## 3 Numerical Results and Discussion
### Verification of Convergence and Correctness
However, in numerical calculation, the truncation value needs to be set as a finite number of terms; in order to ensure the correctness of the calculation results, the truncation value of the Fourier series in the displacement or sound pressure function needs to be increased until the solution results are stable. Therefore, the numerical calculation in Tables 2-4 is carried out here with the aim of finding a more reasonable combination of truncation values for the following examples. In order to fully satisfy the convergence and accuracy, \(M_{\rm c}\times N_{\rm c}\times Q_{\rm c}\) = 4 \(\times\) 4 \(\times\) 4 and \(M_{\rm P}\times N_{\rm P}\) = 18 \(\times\) 18 are selected as the cutoff values in the following numerical examples. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{\(M_{\rm c}\times N_{\rm c}\times Q_{\rm c}\)} & \multirow{2}{*}{\(M_{\rm p}\times N_{\rm P}\)} & \multicolumn{8}{c}{**Modal Order**} \\ \cline{3-10}  & & & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline \(3\times 3\times 3\) & \(10\times 10\) & 83.63 & 113.18 & 132.05 & 144.43 & 174.01 & 185.80 & 212.10 & 221.18 \\ \hline  & \(12\times 12\) & 82.77 & 113.18 & 132.05 & 144.39 & 173.99 & 185.80 & 211.90 & 221.18 \\ \hline  & \(14\times 14\) & 82.04 & 113.19 & 132.03 & 144.36 & 174.01 & 185.80 & 211.65 & 221.18 \\ \hline  & \(16\times 16\) & 81.45 & 113.18 & 132.01 & 144.33 & 173.96 & 185.79 & 211.48 & 221.17 \\ \hline  & \(18\times 18\) & 80.93 & 113.16 & 131.99 & 144.22 & 173.95 & 185.80 & 211.27 & 221.18 \\ \hline \(4\times 4\times 4\) & \(10\times 10\) & 83.59 & 113.18 & 132.03 & 144.42 & 174.00 & 185.80 & 209.37 & 221.17 \\ \hline  & \(12\times 12\) & 82.74 & 113.17 & 132.01 & 144.38 & 173.98 & 185.81 & 209.24 & 221.16 \\ \hline  & \(14\times 14\) & 82.02 & 113.23 & 131.99 & 144.39 & 173.96 & 185.80 & 209.07 & 221.16 \\ \hline  & \(16\times 16\) & 81.42 & 113.13 & 131.97 & 144.31 & 173.98 & 185.80 & 208.97 & 221.15 \\ \hline  & \(18\times 18\) & 80.91 & 113.21 & 131.93 & 144.33 & 173.96 & 185.80 & 208.83 & 221.15 \\ \hline \(5\times 5\times 5\) & \(10\times 10\) & 83.59 & 113.18 & 132.03 & 144.40 & 174.00 & 185.79 & 209.29 & 221.17 \\ \hline  & \(12\times 12\) & 82.73 & 113.18 & 132.01 & 144.38 & 173.98 & 185.79 & 209.16 & 221.17 \\ \hline  & \(14\times 14\) & 82.01 & 113.17 & 131.99 & 144.39 & 173.96 & 185.79 & 208.99 & 221.16 \\ \hline  & \(16\times 16\) & 81.42 & 113.18 & 131.96 & 144.34 & 173.95 & 185.80 & 208.89 & 221.16 \\ \hline  & \(18\times 18\) & 80.91 & 113.18 & 131.93 & 144.32 & 173.94 & 185.80 & 208.75 & 221.15 \\ \hline  & FEM & 82.42 & 113.65 & 132.18 & 144.20 & 173.89 & 185.89 & 206.74 & 221.19 \\ \hline \end{tabular}
\end{table} 
Table 4: Convergence analysis of the coupling system among the composite laminated rotationally stiffened plate and the cylindrical-spherical cavities. Next, we will research the stiffness values of the springs in the coupled system, which specifically include linear springs \(k(k_{u},k_{v},k_{w})\), torsional springs \(K(K_{r},K_{\theta})\), and plate-beam coupling springs \(k_{cp}\) (\(k_{uc}^{\rm CP}\), \(k_{vc}^{\rm CP}\), \(k_{uc}^{\rm CP}\), \(K_{xc}^{\rm CP}\), and \(K_{yc}^{\rm CP}\)). Due to the influence of acoustic cavity coupling in the coupled system, we will not analyze the in-plate coupling springs \(\rm{kc}\) (\(k_{uc}^{\rm P}\), \(k_{vc}^{\rm P}\), \(K_{vc}^{\rm P}\), \(K_{vc}^{\rm P}\)) here. Taking the example of the coupling system among the composite, laminated, rotational plate and the cylindrical-spherical cavities, we will analyze the impact of the boundary-spring stiffness values. Figure 5 provides the variation curves of the first four natural frequencies under different boundary-spring stiffness values.