## 3 Band Structure of Hierarchical Metamaterials
### Effect of Electromechanical Coupling Coefficients and Shunt Circuit Resistances
In Equation (19), the term \(\frac{\theta_{1}^{2}}{C_{1}k_{1}}\) represents the additional stiffness introduced by the electromechanical coupling coefficient, and \(R_{1}C_{1}\) is the decay rate induced by the shunt circuit. The electromechanical coupling coefficient and shunt circuit resistance lead to three possible cases on the solution of \(\omega_{em}\). These cases include the short circuit (when \(R_{1}=0\)), weak coupling, and strong coupling scenarios. When \(R_{1}=0\), the electromechanical resonator tuning frequency becomes identical to that of the undamped resonator tuning frequency, i.e., \(\omega_{em}=\omega_{1}\). In this case, the resonator is purely mechanical. When the system is weakly coupled, the dimensionless stiffness, \(\frac{\theta_{1}^{2}}{C_{1}k_{1}}\), becomes negligible. As a result, Equation (19) provides three roots: \(\omega_{em}=\pm\omega_{1}\) and \(\omega_{em}=-i\frac{1}{R_{1}C_{1}}\). Originally, the undamped resonator did not have an imaginary component. For the weakly coupled resonator, the tuning frequency includes both real and imaginary components. The real part remains the same, so the resonator resonates at the same frequency. However, the added imaginary part introduces attenuation around the tuning frequency. In the third case, the strong coupling of the electrical and mechanical systems alters the stiffness of the resonators, leading to a substantial change in the tuning frequency. For a strongly coupled electromechanical resonator, solving Equation (19) for \(\omega_{em}\) yields three roots. Two of these roots are complex, consisting of both real and imaginary components, while the third root is purely imaginary. The roots of Equation (19) are shown graphically in Figure 4, providing a visual representation of their behavior under varying electromechanical coupling coefficients and the resistance of the shunt circuits for the zero-order hierarchy. The red dotted horizontal lines represents the undamped tuning frequency, \(\omega_{1}\), of the resonator. The real part of the electromechanical resonator tuning frequency increases with both the electromechanical coupling coefficient and the resistance of the shunt circuit. Relative to \(\omega_{1}\), it increases by 3.28% as the coupling coefficient increases, as shown in Figure 4a, and by 8.87% as \(R_{1}\) increases, as shown in Figure 4c. At lower values of the electromechanical coupling coefficients and shunt circuit resistances, the real part of \(\omega_{em}\) aligns with \(\omega_{1}\). The first two imaginary parts are equal and increase as the electromechanical coupling coefficient increases, while the third imaginary part decreases, as shown in Figure 4b. At low shunt circuit resistance, the imaginary component of the third root increases rapidly, which is important for strong vibration attenuation at the tuning frequency. Roots 1 and 3, as depicted in Figure 4c,d, swap positions without affecting the magnitude of the tuning frequency. The tuning frequency analysis of the electromechanical resonator reveals that the weakly coupled electromechanical resonator, shunted with a resistor-loaded circuit, adds only an imaginary component. However, strong electromechanical coupling increases the real part's magnitude and introduces an imaginary part as the coupling coefficient and shunt resistance increase. The imaginary part contributes to the increase in peak attenuation, while the real part shifts the tuning frequency upward and widens the band gap. Now, let us proceed with the wavenumber analysis of hierarchical electromechanical metamaterials. The real part of the wavenumber provides information about the nature of the passband branches. The imaginary part is related to the attenuation constant, offering insight into the damping properties and energy dissipation in the stopband regions. From this point onward, the curve corresponding to the imaginary component of the wavenumber is referred to as the attenuation curve.