Bandgap enhancement of a piezoelectric metamaterial beam shunted with circuits incorporating fractional and cubic nonlinearities

Abstract

Nonlinear metamaterials have been widely studied for their potential to broaden the bandgap for vibration suppression. However, the bandgap widening effect induced by nonlinearity vanishes when the excitation amplitude is insufficient to trigger the nonlinear behavior. In order to broaden the applicable range of excitation amplitude for nonlinear metamaterials, a novel design of a piezoelectric metamaterial beam shunted with combined nonlinear circuits is presented in this work, which incorporates fractional and cubic nonlinearities in circuits to achieve bandgap enhancement under both low and high excitation amplitudes. Theoretical and numerical models are established for predicting the nonlinear dynamics behaviors of the piezoelectric metamaterial beam, and their computational results are in good agreement. The amplitude-dependent bandgap of the metamaterial beam induced by combined nonlinearity is analyzed through the effective bending stiffness of a unit cell and validated through the frequency response of a piezoelectric metamaterial beam. It is found that based on the complementary nature of fractional and cubic nonlinearities in the amplitude-dependent characteristics, combined nonlinearity of the circuits can effectively improve the bandgap width of the beam under both low and high excitation amplitudes. In addition, the effects of the two nonlinearities on the amplitude-dependent characteristics of the bandgap are studied, and the interplay between the two nonlinearities within the combination is analyzed. The results show that the amplitude-dependent characteristics of the combined nonlinear design can be precisely tuned by adjusting the nonlinear coefficients of the circuits. Overall, a piezoelectric metamaterial beam with both fractional and cubic nonlinearities in circuits possesses superior vibration control performance than those with single nonlinear counterparts in circuits.