## 3 Numerical Results and Discussion
### Verification of Convergence and Correctness
The material parameters and the dimensional parameters of the coupled system are the same as those in Table 4.  
As shown in Figure 5a, initially, the torsional spring stiffness value \(K\) is set to zero, and when the linear boundary spring stiffness value k varies in the range from \(10^{-4}\) to \(10^{5}\), the natural frequencies of the coupled system remain at zero. At this point, the laminated plate is considered to have free boundary conditions. As the linear spring stiffness value increases, the natural frequencies of the coupled system also increase. When the linear spring stiffness value k increases to \(10^{12}\), the natural frequencies become relatively stable, and the boundaries of the laminated plate in the coupling system can be considered as simply supported. In Figure 5b, the linear boundary spring stiffness value is kept constant at \(10^{16}\). Because the laminated plate in the coupled system is located between two cavities, the displacement in the torsional direction is already restricted. Therefore, the natural frequencies of the coupled system remain unchanged as the torsional spring stiffness value K increases. It can be assumed that the natural frequencies of the laminated plate in the coupled system remain consistent when the boundary conditions are either clamped or simply supported. Table 6 provides the values of the boundary spring stiffness under different boundary conditions and also lists the values of two elastic boundary condition spring stiffness values besides the classical boundary conditions. Similarly, we research the plate-beam coupling conditions, taking the example of the coupling system among the composite laminated rotationally stiffened plate and the cylindrical-cylindrical cavities. In this case, the material parameters and dimensional parameters are the same as those in Table 3. Figure 6 provides the variation curves of the natural frequencies under different plate-beam coupling spring stiffness values for two boundary conditions. From Figure 6, it can be observed that, when the plate-beam  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{\(M_{\epsilon}\times N_{\epsilon}\times Q_{\epsilon}\)} & \multirow{2}{*}{\(M_{\text{p}}\times N_{\text{p}}\)} & \multicolumn{8}{c}{**Modal Order**} \\ \cline{3-10}  & & & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline \multirow{2}{*}{\(3\times 3\times 3\)} & \(10\times 10\) & 72.86 & 84.98 & 89.62 & 95.60 & 111.71 & 131.07 & 141.42 & 144.26 \\  & \(12\times 12\) & 72.56 & 85.03 & 89.62 & 95.09 & 111.71 & 130.72 & 141.42 & 144.27 \\  & \(14\times 14\) & 72.53 & 84.98 & 89.66 & 94.58 & 111.70 & 130.35 & 141.42 & 144.26 \\  & \(16\times 16\) & 72.52 & 84.96 & 89.62 & 94.18 & 111.70 & 130.08 & 141.42 & 144.26 \\  & \(18\times 18\) & 72.51 & 84.98 & 89.65 & 93.78 & 111.70 & 129.79 & 141.42 & 144.29 \\ \multirow{2}{*}{\(4\times 4\times 4\)} & \(10\times 10\) & 72.86 & 84.95 & 89.63 & 95.60 & 111.71 & 131.07 & 141.42 & 144.27 \\  & \(12\times 12\) & 72.56 & 84.98 & 89.62 & 95.09 & 111.70 & 130.72 & 141.42 & 144.27 \\  & \(14\times 14\) & 72.53 & 85.02 & 89.62 & 94.58 & 111.70 & 130 coupling spring stiffness value \(k_{\text{cp}}\) is less than \(10^{4}\), the natural frequencies remain stable and are smaller than when the stiffness value \(\text{kc}\) is relatively large. In this range, the plateau coupling springs have little effect on the natural frequencies of the coupled system. However, when the stiffness value \(k_{\text{cp}}\) increases to \(10^{11}\), it becomes possible to achieve rigid coupling at the boundary between the composite laminated rotationally stiffened plate and the laminated beam in the coupled system.