Random Vibration Kurtosis Control · Random Vibration Kurtosis Control John G. Van BarenJohn G. Van...
Transcript of Random Vibration Kurtosis Control · Random Vibration Kurtosis Control John G. Van BarenJohn G. Van...
Random Vibration Kurtosis Control
John G. Van BarenJohn G. Van Baren
www.VibrationResearch.com
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Presentation Summary
♦ What is kurtosis?♦ Why are we interested in kurtosis?♦ Kurtosis in the resonance?♦ Kurtosis in the resonance?♦ Papoulis Rule / Central Limit Theorem.♦ Test-Shaker with Resonating bar♦ Test-Shaker with Resonating bar.♦ Control Random ED shaker kurtosis?♦ How does this relate to the real world?♦ How does this relate to the real world?
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The Problem
Definition of the ProblemDefinition of the Problem
♦ Traditional random testing does not always find♦ Traditional random testing does not always find failures that occur during the life of a product.
♦ This is likely because the product experiences♦ This is likely because the product experiences high G forces in actual use that are higher than traditional random testing generates
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Kurtosis
♦ Show of hands: Who has heard of the term♦ Show of hands: Who has heard of the term “Kurtosis” before today?
♦ Definition in terms of statistical momentsMean is the 1st momentMean is the 1 momentVariance is the 2nd momentSkewness is normalized 3rd central momentK t i i li d 4th t l tKurtosis is normalized 4th central moment
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Calculating Kurtosis
♦ The basic function for calculating kurtosisgfor zero-mean data is:
average(data4)average(data2)2average(data2)2
♦ Different people normalize this value in different p pways
As commonly used, Gaussian kurtosis = 3Mi ft E l bt t 3 G i k t i 0Microsoft Excel subtracts 3, so Gaussian kurtosis = 0Others divide by 3, so Gaussian kurtosis = 1
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Traditional Random Testing
♦ Current random testing ♦ Cu e a do es gseeks to achieve a Gaussian distribution
“Normal” distributionConcentrated around meanLow probability of extremeLow probability of extreme valuesKurtosis = 3
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What is Missing on ED Shaker Controllers?
♦ Random vibration controllers have 2 basic “knobs”:♦ Random vibration controllers have 2 basic knobs :Frequency content - Power Spectral Density (PSD)Amplitude level - RMS
♦ Need a third ‘knob’ to adjust the KurtosisAllows adjustment of the PDF (probability density function)Increasing kurtosis = increasing peak levelsIncreasing kurtosis = increasing peak levelsAllows the damage-producing potential of the test to be adjusted independent of the other two controls.
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The Missing Knob
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Kurtosis Control
♦ Objective is to control the amplitude distribution to achieve the higher peaks seen in field datag p
Spectrum is a measure of the frequency contentRMS is a measure of the amplitudeKurtosis is a measure of the “peakiness”p
♦ Solution is use a non-Gaussian vibration and control the Kurtosiscontrol the Kurtosis
♦ This is what we call KurtosionTM
Method to simultaneously control Spectrum, RMS,and KurtosisPatent-pending
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p g
PDF Varies with Kurtosis
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Increased Kurtosis = More Time at Peaks
Kurtosis = 3 is > 3σ0.27% of time
Kurtosis = 4 is > 3σ0.83% of time
K t i 7 i > 3Kurtosis = 7 is > 3σ1.5% of time
Note: 1.5% of a 1 hour test is nearly a full minute above 3σ
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Increased Kurtosis = Higher Peaks
Crest Factor:Crest Factor:The ratio of peak to rms
Note: Crest factor of the 99 994%99.994% probability level is plotted, as this is the 4 times rms (4 sigma) level for a kurtosis=3 random test. Also, this is the typical maximum peak seen on a kurtosis=3 random t t
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test.
Waveform Comparison
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Useful Properties for Kurtosis Control
♦ Set kurtosis independent of RMS.S t k t i ith t ff ti PSD♦ Set kurtosis without affecting PSD.
♦ Increase kurtosis over the full spectrum.D i f th t ll i i t i d♦ Dynamic range of the controller is maintained.
♦ Apply kurtosis even in a resonance.♦ Note Papoulis Rule requirements.
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Papoulis Rule
♦ Papoulis’ Rule states that filtered waveform tends towards Gaussian
♦ Bound is proportional to the 14th root of the filter bandwidth.This is an extremely weak limit. y
♦ Bound is constant across all values of the CDF.Weak limit on the tails of the distribution which give Kurtosis
♦ Practical results of Papoulis’ RuleKurtosis at the resonance gets reduced from the kurtosis of the excitation signalexcitation signalFor practical Q factors, there will still be some significant kurtosis at the resonance.
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Resonating Bar Test Setup
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10,000 Hz Transition Frequency
-21x10
-11x10
01x10at
ion
(G²/
Hz) Acceleration Profile
ity Probability Density Function
200010 100 1000
-41x10
-31x10
Frequency (Hz)
Acc
eler
a
DemandControlch1-head (124)ch2-arm (2537)
0.0000100.0001000.0010000.0100000.100000
roba
bilit
y D
ensi
y y
789
is
Kurtosis v s. TimeK ch1: 6.7K ch2: 3.47
-80 -60 -40 -20 0 20 40 60
Acceleration (G)
Pr
ch1-head (124)ch2-arm (2537)
0 1 2 3 4 5 6123456
Time (Min)
Kur
tos
ch2-arm (2537)
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Time (Min)ch1-head (124)( )
1,000 Hz Transition Frequency
-21 10
-11x10
01x10tio
n (G
²/H
z) Acceleration Profiley Probability Density Function
200010 100 1000
-41x10
-31x10
-21x10
Frequency (Hz)
Acc
eler
at
DemandControlch1-head (124)ch2-arm (2537)
0 0000100.0001000.0010000.0100000.100000
babi
lity
Den
sity Probability Density Function
89
s
Kurtosis vs. TimeK ch1: 6.84K ch2: 3.34
-80 -60 -40 -20 0 20 40 60 80
0.000010
Acceleration (G)
Pro
ch1-head (124)ch2-arm (2537)
0 1 2 3 4 5 61234567
Kur
tosi
s
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Time (Min)ch1-head (124)ch2-arm (2537)
100 Hz transition Frequency
-21x10
-11x10
01x10tio
n (G
²/H
z) Acceleration Profiley Probability Density Function
200010 100 1000
-41x10
-31x101x10
Frequency (Hz)
Acc
eler
at
DemandControlch1-head (124)ch2-arm (2537)
0.0000100.0001000.0010000.0100000.100000
obab
ility
Den
sit Probability Density Function
789
is
Kurtosis vs. TimeK ch1: 7.29K ch2: 4.13
-100 -80 -60 -40 -20 0 20 40 60 80 100
Acceleration (G)
Pro
ch1-head (124)ch2-arm (2537)
0 1 2 3 4 5 6123456
Time (Min)
Kur
tos
ch1 head (124)ch2-arm (2537)
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( )ch1-head (124)
10 Hz Transition Frequency
-21x10
-11x10
01x10on
(G²/H
z) Acceleration Profile
Probability Density Function
200010 100 1000
-41x10
-31x10
21x10
Frequency (Hz)
Acc
eler
ati
DemandControlch1-head (124)ch2-arm (2537)
0.0000100.0001000.0010000.0100000.100000
roba
bilit
y D
ensi
ty
Probability Density Function
10121416
sis
Kurtosis v s. TimeK ch1: 6.38K ch2: 5.98
-150 -100 -50 0 50 100 150
Acceleration (G)
Pr
ch1-head (124)ch2-arm (2537)
0 1 2 3 4 5 602468
10
Time (Min)
Kur
tos
ch1-head (124)ch2-arm (2537)
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Effect of Transition Frequency on PDF
100Probability Density Function for Brass Bar ch.2 (Arm)
Gaussian
2
10-1
ty10000 Hz Transition10 Hz Transition
10-3
10-2
Pro
babi
lity
Den
sit
10-4
-15 -10 -5 0 5 10 15Multiple of Sigma
PDF for the Brass Bar (Arm) Data for tests with Gaussian distribution, Kurtosis = 5 (Transition 10,000 Hz) and Kurtosis = 5 (Transition 10 Hz).
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Effect of Transition Frequency on Kurtosis
5.5Kurtosis vs. Transition Frequency on Brass Bar
Shaker headBrass bar
4
5
4
4.5
Kur
tosi
s
3.5
101
102
103
104
3
Transition Frequency (Hz)
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Control on Resonant Beam
OBJECT TRANSITION FREQUENCY
KURTOSIS SETTING
KURTOSIS HEAD
KURTOSIS ARM
Brass Bar 10-2000 Hz 10000 Hz 5 10.5 3.83
Brass Bar 10-2000 Hz 10000 Hz 5 11.0 3.38
Brass Bar 10-2000 Hz 10000 Hz 5 10.3 3.85
Brass Bar 10-2000 Hz 10 Hz 5 4.99 5.63
Brass Bar 10-2000 Hz 10 Hz 5 4.80 4.69
B B 10 2000 H 10 H 5 5 78 5 0Brass Bar 10-2000 Hz 10 Hz 5 5.78 5.0
Results from Brass Bar Vibrations where the end of the bar was controlled and the head respondedthe head responded.
Lower Transition Frequency allows controller to easily produce the desired kurtosis value at resonance.
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Light Bulb Test - revisited
♦ Previous papers examined the effect of♦ Previous papers examined the effect of increasing kurtosis on failure of light bulbs.
♦ As kurtosis was increased, failure time ,decreased.
♦ Now, we run a test with constant kurtosis, and , ,vary the transition frequency
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Previous Test Kurtosis Results
200
Kurtosis Comparison
Consider the Kurtosis Comparison Chart shown
120
140
160
180Comparison Chart shown here for the 22 bulbs at each kurtosis level.
60
80
100
120
Minutes to Failure
1 2 3
0
20
40
K=7
The time it took to complete a test decreased dramatically as the kurtosis level
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
K=7 K=
5 K=3
Light Bulb Number
K=5K=3
increased.
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K
Previous Test Kurtosis Results (cont)
70
Kurtosis Comparison
Consider the graph of the
50
60
Consider the graph of the Mean Minutes to Failure vs. Kurtosis Level as shown here
30
40
Mean Minutes to Failure
10
20
The increased kurtosis level
K = 7K = 5
K = 3
0
Kurtosis Level
dramatically decreased the mean time it took for the light bulbs to fail.
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New Light Bulb Test
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Minutes to Failure
Gaussian 1.00 X RMS
Kurtosis 5 TF:10000 Hz
Kurtosis 5 TF: 100 Hz
Kurtosis 5 TF: 10 Hz
Kurtosis 5 TF = 1 Hz
32 7 12 6 262 25 12 15 693 30 13 20 8102 37 21 24 1041 11 10 5 241 13 15 12 245 20 17 13 452 27 19 14 1118 19 11 8 236 21 14 8 344 28 15 9 445 32 20 14 10
50 9 22 5 14 9 12 3 5 350.9 22.5 14.9 12.3 5.3
Light bulb failure times at different Transition Frequency values. Note that as the Transition Frequency value decreases the time to failure also decreases.
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Minutes to Failure
Average Time for Light bulb Failure
40
50
60
20
30
40
Minutesto Failure
Gaussian K=5
0
10
K 5TF=10 kHz K=5
TF=100 Hz K=5TF=10 Hz K=5
TF=1 HzVaried Transition Frequency
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What is the Kurtosis of the Real World?
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Kurtosis Values of All Tests
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PDF of a Real Life Environment
Oldsmobile Bravada, dashboard vibrationAs vehicle travels down I-196.
Probability Density Function forProbability Density Function for Gaussian, and actual, and controlled kurtosis
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Random Test with PDF of a Real Life Environment
Spectrum defined by field measured dataTraditional random test with 3 sigma clipping
Same spectrum defined by field measured dataNow with Kurtosis Control set to 5
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Time Waveform Illustrations
60 seconds of some actual field data K=5 7actual field data K 5.7
60 seconds of someTraditional random K=3
60 seconds of K=5.7“Kurtosion” random
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Gravel Road Vibration Waveform
1.5Gravel road
0.5
1
G)
0
Acc
eler
atio
n (G
-1
-0.5
A
0 50 100 150-1.5
Time (sec)
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Gravel Road Spectrum
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Effect of Resonance on Kurtosis
6
6.5Filtered Gravel Road Data, Frequency = 300 Hz
5
5.5
urto
sis
4
4.5
Ku
10-3 10-2 10-1 100 101 1023.5
4
Filter Bandwidth (Hz)
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Conclusions
♦ Papoulis rule, while true, only forces pure Gaussian random on an infinitely narrow resonancerandom on an infinitely narrow resonance.
♦ You can increase the kurtosis of the vibration even at a product’s resonance by paying attention to the transition frequency.
♦ To significantly increase the reliability of your random test you should correlate your kurtosis to real worldtest, you should correlate your kurtosis to real-world measured events.
♦ To significantly accelerate the failure of your product g y y pduring testing, you should increase the kurtosis past the standard of k = 3 that is used today.
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References
♦ Van Baren, John “How to get Random Peaks Back to 120G,” TEST Engineering & Management, August/September 2007.
♦ Van Baren John “Kurtosis: The Missing Dashboard Knob ” TEST♦ Van Baren, John Kurtosis: The Missing Dashboard Knob, TEST Engineering & Management, October/November 2005, pp 14-16.
♦ Van Baren, Philip “The Missing Knob on Your Random Vibration Controller,” Sound and Vibration, October 2005, pp 10-17.S ll d D id O “G ti N G i Vib ti f T ti♦ Smallwood, David O.: “Generating Non-Gaussian Vibration for Testing Purposes,” Sound and Vibration, October 2005, pp 18-23.
♦ Connon, W.H., “Comments on Kurtosis of Military Vehicle Vibration Data,” Journal of the Institute of Environmental Sciences, , ,September/October 1991, pp 38-41.
♦ Steinwolf, A. , “Shaker simulation of random vibration with a high kurtosis value,” Journal of the Institute of Environmental Sciences, May/Jun 1997 pp 33-43May/Jun 1997, pp 33 43.
♦ Steinwolf, A. and Connon W.H., "Limitations on the use of Fourier transform approach to describe test course profiles," Sound and Vibration, Feb. 2005, pp. 12-17.
d h PDF f S d d Vib ti ti l
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♦ www.sandv.com has PDFs of Sound and Vibration articles.
October 2005 Sound and Vibration Magazine
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Oct/Nov 2005 Test Magazine
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August/September 2007 Test Magazine
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Thank You
John G. Van BarenJohn G. Van Baren
www.VibrationResearch.com
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