Towards a Technique for Nonlinear Modal Reduction

  • T. L. Hill
  • A. Cammarano
  • S. A. Neild
  • D. J. Wagg
Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)


In this paper we discuss an analytical method to enable modal reduction of weakly nonlinear systems with multiple degrees-of-freedom. This is achieved through the analysis of backbone curves—the response of the Hamiltonian equivalent of a system—which can help identify internal resonance within systems. An example system, with two interacting modes, is introduced and the method of second-order normal forms is used to describe its backbone curves with simple, analytical expressions. These expressions allow us to highlight which particular interactions are significant, as well as specify the conditions under which they are important. The descriptions of the backbone curves are validated against the results of continuation analysis, and a comparison is also made with the response of the system under various levels of forcing and damping. Finally, we discuss how this technique may be expanded to systems with a greater number of modes.


Backbone curves Second-order normal forms Modal analysis Modal interaction Modal reduction 


  1. 1.
    Ewins DJ (2000) Modal testing: theory, practice, and application. Research Studies Press, Philadelphia, PAGoogle Scholar
  2. 2.
    Shaw SW, Pierre C, Pesheck E (1999) Modal analysis-based reduced-order models for nonlinear structures: an invariant manifold approach. Shock Vib Digest 31:3–16CrossRefGoogle Scholar
  3. 3.
    Pesheck E, Boivin N, Pierre C, Shaw SW (2001) Nonlinear modal analysis of structural systems using multi-mode invariant manifolds. Nonlinear Dynam 25:183–205CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Kerschen G, Peete rs M, Golinval JC, Vakakis AF (2009) Nonlinear normal modes, Part I: A useful framework for the structural dynamicist. Mech Syst Signal Process 23:170–194Google Scholar
  5. 5.
    Peeters M, Vigui R, Srandour G, Kerschen G, Golinval JC (2009) Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques. Mech Syst Signal Process 23:195–216CrossRefGoogle Scholar
  6. 6.
    Doedel EJ, with major contributions from Champneys AR, Fairgrieve TF, Kuznetsov YuA, Dercole F, Oldeman BE, Paffenroth RC, Sandstede B, Wang XJ, Zhang C (2008) AUTO-07P: Continuation and bifurcation software for ordinary differential equations. Concordia University, Montreal.
  7. 7.
    Murdock J (2002) Normal forms and unfoldings for local dynamical systems. Springer, New YorkGoogle Scholar
  8. 8.
    Neild SA, Wagg DJ (2010) Applying the method of normal forms to second-order nonlinear vibration problems. Proc Roy Soc A Math Phys Eng Sci 467:1141–1163CrossRefMathSciNetGoogle Scholar
  9. 9.
    Neild SA (2012) Approximate methods for analysing nonlinear structures. Springer, ViennaGoogle Scholar

Copyright information

© The Society for Experimental Mechanics, Inc. 2014

Authors and Affiliations

  • T. L. Hill
    • 1
  • A. Cammarano
    • 1
  • S. A. Neild
    • 1
  • D. J. Wagg
    • 1
  1. 1.Department of Mechanical EngineeringUniversity of BristolBristolUK

Personalised recommendations