# Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation
## 4 Results and Discussion  
For the purpose of examining the validity of the suggested model of the AFG Euler-Bernoulli nanobeams, a comparison study was conducted. Initially, the frequencies of AFG local single beam (by neglecting the nonlocal scale parameter) were compared to those reported by [PERSON] and [PERSON] [36]. Then, the frequencies of a double isotropic nano beam were compared with those obtained by [PERSON] and [PERSON] [37]. In Table 1, the first two dimensionless frequencies for AFG Euler-Bernoulli beams with clamped edges (C-C) are presented. Moreover, in Table 2, the first four dimensionless frequencies of the suggested model for nonlocal Euler-Bernoulli simply supported double beams were compared with those obtained by [PERSON] and [PERSON] [37]. As observed from these tables, the proposed method could be used to obtain accurate results. Figure 2 shows the influence of the nonlocal scale parameter on the first three dimensionless transverse frequencies of AFG nano CC-CC and SS-SS double beams with \(c_{b_{1}}=c_{b_{2}}=0.1,\ c_{b_{1}}=c_{b_{2}}=0.2,\ K_{G}=10,\ K_{W}=100\), and \(K=100\). The notations CC-CC and SS-SS denote that the two nanobeams were clamped and simply supported at both boundaries, respectively. The nonlocal scale parameter \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(c_{h}\) & \(c_{b}\) & \(\Omega_{1}\) & \(\Omega_{2}\) & \(\Omega_{3}\) & \(\Omega_{4}\) \\ \hline \multirow{2}{*}{\(0\)} & \multirow{2}{*}{\(0\)} & \multirow{2}{*}{\(0.2\)} & Present Study & 8.3569 & 16.4267 & 24.5823 & 28.3600 \\  & & & Ref. [37] & 8.3569 & 16.4267 & 24.5823 & 28.3600 \\ \cline{2-6}  & \multirow{2}{*}{\(0.4\)} & \multirow{2}{*}{\(0.4\)} & Present Study & 6.1456 & 14.5951 & 15.4197 & 20.3228 \\  & & & Ref. [37] & 6.1456 & 14.5951 & 15.4197 & 20.3228 \\ \cline{2-6}  & \multirow{2}{*}{\(0.8\)} & Present Study & 3.6488 & 7.7030 & 11.6787 & 14.6053 \\ \cline{2-6}  & & Ref. [37] & 3.6488 & 7.7030 & 11.6787 & 14.6053 \\ \hline \multirow{2}{*}{\(500\)} & \multirow{2}{*}{\(0.2\)} & Present Study & 8.3569 & 24.5823 & 32.7084 & 40.0536 \\  & & Ref. [37] & 8.3569 & 24.5823 & 32.7084 & 40.0536 \\ \cline{2-6}  & \multirow{2}{*}{\(0.4\)} & Present Study & 6.1456 & 14.5951 & 22.7743 & 30.8121 \\ \cline{2-6}  & & Ref. [37] & 6.1456 & 14.5951 & 22.7743 & 30.8121 \\ \cline{2-6}  & \multirow{2}{*}{\(0.8\)} & Present Study & 3.6488 & 7.7030 & 11.6787 & 15.6308 \\ \cline{2-6}  & & Ref. [37] & 3.6488 & 7.7030 & 11.6787 & 15.6308 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The dimensionless natural transverse frequencies for an isotropic simply supported uniform double nanobeam. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(K\) & \(\mu\) & & & & \\ \hline \multirow{2}{*}{\(100\)} & \multirow{2}{*}{\(0.2\)} & Present Study & 8.3569 & 16.4267 & 24.5823 & 28.3600 \\  & & Ref. [37] & 8.3569 & 16.4267 & 24.5823 & 28.3600 \\ \cline{2-6}  & \multirow{2}{*}{\(0.4\)} & Present Study & 6.1456 & 14.5951 & 15.4197 & 20.3228 \\  & & Ref. [37] & 6.1456 & 14.5951 & 15.4197 & 20.3228 \\ \cline{2-6}  & \multirow{2}{*}{\(0.8\)} & Present Study & 3.6488 & 7.7030 & 11.6787 & 14.6053 \\  & & Ref. [37] & 3.6488 & 7.7030 & 11.6787 & 14.6053 \\ \hline \multirow{2}{*}{\(500\)} & \multirow{2}{*}{\(0.2\)} & Present Study & 8.3569 & 24.5823 & 32.7084 & 40.0536 \\  & & Ref. [37] & 8.3569 & 24.5823 & 32.7084 & 40.0536 \\ \cline{2-6}  & \multirow{2}{*}{\(0.4\)} & Present Study & 6.1456 & 14.5951 & 22.7743 & 30.8121 \\ \cline{2-6}  & & Ref. [37] & 6.1456 & 14.5951 & 22.7743 & 30.8121 \\ \cline{2-6}  & \multirow{2}{*}{\(0.8\)} & Present Study & 3.6488 & 7.7030 & 11.6787 & 15.6308 \\ \cline{2-6}  & & Ref. [37] & 3.6488 & 7.7030 & 11.6787 & 15.6308 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The dimensionless natural transverse frequencies for an AFG C-C tapered single beam. was assumed to be in the interval [0, 1]. It is clear that the frequencies of the CC-CC AFG double nanobeams were greater than that for the SS-SS double beams.