An analytical model of the electromechanical coupling for a piezoelectric stepped buckled beam for energy harvesting applications

Abstract

In this article we intend to present an experimentally validated complex mathematical framework of electromechanical coupling for a piezoelectric stepped Bernoulli–Euler beam for energy harvesting applications, valid for both the pre- and postbuckled states. The proposed model, takes into account the predeformed buckled configuration due to the out-of-plane displacements caused by an axial compressive load higher than the critical one. Moreover, several sources of nonlinearities are also considered such as: inertia, quadratic damping, geometrical and constitutive relations for the piezoelectric material. In addition, an axial elastic restraint as, a non-ideal boundary condition, is assumed at one end of the beam which allows treating intermediate cases between perfectly restricted and unrestricted axial displacements commonly found in experimental realization of axially loaded beams. The spatial discretization of the problem is carried out according to Galerkin Method. Based on a single-mode assumption, the Method of Multiple Scales (MMS) is used to obtain an approximated analytic solution for the harvester. Several experimental tests, based both in the static and dynamic response of the beam, were carried out for a sample formed by a MFC-8504 piezoelectric sheet and bearing steel beam evidencing not only the usefulness of the model here developed, but also the benefits associated with the bistable regime for energy harvesting applications.