## 2 Modeling of the Unified Analysis Model of Vibro-acoustic Characteristics
### Energy Functional
\begin{array}{c}k_{\text{icr}}^{\text{P}} \left(\left.u_{\text{P}}\right|_{\theta=360^{\circ}}-\left.u_{\text{P}}\right|_{ \theta=0}\right)^{2}+k_{\text{c}c}^{\text{P}}\left(\left.v_{\text{P}}\right|_{ \theta=360^{\circ}}-\left.v_{\text{P}}\right|_{\theta=0}\right)^{2}\\ +k_{\text{icr}}^{\text{P}}\left(\left.w_{\text{P}}\right|_{\theta=360^{\circ}}- \left.w_{\text{P}}\right|_{\theta=0}\right)^{2}+k_{\text{c}c}^{\text{P}}\left( \left.\phi_{\text{P}}\right|_{\theta=360^{\circ}}-\left.\phi_{\text{P}}\right|_{ \theta=0}\right)^{2}\\ +K_{\theta}^{\text{P}}\left(\left.\phi_{\text{P}}\right|_{\theta=360^{\circ}}- \left.\phi_{\text{P}}\right|_{\theta=0}\right)^{2}\end{array} \right\}dzdr\] (38) \[U_{\text{B}_{n}\text{-coupling}}=\frac{1}{2}\int_{0}^{\theta}\int_{-l_ {\text{B}}/2}^{h_{\text{P}}/2}\left\{ \begin{array}{c}k_{\text{icr}}^{\text{ bbn}}\left(\left.u_{\text{bend}}\right|_{\theta=360^{\circ}}-\left.u_{\text{bbn}} \right|_{\theta=0}\right)^{2}+k_{\text{c}c}^{\text{bbn}}\left(\left.v_{\text{ bbn}}\right|_{\theta=360^{\circ}}-\left.v_{\text{bbn}}\right|_{\theta=0} \right)^{2}\\ +k_{\text{icr}}^{\text{bbn}}\left(\left.v_{\text{bbn}}\right|_{\theta=360^{\circ}}- \left.w_{\text{bbn}}\right|_{\theta=0}\right)^{2}+k_{\text{c}c}^{\text{bbn}} \left(\left.\phi_{\text{xbn}}\right|_{\theta=360^{\circ}}-\left.\phi_{\text{xbn}} \right|_{\theta=0}\right)^{2}\\ +k_{\text{icr}}^{\text{bbn}}\left(\left.\phi_{\text{bbn}}\right|_{\theta=360^{\circ}}- \left.\phi_{\text{bbn}}\right|_{\theta=0}\right)^{2}\end{array} \right\}dz_{n}\] (39) \[U_{\text{C}_{n}\text{-coupling}}=\int_{0}^{L_{\beta_{n}}}\int_{0}^{L_ {\beta_{n}}}\left\{\begin{array}{c}\left(\frac{\partial p_{n}^{  
\[W_{\text{C}_{n-\text{wall}}}=W_{\text{C}_{n-\text{wall}}}^{1}+W_{\text{C}_{n- \text{wall}}}^{2}+W_{\text{C}_{n-\text{wall}}}^{3}+W_{\text{C}_{n-\text{wall}}}^ {4}+W_{\text{C}_{n-\text{wall}}}^{5}+W_{\text{C}_{n-\text{wall}}}^{6} \tag{48}\]  
in which \(j\) represents the imaginary unit. \(S_{r}\) indicates the region of the \(r\)-th surface of uncoupled acoustic wall, where \(Z_{r}\) is the equivalent acoustic wall impedance. The work performed by the harmonic point force \(F\) on the laminated plate within the composite stiffened plate is expressed as follows:  
\[W_{\text{F}}=\iint\limits_{S}\{f_{u}u_{\text{p}}+f_{v}v_{\text{p}}+f_{w}w_{ \text{p}}+f_{\phi_{r}}\phi_{\text{rp}}+f_{\phi_{\phi}}\phi_{\text{p}}\}rdrd\theta \tag{49}\]  
\[f_{i}=F\delta(r-r_{0})\delta(\theta-\theta_{0}) \tag{50}\]  
where \(f_{i}\) (\(i=u_{j}\), \(v_{j}\), \(w_{j}\), \(\phi_{r}\), \(\phi_{\theta}\)) are functions of the outside load distribution; the location of the harmonic point force \(F\) is at (\(r_{0}\), \(\theta_{0}\)). The work performed by the source of point sound within cavity \(n\) in the coupled system \(W_{\text{Q}_{n}}\) can be expressed as:  
\[W_{\text{Q}_{n}}=\frac{-1}{j\omega}\int_{0}^{L_{xn}}\int_{0}^{L_{ \beta_{n}}}\int_{0}^{L_{xn}}\Bigg{\{}\Bigg{(}P_{n}^{\Omega}(\alpha_{n},\beta_{ n},z_{n})+\sum_{q=1}^{6}P_{n}^{\Theta_{q}}(\alpha_{n},\beta_{n},z_{n})\Bigg{)} \Bigg{\}}Q_{s}^{n}|J_{n}|\cdot H_{\alpha_{n}}H_{\beta_{n}}H_{z_{n}}d\alpha_{n} d\beta_{n}dz_{n} \tag{51}\]  
\[Q_{s}^{n}=\frac{4\pi A}{j\rho c_{s}c_{n}}\delta_{c}(\alpha-\alpha_{e})\delta_{ c}(\beta-\beta_{e})\delta_{c}(z-z_{e}) \tag{52}\]  
in Equation (52), \(Q_{s}^{n}\) is the point sound source's distribution function acting within cavity \(n\). The amplitude of the source of the point sound is denoted as \(A\) (kg/s\({}^{2}\)); its related location of action is (\(a_{e}\), \(\beta_{e}\), \(z_{e}\)). \(\delta_{c}\) represents the three-dimensional Dirac delta function. The wavenumber of sound is represented by \(k\), which is the ratio of circular frequency \(\omega\) to the speed of sound \(c_{n}\): \(k=\omega/c_{n}\).