Abstract.
A three-degree-of-freedom vibro-bounce system is considered. The disturbed map of period one single-impact motion is derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. Dynamical behavior of the system, near the point of codimension two bifurcation, is investigated by using qualitative analysis and numerical simulation. It is found that near the point of Hopf-flip bifurcation there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. The results from simulation show that there exists an interesting torus doubling bifurcation near the codimension two bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transform to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems. Different routes from period one single-impact motion to chaos are observed by numerical simulation.
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References
Whiston, G.S.: Global dynamics of vibro-impacting linear oscillator. Journal of Sound and Vibration, 118(3), 395–429 (1987)
Aidanpää, J.O., Gupta, B.R.: Periodic and chaotic behaviour of a threshold-limited two-degree-of- freedom system. Journal of Sound and Vibration, 165(2): 305–327 (1993)
Jin, D.P., Hu H.Y.: Periodic vibro-impacts and their stability of a dual component system. Acta Mechanica Sinica, 13(4), 366–376 (1997)
Nordmark, A.B.: Non-periodic motion caused by grazing incidence in an impact oscillator. Journal of Sound and Vibration, 145(2): 279–297 (1991)
Hu, H.Y.: Detection of grazing orbits and incident bifurcations of a forced continuous, piecewise- linear oscillator. Journal of Sound and Vibration, 187(3), 485–493 (1994)
Peterka, F.: Bifurcation and transition phenomena in an impact oscillator. Chaos, Solitons & Fractals, 7(10), 1635–1647 (1996)
Hu, H.Y.: Controlling chaos of a periodically forced nonsmooth mechanical system. Acta Mechanica Sinica, 11(3), 251–258 (1995)
Hu, H.Y.: Active control of chaos of mechanical systems. Advances in Mechanics, 26(4), 453–463 (1996) (in Chinese)
Chatterjee, S., Mallik, A.K.: Bifurcations and chaos in autonomous self-excited oscillators with impact damping. Journal of Sound and Vibration, 191(4), 539–562 (1996)
Luo, G.W., Xie, J.H.: Hopf bifurcation of a two- degree-of- freedom vibro-impact system. Journal of Sound and Vibration, 213(3), 391–408 (1998)
Luo, G.W., Xie, J.X.: Bifurcations and chaos in a system with impacts. Physica D, 148: 183–200 (2001)
Budd, C., Dux, F., Cliffe, A.: The effect of frequency and clearance variations on single- degree-of-freedom impact oscillators. Journal of Sound and Vibration, 184(3), 475–502 (1995)
Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. New York: Springer-Verlage, 1998
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The project supported by the National Natural Science Foundation of China (10172042, 50475109) and the Natural Science Foundation of Gansu Province Government of China (ZS-031-A25-007-Z (key item))
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Luo, G., Chu, Y., Zhang, Y. et al. Codimension two bifurcation of a vibro-bounce system. ACTA MECH SINICA 21, 197–206 (2005). https://doi.org/10.1007/s10409-005-0017-y
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DOI: https://doi.org/10.1007/s10409-005-0017-y