## 4 Numerical Results  
A large parametric investigation was performed to check for the sensitivity of the vibration response for the FG annular plates covered by magneto-electro-elastic layers, and surrounded by a Pasternak foundation, for different input parameters, primarily, temperature, moisture, external voltage, different boundary conditions, elastic foundations, thickness ratio, power index and number of grid points. The properties of BaTiO\({}_{3}\)/CoFeO\({}_{4}\), ceramic (alumina) and metal (aluminum) are presented in Tables 1 and 2, in line with Refs. [33] and [27], respectively. At the same time, in the attempt to validate the proposed formulation, a simplified analysis was carried out, first, without considering the external facesheets, the hygrothermal environment and the LW theory, in line with Refs. [6; 35]. The dimensionless frequency \(\omega^{*}\,=\,\omega h\sqrt{\rho/\zeta_{11}}\) was, thus, determined, for different grid point discretization, as reported in Table 3, being \(C_{11}\,=\,E(1-v)/(1+v)(1-2v)\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties** & & \\ \hline \(Q_{11}\) (Pa) & \(2.86\times 10^{11}\) \\ \(Q_{12}\) (Pa) & \(1.73\times 10^{11}\) \\ \(Q_{55}\) (Pa) & \(0.453\times 10^{11}\) \\ \(e_{31}\) (C / m\({}^{2}\)) & \(-4.4\) \\ \(e_{33}\) (C / m\({}^{2}\)) & \(18.6\) \\ \(d_{31}\) (N/ Am) & \(580.3\) \\ \(d_{33}\) (N/ Am) & \(699.7\) \\ \(\in_{11}\) (C\({}^{2}\)/Nm\({}^{2}\)) & \(8.0\times 10^{-11}\) \\ \(\in_{33}\) (C\({}^{2}\)/Nm\({}^{2}\)) & \(9.3\times 10^{-11}\) \\ \(g_{11}\,=\,g_{33}\) & \(3.0\times 10^{-12}\) \\ \(\mu_{11}\) (Ns\({}^{2}\)/C\({}^{2}\)) & \(-590\times 10^{-6}\) \\ \(\mu_{33}\) (Ns\({}^{2}\)/C\({}^{2}\)) & \(157\times 10^{-6}\) \\ \(\alpha_{11}\,=\,\alpha_{22}\) (/K) & \(1\times 10^{-6}\) \\ \(\beta_{11}\) (m\({}^{3}\)/kg) & \(0\) \\ \(\beta_{33}\) (m\({}^{3}\)/kg) & \(1.1\times 10^{-4}\) \\ \(\chi_{1}\,=\,\chi_{3}\) (Cm/ Kg) & \(0\) \\ \(\tau_{1}\,=\,\tau_{3}\)(N/ AmK) & \(6.0\times 10^{-3}\) \\ \(\gamma_{1}\,=\,\gamma_{3}\) (C/ m\({}^{2}\) K) & \(-13.0\times 10^{-5}\) \\ \(v_{1}\,=\,v_{3}\) (N/ Am\({}^{2}\)kg) & \(0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of BaTiO\({}_{3}\)/CoFeO\({}_{4}\)[33]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties** & & \\ \hline \(E_{c}\) (Pa) & \(380\times 10^{9}\) \\ \(\rho_{c}\) (kg/ m\({}^{3}\)) & \(3800\) \\ \(\alpha_{c}\) (/K) & \(7\times 10^{-6}\) \\ \(\beta_{c}\) (wt\(\%\)H\({}_{2}\)O)\({}^{-1}\) & \(0.001\) \\ \(v_{c}\,=\,v_{m}\) & \(0.3\) \\ \(E_{m}\) (Pa) & \(70\) \\ \(\rho_{m}\) (kg/ m\({}^{3}\)) & \(2707\) \\ \(\alpha_{m}\) (/K) & \(23\times 10^{-6}\) \\ \(\beta_{m}\) (wt\(\%\)H\({}_{2}\)O)\({}^{-1}\) & \(0.44\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties of ceramic and metal [27]. The accuracy and convergence of the DQM-based results are summarized in Table 4 in terms of dimensionless frequency for the selected sandwich annular plate, with clamped supports at both edges under different variations in temperature. More specifically, the dimensionless frequency is computed as  
\[\omega^{*}\ =\ \omega r_{i}^{2}\sqrt{\rho_{t}h_{t}/D} \tag{50}\]  
in which  
\[D\ =\ E_{m}h_{t}^{3}/12(1-v_{m}^{2}) \tag{51}\]  
Moreover, in order to verify the free vibration of the annular plate by considering the electromagnetic layer, it is sufficient to consider the material property and geometry used in Ref. [36]. A comparison between the present results and Ref. [36] is shown in Table 5. As can be seen from this table, there is excellent agreement between them. The results are derived for different values of DQM grid points, with a fast rate of convergence, even with a reduced number of grid points equal to eight. Thus, the systematic investigation accounts for the effects of the Winkler and Pasternak coefficients on the dimensionless frequency, as plotted in Figures 2 and 3 vs. the radial ratio of the annular sandwich plates.