Fcn2.m from Dynamics and locomotion of flexible foils in a frictional environment.

Over the past few decades, oscillating flexible foils have been used to study the physics of organismal propulsion in different fluid environments. Here, we extend this work to a study of flexible foils in a frictional environment. When the 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. For friction coefficients lying in a certain range, the transition passes through a sequence of N-periodic and asymmetric states before reaching chaotic dynamics. Resonant peaks are damped and shifted by friction and large heaving amplitudes, leading to bistable states. When the foil is free to locomote, the horizontal motion smoothes the resonant behaviours. For moderate frictional coefficients, steady but slow locomotion is obtained. For large transverse friction and small tangential friction corresponding to wheeled snake robots, faster locomotion is obtained. Travelling wave motions arise spontaneously, and move with horizontal speeds that scale as transverse friction coefficient to the coefficient to and input power that scales as the transverse friction coefficient to the power 5/12. These scalings are consistent with a boundary layer form of the solutions near the foil's leading edge.

近数十年来，振荡柔性翼片（oscillating flexible foils）已被用于探究不同流体环境（fluid environments）下生物推进（organismal propulsion）的物理机制。本研究将此类工作拓展至摩擦环境下的柔性翼片研究。当翼片仅单端受垂荡（heaving）驱动且无法自主运动时，随着垂荡幅值增大或弯曲刚度（bending rigidity）降低，系统动力学将从周期态转变为非周期混沌态。当摩擦系数（friction coefficients）处于特定区间时，该动力学转变过程在进入混沌态前，会依次经历一系列N周期非对称状态。共振峰（resonant peaks）会因摩擦与大垂荡幅值而被抑制并偏移，进而产生双稳态（bistable states）。当翼片可自主运动时，水平运动将平滑共振行为。当摩擦系数适中时，系统可实现稳定但缓慢的推进运动。当对应轮式蛇形机器人（wheeled snake robots）的横向摩擦（transverse friction）系数较大、切向摩擦（tangential friction）系数较小时，系统可实现更快的推进运动。行波运动自发产生，其水平运动速度与横向摩擦系数的某一幂次成正比，而输入功率则与横向摩擦系数的5/12次方成正比。上述标度律与翼片前缘附近解的边界层形式相符。