Summary
Nonlinear resonant interactions between planar waves in a thin circular ring are investigated. It is found that a high-frequency azimuthal wave is unstable against a pair of secondary low-frequency waves. The secondary waves are of two types; either two bending or azimuthal and bending. These are in phase with the primary wave. All three together compose a resonant triad. Such kind of instability causes the stress amplification in the ring. The stress growth constant and the period of energy exchange between the waves are estimated based on analytical solutions to the evolution equations driving the triad. The lowest-order nonlinear approximation analysis predicts stability for bending waves. A good qualitative agreement of the obtained results with some known experimental data is observed.
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References
Grigolyuk, Eu. I.: Nonlinear oscillations and stability of shallow bars and shells (in Russian) Izv. AN SSSR. Mech. Mach.3, 33–68 (1955).
Reissner, E.: On transverse vibration of thin shallow shells. Q. J. Appl. Math.13, 169–180 (1955).
Bolotin, V. V.: Dynamic stability of elastic system (in Russian). Moscow: Gostechizdat 1956.
Kauderer, H.: Nichtlineare Mechanik. Berlin: Springer 1958.
Evensen, D. A., Fulton, R. E.: Some studies on the nonlinear dynamic response of shell-type structures. In: Dynamic stability of structures (Herrmann, G., ed.), pp. 237–254. New York: Pergamon 1966.
Vol'mir, A. S.: Nonlinear dynamics of plates and shells (in Russian). Moscow: Nauka 1972.
Kubenko, V. D., Kovaltchuk, P. S., Krasnopolskaja, T. S.: Nonlinear interaction of flexible modes of oscillations in cylindrical shells (in Russian). Kiev: Naukova dumka 1984.
Lindberg, H. E.: Stress amplification in a ring caused by dynamic instability. Trans. ASME J. Appl. Mech.41, 390–400 (1974).
Goodier, J. N., McIvor, I. K.: The elastic cylindrical shell under nearly uniform radial impulse. Trans. ASME J. Appl. Mech.31, 259–266 (1964).
Kovriguine, D. A., Potapov, A. I.: Nonlinear vibrations of a thin ring. Sov. Appl. Mech.25, 281–286 (1989).
Kovrigin, D. A., Potapov, A. I.: Non-linear resonance interaction of the longitudinal and bending waves in a ring (in Russian). Sov. Phys. Dokl.305, 803–807 (1989).
Lourie, A. I.: Statics of thin-walled elastic shells (in Russian). Moscow-Leningrad: Gostechizdat 1947.
Ginsberg, J. H.: Dynamic stability of transverse axisymmetric waves in circular cylindrical shells. Trans. ASME J. Appl. Mech.41, 77–82 (1974).
Kaup, P. J., Reiman, A., Bers, A.: Space-time evolution of nonlinear three-wave interactions. Interactions in a homogeneous medium. Rev. Modern Phys.51, 275–309 (1979).
Landau, L. D., Lifshits, E. M.: Mechanics, 3rd ed. Oxford: Pergamon Press 1976.
Bretherton, F. P.: Resonant interactions between waves. J. Fluid Mech.20, 457–472 (1964).
Bloembergen, K.: Nonlinear optics. New York, Amsterdam: W. A. Benjamin 1965.
Zakharov, V. E.: Hamiltonian formalism for waves in nonlinear mediums with dispersion (in Russian). Izv. vuzov SSSR. Radiofizika17, 431–453 (1974).
Jahnke-Emde-Lösch: Tafeln Höherer Funktionen, 6. Aufl. Stuttgart: Teubner 1960.
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Kovriguine, D.A., Potapov, A.I. Nonlinear oscillations in a thin ring — I. Three-wave resonant interactions. Acta Mechanica 126, 189–200 (1998). https://doi.org/10.1007/BF01172807
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DOI: https://doi.org/10.1007/BF01172807