Closed-form analytical solution for the shimmy instability of nose-wheel landing gears

Closed-form analytical expressions for the shimmy velocity and shimmy frequency are obtained, considering the lateral dynamics of a three-degree-of-freedom nose-wheel landing gear shimmy model. The analytical solution is based on the observation that the lowest modal frequency of the nose-wheel landing gear on the ground gives a close approximation to the shimmy frequency. Results are obtained for the shimmy frequency and shimmy velocity using these analytical expressions for a range of values of landing gear parameters and compared with the results obtained using an exact solution of the more general formulation. These results show that the analytical expressions can be used as a good first-cut approximation of the critical stability values for use at the early design stage