## 3 Mechanism Analysis for Active Control  
### Model Parameters Assignment and Model Validation  
The geometrical parameters and material properties of the model are listed in Table 1. The structure damping is ignored in the modeling. Thus, a constant internal loss factor \(\eta=0.01\) is introduced into the complex Young's modulus as \(E(1+i\eta)\) to make the simulation results more practical. The ribs are modeled as uniform beam elements with a rectangular cross section. All of the ribs have the same geometrical size of the rectangular cross section. The plane wave has an amplitude of \(P_{0}=1\) Pa, and is an incident at \(\theta=\pi/4\) and \(\alpha=\pi/4\). The controllable point source is located at \((0.1l_{x},0.1l_{y},0.1h)\) in the cavity, in order to excite most of the low frequency modes of the cavity (up to 500 Hz). Generally speaking, the convergence of the model can be achieved in the low frequency range by increasing the number of modes, which can guarantee the accuracy of the solution. Once the solution is convergent at a given frequency, it is also convergent for all frequencies lower than that. For the upper frequency (500 Hz) considered in this research, the numbers of the modes for each subsystem, i.e., the base plate, the rib, and the cavity, are determined as \(M=N=10\), \(N=10\) and \(L=40\). The increment of the radiated sound power of the ribbed plate B is less than 1 dB when the numbers of the mode of the subsystem increase from the case of N2 to N1, which guarantees the accuracy of the model. The variation trend of the radiated sound power of the ribbed plate B with the numbers of the modes gradually increasing is shown in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Geometrical Parameters** & **Value** & **Material Properties** & **Value** \\ \hline The length and width of the base & \(l_{x}=1.2\) m & The material of the & \\ plate A and B & \(l_{y}=0.84\) m & base plates and the ribs & aluminum \\ \hline The thickness of the base & \(h_{a}=0.006\) m & The density of & \\ plate A and B & \(h_{b}=0.008\) m & aluminum & \(\rho=2790\) kg/m\({}^{3}\) \\ \hline The locations of the vertical and horizontal ribs for ribbed plate A & \(x_{a}=0.3\) m & The Young’s modulus & \\  & \(y_{a}=0.21\) m & of aluminum & \(E=7.2\times 10^{10}\) N/m\({}^{2}\) \\ \hline The locations of the vertical and horizontal ribs for ribbed plate B & \(x_{b}=0.9\) m & The Poisson’s ratio of & \\  & \(y_{b}=0.63\) m & aluminum & \(\sigma=0.34\) \\ \hline The size of the rectangular cross section of the rib (wide \(\times\) high) & \(A=0.003\times 0.04\) m\({}^{2}\) & The density of air & \(\rho_{0}=1.21\) kg/m\({}^{3}\) \\ \hline The depth of cavity & \(h=0.2\) m & The sound speed of air & \(c_{0}=344\) m/s \\ \hline The location of the control source & \((0.1l_{x},0.1l_{y},0.1h)\) & The damping ratio of & \\  & & acoustical mode & \(\xi_{l}=0.001\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The geometrical parameters and material properties of the system. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **The Number of the Modes** & **The Radiated Power of the Ribbed Plate B** \\ \hline N1 & \(M=N=10\), \(N=10\), \(L=40\) & 45.8 dB \\ N2 & \(M=N=9\), \(N=9\), \(L=30\) & 45.2 dB \\ N3 & \(M=N=8\), \(N=8\), \(L=20\) & 44.8 dB \\ N4 & \(M=N=6\), \(N=6\), \(L=20\) & 43.5 dB \\ N5 & \(M=N=5\), \(N=5\), \(L=20\) & 37.6 dB \\ N6 & \(M=N=4\), \(N=4\), \(L=20\) & 29.1 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The variation trend of the radiated power of the ribbed plate B (at 500 Hz). For validating the accuracy of the theoretical model, the double-panel structure is also modeled by the Finite Element Software (FES, COMSOL (5.2a, COMSOL Inc., Stockholm, Sweden)). The upper and the lower ribbed plate, and the cavity are modeled using the Volume element. The model is divided by the \"Free tetrahedral mesh\". The element size of the mesh in COMSOL is subdivided into nine grades. The grade of normal is chosen in this study.