The use of normal forms for analysing nonlinear mechanical vibrations

Simon Neild, Alan Champneys, David Wagg, Tom Hill, Andrea Cammarano

Research output: Contribution to journalArticle (Academic Journal)peer-review

45 Citations (Scopus)
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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.
Original languageEnglish
Article number20140404
Number of pages22
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2051
Early online date24 Aug 2015
Publication statusPublished - Sep 2015

Bibliographical note

Date of Acceptance: 23/06/2015


  • nonlinear dynamics
  • normal forms
  • modal analysis
  • cable vibration


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