# Parameter Determination of a Minimal Model for Brake Squeal
## 1 Introduction  
Squeal noise of brake system is one of the difficult problems that need to be solved by automobile manufacturers. It is generally believed that the disc brake squeal is self-excited vibrations induced by friction forces. However, for the mechanisms of self-excited vibrations, the existing literatures proposed a different hypothesis. The first type of literaturesholds that the variation characteristics of the friction coefficient lead to squeal instability, including mechanisms like stick-slip [1; 2; 3] and negative friction-velocity relationship [4; 5]. Nevertheless, the stick-slip mechanism is not sufficient to explain the occurrence of the squeal [6]. As pointed out by [PERSON] et al. [5], stick-slip occurs only at low speeds of a mass driven across a dry surface, while at a sufficiently high speed the mass will be permanently sliding. This proves that the instability of brake discs can be induced by a negative slope of friction-velocity characteristic even without stick-slip motion. The second type of literature indicates that self-excited vibrations can be induced even at a constant friction coefficient, and demonstrates that the generation of squeal is related to the dynamics of the geometric structure [7; 8]. The sprag-slip model introduced in early studies by [PERSON] [7] illustrates the mechanism of squeal in terms of structural geometry, which means that tribological characteristics should not be considered as the only reason for squeal instability [9]. Later, many scholars confirmed the existence of mode-coupling phenomenon [10; 11; 12; 13] in brakesqueal. [PERSON] et al. [11] found that when the friction coefficient is greater than a critical value, namely the Hopf bifurcation point, the oscillation frequencies of two structural modes gradually come closer until they coalesce into a pair of an unstable mode and a stable mode. It is clear that mode coupling explains the squeal phenomenon under a constant friction coefficient instead of a variable friction coefficient. In fact, when mode coupling occurs, the system not only generates unstable oscillations, but simultaneously accompanied by a notable nonlinear variation of friction coefficient. This has an important effect on the dynamic response of the system. Therefore, the stability analysis in this paper is based on the combination of mode coupling and negative slope of friction-velocity characteristics. The modeling method for brake squeal generally includes two categories. One category is to establish the finite element model of brake, then complex eigenvalue analysis (CEA) or transient analysis (TA) is performed for the finite element model [14; 15; 16]. [PERSON] et al. [17] and [PERSON] et al. [18] give a detailed summary of the application of CEA and TA in the study of brake squeal. The other category to investigate squeal is based on minimal models such as the abstract 2-DoF model. [PERSON] et al. [11] proposed a 2-DoF model to explain self-excited vibrations induced by mode-coupling mechanism. [PERSON] et al. [19] adopted a 2-DoF model containing lumped masses representing the disc and the pad respectively, through which the effect of negative slope of friction-velocity characteristics on the oscillation instability was investigated. [PERSON] et al. [20] considered the torsional mode of the brake disc in their 2-DoF model. Later, based on the three models above, [PERSON] [21] put forward a 2-DoF model that further considers vibration of the disc as a rotating elastic plate, so as to associate easily with real disk brake [22]. However, the aforementioned literatures have not given detail descriptions about parameters determination method and its process of the model, and the stability of the model depends greatly on the value of parameters. Since brake components are geometrically complex and are essentially continuous media with infinite DoFs, it is appropriate to use the finite element method with less model simplification and a large number of DoFs [17].