What is Random Vibration? There is a degree of confusion about the different kinds of vibration tests available to the vibration testing engineer. Difficulties encountered usually center on the difference between sinusoidal vibration (sine testing) and random vibration testing. Sinusoidal Vibration (Sine):

Strike a tuning fork or pluck a guitar string and the sound you hear is the result of a single sinusoidal wave produced at a particular frequency (Figure 1). Simple musical tones are sine waves (simple, repetitive, oscillating motion of the air) at a particular frequency. More complicated musical sounds arise from overlaying a number of sine waves of different frequencies at the same time. Sine waves are important in more areas than music. Every substance vibrates and has particular frequencies (resonant frequencies) in which it vibrates with the greatest amplitude. Therefore sine wave vibration is important to help understand how any substance vibrates naturally.

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Figure 1: Representation of a sinusoidal wave. Note its repeatability and predictability.

The vibration testing industry has made good use of sine vibrations to help assess the frequencies at which a particular device under test (DUT) resonates. These resonant frequencies are important to the vibration testing engineer because these resonant frequencies are the frequencies at which the DUT vibrates with the greatest amplitude; and therefore, are the frequencies that are most harmful to the DUT.

Because “real-world” vibrations are not pure sine vibrations, sine testing has a limited place in the vibration testing industry. Part of the usefulness of sine testing is its simplicity, and therefore, it is a good point of entry into the study of vibrations.

Sine testing is used primarily to determine damage to equipment or product. According to Tustin “The best pro-sine reasons are to search for product resonances and then to dwell on one or more of those resonances to: 1. Study modal responses; 2. Determine fatigue life in each mode.”1

Besides testing a product to find and dwell at its resonant frequencies to

determine fatigue life, one might also use sine testing to determine damage to one’s equipment. A sine sweep prior to any shock or random test will identify any resonances of the equipment. Running a sine test after testing a product should produce the same data graphs. Any differences in the sweeps indicates damage to the equipment – perhaps something as simple as a shift in the natural resonant frequencies, possibly suggesting a few loose bolts that need to be tightened.

Random Vibration:

Vibrations found in everyday life scenarios (vehicle on common roadway; rocket

in take-off, or an airplane wing in turbulent airflow) are not repetitive or predictable like the sinusoidal wave. Consider the acceleration waveform for dashboard vibration found in a vehicle traveling on Chicago Drive near Hudsonville, MI (Figure 2). Note that the vibrations are by no means repetitive.

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Figure 2 – Data collected on vehicle dashboard in Hudsonville, MI

Thus, there is an important place in our testing of products for a test that is not repetitive or predictable. Random testing accomplishes this. Random vs. Sine:

Sinusoidal vibration tests are not as helpful as random testing is, because a sine

test essentially consists of a single frequency in time. A random vibration test, on the other hand, consists of all the frequencies in the defined spectrum being sent to the shaker at any given time. Consider Tustin’s description of random vibration. “I’ve heard people describe a continuous spectrum (random vibration, VRC), say 10-2000 Hz, as “1990 sine waves 1 Hz apart”. No. That is close, but not quite correct. . . . (S)ine waves have constant amplitude, cycle after cycle. . . . .Suppose that there were 1990 of them (constant amplitude sine waves, VRC). Would the totality be random? No. For the totality to be random, the amplitude of each slice would have to vary randomly, unpredictably. . . . Unpredictable variations are what we mean by random. Broad-spectrum random vibration contains not sinusoids but rather a continuum of vibrations (with different amplitudes, VRC).”2

Another analogy that is helpful to distinguish the key differences between sine and random vibrations is the spotlight/floodlamp analogy. If one has a spotlight in the backyard it will illuminate brightly a small area of the backyard. If one has a floodlamp it will illuminate a wide range of the backyard but more dimly than the spotlight. A sine

vibration test, similar to the spotlight, will only test one frequency. A random vibration test, on the other hand, similar to the floodlamp, will test a wide range of frequencies. Therefore, in order to accomplish the same degree of testing, multitudes of sine tests would need to be administered, where a simple random test accomplishes all of this in one test. Random vibration testing is, therefore, much more efficient and precise.

Advantages of Random Vibration Testing:

One of the main goals or uses of random vibration testing in industry is to bring a DUT to failure. For example, a company may desire to find out how a particular product may fail because of various environmental vibrations it may be faced with. The company will simulate those vibrations on a shaker and place their product under those conditions. Testing the product to failure will teach the company many important things about their product’s weaknesses and ways to improve the product. Random testing is the key testing method for this kind of application.

Random vibration is also more realistic than sinusoidal vibration testing because

random simultaneously includes all the forcing frequencies and “simultaneously excites all our product’s resonances.”3 Under a sinusoidal test a particular resonant frequency might be found for one part of the device under testing (DUT) and at a different frequency another part of the DUT may hit a resonant frequency. Arriving at resonant frequencies at different times may not cause any kind of failure, but when both resonant frequencies are “hit” at the same time a failure may occur. Random testing will cause both resonances to be “hit” at the same time because all frequencies in the testing range will be forced at the same time. Features of Random Vibration:

Power Spectrum Density (PSD): In order to perform random testing, a random test spectrum must be developed.

Computer software collects real-time data over a time period and combines the data using a spectrum averaging method to produce a statistical approximation of the vibration spectrum. Generally the random vibration spectrum profile is displayed as a power spectrum – a plot of acceleration spectral density (acceleration squared per Hertz) versus frequency (Figure 3). A power spectrum essentially shows which frequencies contain the data’s power.

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Figure 3 – Amount of power per unit (density) of frequency (spectral) as a function of the frequency

The PSD demonstrates how hard the shaker is working. It doesn’t give any direct information about the forces experienced by the DUT. This is important to remember. Since the PSD is the result of an averaging method that produced the statistical approximation of the spectrum, an infinite number of real-time waveforms could have generated such a PSD. Thus, at any time during a test, it is impossible to know specifically from the PSD what forces the DUT is experiencing. The need for the PSD is that it aids the tester in making an appropriate test profile for the shaker that will come close to real-life vibrations that the DUT will experience.

The idea that an infinite number of real-time waveforms could generate a particular PSD can be seen from the following graphs (Figures 4 through 7) produced from data collected at VRC on June 28 and 30, 2005. Note that the PSD spectra formed from the data in both cases is exactly the same, yet generated from different waveforms.

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Figure 4 – Waveform for Body and IP Profile Lightbulb-4 for trial 2005Jun28 – 1330

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Figure 5 – Waveform for Body and IP Profile Lightbulb-4 for trial 2005Jun30 – 1110

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Figure 6 – PSD spectrum for trial 2005Jun28 - 1330

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Figure 7 – PSD spectrum for trial 2005Jun30-1110

Probability Density Function (PDF) Graphs: An examination of the acceleration waveforms (Figures 4 and 5) will indicate that

much of the random vibration acceleration values are nearly the same (± 5 G). However, some of the acceleration values are quite large compared to the normal values. To help illustrate the range of acceleration values, the Power Spectrum Density is converted into an amplitude probability density graph (PDF) (similar to Figure 8). Notice how much of the acceleration values fall near the average acceleration value (represented by 0 Sigma). In fact, much of the vibrations in the “real-world” approximate a Gaussian distribution, that is, a distribution in which the vast majority of the data is in the ± 3 sigma range.

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Figure 8 – Probability density for a Light bulb test using Gaussian distribution (k=3)

There are some “real-life” cases in which there are more high acceleration values

than a Gaussian distribution would show. Unfortunately, most modern techniques covert the PSD into a PDF that assumes that the majority of the data is in the ± 3 sigma range (i.e. Gaussian distribution). This assumption removes from the real-time data the accelerations that were of extremely large magnitude. These higher accelerations, which are present in real-life scenarios, are omitted from the probability density graphs of all those who use the traditional Gaussian distribution method. Consequently, present-day methods of random testing are unrealistic because they fail to take into account these higher level accelerations. Furthermore, random testing with Gaussian distribution will result in a longer time-period to test the product to failure because the higher accelerations responsible for failure have been omitted. Therefore, random testing, for all its advantages over traditional sine testing, has its own disadvantages, and a better method of testing products would prove valuable. Vibration Research Corporation has developed ways to improve upon traditional Gaussian distributions in random testing. This new patent-pending technique is called Kurtosion™. Overall, random testing is an excellent tool for vibration testing. It is more efficient, more precise, and more realistic than sine testing. And, although random testing is not perfectly realistic and can be improved upon, testing industry ought to use random testing extensively in their testing procedures.

Field-Data Replication (FDR):

Development in vibration research has resulted in newer methods that come closer and closer to real-life data replication. Random testing is a great improvement on sine testing but still does not perfectly represent what happens in real-life. In response to this, vibration research companies developed “Sine-on-Random” testing which overlapped sine spectra with random spectra. The goal of this testing is to include some “peaks” that occur in real-life scenarios into the random spectra. This method has been somewhat successful in bringing tests closer to reality.

More recent development has included a method of recording real-life data and

turning it directly into a spectrum to be used in lab. This method, called field-data replication (FDR), is very helpful in accurately representing in a test setting what is happening on the field. This is like “shaped” random in a way, because the spectrum is the same as seen in the real application. This method is good, but also has its shortcomings. It is difficult to find a representative waveform, especially in aerospace applications. When one is obtained, it is representative of a particular situation of the product. Unfortunately, it is probably not representative of the entire life of the product. Gaussian Distribution vs. Kurtosis Distribution: One final helpful distinction is that relating to the probability distributions of a DUT’s vibratory accelerations. As mentioned earlier, a probability distribution shows the reader how the data points compare with the average data point. Most of the data points will center near the average with a number of outliers. Generally, as more data points are collected the probability distribution forms a nice smooth bell-shaped curve. Gaussian distribution is the normal probability distribution of random data. The probability distribution curve takes on the classic “smooth bell-shaped curve”. Consider the Probability Density Function (PDF) graph shown below for a set of data with Gaussian distribution.

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Figure 9 – Probability density for a Light bulb test using Gaussian distribution (k=3)

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. In the previous figure (Figure 9), the data set has a kurtosis value of 3 (Gaussian distribution) and is a smooth curved distribution with few large amplitude outliers. However, Figure 10 shows a data set with a kurtosis value of 5. Note how the “tails” extend further from the mean (indicating large number of outlier data points). The contrast between the PDFs of a Gaussian distribution and a higher kurtosis distribution is clearly seen in Figure 11.

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Figure 10 – Probability density for Light bulb test using Kurtosis Control (k=5)

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Figure 11 – A comparison of kurtosis values 3 and 7. Note how the higher kurtosis value includes higher sigma values (higher peak accelerations).

Therefore, the fundamental difference between a Gaussian distribution and a kurtosis distribution is that, although the two data sets may have the same mean, standard deviation and other properties, yet the Gaussian data set has its data points closely centered on the mean while the kurtosis distribution has larger “tails” further from the mean. Implications of Kurtosis Distribution to Vibration Testing:

The obvious question is “what significance does kurtosis distribution and Gaussian distribution have to vibration testing?” Present-day methods of random testing assume a Gaussian mode of distribution of random data. Modern controllers run random vibration tests with the majority of the RMS values near the mean RMS level, thus vibrating the product only for a short time at peak RMS values. In fact, a Gaussian waveform will instantaneously exceed three times the RMS level only 0.27% of the time. When measuring field data, the situation can be considerably different, with amplitudes exceeding three times the RMS level as much as 1.5% of the time. This difference can be significant, since it has also been reported that most fatigue damage is generated by accelerations in the range of two to four times the RMS level.4 Significantly reducing the amount of time spent near these peak values by using a Gaussian distribution can therefore result in significantly reducing the amount of fatigue damage caused by the test relative to what the product will experience in the real world. Gaussian distribution, therefore, is not very realistic.

A better method of testing products than using the Gaussian distribution of data is to adjust the distribution of data to more closely fit the real-world data by adjusting the kurtosis level. The difference between the Gaussian distribution and a higher kurtosis value is simply the amount of time spent at or near the peak levels. Adjusting the kurtosis level to match the measured field level will result in a more realistic test.

A latest modification in random vibration testing is a closed-loop method of

kurtosis control, developed by Vibration Research Corporation (patent-pending). This method will permit the adjustment of the kurtosis levels while maintaining the same testing profile and spectrum attributes. With this new technique, in addition to the standard random test PDF and RMS parameters, a kurtosis parameter is now defined to produce a test in the lab. This is similar to current random tests but is one step closer to the vibrations measured in the field. This kurtosis parameter can be easily measured from field data in the same manner as the RMS and PDF are currently determined from field data.

1 Tustin, Wayne. Random Vibration and Shock Testing. Equipment Reliability Institute, Santa Barbara, CA, 2005, pg 205.

2 Ibid, pg 234-235. 3 Ibid, pg 224. 4 Connon, W.H., “Comments on Kurtosis of Military Vehicle Vibration Data,”

Journal of the Institute of Environmental Sciences, September/October 1991, pp. 38-41.

Think of it this way: Often, we like to measure something over a period of time, such as how many cars go through an intersection. Then, we can average the number of cars over the length of an hour or over a number of days. Because we're doing this over a period of time, we call this the time domain. You all know that probability and statistics are used for such investigations. In fact, these results would be called a Probability Density.

Random vibration analysis looks at random accelerations or forces over a range of frequencies, which we call the frequency domain. (These random inputs are merely sustaned over a period of time, but are not time-dependent; i.e., the longer the period of time, the better the statistical sampling in the frequency domain.) The range of frequencies is called a spectrum. Therefore, we call these results a Spectral Density. Generically, we use Power Spectral Density*, although this isn't exactly correct. If we are looking at accelerations, we use Acceleration Spectral Density (ASD), and for forces, we use Force Spectral Density (FSD).

The chart shows the area under the curve as gray. This area is actually what we are interested in. If we take the square root of this area, we then have the root mean square value of the acceleration, better known as Grms. This value is what we use in our analysis calculations, for example in our stress calculations. If the area is large due to a high response, then we may have problems; likewise, if the area is small, then we have a small Grms value and we shouldn't have any problems.

Deriving any of these quantities is far beyond the scope of what I'm trying to explain here, so I won't bother. I just hope my simple explanation itself wasn't far beyond the scope of what I am trying to expalin.

* The term Power in Power Spectral Density seems to come from the fact that when random vibration measurements were taken, they were actually recorded electronically and so the power levels were used in the calculations.

http://analyst.gsfc.nasa.gov/ryan/MOLA/random.html