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UNCORRECTED PROOF 1Three-dimensional FE model for the calculation of temperature of a disc brake at 2 temperature-dependent coeffi cients of friction 3A.A.Q1Yevtushenko, A. Adamowicz, P. Grzes 4Faculty of Mechanical Engineering, Bialystok University of Technology (BUT), 45C Wiejska Street, Bialystok 15-351, Poland 5 6 a b s t r a c ta r t i c l ei n f o 7 8Available online xxxx 9 1011 12Keywords: 13Frictional heating 14Temperature 15Pad/disc brake system 16 Temperature-dependent coeffi cient of friction 17Finite element method 18 The three-dimensional transient temperature fi eld of a disc brake generated during a single and a multiple 19 braking process at temperature-dependent and constant coeffi cients of friction was analyzed. The calcula- 20tions were performed for the two materials of a pad (FC-16L and FMC-11) combined with the cast-iron 21 (ChNMKh) disc by using the fi nite element method (FEM). Analytical dependencies of the coeffi cient of fric- 22tion on the temperature for these two friction pairs were obtained on the basis of the experimental data at 23 different values of the contact pressures. It was established that relatively slight fl uctuations of the coeffi cient 24of friction have direct impact on the contact temperature of the disc. The maximum temperature generated 25 during the single braking process at constant coeffi cient of friction in relation to the case incorporating 26 temperature-dependent coeffi cient of friction was underestimated by 14.4% for the friction pair FC-16L/ 27 ChNMKh (increase in the coeffi cient of friction by 23.1%), and overestimated by 4.6% for FMC-11/ChNMKh 28 (decrease in the coeffi cient of friction by 8.4%). 29 2013 Published by Elsevier Ltd. 3031 32 33 341. Introduction 35 The value of the coeffi cient of friction during braking may vary with 36the speed, load, thermophysical properties of materials, physical and 37chemical interactions on the working surfaces, temperature, etc. 1. 38 Hence it is important to establish which of these factors signifi cantly 39affect the relationship between the applied load and the resulting fric- 40tion force. 41A review of investigations of temperature in disc brake and clutch 42 systems by the fi nite element method was given in Ref. 2. It turned 43out that the publications, taking into account the dependence of the 44 coeffi cient of friction on temperature, are not numerous at all. 45An impact of thermomechanical properties of materials and di- 46mensions of the rectangular block sliding over the rigid foundation 47 on the temperature at temperature-dependent coeffi cient of friction 48and wear was studied in Ref. 3. A simulation was carried out by 49using Bouligand-differentiable Newtons method and an optimization 50method. 51 The axisymmetric transient temperature fi elds of the pad and 52the disc generated during a single braking process adopting various 53experimental and theoretical formulas for the heat partition ratio were 54calculatedinRef.4.Thecorrespondingsolutionofanon-linearheatcon- 55 duction problem concerning an infl uence of thermosensitivity of mate- 56rials on the temperature of the pad/disc brake system was obtained in 57 Ref. 5. An effect of variations of the temperature-dependent coeffi - 58cients of friction and thermomechanical wear rate for two materials of 59the pad combined with the cast-iron disc was studied in Ref. 6. 60A series of FEM studies on non-axisymmetric heating of the disc by 61 the heat fl ux moving with the intensity proportional to the specifi c 62 power of friction within an area of contact at constant coeffi cient of 63friction compose Refs. 79. The three-dimensional model for simula- 64tion of non-uniform disc heating within the framework of linear heat 65conduction was developed in Ref. 7. The parallels between the tem- 66perature evolutions of the three-dimensional model and the two- 67dimensional equivalent whose axisymmetric thermal load arose from 68 the average heat fl ux uniformly distributed on the contact surface of 69the disc in the circumferential direction were drawn. Both the temper- 70 atures on the contact surface as well as at the specifi c axial positions of 71the disc of these two models were confronted and compared. It was 72demonstrated that the temperature generated as an effect of axisym- 73metric heating of the disc coincides with the average temperature of 74the three-dimensional model above a certain critical slip speed. 75 An infl uence of the heat transfer coeffi cient corresponding with 76the extreme cooling conditions for automotive application on the tem- 77perature reached during a single and a multiple braking ordered to 78 give equal specifi c power of friction in every of the considered computa- 79tional case was studied in Ref. 8. The determined dependencies of the 80 temperature on the heat transfer coeffi cient revealed linear relationship 81and the slope of these curves was dependent upon the number of brake 82applications. 83Acomparativeanalysisofthermosensitiveandtemperatur 84e-independent materials of the pad/disc brake system was carried International Communications in Heat and Mass Transfer xxx (2013) xxxxxx Communicated by W.J. Minkowycz Corresponding author. E-mail address: .pl (A.A. Yevtushenko). ICHMT-02732; No of Pages 7 0735-1933/$ see front matter 2013 Published by Elsevier Ltd. /10.1016/j.icheatmasstransfer.2012.12.015 Contents lists available at SciVerse ScienceDirect International Communications in Heat and Mass Transfer journal homepage: Please cite this article as: A.A. Yevtushenko, et al., Three-dimensionalFEmodelforthecalculationoftemperatureofadiscbrakeattemperature- dependent coeffi cients of friction, Int. Commun. Heat Mass Transf. (2013), /10.1016/j.icheatmasstransfer.2012.12.015 UNCORRECTED PROOF 85out in Ref. 9. The experimental dependencies of the temperature- 86dependent thermophysical properties of the four materials of the 87pad and four materials of the disc were approximated giving mathe- 88matical formulas applied to the FE three-dimensional model of the 89 disc. An infl uence of variations of these properties both at constant 90and temperature-dependent heat partition ratio incorporating corre- 91sponding constant thermophysical parameters was studied. 92It should be noted that the axisymmetric model to determine the 93temperature in the disc brakes is valid when the overlap factor is 94close to the unity. This article is a generalization of the results of 95Ref. 6 on a three-dimensional case, i.e. for arbitrary values of the 96 overlap factor. The transient 3D temperature fi eld in the disc brake 97 generated at constant and temperature-dependent coeffi cient of fric- 98tion during a single and a multiple braking process is studied by using 99 the fi nite element method. 1002. Statement of the problem 101Let at the initial time moment, a solid disc of a pad/disc brake sys- 102tem rotates with an angular speed 0(Fig. 1). The immovable pads 103are pressed to the outboard and inboard friction surfaces of the disc 104generating constant and uniformly distributed contact pressure p0, 105which resists the movement and the angular speed decreases linearly 106in time t 01 t ts ? ; 0tts:1 107108 109The heat generated due to friction is dissipated through conduc- 110tion within the bodies being in contact and convection from the free 111surfaces of the system. 112Furthermore it is assumed that: 113 the materials of the pad and the disc are isotropic and their 114thermophysical properties are temperature-independent; 115 the coeffi cient of friction depends on temperature; 116 the convective heat exchange with the surrounding air according 117to Newtons law of cooling at the constant and average heat trans- 118 fer coeffi cient h takes place on the exposed surfaces of the disc; 119 radiation mode of heat transfer is ignored; 120 by virtue of the symmetry of the problem about the mid-plane of 121the disc, the computational region is restricted exclusively to the 122half of the entire disc volume with the thickness d. 123Obviously, it should be stated that the temperature dependence of 124 the coeffi cient of friction during braking at the constant and uniform 125 contact pressure has its direct infl uence on the deceleration of the 126disc and consequently may vary the braking time ts. However, this ef- 127fect is not taken into account. 128At abovementioned assumptions the transient temperature distri- 129bution T(r,z,t) inthediscisobtainedfrom thesolution ofthefollowing 130boundary-value problem of heat conduction given in the cylindrical co- 131ordinate system (Fig. 1): 2T r2 1 r T r 1 r2 2T 2 2T z2 1 kd T t T ? ; rdrRd; 02; dbzb0; t 0; 2 132133at the following boundary conditions (Fig. 1):on the contact surface of 134the disc KdT z z0 g t qd; rprRp; 02; 0tts; 1g t ?h TT r;t?; rprRp; 02; t0; ? ? ? ? 3 135136and on the free surfaces of the disc KdT zz0 h TT r;t?; rdrrp; 02; t0; ? ? ?4 Nomenclature c specifi c heat, (J/(kgK) Cheat capacity matrix f coeffi cient of friction g(t)dimensionless function modelling transition of the heating area h heat transfer coeffi cient, (W/(m2K) kthermal diffusivity, (m2/s) Kthermal conductivity, (W/(mK) Kconductivity matrix p0contact pressure, (MPa) q intensity of the heat fl ux, (W/m2) Qlvector of applied linear thermal load Qnvector of applied thermal load that depends on the temperature rradial coordinate r,Rinternal and external radii, respectively, (m) ttime, (s) tsbraking time, (s) Ttemperature, (C) Tambient temperature, (C) T0initial temperature, (C) Ttemperature vector zaxial coordinate Greek symbols heat partition ratio thickness, (m) circumferential coordinate 0cover angle of pad, (deg) density, (kg/m3) relative angular slip speed, (s1) 0initial relative angular slip speed, (s1) Subscripts dindicates pad nnth time step pindicates disc Fig. 1. A diagram of a disc brake with the FE mesh and boundary conditions. 2A.A. Yevtushenko et al. / International Communications in Heat and Mass Transfer xxx (2013) xxxxxx Please cite this article as: A.A. Yevtushenko, et al., Three-dimensionalFEmodelforthecalculationoftemperatureofadiscbrakeattemperature- dependent coeffi cients of friction, Int. Commun. Heat Mass Transf. (2013), /10.1016/j.icheatmasstransfer.2012.12.015 UNCORRECTED PROOF 137138 Kd T rrRd h TT ;z;t?; 02; dz0; t0; ? ? ? ? 5 139140 Kd T rrrd h T ;z;tT?; 02; dz0; t0; ? ? ? ? 6 141142 T zzd 0; rdrRd; 02; t0; z d: ? ? ? ? 7 143144 145At the initial time moment t=0 the disc is heated to the constant 146temperature: T r;z;0 T0; rdrRd; 02; dz0:8 147148 149The dimensionless function g(t) in the condition (3) simulates the 150successive transition of the heating area over the disc contact surface 151 during braking 7,8. The intensity of the heat fl ux qddirected into the 152disc is calculated from the formula qd f T p0 t r; rprRp; 02; 0tts;9 153154where 10 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Kddcd p ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Kddcd p ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Kppcp q:10 155156 157 Justifi cation of the choice of Charrons formula (10) at calculation 158of the heat partition ratio can be found in the article 4. 1593. Finite element analysis 160A considered transient problem of heat conduction with the linear 161and temperature-dependent thermal load leads to the following ma- 162trix formulation C ? d dt Tf g K ? Tf g Qfgl Qfgn;11 163164appeared at the right-hand side vector of applied boundary heat 165 fl uxes Qland Qnstand for independent and dependent on the 166temperature thermal load, respectively. 167A solution to the transient problem with non-linear terms arising 168 from temperature-dependent coeffi cient of friction is obtained by 169using a difference equation approximation to Eq. (11) with a param- 170eter that is adjusted to give a compromise between the stability, ef- 171 fi ciency and high accuracy requirements. In this respect the form of 172the difference equation analogous to the Newmark method is 11 K ? Tf gn1 1 Tf gn ? 1 t C ?Tf gn1 Tf gn ? Qfgln1 1 Qfgln no 1 Qfgn n Q fgn n1; 12 173174where the subscript n stands for the nth time step. The free parameter 175 may be selected in the range 0bb1. Eq. (12) can be rearranged 176giving the iteration algorithm 1 t C ? K ? ? Tf gn1 1 t C ? 1 K ? ? Tf gn Qfgln1 1 Qfgln no 1 Qfgn n Q fgn n1: 13 177178 179The numerical calculations satisfying the abovementioned mathe- 180 matical description were carried out by using MD Nastran fi nite 181element based software package 12. The boundary conditions 182concerning both the cooling and heating of the disc within the friction 183surface were developed separately using the Python programming 184 language. Neglecting an infl uence of the fl uctuations of the coeffi cient 185of friction resulting from the temperature change on the operation 186timeandtheevolutionofaspeedofthedisc,thedimensionlessfunction 187g(t) appeared in Eq. (3) was calculated having in mind exclusively di- 188mensions of thebrake system as well as thedeceleration of the rotating 189 disc (Eq. (1). However an effect of the variations of the coeffi cient of 190friction on the temperature did manifest by the amount of heat propor- 191 tional to an increase (decrease) in the coeffi cient of friction. 192To assure high accuracy of numerical calculations, the numbering 193of elements being comprised within the contact area, corresponded 194 strictly to the boundary heat fl ux simulating non-axisymmetric 195heating due to friction of the stationary pad and the rotating disc. 196Thereby the dimension of the elements in the circumferential direc- 197tion was equal for every path at the given radius. Eventually an area 198of the disc adjoining with an area of contact ( rprRp;02) 199comprised 7200 hex-type eight-node elements in the number of 200360 in circumferential direction and 20 elements in radial direction 201of thickness of 1 mm. The remaining area of the disc was created au- 202tomatically based on the geometry of the disc by means of 71,520 203tet-type four-node elements. Overall the combined FE mesh for nu- 204merical analysis consisted of 78,720 elements and 25,356 nodes 205(Fig. 1). 2064. Numerical analysis 207 The three-dimensional transient temperature fi eld of the disc sub- 208 ject to the non-axisymmetric thermal load was obtained using the fi - 209nite element based software package 12. The modelling technique 210of the problem of the moving heat source acting on the contact sur- 211 face of the disc as the heat fl ux with the intensity proportional to 212 the specifi c power of friction was proposed in Refs. 7,8. In this 213 paper unlike previous calculations an infl uence of the fl uctuations of 214 the coeffi cient of friction resulting from the temperature changes on 215the temperature is taken into account. Owing to the comparison of 216 the temperature fi eld generated during the process at the constant 217 (temperature of 20 C) and temperature-dependent coeffi cient of 218friction, the experimental data determined for different materials 219 and at specifi ed contact pressures were adopted from 13. Based on 220these data the corresponding mathematical formulas adequate for 221the range of temperatures from 20 to 800 C for the friction pair 222FC-16L/ChNMKh f T 0:286 1 0:55102T105 ?2 0:286 1 0:25102T800 ?2at p0 0:39 MPa; 14 223224 f T 0:02 0:288 1 0:7102T95 ?2 0:2 1 0:3102T800 ?2at p0 1:47 MPa; 15 225226and FMC-11/ChNMKh f T 6:35103 0:762 1 0:19102T 180 ?2at p0 0:59 MPa; 16 227228 f T 0:0321 0:642 1 0:19102T 180 ?2 at p0 0:78 MPa; 17 229230 f T 0:0285 0:57 1 0:162102T 2502 ?at p0 1:18 MPa; 18 3A.A. Yevtushenko et al. / International Communications in Heat and Mass Transfer xxx (2013) xxxxxx Please cite this article as: A.A. Yevtushenko, et al., Three-dimensionalFEmodelforthecalculationoftemperatureofadiscbrakeattemperature- dependent coeffi cients of friction, Int. Commun. Heat Mass Transf. (2013), /10.1016/j.icheatmasstransfer.2012.12.015 UNCORRECTED PROOF 231232 f T 0:036 0:48 1 0:15102T 250 ?2 at p0 1:47 MPa; 19 233234were obtained (Fig. 2). The detailed description of the method employed 235todeterminetheseformulaswasgiveninRef.6.Thecorrespondingcon- 236 stant coeffi cients of friction calculated from Eqs. (14) to (19) at tempera- 237ture of 20 C are tabulated in Table 1. It should be noted that considering 238different contact pressures at the same initial velocity, braking time and 239constant deceleration, each computational case corresponds with differ- 240 ent specifi c power of friction. 241Two parallel computational cases incorporating constant and 242 temperature-dependent coeffi cients of friction during a single braking 243process with constant deceleration from the initial speed of the disc 244of 0=88.464 s1to standstillduringthetimets=3.96 s were devel- 245oped 14. Further simulation of a multiple braking was conducted 246according to the diagram shown in Fig. 3, from which it can be seen 247that at the beginning, the single braking process takes place with the 248abovementioned parameters, then the brake is released and the speed 249 increases linearly with time during 15 s. The cycle is repeated fi ve 250times followed by the braking to standstill with the same deceleration 251as in the previous brake applications. 252The