## 4 Dynamic Modeling of the Vibration Isolator
### Analysis of Transmissibility  
Transmissibility is a key factor affecting the performance of a vibration isolator, and is normally represented by displacement at low-frequency vibrations. The transmissibility of the vibration isolator in this study was defined as the ratio of the vibration isolation platform output angle to the basic platform input angle. Although joint friction is dynamic and position-dependent, the rolling friction factor is relative small. Therefore, we could ignore the influence from Coulomb's friction force and only consider viscous damping when solving transmissibility. Note that when \(h=-36.87\) as shown in Figure 2, the structural symmetry along center \(o_{1}\) can be achieved, then we have \(m_{0}h\sin\phi+m_{4}d_{1}\sin\phi+2m_{2}d_{1}\sin\phi-\sqrt{2}m_{1}l_{1}\sin \phi\approx 0\), the system is in quasi zero stiffness state. Assuming the input harmonic excitation of the basic platform is \(\theta=-\Theta\cos(\omega t)\)due to the relationships between the motion state load, rods, and input disturbance torsion, the nonlinear dynamic differential equation of the whole isolator can be obtained as follows:  
\[I_{1}\ddot{\phi}+I_{2}\ddot{\theta}+I_{3}(\ddot{\theta}-\ddot{\phi})+c(\dot{ \phi}-\dot{\theta})+M_{QZS}(\phi-\theta)=I_{\theta}\omega^{2}\Theta\cos(\omega t) \tag{16}\]  
where \(I_{1}\), \(I_{2}\) and \(I_{3}\) are the moments of inertia, \(I_{1}=I_{z0(O)}+I_{z4(O)}\), \(I_{2}=I_{z2(O)}+I_{z6(O)}\), \(I_{3}=I_{z1(O)}+I_{z3(O)}+I_{z5(O)}+I_{z7(O)}\); \(\omega\) is the frequency of input disturbance; \(c\) is the viscous damping coefficient; \(I_{\theta}\) is the equivalent moment of inertia for all the rods with a torsion angle of \(\theta\). Note that \(I_{\theta}\omega^{2}\Theta\cos(\omega t)=I_{2}\ddot{\theta}+I_{3}\ddot{\theta}\), assuming a relative angle of the basic and isolation platforms of \(\vartheta=\phi-\theta\), we obtain the angular velocity and angular acceleration. The differential equation for \(\vartheta\) is written as:  
\[I_{\phi}\ddot{\theta}+c\dot{\phi}  
When \(H=0\), the system frequency ratio is expressed as follows:  
\[\Omega_{d}=\sqrt{\frac{N}{M}} \tag{25}\]  
Assuming that the amplitude of the real-time response is \(Z=\Theta+\rho\), \(|Z|=\sqrt{\Theta^{2}+\rho^{2}+2\theta\rho\cos\alpha}\). \[T_{R}=\frac{\sqrt{\Theta^{2}+\rho^{2}+2\Theta\rho\cos\alpha}}{\dot{\theta}} \tag{26}\]  
where \(\cos\alpha=\frac{3\overline{\lambda}_{1}\rho^{3}}{4\Omega}+\frac{5\overline{ \lambda}_{2}\rho^{5}}{8\Omega}+\frac{35\overline{\lambda}_{3}\rho^{7}}{64 \Omega}-\frac{\rho}{\Theta}\). The characteristics of the vibration isolator are mainly influenced by parameters such as the stiffness coefficient, damping coefficient, and mass. Here, the basic system parameters were set at \(\Theta=0.2\), \(\overline{\lambda}_{1}=2.2\), \(\overline{\lambda}_{2}=-0.18\), and \(\overline{\lambda}_{3}=0.08\). To demonstrate the advantages of this vibration isolation system, we set the amplitude to \(\Theta=0.26\). Figure 7 depicts the relationship of transmissibility with the damping ratio. As shown in Figure 7, both the up-jump frequency, \(\Omega_{u}\), and down-jump frequency, \(\Omega_{d}\), appear when \(\Omega\) is \(<\)1. Regarding transmissibility, the nonlinear vibration isolation system has a greater capacity than the linear system. Furthermore, we can easily observe the strong influence of the damping ratio on the vibration isolation capacity. At an input excitation frequency lower than the up-jump frequency \(\Omega_{u}\), the transmissibility decreases proportionally with the increase of damping ratio. In contrast, if the input excitation frequency exceeds the down-jump frequency \(\Omega_{d}\), performance improves inversely with the increase of damping ratio. In addition, it can be seen from the figure that the damping rate has a great influence on the transmissibility in the resonance region, while it is less affected in the vibration isolation region. This phenomenon shows that the equivalent viscous damping has a great influence on the resonance region of the system.