  Flapping-wing insects, birds, and robots are thought to offset the high power
cost of oscillatory wing motion by using elastic elements for energy storage
and return. Insects possess highly resilient elastic regions in their flight
anatomy that may enable high dynamic efficiency. However, recent experiments
highlight losses due to damping in the insect thorax that could reduce the
benefit of those elastic elements. We performed experiments on, and simulations
of a dynamically-scaled robophysical flapping model with an elastic element and
biologically-relevant structural damping to elucidate the roles of body
mechanics, aerodynamics, and actuation in spring-wing energetics. We measured
oscillatory flapping wing dynamics and energetics subject to a range of
actuation parameters, system inertia, and spring elasticity. To generalize
these results, we derive the non-dimensional spring-wing equation of motion and
present variables that describe the resonance properties of flapping systems:
$N$, a measure of the relative influence of inertia and aerodynamics, and
$\hat{K}$, the reduced stiffness. We show that internal damping scales with
$N$, revealing that dynamic efficiency monotonically decreases with increasing
$N$. Based on these results, we introduce a general framework for understanding
the roles of internal damping, aerodynamic and inertial forces, and elastic
structures within all spring-wing systems.
