## 5 Effects of ABH Parameters and Damping Layer
### Effect of the Damping Layer  
In the case of mechanical force excitation, the adhesion of a small amount of damping layer in the ABH region can significantly improve the adverse effect of the existence of the truncation on the flexural wave aggregation [28]. However, the influence of the damping layer under excitation of diffuse field needs to be further studied. A FEM model of an ABH circular plate attached with a damping layer was established, which is shown in Figure 15. The mesh size of the damping layer is 1 mm and an encryption was applied in specific areas. The damping layer has a thickness of 1 mm and a diameter of 60 mm, which can achieve full coverage of the ABH region. The damping layer is made of 3M ISODAMP TD 1604, and the damping material properties are shown in Table 4.  
Figure 14: Effect of different truncation thickness on TL values of the ABH circular plate: (a) TL values of ABH plates with different truncation thickness; (b) TL value differences compared with the truncation thickness of 0.5 mm. The thickness of the damping layer was selected as 1 mm, 1.5 mm, 2 mm, and 3 mm, respectively, in this study. Firstly, the influence of the thickness of the damping layer on the sound transmission loss of the circular plate embedded with 19 ABHs was calculated, and each ABH region was attached with a damping layer, which is shown in Figure 16. It can be seen from Figure 16 that in the stiffness-controlled region, the thickness variation of the damping layer has almost no effect on the sound transmission loss of the circular plate. However, at the first-order resonance frequency, the larger the thickness of the damping layer is, the smaller the sound transmission loss is, indicating that the increase in the thickness of the damping layer reduces the sound transmission loss at the first-order resonance frequency. In the mass-controlled region, the attaching of the damping layer can significantly improve the sound transmission loss of the circular plate. However, the difference of TL values of the circular plate with the thickness of the damping layer varying from 1 mm to 3 mm is not obvious. At the coincidence frequency, the damping layer can also significantly improve the sound transmission loss. As can be seen from Figure 16b, the sound transmission loss of the ABH circular plate with the thickness of the damping layer of 2 mm is 19.68 dB higher than that without the damping layer. In the damping-controlled region, the attaching of the damping layer can also increase the sound transmission loss of the ABH circular plate at some frequencies, among which the TL of the ABH circular plate with a damping layer thickness of 3 mm is 14.35 dB higher than that of the ABH circular plate without the damping layer at 10 kHz. Figure 17 shows a comparison of TL values of a circular plate embedded with damped 19 ABHs to the TL of a uniform thickness circular plate. As can be seen from Figure 17, below the first-order resonance frequency, the sound transmission loss values of the damped ABH circular plates are lower than that of the uniform thickness circular plate, especially at the first-order resonance frequency, the TL of the ABH circular plate with a damping layer of 3 mm is 23.81 dB lower than that of the uniform \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Young’s Modulus (Pa)** & **Density (kg/m\({}^{3}\))** & **Poisson’s Ratio** & **Loss Factor** \\ \hline Viscoelastic & \(9\times 10^{6}\) & 1812 & 0.45 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Damping material properties. Figure 16: Effect of damping layer thickness on TLs of a circular plate containing 19 ABHs: (**a**) TL values of ABH plates with different thickness of damping layer; (**b**) TL value differences compared with undamped ABH circular plate. Figure 15: FEM model of the ABH circular plate with a damping layer. thickness plate.