## 4 Parameter Analysis
### Effect of Power Exponent on Bandgap Characteristics  
In this section, we investigate the influence of the power exponent \(m\) on the BG characteristics of the AREM. The effect of the power exponent \(m\) on the BG characteristics between the energy bands of the AREM is analyzed while keeping the material parameters and structural parameters constant, as indicated in Figure 9. Figure 9a illustrates the variation in the flexural BG characteristics of the AREM as the power exponent \(m\) increases. As the power exponent \(m\) increases, the starting frequency of the first BG decreases quickly, the cutoff frequency remains unchanged, and the first flexural BG shows a trend of gradually increasing bandwidth and shifting the center frequency to lower frequencies. For the second flexural BG, as the power exponent \(m\) increases, the starting frequency slowly decreases and the cutoff frequency rapidly shifts to lower frequencies. Therefore, the second flexural BG shows a trend of decreasing bandwidth and shifting to lower frequencies. With increasing power exponent \(m\), the broadband of the otherwise narrow third flexural BG gradually grows and gradually shifts to lower frequencies. Figure 9b illustrates the variation in the longitudinal BG characteristics of the AREM with increasing power exponent \(m\). Neither the bandwidth nor the central frequency of the first longitudinal BG changes significantly as the power exponent \(m\) increases. The second longitudinal BG bandwidth gradually increases as the power exponent \(m\) grows, its starting frequency shifts to the low frequency, and the cutoff frequency moves to the high frequency. To better explain the physical mechanism presented by the AREM BG characteristics, the dispersion relation curves were calculated for \(m\) = 2.0, 2.4, and 2.8 respectively, as shown in Figure 10. From Figure 10, it can be noticed that the flatness of the first, second, third, and fourth flexural energy bands increases significantly with the increase in the power exponent \(m\), indicating the enhancement of the structure localization and better LR characteristics. This is due to the increase in \(m\), which enhances the ABH effect in the structure to some extent. Also, it can be found that a new flexural BG is generated below 100 Hz as the power exponent \(m\) grows. This is because the increase in the power exponent \(m\) decreases the stiffness of the AREM. Thus, we can observe from Figure 10 that the third flexural energy band moves rapidly to lower frequencies as the power exponent \(m\) increases, while the second flexural energy band moves very slowly to lower frequencies. This indicates that the second flexural energy band is largely unaffected by the change in stiffness. In contrast, the stiffness is a major factor influencing the change in the third flexural band. Specifically, for the fourth flexural energy band, when the power exponent \(m\) is less than 2.4, the fourth flexural energy band is only able to increase the energy band  
Figure 8: Dispersion relationship graph for AREM with different truncation thickness \(h_{0}\): (**a**–**c**) represent the dispersion relation curves for truncated thicknesses \(h_{0}\) of 0.01, 0.03, and 0.05, respectively. The green lines indicate the curved dispersion relation curves, and the blue lines indicate the longitudinal dispersion relation curves. flatness with the increase in power exponent \(m\). The fourth flexural energy band is not only able to increase the energy band flatness with the increase in power exponent \(m\), but also the energy band gradually moves to the lower frequency. When the power exponent \(m\) is greater than 2.4, the fourth flexural energy band not only increases the energy band flatness with the increase in the power exponent \(m\), but also the energy band gradually moves to the lower frequency. Therefore, when the power exponent \(m\) is less than 2.4, the LR effect is the dominant factor for the energy band change.