  Minimising the energy consumption associated with periodic motion is a
priority common to a wide range of technologies and organisms - among them,
many species of flying insect, for which flapping-wing flight is a
life-essential mode of locomotion. In pursuit of this priority, the following
problem often manifests: how to introduce elasticity into an actuated,
oscillating, system in order to minimise actuator power consumption? Here, we
explore this question in a range of general systems, and find some surprising
answers. For instance, it is widely assumed that, if the system dynamics are
linear, then linear resonant elasticity is the only optimal choice. We show, to
the contrary, that there exist nonlinear elasticities with equivalent
optimality, and provide an elegant method for constructing these elasticities
in general systems. This is a new principle of linear and nonlinear dynamics,
fundamentally altering how questions of energetic optimality in a wide range of
dynamical systems must be approached. Furthermore, we show how this principle
enables new forms of optimal system design, including optimal unidirectional
actuation in nonlinear systems; new tools for the design of optimal biomimetic
propulsion systems; and new insights into the role of structural elasticity in
a range of different organisms.
