## 1 Introduction  
Bridge cables (stay cables and cables) are an important component of long-span bridges and are widely used in modern engineering. With the increase in the length to slenderness ratio of cables and the decrease in transverse stiffness and internal damping, various types of vibration phenomena are easy to occur under the influence of wind loads and moving loads, etc. [1]. The fatigue caused by cable vibration will cause performance degradation of high-strength steel wire, which will not only further reduce the load-bearing capacity and durability of the cable [2], but also bring a severe test to the safe operation and normal use of cable-stayed bridges, and then affect the driving comfort and even bring visual panic to pedestrians, all of which are inseparable from the dynamics analysis of the cable structure. For cables, the statistical analysis of existing engineering cases shows that the average service life of cables is only ten years [3], and the service life of some cables is even less than ten years. Since the cost of repair and replacement of cables is huge, the importance of their dynamics analysis has far exceeded that of static analysis. The analysis of cable dynamic characteristics has become a key issue in the design, performance monitoring and maintenance, and vibration control of such bridge structures during operation. At the same time, the study of such problems can provide important references for bridge design, internal force distribution in the cable and cable system during service life, and monitoring and evaluation of bridge dynamic characteristics. At present, the existing dynamic analysis theory of cable structure can be divided into an analytical method, numerical method, and semi-analytical and semi-numerical method according to the analysis method and solution form. In this paper, the dynamic stiffness method is a semi-analytic and semi-numerical method, which starts from the exact vibration function of the system and obtains the frequency equation and dynamic equation in closed form, while using the Euler beam model to model the cable more in line with the real structure. [PERSON] [4], in 2007, gave a set of frequency equations to describe the cable system with transverse viscous dampers, which can handle articulated and rigid boundaries, but neglects the effects of sag and inclination. To this end, [PERSON] et al. [5] gave closed-form solutions for the transverse dynamic stiffness of Euler beams with small sag, and also studied the variation law of transverse dynamic stiffness in the spatial and frequency domains. Subsequently, [PERSON] [6] further gave a unified characteristic frequency equation for the structure and its numerical solution under the combined influence of transverse force elements and sag effects. At the same time, based on the PSO (particle swarm optimization) optimization algorithm, he provided a multi-level and multi-parameter identification algorithm suitable for stay cables [7]. Further, [PERSON] and [PERSON] [8] used a model of a double-beam system connected by distributed springs to model the composite cable based on the Eulerian beam theory, derived the dynamic stiffness matrix of the model, and studied the influence of parameters such as sag, spring connection layer stiffness, and double-beam flexural stiffness ratio on the modal frequency of the system. At present, for the bare cable system, the existing research work has considered the influence of some factors in the bending stiffness, sag, inclination angle, and internal damping of the cable, but it is not yet possible to consider these factors in a comprehensive manner in an analytical or semi-analytical form. When considering the influence of an elastic support and an elastic embedded boundary at the end of the cable, the existing studies have approximated the vibration function of the cable as the vibration function of the rigidly supported cable. However, the influence of boundary and vibration frequency on the vibration pattern are ignored, so the results are not accurate enough.