Keywords: squeal, brake noise, contact, instability, numerical, linear, nonlinear.

Brake noise is an example of noise caused by vibration induced by friction forces [1]. During brake operation, the friction between the pad and the disc can induce a dynamic instability in the system. Brake squeal can occur in the frequency range between 1 kHz to 20kHz, and over. It is a difficult subject partly because of its strong dependence on many parameters and partly because the mechanical interactions in the brake system are very complicated, including non-linear contact problems at the friction interface. The onset of the squeal phenomenon is due to an unstable behaviour that occurs in linear conditions, during the braking phase. Several authors studied brake systems instability through a complex eigenvalues analysis of the system [2]. The instability is associated to the positive real part of the system eigenvalues. The good agreement with experimental experience proves that squeal instability occurs in linear conditions. Nevertheless, once developed, squeal reaches a new stable state with harmonic oscillations in non-linear conditions; in particular characterized by contact nonlinearities. In such conditions the linear models, utilized for the complex eigenvalues analysis, prove to be useless. In fact, to study the stability of the system through the complex eigenvalues analysis, the friction effects between the pad and the disc are treated introducing linear elements by an asymmetric stiffness matrix. This model of the contact does not account for local detachments and stick phenomena at the contact surface.

This paper presents two different numerical approaches to the problem: i) a linear finite element (FE) model is developed to predict the squeal conditions in function of the main parameters; ii) a non-linear model, that is able to reproduce the contact nonlinearities, is carried out to reproduce the time behavior of the brake system during squeal. The former is preferable to conduct a parametric analysis that allows to characterize the values of the parameters that bring to instability, with a low computing burden. The second uses a specific finite element program, Plast3, appropriate for nonlinear dynamic analyses in the time domain and particularly addressed to contact problems with friction between deformable bodies. This model enables the analysis of the behavior of the contact, by time simulations, in nonlinear conditions once squeal has developed.

The aim of this work is to verify the agreement between the linear and the nonlinear models on the prediction of the squeal instability as a function of the main parameters. Therefore, different values of the parameters are used in the nonlinear model and they are selected to cover both the stable and the unstable regions predicted by the complex eigenvalues analysis.

The analysis of the system during brake simulation is distinguished by two different behaviours: the stable conditions, characterized by no remarkable vibration of the system; and the unstable (squeal) conditions, characterized by an unstable state leading to a limit cycle, with strong vibration of the system. This last case presents a harmonic spectrum of the system vibration and stabilization of the vibration that agrees with the experimental results [3]. It is worth noting that such vibrations, recognized experimentally, are well reproduced by introducing only the nonlinearities due to the contact.

A comparison between the unstable regions obtained by the eigenvalue extraction and by the nonlinear model underlines the validity of the two approaches and the limits of these numerical tools. In fact, both methods point to instability at 3.5 kHz with similar values of the parameters. The same unstable deformed shape is calculated by the modal analysis and by the time simulation. Nevertheless, an over-prediction of the unstable regions, calculated by the complex eigenvalues analysis, is observed with respect to the nonlinear time simulations. The complex eigenvalues analysis predicts all the potential modal instabilities, and does not take into account nonlinear effects. In both cases squeal is detected with the standard Coulomb friction model, without any velocity-dependence of the friction coefficient and without the rise of stick-slip phenomena.

The brake simulations presented in this paper allows the analysis the friction surface, where no experimental data are available during braking tests. This permits the analysis of the "feed-back mechanism" that causes the squeal instability and that takes place at the friction interface. The analysis shows an oscillation of the local contact forces between the leading and trailing edges of the pad that is due to the normal direction component of the bending deformation of the support. The geometry of the models is related to an experimental set-up that will be analysed to validate the models and compare numerical results with the experiments.

1

A. Akay, "Acoustic of friction", Journal of Acoustical Society of America, 111(4), 1525-1548, 2002. doi:10.1121/1.1456514

2

H. Ouyang, W. Nack, Y. Yuan, and F. Chen, "Numerical analysis of automotive disc brake squeal: a review", Int. J. Vehicle Noise and Vibration, Vol. 1, Nos. 3/4 , 207-231, 2005. doi:10.1504/IJVNV.2005.007524

3

F. Massi, O. Giannini, L. Baillet, "Brake squeal as dynamic instability: an experimental investigation", Submitted at the Journal of Acoustical Society of America, 2006. doi:10.1121/1.2228745

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