Non-linear modal analysis of structural systems using invariant manifolds

This dissertation tackles the problem of defining a proper modal analysis procedure for non-linear dynamical systems, in particular for structural systems of relevance to engineering problems. Of particular importance in structural dynamic analysis is the possibility of generating, for such systems, reliable models with a few degrees of freedom only (also called reduced-order models) from a potentially large initial number of degrees of freedom. The primary goal of this dissertation is to determine a systematic procedure for obtaining such reduced-order models, utilizing a small number of appropriately-defined non-linear normal modes of the system, i.e., to develop a modal analysis procedure for non-linear structural systems legitimately allowing for the use of a few non-linear modes only.;For that purpose, motions involving a single non-linear mode or several non-linear modes are described in terms of a finite-dimensional, curved, invariant manifold in the phase space of the system. The dimension of this invariant manifold is twice the number of non-linear modes of interest, and its curvature is due to the influence of all the linear modes on the various non-linear modes. Accordingly, motions occurring on the invariant manifold include the influence of many linear modes but are parametrized by a small number of non-linear modal coordinates. For free response problems, the non-linear modes that are not included in the model are never excited and need not be simulated (this is the invariance property of the manifold). For forced response problems, the invariant manifold is in general time-varying, but utilization of the (time-independent) invariant manifold associated to the unforced system is found to be nearly invariant for low amplitudes of external excitation. The size of the problem is thus effectively reduced to the number of modeled modes only, and the dynamics of the entire system are then recovered from the dynamics of a small number of coupled, non-linear modal oscillators. Non-linear modal interactions between the various modeled non-linear modes are allowed and accounted for, and permit to treat cases with internal resonances automatically.;In practice, an asymptotic determination of the invariant manifold is obtained for weakly non-linear systems in terms of Taylor series expansions. The accuracy of the procedure is increased by determining higher orders of approximation of the manifold using these series, rather than by adding more modes to the model. Demonstration of the potential of the proposed non-linear modal analysis procedure is provided on a case study, for both free and forced responses. It is found that comparable accuracy can be achieved with many fewer non-linear modes using this non-linear modal analysis procedure than with linear modes using a linear modal analysis of the non-linear system. The computational savings brought by this non-linear modal analysis technique are expected to be significant in many practical applications.