# A Rational Numerical Method for Simulation of Drop-Impact Dynamics of Oleo-Pneumatic Landing Gear
## 2 Formulation of Landing Gear Impact Dynamics
\(F_{a}\) and \(F_{d}\) depend on the geometric characteristics of the oleo-pneumatic chamber and can be expressed as functions of the piston's stroke and its time derivative, according to the following equations [7]:  
\[F_{a}=F_{a}(c)=P_{0}A_{0}[L_{o}/(L_{o}-c)]^{\gamma} \tag{3}\]  
\[F_{d}=F_{d}(\dot{c})=\rho_{oil}A_{8}^{3}\dot{c}|\dot{c}|/(2A_{tr}^{2}\tau_{visc}^ {2}) \tag{4}\]  
where \(A_{0}\) and \(P_{0}\) are the air chamber section area and pre-load pressure in the air chamber, respectively; \(L_{0}\) is the initial length of the air chamber (i.e., its longitudinal length at \(t<t_{imp}\)); \(\gamma\) is the polytropic index of gas in the air chamber; \(\rho_{oil}\) and \(c_{visc}\) are the density and viscosity of oil in the hydraulic chamber, respectively; \(A_{8}\) is the section area of the piston; and \(A_{tr}\) is the net section of the orifice of the hydraulic chamber traversed by the oil flow, generally variable with the piston stroke. \(F_{PN}\) basically depends on the tire's mechanical properties and is generally provided by tire manufacturers in the form of experimental curves relating the reaction force to the tire crushing and inflation pressure. For a given tire and inflation pressure, \(F_{PN}\) is therefore considered as a function of the variable \(z_{1}\) only. The tire's overall reaction force is more precisely given by the sum of \(F_{PN}\) and \(F_{d0}\), where \(F_{d0}\) stands for the reaction force due to the deformation of the tire's material (at zero inflating pressure). \(F_{d0}\) is dependent on the rate of deformation of the tire (\(dz_{1}/dt\)) since the tire's material is characterized by viscoelastic behavior. Within the limits of the proposed formulation, \(F_{d0}\) was considered negligible compared to \(F_{PN}\). By imposing the equilibrium of all forces acting along the \(z\)-axis of the reference S (Figure 2), the following equations for the impact dynamics of the system can be written:  
\[\left\{ \begin{array}{ccc}m\ddot{z}_{1}&=&-F_{a}(c)-F_{d}(\dot{c})+F_{PN}(z_ {1})-mg\\ M_{e}\ddot{z}_{2}&=&F_{a}(c)+F_{d}(\dot{c})-M_{e}g\end{array} \right. \tag{5}\]  
where \(M_{e}\) is the effective mass, supposed to be lumped at the head of the cylinder and accounting for the amount of aircraft weight acting on the landing gear element at the moment of impact, and \(m\) is the swinging mass, supposed to be lumped at the middle point of the wheel hub and equal to the sum of the wheel and tire masses, plus one-third of the landing gear leg's mass. Equation (5) are valid also when \(t<t_{imp}\), and basically describe the free fall of masses \(M_{e}\) and \(m\) as two separate bodies;in this case \(F_{a}\), \(F_{d}\) and \(F_{PN}\) are all equal to zero. After the impact, the elastic oscillation of the wheel hub along the \(x\)-axis is governed by the equation:  
\[m\ddot{x}_{1} = -\sigma\dot{x}_{1}-Kx_{1}-\frac{\dot{c}}{|\dot{s}|}\ F_{x} \tag{6}\]  
where \(\sigma\) and \(K\) are the structural (viscous) damping and stiffness exhibited by the landing gear leg along the \(x\)-axis; \(F_{x}\) is the module of the friction force due to the tire contact with the ground, which can be reasonably assumed equal to \(\mu\cdot F_{PN}(z_{1})\), where \(\mu\) is the tire sliding coefficient of friction, and \(s\) is the x-displacement of the point of contact between the tire and the ground (tire slip, Figure 3), given by \((x_{1}+\varphi\rho)\), where \(\rho\) is the distance of the wheel hub axis to the ground, or the difference between the radius of the wheel and the tire crushing. The friction force \(F_{x}\) and displacement \(s\) do not represent two independent variables, provided that they are both called to satisfy the rotational equilibrium equation of the wheel about the hub axis. Under the assumption of negligible bearing friction, the wheel rotational equilibrium may be formulated in the following way:  
\[I\ddot{\varphi} = \rho F_{x} \tag{7}\]with \(I\) equal to the wheel's polar moment of inertia.