## 1 Introduction  
A cable-stayed bridge is a combined structural system of cables, girders, towers, and foundations with excellent spanning capacity [1]. It is one of the main bridge types for long-span bridges [2]. In recent years, with the rapid construction of highway and railway networks around the world, critical projects of the highway and railway routes across rivers, bays, and valleys are often collinear [3]. Thus, a rail-cum-road cable-stayed bridge is increasingly adopted in these cases as it is economical, environmentally friendly, and resource-conserving without having to construct two bridges separately. However, compared with traditional highway or railway cable-stayed bridges, the service condition of the rail-cum-road cable-stayed bridge is more complex as it is under the combined action of trains, vehicles, winds, temperatures, etc. [4]. Hence, its cables, which are the crucial load-bearing components, are usually thicker [5]. A rail-cum-road bridge has a more complex main girder with two layers of decks, and a steel truss main girder with an inverted trapezoidal cross section is widely used to reduce its weight [6]. Many studies have been conducted on the vibration behavior, FE modeling, and fatigue problem of long-span bridges in recent years. [PERSON] et al. [7] conducted a case study on the vortex-induced vibration of a long-span suspension bridge. Their resultsshow that the vortex-induced vibrations will cause bending or torsional single-model vibrations of the main girder. [PERSON] et al. [8] developed an analytical model for the dynamics of wind-vehicle-bridge systems in the time domain with wind, rail vehicles, and bridges modeled as a coupled vibration system. A numerical model of a cable-stayed bridge validated this model. [PERSON] et al. [9] investigated the effect of joint geometry and collar plate size on the structural behavior of the strengthened X-joints subjected to compression based on 138 verified FE models. An accurate fundamental frequency value is a proper indicator for axial load estimation in steel beams, bridge cables, and hangers. [PERSON] et al. [10] derived the frequency functions and mode shape functions for the vibrating cables with various types of boundary conditions, and they pointed out that the sinusoidal components dominantly contribute to the mode shape functions. [PERSON] et al. [11] investigated the effect of prestressing on the beam dynamics. They concluded that the fundamental frequency is practically unaffected by prestressing in thin-walled steel girder bridges with tendons in contact with the cross-section or draped with a deviator. The fatigue behavior of stay cables has been receiving increasing attention as it may markedly decrease the service life of the bridge. Research on the vibration of bridges [12; 13; 14] is also helpful because it is one of the primary excitations for cable vibration, which is the main reason for cable fatigue. To better assess the fatigue life of the cable, a more intensive study on its dynamic response properties is needed. At present, many studies have investigated cables' dynamic responses in highway cable-stayed bridges and the design and construction monitoring of rail-cum-road cable-stayed bridges [15; 16; 17; 18]. However, there is little research on the cables' dynamic responses and the vibration performance of rail-cum-road cable-stayed bridges under complex loads during their service period [19; 20; 21; 22]. In this paper, the cable was separated from the whole bridge structure and the excitations were the variations in its boundary conditions where the cable anchors to the bridge, caused by the vibration of the bridge. Based on the discussion above, this paper focuses on the cables' dynamic response behavior in a long-span rail-cum-road cable-stayed bridge under the train loads. First, in Section 2, a numerical model of cable vibration under double-end excitation was derived by neglecting the bending stiffness, and a numerical solution program was compiled based on MATLAB R2022a software.