## 1 Introduction  
With the continuous improvement in comprehensive national power, the construction layout of China's high-speed railway (HSR) is becoming more and more complete. In order to save land and protect the environment around the railway lines, HSR mostly adopts the form of bridges instead of roads. In order to cross valleys and rivers, HSR adopts a large number of high-pier bridges as support structures of their lines during the design and construction process. The dynamic performance of the ballastless track will affect the dynamic response of the bridge, which will affect safety, and thus the smoothness of operating trains correspondingly. Moreover, the height of the pier affects the dynamic response indices of the train. Ensuring the safety of HSR bridges is a growing concern [1, 2, 3]. Meanwhile, finding a way to swiftly and precisely estimate structural deformation and dynamic response is critical for railway safety assessment, especially in bridge engineering, which accounts for the highest proportion of HSR infrastructure [4, 5]. Vehicles traveling through the bridge, due to the weight of the vehicle itself and the vibration of the driving process, will have an impact on the bridge and thus the bridge's structure by the impact of the role played by vibration. When the vibration amplitude and frequency of the bridge reach a certain limit, this affects the durability of the bridge structure and can even lead to the destruction of the bridge structure due to fatigue, resulting in casualties and property damage. When vehicles are on the bridge at the same time, the bridge road surface can become uneven, and the inherent vibration of the bridge itself will also exacerbate the vibration of the vehicle; this interaction between the vehicle and the bridge, mutual influence, is called the train-track-bridge coupled (TTBC) system by many researchers [6,7]. Therefore, understanding the coupled vibration of TTBC systems is of great value for the safe and smooth operation of HSR. However, the existing literature on vehicle-bridge coupled vibration research predominantly focuses on the coupled vibration response analysis of simply supported beam bridges [8, 9, 10], with fewer studies addressing large-span continuous beam bridges for high-speed railways. To ensure the safe operation of these large-span continuous beam bridges in HSR, it is necessary to study the vehicle-bridge coupled vibration of such structures. Experts and scholars have been exploring the dynamics of vehicle-bridge interactions for a long time. Since [PERSON], many researchers have contributed to the study of vehicle-bridge dynamic responses [11,12]. In the 1970s, [PERSON] et al. [13,14] regarded the vehicle-bridge system as a three-dimensional system, establishing spatial computational models that have had a significant impact on subsequent research. In 2003, [PERSON] [15] conducted a dynamic test on the Antoing Bridge between Paris and Brussels, obtaining natural vibration characteristics such as frequency, mode shapes, and damping of multi-span 50 m prestressed concrete simply supported through beams, as well as the dynamic deflection of the bridge, vertical accelerations of beams and piers, and relative displacements of bearings. In 2005, [PERSON] et al. [16] presented a detailed study of axle coupling dynamics and proposed a comprehensive axle coupling model for a wide range of bridge structures. [PERSON] et al. [17] found that for the response of the bridge and the vehicle, it can be confirmed that a fairly accurate solution can be obtained by considering only the first mode. In 2013, [PERSON] et al. [18] conducted dynamic tests and theoretical calculations on HSR with a design speed of 200-250 km/h, finding that 24 m simply supported beams have a third-order super-harmonic resonance speed of 230.7 km/h, while 32 m simply supported box beams have a second-order super-harmonic resonance speed of 263.4 km/h. In the same year, [PERSON] et al.