## 1 Introduction  
On 1 October 1964, with the opening of the Tokaido Shinkansen in Japan, the world's first high-speed railway was officially put into operation. There has subsequently been a wave of building high-speed railways around the world, and high-speed railways have developed rapidly [1, 2, 3]. Due to the limitations of computational methods, the initial research on vehicle bridge vibration problems mainly focused on independent analysis of vehicle and bridge models, and the models were too simplistic to best reflect the real vehicle bridge dynamic response [4]. The emergence of electronic computers and the development of finite element technology have further promoted the research on vehicle-bridge coupling vibration [5]. [PERSON] [6] used the energy method to derive the motion equation of the dynamic interaction between vehicles and bridges in high-speed railways. [PERSON] et al. [7] first studied the vehicle-bridge coupling vibration system, considering that the vehicle body is a 3-degree of freedom rigid body, and established a vehicle-bridge coupling vibration calculation model. [PERSON]. [8] established a vehicle model consisting of a vehicle body, a bogie, and a wheelset. The train-bridge coupled system was studied using a spring connection between the vehicle body and the bogie, and between the bogie and the wheelset, with the wheelset always in contact with the steel rail. Subsequently, train analysis models constructed with multiple rigid bodies were gradually recognized and adopted by scholars. This paper is also based on this to construct a 38-degree-of-freedom train model. Since the 1960s, there have been multiple high-speed train derailments worldwide, which caused serious casualties and economic losses. One of the reasons for derailment is earthquakes [9; 10; 11]. Therefore, it is necessary to consider seismic effects in the study of the dynamic response of train-track-bridge coupled systems. Currently, many scholars have conducted extensive research on the dynamic response of vehicle-bridge coupling systems under seismic excitation. [PERSON] et al. [12] used the relative motion method to solve seismic influences and analyzed the operational safety of high-speed trains passing over rigid-frame bridges with high piers under seismic influences. [PERSON] et al. [13; 14] studied the dynamic response of trains and bridges under emergency braking during earthquakes, as well as the safety of train operation. [PERSON] [15], [PERSON] [16], [PERSON] [17], [PERSON]. [18] and others used cable-stayed bridges as examples to study the coupling dynamic interaction between trains and cable-stayed bridges under earthquake action, and obtained valuable conclusions. In order to meet the requirements of smooth and stable operation of high-speed trains, land saving, and not affecting ground transportation, the proportion of high-speed railway bridges is constantly increasing. Many studies have shown that the characteristics of bridges significantly affect the dynamic response and safety of train-track-bridge coupled systems. It is necessary to study the effect of bridge characteristics on the train-track-bridge coupled system. Bridge characteristics include bridge structural stiffness, bridge deformation, abutment settlement, pier height, and so on. [PERSON] [19; 20] analyzed the impact of bridge structural stiffness on the dynamic response of the coupling system. Their study shows that when the beam stiffness or lateral stiffness of bridge piers is insufficient, the main dynamic indicators of trains and bridges significantly increase with the decrease in stiffness. [PERSON] et al. [21] found that the damping coefficient of bridges has great influence on the vertical acceleration and mid-span acceleration of vertical and horizontal beams. [PERSON] et al.