Nonlinear Dynamics

, Volume 53, Issue 1–2, pp 129–138 | Cite as

Simply supported elastic beams under parametric excitation

  • In-Soo Son
  • Yuusuke Uchiyama
  • Walter Lacarbonara
  • Hiroshi Yabuno
Original Paper


In this paper, the nonlinear characteristics of the parametric resonance of simply supported elastic beams are investigated. Considering a geometrically exact formulation for unsharable and inextensible elastic beams subject to support motions, the integral-partial-differential equation of motion is obtained. The third-order perturbation of the equation of motion is then determined in a form amenable to an asymptotic treatment. The method of multiple scales is used to obtain the equations that describe the modulation of the amplitude and phase of parametric-resonance motions. The stability and bifurcations of the system are investigated considering, in particular, the frequency-response function. Furthermore, experimental results are shown to confirm the theoretically predicted stability and bifurcations.


Parametric excitation Cosserat theory Nonlinear inertia Nonlinear elastic rod Experimental frequency-response curve Method of multiple scales 


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  1. 1.
    Antman, S.S.: Nonlinear Problems of Elasticity, 2nd edn. New York, Springer (2005) Google Scholar
  2. 2.
    Villaggio, P.: Mathematical Models for Elastic Structures. Cambridge University Press, Cambridge (1997) Google Scholar
  3. 3.
    Cosserat, E., Cosserat, F.: Theorie des corps deformables. Herman, Paris (1909) Google Scholar
  4. 4.
    Starzewski, M.O., Jasiuk, I.: Stress invariance in planar Cosserat elasticity. Proc. R. Soc. Lond. A 451, 453–470 (1995) MATHCrossRefGoogle Scholar
  5. 5.
    Eisley, J.G.: Nonlinear vibration of beams and rectangular plates. Z. Angew. Math. Mech. 15 (1964) Google Scholar
  6. 6.
    Lacarbonara, W., Paolone, A., Yabuno, H.: Modeling of planar nonshallow prestressed beams towards asymptotic solutions. Mech. Res. Commun. 31, 301–310 (2004) MATHCrossRefGoogle Scholar
  7. 7.
    Lacarbonara, W., Yabuno, H.: Refined models of elastic beams undergoing large in-plane motions: Theory and experiment. Int. J. Solids Struct. 43, 5066–5084 (2006) MATHCrossRefGoogle Scholar
  8. 8.
    Dugundji, J., Mukhopadhyay, V.: Lateral bending-torsion vibrations of a thin beam under parametric excitation. ASME J. Appl. Mech. Trans. 9, 693–698 (1973) Google Scholar
  9. 9.
    Saito, H., Sato, K., Otomi, K.: Nonlinear forced vibrations of a beam carrying concentrated mass under gravity. J. Sound Vib. 46(4), 515–525 (1976) CrossRefMATHGoogle Scholar
  10. 10.
    Yabuno, H., Nayfeh, A.H.: Nonlinear normal modes of a parametrically excited cantilever beam. Nonlinear Dyn. 25, 65–77 (2001) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Evensen, J.A., Evan-Iwanowski, R.M.: Effects of longitudinal inertia upon the parametric response of elastic columns. J. Appl. Mech. 33, 141–148 (1996) Google Scholar
  12. 12.
    Nayfeh, A.H., Pai, P.F.: Non-linear non-planar parametric responses of an inextensional beam. Int. J. Non-Linear Mech. 24(2), 139–158 (1989) MATHCrossRefGoogle Scholar
  13. 13.
    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) Google Scholar
  14. 14.
    Anderson, T.J., Nayfeh, A.H., Balachandran, B.: Experimental verification of the importance of the nonlinear curvature in the response of a cantilever beam. J. Vib. Acoust. 118, 21–27 (1996) CrossRefGoogle Scholar
  15. 15.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979) MATHGoogle Scholar
  16. 16.
    Yabuno, H., Koda, T.: Identification of Coulomb friction at supporting points of a hinged-hinged beam. J. Syst. Des. Dyn. 1, 352–361 (2007) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • In-Soo Son
    • 1
  • Yuusuke Uchiyama
    • 2
  • Walter Lacarbonara
    • 3
  • Hiroshi Yabuno
    • 2
  1. 1.Department of Mechanical EngineeringDong-eui UniversityBusanSouth Korea
  2. 2.Graduate School of Systems and Information EngineeringUniversity of TsukubaTsukubaJapan
  3. 3.Dipartimento di Ingegneria Strutturale e GeotecnicaUniversity of Rome La SapienzaRomeItaly

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