# Experimental Study of Vibration Isolation Characteristics of a Geometric Anti-Spring Isolator
## 2 Geometric Anti-Spring Structure
Taking a first order derivative to \(G_{Y}\) with respect to \(y_{L}\), dimensionless dynamic stiffness \(K_{d}\) is obtained (Figure 3b,c). With the decreasing of the nonlinear geometric parameter \(x_{L}\), the nonlinearity of the geometric anti-spring isolator is largely enhanced and the dimensionless lowest dynamic stiffness decreases. When \(x_{L}=0.9027\), the minimum stiffness approaches zero. In order to achieve better vibration isolation performance, it is essential to settle the static equilibrium point of the isolator at  
Figure 3: Dimensionless theoretically numerical result of the geometric anti-spring isolator. (**a**) Dimensionless vertical force. (**b**) Dimensionless vertical stiffness vs. vertical coordinate. (**c**) Dimensionless vertical stiffness vs. vertical force. the minimum stiffness point, which is called the optimal or rated working point. As the nonlinear geometric parameter decreases, the range of displacement for low stiffness is enlarged but the range of the load goes to be narrow. When the nonlinear geometric parameter is small, the change of the load will largely increase the dynamic stiffness of the isolator in low stiffness area. According to the vibration theory, the natural angular frequency of the geometric anti-spring structure can be calculated as follows,  
\[\omega_{Y}^{2}=\frac{1}{F_{Y}/g}\bigg{(}\frac{\partial F_{Y}}{\partial Y_{L}} \bigg{)}_{X_{L}} \tag{5}\]  
And the dimensionless angular frequency is given by:  
\[\frac{L}{g}\omega_{Y}^{2}=\frac{1}{G_{Y}}\bigg{(}\frac{\partial G_{Y}}{ \partial y_{L}}\bigg{)}_{x_{L}} \tag{6}\]  
The dimensionless minimum dynamic stiffness \(K_{d\text{min}}\) and corresponding dimensionless natural angular frequency for different dimensionless horizontal coordinates \(x_{L}\) are shown in Table 2. The corresponding dimensionless vertical coordinate \(y_{0}\) and vertical force \(G_{y_{0}}\) at the rated working point are given at the same time. As the nonlinear geometric parameter decreases, dimensionless minimum dynamic stiffness and natural angular frequency decrease. The dimensionless vertical coordinate increases with the decrement of the nonlinear geometric parameter but vertical force slightly decreases. The carrying capacity of the geometric anti-spring structure reduces with the reduction of the nonlinear geometric parameter. According to the mathematically theoretical analysis results, the relationship between dimensionless restoring force and displacement of the geometric anti-spring isolator for different nonlinear geometric parameters can be accurately fitted with a seven order polynomial function:  
\[G_{Y}(y)=a_{1}y+a_{2}y^{2}+a_{3}y^{3}+a_{4}y^{4}+a_{5}y^{5}+a_{6}y^{6}+a_{7}y ^{7} \tag{7}\]  
Here, \(y=y_{L}-y_{L}|_{G_{Y}=0}\) is the dimensionless vertical displacement of the geometric anti-spring isolator and \(a_{i}(i=1,2,3,4,5,6,7)\) are the fitting coefficients, which are shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(x_{L}\) & \(y_{0}\) & \(G_{y_{0}}\) & \(K_{d\text{min}}\) & \(\  
Figure 4a shows the dimensionless vertical force of the geometric anti-spring isolator for different nonlinear geometric parameters, where the gray solid line represents the mathematical calculated results based on the genetic algorithm and different types of color lines represent the polynomial fitting results. It can be seen that the curve of force vs. displacement of the geometric anti-spring isolator in low-frequency area goes much flatter with the decrement of the geometric parameter. The low dynamic stiffness characteristic is also apparent. However, the static displacement of the geometric anti-spring isolator in the low stiffness area increases as the nonlinear geometric parameter decreases, which means that the static stiffness of the isolator will reduce with the decrement of the nonlinear geometric parameter. Figure 4b shows the dimensionless dynamic stiffness characteristic at the rated working point.