Library / Automotive and vehicle dynamics / VEH-45
Automotive and vehicle dynamics
Wheel angular acceleration under drive and brake torque
Angular acceleration of a driven wheel from the drive torque Td, brake torque Tb, ground force Fx at effective radius re, any other resisting torque Tr and the wheel inertia Jw.
- Ref
- VEH-45
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- ω̇ [rad/s²]
The wheel is a rotating body, so its angular acceleration is the net torque over its inertia: Jw ω̇ = Td − Tb − Fx·re − Tr. The ground force Fx acts at the effective rolling radius re and opposes the drive torque when it is positive forwards. Close to a steady rolling state the ground-force term keeps the wheel near the rolling speed, and a large imbalance spins the wheel up or locks it.
Angular acceleration of a driven wheel from the drive torque Td, brake torque Tb, ground force Fx at effective radius re, any other resisting torque Tr and the wheel inertia Jw. Forward rolling is positive. Do not count rolling resistance twice.
Calculator
- Net torque on the wheel30 N m
- Torque the ground force returns, Fx·re420 N m
- Equivalent rolling acceleration, ω̇·re7 m/s²
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 is accelerating at 20 rad/s², so its speed changes by 2 rad/s every 0.1 s.
- The ground force returns 420 N m of the 500 N m drive torque, which is what pushes the vehicle along; only the rest spins the wheel up.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad/s²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Fx Forward ground force on the tyre | −20,000 to 20,000 N | −4,367 | 4,967 | 9,333 | +800 N | −933 % |
| Td Drive torque | 0 to 5,000 N m | −313.3 | 3,020 | 3,333 | +10% | +167 % |
| Tb Brake torque | 0 to 5,000 N m | −3,287 | 46.67 | 3,333 | +10% | −13.3 % |
| re Effective rolling radius | 0.1 to 1.5 m | −900 | 220 | 1,120 | +10% | −140 % |
| Tr Other resisting torque | 0 to 500 N m | −306.7 | 26.67 | 333.3 | +10% | −3.33 % |
| Jw Wheel inertia | 0.1 to 50 kg m² | 0.6 | 300 | 299.4 | +10% | −9.09 % |
Sweep of Drive torque
| Drive torque (N m) | Wheel angular acceleration (rad/s²) |
|---|---|
| 0 | −313.3 |
| 625 | 103.3 |
| 1,250 | 520 |
| 1,875 | 936.7 |
| 2,500 | 1,353 |
| 3,125 | 1,770 |
| 3,750 | 2,187 |
| 4,375 | 2,603 |
| 5,000 | 3,020 |
Sweep one variable
Variables and how each one contributes
Wheel angular acceleration
What the equation gives you. Everything else feeds it.
Drive torque
The drive torque on the wheel, in N m. Raises the angular acceleration in proportion to 1/Jw. Raising it by 10 % (to 550) moves the result by +167 %, from 20 to 53.33 rad/s².
Brake torque
The brake torque on the wheel, in N m. Lowers the acceleration; large values lead to wheel lock. Raising it by 10 % (to 44) moves the result by −13.3 %, from 20 to 17.33 rad/s².
Forward ground force on the tyre
The forward ground force on the tyre, in N. Opposes wheel spin-up through the moment Fx·re. Raising it by 10 % (to 1,320) moves the result by −140 %, from 20 to −8 rad/s².
Effective rolling radius
The effective rolling radius, in m. A larger radius makes the same ground force a bigger opposing torque. Raising it by 10 % (to 0.385) moves the result by −140 %, from 20 to −8 rad/s².
Other resisting torque
Other resisting torque, such as bearing drag, in N m. Lowers the net torque in direct proportion. Raising it by 10 % (to 11) moves the result by −3.33 %, from 20 to 19.33 rad/s².
Wheel inertia
The wheel and rotating-parts inertia, in kg m². A bigger inertia lowers the angular acceleration for the same net torque. Raising it by 10 % (to 1.65) moves the result by −9.09 %, from 20 to 18.18 rad/s².
Worked example with the values above
- Ground torque: Fx re = 1,200 × 0.35 = 420 N m.
- Net torque: 500 − 40 − 420 − 10 = 30 N m.
- ω̇ = 30 / 1.5 = 20 rad/s².
Assumptions and limits
- A rolling tyre on a flat road with a point contact, small angles where a linearised form is used, and constant properties during the event.
- Velocities resolved in the body or tyre axes named on the page, with the sign convention stated there.
Common mistakes
- Mixing the SAE and ISO sign conventions for slip angle and lateral force.
- Entering steer angle in radians when degrees are expected, or the reverse.
- Using the relaxation-length formula for a step that is not small enough to stay in the linear tyre range.
Statistics and uncertainty
Tyre forces are known only to about 5 % to 15 % in practice, and slip angle and slip ratio are small differences of large velocities, so small velocity errors give large relative errors in slip.
Where it is used
- Computing tyre slip in handling simulations from measured vehicle motion.
- Estimating steering feel and wheel-speed response in control design.
Maths you need
- Rigid-body velocity: v_wheel = v_cg + ω × r
- Slip angle: the angle between a tyre’s heading and its velocity
- Relaxation length: the rolling distance over which tyre force builds up