Nonlinear normal modes for vibratory systems under harmonic excitation

This paper considers the use of numerically constructed invariant manifolds to determine the response of nonlinear vibratory systems that are subjected to harmonic excitation. The approach is an extension of the nonlinear normal mode (NNM) formulation previously developed by the authors for free oscillations, wherein an auxiliary system that models the excitation is used to augment the equations of motion. In this manner, the excitation is simply treated as an additional system state, yielding a system with an extra degree-of-freedom (dof), whose response is known. A reduced-order model for the forced system is then determined by the usual NNM procedure, and an efficient Galerkin-based solution method is used to numerically construct the attendant invariant manifolds. The technique is illustrated by determining the frequency response for a simple 2-dof mass–spring system with cubic nonlinearities, and for a discretized beam model with 12 dof. The results show that this method provides very accurate responses over a range of frequencies near resonances.