## 2 Preamble
### Finite Element Model of the Brake System under Study
It should be noted that previous studies [4; 5] have been performed to validate the mesh refinement on the system under study;
* The four corners of the back plate of the upper pad (pad 1) are connected to a master node which is only allowed to move in the out of plane direction;
* The inner edge of the disc (i.e., cylindrical surface in the \(Z\) direction having a radius of 0.034 m) and the back plate of the lower pad (pad 2) are clamped;
* The hydraulic pressure (\(2\times 10^{5}\) Pa for the present study) is directly applied on the back plate of the upper pad (pad 1);
* The nonlinearities at the friction interface between the pads and the disc are due to the cubic nonlinear terms with the the possibility of contact/no-contact states, such as \[F^{i}_{contact,disc}=\left\{\begin{array}{ll}k_{L}\delta_{i}+k_{NL}\delta_{ i}^{3}&\text{ \ \ \ if \ \ \ \ }\delta_{i}\geq 0\\ 0&\text{\ \ \ otherwise  
where \(F_{contact,disc}^{i}\) defines the normal reaction at the \(i^{\text{th}}\) couple of matching nodes on any interface and \(\delta_{i}\) is the relative normal displacement between one pad and the disc. \(k_{L}\) and \(k_{NL}\) are the linear and nonlinear stiffnesses at the friction interfaces (with \(k_{L}=10^{3}\text{N}\cdot\text{m}^{-1}\) and \(k_{NL}=10^{12}\text{N}\cdot\text{m}^{-3}\)). It should be noted that this formulation has been validated by experimental compression tests [23; 24];
* The normal reaction forces on each pad is classically modeled by \(F_{contact,pad}^{i}=-F_{contact,disc}^{i}\);
* The friction forces are determined with a classical Coulomb's law and the friction coefficient at the different contacts between the two pads and the disc masses is supposed to be constant and equal to \(\mu\). It should be noted that the same coefficient of friction is used for the two frictional interfaces for the sake of simplicity;
* The tangential friction forces are related to the normal reaction forces by \(\mathbf{F}_{friction,disc}^{i}=-\mathbf{F}_{friction,pad}^{i}=\mu\|F_{ contact,disc}^{i}\|\overrightarrow{t_{i}}\) where \(\overrightarrow{t_{i}}\) defines the local orthoradial in-plane direction (i.e., \(\overrightarrow{t_{i}}=\overrightarrow{r_{i}}\times\overrightarrow{z}\));
* the structured mesh uses hexahedrons with linear interpolation. The spatial discretization in the Z direction (i.e., thickness of the disc and pads) is of four nodes for the disc and three nodes for each pad. The spatial discretization on the circumference of the disc is of 150 nodes. The full FEM model has a total of around 50,000 degrees of freedom.  
Additionally, two reduction techniques (i.e., the Craig and Bampton technique (C&B) [25] and Double Modal Synthesis (DMS)) are used on the previous FEM system to condense the number of degrees of freedom. It is recalled that the C&B technique allows us to condense each substructure (i.e., the two pads and the disc), whereas the DMS technique condenses the remaining degrees of freedom at the two frictional interfaces. The interested reader is referred to the following previous studies for more details [20; 21; 22].  
Finally the equations of motion for the reduced brake system are given by  
\[\mathbf{M}\ddot{\mathbf{X}}+\mathbf{C}\dot{\mathbf{X}}+\mathbf{K}\mathbf{X}= \mathbf{F}_{NL}\left(\mathbf{X}\right)+\mathbf{F}_{ext} \tag{2}\]  
where \(\mathbf{X}\) is the generalized displacements vector while the dot denotes derivative with respect to time. \(\mathbf{F}_{NL}\) and \(\mathbf{F}_{ext}\) correspond to the global nonlinear forces at the two interfaces (i.e., \(\mathbf{F}_{NL}=\mathbf{F}_{contact}+\mathbf{F}_{friction}\)) and the hydraulic pressure, respectively. \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) are reduced mass, damping and stiffness matrices, respectively.