Abstract
We have applied the approximation method of statistical linearization and various higher order corrections thereto to the study of a nonlinear oscillator perturbed by Gaussian, delta-correlated noise. We compute the second-order statistics of the response, i.e., the variances, autocorrelation functions, and spectral densities for various forms of the nonlinearity and compare our results with the few more exact calculations which are available in the literature. We show that a very simple modification of statistical linearization, based upon the use of the variance as obtained from the appropriate Fokker-Planck equation, yields results which are in better agreement with the “exact” literature results than either statistical linearization or first-order corrections thereto. This modified method of statistical linearization has the significant advantage of great computational simplicity as compared to other attempts of accurate calculations of second-order statistics of nonlinear stochastic equations now in the literature.
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References
A. B. Budgor,J. Stat. Phys., this issue, preceding paper.
T. K. Caughey,J. Acoust. Soc. Am. 35:1706 (1963);Adv. Appl. Mech. 11:209 (1971).
S. H. Crandall, Nonlinear Vibration Problems,Zagadnienia Drgan Nielinowych 14:39 (1973); Nonlinear Problems in Random Vibrations, presented at the 7th International Conference on Nonlinear Oscillations, Berlin, Sept. 1975.
J. B. Morton and S. Corrsin,J. Stat. Phys. 2:153 (1970).
M. Bixon and R. Zwanzig,J. Stat. Phys. 3:245 (1971).
M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions (National Bureau of Standards Applied Math. Series, No. 55), U.S. Government Printing Office, Washington, D.C. (1968), p. 778.
J. B. Morton, private communications.
T. K. Caughey and J. K. Dienes,J. Appl. Phys. 32:2476 (1961).
S. H. Crandall, S. S. Lee, and J. H. Williams, Jr.,J. Appl. Mech. 41:1094 (1974); S. H. Crandall and S. S. Lee, to appear inIngenieur-Archiv.
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This work was supported in part by the National Science Foundation under Grants MPS 72-04363 and CHE 75-20624.
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Budgor, A.B., Lindenberg, K. & Shuler, K.E. Studies in nonlinear stochastic processes. II. The duffing oscillator revisited. J Stat Phys 15, 375–391 (1976). https://doi.org/10.1007/BF01020340
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DOI: https://doi.org/10.1007/BF01020340