The use of normal forms for analysing nonlinear mechanical vibrations
The use of normal forms for analysing nonlinear mechanical vibrations
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.
Neild, Simon A.
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Champneys, Alan R.
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Wagg, David J.
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Hill, Thomas L.
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Cammarano, Andrea
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28 September 2015
Neild, Simon A.
e11b68bb-ddff-4cac-a8a7-798cc3cc3891
Champneys, Alan R.
47a636dc-28d2-4d57-ad12-578d34661284
Wagg, David J.
7aa7d661-df7e-4ecc-86b1-823d4adaf05f
Hill, Thomas L.
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Cammarano, Andrea
c0c85f55-3dfc-4b97-9b79-e2554406a12b
Neild, Simon A., Champneys, Alan R., Wagg, David J., Hill, Thomas L. and Cammarano, Andrea
(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).
(doi:10.1098/rsta.2014.0404).
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.
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neild-et-al-2015-the-use-of-normal-forms-for-analysing-nonlinear-mechanical-vibrations
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Accepted/In Press date: 23 June 2015
e-pub ahead of print date: 28 September 2015
Published date: 28 September 2015
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Local EPrints ID: 490817
URI: http://eprints.soton.ac.uk/id/eprint/490817
ISSN: 1364-503X
PURE UUID: 04c09f35-61d8-4e19-b3c9-d327655217f0
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Date deposited: 06 Jun 2024 17:08
Last modified: 07 Jun 2024 02:08
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Author:
Simon A. Neild
Author:
Alan R. Champneys
Author:
David J. Wagg
Author:
Thomas L. Hill
Author:
Andrea Cammarano
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