# Effective Frequency Range and Jump Behavior of Horizontal Quasi-Zero Stiffness Isolator
## 2 Isolator Subjected to Base Excitation
### Parametric Study of the Effective Frequency Range  
Figure 8 shows the variation in the effective frequency range of the isolator with the variation of the input amplitude. The \(x\)-axis is the input displacement amplitude, and the \(y\)-axis represents the excitation frequency. This plot was obtained considering 10 values of the input amplitude from 1 to 10 mm and obtaining the amplitude-frequency curve for each value of the input amplitude. Furthermore, the boundary curve for the effective frequency can be obtained based on the amplitude-frequency curves. The effective frequency range is above the boundary curve, i.e., within the white region shown in Figure 8 (the frequency range in which the transmissibility is always less than 1 is referred to as the effective frequency range). It is shown that the effective frequency range becomes narrower with the increase in excitation energies. The lowest effective frequency increases significantly for input amplitudes greater than 8 mm, which is due to the occurrence of the jumping behavior.  
Figure 8: Variations in the effective frequency range of the isolator as a function of base displacement excitation.  
Figure 7: Jump phenomenon observed in the isolator after being subject to an impact load: (**a**) amplitude jump after impact load, (**b**) local view of the low-amplitude steady-state response of the isolator, (**c**) local view of the high-amplitude steady-state response of the isolator. The data plotted were obtained for \(P=0.01\) m, \(\Omega=7\) rad/s, and \(f=10\) N.  
Figure 9a shows the effect of damping on the effective frequency range of the isolator for \(P=0.006\) m; this figure demonstrates that the effective frequency range is enlarged for increasing levels of damping. The effective frequency range changes significantly when the damping is in the range of 6-8 N-s/m. This is because the system exhibits jumping behavior when the damping is small, so the lowest effective frequency of the isolator increases significantly. When the damping is larger than 8 N-s/m, the effective frequency range changes slightly because the system does not exhibit jumping behavior, i.e., the width of the effective frequency range cannot be stably increased by increasing the damping.  
Figure 9b shows the influence of the elongation of the linear springs on the effective frequency range of the isolator. The linear stiffness of the isolator varies with the change in the elongation of the two linear springs. Therefore, the linear natural frequency of the isolator increases with the elongating of the two springs. Consequently, the resonance peak is increased, which leads to the effective frequency range becoming narrower. This finding indicates that the elongation of the springs should not be too large. It is shown from the above analysis that the effective frequency range of the horizontal QZS isolator changes significantly with changes in damping and spring elongation. When the damping is large and the spring elongation is small, the effective frequency range of the isolator is wider. More importantly, the occurrence of the jumping behavior significantly influences the effective frequency range of the isolator.