## 6 Parameter Studies and Discussions  
### Influences of Excitation Amplitude and Damping Ratio on the Transmissibility  
For a QZS isolator with hardening stiffness, the frequency response curve can have a potential trend of bending to the right, which leads to a jump phenomenon in the response amplitude and transmissibility. In the jump frequency range, there exist three possible solutions, namely three values of transmissibility. The intermediate branch of FRCs has been proved unstable in Section 4.3. Therefore, the vibration isolation effect in the jump frequency ranges may be unstable, and the isolation frequency and peak transmissibility (the maximum value of the transmissibility) are two important indexes to be considered for a nonlinear suspension system with hardening stiffness. In the following section, the influences of the excitation amplitude and damping on the transmissibility and isolation frequency are discussed.  
The absolute displacement transmissibility of the QZS-IE system under different excitation amplitudes is shown in Figure 14. The maximum amplitude of transmissibility increases extremely slowly as the excitation amplitude increases. The same case is also found in Figure 15 and the maximum amplitude of transmissibility decreases slowly as the frequency ratio increases. Thus, it can be observed that the QZS-IE system is insensitive to the variations of excitation amplitude and frequency ratio under large excitation and small damping. This insensitivity causes the isolation frequency and peak transmissibility of the QZS-IE system to remain at a lower level than those of the QZS system. In addition, the peaks of the frequency response curve and the transmissibility curve look very \"sharp\" under large excitation and small damping, so it can be concluded that the transmissibility reaches the maximum when the peak response occurs. Therefore, the frequency corresponding to the maximum value of transmissibility is considered to be the isolation frequency, and the calculation of isolation frequencies converted into the calculation of peak response frequency.  
Figure 12: The vibration suppression effects of the seat suspension system (\(u_{0}=0.9\)): **(a)** The displacement transmissibility of the QZS system without inerter elements; **(b)** the displacement transmissibility of the QZS-IE system.  
Figure 13: The vibration suppression effects of the seat suspension system (\(u_{0}=0.4\)): **(a)** The displacement transmissibility of the QZS system without inerter elements; **(b)** the displacement transmissibility of the QZS-IE system.  
Figure 16a,b shows the variation trends of peak response frequency and peak transmissibility of the QZS-IE system and the QZS system with varying damping ratio, and Figure 16c,d shows those with varying excitation amplitude. It can be seen that increasing the damping ratio or decreasing the excitation amplitude causes a relatively noticeable decrease in the peak response frequency and peak transmissibility for the QZS system without inerter element, except that the peak response frequency varies less when increasing the excitation amplitude. Whereas for the QZS-IE system, the peak response frequency and peak transmissibility seem to remain unchanged or vary a little when varying the damping ratio and excitation amplitude; only the peak transmissibility is increased when the damping ratio decreases. As a result, it can be concluded that the peak response frequency and peak transmissibility of the QZS-IE system are much lower than those of the QZS system, especially when under large excitation and small damping.  
Figure 14: The absolute displacement transmissibility of the QZS-IE seat suspension system with different excitation amplitudes (\(\mu=0.4,\lambda=0.2,\zeta=0.05,\varepsilon=2\)).  
Figure 15: The absolute displacement transmissibility of the QZS-IE seat suspension system with different damping ratios (\(\mu=0.4,\lambda=0.2,u_{0}=1.0,\varepsilon=2\)).