## 3 Numerical Results and Discussion
### Verification of Convergence and Correctness
\begin{table}
\begin{tabular}{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{6}{c}{**Modal Order**} \\ \cline{3-10}  & & & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline
3 \(\times\) 3 & 10 \(\times\) 10 & 82.33 & 132.23 & 132.24 & 169.89 & 210.59 & 215.29 & 221.15 & 221.27 \\  & 12 \(\times\) 12 & 81.78 & 132.23 & 132.24 & 169.89 & 210.30 & 215.28 & 221.15 & 221.27 \\  & 14 \(\times\) 14 & 81.32 & 132.23 & 132.29 & 169.86 & 210.01 & 215.28 & 221.14 & 221.27 \\  & 16 \(\times\) 16 & 80.95 & 132.21 & 132.24 & 169.89 & 209.78 & 215.27 & 221.14 & 221.27 \\  & 18 \(\times\) 18 & 80.64 & 132.18 & 132.23 & 169.89 & 209.54 & 215.27 & 221.14 & 221.27 \\
4 \(\times\) 4 & 10 \(\times\) 10 & 82.33 & 132.24 & 132.25 & 169.89 & 210.59 & 215.28 & 221.17 & 221.27 \\  & 12 \(\times\) 12 & 81.78 & 132.23 & 132.24 & 169.90 & 210.30 & 215.28 & 221.14 & 221.27 \\  & 14 \(\times\) 14 & 81.32 & 132.22 & 132.24 & 169.89 & 210.01 & 215.28 & 221.14 & 221.27 \\  & 16 \(\times\) 16 & 80.95 & 132.17 & 132.36 & 169.88 & 209.77 & 215.27 & 221.14 & 221.27 \\  & 18 \(\times\) 18 & 80.64 & 132.29 & 132.33 & 169.88 & 209.54 & 215.27 & 221.24 & 221.27 \\
5 \(\times\) 5 5 & 10 \(\times\) 10 & 82.33 & 132.24 & 132.26 & 169.89 & 210.59 & 215.28 & 221.15 & 221.27 \\  & 12 \(\times\) 12 & 81.78 & 132.23 & 132.24 & 169.89 & 210.30 & 215.28 & 221.15 & 221.27 \\  & 14 \(\times\) 14 & 81.32 & 132.22 & 132.24 & 169.89 & 210.01 & 215.28 & 221.14 & 221.27 \\  & 16 \(\times\) 16 & 80.95 & 132.21 & 132.24 & 169.89 & 209.77 & 215.27 & 221.14 & 221.27 \\  & 18 \(\times\) 18 & 80.64 & 132.15 & 132.23 & 169.88 & 209.54 & 215.27 & 221.14 & 221.27 \\  & FEM & 82.33 & 132.24 & 132.26 & 169.89 & 210.59 & 215.28 & 221.15 & 221.27 \\ \hline \end{tabular}
\end{table} 
Table 3: Convergence analysis of the coupling system among the composite laminated rotationally stiffened plate and the cylindrical-cylindrical cavities. Similarly, from Table 4, it can be observed that the natural frequencies of various modes have essentially converged when \(M_{\rm c}\times N_{\rm c}\times Q_{\rm c}\) is 4 \(\times\) 4 \(\times\) 4 and \(M_{\rm P}\times N_{\rm P}\) is 18 \(\times\) 18. At this point, the maximum error in the natural frequencies of all modes compared to the finite element method results is 1.83%. Table 5 presents the natural frequencies of the coupling system among the composite laminated rotationally stiffened plate and the cylindrical-conical cavities under different plate truncation values and cavity truncation conditions obtained using the present method and the finite element method. The boundaries of the composite laminated rotationally stiffened plate are set to SSSS, the number of stiffeners (\(n\)) is 1, and the coupling system's geometric parameters are as follows: \(R_{1}\) = 0.6 m, \(R_{2}\) = 1.2 m, \(R_{3}\) = 1.8 m, \(\alpha\) = 30\({}^{\circ}\), \(L\) = 2 m, \(h_{\rm p}\) = 0.02 m, \(\theta\) = 90\({}^{\circ}\), \(R_{\rm b1}\) = 1.5 m, \(p_{1}\) = 0.06 m, and \(h_{\rm l}\) = 0.04 m. The material of the laminated plate, laminated beams, layer angles, and the medium in the cavities remain the same as in Tables 3 and 4. It can be observed that the natural frequencies of various modes have essentially converged when \(M_{\rm c}\times N_{\rm c}\times Q_{\rm c}\) is 4 \(\times\) 4 \(\times\) 4 and \(M_{\rm P}\times N_{\rm P}\) is 18 \(\times\) 18. At this point, the maximum error in the natural frequencies of all modes as compared to the finite element method results is less than 3.51%. A continuous function can be expanded by Fourier series, where the truncation value is the Fourier coefficient, which characterizes the smoothness and convergence of the function. According to the Fourier series convergence property and theorem, when the truncation value goes to infinity, the Fourier coefficient goes to zero.