## 3 Mechanism Analysis for Active Control
### Physical Mechanism Analysis  
For the single ribbed double-panel system, one cavity mode can couple with and transmit energy to a set of modes of the radiating base plate. Most of the modes in this set have no geometrical coupling relations with this cavity mode. Essentially, they set up relevance with it through the coupling effects of the rib. This in turn means that the energy contained in one base plate mode is supposed to be transmitted from different cavity modes. Accordingly, the prominent mechanism of active control supresses the base plate modes by rearranging the cavity modes. The suppression and rearrangement mechanisms work together. Due to the complicated coupling relations between the ribbed plate and cavity, the control mechanisms for the orthogonally two ribs stiffened case differ from those of the single ribbed case, and also present more interesting phenomena. Detailed analysis is carried out in a similar way to that of the single ribbed case when the exciting frequency is chosen as the resonant and off-resonant frequencies. (a) On the resonant frequency of cavity  
It is assumed that the exciting frequency is \(f=352\) Hz, where the 6 th (2,1,0) cavity mode resonates. The modal amplitudes of the first twenty cavity modes and base plate B modes before and after control are shown in Figure 6. Many modes of the base plate B are excited. Only (1,2) mode is excited directly by this cavity mode due to their geometrical coupling relations. Other plate modes are all excited by the (1,2) mode in an indirect way through the coupling effects of the vertical and horizontal ribs. For instance, the (2,2) mode is excited indirectly by the coupling effects of the vertical rib. The (4,1) mode should be excited by the (1,1) mode through the coupling effects of the vertical rib, where the prerequisite is that the (1,1) mode should firstly be excited by the (1,2) mode through the coupling effects of the horizontal rib. The (1,3) mode is excited indirectly by the coupling effects of the horizontal rib. Then, it further excites the (2,3) mode by the coupling effects of the vertical rib. In general, one cavity mode can couple with all four types of the base plate modes in direct or indirect ways, due to the coupling effects of the orthogonal two ribs. The coupling relations and energy transmission among the four types of base plate modes are complicated and even difficult to be traced. However, the energy contained in each excited base plate mode comes from one cavity mode. Once the cavity mode is suppressed, all the excited base plate modes are suppressed accordingly, as can be seen in Figure 6b. Only one cavity mode is excited at the resonant frequencies of the cavity, showing that the modal suppression mechanism works effectively. This is formally analogous with that in the single ribbed case. The only difference is that more base plate modes, i.e., all four types of the base plate mode, have participated in the control process. (b) On the resonant frequency of the ribbed plate A  
Three cases, where the exciting frequency is successively fixed at \(f=248\) Hz (Case 1),\(f=161\) Hz (Case 2), and \(f=439\) Hz (Case 3) are considered. The 1 st, 3 rd, and 12 th modes are resonant, respectively. Figure 6: Modal amplitudes of the first twenty modes before and after control (\(f=352\) Hz). (**a**) Cavity modes. (**b**) Base plate B modes. Figure 7 shows the modal amplitudes of the cavity mode and base plate B mode before and after control in these cases. The control mechanism at 248 Hz is also the cavity mode suppression, as shown in Figure 7 (Case 1), which is similar to that of the cavity resonant frequency. Many cavity modes are excited in Cases 2 and 3. Due to the coupling effects of the orthogonal two ribs, the particular coupling relations and energy transmission rule from each excited cavity mode to these excited base plate modes may be difficult to be traced.