The model was comprised of a body placed on a beltdriven by an engine with nonperfect characteristics. Self-excited vibration occursonly for certain velocity values of a belt. The application of various analyticalmethods in the studies on the systems with nonperfect sources of energy is alsopresented in the work [Nayfeh, Mook (1979)].
The vibrations of a plate that rotates as a result of a transverse friction force(curve 4 in Fig. 1.4) are analysed in the work [Nakai (1998)]. In works [Hess,Soom (1991)], damped vibrations of a body with one degree-of-freedom on arigid plane (condition (1.44)) with harmonic perturbations are studied.
The model of an oscillator with friction (curve 1 in Fig. 1.4) for harmonicperturbation is considered in the work [Feeny, Moon (1993)].
In the studies by Warmiński [e.g., Warmiński et al. (2000)] regular and chaoticvibrations of parametric – self-excited systems with perfect and nonperfect sourcesof energy are described.
In the case of machine tools, self-excited vibration makes the work less pre-cise and deteriorates the quality of the obtained surfaces. Therefore, this kind ofvibration has been the object of interest for many researchers [Kauderer (1958),Tondl (1978), Kragelsky, Gitis (1987), Grudziński, Wedman (1998), Grudzińskiet al. (1995), Van De Velde, De Beats (1998), Rozman et al. (1996)].
20 1 Introduction
The stability of an elastic system with a frictional contact is expressed throughthe function of potential energy and dissipation [Mróz (2000)]. Static and dynamicforms of the loss of stability are also determined.
In the analysed models (Equations (1.23), (1.24) for a > 0) chaos mayalso occur. Nowadays, numerous studies are devoted to the problem of chaosin deterministic systems. For instance, one can find in the works [Awrejcewicz(1988), (1996), Awrejcewicz, Mrozowski (1989)] the examples of a self-excitedsystem of van der Pol–Duffing type, excited inertially, where static load is takeninto account.
Let us remember that an irregular and unpredictable evolution of many non-linear systems is called ‘chaos’. Despite this irregularity, chaotic dynamic systemssatisfy deterministic equations that are very sensitive to initial conditions. Forstudying chaotic systems, various analytical and numerical methods can be used[e.g., Awrejcewicz (1988), (1989), (1991), (1996), Awrejcewicz, Holicke (2007),Ott (1993), Baker, Gollub (1996), Nusse, Yorke (1994)]. The reasons for the occur-rence of chaotic vibrations are studied in the work [Awrejcewicz (1996)], wherean extensive bibliography of this field is also provided.
In the system described by Equations (1.23), (1.24) (φ = const), chaotic motionmay occur [Awrejcewicz, Pyryev (2003c)], and the set of parameters for which itoccurs can be determined with the use of the Melnikov method [Melnikov (1963),Guckenheimer, Holmes (1983), Awrejcewicz, Pyryev (2006a), (2006c)]. The basicidea of the Melnikov method is the application of an integrable undisturbed solution(ε = 0, h = 0) of the system of two equations for the solution of a disturbed systemof linear equations. In the works mentioned above, the Melnikov function is builtfor p = 1 for mechanical perturbations with the amplitude ζ and for kinematicalones with the amplitude ζk.
In the work [Litak et al. (1999)], the influence of self-excited and externallyexcited vibrations is studied with the example of the Froude pendulum excitedexternally. Critical values of amplitude and frequency of excitation for which thesystem passes to chaotic motion are determined numerically and on the basis ofMelnikov analytical criterion.
In the work [Awrejcewicz et al. (2002)], the analysis of vibrations of a spinningtriple physical pendulum, whose motion may be regular, chaotic, or hyperchaotic, ispresented. In the work [Pomeau, Manneville (1980)] three types of “intermittent”transition of the system into chaos are determined. The first type is associatedwith the saddle–node bifurcation, the second one related to the subcritical Hopfbifurcation, and the third type connected with doubling of a period.
In the work [Wagg (2003)], the dynamics of a system with one degree-of-freedom described by Equation (1.23) is considered, for Φ(x). Chaotic motion isanalysed, in the case when there is a nonlinear assymetrical conservative force.
A mechanical model of a rolling disc is presented in the framework of non-smooth dynamics and convex analysis [Le Saux et al. (2005)].
Examples of the analysis of regular and chaotic vibrations of self-excitedsystems externally excited are also presented in the works [Guckenheimer, Holmes(1983), Awrejcewicz, Mrozowski (1989), Pyryev (2004)]. The studies [Minorski
1.1 Object of the study 21
(1962), Tondl (1978), Szabelski (1984), (1991), Dao, Dinh (1999)] are devoted tothe analysis of the vibrations of parametric–self-excited systems.
In most of the theoretical works devoted to this issue, periodic, chaotic, orself-excited vibrations caused by friction are the subject of study. In this work, inaddition, tribological phenomena are also studied and modelled.
1.1.4 Contact thermoelasticity
The contact problems have been mainly introduced by the pioneering works [Hertz(1882, 1895)]. In his works, Hertz analyses the problem of pressing two evenelastic surfaces against each other. For the assumption that the main radii of thebodies’ curvature are large in comparison to the characteristic linear size of acontact pad, the problem is reduced to the one that describes the contact of twoelastic half-spaces, which enabled expressing analytically the contact pressure pand the boundary of a contact surface ∂Γc, that separates various types of boun-dary conditions (1.43), (1.44). More complex problems, including the change ofboundary conditions, require solving the integral Fredholm equations describingcontact pressure. In contrast to the mentioned problems, this study considers themodels including boundary conditions that do not change along the surface. Wedo not obtain a Fredholm integral equation describing contact pressure becausethe models are one-dimensional. The model is developed through the analysis ofnonstationary problems and with the variety of boundary conditions taken intoaccount.
Contact problems of the theory of elasticity and of thermoelasticity are thesubject of analysis in several monographs [e.g., Belajev (1945), Mindlin (1949),Shtaerman (1949), Timoshenko, Goodier (1951), Sneddon (1966), de Pater, Kalker(1975), Galin (1976), (1980), Johnson (1985), Aleksandrov, Romalis (1986),Goryacheva, Dobychin (1988), Alexandrov, Pozharskii (2001)].
Analysis of the contact problems including heat generation via friction has beencarried out in the references [Bowden, Ridler (1935), (1936), Blok (1937), Ling(1959),(1973)].
In most of the contact problems, force is used for clamping the bodies (firsttype of problems). In the case of one-dimensional problems, this approach resultsin the fact that constant pressure in the contact pad is known. In this study, theauthors consider contact problems when the close proximity of bodies is takeninto account (second type of problems). In the case of one-dimensional problemssuch perturbations make the pressure in a contact pad an unknown value. What ismore, in this case the problem of elasticity and thermal conduction are coupledby boundary conditions.
Reducing the issue to a one-dimensional problem enabled avoiding many diffi-culties, such as those related to the discussion of possible singularities occurring onthe boundary of the contact area ∂Γc [e.g., Galin (1961), Gladwell (1980), Gladwellet al. (1983)] or, for instance, to the paradox of a “cooled sphere” [Barber (1973),
22 1 Introduction
Kulchytsky-Zhyhailo, Olesiak (2000)]. This paradox occurs when heat flows fromthe body with a higher coefficient of thermal distortion δl = αl(1 + νl)/λl tothe body with a smaller coefficient of thermal distortion. The paradox is to alarge extent related to the assumption of the condition of a perfect thermal contactand a singular distribution of a heat flow. The derived conditions that guaranteethat the paradox of a ‘cooled cylinder’ does not occur are given in the works[Kulchytsky-Zhyhailo, Olesiak (2000)].
We believe that all such ‘paradoxes’ are the effects of imperfect modelling andassumptions resulting from the lack of possibilities of full and adequate modellingof the analysed process in order to obtain any conclusions. Let us remember thateven in the problems without the quality change of boundary conditions on theboundary ∂Γc, but with the bump movement of load (e.g., of the vertical one forthe horizontal one equal to zero), the singularity of shear stress may occur in thepoint of the bump movement of load [Timoshenko, Goodier (1951), Grinchenko,Ulitko (1999)]. In this book, attention is focused on the analysis of quasistationaryproblems, among others, and the thickness of the bodies in contact, their inertia,heat generation, and wear are taken into account.
In the works [Aleksandrov, Annakulova (1990), Zmitrowicz (1987), Moore(1975), Rabinowicz (1965)], the method of construction of general contact con-ditions is presented, and friction, wear, and heat generated by friction withinthe framework of thermomechanical theory are taken into account. In the work[Zmitrowicz (2001)], classical formulation of variational contact problems is gene-ralized for the case of wearing solid bodies, with the layer of wear moleculesbetween the bodies in contact taken into account.
Mechanical (1.43)–(1.46) and thermal (1.47)–(1.49) boundary conditions wereused by the author in the works [Olesiak, Pyryev (1996a), (1996b), Pyryev, Mokryk(1996), Pyryev (2001), (2002)] during the analysis of a one-dimensional problemof thermoelastic contact of bodies, with the relation between heat resistance andcontact pressure or clearance taken into account. It is shown that depending onthe parameters, problems may have one stable stationary solution, three station-ary solutions of which one is an unstable stationary solution, and one unstablestationary solution. In this last case, contact characteristics are characterized byself-excited vibrations [Pyryev (2001)]. Analogical boundary conditions are studiedin the works [Barber, Zhang (1988), Barber, Comninou (1989)].
The analysis of the influence of friction forces that are proportional to Hertzcontact pressure (curve 1 in Fig. 1.4) on the stresses in the case of a problem of asphere slipping along the surface of half space is provided in the work [Hamilton,Goodman (1966)].
The first axial–symmetrical contact problems of thermoelasticity, with heatgeneration taken into account, were solved in the works [Barber (1975), (1976),Generalov et al. (1976)]. In the work [Barber (1976)], one of the bodies slipsalong the surface of the other one with constant velocity. In the works [Barber(1975), Generalov et al. (1976)], one of the bodies rotates around the axis ofsymmetry with constant angular velocity. In both cases, force is used to pressthe bodies. These problems are analysed in more detail in many different works
1.1 Object of the study 23
[e.g., Pauk (1994), Yevtushenko, Kulchytsky-Zhyhailo (1995)]. The studiesmentioned above concern the problems with the assumption of constant velocity ofslipping. In the case of the second type of contact problems, the assumption of theconstant velocity of sliding results in the possibility of occurrence of frictional ther-moelastic instability (FTEI) [Burton et al. (1973), Barber (1976), Morov (1985),Pyryev (1994), (2001), Pyryev et al. (1995), Pyryev, Grilitskiy (1996), Pyryev,Mandzyk (1996), Yevtushenko, Pyryev (1997), Pyryev (2004), Ciavarella, Barber(2005)]. In contrast to these works, the author takes into account the inertia of thebodies in contact, which results in the variable value of the velocity of slipping.The velocity of slipping depends on the mechanical load. The system regulates theamount of generated heat on its own, and in this case FTEI does not occur.
The problems of ThermoElastic Instability (TEI) occurring as a result of vari-ous thermal distortions δl = αl(1 + νl)/λl of the bodies in contact and of therelation between thermal resistance and contact pressure (conditions (1.47)–(1.49)for wτ = 0) were considered by J. Barber in his works [e.g., Barber (1999)].In the studies in which Barber took part, the perturbation method was appliedto analyse the conditions of the occurrence of TEI for two thermoelastic halfspaces in contact, for the thermoelastic layer and half spaces, for two layers, twohalf infinite thin-walled cylinders whose endings are in contact, and the contactproblem of thermoelasticity concerning a long two-layered cylinder.
The problems of thermoelastic instability generated by the relation between thefriction coefficient and the velocity of slipping and contact temperature are studiedin the work [Maksimov (1988)].
The problems of instability of a friction contact are considered in the work [e.g.,Radi et al. (1999)]. In the works [Michalowski, Mróz (1978), Jarzebowski, Mróz(1994), Mróz, Stupkiewicz (1994), Radi et al. (1999)], the problems related to thefriction contact are analysed, without taking tribological processes into account.In the works of Sadowski [e.g., Sadowski (1999)] the thermodynamical nature offriction and of phenomena accompanying it is studied.
Observe that in references [Afferante et al. (2006), Afferrante, Ciavarella (2006),(2007), Yi (2006)] a new form of coupled instability, named Thermo-ElasticDynamic Instability (TEDI), which can occur by interaction between frictionalheating and the natural dynamic modes of sliding bodies, has been introduced.However, in the case of inertial terms neglect (see section 1.1.1), TEDI does notappear.
Modelling of mechanical contact with friction taken into account is conductedwith the use of the laws of thermodynamics of irreversible processes (first andsecond law of thermodynamics, energy balance) and the theory of plasticity inthe work [Klarbring (1990)], with heat generation taken into account in the work[Johansson, Klarbring (1993), Laursen (1999)], and with wear taken into accountin the work [Stromberg et al. (1996)] for small displacements. The example givenin [Stromberg et al. (1996)] concerns the contact of a material point with half-space(heat generation not taken into account). The works mentioned above are basedon the thermodynamic method discussed earlier in the works [Onsager (1931),Nowacki (1962), Ziegler (1963)]. Anisotropic models of friction and wear are
24 1 Introduction
obtained in the works [Mróz, Stupkiewicz (1994), Stromberg (1999)]. Zmitrowicz[Zmitrowicz (1999)] analyses anisotropic models of friction and heat flows. He alsopresents the problem of braking of a half-infinite rod on a thermal-insulated half-space.
The work [Bielski, Telega (2001)] describes the models of friction appliedin geophysics and proposes a new model of friction. In the work [Matysiak,Yevtushenko (2001)] a review of the problems of heat generation during friction ispresented. The discussed problems are divided into stationary, quasistationary, andnonstationary. In the works [Rojek et al. (2001)] a model of adhesion is appliedin the description of the bone–implant interphase. The influence of the productsof wear on the analysed interphase is studied, and the currently applied models ofwear are presented. The work [Stupkiewicz, Mróz (2001)] includes the analysis ofthe most important effects occurring in processes of deformation of fragile contactlayers for cycling loading. New constitutional models for cyclic states are proposed.
The work [Zboiński, Ostachowicz (2001)] presents the summary of the resultsof the three-dimensional analysis of contact in the points of attachment of mov-ing blades in turbo-machines in the elastic and plastic–elastic range. This workfocuses on the formulations of variational contact problems of elasticity andplastic–elasticity, and on their analysis with the use of a finite element method.
Many works [e.g., Willis (1966), Ling, Rice (1966), Conway et al. (1967),Kaczyński, Matysiak (1988), (1993), Matysiak et al. (2000)] are devoted to theproblems of the contact of anisotropic or nonhomogenous bodies. In the work[Ling, Rice (1966)], a two-dimensional quasi-established problem of heat conduc-tion concerning heating of half-spaces with a movable flow of heat is analysed, withthe relation of the coefficients of heat conduction and specific heat to temperature.
The analysis of the temperature on the surface of frictional contact is presentedin the works [Blok (1937), Jaeger (1942), Ling (1959), Kennedy, Ling (1981),Kennedy (1981), Barber, Comninou (1989)]. The temperature of a contact surfaceof the bodies is important in grinding, mechanical polishing, friction welding, andelectric contact, among others.
In the work [Stromberg (1999)], the problem in ℜ2 is analysed concerning thecontact of a thermoelastic plate (l = 2) and a thermally insulated rigid half space(l = 1) in a to-and-fro motion. A Coulomb friction model is assumed (curve 1 inFig. 1.4), as well as an Archard wear model (1.40). The condition of mechanicalcontact is (1.44) and for the initial clamp (g0 < 0). The dynamics of the bodiesin contact is not taken into account there. The problem is solved with the use ofa finite element method.
[Barber, Ciavarella (2000), Tichy, Meyer (2000), Klarbring (1986), Telega(1988)] are important reviews in this field.
The instability of frictional contact is analysed in the work [Radi et al. (1999)].Contact conditions are generalised with the use of the laws of the theory of plas-ticity in the works [Radi et al. (1999), Haraldsson, Wriggers (2000)].
Numerical modelling (a finite element method) of frictional contact is a sub-ject of many works, such as [Laursen, Simo (1993), Laursen, Oancea (1997),Haraldsson, Wriggers (1995), (2000), Wriggers (2002)], and the works [Agelet
1.1 Object of the study 25
de Saracibar, Chiumenti (1999), Gu, Shillor (2001)] take wear into account. Theanalysis of the works published during the last decade in the journal Computer
Methods in Applied Mechanics and Engineering shows that applying a finite ele-ment method in the analysis of contact problems is very useful, which proves thatthis issue is still important.
The authors of this study present a theoretical and numerical analysis of thedynamics of the bodies in contact, taking into account the influence of tribo-logical processes. The subject of this work is therefore mathematical modelling ofphenomena occurring on the contact surface of the bodies moving next to eachother, with frictional heat generation, wear, and the inertia of the bodies taken intoaccount, as well as the solution of certain problems of the mechanics of a solidbody that are related to this modelling.
1.1.5 On some equations governing discontinuous systems
dynamics
Discontinuous dynamical systems arise due to physical discontinuities such as dryfriction, impact, and backlash in mechanical systems or diode elements in electri-cal circuits. Many publications deal with discontinuous systems [Neimark (1978),Filippov (1988), Ibrahim (1994), Kunze (2000), Awrejcewicz and Lamarque(2002)].
Before proceeding we should clarify what we mean by the term nonsmoothsystem: (i) systems described by differential equations with a discontinuous right-hand side (also called Filippov systems [Filippov (1988)], or (ii) systems whichexpose discontinuities in the state, such as impacting systems with velocity rever-sals [Brogliato (1999)].
Because equations including friction modelling possess discontinuities, weconsider the main properties of such a system using a relatively simple Equa-tion (1.23). In what follows we consider the equation governing contact interac-tion with another body moving with the constant velocity ω1 of the followingform,
ϕ = −Φ(ϕ, ϕ, τ) + Ffr(ϕ, ωr, τ), (1.51)
where ωr = ω1 − ϕ(τ) is the relative velocity of the moving bodies, and func-tion Φ(ϕ, ϕ, τ) describes forces acting on the bodies excluding the friction forceFfr(ϕ, ωr, τ).
Owing to Amonton’s assumptions the friction force is defined in the followingway,
Ffr = Sgn(ωr)
⎧⎨⎩
F (|ωr|)p(τ), for ωr = 0,
Fs, for ωr = 0,(1.52)
where F (ωr) denotes the nondimensional kinetic friction coefficient (either of aforce or a moment of forces), Fs = fsp(τ), and fs denotes the static friction
26 1 Introduction
coefficient. Here and in the following we applied the notation F (0+) = fs
(classical Coulomb friction). We use the set-valued sign-function
Sgn(ωr) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
1, for ωr > 0,
[−1, 1], for ωr = 0,
−1, for ωr < 0,
(1.53)
which is set-valued at ωr = 0. The maximum static friction force is denoted byFs. Observe that in the case of zero value velocity of the contacting bodies thefriction force may take arbitrary values from the interval [−Fs, Fs]. The frictionforce magnitude depends on the system load, therefore the system dynamics ismodelled by the following differential inclusion [Filippov (1988), Leine et al.(2000), Kunze (2000), Van de Wouw, Leine (2004)]
ϕ ∈ −Φ(ϕ, ϕ, τ) + Ffr(ϕ, ωr, τ). (1.54)
Owing to Amonton’s assumptions the friction force associated with the consi-dered problem is defined in the following way (a signum model with static frictionpoint),
Ffr =
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
F (|ωr|)sgn(ωr)p(τ), for ωr = 0,
Φ(ϕ, ω1, τ), for ωr = 0, |Φ(ϕ, ω1, τ)| ≤ Fs,
Fssgn(Φ(ϕ, ω1, τ)), for ωr = 0, |Φ(ϕ, ω1, τ)| > Fs.
(1.55)
In this case the equation governing dynamics of the body with friction has thefollowing form (see, for instance, [Finigenko (2001)])
ϕ =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
−Φ(ϕ, ϕ, τ) + F (|ωr|)sgn(ωr)p(τ), for ωr = 0,
0, for ωr = 0, |Φ(ϕ, ω1, τ)| ≤ Fs,
−Φ(ϕ, ω1, τ) + Fssgn(Φ(ϕ, ω1, τ)), for ωr = 0, |Φ(ϕ, ω1, τ)| > Fs.(1.56)
Note that system dynamics is governed by three different equations. The firstand third of them change their form depending on the sign changes of ωr andΦ(ϕ, ω1, τ). The mentioned properties are generated by discontinuities of frictionforce (moment).
The second equation exhibits the case when for certain time instant ωr = 0 andalso for |Φ(ϕ, ω1, τ)| ≤ Fs, one gets ϕ(τ) = 0. The relative body motion vanishesand the body remains in relative rest. In the thus-far described case the actingforces cannot overcome the friction force. In the phase plane (ϕ, ϕ) points beingin relative rest lie on the straight line ϕ = ω1, which divides the phase plane intotwo parts. Points being in rest satisfy the condition |Φ(ϕ, ω1, τ)| ≤ Fs.
1.1 Object of the study 27
The third equation exhibits the case when for a certain time instant τ1 bothrelative velocity ωr = 0 and |Φ(ϕ, ω1, τ)| > Fs are satisfied; then ϕ = 0. The thirdequation yields sgn(ϕ(τ)) = −sgn(Φ(ϕ, ω1, τ)), and hence for τ > τ1 the rela-tive motion exists. The phase curve goes from one to another half-plane of thephase plane depending on sgn(Φ(ϕ, ω1, τ)). Friction in this case is changed by theamount of 2Fs.
Signum model. Friction force (1.55) can be also cast in the form
Ffr =
⎧⎨⎩
F (|ωr|)sgn(ωr)p(τ), for ωr = 0,
min(|Φ(ϕ, ω1, τ)|, Fs)sgn(Φ(ϕ, ω1, τ)), for ωr = 0.(1.57)
The constitutive relation for Ffr is known as the signum model with static frictionpoint. Friction model (1.57) should be understood such that a transition from stickto slip can take place only if |Φ(ϕ, ω1, τ)| exceeds Fs. The mentioned modellinghas been used in references [Leine et al. (2000), Leine, Campen (2002a), (2002b),Batako et al. (2003), Awrejcewicz, Pyryev (2006b)], among others.
The friction force (1.55) can also be described as a sum of static F stfr and
kinematic F dfr [Chin, Chen (1993), Tarng, Cheng (1995)] frictions of the form
Ffr(ϕ, ωr, τ) = F stfr(ϕ, τ) + F d
fr(ωr, τ), (1.58)
where the kinetic friction force F dfr has the following form.
F dfr(ωr, τ) = F (|ωr|)sgn(ωr)p(τ). (1.59)
The static friction force can be cast into the following form,
F stfr(ϕ, τ) = Φ(ϕ, ω1, τ)H+(Z)(1 − |sgn(ωr)|)
− Fssgn(Φ(ϕ, ω1, τ))H+(−Z)sgn(Z)(1 − |sgn(ωr)|), (1.60)
where
Z = Fs − |Φ(ϕ, ω1, τ)|, (1.61)
H+(Z) =
⎧⎨⎩
1, for Z ≥ 0,
0, for Z < 0,sgn(Z) =
⎧⎨⎩{Z/|Z|}, for Z = 0,
0, for Z = 0.(1.62)
The last applied mathematical modelling of friction force (1.60)–(1.62) has beenused in reference [Mostaghel (2005)], but the second term in (1.60) has been omit-ted. In reference [Mostaghel (1999)] the function H+(Z) has been approximatedvia function sgn(Z) through the formula
H+(Z) = 0.5(1 + sgn(Z))(2 − sgn(Z)). (1.63)
28 1 Introduction
Smoothing method. The friction curve is therefore often approximated by asmooth function. One possible approximation for sgn(ωr) is either
sgnε0(ωr) =
⎧⎪⎪⎨⎪⎪⎩
ωr
|ωr|, for |ωr| ≥ ε0,
(2 − |ωr|
ε0
)ωr
ε0, for |ωr| < ε0,
(1.64)
or
sgnε1(ωr) =
2
πarctan
(ωr
ε1
), εm ≪ 1, m = 0, 1. (1.65)
The switch model. The relative velocity ωr will most likely not be exactly zeroin digital computation. Instead, an adjoint switch model [Karnopp (1985), Leineet al. (1998), Andreaus, Casini (2002), Leine, Campen (2002a), (2002b)] is studiedwhich is discontinuous but yields a set of ordinary differential equations. The stateequation for the switch model reads
ϕ = −Φ(ϕ, ϕ, τ) + F (|ωr|)sgn(ωr)p(τ),
for |ωr| > εω, or |Φ(ϕ, ω1, τ)| > Fs, (1.66)
d
dτ
∥∥∥∥ϕϕ
∥∥∥∥ =
∥∥∥∥ω1
ωrωd
∥∥∥∥ ,
for |ωr| < εω, and |Φ(ϕ, ω1, τ)| < Fs, (1.67)
where ωd is the nondimensional free system frequency. The multiplier ωd deter-mines how ‘fast’ the solution is pushed to the centre. The choice of the multiplierωd is somewhat arbitrary [Leine et al. (1998)].
A region of near-zero velocity is defined as ωr < εω (εω ≪ 1). In reference[Awrejcewicz, Pyryev (2002)] ε0 = 10−4 has been applied, whereas in work [Leineet al. (1998)] the following values have been taken: ε1 = 10−6, εω = 10−6.
Continuous friction model. In two of the recent papers [Awrejcewicz et al.(2007), Pyryev at al. (2007)] devoted to mathematical modelling of dry frictionthe so-called ‘continuous friction model’ is proposed and studied using a 1-dofmodel with dry friction. The space (Φ, ωr) (Φ = Φ(ϕ, ϕ, τ)) is divided into thefollowing four regions.
V1 : |ωr| > εω,
V2 : [(0 ≤ ωr ≤ εω) ∩ (Φ > Fs)] ∪ [(−εω ≤ ωr ≤ 0) ∩ (Φ < −Fs)],
V3 : [(0 < ωr ≤ εω) ∩ (Φ < −Fs)] ∪ [(−εω ≤ ωr < 0) ∩ (Φ > Fs)],
V4 : (|ωr| ≤ εω) ∩ (|Φ| ≤ Fs).
1.1 Object of the study 29
Fig. 1.5: The space (Φ - ωr) divided into four regions: V1, V2, V3, and V4.
It is schematically shown in Fig. 1.5. The continuous friction force is defined inthe following way.
Ffr =
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
F (|ωr|) sgn(ωr)p(τ), for V1,
Fs sgn(Φ(ϕ, ϕ, τ)), for V2,
(2A3 − 1)Fs sgn(ωr), for V3,
A3(−Φ(ϕ, ϕ, τ) + Fs sgn(ωr)) + Φ(ϕ, ϕ, τ), for V4,
(1.68)
where
A3 =ω2
r
ε2ω
(3 − 2
|ωr|εω
)(1.69)
is the approximating function.In a continuous friction model the space (Φ, ωr) is divided into regions (switch
model). The friction force is a continuous function on ωr (smoothing methods)and for ωr = 0 the friction force is equal to the forces Φ(ϕ, ϕ, τ) acting on thebodies excluding friction force Ffr(ϕ, ωr, τ) (signum model). In another words,a continuous friction model can be treated as an approximating friction force inswitch instant by a smoothing function taking into consideration the forces actingon the bodies excluding friction force.
Some books deal with nonsmooth dynamics but include only mechanicalor mathematical aspects (see, e.g., [Brogliato (1999), Kunze (2000), Moreau,Panagiotopoulos (1988)]) and neither the physics of such models nor dynamicalbehaviour of the models involving chaos, for example. Some of them include ther-modynamics for nonsmooth systems (e.g., [Fremond (2002)]). Again the approachcan be oriented to modelling of nonsmooth processes mainly with convex or more
30 1 Introduction
sophisticated mathematical analysis. The study is devoted to different systems anddoes not present complex dynamical behaviours.
1.2. Aim and scope
The study comprises the formulation of the models of motion of material systemswith one and two degrees-of-freedom, with tribological processes taken intoaccount. The analysis of both regular and chaotic vibrations is also presentedin this book.
The following objectives are formulated in this monograph.
• Modelling the dynamics of contact systems with one and two degrees-of-freedom with friction, taking into account the inertia of the bodies, heatgenerated by friction, and wear, and the analysis of their influence on the solu-tions of certain problems
• Drawing up the methodology of a nonlinear problem of the motion of thermoe-lastic bodies in contact
• Examining the phenomena of stick-slip motion in self-excited systems, takingtribological processes into account, and determining the influence of the systemparameters on the character of regular vibrations, as well as determining thecritical values of parameters, for which a qualitative change in the system’sbehaviour occurs
• Showing that for external mechanical and harmonic kinematical excitations thesystem may pass to chaotic motion in certain circumstances
• Determining the conditions of the occurrence of frictional thermoelastic insta-bility
• The analysis of the kinetics of contact characteristics during starting, motion,and braking of the bodies in contact
For the determination of regular vibrations of systems, the analytical methodof perturbations is applied, whereas chaotic vibrations are analysed with the useof Melnikov and frequency spectrum methods. The stability of periodic solutionsis determined with the use of numerical methods.
The obtained analytical results are presented for the concrete numerical data,and then they are compared with the results of numerical simulation. Numeri-cal methods are the basis for determining the conditions of transition to chaoticvibrations and bifurcational values of parameters for which this transition takesplace.
Chapter 2
Thermoelastic Contact of Shaft and Bush
in Wear Regime
A classical problem concerning vibration of a friction pair consisting of a rotatingshaft and bush fixed to a frame by massless springs (a simple model of a typicalbraking pad or the so-called Pronny’s brake) has been investigated in references[Andronov et al. (1966), Neimark (1978)]. In works of [Pyryev et al. (1995),Pyryev, Grilitskiy (1995)], the so-called thermoelastic contact between a rotatingcylinder and a fixed noninertial pad has been studied. Further on, a more compli-cated axially symmetric problem of both regular and chaotic self-excited vibrations(caused by friction) and wear of the rotating cylinder and bush (fixed to a frameby springs and viscous damping elements) is investigated.
2.1. Analysed system
An elastic and heat-transferring cylinder (shaft) with radius R1 is inserted intothe bush (full bush or braking pad) with initial compression value of U0hU (t)(hU (t) → 1, t → ∞). The bush located on the shaft has internal radius R1 −U0 (U0/R1 ≪ 1), whereas the cylinder is compressed on the value of U0hU (t)measured in the radial direction.
The initial compression can be realized applying split-bearing bushing lying onthe shaft and initially compressed ([see, e.g. Andronov et al. (1966)]).
A scheme of the analysed system is shown in Fig. 2.1. The bush (pad) is fixedto the frame (housing or base) by springs. It is assumed that in this model the bushis an absolutely rigid body. Furthermore, radial springs are initially compressedand they have stiffness coefficients k1, whereas tangential springs are characterisedby nonlinear stiffness k2, k3 of Duffing type per unit bush (pad) length.
In the tangential direction, the bush (pad) is driven by the damping force mea-sured per bush length unit (c denotes the viscous damping coefficient) and forceF2 = F∗cos(ω′t) is also measured per bush length unit.
It is assumed that (i) the cylinder (shaft) rotates at an angular velocity ϕ1(t)such that centrifugal forces can be neglected in our system; (ii) the cylinder angular
31DOI: 10.1007/978-0-387-09653-7_2, © Springer Science+Business Media, LLC 2009 J. Awrejcewicz and Yu. Pyryev, Nonsmooth Dynamics of Contacting Thermoelastic Bodies,
32 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.1: The analysed system.
velocity ϕ1 = Ω∗ω1(t) (Ω∗, constant having units of angular velocity) is governedby the equation ω1 = ω∗ + ζksinω′t, where ζk is the dimensionless amplitude ofkinematic external excitation and it changes according to rotational motion of thecylinder driven by the moment M = M0hM (t) (hM (t) → 1, t → ∞) measuredper cylinder length unit; (iii) between the bush and cylinder dry friction (per lengthunit) defined by the function Ffr(Vr) occurs, where Vr is the relative velocity ofthe bush and cylinder Vr = ϕ1R1 − ϕ2R1. B1, B2 are the inertial mass momentsof the cylinder and bush also measured per length unit (cylinder, bush).
It is further assumed that the bush transfers heat ideally, and that at the initialtime instant the ambient temperature change is governed by T0hT (t), where T0
is the constant in temperature units, and hT (t) stands for a known dimensionlessfunction of time (hT (t) → 1, t → ∞), and that between the shaft and bushNewton’s heat exchange occurs. The shaft starts to expand, and a contact betweenthe shaft and bush is initiated. Another assumption made is that the dry frictionoccurring between the bush and shaft is defined by the function Ffr(Vr), whereVr = ϕ1R1 − ϕ2R1 is the relative velocity of the shaft and bush.
Owing to Amonton’s assumption, friction force Ffr is the product of normalreaction N(t) and the friction coefficient; that is, Ffr = f(Vr)N(t) is the frictionalforce defining resistance of the two bodies’ movement, and f(Vr) denotes thekinetic friction coefficient.
The so-called Stribeck curve [Kragelsky, Gitis (1987), Ibrahim (1994)] shown inFig. 2.2, has its minimum for Vr = Vmin and for Vr < Vmin we have (f ′(Vr) < 0).
2.2 Mathematical formulation of the problem 33
Fig. 2.2: Kinetic friction coefficient versus relative velocity.
The friction force Ffr yields heat generated by friction on the contact surfaceR = R1 and wear Uw.
Usually it is assumed [see, e.g., (1959)] that the frictional work is transformedto heat energy. However, in practice it means the occurrence of two heat streamsq1 and q2 moving into the insides of the contacting bodies. Let T1(r, t) denotecylinder temperature and let its initial value be Tsm. Furthermore, we assumethat the bush transfers heat perfectly, heat transfer occurring between the cylinderand bush is governed by Newton’s rule, and that temperature of the surroundingmedium changes according to the equation Tsm + T0hT (t).
The formulated problem is defined by the equations that govern dynamics of thebodies in the vicinity of the equilibrium configuration, with angular displacementsϕ1(t), ϕ2(t) and angular velocities ϕ2(t) for the bush and ϕ1(t) for the shaft, withstresses σR(R, t) in the shaft, contact pressure P (t) = N(t)/2πR1 = −σR(R1, t),temperature T1(R, t) of the shaft, and finally with displacement U(R, t) in thedirection along the R axis.
2.2. Mathematical formulation of the problem
2.2.1 Equations for rotational movement of absolutely rigid bush
Let axis Z be the cylinder axis. The equilibrium state of the moments of forceswith respect to the shaft axis yields
34 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.3: Geometry of spring configuration.
B2ϕ2 ∈ Mfr − Me − cR22ϕ2, (2.1)
where Mfr = f(R1(ϕ1−ϕ2))2πR21P (t) denotes the moment of friction force, and
Me = (F ′′−F ′−F2)R2 is the moment of elastic and external forces. In addition,the following geometrical relations hold (Fig. 2.3).
F ′′ = (k2∆l2 + k3(∆l2)3)cosβ2, F ′ = k1∆l1cosβ1.
According to results shown in the figure, one gets
l′′
sin(ϕ2 + Ψ)=
√R2
2 + l22sin(π/2 − β2)
,l′
sin ϕ2=
l1 + R2
sin(π/2 + β1),
and hence
cosβ2 = sin(ϕ2 + Ψ)
√R2
2 + l22l′′
=R2sin ϕ2 + l2cosϕ2
l′′,
cosβ1 =l1 + R2
l′sinϕ2.
Furthermore, we obtain:
l′′ =
√R2
2 + (R22 + l22) − 2R2
√R2
2 + l22cos(ϕ2 + ψ)
=√
2R22 + l22 − 2R2(R2cosϕ2 − l2sin ϕ2),
2.2 Mathematical formulation of the problem 35
l′ =√
R22 + (R2 + l1)2 − 2R2(R2 + l1)cos ϕ2.
Finally, we have
F ′′ = (k2∆l2 + k3(∆l2)3)(R2sinϕ2 + l2cosϕ2)/l′′, ∆l2 = l′′ − l2,
l′′ = l2√
2(R2/l2)2(1 − cosϕ2) + 1 + 2(R2/l2)sin ϕ2,
F ′ = k1∆l1sinϕ2(l1 + R2)/l′, ∆l1 = l0 − l′,
l′ =√
R22 + (R2 + l1)2 − 2R2(R2 + l1)cos ϕ2,
where l0 denotes the radial spring length without initial compression; l′ denotesthe length of the compressed radial spring during bush (pad) rotation with respectto angle ϕ2; l1 denotes the length of the compressed spring, where ϕ2 = 0; l0− l′
is the compression of radial spring length; l′′ is the tangential spring length relatedto bush (pad) rotation with respect to angle ϕ2; l2 is the tangential spring length,where ϕ2 = 0.
Keeping an accuracy of ϕ32, one gets:
sin ϕ2 = ϕ2 − (1/6)ϕ32 + O(ϕ4
2), cosϕ2 = 1 − (1/2)ϕ22 + O(ϕ4
2),
l′ = l1 + (1/2)R2(l1 + R2)/l1ϕ22 + O(ϕ4
2),
l′′ = l2 + R2ϕ2 − (1/6)R2ϕ32 + O(ϕ4
2),
F ′ = k1(l0/l1 − 1)(R2 + l1)ϕ2 + (1/6)(R2 + l1)(1 − (l0/l1)(1 + 3(R2/l1)
+ 3(R2/l1)2))k1ϕ
32 + O(ϕ4
2),
F ′′ = k2R2ϕ2 + (R32k3 − (2/3)R2k2)ϕ
32 + O(ϕ4
2).
Let the initial conditions be
ϕ2(0) = ϕ◦2, ϕ2(0) = ϕ◦
2, (2.2)
and let us introduce the dimensionless parameters
τ =t
t∗, p =
P
P∗, ϕ(τ) = ϕ2(t∗τ), φ(τ) = ϕ1(t∗τ), ϕ◦ = ϕ◦
2, ϕ◦ =ϕ◦
2
Ω∗,
ε =P∗t
2∗2πR2
1
B2, h =
cR22
2B2t∗, ζ =
R2F∗
2πR21P∗
, t2D =B2
k2R22
, χ =t2∗k∗R
22
B2=
t2∗k∗t2Dk2
,
tT = R21/a1, ω0 = ω′t∗, F (φ − ϕ) = f(V∗(φ − ϕ)), V∗ = R1/t∗, (2.3)
36 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
b =
(k3R
42 − 2
3k2R22 − (l1 + R2)R2
(1 − l0
l1
(1 + 3R2
l1+ 3
(R2
l1
)2))k1
6
)t2∗
B2,
a =k1
k∗
(l0l1
− 1
)(1 +
l1R2
)− k2
k∗, (2.4)
where τ is the dimensionless time, tD =√
B2/k2/R2 is the characteristic systemtime related to the bush oscillation period Tp = 2πtD, P∗ is the characteristiccontact pressure at shaft temperature T∗, and V∗ is the characteristic velocity ofthe system, Ω∗ = t−1
∗ .As exhibited by Equation (2.4), parameter a can take arbitrary values (a ∈ ℜ1)
depending on the values of spring parameters. For example, for stretched radialsprings l0/l1 > 1 parameter a > 0, if l0/l1 − 1 > k2k
−11 (1 + l1/R2)
−1. In thecase of nonstretched radial springs one always has a < 0.
Observe that introduction of a spring with stiffness coefficient k1 to our systemallows us to get an unstable system equilibrium which enables the occurrence ofchaotic dynamics. In the case of a lack of radial springs (k1 = 0) the nondi-mensional coefficient standing by ϕ equals a = −k2/k∗, whereas the coefficientstanding by ϕ3 is b = (k3R
42 − (2/3)k2R
22)t
2∗/B2. Introduction of the nonlinear
stiffness coefficient k3 into our system (the Duffing nonlinearity) makes it possi-ble: (i) to take into account positive and negative values of the parameter b; (ii) tointroduce either a stiff (k3 > 0) or weak (k3 < 0) spring characteristic; (iii) to takeinto account physical and geometrical nonlinearity.
The dimensionless equations (inclusion) governing the system dynamics havethe form
ϕ(τ) + 2hϕ(τ) − aχϕ(τ) + bϕ3(τ)
∈ ε[ζcos(ω0τ) + F (φ − ϕ)p(τ)], 0 < τ < ∞, (2.5)
with the initial conditions:
ϕ(0) = ϕ◦, ϕ(0) = ϕ◦ = ω◦. (2.6)
In order to solve the motion equations (inclusion) one needs to know ϕ(τ), φ(τ),and contact pressure p(τ). The latter can be obtained from the thermoelasticityequation that includes also tribological processes.
2.2.2 Thermoelastic shaft
In the analysed case, the inertial terms in the equation of motion can be omittedand the problem may be considered as a quasi-static one. For axially symmetricstress of the shaft, the equations used belong to the theory of thermal stresses for an
2.2 Mathematical formulation of the problem 37
isotropic body, formulated by Nowacki [Nowacki (1962)] and applying cylindricalcoordinates (R, φ, Z):
∂2U(R, t)
∂R2+
1
R
∂U(R, t)
∂R− 1
R2U(R, t) = α1
1 + ν1
1 − ν1
∂T1(R, t)
∂R, (2.7)
∂2T1(R, t)
∂R2+
1
R
∂T1(R, t)
∂R=
1
a1
∂T1(R, t)
∂t, 0 < R < R1, 0 < t < tc, (2.8)
with the attached mechanical
U(0, t) = 0, U(R1, t) = −U0hU (t) + Uw(t), 0 < t < tc, (2.9)
and thermal boundary conditions
λ1∂T1(R1, t)
∂R+ αT (T1(R1, t) − Tsm − T0hT (t)) = (1 − η)f(Vr)VrP (t),
(2.10)
R∂T1(R, t)
∂R
∣∣∣∣R→0
= 0, 0 < t < tc (2.11)
and with the following initial conditions
T1(R, 0) = Tsm, 0 < R < R1. (2.12)
Velocity of the bush (pad) wear is proportional to a certain power of fric-tion force. According to Archard’s assumption [Archard (1953), Aleksandrov,Annakulova (1990), (1992)] we have
Uw(t) = Kw|Vr(t)|P (t). (2.13)
Radial stress σR(R, t) in the cylinder may be found with the use of radialdisplacement U(R, t) and temperature T1(R, t) by the application of the followingformula
σR(R, t) =E1
1 − 2ν1
[1 − ν1
1 + ν1
∂U(R, t)
∂R+
ν1
1 + ν1
U(R, t)
R− α1(T1(R, t) − Tsm)
].
(2.14)The following notation is used: P (t) = N(t)/2πR1 = −σR(R1, t): contact pres-sure; Vr = ϕ1R1 − ϕ2R1, E1: Young’s modulus; ν1: Poisson’s ratio; a1: thermaldiffusivity, α1: thermal expansion coefficient; λ1: thermal conductivity; Kw: wearconstant, tc: time of contact; hT (t): ambient temperature.
Integrating Equation (2.7), with (2.9) and (2.14) taken into account, the contactpressure is determined:
38 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
P (t) =2E1α1
1 − 2ν1
1
R21
R1∫
0
[T1(ξ, t) − Tsm]ξdξ
+E1
(1 − 2ν1)(1 + ν1)R1[U0hU (t) − Uw(t)]. (2.15)
Let us introduce the following dimensionless parameters.
τc =tct∗
, r =R
R1, u =
U
U∗, uw =
Uw
U∗, θ =
T1 − Tsm
T∗, Bi =
αT R1
λ1,
hT (τ) = hT (t∗τ), hU (τ) = hU (t∗τ),
kw =P∗K
wR1
U∗, γ =
2(1 − η)E1α1R21
λ1(1 − 2ν1)t∗, ω =
t∗tT
, (2.16)
where
T∗ =U∗
2α1(1 + ν1)R1, P∗ =
E1U∗
(1 + ν1)(1 − 2ν1)R1=
2α1E1T∗
1 − 2ν1. (2.17)
Note, that physically P∗ denotes the contact pressure between the cylinder andbush with radius R1 during heating up of the cylinder to temperature T∗, or itdescribes the cylinder compression in the radial direction on the amount of U∗.
The thermoelastic problem under consideration takes the following dimension-less form.
∂2θ(r, τ)
∂r2+
1
r
∂θ(r, τ)
∂r=
1
ω
∂θ(r, τ)
∂τ, 0 < τ < τc, 0 < r < 1; (2.18)
∂θ(1, τ)
∂r+ Biθ(1, τ) = γF (φ(τ) − ϕ(τ))(φ(τ) − ϕ(τ))p(τ)
+BiT0
T∗hT (τ), 0 < τ < τc; (2.19)
r∂θ(r, τ)
∂r
∣∣∣∣r→0
= 0, 0 < τ < τc; (2.20)
θ(r, 0) = 0, 0 < r < 1. (2.21)
p(τ) =U0
U∗hU (τ) − uw(τ)+
1∫
0
θ(ξ, τ)ξdξ, 0 < τ < τc; (2.22)
2.2 Mathematical formulation of the problem 39
uw(τ) = kw
τ∫
0
|φ(τ) − ϕ(τ)|p(τ)dτ, 0 < τ < τc. (2.23)
The cylinder radial displacement is
u(r, τ)/r =1
2(1 − ν1)
⎡
⎣ 1
r2
r∫
0
θ(η, t)ηdη −1∫
0
θ(η, t)ηdη
⎤
⎦− U0
U∗hU (τ) + uw(τ).
Observe, that in order to solve the problem defined by Equations (2.18)–(2.23),one should know time-dependent velocities of the bush and shaft. Note also thatproblems (2.5), (2.6), and (2.18)–(2.23) are mutually adjoined and require simul-taneous solution.
2.2.3 Rotational motion of the shaft
Let axis Z coincide with the shaft axis. The moments of force related to the shaftaxis give
B1ϕ1 ∈ M − Mfr, (2.24)
where Mfr = f(Vr)2πR21P (t) denotes the moment of friction force, M is the
moment acting on the shaft, and ϕ1(t) stands for the angle of the shaft position.In order to solve Equation (2.24), the following initial conditions are attached.
ϕ1(0) = ϕ◦1, ϕ1(0) = ϕ◦
1. (2.25)
Introducing the dimensionless parameters
aM =t2∗P∗2πR2
1
B1, m0 =
M0
P∗2πR21
, φ◦ = ϕ◦1, φ◦ =
ϕ◦1
Ω∗, φ = ϕ1(t∗τ),
φ =dφ
dτ= ω1(τ), hM (τ) = hM (t∗τ), F (φ − ϕ) = f(V∗(φ − ϕ)); (2.26)
one gets the dimensionless equation governing the shaft dynamics:
φ(τ) ∈ aM
[m0hM (τ) − F (φ − ϕ)p(τ)
], 0 < τ < ∞, (2.27)
φ(0) = φ◦, φ(0) = φ◦ = ω◦1 . (2.28)
For computation purposes, the multivalued relation sgn(·) is approximated bythe function sgnε0
(·) defined by (1.64), where the regularisation parameter ε0 isa ‘small’ positive real number. The differential inclusions (2.5), (2.27) are thenapproximated by the system of equations.
40 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
It is seen that again contact pressure needs to be defined. It will be foundfollowing the solving procedure for the thermoelastic equation that takes intoaccount the tribologic processes. Note that problems (2.5)–(2.6), (2.18)–(2.23),(2.27), and (2.28) are coupled and require a common solution.
2.2.4 Laplace transform
Let us apply the Laplace transform to Equations (2.18)–(2.23).
{θ(r, s), p(s), uw(s), hU (s), q(s)}
=
∞∫
0
{θ(r, τ), p(τ), uw(τ), hU (τ), q(τ)}e−sτdτ,
where s is the transformation parameter. Taking into account the boundary (2.2.2),(2.20) and the initial conditions (2.21) we obtain
θ(r, s) = sGθ(r, s)q(s), p(s) =U0
U∗hU (s) − uw(s) + sGp(s)q(s),
Gθ(r, s) =I0(sωr)
ωs∆1(s), Gp(s) =
∆2(s)
ωs∆1(s),
∆2(s) =I1(sω)
sω, ∆1(s) = s2
ω∆2(s) + BiI0(sω), sω =
√s
ω, (2.29)
where In(x) = i−nJn(ix) is the modified first-order Bessel function with argu-ment x, and the nonlinear part of boundary problem (2.2.2) has the form
q(τ) = γωF (φ − ϕ)(φ − ϕ)p(τ) + BiωT0
T∗hT (τ). (2.30)
One of the commonly used methods to find inverse integrals of Laplace trans-formation applies the following relation,
{Gp(τ), Gθ(r, τ)} =1
2πi
i∞+cL∫
−i∞+cL
{Gp(s), Gθ(r, s)}esτds,
allowing for computation of the residual sum of function Gθ(r, s)esτ , Gp(s)e
sτ
in the complex plane to the left of straight line s = cL. Using (2.29), applying theinverse Laplace transform, and using the theorem of convolution [Carslaw, Jaeger(1959)], the following function is found.
2.3 Kinematic external shaft excitations 41
p(τ) =U0
U∗hU (τ) − uw(τ) + Biω
T0
T∗
τ∫
0
Gp(τ − ξ)hT (ξ)dξ
+ γω
τ∫
0
Gp(τ − ξ)F (φ − ϕ)p(ξ)(φ − ϕ)dξ, (2.31)
θ(r, τ) = BiωT0
T∗
τ∫
0
Gθ(r, τ − ξ)hT (ξ)dξ
+ γω
τ∫
0
Gθ(r, τ − ξ)F (φ − ϕ)p(ξ)(φ − ϕ)dξ, (2.32)
where
{Gp(τ), Gθ(1, τ)} ={0.5, 1}
Biω−
∞∑
m=1
{2Bi, 2μ2m}
μ2mω(Bi2 + μ2
m)e−µ2
mωτ , (2.33)
and μm are the roots of the characteristic equation (m = 1, 2, 3, . . . ),
Bi J0(μ) − μJ1(μ) = 0. (2.34)
Note that the investigated problem has been transformed to the set of nonlineardifferential Equations (2.5) and (2.27), integral equation (2.31) describing angularvelocities ϕ(τ), φ(τ), and contact pressure p(τ). Temperature is defined by (2.32)and (2.23).
Values of parameters t∗, U∗, k∗ are defined when considering particular casesof the studied problem.
2.3. Kinematic external shaft excitations
We consider first the case of kinematic shaft excitation; that is, we assume that bothrotation velocity ω1(τ) is known and the bush temperature is constant: hT (τ) = 0,hU (τ) = H(τ); H(τ) = 1, τ > 0; H(τ) = 0, τ < 0. The bush (or the pad)is attached to the housing (a frame or base) through springs with stiffness k2
(k1 = k3 = h = 0). Furthermore, it is assumed that U∗ = U0, any arbitraryparameter k∗ = k2 which gives constant a = −1 (see Equation (2.5)). In theconsidered case, Equations (2.27) and (2.28) are neglected.
42 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.4: Block diagram of the system.
2.3.1 Block diagram
The system shown in Fig. 2.1 is presented as a block diagram in Fig. 2.4 (p ≡d/dτ ). One may easily trace various relations and interactions between input andoutput parameters.
2.3.2 Stationary process associated with a constant shaft velocity
rotation
We consider the case when the shaft rotates at constant velocity ω1(τ) = φ(τ) =ω0
1H(τ). The lack of external excitation ζ = ζk = 0 and χ = 1 is assumed. (Thelatter relation yields t∗ =
√B2/(k2R2
2)). In this case t∗ has a physical sense.Namely, it is the period of free bush vibrations.
2.3 Kinematic external shaft excitations 43
Fig. 2.5: Block diagram of the linearized system.
2.3.2.1 Block diagram of the linearized system
The corresponding linearized system is shown in Fig. 2.5.Using the block diagram one can easily derive transmittances Gϕh(s), Gϕω(s),
Gph(s), Gpω(s), Gθh(s), Gθω(s) coupling the disturbance signals h∗U (t), ω∗
1(t)and input signals ϕ∗(t), p∗(t), θ∗(t).
2.3.2.2 Stationary process without wear
The stationary solution to the considered problem (in Equations (2.5) and (2.18)differential terms related to time or omitted) without wear (kw = 0) is governedby the equation
44 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
pst =1
1 − v, θst =
2v
1 − v, ϕst =
γ1ω◦1
1 − v, (2.35)
γ1 =εF (ω◦
1)
ω◦1
, v =γω◦
1F (ω◦1)
2Bi=
E1α1R21ω
◦1F (ω◦
1)
2Biλ1(1 − 2ν1).
The stationary solution (2.35) has a physical meaning (pst > 0), when theinequality v < 1 holds. Observe that the stationary solution (contrary to the func-tion describing a stationary solution) does not have a singularity. When v → 1,a trajectory (unsteady solution) may tend either to periodic motion or it mayincrease exponentially.
The parameter v, as shown in the following, will play an essential role whiledetermining Frictional Thermoelastic Instability (FTEI). It is proportional to theYoung modulus E1, shaft extension coefficient α1, relative velocity Ω∗ω
01R1, body
dimension R1, and friction coefficient F (ω01), but it decreases with an increase of
heat transfer coefficient λ1 and the Biot number Bi.Furthermore, we focus on the analysis of perturbations of the stationary solution
(2.35). Introducing perturbations hU (τ) = 1 + h∗U (τ), ω1(τ) = ω0
1 + ω∗1(τ),
|h∗U | ≪ 1, |ω∗
1 | ≪ 1, the solutions are given in the form
ϕ(r, τ) = ϕst +ϕ∗(τ), θ(r, τ) = θst(r)+ θ∗(r, τ), p(τ) = pst +p∗(τ), (2.36)
where |ϕ∗| ≪ 1, |θ∗| ≪ 1, |p∗| ≪ 1.First, the right-hand sides of Equations (2.5) and the boundary condition (2.2.2)
are linearized giving the equations
ϕ∗(t) + ϕ∗(t) = εF (ω◦1)p∗(τ) + εF ′(ω◦
1)pst(ω∗1 − ϕ∗), 0 < τ < ∞,
(2.37)
ϕ∗(0) = 0, ϕ∗(0) = 0; (2.38)
p∗(τ) = h∗U (τ) +
1∫
0
θ∗(ξ, τ)ξdξ, 0 < τ < ∞; (2.39)
∂2θ∗(r, τ)
∂r2+
1
r
∂θ∗(r, τ)
∂r=
1
ω
∂θ∗(r, τ)
∂τ, 0 < τ < ∞, 0 < r < 1;
(2.40)
∂θ∗(1, τ)
∂r+ Biθ∗(1, τ) = γ[ω◦
1F (ω◦1)p∗ + pst(ω
∗1 − ϕ∗)(F (ω◦
1) + ω◦1F ′(ω◦
1))],
(2.41)
r∂θ∗(r, τ)
∂r
∣∣∣∣r→0
= 0, 0 < τ < ∞; θ∗(r, 0) = 0, 0 < r < 1. (2.42)
2.3 Kinematic external shaft excitations 45
When the Laplace transform is applied to the linear system (2.37)–(2.42), thefollowing solution in the Laplace transform notation is obtained.
{θ∗(r, s), p∗(s), ϕ∗(s), h∗U (s), ω∗
1(s)} =
∞∫
0
{θ∗, p∗, ϕ∗, h∗U , ω∗
1}e−sτdτ.
This gives the solution in the Laplace transform domain
ϕ∗(s) = Gϕh(s)h∗U (s) + Gϕω(s)ω∗
1(s), (2.43)
p∗(s) = Gph(s)h∗U (s) + Gpω(s)ω∗
1(s), (2.44)
θ∗(r, s) = Gθh(r, s)h∗U (s) + Gθω(s)ω∗
1(s), (2.45)
where
{Gϕh(s), Gϕω(s), Gph(s), Gpω(s), Gθh(s), Gθω(s)} =L(s)
∆∗(s), (2.46)
L(s) = {F (ω◦1)∆1(s), pst[γF 2(ω◦
1)∆2(s) + F ′(ω◦1)∆1(s)], Ω2(s)∆1(s),
γε(s2 + 1)∆2(s)(F (ω◦1) + ω◦
1F ′(ω◦1))pst, I0(sω)γω◦
1F (ω◦1)Ω1(s),
γεI0(sω)(s2 + 1)pst(F (ω◦1) + ω◦
1F ′(ω◦1))}.
The characteristic equation of the linearized system is
∆∗(s) = 0, ∆∗(s) = ∆1(s)Ω2(s) − 2Bi v∆2(s)Ω1(s) (2.47)
Ω1(s) = s2 − γ1
1 − vs + 1, Ω2(s) = s2 +
γ2
1 − vs + 1, γ2 = εF ′(ω◦
1).
Observe that the roots sm (Res1 > Res2 > · · · > Resm > · · · , m = 1, 2, 3, . . . )of the characteristic equation (2.47) lie either on the left-hand plane (Res < 0 anda stationary solution is stable) or on the right-hand plane (Res > 0 and a stationarysolution is unstable) of the complex variable s. The parameters separating two half-planes are further referred to as critical ones. Let us analyse the stable stationarysolution. In this case, the characteristic function has the form
∆∗(s) =
∞∑
m=0
( s
ω
)m
dm, (2.48)
46 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.6: Critical values of v versus γ2 (a) and ω (b). (a) Curve 1 – ω = 0.05, curve 2 – ω = 0.1,curve 3 – ω = 1. (b) Curve 1 – γ2 = −0.05, curve 2 – γ2 = −0.08.
dm = (1 − δm1)ω2(d
(1)m−2 − 2Bi vd
(2)m−2)
+ω
1 − v(γ2d
(1)m−1 + 2Bi vγ1d
(2)m−1) + d(1)
m − 2Bi vd(2)m ,
m = 1, 2, . . . ; δmn = 1, m = n; δmn = 0, m = n;
d0 = Bi(1 − v), d(1)m =
Bi + 2m
22m(m!)2, d(2)
m =1
22m+1m!(1 + m)!.
Equation (2.47) has been studied numerically. Figure 2.6a shows the depen-dence of v on γ2 for given values of ω = 0.05; 0.1; 1 (curves 1–3). Figure 2.6billustrates the dependence v on ω for fixed values of γ2 = −0.05;−0.08(curves 1–2). In both cases, Bi = 10 and γ1 = 0.586. The parameters locatedinside the mentioned curves are associated with a stable stationary solution.Observe that decreasing γ2, a stable zone is narrowed owing to v, whereas itis extended owing to an increase of the critical value of ω.
When v = 0 and neglecting heat expansion of the cylinder, we obtain a modelof self-oscillations [Andronov et al. (1966)], with the characteristic equation of thelinearized problem Ω2(s) = 0. In this case, a steady-state solution is stable, whenγ2 > 0. When ω = 0 (the bush is immovable) we obtain a model [Pyryev et al.(1995)] with the characteristic equation ∆1(s) − 2Bi v∆2(s) = 0. In this case,when v > 1, a steady solution is unstable (root s1 > 0). Frictional thermal insta-bility [Pyryev, Grilitskiy (1995)] or thermal explosion [Aleksandrov, Annakulova(1990)] takes place.
The analysis of particular cases of the considered model shows that the steady-state solution is stable when γ2 > 0. This corresponds to Vr > Vmin (see Fig. 2.6).On the other hand, a steady solution is stable when v < 1. The specific parameters
2.3 Kinematic external shaft excitations 47
of the system, the view of Stribeck’s curve, and analysis of the characteristicequation roots (2.47) can help definitely to answer the question of stability of thesteady-state solution. If the steady-state solution (2.35) is unstable, then leavinga transient process it can reach either a stable limit cycle similar to frictionalself-oscillations or it can increase with time (it depends on other nonlinear terms).
2.3.2.3 Analysis of steady-state solution in the presence of wear
The steady-state solution of the problem in the case of the presence of wear(kw = 0) is found as (in Equations (2.5) and (2.18) the terms with time derivativesand uw(τ) = 0 are neglected)
pst = 0, θst = 0, ϕst = 0, uwst = 1. (2.49)
Performing a procedure similar to the previous case, the characteristic equationof the linearized problem is obtained in the vicinity of the steady solution (2.49):
∆∗(s) = 0, ∆∗(s) = (s + kwω◦1)∆1(s) − 2Bi vs∆2(s). (2.50)
To analyse the regions of parameters in which steady-state solution (2.49) is stable,a characteristic function is sought in the form
∆∗(s) =
∞∑
m=0
( s
ω
)m
bm, (2.51)
b0 = kd(1)0 , k = kwω◦
1 ,
bm = kd(1)m + ω(d
(1)m−1 − 2Bi vd
(2)m−1), m = 1, 2, . . . . (2.52)
First, a root of the characteristic equation for small wear k ≪ 1 can be presentedin the form
s1 = − k
1 − v. (2.53)
As can be seen from (2.53), for low wear when v < 1, the steady solution (2.49)is stable and in the opposite case, unstable. In the conditions of wear presencecontact time tc of the system is limited. The material of the bush will wear withtime. That is why we should accurately state the conditions of stability. Parametersof the problem under which roots s1, s2 of the characteristic equation have positivereal part and are complex conjugate, are related to intensive wear. Under theseparameters, the oscillation amplitude increases according to exponential law butthe system contact time is limited. The system will leave the contact faster thanthe contact characteristics will reach critical values (initial assumptions will losesense). Parameters of the problem under which the roots of the characteristicequation s1, s2 have only positive real part are referred to as the critical parameters.
48 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.7: Critical values v versus k for different ω (kw �= 0). Curves 1: ω = 0.05, 2: ω =0.1. Regions under solid curves are stable. Parameter regions between solid and dashed curvescorrespond to intensive wear.
Under these parameters there will be no oscillations but contact characteristics willrise according to the exponential law. The system will be overheated. The rate ofheat expansion is bigger than the wear rate. Figure 2.7 shows the dependence ofcritical value v (solid curves) on parameter k for different values of ω = 0.05, 0.1(curves 1 and 2, respectively).
Taking into account three terms in the decomposition (2.51) the formula forcritical values is derived (roots coincide on real axis Res1 = Res2 > 0, Ims1 =Ims2 = 0),
vcr ≈ 1 +k +
√2Bi kω(4 + Bi)
2Bi ω. (2.54)
Formula (2.54) yields better results for small values of the wear parameter k. Forthe parameters that are situated under the solid curve v < vcr, the system is stable.Moreover, between solid and dashed curves v0 < v < vcr, the time when thesystem is in contact is limited and intensive wear takes place. The formula for thedashed curve is also found (roots pass through an imaginary axis Res1 = Res2 = 0,Ims1 = −Ims2 = 0),
v0 = 1 +k(2 + Bi)
4Bi. (2.55)
Again formula (2.55) yields better results for small values of the wear para-meter k.
2.3.3 Numerical analysis of the transient solution
A numerical analysis of the problem is performed using the Runge–Kutta methodby taking into account the following asymptotes
2.3 Kinematic external shaft excitations 49
Gθ(1, τ) ≈√
1/πτω, Gp(τ) ≈ 1, τ → 0. (2.56)
Dependence of the kinematic friction coefficient on relative velocity (see Fig. 2.2)dependence is approximated by the formula
f(Vr) = Sgn(Vr)
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
fmin + (fs − fmin) exp(−b1|Vr|), for |Vr | < Vmin,
fs, for Vr = 0,
fmin + (fs − fmin) exp(−b1|Vmin|)
+ b2b3(|Vr|−Vmin)2
1+b2(|Vr|−Vmin) , for |Vr | > Vmin,
(2.57)where fs = 0.12, fmin = 0.05, b1 = 140 sm−1, b2 = 10 sm−1, b3 = 2 sm−1,Vmin = 0.035 mc−1. Function sgn(x) is approximated by [Martins et al. (1990)]
sgnε0(x) =
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
1, for x > ε0,(2 − |x|
ε0
)xε0
, for |x| < ε0,
−1, for x < −ε0,
(2.58)
where the regularization parameter ε0 = 0.0001.The results of calculations for different values of the parameter γ = 0, 200, 400
for the steel cylinder (α1 = 14 × 10−6◦C−1, λ1 = 21 W m−1◦C−1, ν1 = 0.3,a1 = 5.9×10−6 m2 s−1, E1 = 19×1010 Pa) and R1 = 0.03 m, Ω∗ω1 = 1 rad s−1,ε = 10, ω = 0.1, Bi = 10, ϕ◦ = 0, ω◦ = 0 are shown in Figs. 2.8 to 2.12. Solidcurves correspond to the case of wear absence kw = 0, dashed – to the case ofwear with the dimensionless wear coefficient kw = 0.1. In this case t∗ = 0.25 s,P∗ = 1.22 × 104 Pa, γ1 = 0.51, γ2 = −196.
In the case of heat expansion absence time evolutions of the dimensionlessspeed ϕ(τ) (curve 1), dimensionless displacement ϕ(τ) (curve 2) of the bush, anddimensionless friction force εF (ω◦
1 − ϕ)p(τ) (dashed curve 3) for γ = 0, (v = 0,s1 = 0.0051 the steady-state solution is unstable) are shown in Fig. 2.8.
Figure 2.9a illustrates the behaviour of the phase trajectory in the phase planewithout taking into account heat expansion of the cylinder γ = 0 and wear of thebush kw = 0 (solid curve) and with wear (dashed curve). In the conditions of wearabsence, a transient solution leads with time to a stable limit cycle with the period6.608. The stick-slip phenomenon takes place.
Let us point out that (as can be seen in this figure) the friction force has a jumpat the instant of time when the bush speed reaches the speed of the cylinder Vr = 0.At that moment the rest friction force changes the sign. Then the dimensionlessfriction force becomes equal to dimensionless displacement and this lasts to themoment when it reaches a maximal value of the static friction force. From thatinstant of time the friction force decreases. After that, it reaches a local minimumwhen the bush acceleration is equal to zero. Then, the friction force increases till it
50 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.8: Dependence of the bush movement dimensionless velocity ϕ (curve 1), dimensionlessdisplacements ϕ (curves 2), and dynamic friction force εF (ω◦
1−ϕ) (curve 3) versus dimensionless
time τ in conditions of heat expansion absence γ = 0. Solid curves: kw = 0, dashed curves:kw = 0.1.
Fig. 2.9: Phase trajectory of bush movement in conditions of heat expansion absence γ = 0(a) and γ = 200 (b). Solid curves: kw = 0, dashed curves: kw = 0.1.
reaches cylinder velocity Vr = 0. The process continues cyclically. In the presenceof wear the friction force will tend to zero with time and the bush will performits own oscillations with the period of 2π.
Time evolutions of the dimensionless speed ϕ (curve 1), dimensionless dis-placement ϕ(τ) (curve 2) of the bush, and dimensionless dynamic friction force
2.3 Kinematic external shaft excitations 51
Fig. 2.10: Dependence of bush dimensionless velocity ϕ (curve 1), dimensionless displacementsϕ (curve 2), and dynamic friction force εF (ω◦
1− ϕ) (curve 3) on dimensionless time τ , for
γ = 200 (a) and γ = 400 (b). Solid curves: kw = 0, dashed curves: kw = 0.1.
εF (ω◦1 − ϕ)p(τ) (dashed curve 3) are shown in Fig. 2.10a for γ = 200 (for
v = 0.51, s1 = 0.0012 the steady-state solution is unstable). Figure 2.9b illus-trates the behaviour of the phase trajectory in the phase plane without taking intoaccount heat expansion of the cylinder γ = 200 and kw = 0 (solid curve) andwith wear (dashed line). As can be seen, in the absence of bush wear, the contactcharacteristics tend with time to the limit cycle with the period equal to 5.5.
Figure 2.10b illustrates time changes of the dimensionless speed ϕ (curve 1),dimensionless displacement ϕ(τ) (curve 2) of the bush, and dimensionless dynamicfriction force εF (ω◦
1 − ϕ)p(τ) (dashed curve 3) for γ = 400 (v = 1.02, s1 =2×10−6). Figure 2.11 illustrates the behaviour of the phase trajectory on the phaseplane without taking into account the heat expansion of the cylinder γ = 400 andkw = 0 (solid curve) and with wear (dashed curve). In the case of wear absencethermoelastic instability takes place. System characteristics do not tend to stablelimit cycles with time but exponentially increase. In that case, the cylinder doesnot have enough time to become cool. Wear presence (dashed curves) leads to lossof the thermoelastic instability.
Time-changing laws for the contact pressure, temperature, and wear are shownin Figs. 2.12 by solid (dashed) curves for the case of wear absence (presence),respectively. Curve 1 corresponds to the case of absence of heat expansion γ = 0,curve 2 to γ = 200, curve 3 to γ = 400. In the last case, contact characteristicsincrease and the cylinder does not have time for cooling down.
Although parameter ω decreases and v < 1, the time necessary for the leadingof contact characteristics on the limit cycle increases. The wear presence leads todecreasing of contact characteristic values and to limiting of system contact time(curves 2 and 3 in Fig. 2.12). With an increase of the parameter γ the system
52 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.11: Phase trajectory of bush movement in conditions for γ = 400. Solid curves: kw = 0,dashed curves: kw = 0.1.
contact time decreases and, correspondingly, the bush wear increases and its valuebecomes bigger than the value of initial deformation caused by an initial stressstate of the cylinder.
A new model of thermoelastic contact is considered, in which changes of thebush movement velocity, contact pressure, friction force, contact temperature, andwear are mutually connected, are considered. Conditions in which frictional self-oscillations arise are investigated. In the wear absence these conditions being at thesame time the conditions of steady-state solution stability are also the conditions ofthe linearized problem. The characteristic equation roots transition from the left-hand side (Res1 < 0) to the right-hand side (Res1 > 0) of the Laplace transformas regards the parameter complex plane associated with the Hopf bifurcation.In conditions of wear presence stick-slip movements disappear with time. Thesystem contact time is limited. The upper bound of the solution stability conditionscan be the conditions of coinciding of first roots of characteristic equation Ims1 =Ims2 = 0, Res1 = Res2 > 0 on the right-hand side of the complex plane. Takinginto account heat expansion of the cylinder and wear of the bush extends regions forthe parameters in which the steady-state solution is stable; that is, heat expansionand wear play the role of stabilising factors.
The possibility of occurrence of self-excited vibrations for the considered systemwith a lack of tribological processes (k1 = 0, k3 = 0, γ = 0) has been pointedout by [Andronov et al. (1966), Neimark (1978)].
2.3 Kinematic external shaft excitations 53
Fig. 2.12: Behaviour of dimensionless contact pressure p (a), dimensionless contact temperatureθ(1, τ) (b), and time evolution of dimensionless wear uw (c) versus dimensionless time τ fordifferent values of γ. Curve 1: γ = 0, 2: γ = 200, 3: γ = 400. Solid curves: kw = 0; dashedcurves: kw = 0.1.
2.3.4 Chaotic motion of the shaft/bush with kinematic external
excitations
In this section, vibrations of a rigid body (bush, pad) attached on the rotatingshaft at nonconstant velocity and initially without taking account of tribologicalprocesses are considered (γ = 0, kw = 0). It is assumed that an arbitrary parameterk∗ = k1(l0/l1 − 1)(1 + l1/R2) − k2, and χ = 1, which yields a = 1, t∗ =√
B2/(k∗R22). In addition, we assume that hU (τ) = H(τ), U∗ = U0, hT (τ) = 0.
In this case, the dimensionless contact pressure p(τ) = 1. In order to predict thebush chaotic dynamics, Melnikov’s technique is applied. The analytical form ofMelnikov’s function is formulated, and then a numerical analysis is performed.
54 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
2.3.4.1 Introduction
Because nonlinear dynamic systems may exhibit either regular or chaotic motions[Awrejcewicz, Holicke (1999),(2007)], one of the recent challenging tasks isfocused on their behaviour control. Such control can be realized, for instance,by applying an external kinematic excitation (perturbation). Although the majorityof the methods aiming at analysis and control of Nonlinear Dynamical Systems(NDS) are realized via numerical algorithms, an analytical treatment seems to bethe most powerful and economical. One of the often-applied approaches that allowsus to calculate the distance between the homoclinic orbits, and give the conditionsof chaos in a nearly Hamiltonian system, is Melnikov’s method [Melnikov (1963)]or the modified Melnikov theory. Despite this method having been widely appliedin the analysis of smooth dynamic systems, its successful application to anal-yse simple dynamic systems with friction has been illustrated only quite recently[Grudziński, Wedman (1998)]. The latter approach has even been extended to studymore complicated regular and chaotic stick-slip dynamics of a rotating shaft with arigid bush in wear and heat transfer conditions. Numerical experiments confirmedwell the analytical prediction based on the analysis of Melnikov’s function.
2.3.4.2 Melnikov function
For the considered case, differential equation (inclusion) (2.5) takes the followingnondimensional form.
ϕ(τ) + 2hϕ(τ) − ϕ(τ) + bϕ3(τ) ∈ εF (ω1 − ϕ), 0 < τ < ∞, (2.59)
ϕ(0) = ϕ◦, ϕ(0) = ϕ◦ = ω◦. (2.60)
Dependence of friction on relative velocity is approximated by the function
F (y) = fsSgn(y) − αy + βy3, where Sgn(y) =
⎧⎨
⎩{y/|y|} for y = 0,
[−1, 1] for y = 0.(2.61)
The dimensionless angular bush velocity is governed by the equation
ω1 = ω∗ + ζksinω0τ, (2.62)
where ζk is the dimensionless amplitude of the kinematic external excitation.In order to apply the combined Melnikov’s and numerical methods, a pertur-
bation of the Hamiltonian system, where the inclusion Sgn(y) occurs, has beenapproximated by continuous perturbation with the application of a small parameter.The multivalued relation Sgn(y) is approximated by the function sgnε0
(y) definedby (2.58), where the regularization parameter ε0 is a ‘small’ positive real number.The differential inclusion (2.59) is then approximated by the equation.
2.3 Kinematic external shaft excitations 55
In the so-called first improvement of Melnikov’s function M(τ0) (see theexpression near ε) for 0 < ε ≪ 1 of the expression representing a distancebetween stable and unstable manifolds of the critical saddle point, a transition ofthe parameter ε0 to zero (ε0 → 0) can be realized. In order to be sure of neglect-ing the so-called second improvement of Melnikov’s function near ε2 (the functionunder the integral includes the differential of the approximated perturbation), thefollowing condition should be satisfied, ε/ε0 ≪ 1. Then if the mentioned conditionis satisfied only the first improvement of Melnikov’s function can be applied toestimate the distance between stable and unstable manifolds of the critical point.
Introducing the new variables
x = ϕ, y = ϕ, (2.63)
the equations of motion can be transformed into the form,
x = p0(x, y) + εp1(x, y, ω0τ, ε),
y = q0(x, y) + εq1(x, y, ω0τ, ε), (2.64)
where
q0(x, y) = x − bx3, q1(x, y, ω0τ, ε) = F (ω∗ + ζksinω0τ − y) − h1y,
p0(x, y) = y, p1(x, y, ω0τ, ε) = 0, h1 = 2h/ε. (2.65)
For a sufficiently small parameter 0 < ε ≪ 1,
x0(τ) =
√2
b
1
cosh(τ), y0(τ) = −
√2
b
sinh(τ)
cosh2(τ), (2.66)
Melnikov’s function is defined by the formula [Awrejcewicz (1996)]
M(τ0) =
+∞∫
−∞
(q0p1 − q1p0)|x=x0(τ−τ0)
y=y0(τ−τ0)
dτ = −+∞∫
−∞
q1p0|x=x0(τ−τ0)
y=y0(τ−τ0)
dτ, (2.67)
where x0(τ), y0(τ) is the solution of a nondisturbed system of equations (ε = 0),which corresponds to the homoclinic orbits, and τ0 is the parameter that charac-terizes positions of the point moving in this orbit. In accordance with Melnikov’stheory, if the function M(τ0) has simple zeros, then for a sufficiently small para-meter ε, the motion governed by system (2.64) can be chaotic.
Introducing the change of variable τ − τ0 = t, Melnikov’s function is
M(τ0) = −+∞∫
−∞
y0(t)[fsSgn(ωr) − αωr + βω3r ]dt, (2.68)
56 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
where the dimensionless relative velocity has the form
ωr(t) = ω∗ + ζksin(ω0(t + τ0)) − y0(t). (2.69)
The substitution of (2.66) and (2.69) into (2.68) gives Melnikov’s function ofthe form
M(τ0) = I(τ0) + J(τ0), (2.70)
where
J(τ0) = 2C + 2ζk
√A2 + B2sin(ω0τ0 + ϕ)
+ 6βζ2k(I220cos2 ω0τ0 + I202sin
2 ω0τ0 − 2ω∗I111sinω0τ0cosω0τ0)
+ 2βζ3k(−I130cos3 ω0τ0 − 3I112sin
2 ω0τ0cosω0τ0),
A = (α − 3βω2∗)I110 − 3βI310, B = 6βω∗I201,
C = βI400 − (α − h1 − 3βω2∗)I200, ϕ = arctan(A/B),
Injk =
∞∫
0
[y0(t)]n[sin(ω0t)]
j [cos(ω0t)]kdt. (2.71)
After integration of Equation (2.71) we obtain
I200 =2
3b, I400 =
8
35b2, I201 =
πω0(2 − ω20)
6b sinh(πω0/2), I110 = − πω0√
2b cosh(πω0/2),
I310 =ω0(11 + 10ω2
0 − ω40)
120b√
2b
{ψ
(1 − iω0
4
)− ψ
(3 − iω0
4
)
+ ψ
(1 + iω0
4
)− ψ
(3 + iω0
4
)},
I130 = − 3πω0
8√
2b
{cot
(π(1 − iω0)
4
)+ cot
(3π(1 − iω0)
4
)
− cot
(π(3 − iω0)
4
)− cot
(π(1 − 3iω0)
4
)},
I112 =πω0 cosh(πω0/2)√2b(1 − 2 cosh(πω0))
, I111 = − πω0√2b cosh(πω0)
,
I220 =πω0(2ω2
0 − 1) + sinh(πω0)
3b sinh(πω0),
2.3 Kinematic external shaft excitations 57
I202 =πω0(1 − 2ω2
0) + sinh(πω0)
3b sinh(πω0), ψ(z) =
Γ ′(z)
Γ (z),
where Ψ(z) denotes the derivative of the natural logarithm of the function Γ (z).In (2.70) the term I(τ0) is defined by the formula
I(τ0) = −fs
+∞∫
−∞
y0(t)Sgn(ωr)dt = 2fs
√2
b
∑
m
sgn(ω′r(tm))
cosh tm, (2.72)
where tm are the roots of the equation
ωr(tm) = ω∗ + ζksin(ω0(tm + τ0)) − y0(tm) = 0, (2.73)
whereas ω′r(t) = ζkω0cos(ω0(t + τ0)) − x0(t) + bx3
0(t).If Melnikov’s function (2.70) changes its sign, then one may expect chaos.
Observe that in our case, Melnikov’s function has a rather complex structure and itsdirect theoretical analysis is not easy. Therefore, in this section our considerationsare limited to the analysis of the function J(τ0) for large values of b and smallvalues of ζk (h1 = 0) of the form
J(τ0) =
(−4
3(α − 3βω2
∗) +16
35
β
b+ O
(1
b2
))1
b
+
[−√
2πω0(α − 3βω2∗)
cosh(πω0/2)cos(ω0τ0) (2.74)
− 2πω0βω∗(ω20 − 2)√
bsinh(πω0/2)sin(ω0τ0) + O
(1
b
)]1√bζk + O(ζ2
k).
Equation (2.74) provides an observation that the function J(τ0) changes itssign (for large b and small ζk) when the following inequality holds
ζ(1)ch < ζk, where ζ
(1)ch =
4cosh(πω0/2)
3√
2bπω0
. (2.75)
Note that for large values of b and small values of ζk we have I(τ0) ≡ 0, andthe remaining terms of J(τ0) in (2.70) are small. The value of the mentionedterms changes its sign when (2.75) holds. For a certain value of ζk, afterwardsreferred to as ζnz , the value of integral I(τ0) is not equal to zero, and starts toplay a dominant role in M(τ0). The function M(τ0) begins to change sign whenζk ≥ ζnz and when I(τ0) is not equal to zero (the function ωr(t) starts to changeits sign). One may find the corresponding estimated value of the parameter ζk withthe use of the following formula,
ζ(2)ch < ζk, where ζ
(2)ch ≈ ζnz = ω∗ − 1/
√2b. (2.76)
58 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.13: Chaotic threshold in the (ζk , b) plane (ε = 0.1, ω0 = 2, ω∗ = 0.4, fs = 0.3, α = 0.3,β = 0.3).
Using both formulas (2.75) and (2.76), the functions ζ(i)ch , i = 1, 2 versus the
parameter b are shown in Fig. 2.13. One may conclude that the region correspon-ding to chaos occurs above the curves.
2.3.4.3 Numerical results
In order to verify the analytical conditions in which chaotic oscillations occur, somenumerical tests have been carried out. The following initial condition x(0) = 0,y(0) = 0 and the following parameters ε = 0.1, ω0 = 2, ω∗ = 0.4, h1 = 0,fs = 0.3, α = 0.3, β = 0.3 are fixed.
Arbitrary values of b. Formula (2.70) holds for any value of b. The numericallyobtained Melnikov’s function M(τ) is shown in Fig. 2.14 for b = 1.
Because Melnikov’s function can change its sign (it has simple zeros) forξch ≈ 3.6, chaos may occur according to Melnikov’s theory. To confirm theanalytical prediction, the bifurcational diagram x(ζk) has been constructed (pro-jection of the Poincare section into the x-axis for ζk ∈ (0, 12) and ζk ∈ (3.5, 4.0)).The obtained results are shown in Fig. 2.15.
Observe that for ζk < ω∗ (ω∗ = 0.4) the periodic motion occurs with freefrequency of the system, which undergoes changes as ζk increases. However, thismotion vanishes when ζk → ω∗. With the increase of the parameter ζk, perioddoubling occurs for ξch ≈ 3.6, and a bifurcation cascade leading to chaos follows.
2.3 Kinematic external shaft excitations 59
Fig. 2.14: Melnikov’s function M(τ0) versus parameter τ0 for ζk = 3.2 (solid curves) and forζk = 3.81 (dashed curves), b = 1.
a) b)
Fig. 2.15: Bifurcation diagrams using ζk as a control parameter; h1 = 0, γ = 0, kw = 0 :(a) ζk ∈ (0, 12); (b) ζk ∈ (3.5, 4.0).
In addition, for the same parameters the Fast Fourier Transform (FFT) of theprocess x(τ) for large values of τ ∈ (τ1, τN ) is reported. The computational resultsof the obtained power spectrum are shown in Fig. 2.16. The following relationsare applied to estimate power spectra.
L(ω) = 20 log |X(ω)|, X(ω) = X(∆ωm) = Xm,
∆ω = 2π/(∆τN), m = 1, 2, 3, . . . , N,
Xm =
N∑
n=1
(xn − x0)e−2πi(n−1)(m−1)N ,
xn = x(n∆τ ), n = 1, 2, 3, . . . , N, x0 =1
N
N∑
n=1
xn.
The following values are fixed during computations: N = 4000 and ∆τ = 0.02.
60 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.16: Power spectrum: (a) ζk = 3.2; (b) ζk = 3.81.
Fig. 2.17: Phase planes and Poincare sections: (a) ζk = 3.2; (b) ζk = 3.81.
Observe that for ζk = 3.2 the periodic-four motion occurs, whereas forζk = 3.81 the motion is chaotic. The phase planes and Poincare sections forζk = 3.2 (Fig. 2.17a) and for ζk = 3.81 (Fig. 2.18b) are shown. The points ofPoincare maps are obtained with respect to the period of kinematic excitation2π/ω0.
Large values of b. In general, for large values of the parameter b the values ofζk responsible for chaos are smaller in comparison with the parameter ω∗. As thenumerical analysis shows, for ζk ∈ [0, ω∗) the system motion is periodic with thefree system frequency ωd. For example, for fixed value of b = 1, the followingvalues of ωd are found; ζk = 0 gives ωd ≈ 0.902; ζk = 0.1 gives ωd ≈ 0.805;ζk = 0.2 gives ωd ≈ 0.675; ζk = 0.3 gives ωd = 1.43175; ζk = 0.4 givesωd = ω0 = 2.
2.3 Kinematic external shaft excitations 61
Fig. 2.18: Function I(τ0) (a) and Melnikov’s function M(τ0) (b) versus the parameters τ0 forζk = 0.165 (solid curves) and ζk = 0.17 (dashed curves); b = 9.
Fig. 2.19: Power spectrum for b = 9: (a) ζk = 0.1; (b) ζk = 0.2.
Numerical analysis of Melnikov’s function is carried out for the parametersassociated with points 1–6 lying in the chaotic area (points 2, 4, 6) and out ofchaos (points 1, 3, 5); see Figs. 2.18, 2.20 and 2.22. For ζk < ζnz , we haveI(τ0) = 0. The obtained values of ζk give ζnz = 0.164 (formula (2.76)) and
ζ(1)ch = 0.58 (formula (2.75)). Figure 2.18 shows the dependence of the function
I(τ0) (a) and Melnikov’s function M(τ0) (b) on the parameter τ0 for point 1 (b = 9,ζk = 0.165; solid curves) and for point 2 (b = 9, ζk = 0.17; dashed curves) fromFig. 2.13. For the mentioned parameters condition (2.75) is satisfied. Althoughfor ζk = 0.165 the function I(τ0) has nonzero values, the function M(τ0) doesnot change its sign yet. The numerical analysis shows that the function M(τ0)
changes its sign for ζ(2)ch ≈ 0.168. Hence, both obtained values of ζnz and ζ
(2)ch are
close to each other. The (numerical) analysis confirms the value of (2.76) used
to obtain the parameter ζ(2)ch responsible for the occurrence of chaos according to
Melnikov’s rule.Numerical analysis of Equation (2.64) is carried out for the parameters in the
vicinity of points 1 and 2 (see Fig. 2.13). Figure 2.19 illustrates the power spectrafor the parameter ζk = 0.1 (it corresponds to the point situated below point 1
62 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.20: I(τ0) (a) and Melnikov’s M(τ0) (b) functions versus the parameter τ0 for ζk = 0.23(solid curves) and for ζk = 0.30 (dashed curves); b = 37.41.
Fig. 2.21: Power spectra for b = 37.41: (a) ζk = 0.23; (b) ζk = 0.29.
and is associated with the quasi-periodic motion of the bush), and for ζk = 0.2(it corresponds to the point lying above point 2 and is associated with chaos).
Note that for b = 37.41 we have ζ(1)ch = ζ
(2)ch (see Fig. 2.13). Figure 2.20
shows I(τ0) (a) and Melnikov’s (b) M(τ0) functions versus parameter τ0 for theparameters associated with point 3 (see Fig. 2.13) (b = 37.41, ζk = 0.23; solidcurves) and with point 4 (b = 37.41, ζk = 0.30; dashed curves). Although thefunction I(τ0) is equal to zero, the function M(τ0) still does not change its sign
(I(τ0) = 0 for ζk < ζ(2)ch ). It begins to change the sign for ζk = 0.30. Therefore,
the numerical tests confirm the values defined by Equations (2.75) and (2.76) and
giving the parameters ζ(1)ch and ζ
(2)ch responsible for chaos occurrence, according
to Melnikov’s theory.The numerical tests of Equations (2.64) are performed for the parameter sets
associated with points 3 and 4 (Fig. 2.13). Figure 2.21 presents the power spectrafor the parameters ζk = 0.23 (point 3 corresponds to the 2π-periodic bush motion)and ζk = 0.29 (point 4 corresponds to the chaotic motion). Chaotic behaviour inthe vicinity of the homoclinic orbits (2.66) is clearly visible for ζk = 0.29.
2.3 Kinematic external shaft excitations 63
Fig. 2.22: I(τ0) (a) and Melnikov’s function M(τ0) (b) versus the parameter τ0 for ζk = 0.14(solid curves) and for ζk = 0.17 (dashed curves); b = 121.
Fig. 2.23: Power spectra for b = 121: (a) ζk = 0.1; (b) ζk = 0.17.
In the next computational step we have taken b = 121. Note that
I(τ0) = 0 for ζ < ζ(2)ch . Formula (2.76) gives ζ
(2)ch = 0.336, whereas formula
(2.75) provides ζ(1)ch = 0.158. Figure 2.22 presents the function I(τ0) (a) and
Melnikov’s function M(τ0) (b) versus τ0 for the parameters associated with points5 (b = 121, ζk = 0.14; solid curves) and 6 (b = 121, ζk = 0.17; dashed curves).In this case I(τ0) = 0 for ζk = 0.14, but the function M(τ0) does not change its
sign (I(τ0) = 0 for ζk < ζ(2)ch ) (it starts to change it for ζk = 0.17). Again nume-
rical computations confirm the applied value of (2.75) to define the parameter ζ(1)ch
responsible for the prediction of analytical chaos.Finally, the numerical analysis of Equation (2.64) is carried out for the para-
meters associated with points 5 and 6 in Fig. 2.13. For the parameter ζk = 0.1 (theinvestigated point lies below the point 5 and corresponds to the periodic motionof the bush) and ζk = 0.17 (point 6 corresponds to the narrow region of chaos)the associated power spectra are shown in Fig. 2.23.
It has been shown that for our investigated nonlinear system with friction theassociated Melnikov’s function can be constructed. As a result, one may control the
64 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
nonlinear dynamics by analytical prediction of either a regular or chaotic systemstate. For some parameter sets the complicated analytical structure of Melnikov’sfunctions can be simplified to provide simple analytical conditions for the occur-rence of chaos.
2.3.4.4 Calculation of Lyapunov exponents
Vibrations of the bush being in thermoelastic contact with the rotating shaft aregoverned by the following nondimensional Equation (2.5),
ϕ(τ) + 2hϕ(τ) − ϕ(τ) + bϕ3(τ) = εF (ω1 − ϕ)p(τ), 0 < τ < ∞, (2.77)
with the initial condition ϕ(0) = ϕ◦, ϕ(0) = ω◦, where the nondimensionalcontact pressure is defined through solutions to Equation (2.31),
p(τ) = hU (τ) − uw(τ) + 2γω
τ∫
0
Gp(τ − ξ)F (ω1 − ϕ)p(ξ)(ω1 − ϕ)dξ. (2.78)
The bush wear uw(τ) and the shaft temperature θ(r, τ) are defined through theEquations (2.23) and (2.32)
uw(τ) = kw
τ∫
0
|ω1 − ϕ(τ)|p(τ)dτ, 0 < τ < τc, (2.79)
θ(r, τ) = γω
τ∫
0
Gθ(r, τ − ξ)F (ω1 − ϕ)p(ξ)(ω1 − ϕ)dξ. (2.80)
In Equations (2.77)–(2.80) the following nondimensional quantities are intro-duced
τ =t
t∗, r =
R
R1, p =
P
P∗, θ =
T1 − Tsm
T∗, ϕ(τ) = ϕ2(t∗τ),
uw =Uw
U∗, ε =
P∗t2∗2πR2
1
B2, h =
cR22
2B2t∗, kw =
P∗KwR1
U∗,
γ =(1 − η)E1α1R
21
λ1(1 − 2ν1)t∗, Bi =
αT R1
λ1, τc =
tct∗
, ω =t∗a1
R21
,
ω0 = ω′t∗, hU (τ) = hU (t∗τ), F (ω1 − ϕ) = f(V∗(ω1 − ϕ)),
2.3 Kinematic external shaft excitations 65
where
V∗ =R1
t∗, t∗ =
√B2
k∗R22
, k∗ = k1
(l0l1
− 1
)(1 +
l1R2
)− k2,
T∗ =U∗
α1(1 + ν1)R1, P∗ =
α1E1T∗
(1 − 2ν1),
and l0 is the nonstretched spring length, l1 is the length of the compressed springfor ϕ2 = 0, (k∗ > 0), E1 is the elasticity modulus, ν1 is the Poisson coefficient,α1 is the coefficient of thermal expansion of the shaft, αT is the heat transfercoefficient, a1 is the thermal diffusivity, λ1 is the heat transfer coefficient, ϕ2(t)is the angle of bush rotation, Kw is the wear coefficient, η denotes the partof heat energy associated with wear η ∈ [0, 1], and tc is the time of contact(0 < t < tc, P (t) > 0).
Note that the stated problem is modelled by both the nonlinear differentialEquation (2.77) and integral equation (2.78) governing rotational velocity ϕ(τ)and contact pressure p(τ). Temperature and wear are defined by Equations (2.80)and (2.79), respectively.
A particular case of our problem is further studied (γ = 0, kw = 0, p(τ) → 1).The dependence of kinematics friction on relative velocity is approximated bythe function F (y) = fssgn(y) − αy + βy3. Because the latter is nonsmooth dueto the presence of the sgn(y) function in the kinematic friction, the methodscommonly used to compute the exponents require smoothness of the vector fieldsas a necessary condition. Nonsmooth systems yield only approximations for theLyapunov exponents, which can be considered valid as long as we do not bothertoo much with the vicinity of the nonsmoothness points [Awrejcewicz, Lamargue(2003)]. The function sgn(y) is approximated by the following one (2.58).
Note that while computing Lyapunov exponents, besides the following equations
x = y, y = x − bx3 + ε[fs sgnε0(vr) − αvr + βv3
r] − εh1y, z = ω0, (2.81)
three additional systems of equations (n = 1, 2, 3) with respect to perturbationsare also solved:
˙x(n) = y(n),
˙y(n) = x(n) − 3bx2x(n) + ε[fsδε0(vr) − α + 3βv2
r]v(n)r − εh1y
(n),
˙z(n) = 0, (2.82)
where x = ϕ(τ), y = ϕ(τ), z = ω0τ , vr = ωk +ζksin z−y, v(n)r = ζkz(n)cos z−
y(n), h1 = 2h/ε, and
66 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
δε0(y) =
⎧⎨⎩
0, |y| > ε0,
(2ε0)(1 − |y|/ε0), |y| < ε0.(2.83)
Twelve equations of system (2.81) and (2.82) are solved using the fourth-orderRunge–Kutta method and Gram–Schmidt reorthonormalisation procedure.
Let x00, y0
0, z00 be initial values of perturbation vectors that are orthonormal.
After time T , an orbit x(τ) reaches the point x1 with the associated perturbationsx1, y1, z1. Then, the so-called Gram–Schmidt reorthonormalization procedure iscarried out and the following new initial set of conditions is formulated.
x01 =
x1
‖x1‖, (2.84)
y01 =
y′1
‖y′1‖
, y′1 = y1 − (y1, x
01)x
01, (2.85)
z01 =
z′1
‖z′1‖, z′1 = z1 − (z1, x
01)x
01 − (z1, y
01)y
01. (2.86)
Next, after time interval T , a new set of perturbation vectors x2, y2, z2 isdefined, which is also reorthonormalized due to the Gram–Schmidt procedure(2.84)–(2.86). This algorithm is repeated M times. Note that (x0
1, y01) = 0,
(x01, z
01) = 0, (y0
1, z01) = 0 and if x = (x, y, z), y = (x1, y1, z1) then
‖x‖ =√
x2 + y2 + z2, and the scalar product (x,y) = xx1 + yy1 + zz1. Finally,a spectrum of three Lyapunov exponents is computed via formulas
λ1 =1
MT
M∑
i=1
ln ‖xi‖, λ2 =1
MT
M∑
i=1
ln ‖y′i‖, λ3 =
1
MT
M∑
i=1
ln ‖z′i‖,
(2.87)where the occurring vectors are taken before the normalization procedure.
2.3.4.5 Numerical analysis
Our numerical computations are carried out for the particular case (γ = 0,kw = 0). The following nondimensional parameters are taken: fs = α = β = 0.3,ω0 = 2, ωk = 0.4, b = 1, ε = 0.1. Numerical analysis is carried out for the bifurca-tion diagram with respect to x versus ζk for ζk ∈ (0, 12) and ζk ∈ (3.5, 4.0)). Theobtained results are shown in Figs. 2.15a and 2.15b for h1 = 0, in Fig. 2.24a forh1 = 0.5 and in Fig. 2.24b for h1 = 1. The Lyapunov exponents in time intervalτ ∈ (1200, 1514) (x0
0 = (1, 0, 0), y00 = (0, 1, 0), z0
0 = (0, 0, 1), T = 0.005,M = 80000, ε0 = 0.01) are computed due to formulas (2.87) for the same valuesof the parameters. In Figs. 2.25a, b, and 2.26a, b dependencies of Lyapunov expo-nents on the control parameter ζk are reported. A study of both Lyapunov exponents
2.3 Kinematic external shaft excitations 67
a) b)
Fig. 2.24: Bifurcation diagrams (a),(b) using ζk as control parameter; γ = 0, kw = 0, b =1 : (a) h1 = 0.5; (b) h1 = 1.
a) b)
Fig. 2.25: Lyapunov exponents (a), (b) using ζk as control parameter; h1 = 0, γ = 0, kw =0, b = 1 : (a) ζk ∈ (0, 12); (b) ζk ∈ (3.5, 4.0).
and bifurcation diagrams implies that chaos begins for (i) ζk = 3.78, for h1 = 0;(ii) for ζk = 3.8, for h1 = 0.5; (iii) for ζk = 4.25, for h1 = 1 (note that the largestLyapunov exponent λ1 is positive). An increase of the parameter h1 responsiblefor damping yields an increase of the amplitude of the bush, where chaos is born.
Note that because our system (2.81) is autonomous, one of the Lyapunovexponents is always zero.
In a general case, numerical analysis is carried out on a steel-made shaft(α1 = 14 · 10−6◦C−1, λ1 = 21 W/(m · ◦C−1), ν1 = 0.3, a1 = 5.9 · 10−6 m2/s,E1 = 19·1010 Pa). Observe that no accounting of tribological processes (h1 = 0.5,ζk = 3.9, γ = 0, kw = 0) yields chaotic dynamics (Fig. 2.27, curve 2). Forh1 = 0.5, ζk = 3.5, γ = 0, kw = 0 regular motion takes place (Fig. 2.27,curve 1). An account of thermal shaft extension (γ = 1.87) removes the chaotic
68 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
a) b)
Fig. 2.26: Lyapunov exponents (a), (b) using ζk as control parameter; γ = 0, kw = 0, b =1 : (a) h1 = 0.5; (b) h1 = 1.
Fig. 2.27: Phase plane of bush motion for h1 = 0.5, kw = 0: curve 1, ζk = 3.5, γ = 0; curve 2,ζk = 3.9, γ = 0; curve 3, ζk = 3.5, γ = 1.87; curve 4, ζk = 3.9, γ = 1.87.
behaviour of our system (Fig. 2.27, curves 3 and 4). For ζk = 3.5 a subharmonicmotion with frequency ω0/2 is obtained (Fig. 2.27, curve 3), whereas for ζk = 3.9periodic motion is exhibited (Fig. 2.27, curve 4).
Owing to an account of wear (kw = 0.01) and neglecting shaft thermal exten-sion (γ = 0), contact pressure tends to zero, whereas cylinder wear approachesU∗(p(τ) → 0, uw(τ) → 1). The nondimensional bush wear is presented inFig. 2.28, curve 1. In addition, in Fig. 2.28, curves 1 and 2 represent time historiesof the nondimensional contact pressure.
2.3 Kinematic external shaft excitations 69
Fig. 2.28: Dimensionless contact pressure p(τ) and wear uw(τ) versus dimensionless time τ :curves 1, ζk = 3.5, γ = 0, kw = 0.01, h1 = 0.5; curves 2, ζk = 3.9, γ = 0, kw = 0.01,h1 = 0.5.
Fig. 2.29: Dimensionless contact pressure p(τ) versus dimensionless time τ : curve 4, ζk = 3.9,γ = 1.87, kw = 0.01, h1 = 0.5.
A simultaneous account of shaft extension and bush wear yields a finite timeof contact between both bodies. For instance, for h1 = 0.5, ζk = 3.9, γ = 1.87,kw = 0.01 contact pressure versus time is exhibited by curve 4 in Fig. 2.29. Thenondimensional time contact interval is τc = 72. For ζk = 3.5 time contact isτc = 65.8. In Fig. 2.30, curves 3 and 4 represent the dependence of nondimen-sional wear on the nondimensional time in a general case. Curve 3 corresponds toh1 = 0.5, ζk = 3.5, γ = 1.87, kw = 0.01, whereas curve 4 is associated withthe parameters: h1 = 0.5, ζk = 3.9, γ = 1.87, kw = 0.01. Owing to heat shaftextension, the wear of the bush is increased 30 times (see curves 4 and 2 inFig. 2.30).
70 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.30: Time history of dimensionless wear uw(τ), h1 = 0.5, kw = 0.01; curve 2, ζk = 3.9,γ = 0; curve 3, ζk = 3.5, γ = 1.87; curve 4, ζk = 3.9, γ = 1.87.
The influence of tribological processes on dynamic behaviour of the analysedsystem in the vicinity of chaos has been illustrated and discussed. An account ofbush wear and neglecting of shaft thermal expansion implies that the contact pres-sure tends to zero, the bush wear approaches the values of the shaft compressing,and bush vibrations are damped.
On the other hand, taking into account the shaft thermal extension and neglect-ing of bush wear results in chaos disappearance and the occurrence of a regularmotion.
In a general case (both shaft thermal extension and bush wear are taken intoaccount), time interval of the contact of two bodies is bounded. With the lack ofcontact, the bush stops due to an extensive wear process.
Experimental investigations focused on chaotic vibrations in the kinematicallyexcited system and modelling a drive–feed system are carried out in reference[Grudziński, Wedman (1998)].
2.3.5 Chaotic motion of the bush subject to mechanical external
excitations
In this section the previous model of the bush lying on the shaft rotating at con-stant velocity is considered (without heat generation; i.e., γ = 0). As assumedearlier k∗ = k1(l0/l1 − 1)(1 + l1/R2) − k2, and χ = 1, which defines a = 1,t∗ =
√B2/(k∗R2
2). We assume also that hU (τ) = H(τ), U∗ = U0, hT (τ) =0. According to the latter assumptions, the dimensionless contact pressure isp(τ) = 1.
2.3 Kinematic external shaft excitations 71
For this case, Melnikov’s function is formulated analytically giving the onsetof chaos. Examples of nonlinear system behaviour are illustrated and testednumerically.
2.3.5.1 Melnikov’s method
For the considered case differential equations (inclusions) (2.5) take the followingform,
ϕ(τ) − ϕ(τ) + bϕ3(τ) ∈ −2hϕ + ε[ζcos(ωτ) + F (ω1 − ϕ)p(τ)], 0 < τ < ∞,(2.88)
ϕ(0) = ϕ◦, ϕ(0) = ω◦, (2.89)
where p(τ) = hU (τ)−uw(τ). Friction versus the relative velocity is approximatedby function (2.61). Consider first the case without wear (kw = 0) and hU (τ) =H(τ). Equations of motion are governed by (2.64), where
q0(x, y) = x − bx3, q1(x, y, ωτ, ε) = ζcos(ωτ) − h1y + F (ω∗ − y),
p0(x, y) = y, p1(x, y, ωτ, ε) = 0, h1 = 2h/ε. (2.90)
In this case, Melnikov’s function (2.67) takes the form
M(τ0) = −+∞∫
−∞
y0(t)[ζcos(ω(t + τ0)) − h1y0(t) + fssgn(ωr) − αωr + βω3r ]dt,
(2.91)and the dimensionless relative velocity is
ωr(t) = ω∗ − y0(t). (2.92)
Taking into account (2.66), (2.92), and (2.91), the following Melnikov’sfunction is defined,
M(τ0) = A − ζBsin(ωτ0), (2.93)
where
A =16β
35b2+ h1
4
3b− 4α
3b+
8βV 2
b2
+ 2fs
√2
b
⎡⎣√
1
2+
√1
4− V 2 −
√1
2−√
1
4− V 2
⎤⎦H
(1
2− V
), (2.94)
72 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
H(x) = 1, x > 0, H(x) = 0, x ≤ 0,
B =2πω√
2bcosh(πω/2), V = ω1
√b
2. (2.95)
Owing to Melnikov’s theory, if the function M(τ0) has simple zeros, then forsufficiently small parameter ε, motion governed by (2.64) should be chaotic. Itoccurs when
|A| < ζB. (2.96)
Owing to wear occurrence kw > 0, after some time, the bush wear amount ofU0 appears (contact pressure p(τ) = 0). Thus, owing to (2.92), the bush starts tovibrate in the chaotic manner when the following condition is satisfied.
h1
ζ<
3√
b
2√
2
πω
cosh(πω/2). (2.97)
Note that the obtained condition is in agreement with that reported by[Guckenheimer, Holmes (1983)]. Furthermore, in the case without damping (see(2.97)), that is, for h1 = 0, a solution to the studied equation (ζ > 0) is alwayschaotic.
2.3.5.2 Numerical analysis
The condition (2.96) is rewritten in a more suitable form
ψ(α, β, fs, b, ω, ω1, h1) < ζ, (2.98)
where ψ(α, β, fs, b, ω, ω1, h1) = |A|/B.Figure 2.31 shows the zones of parameters associated with chaos. According to
Fig. 2.31a, if for chosen parameters (ω, ω1) the dimensionless amplitude lies overthe shown surface, then these parameters provide a chaotic response.
First, ζ plays the role of a control parameter in the case without account of wear.Numerical analysis is carried out for Equations (2.64) for ε = 1, kw = 0. Thebifurcation diagrams are shown in Figs. 2.32a and 2.32b for various values of theparameter b. For b = 1, according to (2.98), the critical dimensionless amplitudevalues ζch = 0.637 for a chaotic threshold are estimated.
For b = 9 the critical value of the control parameter dramatically decreases andachieves ζch = 0.084. The bifurcation diagrams shown in Figs. 2.32 verify theresults obtained through Melnikov’s theory.
When the contact pressure p(τ) = 0 and there is no damping, then accordingto (2.97), chaos always exists and the chaotic zone is decreased with an increaseof b. Fig. 2.33 confirms the latter remark.
In the case of damping occurrence, the bifurcation diagrams for the controlparameter h2 = h1/ζ are also studied. They are given in Fig. 2.34. Critical values
2.3 Kinematic external shaft excitations 73
Fig. 2.31: Chaotic threshold in various parameter spaces: (a) (ψ, ω, ω1) (α = β = fs = 0.3,b = 5, h1 = 0), (b) (ψ, α, β) (fs = 0.3, b = 4, ω = 2, ω1 = 0.4, h1 = 0), (c) (ψ, b, ω1)(α = β = fs = 0.3, ω = 2, h1 = 0), (d) (ψ, b, ω) (α = β = fs = 0.3, ω1 = 0.4, h1 = 0).
Fig. 2.32: Bifurcation diagram for the variable ϕ: (a) b = 1, ζch = 0.637; (b) b = 9, ζch = 0.084;(α = β = fs = 0.3, ω = 2, ω1 = 0.4).
hch of the parameter h2, below which chaotic motion occurs, are computed through(2.97).
Figure 2.35 shows the evolution in time of the dimensionless wear uw forvarious parameters (curves 1, 3, 5, 7 (2, 4, 6, 8) correspond to b = 1 (9)).
74 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.33: Bifurcation diagram for the variable ϕ: (a) b = 1, ζch = 0; (b) b = 9, ζch = 0;(ω = 2, kw �= 0).
Fig. 2.34: Bifurcation diagram for the variable ϕ: (a) b = 1, hch = 0.575; (b) b = 4, hch = 1.15(α = β = fs = 0.3, ω = 2, ω1 = 0.4).
Fig. 2.35: Dimensionless wear versus τ for kw = 0.1, b = 1, ω1 = 0.4 (curves 1); kw = 0.1,b = 9, ω1 = 0.4 (curves 2); kw = 0.01, b = 1, ω1 = 0.4 (curves 3); kw = 0.01, b = 9,ω1 = 0.4 (curves 4); kw = 0.1, b = 1, ω1 = 0.8 (curves 5); kw = 0.1, b = 9, ω1 = 0.8(curves 6); kw = 0.01, b = 1, ω1 = 0.8 (curves 7); kw = 0.01, b = 9, ω1 = 0.8 (curves 8)(α = β = fs = 0.3, ω = 2, ζ = 0.7).
Bush phase trajectory accounted wear is illustrated in Fig. 2.36. In Fig. 2.36cbush dynamics without wear is shown. Figure 2.36b (2.36a) refers to the wearcoefficient kw = 0.01 (kw = 0.1).
2.3 Kinematic external shaft excitations 75
Fig. 2.36: Bush phase trajectory (a) kw = 0.1, b = 1, ω1 = 0.4; (b) kw = 0.01, b = 1,ω1 = 0.4; (c) kw = 0, b = 1, ω1 = 0.4 (α = β = fs = 0.3, ω = 2, ζ = 0.7).
From the mentioned figures, one may conclude that wear essentially influencesa stick-slip motion.
2.3.6 Analysis of the bush motion with wear and cylinder
kinematic excitations
In this section, heat extension is not considered; that is, γ = 0. Additionally, wetake ζ = 0, χ = 1 which gives t∗ =
√B2/(k∗R2
2). We also take hU (τ) = H(τ),U∗ = U0, hT (τ) = 0. Owing to the introduced quantities, a system of equationsand inclusion (2.5), (2.22), (2.23) is given in the form
ϕ(τ) − aϕ(τ) + bϕ3(τ) ∈ εF (ω1 − ϕ)p(τ), 0 < τ < ∞, (2.99)
ϕ(0) = ϕ◦, ϕ(0) = ω◦; (2.100)
p(τ) = hU (τ) − uw(τ), (2.101)
uw(τ) = kw
τ∫
0
|ω1 − ϕ(ξ)|p(ξ)dξ, (2.102)
where
F (ϕ) = fsSgn(ϕ) − αϕ + βϕ, (2.103)
ω1 = ω∗ + ζksin ωτ. (2.104)
The considered problem is reduced to looking for a solution of the nonlin-ear system of differential and integral equations with respect to velocity ϕ andwear uw.
76 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.37: Phase trajectory of bush movement in the conditions of wear of the bush: (a) curve 1,a = 4; 2, a = 2; 3, a = 0.1; (b) curve 1, a = −4; 2, a = −2; 3, a = −0.1.
Numerical analysis. The results of calculations for different values of the para-meter a, ζ for the steel cylinder (E1 = 19 ·1010Pa, ν1 = 0.3) and ω = 2, ω∗ = 0.4,b = 1, α = 0.3, β = 0.3, fs = 0.3, ϕ◦ = 0, ω◦ = 0 are shown in Fig. 2.37.Consider the case hU (τ) = H(τ).
The case when the system behaviour depends on control parameter a in theabsence of external disturbance ζk = 0 has been considered. A numerical analysisof Equations (2.99)–(2.102) has been performed for ε = 1, kw = 0.1. The phasetrajectory for positive values of the parameter a is represented in Fig. 2.37a andfor negative a in Fig. 2.37b.
Also for the same case a dimensionless wear dependence upon dimensionlesstime is shown in Fig. 2.38. As can be seen, wear significantly influences stick-slipmovements. This kind of movement changes with time into harmonic ones. At thesame time, the character of the movements does not play a significant role in wearvalue.
Furthermore the system movement under the external kinematic disturbancewas considered.
In Section 2.3.4.2 Melnikov’s method is applied to analyse chaotic motions ofthe pad driven kinematically. Melnikov’s function is derived when kw = 0 (nowear), and the value of k∗ is found from a = 1.
When Melnikov’s function (2.70) changes sign then the system movement ischaotic.
Calculations were performed for ζk = 3.5, ε = 0.1. Melnikov’s function doesnot change sign. Consequently according to Melnikov’s theory, chaos will notappear. A numerical analysis of Equations (2.99)–(2.102) was performed. Thenthe system trajectory on the phase plane was drawn. In Fig. 2.39a the dashedcurve is typical of systems with friction. Curve 1 in Fig. 2.39a corresponds to thebehaviour of the system with wear kw = 0.1.
2.3 Kinematic external shaft excitations 77
Fig. 2.38: Time evolution of dimensionless wear for different values of the parameter a:(a) curve 1, a = 4; 2, a = 2; 3, a = 0.1; (b) curve 1, a = −4; 2, a = −2; 3, a = −0.1.
Fig. 2.39: Phase trajectory of bush movement in conditions of wear of the bush: (a) solid curve 1:ζk = 3.5, kw = 0.1, dashed curves ζk = 3.5, kw = 0; (b) solid curve 2: ζk = 4.5, kw = 0.1,dashed curves ζk = 4.5, kw = 0.
Furthermore, for calculations we have assumed the following value of the con-trol parameter ζk = 4.5. Melnikov’s function M(τ0) changes sign in this caseand according to Melnikov’s theory chaos should appear. Also for the same casethe numerical analysis of Equations (2.99)–(2.102) was performed. Figure 2.39bshows the phase trajectory of the analysed system. Contrary to the previously anal-ysed movement for ζk = 3.5 now we can see that the system oscillates chaotically(see the dashed curve in Fig. 2.39b).
When wear is taken into account, the system will behave as represented bycurve 2 in Fig. 2.39b. Chaos is absent.
78 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.40: Time evolution of dimensionless wear uw for different values of the parameter ζk.Curve 1: ζk = 3.5; 2: ζk = 4.5.
Also for this case, the dimensionless wear dependence upon dimensionlesstime for different values of the dimensionless amplitude of kinematic distur-bance is shown in Fig. 2.40. It can be seen that wear sufficiently influenceschaotic and harmonic movement. They tend to harmonic movement with time.At the same time the movement character does not play a significant rolein wear.
In the considered work, the equation of self-excited movement of a solid shaftfixed in a rigid bush under external kinematic disturbance in the conditions ofa classical friction model and abrasive wear is developed. It is shown that forthe analysed system in conditions of wear absence the analytical calculation ofMelnikov’s function is possible; that is, we can choose those parameters of thesystem that ensure its regular or chaotic movement. Also a numerical calculationof wear influence on the system oscillation character is performed.
2.3.7 Dynamics with external temperature perturbation
In this section we assume that in the initial time instant both cylinder (shaft)and bush are in free state (without internal excitation, U0 = 0, T∗ = T0, U∗ =2α1(1+ν1)R1T0, P∗ = 2α1E1T0/(1−2ν1)), and dimensionless bush temperatureis governed by the equation hT (τ) = 2(1 − exp(−δτ2)). Owing to heat transfer,the cylinder starts to extend its volume and a contact between cylinder and bushoccurs.
The cylinder rotates with angular velocity Ω(t) = Ω∗ω1(t) such that the cen-trifugal forces can be neglected. We assume that angular speed of the shaft rotationchanges in accordance with ω1 = ω∗ + ζksin ω′t, where ζk is the dimensionless
2.3 Kinematic external shaft excitations 79
Fig. 2.41: Phase plane (a) and Poincare section (b) of bush motion (curves 1: ζk = 3.5, curves 2:ζk = 4.5; solid curves: γ = 1, dashed curves: γ = 0).
amplitude of the kinematic excitation. We take k∗ = k1(l0/l1−1)(1+l1/R2)−k2,and the dependence of friction on relative velocity is approximated by the function(2.61). First, we assume that there is no wear (kw = 0).
In dimensionless form this problem is governed by Equations (2.5), (2.6),(2.22), and (2.31)–(2.33).
In the case when the frictional heat generation is not taken into account (γ = 0),the contact pressure tends to a constant value (p(τ) → 1). As we have alreadymentioned in Section 2.3.4, in this case the bush may vibrate in a chaotic manner.The following parameters are fixed during numerical analysis: ε = 0.1, ω = 2,ω∗ = 0.4, Bi = 10, ω = 0.1, δ = 10, b = 1, fs = 0.3, α = 0.3, and β = 0.3. Thefollowing initial conditions are taken: x(0) = 0 and y(0) = 0. We are going toinvestigate the influence of control parameter ζk on the system dynamics. Curves 1in Figs. 2.41–2.44 correspond to the values of ζk = 3.5, whereas curves 2 cor-respond to ζk = 4.5. Dashed curves in Figs. 2.41–2.44 correspond to the lack offrictional heat generation (γ = 0), whereas solid curves are related to frictionalheat generation exhibited by coefficient γ = 1. Figure 2.41a shows trajectoriesof the analysed system in phase space, whereas Fig. 2.41b presents the Poincaresection.
In the case when the frictional heat generation is not taken into account, thecontact friction as well as the contact pressure tend to constant values p(τ) →1, θ(τ) → 2. Note that our analytical predictions (2.31)–(2.33) give the samevalues.
Bush motion (as shown in Fig. 2.41) is periodic, which is in agreement withanalytical prediction ζk = 3.5 (see dashed curves 1 and point 1 in Fig. 2.41),whereas for parameter ζk = 4.5 it is chaotic (see dashed curves 2 and points 2 inFig. 2.41).
The frictional heat generation causes the occurrence of a period-2 bush motionfor the parameter ζk = 3.5 and period-1 motion for the parameter ζk = 4.5
80 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.42: Time histories of dimensionless contact pressure (a) and contact temperature(b) (curves 1: ζk = 3.5, curves 2: ζk = 4.5; solid curves: γ = 1, dashed curves: γ = 0).
Fig. 2.43: Time histories for τ ∈ (25, 38) of dimensionless contact pressure (a) and contacttemperature (b) (curves 1: ζk = 3.5, curves 2: ζk = 4.5; solid curves: γ = 1, dashed curves:γ = 0).
(see solid curves 1 and 2 in Fig. 2.41a). The contact pressure and the contacttemperature change in time and for τ > 32 and for ζk = 3.5 the dimensionlessperiod is defined by the formula 2Tp = 2 · 2π/ω = 2π (curves 1 in Fig. 2.44).On the other hand, for τ > 180 and for ζk = 4.5 the dimensionless periodTp = 2π/ω = π (see curves 2 in Fig. 2.44).
2.4 External shaft mechanical excitations 81
Fig. 2.44: Time histories for τ ∈ (176, 190) of dimensionless contact pressure (a) and contacttemperature (b) (curves 1: ζk = 3.5, curves 2: ζk = 4.5; solid curves: γ = 1, dashed curves:γ = 0).
2.4. External shaft mechanical excitations
2.4.1 Shaft inertial motion with tribological processes
We analyse a one-dimensional contact model of a thermoelastic inertial shaft witha fixed rigid bush while frictional heat generation and wear take place [Pyryev(2000b)]. The influence of input parameters on the contact characteristics (shaftrotational velocity, contact pressure, temperature, and wear) is studied. A simi-lar model has been examined by [Goryacheva (1988)] (without thermal expansioneffects), with reference to the influence of various wear coefficients on the wearamount. A similar problem for the elastic layer [Pyryev (1994)] was analysedby [Olesiak, Pyryev(1998)], and [Pyryev, Grilitskiy (1996)] and without wearin reference [Pyryev, Grilitskiy (1995)]. Such problems were also studied ear-lier by [Pyryev, Grilitskiy (1996)] (without accounting for body masses) and by[Barber et al. (1985)] in the case of uniform braking. The braking processes,assuming that external loadings are known, have been analysed by [Chichinadzeet al. (1979)]. A quasi-stationary approach to the discussed problem was proposedby [Aleksandrov, Annakulova (1990)].
Here, we assume that a contact zone and its geometry enable a sufficientapproximation of the problem by a one-dimensional model. Thus, variouspeculiarities associated with real frictional behaviour can be included in theanalysis [Chichinadze et al. (1979), Goriaceva (1988), Aleksandrov, Annakulova(1990)].
Next, we assume that k2/k∗ → ∞ and that the bush is fixed; that is, ϕ2 = 0.Taking ω = 1, φ0 = 0 we have t∗ = R2
1/a1. Furthermore, we assume that a frictioncoefficient does not depend on the shaft rotational velocity; that is, F = fs =const. The shaft rotates at the velocity Ω(t) = Ω∗ω1(t). It is assumed also that
82 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
the shaft rotational velocity changes in accordance with the shaft rotational motiondue to input of the moment M0hM (t) (hM → 1, t → ∞) (see Equations (2.24)).
In the dimensionless form, Equations (2.18) to (2.23), (2.27), and (2.28) gov-erning our problem are as follows,
∂2θ(r, τ)
∂r2+
1
r
∂θ(r, τ)
∂r=
∂θ(r, τ)
∂τ, 0 < τ < τc, 0 < r < 1; (2.105)
∂θ(1, τ)
∂r+ Bi[θ(1, τ) − hT (τ)] = γfsωrp(τ) + γfsωrQ(τ); (2.106)
r∂θ(r, τ)
∂r
∣∣∣∣r→0
= 0, 0 < τ < τc; (2.107)
θ(r, 0) = 0, 0 < r < 1. (2.108)
p(τ) = −uw(τ) +
1∫
0
θ(η, τ)ηdη, 0 < τ < τc; (2.109)
uw(τ) = γfsωrξ
τ∫
0
p(η)dη + γfsωrξ
τ∫
0
Q(η)dη, 0 < τ < τc; (2.110)
φ(τ) = ω01 + aM
⎡
⎣m0
τ∫
0
hM (η)dη − fs
τ∫
0
p(η, φ)dη
⎤
⎦ , 0 < τ < τc,
(2.111)
Q(τ) = (φ/ωr − 1)p(τ), (2.112)
where
ξ =kw
γfs=
Kwλ1
2fsα1a1(1 + ν1).
Note that in Equations (2.106), (2.110), and (2.111) ωr denotes an arbitraryconstant enabling separation of the linear system part.
Applying the Laplace transformations [Carslaw, Jaeger (1959)], a solution tothe problem (2.105)–(2.112) has the following form,
θ(r, τ) = ψθ(R, τ) + vd2
dτ2G1(r, τ) ∗ Q(τ), (2.113)
p(τ) = ψp(τ) + vd
dτF2(τ) ∗ Q(τ), (2.114)
uw(τ) = ψu(τ) + vξd
dτF1(τ) ∗ Q(τ), (2.115)
2.4 External shaft mechanical excitations 83
φ(τ) = ψφ(τ) − aMf vF2(τ) ∗ Q(τ), (2.116)
where
ψθ(r, τ) = Bid
dτG2(r, τ) ∗ hT (τ),
ψp(τ) = Bid2
dτ2F3(τ) ∗ hT (τ),
ψu(τ) = Bivξd
dτF3(τ) ∗ hT (τ),
ψφ(τ) = ω◦1 + aM
[m0hM (τ) ∗ H(τ) − f Bi
d
dτF3(τ) ∗ hT (τ)
],
Gn(r, τ) = gn +
∞∑
m=1
Dn(r, sm)
sm∆′(sm)esmτ ,
Fn(τ) = fn +
∞∑
m=1
∆n(sm)
sm∆′(sm)esmτ , n = 1, 2, 3, (2.117)
∆1(sm) = Bi Cm + smSm; ∆2(sm) = smSm − ξ∆1(sm); ∆3(sm) = Sm;
∆(s) = s∆1(s) − v∆2(s); D1(r, s) = CRm; D2(r, s) = (sm + vξ)CR
m;
∆′(sm) = 0.5{(Bi Sm + Cm)(sm + vξ) + 2∆1(sm) − vCm};
CRm = I0(r
√sm), Sm = I1(
√sm)/
√sm, Cm = I0(
√sm),
m = 1, 2, 3, . . .
f1 =1
vξ, f2 = −1
v, f3 =
1
2Bi vξ, g1 =
1
Bi vξ, g2 =
1
Bi, v = γfsωr,
and sm are the roots of the characteristic equation ∆(s) = 0 (m = 1, 2, 3, . . . ).The analysis carried out by [Pyryev, Grilitskiy (1996)] shows that Imsm = 0,Resm < 0 for m = 3, 4, . . . , whereas for m = 1, 2 they lie either in left or righthalf-planes of the complex plane. For 0 < ξ < ξ1 (ξ1 = 1/(1+Bi/2)) and v ≤ v2
the roots are negative, for v2 < v < v1 the root is a complex conjugate withnegative real part, for v1 < v < v3 the root is a complex conjugate with positivereal part, and for v3 < v the root is positive. The approximate relations for vi
(i = 1, 2, 3) are reported by [Pyryev, Grilitskiy (1996)].The quantities p(τ) and φ(τ) that appeared in Equations (2.114) and (2.116)
are governed by a system of nonlinear Volterra–Hammerstein integral equationsof the second-order convolution type [Verlan, Sizikov (1986)]. The numerical
84 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
investigations of the problem are carried out using quadrature and the followingasymptotic estimations.
G1(1, τ) =4τ
3
√τ
π+ O(τ2), G2(1, τ) = 2
√τ
π+ O(τ),
F1(τ) = τ + O(τ1.5), τ → 0,
F2(τ) = (1 − ξ)τ + O(τ1.5), τ → 0,
F3(τ) = 0.5τ2 + O(τ2.5), τ → 0. (2.118)
2.4.1.1 Solution properties
The behaviour of frictional characteristics of the thermoelastic contact is describedby the functions: hT (τ) = (1 − exp(−δτ2))H(τ), hM (τ) = H(τ). Depending onthe parameters, the following famous system dynamic states can be realized.
Behaviour in the initial time instant. Accounting for the Laplace transform, thefollowing asymptotic terms are found for the thermoelastic contact characteristicsin the initial time instant,
θ(1, τ ) = δ16Bi
15τ2
√τ
π+ O(τ3.5),
p(τ) = δBi
3τ3 + O(τ3.5),
uw(τ) = δBi
12γfω◦
1ξτ4 + O(τ4.5),
φ(τ) = ω◦1 + aM
(m0τ − fδ
Bi
12τ4 + O(τ5)
), τ → 0. (2.119)
Shaft motion at constant velocity. When the shaft rotates at constant velocityφ(τ) = ωr = const, the function Q(τ) = 0 and the considered problem is linear.First terms of (2.113)–(2.116) provide solutions to this problem. Note that forv ≥ v3, ξ < ξ1, the so-called frictional instability occurs; that is, the frictionalcontact characteristics increase exponentially. The critical velocity is
v3 = 2Bi1 + ξ
ξ1+
√(1 + ξ
ξ1
)2
− (1 − ξ)2
(1 − ξ)2. (2.120)
If there is no wear (ξ = 0), the dimensionless critical velocity is equal to 2Bi.
2.4 External shaft mechanical excitations 85
Behaviour of a solution when it approaches a stationary solution. Duringwear and conditions f∗ > 1 (f∗ = fs/2m0) there are formally stationary solutionsof nonlinear problem (2.105)–(2.112) defined by (in Equation (2.13), (2.24),and (2.105) differential terms with respect to time are neglected) the parameters:pst = 0.5f−1
∗ , θst = 1, ωst = 0, and uw = 0.5 − pst. Analysis of roots of thelinearized characteristic equation in the vicinity of the stationary solution governedby (2.105)–(2.112) gives Res1 > 0; that is, the stationary solutions are unstableand the inequality τc < ∞ is satisfied.
In the case without wear (ξ = 0), the stationary solution of the nonlinearproblem for f∗ < 1 is
pst = 0.5f−1∗ , θst = 2pst, ωst = 2Bi(1 − f∗)/γfs. (2.121)
The characteristic algebraic equation of the linearized nonlinear problem(2.105)–(2.112) has the form (ξ = 0):
s∆1(s) + (γfsaMm0 − 2Bi s (1 − f1))∆3(s) = 0. (2.122)
It can be shown that if Resm < 0, m = 1, 2, 3, . . . , then the stationary solution(τ → ∞) of the nonlinear problem is stable. Behaviour of a nonlinear problem inthe neighbourhood of the stationary solution is determined for τ → ∞ through theroots s1,2 of the characteristic equation (2.122) which lie closest to the imaginaryaxis. Using the characteristic equation (2.122), the following analytic formuladefines the first two roots
s1,2 = (−B ±√
D)/A, (2.123)
where
B = f∗Bi + γfsamm0/16, D = Bi2f2∗ − γfsamm0(B1 + γfsamm0/768),
A = B1 + f∗Bi/8 + γfsamm0/192, B1 = 1 + Bi/4 + f∗Bi/8.
If the following condition is satisfied,
γfsaMm0 > 384(√
D1 − B1), (2.124)
where D1 = B21 +f2
∗Bi2/192, then the roots (2.123) are complex and the contactcharacteristics exhibit damped oscillation. The imaginary part of the roots definesthe period; that is, Tp = 2π/Ims1.
2.4.1.2 Numerical results
In order to verify the theoretical analysis concerning dynamics of the contactcharacteristics, numerical computations are carried out for the obtained solutionfor different values of the control parameter ξ. A cylinder made from steel is
86 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
examined (ν1 = 0.3, E1 = 190 GPa, α1 = 14 · 10−6◦C−1, λ1 = 21 W m−1◦C−1,a1 = 5.9 · 10−6 m2 s−1) for R1 = 30 mm, δ = 104, Bi = 24, t∗ = 153 s.Depending on the parameters, the considered model can exhibit either braking oracceleration processes further approaching a stationary state.
Acceleration process (f∗ < 1). The problem is dimensionalised for T0 = 5◦ C,ω◦
1 = 15.2, fs = 0.1, m0 = 1/π, aM = 300π and for P∗ = 6.65 · 10−2 MPa,U∗ = 5.46 · 10−6 m. Stationary values (without wear) are as follows: pst = 3.18,θst = 6.37, and ωst = 108.3. Both numerical and analytical analyses show thatduring acceleration and when the contact characteristic approaches a stationarystate and in the absence of wear (the conditions (2.124) hold), damped vibrationsoccur (dashed curves in Fig. 2.45).
Note that the occurrence of even small wear essentially disturbs the contactcharacteristics (see curves 1–4 in Fig. 2.45).
Fig. 2.45: Time histories of the relative pressure p(τ)/pst (a), relative shaft velocity φ(τ)/ωst
(c) and wear uw = Uw(τ)/U∗ (d) versus dimensionless time during accelerating (curves 1:ξ = 0.001, curves 2: ξ = 0.002, curves 3: ξ = 0.003, curves 4: ξ = 0.01, curves 5: ξ = 0.05,dashed curves: ξ = 0).
2.4 External shaft mechanical excitations 87
Figure 2.45 shows the influence of ξ on relative pressure (Fig. 2.45a), relativetemperature (Fig. 2.45b), relative velocity (Fig. 2.45c), and wear (Fig. 2.45d) ondimensionless time.
The period of oscillations of the contact characteristics (curves 1 and 2 inFig. 2.45) is equal to Tos = 1.6. The period estimated analytically through theroots (2.123) is of the amount of 1.8. Results are improved when the roots s1,2
are close to the imaginary axis. Increasing the shaft inertial moment (decrease ofthe parameter ε) causes an increase of the oscillation period. An increase of wearstabilises time history (see curves 5 in Fig. 2.45), whereas time contact interval τc
and wear amount uw decrease. When acceleration of the periodic motion is zero,then the contact pressure achieves its stationary value.
Braking process (f∗ > 1). The following parameters are taken: T0 = 10◦ C,m0 = 0, ω◦
1 = 76.3 · 102, fs = 0.01, aM = 2π · 106 and P∗ = 13.3 · 10−2 MPa,U∗ = 10.9 ·10−6 m. The numerical analysis shows that when the shaft stops and itstemperature achieves the bush temperature, it will not move again (curves 1 and 2 inFig. 2.46). Owing to an increase of the wear coefficient, the braking time (curves 1and 2 in Fig. 2.48) and wear increase, whereas contact pressure and temperaturedecrease. A further increase of the wear coefficient causes disappearance of thecontact zone between the shaft and bush (curves 3 in Fig. 2.46). An increase ofthe shaft inertial moment causes an increase of the braking time.
2.4.2 Inertialess shaft and bush dynamics and frictional heat
generation
In this section (contrary to 2.3), one of the bodies is subject to mechanicalexcitation, which makes us refer to the additional equation of this body. The latternew model opens some new essential questions. Does frictional TEI occur in thiscase, and if not, then how is the system dynamics realized? Is it possible to realizea self-excited motion? What kind of the system motion appears during the brak-ing process? Answers to these questions are given in this section. Note that now,contrary to the case described in Section 2.3, another mechanism of the contactoccurrence between the bodies takes place.
Let us remember that in Section 2.3 bush and shaft are in contact owing tothe introduced initial shaft compression. In this section we study the contactbetween bush and shaft which occurs via thermal shaft extension. In real con-ditions this situation takes place when the shaft thermal extension exceeds thebush heat extension. The mentioned case has been analysed, for instance, in theworks of [Pyryev et al. 1995, Pyryev, Mandzyk 1996, Grilitskiy et al. (1997),(1998)] for an infinite two-layer circular cylinder under friction heating. In thiswork we assume that the shaft transfers heat ideally and hence it is not thermallyextended.
88 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.46: Evolution in time of dimensionless contact pressure p = P/P∗ (a), contact temperatureθ = T/T0 (b), rotational shaft velocity φ/ω◦
1(c) and wear uw = Uw/U∗ in breaking instant
(curves 1: ξ = 0.01, curves 2: ξ = 0.05, curves 3: ξ = 0.1, dashed curves: ξ = 0).
Observe that the system under analysis is described by two characteristic times,that is, the characteristic time related to system oscillations tD (small) and thecharacteristic time related to heat transfer tT (large). If the ratio of these twotime-scales is ‘small,’ then the system can be treated as an uncoupled one.
A numerical verification of the presented analytical estimations of periodicoscillations completes the considerations.
It is assumed that the shaft has the inertial moment B1 and rotates accordingto the rotational motion due to input torque action. It is also assumed that at thebeginning the shaft and bush are not subjected to the external load (T∗ = T0,U∗ = α1(1+ ν1)R1T0, P∗ = α1E1T0/(1−2ν1), γ = α1E1a1/(1−2ν1)λ1, V∗ =R1/tT , tD =
√B2/k2/R2, ω = tD/tT , aM = 2πR2
1t2T P∗/B1, U0 = 0), and the
dimensionless bush temperature changes within the rule hT (τ) = (1−exp(−δτ2)).Note that due to heat transfer the rotating shaft starts to expand and a contactbetween the shaft and bush appears. Consider the case when the bush is coupledwith the housing through springs with the stiffness k2 (k1 = k3 = 0). Owing to an
2.4 External shaft mechanical excitations 89
arbitrary parameter k∗ = k2, which gives the constant a = −1. It is also assumedthat χ = 1 (hence t∗ =
√B2/(k∗R2
2)).The following friction coefficient versus the relative velocity is accounted for
in (2.61).Observe that F (y) = f(V∗y) has a local minimum for ymin =
√α/3β =
Vmin/V∗. The function Sgn(x) is approximated by the function sgnε0(y) of the
form (2.58).
2.4.2.1 Application of the Laplace transform
Applying the Laplace transform to Equations (2.18), (2.22), and (2.23) (seeour earlier consideration described in Section 2.2.4), the following functions areobtained,
p(τ) = 2Bi
τ∫
0
Gp(τ − ξ)hT (ξ)dξ
+ 2γ
τ∫
0
Gp(τ − ξ)F (φ − ϕ)p(ξ)(φ − ϕ)dξ, (2.125)
θ(r, τ) = Bi
τ∫
0
Gθ(r, τ − ξ)hT (ξ)dξ+
+ γ
τ∫
0
Gθ(r, τ −ξ)F (φ − ϕ)p(ξ)(φ − ϕ)dξ, (2.126)
where Gp(τ), Gθ(r, τ) are the known functions (see (2.33)) of the form
{Gp(τ), Gθ(1, τ)} ={0.5, 1}
Bi−
∞∑
m=1
{2Bi, 2μ2m}
μ2m(Bi2 + μ2
m)e−µ2
mτ , (2.127)
and μm (m = 1, 2, 3, . . . ) are the roots of the characteristic Equation (2.33).Note that the considered problem is governed by the system of nonlinear ordi-
nary differential Equations (2.5) and (2.27),
ϕ(τ) + ω−2ϕ(τ) = εF (φ − ϕ)p(τ),
φ = aM [m0hM (τ) − F (φ − ϕ)p(τ)], (2.128)
and the integral equation (2.125) governing the rotational velocities ϕ(τ), φ andthe contact pressure p(τ). The temperature is defined by formulas (2.126).
90 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
2.4.2.2 Stationary dynamics
Observe that in Equation (2.128) the variable φ(τ) does not appear, and hencethe phase space of the bush–shaft system is three-dimensional. Consider the casewhen after a transitional process the cylinder starts to rotate at constant velocityφ(τ) = ωst and the bush does not move ϕ = 0. Then, the steady-state solution ofthe problem under consideration (in Equations (2.128) and (2.18) the differentialterms are omitted) has the following form,
pst =1
1 − v, θst =
1
1 − v, ϕst = ω2εm0, v =
γωstF (ωst)
Bi,
(2.129)where ωst is the solution to the nonlinear equation
F (ωst) =m0
1 + γm0ωst/2Bi. (2.130)
Function F is similar to that defined in (2.61) with (2.130) taken into account,m0 is the applied dimensionless moment, γ is the parameter related to thermaldistortivity (γ = 1.87 for stainless steel, γ = 1.71 for aluminium alloy, γ = 1.61for copper, and γ = 1.38 for titanium alloy), and Bi is the Biot number.
As has been already mentioned (Section 2.3.2.2), the stationary solutions(2.129) have physical meaning only if v < 1.
Figure 2.47 shows a graphical solution of Equation (2.130) for variousparameters m0 and Bi. For a stainless steel shaft (α1 = 14 · 10−6◦C−1,λ1 = 21 W m−1◦C−1, ν = 0.3, a1 = 5.9 mm2 s−1, E1 = 190 GPa) parameterγ = 1.87, and for R1 = 4 · 10−3 m the characteristic time tT = 2.71 s, andV∗ = 1.47 · 10−3 m s−1. Solid curve 1 corresponds to m0 = 0.14, Bi = 10,solid curve 2 corresponds to m0 = 0.1, Bi = 10, solid curve 3 corresponds tom0 = 0.05, Bi = 10, and finally, solid curve 4 corresponds to m0 = 0.14, Bi = 1.The dashed curve represents function F (ωst).
Equation (2.130) may have: one solution ω3st (F ′(ω3
st) > 0) for m0 = 0.14,Bi = 10 (first case); three solutions ω1
st, ω2st, ω3
st (F ′(ω1st) > 0, F ′(ω2
st) < 0,F ′(ω3
st) > 0) for m0 = 0.1, Bi = 10 (second case); one solution ω1st = 0 for
m0 = 0.05, Bi = 10 and for approximation (2.58) ω1st ≈ ε0m0/2fs, F ′(ω1
st) ≈2fs/ε0 (third case); one solution ω2
st (F ′(ω2st) < 0) for m0 = 0.14, Bi = 1 (fourth
case).For a small value of γ/Bi (γ/Bi ≪ 1) (frictional heat generation is neglected
γ = 0) and for m0 ∈ [0, fmin), Equation (2.130) can have one solution ω1st = 0.
It may also have three solutions ω1st, ω2
st, and ω3st, for m0 ∈ (fmin, fs), and one
solution ω3st for m0 ∈ (fs,∞). Equation (2.130) shows that the stationary solution
does not depend on parameter ω. Let us now analyse perturbations of the stationaryprocess (2.130), which are defined by the equation hT (τ) = 1 + h∗
T (τ).A solution is sought in the form
ϕ(τ) = ϕst + ϕ∗(τ), θ(r, τ) = θst(r) + θ∗(r, τ),
2.4 External shaft mechanical excitations 91
Fig. 2.47: Graphical solution to Equation (2.130). Solid curves 1: m0 = 0.14, Bi = 10,2: m0 = 0.1, Bi = 10, 3: m0 = 0.05, Bi = 10, 4: m0 = 0.14, Bi = 1. The dashed curvecorresponds to F (ωst).
p(τ) = pst + p∗(τ), ϕ = ϕ∗(τ), φ = ωst + φ∗(τ), (2.131)
where |ϕ∗| ≪ 1, |θ∗| ≪ 1, |p∗| ≪ 1, |ϕ∗| ≪ 1, |φ∗| ≪ 1.After linearization of the right-hand sides of Equations (2.128) and the boun-
dary condition (2.2.2), the following set of perturbation equations is obtained.
ϕ∗(t) + ω−2ϕ∗(t) = εF (ωst)p∗(τ) + εF ′(ωst)pst(φ
∗ − ϕ∗), 0 < τ < ∞,(2.132)
ϕ∗(0) = 0, ϕ∗(0) = 0; (2.133)
p∗(τ) = 2
1∫
0
θ∗(ξ, τ)ξdξ, 0 < τ < ∞; (2.134)
∂2θ∗(r, τ)
∂r2+
1
r
∂θ∗(r, τ)
∂r=
∂θ∗(r, τ)
∂τ, 0 < τ < ∞, 0 < r < 1;
(2.135)
92 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
∂θ∗(1, τ)
∂r+ Biθ∗(1, τ) = Bi h∗
T + γ[ωstF (ωst)p∗(τ)
+ pst(φ∗ − ϕ∗)(F (ωst) + ωstF
′(ωst))], (2.136)
r∂θ∗(r, τ)
∂r
∣∣∣∣r→0
= 0, 0 < τ < ∞, θ∗(r, 0) = 0, 0 < r < 1,
(2.137)
φ∗(t) = −aM [F (ωst)p∗ + F ′(ωst)pst(φ
∗ − ϕ∗)], 0 < τ < ∞, (2.138)
φ∗(0) = 0, φ∗(0) = 0. (2.139)
Application of the Laplace transformation to the linear system (2.132)–(2.139)of the form
{θ∗(r, s), p∗(s), ϕ∗(s), h∗T (s), φ∗(s)} =
∞∫
0
{θ∗, p∗, ϕ∗, h∗T , φ∗}e−sτdτ,
provides a solution in the form of the Laplace transform. The characteristic equa-tion of the linearized problem has the form
∆∗1(s) = 0, (2.140)
∆∗1(s) = s∆∗(s) + aMpst(ω
2s2 + 1)[β2∆1(s) + 2Bi vβ1∆2(s)],
∆∗(s) = ∆1(s)Ω2(s) − 2Bi v∆2(s)Ω1(s), (2.141)
Ω1(s) = ω2
(s2 − εβ1
1 − vs
)+ 1, Ω2(s) = ω2
(s2 +
εβ2
1 − vs
)+ 1,
β2 = F ′(ωst), β1 =F (ωst)
ωst.
The roots sm (Res1 > Res2 > · · · > Resm > · · · , m = 1, 2, 3, . . . ) ofthe characteristic equation (2.140) may lie in the left-hand part (LHP) Res < 0(a stationary solution is stable) or in the right-hand part (RHP) Res > 0 (a sta-tionary solution is unstable) of the complex plane (s is a complex variable). Theparameters separating the two half-planes are called critical.
If frictional heat generation is not taken into account (γ = 0), the characteristicequation is governed by the following cubic one: sΩ2(s) + β2aM (ω2s2 + 1) = 0.Its roots lie in the RHP of the complex plane if β2 < 0. For a given bush velocity(one may assume aM → 0), the characteristic equation is reduced to the form∆∗(s) = 0, whereas for γ = 0 (the frictional heat generation is not taken intoaccount) it is Ω2(s) = 0.
2.4 External shaft mechanical excitations 93
Let us analyse the stationary stable solution in more detail. The characteristicfunction has the form
∆∗1(s) =
∞∑
m=0
smbm, (2.142)
b0 = aMBi pst(β2 + vβ1), d0 = Bi(1 − v),
b1 =d0 + aMpst[2(2 + Bi)β2 + Bi vβ1]
8,
d1 = 0.5 + Bi[0.25 − 0.125 v + ω2pstε(β2 + v β1)],
bm = dm−1 + aMpst[(d(1)m + ω2d
(1)m−2)β2 + 2Bi v(d(2)
m + ω2d(2)m−2)β1],
dm = d(1)m − 2Bi vd(2)
m + ω2εpst(β2d(1)m−1 + 2Bi vβ1d
(2)m−1)
+ ω2(d(1)m−2 − 2Bi vd
(2)m−2), m = 2, 3, . . .
d(1)m =
Bi + 2m
22m(m!)2, d(2)
m =1
22m+1m!(1 + m)!, m = 0, 1, . . . .
Observe that for small value of aM the roots being sought can be found inEquation (2.142) for m = 2:
s1,2 = 0.5(−b1 ±√
b21 − 4b0b2)/b2. (2.143)
It should be emphasised that for a given shaft velocity (aM → 0), Equa-tion (2.142) provides the condition v > 1 of a frictional TEI. However, when amoment of inertia of the shaft (aM > 0) is taken into account, then the frictionalTEI does not occur. In the latter case, the system itself controls the rotationalshaft velocity by always keeping v < 1. Note that the shaft dynamics significantlyinfluences the values of the characteristic equation roots only if the parameterω2(aM + ε) cannot be considered a ‘small’ one.
Under the assumption of the temperature increase of the surrounding mediumup to T∗ = 5◦C, the stainless steel shaft of the radius R1 = 4 · 10−3 m isfurther investigated (P∗ = 33.2 MPa and tT = 2.71 s). Furthermore, it is assumedthat either the moment M0 = 334 N (m0 = 0.1) or the moment M0 = 468 N(m0 = 0.14) is applied to the shaft. Let either αT = 5.25 · 104 W m−2◦C−1
(Bi = 10) or αT = 5.25 · 103 W m−2◦C−1 (Bi = 1). It is also assumed that theshaft moment of inertia B1 = 245.8 kg · m (aM = 100). On the other hand, it isassumed that the bush moment of inertia B2 = 245.8 kg · m (the internal radiusR2 = 4 · 10−2 m), and the stiffness of the springs k2 = 2.1 · 106 N m−2 (whichgives tD = 0.271 s and ε = 100). The ratio of small tD and large tT time scalesis ω = 0.1.
94 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
In the first case (m0 = 0.14, Bi = 10) there is one solution ω3st = 44.40,
p3st = θ3
st = 2.16, ϕ3st = 0.14 (β1 = 0.14 · 10−2, β2 = 0.12 · 10−2, v = 0.54),
which is stable (the first roots of Equation (2.140) s1,2 = −0.18±10.07i lie in theLHP). The expected ‘period’ of damped oscillations is equal to 2π/Ims1 = 0.62.
In the second case (m0 = 0.1, Bi = 10) there are three solutions. The solutionω3
st = 38.40, p3st = θ3
st = 1.72, ϕ3st = 0.1 (β1 = 0.15 · 10−2, β2 = 0.1 ·
10−2, v = 0.41) is stable (the first roots of Equation (2.140) s1,2 = −0.11 ±10.04i, s3 = −0.53 lie in the LHP). The solution ω2
st = 1.99 (β1 = 4.85 · 10−2,β2 = −0.96 · 10−2, v = 3.58 · 10−2) is unstable (the roots of Equation (2.140)s1,2 = 0.47 ± 9.97i, s3 = 0.87 lie in the RHP). The solution with approximation(2.58) (β1 = 2.4 · 103, β2 = 2.4 · 103, v = 7.8 · 10−7), which corresponds toa periodic motion of the bush and shaft being in a stick (ω1
st ≈ 0) (the roots ofEquation (2.140) s1,2 = −0.52 · 10−4 ± 7.07i lie on the imaginary axis). Observethat in the last case the roots may be found directly from the characteristic equations2 + ω2
0 = 0, where ω0 = 1/(ω√
1 + ε/aM ).In the third case (m0 = 0.05, Bi = 10) there is one solution. The solution
with approximation (2.58) ω1st = 0.21 · 10−4, p1
st = θ1st = 1.0, ϕ1
st = 0.05(β1 = 2.4 · 103, β2 = 2.4 · 103, v = 1.95 · 10−7) corresponds to a periodic motion(the roots of Equation (2.140) s1,2 = −0.52 · 10−4 ± 7.07i are purely imaginary).As in the second case, in the case considered now the roots may also be founddirectly from the characteristic equation s2 + ω2
0 = 0.The steady-state solutions ω1
st correspond to stick conditions with rigid bodytorsional vibrations. Eigenvalues obtained in these cases have a small real partdue to regularization of the step function in the Stribeck approximation. Thesereal parts are spurious effects of the regularization and the real physical behaviourdoes not involve any slip.
In the fourth case (m0 = 0.14, Bi = 1) there is again only one solution. Thesolution ω2
st = 1.41, p2st = θ2
st = 1.37, ϕ2st = 0.14 (β1 = 7.28·10−2, β2 = −1.08·
10−2, v = 0.27) is unstable (the roots of Equation (2.140) s1,2 = 0.745± 10.06i,s3,4 = 0.13±1.36i lie in the RHP). Observe that the roots s1,2 crucially affect theself-excited oscillations as they have the largest real parts and give the oscillationswith the estimated period of Tp = 2π/Ims1 = 0.62, which has been successfullyverified in a numerical way. If there is one unstable solution, the correspondingunsteady-state solution approaches a stick-slip periodic solution.
The most interesting fourth case is characterized by the following limiting cases.For m0 = 0.14, Bi = 1, and when the bush vibrations ω = 0 (fifth case) areneglected, only one solution occurs. The solution ω2
st = 1.41, p2st = θ2
st = 1.37,ϕ2
st = 0.14 (β1 = 7.28 · 10−2, β2 = −1.08 · 10−2, v = 0.27) is unstable (the rootsof Equation (2.140) s1,2 = 0.145± 1.37i lie in the RHP). Note that the roots s1,2
can be well approximated via Equation (2.140) taking into account only the threefirst terms of (2.142) (b0 = 1.2, b1 = −0.043, b2 = 0.615, b3 = 0.0725). Again,they govern the self-excited oscillations with the period Tp = 2π/Ims1 = 4.58.
For m0 = 0.14, Bi = 1, and with no frictional heat generation γ = 0, thereis one solution ω3
st = 86.71, p3st = θ3
st = 1, ϕ3st = 0.14 (β1 = 0.16 · 10−2,
β2 = 0.21 · 10−2, v = 0), which is stable (the first roots of Equation (2.140)
2.4 External shaft mechanical excitations 95
s1,2 = −0.11 ± 10.0i, s3 = −0.216 lie in the LHP). An expected motion has thecharacter of damped oscillations with the period 2π/Ims1 = 0.629 and with thelogarithmic decrement 2πRes1/Ims1 = 0.068.
When comparing the roots of the last and the first case it becomes clear that ifthe frictional heat generation (|Res1|) is taken into account, then the logarithmicoscillation decrement increases. However, the “period” of the damped oscillationsincreases only slightly.
2.4.2.3 Stick-slip process
Let us analyse the system for t → ∞ (hM (τ) = 1), when a stick ωr = φ− ϕ = 0for τ ∈ tst (tst = (τ1, τ2) ∪ · · · (τ2i−1, τ2i) ∪ · · · ), or a slip for τ ∈ tsl (tsl ∈(0, τ1)∪· · · (τ2i, τ2i+1)∪· · · ) occurs. In the first case τ ∈ tst, the bush and cylindermove together, and the governing equation has the form
ϕ(τ)+ω20ϕ(τ) = εm0ω
2ω20 , ω0 = 1/(ω
√1 + ε/aM ), τ ∈ tst. (2.144)
The solution of (2.144) describes a periodic motion
ϕ(τ) = εm0ω2 + C1cos(ω0τ) + C2sin(ω0τ), τ ∈ tst, (2.145)
with the period 2πω√
1 + ε/aM . The cylinder–bush system oscillates periodically,and the bush is subjected to an action of the friction force
F (0)p(τ) = m0 + ω2(C1cos(ω0τ) + C2sin(ω0τ))/(aM + ε). (2.146)
The contact pressure is estimated by the formula
p(τ) = 2Bi
τ∫
0
Gp(τ − ξ)hT (ξ)dξ
+ 2γ
i∑
m=1
τ2m−1∫
τ2m−2
Gp(τ − ξ)F (φ − ϕ)p(ξ)(φ − ϕ)dξ, τ ∈ (τ2i, τ2i+1).
(2.147)
2.4.2.4 Numerical analysis
The numerical analysis of the problem has been carried out using the Runge–Kuttamethod for Equations (2.128) and the quadrature method for Equations (2.125)and (2.126) applying the asymptotic estimations
Gθ(1, τ ) ≈ 2√
τ/π, Gp(τ) ≈ τ, τ → 0. (2.148)
96 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Formula (2.61) has been used to approximate the dependence of the frictionkinematic coefficient on the relative velocity. Numerical calculations have beenexecuted for various values of parameters m0 and Bi, for which an analyticalanalysis has been performed as well.
2.4.2.5 Acceleration process
Assume that at the initial time instant the force moment hM (τ) = 1− exp(−δτ2)acts on the shaft (δ = 100). This moment forces the shaft to rotate with accele-ration. The dimensionless environment temperature is governed by the equationhT (τ) = 1 − exp(−δτ2). Due to the heat transfer the rotating cylinder begins toexpand and eventually comes into contact with the bush. Figs. 2.48–2.51 show theoutcomes of the calculations carried out for ten values of m0 and Bi, and for theinitial conditions ϕ◦ = 0, ω◦ = 0, φ◦ = 0, φ◦ = 0. Curve 1 illustrates the firstcase (m0 = 0.14, Bi = 10, ω = 0.1, γ = 1.87), curve 2 represents the secondcase (m0 = 0.1, Bi = 10, ω = 0.1, γ = 1.87), curve 4 corresponds to the fourthcase (m0 = 0.14, Bi = 1, ω = 0.1, γ = 1.87), and curve 5 presents the fifth case(m0 = 0.14, Bi = 1, ω = 0, γ = 1.87).
Figure 2.48a shows the dependence of the dimensionless angular velocity φ ofthe cylinder (dashed curve) and the bush ϕ (solid curve) on the dimensionlesstime τ for the first and the second of the considered cases is reported. It can beseen that in all cases the system behaviour is in agreement with the analyticalpredictions. In the first case, after certain transitional processes the shaft starts torotate with constant velocity ωst = 44.4. The bush displays damped oscillationswith the period Tp = 0.62. In the second and third case (already for a small forcemoment), the cylinder and bush come into contact and start oscillating periodicallyas one body with the period Tp = 0.89. The stick type oscillations are periodic.
Figures 2.48b,c illustrate the dependence of the dimensionless angular velocityφ of the cylinder (dashed curve) and the bush ϕ (solid curve) on the dimensionlesstime τ for the fourth and the fifth case. In the fourth case, the system shows stick-slip oscillation (Tp = 0.642), and in the fifth case the oscillation exhibited by thesystem is of the thermal stick-slip type (Tp = 4.49). Let us recall that the root ofthe characteristic equation responsible for instability gives the approximated periodTp = 2π/Ims1 = 4.58.
In Figs. 2.49a,b, time histories of both the contact pressure and temperaturefor the considered cases are shown using solid curves 1–5. In cases 4 and 5, thecontact characteristics undergo changes in time.
2.4.2.6 Braking process
It is assumed that at the initial state the shaft rotates at angular velocity φ◦
(hM (τ) = 0). The dimensionless temperature of the bush changes in agreementwith the formula hT (τ) = 1 − exp(−δτ2). Owing to heat transfer, the rotating
2.4 External shaft mechanical excitations 97
Fig. 2.48: Bush dimensionless velocity ϕ (solid curves) and shaft dimensionless velocity φ(dashed curves) versus dimensionless time τ during acceleration for different values of m0 andω. Curve 1: m0 = 0.14, Bi = 10, ω = 0.1, 2: m0 = 0.1, Bi = 10, ω = 0.1, 4: m0 = 0.14,Bi = 1, ω = 0.1, 5: m0 = 0.14, Bi = 1, ω = 0.
cylinder expands and comes into contact with the bush; that is, it starts to brake.The initial conditions are as follows: ϕ◦ = 0, ω◦ = 0, φ◦ = 0, φ◦ = 100. Compu-tational examples for some values of γ are shown in Figs. 2.50 and 2.51. Curve 1corresponds to the case Bi = 10, ω = 0.1, γ = 1.87, whereas curve 2 representsthe case Bi = 10, ω = 0, γ = 1.87 (the bush does not oscillate).
Figure 2.50 shows the dimensionless time histories of the dimensionless angu-lar velocity of the shaft φ (dashed curves) and the bush ϕ (solid curves) duringthe braking process. It is seen (curve 1) that the shaft angular velocity decreases,and the bush undergoes oscillations until the two angular velocities reach the samevalue. Because there is no driving moment and the damping has not been intro-duced, the bush and shaft start to oscillate as one body with the period Tp = 2π/ω0.When the bush dynamics is not taken into account (ω = 0; see curve 2), the shaftvelocity also decreases and finally the shaft stops. A comparison of the results
98 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.49: Dimensionless contact pressure p(τ) (a) and contact temperature θ(τ) = θ(1, τ)(b) versus dimensionless time τ during acceleration for different values of m0, Bi, and ω.Curve 1: m0 = 0.14, Bi = 10, ω = 0.1, 2: m0 = 0.1, Bi = 10, ω = 0.1, 4: m0 = 0.14,Bi = 1, ω = 0.1, 5: m0 = 0.14, Bi = 1, ω = 0.
Fig. 2.50: Time history of the braking pad angular speed ϕ (solid curves) and the shaft speed φ(dashed curves) during braking (m0 = 0) for various values of ω. Curves 1: ω = 0.1, 2: ω = 0.
represented by curves 1 and 2 leads to the conclusion that the shaft braking timefor ω = 0 is smaller than the time interval needed for the shaft and bush to achievea fixed contact (stick) with each other (ω = 0.1). In Figs. 2.51a,b, time historiesof the contact pressure and the temperature are given. Both of the characteristicsincrease in the beginning when γ > 0 (see curves 1 and 2). It can be concludedthat during the braking process the maximal values of the pressure and of thecontact temperature become smaller when the dynamics of the bush is taken intoaccount.
2.4 External shaft mechanical excitations 99
Fig. 2.51: Time history of contact pressure during shaft braking (m0 = 0) for various values ofω. Curves 1: ω = 0.1, 2: ω = 0.
To conclude, one of the classical models of an elastically supported braking padbeing in frictional contact with a rotating shaft during frictional heat generationand heat expansion is analysed with the use of the Stribeck friction model. Thestability analysis of the stationary solutions is followed by the numerically verifiedanalytical estimation of the periodic stick-slip occurrence. It is detected and illus-trated that the stick-slip motion appears in the presence of the driving moment,heat transfer, and thermal expansion of the shaft materials. In addition, numeri-cal calculations illustrating the influence of the parameters on the dynamics andcontact characteristics of the investigated model in the acceleration and brakingprocesses have been performed.
It should be emphasized that, contrary to the results reported in [Pyryev,Grilitskiy (1996)] where already at small wear the frictional TEI occurs (con-tact parameters increase exponentially after the relative speed exceeds the criticalvalue, which is found from the condition v = 1) and either overheating [Pyryevet al. (1995)] or brake heating [Aleksandrov, Annakulova (1990)] may appear, theconsidered system can never be overheated. An increase of the friction force (ormoment of friction force), frictional heat generation, and system overheating iscaused by an increase of the contact pressure while one of the contacting bodiesmoves at constant speed and heat expansion is bounded. In order to keep themotion speed constant, the friction force (or moment of friction force) increases,and consequently, energy is supplied to the system. Although the system heatexpansion is bounded, the contact pressure may increase, which yields an increaseof both the friction force (moment of friction force) and the frictional heat gene-ration. However, the system will not be overheated as the moving body starts tobrake. The heat balance leads to a cooling process, and hence the contact pres-sure and friction force (moment of friction force) start to decrease. This, however,again brings an increase in the relative speed and in the frictional heat generation.
100 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
The described process will repeat again and again, which physically means that thesystem controls itself to avoid overheating. This type of control is called passive.Additionally, it is suggested that the phenomena referred to as ‘stick-slip’ in theclassical terminology be called ‘thermal stick-slip’ instead. This new terminologyseems to be more adequate because a slip relates to a heating process caused by anindependent movement of two bodies, whereas a stick corresponds to the processof cooling that takes place when the relative velocity of two contacting bodiesequals zero. It is worth noting that in the latter case although the bush does notmove, the shaft exhibits thermal stick-slip dynamics.
When the frictional heat generation is not taken into account (γ = 0), thestick-slip oscillations cannot appear. In this case there are two stable stationarysolutions, and one unstable, and hence any trajectory is always attracted by oneof the stable critical points (equilibrium). For γ > 0, thermal stick-slip oscillationcan appear, and then either a short bush oscillation period (Tp = 0.624 for case4) or a long period, being the one of shaft oscillations (Tp = 4.49 for case 5), isachieved. Owing to the dynamics of the considered system, the maximal pressureand temperature values decrease during the braking process.
In other words, in the system where heat expansion of the contacting bodiesis bounded, for the sake of stability it is better to apply the mechanical externalexcitation (considered here) than the kinematic excitation.
2.4.3 Inertial damped dynamics of cylinder and bush and
tribological processes
In this section (contrary to Section 2.4.2) we assume that damping of bushvibrations occurs (h = 0) and its wear is (kw = 0). We assume that the cylinderhas the moment of inertia B1 and rotates according to the applied moment. Wealso assume that in the initial time instant, the cylinder and bush are not driveninternally (U0 = 0, T∗ = T0, U∗ = 2α1(1 + ν1)R1T0, P∗ = 2α1E1T0/(1− 2ν1)),and that dimensionless bush (pad) temperature is changed according to the lawhT (τ) = 2(1 − exp(−δτ2)). Owing to heat exchange the rotating cylinder beginsto increase up to a contact with the bush, and then the wear process begins. Letus consider the case when the bush (bearing) is linked to a basis (frame) bysprings characterized by stiffness k2 (k1 = k3 = 0). Because an arbitrary para-meter k∗ = k2, it gives the following value a = −1. In addition, we assume thatχ = 1 (it provides t∗ =
√B2/(k∗R2
2)).Note that the considered problem is manifested by the system of nonlinear
differential Equations (2.5) and (2.27) and integral equation (2.125) governingangular velocities ϕ(τ), φ(τ) and contact pressure p(τ) values. Temperature isdefined by (2.126), whereas wear by formulas (2.23).
2.4 External shaft mechanical excitations 101
2.4.3.1 Stick-slip process
Consider now how the system behaves for t → ∞ (hM (τ) = 1), when either a stickmotion occurs ωr = φ− ϕ = 0 for τ ∈ tst (tst = (τ1, τ2)∪· · · (τ2i−1, τ2i)∪· · · ),or a slip motion takes place for τ ∈ tsl (tsl ∈ (0, τ1) ∪ · · · (τ2i, τ2i+1) ∪ · · · ).Both bush and shaft for τ ∈ tst do not move relatively to each other. In this casethe process is governed by the equation
ϕ(τ) + 2hω20ϕ(τ) + ω2
0ϕ(τ) = εm0ω20 , ω0 = 1/
√1 + ε/aM , τ ∈ tst.
(2.149)Its solution gives free vibration at frequency ωh = ω0
√1 − ω2
0h2 and with the
period of damped vibrations 2π/ωh:
ϕ(τ) = εm0 + e−hω2
0τ (C1cos(ωhτ) + C2sin(ωhτ)), τ ∈ tst. (2.150)
2.4.3.2 Numerical analysis
Numerical analysis of the stated problem has been carried out using the fourth-order Runge–Kutta method applied to Equations (2.5), (2.6), (2.27), (2.28) andquadrature applied to Equations (2.125) and (2.126) accounting for the asymptoticestimation (2.148).
The dependence of kinematic friction versus the relative velocity (see Fig. 2.2)has been estimated by formula (2.61). The numerical computations are performedfor various parameters kw and γ.
2.4.3.3 Cylinder and bush dynamics during acceleration
Assume that in the initial time instant the force moment hM (τ) = 1− exp(−δτ2)acts on the shaft. This moment forces the shaft to rotate with an acceleration.Dimensionless temperature of the bush is governed by the equation hT (τ) =2(1 − exp(−δτ2)). We take zero as initial conditions: ϕ◦ = 0, ω◦ = 0, φ◦ = 0,φ◦ = 0. Results of computations are given in Figs. 2.52a, 2.53 and 2.54 for a fewvalues of parameter kw and γ.
The following values are taken for a computational purpose: ε = 1, aM = 1,Bi = 1, γ = 20, h = 0.05, ω = 0.1, δ = 10. In Fig. 2.52a dimensionlessangular velocity of the cylinder φ (dashed curves) and bush ϕ (solid curves)versus dimensionless time τ during acceleration (m0 = 0.5) for some parameterskw characterizing bush wear are given.
Figure 2.53 shows time evolutions of contact pressure (a) and temperature(b) during acceleration for the same parameter values. In Fig. 2.54 time evolutionsof dimensionless friction force (Fig. 2.54a) and dimensionless wear (Fig. 2.54b)are presented. In the mentioned figures curves 1 correspond to the case kw = 0.01,whereas curves 2 correspond to kw = 0.1.
102 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.52: Time histories of angular speed of the braking pad ϕ (solid curves) and shaft φ(dashed curves) during acceleration (a) and braking (b) for different values of the parameter kw
(curves 1: kw = 0.01; curves 2: kw = 0.1).
Fig. 2.53: Time histories of contact pressure (a) and contact temperature (b) during accelerationfor different values of the parameter kw (curves 1: kw = 0.01; curves 2: kw = 0.1).
Consider first dynamics for a small value of the wear coefficient kw = 0.01(curve 1). In response to a driven moment action, the shaft starts to rotate (solidcurves). Owing to thermal shaft radial expansion, the contact pressure p increases.To conclude, both dimensionless contact pressure p(τ) (Fig. 2.53a) and dimen-sionless friction force F (φ1− φ2)p(τ) (right-hand side of Equation (2.5)) increasegiving an increase of bush velocity and contact temperature θ(τ) and wear bushuw(τ). For example, in time instant τ = 4.27 (τ = 4.76) the shaft (bush) velocity
2.4 External shaft mechanical excitations 103
Fig. 2.54: Time histories of friction force (a) and wear (b) during acceleration for differentvalues of the parameter kw (curves 1: kw = 0.01; curves 2: kw = 0.1).
starts to decrease (Fig. 2.53a). The maximal values of contact pressure are achievedfor dimensionless time units τ = 8.95 (see curve 1 in Fig. 2.53a). For time instantτ1 = 9.54 the relative sliding velocity of both bodies is equal to zero, and a stickphase begins, which ends for τ2 = 17.5.
In the stick state for τ ∈ (τ1, τ2) the shaft temperature decreases owing to heatexchange, and therefore both contact pressure (see curve 1 in Fig. 2.53a) and fric-tion decrease, but wear does not undergo any changes. Beginning from τ2 = 17.5,a sliding phase appears within the interval of τ ∈ (τ2, τ3), where τ3 = 29.7.In this phase, both an increase and decrease of the shaft velocity are observed,the bush vibrates, and also contact temperature and pressure exhibit oscillatingcharacter (see curves 1 in Fig. 2.53a). Friction accompanied by vibrations alsoincreases the bush wear (Fig. 2.53a). For τ3 = 29.7 the next stick phase occursfor τ ∈ (τ3, τ4), where τ4 = 37.3. It is worth noting that during stick phasesτ ∈ τst = (τ1, τ2) ∪ · · · (τ2i−1, τ2i) ∪ · · · the system velocity oscillates periodi-cally with the period 2π(1 + ε/aM )/
√1 + ε/aM − h2 = 8.89.
Consider now the system dynamics for a larger value of the wear coefficientkw = 0.1 (curve 2). In the beginning of the sliding phase for τ ∈ (0, τ1), whereτ1 = 11.7, all earlier mentioned characteristics of two contacting bodies are similarto the previous case associated with small wear (see curves 2 in Fig. 2.52a). Onlyone sliding phase τ ∈ (τ1, τ2), where τ2 = 13.3, is exhibited. After τ2 = 13.3the bush starts to vibrate and the shaft rotation velocity, as well as contact tem-perature and wear, is increased. The contact pressure approaches zero τc = 50.6in an oscillatory manner (Fig. 2.53a). Beginning from this time instant, the con-tact between two bodies is lost. The shaft starts to rotate with an acceleration,whereas the bush vibrates with the period 2π/
√1 − h2. Zones of sticks are shown
in Fig. 2.52a and are marked by horizontal intervals 1 and 2. One may concludefrom the figure that during acceleration and increase of the dimensionless wear
104 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.55: Time histories of contact temperature for τ ∈ (2, 40) (a) and contact temperaturefor τ ∈ (0, 2) (b) during braking (m0 = 0) for different values of parameter kw (curves 1:kw = 0.01; curves 2: kw = 0.1).
coefficient the previously observed periodic stick-slip motion is substituted by amotion of no contacting bodies (τ > 50.6).
2.4.3.4 Cylinder and bush dynamics during braking
It is assumed that during the initial moment the cylinder rotates at velocity φ◦
(hM (τ) = 0, m0 = 0). Dimensionless temperature of the bush (pad) changes inaccordance with hT (τ) = 2(1 − exp(−δτ2)). Owing to heat transfer, the shaftstarts to extend and begins to keep contact with the bush, and hence brakingoccurs. Initial conditions are as follows: ϕ◦ = 0, ω◦, φ◦ = 0, φ◦ = 4. Com-putational results are shown in Figs. 2.52b, 2.55 and 2.56 for some values ofparameter kw. Curves 1 correspond to the case kw = 0.01, curves 2 correspond tokw = 0.1. In Fig. 2.52b dimensionless dependence of cylinder angular velocity φ(dashed curves) and bush ϕ (solid curves) on time τ during braking is presented.Figure 2.55b shows temperature time evolution for τ ∈ (0, 1.8) and Fig. 2.55afor τ ∈ (1.8, 40). Figures 2.56a and 2.56b illustrate time evolutions of contactpressure and wear, respectively.
Consider first dynamics for a small value of the wear coefficient kw = 0.01(curve 1 in Figs. 2.52b, 2.55, 2.56). As a result of temperature increase in asurrounding medium, temperature of the shaft rotating at dimensionless velocityφ◦
1 = 4 also increases. The shaft and bush start to touch each other, a contact pres-sure increases and achieves its maximal value p = 6.72 for time instant τ = 0.26,and both friction force and bush velocity increase. At τ1 = 0.75 the velocityof two bodies (sliding velocity) will be equal to zero, and a stick phase begin(it ends at τ2 = 5.9). As in the previous case, due to heat exchange the shaft
2.5 Dynamics of contacting bodies with impacts 105
Fig. 2.56: Time histories of contact pressure (a) and wear (b) during braking (m0 = 0) fordifferent values of parameter kw (curves 1: kw = 0.01; curves 2: kw = 0.1).
temperature starts to decrease in the stick phase τ ∈ (τ1, τ2), which causes adecrease of both contact pressure (curve 1 in Fig. 2.56a) and friction force. Thelatter one changes its sign rapidly at τ = 4.57. Wear process is constant duringthe stick phase (Fig. 2.56b), that is, at τ ∈ (τ1, τ2). Zones of sticks are shownin Fig. 2.52b and marked by horizontal intervals 1. Beginning from τ7 = 26.0the stick phase is exhibited, which is observed until damped oscillations (with theperiod 2π(1 + ε/aM)/
√1 + ε/aM − h2 = 8.89) vanish.
Finally, let us consider the braking process for the largest value of the wearcoefficient kw = 0.1 (curve 2). The corresponding stick phases are shown inFig. 2.52b and denoted by horizontal intervals 2. For τc = 33.3 the contact pressureis equal to zero (Fig. 2.56a). The shaft stops, whereas damped bush vibrations areobserved with the period 2π/
√1 − h2.
Note that when the shaft displacement achieves its extreme values (φ = 0, seedashed curves), the friction force changes its sign. In the stick phases τ ∈ τst =(τ1, τ2) ∪ · · · (τ2i−1, τ2i) ∪ · · · the wear process is not observed.
2.5. Dynamics of contacting bodies with impacts
In this section the model of a contact system with heat and wear generated byfriction and/or impacts is studied. The methods and mathematical models of suchsystems applied thus far by others contribute only partially to the description ofcomplex dynamics. First, the analysis of contacting dynamic models omit tribo-logical processes on a contact body surface. Second, the mentioned models do notinclude either the body inertia or impact phenomena usually appearing within thebody clearance. We contribute to the problem by matching both phenomena, which
106 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.57: Analysed system.
improves modelling of dynamic behaviour of contacting bodies. Analysis of bothstick-slip and slip-slip motion exhibited by the system is performed (impactlessbehaviour of this model has already been studied by the authors [Awrejcewicz,Pyryev (2002), (2004a), (2004b)]), among others. Analytically predicted vibro-impact stick-slip and slip-slip dynamics have also been verified numerically.
2.5.1 Mathematical modelling
Attention is focused on modelling of nonlinear dynamics of two bodies consistingof a stiff bush with clearance 2∆ϕ (see Fig. 2.57). The bush is coupled withhousing by springs with stiffness k2 and is mounted on the rotating thermoelasticshaft 1. The following assumptions are taken: (i) the shaft rotates with enoughsmall angular velocity Ω such that centrifugal forces can be omitted; (ii) nonlinearkinetic friction occurs between the bush and the shaft; (iii) heat is generated onthe contacting surface R = R1 due to friction; and (iv) heat transfer betweencontacting bodies is governed by Newton’s law.
2.5.1.1 Equations for shaft rotational movement of an absolutely rigid bush
Let axis Z be a cylinder axis. The equilibrium state of the moments of forces withrespect to the shaft axis gives
B2ϕ2(t) + k2R22ϕ2(t) = f(Vr)2πR2
1P (t), |ϕ2(t)| < ∆ϕ, ϕ2(t) = Ω,(2.151)
ϕ2(t) = 0, |ϕ2(t)| < ∆ϕ, ϕ2(t) = Ω, (2.152)
2.5 Dynamics of contacting bodies with impacts 107
ϕ+2 = −kϕ−
2 , |ϕ2| = ∆ϕ, ϕ−2 ϕ2 > 0, (2.153)
where: Vr = R1Ω − R1ϕ2(t) relative velocity of the contact bodies, k is thecoefficient of restitution, ϕ−
2 (ϕ+2 ) is the bush velocity just before (after) impact,
B2 is the moment of inertia of the bush per length unit, f(Vr) is the kinetic frictioncoefficient depending on relative velocity, and P (t) is the contact pressure. Theinitial value problem is defined in the following way.
ϕ2(0) = ϕ◦2, ϕ2(0) = ω◦
2 . (2.154)
Relation-approximating curve f(Vr) has the following form,
f(Vr) = sgn(Vr)F (|Vr|),
F (Vr) =
⎧⎨⎩
fs − κVr, 0 < Vr ≤ Vmin
fs − κVmin, Vmin < Vr
(2.155)
where fs, κ, Vmin are constant coefficients.
2.5.1.2 Thermoelastic shaft
Inertial terms occurring in the equation of motion are omitted in our study and theproblem may be considered as a quasi-static one. In the case of axially symmetricshaft stresses, the governing equations can be derived using the theory of thermalstresses for an isotropic body [Nowacki (1962)]. Applying cylindrical coordinatesone gets the following set of Equations ((2.7),(2.8)) with the attached mechanical(2.9) and thermal boundary conditions
λ1∂T1(R1, t)
∂R+ αT T1(R1, t) = (1 − η)f(Vr)VrP (t), (2.156)
λ12πR∂T1(R, t)
∂R
∣∣∣∣R→0
= 0, 0 < t < tc, (2.157)
and with the following initial conditions
T1(R, 0) = 0, 0 < R < R1. (2.158)
Velocity of the bush wear is proportional to a certain power of friction force(2.13). Shaft radial stresses σR(R, t) may be found using radial displacementU(R, t) and temperature T1(R, t) from the following formula (2.14).
Upon integration of Equation (2.7) and taking into account (2.9) and (2.14),the contact pressure is
108 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
P (t) =2E1α1
1 − 2ν1
1
R21
R1∫
0
T1(ξ, t)ξdξ
+E1
(1 − 2ν1)(1 + ν1)R1[U0hU (t) − Uw(t)]. (2.159)
2.5.2 Algorithm of solution
Let us introduce the following dimensionless parameters:
τ =t
t∗, τc =
tct∗
, r =R
R1, ϕ(τ) =
ϕ2
∆ϕ, p =
P
P∗, θ =
T1
T∗, uw =
Uw
U0,
α0 =f0
fs, ω1 =
Ωt∗∆ϕ
=1
γ√
α0, μ0 =
1 − α0
α0, η0 =
V0 − Vmin
V0,
ε = μ0γ√
α0 =μ0
ω1, γ =
√2πR2
1P∗fs∆ϕ
B2Ω2, ω2
0 =k2R
22t
2∗
B2, Bi =
αT R1
λ1,
kw =Kw∆ϕE1
(1 − 2ν1)(1 + ν1), γ1 =
(1 − η)E1α1R21f0∆ϕ
λ1(1 − 2ν1)tT, ω =
t∗tT
, Ψ =F
f0,
x =ϕ◦
2
∆ϕ, y =
ω◦2t∗
∆ϕ, f(V0) = f0, hU (τ) = hU (t∗τ), (2.160)
where
t∗ =
√B2∆ϕ
f02πR21P∗
, V∗ =R1∆ϕ
t∗, P∗ =
E1U0
(1 − 2ν1)(1 + ν1)R1,
T∗ =U0
α1(1 + ν1)R1, tT =
R21
a1, V0 = ΩR1.
The dimensionless equations governing dynamics of the analysed system have theform
ϕ(τ) + ω20ϕ(τ) = sgn(ω1 − ϕ)Ψ(ϕ)p(τ), |ϕ(τ)| < 1, ϕ(τ) = ω1,
ϕ(τ) = 0, |ϕ(τ)| < 1, ϕ(τ) = ω1,
ϕ+ = −kϕ−, |ϕ| = 1, ϕ−ϕ > 0,
ϕ(0) = x, ϕ(0) = y, (2.161)
2.5 Dynamics of contacting bodies with impacts 109
where
Ψ(ϕ) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
1 + εω1η0, ϕ < ω1η0, ω1(2 − η0) < ϕ
1 + εϕ, ω1η0 < ϕ < ω1
1 + 2εω1 − εϕ, ω1 < ϕ < ω1(2 − η0)
. (2.162)
In order to solve the motion Equations (2.161) one needs to know contactpressure (2.22) p(τ) and wear (2.23):
p(τ) = hU (τ) − uw(τ) +
1∫
0
θ(ξ, τ)ξdξ, (2.163)
uw(τ) = kw
τ∫
0
|ω1 − ϕ(τ)|p(τ)dτ. (2.164)
The one-dimensional transient heat conduction equation under considerationtakes the following dimensionless form (2.18), whereas the thermal boundaryconditions are
[∂θ(r, τ)
∂r+ Biθ(r, τ)
]
r=1
= γ1ω−1Ψ(ϕ(τ))|ω1 − ϕ(τ)|p(τ),
2πr∂θ(r, τ)
∂r
∣∣∣∣r→0
= 0, (2.165)
and initial conditions are as follows,
θ(r, 0) = 0. (2.166)
Applying an inverse Laplace transform [Carslaw, Jaeger (1959)], the nonlinearproblem governed by Equations (2.18), (2.165), and (2.166) is reduced to thefollowing integral equation of the second kind of Volterra type,
p(τ) = hU (τ) − uw(τ)
+ 2γ1ω−1
τ∫
0
Gp(τ − ξ)Ψ(ϕ(ξ))|ω1 − ϕ(ξ)|p(ξ)dξ. (2.167)
Then the problem is reduced to consideration of Equations (2.161) and (2.167),which yield both dimensionless pressure p(τ) and velocity ϕ(τ). The temperatureis defined by the following formula,
110 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
θ(r, τ) = γ1ω−1
τ∫
0
Gθ(r, τ − ξ)Ψ(ϕ(ξ))|ω1 − ϕ(ξ)|p(ξ)dξ, (2.168)
where Gp(τ) and Gθ(1, τ) are defined by (2.33).Temperature is defined by (2.168). Numerical analysis of the problem is carried
out using the Runge–Kutta method for (2.161) and the quadrature method for(2.167) and (2.168), and taking into account the following asymptotes,
Gθ(1, τ) ≈ 2√
τω/π, Gp(τ) ≈ ωτ, τ → 0. (2.169)
2.5.3 On the periodic motion with impacts
First the case of bush vibrations without tribological processes is studied (γ1 = 0,kw = 0). For this case we have p(τ) = hU (τ). Our system governed by Equa-tions (2.161) may exhibit four different periodic motions. Namely:
(i) Periodic orbit with one impact, where a stick does not appear (Fig. 2.58)(ii) Periodic orbit with one impact, where a stick-slip occurs (Fig. 2.60)(iii) Periodic orbit with two impacts, where a slip of the contacting bodies occurs
(Fig. 2.59)(iv) Periodic orbit with two impacts, where a stick-slip appears (Fig. 2.61).
In what follows we assume that ε ≪ 1, ω20 ≪ 1, and η0 ≤ −1. It means that the
system dynamics is exhibited in the interval (0 < V0 < Vmin), where a decreasingslope of the kinetic friction coefficient is observed.
Fig. 2.58: Phase curve of the bush dynamics with one impact per motion period (ω1 = 2.5,δ = 2, ε = 0.01, x = −0.5, k = 0.9884).
2.5 Dynamics of contacting bodies with impacts 111
Because η0 ≤ −1, then (V0 − Vmin)/V0 ≤ −1, and hence V0 ≤ 0.5Vmin.Observe that if inequality ω2
0 ≪ 1 is violated, then the secular terms occurwhile solving Equation (2.170).
Let us consider the conditions of the first type periodic motion with one impactper period (Fig. 2.58). Our aim is to find an appropriate value of the parameter k(restitution coefficient) to realize the mentioned periodic motion (phase trajectoryAMNA). It is assumed that the phase trajectory part AM starts in the pointA(x, 0), x ∈ (−1, 1). The Cauchy problem associated with Equation (2.161) is asfollows.
ϕ(τ) = 1 + εϕ(τ) − ω20ϕ(τ), ϕ(0) = x, ϕ(0) = 0, x ∈ (−1, 1).
(2.170)A solution being sought is given in the form ϕ(τ) = ϕAM (τ, x, 0),
ϕ(τ) = ωAM (τ, x, 0). Let the point M have coordinates M(1, yM) being achievedby moving point (ϕ, ϕ) at time instant τAM . Therefore our problem is reduced tothe consideration of the following system of nonlinear algebraic equations
1 = ϕAM (τAM , x, 0), yM = ωAM (τAM , x, 0) (2.171)
to be solved with respect to τAM and yM .Because we have assumed ε ≪ 1 and ω2
0 ≪ 1 and assuming ω20 = εδ, we begin
with a solution to the Cauchy problem (2.170) keeping the accuracy of O(ε2) inthe form
ϕAM (τ, x, 0) = x + 0.5τ2 + (1/6)τ3ε − (1/24)τ2(12x + τ2)δε + O(ε2),(2.172)
ωAM (τ, x, 0) = τ + 0.5τ2ε − (1/6)τ(6x + τ2)δε + O(ε2). (2.173)
Therefore, Equations (2.171) take the form
1 = x + 0.5τ2AM + (1/6)τ3
AMε − (1/24)τ2AM (12x + τ2
AM )δε + O(ε2),(2.174)
yM = τAM + 0.5τ2AMε − (1/6)τAM (6x + τ2
AM )δε + O(ε2). (2.175)
A solution to nonlinear Equation (2.174) is sought in the form τAM = AAM +BAMε + O(ε2). Then τ2
AM = A2AM + 2AAMBAMε + O(ε2), τ3
AM = A3AM +
3A2AMBAMε + O(ε2), τ4
AM = A4AM + 4A3
AMBAMε + O(ε2), and the obtainedresults are substituted to (2.174) to yield
1 = x + 0.5(A2AM + 2AAMBAMε) + (1/6)A3
AMε
− (1/24)12xA2AMδε − (1/24)A4
AMδε + O(ε2).
112 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Comparing terms standing by the same power of ε one gets
2(1−x) = A2AM , 0 = AAM (BAM+(1/6)A2
AM )−(1/24)A2AM (12x+A2
AM )δ.
Taking into account the first of the latter two equations one obtains
AAM ≡ τ1 =√
2(1 − x),
whereas the second equation yields
BAM = −(1/6)A2AM − (1/24)AAM (−12x − A2
AM )δ
= −(1/6)τ21 − (1/24)τ1(5τ2
1 − 12)δ.
The solution to Equation (2.174) is
τAM = τ1 − (1/6)τ21 ε − (1/24)τ1(5τ2
1 − 12)δε + O(ε2). (2.176)
Substituting (2.176) into (2.175) one obtains
yM = τ1 + (1/3)τ21 ε + (1/8)τ1(τ
21 − 4)δε + O(ε2). (2.177)
On the other hand, the phase trajectory part NA begins in point N(1, yN). TheCauchy problem associated with Equation (2.161) has the following form,
ϕ(τ) = 1 + εϕ(τ) − ω20ϕ(τ), ϕ(0) = 1, ϕ(0) = yN , (2.178)
where yN = −k yM . Problem (2.178) has the following solution, ϕ(τ) =ϕNA(τ, 1, yN ), ϕ(τ) = ωNA(τ, 1, yN). Because point A has coordinates A(x, 0),it is achieved by moving point (ϕ, ϕ) at time instant τNA. Therefore, the followingnonlinear equations hold
x = ϕNA(τNA, 1, yN), 0 = ωNA(τNA, 1, yN), (2.179)
which define the values τNA and yN being sought.Recall that ε ≪ 1 and ω2
0 ≪ 1, and the Cauchy problem associated withEquation (2.178) is
ϕNA(τ, 1, yN ) = 1 + yNτ + 0.5τ2 + (1/6)τ2(τ + 3yN)ε
− (1/24)τ2(12 + τ2 + 4yNτ)δε + O(ε2), (2.180)
ωNA(τ, 1, yN ) = yN + τ + 0.5τ(τ + 2yN)ε
− (1/6)τ(6 + τ2 + 3yNτ)δε + O(ε2). (2.181)
2.5 Dynamics of contacting bodies with impacts 113
Equations (2.179) take the following form,
x = 1 + yNτNA + 0.5τ2NA + (1/6)τ2
NA(τNA + 3yN)ε
− (1/24)τ2NA(12 + τ2
NA + 4yNτNA)δε + O(ε2), (2.182)
0 = yN + τNA + 0.5τNA(τNA + 2yN )ε
− (1/6)τNA(6 + τ2NA + 3yNτNA)δε + O(ε2). (2.183)
Using Equation (2.183) we express τNA by yN as follows.
τNA = −yN + 0.5y2Nε − (yN − (1/3)y3
N)δε + O(ε2). (2.184)
Substitution of (2.184) into (2.182) yields
yN = −τ1 + (1/3)τ21 ε + (1/8)τ1(4 − τ2
1 )δε + O(ε2), τ1 =√
2(1 − x).(2.185)
Now, substituting (2.185) to (2.184) the quantity τNA is expressed through τ1
in the following way,
τNA = τ1 + (1/6)τ21 ε − (1/24)τ1(5τ2
1 − 12)δε + O(ε2). (2.186)
Taking into account the obtained values (2.177) and (2.185), and equationyN = −k yM , the restitution coefficient being sought is given explicitly in thefollowing form,
kAMNA = 1 − (2/3)τ1ε + O(ε2). (2.187)
Summation of both time intervals τAM and τNA gives the period of case (i):
τAMNA = 2τ1 − (1/12)τ1(5τ21 − 12)δε + O(ε2). (2.188)
Note that the periodic motion (i) takes place for yM < ω1. Furthermore, the useof Equation (2.177) provides the estimation x0 < x < 1, where
x0(ω1) = 1 − (1/2)ω21 + (1/3)ω3
1ε + (1/8)ω21(ω
21 − 4)δε + O(ε2). (2.189)
Let us consider conditions of the occurrence of third-type dynamics (iii)associated with two impacts per motion period (Fig. 2.59). Our aim is to definethe value of parameter k for which the required periodic motion appears (phasetrajectory BMNDB). Let us find the coordinates of point B(−1, yB), where thephase trajectory part BM begins.
We assume that the phase trajectory goes from point A(x.0), x ∈ (−∞,−1)into point B if there is no boundary at x = −1. In this case the Cauchy problemassociated with differential Equation (2.161) takes the form
ϕ(τ) = 1 + εϕ(τ) − ω20ϕ(τ), ϕ(0) = x, ϕ(0) = 0, x ∈ (−∞,−1).
(2.190)
114 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.59: Periodic bush phase orbit with two impacts (ω1 = 2.5, δ = 2, ε = 0.01, x = −1.5,k = 0.9887).
A solution to problem (2.190) is as follows: ϕ(τ) = ϕAB(τ, x, 0), ϕ(τ) =ωAB(τ, x, 0). Because point B is achieved by the moving point (ϕ, ϕ) at timeinstant τAB , one obtains the two following nonlinear equations,
−1 = ϕAB(τAB , x, 0), yB = ωAB(τAB , x, 0) (2.191)
yielding τAB and yB .For ε ≪ 1 and ω2
0 ≪ 1 the Cauchy problem associated with (2.190), keepingthe accuracy order of O(ε2), has a form analogous to that of (2.172) and (2.173).The system of Equations (2.191) takes the following form,
−1 = x + 0.5 τ2AB + (1/6)τ3
ABε − (1/24)τ2AB(12x + τ2
AB)δε + O(ε2), (2.192)
yB = τAB + 0.5τ2ABε − (1/6)τAB(6x + τ2
AB)δε + O(ε2). (2.193)
The solution to Equation (2.192) is as follows.
τAB = τ2 − (1/6)τ22 ε − (1/24)τ2(5τ2
2 + 12)δε + O(ε2),
τ2 =√
−2(1 + x). (2.194)
Substituting (2.194) to (2.193) the following coordinate is found.
yB = τ2 + (1/3)τ22 ε + (1/8)τ2(4 + τ2
2 )δε + O(ε2). (2.195)
Phase trajectory part BM begins at point B(−1, yB). The Cauchy problem ofEquation (2.161) takes the form
2.5 Dynamics of contacting bodies with impacts 115
ϕ(τ) = 1 + εϕ(τ) − ω20ϕ(τ), ϕ(0) = −1, ϕ(0) = yB, (2.196)
where yB is defined by (2.195). A solution to problem (2.196) is as follows,ϕ(τ) = ϕBM (τ,−1, yB), and ϕ(τ) = ωBM (τ,−1, yB). Point M has coordinatesM(1, yM ) and it is achieved by moving point (ϕ, ϕ) at time instant τBM , whichdefines the following set of equations,
1 = ϕBM (τBM ,−1, yB), yM = ωBM (τBM ,−1, yB). (2.197)
A solution to Equations (2.197) gives
τBM = τ1 − τ2 − (2/3)ε + (1/24)(12τ2 + 5τ32
− (32 + 28τ22 + 5τ4
2 )/τ1)δε + O(ε2), (2.198)
yM = τ1 + (1/3)τ21 ε + (1/8)τ2
2 τ1δε + O(ε2). (2.199)
Phase trajectory part ND begins in point N(1, yN), and the Cauchy problemassociated with differential Equation (2.161) takes the form
ϕ(τ) = 1 + εϕ(τ) − ω20ϕ(τ), ϕ(0) = 1, ϕ(0) = yN , (2.200)
where yN = −k yM . Thus, we have the following solution of problem (2.200),ϕ(τ) = ϕND(τ, 1, yN), ϕ(τ) = ωND(τ, 1, yN ). Point D has coordinatesD(−1, yD), where yD = −yB/k and it will be achieved by moving point (ϕ, ϕ)at time instant τND. One obtains the following equations
−1 = ϕND(τND, 1, yN), yD = ωND(τND, 1, yN) (2.201)
yielding the τND and yD being sought. The second equation of (2.201) gives
τND = yD − yN + (1/2)(y2N − y2
D)ε
+ (1/6)(yN − yD)(−6 + 2y2N − y2
D − yNyD)δε + O(ε2). (2.202)
Substituting (2.202) to the first Equation (2.201) and taking into accountyN = −k yM and yB = −k yD one gets
kBMNDB = 1 +τ32 − τ3
1
3(2 + τ22 )
ε + O(ε2). (2.203)
Time interval τND is defined by Equation (2.202) taking into account (2.203),yN = −k yM , and yB = −k yD, as well as (2.193) and (2.199):
116 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
τND = τ1 − τ2 +2
3(2 + τ22 )
(2 − 2τ1τ2 + τ22 )ε
+1
24(2 + τ22 )
(22τ32 + 5τ5
2 − 16τ1 − 5τ42 τ1 + 24τ2 − 18τ2
2 τ1)δε + O(ε2).
(2.204)
Now summing both time intervals τBM and τND one obtains the period associatedwith case (iii) of the form
τBMNDB = 2(τ1 − τ2) −4τ2τ1
3(2 + τ22 )
ε
+(5τ2
2 + 12)τ2τ1 − (5τ42 + 28τ2
2 + 32)
12τ1δε + O(ε2). (2.205)
Note that periodic motion in case (iii) occurs only if yM < ω1. Therefore,taking (2.199) into account, one obtains the following inequality x0 < x < −1,where x0 is defined by (2.189).
Let us define conditions in which the second type motion (ii) occurs; that is,with one impact per period (Fig. 2.60) and with a stick-slip behaviour. Followingprevious approaches we define the value of parameter k responsible for realizationof this motion type (phase trajectory ACMNA). Assuming that the trajectorypart AC begins at A(x, 0), x ∈ (−1, 1), the Cauchy problem for the differentialequation (2.161) takes the form
ϕ(τ) = 1 + εϕ(τ) − ω20ϕ(τ), ϕ(0) = x, ϕ(0) = 0, x ∈ (−1, 1).
(2.206)
Fig. 2.60: Bush phase curve with one impact per motion period in stick-slip conditions (ω1 = 1.5,δ = 2, ε = 0.1, x = −0.3, k = 0.9796, x1 = −0.3359, x0 = −0.1109).
2.5 Dynamics of contacting bodies with impacts 117
We take the following solution ϕ(τ) = ϕAC(τ, x, 0), ϕ(τ) = ωAC(τ, x, 0). Letpoint C have coordinates C(xC , ω1) and let it be achieved by moving point (ϕ, ϕ)at time instant τAC . The following nonlinear equations
xC = ϕAC(τAC , x, 0), ω1 = ωAC(τAC , x, 0) (2.207)
are used to determine τAC and xC .Proceeding in a way similar to the previous analysis we take ε ≪ 1 and ω2
0 ≪ 1,and ω2
0 = εδ to find
τAC = ω1 − 0.5ω21ε + (1/6)ω1(ω
21 − 3τ2
1 + 6)δε + O(ε2), (2.208)
xC = 1 − 0.5(τ21 − ω2
1) − (1/3)ω31ε + (1/8)ω2
1(ω21 − 2τ2
1 + 4)δε + O(ε2).(2.209)
Condition xC ≤ 1 gives the boundary x ≤ x0 (xC |x=x0= 1), where x0 is defined
by (2.189).Phase trajectory NA begins at N(1, yN), and the Cauchy problem for differ-
ential equation (2.161) takes the form
ϕ(τ) = 1 + εϕ(τ) − ω20ϕ(τ), ϕ(0) = 1, ϕ(0) = yN , (2.210)
where yN = −k ω1. The solution to problem (2.210) is: ϕ(τ) = ϕNA(τ, 1, yN ),ϕ(τ) = ωNA(τ, 1, yN ). Point A has coordinates A(x, 0) and it is achieved bymoving point (ϕ, ϕ) at time instant τNA. The following nonlinear set of equations
x = ϕNA(τNA, 1, yN), 0 = ωNA(τNA, 1, yN) (2.211)
gives
τNA = τ1 + (1/6)τ21 ε − (1/24)τ1(5τ2
1 − 12)δε + O(ε2), (2.212)
kACMNA = τ1/ω1 − (1/3)(τ21 /ω1)ε − (1/8)(τ1/ω1)(4 − τ2
1 )δε + O(ε2).(2.213)
A natural limitation k ≤ 1 yields the inequality x ≥ x1, where
x1(ω1) = 1 − 0.5ω21 − (1/3)ω3
1ε + (1/8)ω21(ω
21 − 4)δε + O(ε2). (2.214)
Summation of τAC and τNA as well as of τCM = (1 − xC)/ω1 gives the estima-tion of a period in case (ii):
τACMNA = (τ1 + ω1)2/(2ω1) + (1/6)(τ2
1 − ω21)ε
+ (1/24)(τ1 + ω1)(12 − 5τ21 + ω2
1 − ω1τ1)δε + O(ε2). (2.215)
118 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.61: Bush phase stick-slip curve with two impacts per period (ω1 = 2.5, δ = 2, ε = 0.05,x = −2.0, k = 0.9853, x3 = −2.097, x0 = −1.689).
In the second case periodic motion (−1 ≤ x ≤ 1) occurs when x1 ≤ x ≤ x0,where x0 and x1 are defined by formulas (2.189) and (2.214).
Let us consider now the conditions of appearance of the fourth type of periodicstick-slip motion with two impacts per period (Fig. 2.61). Also in this case weestablish a value of the parameter k being responsible for a stick-slip periodicmotion occurrence (phase trajectory BCMNDB). Let us first find coordinates ofpoint B(−1, yB), where the trajectory BC begins.
We assume that point A(x, 0), x ∈ (−∞,−1) achieves point B if there isno limit x = −1. The Cauchy problem for Equation (2.161) has the form of(2.190). A solution to problem (2.190) has the form ϕ(τ) = ϕAB(τ, x, 0), ϕ(τ) =ωAB(τ, x, 0). Point B is achieved by moving point (ϕ, ϕ) in time instant τAB . Thisobservation enables us to derive nonlinear Equations (2.191) with solutions τAB
and yB defined by (2.194) and (2.195), respectively.Phase trajectory part BC begins at point B(−1, yB). The Cauchy problem for
differential Equation (2.161) has the form of (2.196), where yB is defined by(2.195).
A solution to problem (2.196) has the form: ϕ(τ) = ϕBC(τ,−1, yB), ϕ(τ) =ωBC(τ,−1, yB). Point C has coordinates C(xC , ω1) and it is achieved by movingpoint (ϕ, ϕ) at time instant τBC . Therefore, one gets the following system ofequations
xC = ϕBC(τBC ,−1, yB), ω1 = ωBC(τBC ,−1, yB), (2.216)
2.5 Dynamics of contacting bodies with impacts 119
whose solutions follow
τBC = ω1 − τ2 + (1/6)(τ22 − 3ω2
1)ε
+ (1/24)(12τ2 + 5τ32 − 12τ2
2 ω1 − 24ω1 + 4ω31)δε + O(ε2), (2.217)
xC = −1 − 0.5(τ22 − ω2
1) − (1/3)ω31ε
+ (1/8)ω21(−4 − 2τ2
2 + ω21)δε + O(ε2). (2.218)
On the other hand, the phase trajectory ND begins at point N(1, yN ). TheCauchy problem for differential equation (2.161) has the form
ϕ(τ) = 1 + εϕ(τ) − ω20ϕ(τ), ϕ(0) = 1, ϕ(0) = yN , (2.219)
where yN = −k ω1. The solution of problem (2.219) is: ϕ(τ) = ϕND(τ, 1, yN ),ϕ(τ) = ωND(τ, 1, yN). Point D has coordinates D(−1, yD), where yD = −yB/kand it is achieved by moving point (ϕ, ϕ) at time instant τND. Hence, we havethe following equations
−1 = ϕND(τND, 1, yN), yD = ωND(τND, 1, yN) (2.220)
yielding k and τND. Expressing τND by yD and yN through the formula
τND = yD − yN − 0.5(y2D − y2
N )ε
− (1/6)(yD − yN )(2y2N − y2
D − yNyD − 6)δε + O(ε2), (2.221)
and substituting (2.221) to the first equation of (2.220) and taking into accountyN = −k ω1 and yD = −yB/k, one gets
kBCMNDB =τ0
ω1+
−16τ20 + τ3
2 ω21(τ0 + ω1) − τ2
2 ω21(4 + τ2
0 )
6ω1(τ22 ω2
1 + 2τ20 )
ε
+τ22 τ0ω1(τ
22 + 4)
16(τ22 ω2
1 + 2τ20 )
δε + O(ε2), τ0 =
√2 +
√4 + τ2
2 ω21 ,
(2.222)
and also the following dimensionless time is estimated
τND = τ1 − τ2 + (2/3)(2 − 2τ2τ1 + τ22 )ε/(2 + τ2
2 )
+τ2(22τ2
2 + 5τ42 − 5τ3
2 τ1 + 24 − 18τ2τ1) − 16τ1
24(2 + τ22 )
δε + O(ε2). (2.223)
The natural limitation introduced on the restitution coefficient (k ≤ 1) yieldsx ≥ x2, where
120 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
x2(ω1) = x1(ω1) + (2/3)(ω21 − 4)3/2ε. (2.224)
Now, summing both τBC and τND, as well as τCM = (1 − xC)/ω1 one gets theperiod of stick-slip periodic orbit for case (iv):
τBCMNDB = (τ21 − 4τ2ω1 + 2τ1ω1 + ω2
1)/(2ω1)
+ (1/6)(τ42 − 8τ2τ1 − τ2
2 ω21 + 6τ2
2 − 2ω21 + 8)ε/(2 + τ2
2 )
+ (1/24)(24τ2 + 10τ32 − 8τ1 − 12ω1 + ω3
1 − 5τ22 τ1 − 6τ2
2 ω1)δε + O(ε2).(2.225)
A periodic motion in case (iv) (−∞ ≤ x ≤ −1) takes place if x2 ≤ x ≤ x0,where x0 and x2 are defined by formulas (2.189) and (2.224).
Results of our consideration allow us to give formulas for the coefficient ofrestitution k for a general case of the following case.
k(x, ω1) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
kACMNA, 0 < ω1 < 2 − (4/3)ε, x1 < x < x0
kAMNA, 0 < ω1 < 2 − (4/3)ε, x0 < x < 1
kBCMNDB , 2 − (4/3)ε < ω1 < 2, x1 < x < −1kACMNA, 2 − (4/3)ε < ω1 < 2, −1 < x < x0
kAMNA, 2 − (4/3)ε < ω1 < 2, x0 < x < 1
kBCMNDB , 2 < ω1 < 2 + (4/3)ε, x2 < x < −1kACMNA, 2 < ω1 < 2 + (4/3)ε, −1 < x < x0
kAMNA, 2 < ω1 < 2 + (4/3)ε, x0 < x < 1
kBCMNDB , 2 + (4/3)ε < ω1 < ∞, x2 < x < x0
kBMNDB , 2 + (4/3)ε < ω1 < ∞, x0 < x < −1kAMNA, 2 + (4/3)ε < ω1 < ∞, −1 < x < 1
(2.226)
The obtained results are graphically presented in Fig. 2.62.Observe that the function k(x, ω1) possesses the following values k(x1, ω1) = 1,
0 < ω1 < 2, k(x2, ω1) = 1, 2 < ω1 < ∞, k(1, ω1) = 1 at the boundaries, whereasinside the considered interval it has the following minima
minx∈[x1,1]
k(x, ω1) = k(x0, ω1) = 1 − (2/3)ω1ε, 0 < ω1 < 2 + (4/3)ε, (2.227)
minx∈[x2,1]
k(x, ω1) = k(−1, ω1) = 1 − (4/3)ε, 2 + (4/3)ε < ω1 < ∞, (2.228)
which can be presented in the form
kmin =
⎧⎨⎩
1 − (2/3)ω1ε, 0 < ω1 < 2 + (4/3)ε
1 − (4/3)ε, 2 + (4/3)ε < ω1 < ∞(2.229)
2.5 Dynamics of contacting bodies with impacts 121
Fig. 2.62: Zones of different periodic impact motions (ω2
0= 0.2, δ = 2, ε = 0.1).
Notice that for an arbitrary k∗ ∈ (kmin, 1) there are two values of x∗1, x∗
2
(k(x∗1, ω1) = k(x∗
2, ω1) = k∗). Let us introduce the following intervals
x1 < x∗1 < x0, x0 < x∗
2 < 1, for 0 < ω1 < 2, (2.230)
x2 < x∗1 < x0, x0 < x∗
2 < 1, for 2 < ω1 < 2 + (4/3)ε, (2.231)
x2 < x∗1 < −1, −1 < x∗
2 < 1, for 2 − (4/3)ε < ω1 < ∞. (2.232)
It is not difficult to check that a periodic orbit associated with x∗1 (decreasing
part of the coefficient k(x)) is stable, whereas a periodic orbit associated with x∗2
(increasing part of the coefficient k(x)) is unstable.
2.5.4 Numerical analysis
A numerical analysis has been carried out for the following dimensionless para-meters: ε = 0.1, ω1 = 1.6, δ = 2, ω2
0 = 0.2. Formula (2.229) gives kmin = 0.89.If k∗ = 0.93 (k∗ ∈ (kmin, 1)), then our system exhibits two periodic orbits definedby x∗
1 = −0.33 (stable) case (ii) and x∗2 = 0.44 (unstable) case (i) (see Figs. 2.63
and 2.64).Curves 1 and 2 approach a stable periodic orbit, whereas curve 3 tends to the
stable point (1,0). Note that the dashed curve is associated with an unstable orbit(see Fig. 2.64).
Next, a numerical analysis has been carried out for the following dimensionlessparameters: ε = 0.1, ω1 = 1.6, ω2
0 = 0.2, η0 = −2, ω = 0.1. For x = −0.9 and
122 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.63: Graphical solution of equation k(x, ω1) = k∗.
Fig. 2.64: Phase trajectory of the bush movement for different values of x: curves 1: x = −0.9(0 ≤ x < x∗
1); curves 2: x = 0 (x∗
1< x < x∗
2); curves 3: x = 0.7 (x∗
2< x ≤ 1).
k = 0.93 the corresponding bush phase trajectory for various parameters γ1 andkw is shown in Fig. 2.65.
Time histories of contact pressure, temperature on surface contact, and wearare reported in Fig. 2.66. Curves 1 correspond to the case when γ1 = 0 (lackof heat extension), kw = 0 (lack of bush wear). Curves 2 correspond to the caseof heat transfer lack (γ1 = 0) and kw = 0.02. Curves 3 correspond to the casewhere the shaft heat expansion is taken into account (γ1 = 0.1), but the bush wearis neglected (kw = 0). Curves 4 correspond to the case where both mentionedparameters are taken into account (γ1 = 0.5, kw = 0.02).
In the first case (γ1 = 0, kw = 0), where the tribological processes are not takeninto account the phase curve approaches a stable orbit (curve 1 in Fig. 2.65a). Inthis case the contact pressure is exhibited by curve 1 in Fig. 2.66a. The bushwear (γ1 = 0, kw = 0.02) occurrence decreases the contact pressure (curve 2 inFig. 2.66a), which tends to zero value (the corresponding phase curve is shown inFig. 2.65b). Note that after the wear process, the bush moves in a periodic manner.
2.5 Dynamics of contacting bodies with impacts 123
a) b)
c) d)
Fig. 2.65: Phase trajectory of the bush movement for different values of tribologic parameters:curves 1: γ1 = 0, kw = 0 (without tribologic processes); curves 2: γ1 = 0, kw = 0.02; curves 3:γ1 = 0.1, kw = 0 (with heat generation); curves 4: γ1 = 0.5, kw = 0.02 (with tribologicprocesses).
Bush wear kinematics is shown in Fig. 2.66c (curve 2). An inclusion of the shaftheat expansion (γ1 = 0.1) within the given heat transfer conditions (Bi = 10)yields a periodic change of both contact pressure (curve 3 in Fig. 2.66a) andtemperature (curve 3 in Fig. 2.66b). The phase curve after a transitional processtends to a new stable periodic orbit (curve 3 in Fig. 2.65c). For a general case, (i.e.,where the tribological processes are taken into account (γ1 = 0.5, kw = 0.02)) andfor the given heat transfer conditions (Bi = 10) the obtained results are exhibitedby curves 4 in Figs. 2.65 and 2.66. In this case the bush wear increases owingto the shaft heat extension, and the contact pressure first increases and then ittends to zero (curve 4 in Fig. 2.66a). The contact temperature being changed in anoscillatory manner first increases, but then decreases with decrease of the contact
124 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
a) b)
c)
Fig. 2.66: Time histories of dimensionless contact pressure p(τ) (a), dimensionless contacttemperature θ(1, τ) ≡ θ(τ) (b) and wear uw(τ) (c), versus dimensionless time τ for differentvalues of γ1 and kw; curves 1: γ1 = 0, kw = 0 (without tribologic processes); curves 2: γ1 = 0,kw = 0.02 (heat generation included); curves 3: γ1 = 0.1, kw = 0 (heat generation included,wear omitted); curves 4: γ1 = 0.5, kw = 0.02 (with tribologic processes).
pressure. The bush wear kinematics is exhibited by curve 4 in Fig. 2.66c. Observethat the final wear amount is larger than the initial shaft compression. In this casethe phase curve (after the bush wear) approaches a stable periodic orbit (curve 4in Fig. 2.65d).
In the case when the bush wear is less than the shaft thermal expansion (forinstance, in the case of γ1 = 1, Bi = 10), the contact characteristics increase in anexponential manner with time increase. In the latter case the shaft cannot succeedin making cooling in time.
We have proposed a novel model of vibrations of the bush–shaft system with in-clusion of both impacts and tribological processes occurring on the contact surface.A similar system, however, without impacts, has been studied earlier by the authors
2.5 Dynamics of contacting bodies with impacts 125
and it has been described in references [Awrejcewicz, Pyryev (2002), (2004b)].The occurrence of self-excited vibrations in a more simplified system with a gap(without tribological processes and springs) has also been analysed in reference[Balandin (1993), Balandin et al. (2001)].
Applying the Laplace transform, our problem has been reduced to that of thesystem of one nonlinear differential equation and one second-order Volterra inte-gral equation with respect to the contact pressure. A kernel of the latter equation isthe function of the sliding velocity. We have estimated analytically the restitutioncoefficient for which a periodic orbit occurs assuming small slope of friction char-acteristics. We, among others, have shown various periodic motions exhibited bythe analysed system and we have verified numerically our theoretical considerationsand predictions.
For an arbitrary restitution coefficient k∗ ∈ (kmin, 1) two periodic orbits (stableand unstable) appear on the phase plane. Increase of the parameter k∗ from kmin to1 yields an increase (decrease) of the stable (unstable) periodic orbit. For k∗ = 1the unstable periodic orbit is reduced to the point (1, 0). Decrease of the para-meter k∗ causes the approach of both stable and unstable periodic trajectories. Fork∗ = kmin a bifurcation occurs and a half-stable periodic orbit is born substitutingtwo previous stable and unstable orbits. In other words, for k < kmin a periodicmotion is not exhibited by the studied system.
The final conclusion follows. Tribologic processes have an important impact onthe studied system dynamics, because they may change it even qualitatively.
Observe that our aim was not to investigate either a real engineering system orconstruction. We are rather focused on periodic and chaotic nonsmooth dynamicswith friction and impacts exhibited by relatively simple models shown in Figs. 2.1and 2.57. However, the methodology based on a study of the mentioned simplemodels by analytical (or semi-analytical) methods and numerical methods (neces-sary for the prediction and monitoring of chaotic vibrations) seems to be powerfulfor analysing and understanding the phenomena as well as for designing engineer-ing systems. Even if detailed computations from partial differential equations couldbe a complementary task for applications, the presented approach is profitable forscientists and engineers. Furthermore, the obtained analytical formulas can be usedfor identification of the parameters of the models, or can serve as verification andvalidation of the real engineering objects studied via the finite difference method,finite element method, Bubnov–Galerkin method, and others.
The solid friction occurs during a sliding contact between the moving bodies,and the further taken contact geometry allows us to study body dynamics modelsbeing close to their real behaviour. Although the introduced approach is not directlyapplied to a real engineering object, the studied geometry of joints could be appliedin the analysis of the clamping and interference joints, as well as in various jointswith friction and impact phenomena. The mentioned simple models can be directlyapplied to study the so-called brake pad as well as Pronny’s brake [Andronov et al.(1966)].
126 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
2.6. Stick-slip vibrations continuous friction model
In this section an alternative (novel) continuous friction model is presented andstudied. Advantages of the proposed model with respect to the other friction modelsare illustrated and discussed. As an example of a mechanical system a one degree-of-freedom model with dry friction is considered. The analysed system exhibitsboth regular and nonregular dynamics. The dynamics of the mentioned mechani-cal model is monitored via standard characteristics (trajectories of motion in thesystem’s phase space, and bifurcation diagrams, as well as the Lyapunov expo-nents).
2.6.1 Introduction
Dry friction belongs to one of the most known phenomena exhibited by mechanicalsystems. Its proper mathematical modelling is not an easy task, because in thegeneral case friction force between sliding surfaces is a complex process anddepends on several parameters such as relative velocity of sliding surfaces, normalload, time, and temperature. Several formulations describing dry friction force werepresented in Section 1.1.5. In a signum model the friction force Ffr is definedas a function of the slip velocity ωr of sliding surfaces in the slip phase and asa function of the force Φ(ϕ, ϕ, τ) in the stick phase. This model describes thedry friction phenomenon in a correct and accurate way. Note that an exact valueof zero is rarely computed during numerical simulation and for this reason thementioned ‘signum model’ is equivalent, from a numerical point of view, to theclassical Coulomb model. The dependence of friction force on the sliding relativevelocity based on the ‘signum model’ is not a continuous function for ωr = 0 andstandard numerical procedures devoted to solving differential continuous equationscannot be used. The friction curve is therefore often approximated by a continuousor smooth function (smoothing method) but for ωr = 0 the value of the frictionforce is always equal to zero and the friction force depends on ωr but not onΦ(ϕ, ϕ, τ) in the stick phase, thus equations in the stick mode (static equations)are not correct.
The other so-called ‘switch model’ from a mathematical point of view treatsthe system as three different sets of ordinary differential equations: one for the slipphase, a second for the stick phase, and a third for the transition from stick to slip.At each timestep the state vector is inspected to determine whether the analysedsystem is in the slip, in the stick, or in the transition phase. The conditions forchanging to the stick phase or the slip phase operate as switches between thesystems and a region of stepness parameter εω is very small in comparison torelative sliding velocity. The finite region is necessary for digital computation,because an exact value of zero will rarely be computed. If the relative velocity lieswithin the stick band (phase), friction force is equal to the force Φ(ϕ, ϕ, τ). In theswitch model the acceleration of the body during the stick mode is proportional to
2.6 Stick-slip vibrations continuous friction model 127
Fig. 2.67: Different approximations of friction force in the near-zero relative velocity: signummodel (curve 1), smoothing method (curve 2), switch model (curve 3), and continuous frictionmodel (curve 4).
the relative velocity (with ωd coefficient). The relative velocity, which is directlydependent on the state of the system, cannot be set to zero in the stick phase asthe state vector has to be continuous for ordinary differential equations.
We present a novel continuous friction model taking into account some elementsof the mentioned switch model. A continuous friction model uses friction forceon the basis of the switch model. In other words, in a continuous friction modelthe space (Φ, ωr) is divided into regions (as in the switch model) and frictionforce is a continuous function on ωr (as in the smoothing method), and for ωr = 0the friction force is equal to the force Φ(ϕ, ϕ, τ) (as in the signum model). Forthis reason this continuous friction model can be treated as an approximatingfriction force appearing in the switch model using the continuous functions anddescribing the friction force for ωr = 0 in the correct and accurate way. The curvesdescribing the friction force on the signum model, smoothing model (smoothingmethod), switch model, and continuous friction model in the region |ωr| ≤ εω areplotted in Fig. 2.67.
In Fig. 2.68 the dependencies of friction force as a function of two variablesFfr(ωr, Φ) are plotted for dimensionless parameters Fs = 1 and εω = 0.001.
2.6.2 One degree-of-freedom model
To demonstrate the above continuous friction model and compare results withother friction models a single degree-of-freedom (1-dof) mechanical system withdry friction is considered. This model possesses stick-slip periodic and nonperiodicsolutions. The analysed system is shown in Fig. 2.69.
128 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
a) b)
Fig. 2.68: Friction force Ffr(ωr , Φ) as a function of two variables: (a) switch model, (b) con-tinuous friction model.
Fig. 2.69: The analysed one degree-of-freedom model with dry friction.
The second disc (II) is characterized by linear stiffness k and the mass momentof inertia B and it is fixed to the frame. The second disc is coupled with the drivingfirst disc (I) that is moving with angular velocity ωdr. Between the discs dry frictionoccurs which generates a moment of friction force Mfr (with maximum staticmoment of friction force Ms). In addition, harmonic excitation with amplitudeM0 and circular frequency Ω is added to our model.
The following second-order differential equation of this system is
Bϕ2 + kϕ2 = Mfr + M0 cosΩt. (2.233)
Let us introduce the following similarity coefficients t∗, ϕ∗ and the followingdimensionless parameters
2.6 Stick-slip vibrations continuous friction model 129
τ =t
t∗, ϕ =
ϕ2
ϕ∗, ω = ϕ2t∗ = ω2t∗,
ω2d =
kt2∗B
, ζ =M0t
2∗
Bϕ∗, Fs =
Mst2∗
Bϕ∗, ω0 = Ωt∗, ω1 = ωdrt∗.
Then, in the dimensionless form, the mathematical model reads
ϕ = −ω2dϕ + ζ cosω0τ + Ffr, (2.234)
with the friction model (1.68), where
F (|ωr|)sgn(ωr)p(τ) =Fssgn(ωr)
1 + δ(|ωr| − ε), (2.235)
Φ(ϕ, ϕ, τ) = ω2dϕ − ζ cosω0τ , (2.236)
and the relative angular velocity of the second disc with respect to the first disc isdenoted by ωr = ω1 − ϕ.
This dimensionless dynamical system can be expressed as a set of first-orderordinary differential equations. The governing equations read
⎧⎪⎪⎨
⎪⎪⎩
ϕ = ω,
ω = −ω2dϕ + ζ cosΨ + Ffr,
Ψ = ω0,
(2.237)
where the dot denotes differentiation with respect to dimensionless time τ .The initial parameters of this model taken from the work [Awrejcewicz et al.
(2007)] are: B = 2[kg · m2], k = 10[N · m · rad−1], Ms = 2[N], M0 = 0.5[N],Ω = 2[rad · s−1], ωdr = 0.3[rad · s−1]. For t∗ = 1[s], ϕ∗ = 1[rad] the initialdimensionless parameters of this model are: ω2
d = 5, Fs = 1, ζ = 0.25, ω0 = 2,ω1 = 0.3. The dimensionless parameter δ = 3 and numerical dimensionlessparameters are: time step h = 0.001 and steepness parameter εω = 0.001.
2.6.3 Numerical calculations
In the signum model and switch model friction force is a noncontinuous functionof relative slip velocity and therefore the methods commonly used to compute theLyapunov exponents cannot be applied. A continuous friction model as is stud-ied in this section, does not possess this disadvantage and can be used duringanalysis of the systems, where the Lyapunov exponents are computed by standardprocedures (Equations (2.87)). Note, that while computing Lyapunov exponents,Equations (2.237) and three additional systems of equations with respect to per-turbations are solved.
130 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
a) b)
Fig. 2.70: Phase portraits of the analysed system without harmonic excitation for different modelsof friction: signum model (curve 1), smoothing method (curve 2), switch model (curve 3) andcontinuous friction model (curve 4).
In order to simulate the stick-slip vibrations using other friction models (signummodel, smoothing method ((1.64) approximation), switch model, and continu-ous friction model) Equations (2.237) are solved. In what follows in order tosolve the derived ordinary differential equations the standard numerical algorithmsoften used to study dynamics of lumped mechanical systems can be directly ap-plied. In our case the thus-far mentioned differential equations are solved via thefourth-order Runge–Kutta method (RK4) and the Gram–Schmidt orthonormalisa-tion method. The dynamics of the system is monitored via standard characteristicssuch as time histories in the system’s phase space and bifurcation diagrams as wellas the Lyapunov exponents.
2.6.4 Results
Let us consider first dynamics of the system for ζ = 0, that is, without harmonicexcitation. For this case, the phase portraits obtained with the signum model,smoothing method, switch model, and continuous friction model are shown inFig. 2.70.
The periodic stick-slip oscillations have almost the same sliding velocity at eachmodel (Fig. 2.70a). It is visible too, that in the sticking phase some differences areobserved (Fig. 2.70b). The differences occur in the result of another approximatingfriction force application in the near-zero relative velocity neighbourhood.
The studied mechanical system (with harmonic excitation, ζ = 0.25) possessesboth periodic and nonperiodic solutions. Figures 2.71a and 2.71b present differentbehaviours of the analysed mechanical system in the time interval τ ∈ (200, 500).
It was shown above that in comparison with the other friction models betterresults are obtained for that case using the continuous friction model. It allows usto obtain the same accuracy as in the switch model, but for larger time-step h andsteepness parameter εω. Consequently, the switch model is more expensive thanthe continuous friction model for this case, from the computational point of view.
2.6 Stick-slip vibrations continuous friction model 131
a) b)
Fig. 2.71: Phase portraits of the analysed system with harmonic excitation for various angularvelocities: (a) ω1 = 0.11; (b) ω1 = 0.3.
Fig. 2.72: Phase portraits: switch model (curve 1); continuous friction model (curve 2); contin-uous friction model for εω = 0.01 and h = 0.01 (curve 3).
Figure 2.72 shows a phase portrait of the analysed system for changed parametersω2
d = 10, ζ = 0.5, and w1 = 0.2.In this case, the sliding velocity is almost the same (for both switch and con-
tinuous friction models), but in the sticking phase some differences are observed,too. Contrary to the switch model results, results using the applied continuousfriction model are better. Namely, we have obtained almost exact (high-precision)numerical computations even for larger (10 times) parameter εω and time-step h(curve 3). In addition, the switch model is more expensive than the continuousfriction model from the computational point of view.
Below, the periodic and nonperiodic solutions are detected using a bifurcationdiagram and the Lyapunov exponent identification tools. The bifurcation diagramof the system is shown in Fig. 2.73 with the velocity ω1 as a control parameterand the angle ϕ on the vertical axis. In Fig. 2.74 the bifurcation diagram of thesystem is shown with the ζ as a control parameter and the angle ϕ on the verticalaxis.
132 2 Thermoelastic Contact of Shaft and Bush in Wear Regime
Fig. 2.73: Bifurcation diagram and the largest Lyapunov exponent λ1 of the analysed 1-dofmodel with ω1 as control parameter and ϕ on the vertical axis (ω2
d= 10, ζ = 0.5).
Fig. 2.74: Bifurcation diagram and the largest Lyapunov exponent λ1 of the analysed 1-dofmodel with ζ as control parameter and ϕ on the vertical axis (ω2
d= 5, ω1 = 0.3).
In the same plots dependencies of the largest Lyapunov exponent λ1 versusthe control parameters are also reported. A study of this bifurcation diagramimplies that chaos occurs when the exponent λ1 is positive. One of the com-puted Lyapunov exponents is always negative and second is always equal to zero(not shown in Fig. 2.73 and Fig. 2.74), because the studied system of equations isautonomous.
2.6 Stick-slip vibrations continuous friction model 133
2.6.5 Conclusions
The continuous friction model suitable for simulation of the stick-slip vibrationshas been studied and validated using the one degree-of-freedom mechanical systemwith dry friction. It has been observed that the continuous friction model yieldsengineering-accepted results and possesses some advantages in comparison tothe switch model. Interesting dynamics of the analysed system are reported andanalysed, including stick-slip periodic and chaotic behaviours. One of the impor-tant advantages of the novel continuous friction model is associated with directapplication of the standard numerical procedures devoted to either solving differ-ential equations or to computation of the Lyapunov exponents. The obtained resultshave been compared with those given by the switch model application, and theyindicate better numerical accuracy of the continuous proposed continuous model.This friction model is validated and it gives correct results, even if the numericalsteepness parameter εω is extremely large. It allows obtaining the same accuracyas in the switch model for larger time-step h and steepness parameter εω. Thecontinuous friction model may also be force in any other mechanical systems.
Chapter 3
Thermoelastic Contact of Parallelepiped
Moving Along Walls
In this chapter four models of thermoelastic contact of parallelepiped bodiesare studied. The bodies move vertically between two walls. In the first model(section 3.1), a plate (one body) hangs on a spring and can exhibit vibrations.Walls compress this parallelepiped-like body and move at constant velocity V . Inthe second model (section 3.2), a contact between the plate and nonmovable wallsoccurs owing to their heating and thermal extension. Dynamics of a two degrees-of-freedom system with an account of friction and heating processes is studied(section 3.3). A dynamic 2-dof damper with dry friction and heat generation hasbeen modelled mathematically (section 3.4).
Due to friction occurring on the contacting surface, self-excited vibrationsappear and heat is generated. This section contributes to unification of appliedseparate models (i.e., to match both self-excited and thermoelastic contact models).
If frictional heat generation is neglected, then both models exhibit either smoothfrictional self-vibrations [Kragelsky, Gitis (1987)] or a stick-slip vibration process[see, e.g., Ibrahim (1994)].
In self-excited systems loss of energy is supplied by energy generated from asource with nonvibrating properties. Energy transfer is controlled by dynamics ofthe system.
Models of frictional self-excited vibration types play an important role inmachine tool construction. They have an essential impact on two important qualitycriteria of exploitation of various machine tools: the accuracy and uniformity ofslow displacements. Note that nonuniformities of machine tool elements have aharmful impact on manufactured elements and the smoothness of their motion ishighly required. The behaviour of self-excited vibrations may occur during brak-ing processes while switching on various mechanical junctions (e.g., frictionalclutches), during cutting processes, and so on. Note that during displacements ofheavy bodies, frictional self-excited vibration excitation decreases the velocity ofmoving bodies, and hence also decreases energy requirements needed for carryingout the required work.
Both experimental and theoretical investigations of self-excited vibrations pro-vided the following observations: (1) for Vr < Vmin stick-slip motion may occur,
135DOI: 10.1007/978-0-387-09653-7_3, © Springer Science+Business Media, LLC 2009 J. Awrejcewicz and Yu. Pyryev, Nonsmooth Dynamics of Contacting Thermoelastic Bodies,
136 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
whereas for Vr > Vmin self-excited vibrations do not appear (Vmin denotes thevelocity for which the kinematic friction coefficient is minimal); (2) length ofstick intervals increases with sliding velocity decrease; (3) length of jump dura-tion approaches half of the period of vibrations of a corresponding linear system;(4) stick and slip intervals appear periodically; (5) the increase of sliding velocityinduces a decrease of both jump amplitudes and maximal value of the first displace-ment; (6) the increase of a normal load (contact pressure) causes an increase ofstick interval duration; (7) increase of the system elasticity leads to an increase ofthe period of vibrations and decrease of their amplitude [Kragelsky, Gitis (1987)].
Owing to increase of spring stiffness the considered model can be smoothlytransformed to that of thermoelastic contact of a body with moving walls [Pyryev(1994), Olesiak, Pyryev (1998)]. Similar models have also been studied in refer-ences [Olesiak, Pyryev (2000), Aleksandrov, Annakulova (1990)]. These investi-gations reveal that when the velocity of wall movement V is larger than a certainthreshold value Vcr, the so-called thermoelastic instability, or thermal bang mayoccur [Aleksandrov, Annakulova (1990)]. Thermoelastic instability is manifestedby an exponential increase of the system parameters (temperature and contact pres-sure). Energy produced on a contact surface cannot be absorbed by a surroundingmedium and the body is heated.
3.1. Kinematically driven parallelepiped-type rigid plate
Consider the following one-dimensional model of thermoelastic contact of thebody with a movable surrounding medium. Below, we consider a parallelepiped-like plate (b1 × b2 ×L) that moves vertically and is fixed to a base by a spring oflength l1 and stiffness k1 (Fig. 3.1).
The plate has mass M1 and it may move vertically along the walls in thez1 direction of a rectangular system of coordinates 0x1y1z1. In the beginning,the body is situated at distance Z0 and it has velocity Z0. The walls movingat velocity V play the role of a surrounding medium. The initial value of thedistance between the walls is equal to the plate thickness L. Then, this distance isdecreased according to the formula ε(t) = U0hU (t). As a result of this process,dry friction occurs on the parallelepiped walls X = ±L/2. It is defined by thefunction Ffr(Vr), where Vr is the relative velocity of the plate and walls; thatis, Vr = V − Z . According to Amonton’s assumption, friction force Ffr is equalto the product of the normal reaction component N(t) and friction coefficientFfr = f(Vr)N(t). It characterizes resistance of the bodies during their relativedisplacement; f(Vr) denotes the kinetic friction coefficient (f(−Vr) = −f(Vr))(Fig. 2.2).
The action of friction on the contact surface X = −L/2, X = L/2 generatesheat and wear. We follow the generally accepted assumption [see Ling (1959)] thatfriction work is transformed into heat energy. Furthermore, we assume that wallsideally transform heat and that between the plate and walls heat transfer is governed
3.1 Kinematically driven parallelepiped-type rigid plate 137
Fig. 3.1: Scheme of the plate moving between two movable walls.
by Newton’s law [Carslaw, Jaeger (1959)], and a surrounding medium temperatureis Tsm. Plate surfaces not being in contact with movable walls are thermallyisolated and they have dimension L/b1, L/b2. The mentioned assumptions allowus to introduce a one-dimensional model.
The formulated problem is limited to the determination of plate center move-ment Z(t), plate velocity Z(t), contact pressure
P (t) = N(t)/b1b2 = −σXX(−L/2, t) = −σXX(L/2, t),
plate temperature T1(X, t), displacement U(X, t) in the X direction, and wearUw(t).
3.1.1 Mathematical modelling
In the considered case, the mathematical model is governed by the dynamics of aplate center of the form [Ulitko (1990)]
md2
dt2Z(t) + c(Z(t) − l1) − mg = 2f(Vr)P (t), (3.1)
and the equation governing the theory of thermal stresses for an isotropic body ofthe form [Nowacki (1962)]
138 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
∂
∂X
[∂
∂XU(X, t) − α1
1 + ν1
1 − ν1T1(X, t)
]= 0, (3.2)
∂2
∂X2T1(X, t) =
1
a1
∂
∂tT1(X, t), X ∈ (−L/2, L/2), (3.3)
with the attached mechanical
U
(−L
2, t
)=
U0
2hU (t) − Uw(t), U
(L
2, t
)= −U0
2hU (t) + Uw(t), (3.4)
and heat boundary conditions
±λ1∂T1(±L/2, t)
∂X+ αT (T1(±L/2, t) − Tsm) = f(Vr)VrP (t), (3.5)
and the following initial conditions
T (X, 0) = Tsm, X ∈ (−L/2, L/2), Z(0) = Z0,d
dtZ(0) = Z0.
(3.6)In our further considerations, we assume that velocity of the plate wear is
proportional to a certain power of the friction force. According to the Archardassumption [Archard (1953)] one gets
Uw(t) = Kw |Vr(t)|P (t). (3.7)
Normal stress occurring in the plate is defined by the following formula,
σXX =E1
1 − 2ν1
[1 − ν1
1 + ν1
∂U
∂X− α1T1
]. (3.8)
In the above, the following notation is taken: E1 is the elasticity modulus; ν1,λ1, a1, α1, αT are Poisson coefficients, heat transfer, thermal diffusion, linearheat extension of the plate, and heat transfer from a wall to the plate, respectively;m = M1/b1b2, c = C/b1b2, P (t) = N(t)/b1b2 are the contact pressure.
Integration of Equation (3.2) with Equation (3.8) and boundary conditions (3.4)gives the contact pressure P (t) = −σXX(−L/2, t) = −σXX(L/2, t):
P (t) = P0hU (t) − 2P0
U0Uw(t) +
Eα
L
L/2∫
−L/2
T1(ξ, t)dξ, (3.9)
where
P0 = EU0
L, α = α1
1 + ν1
1 − ν1, E =
E1(1 − ν1)
(1 + ν1)(1 − 2ν1). (3.10)
3.1 Kinematically driven parallelepiped-type rigid plate 139
Let us introduce the following coefficients of similarity
t∗ = 1/ω [s], L∗ = V/ω [m], N∗ = P0L2∗ [N], T∗ = R0V P0[
◦C],
and the following dimensionless parameters
x =X
L∗, z′ =
Z
L∗, τ =
t
t∗, v =
Z(t)
V, z′0 =
Z0
L∗, z′0 =
Z0
V,
u =U
L∗, uw =
Uw
U0, p =
PL2∗
N∗, pst =
PstL2∗
N∗, θ =
T1 − Tsm
T∗, ω =
ω∗
ω,
l =L
L∗, l′ =
l1L∗
, Bi = R0αT , Ω = ωV2
V, γ =
EαV
αT, kw =
KwEV
ωL,
(3.11)
where
R0 =L
λ1, ω =
√c
m, ω∗ =
a1
L2, V2 =
P0
mω∗. (3.12)
In the above, the following notation is applied. P0 is the contact pressureassuming a lack of heat extension; R0 is the heat resistance generated by a heattransfer; and ω is the free frequency of mass center vibrations with a lack of acontact pressure.
In the dimensionless form, the considered mathematical model is
z′(τ) + z′(τ) = 2Ωf(V (1 − z′(τ)))p(τ) + l′ + gt∗/V , (3.13)
∂2
∂x2θ(x, τ) =
1
ωl2∂
∂τθ(x, τ ), x ∈
(− l
2,l
2
), (3.14)
± l∂θ(±l/2, τ)
∂x+ Biθ(±l/2, τ ) = f(V (1 − z′(τ)))(1 − z′(τ))p(τ), (3.15)
θ(x, 0) = 0, z(0) = z′0, z(0) = z′0, (3.16)
where the dimensionless contact pressure is described by the formula:
p(τ) = hU (τ) − 2uw(τ) +γBi
l
l/2∫
−l/2
θ(η, τ)dη. (3.17)
140 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
3.1.2 Laplace transform
In order to solve problem (3.13)–(3.15), the Laplace transform is applied withrespect to time τ . The theorem on ‘convolution’ is applied [Abramowitz, Stegun(1965), Carslaw, Jaeger (1959)] while finding the inverse transform. Finally,we get:
z(τ) + z(τ) = 2Ω[f(V (1 − z(τ)))p(τ) − f(V )pst], (3.18)
z(0) = z0, z(0) = 0,
p(τ) = 1 + 2γωBi
τ∫
0
G(τ − η)f(V (1 − z(η)))(1 − z(η))p(η)dη,
(3.19)
θ(τ) = ω
τ∫
0
g(τ − η)f(V (1 − z(η)))(1 − z(η))p(η)dη, (3.20)
uw(τ) = kw
τ∫
0
|1 − z(η)| p(η)dη, (3.21)
where
z(τ) = z′(τ) − z′st, z0 = z′0 − z′st, z′st = 2Ωf(V )pst + l′ + gt∗/V
g(τ) =∞∑
m=1
∆3(μm)
∆′(μm)exp(−μ2
mωτ),
G(τ) =
∞∑
m=1
∆2(μm)
∆′(μm)exp(−μ2
mωτ ), (3.22)
∆′(μm) = 0.5(2BiSm + Cm − Bi2(Cm − Sm)/μ2m),
∆(μm) = 2BiCm + Sm(Bi2 − μ2m), θ(τ) = θ(−L/L∗, τ) = θ(L/L∗, τ),
∆2(μm) = Sm − BiC0m, ∆3(μm) = 1 + Cm + BiSm,
Cm = cosh(μm), Sm =sinh(μm)
μm, C0
m =Cm − 1
μm,
and sm = −μ2m are the roots of the characteristic equation ∆(s) = 0.
3.1 Kinematically driven parallelepiped-type rigid plate 141
3.1.3 Stationary process subject to kinematic external excitation
Let us first linearize the problem in the vicinity of the stationary solution pst, θst,zst, assuming that
θ(x, τ) = θst(x) + θ∗(x, τ), p(x) = pst + p∗(τ), z(τ) = z∗(τ), (3.23)
where the stationary solutions are
pst =1
1 − γf(V ), θst =
1
Bi
f(V )
1 − γf(V ), zst = 0. (3.24)
The characteristic equation of the linearized system has the form
∆∗(s) = Ω2(s)∆1(s) − 2γBif(V )Ω1(s)∆2(s), (3.25)
where
Ω1(s) = s2 − Ωpstf(V )s + 1, Ω2(s) = s2 + ΩpstV f ′(V )s + 1,
∆1(s) = 2Bi C + S(Bi2 + s/ω), ∆2(s) = S + Bi C0, (3.26)
S = sinh√
s/ω/√
s/ω, C = cosh√
s/ω, C0 = (C − 1)/(s/ω),
γ1 = V2f(V )
V, γ(V ) = V2f
′(V ), v(V ) =V
V1(V )= γf(V ),
V1(V ) =V
γf(V ), V2 =
P0
ω∗m.
The characteristic function is
∆∗(s) =
∞∑
m=0
dmsm, (3.27)
where
d0 = 2Bi(1 + Bi/2)(1 − v(V )),
dm = d(1)m + (1 − δm1)ω
2(d(1)m−2 − 2Bi v(V )d
(2)m−2)
− 2Bi v(V )d(2)m + 2pstω
2(γd(1)m−1 + 2Bi v(V )γ1d
(2)m−1),
d(1)m =
(1
ω
)m [Bi2
(2m + 1)!+
2Bi
(2m)!+
1
(2m − 1)!
],
142 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
d(2)m =
(1
ω
)m [1
(2m + 1)!+
Bi
(2m + 2)!
].
It is worth noting that the series (3.27) for finite value m provides analyticalconditions for stability of a static solution. One may apply the known criteriasuch as the ones proposed by Routh and/or Hurtwitz [Gantmacher (1959)]. Thementioned criteria are based on the determination of zeros of a polynomial withconstant real coefficients with a simultaneous estimation of the right half-plane(RHP) or left half-plane (LHP) of the complex plane. They can be used to derivea first approximation of the critical values of parameters of the considered models.
Let us consider the following limiting cases.
1. In the case when v(V ) = 0 (γ = 0) (i.e., thermal expansion of a parallelepipedis not accounted for), a characteristic equation associated with the perturba-tion equation takes the form Ω2(s) = 0. Roots of the associated characteristicequation will lie in the RHP if f ′(V ) < 0 (γ < 0). In this case a stationarysolution is unstable, and consequently a corresponding nonstationary solutionof the nonlinear problem will move away from the equilibrium state. This kindof motion appears when 0 < V < Vmin. The frequency of vibrations for γ = 0is equal to 1.
2. In the case when ω = 0 (i.e., when the spring stiffness approaches an infinitevalue (unmovable body)), the characteristic equation of the linearized problemis
∆1(s) − 2Bi v(V )∆2(s) = 0. (3.28)
The roots sm, m = 1, 2, 3, . . . of Equation (3.28) lie on the real axis(Imsm = 0). When inequality v(V ) < 1 holds, the roots are negative (sm < 0)and static solutions are stable. For v > 1, root s1 lies in the RHP; that is, the staticsolution is unstable.
For v < 1, an unstationary solution for τ → ∞ tends to a stable solutionof problem (3.24). For v > 1, an unstationary solution increases exponentiallyexp(s1τ), where s1 is estimated by the formula
s1 = −ωd0/d1 = (v − 1)Bi(2 + Bi)/d1. (3.29)
For v > 1 the stable solution loses its physical meaning (pst < 0, θst < 0).A small variation of parameter v in the vicinity of 1 gives the exponential diver-gence of the solution. The latter phenomenon is referred to as the thermoelasticinstability [Pyryev et al. (1995)] or heat explosion [Aleksandrov, Annakulova(1990)].
Note that during the thermoelastic instability, the force required to hold upa uniform motion at velocity V increases, and hence energy supplied from theenvironment increases too. It means physically, that heat generated on the contactsurface cannot be received by the surrounding environment and both bodies areheated.
3.1 Kinematically driven parallelepiped-type rigid plate 143
Fig. 3.2: Zones of parameters associated with an unstable stationary solution.
Next, analysis of the zones of parameters for which the stationary solutions arestable (Fig. 3.2) is carried out. During computations the following parameters arefixed: Bi = 1, γ1 = 0.5.
In Fig. 3.2 zones lying inside the curves correspond to parameters (γ, v) and(ω, v). In Fig. 3.2b the curves correspond to the critical values of the parameters(ω, v) for γ = −0.02 and γ = −0.08.
One may conclude through analysis of the figures that during the decrease of thenegative value of γ, a stable zone with respect to parameter v decreases (narrowingof the zone appears), whereas the critical values of ω increase. For γ < −0.107 asolution for ω = 1, Bi = 1, γ1 = 0.5 is always unstable. Points 1–13 correspondto oscillations at the frequency ω = |Ims1,2|. The oscillation frequency increaseswith increase of v from ω = 1 (heat expansion is not accounted for) to the valueof ω ≈ 4.
3.1.4 Algorithm and solutions
Recall that the analysed problem is governed by the system of two nonlinearequations, that is, differential and integral ones (3.19). The solutions are soughtusing the fourth-order Runge–Kutta method and trapezium approximation.
Solution Analysis. A numerical analysis is carried out for a steel parallelepiped(α = 14 · 10−6 ◦C−1, λ1 = 21 W m−1 ◦C−1, a1 = 5.9 · 10−6 m2 s−1, ν1 = 0.3,E = 19 · 1010 Pa) for the following fixed parameters Bi = 4.76, Ω = 12.8,ω = 0.26 · 10−2, V = 1.5 · 10−2 m s−1, and for various values of both the wearcoefficient kw and distance z0. It is assumed that hU (τ) = H(τ). Dashed curvescorrespond to the case when heat expansion γ = 0.
144 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
Fig. 3.3: Phase portrait (a) and evolution in time of contact temperature (b) for z0 = 15, kw = 0(solid curve: γ = 10; dashed curve: γ = 0).
Fig. 3.4: Phase portrait (a) and evolution in time of contact temperature (b) for z0 = 0.15,kw = 0 (solid curve: γ = 10; dashed curve: γ = 0).
In Figs. 3.3 and 3.4 the contact characteristics of the system determined nu-merically without wear (kw = 0) for various initial conditions z0 = 15, z0 = 0(Fig. 3.3), and z0 = 0.15, z0 = 0 (Fig. 3.4) are shown. Dashed curves in thefigures correspond to the case γ = 0, whereas solid curves refer to γ = 10.
Depending on the initial condition (Figs. 3.3 and 3.4), vibrations approach thestick-slip limit cycle either from its inside or outside. The parallelepiped heatexpansion yields a decrease of vibration amplitude.
Results of the numerical analysis with reference to wear are shown in Fig. 3.5.
Conclusions. To conclude the results obtained in this section, let us mainlyemphasise the study of a new problem of thermoelastic contact of a body with
3.2 Rigid plate dynamics subject to temperature perturbation 145
Fig. 3.5: Phase portrait (a), wear evolution with time (b), contact pressure (c), and contacttemperature (d) for z0 = 0.15 (curve 0: γ = 0, kw = 0.017; curve 1: γ = 10, kw = 0; curve 2:γ = 10, kw = 0.017; curve 3: γ = 10, kw = 0.17).
the moving external medium. The main new features of the problem are exhibitedthrough a body inertia and a dependence of the friction coefficient on the relativevelocity of contacting bodies. The proposed model can be applied to estimate thecontact characteristics of moving tribological systems.
3.2. Rigid plate dynamics subject to temperature perturbation
In the literature there are many examples focused on the analysis of autonomoussystems exhibiting regular nonlinear self-excited vibrations (see Figs. 3.6a,b).Vibrations of a mechanical system modelling woodpecker behaviour (Fig. 3.6a)have been analysed by [Pfeiffer (1984), Leine, Campen (2006)] and also studied inthe monograph by [Awrejcewicz (1996)]. The ‘mechanical woodpecker’ consists
146 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
Fig. 3.6: Models of a woodpecker knocking into a cylinder [Pfeiffer (1984)] (a); an ‘Olędzki’sslider’ [Olędzki, Siwicki (1997)] (b); and ‘frog-type’ system (c).
of a stiff body with the moment of inertia B2 and mass m2, and is coupled toa bush by a spring with stiffness k. The bush mass inertial moment is denotedby B1.
Vibrations of the so-called ‘Olędzki’s slider’ have been studied first in thereference [Olędzki, Siwicki (1997)] (see Fig. 3.6b). If the mass centre S2 of ahorizontal rod is located at a distance larger than b/(2f) (b denotes the sliderlength, f is the friction coefficient), then in static conditions, the rod and slidermotions along the vertical rod–guide axis are not possible. In static conditions thisdistance is
l =
(1 +
m1
m2
)b
2f
√
1 + 2d
b
2δ
b+
(2δ
b
)2
,
where m1 is the slider mass, m2 is the mass of the horizontal rod, d is thecylinder diameter, and δ denotes the small backlash between the cylinder and
3.2 Rigid plate dynamics subject to temperature perturbation 147
bush. However, an infinitely small perturbation of the end of the horizontal rodforces the system to move.
Note that the two models of the slider movement discussed are associated witha stick-slip vibration behaviour.
Despite simplicity of the introduced models, important information is obtained.Namely, it is shown that the static self-braking kinematic pairs can initiate move-ment when vibration appears.
Consider now our new proposed model, which does not have any elastic part,but can exhibit self-excited stick-slip vibrations (Fig. 3.6c). For simplicity, it isfurther referred to as a ‘frog-slider’.
Let us consider a one-dimensional model of the thermoelastic contact of a bodywith a surrounding medium. Assume that this body is represented by a rectangularplate (b1 × b2 × 2L) (Fig. 3.6c). The plate together with a ‘frog’ has mass M1
subject to force F = F∗hF (t) and moves vertically along the walls in direction z1
of the rectangular coordinates 0x1y1z. Initially, the body is situated at distance Z0
and has velocity Z0. The distance between the walls is always equal to the initialplate thickness 2L. The plate moves at nonconstant velocity Z(t).
It is assumed that heat transfer between the plate and the walls is ideal andthe Newton assumptions hold. Initially, the instant temperature is governed by theformula T0hT (t) (hT (t) → 1, t → ∞). It causes a parallelepiped heat extensionin the direction of 0x1, and the body starts to contact the walls. As a result ofthis process a frictional contact occurs on the parallelepiped sides X = ±L.A simple frictional model is applied further; that is, friction force Ffr is theproduct of normal reaction force N(t) and the friction coefficient. It means thatFfr = f(Z)N(t) is the friction force defining resistance of the movement ofbodies. Here, contrary to the assumption made in the reference [Olesiak, Pyryev(2000)], the kinematic friction coefficient f(Z) depends on the relative velocityof the sliding bodies (Fig. 2.2).
The friction force σXZ(X, t) per unit contact surface X = −L, X = L,generates heat. According to Ling’s assumptions [Ling (1959)], the work of frictionforces is transmitted into heat energy. Note that noncontacting plate surfaces areheat-insulated and have the dimensions of L/b1 ≪ 1, L/b2 ≪ 1, which is inagreement with the assumption of our one-dimensional modelling.
Hence, the problem is reduced to the determination of mass plate centre dis-placement Z(t); plate velocity Z(t); contact pressure P (t) = N(t)/b1b2 =−σXX(−L, t) = −σXX(L, t); plate temperature T1(X, t); and displacementU(X, t) in the X-axis direction.
3.2.1 Mathematical formulation
In the considered case, the studied problem is governed by the dynamics of theplate mass centre
mZ(t) = F∗hF (t) − 2f(Z)P (t), (3.30)
148 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
and equations of the heat stress theory for an isotropic body [Nowacki (1962)](3.2), (3.3) with the attached mechanical
U(−L, t) = 0, U(L, t) = 0, (3.31)
heat
− λ1∂T1(−L, t)
∂X+ αT (T1(−L, t) − T0hT (t)) = f(Z)Z(t)P (t), (3.32)
λ1∂T1(L, t)
∂X+ αT (T1(L, t) − T0hT (t)) = f(Z)Z(t)P (t), (3.33)
and initial
T (X, 0) = 0, X ∈ (−L, L), Z(0) = Z0, Z(0) = Z0 (3.34)
conditions. Normal stresses occurring in the plate are defined by (3.8).In the above, the following notation is applied: E1 is the elasticity modulus;
ν1, λ1, a1, α1, αT are Poisson’s ratio, thermal conductivity, thermal diffusivity,thermal expansion, and heat transfer coefficients, respectively; m = M1/b1b2;whereas P (t) = N(t)/b1b2 denotes the contact pressure.
Integration of Equation (3.2) with an account of (3.8) and boundary conditions(3.31) provides the contact pressure P (t) = −σXX(−L, t) = −σXX(L, t):
P (t) = P0 + Eα1
L
L∫
0
T1(ξ, t)dξ. (3.35)
Let us introduce the following similarity coefficients
t∗ = L2/a1 [s], V∗ = a1/L [m s−1], P∗ = T0E1α1/(1 − 2ν1) [N m−2],
and the following dimensionless parameters
x =X
L, τ =
t
t∗, z =
Z
L, p =
P
P∗, θ =
T1
T0, ε1 =
2P∗t2∗
mL, (3.36)
γ =E1α1a1
(1 − 2ν1)λ1, Bi =
LαT
λ1, m0 =
F∗
2P∗, F (z) = f(V∗z). (3.37)
Our problem is modeled using the following dimensionless equations
∂2θ(x, τ)
∂x2=
∂θ(x, τ)
∂τ, x ∈ (−1, 1), τ ∈ (0,∞), (3.38)
z(τ) = ε1(m0hF (τ) − F (z)p(τ)), (3.39)
3.2 Rigid plate dynamics subject to temperature perturbation 149
with attached boundary
[∂θ(x, τ)
∂x± Biθ(x, τ)
]
x=±1
= ±q(τ), (3.40)
and initial conditions
θ(x, 0) = 0, z(0) = z◦, z(0) = z◦, (3.41)
where
q(τ) = BihT (τ) + γF (z)z(τ)p(τ), p(τ) =1
2
1∫
−1
θ(ξ, τ)dξ. (3.42)
3.2.2 Application of the Laplace transform
Applying the Laplace transform
{θ, p, q
}=
∞∫
0
{θ, p, q} e−sτdτ ,
the following is obtained
d2θ
dx2= sθ, (3.43)
[dθ
dx− Biθ
]
x=−1
= −q,
[dθ
dx+ Biθ
]
x=1
= q, (3.44)
p =1
2
1∫
−1
θ(ξ, s)dξ. (3.45)
A solution to Equation (3.43) is sought in the form
θ(x, s) = A(s)S(x, s) + B(s)C(x, s), (3.46)
whereS(x, s) = sinh(
√sx)/
√s, C(x, s) = cosh(
√sx).
The quantities A(s) and B(s) are defined by two boundary value problems (3.44).Finally, we get
θ(x, s) = sq(s)Gθ(x, s), p(s) = sq(s)Gp(s), (3.47)
150 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
where
Gp(x, s) =S(s)
s∆1(s), Gθ(x, s) =
C(x, s)
s∆1(s),
∆1(s) = sS(s) + BiC(s),
S(s) = sinh(√
s)/√
s, C(s) = cosh(√
s). (3.48)
Applying an inverse Laplace transformation, the following system of equationsis obtained,
p(τ) = Bi
τ∫
0
hT (ξ)Gp(τ − ξ)dξ + γ
τ∫
0
F (z)z(ξ)p(ξ)Gp(τ − ξ)dξ, (3.49)
z(τ) = ε1
⎡
⎣m0
τ∫
0
hF (ξ)dξ −τ∫
0
F (z)p(ξ)dξ
⎤
⎦ , (3.50)
which gives dimensionless pressure p(τ) and velocity z(τ). Temperature is definedby the following formula
θ(x, τ) = Bi
τ∫
0
hT (ξ)Gθ(x, τ − ξ)dξ + γ
τ∫
0
F (z)z(ξ)p(ξ)Gθ(x, τ − ξ)dξ,
(3.51)
where
{Gp(τ), Gθ(1, τ)} =1
Bi−
∞∑
m=1
{2Bi, 2μ2
m
}
μ2m[Bi(Bi + 1) + μ2
m]e−µ2
mτ , (3.52)
and μm are the roots of the following characteristic equation
tgμm =Bi
μm, m = 1, 2, . . . . (3.53)
Observe that μm ≈ πm, m → ∞.Note that it is easy to formulate a relation between heat flow and contact pressure
velocity variation in the form
dp(τ)
dτ=
∂θ(τ, x)
∂x
∣∣∣∣x=1
, (3.54)
which is used further to analyse the contact characteristics.In order to carry out numerical analysis, a knowledge of the function (3.52) is
required. The values of function (3.52) for τ → 0 and τ → ∞ are defined using
3.2 Rigid plate dynamics subject to temperature perturbation 151
the theorem [Abramowitz, Stegun (1965)] on the limiting values:
{Gp(τ), Gθ(1, τ)} ≈ 1/Bi, τ → ∞, (3.55)
Gp(τ) ≈ τ, Gθ(1, τ) ≈ 2√
τ/√
π, τ → 0. (3.56)
3.2.3 Stationary process
A stationary solution to the problem is:
pst =1
1 − v, θst =
1
1 − v, v = F (vst)
vstγ
Bi, (3.57)
where vst is the solution of the nonlinear equation
F (vst) =m0
1 + γm0vst/Bi. (3.58)
A graphical solution to Equation (3.58) is presented in Fig. 3.7 for variousparameters m0 and Bi. Recall that for steel γ = 1.87. The right-hand side ofEquation (3.58) is represented by a solid curve for different values of the pa-rameters m0 and Bi. Solid curve 1 is associated with the parameter m0 = 0.14,Bi = 20, solid curve 2 with m0 = 0.1, Bi = 20, solid curve 3 with m0 = 0.1,Bi = 5, and solid curve 4 with m0 = 0.14, Bi = 5. The dashed curve is relatedto the function F (vst).
Fig. 3.7: Graphical solution of Equation (3.58) (solid curves: 1, m0 = 0.14, Bi, 20; 2, m0 = 0.1,Bi = 20; 3, m0 = 0.1, Bi = 5; 4, m0 = 0.14, Bi = 5; dashed curve corresponds to F (vst)).
152 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
At m0 = 0.14, Bi = 20 (first case) Equation (3.58) can have one solutionv3
st (F ′(v3st) > 0), at m0 = 0.1, Bi = 20 (second case) three solutions v1
st,v2
st, v3st (F ′(v1
st) > 0, F ′(v1st) < 0, F ′(v3
st) > 0), and at m0 = 0.1, Bi = 5(third case) one solution v1
st = 0 (with approximation (2.58) v1st ≈ ε0m0/2fs,
F ′(v1st) ≈ 2fs/ε). At m0 = 0.14, Bi = 5 (fourth case) there is one solution v2
st
(F ′(v2st) < 0).
The case of constant friction represented in Fig. 3.7 by dashed horizontal lineF (vst) = fs = const was earlier considered by [Olesiak, Pyryev (2000)], wherevst = Bi(m0/fs − 1)/(m0γ).
Let us introduce a perturbation of the stationary solution (3.57) using thefollowing formulas
z = vst + z∗, p = pst + p∗, θ = θst + θ∗, hT = 1 + h∗T . (3.59)
Owing to the linearization of the right-hand sides of (3.39) and taking intoaccount the boundary condition (3.40), the following linear problem is obtained
∂2θ∗(x, τ)
∂τ2=
∂θ∗(x, τ)
∂τ, (3.60)
z∗(τ) = ε1[−F (vst)p∗(τ) − F ′(vst)pstz
∗], (3.61)
[dθ∗(x, τ)
dx− Biθ∗(x, τ)
]
x=−1
= −q∗(τ), (3.62)
[dθ∗(x, τ)
dx+ Biθ∗(x, τ)
]
x=1
= q∗(τ), (3.63)
where
q∗(τ) = Bih∗T (τ) + γ(vstpst(β1 + β2)z
∗(τ) + vstF (vst)p∗(τ)),
p∗(τ) =1
2
1∫
−1
θ∗(ξ, τ)dξ, β1 =F (vst)
vst, β2 = F ′(vst). (3.64)
Furthermore, applying the Laplace transform, a solution of the problem(3.60)–(3.64) in the transform domain is found. For example, the pressure pertur-bation is
p∗(s) =S(s)Bi(β2pstε1 + s)
∆(s)h∗
T (s), (3.65)
where
∆(s) = (ε1pstβ2 + s)∆1(s) + Biv(ε1pstβ1 − s)S(s) = 0 (3.66)
is the characteristic equation of the linearized problem. The roots sm (Res1 >Res2 > · · · > Resm > · · · , m = 1, 2, 3, . . . ) of the characteristic Equation (3.66)
3.2 Rigid plate dynamics subject to temperature perturbation 153
lie either on the left-hand side of the complex plane Res < 0 (stationary solution isstable) or on the right-hand side of the complex plane Res > 0 (stationary solutionis unstable) of the complex variable s.
The characteristic function ∆(s), has the form of infinite-order polynomial
∆(s) =∞∑
m=0
smbm, (3.67)
where
b0 = ε1pstc0, bm = ε1pstcm + dm−1, m = 1, 2, . . .
c0 = Bi(β2 + vβ1), dm = d(1)m − Bivd(2)
m , cm = β2d(1)m + Bivβ1d
(2)m ,
d(1)m =
2m + Bi
(2m)!, d(2)
m =1
(2m + 1)!.
Observe that during analysis of the roots of the characteristic equation (3.66),the parameter vst represents a solution of the nonlinear Equation (3.58). Further-more, note that in accordance with (3.57) 0 < v < 1. It is easy to prove thatdm > 0 (m = 0, 1, 2, . . . ) and bm > 0, if β2 ≥ 0.
Assuming that the body moves at constant velocity vst = const, the so-calledfrictional thermoelastic instability occurs (Res1 > 0) for v > 1. The latter ischaracterized by an exponential increase of the contact characteristics, and themoving body is overheated.
3.2.4 Constant friction coefficient and nonstationary process
Analysis of the announced problem is carried out for F (z) = fs = const usingthe Runge–Kutta and quadrature methods, and taking into account the asymptoticestimation (3.55), (3.56).
For a steel plate (parameter γ = 1.87) of the parallelepiped shape (α1 =14 · 10−6 ◦C−1, λ1 = 21 W m−1 ·◦ C−1), ν1 = 0.3, a1 = 5.9 · 10−6 m2s−1,E1 = 19 · 1010 Pa), and for L = 0.01 m, Bi = 10, T0 = 5◦C, fs = 0.12 thecomputational results are shown in Fig. 3.8 through 3.10 for ε1 = 1000; 500; 100.Figures 3.8a to 3.10a correspond to body braking (z(0) = z◦ = 169, m0 = 0),whereas Figs. 3.8b–3.10b refer to body acceleration generated by the applied forceF∗ = 9.31 ·106 N m−2 (m0 = 0.14), z◦ = z◦ = 0. In the latter case t∗ = 16.95 s,V∗ = 0.5910−3 m s−1, P∗ = 3.3 · 107 Pa.
Braking process. Dimensionless body velocity z(τ) versus dimensionless timeτ is shown in Fig. 3.8a. Curve 1 corresponds to the dimensionless parameterε1 = 1000 (m = 1.9 · 109 kg m−2), curve 2 corresponds to parameterε1 = 500 (m = 2.8 · 109 kg m−2), and curve 3 refers to parameter ε1 = 100
154 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
Fig. 3.8: Time histories of the body velocity during braking (a) and acceleration (b) for variousvalues of parameter ε1 (curves 1: ε1 = 1000; curves 2: ε1 = 500; curves 3: ε1 = 100).
Fig. 3.9: Time history of contact pressure during braking (a) and acceleration (b) for variousvalues of ε1 (curves 1: ε1 = 1000; curves 2: ε1 = 500; curves 3: ε1 = 100).
(m = 1.9 · 1010 kg m−2). Dimensionless contact pressure p(τ) and dimension-less contact temperature θ(−1, τ) = θ(1, τ) are reported in Figs. 3.9a and 3.10a(curve 1: ε1 = 1000; curve 2: ε1 = 500; curve 3: ε1 = 100).
A body with the initial velocity z◦ = 169 (Z0 = 0.1 m s−1) is stopped by theaction of friction force if the following inequality holds: m0 < fs (F∗ < 2fsP∗).Then the body is cooled, and after reaching wall temperature T0, it starts to moveagain. A reason is that the applied force is smaller than the friction force.
An increase of the body inertia (decrease of parameter ε1) causes an increaseof both braking duration (see Fig. 3.8a) and contact temperature (Fig. 3.10a),and the contact pressure (Fig. 3.9a) is increased. Contrary to the braking process
3.2 Rigid plate dynamics subject to temperature perturbation 155
Fig. 3.10: Time histories of contact temperature during braking (a) and acceleration (b) fordifferent ε1 (curves 1: ε1 = 1000; curves 2: ε1 = 500; curves 3: ε1 = 100).
associated with the constant friction [Chichinadze et al. (1979)], the braking time isnot proportional to the mass body. Both contact temperature and pressure achievetheir maximal values just before the body stops, and then the body temperatureapproaches surrounding medium temperature.
Acceleration process. As Figs. 3.8b to 3.10b show, the contact characteristicsduring acceleration are represented by damped oscillations, which tend to steadystates. According to (3.57), one gets: vst = 6.37, pst = 1.17, and θst = 1.17. Theroots of characteristic equation (3.67) with the maximal real parts (ε1 = 1000) ares1,2 = −1.1±i2.3 for the first case (expected ‘period’ of damped oscillations T =2π/Ims1,2 = 2.7), for the second case (ε1 = 500) s1,2 = −0.99 ± i1.5 (expected‘period’ of damped oscillations Tp = 4.1), and for the third case (ε1 = 100)one gets real root values s1 = −0.49, s2 = −1.33. It means that the contactcharacteristics should change in an aperiodic manner.
The numerical analysis fully confirms our theoretical prediction. In the firstcase the period of damped oscillations of the contact characteristic is Tp = 2.7,whereas in the second case Tp = 4.1. Note that before reaching a stationary state,the body may exhibit stick-slip dynamics (curves 1 in Fig. 3.8b). It is worth notingthat during the acceleration process, owing to an increase of the body inertia(decrease of parameter ε1) both contact pressure (see Fig. 3.9b) and temperature(see Fig. 3.10b) decrease, whereas a ‘period’ of damped oscillations increases (seeFig. 3.8b).
3.2.5 Variable friction coefficient and nonstationary process
In this case the steel parallelepiped plate (α1 = 14 · 10−6◦C−1, λ1 = 21 W m−1 ·◦C−1), ν1 = 0.3, a1 = 5.9 · 10−6 m2 s−1, E1 = 19 · 1010Pa) with L = 0.01 m,
156 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
T0 = 5◦C, z◦ = z◦ = 0 and with nonconstant friction coefficient is studied.One gets t∗ = 16.95 s, V∗ = 0.59 · 10−3m s−1, P∗ = 3.3 · 107 Pa. The functionF (z) = f(V∗z) is defined by formula (2.57).
In the first case (for m0 = 0.14, Bi = 20) one solution v3st = 101.1, p3
st = 2.32,θ3
st = 2.32, (β1 = 0.6·10−3, β2 = 0.4·10−3, v = 0.57) is found. It is always stable(for instance, for ε1 = 800 the roots of Equation (3.67) s1,2 = −1.02 ± 1.57i,whereas for ε1 = 75.45 the roots of Equation (3.67) s1,2 = −0.57 lie on the left-hand side of the complex variable s). In this case after a transitional process, thebody starts to move at constant velocity v3
st = 101.1. The contact characteristicsfor ε1 = 800 achieve their limiting values through damped oscillation process withthe expected ‘period’ Tp = 4, whereas for ε1 < 75.45 the contact characteristicsare overdamped.
In the second case (m0 = 0.1, Bi = 20) three solutions appear. The solutionv3
st = 87.1, p3st = θ3
st = 1.8, θ3st = 2.25, (β1 = 0.6 · 10−3, β2 = 0.3 · 10−3,
v = 0.45) is stable (e.g., for ε1 = 800 the roots of Equation (3.66) s1,2 =−0.93 ± 1.1i lie on the left-hand side of the complex plane s). The solutionv2
st = 5.26 (β1 = 0.2 · 10−1, β2 = −0.4 · 10−2, v = 0.05) can be unstable (e.g.,for ε1 = 800 the root of Equation (3.67) s1 = 2.84). The solution describedthrough approximation (2.58) v1
st = 0.4 ·10−4 (β1 = β2 = 2.4 ·103, v = 4 ·10−7)corresponds to an equilibrium state (roots of Equation (3.67) s1 = −2.2, s2 = −20lie on the left-hand side of the complex plane). In the second case, the contactcharacteristics for ε1 = 800 tend, depending on initial conditions, to one of twostable solutions.
In the third case (m0 = 0.1, Bi = 5) one solution appears. Approximationof solution (2.58) v1
st = 0.4 · 10−4 ≈ 0, p1st = 1.0, θ1
st = 1.0, (β1 = β2 =2.4 · 103, v = 1.5 · 10−6) corresponds to the equilibrium state (for ε1 = 800roots of Equation (3.67) s1 = −1.7, s2 = −16.3 lie on the left-hand side ofthe complex plane). Note that in this case the braking process always occurs (theapplied external force is smaller than the friction force).
In the fourth case (m0 = 0.14, Bi = 5) we have one solution of the form:v2
st = 27.8, p2st = θ2
st = 2.45, (β1 = 0.2 · 10−2, β2 = −0.58 · 10−3, v = 0.59).It is unstable if parameter ε1 is larger than its critical value (ε1 ≥ ε). One mayuse the characteristic function (3.67) for m = 3 to estimate a stability zoneof the stationary solution of (3.57). Recall that one of the conditions for cubiccharacteristic equation roots to lie on the right-hand side of the complex plane isexhibited by the inequality b1b2 − b0b3 < 0. It results in our case in the followinginstability condition
ε1 > ε, ε = (1 − v)(−B −√
B2 − 4AC)/(2A), (3.68)
where A = c1c2−c0c3, B = c1d1+c2d0−c0d2, C = d0d1. In Fig. 3.11 the dashedcurve represents dependence of function ε on parameter Bi. The stability loss curvederived through analysis of the characteristic Equation (3.66) is denoted by thesolid line. Observe that for considered parameters good agreement of unstable zoneestimation by Equations (3.66) and (3.67) for m = 3 is achieved. Furthermore,
3.2 Rigid plate dynamics subject to temperature perturbation 157
Fig. 3.11: Critical parameter ε1 versus Bi.
Fig. 3.12: Dimensionless period Tp versus Bi (a), and dimensional velocity vst versus Bi.
an associated analytical formula is given. An increase of heat taken up (increaseof the dimensionless parameter Bi) causes an increase of critical parameter ε(stable solution zone is increased). Furthermore, for any fixed material body andfor its loading parameters there is a parameter Bi such that a stationary solutionwill be always stable (in the considered case Bi > 8). The period of oscillationversus Bi is shown in Fig. 3.12a. An increase of Bi causes variation of theperiod of oscillations (first it decreases, then increases, and then decreases again).Figure 3.12b shows dependence of a solution to Equation (3.58) on parameterBi. Note that a physical sense of dimensionless stationary velocity vst exists forε1 < ε.
For example, in the fourth case for ε1 = 400 (ε1 < ε) the roots of Equa-tion (3.67) s1,2 = −0.12 ± 1.06i lie on the left-hand side of the complexplane s. A solution is exhibited by ‘periodic’ damped oscillation (expected periodTp = 2π/Ims1 = 5.94). For the critical value ε1 ≈ ε = 587, the roots
158 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
Fig. 3.13: Time history of dimensionless body velocity (a) and friction force (b) for variousvalues of ε1 (curve 1: ε1 = 800; curve 2: ε1 = 586.5; curve 3: ε1 = 400).
Fig. 3.14: Time history of dimensionless contact pressure (a) and temperature (b) for variousvalues of ε1 (curve 1: ε1 = 800; curve 2: ε1 = 586.5; curve 3: ε1 = 400).
s1,2 = ±1.29i lie on the imaginary axis. In this case, the period Tp = 4.87is expected. For ε1 = 800, the roots s1,2 = 0.14 ± 1.5i lie on the right-hand sideof the complex plane s. A stationary solution is unstable, and a limiting stick-slipcycle appears with the expected period Tp = 4.18.
In order to confirm the given conclusions, a numerical analysis is carried out forthe fourth case for Bi = 5 (now ε1 ≈ ε = 586.5), and the computational results areshown in Figs. 3.13–3.14 for a few values of the parameter ε1 = 400; 586.5; 800. InFig. 3.13a, the dependence of dimensionless body velocity z(τ) on dimensionlesstime τ is shown. Curve 1 corresponds to the case when ε1 = 800 (m = 2.4 · 2.4 ·109kg m−22), curve 2 corresponds to ε1 = 586.5 (m = 3.25 · 109kg m−2), and
3.2 Rigid plate dynamics subject to temperature perturbation 159
Fig. 3.15: Tp: periodic oscillations of dimensionless contact characteristics (body displacementz(τ), body velocity z(τ), contact pressure p(τ), contact temperature θ(τ), and friction forceε1F (z)p(τ)) for ε1 = 800.
curve 3 corresponds to ε1 = 400 (m = 4.8 · 1010kg m−2). Figure 3.13b showsthe time history of the dimensionless friction force ε1F (z)p(τ) occurring on theright-hand side of Equation (3.39). Time evolution of both dimensionless contactpressure p(τ) and temperature θ(−1, τ) = θ(1, τ) is illustrated in Figs. 3.14a and3.14b (curve 1: ε1 = 800; curve 2: ε1 = 586.5; curve 3: ε1 = 400).
As Figs. 3.13 and 3.14 show, the contact characteristics either have dampedoscillatory shape (curves 3) approaching stationary states or they are periodic(curves 2) or stick-slip periodic (curves 1). The numerical analysis confirms thetheoretical prediction. It is found that: v2
st = 27.8, p2st = θ2
st = 2.45. In the caseassociated with curves 1, the period of stick-slip contacts Tp = 2.7. In the criticalcase Tp = 4.1, for ε1 = 400 (curve 3) the period of damped oscillations Tp = 2.7.
To facilitate analysis of the contact stick-slip characteristics (ε1 = 800), theassociated periodic orbits are shown in Fig. 3.15.
Note that the period of oscillations T includes slip phase tsl and stick phasetst (T = tsl + tst). The slip phase consists of acceleration (z > 0) and braking.It begins at the time instant when friction force ε1F (z)p(τ) is smaller than forceε1m0 applied to the body. Beginning from this time instant, the body velocityincreases and contact temperature increases, but p(τ) < 0 (according to formula(3.54) the body is cooled). In the time instant corresponding to p(τ) = 0, theheat stream equals zero and the body starts to be heated. Because heat expansionis increased, the contact pressure also increases. The kinetic friction coefficient
160 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
F (z) goes down, the friction force ε1F (z)p(τ) also decreases initially, becausethe contact pressure increases slightly. Further pressure increase causes frictionincrease. When the friction force achieves the value of external force ε1m0, thebody velocity achieves its maximum and braking begins (z < 0) and the kinematicfriction coefficient is decreased. Also, after some slight delay, the contact pressureachieves its maximum. Then the body begins to cool down. When z = 0, the slipphase is finished and the friction force achieves its maximal value. A stick phasebegins. The contact temperature, contact pressure, and friction force decrease.The latter process stops when the frictional force achieves a value of the appliedexternal force. Then the stick-slip process is repeated.
Conclusions. In this section a novel problem of the so-called ‘frog-slider’ mechani-cal system exhibiting frictional thermoelastic contact of a moving body subject toboth constant and nonconstant friction coefficients has been presented and dis-cussed. It has been shown, among others, that in the case of a constant frictioncoefficient, the contact characteristics can achieve their stationary stable valueoscillating processes (the roots of the associated characteristic equation are imagi-nary and they lie on the left-hand side of the complex plane). It is worth notingthat in the case of a nonconstant friction coefficient, the self-excited vibration canappear in our system without an elastic part (stiffness). The last phenomenon iscaused by heating of the body while accelerating, the friction increase, and thenthe braking and cooling of the system. The characteristic changes of both dis-placement (Fig. 3.15) and velocity of the analysed system inspired us to use theexpression: a ‘frog-slider’ system.
3.3. Dynamics of a two degrees-of-freedom system with
friction and heat generation
A novel thermomechanical model of frictional self-excited stick-slip vibrations isproposed. A mechanical system consisting of two masses that are coupled by anelastic spring and moving vertically between two walls is considered. It is assumedthat between masses and walls a Coulomb friction occurs, and stick-slip motion ofthe system is studied. The applied friction force depends on a relative velocity ofthe sliding bodies. Stability of stationary solutions is considered. A computationof contact parameters during heating of the bodies is performed. The possibilityof existence of frictional autovibrations is illustrated and discussed.
Stick-slip motion is intimately related to the nature of frictional phenomenaand is often attributed to the difference between the static and kinematic frictioncoefficients. Even though the topic of friction is a relatively old one and plays animportant role in many practical and engineering applications, surprisingly it isnot as well understood as might be expected.
Research reported in this section extends the authors’ earlier results, whereboth regular and chaotic vibrations in a cylinder–bush system have been analysed
3.3 Dynamics of a two degrees-of-freedom system with friction and heat generation 161
[Awrejcewicz (1996), Awrejcewicz, Pyryev (2002), (2003c), (2004a), (2004b)]. Inaddition, the two degrees-of-freedom system, where a friction force depends on adistance between two masses, has been studied in reference [Wikieł, Hill (2000)].
It is worth noting that the main conditions for occurrence of self-excitedvibrations in the models discussed earlier are associated with a difference betweenstatic and kinematic frictions, and with existence of an elastic coupling in a tri-bomechanical system.
Analysis of various references [Andronov et al. (1966), Awrejcewicz (1996),Awrejcewicz, Pyryev (2002), Kragelsky, Gitis (1987), Martins et al. (1990),Olędzki, Siwicki (1997)] leads to a conclusion that velocity of one of the contactingbodies is always given. The system in the condition of self-excited vibrations takesenergy from a body moving at constant velocity. The self-excited vibrations donot appear when inertia of the contacting bodies is taken into account. The lattercase is considered in this section. It has been shown that owing to heat exten-sion, a body can be periodically heated, braked, cooled, and accelerated. In someconditions, stick-slip self-excited vibrations may also appear.
3.3.1 Statement of the problem
We consider two masses M1 (body 1) and M2 (body 2) which are coupled byan elastic spring as indicated in Fig. 3.16. We assume that the initial length ofthe spring is l0 and that the spring has stiffness k12, which represents the overallelastic properties of the system. We also assume that the masses are constrained bywalls to move only in the vertical direction, and that Z1 and Z2 denote positionsof masses M1 and M2, respectively, as indicated in Fig. 3.16. Let us consider aone-dimensional model of the thermoelastic contact of body 1 with a surrounding
Fig. 3.16: Two coupled masses system.
162 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
medium. Assume that this body 1 is represented by a rectangular plate (l1×l2×2L)(Fig. 3.16). Both bodies are subjected to an action of the forces Fn = Fn
∗ hF (t),n = 1, 2, (hF (t) → 1, t → ∞). At the initial time instant, body 1 (2) is situatedat distance Z◦
1 (Z◦2 ) and its velocity reads Z◦
1 (Z◦2 ). The distance between walls
is always equal to the initial plate thickness 2L.It is assumed that heat conduction between the bodies and the walls obeys
Newton’s law. At an initial instant the temperature is governed by the formulaT0hT (t) (hT (t) → 1, t → ∞). It causes heat extension of the parallelepipedin the direction of 0X , and body 1 starts to contact the walls. As a result of thisprocess, a frictional contact on the parallelepiped sides X = ±L occurs. A simplefrictional model is applied in the further considerations; that is, friction force Ffr
is a product of normal reaction force N(t) and a friction coefficient. That meansthat Ffr = f(Z1)N(t) is the friction force defining resistance of the movement oftwo sliding bodies. Here, owing to the assumption made in references [Andronovet al. (1966), Awrejcewicz, Pyryev (2002), Kragelsky, Gitis (1987)], the kinematicfriction coefficient f(Z1) depends on the relative velocity Vr = Z1 of the slidingbodies (Fig. 2.2).
The friction force σxz(X, t) per unit contact surface X = −L, X = L generatesheat. According to Ling’s assumptions (cf. [Ling (1959)]), the work of the frictionforces is transmitted into heat energy. Note, that the noncontacting plate surfacesare heat-isolated and have the dimensions of L/l1 ≪ 1, L/l2 ≪ 1, which is inagreement with the assumption of our one-dimensional modelling for body 1.Quantities Mn, Fn
∗ , k12 are related to a unit contacting surface.Below, the problem is reduced to determination of the mass plate (body 2)
center displacement Z1(t) (Z2(t)), plate (body 2) velocity Z1(t) (Z2(t)), contactpressure P (t) = N(t)/l1l2 = −σXX(−L, t) = −σXX(L, t), plate temperatureT1(X, t), and displacement U1(X, t) in the direction of the X-axis.
3.3.2 Mathematical problem formulation
In the considered case, the studied problem is governed by two equations of motionin the form
M1Z1(t) + k12(Z1(t) − Z2(t) − l0) = M1g + F 1∗ hF (t) − 2f(Z1)P (t), (3.69)
M2Z2(t) − k12(Z1(t) − Z2(t) − l0) = M2g + F 2∗ hF (t), (3.70)
where Z1, Z2 denote position of both masses as shown in Fig. 3.16; Z1, Z2 denotetheir respective velocities. Equations of the heat stress theory for an isotropicbody 1 [Nowacki (1962)] follow
∂
∂X
[∂
∂XU1(X, t) − α1
1 + ν1
1 − ν1T1(X, t)
]= 0, (3.71)
3.3 Dynamics of a two degrees-of-freedom system with friction and heat generation 163
∂2
∂X2T1(X, t) =
1
a1
∂
∂tT1(X, t), X ∈ (−L, L), (3.72)
and mechanicalU1(−L, t) = 0, U1(L, t) = 0, (3.73)
heat
− λ1∂T1(−L, t)
∂X+ αT (T1(−L, t) − T0hT (t)) = f(Z1)Z1(t)P (t), (3.74)
λ1∂T1(L, t)
∂X+ αT (T1(L, t) − T0hT (t)) = f(Z1)Z1(t)P (t), (3.75)
and initial conditions
T1(X, 0) = 0, X ∈ (−L, L),
Z1(0) = Z◦1 , Z2(0) = Z◦
2 , Z1(0) = Z◦1 , Z2(0) = Z◦
2 (3.76)
are attached. Normal stresses that occur in the plate are defined via the relation
σXX =E1
1 − 2ν1
[1 − ν1
1 + ν1
∂U1
∂X− α1T1
]. (3.77)
In the above, the following notation is applied. E1 is the elasticity modulus; andν1, λ1, a1, α1, αT are Poisson’s ratio, thermal conductivity, thermal diffusivity,thermal expansion and heat transfer coefficients, respectively.
Integration of Equation (3.71), owing to (3.77) and boundary conditions (3.73),yields the contact pressure P (t) = −σXX(−L, t) = −σXX(L, t) cast in theform
P (t) = P0 + Eα1
L
L∫
0
T1(ξ, t)dξ. (3.78)
Let us introduce the following similarity coefficients
t∗ = L2/a1[s], v∗ = a1/L[m/s], P∗ = T0E1α1/(1 − 2ν1)[N/m2],(3.79)
and the following nondimensional parameters
x =X
L, τ =
t
t∗, zn =
Zn
L, p =
P
P∗, θ =
T1
T0, z◦n =
Z◦n
L, z◦n =
Z◦n
v∗,
τnD = t∗/tnD, mn0 = (Mng + Fn)/2P∗, εn = 2P∗t2∗/MnL, n = 1, 2,
164 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
l =l0L
, γ =E1α1a1
(1 − 2ν1)λ1, Bi =
LαT
λ1, F (z) = f(v∗z), (3.80)
where tnD =√
Mn/k12, n = 1, 2.The examined problem is governed by the following nondimensional equations
∂2θ(x, τ)
∂x2=
∂θ(x, τ)
∂τ, x ∈ (−1, 1), τ ∈ (0,∞), (3.81)
z1(τ) + (z1(τ) − z2(τ) − l)τ21D = ε1(m10 − F (z1)p(τ)), (3.82)
z2(τ) − (z1(τ) − z2(τ) − l)τ22D = ε2m20, (3.83)
with both boundaries[∂θ(x, τ)
∂x± Biθ(x, τ)
]
x=±1
= ±q(τ), (3.84)
and initial conditions,
θ(x, 0) = 0, z1(0) = z◦1 , z2(0) = z◦2 , z1(0) = z◦1 , z2(0) = z◦2 (3.85)
where
q(τ) = BihT (τ) + γF (z1)z1(τ)p(τ), p(τ) =1
2
1∫
−1
θ(ξ, τ)dξ. (3.86)
3.3.3 Solution of the problem
Applying an inverse Laplace transform [Carslaw, Jaeger (1959)], our nonlinearproblem governed by Equations (3.81), (3.84), and (3.85) is reduced to the follow-ing integral equation
p(τ) = Bi
τ∫
0
hT (ξ)Gp(τ − ξ)dξ + γ
τ∫
0
F (z1)z1(ξ)p(ξ)Gp(τ − ξ)dξ, (3.87)
which yields both nondimensional pressure p(τ) and velocity z1(τ). The tempera-ture is defined by the following formula
θ(x, τ) = Bi
τ∫
0
hT (ξ)Gθ(x, τ − ξ)dξ + γ
τ∫
0
F (z1)z1(ξ)p(ξ)Gθ(x, τ − ξ)dξ,
(3.88)
3.3 Dynamics of a two degrees-of-freedom system with friction and heat generation 165
where
{Gp(τ), Gθ(1, τ)} =1
Bi−
∞∑
m=1
{2Bi, 2μ2
m
}
μ2m[Bi(Bi + 1) + μ2
m]e−µ2
mτ , (3.89)
and μm are the roots of the following characteristic equation
tgμm =Bi
μm, m = 1, 2, . . . . (3.90)
3.3.4 Steady-state solution analysis
A stationary solution to the problem reads
pst = θst =1
1 − v, v = F (vst)
vstγ
Bi, (3.91)
where vst is the solution to the nonlinear equation
F (vst) =m0
1 + γ m0vst/Bi, m0 = mst
10 + mst20, mst
n0 =Mng + Fn
∗
2P∗.
(3.92)A graphical solution of Equation (3.92) is presented in Fig. 3.17 for various
parameters m0 and Bi. Recall that for steel γ = 1.87.
Fig. 3.17: Graphical solution of Equation (3.92) (solid curves: 1, m0 = 0.15, Bi = 50,γ = 1.87; 2, m0 = 0.1, Bi = 50, γ = 1.87; 3, m0 = 0.1, Bi = 0.5, γ = 1.87; 4, m0 = 0.15,Bi = 0.5, γ = 1.87; 5, m0 = 0.14, γ = 0; 6, m0 = 0.8, γ = 0; 7, m0 = 0.04, γ = 0; dashedcurve corresponds to F (vst)).
166 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
The right-hand side of Equation (3.92) is represented by solid curves fordifferent values of the parameters m0 and Bi. Solid curve 1 is associated withparameters m0 = 0.15, Bi = 50; solid curve 2 with m0 = 0.1, Bi = 50; solidcurve 3 with m0 = 0.1, Bi = 0.5; and solid curve 4 with m0 = 0.15, Bi = 0.5.The dashed curve displays the function F (vst).
For different values of the parameters, Equation (3.92) can have a different
number of solutions: for m0 = 0.15, Bi = 50 (first case) it has one solution v(3)st
(F ′(v(3)st ) > 0); for m0 = 0.1, Bi = 50 (second case) it has three solutions v(1)
st ,
v(2)st , v(3)
st (F ′(v(1)st ) > 0, F ′(v(2)
st ) < 0, F ′(v(3)st ) > 0); and for m0 = 0.1, Bi = 0.5
(third case) one solution v(1)st (F ′(v(1)
st ) > 0). Owing to approximation (2.58) we
have v(1)st ≈ ε0m0/2fs and F ′(v(1)
st ) ≈ 2fs/ε0. For m0 = 0.15, Bi = 0.5 (fourth
case), again one solution exists v(2)st (F ′(v(2)
st ) < 0).Let us introduce a perturbation of the stationary solution (3.91) by means of
the following formulas
zn = vstτ + z∗n, zn = vst + z∗n, n = 1, 2, p = pst + p∗, θ = θst + θ∗,
hT = 1 + h∗T , hF = 1 + h∗
F , |h∗F | ≪ 1, |h∗
T | ≪ 1. (3.93)
Owing to linearization of the right-hand sides of (3.82) and with boundarycondition (3.84) the following linear problem is obtained
∂2θ∗(x, τ)
∂x2=
∂θ∗(x, τ)
∂τ, (3.94)
z∗1(τ) + (z∗1 − z∗2)τ21D = ε1[m
∗10(τ) − F (vst)p
∗(τ) − F ′(vst)pstz∗1 ], (3.95)
z∗2(τ) − (z∗1 − z∗2)τ22D = ε2m
∗20(τ), (3.96)
[dθ∗(x, τ)
dx− Biθ∗(x, τ)
]
x=−1
= −q∗(τ),
[dθ∗(x, τ)
dx+ Biθ∗(x, τ)
]
x=1
= q∗(τ), (3.97)
where
q∗(τ) = Bih∗T (τ) + γ(vstpst(β1 + β2)z
∗1(τ) + vstF (vst)p
∗(τ)),
m∗n0(τ) = Fn
∗ h∗F (τ)/(2P∗), p∗(τ) =
1
2
1∫
−1
θ∗(ξ, τ)dξ,
β1 =F (vst)
vst, β2 = F ′(vst). (3.98)
3.3 Dynamics of a two degrees-of-freedom system with friction and heat generation 167
Furthermore, applying the Laplace transform, a solution of problem(3.94)–(3.97) in the transform domain is found. For example, the Laplace trans-form of the velocity perturbation of body 1 reads
sz∗1(s) = −∆−1(s){ε1β1vstBi(s2 + τ22D)S(s)h∗
T (s)
+(Bi v S(s) − ∆1(s)) [ε1(s2 + τ2
2D)m∗10
(s) + ε2τ21Dm∗
20(s)]}, (3.99)
where
{z∗1(s), h∗
T (s), h∗F (s), m∗
n0(s)}
=
∞∫
0
{z∗1(τ), h∗T (τ), h∗
F (τ), m∗n0(τ)} e−sτdτ .
The characteristic equation of the linearized problem reads
∆(s) = (ε1pstβ2(s2 + τ2
2D) + s(s2 + τ21D + τ2
2D))∆1(s)
+ Bi v(ε1pstβ1(s2 + τ2
2D) − s(s2 + τ21D + τ2
2D))S(s) = 0, (3.100)
where ∆1(s) = sS(s) + BiC(s), S(s) = sinh(√
s)/√
s, C(s) = cosh(√
s).The characteristic function ∆(s), in the form of an infinite-order polynomial
takes the form
∆(s) =
∞∑
m=0
smam, (3.101)
where
a0 = τ22Db0, a1 = τ2
2Db1 + τ21D(d
(1)0 − Bi v),
am = bm−2 + τ22Dbm + τ2
1D(d(1)m−1 − Bi v d
(2)m−1), m = 2, 3, . . . ,
b0 = ε1pstc0, bm = ε1pstcm + dm−1, m = 1, 2, . . . ,
c0 = Bi(β2 + vβ1), dm = d(1)m − Bi vd(2)
m , cm = β2d(1)m + Bi vβ1d
(2)m ,
d(1)m =
2m + Bi
(2m)!, d(2)
m =1
(2m + 1)!m = 0, 1, 2, . . . .
Observe that owing to analysis of the roots of characteristic Equation (3.101),the parameter vst represents a solution to nonlinear Equation (3.92).
If the frictional heat generation is not taken into account (γ = 0), the char-acteristic equations are governed by the following cubic equation: s3 + ε1β2s
2 +(τ2
1D + τ22D)s+ ε1β2τ
22D = 0. Its roots lie in the right-hand part of the complex
plane if β2 < 0.In the case of a perfectly stiff spring (k → ∞), we have τ1D → ∞, τ2D → ∞.
The Laplace transformation of the velocity perturbation of body 1 reads
168 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
sz∗1(s) = −∆−1(s){ε1β1vstBiS(s)h∗
T (s)
+(Bi v S(s) − ∆1(s))(ε1m∗10
(s) + ε2m∗20
(s))}
, (3.102)
where
∆(s) = (ε1pstβ2 + s)∆1(s) + Bi v(ε1pstβ1 − s)S(s) = 0 (3.103)
is the characteristic equation of the linearized problem, and velocity vst is asolution to Equation (3.92). Note that a detailed analysis of roots of Equa-tion (3.103) for β2 = 0 has been carried out in reference [Olesiak, Pyryev(2000)].
In the case when M2 → 0, we have τ2D → ∞, and for F 2∗ = 0 (m∗
20(s) = 0)the Laplace transform of the velocity perturbation of body 1 is defined by Equa-tion (3.102), whereas the characteristic equation is given by (3.103).
In this case, the examination concerns the steel parallelepiped plate (α1 =14 · 10−6◦C−1, λ1 = 21 W/(m · ◦C−1), ν1 = 0.3, a1 = 5.9 · 10−6m2/s, E1 =19 · 1010Pa) with ε1 = 100, ε2 = 900, τ1D = 2, τ2D = 6 and with a nonconstantfriction coefficient. The function F (z) = f(v∗z) is defined by the formula (2.57).
In the first case (for m0 = 0.15, Bi = 50), one solution v(3)st = 21, p
(3)st =
θ(3)st = 1.12 (v = 0.105) is found. It is always stable (the roots of Equation (3.100)
s1,2 = −0.05±6.3 i lie in the left-hand side of the complex variable s). The contactcharacteristics achieve their limiting values through a damped oscillation processwith the expected ‘period’ T = 0.99.
In the second case (m0 = 0.1, Bi = 50), three solutions appear. The solution
v(3)st = 15.94, p
(3)st = θ
(3)st = 1.06, (v = 0.056) is stable (the roots of Equa-
tion (3.100) s1,2 = −0.04±6.32 i lie in the left-hand side of the complex plane s).
The solution v(2)st = 0.41 (v = 0.0015) is unstable (s1 = 3.8, s2,3 = 0.15±6.23 i).
The solution v(1)st ≈ 0 corresponds to an equilibrium state. In the considered case,
the contact characteristics, depending on initial conditions, tend to one of the twostable solutions.
In the third case (m0 = 0.1, Bi = 0.5), there is only one solution, which isstable. Note that in this case, a braking process always occurs (the applied externalforce is smaller than the friction force).
In the fourth case (m0 = 0.15, Bi = 0.5), the only solution that exists is
of the form: v(2)st = 3.01, p
(2)st = θ
(2)st = 2.69, (v = 0.63) and it is unstable
(s1,2 = 0.5±0.68 i, s3,4 = 0.06±6.32 i). If a solution is unstable, then in solvinga nonstationary problem this solution may approach a stable limit cycle or it canbe expressed via oscillations increasing in time (its behaviour depends on othernonlinear terms).
3.3 Dynamics of a two degrees-of-freedom system with friction and heat generation 169
a) b)
Fig. 3.18: Time histories of nondimensional body displacement (a) and velocity (b). Solid curvescorrespond to body 1, dashed curves correspond to body 2.
3.3.5 Numerical analysis of transient solution
Let us consider the fourth case as an example. Figure 3.18a shows the depen-dence of displacement of nondimensional body 1 (body 2) z1(τ) (z2(τ)) versusnondimensional time τ , whereas Fig. 3.18b displays a dependence of velocity ofnondimensional body 1 (body 2) z1(τ) (z2(τ)) versus nondimensional time τ .Solid curves correspond to body 1; dashed curves correspond to body 2. Note thatbody 1 is in a stick-slip state. Zones with stick (z1 = 0) are substituted by zonesof slips.
Evolution of nondimensional contact pressure in time (curve 1) and a tempera-ture on the contact surface (curve 2) is shown in Fig. 3.19.
In this section a new physical and mathematical model of a two degrees-of-freedom system with an account of friction and heating processes is studied. It isassumed that the friction coefficient depends on sliding velocity.
It has been shown that when a heat transfer is not taken into account (γ = 0), thesystem cannot exhibit a stick-slip motion. This potential behaviour of the studiedsystem is displayed by a solid curve in Fig. 3.17. For m0 > fs (solid curve 5)equation F (vst) = m0 has one stable static solution, which attracts a nonstationaryone. For fmin < m0 < fs (solid curve 6) the mentioned equation has three staticsolutions, and one of them is unstable. In this case, a nonstationary solution willbe attracted by one of two static stable solutions. In the cases when m0 < fmin
(solid curve 7) the discussed equation has only one stable solution.In order to realize a stick-slip motion the parameter γ should be positive (see
case 4). In this case, one deals with only one solution, which is unstable and anonstationary solution can be attracted by a limiting cycle. The numerical analysisis in agreement with theoretical prediction of the occurrence of stick-slip dynamicsof our investigated system with friction and heat generation.
170 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
Fig. 3.19: Dimensional contact pressure p (curve 1) and dimensionless contact temperature θ(curve 2) versus dimensionless time τ .
3.4. Tribological dynamical damper of vibrations with
thermoelastic contact
Nonlinear oscillations of mechanical systems are described extensively in a seriesof classical monographs [Den Hartog (1952), Kauderer (1958), Hayashi (1968),Andronov et al. (1966), Bogoliubov et al. (1961), Awrejcewicz (1996), Arnoldet al. (1997)], and some asymptotic methods for solution of equations of non-linear oscillations are presented in well-known books [Andronov et al. (1966),Bogoliubov et al. (1961), Awrejcewicz (1996), Arnold et al. (1997)]. They mainlyaddress classical approaches to study vibrations exhibited by various engineeringsystems.
In many cases in engineering, harmful effects of vibrations are suppressed byinclusion of the so-called dynamic dampers of vibration [Den Hartog (1952)].They have rather a wide spectrum of application and can be used to damp vari-ous longitudinal, torsional, and transversal vibrations of both machines and civilengineering constructions [Giergiel (1990)]. Now, it is well known in engineeringthat in order to avoid harmful effects of resonance, the majority of the externallydriven mechanical systems should be damped.
A drawback of the currently designed vibration dampers is associated with heattransfer to the contacting bodies induced by frictional processes. It causes extensionof the contacting bodies, and a change of contacting pressure and friction oftenresulting in the harmful damper wedging effects.
In this section a one degree-of-freedom system driven by either a force orkinematic excitation is studied. An additional mass is added to the mentionedmechanical system via a special pressing device initiating dry friction occurrenceon the contacting surfaces. Our proposed mathematical model of the preliminary
3.4 Tribological dynamical damper of vibrations with thermoelastic contact 171
described system includes thermal effects that appear on the contacting bodies[Pyryev (1994), Awrejcewicz, Pyryev (2002)]. Note that the damper geometricalproperties, heat transfer between the bodies, and a surrounding medium yielda change of friction on the contacting surface. We focus on a solution to thenonlinear problem of thermal stresses and strongly nonlinear equations governingthe dynamics of the investigated system. Based on the analysis, the directions forproper construction of the mechanical vibration dampers are given.
3.4.1 Mathematical modelling
Below, we consider a damper of torsion vibrations of the so-called Lanchestersystem shown in Fig. 3.20a. In this figure the following notation is introduced:1 denotes a bush for mounting the damper on a shaft; 2 denotes two coupled discsserving as flywheels and freely rotating on bush 3 mounted on shaft 4. The bushis coupled with friction washer 5, where discs are pressed by screw 6. A relativelyweak power tight screw generates small friction forces and weak energy damping.On the other hand, strong power tight screws may eliminate sliding and henceenergy dissipation does not occur.
Figure 3.20b shows a dynamic model of the Lanchester system with dry friction.Body with mass m1 models a fundamental part of the system (shaft 4, bushes 1
a) b)
Fig. 3.20: Torsion damper of the Lanchester system (a) and a model of the problem (b).
172 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
and 3). System vibrations are generated by harmonic force F1 = F0sin ω′0t. The
following notation is applied. m1 is mass; k1 is the elasticity coefficient of the mainsystem constraints; F0, ω′
0 are the amplitude and frequency of the driving force,respectively. The damper (two coupled discs 2 and screws 6 shown in Fig. 3.20a)of mass m2 is added. It is coupled with mass m1 by a pressing system and hencedry friction force Ffr occurs. Assume that the tested damper has the shape ofparallelepiped (2L×L1×L2) moving in direction Z2 along the walls of the mainsystem. The initial value of the distance between the walls, that is, between discs 2and the bush 1 in Fig. 3.20a is equal to the plate thickness 2L (thickness of frictionwashers 5). Then, this distance is decreased according to the formula 2U0hU (t),which is realized via power tight of screws 6, where U0 is a constant larger thanzero, and hU (t) is a known dimensionless function of time (hU (t) → 1 , t → ∞).Then, this distance is decreased according to the formula 2U0hU (t), which isrealized via power tight of screws 6. As a result of this process, dry friction occurson the parallelepiped surfacesX = ±L. It is defined by the function Ffr(Vr),where Vr is the relative velocity of the plate and walls; that is, Vr = Z1 − Z2
(dZi/dt ≡ Zi, i = 1, 2).According to Amonton’s assumption, the friction force Ffr = 2f(Vr)P is
equal to the product of the normal reaction component and friction coefficient,f(Vr) denotes the kinetic friction coefficient (f(−Vr) = −f(Vr)), and we takef(Vr) = fs sgn(Vr). The action of friction on the contact surface X = ±L gene-rates heat. We follow the generally accepted assumption [Ling (1959)] that frictionwork is transformed into heat energy. Furthermore, we assume that walls ideallytransform heat and that between the plate and walls heat transfer is governed byNewton’s law, and a surrounding medium temperature is equal to zero. Plate sur-faces not being in contact with movable walls are thermally isolated and theyhave the dimension L/L1 ≪ 1, L/L2 ≪ 1. The mentioned assumptions allow usto introduce a one-dimensional model. Governing equations of the system pre-sented in Fig. 3.20b have the following form (see [Den Hartog (1952), Pyryev(1994)])
m1Z1 + k1Z1 + Ffr(Vr , Z1) = F1, m2Z2 − Ffr(Vr, Z1) = 0, (3.104)
with the friction model
Ffr(Vr, Z1) =
⎧⎨⎩
2fssgn(Vr)P (t), Vr = 0, slip,
min(Fst, 2fsP (t))sgn(F1 − k1Z1), Vr = 0, stick,(3.105)
whereFst =
m2
m1 + m2|F1 − k1Z1| .
In order to solve Equations (3.104) and (3.105) the knowledge of contact pres-sure P (t) is required. For this purpose the following equation governing the theoryof thermal stresses for an isotropic body [Nowacki (1962)] is solved first,
3.4 Tribological dynamical damper of vibrations with thermoelastic contact 173
∂
∂X
[∂U2(X, t)
∂X− α2
1 + ν2
1 − ν2T2(X, t)
]= 0, (3.106)
∂2T2(X, t)
∂X2=
1
a2
∂T2(X, t)
∂t, X ∈ (−L, L), (3.107)
with the attached mechanical
U2(∓L, t) = ±U0hU (t), (3.108)
and heat boundary conditions
∓λ2∂T2(∓L, t)
∂X+ αT T2(∓L, t) = f(Vr)VrP (t), (3.109)
as well as zero initial conditions.Normal stress occurring in the plate is defined by the following formula
σXX(X, t) =E2
1 − 2ν2
[1 − ν2
1 + ν2
∂U2(X, t)
∂X− α2T2(X, t)
]. (3.110)
In the above, the following notation is taken: E2 is the Young’s modulus of theplate; ν2, λ2, a2, α2, αT are Poisson’s ratio of the plate, thermal conductivity ofthe plate, thermal diffusivity, coefficient of thermal expansion of the plate, and heattransfer coefficient (from the wall to plate), respectively; and P (t) = −σXX(±L, t)denotes contact pressure. Quantities m1, m2, k1, P , F1 are measured per unit ofthe contact surface S = L1 ×L2 of the moving rigid plate and the wall. Note thatwhen the damper is neglected, the considered system is reduced to that with 1-dofwith its natural frequency ω01 =
√k1/m1.
Integration of Equation (3.106) with Equation (3.110) and boundary conditions(3.108) gives the contact pressure P (t) = −σXX(±L, t) of the form
P (t) =E2(1 − ν2)U0hU (t)
(1 + ν2)(1 − 2ν2)L+
E2α2
(1 − 2ν2)L
1
2
L∫
−L
T2(ξ, t)dξ. (3.111)
Motion of the investigated system depends on both ratios Ffr/F0 and ω′0/ω01.
For various values of the Ffr/F0 ratio the system moves with one or more stickswithin half of the period of motion. It should be emphasized that the exact solutionof this problem without tribologic processes and for the case with one stop andwithout stops has been already reported by Den Hartog [Den Hartog (1952)].
Dimensionless differential and integral equations. Let us introduce the follow-ing similarity coefficients
t∗ = 1/ω01[s], L∗ =F0
k1[m],
174 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
N∗ =E2(1 − ν2)U0S
(1 + ν2)(1 − 2ν2)L[N], T∗ =
(1 − ν2)U0
α2(1 + ν2)L[0C], (3.112)
and the following dimensionless parameters
x =X
L, τ =
t
t∗, θ =
T2
T∗, p =
P S
N∗, zn =
Zn
L∗, n = 1, 2, ε =
2N∗fs
F0S,
Bi =αT L
λ2, τT =
t∗tT
, l =L∗
L, γ =
E2α2a2
(1 − 2ν2)λ2
, Ω =Ω1
τT, Ω1 = γl,
ω0 = ω′0/ω01, κ =
C
2m2ω01, μ =
m2
m1, b1 = B1/L∗, b2 = B2/L∗,
f(L∗t−1∗ vr) = fsF (vr), vr = z1 − z2, dzn/dτ ≡ zn, n = 1, 2, (3.113)
where F (vr) = sgn(vr); tT = L2/a is the characteristic time of thermal inertia;Bi is the Biot number. Dimensionless parameter ε represents the friction force, γgoverns the body heat extension, and parameter Ω is responsible for heat generationon the surface contact.
In the dimensionless form the considered mathematical model reads
z1 + z1 + ffr(vr, z1) = sin(ω0τ), μz2 − ffr(vr, z1) = 0, (3.114)
∂2θ(x, τ)
∂x2=
1
τT
∂θ
∂τ, x ∈ (−1, 1), τ ∈ (0,∞), (3.115)
with the friction model
ffr(vr, z1) =
⎧⎨
⎩εF (vr)p(τ), vr = 0, slip,
min(fst, εp(τ))sgn(sin(ω0τ) − z1), vr = 0, stick,(3.116)
wherefst =
μ
1 + μ|sin(ω0τ) − z1| ,
with the following boundary
[∂θ(x, τ)
∂x∓ Biθ(x, τ)
]
x=∓1
= ∓q(τ), (3.117)
and initial conditions
θ(x, 0) = 0, z1(0) = 0, z1(0) = 0, z2(0) = 0, z2(0) = 0, (3.118)
where
q(τ) = ΩF (z1 − z2)p(τ)(z1 − z2), (3.119)
3.4 Tribological dynamical damper of vibrations with thermoelastic contact 175
p(τ) = hU (τ) +1
2
1∫
−1
θ(ξ, τ)dξ. (3.120)
In order to solve problems (3.115) and (3.117), the Laplace transform is appliedwith respect to time τ . The theorem on ‘convolution’ is used [Abramowitz, Stegun(1965)] to find an inverse transform. Finally, we get
p(τ) = hU (τ) + Ω1
τ∫
0
F (z1 − z2)p(ξ)(z1 − z2)Gp(τ − ξ)dξ, (3.121)
θ(x, τ) = Ω1
τ∫
0
F (z1 − z2)p(ξ)(z1 − z2)Gθ(x, τ − ξ)dξ, (3.122)
where
{Gp(τ), Gθ(±1, τ)} =1
τT Bi−
∞∑
m=1
{2Bi, 2μ2
m
}exp(−τT μ2
mτ)
τT μ2m(Bi(Bi + 1) + μ2
m). (3.123)
In the above μm(m = 1, 2, 3, . . . ) are the roots of the characteristic equationtan(μ) = Bi/μ. The functions Gp(τ), Gθ(±1, τ) have the following asymptoticestimations
Gp(τ) ≈ τ, Gθ(±1, τ) ≈ 2√
τ/τT π, τ → 0, (3.124)
{Gp(τ), Gθ(±1, τ)} ≈ 1/(τT Bi), τ → ∞. (3.125)
Observe that the considered problem is reduced to the system of nonlinear differ-ential equations (3.114) and (3.116), and the integral Equation (3.121) describingdimensionless velocities z1 and z2, and the dimensionless contact pressure p(τ).Dimensionless temperature θ(x, τ) is governed by Equation (3.122).
Solution with a lack of heating. We take Ω1 = 0. Because P (t) = N∗/S = P∗,one gets
z1 + z1 + εF (z1 − z2) = sin(ω0τ), (3.126)
z2 − εμ−1F (z1 − z2) = 0. (3.127)
Assuming that the system motion is close to harmonic, one may apply linearisa-tion and use an equivalent viscous damping instead of dry friction. In other words,the equivalent damping is found comparing energy loss in a real AT and in anequivalent AC viscous system during the period Tp = 2π/ω′
0.
176 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
Assuming the harmonic motion
z1(τ) = b11sin(ω0τ) + b12cos(ω0τ) = b1sin(ω0τ + ϕ1),
z2(τ) = b21sin(ω0τ) + b22cos(ω0τ) = b2sin(ω0τ + ϕ2), (3.128)
the relative displacement is
z1 − z2 = dsin(ω0τ + ψ),
whered =
√(b11 − b21)2 + (b12 − b22)2. (3.129)
Comparing works done by the two dampers one gets
AT =
t1+T0∫
t1
2f(Z1 − Z2)P∗(Z1 − Z2)dt,
AC =
t1+T0∫
t1
C(Z1 − Z2)2dt, (3.130)
whereas comparing two works AT = 8P∗fsL∗d and AC = Cπω′0L
2∗d
2 one getsthe following equivalent dimensional
C =8P∗fs
πω′0L∗d
(3.131)
or nondimensional damping coefficient of the form
κ =2ε
πμω0d. (3.132)
Taking into account the equivalent damping coefficient, the governing Equa-tions (3.126) and (3.127) assume the form
z1 + z1 + 2μκ(z1 − z2) = sin(ω0τ), (3.133)
z2 − 2κ(z1 − z2) = 0. (3.134)
A solution to Equations (3.133) and (3.134) is given by (3.128), where thecorresponding dimensionless amplitudes are
b11 =(1 − ω2
0(1 + μ))4κ2 + ω20(1 − ω2
0)
(1 − ω20(1 + μ))24κ2 + ω2
0(1 − ω20)
2, (3.135)
3.4 Tribological dynamical damper of vibrations with thermoelastic contact 177
b12 = − 2μκω30
(1 − ω20(1 + μ))24κ2 + ω2
0(1 − ω20)
2, (3.136)
b21 =(1 − ω2
0(1 + μ))4κ2
(1 − ω20(1 + μ))24κ2 + ω2
0(1 − ω20)
2, (3.137)
b22 = − 2ω0κ(1 − ω20)
(1 − ω20(1 + μ))24κ2 + ω2
0(1 − ω20)
2, (3.138)
b1 =
√ω2
0 + 4κ2
(1 − ω20(1 + μ))24κ2 + ω2
0(1 − ω20)
2, (3.139)
b2 =
√4κ2
(1 − ω20(1 + μ))24κ2 + ω2
0(1 − ω20)
2. (3.140)
According to (3.129) the dimensionless amplitude d is
d =
√ω2
0
(1 − ω20(1 + μ))24κ2 + ω2
0(1 − ω20)
2. (3.141)
According to formula (3.141), d depends on κ, and the equivalent dampingdepends on d (see (3.132)). Solving (3.132), (3.141) one gets
κ =ω0|1 − ω2
0 |2
√ω+ω−
(ω20 − ω+)(ω− − ω2
0). (3.142)
The amplitudes are
b1 =1
ω0|1 − ω20|
√√√√ω20
(1 −
(1 +
2
μ
)(4ε
π
)2)
+2
μ
(4ε
π
)2
,
b2 =4ε
πμω20
(3.143)
where
ω± =ε
(1 + μ)(ε ± ε0), ǫ0 =
π
4
(1 − 1
1 + μ
). (3.144)
For ε < ǫ0 and√
ω+ < ω0, dry friction does not bound resonance amplitude,and for ω0 → 1, b1 → ∞. In the case ǫ0 < ε < π/4 and for
√ω+ < ω0 <
√ω−
dry friction does not bound resonance amplitude either and for ω0 → 1, b1 → ∞.When ε > π/4, we get the frequency interval
√ω+ < ω0 <
√ω−, where the
resonance frequency ω0 = 1 does not appear.For μ → ∞ (m2 → ∞, ǫ0 = π/4) we obtain the case of 1-dof vibration with
friction considered by Den Hartog [Den Hartog (1952)], where
178 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
b1 =1
|1 − ω20 |
√
1 −(
4ε
π
)2
. (3.145)
Observe that for ε < π/4 (b1 is a real number) one gets: ω0 → 1, b1 → ∞.Note that the velocity of body m2 is governed by
z2(τ) =ε
μτ + C+, if z1 − z2 > 0, (3.146)
z2(τ) = − ε
μτ + C−, if z1 − z2 < 0. (3.147)
Constants C+, C− are different for all intervals of motion. Assuming thatonly sliding occurs, the work done within one period by the damping force2fsP∗sgn(Z1 − Z2) is [Den Hartog (1952)]:
AT = 4F0L∗b1ε
√
1 − π2
4
(ε
μω20b1
)2
. (3.148)
An optimal value of friction is obtained from the following equation∂AT /∂ε = 0, if (∂2AT /∂ε2
∣∣ε=εopt
< 0).
The optimal dimensionless friction is
εopt =
√2
πμω2
0b1, (3.149)
and
AT opt =4
πF0L∗b
21μω2
0 . (3.150)
The obtained value corresponds to the maximally damped fundamental system.However, in this case the amplitude b1 should be known (say, from an experiment).
3.4.2 On the heat transfer influence on dynamical damper of
self-vibrations
Numerical analysis of the considered problem is carried out using both numericaland analytical computations. The latter ones include asymptotic estimation (3.124).The function sgn(vr) has been approximated in the following way (2.58).
If heat is not generated by friction (γ = 0), then the contact pressurep(τ) = hU (τ). Let us assume that hU (τ) = H(τ), where H(·) is the Heavisidestep function (H(τ) = 1, τ > 0, H(τ) = 0, τ < 0). The previous analysis(part 3.4.1) indicates that resonance occurs in the system. Taking μ = 0.5, ε = 0.5,
3.4 Tribological dynamical damper of vibrations with thermoelastic contact 179
a) b)
Fig. 3.21: The dimensionless velocity z1(τ) of the fundamental body and velocity z2(τ) of thedamper (a) versus dimensionless time τ during resonance (ω0 = 1); dimensionless dependenceof sliding velocity vr(τ) = z1(τ)− z2(τ) versus time τ (b) (Ω1 = 0, µ = 0.5, ε = 0.5, ω0 = 1).
a) b)
Fig. 3.22: The dependence of the dimensionless displacements z1 of the fundamental body andthe displacements z2 of the damper (a) the dimensionless velocity z1(τ) of the fundamental bodyand the velocity z2(τ) of the damper (b) versus dimensionless time τ for τ ∈ (300, 320), takinginto account heat generation (Ω1 = 0.1, µ = 0.5, ε = 0.5, Bi = 1, τT = 0.1, ω0 = 1).
and using (3.144) one finds ǫ0 = 0.26, ω+ = 0.44, ω− = 1.4. We have ε > ǫ0,√ω+ < ω0 <
√ω− and for ω0 = 1 the system is in resonance. In Fig. 3.21a for a
lack of heat extension (Ω1 = 0), time histories of both dimensionless velocities z1
and z2, and dimensionless relative velocity (Fig. 3.21b) vr = z1− z2 are reported.For a general case, numerical analysis of the considered problem (differen-
tial Equations (3.114), (3.115) and integral Equation (3.121)) is carried outusing the Runge–Kutta method and the method of quadrature with estima-tions (3.124). Temperature on the contact surface is given in formula (3.122).
180 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
a) b)
c) d)
Fig. 3.23: The dimensionless sliding velocity vr(τ) = z1(τ) − z2(τ) versus time τ for τ ∈
(0, 100)(a) and for τ ∈ (300, 320)(b) taking into account heat generation. Time histories ofdimensionless contact pressure p(τ) and contact surface temperature θ(τ) for τ ∈ (0, 100)(c)and for τ ∈ (300, 320)(d), taking into account heat generation (Ω1 = 0.1, µ = 0.5, ε = 0.5,Bi = 1, τT = 0.1, ω0 = 1).
Numerical computations are carried out for various values of the parameters μ,Ω1. Figures 3.22 to 3.24 show results of the numerical analysis for Ω1 = 0.1 andBi = 1, τT = 0.1. In Fig. 3.22a dimensionless displacements z1 of the body withmass m1 and z2 of the damper with mass m2 versus time τ are shown (the sameis done for the velocities in Fig. 3.22b). Figure 3.23 illustrates the dimension-less relative velocities vr of two bodies versus dimensionless time τ . The systemoscillations are out of resonance and they reach periodic attractor (Figs. 3.22,3.23b) with the period of Tp = 2π/ω0 = 2π, whereas the damper undergoes thestick-slip oscillations (Fig. 3.23b). Evolutions of the dimensionless contact pressurep(τ) and the temperature θ(τ) on the contacting surface are shown in Fig. 3.23.
3.4 Tribological dynamical damper of vibrations with thermoelastic contact 181
a) b)
c) d)
Fig. 3.24: The dependence of the dimensionless displacements z1 of the fundamental body andthe displacements z2 of the damper (a) the dimensionless dependence of the sliding velocityvr(τ) = z1(τ) − z2(τ) (b) versus dimensionless time τ for τ ∈ (300, 320), taking into accountheat generation. Time histories of dimensionless contact pressure p(τ) and contact surface tem-perature θ(τ) for τ ∈ (0, 100)(c) and for τ ∈ (300, 320)(d), taking into account heat generation(Ω1 = 0.1, µ = 2, ε = 0.5, Bi = 1, τT = 0.1, ω0 = 1).
Both mentioned characteristics change periodically with the dimensionless periodof Tp = π. The same period has the variable |vr| occurring in (3.119), whichgoverns heat generation on the sliding surface.
In order to investigate how the damper mass influences the system motionwe also investigated our system for μ = 2 (m2 = 2m1). In this case one getsǫ0 = 0.52, ω+ = 0.16. We have ε < ǫ0,
√ω+ < ω0 and for ω0 = 1 the system is
in resonance assuming that the heat extension is omitted. In Fig. 3.24 time evolu-tions of the contact characteristics are reported. In Fig. 3.24a the dependence ofdimensionless displacement z1 and z2 versus dimensionless time is shown, whereas
182 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
a) b)
c) d)
Fig. 3.25: The dependence of the dimensionless displacements z1 of the fundamental body andthe displacements z2 of the damper (a) the dimensionless dependence of the sliding velocityvr(τ) = z1(τ) − z2(τ) (b) versus dimensionless time τ for τ ∈ (800, 820), taking into accountheat generation. Time histories of the dimensionless contact pressure p(τ) and the contact sur-face temperature θ(τ) for τ ∈ (0, 100)(c) and for τ ∈ (800, 820)(d), taking into account heatgeneration (Ω1 = 0.2, µ = 0.5, ε = 0.5, Bi = 1, τT = 0.1, ω0 = 1).
Fig. 3.24b illustrates the corresponding relative velocity. Figures 3.24 show dimen-sionless contact pressure and temperature versus dimensionless time. Note that anincrease of parameter μ causes a decrease of the contact time (see Figs. 3.23band 3.24b) and vibration amplitude decreases (see Figs. 3.22a and 3.24a), but thetemperature amplitude in the periodic state increases (see Figs. 3.23d and 3.24d).
A numerical analysis of results for Ω1 = 0.2, ε = 0.5, Bi = 1, and τT = 0.1is illustrated in Fig. 3.25. An increase of coefficient Ω1 causes an increase oftime τr (the so-called time of passive regulation), when the trajectory achievesperiodic motion. For Ω1 = 0.1 time τr ≈ 250, whereas for Ω1 = 0.2 time
3.4 Tribological dynamical damper of vibrations with thermoelastic contact 183
a) b)
c) d)
Fig. 3.26: Time evolution of nondimensional velocity z1(τ) i z2(τ) (a); nondimensional contactpressure p(τ) and contact surface temperature θ(τ) (b) taking into account heat generation andwear (kw = 0.001). Time evolution of nondimensional velocity z1(τ) i z2(τ) (c); nondimensionalcontact pressure p(τ) and contact surface temperature θ(τ) (d) taking into account wear and heatgeneration (kw = 0.01).
τr ≈ 500. In addition, the time of both bodies being in contact also increases(see Figs. 3.23b and 3.25b). An increase of Ω1 causes a decrease of the relativevelocity amplitude (see Figs. 3.23b and 3.25b).
A dynamic 2-dof damper with dry friction and heat generation has been mod-elled mathematically. The proposed method of solution may also be applied inmodelling of any other nonlinear problem of dynamics of thermoelastic contact-ing bodies. A series of practical results regarding kinetics of the main system andof the dynamic damper are formulated as a result of the analysis of various contactcharacteristics (contact pressure, temperature on the contacting surface).
184 3 Thermoelastic Contact of Parallelepiped Moving Along Walls
It should be emphasized that the dynamic damper with dry friction may notachieve the expected properties. As we have shown, heat generation on the contact-ing surface between the damper and the oscillating body as well as heat expansioneliminate, for certain parameters, resonance phenomena. The real system, in cer-tain conditions, behaves as a self-regulating one; that is, it controls achievementof an optimal contacting pressure. The thermoelastic parallelepiped extends itselfaccording to the conditions of both sliding velocity and heat transfer.
3.4.3 Nonlinear dynamics of a dynamical damper with wear
processes
In this work analysis of influence of the wear coefficient kw on the system (withdynamic damper and friction) motion is carried out. The following nondimensionalparameters are introduced: μ = m2/m1 = 0.5, and ε = 0.5. For dimensionlessfrequency ω0 = ω′
0/ω01 = 1 (ω01 =√
k1/m1) a resonance occurs [Den Har-tog (1952)]. However, taking into account heat plate extension (Ω1 = γl = 0.1,tT = L2/a2 = 0.1, Bi = αT L/λ2 = 1, where γ = E2α2a2λ
−12 /(1 − 2ν2),
l = L∗/L) yields disappearance of the resonance. The computational resultsare shown in Figs. 3.26–3.27 for various wear parameters kw. In Fig. 3.26a thedependence of nondimensional velocity z1(τ) i z2(τ) (zn(τ) = Zn/L∗) versusnondimensional time τ = t/t∗ for nondimensional wear coefficient kw = 0.001(kw = KwN∗L/(U0S)) are reported.
The dependencies of the nondimensional contact pressure p(τ) (p = P S/N∗)and contact surface temperature θ(τ) (θ = T/T∗) versus nondimensional time τare shown in Figs. 3.26b for the wear coefficient kw = 0.001.
a) b)
Fig. 3.27: Time evolution of nondimensional wear uw(τ) taking into account the tribologicprocesses: (a) – kw = 0.001, (b) – kw = 0.01.
3.4 Tribological dynamical damper of vibrations with thermoelastic contact 185
In addition, for kw = 0.001 in Fig. 3.27a the function of nondimensional wearuw(τ) (uw = Uw/U0) against nondimensional time τ is exhibited.
For wear coefficient kw = 0.01 the appropriate contact parameter dependenciesare shown in Figs. 3.26c and 3.27b
Observe that if wear is less than plate heat extension the time of body contactis infinite (Fig. 3.26b), wear increases almost linearly (Fig. 3.27a), temperatureincreases, and compensates wear (Fig. 3.26b). However, if plate wear is largerthan its heat extension the time of body contact is limited (Fig. 3.26d), and thenext resonance phenomenon occurs (Fig. 3.26c).
Chapter 4
Contact Characteristics During
Braking Process
A braking process belongs to nonstationary frictional behaviour, which is associatedwith frictional heat generation. Note that usually temperatures on the rubbingsurfaces achieve large values and play a key role while formulating their wear-resistance criterions.
In brake systems metallic–ceramic and mineral–ceramic materials are widelyused. The mentioned materials are wear- and heat-resistant and they counteract bothchemical bond (frictional contact) and seizing. A frictional cover strap consists ofa thin metallic–ceramic layer put on a copper or steel base. During a brakingprocess, the frictional cover strap is pressed to a body surface (a brake drum,a friction disk, or a wheel rim). The action of friction forces causes conversionof kinematic energy of the moving bodies into heat energy. The brake elementsare overheated and the ability of working conditions of the cover strap are de-creased; that is, the wear increases and the value of the friction coefficient isdecreased. Owing to the described phenomenon, the braking process is extendedand hence a failure may occur. On the basis of the above considerations onemay conclude that heating constraint estimation belongs to one of the most im-portant problems referring to the computation of strength and working time ofbrakes.
Owing to recent computations of brakes with respect to heating, the contactcharacteristics can be defined with a priori given tolerance (braking time, tem-perature, and wear). The mentioned computational techniques are based on thesolution of the system of equations governing Heat Frictional Dynamics (HFD)[Chichinadze et al. (1979), Chichinadze (1995)]. An analytical solution of HFD fortwo uniform homogeneous half-spaces transferring heat is proposed by [Olesiak,Pyryev et al. (1997)], whereas a numerical one is given in references [Yevtushenko,Pyryev (1997), (1998)]. In reference [Pyryev (2004)] a solution of the HFD equa-tions for a three-layered tribomechanical system consisting of three different layersis proposed. The proposed model takes into account the thickness of frictionalelements. It also provides knowledge of the associated heating and wear processesof the tribomechanical system composed of a metallic–ceramic cover strap andmetallic friction disc.
187DOI: 10.1007/978-0-387-09653-7_4, © Springer Science+Business Media, LLC 2009 J. Awrejcewicz and Yu. Pyryev, Nonsmooth Dynamics of Contacting Thermoelastic Bodies,
188 4 Contact Characteristics During Braking Process
A brake temperature model is built [Chichinadze et al. (1979), Chichinadze(1995)], in which a maximum contact temperature is the sum of the average tem-perature of the real frictional surface and the ‘flash’ temperature. The averagetemperature is obtained through analysis of a solution to the one-dimensional heattransfer problem for two contacting bodies assuming that temperature time varia-tion is proportional to an increase of the heat volume. The flash temperature isderived via a solution of the problem dealing with a sliding rod (arbor) along ahalf-space surface assuming that heat transfer intensity is constant in the contactzone. Note that the application of one-dimensional models to solve the heat fric-tional problem is reasonable if in typical heat transfer conditions [Chichinadzeet al. (1979)], the braking time ts is less than 10bl/al (l = 1, 2), where bl, al
are the characteristic dimensions of the friction pair and the temperature wearcoefficient, respectively.
In [Olesiak et al. (1997)] a solution of the heat transfer problem for twosemi-infinite bodies without any limitations on heat stream density is proposed.The cases of constant friction coefficients [Pyryev (2004)] as well as their lineardependence on temperature are accounted for [Olesiak et al. (1997)]. [Yevtushenko,Pyryev (1999b)] found a solution of the heat problem related to nonlinear depen-dence between both friction and temperature coefficients.
4.1. Contact characteristics of three-layer brake models
Let us now consider the problem of friction of a lateral disk brake surface rubbingon a friction surface of a two-layer cover strap subject to external load P (t)(Fig. 4.1). The top layer of thickness H1 (body 1) slides at constant velocity V (t) onthe layer of thickness H2 (body 2) along axis Y of the Cartesian coordinates XY Z
Fig. 4.1: Scheme of a three-layer brake system.
4.1 Contact characteristics of three-layer brake models 189
with the origin situated on the friction surface. Body 2 is attached to nonmovablelayer of thickness H3 (body 3).
Owing to the action of frictional surface on the contact surface X = 0, body 1is braked with a simultaneous generation of heat and wear. The sum of heatstream directions into the first and second layers is equal to the power density offriction forces Q. It is assumed that between the second and third bodies an idealheat contact occurs. The external surface of the layer system keeps zero valuetemperature of the external environment.
Our attention if focused on the derivation of velocity V , temperature Tn,n = 1, 2, 3, and the wear amount Uw in an arbitrary time instant 0 ≤ t ≤ ts,where ts is the braking time.
4.1.1 Mathematical modelling
A decrease of velocity V (t) from the initial value V (0) = V0 to zero duringbraking is defined by the equation
md
dtV (t) = −Ffr, V (0) = V0, t ∈ (0, ts), (4.1)
where M is the mass per contact surface unit, whereas Ffr denotes friction forceper surface unit.
Owing to Amonton’s law Ffr = f(T )P (t), where f(T ) = f0f∗(T ) is the
friction coefficient depending on the temperature. It is worth noting that func-tions f∗(T ) can be (i) decreasing for metallic–ceramic joints; (ii) increased oneswith a minimum or maximum for critical material; or (iii) constant for frictionalmetallo–plastic materials. In a general case, a dependence of friction coefficienton temperature f∗(T ) has the form
f∗(T ) = f1 +f2
(f3(T − Tf ))2 + 1. (4.2)
In the case of rubbing of the metallic–ceramic pair and anti-friction cast iron,the following function is used [Chichinadze et al. (1979)],
f∗(T ) = exp(−λfT ), (4.3)
where f0, λf , Tf , fj (j = 1, 2, 3) are the coefficients defined experimentally[Chichinadze et al. (1979)]. The curves approximating the experimental results areshown in Fig. 4.2a.
Curve 1 corresponds to the case f∗(T ) = 1 (metal–artificial material); curve 2corresponds to the values: f1 = −0.552, f2 = 1.83, f3 = 2.05 · 10−3◦C−1,Tf = 200◦C (plastic); and curve 3 corresponds to f1 = 0.1, f2 = 0.845,f3 = 3.02 · 10−3◦C−1, f3 = 3.02 · 10−3◦C−1, Tf = −100◦C (ceramics).
190 4 Contact Characteristics During Braking Process
Fig. 4.2: Dependence of friction coefficient f∗(T ) (a) and wear kw(T ) (b) on the contacttemperature T .
The following pressure variation in time is assumed
P (t) = P0[1 − exp(−t/tm)][1 + B1sin(B2t/tm)], (4.4)
where tm is the parameter responsible for a loading from 0 to P0; B1 is theamplitude of periodically driven vibrations; B2 denotes the dimensionless periodof vibrations.
The temperature Tn (n = 1, 2, 3) is found using the following nonstationaryheat transfer equation of the form
∂2
∂X2Tn(X, t) =
1
an
∂
∂ tTn(X, t), 0 ≤ t ≤ ts, (4.5)
(0 < X < H1 for n = 1; −H2 < X < 0 for n = 2;
− H2 − H3 < X < −X2 for n = 3),
with the following boundary conditions
λ2∂
∂XT2(0, t) − λ1
∂
∂XT1(0, t) = Q(t), T1(0, t) = T2(0, t),
T1(H1, t) = 0, T2(−H2, t) = T3(−H2, t),
λ2∂
∂XT2(−H2, t) = λ3
∂
∂XT3(−H2, t),
T3(−H2 − H3, t) = 0, 0 ≤ t ≤ ts, (4.6)
4.1 Contact characteristics of three-layer brake models 191
and initial conditions
Tn(X, 0) = 0, n = 1, 2, 3,
where λn are the heat transfer coefficients, and an are the temperature compen-sating coefficients.
The heat stream density Q generated by friction forces on the surface betweendifferent materials is equal to friction work [Aleksandrov, Annakulova (1990)]
Q(t) = (1 − η)f(T )V (t)P (t), 0 ≤ t ≤ ts, (4.7)
where η is the part of friction work devoted to wear.The following friction wear law is applied [Archard (1959), Goryacheva (1988)],
Uw(t) =
t∫
0
f(T )V (t∗)P (t∗)Kw(T )dt∗, (4.8)
where Uw(t) = Uw1 (t) + Uw
2 (t); Uwn is the displacement of the surface working
along the X-axis; Kw = Kw1 +Kw
2 is the wear coefficient; and T (t) = T1(0, t) =T2(0, t) denotes the contact temperature.
Owing to the work of [Chichinadze et al. (1979)], in general one may takeKw(T ) = Kw
0 kw(T ), where
kw(T ) = d0 + d1T (t) +d2
[d3(T (t) − T w1 )]2 + 1
+d4
[d5(T (t) − T w2 )]2 + 1
(4.9)
and T wn , n = 1, 2, dj , j = 0, 1, . . . , 5 are the coefficients found experimentally.
The curves approximating experimental results are shown in Fig. 4.2b. Curve 1corresponds to the case kw(T ) = 0; curve 2 corresponds to the values: d0 = 6,d1 = 0, d2 = 3, d3 = 10−2◦C−1, d4 = 5, d5 = 10−2◦C−1, T w
1 = 100◦C,T w
2 = 800◦C; curve 3 to d0 = 80, d1 = 0, d2 = −72, d3 = 5 ·10−4◦C−1, d4 = 0,T w
1 = 100◦C; curve 4 to d0 = 8, d1 = 0.78 · 10−2◦C−1, d2 = 0, d4 = 0.Let us introduce the following dimensionless parameters
τ = t/tm, τs = ts/tm, δ = t0/tm, q = Q/Q0, q0 = Q0R/T0,
rn = Rn/R, ηn = rn
√Fon, Fon = antm/H2
n, n = 1, 2, 3,(4.10)
and the following dimensionless sought function
v = V/V0, p = P/P0, θ = T/T0,
uw = Uw/U0, θn = Tn/T0, n = 1, 2, 3, (4.11)
192 4 Contact Characteristics During Braking Process
where R = R1 + R2 + R3, Rn = Hn/λn, n = 1, 2, 3,
Q0 = (1 − η)f0V0P0, t0 = mV0/(f0P0) = 2W/Q0,
U0 = f0tmV0P0Kw0 , T0 = Q0
√a1a2tm/π/(λ2
√a1 + λ1
√a2). (4.12)
In the above t0 denotes braking time in the case of constant load action P0, andW = mV 2
0 /2 denotes the initial kinetic energy per unit contact surface, whereasRn is the heat resistance of the layers.
Applying the Laplace transform with respect to time t0 equations of motion(4.1) and boundary conditions (4.5) and (4.6), the following system of HDFequations is obtained.
v(τ) = 1 − δ−1
τ∫
0
f∗(T0θ)p(τ∗)dτ∗, (4.13)
p(τ) = [1 − exp(−τ)][1 + B1sin(B2τ )], θ(τ) = θn(0, τ), n = 1, 2,(4.14)
θn(X, τ ) =
τ∫
0
Gn(X, τ − τ∗)f∗(T0θ)v(τ∗)p(τ∗)dτ∗, (4.15)
uw(τ) =
τ∫
0
f∗(T0θ)v(τ∗)p(τ∗)kw(T0θ)dτ∗, (4.16)
Gn(0, τ) = G(τ) = 2q0
∞∑
m=1
η1tgξ1m(η2tgξ2m + η3tgξ3m)
D(μm)e−µ2
mτ (4.17)
D(μm) = tgξ1mtgξ3m(η3/√
Fo1 + η1/√
Fo3 + η1η3/(η2
√Fo2))
+ tgξ2mtgξ3m(η2/√
Fo3 + η3/√
Fo2 + η1η3/(η2
√Fo1))
+ tgξ1mtgξ2m(η1/√
Fo2 + η2/√
Fo1 + η1η3/(η2
√Fo3)) − 1,
where ξnn = μm/√
Fon, where μm are the roots of the following characteristicequation
η3cosξ1cosξ2sinξ3 + η1cosξ2cosξ3sinξ1
+ η2cosξ1cosξ3sinξ2 − η1η3/η2sinξ1sinξ2sinξ3 = 0, (4.18)
ξn = μ/√
Fon, n = 1, 2, 3.
4.1 Contact characteristics of three-layer brake models 193
Fig. 4.3: Scheme of two-layer braking system.
For the tribological system consisting of two layers (Fig. 4.3), the system ofEquations (4.13)–(4.16) has a similar form; that is, the difference is only mani-fested through another function
Gn(X, τ) = 2
∞∑
m=1
cos(ξxnm) − cotan(ξnm)sin(ξx
nm)
D(μm)e−µ2
mτ , (4.19)
where
D(μm) =λ(1)
Fo21
+λ(2)
Fo22
− (λ(1) + λ(2))cotan(ξ1m)cotan(ξ2m)
μ2m
, (4.20)
ξxnm = (μmX)/(
√FonHn), ξnm = μm/
√Fon,
λ(n) = λnT0/(Hn(1 − η)Q0) n = 1, 2.
In the above, μm denotes the roots of the characteristic equation associated withtwo layers of the form
λ(1)cotan(ξ1) + λ(2)cotan(ξ2) = 0. (4.21)
For small values of time τ or for large values of thickness of layer 1 and 2, thefollowing value of two contacting half-spaces is obtained
Gn(X, τ) = τ−1/2 exp(−X2/(4τtman)). (4.22)
A solution to the system of nonlinear integral equations (4.13)–(4.16) isfound numerically using iterations with an account of trapezoidal approximation[Abramowitz and Stegun (1965)].
194 4 Contact Characteristics During Braking Process
In time instants τ = τi = ih, i = 1, 2, . . . (h, step) one gets
v(j)i = 1 − δ−1
[0.5hΦ
(j−1)i + (1 − δi1)h
i−1∑
m=1
Φ(j−1)m
], (4.23)
θ(j)i = (2/3)
√h[2Φ
(j−1)i v(j−1)
i + Φ(j−1)i−1 v(j−1)
i−1 ]
+ 0.5h(1 − δi1)Φ(j−1)i−1 v(j−1)
i−1 /√
τi − τ1
+ h(1 − δi1)(1 − δi2)
i−2∑
m=1
Φ(j−1)m v(j−1)
m /√
τi − τm, j = 1, 2, . . . ,
(4.24)
θ(1)i = θ
(ji−1)i−1 , v(1)
i = v(ji−1)i−1 , θ0 = 0, v0 = 1,
where
Φ(j−1)i = f∗F (T0θ
(j−1)i )pi, δij = 1, i = j, δij = 0, i = j,
θi = θm(0, τi), vi = v(τt), pi = p(τi).
The iterational process is finished, when for j = ji the following conditionshold
|(θ(ji)i − θ
(ji−1)i )/θ
(ji)i | < εb, |(v(ji)
i − v(ji−1)i )/v(ji)
i | < εb,
where εb is the relative error. The amount of wear is estimated by the relation
uwi = 0.5hΦivik
w(T0θi) + (1 − δi1)hi−1∑
m=1
Φmvmkw(T0θm). (4.25)
4.1.2 Contact characteristics of the metallic–ceramic frictional
strap and the metal disk during braking
The numerical analysis is carried out using Equations (4.13)–(4.16) for the fric-tional pair of alloy cast iron (disc, body 1), metallic–ceramic (frictional strapelement, body 2), and steel 30ChGSA (strap foundation, body 3) for the follow-ing fixed parameters [Chichinadze et al. (1979)]: P0 = 0.98 MPa, tm = 0.2 s,V0 = 30 m s−1, W = 35.4 MN m−1, f0 = 0.8, f1 = 0.1, f2 = 0.845,f3 = 3.02 · 10−3◦C−1, Tf = −100◦C (curve 3 in the Fig. 4.2a), d0 = 6, d1 = 0,d2 = 3, d3 = 10−2◦C−1, d4 = 5, d5 = 10−2◦C−1, T w
1 = 100◦C, T w2 = 800◦C
(curve 2 in Fig. 4.2b), Kw0 = 1 m2 N−1, λ1 = 51 W m−1◦C−1, a1 = 14 mm2 s−1,
4.1 Contact characteristics of three-layer brake models 195
Table 4.1: Definitions of curves for various thickness of layers
CurvesSolid Dashed Dotted
Curve Number H2 = 1 mm H1 = 15 mm H1 = 5 mmH3 = 15 mm H2 = 1 mm H3 = 5 mm
H1, mm H3, mm H2, mm1 15 15 152 10 10 103 5 5 54 1 1 1
λ2 = 34.3 W m−1◦C−1, a2 = 15.2 mm2s−1, λ3 = 37.2 W m−1◦C−1, a3 =10.3 mm2s−1, B1 = 0.
Owing to (4.10)–(4.12) one gets t0 = 3.01 s, δ = 15.05, T0 = 265◦C.Kinetics of the contact temperature T , dimensionless sliding velocity v, and
wear uw during brake are illustrated in Fig. 4.4. The curves characterize variousvalues of body thickness Hn (Table 4.1).
The highest contact temperature is achieved for the minimal thickness values ofthe metallic–ceramic strap, the steel strap, and the disc made from alloy cast iron(Fig. 4.4a). In this case one deals with the longest braking time (Fig. 4.4b) andthe smallest wear (Fig. 4.4c). Increase of the contact temperature is accompaniedby a decrease of wear of the rubbing elements. In this aspect, temperature plays apositive role for the considered tribological system.
For fixed thickness of base and disc layers (H1 = H3 = 5 mm, dotted curves),an increase of thickness of the metallic–ceramic layer causes an increase of contacttemperature, whereas an increase of braking time yields the wear decrease. Hence,minimization of wear of the contacting bodies can be achieved through an increaseof the thickness of the frictional layer of the metallic–ceramic strap.
As the effective measure of element thickness occurred in the considered tri-bological system, the thickness of a hypothetical case is taken, that is, when thecontact temperature differs by 5% in comparison to the case for infinite elements.The computations have shown that the effective thickness of the frictional strapis H2,ef = 2 mm for H1 = H3 = 5 mm; H2,ef = 3 mm for H1 = 5 mm,H3 = 3 mm. Hence, one may conclude that the decrease of thickness of the metal-lic base causes an increase of thickness of the effective metallic–ceramic layer. ForHn > Hn,ef n = 1, 2, 3 one may consider the half-infinite layers and the contactcharacteristics may be found through consideration of rubbing of two half-spaces[Olesiak et al. (1997), Pyryev (2004)].
Below, the exponential dependence of the frictional coefficient on the tem-perature (4.3) is analysed for the following fixed parameters: P0 = 0.98 MPa,f0 = 0.7, λf = 1.5 · 10−3, V0 = 30 m s−1, tm = 0.2 s, W = 35.4 MN m−1,λ1 = 51 W m−1 ◦C−1, a1 = 14 mm2 s−1, λ2 = 34.3 W m−1◦C−1, a2 =15.2 mm2 s−1, λ3 = 37.2 W m−1◦C−1, a3 = 10.3 mm2 s−1, d0 = 6, d1 = 0,d3 = d5 = 0.01◦C−1, d4 = 5, T 2
1 = 100◦C, T w2 = 800◦C (curve 2 in Fig. 4.2b),
B1 = 0.
196 4 Contact Characteristics During Braking Process
Fig. 4.4: Time history of the contact temperature T (τ) (a), dimensionless sliding velocity v(τ)(b), and wear uw(τ) (c) during braking for various layer thicknesses.
The computational results give t0 = 3.44 s, δ = 17.2, and T0 = 232◦C.Numerical results are shown in Fig. 4.5. For constant thickness of friction strap
(H2 = H3 = 5 mm), an increase of disc thickness from 1 mm to 5 mm causedan increase of braking time τs, maximum contact temperature θmax, and time toachieve it τmax (Fig. 4.5a). Simultaneously, wear u = u(τs) quickly decreased. Thelatter results from the fact that the friction coefficient decreases with an increaseof temperature. Further increase of H1 > 5 mm does not influence the variationof the earlier mentioned contact characteristics.
Thickness H2 of the metallic–ceramic layer essentially influences the contactcharacteristics only in the case when thickness H3 of the steel strap base is notgreater than 2 mm (broken curves in Fig. 4.5b). For H3 = 5 mm, the contacttemperature and wear do not change with the increase of metallic–ceramic layerthickness (solid curves in Fig. 4.5b). Hence, one may conclude that a way ofobtaining resistance on the heat action (i.e., stability of the contact characteristics
4.1 Contact characteristics of three-layer brake models 197
Fig. 4.5: Dependencies of dimensionless wear uw (curves 1), braking time τs (2), maximalcontact temperature 10 · θmax (3), and time associated with maximal temperature τmax (4) on(a): disc thickness H1 for H2 = 5 mm, H3 = 5 mm; (b): frictional strap thickness H2 (solidcurves, H1 = 5 mm, H3 = 2 mm; dashed curves, H1 = 5 mm, H3 = 5 mm); (c) strap basethickness H3 for H1 = 5 mm, H2 = 1 mm.
in a wide temperature range) is to cover the steel base of thickness greater than5 mm (Fig. 4.5) by the metallic–ceramic layer (≈1 mm).
Conclusions. During the frictional braking process of the metallic–ceramic layerlying on the steel base (steel 30HGSA) of the frictional pad on the transversal discsurface (cast iron) the following observations have been made.
(i) For the constant thickness of the strap and disc, an increase of thickness ofthe metallic–ceramic layer causes an increase of the contact temperature andwear decrease.
(ii) There are thickness values of the base of the strap and disc such that avariation of the metallic–ceramic layer thickness does not influence the con-tact characteristics; in this case, for computational purposes one may use theresults of HFD for two semi-infinite bodies [Yevtushenko and Pyryev(1998)].
198 4 Contact Characteristics During Braking Process
(iii) For the constant thickness of the frictional strap an increase of the disc thick-ness (braking drum) causes the temperature increase and wear decrease atleast up to a certain extent. For thickness greater than the critical values, themaximal values of temperature and wear do not change.
4.2. Computation of the contact characteristics of the
two-layer brake model
Equations (4.13)–(4.16) and (4.19) served for numerical computations of the con-tact brake characteristics for the frictional pair and the metallic–ceramic material(frictional strap element, body 2) for the parameters: P0 = 0.98 MPa, tm = 0.2 s,V0 = 30 m s−1, W = 35.4 MN m−1, η = 0.5, B1 = 0, f0 = 0.8, f1 = 0.1,f2 = 0.845, f3 = 3.02 · 10−3◦C−1, Tf = −100◦C (curve 3 in Fig. 4.2a), d0 = 6,d1 = 0, d2 = 3, d3 = 10−2◦C−1, d4 = 5, d5 = 10−2◦C−1, T w
1 = 100◦C,T w
2 = 800◦C (curve 2 in Fig. 4.2b), Kw0 = 1 m2N−1, λ1 = 51 W m−1◦C−1),
a1 = 14 mm2 s−1, λ2 = 34.3 W m−1◦C−1), a2 = 15.2 mm2 s−1.From the computations we got t0 = 3.01 s, δ = 15.05, and T0 = 132◦C.Figure 4.6a shows the dependencies of the contact temperature on time τ for
various layer thicknesses H2. The layer thickness is H1 = 15 mm. It is assumedthat the friction coefficient depends on the contact temperature in agreement withcurve 2 in Fig. 4.2a. The corresponding computational results of the sliding velocityand wear are reported in Figs. 4.6b and c.
The dependence of maximal contact temperature Tmax on thickness H2 forthe constant thickness H1 = 15 mm for different types of functions f∗(T ) (seeFig. 4.2a) is shown in Fig. 4.7a. The corresponding computational results of thedimensionless time τs and time τmax are shown in Fig. 4.7b. Numbers of curvesin Fig. 4.7 correspond to the numbers of curves in Fig. 4.2a.
Note that an important brake characteristic is the effective body thickness H∗n.
The latter is defined as a minimum of the rubbing element thickness, which doesnot interact with the maximum of the contact temperature. In other words, Tmax
computed for H∗n with the accuracy of 5% in agreement with the computational
result of Tmax for Hn = ∞. The value of H∗n is obtained from the formula
H∗n = ve
√ants. The parameter ve = 1.75 is due to Fazekas [Fazekas (1953)],
ve = 1.73 is due to Chichinadze [Chichinadze et al. (1979)], whereas ve = 1.29is due to Hasselgruber [Hasselgruber (1963)].
The computational results shown in Fig. 4.7 provide an estimation of ve = 0.8.The largest values of τs and τmax for the fixed thickness H2 during the brakingprocess are obtained for the function f∗(T ) depicted by curve 3 in Fig. 4.2a.From numerical analysis it follows that a decrease of the friction coefficient witha simultaneous temperature increase causes an increase of both ts and tmax. Onthe other hand, the contact temperature and wear decrease.
4.3 Computation of the contact characteristics of the two semi-space brake models 199
Fig. 4.6: Time histories of contact temperature T (τ) (a), dimensionless sliding velocity v(τ)(b) and wear uw(τ) (c) during braking for curve 2 (Fig. 4.2a) at H1 = 15 mm and variousthicknesses H2.
4.3. Computation of the contact characteristics of the two
semi-space brake models
As is known, for different layers of thickness Hn > 10 mm, one may assumethat the contact layers are semi-spaces. In this case the solution is governed by(4.13)–(4.16) and (4.22).
In this section we consider the case of constant friction coefficient f(T ) = f0
and the linear dependence of the wear coefficient versus temperature (informula (4.9) d2 = d4 = 0). Furthermore, a lack of external load (B1 = 0)is assumed. The system of nonlinear Equations (4.13)–(4.16) has been solvedanalytically for the case of two semi-spaces [Yevtushenko and Pyryev (1997)].
200 4 Contact Characteristics During Braking Process
Fig. 4.7: Maximal contact temperature Tmax, dimensionless time τmax, and braking time τs (b)versus thickness H2 for H1 = 15 mm.
4.3.1 Contact temperature and wear during braking
The frictional wear law governed by (4.8) serves for our computational purpose.Assume that the dependence of Kw
n (n = 1, 2) on the temperature is linear[Aleksandrov and Annakulova (1990), Pyryev, Yevtushenko (2000)]
Kwn (t) = Kw
1n(t) + Kw2n(t)βnT (t), βn = (1 + νn)αn/(1 − νn), (4.26)
where T (t) ≡ Tn(0, t) is the contact temperature; and Kwjn are the coefficients
obtained from experimental investigations [Chichinadze et al. (1979)], n, j = 1, 2;νn, αn denote Poisson ratios and heat extension coefficients, respectively. It isassumed that the following load is applied (4.4) when B1 = 0.
Let us introduce, besides parameters (4.10)–(4.12), also the following ones
Kw0 = Kw
11 + Kw12, ε = T0
Kw21 + Kw
22
Kw0
. (4.27)
After integrating the equation of motion (4.13) we obtain the speed-changinglaw during braking
v(τ) = δ−1(1 + δ − τ − exp(−τ)). (4.28)
Upon substitution of Equations (4.14) and (4.28) into Equation (4.15) andintegration we find the dimensionless temperature on the contact surface X = 0in the form
θ(τ) = δ−1[(1 + 2δ)√
τ − 4τ√
τ/3
+ (2τ − 3 − 2δ)FD(√
τ ) +√
2FD(√
2τ)], (4.29)
4.3 Computation of the contact characteristics of the two semi-space brake models 201
where FD(τ) is Douson’s integral [Abramowitz and Stegun (1965)]
FD(τ) = exp(−τ2)
τ∫
0
exp(t2)dt. (4.30)
To calculate FD(T ) we use the formulas
FD(τ) =
∞∑
m=0
(−2τ2)m
(2m + 1)!!, 0 ≤ τ ≤ 3;
FD(τ) =
∞∑
m=0
(2m − 1)!!
(2τ2)m+1, 3 < τ < ∞.
Having an analytical expressions for the speed of v(τ), Equation (4.28) and thecontact temperature θ(τ) Equation (4.29), from Equation (4.16) we find the wearin the form
uw(τ) = uw0 (τ) + εuw
1 (τ), (4.31)
where
uw0 (τ) =
1
δ
[2∑
m=0
bmτm + (1 + δ − τ)exp(−τ) − exp(−2τ)/2
],
uw1 (τ) =
2
δ2
{[a(0) +
3∑
m=0
a(1)m+1τ
m+1/2 + exp(−τ)5∑
m=0
a(2)m τm/2
+√
τ (a(3)0 + a
(3)1 τ) exp(−2τ) + a(4)erf
√τ + a(5)erf
√2τ
]
+ FD(√
τ )
[2∑
m=0
a(6)m τm+exp(−τ)
5∑
m=0
a(7)m τm+(a
(8)0 +a
(8)1 τ) exp(−τ)
]
+ FD(√
2τ)[a(9)0 +a
(9)1 τ +(a
(10)0 +a
(10)1 τ) exp(−τ)+a(11) exp(−2τ)]
},
b0 = −1/2 − δ, b1 = 1 + δ, b2 = −1/2, a(0) = −2 − δ, a(1)1 = −3/4− δ2,
a(1)2 = (1 + 5δ + 2δ2)/3, a
(1)3 = −2(1 + δ)/3, a
(1)4 = 4/21, a
(2)0 = −a(0),
a(2)1 = 1/3 + δ/4 + δ2, a
(2)2 = −a(0), a
(2)3 = −5(1 + 4δ)/12, a
(2)4 = 0,
a(2)5 = 2/3, a
(3)0 = −(1/6 + δ)/2, a
(3)1 = 1/3,
202 4 Contact Characteristics During Braking Process
a(4) =√
π(35/36+11δ/6−δ2)/4, a(7)0 = −(11/4+3δ+δ2)/2, a
(7)1 = −a
(6)1 ,
a(7)2 = −1/2, a
(8)0 = −(7/6 + δ)/3, a
(8)1 = −1/3, a
(9)0 =
√2(1/2 + δ)/4,
a(9)1 =
√2/4, a
(10)0 =
√2(5/3 + δ)/6, a
(10)1 = −
√2/6, a(11) = −
√2/8,
and erf (x) is the probability function [Abramowitz and Stegun (1965)].A numerical analysis is carried out using analytical solution forms (4.28),
(4.29), and (4.31). The following dimensionless quantities δ and ε (4.10), (4.27)serve as initial input parameters.
The parameter 0 < δ < ∞ characterises the ratio of braking time t0 underuniform deceleration to time tm, when the load reaches maximum value and 0 <ε < ∞ characterises the influence of the temperature on wear.
Using the condition v(τs) = 0 from Equation (4.28) we obtain the equation fordimensionless braking time τs = ts/tm,
τs = 1 + δ − exp(−τs). (4.32)
Numerical analysis has also shown that the following approximations can beused,
τs =√
2δ for δ < 0.1, τs = 1 + δ for δ > 1.2. (4.33)
The expressions of Equation (4.33) also imply that for t0 > 1.2tm the brakingtime ts increases relative to time t0 at the value of tm.
An analytical expression of the contact temperature Equation (4.29) allows usto construct such asymptotes
θ(τ) = 4τ√
τ/3[1 − 2τ/5 − 4(3 − δ)τ2/(35δ) + O(τ3)], (4.34)
θ(τ) =√
τ/δ[2(1 + δ) − 4τ/3 − (1 + 2δ)/(2τ) + O(τ−2)]. (4.35)
Comparing approximate values of the temperature, Equations (4.34) and (4.35),with the corresponding accurate values Equation (4.29), we obtain the followingengineering expressions for the maximum dimensionless contact temperature θmax
and for the dimensionless time needed to reach it τmax.
θmax = 4√
τmaxδ−1(1 + δ − 4τmax/3) for δ > 5.5, (4.36)
τmax = 1 + δ/2 for δ > 1.5, (4.37)
θmax = 8τmax√
τmax(1 + 4τmax/15)/9 for δ < 0.1, (4.38)
τmax = 1/4(√
5δ(36 − 7δ) − 5δ)/(3 − δ) for δ < 0.08. (4.39)
Figure 4.9b shows an influence of contact temperature (parameter ε) on wearevolution. When the smallest wear occurs frictional heat is not generated (ε = 0).
4.3 Computation of the contact characteristics of the two semi-space brake models 203
An essential influence of temperature on wear of the contacting bodies appears forε = 3.4.
Numerical results have been computed for the friction couple cast iron/metal–ceramics, for which: P0 = 0.98 · 106 Pa, f = 0.8, V0 = 30 m s−1, tm = 0.2 s,W = 3.54 · 105 N m−1, λ1 = 50.96 W m−1◦C−1, a1 = 14 · 10−6 m2 s−1, λ2 =34.3 W m−1◦C−1, a2 = 15.2 · 10−6 m2 s−1, η = 0, B1 = 0.
Additional computations yield t0 = 3 s, δ = 15, T0 = 264.8◦C. For δ = 15and from Figs. 4.8a and 4.9a one obtains τs = 16, θmax = 3.64, τmax = 8.56.The contact temperature achieves the largest value Tmax = 963.9◦C in the secondtwo of the braking process, that is, for time instant t = 1.7 s.
Numerical analysis of the contact characteristics was carried out and generaldependencies of the friction and wear coefficients on contact temperature wereobtained (see Fig. 4.2). Computational results are displayed in Fig. 4.10 (η = 0.5,T0 = 132◦C). Numbers of curves displayed in Fig. 4.10 correspond to the numbers
Fig. 4.8: Braking time versus parameter δ (a); time history contact temperature (b).
Fig. 4.9: Maximal contact temperature θmax and τmax versus parameter δ during braking (a);time history wear for various parameter values ε (b).
204 4 Contact Characteristics During Braking Process
Fig. 4.10: Contact temperature T (τ) (a), dimensionless velocity v(τ) (b) and wear uw(τ) versusdimensionless time τ . Numbers and curve shapes correspond to the curve numbers from Fig. 4.2.
of curves given in Fig. 4.2a). Shape of the curves shown in Fig. 4.10c correspondsto the shape of curves in Fig. 4.2b.
The analysis leads to a conclusion that for frictional metallic–plastic (curve 1)and plastic (curve 2) materials, the change in time of the contact characteristics issimilar. Therefore, one may assume that the friction coefficient is constant and theanalytical solution (4.28), (4.29) can be applied. Note that the contact temperatureof the braking process has a maximum in the middle of the braking time interval(Fig. 4.10a).
Owing to the increase of parameter δ both maximum temperature θmax and thetime to reach it τmax increase. As Fig. 4.10b shows, a uniformly retarded motionis exhibited during the whole braking process. The negligible deviation from thismotion occurs in the first phase of braking.
Kinetics of wear is illustrated in Fig. 4.10c. Note that in almost the whole brak-ing phase a lapping process occurs when the velocity is constant. Wear achievesits maximum in the last phase of braking.
4.3 Computation of the contact characteristics of the two semi-space brake models 205
Fig. 4.11: Dimensionless contact pressure p(τ) (a), contact temperature T (τ) (b), velocity v(τ)
(c) and wear uw(τ) (d) versus dimensionless time τ . Curve 1: B1 = 0.2, B2 = 15, λf = 0;curve 2: B1 = 0.2, B2 = 15, λf = 1.5 · 10−3◦C−1; curve 3: B1 = 0, λf = 1.5 · 10−3◦C−1.
4.3.2 Contact temperature and wear during braking and
harmonic load excitations
Consider the case of harmonic load input (4.4) and the exponential dependenceof the friction coefficient on temperature (4.3). The contact characteristics of abrake for the frictional pair consist of alloy cast iron (disc, body 1) and metallic–ceramic (budding strap element, body 2) [Chichinadze et al. (1979)] have thefollowing input parameters: P0 = 0.98 MPa, tm = 1.2 s, V0 = 30 m s−1, W =35.4 MN m−1, η = 0.5, f0 = 0.7, λf = 1.5 · 10−3◦C−1, d0 = 6, d1 = 0,d2 = 3, d3 = 10−2◦C−1, d4 = 5, d5 = 10−2◦C−1, T w
1 = 100◦C, T w2 = 800◦C
(curve 2 in Fig. 4.2b), Kw0 = 1 m2 N−1, λ1 = 51 W m−1◦C−1), a1 = 14 m2 s−1,
λ2 = 34.3 W m−1 ◦C−1, a2 = 15.2 m2 s−1. Furthermore, carrying out simplecomputations one gets t0 = 3.44 s, δ = 2.86, T0 = 284◦C. Figure 4.11a showsthree cases of the external load variations. The functions governing the behaviour
206 4 Contact Characteristics During Braking Process
of contact temperature, sliding velocity and wear versus dimensionless time τ forvarious loading types are shown in Figs. 4.11b–d, respectively.
Numerical analysis shows that during the braking process contact pressureoscillations (Fig. 4.11a) influence contact temperature to a lesser extent (Fig. 4.11b),and they do not affect either the velocity behaviour (Fig. 4.11c) or wear (Fig. 4.11d).
The contact temperature T achieves its maximum just before the body’s stop(Fig. 4.11b). The numerical analysis allows us to conclude also that a decrease ofthe friction coefficient caused by temperature increase induces enlargement of tsand tmax, whereas the contact temperature is decreased.
Chapter 5
Thermoelastic Contact of Two Moving Layers
with Friction and Wear
Geometry of the contacting bodies is the same as in Chapter 4. However, thereare essential differences between the two chapters. In Chapter 4 the so-called heatfrictional dynamics (HFD) [Chichinadze et al. (1979)] is studied. One has to studyheat transfer equations with conditions of heat occurrence and heat transfer on thesliding surface. We treat the contact pressure making a contribution to the heatstream as a known quantity. The sliding velocity responsible for heat generation onthe sliding surface is found from the equations of system dynamics. The contactcharacteristics are coupled via friction and wear coefficients and they depend ontemperature. There is no need to solve equations of the theory of elasticity of thecontacting bodies.
In this chapter we study the problem of thermoelastic contact of two slidinglayers. Contrary to considerations presented in the previous chapter, here weassume initial body position. In this problem the contact pressure is not known,and it can be defined by solving a thermoelastic problem of the theory of ther-mal stresses with appropriate boundary conditions. In the case of constant slidingvelocity the obtained problems fit with that of thermoelastic frictional instability.
5.1. Analysed system
A one-dimensional model of thermoelastic contact of two plane parallel layers isshown in Fig. 5.1. The upper layer of thickness L2 and mass m, under the actionof force F = F0ϕF (t), parallel to the plane of contact, slides at velocity v(t) overthe layer of thickness L1. The mass and the force are taken per length unit. Thefriction forces Ffr = f(T )p(t) (f(T ) = fsF (T )) over the contact surface x = L1
cause a deceleration of the upper layer. The braking process is accompanied by thegeneration of heat and wear. The sum of the intensities of heat fluxes directed intolayers is equal to the specific power of friction forces fV (t)P (t). The distancebetween solids U = U0ϕu(t) is known beforehand. Newton’s law of heat exchangeis assumed for fluxes between external surfaces of the layers and the environment.
207DOI: 10.1007/978-0-387-09653-7_5, © Springer Science+Business Media, LLC 2009 J. Awrejcewicz and Yu. Pyryev, Nonsmooth Dynamics of Contacting Thermoelastic Bodies,
208 5 Thermoelastic Contact of Two Moving Layers with Friction and Wear
Fig. 5.1: Geometry of the moving solids.
Temperatures Tn(x, t), n = 1, 2 of the layers, temperature of the contact planeT (t) = T1(L1, t) = T2(L1, t), displacements Un(X, t) along the X-axis, normal
stresses σ(n)XX(X, t), wear of the layers Uw(t), and velocity V(t) are to be found
from the solution. From the mathematical standpoint the considered problem con-sists in solving the system of differential equations of the quasi-static uncoupledthermoelasticity:
∂
∂X
[∂
∂XUn (X, t) − αn
1 + νn
1 − νnTn (X, t)
]= 0, (5.1)
∂2
∂X2Tn (X, t) =
1
an
∂
∂tTn (X, t) , 0 ≤ t ≤ tc, (5.2)
X ∈{
(0, L1), n = 1,
(L1, L1 + L2), n = 2,
and the equation of motion for the upper layer
md
dtV (t) = F0ϕF (t) − f(T )P (t), 0 ≤ t ≤ tc. (5.3)
We assume the following mechanical boundary conditions:
U1(0, t) = 0, U2(L1 + L2, t) = −U0ϕu(t),
σ(1)XX(L1, t) = σ
(2)XX(L1, t), U1(L1, t) − Uw
1 (t) = U2(L1, t) + Uw2 (t), (5.4)
5.1 Analysed system 209
thermal boundary conditions
λ1∂T1(0, t)
∂X− αT
1 T1(0, t) = 0,
λ2∂T2(L1 + L2, t)
∂X+ αT
2 T2(L1 + L2, t) = 0,
λ1∂T1(L1, t)
∂X− λ2
∂T2(L1, t)
∂X= f(T )V (t)P (t),
T1(L1, t) = T2(L1, t) = T (t), (5.5)
and the initial conditions
Tn(X, 0) = 0, n = 1, 2, V (0) = V0. (5.6)
Let us consider the abrasive wear in the form [Aleksandrov, Annakulova (1992)]
Uw(t) = Uw1 (t) + Uw
2 (t) =
t∫
0
Kw(T )V (τ)P (τ)dτ , (5.7)
where Kw(T ) = Kwa K(T ) is the wear coefficient, which depends on contact
temperature.Normal stresses can be found by means of the following Duhamel relations
σ(n)XX =
En
1 − 2νn
[1 − νn
1 + νn
∂Un
∂X− αnTn
], n = 1, 2. (5.8)
In Equations (5.1)–(5.8) we have used the following symbols: En is the Young’smodulus, νn are Poisson’s ratios, an are coefficients of thermal diffusivity, λn arecoefficients of thermal conductivity, αn are coefficients of linear thermal expan-sion, Kw is the coefficient of wear, f is the coefficient of friction, 1/αT
n are thethermal resistances, and tc is the time when the solids are in contact means thatP (t) ≥ 0, and V (t) ≥ 0 for t ∈ (0, tc)).
Making use of the system of Equations (5.1), (5.8), and boundary conditions
(5.4), the contact pressure P (t) = −σ(n)XX(L1, t) can be presented in the form
P (t) = P0
⎡
⎣α1
L1∫
0
T1(ξ, t)dξ + α2
L1+L2∫
L1
T2(ξ, t)dξ − Uw(t) + U0ϕu(t)
⎤
⎦ ,
(5.9)where
P0 =
(L1
E1
+L2
E2
)−1
, αn = αn1 + νn
1 − νn, En =
En(1 − νn)
(1 + νn)(1 − 2νn). (5.10)
210 5 Thermoelastic Contact of Two Moving Layers with Friction and Wear
We introduce the following dimensionless variables,
x =X
L∗, τ =
t
t∗, τc =
tct∗
, v =V
V∗, un =
Un
U0, uw =
Uw
U0, θn =
Tn
T∗,
p =P
P∗, θ =
T
T∗, Ln =
Ln
L∗, an =
an
a∗, λn =
λn
λ∗, Bin =
αTn L∗
λn,
ξ =Kw
a P0L∗
Ω, f0 =
F0
P∗, Ω = fsV∗P∗
L∗
T∗λ∗, a =
t∗P∗
V∗m, ˜αn =
αn
α∗,
where
t∗ =L2∗
a∗, V∗ =
a∗
L∗, P∗ = P0U0, T∗ =
U0
α∗L∗.
Dimensionless equations take the form
∂ 2
∂x2θn(x, τ) =
1
an
∂
∂τθn(x, τ), 0 ≤ τ ≤ τc,
x ∈{
(0, L1), n = 1,
(L1, L1 + L2), n = 2,
∂θ1(0, τ)
∂x− Bi1θ1(0, τ) = 0,
∂θ2(L1 + L2, τ)
∂x+ Bi2θ2(L1 + L2, τ) = 0,
λ1∂θ1(L1, τ)
∂x− λ2
∂θ2(L1, τ)
∂x= ΩF (θT∗)v(τ)p(τ),
θ1(L1, τ) = θ2(L1, τ) = θ(τ), (5.11)
where the dimensionless contact pressure, velocity, and wear are as follows.
p(τ) = ϕu(τ) + ˜α1
L1∫
0
θ1(η, τ)dη + ˜α2
L1+L2∫
L1
θ2(η, τ)dη − uw(τ),
v(τ) = v0 + a
⎡⎣f0
τ∫
0
ϕF (η)dη − fs
τ∫
0
F (θT∗)p(η)dη
⎤⎦ ,
uw(τ) = Ωξ
τ∫
0
K(θT∗)v(η)p(η)dη. (5.12)
5.2 Laplace transform 211
5.2. Laplace transform
A solution of the boundary value problem (Equations (5.11) and (5.12)) canbe obtained by the use of the Laplace integral transforms taken with respect totime τ and expressed in terms of the Laplace transforms of ϕF (τ), ϕu(τ), andq(τ). Because the thicknesses of the moving solids, in fact, are finite, the Laplacetransforms of the solution are the analytical functions of the transform parameters, except a countable set of poles sm.
Performing the inverse Laplace transformation with the help of residuals andconvolution theorems [Carslaw, Jaeger (1959)] one can find the following relations.
θn(x, τ) = ψθn(x, τ)
+ Ω
[q1(τ) ∗
(∂
∂τgn(x, τ) + Ωξgn(x, τ)
)− Ωξq3(τ) ∗ gn(x, τ)
],
p(τ) = ψp(τ) + Ω
[q1(τ) ∗ d2
dτ2G2(τ) − ξq3(τ) ∗ d
dτG1(τ)
],
uw(τ) = ψw(τ)
+ Ω
[ξq3(τ) ∗ d
dτG1(τ) + Ω
(K
Fq1(τ) − q3(τ)
)∗ d
dτG2(τ)
],
v(τ) = ψv(τ)
− afs
[q2(τ) ∗ H(τ) + ΩF
(q1(τ) ∗ d
dτG2(τ) − ξq3(τ) ∗ G1(τ)
)],
q1(τ) = rl(τ)p(τ), l = 1, 2, 3, r1(τ) = F (θT∗)v(τ) − F v,
r2(τ) = F (θT∗) − F , r3(τ) = K(θT∗)v(τ) − K v, (5.13)
where
ψθn(x, τ) = Ωϕu(τ) ∗ ∂
∂τgn(x, τ), ψp(τ) = ϕu(τ) + Ωϕu(τ) ∗ d
dτG3(τ),
ψu(τ) = Ωξϕu(τ) ∗ d
dτG1(τ),
ψv(τ) = v0 + a[f0ϕF (τ) ∗ H(τ) − faF (ϕu(τ) ∗ H(τ) + Ωϕu(τ) ∗ G3(τ))],
g1(L1, τ) = g2(L1, τ) = g(τ) =
∞∑
m=1
D1(sm)D2(sm)
∆′(sm)exp(smτ),
212 5 Thermoelastic Contact of Two Moving Layers with Friction and Wear
Gj(τ) =1
Ωγj
+∞∑
m=1
∆j(sm)
sm∆′(sm)exp(smτ), j = 1, 2, 3,
∆′(sm) =d
ds∆(s)
∣∣∣∣s=sm
, ∆1(s) =1
R1
M1(s)D2(s) +1
R2
M2(s)D1(s),
∆2(s) = αL1 N1(s)D2(s) + αL
2 N2(s)D1(s), ∆3(s) = s∆2(s) − ξ∆1(s),
∆(s) = s∆1(s) − Ω∆3(s), Dn(s) = Cn + BiLnSn,
Nn(s) = Sn + BiLnCn0, Mn(s) = BiLnCn + η2nSn,
Cn = cosh(ηn), Sn =sinh(ηn)
ηn, Cn0 =
1 − Cn
s, ηn =
√s
Fon,
Rn =Lnλ∗
Kn, αL
n =αnLn
α∗, BiLn = BinLn, Fon =
ant∗L2
n
,
Bi1n = 1 + BiLn , Bi2n = 1 + BiLn/2, n = 1, 2,
γ1 = ξ, γ2 = ξb0a−10 , γ3 = −1, Ω = ΩvF , ξ = ξKF−1,
b0 =1
R1
BiL1 Bi12 +1
R2
BiL2 Bi11, a0 = αL1 Bi21Bi12 + αL
2 Bi22Bi11.
The asterisk ‘∗’ denotes the convolution of functions with respect to the timevariable; H(τ) is Heaviside’s step distribution; sm are the roots of the characteristicequation ∆(s) = 0; Resm ≥ Resm+1, m = 1, 2, . . . ; and v, F , K are the arbitraryparameters. They provide a possibility to obtain the limiting cases, and allow foran efficient construction of the numerical algorithm associated with the consideredproblem.
From the analysis of the roots of the characteristic equation, and for smallvalues of parameter ξ, we find that Re(sm) < 0, Im(sm) = 0, for m = 3, 4, . . . ,whereas for m = 1, 2 the roots lie either on the left- or on the right-hand sideof the complex plane s depending on the value of the parameters. For the rootof the greatest value on the right-hand side of the complex plane s we obtains1 = (ξ∗ − ξ)Ω for Ω → ∞, where
ξ∗ = r(αL1
√Fo1 + αL
2
√Fo2), r =
(1
R1
√Fo1
+1
R2
√Fo2
)−1
. (5.14)
If ξ < ξ∗ then for 0 < Ω < v2 roots s1 and s2 are negative. For v2 < Ω < v3
the roots are complex conjugate; moreover when Ω passes v1 then roots s1 ands2 pass from the left-hand side to the right-hand side of the complex plane s. ForΩ > v3 the roots are real and positive. However, for ξ > ξ∗ the roots always lie inthe left half of the complex plane. Let us note that in the case of quasi-stationary
5.3 Algorithm of solutions 213
problems of thermoelasticity the roots of the characteristic equation lie on the realaxis only. In the case when wear is taken into account the roots of the characteristicequation become complex.
Some properties of the function used in solving Equation (5.13) are givenbelow:
G1(τ) = τ + O(τ2), G2(τ) = ξ∗τ2/2 + O(τ3),
G3(τ) = (ξ∗ − ξ)τ + O(τ2),
g(τ) = 2r√
τ/π + O(√
τ3), G3(τ) =d
dτG2(τ) − ξG1(τ). (5.15)
5.3. Algorithm of solutions
Equations (5.13) are integral and nonlinear of the Volterra–Hammerstein type[Verlan, Sizikov (1986)]. A solution is found using an iterative method andapplying the trapezoidal rule. The following input parameters are taken: F = F (0),K = K(0), v = v0. For the time instants τ = τi = i h, i = 1, 2, . . . , where h isthe mesh step, one gets
θ(j)i = ψθi + Ωh
[1
2q1,i−1g
′1 + (1 − δi1)(1 − δi2)
i−2∑
m=1
q1,mg′i−m
+2
3
√1
πhr(2q
(j−1)1,i + q1,i−1) + Ω(1 − δi1)
i−1∑
m=1
(ξq1,m − ξq3,m)gi−m
],
p(j)i =
ψpi + Ωh(1 − δi1)[∑i−1
m=1 q1,mG′′2,i−m − ξ
∑i−1m=1 q3,mG′
1,i−m
]
1 − Ωh(ξ∗r(j−1)1,i − ξr
(j−1)3,i )/2
,
v(j)i = ψvi − afsh
[(1 − δi1)
i−1∑
m=1
q2,m +1
2q(j−1)2,i
+ ΩF (1 − δi1)
(i−1∑
m=1
q1,mG′2,i−m − ξ
i−1∑
m=1
q3,mG1,i−m
)],
uwi = ψui + Ωh
[ξ1
2q3,i + ξ(1 − δi1)
i−1∑
m=1
q3,mG′1,i−m
+ Ω(1 − δi1)
i−1∑
m=1
(K
Fq1,m − q3,m
)G′
2,i−m
],
214 5 Thermoelastic Contact of Two Moving Layers with Friction and Wear
θ(0)i = θ
(ji−1)i−1 , p
(0)i = p
(ji−1)i−1 , v(0)
i = v(ji−1)i−1 , j = 1, 2, . . . ,
θ0 = 0, p0 = 1, uw0 = 0, (5.16)
where
θi = θ(τi), pi = p(τi), uwi = uw(τi),
vi = v(τi), gi = g(τi),
ql,i = ql(τi), q(j−1)l,i = r
(j−1)l,i p
(j−1)i , Gl,i = Gl(τi), l = 1, 2, 3,
r(j−1)1,i = F (θ
(j−1)i T∗)v
(j−1)i − F v, r
(j−1)2,i = F (θ
(j−1)i T∗) − F ,
r(j−1)3,i = K(θ
(j−1)i T∗)v
(j−1)i − Kv,
ψθi = ψθ(τi), ψpi = ψp(τi), ψui = ψu(τi), ψvi = ψv(τi).
The iterative process (5.16) is finished, when for j = ji the following inequali-ties are satisfied
∣∣∣∣∣p(j)i − p
(ji−1)i
p(j)i
∣∣∣∣∣ < ε,
∣∣∣∣∣θ(j)i − θ
(ji−1)i
θ(j)i
∣∣∣∣∣ < ε,
∣∣∣∣∣v(j)
i − v(ji−1)i
v(j)i
∣∣∣∣∣ < ε.
5.4. Solution analysis
Numerical analysis is carried out for the bottom steel layer (α2 = 14 · 10−6◦C−1,λ2 = 21 W m−1 · ◦C, a2 = 5.9 · 10−6 m2 s−1, ν2 = 0.3, E2 = 19 · 1010 Pa)for L2 = 0.02 m, Bi2 = 9.52 and for the upper aluminium layer (α1 = 22 ·10−6◦C−1, λ1 = 173 W m−1 · ◦C, a2 = 6.72 ·10−5 m2 s−1, ν1 = 0.32, E1 = 7.2 ·1010 Pa) for L1 = 0.01 m, Bi1 = 9.25; V0 = 1 m s−1; Ω = 0.36; ξ = 0.7 · 10−4,f0 = 0.17 ·10−3; fs = 0.8; a = 1.3 ·105. The parameters associated with the steellayer serve as the key ones for derivation of the dimensionless equations. In thiscase we have t∗ = 67.8 s, P∗ = 5.7 ·106 Pa, T∗ = 3.57◦C. The external load of thetribological system is taken in the form ϕu(τ) = ϕF (τ) = (1− exp(−δτ2))H(τ)for δ = 100.
Functions F (T ), K(T ) approximate the experimental results of friction andwear coefficients versus temperature [Chichinadze et al. (1979)], and they are asfollows,
F (T ) = r1 +r2
[ r3(T − Tm) ]2 + 1,
K(T ) = d1 +d2
[d3(T − T w1 )]2 + 1
+d4
[d5(T − T w2 )]2 + 1
,
5.5 Frictional thermoelastic instability 215
Fig. 5.2: Friction coefficient (solid curves) and wear (dashed curves) versus contact temperature.
where Tm, T wn , rl, dj are the coefficients obtained in the same way as in the
reference [Chichinadze et al. (1979)].A typical relation of friction coefficient F (T ) (solid curves) and wear K(T )
(dashed curves) versus contact temperature is shown in Fig. 5.2. Curve 1 corre-sponds to the case F (T ) = 1, K(T ) = 1. Solid curve 2 is associated with thevalues r1 = 0.1, r2 = 0.845, r3 = 3.02 · 10−3◦C−1, Tm = −100◦C. Dashedcurve 2 corresponds to the values d1 = 0.75, d2 = 0.375, d3 = 0.01◦C−1,d4 = 0.625, d5 = 0.01◦C−1, T w
1 = 100◦C, T w2 = 800◦C; curve 3 – d1 = 10,
d2 = −9, d3 = 5 · 10−4◦C−1, d4 = 0, T w1 = 100◦C.
Time histories of temperature θ, contact pressure p, sliding velocity v, and wearare shown in Fig. 5.3. Curves 1–3 correspond to the coefficients shown in Fig. 5.2.Owing to numerical analysis, the contact temperature achieves its maximum valuefor the constant values of the friction and wear coefficients. The value of layer wearachieves a maximum for the case of the largest braking time, whereas the frictioncoefficient decreases with an increase of temperature. The contact pressures arealmost the same in the stop instant of the upper layer.
5.5. Frictional thermoelastic instability
In this section a frictional joint model exhibiting a Frictional ThermoElastic In-stability (FTEI) is studied. In particular, our attention is focused on the interactionof nonstationary heat generation and wear, and their influence on the occurrenceof FTEI.
Owing to a fast contact between the bodies, some instabilities of the con-tacting parameters are expected yielding a harmful behaviour of the tribologicalsystem [Barber (1969), Chichinadze et al. (1979), Barber (1999)]. Recall (see
216 5 Thermoelastic Contact of Two Moving Layers with Friction and Wear
Fig. 5.3: Contact temperature θ (a), contact pressure p (b), sliding velocity v (c), and wearuw (d) versus time τ during braking.
Section 2.5.1.1) that FTEI is characterized by an exponential increase of thecontacting characteristics (i.e., temperature, pressure, and wear) assuming that arelative velocity of sliding bodies overcomes a certain threshold velocity.
Experimental verification of TEI has been presented by [Dow, Stockwell(1977)].
Consider a tribological system which may be modelled by two different layers.One of them moves at a constant velocity on the surface of a second fixed layer(Fig. 5.4).
Fig. 5.4: Scheme of the system exhibiting a frictional thermoelastic rubbing instability.
5.5 Frictional thermoelastic instability 217
Fig. 5.5: Positions of roots s1, s2 of the characteristic equation.
The action of frictional forces produces heat and wear. The introduced simplegeometry of contacting bodies enables monitoring of all physical aspects associatedwith the appearance of FTEI.
Observe that here we assume relatively slow variations of mechanical andthermal loading, which allows us to omit inertial terms and hence enablesdirect application of quasi-static equations of thermoelasticity in motion equa-tions (1). However, although this approach allows us to study the so-calledThermoElastic Instability (TEI) in our system, it does not enable a study ofThermoElastic Dynamic Instability (TEDI) (see, e.g., [Afferante et al. (2006),Afferrante, Ciavarella (2006), Yi (2006)]).
The solution obtained earlier for constant friction and wear coefficients areused: F (T ) = 1, K(T ) = 1 (nonlinear functions take the forms q1(τ) = q3(τ) =(v(τ) − v) p(τ) ≡ q(τ), q2(τ) = 0). It is assumed that the layer moves at aconstant velocity v(τ) = v = const. This means that q(τ) = 0 and a solution tothe started problem is governed by the first term of relation (5.13).
Analysis of stability is limited to the investigation of root locations of thecharacteristic equation ∆(s) = 0. For the largest root the following estima-tion holds, s1 ≈ (ξ∗ − ξ)v, v → ∞ (v = Ω, whereas ξ∗ is determined from(5.14)).
Observe that for ξ < ξ∗ and v < v2 roots s1, s2 are negative; for v2 < v < v3
the roots are conjugated; and when v = v1 the real root part changes sign. Forv > v3, the roots are real and positive.
218 5 Thermoelastic Contact of Two Moving Layers with Friction and Wear
Fig. 5.6: Critical parameters v3 (solid curves) and v1 (dashed curves) versus parameter ξ.
For ξ > ξ∗ all roots have a negative real part.In Fig. 5.5, for Bi1 = Bi2 = 10, real (solid curves) and imaginary (dashed
curves) parts of the roots versus dimensionless velocity v for ξ = 0.01 (curve 1)and ξ = 0.1 (curves 2) are shown. For ξ = 0.1 the critical values of v3 (solidcurves) and velocity v1 (dashed curves) ξ for different values Bi are shown inFig. 5.6. Curves 1–4 correspond to the values of Bi = 0.1; 1; 10; ∞.
The characteristic function ∆(s) has the form
∆(s) =∞∑
j=0
sjdj , (5.17)
where the coefficients dj follow
d0 = vξb0, d1 = b0 − v(a0 − ξ(c0 + b1)),
dj = bj−1 − v (aj−1 − ξ (cj−1 + bj)) + cj−2, j = 2, 3, . . .
aj =1
2(2j + 1)!(Fo+
+αL1
√Fo1 + Fo+
−αL2
√Fo2)
+BiL2 αL
1 + BiL1 αL2
2(2j + 2)!Fo−−
√Fo1Fo2
5.5 Frictional thermoelastic instability 219
+
j∑
i=0
{BiL1 BiL2
(2i + 1)!(2j − 2i + 2)!(αL
1 Fo−i2 Foi−j
1 + αL2 Fo−i
1 Foi−j2 )
+1
(2i)!(2j − 2i + 2)!(αL
1 BiL1 Fo−i2 Foi−j
1 + αL2 BiL2 Fo−i
1 Foi−j2 )
};
bj =BiL1 /R1 + BiL2 /R2
2 (2j)!Fo−+ +
BiL1 BiL22 (2j + 1)!
(Fo+
+
√Fo1
R2
+Fo+
−
√Fo2
R1
);
cj =Fo−−
2 (2j + 2)!
(√Fo1
Fo2
BiL1R2
+
√Fo2
Fo1
BiL2R1
)
+1
2 (2j + 1)!
(Fo+
+
R1
√Fo1
+Fo+
−
R2
√Fo2
); j = 0, 1, 2, . . .
Fo± =1√Fo1
± 1√Fo2
, Fo+± = ((Fo+)2j+1 ± (Fo−)2j+1),
Fo−− = ((Fo+)2j+2 − (Fo−)2j+2), Fo−+ = ((Fo+)2j + (Fo−)2j).
Owing to the form of the characteristic function, roots s1, s2 (in the vicinity ofthe origin) can be expressed in an analytical form. Namely, the following analyticalroots estimation holds
v1 =b0
a0 − ξ(b1 + c0),
v2,3 = b0
(a0 + ξ(b1 + c0) ∓
√ξ[a0(b1 + c0) − b0(a1 − ξ(c1 + b2))]
(a0 − ξ(b1 + c0))2 + 4ξb0(a1 − ξ(c1 + b2))
).
(5.18)
In Fig. 5.6 dashed curves correspond to variations of real and imaginary partsof roots s1, s2 computed analytically versus dimensionless velocity v. Three firstterms of the series (5.17) are included in the computations.
Solution properties. Relations defining v1, v2, and v3 enable prediction of thebehaviour of thermoelastic contact characteristics in the conditions of frictionalwear and heat expansion. As an example, the characteristics of thermoleastic con-tact for ϕu(τ) = H(τ) are considered.
In the initial time instant, a solution is taken in the form
p(τ) = 1 + v(ξ − ξ∗)τ, uw(τ) = vξτ,
θ(τ) =2v
(R1
√Fo1)−1 + (R2
√Fo2)−1
√τ
π, τ → 0,
220 5 Thermoelastic Contact of Two Moving Layers with Friction and Wear
where θ(τ) is the contact temperature (θ(τ) = θ1(L1, τ) = θ2(L1, τ)). For ξ < ξ∗(the wear value is lower than heat expansion), and for v < v2 time durability of thecontact τc = ∞, and the contact characteristics tend to stationary value pst = 0,θst(x) = 0, uw
st = 1 with time. The time interval to achieve the mentioned pointsincreases when v → v2. For v2 < v < v3 the time of the contact is bounded. Forv → v3 the maximum values of pressure, temperature, and wear increase.
When velocity v is greater than the critical value v3, both temperature andcontact pressure increase exponentially; that is, a frictional thermoelastic instabilityoccurs (the system does not keep pace with cooling down).
For ξ > ξ∗ (the wear value is greater than heat expansion) and for v < v2, thecontact characteristics also tend to the values of stable equilibrium positions. Forv > v2 the contact time is bounded, although a formally stable solution does exist.
Numerical analysis. Numerical analysis is carried out for the steel and aluminiumlayers for U0 = 10−6 m, Bi1 = Bi2 = 10.
In Fig. 5.7 time histories of contact pressure p(τ) and wear uw(τ) for variousdimensionless velocity v are reported (1, v = 6; 2, v = 8; 3, v = 10; 4, v = 14).Solid curves correspond to the value of ξ = 0.01. In this case v1 = 7.77 (v1 ≈7.84), v3 = 9.1 (v3 ≈ 9.04). Dashed curves correspond to the value of ξ = 0.1;that is, in this case v1 = 8.53 (v1 ≈ 9.4), v3 = 13.5 (v3 = 12.39). In the bracketsthe approximated values for certain characteristic velocities are given, and they arederived from relations (5.18). The numerical analysis confirms that for v > v3 africtional thermoelastic instability occurs (see solid curves 3 and dashed curves 4in Fig. 5.7).
Owing to velocity increase, time duration of the contact decreases, whereasmaximum contact pressure values, contact temperature, and wear increase. For theconstant velocity, an increase of the wear coefficient ξ causes a decrease of the
Fig. 5.7: Time history of the contact pressure p (a) and wear uw (b).
5.5 Frictional thermoelastic instability 221
contact time duration; that is, an intensive wear increase at the beginning occursand yields a decrease of the total amount of wear.
Conclusions. Finally, let us briefly summarize the results obtained in this chapter.The frictional contact of two layers with constant thickness has been investigated.Inertia of the contacting bodies has been included. Both friction and wear depen-dence on the contact temperature as well as the problem of drawing wear of twolayers have been examined. The proposed model can be used for computation ofthe contact characteristics during acceleration and braking processes.
Note that in the case of constant coefficients of friction and wear F (T ) = 1,K(T ) = 1, nonlinear functions take the form q1(τ) = q3(τ) = (v(τ) − v)p(τ) ≡q(τ), q2(τ) = 0. In the case when the layer moves at the constant velocity v(τ) =v = const, q(τ) = 0 and a solution of the corresponding linear problem isgoverned by the first term of series (5.13).
One-layer movement with constant velocity and with constant friction and wearhas been analysed earlier by [Grilitskiy (1996)], whereas a similar but nonlinearmodel has been studied in reference [Aleksandrov, Annakulova (1992)]. Also asimilar problem devoted to consideration of one layer with a constant frictioncoefficient and without (with) wear has been examined by [Pyryev (1994), (2000a)]([Olesiak, Pyryev (1998)]).
In addition, in this chapter the mechanism of occurrence of frictional thermo-elastic instability of the contacting bodies that move at a constant relative velocityis investigated. The proposed model can govern the behaviour of movable contact-type seals, braking pads, or other tribological systems. An equation yielding criticalvelocities is derived. It is illustrated how an account of wear yields an increase ofthe critical speed, and hence occurrence of the frictional thermoelastic instability.In the case when wear is greater than a heat expansion, frictional thermoelas-tic instability does not occur. It is worth noting that a frictional thermoelasticinstability (in the case of wear account) appears when the roots of the characteris-tic equation lie not only in the right half-plane of the Laplace transform variable(v > v1, ξ < ξ∗), but also in the real axis (v > v3).
The results obtained in this chapter can be used in computation of materialstrength of movable joints of machines and mechanisms, to formulate criteriarelated to optimal choice of materials of the frictional pairs, and to investigateheat transfer and wear in transporting mechanisms during braking or accelerationprocesses.
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Index
Amonton’sassumption, 25, 26, 32, 136, 172law, 189
angular velocity, xvii, 7, 31, 32, 78, 96, 97,106, 128, 129
Biot number, xv, 44, 90, 174boundary conditions, 5, 9, 21, 22, 138, 148,
163, 173, 190, 192, 207, 209brake, xii, xiii, 31, 97, 99, 125, 187, 188, 195,
198, 199, 205braking process, 81, 87, 97, 99, 105, 135,
154, 156, 187, 198, 203, 204, 207bush, xv–xvii, 31–33, 35–39, 41, 42, 46, 47,
49–54, 62–65, 67–70, 72, 74–79, 81,87, 88, 90, 92–98, 100–107, 110, 114,122–124
chaos, 1, 20, 29, 57, 58, 60–64, 67, 70–72,76, 77, 132
chaotic dynamics, xi, 20, 36, 53, 67characteristic equation, xvi, xvii, 41, 45–47,
52, 83, 85, 92–94, 96, 140–142, 150,152, 155, 156, 160, 167, 168, 175, 192,212, 213, 217, 221
coefficient of restitution, xv, 107, 120coefficient of viscosity resistance, xvColoumb friction, 13contact
characteristics, 22, 30, 51, 81, 85–87, 96,99, 124, 150, 153, 155, 156, 159, 160,168, 183, 187, 195–199, 203–205, 207,220, 221
pressure, xii, xvi, 9, 12, 14–16, 21–23, 136,138, 139, 145, 148, 150, 154, 155, 159,160, 162, 169, 170, 172, 173, 178, 183,184, 205–207, 209, 215, 216, 220
problem, 9, 21–25surface, xv, xvi, 11, 12, 21, 24, 25, 33, 124,
136, 142, 147, 162, 169, 173, 179–183,189, 192, 200, 207
temperature, xii, 13–16, 23, 52, 80, 81, 88,98, 102–104, 123, 144, 145, 154, 155,159, 160, 190, 191, 195–206, 209, 215,220
cylinder, x, xvi, xvii, 11, 14, 22, 23, 31–33,37–39, 46, 49–52, 68, 75, 76, 78, 85,87, 90, 95–97, 100, 101, 104, 106, 146
deterministic chaos, 1differential equations, x, 1, 9, 19, 25, 71, 113,
115–119, 125, 128, 130, 133, 179dimensionless
amplitude, xvii, 32, 54, 72, 79, 177angular velocity, xvii, 96, 101contact pressure, xvi, 53, 70, 80, 81, 88,
102, 124, 139, 158, 159, 175, 180–182,210
contact temperature, 53, 124, 154, 170displacement, xvi, 8, 49–51, 179–182frequency, xvii, 184moment, xv, 90parameters, 35, 38, 39, 108, 121, 127–129,
139, 148, 174, 191relative velocity, xvii, 56, 71, 179shaft position angle, xviitemperature, xvii, 96time, xvii, 36, 50, 51, 53, 69, 76, 78, 86,
87, 96–98, 101, 103, 119, 124, 129,153, 158, 170, 179–182, 198, 200, 202,204, 205
wear, xv, xvi, 49, 53, 70, 76–78, 101, 103,197
237
238 Index
displacement, ix, xvi, 2–6, 8–10, 14, 16, 23,33, 37, 39, 49, 51, 105, 107, 135, 162,169, 179, 181, 182, 191, 208
dry friction, xi, xii, xv, 12, 13, 18, 19, 25, 28,32, 126–128, 133, 135, 136, 170–172,175, 177, 183, 184
Duhamel–Neumann equations, 5, 8dynamics
2-dof damper, 135, 183analysed system, 108, 133bodies, ix, 5, 24, 25, 33body, 26bush, 98considered system, 100contact characteristics, 85contact systems, 30contacting bodies, xcylinder, 100damper, xii, 170, 183, 184dynamical damper, 184equations, 5feed motion, 11friction force, 50, 51investigated system, xii, 171mechanical systems, 130model, 105, 125, 171modes, 23plate center, 137plate mass center, 147processes, 1rotating shaft, 54system, 1, 20, 54, 130, 135two bodies, 106wheeled vehicles, 13
equation of motion, 5, 6, 8, 36, 55, 107, 162,192, 200, 208
equations of thermoelasticity, 3, 8Euclidean space, 1, 2
Fourier, xvii, 59frequency, xvii, 18–20, 28, 30, 58, 60, 68,
101, 128, 139, 142, 143, 172, 173, 177friction, 1, 2, 9, 11–19, 21–32
coefficient, xv, 12, 13, 18, 23, 25, 26, 32,33, 49, 89, 107, 110, 136, 145–147,153, 155, 156, 159, 160, 162, 168, 172,187–190, 196, 198, 204, 205, 215, 221
force, xvi, 11–14, 18, 19, 22, 25–27, 29,32–34, 37, 39, 49–51, 95, 99, 101–105,107, 126–130, 133, 136, 138, 147, 154,156, 158–162, 168, 171, 172, 174, 187,189, 191, 207
model, 28, 29, 78, 126–131, 133, 172, 174
pair, 31, 188frictional
autovibrations, 160braking process, 197characteristics, 84coefficient, 12, 195contact, 14, 17, 20, 24, 99, 147, 162, 187,
221contact characteristics, 84energy, 16force, 32, 160, 217heat generation, 25, 79, 81, 87, 90, 92, 94,
95, 99, 100, 135, 167, 187heating, 23model, 147, 162pair, 194, 198, 205, 221process, 13, 170self-excited, 135, 160self-oscillations, 47, 52self-vibrations, 135surface, 11–13, 188, 189TEI, 87, 93, 99thermoelastic instability, xiii, 15, 23, 30,
44, 153, 215, 220, 221wear, 219work, 33
heatexpansion, 1, 46, 48–52, 99, 100, 122, 123,
143, 159, 184, 219–221frictional dynamics, 187, 207generation, 1, 15, 21–24, 123, 160, 169,
174, 179–183, 207transfer, xvi, xvii, 31, 44, 54, 65, 78, 88,
99, 104, 106, 123, 136, 138, 139, 147,148, 163, 169–173, 178, 184, 188, 190,191, 207, 221
initial conditions, 1, 20, 35–37, 39, 40, 79,97, 101, 107, 109, 138, 144, 149, 163,164, 168, 173, 174, 191, 209
initial time, 32, 78, 84, 96, 100, 101, 162, 219integral equation, 1, 9, 21, 41, 65, 75, 83, 89,
100, 109, 125, 164, 173, 175, 179, 193
kineticfriction, 9, 12, 18, 25, 27, 32, 110, 136
Lame, xvii, 3Lamb problem, 8Laplace, xvi, 3, 40, 45, 52, 82, 84, 89, 92,
109, 125, 140, 149, 150, 152, 164, 167,168, 175, 192, 211, 221
material, 1
Index 239
mathematicalmodel, ix, 8, 14, 105, 129, 137, 139, 169,
170, 174modelling, x, 25, 27, 28, 126
mechanical boundary conditions, 5, 9, 208Melnikov
criterion, 20function, xi, xvi, 20, 53–59, 61–64, 71,
76–78method, ix, xi, 20, 30, 54, 71, 76rule, 61technique, 53theory, 54, 55, 58, 62, 72, 76, 77
method, ix, x, 1, 9, 12, 15, 17–20, 25, 28–30,40, 54, 65, 105, 125–127, 129, 130,153, 170, 183, 213
models, 2, 3, 11, 13–16, 18–21, 23, 24, 26,27, 127, 171, 172, 187, 188, 198, 199,207, 215, 221
of friction, 24moment of inertia, xv, 93, 100, 128
Newtonassumptions, 147condition, 2
Newton’sheat exchange, 32law, xiii, xv, 106, 137, 162, 172, 207rule, 33
nonlinear, 9, 17, 20, 31, 40, 47, 83, 117, 188,213, 217, 221
algebraic equations, 111differential equation, 41, 65, 89, 100, 125,
175dynamic systems, 54dynamics, 64, 106equation, 90, 111, 112, 114, 117, 118, 143,
151, 153, 165, 167, 171, 199equations of the motion, 4integral equations, 193kinetic friction, 106model, 17, 221oscillations, 170problem, 5, 30, 85, 109, 142, 164, 171, 183self-excited vibrations, 145stiffness coefficient, 36system, 20, 63, 71, 75terms, 168vibrations, 17
nonsmooth, 20, 25, 29, 65, 125nonstationary, x, xiii, 1, 19, 21, 24, 142, 153,
155, 168, 169, 187, 190, 215normal stresses, 208, 209
numerical analysis, ix, xiii, 9, 25, 48, 53, 60,61, 63, 67, 76, 79, 87, 95, 121, 143,144, 150, 155, 158, 159, 169, 179, 182,194, 198, 202, 206, 215, 220
ordinary differential equations, 28, 89, 126,127, 129, 130
period, xi, xvi, 15, 19, 20, 110, 111, 113,116–118, 120, 136, 155–159, 168, 173,175, 178, 180, 181, 190
periodicattractor, 180behaviours, 133change, 123dynamics, xi, 125function, 19manner, 122motion, x, xi, 19, 44, 58, 60, 63, 68, 87,
94, 95, 104, 110, 111, 113, 116, 118,120, 121, 125, 182
occurrence, 99orbit, xi, 110, 120, 121, 123–125, 159oscillations, 88, 130, 157, 159solutions, 1, 30, 94, 127, 130, 131state increases, 182trajectories, 125vibrations, 19, 21
perturbation, xvi, xvii, 9, 19–21, 23, 30, 44,54, 55, 65, 66, 78, 90, 91, 129, 142,145, 147, 152, 166–168
Poisson’s ratios, xvii, 209problem, 1, 5, 8, 9, 11, 12, 14, 15, 17, 18,
20–26, 30, 31, 33, 36, 39–41, 43,46–48, 52, 65, 75, 79, 81, 82, 84, 86,89, 90, 92, 95, 100, 101
Pronny’s brake, 31
quasi-static, ix, 8, 36, 107, 208, 217
radius, xvi, 31, 38, 93relative
displacement, 6, 9, 136, 176error, 194motion, 6, 7, 18, 26, 27pressure, 86, 87rest, 18, 26slip, 18slip velocity, 129speed, 99temperature, 87velocity, xi–xiii, xv, xvi, 16, 18, 25, 27, 28,
32, 33, 44, 49, 54, 56, 65, 71, 79, 86,87, 89, 96, 100, 101, 103, 107, 126,
240 Index
127, 129, 130, 136, 145, 147, 160, 162,172, 180, 182, 183, 216, 221
restitution coefficient, 113, 119, 125Runge–Kutta method, 9, 48, 66, 95, 101, 110,
130, 143, 153, 179
shaft, xv–xvii, 31–33, 36, 39, 41, 42, 44,53, 54, 64, 65, 67–70, 78, 81, 82, 84,86–88, 90, 93, 94, 96–107, 122–124
signum model, 26, 27, 29, 126, 127, 129, 130sliding
bodies, 23, 147, 160, 162, 216contact, 125objects, 12phase, 103speed, 12surfaces, 126, 181, 207velocity, 103, 104, 125, 126, 130, 131, 136,
169, 179–182, 184, 195, 196, 198, 199,206, 207, 215, 216
springs, 31, 36, 41, 88, 93, 100, 106, 125, 135static
conditions, 146equations, 126friction, 12, 18, 27, 161friction coefficient, 25friction force, 14, 26, 27, 49friction point, 26, 27kinematic friction coefficients, 160moment, 128solution, 142, 169
stationary solution, xii, 22, 43–45, 85, 90, 92,99, 100, 141, 143, 151–153, 156, 157,160, 165, 166
stick-slip, 1, 12, 14, 17, 18, 30, 52, 54, 75,76, 94, 96, 99, 100, 104, 106, 110, 116,118, 120, 127, 130, 133, 155, 158–161,169, 180
stress, ix, xvii, 2–4, 8–10, 12, 22, 33, 36, 37,52, 107, 148, 162, 163, 173, 208, 209
Stribeckapproximation, 94curve, 13, 32, 47friction model, 99model, x
switch model, 28, 29, 126–131, 133system of differential equations, 3, 208
thermalboundary conditions, 37, 107, 109, 209conduction, x, 9, 21conductivity, xvii, 37, 148, 163, 173, 209constants, xcontact, 2, 22
contact conditions, 16diffusion, 138diffusivity, xv, 3, 37, 65, 148, 163, 173, 209distortion, 22, 23distortivity, 90effects, 171expansion, x, xvii, 65, 70, 81, 99, 124, 142,
148, 163, 173, 209expansion coefficient, 3, 37extension, 68, 70, 87, 135field, ixinertia, 174loading, 217loads, 3, 8, 17plate extension, xiproblem, xiiprocesses, 3regime, xiiresistance, 23, 209shaft, 67stresses, x, xii, 1, 36, 107, 137, 171, 172,
207thermoelastic
bodies, 3, 5, 30contact, xi, xiii, 22, 31, 52, 64, 84, 135,
136, 144, 147, 160, 161, 170, 183, 207,219
cylinder, x, 14dynamic instability, 23, 217half, 23instability, 23, 51, 136, 142, 217layer, 15, 23plate, 24problem, 38shaft, x, 81, 106
transfers heat, 32, 33, 87tribomechanical system, 15, 18, 187two degrees-of-freedom system, 8, 18, 30,
135, 160, 161, 169
uncoupled thermoelasticity, 3, 208
velocity, ix, xi–xiii, 18, 19, 22, 23, 25, 26,28, 36, 41, 42, 50–54, 65, 70, 75, 81,82, 84, 88, 90, 92, 93, 96, 97, 102–104,107, 109, 126, 135–138, 142, 147, 150,153, 154, 156–162, 164, 167–169, 178,179, 183, 184, 188, 189, 204–208, 210,220, 221
wear, ix–xiii, xvi, 1, 9, 14, 15, 22–25, 30,31, 107, 109, 122–124, 183–185, 187,189–191, 194–196, 198–210, 213,215–217, 220, 221
Index 241
coefficient, xii, xv, 65, 74, 81, 87, 102, 104,105, 184, 185, 188, 191, 199, 203, 207,209, 214, 215, 217, 220
evolution, 202kinematics, 123, 124model, 24parameters, 184
process, ix, 14, 70, 100, 122, 184, 187Winkler
conditions, 2model, 14
Young’s modulus, xv, 37, 209