ON NONLINEAR NORMAL-MODES OF SYSTEMS WITH INTERNAL RESONANCE

A complex-variable il invariant-manifold approach is used to construct the normal modes of weakly nonlinear discrete systems with cubic geom etric nonlinearities and either a one-to-one or a three-to-one interna l resonance. The nonlinear mode shapes are assumed to be slightly curv ed four-dimensional manifolds tangent to the linear eigenspaces of the two modes involved in the internal resonance at the equilibrium posit ion. The dynamics on these manifolds is governed by three first-order autonomous equations. In contrast with the case of no infernal resonan ce, the number of nonlinear normal modes may be more than the number o f linear normal modes. Bifurcations of the calculated nonlinear normal modes are investigated.