## 4 Parameter Analysis  
### Effect of Truncation Thickness on AREM Bandgap Characteristics  
The ABH tip truncation thickness, \(h_{0}\), is an important structural parameter of ABHs. This section analyzes the influence of the truncation thickness at the BG characteristics between the energy bands of the AREM, while keeping the material parameters and lattice constants constant, as illustrated in Figure 7. Figure 7a displays the change in the BG characteristics of the flexural BG of the AREM as the truncation thickness \(h_{0}\) increases. When the truncation thickness \(h_{0}=0.02\) mm, the bandwidth of the first flexure BG is very narrow. With the increase in the truncation thickness \(h_{0}\), the central frequency of the first flexural BG progressively moves to a high frequency, and the bandwidth first decreases and then increases. For the second flexural BG, as the cutoff thickness \(h_{0}\) grows, the bandwidth of the second flexural BG gradually reduces as its lower starting frequency shifts to higher  
Figure 6: Frequency response analysis of interfaces in AREM and WREM structures consisting of 8 cells: (**a**) computational schematic of the corresponding analysis of the frequencies of the different interfaces; (**b**) flexural attenuation at the different interfaces of AREM; (**c**) longitudinal attenuation at the different interfaces of AREM; (**d**) flexural attenuation at the different interfaces of WREM; (**e**) longitudinal attenuation at the different interfaces of WREM. frequencies and its cutoff frequency shifts to lower frequencies. Figure 7b illustrates the influence of the variation in the truncation thickness \(h_{0}\) on the BG characteristics of the longitudinal BG of the AREM. The broadband and central frequency of the first longitudinal BG do not shift significantly with the increase in the truncation thickness \(h_{0}\). For the second longitudinal BG, the broadband of the second longitudinal BG gradually decreases and the central frequency shifts to high frequencies as the cutoff thickness \(h_{0}\) increases. The evolution of the flexural and longitudinal BG characteristics with the cutoff thickness \(h_{0}\) can be determined by the above analysis. From the application point of view, the truncation thickness \(h_{0}\) discussed so far is still small. Therefore, the dispersion relation curves of the AREM structure with the truncation thickness \(h_{0}=0.25\) mm (\(c=0.5\) mm) are calculated. Its flexural BG is under 1100 Hz, the forbidden bandwidth of the flexural BG is 733 Hz, and the percentage of the flexural forbidden band is 66.6%. The forbidden bandwidth of the longitudinal BG is 536.2 Hz, and the proportion of the longitudinal forbidden band is 48.8%. It still has good BG characteristics. To better interpret the physical mechanisms presented by AREM BG characteristics, the dispersion relation curves were calculated for \(h_{0}=0.01\), 0.03, and 0.05, respectively, as presented in Figure 8. It can be noticed from Figure 8 that the flatness of the second and third flexural energy bands in the dispersion curve gradually reduces with the growth of the truncation thickness, indicating that the degree of structural localization is weakened. This is due to the increase in the truncation thickness, which makes the stiffness of the structure increase and at the same time weakens the ABH effect in the AREM. Therefore, the change in the second flexural BG properties is clearly associated with the change in the degree of structural localization. Also, it is increased by the truncated thickness, which makes the AREM stiffness and mass larger, and the stiffness dominates the energy band effect. Therefore, as the truncation thickness increases, the upper cutoff frequency of the first BG and the lower cutoff frequency of the second flexural BG gradually move to higher frequencies. As for the longitudinal BG, we can find that the change in the cutoff thickness has basically no effect on the first longitudinal BG.