## 4 Numerical Results
The thickness and radial ratios are set here as \(h_{t}/h_{c}\ =\ 0.5\) and \(r_{i}/r_{o}\ =\ 0.2\), whereas temperature and moisture variations are assumed to be null, i.e., \(\Delta T\ =\ 0\) and \(\Delta C\ =\ 0\), respectively. As visible in Figures 2 and 3, an increasing radial ratio gradually reduces the value of the frequency. As the Winkler and Pasternak coefficients increase, the dimensionless frequency also increases due to an increased stiffness and stability in the structure for the same radial ratio \(r_{i}/r_{o}\). At the same time, the effect of the Pasternak coefficient is more effective than the Winkler one. This is due to the presence of both normal and transverse shear loads, as considered in the Pasternak foundation but \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(g\) & **Grid Points (N)** & \(\omega^{*}\) & \(\omega^{*}\) & \(\omega^{*}\) \\  & & \(\Delta T\) = \(0[^{\circ}\mathbb{C}]\) & \(\Delta T\) = \(100[^{\circ}\mathbb{C}]\) & \(\Delta T\) = \(200[^{\circ}\mathbb{C}]\) \\ \hline
1 & 5 & 12.4713 & 12.3882 & 12.3046 \\  & 7 & 13.3089 & 13.2251 & 13.1407 \\  & 8 & 13.3076 & 13.2238 & 13.1394 \\  & 10 & 13.3076 & 13.2238 & 13.1394 \\
2 & 5 & 12.0093 & 11.9403 & 11.8710 \\  & 7 & 12.7439 & 12.6747 & 12.6051 \\  & 8 & 12.7437 & 12.6745 & 12.6049 \\  & 10 & 12.7437 & 12.6745 & 12.6049 \\
3 & 5 & 11.7305 & 11.6691 & 11.6074 \\  & 7 & 12.4071 & 12.3456 & 12.2838 \\  & 8 & 12.4070 & 12.3455 & 12.2837 \\  & 10 & 12.4070 & 12.3455 & 12.2837 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Convergence behavior and accuracy of the DQM. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\omega^{*}\)\({}_{1}\) & \(\mathbf{V_{E}}\) &
\begin{tabular}{c} **Dimensionless** \\ **Frequency (\(\omega\))** \\ \(\Omega_{\mathbf{h}}\) = \(-10^{6}\)** \\ \end{tabular} & \(\Omega_{\mathbf{h}}\) = \(0\) & \(\Omega_{\mathbf{h}}\) = \(10^{6}\) \\ \hline Reference & \(-10^{8}\) & 2.88 & 3.08 & 3.25 \\  & \(0\) & 2.85 & 3.06 & 3.23 \\  & \(10^{8}\) & 2.83 & 3.02 & 3.21 \\ \hline Present & \(-10^{8}\) & 2.91 & 3.16 & 3.33 \\  & \(0\) & 2.94 & 3.16 & 3.31 \\  & \(10^{8}\) & 2.87 & 3.10 & 3.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Validation of present work with respect to Ref. [36]. not in the Winkler one, which is described in the normal direction. Figure 4 shows the variation in the dimensionless frequency with the thickness ratio under different external electric potentials while assuming \(k_{w}~{}=~{}0\), \(k_{g}~{}=~{}0\), \(\Delta T~{}=~{}0\) and \(\Delta C~{}=~{}0\). Note that the application of an electric potential can vary the frequency response of the sandwich structure, whereby a negative electric potential decreases the dimensionless frequency, and a positive voltage can yield the opposite effect. The enforcement of negative/positive voltages, indeed, seems to generate an axial compressive/tensile force in the top layer, with a consecutive variation in the overall structural stiffness. This means that the application of an external electric potential is an effective controlling parameter for the behavior of the structure. Figure 5 describes the variation in the dimensionless frequency with the thickness ratios under different external magnetic potentials, \(\lambda\). Based on the plots in this figure, it is clear that a negative magnetic potential decreases the dimensionless frequency under the same thickness ratio assumption, where the contrary occurs under the application of a positive voltage. The influence of a power law index, \(g\), on the dimensionless frequency is illustrated in Figure 6 for different \(r_{i}/r_{o}\) ratios while keeping a null value for \(k_{w}\), \(k_{g}\) in the clamped structure. As visible from the plots in the figures below, the dimensionless frequency decreases monotonically for an increased power index under a fixed \(r_{i}/r_{o}\) ratio, with a clear upward shift of curves, for an increased rational value of \(r_{i}/r_{o}\). Figure 3: Dimensionless frequency versus radial ratio for different Pasternak coefficients.