## 3 Band Structure of Hierarchical Metamaterials
### Effect of Electromechanical Coupling Coefficients and Shunt Circuit Resistances
From this, it is inferred that strong electromechanical coupling merges the passband branches, broadens the bandwidth, and reduces the local resonance peak attenuation in hierarchical metamaterials. The wavenumber components of the strongly coupled hierarchical metamaterial at different resistances of the shunt circuits are detailed in Figure 5d-f. In some cases, the zero-order hierarchy attenuation curve splits into two regions within a single band gap for certain shunt circuit resistance values, as depicted in Figure 5d. At a shunt circuit resistance of \(10^{3}\) ohms, the horizontal black line divides the attenuation curve. With this shunt circuit resistance, the attenuation curve shows an overlapping peak. The separation of the attenuation curve at a shunt circuit resistance of \(10^{5}\) ohms becomes more evident, revealing distinct peaks. However, increasing the resistance value of the shunt circuit reduces the peak attenuation constant. When the resistance of the shunt circuit is \(10^{7}\) ohms, as shown by the blue curve in Figure 5d, the split in the attenuation curve disappears, and the peak attenuation increases once again. In the strongly coupled zero-order hierarchy, along with these effects, increasing the shunt circuit resistance pushes the local resonance band gap boundaries upward. Although an attenuation curve split is observed in the zero-order hierarchy, the real part of the wavenumber indicates the presence of a single local resonance band gap. The width of the local resonance band gap also exhibits a slight increase. The effect of the shunt circuit resistance on the wavenumber components of the first-order hierarchy is demonstrated in Figure 5e,f. The split of the attenuation curves is not observed in the first-order hierarchy; however, the passband branches merge as the resistance of the shunt circuits increases. The higher peak attenuation in the first-order hierarchy indicates stronger damping compared to the zero-order hierarchy. The strongly coupled first-order hierarchy for some shunt circuit resistance values exhibits a higher degree of energy dissipation and resistive damping around the tuning frequencies. The local resonance band gaps and the peak attenuation shift upward as the shunt circuit resistance increases. The results of the wavenumber analysis for the strongly coupled hierarchical system indicate that the resistance of the shunt circuit plays a key role in tuning the band structure in various ways. Furthermore, it is important to note that adjusting the resistance values modifies the electromechanical resonator's tuning frequency, leading to the band gap peak shifting away from the undamped mechanical tuning frequency. To further understand the effect of electromechanical coupling in this system, the response will also be compared to that of a purely mechanical damped system, shown in Figure 5g-i. The simulation of the hierarchical metamaterial with mechanically damped local resonators is carried out by first removing the piezoelectric patches and shunt circuits,followed by the introduction of a mechanical damping force into the governing equations. The equations of motion for mechanically damped resonators are given in Appendix B. The damping ratios of the mechanical dampers are designated as \(\zeta_{1}=\zeta_{2}\). The dispersion curves illustrate that the strongly coupled electromechanical resonators and mechanically damped resonators exhibit a similar band structure pattern in hierarchical metamaterials. Like strongly coupled electromechanical hierarchical metamaterials, the peak attenuation decreases as the value of \(\zeta_{1}\) increases in the mechanically damped hierarchical metamaterials. Although mechanical damping decreases the peak attenuation, it is perfectly aligned with the undamped resonator tuning frequency. In the electromechanical system, as the electromechanical coupling coefficient and the resistance of the shunt circuit increase, the local resonance band gaps shift upward.