## 4 Numerical Results and Discussion
Based on Figure 2, an increased non-local coefficient \(\mu\) yields to a reduced natural frequency, whereas \begin{table}
\begin{tabular}{|c|} \hline Elastic properties \\ \(E=1\) TPa, \(v=0.16\) \\ Density \\ \(\rho=2250\) kg/m\({}^{3}\) \\ Dimensional values \\ \(h=0.34\) nm, \(L_{x}=L_{y}=10\) nm \\ \hline \end{tabular}
\end{table} 
Table 4: Material properties of the viscoelastic nanoplate [36]. \begin{table}
\begin{tabular}{|c|c|} \hline \multicolumn{2}{|c|}{**Natural Frequency (THz)**} \\ \hline
**Present-Non-Local, Navier** & **Non-Local-FSDT** [36] & **Armchair Graphene, MD [36] & \(L_{x}=L_{y}\) (nm)** \\ \hline
**OVFSDT** & & \multicolumn{1}{c|}{} \\ \hline
0.0603135 & 0.0592359 & 0.0595014 & 10 \\
0.0290542 & 0.0284945 & 0.0277928 & 15 \\
0.0154374 & 0.0165309 & 0.0158141 & 20 \\
0.0104078 & 0.0107393 & 0.0099975 & 25 \\
0.0074558 & 0.0075201 & 0.0070712 & 30 \\
0.0055876 & 0.0055531 & 0.0052993 & 35 \\
0.0043356 & 0.0042657 & 0.0041017 & 40 \\
0.0034580 & 0.0033782 & 0.0032614 & 45 \\
0.0028202 & 0.0027408 & 0.0026197 & 50 \\ \hline \end{tabular}
\end{table} 
Table 2: Numerical results based on the OVFSDT, molecular dynamics, and non-local FSDT. \(E=1\) TPa, \(v=0.16\), \(h=0.34\) nm, \(\mu=1.34\) nm\({}^{2}\). \begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \multicolumn{1}{|c|}{\multirow{3}{*}{**Theories**}} & \multicolumn{10}{c|}{**Non-Dimensional Natural Frequency (\(\overline{m}=\omega h/\overline{g}\))**} \\ \cline{2-13}  & \multicolumn{4}{c|}{\(L_{x}/L_{y}=1\)} & \multicolumn{4}{c|}{\(L_{x}/L_{y}=2\)} \\ \cline{2-13}  & \multicolumn{4}{c|}{\(L_{x}/h=10\)} & \multicolumn{4}{c|}{\(L_{x}/L_{y}=20\)} & \multicolumn{4}{c|}{\(L_{x}/h=10\)} & \multicolumn{4}{c|}{\(L_{x}/h=20\)} \\ \cline{2-13}  & \(\mu=0\) & \(\mu=1\) & \(\mu=2\) & \(\mu=0\) & \(\mu=1\) & \(\mu=2\) & \(\mu=0\) & \(\mu=1\) & \(\mu=2\) & \(\mu=0\) & \(\mu=1\) & \(\mu=2\) \\  & nm\({}  
an increasing length scale parameter \(l^{*}\) gets to an increased natural frequency of the nanostructure because of its increased stiffness, under the non-local strain gradient theory. This agrees with findings by [PERSON] and [PERSON] [10]. Moreover, the natural frequency of the flat nanostructure is lower than the corrugated one, for the same fixed non-local coefficient \(\mu\) (Figure 2a) and dimensionless length scale \(l^{*}\) (Figure 2b). Remarkably, all the curves in Figure 2 are almost scalable with the varying geometry and damping parameter. A negligible increase in the natural frequency is also observed for an increasing viscoelastic damping parameter \(g\). Note that the equations based on the nonlocal strain gradient theory revert to the classical mechanics ones, when the length scale and nonlocal parameter are exactly the same (i.e., when \(l\) is equal to \(e_{0}a\)). Based on a comparative evaluation of Figure 2a,b, it is worth noting that the response of the nonlocal parameter \(\mu\) compared to the length scale \(l^{*}\), is more sensitive for each fixed geometry and damping parameters of the nanostructure. Figure 2: Variation of the natural frequency with the nonlocal parameter \(\mu\) (**a**) and the length scale \(l^{*}\) (**b**). A complete overview of the vibrating response of the corrugated nanosheet with varying geometries is plotted in Figures 3 and 4. More specifically, a different number of waves is considered in Figure 3, while varying the amplitude between \(F^{*}=0.05\) and \(F^{*}=0.5\). As visible in Figure 3, an increasing wave amplitude \(F\), for a fixed number of corrugations, leads to an increased frequency and stiffness of the nanoplate. The natural frequency seems to increase linearly up to a threshold value, moving from a flat to a corrugated nanoplate with just one corrugation. This threshold value clearly depends on the corrugation, since it increases for increasing magnitudes \(F\). A meaningless increase of the natural frequencies is also observed in Figure 3 after 1 corrugation, probably because further corrugations do not affect significantly the stiffness of the structure.