Empirical and fully Bayesian approaches for the identification of vibration sources from transverse displacement measurements

This paper introduces the Bayesian regularization applied to the Force Analysis Technique (FAT), a method for identifying vibration sources from displacement measurements. The FAT is based on the equation of motion of a structure instead of a transfer matrix as it is the case for most of inverse problems. This particularity allows the estimation of vibration sources without the need of boundary conditions. Nevertheless, this method is highly sensitive to noise perturbations and needs a careful regularization. Two Bayesian approaches are thus presented. Firstly, the empirical Bayesian regularization which shows better robustness than L-curve and GCV regularizations while keeping a low numerical cost. Secondly, a fully Bayesian procedure using a Markov Chain Monte Carlo (MCMC) algorithm which provides credible intervals on variables of interest besides the automatically regularized vibration source field. In particular, measurement quality can be evaluated by the noise variance estimation and the uncertainties over the source level are quantified for a wide frequency range, with only a unique measurement scan.