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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Mitigating Grazing Bifurcation and Vibro-Impact Instability in Time-Frequency Domain

Journal of Applied Nonlinear Dynamics 5(2) (2016) 169--184 | DOI:10.5890/JAND.2016.06.004

Chi-Wei Kuo; C. Steve Suh

Nonlinear Engineering and Control Lab, Mechanical Engineering Department, Texas A&M University College Station, Texas 77843-3123, U.S.A.

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Impact oscillators are found in many applications. It is common for these applications to undergo the inadvertent state of grazing bifurcation. Vibro-impact incited grazing and route-to-chaos are difficult to control. The Newtonian model of a vibro-impact system rich of complex nonlinear behaviors is considered for the mitigation of impact induced instability and grazing. A novel concept capable of simultaneous control of vibration amplitude in the time-domain and spectral response in the frequency-domain is adopted to formulate a viable control solution. The concept has been demonstrated to be feasible for the control of dynamic instability including bifurcation and route-to-chaos in many nonlinear systems. The developed controller explores wavelet adaptive filters and filtered-x least mean square algorithm to the successful moderation of the grazing and dynamic instability of the non-smooth system. The qualitative behavior of the controlled impact oscillator follows a definitive fractal topology before settling into a stable manifold. The controlled response is categorically quasi-periodic and of the prescribed vibration amplitude and frequency spectrum.


  1. [1]  Peterka, F., (1974), Laws of impact motion of mechanical systems with one degree of freedom. I Theoretical analysis of n-multiple /1/n/-impact motions, ActaTechnicaCSAV, 19(4), 462-473.
  2. [2]  Thompson, J.M.T., and Ghaffari, R., (1983), Chaotic dynamics of an impact oscillator, Physical Review A, 27(3), 1741-1743.
  3. [3]  Holmes, P. J., (1982), The dynamics of repeated impacts with a sinusoidally vibrating table, Journal of Sound and Vibration, 84(2), 173-189.
  4. [4]  Whiston, G. S., (1987), The vibro-impact response of a harmonically excited and preloaded one-dimensional linear oscillator, Journal of Sound and Vibration, 115(2), 303-319.
  5. [5]  Whiston, G. S., (1987), Global dynamics of a vibro-impacting linear oscillator, Journal of Sound and Vibration, 118(3), 395-429.
  6. [6]  Whiston, G. S., (1992), Singularities in vibro-impact dynamics, Journal of Sound and Vibration, 152(3), 427-460.
  7. [7]  Budd, C. J., and Dux, F., (1994), Chattering and related behavior in impacting oscillators, Philosophical Transactions of the Royal Society: Physical and Engineering Sciences, 347(1683), 365-389.
  8. [8]  Budd, C. J., and Dux, F., (1994), Intermittency in impact oscillators close to resonance, Nonlinearity, 7(4), 1191-1224.
  9. [9]  Gegg, B.C., Luo, A.C.J., and Suh, C.S., (2008), Grazing bifurcations of a harmonically excited oscillator moving on a time-varying translation belt, Nonlinear Analysis-Real World Applications, 9(5), 2156-2174.
  10. [10]  Nordmark, A.B., (1991), Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145(2), 279-297.
  11. [11]  NordmarkA.B., (1992), Effects due to low velocity impact in mechanical oscillators, International Journal of Bifurcation and Chaos, 02(03), 597-605.
  12. [12]  Nordmark A. B., (1997), Universal limit mapping in grazing bifurcations, Physical Review E, 55(1), 266-270.
  13. [13]  Fredriksson, M.H., and Nordmark, A.B., (1997), Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators, Proceedings of The Royal Society A-Mathematical Physical and Engineering Sc, 453(1961), 1261-1276.
  14. [14]  Nordmark, A. B., (2001), Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators, Nonlinearity, 14(6), 1517-1542.
  15. [15]  Dankowicz, H., Piiroinen, P., and Nordmark, A.B., (2002), Low-velocity impacts of quasiperiodic oscillations, Chaos Solitons & Fractals, 14(2), 241-255.
  16. [16]  Chillingworth, D.R.J., (2010), Discontinuity geometry for an impact oscillator Dynamical Systems, Dynamical Systems-An International Journal, 17(4), 389-420.
  17. [17]  Chillingworth, D.R.J., (2010), Dynamics of an impact oscillator near a degenerate graze, Nonlinearity, 23(11), 2723-2748.
  18. [18]  Chin, W., Ott, E., Nusse, H. E., and Grebogi, C., (1995), Universal behavior of impact oscillators near grazing incidence, Physics Letters A, 201(2-3), 197-204.
  19. [19]  Gorbikov, S.P., and Men'shenina, A.V., (2007), Statistical description of the limiting set for chaotic motion of the vibro-impact system, Automation and Remote Control, 68(10), 1794-1800.
  20. [20]  Molenaar, J., dewater, W. van, and deWegerand, J., (2000), Grazing impact oscillations, Physical Review E, 62(2), 2030-2041.
  21. [21]  Lamba, H., (1995), Chaotic, regular and unbounded behavior in the elastic impact oscillator, Physica D, 82(1-2), 117-135.
  22. [22]  James, I., Ekaterina, P., Marian, W., and Soumitro, B., (2008), Experimental study of impact oscillator with one-sided elastic constraint, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 366(1866), 679-704.
  23. [23]  James, I., Ekaterina, P., Marian, W., and Soumitro, B., (2009), Invisible grazings and dangerous bifurcations in impacting systems: The problem of narrow-band chaos, Physical Review E, 79(3), 037201.
  24. [24]  James, I., Ekaterina, P., Marian, W., and Soumitro, B., (2010), Bifurcation analysis of an impact oscillator with a one-sided elastic constraint near grazing, Physica D: Nonlinear Phenomena, 239(6), 312-321.
  25. [25]  Ivanov, A.P., (1993), Stabilization of an impact oscillator near grazing incidence owing to resonance, Journal of Sound and Vibration, 162(3), 562-565.
  26. [26]  Lee, J.Y., and Yan, J.J., (2006), Control of impact oscillator, Chaos Solitons & Fractals, 28(1), 136-142.
  27. [27]  Lee, J.Y., and Yan, J.J., (2007), Position control of double-side impact oscillator, Mechanical Systems and Signal Processing, 21(2), 1076-1083.
  28. [28]  Bishop, S.R., Wagg, D.J., and Xu, D., (1998), Use of control to maintain period-1 motions during wind-up or wind-down operations of an impacting driven beam, Chaos Solitons & Fractals, 9(1-2), 261-269.
  29. [29]  Bichri, A., Belhaq, M., and Perret-Liaudet, J, (2011), Control of vibroimpact dynamics of a single-sided Hertzian contact forced oscillator, Nonlinear Dynamics, 63(1-2), 51-60.
  30. [30]  Awrejcewicz, J., Tomczak, K., and Lamarque, C.-H., (1999), Controlling Systems with Impacts, International Journal of Bifurcation and Chaos, 09(03), 547-553.
  31. [31]  Ott, E., Grebogi, C., and Yorke, JA., (1990), Controlling chaos, Physical Review Letters, 64(11), 1196-1199.
  32. [32]  Hassene, G., Safya, B., and Nahla, K., (2014), Control of chaos in an impact mechanical oscillator, ACECS'14 Proceedings, 116-122.
  33. [33]  Silvio, L.T. de, Souza, Ibere L. Caldas, Ricardo, L. Viana, and Jos'e, M. Balthazar, (2008), Control and chaos for vibro-impact and non-ideal oscillators, Journal of Theoretical and Applied Mechanics, 46(3), 641-664.
  34. [34]  Silvio, L.T. de, and Souza, Ibere L. Caldas, (2004), Controlling chaotic orbits in mechanical systems with impacts, Chaos Solitons & Fractals, 19(1), 171-178.
  35. [35]  Silvio, L.T. de, Souza, Ibere L. Caldas, and Ricardo, L. Viana, (2007), Damping control law for a chaotic impact oscillator, Chaos Solitons & Fractals, 32(2), 745-750.
  36. [36]  Suh, C.S. and Liu, M.-K., (2013), Control of cutting vibration and machining instability: a time-frequency approach for precision, micro and nano machining, John Wiley & Sons, Ltd.
  37. [37]  Liu, M.-K.and Suh, C.S., (2012), Simultaneous time-frequency control of bifurcation and chaos, Communications in Nonlinear Science and Numerical Simulation, 17(6), 2539-2550.
  38. [38]  Liu, M.-K., Halfmann, E., and Suh, C.S., (2014), Multi-dimensional time-frequency control of micro-milling instability, Journal of Vibration and Control, 20(5), 643-660.
  39. [39]  Liu, M.-K.and Suh, C.S., (2012), On controlling milling instability and chatter at high speed, Journal of Applied Nonlinear Dynamics, 1(1), 59-72.
  40. [40]  Kryzhevich, S.G., (2008), Grazing Bifurcation and chaotic oscillations of single degree of freedom dynamical systems, Journal of Applied Mathematics and Mechanics, 72(4), 383-390.
  41. [41]  Kuo, S. and Morgan, D., (1996), Active noise control systems: algorithms and DSP implementations, Wiley- Interscience.
  42. [42]  Yang, S., Sheu, G., and Liu, K., (2005), Vibration control of composite smart structures by feedforward adaptive filter in digital signal processor, Journal of Intelligent Material Systems and Structures, 16(9), 773- 779.
  43. [43]  Guan, Y., Lim, T., and Shepard, W., (2005), Experimental study on active vibration control of a gearbox system, Journal of Sound and Vibration, 282(3-5), 713-733.
  44. [44]  Peng, F., Gu, M., and Niemann, H., (2003), Sinusoidal reference strategy for adaptive feedforward vibration control: numerical simulation and experimental study, Journal of Sound and Vibration, 265(5), 1047-1061.
  45. [45]  Hakansson, L., Claesson, I., and Sturesson, P., (1998), Adaptive feedback control of machine-tool vibration based on the filtered-x LMS algorithm, Journal of Low Frequency Noise Vibration and Active Control, 17(4), 199-213.
  46. [46]  Yazdanpanah, A., and Khaki-Sedigh, A., and Yazdanpanah, A., (2005), Adaptive control of chaos in nonlinear chaotic discrete-time systems, 2005 International Conference on Physics and Control, PhysCon 2005, Proceedings, 2005, 913-915.
  47. [47]  Kim, H., Adeli, H., and Asce, F., (2004), Hybrid feedback-least mean square algorithm for structural control, ASCE Journal of Structural Engineering, 130(1), 120-127.