## 2 Theoretical Modeling
### The FGM Plate's Governing Differential Equation and Eigenvalues Computation Using CPT  
Based on [PERSON]'s principle, the governing equation for out of plane free vibration of a FGM plate yields the following equations [1]. \[D_{eff}\
abla^{4}w+I_{0}\frac{\partial^{2}w}{\partial t^{2}}=0 \tag{11}\]  
where \(w(x,y)\) is the transverse displacement, and \(\
abla^{4}=\frac{\partial^{4}}{x^{4}}+2\frac{\partial^{4}}{\partial x^{2} \partial y^{2}}+\frac{\partial^{4}}{y^{4}}\) and \(I_{0}\) and \(D_{eff}\) are the transverse inertia and effective dynamic stiffness of the P-FGM plate, respectively, and are given by  
\[ \begin{array}{l}D_{eff}=\int_{-h/2-z_{0}}^{h/2-z_{0}}z_{0s}^{2}Q_{11}(z_{ns} )dz_{ns}=\int_{-h/2}^{h/2}(z-z_{0})^{2}Q_{11}(z)dz\\ =\frac{12D_{c}}{E_{att}}\bigg{(}\frac{3E_{att}\left(\frac{3E_{att}\left(k^{2}+ k+2\right)+\left(k^{3}+3k^{2}+8k\right)}{12\left(k+1\right)\left(k^{2}+5k+6 \right)}-\frac{z_{0}}{h}\frac{k\left(E_{att}-1\right)}{\left(k+1\right)\left(k +2\right)}+\left(\frac{z_{0}}{h}\right)^{2}\frac{E_{att}+k}{k+1}\bigg{)}\end{array}  \tag{12}\]  
\[I_{0}=\int_{-h/2-z_{0}}^{h/2-z_{0}}\rho(z_{ns})dz_{ns}=\frac{\rho_{c}h}{\rho_{ rat}}\bigg{(}\frac{\rho_{rat}-1}{k+1}+1\bigg{)} \tag{13}\]  
\(D_{eff}\) is dependent upon \(E_{rat}=\frac{E_{c}}{E_{m}}\) and the power-law index '\(k\)' and \(D_{c}=\frac{E_{c}h^{3}}{12\left(1-\
u^{2}\right)}\). \(I_{0}\) is dependent upon \(\rho_{rat}=\frac{\rho_{c}}{\rho_{ran}}\) and the power-law index is '\(k\)'. In the case of harmonic free vibration, \(w(x,y)\) can be written as the superposition of the infinite number of the mode shape function \(\phi(x,y)\) and can be written as  
\[w(x,y)=\sum_{m=1}^{\infty}\sum_{m=1}^{\infty}\phi_{mn}(x,y)\eta_{mn}e^{i\omega t} \tag{14}\]  
where \(\eta_{mn}\) is the modal participation factor representing the contribution of each mode to harmonic response analysis. Because these mode shapes are orthogonal to each other [7; 29; 32; 34; 35], we get,  
\[\int_{0}^{a}\int_{0}^{b}I_{0}\phi_{mn}(x,y)\phi_{jk}(x,y)dxdy\ \left\{ \begin{array}{ll}M_{mn}\,if\ m=j,\ \ n=k\\ 0&\text{other}\end{array} \right. \tag{15}\]  
where \(M_{mn}\) is the modal mass. Seeking a harmonic solution, Equation (11) is reduced in terms of mode shape function \(\phi_{mn}\) and is given by  
\[D_{eff}\
abla^{4}\phi_{mn}(x,y)-\omega_{mn}^{2}I_{0}\phi_{mn}(x,y)=0 \tag{16}\]  
The mode shape function \(\phi_{mn}(x,y)\) is generally the product of the individual beam function [35]. \[\phi_{mn}(x,y)=X_{m}(x)Y_{n}(y) \tag{17}\]  
The two beam functions \(X_{m}(x)\) and \(Y_{n}(y)\) are dependent upon boundary conditions used over the P-FGM plate's edges. For clamped boundary conditions, these functions are given by  
\[X_{m}(x)=cosh\bigg{(}\frac{\lambda_{m}x}{a}\bigg{)}-cos\bigg{(}\frac{\lambda_ {m}x}{a}\bigg{)}-\beta_{m}\bigg{[}sinh\bigg{(}\frac{\lambda_{m}x}{a}\bigg{)}- sin\bigg{(}\frac{\lambda_{m}x}{a}\bigg{)}\bigg{]} \tag{18}\]  
\[Y_{n}(y)=cosh\bigg{(}\frac{\lambda_{n}y}{b}\bigg{)}-cos\bigg{(}\frac{\lambda_ {n}y}{b}\bigg{)}-\beta_{n}\bigg{[}sinh\bigg{(}\frac{\lambda_{n}y}{b}\bigg{)}- sin\bigg{(}\frac{\lambda_{n}y}{b}\bigg{)}\bigg{]} \tag{19}\]  
where  
\[\beta_{i}=\frac{cosh(\lambda_{i})-cos(\lambda_{i})}{sinh(\lambda_{i})-sin( \lambda_{i})}\ \text{for}\ i=m,n \tag{20}\]and \(\lambda_{m}\) & \(\lambda_{n}\) are the roots of Equation (21). \[cosh(\lambda)cos(\lambda)-1=0 \tag{21}\]  
The \((m,n)^{th}\) natural frequency of the P-FGM plate, \(\omega_{mn}\) is given by [35]  
\[\omega_{mn}=\sqrt{\frac{D_{eff}}{I_{0}}}\times\sqrt{\frac{I_{1}I_{2}+2I_{3}I_{4 }+I_{5}I_{6}}{I_{2}I_{6}}} \tag{22}\]  
Considering a combination of individual beam function and boundary condition along the edges of the plate, these \"\(l\)\" terms can be computed. For the present problem, \(I_{i}\) is [32; 35]  
\[