## 3 Mathematical Formulation
### Boundary Conditions
\[D_{x}\frac{d^{4}w_{m}}{dx^{4}}-\alpha_{m}^{2}D_{xy}\frac{d^{2}w_{m}}{dx^{2}}+ \Big{(}\alpha_{m}^{4}D_{y}+\rho h\omega^{2}\Big{)}w_{m}=0 \tag{7}\]0 \[D_{x}\frac{d^{4}w_{m}}{dx^{4}}-\alpha_{m}^{2}D_{xy}\frac{d^{2}w_{m}}{dx^{2}}+ \Big{(}\alpha_{m}^{4}D_{y}+\rho h\omega^{2}\Big{)}w_{m}=0 \tag{7}\]1\[D_{x}\frac{d^{4}w_{m}}{dx^{4}}-\alpha_{m}^{2}D_{xy}\frac{d^{2}w_{m}}{dx^{2}}+ \Big{(}\alpha_{m}^{4}D_{y}+\rho h\omega^{2}\Big{)}w_{m}=0 \tag{7}\]2 \[D_{x}\frac{d^{4}w_{m}}{dx^{4}}-\alpha_{m}^{2}D_{xy}\frac{d^{2}w_{m}}{dx^{2}}+ \Big{(}\alpha_{m}^{4}D_{y}+\rho h\omega^{2}\Big{)}w_{m}=0 \tag{7}\]3 \[D_{x}\frac{d^{4}w_{m}}{dx^{4}}-\alpha_{m}^{2}D_{xy}\frac{d^{2}w_{m}}{dx^{2}}+ \Big{(}\alpha_{m}^{4}D_{y}+\rho h\omega^{2}\Big{)}w_{m}=0 \tag{7}\]4 \[D_{x}\frac{d^{4}w_{m}}{dx^{4}}-\alpha_{m}^{2}D_{xy}\frac{d^{2}w_{m}}{dx^{2}}+ \Big{(}\alpha_{m}^{4}D_{y}+\rho h\omega^{2}\Big{)}w_{m}=0 \tag{7}\]5 \[D_{x}\frac{d^{4}w_{m}}{dx^{4}}-\alpha_{m}^{2}D_{xy}\frac{d^{2}w_{m}}{dx^{2}}+ \Big{(}\alpha_{m}^{4}D_{y}+\rho h\omega^{2}\Big{)}w_{m}=0 \tag{7}\]6 (11) \[D_{x}\frac{d^{4}w_{m}}{dx^{4}}-\alpha_{m}^{2}D_{xy}\frac{d^{2}w_{m}}{dx^{2}}+ \Big{(}\alpha_{m}^{4}D_{y}+\rho h\omega^{2}\Big{)}w_{m}=0 \tag{7}\]7 \[D_{x}\frac{d^{4}w_{m}}{dx^{4}}-\alpha_{m}^{2}D_{xy}\frac{d^{2}w_{m}}{dx^{2}}+ \Big{(}\alpha_{m}^{4}D_{y}+\rho h\omega^{2}\Big{)}w_{m}=0 \tag{7}\]8 \[D_{x}\frac{d^{4}w_{m}}{dx^{4}}-\alpha_{m}^{2}D_{xy}\frac{d^{2}w_{m}}{dx^{2}}+ \Big{(}\alpha_{m}^{4}D_{y}+\rho h\omega^{2}\Big{)}w_{m}=0 \tag{7}\]9 \[r_{1m} =\sqrt{\frac{\alpha_{m}^{2}D_{xy}+\sqrt{\alpha_{m}^{4}\big{(}D_{ xy}^{2}-4D_{x}D_{y}\big{)}-4D_{x}\rho h\omega^{2}}}{2D_{x}}}\]0 \[r_{1m} =\sqrt{\frac{\alpha_{m}^{2}D_{xy}+\sqrt{\alpha_{m}^{4}\big{(}D_{ xy}^{2}-4D_{x}D_{y}\big{)}-4D_{x}\rho h\omega^{2}}}{2D_{x}}}\]1 \[r_{1m} =\sqrt{\frac{\alpha_{m}^{2}D_{xy}+\sqrt{\alpha_{m}^{4}\big{(}D_{ xy}^{2}-4D_{x}D_{y}\big{)}-4D_{x}\rho h\omega^{2}}}{2D_{x}}}\]2 (12) \[r_{1m} =\sqrt{\frac{\alpha_{m}^{2}D_{xy}+\sqrt{\alpha_{m}^{4}\big{(}D_{ xy}^{2}-4D_{x}D_{y}\big{)}-4D_{x}\rho h\omega^{2}}}{2D_{x}}}\]3 \[r_{1m} =\sqrt{\frac{\alpha_{m}^{2}D_{xy}+\sqrt{\alpha_{m}^{4}\big{(}D_{ xy}^{2}-4D_{x}D_{y}\big{)}-4D_{x}\rho h\omega^{2}}}{2D_{x}}}\]4  
The BCs for displacements are:  
\[ \begin{split} x=0,& w_{m}=w_{1};\quad\varnothing_{ ym}=\varnothing_{y1}\\ x=b,& w_{m}=w_{2};\quad\varnothing_{ym}=\varnothing_{ y2}\end{split}  \tag{13}\]  
The BCs for the forces are:  
\[ \begin{split} x=0,& V_{xm}=-V_{1};\quad M_{xxm}=-M_{ 1}\\ x=b,& V_{xm}=V_{2};\quad M_{xxm}=M_{2}\end{split}  \tag{14}\]  
Figure 3: Displacements and force BCs for a plate element. The displacement BCs in Equation (13) are substituting into Equations (8) and (10); the following equations can be obtained as:  
\[ \begin{array}{l}w_{1}=A_{m}+0B_{m}+C_{m}+0D_{m}\\ \varnothing_{y1}=0A_{m}-r_{1m}B_{m}+0C_{m}-r_{2m}D_{m}\\ w_{1}=Ch_{1}A_{m}+Sh_{1}B_{m}+Ch_{2}C_{m}+Sh_{2}D_{m}\\ \varnothing_{y2}=(-r_{1m}Sh_{1})A_{m}-(r_{1m}Ch_{1})B_{m}-(r_{2m}Sh_{2})C_{m}- (r_{2m}Ch_{2})D_{m}\end{array} \]  
where  
\[ \begin{array}{l}Ch_{1}=\cos h(r_{1m}b),\;Cl_{2}=\cos h(r_{2m}b)\\ Sh_{1}=\sin h(r_{1m}b),\;Sh_{2}=\sin h(r_{2m}b)\end{array} \]  
This expression can be rewritten in the matrix form and expressed by Equation (15). \[ \begin{bmatrix}w_{1}\\ \varnothing_{1}\\ w_{2}\\ \varnothing_{2}\end{bmatrix} = \begin{bmatrix}1&0&1&0\\ 0&-r_{1m}&0&r_{2m}\\ Ch_{1}&Sh_{1}&Ch_{2}&Sh_{2}\\ -r_{1m}Sh_{1}&-r_{1m}Ch_{1}&-r_{2m}Sh_{2}&-r_{2m}Ch_{2}\end{bmatrix}\begin{array} []{l}A_{m}\\ B_{m}\\ C_{m}\\ D_{m}\end{array}  \tag{15}\]  
i.e.,  
\[[\delta]=[A]\{C\} \tag{16}\]  
Similarly, the force BCs are applied; substituting Equation (14) into Equations (11) and (12), the following matrix relationship can be developed and expressed by Equation (17). \[ \begin{bmatrix}V_{1}\\ M_{1}\\ V_{2}\\ M_{2}\end{bmatrix} = \begin{bmatrix}0&R_{1}&0&R_{2}\\ L_{1}&0&L_{2}&0\\ -R_{1}Sh_{1}&-R_{1}Ch_{1}&-R_{2}Sh_{2}&-R_{1}Ch_{2}\\ -L_{1}Ch_{1}&-L_{1}Sh_{1}&-L_{2}Ch_{2}&-L_{2}Sh_{2}\end{bmatrix}\begin{array} []{l}A_{m}\\ B_{m}\\ C_{m}\\ D_{m}\end{array} .