## 3 Mathematical Formulation
### Boundary Conditions  
The natural boundary conditions (BCs) are obtained by applying [PERSON]'s principle, given by Equation (5). \[ \begin{array}{l}V_{x}=D_{x}\frac{\partial^{2}w^{0}}{\partial x^{3}}+\left( \frac{D_{xy}}{2}-\frac{h^{3}}{6}Q_{66}\right)\frac{\partial^{2}w^{0}}{ \partial x\partial y^{2}}\\ M_{xx}=D_{x}\frac{\partial^{2}w^{0}}{\partial x^{2}}-\frac{h^{3}}{12}Q_{12} \frac{\partial^{2}w^{0}}{\partial y^{2}}\end{array}  \tag{5}\]  
where \(V_{x}\), \(M_{xx}\) indicate the shear force and bending moments of the plate. **4. Formulation of Dynamic Stiffness (DS) Matrix with Levy Solution**  
The generalized differential Equation (3) is solved by applying force and displacement boundary conditions to develop the DS matrix. A Levy solution is implemented to solve Equation (3), and to satisfy the BCs given by Equation (5), it is sought in the given form as presented in Equation (6) [33]. \[w^{0}(x,y,t)=\sum_{m=1}^{\infty}w_{m}(x)e^{i\omega t}\sin(\alpha_{m}y)\text{ with }\alpha_{m}=\frac{m\pi}{L}(m=1,2,\ldots\ldots\ldots\infty) \tag{6}\]  
where \(\omega\) represents the unknown frequency. A generalized fourth-order ordinary differential equation is determined by substituting Equation (6) into Equation (3); it can be expressed as  
\[D_{x}\frac{d^{4}w_{m}}{dx^{4}}-\alpha_{m}^{2}D_{xy}\frac{d^{2}w_{m}}{dx^{2}}+ \Big{(}\alpha_{m}^{4}D_{y}+\rho h\omega^{2}\Big{)}w_{m}=0 \tag{7}\]  
The developed Equation (7) produces the standard four roots; based on its nature, there are only two feasible solutions obtained and given by cases 1 and 2. Case 1. \(\alpha_{m}^{2}\geq\sqrt{\alpha_{m}^{4}\Big{(}D_{xy}^{2}-4D_{x}D_{y}\Big{)}}-4D _{x}\rho h\omega^{2}\)  
In the above case, all roots are real (\(r_{1m},-r_{1m},r_{2m},-r_{2m}\)) and can be expressed as  
\[r_{1m} =\sqrt{\frac{\alpha_{m}^{2}D_{xy}+\sqrt{\alpha_{m}^{4}\big{(}D_{ xy}^{2}-4D_{x}D_{y}\big{)}-4D_{x}\rho h\omega^{2}}}{2D_{x}}}\] \[r_{2m} =\sqrt{\frac{\alpha_{m}^{2}D_{xy}-\sqrt{\alpha_{m}^{4}\big{(}D_{ xy}^{2}-4D_{x}D_{y}\big{)}-4D_{x}\rho h\omega^{2}}}{2D_{x}}}\] \[r_{1m} =\sqrt{\frac{\alpha_{m}^{2}D_{xy}+\sqrt{\alpha_{m}^{4}\big{(}D_{ xy}^{2}-4D_{x}D_{y}\big{)}-4D_{x}\rho h\omega^{2}}}{2D_{x}}}\] \[r_{2m} =\sqrt{\frac{\alpha_{m}^{2}D_{xy}-\sqrt{\alpha_{m}^{4}\big{(}D_{ xy}^{2}-4D_{x}D_{y}\big{)}-4D_{x}\rho h\omega^{2}}}{2D_{x}}}\]  
The solution is given by Equation (8). \[w_{m}(x)=A_{m}\text{cos}\,h(r_{1m}x)+B_{m}\text{sin}\,h(r_{1m}x)+C_{m}\text{ cos}\,h(r_{2m}x)+D_{m}\text{sin}\,h(r_{2m}x) \tag{8}\]  
where \(A_{m},\ B_{m},\ C_{m},\) and \(D_{m}\) are constants. Case 2. \(\alpha_{m}^{2}\leq\sqrt{\alpha_{m}^{4}\Big{(}D_{xy}^{2}-4D_{x}D_{y}\Big{)}-4D _{x}\rho h\omega^{2}}\)  
In the above case are two real roots and two imaginary roots (\(r_{1m},-r_{1m},ir_{2m},-ir_{2m}\)), and can be expressed as  
\[r_{1m} =\sqrt{\frac{\alpha_{m}^{2}D_{xy}+\sqrt{\alpha_{m}^{4}\big{(}D_{ xy}^{2}-4D_{x}D_{y}\big{)}-4D_{x}\rho h\omega^{2}}}{2D_{x}}}\] \[r_{2m} =\sqrt{\frac{-\alpha_{m}^{2}D_{yy}+\sqrt{\alpha_{m}^{4}\big{(}D_{ xy}^{2}-4D_{x}D_{y}\big{)}-4D_{x}\rho h\omega^{2}}}{2D_{x}}}\]  
The solution is given by Equation (9). \[w_{m}(x)=A_{m}\cos h(r_{1m}x)+B_{m}\sin h(r_{1m}x)+C_{m}\cos(r_{2m}x)+A_{m}\cos (r_{2m}x) \tag{9}\]  
For case 1, the formulation of the DS matrix is explained below. A similar pattern is implemented for case 2, but is not explained here for brevity. By applying Equations (6) and (8), the displacement \((w^{0})\), bending rotation \((\varnothing_{y})\), shear force \((V_{x})\), and moment \((M_{xx})\) are determined and can be given by Equations (10)-(12).