Dynamical systems subject to random excitations exhibit non-linear behavior for sufficiently large motion. In the case of 1 d.o.f. models, the multiple time scale method has been extensively utilized in the framework of non-linear deterministic analysis to obtain two averaged first-order differential equations describing the slow time scale modulation of amplitude and phase response. In this paper a stochastic frequency-response relationship has been derived involving the response amplitude statistics and the input power spectral density. A low-intensity noise has been assumed to separate the strong mean motion from its weak fluctuations. The moment differential equations of phase and amplitude have been derived and a linearization technique applied to evaluate the second order statistics. The theory has been validated through digital simulations on a Duffing-Rayleigh oscillator and on a cantilever beam with tip force.

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