## 4 Simulation Study
### Simulation Results and Discussion  
Figure 8 illustrates the response of the hybrid isolation system under harmonic wave conditions, where the seismic load is a harmonic wave with an amplitude \(\mid\)A \(\mid\) = 1 and a frequency of 1 Hz. Figure 8a shows the displacement response of the hybrid isolation system under harmonic waves. The displacement and acceleration of the superstructure are significantly lower than those of the substructure. The peak displacement of the superstructure is 0.02244 m, while the peak displacement of the substructure reaches 0.3354 m. Figure 8b presents the acceleration response under harmonic waves. The acceleration of the superstructure is much lower than that of the substructure, with the peak acceleration of 0.2871 m/s\({}^{2}\) for the superstructure and 4.6016 m/s\({}^{2}\) for the substructure. Figure 8c displays the control force applied by the PID controller to the superstructure under harmonic waves. Figure 8d shows the real-time variations of the PID controller's three parameters throughout the control process. The peak isolation rate of the hybrid isolation system under harmonic waves is 93.3%. \begin{table}
\begin{tabular}{c c c} \hline
**Controller Parameter** & **Lower Value** & **Upper Value** \\ \hline K\({}_{\text{P}}\) & 0 & 50 \\ K\({}_{\text{i}}\) & 0 & 50 \\ K\({}_{\text{d}}\) & 0 & 50 \\ \hline \end{tabular}
\end{table} 
Table 6: Parameters of PID controllers. Figure 8: _Cont._ \begin{table}
\begin{tabular}{c c} \hline
**GA Parameters** & **Value** \\ \hline Population Size & 50 \\ Crossover Probability & 0.8 \\ Mutation Probability & 0.1 \\ Max Generations & 20 \\ \hline \end{tabular}
\end{table} 
Table 5: Parameters of GA. Figure 9 presents a comparison of displacement under harmonic wave conditions, with and without the control force applied to the substructure. Figure 9a shows the upper layer displacement comparison under harmonic waves. When the PID controller applies a control reaction force to the lower layer, the upper layer's maximum displacement reaches 0.02244 m. When no control reaction force is applied to the lower layer, the upper layer's maximum displacement is 0.01166 m. Compared to the case without a control reaction force, applying the control reaction force increases the peak displacement by 92.4%. Figure 9b depicts the lower layer displacement comparison under harmonic waves. When the PID controller applies a control reaction force to the lower layer, the lower layer's maximum displacement is 0.3354 m. Without the control reaction force, the lower layer's maximum displacement is 0.09166 m. Applying the control reaction force results in a 238.2% increase in peak displacement. Moreover, when the PID controller applies the control reaction force to the lower layer, the system's overall motion period is extended by 1.3 to 1.4 times compared to the case without the control reaction force. Figure 10 illustrates the response of the hybrid isolation system under harmonic wave conditions, where the seismic load is a harmonic wave with an amplitude \(\mid\)A\(\mid\) = 0.5 and a frequency of 10 Hz. Figure 10a shows the displacement response of the hybrid isolation system under harmonic waves. The displacement and acceleration of upper layer are significantly lower than those of the lower layer. The peak displacement of the upper layer is 0.0008818 m, while the peak displacement of the lower layer reaches 0.009771 m. Figure 10b presents the acceleration response under harmonic waves. The upper layer's  
Figure 8: Response in a simple harmonic state with an amplitude of 1 and a frequency of 1 Hz: (**a**) displacement response; (**b**) acceleration response; (**c**) control force response; (**d**) PID parameter response. Figure 9: Comparison of the displacement of the control force applied to the lower layer: (**a**) comparison of the displacement of the upper layer; (**b**) comparison of the displacement of the lower layer.