Nonlinear Normal Modes in a System with Nonholonomic Constraints

We investigate the dynamics of a system involving the planar motion of a rigid body which is restrained by linear springs and which possesses a skate-like nonholonomic constraint known as C̆aplygin’s sleigh. It is shown that the system can be reduced to one with 2 2 degrees of freedom. The resulting phase flow is shown to involve a curve of nonisolated equilibria. Using second-order averaging, the system is shown to possess two families of nonlinear normal modes (NNMs). Each NNM involves two amplitude parameters. The structure of the NNMs is shown to depart from the generic form in the neighborhood of a 1:1 internal resonance.