# Coupled Vibration Analysis of Multi-Span Continuous Cable Structure Considering Frictional Slip
###### Abstract  
As important load-bearing structures, suspension cables have been widely used in suspension bridges, engineering repeways, cable suspension systems and other special equipment. Their dynamic problems have always been a research hotspot. Especially for complex cable systems such as engineering repeways and cable lifting equipment, there will be moving loads acting on multi-span continuous friction-slip cable structures, resulting in nonlinear coupled vibration. Therefore, few scholars have studied how to calculate the nonlinear coupling vibration effect between such moving loads and multi-span continuous cables considering friction slip. Therefore, this paper proposes the use of the combination of the direct stiffness method and the Newmark-\(\beta\) integration method to solve the nonlinear system of equations of motion, which can be derived from the coupled vibration response between the moving load and the main cable. The corresponding calculation program is prepared. Combined with the dynamic load test and simulation results of engineering cases, the correctness and reasonableness of the coupled vibration equations and the program can be verified through comparative analysis. The results show that the calculation results of the self-programmed program are in good agreement with the dynamic load test results, in which the maximum error of the vertical displacement in the span is \(-4.40\%\) and \(0.86\%\), and the error of the static calculation reaches \(-13.90\%\). The impact effect is more obvious when hoisting the weight out of the pulling cable, in which the impact coefficient of the main cable can be up to \(2.0\). The impact coefficient of the deviation of the cable tower is \(4.0\). During the traveling process of the moving load, the vertical downward deflection of the main cable at the action point is the largest, and the upward deflection is in the region of \(0.2\)-\(0.8\)L from the action point. frictional slip; nonlinear coupled vibration; Newmark-\(\beta\); dynamic load test; moving load +
Footnote †: journal: Applied Sciences  
When the crane is traveling with a load, the main cable under its action will be forced to vibrate, resulting in a load greater than the static load under the action of deformation and internal force; this phenomenon is the \"coupling vibration\" effect. Since the main cable is pulley-supported at the top of the cable tower (as shown in Figure 3), the main cable will produce relative sliding at the saddle when the crane is traveling. However, the friction between the main cable and the saddle will lead to longitudinal deflection of the cable tower. So, the forced vibration of the main cable will be different from the single cable vibration. Flexible suspension cables are susceptible to complex coupled vibrations under moving load excitation. The coupled vibration response between the main cable and the moving load can be analyzed with reference to the theory of coupled vibration of a vehicle and bridge, but they are different. The differences are as follows: First, the bridge structure is rigid, with strong bending performance, while the cable structure is a flexible material with strong geometric nonlinearity. Second, the boundary conditions are different. When analyzing the coupled vibration of a vehicle and bridge, the bridge structure is generally constrained to the longitudinal displacement. For the coupled vibration of a moving load and cable, the two ends are in a relative slip. Therefore, in order to investigate the response  
Figure 1: Traveling crane. Figure 3: Fixed saddle. Figure 2: Cable hoisting system. mechanism of coupled vibration, two major difficulties need to be solved. One is the problem of geometric nonlinearity and relative slip of the cable structure, and the other is the dynamic characteristics of the cable structure. In order to model the sliding friction action of cable structures, many scholars have also proposed their own theories and insights. [PERSON] et al.