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Nonlinear Dynamics

, Volume 67, Issue 1, pp 735–753 | Cite as

Using the delayed feedback control and saturation control to suppress the vibration of the dynamical system

  • Y. Y. Zhao
  • J. Xu
Original Paper

Abstract

In the present paper, the delayed feedback control is applied to suppress or stabilize the vibration of the primary system in a two degree-of-freedom dynamical system with parametrically excited pendulum. The case of a 1:2 internal resonance between pendulum and primary system is studied. The method of multiple scales is applied to obtain second-order approximations of the response of the system. The system stability and bifurcations of equilibrium point of the averaged equations are computed. It is shown that the delayed feedback control can be used to suppress the vibration or stabilize the system when the saturation control is invalid. The vibration of the primary system can be suppressed by the delayed feedback control when the original system is in the single-mode motion. The effect of gain and delay on the vibration suppression is discussed. As the delay varies at a fixed value of the gain, the vibration of the primary system can be suppressed at some values of the delay. The vibration suppression performance of the system is improved at a large value of the gain. The vibration of the primary system could be suppressed about 56% compared with the original system by choosing the appropriate values of gain and delay. The delayed feedback control also can be used to stabilize the system when the original system is unstable. The gain and delay could be chosen as the controlling parameters. Numerical simulation is agreement with the analytical solutions well.

Keywords

Vibration suppression Delay Dynamical vibration absorber Delayed feedback control 

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References

  1. 1.
    Nayfeh, A., Mook, D., Marshall, L.: Nonlinear coupling of pitch and roll modes in ship motion. J. Hydronaut. 7(4), 145–152 (1973) CrossRefGoogle Scholar
  2. 2.
    Haddow, A., Barr, A., Mook, D.: Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure. J. Sound Vib. 97(3), 451–473 (1984) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Oueini, S., Golnaraghi, M.F.: Experimental implementation of the internal resonance control strategy. J. Sound Vib. 191(3), 377–396 (1996) CrossRefGoogle Scholar
  4. 4.
    Oueini, S., Nayfeh, A.: Analysis and application of a nonlinear vibration absorber. J. Vib. Control 6(7), 999–1016 (2000) CrossRefGoogle Scholar
  5. 5.
    Pai, P., Wen, B., Naser, A., Schultz, M.: Structural vibration control using PZT patches and non-linear phenomena. J. Sound Vib. 215, 273–296 (1998) CrossRefGoogle Scholar
  6. 6.
    Pratt, J., Oueini, S., Nayfeh, A.: A Terfenol-D nonlinear vibration absorber. J. Intell. Mater. Syst. Struct. 10, 29–35 (1999) Google Scholar
  7. 7.
    Li, J., Hua, H.X., Shen, R.Y.: Saturation-based active absorber for a non-linear plant to a principal external excitation. Mech. Syst. Signal Process. 21, 1489–1498 (2007) CrossRefGoogle Scholar
  8. 8.
    Haxton, R., Barr, A.: The autoparametric vibration absorber. J. Eng. Ind. 94, 119–125 (1972) CrossRefGoogle Scholar
  9. 9.
    Banerjee, B., Bajaj, A.K., Davies, P.: Resonant dynamics of an autoparametric system: A study using higher order averaging. Int. J. Non-Linear Mech. 31, 21–39 (1993) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bajaj, A.K., Chang, S.I., Johnson, J.M.: Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system. Nonlinear Dyn. 5, 433–457 (1994) CrossRefGoogle Scholar
  11. 11.
    Song, Y., Sato, H., Iwata, Y., Komatsuzaki, T.: The response of a dynamic vibration absorber system with a parametrically excited pendulum. J. Sound Vib. 259(4), 747–759 (2003) CrossRefGoogle Scholar
  12. 12.
    Zhao, Y.Y., Xu, J.: Effects of delayed feedback control on nonlinear vibration absorber system. J. Sound Vib. 308, 212–230 (2007) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Olgac, N., Holm-Hansen, B.T.: A novel active vibration absorption technique: delayed resonator. J. Sound Vib. 176(1), 93–104 (1996) CrossRefGoogle Scholar
  14. 14.
    Olgac, N., Jalili, N.: Modal analysis of flexible beams with delayed resonator vibration absorber: theory and experiments. J. Sound Vib. 218(2), 307–331 (1998) CrossRefGoogle Scholar
  15. 15.
    Olgac, N., Hosek, M.: A new perspective and analysis for regenerative machine tool chatter. Int. J. Mach. Tools Manuf. 38, 783–798 (1998) CrossRefGoogle Scholar
  16. 16.
    Jalili, N., Olgac, N.: Multiple delayed resonator vibration absorbers for mul-degree-of-freedom mechanical structures. J. Sound Vib. 223(4), 567–585 (1999) CrossRefGoogle Scholar
  17. 17.
    Filipovic, D., Olgac, N.: Delayed resonator with speed feedback-design and performance analysis. Mechatronics 12, 393–413 (2002) CrossRefGoogle Scholar
  18. 18.
    Olgac, N., Sipahi, R.: An exact method for the stability analysis of time-delayed linear time-invariant (LTI) system. IEEE Trans. Autom. Control 47(5), 793–797 (2002) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhao, Y.Y., Xu, J.: Effects of delayed feedback control on vibration suppression in an autoparametric dynamical absorber. Acta Mech. Solida Sin. 28(4), 347–354 (2007) Google Scholar
  20. 20.
    Zhao, Y.Y., Xu, J.: Mechanism analysis of delayed nonlinear vibration absorber. Acta Mech. Sin. 40(1), 98–106 (2008) MathSciNetGoogle Scholar
  21. 21.
    Wirkus, S., Rand, R.: The dynamics of two coupled van der Pol oscillators with delay coupling. Nonlinear Dyn. 30, 205–221 (2002) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.School of Aircraft EngineeringNanchang Hangkong UniversityNanchangChina
  2. 2.School of Aerospace Engineering and Applied MechanicsTongji UniversityShanghaiChina

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