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We consider the theoretical maximum extractable average power from an energyharvesting device attached to a vibrating table which provides a unidirectionaldisplacement $A\sin(\omega t)$. The total mass of moving components in thedevice is $m$. The device is assembled in a container of dimension $L$,limiting the displacements and deformations of components within. The masses inthe device may be interconnected in arbitrary ways. The maximum extractableaverage power is bounded by $\frac{mLA\omega^3}{\pi}$, for motions in 1, 2, or3 dimensions; with both rectilinear and rotary motions as special cases; andwith either single or multiple degrees of freedom. The limiting displacementprofile of the moving masses for extracting maximum power is discontinuous, andnot physically realizable. But smooth approximations can be nearly as good:with $15$ terms in a Fourier approximation, the upper limit is $99$\% of thetheoretical maximum. Purely sinusoidal solutions are limited to $\frac{\pi}{4}$times the theoretical maximum. For both single-degree-of-freedom linearresonant devices and nonresonant whirling devices where the energy extractionmimics a linear torsional damper, the maximum average power output is$\frac{mLA\omega^3}{4}$. Thirty-six experimental energy harvesting devices inthe literature are found to extract power amounts ranging from $0.0036$\% to$29$\% of the theoretical maximum. Of these thirty-six, twenty achieve lessthan 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 consideredexcellent, and extraction above $\frac{0.3 mLA\omega^3}{\pi}$ may be consideredchallenging.

Performance limit for base-excited energy harvesting, and comparison with experiments