Time-discretized moments of a randomly excited uncertain linear oscillator
Abstract
Uncertainty is present in all physical systems to some degree. The effect of uncertainty is to make the analysis of a linear structure nonlinear in nature. In conjunction with this effect, the response of a linear oscillator to Gaussian excitation becomes non-Gaussian when its stiffness is uncertain. For large uncertainties, the response and statistical moments are affected significantly. The objective of this thesis is to investigate the actual (not the approximate) behavior of a simple uncertain system. A single-degree-of-freedom damped linear oscillator having uncertain stiffness, and subjected to stationary mean-zero Gaussian white-noise excitation, is considered. General mathematical relations for a total of twenty-nine different statistical moments involving its response, stiffness, and load are derived, numerically integrated, and verified with simulation. The mathematical relations are based on the law of total probability. The moment equations are derived in both continuous and time-discretized form. The time-discretized versions allow the study of various aspects of step-by-step solution procedures employed by several existing stochastic finite element methods which have been proposed by the author and others. The effect of periodic enforcement of a Gaussian response approximation is isolated. It is shown mathematically that as the integration step size becomes small, the moments computed using such an approximation approach those of a deterministic oscillator having a stiffness equal to the mean of its distribution. It is found that the process of time discretization introduces artificial zeros in the covariance between the same-time load and velocity at the initial instant of each discrete time step. Time-discretized moment equations are also studied using two discretized loading functions meant to emulate white noise. The relative convergence properties of the approximate moments found using these models are found mathematically, and the effect of uncertainty on the allowable step size is briefly addressed. Two moments are noted which undergo discontinuous step-to-step jumps when one of the approximate load models is used.
Degree
Ph.D.
Advisors
Yao, Purdue University.
Subject Area
Civil engineering
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