Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Random excitation of nonlinear coupled oscillators

  • 184 Accesses

  • 9 Citations


This paper presents the experimental results of random excitation of a nonlinear two-degree-of-freedom system in the neighborhood of internal resonance. The random signals of the excitation and response coordinates are processed to estimate the mean squares, autocorrelation functions, power spectral densities, and probability density functions. The results are qualitatively compared with those predicted by the Fokker-Planck equation together with a non-Gaussian closure scheme. The effects of system damping ratios, nonlinear coupling parameter, internal detuning ratio, and excitation spectral density level are considered in both studies except the effect of damping ratios is not considered in the experimental investigation. Both studies reveal similar dynamic features such as autoparametric absorber effect and stochastic instability of the coupled system. The experimental results show that the autocorrelation function of the coupled system has the feature of ultra narrow band process and degenerates to a periodic one as the internal detuning departs from the exact internal resonance condition. The measured probability density functions of the response of the main system suggests that the Gaussian representation is sufticient as long as the excitation level is relatively low in the neighborhood of the system internal resonance condition.

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


  1. 1.

    Bolotin, V. V.,The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, California, 1964.

  2. 2.

    Schmidt, G.,Parametererregte Schwingungen, VEB, Deutscher Verlag der Wissenschaften, Berlin. 1975.

  3. 3.

    Evan-Iwanowski, R. M.,Resonance Oscillations in Mechanical Systems, Elsevier, New York, 1976.

  4. 4.

    Nayfeh, A. H. and Mook, D. T.:Nonlinear Oscillations, Wiley-Interscience, New York, 1979.

  5. 5.

    Schmidt, G. and Tondl, A.,Nonlinear Vibrations. Cambridge University Press. London, 1986.

  6. 6.

    Minorsky, N.,Nonlinear Oscillation. Van Nostrand. New York, 1962.

  7. 7.

    Nayfeh, A. H., Mook, D. T., and Marshall, L. R., ‘Nonlinear coupling of pitch and roll modes in ship motions’,Journal of Hydrodynamics 7, 1973, 145–152.

  8. 8.

    Haddow, A. G., Barr, A. D. S., and Mook, D. T., ‘Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure”,Journal of Sound and Vibration 97, 1984, 451–473.

  9. 9.

    Bux, S. L. and Roberts, J. W., ‘Nonlinear vibratory interaction in systems of coupled beams”.Journal of Sound and Vibration 104, 1986, 497–520.

  10. 10.

    Gray, A. H. Jr. and Caughey, T. K., ‘A controversy in problems involving random parametric excitation’.Journal of Mathematical Physics 44 (3), 288–296.

  11. 11.

    Ariaratnam, S. T., ‘Bifurcation in nonlinear stochastic systems’, in P. H. Holmes (ed.)New Developments in Nonlinear Problems in Dynamics, SIAM, Philadelphia, 1980, 470–474.

  12. 12.

    Ibrahim, R. A., Soundararajan, A., and Heo, H., ‘Stochastic response of nonlinear dynamic systems based on non-Gaussian closure’,ASME Journal of Applied Mechanics 52(4), 1985, 96–970.

  13. 13.

    Sun, J. Q. and Hsu, C. S., ‘Cumulant-neglect closure method for nonlinear systems under random excitation’,ASME Journal of Applied Mechanics 54, 1987, 649–655.

  14. 14.

    Bogdanoff, J. L. and Citron, S. J., ‘On the stabilization of the inverted pendulum’, in Proceedings of the 9th Midwestern Mechanics Conference.Development in Mechanics 3(1), 1965, 3–15.

  15. 15.

    Bogdanoff, J. L. and Citron, S. J., ‘Experiments with an inverted pendulum subjected to random parametric excitation’,Journal of Acoustical Society of America 38(9), 1965, 447–452.

  16. 16.

    Dalzell, J. F., ‘Exploratory studies of liquid behavior in randomly excited tanks: longitudinal excitation’, Southwest Research Institute, San Antonio, Texas, Technical Report No. 1, 1967.

  17. 17.

    Ibrahim, R. A. and Heinrich, R. T., ‘Experimental investigation of liquid sloshing under parametric random excitation’.ASME Journal of Applied Mechanics 55(2), 1988, 467–573.

  18. 18.

    Roberts, J. W., Random excitation of a vibratory system with autoparametric interaction’,Journal of Sound and Vibration 69, 101–116.

  19. 19.

    Ibrahim, R. A. and Sullivan, D. G., ‘Experimental investigation of structural autoparametric interaction under random excitation’,AIAA Journal 28(2), 1990.

  20. 20.

    Ibrahim, R. A., Evans, M., and Yoon, Y. J., ‘Experimental investigation of random excitation of nonlinear systems with autoparametric coupling’.Structural Safety 6, 1989, 161–176.

  21. 21.

    Ibrahim, R. A.,Parametric Random Vibration, John Wiley & Sons. New York, 1985.

  22. 22.

    Vendat, J. S. and Piersol, A. G.,Random Data: Analysis and Measurement Procedures. John Wiley & Sons. New York, 1971.

  23. 23.

    Evans, M. G., ‘Experimental investigation of nonlinear systems under wide band random excitation’, M.S. Thesis, Wayne State University, Department of Mechanical Engineering, 1989.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ibrahim, R.A., Yoon, Y.J. & Evans, M.G. Random excitation of nonlinear coupled oscillators. Nonlinear Dyn 1, 91–116 (1990). https://doi.org/10.1007/BF01857587

Download citation

Key words

  • internal resonance
  • random vibrations
  • non-Gaussian closure experiments