Dynamics and control of band gaps in a mass-in mass metamaterial model with an extra attached mass

It is shown that the continuum limit of the metamaterial mass-in-mass model with additional attached mass describes not only the appearance of the additional band gap but also variations in the width and the position of the band gaps. These variations are governed by the key parameter-the stiffness ratio of the attached masses. Numerical study of periodic boundary excitation of the harmonic waves reveals suppression of the harmonic waves for the frequencies lying inside the both band gap areas. Also it is found that harmonic waves recover differently for the frequencies below and above the band gap values. The control mechanism is developed based on the abrupt variation of the stiffness ratio. It gives rise to the arising of phase shift of the wave, its suppression or recovery of the previously suppressed harmonic wave. Nonlinear long wavelength generalization of the model results in obtaining the model equation whose coefficients differ from those of the usual nonlinear mass-in-mass model. It gives rise to propagation of the localized wave with another amplitude and velocity.

Automated summary

The continuum limit of the metamaterial mass-in-mass model with additional attached mass reveals the appearance, width, and position variations of the band gaps, which are governed by the stiffness ratio of the attached masses. Numerical study shows the suppression of harmonic waves within the band gap areas and different recovery patterns for frequencies below and above the band gap values. A control mechanism based on the abrupt variation of the stiffness ratio is developed to induce phase shift, suppression, or recovery of suppressed harmonic waves. The nonlinear long wavelength generalization of the model leads to the propagation of localized waves with different amplitude and velocity.