Abstract
A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations.
Highlights
A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points
The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-orderin-time differential equations
The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping
Summary
The problem addressed here is how to extend the wellestablished notion of normal modes of linear vibration systems to nonlinear systems in a mathematically consistent way that allows for practical implementation. The concept of the NNM applies well to the case where there is no forcing or damping, so that P = D = 0 in (1.1) and N is a pure function of x In this case, Shaw & Pierre [9,10] argued that an NNM is just an invariant manifold composed of periodic solutions, whose frequency in the limit of amplitude tending to zero is the same as the linear mode. Application of the harmonic balance method to the derived normal form, though, is shown to lead to remarkably powerful predictions for resonant responses, leading to so-called backbone curves in structural analysis, for mode switching in complex multimodal responses and for analytically finding NNMs. The rest of this paper is outlined as follows.
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