Library / Automotive and vehicle dynamics / VEH-36
Automotive and vehicle dynamics
Longitudinal centre of gravity from scale readings
Distance of the centre of gravity behind the front axle, from the wheelbase L and the total scale readings under the front axle Wf and the rear axle Wr (left plus right, same units).
- Ref
- VEH-36
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- a [m]
With the vehicle level and stationary, taking moments about the front axle gives (Wf + Wr)·a = Wr·L, so the centre of gravity lies a = L·Wr/(Wf + Wr) behind the front axle. The scale readings must be the total under each axle, with the same units for both, and the vehicle must be on level ground with the suspension settled.
Distance of the centre of gravity behind the front axle, from the wheelbase L and the total scale readings under the front axle Wf and the rear axle Wr (left plus right, same units). The vehicle is stationary, level and carries no external load.
Calculator
- Share of the weight on the front axle, Wf/(Wf + Wr)60 %
- Distance ahead of the rear axle, L − a1.8 m
- Total vehicle weight, Wf + Wr15,000 N
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 centre of gravity lies 1.2 m behind the front axle, which is 40 % of the wheelbase.
- With 60 % of the weight on the front axle, the centre of gravity is nearer the front axle.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Wheelbase | 0.5 to 10 m | 0.2 | 4 | 3.8 | +10% | +10 % |
| Wr Rear-axle scale reading | 100 to 1,000,000 N | 0.03297 | 2.973 | 2.94 | +10% | +5.77 % |
| Wf Front-axle scale reading | 100 to 1,000,000 N | 0.01789 | 2.951 | 2.933 | +10% | −5.66 % |
Sweep of Rear-axle scale reading
| Rear-axle scale reading (N) | CG distance behind the front axle (m) |
|---|---|
| 100 | 0.03297 |
| 316.2 | 0.1018 |
| 1,000 | 0.3 |
| 3,162 | 0.78 |
| 10,000 | 1.579 |
| 31,620 | 2.335 |
| 100,000 | 2.752 |
| 316,200 | 2.917 |
| 1,000,000 | 2.973 |
Sweep one variable
Variables and how each one contributes
CG distance behind the front axle
What the equation gives you. Everything else feeds it.
Wheelbase
The wheelbase, in m. A longer wheelbase moves the centre of gravity further from the front axle in direct proportion. Raising it by 10 % (to 3.3) moves the result by +10 %, from 1.2 to 1.32 m.
Front-axle scale reading
The total scale reading under the front axle, in N. A heavier front axle moves the centre of gravity forward and shrinks a. Raising it by 10 % (to 9,900) moves the result by −5.66 %, from 1.2 to 1.132 m.
Rear-axle scale reading
The total scale reading under the rear axle, in N. A heavier rear axle moves the centre of gravity backwards and raises a. Raising it by 10 % (to 6,600) moves the result by +5.77 %, from 1.2 to 1.269 m.
Worked example with the values above
- Total weight: Wf + Wr = 9,000 + 6,000 = 15,000 N.
- Rear share: Wr/(Wf + Wr) = 6,000 / 15,000 = 0.4.
- a = L × share = 3 × 0.4 = 1.2 m.
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