## 4 Simulation Study
### Simulation Results and Discussion
acceleration is much higher than that of the lower layer, with a maximum acceleration of 0.7291 m/s\({}^{2}\) for the upper layer and 0.5140 m/s\({}^{2}\) for the lower layer. Figure 10c displays the control force applied by the PID controller to the superstructure under harmonic waves. Figure 10d 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 89.2%. Figure 11 presents a comparison of displacement under harmonic wave conditions, with and without the control force applied to the lower layer. Figure 11a shows the upper layer displacement comparison under harmonic waves. When the PID controller applies a control reaction force to the lower layer, the peak displacement of the upper layer reaches 0.0008800 m. When no control reaction force is applied to the lower layer, the peak displacement of the upper layer is 0.0004800 m. Compared to the case without a control reaction force, applying the control reaction force increases the peak displacement by 83.3%. Figure 11b depicts the lower layer displacement comparison under harmonic waves. When the PID controller applies a control reaction force to the lower layer, the peak displacement of the lower layer is 0.009771 m. Without the control reaction force, the peak displacement of the lower layer is 0.002432 m. Applying the control reaction force results in a 302.1% 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 1.5 times compared to the case without the control reaction force. Figure 10: Response in a simple harmonic state with an amplitude of 0.5 and a frequency of 10 Hz: (**a**) displacement response; (**b**) acceleration response; (**c**) control force response; (**d**) PID parameter response. Figure 12 illustrates the displacement response of the hybrid isolation system under random wave conditions. Figure 12a shows the displacement response under the Tangshan wave. In this scenario, the peak displacement of the upper layer is 0.009986 m, while the peak displacement of the lower layer reaches 0.18225 m. The isolation efficiency of the hybrid isolation system under the Tangshan wave is 94.5%. Figure 12b presents the displacement response under the Superstition Mountain wave. For this case, the peak displacement of the upper layer is 0.005991 m, and the peak displacement of the lower layer is 0.08173 m. The isolation efficiency of the hybrid isolation system under the Superstition Mountain wave is 92.6%. Figure 13 illustrates the acceleration response of the hybrid isolation system under random wave conditions. Figure 13a shows the acceleration response under the Tangshan wave. The peak acceleration of the upper layer is 0.3007 m/s\({}^{2}\), while the peak acceleration of the lower layer reaches 1.6292 m/s\({}^{2}\). Figure 13b presents the acceleration response under the Superstition Mountain wave. In this scenario, the peak acceleration of the upper layer is 1.0999 m/s\({}^{2}\), and the peak acceleration of the lower layer is 2.2751 m/s\({}^{2}\). Figure 14 illustrates the control forces of the hybrid isolation system under random wave conditions. Figure 14a shows the control forces under the Tangshan wave. Figure 14b presents the control forces under the Superstition Mountain wave. Figure 11: 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. Figure 12: Displacement response in a random wave state: (**a**) displacement response of Tangshan wave; (**b**) displacement response of Superstition Mountain wave. Figure 15 illustrates the variation of PID parameters in the hybrid isolation system under random wave conditions. Figure 15a shows the PID parameter variation under the Tangshan wave. Figure 15b presents the PID parameter variation under the Superstition Mountain wave.