Library / Automotive and vehicle dynamics / VEH-49
Automotive and vehicle dynamics
No-side-friction speed on a banked curve
Speed at which the inward component of the road normal force alone supplies the centripetal force on a banked curve of horizontal radius R and bank angle θ, so no side friction is needed.
- Ref
- VEH-49
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- U [m/s]
On a banked curve of bank angle θ, the road normal force N has a horizontal component N sin θ pointing to the centre of the curve, and N cos θ = mg vertically. With no side friction the centripetal requirement mU²/R = N sin θ gives U = √(gR tan θ). Above this speed friction must act down the slope; below it friction must act up the slope.
Speed at which the inward component of the road normal force alone supplies the centripetal force on a banked curve of horizontal radius R and bank angle θ, so no side friction is needed. Ideal point mass in steady motion.
Calculator
- Speed in kilometres per hour35.66 km/h
- Lateral acceleration, g tan θ0.981 m/s²
- Bank angle for 20 m/s at this radius, arctan(400/(gR))22.18 °
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 curve needs no side friction at 9.905 m/s (35.66 km/h), at a lateral acceleration of 0.981 m/s².
- Doubling the radius raises the speed only by 41.4 %; doubling the speed needs four times the radius, or a bank angle whose tangent is four times larger.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| R Horizontal path radius | 5 to 2,000 m | 2.215 | 44.29 | 42.08 | +10% | +4.88 % |
| θ Bank angle | 0.1 to 45 ° | 1.308 | 31.32 | 30.01 | +10% | +4.92 % |
| g Gravitational acceleration | 1 to 25 m/s² | 3.162 | 15.81 | 12.65 | +10% | +4.88 % |
Sweep of Bank angle
| Bank angle (°) | No-friction speed (m/s) |
|---|---|
| 0.1 | 1.308 |
| 5.712 | 9.906 |
| 11.32 | 14.02 |
| 16.94 | 17.28 |
| 22.55 | 20.18 |
| 28.16 | 22.92 |
| 33.77 | 25.61 |
| 39.39 | 28.38 |
| 45 | 31.32 |
Sweep one variable
Variables and how each one contributes
No-friction speed
What the equation gives you. Everything else feeds it.
Gravitational acceleration
The gravitational acceleration, in m/s². Raises the speed as √g. Raising it by 10 % (to 10.79) moves the result by +4.88 %, from 9.905 to 10.39 m/s.
Horizontal path radius
The horizontal radius of the path, in m. A larger radius raises the speed as √R. Raising it by 10 % (to 110) moves the result by +4.88 %, from 9.905 to 10.39 m/s.
Bank angle
The bank angle, in degrees. A steeper bank raises the speed with √(tan θ), which grows rapidly as the angle approaches 90°. Raising it by 10 % (to 6.282) moves the result by +4.92 %, from 9.905 to 10.39 m/s.
Worked example with the values above
- Tangent: tan θ = tan 5.7106° = 0.1.
- Product: g R tan θ = 9.81 × 100 × 0.1 = 98.1 m²/s².
- U = √98.1 = 9.9045 m/s.
Assumptions and limits
- A rigid vehicle on a flat road in steady or slowly varying motion, small angles and a linear single-track (bicycle) model where stated.
- Level road, no aerodynamic yaw moment and no roll effects on the axle forces unless the page says so.
Common mistakes
- Mixing up a and b: a is measured to the front axle and b to the rear axle.
- Using the track-averaged force for an axle without checking the total.
- Entering the path radius without its sign, which flips the sideslip sign.
Statistics and uncertainty
Yaw inertia and cornering stiffness are usually known to 10 % to 20 %, and steady sideslip is the small difference of two similar terms, so it is more sensitive to errors than either term alone.
Where it is used
- Splitting cornering force between axles and checking handling balance.
- Yaw-control and torque-vectoring design, and banked test-track layout.
Maths you need
- Newton’s second law: F = ma, for force and for moment
- Single-track model: two tyres lumped as one per axle
- Centripetal acceleration: a = U²/R