## 5 Conclusions  
This study explored the band structure of hierarchical metamaterials containing electromechanical resonators shunted to resistor-loaded circuits for vibration attenuation and energy harvesting applications. The outward and inward hierarchical metamaterials were designed by configuring multiple electromechanical resonators hierarchically and integrating shunt circuits. The influence of the electromechanical coupling coefficients, resistances of the shunt circuit, and resonator mass ratios on the band structure of the zero-order and first-order hierarchies was examined. The dispersion of hierarchical metamaterials designed with strongly coupled electromechanical resonators shunted to an electric circuit was compared to that of hierarchical metamaterials with a resonator connected by springs and pure mechanical dampers. Finally, the displacement transmissibility and normalized total power output for finite-chain hierarchical configurations are analyzed. The study reveals that the local resonance band gaps of hierarchical electromechanical metamaterials are a function of configuration type, mechanical and electrical tuning parameters. The effects of electrical parameters are dependent on the coupling conditions. The first-order hierarchical configurations offer a broad total band gap width and a broad bandwidth compared to the zero-order hierarchy for the same total mass. The role of \(\beta_{1}\) in tuning the band structure of the first-order hierarchical metamaterials is paramount. The strongly and weakly coupled outward configuration generates a symmetric band structure about \(\beta_{1}=0.5\), allowing the band structure to be tuned on only one side of \(\beta_{1}\). In contrast, the strongly and weakly coupled inward configuration produces a lower frequency first band gap and a broad second band gap when \(\beta_{1}\) is much less than 0.5. In addition to the mechanical tuning parameter \(\beta_{1}\), the band structure of a strongly coupled hierarchical metamaterial is affected by the electromechanical coupling coefficients and resistance of the shunt circuits. As the coupling coefficient increases, the bandwidth of strongly coupled hierarchical metamaterials increases, and the passbands merge into a single curve. The width and location of the band gaps are substantially influenced by the resistance of the shunt circuit. The finite chain transmissibility showed that resistance of the shunt circuit significantly affects boundaries of high-frequency passbands. In weakly coupled hierarchical metamaterials, the band structure is unaffected by the electromechanical coupling coefficient and shunt circuit resistance, enabling energy harvesting without altering the band structure. Electromechanical damping showed strong peak attenuation and a shift of the band gap location compared to pure mechanical damping. The outer mass displacement transmissibility identified that the vibration attenuation performance of first-order hierarchical metamaterials is larger than the zero-order hierarchy. The inward configuration is effective for attenuating relatively lower and higher frequency waves, whereas the outward configuration is more suitable for achieving wave control where the inward configuration does not. The normalized total power output demonstrates that energy harvesting is strongly influenced by the configuration type and electromechanical tuning parameters, requiring careful selection of both to optimize the performance. From this, it is concluded that hierarchical electromechanical metamaterials provide a broader total band gap width, increased bandwidth, and a highly tunable band structure for vibration attenuation and energy harvesting applications, outperforming single-resonator metamaterials with the same total mass. Conceptualization, A.A.M., J.L. and O.B. ; methodology, A.A.M., J.L. and O.B. ; software, A.A.M. and J.L. ; formal analysis, A.A.M. ; investigation, A.A.M., J.L. and O.B. ; resources, O.B. ; data curation, A.A.M. ; writing--original draft preparation, A.A.M.