A NEW GALERKIN-BASED APPROACH FOR ACCURATE NON-LINEAR NORMAL MODES THROUGH INVARIANT MANIFOLDS

A method for producing accurate reduced order models of non-linear vibratory systems is presented based on the invariant manifold description of non-linear normal modes (NNM). This approach makes use of polar co-ordinates to obtain equations which govern the geometry of the invariant manifold. These equations are discretized through a series expansion and Galerkin projection over a chosen amplitude and phase domain, yielding non-linear equations in the expansion coefficients. These equations, when solved numerically, yield an invariant manifold which is accurate to the degree of the expansion, and devoid of the limitations which plague typical asymptotic solutions. Such Galerkin-based solutions may be used to generate accurate reduced-order models for large-amplitude, strongly non-linear motions. This procedure is illustrated using two non-linear examples, a two degree-of-freedom oscillator, and a finite element beam model. The solution convergence and manifold geometry are discussed and the resultant reduced-order models are shown to possess exceptional accuracy over large amplitude ranges. This approach allows the full potential of the invariant manifold formulation to be reached, and is suitably general for application to a wide variety of non-linear systems.