## 2 Basic Equations
\begin{array}{ll}c_{n}=&\frac{2}{b}\int_{0}^{b}w|_{x=0}\sin\frac{\lambda_{n}y}{2 }dy\;,\;d_{n}=\frac{2}{b}\int_{0}^{b}\frac{\partial w}{\partial x}\Big{|}_{x=a} \sin\frac{\lambda_{n}y}{2}dy\\ e_{n}=&\frac{2}{b}\int_{0}^{b}\frac{\partial^{2}w}{\partial x^{2}}\Big{|}_{x=0} \sin\frac{\lambda_{n}y}{2}dy\;,\;f_{n}=\frac{2}{b}\int_{0}^{b}\frac{\partial^{ 2}w}{\partial x^{3}}\Big{|}_{x=a}\sin\frac{\lambda_{n}y}{2}dy\\ g_{m}=&\frac{2}{a}\int_{0}^{a}w|_{y=0}\sin\frac{\delta_{n}x}{2}dx\;,\;h_{m}= \frac{2}{a}\int_{0}^{a}\frac{\partial w}{\partial y}\Big{|}_{y=b}\sin\frac{ \delta_{n}x}{2}dx\\ i_{m}=&\frac{2}{a}\int_{0}^{a}\frac{\partial^{2}w}{\partial y^{2}}\Big{|}_{y =0}\sin\frac{\delta_{n}x}{2}dx\;,\;j_{m}=\frac{2}{a}\int_{0}^{a}\frac{ \partial^{2}w}{\partial y^{3}}\Big{|}_{y=b}\sin\frac{\delta_{n}x}{2}dx\\ k_{m}=&\frac{2}{a}\int_{0}^{a}\frac{\partial^{2}w}{\partial x^{2}}\Big{|}_{y =0}\sin\frac{\delta_{n}x}{2}dx\;,\;h_{n}=\frac{2}{b}\int_{0}^{b}\frac{ \partial^{2}w}{\partial y^{2}}\Big{|}_{x=0}\sin\frac{\lambda_{n}y}{2}dy\end{array}  \tag{11}\]  
Substituting the assumed \(w(x,y)\) and its fourth high-order derivatives, that appeared in Equations (8)-(10), into Equation (3) yields the following expression for the free vibration governing equation:  
\[ \begin{array}{ll}\sum_{m=1,3}^{\infty}\sum_{n=1,3}^{\infty}\left\{\begin{array} []{ll}\left[\left(\frac{\phi_{x}^{2}}{4}+\frac{\lambda_{n}^{2}}{4}\right)^{2} -\frac{\rho h\omega^{2}}{D}\right]w_{mn}\\ +\frac{2}{a}\bigg{[}(-1)^{\frac{m-1}{2}}f_{n}+\frac{\delta_{m}}{2}e_{n}-(-1)^{ \frac{m-1}{2}}\left(\frac{\delta_{m}}{2}\right)^{2}\!d_{n}-\left(\frac{\delta _{m}}{2}\right)^{3}\!c_{n}\bigg{]}\\ +\frac{2}{b}\bigg{[}(-1)^{\frac{n-1}{2}}j_{m}+\frac{\lambda_{n}}{2}i_{m}-(-1) ^{\frac{n-1}{2}}\left(\frac{\lambda_{n}}{2}\right)^{2}\!h_{m}-\left(\frac{ \lambda_{n}}{2}\right)^{3}\!g_{m}\bigg{]}\\ +\frac{8}{ab}(-1)^{\frac{m-1}{2}}\bigg{[}(-1)^{\frac{n-1}{2}}\frac{\partial^{2 }w}{\partial x\partial y}\bigg{|}_{x=a,y=b}+\frac{\lambda_{n}}{2}\frac{ \partial w}{\partial x}\bigg{|}_{x=a,y=0}\bigg{]}\\ -\frac{4}{a}\bigg{[}(-1)^{\frac{m-1}{2}}\Big{(}\frac{\lambda_{n}}{2}\Big{)}^{ 2}\!d_{n}-\frac{2}{b}\frac{\delta_{m}}{2}l_{n}\bigg{]}-\frac{4}{b}\Big{(} \frac{\delta_{m}}{2}\Big{)}^{2}\Big{[}(-1)^{\frac{n-1}{2}}h_{m}+\frac{\lambda _{n}}{2}\delta_{m}\bigg{]}\end{array}\right.\end{array} \right\}\sin\frac{ \alpha_{m}x}{2}\sin\frac{\beta_{n}y}{2}\;=0 \tag{12}\]  
It is obvious that \(\left.\frac{\partial^{2}w}{\partial x\partial y}\right|_{x=a,y=b}\) is zero since the zero-twisting moment at free corner. Similarly, due to the zero deflection at rotationally-restrained edges (\(x=0\), \(y=0\)) and zero effective shear force at free edges (\(x=a\), \(y=b\)), it is easy to derive that  
\[\frac{\partial^{2}w}{\partial y^{2}}\Big{|}_{x=0}=0\;,\;\;\frac{\partial w}{ \partial x}\Big{|}_{y=0}=\left.\frac{\partial^{2}w}{\partial x^{2}}\right|_{y=0}=0 \tag{13}\]  
\[c_{n}=0\;,\;l_{n}=0\;,\;g_{m}=0\;,\;k_{m}=0\;, \tag{14}\]  
\[f_{n}=(2-\mu)\bigg{(}\frac{\lambda_{n}}{2}\bigg{)}^{2}d_{n}\;,\;j_{m}=(2-\mu) \bigg{(}\frac{\delta_{m}}{2}\bigg{)}^{2}h_{m} \tag{15}\]  
It is clearly that \(d_{n}\) and \(h_{m}\) are the Fourier coefficients of the slope along the free edges. Obviously \(-D\cdot i_{m}\) and \(-D\cdot e_{n}\) are the Fourier coefficients of bending moments along the rotationally restrained edges, the corresponding bending moments can be directly acquired by the following formula:  
\[\left.M_{x}\right|_{x=0}=\left.-D\frac{\partial^{2}w}{\partial x^{2}}\right|_{x =0}=-D\sum_{n=1,3}^{\infty}e_{n}\sin\frac{\lambda_{n}y}{2}\;,\;\;\left.M_{y} \right|_{y=0}=\left.-D\frac{\partial^{2}w}{\partial y^{2}}\right|_{y=0}=-D \sum_{n=1,3}^{\infty}i_{m}\sin\frac{\delta_{m}x}{2} \tag{16}\]  
One can obtain the following simplified expression for the free vibration governing equations by substituting Equations (13)-(15) into the Equation (12), which is shown below:  
\[\sum_{m=1,3}^{\infty}\sum_{n=1,3}^{\infty}\left\{