Skip to main content
SpringerLink
Log in

Stability and bifurcation dynamics for a nonlinear controlled system subjected to parametric excitation

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

Through both analytical and numerical approaches, stability and bifurcation dynamics are studied for a nonlinear controlled system subjected to parametric excitation. The controlled system is a typical case of a two-degree-of-freedom system composed of a parametrically excited pendulum and its driving device. Three types of critical points for the modulation equations are considered near the principle resonance and internal resonance, which are characterized by a double zero and two negative eigenvalues, a double zero and a pair of purely imaginary eigenvalues, and two pairs of purely imaginary eigenvalues, respectively. With the aid of normal form theory, the stability regions for the initial equilibrium solutions and the critical bifurcation curves are obtained analytically, which exhibit some new dynamical behaviors. A time integration scheme is used to find the numerical solutions for these bifurcations cases, which confirm these analytical predictions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Marghitu, D.B., Diaconescu, C., Boghiu, D.: Fuzzy-logic control of parametrically excited impacting flexible system. Arch. Appl. Mech. 68, 259–270 (1998)

    Article  MATH  Google Scholar 

  2. Bucklaew, T.P., Liu, C.S.: Pitchfork-type bifurcations in a parametrically excited, PD-controlled pendulum or manipulator. J. Sound Vib. 247, 655–672 (2001)

    Article  Google Scholar 

  3. Amer, Y.A., Bauomy, H.S., Sayed, M.: Vibration suppression in a twin-tail system to parametric and external excitations. Comput. Math. Appl. 58, 1947–1964 (2009)

    Article  MATH  Google Scholar 

  4. El-Badawy, A.A., Mehrez, M.W., Ali, A.R.: Nonlinear modeling and control of flexible-link manipulators subjected to parametric excitation. Nonlinear Dyn. 62, 769–779 (2010)

    Article  MATH  Google Scholar 

  5. Pratiher, B.: Vibration control of a transversely excited cantilever beam with tip mass. Arch. Appl. Mech. 82, 31–42 (2012)

    Article  MATH  Google Scholar 

  6. Bi, Q.S., Zhang, R., Zhang, Z.D.: Bifurcation mechanism of bursting oscillations in parametrically excited dynamical system. Appl. Math. Comput. 243, 482–491 (2014)

    MATH  Google Scholar 

  7. Vázquez, C., Collado, J., Fridman, L.: Super twisting control of a parametrically excited overhead crane. J. Frankl. Inst. 351, 2283–2298 (2014)

    Article  MathSciNet  Google Scholar 

  8. Nguyen, Q.C., Krylov, S.: Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam. J. Sound Vib. 338, 91–104 (2015)

    Article  Google Scholar 

  9. Maccari, A.: Vibration control for parametrically excited Liénard systems. Int. J. Non-Linear Mech. 41, 146–155 (2006)

    Article  MATH  Google Scholar 

  10. Ying, Z.G., Ni, Y.Q., Ko, J.M.: Parametrically excited instability analysis of a semi-actively controlled cable. Eng. Struct. 29, 567–575 (2007)

    Article  Google Scholar 

  11. Alhazza, K.A., Daqaq, M.F., Nayfeh, A.H., Inman, D.J.: Non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control. Int. J. Non-Linear Mech. 43, 801–812 (2008)

    Article  MATH  Google Scholar 

  12. Han, W., Jin, D.P., Hu, H.Y.: Dynamics and stability of a non-linear controlled system subject to parametric excitation. Acta Mech. Sin. 35, 303–309 (2003)

    Google Scholar 

  13. Holmes, M.H.: Singular Perturbation Theory. Springer, Berlin (2005)

    Google Scholar 

  14. Engelberg, S.: A Mathematical Introduction to Control Theory, pp. 75–89. World Scientific, Singapore (2005)

    MATH  Google Scholar 

  15. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    Book  MATH  Google Scholar 

  16. Yu, P., Huseyin, K.: Static and dynamic bifurcations associated with a double zero eigenvalues. Dyn. Stab. Syst. 1, 73–86 (1986)

    MathSciNet  MATH  Google Scholar 

  17. Yu, P., Huseyin, K.: A perturbation analysis of interactive static and dynamic bifurcations. IEEE Trans. Auto Control 33, 28–41 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yu, P., Bi, Q.: Analysis of non-linear dynamics and bifurcations of a double pendulum. J. Sound Vib. 217, 691–736 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yu, P.: Symbolic computation of normal forms for resonant double Hopf bifurcations using a perturbation technique. J. Sound Vib. 247, 615–632 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yu, P.: Analysis on double Hopf bifurcation using computer algebra with the aid of multiple scales. Nonlinear Dyn. 27, 19–53 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors greatly appreciate the anonymous reviews for their insightful comments and suggestions for further improving the quality of this work. This research was supported by National Natural Science Foundation of China (11602134, 11501055), Natural Science Foundation of Shanghai (16ZR1413700), Natural Science Foundation of Shanghai University of Engineering Science (nhrc-2015-06) and National Research Foundation for the Doctoral Program of Higher Education of China (20133218110025).

Author information

Authors and Affiliations

  1. School of Fundamental Studies, Shanghai University of Engineering Science, Shanghai, 201620, People’s Republic of China

    Weiqin Yu, Na Li, Tianbo Wang & Shouwei Zhao

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Fangqi Chen

Authors
  1. Weiqin Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fangqi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Na Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tianbo Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Shouwei Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weiqin Yu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yu, W., Chen, F., Li, N. et al. Stability and bifurcation dynamics for a nonlinear controlled system subjected to parametric excitation. Arch Appl Mech 87, 479–487 (2017). https://doi.org/10.1007/s00419-016-1205-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-016-1205-x

Keywords

Search

Navigation