## 2 Basic Theory of Cable Dynamics Analysis
### Basic Equation and Solution
\right\} \tag{7}\]  
where \(\xi=\frac{EI}{I}\) and \(G=\left\{ \begin{array}{c}\mathbf{\Phi}(\xi|_{=0})&\mathbf{\Phi}^{\prime}( \xi|_{=0})&\mathbf{\Phi}(\xi|_{=l})&\mathbf{\Phi}^{\prime}(\xi|_{-l})\end{array}  \right\}^{T}\), the specific display expressions are shown in Appendix A. The constant vector \(\left\{ \begin{array}{ccc}A_{1}&A_{2}&A_{3}&A_{4}\end{array} \right\}^{T}\) can be expressed by Equation (8). \[\left\{ \begin{array}{c}A_{1}\\ A_{2}\\ A_{3}\\ A_{4}\end{array} \right\}=\xi\mathbf{G}^{-1}\Bigg{\{} \begin{array}{c}\alpha_{a }\\ \theta_{a}l\\ \alpha_{b}\\ \theta_{b}l\end{array} \right\} \tag{8}\]  
The relationship between the nodal forces and displacements of the cable is:  
\[\left\{ \begin{array}{c}V(x,t)=(EI\frac{\partial^{3}\varphi}{\partial x^{2}} -H\frac{\partial\varphi}{\partial x})e^{i\omega t}=l(\varphi^{\prime\prime \prime}(\xi)-\gamma^{2}\varphi^{\prime}(\xi))e^{i\omega t}\\ M(x,t)=EI\frac{\partial^{2}\varphi}{\partial x^{2}}e^{i\omega t}=l^{2}\varphi^{2 }(\xi)e^{i\omega t}\end{array} \right. \tag{9}\]  
Taking into account the differences in symbolic conventions, the forces and bending moments at both ends of the cable are expressed as:  
\[\left\{ \begin{array}{c}V(0,t)\\ M(0,t)\\ V(l,t)\\ M(l,t)\end{array} \right\}=\left\{ \begin{array}{c}V_{a}\\ M_{a}\\ V_{b}\\ M_{b}\end{array} \right\}e^{i\omega t} \tag{10}\]Combined with the vibration function expressions in Equations (6) and (8), the following equation can be obtained:  
\[\left\{ \begin{array}{l}V_{a}\quad M_{a}/l\quad V_{b}\quad M_{b}/l\end{array}  \right\}^{T}=l_{\xi}\mathbf{D}\mathbf{G}^{-1}\left\{ \begin{array}{l}\alpha_{a }\quad\theta_{a}l\quad\alpha_{b}\quad\theta_{b}l\end{array} \right\}^{T} \tag{11}\]  
where \(\mathbf{D}=\left\{ \begin{array}{l}\[PERSON]^{\prime\prime}(\xi|_{=0})-\gamma^{2} \[PERSON]^{\prime}(\xi|_{=0})\\ -\[PERSON]^{\prime\prime}(\xi|_{=0})\\ \[PERSON]^{\prime\prime}(\xi|_{=1})-\gamma^{2}\[PERSON]^{\prime}(\xi|_{=1})\\ -\[PERSON]^{\prime\prime}(\xi|_{=1})\end{array} \right\}\), the specific explicit expression is shown in the Appendix A. Thus, the dynamic stiffness matrix is:  
\[\mathbf{K}_{1}=l_{\xi}\mathbf{D}\mathbf{G}^{-1} \tag{12}\]  
where \(\mathbf{K}_{1}=\frac{EI}{l^{3}}\left( \begin{array}{cccc}k_{11}&k_{12}&k_{13 }&k_{14}\\ k_{21}&k_{22}&k_{23}&k_{24}\\ k_{31}&k_{32}&k_{33}&k_{34}\\ k_{41}&k_{42}&k_{43}&k_{44}\end{array} \right)=\frac{EI}{l^{3}}\left( \begin{array} []{cccc}k_{11}&k_{12}&k_{13}&k_{14}\\ k_{12}&k_{22}&-k_{14}&k_{24}\\ k_{13}&-k_{14}&k_{11}&-k_{12}\\ k_{14}&k_{24}&-k_{12}&k_{22}\end{array} \right)\), and the display expressions for each element in \(\mathbf{K}_{1}\) are shown in the Appendix A. The following equation is obtained by boundary conditions of the cable:  
\[\frac{EI}{l^{3}}\left( \begin{array}{cccc}k_{11}&k_{12}&k_{13}&k_{14}\\ k_{12}&k_{22}&-k_{14}&k_{24}\\ k_{13}&-k_{14}&k_{11}&-k_{12}\\ k_{14}&k_{24}&-k_{12}&k_{22}\end{array} \right)\left\{ \begin{array}{l}a_{a} \\ \theta_{a}l\\ a_{b}\\ \theta_{b}l\end{array} \right\}=\left\{ \begin{array}{l}k_{A}{}^{a}\alpha_{a }\\ k_{A}{}^{b}\theta_{a}l\\ k_{B}{}^{a}\alpha_{b}\\ k_{B}{}^{b}\theta_{b}l\end{array} \right\} \tag{13}\]  
Carrying out matrix transformation of Equation (13) leads to:  
\[\frac{EI}{l^{3}}\left( \begin{array}{cccc}k_{11}-\frac{\beta}{EI}k_{A}{}^{a}& k_{12}&k_{13}&k_{14}\\ k_{12}&k_{22}-\frac{\beta}{EI}k_{A}{}^{a}\theta&-k_{14}&k_{24}\\ k_{13}&-k_{14}&k_{11}-\frac{\beta}{EI}k_{B}{}^{a}&-k_{12}\\ k_{14}&k_{24}&-k_{12}&k_{22}-\frac{\beta}{EI}k_{B}{}^{a}\end{array} \right) \left\{ \begin{array}{l}\alpha_{a}\\ \theta_{a}l\\ \alpha_{b}\\ \theta_{b}l\end{array} \right\}=\left\{ \begin{array}{l}0\\ 0\\ 0\\ 0\end{array} \right\} \tag{14}\]  
Then, the following is obtained:  
\[\mathbf{K}(\omega)\left\{ \begin{array}{l}\alpha_{a}\\ \theta_{a}l\\ \alpha_{b}\\ \theta_{b}l\end{array} \right\}=\left\{ \begin{array}{l}0\\ 0\\ 0\\ 0\end{array}