A Bayesian updating procedure for the electromechanical properties of piezoelectric energy harvesters

In the last decade, several numerical and analytic procedures have been proposed to predict the dynamic behavior of piezoelectric energy har- vesters (PEHs). Nevertheless, PEHs present characteristics that are di ffi cult to control in their manufacturing process, for example the electromechanical properties of the materials present variations up to 20% of their nominal val- ues. In that sense, the use of deterministic models to obtain accurate predictions implies to have full information about the geometry and the electromechanical properties. This work introduces a procedure to update the electromechani- cal properties of PEHs based on Bayesian updating techniques. The procedure requires the use of: (i) a predictive model, (ii) a prior multivariate probabilis- tic density function for the electromechanical properties, and (iii) experimen- tal measurements of the harvester response. The mode of the updated elec- tromechanical properties is identified adopting a Maximum a Posteriori esti- mate while the probability density function associated is obtained by applying a Laplace’s asymptotic approximation. The procedure is exemplified using the experimental characterization of 20 nominally identical PEHs. Results show the capability of the procedure to update not only the electromechanical proper- ties of each PEH but also the characteristics of the whole sample of harvesters (mandatory information for design purposes).