# Design, Experiment, and Improvement of a Quasi-Zero-Stiffness Vibration Isolation System
## 5 The Improvement of the QZS System
### Dynamic Analysis for the Improved System  
It can be seen from the previous analysis that the improved QZS system had a better isolation performance. In this part, the force transmissibility was the main consideration of the dynamic analysis. From the study of [PERSON] et al. [16] on QZS's transmissibility, the fitting stiffness equation of the improved QZS system can be given as:  
\[K_{QZS}=\alpha_{1}x^{2}+\alpha_{2}x^{4}. \tag{26}\]  
For the QZS system, the dynamic equation of the system in a non-dimensional form can be given as:  
\[\hat{y}^{\prime\prime}+2\zeta\hat{y}^{\prime}+\chi_{1}\hat{y}^{3}+\chi_{2}\hat{ y}^{5}=\overline{F}_{0}\cos(\Omega\tau+\theta), \tag{27}\]  
where \(\hat{y}=\frac{y}{y_{s}}\), \(\Omega=\frac{\omega}{\omega_{n}}\), \(\omega_{n}=\sqrt{\frac{k_{v}}{m}}\), \(\chi_{1}=\frac{\alpha_{1}y_{2}^{2}}{3k_{v}}\), \(\chi_{2}=\frac{\alpha_{2}y_{3}^{4}}{3k_{v}}\) and \(\overline{F}_{0}=\frac{F_{0}}{k_{v}y_{s}}\); \(y\) is the displacement of the mass; \(y_{s}\) is the static displacement of the linear spring; \(k_{v}\) is the stiffness coefficient of the positive spring; and \(F_{0}\) is the exciting force. Figure 17: 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 18: The comparison between the non-improved system and the improved QZS system regarding the stiffness–displacement characteristic. According to the solution method used in Section 3, the force transmissibility of the improved system is defined as:  
\[T=\frac{\sqrt{\left(2\zeta\gamma\Upsilon\Omega\right)^{2}+{H^{\prime}}^{2}}}{ \overline{F}_{0}}, \tag{28}\]  
where \({H^{\prime}}^{2}=\frac{3}{4}\chi_{1}Y^{3}+\frac{5}{8}\chi_{2}Y^{5}\) and \(\tan\varphi=\frac{2\zeta Y\Omega}{H^{\prime}}\); it is assumed that the time domain response of the whole system is \(\hat{y}(\tau)=Y\cos(\Omega\tau)\). From Equation (28), the expression of the force transmissibility of the improved system was similar to that of the non-improved system, and they were all associated with the damping ratio and the excitation amplitude. Comparisons between the non-improved and the improved force transmissibility with the different damping ratio can be seen in Figure 19. The fit coefficients of the improved QZS system were \(\alpha_{1}=-8.154\times 10^{-4}\) and \(\alpha_{2}=2.389\times 10^{-4}\), and the excitation amplitude was set to \(\overline{F}_{0}=0.30\). It can be seen that when the damping ratio was \(\zeta=0.05\), the force transmissibility of the non-improved QZS system was bigger than that of the improved system in the latter half of the jumping range, but the jump-down frequency decreased. The result of the improvement was increasingly obvious when increasing the damping ratio. Figure 19 shows that after improvement, the peak of the force transmissibility and the jump-down frequency reduced significantly. Therefore, the improved quasi-zero-stiffness system had a better performance for low-frequency vibration isolation. Figure 20 illustrates the comparison between the non-improved system and the improved system with the same excitation amplitude. The fit coefficients of the improved QZS system were \(\alpha_{1}=-8.154\times 10^{-4}\) and \(\alpha_{2}=2.389\times 10^{-4}\), and the damping ratio was set to \(\zeta=0.05\). When the amplitude of excitation was \(\overline{F}_{0}=0.15\), the peak of the force transmissibility of the improved system was lower than that of the non-improved system and the jump-down frequency of the improved system was also lower than that of the non-improved system. With an increased excitation amplitude, the difference between the peaks of the two curves became increasingly small, but the jump-down frequency of the improved system was still lower than that of the non-improved system.