Dynamic instability of a thin circular plate with friction interface and its application to disc brake squeal

Abstract

The mathematical formulation for determining the dynamic instability due to transverse doublet modes in the self-excited vibration of a thin annular plate is presented in this paper. An analytical approach is developed to obtain the stability results from the eigenvalue problem of a stationary disc with a finite contact area. The approach uses the eigenfunctions of transverse doublet modes in classical plate theory and establishes the formulation of modal instability due to the modal-interaction of a doublet mode pair. The one-doublet mode model of a disc and a discrete model equivalent to the one-doublet mode model are proposed for providing a more fundamental understanding of the onset of squeal. The analytical models are validated through a comparison of results from a modal expansion model obtained from finite element component models. Throughout the analytical investigation, the pad arc length is found to be a critical design parameter in controlling squeal propensity.

Introduction

Friction is the source of squeal noise in a brake system. Friction force on contact interface produces non-conservative work that can eventually lead to unstable oscillations. An effective model for friction-induced vibration is constructed from the system equations of motion using a linearized friction and contact stiffness model. The dynamic instability near a steady sliding equilibrium position can be determined from the appearance of the positive real parts of eigenvalues of the linearized model as extensively reviewed in the review article [1].

In mathematical interpretation, the non-conservative frictional work produces a non-symmetric stiffness matrix in the linearized equations of motion that is necessary for the appearance of eigenvalues with positive real parts. The non-symmetric elements in the stiffness matrix of brake system model can arise from the non-conservative nature of follower forces and friction-couples. Mottershead and Chan [2] used frictional follower loads to study the flutter instability of doublet modes of a finite element non-rotating disc model. Also, Mottershead [3] extensively introduced several follower force friction models in his review article. However, the influence of follower forces on squeal propensity has been seen to be marginal. Flint and Hulten [4] developed a stationary disc brake model with follower force and friction-couple components and concluded that the relative effect of follower forces was negligible. Heilig and Wauer [5] developed a simplified rotating disc brake model with two different contact models, global contact and local gradient contact models, where the local gradient contact model describes the direction change of the contact forces corresponding to the disc deformation. Therefore, the contact forces of the local gradient contact model are follower forces. From their numerical results, they also concluded “the differences in the results between the global and local gradient contact modeling are marginal (<0.1%) and not visible in the plots”. Ouyang and Mottershead [6] have shown that the follower force term is many times smaller than the friction-couple term in disc brake system.

In contrast, the friction-couple mechanism is considered to be a significant brake squeal leading factor in producing vibrations. Since the friction-couple mechanism is derived from the undeformed contact kinematics (or global contact model), the global contact model is sufficient in describing the contact mechanics of the brake system. Ouyang and his coworkers [7] provided an extensive review on the automotive disc brake squeal analysis described by the friction-couple mechanism in a linearized set of equations of motion.

One of the primary sources of nonlinearities in brake system models is the nonlinear load-deflection behavior of the friction material. A small change of preload can lead to a significant change in the linearized contact stiffness. Vanderlugt [8] in his experimental work showed that the vibration modes leading to squeal can be modified as contact stiffness changes with respect to brake pressure variation. This implies that the stability of each mode depends on contact stiffness variation. In order to investigate the modal stability behavior and its stability boundary from the linearized equations of motion, Chowdhary et al. [9] treated contact stiffness as a system parameter and solved the eigenvalue sensitivity problem of disc brake system on the dynamic instability of the steady-sliding response. Their contribution to brake squeal research has been to show the role of mode-coupling and mode-merging on the stability boundaries in the stiffness-friction plane. Later, Huang et al. [10] presented the qualitative relations between mode-coupling and mode-merging and showed that the compatibility of mode shapes needed for mode coupling is one of the factors dictating the onset of squeal in drum brake system. However, numerical results from their analysis did not establish the generalized squeal theory associated with modal interaction.

In the present study, a simplified mathematical disc model is constructed on the basis of the physical disc brake system that was studied experimentally and numerically by Vanderlugt [8]. Since the unstable modes on squeal frequencies were found to be disc doublet modes in his work, the one-doublet mode model of a thin annular plate representing a brake rotor is developed and investigated. For the practical purpose, the system parameters and the component disc natural frequencies are obtained from the measured data and the finite element (FE) analysis in Vanderlugt's work [8]. The modal results obtained from the finite element method showed the non-coincidence of the two natural frequencies in a doublet mode. The frequency separation of the component disc mode pair will be the focus of this study and the corresponding modal stability boundary will be solved with respect to contact stiffness variation.

The main objective of the work presented in this paper is to analytically describe the modal instability of disc brake system. This work provides the formulation for the squeal onset of brake geometry that has not been addressed in any previous papers. The essence of the formulation is to provide the fundamental design concept of reducing squeal occurrence in disc brake system. Also, this work provides the physical interpretation for the formulation of squeal onset. For better physical interpretation, a discrete model representing the modification of Hoffmann's model [11] is introduced, where this simplified model is shown to be mathematically equivalent to the doublet mode model.