On the optimal planform of a cantilever unimorph piezoelectric vibrating energy harvester

This work relates to piezoelectric vibrating energy harvesters, that are constructed from a unimorph cantilever with a massive edge block. The dynamic response of the cantilever is considered when it is excited into vibrations at its natural frequency, where its deformation amplitude is maximal. The optimal response of such a harvester is achieved when the amplitude of the axial strain in the piezoelectric layer, is uniform. Practical technological considerations dictate the thickness of the unimorph, but its planform geometry (i.e. the vareation of the width along the cantilever) is a design choice. The optimal planform of such a unimorph cantilever has been the focus of many previous studies, which included extensive simulations and experimental investigations. In these previous studies it was concluded that the optimal planform is a trapeze, where the cantilever tapers from its clamped edge towards the edge block. However, to date, no model with explicit predictive capabilities was proposed. In the present study we derive an analytic expression of the planform, that ensures a uniform axial strain over the top surface of a cantilever unimorph with an edge block. Our analysis provides a rational explanation why a trapeze planform is optimal, and provides an explicit functional form of the optimal geometrical parameters of this planform. The predictive capabilities of our model are validated by comparison to finite element simulations.