Road profile mathematical modeling

The road is the most intensive source of excitation of the vehicle. The vehicle vertical dynamic behaviour (oscillations) depends on a series of factors, most of them related to the road: length, height, shape, irregularities frequency, etc., and on the vehicle speed. Also, the chassis parameters have a big influence on the vehicle’s vertical dynamics.

Every road has a profile of irregularities (small up and downs), which can be periodic or random (stochastic). Most of the real roads have a random profile or irregularities. For vertical dynamics study we can simply the road profile to match a sinusoidal signal or a periodic signal.

Legend:
h [m] – road height
h₀ [m] – road irregularity amplitude
L [m] – road irregularity length
T [s] – road irregularity period
t [s] – time
x [m] – distance

We can consider that the road has a sinusoidal shape on the direction of the travel. In this case, the profile is function of the length (L) and the amplitude (h₀) of the road irregularities, for a given distance (x). We can also consider that the repetition of the road irregularities has a period (T) for a give time (t).

We can describe the road profile with a sine function:

\[h(x)=h_0 sin(\omega t + \epsilon)\]

where:
ε [rad] – is the phase offset
ω [rad/s] – is the angular frequency

The angular frequency (ω) can be expressed function on the road irregularity length (L) and vehicle speed (v):

\[\omega = \frac{2 \pi v}{L}\]

Using Xcos (from Scilab) we can design a model that generates the sinusoidal profile of the road. The output (road height profile in time) can be used for quarter car mathematical models, to simulate the oscillations of the wheel and chassis for a sinusoidal road input.

The time (t) is generated with a Time block. For the angular frequency (ω) we use a Gain block, with the parameter set to 1. The value can be replace with a function which takes into account the irregularity length (L) and vehicle speed (v).

The amplitude (h₀) is set to a constant 0.02 m, also inside a Gain block..

The model generates a sinusoidal signal the road height function of time:

\[h(t) = h_0 sin(\omega t + \epsilon)\]

The simulation is run for 10 seconds. The h(t) signal is in the plot below.

In the table below you can find some statistical data for road irregularity length and amplitude, for different types of roads. You can use them to generate different types of road profiles, for different simulation scenarios.

Road type	h0 [mm]	L [m]
Motorway / Highway	10 – 20	10 – 15
Urban roads (asphalt concrete)	10 – 20	1.0 – 2.0
Pavement roads (cobbles)	30 – 40	0.15 – 0.30
Offroad	50 – 70	0.10 – 0.15

Most of the time, the profile of a real road is different than a perfect sinusoidal shape. A more realistic road profile can be generated using a periodic signal as a sum of several harmonics. The mathematical model can be represented as a Fourier series which is a sum of sine functions with different frequencies and amplitudes.

\[h(t)=b_0 + b_1 sin(\omega t + \epsilon_1)+ b_2 sin(2 \omega t + \epsilon_2) + …+ b_n sin(n \omega t + \epsilon_n)\]

where:
b₀, b₁, b₂, … , b_n [m] – are the amplitudes of the harmonic components (known also as Fourier coefficients)
ε₁, ε₂, … , ε_n [rad] – are the phase offsets for each harmonic

Using Xcos (from Scilab) we can model the above equation of the road profile height function of time h(t), which can be further used in more complex wheel dynamics studies.

For simplification, in this model, we considered that all the phase offsets are 0, b₀ is 0 and ω is 1. The order of the harmonics is 6.

Time Time block generates a time signal which is duplicated six times. The calculation of the harmonics is done on a vector basis, at the end all six harmonics are summed together giving the final periodic signal h(t). The plot of the 10 seconds of simulation is in the image below.

In reality, most of the time, road irregularities can not be expressed as a periodic function. The road height profile in time is random (stochastic). Adding more harmonics to the periodic signal, represented as a Fourier series, could improve the representative of the signal, but still it will not be random.

These mathematical models for the road height function of time will be used in future articles, as road input signals for wheel dynamics study models (e.g. quarter car model).

For any questions or observations regarding this tutorial please use the comment form below.

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