## 3 Numerical Results and Discussion
### Free Vibration Analysis  
This part focuses on the free vibration of the coupled system between the composite laminated rotationally stiffened plate and the cylindrical-cylindrical, spherical-cylindrical, and conical-cylindrical cavities, studying the impact of relevant factors affecting the coupled system's vibro-acoustic characteristics. Figure 10 depicts the variation curve of the coupled system's natural frequencies between the composite laminated rotationally stiffened plate and the cylindrical-cylindrical cavity at different rotation angles. In this case, the number of stiffeners in the composite laminated rotationally stiffened plate is \(n=2\). The geometric parameters determined of the coupling system are: \(R_{1}=0.9\) m, \(R_{2}=1.8\) m, \(L_{1}=1\) m, \(L_{2}=2\) m, \(h_{\text{P}}=0.03\) m, \(R_{\text{b1}}=1.2\) m, \(b_{1}=0.08\) m, \(h_{1}=0.06\) m, \(R_{\text{b2}}=1.5\) m, \(b_{2}=0.08\) m, and \(h_{2}=0.06\) m. The material of the laminated plate and stiffeners is glass epoxy resin, with a layer angle of \([90^{\circ}/0^{\circ}/90^{\circ}]\), and the medium in the cavities is air. The boundary conditions are defined as CCCC. As shown in Figure 10, the natural frequencies of the coupled system decrease as the rotation angle \(\vartheta\) increases, but this trend changes when the rotation angle \(\vartheta=360^{\circ}\) is reached. Table 7 provides the first eight natural frequencies of the coupled system at different rotation angles. It can be observed that the natural frequencies at \(\vartheta=360^{\circ}\) are roughly the same as those at \(\vartheta=180^{\circ}\). In the case of coupling between two sound walls, a closed loop is formed, resulting in repeated modes and causing an increase in the natural frequencies of all orders. These conditions explain the reason for the increase in natural frequencies at \(\vartheta=360^{\circ}\), as shown in Figure 10.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Rotation Angle \(\vartheta\)**} & \multicolumn{8}{c}{**Modal Order**} \\ \cline{2-9}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline
45\({}^{\circ}\) & 85.00 & 155.58 & 155.58 & 169.99 & 170.00 & 177.30 & 192.19 & 1  
CCCC. From Figure 8, it is evident that the natural frequencies of the coupled system at the same order decrease as the heights \(L_{1}\) and \(L_{2}\) of cylindrical cavities 1 and 2 increase. The parametric study of the coupling system among the composite laminated rotationally stiffened plate and the spherical-cylindrical cavity considers both the height \(L\) of the cylindrical cavity and the top angle \(\phi\) of the spherical cavity. Figure 12 illustrates the trends of the first four natural frequencies of the coupled system between the composite laminated rotationally stiffened plate and the spherical-cylindrical cavity as \(\phi\) and \(L\) vary. In this example, the number of stiffeners is \(n=2\), the calculated parameters of geometry are as follows: \(R_{1}=0.8\) m, \(R_{2}=1.7\) m, \(\phi=120^{\circ}\), \(\phi_{2}=90^{\circ}\), \(h_{\rm p}=0.02\) m, \(R_{\rm b1}=1.1\) m, \(b_{1}=0.05\) m, \(h_{1}=0.05\) m, \(R_{\rm b2}=1.4\) m, \(b_{2}=0.05\) m, and \(h_{2}=0.05\) m.  
The material used is graphite epoxy resin with a layer angle of [90\({}^{\circ}\)/0\({}^{\circ}\)/90\({}^{\circ}\)], and the medium in the cavities is air. The boundary conditions are set as CCCC. As shown in Figure 12, the natural frequencies of the coupled system in the same order decrease as the height \(L\) of the cylindrical cavity increases. The top angle \(\phi\) of the spherical cavity also generally causes a decrease in the system's natural frequencies, but the effect within the range of 0\({}^{\circ}\) to 90\({}^{\circ}\) is relatively small.