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An analytical approach for detecting isolated periodic solution branches in weakly nonlinear structures

An analytical approach for detecting isolated periodic solution branches in weakly nonlinear structures
An analytical approach for detecting isolated periodic solution branches in weakly nonlinear structures
This paper considers isolated responses in nonlinear systems; both in terms of isolas in the forced responses, and isolated backbone curves (i.e. the unforced, undamped responses). As isolated responses are disconnected from other response branches, reliably predicting their existence poses a significant challenge. Firstly, it is shown that breaking the symmetry of a two-mass nonlinear oscillator can lead to the breaking of a bifurcation on the backbone curves, generating an isolated backbone. It is then shown how an energy-based, analytical method may be used to compute the points at which the forced responses cross the backbone curves at resonance, and how this may be used as a tool for finding isolas in the forced responses. This is firstly demonstrated for a symmetric system, where an isola envelops the secondary backbone curves, which emerge from a bifurcation. Next, an asymmetric configuration of the system is considered and it is shown how isolas may envelop a primary backbone curve, i.e. one that is connected directly to the zero-amplitude solution, as well as the isolated backbone curve. This is achieved by using the energy-based method to determine the relationship between the external forcing amplitude and the positions of the crossing points of the forced response. Along with predicting the existence of the isolas, this technique also reveals the nature of the responses, thus simplifying the process of finding isolas using numerical continuation.
0022-460X
150-165
Hill, T.L.
96922dda-e993-4889-a519-5f89fe6ebca8
Neild, S.A.
e11b68bb-ddff-4cac-a8a7-798cc3cc3891
Cammarano, A.
c0c85f55-3dfc-4b97-9b79-e2554406a12b
Hill, T.L.
96922dda-e993-4889-a519-5f89fe6ebca8
Neild, S.A.
e11b68bb-ddff-4cac-a8a7-798cc3cc3891
Cammarano, A.
c0c85f55-3dfc-4b97-9b79-e2554406a12b

Hill, T.L., Neild, S.A. and Cammarano, A. (2016) An analytical approach for detecting isolated periodic solution branches in weakly nonlinear structures. Journal of Sound and Vibration, 379, 150-165. (doi:10.1016/j.jsv.2016.05.030).

Record type: Article

Abstract

This paper considers isolated responses in nonlinear systems; both in terms of isolas in the forced responses, and isolated backbone curves (i.e. the unforced, undamped responses). As isolated responses are disconnected from other response branches, reliably predicting their existence poses a significant challenge. Firstly, it is shown that breaking the symmetry of a two-mass nonlinear oscillator can lead to the breaking of a bifurcation on the backbone curves, generating an isolated backbone. It is then shown how an energy-based, analytical method may be used to compute the points at which the forced responses cross the backbone curves at resonance, and how this may be used as a tool for finding isolas in the forced responses. This is firstly demonstrated for a symmetric system, where an isola envelops the secondary backbone curves, which emerge from a bifurcation. Next, an asymmetric configuration of the system is considered and it is shown how isolas may envelop a primary backbone curve, i.e. one that is connected directly to the zero-amplitude solution, as well as the isolated backbone curve. This is achieved by using the energy-based method to determine the relationship between the external forcing amplitude and the positions of the crossing points of the forced response. Along with predicting the existence of the isolas, this technique also reveals the nature of the responses, thus simplifying the process of finding isolas using numerical continuation.

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Accepted/In Press date: 17 May 2016
e-pub ahead of print date: 10 June 2016
Published date: 24 June 2016

Identifiers

Local EPrints ID: 490792
URI: http://eprints.soton.ac.uk/id/eprint/490792
ISSN: 0022-460X
PURE UUID: 7ceba429-0b06-4d5e-8de7-027dddb7e876
ORCID for A. Cammarano: ORCID iD orcid.org/0000-0002-8222-8150

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Date deposited: 06 Jun 2024 16:49
Last modified: 07 Jun 2024 02:08

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Contributors

Author: T.L. Hill
Author: S.A. Neild
Author: A. Cammarano ORCID iD

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