## 2 Damping Force Characteristics of MREA under Impact Load
### Damping Force Model of the MREA  
Compared with the effective cross-sectional area of the piston, the annular gap is quite small so that the MR fluid flow velocity in the annular gap is amplified relative to piston velocity. When the piston velocity becomes large, the MR fluid flow inside the gap is complex. [PERSON] et al. [33] established a Bingham-Plastic-Minor (BPM) model to describe the damping force characteristic of the MREA under an impact load. This model uses the Reynolds number of MR fluid in an annular gap to analyze the motion state of MR fluid. The damping force can be described as  
\[F=\left(\Delta P_{\eta}+\Delta P_{ml}\right)A_{p}+\left(\Delta P_{\tau}A_{p}+F _{f}\right)sign(v) \tag{1}\]  
where \(\Delta P_{\eta}\), \(\Delta P_{ml}\), and \(\Delta P_{\tau}\) are the pressure drops due to viscous flow, minor losses, and the MR effect, respectively. \(A_{p}\) denotes the effective area of the piston. \(F_{f}\) is the friction force of the MREA, which can be measured by experiment. \(v\) is the piston velocity. The pressure drop due to the MR effect depends on the magnetic field in the annular gap. However, the fluid viscous pressure drop and minor loss pressure drop are a consequence of fluid flow. Ignoring friction force, the field-off pressure drop can be expressed as  
\[\Delta P_{field-off}=\Delta P_{\eta}+\Delta P_{ml} \tag{2}\]  
When the current is applied in the MREA, the pressure drop of the MREA can be expressed as  
\[\Delta P_{field-on}=\Delta P_{field-off}+\Delta P_{\tau} \tag{3}\]  
The pressure drop caused by viscous flow in the MREA is  
\[\Delta P_{\eta}=\frac{\rho}{2}V_{s}^{2}\frac{fL}{2d} \tag{4}\]  
where\(f\) is the Darcy friction coefficient relating to the fluid flow state in the annular gap. Here, \(\rho\) is the density of the MR fluid, \(L\) is the total length of the annular gap, \(d\) is the width  
Figure 3: Magnetic flux density with various applied current. Figure 2: Magnetic analysis results with 2 A applied current. of the annular gap, and \(V_{g}\) is the mean fluid velocity. Note that \(V_{g}=A_{p}V_{p}/A_{g}\), where \(V_{p}\) is the piston absolute velocity, \(V_{p}=\left|\,v\,\right|.\)\(A_{g}\) is the cross-sectional area of the annular gap, given by  
\[A_{g}=\frac{\pi}{4}\Big{[}(D_{in}+d)^{2}-D_{in}^{2}\Big{]} \tag{5}\]  
The Reynolds number of the MR fluid in annular gap can be expressed as  
\[Re=\frac{2\rho V_{g}d}{\eta} \tag{6}\]  
where \(\eta\) is the viscosity of the MR fluid. The MR fluid flow state in the annular gap is related to the Reynolds number. The Reynolds number is proportional to the velocity of the piston. When the piston velocity of the MREA is sufficiently low so that the Reynolds number is \(Re\leq 2000\), the MR fluid is in laminar flow state. The Darcy friction coefficient can be expressed as  
\[f=\frac{96}{Re} \tag{7}\]  
When the Reynolds number is \(2000\leq Re\leq 4000\), the MR fluid in an annular gap is in a transitional flow state. The Darcy friction coefficient can be expressed as  
\[f=(1-\alpha)\frac{96}{2000}+\frac{\alpha}{\left\{1.8\log_{10}\left[\left( \frac{\epsilon}{3.7D_{h}}\right)^{1.11}+\frac{6.9}{4000}\right]\right\}^{2}} \tag{8}\]  
where \(\alpha\) is a constant and \(\alpha=(Re-2000)/(4000-2000)\). Here, \(\epsilon\) is the average pipe wall roughness and was chosen to be 0.016 mm based on the wall roughness for drawn tubing given by [PERSON] et al. (2000). \(D_{h}\) is the hydraulic diameter, and \(D_{h}=2d\). When the velocity of a piston rod is so large that the Reynolds number is \(Re\geq 4000\), the MR fluid in an annular gap is in a turbulent flow state. The Darcy friction coefficient can be expressed as  
\[\Delta P_{field-off}=\Delta P_{\eta}+\Delta P_{ml} \tag{2}\]0  
The pressure drop due to minor losses contains four components as in \[\Delta P_{field-off}=\Delta P_{\eta}+\Delta P_{ml} \tag{2}\]1  
1. Entrance effect for flow from region A1 to A2;
2. Exit effect for flow from region A2 to A3.