## 2 Dynamic Model  
### Model Description  
The dual-stage vibration isolation system with the bio-inspired vibration isolator (BI-DSVI) is illustrated in Figure 1. The control object \(m_{u}\) (which needs vibration isolation from the base excitation \(z_{01},\ z_{02},\ z_{03},\ z_{04}\)) is connected to the intermediate mass \(m_{d}\) with four linear isolators (each with linear spring \(k_{u}\) and linear damping \(c_{u}\)). The intermediate mass is supported on the base by four bio-inspired vibration isolators, each containing a linear damping \(c_{d}\) and a bio-inspired structure [6; 7], as shown in Figure 1a. \(l_{d}\) is the length of the connecting rods, \(k_{d}\) is the stiffness of the transverse spring between the connecting joints, and the initial assembly angle is \(\varphi_{d0}\). The absolute displacements of the control object and the intermediate mass are \(z_{u}\) and \(z_{d}\), respectively. The rotation angles of the control object around its horizontal inertia principal axes paralleling the \(x,y\) axes are \(\alpha\) and \(\beta\), and the rotation angles of the intermediate mass around its horizontal inertia principal axes paralleling the \(x,y\) axes are \(\psi\) and \(\theta\). To simplify the problem, only the translational motion along the height direction of the isolator (i.e., the direction of the coordinate axis \(z\)) and the rotational motion around the horizontal coordinate axes are considered. Thus, there are six generalized coordinates in the dynamic model: \(z_{u},\psi,\theta,z_{d},\alpha,\beta\).  
\[B_{2}(z_{d},\psi,\theta) = \frac{l_{d}cos\varphi_{d0}}{\sqrt{l_{d}^{2}-h_{2}^{2}}}-h_{2} \tag{8}\] \[B_{3}(z_{d},\psi,\theta) = \frac{l_{d}cos\varphi_{d0}}{\sqrt{l_{d}^{2}-h_{3}^{2}}}-h_{3}\] (9) \[B_{4}(z_{d},\psi,\theta) = \frac{l_{d}cos\varphi_{d0}}{\sqrt{l_{d}^{2}-h_{4}^{2}}}-h_{4} \tag{10}\]  
where  
\[\left\{\begin{array}{l}h_{1}=\left[z_{d}-\psi\big{(}L_{y}-a\big{)}-\theta (L_{x}-b)-z_{01}\right]/4,\\ h_{2}=\left[z_{d}+\psi a-\theta(L_{x}-b)-z_{02}\right]/4,\\ h_{3}=\left(z_{d}+\psi a+\theta b-z_{03}\right)/4,\\ h_{4}=\left[z_{d}-\psi\big{(}L_{y}-a\big{)}+\theta b-z_{04}\right]/4.\end{array}\right. \tag{11}\]  
We assume that the base excitation is \(\tilde{z}_{0i}=\left.Acos(2\pi ft+\phi_{0i}),i=1,2,3,4\), where \(A\) is the excitation amplitude, \(f\) is the excitation frequency, and \(\phi_{0i}\) is the initial phase. We define the vibration transmissibility as  
\[T=\frac{\left|\tilde{z}_{u}\right|}{A} \tag{12}\]  
where \(\mid\mid\) represents the amplitude. In other words, the smaller the vibration transmissibility \(T\), the better the vibration isolation performance. By changing the initial phase values at the four base positions, different types of base excitation can be achieved. For instance, \(\phi_{01}=\phi_{02}=0,\ \phi_{03}=\phi_{04}=\pi\) means the rotation excitation of the base around the \(y\) axis, and \(\phi_{01}=\phi_{04}=0,\ \phi_{02}=\phi_{03}=\pi\) means the rotation excitation of the base around the \(x\) axis.  
In the following study of this paper, a sweep signal with amplitude \(A=1\) m/s\({}^{2}\) will be used as the base excitation, in which excitation frequency \(f\) is linearly increased with the rate 0.01 Hz/s from 2 Hz to 12 Hz. The dynamic Equations (1)-(6) will be numerically solved by the 4 th-order Runge-Kutta method with a 0.01 s time step and the initial conditions are \(\left[0\ 0\ 0\ 0\ 0\right]\). The numerical solution is verified by [PERSON] in the Appendix A.