Car Disc Brake Squeal - Michael I Friswell

Materials Science Forum Vols. 440-441 (2003) pp. 269-276 online at http://www.scientific.net © (2003) Trans Tech Publications, Switzerland Citation & Copyright (to be inserted by the publisher )

Car Disc Brake Squeal: Theoretical and Experimental Study Q Cao1, M I Friswell2, H Ouyang1, J E Mottershead1 and S James1 1

Department of Engineering, University of Liverpool, Liverpool L69 3GH, UK

2

Department of Aerospace Engineering, University of Bristol, Bristol BS8, UK

Keywords: Disc brake squeal, moving load, analytic methods, finite elements, experiments

Abstract. This paper presents a numerical method for the calculation of the unstable frequencies of a car disc brake and the analysis procedure. The stationary components of the disc brake are modelled using finite elements and the disc as a thin plate. This approach facilitates the modelling of the disc brake squeal as a moving load problem. Some uncertain system parameters of the stationary components and the disc are tuned to fit experimental results. A linear, complex-valued, asymmetric eigenvalue formulation is derived for disc brake squeal. Predicted unstable frequencies are compared with experimentally established squeal frequencies of a realistic car disc brake. Introduction Friction-induced vibration and noise emanating from car disc brakes is a source of discomfort and leads to customer dissatisfaction. The high frequency noise above 1kHz, known as squeal, is most annoying and is very difficult to eliminate. Akay [1] recently reviewed friction-induced noise, including car disc brake squeal. There exists an extensive literature on theoretical and experimental investigations of disc brake squeal. To appreciate the problem, reviews [2-5] conducted at different periods are recommended here. Kinkaid et al.’s recent thorough survey [6] deserves a special mention. A car disc brake system consists of a rotating disc and stationary (non-rotating) pads, carrier bracket, calliper and mounting pins. The pads are loosely housed in the calliper and located by the carrier bracket. The calliper itself is allowed to slide fairly freely along the two mounting pins in a floating calliper design. A typical floating-type vented disc brake system is shown in Figure 1.

Figure 1. A car disc brake of floating calliper design The disc is mounted to a car wheel and thus rotates at the same speed as the wheel. When the disc brake is applied, the two pads are brought into contact with the disc surfaces. Most of the kinetic energy of the travelling car is converted to heat through friction. But a small part of it converts into sound energy and generates noise. A squealing brake is difficult and expensive to cure. Preferably the noise issue should be resolved at the design stage.

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Title of Publication (to be inserted by the publisher)

In parallel with experimental study, modelling of disc brakes and simulating their dynamics and acoustic behaviour is an efficient way of understanding the squeal mechanism and designing quiet disc brakes. The standard technology in the car manufacturing industry is to use large finite element models. The majority of recent finite element analyses also contain a static contact analysis of the contact area and the pressure distribution of the disc/pads interface [7-9]. Some even studied the dynamic contact between the disc and the pads [10]. In a disc brake system, the disc rotates past the stationary pads housed by the carrier bracket and the calliper. As a result, the pads mate with different spatial area of the disc as vibration proceeds. This is a moving load problem. Because disc brakes tend to squeal at low speeds, the moving load nature of the problem has been omitted until very recently. Ouyang et al. [11] conceptually divided a disc brake into the rotating disc and the non-rotating, stationary components, and put forward an analytical-numerical combined approach for analysing disc brake vibration and squeal. They cast the disc vibration and squeal as a moving load problem and derived a highly nonlinear eigenvalue formulation [9]. A linear eigenvalue formulation is established and predicted numerical results are compared with experimental results in this paper. Equation of Motion of the Disc The low frequency squeal was defined as the range of frequency below the first in-plane squeal frequency [12], which is the topic of this paper. As such, the in-plane motion of the disc is omitted. Subsequently, the disc is approximated as an annular plate. It is subjected to normal and tangential friction forces at the disc/pads interface. The equation of transverse motion w(r ,q , t ) of the disc is

rh

¶2w ¶w 1 j ¶ +c + DÑ 4 w = { pi (t )d(q - q i - W t ) + [ M i (t )d(q - q i - W t )]}d(r - ri ) , 2 ¶t ¶t r i =1 r¶q

(1)

where r , h, c and D are the specific density, the thickness, the viscous damping and the flexural rigidity of the disc. m i is the friction coefficient, p i the normal force and M i = m i pi h / 2

(2)

is the bending couple produced by the friction force, all on the ith contact node (ri ,q i , 0) at the disc/pads interface. Friction manifested in this way was first proposed by North for a rigid disc model [2] and extended by Hulten and Flint [13] to a beam model of the disc. m i can be a function of disc speed W (in radians per second) and may vary from node to node. It is taken as a constant in this paper. d is the Dirac delta function. Other symbols are explained in the Appendix. The solution of Eq.1 can be expressed as ¥

w(r ,q , t ) =

¥

y mn (r ,q )q mn (t ) , m = 0 n = -¥

y mn (r ,q ) =

Rmn (r )

rhb 2

exp(inq ) .

(3)

where i = - 1 . Rmn (r ) is a linear combination of Bessel functions that satisfy the boundary conditions at the inner radius a and the outer radius b of the disc. The mode functions satisfy the ortho-normality conditions of

Materials Science Forum Vols. 440-441

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2p b

3

2p b 2 Dy kl Ñ 4y mn rdrdq = w mn d km d ln .

rhy kly mn rdrdq = d km d ln , 0 a

0

(4)

a

where d km and dln are the Kronecker delta, and the overhead bar denotes complex conjugation. Substituting Eq.2 and Eq.3 into Eq.1 and making use of Eq.4 yields

q&&kl + 2xw kl q& kl + w kl2 q kl = -

j

1

rhb 2

Rkl (ri ) exp[-il (q i + W t )] (1 i

mi h 2ri

il ) pi (l = 0, ± 1, ± 2, ...) . (5)

Eq.5 may be re-written in matrix form as d2 d D = diag[ 2 + 2xw kl + w kl2 ] , dt dt

Dq = -diag[exp(-ilW t )]S ¢ H p ,

(6)

where the elements of matrix S ¢ are S ¢(i, m) =

Rkm (ri )

rhb 2

exp(imq i )(1 +

mi h 2ri

im ) ,

(7)

and diag[exp(-ilW t )] is a diagonal matrix with the diagonal elements following the sequence of l=0, ±1, ±2, …, representing the number of diameters in an unloaded disc mode. Equation of Motion of the Stationary Components The stationary components of a disc brake, as seen from Figure 1, are very complicated in geometry. An industrial finite element model of the stationary components of a disc brake may have over 100,000 degrees-of-freedom. To increase computational efficiency, substructuring or superelements are used. Another complexity of a disc brake system is the presence of a number of contact interfaces between any two stationary components because of the way they are assembled. The interface between the pads and the disc is the most important one and should be modelled carefully. When the brake is applied, the asperities on the disc surface and on the pad surfaces are squashed under high local pressure and hence form a layer of material that is different from the disc and pad materials and is represented by a thin layer of solid elements of orthotropic material properties (a very high Young’s modulus in the normal direction and small Young’s moduli in the other two the tangential directions). These Young’s moduli may vary from element to element and are assumed to depend on the local static pressures determined through a nonlinear, static contact analysis. Some other works, for example [7, 8], used spring elements for this contact. In the present contact analysis, the disc is assumed to be rigid and the pads are allowed to slide on the disc surface. Once the pressure distribution is found, the local, element-wise Young’s moduli can be determined if the pressure-dependence of the pad material properties is known. When the stationary components of a disc brake are considered as a separate part of a disc brake, the disc and pads interface become a free boundary with unknown forces and displacements. The equation of motion of the reduced finite element model of the stationary components is M&x& + Cx& + Kx = f

(8)

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The matrices in the above equation are partitioned into two parts, the one for the contact nodes at the disc/pads interface and the one for the other nodes. The displacements at these contact nodes are further divided into three subsets for u , v and w displacements respectively. Eq.8 then becomes D p ,uu D p,vu D p ,wu D op,u

D p,uv D p ,vv D p , wv D op,v

D p ,uw D p,vw D p ,ww D op , w

D po ,u D po ,v D po, w D op

up vp = wp xo

0 p p 0

(9)

where D a ,b = M a ,b

d2 d + C a ,b + K a , b , 2 dt dt

= diag[ m ] ,

(10)

and the subscript ‘a’ designates nodes at disc/pads interface or other places, and the subscript ‘b’ the displacements in the three normal directions, respectively. Multiplying the 3rd row in Eq.9 by m and subtracting it from the 2nd row gives D p ,uu D p ,vu - D p , wu D op,u

D p ,uv D p ,vv - D p , wv D op,v

D p ,uw D p ,vw - D p , ww D op, w

D po ,u D po ,v - D po , w D op

up

0 vp = 0 wp 0 xo

(11)

Dynamics of the Whole Disc Brake System As mentioned before, the disc rotates past stationary pads. This moving frictional contact must be duly considered. Therefore the vibration and squeal of disc brakes should be cast as a moving load problem. Analytical solutions of many simple moving load problems were summarised by Fryba [14]. The vibration and dynamic stability of discs, including moving loads, were reviewed by Mottershead [15]. Since in this investigation the interest is in the dynamics of the disc in the very low speed range, gyroscopic and centrifugal effects are very small and can be safely omitted. A cylindrical coordinate system fixed to the centre of the plate is used to describe its transverse vibration. The stationary components can be thought as moving relative to the disc. At the disc/pads interface, it is assumed that the w-displacements of the pads equal the transverse deflections of the annular plate. Taking into account the relative motion of the disc at constant rotating speed, the displacement continuity condition at the disc/pads interface (when the pads are moving in the q direction) is

w p = {w(r1 , q1 + W t , t ), w(r2 , q 2 + W t , t ), ......, w(r j , q j + W t , t )}T =Sdiag[exp(inW t )]q

(12)

where the elements of S are S(i, m) =

Rkm (ri )

rhb 2

exp(imq i )

(13)


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Eq.6, Eq.11 and Eq.12 provide the means of solving u p , v p , w p and q . A close examination of them reveals that the diagonal (exponential in the time domain) matrices in these two equations happen to be an inverse to each other. Moreover, the left-hand side differential operator of Eq.6 is also a diagonal matrix. This finding affords a new way of solving these complicated equations. Let q = diag[exp(-inW t )]y

(14)

where y is a new vector. Substituting Eq.14 into Eq.6 transforms Eq.6 into D¢y = -S ¢ H p ,

D¢ = diag[

d2 d + 2(xw kl - ilW ) + (w kl2 - i 2xw kl lW - l 2 W 2 )] 2 dt dt

(15)

Further mathematical manipulation leads to D p,uu D p ,vu - D p, wu D op ,u S ¢ H D p, wu

D p ,uv D p,vv - D p, wv D op,v S ¢ H D p , wv

D p,uw S D po ,u (D p,vw - D p, ww )S D po ,v - D po , w D op , w S D op H H D¢ + S ¢ D p, ww S S ¢ D po , w

up vp = 0. y xo

(17)

Eq.17 should be converted to a system of simultaneous 1st-order differential equations so that the system eigenvalues can be solved. The eigenvalue problem involves complex-valued, asymmetric matrices. Due to the page number restriction, the detailed derivation is not presented in the paper. Numerical Analysis and Discussion The disc studied has the following dimensions and properties: a = 0.045m , b = 0.133m , h = 0.012m , n = 0.211 , r = 7200kg × m -3 , Young’s modulus E = 1.2 ´ 10 5 MPa . Thirty-five natural frequencies of the disc are calculated by an analytical method, of which the first eleven distinct ones are given in Table 1. Among the thirty-five frequencies, thirty-two are double frequencies while three are single frequencies, including the fundamental frequency. Thirty-five disc modes are involved in the subsequent computation of the eigenvalues of the whole disc brake system. The material properties of the stationary components, except those of the pads, are summarised in Table 2. k, l

0, 0

w kl

1087

1, 0 0, ±1 0, ±2 0, ±3 0, ±4 0, ±5 0, ±6 1, ±1 1090 1279 1957 3130 4704 6622 7028 7295 Table 1. Numerical natural frequencies [Hz] of the disc

1, ±2 8115

0, ±7 8859

Components

Young’s modulus [GPa]

Poisson’s ratio

Density [kg × m -3 ]

Calliper

188

0.3

7100

Carrier

170

0.3

7564

Back plate

210 0.3 Table 2. Material data of the stationary components

7850

274


6


The pads are of nonlinear, viscoelastic material with orthotropic properties. Here they are taken to be linear, elastic and orthotropic but their Young’s modulus depends on the piston line pressure. This dependency is worked out indirectly by comparing the finite element results with the experimental results of the pads. The Young’s modulus is between 5.4 GPa and 10.8 GPa. In addition to the above material properties, there are other parameters which are not readily available, for example, the stiffness of the brake fluid. These less certain parameters are tuned using experimental results. This is a time-consuming process but is considered necessary for building a reliable model and getting credible numerical results. The system eigenvalues with positive real parts s obtained from the numerical analysis indicate possible squeal frequencies in practice. Figure 2 presents noise indices (defined as

s 2

[16])

s +w2 versus frequencies at two disc speeds of W = 0 (representing static friction) and W = 5rad/s . The parameter values used in the above calculation are 0.1% damping coefficient for the stationary components, disc damping of x = 0 and a rather high constant friction coefficient of mi =0.7. 0.07

noise index

0.06 0.05 0.04

W=0

0.03

W=5

0.02 frequency [Hz]

0.01 0 1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 2. Noise index versus frequency at two disc speeds [rad/s]

Experiments on disc brake squeal have been conducted at different piston line pressures and/or slightly different rotating speeds using twelve non-contact displacement transducers. The results are shown in Figure 3. The authors’ detailed experimental work on disc brake squeal is reported in [17].

Amplitude - mm

1 .E -0 3

1 .E -0 4

F req u en cy 1 .E -0 5 1000 2000 3000 4000 5000 6000 7000 8000 F ig u re 3 . S q u e a l te s t re s u lts fo r th e M e rc e d e s v e n te d d is c b ra k e a t L iv e rp o o l


275


It can be seen that most experimental squeal frequencies are predicted by numerical unstable frequencies. There are two exceptions. One is that there is no squeal frequency around 5kHz, but a few numerical unstable frequencies are predicted there. The other is the squeal frequency at 7.8kHz, which is not predicted. Squeal noise is more likely to appear in the frequency ranges of 2.1-4.2kHz and 5.6-7.0kHz. The noise indices of predicted unstable frequencies do not correlate so well with the amplitudes of experimental squeal frequencies. Squeal noise is strong at 4.2kHz, 2.1kHz and 5.9kHz, while numerical unstable frequencies have greater noise indices at 6.8-6.9kHz, 1.4kHz and 4.9kHz. It is always difficult to predict squeal frequencies. Predicting noise level is beyond reach at present. There is some noticeable difference between noise behaviour when disc is not rotating and when rotating. By comparing the noise indices at W = 0 and W = 5 shown in Figure 2, it can be seen that the real parts of most unstable frequencies increase when the disc rotates at 5 rad/s, indicating that the disc rotation can make a disc brake system more unstable in general. This shows that disc brake squeal modelled as a moving load problem is indeed a useful development. Summary This paper presents a method for modelling car disc brakes and predicting squeal frequencies. The disc brake is treated as a moving load problem consisting of two parts: the rotating disc and the stationary components, which are dealt with respectively by a classical analysis of a differential equation and the finite element method. A plausible squeal mechanism for inducing dynamic instability is incorporated into the model. The unstable frequencies of the disc brake are obtained from a linear, complex-valued, asymmetric eigenvalue formulation. The predicted unstable frequencies show a reasonably good agreement with the experimentally established squeal frequencies, though the magnitudes of the positive real parts of the predicted eigenvalues do not correlate so well with the amplitudes of experimental squeal frequencies. Acknowledgement The authors are grateful for the support by TRW Automotive. References [1] Akay, A. Acoustics of friction. J. Acoust. Soc. Am., Vol. 111 (2002), p. 1525-1548 [2] North, N. R. Disc Brake squeal. Proc. IMechE, C38/76 (1976), p. 169-176 [3] Crolla, D. A. and Lang, A. M. Brake noise and vibration ¾ the state of the art. In Vehicle Technology, No.18 in Tribology Series, ed. Dawson, D., Taylor, C. M. and Godet, M. (Pro. Eng. Pub., 1991), p. 165-174 [4] Nishiwaki, M. R Review of Study on Brake Squeal. Jap. Soc. Auto. Eng. Rev., Vol. 11 (1990), p. 48-54 [5] Yang, S. and Gibson, R. F. Brake vibration and noise: review, comments and proposals. Int. J. Mater. Product Tech., Vol. 12 (1997), p. 496-513 [6] Kinkaid, N. M., O’Reilly, O M and Papadopoulos, P. Automotive disc brake squeal: a review. J. Sound Vib., to be published in 2003. [7] Lee, Y. S., Brooks, P. C., Barton, D. and C., Crolla, D. A. A study of disc brake squeal propensity using parametric finite element model. In IMechE Conf. Trans., European Conf. On Noise and Vibration, 12-13 May 1998, p. 191-201.

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[8] Nack, W. V. Brake squeal analysis by the finite element method. Int. J. Vehicle Des., Vol. 23 (2000), p. 263-275 [9] Ouyang, H. and Mottershead, J. E. A moving-load model for disc-brake stability analysis. Trans. ASME, J. Vib. Acoust., Vol. 125 (2003), p. 53-58 [10] Hu, Y. and Nagy, L. I. Brake squeal analysis using nonlinear transient finite element method. SAE Paper 971610, 1997 [11] Ouyang, H., Mottershead, J. E., Brookfield, D. J., James, S. and Cartmell, M. P. A methodology for the determination of dynamic instabilities in a car disc brake. Int. J. Vehicle Design, Vol. 23 (2000), p. 241-262 [12] Dunlap, K. B, Riehle, M. A. and Longhouse R. E. An investigative overview of automotive disc brake noise. SAE Paper 1999-01-0142, 1992. [13] Hulten, J. O. and Flint, J. An assumed modes approach to disc brake squeal analysis. SAE Paper 1999-01-1335, 1999. [14] Fryba, L. Vibration of Solids and Structures under Moving Loads. Noordhoff, Groningen, 1972. [15] Mottershead, J. E. Vibration and friction-induced instability in discs. Shock Vib. Dig., Vol 30 (1998), p. 14-31 [16] Yuan, Y. A study of the effects of negative friction-speed slope on brake squeal. Proc. 1995 ASME Des. Eng. Conf., Boston, Vol.3, Part A, p. 1135-1162 [17] James, S., Brookfield, D., Ouyang, H., and Motterhsead, J. E. Disc brake squeal – an experimental approach. Proc. 5th Int Conf on Modern Practice in Stress and Vibration Analysis, 911 Sept 2003, University of Glasgow, Scotland. Appendix C, K, M damping, mass and stiffness matrices of the stationary components. f force vector for the finite element model of the stationary components. j number of contact nodes at the disc/pads interface. k, l (m, n) number of nodal circles and number of nodal diameters in the mode of the bolted disc. p force vector consisting of all p i . q kl modal co-ordinate for k nodal circles and l nodal diameters for the disc. q modal co-ordinate vectors for the disc and the stationary components. r , q , z radial, circumferential and axial co-ordinates of the cylindrical co-ordinate system. t time. u,v,w displacements in the r , q and z directions respectively. u p , v p , w p u,v,w displacement vectors of the pad nodes at the disc/pads interface. x xo

n x w kl w

displacement vector corresponding to f. displacement vector for the nodes other than the contact nodes at the disc/pads interface. Poisson’s ratio of the disc. damping coefficient of the disc. undamped natural frequency corresponding to q kl . the imaginary part of a system eigenvalue (frequency).