Vanishing singularity in hard impacting systems

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July 2011, 16(1): 319-332. doi: 10.3934/dcdsb.2011.16.319

Vanishing singularity in hard impacting systems

Soumya Kundu ^1, , Soumitro Banerjee ^2, and Damian Giaouris ^3,

1.	Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, United States
2.	Department of Physical Sciences, Indian Institute of Science Education & Research, Mohanpur-741252, Nadia, West Bengal, India
3.	School of Electrical, Electronic and Computer Engineering, Newcastle University, NE1 7RU, England, United Kingdom

Received February 2010 Revised June 2010 Published April 2011

Abstract
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It is known that the Jacobian of the discrete-time map of an impact oscillator in the neighborhood of a grazing orbit depends on the square-root of the distance the mass would have gone beyond the position of the wall if the wall were not there. This results in an infinite stretching of the phase space, known as the square-root singularity. In this paper we look closer into the Jacobian matrix and find out the behavior of its two parameters---the trace and the determinant, across the grazing event. We show that the determinant of the matrix remains invariant in the neighborhood of a grazing orbit, and that the singularity appears only in the trace of the matrix. Investigating the character of the trace, we show that the singularity disappears if the damped frequency of the oscillator is an integral multiple of half of the forcing frequency.

Keywords: nonsmooth system, Impact oscillator, grazing, border collision bifurcation.

Mathematics Subject Classification: Primary: 37G15, 34K18; Secondary: 34C2.

Citation: Soumya Kundu, Soumitro Banerjee, Damian Giaouris. Vanishing singularity in hard impacting systems. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 319-332. doi: 10.3934/dcdsb.2011.16.319

References:

[1]	S. W. Shaw and P. J. Holmes, A periodically forced piecewise linear oscillator, Journal of Sound & Vibration, 90 (1983), 129-155. doi: 10.1016/0022-460X(83)90407-8.
[2]	F. Peterka and J. Vacik, Transition to chaotic motion in mechanical systems with impacts, Journal of Sound and Vibration, 154 (1992), 95-115. doi: 10.1016/0022-460X(92)90406-N.
[3]	B. Blazejczyk-Okolewska and T. Kapitaniak, Co-existing attractors of impact oscillator, Chaos, Solitons & Fractals, 9 (1998), 1439-1443. doi: 10.1016/S0960-0779(98)00164-7.
[4]	S. Lenci and G. Rega, A procedure for reducing the chaotic response region in an impact mechanical system, Nonlinear Dynamics, 15 (1998), 391-409. doi: 10.1023/A:1008209513877.
[5]	D. J. Wagg, G. Karpodinis and S. R. Bishop, An experimental study of the impulse response of a vibro-impacting cantilever beam, Journal of Sound & Vibration, 228 (1999), 243-264. doi: 10.1006/jsvi.1999.2318.
[6]	E. K. Ervin and J. A. Wickert, Experiments on a beam-rigid body structure repetitively impacting a rod, Nonlinear Dynamics, 50 (2007), 701-716. doi: 10.1007/s11071-006-9180-3.
[7]	J. Ing, E. Pavlovskaia and M. Wiercigroch, An experimental study into the bilinear oscillator close to grazing, In "International Symposium on Nonlinear Dynamics, Journal of Physics: Conference Series," 96 (2007), 012119.
[8]	A. B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145 (1991), 279-297. doi: 10.1016/0022-460X(91)90592-8.
[9]	A. B. Nordmark, Universal limit mapping in grazing bifurcations, Phys. Rev. E, 55 (1997), 266-270. doi: 10.1103/PhysRevE.55.266.
[10]	M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "Piecewise-smooth Dynamical Systems: Theory and Applications," Springer Verlag (Applied Mathematical Sciences), London, 2008.
[11]	Y. Ma, M. Agarwal and S. Banerjee, Border collision bifurcations in a soft impact system, Physics Letters A, 354 (2006), 281-287. doi: 10.1016/j.physleta.2006.01.025.
[12]	Y. Ma, J. Ing, S. Banerjee, M. Wiercigroch and E. Pavlovskaia, The nature of the normal form map for soft impacting systems, International Journal of Nonlinear Mechanics, 43 (2008), 504-513. doi: 10.1016/j.ijnonlinmec.2008.04.001.
[13]	J. Ing, E. Pavlovskaia, M. Wiercigroch and S. Banerjee, Experimental study of impact oscillator with one-sided elastic constraint, Philosophical Transactions of the Royal Society of London, Part A, 366 (2008), 679-704.
[14]	R. I. Leine and H. Nijmeijer, "Dynamics and Bifurcations in Non-Smooth Mechanical Systems," Springer Verlag, Berlin, 2004.
[15]	S. Banerjee and C. Grebogi, Border collision bifurcations in two-dimensional piecewise smooth maps, Physical Review E, 59 (1999), 4052-4061. doi: 10.1103/PhysRevE.59.4052.
[16]	S. Banerjee, P. Ranjan and C. Grebogi, Bifurcations in two-dimensional piecewise smooth maps -- theory and applications in switching circuits, IEEE Transactions on Circuits and Systems-I, 47 (2000), 633-643. doi: 10.1109/81.847870.
[17]	H. Dankowicz and F. Svahn, On the stabilizability of near-grazing dynamics in impact oscillators, Int. J. Robust & Nonlinear Control, 17 (2007), 1405-1429. doi: 10.1002/rnc.1252.

show all references

References:

[1]	S. W. Shaw and P. J. Holmes, A periodically forced piecewise linear oscillator, Journal of Sound & Vibration, 90 (1983), 129-155. doi: 10.1016/0022-460X(83)90407-8.
[2]	F. Peterka and J. Vacik, Transition to chaotic motion in mechanical systems with impacts, Journal of Sound and Vibration, 154 (1992), 95-115. doi: 10.1016/0022-460X(92)90406-N.
[3]	B. Blazejczyk-Okolewska and T. Kapitaniak, Co-existing attractors of impact oscillator, Chaos, Solitons & Fractals, 9 (1998), 1439-1443. doi: 10.1016/S0960-0779(98)00164-7.
[4]	S. Lenci and G. Rega, A procedure for reducing the chaotic response region in an impact mechanical system, Nonlinear Dynamics, 15 (1998), 391-409. doi: 10.1023/A:1008209513877.
[5]	D. J. Wagg, G. Karpodinis and S. R. Bishop, An experimental study of the impulse response of a vibro-impacting cantilever beam, Journal of Sound & Vibration, 228 (1999), 243-264. doi: 10.1006/jsvi.1999.2318.
[6]	E. K. Ervin and J. A. Wickert, Experiments on a beam-rigid body structure repetitively impacting a rod, Nonlinear Dynamics, 50 (2007), 701-716. doi: 10.1007/s11071-006-9180-3.
[7]	J. Ing, E. Pavlovskaia and M. Wiercigroch, An experimental study into the bilinear oscillator close to grazing, In "International Symposium on Nonlinear Dynamics, Journal of Physics: Conference Series," 96 (2007), 012119.
[8]	A. B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145 (1991), 279-297. doi: 10.1016/0022-460X(91)90592-8.
[9]	A. B. Nordmark, Universal limit mapping in grazing bifurcations, Phys. Rev. E, 55 (1997), 266-270. doi: 10.1103/PhysRevE.55.266.
[10]	M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "Piecewise-smooth Dynamical Systems: Theory and Applications," Springer Verlag (Applied Mathematical Sciences), London, 2008.
[11]	Y. Ma, M. Agarwal and S. Banerjee, Border collision bifurcations in a soft impact system, Physics Letters A, 354 (2006), 281-287. doi: 10.1016/j.physleta.2006.01.025.
[12]	Y. Ma, J. Ing, S. Banerjee, M. Wiercigroch and E. Pavlovskaia, The nature of the normal form map for soft impacting systems, International Journal of Nonlinear Mechanics, 43 (2008), 504-513. doi: 10.1016/j.ijnonlinmec.2008.04.001.
[13]	J. Ing, E. Pavlovskaia, M. Wiercigroch and S. Banerjee, Experimental study of impact oscillator with one-sided elastic constraint, Philosophical Transactions of the Royal Society of London, Part A, 366 (2008), 679-704.
[14]	R. I. Leine and H. Nijmeijer, "Dynamics and Bifurcations in Non-Smooth Mechanical Systems," Springer Verlag, Berlin, 2004.
[15]	S. Banerjee and C. Grebogi, Border collision bifurcations in two-dimensional piecewise smooth maps, Physical Review E, 59 (1999), 4052-4061. doi: 10.1103/PhysRevE.59.4052.
[16]	S. Banerjee, P. Ranjan and C. Grebogi, Bifurcations in two-dimensional piecewise smooth maps -- theory and applications in switching circuits, IEEE Transactions on Circuits and Systems-I, 47 (2000), 633-643. doi: 10.1109/81.847870.
[17]	H. Dankowicz and F. Svahn, On the stabilizability of near-grazing dynamics in impact oscillators, Int. J. Robust & Nonlinear Control, 17 (2007), 1405-1429. doi: 10.1002/rnc.1252.

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