A nonlinear resonator with inertial amplification for very low-frequency flexural wave attenuations in beams

Although elastic metamaterials in a subwavelength scale can control macroscopic waves, it is still a big challenge to attenuate elastic waves at very low frequency (a few tens Hz). The main contribution of this paper is to develop a high-static-low-dynamic-stiffness (HSLDS) resonator with an inertial amplification mechanism (IAM), which is able to create a much lower band gap than a pure HSLDS resonator. The nonlinear characteristics of a locally resonant (LR) beam attached with such new resonators are also explored. The band gap of this LR-IAM beam is revealed by employing transfer matrix method and validated by numerical simulations using Galerkin discretization. It is shown that a very low-frequency band gap can be formed by tuning the net stiffness of the resonator towards an ultra-low value. In addition, the nonlinearity, arising from the restoring force of the resonator, the damping force and effective inertia of the IAM, gives rise to an intriguing feature of amplitude-dependent wave attenuation, which could potentially act as a switch or filter to manipulate flexural waves.