# A Rational Numerical Method for Simulation of Drop-Impact Dynamics of Oleo-Pneumatic Landing Gear
## 3 Drop Test Simulation and Results
A new value of effective mass \(M_{e,2}\) was then recalculated by Equation (11), this time corresponding to \(d=d_{2}\); the input parameters \(M_{e,2}\), \(d_{2}\), were used for a new drop simulation, leading to a new output value \(d_{3}\). The process was repeated until the difference, \(\Delta d\), between the values of \(d\) obtained in two consecutive iterations were lower than 0.005 m.  
After three iterations, the convergence of the process was reached, resulting in \(\Delta d=0.003\) m, \(d=0.360\) m, and \(M_{e}=1083\) kg. Obtained values for \(d\) and \(M_{e}\) were in full agreement with those reported in the drop test documentation of the reference landing gear (\(d=0.45\) m, \(M_{e}=1090\) kg). At each iteration, the drop dynamics equations were integrated starting from the initial conditions (listed in Table 1), considering a time step of \(5\times 10^{-4}\) s over an interval of 1 s.  
The complete set of data related to the reference landing gear and used to run the simulations is reported in Appendix B. The output related to the last iteration was considered relevant for the drop test under investigation, as it led to coherent values of \(M_{e}\) and \(d\). The time histories obtained for the variables \(z_{1}\), \(z_{2}\), and \(z_{0}\) are reported in Figure 7. \begin{table}
\begin{tabular}{c c c} \hline \(z_{1}\) (0) & = & \(h=0.475\) m (\(\to t_{imp}=(2\) h/g\()^{0.5}=0.31\) s) \\ \(z_{2}\) (0) & = & \(h+L^{*}+L_{1}+L_{0}=(0.475+0.329+0.287+0.165)=1.256\) m \\ \(x_{1}(t_{imp})\) & = & 0 \\ \(\phi\) (0) & = & 0 \\ \(\dot{z}_{1}(0)\) & = & 0 \\ \(\dot{z}_{2}(0)\) & = & 0 \\ \(\dot{x}_{1}(t_{imp})\) & = & \(V_{0}=45.276\) m/s (landing speed, equal to 1.2’\(V_{s0}\), where \(V_{s0}\) is the aircraft stall speed with flaps down) \\ \(\dot{\phi}(t_{imp})\) & = & \(V_{0}/R=154.65\) rad/s, where R is the nominal radius of the tire (equal to 0.254 m for the reference gear unit) \\ \hline \end{tabular}
\end{table} 
Table 1: Initial conditions used for integration of drop dynamic equations. Figure 7: Time histories of vertical displacements \(z_{1}\), \(z_{0}\), and \(z_{2}\). Before the impact (\(t<t_{imp}\)), the landing gear and drop mass are in free fall and all the variables show the same parabolic trend versus time. Therefore, the piston stroke is identically null (Figure 8) and no work is done by the shock absorber. After the impact, the piston stroke increases up to its maximum value; in this time interval, \(z_{1}\), \(z_{0}\), and \(z_{2}\) continue to decrease coherently to what is expected during the shock absorber's compression. Their trends are now different and no longer parabolic, as they are influenced by the shock absorber dynamics and the crushing of the tire. The tire's crushing effect is more evident in the wheel hub vertical displacement (\(z_{1}\)) and becomes less dominant for \(z_{0}\) and \(z_{2}\) due to the shock absorber's damping action. When the piston stroke decreases, the shock absorber starts its extension phase and the displacements \(z_{1}\), \(z_{0}\), and \(z_{2}\) consequently increase. During this phase, the vertical displacement trends are practically affected only by the shock absorber's dissipation force. This is even more evident if we consider the time histories of the effective mass acceleration (\(\tilde{z}_{2}(t)\)) (Figure 9) and the overall shock absorber force \(F_{sat}(t)=(F_{a}(t)+F_{d}(t))\) (Figure 10). Figure 8: Time history of piston stroke. Figure 9: Time history of effective mass acceleration along vertical axis. Since the effect of \(F_{PN}\) on \(\ddot{z}_{2}\) is negligible along the entire piston stroke, the effective mass experiences an enforced motion that is practically driven only by the effective mass weight and overall shock absorber force. Thus, as shown by the results plotted in Figures 9 and 10, the time histories of \(\ddot{z}_{2}(t)\) and \(F_{Sat}(t)\) have similar trends, represented by the linear relation \(M_{\mathrm{c}}\ddot{z}_{2}(t)\approx F_{Sat}(t)-M_{\mathrm{c}}g\).