Abstract
Parametric resonance may arise in a mechanical system, the Excited System, in which one of the forces is varying periodically. The classical example is a pendulum with a suspension point which moves harmonically in the vertical direction. We shall discuss a fairly general one degree of freedom, parametrically excited system in section 2. This system is a dissipative version of the study by Broer and Vegter (1992). In section 3 we consider an autoparametric two degrees of freedom system which is in some sense a generalisation of section 2. Such a system admits a richer bifurcation structure and chaotic dynamics.
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References
Arnold V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, 1984.
Broecker Th., Lander L., Differentiable Germs and Catastrophes (LMS Lecture Notes vol 11), Cambridge University Press, 1975.
Broer H.W., Vegter G., Bifurcational aspects of parametric resonance, Dynamics reported: new series vol. 1: Expositions in dynamical systems, eds. C.K.R.T. Jones, U. Kirchgraber and H.O. Walther, Springer, 1992.
Cartmell, M., Introduction to Linear, Parametric and Nonlinear Vibrations, Chapman and Hall, London, 1990.
Golubitsky M., Schaeffer D.G., Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York, 1983.
Guckenheimer J., Holmes P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vectorfields, Springer-Verlag, New York. 1983.
Guckenheimer J., On a codimension two bifurcation. Dynamical Systems and Tubulence, Warwick 1980, Springer LNM 898.
Hoveijn I., Aspects of Resonance in Dynamical Systems. Thesis, Utrecht University, 1992.
Iooss G., Global characterization of the normal form for a vectorfield near a closed orbit., Journal of Differential Equations 76, 47–76. (1988).
Khibnik A.I., Kuznetsov Yu.A., Levitin V.V., Nikolaev E.V., LOCBIF, CAN Expertise Centre, Amsterdam, 1992.
Nabergoj, R., Tondl, A., Simulation of parametric ship rolling: effects of hull bending and torsional elasticity, Nonlinear Dynamics, 5, 1993.
Nayfeh A.H., Mook D.T., Nonlinear Oscillations, Wiley Interscience, New York. (1979)
Schmidt G., Tondl A., Non-linear Vibrations, Cambridge University Press, Cambridge, 1986.
Poston T., Stewart I., Catastrophe Theory, Pitman, London, 1978.
Ruijgrok M., Studies in Parametric and Autoparametric Resonance, Thesis, Utrecht University, 1995.
Silnikov L.P., A case of the existence of a denumerable set of periodic motions, Sov. Math. Dokl., 6, 63–71, 1965.
Svoboda, R., Tondl, A., Verhulst, F., Autoparametric resonance by coupling of linear and nonlinear systems, International Journal of Non-Linear Mechanics, 123, 1992.
Takens F., Singularities of vectorfields, Publ. Math. I.H.E.S., 43, 1974.
Tondl, A., On the stability of a rotor system. Acta Technica CSAV, 36, 331–338, 1991.
Tondl, A., A contribution to the analysis of autoparametric systems, Acta Technica CSAV, 37, 735–758, 1992.
Tondl A., Elastically mounted body in cross flow with an attached pendulum, Proceedings 14th Biennial ASME Conference Mechanical Vibration and Noise, Albuquerque, 1993.
Tondl, A., Nabergoj, R., Model simulation of parametrically excited ship rolling, Nonlinear Dynamics, 1, 131–141, 1990.
Tondl, A., Nabergoj, R., Simulation of parametric ship hull and twist oscillations, Nonlinear Dynamics, 3, 41–56, 1992.
Tresser C., About some theorems of L.P. Silnikov, Ann. Inst. H. Poincare, 40, 440–461, 1984.
Verhulst F., Nonlinear Differential Equations and Dynamical Systems, revised edition, Springer-Verlag, New York, (1996).
Verhulst F. and Tondl A., Autoparametric resonance by self-excitation, to be published, 1995.
Wiggins S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.
Yakubovich V.A.. Starzhinskii V.M., Linear differential equations with periodic coefficients, Vols. I and II, Wiley, New York, 1975.
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Ruijgrok, M., Verhulst, F. (1996). Parametric and autoparametric resonance. In: Broer, H.W., van Gils, S.A., Hoveijn, I., Takens, F. (eds) Nonlinear Dynamical Systems and Chaos. Progress in Nonlinear Differential Equations and Their Applications, vol 19. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7518-9_13
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DOI: https://doi.org/10.1007/978-3-0348-7518-9_13
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