Abstract: Magnetic force has typical nonlinear characteristics, and magnetic springs have received extensive attention from scholars for their excellent vibration damping properties. The nonlinear magnetic force expression of the equivalent single cell is derived by analyzing the static properties of the magnetic spring and fitting and combining the polynomial magnetic force-displacement curves. Then the magnetic springs are connected in series to form a periodic structure with local resonance beams attached. The periodic structure is equivalent to a spring-vibrator system with typical nonlinear characteristics, and its dispersion equations in different states are organically unified by theoretical derivation of formula using the perturbation approach. From the theoretical point of view, the mechanical evolution of the bandgap characteristics at different amplitudes is elucidated. That is, the local resonance bandgap is not only related to the mass ratio and stiffness ratio, but also related to the excitation amplitude and the soft and hard nonlinear stiffness. The soft nonlinear stiffness, large mass ratio, large stiffness ratio and large amplitude are favorable to broaden the width of the local resonance bandgap. The local resonance bandgap of the periodic structure is verified numerically and experimentally. New ideas are provided for the study of bandgap characteristics of nonlinear structures.