## 2 Modeling of the Unified Analysis Model of Vibro-acoustic Characteristics
### Energy Functional
\begin{array}{c}\left( \left[\left(I_{11}^{n}\right)^{2}+\left(I_{21}^{n}\right)^{2}\right]\left(\frac {\partial p_{n}^{\Omega}}{H_{n}\partial a_{n}}+\sum\limits_{q=1}^{6}\frac{ \partial p_{n}^{\Omega}}{H_{n}\partial a_{n}}\right)\right)^{2}+\\ \\ \left(I_{11}^{n}\cdot I_{21}^{n}\right)\left(\frac{\partial p_{n}^{\Omega}}{H_{ n}\partial a_{n}}+\sum\limits_{q=1}^{6}\frac{\partial p_{n}^{\Omega}}{H_{n} \partial a_{n}}\right)\left(\frac{\partial p_{n}^{\Omega}}{H_{n}\partial a_{n}}+ \sum\limits_{q=1}^{6}\frac{\partial p_{n}^{\Omega}}{H_{n}\partial a_{n}}\right) +\\ \\ \left(\left[\left(I_{12}^{n}\right)^{2}+\left(I_{22}^{n}\right)^{2}\right]\left( \frac{\partial p_{n}^{\Omega}}{H_{n}\partial a_{n}}+\sum\limits_{q=1}^{6}\frac {\partial p_{n}^{\Omega}}{H_{n}\partial a_{n}}\right)\right)^{2}+\\ \\ \left(I_{12}^{n}\cdot I_{22}^{n}\right)\left(\frac{\partial p_{n}^{\Omega}}{H_{ n}\partial a_{n}}+\sum\limits_{q=1}^{6}\frac{\partial p_{n}^{\Omega}}{H_{n} \partial a_{n}}\right)\left(\frac{\partial p_{n}^{\Omega}}{H_{n}\partial a_{n}} +\sum\limits_{q=1}^{6}\frac{\partial p_{n}^{\Omega}}{H_{n}\partial a_{n}} \right)+\\ \\ \left[\frac{\partial p_{n}^{\Omega}}{H_{n}\partial a_{n}}\phi_{n}+\frac{6}{q=1} \frac{\partial p_{n}^{\Omega}}{H_{n}\partial a_{n}}\right]^{2}\end{array} \begin{cases}|J_{n}|\cdot H_{n_{n}}H_{n_{n}}H_{n_{n}}dx_{n}d\beta_{n}dz_{n}\end{cases}  \tag{34}\]  
Several total potential energies within the coupled system \(U_{\text{P}}\), \(U_{\text{B}_{n}}\) and \(U_{\text{C}_{n}}\) can be written as:  
\[U_{\text{P}}=U_{\text{stretch}}+U_{\text{s-b}}+U_{\text{bend}}= \frac{1}{2}\int_{0}^{R_{\text{P}}}\int_{0}^{\theta}\left\{ \begin{array}{c}N_{ P}^{\text{P}}\varepsilon_{r}^{00}+N_{r}^{\text{P}}\varepsilon_{r}^{0}+N_{p}^{ \text{P}}\varepsilon_{r}^{0}+N_{p}^{\text{P}}\eta_{r}^{0}+M_{r}^{\text{P}} \chi_{r}^{\text{P}}+\\ M_{\theta}^{\text{P}}\kappa_{\theta}^{\text{P}}+M_{r\theta}^{\text{P}}\chi_{r \theta}^{\text{P}}+Q_{r}^{\text{P}}\gamma_{r}^{\text{P}}^{0}+Q_{r}^{\text{P}} \delta_{r\theta}^{\text{P}}\gamma_{r}^{0}+Q_{r}^{\text{P}}\delta_{r\theta}^{ \text{P}}\gamma_{\theta z}^{0}\end{array} \right\}rdrd\theta \tag{35}\] \[U_{\text{B}_{n}}=\frac{1}{2}\int_{0}^{\theta}\left\{\begin{array}{c}A _{22}\left(\frac{\partial p_{\text{B}_{n}}}{R_{\text{bend}}\theta}+\frac{v_{ \text{B}_{n}}}{R_{\text{bend}}}\right)^{2}+2A_{26}\left(\frac{\partial p_{ \text{B}_{n}}}{R_{\text{bend}}\theta}\right)\left(\frac{\partial v_{\text{B}_ {n}}}{R_{\text{bend}}\theta}+\frac{v_{\text{B}_{n}}}{R_{\text{bend}}}\right) \left(\phi_{\text{xbn}}\right)\\ +\overline{\kappa_{s}}A_{44}\left(\phi_{\text{xbn}}\right)^{2}+2\overline{ \kappa_{s}}A_{4}\left(\frac{\partial v_{\text{xbn}}}{R_{\text{bend}}\theta} \right)\frac{v_{\text{xbn}}}{R_{\text{bend}}\theta}+\phi_{\text{xbn}}\right) \left(\phi_{\text{xbn}}\right)\] (36) \[U_{\text{B}_{n}}=\frac{1}{2\rho_{\text{C}_{n}}c_{n}^{2}}\int_{0}^{L_ {n_{n}}}\int_{0}^{L_{\beta_{n}}}\int_{0}^{L_{\beta_{n}}}\left\{P_{n}^{\Omega}( \alpha_{n},\beta_{n},z_{n})+\sum\limits_{q=1}^{6}P_{n}^{\Theta_{q}}(\alpha_{n}, \beta_{n},z_{n})\right\}^{2}|J_{n}|\cdot H_{\alpha_{n}}H_{\beta_{n}}H_{\alpha_{n} }d\beta_{n}dz_{n} \tag{37}\]  
\(\rho_{\text{P}}^{\text{k}}\) represents the material density of the kth layer of the laminated plate, \(\rho_{\text{bend}}^{\text{k}}\) represents the material density of the kth layer of the laminated beam, \(\rho_{\text{C}_{n}}\) represents the density of the acoustic medium inside cavity \(n\), \(c_{n}\) represents the speed at which acoustic waves propagate within cavity \(n\), and \(\omega\) represents the circular frequency of the rotationally acoustic cavity. When \(\vartheta=360^{\circ}\), the potential energy of coupling within the coupled system \(U_{\text{P-coupling}}\), the coupled potential energy of the nth laminated curved beam \(U_{\text{B}_{n}\text{-coupling}}\), and the coupled potential energy of the \(n\)th sound cavity \(U_{\text{C}_{n}\text{-coupling}}\) can be written as:  
\[U_{\text{P-coupling}}=\frac{1}{2}\int_{0}^{R_{\text{P}}}\int_{-l_{ \text{P}}/2}^{h_{\text{P}}/2}\left\{