## 1 Introduction  
In the past two decades, as a new type of passive vibration reduction and noise attenuation technology, the acoustic black hole (ABH) was studied a lot in terms of theoretical development, modeling and simulation, and analysis of vibration and acoustic characteristics, and it is expected to provide a new solution to the problems of vibration and noise in engineering [1, 2, 3, 4, 5]. [PERSON] found in a thin wedge-shaped structure that if the thickness of the structure is reduced in a certain power exponential form (the power exponent is not less than 2), the velocity of the flexural wave in the structure will gradually decrease as the thickness of the structure decreases [6]. Ideally, the velocity of the flexural wave can be reduced to zero to achieve zero reflection. [PERSON] first proposed the concept of an \"acoustic black hole\" to describe the above phenomenon and applied the ABH to beam and plate structures. [PERSON] used the geometric acoustic method [7, 8] and experimental method [9] to study the characteristics of ABH structure, such as flexural wave propagation, vibration suppression, and sound radiation reduction. The ABH structure can achieve the concentration and manipulation effects of flexural waves in solid structures in a wide frequency range, thereby achieving vibration reduction and noise attenuation; in addition, it also contributes to the light-weighting of the structures [10]. There is much research on the vibration characteristics of the ABH structures. [PERSON] et al. established a semi-analytical model by using Mexican wavelet function to fit the sharply changing wave group at the edge of the ABH to analyze the vibration characteristics of one-dimensional ABH structure and studied the influences of damping layer thickness and damping layer stiffness on the dynamic response of the ABH structure [11; 12]. [PERSON] et al. established a dynamic model of the truncated one-dimensional ABH beam containing passive constrained viscoelastic layers (PCVL) using the semi-analytical method, and both the analysis and the experimental results show that the truncated one-dimensional ABH beam with PCVL can achieve better vibration attenuation effects than the one covered with a viscoelastic damping layer [13]. [PERSON] et al. also analyzed the influence of ABH parameters on the vibration characteristics of the two-dimensional ABH circular plate based on the semi-analytical method [14]. Additionally, they studied the vibration response of the new ring-shaped two-dimensional ABH, and the results showed that the ring-shaped ABH exhibited excellent vibration isolation performance over the entire frequency range [15]. [PERSON] et al. established a semi-analytical model to analyze the vibration of a two-dimensional ABH plate and used the Daubechies wavelet function to express the lateral displacement of the plate in enhancing the effectiveness of the Rayleigh-Ritz method [16]. Based on the approximate analytical method, [PERSON] et al. analyzed the frequency response characteristics of a square plate containing a two-dimensional square ABH [17] and a circular plate containing a two-dimensional circular ABH [18] and investigated the influence of the damping layer. The result shows that the cross mobility reaches 17 dB attenuation at the resonance peaks. [PERSON] arranged multiple two-dimensional ABHs in a thin plate with a certain periodicity and analyzed the vibration response of the plate by establishing a finite element model [19]. [PERSON] et al. evaluated the vibration performance of the thin-walled ABH structure under transient and steady-state excitation through numerical simulation and experimental methods [20], and both the numerical and experimental results show that the proposed structure can achieve an efficient and broadband vibration attenuation performance. [PERSON] et al.