## 3 Band Structure of Hierarchical Metamaterials
### Effect of Resonator Mass Ratios  
Next, the effect of electromechanical resonator masses is analyzed through \(\beta_{1}\) and \(\eta\). Analyzing the effect of the resonator masses on the band structure of hierarchical metamaterials is fundamental because it directly influences the system's dynamic properties. The masses of the resonators, along with stiffness determine the tuning frequency of the local resonance band gap. For the design of hierarchical metamaterials, understanding the distribution of resonator mass helps predict and optimize the emergence of multiple  
Figure 6: Real and imaginary wavenumber components of weakly coupled hierarchical metamaterials; the effect of electromechanical coupling coefficients, \(\beta_{1}=0.5\) and \(R_{1}=R_{2}=10^{4}\) ohms: (**a**) zero-order, (**b**) outward, and (**c**) inward. band gaps. In this analysis, the electromechanical local resonator mass for the zero-order configuration is \(m_{1}=\eta m_{0}\). To maintain a constant local resonator mass between hierarchies, the electromechanical resonator masses for the first-order configurations are \(m_{1}=\eta\beta_{1}m_{0}\), and \(m_{2}=\eta(1-\beta_{1})m_{0}\). To isolate individual effects, \(\eta\) is kept constant at \(\eta=0.75\) when \(\beta_{1}\) is varied, and \(\beta_{1}\) is kept constant at \(\beta_{1}=0.85\) when \(\eta\) is varied. The effect of \(\eta\) follows a similar trend for strongly coupled and weakly coupled systems, except that stronger electromechanical coupling leads to a greater increase in the band structure. The effect of \(\eta\) was exclusively investigated for the weakly coupled system to highlight its characteristics. The graphs in Figure 7 illustrate the influence of the total electromechanical resonator mass ratio, \(\eta\), on the band structure. Increasing the value of \(\eta\) lowers the upper limit of the first passband in both hierarchies. A higher value of \(\eta\) increases the width of the first local resonance band gap for both zero-order and first-order hierarchies. The second band gap shifts upward, with the width increasing for the outward configuration and decreasing for the inward configuration. As \(\eta\) increases, the lower limit of the second passband and the third passband of the outward configuration shift upward, as shown in Figure 7b. The third passband of the inward configuration remains nearly flat for any \(\beta_{1}\) and is minimally affected by \(\eta\), as seen in Figure 7c. In the first-order hierarchy, \(\eta\) allows tuning the boundaries of all three passbands in the outward configuration. However, in the inward configuration, only the first and second passbands can be tuned, while the third remains unaffected. A significant drawback of increasing \(\eta\) is the associated increase in the total mass of the metamaterial. Therefore, the effect of \(\beta_{1}\) becomes particularly interesting and important, as it does not alter the total mass, making it a key focus. The effect of \(\beta_{1}\) on the dispersion of the weakly and strongly coupled first-order hierarchy is shown in Figure 8. The band structure of the outward configuration, as demonstrated in Figure 8a,b, is symmetric with respect to \(\beta_{1}=0.5\). In the outward configuration, the distinction between \(m_{1}\) and \(m_{2}\) is arbitrary, and the physical system mirrors itself at values above and below \(\beta_{1}=0.5\). As such, the dynamic response of the system is identical for \(\beta_{1}\) values equally far from \(0.5\). For example, in Figure 8a, the same dispersion curves are obtained at \(\beta_{1}=0.15\) and \(\beta_{1}=0.85\). The symmetrical nature seen in the outward configuration can offer the opportunity to tailor the band structure on one side of \(\beta_{1}=0.5\). As the value of \(\beta_{1}\) diverges from \(0.5\), the upper limit of the first passband decreases, leading to an increase in the width of the first band gap. This helps to push the boundary of the first local resonance band gap to the lower frequency, which is desirable.