Abstract
The behavior of a nonlinear, non-Hamiltonian system in the postcritical (flutter) domain is studied with special attention to the influence of initial conditions on the properties of attractors situated at a certain point of the control parameter space. As a prototype system, an elastic panel is considered that is subjected to a combination of supersonic gas flow and quasistatic loading in the middle surface. A two natural modes approximation, resulting in a four-dimensional phase space and several control parameters is considered in detail. For two fixed points in the control parameter space, several plane sections of the four-dimensional space of initial conditions are presented and the asymptotic behavior of the final stationary responses are identified. Amongst the latter there are stable periodic orbits, both symmetric and asymmetric with respect to the origin, as well as chaotic attractors. The mosaic structure of the attraction basins is observed. In particular, it is shown that even for neighboring initial conditions can result in distinctly different nonstationary responses asymptotically approach quite different types of attractors. A number of closely neighboring periodic attractors are observed, separated by Hopf bifurcations. Periodic attractors also are observed under special initial conditions in the domains where chaotic behavior is usually expected.
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References
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
Wiggins, S., Introduction to Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.
Moon, F. C., Chaotic Vibrations. An Introduction for Applied Scientists and Engineers, Wiley, New York, 1987.
Bolotin, V. V., ‘Nonlinear flutter of plates and shells’, Inzhenerny Sbornik 28, 1960, 55–75 [in Russian].
Bolotin, V. V., Nonconservative Problems of the Theory of Elastic Stability, Fizmatgiz, Moscow, 1961 [in Russian]. English translation: Pergamon, Oxford, 1963.
Volmir, A. S., Stability of Deformable Systems, Fizmatgiz, Moscow, 1967 [in Russian].
Dowell, E. H., ‘Aeroelastic stability of plates and shells: An innocent’s guide to the literature’, in Instability of Continuous Systems, H. Leipholz (ed.), Springer-Verlag, Berlin, 1971, pp. 65–77.
Dowell, E. H., ‘Flutter of a buckled plate as an example of chaotic motion of a deterministic autonomous system’, Journal of Sound and Vibration 85(3), 1982, 330–344.
Simitses, G. J., An Introduction to the Elastic Stability of Structures, Robert E. Kriger, Malabar, FL, 1986.
Leipholtz, H., Stability Theory, Academic Press, New York, 1970.
Sugiyama, Y., Katayama, K., and Kinoi, S., ‘Flutter of cantilivered column under rocket thrust’, Journal of Aerospace Engineering 8(1), 1995, 9–15.
Crandall, S. H., ‘The effect of damping on the stability of gyroscopic pendulums’, Zeitschrift für Angewandte Mathematik und Physik 46, 1995, 761–780.
Ibrahim, R. A., ‘Friction-induced vibration, chatter, squeal, and chaos, Part I: Mechanics of contact of friction’, ASME Applied Mechanics Review 47(7), 1994, 209–226.
Ibrahim, R. A., ‘Friction-induced vibration, chatter, squeal, and chaos, Part II: Dynamics and modelling’, ASME Applied Mechanics Review 47(7), 1994, 227–253.
Simpson, T. A. and Ibrahim, R. A., ‘Nonlinear friction-induced vibration in water-lubricated bearings’, Journal of Vibration and Control 2(1), 1996, 87–113.
Kounadis, A. N., ‘On the failure of static stability analyses of nonconservative systems in regions of divergence instability’, International Journal of Solids and Structures 31(15), 1994, 2099–2120.
Kounadis, A. N., ‘Nonlinear stability and dynamic buckling of autonomous dissipative systems’, Zeitschrift f ür Angewandte Mathematik und Mechanik 75(4), 1995, 283–293.
Kounadis, A. N. and Krätzig, W. B. (eds.), Nonlinear Stability of Structures. Theory and Computational Techniques, Springer-Verlag, Wien, 1995.
Chatterjee, S. and Mallik, A. K., ‘Bifurcations and chaos in autonomous self-excited oscillators with impact damping’, Journal of Sound and Vibration 191(4), 1996, 539–562.
Arnold, V. I., Dynamical Systems, Springer-Verlag, New York, 1988.
Arrowsmith, D. K. and Place, C. M., An Introduction to Dynamical Systems, Cambridge University Press, Cambridge, MA, 1990.
Virgin, L.N. and Dowell, E. H., ‘Nonlinear aeroelasticity and chaos’, in Computational Nonlinear Mechanics in Aerospace Engineering, S. N. Atluri (ed.), AIAA, Washington, 1992, pp. 531–546.
Bolotin, V. V., ‘Postcritical dynamics of nonconservative systems’, in Proceedings of the 3rd European Conference on Structural Dynamics: EURODYN’96 Conference, G. Augusti, C. Borri, and P. Spinelli (eds.), Balkema, Rotterdam, 1996, pp. 357–362.
Bolotin, V. V., Grishko A. A., and Petrovsky, A. V., ‘On the influence of damping on the postcritical behavior of essentially nonpotential systems’, Izvestiya RAN, Mechanika Tverdogo Tela (MTT) 2, 1994, 154–167 [in Russian].
Bolotin, V. V., Petrovsky, A. V., and Grishko A. A., ‘Secondary bifurcation and global instability of an aeroelastic nonlinear system in the divergence domain’, Journal of Sound and Vibration 191(3), 1996, 431–451.
Fomenko, A. T., Visual Geometry and Topology. Mathematical Images in the Real World, MSU Publishers, Moscow, 1992 [in Russian].
Roberts, J. B. and Spanos, P. D., Random Vibrations and Statistical Linearization, Wiley, New York, 1990.
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Bolotin, V.V., Grishko, A.A., Kounadis, A.N. et al. Influence of Initial Conditions on the Postcritical Behavior of a Nonlinear Aeroelastic System. Nonlinear Dynamics 15, 63–81 (1998). https://doi.org/10.1023/A:1008204409853
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DOI: https://doi.org/10.1023/A:1008204409853