Response of nonlinear systems in probability domain using stochastic averaging

Determination of probability density function (PDF) of the response for strongly nonlinear single-degree-of-freedom system subjected to both multiplicative and additive random excitations using stochastic averaging technique is faced with few difficulties. Firstly, the size of excitations should be small such that the response of the system converges weakly to a Markov process. Secondly, the excitations should preferably be broad banded so that the analytical results are reasonably accurate and finally, the nonlinear functions should be integrable if closed-formed expressions for the result are desired. The above issues are examined in the paper with a view to show (a) limiting values of the size parameter of excitations for which stochastic averaging technique can be applied to obtain reasonable estimates of PDF and mean square value of response, (b) the effect of the nature of the frequency contents of excitation on the response and (c) the use of the stochastic averaging method employing generalized harmonic functions for nonintegrable functions representing the dynamic system. It is shown that the stochastic averaging procedure provides results which compare very well with those obtained from simulation analysis not only for wide band excitations, but also for narrow band excitations for a wide range of the size parameter of excitation. Further, the procedure can be easily implemented for highly irregular functions by employing a numerical scheme with FFT.