Criteria for chaotic transient oscillations in a model of driven buckled beams

• Autorzy: Szemplińska-Stupnicka W. , Tyrkiel E.
• Języki publikacji: EN

Abstrakty

EN

The single-mode equation of motion of a class of buckled beams is considered, and the attention is focused on the phenomena of irregular, unpredictable transient oscillations which are obserwed in the region of the nonlinear resonsnce hysteresis. This type of transient motion may be dangerous in engineering dynamics, because it may last very long and is defined neither by the coefficient of damping nor by the magnitude of perturbation. While the steady-state chaotic motion has been studied extensively in the recent literature, little attention was paid to the chaotic transients. In the paper the criteria for transient chaos, i.e. the domain of the system control parameter values, where the chaotic transient motion can occur, are determined. The criteria are based on the theoretical concept of global bifurcations, and are estimated numerically.

Słowa kluczowe

EN

chaotic motion; oscillator; beam

PL

ruch chaotyczny; oscylator; wiązka

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Computer Assisted Mechanics and Engineering Sciences
• Rocznik: 1999
• Tom: Vol. 6, No. 1
• Strony: 63--82
• Opis fizyczny: Bibliogr. 27 poz., il., wykr.

Twórcy

autor

Szemplińska-Stupnicka W.

• Institute of Fundamental Technological Research PAS, ul. Świętokrzyska 21, 00-049 Warsaw, Poland

autor

Tyrkiel E.

• Institute of Fundamental Technological Research PAS, ul. Świętokrzyska 21, 00-049 Warsaw, Poland

Bibliografia

• [1] M.J. Feigenbaum. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19: 25-52, 1978.
• [2] C. Grebogi, E. Ott and J.A. Yorke. Crises, sudden changes in chaotic attractors and transient chaos. Physica D7:181-200, 1983.
• [3] J. Guckenheimer and P.J. Holmes. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983.
• [4] Ch. Hayashi. Nonlinear Oscillations in Physical Systems. Princeton University Press, Princeton, N.J., 1985.
• [5] P.J. Holmes. A nonlinear oscillator with a strange attractor. Phil. Trans. of the Royal Soc. Lond., A292(1394):419-448, 1979.
• [6] N.C. Huang and W. Nachbar. Dynamic snap-through of imperfect viscoelastic shallow arches. Trans. ASME J. Applied Mech., 35:286-297, 1968. |
• [7] E.N. Lorenz. Deterministic non-periodic flow, J. Atmos. Sci., 20:130-141, 1963.
• [8] S.W. McDonald, C. Grebogi, E. Ott and J.A. Yorke. Fractal basin boundaries. Physica D17: 125-153, 1985.
• [9] F.C. Moon. Experiments on chaotic motion of a forced nonlinear oscillator - strange attractors. ASME J. of Applied Mechanics, 47:638-644, 1980.
• [10] F.C. Moon. Experimental models for strange attractors: vibrations in elastic systems. In: New Approach to Nonlinear Problems in Dynamics, SIAM, 487-495, 1980.
• [11] F.C. Moon. Chaotic vibrations. John Willey & Sons, New York, 1987.
• [12] F.C. Moon and P.J. Holmes. A magnetoelastic strange attractor. J. Sound and Vibrations, 65(2):275-296, 1979.
• [13] H.E. Nusse and J.A. Yorke. Dynamics: Numerical Explorations. Springer-Verlag, New York, 1994.
• [14] E. Ott. Chaos in Dynamical Systems. Cambridge University Press, Cambridge, 1993.
• [15] M.S. Soliman. Jumps to resonance: Long chaotic transients, unpredictable outcome, and the probability of restabilization. ASME J. Applied Mechanics, 60:669-676, 1993.
• [16] H.B. Stewart and Y. Ueda. Catastrophes with indeterminate outcome. Proc. Roy. Soc. Lond., A432:113-123, 1991.
• [17] W. Szemplińska-Stupnicka and J. Rudowski. Steady-states in the twin-well potential oscillator: Computer simulations and approximate analytical studies. CHAOS, Int. J. Nonlinear Science, 3(3):375-385, 1993.
• [18] W. Szemplińska-Stupnicka and K.L. Janicki. Basin boundary bifurcations and boundary crisis in the twin-well Duffing oscillator: scenarios related to the saddle of the large resonant orbit. Int. J. Bifurcation and Chaos, 7(1):129-146, 1997.
• [19] W. Szemplińska-Stupnicka and E. Tyrkiel. Sequences of global bifurcations and the related outcomes after crisis of the resonant attractor in a nonlinear oscillator. Int. J. Bifurcation and Chaos, 7 (11), 1997.
• [20] W. Szemplińska-Stupnicka and E. Tyrkiel. Sequences of global bifurcations and multiple chaotic transients in a mechanical driven oscillator, to be published in the Proceedings of the IUTAM Symposium CHAOS’97, Kluwer Academic Publishers, Dordrecht/Boston/London, 1997.
• [21] J.M.T. Thompson and H.B. Stewart. Nonlinear Dynamics and Chaos. John Wiley & Sons, Chichester, 1986.
• [22] J.M.T. Thompson, H.B. Stewart and Y. Ueda. Safe, explosive and dangerous bifurcations in dissipative dynamical systems. Phys. Rev., E 49(2): 1019-1027, 1994.
• [23] W.Y. Tseng and J. Dugundji. Nonlinear vibrations of a buckled beam under harmonic excitation. ASME J. Applied Mechanics, 36: 467-476, 1971.
• [24] E. Tyrkiel. Numerical exploration of post-bifurcation unpredictable outcomes in a nonlinear driven oscillator. In: Proc. of the XIII Polish Conference on Computer Methods in Mechanics, 1315-1322, Poznań University of Technology, Poznań, Poland, 1997.
• [25] Y. Ueda, S. Yoshida, H.B. Stewart and J.M.T. Thompson. Basin explosions and escape phenomena in the twin- well Duffing oscillator: compound global bifurcations organizing behavior. Phil. Trans. Roy. Soc. London, A332: 169-186, 1990.
• [26] S. Wiggins. Global Bifurcations and Chaos: Analytical Methods. Springer-Verlag, New York, 1988.
• [27] S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, 1990.