Library / Automotive and vehicle dynamics / VEH-15
Automotive and vehicle dynamics
Ideal front brake force fraction
The share of the total braking force that the front axle should supply so that front and rear tyres are used equally hard.
- Ref
- VEH-15
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- βf [%]
b is the distance from the centre of gravity to the rear axle, h the height of the centre of gravity, L the wheelbase and ab the design deceleration in g. The ideal front fraction is the front axle load divided by the weight, βf ≈ Fz,f / m g, which uses front and rear tyres equally hard if both axles have the same friction coefficient. Because weight moves forward as the deceleration rises, the ideal fraction rises with the deceleration. The result is shown as a percentage of the total braking force.
The share of the total braking force that the front axle should supply so that front and rear tyres are used equally hard. It rises with deceleration because weight moves forward.
Calculator
- Static front share, b / L51.85 %
- Ideal rear share31.85 %
- Shift of braking force to the front compared with static16.3 percentage points
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 rest the front axle carries 51.85 % of the weight, but braking at 0.8 g needs 68.15 % of the braking force at the front. At 0.2 g the ideal share is only 55.93 %.
- A fixed brake bias can match the ideal curve at one deceleration only. Road cars set it so that the front wheels lock first, which is stable, and modern systems use electronic brake distribution to follow the ideal curve more closely.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (%) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Distance from the centre of gravity to the rear axle | 0.9 to 1.8 m | 49.63 | 82.96 | 33.33 | +10% | +7.61 % |
| L Wheelbase | 2.2 to 3.5 m | 52.57 | 83.64 | 31.06 | +10% | −9.09 % |
| ab Design deceleration | 0.1 to 1.2 g | 53.89 | 76.3 | 22.41 | +10% | +2.39 % |
| h Centre of gravity height | 0.4 to 0.9 m | 63.7 | 78.52 | 14.81 | +10% | +2.39 % |
Sweep of Design deceleration
| Design deceleration (g) | Ideal front brake fraction (%) |
|---|---|
| 0.1 | 53.89 |
| 0.2375 | 56.69 |
| 0.375 | 59.49 |
| 0.5125 | 62.29 |
| 0.65 | 65.09 |
| 0.7875 | 67.89 |
| 0.925 | 70.69 |
| 1.063 | 73.5 |
| 1.2 | 76.3 |
Sweep one variable
Variables and how each one contributes
Ideal front brake fraction
What the equation gives you. Everything else feeds it.
Distance from the centre of gravity to the rear axle
The distance from the centre of gravity to the rear axle, in m. Raises the static share on the front axle in proportion, and so the ideal front brake fraction. Raising it by 10 % (to 1.54) moves the result by +7.61 %, from 68.15 to 73.33 %.
Centre of gravity height
The height of the centre of gravity, in m. Raises the weight transfer, so a taller vehicle needs more front bias at the same deceleration. Raising it by 10 % (to 0.605) moves the result by +2.39 %, from 68.15 to 69.78 %.
Wheelbase
The wheelbase, in m. Lowers the ideal front fraction in inverse proportion, because a longer wheelbase spreads the weight more evenly. Raising it by 10 % (to 2.97) moves the result by −9.09 %, from 68.15 to 61.95 %.
Design deceleration
The design deceleration, in g. The ideal front share is evaluated at this value. Raises the weight transferred and so the ideal front share. The bias that is ideal at 0.8 g is too high at light braking. Raising it by 10 % (to 0.88) moves the result by +2.39 %, from 68.15 to 69.78 %.
Worked example with the values above
- Weight transfer term: (ab/g) h = 0.8 × 0.55 = 0.44 m.
- Effective distance: b + (ab/g) h = 1.4 + 0.44 = 1.84 m, over the wheelbase L = 2.7 m.
- βf = 1.84 / 2.7 = 0.6815, which is 68.15 %.
Assumptions and limits
- Front and rear tyres have the same friction coefficient, and braking is steady on a level road.
- No aerodynamic forces, and the loads are those of a rigid vehicle without pitching motion.
Common mistakes
- Choosing a bias from the static weight distribution. This is too far to the rear at any real braking level.
- Putting more braking on the rear axle than the ideal fraction, which locks the rear wheels first and can cause a spin.
Statistics and uncertainty
The ideal fraction follows from the geometry, so its uncertainty comes from the height of the centre of gravity and the load condition, which change with passengers and cargo. A 10 % error in h shifts the ideal share by about 1.6 percentage points at 0.8 g. Because the ideal curve moves with load, brake systems are normally set with a margin to the safe, front-biased side.
Where it is used
- Setting the brake bias of a car or truck.
- Checking whether a load or a change of centre of gravity height makes the existing bias unsafe.
Maths you need
- Weight transfer: braking moves load to the front axle
- Fractions and percentages: share of a total