## 5 Conclusions  
In this study, a vertical vibration analysis model for a vehicle-track-bridge coupled system was developed based on the FE-SEA hybrid method. Taking short-wavelength irregularities as high-frequency excitation, the vibration responses of the rail and track slab were analyzed. To address the adverse effect of TIDs on track slab vibration, a multi-objective optimization model was established. The following conclusions are drawn:  
Figure 8: Comparison diagram of optimal control results. (**a**) Vertical acceleration of the track slab. (**b**) Vertical acceleration of the track slab. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**ID**} & \(b_{t}\) & \(k_{t}\) & \(c_{t}\) & \(a_{rm}\)/ & \(a_{rRMS}\)/ & \(a_{rst}\)/ & \multirow{2}{*}{\(\eta_{r}\)} & \(a_{sm}\)/ \\  & **(kg)** & **(N/m)** & **(N-s/m)** & \(\eta_{arm}\) & \(\eta_{arRMS}\) & \(\eta_{arsst}\) & & **Reduction** \\ \hline
1 & 3.4464 & \(1.1973\times 10^{8}\) & 175  
By comparing the results and computational efficiency of the conventional finite element method (FEM), the FE-SEA hybrid method was shown to provide both high accuracy and computational efficiency for analyzing the vertical vibration of track-bridge structures. This significantly reduces the computational cost of subsequent TID parameter optimization. It was also found that short-wavelength irregularities as high-frequency excitation induce large acceleration but relatively small displacement in the rail. 2. The proposed numerical model for the vehicle-track-bridge system enables the effects of TIDs on the track structure to be incorporated conveniently in matrix form. The model can be easily extended to other types of vibration dampers by modifying the corresponding parameters and matrices. This provides a new design approach and theoretical foundation for vibration control analysis in vehicle-track-bridge coupled systems. 3. Installing TIDs between the rail and the track slab effectively suppresses rail vibration. When the TID's working frequency approaches the first pinned-pinned resonance frequency of the rail, the control performance against high-frequency rail vibration caused by short-wavelength irregularities is optimal. Under this condition, the rail acceleration amplitude can be reduced by 16.37%, without increasing the RMS value or stabilization time of the rail acceleration. However, the same TID parameters result in a significant increase in the vibration acceleration amplitude of the track slab, with a maximum increase of 11.22%, indicating an adverse effect. 4. Enhancing the vibration control effect of TIDs on the rail generally weakens their effect on the track slab. Using the proposed multi-objective optimization model and a genetic algorithm, the optimal TID design parameters were determined as \[b_{t}=3.4464\ \mathrm{kg},k_{t}=1.1973\times 10^{8}\ \mathrm{N/m},c_{t}=1750.5\ \mathrm{N \cdot s/m}.\] Under these parameters, the rail and track slab acceleration amplitudes were reduced by 16.43% and 5.47%, respectively. The comprehensive rail control rate was 12.45%, effectively mitigating the adverse effect of increased track slab vibration caused by TID-based rail vibration suppression. 5. The optimally tuned TID also reduced the vertical displacement and vertical acceleration amplitudes of the car body by 14.22% and 47.5%, respectively. This indicates that the TID vibration control parameters proposed in this study not only suppress rail vibration but also significantly improve the dynamic performance of the vehicle, thereby enhancing passenger ride comfort. This study focuses on a two-dimensional vertical vibration model and utilizes a multi-objective optimization framework to address the adverse effects of tuned inerter dampers (TIDs) in controlling the vertical vibrations of track-bridge structures, offering a new design approach and theoretical guidance. The analysis is limited to a single train speed and does not consider the full three-dimensional dynamic response of the structure.