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. Abramian
  • W. T. van Horssen
  • S. A. Vakulenko
Original Paper

Abstract

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.

Keywords

Time-varying mass Beam  Small rigidity Internal resonances 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • A. K. Abramian
    • 1
  • 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

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