## 2 Modeling of the Unified Analysis Model of Vibro-acoustic Characteristics
### Energy Functional
Once the energy equation of the coupling system has been obtained, it can be solved using the Rayleigh-Ritz method [25; 26]:  
\[\frac{\partial L_{\text{P}}}{\partial\textbf{P}_{mn}}=\frac{\partial T_{\text {P}}}{\partial\textbf{P}_{mn}}-\frac{\partial U_{\text{P}}}{\partial\textbf{P }_{mn}}-\frac{\partial U_{\text{P}\text{-coupling}}}{\partial\textbf{P}_{mn}} -\frac{\partial U_{SP}}{\partial\textbf{P}_{mn}}-\frac{\partial W_{\text{P} \text{R}\text{C}_{1}}}{\partial\textbf{P}_{mn}}-\frac{\partial W_{\text{P} \text{R}\text{C}_{1}}}{\partial\textbf{P}_{mn}}-\frac{\partial W_{\text{P} \text{R}\text{C}_{2}}}{\partial\textbf{P}_{mn}}+\frac{\partial W_{\text{P}}}{ \partial\textbf{P}_{mn}}=\textbf{0} \tag{53}\]  
\[\frac{\partial L_{\text{B}_{n}}}{\partial\textbf{Q}_{l}}=\frac{\partial T_{ \text{B}_{n}}}{\partial\textbf{Q}_{l}}-\frac{\partial U_{\text{B}_{n}}}{ \partial\textbf{Q}_{l}}-\frac{\partial U_{\text{B}_{n}\text{-coupling}}}{ \partial\textbf{Q}_{l}}-\frac{\partial U_{\text{B}_{n}\text{-coupling}}}{ \partial\textbf{Q}_{l}}-\frac{\partial W_{\text{P}\text{R}\text{B}_{n}}}{ \partial\textbf{Q}_{l}}=\textbf{0} \tag{54}\]  
\[\frac{\partial L_{\text{C}_{1}}}{\partial\textbf{A}_{m_{n}l_{l}}}=\frac{ \partial T_{\text{C}_{1}}}{\partial\textbf{A}_{m_{n}l_{l}}}-\frac{\partial U_{ \text{C}_{1}}}{\partial\textbf{A}_{m_{n}l_{l}}}-\frac{\partial U_{\text{C}_{1} \text{-coupling}}}{\partial\textbf{A}_{m_{n}l_{l}}l_{l}}-\frac{\partial W_{ \text{P}\text{R}\text{C}_{1}}}{\partial\textbf{A}_{m_{n}l_{l}}l_{l}}+\frac{ \partial W_{\text{C}_{1}\text{-wall}}}{\partial\textbf{A}_{m_{n}l_{l}}l_{l}}+ \frac{\partial W_{\text{Q}_{l}}}{\partial\textbf{A}_{m_{n}l_{l}}l_{l}}= \textbf{0} \tag{55}\]  
\[\frac{\partial L_{\text{C}_{2}}}{\partial\textbf{B}_{m_{n}l_{l}}l_{l}}=\frac{ \partial T_{\text{C}_{2}}}{\partial\textbf{B}_{m_{n}l_{l}}l_{l}}-\frac{ \partial U_{\text{C}_{2}}}{\partial\textbf{B}_{m_{n}l_{l}}l_{l}}-\frac{\partial U _{\text{C}_{2}\text{-coupling}}}{\partial\textbf{B}_{m_{n}l_{l}}l_{l}}-\frac{ \partial W_{\text{P}\text{R}\text{C}_{2}}}{\partial\textbf{B}_{m_{n}l_{l}}l_{l}}+ \frac{\partial W_{\text{C}_{2}\text{-wall}}}{\partial\textbf{B}_{m_{n}l_{l}}l_{l }}+\frac{\partial W_{\text{Q}_{2}}}{\partial\textbf{B}_{m_{n}l_{l}}l_{l}}= \textbf{0} \tag{56}\]  
\[\textbf{P}_{mn}= \begin{bmatrix}\textbf{A}_{mn}&\textbf{B}_{mn}&\textbf{C}_{mn}& \textbf{D}_{mn}&\textbf{E}_{mn}\end{bmatrix} ^{T} \tag{57}\]  
\[\textbf{Q}_{l}= \begin{bmatrix}\textbf{A}_{l}&\textbf{B}_{l}&\textbf{C}_{l}&\textbf {D}_{l}&\textbf{E}_{l}\end{bmatrix} ^{T} \tag{58}\]  
where \(\textbf{P}_{mn}\) represents the two-dimensional unknown Fourier coefficient matrix of the laminated plate, and \(\textbf{Q}_{l}\) represents the one-dimensional unknown Fourier coefficient matrix of the laminated beam. For an acoustic cavity with an impedance wall, its characteristic values are generally complex, forming a complex modal problem, which, in physics, represents the system undergoing damped oscillations. In the complex characteristic values, the real part corresponds to the natural circular frequency of the acoustic cavity, and the imaginary part represents modal decay. Furthermore, the imaginary part is a relatively small number. Therefore, in this study of natural frequencies, only the structure-acoustic coupling system under a rigid wall is considered.