A unified nonlinear model of piezoelectric cantilever beams with complex geometries for energy harvesting applications

Abstract It is generally known that piezoelectric material is inherently nonlinear. This nonlinearity is more apparent at high vibration amplitudes, which are generally present in a number of energy harvesting applications such as high-power machine tools, subway vehicles, aircraft engines and rotors. The effect of nonlinearity in piezoelectric energy harvesters (PEHs) implemented for these applications needs to be assessed and well understood. The purpose of this article is to model the material nonlinearities for a piezoelectric cantilever consisting of complex geometries. The geometries modeled in this work include morphological configurations (unimorph and bimorph), tapered beam profiles, partial piezoelectric layer coverage and tip mass. Within the existing literature, several geometry-specific nonlinear models are provided for the piezoelectric cantilever which makes the study of complex designs difficult. In this work, a generalized method is developed by incorporating the transfer matrix method. The nonlinear governing equations of the piezoelectric cantilever are generalized to include higher-order nonlinear terms until n th order. The governing equations are derived by utilizing the nonlinear piezoelectric constitutive equations in the extended Hamilton’s principle and a numerical Ordinary Differential Equation solver is used to obtain the PEH output. The nonlinear model is validated using finite element model, literature and experiments conducted on a bimorph PEH sample. It is found that nonlinear response of PEH is attributed to material nonlinearity and the contribution of higher-order nonlinear terms is significant.