A New Stationary PDF Approximation for Non-linear Oscillators

A new method is presented for approximating the stationary probability density 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 function (PDF) from a parameterized class of trial non-Gaussian PDFs by minimizing a weighted norm of the Fokker–Planck–Kolmogorov equation error. The proposed procedure yields simple expressions in terms of one-dimensional integrals for determining desired probabilistic characteristics of the system response, 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 equivalent 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 reliability of the system. In contrast, the equivalent non-linearization and the dissipation energy balancing methods, known to provide the most accurate estimates for the mean-square response, give very poor estimates of the mean outcrossing rates as the number of level outcrossings decreases.