Geometrical nonlinearities and shape effects in electromechanical models of piezoelectric bridge structures

Fredrik Ohlsson,Cristina Rusu,Pontus Johannisson

doi:10.1007/s40095-021-00395-z

Abstract

We consider nonlinear shape effects appearing in the lumped electromechanical model of a bimorph piezoelectric bridge structure due to the interaction between the electromechanical constitutive model and the geometry of the structure. At finite proof-mass displacement and electrode voltage, the shape of the beams is no longer given by Euler-Bernoulli theory which implies that shape effects enter in both the electrical and mechanical domains and in the coupling between them. Accounting for such effects is important for the accurate modelling of, e.g., piezoelectrical energy harvesters and actuators in the regime of large deflections and voltages. We present a general method, based on a variational approach minimizing the Gibbs enthalpy of the system, for computing corrections to the nominal shape function and the associated corrections to the lumped model. The lowest order correction is derived explicitly and is shown to produce significant improvements in model accuracy, both in terms of the Gibbs enthalpy and the shape function itself, over a large range of displacements and voltages. Furthermore, we validate the theoretical model using large deflection finite element simulations of the bridge structure and conclude that the lowest order correction substantially improve the model, obtaining a level of accuracy expected to be sufficient for most applications. Finally, we derive the equations of motion for the lowest order corrected model and show how the coupling between the electromechanical properties and the geometry of the bridge structure introduces nonlinear interaction terms.

Highlights

The application of nonlinear mechanical structures in inertial energy harvesting using piezoelectric transduction is a growing field of research motivated by the potential advantages over linear resonator systems in terms of bandwidth, durability and amplitude [1,2,3,4,5]
We extend the analysis to a piezoelectric bridge structure by including electromechanical effects in the description of the system
Vem = ∫ d w and we have introduced the constant dA e31 = 2We31, h accounting for the geometry and electromechanical properties of the structure

Summary

Introduction

The application of nonlinear mechanical structures in inertial energy harvesting using piezoelectric transduction is a growing field of research motivated by the potential advantages over linear resonator systems in terms of bandwidth, durability and amplitude [1,2,3,4,5]. The purpose of computing corrections to the shape function is to obtain an improved estimate compared to the lowest order N = 3 function, which we have seen describes the pure bending situation at infinitesimal displacements and applied voltages and is not expected to properly account for large deformations of the doubly clamped structure or large electrical fields. It is straightforward to verify numerically that for the range of and considered here, the discriminant of the cubic Eq (44) is negative, implying that the existence of a single real root which guarantees the uniqueness of the numerical solution While it is clear from the results presented in the previous section that the N = 5 model provides a (substantial) improvement compared to the N = 3 model, it remains to validate the variational approach in comparison with the physical systems it is meant to describe, and to quantify the improvement obtained by including the N = 5 model corrections in this context. For a purely resistive load, the only contribution to the kinetic energy is mechanical

Rq 2 2

Findings

Conclusion