Library / Terramechanics and off-road mobility / TER-46
Terramechanics and off-road mobility
Measured longitudinal centre of track pressure
Position of the centre of pressure along a track or contact patch, from cell-average pressures p, cell areas A and cell centroids x on a common longitudinal datum.
- Ref
- TER-46
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- xcp [m]
The centre of pressure of a pressure map is the pressure-weighted mean position, x_cp = Σ xi·pi·Ai / Σ pi·Ai, where the forces pi·Ai on the cells act at their centroids xi. It gives the line of action of the resultant normal force, and in equilibrium it must lie under the centre of gravity (plus any pitch moment from the drawbar) for a stationary vehicle.
Position of the centre of pressure along a track or contact patch, from cell-average pressures p, cell areas A and cell centroids x on a common longitudinal datum. Four cells can be entered; set the pressure and area of an unused cell to zero. Pressures are non-negative.
Calculator
- Total normal force, Σ pi Ai24,000 N
- Mean pressure over the active cells20,000 Pa
- Moment about the datum, Σ xi pi Ai44,000 N 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 resultant of 24,000 N acts at 1.833 m, beyond the midpoint of the instrumented cells at 1.5 m.
- The pressure rises along the patch, so the load is carried ahead of the middle; a roadwheel pattern like this suggests the track is pitched nose-down.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| x3 Centroid position, cell 3 | 0 to 10 m | 0.5833 | 5.583 | 5 | +10% | +6.82 % |
| x2 Centroid position, cell 2 | 0 to 10 m | 1.333 | 4.667 | 3.333 | +10% | +2.73 % |
| x1 Centroid position, cell 1 | 0 to 10 m | 1.75 | 3.417 | 1.667 | +10% | +0.455 % |
| p1 Pressure, cell 1 | 1,000 to 1,000,000 Pa | 0.5762 | 2.069 | 1.492 | +10% | −1.19 % |
| p3 Pressure, cell 3 | 1,000 to 1,000,000 Pa | 1.21 | 2.461 | 1.251 | +10% | +1.73 % |
| A3 Area, cell 3 | 0.01 to 5 m² | 1.199 | 2.401 | 1.202 | +10% | +1.73 % |
| A1 Area, cell 1 | 0.01 to 5 m² | 0.9571 | 2.092 | 1.135 | +10% | −1.19 % |
| p2 Pressure, cell 2 | 1,000 to 1,000,000 Pa | 1.519 | 1.988 | 0.4686 | +10% | −0.587 % |
| A2 Area, cell 2 | 0.01 to 5 m² | 1.569 | 1.994 | 0.4249 | +10% | −0.587 % |
| x4 Centroid position, cell 4 (unused if A₄ = 0) | 0 to 10 m | 1.833 | 1.833 | 0 | +10% | 0 % |
| p4 Pressure, cell 4 | 0 to 1,000,000 Pa | 1.833 | 1.833 | 0 | +10% | 0 % |
| A4 Area, cell 4 | 0 to 5 m² | 1.833 | 1.833 | 0 | +10% | 0 % |
Sweep of Pressure, cell 3
| Pressure, cell 3 (Pa) | Centre of pressure (m) |
|---|---|
| 1,000 | 1.21 |
| 2,371 | 1.264 |
| 5,623 | 1.377 |
| 13,340 | 1.577 |
| 31,620 | 1.851 |
| 74,990 | 2.119 |
| 177,800 | 2.308 |
| 421,700 | 2.411 |
| 1,000,000 | 2.461 |
Sweep one variable
Variables and how each one contributes
Centre of pressure
What the equation gives you. Everything else feeds it.
Centroid position, cell 1
The centroid position of cell 1, in m. Moves the centre of pressure towards cell 1 in proportion to its force. Raising it by 10 % (to 0.55) moves the result by +0.455 %, from 1.833 to 1.842 m.
Pressure, cell 1
The pressure on cell 1, in Pa. Raises cell 1’s weight in the average. Raising it by 10 % (to 11,000) moves the result by −1.19 %, from 1.833 to 1.811 m.
Area, cell 1
The area of cell 1, in m². Raises cell 1’s weight in the average. Raising it by 10 % (to 0.44) moves the result by −1.19 %, from 1.833 to 1.811 m.
Centroid position, cell 2
The centroid position of cell 2, in m. Moves the centre of pressure towards cell 2. Raising it by 10 % (to 1.65) moves the result by +2.73 %, from 1.833 to 1.883 m.
Pressure, cell 2
The pressure on cell 2, in Pa. Raises cell 2’s weight in the average. Raising it by 10 % (to 22,000) moves the result by −0.587 %, from 1.833 to 1.823 m.
Area, cell 2
The area of cell 2, in m². Raises cell 2’s weight in the average. Raising it by 10 % (to 0.44) moves the result by −0.587 %, from 1.833 to 1.823 m.
Centroid position, cell 3
The centroid position of cell 3, in m. Moves the centre of pressure towards cell 3. Raising it by 10 % (to 2.75) moves the result by +6.82 %, from 1.833 to 1.958 m.
Pressure, cell 3
The pressure on cell 3, in Pa. Raises cell 3’s weight in the average. Raising it by 10 % (to 33,000) moves the result by +1.73 %, from 1.833 to 1.865 m.
Area, cell 3
The area of cell 3, in m². Raises cell 3’s weight in the average. Raising it by 10 % (to 0.44) moves the result by +1.73 %, from 1.833 to 1.865 m.
Centroid position, cell 4 (unused if A₄ = 0)
The centroid position of cell 4, in m (unused if A₄ = 0). Has an effect only if the cell’s force is not zero. Raising it by 10 % (to 0) moves the result by +0 %, from 1.833 to 1.833 m.
Pressure, cell 4
The pressure on cell 4, in Pa. Raises cell 4’s weight if its area is not zero. Raising it by 10 % (to 0) moves the result by +0 %, from 1.833 to 1.833 m.
Area, cell 4
The area of cell 4, in m² (set to zero if unused). Raises cell 4’s weight if its pressure is not zero. Raising it by 10 % (to 0) moves the result by +0 %, from 1.833 to 1.833 m.
Worked example with the values above
- Cell forces pi Ai: 4,000, 8,000, 12,000 and 0 N; total 24,000 N.
- Moment about the datum: Σ xi pi Ai = 44,000 N m.
- x_cp = 44,000 / 24,000 = 1.8333 m.
Assumptions and limits
- A rigid, flat track patch or cone in homogeneous soil, with a linear pressure profile where stated.
- Constant soil strength and deformation length over the test, and the same rate and cone for comparisons.
Common mistakes
- Using the linear pressure formula beyond the one-sixth limit on the load offset.
- Mixing the left and right sign convention for yaw moment.
- Including rod friction in the cone force.
Statistics and uncertainty
Pressure cells and penetrometers carry 5 % to 10 % uncertainty, and the penetration or shear work depends on rate and soil state. Compare only tests run under the same conditions.
Where it is used
- Checking track pressure distribution, thrust sharing and steering demand.
- Comparing soil strength and shear energy between test beds.
Maths you need
- Linear pressure distribution: p = W/A (1 ± 6e/L)
- Work as an integral: W = ∫ F dz
- Moment balance: the sum of moments about the vehicle centre is zero