Nonlinear Dynamics

, Volume 78, Issue 1, pp 449–459 | Cite as

On oscillations of a beam with a small rigidity and a time-varying mass

  • A. K. AbramianEmail author
  • W. T. van Horssen
  • S. A. Vakulenko
Original Paper


In this paper, we show that oscillations of an Euler–Bernoulli beam with a small rigidity and with a time varying mass can lead to a resonance, which involves a large number of modes. This effect can induce a stability loss. The corresponding equations are complicated, in particular, in the nonlinear case with an external excitation. To analyze these equations, a new asymptotic method (which has a variational nature and is based on energy estimates) was suggested and applied. This method allows us to investigate the stability problem and to find how the system stability depends on the beam parameters. The number of modes involved in a resonance can be computed with the help of suggested explicit formulas. The effect of modal interactions for a problem with an external excitation term \(\rho _0 u_\mathrm{t} - \rho _1 u_\mathrm{t}^3\) in the equation describing the beam displacement, where \(\rho _0\) and \(\rho _1\) are some positive coefficients, was studied. This type of cubic nonlinearity can model a wind force acting on the beam.


Time-varying mass Beam  Small rigidity Internal resonances 



This work is supported by a grant of the Dutch Organization for Scientific Research NWO.


  1. 1.
    Irschik, H., Holl, H.J.: Mechanics of variable-mass systems-Part1: Balance of mass and linear momentum. Appl. Mech. Rev. 57(1–6), 145–160 (2004)CrossRefGoogle Scholar
  2. 2.
    van der Burgh, A.H.P., Abramian, A.K.: A new model for the study of rain-wind-induced vibrations of a simple oscillator. Int. J. Non-linear Mech. 41, 345–358 (2006)CrossRefzbMATHGoogle Scholar
  3. 3.
    Pischanskyy, O.V., van Horssen, W.T.: On the nonlinear dynamics of a single degree of freedom oscillator with a time-varying mass. J. Sound Vib. 331, 1887–1897 (2012)CrossRefGoogle Scholar
  4. 4.
    van Horssen, W.T., Abramian, A.K.: On the free vibrations of an oscillator with a periodically time-varying mass. J. Sound Vib. 298, 1166–1172 (2006)CrossRefzbMATHGoogle Scholar
  5. 5.
    Hikami, Y., Shiraishi, N.: Rain-wind induced vibrations of cables in cable stayed bridges. J. Wind Engn. Ind. Aerodyn. 29, 409–418 (1988)CrossRefGoogle Scholar
  6. 6.
    Yamaguchi, H.: Analytical study on growth mechanism of rain vibrations of cables. J. Wind Eng. Ind. Aerodyn. 33, 73–80 (1990)CrossRefGoogle Scholar
  7. 7.
    Gu, M., et al.: Experimental and theoretical simulations on wind-rain-induced vibration of 3-D rigid stay cables. J. Sound Vib. 320, 184–200 (2009)CrossRefGoogle Scholar
  8. 8.
    Peil, U., Dreyer, O.: Rain-wind induced vibrations of cables in laminar and turbulent flow. J. Wind Struct. 10(1), 83–97 (2007)CrossRefGoogle Scholar
  9. 9.
    Phelan, R.S., et al.: Full-scale measurements to investigate rain-wind induced cable-stay vibration and its mitigation. J. Bridge Eng. 11, 293–304 (2006)CrossRefGoogle Scholar
  10. 10.
    Alekseenko, S.V., et al.: Rivulet flow of liquid on the outer surface of inclined cylinder. J. Appl. Tech. Phys. 38(4), 649–653 (1997)CrossRefGoogle Scholar
  11. 11.
    Abramian, A., Vakulenko, S.: Oscillations of a beam with time varying mass. Nonlinear Dyn. 63(1–2), 135–147 (2010)Google Scholar
  12. 12.
    Abramian, A.K., van Horssen, W.T., Vakulenko, S.: Nonlinear vibrations of a beam with time-varying rigidity and mass. Nonlinear Dyn. 71(1–2), 291–312 (2013)CrossRefzbMATHGoogle Scholar
  13. 13.
    Andrianov, I.V., van Horssen, W.T.: On the transversal vibrations of a conveyor belt: applicability of simplified models. J. Sound Vib. 313, 822–829 (2008) Google Scholar
  14. 14.
    Nayfeh, Ali H.: Introduction to Perturbation Techniques. Wiley, New York (1980)Google Scholar
  15. 15.
    Verhulst, F.: Nonlinear Differential Equations and Dynamical Syatems, 2nd edn. Springer, Berlin, Heidelberg, NewYork (1996)CrossRefGoogle Scholar
  16. 16.
    Bogolyubov, N.N., Mitropol’skii, YuA: Asymptotic methods in the theory of nonlinear oscillations. Gordon and Breach, Delhi (1961)Google Scholar
  17. 17.
    Andrianov, I.V., Manevitch, L.I.: Asymptotology Ideas, Methods, and Applications. Kluwer, Dodrecht, Boston, London (2002)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • A. K. Abramian
    • 1
    Email author
  • W. T. van Horssen
    • 2
  • S. A. Vakulenko
    • 1
    • 3
  1. 1.Institute of Problems in Mechanical EngineeringRussian Academy of SciencesSt. PetersburgRussia
  2. 2.Department of Applied Mathematical Analysis, Faculty EEMCSDelft University of Technology DelftThe Netherlands
  3. 3.Saint Petersburg National Research University of Information Technologies, Mechanics and OpticsSt. PetersburgRussia

Personalised recommendations