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Over the past few decades, oscillating flexible foils have been used to studythe physics of organismal propulsion in different fluid environments. Here weextend this work to a study of flexible foils in a frictional environment. Whenthe foil is oscillated by heaving at one end but not allowed to locomotefreely, the dynamics change from periodic to non-periodic and chaotic as theheaving amplitude is increased or the bending rigidity is decreased. Forfriction coefficients lying in a certain range, the transition passes through asequence of $N$-periodic and asymmetric states before reaching chaoticdynamics. Resonant peaks are damped and shifted by friction and large heavingamplitudes, leading to bistable states.  When the foil is allowed to locomote freely, the horizontal motion smoothesthe resonant behaviors. For moderate frictional coefficients, steady but slowlocomotion is obtained. For large transverse friction and small tangentialfriction corresponding to wheeled snake robots, faster locomotion is obtained.Traveling wave motions arise spontaneously, and and move with horizontal speedthat scales as transverse friction to the 1/4 power and input power that scalesas transverse friction to the 5/12 power. These scalings are consistent with aboundary layer form of the solutions near the foil's leading edge.

Dynamics and locomotion of flexible foils in a frictional environment