## 4 Numerical Results  
In this section, we present the numerical results from a large parametric investigation into the nonlinear vibration behavior of thick sandwich beams reinforced with GPLs. To check for the accuracy of the proposed model, our numerical results obtained for FG GPL-reinforced beams are compared with those available from the literature. In this regard, the thickness of the core layer in the present model is assumed to be zero (\(h_{\mathrm{c}}=0\)). In Tables 1 and 2, the dimensionless free linear (\(\overline{\omega}_{L}=\omega_{L}\times L\times\sqrt{m_{10}/A_{10}}\)) and nonlinear (\(\overline{\omega}_{NL}=\omega_{NL}\times L\times\sqrt{m_{10}/A_{10}}\)) frequencies are provided for a simply supported, GPL-reinforced beam and compared with numerical results reported by [PERSON] et al. [30]. The reference model is based on a Timoshenko beam theory, whereby two different patterns are considered for validation purposes. The following properties are assumed for the beam: \(E_{M}=2.85\) GPa, \(\rho_{M}=1200\)Kg/m\({}^{3}\), \(E_{\mathrm{GPL}}=1.01\) TPa, \(\rho_{\mathrm{GPL}}=1062.5\)Kg/m\({}^{3}\), \(w_{\mathrm{GPL}}=1.5\)\(\upmu\)m, \(l_{\mathrm{GPL}}=2.5\)\(\upmu\)m, and \(l_{\mathrm{GPL}}=1.5\) nm. In Table 3, a comparison has been attempted between results from the present formulation for the first four dimensionless frequencies (\(\overline{\omega}_{L}=\omega_{L}\times\left(L^{2}/h\right)\sqrt{\rho/E_{11}}\)) of simply supported orthotropic beams and those from the literature, based on different higher-order shear deformation theories [59; 60; 61]. The material properties are assumed to be \(E_{11}=144.9\) GPa, \(E_{22}=9.65\) GPa, \(G_{12}=G_{13}=4.14\) GPa, \(G_{23}=3.45\) GPa, \(\rho=1389.23\)Kg/m\({}^{3}\), and \(\
u_{12}=\
u_{21}=0.3\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Pattern**} & \multirow{2}{*}{**Reference**} & \multicolumn{4}{c}{**Mode**} \\ \cline{3-5}  & & **1** & **2** & **3** \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} UD \\ (\(\kappa=0\)) \\ \end{tabular} } & [PERSON] et al. [30] & 0.21542 & 0.85226 & 1.88292 \\ \cline{2-5}  & Present & 0.23482 & 0.92903 & 2.05352 \\ \hline \multirow{2}{*}{
\begin{tabular}{c} FG-X \\ (\(\kappa=1\)) \\ \end{tabular} } & [PERSON] et al. [30] & 0.25853 & 1.01309 & 2.20666 \\ \cline{2-5}  & Present & 0.26759 & 1.05199 & 2.30258 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: First three dimensionless natural frequencies (\(\overline{\omega}_{L}=\omega_{L}\times L\times\sqrt{m_{10}/A_{10}}\)) of simply supported laminated beams reinforced with GPLs (\(L/h_{t}=20\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Reference**} & \multicolumn{4}{c}{**Mode**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** \\ \hline [PERSON] et al. [61] & 2.3100 & 6.9538 & 11.9707 & 17.0393 \\ \hline [PERSON] and [PERSON] [60] & 2.3188 & 7.0204 & 12.0894 & 17.3139 \\ \hline [PERSON] and [PERSON] [59] & 2.3198 & 7.0091 & 12.1250 & 17.2949 \\ \hline Present & 2.4038 & 7.2110 & 12.3975 & 17.6279 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the first four dimensionless natural frequencies (\(\overline{\omega}_{L}=\omega_{L}\times\left(L^{2}/h\right)\sqrt{\rho/E_{11}}\)) of orthotropic thick beams based on different higher-order shear deformable beam theories (\(L/h_{t}=10\)). As is clearly visible in Table 3, the numerical results based on the proposed model agree very well with predictions from the literature based on other shear deformable models. It seems that higher-order-models available in the literature [59; 60; 61] get more conservative results than our formulation. At the same time, the proposed multiple timescale approach proves to be an efficient analytical tool to solve nonlinear systems in a very easy and straightforward manner. In the benchmark Tables 4 and 5, we report the numerical results in terms of the first-order nonlinear frequency and nonlinear-to-linear frequency ratio for a GPL-reinforced sandwich beam.