# Parametric Analysis on Landing Gear Strut Friction of Light Aircraft for Touchdown Performance
## 2 Landing Gear Modeling and Validating  
### Landing Gear Dynamics Model  
In this subsection, the aircraft half-axle main landing gear (MLG) is selected as the research object, and a two-degrees-of-freedom spring damping model is established, as shown in Figure 1. The aircraft landing buffer response is simplified as the dynamic response of two masses under spring damping constraint in this model [7]. The two masses are the sprung mass and unsprung mass. The sprung mass is the airframe structure and the structural mass above the outer cylinder of the shock absorber, and the unsprung mass is the other structure of the landing gear below the shock absorber piston rod. The oleo-pneumatic shock absorber consists of upper and lower chambers separated by orifices and the metering pin. The upper part of the upper chamber is filled with pressurized nitrogen to provide a gas spring, and the other spaces of the two chambers are filled with oil to provide oil damping. The tire force acting point of the half-axle landing gear is not on the strut axis, which will augment the normal force in the bearing area. In this way, the half-axle landing gear can fully reflect the impact of the friction load on the landing buffer. Figure 1: Schematic of the landing gear with two degrees of freedom. According to the mathematical model shown in Figure 1, the dynamic equilibrium governing equation of motion for the main landing gear is written as  
\[\left\{ \begin{array}{l}m_{1}\ddot{z}_{1}=m_{1}g-F_{a}-F_{h}-F_{f},\\ m_{2}\ddot{z}_{2}=-F_{V}+m_{2}g+F_{a}+F_{h}+F_{f},\end{array} \right. \tag{1}\]  
where \(m_{1}\) and \(m_{2}\) indicate the sprung mass and unsprung mass, respectively; \(z_{1}\) and \(z_{2}\) denote the vertical displacement of the sprung mass and unsprung mass, respectively; \(F_{a}\), \(F_{h}\), and \(F_{f}\) represent the gas spring stiffness force, oil damping force, and friction force in the absorber strut, respectively. \(F_{V}\) is the ground vertical force acting on the tire. The gas spring stiffness force is related to initial gas volume closely, the equation can be expressed as  
\[F_{a}=A_{a}\bigg{[}P_{0}\bigg{(}\frac{V_{0}}{V_{0}-A_{a}S}\bigg{)}^{\gamma}-P_ {atm}\bigg{]}, \tag{2}\]  
where \(A_{a}\) is the gas compressed area, \(P_{0}\) is the gas initial pressure, \(V_{0}\) is the gas initial volume, \(S\) is the shock absorber stroke, \(\gamma\) is the gas polytropic exponent, and \(P_{atm}\) is the atmospheric pressure. The oil damping force is proportional to stroke slide velocity squared and parameters of the orifice, which is presented by the equation  
\[F_{h}=\rho\frac{A_{h}^{3}}{2C_{d}^{2}A_{d}^{2}}\dot{s}|\dot{s}|, \tag{3}\]  
where \(\rho\) is the mass density of oil, \(A_{h}\) is the oil compressed area of the absorber, \(A_{d}\) is the orifice bypass area, \(C_{d}\) is the oil damping discharge coefficient, and \(\dot{S}\) is the stroke slide velocity. The friction forces in the strut contain the journal friction force and seal friction force. The journal friction force is induced by the normal force acting on the upper and lower bearing area. The seal friction force result from the friction of internal seals in the shock absorber depends on the internal gas pressure. The friction forces in the shock strut are described by equation  
\[F_{f}=F_{nf}+F_{sf}, \tag{4}\]  
where \(F_{nf}\) is the journal friction force and \(F_{sf}\) is the seal friction force. The seal friction force depends on the internal gas pressure [11] and is expressed as  
\[F_{sf}=-\mu_{sf}F_{a}\text{sgn}(\dot{s}), \tag{5}\]  
where \(\mu_{sf}\) is the seal friction coefficient and sgn is the signum function. The journal friction force is the product of the friction coefficient and the normal force. Due to the poor lubricating properties of hydraulic oil and the shape of the bearing surfaces in the shock absorber, it can be assumed that the shock-strut is under a dry friction condition [7].