# 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
The vibration modes of special points in the real part curve of the complex energy band in Figure 2c are calculated, as shown in Figure 5. The vibration mode at the initial frequency A3 (346 Hz) of the first flexural bandgap also presents a local resonance mode. At the cut-off frequency point B3 (1197 Hz), the vibration mode is mainly distributed in the whole region of the second-stage ABH, and all the sub-ABHs of the third-stage show an ABH effect. Observing the imaginary part of the complex energy band in Figure 2c, the semicircular arc of the imaginary part corresponding to the first flexural band gap also appears smooth and asymmetric. Therefore, the first flexural wave band gap is also produced by the complex coupling of local resonance and Bragg scattering and shows an ABH effect. The initial frequency point C3 (1383 Hz) and the cut-off frequency point D3 (1584 Hz) of the second flexural band gap, the vibrational modes show a \"swing\" vibration pattern dominated by the ABH area of the first stage. And the imaginary part of the complex energy band of the second flexural band gap in Figure 2c is also smooth and asymmetric [37], and it is the coupling mechanism of the local resonance and the Bragg scattering. The E3 (1207 Hz) and F3 (1386 Hz) vibration modes at the starting and stopping frequency points of the first longitudinal wave bandgap show local inward-phase vibration modes due to the stretching of the single cell in the one-dimensional x-direction, which are characteristic of the local resonance mechanism [39]. Figure 4: Vibration modal diagrams of special points in the real part of the complex energy band of SD2-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. Further, by comparing the size of the semicircular arc of the imaginary part of the complex energy band in Figure 2a-c, it is found that the size of the semicircular arc of the imaginary part corresponding to the first flexural wave bandgap increases gradually with the increase in the number of stages N, indicating that the attenuation intensity of the first flexural wave bandgap increases gradually [40]. For investigating the attenuation enhancement mechanism of the elastic wave within the band gap, the evanescent modes at special points in the imaginary part of the complex energy bands of Figure 2a-c are computed, respectively, as shown in Figure 6. Observing the evanescent mode at point G1, the maximum vibrational displacement is located at the tip of the non-ABH, and no ABH effect occurs at this time. While for the evanescent mode at point G2, the maximum vibrational displacement of the second-stage ABHs is concentrated at the tips of multiple sub-ABHs, which generates multiple ABH effects. For the evanescent mode at point G3, vibrational aggregation exists at the tips of multiple sub-ABHs at the third-stage ABHs and at the tips of some of the second-stage ABHs, and the number of sub-ABHs producing the ABH effect is significantly increased. Therefore, for SDM-ABH metamaterials, as the number of stages increases, the number of sub-ABH structures exhibiting ABH effects increases, resulting in a substantial increase in the attenuation strength of the first flexural wave bandgap. In this paper, the SDM-ABH metamaterial structures are compared with the previously structures, while the structures have similar lattice constants and the same material properties between them, as shown in Table 1. The comparison reveals that SD2-ABH and SD3-ABH metamaterials have ultra-wide first flexural wave bandgap with generalised widths of \(\gamma=2.245\) and \(\gamma=2.459\), respectively, which are also superior to the other two structures of Table 1. This is of great significance for broadening the low-frequency damping.