## 4 Numerical Results
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Items**} & \multicolumn{6}{c}{**Grid Density**} \\ \cline{2-7}  & **9\(\times\)9** & **13\(\times\)13** & **17\(\times\)17** & **21\(\times\)21** & **25\(\times\)25** \\ \hline \multirow{2}{*}{Linear} & \(\hat{\omega}_{1}\)(Hz) & 37.945 & 33.505 & 31.679 & 31.250 & 31.191 \\  & \(\hat{\omega}_{1}^{rel}\)(\%) & 21.65 & 7.42 & 1.56 & 0.19 & - \\ \hline \multirow{2}{*}{Nonlinear} & \(\omega_{NL}/\omega_{L}\) (\(w_{\max}/h=1.0\)) & 1.157 & 1.145 & 1.166 & 1.160 & 1.163 \\  & \((\omega_{NL}/\omega_{L})^{rel}\)(\%) & \(-1.81\) & \(-1.55\) & 0.26 & \(-0.26\) & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Sensitivity of \(\hat{\omega}_{1}\) and \(\omega_{NL}/\omega_{L}\) of a cracked isotropic cylindrical panel to the grid density (\(L/R=2,\;S/L=1,\;R/h=20,\;a/S=0.7,\;\alpha=45^{\circ}\), CCCC). Figure 9a represents the variation in frequency ratio \(\omega_{NL}/\omega_{L}\) to the GPL mass fraction \(g_{GPI}^{*}\) for the FG-O GPLRC cylindrical panel with a central crack. The inclination angle and relative length of a crack are set at \(30^{0}\) and \(0.3\). Each nonlinear free vibration analysis for five values of \(w_{c}^{max}/h\) was completed within five minutes on an Intel Pentium PC with a dual core CPU (1.86 GHz). It is observed that the frequency ratio uniformly increases in proportion to the GPL mass fraction because the mass fraction increase in GPLs with a higher elastic modulus leads to an increase in panel stiffness. Figure 9b represents the influence of the GPL distribution pattern on the frequency ratio \(\omega_{NL}/\omega_{L}\), for which the GPL mass \(g_{GPI}^{*}\) was set to \(0.4\%\), while three different GPL distribution patterns were additionally considered. It is found that the magnitude order of frequency ratio \(\omega_{NL}/\omega_{L}\) is FG-X, FG-O, FG-U, and FG-\(\Lambda\). This trend is slightly different from the magnitude order of linear fundamental frequency \(\hat{\omega}_{1}\) given in Table 3, where FG-O shows the lowest level. In the frequency ratio \(\omega_{NL}/\omega_{L}\), FG-O shows the second highest level, which is caused by the fact that the magnitude of \(\omega_{NL}/\omega_{L}\) is not determined by the magnitudes of \(\omega_{NL}\) and \(\omega_{L}\) but the relative value of these two. This trend was also observed from Figure 8a, where larger inclination crack angles showed lower frequency ratios \(\omega_{NL}/\omega_{L}\) even though the linear fundamental frequency was higher at larger inclination crack angles. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Crack Length**} & \multirow{2}{*}{**Crack Angle**} & \multicolumn{4}{c}{**SPL Distribution**} \\  & & \(\alpha\)(_deg_) & **FG-U** & **FG-O** & **FG-X** & **FG-\(\Lambda\)** \\ \hline \multirow{4}{*}{0.3} & 0 & 50.0201 & 45.2187 & 51.9930 & 48.3275 \\  & 30 & 50.1616 & 45.8623 & 52.2722 & 48.9641 \\  & 60 & 50.2431 & 46.8017 & 52.3461 & 49.4227 \\  & 90 & 50.3914 & 46.8707 & 52.4892 & 49.5481 \\ \hline \multirow{4}{*}{0.6} & 0 & 47.8834 & 44.4466 & 49.7297 & 47.1588 \\  & 30 & 48.2756 & 44.8122 & 50.2369 & 47.7287 \\ \cline{1-1}  & 60 & 48.3421 & 45.0171 & 50.4532 & 47.8063 \\ \cline{1-1}  & 90 & 49.0014 & 45.8415 & 50.9793 & 48.2326 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Non-dimensional first frequencies \(\hat{\omega}_{1}\) of non-porous FG-GPLRC cylindrical panels with a central crack (\(L/R=2\), \(S/L=1\), \(R/h=20\), \(g_{GPI}^{*}=0.4\%\), CCCC). Next, the nonlinear free vibration of the cracked cylindrical panel between GPLRC and CNTRC was compared by keeping the material properties of the matrix unchanged.