General scaling relations for locomotion in granular media

Inspired by dynamic similarity in fluid systems, we have derived a general dimensionless form for locomotion in granular materials, which is validated in experiments and discrete element method (DEM) simulations. The form instructs how to scale size, mass, and driving parameters in order to relate dynamic behaviors of different locomotors in the same granular media. The scaling can be derived by assuming intrusion forces arise from resistive force theory or equivalently by assuming the granular material behaves as a continuum obeying a frictional yield criterion. The scalings are experimentally confirmed using pairs of wheels of various shapes and sizes under many driving conditions in a common sand bed. We discuss why the two models provide such a robust set of scaling laws even though they neglect a number of the complexities of granular rheology. Motivated by potential extraplanetary applications, the dimensionless form also implies a way to predict wheel performance in one ambient gravity based on tests in a different ambient gravity. We confirm this using DEM simulations, which show that scaling relations are satisfied over an array of driving modes even when gravity differs between scaled tests.

Figure 1

(a) General problem: driving an arbitrarily shaped wheel of width $D$ and rotational velocity $\omega$, carrying a mass $M$ under gravity $g$. Blowup of a surface element shown. (b),(c) The two wheel shapes used in our paper and their corresponding definitions of $L$.

Figure 2

Experimental apparatus. The wheel rotates in the counterclockwise direction. A constant force “spring” and a pulley are used to alter the wheel's effective gravity.

Figure 3

The three different wheel pairs tested: pair A (a), pair B (b), and pair C (c).

Figure 4

Time-averaged nondimensional power and velocity values for pair A (a), pair B (b), and pair C (c). Each color in each plot represents a different mass pair selection and the corresponding four points of each color represent different rotational velocity pairings. The solid line is the model prediction of $\frac{\langle {\tilde{P}}_b\rangle }{\langle {\tilde{P}}_s\rangle }=\frac{\langle {\tilde{V}}_b\rangle }{\langle {\tilde{V}}_s\rangle }=1$.

Figure 5

Nondimensional power trajectories for two pairs of lug-wheel experiments over a single cycle (${90}^{\circ }$ of rotation). One pair is shown in solid lines, while the other is shown in dotted lines.

Figure 6

Walking mode ($\tilde{\rho }=44.4$, $\tilde{g}=14.8$). Small test inputs: $g_s=9.8\frac{\text{m}}{{\text{s}}^2}$, $L_s=0.0168\text{m}$, $M_s=12.7\times {10}^{-6}\text{kg}$, ${\omega }_s=6.28\frac{\text{rad}}{\text{s}}$. Big test inputs: $g_b=39.2\frac{\text{m}}{{\text{s}}^2}$, $L_b=0.0224\text{m}$, $M_b=22.4\times {10}^{-6}\text{kg}$, ${\omega }_b=10.9\frac{\text{rad}}{\text{s}}$. Both tests use the same grains. (a) Snapshot of the small and big systems at a common value of $\tilde{t}$. (b),(c) Time dependence of power and velocity in dimensionless form during locomotion process. Blue lines are for the big test and red lines are for the small test.

Figure 7

Trudging mode ($\tilde{\rho }=4.44$, $\tilde{g}=14.8$). Small test inputs: $g_s=9.8\frac{\text{m}}{{\text{s}}^2}$, $L_s=0.0168\text{m}$, $M_s=127\times {10}^{-6}\text{kg}$, ${\omega }_s=6.28\frac{\text{rad}}{\text{s}}$. Big test inputs: $g_b=39.2\frac{\text{m}}{{\text{s}}^2}$, $L_b=0.0224\text{m}$, $M_b=224\times {10}^{-6}\text{kg}$, ${\omega }_b=10.9\frac{\text{rad}}{\text{s}}$. Both tests use the same grains. (a) Snapshot of the small and big systems at a common value of $\tilde{t}$. (b),(c) Blue lines are for the big test and red lines are for the small test.

Figure 8

Skipping mode ($\tilde{\rho }=13.4$, $\tilde{g}=0.148$). Small test inputs: $g_s=9.8\frac{\text{m}}{{\text{s}}^2}$, $L_s=0.0168\text{m}$, $M_s=42.2\times {10}^{-6}\text{kg}$, ${\omega }_s=62.8\frac{\text{rad}}{\text{s}}$. Big test inputs: $g_b=39.2\frac{\text{m}}{{\text{s}}^2}$, $L_b=0.0224\text{m}$, $M_b=75.0\times {10}^{-6}\text{kg}$, ${\omega }_b=109\frac{\text{rad}}{\text{s}}$. Both tests use the same grains. (a) Snapshot of the small and big systems at a common value of $\tilde{t}$. (b),(c) Blue lines are for the big test and red lines are for the small test.