title: The use of normal forms for analysing nonlinear mechanical vibrations.
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.
authors: Faculty of Engineering, University of Bristol, Bristol BS8 1TR, UK simon.neild@bristol.ac.uk.
authors: Simon A
authors: SA
authors: Neild
authors: Faculty of Engineering, University of Bristol, Bristol BS8 1TR, UK.
authors: Alan R
authors: AR
authors: Champneys
authors: Department of Mechanical Engineering, University of Sheffield, Sheffield S1 3JD, UK.
authors: David J
authors: DJ
authors: Wagg
authors: Faculty of Engineering, University of Bristol, Bristol BS8 1TR, UK.
authors: Thomas L
authors: TL
authors: Hill
authors: Faculty of Engineering, University of Bristol, Bristol BS8 1TR, UK.
authors: Andrea
authors: A
authors: Cammarano
keywords: cable vibration; modal analysis; nonlinear dynamics; normal forms