# Continuous Rotor Dynamics of Multi-Disc and Multi-Span
## 2 Theory
### Analytical Solution
\begin{array}{cccc}-\pi m_{1u}w^{2}e_{1}(\sin\alpha_{1}-i\cdot\cos \alpha_{1})G_{u}(q_{1d},q_{1d})-\cdots\cdots-\pi m_{mu}w^{2}e_{m}(\sin\alpha_{m }-i\cdot\cos\alpha_{m})G_{u}(q_{1d},q_{md})\\ \vdots\\ -\pi m_{1u}w^{2}e_{1}(\sin\alpha_{1}-i\cdot\cos\alpha_{1})G_{u}(q_{md},q_{1d})- \cdots\cdots-\pi m_{mu}w^{2}e_{m}(\sin\alpha_{m}-i\cdot\cos\alpha_{m})G_{u}(q _{md},q_{md})\\ -\pi m_{1u}w^{2}e_{1}(\cos\alpha_{1}+i\cdot\sin\alpha_{1})G_{v}(q_{1d},q_{1d})- \cdots\cdots-\pi m_{mu}w^{2}e_{m}(\cos\alpha_{m}+i\cdot\sin\alpha_{m})G_{v}(q _{1d},q_{md})\\ \vdots\\ -\pi m_{1u}w^{2}e_{1}(\cos\alpha_{1}+i\cdot\sin\alpha_{1})G_{v}(q_{md},q_{1d})- \cdots\cdots-\pi m_{mu}w^{2}e_{m}(\cos\alpha_{m}+i\cdot\sin\alpha_{m})G_{v}(q _{md},q_{md})\\ -\pi m_{1u}w^{2}e_{1}(\sin\alpha_{1}-i\cdot\cos\alpha_{1})G_{u}(q_{1b},q_{1d})- \cdots\cdots-\pi m_{mu}w^{2}e_{m}(\sin\alpha_{m}-i\cdot\cos\alpha_{m})G_{u}(q _{1b},q_{md})\\ -\pi m_{1u}w^{2}e_{1}(\sin\alpha_{1}-i\cdot\cos\alpha_{1})G_{u}(q_{2b},q_{1d})- \cdots\cdots-\pi m_{mu}w^{2}e_{m}(\sin\alpha_{m}-i\cdot\cos\alpha_{m})G_{u}(q _{2b},q_{md})\\ \vdots\\ -\pi m_{1u}w^{2}e_{1}(\sin\alpha_{1}-i\cdot\cos\alpha_{1})G_{u}(q_{nb},q_{1d})- \cdots\cdots-\pi m_{mu}w^{2}e_{m}(\sin\alpha_{m}-i\cdot\cos\alpha_{m})G_{u}(q _{nb},q_{md})\\ -\pi m_{1u}w^{2}e_{1}(\cos\alpha_{1}+i\cdot\sin\alpha_{1})G_{v}(q_{1b},q_{1d})- \cdots\cdots-\pi m_{mu}w^{2}e_{m}(\cos\alpha_{m}+i\cdot\sin\alpha_{m})G_{v}(q _{1b},q_{md})\\ -\pi m_{1u}w^{2}e_{1}(\cos\alpha_{1}+i\cdot\sin\alpha_{1})G_{v}(q_{2b},q_{1d})- \cdots\cdots-\pi m_{mu}w^{2}e_{m}(\cos\alpha_{m}+i\cdot\sin\alpha_{m})G_{v}(q _{2b},q_{md})\\ \vdots\\ -\pi m_{1u}w^{2}e_{1}(\cos\alpha_{1}+i\cdot\sin\alpha_{1})G_{v}(q_{nb},q_{1d})- \cdots\cdots-\pi m_{mu}w^{2}e_{m}(\cos\alpha_{m}+i\cdot\sin\alpha_{m})G_{v}(q _{nb},q_{md})\end{array} \right]\]\[H_{2}=\left[ \begin{array}{cccc}H_{11}&H_{12}&H_{13}&H_{14}\\ H_{21}&H_{22}&H_{23}&H_{24}\\ H_{31}&H_{32}&H_{33}&H_{34}\\ H_{41}&H_{42}&H_{43}&H_{44}\end{array} \right]_{2(m+n)\times 2(m+n)}-\left[ \begin{array}{cccc}\frac{L^{2}}{ET}&&&\\ &\ddots&&\\ &&\ddots&\\ &&&\frac{L^{2}}{ET}\end{array} \right]_{2(m+n)\times 2(m+n)}\]  
The sub-matrixes of \(H_{2}\) are in Equations (A3)-(A18) in Appendix A. According to Equation (21), \(U_{1d}\ldots U_{md}\), \(V_{1d}\ldots V_{md}\), \(U_{1b}\ldots U_{nb}\), \(V_{1b}\ldots V_{nb}\) can be expressed as follows:  
\[\left[ \begin{array}{cccc}U_{1d}\cdots U_{md}&V_{1d}\cdots V_{md}&U_{1b} \cdots U_{nb}&V_{1b}\cdots V_{nb}\end{array} \right]=H_{2}^{-1}\cdot H_{1} \tag{22}\]  
Hence, as shown in Figure 3, the dimensionless unbalance response in frequency domain can be calculated using Equations (17) and (18) in which the unknown Green's coefficients can be calculated based on Equation (19) and the dimensionless unbalance response of the bearings and discs can be calculated using Equation (22). Thus, the unbalance response in frequency domain can be expressed as following. \[Y(z)=U(q)L \tag{23}\]  
\[X(z)=V(q)L \tag{24}\]  
Change \(Y(z)\) and \(X(z)\) from frequency domain to time domain and the following are obtained:  
\[y(z,t)=\frac{\text{abs}(Y(z))}{\pi}\cos(wt+\text{angle}(Y(z))) \tag{25}\]  
\[x(z,t)=\frac{\text{abs}(X(z))}{\pi}\cos(wt+\text{angle}(X(z))) \tag{26}\]  
where \(\text{abs}()\) is the amplitude of a complex number; \(\text{angle}()\) is the angle of a complex number. Figure 3: Flowchart of the proposed CRDAM.