## 4 Transmissibility
The first-order hierarchy transmits the high-frequency third passband, which is not possible in the zero-order hierarchy, with a narrower bandwidth in the inward configuration compared to the outward configuration. In addition to the effect of \(\beta_{1}\), the outer mass displacement transmissibility of strongly coupled hierarchical metamaterials is affected by electromechanical parameters. Strongly coupled hierarchical metamaterials effectively suppress higher frequency resonance peaks, as demonstrated in Figure 11d-f. The strongly coupled system significantly reduces the local resonance peaks of the third passband in first-order configurations. However, the second passband exhibits higher transmissibility, and the first local resonance band gap shows weaker attenuation in the strongly coupled inward configuration. The decrease in peak displacement transmissibility with increasing frequency in strongly coupled electromechanical systems indicates that strong electromechanical coupling is more effective at damping  
Figure 11: The displacement transmissibilities of hierarchical metamaterial at different system parameters. Weakly coupled electromechanical resonator (**a**) \(\beta_{1}=0.15\), (**b**) \(\beta_{1}=0.5\), and (**c**) \(\beta_{1}=0.85\). Strongly coupled electromechanical resonator (**d**) \(\beta_{1}=0.15\), (**e**) \(\beta_{1}=0.5\), and (**f**) \(\beta_{1}=0.85\). higher frequency vibrations. The outward configuration shows relatively minimal peaks and deep valleys. The curves seen in Figure 12 illustrate the normalized total power output of the shunt circuits in the last unit cell at a shunt circuit resistance of \(10^{4}\) ohms. The zero-order hierarchy harvests energy at low frequencies, and in this case, it is limited to a normalized frequency of 2.5. The first-order hierarchy harvests energy at frequencies higher than those of the zero-order hierarchy. The normalized power output increases as the electromechanical coupling coefficient transitions from weak to strong coupling. In the strongly coupled scenario, the maximum peaks are observed at lower frequencies and gradually flatten as the normalized frequency increases. For harvesting energy from lower frequency vibrations, the inward configuration is more effective, whereas the outward configuration excels across a wider frequency range. Their ability to capture energy across a wide range of frequencies makes them preferable for broadband energy harvesting. When designing a vibration attenuator and energy harvester, optimizing performance across different system parameters is essential [57]. The vibration attenuation and energy harvesting performance of each configuration are optimized for shunt circuit resistance in the strong coupling case. Optimizing shunt circuit resistance in strongly coupled metamaterials is necessary for vibration attenuation and energy harvesting. This is because shunt circuit resistances in strong coupling significantly dampen the local resonance peak attenuation and directly impact power estimation. Based on the results illustrated in Figure 13, the effect of shunt circuit resistance can be categorized into three groups. The first group corresponds to a shunt circuit resistance less than \(10^{3}\) ohms characterized by narrow bandwidths and deep, narrow local resonance attenuation regions. Figure 12: The normalized total power output of hierarchical metamaterial at different system parameters. Weakly coupled electromechanical resonator (**a**) \(\beta_{1}=0.15\), (**b**) \(\beta_{1}=0.5\), and (**c**) \(\beta_{1}=0.85\). Strongly coupled electromechanical resonator (**d**) \(\beta_{1}=0.15\), (**e**) \(\beta_{1}=0.5\), and (**f**) \(\beta_{1}=0.85\). The second group includes resistance between \(10^{3}\) and \(10^{5}\) ohms where the damping effect becomes more pronounced. A shallow first local resonance attenuation valley is followed by a strongly damped high frequency resonance peak.