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Vanishing singularity in hard impacting systems
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 |
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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