## 2 Theory
### Algorithm II for Identification of Bearings' Main and Cross-Coupled Coefficients
\begin{array}{c}k_{n\text{-}xy}+i\cdot w\cdot c_{n\text{-}xy}+\left( k_{n\text{-}xx}+i\cdot w\cdot c_{n\text{-}xx}\right)\cdot\frac{V_{nb}}{U_{nb}}= \frac{H_{4}(m,1)}{-L\cdot U_{nb}}\\ k_{n\text{-}xy}+i\cdot w^{\prime}\cdot c_{n\text{-}xy}+\left(k_{n\text{-}xx}+i \cdot w^{\prime}\cdot c_{n\text{-}xx}\right)\cdot\frac{V_{nb}{}^{\prime}}{U_{ nb}}=\frac{H_{4}(m,1)}{-L\cdot U_{nb}{}^{\prime}}\end{array} \right. \tag{15}\]  
Therefore, according to Equation (14), Equation (16) is obtained. It can be used to calculate the stiffness and damping coefficients of each bearing in the y direction. According to Equation (15), Equation (17), which can be used for estimating the coefficients of each bearing in the x direction, is obtained. According to Equations (16) and (17), there should be m + n unbalance responses in both x and y directions to obtain the matrices \(H_{2}\), \(H_{2}{}^{\prime}\), \(H_{4}\) and \(H_{4}{}^{\prime}\). If a bearing's coefficients should be identified, its unbalance responses in both x and y directions are required. Hence, there should be m + n + 1 measured unbalance responses in the two orthogonal directions. To identify all the bearings' coefficients, the unbalance responses of all bearings must be measured. \[\left[ \begin{array}{c}k_{1:yy}\\ k_{1:yx}\\ c_{1:yy}\\ c_{1:yx}\end{array} \right]=\left[ \begin{array}{cccc}Re\Big{(}\frac{U_{\rm II}}{V_{ \rm II}}\Big{)}&1&w\cdot Im\Big{(}\frac{U_{\rm II}}{V_{\rm II}}\Big{)}&0\\ Im\Big{(}\frac{U_{\rm II}}{V_{\rm II}}\Big{)}&0&w\cdot Re\Big{(}\frac{U_{\rm II }}{V_{\rm II}}\Big{)}&w\\ Re\Big{(}\frac{U_{\rm II}}{V_{\rm II}}\Big{)}&1&w^{\prime}\cdot Im\Big{(}\frac{U_ {\rm II}}{V_{\rm II}}\Big{)}&0\\ Im\Big{(}\frac{U_{\rm II}}{V_{\rm II}}\Big{)}&0&w^{\prime}\cdot Re\Big{(}\frac{U_ {\rm II}}{V_{\rm II}}\Big{)}&w^{\prime}\end{array} \right]^{-1}\cdot\left[ \begin{array}{c}Re\Big{(}\frac{-H_{2}(m+1,1)}{L\cdot V_{\rm II}}\Big{)}\\ Im\Big{(}\frac{-H_{2}(m+1,1)}{L\cdot V_{\rm II}}\Big{)}\\ Re\Big{(}\frac{-H_{2}^{\prime}(m+1,1)}{L\cdot V_{\rm II}}\Big{)}\\ Im\Big{(}\frac{-H_{2}^{\prime}(m+1,1)}{L\cdot V_{\rm II}}\Big{)}\end{array} \right] \tag{16}\]  
\[\left[ \begin{array}{c}k_{n:yy}\\ k_{n:yx}\\ c_{n:yy}\\ c_{n:yy}\end{array} \right]=\left[ \begin{array}{cccc}Re\Big{(}\frac{U_{\rm II }}{V_{\rm II}}\Big{)}&1&w\cdot Im\Big{(}\frac{U_{\rm II}}{V_{\rm II}}\Big{)}&0 \\ Im\Big{(}\frac{U_{\rm II}}{V_{\rm II}}\Big{)}&0&w\cdot Re\Big{(}\frac{U_{\rm II }}{V_{\rm II}}\Big{)}&w\\ Re\Big{(}\frac{U_{\rm II}}{V_{\rm II}}\Big{)}&1&w^{\prime}\cdot Im\Big{(}\frac{U_ {\rm II}}{V_{\rm II}}\Big{)}&0\\ Im\Big{(}\frac{U_{\rm II}}{V_{\rm II}}\Big{)}&0&w^{\prime}\cdot Re\Big{(}\frac{U_ {\rm II}}{V_{\rm II}}\Big{)}&w^{\prime}\end{array} \right]^{-1}\cdot\left[ \begin{array}{c}Re\Big{(}\frac{-H_{2}(m,1)}{L\cdot V_{\rm II}}\Big{)}\\ Im\Big{(}\frac{-H_{2}(m,1)}{L\cdot V_{\rm II}}\Big{)}\\ Re\Big{(}\frac{-H_{2}^{\prime}(m,1)}{L\cdot V_{\rm II}}\Big{)}\\ Im\Big{(}\frac{-H_{2}^{\prime}(m,1)}{L\cdot V_{\rm II}}\Big{)}\end{array} \right]\]  
\[\left[ \begin{array}{c}k_{1:xx}\\ k_{1:xy}\\ c_{1:xx}\\ c_{1:xy}\end{array} \right]=\left[ \begin{array}{cccc}Re\Big{(}\frac{V_{\rm II }}{U_{\rm II}}\Big{)}&1&w\cdot Im\Big{(}\frac{V_{\rm II}}{U_{\rm II}}\Big{)}&0 \\ Im\Big{(}\frac{V_{\rm II}}{V_{\rm II}}\Big{)}&0&w\cdot Re\Big{(}\frac{V_{\rm II }}{V_{\rm II}}\Big{)}&w\\ Re\Big{(}\frac{V_{\rm II}}{V_{\rm II}}\Big{)}&1&w^{\prime}\cdot Im\Big{(}\frac{V_ {\rm II}}{V_{\rm II}}\Big{)}&0\\ Im\Big{(}\frac{V_{\rm II}}{V_{\rm II}}\Big{)}&0&w^{\prime}\cdot Re\Big{(}\frac{V_ {\rm II}}{V_{\rm II}}\Big{)}&w^{\prime}\end{array} \right]^{-1}\cdot\left[ \begin{array}{c}Re\Big{(}\frac{H_{4}(m+1,1)}{-L\cdot U_{\rm II}}\Big{)}\\ Im\Big{(}\frac{H_{4}(m+1,1)}{-L\cdot U_{\rm II}}\Big{)}\\ Re\Big{(}\frac{H_{4}^{\prime}(m+1,1)}{-L\cdot U_{\rm II}}\Big{)}\\ Im\Big{(}\frac{H_{4}^{\prime}(m+1,1)}{-L\cdot U_{\rm II}}\Big{)}\end{array} \right] \tag{17}\]  
where \(Re()\) is the real part of a complex number and \(Im()\) is the imaginary part of a complex number.