# Experimental Study of Vibration Isolation Characteristics of a Geometric Anti-Spring Isolator
## 2 Geometric Anti-Spring Structure  
The geometric anti-spring isolator (Figure 1a) consists of several quasi-trapezoid cantilever blade springs. The bottoms of blade springs are fixed to a massive base with V-type fixtures and their vertices are connected to each other through a central load plate. Both the bottoms and the vertices of the blade spring are kept at a constant angle to the horizontal direction. The central load plate connected with the mass load is limited to move along the vertical direction. The shape of cantilever blade spring is shown in Figure 1d. The effective length of spring is \(L\), and the width of the blade spring is represented by \(\omega(l)\), with \(l\in[0,L]\). The shape function of the blade spring is  
\[\gamma(\xi)=\frac{w(l)}{w(0)}=[c_{1}+c_{2}\cos(\beta\xi)+c_{3}\sin(\beta\xi)] ^{-1} \tag{1}\]  
where \(c_{1}=-0.377\), \(c_{2}=1.377\), \(c_{3}=0.195\) and \(\beta=1.361\). This particular shape is chosen to achieve a uniform inner stress distribution [25]. Meanwhile, total stress of the spring will be the smallest in comparison to that with other shapes when the vibration isolation structure works at the low stiffness area. The blade spring has a constant thickness, which is much smaller than both the length and the width of the spring. According to the geometric symmetry, the mechanical property of the geometric anti-spring structure can be obtained through mechanical analysis of single blade spring. Figure 1e shows a one dimension mechanical model of the blade spring, in which the influence of thickness is ignored. Rectangular coordinate system O-(X,Y), of which the origin O is sited at the bottom of the blade spring, X axis represents the horizontal direction and Y axis is for the vertical direction, is established to analyze the mechanical property of the blade spring. The variable \(l\) is the curvilinear coordinate along the length direction of the blade spring. It is convenient to define \(\theta(l)\) as the angle between the tangent of curvilinear and the positive direction of X axis of the rectangular coordinate system. The quantity \(F_{X}\) represents horizontal constraint force and the quantity \(F_{Y}\) represents vertical payload applied on the vertex of the blade spring. The relationship between the dimensionless force \(G_{i}\) and actual force \(F_{i}\) is shown as follows. \[G_{i}=\frac{12L^{2}}{Ed^{2}w(0)}F_{i},\ i=X,Y \tag{2}\]  
Here, \(E\) is [PERSON]'s modulus of the material of blade spring. Dimensionless vertical coordinate \(y_{L}\) and dimensionless horizontal coordinate \(x_{L}\) of blade spring's vertex are given by  
\[ \begin{array}{l}x_{L}=\frac{X}{X}=\int_{0}^{1}\cos\xi d\xi\\ y_{L}=\frac{Y}{L}=\int_{0}^{1}\sin\xi d\xi\end{array}  \tag{3}\]  
Here, \(\xi=l/L\) is dimensionless arc length. According to the literature [25], the mechanical properties of the blade spring can be described by the first order differential boundary equations as follows:  
\[\left\{ \begin{array}{c}\frac{dp}{d\zeta}=G_{X}\sin\theta(\xi)-G_{Y}\cos\theta( \xi)\\ \frac{d\theta}{d\xi}=\gamma(\xi)p\end{array} \right. \tag{4}\]  
Here, \(p\) is an intermediate parameter. Boundary conditions are \(\theta(0)=\pi/4\), \(\theta(L)=-\pi/6\). In general, the exact analytical solution of Equation (4) cannot be obtained theoretically with an explicit expression formulation. The dimensionless horizontal coordinate \(x_{L}\) of spring's vertex is the key parameter to distinguish the mechanical characteristic of the geometric anti-spring isolator under different work state. For the sake of simplicity, the dimensionless horizontal coordinate of the vertex of the spring is referred to the nonlinear geometric parameter in this paper. Utilizing a MATLAB program based on genetic algorithm, the approximately numerical solution of Equation (4) was obtained. The flow chart of the algorithm to solve the boundary problem of Equation (4) is shown in Figure 2. This solution algorithm transforms the differential boundary problem to an optimization problem.