title: UNCERTAINTY QUANTIFICATION AND MANAGEMENT IN OPERATIONAL MODAL ANALYSIS WITH MULTIPLE SETUPS
Creators: name: Xie, Y
Publisher: name: University of Liverpool Repository
Year: 2019

description: Operational modal analysis (OMA) aims at identifying the structural modal properties (e.g., natural frequencies, damping ratios and mode shapes) using ambient vibration data that is measured when the structure is under its working condition. Due to its economy in implementation, OMA has attracted considerable attention in field testing of civil engineering structures. Considering practical constraints (e.g., budget) in field tests, a multiple-setup strategy is often employed to measure a large number of degrees of freedom with a limited number of sensors. In ambient vibration tests, the input loading is not measured. Because of this, the identification uncertainty of modal parameters can become significant. Different test configurations can yield different levels of uncertainty. It is therefore desirable to quantitatively assess the identification uncertainty and investigate how it is related to test configuration. For planning purpose, especially for multiple setups, it is desirable to have an uncertainty-based assessment for a given test configuration. Motivated by the above concerns, the research in this thesis aims at quantifying and managing identification uncertainties in OMA with multiple setups. Based on a Bayesian modal identification framework, uncertainty quantification is first addressed by investigating the computation of the ‘posterior covariance matrix’ from the inverse of the Hessian of the negative log-likelihood function of modal parameters. Difficulties arise in deriving the Hessian matrix since the modal parameters are subjected to constraints, e.g., the mode shape is subjected to a scaling constraint. A theoretical framework is developed for evaluating the Hessian of a function under general constraints. The theory is applied to Bayesian OMA with single setup and multiple setups, where new analytical expressions for calculating the posterior covariance matrix are derived. Numerical examples are also provided to validate the proposed theory. The uncertainty management of multiple setups is addressed by investigating the leading order behaviour of the posterior covariance matrix. For sufficiently long data, the posterior covariance matrix is asymptotically equal to the inverse of the Fisher Information Matrix (FIM). A closed-form asymptotic expression for the FIM is derived with small damping and high modal signal-to-noise ratio. Leveraging on the asymptotic decoupling of modal parameters, the dimension of the FIM can be reduced, making the inverse algebraically manageable. This leads to closed-form expressions of the leading order posterior coefficient of variation (standard deviation / mean) of the modal parameters. The collection of these results is referred as ‘uncertainty law’ of multiple setups, which reveals how identification uncertainty is related to test configuration. Investigation into the uncertainty law provides engineering guidance for the experimental design of multiple setups. Illustrative examples with synthetic, laboratory and field test data are presented to validate the proposed theory. The uncertainty law is applied for multiple-setup ambient vibration test of an office building and a suspension footbridge, where the test configuration is quantitatively assessed from an uncertainty point of view.