# A Single-Phase Lightweight Double-Leaf Multi-Stage Acoustic Black Hole Model of Metamaterial
## 3 Calculation Results and Discussion
### Complex Energy Band Structure and Vibration Modes
Among them are the following: the first flexural wave bandgap (327-1061 Hz), with a generalised width of the bandgap is \(\gamma=2.245\); the second flexural wave bandgap (1395-1553 Hz), with a generalised width of \(\gamma=0.113\); the third flexural wave bandgap (1832-2000 Hz), with a generalised width of \(\gamma=0.092\); and the first longitudinal wave bandgap (1238-1402 Hz), with a generalised width of \(\gamma=0.132\). At this time, in the 0-2000 Hz range, the percentage of SD2-ABH metamaterials in the bandgap is increased to 60.9%, while their weights are 10% lighter than those of SD1-ABH metamaterials. For analysing the formation mechanism of the SD2-ABH metamaterial bandgap, the vibrational modes at special points of the real part of the curve in the complex energy band of Figure 2 are calculated, as shown in Figure 4. It is found that the vibrational modes at the point A2 (327 Hz), the initial frequency of the first flexural wave bandgap, also present local antisymmetric flexural modes, and it is found that the vibrational modes at the point B2 (1061 Hz) at the cut-off frequency are mainly concentrated in the non-tip region of the first-stage ABH, whereas all the sub-ABHs of the second-stage show energy aggregation, which at this time presents the coupling of local resonance modes and the ABH effect. Moreover, the torsional vibrational displacements of the two sides of the unit cell are larger, and the complex multiple elastic scattering phenomenon begins to intensify. Observing the imaginary part of the complex energy band in Figure 2b, it is found that the semicircular arc of the imaginary part corresponding to the first flexural wave bandgap still exhibits the typical shape characteristics of local resonance and Bragg scattering coupling [37]. Thus, the first flexural wave bandgap of the SD2-ABH metamaterials is generated by a complex coupling of ABH effect, local resonance, and Bragg scattering. The vibration modes of the initial frequency point C2 (1395 Hz) and the cut-off frequency point D2 (1553 Hz) of the second flexural wave bandgap are focused on the whole region of the first-stage ABH of the cell, which show the local in-phase vibration modes. In addition, the both unit cells of the C2 point also produce small torsional vibration, showing multiple elastic scattering phenomena, so the second flexural wave bandgap also indicates the typical properties of local resonance and Bragg scattering coupling [37]. For the longitudinal wave bandgap starting and stopping frequencies for the points E2 (1238 Hz) and F2 (1402 Hz), respectively, the vibrational modes are centrally distributed in the whole region of the first-stage ABH of the single cell due to the compression of the two sides of the single cell in the one-dimensional x-direction, which show local outward anti-phase  
Figure 3: Vibration modal diagrams of special points in the real part of the complex energy band of SD1-ABH metamaterial, where the direction and length of each arrow indicate the direction and magnitude of the displacement of the medium at the starting point of the arrow, respectively. vibration modes. Therefore, observing the imaginary part of the complex energy band in Figure 2b, the evanescent wave attenuation region corresponding to the first longitudinal wave bandgap shows a sharp tip shape, and the attenuation undergoes an abrupt change process, which is characteristic of the local resonance mechanism [38]. The structure is SD3-ABH metamaterial when N = 3, and the complex energy band structure is shown in Figure 2c. Observing the real part of the complex energy band in Figure 2c, there are three flexural wave bandgaps (346-1197 Hz, 1383-1584 Hz, and 1954-2000 Hz) and one longitudinal wave bandgap (1207-1386 Hz) in 0-2000 Hz. In this situation, the bandgap percentage increases to 63.7% and the weight is reduced by 13% compared with the SD1-ABH metamaterial.