# Coupled Vibration Analysis of Multi-Span Continuous Cable Structure Considering Frictional Slip
## 2 Nonlinear Coupled Vibration Model  
### Moving Load Model  
The main cable of the cable hoisting system can slide at the cable saddle at the top of the cable tower, and the crane only operates with a load in the mid-span. So, this paper only considers the coupling vibration between the crane and the main cable in the mid-span, in order to establish the dynamic control equations of the crane-cable coupling vibration. The computational model of the main cable under the action of the crane traveling with a load is shown in Figure 4. In the figure, points A and D are fixed, and the main cable can produce \(x\)-direction sliding at points B and C. The main cable can also slide in the \(x\)-direction. In order to further establish a model close to the actual cables, the hoisting cables between the crane and the hoisting section are considered as springs and damping devices, so as to establish the model as shown in the figure below. The moving load on the main cables consists of the two traveling crane masses \(m_{1}\) and \(m_{2}\) and the hoisting section \(m_{3}\). The mass moment of inertia \(I_{3}\) is due to the fact that \(m_{3}\) is connected between two axes at a certain distance, which generates the mass moment of inertia \(I_{3}\). The two cranes are connected to the hoisting arch section by the damper \(c_{1}\), spring \(k_{1}\), damper \(c_{2}\) and spring \(k_{2}\). The distance between the two traveling cranes is \(a\). It is assumed that the cranes do not separate from the main cables during the moving process. Let the modulus of elasticity of the main cable be \(E\), the area be \(A\), the damping coefficient be \(C\) and the mass per unit length of the cable be \(m\).  
Since the speed of the crane is very small when it is traveling with a load, it is assumed that the crane travels at a constant speed with a speed \(V\) and moves a distance \(x=Vt\), and the main cable produces only a vertical dynamic deflection of \(u_{b}\) without bending deformation. The vertical dynamic displacement of the hoisting arch section \(m_{3}\) is \(h_{3}\), the rotational displacement \(\theta\), and the vertical dynamic displacemen  
Figure 4: Calculation model of the main cable under the action of crane with a load movement.  
are \(h_{1}\) and \(h_{2}\). Then, the force acting on the undersprung mass \(m_{3}\) has the inertial force \(P_{2}=(m_{3}+I_{3})\ddot{u}_{v}\). The forces of \(m_{1}\) and \(m_{2}\) include inertial forces \(P_{1}=(m_{1}+m_{2})\ddot{u}_{v}\), elastic forces generated by hoisting cables \(P_{S}=(k_{1}+k_{2})(u_{v}-u_{b})\big{|}_{x=Vt}\), damping forces generated by hoisting cables \(P_{D}=(c_{1}+c_{2})\big{(}\ddot{u}_{v}-\ddot{u}_{b})\big{|}_{x=Vt}\). The equation of motion for \(m_{3}\) can be obtained from the force balance of \(m_{3}\) as [27]  
\[[M_{v}]\big{\{}\ddot{u}_{v}\big{\}}+[C_{v}]\big{\{}\dot{u}_{v}\big{\}}+[K_{v}] \big{\{}u_{v}\big{\}}=0 \tag{1}\]  
where \(M_{v}\) is the mass matrix of the moving load model, \(C_{v}\) is the damping matrix of the moving load model and \(K_{v}\) is the stiffness matrix of the moving load model.  
\[[K_{v}]=\begin{bmatrix}k_{1}+k_{2}&(k_{1}-k_{2})\frac{a}{2}&-k_{1}&-k_{2}\\ (k_{1}-k_{2})\frac{a}{2}&(k_{1}+k_{2})\big{(}\frac{a}{2}\big{)}^{2}&-k_{1} \frac{a}{2}&k_{2}\frac{a}{2}\\ -k_{1}&-k_{1}\frac{a}{2}&k_{1}&0\\ -k_{2}&k_{2}\frac{a}{2}&0&k_{2}\end{bmatrix}\]