## 2 Mathematical Modeling for Aircraft MR Shock Absorber
### Pressure Drop and Damping Force under Compression Conditions  
When the piston is compressed, the fluid flows through the flow path, as shown in Figure 4a. Designed to close the recoil valve during piston compression, no fluid flows on the outer wall of the valve, including the bypass. That is, the pressure drop is generated by the velocity profile for off-state shown in Figure 4c because the fluid flows only through the orifice inside the valve. Derived from the Navier-Stokes equation, the pressure drops for the major and minor loss in the case of compression, \(\Delta P_{major}\) and \(\Delta P_{minor}\), are given by:  
\[\Delta P_{major}=C_{1}\cdot Q_{o} \tag{8}\]  
\[\Delta P_{minor}=C_{2}\cdot Q_{o}{}^{2} \tag{9}\]  
where \(C_{1}\) and \(C_{2}\) are the major and minor pressure loss coefficients, respectively, and \(Q_{o}\) represents the flow rate into the orifice. \(Q_{o}\) is equal to \(Q\) under compressed conditions because the recoil valve is closed so that the fluid flows only through the orifice. The viscosity causes the major loss, and the inertia of MR fluid causes the minor loss due to the density of the fluid. Therefore, the minor loss cannot be ignored in a system with large fluid inertia such as an aircraft shock absorber, which has a high stroke velocity. \(C_{1}\) and \(C_{2}\) are calculated as follows:  
\[C_{1}=\frac{1}{8}f\cdot Re\frac{\mu\cdot L}{\pi D_{o}\cdot t_{o}{}^{3}} \tag{10}\]  
\[C_{2}=\frac{1}{2}\Sigma k_{o,c}\frac{\rho}{(\pi D_{o}\cdot t_{o})^{2}} \tag{11}\]  
where \(f\) is the friction factor for laminar flow; \(L\) is the orifice and bypass length of the MR valve; and \(\Sigma k_{o,c}\) is the total loss coefficient, the sum of loss coefficients determined by the geometric shape of the flow path for orifice flow under compression conditions. The  
Figure 3: MR valve configuration and pressure drop in the annular orifice under compression condition. product of friction factor and Reynolds number, \(f\cdot Re\), in Equation (10) can be calculated by the equation of geometry ratio for the annular orifice [18]:  
\[f\cdot Re=\frac{128{t_{o}}^{2}}{{D_{o}}^{2}+{t_{o}}^{2}-2D_{o}\cdot t_{o}/(\ln(D _{o}+t_{o})-\ln(D_{o}-t_{o}))} \tag{12}\]  
As shown in Figure 3, the total loss coefficient from the MR valve is calculated by the sum of the loss coefficient of the sudden contraction, the developing region, and the sudden expansion. MR fluid can be modeled using the Bingham-plastic-viscous model. Therefore, in the off-state case, the pressure drop is originated by the dynamic viscosity and the initial yield stress of MR fluid. In the case of on-state, the magnetic poles generate an additional pressure drop due to the magnetic field, which is defined as the pressure drop due to yield stress. Assuming that the initial yield stress that acts without the magnetic field is infinitesimal, and the core is designed so that all magnetic fields in the magnetic poles act at similar levels depending on the input current, then the pressure drop for the yield stress, \(\Delta P_{y}\), can be expressed by the following simple equation:  
\[\Delta P_{y}=\left(c(\overline{H})\frac{\tau_{y}(\overline{H})}{t_{o}}\sum_{ j}^{k}L_{p,j}+3.07\frac{\tau_{0}}{t_{o}}\left(L-\sum_{j}^{k}L_{p,j}\right) \right)\text{sgn}(\hat{s}) \tag{13}\]  
Figure 4: Fluid flow in compression and rebound conditions: (**a**) flow path in compression, valve closed; (**b**) flow path in rebound, valve open; (**c**) fluid velocity profile of the orifice: annular Poiseuille flow (APF); (**d**) fluid velocity profile of the bypass: annular Poiseuille–Couette flow (APCF). where \(\tau_{0}\) is the initial yield stress at zero magnetic field, characteristic of the Bingham-plastic-viscous model; \(\tau_{y}(\overline{H})\) is the yield stress, a function of magnetic intensity; \(\overline{H}\) represents the mean value of magnetic intensity at all magnetic poles; \(k\) is the total number of magnetic poles; and \(L_{p,j}\) is the length of the \(j\)-th magnetic pole;'sgn' denotes signum function.