Dynamics and locomotion of flexible foils in a frictional environment

Flexible foils have long been used as a generic model for the bodies and appendages of organisms (e.g. fish and snakes) in locomotion. To understand the range of possible dynamics--including efficient locomotion--in a terrestrial environment or granular medium, we have used analysis and computations to study flexible foils moving under frictional forces. When a flexible foil is oscillated by heaving at one end but is not free to locomote, the dynamics change from periodic to non-periodic and chaotic as the heaving amplitude increases or the bending rigidity decreases. Resonant peaks and bistable states are observed. When the foil is free to locomote, the horizontal motion smoothes the resonant peaks. Locomotion is steady but slow at moderate frictional coefficients, and faster at larger transverse friction and small tangential friction corresponding to wheeled snake robots. Here traveling wave motions arise spontaneously, and move with horizontal speeds that scale as transverse friction coefficient to the power 1/4 and input power that scales as the transverse friction coefficient to the power 5/12. These scalings correspond to boundary layers near the foil’s leading edge.