## 3 Band Structure of Hierarchical Metamaterials
### Effect of Resonator Mass Ratios
The interaction between coupled local resonators in the inward configuration is responsible for the formation of two distinct local resonance band gaps for all values of \(\beta_{1}\). The undamped tuning frequencies of the first-order inward configuration's local resonators are the eigenvalues of the characteristic equation given in Equation (16). The undamped eigenvalues of the resonators align with the peak attenuation of the weakly coupled local resonance band gap for the inward configuration, as depicted in Figure 9d-f. The effect of \(\beta_{1}\) on the peak attenuation is negligible, but it significantly alters the boundaries of the passbands. The coupled motion of local resonators offers different tuning frequencies, causing the inward configuration to produce two local resonance band gaps at any \(\beta_{1}\). This is clearly demonstrated in Figure 9e. When \(\beta_{1}=0.5\), both resonators are identical, but there are still two distinct band gaps, unlike in the outward case in Figure 9b. Figure 9: Real and imaginary wavenumber components of weakly coupled systems. (**a**) outward at \(\beta_{1}=0.1\), (**b**) outward \(\beta_{1}=0.5\), (**c**) outward at \(\beta_{1}=0.9\), (**d**) inward \(\beta_{1}=0.1\), (**e**) inward \(\beta_{1}=0.5\), (**f**) inward \(\beta_{1}=0.9\). The band structure enhancement of the weakly coupled first-order hierarchical metamaterial due to \(\beta_{1}\) is summarized in Figure 10. To compare the band gap width of the first-order configurations with the zero-order hierarchy, the total width of these gaps must be the sum of all individual band gap widths. The first-order hierarchy opens two local resonance band gaps, with the total width of the local resonance band gap being the sum of the first and second band gaps. The total band gap width ratio is defined as the ratio of the total band gap width to the upper limit frequency of the last passband. Due to the absence of clearly separated band gaps in strongly coupled systems, the total band gap width ratio is not performed for strongly coupled systems. In the zero-order hierarchy, the total band gap width ratio is defined as the ratio of the band gap width to the upper limit frequency of the second passband. This ratio remains constant regardless of the variation of \(\beta_{1}\). In the first-order hierarchy, the total band gap width ratio is the ratio of the total band gap width to the upper limit frequency of the third passband. The inward configuration's total band gap width ratio is lower than the outward configuration's total band gap width ratio when \(\beta_{1}\) is greater than 0.4. For \(\beta_{1}<0.4\), the total band gap width ratio of the inward configuration exceeds the outward configuration's total band gap width ratio. The inward configuration enhances the zero-order configuration's band structure by a minimum of 40.95%, while the outward configuration achieves at least 55.17% enhancement, as illustrated in Figure 10. Moreover, the total band gap width ratios of the first-order configurations are larger than the zero-order hierarchy. This implies that, relative to the zero-order hierarchy, the first-order hierarchy produces a larger total band gap width and increased bandwidth.