Nonlinear Random Vibration Analysis: A Bayesian Nonparametric Approach

Random vibration analysis aims to estimate the response statistics of dynamical systems subject to stochastic excitations. Stochastic differential equations (SDEs) that govern the response of general nonlinear systems are often complicated, and their analytical solutions are scarce. Thus, a range of approximate methods and simulation techniques have been developed. This paper develops a hybrid approach that approximates the governing SDE of nonlinear systems using a small number of response simulations and information available a priori. The main idea is to identify a set of surrogate linear systems such that their response probability distributions collectively estimate the response probability distribution of the original nonlinear system. To identify the surrogate linear systems, the proposed method integrates the simulated responses of the original nonlinear system with information available a priori about the number and parameters of the surrogate linear systems. There will be epistemic uncertainty in the number and parameters of the surrogate linear systems because of the limited data. This paper proposes a Bayesian nonparametric approach, called a Dirichlet Process Mixture Model, to capture these uncertainties. The Dirichlet process models the uncertainty over an infinite-dimensional parameter space, representing an infinite number of potential surrogate linear systems. Specifically, the proposed method allows the number of surrogate linear systems to grow indefinitely as the nonlinear system observed dynamic unveil new patterns. The quantified uncertainty in the estimates of the unknown model parameters propagates into the response probability distribution. The paper then shows that, under some mild conditions, the estimated probability distribution approaches, as close as desired, to the original nonlinear system's response probability distribution. As a measure of model accuracy, the paper provides the convergence rate of the response probability distribution. Because the posterior distribution of the unknown model parameters is often not analytically tractable, a Gibbs sampling algorithm is presented to draw samples from the posterior distribution. Variational Bayesian inference is also introduced to derive an approximate closed-form expression for the posterior distribution. The paper illustrates the proposed method through the random vibration analysis of a nonlinear elastic and a nonlinear hysteretic system.