An analytical perspective on Bayesian uncertainty quantification and propagation in mode shape assembly

Abstract

Assembling local mode shapes identified from multiple setups to form global mode shapes is of practical importance when the degrees of freedom (dofs) of interest are measured separately in individual setups or when one expects to exploit the computational autonomous capabilities of different setups in full-scale operational modal test. The Bayesian mode assembly methodology was able to obtain the optimal global mode shape as well as the associated uncertainties by taking the inverse of the analytically derived Hessian matrix of the negative log-likelihood function (NLLF) (Yan and Katafygiotis, 2015) [1]. In this study, we investigate how the posterior uncertainties existing in the local mode shapes obtained from different setups propagate into the global mode shapes in an explicit manner by borrowing a novel approximate analysis strategy. The explicit closed-form approximation expressions are derived to investigate the effects of various data parameters on the posterior covariance matrix of the global mode shapes. Such quantitative relationships, connecting the posterior uncertainties with global mode shapes and the data information, offer a better understanding of uncertainty propagation over the process of mode shape assembly. The posterior uncertainty of the global mode shapes is inversely proportional to ‘normalized data length’ and the ‘frequency bandwidth factor’, and propositional to ‘noise-to-environment’ ratio and damping ratio. Validation studies using field test data measured from the Metsovo bridge located in Greece provide a practical verification of the rationality of the theoretical findings of uncertainty quantification and propagation analysis in Bayesian mode shape assembly.