Nonlinear Dynamics

, Volume 77, Issue 1–2, pp 311–320 | Cite as

Bifurcations of backbone curves for systems of coupled nonlinear two mass oscillator

  • A. CammaranoEmail author
  • T. L. Hill
  • S. A. Neild
  • D. J. Wagg
Original Paper


This paper considers the dynamic response of coupled, forced and lightly damped nonlinear oscillators with two degree-of-freedom. For these systems, backbone curves define the resonant peaks in the frequency–displacement plane and give valuable information on the prediction of the frequency response of the system. Previously, it has been shown that bifurcations can occur in the backbone curves. In this paper, we present an analytical method enabling the identification of the conditions under which such bifurcations occur. The method, based on second-order nonlinear normal forms, is also able to provide information on the nature of the bifurcations and how they affect the characteristics of the response. This approach is applied to a two-degree-of-freedom mass, spring, damper system with cubic hardening springs. We use the second-order normal form method to transform the system coordinates and identify which parameter values will lead to resonant interactions and bifurcations of the backbone curves. Furthermore, the relationship between the backbone curves and the complex dynamics of the forced system is shown.


Backbone curve Bifurcation Nonlinear oscillator  Second-order normal form method 


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • A. Cammarano
    • 1
    Email author
  • T. L. Hill
    • 1
  • S. A. Neild
    • 1
  • D. J. Wagg
    • 2
  1. 1.Department of Mechanical EngineeringUniversity of BristolBristol UK
  2. 2.Department of Mechanical EngineeringUniversity of SheffieldSheffield UK

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