A frequency response notion comparable to the classical Bode gain and phase response for linear time invariant (LTI) systems has not been developed for linear time periodic (LTP) systems. In this paper, fundamental input and output signal spaces are identified that lead to a one-to-one map and a linear operator (transfer function). The LTP frequency response, including a characterization of gain, phase and their directional properties, is then presented in terms familiar to the multivariable LTI control theory.

High‐strain, high‐force mechanical actuation technologies are desirable for numerous applications ranging from microelectromechanical systems (MEMS) to large‐scale “smart structures” that are able to change shape to optimize performance. Here we show that electrochemical intercalation of inorganic compounds of high elastic modulus offers a low‐voltage mechanism (less than 5 V) with intrinsic energy density approaching that of hydraulics and more than a hundred times greater than that of existing field‐operated mechanisms, such as piezostriction and magnetostriction. Exploitation of the reversible crystallographic strains (several percent) of intercalation compounds while under high stress is key to realization of the available energy. Using a micromachined actuator design, we test the strain capability of oriented graphite due to electrochemical lithiation under stresses up to 200 MPa. We further demonstrate that simultaneous electrochemical expansion of the LiCoO2/graphite cathode/anode couple can be exploited for actuation under stresses up to ∼ 20 MPa in laminated macroscopic composite actuators of similar design to current lithium‐ion batteries. While the transport‐limited actuation mechanism of these devices results in intrinsically slower actuation compared to most ferroic materials, we demonstrate up to 6.7 mHz (150 s) cyclic actuation in a laminated actuator designed for a high charge/discharge rate. The potential for a new class of high‐strain, high‐force, moderate‐frequency actuators suitable for a broad range of applications is suggested.

The Betz criterion for minimum induced loss is used to compute the optimal circulation distribution along the span of flapping wings in fast forward flight. In particular, we consider the case where flapping motion is used to generate both lift (weight support) and thrust. The Betz criterion is used to develop two different numerical models of flapping. In the first model, which applies to small-amplitude harmonic flapping motions, the optimality condition is reduced to a one-dimensional integral equation which we solve numerically. In the second model, which applies to large-amplitude periodic flapping motions, the optimal circulation problem is reduced to solving for the flow over an infinitely long wavy sheet translating through an inviscid fluid at rest at infinity. This three-dimensional flow problem is solved using a vortex-lattice technique. Both methods predict that the induced power required to produce thrust decreases with increasing flapping amplitude and frequency. Using the large-amplitude theory, we find that the induced power required to produce lift increases with flapping amplitude and frequency. Therefore, an optimum flapping amplitude exists when the flapping motion of wings must simultaneously produce lift and thrust.