Hierarchical Bayesian modeling framework for model updating and robust predictions in structural dynamics using modal features

The hierarchical Bayesian modeling (HBM) framework has recently been developed to tackle the uncertainty quantification and propagation in structural dynamics inverse problems. This new framework characterizes the ensemble variability of structural parameters observed over multiple datasets together with the estimation uncertainty obtained based on the discrepancy between the measured and model outputs. The present paper expands on this framework, developing it further for model inference based on modal features. It generalizes the HBM framework by considering an additional hyper distribution to characterize the uncertainty of prediction error variances across different datasets. Moreover, asymptotic approximations are integrated into the HBM framework to simplify the computation of the posterior distribution of hyper-parameters, providing insights on different sources of uncertainties and the relation of the estimates of the hyper-parameters with the parameter variability and estimation uncertainties. Conditions are presented under which the approximations are expected to be accurate. Introducing the HBM formulation is beneficial, particularly for the propagation of uncertainty based on both structural and prediction error parameters providing reasonable uncertainty bounds. The posterior uncertainty of the structural and prediction error parameters is propagated to estimate data-informed output quantities of interests, including failure probabilities, which offers robustness to the variability over datasets. The proposed approximations are tested and verified using simulated and experimental examples. The effects of the uncertainty due to dataset variability and the prediction error uncertainty are illustrated in these examples.