# Coupled Vibration Analysis of Multi-Span Continuous Cable Structure Considering Frictional Slip
## 3 Algorithms for Solving Nonlinear Systems of Time-Varying Equations  
In a cable system with a single hoisting span of twin towers, the crane with a load only operates in the middle hoisting span. So, only the nonlinear coupled vibration response between the mid-span cables and the crane is analyzed. It is assumed that the main cables are able to slide in the position of the cable tower. Based on the Newmark-\(\beta\) direct integration method combined with the dynamic stiffness method, the nonlinear system of equations of motion is solved. The solution method and steps, for which the flow chart of crane-cable coupled vibration refinement analysis is shown in Figure 9, are as follows:  
1. Firstly, select the Newmark parameters: \(\beta\) and \(\gamma\). It is important to select the two parameters correctly, and the analysis shows that when \(\gamma\geq 1/2\) and \(\beta\geq\gamma/2\) are taken, the Newmark\(-\beta\) method is not able to be conditionally stabilized. Usually, \(\gamma=1/2\) is selected, and then \(\beta\) is adjusted to achieve the purpose of different corrections to  
Figure 9: Flow chart of refined analysis of crane–cable coupling vibration. the acceleration. When \(\gamma=1/2\) and \(\beta=\gamma/2\), it is linear acceleration method; when \(\gamma=1/2\) and \(\beta=1/6\), it is average acceleration method. 2. Time discretization: The studied motion time \(n\) is divided into equal parts with time step \(\Delta t\); the discrete points are 0, 1, 2, \(\ldots\), \(n-1\), \(n\). The time corresponding to the discrete points is \(t_{j}=j\Delta t\) (\(j=0\), 1, 2, \(\ldots\), \(n-1\), \(n\)), with a start time \(t_{0}=0\) and a termination time \(t_{m}=n\Delta t\). 3. Each material parameter of the cable structure and the crane model (including the unstressed cable length \(s_{0}\) of the cables) is entered and stored. 4. The whole cable sling is divided into n nodes and m elements, in which the mid-span cable is divided into m\(-\)2 two-node catenary cable elements. The anchorage point of the side span, the top node of the cable tower and the node of the mid-span cable closest to the cable tower are divided into three-node catenary cable elements, with a total of two for the left and right side spans. 5. Call the model cell of the crane with a load and calculate its mass matrix \(\mathbf{M}_{\rm v}\), stiffness matrix \(\mathbf{K}_{\rm v}\) and damping matrix \(\mathbf{C}_{\rm v}\). 6. Call the two-node catenary cable element and the three-node catenary cable element and calculate their mass matrix, stiffness matrix and damping matrix, and then assemble them into the total mass matrix \(\mathbf{M}_{\rm b}\), total stiffness matrix \(\mathbf{K}_{\rm b}\) and damping matrix \(\mathbf{C}_{\rm b}\) of the cable structure. 7. The main cable model is coupled and assembled with the crane model to form the total mass matrix \(\mathbf{M}\), the total stiffness matrix \(\mathbf{K}\) and the damping matrix \(\mathbf{C}\). 8. Calculate the total nodal force vector \(\mathbf{F}_{\rm int}\) and the total external load vector (weight carried by the crane) \(\mathbf{F}_{\rm g}\) for the cable structure. 9. Process the boundary conditions to solve for the nodal displacements \(\{u_{\rm b}\}\) of the cable structure and the displacements \(\{u_{\rm v}\}\) of the crane model, and compose the total displacement vector \(\{x_{i}\}\). 10. Calculate the initial motion parameters for the time step obtained from the previous time step: \(\{x\}_{j}\), \(\{\dot{x}\}_{j}\), \(\{\ddot{x}\}_{j}\). When \(j=0\), the time step is obtained from the initial conditions: \(\{x\}_{0}\), \(\{\dot{x}\}_{0}\), \(\{\ddot{x}\}_{0}\), \(\{\ddot{x}\}_{0}^{-}=[m]^{-1}\{\{f\}_{0}-[c]\,\{\dot{x}\}_{0}-[k]\,\{x\}_{0}\}\). 11. Calculate the time step termination motion parameters: 1 Calculate the equivalent stiffness matrix and the equivalent load increment vector using Equation (20): \[