## 4 Vibration Control Analysis of Track-Bridge Structure Based on TIDs
### Multi-Objective Optimization of TID Design Parameters  
As revealed in the previous section, when a TID achieves optimal control performance for rail vibration, it may introduce a significant adverse effect on the vibration of the track slab. Therefore, it is necessary to comprehensively consider the vibration reduction effects on both the rail and the track slab when designing reasonable TID parameters. To obtain the optimal design parameters of the TID, this section treats the TID parameters as decision variables and takes the rail acceleration amplitude \(a_{rm}\) and the track slab acceleration amplitude \(a_{sm}\) as dual objective functions. Meanwhile, two additional rail vibration control indicators--namely, the root-mean-square value \(a_{rRMS}\) and the stabilization time \(a_{rst}\)--are considered as constraints to ensure that they do not exceed the uncontrolled results. Based on the optimal parameter ranges identified in Section 4.2, an appropriate parameter search space is defined, and a multi-objective optimization control model is formulated as follows:  
\[\min\{a_{rm},a_{sm}\}\]  
Figure 6: Adverse effect of TID on track slab vibration acceleration amplitude. \[s.t.\left\{ \begin{array}{l}a_{rm}<a_{rm}<a_{rm}\cr a_{rm}<a_{rm}\cr a_{rm}<a_{rm} \cr a_{rRMS}<a_{rRMS0}\cr a_{rstd}<a_{rstd0}\cr 1\rm{kg}\leq b_{t}\leq 10\rm{kg} \cr 10^{7}\rm{N}\cdot\rm{m}^{-1}\leq k_{t}\leq 8\times 10^{8}\rm{N}\cdot\rm{m}^{-1} \cr 10\rm{N}\cdot\rm{s}\cdot\rm{m}^{-1}\leq c_{t}\leq 5\times 10^{3}\rm{N}\cdot \rm{s}\cdot\rm{m}^{-1}\end{array} \right. \tag{40}\]  
Here, \(a_{rm}\), \(a_{rm}\),\(a_{rm}\), \(a_{rRMS0}\), and \(a_{rstd0}\) denote the corresponding values in the uncontrolled case. The computational efficiency of solving this model primarily depends on the speed at which the two objective functions are evaluated, as each feasible solution requires solving for both the rail and track slab acceleration amplitudes. Thanks to the high efficiency of the proposed FE-SEA hybrid method, the model is solved using the multi-objective genetic algorithm (MOGA) toolbox in MATLAB2018(a). The Pareto front is obtained, and based on a trade-off strategy, six tentative optimal solutions are selected, as shown in Figure 7. For each tentative optimal solution, the corresponding TID parameters are extracted, and the values and reduction rates of all control indicators are calculated. The comprehensive control rate of rail vibration is defined  
\[\eta_{r}=(\eta_{arm}+\eta_{arRMS}+\eta_{arst})/3 \tag{41}\]  
where \(\eta_{arm}\), \(\eta_{arRMS}\), and \(\eta_{arst}\) denote the reduction rates of the rail acceleration amplitude, RMS acceleration, and stabilization time, respectively. The results are summarized in Table 3.  
As shown in Figure 7, when the rail acceleration amplitude decreases, the acceleration amplitude of the track slab gradually increases. This further demonstrates that improving the vibration control performance of the TID on the rail may lead to a deterioration in its control effect on the track slab. According to Table 3, Solutions 1 and 3 exhibit the highest and identical comprehensive control rates for rail vibration. Since the acceleration amplitude of the rail is lower in Solution 1, it is selected as the final optimal solution in this study. This solution eliminates the adverse effect of the TID on the track slab while controlling rail vibration, resulting in a 5.47% reduction in the acceleration amplitude of the track slab. Figure 7: Pareto front of multi-objective optimization. Using the TID parameters in Solution 1, the time-history curves of rail and track slab acceleration are plotted, as shown in Figure 8.  
As shown in Figure 8, under the control of the TID with optimal parameters, not only is the acceleration amplitude of the rail vibration reduced, but the overall acceleration response is also effectively suppressed.