## 1 Introduction
### Calculation of the Transfered Power Flow of the Coupling System
However, the effect of \(g_{11}^{(1)}\) on the optimization of system sound insulation in the high-frequency region is not obvious. In this simulation, the significant effective sound insulation frequency range is about 1-400 Hz with proper assignment of \(g_{11}^{(1)}\). Combining Figure 2c-f, when \(g_{11}^{(1)}<0\) or \(1\times 10^{7}<g_{11}^{(1)}<0.2\times 10^{9}\) or \(g_{11}^{(1)}>0.8\times 10^{9}\), the first resonance peak is basically eliminated, which indicates that the sound insulation effect of the system is well. When \(0.2\times 10^{9}<g_{11}^{(1)}<0.8\times 10^{9}\), 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 \(0.5\times 10^{9}<g_{11}^{(1)}<0.6\times 10^{9}\). This indicates that the \(g_{11}^{(1)}\) at this point exacerbates the resonance effect and deteriorates the system's sound insulation. In summary, the absolute displacement 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. However, this control method is not ideal for system's sound insulation in the high-frequency region, and even some inappropriate values of feedback parameters can deteriorate the system's high-frequency sound insulation instead. As shown in Figure 3a,b, the overall \(P_{\text{tr}}\) curve shows a gradual sinking trend as \(\left|g_{11}^{(2)}\right|\) increases. In the low- and medium-frequency regions, the first resonance peak gradually shifts to the left with the increase of \(\left|g_{11}^{(2)}\right|\), and its peak is also reduced. In the high-frequency region, \(g_{11}^{(2)}\) does not make the problem solved, where the system's higher-order modes are excited and trigger a large number of peaks. This indicates that properly assigned \(g_{11}^{(2)}\) can optimize the system's sound insulation in the low- and medium-frequency region. However, the effect of \(g_{11}^{(2)}\) on the optimization of system sound insulation in the high-frequency region is not obvious. In this simulation, the significant effective sound insulation frequency range is about 1-400 Hz with proper assignment of \(g_{11}^{(2)}\). Combining Figure 3c-f, when \(g_{11}^{(2)}-0.1\times 10^{5}\) or \(g_{11}^{(2)}>0.1\times 10^{5}\), the first resonance peak is basically eliminated, which indicates that the sound insulation effect of the system is well. When \(-0.8\times 10^{5}<g_{11}^{(2)}<0.1\times 10^{5}\) (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. In summary, the absolute velocity feedback control method plays a significant role for the system's sound insulation in the low- and medium-frequency domain. Almost all feedback parameter values are validly taken and can optimize the system's sound insulation in the low and medium frequency region. However, this control method is not ideal for system's sound insulation in the high-frequency region. As shown in Figure 4a,b, the overall \(P_{\text{tr}}\) curve shows a gradual sinking trend as \(\left|g_{11}^{(3)}\right|\) increases. In the low- and medium-frequency regions, the first resonance peak gradually shifts to the left with the increase of \(\left|g_{11}^{(3)}\right|\), and its peak is also reduced. In the high-frequency region, the number of large peaks triggered by the excitation of the higher-order modes of the system is reduced. This indicates that a properly assigned \(g_{11}^{(3)}\) has a significant optimization effect on the system's sound insulation in both the low and medium-frequency regions as well as the high-frequency region.