Parametric Random Excitation. I: Exponentially Correlated Parameters

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Abstract
The problem of a simple harmonic oscillator under parametric and nonhomogeneous random excitation is investigated. The governing differential equation is recast as an integral equation, and the solution is taken to be in the form of an infinite sum. The specific case considered here assumes stiffness and force to be modeled as stationary, Gaussian, nonwhite random processes. An iterative solution scheme is set up, and a symbolic manipulation code MACSYMA is used to generate the first three nonzero terms in the series. Approximate expressions are derived for the mean value and spectral density of the oscillator response. In order to obtain some measures for the validity of the method, a Monte Carlo simulation is performed. An important conclusion is that a one‐term “correction” can provide a good measure of the effect of system randomness on output statistics. In a companion paper (5), these methods are used to derive exact closed‐form solutions for the case where the system parameter is a white‐noise process. This fact, that an exact solution is possible for white‐noise parameters, will be very useful in those instances where the physical processes can be reasonably approximated as white noise. To the best knowledge of the writers, this extensive analytical/numerical simulation study has not appeared in the literature.