## 3 Results and Discussions
### Convergence Studies  
According to the above theoretical analysis, the calculation results of this method are mainly related to the rigidity of the constrained spring and the item numbers of the Fourier series. Therefore, this section will focus on the relationship between these factors and the convergence of the present method. It should be noted that the calculation parameters used in the following numerical calculations are the same as those used in the previous \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Boundary** & **Mode** & **[PERSON] et al.** & **[PERSON] et al.** & **[PERSON] et al.** & **Present Results** & **Error** \\
**Condition** & & [19] & [21] & [27] & **(M = 10)** & **Error** \\ \hline \multirow{4}{*}{\begin{tabular}{c} Clamped \\ ends(CC–CC) \\ \end{tabular} } & 1 & 264.99 & 265.44 & 265.21 & 266.02 & \(-\)0.22\% \\  & 2 & 693.24 & 695.72 & 694.48 & 699.10 & \(-\)0.49\% \\  & 3 & 1278.8 & 1286.1 & 1282.2 & 1295.25 & \(-\)0.71\% \\  & 4 & 1977.7 & 1993.7 & 1986.9 & 2014.54 & \(-\)1.05\% \\  & 5 & 2756.9 & 2786.3 & 2764.5 & 2817.91 & \(-\)1.13\% \\ \hline \multirow{4}{*}{\
\begin{tabular}{c} Simply \\ supported \\ ends(SS–SS) \\ \end{tabular} } & 1 & 121.37 & 121.42 & 121.33 & 121.42 & 0.00\% \\  & 2 & 471.66 & 472.30 & 472.1 & 472.31 & 0.00\% \\  & 3 & 1014.9 & 1017.9 & 1014.6 & 1017.95 & 0.00\% \\  & 4 & 1705.6 & 1714.2 & 1702.9 & 1714.18 & 0.00\% \\  & 5 & 2500.0 & 2518.4 & 2506.3 & 2518.40 & 0.00\% \\ \hline \multirow{4}{*}{
\begin{tabular}{c} Clamped\(-\)free \\ ends(C–F) \\ \end{tabular} } & 1 & 43.468 & 43.572 & 43.63 & 43.48 & 0.21\% \\  & 2 & 265.01 & 266.40 & 265.76 & 265.49 & 0.34\% \\ \cline{1-1}  & 3 & 712.04 & 718.13 & 715.61 & 715.08 & 0.42\% \\ \cline{1-1}  & 4 & 1321.2 & 1337.5 & 1333.7 & 1331.15 & 0.48\% \\ \cline{1-1}  & 5 & 2051.8 & 2085.7 & 2033.1 & 2074.36 & 0.54\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the natural frequencies for the lattice sandwich beam with a pyramidal lattice core (Hz). section. Theoretically, as the number of Fourier series terms increases, the numerical results will become more and more accurate. Therefore, as long as the calculation cost allows it, the present method can be considered to have arbitrary accuracy. However, in the actual calculation process, the Fourier series with infinite terms is often truncated to a finite number of terms M, which ensures the acceptable accuracy of the calculation results. To investigate the relationship between the item numbers of the Fourier series and the convergence of the present method, the frequency parameter \(\Omega=\omega L^{2}\sqrt{\rho A/EI}\) is firstly introduced into this study. Taking the sandwich beam with a pyramidal lattice core under the SS-SS boundary condition as a special case, the first six frequency parameters \(\Omega=\omega L^{2}\sqrt{\rho A/EI}\) of the sandwich beam are calculated as the increase of the item numbers (represented by M). The comparison results are shown in Table 2. The results show that the lower the frequency is, the easier the convergence is. When the model sequence (represented by m) is less than four, the calculation results can rapidly converge at M = 6, and a four-digit calculation precision can be obtained. As the model sequence increases, the number of items that need to be converged also increases. The more items there are, the more accurate the calculation results will be. When the model sequence is greater than 4, the calculation results are plotted in Figure 4a,b, respectively. As shown in Figure 4a,b, when the number of terms is greater than 10, the results converge stably to all frequencies. Therefore, in order to obtain sufficient accuracy while controlling the calculation cost, all Fourier series will be truncated to M = 10 in the following numerical calculations.