Parametric instabilities for vibratory energy harvesting under harmonic, time-varying frequency, and random excitations

This effort investigates and evaluates the prospect of using parametric instabilities for vibratory energy harvesting. To that end, we consider a parametrically-excited piezoelectric cantilever beam and study its performance as an energy harvester under (i) fixed-frequency harmonic excitations, (ii) time-varying frequency excitations, and (iii) band-limited Gaussian noise. In the case of fixed-frequency excitations, we use the Method of Multiple Scales to obtain approximate analytical expressions for the steady-state response amplitude and instantaneous output power in the vicinity of the first principle parametric resonance. We show that the electromechanical coupling and load resistance play an important role in determining the output power and characterizing the bandwidth of the harvester. Specifically, we demonstrate that the region of parametric instability wherein energy can be harvested shrinks as the coupling coefficient increases, and that there exists a coupling coefficient beyond which the peak power decreases. We also show that there is a critical excitation level below which no energy can be harvested. The magnitude of this critical excitation increases with the coupling coefficient and is maximized for a given electric load resistance. Theoretical findings were compared to experimental data showing good agreement and reflecting the general physical trends.;In the case of time-varying frequency excitations, we consider two beams of different nonlinear behaviors: one exhibiting a softening response while the other exhibiting hardening characteristics. We show that, for both beams, the bandwidth of the harvester decreases with increasing frequency sweep rate and that the instantaneous peak power during a sweep cycle decreases and shifts in the direction of the sweep. Furthermore, experimental findings illustrate that the average output power of the harvester is significantly higher when the sweep is in the direction in which the steady-state principle parametric resonance curves of the beams bend. Also, as the frequency sweep rate increases, the average output power decreases until beyond a threshold sweep rate where no power can be harvested.;Based on the preceding conclusions, we introduce the new concept of a Softening-Hardening Hysteretic Harvester (SHHH), which is designed to scavenge energy efficiently from an excitation source whose frequency varies with time around a center frequency. Introductory experimental investigation on the SHHH illustrated that this concept produces more power than either a softening or a hardening beam alone.;Finally, in an effort to duplicate real-world scenarios under which energy harvesting occurs, both the hardening and the softening beam were subjected to parametric, band-limited, random Gaussian excitations and their performance in scavenging energy under different excitation bandwidths was evaluated. We observed that, under narrow bandwidth excitations (on the order of the harvester's steady-state bandwidth) and regardless of the beam's nonlinear characteristics, the parametric instability was activated for the length of the experiment. However, the average output power was very low (on the order of micro-Watts under excitations having a variance of 1.5 g). The power decreased even further as the bandwidth of the excitation was increased.