## 2 Mathematical Modeling for Aircraft MR Shock Absorber
### MR Valve Design Considering Pressure Drop of Major and Minor Losses  
Figure 3 shows the configuration of the MR valve, the magnetic field direction for the input current, and the pressure drop curves inside the orifice along the \(z\)-axis. MR valves are divided into entry and MR cores. The MR core is a type of electromagnet that causes the MR fluid to form a magnetic pole by the electric current input of the solenoid coil that is wound around the MR core. The region where the magnetic field path overlaps the MR fluid in the orifice is defined as the magnetic pole. Low carbon steel is mainly used for the MR core to form the path of a magnetic field well. In Figure 3, the state in which the current is applied to the coil is described as on-state, and the state with no current is described as  
Figure 2: Schematic diagram of MR shock absorber: (**a**) configuration; (**b**) free body diagram. off-state. The MR core designed in this work has three solenoid coils, and magnetic poles are placed in four locations. In Figure 2b, the pressure of \(P_{1}\) and \(P_{2}\) exist on both sides of the MR valve; the pressure in the region smaller than 0 is \(P_{1}\) and the pressure in the region larger than \(L\) is \(P_{2}\), based on the z-axis in the graph of Figure 3. The pressure curve for the \(z\)-axis in the figure represents the total pressure drop under the condition that the piston compresses, where \(\Delta P_{major}\) and \(\Delta P_{minor}\) are hydraulic pressure drops, considering the major and minor loss, respectively. \(\Delta P_{y}\) indicates the pressure drop generated by the yield stress of the MR fluid. Such a pressure drop produces the flow rate of the MR fluid, and the flow rate, \(Q\), is calculated as a function of the piston stroke velocity as follows:  
\[Q=\frac{\pi}{4}{D_{1}}^{2}\cdot\hat{s} \tag{5}\]  
where \(\hat{s}\) is the stroke velocity of the piston. The signs of the stroke velocity and flow rate are set to positive during piston compression and negative during piston rebound. For precise control of controllable force, the fluid must be of a laminar flow inside the MR valve, and the magnetic pole should be located in the fully developed region [4]. When operating at low piston velocity, it is common to design without considering the developing region in Figure 3. However, a general aircraft shock absorber has to dissipate the impact energy generated within 1 s at a sink speed (descent rate) of 3.05 m/s, so its operating speed must be rapid [15]. Therefore, when designing an MR shock absorber for the aircraft landing gear system, the auxiliary part must be installed to locate the core in the fully developed region. The name of this auxiliary part is 'Entry', provided in Figures 2a and 3. The Reynolds number in the annular orifice must be calculated in order to determine the entry length for the annular orifice. The Reynolds number, \(Re\), for the annular pipe, taking into account the hydraulic diameter and wetted perimeter, is given by:  
\[Re=\frac{2\rho\cdot Q}{\pi\mu\cdot D_{o}} \tag{6}\]  
where \(D_{o}\) is the orifice mean circumference (diameter) and \(\rho\) and \(\mu\) are density and dynamic viscosity of the MR fluid, respectively; the Reynolds number for the annular pipe must be less than 2000 for laminar flow [16]. In the case of laminar flow, the entry length, \(L_{e}\), can be calculated using the following empirical formula [17]:  
\[L_{e}=0.0322\text{max}(Re)\cdot t_{o} \tag{7}\]  
where \(t_{o}\) is orifice gap size. The length of the entry for the MR valve can be determined with the formula. The recoil valve in Figure 2a performs the same function as the check valve and implements asymmetry of the damping force during compression and rebound. Figure 4 shows the flow paths and the orifice and bypass velocity profiles under compressive and rebounding conditions. The recoil valve is closed by flow, only through the orifice during compression (Figure 4a). The recoil valve is opened by parallel flow through the orifice and bypass during rebound (Figure 4b).