## 2 Preamble
### Stability Analysis, Transient and Self-Excited Vibrations  
Two main strategies exist to predict squeal. The first one corresponds to the well-known Complex Eigenvalue Analysis (CEA) which is to determine the stability of the equilibrium point of the system. Even if this first step is essential to estimate the propensity of instability events in a design process, the stability analysis is often over- or under-predictive [21] due to the fact that all instabilities predicted by the CEA do not necessarily participate in the nonlinear self-sustaining dynamic response and moreover some frequencies of the dynamic response might not be identified by the CEA. This is the reason why a prediction of the vibratory regime and a self-sustained vibration analysis are generally performed as a second step. The prediction of instabilities is based on the CEA of the system linearized at the static sliding equilibrium point \(\mathbf{X}_{0}\). This equilibrium point is determined by solving the nonlinear static equation  
\[\left(\mathbf{K}+\mathbf{K}_{\mu}\right)\mathbf{X}_{0}=\mathbf{F_{n1}}\left( \mathbf{X}_{0}\right) \tag{7}\]  
Therefore the eigenvalue analysis of the characteristic linearized equation at the equilibrium point \(\mathbf{X}_{0}\) can be rewritten as follows:  
\[\det\left(\lambda^{2}\mathbf{M}+\lambda\mathbf{C}+\mathbf{K}+\mathbf{K}_{\mu} -\mathbf{J_{n1}}\left(\mathbf{X}_{0}\right)\right)=0 \tag{8}\]  
where \(\mathbf{J_{n1}}\left(\mathbf{X}_{0}\right)\) corresponds to the Jacobian matrix of nonlinear efforts \(\mathbf{F_{n1}}\) at the equilibrium point. \(\lambda\) define the complex eigenvalues of the system. If all eigenvalues have negative real parts, the system is stable. If at least one eigenvalue has positive real part, the system is unstable. The imaginary part of the associated positive eigenvalue defines the pulsation of the associated unstable mode. Figure 2 displays the evolution of real parts and frequencies as a function of the friction coefficient \(\mu\) and the associated stability analysis of the mechanical system in the complex plan. The curves highlight the classical mode coupling: as the friction coefficient increases, the imaginary parts increase or decrease slowly. For \(\mu=0.34\), the first Hopf bifurcation point occurs with the appearance of a coalescence pattern (i.e., coupling between two stable and unstable modes around 10 Hz). For \(0.34~{}<~{}\mu~{}<~{}0.48\), the system has one instability (i.e., one mode with a positive real part). For \(\mu>0.48\), a new Hopf bifurcation point occurs with the appearance of a second coalescence pattern around 6.6 Hz and the mechanical system has two instabilities. It can be noted that after each Hopf bifurcation point the frequencies of the two modes remain close and their associated real parts become opposite. For the reader comprehension, the evolution of the real part and frequency of each eigenvalue in regard to the friction coefficient is defined by a specific color: blue for the 1 st mode; yellow for the 2 nd mode; orange for the 3 rd mode; and purple for the 4 th mode. Due to the appearance of an unstable equilibrium point, nonlinear transient and steady-state self-excited friction-induced vibrations can be generated. For the sake of brevity, only the two cases that will be considered throughout the paper are shown and discussed. Figures 3 and 4 illustrate two classical transient and self-excited vibrations for different friction coefficients (i.e., \(\mu=0.4\) and \(\mu=0.7\), respectively) and the associated limit cycles amplitudes. It can be observed that for some cases a fast increase of the transient oscillations (i.e., the initial growing transient oscillations with the highest amplitudes) is followed by a slight decrease until the nonlinear periodic motions (see for example Figures 4 for \(X_{1}\)). Moreover a complex nonlinear behavior with quasi-periodic motion is observed for \(\mu=0.7\) (see for example the limit cycles \(\left(Y_{1},\dot{Y}_{1}\right)\) and \(\left(Y_{2},\dot{Y}_{2}\right)\) on Figure 4).