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disc brake system at temperature-dependent coefficients of friction and wear
International Communications in Heat and Mass Transfer 39 (2012) 1045–1053
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Axisymmetric FEA of temperature in a pad/disc brake system at temperature-dependent coefficients of friction and wear☆ A.A. Yevtushenko ⁎, 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 28 July 2012 Keywords: Frictional heating Temperature Pad/disc brake system Temperature-dependent coefficient of friction Wear Finite element method
a b s t r a c t Robust braking results in heat generation whose effects may have considerable impact on the parameters of the process such as wear rate and coefficient of friction. Fluctuations of the latter disagree with essential operational and braking requirements. Finite element analysis (FEA) of a single braking process for axisymmetric heat conduction problem of friction in a pad/disc brake system in the present article was carried out. Two materials of the pad FC-16L (retinax A) and FMC-11 (metal ceramic) and one material of the disc ChNMKh (cast iron) were analysed. Experimental dependencies of the coefficient of friction and wear rate on the temperature under specified contact pressures for these two friction pairs were approximated and applied to FE contact model. The temperature and wear evolutions on the contact surface of the pad/disc brake system obtained for constant and temperature-dependent abovementioned coefficients were confronted and compared. Mutual correlations of the obtained results with the studied materials were discussed. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The magnitude and fluctuations of the coefficient of friction during braking depend on the mechanical and thermophysical properties of materials, speed, load, temperature regimes of frictional contact as well as the nature of physical and chemical processes [1]. The process of braking results in continuous and interdependent changes in both external factors (speed, load, temperature) and the properties of subsurface layers of materials and the intensity of frictional contact processes [2]. Thus it is very important to distinguish factors which have decisive impact on the coefficient of friction. To evaluate the variations of the coefficient of friction during braking the most convenient is its dependence on the speed, temperature and temperature gradient (the load at braking is practically constant) [3]. The abovementioned quantities can be directly measured during the process, or obtained from calculations based on the experimental data. At the same time structural changes and the intensity of the processes practically do not affect the temperatures and their gradients [4]. Having regard to the time, process of braking can be divided into four periods [5] (Fig. 1). The duration of the first 0 ≤ t b t1 is determined from the time of an increase in the load acting on the contacting bodies. According to the experimental data, the time from 0 to t1 equals about 2% of the total braking time ts. This period is characterized by the high ☆ 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.2012.07.025
speed of sliding, the high flash temperature occurred on microcontacts, the low temperature of contact surface of friction and low temperature gradients normal to the contact surface. The volumetric temperature does not reveal any changes. During the second period t1 ≤t b t2 the speed remains high, the load reaches nominal value, the flash temperature passes through a maximum point, the temperature and its gradient on the contact surface are high, and the volumetric temperature increases. In the interval of time t2 ≤ t b t3 the speed is still quite high, the load is constant, the flash temperature decreases, the surface temperature reaches maximum value, and the temperature gradient begins to decrease as the volumetric temperature increases. In the last, fourth period of braking (t3 ≤ t b ts) the speed is close to zero, the load is constant, the flash temperature practically equals zero, the temperature on the contact surface smoothly decreases and brings nearer the volumetric temperature, and the temperature gradient is very small. Duration of this period ranges from 3% to 5% from general time of braking. The value of coefficient of friction in the first period is typically determined pffiffiffiffi using the speed, i.e. the flash temperature, which is proportional to V [6]. In the second period — using the mean temperature on the surface of friction and the flash temperature at the prevailing value of the first. In the third and the final period of braking the value of the coefficient of friction is determined by means of temperature on the contact surface. Therefore, with an increase in temperature an influence of the speed and the temperature gradient on the coefficient of friction decreases and the magnitude and character of variation during braking are mainly determined by the temperature of the frictional contact. To take into account
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Nomenclature c [C] f f0 h I(T) It(T) k K [K] l n p0 {Q(T)} r r, R t t1, t2, t3 ts T T∞ T0 {T}o n T_ V z
Specific heat, (J/(kgK)) Specific heat (or capacitance) matrix Coefficient of friction Coefficient of friction at temperature of 20 °C Heat transfer coefficient, (W/(m 2K)) Temperature-dependent coefficient of wear rate, (μg/ (Nm)) Temperature-dependent wear, (mg) Thermal diffusivity, (m 2/s) Thermal conductivity, (W/(mK)) Conductivity matrix Characteristic dimension of finite element, (m) Iteration number Contact pressure, (MPa) Vector of heat loads that depends on temperature Radial coordinate Internal and external radius, respectively, (m) Time, (s) Characteristic moments in time during single braking process, (s) Braking time, (s) Temperature, (°C) Ambient temperature, (°C) Initial temperature, (°C) Temperature vector Vector of time derivative of vector of nodal temperature {T} Sliding speed, (m/s) Axial coordinate
Greek symbols δ Thickness, (m) η Fluctuation of the coefficient of friction, η = fmin/fmax θ Circumferential coordinate θ0 Cover angle of pad, (deg) ρ Density, (kg/m 3) ω Angular speed of the disc, (s −1) ω0 Initial angular speed of the disc, (s −1)
Subscripts p Indicates pad d Indicates disc
Superscripts ± values, obtained at the approach to plane z = 0 along positive (+) or negative (−) direction of the axis Oz.
the temperature dependence of the coefficient of friction in the calculations concerned the frictional heating systems, the corresponding relationships can be derived from the experimental data [7,8]. The analytical method for solving one-dimensional thermal problems of friction during braking for tribosystem consisting of two semi-spaces incorporating the dependence of the coefficients of friction and wear on the temperature was proposed in Refs. [9,10]. The development of this method to the case of frictional contact between two strips
was obtained in Ref. [11] and for a three-strip tribosystem — in Ref. [12]. A generalization of these results can be found also in Ref. [13]. In recent times, the more active simulations of friction heating in brake systems are being carried out employing numerical methods, in particular the finite element method (FEM) [14]. The thermomechanical model based on the finite element method, taking into account the temperature-dependent friction and wear coefficients during uniform sliding of the elastic rectangular block over the rigid support was proposed in Ref. [15]. The numerical solution of two-dimensional transient problem of heat conduction for tribosystem consisting of a strip of finite length, sliding between the surfaces of the two shorter strips, with thermal resistance of the contact surfaces and the temperature-dependent friction coefficient, was obtained in Ref. [16]. The axisymmetric FE model to study the transient temperature field in a pad/disc tribosystem with constant thermophysical properties of materials was proposed in Refs. [17–20]. A generalization of this model to the case of thermosensitive materials of the pad and the disc was carried out in Ref. [21]. The proposed article is a synthesis of that model for the case, when the coefficient of friction depends on the temperature. In addition, it is assumed that the wear is temperature-dependent, as well. 2. Statement of the problem The process of frictional heating in a pad/disc tribosystem during a single braking is considered (Fig. 2). The coefficient of mutual overlap of the pad and the disc is close to the unit. At the initial moment of time t = 0 the pads located on the outboard and inboard surfaces of the rotating with constant angular speed ω0 disc are pressed against generating the constant and uniform contact pressure p0 on the working surfaces. As a result of friction, the speed of rotation of the disc begins to decrease linearly with time: t ωðt Þ ¼ ω0 1− ; 0≤t ≤t s ; ts
ð1Þ
and on the contact surfaces the heat is generated, causing consequently heating of the elements of friction pair. The sum of the intensities of heat fluxes directed from the surface of friction inside the pads and the disc is equal to the specific power of friction. It is assumed that the thermal resistance on the surfaces of contact is negligible and therefore temperatures of the pads and the disc on these surfaces are equal. The free surfaces of the pads and the disc are convectively cooled with the same constant heat transfer coefficient h being average value for the analysed braking process. The thermophysical properties of materials of the pads and the disc are constant whereas the coefficient of friction f depends on the temperature T. It should be noted that the nature of change of the coefficient of friction during braking caused by an increase in temperature in the area of contact affects in fact evolution of a speed and in consequence, time of duration of the process. That effect in the considered study was omitted. The considered tribosystem is symmetric regarding the mid-plane of the disc. Therefore the system consisting of one pad sliding over the contact surface of the half of the entire disc will be considered. The opposite (median) surface of the disc is assumed to be thermally insulated. All of the values and parameters which refer to the pad and the disc in the further considerations will have subscripts p and d, respectively. The considered tribosystem is referred to the cylindrical system with coordinates (r, z) (Fig. 2). Then according to the abovementioned assumptions the distribution of the transient temperature in the pad and in the disc will be found from the solution of the following axisymmetric boundary-value problem of heat conduction: ∂2 T 1 ∂T ∂2 T 1 ∂T þ ¼ þ ; r p ≤ r ≤Rp ; 0 < z < δp ; t > 0; ∂r 2 r ∂r ∂z2 kp ∂t
ð2Þ
A.A. Yevtushenko, P. Grzes / International Communications in Heat and Mass Transfer 39 (2012) 1045–1053
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h i ∂T K p ¼ h T ∞ −T r; δp ; t ; r p ≤ r ≤ Rp ; t ≥ 0; ∂z z¼δ
ð8Þ
Kd
∂T ¼ h½T ∞ −T ðr; 0; t Þ; r d ≤ r ≤ r p ; t ≥ 0; ∂z z¼0
ð9Þ
Kd
∂T ¼ h½T ∞ −T ðRd ; z; t Þ; − δd ≤ z ≤ 0; t ≥ 0 ∂r r¼Rd
ð10Þ
Kd
∂T ¼ h½T ðr d ; z; t Þ−T ∞ ; − δd ≤ z ≤ 0; t ≥ 0 ∂r r¼rd
ð11Þ
p
Fig. 1. Typical changes of the sliding speed, temperature and temperature gradient during single braking [5].
∂2 T 1 ∂T ∂2 T 1 ∂T þ ¼ þ ; r d ≤ r ≤Rd ; −δd < z < 0; t > 0; ∂r2 r ∂r ∂z2 kd ∂t þ − T r; 0 ; t ¼ T ðr; 0 ; t Þ; r p ≤ r ≤Rp ; 0 ≤t ≤t s ; ∂T ∂T K d − −K p þ ¼ f ðT Þrωðt Þp0 ; r p ≤ r ≤ Rp ; 0 ≤ t ≤ t s ; ∂z z¼0 ∂z z¼0
ð3Þ
ð12Þ
T ðr; z; 0Þ ¼ T 0 ; r p ≤ r ≤ Rp ; 0 ≤ z ≤ δp
ð13Þ
T ðr; z; 0Þ ¼ T 0 ; r d ≤ r≤ Rd ; − δd ≤ z ≤ 0
ð14Þ
Frictional contact of the pads and the disc is accompanied by wear processes. The main influence on the wear for intensive process of braking with a significant load has the heat manifesting by the temperature on the contact surface [22]. The change of the wear It [mg] during braking has the form [23]: t
It ðT Þ ¼ ∫ IðT Þ f ðT Þ p0 V ðτ Þ Ak dτ; 0 ≤ t ≤ t s ; ð4Þ
ð5Þ
Kp
h i ∂T ¼ h T r p ; z; t −T ∞ ; 0 ≤ z ≤ δp ; t ≥ 0 ∂r r¼rp
ð6Þ
Kp
h i ∂T ¼ h T ∞ −T Rp ; z; t ; 0 ≤ z ≤ δp ; t ≥ 0 ∂r r¼Rp
ð7Þ
Fig. 2. A schematic diagram of a disc brake.
∂T ¼ 0; r d ≤ r ≤ Rd ; t ≥ 0 ∂z z¼−δd
ð15Þ
0
where I(T) [μg/(Nm)] is the temperature-dependent coefficient of wear rate, V(t) is the time-dependent speed of sliding of the pad over the rubbing path of the disc. Analysis of experimental data presented in publications [2,8,22,23] showed that the dependence of coefficient of friction and wear rate on the temperature may be either increasing or decreasing, with one or two maxima or minima (Fig. 3). For approximation of these types of experimental curves the function was developed in the form [7]: f ðT Þ ¼ f 1 þ
f f h 2 i2 þ h 4 i2 1 þ f 3 T−T f 1 1 þ f 5 T−T f 2
Fig. 3. Typical dependencies of coefficient of friction during braking [7].
ð16Þ
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where the constants f1,…, f5, Tf1, Tf2 are chosen taking into account the nature of variation of the curves shown in Fig. 3. Some general recommendations on selection of these constants are as follows: 1) Constant function (curve 1) — f1 > 0, f2 = f4 = 0; 2) Monotonically decreasing function (curve 2) — f1 > 0, f2 > 0, f4 = 0, Tf1 b 0; 3) Decreasing function with two local maxima (curve 3) — f1 > 0, f2 > 0, f4 b f2, Tf1 > 0, Tf2 > 0; 4) Monotonically increasing function (curve 4) — f1 > 0, f2 b 0, f4 = 0, Tf1 = 0; 5) Increasing function with two local maxima (curve 5) — f1 > 0, f2 > 0, f4 > f2, Tf1 > 0, Tf2 > 0; 6) Function with one local maximum and the minimum (curve 6) — f1 > 0, f2 > 0, f4 > > f2, Tf1 > 0,Tf2 > 0. In all the abovementioned cases f3 > 0 and f5 > 0. 3. Temporal integration The finite element code MD Nastran provides a convenient means of numerical modelling of the frictional heating during braking [24]. A system with an infinite number of unknowns (the temperature and its gradients at every location of the brake) can be transformed into one which has a finite number of unknowns related to each other by elements of finite size. In the finite element formulation by means of linear Galerkin spatial approximation, the boundary-value heat conduction problems (2)–(14) can be written as [25]: n o ½C T_ þ fKg fTg ¼ fQ ðTÞg;
ð17Þ
where {Q(T)} is the temperature-dependent vector containing the boundary conditions (4)–(14). The nonlinear ordinary differential equations of type of Eq. (17) are usually solved numerically either by means of explicit or implicit methods [26]. But the most effective is the scheme, known as a predictor–corrector method, when on a stage of prediction the initial approximation is determined by means of the explicit algorithm and on the stage of correction the iterations are performed using an implicit scheme. Due to simplicity, unconditional stability and robustness the first-order time backward Euler stepping approach was used for temporary discretization [27]: n o fTgtþΔt ¼ fTgt þ Δt T_
tþΔt
;
ð18Þ
where {T}t and {T}t + Δt are the values of the temperature vector {T} at the time moments t and t + Δt, respectively; Δt is the time step. Substituting the relation (18) into the Eq. (17) we obtain the following system of algebraic equation with respect to {T}t + Δt: ð½C þ Δt ½KÞfTgtþΔt ¼ ½C fTgt þ Δt fQ ðTÞgtþΔt :
ð19Þ
The system of nonlinear algebraic Eq. (19) will be solved by a simple iteration method. If the approximation {T}t+Δt(n)for the iteration n is known, then for the iterations n + 1 we have the following linearized system: ðnþ1Þ
ðnÞ
ð½C þ Δt ½KÞfTgtþΔt ¼ ½C fTgt þ Δt fQ ðTÞgtþΔt ; n ¼ 0; 1; 2; …; n max :
ð20Þ
Note that, regardless of the initial approximation {T}t+Δt(0) the iteration {T}t+Δt(nmax) converges to the solution {T}t+Δt, if a user-defined maximum number of iterations (the same at each time step) nmax →∞ [28]. However at the fixed value nmax ≥1 the approximation {T}t+Δt(nmax) will be better; the size of the step time Δt and the initial error (0) {T}t+Δt −{T}t+Δt are smaller.
To determine the initial approximation the explicit Euler scheme was used: n o fTgtþΔt ¼ fTgt þ Δt T_ ; t
ð21Þ
on the basis of which we have: ð0Þ
½CfTgtþΔt ¼ ð½C−Δt ½KÞfTgt þ Δt fQ ðTÞgt :
ð22Þ
The accuracy of the solution obtained using such a scheme is O(Δt 2). The maximum number of iterations per time step has been limited to 50. If the number of iterations exceeds this number, then the step size decreased by a quarter. Additional algorithm details, including evaluation of the local temporal truncation errors can be found, for example, in Ref. [29]. Since the considered heat conduction problem of friction is axisymmetric the geometrical basis for numerical calculations were two rectangles (pad and half of the disc) in coordinates (r, z) (cross section) representing the pad and the disc on which 2D axisymmetric solid triangular three-node finite elements were constructed. The overall number of elements was equal 6994 elements and 3711 nodes, where 2850 elements and 1536 nodes came for the disc, and 4144 elements and 2175 nodes came for the pad (Fig. 4). In order to accomplish the boundary conditions within the area of contact (Eqs. (4) and (5)), 75 of the multi point constraints (MPC) were constructed combining every corresponding two nodes on the opposite contact surfaces of the pad and the disc, respectively. The characteristic dimension of one specific element differed in the pad/disc FE model according to the coordinates of the system. In radial direction dimension of the side of one triangle (parallel to coordinate r) was equal within the entire length of the pad and the disc l = 0.5 mm. In axial direction z, the mesh division was the finest in the location of the contact surface and increased linearly with the distance from z = 0 both for the disc and for the pad. The characteristic dimension varied from the smallest value l = 0. 25 mm to 0. 5 mm at the furthest location from the contact surface. These values were equal for the disc and for the pad. Hence the time step for numerical 2 calculations (Δt = 0.1lmin k −1) was calculated for greater value of the thermal diffusivity (a pad or a disc material). 4. Numerical analysis To perform numerical calculations two friction pairs were selected: FC-16L (retinax A)-pad/ChNMKh (cast iron)-disc and FMC-11 (metal ceramic)-pad/ChNMKh (cast iron)-disc. These friction pairs are widely distributed in mechanical engineering and friction materials. Their thermophysical properties and the nature of physical and chemical processes occurring during sliding contact represent two completely different classes of frictional materials (Table 1). The experimental dependencies of the coefficient of friction and wear rate on the temperature for the asbestos friction material (AFM) with a resin binder FC-16L and sintered powder material FMC-11 for dry friction with the cast-iron ChNMKh were presented in Ref. [7]. These relationships take into account the effect of pressure, i.e. the series of curves are obtained with different p0, which cover a range of specific loads occurring during friction or maximum permissible for a given pair. The curves approximating experimental data [7] by means of function (16) are shown in Fig. 5. The values of constants f1,…, f5, Tf1, and Tf2 for the coefficient of friction f(T) and I1,…,I5, TI1, and TI2 for the coefficient of wear rate I(t) are given in Table 2. The curves in Fig. 5 are constructed for the following values of pressure p0: 1– 0.39 MPa, 2–1.47 MPa, 3–0.59 MPa; 4–0.78 MPa; 5–1.18 MPa; and 6–1.47 MPa.
A.A. Yevtushenko, P. Grzes / International Communications in Heat and Mass Transfer 39 (2012) 1045–1053
Fig. 4. Axisymmetric FE model of a pad/disc brake system with the boundary conditions.
The most important determining factor for sliding contact of AFM is the temperature, which affects the change of mechanical properties of friction materials as a consequence of increased intensity of thermal motion of molecules and the initiation of various physical and chemical phenomena [30]. The thermal degradation of binders included in the AFM and the resulting effects generated by friction arise at temperatures higher than 150 °С and intensively developed in the range from 200 to 300 °C. This also confirms dependencies of the coefficient of friction on the temperature for the FC-16L/ChNMKh couple, which are presented in Fig. 5a. The region of the reduction can be clearly seen between 200 and 300 °С. This can be explained by the fact that the products of destruction of the binder (resin, rubber) are the liquid fraction, which congregate on the frictional contact and act as a Table 1 Thermophysical properties of materials of the pad and the disc at 20 °C [7]. Material
K [W/(mK)]
ρ [kg/m3]
c [J/(kg K)]
k × 106[m2/s]
ChNMKh FC-16L FMC-11
51 0.79 34.3
7100 2500 4700
500 961 500
14.4 0.33 14.6
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lubricant, reducing the coefficient of friction. Therefore, the AFM on the resin coupler (including FC-16L) has such technical flaws as low and unstable coefficient of friction and high levels of residual stresses that cause disruption of the continuity of the material and bad compatibility with the metal base of a brake. Wide application of AFM is caused by their good stability of wear, as well as low cost and simplicity of manufacturing. In the modern highly loaded disc brakes such as aircraft brakes, the friction material made by powder metallurgy — the friction metal-ceramic materials (FMC) are used. These materials are composed of sintered friction layer and steel substrate. These include, in particular, the sintered material on iron substrate FMC-11. The percentage of the components in this material is: 45% Fe, 15% Cu, 9% C, 3% SiO2, 3% asbestos, 6% BaSO4 and other additives [31]. With increasing temperature, the coefficient of friction of the FMC-11 decreases almost linearly (Fig. 5b). However, at temperatures from 200 to 400 °C that are critical for most of the AFM, the value f even at the maximum pressure p0 = 1.47 MPa is at a level which allows to provide sufficient efficiency of braking. Operation parameters of braking and dimensions of a brake system for numerical calculations are listed in Table 3. The calculations at the mean radius of the braking path (r = 95 mm) were carried out during single braking with linearly decreasing speed of the relative sliding at constant (T = 20 °C, solid lines) and temperature-dependent (dashed lines) coefficients of friction and wear (Figs. 6–9). The numbering of the curves in Figs. 6–9 corresponds to the numbering of the curves in Fig. 5. The temperature evolutions on the pad/disc interface and the changes obtained from Eq. (16) of the coefficient of friction in time for friction pair FC-16L/ChNMKh are shown in Fig. 6. The corresponding results for the FMC-11/ChNMKh are shown in Fig. 7. The temperature values calculated for friction pair FC-16L/ChNMKh at relatively low contact pressure p0 = 0.39 MPa (curve 1, Fig. 6a) and temperature-dependent coefficient of friction f(T) are slightly higher, than those obtained at constant friction coefficient f0 ≡ f(20 °C). An increase in the contact pressure from p0 = 0.39 MPa to p0 = 1.47 MPa generates an inverse situation — after a relatively short time t = 0.64 s (T = 206.3 °C) the temperature calculated at temperature-dependent coefficient of friction, becomes lower than that found at constant value of f0 (curve 2, Fig. 6a). The maximum temperature difference ΔT = 55.2 °C is reached at t = 3.16s. Such behaviour of the temperature calculated at temperature-dependent and constant coefficient of friction can be explained by different characters of changes in braking time of friction pair FC-16L/ChNMKh for these two established values of pressure (Fig. 6b). The coefficient of friction f0 =0.294 at pressure p0 = 0.39 MPa first monotonically increases up to nominal value f = 0.36 (T = 114.2 °C) attained at t = 1.33 s, and remains on the same level to standstill (curve 1, Fig. 6b). At the considerably higher pressure p0 = 1.47 MPa the coefficient of friction with the beginning of braking first increases rapidly from the value f0 =0.277 to the maximum value fmax = f(0. 063 s) =0.345 (T = 96.3 °C), and then decreases to the value of fmin = f(3.02 s)= 0.185 (curve 2, Fig. 6b). A decrease in 0.16 equals to 46%. Further up to standstill slight increase in f may be observed. Comparing behaviours of curves from Fig. 6, it can be seen that for the established value of contact pressure, an increase (decrease) in the coefficient of friction compared with the constant value f0 causes an increase (decrease) in the amount of the generated heat on the pad/ disc interface, and in consequence either higher or lower temperatures on it. For the given value of pressure (FMC-11/ChNMKh), the coefficient of friction decreases almost linearly with the increase in temperature (Fig. 5b). Consideration in calculations for the f(T) results in lower temperatures compared with values of temperature obtained with constant coefficient of friction f0 (Fig. 7a). At the beginning of braking the coefficient of friction decreases and reaches minimum value (Fig. 7b) whereas at that moment the temperature reaches maximum value (Fig. 7a).
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Fig. 5. Mathematical approximations (Eq. (16)) of experimental dependencies of coefficients of friction f and wear rate I [μg/(Nm)] on temperature for cast-iron disc combined with the pad made of FC-16L (a, c) and the pad made of FMC-11 (b, d).
Table 2 Coefficients in formula (16). f
p0, MPa
I ChNMKh/FC-16L
0.39 1.47
ChNMKh/FMC-11
0.59 0.78 1.18 1.47
f1
f2
μg I1, Nm
I2,
0 9.72 0.02 0 6.35 ⋅ 10−3 2.16 0.0321 2.64 0.0285 3.045 0.036 4.95
μg Nm
0.286 57.6 0.288 72 0.762 3.69 0.642 3.84 0.57 4.2 0.48 3.6
f3, °C−1
Tf1, °C
f4
I3, °C−1
TI1, °C
I4,
−2
0.55 ·10 0.6 ·10−2 0.7 ·10−2 0.8 ·10−2 0.19 ·10−2 0.4 ·10−2 0.19 ·10−2 0.4 ·10−2 0.162 ·10−2 0.41 ·10−2 0.15 ·10−2 0.5 ·10−2
105 800 95 850 −180 100 −180 100 −250 120 −250 105
f5, °C−1 μg Nm
0.286 −10.8 0.2 12 0 4.5 0 4.32 0 5.46 0 5.535
Tf2, °C
I5, °C−1
TI2, °C −2
0.25 ⋅ 10 0.4 ⋅ 10−2 0.3 ⋅ 10−2 1 ⋅ 10−2 0 0.6 ⋅ 10−2 0 0.5 ⋅ 10−2 0 0.5 ⋅ 10−2 0 0.62 ⋅ 10−2
800 300 800 160 0 750 0 750 0 780 0 790
A.A. Yevtushenko, P. Grzes / International Communications in Heat and Mass Transfer 39 (2012) 1045–1053 Table 3 Operation parameters and dimensions of the disc and the pad [18]. Items
Disc
Pad
Inner radius, r [mm] Outer radius, R [mm] Thickness, δ [mm] Initial angular speed, ω0 [s−1] Braking time, ts [s] Heat transfer coefficient, h [W / (m2K)] Initial temperature, T0 [°C] Ambient temperature, T∞ [°C]
66 113.5 5.5 88.464 3.96 60 20 20
76.5 10
1051
The change of thermomechanical wear It [mg] (Eq. (16)) and coefficient of wear rate I [μg/(Nm)] versus braking time is shown in Figs. 8 and 9. The results, plotted with solid lines were obtained incorporating constant f0, whereas dashed lines denote values calculated taking into account its temperature dependence. During numerical calculations the dependencies f(T) and I(T) shown in Fig. 5 were taken. The thermomechanical wear of the working surfaces increases monotonically and nonlinearly with time, reaching maximum value at the moment of standstill. For the friction pair FC-16L/ ChNMKh an influence of thermal sensitivity of the coefficients of friction and wear become significant at p0 = 1.47 MPa (curve 2, Fig. 8a). It can be seen that the wear It at p0 = 1.47 MPa, obtained taking into account the constant coefficient of friction is underestimated by
Fig. 6. Evolutions of temperature on the contact surface of the disc (a), and change of the coefficient of friction (b) during braking for the cast-iron disc and the pad made of FC-16L.
Fig. 7. Evolutions of temperature on the contact surface of the disc (a), and change of the coefficient of friction (b) during braking for the cast-iron disc and the pad made of FMC-11.
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Fig. 8. Evolutions of wear It [mg] (a) and coefficients of wear rate I [μg/(Nm)] (b) during braking for the cast-iron disc and the pad made of FC-16L.
22% at the stop time moment. For the pressure p0 = 0.39 MPa the wear It at p0 = 1.47 MPa, obtained taking into account the constant coefficient of friction is underestimated by 4% (curve 1, Fig. 8a). Also for the FMC-11/ChNMKh the wear It for the constant coefficients of friction is underestimated. For the pressures p0 = 0.59; 0.78; 1.18 and 1.47 MPa these values are equal 3.6, 3.2, 4.0, and 3.9%, respectively. At the given value of pressure neglecting the temperature dependence of the coefficient of friction leads to underestimation of wear for both of the considered friction pairs. The corresponding evolutions of the coefficients of wear rate I [μg/ (Nm)] during braking are shown in Figs. 8b and 9b. During short time
from the brake application these coefficients reach maximum value, after which start to decrease attaining values occurring at the half of the braking time.
5. Conclusions In the paper FE simulation of the single braking process of transient frictional heating during braking in the pad/disc system taking into account dependence of coefficients of friction and intensity of thermomechanical wear on the temperature was carried out. The
Fig. 9. Evolutions of wear It [mg] (a) and coefficients of wear rate I [μg/(Nm)] (b) during braking for the cast-iron disc and the pad made of FMC-11.
A.A. Yevtushenko, P. Grzes / International Communications in Heat and Mass Transfer 39 (2012) 1045–1053
numerical calculations included two different (retinax or metal ceramic) pad materials and one (cast iron) material of the disc. For the friction pair FC-16L/ChNMKh it was established that an influence of thermal sensitivity of the coefficients of friction on the temperature and intensity of wear of the friction surface should be taken into account at considerable (p0 = 1.47 MPa) values of pressure, thereby considerable amount of the heat generated on that surface. The change of the coefficient of friction in time for the friction pair depending on the contact pressure is different, namely at p0 = 0.39 MPa an increase in friction coefficient f with braking time may be observed (curve 1, Fig. 6b). At the pressure p0 = 1.47 MPa first rapid increase in friction coefficient takes place, then quick reduction and transition to the approximately even level (curve 2, Fig. 6b). Fluctuation of the coefficient of friction η = fmin/fmax for those two values of the pressure equals η = 0.54 and 0.82, respectively, which means that η increases with the increase in pressure. For the friction pair FMC-11/ChNMKh the temperature difference calculated at constant and temperature-dependent coefficients of friction is considerable even at low pressure (i.e. 0.59 MPa) and increases with the increase in pressure. The coefficients of friction during braking decrease nonlinearly, and the values of their fluctuations equal η =0.79, 0.77, 0.74, 0.75 for the values of pressure p0 =0.59 MPa; 0.78 MPa; 1.18 MPa and 1.47 MPa, respectively. During braking the thermomechanical wear increases monotonically reaching its maximal value at the moment of standstill. An influence of the dependence of the coefficients of friction and wear on the temperature should be taken into account for the friction pair FC-16L/ ChNMKh at high (> 1.5 MPa) values of pressure. The carried out numerical analysis confirms known from the outcomes of experimental tests that for the considered friction pairs the change of pressure leads to the change of the specific power of friction, and in consequence the corresponding temperature and wear variations — with the increase in pressure the coefficient of friction decreases, and the intensity of thermomechanical wear increases [32].
[6] [7] [8] [9] [10] [11] [12]
[13] [14]
[15] [16]
[17]
[18] [19]
[20]
[21]
[22]
[23]
Acknowledgement
[24] [25]
The present article is financially supported by the National Science Centre in Poland (research project No 2011/01/B/ST8/07446).
[26]
References
[28]
[1] A.V. Chichinadze, Calculation and Investigation of External Friction During Braking, Nauka, Moscow, 1967. (in Russian). [2] I.P. Koriauli, A.V. Chichinadze, Temperature variation of the coefficient of friction, in: Problem of Non-Stationary Friction in Machines, Apparatus and Vehicles, Nauka, Moscow, 1978, pp. 217–219, (in Russian). [3] A.V. Chichinadze, E.V. Zinoviev, To assess the importance of some from the defining parameters of friction during braking, in: Optimal Use of Friction Materials in Friction Units of Machines, Nauka, Moscow, 1973, pp. 63–69, (in Russian). [4] A.V. Chichinadze, Z.V. Ignatieva, The account of variation of frictional pairs at calculation of temperature of a surface of friction, Journal of Machinery Manufacture and Reliability (4) (1971) 11–17 (in Russian). [5] A.G. Ginzburg, A.V. Chichinadze, Investigation of the dependence of a coefficient of friction on temperature at non-stationary regime of friction, in: Calculation
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and Simulation of a Mode of Operation of Brake and Friction Devices, Nauka, Moscow, 1974, pp. 30–35, (in Russian). H. Blok, The flash temperature concept, Wear 6 (6) (1963) 483–494. A.V. Chichinadze, R.M. Matveevski, E.P. Braun, Materials in Tribotechnics Non-Stationary Processes, Nauka, Moscow, 1986. (in Russian). M. Krupa, Influence of temperature on value of friction coefficient in friction brakes, Scientific Papers of Silesian University of Technology, Transport 64 (2008) 151–157. Z. Olesiak, Yu. Pyryev, A.A. Yevtushenko, Determination of temperature and wear during braking, Wear 210 (1–2) (1997) 120–126. O.O. Evtushenko, Yu.O. Pyr'ev, Calculation of the contact temperature and wear of frictional elements of brakes, Materials Science 34 (2) (1998) 249–254. Yu. Pyryev, A. Yevtushenko, The influence of the brakes friction elements thickness on the contact temperature and wear, Heat and Mass Transfer 36 (4) (2000) 319–323. O.O. Evtushenko, Yu.O. Pyr'ev, Temperature and wear of the friction surfaces of a cermet patch and metal disc in the process of braking, Materials Science 36 (2) (2000) 218–223. J. Awrejcewicz, Yu. Pyr'yev, Nonsmooth Dynamics of Contacting Thermoelastic Bodies, Springer, New York, 2009. 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. D. Thuresson, Influence of material properties on sliding contact braking applications, Wear 257 (5–6) (2004) 451–460. J. Voldřich, Š. Morávka, J. Študent, Transient temperature field in intermittent sliding contact at temperature dependent coefficient of friction, in: Proceedings of the Twenty Second Computational Mechanics Conference, Nečtiny, Czech Republic, 2006, pp. 697–704. F. Talati, S. Jalalifar, Investigation of heat transfer phenomena in a ventilated disk brake rotor with straight radial rounded vanes, Journal of Applied Sciences 20 (8) (2008) 3583–3592. F. Talati, S. Jalalifar, Analysis of heat conduction in a disk brake system, Heat and Mass Transfer 45 (8) (2009) 1047–1059. H. Mazidi, S. Jalalifar, S. Jalalifar, J. Chakhoo, Mathematical modeling of heat conduction in a disk brake system during braking, Asian Journal of Applied Sciences 4 (2) (2011) 119–136. 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. 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. A.G. Ginzburg, A.V. Chichinadze, To calculation of wear during braking using the equations of thermal dynamics of friction, in: Friction and Wear of Friction Materials, Nauka, Moscow, 1977, pp. 26–30, (in Russian). A.V. Chichinadze, E.D. Braun, A.G. Ginsburg, E.V. Ignat'eva, Calculation, Test and Selection of Frictional Couples, Nauka, Moscow, 1979. (in Russian). MSC.Nastran, Thermal Analysis User's Guide, 2002. R.W. Lewis, P. Nithiarasu, K.N. Seetharamu, Fundamentals of the Finite Element Method for Heat and Fluid Flow, John Wiley & Sons, New York, 2004. C.W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, first ed. Prentice-Hall, Englewood Cliffs, New Jersey, 1971. G.E. Forsythe, M.A. Malcolm, C.B. Moler, Computer Methods for Mathematical Computation, first ed. Prentice-Hall, Englewood Cliffs, New Jersey, 1977. A. Krupowicz, Numerical Methods for Initial Problems of the Ordinary Differential Equations, PWN, Warsaw, 1986. D.N. Kavetski, P. Binning, S.W. Sloan, Adaptive backward Euler time stepping with truncation error control for numerical modeling of unsaturated fluid flow, International Journal for Numerical Methods in Engineering 53 (6) (2002) 1301–1322. S. Kim, H.T. Sapronov, M.V. Malyutin, Investigation of temperature fields of different friction pairs used in heavily loaded multi-disc brakes, in: Friction and Wear of Friction Materials, Nauka, Moscow, 1977, pp. 72–79, (in Russian). V.A. Balakin, V.P. Sergienko, Heat Calculations of Brakes and Friction Joints, MPRI NASB, Gomel, 1999. (in Russian). A.V. Chichinadze, E.V. Zinoviev, Physical–chemical mechanics of contact interaction and friction of the asbestos frictional plastics in a pair with metals, in: Thermal Dynamics and Simulation of External Friction, Nauka, Moscow, 1975, pp. 117–173, (in Russian).
Contents lists available at SciVerse ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Axisymmetric FEA of temperature in a pad/disc brake system at temperature-dependent coefficients of friction and wear☆ A.A. Yevtushenko ⁎, 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 28 July 2012 Keywords: Frictional heating Temperature Pad/disc brake system Temperature-dependent coefficient of friction Wear Finite element method
a b s t r a c t Robust braking results in heat generation whose effects may have considerable impact on the parameters of the process such as wear rate and coefficient of friction. Fluctuations of the latter disagree with essential operational and braking requirements. Finite element analysis (FEA) of a single braking process for axisymmetric heat conduction problem of friction in a pad/disc brake system in the present article was carried out. Two materials of the pad FC-16L (retinax A) and FMC-11 (metal ceramic) and one material of the disc ChNMKh (cast iron) were analysed. Experimental dependencies of the coefficient of friction and wear rate on the temperature under specified contact pressures for these two friction pairs were approximated and applied to FE contact model. The temperature and wear evolutions on the contact surface of the pad/disc brake system obtained for constant and temperature-dependent abovementioned coefficients were confronted and compared. Mutual correlations of the obtained results with the studied materials were discussed. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The magnitude and fluctuations of the coefficient of friction during braking depend on the mechanical and thermophysical properties of materials, speed, load, temperature regimes of frictional contact as well as the nature of physical and chemical processes [1]. The process of braking results in continuous and interdependent changes in both external factors (speed, load, temperature) and the properties of subsurface layers of materials and the intensity of frictional contact processes [2]. Thus it is very important to distinguish factors which have decisive impact on the coefficient of friction. To evaluate the variations of the coefficient of friction during braking the most convenient is its dependence on the speed, temperature and temperature gradient (the load at braking is practically constant) [3]. The abovementioned quantities can be directly measured during the process, or obtained from calculations based on the experimental data. At the same time structural changes and the intensity of the processes practically do not affect the temperatures and their gradients [4]. Having regard to the time, process of braking can be divided into four periods [5] (Fig. 1). The duration of the first 0 ≤ t b t1 is determined from the time of an increase in the load acting on the contacting bodies. According to the experimental data, the time from 0 to t1 equals about 2% of the total braking time ts. This period is characterized by the high ☆ 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.2012.07.025
speed of sliding, the high flash temperature occurred on microcontacts, the low temperature of contact surface of friction and low temperature gradients normal to the contact surface. The volumetric temperature does not reveal any changes. During the second period t1 ≤t b t2 the speed remains high, the load reaches nominal value, the flash temperature passes through a maximum point, the temperature and its gradient on the contact surface are high, and the volumetric temperature increases. In the interval of time t2 ≤ t b t3 the speed is still quite high, the load is constant, the flash temperature decreases, the surface temperature reaches maximum value, and the temperature gradient begins to decrease as the volumetric temperature increases. In the last, fourth period of braking (t3 ≤ t b ts) the speed is close to zero, the load is constant, the flash temperature practically equals zero, the temperature on the contact surface smoothly decreases and brings nearer the volumetric temperature, and the temperature gradient is very small. Duration of this period ranges from 3% to 5% from general time of braking. The value of coefficient of friction in the first period is typically determined pffiffiffiffi using the speed, i.e. the flash temperature, which is proportional to V [6]. In the second period — using the mean temperature on the surface of friction and the flash temperature at the prevailing value of the first. In the third and the final period of braking the value of the coefficient of friction is determined by means of temperature on the contact surface. Therefore, with an increase in temperature an influence of the speed and the temperature gradient on the coefficient of friction decreases and the magnitude and character of variation during braking are mainly determined by the temperature of the frictional contact. To take into account
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Nomenclature c [C] f f0 h I(T) It(T) k K [K] l n p0 {Q(T)} r r, R t t1, t2, t3 ts T T∞ T0 {T}o n T_ V z
Specific heat, (J/(kgK)) Specific heat (or capacitance) matrix Coefficient of friction Coefficient of friction at temperature of 20 °C Heat transfer coefficient, (W/(m 2K)) Temperature-dependent coefficient of wear rate, (μg/ (Nm)) Temperature-dependent wear, (mg) Thermal diffusivity, (m 2/s) Thermal conductivity, (W/(mK)) Conductivity matrix Characteristic dimension of finite element, (m) Iteration number Contact pressure, (MPa) Vector of heat loads that depends on temperature Radial coordinate Internal and external radius, respectively, (m) Time, (s) Characteristic moments in time during single braking process, (s) Braking time, (s) Temperature, (°C) Ambient temperature, (°C) Initial temperature, (°C) Temperature vector Vector of time derivative of vector of nodal temperature {T} Sliding speed, (m/s) Axial coordinate
Greek symbols δ Thickness, (m) η Fluctuation of the coefficient of friction, η = fmin/fmax θ Circumferential coordinate θ0 Cover angle of pad, (deg) ρ Density, (kg/m 3) ω Angular speed of the disc, (s −1) ω0 Initial angular speed of the disc, (s −1)
Subscripts p Indicates pad d Indicates disc
Superscripts ± values, obtained at the approach to plane z = 0 along positive (+) or negative (−) direction of the axis Oz.
the temperature dependence of the coefficient of friction in the calculations concerned the frictional heating systems, the corresponding relationships can be derived from the experimental data [7,8]. The analytical method for solving one-dimensional thermal problems of friction during braking for tribosystem consisting of two semi-spaces incorporating the dependence of the coefficients of friction and wear on the temperature was proposed in Refs. [9,10]. The development of this method to the case of frictional contact between two strips
was obtained in Ref. [11] and for a three-strip tribosystem — in Ref. [12]. A generalization of these results can be found also in Ref. [13]. In recent times, the more active simulations of friction heating in brake systems are being carried out employing numerical methods, in particular the finite element method (FEM) [14]. The thermomechanical model based on the finite element method, taking into account the temperature-dependent friction and wear coefficients during uniform sliding of the elastic rectangular block over the rigid support was proposed in Ref. [15]. The numerical solution of two-dimensional transient problem of heat conduction for tribosystem consisting of a strip of finite length, sliding between the surfaces of the two shorter strips, with thermal resistance of the contact surfaces and the temperature-dependent friction coefficient, was obtained in Ref. [16]. The axisymmetric FE model to study the transient temperature field in a pad/disc tribosystem with constant thermophysical properties of materials was proposed in Refs. [17–20]. A generalization of this model to the case of thermosensitive materials of the pad and the disc was carried out in Ref. [21]. The proposed article is a synthesis of that model for the case, when the coefficient of friction depends on the temperature. In addition, it is assumed that the wear is temperature-dependent, as well. 2. Statement of the problem The process of frictional heating in a pad/disc tribosystem during a single braking is considered (Fig. 2). The coefficient of mutual overlap of the pad and the disc is close to the unit. At the initial moment of time t = 0 the pads located on the outboard and inboard surfaces of the rotating with constant angular speed ω0 disc are pressed against generating the constant and uniform contact pressure p0 on the working surfaces. As a result of friction, the speed of rotation of the disc begins to decrease linearly with time: t ωðt Þ ¼ ω0 1− ; 0≤t ≤t s ; ts
ð1Þ
and on the contact surfaces the heat is generated, causing consequently heating of the elements of friction pair. The sum of the intensities of heat fluxes directed from the surface of friction inside the pads and the disc is equal to the specific power of friction. It is assumed that the thermal resistance on the surfaces of contact is negligible and therefore temperatures of the pads and the disc on these surfaces are equal. The free surfaces of the pads and the disc are convectively cooled with the same constant heat transfer coefficient h being average value for the analysed braking process. The thermophysical properties of materials of the pads and the disc are constant whereas the coefficient of friction f depends on the temperature T. It should be noted that the nature of change of the coefficient of friction during braking caused by an increase in temperature in the area of contact affects in fact evolution of a speed and in consequence, time of duration of the process. That effect in the considered study was omitted. The considered tribosystem is symmetric regarding the mid-plane of the disc. Therefore the system consisting of one pad sliding over the contact surface of the half of the entire disc will be considered. The opposite (median) surface of the disc is assumed to be thermally insulated. All of the values and parameters which refer to the pad and the disc in the further considerations will have subscripts p and d, respectively. The considered tribosystem is referred to the cylindrical system with coordinates (r, z) (Fig. 2). Then according to the abovementioned assumptions the distribution of the transient temperature in the pad and in the disc will be found from the solution of the following axisymmetric boundary-value problem of heat conduction: ∂2 T 1 ∂T ∂2 T 1 ∂T þ ¼ þ ; r p ≤ r ≤Rp ; 0 < z < δp ; t > 0; ∂r 2 r ∂r ∂z2 kp ∂t
ð2Þ
A.A. Yevtushenko, P. Grzes / International Communications in Heat and Mass Transfer 39 (2012) 1045–1053
1047
h i ∂T K p ¼ h T ∞ −T r; δp ; t ; r p ≤ r ≤ Rp ; t ≥ 0; ∂z z¼δ
ð8Þ
Kd
∂T ¼ h½T ∞ −T ðr; 0; t Þ; r d ≤ r ≤ r p ; t ≥ 0; ∂z z¼0
ð9Þ
Kd
∂T ¼ h½T ∞ −T ðRd ; z; t Þ; − δd ≤ z ≤ 0; t ≥ 0 ∂r r¼Rd
ð10Þ
Kd
∂T ¼ h½T ðr d ; z; t Þ−T ∞ ; − δd ≤ z ≤ 0; t ≥ 0 ∂r r¼rd
ð11Þ
p
Fig. 1. Typical changes of the sliding speed, temperature and temperature gradient during single braking [5].
∂2 T 1 ∂T ∂2 T 1 ∂T þ ¼ þ ; r d ≤ r ≤Rd ; −δd < z < 0; t > 0; ∂r2 r ∂r ∂z2 kd ∂t þ − T r; 0 ; t ¼ T ðr; 0 ; t Þ; r p ≤ r ≤Rp ; 0 ≤t ≤t s ; ∂T ∂T K d − −K p þ ¼ f ðT Þrωðt Þp0 ; r p ≤ r ≤ Rp ; 0 ≤ t ≤ t s ; ∂z z¼0 ∂z z¼0
ð3Þ
ð12Þ
T ðr; z; 0Þ ¼ T 0 ; r p ≤ r ≤ Rp ; 0 ≤ z ≤ δp
ð13Þ
T ðr; z; 0Þ ¼ T 0 ; r d ≤ r≤ Rd ; − δd ≤ z ≤ 0
ð14Þ
Frictional contact of the pads and the disc is accompanied by wear processes. The main influence on the wear for intensive process of braking with a significant load has the heat manifesting by the temperature on the contact surface [22]. The change of the wear It [mg] during braking has the form [23]: t
It ðT Þ ¼ ∫ IðT Þ f ðT Þ p0 V ðτ Þ Ak dτ; 0 ≤ t ≤ t s ; ð4Þ
ð5Þ
Kp
h i ∂T ¼ h T r p ; z; t −T ∞ ; 0 ≤ z ≤ δp ; t ≥ 0 ∂r r¼rp
ð6Þ
Kp
h i ∂T ¼ h T ∞ −T Rp ; z; t ; 0 ≤ z ≤ δp ; t ≥ 0 ∂r r¼Rp
ð7Þ
Fig. 2. A schematic diagram of a disc brake.
∂T ¼ 0; r d ≤ r ≤ Rd ; t ≥ 0 ∂z z¼−δd
ð15Þ
0
where I(T) [μg/(Nm)] is the temperature-dependent coefficient of wear rate, V(t) is the time-dependent speed of sliding of the pad over the rubbing path of the disc. Analysis of experimental data presented in publications [2,8,22,23] showed that the dependence of coefficient of friction and wear rate on the temperature may be either increasing or decreasing, with one or two maxima or minima (Fig. 3). For approximation of these types of experimental curves the function was developed in the form [7]: f ðT Þ ¼ f 1 þ
f f h 2 i2 þ h 4 i2 1 þ f 3 T−T f 1 1 þ f 5 T−T f 2
Fig. 3. Typical dependencies of coefficient of friction during braking [7].
ð16Þ
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where the constants f1,…, f5, Tf1, Tf2 are chosen taking into account the nature of variation of the curves shown in Fig. 3. Some general recommendations on selection of these constants are as follows: 1) Constant function (curve 1) — f1 > 0, f2 = f4 = 0; 2) Monotonically decreasing function (curve 2) — f1 > 0, f2 > 0, f4 = 0, Tf1 b 0; 3) Decreasing function with two local maxima (curve 3) — f1 > 0, f2 > 0, f4 b f2, Tf1 > 0, Tf2 > 0; 4) Monotonically increasing function (curve 4) — f1 > 0, f2 b 0, f4 = 0, Tf1 = 0; 5) Increasing function with two local maxima (curve 5) — f1 > 0, f2 > 0, f4 > f2, Tf1 > 0, Tf2 > 0; 6) Function with one local maximum and the minimum (curve 6) — f1 > 0, f2 > 0, f4 > > f2, Tf1 > 0,Tf2 > 0. In all the abovementioned cases f3 > 0 and f5 > 0. 3. Temporal integration The finite element code MD Nastran provides a convenient means of numerical modelling of the frictional heating during braking [24]. A system with an infinite number of unknowns (the temperature and its gradients at every location of the brake) can be transformed into one which has a finite number of unknowns related to each other by elements of finite size. In the finite element formulation by means of linear Galerkin spatial approximation, the boundary-value heat conduction problems (2)–(14) can be written as [25]: n o ½C T_ þ fKg fTg ¼ fQ ðTÞg;
ð17Þ
where {Q(T)} is the temperature-dependent vector containing the boundary conditions (4)–(14). The nonlinear ordinary differential equations of type of Eq. (17) are usually solved numerically either by means of explicit or implicit methods [26]. But the most effective is the scheme, known as a predictor–corrector method, when on a stage of prediction the initial approximation is determined by means of the explicit algorithm and on the stage of correction the iterations are performed using an implicit scheme. Due to simplicity, unconditional stability and robustness the first-order time backward Euler stepping approach was used for temporary discretization [27]: n o fTgtþΔt ¼ fTgt þ Δt T_
tþΔt
;
ð18Þ
where {T}t and {T}t + Δt are the values of the temperature vector {T} at the time moments t and t + Δt, respectively; Δt is the time step. Substituting the relation (18) into the Eq. (17) we obtain the following system of algebraic equation with respect to {T}t + Δt: ð½C þ Δt ½KÞfTgtþΔt ¼ ½C fTgt þ Δt fQ ðTÞgtþΔt :
ð19Þ
The system of nonlinear algebraic Eq. (19) will be solved by a simple iteration method. If the approximation {T}t+Δt(n)for the iteration n is known, then for the iterations n + 1 we have the following linearized system: ðnþ1Þ
ðnÞ
ð½C þ Δt ½KÞfTgtþΔt ¼ ½C fTgt þ Δt fQ ðTÞgtþΔt ; n ¼ 0; 1; 2; …; n max :
ð20Þ
Note that, regardless of the initial approximation {T}t+Δt(0) the iteration {T}t+Δt(nmax) converges to the solution {T}t+Δt, if a user-defined maximum number of iterations (the same at each time step) nmax →∞ [28]. However at the fixed value nmax ≥1 the approximation {T}t+Δt(nmax) will be better; the size of the step time Δt and the initial error (0) {T}t+Δt −{T}t+Δt are smaller.
To determine the initial approximation the explicit Euler scheme was used: n o fTgtþΔt ¼ fTgt þ Δt T_ ; t
ð21Þ
on the basis of which we have: ð0Þ
½CfTgtþΔt ¼ ð½C−Δt ½KÞfTgt þ Δt fQ ðTÞgt :
ð22Þ
The accuracy of the solution obtained using such a scheme is O(Δt 2). The maximum number of iterations per time step has been limited to 50. If the number of iterations exceeds this number, then the step size decreased by a quarter. Additional algorithm details, including evaluation of the local temporal truncation errors can be found, for example, in Ref. [29]. Since the considered heat conduction problem of friction is axisymmetric the geometrical basis for numerical calculations were two rectangles (pad and half of the disc) in coordinates (r, z) (cross section) representing the pad and the disc on which 2D axisymmetric solid triangular three-node finite elements were constructed. The overall number of elements was equal 6994 elements and 3711 nodes, where 2850 elements and 1536 nodes came for the disc, and 4144 elements and 2175 nodes came for the pad (Fig. 4). In order to accomplish the boundary conditions within the area of contact (Eqs. (4) and (5)), 75 of the multi point constraints (MPC) were constructed combining every corresponding two nodes on the opposite contact surfaces of the pad and the disc, respectively. The characteristic dimension of one specific element differed in the pad/disc FE model according to the coordinates of the system. In radial direction dimension of the side of one triangle (parallel to coordinate r) was equal within the entire length of the pad and the disc l = 0.5 mm. In axial direction z, the mesh division was the finest in the location of the contact surface and increased linearly with the distance from z = 0 both for the disc and for the pad. The characteristic dimension varied from the smallest value l = 0. 25 mm to 0. 5 mm at the furthest location from the contact surface. These values were equal for the disc and for the pad. Hence the time step for numerical 2 calculations (Δt = 0.1lmin k −1) was calculated for greater value of the thermal diffusivity (a pad or a disc material). 4. Numerical analysis To perform numerical calculations two friction pairs were selected: FC-16L (retinax A)-pad/ChNMKh (cast iron)-disc and FMC-11 (metal ceramic)-pad/ChNMKh (cast iron)-disc. These friction pairs are widely distributed in mechanical engineering and friction materials. Their thermophysical properties and the nature of physical and chemical processes occurring during sliding contact represent two completely different classes of frictional materials (Table 1). The experimental dependencies of the coefficient of friction and wear rate on the temperature for the asbestos friction material (AFM) with a resin binder FC-16L and sintered powder material FMC-11 for dry friction with the cast-iron ChNMKh were presented in Ref. [7]. These relationships take into account the effect of pressure, i.e. the series of curves are obtained with different p0, which cover a range of specific loads occurring during friction or maximum permissible for a given pair. The curves approximating experimental data [7] by means of function (16) are shown in Fig. 5. The values of constants f1,…, f5, Tf1, and Tf2 for the coefficient of friction f(T) and I1,…,I5, TI1, and TI2 for the coefficient of wear rate I(t) are given in Table 2. The curves in Fig. 5 are constructed for the following values of pressure p0: 1– 0.39 MPa, 2–1.47 MPa, 3–0.59 MPa; 4–0.78 MPa; 5–1.18 MPa; and 6–1.47 MPa.
A.A. Yevtushenko, P. Grzes / International Communications in Heat and Mass Transfer 39 (2012) 1045–1053
Fig. 4. Axisymmetric FE model of a pad/disc brake system with the boundary conditions.
The most important determining factor for sliding contact of AFM is the temperature, which affects the change of mechanical properties of friction materials as a consequence of increased intensity of thermal motion of molecules and the initiation of various physical and chemical phenomena [30]. The thermal degradation of binders included in the AFM and the resulting effects generated by friction arise at temperatures higher than 150 °С and intensively developed in the range from 200 to 300 °C. This also confirms dependencies of the coefficient of friction on the temperature for the FC-16L/ChNMKh couple, which are presented in Fig. 5a. The region of the reduction can be clearly seen between 200 and 300 °С. This can be explained by the fact that the products of destruction of the binder (resin, rubber) are the liquid fraction, which congregate on the frictional contact and act as a Table 1 Thermophysical properties of materials of the pad and the disc at 20 °C [7]. Material
K [W/(mK)]
ρ [kg/m3]
c [J/(kg K)]
k × 106[m2/s]
ChNMKh FC-16L FMC-11
51 0.79 34.3
7100 2500 4700
500 961 500
14.4 0.33 14.6
1049
lubricant, reducing the coefficient of friction. Therefore, the AFM on the resin coupler (including FC-16L) has such technical flaws as low and unstable coefficient of friction and high levels of residual stresses that cause disruption of the continuity of the material and bad compatibility with the metal base of a brake. Wide application of AFM is caused by their good stability of wear, as well as low cost and simplicity of manufacturing. In the modern highly loaded disc brakes such as aircraft brakes, the friction material made by powder metallurgy — the friction metal-ceramic materials (FMC) are used. These materials are composed of sintered friction layer and steel substrate. These include, in particular, the sintered material on iron substrate FMC-11. The percentage of the components in this material is: 45% Fe, 15% Cu, 9% C, 3% SiO2, 3% asbestos, 6% BaSO4 and other additives [31]. With increasing temperature, the coefficient of friction of the FMC-11 decreases almost linearly (Fig. 5b). However, at temperatures from 200 to 400 °C that are critical for most of the AFM, the value f even at the maximum pressure p0 = 1.47 MPa is at a level which allows to provide sufficient efficiency of braking. Operation parameters of braking and dimensions of a brake system for numerical calculations are listed in Table 3. The calculations at the mean radius of the braking path (r = 95 mm) were carried out during single braking with linearly decreasing speed of the relative sliding at constant (T = 20 °C, solid lines) and temperature-dependent (dashed lines) coefficients of friction and wear (Figs. 6–9). The numbering of the curves in Figs. 6–9 corresponds to the numbering of the curves in Fig. 5. The temperature evolutions on the pad/disc interface and the changes obtained from Eq. (16) of the coefficient of friction in time for friction pair FC-16L/ChNMKh are shown in Fig. 6. The corresponding results for the FMC-11/ChNMKh are shown in Fig. 7. The temperature values calculated for friction pair FC-16L/ChNMKh at relatively low contact pressure p0 = 0.39 MPa (curve 1, Fig. 6a) and temperature-dependent coefficient of friction f(T) are slightly higher, than those obtained at constant friction coefficient f0 ≡ f(20 °C). An increase in the contact pressure from p0 = 0.39 MPa to p0 = 1.47 MPa generates an inverse situation — after a relatively short time t = 0.64 s (T = 206.3 °C) the temperature calculated at temperature-dependent coefficient of friction, becomes lower than that found at constant value of f0 (curve 2, Fig. 6a). The maximum temperature difference ΔT = 55.2 °C is reached at t = 3.16s. Such behaviour of the temperature calculated at temperature-dependent and constant coefficient of friction can be explained by different characters of changes in braking time of friction pair FC-16L/ChNMKh for these two established values of pressure (Fig. 6b). The coefficient of friction f0 =0.294 at pressure p0 = 0.39 MPa first monotonically increases up to nominal value f = 0.36 (T = 114.2 °C) attained at t = 1.33 s, and remains on the same level to standstill (curve 1, Fig. 6b). At the considerably higher pressure p0 = 1.47 MPa the coefficient of friction with the beginning of braking first increases rapidly from the value f0 =0.277 to the maximum value fmax = f(0. 063 s) =0.345 (T = 96.3 °C), and then decreases to the value of fmin = f(3.02 s)= 0.185 (curve 2, Fig. 6b). A decrease in 0.16 equals to 46%. Further up to standstill slight increase in f may be observed. Comparing behaviours of curves from Fig. 6, it can be seen that for the established value of contact pressure, an increase (decrease) in the coefficient of friction compared with the constant value f0 causes an increase (decrease) in the amount of the generated heat on the pad/ disc interface, and in consequence either higher or lower temperatures on it. For the given value of pressure (FMC-11/ChNMKh), the coefficient of friction decreases almost linearly with the increase in temperature (Fig. 5b). Consideration in calculations for the f(T) results in lower temperatures compared with values of temperature obtained with constant coefficient of friction f0 (Fig. 7a). At the beginning of braking the coefficient of friction decreases and reaches minimum value (Fig. 7b) whereas at that moment the temperature reaches maximum value (Fig. 7a).
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A.A. Yevtushenko, P. Grzes / International Communications in Heat and Mass Transfer 39 (2012) 1045–1053
Fig. 5. Mathematical approximations (Eq. (16)) of experimental dependencies of coefficients of friction f and wear rate I [μg/(Nm)] on temperature for cast-iron disc combined with the pad made of FC-16L (a, c) and the pad made of FMC-11 (b, d).
Table 2 Coefficients in formula (16). f
p0, MPa
I ChNMKh/FC-16L
0.39 1.47
ChNMKh/FMC-11
0.59 0.78 1.18 1.47
f1
f2
μg I1, Nm
I2,
0 9.72 0.02 0 6.35 ⋅ 10−3 2.16 0.0321 2.64 0.0285 3.045 0.036 4.95
μg Nm
0.286 57.6 0.288 72 0.762 3.69 0.642 3.84 0.57 4.2 0.48 3.6
f3, °C−1
Tf1, °C
f4
I3, °C−1
TI1, °C
I4,
−2
0.55 ·10 0.6 ·10−2 0.7 ·10−2 0.8 ·10−2 0.19 ·10−2 0.4 ·10−2 0.19 ·10−2 0.4 ·10−2 0.162 ·10−2 0.41 ·10−2 0.15 ·10−2 0.5 ·10−2
105 800 95 850 −180 100 −180 100 −250 120 −250 105
f5, °C−1 μg Nm
0.286 −10.8 0.2 12 0 4.5 0 4.32 0 5.46 0 5.535
Tf2, °C
I5, °C−1
TI2, °C −2
0.25 ⋅ 10 0.4 ⋅ 10−2 0.3 ⋅ 10−2 1 ⋅ 10−2 0 0.6 ⋅ 10−2 0 0.5 ⋅ 10−2 0 0.5 ⋅ 10−2 0 0.62 ⋅ 10−2
800 300 800 160 0 750 0 750 0 780 0 790
A.A. Yevtushenko, P. Grzes / International Communications in Heat and Mass Transfer 39 (2012) 1045–1053 Table 3 Operation parameters and dimensions of the disc and the pad [18]. Items
Disc
Pad
Inner radius, r [mm] Outer radius, R [mm] Thickness, δ [mm] Initial angular speed, ω0 [s−1] Braking time, ts [s] Heat transfer coefficient, h [W / (m2K)] Initial temperature, T0 [°C] Ambient temperature, T∞ [°C]
66 113.5 5.5 88.464 3.96 60 20 20
76.5 10
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The change of thermomechanical wear It [mg] (Eq. (16)) and coefficient of wear rate I [μg/(Nm)] versus braking time is shown in Figs. 8 and 9. The results, plotted with solid lines were obtained incorporating constant f0, whereas dashed lines denote values calculated taking into account its temperature dependence. During numerical calculations the dependencies f(T) and I(T) shown in Fig. 5 were taken. The thermomechanical wear of the working surfaces increases monotonically and nonlinearly with time, reaching maximum value at the moment of standstill. For the friction pair FC-16L/ ChNMKh an influence of thermal sensitivity of the coefficients of friction and wear become significant at p0 = 1.47 MPa (curve 2, Fig. 8a). It can be seen that the wear It at p0 = 1.47 MPa, obtained taking into account the constant coefficient of friction is underestimated by
Fig. 6. Evolutions of temperature on the contact surface of the disc (a), and change of the coefficient of friction (b) during braking for the cast-iron disc and the pad made of FC-16L.
Fig. 7. Evolutions of temperature on the contact surface of the disc (a), and change of the coefficient of friction (b) during braking for the cast-iron disc and the pad made of FMC-11.
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A.A. Yevtushenko, P. Grzes / International Communications in Heat and Mass Transfer 39 (2012) 1045–1053
Fig. 8. Evolutions of wear It [mg] (a) and coefficients of wear rate I [μg/(Nm)] (b) during braking for the cast-iron disc and the pad made of FC-16L.
22% at the stop time moment. For the pressure p0 = 0.39 MPa the wear It at p0 = 1.47 MPa, obtained taking into account the constant coefficient of friction is underestimated by 4% (curve 1, Fig. 8a). Also for the FMC-11/ChNMKh the wear It for the constant coefficients of friction is underestimated. For the pressures p0 = 0.59; 0.78; 1.18 and 1.47 MPa these values are equal 3.6, 3.2, 4.0, and 3.9%, respectively. At the given value of pressure neglecting the temperature dependence of the coefficient of friction leads to underestimation of wear for both of the considered friction pairs. The corresponding evolutions of the coefficients of wear rate I [μg/ (Nm)] during braking are shown in Figs. 8b and 9b. During short time
from the brake application these coefficients reach maximum value, after which start to decrease attaining values occurring at the half of the braking time.
5. Conclusions In the paper FE simulation of the single braking process of transient frictional heating during braking in the pad/disc system taking into account dependence of coefficients of friction and intensity of thermomechanical wear on the temperature was carried out. The
Fig. 9. Evolutions of wear It [mg] (a) and coefficients of wear rate I [μg/(Nm)] (b) during braking for the cast-iron disc and the pad made of FMC-11.
A.A. Yevtushenko, P. Grzes / International Communications in Heat and Mass Transfer 39 (2012) 1045–1053
numerical calculations included two different (retinax or metal ceramic) pad materials and one (cast iron) material of the disc. For the friction pair FC-16L/ChNMKh it was established that an influence of thermal sensitivity of the coefficients of friction on the temperature and intensity of wear of the friction surface should be taken into account at considerable (p0 = 1.47 MPa) values of pressure, thereby considerable amount of the heat generated on that surface. The change of the coefficient of friction in time for the friction pair depending on the contact pressure is different, namely at p0 = 0.39 MPa an increase in friction coefficient f with braking time may be observed (curve 1, Fig. 6b). At the pressure p0 = 1.47 MPa first rapid increase in friction coefficient takes place, then quick reduction and transition to the approximately even level (curve 2, Fig. 6b). Fluctuation of the coefficient of friction η = fmin/fmax for those two values of the pressure equals η = 0.54 and 0.82, respectively, which means that η increases with the increase in pressure. For the friction pair FMC-11/ChNMKh the temperature difference calculated at constant and temperature-dependent coefficients of friction is considerable even at low pressure (i.e. 0.59 MPa) and increases with the increase in pressure. The coefficients of friction during braking decrease nonlinearly, and the values of their fluctuations equal η =0.79, 0.77, 0.74, 0.75 for the values of pressure p0 =0.59 MPa; 0.78 MPa; 1.18 MPa and 1.47 MPa, respectively. During braking the thermomechanical wear increases monotonically reaching its maximal value at the moment of standstill. An influence of the dependence of the coefficients of friction and wear on the temperature should be taken into account for the friction pair FC-16L/ ChNMKh at high (> 1.5 MPa) values of pressure. The carried out numerical analysis confirms known from the outcomes of experimental tests that for the considered friction pairs the change of pressure leads to the change of the specific power of friction, and in consequence the corresponding temperature and wear variations — with the increase in pressure the coefficient of friction decreases, and the intensity of thermomechanical wear increases [32].
[6] [7] [8] [9] [10] [11] [12]
[13] [14]
[15] [16]
[17]
[18] [19]
[20]
[21]
[22]
[23]
Acknowledgement
[24] [25]
The present article is financially supported by the National Science Centre in Poland (research project No 2011/01/B/ST8/07446).
[26]
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