Objective <p>This study aims to delve into the functional gradient (FG) acoustic black hole and its vibration characteristics. Based on the nonlocal strain gradient theory, we constructed a comprehensive mathematical model focusing on size-dependent functional gradient acoustic black holes (FG-ABH) and conducted a detailed analysis.</p> Methods <p>The nonlocal strain gradient theory was employed to build the comprehensive mathematical model of FG-ABH, and equations for the wavenumber and reflection coefficient ratio were derived; the Rayleigh-Ritz method was used to calculate the natural frequencies and vibration modes of FG-ABH.</p> Findings <p>The research results show that increasing μ significantly reduces α, indicating a stronger energy constraint. Additionally, a higher material power-law index κ also reduces α. Changes in μ have a significant impact on the effective stiffness of FG-ABH, while changes in κ alter the volume fraction of alumina, thereby affecting the natural frequencies and vibration modes.</p> Practical Implications <p>This study provides important theoretical insights and practical guidance for the design and optimization of micro-nano structured acoustic black holes.</p> Value of the Paper <p>In previous research, we conducted a preliminary analysis of the scale dependence of microstructured acoustic black holes. However, we did not thoroughly explore the functional gradient (FG) acousticblack hole and its vibration behavior.</p>