Dimension-reduced FPK equation for structures excited by filtered noises

Abstract

Stochastic engineering dynamic actions such as earthquakes and strong wind can be regarded as colored noise with some certain power spectral density functions. By inserting filtering equations the response of the original system could be inverted to a Markov process because most colored noises can be generated through filtering white noises, thus making it possible to adopt the method of FPK equation and other related methods. However, the large dimension of the systems lead to great challenge in the solution of related high-dimensional FPK equations. For this purpose, the present paper proposed a simplified method for the extended system by integrating the highdimensional FPK equation and establishing equivalent drift coefficients, thus resulting in a dimensionreduced FPK equation. The Kanai-Tajimi power spectral density model is used as an example. Inserting the estimated equivalent drift coefficients into the dimension-reduced FPK equation and solving it by the finite difference method leads to the PDF of response of the systems. Numerical examples are illustrated. The method established can be extended to multiplicative noises.