A unified theoretical framework of piezoelectric energy harvesters: Euler–Bernoulli, Timoshenko and Reddy beam models with the high-order electric field assumption

A generalized framework for the electromechanical model of piezoelectric energy harvesters is presented with high-order electric field assumption. This assumption enables the accurate description the non-uniform distribution of the electric field across the thickness of piezoelectric layers. The Euler–Bernoulli, Timoshenko and Reddy’s assumptions for displacements are included in the general form and handled in a uniform manner. In consideration of the axial displacement variable and the electromechanical coupling, the governing equations and boundary conditions are derived through the application of Hamilton's principle. These equations are solved by the differential quadrature method. The outcomes demonstrate that the proposed theory is capable of accurately predicting the outcomes of the finite element analysis. Subsequently, the limitations of different beam models are also discussed. The necessity of the non-uniform electric field assumption becomes more pronounced when the harvester is excited by random vibrations. Compared to the results obtained with the uniform electric field assumption, the use of a non-uniform electric field is capable of correcting 15% of the voltage amplitude.