# Experimental Study of Vibration Isolation Characteristics of a Geometric Anti-Spring Isolator
## 3 Mathematical Simulation  
In order to theoretically analyze the vibration isolation performance of the geometric anti-spring isolator, a mathematical simulation based on a four order Runge-Kutta algorithm is carried out. When subjected to base excitation \(v=B\sin(\omega t)\), the governing equation of motion in the geometric anti-spring isolator is  
\[\mathrm{m}\ddot{z}+c\dot{z}+G-F(D_{st}-z)=-m\ddot{v} \tag{8}\]  
\[F(y)=\frac{Ed^{3}w(0)}{12L^{2}}G_{Y}(y)\]  
where \(z=u-v\) represents the relative motion between the excitation displacement \(v\) of base and the displacement \(u\) of mass. \(W=mg\) is the load weight. \(D_{st}\) is the static displacement. The relationship between the load weight and the static displacement is given by \(W=F(D_{st})\). Figure 4: Dimensionless vertical force (**a**) and stiffness characteristic at the optimal working point (**b**) of the geometric anti-spring isolator. Introducing the dimensionless parameters  
\[\hat{z}=\frac{z}{D_{st}},\ \omega_{e}^{2}=\frac{W/D_{st}}{m}=\frac{g}{D_{st}},\ \tau=\omega_{e}t,\ \zeta=\frac{c}{2m\omega_{e}},\ r(\hat{z})=1-\frac{F[\mathrm{D}_{st}(1-\hat{z})]} {W},\ \hat{B}=\frac{B}{D_{st}},\ \Omega=\frac{\omega}{\omega_{e}}\]  
Equation (8) can be written as  
\[\hat{z}^{\prime\prime}+2\zeta\hat{z}^{\prime}+r(\hat{z})=\hat{\mathrm{B}}\Omega ^{2}\sin(\Omega\tau) \tag{9}\]  
where \(\hat{z}^{\prime\prime}\) and \(\hat{z}^{\prime}\) denote the second and first derivative of the dimensionless relative displacement \(\hat{z}\) with respect to the dimensionless time \(\tau\). The intermediate variable \(\omega_{e}\) is the angular frequency of the equivalent linear isolator. For the equivalent linear vibration isolation system, the dimensionless initial isolation frequency \(\Omega_{init}\) is \(\sqrt{2}\). Figure 5 shows the Matlab/Simulink program (Matlab 2012a) used for solving the nonlinear dynamic differential equation (Equation (9)) of motion of the geometric anti-spring isolator. The model of nonlinear restoring force (Figure 5b) is set up based on Equation (7). Giving the initial state conditions (initial displacement and initial velocity), time domain solution of Equation (9) can be obtained using four order Runge-Kutta algorithm. Calculating the ratio of the absolute acceleration amplitude \(|\hat{u}^{\prime\prime}|\) of the load mass to the excitation acceleration amplitude \(|\hat{\phi}^{\prime\prime}|=\hat{B}\Omega^{2}\), the acceleration transmissibility of the geometric anti-spring isolator can be obtained. Here, the effects of the nonlinear geometric parameter, excitation displacement amplitude, static payload weight and viscous damping ratio on the transmissibility of the geometric anti-spring isolator are considered. The transmissibility curves of the geometric anti-spring isolator are shown in Figure 6. Due to the existence of the nonlinearity, the response of the geometric anti-spring isolator under certain excitation levels is dependent on the initial condition. The transmissibility curves of the geometric anti-spring isolator for the situation of gradually increasing frequency and decreasing frequency are different when the nonlinearity is strong, which are distinguished by different types of lines. Figure 5: The mathematical simulation program. (**a**) The simulation model under harmonic displacement excitation. (**b**) The simulation model of the nonlinear restoring force. From Figure 6a, it can be seen that as the nonlinear geometric parameter grows, the resonance frequency of the geometric anti-spring isolator increases and the nonlinearity enhances as well. When \(x_{L}\) is larger than 0.93, the transmissibility curves in the resonance area do not follow the same path. When the excitation frequency is gradually accumulated, a downward jumping phenomena is observed (as is shown by the solid line). When the excitation frequency is gradually reduced, an upward jumping phenomenon is observed (as is shown by the dotted line). The geometric anti-spring isolator shows a hardening-type stiffness characteristic at the rated load.