Dimensional analysis of spring-wing systems reveals performance metrics for resonant flapping-wing flight

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: the
Weis-Fogh number, a measure of the relative influence of inertia and
aerodynamics, and the reduced stiffness. We show that internal damping
scales with Weis-Fogh number, revealing that dynamic efficiency
monotonically decreases with increasing Weis-Fogh number. 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.