# Analysis of the Vibration Characteristics of Ballastless Track on Bridges Using an Energy Method
## 1 Introduction  
With the rapid development of the high-speed railway (HSR) system, the length of HSRs in service in China has exceeded 35,000 km, and the convenience brought has been obvious to all. However, the induced vibration and noise have become increasingly acute. Faced with these problems, numerous researchers have worked to address them. [PERSON] et al. [1] proposed a vehicle-track coupled model in the frequency domain and utilized the compliance method to investigate the dynamic compliance for the vehicle-track system. Further, the structure-borne noises of a viaduct were analyzed via the finite element method (FEM). [PERSON] et al. [2] established a coupled model of the HSR simply supported beam bridge-track structure system (HSRBTS), in which the effect of shear deformation was taken into account. Moreover, the natural vibration characteristics of HSRBTS under different interlayer stiffness and lengths of rails at different subgrade sections were evaluated by means of the analytic method put forward in that paper. [PERSON] et al. [3] carried out in situ testing on the environmental vibration response of the surrounding buildings and the ground caused by HSR trains, and the spectrum analysis method was used to study the vibration characteristics of the surrounding buildings and ground. Finally, the environmental vibration levels of HSRs were analyzed and evaluated. [PERSON] et al. [4] used the FEM to determine the seismic performance levels of four bridges basedon ambient vibration testing. For the vibration testing, they selected the Ankara-Sivas HSR line for analytical and experimental studies. To sum up, the vibration responses of the bridge were investigated by various methods. Ballastless track systems have been widely adopted in the HSRs, such as the China Railways Track System (CRTS), and have the advantages of longer service life, lower cost of maintenance, better control of the track geometry, etc. For the ballastless track on bridges, laying a slab mat layer between the track and the bridge is an effective measure to reduce vibration. In all types of the CRTSs, the CRTS-II track, which consists of a track slab, CA (cement asphalt) mortar, base slab, and so on [5], has been widely employed in the HSR lines Beijing-Tianjin, Beijing-Shanghai, Beijing-Shijiazhuang, etc. Further, it has shown good performance overall. For studying the vibration and noise induced by the HSRs, many scholars and researchers have focused on those induced in the CRTS-II track system. [PERSON] et al. [6] developed a coupled train-track-soil model, in which the multibody theory, FEM, and perfectly matched layers method were combined together to investigate the ground vibrations in HSR systems using the CRTS-II track. [PERSON] et al. [7] established a frequency domain model to investigate the frequency response of the CRTS-II track. The vibrations of a vehicle-track coupled model of HSRs were studied. [PERSON] et al. [8] developed a vehicle-track-subgrade coupled model based on the dynamic Biot's equations to investigate the vibration responses of the coupled model, in which fluid-solid interaction was considered. [PERSON] et al. [9] established a vehicle-track-bridge-soil coupled model to analyze the influence of elastic bearings on ground vibrations when vehicles pass over the bridge. [PERSON] et al. [10] established a vehicle-track-subgrade coupled model based on the vehicle-track coupling theory to investigate the effects of mortar debonding on the vibration properties of the CRTS-II track. The influences of different debonding lengths on the dynamic responses of the vehicle-track system were also researched by means of the FEM. [PERSON] and [PERSON] [11] presented a 3D model based on the FEM to simulate interface damage between the track slab and CA mortar of the CRTS-II track, and they also established a vehicle-track coupled model to study the effects of interface damage of the slab track on vibrations.