Dynamics of mechanical and electromechanical locally resonant metastructures

Abstract

Locally resonant (LR) elastic/acoustic metamaterials enable bandgap formation at wavelengths much longer than the lattice parameter for low frequency vibration/sound attenuation. Mechanical LR metamaterials are based on unit cells hosting mass-spring elements as resonating components, whereas electromechanical LR metamaterials leverage piezoelectric unit cells shunted to inductive circuits. Existing modeling and analysis approaches for LR metamaterials have been mostly focused on band structure dispersion analysis for waves propagating in an infinite metamaterial that comprises a perfectly periodic lattice arrangement. By neglecting the effects of boundary conditions, these models are only valid for high-frequency applications, and cannot easily be extended to LR metamaterials operating at low frequencies. Thus, for real-world implementation of LR metamaterial concepts, the formation of LR bandgaps in infinite metamaterials must be reconciled with the interactions between the local resonators and the vibrational modes of finite structures. Furthermore, because LR bandgaps do not rely on phenomena such as Bragg scattering, it is not necessary to assume a periodic lattice arrangement. To address these research questions, this work establishes a general modal analysis framework for analyzing both mechanical and electromechanical LR metastructures (i.e. LR metamaterial-based finite structures with specified boundary conditions), which explicitly obtains the LR bandgap in closed form under the assumption of an infinite number of resonators. Because the analysis is based on a finite structure, it leads to new insights regarding the required number of resonators, resonator placement, and mode shapes of the metastructure, while still yielding the intended metamaterial-type performance for a sufficient number of resonators. Theoretical and experimental developments are presented for (1) general mechanical metastructures (spanning from Euler-Bernoulli beams to Kirchhoff plates); (2) electromechanical metastructures based on piezoelectric bimorphs; (3) hybrid mechanical-electromechanical metastructures that leverage LR dynamics; (4) programmable electromechanical metastructures leveraging synthetic impedance shunt circuits; and (5) space- and time-modulated electromechanical metamaterials. It is shown that the bandgap size in the mechanical LR metastructures depends on the added mass due to the resonators, while the bandgap in electromechanical LR metastructures depends on the system-level electromechanical coupling. A root locus-based pole/zero placement method is developed to design more general shunt circuits for electromechanical metastructures to obtain any desired band structure, and experimental validations are presented for a “programmable" piezoelectric bimorph beam utilizing digital synthetic impedance circuits. Lastly, a fully coupled electromechanical framework is developed for piezoelectric metamaterials with spatiotemporal periodic shunt impedance, demonstrating the capability of these structures to break wave reciprocity.