# Coupled Vibration Analysis of Multi-Span Continuous Cable Structure Considering Frictional Slip
## 5 Conclusions  
To study the nonlinear vibration response of a multi-span continuous cable structure under moving loads considering friction slip, this paper proposes using the direct stiffness method to establish the main cable subsystem model and the moving load model. The coupled vibration control equations are derived through the equilibrium relationship between the moving load and the main cables, and then the dynamics equations are solved based on the Newmark\(-\beta\) direct integration method, so as to obtain the dynamic response of the two. Simulation results were considered for comparison, and the following conclusions were obtained:  
1. Comparing the calculation results of the self-programmed program, the simulation model results and the dynamic load test values, it can be seen that the calculated results of the first two are in good agreement with the measured values. The errors of the vertical displacement in the span calculated by both of them are 0.86% and \(-\)4.40%, but the error reaches \(-\)13.90% when the coupled vibration response is not taken into account. This can prove the correctness and rationality of the deduced coupled vibration control equations and programming. It is necessary to consider the coupled vibration of the moving loads and the main cable during the design of the main cable.
2. Through the analysis of the impact effect of the cable hoisting system when hoisting weights out of the lifting cable, it can be seen that the instantaneous hoisting of weights out of the coupling vibration generated by the impact effect is very obvious. The impact coefficient of the deflection of the main cable is 2.0. The impact coefficient of the force of the main cable is 2.16. The impact coefficient of the cable tower deflection can be up to 4.0.
3. By establishing the vertical displacement-time-span three-dimensional surface diagram of the main cable, it can be seen that during the traveling process of the overhead crane with a load, the vertical downward deflection of the main cable at the point of action is the largest. But the other parts of the cable will show an upward deflection region; the upward deflection region is mainly concentrated in the interval from the  
Figure 18: Relative values of mid–span displacement and cable force versus time and load parameters: (**a**) top view of relative values of mid–span displacements, (**b**) top view of relative values of mid–span cable forces.  
point of action of 0.5L-0.8L. The closer to the middle of the span, the smaller the upward deflection region. When the crane is traveling away from the main cable of the middle span, the side-span cable is the first one that vibrates downward and reaches the peak value.
4. By comparing the relative values of span displacement and cable force with time, crane speed and load parameters, it can be seen that the crane speed is proportional to the vertical displacement in the span. The cable force in the span will show an obvious rapid increase when it exceeds the design speed. When the load exceeds 0.7 times the design load, the vertical displacement will increase rapidly. The cable force in the span will increase rapidly when the load exceeds 1.5 times the design load, and the moment of reaching the peak value will be faster.  
Z.T.: project administration, funding acquisition, conceptualization, formal analysis; B.X.: methodology, formal analysis, investigation, data curation, writing--original draft, writing--review and editing. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China [52078058] and the Hunan Provincial Natural Science Foundation Project of China [2022 JJ50323].  
Not applicable.  
All data that support the findings of this study are available from the corresponding author upon reasonable request.  
The authors declare no conflicts of interest.