## 2 Stochastic Complex Eigenvalue Analysis of Brake System
### Stochastic Complex Eigenvalue Analysis under Uncertainties  
If some structure parts have parametric uncertainties, they can be modeled by introducing additional terms to the deterministic system. Let \(\Theta\), \(\mathcal{F}\), \(\mathcal{P}\) denotes the probability spaces, where \(\Theta\), \(\mathcal{F}\) and \(\mathcal{P}\) are the sample space, \(\sigma\)-algebra, and probability measure of \(M\) random variables \(\zeta=(\zeta_{1},\zeta_{2},\ldots\zeta_{M})\), respectively. If the uncertainty is modeled as a spatially-distributed field, the random field can be represented by spectral decomposition using a truncated Karhunen-Loeve (KL) expansion [20]. Based on the modeled random variable, each system matrix is represented as a combination of the deterministic and random matrix as follows:  
\[\mathbf{M}(\zeta)=\mathbf{M}_{0}+\sum_{i=1}^{n_{1}}\mathbf{M}_{i}(\zeta)\,\ \ \ \mathbf{C}(\zeta)\ =\mathbf{C}_{0}+\sum_{i=1}^{n_{2}}\mathbf{C}_{i}(\zeta)\,\ \ \ \mathbf{K}(\zeta)=\mathbf{K}_{0}+\sum_{i=1}^{n_{3}}\mathbf{K}_{i}(\zeta) \tag{7}\]  
where \(\mathbf{M}(\zeta)\), \(\mathbf{C}(\zeta)\), and \(\mathbf{K}(\zeta)\) are random mass, damping, and stiffness matrix. The subscripts zero and \(i\) represent the deterministic and \(i\)-th random matrix, and \(n_{1}\), \(n_{2}\), and \(n_{3}\) indicate the number of random variables in each system matrix, respectively. Utilizing the obtained random system matrices, the stochastic eigenvalue problem can be expressed as  
\[\left[\mathbf{K}(\zeta)+\lambda\ _{i}(\zeta)\mathbf{C}(\zeta)+\lambda\ _{i}^{2}(\zeta)\mathbf{M}(\zeta)\right]\mathbf{\Phi}_{i}(\zeta)\ \ =0 \tag{8}\]  
where \(\lambda\ _{i}(\zeta)\), \(\mathbf{\Phi}_{i}(\zeta)\) are the \(i\)-th random eigenvalue and eigenvector.  
Prior to describing the proposed method for the stochastic eigenvalue problem, the response due to the friction coefficient is computed. In the absence of damping, the first three unstable eigenvalue pairs within the friction coefficient interval \([0,1]\) are shown in Figure 4.  
It can be seen from the results that after the critical friction coefficient, the two real eigenvalues are separated (bifurcation), while the imaginary parts coalesce (mode-coupling). Near the critical point, the first derivative of eigenvalues may not be continuous and, after this point, the response patterns are changed. Also, this coalescence pattern can be changed by the effect of damping [3]. Let \(c_{k}=2\chi_{k}a_{k}\) be the structure damping associated with an unstable eigenvalue pair, where \(\omega_{k}\) is the natural frequency without friction and \(\chi_{k}\) is the modal damping ratio. If damping is equal on the two coupling modes  
Figure 4: Evolution of the first three unstable eigenvalue pairs according to the friction coefficient. (a) Real part; (b) Imaginary part.  
(i.e., \(c_{1}=c_{2}\)), it will lower the real part curves, whereas the coalescence point is still singular. However, if damping is nonequally distributed (i.e., \(c_{1}\
eq c_{2}\)), the coalescence is not perfect, and the imaginary parts will not be the same at a critical point. In this paper, only singular-coalescence point cases (e.g., undamped and equally damped) are considered.