On the Bayesian model updating based on model reduction using complex modal data for damage detection

An approach for the Bayesian model updating of a linear dynamic system using complex modal data identified from dynamic test data is proposed in this paper. Very few works have utilized complex modal data for Bayesian model updating, and these works consider only the Most Probable Value (MPV) of the modal parameters as the modal data. On the other hand, the present work considers the posterior uncertainties of the modal parameters along with the MPVs as the modal data. Additionally, dynamic reduction has been applied to downsize the full system model to a reduced model with master Degrees of Freedom (DOFs) of the reduced model confined to the observed DOFs. Dynamic reduction facilitates updating of structural parameters without the requirement of mode matching. Additional uncertain parameters in the form of system modal frequencies, damping ratios, and partial mode shapes are introduced to establish a link between the structural model parameters and the modal data through the eigenvalue equation. Detailed formulation leading to the development of the posterior Probability Density Function (PDF) is presented, and a new Metropolis-within-Gibbs (MWG) sampler is proposed to simulate samples from the posterior PDF. Furthermore, a formulation is presented for evaluating the probability of damage based on the posterior samples obtained using the proposed approach from the structure's undamaged and possibly-damaged state. The proposed approach is validated and demonstrated comprehensively through simulated examples and an experimental study. Lastly, the performance of the proposed approach is compared with an alternative approach which is developed by integrating out the partial system mode shape from the posterior PDF.