## 3 Results
### Performance Analysis of the ERS-TQZS VI  
The accuracy of the analytical solution has been validated; the next step is to determine the advantages of the ERS-TQZS VI system over existing methods, i.e., TQZS VI and the two-stage linear vibration isolator (TL VI). For this, we compare the force transmissibility curve of the ERS-TQZS VI with the TQZS VI and the TL VI. For a given frequency ratio \(\Omega\), the amplitudes \(\widehat{Z}_{11}\), \(\widehat{Z}_{12}\), \(\widehat{Z}_{21}\), \(\widehat{Z}_{22}\), \(\widehat{t}_{31}\), and \(\widehat{t}_{32}\) are determined by numerically solving a system of six nonlinear equations, as described in Equation (6). The transmitted force can be expressed as  
\[f_{t}=2\mu\zeta_{2}\widehat{\hat{z}}_{2}+\alpha_{2}\widehat{z}_{2}+\gamma_{2} \widehat{z}_{2}^{3} \tag{7}\]  
Subsequently, utilizing the aforementioned equations, the force transmissibility of ERS-TQZS VI system can be derived as:  
\[T_{F}=\frac{F_{t}}{\widehat{F}_{e}}=\left(\left[\frac{3}{4}\gamma_{2}\widehat{ Z}_{22}^{3}+2\mu\zeta_{2}\widehat{Z}_{21}\Omega+(\frac{3}{4}\gamma_{2} \widehat{Z}_{21}^{2}+\alpha_{2})\widehat{Z}_{22}\right]^{2}+\left[\frac{3}{4} \gamma_{2}\widehat{Z}_{21}^{3}-2\mu\zeta_{2}\widehat{Z}_{22}\Omega+(\frac{3}{ 4}\gamma_{2}\widehat{Z}_{22}^{2}+\alpha_{2})\widehat{Z}_{21}\right]^{2}\right) ^{\frac{1}{2}}\left/\widehat{F}_{e}\right. \tag{8}\]  
The comparison with TQZS VI and TL VI are shown in Figure 3(a). The mechanical and electrical parameters of ERS-TQZS VI, TQZS VI and TL VI are \(\zeta_{1}=\zeta_{2}=0.05\), \(\mu=0.2\)\(\widehat{F}_{e}=0.005\), \(l=0.8\), \(\widehat{C}_{d}=0.025\), \(\widehat{C}_{e}=50\), \(\widehat{R}=14\), \(\widehat{C}=1\). Figure 4 reveals a notable reduction in the initial isolation frequency of the ERS-TQZS VI system of 55.0% (from 1.02 to 0.45) when compared to the TL VI system. These findings suggest that the ERS-TQZS VI successfully extends the frequency range for vibration attenuation towards lower frequencies, effectively mitigating the vibrations of such frequencies. Moreover, compared to the TL VI and TQZS VI systems, the resonance peak is reduced by 26.5% and 15.5%, respectively, which implies that the ERS-TQZS VI can considerably attenuate resonance peaks of low-frequency vibrations. Meanwhile, the resonance frequency is reduced by 7.2%, compared to that of the TQZS VI system. It should be noted that the ERSD component in TQZS VI provides better low-frequency vibration control than the conventional TQZS VI with viscous damping. Furthermore, the ERS-TQZS VI system can not only effectively attenuate low-frequency vibrations but can also harvest energy from such vibrations, as depicted in Figure 4b. We introduce the following output power expression to define the harvested energy of the ESD-TQZS VI as follows:  
\[P_{ESD}=\widehat{I}_{e}^{2}\widehat{R}=\left(\sqrt{\widehat{I}_{31}^{2}+ \widehat{I}_{32}^{2}}\right)^{2}\widehat{R} \tag{9}\]  
From Figure 4b, it can be observed that a notable disparity in output power exists between the resonant and anti-resonant frequencies. The significantly higher output power at the resonant frequency can be attributed to the considerably larger displacement amplitude response of the ERS-TQZS VI compared to the anti-resonant frequency. Moreover, in the energy harvesting range of the ERS-TQZS VI, the output power reaches the maximum value at the resonant frequency. The findings suggest that the ERS-TQZS VI system can not only effectively suppress resonance peaks of low-frequency vibration but can also convert vibrational energy into electrical energy. Furthermore, the ERS-TQZS VI system exhibits the ability to harvest vibrational energy within a broad frequency range. More importantly, we emphasize that the main purpose of the ERS-TQZS VI system is to suppress low-frequency vibration while simultaneously maximizing vibrational energy harvesting by optimizing the ERS-TQZS VI to balance between these two functions.