## 2 Modeling of the Unified Analysis Model of Vibro-acoustic Characteristics
### Energy Functional
Equations (53)-(58) can be transformed into matrix form:  
\[(\textbf{K}_{P}-\omega^{2}\textbf{M}_{\text{P}})P_{mn}+\textbf{C}_{\text{B}_{n} \text{,\&P}}\textbf{Q}_{l}+\textbf{C}_{\text{C}_{1}\text{,\&P}}A_{m_{n}l_{l}}- \textbf{C}_{\text{C}_{2}\text{,\&P}}\textbf{B}_{m_{n}l_{l}l_{l}}=\textbf{F} \tag{59}\]\[(\mathbf{K}_{\text{B}_{n}}-\omega^{2}\mathbf{M}_{\text{B}_{n}})\mathbf{Q}_{l}+ \mathbf{C}_{\text{P}\&\text{B}_{n}}\mathbf{P}_{mn}=\mathbf{0} \tag{60}\] \[(\mathbf{K}_{\text{c}_{1}}-\omega\mathbf{Z}_{\text{c}_{1}}-\omega^ {2}\mathbf{M}_{\text{c}_{1}})\mathbf{A}_{m_{n}n_{l}l_{l}}+\omega^{2}\mathbf{C}_ {\text{P}\&\text{C}_{1}}\mathbf{P}_{mn}=\mathbf{Q}_{1}\] (61) \[(\mathbf{K}_{\text{c}_{2}}-\omega\mathbf{Z}_{\text{c}_{2}}-\omega^ {2}\mathbf{M}_{\text{c}_{2}})\mathbf{B}_{m_{n}n_{l}l_{l}}-\omega^{2}\mathbf{C} _{\text{P}\&\text{C}_{2}}\mathbf{P}_{mn}=\mathbf{Q}_{2} \tag{62}\]  
where \(\mathbf{K}_{\text{P}}\), \(\mathbf{K}_{\text{B}_{n}}\) and \(\mathbf{K}_{\text{c}_{n}}\) indicate the laminated plate's stiffness matrices, the \(n\)th laminated beam, and the \(n\)th acoustic cavity, respectively. \(\mathbf{M}_{\text{P}}\), \(\mathbf{M}_{\text{B}_{n}}\), and \(\mathbf{M}_{\text{c}_{n}}\) are the laminated plate's mass matrices, the nth laminated beam, and the \(n\)th acoustic cavity, respectively. \(\mathbf{C}_{\text{B}_{n}\&\text{P}}\) denotes the coupling matrix between the \(n\)th laminated beam and the laminated plate, \(\mathbf{C}_{\text{P}\&\text{B}_{n}}=\mathbf{C}_{\text{B}_{n}\&\text{P}}^{T}\), while \(\mathbf{Z}_{\text{c}_{n}}\) is the matrix of impedance of the \(n\)th cylindrical cavity. \(\mathbf{C}_{\text{C}_{n}\&\text{P}}\) represents the acoustic-structural coupling matrix within the laminated plate and the \(n\)th cylindrical acoustic cavity, while \(\mathbf{C}_{\text{P}\&\text{C}_{n}}=\mathbf{C}_{\text{C}_{n}\&\text{P}}^{T}\). When \(\mathbf{F}_{n}\) and \(\mathbf{Q}_{n}\) are set to 0, meaning that no point force or point sound source are introduced, the system of equations for solving the natural frequencies and mode shapes of the coupled system between the composite laminated rotationally stiffened plate and acoustic cavities can be obtained. Eventually, the natural frequencies \(\omega\) and their corresponding modes \(\mathbf{G}\) of the coupled system can be determined. By substituting the source of the point sound and the harmonic point force into Equations (63)-(65), the steady-state response of the coupling system could be acquired. The detail expressions are given in Appendix A.  
\[(\mathbf{R}-\omega\mathbf{S})\mathbf{G}=\mathbf{0} \tag{63}\]  
\[\mathbf{R}=\left( \begin{array}{cccc}\mathbf{K}_{\text{P}}&\mathbf{C}_{\text {P}\&\text{E}\&\mathbf{C}_{\text{P}\&\text{C}_{1}}&-\mathbf{C}_{\text{P}\& \text{C}_{2}}\\ \mathbf{C}_{\text{B}\&\text{E}\&\mathbf{K}_{\text{B}}&\mathbf{0}&\mathbf{0}& \mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{K}_{\text{C}_{1}}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{K}_{\text{C}_{2}}\end{array} \right) \tag{64}\]  
\[\mathbf{S}=\left( \begin{array}{cccc}\mathbf{M}_{\text{P}}&\mathbf{0}& \mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{M}_{\text{B}}&\mathbf{0}&\mathbf{0}\\ \mathbf{C}_{\text{C}_{1}\&\text{E}\text{P}}&\mathbf{0}&\mathbf{M}_{\text{C}_{ 1}}&\mathbf{0}\\ -\mathbf{C}_{\text{C}_{2}\&\text{E}\text{P}}&\mathbf{0}&\mathbf{0}&\mathbf{M}_{ \text{C}_{2}}\end{array} \right) \tag{65}\]