## 5 Effects of ABH Parameters and Damping Layer
### Effect of the Damping Layer
At most frequencies of the mass-controlled region and the coincidence frequency, the sound insulation performance of the circular plate embedded with damped 19 ABHs is better than that of the uniform circular plate. In the damping-controlled region, the TL values of the circular plates embedded with damped 19 ABHs are improved at some frequencies, while reduced at other frequencies compared with that of the uniform circular plate. The effect of the thickness of the damping layer on the circular plate embedded with single ABH was further studied, as shown in Figure 18. It can be seen from Figure 18 that, below the coincidence frequency, there is almost no change in the sound transmission loss of the circular plate with damped ABH when the thickness of the damping layer varying from 1 mm to 3 mm, indicating that increasing the thickness of the damping layer could not significantly increase the TL at this frequency interval. At the first-order resonance frequency, all of the TL values of the circular plates with single damped ABH are lower than that of the circular plate with single undamped ABH. As can be seen from Figure 18b, the TL the circular plate with single damped ABH is much higher than that of the circular plate with single undamped ABH at the coincidence frequency, among which the difference between the TL of the ABH circular plate with a damping thickness of 1.5 mm and the circular plate with single undamped ABH is 10.74 dB. It also can be concluded that, in the damping-controlled region, the TL of the circular plate with single damped ABH is lower than that of the ABH circular plate without the damping layer at most frequencies when the thickness of the damping layer is 1 mm and 2 mm. The ABH circular plates with a damping thickness of 1.5 mm and 3 mm seem to have a better sound insulation performance in the damping-controlled region. Figure 17: Effect of damping layer thickness on TL of a circular plate containing 19 ABHs (compared with a uniform circular plate). Figure 18: Effect of damping layer thickness on TL of a circular plate embedded with single ABH: (**a**) TL values of ABH plates with different thickness of damping layer; (**b**) TL value differences compared with undamped ABH circular plate. The sound transmission loss of a circular plate containing single damped ABH with different damping layer thicknesses was compared with that of a uniform thickness circular plate, and the results are shown in Figure 19. It can be seen from Figure 19 that below the first-order resonance frequency, the damping layer can improve the sound transmission loss of the circular plate embedded with single damped ABH, which is different from the case of the circular plate containing 19 ABHs. The TL of the circular plate with single damped ABH is 19.34 dB larger than that of the uniform circular plate at the first-order resonance frequency. In the mass-controlled region, single ABH with a damping layer can increase the TL of the circular plate at some frequencies, but at other frequencies the TL is reduced compared with that of the uniform circular plate. At the coincidence frequency, the TLs of circular plates containing single damped ABH are significantly lower than that of the uniform thickness circular plate. In the damping-controlled region, the sound transmission loss of circular plates containing single damped ABH can achieve a significant sound insulation effect only at 8 kHz compared with that of the uniform thickness circular plate, and at this time, the TL of the circular plate containing single damped ABH with a damping layer thickness of 3 mm is 20.83 dB higher than that of a uniform thickness circular plate. The modal loss factor can reveal the mechanism of vibration suppression and radiated sound attenuation of acoustic black hole structures [29; 30].