# Vibration analysis metrics: Kurtosis and Skewness

In addition to quantitative metrics such as peak, peak-to-peak, and RMS, which we have commented here on the blog before, it is possible to extract metrics regarding the shape and asymmetry of a waveform signal.

Since a digital signal is made up of discrete points in time, we can treat any vibration signal as a statistical distribution of acceleration, velocity, or displacement points.

## Vibration signals and the Gauss Curve

You have probably heard of the normal, or Gaussian, distribution. This is one of the most widely used probability distributions for modelling natural phenomena.

Analyzing the following figure, the widest range of this distribution, corresponding to 3 sigmas, represents about 99.74% of the values of a signal. This region is known as the natural range of variation of the process.

In statistics, it is understood that a process always has a certain degree of variability, that is, it operates within a range of values, with a certain variation. If the process is stable, that means that the variation will occur within this range of values.

Applying this concept to machine vibration, if we have a random signal in acceleration, we will have a normal distribution similar to the following:

## Kurtosis and Skewness

The metrics of Kurtosis and Skewness (also translated as distortion or asymmetry) are two important statistical parameters. These parameters are used to qualitatively describe the shape of a vibration signal.

In the context of vibration analysis, these parameters are used to quantify the peaks or flattening of a signal, as well as the degree of symmetry around its mean value.

### Kurtosis

The kurtosis in a vibration signal can be interpreted graphically by looking at the distribution of the signal values, as shown in the following figure.

The distribution shape of a signal with a normal or random distribution, as showed earlier, has a kurtosis value equal to 3.

A distribution with a higher kurtosis will have a more tapered shape, with a concentration of values around its mean. On the other hand, a distribution with a lower kurtosis will have a flat shape, with values spread over a larger range.

Shortly, kurtosis expresses how similar the values of a signal are. Therefore, in case of a signal with many distinct peaks or impacts, the kurtosis will be higher.

Let’s look at some examples:

In a new bearing, due to the lack of defects, a rather random signal is expected, such as the one shown in the following waveform. In this case, as we have already seen, the kurtosis value will be close to 3.

As defects in a bearing emerge and evolve, the impact signals directly modify signal distribution, this leads to a change in the kurtosis value. As we can see below, the signal has a high kurtosis value of 10.

In all cases, from the waveform, it is possible to extract the kurtosis in terms of acceleration, velocity, displacement, and envelope, making the analysis even richer and more pertinent.

### More examples of kurtosis applications

Time comparisons are also valid for identifying defect evolutions. Take the following example, an exciter of a vibrating screen that had a bearing defect (inner race), as illustrated in the spectral envelope below:

When we look at the kurtosis metric in time acceleration for this monitoring point, focusing on a high frequency band (1000 Hz to 6400 Hz) we see that the value has increased significantly. As the bearing defect evolves, we can notice the increased presence of peaks in the waveform signal.

It is important to emphasize that these waveform parameters do not replace spectral analysis; in fact, they are complementary. The best vibration analysis will therefore be done by crossing all techniques for a better diagnosis.

### Skewness

Another parameter that can be used in the perception of an asset’s health is skewness. As mentioned earlier, it is the distortion or asymmetry of a signal, and it can also be interpreted graphically by looking at the shape of the signal’s distribution.

Thus, a distribution with positive asymmetry (positive values more prominent in the waveform) will have a longer ‘tail’ to the right of the mean, while a distribution with negative asymmetry (negative values more prominent in the waveform) will have a longer ‘tail’ to the left of the mean.

Going back to the example of the defective bearing signal that we analyzed earlier, if the signal were fully symmetric, the value of the skewness would equal zero.

However, as we can see from the waveform, the signal is trending downward, toward the negative half of the graph. This characteristic of the signal causes the skewness value to be negative (-0.69) and, as can be seen in the distribution curve, the signal is shifted to the left. This may be a result of the bearing’s defect region or even the load distribution at the location.

Another example of a failure that can be detected with skewness analysis is rotor rubbing. It is generally characterized by a signal similar to the looseness signal; however, this signal has a certain level of truncation in the waveform, as can be seen below.

This is due to the contact of the rotating part with the stationary part of the equipment. In this example, you can also see that the asymmetry of the signal is tending towards more negative levels, leading to a skewness of -1.48 as can be seen in the signal statistics table.

Thus, any waveforms that have a non-symmetric shape with respect to the zero point of vibration tend to have sharp skewness values.

Knowing metrics such as kurtosis and skewness contributes to richer and more assertive signal analysis, especially in vibration signals. This and other tools are available in the DynaPredict Web Platform to contribute to your analyses.

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