Hydrodynamics of biological propulsion leveraging parametric stiffness modulation

Abstract

Animals are exceptional swimmers, often exceeding our technological capabilities for moving quickly, efficiently, maneuverably, and stealthily through water. Natural swimmers have many tools to help them swim, including optimized fin shapes, unique surface textures, and active muscle action. However, it is unknown to what extent animals dynamically use their muscles throughout their swimming gait, specifically whether they change the propulsor stiffness on the time scale of their oscillations. We seek to hydrodynamically show that there is enhanced swimming performance in oscillating propulsors if the stiffness were modulated throughout the gait cycle, particularly enhancing thrust and maneuverability. We use small-amplitude inviscid theory and Floquet theory to study the swimming performance and maneuverability of flexible flapping plates with time-periodic flexibility. Our investigation compares plates with constant and time-periodic stiffness across a range of mean plate stiffness values, oscillating stiffness amplitudes, and phase relationships for isolated heaving, isolated pitching, and combined leading edge kinematics. We examine both the natural response of the system and its behavior under various input actuations, with particular attention to stiffness oscillations at twice the kinematic frequency (to maintain symmetric motion) and at the same frequency (to generate net side forces). The Floquet exponents are calculated for a range of time-periodic systems, and their initial coalescence locations are determined using geometric arguments. Our analysis reveals that time-periodic stiffness enhances thrust for specific frequency values related to both the driving frequency and the structural natural frequency, with the phases of the independent frequencies significantly affecting the long-term system dynamics. For maneuvering applications, we explore two primary control strategies: pitch bias and asymmetric oscillating stiffness. Pitch bias inherently provides a nonzero lift coefficient through an average nonzero angle of attack, with behavior following thin airfoil theory superimposed on the oscillating system for rigid plates, though flexibility attenuates the added lift. Asymmetric stiffness oscillations create a "power stroke" effect, generating high positive (negative) lift during one portion of the motion and low negative (positive) lift during the other. We demonstrate that oscillating flexibility can generate equal or higher average lift compared to pitch bias alone, with stiffness oscillations up to 50\%. Ultimately, combining pitch bias with asymmetric stiffness achieves a lift coefficient of C_L = 3.6, exceeding the performance of either isolated approach.