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UNCORRECTED PROOF1Three-dimensional FE model for the calculation of temperature of a disc brake at2temperature-dependent coefficients of friction3A.A.Q1Yevtushenko, A. Adamowicz, P. Grzes4Faculty of Mechanical Engineering, Bialystok University of Technology (BUT), 45C Wiejska Street, Bialystok 15-351, Poland56a b s t r a c ta r t i c l ei n f o78Available online xxxx9 101112Keywords:13Frictional heating14Temperature15Pad/disc brake system16Temperature-dependent coefficient of friction17Finite element method18The three-dimensional transient temperature field of a disc brake generated during a single and a multiple19braking process at temperature-dependent and constant coefficients 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-iron21(ChNMKh) disc by using the finite element method (FEM). Analytical dependencies of the coefficient of fric-22tion on the temperature for these two friction pairs were obtained on the basis of the experimental data at23different values of the contact pressures. It was established that relatively slight fluctuations of the coefficient24of friction have direct impact on the contact temperature of the disc. The maximum temperature generated25during the single braking process at constant coefficient of friction in relation to the case incorporating26temperature-dependent coefficient of friction was underestimated by 14.4% for the friction pair FC-16L/27ChNMKh (increase in the coefficient of friction by 23.1%), and overestimated by 4.6% for FMC-11/ChNMKh28(decrease in the coefficient of friction by 8.4%).29 2013 Published by Elsevier Ltd.30313233341. Introduction35The value of the coefficient of friction during braking may vary with36the speed, load, thermophysical properties of materials, physical and37chemical interactions on the working surfaces, temperature, etc. 1.38Hence it is important to establish which of these factors significantly39affect the relationship between the applied load and the resulting fric-40tion force.41A review of investigations of temperature in disc brake and clutch42systems by the finite element method was given in Ref. 2. It turned43out that the publications, taking into account the dependence of the44coefficient 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 foundation47on the temperature at temperature-dependent coefficient of friction48and wear was studied in Ref. 3. A simulation was carried out by49using Bouligand-differentiable Newtons method and an optimization50method.51The axisymmetric transient temperature fields of the pad and52the disc generated during a single braking process adopting various53experimental and theoretical formulas for the heat partition ratio were54calculatedinRef.4.Thecorrespondingsolutionofanon-linearheatcon-55duction problem concerning an influence of thermosensitivity of mate-56rials on the temperature of the pad/disc brake system was obtained in57Ref. 5. An effect of variations of the temperature-dependent coeffi-58cients of friction and thermomechanical wear rate for two materials of59the pad combined with the cast-iron disc was studied in Ref. 6.60A series of FEM studies on non-axisymmetric heating of the disc by61the heat flux moving with the intensity proportional to the specific62power of friction within an area of contact at constant coefficient of63friction compose Refs. 79. The three-dimensional model for simula-64tion of non-uniform disc heating within the framework of linear heat65conduction 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 from68the average heat flux uniformly distributed on the contact surface of69the disc in the circumferential direction were drawn. Both the temper-70atures on the contact surface as well as at the specific axial positions of71the disc of these two models were confronted and compared. It was72demonstrated that the temperature generated as an effect of axisym-73metric heating of the disc coincides with the average temperature of74the three-dimensional model above a certain critical slip speed.75An influence of the heat transfer coefficient corresponding with76the extreme cooling conditions for automotive application on the tem-77perature reached during a single and a multiple braking ordered to78give equal specific power of friction in every of the considered computa-79tional case was studied in Ref. 8. The determined dependencies of the80temperature on the heat transfer coefficient revealed linear relationship81and the slope of these curves was dependent upon the number of brake82applications.83Acomparativeanalysisofthermosensitiveandtemperatur84e-independent materials of the pad/disc brake system was carriedInternational Communications in Heat and Mass Transfer xxx (2013) xxxxxx Communicated by W.J. Minkowycz Corresponding author.E-mail address: a.yevtushenko.pl (A.A. Yevtushenko).ICHMT-02732; No of Pages 70735-1933/$ see front matter 2013 Published by Elsevier Ltd./10.1016/j.icheatmasstransfer.2012.12.015Contents lists available at SciVerse ScienceDirectInternational Communications in Heat and Mass Transferjournal homepage: /locate/ichmtPlease cite this article as: A.A. Yevtushenko, et al., Three-dimensionalFEmodelforthecalculationoftemperatureofadiscbrakeattemperature-dependent coefficients of friction, Int. Commun. Heat Mass Transf. (2013), /10.1016/j.icheatmasstransfer.2012.12.015UNCORRECTED PROOF85out in Ref. 9. The experimental dependencies of the temperature-86dependent thermophysical properties of the four materials of the87pad and four materials of the disc were approximated giving mathe-88matical formulas applied to the FE three-dimensional model of the89disc. An influence of variations of these properties both at constant90and temperature-dependent heat partition ratio incorporating corre-91sponding constant thermophysical parameters was studied.92It should be noted that the axisymmetric model to determine the93temperature in the disc brakes is valid when the overlap factor is94close to the unity. This article is a generalization of the results of95Ref. 6 on a three-dimensional case, i.e. for arbitrary values of the96overlap factor. The transient 3D temperature field in the disc brake97generated at constant and temperature-dependent coefficient of fric-98tion during a single and a multiple braking process is studied by using99the finite element method.1002. Statement of the problem101Let 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 pads103are pressed to the outboard and inboard friction surfaces of the disc104generating constant and uniformly distributed contact pressure p0,105which resists the movement and the angular speed decreases linearly106in time t 01tts?; 0tts:1107108109The heat generated due to friction is dissipated through conduc-110tion within the bodies being in contact and convection from the free111surfaces of the system.112Furthermore it is assumed that:113 the materials of the pad and the disc are isotropic and their114thermophysical properties are temperature-independent;115 the coefficient of friction depends on temperature;116 the convective heat exchange with the surrounding air according117to Newtons law of cooling at the constant and average heat trans-118fer coefficient 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 of121the disc, the computational region is restricted exclusively to the122half of the entire disc volume with the thickness d.123Obviously, it should be stated that the temperature dependence of124the coefficient of friction during braking at the constant and uniform125contact pressure has its direct influence on the deceleration of the126disc 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 ofthefollowing130boundary-value problem of heat conduction given in the cylindrical co-131ordinate system (Fig. 1):2Tr21rTr1r22T22Tz21kdTt T?; rdrRd; 02; dbzb0; t 0;2132133at the following boundary conditions (Fig. 1):on the contact surface of134the discKdTzz0g t qd; rprRp; 02; 0tts;1g t ?h TT r;t?; rprRp; 02; t0;?3135136and on the free surfaces of the discKdTzz0 h TT r;t?; rdrrp; 02; t0;?4Nomenclaturecspecific heat, (J/(kgK)Cheat capacity matrixfcoefficient of frictiong(t)dimensionless function modelling transition of theheating areahheat transfer coefficient, (W/(m2K)kthermal diffusivity, (m2/s)Kthermal conductivity, (W/(mK)Kconductivity matrixp0contact pressure, (MPa)qintensity of the heat flux, (W/m2)Qlvector of applied linear thermal loadQnvector of applied thermal load that depends on thetemperaturerradial coordinater,Rinternal and external radii, respectively, (m)ttime, (s)tsbraking time, (s)Ttemperature, (C)Tambient temperature, (C)T0initial temperature, (C)Ttemperature vectorzaxial coordinateGreek symbolsheat partition ratiothickness, (m)circumferential coordinate0cover angle of pad, (deg)density, (kg/m3)relative angular slip speed, (s1)0initial relative angular slip speed, (s1)Subscriptsdindicates padnnth time steppindicates discFig. 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) xxxxxxPlease cite this article as: A.A. Yevtushenko, et al., Three-dimensionalFEmodelforthecalculationoftemperatureofadiscbrakeattemperature-dependent coefficients of friction, Int. Commun. Heat Mass Transf. (2013), /10.1016/j.icheatmasstransfer.2012.12.015UNCORRECTED PROOF137138KdTrrRd h TT ;z;t?; 02; dz0; t0;?5139140KdTrrrd h T ;z;tT?; 02; dz0; t0;?6141142Tzzd 0; rdrRd; 02; t0; z d:?7143144145At the initial time moment t=0 the disc is heated to the constant146temperature:T r;z;0 T0; rdrRd; 02; dz0:8147148149The dimensionless function g(t) in the condition (3) simulates the150successive transition of the heating area over the disc contact surface151during braking 7,8. The intensity of the heat flux qddirected into the152disc is calculated from the formulaqd f T p0 t r; rprRp; 02; 0tts;9153154where 10 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKddcdpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKddcdpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKppcpq:10155156157Justification of the choice of Charrons formula (10) at calculation158of the heat partition ratio can be found in the article 4.1593. Finite element analysis160A considered transient problem of heat conduction with the linear161and temperature-dependent thermal load leads to the following ma-162trix formulationC ?ddtTf g K ? Tf g Qfgl Qfgn;11163164appeared at the right-hand side vector of applied boundary heat165fluxes Qland Qnstand for independent and dependent on the166temperature thermal load, respectively.167A solution to the transient problem with non-linear terms arising168from temperature-dependent coefficient of friction is obtained by169using a difference equation approximation to Eq. (11) with a param-170eter that is adjusted to give a compromise between the stability, ef-171ficiency and high accuracy requirements. In this respect the form of172the difference equation analogous to the Newmark method is 11K ? Tf gn1 1 Tf gn?1tC ?Tf gn1 Tf gn? Qfgln1 1 Qfglnno 1 Qfgnn Qfgnn1;12173174where the subscript n stands for the nth time step. The free parameter175 may be selected in the range 0bb1. Eq. (12) can be rearranged176giving the iteration algorithm1tC ? K ?Tf gn11tC ? 1 K ?Tf gn Qfgln1 1 Qfglnno 1 Qfgnn Qfgnn1:13177178179The numerical calculations satisfying the abovementioned mathe-180matical description were carried out by using MD Nastran finite181element based software package 12. The boundary conditions182concerning both the cooling and heating of the disc within the friction183surface were developed separately using the Python programming184language. Neglecting an influence of the fluctuations of the coefficient185of friction resulting from the temperature change on the operation186timeandtheevolutionofaspeedofthedisc,thedimensionlessfunction187g(t) appeared in Eq. (3) was calculated having in mind exclusively di-188mensions of thebrake system as well as thedeceleration of the rotating189disc (Eq. (1). However an effect of the variations of the coefficient of190friction on the temperature did manifest by the amount of heat propor-191tional to an increase (decrease) in the coefficient of friction.192To assure high accuracy of numerical calculations, the numbering193of elements being comprised within the contact area, corresponded194strictly to the boundary heat flux simulating non-axisymmetric195heating 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 area198of the disc adjoining with an area of contact ( rprRp;02)199comprised 7200 hex-type eight-node elements in the number of200360 in circumferential direction and 20 elements in radial direction201of thickness of 1 mm. The remaining area of the disc was created au-202tomatically based on the geometry of the disc by means of 71,520203tet-type four-node elements. Overall the combined FE mesh for nu-204merical analysis consisted of 78,720 elements and 25,356 nodes205(Fig. 1).2064. Numerical analysis207The three-dimensional transient temperature field of the disc sub-208ject to the non-axisymmetric thermal load was obtained using the fi-209nite element based software package 12. The modelling technique210of the problem of the moving heat source acting on the contact sur-211face of the disc as the heat flux with the intensity proportional to212the specific power of friction was proposed in Refs. 7,8. In this213paper unlike previous calculations an influence of the fluctuations of214the coefficient of friction resulting from the temperature changes on215the temperature is taken into account. Owing to the comparison of216the temperature field generated during the process at the constant217(temperature of 20 C) and temperature-dependent coefficient of218friction, the experimental data determined for different materials219and at specified contact pressures were adopted from 13. Based on220these data the corresponding mathematical formulas adequate for221the range of temperatures from 20 to 800 C for the friction pair222FC-16L/ChNMKhf T 0:2861 0:55102T105?20:2861 0:25102T800?2at p0 0:39 MPa;14223224f T 0:02 0:2881 0:7102T95?20:21 0:3102T800?2at p0 1:47 MPa;15225226and FMC-11/ChNMKhf T 6:351030:7621 0:19102T 180?2at p0 0:59 MPa;16227228f T 0:0321 0:6421 0:19102T 180?2at p0 0:78 MPa; 17229230f T 0:0285 0:571 0:162102T 2502?at p0 1:18 MPa;183A.A. Yevtushenko et al. / International Communications in Heat and Mass Transfer xxx (2013) xxxxxxPlease cite this article as: A.A. Yevtushenko, et al., Three-dimensionalFEmodelforthecalculationoftemperatureofadiscbrakeattemperature-dependent coefficients of friction, Int. Commun. Heat Mass Transf. (2013), /10.1016/j.icheatmasstransfer.2012.12.015UNCORRECTED PROOF231232f T 0:036 0:481 0:15102T 250?2at p0 1:47 MPa;19233234were obtained (Fig. 2). The detailed description of the method employed235todeterminetheseformulaswasgiveninRef.6.Thecorrespondingcon-236stant coefficients of friction calculated from Eqs. (14) to (19) at tempera-237ture of 20 C are tabulated in Table 1. It should be noted that considering238different contact pressures at the same initial velocity, braking time and239constant deceleration, each computational case corresponds with differ-240ent specific power of friction.241Two parallel computational cases incorporating constant and242temperature-dependent coefficients of friction during a single braking243process with constant deceleration from the initial speed of the disc244of 0=88.464 s1to standstillduringthetimets=3.96 s were devel-245oped 14. Further simulation of a multiple braking was conducted246according to the diagram shown in Fig. 3, from which it can be seen247that at the beginning, the single braking process takes place with the248abovementioned parameters, then the brake is released and the speed249increases linearly with time during 15 s. The cycle is repeated five250times followed by the braking to standstill with the same deceleration251as in the previous brake applications.252The dimensions of the pad/disc system are listed in Table 2. The253heat transfer coefficient is equal h=60 W/(m2K) both for the single254and the multiple braking having its main purpose in estimation of the255temperature difference calculated at the temperature-dependent and256constant coefficients of friction. The value of h parameter was derived257from the process of a single braking with constant deceleration of a258vehicle from initial speed of 100 km/h (0=88.464 s1) to stand-259still 14, however other calculations using the same materials and di-260mensions of the pad and the disc demonstrated that for extreme261cooling conditions in the range of the heat transfer coefficient varying262from 0 to 100 W/(m2K) for fivefold braking with constant speed263proceeded during 55 s, the temperature difference was relatively264small and equal T=69.4 C 8.265Initially, the disc has a uniform temperature T0=20 C throughout266its body, and the ambient temperature is equal to T=20 C. The267thermophysical properties of the materials of the disc and the pad are268constant and listed in Table 3.269It was assumed that the entire mechanical energy is converted into270heat at the pad/disc interface (z=0). This axial position was chosen271asthe sourceof heating, therebydeterminant of thetemperature distri-272bution of the pad/disc system. Furthermore both the circumferential273and radial positions were fixed for the thermal analysis. The point r=274113.5 mm, z=0, =0 being the maximum radius of the rubbing275path of the disc equal to the pad maximum radius (approximately theFig. 2. Temperature dependencies of the coefficients of friction in the range from 20 to800 C 13.Table 1t1:1t1:2Coefficients of friction calculated from Eqs. (14)(19) at temperature of 20 C.t1:3Friction pairPressure, p0MPaCoefficient of friction f attemperature of 20 Ct1:4FC-16L/ChNMKh0.390.294t1:51.470.277t1:6FMC-11/ChNMKh0.590.672t1:70.780.593t1:81.180.507t1:91.470.448Fig. 3. Evolutions of the dimensionless angular speed of the disc during multiple braking.Table 2t2:1t2:2Dimensions of the pad and the disc 14.t2:3ItemsDiscPadt2:4Inner radius, r mm6676.5t2:5Outer radius, R mm113.5t2:6Thickness, mm5.510t2:7Pad arc length, 0deg64.54A.A. Yevtushenko et al. / International Communications in Heat and Mass Transfer xxx (2013) xxxxxxPlease cite this article as: A.A. Yevtushenko, et al., Three-dimensionalFEmodelforthecalculationoftemperatureofadiscbrakeattemperature-dependent coefficients of friction, Int. Commun. Heat Mass Transf. (2013), /10.1016/j.icheatmasstransfer.2012.12.015UNCORRECTED PROOF276highest possible temperature within the disc volume due to the maxi-277mum slip speed) (see Table 2) will be considered.278The temperature evolutions on the friction surface of the disc (r=279113.5 mm, =0) during the single braking process for the cast-iron280(ChNMKh) disc combined with the pad made of FC-16L are shown in281Fig. 4. The grey lines correspond with the calculations carried out at282the temperature-dependent coefficient of friction, whereas the black283lines represent braking at the constant coefficient of friction corre-284sponding with the temperature of 20 C. Such annotation will be285remained in every of the presented further figures. Two successive286stages can be defined from the depicted temperature curves, namely287the heating and the cooling phases resulted directly from the transi-288tion of the heating area (the pad contact surface) over the fixed ana-289lyzed spot within the rubbing path of the disc followed by the cooling290during when there is no contact with the pad. The general nature of291these evolutions reveals a rapid increase in temperature at the begin-292ning of the process, then the maximum value is reached and eventu-293ally the temperature decreases slightly. The phenomenon is naturally294connected with the relationship between the magnitude of the ther-295mal load and its simultaneous absorption through conduction and296convection, however the latter is negligible during normal single297braking conditions. Two applied contact pressures p0=0.39 MPa298and p0=1.47 MPa result in higher temperatures for the case incor-299porating the temperature-dependent coefficient of friction (grey300lines) which agrees well with the dependencies shown for that fric-301tion pair in Fig. 2a. The maximum attained temperature is equal to302T=112.3 C (t=2.727 s). The maximum temperature difference be-303tween the case with the temperature-dependent and constant coeffi-304cient of friction at p0=1.47 MPa T=16.2 C occurs at the time305moment t=2.727 s. At the pressure p0=0.39 MPa the maximum306temperature difference is clearly lower and is equal to T=1.2 C.307It may be observed that when the coefficient of friction increases308from the temperature of 20 C to the maximum attained temperature309of 112.3 C by 23.1% at p0=1.47 MPa (Fig. 2a), the related to that310case temperature calculated at constant coefficient of friction (T=31196.1 C) is underestimated by 14.4%.312The corresponding temperature evolutions for the single braking313process on the contact surface of the disc (r=113.5 mm, =0) for314the friction pair FMC-11/ChNMKh are shown in Fig. 5. In addition315the calculated temperatures were confronted with the temperature316curve corresponding with the same input parameters and at constant317coefficient of friction f=0.5 from the article 9. Naturally higher318value of the coefficient of friction than that used in this study results319in adequately higher temperature.320However anorderoftheappearedtemperatures of 100 C (Fig. 5) at321the pressure p0=1.47 MPa is the same as for the friction pair FC-16L/322ChNMKh (Fig. 4), in this case the maximum temperature difference323is definitely lower and equals T=4.4 C. The time when this324temperature (101.5 C at constant coefficient of friction and 97.1 C at325temperature dependent coefficient of friction) is reached (t=2.727 s)326and is the same as for the friction pair FMC-11/ChNMKh (Fig. 4).Q2327It may be also observed that the temperature difference between328the case with the temperature-dependent and constant coefficient of329friction decreases successively with the decrease in contact pressure330since the level of temperatures decreases. It may be estimated that331when the coefficient of friction decreases from the temperature of33220 C to the maximum attained temperature of 97.1 C by 8.4% (p0=3331.47 MPa),therelatedtothatcasetemperaturereachedattheconstant334coefficient of friction is overestimated by 4.6%.335Fig. 6 shows the temperature evolutions obtained during the multi-336ple braking process for the disc made of ChNMKh and the pad made of337FC-16L according to the diagram of the applied angular speed shown338in Fig. 3. Two temperature curves represent the values calculated at339temperature-dependent coefficient of friction (grey lines) as well as at340constant coefficient of friction corresponding with the temperature of34120 C (black lines). As it can be seen that the temperature for the342temperature-dependent coefficient of friction at the beginning of343the process is higher than that obtained at constant coefficient of344friction and the temperature difference increases with an increase inFig. 4. Evolutionsofthenodaltemperatureonthecontactsurfaceofthedisc(r=113.5 mm,0=0) during the single braking process at constant (black lines) and temperature-dependent (grey lines) coefficients of friction (pad FC-16L/disc ChNMKh).Fig. 5. Evolutions of the nodal temperature on the contact surface of the disc (r=113.5 mm, 0=0) during the single braking process at constant (black lines) andtemperature-dependent (grey line) coefficients of friction (pad FMC-11/disc ChNMKh).Table 3t3:1t3:2Thermophysical properties of materials of the pad and the disc at temperature of 20 Ct3:313.t3:4MaterialK W/(mK)k106m2/sc J/(kgK) kg/m3t3:5ChNMKh5114.45007100t3:6FC-16L0.790.339612500t3:7FMC-1134.314.650047005A.A. Yevtushenko et al. / International Communications in Heat and Mass Transfer xxx (2013) xxxxxxPlease cite this article as: A.A. Yevtushenko, et al., Three-dimensionalFEmodelforthecalculationoftemperatureofadiscbrakeattemperature-dependent coefficients of friction, Int. Commun. Heat Mass Transf. (2013), /10.1016/j.icheatmasstransfer.2012.12.015UNCORRECTED PROOF345temperature up to the time of about 20 s, then remains on the same346level to the time moment t40 s, and eventually decreases. This effect347agrees with the temperature dependence of the coefficient of friction348shown in Fig. 2a whosevalueincreaseswithanincrease intemperature349to the value of 96.3 C (maximum) and then decreases. The maximum350temperature difference equals T=26.3 C and is attained at the351time moment t=56.9 s.352Thetemperatureevolutionsonthecontactsurfaceofthediscfor the353friction pair FMC-11/ChNMKh calculated during the multiple braking354process are shown in Fig. 7. Unlike the corresponding results obtained355for the friction pair ChNMKh/FC-16L, these outcomes reveal an approx-356imately linear increase in temperature difference throughout the entire357process of the multiple braking, which corresponds with the depen-358dence of the coefficient of friction on the temperature and approved359its direct influence on the resulting temperature. The maximum360temperature difference equals 47.6 C and is attained at the end of the361multiple braking.3625. Summary and conclusions363In this study a simulation of a braking process for a pad/disc brake364system was carried out. A three-dimensional temperature field of365a disc subject to non-axisymmetric thermal load at the constant366and temperature-dependent coefficients of friction was obtained by367using the finite element method. Both the single and the multiple368braking processes with the constant deceleration were studied. The369calculated temperature evolutions at the temperature-dependent co-370efficients of friction on the contact surface of the disc were confronted371with the corresponding values determined at the constant coefficients372of friction (at temperature of 20 C). It was established that:373 for the friction pair FC-16L/ChNMKh at p0=1.47 MPa when the374coefficient of friction increases from the temperature of 20 C to375the maximum attained temperature of the friction surface of the376disc (T=112.3 C, grey line, Fig. 4) at time moment t=2.727 s by37723.1%, the related to that case temperature calculated at constant378coefficient of friction T=96.1 C is underestimated by 14.4%;379 for the friction pair FMC-11/ChNMKh at p0=1.47 MPa when the380coefficient of friction decreases from the temperature of 20 C to381the maximum attained temperature of the contact surface of the382disc (T=97.1 C, grey line, Fig. 5) at time moment t=2.727 s by3838.4%, the related to that case temperature calculated at constant384coefficient of friction (T=101.5 C) is overestimated by 4.6%;385 under the conditions of the multiple braking both for FC-16L/386ChNMKh (Fig. 6) as well as FMC-11/ChNMKh (Fig. 7), the tempera-387ture evolutions determined on the friction surface of the disc agree388with the established temperature dependencies of the coefficients389of friction (Fig. 2a,b).390Obviously aiming to attain low temperatures reached during brak-391ing the coefficient of friction should be low, this however is in conflict392with the requirements concerning the highest moment of friction. The393carried out calculations exhibited that at relatively low temperatures394(T100 C) an influence of the coefficient of friction on the resulting395temperature is apparent.396Acknowledgement397The present article is financially supported by the National Science398Centre in Poland (research project no 2011/01/B/ST8/07446).399References4001 A.V. Chichinadze, Calculation and Investigation of External Friction During Braking,401Nauka, Moscow, 1967. (in Russian).4022 A.A. Yevtushenko, P. Grzes, FEM-modeling of the frictional heating phenomenon403in the pad/disc tribosystem (a review), Numeric