## 1 Introduction  
Acoustic black holes (ABHs) are a unique type of acoustic structure, defined by a continuous reduction in material thickness along the length of the structure, following a power-law distribution. This design leads to a gradual slowing down of the phase velocity of waves as they move towards the end of the ABH. In theory, when the thickness reaches zero, the wave velocity should drop to zero as well, preventing the waves from spreading further and achieving nearly perfect sound isolation [1, 2, 3]. The applications of ABHs are broad, including reducing mechanical vibrations, enhancing the efficiency of acoustic devices, and serving as energy conversion devices that convert kinetic energy into electrical energy [4, 5, 6, 7, 8, 9]. While ABHs are lauded for their exceptional theoretical capabilities, the challenge of crafting a perfect ABH for real-world applications is fraught with difficulties. These challenges stem from not just the constraints of manufacturing accuracy and the inherent properties of materials, but also from the requirement to preserve a certain level of truncation thickness, which can negatively impact the system's efficiency. Two main strategies have been recognized for using ABHs to reduce vibrations and noise. The first strategy involves altering the geometric design of a critical structural component, such as a beam [10; 11; 12], plate [13; 14; 15], rotating disk [16], or shell [17], to create a segment that can accommodate an ABH. Although integrating ABHs can lead to weight reduction, implementing this approach in practical settings is complicated by the potential compromise to the structural strength. To successfully integrate ABH technology into practical scenarios, it is crucial to develop practical strategies that address current limitations. An alternative and promising approach is to incorporate an ABH layer on the outside of the main structure, rather than integrating it internally. This idea eliminates the need to change the thickness of the main structure, thus minimizing any risks to its structural integrity. The resulting hybrid structure exhibits enhanced vibration damping capabilities, which can significantly improve its operational performance [18; 19; 20; 21]. However, the additional ABHs discussed above are typically considered as separate or multiple absorbers, without considering their potential for periodic arrangement. In reality, the periodic arrangement of additional ABHs offers the unique advantage of activating both localized resonance and Bragg scattering bandgaps simultaneously. Furthermore, their strong damping characteristics lead to a substantial reduction in vibrations across the passbands [22; 23; 24]. Given the intricate composite structure found in additional ABHs, the existing literature suggests that the finite element method (FEM), when combined with experimental studies, is virtually the sole technique that has been utilized. In fact, studying individual ABH units within periodic structures is very helpful for enhancing the overall damping characteristics and is still worth further investigation. To address the challenge of truncation, studies have indicated that the application of a thin layer of viscoelastic damping material to the ABH region is crucial for energy extraction from the system [3], as supported by further research [25; 26; 27]. Recent investigations have revealed that the incorporation of damping layers can significantly enhance the acoustic emission efficiency of an ABH panel, primarily due to the additional rigidity these layers offer [28]. The inclusion of damping layers is essential for achieving the optimal ABH effect, which can operate independently [29] or in tandem with other constraints [30]. However, excessive use of damping layers has been identified as potentially detrimental. For example, the addition of a coating layer could introduce excessive stiffness to the underlying structure, thereby altering the expected ABH effect [31].