Abstract
In impacting mechanical systems, a large-amplitude chaotic oscillation develops close to the grazing condition, and it is explained in terms of “square root singularity” in the discrete-time model. It has been shown that this singularity vanishes when the forcing frequency is an integer multiple of twice the natural frequency of the system. It has also been shown that a dangerous border collision bifurcation occurs at the grazing condition causing divergence of the orbit to the unstable manifold of a preexisting saddle fixed point. In this paper, we show that the onset of large-amplitude chaotic oscillation can be avoided over finite ranges of the forcing frequency, because the unstable periodic orbits that cause the dangerous border collision bifurcation move away and do not coexist with the main orbit over a wide range of the forcing frequency.
Similar content being viewed by others
References
Banerjee, S., Ing, J., Pavlovskaia, E., Wiercigroch, M., Reddy, R.K.: Invisible grazings and dangerous bifurcations in impacting systems: the problem of narrow-band chaos. Phys. Rev. E 79, 037201 (2009)
Blazejczyk-Okolewska, B., Kapitaniak, T.: Co-existing attractors of impact oscillator. Chaos Solitons Fract. 9(8), 1439–1443 (1998)
Budd, C.: Grazing in impact oscillators. In: Branner, B., Hjorth, P. (eds.) Real and Complex Dynamical Systems, pp. 47–64. Kluwer, Dordrecht (1995)
Budd, C., Dux, F.: Chattering and related behaviour in impacting oscillators. Philos. Trans. R. Soc. 347, 365–389 (1994)
Chin, W., Ott, E., Nusse, H.E., Grebogi, C.: Grazing bifurcations in impact oscillators. Phys. Rev. E 50(6), 4427–4444 (1994)
de Weger, J., van de Water, W., Molenaar, J.: Grazing impact oscillations. Phys. Rev. E 62(2), 2030–2041 (2000)
Ganguli, A., Banerjee, S.: Dangerous bifurcation at border collision: When does it occur? Phys. Rev. E 71(5), 057,202 (2005)
Hassouneh, M.A., Abed, E.H., Nusse, H.E.: Robust dangerous border-collision bifurcations in piecewise smooth systems. Phys. Rev. Lett. 92(7), 070,201 (2004)
Ing, J., Pavlovskaia, E., Wiercigroch, M.: Dynamics of a nearly symmetrical piecewise oscillator close to grazing incidence: modelling and experimental verification. Nonlinear Dyn. 46, 225–238 (2006)
Ing, J., Pavlovskaia, E.E., Wiercigroch, M., Banerjee, S.: Experimental study of impact oscillator with one sided elastic constraint. Philos. Trans. R. Soc. 366, 679–704 (2008)
Kundu, S., Banerjee, S., Giaouris, D.: Vanishing singularity in hard impacting systems. Discrete Contin. Dyn. Syst. Ser. B 16(1), 319–332 (2011)
Kundu, S., Banerjee, S., Ing, J., Pavlovskaia, E., Wiercigroch, M.: Singularities in soft-impacting systems. Phys. D 241(5), 553–565 (2012)
Leine, R.I., Nijmeijer, H.: Dynamics and Bifurcations in Non-Smooth Mechanical Systems. Springer, Berlin (2004)
Ma, Y., Agarwal, M., Banerjee, S.: Border collision bifurcations in a soft impact system. Phys. Lett. A 354(4), 281–287 (2006)
Ma, Y., Ing, J., Banerjee, S., Wiercigroch, M., Pavlovskaia, E.: The nature of the normal form map for soft impacting systems. Int. J. Nonlinear Mech. 43(4), 504–513 (2008)
Mandal, K., Chakraborty, C., Abusorrah, A., Al-Hindawi, M., Al-Turki, Y., Banerjee, S.: An automated algorithm for stability analysis of hybrid dynamical systems. Eur. Phys. J. Spec. Top. 222(3–4), 757–768 (2013)
Molenaar, J., de Weger, J.G., van de Water, W.: Mappings of grazing-impact oscillators. Nonlinearity 14(2), 301 (2001)
Nordmark, A.B.: Non-periodic motion caused by grazing incidence in an impact oscillator. J. Sound Vib. 145(2), 279–297 (1991)
Nordmark, A.B.: Universal limit mapping in grazing bifurcations. Phys. Rev. E 55(1), 266–270 (1997)
Pavlovskaia, E., Wiercigroch, M.: Analytical drift reconstruction for visco-elastic impact oscillators operating in periodic and chaotic regimes. Chaos Solitons Fract. 19, 151–161 (2004)
Pavlovskaia, E., Wiercigroch, M., Grebogi, C.: Two-dimensional map for impact oscillator with drift. Phys. Rev. E 70, 036201 (2004)
Shaw, S.W., Holmes, P.J.: A periodically forced piecewise linear oscillator. J. Sound Vib. 90(1), 129–155 (1983)
Thota, P., Dankowicz, H.: Continuous and discontinuous grazing bifurcations in impacting oscillators. Phys. D 214, 187–197 (2006)
Acknowledgments
SB acknowledges financial support from the Science and Engineering Research Board, Department of Science and Technology, Government of India, in the form of J. C. Bose Fellowship, Project No. SB/S2/JCB-023/2015.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Suda, N., Banerjee, S. Why does narrow band chaos in impact oscillators disappear over a range of frequencies?. Nonlinear Dyn 86, 2017–2022 (2016). https://doi.org/10.1007/s11071-016-3011-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-016-3011-y