Abstract
In many cases of engineering interest it has become quite common to use stochastic processes to model loadings resulting from earthquake, turbulent winds or ocean waves. In these circumstances the structural response needs to be adequately described in a probabilistic sense, by evaluating the cumulants or the moments of any order of the response (see e.g. [1, 2]). In particular, for linear systems excited by normal input, the response process is normal too and the moments or the cumulants up to the second order fully characterize the probability density function of both input and output processes. Many practical problems involve processes which are approximately normal and the effect of the non-normality can often be regarded as negligible. This explains the popularity of second order analyses.
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References
Stratonovich, R.L.: Topics in the theory of random noise, Gordon and Breach, New York, N.Y. 1963.
Lin, Y.K.: Probabilistic theory of structural dynamics, McGraw-Hill, New York, N.Y. 1967.
Itô, K.: On a formula concerning stochastic differential, Nagoya Mathematical Journal, 3, (1951), 55–65.
Gardiner, C.W.: Handbook of stochastic methods, Springer-Verlag, Berlin 1990.
Jazwinski, A.H.: Stochastic processes and filtering theory, Academic Press, New York, N.Y. 1973.
Srinvasan, S.K.: Stochastic integrals, SM Archives, 3, (1978), 325–379.
Stratonovich, R.L.: A new form of representing stochastic integrals and equations, J. SIAM Control, 4, (1966), 362–371.
Wong, E., and Zakai, M.: On the relation between ordinary and stochastic differential equations, International Journal of Engineering Sciences, 3, (1965), 213–229.
To, C.W.S.: On the dynamic systems distributed by random parametric excitation, Journal of Sound and Vibration, 123, (1988), 387–390.
Tung C.C.: Random response of highway bridges to vehicle loads, Journal of the Engineering Mechanics Division, ASME, 93, (1967), 79–84.
Liepmann, H.W.: On the application of statistical concepts to the buffeting problem, Journal of Aeronautical Sciences, 19, (1952), 793–800.
Cornell, C.A.: Stochastic process models in structural engineering, Stanford University, Technical Report 34, 1964.
Merchant, D.H.: A stochastic model of wind gusts, Stanford University, Technical Report 48.
Lutes, L.D.: Cumulants of stochastic response for linear systems, Journal of Engineering Mechanics, ASCE, 112, 10, (1986), 1062–1075.
Lutes, L.D., and Hu, S.L.J.: Non-normal stochastic response of linear systems, Journal of Applied Mechanics, 112, 2, (1986), 127–141.
Di Paola, M. and Muscolino, G.: Non stationary probabilistic response of linear systems under non-Gaussian input, in: Computational Stochastic Mechanics, (Ed. P.D. Spanos), (1991), 293–302.
Feller, W.: On the integro-differential equations of completely discontinuous Markov processes, Trans. Amer. Math. Soc., 48, 3, 1948.
Roberts, J.B.: System response to random impulses, Journal of Sound and Vibration, 24, (1972), 23–34.
Iwankiewicz, R., Nielsen, S.R.K., and Christensen, P.: Dynamic response of nonlinear systems to Poisson distributed pulse train: Markov approach, in: Nonlinear Structural Systems Under Random Condition, (Eds. Casciati, F., Elishakoff, I., and Roberts, J.B. ), (1990), 223–238.
Iwankiewicz, R., Nielsen, S.R.K.: Dynamic response of non-linear systems to Poisson distributed random pulses, Journal of Sound and Vibration, 156, 3, (1992), 407–423.
Di Paola, M., and Falsone, G.: Stochastic dynamics of non-linear systems driven by non-normal delta-correlated processes, Journal of Applied Mechanics, 60, (1993), 141–148.
Ibrahim, R.A.: Parametric random vibration, Research Studies Press, Letchworth, England 1985.
Casciati, F. and Favarelli, L: Fragility analysis of complex structural systems, Research Studies Press, Taunton 1991.
Soong, T.T. and Grigoriu, M.: Random vibration of mechanical structural systems, Prentice Hall, Englewood Cliffs 1993.
Snyder, D.L.: Random point processes, John Wiley, New York 1975.
Lin, Y.K.: Nonstationary excitation and response in linear systems treated as sequences of random pulses, Journal of the Acoustical Society of America, 38, (1965), 453–460.
Parzen, E.: Modern probability theory and its applications, John Wiley & Sons, New York 1960.
Loeve, M.: Probability theory, Van Nostrand-Reinhold, Princeton, New Jersey 1963.
Doob, J.L.: Stochastic processes, John Wiley & Sons, New York, N.Y. 1953.
Soong, T.T.: Random differential equations, Science and Engineering, Academic Press 1973.
Ibrahim, R.A., Soundararajan, A., and Heo, H.: Stochastic response of non-linear dynamics system based on a non Gaussian closure, Journal Applied Mechanics, 52, (1985), 965–970.
Wu, W.F., and Lin, Y.K.: Cumulant closure for non-linear oscillators under random parametric and external excitation, International Journal Nonlinear Mechanics, 19, (1984), 349–362.
Di Paola, M.: Moments of non-linear systems, in Probabilistic Methods in Civil Engineering, (Ed. P.D. Spanos), Proc. 5-th ASCE Spec. Conf., Blacksburg, Virginia, 1988, 285–288.
Di Paola, M. and Muscolino, G.: Differential moment equation of FE modelled structures with geometrical non-linearities, Int. Journal of Non-linear Mechanics, 25, 4, (1990), 363373.
Di Paola, M., Falsone, G., and Pirrotta, A.: Stochastic response analysis of non-linear systems under Gaussian input, Probabilistic Engineering Mechanics, 7, (1992), 15–21.
Brewer, J.W.: Kronecker products and matrix calculus in system theory, IEEE Transactions on Circuits and Systems, 25, 9, (1978), 772–781.
Arnold, L.: Stochastic differential equations: theory and applications, John Wiley & Sons, New York, N.Y. 1973.
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Di Paola, M. (1993). Stochastic Differential Calculus. In: Casciati, F. (eds) Dynamic Motion: Chaotic and Stochastic Behaviour. International Centre for Mechanical Sciences, vol 340. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2682-0_2
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DOI: https://doi.org/10.1007/978-3-7091-2682-0_2
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