## 3 Band Structure of Hierarchical Metamaterials
### Effect of Electromechanical Coupling Coefficients and Shunt Circuit Resistances
A more significant increase is observed in the second band gap width for the first-order configuration, reaching 51.87% and 193.80% for the outward and inward configurations, respectively. The strong electromechanical coupling appears to have a stronger effect at higher frequencies. In contrast to the strong electromechanical coupling, weak electromechanical coupling does not produce variation in the band structure, as illustrated in Figure 2d,e for both the zero-order and first-order hierarchies. Furthermore, the dispersion relationship of the zero-order configuration agrees well with the dispersion of the electromechanical metamaterial with a single resonator studied by [PERSON] et al. [47]. Figure 3 shows the dispersion curves of hierarchical metamaterials for strongly and weakly coupled systems, evaluated at different resistance values of the shunt circuit. The resistances of the shunt circuit show a noticeable impact on the band structure of strongly coupled hierarchical metamaterials, as evidenced in Figure 3a-c. However, it exhibits minimal effect on the band structures for the weakly coupled system, as shown in Figure 3d-f. This highlights how the coupling strength dictates the sensitivity of the system's band structure to changes made to the shunt circuit resistance values. For a relatively strong electromechanical coupling, such as 10\({}^{-1}\) N/V, the second and third passbands of the first-order configurations are affected by the resistances of the shunt circuit. Although the effect on the first passband at lower frequencies is almost negligible, an influence is observed at higher frequencies. Under strong electromechanical coupling, with a shunt circuit resistance of 10\({}^{5}\) ohms, the band gap width increases by 15.74% for the  
Figure 3: The effect of shunt circuit resistances on the band structure of strongly coupled systems at \(\theta_{1}=0.3\) N/V: (**a**) zero-order, (**b**) outward, and (**c**) inward configurations. The effect of shunt circuit resistances on the band structure of weakly coupled systems at \(\theta_{1}=10^{-10}\) N/V: (**d**) zero-order, (**e**) outward, and (**f**) inward configurations at \(\eta=0.75\) and \(\beta_{1}=0.5\). zero-order configuration, and by 30.55% and 11.51% in the first band gap for the outward and inward configurations, respectively. Additionally, the second band gap width increases for the first-order configurations, reaching 10.11% and 47.50% for the outward and inward configurations, respectively. The band structure is significantly affected when the resistance of the shunt circuits is between \(10^{3}\) and \(10^{5}\) ohms. Within this range, increasing the shunt circuit resistance leads to a band gap width expansion. The impact of the shunt circuit resistances on the band structure primarily depends on the electromechanical coupling coefficients. The band structure of strongly coupled hierarchical metamaterials with lower resistance values is similar to that observed in weakly coupled scenarios. To obtain a detailed understanding of the band structure and the attenuation level of hierarchical metamaterials, it is crucial to study the real and imaginary wavenumber components. However, before analyzing the wavenumber components, let us first examine how the tuning frequencies of the electromechanical resonators behave as a function of the electromechanical coupling coefficient and the shunt circuit resistance. For simplicity, this analysis focuses on the zero-order hierarchy. The tuning frequency of the electromechanical resonator, \(\omega_{em}\), for the zero-order configuration (\(n=1\)), is obtained by solving the determinant of Equations (17) and (18). By equating the determinant to zero and applying mathematical manipulations, Equation (19) is obtained. \[\omega_{1}^{2}-\omega_{em}^{2}-iR_{1}C_{1}\Bigg{(}\Bigg{(}1+\frac{\theta_{1} ^{2}}{C_{1}k_{1}}\Bigg{)}\omega_{1}^{2}\omega_{em}-\omega_{em}^{3}\Bigg{)}=0 \tag{19}\]  
Here, \(\omega_{1}=\sqrt{\frac{k_{1}}{m_{1}}}\), and \(\omega_{em}\) is the electromechanical resonator tuning frequency.