A semi-active metamaterial beam with electromagnetic quasi-zero-stiffness resonators for ultralow-frequency band gap tuning

Introducing a negative-stiffness mechanism (NSM) into a traditional linear resonator to form a high-static-low-dynamic-stiffness (HSLDS) resonator is an ideal way to create a low-frequency band gap. However, with the decrease in frequencies of the band gap, the band width narrows, which could hinder the application of the metamaterials for attenuating ultralow-frequency elastic waves. In this paper, a regulatory mechanism (RM) constituted by an electrically charged coil and a magnet ring is introduced into an HSLDS resonator to devise a semi-active quasi-zero-stiffness (QZS) resonator. With these semi-active resonators attached onto a beam periodically, a semi-active metamaterial beam (meta-beam) is realized. The expressions of both the restoring force and the stiffness of the semi-active resonator are derived firstly, and then the theoretical dispersion relation and the band structure are obtained by the transfer matrix method. Finally, by establishing and then numerically solving the equation of motion of the semi-active meta-beam, the wave transmissibility is acquired and utilized to validate the theoretically predicted band structure. The analytical and numerical results show that the band gap can be effectively tuned by the RM, which enables excellent wave manipulation in an ultralow and wide frequency range.