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 proceedingConference Contribution (Conference Proceeding)

5 Citations (Scopus)
330 Downloads (Pure)


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 languageEnglish
Title of host publicationNonlinear Dynamics, Volume 1
Subtitle of host publicationProceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015
EditorsGaëtan Kerschen
Number of pages9
ISBN (Electronic)9783319152219
ISBN (Print)9783319152202
Publication statusPublished - 14 Aug 2015
Event33rd IMAC Conference and Exposition on Structural Dynamics, 2015 - Orlando, FL, United States
Duration: 2 Feb 20155 Feb 2015

Publication series

NameConference Proceedings of the Society for Experimental Mechanics Series
ISSN (Print)2191-5644


Conference33rd IMAC Conference and Exposition on Structural Dynamics, 2015
Country/TerritoryUnited States
CityOrlando, FL


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


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