## 3 The SGC Post-Buckling Vibration Isolator
### Modeling  
Since the vibration separator includes 16 post-buckling springs and its vibration displacement is limited to a small range, the damping coefficient of the system is constant, and the elastic recovery is 16P according to Equation (11). The vibration isolation system  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{E}\) (\(\mathbf{\mathrm{Pa}}\)) & \(\mathbf{D}\) (\(\mathbf{\mathrm{m}}\)) & \(\mathbf{K_{1}}\) & \(\mathbf{K_{2}}\) & \(\mathbf{K_{3}}\) & \(\mathbf{P_{0}}\) (\(\mathbf{\mathrm{N}}\)) \\ \hline \multirow{6}{*}{\(3.5\times 10^{4}\)} & 0.02 & 2.1211 & 4.7546 & 420.2 & 0.330723 \\  & 0.025 & 2.7272 & 6.1134 & 540.27 & 0.807425 \\  & 0.03 & 3.3333 & 7.4715 & 660.34 & 1.674255 \\  & 0.035 & 3.9391 & 8.83 & 780.35 & 3.10176 \\  & 0.04 & 4.5453 & 10.189 & 900.44 & 5.29161 \\ \(3.5\times 10^{4}\) & & 2.1211 & 4.7546 & 420.2 & 0.330723 \\ \(4.5\times 10^{4}\) & & 5.1783 & 11.608 & 1025.9 & 0.4252194 \\ \(5.5\times 10^{4}\) & 0.02 & 10.737 & 24.069 & 2127.1 & 0.519715 \\ \(6.5\times 10^{4}\) & & 19.892 & 44.591 & 3940.7 & 0.614197 \\ \(7.5\times 10^{4}\) & & 33.938 & 76.077 & 6723.3 & 0.708697 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The stiffness coefficient and initial load.  
Figure 6: Comparison of spring recovery force curves under different parameters. (**a**) E = \(3.5\times 10^{4}\) Pa; (**b**) D = 0.02 m.  
Figure 7: The physical model of vibration isolator.  
model shown in Figure 8 mainly blocks the vibration propagation from the foundation to the system.  
The kinetic differential equation of the SGC vibration isolation system can be written as  
\[m\ddot{x}+c(\dot{x}-\dot{u})+k_{1}(x-u)+k_{2}(x-u)^{2}+k_{3}(x-u)^{3}=0 \tag{12}\]  
by using \(z=x-u\), the dynamic equation is expressed as  
\[m\ddot{z}+c\dot{z}+k_{1}z+k_{2}z^{2}+k_{3}z^{3}=-m\ddot{u} \tag{13}\]  
where \(m\) is mass, \(c\) is damping coefficient, \(k_{1}\), \(k_{2}\), \(k_{3}\) are stiffness coefficients, \(x\) is vibration displacement,\(f\) is force excitation, and \(u\) is foundation displacement excitation.  
According to the harmonic balance method (HBM), when the base is motivated by a sinusoidal function, such as  
\[\dddot{u}=\Lambda\cos(\omega t+\gamma) \tag{14}\]  
the solution of the Equation (13) can be expressed in the following form:  
\[z=b_{0}+b_{1}\cos\omega t \tag{15}\]  
where \(\Lambda\) is the base acceleration excitation amplitude, \(\omega\) is the circle frequency, \(t\) is the time, \(\gamma\) is the phase, \(b_{0}\) is the nonlinear response constant, and \(b_{1}\) is the response harmonic.  
Substituting Equations (14) and (15) into Equation (13), the amplitude frequency response characteristics of vibration isolation system is obtained as  
\[(c\omega b_{1})^{2}+\left(\begin{array}{c}-m\omega^{2}b_{1}+k_{1}b_{1}\\ +2k_{2}b_{0}b_{1}+\frac{3}{4}k_{3}b_{1}{}^{3}\end{array}\right)^{2}=(-m\Lambda )^{2} \tag{16}\]  
where the phase satisfies the following formula  
\[\tan\gamma=\frac{c\omega}{-m\omega^{2}+k_{1}+2k_{2}b_{0}+\frac{3}{4}k_{3}b_{1} {}^{2}} \tag{17}\]  
and the damping coefficient is approximately taken as \(c\approx 2\zeta\sqrt{mk_{1}}\); \(\zeta\) is the viscous damping ratio.  
The numerical solution of the Runge-Kutta method (RKM) [30] is used to verify the accuracy of the HBM, and the parameter of vibration isolation system is selected as: \(m=0.529\) kg; \(\zeta=0.3\); \(k_{1}=33.9376\); \(k_{2}=76.0736\); \(k_{3}=6723.2\); \(\Lambda=0.05\) m/s\({}^{2}\); \(\omega=7.262\) Hz. The analytical and numerical results of the vibration isolation system are shown in Figure 9, and the approximate analytical solution obtained by the HBM agrees well with the numerical solution obtained by the RKM.  
Figure 8: Passive vibration isolation model.