## 2 Theory
### Algorithm I for Identification of Main Stiffness and Damping Coefficients of Rolling Bearings
\(q_{1s},\ldots,q_{ms}\) and \(q_{(m+1)s},\ldots,q_{(m+n)s}\) are their dimensionless values of locations on the shaft excluding locations of all discs and bearings; \(U(q_{1s}),\ldots,\)\(U(q_{ms})\) and \(U\big{(}q_{(m+1)s}\big{)},\ldots,\)\(U\big{(}q_{(m+n)s}\big{)}\) are the measured dimensionless unbalance responses in the frequency domain in the y direction of locations excluding all discs and bearings; \(V(q_{1s}),\ldots,\)\(V(q_{ms})\) and \(V\big{(}q_{(m+1)s}\big{)},\ldots,\)\(V\big{(}q_{(m+n)s}\big{)}\) are the measured dimensionless unbalance responses in the frequency domain in the y direction of locations excluding all discs and bearings; \(k_{1s\cdot xx},\)\(k_{1s\cdot yy},\ldots,\)\(k_{ns\cdot xx},\)\(k_{ns\cdot yy}\) are the main complex coefficients; \(k_{1s\cdot xy},\)\(k_{1s\cdot yx},\)\(\ldots,\)\(k_{ns\cdot xy},\)\(k_{ns\cdot yx}\) are the cross-coupled complex coefficients defined in Equation (3); \(H_{1}\) and \(H_{3}\) are the \((m+n)\times(m+n)\) matrices defined in Equations (4) and (5) and can be calculated using [PERSON]'s functions \(G_{u}(q,q_{i})\) and \(G_{v}(q,q_{i})\) as long as the locations of the measuring points, the bearings and the discs are known; \(q=z/L\); \(z\) is the axial position of the shaft; \(q_{i}\) is the dimensionless location of the bearings and discs. \[\left\{ \begin{array}{c}k_{1s\cdot xx}=k_{1\cdot xx}+i\cdot w\cdot c_{1\cdot xx }\\ k_{1s\cdot xy}=k_{1\cdot xy}+i\cdot w\cdot c_{1\cdot xy}\\ k_{1s\cdot yy}=k_{1\cdot yy}+i\cdot w\cdot c_{1\cdot yy}\\ k_{1s\cdot yx}=k_{1\cdot yx}+i\cdot w\cdot c_{1\cdot yx}\end{array} \right., \ \ \cdots,\ \ \left\{ \begin{array}{c}k_{ns\cdot xx}=k_{n\cdot xx}+i\cdot w\cdot c_{n \cdot xx}\\ k_{ns\cdot xy}=k_{n\cdot xy}+i\cdot w\cdot c_{n\cdot xy}\\ k_{ns\cdot yx}=k_{n\cdot yy}+i\cdot w\cdot c_{n\cdot yy}\\ k_{ns\cdot yx}=k_{n\cdot yx}+i\cdot w\cdot c_{n\cdot yx}\end{array} \right. \tag{3}\]\[H_{1}=\left[ \begin{array}{cccccc}G_{u}(q_{1s},q_{1d})&\cdots&G_{u}(q_{1s},q_{md})&G _{u}(q_{1s},q_{1b})&\cdots&G_{u}(q_{1s},q_{nb})\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ G_{u}(q_{ms},q_{1d})&\cdots&G_{u}(q_{ms},q_{md})&G_{u}(q_{ms},q_{nb})&\cdots&G_ {u}(q_{ms},q_{nb})\\ G_{u}\Big{(}q_{(m+1)s},q_{1d}\Big{)}&\cdots&G_{u}\Big{(}q_{(m+1)s},q_{md}\Big{)} &G_{u}\Big{(}q_{(m+1)s},q_{1b}\Big{)}&\cdots&G_{u}\Big{(}q_{(m+1)s},q_{nb}\Big{)} \\ \vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\ G_{u}\Big{(}q_{(m+n)s^{\prime}},q_{1d}\Big{)}&\cdots&G_{u}\Big{(}q_{(m+n)s^{ \prime}},q_{md}\Big{)}&G_{u}\Big{(}q_{(m+n)s^{\prime}},q_{1b}\Big{)}&\cdots&G _{u}\Big{(}q_{(m+n)s^{\prime}},q_{nb}\Big{)}\\ \end{array} \right]_{(m+n)\times(m+n)} \tag{4}\]  
\[H_{3}=\left[ \begin{array}{cccccc}G_{v}(q_{1s},q_{1d})&\cdots&G_{v}(q_{1s},q_ {md})&G_{v}(q_{1s},q_{1b})&\cdots&G_{v}(q_{1s},q_{nb})\\ G_{v}(q_{ms},q_{1d})&G_{v}(q_{ms},q_{md})&G_{v}(q_{ms},q_{1b})&G_{v}(q_{ms},q_ {nb})\\ G_{v}\Big{(}q_{(m+1)s},q_{1d}\Big{)}&G_{v}\Big{(}q_{(m+1)s},q_{md}\Big{)}&G_ {v}\Big{(}q_{(m+1)s},q_{1b}\Big{)}&G_{v}\Big{(}q_{(m+1)s},q_{nb}\Big{)}\\ G_{v}(q_{ns},q_{1d})&G_{v}(q_{ns},q_{md})&G_{v}(q_{ns},q_{nb})&G_{v}(q_{ns},q_ {nb})\\ \end{array} \right]_{(m+n)\times(m+n)} \tag{5}\]  
where \(q_{1d}=z_{1d}/L,\cdots\), \(q_{md}=z_{md}/L\), \(q_{1b}=z_{1b}/L,\cdots\), \(q_{nb}=z_{nb}/L\); \(z_{1d},\cdots\), \(z_{md}\) are the z coordinate positions of each disc; \(z_{1b},\cdots z_{nb}\) are the z coordinate positions of each bearing; \(G_{u}(q,q_{1d}),\cdots\), \(G_{u}(q,q_{md})\), \(G_{u}(q,q_{1b}),\cdots\), \(G_{u}(q,q_{nb})\), \(G_{v}(q,q_{1d}),\cdots\), \(G_{v}(q,q_{md})\), \(G_{v}(q,q_{1b}),\cdots\), \(G_{v}(q,q_{mb})\) are Green's coefficients, which can be calculated using Green's functions; Green's functions \(G_{u}(q,q_{i})\) and \(G_{v}(q,q_{i})\) can be found in reference [24]. According to Equation (1), Equation (6) can be obtained. \[\left\{ \begin{array}{c}-L\cdot k_{1s\cdot yx}\cdot V_{1b}-L\cdot k_{1s\cdot yy }\cdot U_{1b}=H_{2}(m+1,1)\\ \vdots\\ -L\cdot k_{ns\cdot yx}\cdot V_{nb}-L\cdot k_{ns\cdot yy}\cdot U_{nb}=H_{2}(m+ n,1)\\ \end{array} \right.