## 3 Results and Discussions
### Effects of Boundary Conditions  
To investigate the influence of different boundary conditions on the vibration characteristics of lattice sandwich beams, twenty-one different boundary conditions are first obtained from Table 3 by free combination. They are divided into three different groups. The first group includes F-F, SS-SS, C-C, SS-F, C-F and SS-C, which represent classical boundary conditions. The second one includes E1-E1, E--E2, E3-E3, E1-E2, E1-E3 and E2-E3, which represent elastic boundary conditions. The third one includes E1-F, E2-F, E3-F, E1-C, E2-C, E3-C, E1-SS, E2-SS and E3-SS, which represent arbitrary boundary conditions. To clarify the meaning of the above symbols, the symbol E--SS will be described in detail as an example; that is, at \(x=0\) and \(x=L\), the boundary supports of the sandwich beam are elastic supports and simple supports, respectively. In this case, \(k_{0}=10^{6}\), \(K_{0}=10^{8}\) at \(x=0\) and \(k_{L}=10^{14}\), \(K_{L}=0\) at \(x=L\). Based on these three different groups of boundary conditions, the first six order dimensionless frequency parameters (\(\Omega\)) of lattice sandwich beams under the corresponding boundary conditions are shown in Tables 4-6, respectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Boundary Conditions**} & \multicolumn{6}{c}{**Mode Sequence**} \\ \cline{2-7}  & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline F–F & 0.0000 & 0.0019 & 4.6697 & 12.4591 & 23.3257 & 36.4479 \\ SS–SS & 2.0689 & 8.0478 & 17.3451 & 29.2085 & 42.9118 & 57.8562 \\ C–C & 4.5327 & 11.9122 & 22.0701 & 34.3264 & 48.0153 & 62.8635 \\ SS–F & 0.0009 & 3.2241 & 10.1398 & 20.2331 & 32.7395 & 46.9466 \\ C–F & 0.7408 & 4.5238 & 12.1844 & 22.6818 & 35.3457 & 49.5428 \\ SS–C & 3.1828 & 9.9286 & 19.7000 & 31.7739 & 45.5013 & 60.3664 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Frequency parameters \(\Omega\) for lattice sandwich beams with several classical boundary conditions. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Boundary Conditions**} & **F** & **C** & **SS** & **E1** & **E2** & **E3** \\ \hline k\(i\) (i = 0, L) & 0 & \(10^{14}\) & \(10^{14}\) & \(10^{5}\) & \(10^{6}\) & \(10^{7}\) \\ K\(i\) (i = 0, L) & 0 & \(10^{14}\) & 0 & \(10^{7}\) & \(10^{8}\) & \(10^{9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The values of spring rigidity under arbitrary boundary conditions. Comparing the data in Tables 4-6, it can be clearly concluded that the influence of boundary conditions on structural vibration is very important. When the boundary conditions are different, there will be a large difference in the frequency parameters (\(\Omega\)) of the lattice sandwich. As the rigidity of the restraint spring at both ends of the lattice sandwich beam continues to increase, its frequency parameters (\(\Omega\)) also increase. Among them, the frequency parameters (\(\Omega\)) are the largest under the lamped ends. These results have accumulated data for discussing the vibration characteristics of lattice sandwich beams under arbitrary boundary conditions. In order to further study the influence of boundary conditions on the vibration characteristics of lattice sandwich beams, the mode shapes of lattice sandwich beams under different boundary conditions are also plotted. To facilitate this study, one of the three different boundary conditions mentioned above is selected as the representative, namely, SS-SS, E3-E3 and E3-SS, and its mode shape is drawn in Figures 6-8, respectively. Comparing Figures 6-8, it can be seen that the mode shapes of the structure under the three boundary conditions are different. The boundary conditions have a significant influence on the mode shape of the structure. Under the classical boundary conditions (e.g., SS-SS), the