## 3 Effect of Damping Layers in ABH Array on Energy Dissipation and Sound Radiation of Plate
### Energy Dissipation in Plate Embedded with ABH Array with Damping Layers  
Figure 14a shows the plate embedded with 5 ABHs coated with damping layers. Sections A and B are cross-sections of the plate. Vibration energy can be expressed as average power flow at the given section per unit time, as defined in Equation (5). Consequently, to evaluate the energy dissipation characteristics of the plate embedded with the ABH array with different damping layers, the difference of average power flow between section A (before ABHs) and section B (after ABHs) is calculated. For the plate embedded with the ABH array, shown as Figure 3b with varying thickness damping layers, uniform thickness damping layers and without damping layers under both BC 1 and BC 2, the difference of vibration energy per time unit (average power flow) between section A and B is calculated in frequency domain, respectively. Additionally, the vibration energy difference of a uniform plate with the same varying thickness damping layers with BC 1 is also calculated as a reference. Results are shown in Figures 15 and 16. Both show that the ABH plate without damping layers has the highest peaks, which indicates that more energy is gathered within the ABH areas between the two sections. Moreover, peak values of energy difference decrease obviously after the application of damping layers because of the energy dissipation effect of damping materials. As a result, a larger difference means a smaller dissipation effect by the damping layers. Figure 16: The vibration energy difference of ABH plates embedded with different damping configurations with BC 2. Figure 15: The vibration energy difference of ABH plates embedded with different damping configurations and a reference plate with BC 1. Figure 15 shows that every curve has two peaks, and the vibration energy difference between sections A and B of the plates embedded with the ABH array with different damping layers are obviously higher than that of the uniform plate with damping layers in the range of 980-1060 Hz. This indicates that the energy dissipation efficiency of damping materials is obviously improved because of the ABHs, particularly in the range of 1020-1040 Hz. Besides, for plates embedded with the ABH array, the peaks occur at different frequencies for different damping configurations. This indicates that the damping materials may change the energy distribution of plate embedded with the ABH array in the frequency domain to some extent and reduce the frequency at which the ABH array starts to work. Moreover, due to the energy dissipation of damping layers, the peak energy difference values of the ABH plate with uniform damping layers and the ABH plate with varying thickness damping layers are smaller than that of the ABH plate without damping layers, which means that some energy is dissipated by damping materials. The peak value of the ABH plate with uniform damping layers is smaller than that of the ABH plate with varying thickness damping layers, which indicates that the uniform damping layers have a better energy dissipation effect than varying thickness damping layers in this case. Similar conclusions can be drawn from Figure 16. First, similar to the ABH plate with BC 1, every curve has two peaks and the peak values ranging from larger to small is an ABH plate without damping layers, an ABH plate with varying thickness damping layers and an ABH plate with uniform damping layers, respectively. Besides, the curves also show that damping layers change the energy distribution in the frequency domain and reduce the frequency where the ABH starts to work. Moreover, as the boundary condition is changed, the frequency corresponding to the first peak is lower than that with BC 1 and the frequency corresponding to the second peak value is higher than that with BC 1, which means that boundary condition can also influence the energy distribution in the frequency domain.