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On the determination of the power spectrum of randomly excited oscillators via stochastic averaging: An alternative perspective

Abstract : An approximate formula which utilizes the concept of conditional power spectral density (PSD) has been employed by several investigators to determine the response PSD of stochastically excited nonlinear systems in numerous applications. However, its derivation has been treated to date in a rather heuristic, even "unnatural" manner, and its mathematical legitimacy has been based on loosely supported arguments. In this paper, a perspective on the veracity of the formula is provided by utilizing spectral representations both for the excitation and for the response processes of the nonlinear system: this is done in conjunction with a stochastic averaging treatment of the problem. Then, the orthogonality properties of the monochromatic functions which are involved in the representations are utilized. Further, not only stationarity but ergodicity of the system response are invoked. In this context, the nonlinear response PSD is construed as a sum of the PSDs which correspond to equivalent response amplitude dependent linear systems. Next, relying on classical excitation-response PSD relationships for these linear systems leads, readily, to the derivation of the formula for the determination of the PSD of the nonlinear system. Related numerical results are also included.
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P.D. Spanos, I.A. Kougioumtzoglou, Christian Soize. On the determination of the power spectrum of randomly excited oscillators via stochastic averaging: An alternative perspective. Probabilistic Engineering Mechanics, Elsevier, 2011, 26 (1), pp.10-15. ⟨10.1016/j.probengmech.2010.06.001⟩. ⟨hal-00684306⟩

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