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Parametric and Direct Resonances in a Base-Excited Beam Carrying a Top Mass

  • Rob H. B. FeyEmail author
  • Niels J. Mallon
  • C. Stefan Kraaij
  • Henk Nijmeijer
Chapter
  • 1.4k Downloads

Abstract

In this chapter, nonlinear resonances in a coupled shaker-beam-top mass system are investigated both numerically and experimentally. The imperfect, vertical beam carries the top mass and is axially excited by the shaker at its base. The weight of the top mass is below the beam’s static buckling load. A semi-analytical model is derived for the coupled system. In this model, Taylor-series approximations are used for the inextensibility constraint and the curvature of the beam. The steady-state behavior of the model is studied using numerical tools. In the model with a single beam mode, parametric and direct resonances are found, which affect the dynamic stability of the structure. The model with two beam modes not only shows an additional second harmonic resonance, but also reveals some extra small resonances in the low-frequency range, some of which can be identified as combination resonances. The experimental steady-state response is obtained by performing a (stepped) frequency sweep-up and sweep-down with respect to the harmonic input voltage of the amplifier-shaker combination. A good correspondence between the numerical and experimental steady-state responses is obtained.

Keywords

Period Doubling Bifurcation Beam Mode Subharmonic Resonance Generalize DOFs Slender Beam 
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

Acknowledgements

This research is supported by the Dutch Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs (STW project EWO.5792).

References

  1. Amabili(2008).
    Amabili, M.: Nonlinear vibrations and stability of shells and plates. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  2. Anderson et al(1996)Anderson, Nayfeh, and Balachandran.
    Anderson, T., Nayfeh, A., Balachandran, B.:Experimental verification of the importance of the nonlinear curvature in the response of a cantilever beam. ASME J Vibr and Acoustics 118(1):21–27 (1996)CrossRefGoogle Scholar
  3. Doedel et al(1998)Doedel, Paffenroth, Champneys, Fairgrieve, Kuznetsov, Oldeman, Sandstede, and Wang.
    Doedel, E., Paffenroth, R., Champneys, A., Fairgrieve, T., Kuznetsov, Y., Oldeman, B., Sandstede, B., Wang, X.: AUTO97: Continuation and bifurcation software for ordinary differential equations (with HOMCONT). Technical Report, Concordia University (1998)Google Scholar
  4. Gibson(2001).
    Gibson, C.: Elementary geometry of differentiable curves: An Undergraduate Introduction. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  5. Mallon(2008).
    Mallon, N.: Dynamic stability of thin-walled structures: a semi-analytical and experimental approach. PhD thesis, Eindhoven University of Technology (2008)Google Scholar
  6. Mallon et al(2006)Mallon, Fey, and Nijmeijer.
    Mallon, N., Fey, R., Nijmeijer, H.: Dynamic stability of a base-excited thin beam with top mass. In: Proc. of the 2006 ASME IMECE, Nov. 5-10, Paper 13148, Chicago, USA, pp 1–10 (2006)Google Scholar
  7. McConnell(1995).
    McConnell, K.: Vibration testing, Theory and Practice. Wiley, Chichester (1995)zbMATHGoogle Scholar
  8. Mettler(1962).
    Mettler, E.: Dynamic buckling. In: Handbook of engineering mechanics (Flügge, W., ed.). McGraw-Hill, London (1962)Google Scholar
  9. Nayfeh and Pai(2004).
    Nayfeh, A., Pai, P.: Linear and nonlinear structural mechanics. Wiley-VCH, Weinheim (2004)CrossRefGoogle Scholar
  10. Noijen et al(2007)Noijen, Mallon, Fey, Nijmeijer, and Zhang.
    Noijen, S., Mallon, N., Fey, R., Nijmeijer, H., Zhang, G.: Periodic excitation of a buckled beam using a higher order semi-analytic approach. Nonlinear Dynamics 50(1-2):325–339 (2007)CrossRefGoogle Scholar
  11. Preumont(2006).
    Preumont, A.: Mechatronics, dynamics of electromechanical and piezoelectric systems. Springer, Dordrecht (2006)zbMATHGoogle Scholar
  12. Ribeiro and Carneiro(2004).
    Ribeiro, P., Carneiro, R.: Experimental detection of modal interactions in the non-linear vibration of a hinged-hinged beam. J of Sound and Vibr 277(4-5):943–954 (2004)CrossRefGoogle Scholar
  13. Son et al(2008)Son, Uchiyama, Lacarbonara, and Yabuno.
    Son, I. S., Uchiyama, Y., Lacarbonara, W., Yabuno, H.: Simply supported elastic beams under parametric excitation. Nonlinear Dynamics 53:129–138 (2008)MathSciNetCrossRefGoogle Scholar
  14. Thomsen(2003).
    Thomsen, J.: Vibrations and stability; advanced theory, analysis, and tools, second edition. Springer Verlag, Berlin (2003)zbMATHGoogle Scholar
  15. TueDACS(2008).
    TueDACS: TUeDACS Advanced Quadrature Interface (2008)Google Scholar
  16. Verbeek et al(1995)Verbeek, de Kraker, and van Campen.
    Verbeek, G., de Kraker, A., van Campen, D.: Nonlinear parametric identification using periodic equilibrium states. Nonlinear Dynamics 7:499–515 (1995)CrossRefGoogle Scholar
  17. Virgin(2007).
    Virgin, L.: Vibration of axially loaded structures. Cambridge University Press, New York (2007)CrossRefGoogle Scholar
  18. Yabuno et al(1998)Yabuno, Ide, and Aoshima.
    Yabuno, H., Ide, Y., Aoshima, N.: Nonlinear analysis of a parametrically excited cantilever beam (effect of the tip mass on stationary response). JSME Int J 41(3):555–562 (1998)CrossRefGoogle Scholar
  19. Yabuno et al(2003)Yabuno, Okhuma, and Lacarbonara.
    Yabuno, H., Okhuma, M., Lacarbonara, W.: An experimental investigation of the parametric resonance in a buckled beam. In: Proceedings of the ASME DETC’03, Chicago, USA, pp 2565–2574 (2003)Google Scholar
  20. Zavodney and Nayfeh(1989).
    Zavodney, L., Nayfeh, A.: The non-linear response of a slender beam carrying a lumped mass to a principal parametric excitation: theory and experiment. Int J Non-Linear Mech 24(2):105–125 (1989)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Rob H. B. Fey
    • 1
    Email author
  • Niels J. Mallon
    • 2
  • C. Stefan Kraaij
    • 3
  • Henk Nijmeijer
    • 1
  1. 1.Department of Mechanical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Centre for Mechanical and Maritime StructuresTNO Built Environment and GeosciencesDelftThe Netherlands
  3. 3.IHC Lagersmit BVKinderdijkThe Netherlands

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