## 2 Theory
### Single Direction Algorithm  
Identification of rotor unbalance amplitude \(m_{jib}\cdot e_{j}\) and angle \(\alpha_{j}\) is the inverse problem of the CRDAM. According to Equations (1) and (2), the relationships between unbalance response and rotor unbalance and the bearing coefficients are non-linear transcendental, although the eight stiffness and damping coefficients of the bearing are linearized. As the number of bearings and discs are unknown, the numbers of equations and unknown variables are also unknown. Moreover, rotor unbalance and bearing coefficients are coupled together according to Equations (1) and (2). Although enough equations can be obtained by using the unbalance responses as inputs and considering rotor unbalances and bearings coefficients as unknown variables, they cannot be solved directly to obtain the unknown variables (amplitude and angle of rotor unbalance) due to the non-linear transcendental equations, the unfixed number of equations, and the unknown variables. Hence, it is proposed that Equation (1) is written in matrix form as follows:  
\[\frac{U(q)\cdot EI}{L^{2}}=\left[ \begin{array}{c}G_{\alpha}(q,q_{\alpha l}) \\ \vdots\\ G_{\alpha}(q,q_{\alpha l})\\ G_{\alpha}(q,q_{\alpha l})\\ \vdots\\ G_{\alpha}(q,q_{\alpha l})\\ \end{array} \right]^{T}\left[ \begin{array}{c}\pi\cdot w^{2}\cdot m_{1u}e_{1} \cdot(\sin\alpha_{1}-i\cdot\cos\alpha_{1})+L\cdot w^{2}\cdot m_{1d}U_{ld}\\ \vdots\\ \pi\cdot w^{2}\cdot m_{mn}e_{n}\cdot(\sin\alpha_{m}-i\cdot\cos\alpha_{m})+L \cdot w^{2}\cdot m_{md}U_{md}\\ -L\cdot k_{1 xy}\cdot V_{1b}-L\cdot k_{1 xy}\cdot U_{1b}\\ \vdots\\ -L\cdot k_{\text{ss}yx}\cdot V_{\text{sh}}-L\cdot k_{\text{ss}yy}\cdot U_{lab} \\ \end{array} \right]_{m+n} \tag{3}\]  
\(m+n\) equations can be obtained based on Equation (3) by using \(m+n\) unbalance responses as inputs. There are two unknown variables (amplitude and angle) for a disc's unbalance. Moreover, the eight coefficients of a bearing are also unknown. Hence, there are \(2\cdot m+8\cdot n\) unknown variables for a rotor with \(m\) discs and \(n\) bearings, while only \(m+n\) equations exist. The greatest difficulty is due to the fact that the relationship among these variables is non-linear transcendental. A direct solution is not feasible. Therefore, by writing the \(m+n\) equations in matrix form, Equation (4) can be obtained as the following:  
\[\frac{EI}{L^{2}}\left[ \begin{array}{c}U(q_{1s})\\ \vdots\\ U(q_{(m)+1s})\\ \vdots\\ U(q_{(m+n)s})\\ \end{array} \right]=H_{1}\cdot\left[ \begin{array}{c}\pi\cdot w^{2}\cdot m_{1u} e_{1}\cdot(\sin\alpha_{1}-i\cdot\cos\alpha_{1})+L\cdot w^{2}\cdot m_{1u}U_{ld}\\ \vdots\\ \pi\cdot w^{2}\cdot m_{mn}e_{m}\cdot(\sin\alpha_{m}-i\cdot\cos\alpha_{m})+L \cdot w^{2}\cdot m_{md}U_{md}\\ -L\cdot k_{1 xy}\cdot V_{1b}-L\cdot k_{1 xy}\cdot U_{1b}\\ \vdots\\ -L\cdot k_{\text{ss}yx}\cdot V_{\text{sh}}-L\cdot k_{\text{ss}yy}\cdot U_{lab} \\ \end{array} \right]_{m+n} \tag{4}\]  
where \(q_{1s},\ldots,q_{ms}\) and \(q_{(m+1)s},\ldots,q_{(m+n)s}\) are the dimensionless values of locations on the shaft excluding the 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; the number of \(U(q_{1s}),\ldots,\)\(U(q_{ms})\) is \(m\) and the number of \(U\Big{(}q_{(m+1)s}\Big{)},\ldots,\)\(U\Big{(}q_{(m+n)s}\Big{)}\) is \(n\); \(H_{1}\) is a  
Figure 1: A multi-disc and multi-span rotor-bearing system. \((m+n)\times(m+n)\) matrix and can be calculated using [PERSON]'s functions \(G_{u}(q,q_{i})\) as long as the locations of the measuring points, the bearings, and the discs are known; \(H_{1}\) is as follows:  
\[H_{1}=\left[