## 5 Numerical Examples and Results
### Free Vibration Analysis  
In order to properly evaluate the behaviour of FGM plates, in addition to the results of the static analysis, an analysis of the behaviour of plates in a dynamic environment is also required. This section presents the results of free vibrations of FGM plates placed on an elastic foundation for different values of Winkler coefficient k\({}_{0}\), Pasternak coefficient k\({}_{1}\), and power law index p. As with the results of the thermal analysis described above, the verification of the developed and implemented theoretical results based on the newly introduced shape function was performed through a tabular representation of the obtained results, in comparison with the results from the literature. The procedure verified in this way was used to obtain other results for plates of different gradient structure. Based on the results of the dynamic analysis, appropriate interpretations and the comments were provided, and certain conclusions made. In order to display the numerical values of natural frequencies, for rectangular and square FGM plates, it is necessary to normalize the obtained values according to:  
\[\widetilde{\omega}=\omega\mathrm{h}\sqrt{\frac{\rho_{m}}{E_{m}}}. \tag{34}\]  
Tables 5 and 6 show the non-dimensional natural frequencies \(\widetilde{\omega}\) of rectangular (a/b = 0.5) and square (a/b = 1) plates resting on an elastic foundation for different values of Winkler coefficient (k\({}_{0}\)), Pasternak coefficient (k\({}_{1}\)) and index \(p\). In order to observe the impact of the elastic foundation, the values of k\({}_{0}\) = 0 and k\({}_{1}\) = 0 were first taken, introducing one coefficient at the time in order to determine which of the two coefficients has a greater impact. By analyzing the results, it is evident that the impact of the coefficient k\({}_{1}\) on the \(\widetilde{\omega}\) is far greater than the impact of the coefficient k\({}_{0}\). Figure 8: Effect of the Winkler coefficient k\({}_{0}\) and Pasternak coefficient k\({}_{1}\) on the critical buckling temperature \(\Delta\mathrm{t}_{\mathrm{cr}}\) for case of nonlinear temperature change across plate thickness: (a) a/h = 10, a/b = 1, s = 3, k\({}_{1}\) = 10; (b) a/h = 10, a/b = 1, k\({}_{0}\) = 10. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & & \(\widetilde{\omega}\) & \\ \cline{3-8}
**a/b** & **k\({}_{0}\)** & **k\({}_{1}\)** & **Theory** & \multicolumn{4}{c}{**a/h = 5**} & \\ \cline{3-8}  & & & & & \(p\) = 0 & \(p\) = 1 & \(p\) = 5 & \(p\) = 10 \\ \hline  & & & Present study & 6.761 & 5.2016 & 4.3761 & 4.206 \\  & & & No. 1 & 6.7609 & 5.2015 & 4.3757 & 4.2058 \\  & & & No. 2 & 6.7609 & 5.2015 & 4.3757 & 4.2058 \\
0.5 & 0 & 0 & No. 3 & 6.7616 & 5.202 & 4.3733 & 4.205 \\  & & & No. 4 & 6.775 & 5.2108 & 4.3753 & 4.2136 \\  & & & No. 5 & 6.7609 & 5.2015 & 4.3757 & 4.2058 \\  & & & No. 6 & 6.7628 & 5.2027 & 4.3832 & 4.211 \\  & & & No. 7 & 6.7636 & 5.2033 & 4.3722 & 4.2055 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Non-dimensional natural frequencies (\(\widetilde{\omega}\)) of rectangular FGM plates placed on an elastic foundation for different values of the Winkler coefficient k\({}_{0}\). Pasternak coefficient k\({}_{1}\), and power law index p (a/b = 0.5, a/h = 5, m = 1, n = 1). Figure 9 shows the diagrams of the non-dimensional natural frequencies \(\widetilde{\omega}\) of the plates resting on an elastic foundation for different ratios a/h, a/b, and the values of the coefficients k\({}_{0}\), k\({}_{1}\), and index p. Figure 9a shows the impact of the Winkler coefficient (k\({}_{0}\)) and Pasternak coefficient (k\({}_{1}\)) on the values of natural frequencies for the first oscillation mode. It can be clearly observed that the introduction of the coefficient k\({}_{0}\) leads to very small changes in the \(\widetilde{\omega}\) values in comparison to the case of absence of an elastic foundation (k\({}_{0}\) = 0 and k\({}_{1}\) = 0).