# A Single-Phase Lightweight Double-Leaf Multi-Stage Acoustic Black Hole Model of Metamaterial
## 4 Discussion of Geometric Parameter on Complex Band Structure  
### Effect of Truncated Thickness \(t\) on Band Gap  
The truncated thickness \(t\) is an important structural parameter for acoustic black holes. Keeping other parameters constant, the effect of \(t\) on the first flexural wave bandgap characteristics of an SD2-ABH metamaterial is investigated, as shown in Figure 9. Observing Figure 9, it is found that the first flexural wave band gap initial frequency increases gradually with the increase in \(t\), which is caused by the increase in structural stiffness because of the increase in \(t\). The cut-off frequency of the first flexural wave band gap firstly increases and then decreases with the increase in \(t\), and the maximum value is when the truncated thickness \(t=2\) mm. The band structure diagram in the real part of the complex energy band and the vibrational modes at the cut-off frequency of SD2-ABH metamaterials with different truncation thicknesses are calculated, respectively, as shown in Figure 10. Observing the vibration modes at the cut-off frequency of the first bending band gap in Figure 10, it is found that when \(t<2\) mm, the effect of ABH in the structure gradually decreases with the increase in \(t\), and thus, the cut-off frequency moves to the high frequency. When \(t\geq 2\) mm, the ABH effect disappears, and the vibration modes change to the local resonance of the whole region of the first-stage ABH. Because of the increase in truncation thickness \(t\), the resonance mass of the whole region of the first-stage ABH increases, and the cut-off frequency decreases. As a result, the first flexural wave band gap bandwidth firstly increases and secondly decreases with the truncation thickness \(t\).  
Figure 8: Vibration displacement field of the first period of the finite period construction: (**a**) represents the vibration displacement field at the frequency of 273 Hz in Figure 7b; (**b**) represents the vibration displacement field at the frequency of 411 Hz in Figure 7c; (**c**) represents the vibration displacement field at the frequency of 476 Hz in Figure 7d.  
Figure 9: Effect of truncation thickness \(t\) on the bandgap of the first flexural wave of an SD2-ABH metamaterial. Symbols indicate band gap boundaries and red coloured lines between symbols indicate band gap widths.  
The effect of truncation thickness t on the first flexural wave bandgap characteristics of SD3-ABH metamaterials is shown in Figure 11. Figure 11 reveals that the same variation trend as that of the SD2-ABH metamaterial is presented. The special point vibrational modal analysis of the real part of the complex energy band is further calculated as shown in Figure 12. Observing Figure 12, it is found that for SD3-ABH metamaterials, the stiffness increases with the increase in t, leading to a gradual increase in the initial frequency of the first flexural bandgap; the vibrational modes at the cut-off frequency of the first flexural bandgap change from the coupling of the local resonance and the ABH effect to the local resonance modes in the overall region of the first-stage ABH in the same way.  
Figure 10: Vibration modes of the real part of the complex energy band with different truncation thicknesses as well as special points.  
Figure 11: Effect of truncation thickness t on the bandgap of the first flexural wave of an SD3-ABH metamaterial. Symbols indicate band gap boundaries and red coloured lines between symbols indicate band gap widths.  
Figure 12: Vibration modes of the real part of the complex energy band with different truncation thicknesses as well as special points.