Parametrically amplified Mathieu-Duffing nonlinear energy harvesters

Abstract

The steady-state response of a nonlinear piezoelectric energy harvester subjected to external and parametric excitation is investigated based on the Mathieu-Duffing nonlinear oscillator model. The parametric excitation is introduced to amplify the external harmonic excitation and extend the capabilities of the nonlinear piezoelectric energy harvester device. To obtain the approximated solution of the nonlinear periodic responses for displacement and electrical voltage of the energy harvester, the incremental harmonic balance method in combination with the path-following technique is adopted. It is assumed that the proposed nonlinear model consists of cubic and quadratic nonlinearity, where parametric amplification appears in the form of a trigonometric function. The frequency is tuned as one-to-one and the one-to-two ratio between external and parametric excitation. The effects of quadratic and cubic nonlinearity as well as parametric amplification are studied in detail, and their incredible properties to extend harvester application performance is illustrated. It is explicitly demonstrated that for some particular combination of the system parameters, vibration amplitudes and harvested power can be amplified up to three or five times in comparison to the classical broadband nonlinear energy harvester based on the forced Duffing oscillator. This extraordinary amplification shown to be a key motivation to realize the proposed concept in practice. The presence of combined quadratic and cubic nonlinearities resulted in both hardening and softening spring behavior and leading to the appearance of coexisting periodic solutions in the amplitude-frequency responses. Periodic orbits obtained by the proposed methodology are verified with the results from direct numerical integration and fine agreement is demonstrated. Moreover, a significant influence of the parametric amplification on the instantaneous power is revealed in time response diagrams, thus showing better performance of the proposed energy harvester system.