Effect of elastic restraints in the modeling of prestressed piezoelectric energy harvesters

Abstract

Prestressed piezoelectric energy harvesters (PEH) provide benefits referred to higher generation levels and scavenge energy from lower frequency sources. In the study of axially loaded beams subject to lateral base forces, it is common to assume that the axial displacement is of a lower order than the lateral one. When the beam is slender enough, it is well known from experimental and numerical results, that this hypothesis is satisfied. Although this relation arises naturally from reducing the axial equation, diverse approaches can be found by applying different boundary conditions for the axial problem. Among several combinations, two ideal extreme cases can be distinguished: assuming that the beam is fixed at one end and (a) loaded at the other one; or (b) considering the prescribed displacement produced by the same load, according to the linear static problem. Even though both approaches seem similar, the final expressions of the nonlinear geometric stiffness are drastically modified, which is reflected in the disagreement with the predicted hardening behavior as the forces increase. In order to identify the effect of these assumptions in the dynamic response, the equations of motion are derived and solved for both cases. On the other hand, as an intermediate and non-ideal case (c), an elastic restraint on the end of the beam where the load is applied is proposed. For this last one, both assumptions (a) and (b) can be recovered by choosing the appropriate combination of the elastic restraint k and a proposed dimensionless parameter. The experimental findings are compared with numerical predictions on the voltage of different PEH samples that show the need for the elastic constraint to reproduce the experimental data from our proposed device.