Research Article
Finite element solution of the three-dimensional system of equations of heat dynamics of friction and wear during single braking
Advances in Mechanical Engineering 2018, Vol. 10(11) 1–15 Ó The Author(s) 2018 DOI: 10.1177/1687814018808643 journals.sagepub.com/home/ade
Piotr Grzes
Abstract A three-dimensional numerical model of a disk brake to study temperature on a discrete contact of rough surfaces has been developed. It includes the system of equations formulated based on thermotribological postulates of heat dynamics of friction and wear with mutual influence of contact pressure, velocity, properties of materials, and temperature. Two approaches of calculation of the flash temperature and its influence on the maximum temperature during a single braking application were studied. Changes in the contact temperature, sliding velocity, and the thermomechanical wear during braking were shown and discussed. It was found that two of the examined variants of calculation of the flash temperature agree well for the three considered materials of the brake pads combined with the cast iron disk, at each initial sliding velocity in the range from 5 to 20 m s21. Keywords Disk brake, friction, flash temperature, thermosensitive materials, finite element method
Date received: 5 June 2018; accepted: 1 October 2018 Handling Editor: Noel Brunetiere
Introduction Modeling of non-stationary processes of frictional heat generation with interdependent temperature, coefficient of friction, sliding velocity, contact pressure, and wear requires taking into account complex mathematical and physical aspects. Mechanisms of interactions between the operating parameters and the phenomenon of formation of a micro-contact on the friction surfaces were analyzed with reference to models of frictional heating proposed by Blok, Jaeger, Archard, and Kuhlmann-Wilsdorf in the review article.1 These solutions concern mainly specific geometrical objects, which are difficult to implement in real friction couples. The comprehensive research on thermal processes in brakes and clutches with the development of new materials and methods for their testing was carried out by Chichinadze and his co-workers. The experimental
studies and mathematical modeling contributed to formulate theoretical fundamentals of the system of integral–differential and algebraic equations and further simulation of the friction and wear with the interdependent governing parameters.2,3 In the case of a single braking, the main point of such system is hypothesis of the dependence of a coefficient of friction on a maximum temperature Tmax , being the sum of a mean temperature Tm of a nominal area of contact and a flash temperature Tf of a real area of
Faculty of Mechanical Engineering, Bialystok University of Technology (BUT), Bialystok, Poland Corresponding author: Piotr Grzes, Faculty of Mechanical Engineering, Bialystok University of Technology (BUT), 45C Wiejska Street, 15-351 Bialystok, Poland. Email:
[email protected]
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
2 contact.3 Most of the known solutions of the heat dynamics of friction and wear (HDFW) systems of equations are based on the analytical solutions of the boundary value heat conduction problems related to the determination of the mean temperature Tm of friction surfaces and the flash temperature Tf .4 A model for calculating Tm is a one-dimensional (1D) boundary value problem of heat conduction for two sliding semi-spaces or layers with variable during the process (in general linearly decreasing) in the friction power density.5–7 The analytical or analytical–numerical solutions of the boundary value heat conduction problems for semispaces with the moving local area of frictional heating8– 10 are used to estimate the flash temperature Tf . Both solutions are linear—in their formulation, constant value of the coefficient of friction and properties of materials, corresponding to the initial temperature, are used. These constants are determined from the empirical formulas for a bulk temperature TV , averaged over time.11 Based on the 1D HDFW systems of equations, approximate estimations of the maximum temperature Tmax can be obtained. These solutions do not also take into account the spatial distribution of the temperature field, changes in properties of materials during frictional heating, and convective cooling of a given braking system. In the most cases of the known numerical computations using the finite element method (FEM) of the three-dimensional (3D) HDFW systems of equations, mutual relationship of a sliding velocity, temperature, and thermosensitive materials of pads and a disk is neglected.12–14 This is mainly due to the fact that an emphasis is laid on examining other important parameters of the braking process, that is, heat dissipation from the ventilated brake disk12 or thermal stresses, taking into account non-uniform contact pressure distribution.13 A review of the studies on numerical modeling of the thermal stress in disk brakes and clutches was reported previously.14 The most general axisymmetric (two-dimensional (2D)) HDFW system of equations for the description of frictional heating during the simultaneous operation of a disk brake were formulated and solved using FEM in previous studies.15,16 A detailed explanation of the structure of the HDFW systems and a background of their evolution was reported in Grzes.16 The 3D HDFW comprehensive system of equations has not so far been considered. A simplified 3D variant of the HDFW system for a disk brake, at constant properties of materials, and taking into account the flash temperature Tf , was developed in Yevtushenko and Grzes,17 and using thermosensitive materials in Yevtushenko and Grzes.18 The above-mentioned investigations16–18 have demonstrated that taking into account the change in the coefficient of friction affects the braking time and slightly changes the peak value of the mean temperature.
Advances in Mechanical Engineering This study is a generalization of the 2D HDFW equations16 to the 3D problem. It includes the following: 1.
2.
3.
4.
Initial value problem for the equation of motion with time-varying contact pressure during braking; Spatial non-linear thermal problem of friction for determining the mean temperature Tm on the nominal area of contact of pads with a brake disk. It contains a frictional heating on the nominal area of contact, convection cooling of the free surfaces, as well as the thermosensitive materials of the pads and the disk; Formulas for calculation of the flash temperature Tf on the real area of contact. Two variants were examined. In the first, the temperature dependence of the hardness of material and parameters of microgeometry of the rotor (harder material) were taken into account (variant 1), whereas variant 2 is a functional relationship of the flash temperature and the mean temperature; Formulas for calculating the wear rate of the material of the pads and the disk with temperature-dependent coefficient of wear rate.
Mutual relationship of the above-mentioned problems is based on the assumption of dependence of the coefficient of friction f (at each time step) on the maximum temperature of the friction surfaces Tmax = Tm + Tf .1
Sliding velocity in a pad–disk tribosystem Consider a braking system consisting of a solid disk of a thickness 2dd and two pads of a thickness dp (Figure 1).19 The disk is attached to the wheel, which rotates about its axis of symmetry with the velocity v0 = V0 =Rw , where V0 is the linear velocity of the vehicle and Rw is the outer radius of the wheel. When the brake is applied t = 0, the clamping force exerts pressure on the pads, and due to friction the angular velocity of the disk v decreases from the initial value v0 to zero at the stop t = ts . Friction on the contact surfaces of the sliding components transforms kinetic energy into heat and their temperature increases from the initial value T0 . On the free surfaces of the pads and the disk, convection with the constant average value of the heat transfer coefficient h takes place. Materials of the friction couple are thermosensitive, namely, thermal conductivity Kd, p and specific heat cd, p vary with temperature. From here on, the subscripts p and d will refer to the pads and the disk, respectively.
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Figure 1. FE mesh of the pad–disk tribosystem and schematic diagram of the discrete contact of rough surfaces.19
According to the HDFW conception, change in the maximum temperature Tmax on the friction surfaces can be represented as the sum of the mean temperature Tm on the nominal contact region G and the flash temperature Tf on the real area of contact103,4 Tmax (t) = Tm (t) + Tf (t),
0 ł t ł ts
ð1Þ
and the coefficient of friction f depends on Tmax (equation (1)) as follows f ½Tmax (t) = f0 f ½Tmax (t),
f0 [f (T0 ), 0 ł t ł ts
ð2Þ
The assumptions in equations (1)–(3), regarding temperature dependence of the coefficient of friction, allow investigating the mutual influence of temperature and angular velocity during braking. If the contact pressure p increases monotonically over time, from zero to the nominal value p0 according to the formula22,23 p(t) = p0 p (t), p (t) = 1 et=tm ,
f ½Tmax (t) = f1 +
I0 2
½Tmax (t) Tf1 f3 + 1 f4 + 2 ½Tmax (t) Tf2 f5 + 1
ð3Þ
where fi (i = 1, 2, 3, 4, 5) and Tfj (j = 1, 2) are the known coefficients of approximation of the experimental data for a specified friction couple.20,21
ð4Þ
the change in the angular velocity v during braking will be determined from the solution of the initial value problem for the equation of motion
where f2
0 ł t ł ts
dv(t) = 2M(t), dt
0 ł t ł ts , v(0) = v0
ð5Þ
where I0 =
2W0 v20
ð6Þ
is the moment of inertia of the friction couple with the initial kinetic energy W0
4
Advances in Mechanical Engineering M(t) = M0 M (t), M0 = f0 p0 req Aa ,
ð7Þ
M (t) = f ½Tmax (t)p (t) is the braking torque 1 rp 0:5u0 rm e2 1+ with e = , sin (0:5u0 ) 3 1 + rp rp rp + Rp rp = , rm = Rp 2 req =
ð8Þ
is the equivalent radius of the contact region G,24 and rp and Rp are the inner and outer radii of the pad (nominal area of contact), respectively. It is should be noted that for calculations of temperature in the pad–disk tribosystem, typically the following formula is used25 req =
2 (R3p rp3 ) 3 (R2p rp2 )
ð9Þ
which is a special case of formula (8), at small values of the cover angle u0 . The solution of the initial value problem (equations (4)–(8)) has the form v(t) = v0 v (t),
0 ł t ł ts
ð10Þ
where v (t) = 1
1 F (t), F (t) = ts0
ðt
M (t)dt, ts0 =
I0 v0 2M0
0
ð11Þ Taking into account the solution (equations (10) and (11)), from the condition of a stop v(ts ) = 0, the functional equation for the calculation of the total braking time ts is obtained F (ts ) = ts0
ð12Þ
It should be noted that when the contact pressure p (equation (4)) reaches the nominal value p0 instantly (tm ! 0), and the coefficient of friction is constant f = f0 ; then from formulas (7) and (11), M (t) = 1, F (t) = t, v (t) = 1 t=ts0 , and 0 ł t ł ts0 . Thus, the parameter ts0 (equation (11)) represents the time to a stop at braking with constant deceleration.
Temperature on the nominal area of contact Due to the geometric and load symmetry of the analyzed braking system, only one pad sliding on a disk of a thickness dd , with adiabatic condition on a surface in the midplane, will be taken into consideration. In a
cylindrical coordinate system (r, u, z), the pad and the disk occupy the regions Op = frp ł r ł Rp , 0:5u0 ł u and Od = frd ł r ł Rd , ł 0:5u0 , 0 ł z ł dp g 0 ł u ł 2p, dd ł z ł 0g, respectively (Figure 1). The nominal contact region is denoted G= frp ł r ł Rp = Rd , 0:5u0 ł u ł 0:5u0 , z = 0g, and its area is equal to Aa = 0:5(R2p rp2 )u0 . To determine the change in the mean temperature during braking 1 Tm (t) = Aa
0:5u ð 0 Rðp
T (r, u, 0, t)rdrdu,
0\t ł ts
ð13Þ
0:5u0 rp
the distribution of temperature T (r, u, 0, t) on the contact region G has to be known. Assuming the perfect thermal contact of friction between the pads and the disk, the distribution of the transient 3D temperature field T (r, u, z, t) in the pad–disk tribosystem will be found from the solution of the following non-linear boundary value heat conduction problem 1∂ ∂T 1 ∂ ∂T Kp (T)r + 2 Kp (T ) r ∂r ∂r r ∂u ∂u ∂ ∂T ∂T Kp (T) = rp cp (T ) + ∂z ∂z ∂t rp \r\Rp , juj\0:5u0 , 0\z\dp , 0\t ł ts 1∂ ∂T 1 ∂ ∂T Kd (T )r + 2 Kd (T ) r ∂r ∂r r ∂u ∂u ∂ ∂T ∂T ∂T Kd (T ) = rd cd (T ) + v(t) + ∂z ∂z ∂t ∂u rd \r\Rd , 0 ł u ł 2p, dd \z\0, 0\t ł ts ∂T ∂T Kp (T ) = q(r, t), Kd (T) ∂z z = 0 ∂z z = 0+
ð14Þ
ð15Þ
ð16Þ
(r, u) 2 G, 0\t ł ts T (r, u, 0+ , t) = T(r, u, 0 , t),
(r, u) 2 G, 0\t ł ts ð17Þ
∂T Kp (T) = h½T(rp , u, z, t) T0 , ∂r r = rp
ð18Þ
juj\0:5u0 , 0\z\dp , 0\t ł ts ∂T Kp (T ) = h½T0 T (Rp , u, z, t), ∂r r = Rp
ð19Þ
juj\0:5u0 , 0\z\dp , 0\t ł ts 1 ∂T Kp (T ) = h½T (r, 0:5u0 , z, t) T0 r ∂u u = 0:5u0 rp \r\Rp , 0\z\dp , 0\t ł ts
ð20Þ
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Kp (T )
1 ∂T = h½T0 T (r, 0:5u0 , z, t) r ∂u u = 0:5u0
rp \r\Rp , 0\z\dp , 0\t ł ts ∂T = h½Ta T (r, u, dp , t), Kp (T ) ∂z z = dp
q(r, t) = f ½Tmax (t)p(t)rv(t),
ð21Þ
ð22Þ
rp \r\Rp , juj\0:5u0 , 0\t ł ts ∂T Kd (T ) = h½T (rd , u, z, t) T0 , ∂r r = rd
ð23Þ
0 ł u ł 2p, dd \z\0, 0\t ł ts ∂T = h½T0 T (Rd , u, z, t), Kd (T ) ∂r r = Rd
ð24Þ
0 ł u ł 2p, dd \z\0, 0 ł t ł ts ∂T = h½T0 T (r, u, 0 , t), Kd (T ) ∂z z = 0
ð25Þ
rd \r\rp , 0 ł u ł 2p, 0\t ł ts ∂T = h½T0 T (r, u, 0 , t), Kd (T ) ∂z z = 0
ð26Þ
rp \r\Rp , juj.0:5u0 , 0\t ł ts ∂T = 0, rd \r\Rd , 0 ł u ł 2p, 0\t ł ts ð27Þ ∂z z = dd T (r, u, z, 0) = T0 ,
(r, u, z) 2 Op [ Od
ð28Þ
where (0) (0) Kd, p (T ) = Kd, p Kd, p (T ), Kd, p [Kd, p (T0 ), (0) cd, p (T ) = c(0) d, p cd, p (T ), cd, p [cd, p (T0 )
(5) (T TK2 )Kd, p
i2
ð29Þ
ð30Þ
cd, p (T ) = c(1) i2 d, p + h (T Tc1 )c(3) +1 d, p c(4) d, p (T Tc2 )c(5) d, p
Surface asperities of the sliding bodies are subject to elastic, elastic–plastic, and plastic mechanisms of deformation.26,27 The temperature of discrete zones of rough surfaces during frictional heating is termed the flash temperature.9 During the plastic deformation, one of the materials of the friction couple has noticeably smaller hardness, rigidity, and roughness. Usually, it is a material of brake pads. Plastic micro-contact during braking is the most likely, due to a significant amount of mechanical energy being converted into heat in a short period of time. The parameters characterizing the flash temperature are the real area of contact Ar and the diameter of an average spot of a real contact dr . In the case of plastic mechanism of deformation, their changes are determined using the following formulas28 p(t)Aa , 0 ł t ł ts ð33Þ HBp ½Tm (t) 1 2n1 8rav hmax 2 p(t) dr (t) = , 0 ł t ł ts ð34Þ n HBp ½Tm (t)b0 Ar (t) =
HBp ½Tm (t) = HB(0) p HBp ½Tm (t),
i2
HBp ½Tm (t) = HB(1) p + n +n
+1
c(2) d, p
+h
Flash temperature
ð35Þ
HB(0) p [HBp (T0 ), 0 ł t ł ts
(1) Kd, i2 p (T ) = Kd, p + h (3) (T TK1 )Kd, +1 p
+h
where the dependence of the coefficient of friction f on the maximum temperature Tmax (equation (1)) has the form of equations (2) and (3).
where the Brinell hardness of the pad material has the form
(2) Kd, p
(4) Kd, p
rp ł r ł Rp , 0\t ł ts ð32Þ
HB(2) p ½Tm (t) THB1 HB(3) p HB(4) p
½Tm (t) THB2 HB(5) p
o2
o2
+1
+1 ð36Þ
ð31Þ
+1
(i) (i) Kd, p , cd, p (i = 1, 2, 3, 4, 5) and TKj , Tcj (j = 1, 2) are the coefficients approximating the experimental data on the temperature dependence of the thermal conductivity and specific heat for the materials of the disk and the pads. The specific friction power appeared in the boundary condition (equation (16)) is equal to15,16
with HP(i) p (i = 1, 2, 3, 4, 5) and THPj (j = 1, 2) being known coefficients of approximation of the experimental data for the pad material, p the contact pressure (equation (4)), rav and hmax the mean roughness radius and the maximum roughness height of the rigid element, respectively, and b0 and n being the parameters of the rigid element reference surface curve (usually a disk).29 The formulas for calculating the flash temperature during braking were obtained using the monographs.2,24 With regard to the taken notation, those formulas can be written as
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Advances in Mechanical Engineering
Tf (t) =
1:707Q(t)dr (t) , D(t)Ar (t)
0 ł t ł ts
ð37Þ
where Q(t) = q(req , t)Aa
ð38Þ
n o12 D(t) = 4Kd ½Tm (t) + pdr (t)Veq (t)Kp ½Tm (t)cp ½Tm (t)rp ð39Þ Veq (t) = v(t)req
ð40Þ
and the specific friction power q at the equivalent radius req (equation (8)) is calculated from formula (32), the change in angular velocity v over time is presented in formulas (10)–(12). The evolutions of the real area of contact Ar and the diameter of an average spot of the real contact dr are obtained using formulas (33)–(36). The calculation of temperature on the real contact spot depends on the correctness of the estimation of the individual parameters included in formulas (33)–(40). It can be seen that the flash temperature is determined not only by the parameters of friction and properties of materials, but also by the characteristics of contacting surfaces. It is known that the calculation of flash temperature without taking into account the temperature dependence of the thermophysical and mechanical properties of materials leads to incorrect estimation of Tf , hence also Tmax (equation (1)).30 The values of thermal conductivity Kp, d , specific heat cp, d , and hardness HBp in formulas (33)–(40) were taken at the mean temperature Tm (equation (13)) of the area of contact and are determined from the solution of the boundary value problem (equations (14)–(32)). Use of the current values of physical properties of the materials, corresponding to the mean temperature Tf , in the calculation of the flash temperature is obviously possible. Then the data of the current values of characteristics of the microgeometry of the sliding surfaces, practically do not appear. Usually, for calculating Tf , the data obtained from the investigation of the contact surface of the component after the end of the process of frictional sliding are used. This assumes that the surface roughness, which has established after the break-in period, does not change during friction.31 However, it may be acceptable only for the stationary regime of friction. For non-stationary frictional heating, which takes place during braking, the roughness of the rubbing surfaces can be noticeably changed and consequently the characteristics of microgeometry in formulas (33)–(40). According to the hypothesis of summation of the temperature on frictional contact (equation (1)), the value of the flash temperature Tf depends on the mean surface temperature Tm , which affects the mechanical and thermophysical properties of the contacting asperities, and therefore the size of
the real area of contact. The experimental studies have shown that the flash temperature decreases with an increase in the mean temperature.32 It was found that the relationship between Tf and Tm has the form3 Tf (t) = B(t) exp½C(t)Tm (t),
0 ł t ł ts
ð41Þ
where the dependencies of the parameters B and C on the contact pressure p (equation (4)) and the sliding velocity Veq (equation (40)) were obtained in Mamchegov et al.33 It was shown that these relations can be approximated in the following form3
p(t)Veq (t) B(t) = l1
13 , C(t) =
l2 + l3 Veq (t)
ð42Þ
where li (i = 1, 2, 3) are the known coefficients. The use of formulas (41) and (42) allows reducing significantly the calculation time of flash temperature, since having the coefficients li (i = 1, 2, 3) eliminates the introduction of the dependencies of the mechanical and thermophysical properties of materials, as well as changes in parameters of microgeometry of the friction surface during braking. The calculations based on formulas (41) and (42) were verified experimentally by the character of the structural transformations in a thin subsurface layer, and their good convergence has been established in the assessment of the maximum temperature Tmax (equation (1)).34
Wear The change in braking time of the amount of wear I of the friction surfaces of the pad–disk system was calculated by the formula35 ðt I(t) =
Iw ½Tmax (t)Q(t)dt,
0 ł t ł ts
ð43Þ
0
where Q is the friction power (equation (38)), whereas the dependence of the wear rate Iw (mg N1 m1 ) on the maximum temperature Tmax (equation (1)) has the form Iw ½Tmax (t) = Iw(0) Iw ½Tmax (t),
Iw(0) [Iw (T0 ), 0 ł t ł ts ð44Þ
Iw ½Tmax (t) = Iw(1) + n
Iw(2) ½Tmax (t) TI1 Iw(3) Iw(4)
o2
+1 ð45Þ
+n o2 ½Tmax (t) TI2 Iw(5) + 1 General recommendations for determining the coefficients Iw(i) (i = 1, 2, . . . , 5) and TIj (j = 1, 2), can be found in a previous study.36
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Table 1. Operating parameters for thermal analysis. Parameter
Value
Mechanical work per one braking system Ws (kJ) Initial velocity on the equivalent radius Veq (m s21) Nominal contact pressure p0 (MPa) Heat transfer coefficient h (W m22 K21) Initial/ambient temperature T0 (°C) Rise time tm (s) Inner/outer radius of the disk rd/Rd (mm) Inner/outer radius of the pad rp/Rp (mm) Cover angle of the pad u0 (rad)
50.7 5 1.47 60 20 0.5 66/113.5 76.5/113.5 1.126
Some details of calculations All the three problems of the HDFW system of equations, namely, the initial value problem for the equation of motion (5) and its solution (equations (10)–(12)), the boundary value problem of heat conduction (equations (14)–(32)) for determining the mean temperature Tm (equation (13)), and the formulas (33)–(40) or (41) and (42) for the calculation of flash temperature Tf , are interrelated through the coefficient of friction f (equations (2) and (3)), dependent on the maximum temperature Tmax = Tm + Tf (equation (1)). The solution of the HDFW system of equations, for the pad–disk friction couple, was obtained using the implicit time-stepping algorithm—a backward differentiation formula (BDF) with an adaptive time step.37 The algorithm seeks the solutions at up to five previous time steps to find the solution at the next time step. The solution of the boundary value non-linear heat conduction problem (equations (14)–(32)) was obtained using the FEM implemented in COMSOL MultiphysicsÒ version 5.3 software.37 The 3D FE mesh of the pad–disk braking system used in thermal analysis is shown in Figure 1. The computational region was divided into 8520 second-order hexahedral elements (1320 elements for the pad and 7200 elements for the disk). The number of degrees of freedom (DoFs) solved was equal to 308343. The minimum element size was equal to 0.917 mm and the maximum element size was 6.02 mm. The dimension of elements in the z-direction decreased with the distance from the contact surface z = 0, both for the disk (four elements) and for the pad (six elements). The ratios of the largest to the smallest element were 2 and 4 for the disk and the pad, respectively. In the radial direction, also non-uniform distribution of elements was set. The strict time step Dt for the results shown was equal to 0:05 s, except for the temperature T from Figure 4, for which the time step was equal to Dt = 0:002 s. In order to calculate the flash temperature Tf (equation (37)), it was necessary to have the friction power Q
203 10
456.7 15
812 20
(equation (38)), which depends on the coefficient of friction f (Tmax ) = f (Tm + Tf ). Therefore, for calculations of the coefficient of friction on the nth time increment tn = tn1 + Dt (n = 1, 2, . . . ), the iterative procedure (n) ) = f0 f (Tm(n) + Tf(n1) ) with f = f0 at t = t0 = 0 f (Tmax has been established. Based on this relationship, the software finds the solution of the initial value problem for the equations of motion ((10)–(12)). The calculations were performed until reaching the braking time ts through checking the condition v(ts ) = 0.
Numerical analysis The procedure outlined in the previous sections is applied to the simulation of the braking process under the operating parameters shown in Table 1. Three different pad materials—the cermet FMC-11 (64% Fe, 15% Su, 3% SiO2, 6% BaSO4, 3% of asbestos, and 9% graphite), MCV-50 (64% Fe, 10% Su, 5% V4S, 5% SiC, 5% FeSO4, and 3% of asbestos), and the FC16L Retinax (produced on the basis of phenol– formaldehyde resins and reinforced by a wire with the addition of the brass shavings)—combined with the cast iron ChNMKh brake disk were examined. The (0) (0) values of the thermophysical Kd, p , cd, p (equation (29)) (0) and mechanical HBp (equation (35)) properties of the materials at the initial temperature T0 are listed in Table 2. In the same table, the approximation coeffi cients, for the dimensionless functions Kd, p (T ) (equa tion (30)), cd, p (T) (equation (31)), and HBp (T ) (equation (36)) are presented. The graphs of these functions are shown in Figure 2. One should note that the thermophysical properties of the above-mentioned materials, except for the FC-16L Retinax, are temperature dependent. The density rd, p values are 7100, 4700, 5300, and 2500 kg m23 for ChNMKh, FMC-11, MCV50, and FC-16L, respectively. The components of the HDFW system of equations, essential to perform calculations, are the temperature dependencies of the coefficients of friction f (T ) (equation (2)) and the wear rate Iw (T ) (equation (44)). The values of these parameters at the initial temperature f0 and Iw(0) , as well as the coefficients in dimensionless
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Advances in Mechanical Engineering
Table 2. Coefficients in the approximation formula for the dependencies of the thermophysical properties of materials Kd, p , cd, p and Brinell hardness HBp on temperature T. K
K(0) (W m1 8C1 )
K(1)
K(2)
K(3) (8C1 )
TK1 (8C)
K(4)
K(5) (8C1 )
TK2 (8C)
FMC-11 MCV-50 FC-16L ChNMKh
35 30.78 0.79 52.17
1.125 –0.0621 – 2:3706
–0.6393 0.5717 – 4.223
2:33103 5:433106 – 0:1963103
900 –2037 – –2543
0 0.544 – 0
0 2:763103 – 0
0 –100 – 0
c
c(0) (J kg1 8C1 )
c(1)
c(2)
c(3) (8C1 )
Tc1 (8C)
c(4)
c(5) (8C1 )
TK2 (8C)
FMC-11 MCV-50 FC-16L ChNMKh
478.9 397.3 961 444.6
0.7748 –0.786 – –0.8461
0.7385 2.759 – 6.598
3:53103 0:5833103 – 0:5713103
1059 1306 – 4903
0.4992 0.02 – 1.372
2:63103 4:723103 – 1:23103
573 44 – 443
HB
HB(0) (MPa)
HB(1)
HB(2)
HB(3) (8C1 )
THB1 (8C)
HB(4)
HB(5) (8C1 )
THB2 (8C)
FMC-11 MCV-50 FC-16L
1372.9 1225.8 392.3
–0.926 –0.5665 –0.686
0.823 1.08 1.271
2:343103 0:5943103 6:623104
546 –246 362
2.02 0.553 41.7
2:023103 3:73103 5:83102
–233 –57 –141
Table 3. Coefficients in the formula for the dependencies of the coefficient of friction f and wear rate I on the temperature T at the nominal contact pressure p0 = 1:47 MPa. 1 1 I(0) m ) w (mg N
I(1) w
I(2) w
1 I(3) w (8C )
TI1 (8C)
I(4) w
1 I(5) w (8C )
TI2 (8C)
FMC-11 MCV-50 FC-16L
0.839 0.543 1.479
0.602 0.282 0
0.437 1.877 3.412
53103 0:43102 0:83102
105 –300 850
0.672 1.877 0.93
6:23103 23102 0:43102
790 870 0
f
f0
f1
f2
f3 (°C21)
Tf1 (8C)
f4
f5 (°C21)
Tf2 (8C)
FMC-11 MCV-50 FC-16L
0.45 0.45 0.28
8:043102 0.08 0.072
1.071 0.26 1.041
1:53103 2:163103 73103
–250 –167 95
0 0.73 0.723
0 1:7563103 33103
0 –106 800
functions f (T ) (equation (3)) and Iw (T ) (equation (45)) for the three friction couples analyzed, are presented in Table 3. The functions f (T ) and Iw (T ) are plotted in Figure 3. These dependencies are usually obtained as a result of processing of the experimental data from the experimental studies on the heat stability of the friction couples.20 The evolution of the mechanical work was calculated by the formula ðt W (t) = 2 Q(t)dt,
0 ł t ł ts
ð46Þ
0
where Q(t) is the current friction power (equation (38)). In order to compare the obtained results, the mechanical work was kept constant (Ws [W (ts ) = const), for all of the three friction couples and the applied input parameters. The calculations were conducted at the four
(0) initial velocities Veq [Veq (0), and the corresponding mechanical work done per one braking system Ws is shown in Table 1. The flash temperature Tf was calculated using two different approaches. The first of them (variant 1) is based on the formulas (33)–(40) and uses the following parameters of roughness of the friction surface of the rigid material (cast iron disk): b0 = 1, n = 2:1, hmax = 2:5 mm, and rav = 450 mm.2 The corresponding values of the coefficients li (i = 1, 2, 3), which were used in the calculations according to formulas (41) and (42) (variant 2) are presented in Table 4.2,33 The results of calculations, shown in Figures 4–8 for each of the three analyzed friction couples, obtained using variant 1 are marked with solid lines, whereas the dashed lines correspond to those obtained using variant 2. When analyzing the obtained results, it should be taken into account the fact that the set of dimensionless time t=ts , in Figures 4, 5, 6, and 8, requires specific approach, for
Grzes
9 Table 4. Coefficients li (i = 1, 2, 3) in formulas (41) and (42) to calculate the flash temperature (variant 2) for the three pad materials combined with cast iron disk. Pad material
l1 (W m2 8 C3 )
l2 (m s1 8 C1 )
l3 (8 C1 )
FMC-11 MCV-50 FC-16L
0.981 1.177 4.609
2 3 10–2 2 3 10–2 4 3 10–2
1.9 3 10–3 1.9 3 10–3 2.0 3 10–3
Table 5. Peak values of temperature Tf at two variants of their calculation for the three studied pairs of materials and four initial (0) velocities Veq = 5, 10, 15, 20 m s1 . Peak value of temperature Tf (8 C)
Figure 2. Dependencies of the dimensionless thermal conductivity Kd, p, specific heat cd, p , and Brinell hardness HBp on the temperature T for cast iron ChNMKh (solid line), FMC-11 (dashed line), MCV-50 (dotted line), and FC-16L (dash-dot line).
Figure 3. Dependencies of the dimensionless coefficient of friction f and coefficient of wear Iw on the temperature T for the friction couple consisting of the cast iron disk and pad made of FMC-11 (dashed line), MCV-50 (dotted line), and FC-16L (dash-dot line).
example, the maximum values reached can occur at different moments of braking time. The flash temperature increases rapidly at the beginning of the braking process and reaches the maximum value, approximately at the same time for each of the three pad materials, and both variants of calculation of Tf (Figure 4). However, for the second variant, the flash
Pad material
FMC-11
MCV-50
FC-16L
Variant of Tf
1
2
1
2
1
(0) Veq = 5 m s1 (0) Veq = 10 m s1 (0) = 15 m s1 Veq (0) Veq = 20 m s1
154 234.2 288.8 330.4
143.9 188.9 215.1 234
141.9 215.5 264.5 301.5
135 74 70 177.4 118.7 97.8 202.7 148.2 116.2 220.2 170.8 129.5
2
temperature decreases more rapidly after reaching the (0) maximum. When increasing the initial velocity Veq , the dimensionless time t=ts of reaching the peak value of the flash temperature decreases. It shows that for longer braking durations the flash temperature becomes less important. The time of reaching the peak value of Tf for all of the studied velocities and materials is in the range t = 0:25 0:5 s. Selection of one out of the two variants considered affects the peak values of the flash temperature Tf , particularly at high initial velocities and for harder materials of the pads (Table 5). It was also found that all of the obtained results are similar for the pads made of cermet materials (FMC-11, MCV-50). This stems directly from comparable properties of materials and their dependencies on temperature (Figures 2 and 3). Evolutions of the mean temperature Tm , determined by formula (13) on the basis of the transient temperature field T (r, u, z, t), are shown in Figure 5. In addition, changes in temperature for the fixed point on the friction surface of the brake disk T(r = req = 96:2 mm, (0) = 10 m s1 are preu = 0, z = 0) at the velocity of Veq sented. The mean temperature change Tm , determined (0) =5 by this analysis at the linear initial velocities of Veq 1 and 10 m s , agrees with the corresponding results of calculations carried out in Yevtushenko and Grzes,17 obtained for the same dimensions of the braking system, but constant properties of materials, coefficient of friction dependent on the mean temperature Tm , and a nominal contact pressure of 1.47 MPa. In that study,
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Figure 4. Changes in the flash temperature V0 = 100 km h 1 on the contact surface during braking for cast iron disk and the pad made of (a) FMC-11, (b) MCV-50, and (c) FC-16L. Solid line represents the calculation of Tf using variant 1 and dashed line using variant 2.
the peak value of the mean temperature, at the ana(0) = 8:51 m s1 (V0 = 100 km h1 lyzed initial velocity Veq at the wheel radius Rw = 0:314 m), was approximately equal to Tm ’ 1408C. As shown in Figure 5, an increase in the mean temperature for each studied velocity is smooth. This is partly an effect of the applied contact pressure and a result of change in the specific friction power.38 An influence of the flash temperature Tf on the mean temperature Tm is insignificant—for any initial velocity, the mean temperature for the two variants considered almost coincides. A slight difference is more evident at the highest of the studied initial velocities (0) = 20 m s1 . It can be seen that the mean temperaVeq tures obtained using the cermet pads are almost equal (Figure 5(a) and (b)), whereas the maximum value of Tm and its change during braking vary for the pads
made of FC-16L Retinax (Figure 5(c)). It is also necessary to note that at low initial sliding velocities the maximum values of the mean temperatures Tm for the cermet pads are smaller than the corresponding peak values of the flash temperatures. Probably, this is due to the fact that the hardness of the cermet materials (FMC-11, MCV-50) is considerably (almost three times) higher than that of FC-16L Retinax. The evolutions of the maximum temperature Tmax = Tm + Tf during braking are shown in Figure 6. The maximum temperature is more ‘‘sensitive’’ to the selection of the computational variant of the flash temperature Tf than the mean temperature Tm on the pad– disk contact surfaces. For the specified variant of calculation of the flash temperature, the difference between the corresponding values of the mean and maximum temperatures is the most significant at the initial stage
Grzes
11
Figure 5. Changes in the mean temperature Tm on the contact surface during braking for cast iron disk and the pad made of (a) FMC-11, (b) MCV-50, and (c) FC-16L. Solid line represents the calculation of Tf using variant 1 and dashed line using variant 2.
of braking, when the flash temperature is the highest. With the increase in the braking time and the heating of the rubbing surface, this difference decreases. The corresponding results (Figures 4–6) obtained using the analytical methods for the simplified geometrical model of the shoe brake were obtained using the monograph.19 The drum was made of steel 30KhGSA with similar thermophysical properties to cast iron ChNMKh used in this study, and the shoe was made of FC-16L Retinax. It was found that the peak value of the flash temperature at braking from the initial velocity of 25 m s1 (braking time 3 s, contact pressure 0.5 MPa) was equal to 450°C. This value was three times higher than the mean temperature Tm of the nominal contact region of the drum ( ’ 1508C) and two times higher than the corresponding value for the shoe ( ’ 2508C). The changes in the velocity Veq and the mechanical work W (equation (46)) occurred during braking at
(0) Veq = 10 m s1 are shown in Figure 7. Except for the short time interval at the beginning, Veq decreases almost linearly. It can also be seen that an effect of flash temperature on the duration of braking is evident. The braking time for the first variant of calculations of the flash temperature is longer than that for the second by 8% when the pads are made of FC-16L Retinax and by 6% for cermet materials (FMC-11 and MCV-50). According to the formulated initial value problem for the equation of motion for each studied pair of materials and at the given initial velocity, the mechanical work is equal to W0 = 203 kJ, whereas the braking time varies. This approves the carried out comparisons of the obtained results. The differences in braking durations for the examined materials stem from the applied changes in the coefficient f ½Tmax (t) = f0 f ½Tmax (t), where f0 in this study is 0.45 (cermet) and 0.28 (Retinax A) (Table 3). Other approach, as reported in Belhocine
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Advances in Mechanical Engineering
Figure 6. Changes in the maximum temperature Tmax on the contact surface during braking for cast iron disk and the pad made of (a) FMC-11, (b) MCV-50, and (c) FC-16L. Solid line represents the calculation of Tf using variant 1 and dashed line using variant 2.
and Omar,39 is to find the coefficient of friction from complex relations and methods (e.g. Lagrange multipliers) implemented in thermomechanical analysis. Changes in the wear rate I (equations (43)–(45)) during braking are shown in Figure 8. The amount of wear increases monotonically over time, reaching a maximum value at the stop. By comparing all the solid and dashed curves in Figure 8, it may be easily observed that the selection of the variant for the calculation of flash temperature affects wear, both for the cermet and Retinax pads. The least value of wear over the braking duration was obtained for the cermet MCV-50 pads and was the highest for the pads made of FC-16L Retinax. This corresponds with the values of the initial coefficients of wear rate Iw(0) (equation (44)) shown in Table 3. In the case of the pads made of Retinax, the mass of the material worn at the stop t = ts , found using variant 1, is 5.5% lower than the corresponding mass obtained by means of variant 2.
Summary and conclusion Computer simulations of a single braking to study the flash temperature, found based on two Chichinadze’s methods, as well as mutual influence with the braking parameters, were carried out. The proposed spatial model of the disk brake is an extension of the axisymmetric 2D model using the first computational variant of the flash temperature (equations (33)–(40)).16 The analysis of the three different materials of the brake (0) in the range from 5 pads and four initial velocities Veq 21 to 20 m s , covering the majority of cases occurring under real conditions of braking of motor vehicles, allowed formulating the following conclusions: 1.
The flash temperature (variants 1 and 2) calculated on the basis of the spatial system differs slightly compared to the corresponding results obtained by means of the axisymmetric model.16 As observed, the flash temperature has significant influence on the hardness of the material.
Grzes
13
3.
4.
Figure 7. Evolutions of the sliding velocity Veq and the (0) mechanical work W for Veq = 10 m s1. Solid line represents the calculation of Tf using variant 1 and dashed line using variant 2.
5.
6.
7.
Figure 8. I of the pad and the disk working surfaces for (0) Veq = 10 m s1 . Solid line represents the calculation of Tf using variant 1 and dashed line using variant 2.
2.
In the case of brake pads made of FMC-11 and MCV with the Brinell hardness values of HB(0) p = 1372:9 and 1225.8 MPa, respectively, compared to FC-16L material with a hardness of HB(0) p = 392:3 MPa, it led to an approximately two times higher flash temperature, at each of the initial sliding velocities examined. (0) ø10 m s1 of the For the sliding velocity of Veq two analyzed cermet materials (FMC-11 and MCV-50), the flash temperature is lower than the average temperature of the nominal contact region. Below that velocity, Tf exceeds an
average temperature Tm . However, for the softer material (FC-16L), Tf is less important. Temperature changes (Figures 4–6), at the velo(0) = 5 m s1 , are nearly coincident for city of Veq all the materials. Above that velocity, the differences increase to around ’ 29% at (0) = 20 m s1 . Veq The best agreement of the flash temperatures (variants 1 and 2) was obtained at the beginning of the heating process until reaching the maximum and at the moment of stopping. Approximately in the middle range of the braking time ts , the flash temperature calculated using variant 1 is higher than the corresponding value calculated by the second variant, for each velocity and each material. The peak values of the mean Tm and the reached maximum Tmax temperature calculated by means of the two different approaches vary slightly when the initial velocity is high (ø10 m s1 ). Since the maximum flash temperature is reached at the beginning unlike the mean dominant temperature, the resulting maximum temperature remains unchanged even for greater difference in the flash temperature. Durations of braking determined using both (0) = 10 m s1 differ by variants calculated for Veq 6% for cermet and by 8% for Retinax material. The impact of the method of calculation of Tf on the amount of wear, at the velocity of (0) = 10 m s1 , is the most noticeable at the Veq end of braking for the pad made of FC-16L Retinax. This is a complex relation of several parameters and properties of materials but the most direct are the function of temperature dependence of wear Iw ½Tmax (t) and the constant value of Iw(0) = 1:479 mg N1 m1 corresponding to the temperature of 20°C. As shown in Table 3, for cermet material Iw(0) is equal to for FMC-11 and 0:839 mg N1 m1 0:543 mg N1 m1 for MCV-50.
The carried out calculations are an attempt to gain a better understanding of the influence of flash temperature on the frictional heating during the braking process. In reviewing the temperature curves shown in Figure 4, it might be concluded that the empirical formulas based on the relationship between the flash temperature on the real area of contact and the mean temperature of the nominal area of contact are fully justified in estimating the maximum temperature of the pad–disk tribosystem. Experimental methods of obtaining such formulas for the single braking, including the construction of the nomograms, were presented in the literature.2,33,34
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Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was performed within the framework of the research project No. 2015/19/D/ST8/00837 financed by the National Science Centre, Poland.
ORCID iD Piotr Grzes
https://orcid.org/0000-0002-8283-8418
References 1. Bogdanovich PN and Tkachuk DV. Thermal and thermomechanical phenomena in sliding contact. J Frict Wear 2009; 30: 153–163. 2. Chichinadze AV, Braun ED, Ginsburg AG, et al. Designing, testing and selecting frictional couples. Moscow: Nauka, 1979 (in Russian). 3. Chichinadze AV. Processes in heat dynamics and modelling of friction and wear (dry and boundary friction). Tribol Int 1995; 28: 55–58. 4. Chichinadze AV. Theoretical and practical problems of thermal dynamics and simulation of the friction and wear of tribocouples. J Frict Wear 2009; 30: 199–215. 5. Ginzburg AG. Theoretical and experimental bases for calculation of the single braking process with help of the system of equations of heat dynamics of friction, optimal use of friction materials in friction units of machines. Moscow: Nauka, 1973, pp 93–105 (in Russian). 6. Awrejcewicz J and Pyryev Yu. Nonsmooth dynamics of contacting thermoelastic bodies. New York: Springer, 2009. 7. Yevtushenko AA and Kuciej M. One-dimensional thermal problem of friction during braking: the history of development and actual state. Int J Heat Mass Trans 2012; 55: 4118–4153. 8. Jaeger JC. Moving sources of heat and the temperature of sliding contacts. J Proc R Soc NSW 1942; 76: 203–224. 9. Blok H. The flash temperature concept. Wear 1963; 6: 483–494. 10. Kennedy FE. Frictional heating and contact temperatures. In: Bhushan B (ed.) Modern tribology handbook. Boca Raton, FL: CRS Press, 2001, pp.235–272. 11. Chichinadze AV, Kozhemyakina VD and Suvorov AV. Method of temperature-field calculation in model ring specimens during bilateral friction in multidisk aircraft brakes with the IM-58-T2 new multipurpose friction machine. J Frict Wear 2010; 31: 23–32. 12. Belhocine A and Omar WZW. CFD analysis of the brake disc and the wheel house through air flow: predictions of surface heat transfer coefficients (STHC) during braking operation. J Mech Sci Technol 2018; 32: 481–490.
13. Belhocine A. FE prediction of thermal performance and stresses in an automotive disc brake system. Int J Adv Manuf Technol 2017; 89: 3563–3578. 14. Yevtushenko AA, Grzes P and Adamowicz A. Numerical analysis of thermal stresses in disk brakes and clutches (a review). Numer Heat Transfer Part A 2015; 67: 170–188. 15. Yevtushenko AA and Grzes P. Maximum temperature in a three-disc thermally nonlinear braking system. Int Commun Heat Mass Transfer 2015; 68: 291–298. 16. Grzes P. Determination of the maximum temperature at single braking from the FE solution of heat dynamics of friction and wear system of equations. Numer Heat Transfer Part A 2017; 71: 737–753. 17. Yevtushenko AA and Grzes P. 3D FE model of frictional heating and wear with a mutual influence of the sliding velocity and temperature in a disc brake. Int Commun Heat Mass Transfer 2015; 62: 37–44. 18. Yevtushenko AA and Grzes P. Mutual influence of the sliding velocity and temperature in frictional heating of the thermally nonlinear disc brake. Int J Therm Sci 2016; 102: 254–262. 19. Chichinadze AV (ed.). Polymers in friction assemblies of machines and devices: a handbook. New York: Allerton Press, Inc., 1984, p.280. 20. Chichinadze AV, Matveevski RM and Braun EP. Materials in tribotechnics non-stationary processes. Moscow: Nauka, 1986 (in Russian). 21. Yevtushenko AA and Grzes P. Axisymmetric finite element model for the calculation of temperature at braking for thermosensitive materials of a pad and a disc. Numer Heat Transfer Part A 2012; 62: 211–230. 22. Yevtushenko AA, Kuciej M, Och E, et al. Effect of the thermal sensitivity in modeling of the frictional heating during braking. Adv Mech Eng 2016; 8: 1–10. 23. Gao CH and Lin XZ. Transient temperature field analysis of a brake in a non-axisymmetric three-dimensional model. J Mater Proc Tech 2002; 129: 513–517. 24. Chichinadze AV. Estimation and investigation of external friction during braking. Moscow: Nauka, 1967 (in Russian). 25. Wauer J and Schweizer B. Dynamics of rotating thermoelastic disks with stationary heat source. Appl Math Comput 2010; 215: 4272–4279. 26. Kragelsky IV and Demkin NB. Contact area of rough surfaces. Wear 1960; 3: 170–187. 27. Bhushan B. Introduction to tribology. 2nd ed. New York: Wiley, 2013. 28. Demkin NB. Contact of rough surfaces. Moscow: Nauka, 1970 (in Russian). 29. Demkin NB, Korotkov MA and Alekseev VM. Method of calculating the characteristics of frictional contact, calculation and simulation of working mode of brake and friction devices. Moscow: Nauka, 1974, pp.5–15 (in Russian). 30. Powell DI and Earles SWE. An assessment of surface temperature predictions in the high speed sliding of unlubricated SAE 1113 steel surfaces. Trans ASLE 1972; 15: 103–112. 31. Kragelsky IV, Dobychin MN and Kombalov VS. Friction and wear: calculation methods. Oxford: Pergamon, 1982. 32. Earles SWE, Hayler MG and Powell DG. A comparison of surface temperature theories and experimental results
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for high speed dry sliding. Trans ASLE 1971; 14: 135–143. Mamchegov MA, Zinovyeva ZV and Ginzburg AG. Questions of calculation of the maximum temperature at nonstationary friction with intensive heat generation, friction and wear. Moscow: Nauka, 1977, pp.46–53 (in Russian). Mamchegov MA. Determination of the maximum temperature of the sliding contact. Mashinostroeniye 1977; 1: 107–112 (in Russian). Ginzburg AG and Chichinadze AV. To calculation of wear during braking using the equations of thermal dynamics of friction, friction and wear of friction materials. Moscow: Nauka, 1977, pp.26–30 (in Russian). Yevtushenko AA and Grzes P. Axisymmetric FEA of temperature in a pad/disc brake system at temperaturedependent coefficients of friction and wear. Int Commun Heat Mass Transfer 2012; 39: 1045–1053. COMSOL MultiphysicsÒv. 5.3. www.comsol.com. COM SOL AB, Stockholm, Sweden. Yevtushenko AA, Kuciej M and Topczewska K. Analytical model for investigation of the effect of friction power on temperature in the disk brake. Adv Mech Eng. Epub ahead of print 11 December 2017. DOI: 10.1177/ 1687814017744095 Belhocine A and Omar WZW. Three-dimensional finite element modeling and analysis of the mechanical behavior of dry contact slipping between the disc and the brake pads. Int J Adv Manuf Technol 2017; 88: 1035–1051.
Iw Iw(0) K K (0) M M0 p p0 q Q r rav req t tm ts ts0 T T0 Tf Tm
Appendix 1
Tmax
Notation
Veq
Aa Ac Ar b0 c c(0) dr f f0 h hmax HB HB(0) I I0
nominal area of contact (m2) contour area of contact (m2) real area of contact (m2) parameter of the rigid element reference surface curve temperature-dependent specific heat capacity (J kg21 K21) specific heat capacity at the initial temperature (J kg21 K21) diameter of an average spot of real contact (m) coefficient of friction coefficient of friction at initial temperature heat transfer coefficient (W m22 K21) maximum height of the asperities (m) temperature-dependent Brinell hardness (MPa) Brinell hardness at the initial temperature (MPa) wear rate (kg) moment of inertia of the friction couple (kg m2)
(0) Veq
Ws z G d u u0 n r v v0 O
temperature-dependent coefficient of wear rate (mg N21 m21) temperature-dependent coefficient of wear at the initial temperature (mg N21 m21) temperature-dependent thermal conductivity (W m21 K21) thermal conductivity at the initial temperature (W m21 K21) braking torque from Aa (N m) initial braking torque from Aa (N m) contact pressure (MPa) nominal contact pressure (MPa) specific friction power (W m22) friction power (W) radial coordinate mean roughness radius (m) equivalent radius of the contact region (m) time (s) rise time (s) braking time (s) braking time at constant deceleration (s) temperature (°C) initial/ambient temperature (°C) flash temperature in the real area of contact (°C) mean temperature on the friction surface (°C) maximum temperature on the friction surface (°C) linear velocity on the equivalent radius (m s21) initial linear velocity on the equivalent radius (m s21) mechanical work per one braking system (J) axial coordinate contact region thickness of the analyzed region (m) circumferential coordinate cover angle of the pad (rad) parameter of the rigid element reference surface curve density (kg m23) angular velocity of the disk (rad s21) initial angular velocity of the disk (rad s21) computational region
Subscripts d p
disk pad
