Analytical Investigation of Electromechanical Hierarchical Metamaterials for Vibration Attenuation and Energy Harvesting

Abstract

This work presents a theoretical study of outward and inward hierarchical metamaterials. Hierarchically configured multiple electromechanical resonators with shunt circuits are implemented, maintaining the same overall mass as that of a comparable single resonator metamaterial. The governing equations of motion for the outward and inward hierarchical configurations are derived. Dispersion relations are determined for each configuration with varying system parameters to identify key design parameters and assess their impact on the system’s dynamic behavior. Furthermore, outer mass displacement transmissibility and normalized total power output of finite chain hierarchical metamaterials are compared to observe vibration attenuation and energy harvesting capacity. The results reveal that the band structure of the hierarchical electromechanical metamaterials depends on the configuration type, the resonator masses, the electromechanical coupling coefficient, and the resistance of the shunt circuit. The first-order hierarchy offers a greater total band gap width, increased bandwidth, and greater flexibility in tuning the band structure. Finite chain transmissibility analysis demonstrates that, compared to the baseline performance of the zero-order hierarchy, the first-order hierarchy exhibits superior abilities in vibration attenuation and energy harvesting for the same total mass. The ideal design requires careful consideration of the resonator masses and their configuration, electromechanical coupling coefficient, and resistance of the shunt circuits. This theoretical work provides a foundation for designing lightweight hierarchical metamaterials for simultaneous vibration attenuation and energy harvesting.