The bandgap adjustment and expansion is an important research direction in acoustic metamaterial fields. This paper designs a kind of acoustic metamaterial composed of periodically arranging oscillator unit cells, and proposes an effective way to adjust the central frequencies of the bandgap and widen the frequency range of the bandgap. First, we design an acoustic metamaterial unit cell constructed by the base structure, a mass block, and bending elastic beams connecting the mass block with the base structure; then, study the effect of pre-stresses applying to the bending elastic beams on the bandgap characteristics of the acoustic metamaterials composed of oscillator unit cells; finally, construct a kind of periodic acoustic metamaterial by combing three groups of oscillator unit cells with different pre-stresses, and realize the effective expansion of the bandgap. Our study shows that, this kind of periodic acoustic metamaterial has bandgap characteristics of low-frequency and broadband; applying pre-stresses to the bending elastic beams can make the bandgap shift towards low frequencies but will result in a narrow bandgap; combining oscillator unit cells with different pre-stresses can effectively widen the bandgap, overcoming the limitation of narrow bandgap due to applying pre-stresses.