Lunar Rover Wheel on Loose Soil

What is modelled

Loose soil is not a floor. The wheel sinks until the pressure under it, which grows with depth, carries the load. Pushing the soil down costs energy (compaction resistance). To move forward the wheel must also shear the soil, which gives thrust that rises with slip and saturates at the soil strength. What is left after the compaction resistance is the drawbar pull the rover can use to climb or tow.

Equations

• Pressure under the wheel p = (kc/b + kφ) zn (Bekker), integrated over the arc to balance the load.
• Compaction resistance Rc = b (kc/b + kφ) z0n+1 / (n+1).
• Shear τ = (c + σ tanφ)(1 − e−j/K), thrust H = (A c + W tanφ)[1 − (K/iL)(1 − e−iL/K)] (Janosi and Shirley).
• Drawbar pull DP = H − Rc.

Checks run in the page

• The vertical pressure integral against the wheel load.
• Compaction resistance by integration against the textbook form.
• The shear stress integral against the closed form.
• Zero thrust at zero slip and the Coulomb limit at very high slip.

Honest simplifications

The wheel is rigid with a uniform load, the soil is homogeneous with a fixed sinkage exponent of 1, bulldozing, grouser shape, the rear wheels running in the front wheel ruts and slope are ignored, and the stress under the wheel is assumed uniform for the shear integral. The lunar soil constants are typical published values, not a measurement of any site.

• Press Same rover on Earth: the load grows six times and the wheel sinks much deeper.
• Press Wide wheels and compare the sinkage with the narrow ones.
• Move the slip slider and find the slip where the drawbar pull is largest.