Mean and Variability Response Functions for Stochastic Systems Under Dynamic Excitation

Abstract eng:
The concept of the so called Variability Response Function (VRF), recently proposed for statically determinate and indeterminate stochastic systems [1, 2], is extended in this work to linear dynamic stochastic systems. An integral form for the variance of the dynamic response of stochastic systems is considered, involving a Dynamic VRF (DVRF) and the spectral density function of the stochastic field modeling the uncertain system properties. As in the case of static systems, the independence of the DVRF to the spectral density and the marginal probability density function of the stochastic field modeling the uncertain parameters is validated using brute-force Monte Carlo simulations as well as a series of different moving power spectral density functions for the calculation of the DVRF. The uncertain system property considered is the inverse of the elastic modulus (flexibility). It is demonstrated that DVRF is a function of the standard deviation of the stochastic field modeling flexibility. The same integral expression can be used to calculate the mean response of a dynamic system using the concept of the so called Dynamic Mean Response Function (DMRF), which is a function similar to the DVRF [3]. These integral forms can be used to efficiently compute the mean and variance of the transient system response at any time of the dynamic response together with spectral-distribution-free upper bounds. They also provide an insight into the mechanisms controlling the dynamic mean and variability response. In this work this methodology is effectively utilized to estimate the stochastic dynamic response of a single degree of freedom system subjected to a) sinusoidal load at the end of its length and b) El Centro earthquake. In both cases results are drawn for different values of the stochastic field standard deviation and for various Gaussian and non-Gaussian probability distributions.

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