Random vibration testing has been around for quite some time. But traditional random vibration testing is Gaussian in nature. The averaging method used in random vibration testing produces a Gaussian distribution of data in which the highest peak accelerations are approximately +/- 3 times the average acceleration. Gaussian distribution has been universally accepted as a legitimate averaging technique, in part, because it has been assumed that “real-world” data is Gaussian.
However, several studies have determined that “real-world” data in actually non-Gaussian. Since this is true, the question is how can one make a random vibration test more realistic?
What is Gaussian Distribution?
With the use of statistics, one can find a number of interesting things about a set of collected data. For example, one can easily compute the mean and the standard deviation of a data set – statistical concepts familiar to most people communicating the average of the data set and the range in which most of the data points fall. But a less familiar statistical concept is the kurtosis of the data set. Kurtosis is a measure of the “peakyness” of the probability distribution of the data. For example, a high kurtosis value indicates the data is distributed with some very large outlier data points, while a low kurtosis value indicates most data points fall near the mean with few and small outlier data points.
If one evaluates a random data set, one will find that the data points will fall within +/- 3 standard deviations of the average (mean) (Figure 1). This is defined as a Gaussian distribution (kurtosis = 3).
Figure 1: Random vibration test waveform and Probability Density Function (PDF) for a Gaussian data set.
Modern controllers, utilizing a Fourier transform calculation, produce a Gaussian waveform. The Fourier transform calculation of the original Gaussian waveform produces results that have a Gaussian distribution. It must be understood, therefore, that traditional random vibration testing averages the data in such a way as to produce a Gaussian distribution of data.
What is Non-Gaussian (Kurtosis) Distribution?
In the PDF below, note the comparison between the two data sets. In one, a data set has a kurtosis value of 3 (Gaussian distribution) and is a smooth curved distribution with few large amplitude outliers. The other however, is a data set with a kurtosis value of 7 (Non-Gaussian distribution). Note how the “tails” extend further from the mean (indicating large number of outlier data points) (Figure 2).
Figure 2: PDF plot of a Gaussian data set (k=3) and a Non-Gaussian data set (k=7)
Non-Gaussian distribution (kurtosis > 3) of data is a distribution of the data so that there are many more points that are far from the mean than are found in a Gaussian distribution.
The question that is pertinent to vibration testing is “Can a test be produced that has a non-Gaussian distribution, while maintaining the same average energy and the same power spectral density (PSD)?”
Suppose the following table represents tests scores from a class of 26 students. Suppose the first row represents the first test the students took (average 79.5) and the second row represents the second test the students took (average also 79.5). The first data set, if plotted as a histogram, produces a Gaussian bell-shaped curve, while the second data set, if plotted as a histogram, produces a non-Gaussian curve (Figure 3).
Figure 3: Gaussian (Test Scores #1) and Non-Gaussian distribution of test scores (Test Scores #2)
As you can tell, data can have the same average but a different distribution around that average. Consequently, in a data set from a vibration test, if one could take some of the energy away from the middle sections of the PDF tails (+/- 1 sigma from the average) and distribute that energy at +/- 4 sigma, then the extreme tails will contain more energy than before while keeping the total energy under the curve the same. By doing so, one can distribute the data in ways other than the traditional Gaussian distribution.
When should I add Kurtosion to my Random Vibration Test?
Since “real-world” data is not always Gaussian, it is wise to adjust your test settings so that the distribution of data as processed by the controller is a Non-Gaussian distribution (while maintaining the same PSD (energy of the test at each frequency). As demonstrated above, this is possible. It requires an increase in the kurtosis value of the data set.
There are two situations in which one would want to utilize Kurtosion (a patented method to increase the kurtosis of the test). First, if an engineer wants to have the test be more representative of real-life vibrations, the test engineer should adjust the kurtosis of the test to match the measured kurtosis value of the real-life data. Secondly, if an engineer is trying to fatigue an item to the point that it breaks, one can bring the failure mode more quickly by running the test at a higher kurtosis setting. This is a valid technique because the same amount of energy will be applied to the product (not making the shaker work harder) as before, but the higher acceleration values will be present in the test. These large acceleration values are what will break the product. Since more of them are present in the test than in the Gaussian situation, the test will cause failure sooner than normal.
In summary, Kurtosion should be added to a Random vibration test:
- When the vibration test needs to be more realistic to real-life situations
- When the desire is to use a vibration test to more quickly break a product without using more energy (working the shaker harder)
