Keywords: squeal, nonlinear stochastic dynamics, state representation, vector ARMA method.

Since the early 20th century, many investigators have examined the problem of disc brake dynamics with experimental, theoretical and numerical techniques. As reviewed by Kinkaid et al. in 2003 [1], this work has shed some light on the physical phenomena leading to squeal. For example studies of nonlinear modes and stability of braking systems have led to estimate the propensity of a braking system to squeal [2,3]. Tribological studies have led to the emphasis on the importance of friction and disc-pad interface or third body behaviour on the onset of squeal. More recently, very accurate finite element modelling of disc-pad contact [4] have permit to quantify the influence of several parameters as relative speed, normal pressure or geometrical imperfections of the disc surface. These models have also shown the complexity of the dynamical phenomena involved in the contact zone during the squealing phase which vary from "quiet sliding" to stick-slip and stick-slip-separation.

One striking point when reading the literature is that whereas many authors refer to the stochastic or random nature of the squealing process only few articles are dedicated to its study from a stochastic point of view. Ibrahim and his co-workers at Wayne State University have developed stochastic models of the friction induced vibration of disc brakes [5,6]. This work shows the potential of the stochastic analysis to treat friction-induced vibration problems. Nevertheless it is based on a relatively simple mechanical model which can not embrace the complexity of the phenomena.

Faced with this complexity and with the only partial understanding of the squealing process our idea is to develop an approach in order to analyze the phenomena and build a stochastic model using experimental data.

The first step of the proposed approach is to use statistical tools in order to better understand and thereafter better model the squealing and the other different vibration regimes occurring in systems where friction induced vibration are present. An important question to answer is to find macroscopic parameters or a criterion based on these macroscopic parameters able to characterize the squealing phase. The stochastic analysis is performed using the system response which is partially known through experimental values. The tools are the power spectral density (PSD) and the coherence function (CF). This analysis reveals an interesting correlation between the measured parameters and comparisons between their spectral content in squealing and non squealing regimes are made. It also points out the potential of a proposed parameter called "dissipative squealing power".

The second consists in building a simple stochastic model of this process in the light of the stochastic analysis. In this context, we propose a stochastic modelling of spectral linearization type based on the vector ARMA processes theory using the natural link between vector ARMA representations and state representations. This method, previously validated on nonlinear oscillators [7], is used to model the measured parameters. The modelling is performed in different conditions where intermittent squealing occurs. A first attempt is made and its efficiency is discussed. This discussion is the starting point toward a new type of stochastic model especially developed to suit to the squealing problem in its entirety.

1

N.M. Kinkaid, O.M. O'Reilly and P. Papadopoulos, "Automotive disc brake squeal", Journal of Sound and Vibration, 267(1), 105-166, 2003. doi:10.1016/S0022-460X(02)01573-0

2

S.W.E. Earles and C.K. Lee, "Instabilities arising from frictional interaction of a pin-disk system resulting in noise generation", ASME Journal of Engineering for Industry, 98(1), 81-86, 1976.

3

J.-J. Sinou, O. Dereure, G.-B. Mazet, F. Thouverez and L. Jezequel, "Friction-induced vibration for an aircraft brake system-Part 1: Experimental approach and stability analysis", International Journal of Mechanical Sciences, 48(5), 536-554, 2006. doi:10.1016/j.ijmecsci.2005.12.002

4

L. Baillet, S. D'Errico and B. Laulagnet, "Understanding the occurrence of squealing noise using the temporal finite element method", Journal of Sound and Vibration, 292(3-5), 443-460, 2006. doi:10.1016/j.jsv.2005.08.001

5

S.L. Qiao and R.A. Ibrahim, "Stochastic dynamics of systems with friction-induced vibration", Journal of Sound and Vibration, 223(1), 115-140 ,1999. doi:10.1006/jsvi.1998.2099

6

S.L. Qiao, D.M. Beloiu and R.A. Ibrahim, "Deterministic and Stochastic Characterization of Friction-Induced vibration of Disc Brakes", Nonlinear Dynamics, 36, 361-378, 2004. doi:10.1023/B:NODY.0000045512.75470.f4

7

D. Daucher, M. Fogli and D. Clair, "Modeling of complex dynamical behaviours using a state representation technique based on a vector ARMA approach", Probabilistic Engineering Mechanics, 21(1), 73-80, 2006. doi:10.1016/j.probengmech.2005.07.003

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