The use of normal forms for analysing nonlinear mechanical vibrations

Neild, S. A., Champneys, A. R., Wagg, D. J., Hill, T. L. and Cammarano, A. (2015) The use of normal forms for analysing nonlinear mechanical vibrations. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 373(2051), 20140404. (doi: 10.1098/rsta.2014.0404) (PMID:26303917) (PMCID:PMC4549939)

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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Cammarano, Dr Andrea
• Authors: Neild, S. A., Champneys, A. R., Wagg, D. J., Hill, T. L., and Cammarano, A.
• College/School: College of Science and Engineering > School of Engineering > Systems Power and Energy
• Journal Name: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
• Publisher: Royal Society
• ISSN: 1364-503X
• ISSN (Online): 1471-2962
• Published Online: 24 August 2015
• Copyright Holders: Copyright © 2015 The Authors
• First Published: First published in Royal Society of London Philosophical Transactions A: Mathematical, Physical and Engineering Sciences 373(2051):20140404
• Publisher Policy: Reproduced under a Creative Commons License