## 2 Preamble  
### Mechanical System under Study  
In order to analyze multi-instabilities and nonlinear friction-induced vibrations, a phenomenological lumped self-excited model was originally proposed by [PERSON] and [PERSON] [16] and then used to test new numerical strategies to predict brake squeal by [PERSON] et al. [17; 18]. This four-degrees-of-freedom model (see Figure 1) corresponds to an extension of the two-degrees-of-freedom model proposed by [PERSON] [19; 20]. As a reminder, this model is classically used to analyze the problems of self-exciting mechanism with constant friction related to drum brake squeal instabilities. The proposed model consists of two masses \(m_{1}\) and \(m_{2}\) connected to moving belts by means of four dampers and four plates supported by four different elastic springs (whose two have a cubic nonlinearity). For the sake of simplicity, the following assumptions are adopted in the present study:  
* The bands and the masses are supposed to be permanently in contact because of a preload applied to the system;
* The friction forces are determined with a classical Coulomb's law and the friction coefficient at the different contacts is supposed to be constant and equal to \(\mu\). The tangential friction forces \(F_{t}\) are related to the normal forces \(F_{n}\) by the following formula: \(F_{t}=\mu F_{n}\);
* The stick-slip phenomenon is not taken into consideration;
* The belt runs at a constant speed and the relative velocity between the bands and the masses are positive so that the tangential friction forces \(F_{t}\) do not change.  
Thus, the dynamic equation of the system is:  
\[\mathbf{M}\ddot{\mathbf{X}}+\mathbf{C}\dot{\mathbf{X}}+\left(\mathbf{K}+ \mathbf{K}_{\mu}\right)\mathbf{X}=\mathbf{F}_{\mathbf{nl}}\left(\mathbf{X}\right) \tag{1}\]where \(\mathbf{X}=\left[X_{1}\ Y_{1}\ X_{2}\ Y_{2}\right]^{\mathrm{T}}\) defines the vector of the four degrees-of-freedom of the mechanical system under study. \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) correspond to the mass, damping and stiffness matrices given by :  
\[\mathbf{M}=\begin{bmatrix}m_{1}&0&0&0\\ 0&m_{1}&0&0\\ 0&0&m_{2}&0\\ 0&0&0&m_{2}\end{bmatrix} \tag{2}\]  
\[\mathbf{C}=\begin{bmatrix}c_{X_{1}}+c_{a}&0&-c_{a}&0\\ 0&c_{Y_{1}}&0&0\\ -c_{a}&0&c_{X_{2}}+c_{a}&0\\ 0&0&0&c_{Y_{2}}\end{bmatrix} \tag{3}\]  
\[\mathbf{K}=\begin{bmatrix}k_{X_{1}}+k_{a}&0&-k_{a}&0\\ 0&k_{Y_{1}}&0&0\\ -k_{a}&0&k_{X_{2}}+k_{a}&0\\ 0&0&0&k_{Y_{2}}\end{bmatrix} \tag{4}\]  
The vector of the nonlinear efforts \(\mathbf{F_{nl}}\left(\mathbf{X}\right)\) and the stiffness contribution \(\mathbf{K}_{\mu}\) due to friction are defined as follows:  
\[\mathbf{K}_{\mu}=\begin{bmatrix}0&\mu_{2}\,k_{Y_{1}}&0&0\\ -\mu_{1}\,k_{X_{1}}&0&0&0\\ 0&0&0&\mu_{2}\,k_{Y_{2}}\\ 0&0&-\mu_{3}\,k_{X_{2}}&0\end{bmatrix} \tag{5}\]  
\[\mathbf{F_{nl}}\left(\mathbf{X}\right)=\begin{bmatrix}-k_{X_{1}}^{NL}\,X_{1}^ {3}&\mu_{1}\,k_{X_{1}}^{NL}\,X_{1}^{3}&-k_{X_{2}}^{NL}\,X_{2}^{3}&\mu_{3}\,k_ {X_{2}}^{NL}\,X_{2}^{3}\end{bmatrix}^{T} \tag{6}\]  
For the rest of the study, the following sets of parameters are used [16]: \(m_{1}=m_{2}=1\) kg; \(c_{X_{1}}=c_{X_{2}}=c_{Y_{1}}=c_{Y_{2}}=c_{a}=1\)Ns/m; \(k_{X_{1}}=6000\)N/m; \(k_{X_{2}}=k_{Y_{1}}=3000\)N/m; \(k_{Y_{2}}=1000\)N/m; \(k_{a}=100\)N/m; \(k_{X_{1}}^{NL}=k_{X_{2}}^{NL}=10^{6}\)N/m3. Moreover the three coefficients of friction are assumed to be equal (i.e., \(\mu_{1}=\mu_{2}=\mu_{3}\)). The values of all of these parameters have been chosen:  
* To allow realistic results for the physical quantities observed (instability frequencies, vibration amplitudes, velocities, etc.) and thus test the relevance of the control in a context close to the engineering problem in the real world;
* To validate the control for a mechanical system subjected to more or less complex nonlinear behaviors (with mono or multi-instabilities in the presence of friction).