Skip to main content
SpringerLink
Log in

Response of nonlinear oscillator under narrow-band random excitation

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

The principal resonance of Duffing oscillator to narrow-band random parametric excitation was investigated. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response were studied by means of qualitative analyses. The effects of damping, detuning, bandwidth and magnitudes of deterministic and random excitations were analyzed. The theoretical analyses were verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increases, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady state solutions.

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

Similar content being viewed by others

References

  1. Den Hartog J P.Mechanical Vibrations[M]. Fourth edition. New York: McGraw-Hill, 1956.

    Google Scholar 

  2. Lyon R H, Heckl M, hazelgrove C B. Response of hard-spring oscillator to narrow-band excitation [J].Journal of the Acoustical Society of America, 1961,33(4): 1404–1411.

    Article  Google Scholar 

  3. Richard K, Anand G V. Nonlinear resonance in strings under narrow band random excitation-part I: planar response and stability[J].Journal of Sound and Vibration, 1983,86(8): 85–98.

    Article  MATH  MathSciNet  Google Scholar 

  4. Davies H G, Nandall D. Phase plane for narrow band random excitation of a Duffing oscillator[J].Journal of Sound and Vibration, 1986,104(2): 277–283.

    Article  MathSciNet  Google Scholar 

  5. Iyengar R N. Response of nonlinear systems to narrow-band excitation[J].Structural Safety, 1989,6(2): 177–185.

    Article  Google Scholar 

  6. Lennox W C, Kuak Y C. Narrow band excitation of a nonlinear oscillator[J].Journal of Applied Mechanics, 1976,43(2): 340–344.

    MATH  Google Scholar 

  7. Grigoriu M. Probabilistic analysis of response of Duffing oscillator to narrow band stationary Gaussian excitation[A]. In: E H Dowell Ed.Proceedings of First Pan-American Congress of Applied Mechanics[C]. Rio de Janeiro: Brazil, 1989.

  8. Davis H G, Liu Q. The response probability density function of a Duffing oscillator with random narrow band excitation[J].Journal of Sound and Vibration, 1990,139(1): 1–8.

    Article  MathSciNet  Google Scholar 

  9. Kapitaniak T. Stochastic response with bifurcations to non-linear Duffing's oscillator[J].Journal of Sound and Vibration, 1985,102(3): 440–441.

    Article  MathSciNet  Google Scholar 

  10. Fang T, Dowell E H. Numerical simulations of jump phenomena in stable Duffing systems[J].International Journal of Nonlinear Mechanics, 1987,22(2): 267–274.

    Article  MathSciNet  Google Scholar 

  11. Zhu W Q, Lu M Q, Wu Q T. Stochastic jump and bifurcation of a Duffing oscillator under narrow-band excitation[J].Journal of Sound and Vibration, 1993,165(2): 285–304.

    Article  MATH  Google Scholar 

  12. Wedig W V. Invariant measures and Lyapunov exponents for generalized parameter fluctuations [J].Structural Safety, 1990,8(1):13–25.

    Article  Google Scholar 

  13. Nayfeh A H.Introduction to Perturbation Techniques[M]. New York: Wiley, 1981.

    Google Scholar 

  14. Rajan S, Davies H G. Multiple time scaling of the response of a Duffing oscillator to narrow-band excitations[J].Journal of Sound and Vibration, 1988,123(3): 497–506.

    MathSciNet  Google Scholar 

  15. Nayfeh A H, Serhan S J. Response statistics of nonlinear systems to combined deterministic and random excitations[J].International Journal of Nonlinear Mechanics, 1990,25(5): 493–509.

    Article  MATH  MathSciNet  Google Scholar 

  16. RONG Hai-wu, XU Wei, FANG Tong. Principal response of Duffing oscillator to combined deterministic and narrow-band random parametric excitation[J].Journal of Sound and Vibration, 1998,210(4): 483–515.

    Article  Google Scholar 

  17. ZHU Wei-qiu.Random Vibration[M]. Beijing: Science Press, 1992. (in Chinese)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Foshan University, 528000, Foshan, Guangdong, P. R. China

    Rong Hai-wu & Wang Xiang-dong

  2. The State Key Laboratory of Vibration, Shock and Noise, Shanghai Jiaotong University, 200030, Shanghai, P. R. China

    Rong Hai-wu & Meng Guang

  3. Department of Applied Mathematics, Northwestern Polytechnical University, 710072, Xi'an, P. R. China

    Xu Wei & Fang Tong

Authors
  1. Rong Hai-wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wang Xiang-dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Meng Guang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xu Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Fang Tong
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by ZHANG Shi-sheng and LI Li

Foundation items: the National Natural Science Foundation of China (10072049, 19972054); the Natural Science Foundation of Guangdong Province (000017); the Open Fund of the State Key Laboratory of Vibration, Shock and Noise of Shanghai Jiaotong University (VSN-2002-04)

Biography: RONG Hai-wu (1966 ∼), Associate Professor, Doctor (E-mail: ronghw@foshan.net)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hai-wu, R., Xiang-dong, W., Guang, M. et al. Response of nonlinear oscillator under narrow-band random excitation. Appl Math Mech 24, 817–825 (2003). https://doi.org/10.1007/BF02437814

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02437814

Key words

Chinese Library Classification

2000 MR Subject Classification

Search

Navigation