By F. Casciati (eds.)
The chapters of this ebook have been written by means of structural engineers. The strategy, for that reason, isn't aiming towards a systematic modelling of the reaction yet to the definition of engineering systems for detecting and averting undesired phenomena. during this experience chaotic and stochastic behaviour might be tackled in an identical demeanour. This element is illustrated in bankruptcy 1. Chapters 2 and three are solely dedicated to Stochastic Dynamics and canopy single-degree-of-freedom platforms and impression difficulties, respectively. bankruptcy four offers info at the numerical instruments beneficial for comparing the most indexes worthy for the class of the movement and for estimating the reaction chance density functionality. bankruptcy five offers an outline of random vibration equipment for linear and nonlinear multi-degree-of-freedom structures. The randomness of the cloth features and the suitable stochastic versions ar thought of in bankruptcy 6. bankruptcy 7, ultimately, bargains with huge engineering sytems less than stochastic excitation and makes it possible for the stochastic nature of the mechanical and geometrical properties.
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Additional info for Dynamic Motion: Chaotic and Stochastic Behaviour
Probabilistic Methods in Structural Engineering, Chapman & Hall, London 1984. , Chaos in Classical and Quantum Mechanics, Springer Verlag, New York 1990.  Bontempi F. : Chaotic Motion and Stochastic Excitation, Nonlinear Dynamics, (1993). : Elements of Vibration Analysis, McGraw Hill, New York 1986. : Necessary and Sufficient Conditions for Asymptotic Stability of Linear Stochastic Systems, Theor. Prob. , 12 (1967), 144-147. : A Multiplicative Ergodic Theorem, Lyapunov Characteristic Numbers for Dynamical Systems, Trans.
Many practical problems involve processes which are approximately nonnal and the effect of the non-normality can often be regarded as negligible. This explains the popularity of second order analyses. 30 M. DiPaola 'or linear or non-linear systems driven by normal delta-correlated processes the Ito stochastic ifferential calculus [3, 5] represents the main tool for evaluating the response in terms of statistical 1oments. A paper by Srinvasan  reviews the evolution of the researches on the stochastic 1tegrals and on the problems connected to their applications.
From equations (1), (2) and (4), it is evident that the complete probabilistic description of the stochastic process X(t) at a fixed time t can be obtained indifferently by means of the knowledge of its probability density function or of its characteristic function or of its moments and/or cumulants of every order. Many physical problems have small values of higher order cumulants and, for normal (Gaussian) processes they are exactly zero for order greater than two, thus in these cases the first cumulant (mean value) and the second cumulant also called variance fully define the stochastic process X(t) from a probabilistic point of view at a fixed time t A complete probabilistic description of a stochastic process X(t) can be obtained by means of the knowledge of the infmite hierarchy of the joint probability density functions (7) where (8) In equation (8) the apex T means transpose and Xk is the random variable obtained by X(t) at the time instants tk, that is Xk = X(tk).
Dynamic Motion: Chaotic and Stochastic Behaviour by F. Casciati (eds.)