## 5 Flexible Body Dynamics Based on Rayleigh-Ritz Shape Matrix
\begin{array}{cccccccc}\frac{\partial{\Gamma_{1}}}{\partial{ \sigma_{1}}^{4}}&\frac{\partial{\Gamma_{1}}}{\partial{\sigma_{1}}^{3}}&\frac{ \partial{\Gamma_{1}}}{\partial{\sigma_{0,t}}^{2}}&\frac{\partial{\Gamma_{1}}}{ \partial{\sigma_{1}}^{2}}&\frac{\partial{\Gamma_{1}}}{\partial{\sigma_{0,t}} ^{2}}&\dots&\frac{\partial{\Gamma_{1}}}{\partial{\sigma_{0,t}}^{4}}&\frac{ \partial{\Gamma_{1}}}{\partial{\sigma_{0,t}}^{4}}&\frac{\partial{\Gamma_{1}}}{ \partial{\sigma_{0,t}}^{4}}\\ \frac{\partial{\Gamma_{2}}}{\partial{\sigma_{1}}^{4}}&\frac{\partial{\Gamma_{ 2}}}{\partial{\sigma_{1}}^{5}}&\frac{\partial{\Gamma_{2}}}{\partial{\sigma_{0, t}}^{6}}&\frac{\partial{\Gamma_{2}}}{\partial{\sigma_{0,t}}^{2}}&\frac{\partial{ \Gamma_{2}}}{\partial{\sigma_{0,t}}^{2}}&\dots&\frac{\partial{\Gamma_{2}}}{ \partial{\sigma_{0,t}}^{4}}&\frac{\partial{\Gamma_{5}}}{\partial{\sigma_{0,t}} ^{4}}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ \frac{\partial{\Gamma_{9}}}{\partial{\sigma_{0,t}}^{4}}&\frac{\partial{\Gamma_{ 9}}}{\partial{\sigma_{0,t}}^{5}}&\frac{\partial{\Gamma_{9}}}{\partial{\sigma_{ 0,t}}^{6}}&\frac{\partial{\Gamma_{9}}}{\partial{\sigma_{0,t}}^{4}}&\frac{ \partial{\Gamma_{9}}}{\partial{\sigma_{0,t}}^{2}}&\dots&\frac{\partial{\Gamma_{ 9}}}{\partial{\sigma_{0,t}}^{4}}&\frac{\partial{\Gamma_{9}}}{\partial{\sigma_{0,t}}^{4}}&\frac{\partial{\Gamma_{9}}}{\partial{\sigma_{0,t}}^{4}}\\ \end{array} \right] \tag{86}\]  
Finally, the assembled equations of motion are given by:  
\[\left[ \begin{array}{cc}M_{rr}&M_{ff}\\ M_{fr}&M_{ff}\\ \end{array} \right]+\left[ \begin{array}{cc}0&0\\ 0&K_{ff}\\ \end{array} \right]+\left[ \begin{array}{cc}{B_{r}}^{T}\\ {B_{f}}^{T}\\ \end{array} \right]\lambda=\left[ \begin{array}{c}(Q_{v})_{r}\\ (Q_{v})_{f}\\ \end{array} \right]+\left[ \begin{array}{c}(Q_{g})\\ (Q_{g})_{f}\\ \end{array} \right]+\left[ \begin{array}{c}(Q_{e})_{r}\\ (Q_{e})_{f}\\ \end{array} \right] \tag{88}\]  
To solve the system DAEs, the coordinate partitioning method was applied as in the rigid body dynamics presented earlier. In this case, the coordinates are partitioned into the following dependent and independent coordinates which are indicated by subscripts \(d\) and \(i\), respectively:  
\[q=\left[ \begin{array}{c}q_{d(9x1)}\\ q_{i(15x1)}\\ \end{array} \right]=\left[ \begin{array}{cccccccccccc}{r_{o_{1,x}}}^{1}&{r_{o_{1, y}}}^{1}&{\alpha_{1}}&{r_{o_{2,x}}}^{2}&{r_{o_{2,y}}}^{2}&{\alpha_{3}}&{r_{o_{3,x}}}^{3}&{r_{o_{3,y}}}^{3}&{ \alpha_{3}}\\ \end{array} \right]^{T}\right] \tag{89}\]  
Using again the native names of the rigid coordinates and identifying zero coordinates, we can write Equation (89) as:  
\[q=\left[ \begin{array}{c}q_{d(9x1)}\\ q_{i(15x1)}\\ \end{array} \right]=\left[\begin{array}{cccccccccccc}{\left[ \begin{array}{cccccccc}{x_{1}}&0& {\alpha_{1}}&{x_{2}}&0&{\alpha_{3}}&{x_{3}}&0&{\alpha_{3}}\\ \end{array} \right]^{T}\right]} \tag{90}\]  
The descriptor form of the DAEs is:  
\[\left[ \begin{array}{cc}M&B^{T}\\ \end{array} \right]\left[ \begin{array}{c}\ddot{q}\\ \lambda\\ \end{array} \right]=\left[ \begin{array}{c}Q_{v}+Q_{g}+Q_{e}-Kq\\ -\ddot{B}q\\ \end{array} \right] \tag{91}\]which can be partitioned into the following:  
\[\left[ \begin{array}{ccc}M_{dd}&M_{di}&B_{d}{}^{T}\\ M_{id}&M_{ii}&B_{i}{}^{T}\\ B_{d}&B_{i}&0\end{array} \right]\left[ \begin{array}{c}\ddot{q}_{d}\\ \ddot{q}_{i}\\ \lambda\end{array} \right]=\left[ \begin{array}{c}(Q_{v})_{d}+\left(Q_{g} \right)_{d}+(Q_{e})_{d}-K_{d}q\\ (Q_{v})_{i}+\left(Q_{g}\right)_{i}+(Q_{e})_{i}-K_{i}q\\ -B\ddot{q}\end{array}