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)

[img]
Preview
Text
111004.pdf - Published Version
Available under License Creative Commons Attribution.

762kB

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

University Staff: Request a correction | Enlighten Editors: Update this record