Equivalent linearization method using Gaussian mixture (GM-ELM) for nonlinear random vibration analysis

摘要

A new equivalent linearization method is developed for nonlinear random vibration analysis. The method employs a Gaussian mixture distribution model to approximate the probabilistic distribution of a nonlinear system response. The parameters of the Gaussian mixture model are estimated by an optimization algorithm which requires a few rounds of dynamic analysis of the nonlinear system. Due to properties of the Gaussian mixture distribution model, the proposed Gaussian mixture based equivalent linearization method (GM-ELM) can decompose the non-Gaussian response of a nonlinear system into multiple Gaussian responses of linear single-degree-of-freedom oscillators. Using a probabilistic combination technique, the linear system of GM-ELM can provide the response probability distribution equal to the Gaussian mixture estimation of the nonlinear response distribution. Using the linear system of GM ELM in conjunction with linear random vibration theories, response statistics such as the mean up crossing rate and first-passage probability of the nonlinear system can be conveniently computed. In order to facilitate applications of GM-ELM in earthquake engineering practice, a response spectrum formula is also proposed to compute the mean peak response of the nonlinear system by using the elastic response spectra representing the peak responses of the linear single-degree-of-freedom oscillators. Finally, two numerical examples are presented to illustrate and test GM-ELM. The analysis results obtained from GM-ELM are compared with those obtained from the conventional ELM and Monte Carlo simulation. The supporting source code and data are available for download at https://github.com/ziqidwang/GitHub-GM-ELM-code.git. (C) 2016 The Authors. Published by Elsevier Ltd.