# Continuous Rotor Dynamics of Multi-Disc and Multi-Span
## 2 Theory
### Analytical Solution  
According to the fourier transform of Equations (11) and (12) on the variable t and considering that the frequency of the unbalance response is equal to the rotation frequency \(w\), the dimensionless unbalance responses in frequency domain satisfy Equations (15) and (16). \[ \begin{array}{l}\frac{\partial^{4}U}{\partial q^{2}}-\frac{m:w^{2}L^{4}}{EI} U+\frac{Lu_{1}L^{2}w^{2}}{EI}\frac{\partial^{2}U}{\partial q^{2}}-\frac{i \omega^{2}L^{2}}{EI}\frac{\partial^{2}V}{\partial q^{2}}\\ =\big{\{}\big{[}m_{1u}w^{2}e_{1}\tau(\sin\alpha_{1}-i\cdot\cos\alpha_{1})+w^{ 2}m_{1d}LU_{1d}\big{]}\delta(q-q_{1d})\\ +\cdots\ldots\\ +\big{[}m_{mm}w^{2}e_{m}\tau(\sin\alpha_{m}-i\cdot\cos\alpha_{m})+w^{2}m_{md} LU_{md}\big{]}\delta(q-q_{md})\\ -L\big{(}k_{1b\cdot yx}V_{1b}+k_{1b\cdot yy}U_{1b}+i\cdot wc_{1b\cdot yx}V_{1b}+i \cdot wc_{1b\cdot yy}U_{1b}\big{)}\delta(q-q_{1b})\\ -L\big{(}k_{2b\cdot yx}V_{2b}+k_{2b\cdot yy}U_{2b}+i\cdot wc_{2b\cdot yx}V_{2b }+i\cdot wc_{2b\cdot yy}U_{2b}\big{)}\delta(q-q_{2b})\\ +\cdots\cdots\\ -L\Big{(}k_{nb\cdot yx}V_{nb}+k_{nb\cdot yy}U_{nb}+i\cdot wc_{nb\cdot yx}V_{nb }+i\cdot wc_{nb\cdot yy}U_{nb}\Big{)}\delta(q-q_{nb})\Big{\}}\frac{L^{2}}{EI} \\ \frac{\partial^{4}V}{\partial q^{4}}-\frac{w^{2}m_{1d}^{4}}{EI}V+\frac{w^{2}m_ {1d}^{2}}{EI}\frac{\partial^{2}V}{\partial q^{2}}+\frac{i\cdot w^{2}l_{1d}^{2 }}{EI}\frac{\partial^{2}U}{\partial q^{2}}\\ =\big{\{}\big{[}m_{1u}w^{2}e_{1}\tau(\cos\alpha_{1}+i\cdot\sin\alpha_{1})+w^{ 2}m_{1d}LV_{1d}\big{]}\delta(q-q_{1d})\\ +\cdots\cdots\\ +\big{[}m_{mm}w^{2}e_{m}\tau(\cos\alpha_{m}+i\cdot\sin\alpha_{m})+w^{2}m_{md} LV_{md}\big{]}\delta(q-q_{md})\\ -L\big{(}k_{1b\cdot xx}V_{1b}+k_{1b\cdot yy}U_{1b}+i\cdot wc_{1b\cdot xx}V_{1b }+i\cdot wc_{1b\cdot xy}U_{1b}\big{)}\delta(q-q_{1b})\\ -L\Big{(}k_{2b\cdot xx}V_{2b}+k_{2b\cdot xy}U_{2b}+i\cdot wc_{2b\cdot xx}V_{2b }+i\cdot wc_{2b\cdot xy}U_{2b}\Big{)}\delta(q-q_{2b})\\ -\cdots\cdots\\ -L\Big{(}k_{nb\cdot xx}V_{nb}+k_{nb\cdot xy}U_{nb}+i\cdot wc_{nb\cdot xx}V_{nb }+i\cdot wc_{nb\cdot xy}U_{nb}\Big{)}\delta(q-q_{nb})\Big{\}}\frac{L^{2}}{EI} \end{array}  \tag{16}\]  
where \(U\) and \(V\) are the dimensionless unbalance responses in frequency domain in \(x\) and \(y\) directions. \(U(q)\) and \(V(q)\) can be expressed using Green Function as following. \[U(q)=\frac{L^{2}}{EI}\left[ \begin{array}{c}\pi m_{1u}w^{2}e_{1}(\sin\alpha_{1 }-i\cdot\cos\alpha_{1})+w^{2}m_{1d}LU_{1d}\\ \vdots\\ \pi m_{mm}w^{2}e_{m}(\sin\alpha_{m}-i\cdot\cos\alpha_{m})+w^{2}m_{md}LU_{md}\\ -L\big{(}k_{1b\cdot xy}V_{1b}+k_{1\cdot yy}U_{1b}+i\cdot wc_{1\cdot yx}V_{1b}+i \cdot wc_{1\cdot yy}U_{1b}\big{)}\\ \vdots\\ -L\big{(}k_{n\cdot yx}V_{nb}+k_{n\cdot yy}U_{nb}+i\cdot wc_{n\cdot yx}V_{nb}+i \cdot wc_{n\cdot yy}U_{nb}\big{)}\\ \end{array} \right]^{T}\left[ \begin{array}{c}G_{n}(q,q_{1d})\\ \vdots\\ G_{n}(q,q_{md})\\ G_{n}(q,q_{nb})\\ \vdots\\ G_{n}(q,q_{nb})\\ \end{array} \right] \tag{17}\]  
\[V(q)=\frac{L^{2}}{EI}\left[ \begin{array}{c}\pi m_{1u}w^{2}e_{1}(\cos\alpha_{1 }+i\cdot\sin\alpha_{1})+w^{2}m_{1d}LV_{1d}\\ \vdots\\ \pi m_{mm}w^{2}e_{m}(\cos\alpha_{m}+i\cdot\sin\alpha_{m})+w^{2}m_{md}LV_{md}\\ -L\big{(}k_{1x\cdot xy}V_{1b}+k_{1\cdot xy}U_{1b}+i\cdot wc_{1\cdot xx}V_{1b} +i\cdot wc_{1\cdot yy}U_{1b}\big{)}\\ \vdots\\ -L\big{(}k_{n\cdot xx}V_{nb}+k_{n\cdot xy}U_{nb}+i\cdot wc_{n\cdot xy}V_{nb}+i \cdot wc_{n\cdot xy}U_{nb}\big{)}\\ \end{array} \right]^{T}\left[ \begin{array}{c}G_{n}(q,q_{1d})\\ \vdots\\ G_{n}(q,q_{md})\\ G_{n}(q,q_{nb})\\ G_{n}(q,q_{nb})\\ \end{array} \right]_{(m\
eq n)\times 1}\\ -L\big{(}k_{n\cdot xx}V_{nb}+k_{n\cdot xy}U_{nb}+i\cdot wc_{n\cdot xy}V_{nb}+i \cdot wc_{n\cdot xy}U_{nb}\big{)}\\ \end{array}\right]_{(m\
eq n)\times 1}\\ \end{array  
\(V_{1b}\), \(V_{2b}\) and \(V_{nb}\) represent the dimensionless unbalance response of the bearings and discs in the frequency domain. The above-mentioned Green's coefficients can be calculated using Equation (19) which is derived in reference [33]. \[\left[ \begin{array}{c}G_{u}(q,q_{i})\\ G_{v}(q,q_{i})\end{array} \right]=\] \[\left[