High order analysis of a nonlinear piezoelectric energy harvesting of a piezo patched cantilever beam under parametric and direct excitations

In this paper, the nonlinear dynamics of a piezoelectric beam under simultaneous parametric and external excitation is studied. The energy sensor considered is composed of a substrate and a piezoelectric layer used as a vibratory sensor located in the middle of the beam. The model thus constituted takes into account the electromechanical nonlinearities. Based on the Galerkin's method, a strongly coupled nonlinear system is obtained. The multiple scales method is used to determine the analytical expressions of the deflection of the beam, the output voltage generated by the piezoelectric patch and the power harvested around the resonant frequency of the structure. The nonlinear characteristics of the energy harvester are explored under parametric and direct excitations. Analytical formulations of the deflection and the output voltage have been provided leading to investigate new insights into the effects of certain parameters such as the electromechanical coupling coefficient, the thickness and width of the elastic beam and the piezoelectric patch. In particular the position of the piezoelectric patch on the performance of the energy harvester is studied. The results show that for large values of the damping coefficient and thickness, the frequency-amplitude response curves of the energy harvester decrease. On the other hand, the frequency response curves of the displacement and voltage increase for large values of the excitation width and amplitude. We also note the fact that the length ratio influences not only the vibration amplitudes but also the output voltage. The deflection amplitudes and the output voltage obtained by the elaborated second-order multiple scale method are higher than those obtained by the first order, these results show the importance of nonlinearities in the dynamic study of vibrational energy harvesting systems by piezoelectric materials.