## 3 Results and Discussions
### Convergence Studies
To study the influence of the restraint spring rigidity at both ends of lattice sandwich beams on the convergence of the present method, Figure 5 plots the relationship between the first six order-dimensionless frequency parameters (\(\Omega\)) of lattice sandwich beams and the restraint springs' rigidity. During the numerical calculation process, the rigidity of the restraint springs is set in two cases. In one case, based on the simply supported ends (SS-SS), the rigidity of the rotational springs (\(K_{i}\), \(i\) = 0, \(L\)) gradually increases from 10\({}^{0}\) to \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**M**} & \multicolumn{6}{c}{**Model Sequence**} \\ \cline{2-7}  & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline
6 & 2.0689 & 8.0478 & 17.3451 & 29.2085 & 42.9238 & 58.0482 \\
8 & 2.0689 & 8.0478 & 17.3451 & 29.2085 & 42.9145 & 57.8734 \\
10 & 2.0689 & 8.0478 & 17.3451 & 29.2085 & 42.9128 & 57.8594 \\
12 & 2.0689 & 8.0478 & 17.3451 & 29.2085 & 42.9118 & 57.8562 \\
14 & 2.0689 & 8.0478 & 17.3451 & 29.2085 & 42.9117 & 57.8560 \\
16 & 2.0689 & 8.0478 & 17.3451 & 29.2085 & 42.9117 & 57.8559 \\ \hline \end{tabular}
\end{table} 
Table 2: Convergence analysis of \(\Omega=\omega L^{2}\sqrt{\rho A/EI}\) for a SS–SS sandwich beam with a pyramidal lattice core, as M increases. Figure 4: Frequency parameters \(\Omega\) for different numbers of items in the lattice sandwich beam with a pyramidal lattice core. \(10^{14}\). The corresponding results are plotted in Figure 5a,b. In the other case, the rigidity of the translational springs (\(k_{i}\), \(i=0\), \(L\)) gradually changes from \(10^{0}\) to \(10^{14}\), when that of the rotational springs is fixed at \(10^{14}\). The corresponding results are plotted in Figure 5c,d. As shown in Figure 5, the first six order-dimensionless frequency parameters (\(\Omega\)) curve of lattice sandwich beams have the same change rule. As the rigidity for the translational and rotational springs continues to increase, the dimensionless frequency parameters (\(\Omega\)) curve of the lattice sandwich beam is a straight line within a certain interval (rigidity changes from \(10^{0}\) to \(10^{4}\)), and then it increases sharply within a certain range of rigidity (from \(10^{4}\) to \(10^{10}\)) to eventually gradually approach a straight line before remaining unchanged in the range of \(10^{10}\)-\(10^{14}\). It can be clearly seen that the convergence of the calculation results can be guaranteed when the rigidity value of all types is either more than \(10^{10}\) or less than \(10^{4}\). The analysis result shows that when the rigidity of the restraint springs is greater than \(10^{10}\), it can be regarded as a clamped boundary condition. Afterwards, the increasing rigidity of the restraint springs has little effect on the natural frequency of the lattice sandwich beam. Therefore, the rigidity value equal to \(10^{14}\) can be used for simulating infinite rigidity during the numerical calculations. When the rigidity of the restraint springs is less than \(10^{4}\), the support effect of the restraint spring is very weak. At that time, it has the same effect as if the rigidity value was equal to zero. It can be regarded as a free boundary condition. Therefore, it is reasonable to use \(10^{0}\) to simulate an extremely small rigidity in numerical calculations. These instructions indicate that the constraint spring rigidity set in the previous section is very reasonable for numerical verification. When the rigidity of the restraint springs is between \(10^{4}\) and \(10^{10}\), it can be regarded as an elastic boundary condition. Figure 5: Relationships of frequency parameters \(\Omega\) with restraint spring stiffness of the lattice sandwich beam with a pyramidal lattice core. For the convenience of subsequent research, the applicable boundary conditions in the calculation are shown in Table 3, where the symbols F, C and SS, respectively, represent free boundary conditions, clamping boundary conditions and simply supported boundary conditions.