Modelling-control of shimmy oscillations in aircraft landing gear and application design

Abstract

In aviation, the landing gear is a critical component that ensures the aircraft's stability and safety during ground operations. There are two fundamental types of landing gear configurations: Tail-Wheel and Tricycle. The Tail-Wheel setup consists of two main landing gears at the front and a single tail gear at the rear. Conversely, the Tricycle arrangement features a single nose landing gear complemented by two main landing gears at the rear. One of the significant challenges in landing gear design is the phenomenon known as shimmy. Shimmy refers to an oscillatory motion that combines lateral and yaw movements of the landing gear. This motion results from the complex interaction between the tire dynamics and the structural characteristics of the landing gear. Both nose and main landing gears can exhibit shimmy oscillations; however, the nose gear in Tricycle configurations and the tail gear in Tail-Wheel setups are particularly susceptible. The prevalence of shimmy oscillations has led to extensive research, with most studies concentrating on the nose or tail landing gears. Shimmy is characterized by self excited oscillations propelled by the aircraft's forward movement. The oscillation amplitude can range from minor disturbances affecting comfort and visibility to intense vibrations that may cause structural damage or even catastrophic failure. To mitigate these risks, a comprehensive modelling and analysis of the landing gear's dynamic behavior and structural integrity are imperative. This approach enables the evaluation of the landing gear design and facilitates the implementation of necessary modifications at an early development stage. For analytical purposes, a mathematical model of the landing gear is derived using the Lagrange equation. This model includes a third-order, 1-degree-of-freedom system representing the yaw motion, and a more complex fifth-order, 2-degree-of-freedom system accounting for both yaw and lateral movements. Subsequently, the Routh Hurwitz criteria and the coefficients of the characteristic equation are employed to conduct a linear stability analysis. Following the derivation of the mathematical equations, various simulations are executed using MATLAB and Simulink. Once the mathematical models and simulation frameworks are established and validated, control strategies such as Linear Quadratic Regulator (LQR) controller design is applied to both models. To streamline the analysis process, a specialized application named LaGeSh has been developed. This application allows for the adjustment of multiple parameters, including tire characteristics, slip angle, caster length, force, moments of inertia, and aircraft speed. These modifications enable a practical assessment of the aircraft's susceptibility to shimmy vibrations and facilitate the extraction of controller parameters based on the input values. Moreover, LaGeSh can generate comparative graphs, such as caster length versus aircraft speed and force versus aircraft speed, to identify the most optimal parameter configurations. As a result this study not only elaborates on the technical aspects of landing gear dynamics but also provides a comprehensive overview of the analytical methods and tools used in the study of shimmy oscillations.