Nonlinear Normal Modes in a System with Nonholonomic Constraints

We investigate the dynamics of a system involving the planar motionof a rigid body which is restrained by linear springs and whichpossesses a skate-like nonholonomic constraint known as aplygin'ssleigh. It is shown that the system can be reduced to one with 2 degrees of freedom. The resulting phase flow is shownto involve a curve of nonisolated equilibria. Using second-orderaveraging, the system is shown to possess two families of nonlinearnormal modes (NNMs). Each NNM involves two amplitude parameters.The structure of the NNMs is shown to depart from the generic formin the neighborhood of a 1:1 internal resonance.