what is random vibration

7/31/2019 What is Random Vibration

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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 musicaltones are sine waves (simple, repetitive, oscillating motion of the air) at a particular

frequency. More complicated musical sounds arise from overlaying a number of sinewaves of different frequencies at the same time. Sine waves are important in more areas

than music. Every substance vibrates and has particular frequencies (resonantfrequencies) in which it vibrates with the greatest amplitude. Therefore sine wave

vibration is important to help understand how any substance vibrates naturally.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

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Sine Wave: Displacement vs. Time

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


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The vibration testing industry has made good use of sine vibrations to help assessthe frequencies at which a particular device under test (DUT) resonates. These resonant

frequencies are important to the vibration testing engineer because these resonantfrequencies 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 alimited 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 onesequipment. 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 samedata 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 afew loose bolts that need to be tightened.

Random Vibration:

Vibrations found in everyday life scenarios (vehicle on common roadway; rocketin 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 foundin a vehicle traveling on Chicago Drive near Hudsonville, MI (Figure 2). Note that the

vibrations are by no means repetitive.


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0 50 100 150 200 250 300-2.5

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Time (sec)

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Acceleration Waveform: Dashboard Vibration: Real Test 2

Chicago Dr.: Real Data: 080205

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 repetitiveor 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 theother hand, consists of all the frequencies in the defined spectrum being sent to the shaker

at any given time. Consider Tustins description of random vibration. Ive heard peopledescribe 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 haveconstant 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 berandom, the amplitude of each slice would have to vary randomly, unpredictably. . . .

Unpredictable variations are what we mean by random. Broad-spectrum randomvibration 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 sineand 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 floodlampit will illuminate a wide range of the backyard but more dimly than the spotlight. A sine


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vibration test, similar to the spotlight, will only test one frequency. A random vibrationtest, 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 testswould 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 aDUT 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 companywill 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 theirproducts 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 becauserandom simultaneously includes all the forcing frequencies and simultaneously excites

all our products resonances.3

Under a sinusoidal test a particular resonant frequencymight 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 resonantfrequencies 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 causeboth 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 vibrationspectrum. Generally the random vibration spectrum profile is displayed as a power

spectrum a plot of acceleration spectral density (acceleration squared per Hertz) versusfrequency (Figure 3). A power spectrum essentially shows which frequencies contain the

datas power.


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2001 10 100

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Frequency (Hz)

Acceleration(G/Hz)

Acceleration P rofile

Demand

Control

Figure 3 Amount ofpowerper unit (density) of frequency (spectral) as a function of thefrequency

The PSD demonstrates how hard the shaker is working. It doesnt 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 havegenerated 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 comeclose to real-life vibrations that the DUT will experience.

The idea that an infinite number of real-time waveforms could generate aparticular 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 fromthe data in both cases is exactly the same, yet generated from different waveforms.


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0 100 200 300 400 500 600 700 800-15

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Acceleration Waveform

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

0 100 200 300 400 500 600 700 800-15

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Acceleration Waveform

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


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5 10 100 1000

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/Hz)

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Figure 6 PSD spectrum for trial 2005Jun28 - 1330

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Demand

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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 therange of acceleration values, the Power Spectrum Density is converted intoan 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). Infact, 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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-5 -4 -3 -2 -1 0 1 2 3 4 510

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Sigma

ProbabilityDensity

Probability Density Function for K = 3

2005Jun30 -1117k3

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 valuesthan 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 theaccelerations that were of extremely large magnitude. These higher accelerations, whichare present in real-life scenarios, are omitted from the probability density graphs of all

those who use the traditional Gaussian distribution method. Consequently, present-daymethods of random testing are unrealistic because they fail to take into account these

higher level accelerations. Furthermore, random testing with Gaussian distribution willresult 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 allits advantages over traditional sine testing, has its own disadvantages, and a better

method of testing products would prove valuable. Vibration Research Corporation hasdeveloped 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 moreefficient, 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 userandom testing extensively in their testing procedures.


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Field-Data Replication (FDR):

Development in vibration research has resulted in newer methods that comecloser 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 whichoverlapped sine spectra with random spectra. The goal of this testing is to include somepeaks 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 andturning 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 ishappening 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 itsshortcomings. It is difficult to find a representative waveform, especially in aerospace

applications. When one is obtained, it is representative of a particular situation of theproduct. 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 aDUTs 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 pointswill 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. Theprobability distribution curve takes on the classic smooth bell-shaped curve. Consider

the Probability Density Function (PDF) graph shown below for a set of data withGaussian distribution.


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-5 -4 -3 -2 -1 0 1 2 3 4 510

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Sigma

ProbabilityDensity


2005Jun30 -1117k3

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 standarddeviation 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 lessfamiliar 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 alow kurtosis value indicates most data points fall near the mean with few and smalloutlier 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 amplitudeoutliers. 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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-8 -6 -4 -2 0 2 4 6 810

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ProbabilityDensity


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

-8 -6 -4 -2 0 2 4 6 8

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Multiple of Sigma

ProbabilityDensity

Probability Density Function

Kurtosis = 3

Kurtosis = 7

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


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Therefore, the fundamental difference between a Gaussian distribution and akurtosis 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 closelycentered 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 andGaussian distribution have to vibration testing? Present-day methods of random testing

assume a Gaussian mode of distribution of random data. Modern controllers run randomvibration 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 Gaussianwaveform 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 amplitudesexceeding 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 byaccelerations 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 cantherefore 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 thekurtosis 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 thekurtosis 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). Thismethod will permit the adjustment of the kurtosis levels while maintaining the same

testing profile and spectrum attributes. With this new technique, in addition to thestandard 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 tothe 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 fielddata.

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.

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