Piezoelectric Energy Harvesting from a Pitch-Plunge Aerofoil in Compressible Flow, the Euler Full-Order Model, Strip Theory and the Reduced Models Compared

Abstract

A piezoelectric transducer embedded in a pitch-plunge aerofoil with a finite-mass trailingedge flap converts the limit-cycle oscillation that follows flutter into electrical power. The design findings for such a harvester have so far been obtained with incompressible strip theory, whose aerodynamic state is a handful of lag variables. Here the same section, the same transducer and the same test cases are coupled instead to the two-dimensional compressible Euler equations, giving a full-order model of twelve thousand states, and the two aerodynamic models are placed side by side on the flutter boundary, on the limit cycle and on the harvested power, with the strip-theory results compared against higher fidelity aerodynamic modelling provided by CFD. The degree of freedom carrying the transducer remains the first-order design variable, and the ranking of the mountings is unchanged and wider, while strip theory is found to under-predict the harvested power of the compressible section and to overstate the effect of the transducer on stability where the electrical time constant meets the flutter frequency. A nonlinear reduced model of four states, built on eigenvectors of the coupled Jacobian at the velocity of interest, reproduces the frequency of the cycle, its pitch amplitude, the transducer voltage and the mean power, and misstates the proportion of plunge to pitch in the orbit and its phase. That error is shown to be independent of the amplitude of the cycle and to survive enlargement of the basis, which places it in the retained subspace rather than in the order of the expansion, and suggests that such models should be judged on the shape of the orbit rather than on a single amplitude.