## 1 Introduction
### Calculation of the Transfered Power Flow of the Coupling System
Simulation Calculation and Analysis of Active Feedback Sound Insulation Strategy in Frequency Domain  
In this paper, the average energy input \(P_{\rm tr}\) in the acoustic cavity is calculated by a MATLAB environment simulation and used as an evaluation index of the sound insulation effect. In order not to lose the generality of the system, the active sound insulation units are laid out symmetrically, and the number of arrangements is four. As mentioned in Section 2.2, the time lag \(\omega\tau_{l}\prime\) of the active actuator can be neglected in the simulation due to its extremely small value. To simplify the simulation and analysis, all active actuators are used with the same specifications, so that the electrical time constant \(\tau_{1}=\tau_{2}=\tau_{3}=\tau_{4}=\tau\), the amplifier gain parameter \(k_{\rm a1}=k_{\rm a2}=k_{\rm a3}=k_{\rm a4}=k_{\rm a}\), and the control circuit resistance \(R_{1}=R_{2}=R_{3}=R_{4}=R\). Additionally, \(g_{rs}^{(k)}=0\) (\(r,s=1,2,3,4\); \(k=1,2,\dots,6\)), except for \(g_{11}^{(k)},g_{22}^{(k)},g_{33}^{(k)}\), and \(g_{44}^{(k)}\). Let \(K=k_{\rm a}/(1+\mathrm{j}\omega\tau)\). Then, Equation (32) can be simplified as follows:  
\[\mathbf{G}_{k}=\frac{K}{R}\Bigg{[} \begin{matrix}\delta_{11}^{(k)}&&\\ &\delta_{22}^{(k)}&&\\ &&\delta_{33}^{(k)}&\\ &&\delta_{44}^{(k)}\end{matrix} \Bigg{]} \tag{47}\]  
From Equation (47), the feedback parameter matrices \(\mathbf{G}_{1}\), \(\mathbf{G}_{2}\), \(\mathbf{G}_{3}\), \(\mathbf{G}_{4}\), \(\mathbf{G}_{5}\) and \(\mathbf{G}_{6}\) are only linearly related to \(g_{11}^{(k)},g_{22}^{(k)},g_{33}^{(k)}\), and \(g_{44}^{(k)}\). To further facilitate variable analysis, let all the actuators of the system have the same feedback signal controller gain at their respective installation points, that is \(g_{11}^{(k)}=g_{22}^{(k)}=g_{33}^{(k)}=g_{44}^{(k)}\). The settings of the system's basic parameters are given here, as shown in Table 1.  
To completely represent the effect of the continuous variation of the feedback parameter values in the frequency domain on the average energy transfer \(P_{\text{tr}}\) in the acoustic cavity, the \(P_{\text{tr}}\) values in the frequency domain range from 0 to 1000 Hz for different values of the feedback parameters are calculated numerically to generate the comparison graphs in this section. The 3D image of surf (X, Y, Z) and its corresponding plane three views are generated by simulation, where the feedback parameter value \(g_{11}^{(k)}\) --frequency \(Fre\) --average energy input \(P_{\text{tr}}\) in the acoustic cavity is the X-Y-Z axis of the 3D image. The 2D images and 3D images are analyzed by comprehensive comparison, as shown in Figures 2-7. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Vibration Isolation System** & **Basic Parameters** & **Symbols** & **Units** &
\begin{tabular}{c} **Numerical** \\ **Values** \\ \end{tabular} & **Remarks** \\ \hline \multirow{4}{*}{Rectangular chamber} & Length & \(L_{x}\) & m & 0.4 & \\  & Width & \(L_{y}\) & m & 0.6 & \\  & Height & \(L_{z}\) & m & 0.7 & \\  & Density of the medium & \(\rho_{0}\) & kg/m\({}^{3}\) & 1000 & \\  & Sound velocity & \(\epsilon_{0}\) & m/s & 1500 & \\ \hline \multirow{4}{*}{Double-layer structure} & Thickness of each layer & \(h\) & cm & 0.5 & Stainless steel plate \\  & [PERSON]’s modulus of elasticity & \(E\) & N/m\({}^{2}\) & \(2.0\times 10^{11}\) & \\ \cline{1-1}  & Density & \(\rho\) & kg/m\({}^{3}\) & \(7.8\times 10^{3}\) & \\ \cline{1-1}  & Poisson’s ratio & \(\mu\) & & 0.28 & \\ \cline{1-1}  & Damping ratio & \(\xi\) & & 0.01 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Setting of system’s basic parameters.