Abstract
We consider the theoretical maximum extractable average power from an energy harvesting device attached to a vibrating table which provides a unidirectional displacement \(A\sin (\omega t)\). The total mass of moving components in the device is m. The device is assembled in a container of dimension L. The masses in the device may be interconnected in arbitrary ways. The maximum extractable average power is bounded by \(\frac{mLA\omega ^3}{\pi }\), for motions in 1, 2, or 3 dimensions; with both rectilinear and rotary motions as special cases; and with either single or multiple degrees of freedom. The limiting displacement profile of the moving masses for extracting maximum power is discontinuous, and not physically realizable. But smooth approximations can be nearly as good: With 15 terms in a Fourier approximation, the upper limit is 99% of the theoretical maximum. For both single-degree-of-freedom linear resonant devices and nonresonant devices where the energy extraction mimics a linear torsional damper, the maximum average power output is \(\frac{mLA\omega ^3}{4}\). Thirty-six experimental energy harvesting devices in the literature are found to extract power amounts ranging from 0.0036 to 29% of the theoretical maximum. Of these thirty-six, twenty achieve less than 2% and three achieve more than 20%. We suggest, as a figure of merit, that energy extraction above \(\frac{0.2 mLA\omega ^3}{\pi }\) may be considered excellent, and extraction above \(\frac{0.3 mLA\omega ^3}{\pi }\) may be considered challenging.
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Notes
An example may help. Suppose we have a long vertical vibrating pipe in the ocean, and on this pipe we attach an energy harvesting device for some sensing application. The harvesting device changes the local hydrodynamics and may slightly change the local fluid forces experienced by the pipe, but those forces are external and not included in our study of the efficiency of the energy harvester.
Other arguments are possible. We could note that \(|y_1|\) is bounded by L/2 (with nondimensionalization, \(L=1\)), and offers an infinite sequence of turning points; and we could choose a sequence of T values to coincide with those turning points.
Technically, it is a supremum and not a maximum; the term “maximum” is used informally. In practical terms, even 30% of this value will be seen to be excellent.
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Acknowledgements
AC thanks Marcelo Savi, Atanu Mohanty and Sumit Basu for discussions. CPV thanks Thomas Uchida and Ajinkya Desai for helpful comments on earlier drafts.
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Tiwari, S., Vyasarayani, C.P. & Chatterjee, A. Performance limit for base-excited energy harvesting, and comparison with experiments. Nonlinear Dyn 103, 197–214 (2021). https://doi.org/10.1007/s11071-020-06145-w
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DOI: https://doi.org/10.1007/s11071-020-06145-w