Mathematical modelling of brake noise vibrations using spectral methods

Using holographic techniques it has become possible not only to perform a modal investigation of disc brake squeal but also to examine the actual evolution in time of the wave motion on the disc surface. This applies to both out-of-plane and in-plane displacements. In order to mathematically model this disc wave motion it was decided to employ a method that is used in the numerical solution of partial differential equations (pdes) , namely spectral collocation. This technique approximates the solutions of pdes by trigonometric or Chebyshev polynomials so that spatial differentiation can be performed by differentiation matrices. Time evolution is carried out by standard time-stepping techniques. In the absence of strong shocks and for regularly shaped regions the method is particularly efficient and has been successfully used in geophysics and meteorology. This work investigates its application in elastodynamics where it can be used to model moving contact problems. INTRODUCTION Brake noise (squeal) is notoriously difficult to predict by mathematical modelling [1]. Finite element models can indicate possible frequencies at which noise can occur, but have been unsatisfactory in providing a deeper understanding of the mechanism involved. Dynamic models using finite elements have proved to be very expensive in computer time. Experimental techniques, especially holographic interferometry, have made it possible to build up a considerable amount of knowledge about the types of vibration that occur in a noisy brake. Using electronic triggering devices it has been possible to take a series of holograms at different times during the period of vibration when a brake is continuously emitting noise (see Figure 1). This provides a picture of the waveform on the surface of a vibrating brake disc. It has also been possible using mirrors to take three holographic images at different angles, thus enabling both in-plane and out-of-plane motion of a brake disc to be investigated. Significantly large in-plane components are often observed. From this information animations can be constructed showing the real vibration of a brake system as it generates noise [2,3]. Such detailed experimental information has motivated the search for modelling techniques that can provide dynamic models without being prohibitive on computing time. One possible approach is the use of Spectral methods [4,5,6]. With smooth data and for simple domains such methods are particularly efficient for the numerical solution of partial differential equations (PDEs) and can often satisfactorily include nonlinearities. The method has proved very successful in models that normally involve considerable computation time in areas such as meteorology and geophysics. If the independent variables are approximated by polynomials or truncated trigonometric series, numerical differentiation can be carried out by using standard differentiation matrices. The technique is usually highly accurate and is also relatively easy to implement in computer programs. A very good exposition of the method, including application to a range of ordinary and partial differential equations with efficient Matlab code, is available in [4]. Figure 1: Holograms of disc brake generating noise taken at different times during a cycle of vibration. Fringe lines can be regarded as contour lines of out-of-plane displacement field showing the existence of travelling waves. OUTLINE OF SPECTRAL METHOD To illustrate the type of spectral method that can be used to solve elasticity problems it is perhaps easiest to start with a simple ordinary differential equation (ODE): ) ( ) ( 2 2 x f x u dx d = , , u (1) a u = − ) 1 ( b = + ) 1 ( The basic idea is to replace with an n th degree polynomial that is determined by its values at n+1 collocation points . It is conventional to take and but clearly scaling is ) (x u n x ,..., 1 ) (x p 1 + = x x , 0 0 x 1 − = n x possible. Rather than take the points to be equally spaced between –1 and +1, the method supposes them to be so-called Chebyshev points, that is the points given by the formula n k xk π cos =