A novel composite structure composed of acoustic black hole (ABH) and piezoelectric shunt damping (PSD) unit is proposed in this paper. For the ABH-PSD composite structure, piezoelectric patches connected with shunting circuits are stuck on the surface of the ABH. The equivalent elastic modulus of the piezoelectric patch with shunting circuits is derived according to the piezoelectric equation. The finite element method (FEM) is used to analyze the displacement transmission characteristics of the ABH-PSD composite structure. Results show that the displacement transfer response of the structure at the corresponding bandgap is significantly attenuated. The electromagnetic oscillation bandgap can be regulated by changing the parameters of electrical components. The ABH-PSD composite structure proposed in this paper provides a new method for the bandgap regulation and vibration suppression of thin- walled structures.