# A Rational Numerical Method for Simulation of Drop-Impact Dynamics of Oleo-Pneumatic Landing Gear
## 3 Drop Test Simulation and Results  
A set of intercommunicating MATLAB(r)[19] routines was implemented and arranged in a compact software environment to automatically solve the differential equations governing the drop dynamics of the landing gear according to the approaches described in Section 2. The tool consists of three main modules (Figure 5):  
* An input module, which reads and processes the input data of the problem, in terms of both the landing gear type (main or nose) and characteristics. * A solver module, which implements the Runge-Kutta method [20] to integrate the system of differential equations numerically. * An output module, which prints the time histories of the most relevant physical parameters and plots practical diagrams to visually support the result analysis. To check the reliability of the proposed numerical approach and its implementation in the software tool, a drop test of a reference landing gear was simulated, and the results were compared with already available experimental data. The main landing gear of a large UAV was selected as a reference; the name and pictures of the aircraft are here omitted for confidentiality reasons. To properly frame the application scenario, it is sufficient only to mention that the UAV had a maximum landing weight (_MLW_) of 3040 kg and a wing surface area (\(S\)) of 21 m\({}^{2}\), and was designed in compliance with NATO-STANAG 4671 airworthiness requirements [21]. Among the tests performed to qualify the main landing gear, the limit drop (paragraph 725 of [21]) was selected as a benchmark case for the drop simulation code. Figure 5: Main modules and structure of drop test simulation tool. According to the limit drop test requirements, the drop height must be equal to:  
\[h=0.0132\sqrt{\frac{MLW_{[\text{Kg}]}\cdot\mathcal{S}[\frac{m}{\sigma^{2}}]}{S_{[ \text{m}^{2}]}}} \tag{10}\]  
and not lower than 0.234 m or greater than 0.475 m.  
The landing gear must be dropped while carrying an effective mass equal to:  
\[M_{e}=M\frac{h+(1-L)d}{h+d} \tag{11}\]  
where \(M_{e}\) is the effective mass to be used in the drop test; \(h\) is the specified drop height; \(d\) is the deflection under the impact of the tire, plus the vertical component of the wheel hub travel relative to the drop mass at the instant of time when the piston stroke reaches its maximum value; \(M\) is equal to the static weight on the main landing gear with the UAV in level attitude with the nose wheel clear; and \(L\) is the ratio of assumed wing lift to UAV weight (not greater than 2/3). Replacing \(MLW=3040\) kg and \(S=21\) m\({}^{2}\) in Equation (10), the reference landing gear's drop height was equal to 0.497 m and therefore was limited to 0.475 m per regulatory requirements. The effective mass to be used in the drop test depends on the parameter \(d\), which is an output of the drop itself; for this reason, an iterative process, based on trial drops, is usually carried out to determine the couple (\(M_{e}\), \(d\)) satisfying Equation (11) with a given level of approximation (Figure 6a). Figure 6: Iterative process for determining effective mass: (**a**) trial drop tests and (**b**) numerical method. Although the \(M_{e}\) and \(d\) values used during the limit drop tests were already available from the landing gear's test reports, they were calculated by simulating the trial drop procedure usually carried out before the limit drop test (Figure 6b); this was clearly done to get a first proof of validation for the proposed numerical formulations. An initial (trial) value, \(d_{1}\), was imposed on \(d\), and then, the corresponding effective mass \(\mathrm{M_{e,1}}\) was obtained by Equation (11); L was imposed as equal to \(2/3\), and \(M\), as equal to half of the \(MLW\) (\(M=1520\) kg). Drop dynamics equations were solved, thus giving in output the effective value \(d_{2}\) of parameter \(d\).