Interpreting the forced responses of a two-degree-of-freedom nonlinear oscillator using backbone curves

Tom L Hill, Andrea Cammarano, Simon A Neild, David J Wagg

Research output: Contribution to journalArticle (Academic Journal)peer-review

60 Citations (Scopus)
431 Downloads (Pure)


In this paper the backbone curves of a two-degree-of-freedom nonlinear oscillator are used to interpret its behaviour when subjected to external forcing. The backbone curves describe the loci of dynamic responses of a system when unforced and undamped, and are represented in the frequency–amplitude projection. In this study we provide an analytical method for relating the backbone curves, found using the second-order normal form technique, to the forced responses. This is achieved using an energy-based analysis to predict the resonant crossing points between the forced responses and the backbone curves. This approach is applied to an example system subjected to two different forcing cases: one in which the forcing is applied directly to an underlying linear mode and the other subjected to forcing in both linear modes. Additionally, a method for assessing the accuracy of the prediction of the resonant crossing points is then introduced, and these predictions are then compared to responses found using numerical continuation.
Original languageEnglish
Pages (from-to)276–288
Number of pages13
JournalJournal of Sound and Vibration
Early online date21 Apr 2015
Publication statusPublished - 4 Aug 2015

Bibliographical note

Date of Acceptance: 14/05/2015


  • backbone curve
  • coupled nonlinear oscillator
  • internal resonance
  • numerical continuation
  • forced response


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