Drag forces on objects moving on elastically deformable membranes

To develop a resistive force theory (RFT) type model of the dynamics of a wheeled vehicle locomoting on a deformable spandex membrane (discussed in Li et al., PNAS, 2022), we investigate drag forces on a wheel moving at constant speed across the surface. We are interested in how the parallel and perpendicular components of the drag differ (i.e., anisotropy) and depend on the speed as well as the angle of attack, the angle between velocity and orientation of the wheel. We use a robot arm to drag a fixed and a freely rotating wheel (4 cm in diameter) across the membrane at varying speeds (1 to 50 cm/s) and angles of attack (0 to 90 degrees) while monitoring drag force components (perpendicular and parallel to the wheel axis) and their magnitudes. Experiments reveal that the fixed wheel's drag exhibits a sublinear dependence on speed. For speeds greater than 30 cm/s, drag forces fluctuate due to stick-slip-type motion, where the membrane wrinkles and bunches under the wheel. The rotating wheel experiences speed-independent drag for zero angle of attack and rate-dependent drag forces for non-zero angle of attack. The fixed wheel's drag force is linear with the angle of attack in the parallel and perpendicular directions. The drag force on the rotating wheel remains small until a 60-degree angle of attack. between 60 and 80 degrees, the drag force starts to fluctuate due to stick-slip-type motion. For attack angles greater than 80 the parallel drag drops to zero, and the perpendicular drag becomes steady and non-zero.