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Minimising the energy consumption associated with periodic motion is apriority common to a wide range of technologies and organisms - among them,many species of flying insect, for which flapping-wing flight is alife-essential mode of locomotion. In pursuit of this priority, the followingproblem often manifests: how to introduce elasticity into an actuated,oscillating, system in order to minimise actuator power consumption? Here, weexplore this question in a range of general systems, and find some surprisinganswers. For instance, it is widely assumed that, if the system dynamics arelinear, then linear resonant elasticity is the only optimal choice. We show, tothe contrary, that there exist nonlinear elasticities with equivalentoptimality, and provide an elegant method for constructing these elasticitiesin general systems. This is a new principle of linear and nonlinear dynamics,fundamentally altering how questions of energetic optimality in a wide range ofdynamical systems must be approached. Furthermore, we show how this principleenables new forms of optimal system design, including optimal unidirectionalactuation in nonlinear systems; new tools for the design of optimal biomimeticpropulsion systems; and new insights into the role of structural elasticity ina range of different organisms.

Conditions for energetically-optimal elasticity and their implications for biomimetic propulsion systems