# Influence of Material-Dependent Damping on Brake Squeal in a Specific Disc Brake System
## 1 Introduction  
Eigenvalue analyses to predict the stability of physical systems exposed to nonconservative forces is a current topic in engineering applications. The eigenvalues of such systems exhibit behavior that depends on the amplitudes of the nonconservative forces acting on them. These dependencies lead to effects such as semisimple and non-semisimple 1:1 resonance of the system [1]. This phenomenon is also known as Hamiltonian-Hopf bifurcation [2] or Krein Kollision [3] and, in mechanical engineering terminology, known as a mode coupling, causing flutter-type instability in mechanical systems. This phenomenon is well-known in various mechanical applications where nonconservative forces play significant roles, for example, in rotary machinery with viscous damping in the shaft bearings [4], in aerodynamic systems exposed to wind flow [5], in hydrodynamic systems loaded with hydrodynamic forces [6], and in disc brake systems where friction forces are acting between a pad and a disc [7]. Nonconservative friction force into a brake system produces an undesirable phenomenon known as brake squeal. Brake squeal is a complex problem caused by the nonconservative nature of friction force, which desymmetrizes the coefficient matrixes of mechanical systems [8]. Several different mechanisms causing brake squeal have been studied in the past decades [9, 10], but the most significant contributor to brake squeal was identified to be mode coupling. The unstable behavior of mechanical systems can be caused by dissipation-induced instability, which was previously discussed in the work of [PERSON] et al. [11] in the caseof brake systems. Dissipation-induced instability of disc brakes was later discussed in [12] through finite element (FE) analyses on a complex brake system. The authors reported changes in coalescence patterns when two coalescing modes are differently damped by modification of the damping matrix. Damping values were, however, artificially chosen without evidence, so the question remains about the significance of this effect in real systems. In general, the destabilization effect was first observed by [PERSON] [13] when studying the stability of elastic rotors exposed to friction in bearings. Vibration energy in brake systems is mainly dissipated due to material damping in system components and friction-induced damping between a pad and a disc [14]. Regarding the squeal prediction procedures, the FE method and complex eigenvalue analysis (CEVA) are commonly used to predict unstable vibration modes of a system [15]. Several research studies reported that the simplification of damping in the form of Rayleigh damping can be acceptable for conservative systems, but in nonconservative circulatory systems, like disc brakes, stability is produced not only by the magnitude of damping but also by damping distribution over the structure. Since the friction material (FM) and the disc have damping that is an order of magnitude different [16; 17] and system damping is thus far from Raleigh, it is appropriate to expect dissipation-induced instability in such systems. [PERSON] et al. [18] studied the modification of damping properties of the FM by adding thermoplastic elastomers into a FM composition. The possibility of changing the damping properties of the FM appears to be a method to control dissipation-induced instability. In FE models, system stability depends not only on the magnitude of damping but also on the structure of the damping coefficient matrix. [PERSON] et al. [19] thoroughly demonstrated, on simplified examples, the influence of the damping matrix structure on disc brake and paper calender system stability. Material-dependent damping optimization of the full industrial FE model of disc brakes is well-documented in a book [20]. However, the model does not consist of a physically reasonable damping factor for the FM; thus, the effect of dissipation-induced instability due to different FMs and disc damping cannot be fully assessed from the study.