A Bayesian identification framework for stochastic nonlinear dynamic systems based on a new likelihood approximation

Abstract

The paper introduces a Bayesian framework in conjunction with Markov Chain Monte Carlo (MCMC) sampling for parameter estimation in nonlinear stochastic dynamic systems. The proposed methodology is applied to both numerical and experimental cases. The paper commences by introducing Bayesian inference and its constituents: the likelihood function, prior distribution, and posterior distribution. Resonant decay method is employed to extract backbone curves, which capture the nonlinear behavior of the system. A mathematical model based on a single degree of freedom (SDOF) system is formulated, and backbone curves are obtained from time response data. Subsequently, MCMC sampling is employed to estimate the parameters using both numerical and experimental data. The obtained results demonstrate the convergence of the Markov chain, present parameter trace plots, and provide estimates of posterior distributions of updated parameters along with their uncertainties. Experimental validation is performed on a cantilever beam system equipped with permanent magnets and electromagnets. The proposed methodology demonstrates promising results in accurately estimating parameters of stochastic nonlinear dynamical systems. Through the use of proposed likelihood functions grounded in backbone curves, the probability distributions of both linear and nonlinear parameters are simultaneously identified. Based on this view, the necessity to segregate stochastic linear and nonlinear model updating is eliminated.