## 1 Introduction  
In the last decade, acoustic black holes (ABHs) have become a research hotspot in the field of vibration and noise reduction due to their excellent modulation effect on elastic waves. ABHs realize the asymptotic change of structural impedance by power-law tailoring the structure, and then realize the energy aggregation and manipulation of waves [1]. Research scholars have studied ABHs from various perspectives, such as vibration control, acoustic radiation reduction, and energy harvesting [2, 3]. The most important feature of ABHs is the modulation of flexural waves [4]. The ABH changes the phase and group velocities of flexural waves in the structure through the change in structural impedance, which realizes the aggregation of waves in the local region of the structure. Conventional ABH units can dissipate vibrations very effectively at high frequencies [5], but limited by modern manufacturing process levels and structural features of ABHs, the ABH truncation thickness and ABH cutoff frequency prevent ABH units from producing effective attenuation of low-frequency vibrations [6]. To improve the damping and noise reduction capability of ABH structures in the low frequency range, [PERSON] and [PERSON] [7] first proposed the ABH elastic metamaterial structure by arranging the double-leaf-type ABH structure on a beam in a periodic pattern, thus generating multiple bandgaps (BGs) in the frequency range of 0-30 kHz. Subsequently, [PERSON] and [PERSON] [8] proposed an embedded double-leaf ABH metamaterial plate to achieve the directional attenuation of flexural waves in the low frequency range in a plate-type structure. [PERSON] et al. [9, 10, 11, 12] presented a two-lobe ABH beam structure with a spatial V-bend, a one-dimensional (1D)two-lobe ABH beam with nested damping units, and a 1D two-lobe ABH beam with a periodic array of large and small units, all of which have some vibration attenuation at low frequencies. [PERSON] et al. [13] investigated two-dimensional (2D) periodic ABHs and found that the 2D periodic arrangement of the square lattice thin plates embedded in ABHs have dispersion properties such as bi-refraction and zero-group velocity in the fundamental mode. [PERSON] et al. [14] applied embedded ABH features to panel structures and found that 2D ABH arrays can reduce the vibration and acoustic radiation from beams and plates. Although the above work obtained a low-frequency BG, the width of the BG needs to be further widened. This is because the ABH structure is limited by its cutoff frequency, and it is difficult to extend its structural BG to the low frequency range. Although there are BGs that extend into the low frequency range, the BG width is narrow, and practical application scenarios are limited by the narrow bandwidth. Therefore, widening the width of the low-frequency BG of ABH structures is of great importance for the study of ABHs. Radial elastic metamaterials are ring-like structures with omnidirectional BG properties, and the structures are periodically arranged in the radial direction [15; 16; 17]. [PERSON] et al. [18] studied a 2D column-shell circular plate-type structure and obtained a two-component disk-like structure with a complete BG by adding scatterer material in the radial and circumferential directions to the substrate. [PERSON] et al. [19] studied a two-layer radial phononic crystal plate with a low-broadband BG that could significantly modulate the omnidirectional BG by crystal slip in the radial direction and perpendicular to the plate thickness. The high efficiency, frequency selectivity, and omnidirectional wave shielding characteristics of radial elastic metamaterials for elastic wave attenuation provide superior wave shielding and modulation characteristics compared to conventional metamaterials [20], making them have great potential applications in the field of ultra-low-frequency vibration/shock reduction.