## 3 Results and Discussions
### Effects of Boundary Conditions
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Inclination**} & \multicolumn{5}{c}{**Arbitrary Boundary Conditions**} \\ \cline{2-6}  & **SS–SS** & **C–C** & **E3–E3** & **E3–SS** & **E3–C** \\ \hline
15 & 127.2862 & 213.6764 & 178.1381 & 154.7157 & 192.9505 \\
20 & 132.3566 & 251.2956 & 196.1763 & 167.8809 & 217.1953 \\
25 & 132.8548 & 268.8284 & 202.2133 & 171.8793 & 226.3097 \\
30 & 131.8481 & 276.0647 & 203.3529 & 172.2663 & 228.8377 \\
40 & 128.7111 & 277.4762 & 200.6993 & 169.5438 & 226.8882 \\
45 & 127.1033 & 275.5196 & 198.5928 & 167.6779 & 224.6977 \\
55 & 124.1497 & 269.3705 & 194.0418 & 163.8221 & 219.5808 \\
65 & 121.4531 & 259.7725 & 188.8268 & 159.6357 & 213.2088 \\
75 & 117.8059 & 237.8299 & 179.1155 & 152.2906 & 200.3804 \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Comparisons of the natural frequencies of 3D–Kagome truss-core sandwich beams with different inclination angles under arbitrary boundary conditions (Hz). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Inclination**} & \multicolumn{5}{c}{**Arbitrary Boundary Conditions**} \\ \cline{2-6}  & **SS–SS** & **C–C** & **E3–E3** & **E3–SS** & **E3–C** \\ \hline
15 & 116.4407 & 165.7399 & 148.9525 & 132.5366 & 156.6238 \\
20 & 130.5388 & 217.1402 & 181.8840 & 158.1508 & 196.6915 \\
25 & 135.9736 & 251.1696 & 199.1338 & 170.9350 & 219.4220 \\
30 & 137.3951 & 270.8547 & 206.9229 & 176.3563 & 230.5770 \\
40 & 134.6366 & 283.5999 & 208.1194 & 176.2027 & 234.4181 \\
45 & 131.3832 & 281.3719 & 204.3654 & 172.7486 & 230.7946 \\
55 & 120.9219 & 263.9603 & 189.4050 & 159.8195 & 214.5292 \\
65 & 102.8408 & 226.6169 & 161.6207 & 136.2611 & 183.3229 \\
75 & 72.5552 & 160.6354 & 114.2081 & 96.2506 & 129.6388 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Comparisons of the natural frequencies of tetrahedral truss-core sandwich beams with different inclination angles under arbitrary boundary conditions (Hz). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Inclination**} & \multicolumn{5}{c}{**Arbitrary Boundary Conditions**} \\ \cline{2-6}  & **SS–SS** & **C–C** & **E3–E3** & **E3–SS** & **E3–C** \\ \hline
15 & 127.2862 & 213.6764 & 178.1381 & 154.7157 & 192.9505 \\
20 & 132.3566 & 251.2956 & 196.1763 & 167.8809 & 217.1953 \\
25 & 132.8548 & 268.8284 & 202.2133 & 171.8793 & 226.3097 \\
30 & 131.8481 & 276.0647 & 203.3529 & 172.2663 & 228.8377 \\
40 & 128.7111 & 277.4762 & 200.6993 & 169.5438 & 226.8882 \\
45 & 127.1033 & 275.5196 & 198.5928 & 167.6779 & 224.6977 \\
55 & 124.1497 & 269.3705 & 194.0418 & 163.8221 & 219.5808 \\
65 & 121.4531 & 259.7725 & 188.8268 & 159.6357 & 213.2088 \\
75 & 117.8059 & 237.8299 & 179.1155 & 152.2906 & 200.3804 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Comparisons of the natural frequencies of pyramidal truss-core sandwich beams with different inclination angles under arbitrary boundary conditions (Hz). As shown in Tables 13-15 and Figures 17-19, the variation trend of the free vibration frequencies of the structures with different inclination angles is the same under different boundary conditions. In this case, the free vibration frequency of the structure first increases and then decreases. The pyramidal truss core and tetrahedral truss core changes in the same manner. However, for the 3D-Kagome truss core, the mechanism that causes the natural frequency of the structure to change is indeed different. As shown in Figures 20 and 21, as the inclination angle increases, the equivalent density of the 3D-Kagome truss core increases very slowly, but its equivalent transverse shear modulus first increases and then decreases. It is for this reason that, compared with the pyramidal and tetrahedral truss core, the free vibration frequencies of the structure with 3D-Kagome truss core decrease very slowly as the inclination of the truss increases from \(15^{\circ}\) to \(75^{\circ}\), as shown in Figures 17-19.