A Study of the Modal Interaction Amongst Three Nonlinear Normal Modes Using a Backbone Curve Approach

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


In this paper, a three degree-of-freedom oscillator with cubic elastic nonlinearities is considered. For this system, the natural frequencies of its underlying linear modes are set to be approximately equal so that \(\omega _{n1}:\omega _{n2}:\omega _{n3} \approx 1:1:1\). As a result, the nonlinear normal modes in the system are able to potentially interact with each other. In this study, the underlying unforced and undamped system is considered. The second-order normal forms technique is used to estimate the backbone curves of the system, which give information on the frequency and modal response amplitudes and phases. Then, through choosing the activate modes and their specific phase differences, the single-, double- and triple-mode backbone curves are computed. The results show the effect of nonlinear multi-mode interactions on the dynamic response of nonlinear oscillators. These insights will be beneficial when considering how a structure will respond and for the system identification of nonlinear multi-degree-of-freedom systems.


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



The authors would like to acknowledges the support of the Engineering and Physical Sciences Research Council. S.A.N is supported by EPSRC Fellowship EP/K005375/1. D.J.W is supported by EPSRC grant EP/K003836/1.


  1. 1.
    Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Arnol’d, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations, vol. 250. Springer Science & Business Media (2012)Google Scholar
  3. 3.
    Cammarano, A., Hill, T., Neild, S., Wagg, D.: Bifurcations of backbone curves for systems of coupled nonlinear two mass oscillator. Nonlinear Dyn. 77(1–2), 311–320 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hill, T., Cammarano, A., Neild, S., Wagg, D.: Interpreting the forced responses of a two-degree-of-freedom nonlinear oscillator using backbone curves. J. Sound Vib. 349, 276–288 (2015)CrossRefGoogle Scholar
  5. 5.
    Hill, T.L., Cammarano, A., Neild, S.A., Wagg, D.J.: Out-of-unison resonance in weakly nonlinear coupled oscillators. Proc. Roy. Soc. Lond. A Math. Phys. Eng. Sci. 471, 20140659 (2015)Google Scholar
  6. 6.
    Jezequel, L., Lamarque, C.H.: Analysis of non-linear dynamical systems by the normal form theory. J. Sound Vib. 149(3), 429–459 (1991)CrossRefGoogle Scholar
  7. 7.
    Kerschen, G., Kowtko, J.J., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Theoretical and experimental study of multimodal targeted energy transfer in a system of coupled oscillators. Nonlinear Dyn. 47(1–3), 285–309 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F.: Nonlinear normal modes, part i: a useful framework for the structural dynamicist. Mech. Syst. Signal Process. 23(1), 170–194 (2009)CrossRefGoogle Scholar
  9. 9.
    Lewandowski, R.: On beams membranes and plates vibration backbone curves in cases of internal resonance. Meccanica 31(3), 323–346 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Liu, X., Cammarano, A., Wagg, D., Neild, S., Barthorpe, R.: N-1 modal interactions of a three-degree-of-freedom system with cubic elastic nonlinearities. Nonlinear Dyn. 1–15 (2015)Google Scholar
  11. 11.
    Liu, X., Cammarano, A., Wagg, D., Neild, S., Barthorpe, R.: Nonlinear modal interaction analysis for a three degree-of-freedom system with cubic nonlinearities. Nonlinear Dyn. 1, 123–131 (2016)MathSciNetGoogle Scholar
  12. 12.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)zbMATHGoogle Scholar
  13. 13.
    Neild, S.A., Wagg, D.J.: Applying the method of normal forms to second-order nonlinear vibration problems. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 467, 1141–1163 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pierre, C., Jiang, D., Shaw, S.: Nonlinear normal modes and their application in structural dynamics. Math. Probl. Eng. 2006 (2006)Google Scholar
  15. 15.
    Rega, G., Lacarbonara, W., Nayfeh, A., Chin, C.: Multiple resonances in suspended cables: direct versus reduced-order models. Int. J. Non-Linear Mech. 34(5), 901–924 (1999)CrossRefzbMATHGoogle Scholar
  16. 16.
    Touzé, C., Amabili, M.: Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modelling of harmonically forced structures. J. Sound Vib. 298(4), 958–981 (2006)CrossRefGoogle Scholar
  17. 17.
    Touzé, C., Thomas, O., Chaigne, A.: Asymmetric non-linear forced vibrations of free-edge circular plates. part 1: theory. J. Sound Vib. 258(4), 649–676 (2002)Google Scholar
  18. 18.
    Wagg, D., Neild, S.: Nonlinear Vibration with Control. Springer, Berlin (2014)zbMATHGoogle Scholar

Copyright information

© The Society for Experimental Mechanics, Inc. 2016

Authors and Affiliations

  • X. Liu
    • 1
    Email author
  • A. Cammarano
    • 2
  • D. J. Wagg
    • 1
  • S. A. Neild
    • 3
  1. 1.Department of Mechanical EngineeringUniversity of SheffieldSheffieldUK
  2. 2.School of EngineeringUniversity of GlasgowGlasgowUK
  3. 3.Department of Mechanical EngineeringUniversity of BristolBristolUK

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