Abstract
A relatively straightforward formulation is presented for deriving the differential equations governing the evolution of the response cumulants and moments of a dynamical system. This is a very general framework which applies to linear and nonlinear systems subjected to external and multiplicative non-Gaussian, delta-correlated processes. This formulation provides an alternative to both the partial differential Fokker Planck equation that has sometimes been used in deriving moment or cumulant equations, and the differential (as opposed to derivative) relationships of the It8 calculus. It is believed that many analysts may find the technique used here to be more obvious than the alternatives, since the derivative relationships for the stochastic process are of the same form as in the more familiar ordinary differential equations.
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References
Crandall, S. H. (1980). “Non-Gaussian closure for random vibration excitation,” International Journal of Non-Linear Mechanics, 15, 303–313.
Di Paola, M. and Falsone, G. (1993). “Stochastic dynamics of nonlinear systems driven by non-normal delta-correlated processes,” Journal of Applied Mechanics, ASME, 60, 141–148.
Di Paola, M. and Falsone, G. (1997a). “Higher order statistics of the response of MDOF linear systems excited by linearly parametric white noises and external excitations,” Probabilistic Engineering Mechanics, 12(3), 179–188.
Di Paola, M. and Falsone, G. (1997b). “Higher order statistics of the response of MDOF systems under polynomials of filtered normal white noises,” Probabilistic Engineering Mechanics, 12(3), 189–196.
Di Paola, M., Falsone, G. (1994). “Stochastic dynamics of MDOF structural systems under non-normal filtered inputs,” Probabilistic Engineering Mechanics, 9(4), 265–272.
Di Paola, M., Falsone, G., and Pirotta, A. (1992). “Stochastic response analysis of nonlinear systems under Gaussian inputs,” Probabilistic Engineering Mechanics, 7, 15–21.
Di Paola, M. and Muscolino, G. (1990). “Differential moment equations of FE modeled structures with geometrical non-linearities,” International Journal of Non-Linear Mechanics, 25, 363–373.
Falsone, G. (1994). “Cumulants and correlations for nonlinear systems under non-stationary deltacorrelated processes,” Probabilistic Engineering Mechanics, 9(3), 157–165.
Grigoriu, M. and Waisman, F. (1997). “Linear systems with polynomials of filtered Poisson processes,” Probabilistic Engineering Mechanics, 12(2), 97–103.
Hasofer, A.M., and Grigoriu, M. (1995). “A new perspective on the moment closure method,” Journal of Applied Mechanics, 62(2), 527–532.
Hu, S-L.J. (1993). `Responses of dynamic systems excited by non-Gaussian pulse processes,“ Journal of Engineering Mechanics, ASCE, 119(9), 1818–1827.
Hu, S-L.J. (1995). “Parametric random vibrations under non-Gaussian pulse processes,” Journal of Engineering Mechanics, ASCE, 121(12), 1366–1371.
Hu, S-L.J. (1997). “Response cumulant equations for dynamic systems under delta-correlated processes,” Journal of Engineering Mechanics, ASCE, 123(2), 174–177.
Lin, Y.K. (1967). Probabilistic Theory of Structural Dynamics, McGraw-Hill Book Company, New York. Reprinted in 1976 by Krieger Publishing Co., Melbourne, FL.
Lutes, L.D. (1986). “State space analysis of stochastic response cumulants,” Probabilistic Engineering Mechanics, 1(2), 94–98.
Lutes, L.D. and Chen, D.0-K. (1992). “Stochastic response moments for linear systems,” Probabilistic Engineering Mechanics, 7(3), 165–173.
Lutes, L.D. and Sarkani, S. (1997). Stochastic Analysis of Structural and Mechanical Vibrations, Prentice-Hall, Englewood Cliffs, NJ..
Papadimitriou, C. and Lutes L.D. (1994). “Approximate analysis of higher cumulants for MDF random vibration,” Probabilistic Engineering Mechanics, 9(1), 71–82.
Papadimitriou, C. and Lutes L.D. (1996). “Stochastic cumulant analysis of MDOF systems with polynomial-type nonlinearities,” Probabilistic Engineering Mechanics, 11(1), 1–13.
Papadimitriou, C., Katafygiotis, L.S. and Lutes, L.D. (1999). Response cumulants of nonlinear systems subject to external and multiplicative excitations, Probabilistic Engineering Mechanics, in press.
Spanos, P-T.D. (1983). “Spectral moments calculation of linear system output,” Journal of Applied Mechanics, ASME, 50, 901–903.
Stratonovich, R.L. (1963). Topics in the Theory of Random Noise. Translated from the Russian by Silverman, R.A., Gordon and Breach, New York.
Wojtkiewicz, S.F., Spencer, B.F., and Bergman, L.A. (1996). “On the cumulant-neglect closure method in stochastic dynamics,” International Journal of Non-Linear Mechanics, 31(5), 657–684.
Wu, W.F. and Lin, Y.K. (1984). “Cumulant-neglect closure for non-linear oscillators under random parametric and external excitations,” International Journal of Non-Linear Mechanics, 19, 349–362.
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Lutes, L.D., Papadimitriou, C. (2001). Finding Response Cumulants for Nonlinear Systems With Multiplicative Excitations. In: Narayanan, S., Iyengar, R.N. (eds) IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics. Solid Mechanics and its Applications, vol 85. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0886-0_10
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DOI: https://doi.org/10.1007/978-94-010-0886-0_10
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