Constructing Backbone Curves from Free-Decay Vibrations Data in Multi-Degrees of Freedom Oscillatory Systems

  • Mattia CenedeseEmail author
  • George Haller
Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)


Backbone curves are often the best representation of the nonlinear behavior for the vibrations of mechanical systems. Several approaches for obtaining them are present in literature, either analytical, numerical or experimental ones. However, they often make assumptions that unavoidably limit the range of applicability, such as the dynamics of the underlying conservative system and the modeling of damping terms. Here, we describe a mathematical theory and the corresponding numerical methodology that is able to rigorously extract backbone curves from free-decay vibrations data and that can overcome some of the main limitations of existing methods. We illustrate our findings with synthetic and real experiment vibration measurements.


Backbone curves Nonlinear vibrations Damped vibrations Nonlinear system identification Spectral submanifolds 


  1. 1.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley (2007)Google Scholar
  2. 2.
    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
  3. 3.
    Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control. 55(3), 531–534 (1992)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Szalai, R., Ehrhardt, D., Haller, G.: Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations. Proc. Roy. Soc. Lond A. 473(2202), 32 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems, Volume 59 of Applied Mathematical Sciences, 2nd edn. Springer-Verlag, New York (2007)zbMATHGoogle Scholar
  6. 6.
    Doedel, R.C., Paffenroth, E.J., Champneys, A.R., Fairgrieve, T.F., Kutnetsov, Y.A., Oldeman, B.E., Sandstede, B., Wang, X.J.: Auto2000: Continuation and Bifurcation Software for Ordinary Differential EquationsGoogle Scholar
  7. 7.
    Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: Matcont: a Matlab package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29(2), 141–164 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dankowicz, H., Schilder, F.: Recipes for Continuation. Society for Industrial and Applied Mathematics (2013)Google Scholar
  9. 9.
    Neild, S.A., Wagg, D.J.: Applying the method of normal forms to second-order nonlinear vibration problems. Proc. Roy. Soc. Lond. A. 467(2128), 1141–1163 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Renson, L., Kerschen, G., Cochelin, B.: Numerical computation of nonlinear normal modes in mechanical engineering. J. Sound Vib. 364, 177–206 (2016)CrossRefGoogle Scholar
  11. 11.
    Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.C.: Nonlinear normal modes, part II: Toward a practical computation using numerical continuation techniques. Mech. Syst. Signal Process. 23(1), 195–216 (2009). Special Issue: Nonlinear Structural DynamicsCrossRefGoogle Scholar
  12. 12.
    Noël, J.P., Kerschen, G.: Nonlinear system identification in structural dynamics: 10 more years of progress. Mech. Syst. Signal Process. 83, 2–35 (2017)CrossRefGoogle Scholar
  13. 13.
    Peeters, M., Kerschen, G., Golinval, J.C.: Dynamic testing of nonlinear vibrating structures using nonlinear normal modes. J. Sound Vib. 330(3), 486–509 (2011)CrossRefGoogle Scholar
  14. 14.
    Peeters, M., Kerschen, G., Golinval, J.C.: Modal testing of nonlinear vibrating structures based on nonlinear normal modes: experimental demonstration. Mech. Syst. Signal Process. 25(4), 1227–1247 (2011)CrossRefGoogle Scholar
  15. 15.
    Londoño, J.M., Neild, S.A., Cooper, J.E.: Identification of backbone curves of nonlinear systems from resonance decay responses. J. Sound Vib. 348, 224–238 (2015)CrossRefGoogle Scholar
  16. 16.
    Renson, L., Gonzalez-Buelga, A., Barton, D.A.W., Neild, S.A.: Robust identification of backbone curves using control-based continuation. J. Sound Vib. 367, 145–158 (2016)CrossRefGoogle Scholar
  17. 17.
    Haller, G., Ponsioen, S.: Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction. Nonlinear Dynam. 86(3), 1493–1534 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Breunung, T., Haller, G.: Explicit backbone curves from spectral submanifolds of forced-damped nonlinear mechanical systems. Proc. Roy. Soc. Lond. A. 474(2213), (2018)Google Scholar
  19. 19.
    Avitabile, P.: Modal Testing: A Practitioner’s Guide. Wiley (2017)Google Scholar
  20. 20.
    Billings, S.A.: Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains. Wiley (2013)Google Scholar

Copyright information

© Society for Experimental Mechanics, Inc. 2020

Authors and Affiliations

  1. 1.Institute for Mechanical SystemsETH ZürichZürichSwitzerland

Personalised recommendations