## 2 Damping Force Characteristics of MREA under Impact Load
### Experimental Setup for Damping Force  
The test setup of the MREA yield force is shown in Figure 4. The test setup was based on an electro-servo-hydraulic fatigue testing machine. One side of the MREA was attached to a linear variable differential transformer (LVDT). The opposite side was attached to a load cell to measure the damper force. The electromagnetic coils in the MREA were activated by a power supply. As shown in Equation (1), in addition to the viscous damping force \(F_{v}\) and yield force \(F_{MR}\), the output force of the MREA also includes the friction force \(F_{f}\) produced by the MREA seal on the piston rod. That is to say, \(F=F_{v}+F_{MR}+F_{f}\). The viscous damping force \(F_{v}\) is related to the piston velocity, while the yield force \(F_{MR}\) and the friction force \(F_{f}\) are independent to the piston velocity. In order to measure the yield force under different applied currents, the piston should work at a quite low velocity for a long time. The MREA is excited using a low frequency triangular wave displacement. In this case, the viscous damping force \(F_{v}\) can be approximate considered as 0. The friction force \(F_{f}\) can be measured at 0 A applied current. Figure 5 shows the tested damping force response of the MREA under triangular wave displacement excitation. The excitation frequency and amplitude are 0.05 Hz and 20 mm, respectively. The velocity of piston rod is \(v=4\)/z = 0.004 m/s. The damping force of the MREA can be approximated as frictional force \(F_{f}\) at zero applied current condition. The friction force of the MREA developed in this study is \(F_{f}\)= 0.16 kN (\(I\) = 0 A in Figure 5). When the applied current is \(I\) = 1 A, the damping force is 1.6 kN. With the increase in applied current, the damping force of the MREA also increases monotonically. The damping force of the MREA is 3.96 kN at the maximum applied current of 4 A.  
Figure 4: Experimental setup for the yield force of MREA. Figure 5: The MREA yield force characteristic under triangular excitation. The yield force can be expressed as the difference between the total damping force and the friction force, that is,  
\[F_{MR}=F-F_{f} \tag{13}\]  
Therefore, the relationship between yield force and applied current can be depicted in Figure 6. The mathematical relationship between yield force and current can be fitted to a polynomial, as shown in Equation (14). In Figure 6, the R-squared value is R\({}^{2}\) = 0.9974, which shows the curve fits well. \[F_{MR}=0.052I^{4}-0.4016I^{3}+0.7781I^{2}+1.0299I \tag{14}\]  
The highspeed impact test was implemented on CEAST 9350 Drop Tower Impact Setup produced by INSTRON company, Boston, USA as shown in Figure 7. The impact setup contained a weight system and a data acquisition system. The weight system consisted of hammer, house separator, and combinable mass. The total mass of the weight system was the drop mass. The data acquisition system included data acquisition, velocity sensor, and load cell, which was used to measure hammer velocity and impact load. The damping force and piston velocity of the MREA can be directly read on the computer. Figure 8 shows the damping force characteristic of the MREA under different piston velocities. The experimental observations are compared with BPM model. It can be seen the damping force predicted by BPM model agreed well with the experimental data. The BPM model can describe the damping force behavior accurately under high-speed impact. The damping force of the MREA at the applied current of 4 A was much larger than that at zero current. The damping force measured when the piston velocity was 5.0 m/s at the applied current of 4 A was 6.99 kN, which was about 2 times greater than that with no current supplied (3.49 kN). The damping force increased quadratically with piston velocity, as shown in Figure 8. A quadratic polynomial between damping force and piston velocity at 0 A applied current is shown in Equation (15), which, from Figure 8, accurately represents the measured data.