## 2 Basic Theory of Cable Dynamics Analysis  
### Basic Equation and Solution  
(1) Basic assumptions and vibration control differential equations. Taking a single cable as the object of analysis, its simplified model is shown in Figure 1.  
In order to study the vibration of the single cable shown in Figure 1, the following reasonable assumptions need to be made: (a) the material property of cable is linear elasticity and the cable is always within the range of linear elasticity during the vibration; (b) the strain of the cable is small and its cross-sectional size does not change; (c) the change of the internal tension of the cable along the length of the cable is not considered; (d) the vibration of the cable only occurs in the transverse plane and its vertical axial vibration is neglected; (e) the level of stresses under loads and its influence on vibration in the cable is neglected; and (f) consider the bending stiffness of the cable. Based on the above basic assumptions, the vibration control differential equations [14] for a transverse unloaded vertical cable with consideration of flexural stiffness in the absence of damping are obtained as follows:  
\[EI\frac{\partial^{4}u(x,t)}{\partial x^{4}}+m\frac{\partial^{2}u(x,t)}{ \partial t^{2}}-H\frac{\partial^{2}u(x,t)}{\partial x^{2}}=0 \tag{1}\]  
where, \(EI\) is the bending stiffness of the cable; \(u(x,t)\) is the transverse displacement function of the cable; \(H\) is the cable force; \(x\) is the distance from the upper boundary of the cable; and \(m\) is the mass per unit length of the cable. (2) Frequency equation of the cable system  
According to the dynamic stiffness theory (DSM) [15], the transverse displacement function of the cable is assumed:  
\[u(x,t)=\varphi(x)e^{i\omega t} \tag{2}\]  
where \(\omega\) is the circular frequency (rad/s) and \(t\) is the time. Substituting Equation (2) into Equation (1) and the following equation can be obtained:  
\[EI\frac{d^{4}\varphi(x)}{dx^{4}}-m\omega^{2}\varphi(x)-H\frac{d^{2}\varphi(x)} {dx^{2}}=0 \tag{3}\]  
Figure 1: Analysis model of a single cable. Introducing dimensionless parameters, \(\xi=\frac{\chi}{\tau}\), \(\varphi(\xi)=\frac{\varphi(\chi)EI}{I}\), where \(l\) is the calculated length of the cable, the dimensionless control differential equation of the system can be obtained by Equation (3):  
\[\varphi(\xi)^{\prime\prime\prime}-\gamma^{2}\varphi(\xi)^{\prime\prime}- \widetilde{\omega}^{2}\varphi(\xi)=0 \tag{4}\]  
where, \(\gamma^{2}=\frac{Hl^{2}}{EI}\) is the ratio of the cable axial force to bending stiffness of the cable and \(\widetilde{\omega}=\frac{\omega l^{2}}{\sqrt{EI/m}}\) is the dimensionless frequency of the cable. Thus, the general solution of Equation (4) can be expressed as:  
\[\varphi(\xi)=A_{1}e^{-p\xi}+A_{2}e^{-p(1-\xi)}+A_{3}\cos(q\xi)+A_{4}\sin(qx) \tag{5}\]  
where \( \begin{array}{c}p\\ q\\ \omega_{0}=\frac{\pi}{I\sqrt{H/m}}\end{array} \) is the fundamental frequency of the cable. The matrix expression of the general solution (5) is:  
\[\varphi(\xi)=\mathbf{\Phi}(\xi)\big{\{} \begin{array}{ccc}A_{1}&A_{2}&A_{3}& A_{4}\end{array} \big{\}}^{T} \tag{6}\]  
where \(\mathbf{\Phi}(\xi)=\left\{ \begin{array}{ccc}e^{-p\xi}&e^{-p(1-\xi)}&\cos(q \xi)&\sin(qx)\end{array} \right\}\). According to the relationship between the nodal force and the displacement function of the cable, the nodal displacement of the system represented by the vibration mode function can be obtained. \[\left\{ \begin{array}{c}a_{a}\\ \theta_{a}l\\ \alpha_{b}\\ \theta_{b}l\end{array} \right\}=\frac{l^{4}}{EI}\left\{ \begin{array}{c} \varphi(\xi|_{-0})\\ \varphi^{\prime}(\xi|_{-l})\\ \varphi^{\prime}(\xi|_{-l})\\ \varphi^{\prime}(\xi|_{-1})\end{array} \right\}=\frac{l^{4}}{EI}\left\{ \begin{array}{c}\mathbf{\Phi}(\xi|_{-0})\\ \mathbf{\Phi}^{\prime}(\xi|_{-0})\\ \mathbf{\Phi}(\xi|_{-1})\\ \mathbf{\Phi}^{\prime}(\xi|_{-1})\end{array} \right\}\left\{ \begin{array}{c} A_{1}\\ A_{2}\\ A_{3}\\ A_{4}\end{array} \right\}=\frac{1}{\xi}\mathbf{G}\left\{ \begin{array}{c}A_{1} \\ A_{2}\\ A_{3}\\ A_{4}\end{array}