Mathematical modeling and analysis of a meta-plate for very low-frequency band gap

Abstract

• A meta-plate with high-static-low-dynamic stiffness local resonators is proposed. • Mathematical modeling of the meta-plate is established and solved by plane-wave expansion method. • The band gaps are predicted analytically and validated by numerical simulations. • The meta-plate with HSLD resonators can generate very low-frequency band gaps. Aiming to attenuate very low-frequency flexural wave in a thin plate, this paper proposes a meta-plate model by periodically attaching high-static-low-dynamic-stiffness (HSLDS) resonators onto the thin plate. The HSLDS resonator consists of a linear spring with a negative stiffness mechanism (NSM) in parallel, and thus its stiffness can be adjusted to any low values within a range from zero to the stiffness of the linear spring. Using the plane-wave expansion (PWE) method, and considering the linearized stiffness of the resonator, the dispersion relation of the meta-plate can be derived and the band structure can be obtained. A dynamic model of the meta-plate with the original nonlinear stiffness under external excitations is also established. The numerical simulations are carried out by the Galerkin method to evaluate the band structure and study the propagation of the flexural wave along the meta-plate with different azimuth angles. The analytical results exhibit good agreement with the numerical ones, which indicates that the proposed meta-plate can create a very low-frequency band gap.