Uncertainty quantification in Bayesian operational modal analysis with multiple modes and multiple setups

Abstract

In full-scale ambient vibration tests, multiple setups are often performed for measurement when it is demanded to obtain a detailed mode shape with more measured degrees of freedom than the available number of synchronous data channels. In a previous work, a Bayesian operational modal analysis framework for the general case of multiple modes identified with ambient data from multiple setups was developed, together with an Expectation-Maximisation algorithm for efficiently calculating the most probable value (MPV) of modal parameters. Complementing the previous effort, this work investigates the posterior uncertainty of the modal parameters in terms of their posterior covariance matrix. Mathematically, the posterior covariance matrix is equal to the inverse of the Hessian of negative log-likelihood function at the MPV. The computational issues are investigated and analytical expressions for the Hessian matrix are derived, allowing the covariance matrix to be determined efficiently and accurately without resorting to the finite difference method. The proposed algorithm is verified with synthetic data, where the Bayesian and frequentist statistics are compared, and the effect of reference location is investigated. The developed computational tools are applied to investigate identification uncertainty with field data, where associated practical issues are also discussed.