The growing demand for low-frequency, broadband vibration and noise suppression technologies in next-generation mechanical equipment has become increasingly urgent. Effective negative mass locally resonant structures represent one of the most paradigmatic classes of acoustic metamaterials. Their unique elastic wave bandgaps enable efficient suppression of low-frequency vibrations, while inherent nonlinear effects provide significant potential for the design and tunability of these bandgaps. To achieve ultra-low-frequency and ultra-broadband vibration attenuation, this study employs Duffing oscillators exhibiting negative-stiffness characteristics as structural elements, establishing a bistable nonlinear acoustic-metamaterial mechanical model. Subsequently, based on the effective negative mass local resonance theory, the perturbation solution for the dispersion curves is derived using the perturbation method. Finally, the effects of mass ratio, stiffness ratio, and nonlinear term on the starting and cutoff frequencies of the bandgap are analyzed, and key geometric parameters influencing the design of ultra-low vibration reduction bandgaps are comprehensively investigated. Subsequently, the influence of external excitation amplitude and the nonlinear term on bandgap formation is analyzed using numerical computation methods. Finally, effective positive mass, negative mass, and zero-mass phenomena within distinct frequency ranges of the bandgap and passband are examined to validate the theoretically derived results. The findings demonstrate that, compared to a positive-stiffness system, the bandgap of the bistable nonlinear acoustic metamaterial incorporating negative-stiffness Duffing oscillators shifts to higher frequencies and widens by a factor of $\sqrt{2}$. The external excitation amplitude F changes the bandgap starting frequency and cutoff frequency. As F increases, the starting frequency rises while the cutoff frequency decreases, resulting in a narrowing of the bandgap width. Within the frequency range bounded by the bandgap starting frequency and cutoff frequency, the region between the resonance frequency and cutoff frequency corresponds to an effective negative mass state, whereas the region between the bandgap starting frequency and resonance frequency exhibits an effective positive mass state. Critically, the bandgap encompasses both effective positive mass and negative mass regions, wherein vibration propagation is suppressed. Concurrently, a zero-mass state emerges within this structure, with its frequency precisely coinciding with the bandgap cutoff frequency. This study provides a theoretical foundation and practical guidelines for designing nonlinear acoustic metamaterials targeting ultra-low-frequency and ultra-broadband vibration and noise mitigation.

Traditional acoustic metamaterials with inherently fixed bandgaps face difficulty in satisfying the dynamic requirements of vibration isolation and vibration attenuation under varying operating conditions, which limits their practical engineering applications. Nonlinear acoustic metamaterials based on effective-negative-mass (ENM) mechanisms provide a promising approach for achieving adaptive vibration attenuation through bandgap regulation. In this study, a temperature-tunable nonlinear local-resonance acoustic metamaterial incorporating shape-memory-alloy (SMA) helical-spring oscillators is proposed. A positive-stiffness nonlinear acoustic metamaterial model is established by combining a nonlinear Duffing oscillator with SMA springs. Based on the ENM local-resonance theory, the governing equations are nondimensionalized, and the first-order perturbation solutions of the acoustic and optical dispersion branches are obtained using the perturbation method. The effects of temperature variation and stiffness ratio on the dispersion characteristics and the starting and cut-off frequencies of the bandgap are theoretically analyzed. The theoretical results show that, as the temperature decreases, the temperature-induced phase transformation of SMA reduces the stiffness of the SMA springs, causing both the starting and cut-off frequencies to decrease and the bandgap to shift toward lower frequencies and become narrower. Furthermore, a five-period finite structure model is established, and the numerical vibration-transmission results verify the theoretical predictions of temperature-dependent bandgap tuning. The dynamic responses further reveal that effective-positive-mass and ENM regions coexist within the bandgap, while a zero-effective-mass state occurs near the bandgap cut-off frequency. These findings demonstrate that the proposed SMA-based nonlinear acoustic metamaterial can actively regulate the low-frequency vibration attenuation range through temperature control, providing a theoretical basis for the design of adaptive acoustic metamaterials for vibration isolation and noise reduction.