## 4 Numerical Results and Discussion  
In this section, we check first the accuracy of the proposed OVFSDT-based approach by means of a comparative evaluation between our numerical predictions and those available from the literature. More specifically, due to a complete lack of results about corrugated nanoplates, we here compare our results with those ones obtained for flat nanosheets. Among a large variety of possible approaches adopted in literature for the vibration study of graphene plates, the molecular dynamics and the non-local GDQ-based FSDT, as proposed in [36], have been here selected as efficient and accurate methods for comparative purposes. The comparative results are summarized in Tables 1 and 2, for a grapheme nanoplate with a zigzag hexagonal arrangement, or an armchair arrangement, respectively, assuming a Young's modulus \(E=1\) TPa, a [PERSON]'s ratio \(v=0.16\), a constant thickness \(h=0.34\) nm, two different non-local parameters \(\mu=1.41\) nm\({}^{2}\) and \(\mu=1.34\) nm\({}^{2}\). As visible in these two tables, the very good agreement between results reflects the accuracy of the proposed formulation to approach a similar problem, provided that non-local parameters are adequately selected. Note that few discrepancies between results can be mainly related to the different solution method of the governing equations, e.g., a Navier's approach instead of a GDQ. These differences become even more negligible for increasing lengths \(L_{x}=L_{y}\) as visible in Tables 1 and 2. Some possible effects can be also related to the selected plate theory, as evaluated comparatively in Table 3 according to a CPT, FSDT, or a Third-order Shear Deformation Theory (TSDT) [23], as well as according to a three-dimensional elasticity [37] and the S-FSDT [38]. More specifically, the dimensionless natural frequency \(\overline{\omega}=\omega h\sqrt{\rho/G}\) is computed for two different length-ratios \(L_{x}/L_{y}\) (i.e., \(L_{x}/L_{y}=1\) and \(2\), respectively), where two different length-to-thickness ratios \(L_{x}/h=10\) and \(L_{x}/h=20\) are assumed, while varying the non-local parameter \(\mu=0,1,2\). Based on Table 3, our results resemble very well those presented in [23; 37; 38], especially when treating thin plates. A general decrease in the natural frequency is also observed for an increasing non-local parameter, and a fixed geometry, independently of the selected plate theory. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Natural Frequency (THz)**} \\ \hline
**Present-Non-Local, Navier** & & & \\ \hline
**OVFSDT** & & & \\ \hline
0.0592339 & 0.0584221 & 0.0587725 & 10 \\
0.0287035 & 0.0282888 & 0.0273881 & 15 \\
0.0153208 & 0.0164593 & 0.0157524 & 20 \\
0.0103514 & 0.0107085 & 0.0099480 & 25 \\
0.0074258 & 0.0075049 & 0.0070655 & 30 \\
0.0055703 & 0.0055447 & 0.0052982 & 35 \\
0.0043250 & 0.0042608 & 0.0040985 & 40 \\
0.0034512 & 0.0033751 & 0.0032609 & 45 \\
0.0028157 & 0.0027388 & 0.0026194 & 50 \\ \hline \hline \end{tabular} GDQ: Generalized Differential Quadrature.  
\end{table} 
Table 1: Numerical results based on the one-variable first-order shear deformation plate theory (OVFSDT), molecular dynamics, and non-local FSDT. \(E=1\) TPa, \(v=0.16\), \(h=0.34\) nm, \(\mu=1.41\) nm\({}^{2}\). After this preliminary check about the consistency of the proposed OVFSDT, we continue to apply this approach for a corrugated viscoelastic nanoplate with mechanical and geometrical properties, as summarized in Table 4.  
Figure 2 shows the variation of the natural frequency with the non-local parameter \(\mu\) (Figure 2a) and dimensionless length scale \(l^{*}=l/h\) (Figure 2b) within the proposed formulation, where both flat and curved geometries are compared in the presence or not of the viscoelastic damping parameter, namely \(g=0.5\)\(\,\mathrm{Ns/m^{2}}\) or \(g=0\)\(\,\mathrm{Ns/m^{2}}\). The nanoplate features two sinusoidal corrugations with amplitude \(F=0.5h\) (i.e., dimensionless amplitude \(F^{*}=F/h=0.5\)) and semi-length \(c=0.25L_{x}\).