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

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.