Advertisement

Nonlinear Dynamics

, Volume 93, Issue 1, pp 63–78 | Cite as

Forecasting bifurcations of multi-degree-of-freedom nonlinear systems with parametric resonance

  • Shiyang Chen
  • Bogdan Epureanu
Original Paper
  • 218 Downloads

Abstract

The location of bifurcation points and bifurcation diagrams is important for the understanding of a nonlinear system with parametric resonance. This paper presents a model-less method to predict bifurcations of slightly damped multi-degree-of-freedom nonlinear systems with parametric resonance using transient recovery data in the pre-bifurcation regime. This method is based on the observation that the envelope amplitude of a decaying response in the pre-bifurcation regime recovers more slowly to the equilibrium as the system becomes closer to the bifurcation. Data obtained from both simulations and experiments are used to forecast the location of the bifurcation point and the bifurcation diagram. Forecasting results demonstrate that the method can be used to predict the bifurcation accurately under a set of specific assumptions.

Keywords

Parametric resonance Forecasting bifurcations Model-less approach 

Notes

Acknowledgements

This research was supported by the National Institute of General Medical Sciences of the National Institutes of Health under Award Number U01GM110744. The content is solely the responsibility of the authors and does not necessarily reflect the official views of the National Institutes of Health.

References

  1. 1.
    Oropeza-Ramos, A.L., Turner, L.K.: Parametric resonance amplification in a MEMGyroscope. In: 2005 IEEE Sensors, p. 4. IEEE (2005)Google Scholar
  2. 2.
    Akhmedov, E.K.: Parametric resonance of neutrino oscillations and passage of solar and atmospheric neutrinos through the earth. Nucl. Phys. B 538(1), 25–51 (1999)CrossRefGoogle Scholar
  3. 3.
    Lednev, V.V.: Possible mechanism for the influence of weak magnetic fields on biological systems. Bioelectromagnetics 12(2), 71–75 (1991)CrossRefGoogle Scholar
  4. 4.
    Chen, S., Epureanu, B.: Regular biennial cycles in epidemics caused by parametric resonance. J. Theor. Biol. 415, 137–144 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Shin, Y., Belenky, V., Paulling, J., et al.: Criteria for parametric roll of large containerships in longitudinal seas. Trans. Soc. Nav. Archit. Mar. Eng. 112, 14–47 (2004)Google Scholar
  6. 6.
    Pellicano, F., Fregolent, A., Bertuzzi, A., Vestroni, F.: Primary and parametric non-linear resonances of a power transmission belt: experimental and theoretical analysis. J. Sound Vib. 244(4), 669–684 (2001)CrossRefGoogle Scholar
  7. 7.
    Daqaq, M.F., et al.: Investigation of power harvesting via parametric excitations. J. Intell. Mater. Syst. Struct. 20(5), 545–557 (2009)CrossRefGoogle Scholar
  8. 8.
    Bulian, G., Francescutto, A., Lugni, C.: On the nonlinear modeling of parametric rolling in regular and irregular waves. Int. Shipbuild. Prog. 51(2), 173 (2004)Google Scholar
  9. 9.
    Spina, D., Valente, C., Tomlinson, G.R.: A new procedure for detecting nonlinearity from transient data using the gabor transform. Nonlinear Dyn. 11(3), 235–254 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kerschen, G., et al.: Past, present and future of nonlinear system identification in structural dynamics. Mech. Syst. Signal Process. 20(3), 505–592 (2006)CrossRefGoogle Scholar
  11. 11.
    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
  12. 12.
    Lim, J., Epureanu, B.I.: Forecasting a class of bifurcations: theory and experiment. Phys. Rev. E 83(1), 016203 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    D’Souza, K., Epureanu, B.I., Pascual, M.: Forecasting bifurcations from large perturbation recoveries in feedback ecosystems. PLoS ONE 10(9), e0137779 (2015)CrossRefGoogle Scholar
  14. 14.
    Ghadami, A., Epureanu, B.I.: Bifurcation forecasting for large dimensional oscillatory systems: forecasting flutter using gust responses. J. Comput. Nonlinear Dyn. 11(6), 061009 (2016)CrossRefGoogle Scholar
  15. 15.
    Scheffer, M., Carpenter, S.R., Lenton, T.M., Bascompte, J., Brock, W., Dakos, V., Van De Koppel, J., Leemput, I.A.V.D., Levin, S.A., Nes, E.H.V., et al.: Anticipating critical transitions. Science 338(6105), 344 (2012)CrossRefGoogle Scholar
  16. 16.
    Scheffer, M., Bascompte, J., Brock, W.A., Brovkin, V., Carpenter, S.R., Dakos, V., Held, H., van Nes, E.H., Rietkerk, M., Sugihara, G.: Early-warning signals for critical transitions. Nature 461(7260), 53 (2009)CrossRefGoogle Scholar
  17. 17.
    Kéfi, S., Dakos, V., Scheffer, M., Van Nes, E.H., Rietkerk, M.: Early warning signals also precede non-catastrophic transitions. Oikos 122(5), 641–648 (2013)CrossRefGoogle Scholar
  18. 18.
    Lenton, T.M.: Early warning of climate tipping points. Nat. Clim. Change 1(4), 201 (2011)CrossRefGoogle Scholar
  19. 19.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, Boulder (2014)zbMATHGoogle Scholar
  20. 20.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)zbMATHGoogle Scholar
  21. 21.
    Nayfeh, A.: The response of multidegree-of-freedom systems with quadratic non-linearities to a harmonic parametric resonance. J. Sound Vib. 90(2), 237–244 (1983)CrossRefzbMATHGoogle Scholar
  22. 22.
    Nayfeh, A.H., Balachandran, B.: Modal interactions in dynamical and structural systems. Appl. Mech. Rev. 42(11S), S175–S201 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (2013)zbMATHGoogle Scholar
  24. 24.
    Miles, J.: Parametric excitation of an internally resonant double pendulum. Zeitschrift für angewandte Mathematik und Physik ZAMP 36(3), 337–345 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of MichiganAnn ArborUSA

Personalised recommendations