## 6 Conclusions  
1. For the BT, the LT, and the USBT, if the minimum wavelength of the bridge deformation exceeds 6, 10, and 16 m, respectively, the rail deformation can accurately mirror the bridge deformation. Otherwise, it becomes essential to establish a comprehensive track-bridge model to accurately capture the rail deformation.
2. The minimum wavelength of the long-span, cable-stayed bridge tends to decrease with an increase in the number of the auxiliary pier and a decrease in the cable-stay stiffness ratio under complex loads. While the addition of the auxiliary pier in long-span, cable-stayed bridges can effectively minimize the mid-span displacements of the bridge's tower and the main span, it affects the smoothness of long waves in the vicinity of the bridge pier. The vertical deformation wavelength range for long-span, cable-stayed bridges with main spans ranging from 200 m to 600 m is observed to be between 21.38 m and 1270.74 m.
3. The deformation of the cable-stayed bridge is primarily influenced by wavelengths greater than 200 m under the influence of overall loads such as the temperature and overall settlement. However, during the ingress and egress of trains, as well as local loads like the single auxiliary pier's settlement, there may be the vertical deformation with the wavelength of approximately 200 m near the pier, impacting the overall smoothness of the bridge's long waves.
4. Based on the track-bridge mapping relationship and the wavelength's range of the vertical deformation for the cable-stayed bridge, the comparison was made between the train's vibration of the TBSIM model and the TTBSIM model under various excitations. The simulation results are found to be essentially consistent. The impact of the vertical deformation of the long-span, cable-stayed bridge on the train's vibration can be accurately calculated using the train-bridge dynamic coupling model. In this model, the track is simplified and considered as the second-phase's constant load, contributing to improved computational efficiency.  
Figure 22: Train vibration response under settlement and deformation excitation of auxiliary pier.  
**Author Contributions:** Methodology and formal analysis, Z.P. and M.G.; software, G.L.; formal analysis, J.Y.; writing--review and editing, F.Y. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the Science and Technology Research and Development Project of China National Railway Co., Ltd., grant number K2021G020, and the Science and Technology Research and Development Project of China Railway Research Institute Group Co., Ltd., grant number 2023 YG020.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available upon request from the corresponding author. The data are not publicly available due to privacy concerns.  
**Conflicts of Interest:** The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. [PERSON], [PERSON], [PERSON], and [PERSON] are employees of Infrastructure Inspection Research Institute, China Academy of Railway Sciences Co., Ltd., who provided funding and technical support for the work. The funder had the following involvement with the study: Methodology and formal analysis, M.G. ([PERSON]); software, G.L. ([PERSON]); formal analysis, J.Y. ([PERSON]); writing--review and editing, F.Y. ([PERSON]).