## 2 Preamble
### Basic Results for the Stability Analysis and the Transient Nonlinear Dynamics  
The main aim of this section is to provide some basic results concerning the stability of the brake system under study, as well as the nonlinear self-sustaining dynamic response if the system is unstable.  
First of all the stability of the system can be determined by using the well-known Complex Eigenvalue Analysis (CEA). Stability is determined by considering the real part of eigenvalues for the characteristic linearized equation at the equilibrium point. If all eigenvalues have negative real parts, the system is stable. If at least one eigenvalue has a positive real part, the system is unstable. The imaginary part of the associated positive eigenvalue defines the angular frequency of the unstable mode. Table 2 summarizes numerical results concerning the stability analysis of the original or reduced FEM system versus the friction coefficient (all other parameters are considered fixed). For \(\mu<0.67\), the system is stable. For \(0.67<\mu<0.76\), the system has one instability, and for \(0.76<\mu<1\), the system has two instabilities with the appearance of a second coalescence pattern. It should be noted that the convergence study versus the stability analysis for the C&B and DMS reductions is out  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter** & **Unit** & **Disc** & **Pads** \\ \hline Thickness (Z direction) & m & 0.02 & 0.0128 \\ Outer radius & m & 0.151 & 0.147 \\ Inner radius & m & 0.034 & 0.091 \\ Young modulus E & Pa & \(1.25\times 10^{11}\) & \(2\times 10^{9}\) \\ Poisson’s ratio & & 0.3 & 0.1 \\ Density & kg\(\cdot\)m\({}^{-3}\) & 7200 & 2500 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Model parameters.  
Figure 1: Simplified brake system under study (**a**) Finite Element Model (**b**) Position of the selected point for the description of the nonlinear dynamic response.  
of the scope of the present paper. The interested reader is referred to the previous studies [20; 21] for more details.  
Then, Figure 2 illustrates the nonlinear self-sustaining dynamic response of the system and the spectrograms for \(\mu=0.8\) at the frictional interface by using the full original model. The position of the selected point is given in Figure 1. For t = [0; 0.2] s, a classical exponential increase in the solution from the perturbed equilibrium is shown. Then a transient nonlinear behavior appears (for t = [0.2; 0.5] s) with the predominant presence of the second unstable mode \(f_{2}\) and its harmonic \(2f_{2}\). Finally a stabilized regime is obtained for t > 0.8 s with the appearance of the first unstable mode \(f_{1}\) and harmonic components \(2f_{1}\), \(3f_{1}\) and \(4f_{1}\). The presence of the harmonic combination \(-f_{1}+f_{2}\) for t = [0.4; 0.6] s is also observed, which illustrates the potential complex transient nonlinear behavior of the brake system under study. As a reminder, the appearance of such nonlinear components illustrates the limitations of the stability analysis that may lead to an underestimation of the frequency components during squeal events.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Friction Coefficient** & **Number of Instabilities** & **Frequency of Unstable Mode(s)** \\ \hline \(\mu<0.67\) & 0 & - \\ \(0.67<\mu<0.76\) & 1 & \(f_{1}=857\) Hz \\ \(0.76<\mu<1\) & 2 & \(f_{1}=857\) Hz and \(f_{2}=2223\) Hz \\ \hline \hline \end{tabular}
\end{table}
Table 2: Stability analysis of the system versus the friction coefficient.  
Figure 2: _Cont._