## 4 Equations of Motion of FG Plate Placed on Elastic Foundation  
As the subject of this chapter is linear dynamic analysis, the kinematic relations of displacements and deformations are defined under assumptions of small deformations. Since the total potential energy \(\Pi\) is represented as the sum of the strain energy of the plate and the potential energy of the elastic foundation (10), for the application of the [PERSON]'s principle it is necessary to further define the kinetic energy:  
\[\mathrm{K}=\frac{1}{2}\int\limits_{-h/2}^{h/2}\int\limits_{A}\rho(z)\left[ \left(\frac{\partial u}{\partial t}\right)^{2}+\left(\frac{\partial v}{ \partial t}\right)^{2}+\left(\frac{\partial w}{\partial t}\right)^{2}\right] \mathrm{d}Adz, \tag{22}\]  
wherein:  
\(\rho(z)\)-material density in an arbitrary cross-section z,  
As can be seen, as a consequence of the gradient structure of the plate material, the material density represents a function of the z coordinate. The change of density in the direction of z-axis is defined in accordance to the power-law distribution as:  
\[\rho(z)=\rho_{m}+\rho_{cm}\left(\frac{1}{2}+\frac{z}{h}\right)^{P},\;\;\rho_{cm }=\rho_{c}-\rho_{m}, \tag{23}\]  
By substituting the strain energy of the plate (8), the potential energy of the elastic foundation (9) and the kinetic energy (22) into the [PERSON]'s principle:  
\[ \begin{array}{l}\delta\int\limits_{t_{1}}^{t_{1}}[\mathrm{K}-(U_{s}+U_{e})] \mathrm{d}t=\int\limits_{t_{1}}^{t_{2}}\left[-\int\limits_{A}(N_{xx}\varepsilon _{xx}^{(0)}+N_{yy}\varepsilon_{yy}^{(0)}+N_{xy}\gamma_{xy}^{(0)}+M_{xx}k_{xx}^ {(0)}+M_{yy}k_{yy}^{(0)}+M_{xy}k_{xy}^{(0)}\right.\\ \left.+P_{xx}k_{xx}^{(1)}+P_{yy}k_{yy}^{(1)}+P_{xy}k_{xy}^{(1)}+R_{x}k_{xz}^{( 2)}+R_{y}k_{yz}^{(2)})dA\\ -\int\limits_{A}\left\{k_{0}w\delta w+k_{1}\left(\frac{\partial w}{\partial x }\frac{\partial sw}{\partial x}+\frac{\partial w}{\partial y}\frac{\partial sw }{\partial y}\right)\right\}\mathrm{d}A\\ +\int\limits_{-h/2}^{h/2}\int\limits_{A}\rho(z)\left(\frac{\partial u}{ \partial t}\frac{\partial su}{\partial t}+\frac{\partial v}{\partial t}\frac{ \partial sv}{\partial t}+\frac{\partial w}{\partial t}\frac{\partial sw}{ \partial t}\right)dAdz\right]\mathrm{d}t=0.\end{array}  \tag{24}\]  
By substituting the strain components expressed by assumed displacement forms as well as by applying the calculus of variations and group the members with the \(\delta u_{0}\), \(\delta v_{0}\), \(\delta w_{0}\), \(\delta\theta_{x}\) and \(\delta\theta_{y}\), the equations of motion are obtained:  
\[ \begin{array}{l}\delta u_{0}:\;\frac{\partial N_{xx}}{\partial x}+\frac{ \partial N_{xy}}{\partial y}=I_{1}\ddot{u}_{0}-I_{2}\frac{\partial\ddot{w}_{0} }{\partial x}+I_{4}\ddot{\theta}_{x},\\ \delta v_{0}:\;\frac{\partial N_{yy}}{\partial y}+\frac{\partial N_{xy}}{ \partial x}=I_{1}\ddot{v}_{0}-I_{2}\frac{\partial\ddot{w}_{0}}{\partial y}+I_ {4}\ddot{\theta}_{y},\\ \delta w_{0}:\;\frac{\partial^{2}M_{xx}}{\partial x^{2}}+2\frac{\partial^{2}M _{xy}}{\partial x\partial y}+\frac{\partial^{2}M_{yy}}{\partial y^{2}}-k_{0} w_{0}+k_{1}\left(\frac{\partial^{2}w_{0}}{\partial x^{2}}+\frac{\partial^{2}w_{0}}{ \partial y^{2}}\right)=I_{1}\ddot{w}_{0}+I_{2}\left(\frac{\partial\ddot{w}_{0 }}{\partial x}+\frac{\partial\ddot{v}_{0}}{\partial y}\right)\\ -I_{3}\left(\frac{\partial^{2}\ddot{w}_{0}}{\partial x^{2}}+\frac{\partial^{2} \ddot{w}_{0}}{\partial y^{2}}\right)+I_{5}\left(\frac{\partial\ddot{w}_{0}}{ \partial x}+\frac{\partial\ddot{v}_{0}}{\partial y}\right),\\ \delta\theta_{x}:\;\frac{\partial P_{xx}}{\partial x}+\frac{\partial P_{yy}}{ \partial y}-R_{x}=I_{4}\ddot{u}_{0}-I_{5}\frac{\partial\ddot{w}_{0}}{\partial x }+I_{6}\ddot{\theta}_{x},\\ \delta\theta_{y}:\;\frac{\partial P_{yy}}{\partial x}+\frac{\partial P_{yy}}{ \partial y}-R_{y}=I_{4}\ddot{v}_{0}-I_{5}\frac{\partial\ddot{w}_{0}}{\partial y }+I_{6}\ddot{\theta}_{y}.\end{array}  \tag{25}\]where \(\rm I_{i}\) (\(i=1,2,3,4,5\,i\,6\)) are terms due to inertia defined as:  
\[