In Theory

Permanent waves: road power spectral density

In the last post, we saw how bigger wheels offer better performance when riding over obstacles. In the next installment of the series, we’ll investigate if the same can be said about road roughness. But to do so in an exhaustive way, we will need two concepts that are quite interesting on their own. I don’t want to cramp everything into a single article, as it would make it unnecessarily long and steal some of the spotlight from the main topic. I thus decided to dedicate a separate piece to each.

Explain it to me like I’m Michael Scott

Regardless if you are a motorist or not, you must know from everyday experience that no road is perfectly flat. Instead, a typical road displays changes in elevation, and on various scales. On the largest of those, a road follows the topography of the surrounding terrain, raising and falling with the ground below. Standing from afar, you’d be hard pressed to notice anything else. Zooming in, you’d start noticing smaller undulations, bumps, cracks, and potholes. Come even closer, and suddenly what seemed to be a smooth surface unfolds into hills and valleys formed by the aggregates comprising the asphalt. Now get a microscope out, and you’d see even finer details.

Every road displays surface elevation changes on different scales, from kilometers to microns and below.

Notice that the magnitude of each of those features changes with the scale. For instance, road elevation can change dozens of meters over a kilometer due to terrain changes. On the other hand, bumps are rarely bigger than a few centimeters, while aggregate grains are of the order of millimeters.

If we wanted to describe the relationship between the feature scale and their magnitude for a particular road, we could do so in various ways, one of which is by defining its power spectral density (or just spectral density). The name might be slightly misleading, as it has no relation to actual power. However, it tells us a lot about road quality and its influence on passengers and safety issues (more on that later). For instance, a road that has undulations of the order of several meters will likely be despised by people prone to motion sickness. On the other hand, elevation changes happening over millimeters and less will greatly affect tire grip and noise levels. All that can be read from power spectral density graphs, as we shall see below.

A formal approach

Imagine a probe following a road. Let’s denote by x the position of the probe along the road (measured horizontally), and by y(x) the reported height. The probe is assumed to be so precise, that it is able to pick even the smallest height differences, even down to the surface porosity and below.

We now define the autocorrelation function:

    \[\bar{P}(\bar{x}) = \frac{1}{L} \int_{-\frac{L}{2}}^{\frac{L}{2}} y(x)y(x+\bar{x}) \, dx\]

where L is the length of the road.

The above formula assumes that the origin of the frame of reference in which we measure the road is at its halfway. This is convenient if we wanted to analyze a road of infinite length (as some mathematicians might be tempted to do), in which case we simply consider the limit of L \rightarrow \infty.

A power spectral density for a road is defined simply as a Fourier transform of the autocorrelation function:

    \[S(k) = \int_{-\infty}^{\infty}\bar{P}(\bar{x}) e^{-2 \pi i k \bar{x}} \, d \bar{x}\]

where k is the wavenumber, expressed in cycles per unit of length (usually cycles per meter). Also note that in the SI system, the unit of S(k) is m^2/(cycle/m).

But why is it defined like that, and where does the word ‘power’ come from? The answer to both questions can be found in the signal processing theory. Imagine that instead of road height, we considered e.g. a radio transmission from a local broadcaster. It turns out that in this case, the radio signal power is distributed across different frequencies proportionally to the spectral power density as defined above, hence the name.

Real roads and ISO profiles

If the goal of introducing the spectral power density was to describe the road quality, why can’t we just use a Fourier transform? After all, it, too, gives us information on which wavelengths contribute to the road surface and to what degree. The problem, however, is that it conveys too much information. Indeed, knowing the road Fourier transform we could perform its inverse and recreate the exact road profile. In this way, we could just as well not bother with it at all.

The spectral density function, on the other hand, gives an overview of the wavelengths present in the road profile, but without stating the phase shifts between them. The reasoning behind it is that road surfaces will be perceived as similar by humans as long as their spectral densities are the same. Although one can argue that this might not always be the case, it is true in most real-world scenarios. And, after all, it’s better to have a measure that might not be perfect than nothing at all.

This is exactly the approach that the International Organization for Standardization took when it published its standard no. 8608. Their research showed that the spectral density of most roads can be characterized by the same function of the following form:

    \[S(k) = a \left(\frac{k_0}{k}\right)^2\]

where a is a constant and k_0 is a constant equl to 0.1 cycles/m. Additionally, ISO 8608 specifies value ranges for a, dividing roads into eight different categories, A (smoothest) to H (roughest), with each consecutive class being twice as bumpy as the preceeding one.

Power spectral density for different road classes as defined in ISO 8608 standard.
Power spectral density for different road classes as defined in ISO 8608 standard. Dashed lines separate different classes, while solid ones denote geometric mean values.

Power spectral density and comfort

Riding on an uneven road causes vibrations which are transmitted to the vehicle and, in turn, its occupants. If the passengers are exposed to vibrations long enough, they will start to feel discomfort. The onset depends on vibration frequency and amplitude and differs somewhat between individuals. In general, however, the bigger the amplitude, the less time it takes for an oscillation to become unbearable.

Roadgoing vehicles can be ridden for hours at a time, therefore comfort is, unsurprisingly, a major design consideration (even for race cars). Spectral density analysis is a powerful tool to assess vehicle performance in this regard. They allow us to directly estimate the spectra of vibrations a car or motorcycle is subject to depending on the travel velocity. Indeed, this amounts to simply rescaling the axes on the road power spectral density graphs. In turn, suspension transfer functions can be measured and compared between designs. Furthermore, it’s possible to compare performance without necessarily using the same road in the same physical location, driving the costs down.

As a final remark, note that even though the power spectra defined by ISO 8608 theoretically span all wavenumbers from 0 to infinity, in practice, we can safely omit everything above a certain number. First, the distance between the atoms in the asphalt is of the order of 10^{-10} m, so according to the Nyquist-Shannon theorem, higher frequencies cannot be represented in the road profile. But even at much smaller wavenumbers, the oscillation amplitudes are much below the threshold of human perception.

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