## 2 Theory  
### Algorithm I for Identification of Main Stiffness and Damping Coefficients of Rolling Bearings  
Aiming at a rotor with m discs and n bearings shown in Figure 1, the continuous rotor dynamic analysis method (CRDAM) is proposed in reference [25]. Based on CRDAM, the unbalance response can be expressed as the function of the position, rotor unbalances, the bearings' stiffness and damping coefficients. Accordingly, the inverse problem, which is identifying rotor unbalances, is solved in reference [26]. The matrix method is proposed to solve the problems that: the equations built based on CRDAM are non-linear transcendental, there are too many unknown variables in the equations, and the rotor unbalances and bearings' coefficients are coupling together. The following equations, which eliminate the coupling between the coefficients and the rotor unbalances, are proposed in reference [26]. \[\left[ \begin{array}{c}\pi\cdot w^{2}\cdot m_{1u}e_{1}\cdot(\sin\alpha_{1}-i \cdot\cos\alpha_{1})+L\cdot w^{2}\cdot m_{1d}U_{1d}\\ \vdots\\ \pi\cdot w^{2}\cdot m_{mu}e_{m}\cdot(\sin\alpha_{m}-i\cdot\cos\alpha_{m})+L \cdot w^{2}\cdot m_{md}U_{md}\\ -L\cdot k_{1s\cdot yx}\cdot V_{1b}-L\cdot k_{1s\cdot yy}\cdot U_{1b}\\ \vdots\\ -L\cdot k_{ns\cdot yx}\cdot V_{nb}-L\cdot k_{ns\cdot yy}\cdot U_{nb}\end{array}  \right]_{m+n}=\frac{EI}{L^{2}}\cdot{H_{1}}^{-1}\left[ \begin{array}{c}U(q_{1s}) \\ \vdots\\ U(q_{ms})\\ \vdots\\ U\big{(}q_{(m+1)s}\big{)}\\ \vdots\\ U\big{(}q_{(m+n)s}\big{)}\end{array} \right]=H_{2} \tag{1}\]  
\[\left[ \begin{array}{c}\pi\cdot w^{2}\cdot m_{u1}\cdot e_{1}\cdot(\cos\alpha_{ 1}+i\cdot\sin\alpha_{1})+w^{2}\cdot m_{1d}L\cdot V_{1d}\\ \vdots\\ \pi\cdot w^{2}\cdot m_{mu}\cdot e_{m}\cdot(\cos\alpha_{m}+i\cdot\sin\alpha_{m} )+w^{2}\cdot m_{md}L\cdot V_{md}\\ -L\cdot k_{1s\cdot xy}U_{1b}-L\cdot k_{1s\cdot xx}V_{1b}\\ \vdots\\ -L\cdot k_{ns\cdot xy}U_{nb}-L\cdot k_{ns\cdot xx}V_{nb}\end{array} \right]_{m+n}= \frac{EI}{L^{2}}\cdot{H_{3}}^{-1}\left[ \begin{array}{c}V(q_{1s})\\ \vdots\\ V(q_{ms})\\ \vdots\\ V(q_{ns})\end{array} \right]=H_{4} \tag{2}\]  
where \(w\) is the rotation frequency; \(L\) is the length of the shaft; \(E\) is the elastic modulus of the shaft; \(I\) is the diametric shaft cross-sectional geometric moment of inertia; \(m_{1u}\cdot m_{mu}\) are the eccentric masses of \(\#1\cdot\cdot\cdot m\) disc; \(e_{1}\cdot\cdot e_{m}\) are the eccentric distance of \(\#1\cdot\cdot\cdot m\) disc; \(\alpha_{1}\cdot\cdot\cdot m_{m}\) are the eccentric angles, which are defined as the angles between the x-axis and the disc's eccentric position in the rotation direction; \(m_{1d}\cdot\cdot m_{md}\) are the masses of \(\#1\cdot\cdot\cdot m\) disc; \(U_{1d}\cdot\cdot\cdot U_{md}\) represent the dimensionless unbalance response of each disc in the frequency domain in the y direction; \(V_{1d}\cdot\cdot\cdot V_{md}\) represent the dimensionless unbalance response of each disc in the frequency domain in the x direction; \(U_{1b}\cdot\cdot\cdot U_{nb}\) represent the dimensionless unbalance response of each bearing in the frequency domain in the y direction; \(V_{1b}\cdot\cdot\cdot V_{nb}\) represent the dimensionless unbalance response of each bearing in the frequency domain in the x direction.