## 4 Vibration Control Analysis of Track-Bridge Structure Based on TIDs
### Analysis of TID Control Effectiveness on Rail Vibration  
According to the results in Section 3.3, when short-wavelength irregularities are used as high-frequency excitation input to the rail, they generate a significant acceleration response but relatively small displacement response. Therefore, the primary goal of TID implementation is to reduce the acceleration response of the rail. First, the control effectiveness of TIDs with different parameters on the acceleration amplitude of the rail (\(a_{rm}\)) is analyzed. Meanwhile, the variations in the root-mean-square value and stabilization time of the rail acceleration (these control indices are, respectively, denoted as \(a_{rRMS}\) and \(a_{rst}\)) are evaluated when the acceleration amplitude reaches its minimum. The stabilization time is defined as the time when the acceleration decreases to 5% of its peak value and remains below that value thereafter. Considering installation cost and engineering feasibility, the parameter ranges for TID, \(b_{t}\), \(k_{t}\), and \(c_{t}\), are selected as \([1,10]\), \([10^{4},10^{9}]\), and \([10,10^{4}]\), respectively. The step size for \(b_{t}\) is set to 1; for \(k_{t}\), nine values are evenly selected within each exponential segment; and for \(c_{t}\), four values are selected with equal logarithmic intervals, and the units are kg, N/m, and N\(\cdot\)s/m, respectively. By substituting these parameters into Equation (39), the rail acceleration amplitude is calculated. The optimal value of \(a_{rm}\) is selected for each \(b_{t}\), and the corresponding \(a_{rRMS}\) and \(a_{rst}\) time are obtained and compared to the uncontrolled case, as shown in Figure 5. The optimal TID parameters for each are listed in Table 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & \multicolumn{5}{c}{**Values**} \\ \hline \(b_{t}(\mathrm{kg})\) & 1 & 2 & 3 & 4 & 5 \\ \(k_{t}(\mathrm{N}\,/\,\mathrm{m})\) & \(2\times 10^{7}\) & \(1\times 10^{8}\) & \(1\times 10^{8}\) & \(7\times 10^{8}\) & \(2\times 10^{8}\) \\ \(c_{t}(\mathrm{N}\cdot\mathrm{s}\,/\,\mathrm{m})\) & 10 & 10 & \(10^{3}\) & 10 & \(10^{3}\) \\ \(b_{t}(\mathrm{kg})\) & 6 & 7 & 8 & 9 & 10 \\ \(k_{t}(\mathrm{N}\,/\,\mathrm{m})\) & \(7\times 10^{8}\) & \(3\times 10^{8}\) & \(6\times 10^{7}\) & \(4\times 10^{7}\) & \(3\times 10^{8}\) \\ \(c_{t}(\mathrm{N}\cdot\mathrm{s}\,/\,\mathrm{m})\) & 10 & \(10^{3}\) & 10 & 10 & \(10^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimized design parameters of the TID. Results Analysis:
* As shown in Figure 5a, TIDs with different inertance coefficients can all reduce the acceleration amplitude of the rail to varying degrees when their respective optimal control parameters are applied. When \(b_{t}=5\), the global minimum value of \(a_{rm}\) is achieved, decreasing from 1005.6 m/s\({}^{2}\) to 841.0 m/s\({}^{2}\), representing a reduction rate of 16.37%. At this point, the optimal TID stiffness coefficient is \(2\times 10^{8}\) N/m and the damping coefficient is \(10^{3}\) N \(\cdot\) s/m. The corresponding working frequency of the TID is calculated to be 1006.6 Hz. This indicates that when the working frequency of the TID is close to the first pinned-pinned resonance frequency of the rail, the TID can effectively suppress high-frequency vibrations induced by high-frequency excitation. However, if the damping is too small or too large, the control effectiveness of the TID will be diminished. * From Figure 5b,c, in combination with the parameter values listed in Table 2, it is observed that when the inertance coefficient \(b_{t}\) is too small or too large, the root-mean-square acceleration \(a_{rRMS}\) may become comparable to or even exceed the uncontrolled case. In addition, when the damping coefficient is too small, the stabilization time \(a_{rst}\) increases. This indicates that appropriately increasing the damping coefficient can reduce the acceleration stabilization time.