## 3 Mechanism Analysis for Active Control
### Physical Mechanism Analysis
Even for the cavity and plate mode pair that satisfies the geometrical coupling relation (expressed in the coupling coefficient \(\mathbf{L}_{b}(i_{plate-mode},i_{cavity-mode})\
eq 0\)), the part of energy contained in the plate mode contributed from this cavity mode is supposed to be transmitted both in direct and indirect ways. In general, each cavity mode can couple with and transmit energy to any one of the base plate modes. The mechanism of active control can be interpreted by analyzing the variation of energy contributions of different cavity modes to each base plate mode in controlled conditions. According to Equation (15) in Ref. [8], the modal amplitudes of the base plate B can be expressed as:  
\[\mathbf{q}_{b}=\left(\mathbf{K}_{b}-\omega^{2}\mathbf{M}_{b}\right)^{-1}( \mathbf{L}_{b})^{\mathrm{T}}\mathbf{P}=\mathbf{Z}\mathbf{P} \tag{2}\]  
where \(\mathbf{Z}=\left(\mathbf{K}_{b}-\omega^{2}\mathbf{M}_{b}\right)^{-1}(\mathbf{ L}_{b})^{\mathrm{T}}\) is defined as the transfer impedance matrix. For the energy contained in each base plate mode, its contribution from each cavity mode can be calculated using Equation (2). These contributions can be represented by the value of \(z(i,j)P(i)\) which are expressed as a complex value (amplitude [phase]) to illustrate their superposition or cancellation effects. In Case 2, the energy contained in the first, second and fourth base plate B modes, which are contributed from the first five cavity modes, are listed in Table 6. In Case 3, the energy contained in the third to eighth modes of the base plate B that are contributed from the first ten modes of the cavity are also listed in Table 6.  
In Case 2, it is noteworthy that the energy contained in these excited base plate modes are all mainly transmitted from the second cavity mode (1,0,0), and only a little from the (0,1,0) mode. Although the (1,0,0) mode is weakly excited with small amplitude, as shown in Figure 7c, its energy contribution is predominant due to the direct strongly coupling effects between this mode and the (2,1) mode, seen in Table 6 (Case 2). Whereas, although the 3 rd (0,1,0) cavity mode is strongly excited, its energy contribution is insignificant. The reason is that the energy from this cavity mode to these base plate modes is transmitted through complicated indirect coupling path. In a controlled condition, the (1,0,0) cavity mode is firstly suppressed to a certain degree. Then, the quantity of its energy contribution to the base plate modes is considerable with that of the (0,1,0) mode. But, the phase of the (1,0,0) mode is just adjusted to have a 180\({}^{\circ}\) phase difference (opposite phase) with that of the (0,1,0) mode. The energy contributions from these two cavity modes further cancel each other to achieve the suppression of the modes of the base plate B. The cavity modal suppression and rearrangement work together to achieve the suppression of the modes of the base plate B, as shown in Figure 7d and Table 6 (Case 2). Besides, the control process for different base plate modes is nearly the same. The energy contained in these plate modes is almost completely transmitted from the same energy source of a group of excited cavity modes. Hence, the control effects will be delivered successively to all of these excited base plate modes, as shown in Table 6 (Case 2). The situation in Case 3 is basically the same as that in Case 2. The energy contained in these intensively excited (1,2), (3,1), (2,2), (3,2), (4,1), and (1,3) plate modes is mainly transmitted from the (3,0,0) and (1,2,0) cavity modes. The phase difference of the two parts of energy contribution from these two cavity modes is small before control. Hence, the two parts of energy contribution to base plate modes approach complete superposition. Since the (1,2,0) mode is intensively excited with a highest modal amplitude, the energy contribution from this mode is more than that from the (3,0,0) mode. The (1,2,0) mode is firstly suppressed to a certain degree in a controlled condition.