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1、Frictionally excited thermoelastic instability in disc brakesTransient problem in the full contact regimeAbstractExceeding the critical sliding velocity in disc brakes can cause unwanted forming of hot spots, non-uniform distribution of contact pressure, vibration, and also, in many cases, permanent

2、 damage of the disc. Consequently, in the last decade, a great deal of consideration has been given to modeling methods of thermo elastic instability (TEI), which leads to these effects. Models based on the finite element method are also being developed in addition to the analytical approach. The an

3、alytical model of TEI development described in the paper by Lee and Barber Frictionally excited thermo elastic instability in automotive disk brakes. ASME Journal of Tribology 1993;115:60714 has been expanded in the presented work. Specific attention was given to the modification of their model, to

4、catch the fact that the arc length of pads is less than the circumference of the disc, and to the development of temperature perturbation amplitude in the early stage of breaking, when pads are in the full contact with the disc. A way is proposed how to take into account both of the initial non-flat

5、ness of the disc friction surface and change of the perturbation shape inside the disc in the course of braking.Keywords: Thermo elastic instability; TEI; Disc brake; Hot spots1. IntroductionFormation of hot spots as well as non-uniform distribution of the contact pressure is an unwanted effect emer

6、ging in disc brakes in the course of braking or during engagement of a transmission clutch. If the sliding velocity is high enough, this effect can become unstable and can result in disc material damage, frictional vibration, wear, etc. Therefore, a lot of experimental effort is being spent to under

7、stand better this effect (cf. Refs.) or to model it in the most feasible fashion. Barber described the thermo elastic instability (TEI)as the cause of the phenomenon. Later Dow and Burton and Burton et al. introduced a mathematical model to establish critical sliding velocity for instability, where

8、two thermo elastic half-planes are considered in contact along their common interface. It is in a work by Lee and Barber that the effect of the thickness was considered and that a model applicable for disc brakes was proposed. Lee and Barbers model is made up with a metallic layer sliding between tw

9、o half-planes of frictional material. Only recently a parametric analysis of TEI in disc brakes was made or TEI in multi-disc clutches and brakes was modeled. The evolution of hot spots amplitudes has been addressed in Refs. Using analytical approach or the effect of intermittent contact was conside

10、red. Finally, the finite element method was also applied to render the onset of TEI (see Ref.).The analysis of nonlinear transient behavior in the mode, when separated contact regions occur, is even accomplished in Ref. As in the case of other engineering problems of instability, it turns out that a

11、 more accurate prediction by mathematical modeling is often questionable. This is mainly imparted by neglecting various imperfections and random fluctuations or by the impossibility to describe all possible influences appropriately. Therefore, some effort aroused to interpret results of certain expe

12、riments in addition to classical TEI (see, e.g.Ref).This paper is related to the work by Lee and Barber 7.Using an analytical approach, it treats the inception of TEI and the development of hot spots during the full contact regime in the disc brakes. The model proposed in Section 2 enables to cover

13、finite thickness of both friction pads and the ribbed portion of the disc. Section 3 is devoted to the problems of modeling of partial disc surface contact with the pads. Section 4 introduces the term of thermal capacity of perturbation emphasizing its association with the value of growth rate, or t

14、he sliding velocity magnitude. An analysis of the disc friction surfaces non-flatness and its influence on initial amplitude of perturbations is put forward in the Section 5. Finally, the Section 6 offers a model of temperature perturbation development initiated by the mentioned initial disc non-fla

15、tness in the course of braking. The model being in use here comes from a differential equation that covers the variation of thethermal capacity during the full contact regime of the braking.2. Elaboration of Lee and Barber modelThe brake disc is represented by three layers. The middle one of thickne

16、ss 2a3 stands for the ribbed portion of the disc with full sidewalls of thickness a2 connected to it. The pads are represented by layers of thickness a1, which are immovable and pressed to each other by a uniform pressure p. The brake disc slips in between these pads at a constant velocity V.We will

17、 investigate the conditions under which a spatially sinusoidal perturbation in the temperature and stress fields can grow exponentially with respect to the time in a similar manner to that adopted by Lee and Barber. It is evidenced in their work 7 that it is sufficient to handle only the antisymmetr

18、ic problem. The perturbations that are symmetric with respect to the midplane of the disc can grow at a velocity well above the sliding velocity V thus being made uninteresting.Let us introduce a coordinate system （x1; y1）fixed to one of the pads (see Fig. 1) the points of contact surface between th

19、e pad and disc having y1 = 0. Furthermore, let a coordinate system （x2; y2） be fixed to the disc with y2=0 for the points of the midplane. We suppose the perturbation to have a relative velocity ci with respect to the layer i, and the coordinate system （x; y）to move together with the perturbated fie

20、ld. Then we can writeV = c1 -c2; c2 = c3; x = x1 -c1t = x2 -c2t,x2 = x3; y = y2 =y3 =y1 + a2 + a3.We will search the perturbation of the uniform temperature field in the form and the perturbation of the contact pressure in the formwhere t is the time, b denotes a growth rate, subscript I refers to a

21、 layer in the model, and j =-1is the imaginary unit. The parameter m=m（n）=2pin/cir =2pi/L, where n is the number of hot spots on the circumference of the disc cir and L is wavelength of perturbations. The symbols T0m and p0m in the above formulae denote the amplitudes of initial non-uniformities (e.

22、g. fluctuations). Both perturbations (2) and (3) will be searched as complex functions their real part describing the actual perturbation of temperature or pressure field.Obviously, if the growth rate b0, the initial fluctuations are damped. On the other hand, instability develops ifB0.2.1. Temperat

23、ure field perturbationHeat flux in the direction of the x-axis is zero when the ribbed portion of the disc is considered. Next, let us denote ki = Ki/Qicpi coefficient of the layer i temperature diffusion. Parameters Ki, Qi, cpi are, respectively, the thermal conductivity, density and specific heat

24、of the material for i =1,2. They have been re-calculated to the entire volume of the layer (i = 3) when the ribbed portion of the disc is considered. The perturbation of the temperature field is the solution of the equationsWith and it will meet the following conditions:1，The layers 1 and 2 will hav

25、e the same temperature at the contact surface2，The layers 2 and 3 will reach the same temperature and the same heat flux in the direction y待添加的隐藏文字内容2,3，Antisymmetric condition at the midplane The perturbations will be zero at the external surface of a friction pad (If, instead, zero heat flux throu

26、gh external surface has been specified, we obtain practically identical numerical solution for current pads).If we write the temperature development in individual layers in a suitable form we obtain where and2.2. Thermo elastic stresses and displacementsFor the sake of simplicity, let us consider th

27、e ribbed portion of the disc to be isotropic environment with corrected modulus of elasticity though, actually, the stiffness of this layer in the direction x differs from that in the direction y. Such simplification is, however, admissible as the yielding central layer 3 practically does not take e

28、ffect on the disc flexural rigidity unlike full sidewalls (layer 2). Given a thermal field perturbation, we can express the stress state and displacements caused by this perturbation for any layer. The thermo elastic problem can be solved by superimposing a particular solution on the general isother

29、mal solution. We look for the particular solution of a layer in form of a strain potential. The general isothermal solution is given by means of the harmonic potentials after Green and Zerna (see Ref. 18) and contains four coefficients A, B, C, D for every layer. The related displacement and stress field components are written out in the Appendix A.