## 3 Mechanism Analysis for Active Control
### The Resonant Characteristics of the Ribbed Plate  
The mode shapes and resonant frequencies of the orthogonally ribbed plate can be obtained by resolving the eigenvalues and eigenvectors of the homogeneous equations of Equations (14) and (15) in Ref. [8]. The theoretically calculated resonant frequencies below 500 Hz and also the results obtained by FEM method are listed in Table 3. The resonant frequencies of the cavity below 500 Hz are also shown in Table 4 to assist with the analysis.  
The mode shapes of the first ten modes of the ribbed plate B are plotted in Figure 4. The formation of the resonant mode of the orthogonally two ribs stiffened plate is similar to that of the single ribbed case. For the single ribbed plate, the modes of the base plate, of which the modal index along the rib direction are identical, couple with each other and have mutual energy transmission due to the coupling effects of the rib [8]. The overall vibrational superposition of this set of modes forms the resonant mode of the ribbed plate. For the orthogonally two ribs stiffened case, additional coupling effects induced by the horizontal rib are introduced. The coupling effects are produced in the same manner that the base plate modes, of which the modal index along the horizontal rib direction are identical, can couple with each other. Simultaneously taking the coupling effects of the horizontal and vertical ribs into considerations, it can be predicted that all base plate modes can couple with each other. One mode can always couple with any other mode through a complex coupling relation and energy transmission process. Therefore, the set of modes for constituting the resonant mode of the orthogonally ribbed plate is supposed to contain all four types of the base plate mode. In this set, one or two modes with high modal amplitudes dominate. The first twenty base plate modes are also shown in Table 5 for aiding the analysis.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multicolumn{11}{c}{**Resonant Frequency (Hz)**} \\ \hline \multirow{2}{*}{Modal index} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \cline{2-11}  & (0,0,0) & (1,0,0) & (0,1,0) & (1,1,0) & (2,0,0) & (2,1,0) & (0,2,0) & (3,0,0) & (1,2,0) & (3,1,0) & (2,2,0) \\ \hline Frequency & 0 & 143 & 205 & 250 & 287 & 352 & 410 & 430 & 434 & 476 & 500 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The resonant frequencies of the cavity below 500 Hz.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{11}{c}{**Category**} \\ \hline \multicolumn{11}{c}{**Modal Sequence**} & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** & **14** \\ \hline \multirow{3}{*}{Ribbed plate A} & Analytical & 73 & 132 & 161 & 201 & 248 & 261 & 278 & 347 & 358 & 384 & 430 & 439 & 471 & 489 \\ \cline{2-11}  & COMSOL & 76 & 135 & 165 & 204 & 250 & 264 & 280 & 347 & 361 & 383 & 430 & 439 & 472 & 493 \\ \hline \multirow{3}{*}{Ribbed plate B} & Analytical & 90 & 158 & 203 & 238 & 308 & 333 & 364 & 425 & 467 & 473 & & & & \\ \cline{2-11}  & COMSOL & 92 & 161 & 206 & 239 & 310 & 333 & 365 & 419 & 467 & 477 & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: The resonant frequencies of the ribbed plate A and B below 500 Hz.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \multicolumn{11}{c}{**Modal Index**} \\ \hline \multirow{2}{*}{Modal index} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \cline{2-11}  & (1,1) & (1,2) & (3,1) & (2,2) & (3,2) & (4,1) & (1,3) & (2,3) & (4,2) & (5,1) & (3,3) & (5,2) & (4,3) & (1,4) \\ \hline
16 & 17 & 18 & 19 & 20 & & & & & & & & & & \\ \hline (6,1) & (2,4) & (3,4) & (5,3) & (6