## 1 Introduction
### Calculation of the Transfered Power Flow of the Coupling System
In this simulation, the significant effective sound insulation frequency range is 1-1000 Hz with proper assignment of \(g_{11}^{(3)}\). Combining Figure 4c-f, when \(g_{11}^{(3)}<-3000\) or \(g_{11}^{(3)}>1000\), the first resonance peak is basically eliminated, which indicates that the sound insulation effect of the system is well. When \(-3000<g_{11}^{(3)}<1000\) (except 0), the first resonance peaks still exist, but they all move to the lower-frequency region, which indicates that the system has a certain sound insulation effect. However, from the peak perspective, the first resonance peak is significantly raised at \(-2700<g_{11}^{(3)}<-2600\). This indicates that the \(g_{11}^{(3)}\) at this point exacerbates the resonance effect and deteriorates the system's sound insulation. In summary, the absolute acceleration feedback control method plays a significant role for the system's sound insulation in the low- and medium-frequency domain. The vast majority of the feedback parameter values are within the effective range of values and can optimize the system's sound insulation in the low- and medium-frequency region. At the same time, this control method produces a significant effect on the system's high-frequency sound insulation. In addition, the average energy input amplitude-frequency response of the system is most sensitive under the absolute acceleration feedback control method compared to the absolute displacement and velocity feedback control methods. As shown in Figure 5a,b, the overall \(P_{\text{tr}}\) curve has a slight rightward shift as \(\left|g_{11}^{(4)}\right|\) increases. In the high-frequency region, \(g_{11}^{(4)}\) does not make the problem solved, where the system's higher-order modes are excited and trigger a large number of peaks, and even the \(P_{\text{tr}}\) value increases slightly. From the point of view of resonant frequencies, improperly assigned \(g_{11}^{(4)}\) can slightly deteriorate the system's sound insulation in the low- and medium-frequency regions. From the point of view of the average energy input value, improperly assigned \(g_{11}^{(4)}\) can also deteriorate the system's sound insulation in the high-frequency region. In this simulation, the significant effective sound insulation frequency range is about 1-400 Hz with proper assignment of \(g_{11}^{(4)}\). Combining Figure 5c,f, in the narrow region on the right side very close to \(g_{11}^{(4)}=0\), the first resonance peak is basically eliminated, which indicates that the sound insulation effect of the system is well. Outside this narrow region, although the peak is reduced compared to that at \(g_{11}^{(4)}=0\), it is not very significant. In addition to the first resonance peak, the second and third resonance peaks are still present. Therefore, the sound insulation effect of the system is not ideal at this time. In summary, the effective range of the feedback parameters for the relative displacement feedback control method is narrow. Within the effective range of values, it can improve the system's low- and medium-frequency sound insulation capability. However, outside the effective range, it does not play a role in the system's low- and medium-frequency sound insulation, and even plays a deteriorating role for the system's high-frequency sound insulation. On the whole, the sound insulation effect under the relative displacement feedback control method is worse than that of the absolute displacement feedback control method. As shown in Figure 6a,b, the overall \(P_{\text{tr}}\) curve has a slight rightward shift as \(\left|g_{11}^{(5)}\right|\) increases. In the high-frequency region, \(g_{11}^{(5)}\) does not make the problem solved, where the system's higher-order modes are excited and trigger a large number of peaks, and even the \(P_{\text{tr}}\) value increases slightly. From the point of view of resonant frequencies, improperly assigned \(g_{11}^{(5)}\) can slightly deteriorate the system's sound insulation in the low- and medium-frequency regions.