Laplace domain method for evaluating mean-square response of simple oscillators to nonstationary excitation

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Evaluating mean-square response of linear systems to nonstationary excitation has been extensively studied, but exact closed-form solutions under various assumptions have been rarely available. This paper derives exact closed-form solutions for the mean-square response of simple oscillators subjected to nonstationary excitation which is formulated as the multiplication of a stationary excitation characterized by an arbitrary spectral density function (PSD) and an envelope function being the sum of several exponential functions. Special attention is given to the envelope function as the sum of two exponential functions because of its practical importance. While analytically evaluating the nonstationary mean square response of a SDOF (single-degree-of-freedom) system subjected to a nonstationary stochastic excitation requires carrying out a triple integral, traditional methods would conduct tedious calculus in the time, frequency or time–frequency domain. In contrast, the proposed method – much more efficient than traditional methods – primarily operates in the Laplace domain (complex plane) based on pole–residue formulations. Another advantage of the proposed pole–residue method is that meaningful physical and mathematical insights could be gained in the solution procedure. For demonstrating the effectiveness and versatility of the proposed method, two cases of the input PSD function are studied: (1) a modulated white noise process, and (2) a modulated correlated process. The correctness of the exact closed-form solutions is verified numerically by Monte Carlo simulations.

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Probabilistic Engineering Mechanics