## 3 Analysis of Results and Mechanisms
### Analysis of Very Low-Frequency Wideband Mechanisms
Therefore, the production of the first flexural BG is associated with both the LR mechanism and the BS mechanism. In addition, in the vibration mode at point A\({}_{1}\), the deformation at the center of the ABH contour is near zero, which means the attenuation of the low-frequency flexural wave is the coupling between the vibration modes rather than caused by the ABH effect, and its effect should be more reflected at the middle and high frequencies [9; 24; 25; 26]. In the vibration mode at point A\({}_{2}\), the deformation at the center of the ABH contour is dominant. This means that the attenuation of the flexural wave at point A\({}_{2}\) is caused by the ABH effect of the ABH contour. At the point B\({}_{1}\) on the first longitudinal BG upper boundary, the AREM single cell as a whole vibrates in the radial direction, the vibration displacement on the right side of the structure is larger, and the structure exhibits an LR mode. The combined force formed by this mode in the radial direction couples with the long-wave traveling waves propagating in the structure, resulting in the generation of this local resonant BG. At the same time, the imaginary part of the first longitudinal BG in Figure 2 also shows an obvious asymmetry, which is the result of the interference of the Fano-like phenomenon characteristic of the LR BG, one of the typical features of the LR BG [27; 28]. All the above findings indicate that the first longitudinal BG in the AREM is related to the LR mechanism. At the point B\({}_{2}\) of the lower boundary of the second longitudinal BG, the AREM single cell shows a symmetric longitudinal vibration mode in the radial direction where both sides of the ABH center are stretched outward. And at the point B\({}_{3}\) of the lower boundary of the second longitudinal BG, the AREM structure as a whole is a longitudinal vibration mode with one side being stretched and the other side being compressed, and the structure shows an LR mode. In addition, the longitudinal wavelength \(\lambda=c_{t}/f=0.012\) m corresponding to the center frequency of the second longitudinal BG is similar to the lattice constant a. In addition, by viewing 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 second longitudinal BG shows a semicircular shape, indicating that the wave attenuation performance within the BG varies with the smoothness of the frequency. This phenomenon indicates that the generation of the second longitudinal direction is also related to both the LR mechanism and the BS mechanism. To further investigate the important role of the ABH effect in the AREM structure, the points with similar BG characteristics in the BGs of the AREM and WREM were compared and analyzed. The eigenmodes and displacement vector fields of the special points at the upper and lower edges of the second flexural BG of the WREM structure in Figure 3 are shown in Figure 5. As can be seen in Figures 4 and 5, the modes at points A\({}_{1}\) and C\({}_{1}\) both exhibit antisymmetric flexural vibration affected by rotational motion, and at the two 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 LR modes. The difference is that the AREM has less stiffness compared to the WREM, and the mode at point A\({}_{1}\) has a higher degree of energy localization. Therefore, the frequency of the mode at point A\({}_{1}\) is lower than that at point C\({}_{1}\), the vibration mode at point A\({}_{2}\) shows the up-and-down motion of the center of the structure, and the displacement amplitude of the rotational motion on both sides is much smaller than that of the center of the structure. The vibration mode at point C\({}_{2}\) is the up-and-down motion of the center of the structure and the rotational motion of both sides of the structure.