## 1 Introduction
The influence of added mass is particularly relevant when the damping layer thickness is substantial compared to the ABH wedge's thickness near the tip area. Furthermore, applying a coating to the central part of an indentation in a less-than-ideal ABH plate, designed for a specific ABH thickness profile, could lead to an increase in the minimum thickness, thereby disrupting the intended energy concentration. By using laser-induced excitation combined with wave analysis techniques, the optimal damping layer thickness that minimizes wave reflections was determined experimentally [32]. It was found that an excessive amount of coating could counteract or even damage the intended energy focusing due to the dynamic disturbances it induces within the host structure, leading to a decrease in energy dispersion efficiency. These findings underscore the significance of a deliberate and strategic design approach for the placement of damping layers, which is vital for maintaining a balance between the opposing yet interconnected effects of the layers: enhancing damping on one hand, and increasing wave reflection on the other. Investigations have demonstrated that the effectiveness of damping layers in dissipating energy is significantly influenced by their geometric properties, material characteristics, and placement [33; 34]. To attain a more accurate representation of practical scenarios, it is imperative to account for the intricate interplay between the damping layer and the wedge with a power-law profile. The Ross-Unar-Kerwin (RUK) model [35] is commonly utilized to evaluate the impact of thin damping layers, assuming that their thickness is considerably less than that of the wedge. However, in real-world applications, even the thinnest damping layer's thickness might be similar to the wedge's tip. A significant concern is the increasing relevance of the added mass effect, which has been recognized in previous studies for layers with uniform thickness [36]. Enhancing the performance of a damping layer might require precise adjustments to its positioning and setup. This necessitates a more flexible methodology to address the layer's additional mass and stiffness implications. In [29], a semi-analytical technique was devised to examine the impact of varying the thickness of damping layers with constant mass on the system's damping loss factor across different linear distributions. The research indicated that altering the thickness of damping layers could offer a distinct avenue for optimizing energy dissipation efficiency. However, there remains a void in the field as comprehensive studies on this topic and the creation of critical optimization tools have not been adequately addressed. The present study zeroes in on an individual ABH element and utilizes a backpropagation (BP) neural network for an in-depth examination of the effects exerted by damping layers. Renowned in artificial intelligence circles, the BP algorithm boasts exceptional error tolerance, formidable learning capabilities, and steadfast performance. It stands out in managing intricate nonlinear dynamics, showcasing its adaptability and applicability across diverse scenarios [37; 38]. The investigation commences with the construction of a mathematical model for a symmetric ABH beam, leveraging Lagrangian mechanics. Subsequently, a semi-analytical solution is derived through the Gaussian expansion method (GEM) [17], validated by aligning it with FEM outcomes. A function is introduced to delineate the thickness gradient of damping layers. Following this, a BP neural network, informed by the semi-analytical solution data, is tasked with refining the damping layers' positioning, thickness, and geometry. The findings indicate that the damping layer's geometry has a more intricate impact than previously acknowledged. The BP algorithm emerges as a potent instrument for amplifying the performance analysis of ABH structures amidst a multitude of variables.