## 2 Theoretical Formula
### Vibration Equation
\begin{array}{cc} \overline{Q}_{11}^{(k)}&\overline{Q}_{12}^{(k)}\\ \overline{Q}_{12}^{(k)}&\overline{Q}_{22}^{(k)}\end{array} \right]\left\{ \begin{array} []{l}\alpha_{x}\\ \alpha_{y}\end{array} \right\}^{(k)}\Delta Th_{k} \tag{19}\]  
\[\left\{ \begin{array}{l}N_{x}^{T}\\ N_{y}^{T}\end{array} \right\}=\sum\limits_{k=l,b}\left\{ \begin{array}{l} \sigma_{x}^{T}\\ \sigma_{y}^{T}\end{array} \right\}^{(k)}h_{k} \tag{20}\]  
\[\left\{ \begin{array}{l}N_{x}^{T}\\ N_{y}^{T}\end{array} \right\}=\int_{-h_{c}/2}^{h_{c}/2}\left[ \begin{array}{ cc}\overline{Q}_{11}^{(c)}(z)&\overline{Q}_{12}^{(c)}(z)\\ \overline{Q}_{12}^{(c)}(z)&\overline{Q}_{22}^{(c)}(z)\end{array} \right] \left\{ \begin{array}{l}\alpha_{x}\\ \alpha_{y}\end{array} \right\}^{(c)}\Delta Tdz \tag{21}\]  
For the simply supported square/rectangle functionally gradient plate, the boundary conditions and the state variables are assumed as  
\[\left\{ \begin{array}{l}w=\varphi_{y}=0\;at\;x=0,L_{a}\\ w=\varphi_{x}=0\;at\;y=0,L_{b}\end{array} \right. \tag{22}\]  
\[\left\{ \begin{array}{l}\varphi_{x}(x,y,t)=\sum\limits_{m=1}^{\infty}\sum \limits_{n=1}^{\infty}\Phi_{xmm}cos\frac{m\pi x}{l_{a}}sin\frac{n\pi y}{l_{b} }e^{j\omega t}\\ \varphi_{y}(x,y,t)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\Phi_ {ymn}sin\frac{m\pi x}{l_{a}}cos\frac{n\pi y}{l_{b}}e^{j\omega t}\\ w(x,y,t)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}W_{mn}sin\frac {m\pi x}{l_{a}}sin\frac{n\pi y}{l_{b}}e^{j\omega t}\end{array} \right. \tag{23}\]  
where \(\Phi_{xmm}\), \(\Phi_{ymn}\) and \(W_{mn}\) are the Fourier expansion coefficients of the solution, \(m\) and \(n\) are the half wave numbers in the \(x\) and \(y\) directions, respectively, and \(\omega\) is the circular frequency. By substituting Equation (23) into Equations (14)-(16), the following equation can be obtained as  
\[\left([K]-\omega^{2}[M]\right)\left\{U\right\}=\left\{F\right\} \tag{24}\]  
where \(\left\{U\right\}=\left\{\Phi_{xmm},\Phi_{ymn},W_{mm}\right\}^{T}\), \(\left\{F\right\}=\left\{0,0,q\right\}^{T}\) and \(\left[K\right]\), \(\left[M\right]\) can be written as  
\[\left[K\right]=\left[ \begin{array}{ccc}K_{11}&K_{12}&K_{13}\\ K_{12}&K_{22}&K_{23}\\ K_{13}&K_{23}&K_{33}\end{array} \right],\;\left[M\right]=\left[ \begin{array}[] {ccc}m_{11}&m_{12}&m_{13}\\ m_{21}&m_{22}&m_{23}\\ m_{31}&m_{32}&m_{33}\end{array} \right] \tag{25}\]  
in which \(K_{ij}\), \(m_{ij}\) are given as  
\[ \begin{array}{l}K_{11}=\left(D_{11}-\frac{8}{3h^{2}}F_{11}+\frac{16}{9h^{2 }}H_{11}\right)\left(\frac{m\pi}{l_{a}}\right)^{2}-\left(D_{66}-\frac{4}{3h^{2 }}F_{66}+\frac{16}{9h^{2}}H_{66}\right)\left(\frac{m\pi}{l_{b}}\right)^{2}\\ +\left(A_{44}-\frac{8}{h^{2}}D_{44}+\frac{16}{h^{2}}F_{44}\right)\\ \\ K_{12}=K_{21}=\left(D_{12}-\frac{8}{3h^{2}}F_{12}+D_{66}-\frac{8}{3h^{2}}F_{66 }+\frac{16}{9h^{4}}H_{66}\right)\left(\frac{m\pi}{l_{a}}\right)\left(\frac{m \pi}{l_{b}}\right)\\ \\ K_{13}=K_{31}=\left(-\frac{4}{3h^{2}}F_{11}+\frac{16}{9h^{4}}H_{11}\right) \left(\frac{m\pi}{l_{a}}\right)^{3}\\ +\left(-\frac{4}{3h^{2}}F_{11}+\frac{16}{9h^{4}}H_{12}-\frac{8}{3h^{2}}F_{66 }+\frac{32}{9h^{4}}H_{66}\right)\left(\frac{m\pi}{l_{a}}\right)\left(\frac{m \pi}{l_{b}}\right)^{2}\\ +\left(A_{44}-\frac{8}{h^{2}}D_{44}+\frac{16}{h^{4}}F_{44}\right)\left(\frac{m \pi}{l_{a}}\right)\\ \\ K_{22}=\left(D_{22}-\frac{8}{3h^{2}}F_{22}+\frac{16}{9h^{4}}H_{11}\right) \left(\frac{m\pi}{l_{b}}\right)^{2}+\left(D_{66}-\frac{4}{3h^{2}}F_{66}+\frac{ 16}{9h^{4}}H_{66}\right)\left(\frac{m\pi}{l_{a}}\right)^{2}\\ +\left(A_{55}-\frac{8}{h^{2}}D_{55}+\frac{16}{h^{4}}F_{55}\right)\end{array}  \tag{29}\]\[K_{23}=\left(-\frac{4}{3\hbar^{2}}F_{11}+\frac{16}{9\hbar^{4}}H_{12}-\frac{8}{3 \hbar^{2}}F_{66}+\frac{32}{9\hbar^{2}}H_{66}\right)\left(\frac{m\pi}{L_{a}} \right)^{2}\left(\frac{m\pi}{L_{b}}\right) \tag{30}\]  
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