# A Rational Numerical Method for Simulation of Drop-Impact Dynamics of Oleo-Pneumatic Landing Gear
## 2 Formulation of Landing Gear Impact Dynamics
Well-performing, and well-lubricated, bearings are commonly used in aeronautics; consequently, and within the limits of approximation of the proposed formulation, the torque due to bearing friction can be reasonably considered negligible compared to that induced by the friction force between the tire and the ground. As a whole, Equations (5)-(7) define the system of nonlinear and coupled differential equations whose solution describes the impact dynamics of the landing gear in terms of the time-dependent state variables \(x_{1},z_{1},z_{2}\), and \(\varphi\). It is worth noting that the expressions of the stroke given by (2) and the vertical equilibrium given by (5) are valid for the specific landing gear arrangement depicted in Figure 2.  
This arrangement is typical of main landing gear and is characterized by a hinged connection between the fork, the piston, and the cylinder, usually referred to as a _trailing link_. The cylinder's inclination angle with respect to the vertical axis \(z\) is null or negligible. In some other arrangements generally adopted for nose landing gear, the fork and piston move together with no relative rotation between them, and the movement occurs along a direction that is inclined with respect to the normal to the ground (Figure 4). In such cases, Equations (2) and (5) can be easily rewritten following the same approach that has been used so far. Referring to the typical nose landing gear arrangement in Figure 4 (telescopic landing gear), we can immediately observe that:  
\[z_{1}(t)+L_{1}+L_{0}-c(t)\cdot cos\theta=z_{2}(t)\]  
and then,  
\[c(t)=c((z_{1}(t),z_{2}(t))=(z_{1}(t)-z_{2}(t)+L_{1}+L_{0})/cos\theta. \tag{8}\]  
By imposing the equilibrium along the vertical axis \(z\), we can finally obtain:  
\[\left\{ \begin{array}{rcl}m\ddot{z}_{1}&=&[-F_{a}(c)-F_{d}(\dot{c})]cos\theta +F_{PN}(z_{1})-mg\\ M_{e}\ddot{z}_{2}&=&[F_{a}(c)+F_{d}(\dot{c})]cos\theta-M_{e}g\end{array} \right.. \tag{9}\]  
Equations (6) and (7) have general validity and can therefore be used together with (8) and (9) to complete the set of equations governing the impact dynamics of the landing gear arrangement of Figure 4.  
Figure 3: Graphical sketch explaining relationships among variables \(s,x_{1}\), and \(\varphi\). In conclusion, the proposed simulation approach leads to two different formulations for the equations of drop impact dynamics, depending on the type of landing gear under investigation. Equations (5)-(7), and the expression of the stroke given by (2) apply to landing gear with a trailing link arrangement and vertical leg (typical configuration of main landing systems), while Equations (6), (7), and (9) combined with Equation (8) for the stroke apply to nose landing gear, generally characterized by the configuration depicted in Figure 4.  
During the preliminary landing gear design stage, a geometric layout of the item is assumed in combination with a wheel type. Based on these assumptions and the drop-test prescriptions of the reference airworthiness regulations, all the input data required to run the simulation are available at the preliminary design stage, with the exception of \(\sigma\) and \(K\) (used in Equation (6)). Nevertheless, first trial values may be assigned to \(\sigma\) and \(K\) by using the structural similitude of landing gear belonging to the same category (i.e., suitable for installation on aircraft in the same category of MTOW). It can be observed that the natural frequency of the (main or nose) landing gear's fore and at blending mode does not significantly change over different landing gear models in the same category.