Abstract This study investigates the dynamics of locally resonant (LR) nonlinear metamaterials with programmable properties. For a finite nonlinear metamaterial consisting of an array of periodically attached resonators on a cantilever beam with nonlinear stiffness, changes in the overall system parameters due to local resonators are investigated. The local resonators effectively change the beam’s linear and nonlinear stiffness in a specific frequency range, and this change can be tailored through reconfiguring the resonator parameters. To determine the relationship between the resonator parameters and the stiffness of the metamaterial-beam, a simplified reduced-order model (ROM) with nonlinear stiffness is developed for the metamaterial beam. The ROM can capture the behavior of the metamaterial beam in a certain frequency range. Analyzing the response characteristics of the metamaterial-beam, such as resonance peaks, and the amplitude-frequency dependence within this frequency range, parameters of the ROM are determined. The Harmonic Balance Method (HBM) is applied to solve the coupled beam-resonator equations, and derive the backbone curves of the undamped and unforced system. Then, the changes in the effective natural frequency and the cubic stiffness of the reduced-order model were investigated. The frequency response of the ROM is compared to that of the metamaterial beam to validate that the ROM can capture the nonlinear behavior of the metamaterial beam within the specified frequency range. This provides a solid theoretical foundation for the realization of nonlinear metamaterials with programmable properties.
Wu et al. (2026) studied this question.