Bayesian modal updating using complete input and incomplete response noisy measurements

The problem of identification of the modal parameters of a structural model using complete input and incomplete response time histories is addressed. It is assumed that there exist both input error (due to input measurement noise) and output error (due to output measurement noise and modeling error). These errors are modeled by independent white noise processes, and contribute towards uncertainty in the identification of the modal parameters of the model. To explicitly treat these uncertainties, a Bayesian framework is adopted and a Bayesian time-domain methodology for modal updating based on an approximate conditional probability expansion is presented. The methodology allows one to obtain not only the optimal (most probable) values of the updated modal parameters but also their uncertainties, calculated from their joint probability distribution. Calculation of the uncertainties of the identified modal parameters is very important if one plans to proceed with the updating of a theoretical finite-element model based on these modal estimates. The proposed approach requires only one set of excitation and corresponding response data. It is found that the updated probability density function (PDF) can be well approximated by a Gaussian distribution centered at the optimal parameters at which the posterior PDF is maximized. Numerical examples using noisy simulated data are presented to illustrate the proposed method.