## 1 Introduction  
Over the past decade, the structure of the acoustic black hole (ABH) has been a frontier topic in the field of sound and vibration, which is a novel vibration and noise reduction technique [1; 2; 3], and it can also provide new methods for improving the vibration and noise performance of machines or components in engineering [4; 5]. The ABH structure may accomplish flexural wave manipulation and concentration effects for solid structures in broadband, resulting in vibration reduction and noise attenuation, as well as contributing to structural light-weighting [6]. The acoustic black hole structure has a relatively simple geometric structure, and is easy to manufacture, which can be easily embedded into existing mechanical components or mechanical systems in engineering. [PERSON] derived the motion equation of the flexural wave in a thin wedge-shaped structure, which can be called one-dimension acoustic black hole, and found that the wave velocity in the structure can be reduced to zero to achieve unreflecting of the flexural wave in an ideal case [7; 8]. [PERSON] also carried out experiments to investigate the vibrational and acoustical properties of ABH structures [9; 10]. The existing research on ABHs is more concentrated on the vibrational characteristics of different forms of acoustic black hole structures [11; 12; 13; 14], while the acoustical and vibro-acoustic characteristics of ABH structures are also an important research direction. [PERSON] et al. conducted experiments and found that the noise in the engine compartment will be decreased by 6.5 dB after the car engine housing is embedded with the two-dimensional ABH [15]. [PERSON] investigated the influence of damping layer diameter on the emitted sound of an aluminum plate with 20 two-dimensional damped ABHs throughout the whole frequency range [16]. [PERSON] also carried out experimental and numerical study on the transmission loss (TL) of plates embedded with ABHs [17], and used a wavenumber transformation analysis method to study the vibro-acoustic coupling characteristics of the ABH system and calculated the radiation efficiency of the embedded ABH plates [18]. [PERSON] et al. developed a 2D daubechies wavelet model for predicting the sound radiation from plates embedded with a circular ABH [19] and proposed a topology optimization method for optimizing the layout of the damping layer of ABH plates to decrease sound radiation [20]. [PERSON] et al. used the transfer matrix method (TMM) to develop analytical solutions for sound radiation of a beam embedded with an ABH characteristic [21]. [PERSON] et al. developed an acoustic radiation FEM model that takes plate-cavity coupling into account, and asserted that the damping enhancement and coupling strength decreasing between the plate and the cavity are two strategies for reducing intracavity noise [22]. [PERSON] et al. introduced a wavenumber domain approach (WNDM) for analyzing the vibro-acoustic coupling characteristics and inner noise cancellation mechanism of a pentahedral cavity enclosed by an elastic plate with a 2D ABH [23]. [PERSON] et al. investigated the advantages of embedding ABHs on plates for noise suppression between cavities by using statistical modal energy distribution analysis (SmEdA) to assess the transmission loss of ABH panels between two air cavities [24]. [PERSON] also developed a semi-analytical approach for calculating the acoustic power, radiation efficiency, and far-field acoustic pressure of ABH shells, and compared them to those of uniform thickness shells [25]. [PERSON] et al. used FEM calculations to evaluate the sound radiation of a finite plate with four tunneled double-leaf ABH cells in order to minimize low-frequency sound radiation [26]. To increase low frequency performance, [PERSON] et al. evaluated the impact of ABHs implanted in plate-like structures on both vibration and structural radiated sound [27]. [PERSON] et al.