## 6 Conclusions  
Using a buckled LS fixed at both ends as a negative stiffness element can achieve a large negative stiffness. When combined with linear positive stiffness springs in a parallel arrangement, it forms a QZS system that effectively isolates low-frequency vibration excitations. In the calculation of negative stiffness values, the elastic potential energy generated by the elastic deformation of the LS and springs in both the horizontal and vertical directions is used to establish the force equilibrium equation of the quasi-zero stiffness system, obtaining an analytical solution for the negative stiffness value.  
The accuracy of the negative stiffness value is verified using a finite element simulation. Static structural simulations are conducted to obtain the negative stiffness values of LSs with different lengths, widths, thicknesses, and compressions. By comparing data from 81 sets of finite element simulations with theoretical calculations, corrections are made to the analytical solution, significantly improving the accuracy of the negative stiffness values. Through the verification of experiments and finite element simulations, it was determined that the accuracy of the negative stiffness calculation formula is sufficient to meet the practical requirements. According to the dynamic deflection curve obtained during the derivation of the negative stiffness formula, the stress distribution of the compression buckling leaf springs in the form of fixed at both ends was deduced and verified by using finite element simulation, and the overall trend of the change is in line with the results of the analytical solution of the stress distribution in comparison with the value of the maximum stress. The simulation and experimental results demonstrate that the QZS system with flexural leaf spring structure has a certain vibration isolation performance, in which the random vibration simulation results show that the QZS isolation system significantly isolates the vibration excitation of the crystal cooling system. Therefore, the QZS system is suitable for the vibration isolation of ultra-high-stability DCM frames. It should be noted that QZS vibration isolation is a passive vibration isolation device, its dynamics are similar to those of traditional passive vibration isolation devices, and there are obvious characteristics of the natural frequency, so the design of quasi-zero stiffness vibration isolation needs to consider the vibration excitation characteristics of the application scenario, so that the natural frequency avoids the strong excitation energy frequency range.  
Conceptualization, X.G. and S.L.; methodology, S.L.; software, S.L. and K.C.; validation, X.G., Q.L. and Y.S.; formal analysis, X.G.; investigation, Y.S.; resources, X.G. and Q.L.; data curation, A.L.; writing--original draft preparation, S.L. and W.Z.; writing--review and editing, X.G.  
Figure 20: Results of vibration isolation experiments: (**a**) time-domain signal, (**b**) Fourier transform of time-domain signal, (**c**) transmissibility.  
and Y.S.; visualization, W.S. and Y.B.; supervision, X.G.; project administration, X.G. and Q.L.; funding acquisition, X.G. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (No. 62375261, 61974142) and the National Key Research and Development Program of China (No. 2023 YFA1608603).  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.