A piezoelectric beam model with geometric, material and damping nonlinearities for energy harvesting

Abstract

To predict electrical generation in piezoelectric small-scale beam energy harvesting devices, it is important to have a complete mathematical model that captures the different associated phenomena. In the literature, some authors propose several alternatives of non-linear mathematical formulations, with non-linearities coming from different physical aspects. All these formulations present good aptitudes to predict the nonlinear behavior of the system under different values of accelerations, geometry and boundary conditions. At the same time, they do not represent a unified general proposal for modeling multimodal energy harvesting devices of any type of mode generation and boundary conditions at large excitations. In this sense, this paper presents a mathematical description of inextensional nonlinear Euler-Bernoulli piezoelectric beams that combines the best contributions of the literature to the voltage generation of multimodal nonlinear piezoelectric energy harvesters (geometric, material and damping non-linearities). The developed analytical model yields a total set of N+ 1 ordinary differential equations for the first N modes and for the output voltage. However, direct solution of this ordinary nonlinear differential system of N equations is computationally costly. Instead, a reduced algebraic system of 2 algebraic equations is proposed applying the method of averaging. Its main advantage is that it makes more suitable and computationally economical for the implementation of a parameter identification process involving any number of piezoelectric inserts (unimorph or bimorph) and mode of generation (d33 or d31). Two types of validations are presented for some selected physical systems to test the validity of the assumptions: a numerical one, by the direct integration of the equations of motion and an experimental one. A final comparison between the results demonstrates the importance of the having a unified nonlinear model to predict the generated voltage in multimodal energy harvesters.