Library / Automotive and vehicle dynamics / VEH-32
Automotive and vehicle dynamics
Wheel-hop frequency
Approximate natural frequency of the wheel and axle bouncing on the combined suspension and tyre springs, treating the body as stationary.
- Ref
- VEH-32
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- fhop [Hz]
kw is the suspension stiffness at the wheel (the wheel rate), kt the vertical tyre stiffness and mu the unsprung mass at one corner: the wheel, tyre, brake and part of the suspension links. The wheel hops on the combination of the two springs working in parallel against the body, which is treated as stationary because it is much heavier and moves slowly at this frequency. The result is the wheel-hop frequency in Hz.
Approximate natural frequency of the wheel and axle bouncing on the combined suspension and tyre springs, treating the body as stationary. It is much higher than the body ride frequency.
Calculator
- Total spring stiffness, kw + kt225 kN/m
- Period of the hop motion83.78 ms
- Ratio of tyre stiffness to suspension stiffness8
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.
- With a 40 kg unsprung mass the wheel hops at 11.94 Hz, between 8 and 9 times the ride frequency of 1.4 Hz. This separation is what lets the body and the wheel be treated as separate in the quarter-car model.
- Cutting the unsprung mass to 30 kg raises the hop frequency to 13.78 Hz; raising it to 60 kg lowers it to 9.746 Hz. A lower hop frequency moves it towards road input from rough surfaces, which is why heavy wheels and brakes hurt grip on bumpy roads.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| mu Unsprung mass at one corner | 20 to 100 kg | 7.549 | 16.88 | 9.332 | +10% | −4.65 % |
| kt Vertical tyre stiffness | 150 to 350 kN/m | 10.53 | 15.41 | 4.883 | +10% | +4.35 % |
| kw Suspension stiffness at the wheel | 10 to 80 kN/m | 11.53 | 13.32 | 1.784 | +10% | +0.554 % |
Sweep of Unsprung mass at one corner
| Unsprung mass at one corner (kg) | Wheel-hop frequency (Hz) |
|---|---|
| 20 | 16.88 |
| 30 | 13.78 |
| 40 | 11.94 |
| 50 | 10.68 |
| 60 | 9.746 |
| 70 | 9.023 |
| 80 | 8.44 |
| 90 | 7.958 |
| 100 | 7.549 |
Sweep one variable
Variables and how each one contributes
Wheel-hop frequency
What the equation gives you. Everything else feeds it.
Suspension stiffness at the wheel
The suspension stiffness at the wheel (spring rate times motion ratio squared), in kN/m. Raises the frequency with the square root of the total stiffness, but the tyre is much stiffer and dominates. Raising it by 10 % (to 27.5) moves the result by +0.554 %, from 11.94 to 12 Hz.
Vertical tyre stiffness
The vertical stiffness of the tyre, in kN/m. The dominant spring: raising the tyre stiffness raises the hop frequency, by about the square root of the change of the total. Raising it by 10 % (to 220) moves the result by +4.35 %, from 11.94 to 12.46 Hz.
Unsprung mass at one corner
The unsprung mass at one corner, in kg: typically 30 to 60 kg for a car. The frequency falls with the square root of the mass, so lighter wheels and brakes raise it and improve tyre contact. Raising it by 10 % (to 44) moves the result by −4.65 %, from 11.94 to 11.38 Hz.
Worked example with the values above
- Total stiffness: kw + kt = 25 + 200 = 225 kN/m = 225,000 N/m.
- Divide by the unsprung mass: (kw + kt) / mu = 225,000 / 40 = 5,625 s⁻², whose square root is 75 rad/s.
- fhop = 75 / 2π = 11.94 Hz.
Assumptions and limits
- The sprung mass is stationary, so the wheel moves on the two springs in parallel, with damping neglected.
- Linear stiffnesses and a constant unsprung mass, giving a single natural frequency.
Common mistakes
- Using the spring stiffness instead of the wheel rate for kw.
- Using the total vehicle mass in place of the unsprung mass at one wheel.
Statistics and uncertainty
The tyre stiffness is the main uncertainty, because it changes with pressure (about 5 to 10 % per 0.2 bar) and speed, and the effective unsprung mass depends on how much of the suspension links is included, which is known to about 10 %. The frequency therefore has an uncertainty of about 5 %. Measured values from a shaker rig are better than this estimate when accuracy is important.
Where it is used
- Avoiding wheel-hop resonance by tuning dampers and the unsprung mass.
- Estimating the frequency range that the suspension damper must control.
Maths you need
- Springs in parallel: stiffnesses add
- Natural frequency: f = (1/2π) √(k / m)