Structural Engineering and Mechanics

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Stochastic analysis of external and parametric dynamical systems under sub-Gaussian Levy white-noise

  • Received : 2005.09.15
  • Accepted : 2007.12.28
  • Published : 2008.03.10
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Abstract

In this study stochastic analysis of non-linear dynamical systems under ${\alpha}$-stable, multiplicative white noise has been conducted. The analysis has dealt with a special class of ${\alpha}$-stable stochastic processes namely sub-Gaussian white noises. In this setting the governing equation either of the probability density function or of the characteristic function of the dynamical response may be obtained considering the dynamical system forced by a Gaussian white noise with an uncertain factor with ${\alpha}/2$- stable distribution. This consideration yields the probability density function or the characteristic function of the response by means of a simple integral involving the probability density function of the system under Gaussian white noise and the probability density function of the ${\alpha}/2$-stable random parameter. Some numerical applications have been reported assessing the reliability of the proposed formulation. Moreover a proper way to perform digital simulation of the sub-Gaussian ${\alpha}$-stable random process preventing dynamical systems from numerical overflows has been reported and discussed in detail.

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References

  1. Cai, C.Q. and Lin, Y.K. (1995), Probabilistic Structural Dynamics: Advanced Theory and Applications, McGraw-Hill, NY.
  2. Carpinteri, A., Chiaia, B. and Cornetti, P. (2003), "On the mechanics of quasi brittle-materials with a fractal microstructure", Eng. Frac. Mech., 70, 2321-2349. https://doi.org/10.1016/S0013-7944(02)00220-5
  3. Chechkin, A., Gonchar, V, Klafter, J., Metzler, R. and Tanatarov, L. (2002), "Stationary states of nonlinear oscillators driven by Levy noise", Chem. Phys., 284, 233-251. https://doi.org/10.1016/S0301-0104(02)00551-7
  4. Di Paola, M. and Failla, G (2005), "Stochastic response of linear and nonlinear systems to a-stable Levy white noises", Prob. Eng. Mech., 20(2), 128-135. https://doi.org/10.1016/j.probengmech.2004.12.001
  5. Di Paola, M. and Sofi, A. (2008), "Linear and nonlinear systems under sub-Gaussian (a-stable) input", Prob. Eng. Mech. (submitted).
  6. Di Paola, M., Pirrotta, A. and Zingales, M. (2007), "Ito calculus extended to systems driven by alpha-stable Levy white noises (A novel clip on the tails of Levy motion)", Int. J. Nln. Mech., 42, 1046-1054. https://doi.org/10.1016/j.ijnonlinmec.2007.07.001
  7. Falsone, G (1994), "Cumulants and correlations for linear systems under non-stationary delta-correlated processes", Prob. Eng. Mech., 9, 157-165. https://doi.org/10.1016/0266-8920(94)90001-9
  8. Grigoriu, M. (1995a), "Linear systems subject to non-Gaussian a-stable processes", Prob. Eng. Mech., 10, 23-34. Grigoriu, M. (1995b), "Linear and nonlinear systems with non-Gaussian white noise input", Prob. Eng. Mech., 10, 171-179. https://doi.org/10.1016/0266-8920(95)00014-P
  9. Grigoriu, M. (2000), "Equivalent linearization for systems driven by Levy white noise", Prob. Eng. Mech., 15, 285-190.
  10. Grigoriu, M. (2004), "Characteristic function equations for the state of dynamic systems with Gaussian, poisson and Levy white noise", Prob. Eng. Mech., 449-461.
  11. Hilfer, R. (ed.) (2000), Fractional Calculus in Phyisics, World Scientific, Singapore.
  12. Lin, YK. (1976), Probabilistic Theory of Structural Dynamics, Kriegher, FL.
  13. Samko, S.G, Kilbas, A.A. and Marichev, O. I. (1993), Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach Science Publishers, Amsterdam, NL.
  14. Samorodnitsky, GG and Grigoriu, M. (2003), "Tails of solutions of certain nonlinear stochastic differential equations driven by heavy tailed Levy motions", Stoch. Proc. Appl., 105, 69-97. https://doi.org/10.1016/S0304-4149(03)00002-4
  15. Samorodnitsky, Cz and Taqqu, M.S. (1994), Stable non-Gaussian Random Processes, Stochastic Models with Infinite Variance, Chapman and Hall, UK.
  16. Sokolov, I.M., Chechkin, A.V and Klafter, J. (2004), "Fractional diffusion equation for a power-law-truncated Levy process", Phys. A, 336, 245-251. https://doi.org/10.1016/j.physa.2003.12.044
  17. Stratonovich, R.L. (1967), Topics in the Theory of Random Noise, Gordon & Breach, NY.