Controlling bifurcation and chaos of a plastic impact oscillator. (English) Zbl 1163.34343

Luo, G. W.; Lv, X. H.

Nonlinear Anal., Real World Appl. 10, No. 4, 2047-2061 (2009).

Summary: A two-degree-of-freedom plastic impact oscillator is considered. Based on the analysis of sticking and non-sticking impact motions of the system, we introduce a three-dimensional impact Poincaré map with dynamical variables defined at the impact instants. The plastic impacts complicate the dynamic responses of the impact oscillator considerably. Consequently, the piecewise property and singularity are found to exist in the three-dimensional map. The piecewise property is caused by the transitions of free flight and sticking motions of two masses immediately after impact, and the singularity of the map is generated via the grazing contact of two masses and the instability of their corresponding periodic motions. The nonlinear dynamics of the plastic impact oscillator is analyzed by using the Poincaré map. The simulated results show that the dynamic behavior of this system is very complex under parameter variation, varying from different types of sticking or non-sticking periodic motions to chaos. Suppressing bifurcation and chaotic-impact motions is studied by using an external driving force, delay feedback and damping control law. The effectiveness of these methods is demonstrated by the presentation of examples of suppressing bifurcations and chaos for the plastic impact oscillator.

MSC:

34C15

Nonlinear oscillations and coupled oscillators for ordinary differential equations

Keywords:

vibration; impact; periodic motion; bifurcation; controlling chaos

References:

[1]	Pavlovskaia, E. E.; Wiercigroch, M.; Grebogi, C., Two-dimensional map for impact oscillator with drift, Physical Review E, 70, 036201 (2004)
[2]	de Souza, S. L.T.; Caldas, I. L.; Viana, R. L., Basins of attraction changes by amplitude constraining of oscillators with limited power supply, Chaos, Solitons & Fractals, 26, 4, 1211-1220 (2005) · Zbl 1093.37516
[3]	Wagg, D. J., Multiple non-smooth events in multi-degree-of-freedom vibro-impact systems, Nonlinear Dynamics, 43, 1-2, 137-148 (2006) · Zbl 1138.70341
[4]	Nordmark, A. B., Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145, 2, 279-297 (1991)
[5]	Ivanov, A. P., Bifurcation in impact systems, Chaos, Solitons & Fractal, 7, 10, 1615-1634 (1996) · Zbl 1080.37570
[6]	Murphy, K. D.; Morrison, T. M., Grazing instabilities and post-bifurcation behavior in an impacting string, Journal of the Acoustical Society of America, 111, 2, 884-892 (2002)
[7]	Pavlovskaia, E. E.; Wiercigroch, M.; Grebogi, C., Modeling of an impact system with a drift, Physical Review E, 64, 056224 (2001)
[8]	Wiercigroch, M.; Neilson, R. D.; Player, M. A., Material removal rate prediction for ultrasonic drilling of hard materials using an impact oscillator approach, Physics Letters A, 259, 2, 91-96 (1999)
[9]	Wagg, D. J., Periodic sticking motion in a two-degree-of-freedom impact oscillator, International Journal of Non-Linear Mechanics, 40, 8, 1076-1087 (2005) · Zbl 1349.74283
[10]	Luo, A. C.J.; Gegg, B. C., Stick and non-stick periodic motions in periodically forced oscillators with dry friction, Journal of Sound and Vibration, 291, 1-2, 132-168 (2006) · Zbl 1243.70025
[11]	Meijaard, J. P.; de Pater, A. D., Railway vehicle systems dynamics and chaotic vibrations, International Journal of Non-Linear Mechanics, 24, 1, 1-17 (1989) · Zbl 0672.70030
[12]	Karpenko, E. V.; Wiercigroch, M.; Pavlovskaia, E. E.; Cartmell, M. P., Piecewise approximate analytical solutions for a Jeffcott rotor with a snubber ring, International Journal of Mechanical Sciences, 44, 3, 475-488 (2002) · Zbl 0993.70500
[13]	Quinn, D. D., The dynamics of two parametrically excited pendula with impacts, International Journal of Bifurcation and Chaos, 15, 6 (1975-1988) · Zbl 1092.70515
[14]	Quinn, D. D.; Bairavarasu, K., Near-simultaneous impacts, International Journal of Impact Engineering, 32, 6, 889-904 (2006)
[15]	Luo, A. C.J.; Chen, L., Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts, Chaos, Solitons & Fractals, 24, 2, 567-578 (2005) · Zbl 1135.70312
[16]	Han, P. R.S.; Luo, A. C.J.; Deng, W., Chaotic motion of a horizontal impact pair, Journal of Sound and Vibration, 181, 2, 231-250 (1995) · Zbl 1237.70028
[17]	Bapat, C. N., The general motion of an inclined impact damper with friction, Journal of Sound and Vibration, 184, 3, 417-427 (1995), M.R. Duncan · Zbl 0982.70521
[18]	Duncan, M. R.; Wassgren, C. R.; Krousgrill, C. M., The damping performance of a single particle impact damper, Journal of Sound and Vibration, 286, 1-2, 123-144 (2005)
[19]	Blazejczyk-Okolewska, B.; Czolczynski, K.; Kapitaniak, T., European Journal of Mechanics A-Solids, 23, 3, 517-537 (2004) · Zbl 1060.70514
[20]	Blazejczyk-Okolewska, B.; Czolczynski, K.; Kapitaniak, T., Determination of geometrical conditions of assembly and impacts in classified types of mechanical systems with impacts, European Journal of Mechanics, A-Solids, 24, 2, 277-291 (2005) · Zbl 1082.70011
[21]	Hu, H. Y., Controlling chaos of a periodically forced nonsmooth mechanical system, Acta Mechanica Sinica, 11, 3, 251-258 (1995) · Zbl 0855.70016
[22]	de Souza, S. L.T.; Caldas, I. L., Controlling chaotic orbits in mechanical systems with impacts, Chaos, Solitons and Fractals, 19, 1, 171-178 (2004) · Zbl 1086.37045
[23]	de Souza, S. L.T.; Wiercigroch, M.; Caldas, I. L.; Balthazar, J. M., Suppressing grazing chaos in impacting system by structural nonlinearity, Chaos, Solitons & Fractals (2007)
[24]	de Souza, S. L.T.; Caldas, I. L.; Viana, R. L., Damping control law for a chaotic impact oscillator, Chaos, Solitons and Fractals, 32, 2, 745-750 (2007)
[25]	de Souza, S. L.T.; Caldas, I. L.; Viana, R. L., Impact dampers for controlling chaos in systems with limited power supply, Journal of Sound and Vibration, 279, 3-5, 955-967 (2005)
[26]	Lee, J. Y.; Yan, J. J., Control of impact oscillator, Chaos, Solitons and Fractals, 28, 1, 136-142 (2006) · Zbl 1140.70475
[27]	Lee, June-Yule; Yan, Jun-Juh, Position control of double-side impact oscillator, Mechanical Systems and Signal Processing, 21, 1076-1083 (2007)
[28]	Zinjade, P. B.; Mallik, A. K., Impact damper for controlling friction-driven oscillations, Journal of Sound and Vibration, 306, 1-2, 238-251 (2007)
[29]	Zhao, W. L.; Wang, L. Z., Random Fatigue and Clearance Nonlinearity of Mechanical Vibrating Systems (2006), Science publishing house: Science publishing house Beijing, China
[30]	Luo, A. C.J., On flow switching bifurcations in discontinuous dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 12, 1, 100-116 (2007) · Zbl 1102.37014
[31]	Wiercigroch, M.; Sin, V. W.T., Experimental study of a symmetrical piecewise based-excited oscillator, ASME Journal of Applied Mechanics, 65, 3, 657-663 (1998)
[32]	Shaw, S. W.; Holmes, P. J., Periodically forced linear oscillator with impacts: Chaos and long-period motions, Physical Review Letters, 51, 857-894 (1983)
[33]	Shaw, S. W.; Holmes, P. J., A periodically forced impact oscillator with large dissipation, Journal of Applied Mechanics, 50, 849-857 (1983) · Zbl 0539.70032
[34]	Wagg, D. J., Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator, Chaos, Solitons and Fractals, 22, 541-548 (2004) · Zbl 1116.70334
[35]	Luo, A. C.J.; Gegg, B. C., Stick and non-stick periodic motions in periodically forced oscillators with dry friction, Journal of Sound and Vibration, 291, 1-2, 132-168 (2006) · Zbl 1243.70025
[36]	Xie, J. H., The mathematical model for the impact hammer and global bifurcations, Acta Mechanica Sinica, 29, 4, 456-463 (1997)