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Three-dimensional FE model for the calculation of temperature of a disc brake at temperature-dependent coefficients of friction
International Communications in Heat and Mass Transfer 42 (2013) 18–24
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Three-dimensional FE model for the calculation of temperature of a disc brake at temperature-dependent coefficients of friction☆ A.A. Yevtushenko ⁎, A. Adamowicz, P. Grzes Faculty of Mechanical Engineering, Bialystok University of Technology (BUT), 45C Wiejska Street, Bialystok 15-351, Poland
a r t i c l e
i n f o
Available online 2 January 2013 Keywords: Frictional heating Temperature Pad/disc brake system Temperature-dependent coefficient of friction Finite element method
a b s t r a c t The three-dimensional transient temperature field of a disc brake generated during a single and a multiple braking process at temperature-dependent and constant coefficients of friction was analyzed. The calculations were performed for the two materials of a pad (FC-16L and FMC-11) combined with the cast-iron (ChNMKh) disc by using the finite element method (FEM). Analytical dependencies of the coefficient of friction on the temperature for these two friction pairs were obtained on the basis of the experimental data at different values of the contact pressures. It was established that relatively slight fluctuations of the coefficient of friction have direct impact on the contact temperature of the disc. The maximum temperature generated during the single braking process at constant coefficient of friction in relation to the case incorporating temperature-dependent coefficient of friction was underestimated by 14.4% for the friction pair FC-16L/ ChNMKh (increase in the coefficient of friction by 23.1%), and overestimated by 4.6% for FMC-11/ChNMKh (decrease in the coefficient of friction by 8.4%). © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The value of the coefficient of friction during braking may vary with the speed, load, thermophysical properties of materials, physical and chemical interactions on the working surfaces, temperature, etc. [1]. Hence it is important to establish which of these factors significantly affect the relationship between the applied load and the resulting friction force. A review of investigations of temperature in disc brake and clutch systems by the finite element method was given in Ref. [2]. It turned out that the publications, taking into account the dependence of the coefficient of friction on temperature, are not numerous at all. An impact of thermomechanical properties of materials and dimensions of the rectangular block sliding over the rigid foundation on the temperature at temperature-dependent coefficient of friction and wear was studied in Ref. [3]. A simulation was carried out by using Bouligand-differentiable Newton's method and an optimization method. The axisymmetric transient temperature fields of the pad and the disc generated during a single braking process adopting various experimental and theoretical formulas for the heat partition ratio were calculated in Ref. [4]. The corresponding solution of a non-linear heat conduction problem concerning an influence of thermosensitivity of materials on the temperature of the pad/disc brake system was obtained in ☆ Communicated by W.J. Minkowycz ⁎ Corresponding author. E-mail address: [email protected] (A.A. Yevtushenko). 0735-1933/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2012.12.015
Ref. [5]. An effect of variations of the temperature-dependent coefficients of friction and thermomechanical wear rate for two materials of the pad combined with the cast-iron disc was studied in Ref. [6]. A series of FEM studies on non-axisymmetric heating of the disc by the heat flux moving with the intensity proportional to the specific power of friction within an area of contact at constant coefficient of friction compose Refs. [7–9]. The three-dimensional model for simulation of non-uniform disc heating within the framework of linear heat conduction was developed in Ref. [7]. The parallels between the temperature evolutions of the three-dimensional model and the twodimensional equivalent whose axisymmetric thermal load arose from the average heat flux uniformly distributed on the contact surface of the disc in the circumferential direction were drawn. Both the temperatures on the contact surface as well as at the specific axial positions of the disc of these two models were confronted and compared. It was demonstrated that the temperature generated as an effect of axisymmetric heating of the disc coincides with the average temperature of the three-dimensional model above a certain critical slip speed. An influence of the heat transfer coefficient corresponding with the extreme cooling conditions for automotive application on the temperature reached during a single and a multiple braking ordered to give equal specific power of friction in every of the considered computational case was studied in Ref. [8]. The determined dependencies of the temperature on the heat transfer coefficient revealed linear relationship and the slope of these curves was dependent upon the number of brake applications. A comparative analysis of thermosensitive and temperatur e-independent materials of the pad/disc brake system was carried
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Nomenclature c [C] f g(t) h k K [K] p0 q {Q} l {Q} n r r, R t ts T T∞ T0 {T} z
specific heat, (J/(kgK)) heat capacity matrix coefficient of friction dimensionless function modelling transition of the heating area heat transfer coefficient, (W/(m 2K)) thermal diffusivity, (m 2/s) thermal conductivity, (W/(mK)) conductivity matrix contact pressure, (MPa) intensity of the heat flux, (W/m 2) vector of applied linear thermal load vector of applied thermal load that depends on the temperature radial coordinate internal and external radii, respectively, (m) time, (s) braking time, (s) temperature, (°C) ambient temperature, (°C) initial temperature, (°C) temperature vector axial coordinate
Greek symbols γ heat partition ratio δ thickness, (m) θ circumferential coordinate θ0 cover angle of pad, (deg) ρ density, (kg/m 3) ω relative angular slip speed, (s −1) ω0 initial relative angular slip speed, (s −1)
Subscripts d indicates pad n nth time step p indicates disc
out in Ref. [9]. The experimental dependencies of the temperaturedependent thermophysical properties of the four materials of the pad and four materials of the disc were approximated giving mathematical formulas applied to the FE three-dimensional model of the disc. An influence of variations of these properties both at constant and temperature-dependent heat partition ratio incorporating corresponding constant thermophysical parameters was studied. It should be noted that the axisymmetric model to determine the temperature in the disc brakes is valid when the overlap factor is close to the unity. This article is a generalization of the results of Ref. [6] on a three-dimensional case, i.e. for arbitrary values of the overlap factor. The transient 3D temperature field in the disc brake generated at constant and temperature-dependent coefficient of friction during a single and a multiple braking process is studied by using the finite element method.
Fig. 1. A diagram of a disc brake with the FE mesh and boundary conditions.
generating constant and uniformly distributed contact pressure p0, which resists the movement and the angular speed decreases linearly in time t ωðt Þ ¼ ω0 1− ; 0≤t≤t s : ts
ð1Þ
The heat generated due to friction is dissipated through conduction within the bodies being in contact and convection from the free surfaces of the system. Furthermore it is assumed that: − the materials of the pad and the disc are isotropic and their thermophysical properties are temperature-independent; − the coefficient of friction depends on temperature; − the convective heat exchange with the surrounding air according to Newton's law of cooling at the constant and average heat transfer coefficient h takes place on the exposed surfaces of the disc; − radiation mode of heat transfer is ignored; − by virtue of the symmetry of the problem about the mid-plane of the disc, the computational region is restricted exclusively to the half of the entire disc volume with the thickness δd. Obviously, it should be stated that the temperature dependence of the coefficient of friction during braking at the constant and uniform contact pressure has its direct influence on the deceleration of the disc and consequently may vary the braking time ts. However, this effect is not taken into account. At abovementioned assumptions the transient temperature distribution T(r,z,θ,t) in the disc is obtained from the solution of the following boundary-value problem of heat conduction given in the cylindrical coordinate system (Fig. 1): 2
2
2
∂ T 1 ∂T 1 ∂ T ∂ T þ þ þ ∂r 2 r ∂r r 2 ∂θ2 ∂z2 1 ∂T ∂T ¼ þω ; r d ≤r≤Rd ; 0≤θ≤2π; −δd bzb0; t > 0; kd ∂t ∂θ
ð2Þ
at the following boundary conditions (Fig. 1):on the contact surface of the disc Kd
g ðt Þ⋅qd ; r p ≤r≤Rp ; 0≤θ≤2π; 0≤t≤t s ; ∂T ¼ ð3Þ ½1−g ðt Þh½T ∞ −T ðr; θ; t Þ; r p ≤r≤Rp ; 0≤θ≤2π; t≥0; ∂z z¼0
2. Statement of the problem and on the free surfaces of the disc Let at the initial time moment, a solid disc of a pad/disc brake system rotates with an angular speed ω0 (Fig. 1). The immovable pads are pressed to the outboard and inboard friction surfaces of the disc
Kd
∂T ¼ h½T ∞ −T ðr; θ; t Þ; r d ≤r≤r p ; 0≤θ≤2π; t≥0; ∂z z¼0
ð4Þ
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Kd
∂T ¼ h½T ∞ −T ðθ; z; t Þ; 0≤θ≤2π; −δd ≤z≤0; t≥0; ∂r r¼Rd
ð5Þ
Kd
∂T ¼ h½T ðθ; z; t Þ−T ∞ ; 0≤θ≤2π; −δd ≤z≤0; t≥0; ∂r r¼rd
ð6Þ
∂T ¼ 0; r d ≤r≤Rd ; 0≤θ≤2π; t≥0; z ¼ −δd : ∂z z¼−δd
ð7Þ
At the initial time moment t = 0 the disc is heated to the constant temperature: T ðr; θ; z; 0Þ ¼ T 0 ; r d ≤r≤Rd ; 0≤θ≤2π; −δd ≤z≤0:
ð8Þ
The dimensionless function g(t) in the condition (3) simulates the successive transition of the heating area over the disc contact surface during braking [7,8]. The intensity of the heat flux qd directed into the disc is calculated from the formula qd ¼ γf ðT Þp0 ωðt Þr; r p ≤r≤Rp ; 0≤θ≤2π; 0≤t≤t s ;
ð9Þ
where [10] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ρd c d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : γ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d q K d ρd cd þ K p ρp cp
ð10Þ
Justification of the choice of Charron's formula (10) at calculation of the heat partition ratio γ can be found in the article [4]. 3. Finite element analysis A considered transient problem of heat conduction with the linear and temperature-dependent thermal load leads to the following matrix formulation ½C
d l n fTg þ ½KfTg ¼ fQ g þ fQ g ; dt
ð11Þ
appeared at the right-hand side vector of applied boundary heat fluxes {Q} l and {Q} n stand for independent and dependent on the temperature thermal load, respectively. A solution to the transient problem with non-linear terms arising from temperature-dependent coefficient of friction is obtained by using a difference equation approximation to Eq. (11) with a parameter β that is adjusted to give a compromise between the stability, efficiency and high accuracy requirements. In this respect the form of the difference equation analogous to the Newmark β method is [11] 1 ½K βfTgnþ1 þ ð1−βÞfTgn þ ½C fTgnþ1 −fTgn Δt n o l l n n ¼ βfQ gnþ1 þ ð1−βÞfQ gn þ ð1 þ βÞfQ gn −βfQ gn−1 ;
4. Numerical analysis The three-dimensional transient temperature field of the disc subject to the non-axisymmetric thermal load was obtained using the finite element based software package [12]. The modelling technique of the problem of the moving heat source acting on the contact surface of the disc as the heat flux with the intensity proportional to the specific power of friction was proposed in Refs. [7,8]. In this paper unlike previous calculations an influence of the fluctuations of the coefficient of friction resulting from the temperature changes on the temperature is taken into account. Owing to the comparison of the temperature field generated during the process at the constant (temperature of 20 °C) and temperature-dependent coefficient of friction, the experimental data determined for different materials and at specified contact pressures were adopted from [13]. Based on these data the corresponding mathematical formulas adequate for the range of temperatures from 20 to 800 °C for the friction pair FC-16L/ChNMKh f ðT Þ ¼
0:286 2 1 þ 0:55⋅10−2 ðT−105Þ 0:286 þ 2 at p0 ¼ 0:39 MPa; 1 þ 0:25⋅10−2 ðT−800Þ
f ðT Þ ¼ 0:02 þ ð12Þ
where the subscript n stands for the nth time step. The free parameter β may be selected in the range 0 b β b 1. Eq. (12) can be rearranged giving the iteration algorithm
1 1 ½C þ β½K fT gnþ1 ¼ ½C−ð1−βÞ½K fTgn Δt Δt n o l þ βfQ gnþ1 þ ð1−βÞfQ gln þ ð1 þ βÞfQ gnn −βfQ gnn−1 :
language. Neglecting an influence of the fluctuations of the coefficient of friction resulting from the temperature change on the operation time and the evolution of a speed of the disc, the dimensionless function g(t) appeared in Eq. (3) was calculated having in mind exclusively dimensions of the brake system as well as the deceleration of the rotating disc (Eq. (1)). However an effect of the variations of the coefficient of friction on the temperature did manifest by the amount of heat proportional to an increase (decrease) in the coefficient of friction. To assure high accuracy of numerical calculations, the numbering of elements being comprised within the contact area, corresponded strictly to the boundary heat flux simulating non-axisymmetric heating due to friction of the stationary pad and the rotating disc. Thereby the dimension of the elements in the circumferential direction was equal for every path at the given radius. Eventually an area of the disc adjoining with an area of contact ( r p ≤r≤Rp ; 0≤θ≤2π ) comprised 7200 hex-type eight-node elements in the number of 360 in circumferential direction and 20 elements in radial direction of thickness of 1 mm. The remaining area of the disc was created automatically based on the geometry of the disc by means of 71,520 tet-type four-node elements. Overall the combined FE mesh for numerical analysis consisted of 78,720 elements and 25,356 nodes (Fig. 1).
0:288
2 1 þ 0:7⋅10−2 ðT−95Þ 0:2 þ 2 at p0 ¼ 1:47 MPa; 1 þ 0:3⋅10−2 ðT−800Þ
ð15Þ
and FMC-11/ChNMKh f ðT Þ ¼ 6:35⋅10
−3
þ
0:762 2 1 þ 0:19⋅10−2 ðT þ 180Þ
at p0 ¼ 0:59 MPa ; ð16Þ
ð13Þ
The numerical calculations satisfying the abovementioned mathematical description were carried out by using MD Nastran finite element based software package [12]. The boundary conditions concerning both the cooling and heating of the disc within the friction surface were developed separately using the Python programming
ð14Þ
f ðT Þ ¼ 0:0321 þ
0:642 2 1 þ 0:19⋅10−2 ðT þ 180Þ
f ðT Þ ¼ 0:0285 þ
0:57 1 þ 0:162⋅10−2 ðT þ 250Þ2
at p0 ¼ 0:78 MPa ; ð17Þ
at p0 ¼ 1:18 MPa; ð18Þ
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f ðT Þ ¼ 0:036 þ
0:48 2 1 þ 0:15⋅10−2 ðT þ 250Þ
at p0 ¼ 1:47 MPa;
ð19Þ
were obtained (Fig. 2). The detailed description of the method employed to determine these formulas was given in Ref. [6]. The corresponding constant coefficients of friction calculated from Eqs. (14) to (19) at temperature of 20 °C are tabulated in Table 1. It should be noted that considering different contact pressures at the same initial velocity, braking time and constant deceleration, each computational case corresponds with different specific power of friction. Two parallel computational cases incorporating constant and temperature-dependent coefficients of friction during a single braking process with constant deceleration from the initial speed of the disc of ω0 = 88.464 s −1 to standstill during the time ts =3.96 s were developed [14]. Further simulation of a multiple braking was conducted according to the diagram shown in Fig. 3, from which it can be seen
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Table 1 Coefficients of friction calculated from Eqs. (14)–(19) at temperature of 20 °C. Friction pair
Pressure, p0 [MPa]
Coefficient of friction f at temperature of 20 °C
FC-16L/ChNMKh
0.39 1.47 0.59 0.78 1.18 1.47
0.294 0.277 0.672 0.593 0.507 0.448
FMC-11/ChNMKh
that at the beginning, the single braking process takes place with the abovementioned parameters, then the brake is released and the speed increases linearly with time during 15 s. The cycle is repeated five times followed by the braking to standstill with the same deceleration as in the previous brake applications. The dimensions of the pad/disc system are listed in Table 2. The heat transfer coefficient is equal h = 60 W/(m 2K) both for the single and the multiple braking having its main purpose in estimation of the temperature difference calculated at the temperature-dependent and constant coefficients of friction. The value of h parameter was derived from the process of a single braking with constant deceleration of a vehicle from initial speed of 100 km/h (ω0 = 88.464 s −1) to standstill [14], however other calculations using the same materials and dimensions of the pad and the disc demonstrated that for extreme cooling conditions in the range of the heat transfer coefficient varying from 0 to 100 W/(m 2K) for fivefold braking with constant speed proceeded during 55 s, the temperature difference was relatively small and equal ΔT = 69.4 °C [8]. Initially, the disc has a uniform temperature T0 = 20 °C throughout its body, and the ambient temperature is equal to T∞ = 20 °C. The thermophysical properties of the materials of the disc and the pad are constant and listed in Table 3. It was assumed that the entire mechanical energy is converted into heat at the pad/disc interface (z = 0). This axial position was chosen as the source of heating, thereby determinant of the temperature distribution of the pad/disc system. Furthermore both the circumferential and radial positions were fixed for the thermal analysis. The point r = 113.5 mm, z = 0, θ = 0° being the maximum radius of the rubbing path of the disc equal to the pad maximum radius (approximately the
Fig. 3. Evolutions of the dimensionless angular speed of the disc during multiple braking.
Table 2 Dimensions of the pad and the disc [14].
Fig. 2. Temperature dependencies of the coefficients of friction in the range from 20 to 800 °C [13].
Items
Disc
Pad
Inner radius, r [mm] Outer radius, R [mm] Thickness, δ [mm] Pad arc length, θ0 [deg]
66 113.5 5.5
76.5 10 64.5
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highest possible temperature within the disc volume due to the maximum slip speed) (see Table 2) will be considered. The temperature evolutions on the friction surface of the disc (r = 113.5 mm, θ = 0°) during the single braking process for the cast-iron (ChNMKh) disc combined with the pad made of FC-16L are shown in Fig. 4. The grey lines correspond with the calculations carried out at the temperature-dependent coefficient of friction, whereas the black lines represent braking at the constant coefficient of friction corresponding with the temperature of 20 °C. Such annotation will be remained in every of the presented further figures. Two successive stages can be defined from the depicted temperature curves, namely the heating and the cooling phases resulted directly from the transition of the heating area (the pad contact surface) over the fixed analyzed spot within the rubbing path of the disc followed by the cooling during when there is no contact with the pad. The general nature of these evolutions reveals a rapid increase in temperature at the beginning of the process, then the maximum value is reached and eventually the temperature decreases slightly. The phenomenon is naturally connected with the relationship between the magnitude of the thermal load and its simultaneous absorption through conduction and convection, however the latter is negligible during normal single braking conditions. Two applied contact pressures p0 = 0.39 MPa and p0 = 1.47 MPa result in higher temperatures for the case incorporating the temperature-dependent coefficient of friction (grey lines) which agrees well with the dependencies shown for that friction pair in Fig. 2a. The maximum attained temperature is equal to T = 112.3 °C (t = 2.727 s). The maximum temperature difference between the case with the temperature-dependent and constant coefficient of friction at p0 = 1.47 MPa ΔT = 16.2 °C occurs at the time moment t = 2.727 s. At the pressure p0 = 0.39 MPa the maximum
temperature difference is clearly lower and is equal to ΔT = 1.2 °C. It may be observed that when the coefficient of friction increases from the temperature of 20 °C to the maximum attained temperature of 112.3 °C by 23.1% at p0 = 1.47 MPa (Fig. 2a), the related to that case temperature calculated at constant coefficient of friction (T = 96.1 °C) is underestimated by 14.4%. The corresponding temperature evolutions for the single braking process on the contact surface of the disc (r = 113.5 mm, θ = 0°) for the friction pair FMC-11/ChNMKh are shown in Fig. 5. In addition the calculated temperatures were confronted with the temperature curve corresponding with the same input parameters and at constant coefficient of friction f = 0.5 from the article [9]. Naturally higher value of the coefficient of friction than that used in this study results in adequately higher temperature. However an order of the appeared temperatures of 100 °C (Fig. 5) at the pressure p0 = 1.47 MPa is the same as for the friction pair FC-16L/ ChNMKh (Fig. 4), in this case the maximum temperature difference is definitely lower and equals ΔT = 4.4 °C. The time when this temperature (101.5 °C at constant coefficient of friction and 97.1 °C at temperature dependent coefficient of friction) is reached (t = 2.727 s) and is the same as for the friction pair FMC-11/ChNMKh (Fig. 4). It may be also observed that the temperature difference between the case with the temperature-dependent and constant coefficient of friction decreases successively with the decrease in contact pressure since the level of temperatures decreases. It may be estimated that when the coefficient of friction decreases from the temperature of 20 °C to the maximum attained temperature of 97.1 °C by 8.4% (p0 = 1.47 MPa), the related to that case temperature reached at the constant coefficient of friction is overestimated by 4.6%. Fig. 6 shows the temperature evolutions obtained during the multiple braking process for the disc made of ChNMKh and the pad made of FC-16L according to the diagram of the applied angular speed shown in Fig. 3. Two temperature curves represent the values calculated at temperature-dependent coefficient of friction (grey lines) as well as at constant coefficient of friction corresponding with the temperature of 20 °C (black lines). As it can be seen that the temperature for the temperature-dependent coefficient of friction at the beginning of the process is higher than that obtained at constant coefficient of friction and the temperature difference increases with an increase in
Fig. 4. Evolutions of the nodal temperature on the contact surface of the disc (r=113.5 mm, θ0 =0°) during the single braking process at constant (black lines) and temperaturedependent (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, θ0 = 0°) during the single braking process at constant (black lines) and temperature-dependent (grey line) coefficients of friction (pad FMC-11/disc ChNMKh).
Table 3 Thermophysical properties of materials of the pad and the disc at temperature of 20 °C [13]. Material
K [W/(mK)]
k × 106 [m2/s]
c [J/(kgK)]
ρ [kg/m3]
ChNMKh FC-16L FMC-11
51 0.79 34.3
14.4 0.33 14.6
500 961 500
7100 2500 4700
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process of the multiple braking, which corresponds with the dependence of the coefficient of friction on the temperature and approved its direct influence on the resulting temperature. The maximum temperature difference equals 47.6 °C and is attained at the end of the multiple braking. 5. Summary and conclusions In this study a simulation of a braking process for a pad/disc brake system was carried out. A three-dimensional temperature field of a disc subject to non-axisymmetric thermal load at the constant and temperature-dependent coefficients of friction was obtained by using the finite element method. Both the single and the multiple braking processes with the constant deceleration were studied. The calculated temperature evolutions at the temperature-dependent coefficients of friction on the contact surface of the disc were confronted with the corresponding values determined at the constant coefficients of friction (at temperature of 20 °C). It was established that:
Fig. 6. Evolutions of the nodal temperature on the contact surface of the disc (r = 113.5 mm, θ0 = 0°) during the multiple braking process at constant (black line) and temperature-dependent (grey line) coefficient of friction (pad FC-16L/disc ChNMKh).
temperature up to the time of about 20 s, then remains on the same level to the time moment t ≈ 40 s, and eventually decreases. This effect agrees with the temperature dependence of the coefficient of friction shown in Fig. 2a whose value increases with an increase in temperature to the value of 96.3 °C (maximum) and then decreases. The maximum temperature difference equals ΔT = 26.3 °C and is attained at the time moment t = 56.9 s. The temperature evolutions on the contact surface of the disc for the friction pair FMC-11/ChNMKh calculated during the multiple braking process are shown in Fig. 7. Unlike the corresponding results obtained for the friction pair ChNMKh/FC-16L, these outcomes reveal an approximately linear increase in temperature difference throughout the entire
− for the friction pair FC-16L/ChNMKh at p0 = 1.47 MPa when the coefficient of friction increases from the temperature of 20 °C to the maximum attained temperature of the friction surface of the disc (T = 112.3 °C, grey line, Fig. 4) at time moment t = 2.727 s by 23.1%, the related to that case temperature calculated at constant coefficient of friction T = 96.1 °C is underestimated by 14.4%; − for the friction pair FMC-11/ChNMKh at p0 = 1.47 MPa when the coefficient of friction decreases from the temperature of 20 °C to the maximum attained temperature of the contact surface of the disc (T = 97.1 °C, grey line, Fig. 5) at time moment t = 2.727 s by 8.4%, the related to that case temperature calculated at constant coefficient of friction (T = 101.5 °C) is overestimated by 4.6%; − under the conditions of the multiple braking both for FC-16L/ ChNMKh (Fig. 6) as well as FMC-11/ChNMKh (Fig. 7), the temperature evolutions determined on the friction surface of the disc agree with the established temperature dependencies of the coefficients of friction (Fig. 2a,b). Obviously aiming to attain low temperatures reached during braking the coefficient of friction should be low, this however is in conflict with the requirements concerning the highest moment of friction. The carried out calculations exhibited that at relatively low temperatures (T ≈ 100 °C) an influence of the coefficient of friction on the resulting temperature is apparent. Acknowledgement The present article is financially supported by the National Science Centre in Poland (research project no 2011/01/B/ST8/07446). References
Fig. 7. Evolutions of the nodal temperature on the contact surface of the disc (r = 113.5 mm, θ0 = 0°) during the multiple braking process at constant (black line) and temperature-dependent (grey line) coefficient of friction (pad FMC-11/disc ChNMKh).
[1] A.V. Chichinadze, Calculation and Investigation of External Friction During Braking, Nauka, Moscow, 1967. (in Russian). [2] A.A. Yevtushenko, P. Grzes, FEM-modeling of the frictional heating phenomenon in the pad/disc tribosystem (a review), Numerical Heat Transfer, Part A Applications 58 (3) (2010) 207–226. [3] D. Thuresson, Influence of material properties on sliding contact braking applications, Wear 257 (5–6) (2004) 451–460. [4] A. Yevtushenko, P. Grzes, Finite element analysis of heat partition in a pad/disc brake system, Numerical Heat Transfer, Part A Applications 59 (7) (2011) 521–542. [5] A.A. Yevtushenko, P. Grzes, Axisymmetric finite element model for the calculation of temperature at braking for thermosensitive materials of a pad and a disc, Numerical Heat Transfer, Part A Applications 62 (3) (2012) 211–230. [6] A.A. Yevtushenko, P. Grzes, Axisymmetric FEA of temperature in a pad/disc brake system at temperature-dependent coefficients of friction and wear, International Communications in Heat and Mass Transfer 39 (8) (2012) 1045–1053. [7] A. Adamowicz, P. Grzes, Analysis of disc brake temperature distribution during single braking under non-axisymmetric load, Applied Thermal Engineering 31 (6–7) (2011) 1003–1012. [8] A. Adamowicz, P. Grzes, Influence of convective cooling on a disc brake temperature distribution during repetitive braking, Applied Thermal Engineering 31 (14–15) (2011) 2177–2185.
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[9] A. Adamowicz, P. Grzes, Three-dimensional FE model for calculation of temperature of a thermosensitive disc, Applied Thermal Engineering 50 (1) (2013) 572–581. [10] F. Charron, Partage de la chaleur entre deux corps frottants, Publ. scient. et techn. Ministere air., 182, 1943. [11] H.-P. Lee, Nastran Thermal Analyzer—Theory and Application Including a Guide to Modeling Engineering Problems, vol. 1, National Aeronautics and Space Administration, Washington, D.C., 1977
[12] M.S.C. Nastran, Thermal Analysis User's Guide, 2002. [13] A.V. Chichinadze, R.M. Matveevski, E.P. Braun, Materials in Tribotechnics Non-stationary Processes, Nauka, Moscow, 1986. (in Russian). [14] F. Talati, S. Jalalifar, Analysis of heat conduction in a disk brake system, Heat and Mass Transfer 45 (8) (2009) 1047–1059.
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Three-dimensional FE model for the calculation of temperature of a disc brake at temperature-dependent coefficients of friction☆ A.A. Yevtushenko ⁎, A. Adamowicz, P. Grzes Faculty of Mechanical Engineering, Bialystok University of Technology (BUT), 45C Wiejska Street, Bialystok 15-351, Poland
a r t i c l e
i n f o
Available online 2 January 2013 Keywords: Frictional heating Temperature Pad/disc brake system Temperature-dependent coefficient of friction Finite element method
a b s t r a c t The three-dimensional transient temperature field of a disc brake generated during a single and a multiple braking process at temperature-dependent and constant coefficients of friction was analyzed. The calculations were performed for the two materials of a pad (FC-16L and FMC-11) combined with the cast-iron (ChNMKh) disc by using the finite element method (FEM). Analytical dependencies of the coefficient of friction on the temperature for these two friction pairs were obtained on the basis of the experimental data at different values of the contact pressures. It was established that relatively slight fluctuations of the coefficient of friction have direct impact on the contact temperature of the disc. The maximum temperature generated during the single braking process at constant coefficient of friction in relation to the case incorporating temperature-dependent coefficient of friction was underestimated by 14.4% for the friction pair FC-16L/ ChNMKh (increase in the coefficient of friction by 23.1%), and overestimated by 4.6% for FMC-11/ChNMKh (decrease in the coefficient of friction by 8.4%). © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The value of the coefficient of friction during braking may vary with the speed, load, thermophysical properties of materials, physical and chemical interactions on the working surfaces, temperature, etc. [1]. Hence it is important to establish which of these factors significantly affect the relationship between the applied load and the resulting friction force. A review of investigations of temperature in disc brake and clutch systems by the finite element method was given in Ref. [2]. It turned out that the publications, taking into account the dependence of the coefficient of friction on temperature, are not numerous at all. An impact of thermomechanical properties of materials and dimensions of the rectangular block sliding over the rigid foundation on the temperature at temperature-dependent coefficient of friction and wear was studied in Ref. [3]. A simulation was carried out by using Bouligand-differentiable Newton's method and an optimization method. The axisymmetric transient temperature fields of the pad and the disc generated during a single braking process adopting various experimental and theoretical formulas for the heat partition ratio were calculated in Ref. [4]. The corresponding solution of a non-linear heat conduction problem concerning an influence of thermosensitivity of materials on the temperature of the pad/disc brake system was obtained in ☆ Communicated by W.J. Minkowycz ⁎ Corresponding author. E-mail address: [email protected] (A.A. Yevtushenko). 0735-1933/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2012.12.015
Ref. [5]. An effect of variations of the temperature-dependent coefficients of friction and thermomechanical wear rate for two materials of the pad combined with the cast-iron disc was studied in Ref. [6]. A series of FEM studies on non-axisymmetric heating of the disc by the heat flux moving with the intensity proportional to the specific power of friction within an area of contact at constant coefficient of friction compose Refs. [7–9]. The three-dimensional model for simulation of non-uniform disc heating within the framework of linear heat conduction was developed in Ref. [7]. The parallels between the temperature evolutions of the three-dimensional model and the twodimensional equivalent whose axisymmetric thermal load arose from the average heat flux uniformly distributed on the contact surface of the disc in the circumferential direction were drawn. Both the temperatures on the contact surface as well as at the specific axial positions of the disc of these two models were confronted and compared. It was demonstrated that the temperature generated as an effect of axisymmetric heating of the disc coincides with the average temperature of the three-dimensional model above a certain critical slip speed. An influence of the heat transfer coefficient corresponding with the extreme cooling conditions for automotive application on the temperature reached during a single and a multiple braking ordered to give equal specific power of friction in every of the considered computational case was studied in Ref. [8]. The determined dependencies of the temperature on the heat transfer coefficient revealed linear relationship and the slope of these curves was dependent upon the number of brake applications. A comparative analysis of thermosensitive and temperatur e-independent materials of the pad/disc brake system was carried
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Nomenclature c [C] f g(t) h k K [K] p0 q {Q} l {Q} n r r, R t ts T T∞ T0 {T} z
specific heat, (J/(kgK)) heat capacity matrix coefficient of friction dimensionless function modelling transition of the heating area heat transfer coefficient, (W/(m 2K)) thermal diffusivity, (m 2/s) thermal conductivity, (W/(mK)) conductivity matrix contact pressure, (MPa) intensity of the heat flux, (W/m 2) vector of applied linear thermal load vector of applied thermal load that depends on the temperature radial coordinate internal and external radii, respectively, (m) time, (s) braking time, (s) temperature, (°C) ambient temperature, (°C) initial temperature, (°C) temperature vector axial coordinate
Greek symbols γ heat partition ratio δ thickness, (m) θ circumferential coordinate θ0 cover angle of pad, (deg) ρ density, (kg/m 3) ω relative angular slip speed, (s −1) ω0 initial relative angular slip speed, (s −1)
Subscripts d indicates pad n nth time step p indicates disc
out in Ref. [9]. The experimental dependencies of the temperaturedependent thermophysical properties of the four materials of the pad and four materials of the disc were approximated giving mathematical formulas applied to the FE three-dimensional model of the disc. An influence of variations of these properties both at constant and temperature-dependent heat partition ratio incorporating corresponding constant thermophysical parameters was studied. It should be noted that the axisymmetric model to determine the temperature in the disc brakes is valid when the overlap factor is close to the unity. This article is a generalization of the results of Ref. [6] on a three-dimensional case, i.e. for arbitrary values of the overlap factor. The transient 3D temperature field in the disc brake generated at constant and temperature-dependent coefficient of friction during a single and a multiple braking process is studied by using the finite element method.
Fig. 1. A diagram of a disc brake with the FE mesh and boundary conditions.
generating constant and uniformly distributed contact pressure p0, which resists the movement and the angular speed decreases linearly in time t ωðt Þ ¼ ω0 1− ; 0≤t≤t s : ts
ð1Þ
The heat generated due to friction is dissipated through conduction within the bodies being in contact and convection from the free surfaces of the system. Furthermore it is assumed that: − the materials of the pad and the disc are isotropic and their thermophysical properties are temperature-independent; − the coefficient of friction depends on temperature; − the convective heat exchange with the surrounding air according to Newton's law of cooling at the constant and average heat transfer coefficient h takes place on the exposed surfaces of the disc; − radiation mode of heat transfer is ignored; − by virtue of the symmetry of the problem about the mid-plane of the disc, the computational region is restricted exclusively to the half of the entire disc volume with the thickness δd. Obviously, it should be stated that the temperature dependence of the coefficient of friction during braking at the constant and uniform contact pressure has its direct influence on the deceleration of the disc and consequently may vary the braking time ts. However, this effect is not taken into account. At abovementioned assumptions the transient temperature distribution T(r,z,θ,t) in the disc is obtained from the solution of the following boundary-value problem of heat conduction given in the cylindrical coordinate system (Fig. 1): 2
2
2
∂ T 1 ∂T 1 ∂ T ∂ T þ þ þ ∂r 2 r ∂r r 2 ∂θ2 ∂z2 1 ∂T ∂T ¼ þω ; r d ≤r≤Rd ; 0≤θ≤2π; −δd bzb0; t > 0; kd ∂t ∂θ
ð2Þ
at the following boundary conditions (Fig. 1):on the contact surface of the disc Kd
g ðt Þ⋅qd ; r p ≤r≤Rp ; 0≤θ≤2π; 0≤t≤t s ; ∂T ¼ ð3Þ ½1−g ðt Þh½T ∞ −T ðr; θ; t Þ; r p ≤r≤Rp ; 0≤θ≤2π; t≥0; ∂z z¼0
2. Statement of the problem and on the free surfaces of the disc Let at the initial time moment, a solid disc of a pad/disc brake system rotates with an angular speed ω0 (Fig. 1). The immovable pads are pressed to the outboard and inboard friction surfaces of the disc
Kd
∂T ¼ h½T ∞ −T ðr; θ; t Þ; r d ≤r≤r p ; 0≤θ≤2π; t≥0; ∂z z¼0
ð4Þ
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Kd
∂T ¼ h½T ∞ −T ðθ; z; t Þ; 0≤θ≤2π; −δd ≤z≤0; t≥0; ∂r r¼Rd
ð5Þ
Kd
∂T ¼ h½T ðθ; z; t Þ−T ∞ ; 0≤θ≤2π; −δd ≤z≤0; t≥0; ∂r r¼rd
ð6Þ
∂T ¼ 0; r d ≤r≤Rd ; 0≤θ≤2π; t≥0; z ¼ −δd : ∂z z¼−δd
ð7Þ
At the initial time moment t = 0 the disc is heated to the constant temperature: T ðr; θ; z; 0Þ ¼ T 0 ; r d ≤r≤Rd ; 0≤θ≤2π; −δd ≤z≤0:
ð8Þ
The dimensionless function g(t) in the condition (3) simulates the successive transition of the heating area over the disc contact surface during braking [7,8]. The intensity of the heat flux qd directed into the disc is calculated from the formula qd ¼ γf ðT Þp0 ωðt Þr; r p ≤r≤Rp ; 0≤θ≤2π; 0≤t≤t s ;
ð9Þ
where [10] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ρd c d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : γ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d q K d ρd cd þ K p ρp cp
ð10Þ
Justification of the choice of Charron's formula (10) at calculation of the heat partition ratio γ can be found in the article [4]. 3. Finite element analysis A considered transient problem of heat conduction with the linear and temperature-dependent thermal load leads to the following matrix formulation ½C
d l n fTg þ ½KfTg ¼ fQ g þ fQ g ; dt
ð11Þ
appeared at the right-hand side vector of applied boundary heat fluxes {Q} l and {Q} n stand for independent and dependent on the temperature thermal load, respectively. A solution to the transient problem with non-linear terms arising from temperature-dependent coefficient of friction is obtained by using a difference equation approximation to Eq. (11) with a parameter β that is adjusted to give a compromise between the stability, efficiency and high accuracy requirements. In this respect the form of the difference equation analogous to the Newmark β method is [11] 1 ½K βfTgnþ1 þ ð1−βÞfTgn þ ½C fTgnþ1 −fTgn Δt n o l l n n ¼ βfQ gnþ1 þ ð1−βÞfQ gn þ ð1 þ βÞfQ gn −βfQ gn−1 ;
4. Numerical analysis The three-dimensional transient temperature field of the disc subject to the non-axisymmetric thermal load was obtained using the finite element based software package [12]. The modelling technique of the problem of the moving heat source acting on the contact surface of the disc as the heat flux with the intensity proportional to the specific power of friction was proposed in Refs. [7,8]. In this paper unlike previous calculations an influence of the fluctuations of the coefficient of friction resulting from the temperature changes on the temperature is taken into account. Owing to the comparison of the temperature field generated during the process at the constant (temperature of 20 °C) and temperature-dependent coefficient of friction, the experimental data determined for different materials and at specified contact pressures were adopted from [13]. Based on these data the corresponding mathematical formulas adequate for the range of temperatures from 20 to 800 °C for the friction pair FC-16L/ChNMKh f ðT Þ ¼
0:286 2 1 þ 0:55⋅10−2 ðT−105Þ 0:286 þ 2 at p0 ¼ 0:39 MPa; 1 þ 0:25⋅10−2 ðT−800Þ
f ðT Þ ¼ 0:02 þ ð12Þ
where the subscript n stands for the nth time step. The free parameter β may be selected in the range 0 b β b 1. Eq. (12) can be rearranged giving the iteration algorithm
1 1 ½C þ β½K fT gnþ1 ¼ ½C−ð1−βÞ½K fTgn Δt Δt n o l þ βfQ gnþ1 þ ð1−βÞfQ gln þ ð1 þ βÞfQ gnn −βfQ gnn−1 :
language. Neglecting an influence of the fluctuations of the coefficient of friction resulting from the temperature change on the operation time and the evolution of a speed of the disc, the dimensionless function g(t) appeared in Eq. (3) was calculated having in mind exclusively dimensions of the brake system as well as the deceleration of the rotating disc (Eq. (1)). However an effect of the variations of the coefficient of friction on the temperature did manifest by the amount of heat proportional to an increase (decrease) in the coefficient of friction. To assure high accuracy of numerical calculations, the numbering of elements being comprised within the contact area, corresponded strictly to the boundary heat flux simulating non-axisymmetric heating due to friction of the stationary pad and the rotating disc. Thereby the dimension of the elements in the circumferential direction was equal for every path at the given radius. Eventually an area of the disc adjoining with an area of contact ( r p ≤r≤Rp ; 0≤θ≤2π ) comprised 7200 hex-type eight-node elements in the number of 360 in circumferential direction and 20 elements in radial direction of thickness of 1 mm. The remaining area of the disc was created automatically based on the geometry of the disc by means of 71,520 tet-type four-node elements. Overall the combined FE mesh for numerical analysis consisted of 78,720 elements and 25,356 nodes (Fig. 1).
0:288
2 1 þ 0:7⋅10−2 ðT−95Þ 0:2 þ 2 at p0 ¼ 1:47 MPa; 1 þ 0:3⋅10−2 ðT−800Þ
ð15Þ
and FMC-11/ChNMKh f ðT Þ ¼ 6:35⋅10
−3
þ
0:762 2 1 þ 0:19⋅10−2 ðT þ 180Þ
at p0 ¼ 0:59 MPa ; ð16Þ
ð13Þ
The numerical calculations satisfying the abovementioned mathematical description were carried out by using MD Nastran finite element based software package [12]. The boundary conditions concerning both the cooling and heating of the disc within the friction surface were developed separately using the Python programming
ð14Þ
f ðT Þ ¼ 0:0321 þ
0:642 2 1 þ 0:19⋅10−2 ðT þ 180Þ
f ðT Þ ¼ 0:0285 þ
0:57 1 þ 0:162⋅10−2 ðT þ 250Þ2
at p0 ¼ 0:78 MPa ; ð17Þ
at p0 ¼ 1:18 MPa; ð18Þ
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f ðT Þ ¼ 0:036 þ
0:48 2 1 þ 0:15⋅10−2 ðT þ 250Þ
at p0 ¼ 1:47 MPa;
ð19Þ
were obtained (Fig. 2). The detailed description of the method employed to determine these formulas was given in Ref. [6]. The corresponding constant coefficients of friction calculated from Eqs. (14) to (19) at temperature of 20 °C are tabulated in Table 1. It should be noted that considering different contact pressures at the same initial velocity, braking time and constant deceleration, each computational case corresponds with different specific power of friction. Two parallel computational cases incorporating constant and temperature-dependent coefficients of friction during a single braking process with constant deceleration from the initial speed of the disc of ω0 = 88.464 s −1 to standstill during the time ts =3.96 s were developed [14]. Further simulation of a multiple braking was conducted according to the diagram shown in Fig. 3, from which it can be seen
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Table 1 Coefficients of friction calculated from Eqs. (14)–(19) at temperature of 20 °C. Friction pair
Pressure, p0 [MPa]
Coefficient of friction f at temperature of 20 °C
FC-16L/ChNMKh
0.39 1.47 0.59 0.78 1.18 1.47
0.294 0.277 0.672 0.593 0.507 0.448
FMC-11/ChNMKh
that at the beginning, the single braking process takes place with the abovementioned parameters, then the brake is released and the speed increases linearly with time during 15 s. The cycle is repeated five times followed by the braking to standstill with the same deceleration as in the previous brake applications. The dimensions of the pad/disc system are listed in Table 2. The heat transfer coefficient is equal h = 60 W/(m 2K) both for the single and the multiple braking having its main purpose in estimation of the temperature difference calculated at the temperature-dependent and constant coefficients of friction. The value of h parameter was derived from the process of a single braking with constant deceleration of a vehicle from initial speed of 100 km/h (ω0 = 88.464 s −1) to standstill [14], however other calculations using the same materials and dimensions of the pad and the disc demonstrated that for extreme cooling conditions in the range of the heat transfer coefficient varying from 0 to 100 W/(m 2K) for fivefold braking with constant speed proceeded during 55 s, the temperature difference was relatively small and equal ΔT = 69.4 °C [8]. Initially, the disc has a uniform temperature T0 = 20 °C throughout its body, and the ambient temperature is equal to T∞ = 20 °C. The thermophysical properties of the materials of the disc and the pad are constant and listed in Table 3. It was assumed that the entire mechanical energy is converted into heat at the pad/disc interface (z = 0). This axial position was chosen as the source of heating, thereby determinant of the temperature distribution of the pad/disc system. Furthermore both the circumferential and radial positions were fixed for the thermal analysis. The point r = 113.5 mm, z = 0, θ = 0° being the maximum radius of the rubbing path of the disc equal to the pad maximum radius (approximately the
Fig. 3. Evolutions of the dimensionless angular speed of the disc during multiple braking.
Table 2 Dimensions of the pad and the disc [14].
Fig. 2. Temperature dependencies of the coefficients of friction in the range from 20 to 800 °C [13].
Items
Disc
Pad
Inner radius, r [mm] Outer radius, R [mm] Thickness, δ [mm] Pad arc length, θ0 [deg]
66 113.5 5.5
76.5 10 64.5
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highest possible temperature within the disc volume due to the maximum slip speed) (see Table 2) will be considered. The temperature evolutions on the friction surface of the disc (r = 113.5 mm, θ = 0°) during the single braking process for the cast-iron (ChNMKh) disc combined with the pad made of FC-16L are shown in Fig. 4. The grey lines correspond with the calculations carried out at the temperature-dependent coefficient of friction, whereas the black lines represent braking at the constant coefficient of friction corresponding with the temperature of 20 °C. Such annotation will be remained in every of the presented further figures. Two successive stages can be defined from the depicted temperature curves, namely the heating and the cooling phases resulted directly from the transition of the heating area (the pad contact surface) over the fixed analyzed spot within the rubbing path of the disc followed by the cooling during when there is no contact with the pad. The general nature of these evolutions reveals a rapid increase in temperature at the beginning of the process, then the maximum value is reached and eventually the temperature decreases slightly. The phenomenon is naturally connected with the relationship between the magnitude of the thermal load and its simultaneous absorption through conduction and convection, however the latter is negligible during normal single braking conditions. Two applied contact pressures p0 = 0.39 MPa and p0 = 1.47 MPa result in higher temperatures for the case incorporating the temperature-dependent coefficient of friction (grey lines) which agrees well with the dependencies shown for that friction pair in Fig. 2a. The maximum attained temperature is equal to T = 112.3 °C (t = 2.727 s). The maximum temperature difference between the case with the temperature-dependent and constant coefficient of friction at p0 = 1.47 MPa ΔT = 16.2 °C occurs at the time moment t = 2.727 s. At the pressure p0 = 0.39 MPa the maximum
temperature difference is clearly lower and is equal to ΔT = 1.2 °C. It may be observed that when the coefficient of friction increases from the temperature of 20 °C to the maximum attained temperature of 112.3 °C by 23.1% at p0 = 1.47 MPa (Fig. 2a), the related to that case temperature calculated at constant coefficient of friction (T = 96.1 °C) is underestimated by 14.4%. The corresponding temperature evolutions for the single braking process on the contact surface of the disc (r = 113.5 mm, θ = 0°) for the friction pair FMC-11/ChNMKh are shown in Fig. 5. In addition the calculated temperatures were confronted with the temperature curve corresponding with the same input parameters and at constant coefficient of friction f = 0.5 from the article [9]. Naturally higher value of the coefficient of friction than that used in this study results in adequately higher temperature. However an order of the appeared temperatures of 100 °C (Fig. 5) at the pressure p0 = 1.47 MPa is the same as for the friction pair FC-16L/ ChNMKh (Fig. 4), in this case the maximum temperature difference is definitely lower and equals ΔT = 4.4 °C. The time when this temperature (101.5 °C at constant coefficient of friction and 97.1 °C at temperature dependent coefficient of friction) is reached (t = 2.727 s) and is the same as for the friction pair FMC-11/ChNMKh (Fig. 4). It may be also observed that the temperature difference between the case with the temperature-dependent and constant coefficient of friction decreases successively with the decrease in contact pressure since the level of temperatures decreases. It may be estimated that when the coefficient of friction decreases from the temperature of 20 °C to the maximum attained temperature of 97.1 °C by 8.4% (p0 = 1.47 MPa), the related to that case temperature reached at the constant coefficient of friction is overestimated by 4.6%. Fig. 6 shows the temperature evolutions obtained during the multiple braking process for the disc made of ChNMKh and the pad made of FC-16L according to the diagram of the applied angular speed shown in Fig. 3. Two temperature curves represent the values calculated at temperature-dependent coefficient of friction (grey lines) as well as at constant coefficient of friction corresponding with the temperature of 20 °C (black lines). As it can be seen that the temperature for the temperature-dependent coefficient of friction at the beginning of the process is higher than that obtained at constant coefficient of friction and the temperature difference increases with an increase in
Fig. 4. Evolutions of the nodal temperature on the contact surface of the disc (r=113.5 mm, θ0 =0°) during the single braking process at constant (black lines) and temperaturedependent (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, θ0 = 0°) during the single braking process at constant (black lines) and temperature-dependent (grey line) coefficients of friction (pad FMC-11/disc ChNMKh).
Table 3 Thermophysical properties of materials of the pad and the disc at temperature of 20 °C [13]. Material
K [W/(mK)]
k × 106 [m2/s]
c [J/(kgK)]
ρ [kg/m3]
ChNMKh FC-16L FMC-11
51 0.79 34.3
14.4 0.33 14.6
500 961 500
7100 2500 4700
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process of the multiple braking, which corresponds with the dependence of the coefficient of friction on the temperature and approved its direct influence on the resulting temperature. The maximum temperature difference equals 47.6 °C and is attained at the end of the multiple braking. 5. Summary and conclusions In this study a simulation of a braking process for a pad/disc brake system was carried out. A three-dimensional temperature field of a disc subject to non-axisymmetric thermal load at the constant and temperature-dependent coefficients of friction was obtained by using the finite element method. Both the single and the multiple braking processes with the constant deceleration were studied. The calculated temperature evolutions at the temperature-dependent coefficients of friction on the contact surface of the disc were confronted with the corresponding values determined at the constant coefficients of friction (at temperature of 20 °C). It was established that:
Fig. 6. Evolutions of the nodal temperature on the contact surface of the disc (r = 113.5 mm, θ0 = 0°) during the multiple braking process at constant (black line) and temperature-dependent (grey line) coefficient of friction (pad FC-16L/disc ChNMKh).
temperature up to the time of about 20 s, then remains on the same level to the time moment t ≈ 40 s, and eventually decreases. This effect agrees with the temperature dependence of the coefficient of friction shown in Fig. 2a whose value increases with an increase in temperature to the value of 96.3 °C (maximum) and then decreases. The maximum temperature difference equals ΔT = 26.3 °C and is attained at the time moment t = 56.9 s. The temperature evolutions on the contact surface of the disc for the friction pair FMC-11/ChNMKh calculated during the multiple braking process are shown in Fig. 7. Unlike the corresponding results obtained for the friction pair ChNMKh/FC-16L, these outcomes reveal an approximately linear increase in temperature difference throughout the entire
− for the friction pair FC-16L/ChNMKh at p0 = 1.47 MPa when the coefficient of friction increases from the temperature of 20 °C to the maximum attained temperature of the friction surface of the disc (T = 112.3 °C, grey line, Fig. 4) at time moment t = 2.727 s by 23.1%, the related to that case temperature calculated at constant coefficient of friction T = 96.1 °C is underestimated by 14.4%; − for the friction pair FMC-11/ChNMKh at p0 = 1.47 MPa when the coefficient of friction decreases from the temperature of 20 °C to the maximum attained temperature of the contact surface of the disc (T = 97.1 °C, grey line, Fig. 5) at time moment t = 2.727 s by 8.4%, the related to that case temperature calculated at constant coefficient of friction (T = 101.5 °C) is overestimated by 4.6%; − under the conditions of the multiple braking both for FC-16L/ ChNMKh (Fig. 6) as well as FMC-11/ChNMKh (Fig. 7), the temperature evolutions determined on the friction surface of the disc agree with the established temperature dependencies of the coefficients of friction (Fig. 2a,b). Obviously aiming to attain low temperatures reached during braking the coefficient of friction should be low, this however is in conflict with the requirements concerning the highest moment of friction. The carried out calculations exhibited that at relatively low temperatures (T ≈ 100 °C) an influence of the coefficient of friction on the resulting temperature is apparent. Acknowledgement The present article is financially supported by the National Science Centre in Poland (research project no 2011/01/B/ST8/07446). References
Fig. 7. Evolutions of the nodal temperature on the contact surface of the disc (r = 113.5 mm, θ0 = 0°) during the multiple braking process at constant (black line) and temperature-dependent (grey line) coefficient of friction (pad FMC-11/disc ChNMKh).
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