Drastic bandwidth enhancement of bistable energy harvesters: Study of subharmonic behaviors and their stability robustness

Abstract

In order to provide a serious alternative to chemical batteries for the energy supply of isolated sensors, bistable generators have been enthusiastically highlighted in recent years for their ability to harvest vibration energy on a wider frequency range compared to linear generators. Nevertheless, these bistable harvesters are generally characterized through a frequency sweep which does not reveal all the steady-state behaviors they can reach and therefore their full energy harvesting potential. Among such behaviors, subharmonic motions are hidden by this classical characterization and therefore had not received a lot of attention. This study proposes an original complete analytical analysis of subharmonic orbits for energy harvesting to predict their contribution to the global bandwidth of bistable generators. In addition, a new criterion, referred as stability robustness, is introduced to estimate the sensitivity of those behaviors to disturbances of different levels, allowing to finely and accurately estimate suitable behaviors for energy harvesting purposes in realistic conditions (behaviors easy to reach and maintain in time). Experimental results conducted with a buckled beams based electromagnetic generator confirm the pertinence of this criterion showing good agreement with the analytical predictions. Subharmonic behaviors finally appears, both theoretically and experimentally, to be of significant interest, as exploiting them leads to a 180% increase of the global operating frequency range of the considered bistable energy harvester, for which more than 100 µW are generated on a 70 Hz bandwidth at 0.5 g.

Orbit jump in bistable energy harvesters through buckling level modification

“…Several different behaviors can coexist on the same frequency range, solely determined by its initial conditions. Some of those behaviors are interesting for energy harvesting, namely the common harmonic 1 high orbit (for which the mass moves from one stable position to another at the same frequency as the excitation) [8,9] and the subharmonic 3 high orbit (for which the mass moves from one stable position to another at a frequency 3 times lower than the excitation), as reported by the authors in previous studies [10,11]. However, some other behaviors, namely the low orbits (for which the mass oscillate around one stable position), are not interesting for energy harvesting as they induce low amplitude displacements and thus low energy levels.…”

“…On the opposite, subharmonic 3 high orbit is more robust and its frequency range is less affected. Subharmonic 3 orbit then appears to be relevant for energy harvesting purposes as it greatly enhances the frequency range on which the bistable harvester proposes a high orbit, as reported in [11]. Figure 3 shows promising theoretical results for the bistable harvester which has a continuous frequency band of 63 Hz (from 15 Hz to 78 Hz) for which a robust high orbit exists making it ideal for real condition wideband vibration energy harvesting as it can be adapted to non-constant vibrations.…”

“…It can be noted that the stability robustness of even subharmonic high orbits is low compared to odd subharmonic high orbits which confirms the difficulty of observing them experimentally. The stability robustness of harmonic 1 and odd subharmonic high orbits decreases as they approach their theoretical cutting frequency happening when the phase shown in Figure 3(b) reaches π/2 [11,25]. It is interesting to note that the frequency Figure 3: Bistable harvester spectra analytically obtained for a sinusoidal acceleration of magnitude 2.5 m/s 2 : (a) amplitude of stable orbits (b) phase of the excitation displacement when the position of the mass reaches a maximum for stable high orbits (c) stability robustness for stable high orbits (d) average harvested power for stable orbits.…”

“…Those orbits, defined on restricted frequency ranges, are more interesting for energy harvesting as they are more suitable in realistic conditions. To estimate this sensitivity to external disturbances, the authors recently introduced an analytic criterion called the stability robustness [11] that will be used through this study.…”

“…For low stability robustness, the stable high orbits will no longer be considered as interesting for energy harvesting because too sensitive to disturbances for being maintainable over time. Figure 3 shows the bistable harvester analytical frequency responses obtained with the parameters listed in Table 1 and following the procedure detailed in [11]. Four different spectra are presented: the amplitude of stable orbits, the phase of the excitation displacement when the position of the mass reaches a maximum for stable high orbits, the stability robustness for stable high orbits, and the mean harvested power for stable orbits.…”