## 3 Results and Discussions
### Effects of Boundary Conditions
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Panel thicknesses**} & \multicolumn{5}{c}{**Arbitrary Boundary Conditions**} \\ \cline{2-6}  & **SS–SS** & **C–C** & **E3–E3** & **E3–SS** & **E3–C** \\ \hline
0.5 & 121.4175 & 266.0154 & 190.4328 & 160.6320 & 215.8123 \\
1 & 137.7297 & 292.6634 & 168.9821 & 155.7359 & 201.3014 \\
1.5 & 146.5784 & 302.9722 & 150.4007 & 145.0440 & 185.5933 \\
2 & 153.0482 & 308.5381 & 136.3312 & 135.3830 & 173.7080 \\
2.5 & 158.4541 & 312.3427 & 125.4501 & 127.4007 & 164.9065 \\
3 & 163.2901 & 315.3960 & 116.7719 & 120.8833 & 158.3260 \\
3.5 & 167.7836 & 318.1906 & 109.6617 & 115.5383 & 153.3543 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of the natural frequencies of pyramidal truss-core sandwich beams of different panel thicknesses under arbitrary boundary conditions (Hz). Figure 10: Effect of the panel thickness on the first natural frequency of the pyramidal truss-core sandwich beam under arbitrary boundary conditions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Core Height**} & \multicolumn{5}{c}{**Arbitrary Boundary Conditions**} \\ \cline{2-6}  & **SS–SS** & **C–C** & **E3–E3** & **E3–SS** & **E3–C** \\ \hline
5 & 32.9684 & 74.7245 & 70.7994 & 50.4886 & 72.6211 \\
10 & 76.0519 & 170.5798 & 142.6226 & 110.1543 & 154.1021 \\
15 & 121.4175 & 266.0154 & 190.4328 & 160.6320 &It can be seen from Table 7 and Figure 9 that, for different boundary conditions, the free vibration frequency of the structure decreases with the increase of the cell number. The main reason is that as the beam length continues to increase, the rigidity of the structure decreases due to the increase in its weight. Combining Table 8 and Figure 10, it can be known that the free vibration frequencies of the structure will become larger due to the increase of the panel thickness under C-C and SS-SS boundary conditions, but it will become smaller as the panel thickness increases under E3-E3, E3-SS and E3-C boundary conditions. The main reason is that when the boundary conditions are C-C and SS-SS, the rigidity of the restraint springs at both ends of the lattice sandwich beam is sufficiently large. In this case, the increase in the bending rigidity of the structure is greater than that in the structural weight. As a result, the free vibration frequencies of the structure will increase as the panel thickness increases. However, when the boundary conditions are E3-E3, E3-SS and E3-C, the overall rigidity of the restraint springs at boundary supports is relatively small. In this case, the increase in structural weight is dominant, but the increase in structural bending rigidity is relatively small. Hence, the free vibration frequencies of the structure will decrease as the panel thickness increases. These statements indicate that the boundary support conditions have an important effect on the free vibration frequency of lattice sandwich beams, especially under the elastic boundary conditions. It can be seen from Table 9 and Figure 11 that, under different boundary conditions, with the increase of core height, the free vibration frequency of lattice sandwich beams will increase. The main reason is that as the core height continues to increase, the equivalent density of the truss core will decrease, resulting in a decline in the weight of the structure and an increase in its natural frequency. In addition, comparing Figures 10 and 11, under the classical boundary conditions such as C-C and SS-SS, it is more effective to change the free vibration frequencies of lattice sandwich beams by changing the core height than by changing the panel thickness because, when the height of the core layer increases, not only is the weight of the structure reduced, but also the moment of inertia of the structure increases.