# Experimental Study of Vibration Isolation Characteristics of a Geometric Anti-Spring Isolator
## 2 Geometric Anti-Spring Structure
The horizontal constraint force \(G_{X}\) is unknown in Equation (4). The objective of the optimization algorithm is to gain the horizontal constraint force \(G_{X}\) of the blade spring under certain a vertical load and a nonlinear geometric parameter. The procedure of the solution algorithm is given as follow. **Step 1**: Giving an initial assuming value \(G_{X}\) of horizontal constraint force for the given vertical load \(G_{Y}\) and nonlinear geometric parameter \(x_{L}\). **Step 2**: Solve the boundary problem mathematically and get the horizontal coordinate \(x_{s}\) of the vertex of the spring. **Step 3**: Calculate the relative error between \(x_{s}\) and \(x_{L}\). If the calculated error is less than the required accuracy e (=1 \(\times\) 10\({}^{-9}\)), record the horizontal force \(G_{X}\) and go to Step 5. Otherwise, go to Step 4. **Step 4**: Update the assuming value of the horizontal constraints force \(G_{X}\) using the genetic algorithm function ga() given by Matlab software (Matlab 2012a) and repeat Steps 2 and 3. **Step 5**: Solve the boundary problem mathematically under the given vertical force \(G_{Y}\) and the recorded optimal horizontal force \(G_{X}\). And get the vertical coordinate \(y_{L}\) of the vertex of the spring. Figure 1: The model of the geometric anti-spring isolator. (**a**) Schematic of the geometric anti-spring isolator. (**b**) Photograph of the geometric anti-spring isolator. (**c**) Cross-section diagrams of the geometric anti-spring isolator. (**d**) The shape of the cantilever blade spring. (**e**) One dimension mechanical model of the blade spring. In this optimization problem, the number of variables for the genetic algorithm function ga(0) is 1. Some key arguments in the options of the genetic algorithm function ga() are listed in Table 1. The other arguments are the defaults. In the solution procedure, when the geometric parameter is larger than 0.9027, optimization terminates when the average change in the fitness value is less than the options.TolFun. The maximum iteration generation is less than 100. The optimization algorithm is time-saving. There is no multi-value solution obtained, which means that the vertical force \(G_{Y}\) and the vertical coordinate \(y_{L}\) of the blade spring's vertex are one to one correspondence and no negative stiffness appears. According to the literature [25], the negative stiffness appears when the nonlinear geometric parameter is smaller than 0.9027. And for the situation that the geometric parameter is less than 0.9027, the time of optimization progress is long when the nondimensional vertical force is close to \(-1.75\) and the multi-value solution can not be achieved. Therefore, the negative stiffness characteristics of the geometric anti-spring \begin{table}
\begin{tabular}{c c} \hline \hline
**Options** & **Value** \\ \hline PlotFcns & @gaplotbestf \\ Generations & 200 \\ CrossoverFraction & 0.2 \\ InitialPopulation & \(-2.0\) \\ MutationFcn & @mutationadaptfeasible \\ TolFun & \(1\times 10^{-12}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The optional parameters of genetic algorithm function ga(0). Figure 2: The flow chart of the genetic algorithm to solve the boundary problem. isolator when the nonlinear geometric parameter is smaller than 0.9027 are not discussed in this paper. This will be studied in the future work. Figure 3a depicts the relationship between dimensionless vertical force \(G_{Y}\) and dimensionless vertical coordinate \(y_{L}\) of spring's vertex. As the nonlinear geometric parameter \(x_{L}\) decreases, the curve of dimensionless vertical force \(G_{Y}\) vs. dimensionless vertical coordinate \(y_{L}\) changes from an approximately straight line to a nonlinear curve. When the dimensionless vertical coordinate \(y_{L}\) approaches to 0.1, dimensionless vertical force \(G_{Y}\) grows slowly, which means that the geometric anti-spring isolator possesses very low local stiffness in these load range.