Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities

Xu Liu; Andrea Cammarano; David J Wagg; Simon A Neild; R. J. Barthorpe

doi:10.1007/978-3-319-15221-9_10

Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities

Xu Liu, Andrea Cammarano, David J Wagg, Simon A Neild, R. J. Barthorpe

Research output: Chapter in Book/Report/Conference proceeding › Conference Contribution (Conference Proceeding)

5 Citations (Scopus)

271 Downloads (Pure)

Abstract

The majority of work in the literature on modal interaction is based on two degree-of-freedom nonlinear systems with cubic nonlinearities. In this paper we consider a three degree-of-freedom system with nonlinear springs containing cubic nonlinear terms. First the undamped, unforced case is considered. Specifically the modal interaction case that occurs when all the underlying linear modal frequencies are close is considered (i.e. ωn1:ωn2:ωn3≃1:1:1). In the case considered, due to the symmetry of the system, the first mode is linear and not coupled with the other two modes. The analysis is carried out by using a normal form transformation to obtain the nonlinear backbone curves of the undamped, unforced response. In addition, the frequency response function (FRF) of the corresponding lightly damped and harmonically forced system obtained from the continuation software AUTO-07p is compared with the backbone curves to show its validity for predicting the nonlinear resonant frequency and amplitude. A comparison of the results gives an insight into how modal interactions in the forced-damped response can be predicted using just the backbone curves, and how this might be applied to predict resonant responses of multi-modal nonlinear systems more generally.

Original language	English
Title of host publication	Nonlinear Dynamics, Volume 1
Subtitle of host publication	Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015
Editors	Gaëtan Kerschen
Publisher	Springer
Pages	123-131
Number of pages	9
ISBN (Electronic)	9783319152219
ISBN (Print)	9783319152202
DOIs	https://doi.org/10.1007/978-3-319-15221-9_10
Publication status	Published - 14 Aug 2015
Event	33rd IMAC Conference and Exposition on Structural Dynamics, 2015 - Orlando, FL, United States Duration: 2 Feb 2015 → 5 Feb 2015

Publication series

Name	Conference Proceedings of the Society for Experimental Mechanics Series
Publisher	Springer
ISSN (Print)	2191-5644

Conference

Conference	33rd IMAC Conference and Exposition on Structural Dynamics, 2015
Country	United States
City	Orlando, FL
Period	2/02/15 → 5/02/15

Keywords

Backbone curve
3-DoF nonlinear oscillator
Nonlinear modal interaction
Cubic nonlinearity
Second-order normal form method

Access to Document

10.1007/978-3-319-15221-9_10

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Accepted author manuscript, 357 KB

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Cite this

Liu, X., Cammarano, A., Wagg, D. J., Neild, S. A., & Barthorpe, R. J. (2015). Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities. In G. Kerschen (Ed.), Nonlinear Dynamics, Volume 1: Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015 (pp. 123-131). (Conference Proceedings of the Society for Experimental Mechanics Series). Springer. https://doi.org/10.1007/978-3-319-15221-9_10

Liu, Xu ; Cammarano, Andrea ; Wagg, David J ; Neild, Simon A ; Barthorpe, R. J. / Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities. Nonlinear Dynamics, Volume 1: Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015. editor / Gaëtan Kerschen. Springer, 2015. pp. 123-131 (Conference Proceedings of the Society for Experimental Mechanics Series).

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title = "Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities",

abstract = "The majority of work in the literature on modal interaction is based on two degree-of-freedom nonlinear systems with cubic nonlinearities. In this paper we consider a three degree-of-freedom system with nonlinear springs containing cubic nonlinear terms. First the undamped, unforced case is considered. Specifically the modal interaction case that occurs when all the underlying linear modal frequencies are close is considered (i.e. ωn1:ωn2:ωn3≃1:1:1). In the case considered, due to the symmetry of the system, the first mode is linear and not coupled with the other two modes. The analysis is carried out by using a normal form transformation to obtain the nonlinear backbone curves of the undamped, unforced response. In addition, the frequency response function (FRF) of the corresponding lightly damped and harmonically forced system obtained from the continuation software AUTO-07p is compared with the backbone curves to show its validity for predicting the nonlinear resonant frequency and amplitude. A comparison of the results gives an insight into how modal interactions in the forced-damped response can be predicted using just the backbone curves, and how this might be applied to predict resonant responses of multi-modal nonlinear systems more generally.",

keywords = "Backbone curve, 3-DoF nonlinear oscillator, Nonlinear modal interaction, Cubic nonlinearity, Second-order normal form method",

author = "Xu Liu and Andrea Cammarano and Wagg, {David J} and Neild, {Simon A} and Barthorpe, {R. J.}",

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Liu, X, Cammarano, A, Wagg, DJ, Neild, SA & Barthorpe, RJ 2015, Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities. in G Kerschen (ed.), Nonlinear Dynamics, Volume 1: Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015. Conference Proceedings of the Society for Experimental Mechanics Series, Springer, pp. 123-131, 33rd IMAC Conference and Exposition on Structural Dynamics, 2015, Orlando, FL, United States, 2/02/15. https://doi.org/10.1007/978-3-319-15221-9_10

Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities. / Liu, Xu; Cammarano, Andrea; Wagg, David J; Neild, Simon A; Barthorpe, R. J.

Nonlinear Dynamics, Volume 1: Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015. ed. / Gaëtan Kerschen. Springer, 2015. p. 123-131 (Conference Proceedings of the Society for Experimental Mechanics Series).

Research output: Chapter in Book/Report/Conference proceeding › Conference Contribution (Conference Proceeding)

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T1 - Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities

AU - Liu, Xu

AU - Cammarano, Andrea

AU - Wagg, David J

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AU - Barthorpe, R. J.

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N2 - The majority of work in the literature on modal interaction is based on two degree-of-freedom nonlinear systems with cubic nonlinearities. In this paper we consider a three degree-of-freedom system with nonlinear springs containing cubic nonlinear terms. First the undamped, unforced case is considered. Specifically the modal interaction case that occurs when all the underlying linear modal frequencies are close is considered (i.e. ωn1:ωn2:ωn3≃1:1:1). In the case considered, due to the symmetry of the system, the first mode is linear and not coupled with the other two modes. The analysis is carried out by using a normal form transformation to obtain the nonlinear backbone curves of the undamped, unforced response. In addition, the frequency response function (FRF) of the corresponding lightly damped and harmonically forced system obtained from the continuation software AUTO-07p is compared with the backbone curves to show its validity for predicting the nonlinear resonant frequency and amplitude. A comparison of the results gives an insight into how modal interactions in the forced-damped response can be predicted using just the backbone curves, and how this might be applied to predict resonant responses of multi-modal nonlinear systems more generally.

AB - The majority of work in the literature on modal interaction is based on two degree-of-freedom nonlinear systems with cubic nonlinearities. In this paper we consider a three degree-of-freedom system with nonlinear springs containing cubic nonlinear terms. First the undamped, unforced case is considered. Specifically the modal interaction case that occurs when all the underlying linear modal frequencies are close is considered (i.e. ωn1:ωn2:ωn3≃1:1:1). In the case considered, due to the symmetry of the system, the first mode is linear and not coupled with the other two modes. The analysis is carried out by using a normal form transformation to obtain the nonlinear backbone curves of the undamped, unforced response. In addition, the frequency response function (FRF) of the corresponding lightly damped and harmonically forced system obtained from the continuation software AUTO-07p is compared with the backbone curves to show its validity for predicting the nonlinear resonant frequency and amplitude. A comparison of the results gives an insight into how modal interactions in the forced-damped response can be predicted using just the backbone curves, and how this might be applied to predict resonant responses of multi-modal nonlinear systems more generally.

KW - Backbone curve

KW - 3-DoF nonlinear oscillator

KW - Nonlinear modal interaction

KW - Cubic nonlinearity

KW - Second-order normal form method

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DO - 10.1007/978-3-319-15221-9_10

M3 - Conference Contribution (Conference Proceeding)

SN - 9783319152202

T3 - Conference Proceedings of the Society for Experimental Mechanics Series

SP - 123

EP - 131

BT - Nonlinear Dynamics, Volume 1

A2 - Kerschen, Gaëtan

PB - Springer

T2 - 33rd IMAC Conference and Exposition on Structural Dynamics, 2015

Y2 - 2 February 2015 through 5 February 2015

ER -

Liu X, Cammarano A, Wagg DJ, Neild SA, Barthorpe RJ. Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities. In Kerschen G, editor, Nonlinear Dynamics, Volume 1: Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015. Springer. 2015. p. 123-131. (Conference Proceedings of the Society for Experimental Mechanics Series). https://doi.org/10.1007/978-3-319-15221-9_10