# Design, Experiment, and Improvement of a Quasi-Zero-Stiffness Vibration Isolation System
## 3 Dynamic Analysis of the QZS System  
When considering the QZS dynamic response of the entire system, it is inconvenient to directly use Equation (18) for dynamic analysis; instead, a nonlinear curve should be fitted.  
The equation of the force of the QZS of the vibration isolation system using the fitted curve can be expressed as Equation (19). The parameters \(\gamma_{1}\) and \(\gamma_{2}\) depict the nonlinearity of the force, and the force is as follows:  
\[F_{QZS}(x)=\gamma_{1}x^{3}+\gamma_{2}x^{5}. \tag{19}\]  
Assuming that the exciting force is \(f=F_{0}\cos(\omega t+\theta)\), the dynamic equation of the QZS system can be given as:  
\[m\ddot{y}+c\dot{y}+\gamma_{1}y^{3}+\gamma_{2}y^{5}=F_{0}\cos(\omega t+\theta). \tag{20}\]  
For the QZS system, Equation (20) can be written in a non-dimensional form as:  
\[\ddot{y}^{\prime\prime}+2\xi\dot{y}^{\prime}+\alpha\dot{y}^{3}+\beta\dot{y}^{5 }=\overline{F}_{0}\cos(\Omega\tau+\theta), \tag{21}\]  
where \(\dot{y}=\frac{y}{y_{s}}\), \(\Omega=\frac{\omega}{\omega_{n}}\), \(\tau=\omega_{n}t\), \(\omega_{n}=\sqrt{\frac{k_{v}}{m}}\), \(\xi=\frac{c}{2\sqrt{m\omega_{c}}}\), \(\alpha=\frac{\gamma_{1}y^{2}}{k_{v}}\), \(\beta=\frac{\gamma_{2}y^{4}}{k_{v}}\), \(\overline{F}_{0}=\frac{F_{0}}{k_{v}y_{s}}\) and \(\dot{y}^{\prime}=\frac{dg}{d\overline{x}}\). Note that \(y_{s}\) is the static displacement of the linear spring and \(k_{v}\) is the stiffness coefficient of the positive spring.  
Assuming that the response of the whole system is \(\dot{y}=Y\cos(\Omega\tau)\) and substituting it into Equation (21), the frequency-amplitude can be given as below based on the harmonic balance method:  
\[\left(Y\Omega^{2}+\frac{3}{4}\alpha Y^{3}+\frac{5}{8}Y^{5}\right)+\left(2\zeta Y \Omega\right)^{2}=\overline{F}_{0}^{2}. \tag{22}\]  
Equation (22) can be solved to obtain:  
\[\Omega_{1,2}=\sqrt{\frac{H}{Y}-2\xi^{2}\pm\frac{1}{Y}\sqrt{4\xi^{4}Y^{2}-4\xi ^{2}HY+\overline{F}_{0}^{2}}}. \tag{23}\]  
where \(H^{2}=\frac{3}{4}\alpha Y^{3}+\frac{5}{8}\beta Y^{5}\).  
Figure 7: The force versus displacement and the stiffness versus displacement of the QZS system: (**a**) the relationship between the force and displacement of the QZS system, and (**b**) the relationship between the stiffness and displacement of the QZS system.  
The non-dimensional form of the force transmitted through the QZS system and the simplified equation can be given as:  
\[\overline{F}_{0}=2\xi\dot{\hat{y}}+\alpha y^{3}+\beta y^{5}\approx-2\xi Y\Omega \sin\Omega\tau+H\cos\Omega\tau=\sqrt{\left(2\xi Y\Omega\right)^{2}+H^{2}}\cos( \Omega\tau+\phi), \tag{24}\]  
where \(\textit{tan}\phi=\frac{2\xi Y\Omega}{H}\).  
The force transmissibility is given as:  
\[T=\frac{\overline{F}_{t}}{\overline{F}_{0}}=\frac{\sqrt{\left(2\xi Y\Omega \right)^{2}+H^{2}}}{\overline{F}_{0}}. \tag{25}\]  
As shown in Figure 8a, when the damping ratio was increased gradually, the phenomenon of the curve bending to the right was increasingly more inconspicuous and the strongest transmissibility was cut down, which means that the jump phenomenon became inconspicuous. Increasing the damping ratio was beneficial for reducing the initial frequency of vibration isolation as the jump-down frequency degraded in the process of increasing the damping ratio. However, the cost of increasing the damping ratio was that the force transmissibility became gradually larger in the high-frequency range of vibration isolation.  
As shown in Figure 8b, the jump-down frequency and the force transmissibility were larger for the bigger excitation amplitudes, such that the initial frequency of vibration isolation did not meet the requirement of low-frequency vibration isolation. However, the influence of changing the excitation amplitude was not obvious in the high-frequency range of vibration isolation.