Library / Terramechanics and off-road mobility / TER-40
Terramechanics and off-road mobility
Wheel entry angle from geometric sinkage
Angle from the downward vertical to the point where a rigid wheel of radius r first meets the soil, from the geometric sinkage z measured from the undisturbed surface.
- Ref
- TER-40
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- θf [rad]
A rigid wheel of radius r sunk a depth z below the undisturbed surface meets the soil where its rim crosses the surface. The vertical distance from the centre to that point is r − z = r cos θf, so the entry angle from the downward vertical is θf = arccos(1 − z/r). It uses the geometric sinkage, not the plate sinkage, and assumes the soil does not heave in front of the wheel.
Angle from the downward vertical to the point where a rigid wheel of radius r first meets the soil, from the geometric sinkage z measured from the undisturbed surface. It uses 0 ≤ z ≤ r, the shallow-contact geometry.
Calculator
- Entry angle in degrees36.87 °
- Horizontal contact length, r sin θf0.12 m
- Arc length of contact, r θf0.1287 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The wheel enters the soil at 0.6435 rad (36.9°) from the vertical, giving a contact arc of 0.1287 m.
- The entry angle grows as √(2z/r) for small sinkage, so halving the sinkage lowers it by only about 30 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad) | Local step | Change in result |
|---|---|---|---|---|---|---|
| r Wheel radius | 0.04 to 1.5 m | 0.2315 | 1.571 | 1.339 | +10% | −4.81 % |
| z Geometric sinkage | 0.001 to 0.04 m | 0.1 | 0.6435 | 0.5435 | +10% | +5.07 % |
Sweep of Geometric sinkage
| Geometric sinkage (m) | Entry angle (rad) |
|---|---|
| 0.001 | 0.1 |
| 0.005875 | 0.243 |
| 0.01075 | 0.3294 |
| 0.01563 | 0.3979 |
| 0.0205 | 0.4567 |
| 0.02538 | 0.5092 |
| 0.03025 | 0.5572 |
| 0.03513 | 0.6017 |
| 0.04 | 0.6435 |
Sweep one variable
Variables and how each one contributes
Entry angle
What the equation gives you. Everything else feeds it.
Geometric sinkage
The geometric sinkage of the wheel, in m. A deeper sinkage raises the entry angle, rapidly at first and then more slowly. Raising it by 10 % (to 0.044) moves the result by +5.07 %, from 0.6435 to 0.6761 rad.
Wheel radius
The wheel radius, in m. A larger wheel has a smaller entry angle at the same sinkage, so its contact is shallower. Raising it by 10 % (to 0.22) moves the result by −4.81 %, from 0.6435 to 0.6126 rad.
Worked example with the values above
- Ratio: z/r = 0.04 / 0.2 = 0.2.
- Cosine: 1 − z/r = 0.8.
- θf = arccos(0.8) = 0.6435 rad.
Assumptions and limits
- A rigid wheel of constant radius and width, with stresses uniform across the width and constant driving slip.
- Contact angle measured from the downward vertical, with angles increasing from exit to entry, and an arc without soil bulging.
Common mistakes
- Entering angles in degrees instead of radians.
- Ordering the nodes from entry to exit, which reverses the sign of the integral.
- Mixing the normal-stress and shear-stress sign conventions.
Statistics and uncertainty
Stress sensors on a wheel rim read to perhaps 5 % to 10 %, and the trapezoidal sum over five nodes is coarse, so a smooth stress peak between nodes can be missed. Add nodes where stress changes quickly.
Where it is used
- Checking measured wheel loads and thrusts against force-cell readings.
- Building wheel–soil models from measured stress data.
Maths you need
- Contact angle: angle at the wheel centre from the vertical to a contact point
- Trapezoidal rule: area under sampled data
- Janosi–Hanamoto shear: τ = τmax(1 − e^(−j/K))