Non-linear modal analysis of the forced response of structural systems

A non-linear modal analysis procedure is presented for the forced response of non-linear structural systems. It utilizes the notion of invariant manifolds in the phase space, which was recently used to define non-linear normal modes and the corresponding non-linear modal analysis for unforced vibratory systems. For harmonic forcing, a similar procedure could be formulated, simply by augmenting the size of the free vibration problem. However, in order to accommodate general, non-harmonic external excitations, the invariant manifolds associated with the unforced system are used herein for the forced response analysis. The procedure allows one to generate reduced-order models for the forced analysis of structural systems. Although strictly speaking the invariance property is violated, good results are obtained for the case study considered. In particular, it is found that fewer non-linear modes than linear modes are needed to perform a forced modal analysis with the same accuracy. For systems with small and/or diagonal damping, approximate invariant manifolds are determined, which are shown to yield good results for both the unforced and forced responses.