Uncertainty quantification in fast Bayesian modal identification using forced vibration data considering the ambient effect

•Formulations of the posterior uncertainty using forced and ambient data were developed rigorously.•Posterior uncertainty can be quantified analytically without resorting to finite difference.•Synthetic examples were designed to verify the method for both MPV and associated posterior uncertainty.•Proposed method was applied successfully in a series of field vibration tests of a footbridge.•Dynamic characteristics of the footbridge were investigated with different styles of input. A Bayesian framework for modal identification using forced vibration data considering the ambient effect has been developed and a fast algorithm has been proposed to identify the modal parameters efficiently in Ni and Zhang (2018). Due to the existence of environmental noise, modeling error, etc., the associated posterior uncertainty of modal parameters has also attracted increasing attention. In this work, the posterior uncertainty is investigated in terms of its posterior covariance matrix. Based on the negative log-likelihood function (NLLF) constructed in Ni and Zhang (2018), the covariance matrix is derived and it is equal to the inverse of the Hessian matrix of the NLLF with respect to the identified modal parameters. The computational difficulty to determine the covariance matrix is discussed, and analytical formulation is derived to obtain the covariance matrix instead of the finite difference method. Two examples are used to illustrate the proposed method. The first one is a simulated bridge that is used to verify the proposed method. The effects of noise and ambient environment are studied to investigate the variation of the posterior uncertainty. In the second case, the proposed method is applied to a footbridge, where a series of shaker tests was carried out to provide both chirp excitation and known pseudorandom excitation to this structure. The results obtained using two kinds of excitations were compared and investigated.