# A Single-Phase Lightweight Double-Leaf Multi-Stage Acoustic Black Hole Model of Metamaterial
## 3 Calculation Results and Discussion
### Vibration Transmission and Vibration Displacement Field  
The complex energy band structure is computed from the metamaterial with infinite periodic construction, and there is no propagatable wave in the bandgap. However, in practical applications, the metamaterial consists of finite periodic structure, and the propagatable wave in the bandgap is suppressed [41]. Considering the practical application, as in Figure 7a, a metamaterial consisting of 12 unit cells is used to apply flexural wave excitation along the y-direction and longitudinal wave excitation along the x-direction at the specified boundary on the left side, respectively, to pick up the displacement response on the rightmost side. Also, disregarding the influence of reflected waves on its results, a perfect matching layer is added on each side to better characterise the attenuation of the propagatable waves. The displacement frequency response function is calculated as follows: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Reference** & **Lattice Constant (mm)** & **Materials** & **The First Flexural Bandgap Range** & **Generalised Width** \\ \hline [14] & 120 & & 366–922 & 1.519 \\ \hline [15] & & & 179–317 & 0.771 \\ \hline SD2-ABH & 140 & & 327–1061 & 2.245 \\ \hline SD3-ABH & & & 346–1197 & 2.459 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: BG comparison of similar structures. Figure 6: Evanescent mode diagrams of special points in the imaginary part of the complex energy band of the SDM-ABH metamaterial. (**a**) SD1-ABH metamaterial evanescent mode. (**b**) SD2-ABH metamaterial evanescent mode; (**c**) SD3-ABH metamaterial evanescent mode. \[W=20 log_{10}\left(\frac{u_{out}}{u_{in}}\right) \tag{13}\]  
In Equation (13), \(u_{out}\) and \(u_{in}\) are the pickup displacement and excitation displacement, respectively. In addition, for the calculation of the displacement frequency response function, a dense meshing is used, and in order to ensure that the structure has the maximum degree of freedom, the maximum mesh unit size is set to 0.0252, which is also much smaller than one tenth of the elastic wave length. The frequency response functions of the flexural and longitudinal waves of the finite-period construction at N = 1, 2, and 3 were calculated using the displacement transfer rate, as shown in Figure 7b-d, respectively. Observing Figure 7, it is found that the suppressed range of propagable waves in the finite- period structure is basically consistent with the band gap in the complex energy band structure in Figure 2. Further observation of Figure 7 reveals that the width of the first flexural wave bandgap of the three structures increases dramatically with the increase in the number of stages. And the maximum attenuation within the bandgap increases to 1.4 and 1.6 times more than that of the SD1-ABH metamaterials (89 dB), respectively, due to the increase in the number of sub-ABHs presenting an ABH effect. The vibrational displacement field within the band gap of the first flexural wave of the first period in Figure 7a is calculated when N = 1, 2, and 3, as shown in Figure 8a-c, respectively. As can be observed in Figure 8, it is found that the vibrational displacement of the flexural wave at the frequency within the bandgap attenuation from the excitation value of 1 m to 0.443 m, 0.341 m, and 0.295 m occurs, respectively, when N = 1, 2, and 3. In Figure 8b, the SD2-ABH metamaterial undergoes a mutation at the tip of the second-stage ABH due to the ABH effect displacement attenuation. In Figure 8c, the SD3-ABH metamaterial undergoes a mutation at the tip of the third-stage ABH hole and at the tip of the second-stage ABH due to the ABH effect displacement attenuation.