Nonlinear modeling and sensitivity analysis of}{oleo-pneumatic strut landing gear systems for aircraft landing performance

Abstract

Landing gear systems support the aircraft throughout ground operations and play a crucial role in absorbing impact forces at touchdown. In today`s modern landing gears, mostly oleo-pneumatic struts are utilized as the shock-absorber component in the landing gear system since it has the maximum efficiency to weight ratio and the maximum shock absorber efficiency. The aim of this study is to generate a Simulink-model for the oleo-pneumatic strut landing gear non-linear dynamics and conduct a sensitivity analysis to evaluate the effects of oleo-pneumatic strut landing gear design parameters on the key performance criteria. The vertical motion of a landing gear system is analytically modeled as a double mass-spring-damper system, capturing the dynamics of both the aircraft and the landing gear system as separate masses: upper or sprung mass and lower or unsprung mass, respectively. The forces acting on the upper mass are accounts for the forces transmitted through the shock strut, the normal reaction force component along the strut axis -applicable when the landing gear has an inclination angle (rake angle) - the aerodynamic lift acting upward on the aircraft, and its weight. Similarly, the lower mass experiences the downward restoring forces from the strut and the strut's normal axis, as well as the upward tire-ground reaction force and its own weight. The aircraft touches the ground with its tires that refers the lower mass in the dynamic model and the tire-ground interaction is modeled as one spring. The stiffness characteristics of a tire is usually obtained by experimental data and it can be defined with an exponential curve using tire stiffness related coefficients and the tire diameter. The upper and lower masses are connected to each other with the oleo-pneumatic strut, generally referring to a piston-cylinder assembly and the cylinder. During aircraft landing operation, the oleo strut compresses under load or extends as the load is relieved. It is usually divided into two chambers and these two chambers are connected with an orifice. The upper chamber is typically filled with gas like nitrogen or air and the lower chamber is filled with highly viscous fluids like oil. As the oleo-strut compresses or extends, the damping force in the oleo strut is generated by the flow of viscous fluid into the orifice and the spring force is produced by the compressed air in the upper chamber. The friction forces are also generated by the moving parts in the oleo-pneumatic strut assembly. The proper characterization and constant tracking of the stroke and stroke velocity of the oleo pneumatic shock strut is an essential component of dynamic modeling. This is because these variables influence the strut's response to impact and dictate the force transmission profile that is produced as a result of the impact. The vertical displacement difference between the upper mass and the lower mass is defined as the strut stroke. Similarly, the stroke velocity is the relative vertical velocity of the upper and lower masses. The lift also acts on the aircraft wings because of the horizontal speed during the aircraft landing operation. The lift is usually taken equal to the total weight of the aircraft at touchdown. After touchdown, the lift decreases as the horizontal speed decreases during the landing operation. The reduction of horizontal velocity is represented by a formula that incorporates wheel braking, thrust (or thrust reverser if relevant), and aerodynamic forces: drag and lift. The aerodynamic forces are determined using the aircraft's horizontal velocity, which is derived from the deceleration of that velocity. This introduces additional complexity and non-linearity to the system. By defining constant parameters (e.g., dimensional properties, friction coefficients, tire stiffness) and variable factors (e.g., stroke, stroke velocity), the spring and damping characteristics of the oleo-pneumatic shock strut and tire are established as well as the deceleration in horizontal velocity and consequently decrease in lift force. Considering all the variants on the spring effects, damping characteristics and forces caused by the friction, the vertical motion of a landing gear system demonstrates a complex and non-linear behavior. This study develops a nonlinear Simulink model to simulate the vertical dynamics of an aircraft landing gear system. The model integrates the equations of motion for both the upper and lower masses and is validated by experimental data and reference model from NACA-TN-2755. Two validation cases are taken into account with different touchdown horizontal velocities (a.k.a sink rate) and the higher sink rate results in a tire-bottoming condition, which occurs once tire's vertical deflection goes beyond its allowed deformation limit and this changes the stiffness regime of the tire. The validation reveals a robust correlation between the model and actual behavior, affirming its capacity to effectively simulate landing gear dynamics within the Simulink environment. Comparing with the reference model presented in NACA-TN-2755, the developed simulink model exhibits maximum 6.76\% error difference for the validation case with no tire bottoming and 8.64\% error difference for the validation case with tire bottoming. The maximum error differences mostly pertains to the overall force acting on the aircraft's mass. The created Simulink model demonstrates comparable or superior performance to the reference model for outputs such as aircraft displacement, tire displacement, stroke, and stroke velocity. The primary performance criteria for landing gear design encompass shock absorber efficiency, the overall post touch down duration necessary for the system to achieve stabilization, and the peak vertical force exerted on the aircraft. A One-at-a-Time (OAT) sensitivity analysis is conducted to evaluate the impact of landing gear design parameters on determined performance criteria by utilizing a baseline model derived from the North American T-6 Texan aircraft. To comprehend the relationship between the parameters and the performance requirements, linearity is initially established for each parameter using the coefficient of determination ($R^2$). Sensitivity analysis is conducted solely for the parameters that have a significant correlation with the performance criteria, utilizing the sensitivity index. The simultaneous application of the coefficient of determination and sensitivity index facilitates the assessment of both the existence of a relationship between the design parameter and performance criteria, as well as the magnitude of this relationship. The sensitivity analysis results indicate that the areas of the hydraulic and air chambers have tangible influence on shock absorber efficiency. Hydraulic damping parameters are the primary factors affecting the required total time for the system to be balanced. The maximum vertical force acting on the aircraft mass is predominantly governed by the sink rate but is also strongly affected by the hydraulic damping characteristics. Additionally, air chamber pressure and area moderately impact the maximum vertical force acting on the aircraft mass. On the other hand, factors such as the initial air volume, gas polytropic index, rake angle, and friction coefficients exhibit do not show concrete impact on the chosen performance criteria. Finally, although engine thrust alone does not markedly impact landing gear performance criteria, the deployment of thrust reverser may influence the maximum force exerted on the landing gear strut and alter the load cycle on it during landing. In conclusion, validation of the model using experimental data shows that the generated Simulink model faithfully depicts real-world behavior by means of simulated reactions and sufficiently captures the non-linear dynamics of the landing gear system. Focusing on specified performance criteria: shock absorber efficiency, total stabilization time post-touchdown, and maximum vertical force exerted on the aircraft, this paper offers a comprehensive sensitivity analysis of the design parameters of oleo-pneumatic strut landing gear systems.