## 3 Eigenvalue and Forced Response Analysis  
### Model Order Reduction  
Directly employing the large order model in conducting nonlinear vibration analysis usually requires huge time in computation, as a large number of iteration steps are needed to reach the converged solutions. In order to solve this issue, model order reduction is conducted before further analysis. In this paper, the free-interface complex component mode synthesis method is employed. First, Equation (9) is converted into the first-order state space form  
\[\Big{(}\mathbf{A}+\mathbf{A}_{n}e^{-j\Omega t}+\mathbf{A}_{p}e^{j\Omega t} \Big{)}\dot{\mathbf{X}}+\Big{(}\mathbf{B}+\mathbf{B}_{n}e^{-j\Omega t}+ \mathbf{B}_{p}e^{j\Omega t}\Big{)}\mathbf{X}=\mathbf{F}_{ext}+\mathbf{F}_{crk}( \mathbf{X}) \tag{10}\]  
where \(\mathbf{A}\) and \(\mathbf{B}\) are referred to as the state mass and stiffness matrices, respectively;  
\[\mathbf{A}=\left[\begin{array}{cc}0&\mathbf{M}\\ \mathbf{M}&\Omega\mathbf{C}_{cor}+\mathbf{C}_{bc}\end{array}\right]\ \mathbf{B}=\left[\begin{array}{cc}-\mathbf{M}&0\\ 0&\mathbf{K}_{s}-\Omega^{2}\mathbf{K}_{d}+\mathbf{K}_{bc}\end{array}\right] \tag{11}\]  
\(\mathbf{A}_{n}\), \(\mathbf{A}_{p}\), \(\mathbf{B}_{n}\) and \(\mathbf{B}_{p}\) denotes the matrices regarding the anisotropic journal bearings;  
\[\mathbf{A}_{n}=\left[\begin{array}{cc}0&0\\ 0&\mathbf{C}_{bn}\end{array}\right]\ \mathbf{A}_{p}=\left[\begin{array}{cc}0&0\\ 0&\mathbf{C}_{bp}\end{array}\right]\ \mathbf{B}_{n}=\left[\begin{array}{cc}0&0\\ 0&\mathbf{K}_{bn}\end{array}\right]\ \mathbf{B}_{p}=\left[\begin{array}{cc}0&0\\ 0&\mathbf{K}_{bp}\end{array}\right] \tag{12}\]  
\(\mathbf{X}\) denotes the state displacement vector; \(\mathbf{F}_{ext}\) and \(\mathbf{F}_{crk}(\mathbf{X})\) are the state force vectors;  
\[\mathbf{X}=\left\{\begin{array}{c}\dot{\mathbf{u}}\\ \mathbf{u}\end{array}\right\}\!,\quad\mathbf{F}_{ext}=\left\{\begin{array}{c} 0\\ \mathbf{f}_{ext}\end{array}\right\}\!,\quad\mathbf{F}_{crk}(\mathbf{X})=\left\{ \begin{array}{c}0\\ \mathbf{f}_{crk}(\mathbf{u})\end{array}\right\} \tag{13}\]  
In order to simulate the breathing behaviors of the crack, the DOFs of the nodes on the crack surfaces are retained in the reduced order model. Besides, the DOFs of the nodes for applying external forces and connecting journal bearings are also retained to enhance the analysis accuracy. The other DOFs are represented by the generalized coordinates to reduce the order of the model. The coordinate transformation of the free-interface complex component mode synthesis method can be written as  
\[\mathbf{X}=\left[\begin{array}{cc}\mathbf{\Phi}_{d}&\mathbf{\Phi}_{n}\end{array} \right]\left\{\begin{array}{c}\mathbf{q}_{d}\\ \mathbf{q}_{n}\end{array}\right\}=\mathbf{\Phi}_{rel}\mathbf{q} \tag{14}\]  
where \(\mathbf{\Phi}_{d}\) is referred to as the attachment mode matrix; \(\mathbf{\Phi}_{n}\) denotes the truncated eigenvector matrix; \(\mathbf{q}_{d}\) and \(\mathbf{q}_{n}\) are the generalized coordinates. The attachment mode matrix \(\mathbf{\Phi}_{d}\) and the eigenvector matrix \(\mathbf{\Phi}_{n}\) are generated by the time-invariant terms of Equation (10).  
Then, the governing equations of the reduced order model can be expressed as  
\[\Big{(}\mathbf{\widetilde{A}}+\mathbf{\widetilde{A}}_{n}e^{-j\Omega t}+\mathbf{ \widetilde{A}}_{p}e^{j\Omega t}\Big{)}\dot{\mathbf{q}}+\Big{(}\mathbf{ \widetilde{B}}+\mathbf{\widetilde{B}}_{n}e^{-j\Omega t}+\mathbf{\widetilde{B} }_{p}e^{j\Omega t}\Big{)}\mathbf{q}=\mathbf{\widetilde{F}}_{ext}+\mathbf{ \widetilde{F}}_{crk}(\mathbf{q}) \tag{15}\]  
where  
\[\begin{array}{c}\mathbf{\widetilde{A  
where \(\mathbf{\Psi}_{rel}\) denotes the left coordinate transformation matrix, which is similar to \(\mathbf{\Phi}_{red}\) and is computed by the adjoint equation of the time-invariant terms of Equation (10).