## 4 Numerical Results
It is seen that the frequency ratio \(\omega_{NL}/\omega_{L}\) becomes saturated such that FG-O and FG-\(\Lambda\) approach the upper bounded value while FG-U and FG-X approach the lower bounded value. Thus, it has been justified that the nonlinearity intensity in the free vibration becomes insensitive to \(V_{GPL}^{*}\) and \(V_{CNT}^{*}\) when the amount of GPLs and CNTs reaches a critical value. Figure 11b represents the variation in \(\omega_{NL}/\omega_{L}\) to the CNT volume fraction when CNTs are aligned in the circumferential direction. The relative order in the magnitude of \(\omega_{NL}/\omega_{L}\) is the same as that of \(\alpha_{CNT}=90^{\circ}\), shown in Figure 10a, but the magnitude of \(\omega_{NL}/\omega_{L}\) and its sensitivity to \(V_{CNT}^{*}\) are shown to be relatively larger than those of CNTRC with \(\alpha_{CNT}=90^{\circ}\). It is noticed that the \(\omega_{NL}/\omega_{L}\) of \(V_{CNT}^{*}=0.17\) is smaller than the \(\omega_{NL}/\omega_{L}\) of \(V_{CNT}^{*}=0.12\) regardless of the CNT alignment angle. It is of course attributed to the fact that both the \(\omega_{L}\) and \(\omega_{NL}\) of \(V_{CNT}^{*}=0.17\) are larger those of \(V_{CNT}^{*}=0.12\) but their relative ratio can be smaller than that of \(V_{CNT}^{*}=0.12\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Type**} & \multirow{2}{*}{**Volume Fraction \(V_{GPL}^{*}\)=\(V_{CNT}^{*}\) (\(\delta_{GPL}^{*}\))**} & \multirow{2}{*}{**0.12 (10.75\%)**} & \multirow{2}{*}{**0.17 (15.32\%)**} & \multirow{2}{*}{**0.28 (25.57\%)**} \\ \cline{3-3} \cline{5-5}  & & & & \\ \hline \multicolumn{2}{c}{GPLRC} & 216.31 & 258.31 & 335.64 \\ \hline \multirow{2}{*}{CNTRC} & \(\alpha_{CNT}=90^{0}\) & 39.19 & 50.64 & 55.55 \\  & \(\alpha_{CNT}=0^{0}\) & 56.03 & 72.20 & 80.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The non-dimensional linear first frequencies \(\hat{\omega}_{1}\) (FG-X, \(a/S=0.3\), \(\alpha=30^{\circ}\)). Figure 10: Comparison of nonlinear-to-linear frequency ratios \(\omega_{NL}/\omega_{L}\) between GPLRC and CNTRC (FG-X, \(a/S=0.3\), \(\alpha=30^{\circ}\)): (a) CNTRC; (b) GPLRC. Next, the influence of porosity on the nonlinear free vibration of a cracked FG-GPLRC cylindrical panel was investigated. The mass fraction \(g^{*}_{GPL}\) and distribution pattern are 0.4% and FG-U, and the inclination angle \(\alpha\) and relative length \(a/S\) of the crack are 30\({}^{\circ}\). Three different porosity distributions and the five porosity parameters were considered. Figure 11(a) comparatively represents the frequency ratios \(\omega_{NL}/\omega_{L}\) of an FG-U GPLRC cylindrical panel with the center-biased porosity distribution (PD_1). It is observed that the frequency ratio becomes uniformly smaller with the increase in \(e_{0}\) until \(w^{max}_{c}/h=0.2\), but thereafter, it uniformly increases in proportion to \(e_{0}\). However, the extent of decrease becomes smaller as the porosity parameter \(e_{0}\) increases. Both \(\omega_{L}\) and \(\omega_{NL}\) decrease proportionally to \(e_{0}\) because the increase in porosity decreases the structural stiffness [54]. But, the frequency ratio \(\omega_{NL}/\omega_{L}\) may increase with increasing \(e_{0}\) because the denominator \(\omega_{L}\) decreases in proportion to \(e_{0}\). Figure 11(b) compares the plots of \(\omega_{NL}/\omega_{L}\) for three different porosity distributions, where the center-biased distribution (PD_1) shows the highest level while the outer-biased distribution (PD_2) leads to the lowest level. This is because the structural stiffness increases as the porosity becomes biased towards the mid-surface of panel for the same amount of porosity. However, the relative difference in \(\omega_{NL}/\omega_{L}\) among the three porosity distributions becomes different when the GPL distribution pattern is changed, as will be seen next. Figure 12(a)-d comparatively represent the differences in \(\omega_{NL}/\omega_{L}\) between three porosity distributions for FG-U, G-O, FG-X, and FG-\(\wedge\).