Three-element model of frictional heating during braking with contact thermal resistance and time-dependent pressure

International Journal of Thermal Sciences 50 (2011) 1116e1124

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Three-element model of frictional heating during braking with contact thermal resistance and time-dependent pressure A.A. Yevtushenko*, M. Kuciej, O. Yevtushenko* Department of Applied Mechanics and Informatics, Faculty of Mechanical Engineering, Bialystok University of Technology, 45C Wiejska Street, Bialystok, 15-351 Poland

a r t i c l e i n f o

a b s t r a c t

Article history: Received 7 January 2010 Received in revised form 14 September 2010 Accepted 16 November 2010 Available online 18 December 2010

The solution to a thermal problem of friction during braking for a three-element tribosystem disc/pad/ caliper with time-dependent specific power of friction and heat transfer through a contact surface has been obtained. The influence of duration of increase in pressure (from zero at the initial moment of time to nominal value at the moment of a stop) and the Biot number on the temperature for such materials as cast iron disc/metaleceramic pad/steel caliper has been studied. Ó 2010 Elsevier Masson SAS. All rights reserved.

Keywords: Frictional heating Temperature Braking Heat transfer Thermal resistance Pressure

1. Introduction Basic elements of disc brakes are a cast-iron disc which rotates with the wheel, friction material (brake pads) and a caliper fixed to the steering knuckle (Fig. 1). When the braking process occurs, the hydraulic pressure forces the piston, and, therefore, pads and disc brake are in sliding contact. Set up force resists the movement and the vehicle slows down or eventually stops. Friction between disc and pads always opposes motion and the heat is generated due to conversion of the kinetic energy. However, friction surface is exposed to the enlarged air flow during drag braking and the heat is dissipated. The knowledge of a temperature regime of these elements is important to estimate efficiency and reliability of work of frictional materials of brakes in the given conditions of operation. The temperature appreciably determines the friction and wear characteristics of materials of brakes [1e3], the structural transformations within them [4], the intensity of the processes of physical and chemical mechanics on frictional contact [5]. Experimental determination of temperature of a surface of contact concerning authentic objects in most cases causes significant technical difficulties and is connected with essential material and time expenses [6,7]. Therefore, theoretical (analytical or numerical) definition of a temperature regime of elements of friction couple * Corresponding authors. Tel.: þ48 085 746 92 00; fax: þ48 085 746 92 10. E-mail address: [email protected] (A.A. Yevtushenko). 1290-0729/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2010.11.009

obtained by the solution of a corresponding thermal problem of friction during braking attracts the great interest. The problem of heat conductivity at friction is formulated as follows: to find distribution of temperatures in elements of friction pair when a contact surface is heated by heat flux and on the free surfaces the heat transfer to the environment takes place. Mathematically, this problem can be formulated as a boundaryvalue problem for one or several heat conductivity equations of parabolic type. Usually, on the contact surface two boundary conditions are set. The first equates the sum of intensities of heat fluxes directed inside the body to specific power of friction [8]. In the second e boundary condition affirms, that the difference in the intensity of heat flux on the surface of contact is proportional to the difference of contact temperatures. The coefficient of proportionality is the contact heat transfer coefficient (the value, inversely proportional to the thermal resistance of contact). The thermal contact of bodies, taking into account the jump of their temperatures on the surface of contact, is called incomplete or imperfect [9,10]. In the special case when the heat transfer coefficient tends to infinity (or the contact thermal resistance tends to zero), the second of these conditions is reduced to the condition of equality of temperatures on the contact surface. Together with the first boundary condition this condition of equality of contact temperatures constitute a system of boundary conditions of perfect thermal contact [11]. In the case of inhomogeneous bodies, at the presence of the thermal resistance on the contact, depending on the pressure

A.A. Yevtushenko et al. / International Journal of Thermal Sciences 50 (2011) 1116e1124

Nomenclature Bi ¼ hd/Ks Biot number d thickness of the strip, m f coefficient of friction h coefficient of thermal conductivity of contact, W/(m2K) K coefficient of heat conduction, W/(mK) Kt* ¼ Kt =Ks Kf* ¼ Kf =Ks k coefficient of thermal diffusivity, m2/s k*t ¼ kt =ks k*f ¼ kf =ks p pressure, Pa maximal value of pressure, Pa p0 q0 ¼ fV0p0 intensity of the frictional heat flux (the friction power), W/m2 T temperature,  C T0 ¼ q0d/Ks temperature scaling factor T* ¼ T/T0 dimensionless temperature t time, s

(the oxidation film, the products of wear), at the complex heat transfer conditions on the boundaries with environmental, the solution to the thermal problem of friction during braking is possible only by numerical methods [12,13]. Additionally, in many cases the boundary conditions are difficult for defining in engineering practice [14]. Therefore, at statement of a thermal problem of friction some simplifications are possible. The process of braking is a nonstationary process of friction, in most cases short-term. Experimental study of temperature fields of friction brakes of various designs with different materials have shown that in the absence of forced cooling systems and the Fourier numbers less than ten, the heat transfer to the environment during braking in calculating the temperature can be neglected. According to the experimental data of articles [15e17], in such conditions the heat transfer during braking is less than 5% of the total amount of heat generated by the friction. Comparison of temperature values obtained from analytical solutions of thermal problems of friction, with the known results of

tmax tm ts0 ts V V0 z

1117

time, when maximal temperature is reached, s duration of the increase of the pressure from zero to maximum value, s duration of braking in the case of constant pressure, s time of braking, s sliding speed, m/s initial sliding speed, m/s spatial coordinate, m

Greek symbols s ¼ kst/d2 dimensionless time smax ¼ kstmax/d2 sm ¼ kstm/d2 s0s ¼ ks ts0 =d2 ss ¼ ksts/d2 z ¼ z/d dimensionless coordinate Indexes f s t

bottom semi-space strip upper semi-space

experimental investigations of temperature regimes of the brakes showed that for the calculation of the temperature can be used the solution of the one-dimensional transient heat conduction problem [18e24]. In this problem distribution of heat only on a normal to a surface of friction is considered. Such approach has allowed to create the general methodology for calculating the temperature fields for the various types of the brakes [25]. It has been well proved from the point of view of simplicity of final formulae and sufficient accuracy of calculation. Reviews of analytical methods of the solution to thermal problems of friction during braking are given in articles [26,27], and of FEM-solutions e in article [28]. Previously, the authors obtained the exact solution to thermal problems of friction for three-element tribosystem (disc/pad/caliper) at the constant pressure and under perfect [29] or imperfect [30] conditions of thermal contact. In present paper, the corresponding solution for the same tribosystem with time-dependent pressure and at imperfect thermal contact is investigated. 2. Problem formulation The problem of contact interaction of two semi-spaces is considered, where one of them is homogeneous (the disc) and the other is a semi-infinite foundation (the caliper) with a surface covered by a strip (the pad) of thickness d. The time-dependent normal pressure p (t) ¼ p0p*(t), 0  t  ts in the direction of the z-axis of the Cartesian system of coordinates Oxyz is applied to the infinity in semi-spaces (Fig. 2). In addition, the top semi-space slides with the speed V (t) ¼ V0V*(t), V0 hVð0Þ, 0  t  ts in the direction of the y-axis on the surface of the strip. Due to friction, the heat is generated on a surface of contact z ¼ 0, and the elements are heated. It is assumed, that: 1) the sum of heat fluxes, directed from a surface of contact inside each bodies, is equal to specific friction power

qðtÞ ¼ q0 q* ðtÞ; q0 ¼ fp0 V0 ; q* ðtÞ ¼ p* ðtÞV * ðtÞ; 0  t  ts ;

Fig. 1. Front disc brake of the passenger’s car.

(1)

2) through the contact surface of the top semi space and the strip the heat transfer takes place with a constant coefficient of thermal conductivity of contact h;

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A.A. Yevtushenko et al. / International Journal of Thermal Sciences 50 (2011) 1116e1124

Ts* ðz; 0Þ ¼ 0;

1  z  0;

(12)

Tf* ðz; 0Þ ¼ 0;

N < z  1;

(13)

where

Kf kf z ks t ks ts Kt ; ss ¼ 2 ; Kf* ¼ ; Kt* ¼ ; k*f ¼ ; Ks Ks ks d d d2 k hd t ; ð14Þ k*t ¼ ; Bi ¼ Ks ks

z ¼ ; s ¼

Tt* ¼

3) the perfect heat contact between the strip and the foundation takes place; 4) the wear of the contact surface is negligible. Let us find the distribution of temperature fields in the elements of tribosystem. Further, all values and the parameters concerning top semi-space, strip and foundation will have bottom indexes “t”, “s” and “f”, respectively (Fig. 2). The transient temperature fields Tt,s,f(z,t) can be found from the solution to the following transient boundary-value heat conduction problem of friction:

vz

2

v2 Ts* ðz; sÞ vz

2

v2 Tf* ðz; sÞ vz

2

¼

1 vTt* ðz; sÞ ; vs k*t

¼

vTs* ðz; sÞ ; vs

¼

* 1 vTf ðz; sÞ ; vs k*f

0 < z < N; 0 < s < ss ;

1 < z < 0; 0 < s < ss ;

N < z < 1; 0 < s < ss ;



*
vTs*

* vTt
 K ¼ q* ðsÞ; t vz
z¼0 vz
z¼0þ

(2)

(3)

(4)

(5)



*
  vTs*

* vTt
þ K ¼ Bi Tt* ð0; sÞ  Ts* ð0; sÞ ; 0  s  ss ; t vz
z¼0 vz
z¼0þ

(6)

Ts* ð1; sÞ ¼ Tf* ð1þ; sÞ;

(7)



vTf*
vTs*


¼ Kf* ;
vz z¼1þ vz
z¼1 Tt* ðz; sÞ/0; z/N;

Tt* ðz; 0Þ ¼ 0;

0  s  ss ;

0  s  ss ;

0  s  ss ;

Tf* ðz; sÞ/0; z/  N;

0  s  ss ;

0  z < N;

s

p* ðsÞ ¼ 1  esm ;

(8)

(9)

0  s  ss ;

(16)

where sm is the dimensionless time of the increase of the pressure from zero to the maximal value p0. At the known temporal profile of the pressure (9) the evolution of the sliding speed takes the form [22]

V * ðsÞ ¼ 1 

s sm *  p ðsÞ; s0s s0s

0  s  ss

(17)

where s0s is the dimensionless duration of braking in the case of constant pressure p(t) ¼ p0. In the limiting case sm / 0 from equations (16) and (17), it follows that

p* ðsÞ ¼ 1; V * ðsÞ ¼ 1 

s ; s0s

0  s  s0s :

(18)

Thus, in the case of constant pressure p(t) ¼ p0 the sliding speed decreases from initial value V0 to zero at the stop time moment linearly with time (braking with constant deceleration). Using the stop condition V(ts) ¼ 0 (V*(ss) ¼ 0), from expression (10) we find the non-linear equation for the dimensionless braking time ss:

1 0  s  ss ;

(15)

During braking the function p* ðtÞhp* ðsÞ in the equation (1) takes the form [14]

Fig. 2. Scheme of a three-element brake system.

v2 Tt* ðz; sÞ

Tf Tt Ts q d ; Ts* ¼ ; Tf* ¼ ; T0 ¼ 0 : Ks T0 T0 T0

ss sm *  p ðss Þ ¼ 0: s0s s0s

(19)

Numerical analysis has shown that equation (19) for given values of the dimensionless input parameters sm and s0s always has one root. The value of this root gives us a dimensionless time of braking ss. Additionally, it is established that for values s0s  0:6, dependences of dimensionless time of braking ss on the dimensionless time sm are linear. For example, at s0s ¼ 1 we obtain ss ¼ 0.99sm þ 1.0. Approximation with the same error for s0s ¼ 0:5 has the form ss ¼ 0:437s2m þ 1:0609sm þ 0:4986. The evolutions of the dimensionless contact pressure p*(s) (16) and sliding speed (17) for times of braking ss found from equation (19) at s0s ¼ 1 are shown in Fig. 3. Taking the formulae (1), (16) and (17) into account, we present the dimensionless intensity of the heat flux q*(s)in the boundary condition (5) as

  s sm 1  0 þ 0 p* ðsÞ p* ðsÞ;

(10)

q* ðsÞ ¼

(11)

where the dimensionless time of braking ss is found from the equation (19).

ss

ss

0  s  ss ;

(20)

A.A. Yevtushenko et al. / International Journal of Thermal Sciences 50 (2011) 1116e1124

1119

2

3

6ð1 þ zÞx7 Gf ðz; xÞ ¼ ½BiDR ðxÞ þ 3t xDI ðxÞcos4 qffiffiffiffiffi 5 þ ½BiDI ðxÞ k*f 3 2 6ð1 þ zÞx7  3t xDR ðxÞsin4 qffiffiffiffiffi 5; k*f

N < z  1;

aðxÞ ¼ Bi cosx  xsinx; bðxÞ ¼ 3f ðBi sinx þ xcosxÞ; Pðs; xÞ ¼

 2 1

1  ex s ; 2 x

f t 3t ¼ qffiffiffiffiffi ; 3f ¼ qffiffiffiffiffi;

(31)

k*f

k*t

(29)

(30)

K*

K*

ð28Þ

By Fourier’s law the dimensionless intensity of the heat fluxes in the top semi-space, the strip and the foundation are connected with corresponding dimensionless temperatures:

Fig. 3. Evolution of the dimensionless pressure p*(s) and sliding speed V*(s) during braking for different values of the dimensionless time of pressure increasing sm.

q*t ðz; sÞ ¼ Kt* q*s ðz; sÞ ¼

3. Solution to the problem at uniform frictional heating First, we construct a solution to the boundary-value problem of heat conduction (2)e(13) at constant power of friction force, when in the right side of the boundary condition (5) we have q*(s) ¼ 1. In this case, by applying the integral Laplace transform in dimensionless time s [31], * T t;s;f ðz; pÞ

ZN ¼

* Tt;s;f ðz; sÞ eps ds

(21)

vTs* ðz; sÞ ; vz

q*f ðz; sÞ ¼ Kf*

2

* Tt;s;f ðz; sÞ ¼

p

ZN 0

Gt;s;f ðz; sÞ Pðs; xÞdx; DðxÞ

s0

(22)

where

DðxÞ ¼ D2R ðxÞ þ D2I ðxÞ; 

(23)



DR ðxÞ ¼ Bi 3t þ 3f cosx  23t xsinx; 

(24)



DI ðxÞ ¼ Bi 1 þ 3t 3f sinx þ 23t 3f cosx;

(25)

vTf* ðz; sÞ vz

0  z < N; s  0;

1  z  0; s  0;

;

1

B zx C Gt ðz; xÞ ¼ ½aðxÞ DR ðxÞ þ bðxÞDI ðxÞcos@qffiffiffiffiffiA þ ½bðxÞDR ðxÞ k*t 0 1 B zx C  aðxÞDI ðxÞsin@qffiffiffiffiffiA; k*t

0  z < N;

ð26Þ

(33)

N < z  1; s  0:

q*t;s;f ðz; sÞ ¼ q0t;s;f 

2

p

ZN 0

Qt;s;f ðz; xÞ x2 s dx; e xDðxÞ

(34)

s  0:

(35)

where

q0t ¼

3t ; 3t þ 3f

q0s ¼ q0f ¼

3f ; 3t þ 3f

(36)

8 0 1 > < B zx C Qt ðz; xÞ ¼ 3t ½aðxÞDI ðxÞ  bððxÞDR ðxÞcos@qffiffiffiffiffiA þ ½bðxÞDI ðxÞ > : 0 19 k*t > = B zx C þaðxÞDR ðxÞsin@qffiffiffiffiffiA ; 0  z < N; (37) > k*t ;   Qs ðz; xÞ ¼ 3f BiDI ðxÞ  3t 3f xDR ðxÞ cos½ð1 þ zÞx  ½BiDR ðxÞ þ3t xDI ðxÞsin½ð1 þ zÞx;

0

(32)

Taking the relations (32)e(34) into account, after differentiating the temperatures (22)e(31) with respect to z, we obtain the intensities of the heat fluxes in the form:

0

we obtain the dimensionless temperatures in the following form:

vTt* ðz; sÞ ; vz

1  z  0;

(38)

8 2 3 > < 6ð1 þ zÞx7 Qf ðz; xÞ ¼ 3f ½BiDI ðxÞ  3t xDR ðxÞcos4 qffiffiffiffiffi 5  ½BiDR ðxÞ > : k*f 2 39 > = 6ð1 þ zÞx7 þ3t xDI ðxÞsin4 qffiffiffiffiffi 5 ; N < z  1: (39) > ; k* f

Gs ðz; xÞ ¼ ½BiDR ðxÞ þ 3t xDI ðxÞcos½ð1 þ zÞx þ 3f ½BiDI ðxÞ  3t xDR ðxÞsin½ð1 þ zÞx;

1  z  0;

ð27Þ

We note that the values of q0t;s;f (36) are the value of the intensities of the heat fluxes in steady state. In contrast to the heat flux, the temperature (22) does not reach the stationary state.

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On the friction surface z ¼ 0 from the formulae (37) and (38) results

 h

i Qs ð0;xÞþQt ð0þ;xÞ ¼ Bi 3t þ 3f cosx23t xsinx DI ðxÞ  i h

 Bi 1þ 3t 3f sinxþ23t 3f xcosx DR ðxÞ ¼ DR ðxÞDI ðxÞ DI ðxÞDR ðxÞ ¼ 0; (40) and from equations (35) and (36) we find q*s ð0þ; sÞþq*t ð0; sÞ ¼ 1; s  0, which means that boundary conditions (5) are satisfied. Difference in temperature and intensity of heat flux on the contact surface is found from formulae (22) and (35) in the form

Tt* ð0þ; sÞ  ZN ¼

2

p

0

Ts* ð0; sÞ

b * ðz; sÞ (22)e(31) to the right Substituting the temperature T t;s;f part of equation (47) and changing the order of the integration, we obtain * Tt;s;f ðz; sÞ ¼

s  0;

q*s ð0; sÞ  q*t ð0þ; sÞ N 3f  3t 2 Z ½Qs ð0; xÞ  Qt ð0þ; xÞ x2 s  dx; ¼ e 3f þ 3t p xDðxÞ

ð41Þ

Pm ðs; xÞ ¼

s  0; ð42Þ

(43)

i dx   3f  3t DI ðxÞcosx  1  3t 3f DR ðxÞsinx xDðxÞ 

Gs ð1; xÞ ¼ BiDR ðxÞ þ 3t xDI ðxÞ ¼ Gf ð1; xÞ;

(45)

Qs ð1; xÞ ¼ 3f ½BiDI ðxÞ  3t xDR ðxÞ ¼ Qf ð1; xÞ;

(46)

i.e., the boundary conditions (7) and (8) are satisfied. 4. The solution to the thermal problem of friction during braking The solution to a boundary-value problem of heat conductivity (2)e(8) in a case when the function q*(s) on the right side of a boundary condition (5) has the form (20), we find using the Duhamel’s theorem [33]

Zs ¼ 0

v b* q* ðsÞ T ðz; s  sÞ ds; vs t;s;f

0  s  ss ;

sm I ðs; xÞ; s0s 2

0  x < N; 0  s  ss ;

(50)

Zs  1

s

s0s



2 p* ðsÞex ðssÞ ds;

ð51Þ

0

Inserting the dimensionless pressure p*(s) (16) into the righthand side of the equations (51), after integrating we find

(44)

Calculations of the integral on the left side of the equation (44) by procedure QAGI from a package QUADPACK [32] has shown that this equality is carried out identically. On the interface z ¼ 1 from equations (27), (28) and (38), (39) it follows

* Tt;s;f ðz; sÞ

I1 ðs; xÞ ¼



3f  3t  : ¼ 2 Bi 3f þ 3t

(49)

where

I1 ðs; xÞ ¼



(48)

functions q*(s), P(s,x) have the form (20) and (30), accordingly. Taking the form of the dimensionless intensity of a heat flux q*(s) (20) and the function P(s,x) (30) into account, the function Pm(s,x) (49) can be written as

Zs h i2 2 p* ðsÞ ex ðssÞ ds: I2 ðs; xÞ ¼

Substituting the equations (41)e(43) into boundary condition (6), we obtain the relationship

p

0  s  ss ;

v q* ðsÞ Pðs  s; xÞds: vs

0

Gt ð0þ; xÞ  Gs ð0; xÞ ¼

0

Zs

Pm ðs; xÞ ¼ I1 ðs; xÞ þ

where, taking the formulae (26), (27) and (37), (38) into account, we have

x ½Qs ð0; xÞ  Qt ð0þ; xÞ Bi   ¼ x 3f  3t DI ðxÞcosx     1  3t 3f DR ðxÞsinx :

0

Gt;s;f ðz; xÞ Pm ðs; xÞdx; DðxÞ

where

0

ZNh 

p

0

½Gt ð0þ; xÞ  Gs ð0; xÞ 2  1  ex s dx; x2 DðxÞ

ZN

2

I2 ðs; xÞ ¼

  1

1 1

x2 s x2 s s  1  e  1  e s0s x2 x2 x2 ( i h s s s 1 1  sm x2 s   þ 0  2 1 esm e  2 1 e ss x sm x sm ) h s i 1 sm x2 s ;  e 2 e x2 s1 m  i h s 2 2 1

2 1  ex s   2 1  esm  ex s 2 x x sm i h s 2 1  e2sm  ex s : þ 2 x 2s1 m

(52)

(53)

Substituting the functions In(s,x), n ¼ 1,2 (52) and (53) into equation (50) we obtain



  sm 1 1 2sm x2 s 1  e  1þ 0 Pm ðs; xÞ ¼ 1 þ 0 þ 0 2 s s s s x x2 ss



s 2 1 1 sm    e þ 0 2  ex s ss x  s1 x2  s1 m m 

s s sm s 2  e2sm  ex s þ  esm þ 0 2 1 s0s x2  s1 ss x  2sm m 

s ; s0s x2

0  x < N; 0  s  ss :

ð54Þ

In the limiting case of braking with constant deceleration at

(47)

b * ðz; sÞ have the form where the dimensionless temperatures T t;s;f (22) with integrands (23)e(31), and the dimensionless braking time ss is the root of the non-linear equation (19).

sm / 0 from formula (54) we obtain P0 ðs;xÞ ¼ Pðs;xÞ 

1 ½s  Pðs;xÞ; 0  x < N; 0  s  ss ; x2 s0s

where function P(s,x) has the form (30).

(55)

A.A. Yevtushenko et al. / International Journal of Thermal Sciences 50 (2011) 1116e1124

1121

5. Numerical analysis Calculations are made for the top semi-space (the disc) made from cast iron ChNMKh (Kt ¼ 51 W m K, kt ¼ 14  106 m s), the metal-ceramic strip (the pad) FMK-11 (Ks ¼ 34.3 W m1 K1, ks ¼ 15.2  106 m2 s1) and the foundation (the caliper) from steel 30KhGSA (Kf ¼ 37.2 W m1 K1, kf ¼ 10.3  106m2 s1). Such a friction pair is used in brakes of planes [34]. The calculations are carried out according to the following scheme: 1) set the initial values of the input dimensionless parameters of the problem: z, s, s0s , sm, Kt;* f , k*t; f , Bi; 2) the dimensionless time of braking ss may be obtained from solution of the nonlinear functional equation (19), 3) the dimensionless temperature in each element of tribosystem is defined from solution (48). Integral in this formula is calculated using the procedure QAGI from a package of numerical integration QUADPACK [32]. The main purpose of this article is to study the influence of the two dimensionless parameters on temperature distribution: the dimensionless time sm of the increase of the dimensionless pressure p*(s) (16) from zero up to one, and the Biot number Bi, which characterizes the heat transfer through the surface of contact between the strip and the top semi-space. We note that in a limiting case Bi / N we should obtain such results as presented in article [30], where the influence of such input parameters as the * , k* (14) and the thickness of the strip d, the physical constants Kt;f t;f time ts0 ðs0s Þ of braking with constant deceleration on the temperature in tribosystem under consideration has been studied. The evolutions of the dimensionless temperature T* (48) on a contact surface z ¼ 0 during braking for three values of Biot number are shown in Fig. 4. Evolution of the contact temperature during braking consists of two phases. At the first stage, at the beginning of braking the temperature on a surface of contact * in the moment of sharply rises, reaches the maximal value Tmax time smax. After that, the second phase follows, when the temperature decreases monotonically to its minimum value at the time of the stop ss. For large values of Biot number (Bi / N) the thermal contact of the pad and the disc becomes perfect, i.e. temperature of these elements on the contact surface are the same (Fig. 4a). Reduction of Biot number leads to jump in temperatures on the surface of contact (Fig. 4b). The greatest jump of temperatures of the top semi-space and the strip on the contact surface are noticed for the small values of Biot number (Bi / 0) (Fig. 4c). At the fixed value of Biot number the largest value of the contact temperature is reached during braking with the constant deceleration (sm ¼ 0), when pressure almost instantly reaches the maximal value. This conclusion, based on the data presented in Fig. 4, also confirms the results shown in the Fig. 5. The increase in duration of achieving the maximal value by pressure (an increase of the dimensionless parameter sm), leads to a decrease in the maximum temperature on the contact surfaces of the pad and the disc. The dimensionless time smax, when the temperature reaches * increases almost linearly with an increase in a maximum value Tmax parameter sm (Fig. 6). In the presence of thermal resistance, at the surface contact of the pad and the disc, for a fixed value of the parameter sm, the maximum temperature is first reached on the pad’s surface and then on the disk’s (smax of the pad is less than the disc). This “effect of delay” is visible also in Fig. 4b, c. Dependences of the maximal contact temperatures of the top semi-space and strip on the Biot number are shown in Fig. 7. We see that the contact temperature is greater in that element of friction pair, which has a smaller coefficient of thermal conductivity. Such

Fig. 4. Evolution of the dimensionless contact temperature T*(0,s) at several values of the dimensionless time of pressure increasing sm and for three values of the Biot number: a) Bi / N; b) Bi ¼ 5; c) Bi / 0.

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Fig. 6. Dependence of the dimensionless time smax, when maximal dimensionless * is reached, on the dimensionless parameter sm for three values of the temperature Tmax Biot number Bi.

element is the metal-ceramic FMK-11 strip. With the increase in the Biot number, the jump of temperatures decreases. Even at Bi ¼ 100 the jump of temperatures is still significant. The above numerical analysis of the problem has been obtained for the dimensionless temperature T* (15). It allows us to define the basic principles of its evolution and dependences on dimensionless time of increase in pressure sm, and the Biot number Bi. To compare the received results with experimental values of the maximal temperature, we have calculated the evolution of temperature according to data of the monograph [14]: f ¼ 0.7, d ¼ 5 mm, p0 ¼ 1 MPa, V0 ¼ 30 m s, tm ¼ 0 s, ts ¼ ts0 ¼ 3:44 s, and the initial temperature was equal 20  C. Evolution of temperature for these values is shown in Fig. 8. We see, that when the values of Biot number Bi / N, the maximal temperature on the contact surface is equal 741  C. This theoretical value coincides with experimental value 760  C from the monograph [14]. If we take into account the heat transfer through the surface of contact (Bi ¼ 5), then the

* Fig. 5. Dependence of the maximal dimensionless temperature Tmax on the dimensionless parameter sm for three values of the Biot number: a) Bi / N; b) Bi ¼ 5; c) Bi / 0.

* Fig. 7. Dependence of the maximal dimensionless temperature Tmax on the Biot number Bi at the value of the dimensionless time of pressure increasing sm ¼ 0.1.

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shown that the proposed model can reliably estimate the maximum temperature for the tribosystem cast iron/cermet/steel during braking with constant deceleration. The most accurate theoretical result has been obtained taking into account the influence of thermal resistance of the contact surface. But in the case of equality of temperatures of the pad and the disc on the contact surface error (19  C) is quite acceptable. The obtained results can be applied to finding the transient stress fields of the brake under an emergency braking [35], to theoretical study of hot spots in disc brakes [36], to the cooling of brake discs [37], etc.

References

Fig. 8. Evolution of the contact temperature T(0,t) at braking with uniform retardation (tm ¼ 0) under perfect (Bi / N) and imperfect (Bi ¼ 5) thermal contact of the cast iron disc and metal-ceramic pad with the steel caliper.

calculated maximal temperature on a surface of the top semi-space will be equal 757  C.

6. Conclusions The mathematical model of calculation of the temperature in the disc/pad/caliper tribosystem has been proposed. The process of frictional heating during braking in the disc/pad/caliper system is modeled using the transient thermal problem of friction for the three-element system: the top semi-space (the disc), the strip (the pad) and the foundation (the caliper). The main difference in the formulation of this one-dimensional boundary-value problem of heat conduction for such tribosystem from the previously obtained solutions [29,30] is that the dependence of the contact pressure on time and the thermal resistance on the contact surface have been taken into account. The numerical analysis has been executed for frictional materials such as cast iron disc, the metal-ceramic pad and the steel caliper. It is established, that the contact temperature essentially depends on value of input parameter tm (dimensionless sm), which characterizes duration of achievement by time-dependent pressure p(t) the maximal value p0. The maximal temperature occurs at braking with constant deceleration, when the pressure instantly reaches its maximum value (sm ¼ 0). With increase of the parameter sm, the temperature at the contact surface decreases. It is known, that with temperature on the surface of friction, the magnitude of wear of interacting surfaces [3] is associated. We can therefore say that the smooth braking is the most desirable. This is a well-known practical result; we just gave it a theoretical foundation. Influence of the dimensionless parameter e the Biot number Bi, describing a heat transfer through the contact surface, on temperature fields in the elements of tribosystem has been studied, too. The main result obtained in this article is the definition for the investigated class of materials and input parameters of the limit values of Biot number, in which it is possible to neglect the influence of heat transfer through the contact surface on the temperature, and use the solutions of articles [29,30]. For tribosystem under consideration the limiting value of the Biot number is equal, approximately, 100. The comparative analysis of the theoretical values of the maximal temperature obtained by means of given model with corresponding its experimental values has been led. It has been

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