## 2 Hybrid Vibration Isolation System
### Rubber Vibration Isolator  
The rubber vibration isolator was fixed on the base of the hybrid vibration isolation system, with the upper end connected in parallel with the electromagnetic actuator, and the magnetorheological damper installed inside. The structure of the rubber vibration isolator is shown in Figure 11 below. The dynamic characteristics of the rubber vibration isolator are related to temperature, excitation frequency and acceleration, which directly affect the isolation effect. Rubber vibration isolators react with different displacements when excitation forces of the same amplitude and different frequencies are applied, which is the dynamic stiffness  
Figure 8: Nephograms of the magnetic induction intensity and magnetic induction line of the electromagnetic actuator under different frequencies and current excitations. characteristic, \(K_{dyn}\), defined as the dynamic stiffness of the rubber vibration isolator, and its expression is  
\[K_{dyn}(f)=\frac{F(f)}{X(f)} \tag{1}\]  
which describes the transfer function from the excitation force to the displacement of the rubber vibration isolator. In addition, the damping lag angle, \(\theta_{delta}\), is also an important parameter to measure the effect of the passive vibration isolation, and it describes the lag characteristics of the rubber vibration isolator after excitation. The axial dynamic stiffness and the damping lag angle characteristics of the rubber vibration isolator were tested on the MTS 831 dynamic performance testing machine. The MTS testing machine can provide 10 kN excitation in the frequency band of 0.01-1000 Hz. The integrated vibration isolator was installed on the fixture; the upper and lower ends of the testing machine are, respectively, the excitation rod and the test rod; and a displacement sensor and a force sensor are installed at the lower end. The test fixture is shown in Figure 12.  
At room temperature, 0.5 g of excitation force was applied to the rubber vibration isolator, and swept frequency excitation was performed in the range of 0-200 Hz at a rate of 1 oct/min, and the dynamic stiffness curve and damping lag angle curve of the rubber vibration isolator were obtained, as shown in Figures 13 and 14, respectively. The dynamic stiffness curve first decreased and then increased, and then it reached the minimum value at 18 Hz, suggesting that the displacement was maximum under the same amplitude excitation force; that is, the intrinsic frequency of the rubber vibration isolator was reached. The damping lag angle curve first increased, and then it reached the peak value near the intrinsic frequency of 18 Hz, and then it decreased with the increase in the excitation frequency. The test shows that the dynamic stiffness and damping lag angle are good in the low-frequency range, with high dynamic stiffness and large damping, but the dynamic stiffness remains large at high frequency. At this time, the electromagnetic actuator is required to output the electromagnetic force to reduce the displacement of the rubber vibration isolator and the dynamic stiffness of the system, thereby improving the vibration isolation performance. Figure 9: Magnetic output diagram of the electromagnetic actuator under different frequencies and current excitations. Figure 10: Magnetic flux density distribution nephograms and magnetic vector distribution nephograms under three working conditions: **(a)** 12 Hz magnetic flux density distribution nephogram; **(b)** 12 Hz magnetic vector distribution nephogram; **(c)** 50 Hz magnetic flux density distribution nephogram; **(d)** 50 Hz magnetic vector distribution nephogram; **(e)** 300 Hz magnetic flux density distribution nephogram; **(f)** 300 Hz magnetic vector distribution nephogram. Figure 11: Structural diagram of the rubber vibration isolator. Figure 12: Test fixture for the rubber vibration isolator. Figure 13: Test curve of dynamic stiffness. In order to prevent overflow, it is necessary to restrict the output signal of the controller.