## 3 Actuators/Sensors Quantity Configuration
Under the same control excitation force, the modal frequency point with a larger eigenvalue can provide more control energy. Similarly, the modal frequency where the larger eigenvalue is located is the sensitive response position of the system vibration. To sum up, according to the eigenvalues in the energy correlation matrix \(G_{2m\times 2m}\), the \(n\)-order modal frequency that satisfies a certain energy ratio of \(\delta\) can be selected to realize the active control of acoustic and vibration of the structure. Considering different vibration systems, the value of \(\delta\) is generally 90% or higher as follows  
\[\frac{\sum_{i=1}^{n}\lambda_{i}}{\sum_{i=1}^{m}\lambda_{i}}>\delta \tag{10}\]  
The ranking calculation obtains the minimum value of \(n\) when the above equation is satisfied. Consequently, it can be obtained that under certain operating conditions, the proportion of system vibration response energy exceeds \(\delta\) minimum number of modes. This quantity is the minimum number of control modes to achieve efficient active sound and vibration suppression for a vibration system. \[\frac{\sum_{i=1}^{n}E_{i}}{\sum_{i=1}^{m}E_{i}}>\delta \tag{11}\]  
The size of \(\lambda_{i}\) in the above formula corresponds to the proportion of the energy contribution of the \(i\)-th modal force of the structure to the vibration response of the system. For the multidegree of freedom and irregular large-space complex structures involved in this paper, \(\lambda_{i}\) is not easy to obtain by computational simulation. Based on the existing conclusions, applying equal-intensity excitation in a certain frequency range to obtain the vibration energy of the vibration system at each modal frequency, and adopting the same principle to achieve sorting and statistics, as shown in Equation (11). \(E_{i}\) corresponds to the \(i\)-th vibration energy from large to small at each modal position, and its value is the sum of the kinetic energy and elastic strain energy of the structure at the modal position. Using this evaluation index, the reasonable configuration of the number of sensors/actuators in complex model control can be realized, and the optimal control order and corresponding modal frequency of the investigated structure under active acoustic and vibration control can be determined. Base on the modal shape analysis, in accordance with the above structural model and excitation loading conditions, the frequency domain simulation analysis is carried, and the modal energy value of each frequency point is obtained through the acceleration surface integral. The results are shown in Figure 3, the modal energy is composed of elastic energy and kinetic energy. The above energy data are statistically transformed and filtered to obtain the modal energy distribution of each order as shown in Figure 4.  
From Figure 4, in the 0-200 Hz frequency band, after applying a specific excitation, the various order modes of the body structure can be excited to a certain extent, among which the modal frequency positions correspond to 20 Hz, 45 Hz, 75 Hz and 157 Hz. Resonance energy is generated, and its mode shape is shown in Figure 4, which is close to the previous modal shape analysis results. Simultaneously, combined with Equation (11), the modal energy results are sorted and processed, and the energy proportions of multiorder modes are obtained as shown in Table 1. \begin{table}
\begin{tabular}{c c c} \hline
**Eigenfrequency** & **Corresponding Energy** & **Energy Ratio** \\ \hline
157 & 40.563 & 65.9580\% \\
87 & 10.4412 & 16.9781\% \\
45 & 3.58916 & 5.8362\% \\
20 & 1.65848 & 2.6968\% \\
116 & 1.55857 & 2.5343\% \\
143 & 1.06384 & 1.7299\% \\
172 & 0.96976 & 1.5769\% \\
15 & 0.60876 & 0.9899\% \\
87 & 0.555772 & 0.9037\% \\
191 & 0.48967 & 0.7962\% \\ \hline \end{tabular}
\end{table} 
Table 1: Energy ratio of each order mode. Figure 4: Mode energy distribution of each order: (**a**) before treatment; (**b**) after treatment. Figure 3: Surface integral results of elastic energy and kinetic energy under working load excitation.