## 4 Parameter Study
### Influence of the Base Excitation on System Response  
Considering the effect of the base rotation excitation on the control object \(m_{u}\), we define the vibration transmissibility \(T_{v}\) as  
\[T_{v}=\frac{\left|\tilde{z}_{v}\right|}{A} \tag{15}\]  
where \(\tilde{z}_{v}=\tilde{z}_{u}-\frac{l_{d}}{A}\alpha-\frac{B_{u}}{2}\beta\) represents the acceleration along the \(z\) axis of a vertex of the control object \(m_{u}\). Figure 7a shows the vibration transmissibility \(T\) under the three different base excitations investigated in Section 3. The structural parameters of the bio-inspired isolators are taken from the optimal values obtained in Table 2 and the other parameters are the same as in Table 1. When the base excitation phase is 0, the whole system displays upward base acceleration, and the vibration transmissibility \(T\) is relatively large. When inverse-phase base excitation is applied, the vibration transmissibility \(T\) is greatly reduced. To clearly describe the influence of the base rotation excitation on the control object, the comparison curves of \(T\) and \(T_{v}\) are shown in Figure 7b. It can be seen that \(T\) and \(T_{v}\) coincide completely, which means that the motion of the vertex of the control object in the \(z\) direction is consistent with \(z_{u}\). This indicates that the bio-inspired isolator used in the BI-DSVI can well isolate the disturbance due to the base rotational motion. Figure 7: (**a**) The vibration transmissibility \(T\) under three different base excitations; (**b**) the vibration transmissibility comparison of \(T\) and \(T_{v}\) when \(\varphi_{01}=\varphi_{04}=0\), \(\varphi_{02}=\varphi_{03}=\pi\). To develop insight into the dynamic behaviors of the BI-DSVI in vibration isolation under different base excitations, the responses at frequency 2.62 Hz (the first resonance frequency of the BI-DSVI system) are investigated. As shown in Figure 8a,b, under the excitation of the base rolling motion, the vibration acceleration amplitude of the control object along the \(z\) direction is reduced significantly, which results in the lower vibration transmissibility \(T\) at frequency 2.62 Hz shown in Figure 7a. Figure 8c plots the phase trajectory of the BI-DSVI under two different base excitations. After reaching the steady state, the first excitation condition (\(\phi_{01}=\phi_{02}=\phi_{03}=\phi_{04}=0\)) exhibits periodic motion, while the second excitation condition (\(\phi_{01}=\phi_{04}=0,\ \phi_{02}=\phi_{03}=\pi\)) exhibits periodic motion in which the vibration amplitude varies in a small range. It can be obtained from the above results that the base rotational motion has little influence on the translational motion along the \(z\) direction of the control object. In other words, the bio-inspired isolator can effectively isolate the vibration caused by the base rotation excitation. The dual-stage vibration isolation system is usually sensitive to base excitation with the natural frequency of the bottom vibration isolators. On this account, a simulation for a single bio-inspired isolator with an oscillator mass is conducted to find the natural frequency of the bio-inspired isolators. In the above studies, the intermediate mass and the control object mass are 12 kg in total--that is, the oscillator mass should be 3 kg. According to the optimal structural parameters and the oscillator mass, the vibration transmissibility of the single bio-inspired isolation system is calculated and is shown in Figure 9a. It can be seen that the natural frequency of the bio-inspired isolator is 1.72 Hz. Figure 9b-d show the time history results, FFT results, and phase plots of the BI-DSVI under two different base excitations \(\phi_{01}=\phi_{02}=\phi_{03}=\phi_{04}=0\) and \(\phi_{01}=\phi_{04}=0,\ \phi_{02}=\phi_{03}=\pi\) at frequency 1.72 Hz. The results present similar dynamic behaviors to the base excitations with 2.62 Hz.