## 6 Conclusions  
This paper investigates the impact of changing the geometric characteristics of damping layers on the energy dissipation capabilities of a symmetric ABH beam. The utilization of the BP algorithm is proposed to optimize the shape variation of the damping layer. Initially, a semi-analytical solution for the ABH beam's vibration behavior is derived using the GEM, and the accuracy of this solution is confirmed by comparing it against results from the FEM. Next, leveraging the semi-analytical solution, an analysis of the impact of damping layer parameters, including position, thickness, and shape, on the ABH effect is conducted. Under the premise of maintaining a constant mass of the damping layer, a thickness variation function with a good energy dissipation effect is preliminarily selected. Subsequently, a BP neural network is trained based on the database generated by the semi-analytical solution to optimize the thickness variation function. This study reveals that the damping layer's geometry has a more intricate impact on performance than previously understood. The BP algorithm stands out as a powerful tool for enhancing the performance evaluation of ABH structures amidst a variety of factors. The optimization analysis conducted using the BP neural network suggests that achieving an effective ABH effect does not require the application of a damping layer across the entire ABH section. The optimal configuration involves maximizing the damping layer thickness at the ABH tip, followed by a rapid decrease in thickness as one moves away from the tip. It is deemed unnecessary to apply a damping layer in areas that are far from the tip.  
By appropriately stretching the ABH beam along the truncation thickness while keeping the thickness constant, this paper also optimizes the thickness of the damping layer for the modified ABH beam structure. This study finds that with the same mass of damping layer attached, the modified ABH beam structure can exhibit better energy dissipation effects. This is a novel approach that further enhances the energy dissipation performance of ABH beams without reducing the truncation thickness. Based on the research approach of this paper, the length of the stretched section in the ABH beam can be further optimized using the BP algorithm to achieve better energy dissipation effects. The research in this paper helps to gain a deeper understanding of the impact of the damping layer on the energy dissipation effect of ABH beams.  
Conceptualization, B.Z.; methodology, B.Z.; software, J.Z.; investigation, L.O. and J.Z.; writing--original draft preparation, L.O. and J.Z.; writing--review and editing, L.O. and  
Figure 20: The comparison of the impact on the ABH effect by the thicknesses of the damping layer set according to functions \(h_{dp}(x)\) and \(h_{412}(x)\), as defined in Equations (34) and (35), respectively.  
B.Z.; visualization, J.Z.; supervision, B.Z.; project administration, B.Z.; funding acquisition, B.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China grant number 11672185.  
Not applicable  
Data Availability Statement: Data are contained within the article.  
The authors declare no conflicts of interest.