On the Use of the Multiple Scale Harmonic Balance Method for Nonlinear Energy Sinks Controlled Systems

Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 168)

Abstract

The Multiple Scale Harmonic Balance Method (MSHBM) is discussed here for several paradigmatic systems (primary structures) equipped with a Nonlinear Energy Sink (NES). This is a small-mass oscillator with essentially nonlinear stiffness, used for passive control purpose. The method permits to overcome the difficulties inherent to standard perturbation methods, which occur as a consequence of the nonlinearizable nature of the NES equation. It combines the Multiple Scale Method and the Harmonic Balance Method to furnish Amplitude Modulation Equations ruling the slow asymptotic dynamics of the augmented system. The MSHBM is illustrated here for a general, internally non-resonant, multi d.o.f. structure equipped with a NES and under multiple concurrent actions, namely steady wind inducing Hopf bifurcation , and 1:1 and 1:3 resonant harmonic forces. The relevant Amplitude Modulation Equations are specialized for simpler cases, where the single contributions of each external action is considered separately. The effect of the NES on the dynamics of the system is discussed for each case and numerical results are displayed.

Keywords

Hopf Bifurcation Main System Nonlinear Energy Sink Subharmonic Resonance Galerkin Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

This work was granted by the Italian Ministry of University and Research (MIUR), under the PRIN10-12 program, project No. 2010MBJK5B.

References

  1. 1.
    Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems I. Springer, New York (2008)Google Scholar
  2. 2.
    Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems II. Springer, New York (2008)Google Scholar
  3. 3.
    Maniadis, P., Kopidakis, G., Aubry, S.: Classical and quantum targeted energy transfer between nonlinear oscillators. Physica D 188, 153–177 (2004)CrossRefADSMATHGoogle Scholar
  4. 4.
    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, 285–309 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Panagopoulos, P.N., Gendelman, O., Vakakis, A.F.: Robustness of nonlinear targeted energy transfer in coupled oscillators to changes of initial conditions. Nonlinear Dyn. 47, 377–387 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Aubry, S., Kopidakis, G., Morgante, A.M., Tsironis, G.P.: Analytic conditions for targeted energy transfer between nonlinear oscillators or discrete breathers. Physica B: Phys. Conden. Matter 296, 222–236 (2001)CrossRefADSGoogle Scholar
  7. 7.
    Tsakirtzis, S., Panagopoulos, P.N., Kerschen, G., Gendelman, O., Vakakis, A.F., Bergman, L.A.: Complex dynamics and targeted energy transfer in linear oscillatorscoupled to multi-degree-of-freedom essentially nonlinear attachments. Nonlinear Dyn. 48, 285–318 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Guckenheimer, J., Wechselberger, M., Young, L.-S.: Chaotic attractors of relaxation oscillators. Nonlinearity 19, 701–720 (2006)MathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Guckenheimer, J., Hoffman, K., Weckesser, W.: Bifurcations of relaxation oscillations near folded saddles. Int. J. Bifurcat. Chaos 15, 3411–3421 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gendelman, O.V., Starosvetsky, Y., Feldman, M.: Attractors of harmonically forced linear oscillator with attached nonlinear energy sink. Part I: description of response regimes. Nonlinear Dyn. 51, 31–46 (2008)CrossRefMATHGoogle Scholar
  11. 11.
    Starosvetsky, Y., Gendelman, O.V.: Response regimes of linear oscillator coupled to nonlinear energy sink with harmonic forcing and frequency detuning. J. Sound Vib. 315, 746–765 (2008)CrossRefADSGoogle Scholar
  12. 12.
    Vaurigaud, B., Savadkoohi, A.T., Lamarque, C.-H.: Targeted energy transfer with parallel nonlinear energy sinks. Part I: design theory and numerical results. Nonlinear Dyn. 66(4), 763–780 (2011)Google Scholar
  13. 13.
    Savadkoohi, A.T., Vaurigaud, B., Lamarque, C.-H., Pernot, S.: Targeted energy transfer with parallel nonlinear energy sinks. Part II: theory and experiments. Nonlinear Dyn. 67(1), 37–46 (2012)CrossRefMATHGoogle Scholar
  14. 14.
    Lamarque, C.-H., Gendelman, O.V., Savadkoohi, A.T., Etcheverria, E.: Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink. Acta Mechanica 221, 175–200 (2011)CrossRefGoogle Scholar
  15. 15.
    Gendelman, O.V., Vakakis, A.F., Bergman, L.A., McFarland, D.M.: Asymptotic analysis of passive nonlinear suppression of aeroelastic instabilities of a rigid wing in subsonic flow. SIAM J. Appl. Math. 70(5), 1655–1677 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Vaurigaud, B., Manevitch, L.I., Lamarque, C.-H.: Passive control of aeroelastic instability in a long span bridge model prone to coupled flutter using targeted energy transfer. J. Sound Vib. 330, 2580–2595 (2011)Google Scholar
  17. 17.
    Manevitch, L.: The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables. Nonlinear Dyn. 25, 95–109 (2001)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gendelman, O.V.: Targeted energy transfer in systems with non-polynomial nonlinearity. J. Sound Vib. 315, 732–745 (2008)CrossRefADSGoogle Scholar
  19. 19.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)MATHGoogle Scholar
  20. 20.
    Jiang, X., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Steady state passive nonlinear energy pumping in coupled oscillators: theoretical and experimental results. Nonlinear Dyn. 33, 87–102 (2003)CrossRefMATHGoogle Scholar
  21. 21.
    Malatkar, P., Nayfeh, A.H.: Steady-state dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator. Nonlinear Dyn. 47, 167–179 (2007)CrossRefMATHGoogle Scholar
  22. 22.
    Luongo, A., Zulli, D.: Dynamic analysis of externally excited NES-controlled systems via a mixed Multiple Scale/Harmonic Balance algorithm. Nonlinear Dyn. 70(3), 2049–2061 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Luongo, A., Zulli, D.: Aeroelastic instability analysis of NES-controlled systems via a mixed Multiple Scale/Harmonic Balance Method. J. Vib. Control 20(13), (2014)Google Scholar
  24. 24.
    Zulli, D., Luongo, A.: Nonlinear energy sink to control vibrations of an internally nonresonant elastic string. Meccanica (2014). doi: 10.1007/s11012-014-0057-0 Google Scholar
  25. 25.
    Doedel, E.J., Oldeman, B.E.: AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equation (2012). http://cmvl.cs.concordia.ca/auto/
  26. 26.
    Nayfeh, S.A., Nayfeh, A.H., Mook, D.T.: Nonlinear response of a taut string to longitudinal and transverse end excitation. J. Vib. Control 1(3), 307–334 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.M&MoCS—University of L’AquilaL’Aquila (AQ)Italy

Personalised recommendations