## 5 Conclusions  
This paper establishes the vehicle-track-bridge coupled dynamics model to systematically investigate the transmission patterns of vibration power flow induced by train operations within the track structure. The principal findings of this study can be summarized as follows.  
1. The vibration energy caused by train operation in the low-frequency band is mainly concentrated in the integral ballast bed and the bridge roof and bottom plate of the box girder bridge. Most of the vibration energy will accumulate in the ballast bed. The vibration energy in the middle- and high-frequency band is mainly concentrated in the track position.
2. The fastener system, as the main vibration reduction component of the ordinary integral ballast, has a greater impact on the power flow in the low frequency range. Strategically reducing the stiffness of fasteners can effectively attenuate the transmission of low-frequency vibrations to the bridge, consequently leading to a reduction in the low-frequency vibrations experienced by the bridge structure. In the range of middle and low frequency, the accumulation of vibration energy of ballast decreases with the increase in damping of the fastener. However, with the increase in frequency, the accumulation state of vibration energy of the ballast bed and bridge will be intensified.
3. The fastener is the vibration reduction component of the ordinary integral ballast bed. In order to reduce the vibration and noise of the bridge on the bridge line, it is recommended to set the stiffness and damping of the fastener to 40 kN/mm and 50 kN\(\cdot\)s/m, respectively.  
Based on the above conclusions, future studies can apply the vibration power flow research method to different forms of urban rail transit track structures, such as flexible long sleeper tracks, steel-spring floating-plate tracks, and trapezoidal sleeper tracks. Vibration power flow studies can also be conducted on other modes of rail transport, such as high-speed railways, intercity railways, and urban light railways.  
X.C., L.Y. and X.Z.: methodology. P.L.: software. J.X.: validation. X.C. and L.Y.: formal analysis. L.Y.: data curation. X.Z., L.Y. and P.L.: writing--original draft preparation. X.C. and J.X.: writing--review and editing. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (NSFC) grant number 52362049; the Natural Science Foundation of Gansu Province, grant numbers 2 JR11 RA152 and 22 JR5 RA344; The Joint Innovation Fund Project of Lanzhou Jiaotong University and Tianjin University, grant number 2022063; The Young Doctor Foundation Education Department of Gansu Province, grant number 2023 QB-037; Lanzhou Science and Technology Planning Project, grant number 2022-ZD-131; Gansu Province Outstanding Graduate Innovation Star Project, grant number 2023 CXZX-517.  
Not applicable.  
The data used to support the findings of this study are available from the corresponding author upon request. The data are not publicly available due to privacy.  
Author [PERSON] is employed by the CRRC Zhuzhou Institute Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.