扇贝爬行：基于姿态的单自由度运动
Crawling scallop: Friction-based locomotion with one degree of freedom
摘要：
基于流体的运动在低雷诺数的扇贝定理的约束，这表明，身体运动学相同的时间反演对称（特别是，那些与一个单一的自由度）不能显示运动的平均值。该定理的含义自然迫使人们问，在不同的环境中，运动是否存在类似的对称性约束。在这项工作中，我们考虑运动沿一个表面的力量所描述的各向同性库仑摩擦。为了解决是否与一个单一的自由度的运动可以导致运输，我们分析了一个模型系统组成的两个机构的间隔距离经历周期性的时间变化。两体系统的行为完全由它们分离的运动学规范、摩擦力和每个物体的质量决定。我们表明，扇贝定理的约束可以逃脱在摩擦介质中，如果同时满足两个不对称的条件：每个机构对表面的摩擦力必须是不同的，身体的身体分离的时间变化必须在时间上不对称地变化（所以快-慢或慢-快的伸展-收缩阶段）。我们的结果证明数值和解释使用渐近展开。
英文摘要：
Fluid-based locomotion at low Reynolds number is subject to the constraints of the scallop theorem, which dictate that body kinematics identical under a time-reversal symmetry (in particular, those with a single degree of freedom) cannot display locomotion on average. The implications of the theorem naturally compel one to ask whether similar symmetry constraints exist for locomotion in different environments. In this work we consider locomotion along a surface where forces are described by isotropic Coulomb friction. To address whether motions with a single degree of freedom can lead to transport, we analyze a model system consisting of two bodies whose separation distance undergoes periodic time variations. The behavior of the two-body system is entirely determined by the kinematic specification of their separation, the friction forces, and the mass of each body. We show that the constraints of the scallop theorem can be escaped in frictional media if two asymmetry conditions are met at the same time: the frictional forces of each body against the surface must be distinct and the time-variation of the body-body separation must vary asymmetrically in time (so quick-slow or slow-quick in the extension-contraction phases). Our results are demonstrated numerically and interpreted using asymptotic expansions.
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