## 4 Simulation Study
### Simulation Results and Discussion
Figure 16 presents a comparison of displacement in the hybrid isolation system under the Tangshan wave, with and without applying control force to the lower layer. Figure 16a  
Figure 14: Control forces in a random wave state: (**a**) control forces of Tangshan wave; (**b**) control forces response of Superstition Mountain wave. Figure 13: Acceleration response in a random wave state: (**a**) acceleration response of Tangshan wave; (**b**) acceleration response of Superstition Mountain wave. Figure 15: PID parameters in a random wave state: (**a**) PID parameters of Tangshan wave; (**b**) PID parameters response of Superstition Mountain wave. shows the upper layer displacement comparison. When the PID controller applies a reaction force to the lower layer, the peak upper layer displacement is 0.009986 m. Without applying the reaction force to the lower layer, the peak upper layer displacement is 0.005181 m. This indicates that the peak displacement increases by 92.6% when the reaction force is applied. Figure 16b illustrates the lower layer displacement comparison. When the PID controller applies a reaction force to the lower layer, the peak lower layer displacement is 0.18225 m. Without applying the reaction force, the peak lower layer displacement is 0.08342 m, which corresponds to a 118.4% increase in peak displacement when the reaction force is applied. Overall, when the PID controller applies control force to the lower layer, the entire system's motion period extends by 1.3-1.5 times compared to the case without the control force. Figure 17 presents the displacement comparison in the hybrid isolation system under the Superstition Mountain wave, both with and without applying control force to the lower layer. Figure 17a shows the upper layer displacement comparison. When the PID controller applies a reaction force to the lower layer, the peak upper layer displacement is 0.005997 m. Without applying the reaction force, the peak upper layer displacement is 0.002967 m. This results in a 102.1% increase in peak displacement when the reaction force is applied. Figure 17b illustrates the lower layer displacement comparison. When the PID controller applies a reaction force to the lower layer, the peak lower layer displacement is 0.0817 m. Without applying the reaction force, the peak lower layer displacement is 0.022356 m, leading to a 265.4% increase in peak displacement when the reaction force is applied. Overall, when the PID controller applies control force to the lower layer in the hybrid isolation system, the system's motion period is extended by 1.3-1.5 times compared to when the reaction force is not applied. Figure 16: Comparison of the displacements of Tangshan waves: (**a**) comparison of upper layer displacements; (**b**) comparison of lower layer displacements. Figure 8:0 Comparison of the displacements of Superstition Mountain waves: (**a**) comparison of upper layer displacements; (**b**) comparison of lower layer displacements. In the aforementioned simulation analyses, the hybrid isolation system achieved a peak isolation efficiency of 90% when applying a control reaction force to the lower layer, demonstrating excellent seismic isolation performance. When no reaction force was applied to the lower layer, the peak isolation efficiency was around 85%. By considering the reaction force, the simulation becomes more realistic and improves the isolation efficiency. However, compared to the scenario where no control reaction force is applied, applying the control reaction force increases the upper layer displacement by approximately 95%. For the lower layer, the displacement is amplified by approximately 250% under harmonic and Superstition Mountain wave conditions. Additionally, when the hybrid isolation system applies control force to the lower layer, the motion period is extended by 1.3-1.5 times. While a longer period typically enhances the isolation performance [30], it also results in larger displacement responses.