## 5 Effects of ABH Parameters and Damping Layer
### Effect of the ABH Number  
The circular plates with ABH numbers of 7, 13, and 19 were selected for the TL analysis. The parameters of the ABH region are the same as those in Table 1, and the models are shown in Figure 11. Except for the ABH at the center of the circular plate, the rest of the ABHs are evenly distributed along the circumference, and the distance from the center of the ABHs to the center of the circular plate is indicated in Figure 11. For the plate with a number of ABHs of 7, the distance between the center of the central ABH and other six ABH centers is 80 mm; for the plate with a number of ABHs of 13, the distance between the center of the central ABH and other 12 ABH centers is 160 mm; for the plate with a number of ABHs of 19, the distances between the center of the central ABH and other 18 ABH centers are 80 mm and 160 mm, respectively. The sound transmission loss values of circular plates embedded with different numbers of ABHs are shown in Figure 12. It can be seen from Figure 12 that, below the first-order resonance frequency, the TL of the circular plate with the ABH number of 19 is significantly smaller than those of the ABH number of 1, 7, and 13, and the circular plate with ABH number 1 has the largest sound transmission loss. However, at the first-order resonance frequency, the plates ranked by the sound transmission loss values are 19 ABH, 7 ABH, and 13 ABH, respectively, and the sound transmission loss is determined by the system damping at this frequency. In the mass-controlled region, the increase in the number of ABHs does not improve the TL of the plate as a whole, but at some specific frequencies, the TL is improved compared with that of the ABH number of 1, such as 800 Hz and 1250 Hz. While at the coincidence frequency, the TLs of circular plates with the ABH number of 7, 13, 19 are both smaller than that of the plate with the ABH number of 1, indicating that increasing the ABH number at the coincidence frequency cannot improve the sound transmission loss of the circular plate. It also can be  
Figure 11: Circular plates embedded with different numbers of ABHs: (**a**) 7 ABHs; (**b**) 13 ABHs; (**c**) 19 ABHs. Figure 10: Effect of the orientation of the ABH region on the TL of the circular plate. concluded that in the damping-controlled region, the increase in the number of ABHs is also not able to increase the TL of the circular plate. The sound transmission loss values of circular plates with different ABH numbers were further compared with that of a circular plate of uniform thickness, and the results are shown in Figure 13. It can be seen from Figure 13 that the sound insulation performance of the circular plate with the ABH number of 1 is the best at the first-order resonance frequency and in the stiffness-controlled region. The TL of the circular plate with the ABH number of 1 is 21.13 dB larger than that of the uniform circular plate at 315 Hz. At the first-order resonance frequency, 630 Hz and 8000 Hz, the circular plate embedded with ABHs can significantly improve the sound insulation performance compared with the uniform thickness circular plate, but the increase of the number of ABHs at this time does not necessarily improve the sound transmission loss of the circular plate. To conclude, the embedding of the ABH structure into a circular plate can increase the sound transmission loss at some specific frequencies, but it cannot improve the sound transmission loss of circular plate in the entire frequency band. The increase in the number of ABHs is not able to increase the sound transmission loss at some specific frequencies, but rather reduces the sound transmission loss. The circular plate with single ABH exhibits good sound insulation effect at most frequencies. In applications, the appropriate number of ABHs should be selected according to the main frequency of the noise source. Figure 12: TL of circular plates with different ABH numbers: (**a**) TL values of plates with ABH number of 7, 13, and 19; (**b**) TL value differences compared with single ABH plate.