## 4 Conclusions  
In this paper, a plate model embedded with the array of five two-dimensional ABHs are established. Moreover, the energy propagation characteristics arising from ABH are analyzed by using FEM and power flow method. Additionally, the energy focalization effect is studied quantitatively by calculating average vibration power density at different plate sections. Further, the effects of damping materials and different boundary conditions on model energy dissipation are also investigated. Finally, the effects of damping layers on radiated sound power are studied. Conclusions are reached as follows:  
1. In the low frequency range of 50-180 Hz, plate embedded with the ABH array has a better energy-gathering performance than the plates embedded with a large ABH or a single ABH. Figure 17: The acoustic-ABH plate coupling model. Figure 18: Sound pressure level of plates embedded with ABH array with/without damping layers. 2. For the plate embedded with five ABHs in this paper, the power flow density at the ABH central area is much higher than that of other areas. However, under certain excitation force, the energy focalization effect of the plate embedded with the ABH array is related to the positions of ABH pits. 3. The average vibration power density ratios at different circular sections in ABH are greater than 1, and the maximum value can reach hundreds when excitation frequency is near the natural frequency of plate, which indicates a significant energy focalization effect. In addition, the energy focalization effect is not obvious when the radius of the circular section is larger than 0.03 m, which provides a reference for using damping layers more effectively. 4. For the ABH plate in this paper, damping materials can change the plate energy distribution in the frequency domain and reduce the frequency where ABH starts to work. Results also show that the uniform damping layers have a better energy dissipation effect than varying thickness damping layers. 5. The radiated sound power level of the ABH plate with damping layers is reduced by 1-4 dB compared with the ABH plate without damping layers. In this paper, only the characteristics of the plate embedded with the ABH array in a certain excitation frequency band were investigated. However, in fact, the analysis method in this paper can be used in any excitation frequency band. In addition, experimental test to verify the vibration and noise control performance of the plate embedded with the ABH array is ongoing. Future work will explore the optimization of the ABH array for optimal noise and vibration control and weight minimization, including the ABH position, size, and ABH cell parameters. Conceptualization, X.L., Y.S. and T.H. ; methodology, X.L. ; software, H.L. ; validation, H.L., Y.B. and J.Y. ; formal analysis, H.L. ; investigation, H.L. ; resources, Y.B. ; data curation, H.L. ; writing--original draft preparation, H.L. ; writing--review and editing, X.L. ; visualization, H.L. ; supervision, X.L. ; project administration, H.L. ; funding acquisition, X.L. All authors have read and agreed to the published version of the manuscript. This research was funded by the Natural Science Foundation of China, grant number 11972003. Not applicable. The authors would like to thank the financial support provided by the Natural Science Foundation of China (Grant No. 11972003). The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Appendix AFigure A2: Power flow transmission graphs at an excitation frequency of 1050 Hz in x–y plane: (**a**) ABH(i) with BC 1, (**b**) ABH(ii) with BC 1, (**c**) ABH(iii) with BC 1, (**d**) ABH(i) with BC 2, (**e**) ABH(ii) with BC 2, (**f**) ABH(iii) with BC 2.