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Nonlinearly damped systems under simultaneous broad-band and harmonic excitations

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Abstract

The probability distribution of the response of a nonlinearly damped system subjected to both broad-band and harmonic excitations is investigated. The broad-band excitation is additive, and the harmonic excitations can be either additive or multiplicative. The frequency of a harmonic excitation can be either near or far from a resonance frequency of the system. The stochastic averaging method is applied to obtain the Itô type stochastic differential equations for an averaged system described by a set of slowly varying variables, which are approximated as components of a Markov vector. Then, a procedure based on the concept of stationary potential is used to obtain the exact stationary probability density for a class of such averaged systems. For those systems not belonging to this class, approximate solutions are obtained using the method of weighted residuals. Application of the exact and approximate solution procedures are illustrated in two specific cases, and the results are compared with those obtained from Monte Carlo simulations.

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References

  1. Ivengar, R. N. and Manohar, C. S., ‘Van der Pol's oscillator under combined periodic and random excitations’, in Nonlinear Stochastic Dynamics Engineering Systems, F. Ziegler and G. I. Schueller, eds., Springer-Verlag, Berlin, Heidelberg, 1988, 69–85.

    Google Scholar 

  2. Iyengar, R. N., ‘Multiple response moments and stochastic stability of a nonlinear system’, in Stochastic Structural Dynamics, Progress in Theory and Applications, S. T. Ariaratnam, G. I. Schueller, and I. Elishakoff, eds., Elsevier Applied Science, New York, 1988, 159–172.

    Google Scholar 

  3. Nayfey, A. H. and Serhan, S. J., ‘Response statistics of non-linear systems to combined deterministic and random excitations’, International Journal of Non-Linear Mechanics 25, 1990, 493–509.

    Google Scholar 

  4. Bogoliubov, N. and Mitropolski, A., Asymptotic Methods in the Theory of Non-Linear Oscillators, Gordon & Breach, New York, 1961.

    Google Scholar 

  5. Stratonovich, R. L., Topics in the Theory of Random Noise, Vol. 1, Gordon and Beach, New York, 1963.

    Google Scholar 

  6. Stratonovich, R. L., Topics in the Theory of Random Noise, Vol. 2, Gordon and Beach, New York, 1967.

    Google Scholar 

  7. Khaz'minskii, R. Z., ‘A limit theorem for the solution of differential equations with random right hand sides’, Theory of Probability and Applications 2, 1966, 390–405.

    Google Scholar 

  8. Ariaratnam, S. T. and Tam, D. S. F., ‘Parametric random excitation of a damped Mathieu oscillator’, ZAMM 56, 1976, 449–452.

    Google Scholar 

  9. Ariaratnam, S. T. and Tam, D. S.F., ‘Moment stability of coupled linear systems under combined and stochastic excitation’, in Stochastic Problems in Dynamics, B. L. Clarkson, ed., Pitman, 1977, 90–103.

  10. Dimentberg, M. F., Statistical Dynamics of Nonlinear and Time-Varying Systems, Research Studies Press LTD., Willy, New York, 1988.

    Google Scholar 

  11. Zhu, W. Q. and Huang, T. C., ‘Dynamic instability of liquid free surface in a container with elastic bottom under combined harmonic and stochastic longitudinal excitations’, in Random Vibrations, T. C. Huang and P. D. Spanos, eds., ASME, AMD 65, 1984, 195–220.

  12. Ahn, N. D., ‘Integrability condition for the averaged Kolmogorov-Fokker-Planck equations in theory of random vibrations’, translated from Prikladnaya Mekhanika 21, 1985, 92–96.

    Google Scholar 

  13. Ahn, N. D., ‘On the study of random oscillations in non-autonomous mechanical systems using the Fokker-Planck-Kolmogorov equations’, PMM U. S. S. R. 49, 1985, 392–397.

    Google Scholar 

  14. Ahn, N. D., ‘Random oscillations in a van der Pol system subjected to periodic and random excitations’, Soviet Math. Dokl. 34, 1987, 67–70.

    Google Scholar 

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

    Google Scholar 

  16. Finlayson, B. A., The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972.

    Google Scholar 

  17. Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.

    Google Scholar 

  18. Cai, G. O. and Lin, Y. K., ‘A new approximate solution technique for randomly excited non-linear oscillators’, International Journal of Non-Linear Mechanics 23, 1988, 409–420.

    Google Scholar 

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Authors and Affiliations

  1. Center for Applied Stochastics Research, Florida Atlantic University, 33431, Boca Raton, FL, U. S. A.

    G. O. Cai & Y. K. Lin

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  1. G. O. Cai
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  2. Y. K. Lin
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Cai, G.O., Lin, Y.K. Nonlinearly damped systems under simultaneous broad-band and harmonic excitations. Nonlinear Dyn 6, 163–177 (1994). https://doi.org/10.1007/BF00044983

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  • DOI: https://doi.org/10.1007/BF00044983

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