# Design, Experiment, and Improvement of a Quasi-Zero-Stiffness Vibration Isolation System
## 5 The Improvement of the QZS System  
### Ways to Improve Under a Wide Range of Loads  
Based on the discussion in Section 2.2, it was found that the geometrical parameters of the magnetic spring affected the characteristics of the negative stiffness. For example, the height of the ring magnets affected the stiffness near the equilibrium position. As shown in Figure 5b, if the height of the inner ring magnet was kept constant and only the height of the outer ring magnet was increased, the number of peaks changed from 1 to 2 and the distance between the two peaks became increasingly larger. The nonlinearity of the curve changed from strong to weak and then became strong in the process of increasing the height of the outer ring magnet. Therefore, a practical way to improve the stiffness near the equilibrium position of the quasi-zero system was to change the height of the ring magnets, which enlarged the low stiffness range. This was the first method used for improvement.  
Another method used for improvement was designing a nonlinear spring that was parallel with the linear spring to be the new positive stiffness system. The principle is shown in Figure 14. Here, flexible bars were used to design the nonlinear spring. The finite element software ANSYS 16.0 (ANSYS, Inc. USA)was used to investigate the effects of these parameters to find out how these geometrical parameters affected the nonlinear stiffness, and the structural stiffness of the flexible bars was comparatively studied in accordance with the calculation results of the plane beam element. Figures 15-17 illustrate the different kinds of cross-section of bars that were used to construct the nonlinear spring.  
Figure 12: The transmissibility comparison between the two systems.  
Figure 13: The jumping phenomenon of the QZS system.  
As shown in Figures 15 and 16, the nonlinear spring of the first two methods had certain soft spring properties. However, when it came to the QZS isolation system, the negative spring made the stiffness of the entire system unbalanced near the equilibrium position. In Figure 17, the curve of stiffness for the nonlinear spring went down, then up. It had the characteristic of symmetry near the 30 mm position, which depicted a better non-linear property. Therefore, a third nonlinear spring was adopted to improve the positive stiffness system and a displacement of 30 mm was used as the equilibrium position. This was the second method used for improvement.  
Figure 16: The shape of the cross-section was a rectangle: (**a**) the shape of a nonlinear spring and (**b**) the stiffness of the nonlinear spring for finite element analysis.  
Figure 14: The second way to improve the QZS system: (**a**) the original QZS system and (**b**) the improved QZS system.  
Figure 15: The shape of the cross-section was round: (**a**) the shape of a nonlinear spring and (**b**) the stiffness of the nonlinear spring for finite element analysis.  
Based on the improvement of the QZS isolation system, with the combination of the first method used for improvement by changing geometrical parameters of magnetic spring, and the second method used for improvement by using a non-linear spring as mentioned above, the result is shown in Figure 18. It can be seen that, after improvement, the stiffness in the interval \(-3\) to \(3\) mm was very close to \(0\), which shows that the QZS after improvement had a better isolation performance for low frequencies with a small amplitude excitation.