Skip to main content
SpringerLink
Log in

Nonlinear Random Response of Ocean Structures Using First- and Second-Order Stochastic Averaging

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper deals with the dynamic response of nonlinear elastic structure subjected to random hydrodynamic forces and parametric excitation using a first- and second-order stochastic averaging method. The governing equation of motion is derived by using Hamilton's principle, taking into account inertia and curvature nonlinearities and work done due to hydrodynamic forces. Within the framework of first-order stochastic averaging, the system response statistics and stability boundaries are obtained. Unfortunately, the effects of nonlinear inertia and curvature are not reflected in the final results, since the contribution of these nonlinearities is lost during the averaging process. In the absence of hydrodynamic forces, the method fails to give bounded response statistics, and the analysis yields stability conditions. It is the second-order stochastic averaging which can capture the influence of stiffness and inertia nonlinearities that were lost in the first-order averaging process. The results of the second-order averaging are compared with those predicted by Gaussian and non-Gaussian closures and by Monte Carlo simulation. In the absence of parametric excitation, the non-Gaussian closure solutions are in good agreement with Monte Carlo simulation. On the other hand, in the absence of hydrodynamic forces, second-order averaging gives more reliable results in the neighborhood of stochastic bifurcation. However, under pure parametric random excitation, the stochastic averaging and Monte Carlo simulation predict the on-off intermittency phenomenon near bifurcation point, in addition to stochastic bifurcation in probability.

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

    Google Scholar 

  2. Roberts, J. B. and Spanos, P.-T. D., ‘Stochastic averaging: An approximate method of solving random vibration problems’, International Journal of Non-Linear Mechanics 21(2), 1986, 111–134.

    Google Scholar 

  3. Red-Horse, J. R. and Spanos, P. D., ‘A generalization to stochastic averaging in random vibration’, International Journal of Non-Linear Mechanics 27(1), 1992, 85–101.

    Google Scholar 

  4. Caughey, T. K, ‘Equivalent linearization technique’, Journal of Acousical Society of America 35, 1963, 1706–1711.

    Google Scholar 

  5. Spanos, P. D., ‘Stochastic linearization in structural dynamics’, ASME Applied Mechanics Reviews 34, 1980, 1–8.

    Google Scholar 

  6. Crandall, S. H., ‘Perturbation technique for random vibration of nonlinear systems’, Journal of Acousical Society of America 35(11), 1963, 1700–1705.

    Google Scholar 

  7. Schetzen, M., The Volterra and Wiener Theories of Nonlinear Systems, John Wiley & Sons, New York, 1980.

    Google Scholar 

  8. Spanos, P.-T. D. and Chen, T. W., ‘Response of a dynamical system to flow-induced load’, International Journal of Non-Linear Mechanics 15, 1980, 115-126.

    Google Scholar 

  9. Lipsett, A. W., ‘Nonlinear structural response in random waves’, ASCE Journal of Structural Engineering 112, 1986, 2416–2429.

    Google Scholar 

  10. Eatock Taylor, R. and Rajagopalan, A., ‘Dynamics of offshore structures: Parts 1 and 2’, Journal of Sound and Vibration 83, 1982, 401–431.

    Google Scholar 

  11. Moshchuk, N. K., Ibrahim, R.A., and Khasminskii, R. Z., ‘Response statistics of ocean structures to nonlinear hydrodynamic loading, Part I: Gaussian ocean waves’, Journal Sound and Vibration 184(4), 1995, 681–701.

    Google Scholar 

  12. Quek, S., Li, X., and Koh, C., ‘Stochastic response of jack-up platform by the method of statistical quadratization’, Applied Ocean Research 16(2), 1994, 113–122.

    Google Scholar 

  13. Donley, M. and Spanos, P., ‘Stochastic response of a tension leg platform to viscous and potential drift forces’, in Proceedings of the 11th International Conference on ‘Offshore Mechanics and Arctic Engineering', Vol. 2, 1992, pp. 325–334.

    Google Scholar 

  14. Shinozuka, M., Yun, C., and Vaicaitis, R., ‘Dynamic analysis of fixed offshore structures subjected to wind generated waves’, ASCE Journal of Structural Mechanics 5(2), 1977, 135–146.

    Google Scholar 

  15. Spanos, P.-T. D. and Hansen, J. E., ‘Linear prediction theory for digital simulation of sea waves’, ASME Journal of Energy Resources Technology 103, 1981, 43–249.

    Google Scholar 

  16. Lin, N. K. and Hartt, W. A., ‘Time series simulations of wide-band spectra for fatigue tests of off-shore structures’, ASME Journal of Energy Resources Technology 106, 1984, 468–472.

    Google Scholar 

  17. Thampi, S. K. and Niedzwecki, M., ‘Parametric and external excitation of marine risers’, ASCE Journal of Engineering Mechanics 118, 1992, 942–960.

    Google Scholar 

  18. Stratonovich, R. L., Topics in the Theory of Random Noise, Vol. 1, Gordon & Breach, New York, 1963.

    Google Scholar 

  19. Roberts, J. B., ‘The energy envelope of a randomly excited nonlinear oscillator’, Journal of Sound and Vibration 60(2), 1978, 177–185.

    Google Scholar 

  20. Roberts, J. B., ‘Averaging methods in random vibration’, Technical University of Denmark, No. 245, Lyngby, 1989.

  21. Dimentberg, M. F. and Menyailov, A. I., ‘Response of a single-mass vibroimpact system to white noise random excitation’, Zeitschrift für angewandte Mathematik und Mechanik (ZAMM) 59, 1979, 709–716.

    Google Scholar 

  22. Dimentberg, M. F., ‘Oscillations of a system with nonlinear stiffness under simultaneous and parametric random excitations’, Mechanics of Solids (Mekh. Tver. Tela) 15(5), 1980, 42–45.

    Google Scholar 

  23. Zhu, W. Q., ‘Stochastic averaging methods in random vibrations’, ASME Applied Mechanics Review 41, 1988, 189–199.

    Google Scholar 

  24. Baxter, G. K., ‘The non-linear response of mechanical systems to parametric random excitation’, Ph.D. Thesis, University of Syracuse, 1971.

  25. Schmidt, G., ‘Vibrations caused by simultaneous random forced and parametric excitations’, Zeitschrift für angewandte Mathematik und Mechanik (ZAMM) 60, 1981, 409–419.

    Google Scholar 

  26. Naprstek, J., ‘Solution of random vibrations of nonlinear systems by means of Markov process’, Acta Technica CSAV 21, 1976, 302–345.

    Google Scholar 

  27. Iwan, W. D. and Spanos, P. D., ‘Response envelope statistics for nonlinear oscillators with random excitation’, ASME Journal of Applied Mechanics 45, 1978, 170–174.

    Google Scholar 

  28. Roberts, J. B. and Spanos, P. D., ‘Stochastic averaging: An approximate method of solving random vibration problems’, International Journal of Non-Linear Mechanics 21, 1986, 111–134.

    Google Scholar 

  29. Roberts, J. B., ‘The energy envelope of a randomly excited nonlinear oscillator’, Journal of Sound and Vibration 60, 1978, 177–185.

    Google Scholar 

  30. Zhu, W. Q., ‘Stochastic averaging of the energy envelope of nearly Lyapunov systems’, in 'Random Vibrations and Reliability’, K. Hennig (ed.), Academie Verlag, Berlin, 1983, pp. 347–357.

    Google Scholar 

  31. Red-Horse, J. R. and Spanos P. D., ‘A generalization to stochastic averaging in random vibration’, International Journal of Non-Linear Mechanics 27(1), 1992, 85–101.

    Google Scholar 

  32. Hijawi, M., Moshchuk, N., and Ibrahim, R. A., ‘Unified second-order stochastic averaging approach’, ASME Journal of Applied Mechanics, 1997, in print.

  33. Khasminskii, R., ‘A limit theorem for the solutions of differential equations with random right-hand sides’, Theory of Probability and Its Applications 11, 1966, 390–405.

    Google Scholar 

  34. Khasminskii, R. Z., ‘On averaging principle for Ito stochastic differential equations’, Cybernetics (Kybernetika Cislo 3, Rocnik) 3, 1968, 260–279 [in Russian].

    Google Scholar 

  35. Pierson, W. J. and Moskowitz, L., ‘A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii’, Journal of Geophysical Research 69, 1964, 5181–5190.

    Google Scholar 

  36. Moshchuk, N. and Ibrahim, R.A., ‘Response statistics of ocean structures to nonlinear hydrodynamic loading, Part II: Non-Gaussian ocean waves’, Journal of Sound and Vibration, 1996, in print.

  37. Gradshteyn, I. and Ryzhik I., Tables of Integrals, Series, and Products, Academic Press, New York, 1980.

    Google Scholar 

  38. Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1970.

    Google Scholar 

  39. Ibrahim, R. A. and Heinrich, R. T., ‘Experimental investigation of liquid sloshing under parametric random excitation’, ASME Journal of Applied Mechanics 55, 1988, 467–473.

    Google Scholar 

  40. Horsthemke, W. and Lefever, R., Noise-Induced Transitions, Theory and Applications in Physics, Chemistry, and Biology, Springer-Verlag, New York, 1984.

    Google Scholar 

  41. Horsthemke, W. and Lefever, R., ‘Noise-induced transitions’, Chapter 8 in Noise in Nonlinear Dynamical Systems, Vol. 2, F. Moss and P. V. E. McClintock (eds.), Cambridge University Press, Cambridge, 1989, pp. 179–208.

    Google Scholar 

  42. Ibrahim, R. A., ‘Nonlinear random vibration: Experimental results’, ASME Applied Mechanics Reviews 44(10), 1991, 423–446.

    Google Scholar 

  43. Graham, R. and Schenzle, A., ‘Stabilization by multiplicative noise’, Physical Review A 26(3), 1982, 1676–1685.

    Google Scholar 

  44. Zeghlache, H., Mandel, P., and Van den Broeck, C., ‘Influence of noise on delayed bifurcations’, Physical Review A 40(1), 1989, 286–294.

    Google Scholar 

  45. Stocks, N. G., Manella, R., and McClintock, P. V. E., ‘Influence of random fluctuations on delayed bifurcations, II: The cases of white and colored additive and multiplicative noise’, Physical Review A 42(6), 1990, 3356–3362.

    Google Scholar 

  46. Billah, K. Y. R. and Shinozuka, M, ‘Numerical method for colored-noise generation and its application to a bistable system’, Physical Review A 42(12), 1990, 7492–7495.

    Google Scholar 

  47. Billah, K. Y. R. and Shinozuka, M., ‘Stabilization of a nonlinear system by multiplicative noise’, Physical Review A 44(8), 1991, 4779–4781.

    Google Scholar 

  48. Yoon, Y. J. and Ibrahim, R. A., ‘Parametric random excitation of nonlinear coupled oscillators’, Nonlinear Dynamics 8, 1996, 385-413.

    Google Scholar 

  49. Platt, N., Spiegel, E. A., and Tresser, C., ‘On-off intermittency: A mechanism for bursting’, Physical Review Letters 70(3), 1993, 279–282.

    Google Scholar 

  50. Platt, N., Hammel, S. M., and Heagy, J. F., ‘Effects of additive noise on on-off intermittency’, Physical Review Letters 72(22), 1994, 3498–3501.

    Google Scholar 

  51. Heagy, J. F., Platt, N., and Hammel, S. M., ‘Characterization of on-off intermittency’, Physical Review E 49(2), 1994, 1140–1150.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Wayne University, Detroit, MI, 48202, U.S.A

    M. Hijawi, R. A. Ibrahim & N. Moshchuk

Authors
  1. M. Hijawi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. A. Ibrahim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. Moshchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hijawi, M., Ibrahim, R.A. & Moshchuk, N. Nonlinear Random Response of Ocean Structures Using First- and Second-Order Stochastic Averaging. Nonlinear Dynamics 12, 155–197 (1997). https://doi.org/10.1023/A:1008299615084

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008299615084

Search

Navigation