## 2 Mathematical Modeling for Aircraft MR Shock Absorber
### Pressure Drop and Damping Force under Rebound Conditions  
In the first piston compression during landing, most of the impact energy is dissipated in the shock absorber. Therefore, the hydraulic force during rebound can ignore the minor loss term because the stroke speed is slower than that during piston compression. However, in this work, the effect of minor loss, even in rebound cases, is analyzed. A lower damping force is required for safe landing, which is calculated by comparing the compression case to achieve a high-speed stroke during rebound. The bypass flow path is constructed to implement the lower damping force so that the fluid can flow in parallel, as shown in Figure 4b. When the MR shock absorber rebounds, the parallel flow occurs. Because the recoil valve is opened and fluid flows simultaneously through the orifice and bypass, the flow velocity profiles at the orifice in Figure 4c and bypass in Figure 4d generate the pressure drop. Based on Equations (8)-(14), during the piston rebound, the total pressure drop, \(\Delta P_{o,r}\), acting from the orifice by hydraulics and yields is as follows:  
\[\Delta P_{o,r}=C_{1}\cdot Q_{o}-C_{3}\cdot{Q_{o}}^{2}-3.07\left(\frac{\tau_{ y}(\overline{H})-\tau_{0}}{t_{o}}\frac{k}{j}L_{p,j}+\frac{\tau_{0}}{t_{o}}L\right) \tag{19}\]  
where \(C_{3}\) is the minor pressure loss coefficient for orifice flow, expressed similarly to Equation (11), as follows:  
\[C_{3}=\frac{1}{2}\Sigma k_{o,r}\frac{\rho}{\left(\pi D_{o}\cdot t_{o}\right)^{ 2}} \tag{20}\]where \(\Sigma k_{o,r}\) is the total loss coefficient for the orifice flow under rebound conditions. Because the magnetic field does not act on bypass, the total pressure drop, \(\Delta P_{b,r}\) from the bypass can be expressed by considering only the pressure loss of hydraulic and initial yield stress:  
\[\Delta P_{b,r}=C_{4}\cdot(Q-Q_{o})-C_{5}\cdot(Q-Q_{o})^{2}-3.07\frac{\tau_{0}}{ t_{b}}L \tag{21}\]  
where \(t_{b}\) is the bypass gap size; \(C_{4}\) and \(C_{5}\) are, respectively, the major and minor pressure loss coefficients for APCF in the bypass. Because MR fluid flows parallel through the orifice and bypass, the coefficients of the major and minor pressure loss of the bypass must be considered. Additionally, APCF at the bypass must be considered because APF and annular Couette flow are formed simultaneously by the relative motion of the stroke from the inner wall of the main strut and MR valve. Figure 4d illustrates the APCF well. The coefficients of major and minor pressure loss for the APCF, \(C_{4}\), and \(C_{5}\), are as follows:  
\[C_{4}=\frac{12\mu\cdot L}{\pi D_{b}\cdot t_{b}}^{3}-\frac{24\mu\cdot L}{(\pi D _{1}\cdot t_{b})^{2}} \tag{22}\]  
\[C_{5}=\frac{1}{2}\Sigma k_{b,r}\frac{\rho}{(\pi D_{b}\cdot t_{b})^{2}} \tag{23}\]  
where \(D_{b}\) is the bypass mean circumference and \(\Sigma k_{b,r}\) is the total loss coefficient for bypass flow under rebound conditions. The pressure drop generated in the orifice and bypass during rebound is the same; Equation (19) is equal to Equation (21). Through some mathematical expansion, the flow rate of the orifice, \(Q_{o,r}\), and the total pressure drop, \(\Delta P_{r}\) for piston rebound are calculated as:  
\[Q_{o,r}= \begin{cases}a_{1}+\sqrt{{a_{1}}^{2}+a_{2}}\text{sgn}(C_{5}-C_{3})&, \ a_{2}<0\\ 0&,\ a_{2}\geq 0\end{cases}  \tag{24}\]  
\[\Delta P_{r}=C_{4}(Q-Q_{o,r})-C_{5}(Q-Q_{o,r})^{2}-3.07\frac{\tau_{0}}{t_{b}}L \tag{25}\]  
In the above, the variables \(a_{1}\) and \(a_{2}\) are used to simplify the expression and are defined as follows:  
\[a_{1}:=\frac{2C_{5}\cdot Q-C_{1}-C_{4}}{2(C_{5}-C_{3})} \tag{26}\]  
\[a_{2}:=\frac{C_{4}\cdot Q-C_{5}\cdot Q^{2}}{C_{5}-C_{3}}+\frac{3.07}{C_{5}-C_ {3}}\left(\frac{\tau_{y}(\overline{H})-\tau_{0}}{t_{o}}\sum\limits_{j}^{k}L_{ p,j}+\frac{t_{b}-t_{o}}{t_{b}\cdot t_{o}}\tau_{0}\cdot L\right) \tag{27}\]  
If the pressure drop caused by the flow rate is smaller than the pressure drop caused by the yield stress, depending on the magnetic field, a block-up phenomenon occurs in the orifice [4].