## 3 Band Structure of Hierarchical Metamaterials
### Effect of Electromechanical Coupling Coefficients and Shunt Circuit Resistances
The red dotted horizontal lines in Figures 5 and 6 are the undamped tuning frequencies of the resonators. The wavenumber components of the strongly coupled first-order configurations are generated at \(\beta_{1}=0.5\). Figure 5a-c show the wavenumber components of the zero-order and first-order hierarchies under the strong electromechanical coupling conditions, as determined by the inversion method [53]. The local resonance band gaps are characterized by extended peaks, while  
Figure 4: The effect of the electromechanical coupling coefficient and shunt circuit resistance on the tuning frequency of a strongly coupled electromechanical resonator in the zero-order hierarchy, with \(\eta=0.75\), \(R_{1}=10^{4}\) ohms and \(\theta_{1}=3\times 10^{-1}\) N/V: (**a**) effect of \(\theta_{1}\) on the real part, (**b**) imaginary part as a function of \(\theta_{1}\), (**c**) effect of \(R_{1}\) on the real part and (**d**) imaginary part as a function of \(R_{1}\). the Bragg scattering band gap is characterized by the semi-elliptical band gap above the local resonance band gaps. At \(\theta_{1}=10^{-2}\) N/V, as illustrated by the blue curves, there are distinct and clearly separated passband branches for all configurations. The characteristics of the wavenumber components of the zero-order hierarchy at this electromechanical coupling coefficient are similar to those of the undamped dispersion of the locally resonant metamaterial with a single resonator reported by [PERSON] et al. [10]. The authors established that metamaterials with undamped local resonators generate passbands distinguished by well separated mode branches. The wavenumber components of strongly coupled hierarchical metamaterials at low electromechanical coupling coefficients reveal that energy dissipation, or damping, is too small to affect the passbands. The strong electromechanical coupling coefficients introduce energy dissipation in hierarchical metamaterials due to their effect on the tuning frequency. Energy dissipation effectively reduces the resonance peaks and moderately broadens the bandwidth of the  
Figure 5: Real and imaginary wavenumber components of strongly coupled hierarchical metamaterials: The effect of the electromechanical coupling coefficients at \(\beta_{1}=0.5\) and \(R_{1}=10^{4}\) ohms: (**a**) zero-order, (**b**) outward, and (**c**) inward. The effect of shunt circuit resistances, \(\theta_{1}=0.3\) N/V: (**d**) zero-order, (**e**) outward, and (**f**) inward. The effect of mechanical damper: (**g**) zero-order, (**h**) outward, and (**i**) inward. passbands. At higher electromechanical coupling coefficients, the horizontal lines (resonance branches) seen on the real wavenumber component transform into descending curves. This effect results from energy dissipation as a result of the strong electromechanical coupling. Previously, the horizontal lines were aligned with the local resonance peak attenuation at a lower electromechanical coupling coefficient. Further increasing the electromechanical coupling coefficient in a strong electromechanical coupling scenario eventually merges the separated passband branches into a single curve, and the distinct behaviors of these modes blend. The black curve in Figure 5 illustrates that the merged passband exhibits a wider bandwidth compared to the separated branches. The local resonance peaks decrease as the electromechanical coupling coefficient increases further for each case. This phenomenon shows that as the electromechanical coupling coefficient increases, the energy dissipation becomes more pronounced in hierarchical metamaterials. Higher energy dissipation causes the peak attenuation constant at the center of the local resonance band gap to be reduced, while the bandwidth increases. Moreover, strong electromechanical coupling alters the resonator's tuning frequency, causing shifts in the local resonance band gap location.