Bayesian-based response expansion and uncertainty quantification using sparse measurement sets

•Response expansion formulated within a Bayesian framework to quantify uncertainty.•Expansion in frequency domain enables use of frequency response functions.•Regularization enables use of the model’s entire set of modes in the expansion.•Extension to power spectral densities enables application to random vibration. Systems subjected to dynamic loads often require monitoring of their vibrational response, but limitations on the total number and placement of the measurement sensors can hinder the data-collection process. This paper presents an indirect approach to estimate a system’s full-field dynamic response, including all uninstrumented locations, using response measurements from sensors sparsely located on the system. This approach relies on Bayesian inference that utilizes a system model to estimate the full-field response and quantify the uncertainty in these estimates. By casting the estimation problem in the frequency domain, this approach utilizes the modal frequency response functions as a natural, frequency-dependent weighting scheme for the system mode shapes to perform the expansion. This frequency-dependent weighting scheme enables an accurate expansion, even with highly correlated mode shapes that may arise from spatial aliasing due to the limited number of sensors, provided these correlated modes do not have natural frequencies that are closely spaced. Furthermore, the inherent regularization mechanism that arises in this Bayesian-based procedure enables the utilization of the full set of system mode shapes for the expansion, rather than any reduced subset. This approach can produce estimates when considering a single realization of the measured responses, and with some modification, it can also produce estimates for power spectral density matrices measured from many realizations of the responses from statistically stationary random processes. A simply supported beam provides an initial numerical validation, and a cylindrical test article excited by acoustic loads in a reverberation chamber provides experimental validation.