ib

ka S

Received 6 September 2010

Keywords:BrakingPad/disc systemFrictional heatingMoving heat sourceHeat conductionFinite element method

andmembers of the disc brake system basing on two- and three-dimensional FE modelling techniques and

bers o

hazardous environments such as coal mines [1e4].Complexity of the friction and wear processes state major dif-

culty of formulating universal physical model to determine criticaloperation conditions for specied case of braking action. Exactanalytical solutions of temperature of friction pair may be obtainedwith restriction to semi-spaces, plane parallel strip or semi-planes.

heating systems should be noticed.Rotating systems such as disc brakes in which pads cover solely

the segment of rubbing path of a disc, are intrinsically submitted tonon-axisymmetric thermal load. Simplications of a real three-dimensional modelling techniques into two-dimensionalityrelating to the heat rate uniformly distributed in circumferentialdirection were so far accomplished [7e11]. In point of fact theyenter simplications of three-dimensional process of heating,which is omitted in systems where the friction surface of a body

* Corresponding author. Tel.: 48 85 746 93 12; fax: 48 85 746 92 10.

Contents lists availab

a

sev

Applied Thermal Engineering 31 (2011) 1003e1012E-mail address: p.grzes@doktoranci.pb.edu.pl (P. Grzes).kinetic energy conversion into heat at the pad/disc interface. Theincrease of friction moment is a limited quantity and depends onthe coefcient of friction, radius of rubbing path, and forces that acton the pads. The process of slipping leads the increase of temper-ature, whereas its peak value is one of the most crucial factor in thecourse of action to occur. The temperature on the contact surfacesof the tribosystem during emergency braking intensied bysignicant thermal load due to frictional forces as well as the highvelocity of the process is, in particular, important to predict in

gular prismand circular source on a rotating cylinderwereproposedin article [5]. The temperature and the thermal constriction resis-tance as a function of geometric characters and velocity weredetermined. The temperature and the thermal stresses of the pad(the strip) sliding with the constant retardation on a surface of thedisc (the semi-space) both during heating and after the moment ofstandstill were studied [6]. However these geometric congurationsmay correlate with actual engineering applications, absence of theexact solutions, primarily application of nite areas of frictional1. Introduction

The sliding contact of the mem1359-4311/$ e see front matter 2010 Elsevier Ltd.doi:10.1016/j.applthermaleng.2010.12.016second, the three-dimensional rotor is subjected to the non-axisymmetric thermal load to simulaterealistic thermal behaviour of the brake action. Operation conditions, thermo-physical properties ofmaterials and dimensions of the brake system were adopted from the real representation of the brakingprocess of the passenger vehicle. Arbitrarily selected four values of the velocities at the moment of brakeengagement were applied to the models so as to investigate theirs inuence on the obtained solutions ofthe temperature evolutions on the contact surface of the disc volume referring to two separated niteelement analysis. The large amount of heat generated at the pad/disc interface during emergency brakingindisputably evokes non-uniform temperature distributions in the domain of the rotor, whereas the padelement is constantly heated during mutual sliding. The obtained results of the original code of three-dimensional modelling technique implemented to the conventional FE software revel high agreementwith the solution of simplied process of friction heating.

2010 Elsevier Ltd. All rights reserved.

f disc brake results in

Typically the heat ux condition is applied at the region of contact.The three-dimensional temperature distributions of a moving heatsource problem with a rectangular and elliptic source on a rectan-where the intensity of heat ux was assumed to be uniformly distributed on the friction surface of discduring braking process, and the heat is transferred exclusively in axial direction, whereas during theAccepted 6 December 2010Available online 15 December 2010

complexity of the phenomenon. First step of the analysis based on the previously developed modelAnalysis of disc brake temperature distrunder non-axisymmetric load

Adam Adamowicz, Piotr Grzes*

Faculty of Mechanical Engineering, Bialystok University of Technology (BUT), 45C Wiejs

a r t i c l e i n f o

Article history:

a b s t r a c t

This paper aims to study

Applied Therm

journal homepage: www.elAll rights reserved.ution during single braking

treet, Bialystok 15-351, Poland

compare the temperature distributions caused by mutual sliding of two

le at ScienceDirect

l Engineering

ier .com/locate/apthermeng

T0 initial temperature, C{T} temperature vectorV velocity of the vehicle, km/hV0 initial velocity of the vehicle, km/hDx the mesh size (smallest element dimension), mz axial coordinate, m

Greek symbolsg heat partition ratiod thickness3 coefcient of thermal activityq circumferential coordinate, degr density, kg/m3

f0 cover angle of pad, degu angular velocity, 1/su0 initial angular velocity, 1/s

rmal Engineering 31 (2011) 1003e1012and counterbody is equal aircraft brakes and clutch systems [12].Two models of heat dissipation utilizing axisymmetric arrange-ment of a disc brake: namely macroscopic and microscopic modelwere implemented in articles [7,8]. In the macroscopic model rstlaw of thermodynamics has been taken into account and formicroscopic model various characteristics such as braking time,velocity of the vehicle, thermo-physical properties of materials,contact pressure, and dimensions of a real disc brake assembly havebeen studied. Greens functions were used to determine tempera-ture distributions in the disc and pad volume [8].

Formulation of the heat ux activity during frictional heatingindependent of circumferential coordinate q may cause unrealisticcontact conditions and falsify actual, elastic distortions. In order tosimulate reasonable emergency braking process, the three-dimensional FE model assuming nonlinear pressure distributionand angular velocity variability was proposed in article [13]. The

Nomenclature

c specic heat, J/kg K[C] heat capacity matrixf coefcient of frictionh heat transfer coefcient, W/(m2 K)k thermal diffusivity, m2/sK thermal conductivity, W/(m K)[K] conductivity matrixp pressure, MPap0 contact pressure, MPaq intensity of heat ux, W/m2

r inner radius, mR outer radius, m{R} heat source vectort time, sts braking time, st0s time of braking with constant deceleration, sDt time step, sT temperature, CTN ambient temperature, C

A. Adamowicz, P. Grzes / Applied The1004thermo-physical properties of materials independent on tempera-ture have been used.

Operation of disc brake above the certain range of the velocitymay lead to thermoelastic distortions and in consequence to non-uniform pressure distribution due the interchanged moments ofcontact and its absence during rotation, known as thermoelasticinstability (TEI) [14]. The upwind scheme in nite elementformulation to prevent possible perturbations owing high Pecletnumber was developed [15].

The conventional nite element method is well adopted instationaryproblems, however three-dimensionalmodellingof partsbeing inmotion imposes very nemesh due to high values of Pecletnumber, which determines the range of the velocity, above whichoscillations may occur. The hybrid method combining the niteelement method and the fast Fourier transform (FFT) technique, asan alternative approach in order to reduce computational timewithout loss of the temperaturealterationsowing the circumferenceof a disc brake was used [3,4,9,10,15,16]. The temperature distribu-tions during different operation conditions were presented. Thereview of FEM-solutions of thermal problems of friction duringbraking is given in the article of Yevtushenko and Grzes [17].

In order to predict the temperature on the contact surfaces ofelements of disc brake, experimental examinations includinginfrared techniques such as two colour pyrometry [18], infraredmapping [3,4] as well as thermocouples [1,19,20] were developed.In this paper three-dimensional nite element analysis regardingmovable behaviour of the disc brake system was developed andcompared with the two-dimensional modelling of frictional heatingproblemderived from previous authors study [11]. In order to assureaccuracy of the solution several nite element meshes of the twospecied models of the real disc brake was tested. Investigationcomprises dissimilar evolutions of the external loadduringmountaindescent with constant velocity and application of single, emergencybraking to standstill. For the purpose of comparison of obtainedresults, dimensions of the disc brake, operation conditions andthermo-physical material properties were adopted from the studydeveloped previously [8]. Special concern is focused on the descrip-tionof theFEmodelling techniqueof themovingheat sourceproblemcorresponding to axial conguration of the same phenomenon.

Subscriptsd indicates discp indicates padw indicates wheel2. Statement of the problem

The disc brake system comprises in the majority two elements:rotating axisymmetric disc and immovable non-axisymmetric pad(Fig. 1). When the braking process occurs, the hydraulic pressureforces the piston and therefore pads and disc brake are in sliding

Fig. 1. A schematic diagram with three-dimensional nite element mesh of a pad/discbrake system.

ermapad in operation is equal to the apparent surface in the slidingmotion. The contact pressure is uniformly distributed over allfriction surfaces hence the heat generation of the midplane isconsidered as symmetric;

(3) The average of the intensity of heat ux into disc on the contactarea equals [21]:

qdr; q; tjzdd 1 g fprut; rp r Rp;0 q 2p; 0 t ts; 1

qpr;q;tjzdp gfprut; rp rRp; 0 qf0; 0 t ts; (2)

(4) The heat partitioning factor representing the fraction of fric-tional heat ux entering the pad has the following form [22]

g 11 3; (3)

where

3 Kdkp

pKp

kd

p ; (4)is the thermal activity coefcient [23]

(5) The frictional heat due to Newtons law has been dissipated toatmosphere on the other surfaces. The heat transfer coefcienth is constant during simulation of braking process;

(6) Radiation is neglected by virtue of short braking time andhence relatively low temperature;

(7) The wear on the contact surface is negligible.

In the three-dimensional model of solid disc, single surface of itssymmetry in axial direction is insulated owing nature of consideredphenomenon of heating. On both, the external, internal surface offactory outcomes. Thus primarily relevant in the present study wasto examine proposed technique of moving heat source modellingproblem. The solid disc brake was analyzed, where the dimensions,operating parameters and properties of materials were adoptedfrom the study of Talati and Jalalifar [8].

For both types of disc models it has been assumed as follows:

(1) Material properties are isotropic and independent of thetemperature;

(2) The nominal surface of contact between the disc brake and thecontact. The friction at pad/disc interface resists the movement andthe vehicle slows down, remains at the same level of the velocityduring mountain descents or eventually, stops. The frictionbetween disc and pads always opposes motion and the heat isgenerated due to conversion of the kinetic energy, whose portion isdissipated by convection to the atmosphere in accordance toNewtons law. However radiation as a third type of heat exchangealways takes place, owing its negligibly amount is omitted in themodelling of the presented phenomenon.

In this paper non-axisymmetric thermal load due to the fric-tional heat generated during the single braking process imple-mented in the three-dimensional model is investigated to compareobtained solution of the temperature evolution on the disc frictionsurface with the two-dimensional representation of the constantheating studied previously [19] and to answer if there is theaccurate range of the velocity under which the uniform heat uxratio upon the circumference of the disc may results in the satis-

A. Adamowicz, P. Grzes / Applied Ththe disc and contact surface free from friction, the convectionconditions are prescribed due to the Newtons law of cooling. In thezone of temporary contact of the pad and disc, the thermal ux isassigned, which differs in the area of disc at any instant of brakingtime corresponding to the components of the intensity of heat uxproduct Eq. (2). The contact pressure p0 is constant during theanalysis, whereas the velocity for the rst case of the analysisdecreases linearly with time

ut u0 1 t

t0s

!; 0 t t0s ; (5)

and during the second constant value of the velocity is assumed.

3. Mathematical model

In order to determine the temperature distributions, bothanalytical and numerical techniques have been employed. Thestarting point of the analysis of the temperature elds in the discvolume, is the parabolic heat conduction equation given in thecylindrical coordinate system (r, q, z) [24]

v2Tvr2

1rvTvr

1r2

v2T

vq2 v

2Tvz2

1kd

vTvt

uvTvq

; rd r Rd;

0 q 2p; 0 < z < dd; t > 0 6

The boundary and initial conditions of non-stationary problemare established as follows (Fig. 1)

KdvTvz

z0 qdr; q; t; rp r Rp; 0 q 2p;0 t ts; G 7

KdvTvz

z0 hTN Tr; q; t; rd r rp; 0 q 2p;t 0; U1 8

KdvTvr

rRd hTN Tq; z; t; 0 q 2p; 0 z dd;t 0; U2 9

KdvTvr

rrd hTN Tq; z; t; 0 q 2p; 0 z dd;t 0; U3 10

vTvz

zdd 0; rd r Rd; 0 q 2p; t 0; U4 (11)Tr; q; z;0 T0; rd r Rd; 0 q 2p; 0 z dd (12)

4. FE formulation

The object of this section is to develop approximate time-step-ping procedures for axisymmetric transient governing equations.Using Galerkins approach the following matrix form of the Eq. (6)is formulated [25]

CdfTgdt

KfTg fRg (13)

In order to solve the ordinary differential equation (13) the

l Engineering 31 (2011) 1003e1012 1005Crank-Nicolson method was used. Based on the assumption that

were partially coated, the elements for the pad were simulta-neously uncovered. The process was repeated and the time of padimaginary contact area with the constant number of elementsduring computations was successively longer compatibly to therate of deceleration until standstill. In the case of braking withconstant velocity, the time of heating phase of three-dimensionalmodel equals f0=2p of time of one rotation, whereas cooling phaselasts longer due to angular dimension of pad element and equals1 f0=2p of time of one rotation of the wheel.

The thermal ux entering the disc acted in the shape of theintensity of heat ux applied to three-dimensional nite elementsin the area of pad operation during braking. Instead of automaticmesh generation capabilities, the original programming of thebuilt-in commends of nite element software covering the algo-rithm of moving heat source described above, to assure correctnessof the boundary conditions prescribed to specic elements, in

rmal Engineering 31 (2011) 1003e1012temperature {T}t and {T}tDt at time t and t Dt respectively, thefollowing relation is specied

1Dt

fTgtDtfTgt 1 bdTdt

tbdTdt

tDt

(14)

Substituting Eq. (14) to Eq. (13) we obtain the following implicitalgebraic equation

C bDtKfTgtDt C 1 bKDtfTgt 1 bDtfRgtbDtfRgtDt 15

where b is the factor which ranges from 0.5 to 1 and is given todetermine an integration accuracy and stable scheme.

The transientnite element analysiswas developedusing theMDPatran/MD Nastran software package [26,27]. The nite elementmesh of the disc model chosen for the analysis is illustrated in Fig. 1.The accuracyof the solutionwasobtainedby testingdifferent grids ofnite elements, all of which had its own global number of elementsdue to specic division in circumferential, radial, and axial direction.The investigated, individual grids at the initial phase of the compu-tations consisted of the 180, 240, 360, 450, 540 elements in thecircumference,10,15, 20, 25, 30 in the radial direction of the rubbingpath, and 3, 3, 4, 4, 5 elements in the axial direction, respectively. Thecalculationsof transient temperatureof the rotorwere carriedout forthe braking process with constant deceleration from the initialvelocity of 25km/h. Themeshof thenite elementswas selecteddueto the difference of the obtained peak values of temperature relatingto the nest mesh (model with the 540 elements in the circumfer-ence). The FE model of disc employed for the transient analysisconsisted of 43,200 eight-node hexagonal elements e HEX8 (360elements in the circumference and 4 in axial direction) and 33,693nodes was used in the thermal analysis. As the mesh should becapable to reproduce the rapid temperature variations in theimmediate vicinity of the contact surface, the size of the niteelement increased with the distance from the region of generatedsurface of friction. To avoid inaccurate or unstable results, a properxed time step associated with spatial mesh size is essential [26].

Dt Dx2 rdcd10Kd

(16)

In order to simulate moving heat source problem in the processof emergency braking, avoiding inaccuracies and oscillations tooccur due to Peclet number which in presented case markedlyexceeds the critical value of Pe 2, time-stepping procedure cor-responding to the relative pad/disc location was developed. Theknown amount of the intensity of heat ux entering the disc atsucceeding instants of time, determined from the product of radialdistance from the axis of disc on the friction surface, the contactpressure, velocity of the vehicle with the rate of deceleration Eq. (5)and friction coefcient was implemented to the FE models and theproblem was solved based on the programming technique imple-mented to the commercial FEM programme [26,27]. In conse-quence spatial scheme of heating issue was accomplished and timedependent boundary conditions due to rotating pad activity wereestablished. At the beginning of the process, after the brakeengagement the amount of heat (Eq. (1)) was applied to theselected nite elements of pad area of the three-dimensionalmodel. At the next time step, smaller than computed from the Eq.(16), the corresponding motion of heat source (brake pad) wascalculated and displaced to the adjacent elements according tomutual sliding direction of the members of braking system. Thisprocess was modelled by the function, which imitated the processof covering of elements of the model during relative motion of

A. Adamowicz, P. Grzes / Applied The1006rotating disc and xed pad. While elements near by the front of padparticular, on the contact surface of disc was developed.

5. Results and discussion

In this paper temperature distributions of the disc brake withoutpad have been investigated. The disc rotor is subjected to high non-axisymmetric thermal loadwhichmay lead to non-uniformpressureandtemperaturedistributions. Therefore three-dimensional analysisfacilitates to examine temperature alterations in the circumferenceand theirs inuence on the area inside the disc. Both convection andconduction have been analyzed. Particularly conduction wasconsidered to be the most important mode of heat transfer.

In order to validate proposed transient numerical analysis twodifferent types of the FE models were investigated, namely two-and three-dimensional conguration [8]. The part of presentedtemperature evolutions for two-dimensional model (braking withthe constant retardation from the velocity of V0 100 km/h)originates from previous authors study [11]. The transient solutionwas performed for four selected initial velocities and relateddurations of braking process with constant deceleration. For thecase of braking with constant velocity, exclusively the action ofV 100 km/h (ts 3.96 s) was tested. Material properties andoperation conditions adopted in the analysis were the same forboth types of FE models and are given in Table 1.

In Fig. 2 numerical solutions of three-dimensional (continuousline) transient analysis of the disc contact surface temperatureevolutions for specied radii ofdisc andposition in the circumferenceq 0 confronted with the results obtained from the two-dimen-sional analysis (dashed line) are shown. In order to illustrate effect offrictionallyexcitedheatingover the frictionsurface, eachofdescribedgures covers four characteristic points of the position along the

Table 1Thermo-physical properties of materials, dimensions and operation conditions forthe transient analysis (from Talati and Jalalifar [8]).

Items Disc Pad

Thermal conductivity, K [W/(m K)] 43 12Heat capacity, c [J/(kg K)] 445 900Density, r [kg/m3] 7850 2500Inner radius, r [mm] 66 76.5Outer radius, R [mm] 113.5Cover angle of pad, 40 64.5Thickness, d [mm] 5.5 10Radius of the wheel, Rw [mm] 314Initial velocity of the vehicle, V0 [km/h] 100 75 50 25Time of braking, ts [s] 3.96 2.97 1.98 0.99Pressure, p0 [MPa] 3.17Coefcient of friction, f 0.5Heat transfer coefcient, h [W/(m2 K)] 60Initial temperature, T0 [C] 20

Ambient temperature, TN [C] 20

Fig. 3. Evolutions of temperature on the contact surface of the disc brake duringbraking from the initial velocity V0 75 km/h at selected radial locations for three-

ermal Engineering 31 (2011) 1003e1012 1007radius, namely the external radius of the disc Rd, the mean radius ofrubbing path rm, theminimal radius of pad rp, and the internal radiusof the disc rd. The temperature curves directly correspond with thedivers representations of the disc brake model congurations. It isnoticeable, that the value of temperature in each case of axisym-metric heating of disc rapidly rises at the beginning of brakingprocess, reaches its maximal value, then decreases to the lower leveland eventually stops, which is coherent with the studies [7,8,11,13].However uctuations of temperatures have a presence in the solu-tion of three-dimensional model of frictional heating, the approxi-

Fig. 2. Evolutions of temperature on the contact surface of the disc brake duringbraking from the initial velocity V0 100 km/h at selected radial locations for three-(solid curves) and two-dimensional (dashed curves) models.

A. Adamowicz, P. Grzes / Applied Thmated values remain the approximate conrming the stability of theFE modelling in the two-dimensionality. The temperature curvesexpose saw-shaped character, which stems from the mutual rota-tional motion of the disc over the xed pad [13,20]. The presentedtemperature evolution is obtained for certain, xed spot on thecircumference of a disc, therefore periods of heating and coolingphases may be distinguished. When the specic nite element oftemperature calculations on the contact surface of disc is covered bypad (heating phase) the increase of temperature is noticeablebecause of accumulation of the frictional heat. On the contrarywhenthe pad is out of considered spot on the rubbing path, the coolingconditions according to Newtons law are established and thetemperature decreases. Each revolution of the wheel strictly corre-sponds to one cycle of heating and cooling state. It is evident, that thetemperaturedistributioncorrelates intermediately to the intensityofheat ux entering the disc, whose value in the plane model linearlydecreases with time until the standstill, whereas spatial represen-tation accessorily complies non-continuous heating of disc over thecircumference. In the solutionof two-dimensionalmodel the highesttemperature T 227.94 C is reached at the radial positionr113.5mm, after time t3.022 s,whereas thehighest temperatureT 259.34 C of fully three-dimensional disc, occurs at the sameradius r 113.5 mm after time t 2.688 s. The discrepancy oftemperatures is lowerat theendof theprocessandequalsT3.52 C.At the radial locationof76.5mmthehighest temperatureof two- andthree-dimensional FE model equals T 98.98 C and T 108.31 C,respectively. Exclusively at the internal surface of disc r 66 mm inboth FE models the highest value of temperatures (T 47.32 C andT 47.43 C) is attained at the end of the braking process. Themaximal temperature at the radii of 76.5, 95 and113.5mmof the 3-Dmodel occurs at the same time t 2.688 s, whereas identical radialpositions of axisymmetric case gives the solutions of time equalledt 3.36 s, t 3.098 s and t 3.022 s respectively.

The temperature evolutions on the contact surface of discconditioned by the obtained results of two types of braking processsimulations from the initial velocity V0 75 km/h are shown inFig. 3. In the two-dimensional model the temperature curves ofsurface of friction continuously alter with time analogously as wasduring braking from V0 100 km/h (Fig. 2). In the spatial model, the

(solid curves) and two-dimensional (dashed curves) models.increase of temperature is noticeable after the moment without

Fig. 4. Evolutions of temperature on the contact surface of the disc brake duringbraking from the initial velocity V0 50 km/h at selected radial locations for three-(solid curves) and two-dimensional (dashed curves) models.

sliding contact at the specied location in the circumference q 0,then the maximal temperature is attained, and succeeding periodof pad absence effects with its rapid descend. The nature of therepeated heating and cooling states (Figs. 2e5) indicates two typesof temperature curves, the rst, describing period of heating is theconcave curve, the second part of one rotation of disc is describedby the convex curve, which is caused by the extortion of frictionallygenerated heat impulse and its absence after the pad transitionwith the convective cooling. The time of these periods differs due tothe velocity of braking and is constant at its specied value at eachradial location on the friction surface. The temperature curves atthe radii of 66, 76.5 mm almost coincide near the time of full stopwith both solutions owing complexity of the model, whereas thediscrepancy of temperatures during the action and after standstilloverlaps less at radii r 95 and r 113.5 mm.

In Fig. 4 temperature evolutions on the contact surface duringbraking from the initial velocity of 50 km/h are presented. Thesimilar pattern of temperature progress with regard to Figs. 2 and 3may be observed. The averages of temperatures curves of 3D-modelagree highlywith the results obtained from themodel drawn on theintensity of heat ux uniformly distributed in the circumference ofthe disc. The maximal temperature reached during barking from50 km/h equals T 81.86 C and T 112.54 C for two- and three-dimensional model respectively. The seventh rotation ends at theposition of pad covering the tested location in the circumferenceq 0 (twenty eighth rotation in Fig. 2). Therefore the temperatureafter the full stop is closer to the value obtained in the axisymmetricproblem of frictional heating. Relating to the radial location of thepresented temperaturecurves, proportionof thedistance from zaxis

Fig. 5. Evolutions of temperature on the contact surface of the disc brake duringbraking from the initial velocity V0 25 km/h at selected radial locations for three-(solid curves) and two-dimensional (dashed curves) models.

Fig. 6. Temperature distributions on the contact surface at the moment of standstill of brakV0 25 km/h for three- (solid curves) and two-dimensional (dashed curves) models.

A. Adamowicz, P. Grzes / Applied Thermal Engineering 31 (2011) 1003e10121008ing from the initial velocity: (a) V0 100 km/h, (b) V0 75 km/h, (c) V0 50 km/h, (d)

is not equalled to the corresponding values of temperature. Thisphenomenonmay be attributed to the contact surface of disc whichis situated on itsmargin, thus the area beneath rubbing path absorbsmore heat during action and temperature is adequately lower.

The evolutions of temperatures during braking from the lowesttested velocity V0 25 km/h are shown in Fig. 5. It may be observedthat only one rotationwas accomplished within sliding process. Theplotted curves at each radius reveal signicant disagreement of thetwopresented solutions. Themaximal value of temperature of three-dimensional model attained in the action, at radius of 113.5 mmequals T 64.73 C, whereas in opposite approach of modelling,temperature equals T 41.21 C. Such a spread of results, relating tothe simplied process of heating may mislead the actual effecttemperature variations. Themoment,when the highest temperatureoccurs evidently depends on the investigated location in thecircumference, and inparticular casemay be identical to the solutionof two-dimensional model. At the radius of 66 mm the temperatureremains unchanged in both cases.

Fig. 6 depicts the temperature elds on contact surface in thecircumference at the moment of standstill for selected radial loca-tions and different initial velocities: a) V 100 km/h, b) V 75 km/h,c) V 50 km/h, d) V 25 km/h. The temperature curves of three-dimensional model are plotted with regard to the constant temper-ature of two-dimensional FEmodel. In fact temperature distributionof axialmodel in Fig. 6 shouldbe illustrated as a point, but to facilitateclarity straight line (dashed) is used. It may be observed that thetemperature rises when the pad passes specied position on thefriction surface of disc and decreases to the level beneath the distri-bution of two-dimensional event. The highest calculated range of

amplitude of temperature occurs on the external edge of disc in eachcase of braking. For r 76.5 mm the temperature is more smoothunder pad transition, whereas on the edge of external surface isalmost constant in the circumference because of the distance fromthe rubbing path. The presented plots of temperature drawn alongthe circumferential direction which correspond to the articles ofFloquet and Dubourg [9] and Cho and Ahn [16].

The average temperature of spatial problem during brakingfrom V 25 km/h (Fig. 6d) at the radius r 113.5, 95, 76.5 and66 mm equals T 35.59, 33.47, 25.74, 20.09 C respectively,whereas temperature at the end of braking of 2D model equalsT 35.65, 33.52, 25.65, 20.13 C, therefore when the initial angularvelocity equals u0 22.116 s1 (V0 25 km/h), the mean temper-ature coincides in each case of the solution with the relative errorlower than 0.5%, whereas for the initial velocity u0 44.232,66.348, 88.464 s1 (V0 50, 75, 100 km/h) equals 1, 2, and 3%(Fig. 6aec), respectively. However this arithmetic mean of spatialdistribution of temperature is not able to include realistic responseof material heating of disc during process of braking. The level oftemperature in each case of brake engagement owing differentinitial velocities corresponds to temperatures at the moment ofstandstill presented in Figs. 2e5. The temperature distributionsexpose importance of place under examination in the circumfer-ence of spatial model and its parallel time.

Fig. 7 shows the temperature distributions that evolved on theexternal radius of disc (r 113.5mm) at different locations in depthduring braking from the previously selected initial velocities of thevehicle: a) V0 100 km/h, b) V0 75 km/h, c) V0 50 km/h, d)V0 25 km/h. The solutions of spatial model are confronted with

A. Adamowicz, P. Grzes / Applied Thermal Engineering 31 (2011) 1003e1012 1009Fig. 7. Evolutions of the disc temperature at different axial distances from the contact surfac(b) V0 75 km/h, (c) V0 50 km/h, (d) V0 25 km/h for three- (solid curves) and two-die at the radius of 113.5 mm during braking from the initial velocity: (a) V0 100 km/h,mensional (dashed curves) models.

the axisymmetric representation. The permanent rise of tempera-ture until attainment of its maximal value and slightly descendafter, near the moment of full stop on the contact surface(z 0 mm) and for z 1, 2 mm is noticeable. The bigger distance(z 3.5, 5.5 mm) results in constant increase of temperature untilthe end of the process. The character of evolution of temperature onthe axial position z 1, 2 mm slightly differs from the trace at theposition of 0 mm, whereas temperature on the depth of 2, 3.5,5.5 mm evolves almost identically to the curves generated inaxisymmetric model. The presented evolutions of temperature areplotted for the specied location in the circumference, thusagreement of the results owing type of the frictionally excitedheating process strongly depends on the position of testing as well.Nevertheless chosen point q 0 in the circumference of the three-dimensional model during braking from V 100 km/h (Fig. 7a) andV 50 km/h (Fig. 7c) almost overlapped selected axial distancesfrom contact surface of two-dimensional model. It stems from thefact that during the immediate moment of standstill, pad covers thespot of disc under examination and therefore causes slight rise oftemperature sufcient to improve agreement of plane and spatialsolution of heating. It has to be noticed that in the two-dimensionalmodel convective terms on the friction surface have been neglectedand constant heating with the same value of thermal ux in thecircumference during braking was established. All of the temper-ature curves in Fig. 7 which represents two-dimensional model onthe depth of z 2, 3.5, 5.5 mm exceed values of the related

time ts obtained during braking from the initial velocityV0 100 km/h is shown. The circumferential location of presentedtemperature curves was chosen due to the disc/pad related positionnext to sliding pad. The temperature for the particular distancesfrom the axis of the rotor correlates with presented curves plottedversus braking time for four characteristic radii (Fig. 2). The value oftemperature on the internal surface of disc coincides within bothconguration of the model of frictional heating at any step of time.At the end of braking process, the temperature on the frictionsurface along the radius is approximate. Nevertheless for the timet 25%ts, 50%ts and 75%ts the temperature in the contact zoneobtained from spatial model highly exceeds the correspondingvalues of the two-dimensional phenomenon, which stems from thefact that during braking with linear decrease of time the padinuences directly the level of temperature until themoment of fullstop.

In Fig. 9 the temperature evolutions on the contact surface ofdisc versus time during braking with the constant velocity of100 km/h are illustrated. The temperature of the two-dimensionalsolution at the position r 76.5, 95, and 113.5 mm rapidly rises atthe beginning of action, then linear increase is noticeable untilstandstill, whereas in case of spatial model the delay of tempera-ture variations is may be seen at the beginning of braking, afterwhich impulse nature of heating takes place. The process of heatingrelating to the average of temperature highly agrees withindifferent FE model of the same phenomenon. The maximal value of

A. Adamowicz, P. Grzes / Applied Thermal Engineering 31 (2011) 1003e10121010temperature evolutions of spatial model during braking process.However temporary peak values of temperature during pad passingare higher than smoothed evolution of temperature of uniformheating in the circumference, owing the average amount oftemperature of spatial model its inuence on heating is lowered.The temperature evolutions of braking from the angular velocityu0 22.116 s1 (Fig. 7d) conrm that for that case the solutions ofheat transfer in disc brake are obviously dissimilar when considertwo- and three-dimensional model.

In Fig. 8 the temperature distribution on the contact surfacealong the radius of disc at four different moments of time braking

Fig. 8. Temperature distributions on the friction surface versus radial direction during

braking from the initial velocity V0 100 km/h for three- (solid curves) and two-dimensional (dashed curves) models.temperature on the contact surface is reached at the external radiusof disc at the end of braking for two-dimensional modelT 494.35 C and for the last pad rotation at the time of t 3.906 s,T 526.63 C. The similar approach of simulation of brakingprocess with constant velocity using three-dimensional model hasbeen investigated [2]. Assumptions regardless circumferentialconductive ux were made with the assessment of the enterederror.

The temperature variations at different locations in depth fromthe disc/pad interface z 0 mm to the surface of symmetry of discz 5.5 mm are shown in Fig. 10. With a distinction to the case of

Fig. 9. Evolutions of temperature on the contact surface of the disc brake duringbraking with constant velocity V 100 km/h at selected radial locations for the two-

and three-dimensional problem for three- (solid curves) and two-dimensional (dashedcurves) models.

Fig. 10. Evolutions of the disc temperature at different axial distances from the contactsurface at the radius of 113.5 mm during braking with constant velocity V 100 km/hfor three- (solid curves) and two-dimensional (dashed curves) models.

A. Adamowicz, P. Grzes / Applied Thermabraking with linearly decreased velocity of the vehicle (Fig. 7a)braking with the constant velocity results in the increase oftemperature after the initial moment of time nearly linear until fullstop. The temperatures of two-dimensional model at the positionz1,2,3.5,5.5 mm are higher at any instant of braking time.

The temperature distributions on the friction surface along thecircumference of disc for the case of braking with constant velocityof 100 km/h are shown in Fig. 11. It is clearly noticeable for r 76.5,

95, 113.5 mm that the temperature on the contact surface corre-sponds to pad transition over the rotational disc.

Fig. 11. Temperature distributions on the contact surface at the moment of standstill,process of braking with the constant velocity V 100 km/h for three- (solid curves)and two-dimensional (dashed curves) models.Fig. 12 depicts the corresponding temperature distributions onthe contact surface of disc brake for selected moments of time. Thetemperature prole for the time equalled 0.25 of ts has the longestlinear section in the middle of braking path. At the subsequentmoments of time this spot is more rounded. In the contrary to Fig. 8,this case of braking with constant velocity V 100 km/h lasting3.96 s results in aligned respective plots of temperature elds dueto the two- and three-dimensional description of the analyzed

Fig. 12. Temperature distributions on the friction surface versus radial direction duringbraking with the constant velocity V 100 km/h for three- (solid curves) and two-dimensional (dashed curves) models.

l Engineering 31 (2011) 1003e1012 1011phenomenon.

6. Conclusions

In this paper three-dimensional nite element analysis wascarried out for temperature distributions assessment in disc brakesystem during single braking. The disc rotor was examined withoutpad presence. The heat conductivity problem was divided into twocases of different congurations of the disc brake FE models owingcomplexity of the problem.

From the obtained results we can conclude, that the tempera-ture of disc on the contact surface of two-dimensional model andaveraged solution of spatial solution during braking with theconstant deceleration sharply rises at the beginning of the process,reached its maximal value and eventually stops on the lower level,whereas if the velocity of the vehicle is constant the temperatureafter the initial moment of time increases approximately linearly.

The character of temperature evolution on the contact surface ofdisc and its inuence in depth reveals high coincidence with regardto the three-dimensional model and simplied two-dimensionalrepresentation of the considered problem. Therefore validation ofthe outcomes of previously conducted study of frictional heating ofdisc with uniformly distributed heat ux has been made.

Fully three-dimensional analysis under non-axisymmetricthermal load provides information of realistic behaviour oftemperature alterations distinguishing period of heating (concavecurve) and cooling (convex curve) in the selected spot on the fric-tion surface during both single braking to full stop and brakingduring mountain descent with the constant velocity.

Based on the investigated individual cases of single brakingfrom the different initial velocities, it may be observed that thecompatibility of two- and three-dimensional model lowers withthe decrease of the velocity of the vehicle. The above axisymmetricsolution of the temperature elds of disc indicates that the solu-tion is reliable if the angular velocity of disc exceeds u0 44.232 s1.

The developed nite element analysis of friction heating of discduring emergency braking has conrmed the solution in the two-dimensionality feasible further to carry the fully transient simula-tionwith the time dependent material properties and coefcient offriction due to adequately low computer storage requirements.

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Analysis of disc brake temperature distribution during single braking under non-axisymmetric loadIntroductionStatement of the problemMathematical modelFE formulationResults and discussionConclusionsReferences