An Asymptotic Method for Predicting Amplitudes of Nonlinear Wheel Shimmy

An asymptotic method involving the multiple-time-scale perturbation technique is presented for nonlinear stability analysis of landing gear wheel shimmy models that include a velocity-squared damper. General expressions for the limit cycle amplitude and frequency are obtained, with the stability of the limit cycles determined by the sign of a computed coefficient. It is found that for positive values of this coefficient, a stable limit cycle exists for velocities exceeding a critical value and that complete stability ensues for velocities less than the critical velocity. For negative values of the coefficient, an unstable limit cycle exists for less-than-critical velocities, and the system is unstable for velocities exceeding the critical value. A simple shimmy model with a nonlinear damper is analyzed, and the limit cycle results obtained by the perturbation solution are shown to be in agreement with those calculated by numerical integration of the equations of motion.