Nonlinear Dynamics

, Volume 88, Issue 3, pp 1769–1784 | Cite as

Experimental investigation and analytical description of a vibro-impact NES coupled to a single-degree-of-freedom linear oscillator harmonically forced

  • Giuseppe Pennisi
  • Cyrille Stephan
  • Etienne Gourc
  • Guilhem Michon
Original Paper


In this paper, the dynamics of a system composed of a harmonically forced single-degree-of-freedom linear oscillator coupled to a vibro-impact nonlinear energy sink (VI-NES) is experimentally investigated. The mass ratio between the VI-NES and the primary system is about \(1\%\). Depending on the external force’s amplitude and frequency, either a strongly modulated response (SMR) or a constant amplitude response (CAR) is observed. In both cases, an irreversible transfer of energy occurs from the linear oscillator toward the VI-NES: process known in the literature as passive targeted energy transfer. Furthermore, the problem is analytically studied by using the method of multiple scales. The obtained slow invariant manifold shows the existence of a stable and of an unstable branch of solutions, as well as of an energy threshold (a saddle-node bifurcation) for the solutions to appear. Subsequently, the fixed points of the problem are calculated. When a stable fixed point is reached, the system is naturally drawn to it and a CAR is established, whereas when no stable point is attained, the system exhibits a SMR regime. Finally, a good correlation between the experimental and the analytical results is presented.


Structural dynamics Nonlinear dynamics Vibrations Nonlinear energy sink Experimental dynamics 


  1. 1.
    Gendelman, O.: Transition of energy to a nonlinear localized mode in a highly asymmetric system of two oscillators. Nonlinear Dyn. 25, 237–253 (2001)CrossRefMATHGoogle Scholar
  2. 2.
    Vakakis, A.F., Gendelman, O.V., Manevitch, L.I., McCloskey, R.: Energy pumping in nonlinear mechanical oscillators part ii resonance capture. J. Appl. Mech. Trans. ASME 68, 42–48 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Vakakis, A., Gendelman, O., Manevitch, L., McCloskey, R.: Energy pumping in nonlinear mechanical oscillators i: dynamics of the underlying hamiltonian system. J. Appl. Mech.Trans. ASME 68, 34–41 (2001)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Manevitch, L., Gourdon, E., Lamarque, C.: Parameters optimization for energy pumping in strongly nonhomogeneous 2 dof system. Chaos Solitons Fractals 31(4), 900–911 (2007)CrossRefGoogle Scholar
  5. 5.
    Kerschen, G., Lee, Y.S., Vakakis, A.F., McFarland, D.M., Bergman, L.A.: Irreversible passive energy transfer in coupled oscillators with essential nonlinearity. SIAM J. Appl. Math. 66(2), 648–679 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    McFarland, D., Bergman, L., Vakakis, A.: Experimental study of non-linear energy pumping occurring at a single fast frequency experimental study of non-linear energy pumping occurring at a single fast frequency. Int. J. Non-Linear Mech. 40(6), 891–899 (2005)CrossRefMATHGoogle Scholar
  7. 7.
    Gourdon, E., Alexander, N., Taylor, C., Lamarque, C., Pernot, S.: Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results. J. Sound Vib. 300(3), 522–551 (2007)CrossRefGoogle Scholar
  8. 8.
    Kerschen, G., Kowtko, J., McFarland, D., Bergman, L., Vakakis, A.: Theoretical and experimental study of multimodal targeted energy transfer in a system of coupled oscillators. Nonlinear Dyn. 47(1), 285–309 (2007)MathSciNetMATHGoogle Scholar
  9. 9.
    Starosvetsky, Y., Gendelman, O.: Strongly modulated response in forced 2 dof oscillatory system with essential mass and potential asymmetry. Physica D 237(13), 1719–1733 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gourc, E., Michon, G., Seguy, S., Berlioz, A.: Experimental investigation and design optimization of targeted energy transfer under periodic forcing. J. Vib. Acoust. 136, 021021 (2014)CrossRefGoogle Scholar
  11. 11.
    Gendelman, O., Bar, T.: Bifurcations of self-excitation regimes in a van der pol oscillator with a nonlinear energy sink. Physica D 239(3), 220–229 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lee, Y.S., Vakakis, A.F., Bergman, L., McFarland, D., Kerschen, G.: Suppressing aeroelastic instability using broadband passive targeted energy transfers, part 1: Theory. AIAA J. 45(3), 693–711 (2007)CrossRefGoogle Scholar
  13. 13.
    Lee, Y.S., Kerschen, G., McFarland, D., Hill, W.J., Nichkawde, C., Strganac, T., Bergman, L., Vakakis, A.F.: Suppressing aeroelastic instability using broadband passive targeted energy transfers, part 2: Experiments. AIAA J. 45(10), 2391–2400 (2007)CrossRefGoogle Scholar
  14. 14.
    Gendelman, O., Vakakis, A., Bergman, L., McFarland, D.: 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
  15. 15.
    Vaurigaud, B., Manevitch, L., Lamarque, C.: Passive control of aeroelastic instability in a long span bridge model prone to coupled flutter using targeted energy transfer. J. Sound Vib. 330(11), 2580–2595 (2011)CrossRefGoogle Scholar
  16. 16.
    Gendelman, O.: Targeted energy transfer in systems with non-polynomial nonlinearity. J. Sound Vib. 315, 732–745 (2008)CrossRefGoogle Scholar
  17. 17.
    Gendelman, O., Lamarque, C.: Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium. Chaos Solitons Fractals 24, 501–509 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Georgiadis, F., Vakakis, A.F., McFarland, D.M., Bergman, L.A.: Shock isolation through passive energy pumping caused by non-smooth nonlinearities. Int. J. Bifurc. Chaos 15, 1–13 (2005)CrossRefGoogle Scholar
  19. 19.
    Karayannis, I., Vakakis, A.F., Georgiadis, F.: Vibro-impact attachments as shock absorbers. Proc. Inst. Mech. Eng. J. Mech. Eng. Sci. 222(222), 1899–1908 (2008)CrossRefGoogle Scholar
  20. 20.
    Lee, Y.S., Nucera, F., Vakakis, A.F., McFarland, D.M., Bergman, L.A.: Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments. Physica D 238, 1868–1896 (2009)CrossRefMATHGoogle Scholar
  21. 21.
    Nucera, F., Lo Iacono, F., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: Part ii. experimental results. J. Sound Vib. 313, 57–76 (2008)CrossRefGoogle Scholar
  22. 22.
    Gendelman, O.: Analytic treatment of a system with a vibro-impact nonlinear energy sink. J. Sound Vib. 331, 4599–4608 (2012)CrossRefGoogle Scholar
  23. 23.
    Gendelman, O.V., Alloni, A.: Dynamics of forced system with vibro-impact energy sink. J. Sound Vib. 358, 301–314 (2015)Google Scholar
  24. 24.
    Gourc, E., Michon, G., Seguy, S., Berlioz, A.: Theoretical and experimental study of an harmonically forced vibro-impact nonlinear energy sink. J. Vib. Acoust. 137, 031008 (2014)Google Scholar
  25. 25.
    Nayfeh, A.H.: Perturbation Methods. Wiley, New York (2008)Google Scholar
  26. 26.
    Li, G.X., Rand, R.H., Moon, F.C.: Bifurcations and chaos in a forced zero-stiffness impact oscillator. J. Nonlinear Mech. 25(4), 417–432 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Giuseppe Pennisi
    • 1
  • Cyrille Stephan
    • 2
  • Etienne Gourc
    • 3
  • Guilhem Michon
    • 4
  1. 1.ONERA The French Aerospace Lab, Université de Toulouse, ICAISAEChâtillonFrance
  2. 2.ONERA The French Aerospace LabChâtillonFrance
  3. 3.Université de Toulouse, ICAINSAToulouseFrance
  4. 4.Université de Toulouse, ICAISAEToulouseFrance

Personalised recommendations