  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 $L$,
limiting the displacements and deformations of components within. 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. Purely sinusoidal solutions are limited to $\frac{\pi}{4}$
times the theoretical maximum. For both single-degree-of-freedom linear
resonant devices and nonresonant whirling 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.
