yevtushenko (2010) the fem-modeling of the frictional heating phenomenon in the pad-disc tribosystem...

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Numerical Heat Transfer, Part A:Applications: An International Journal ofComputation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unht20

The FEM-Modeling of the FrictionalHeating Phenomenon in the Pad/DiscTribosystem (A Review)A. A. Yevtushenko a & P. Grzes aa Faculty of Mechanical Engineering , Bialystok University ofTechnology (BUT) , Bialystok, PolandPublished online: 23 Aug 2010.

To cite this article: A. A. Yevtushenko & P. Grzes (2010) The FEM-Modeling of the FrictionalHeating Phenomenon in the Pad/Disc Tribosystem (A Review), Numerical Heat Transfer, Part A:Applications: An International Journal of Computation and Methodology, 58:3, 207-226, DOI:10.1080/10407782.2010.497312

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THE FEM-MODELING OF THE FRICTIONAL HEATINGPHENOMENON IN THE PAD/DISC TRIBOSYSTEM(A REVIEW)

A. A. Yevtushenko and P. GrzesFaculty of Mechanical Engineering, Bialystok University of Technology(BUT), Bialystok, Poland

This article is concerned with frictional heating phenomenon playing a central role in disc

brake (clutch) systems performance. It was of great interest to carry the description of

existing numerical solutions of the problem. The level of the calculations was categorized

into three problems. The rst is a common approach, drawn primarily on the assumption

that the heat ow rate generated at the disc/pad interface is a known quantity. Heat gen-

eration was substituted by the intensity of heat ux depending on contact pressure, coef-

cient of friction and evolution of angular velocity, and radius of the disc product. A

more complex description of the braking action was given by the assumption of mutual

relation between mating parts of the disc brake system. In this case, the thermoelasticity

theory was applied to the model. The facts of existing critical speed is established, in which

hot spots caused mainly by contact pressure variations frictionally-excited thermoelastic

instability phenomenon entered commonly into consideration.

1. INTRODUCTION

Disc brakes are apparatus used to stop, decelerate, or to enable the motion of avehicle. The increase of friction moment is limited by the diameter of contact surface,coefcient of friction, pressure per unit area, and cover angle of pad. Nevertheless,the peak temperature value is one of the most crucial factors of the braking process.When the braking process occurs, the kinetic energy is converted into heat energyand parameters of the process (velocity, contact pressure, coefcient of friction,and thermophysical properties of materials) vary with time. The operation of thetemperature above a certain critical value may cause undesirable effects such asbrake fade, low frequency vibrations, thermal cracks, and premature wear. Predic-tion of temperature distribution during the braking process can be obtained by usingexperimental tests or theoretical calculations. Reliable outcomes of the experimentalinvestigations are the validation of numerical solutions. However, special effort toperform identical conditions of the operation is required. An accuracy of determin-ing peak value of temperature at the initial stage of the design process of a specied

Received 9 April 2010; accepted 12 May 2010.

Address correspondence to A. A. Yevtushenko, Faculty of Mechanical Engineering, Technical

University of Bialystok (BTU), 45C Wiejska Street, Bialystok 15-351, Poland. E-mail: a.yevtushenko@

pb.edu.pl

Numerical Heat Transfer, Part A, 58: 207226, 2010

Copyright # Taylor & Francis Group, LLCISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2010.497312

207

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brake system facilitates to avoid unwanted shortcoming. Analytical calculationsconsidering frictional heating systems require semi-spaces or plane parallel stripand therefore, application of a nite region is unfeasible. Hence, it is extremely dif-cult to obtain an exact solution of the problem, which is an impact of complexgeometry of the disc brake system. Reviews of analytical methods of the solutionof thermal problems of friction during braking are given in the monographs [13]and in articles [4, 5]. Owing this, the use of the nite-element method (FEM) has

NOMENCLATURE

c specic heat, J=kgK

[CT] capacity matrix

[D] elasticity matrix

d dimension, m

f coefcient of friction

h heat transfer coefcient, W=m2K

k thermal diffusivity, m2=s

K thermal conductivity, W=mK

[K] stiffness matrix

[KHT] conductivity matrix

p contact pressure, MPa

{Pf} body force vector

{Ps} surface traction vector

{PDT} thermal load vector

q intensity of heat ux, W=m2

q0 initial intensity of heat ux, W=m2

r radial coordinate, m

{R} heat source vector

f~RRg Fourier integral transform of heat sourcevector with respect to the h coordinate

t time, s

ts braking time, s

T temperature, CT temperature at the initial moment of

time, C

T1 ambient temperature, C{T} temperature vector

f~TTg Fourier integral transform oftemperature vector with respect to the

h coordinateu displacement, m

{U} vector of nodal displacements

V linear velocity at any instant, m=s

z axial coordinate, m

{a} thermal expansion coefcient vector

c heat partition ratio{e} strain vector

{e0} initial strain vector

h circumferential coordinate, radq density, kg=m3

r stress, MPa{r} stress vector

rz normal stresses on the disc=padinterface, MPa

x angular velocity, 1=s

Indexes

z z coordinate

d disc

p pad

Figure 1. Feedback process for thermoelastic instability phenomenon [54].

208 A. A. YEVTUSHENKO AND P. GRZES

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made a signicant contribution to the area of the heat dissipation problem in discbrakes or dry=wet clutches. However, it must be noted that in the analytical ornumerical approximation of the problem, simplication of the process may falsifythe results and a cautious conclusion is necessary.

In this article, the review of FEM applications of frictional heating phenom-enon concerning disc brakes and clutches systems is presented. The aim was tocompare different techniques of temperature estimation developed up to now withreference to two- and three-dimensional representation of the models. Thermalanalysis, thermoelasticity contact problem, and frictionally-excited thermoelasticinstability (TEI) phenomenon were illustrated individually.

2. HEAT CONDUCTION PROBLEM

The work done during the braking process or after clutch engagementproduces heat energy due to the rate of deceleration. Estimation of temperaturedistribution in the operation may provide an important information of the mutualrelationship between friction, wear, and temperature.

The temperature elds and the thermal stresses in the discs of a multi-disc wetclutch after engagement were investigated by Zagrodzki [6]. An assumption wasmade that the interface conditions are independent of the temperature. Thermaland mechanical phenomenon in each disc were stated to be symmetric in referenceto the discs midplanes. The contact pressure was uniformly distributed over therubbing surfaces and established as independent of thermoelastic deformations offriction materials [7]. The axisymmetric model due to the intensity of heat ux uni-formly distributed in the circumference and an axial symmetry of friction materialwas proposed. An assumption was made that the intensity of heat ux distributedinto friction surfaces with respect to radial position of discs is a known quantityequal to the specic power of friction [8].

qr; t fpxtr; 0 t ts 1

The transient heat conduction boundary-value problem for parabolic equation,

q2Tqr2

1r

qTqr

q2T

qz2 1k

qTqt

2

was solved numerically using the nite-difference method. In order to obtain rstorder heat transfer differential equations, the standard Crank-Nicolson methodwas employed [9]. The corresponding uncoupled quasi-static thermoelastic problemwas accomplished using FEM, where triangular cross-sectioned axisymmetric ele-ments with appropriate division of grid have been used. Two various materials ofthe disc were considered. The discs with a steel core, lined with sintered bronze usedas friction material layers were submitted to complex analysis, including thermalstresses due to frequent service failure. The discs, entirely made of steel, were exclus-ively used in the heat conduction analysis.

FRICTIONAL HEATING IN THE PAD/DISC TRIBOSYSTEM 209

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Disc brakes are subjected to high thermal loads during emergency brakingaction, drag braking, and repetitive cycles of operation. Multiple brake engagement,especially, without a period of forced or natural cooling may evoke undesirableeffects between mating parts of the disc brake system. Variations of temperaturesabove critical, acceptable value lead to thermal deformations, low frequency vibra-tions, uctuation of friction coefcient, and brake fade. Temperature distributionsduring repeated brake cycle were determined by Ramachandra Rao et al. [10].Material properties were assumed to be independent, on temperature. The startingpoint of the analysis was the parabolic heat conduction equation (Eq. 2). The heatpartition ratio imposed division of heat entering the disc and pad, respectively,and was given as follows [11].

c Kpqpcp

pKpqpcp

p Kdqdcdp 3The amount of heat was calculated from the energy balance relation, where

total kinetic energy is assumed to be converted into heat. Prevailing heat wasattached to the disc material; thus, pad volume was neglected for the analysis.Two cases of computations were developed using conventional methods of heatinput and the clock mechanism technique. In the conventional analysis, kineticenergy was calculated at each time step resulting in temperature distribution in a discarea. Additionally, two periods in terms of this calculations were distinguished: brak-ing on with load of frictional heating and braking off time, where cooling by convec-tion and radiation was assumed. In the second case of the analysis, braking actionwas divided with reference to periodic pad activity. In the time of pad presence, heatentered the disc. After, heating was switched off and cooling, due to convection, wasestablished. Hence, heating and cooling periods were included in braking actionopposite the rst step of the proposed conventional time stepping procedure. Thepresented clock mechanism provides relatively detailed description of disc brakeoperation in frictional heating terms. For validation purposes of the proposednite-element analysis, experimental tests were conducted. A sophisticated double-ended micro processor with rubbing type of thermocouple measured temperaturedistributions. Tips of the thermocouples tted at both sides of the brake at meanradius of the rubbing path.

The disc brake system is designed for determined operation conditions. In thecourse of brake action, both the ability to conduct heat corresponding to thermaldiffusivity and specic heat capacity of the given material, and dimensions of the discbrake with its special observation to mass advantages should be considered. Castiron disc brakes are widely used in passenger vehicles for numerous advantages,including relatively low costs, ability to resist cracking, and its high thermal conduc-tivity. However, drawbacks such as susceptibility to corrosion or weight featuresrelated to lightweight metal matrix composites are noticeable. Grieve et al. [12]compared the application of different materials, namely cast iron and aluminium,in a front disc brake system of a typical small-medium passenger car. Thermalnite-element analysis was developed as an efcient method in terms of peak tempe-rature estimation using different materials where chemical composition, dimensionsof a brake system, and thermophysical and strength properties were distinguished. A


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Taguchi technique [13] was applied to estimate major critical factors of suitabledesign of lightweight components of automotive disc brake systems. Among differ-ent materials used in brake rotors, especially aluminium metal matrix composite(Al-MMC), received attention. Three separate tests were conducted. First, a studyof vehicle mass in different simulation cases of brake operation, Alpine descent,and emergency high-speed autobahn stop. Second, was to nd the most inuentialfactors to be considered for the design of a prototype Al-MMC disc brake. Finally,was detailed parametric study to examine the effect of the disc thickness and ventswidth.

Finite-element analysis has proven to be one of the most efcient approachesto solve the problem of frictional heating in sliding systems. Nevertheless disadvan-tages have presence considering the transient character of the braking problem. Ahigh Peclet number imposes irrational mesh division of the modeled parts of the sys-tem. As a result, large computer storage is indispensable. The complex procedure toavoid the above shortcomings was proposed by Kennedy et al. [14]. The authorsextended a nite-element program developed earlier to handle transient cases andtemperature-dependent thermal properties of materials. A value of 0.5 or slightlyhigher for a weight parameter of the Crank-Nicolson method previously recom-mended as relevant for maximum solution accuracy, was validated in this article.The nite-element method for temperature assessment due to frictional heatingwas taken into consideration for one- and two-dimensional objects. Revision of thesein order to avoid redundant perturbations was developed. An attempt to solve pre-viously unsolved problems in heat generation of slipping bodies was made. Severaltechniques of two-dimensional studies were examined by Carslaw [15], Heinrich[16], Baliga [17], Finn [18], ONeill [19], Brooks [20], and Kanarachos [21].

The disc brake is subjected to nonaxisymmetric thermal load caused by rota-tional motion of the disc corresponding to xed pads. The exchange of large amountsof kinetic energy into thermal energy imposes nonuniform temperature distribution,which is rmly noticeable above a specic value of angular velocity of disc brake.When the movable lining material exceeds a Peclet number of Pe 2, oscillationsmay occur. However, for a detailed description of the heat generation phenomenon,the disc brake model is susceptible to be considered in three-dimensionality. In thecourse of spatial usage of the disc brake system, simplication methods have also beenemployed to examine temperature elds variable in the circumference of the contactsurface. It is known that simplication of the intensity of heat ux assumed to be uni-form in the circumferential direction for time shortage unfortunately yields to incor-rect outcomes. The nite-element method (FEM) is very well adapted for xed parts,but three-dimensional FE modeling necessitates a more exacting approach. For thatreason, Floquet and Duborg [22] proposed new method to attain three-dimensionaldistribution of temperature with a moving heat source and pad inuence. Athree-dimensional model with transient nonaxisymmetric operation of the heat uxdue to variable velocity and spatial effects with signicant pad=disc interface con-ditions of thermoelastic behavior was developed. In order to obtain temperature dis-tribution, fast Fourier transform technique (FFT) utilizing the nite-element methodwith sufciently short computational time was applied. Three steps of the analysiswere conducted. First, was a focus on three-dimensional transient analysis with vari-able velocity effects. The second step included complex thermoelastic analysis with


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the presence of a pad. Eventually, the third step was comprised of a thermal crazinginvestigation. The heat conduction equation was given as follows.

q2Tqr2

1r

qTqr

q2T

qz2 1r2q2T

qh2 1kd

qTqt

x qTqh

4

Both convection and radiation were taken into account corresponding to theheat transfer coefcient h, which remained constant during each time step. The heatux was constant in the radial direction, and variable versus the circumferentialcoordinate and time. Authors concluded that FFT-FEM is 225 time faster thanthe classical nite-element method using a fully three-dimensional model. Therefore,FEM appeared unsuitable in the case under examination. In order to validate theabove technique, simple models with different boundary conditions comprising velo-city variety and nonaxisymmetric effects of heat ux were tested. Validation casesalso had analytical solutions extracted.

Later [23], the same authors extended the FFT-FEM model [22] to determinethe temperature eld in a ventilated disc brake assembly, where geometry is spatiallyperiodic but nonaxisymmetric. A three-dimensional numerical method for the pre-diction of transient temperatures reached during realistic braking operations in aventilated disc brake assembly is presented in this article. This layered FFT-FEMnumerical method takes advantage of the high Peclet number skin effect to decouplethe moving boundary contact problem from the spatially periodic but nonaxisym-metric cooling associated with the vents. Validation tests focusing on the continuityand the accuracy of the temperature variations at the interface between the FFT-FEM and 3-D nite-element methods are presented for the layered FFT-FEMmethod. Comparisons with analytical calculations show very good agreement. Therst problem is modeled according to the FFT-FEM method developed previouslyby the authors [22]. This technique succeeded in using high Peclet numbers foraxisymmetric moving solids submitted to general loading conditions. No numericalconvergence problems were encountered, and relatively short computer time wasrequired. The second problem is modeled according to a traditional three-dimensional nite-element technique. The layered FFT-FEM results were comparedwith the analytical solution. Different convection conditions on hub cap tracks,vents, and elsewhere were established. It was assumed that the properties of materi-als vary with temperature. The disc was divided into three parts: two hub caps anda central part.

A hybrid scheme that combines the fast Fourier transform technique andnite-element method is an efcient method of solving an axisymmetric object sub-jected to nonaxisymmetric load [22, 23]. Although, inverse Fourier transform of theproblem ought to be used. Numerical solution at each frequency of time domain iswell adopted in thermal problems, whereas unfavorable features of this method arenoticeable in coupled thermoelastic analysis. The nonaxisymmetric arrangement ofthe disc brake system in transient nite-element analysis was used by Gao and Lin[24]. Nonlinear operation conditions, including rotating speed of a disc andevolution of the contact pressure for more realistic simulation, were assumed.

Complex behavior of the frictional pair of a disc brake system necessitatesspecial observation of the experimental led. Qi and Day [25] carried out analysis


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of temperature expansion phenomenon due to frictional heating in disc brakes. Thetechnique of the exposed thermocouple interface temperature measurement was usedin the investigation of the characteristics that inuence temperature rise in theimposed conditions. The above procedure of the temperature distribution assessmentis an efcient method developed previously [26]. Additionally, convectional embed-ded thermocouple technique commonly used to examine disc temperature wasemployed to confront system automatic control and failure diagnosis. For a simula-tion cause of the experimental results, the nite-element modeling technique was pro-posed as a relevant method of pad=disc interface temperature assessment. Theauthors focused on phenomenon of the real and apparent area of contact betweentwo sliding bodies due to topography of the friction surface. An attempt was madeto determine the range of the affection on its performance, including temperaturegrowth and wear presence. Statistical design of experiments (DoE) was used indetailed analysis of the characteristics in the braking process. A four-factor,two-level full factorial orthogonal design was conducted. Two different frictionalmaterials were selected. New and used pads have been considered in the analysis.The major outputs of the DoE were temperature and the coefcient of friction.

The heat transfer problem in a ventilated disc brake with straight radialrounded vanes was investigated by Talati and Jalalifar [27]. Two models of frictionalheatingmacroscopic and microscopicwere examined. In the macroscopic model,the rst law of thermodynamics was taken into account, and for the microscopicmodel, various characteristics such as duration of braking, thermophysical proper-ties of materials, and dimensions of a disc brake system have been studied. Both discand pad volume were investigated separately to evaluate temperature distributions.The conduction heat transfer was investigated using FEM [27]. Two cases of contactpressure distribution were considered: uniform pressure and uniform wear. Experi-mental correlations and numerical computation results [29] were used for thedescription of the heat convection trough considered a type of SRV-R rotor. Theresults of the numerical analysis were also veried by experimental investigations.Problem of frictional heating in a solid disc brake assuming uniformity of the inten-sity of heat ux was solved analytically using Greens function approach by Talatiand Jalalifar [28].

3. THERMOELASTICITY

In order to gure out acceptable outcomes of a frictional heating problem,both thermal and elastic problems should be considered. Typically, coupled analysisof disc brakes and clutches aim to simulate deformation of contacting bodies causedby high thermal loads. Nonuniformity of contact pressure distribution over the rub-bing surface both in circumferential and radial direction is unavoidable. Such aphenomenon may be evoked at the initial moment of the engagement action, owingdifferent characters of the piston and nger side of the caliper. Even if the initial dis-similarities between the pressure distributions are relatively insignicant, they mayunfortunately become noticeable when the thermal deformations appear.

The energy transformation during the braking process using FEM was studiedby Day and Newcomb [30]. Disc brake performance and temperature analysis havebeen taken into account. Contact pressure and temperature distribution with respect


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to wear increment was calculated. It was concluded, that wear debris generatedunder braking conditions have a signicant effect upon heat transfer from the inter-face. However, several assumptions of real behavior of two sliding bodies in contacthave been neglected, including chemical reaction energy interchange. Contact press-ure, temperature, and wear distributions obtained in the numerical analysis wereconfronted with observed and measured experimental outcomes from an annular testrig. The pressure was assumed to remain constant during braking. The intensity ofheat ux depended on the product of velocity and interface pressure at eachincrement time of the transient analysis. The analysis of contact pressure was per-formed using the gap force method. Two parts of the friction pair were modeled sep-arately. Connection between individual nite element of the friction interface wasdeveloped by equal and nodal gap forces. The gap forces were calculated using amethod of deection coefcients in the iterative solution of the dynamic interfacepressure. In the normal direction, only compressive forces are enabled and frictionalforces are adapted in the tangential directions computed according to Amontonslaw. The coefcient of friction was assumed to remain constant during analysis.The annular disc has been modeled in a two-dimensional axisymmetric congur-ation. The contact conditions of the mating pair were determined at each time stepof the analysis by the amount of wear estimated from nodal temperature and press-ure values [31]. The convective terms of heat transfer were assumed on the free sur-faces of the model using a constant value of 100W=(m2K) of the heat transfercoefcient h derived from empirical correlations. The sum of the intensities of heatuxes into the pad and the disc are given by the following formula.

qt q0 1 tts

; 0 t ts 5

Assuming equal temperatures between surfaces in contact, the heat partitionfactor c was obtained from Eq. (3). First approximation of the surface temperaturerise was obtained from the Fazekas [32] known formula.

T t 2q0kpt

pKp

p

p 1 2t3ts

; 0 t ts 6

and for longer periods of braking time, the following form of equation for steel discbrake should be used

T t q0kdKdd

t 1 t2ts

d

2

3kd1 t

ts

d

4

45kd2ts

; 0 t ts 7

This equation omits inuence of heat loss and any effect of interface contactresistance.

Transient thermomechanical behavior of multidisc wet clutches and brakes toanswer the question if the assumptions concerning the uniform distribution of nor-mal pressure is valid and when it is reasonable was examined by Zagrodzki [33]. Thephenomenon of wear in short periods of clutch (brake) engagement and=or lubri-cated environment is not able to vary signicantly contact conditions. Hence, the


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inuence of wear presence has been neglected. This study established simplicationof the model, which evokes axisymmetric elds of temperature distributions. Linearheat conduction was prescribed in each subregion. Including iterative order of thegoverning heat transfer Eq. (2) for the ith subregion was solved. The boundary con-ditions have been specied with respect to Newtons cooling law to sequential sub-regions. The heat ux qij in the contact region S

qij was an unknown variable, and

is dened as follows.

KiqTiqz

Sqij

hT1 TijShi 8

where the heat transfer coefcient h varies including different fragments of the sur-face. It was essential to establish two additional conditions in the contact area. First,was the condition of equality of temperatures.

TijSqij

TjSqji

9

Second, was the law of the energy conservation.

TijSqij TjSqji

fpijt; rV 10

where pij is the contact pressure, depending on the radial coordinate r and on thetime t. On the free surfaces S

hi , the following conditions have been assumed.

KiqTiqn

Shi

hT1 TijShi 11

At the initial moment of time, the temperature was equal.

Ti0; r; z T0i r; z 12In order to determine temperature distribution of this transient heat conduc-

tion problem, the ne mesh element was essential. Moreover, when the iterativemethod of the given problem is employed, then a relatively short time is neededfor the calculations. The real regions of the piston and plate were substituted by rec-tangles, which facilitate adoption of the nite-difference method. In the next step ofthe study, the Crank-Nicolson method was selected as an unconditionally stablescheme. To solve the nite-difference equations, the Gauss-Seidel stationary iterativemethod was used.

In the mechanical system it is assumed that each subregion is linearly elasticand the elastic phenomena was considered as a static case. Two types of forces affectsystem: the hydraulic pressure acting on pads and frictional stresses resulting fromthe rotation of the disc. The axial component of displacements on the contact surfacewas equal

uz;iSqij

uz;jSqji

13


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at the condition that the normal stresses rz on the interface are compressive.

rzjSqij pij < 0 14

Three levels of the disc brake thermal behavior can be distinguished; namely,bulk, macro and micro [34,35]. At the beginning of the disc brake design process,bulk examination is the most suitable and efcient method of temperature predictionin terms of general thermal dissipation. However, macroscopic scale helps to deter-mine more sophisticated effects including thermal effects, related specically to hotspots, banding, and possible rotor surface damage. The railway disc brake systemwith reference to thermal loads as a dominant in this type of brake systems studywas analyzed by Tirovic [36]. Ventilated disc brake area in the section of 7.5 atthe circumference containing half of the vane thickness and half of the disc areabetween vanes was examined without pad presence. Such a simplication, was anissue of generation appropriate results with relatively low time coast. The materialproperties were assumed to be dependent on temperature. Plastic deformation havebeen neglected. However, this study aimed to be preliminary to the subsequentinvestigations addressing macro effects on the rubbing surface. Careful examinationby comparing the stress and temperature outcomes with the yield stress as a functionof temperature was referred to be essential in the course of the considered brakingprocess.

Powerful thermal loads upon repeated braking action with, high rate of decel-eration may be revealed in areas of signicant plastic strain accumulation of discbrake material. The thermomechanical behavior of a ventilated disc brake of a pass-enger car was examined by Koetniyom et al. [37]. For convenience,temperature-dependent material properties excluding density were derived fromexperimental investigations on samples cut from real cast iron disc. Special effortwas made in the application of yield properties of cast iron with distinction in tensionand compression state. The cyclic strain-stress response of the cast iron at differentvalues of temperature was examined. The authors introduced a back-vented highcarbon cast iron disc (air ow passes mainly from the inner side of the rotor)developed by Rover Cars. This type of rotor was compared with the conventionaltype, where thermal cracks were potentially more susceptible to occur. One sectorof the disc brake was analyzed, represented by a unique shape repeated in the cir-cumference. Calculations of the heat ux amount was based on the energy transferconsidering both mass and initial velocity of the vehicle. Moreover, correction fac-tors due to unequal distribution of the effort at the front and rear axle and the energyloss due to aerodynamic drag on the vehicle were employed. Both convection andradiation have been considered in the analysis. The heat transfer coefcient wasadopted from Grieve et al. [12].

Prediction of temperature rise is the main problem in the course of brakingoperation, including material degradation such as wear, phase transformation, ther-mal cracks, etc. Disc brake systems in the aircraft application are subjected to suf-ciently high temperatures in the range of 7001500C, due to extremely largeamounts of energy transformation in a relatively short time. Contravention ofthese may affect adjacent elements of the brake system: the tire zone near the hotspots, bearings, brake uid vaporization, hydraulic cylinder, and piston. Therefore,


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importance of temperature determination is indisputable. An aircraft landing gearsubassembly was examined by Wolejsza et al. [38]. The coupled thermomechanicalanalysis of the wheel-brake assembly was developed. The authors used a multi-faceted approach, both the numerical solutions of the continuity model by meansof differential equations and results from FEM analysis were obtained and com-pared. The heat generated due to friction in the analysis was given as follows.

qr; t f tpr; trxt; 0 t ts 15The axisymmetric nite-element model was used to simulate the most importantphysical phenomenon without overloading computational resources. Both freeconvection and radiation at the outer surfaces of the assembly have been assumed.The value of friction coefcient and angular velocity used in the analysis was derivedfrom experimental data.

The transient coupled thermomechanical analysis of disc brake using FFT-FEM was performed by Cho and Ahn [39]. It was assumed that on the disc=padcontact zone the intensity of heat ux is prescribed due to frictional heating. Inthe region without frictional contact at the disc=pad interface side, the convectionboundary conditions took place. The outer side of the disc at the circuit wasinsulated; whereas, at the hub area of the disc temperature was specied. Amongthe variety of numerical techniques for solving of differential equation of parabolictype (4), the fast Fourier transform was selected considering its advantages ofefciently less computational time related to standard FEM. The FFT techniquereduces three-dimensional partial differential equations into two-dimensional inthe periodic domain of the disc including circumferential variety. In order to elimin-ate the second order of the partial differential equation, a weak form of theGalerkin-type technique was used. The matrix form of equation was obtained.

CT df~TTgdt

KHT f~TTg f~RRg 16

The Crank-Nicolson method due to time integration of the differentialequations system (16) was adopted. It should be noted that the pad had to beconsidered in a fully three-dimensional system. Hence, extracted Eq. (16) areappropriate only for the disc. Pads were supposed to be analyzed excluding FFTreduction.

The constitutive equation for an elastic solid under mechanical loading andthermal expression has the following form.

frg D feg fe0g 17The complex analysis of thermoelastic behavior as signicant in the estimation

of the disc brake or clutch temperature and pressure distribution was performed byZhaoa et al. [40]. In addition, the inuence of material properties has been analyzed.An axisymmetric FE model including carbon-carbon composite materials in thecomputations have been used. Both thermal and elastic problems were developedin coupled analysis. A heat conduction equation for each body were extracted in


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the cylindrical coordinate system. Using Galerkins approach, the governing heatconduction equation was described in the following matrix form.

CT dfTgdt

KHT fTg fRg 18

The solution of the rst order differential Eq. (18) was obtained using the mostcommon Crank-Nicolson method with a weight parameter, which ranges from 0.5 to1, and was chosen from the conditions of achievement of necessary accuracy of inte-gration and stable scheme.

In order to obtain the equation of equilibrium for coupled analysis includingthe thermal and elastic problems, the variational principle was used. The thermoelas-tic equations have the following matrix form.

KfUg fPf g fPrg fPDTg 19Complex phenomenon of thermomechanical behavior in disc brakes by using

three-dimensional macrostructural model in nite-element analysis was developedby Dufrenoy [41]. The distributions of the temperature as a main problem in thermalexpansion analysis was determined. The tribological actions such as wear, distor-tions, and coefcient of friction variations were considered. The highly transientnature of the problem was taken into account:

. Speed decreased linearly

. Thermal and mechanical material properties of the disc and pad varied withtemperature

. Value of thermal resistance was indirectly found from special experiment measure-ments and varied during analysis procedure

. Convection and radiation heat transfer on free surfaces took place

The temperature distribution in the disc and pads were calculated separately.The disc has been made of 28CrMoV5 08 steel manufactured by a forging process.The pad material was sintered Fe-CuSn metal matrix composite reinforced byceramic particles. Numerical and experimental results were compared. For thispurpose, the thermocouple and thermographs techniques were employed.

Assuming a wide variety of operating characteristics and geometrical dimen-sions of two bodies in sliding contact, the temperature of the interface and contactpressure becomes the most important factors of its performance. The complexnature of the frictional systems behavior imposes special concern of mutualinteraction of thermomechanical behavior. This is achieved by using optimization(multilevel coordinate searchMCS) and two level ordinary design of experiments(DoE) [42]. In the physical model, a global approach was made in order to assessthe general inuence of heat generation in sliding systems (drum brakes, discbrakes, clutches) instead of limitation to its specic type and dimensions. In thisarticle, a rectangular block sliding over rigid support represents the frictionmaterial. The support was established to be rigid and thermally conductive. Separ-ation of heat ux on the surface friction was governed by a heat partitioning


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factor. It was assumed that the coefcient of friction and the coefcient of wear inArchards law depends on temperature. The wide ranges of the material properties,such as Youngs modulus, coefcient of friction, thermal expansion, specic heat,thermal conductivity, wear coefcient, and length and thickness of sliding blockhave been investigated. Global optimization was conducted corresponding to maxi-mum temperature during 5 s time of sliding with speed of 10ms1 and at 10 kNforce presses on the top surface of block. The task was to nd the optimum valueof the design parameters, and a low temperature was desirable. The FE model ofthe sliding process consisted of 595 four-node rectangular bilinear elements. Atrapezoidal integration method was used for the contact elements conditions.

A detailed description to predict frictional materials wear was presented byAbuBakar and Ouyang [43]. Both experimental and numerical investigations wereperformed. Disc brake assembly was employed in the experiment to estimate contactpressure distribution, essential to subsequent wear prediction. Three stages of differ-ent durations of the braking process were executed under a brake pressure of 1MPaand at a rotational speed of 6 s1. The stages lasted 10, 10, and 60 minutes, respect-ively. At the end of each stage, a contact test of 2.5MPa in stationary conditions wasdeveloped. Results were compared with FEA. A three-dimensional detailed FEmodel consisted of a disc, a nger, piston pads, a caliper, a carrier, two bolts, andtwo guide pins. Linear gauge and node mapping in the experiment in order to deter-mine node positions from FE model was used. Fixed boundary conditions at the boltholes of the disc were imposed. Rhhes [44] formula of nonlinear wear was adaptedand modied to facilitate numerical computations. The surface contact topographyor surface roughness due to wear and its inuence on the brake squeal was estimated.Results obtained in the previous analysis enables to develop brake squeal test. Fromthe two typical types of the squeal noise prediction, transient dynamic analysis andcomplex eigenvalue analysis; the latter because of its advantages was selected. Thefollowing assumptions were conceived;

Nonlinear static analysis for applying brake-line pressure Nonlinear static analysis to impose rotation of the disc Normal mode analysis to extract natural frequencies and modes of undamped

system Complex eigenvalue analysis that incorporates the effect of contact stiffness and

friction coupling

It is established that there is an unstable frequency of 4.2 kHz with positive realparts towards the end of braking, and it is obtained in the third stage of simulation.These numerical results agree well with the experiments, where an unstable frequencyequals 4 kHz.

Reibenschuh [45] investigated 1=12th part of the unused (brand new) discbrake and one with symptoms of wear within two major simulation types of thebraking process (descent driving and braking to a standstill). A ventilated type ofthe rotor was considered. In order to evaluate deformation, stress and temperaturedistribution of the load during braking was divided into three groups: centrifugalload at a constant temperature, centrifugal and thermal loads combined duringmountain descent action, and centrifugal and thermal loads combined during


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braking downhill to a permanent stop. The amount of heat ux was calculated fromthe known relation of energy balance in an individual case of the analysis.

4. THERMOELASTIC INSTABILITY

Sliding systems such as disc brakes and clutches inherently produce a signi-cant amount of heat due to a rotating motion of the disc against frictional material.The location of pressure and temperature distribution in the circumference offriction surfaces above critical value of the speed cause hot spots to appear [4648].Therefore, deformations of the rotor material may be the consequences. Thisphenomenon is known as frictionally excited thermoelastic instability TEI. It is cru-cial to predict the safety range of the operation of considering a sliding system. In thestudy in reference [49], the sinusoidal perturbation method of the stability of contactbetween two half-planes with single interface was used. Lee and Barber [50] extendedthis method to the case of a layer between two half-planes. The problem wasdeveloped in two-dimensionality with perturbations in the stress and temperatureelds which grow exponentially with time. The authors aimed to examine the dis-crepancy of the solutions where the critical speed wasnt found to agree well withthe practical solutions of the problem. The symmetric and antisymmetric approachesabout the midplane of the layer was used to investigate whether the unstable pertur-bations will exhibit the same symmetry. In the study, the disc-represented modelowing nite thickness with its typical material properties. Pads were considered ashalf-planes which acted upon the disc with uniform pressure. Accuracy was animpact of thermal conductivity of the pad material and therefore errors are supposedto be satisfactorily low. However, a slight migration process of the perturbations ofthe friction material performs an important part due to the reduction of thermalexpansion caused by perturbations in heat input and, hence, magnies critical speed.The seminal Lee and Barbers [50] article based on Burton et al.s [49] sinusoidalperturbation method of the stability of contact between two half-planes, laid thefoundation for subsequent investigations of frictionally excited thermoelastic insta-bility phenomenon.

Yi et al. [51] explored the effect of geometry for TEI phenomenon based on Duet al.s model [52]. Burtons perturbation method was employed to examine the stab-ility of contact. It was assumed that the dominant perturbation has a real growthrate. Only one element of the sliding system was established to conduct heat. Thisfact led to the solution where the most general perturbation was independent ofthe coordinate of sliding direction. Yi et al. [51] modied Dus method by usingFourier decomposition to the case of the eigenvalue problem.

High efciency of the nite-element analysis of linear perturbations on theconstant speed of solution was proven by Yi et al. [53, 54]. The analysis of the ther-momechanical feedback process due to frictional heating in a disc brake system wasperformed by Yi et al. [53]. The FFT technique was employed in order to calculatethe exponential growth rate for different hot spot numbers and critical speed. Thefeedback process was described as follows.

The above scheme is commonly used in the process responsible for the thermo-elastic instability mechanism in sliding systems. The authors associated the numberof hot spots with the critical value Vcr of the sliding speed. However, it was affected


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by the shape of the disc brake. The temperature measurements of the noisy brakesindicated nonuniform distributions in the circumferential direction. The unstableinteraction between thermoelastic distortion and heat generation revealed approxi-mately regular expansion of the hot spots in the circumference of the surface in con-tact. For the validation purposes of the numerical scheme of the thermoelasticsolution, experimental investigations were developed. Results shows that for eachbrake system tested in this case, hot spots were explicitly able to be identied, parti-cularly when the sliding speed was relatively high. Theoretical predictions for allwavenumber of critical speed agree well with experimental estimations. It was con-cluded that the FEM enables to predict critical speed for TEI phenomenon in discbrakes, and simplies alterations of the preliminary design process.

In aiming lubricant environments, such as clutches, simplications were con-sidered neglecting the effect of uid layer in most of the solutions of thermoelasticinstability phenomenon. However, experimental tests validated the proposed schemefor dry discs or clutches with sufcient accuracy [53].

Numerous studies have thus far focused on determining the critical speed ofthermoelastic instability phenomenon in clutches or brakes systems towards explo-ration of the unstable behavior which, was an impact of typical applications, wherecritical speed is commonly exceeded [5557].

The complex analysis of thermoelastic behavior, including TEI phenomenon,as being signicant in the investigation of the disc brake temperature and pressuredistribution has been executed by Choi and Lee [58, 59]. In addition, the inuenceof the material properties on the TEI has been analyzed. Based on numerical results,the carboncarbon materials with expected excellent characteristics were compared.

The thermoelastic instability phenomenon of the mine winder disc brake wasstudied by Scieszka and Zolnierz [60, 61]. A wide variety of the parameters usedin numerical analysis were adopted from numerous examinations comprising infra-red mapping. The computations were conrmed in the experimental investigations.The value of the kinetic and static coefcient of friction at any instant of time relatedto the temperature were approximately conducted from the tribological character-istic of the friction couple. It was of great interest to elaborate the FE model withadequate boundary and operation conditions during the braking process. Takinginto account, various mechanical, material, and parameters describing efciency ofthe braking system, the inuence of special characteristics of the mine winder havebeen analyzed utilizing a critical value term related to axial distortions of the rotor[62]. An attempt was made to determine a stable range of the operation. In the infra-red mapping investigation of the mine winder disc brake the AGEMA 880LWBequipped with a special lens was used. In order to overcome unreasonable nite-element division, a hybrid method with the fast Fourier transform technique wasemployed [63]. It was concluded, that a major inuence on the critical speed valuehas the coefcient of thermal diffusivity of the disc and Youngs modulus relatedto the thickness of the disc.

The nite-element method for the foregoing implementations imposesirrational timeintensive computations due to the complexity of the problem. Liand Barber [64] used the fast speed expansion method for transient solution of thethermoelastic contact problem as an alternative, and relied on the statement of thetemperature and stress elds as modal expansion. A specied number of critical


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speed with important interpolation process were used to achieve more efcient cal-culations of eigenvalue analysis. Convergence tests were developed and presentedby an exemplar of a multidisc clutch.

Yi [65] emphasized an importance considering TEI with association to inertialterms. Investigation of structural vibration almost always omits the fact of thermaleffect occurrence, and TEI studies disregard dynamic instabilities associated withhigh frequency oscillations. Yi adapted Affrenate et al.s [66] model to investigatea signicant effect of coupling thermoelastic and dynamic instabilities, referred toas thermoelastodynamic instability (TEDI). Results obtained in the analysis, showedthat some modes perpetually grow beneath the critical value of sliding speed.

Zagrodzki [67] made a distinction of the thermoelastic process: the back-ground process of the contact pressure results from the external loads and the per-turbed process is a result of the initial condition. The background process does notaffect the stability of the linear systems hence, it has an absence in many studies[50, 53, 68]. A similar approach was executed in transient analysis in references[69, 70], where the background process wasnt taken into consideration. The mech-anism of exciting unstable modes was examined by Zagrodzki and Zhao [71] in thecase of a wet multidisc clutch considering different geometry of an external load,socalled nger piston. The hot spotting problem using simulations and experi-mental tests was identied.

5. CONCLUSION

The FEM solutions of the heat conduction and thermoelastic contact problemof disc brake (clutch) systems in the course of temperature estimation give acceptableresults compared to the outcomes derived from experimental tests. However,different approaches of modeling of the heat generation problem impose specieddifculties of inference.

The heat partitioning factor used in thermal analysis plays a central roledepending on the particular selection among a wide variety of formulas, whichmay differ markedly. It stems from the fact that division of, heat entering the discand pad, respectively, is a result that depends on interface conditions, while in thecomputations it is an articial partitioning factor of heat ux assumed a priori atthe beginning of the simulation of the braking process.

A hybrid method combining fast Fourier transform and the nite-elementmethod help to determine complete temperature distribution within the circumfer-ence of the disc brake using two-dimensional models. The coupling of these techni-ques save time in calculations upon conventional nite-element analysis withnonaxisymmetric thermal loads. However, investigations considering the movingheat source place an emphasis on the importance of resultant fatigue stresses andthermal cracking phenomenon; the thermoelastic contact problem of friction pairusing FFT-FEM is unfavorable to solve.

An extensive body of literature of frictional heating phenomenon provides awide variety of model description complying with physical and mathematicalcharacters. Nonetheless, the lack of the thermoelastic contact problem with its fullytransient behavior, including temperature the dependent material properties andcoefcient of friction is noticeable.


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Prediction of wear during analysis of temperature distribution is essential onlyin terms of relatively long repetitive braking application, mainly in thermoelasticanalysis; whereas, contact conditions are strongly dependent on the topography ofrubbing surfaces and may rmly affect obtained results.

Signicant trends aiming at a detailed investigation of the frictionally excitedthermoelastic instability problem are perceptible. So far, satisfactorily describedstable behavior with the conventional perturbation approach is gradually extendedto the case of instability referenced to its practical use in the disc brakes and clutchsystems operating above critical speed.

The FEM apparatus has proven to be a powerful instrument in the use offrictional heating phenomenon modeling, helping to overcome analytical solutionslimited to semi-spaces or plane parallel strip.

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