Parametric amplification in the context of vibratory energy harvesting

In this manuscript, we propose a technique to harvest energy from excitation sources that possess two frequency components: a fundamental component with large energy content, and a super-harmonic component with smaller energy content at twice the fundamental component. Excitations of this nature are common in the environment due to inherent nonlinearities in the dynamics of the excitation source. Normally, two separate energy harvesters are needed to extract the energy at each frequency; however, this paper discusses a single cantilevered piezoelectric vibratory energy harvester (VEH) that exploits the parametric amplification phenomenon to scavenge energy from both frequencies by varying the tilt angle between the axis of the harvester and the direction of the excitation. To investigate the efficacy of the proposed concept, the equations governing the electromechanical dynamics of the harvester are derived. The resulting partial differential equations and associated boundary conditions are then reduced to a single-mode Galerkin based reduced-order model. Analytical expressions for the steady-state output power across a purely resistive load are obtained using the method of multiple scales. Results indicate that percentage improvement in the output power depends on the excitation's parameters, the tilt angle, and the mechanical damping ratio. It is observed that there is an optimal tilt angle at which the flow of energy from the environment to the electric load is maximized. Furthermore, when the mechanical damping ratio is small, significant enhancement in the output power is attainable even when the magnitude of the super-harmonic is very small when compared to the fundamental component. Such findings reveal that, under certain conditions, parametric amplification can be utilized to enhance the output power of a VEH especially for micro-scale applications where the damping ratio can be easily controlled. Experimental results are presented to validate the theoretical concepts.