## 2 Stochastic Complex Eigenvalue Analysis of Brake System  
### Finite Element Model of the Brake System  
The finite element model of the simplified brake system used in this study is illustrated in Figure 1. This model is taken from [15] and the parameters are listed in Table 1.  
Figure 1: The finite element model of the simplified brake system. To discretize the given geometry, low order hexahedral elements are applied, and a total of 10,692 nodes and 6820 elements are used. The total DOF of the given model is 32,076. When applying the boundary conditions, the disc's inner surface and in-plane of the pad are fixed. In implementing the contact algorithm at the interface, the mortar method [18] with element-based integration was adopted. Let us define the pad and disc interface as a slave (nonmortar) and master (mortar) segment. At each integration point of the slave segment, orthogonal projection is applied to find the master segment and its natural coordinate. A graphical interpretation is presented in Figure 2, this relationship may be formulated with the following nonlinear equations. \[ \begin{array}{l}\mathbf{x}_{c,\xi}^{1}(\xi_{c},\eta_{c})\cdot\left(\mathbf{x }_{c}^{2}-\mathbf{x}_{c}^{1}(\xi_{c},\eta_{c})\right)=0\\ \mathbf{x}_{c,\eta}^{1}(\xi_{c},\eta_{c})\cdot\left(\mathbf{x}_{c}^{2}-\mathbf{ x}_{c}^{1}(\xi_{c},\eta_{c})\right)=0\end{array}  \tag{1}\]  
where \(\mathbf{x}_{c}^{2},\ \mathbf{x}_{c}^{1}(\xi_{c},\eta_{c})\) are the contact point of the slave and master segment. Since Equation (1) is nonlinear, the iterative procedure based on [PERSON]'s method is applied. The associated linearization can be expressed as  
\[\left[ \begin{array}{c}\mathbf{x}_{c,\xi}^{1}\cdot\mathbf{x}_{c,\xi}^{1}- \mathbf{x}_{c,\xi\xi}^{1}\cdot\left(\mathbf{x}_{c}^{2}-\mathbf{x}_{c}^{1} \right)\ \ \ \mathbf{x}_{c,\xi}^{1}\cdot\mathbf{x}_{c,\eta}^{1}-\mathbf{x}_{c,\xi\eta}^{1} \cdot\left(\mathbf{x}_{c}^{2}-\mathbf{x}_{c}^{1}\right)\\ \mathbf{x}_{c,\xi}^{1}\cdot\mathbf{x}_{c,\eta}^{1}-\mathbf{x}_{c,\xi\eta}^{1} \cdot\left(\mathbf{x}_{c}^{2}-\mathbf{x}_{c}^{1}\right)\ \ \ \mathbf{x}_{c,\eta}^{1}-\mathbf{x}_{c,\eta\eta}^{1} \cdot\left(\mathbf{x}_{c}^{2}-\mathbf{x}_{c}^{1}\right)\end{array} \right] \left[ \begin{array}{c}\Delta\xi_{c}\\ \Delta\eta_{c}\end{array} \right]=\left[ \begin{array}{c}\mathbf{x}_{c,\xi}^{1} \cdot\left(\mathbf{x}_{c}^{2}-\mathbf{x}_{c}^{1}\right)\\ \mathbf{x}_{c,\eta}^{1}\cdot\left(\mathbf{x}_{c}^{2}-\mathbf{x}_{c}^{1} \right)\end{array} \right] \tag{2}\]  
where the row vectors \(\mathbf{x}_{c,\xi}^{1},\ \mathbf{x}_{c,\eta}^{1}\) are derivative of \(\mathbf{x}_{c}^{1}\) with respect to \(\xi,\ \eta,\) and \(\mathbf{x}_{c,\xi\xi}^{1},\ \mathbf{x}_{c,\xi\eta}^{1},\ \mathbf{x}_{c,\xi\eta}^{1},\) and \(\mathbf{x}_{c,\eta\eta}^{1}\) are second derivative of \(\mathbf{x}_{c}^{1}\). In determining the contact status, this paper assumes that all integration points at the interface are in contact and tangentially sliding. Utilizing the linear force relationship and Coulomb's friction law, the normal and tangential contact force are modeled as follows:  
\[ \begin{array}{l}F_{n}=k_{L}\left(u_{p}-u_{d}\right)\\ F_{t}=\mu F_{n}\end{array}  \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline
**Parameters** & **Disc** & **Pad** \\ \hline Young’s modulus \(E\) (N/m\({}^{2}\)) & \(125\times 10^{9}\) & \(2\times 10^{9}\) \\ Poisson ratio \(\
u\) & 0.3 & 0.1 \\ Density \(\rho\) (kg/m\({}^{3}\)) & 7200 & 2500 \\ Inner radius (m) & 0.034 & 0.091 \\ Outer radius (m) & 0.151 & 0.147 \\ Thickness (m) & 0.019 & 0.0128 \\ \hline \end{tabular}
\end{table} 
Table 1: Model properties of the simplified brake system. Figure 2: Orthogonal projection between slave integration point and master segment. where \(F_{n},~{}~{}F_{t}\) are normal and tangential contact force, and \(k_{L},~{}~{}u_{p},~{}~{}u_{d}\), and \(\mu\) are normal contact stiffness, displacement of the pad and disc, and the friction coefficient, respectively. Throughout this paper, normal contact stiffness, i.e., \(k_{L}=10^{16}\), is used.