New Insights on the Application of Moment Closure Methods to Nonlinear Stochastic Systems

  • S. F. Wojtkiewicz
  • B. F. SpencerJr.
  • L. A. Bergman
Part of the Solid Mechanics and its Applications book series (SMIA, volume 47)


The cumulant-neglect closure method is briefly outlined and subsequently applied to two Duffing systems, one exhibiting a unimodal and the other a bimodal response probability density function. The closure results are compared at stationarity to the exact solution over a broad range of parameters, and some connections are drawn between the accuracy of cumulant-neglect closure and the choice of system parameters. Finally, characteristic equations governing all stationary solutions of the closed system of moments equations are obtained, and the stability of the resulting solutions is ascertained. From this analysis, it is determined whether the closure results are physically consistent; that is, if the stationary closure results can be reached by letting the system evolve from arbitrary initial conditions.


Moment Equation Random Vibration Closure Result Closure Scheme Duffing Oscillator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bergman, L.A., Wojtkiewicz, S.F., Johnson, E.A., and Spencer, Jr., B.F. (1995) Some Reflections on the Efficacy of Moment Closure Methods, in P.D. Spanos (ed.), Computational Stochastic Dynamics, A.A. Balkema, Rotterdam, 87–95Google Scholar
  2. 2.
    Spencer, B.F., Jr. and Bergman, L.A. (1993) On the Numerical Solution of the Fokker-Planck Equation for Nonlinear Stochastic Systems, Nonlinear Dynamics 4, 357–372CrossRefGoogle Scholar
  3. 3.
    Wojtkiewicz, S.F., Bergman, L.A., and Spencer, Jr., B.F. (1995) On the Cumulant-Neglect Closure Method in Stochastic Dynamics, International Journal of Nonlinear Mechanics, submitted for publicationGoogle Scholar
  4. 4.
    Caughey, T.K., (1971) Nonlinear Theory of Random Vibrations, in Chia-Shun Yih, (ed.), Advances in Applied Mechanics, Academic Press, New York, 11,209–253Google Scholar
  5. 5.
    Wu, W.F. and Lin, Y.K. (1984) Cumulant-Neglect Closure for Nonlinear Oscillators Under Parametric and External Excitations. International Journal of Nonlinear Mechanics, 19, 349–362MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ibrahim, R.A. (1985) Parametric Random Vibration Research Studies Press, Great BritainzbMATHGoogle Scholar
  7. 7.
    Pawleta, M. and Socha, L. (1990) Cumulant-Neglect Closure of Nonstationary Solutions of Stochastic System, Journal of Applied Mechanics, 57 ,776–779MathSciNetCrossRefGoogle Scholar
  8. 8.
    Sun, J.-Q. and Hsu, C.S. (1987) Cumulant-Neglect Closure Method for Nonlinear Systems Under Random Excitations, Journal of Applied Mechanics, 54 ,649–655zbMATHCrossRefGoogle Scholar
  9. 9.
    Fan, F.G. and Ahmadi, G. (1990) On Loss of Accuracy and Non-Uniqueness of Solutions Generated by the Equivalent Linearization and Cumulant-Neglect Methods, Journal of Sound and Vibration, 137 :3,385–401MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gardiner, C.W. (1983) Handbook of Stochastic Methods, Springer Verlag, HeidelbergzbMATHGoogle Scholar
  11. 11.
    Bolotin, V.V. (1984) Random Vibrations of Elastic Systems, Martinus Nijhoff, The HaguezbMATHGoogle Scholar
  12. 12.
    Soong, T. and Grigoriu, M. (1993) Random Vibration of Mechanical and Structural Systems, Prentice Hall, Englewood Cliffs, New JerseyGoogle Scholar
  13. 13.
    Lin, Y.K. and Cai, G.Q. (1995) Probabilistic Structural Dynamics: Advanced Theory and Applications, McGraw Hill, New YorkGoogle Scholar
  14. 14.
    Char, B.W., Geddes, K.O., Gonnet, G.H., Monagan, M.B., and Watt, S.M., (1991) MAPLE V Language Reference Manual, Springer-Verlag, New YorkzbMATHGoogle Scholar
  15. 15.
    Roberts, J.B. and Spanos, P.D. (1990) Random Vibration and Statistical Linearization., Wiley, New YorkzbMATHGoogle Scholar
  16. 16.
    Langley, R.S. (1988) An Investigation of Multiple Solutions Yielded by the Equivalent Linearization Technique, Journal of Sound and Vibration, 127 :2, 271–281MathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • S. F. Wojtkiewicz
    • 1
  • B. F. SpencerJr.
    • 2
  • L. A. Bergman
    • 3
  1. 1.Department of Aeronautical and Astronautical EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Civil Engineering and Geological SciencesUniversity of Notre DameNotre DameUSA
  3. 3.Department of Aeronautical and Astronautical EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations