## 3 Numerical Results and Discussion
### Verification of Convergence and Correctness
This study conducted experiments using a coupled system between a circular stiffened plate and a cylindrical cavity, which is relatively straightforward to implement, to validate the correctness of the unified analysis model of the vibro-acoustic characteristics of the coupled system between a composite laminated rotationally stiffened plate and acoustic cavities presented in this paper. The experimental setup included an LC02 impact hammer, a 3A105 force sensor, a DH5857-1 charge amplifier, a 1A116E accelerometer sensor, and a DH5922D dynamic signal testing and analysis system, as well as a C-35 Bluetooth speaker, an MPA201 loudspeaker, a DH40200 shaker, and a YE5872A power amplifier. The experiment was conducted in an acoustic chamber to minimize the influence of external environmental noise on the test results. Data were collected by opening the Bluetooth speaker to provide a point source excitation for the cylindrical-cylindrical cavity system. A JZK-20 shaker was secured above the circular stiffened plate and connected to the YE5872A power amplifier. The DH5922D dynamic signal testing and analysis system was also connected to the power amplifier for control of the shaker. Data collection commenced after setting the point force amplitude and phase within the system. The amplitude and phase of the point force could be adjusted using the knobs on the power amplifier. After collecting specific frequency sound pressure values under point force and ambient noise excitations, the two sets of values were subtracted. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Boundary Condition**} & \multicolumn{6}{c}{**The Stiffness Value of the Boundary Spring**} \\ \cline{2-6}  & \(k_{u}\) & \(k_{\sigma}\) & \(k_{w}\) & \(K_{r}\) & \(K_{\theta}\) \\ \hline C & \(10^{16}\) & \(10^{16}\) & \(10^{16}\) & \(10^{16}\) & \(10^{16}\) \\ S & \(10^{16}\) & \(10^{16}\) & \(10^{16}\) & \(0\) & \(0\) \\ F & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) \\ E\({}^{1}\) & \(10^{8}\) & \(10^{8}\) & \(10^{8}\) & \(10^{8}\) & \(10^{8}\) \\ E\({}^{2}\) & \(10^{6}\) & \(10^{6}\) & \(10^{6}\) & \(0\) & \(0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The stiffness value of the boundary spring under different boundary conditions. Figure 5: Variation curves of the natural frequencies of the coupled system between the composite, laminated rotational plate and the cylindrical-spherical cavities under different boundary spring stiffness values. The coupled system between the circular stiffened plate and the cylindrical cavity consisted of a clamped Q235 steel circular stiffened plate and an acrylic cylindrical cavity. The circular plate was connected with top and bottom plates using 32 evenly spaced \(\Phi\)16 bolts to create a clamped boundary condition. The structural dimensions of the stiffened circular plate used in the experiment are as follows: \(R_{1}\) = 0 m, \(R_{2}\) = 0.265 m, \(h_{\rm P}\) = 0.005 m, \(R_{\rm b1}\) = 0.088 m, \(R_{\rm b2}\) = 0.177 m, \(b_{1}\) = \(b_{2}\) = 0.02 m, \(h_{1}\) = \(h_{2}\) = 0.01 m, and \(\vartheta\) = 360\({}^{\circ}\). The circular plate was made of either Q235 steel or carbon fiber composite material. The material parameters of Q235 steel are: \(E\) = 216 GPa, \(\mu\) = 0.3, and \(\rho_{\rm P}\) = 7800 kg/m\({}^{3}\). The material parameters of the carbon fiber composite material are: \(E_{1}\) = \(E_{2}\) = 77.8 GPa, \(G_{1}\) = \(G_{2}\) = \(G_{3}\) = 77.8 GPa, \(\mu\) = 0.26, and \(\rho_{\rm P}\) = 7800 kg/m\({}^{3}\). The layer angle of the composite material is [0\({}^{\circ}\)/90\({}^{\circ}\)/0\({}^{\circ}\)/90\({}^{\circ}\)/0\({}^{\circ}\)]. The main body of the cylindrical-cylindrical cavity coupling system was constructed from a 10 mm thick transparent acrylic plate and comprises a cylindrical shell and two circular plates. The effective dimensions of the cavity coupling system are as follows: \(R_{1}\) = 0 m, \(R_{2}\) = 0.265 m, \(L\) = 0.4 m, and \(\vartheta\) = 360\({}^{\circ}\).