Determination of nonlinear normal modes of a planar nonlinear system with a constraint condition

To demonstrate in what a way nonlinear normal modes may influence or be used to predict system responses, this paper derives analytically the nonlinear normal modes with a constraint condition from the free vibration equation of the non-conservative nonlinear subsystem of a piecewise smooth system. Since the nonlinear subsystem describes the planar motion of an isotropic rotor with cross-coupling stiffness in contact with its stator, the nonlinear normal modes reflecting the two potential whirling directions within a plane possess both positive and negative modal frequencies. The results of this paper show that the stable and the unstable nonlinear normal modes affect the responses of the whole piecewise smooth system in different ways. Moreover, the nonlinear normal modes can also be used to predict some important characteristics of the self-excited backward whirl motion of the piecewise smooth system.