# Coupled Vibration Analysis of Multi-Span Continuous Cable Structure Considering Frictional Slip
###### Abstract
developed a three-dimensional cable element considering frictional slip based on the theory of catenaries, which provided research ideas for subsequent scholars, but only for two-node catenary cable elements, which are not adapted to multi-node continuous cable structures [3]. [PERSON] et al. proposed a new multi-node sliding catenary cable element to solve the slipping problem that exists in the main cables of the cable hosting system. But the element does not take into account the friction effect of the cables [4]. [PERSON] et al. proposed an accurate finite element method for continuous cable structures based on the suspension line theory and the Euler-Eytelwein equation with consideration of sliding friction, but they did not consider temperature effects [5; 6]. [PERSON] et al. proposed a finite element formulation for cable structures that takes into account the effects of temperature and sliding friction. The equation overcomes the limitations of existing methods that ignore or approximate friction, pulley size, temperature and geometric nonlinearities [7]. For the dynamic response of suspension cables under external load excitation, scholars have carried out a large number of experimental and theoretical studies. [PERSON] and [PERSON] put forward the linear theory of free vibration by conducting free vibration experiments on a single-curved suspension structure [8; 9]. However, the suspension cables themselves have strong geometric nonlinearities, and the nonlinearities have a large influence on the vibration of the cables. So, the geometric nonlinearities of the suspension cables cannot be ignored in the research process. [PERSON] et al. started to extend the linear vibration study of the suspension cables to the geometric nonlinearities by utilizing the Galyokin method [10]. Subsequently, researchers used a variety of methods to study the free vibration, nonlinear periodic and semi-periodic vibration characteristics of suspension cables [11; 12; 13; 14]. In this aspect of research, the first use of a multi-degree-of-freedom discrete model with a linear modal deviation function was [PERSON]'s program applied to partial differential control equations and the use of multiple scaling methods to obtain the expected vibration response. The above solution of nonlinear vibration equations for suspension structures is only applicable in the case of weak nonlinearity. In order to analyze the strong nonlinear vibration effects of suspension cable structures, researchers used vibration differential equations and finite elements combined with the direct integration method. [PERSON] and [PERSON] analyzed the vibration response of suspension cables due to dynamic loads by using the Newmark direct integration combined with the Newton-Raphson iteration method [15], [PERSON] and [PERSON] discretized the cable structure into a two-degree-of-freedom system and analyzed the vibration response of the structure under nonlinear excitation by the MCS method [16]. But this method is prone to convergence difficulties. In addition, the virtual force method [17; 18] and the incremental superposition technique [19; 20] have also been applied to the nonlinear vibration response analysis of suspension cables. [PERSON] and [PERSON] obtained the vibration differential equations of a cable structure in the undamped case and solved the intrinsic frequency and characteristic equations of the cable structure by using transcendental equations [21]. [PERSON] et al. analyzed the effect of nonlinear coupled vibration using a multiscale model order reduction strategy, which solved the problem of error and convergence difficulty brought by the finite method [22]. However, these studies all analyzed single cable structures for multi-span continuous complex cable systems. [PERSON] et al. proposed using the dynamic stiffness method to calculate the vibration equations of a cable structure in the damped case and established the relationship between the damped frequency and the undamped frequency [23; 24; 25; 26].