## 2 Mathematical Modeling for Aircraft MR Shock Absorber  
### Damping Force of MR Shock Absorber  
Figure 1 shows the basic configuration of an aircraft landing gear system and a lumped diagram of the MR shock absorber. For MR shock absorbers, not only the pneumatic and hydraulic force of the oleo-strut, but also the controllable MR force, are added. MR force can change the damping force depending on the input current generated by the controller, which is effective for landing performance. There are several different types of MR shock absorber that can be devised for landing gear systems. In this work, a cylindrical tube type, shown in Figure 2, is proposed.  
The left of the diagram is the direction of the aircraft fuselage, and the right is the direction of the ground where the tire and wheel are located. The inside of the shock absorber is divided into an upper chamber, a lower chamber, and a gas chamber, with the MR valve and separator as boundaries. The upper and lower chambers are filled with MR fluid, and the gas chamber can be filled with air or nitrogen. Figure 2b shows a free-body diagram of the internal pressure and the external force acting on the MR shock absorber. From the figure, the main strut of the MR shock absorber is fixed to the aircraft fuselage, and the piston makes a relative motion depending on the stroke displacement, \(s\), by the ground reaction. The external force in the direction of the piston axis is called a strut force, \(F_{S}\). When the strut force is applied, the MR fluid flows through the MR valve based on the relative motion of the piston, and the damping force is generated to resist the strut force. In addition, the flow causes a change in the amount of fluid in the lower chamber, and the separator induces a pressure change in the gas chamber while making relative movements inside the piston to compensate for the changing fluid volume and the pneumatic pressure increases. The upper, lower, and gas chamber pressures are defined as \(P_{1}\), \(P_{2}\), and \(P_{gas}\), respectively. The MR shock absorber is a form in which a damper and a spring are combined. In other words, the difference between \(P_{1}\) and \(P_{2}\) generates a damping force responsible for the damper's function, and \(P_{gas}\) generates the pneumatic pressure, which acts as the gas spring. The difference between \(P_{1}\) and \(P_{2}\) is the pressure drop, \(\Delta P\), which can be expressed as follows:  
\[\Delta P=P_{1}-P_{2} \tag{1}\]  
where \(P_{1}\) and \(P_{2}\) are the pressure of the upper and lower chambers, respectively. Using Equation (1) and the condition that \(P_{gas}\) is equal to \(P_{2}\) in the quasi-equilibrium state,  
Figure 1: Configuration of aircraft landing gear and MR shock absorber.  
the strut force, \(F_{s}\), the damping force, \(F_{d}\), and the pneumatic force, \(F_{gas}\), are calculated, respectively, as follows:  
\[F_{s}=F_{d}+F_{gas} \tag{2}\]  
\[F_{d}=\frac{\pi}{4}D_{1}{}^{2}\cdot\Delta P \tag{3}\]  
\[F_{gas}=\frac{\pi}{4}D_{2}{}^{2}\cdot P_{gas} \tag{4}\]  
where \(D_{1}\) is the inner diameter of the main strut, \(D_{2}\) is the outer diameter of the piston, and \(P_{gas}\) is the pneumatic pressure. Due to the volume ratio of the separator and the piston displacements, it is noted that the pneumatic force is expressed as the product of the pneumatic pressure and the cross-sectional area of the outer diameter of the piston, not the inner diameter. In this study, the description of the pneumatic pressure is omitted because the pressure drop for the major and minor losses will be focused on.