Library / Automotive and vehicle dynamics / VEH-59
Automotive and vehicle dynamics
Road-to-body displacement transfer function
Magnitude of the complex sprung-mass displacement per unit harmonic road displacement at angular frequency ω, for the linear two-mass model without tyre damping.
- Ref
- VEH-59
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- |H|
With a harmonic road input zr = Re[Z e^(iωt)], the body response is a complex multiple H(iω) = kt(cs·iω + ks)/D, where D = (ms s² + cs s + ks)(mu s² + cs s + ks + kt) − (cs s + ks)² and s = iω. The magnitude shows the resonance at the body bounce frequency, the dip between, and the wheel-hop resonance, and is the transmissibility of the road displacement to the body.
Magnitude of the complex sprung-mass displacement per unit harmonic road displacement at angular frequency ω, for the linear two-mass model without tyre damping. D(s) = (ms s² + cs s + ks)(mu s² + cs s + ks + kt) − (cs s + ks)². The real part, imaginary part and phase are in the derived rows.
Calculator
- Real part of H−0.2794
- Imaginary part of H−1.308
- Phase of H−102.1 °
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 1.59 Hz the body moves 1.337 times the road amplitude, with a phase of −102°.
- The response is above 1, so this frequency lies near the body resonance and the suspension is amplifying the road. The sweep chart shows the full curve against each input.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| ks Suspension spring rate | 1,000 to 200,000 N/m | 0.4718 | 3.461 | 2.989 | +10% | +13.8 % |
| ω Angular frequency | 0.1 to 200 rad/s | 0.002658 | 2.272 | 2.27 | +10% | −21.5 % |
| ms Sprung mass | 50 to 5,000 kg | 0.03827 | 1.59 | 1.552 | +10% | −11.9 % |
| kt Tyre vertical stiffness | 20,000 to 1,000,000 N/m | 0.7212 | 1.343 | 0.6217 | +10% | +0.156 % |
| mu Unsprung mass | 5 to 500 kg | 1.309 | 1.856 | 0.5468 | +10% | +0.246 % |
| cs Suspension damping | 100 to 10,000 N s/m | 1.225 | 1.418 | 0.1928 | +10% | −0.692 % |
Sweep of Angular frequency
| Angular frequency (rad/s) | Magnitude of the transfer function |
|---|---|
| 0.1 | 1 |
| 0.2586 | 1.001 |
| 0.6687 | 1.008 |
| 1.729 | 1.054 |
| 4.472 | 1.448 |
| 11.56 | 0.9302 |
| 29.91 | 0.2125 |
| 77.34 | 0.0869 |
| 200 | 0.002658 |
Sweep one variable
Variables and how each one contributes
Magnitude of the transfer function
What the equation gives you. Everything else feeds it.
Tyre vertical stiffness
The vertical tyre stiffness, in N/m. Raises the numerator and the response of the whole system at all frequencies. Raising it by 10 % (to 176,000) moves the result by +0.156 %, from 1.337 to 1.339.
Suspension damping
The suspension damping, in N s/m. Reduces the resonance peaks but raises the response at high frequency, where the damper transmits force. Raising it by 10 % (to 1,320) moves the result by −0.692 %, from 1.337 to 1.328.
Suspension spring rate
The suspension spring rate, in N/m. Raises the body frequency and the response below it. Raising it by 10 % (to 17,600) moves the result by +13.8 %, from 1.337 to 1.522.
Sprung mass
The sprung mass, in kg. A heavier body lowers the resonance and the response above it. Raising it by 10 % (to 275) moves the result by −11.9 %, from 1.337 to 1.179.
Unsprung mass
The unsprung mass, in kg. Mostly affects the wheel-hop resonance. Raising it by 10 % (to 44) moves the result by +0.246 %, from 1.337 to 1.341.
Angular frequency
The angular frequency of the road input, in rad/s. Moves along the response curve: low values give about 1, the body resonance gives a peak, and high values give a falling response. Raising it by 10 % (to 11) moves the result by −21.5 %, from 1.337 to 1.05.
Worked example with the values above
- Factors at ω = 10: first = −9,000 + 12,000i, second = 172,000 + 12,000i.
- Numerator: kt(ks + i cs ω) = 2.56×109 + 1.92×109i.
- H = −0.27945 − 1.3078i, so |H| = 1.3373.
Assumptions and limits
- A linear two-mass model with a point-contact tyre that never leaves the road and with no tyre damping.
- Displacements measured from static equilibrium, so gravity cancels, and a sinusoidal or sampled input as stated.
Common mistakes
- Entering the frequency in hertz instead of rad/s.
- Adding tyre damping that the formula does not contain, and then expecting the response to match.
- Reading the magnitude as the acceleration transmissibility: this is displacement transfer.
Statistics and uncertainty
Tyre stiffness varies by 10 % or more with pressure, load and speed, and damper behaviour is usually nonlinear, so computed responses give trends and frequency estimates rather than exact amplitudes.
Where it is used
- Ride, comfort and road-holding studies of passenger and off-road vehicles.
- Choosing spring and damper values for a quarter-car model.
Maths you need
- Newton’s second law: m z̈ = Σ forces
- Eigenvalue problem: natural frequencies from the system matrices
- Complex frequency response: amplitude and phase at a driven frequency