# Effective Frequency Range and Jump Behavior of Horizontal Quasi-Zero Stiffness Isolator
## 3 Isolator Installed on Linear Structure
### Jump Behavior  
The accuracy of the derivation shown above was also verified here by comparing the semi-analytical and numerical solutions, and the results are shown in Figure 11a,b. The parameters of the isolator are the same as those adopted in the previous section; the mass, stiffness, and damping of the primary structure are \(m=100\) kg, \(k_{0}=40000\) N/m, and \(\lambda=10\) N\(\cdot\)s/m. The paper aims to demonstrate the common problems of the quasi-zero stiffness isolators, and thus the parameters of the primary structure are not taken from actual engineering. In essence, the behaviors which are the qualitatively same as those shown in the paper can be obtained as long as the excitation is appropriate even if the parameters of the primary structure are changed. Figure 11: _Cont._  
Figure 11 shows the amplitude-frequency response of the system for different excitation amplitudes. It is found that the primary structure only has one resonance band, and the isolator has two resonance bands for \(p=0.5,\,1,\,\) and \(\,2\,\)N. The system produces unstable responses around the resonance frequency after the excitation amplitude is increased to a certain threshold. In addition, the response of the system at \(p=3\,\)N is significantly different from the system excited by a relatively small force. It is shown that the multiple solutions occur in a frequency band that is lower than the resonance frequency of the primary structure. The characteristics of the multiple solutions in the two-degree-of-freedom vibration isolation system are similar to those observed in systems with nonlinear energy sinks [36; 37; 38]. Moreover, it is noted that the secondary oscillator does not introduce a new resonance band in the primary oscillator, even for excitation amplitudes that are extremely large, which is a beneficial characteristic for nonlinear energy sinks. The realization of a purely nonlinear stiffness (that is, the linear stiffness is exactly equal to zero) is difficult to achieve in practice; in practical situations, a small positive or negative linear stiffness inevitably arises. The small positive linear stiffness only slightly affects the response of the primary structure, which demonstrates that the configurations of the QZS vibration isolator can be modified to the QZS vibration absorber according to the theory of nonlinear energy sink conveniently by amending parameters. In order to describe the effective frequency range of the isolator directly, the amplitude ratio of the isolator and primary structure was obtained. The isolator is effective in the frequency range in which the amplitude ratio is smaller than one. The isolator loses efficacy in the frequency range slightly higher than zero-frequency as the excitation amplitude is small, as shown in Figure 12a. However, the isolator can produce large amplitude vibration in the resonance band of the primary structure. With the increase in the excitation amplitude, the amplitude of the isolator exceeds that of the primary structure in the resonance band of the primary structure, the isolator acts as a vibration absorber in these cases, as shown in Figure 12b,c. Moreover, the system produces chaotic responses in the resonance bands of the primary structure; the solutions are unstable in these regimes. A typical chaotic response is depicted in Figure 13.  
Figure 11: Amplitude–frequency responses of vibration isolation system. The figure shows systems with (**a**) \(p=0.5\,\)N, (**b**) \(p=1\,\)N, (**c**) \(p=2\,\)N, and (**d**) \(p=3\,\)N.  
In Figure 12d, there exist multiple solutions in three frequency ranges. In the lowest frequency range, the curve is similar to the counterpart considered in Figure 4b; the jump phenomenon is also similar to that observed in the previous section. In the second frequency range (9.96-18.84 rad/s), two stable solutions exist, which is a requirement for the jumping behavior. When the system has an additional impact load imposed on it, the jump can occur, as shown in Figures 14 and 15.