# Free Vibration Analysis of Functionally Graded Sandwich Plates with a Homogeneous Core
## 4 Numerical Results
### Free Vibration Characteristics
With the increasing value of volume fraction index, \(n\), two FGM layers in the softcore sandwich plate became ceramic-rich, while those in the hardcore sandwich plate became metal-rich. This fact can be confirmed from the volume fraction distributions shown in Figure 2. However, in both softcore and hardcore cases, it was observed that the calibrated fundamental frequency increased in proportion to the width-to-thickness ratio, \(b/h\). Figure 6: Calibrated fundamental frequencies, \(\overline{\omega}\), for functionally graded sandwich plates with homogeneous softcores: (**a**) (2-2-1), (**b**) (6-6-1). Figure 7: Calibrated fundamental frequencies, \(\overline{\omega}\), for functionally graded sandwich plates with homogeneous hardcores: (**a**) (2-2-1), (**b**) (6-6-1). Tables 5 and 6 present the calibrated fundamental frequencies versus the relative thickness \(R_{3}=H_{3}/(H_{1}+H_{2}+H_{3})\) of the top facesheet of softcore and hardcore FG sandwich plates, respectively. The 8-8-1 and 16-16-1 configurations were included to examine the limits of \(\overline{\omega}\) as \(R_{3}\) tends to zero. The width-to-thickness ratio, \(b/h\), of the sandwich plates was set by 10. The results are comparatively represented in Figure 9a,b, where the value, \(\overline{\omega}\), of the \(A\mathscr{E}\)-\(A\mathscr{E}_{2}O_{3}\) bi-material plate is 1.259. The curves of the softcore cases were lower-bounded because the amount of \(Ni\) slightly increased as \(R_{3}\) became smaller. Conversely, those of hardcore cases were upper-bounded as \(R_{3}\) became smaller, because the slight increase of the amount of \(A\mathscr{E}_{2}O_{3}\). However, in both softcore and hardcore cases, the calibrated fundamental frequencies tended to 1.259 of \(A\mathscr{E}\)-\(A\mathscr{E}_{2}O_{3}\) bi-material plate as the volume fraction index \(n\) tend to \(+\infty\). This is because the \(H_{1}-H_{2}-H_{3}(H_{1}=H_{2})\) of the FG sandwich plates approached that of the \(A\mathscr{E}\)-\(A\mathscr{E}_{2}O_{3}\) bi-material plates as \(H_{1}\) and \(n\) tended to \(+\infty\), regardless of the material type of core layer. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\(n\)} & \multicolumn{8}{c}{**Hardcore**} \\ \cline{2-7}  & **1-1-1** & **2-2-1** & **3-3-1** & **4-4-1** & **8-8-1** & **16-16-1** \\ \hline
0 & 1.85670 & 1.85670 & 1.85670 & 1.85670 & 1.85670 & 1.85670 \\
0.5 & 1.54887 & 1.58688 & 1.59295 & 1.60389 & 1.62626 & 1.63537 \\
1 & 1.38588 & 1.43531 & 1.45256 & 1.46875 & 1.49820 & 1.52534 \\
5 & 1.08127 & 1.14988 & 1.18662 & 1.21352 & 1.25575 & 1.28786 \\
10 & 1.03080 & 1.10044 & 1.14199 & 1.17248 & 1.21105 & 1.25421 \\ \hline \end{tabular}
\end{table} 
Table 6: _The facesheet limits_ of the calibrated fundamental frequencies, \(\overline{\omega}\), for simply supported square FG sandwich plates with homogeneous hardcore \(\mathrm{s}(b/h=10\) ). Figure 8: _The gradient limits_ of calibrated fundamental frequency, \(\overline{\omega}\), to the volume fraction index, \(n\) for: (**a**) softcore cases, (**b**) hardcore cases. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\(n\)} & \multicolumn{8}{c}{**Softcore**} \\ \cline{2-7}  & **1-1-1** & **2-2-1** & **3-3-1** & **4-4-1** & **8-8-1** & **16-16-1** \\ \hline
0 & 0.93464 & 0.93464 & 0.93464 & 0.93464 & 0.93464 & 0.93464 \\
0.5 & 1.45678 & 1.41018 & 1.37057 & 1.34075 & 1.27588 & 1.23348 \\
1 & 1.60105 & 1.54583 & 1.49576 & 1.45646 & 1.36884 & 1.30447 \\
5 & 1.77674 & 1.71733 & 1.65475 & 1.60164 & 1.47473 & 1.38047 \\
10 & 1.80162 & 1.74416 & 1.67827 & 1.62098 & 1.48510 & 1.38334 \\ \hline \end{tabular}
\end{table} 
Table 5: _The facesheet limits_ of the calibrated fundamental frequencies, \(\overline{\omega}\), for simply supported square FG sandwich plates with homogeneous softcores (\(b/h=10\) ). Table 7 presents the calibrated fundamental frequencies, \(\overline{\omega}\), of simply supported FG sandwich plates with respect to the aspect ratio \(a/b\) for various values of \(R_{3}=H_{3}/(H_{1}+H_{2}+H_{3})\).