The superposition theorem is the cornerstone of linear system theory, which enables the study of the dynamic behavior of linear systems, both analytically and numerically. It is precisely this theorem which, by allowing modal analysis, capitalizes on the concept of normal modes of motions and renders them so important. By contrast, it is also precisely such a manner of recombining general motions from individual modal co-ordinates that is missing for non-linear dynamic systems (in particular, structural systems, on which this line of work is focused). Consequently, although the concept of non-linear normal modes of vibrations is well established for general, vibratory, non-linear, structural systems (see references [1][2][3][4][5][6][7]), their use in structural dynamics is restricted to very particular motions; namely, those involving a single mode response. Indeed, as superposition of individual modal responses cannot be performed, general motions of a non-linear structural system cannot usually be easily determined (except possibly for systems with similar non-linear modes). Such motions are typically handled by projecting the non-linear system on to a basis consisting of the modes of the linearized system.
However, it is possible to extend the definition of non-linear normal modes in a manner that enables a non-linear modal analysis of the free response of non-linear systems. By encompassing all modes of interest in a single multi-mode invariant manifold in the phase space-rather than separate single mode manifolds-one can create the framework needed to perform a reduced order modal analysis of non-linear systems, where interactions between the modelled modes are accounted for, while contaminations of (and from) the non-modelled modes is prevented-much like linear modal analysis of linear systems. This methodology is also perfectly suitable for dynamic (structural) systems with internal resonances. In this case, prior knowledge of any potential or existing internal resonances is not even required, as the procedure will exhibit them automatically. The number of equations to be simulated is the same as that of modelled modes in the multi-mode manifold. This non-linear modal analysis should be compared to performing a linear modal analysis of the non-linear system-a commonly employed technique-which often results in simulating a large number of equations to capture the various interactions between the linear modal co-ordinates. Also, the linear modal analysis of non-linear systems may completely overlook the occurrence of internal resonances-if internally resonant modes are not modelled simultaneously, numerical simulations can still be performed with potentially qualitatively inexact results-while the anomaly would be signalled by the present non-linear modal analysis procedure.