Library / Automotive and vehicle dynamics / VEH-37
Automotive and vehicle dynamics
Axle reactions on a longitudinal grade
Front-axle normal reaction of a vehicle facing uphill on a grade of angle θ, with b the distance along the road from the rear axle to the centre of gravity and h its height normal to the road.
- Ref
- VEH-37
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Nf [N]
On a grade of angle θ, the weight resolves into W cos θ normal to the road and W sin θ along it. Taking moments about the rear contact patch gives Nf·L = W(b cos θ − h sin θ), where b is the distance along the road from the rear axle to the centre of gravity and h its height normal to the road. The second term is the load transferred to the rear by the slope. A front reaction that reaches zero marks the onset of tipping.
Front-axle normal reaction of a vehicle facing uphill on a grade of angle θ, with b the distance along the road from the rear axle to the centre of gravity and h its height normal to the road. Static pitch equilibrium with no hitch or aerodynamic load. A negative reaction means the wheels have lost contact.
Calculator
- Rear-axle reaction, W[(L − b)cos θ + h sin θ]/L5,000 N
- Total normal reaction, W cos θ8,000 N
- Grade at which the front reaction reaches zero, arctan(b/h)71.57 °
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.
- At 36.87° the front axle carries 3,000 N, which is 37.5 % of the normal load W cos θ.
- The front wheels lift off at 71.57°, set only by the ratio b/h, which is why a high centre of gravity is the first thing to lower for steep-slope work.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| W Vehicle weight | 100 to 1,000,000 N | 30 | 300,000 | 300,000 | +10% | +10 % |
| b Distance from rear axle to CG along the road | 0.1 to 10 m | −733.3 | 25,670 | 26,400 | +10% | +13.3 % |
| L Wheelbase | 0.5 to 10 m | 900 | 18,000 | 17,100 | +10% | −9.09 % |
| h CG height above the road | 0.05 to 5 m | −6,000 | 3,900 | 9,900 | +10% | −3.33 % |
| θ Grade angle | 0 to 60 ° | 1,057 | 5,000 | 3,943 | +10% | −9.5 % |
Sweep of Grade angle
| Grade angle (°) | Front-axle reaction (N) |
|---|---|
| 0 | 5,000 |
| 7.5 | 4,740 |
| 15 | 4,398 |
| 22.5 | 3,982 |
| 30 | 3,497 |
| 37.5 | 2,952 |
| 45 | 2,357 |
| 52.5 | 1,722 |
| 60 | 1,057 |
Sweep one variable
Variables and how each one contributes
Front-axle reaction
What the equation gives you. Everything else feeds it.
Vehicle weight
The vehicle weight, in N. Raises the front reaction in direct proportion. Raising it by 10 % (to 11,000) moves the result by +10 %, from 3,000 to 3,300 N.
Distance from rear axle to CG along the road
The distance from the rear axle to the centre of gravity along the road, in m. A larger b puts the centre of gravity nearer the front and raises the front reaction. Raising it by 10 % (to 1.65) moves the result by +13.3 %, from 3,000 to 3,400 N.
CG height above the road
The height of the centre of gravity above the road, in m. A higher centre of gravity transfers more load to the rear on a slope and lowers the front reaction. Raising it by 10 % (to 0.55) moves the result by −3.33 %, from 3,000 to 2,900 N.
Wheelbase
The wheelbase, in m. A longer wheelbase shares the moment over a bigger lever arm and lowers the front reaction for a given moment. Raising it by 10 % (to 3.3) moves the result by −9.09 %, from 3,000 to 2,727 N.
Grade angle
The grade angle, in degrees, with the vehicle facing uphill. A steeper grade lowers the cosine term and raises the sine term, so the front load falls and reaches zero at tan θ = b/h. Raising it by 10 % (to 40.56) moves the result by −9.5 %, from 3,000 to 2,715 N.
Worked example with the values above
- Trig terms: cos θ = 0.8, sin θ = 0.6.
- Moment term: b cos θ − h sin θ = 1.5 × 0.8 − 0.5 × 0.6 = 0.9 m.
- Nf = W × 0.9 / L = 10,000 × 0.9 / 3 = 3,000 N.
Assumptions and limits
- A rigid vehicle in static equilibrium, with no braking, driving or aerodynamic loads unless the page names one.
- Flat tyres contact patches treated as single points on a plane road, and suspension deflection that does not move the centre of gravity.
Common mistakes
- Measuring the centre-of-gravity height or hitch height from the wrong datum (axle centre instead of the road).
- Mixing the distance from the front axle with the distance from the rear axle.
- Using the vehicle weight where the road-normal component W cos θ is needed.
Statistics and uncertainty
Axle scales read to about 0.5 % to 1 %, but the geometry inputs (wheelbase, centre-of-gravity height, hitch height) are often known only to a few centimetres, and on short wheelbases that is a visible fraction of the result.
Where it is used
- Setting axle loads, tongue weight and drawbar loads for tractors and trailers.
- Checking tip-up and wheel lift-off limits before field tests.
Maths you need
- Static equilibrium: forces and moments each sum to zero
- Moment about a contact point: a force times its perpendicular distance
- Centre of gravity: the point where the weight acts