# Effective Frequency Range and Jump Behavior of Horizontal Quasi-Zero Stiffness Isolator
## 4 Conclusions  
The jump behavior caused by the strong nonlinearity of the horizontal QZS isolator is a primary reason for its failure. The dynamics of the isolator were studied to grasp the effective frequency range and failure mechanism using the complexification-averaging method and the Runge-Kutta method. The accuracy of the semi-analytical results was verified by comparing the results obtained from the two methods. The amplitude-frequency response curves of the system were presented.  
For small excitation amplitudes, the isolator exhibits good vibration isolation performance. With an increase in the excitation amplitude, the systems exhibit jump behavior  
Figure 16: Variations in the effective frequency range of the isolator as a function of the harmonic excitation imposed on the primary structure.  
Figure 17: The influences of isolator parameters on the effective frequency range. (**a**) Influence of damping, (**b**) influence of elongation of linear springs. Results are shown for \(p=2\) N.  
around the natural frequency of the isolator and then increase the lowest effective frequency of the isolator. In the two-degree-of-freedom system, the isolator loses efficacy in the resonance band of the primary structure when the excitation exceeds a certain threshold. Furthermore, the two-degree-of-freedom system generates higher branches of response after the excitation reaches a second threshold, which leads to a narrower effective frequency range between the two natural frequencies of the system. In addition, the influences of the isolator parameters on the effective frequency range were studied. The results show that increases in the damping of the system can effectively eliminate the jump phenomena and expand the effective frequency range. It is also found that decreases in the elongation of the two linear springs reduce the lowest effective frequency, but such changes do not significantly affect the jump behavior.  
Low-frequency and broadband vibration isolation are important issues in engineering practices. The jump phenomenon is the biggest obstacle to achieving these two goals. This study can deepen the understanding of the jump behaviors induced by the horizontal QZS isolator and provide design guidance for this kind of isolator to avoid jumps. However, the factors that affect the dynamic characteristics of the isolator in actual engineering are not considered. In the future, the jump behaviors that lead to the narrowing of the effective frequency range should be investigated under the full consideration of the engineering environments.  
Conceptualization, J.C.; methodology, J.C.; validation, L.X. and K.D.; formal analysis, L.X.; investigation, L.X., K.D., H.H., and H.Z.; writing--original draft, L.X., K.D., H.H., and H.Z.; writing--review and editing, J.C.; supervision, J.C.; funding acquisition, J.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the College Students Innovation and Entrepreneurship Training Program of Tianjin (No. 202110060017), and the National Natural Science Foundation of China (No. 11872274).  
Not applicable.  
All data included in this study are available from the corresponding author upon reasonable request.  
We sincerely thank the reviewers for their helpful and constructive suggestions and the editors for their careful and patient work.  
The authors declare that they have no conflicts of interest.