Probability Density Evolution Method of Nonlinear Random Vibration Analysis

In order to reveal the applicability of the probability density evolution method in nonlinear random vibration analysis, a comparative research of the probability density evolution method and the classical nonlinear random vibration analysis was carried out by investigating the nonlinear responses of a class of randomly base-driven Duffing oscillators using the probability density evolution method (PDEM), the adaptive polynomial chaos expansion (APCE) and the Monte Carlo simulation (MCS). A physically based stochastic ground motion model was employed, and represented by a Karhunen-Love expansion in the application of the APCE. This discrete representation can be viewed as a projection of the physical vector space into the Gaussian vector space. Numerical results reveal that the solution processes of the three approaches are identical to weakly nonlinear systems, while they are approximately identical to strongly nonlinear systems though errors resulted from numerical techniques and artificial truncations are amplified, indicating that the solution of the PDEM is equivalent to that of the classical nonlinear random vibration analysis in the mean-square sense. The PDEM, moreover, goes a step further than the classical nonlinear random vibration analysis since the probability density function of responses and the dynamic reliability of systems can be simultaneously provided by the PDEM. The other methods, however, need much more computational efforts to obtain high order statistics of responses.