Influence of nonlinear subunits on the resonance frequency band gaps of acoustic metamaterial

Recently, a significant attention has been directed toward so called 'acoustic metamaterials' which have large similarity with already-known 'electromagnetic metamaterials' which are applied for elimination of the electromagnetic waves. The stop of electromagnetic waves is realized with the negative refractive index, negative permittivity and negative permeability. Motivated by the mathematical analogy between acoustic and electromagnetic waves, the acoustic metamaterials are introduced. It was asked the material to have negative effective mass. To obtain the negative effective mass, the artificial material, usually composite, has to be designed. The basic unit is a vibration absorber which consists of a lumped mass attached with a spring to the basic mechanical system. The purpose of the unit is to give a band gap where some frequencies of acoustic wave are stopped. We investigated the nonlinear mass-in-mass unit excited with any periodic force. Mathematical model of the motion is a system of two coupled strongly nonlinear and nonhomogeneous second-order differential equations. The solution of equations is assumed in the form of the Ateb (inverse beta) periodic function. The frequency of vibration is obtained as the function of the parameters of the excitation force. The effective mass of the system is also determined. Regions of negative effective mass are calculated. For these values the motion of the forced mass stops. It is concluded that the stop frequency gaps are much wider for the nonlinear than for the linear system. Based on the obtained parameter values, the acoustic metamaterial could be designed.