# A Rational Numerical Method for Simulation of Drop-Impact Dynamics of Oleo-Pneumatic Landing Gear
## 2 Formulation of Landing Gear Impact Dynamics  
The landing gear was modeled as an elastic system with four degrees of freedom, adequately selected to effectively capture the impact dynamics. The following Lagrangian coordinates were defined with reference to the Cartesian system \(S\) depicted in Figure 2:  
* \(x_{1},z_{1}\): \(x\)- and \(z\)-coordinates of point \(M\) located at the center of the wheel hub;
* \(z_{2}\): \(z\)-coordinate of the cylinder head; and
* \(\varphi\): wheel rotation angle. According to the scheme of Figure 2, the impact dynamics was roughly divided into three subsequent time phases:  
Figure 2: Phases of landing gear drop and main parameters involved in impact dynamics. 1. Free fall, on the left side of the figure, during which the landing gear is approaching the ground;
2. Impact, at the center of the figure, occurring at the single instant \(t_{imp}\), when the wheel touches the ground; and
3. Shock absorption, on the right side of the figure, during which the landing gear works to dampen the impact-induced acceleration. If \(c\) indicates the stroke of the piston and \(z_{0}\) the z-coordinate of point \(O\) located at the base of the piston corresponding to the piston-fork joint, at any instant of time \(t\), the result is:  
\[z_{O}(t)+L_{1}+L_{0}-c(t)=z_{2}(t) \tag{1}\]  
where \(L_{0}\) and \(L_{1}\) indicate the longitudinal dimension of the air chamber and piston, respectively. Equation (1) can be easily rearranged to express the stroke as a function of variables \(z_{0}\) and \(z_{2}\):  
\[c(t)=z_{O}(t)-z_{2}(t)+L_{1}+L_{0} \tag{2}\]  
Although not evident in Equation (2), the stroke is dependent only on Lagrangian coordinates \(z_{1}\) and \(z_{2}\) (\(c(t)=c(z_{1}(t),z_{2}(t))\). \(z_{0}\) is a dummy variable used only to express the stroke in a more intuitive and compact manner; this variable is linked to \(z_{1}\) and \(z_{2}\) by a nonlinear equation, which is derived in Appendix A by means of simple geometric and kinematic considerations. During free fall, the stroke is identically equal to zero. At the origin of the investigation time frame (\(t=0\)), we have:  
\(z_{1}(0)=h\), where \(h\) is the drop height,  
\(z_{2}(0)=h+L^{*}+L_{1}+L_{0}\), where \(L^{*}\) is the vertical distance between points \(O\) and \(M\) when \(t\leq t_{imp}\),  
\(z_{0}(0)=h+L^{*}\),  
and, according to Equation (2):  
\(c(0)=z_{O}(0)-z_{2}(0)+L_{1}+L_{0}=h+L^{*}-h-L^{*}-L_{1}-L_{0}+L_{1}+L_{0}=0\). If \(0<t<t_{imp}=\sqrt{2h/g}\), then:  
\(z_{1}(t)=z_{1}(0)-\frac{1}{2}gt^{2}\),  
\(z_{2}(t)=z_{2}(0)-\frac{1}{2}gt^{2}\),  
\(z_{0}(t)=z_{0}(0)-\frac{1}{2}gt^{2}\),  
and, again,  
\(c(t)=z_{O}(0)-\frac{1}{2}gt^{2}-z_{2}(0)+\frac{1}{2}gt^{2}+L_{1}+L_{0}=h+L^{* }-h-L^{*}-L_{1}-L_{0}+L_{1}+L_{0}=0\). The stroke is also identically null at the instant of impact; when \(t=t_{imp}=\sqrt{2h/g}\), the following occurs:  
\(z_{1}(t_{imp})=0\),  
\(z_{2}(t_{imp})=L^{*}+L_{1}+L_{0}\),  
\(z_{0}(t_{imp})=L^{*}\),  
and, finally,  
\(c(t_{imp})=z_{O}(t_{imp})-z_{2}(t_{imp})+L_{1}+L_{0}=L^{*}-L^{*}-L_{1}-L_{0}+L _{1}+L_{0}=0\). When \(t>t_{imp}\), the piston's stroke is no longer equal to zero; the motion of the piston into the cylinder generates vertical reaction forces \(F_{a}\) and \(F_{d}\) due to the compression of gas and the flow of oil into the oleo-pneumatic chamber, respectively. In addition, a further reaction force, \(F_{PN}\), arises along the vertical direction due to the compression of gas in the tire.