Research Papers

Interaction Between Coexisting Orbits in Impact Oscillators

[+] Author and Article Information
Narasimha Suda

Department of Physical Sciences,
Indian Institute of Science Education and Research Kolkata,
Mohanpur 741246, India
e-mail: ssuda555@gmail.com

Soumitro Banerjee

Department of Physical Sciences,
Indian Institute of Science Education and
Research Kolkata,
Mohanpur 741246, India
e-mail: soumitro@iiserkol.ac.in

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 1, 2016; final manuscript received May 1, 2017; published online September 7, 2017. Assoc. Editor: Przemyslaw Perlikowski.

J. Comput. Nonlinear Dynam 12(6), 061015 (Sep 07, 2017) (4 pages) Paper No: CND-16-1596; doi: 10.1115/1.4036712 History: Received December 01, 2016; Revised May 01, 2017

Impact oscillators exhibit an abrupt onset of chaos close to grazing due to the square-root singularity in their discrete time maps. In practical applications, this large-amplitude chaotic vibration needs to be avoided. It has been shown that this can be achieved if the ratio of the natural frequency of the oscillator ω0 and the forcing frequency is an even integer. But, in practice, it is difficult to set a parameter at such a precise value. We show that in systems with square-root singularity (prestressed impacting surface), there exists a range of ω0 around the theoretical value over which the chaotic orbit does not occur, and that this is due to an interplay between the main attractor and coexisting orbits. We show that this range of forcing frequency has exponential dependence on the amount of prestress as well as on the stiffness ratio of the springs.

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Nordmark, A. B. , 1991, “ Non-Periodic Motion Caused by Grazing Incidence in an Impact Oscillator,” J. Sound Vib., 145(2), pp. 279–297. [CrossRef]
Nordmark, A. B. , 1997, “ Universal Limit Mapping in Grazing Bifurcations,” Phys. Rev. E, 55(1), pp. 266–270. [CrossRef]
Chin, W. , Ott, E. , Nusse, H. E. , and Grebogi, C. , 1994, “ Grazing Bifurcations in Impact Oscillators,” Phys. Rev. E, 50(6), pp. 4427–4444. [CrossRef]
Ma, Y. , Agarwal, M. , and Banerjee, S. , 2006, “ Border Collision Bifurcations in a Soft Impact System,” Phys. Lett. A, 354(4), pp. 281–287. [CrossRef]
Molenaar, J. , de Weger, J. G. , and van de Water, W. , 2001, “ Mappings of Grazing-Impact Oscillators,” Nonlinearity, 14(1), pp. 301–321. [CrossRef]
de Weger, J. , van de Water, W. , and Molenaar, J. , 2000, “ Grazing Impact Oscillations,” Phys. Rev. E, 62(2), pp. 2030–2041. [CrossRef]
Ma, Y. , Ing, J. , Banerjee, S. , Wiercigroch, M. , and Pavlovskaia, E. , 2008, “ The Nature of the Normal Form Map for Soft Impacting Systems,” Int. J. Nonlinear Mech., 43(4), pp. 504–513. [CrossRef]
Budd, C. , 2013, “ Grazing in Impact Oscillators,” Real and Complex Dynamical Systems, Vol. 464, B. Branner and P. Hjorth , eds., Springer Science & Business Media, Dordrecht, The Netherlands, pp. 47–64. [CrossRef]
Budd, C. , and Dux, F. , 1994, “ Chattering and Related Behaviour in Impacting Oscillators,” Philos. Trans. R. Soc., 347(1683), pp. 365–389. [CrossRef]
Kundu, S. , Banerjee, S. , and Giaouris, D. , 2011, “ Vanishing Singularity in Hard Impacting Systems,” Discrete Contin. Dyn. Syst., Ser. B, 16(1), pp. 319–332. [CrossRef]
Kundu, S. , Banerjee, S. , Ing, J. , Pavlovskaia, E. , and Wiercigroch, M. , 2012, “ Singularities in Soft-Impacting Systems,” Physica D, 241(5), pp. 553–565. [CrossRef]
Pavlovskaia, E. , Wiercigroch, M. , and Grebogi, C. , 2004, “ Two-Dimensional Map for Impact Oscillator With Drift,” Phys. Rev. E, 70(3), p. 036201. [CrossRef]
Pavlovskaia, E. , and Wiercigroch, M. , 2004, “ Analytical Drift Reconstruction for Visco-Elastic Impact Oscillators Operating in Periodic and Chaotic Regimes,” Chaos, Solitons Fractals, 19(1), pp. 151–161. [CrossRef]
Blazejczyk-Okolewska, B. , and Kapitaniak, T. , 1998, “ Co-Existing Attractors of Impact Oscillator,” Chaos, Solitons Fractals, 9(8), pp. 1439–1443. [CrossRef]
Ing, J. , Pavlovskaia, E. , and Wiercigroch, M. , 2006, “ Dynamics of a Nearly Symmetrical Piecewise Oscillator Close to Grazing Incidence: Modelling and Experimental Verification,” Nonlinear Dyn., 46(3), pp. 225–238. [CrossRef]
Ing, J. , Pavlovskaia, E. E. , Wiercigroch, M. , and Banerjee, S. , 2008, “ Experimental Study of Impact Oscillator With One Sided Elastic Constraint,” Philos. Trans. R. Soc., 366(1866), pp. 679–704. [CrossRef]
Hassouneh, M. A. , Abed, E. H. , and Nusse, H. E. , 2004, “ Robust Dangerous Border-Collision Bifurcations in Piecewise Smooth Systems,” Phys. Rev. Lett., 92(7), p. 070201. [CrossRef] [PubMed]
Ganguli, A. , and Banerjee, S. , 2005, “ Dangerous Bifurcation at Border Collision: When Does it Occur?,” Phys. Rev. E, 71(5), p. 057202. [CrossRef]
Banerjee, S. , Ing, J. , Pavlovskaia, E. , Wiercigroch, M. , and Reddy, R. K. , 2009, “ Invisible Grazings and Dangerous Bifurcations in Impacting Systems: The Problem of Narrow-Band Chaos,” Phys. Rev. E, 79(3), p. 037201. [CrossRef]
Suda, N. , and Banerjee, S. , 2016, “ Why Does Narrow Band Chaos in Impact Oscillators Disappear Over a Range of Frequencies?,” Nonlinear Dyn., 86(3), pp. 2017–2022. [CrossRef]
Mandal, K. , Chakraborty, C. , Abusorrah, A. , Al-Hindawi, M. , Al-Turki, Y. , and Banerjee, S. , 2013, “ An Automated Algorithm for Stability Analysis of Hybrid Dynamical Systems,” Eur. Phys. J. Spec. Top., 222(3–4), pp. 757–768. [CrossRef]


Grahic Jump Location
Fig. 1

Schematic diagram of the impact oscillator

Grahic Jump Location
Fig. 2

Bifurcation diagrams for m = 2.20, with amplitude F0 as the bifurcation parameter, and the displacement x as the variable observed at the positive-slope zero crossings of the sinusoidal input signal. (a) Brute-force bifurcation diagram and (b) showing the evolution of the unstable period-3 orbits. The other parameters are ζ = 0.01, β = 8, e = 1.26, and e1 = 0.05. Blue and black colors correspond to period-1 stable and unstable orbits, respectively, red color denotes unstable period-3, and green color denotes stable period-3 orbits (see color figure online).

Grahic Jump Location
Fig. 3

Bifurcation diagrams for m = 2.10. (a) Brute-force bifurcation diagram and (b) showing the evolution of the unstable period-3 orbits. The other parameters and the color coding are the same as in Fig. 2. In addition, the light blue color indicates the region where the table (green) orbit becomes unstable (see color figure online).

Grahic Jump Location
Fig. 4

Figure for critical values m for different values of β with ζ = 0.01, e = 1.26, and e1 = 0.1

Grahic Jump Location
Fig. 5

Figure for critical values m for different values of e1 with ζ = 0.01, e = 1.26, and β = 10



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