  Inviscid computational results are presented on a self-propelled virtual body
combined with an airfoil undergoing pitch oscillations about its leading-edge.
The scaling trends of the time-averaged thrust forces are shown to be predicted
accurately by Garrick's theory. However, the scaling of the time-averaged power
for finite amplitude motions is shown to deviate from the theory. Novel
time-averaged power scalings are presented that account for a contribution from
added-mass forces, from the large-amplitude separating shear layer at the
trailing-edge, and from the proximity of the trailing-edge vortex. Scaling laws
for the self-propelled speed, efficiency and cost of transport ($CoT$) are
subsequently derived. Using these scaling relations the self-propelled metrics
can be predicted to within 5% of their full-scale values by using parameters
known a priori. The relations may be used to drastically speed-up the design
phase of bio-inspired propulsion systems by offering a direct link between
design parameters and the expected $CoT$. The scaling relations also offer one
of the first mechanistic rationales for the scaling of the energetics of
self-propelled swimming. Specifically, the cost of transport is shown to scale
predominately with the added mass power. This suggests that the $CoT$ of
organisms or vehicles using unsteady propulsion will scale with their mass as
$CoT \propto m^{-1/3}$, which is indeed shown to be consistent with existing
biological data.
