## 2 Formulation and Methodology
 \tag{18}\]where  
\[\Omega_{mn}=D_{x}\alpha_{m}^{4}+2 Ha\alpha_{m}^{2}\beta_{n}^{2}+D_{y}\beta_{n}^{4} \tag{19}\]  
Taking the inverse FCT of Equation (18), it is obtained  
\[\bar{w}(m,y)=\frac{2}{b}\sum_{n=0}^{\infty}\varepsilon_{n}\hat{w}(m,n)\cos \beta_{n}y \tag{20}\]  
Taking second-order derivative of Equation (20) with respect to \(y\) and applying the Stokes' transformation [25] gives  
\[\bar{w}_{,yy}\left(m,y\right)=\frac{2}{b}\sum_{n=0}^{\infty}\varepsilon_{n} \Bigg{\{}\left[(-1)^{n}\bar{w}_{,yy}\left(m,b\right)-\bar{w}_{,yy}\left(m,0 \right)\right]-\beta_{n}^{2}\hat{w}(m,n)\Bigg{\}}\cos\beta_{n}y \tag{21}\]  
Substituting Equations (7c) and (20) into Equation (21) gives  
\[\sum_{n=0}^{\infty}\varepsilon_{n}\Bigg{\{}\left[(-1)^{n}\bar{w}_{,yy}\left(m,b\right)-\bar{w}_{,yy}\left(m,0\right)\right]-(\
u_{x}\alpha_{m}^{2}+\beta_{ n}^{2})\hat{w}(m,n)\Bigg{\}}=0 \tag{22a}\] \[\sum_{n=0}^{\infty}(-1)^{n}\varepsilon_{n}\Bigg{\{}\left[(-1)^{n}\bar{w}_{,yy} \left(m,b\right)-\bar{w}_{,yy}\left(m,0\right)\right]-(\
u_{x}\alpha_{m}^{2}+ \beta_{n}^{2})\hat{w}(m,n)\Bigg{\}}=0 \tag{22b}\]  
For numerical calculations, the infinite series in Equation (22) should be truncated to be finite terms, \(N\). Then, it can be obtained expressions for \(\bar{w}_{,yy}\left(m,0\right)\) and \(\bar{w}_{,yy}\left(m,b\right)\) by solving Equation (22). They are expressed as  
\[\bar{w}_{,yy}\left(m,0\right) =\sum_{n=0}^{N}\frac{(-1)^{n+N}-(2N+1)}{2N(N+1)}\varepsilon_{n}( \
u_{x}\alpha_{m}^{2}+\beta_{n}^{2})\hat{w}(m,n) \tag{23a}\] \[\bar{w}_{,yy}\left(m,b\right) =\sum_{n=0}^{N}\frac{(-1)^{n}(2N+1)-(-1)^{N}}{2N(N+1)}\varepsilon_ {n}(\
u_{x}\alpha_{m}^{2}+\beta_{n}^{2})\hat{w}(m,n) \tag{23b}\]  
Then, taking the inverse finite sine transform of Equation (18) with respect to \(x\) yields  
\[\hat{w}(x,n)=\frac{2}{a}\sum_{m=1}^{\infty}\hat{w}(m,n)\sin\alpha_{m}x \tag{24}\]  
Taking the derivative of Equation (24) with respect to \(x\) and using [PERSON]'s transformation [25], it is found  
\[\hat{w}_{,xx}\left(x,n\right)=\frac{2}{a}\sum_{m=1}^{\infty}\alpha_{m}\hat{w}( m,n)\cos\alpha_{m}x \tag{25}\]  
Substituting Equations (7a) and (7b) into Equation (25) and replacing constants \(R_{x0}\) and \(R_{xa}\) by the corresponding rotational fixity factors \(r_{x0}\) and \(r_{xa}\) based in Equations (6), it yields  
\[\hat{w}_{,xx}\left(0,n\right) =\frac{6r_{x0}}{a^{2}(1-r_{x0})}\sum_{m=1}^{\infty}\alpha_{m}\hat {w}(m,n) \tag{26a}\] \[\bar{w}(m,y)=\frac{2}{b}\sum_{n=0}^{\infty}\varepsilon_{n}\hat{w}(m,n)\cos \beta_{n}y \tag{20}\]0At last, substituting expressions of \(\hat{w}_{,xx}\left(0,n\right)\), \(\hat{w}_{,xx}\left(a,n\right)\), \(\hat{w}_{,yy}\left(m,0\right)\) and \(\hat{w}_{,yy}\left(m,b\right)\) in Equations (23) and Equations (26) into Equation (18), it yields  
\[ \begin{split}&\Omega_{mn}\hat{W}(m,n)+\frac{2\alpha_{m}D_{x}}{a^{2}} \sum_{i=1}^{M}\left[(-1)^{i+m}\frac{3r_{xa}}{1-r_{xa}}+\frac{3r_{x0}}{1-r_{x0} }\right]\alpha_{i}\hat{W}(i,n)\\ &+D_{y}(\
u_{x}\alpha_{m}^{2}+\beta_{n}^{2})\sum_{j=0}^{N}\frac{( -1)^{n+N}+(-1)^{j+N}-[(-1)^{j+n}+1](2N+1)}{2N(N+1)}\varepsilon_{j}(\
u_{x} \alpha_{m}^{2}+\beta_{j}^{2})\hat{W}(m,j)\\ &=\omega^{2}\rho h\hat{W}(m,n)\end{split}  \tag{27}\]  
It can be found that Equation (27) is different from formulations for the bending problems of plates which were presented in [21,22]. If adopting the formulations for bending problems, frequencies should be acquired by solving a highly nonlinear equation which would be quite difficult. However, the present Equation (27) is an eigenvalue problem which is much easier to solve. It should be noted that all the series expansions are truncated herein to finite number \(M\) for \(m\) and \(N\) for \(n\) while the upper limit of summation is theoretically specified as infinity. For each combination of \(M\) and \(N\), Equation (27) produces \(M\times(N+1)\) equations with \(M\times(N+1)\) unknown coefficients.