## 2 Analysis of Cable Vibration under Double-End Excitation
### Theoretical Model  
Taking the completion state of a cable-stayed bridge as the initial state, the cable is under the action of both initial tension stress and gravity. Figure 1 shows the calculation diagram of the cable. To simplify the expression of the cable's vibration formulation, we transformed the \(x\)-\(y\)-\(z\) coordinate system to the \(u\)-\(v\)-\(w\) coordinate system, in which \(v_{s}\) denotes the initial sag, \(u\) represents the axial displacement, \(v\) means the in-plane lateral displacement, and \(w\) denotes out-plane vertical displacement. The conversion formulations of the two coordinate systems are given in Equation (1). \[\left\{ \begin{array}{l}u=ysin\varphi+xcos\varphi\\ v=ycos\varphi-xsin\varphi\\ w=z\end{array} \right. \tag{1}\]  
To derive the vibration equations of the cable, four basic assumptions are given [23]:  
* The bending and torsional stiffness of the cable are ignored;
* The density of the cable remains constant along the \(u\)-axis;
* The stress of the cable is always within the elastic limit;
* The initial curve of the sagged cable is approximated as a parabola. The first assumption is made under the consideration that the length of the cable is much larger than its diameter, and its slenderness ratio is usually less than 0.001 for the long-span cable-stayed bridge. The third assumption holds for the long-span bridge because the cable material is high-strength structural steel with an ultimate tensile strength of over 1570 megapascals. The cables' stresses during service are usually within one-third of the ultimate tensile strength. It should be noticed that the boundary conditions of the cable are the pure force boundary conditions, which means the cable is under the control of the force on its two ends. Based on these assumptions above, the force analysis of the cable's microsegment can be conducted in the \(u\)-\(v\)-\(u\) coordinate system, as shown in Figure 2.  
The partial differential equations can be derived based on D'Alembert's principle [24], as shown in Equation (2). \[\left\{ \begin{array}{l}\frac{\partial}{\partial s}\Big{[}\big{(}T_{s}+T_{d} \big{)}.\frac{\partial(v_{s}+v)}{\partial s}\Big{]}=m:\frac{\partial^{2}v}{ \partial t^{2}}+c_{v}.\frac{\partial v}{\partial t}+mgcos\varphi+f_{v}(u,t)\\ \frac{\partial}{\partial s}\Big{[}\big{(}T_{s}+T_{d}\big{)}.\frac{\partial v }{\partial s}\Big{]}=m:\frac{\partial^{2}v}{\partial t^{2}}+c_{v}.\frac{ \partial v}{\partial t}+f_{w}(u,t)\end{array} \right. \tag{2}\]  
Figure 1: Calculation diagram of the cable. Figure 2: Force analysis of cable’s microsegment: (**a**) in-plane diagram; (**b**) out-plane diagram. where \(T_{s}\) is the initial tension; \(T_{d}\) is the tension caused by the cable vibration; \(v_{s}\) is the initial sag; \(c_{v}\) and \(c_{w}\) are the in-plane and out-plane damping coefficients; \(f_{v}(u,\,t)\) and \(f_{w}(u,\,t)\) are the in-plane and out-plane exciting forces; and \(m\) is the linear density of the cable. Assuming that the initial arc length of the microssegment is \(ds_{0}\), and it becomes \(ds\) during the cable vibration, then the two arc lengths can be expressed as Equations (3) and (4), respectively, by ignoring the axial deformation:  
\[ds_{0}^{2}=du^{2}+dv_{s}^{2} \tag{3}\]  
\[ds^{2}=du^{2}+\left(dv_{s}+dv\right)^{2}+dw^{2} \tag{4}\]  
Then, the dynamic strain of the cable can be deduced as Equation (5). \[ \begin{split}\varepsilon=\frac{ds-ds_{0}}{ds_{0}}=\frac{\sqrt{du^ {2}+\left(dv_{s}+dv\right)^{2}+dw^{2}}-ds_{0}}{ds_{0}}\\ =\frac{\sqrt{ds_{0}^{2}+2 dvdv_{s}+dv^{2}+dw^{2}-ds_{0}}}{ds_{0}} \\ \approx\frac{dv}{ds_{0}}\frac{dv_{s}}{ds_{0}}+\frac{1}{2}\Bigg{[} \left(\frac{dv}{ds_{0}}\right)^{2}+\left(\frac{dw}{ds_{0}}\right)^{2}\Bigg{]} \end{split}  \tag{5}\]  
The dynamic tension \(T_{d}\) can be decomposed into two components: the dynamic tension \(T_{v}\) resulting from cable vibration-induced deformation and the dynamic tension \(T_{m}\) arising from changes in the boundary conditions of the cable.