General Scaling Relations in Granular Locomotion

We have identified a family of geometrically-general scaling relations for granular locomotion. They instruct how to conduct an experiment with a reference locomotor and predict the behavior of another locomotor of the same shape but with different size, mass, rotation speed, and gravity. The scalings derive from reduced-order granular continuum equations and are confirmed with numerous experiments and discrete particle tests. We see this effort as analogous to scaling laws in fluid dynamics, which enable, for example, evaluation of a large wing using carefully down-scaled tests in a wind tunnel. Here, the goal might be to predict the preformance of an off-road truck using a toy truck in a sandbox. Importantly and conveniently, our proposed scaling laws allow the same granular media to be used in relating a scaled pair of experiments; no new sand must be manufactured to conduct a scaled test.

Above: Two discrete particle simulations demonstrating the proposed locomotive scaling relation. The wheels are bar-shaped and both drive in the same grains. The scaled-up wheel is twice the size of the reference, has four times the mass, and spins sqrt(2) slower, but, according to our theory, is scale-equivalent to the reference test. Qualitatively, one can see the scaling in action by observing the scaled-up wheel drives like the reference; they both "skip" through the grains, kicking up material behind them as they push forward. Quantitatively, it can be shown that outputs of the scaled-up test such as mean driving speed and power consumption are accurately predicted by scaling the outputs of the reference test. Confirmation of the scaling relation in numerous cases and under various locomotive modes is shown in Slonaker et al PRE 2017.