## 2 Modeling of the Unified Analysis Model of Vibro-acoustic Characteristics
### Stress-Strain and Displacement Relations  
According to the relevant knowledge of elastic mechanics, normal and shear strain at any location on the composite, laminated plate [24] or the laminated beam can be defined in terms of mid-surface strains and curvature changes:  
\[\left\{ \begin{array}{l}\varepsilon_{r}^{\text{P}}=\varepsilon_{r}^{\text{P}0}+z \chi_{r}^{\text{P}}\\ \varepsilon_{\theta}^{\text{P}}=\varepsilon_{\theta}^{\text{P}0}+z\chi_{\theta}^{ \text{P}}\end{array} \right. \tag{14}\]  
\[\left\{ \begin{array}{l}\gamma_{r\theta}^{\text{P}}=\gamma_{r\theta}^{\text{P}0}+z \chi_{r\theta}^{\text{P}}\\ \gamma_{rz}^{\text{P}}=\gamma_{rz}^{\text{P}0}\\ \gamma_{\theta z}^{\text{P}}=\gamma_{\theta z}^{\text{P}0}\end{array} \right. \tag{15}\]  
The detail expressions are given in Appendix A. [PERSON]'s law may be used to determine the stress-strain relationship between the laminated plate and the laminated curved beam at the \(k\)-th layer:  
\[\left\{ \begin{array}{l}\sigma_{r}^{\text{P}}\\ \sigma_{\theta}^{\text{P}}\\ \tau_{r\theta}^{\text{P}}\\ \tau_{rz}^{\text{P}}\end{array} \right\}=\left[\begin{array}{cccc}\overline{Q_{11}^{k} }&\overline{Q_{12}^{k}}&\mathbf{0}&\mathbf{0}&\overline{Q_{16}^{k}}\\ \overline{Q_{21}^{k}}&\overline{Q_{22}^where \(\overline{Q_{ij}^{k}}\) (\(i,j=1,\,2,\,\ldots,\,6\)) is the relevant stiffness coefficient, which can be obtained from the following equation:  
\[ \begin{bmatrix}\overline{Q_{11}^{k}}&\overline{Q_{12}^{k}}&\mathbf{0}&\mathbf{ 0}&\overline{Q_{16}^{k}}\\ \overline{Q_{21}^{k}}&\overline{Q_{22}^{k}}&\mathbf{0}&\mathbf{0}&\overline{Q _{26}^{k}}\\ \mathbf{0}&\mathbf{0}&\overline{Q_{44}^{k}}&\overline{Q_{45}^{k}}&\mathbf{0} \\ \mathbf{0}&\mathbf{0}&\overline{Q_{54}^{k}}&\overline{Q_{55}^{k}}&\mathbf{0} \\ \overline{Q_{61}^{k}}&\overline{Q_{62}^{k}}&\mathbf{0}&\mathbf{0}&\overline{Q _{66}^{k}}\end{bmatrix} =\mathbf{T} \begin{bmatrix}Q_{11}^{k}&Q_{12}^{k}& \mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathcal{Q_{21}^{k}}&\mathcal{Q_{22}^{k}}&\mathbf{0}&\mathbf{0}&\mathbf{0} \\ \mathcal{Q_{21}^{k}}&\mathcal{Q_{22}^{k}}&\mathbf{0}&\mathbf{0}&\mathbf{0} \\ \mathbf{0}&\mathbf{0}&\mathcal{Q_{44}^{k}}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathcal{Q_{55}^{k}}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathcal{Q_{66}^{k}}\end{bmatrix} \mathbf{T}^ {T} \tag{18}\]  
In Equation (16), \(\mathbf{T}\) is the transformation matrix, defined as follows; the included angle between the main direction and the \(r\) direction is called \(\theta_{k}\) and is the layer angle:  
\[\mathbf{T}= \begin{bmatrix}\cos^{2}\theta_{k}&\sin^{2}\theta_{k}&\mathbf{0}& \mathbf{0}&-2\sin\theta_{k}\cos\theta_{k}\\ \sin^{2}\theta_{k}&\cos^{2}\theta_{k}&\mathbf{0}&\mathbf{0}&2\sin\theta_{k} \cos\theta_{k}\\ \mathbf{0}&\mathbf{0}&\cos\theta_{k}&\sin\theta_{k}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&-\sin\theta_{k}&\cos\theta_{k}&\mathbf{0}&\mathbf{0} \\ \sin\theta_{k}\cos\theta_{k}&-\sin\theta_{k}\cos\theta_{k}&\mathbf{0}&\mathbf{0 }&\cos^{2}\theta_{k}-\sin^{2}\theta_{k}\end{bmatrix}  \tag{19}\]  
in which \(Q_{ij}^{k}\) represents the \(k\) layer's material coefficient of the \(k\) layer of the laminated plate (laminated curved beam), the value of which may be achieved using the engineering constant of the \(k\) layer of the laminated plate (laminated curved beam):  
\[ \begin{array}{c}Q_{11}^{k}=\frac{E_{1}}{1-\mu_{12}\mu_{21}}\ \ Q_{12}^{k}= \frac{\mu_{12}E_{2}}{1-\mu_{12}\mu_{21}}=Q_{21}^{k}\ Q_{22}^{k}=\frac{E_{2}}{1 -\mu_{12}\mu_{21}}\\ Q_{44}^{k}=G_{23}\ Q_{55}^{k}=G_{13}\ Q_{66}^{k}=G_{12}\end{array}  \tag{20}\]  
where \(E_{1}\) and \(E_{2}\) are [PERSON]'s modulus; \(G_{12}\), \(G_{13}\), and \(G_{23}\) are the shear moduli. According to the symmetry of the flexible matrix, the engineering constants of the orthotropic materials satisfy the following equations. The relationship between [PERSON]'s ratio \(\mu_{12}\) and \(\mu_{21}\) is as follows: \(\mu_{12}E_{2}=\mu_{21}E_{1}\).