Nonlinear Response of SDOF systems under combined Deterministic and Random Excitations

Dynamical systems subjected to random excitations exhibit non-linear behavior for sufficiently large motion. 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 the multiple time scale method, opportunely modified to take properly into account the correlation structure of the stochastic input process, is adopted to derive a stochastic frequency-response relationship involving the response amplitude statistics and the input power spectral density. A low-intensity noise is assumed to separate the strong mean motion from its weak fluctuations. The moment differential equations of phase and amplitude are derived and a linearization technique applied to evaluate the second order statistics. The theory is validated through digital simulations on a nonlinear single degree of freedom model for the transversal oscillation of a cantilever beam with tip force and to a Duffing-Rayleigh oscillator, to analyze non-linear damping effects.