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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 …