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.
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References
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)
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)
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)
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)
Hikami, Y., Shiraishi, N.: Rain-wind induced vibrations of cables in cable stayed bridges. J. Wind Engn. Ind. Aerodyn. 29, 409–418 (1988)
Yamaguchi, H.: Analytical study on growth mechanism of rain vibrations of cables. J. Wind Eng. Ind. Aerodyn. 33, 73–80 (1990)
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)
Peil, U., Dreyer, O.: Rain-wind induced vibrations of cables in laminar and turbulent flow. J. Wind Struct. 10(1), 83–97 (2007)
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)
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)
Abramian, A., Vakulenko, S.: Oscillations of a beam with time varying mass. Nonlinear Dyn. 63(1–2), 135–147 (2010)
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)
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)
Nayfeh, Ali H.: Introduction to Perturbation Techniques. Wiley, New York (1980)
Verhulst, F.: Nonlinear Differential Equations and Dynamical Syatems, 2nd edn. Springer, Berlin, Heidelberg, NewYork (1996)
Bogolyubov, N.N., Mitropol’skii, YuA: Asymptotic methods in the theory of nonlinear oscillations. Gordon and Breach, Delhi (1961)
Andrianov, I.V., Manevitch, L.I.: Asymptotology Ideas, Methods, and Applications. Kluwer, Dodrecht, Boston, London (2002)
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This work is supported by a grant of the Dutch Organization for Scientific Research NWO.
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Abramian, A.K., van Horssen, W.T. & Vakulenko, S.A. On oscillations of a beam with a small rigidity and a time-varying mass. Nonlinear Dyn 78, 449–459 (2014). https://doi.org/10.1007/s11071-014-1451-9
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DOI: https://doi.org/10.1007/s11071-014-1451-9