## 3 Analysis of Results and Mechanisms
### Analysis of Very Low-Frequency Wideband Mechanisms  
In this section, the eigenmode and displacement vector fields are calculated for the edge-specific points of the flexural BG and the longitudinal BG in the AREM dispersion relation curve of Figure 2, as shown in Figure 4. It is utilized to investigate the physical mechanism of the AREM ultra-low-frequency broadband characteristics. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Reference** & \begin{tabular}{c} **Lattice** \\ **Constant** \\ **(mm)** \\ \end{tabular} & **Materials** & \begin{tabular}{c} **Bandgap** \\ **Range** \\ **(Hz)** \\ \end{tabular} & \
\begin{tabular}{c} **Percentage of** \\ **Bandgap Below** \\ **5000 HZ** \\ \end{tabular} &
\begin{tabular}{c} **Minimum** \\ **Transmission** \\ **(dB)** \\ \end{tabular} \\ \hline \multirow{4}{*}{[11]} & \multirow{4}{*}{120} & \multirow{4}{*}{120} & 366–922 & \multirow{4}{*}{69.1\%} & \multirow{4}{*}{\(-\)40} \\ \cline{5-6}  & & & 1497–4135 & & \\ \cline{5-6}  & & & 4291–4551 & & \\ \cline{5-6}  & & & 6326–7012 & & \\ \cline{5-6}  & & & 9239–10,000 & & \\ \hline \multirow{4}{*}{[10]} & \multirow{4}{*}{230} & \multirow{4}{*}{steel} & 249–967 & \multirow{4}{*}{64.2\%} & \multirow{4}{*}{\(-\)50} \\ \cline{5-6}  & & & 1751–3220 & & \\ \cline{5-6}  & & & 3304–3761 & & \\ \cline{5-6}  & & & 3992–4558 & & \\ \hline \multirow{4}{*}{[9]} & \multirow{4}{*}{220} & \multirow{4}{*}{steel} & 176–218 & \multirow{4}{*}{73.7\%} & \multirow{4}{*}{\(-\)70} \\ \cline{5-6}  & & & 304–937 & & \\ \cline{5-6}  & & & 1171–2658 & & \\ \cline{5-6}  & & & 2770–3866 & & \\ \cline{5-6}  & & & 3886–4314 & & \\ \cline{5-6}  & & & 452–543 & & \\ \cline{5-6}  & & & 1116–12,576 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: BG comparison of similar structures. Figure 4: Eigenmode and displacement vector fields at special points A\({}_{1}\) and A\({}_{2}\) are the lower and upper boundary modes of the first flexural BG of the AREM structure; B\({}_{1}\) is the upper boundary mode of the first longitudinal BG of the AREM structure; B\({}_{2}\) and B\({}_{3}\) are the lower and upper boundary modes of the second longitudinal BG of the AREM structure. As can be seen from Figure 4, at the point A\({}_{1}\), the lower boundary of the first flexural BG, the vibration mode exhibits antisymmetric flexural vibration influenced by rotational motion, which is caused by complicated multiple elastic scattering of traveling waves between periodic structures, which is a typical Bragg scattering (BS) phenomenon [22]. At both ends of the whole structure, the antisymmetric flexural vibration modes are opposite, the vibration displacement of the part further away from the rotation center is larger, and the vibration modes exhibit a local resonance (LR) mode. At the point A\({}_{2}\) on the first flexural BG upper boundary, the vibration mode is shown as a two-sided symmetric flexural vibration influenced by rotational motion, with the center of the structure moving upward parallel to the \(Z\) direction, and the vibration displacement of the center of the structure is the largest, showing an LR mode. It is noteworthy that the structural LR modes appear in both points A\({}_{1}\) and A\({}_{2}\). Further, by means of Figure 2 and characteristics of a BS-type phonon crystal, it can be viewed that the elastic wave wavelength \(\lambda=c_{t}/f=0.0089\) m = 8.90 mm corresponding to the center frequency of the first flexural BG is in an order of magnitude with the lattice constant \(a\). In addition, by observing the dispersion relation curve of the imaginary part of the AREM in Figure 2, it can be seen that the imaginary part corresponding to the first flexural BG shows a semicircular shape, which indicates that the wave attenuation performance within the BG varies with the smoothness of the frequency, and because the width of the BG is larger compared to that of the single LR mechanism or the BS mechanism. This is a characteristic of the wave attenuation performance of the coupled BG of the LR and BS mechanisms [23].