## 6 Numerical Analysis of Test
### Time Domain Analysis of TMD Installed on the Powerhouse Structure  
In this section, the interaction mechanism of structure-TMD system and the influence law of different TMD control strategies was studied through a time domain analysis. The response of TMD and structure to high-frequency vibration is shown in Figure 13. The results show that the vibration velocity of the TMD system was much higher than that of the structure. Through further analysis, it was found that the velocity direction of the TMD was always opposite to the vibration direction of the structure, which further verified the working principle of the TMD. By observing the time history curve of the non-optimal parameter TMD in the previous section, it can be found that TMD moves in the same direction as the structure in part of the phase, thus increasing the structural response. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Response**} & \multicolumn{2}{c}{**Displacement (10\({}^{-4}\) m)**} & \multicolumn{2}{c}{**Velocity (10\({}^{-2}\) m/s)**} & \multicolumn{2}{c}{**Acceleration (g)**} \\ \cline{2-7}  & **Top** & **Bottom** & **Top** & **Bottom** & **TMD** & **Top** & **Bottom** \\ \hline Original & 1.20 & 1.08 & 5.34 & 6.00 & - & 3.47 & 4.21 \\ \(\gamma=0.5\) & 1.21 & 1.17 & 5.39 & 6.83 & 2.80 & 3.49 & 4.84 \\ \(\gamma=0.75\) & 1.21 & 1.16 & 5.52 & 6.80 & 7.12 & 3.56 & 4.83 \\ \(\gamma=\) optimal & 1.07 & 0.87 & 3.90 & 4.20 & 53.65 & 2.47 & 2.76 \\ \(\gamma=1.25\) & 1.15 & 1.17 & 4.92 & 6.57 & 13.46 & 3.12 & 4.63 \\ \(\gamma=1.5\) & 1.17 & 1.17 & 5.08 & 6.70 & 9.05 & 3.26 & 4.73 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Structural responses with different tuning frequency ratios. Figure 12: TMD system velocity time history curves at different tuning ratios. Figure 13: Comparison of structure and TMD velocity time-history curve. under different control strategies. This section uses a harmonic response analysis to analyze the spectral characteristics of the structure under different control strategies. Figure 15a shows that the vibration amplitude of the top layer of the structure was greatly reduced due to the addition of the TMD device, indicating that the vibration damping performance of the structure with the additional TMD was better. Compared with the uncontrolled structure, the amplitude and response of the damping structure with the additional TMD under excitation frequency reached the minimum, which indicates that the parameter setting of the TMD was reasonable and effective. Since the parameters of the TMD did not change, it was observed that the control bandwidth was the same, while the vibration amplitude of the structure was small when the TMD was arranged at the bottom layer. This makes the bottom layer the maximum position of the vibration mode. As a result, the underlying TMD layout of the structure was more efficient and significant. Figure 15b, c shows that as the number of TMD increased, the vibration reduction performance improved. More importantly, the width of the damper tuning band increased gradually, but the control effect decreased gradually in the whole band. In other words, multiple TMDs can increase the control bandwidth at the cost of loss control effect, but for a simple harmonic excitation load, multiple TMDs can still ensure the best control effect in the control band. For the four TMD control strategies in Figure 15d, all three cases show a good control effect, among which 4 MTMT-B had a better control effect and robustness. When the external excitation is unstable, the robustness of multiple TMDs can ensure the vibration damping performance remains unchanged. If replaced by a single TMD, the unstable excitation may cause the controller to fail and, even further, increase the structural response. Therefore, multi-TMD coordination to improve robustness is particularly important in practical engineering.