## 3 Results and Discussions
### Effects of Boundary Conditions
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**Radius of**} & \multicolumn{5}{c}{**Arbitrary Boundary Conditions**} \\ \cline{2-6} \multicolumn{1}{c}{**the Truss (mm)**} & **SS–SSS** & **C–C** & **E3–E3** & **E3–SS** & **E3–C** \\ \hline
0.4 & 139.0901 & 267.8756 & 208.1397 & 177.9489 & 230.7810 \\
0.6 & 136.8992 & 284.5443 & 210.5608 & 178.4970 & 236.6766 \\
0.8 & 129.8016 & 279.3969 & 202.2866 & 170.9090 & 228.6271 \\
1 & 121.4175 & 266.0154 & 190.4328 & 160.6320 & 215.8123 \\
1.2 & 112.8727 & 249.7781 & 177.6603 & 149.7228 & 201.6419 \\
1.4 & 104.6991 & 233.1444 & 165.1523 & 139.1051 & 187.6189 \\
1.6 & 97.1334 & 217.1685 & 153.4348 & 129.1895 & 174.4136 \\
1.8 & 90.2537 & 202.3523 & 142.7060 & 120.1265 & 162.2863 \\
2 & 84.0553 & 188.8378 & 132.9978 & 111.9348 & 151.2920 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparisons of the natural frequencies of pyramidal truss-core sandwich beams with different truss radius under arbitrary boundary conditions (Hz). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**Radius of**} & \multicolumn{5}{c}{**Arbitrary Boundary Conditions**} \\ \cline{2-6} \multicolumn{1}{c}{**the Truss (mm)**} & **SS–SSS** & **C–C** & **E3–E3** & **E3–SS** & **E3–C** \\ \hline
0.4 & 137.4313 & 242.6776 & 197.1684 & 170.1449 & 215.5373 \\
0.6 & 139.8083 & 276.1559 & 210.7334 & 179.5651 & 234.8861 \\
0.8 & 136.6687 & 284.5942 & 210.3639 & 178.2960 & 236.5277 \\
1 & 131.3832 & 281.3776 & 204.3654 & 172.7485 & 230.7951 \\
1.2 & 125.1822 & 272.4854 & 195.8814 & 165.3260 & 221.7693 \\
1.4 & 118.6770 & 261.0067 & 186.3849 & 157.1637 & 211.3455 \\
1.6 & 112.2116 & 248.4577 & 176.6567 & 148.8691 & 200.5202 \\
1.8 & 105.9816 & 235.7920 & 167.1272 & 140.7789 & 189.8390 \\
2 & 100.0916 & 223.4619 & 158.0284 & 133.0740 & 179.5965 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Comparisons of the natural frequencies of tetrahedral truss-core sandwich beams with different truss radii under arbitrary boundary conditions (Hz). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**Radius of**} & \multicolumn{5}{c}{**Arbitrary Boundary Conditions**} \\ \cline{2-6} \multicolumn{1}{c}{**the Truss (mm)**} & **SS–SSS** & **C–C** & **E3–E3** & **E3–SS** & **E3–C** \\ \hline
0.4 & 139.0700 & 255.8957 & 203.3113 & 174.5988 & 223.8548 \\
0.6 & 138.8832 & 281.7456 & 211.6004 & 179.8111 & 236.9097 \\
0.8 & 133.8783 & 283.8063 & 207.4455 & 175.5245 & 233.8970 \\
1 & 127.1032 & 275.5349 & 198.5927 & 167.6778 & 224.6977 \\
1.2 & 119.7316 & 262.9310 & 187.9493 & 158.5026 & 213.0750 \\
1.4 & 112.3501 & 248.7202 & 176.8671 & 149.0480 & 200.7555 \\
1.6 & 105.2674 & 234.2966 & 166.0278 & 139.8470 & 188.6031 \\
1.8 & 98.6351 & 220.3493 & 155.7685 & 131.1625 & 177.0474 \\
2 & 92.5132 & 207.2480 & 146.2359 & 123.1070 & 166.2793 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Comparisons of the natural frequencies of 3D–Kagome truss-core sandwich beams with different truss radii under arbitrary boundary conditions (Hz). Figure 14: Effect of the truss radius on the first natural frequency of the 3D–Kagome truss-core sandwich beam under arbitrary boundary conditions. Figure 12: Effect of the truss radius on the first natural frequency of the pyramidal truss-core sandwich beam under arbitrary boundary conditions. Figure 13: Effect of the truss radius on the first natural frequency of the tetrahedral truss-core sandwich beam under arbitrary boundary conditions.