Abstract
A nonlinear time-varying one-degree-of-freedom system, which is used for the modelling of the buckling of a loaded beam in Euler’s problem, is considered. For a slowly changing load, the deterministic approach in this problem fails if the trajectories pass through the separatrix. An expression for the probability of possible outcomes of the evolution of the oscillations is obtained. The analytical and numerical results are compared.
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References
Euler, L., Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Additamentum 1: De curvis elasticis, Lausanne and Geneva: 1744; from Opera omnia, Ser. 1, vol. 24, Zürich: Orell Füssli, 1952.
Arnold, V. I., Kozlov, V. V., and Neishtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, Berlin: Springer-Verlag, 1988.
Avramov, K. V., Bifurcations of parametric oscillations of beams with three equilibria, Acta Mech., 2003, vol. 164, pp. 115–138.
Bolotin, V. V., The dynamic stability of elastic systems, San-Francisco: Holden-Day, 1964.
Gradshteyn, I. S. and Ryzhik, I. M., Tables of Integrals, Series, and Products, Orlando, Florida: Academic Press, 2000.
Mettler, E., Nichtlineare Schwingungen und kinetische Instabilität bei Saiten und Stäben, Archive of Applied Mechanics, 1955, vol. 23, pp. 354–364.
Mettler, E. and Weidenhammer, F., Der axial pulsierend belastete Stab mit Endmasse, ZAMM, 1956, vol. 36, no. 7/8, pp. 284–287.
Neishtadt, A. I., Probability phenomena due to separatrix crossing, CHAOS, 1991, vol. 1, pp. 42–48.
Neishtadt, A. I., Passage through a separatrix in a resonance problem, Prikl. Mat. Mech., 1975, vol. 39, no. 4, pp. 621–632, [translation in J. Appl. Math. Mech.].
Neishtadt, A. I., Artemyev, A. V., and Zelenyi, L. M., Regular and chaotic charged particle dynamics in low frequency waves and role of separatrix crossings, Regul. Chaotic Dyn., 2010, vol. 15, no. 4–5, pp. 564–574.
Neishtadt, A. I. and Pivovarov, M. L., Separatrix crossing in dynamics of dual-spin spacecraft, Prikl. Mat. Mech., 2000, vol. 64, no. 5, pp. 741–746 [translation in J. Appl. Math. Mech.].
Rao, S., Vibration of continuous systems, Mexico: John Wiley & Sons, 2007.
Stoker, J. J., Nonlinear Vibrations in Mechanical and Electrical Systems, New York: Wiley, 1992.
Timoshenko, S. and Gere, M., Theory of elastic stability, New York: Dover Publications, 2009.
Weidenhammer, F., Das Stabilitätsverhalten der nichtlinearen Biegeschwingungen des axial pulsierend belasteten Stabes, Archive of Applied Mechanics, 1956, vol. 24, pp. 53–68.
Ziegler, H., Principles of Structural Stability, Basel: Birhaeuser-Verlag, 1977.
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Pivovarov, M. On the probability of the outcomes in buckling of an elastic beam. Regul. Chaot. Dyn. 17, 506–511 (2012). https://doi.org/10.1134/S1560354712060032
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DOI: https://doi.org/10.1134/S1560354712060032