The effect of different shapes of cantilever beam in piezoelectric energy harvesters on their electrical output

In this work, for an arbitrary shape (i.e. the width can be described by an arbitrary function) beam, analytical models for vibrations and electrical output are established. And experiments are conducted to verify the models. In fact, for a rectangular beam, it can be found both theoretically and experimentally that the total output charges are independent of the beam width. For an arbitrary shape beam, parameters, which could be sensitive for obtaining a satisfied energy harvesting, could be learned by means of the analytical model in this paper. At the same time, the optimal shapes presented by Ayed et al. (J Intell Mater Syst Struct 25(2):174–186, 2014) and Tabatabaei et al. (Microsyst Technol 22:2435–2446, 2016) have been proved and supported by the analytical model in this paper.