## 2 Theory  
### Revisiting the CRDAM  
There are \(m\) discs and \(n\) bearings in the rotor shown in Figure 1. The unbalance response of any position on the rotor shaft can be expressed as a function of position, rotor unbalance, and bearing stiffness and damping coefficients, according to the CRDAM. Equations (1) and (2) represent the dimensionless form of the unbalance response in the frequency domain.  
\[\begin{array}{l}U(q)=\left\{\begin{array}{c}\sum\limits_{j=1}^{m}\left[ \pi\cdot m_{j\mu}\cdot w^{2}\cdot e_{j}\cdot\left(\sin\alpha_{j}-j\cdot\cos \alpha_{j}\right)+w^{2}\cdot m_{j\mu}L\cdot U_{j\mu}\right]\cdot G_{u}\left(q, q_{j\mu}\right)\\ -\sum\limits_{j=1}^{n}L\left(k_{jj\cdot yx}\cdot V_{jj\mu}+k_{jj\cdot yy} \cdot U_{jj\mu}+i\cdot w\cdot c_{jj\cdot yx}\cdot V_{jj\mu}+i\cdot w\cdot c_{ jj\cdot yy}\cdot U_{jj\mu}\right)G_{u}\left(q,q_{jj\mu}\right)\right\}\frac{l^{2}}{E-T} \\ \\ V(q)=\left\{\begin{array}{c}\sum\limits_{j=1}^{m}\left[\pi\cdot m_{j\mu} \cdot w^{2}\cdot e_{j}\cdot\left(\cos\alpha_{j}+i\cdot\sin\alpha_{j}\right)+w ^{2}\cdot m_{j\mu}L\cdot V_{j\mu}\right]G_{v}\left(q,q_{j\mu}\right)\\ -\sum\limits_{j=1}^{n}L\left(k_{jj\cdot xx}\cdot V_{jj\mu}+k_{jj\cdot xy} \cdot U_{jj\mu}+i\cdot w\cdot c_{jj\cdot xx}\cdot V_{jj\mu}+i\cdot w\cdot c _{jj\cdot xy}\cdot U_{jj\mu}\right)G_{v}\left(q,q_{jj\mu}\right)\right\}\frac{l^ {2}}{E-T}\end{array} \tag{2}\]  
where \(U(q)\) and \(V(q)\) are the dimensionless unbalance response in the frequency domain along the \(y\)-axis and \(x\)-axis, respectively; \(m_{j\mu}\) are the eccentric masses of \(\#j\) disc; \(e_{j}\) is the eccentric distance of \(\#j\) disc; \(\alpha_{j}\) are the eccentric angles, which are defined as the angles between the \(x\)-axis and the eccentric position in the direction of rotation; \(m_{j\mu}\) are the masses of \(\#j\) disc; \(w\) is the rotation frequency; \(L\) is the length of the shaft; \(E\) is the elastic modulus of the shaft; \(l\) is the diametric shaft cross-sectional geometric moment of inertia; \(k_{jj\cdot xx}\), \(k_{jj\cdot xy}\), \(k_{jj\cdot yx}\), and \(k_{jj\cdot yy}\) are the stiffness coefficients of \(\#jj\) bearings respectively; \(c_{jj\cdot xx}\), \(c_{jj\cdot xy}\), \(c_{jj\cdot yx}\), and \(c_{jj\cdot yy}\) are the damping coefficients of \(\#jj\) bearing, respectively; \(z\) is the axial position of the shaft; \(z_{j\mu}\) are the z coordinate positions of each disc; \(z_{j\mu}\) is the \(z\) coordinate position of each bearing; \(q_{j\mu}\), and \(q_{jj\mu}\) are their dimensionless values; \(q=z/L\), \(q_{j\mu}=z_{j\mu}/L\), \(q_{jj\mu}=z_{jj\mu}/L\); \(G_{u}\left(q,q_{j\mu}\right)\), \(G_{u}\left(q,q_{j\mu}\right)\), \(G_{v}\left(q,q_{j\mu}\right)\), and \(G_{v}\left(q,q_{j\mu}\right)\) are [PERSON]'s coefficients, which can be calculated using [PERSON]'s functions \(G_{u}(q,q_{i})\) and \(G_{v}(q,q_{i})\); and [PERSON]'s functions \(G_{u}(q,q_{i})\) and \(G_{v}(q,q_{i})\) can be found in reference [30]. Further, \(U_{j\mu}\) represents the dimensionless unbalance response of each disc in the frequency domain in the \(y\) direction; \(V_{j\mu}\) represents the dimensionless unbalance response of each disc in the frequency domain in the \(x\) direction; \(U_{jib}\) represents the dimensionless unbalance response of each bearing in the frequency domain in the \(y\) direction; \(V_{jib}\) represents the dimensionless unbalance response of each bearing in the frequency domain in the \(x\) direction.