A new method is presented for approximating the stationary probability dens
ity function of the response of a general class of non-linear single-degree
-of-freedom dynamical systems subjected to additive stochastic white noise
excitation. The method is based on finding the best probability density fun
ction (PDF) from a parameterized class of trial non-Gaussian PDFs by minimi
zing a weighted norm of the Fokker-Planck-Kolmogorov equation error. The pr
oposed procedure yields simple expressions in terms of one-dimensional inte
grals for determining desired probabilistic characteristics of the system r
esponse, such as moments and mean outcrossing rates. Examples illustrating
the applicability and accuracy of the method include a system modeling the
rolling motion of a ship and a Duffing oscillator with non-linear damping.
Comparisons are made with some other approximate methods, including equival
ent linearization, partial linearization, equivalent non-linearization, and
dissipation energy balancing methods, that show that the new method yields
substantially improved estimates for the expected outcrossing rates of the
response. These outcrossing rates are crucial for evaluating the reliabili
ty of the system. In contrast, the equivalent non-linearization and the dis
sipation energy balancing methods, known to provide the most accurate estim
ates for the mean-square response, give very poor estimates of the mean out
crossing rates as the number of level outcrossings decreases. (C) 2000 Else
vier Science Ltd. All rights reserved.