## 2 Structure and Principle of the Seat Suspension System
### Dynamic Model of the MRF Damper
#### 2.2.2 Dynamic Model of the MRF Damper  
To fully utilize the semi-active characteristics of the MRF damper, we needed to build an accurate dynamic model to simulate the real-time performance of semi-active control. The Bouc-Wen model is a mathematical model that was proposed by [PERSON] and [PERSON] [20] to describe the hyperbolic hysteretic characteristics of an MRF damper. It is composed of a spring, a damper, and a hysteretic system, in parallel. The damping force of the MRF damper can be obtained by the following equation:  
\[\left\{\begin{array}{l}F=c_{0}\dot{x}+k_{0}(x-x_{0})+\alpha z\\ \dot{z}=-\gamma\big{|}\dot{x}|z|^{n-1}-\beta\dot{x}|z|^{n}+A\dot{x}\end{array}\right. \tag{1}\]  
where \(F\) is the damping force; \(c_{0}\) is the viscosity coefficient; \(k_{0}\) is the stiffness coefficient; \(x\) and \(\dot{x}\) are the relative displacement and relative velocity at both ends of the damper, respectively; \(x_{0}\) is the relative equilibrium displacement; and \(c_{0}\dot{x}\) is the output force of the damping element. \(k_{0}(x-x_{0})\) represents the output force of the spring element, and \(\alpha z\) is the output force of the hysteretic system; these three parts together constitute the damping force of the MRF damper. Furthermore, \(\alpha\) is the proportion adjustment parameter of the hysteretic force in the damping force, \(z\) is the hysteretic variable, \(n\) is the smoothing coefficient, and \(\gamma\), \(\beta\), and \(A\) are parameters that are used to adjust the smoothness of the model, and the linear characteristics from the yield area to the post-yield area of the curve.  
Figure 4: Damping characteristic diagrams of the MRF damper. (**a**) 10 mm–2 Hz displacement–force diagram; (**b**) 10 mm–2 Hz velocity–force characteristic diagram.  
Figure 3: Dynamics test of the MRF damper. (**a**) E10000 electronic dynamic testing machine; (**b**) MRF damper installed on the test machine.  
In the Bouc-Wen model, eight parameters need to be identified, for which the authors of this paper used the Simulink Design Optimization toolbox. These parameters have different effects on the model. [PERSON] found that \(\gamma\) and \(\beta\) in the model only affect the shape of the hysteretic curve, but not the damping force, and \(n\) only affects the smoothness of the transition curve from the elastic zone to the plastic zone [21]. It can be seen from Figure 3 that the magnitude of current has a significant influence on the amplitude of the damping force. According to the preliminary identification, the values of parameters \(\alpha\) and \(c0\) significantly vary with a change in current; thus, it was assumed that there is a specific relationship between \(\alpha\), \(c0\), and the current. The other parameters were fixed as constants. The determined model parameters are shown in Table 1.  
In Table 1, \(I\) is the input current. The identification results were plotted and compared with the test results in Figure 5. In Figure 5, the experimental data are shown as solid lines, and the simulation results are shown as dotted lines.  
Figure 5 shows that the calculated displacement-force curve and the velocity-force curve calculated with the Bouc-Wen model identified in this article were consistent with the test results.