## 2 Theory
### Algorithm II for Identification of Bearings' Main and Cross-Coupled Coefficients  
Although the algorithm provided by Equations (8) and (11) is suitable to identify the main stiffness and damping coefficients of each bearing in a rolling-bearing rotor with m discs and n bearings, it cannot be used for estimating both the main and the cross-coupled coefficients of journal bearings. The reason is that the cross-coupled stiffness and damping coefficients of journal bearings are very big and cannot be ignored. Another set of equations is required in this case. However, the second equation set cannot be simply built just by adding more measuring unbalance responses because the slop \(U_{ub}/V_{ub}\) can be proved constant for a single-span and single-disc rotor when the rotating speed is maintained at a fixed speed [27]. Moreover, using double measuring points can make the measuring system complex and costly. Although the values of the stiffness and damping coefficients change with the rotating speed, the changing relationship is extremely gradual. Hence, the bearing coefficients can be considered unchangeable when the rotating speed is slightly modified. By slightly changing the rotating speed from \(w\) to \(w^{\prime}\), the second set of Equations (12) and (13) are obtained according to Equations (7) and (10). \[\left\{ \begin{array}{c}k_{1\text{-}yx}+k_{1\text{-}yy}\cdot\frac{U_{1b}{}^{ \prime}}{V_{1b}}=\frac{-H_{2}{}^{\prime}(m+1,1)}{L\cdot V_{1b}{}^{\prime}}\\ \vdots\\ k_{ns\text{-}yx}+k_{ns\text{-}yy}\cdot\frac{U_{1b}{}^{\prime}}{V_{nb}}=\frac{- H_{2}{}^{\prime}(n+1,1)}{L\cdot V_{nb}{}^{\prime}}\end{array} \right. \tag{12}\]  
\[\left\{ \begin{array}{c}k_{1\text{-}xy}+k_{1\text{-}xx}\cdot\frac{V_{1b}{}^{ \prime}}{U_{1b}{}^{\prime}}=\frac{H_{4}{}^{\prime}(m+1,1)}{-L\cdot U_{1b}{}^{ \prime}}\\ \vdots\\ k_{ns\text{-}xy}+k_{ns\text{-}xx}\cdot\frac{V_{1b}{}^{\prime}}{U_{1b}{}^{ \prime}}=\frac{H_{4}{}^{\prime}(n,1)}{-L\cdot U_{nb}{}^{\prime}}\end{array} \right. \tag{13}\]  
where \(U_{1b}{}^{\prime}\), \(U_{2b}{}^{\prime}\),..., \(U_{nb}{}^{\prime}\) represent the dimensionless unbalance responses of each bearing in the frequency domain in the y direction when the rotating frequency is changed to \(\omega^{\prime}\); \(V_{1b}{}^{\prime}\), \(V_{2b}{}^{\prime}\),..., \(V_{nb}{}^{\prime}\) represent the dimensionless unbalance responses of each bearing in the frequency domain in the x direction when the rotating frequency is changed to \(\omega^{\prime}\). \(H_{2}{}^{\prime}\) and \(H_{4}{}^{\prime}\) can also be calculated based on CRDAM using the m + n measuring unbalance responses when the rotating frequency is changed to \(\omega^{\prime}\). By expanding the complex coefficients in Equations (12) and (13) to form the imaginary part and real part, the following can be obtained  
\[\left\{ \begin{array}{c}k_{1\text{-}yx}+i\cdot w\cdot c_{1\text{-}yx}+\left( k_{1\text{-}yy}+i\cdot w\cdot c_{1\text{-}yy}\right)\cdot\frac{U_{1b}}{V_{1b}}= \frac{-H_{2}(m+1,1)}{L\cdot V_{1b}}\\ k_{1\text{-}yx}+i\cdot w^{\prime}\cdot c_{1\text{-}yx}+\left(k_{1\text{-}yy}+i \cdot w^{\prime}\cdot c_{1\text{-}yy}\right)\cdot\frac{U_{1b}}{V_{1b}}=\frac{ -H_{2}^{\prime}(m+1,1)}{L\cdot V_{1b}}\\ \vdots\\ k_{n\text{-}yx}+i\cdot w^{\prime}\cdot c_{n\text{-}yx}+\left(k_{n\text{-}yy}+ i\cdot w^{\prime}\cdot c_{n\text{-}yy}\right)\cdot\frac{U_{nb}}{V_{nb}}=\frac{-H_{2} (m+1,1)}{L\cdot V_{nb}{}^{\prime}}\end{array} \right. \tag{14}\]  
\[\left\{\begin{array}{c}k_{1\text{-}xy}+i\cdot w\cdot c_{1\text{-}xy}+\left( k_{1\text{-}xx}+i\cdot w\cdot c_{1\text{-}xx}\right)\cdot\frac{V_{1b}}{U_{1b}}= \frac{H_{4}(m+1,1)}{-L\cdot U_{1b}}\\ k_{1\text{-}xy}+i\cdot w^{\prime}\cdot c_{1\text{-}xy}+\left(k_{1\text{-}xx}+ i\cdot w^{\prime}\cdot c_{1\text{-}xx}\right)\cdot\frac{V_{1b}{}^{\prime}}{U_{1b}}= \frac{H_{4}(m+1,1)}{-L\cdot U_{1b}{}^{\prime}}\\ \vdots\\ \left\{