# Free Vibration Analysis of Functionally Graded Sandwich Plates with a Homogeneous Core
## 4 Numerical Results
### Free Vibration Characteristics  
Next, the free vibration characteristics of FG sandwich plates with homogeneous cores were investigated with respect to the major parameters of the plates. For this parametric experiment, the (3,3,2) hierarchical model was used because its convergence to the 3-D elasticity was verified through the previous verification experiment. Figure 6a,b represent the calibrated fundamental frequencies versus the thickness-to-width ratios of (2-2-1) and (6-6-1) FG sandwich plates with homogeneous softcores. The bottom curve of the metal is the case of \(n=0\), while the top curve of the ceramic corresponds to \(n=+\infty\). It was observed that the fundamental frequency increased in proportion to the volume fraction index, \(n\). Referring to Figure 2, the relative amount of ceramic present in the plate increased in proportion to the volume fraction index, \(n\). Furthermore, the ceramic was much stiffer than the metal, even though its density was greater than the metal's, as given in Table 1. For this reason, the sandwich plate became stiffer with the increasing value of the volume fraction index, \(n\). Meanwhile, from Figure 6b, it may be seen that the frequencies of the center five curves became lower. This is because the relative amount of ceramic became smaller as the pseudo thickness ratio was changed from (2-2-1) to (6-6-1). In other words, the relative thickness ratio of metal core increased from 2/5 (= 40%) to 6/13 (= 46%). Figure 7a,b represent the curves of FG sandwich plates with homogeneous hardcores. Because the core was ceramic, the bottom and top curves correspond to \(n=+\infty\) and \(n=0\), respectively. Contrary to the previous softcore, the fundamental frequency decreased in proportion to the volume fraction index, \(n\). This is because the sandwich plate with a hardcore became softer with the increasing value of the volume fraction index, \(n\). The reason for this can be explained in the opposite manner to the previous softcore. Meanwhile, from Figure 7b, one can see that the frequencies of five curves in the middle became higher. This is because the relative amount of ceramic became larger as the pseudo thickness ratio was changed from (2-2-1) to (6-6-1). \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{_b/h_} & \multirow{2}{*}{**Method**} & \multicolumn{5}{c}{_n_} \\ \cline{3-6}  & & **0** & **0.5** & **1** & **5** & **10** \\ \hline \multirow{3}{*}{5} & Bennoun [10] & 1.67949 & 1.44770 & 1.31762 & 1.06611 & 1.02325 \\  & Present & 1.69856 & 1.46343 & 1.32876 & 1.06944 & 1.02370 \\  & (Relative error) & (1.135\%) & (1.087\%) & (0.845\%) & (0.312\%) & (0.044\%) \\ \hline \multirow{3}{*}{10} & Bennoun [10] & 1.82854 & 1.55146 & 1.40103 & 1.11900 & 1.07233 \\  & Present & 1.85670 & 1.58688 & 1.43531 & 1.14988 & 1.10044 \\  & (Relative error) & (1.540\%) & (2.283\%) & (2.454\%) & (2.760\%) & (2.621\%) \\ \hline \multirow{3}{*}{100} & Bennoun [10] & 1.88978 & 1.59246 & 1.43337 & 1.13887 & 1.09070 \\  & Present & 1.95360 & 1.67033 & 1.51270 & 1.21815 & 1.16698 \\ \cline{1-1}  & (Relative error) & (3.377\%) & (4.890\%) & (5.535\%) & (7.961\%) & (6.994\%) \\ \hline \end{tabular}
\end{table} 
Table 4: Comparison of calibrated fundamental frequencies, \(\overline{\omega}\), for simply supported square FG sandwich plates with homogeneous hardcores (2-2-1). Figure 8a,b represent the calibrated fundamental frequencies versus the volume fraction index, \(n\), for (2-2-1) FG sandwich plates with homogenous cores. It is seen that the curves converged to the limits, both the fundamental frequencies of ceramic plates for softcore cases and those of metal plates for hardcore cases. The curves were upper-bounded for the softcore cases while those of the hardcore cases are lower-bounded. This limit characteristic, due to the material gradient in the FGM layers, could be confirmed from Figures 6a and 7a.