## 2 Modeling of the Unified Analysis Model of Vibro-acoustic Characteristics
### Energy Functional  
The energy equation of the coupled system of composite laminated rotationally stiffened plate and acoustic cavities can be derived, with the expression of the Lagrangian equation of the coupled system, as follows:  
\[L_{\rm P}=T_{\rm P}-U_{\rm P}-U_{\rm P\text{-coupling}}-U_{SP}-W_{\rm P\&B_{n} }-W_{\rm P\&C_{1}}-W_{\rm P\&C_{2}}+W_{\rm F} \tag{26}\]  
\[L_{\rm B_{n}}=T_{\rm B_{n}}-U_{\rm B_{n}}-U_{\rm B_{n}\text{-coupling}}-W_{\rm P \&B_{n}} \tag{27}\]  
\[L_{\rm C_{1}}=T_{\rm C_{1}}-U_{\rm C_{1}}-U_{\rm C_{1}\text{-coupling}}-W_{\rm P \&C_{1}}+W_{\rm C_{1}\text{-wall}}+W_{\rm Q_{1}} \tag{28}\]  
\[L_{\rm C_{2}}=T_{\rm C_{2}}-U_{\rm C_{2}}\text{-coupling}-W_{\rm P\&C_{2}}+W_{ \rm C_{2}\text{-wall}}+W_{\rm Q_{2}} \tag{29}\]  
where \(T_{\rm P}\), \(T_{\rm B_{n}}\), \(T_{\rm C_{n}}\) and, respectively, represent the entire kinetic energy of the composite, laminated plate, the nth laminated beam, and the nth cavity. \(U_{\rm P}\), \(U_{\rm B_{n}}\), and \(U_{\rm C_{n}}\) represent the entire potential energy of the laminated plate, the nth laminated beam, and the nth cavity, respectively. \(U_{\rm P\text{-coupling}}\), \(U_{\rm B_{n}\text{-coupling}}\), and \(U_{\rm C_{n}\text{-coupling}}\) denote the coupling potential energy of the laminated plate, the nth laminated beam, and the nth cavity when \(\vartheta=360\). \(U_{SP}\) represents the potential energy due to the boundary springs of the laminated plate. \(W_{\rm P\&B_{n}}\) is the coupling potential energy between the laminated plate and the nth laminated beam; \(W_{\rm P\&C_{1}}\) and \(W_{\rm P\&C_{2}}\) represent the coupling potential energy between the laminated plate and cavity 1 and cavity 2, respectively. \(W_{\rm C_{n}\text{-wall}}\) indicates the impedance potential energy generated by the impedance wall surface within the nth cavity. \(W_{\rm F}\) indicates the work performed by the harmonic point force F on the composite laminated rotationally stiffened plate. \(W_{\rm Q_{n}}\) indicates the work performed by the monopole-point sound source within the coupling system's nth cavity. Several total kinetic energies within the coupled system \(T_{\rm P}\), \(T_{\rm B_{n}}\), and \(T_{\rm C_{n}}\) can be written as:  
\[T_{\rm P}=\frac{1}{2}\int_{0}^{R_{\rm P}}\int_{0}^{\vartheta}\left\{ \begin{array} []{l}I_{\rm p0}\Big{(}\frac{\partial u_{\rm bm}}{\partial t}\Big{)}^{2}+2I_{ \rm P1}\Big{(}\frac{\partial u_{\rm p}}{\partial t}\Big{)}\Big{(}\frac{ \partial\phi_{\rm bm}}{\partial t}\Big{)}+I_{\rm p2}\Big{(}\frac{\partial\phi _{\rm bm}}{\partial t}\Big{)}^{2}+I_{\rm p0}\Big{(}\frac{\partial\varphi_{\rm b m }}{\partial t}\Big{)}^{2}\\ +2I_{\rm P1}\Big{(}\frac{\partial\varphi_{\rm bm}}{\partial t}\Big{)}\Big{(} \frac{\partial\phi_{\rm bm}}{\partial t}\Big{)}+I_{\rm P2}\Big{(}\frac{ \partial\phi_{\rm bm}}{\partial t}\Big{)}^{2}+I_{\rm p0}\Big{(}\frac{\partial \omega_{\rm bm}}{\partial t}\Big{)}^{2}\end{array} \right\}(r+R_{1})drd\theta \tag{30}\]  
\[I_{\rm p0}=\sum_{k=1}^{N_{l}}\int_{Z_{k}}^{Z_{k+1}}\rho_{\rm P}^{k}dz\ \ I_{\rm p1}=\sum_{k=1}^{N_{l}}\int_{Z_{k}}^{Z_{k+1}}\rho_{\rm P}^{k}\cdot zdz \ \ I_{\rm p2}=\sum_{k=1}^{N_{l}}\int_{Z_{k}}^{Z_{k+1}}\rho_{\rm P}^{k}\cdot z^{2}dz \tag{31}\]  
\[I_{\rm bm0}=\sum_{k=1}^{N_{l}}\int_{Z_{k}}^{Z_{k+1}}\rho_{\rm bm}^{k}dz\ \ I_{\rm p1}=\sum_{k=1}^{N_{l}}\int_{Z_{k}}^{Z_{k+1}}\rho_{\rm bm}^{k}\cdot z _{n}dz\ \ I_{\rm bm2}=\sum_{k=1}^{N_{l}}\int_{Z_{k}}^{Z_{k+1}}\rho_{\rm bm}^{k}\cdot z _{n}^{2}dz_{n} \tag{33}\]\[T_{\text{C}_{n}}=\frac{1}{2\rho_{\text{C}_{n}}\omega^{2}}\int_{0}^{L_{n_{n}}} \int_{0}^{L_{n_{n}}}\int_{0}^{L_{n_{n}}}\begin{cases}