## 4 Numerical Results
Figure 6a,b comparatively represent the effect of crack inclination angle on the second natural modes for \(a/S=0.3\) and 0.6, respectively, when the ceramic power-law index \(r\) is 0. The reason for showing the second mode is because the fundamental mode does not produce any apparent change with respect to the crack angle. It is seen that the mode shape becomes separated from the left and right as \(\alpha\) goes to zero, while it becomes similar to the non-separated single plate vibration mode as \(\alpha\) increases. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Items**} & \multicolumn{6}{c}{**Grid Density**} \\ \cline{2-7}  & **13\(\times\)13** & **15\(\times\)15** & **17\(\times\)17** & **19\(\times\)19** & **21\(\times\)21** & **23\(\times\)23** & **25\(\times\)25** \\ \hline \(\hat{\omega}_{1}(\mathrm{Hz})\) & 296.501 & 288.927 & 282.838 & 277.882 & 273.829 & 270.491 & 267.708 \\ \(\hat{\omega}_{1}^{rel}(\%)\) & 9.70 & 7.93 & 5.65 & 3.80 & 2.27 & 1.04 & - \\ \hline \end{tabular}
\end{table} 
Table 1: Dependence of non-dimensional first frequency of an isotropic intact cylindrical panel on the grid density (\(L/R=2,S/L=1,R/h=20\), CCCC). \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**\(\mathbf{\alpha}\) (deg)**} & \multicolumn{6}{c}{**r**} \\ \cline{3-6}  & & **0** & **0.2** & **0.5** & **1** & **5** \\ \hline \multirow{4}{*}{PFF [36]} & 0 & 269.69 & 243.42 & 215.53 & 187.67 & 144.15 \\  & 30 & 274.85 & 249.43 & 223.17 & 196.01 & 146.73 \\  & 60 & 275.50 & 250.00 & 223.67 & 196.95 & 147.04 \\  & 90 & 275.73 & 250.20 & 223.84 & 197.09 & 147.16 \\ \hline \multirow{4}{*}{Present} & 0 & 271.35 & 247.84 & 222.83 & 196.75 & 145.44 \\  & 30 & 273.25 & 249.38 & 223.79 & 196.96 & 146.03 \\ \cline{1-1}  & 60 & 275.67 & 251.18 & 224.48 & 196.32 & 147.09 \\ \cline{1-1}  & 90 & 277.14 & 251.87 & 224.69 & 196.74 & 148.02 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of first frequencies of ceramic–metal FG cracked cylindrical panels (\(L/R=2,S/L=1,R/h=20\), \(a/S=0.3\), CCCC). The present method is also verified through two nonlinear free vibration analyses. One is an isotropic intact cylindrical panel under the simply-supported boundary condition with the geometry and material data given by \(S/L=1\), \(S/h=10\), \(R/h=0\), \(E_{m}=68.95\) GPa, \(v_{m}=0.3\), and \(\rho_{m}=1150\) kg/m\({}^{3}\). The nonlinear-to-linear frequencies \(\omega_{NL}/\omega_{L}\) were computed for six different values of amplitude-thickness ratio \(w_{max}/h\), and the frequency-amplitude plot is compared in Figure 7a. The reference solution by [PERSON] [49] was solved by applying the fourth-order Runge-Kutta method to the FSDT. Meanwhile, the other reference solution by [PERSON] and [PERSON] [50] was analytically solved by applying a two-step perturbation method to the HSDT. The von Karman-type geometric nonlinearity was adopted to both solutions, as for the present study. One can clearly see that the three methods are in excellent agreement such that the maximum relative difference in \(\omega_{NL}/\omega_{L}\) between the present method and that of [PERSON] [49] is 1.149% at \(w_{max}/h=1.0\). Figure 6: Variation in the second-mode shapes of a clamped FG-GPLRC cylindrical panel to the crack inclination angle (\(\alpha=0^{\circ},30^{\circ}\), \(60^{\circ}\) and \(90^{\circ}\)) for \(r=0\) (**a**) at \(a/S=0.3\) and (**b**) at \(a/S=0.6\). Figure 7: Comparison of frequency–amplitude curves of isotropic cylindrical panels: (**a**) non-porous; (**b**) porous (PD 1) [49; 50]. The other nonlinear example is an intact porous isotropic cylindrical panel with the geometric and material data given by \(L=S=1.0m\), \(S/h=10\), \(R/S=5\), \(E_{m}=200\) GPa, \(\
u_{m}=0.3\), and \(\rho_{m}=7850\) kg/m\({}^{3}\). The porosity distribution is PD_1 shown in Figure 2 with the porosity coefficient \(e_{0}=0.6\), and the four sides of panel are clamped.