<p>In recent years, acoustic black hole (ABH) structures, as a new type of acoustic structure, can effectively reduce the propagation speed of bending waves in the structure. Due to the influence of high amplitude vibrations, ABH structures are more susceptible to damage, such as through cracks in parts. The detection of structural cracks has always been an important issue in engineering. Therefore, this paper combines the symplectic wave propagation method and linear spring method to calculate the forced vibration response of the cracked variable thickness structures of beams and plates, such as ABH beams and ABH plates. The effectiveness of the proposed method is verified by comparing it with the finite element method. Research has shown that compared to the finite element method, the method proposed in this paper not only has higher computational accuracy, but also higher computational efficiency. Then, the experiment is conducted on the transverse transmission rate of the ABH beam with a crack, indicating that this method still has good reliability in practical engineering applications. Finally, the forced vibration response of ABH structures with different crack parameters are obtained by changing the depth, width, location and number of cracks. Through parametric analysis, the properties of cracks can be determined by the peak and phase information of the frequency response curve, providing a new approach for crack detection in engineering applications.</p>