## 3 Eigenvalue and Forced Response Analysis
### Eigenvalue Analysis  
Based on the obtained reduced order model, the eigenvalues and the force dynamic response of the cracked rotor-bearing system can be computed efficiently. The eigenvalues can be used to generate the Campbell diagram, which is an important tool in designing rotating machinery. Besides, the real parts of the eigenvalues can reflect the stability of the system. Different from the analysis of conventional structures, the governing equation of the anisotropic cracked rotor-bearing system contains periodically time-variant terms. With respect to this issue, the [PERSON]'s method is employed to obtain the eigenvalue equation of Equation (15). First, the generalized coordinates \(\mathbf{q}\) are expressed as the product of a periodical function \(\mathbf{\Phi}(t)\) and an exponential function  
\[\mathbf{q}=\mathbf{\Phi}(t)e^{\lambda t},\quad\mathbf{\Phi}(t)=\sum_{h=- \infty}^{+\infty}\overline{\phi}_{h}e^{j2h\Omega t} \tag{17}\]  
where \(\lambda\) represents the eigenvalue variable; \(\overline{\phi}_{h}\) denotes the mode shape of the \(h\)th harmonic.  
By ignoring the force vectors at the right part of Equation (15) and substituting Equation (17) into Equation (15), the eigenvalue equation can be expressed as  
\[\lambda\Big{(}\mathbf{\overline{A}}+\mathbf{\overline{A}}_{nt}e^{-j2\Omega t }+\mathbf{\overline{A}}_{p}e^{j2\Omega t}\Big{)}\!\sum_{h=-\infty}^{+\infty} \overline{\phi}_{h}e^{j2h\Omega t}+\sum_{h=-\infty}^{+\infty}\Big{[}\mathbf{ \overline{B}}_{h}+\mathbf{\overline{B}}_{n,h}e^{-j2\Omega t}+\mathbf{ \overline{B}}_{p,h}e^{j2\Omega t}\Big{]}\!\overline{\phi}_{h}e^{j2h\Omega t}=0 \tag{18}\]  
where  
\[\begin{array}{rl}\mathbf{\overline{B}}_{h}&=j2h\Omega\mathbf{\overline{A}}+ \mathbf{\overline{B}}\\ \mathbf{\overline{B}}_{n,h}&=j2h\Omega\mathbf{\overline{A}}_{n}+\mathbf{ \overline{B}}_{n}\\ \mathbf{\overline{B}}_{p,h}&=j2h\Omega\mathbf{\overline{A}}_{p}+\mathbf{ \overline{B}}_{p}\end{array} \tag{19}\]  
Equation (18) consists of a series of algebraic equations regarding the different harmonics. By balancing the terms in Equation (18), the equation for the \(h\)th harmonic can be expressed as  
\[\lambda\Big{(}\mathbf{\overline{A}}\overline{\phi}_{h}+\mathbf{\overline{A}}_ {nt}\overline{\phi}_{h+1}+\mathbf{\overline{A}}_{p}\overline{\phi}_{h-1}\Big{)} +\Big{(}\mathbf{\overline{B}}_{h}\overline{\phi}_{h}+\mathbf{\overline{B}}_{n, h+1}\overline{\phi}_{h+1}+\mathbf{\overline{B}}_{p,h-1}\overline{\phi}_{h-1} \Big{)}=0 \tag{20}\]  
By assembling the equations of each harmonic, the overall eigenvalue equation can be written as  
\[\Big{(}\lambda\mathbf{\hat{A}}+\mathbf{\hat{B}}\Big{)}\mathbf{\hat{\Phi}}=0 \tag{21}\]where  
\[\hat{\mathbf{A}}=\left[\begin{array}{cccccccc}\ddots&\vdots&\vdots&\vdots& \vdots&\vdots&\vdots&\vdots&\vdots\\ \cdots&\overline{\mathbf{A}}&\overline{\mathbf{A}}_{n}&0&0&0&\cdots\\ \cdots&\overline{\mathbf{A}}_{p}&\overline{\mathbf{A}}&\overline{\mathbf{A}}_ {n}&0&0&\cdots\\ \cdots&0&\overline{\mathbf{A}}_{p}&\overline{\mathbf{A}}&\overline{\mathbf{A}}_ {n}&0&\cdots\\ \cdots&0&0&\overline{\mathbf{A}}_{p}&\overline{\mathbf{A}}&\overline{\mathbf{A} }_{n}&0&\cdots\\ \cdots&0&0&\overline{\mathbf{A}}_{p}&\overline{\mathbf{A}}&\overline{\mathbf{A }}_{n}&\cdots\\ \cdots&0&0&0&\overline{\mathbf{A}}_{p}&\overline{\mathbf{A}}&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right] \tag{22}\]  
In order to solve the infinitely dimensional eigenvalue equation, necessary truncation to the equation is needed. In real applications, a satisfactory approximation for the eigenvalues can be reached by retaining only several harmonics of low orders. After the truncation, Equation (21) becomes finitely dimensional and the eigenvalues can be obtained. As the nonlinear breathing effects of crack are not considered in the eigenvalue equation, the obtained eigenvalues are actually that of the anisotropic rotor-bearing system with an open crack. Nevertheless, they can also be used to reflect the effects of the anisotropy in journal bearings on the system's natural vibration.