This study presents the application of Bayesian Expectation-Maximization (BEM) methodology to coupled state-input-parameter estimation in both linear and nonlinear structures. The BEM is built upon a Bayesian foundation, which utilizes the EM algorithm to deliver accurate estimates for latent states, model parameters, and input forces while updating noise characteristics effectively. This feature allows for quantifying associated uncertainties using response-only measurements. The proposed methodology is equipped with a recursive backward-forward Bayesian estimator that provides smoothed estimates of the state, input, and parameters during the Expectation step. Next, these estimates help calculate the most probable values of the noise parameters based on the observed data. This adaptive approach to the coupled estimation problem allows for real-time quantification of estimation uncertainties, whereby displacement, velocity, acceleration, strain, and stress states can be reconstructed for all degrees-of-freedom through virtual sensing. Through numerical examples, it is demonstrated that the BEM accurately estimates the unknown quantities based on the measured quantities, not only when a fusion of displacement and acceleration measurements is available but also in the presence of acceleration-only response measurements.
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