## 5 Vibration Suppression Effect  
The vibration suppression effects of the seat suspension system are investigated in this section. The maximum absolute displacement transmissibility for the uncontrolled system (\(\zeta=0.05\)) and the controlled system is denoted as \(T_{uc}\) and \(T_{c}\), respectively. The reduction percentage of the maximum absolute displacement transmissibility for the controlled system is:  
\[Q_{T}=\frac{T_{uc}-T_{c}}{T_{uc}}100\% \tag{63}\]  
The frequency ratio at the maximum resonance response peak for the uncontrolled system (\(\zeta=0.05\)) and the controlled system is denoted as \(\Omega_{uc}\) and \(\Omega_{c}\), respectively. The reduction percentage of the frequency at the maximum resonance response peak for the controlled system is:  
\[Q_{\Omega}=\frac{\Omega_{uc}-\Omega_{c}}{\Omega_{uc}}100\% \tag{64}\]  
Figure 9: Displacement time histories. Figure 11 shows the comparisons of the vibration suppression effects between the QZS system without inerter element and the QZS-IE system when the excitation amplitude is \(u_{0}=1.0\). It can be observed that the frequency response curves of both systems have obvious nonlinear characteristics, especially at the maximum resonance response peak. The vibration suppression effect can be severely affected by the strong nonlinear characteristics. As shown in Figure 11a, for the quasi-zero-stiffness system, the reduction percentage of the maximum displacement transmissibility is 42.7%. The reduction percentage of the frequency at the maximum resonance peak is only 10.5%. However, for the QZS-IE system, Figure 11b shows that the reduction percentages of the maximum displacement transmissibility and frequency at the maximum resonance peak are, respectively, 56.3% and 20.6%. Obviously, it is concluded that the QZS-IE system has significant vibration suppression advantages for the maximum resonance response peak. In addition, in the high frequency region, the curves of the QZS system, the QZS-IE system, and the uncontrolled system are not coincident. The vibration suppression effect of the displacement transmissibility of the QZS-IE system is slightly worse, especially for a frequency ratio greater than 1.25; the displacement transmissibility of the QZS-IE system is slightly higher than that of the QZS system and the uncontrolled system. Next, the influence of the excitation amplitude on the vibration suppression effect is analyzed. When \(u_{0}=0.9\) and \(u_{0}=0.4\), the comparisons of the vibration suppression effects between the QZS system and the QZS-IE system are shown in Figures 12 and 13, respectively. It can be found that with a decrease in excitation amplitude, the curve changes of the two systems have the same trend when the frequency ratio is about larger than 0.9. Moreover, the value of \(Q_{T}\) and \(Q_{\Omega}\) for both systems increases as the excitation amplitude decreases. For the QZS system, the displacement transmissibility curve bends to the right, which reduces the effective isolation region. In contrast, for the QZS-IE system, the bending trend of the displacement transmissibility curve is considerably weakened, thereby, resulting in a much wider isolation frequency range as well as a lower peak transmissibility. Meanwhile, with a decrease in the excitation amplitude, the differences of \(Q_{T}\) between the QZS-IE system and the QZS system are, respectively, 13.6%, 12.6%, and 11.5%, and the differences of \(Q_{\Omega}\) are 9.5%, 11%, and 16%, respectively. Therefore, this illustrates that for the maximum displacement transmissibility, the larger the excitation amplitude is, the greater the advantage of the QZS-IE system in vibration suppression performance. For the frequency at the maximum resonance response peak, although the difference value variation is the opposite, the changing gradient gradually decreases and the vibration suppression performance is not seriously affected. The QZS-IE system still has better performance advantages.