An optimization problem for maximum vibration suppression in reconfigurable one dimensional metamaterials

Acoustic metamaterials have already been shown to be effective for vibration reduction and control. Local resonances in the metamaterial cause waves at frequencies within band gaps to become evanescent, thus preventing wave propagation through the material. Active and adaptable local resonances enables the band gaps to be shifted in frequency and increased in bandwidth. Since metamaterial local resonances are usually composite, methods to specify optimal component configurations are helpful for passive metamaterials and almost necessary for adaptable metamaterials, where the metamaterial must be reconfigured for optimal performance at various frequency ranges. To assess band gap locations and bandwidths for metamaterials, a wavenumber spectrum is commonly computed. Commonly, a parameter study of adaptable unit cell variables will be performed to assess optimal configurations of adaptable metamaterials. In this paper, the complex wavenumber is proposed as a direct optimization objective for reconfiguration of active adaptable acoustic metamaterials for maximum vibration suppression at a frequency range of choice. By directly maximizing the imaginary part of the wavenumber, associated with wave attenuation, the unit cell configuration maximum vibration suppression can be obtained for an operating frequency of choice. Additionally, since the optimization problem requires constraints for feasible solutions and the example active piezoelectric metamaterial system shown here is electrically unstable at some configurations, we also explore an experimental method for bounding the optimization problem. Numerical results of the optimization problem are presented.