Random Vibration Kurtosis Control John G. Van Baren
[email protected] www.VibrationResearch.com
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Random Kurtosis Page 1 October 2007 - Detroit
Presentation Summary ♦ What is kurtosis? ♦ Why are we interested in kurtosis? ♦ Kurtosis in the resonance? ♦ Papoulis Rule / Central Limit Theorem. ♦ Test-Shaker with Resonating bar bar. ♦ Control Random ED shaker kurtosis? ♦ How does this relate to the real world?
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Random Kurtosis Page 2 October 2007 - Detroit
The Problem Definition of the Problem ♦ Traditional random testing does not always find failures that occur during the life of a product. ♦ 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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Random Kurtosis Page 3 October 2007 - Detroit
Kurtosis
♦ Show of hands: Who has heard of the term “Kurtosis” before today? ♦ Definition in terms of statistical moments ¾Mean is the 1st moment ¾Variance is the 2nd moment ¾Skewness is normalized 3rd central moment ¾K t i is ¾Kurtosis i normalized li d 4th central t l momentt
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Random Kurtosis Page 4 October 2007 - Detroit
Calculating Kurtosis ♦ The basic function for calculating g kurtosis for zero-mean data is: average(data4) average(data2)2 ♦ Different p people p normalize this value in different ways ¾ As commonly used, Gaussian kurtosis = 3 ¾ Microsoft Mi ft E Excell subtracts bt t 3, 3 so G Gaussian i kkurtosis t i =0 ¾ Others divide by 3, so Gaussian kurtosis = 1
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Random Kurtosis Page 5 October 2007 - Detroit
Traditional Random Testing ♦ Current Cu e random a do testing es g seeks to achieve a Gaussian distribution ¾ “Normal” distribution ¾ Concentrated around mean ¾ Low probability of extreme values ¾ Kurtosis = 3
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Random Kurtosis Page 6 October 2007 - Detroit
What is Missing on ED Shaker Controllers?
♦ Random vibration controllers have 2 basic “knobs”: knobs : ¾ Frequency content - Power Spectral Density (PSD) ¾ Amplitude level - RMS
♦ Need a third ‘knob’ to adjust the Kurtosis ¾ Allows adjustment of the PDF (probability density function) ¾ Increasing kurtosis = increasing peak levels ¾ Allows the damage-producing potential of the test to be adjusted independent of the other two controls.
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Random Kurtosis Page 7 October 2007 - Detroit
The Missing Knob
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Kurtosis Control ♦ Objective is to control the amplitude distribution to achieve the higher g p peaks seen in field data ¾ Spectrum is a measure of the frequency content ¾ RMS is a measure of the amplitude p ¾ Kurtosis is a measure of the “peakiness”
♦ Solution is use a non-Gaussian vibration and control the Kurtosis ♦ This is what we call KurtosionTM
¾ Method to simultaneously control Spectrum, RMS, and Kurtosis p g ¾ Patent-pending
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Random Kurtosis Page 9 October 2007 - Detroit
PDF Varies with Kurtosis
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Random Kurtosis Page 10 October 2007 - Detroit
Increased Kurtosis = More Time at Peaks Kurtosis = 3 is > 3σ 0.27% of time Kurtosis = 4 is > 3σ 0.83% of time Kurtosis K t i = 7 iis > 3 3σ 1.5% of time
Note: 1.5% of a 1 hour test is nearly a full minute above 3σ
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Random Kurtosis Page 11 October 2007 - Detroit
Increased Kurtosis = Higher Peaks
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 test.
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Waveform Comparison
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Useful Properties for Kurtosis Control
♦ Set kurtosis independent of RMS. ♦ Set S t kurtosis k t i without ith t affecting ff ti PSD PSD. ♦ Increase kurtosis over the full spectrum. ♦ Dynamic D i range off th the controller t ll iis maintained. i t i d ♦ Apply kurtosis even in a resonance. ♦ Note Papoulis Rule requirements.
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Random Kurtosis Page 14 October 2007 - Detroit
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 y weak limit. ♦ Bound is constant across all values of the CDF. Weak limit on the tails of the distribution which give Kurtosis ♦ Practical results of Papoulis’ Rule ¾ Kurtosis at the resonance gets reduced from the kurtosis of the excitation signal ¾ For practical Q factors, there will still be some significant kurtosis at the resonance.
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Random Kurtosis Page 15 October 2007 - Detroit
Resonating Bar Test Setup
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Random Kurtosis Page 16 October 2007 - Detroit
Accelera ation (G²/Hz)
10,000 Hz Transition Frequency Acceleration Profile 1x10 0 1x10-1 1x10-2 1x10-3 1x10-4
Prrobability Densiity
ch2-arm (2537) ch1-head (124) Control Demand
10
100
1000
2000
Frequency (Hz)
Probability y Density y Function 0.100000 0.010000 0.001000 0.000100 0.000010 -80
-60
-40
-20
Kurtos is ch2-arm ((2537)) ch1-head (124)
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0
20
40
60
Acceleration (G)
ch2-arm (2537) ch1-head (124)
9 K ch1: 6.7 8 K ch2: 3.47 7 6 5 4 3 2 1 0
Kurtosis v s. Time
1
2
3
4
5
6
Time (Min)
Random Kurtosis Page 17 October 2007 - Detroit
Accelerattion (G²/Hz)
1,000 Hz Transition Frequency Acceleration Profile 1x10 0 1x10-1 1 10-2 1x10 1x10-3 1x10-4 10 (2537)
100
Probability Density y
ch2-arm ch1-head (124) Control Demand
1000
2000
60
80
Frequency (Hz)
Probability Density Function 0.100000 0.010000 0.001000 0.000100 0 000010 0.000010 -80
-60
-40
-20
Kurtosis s ch2-arm (2537) ch1-head (124)
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0
20
40
Acceleration (G)
ch2-arm (2537) ch1-head (124)
9 8 7 6 5 4 3 2 1
Kurtosis v s. Time
K ch1: 6.84 K ch2: 3.34
0
1
2
3
4
5
6
Time (Min)
Random Kurtosis Page 18 October 2007 - Detroit
Accelerattion (G²/Hz)
100 Hz transition Frequency Acceleration Profile 1x10 0 1x10-1 1x10-2 1x10-3 1x10-4
Pro obability Densit y
ch2-arm (2537) ch1-head (124) Control Demand
10
100
1000
2000
Frequency (Hz)
Probability Density Function 0.100000 0.010000 0.001000 0.000100 0.000010 -100
-80
-60
-40
-20
Kurtos is www.vibrationresearch.com
20
40
60
80
100
Acceleration (G)
ch2-arm (2537) ch1-head (124)
ch2-arm (2537) ch1 head (124) ch1-head
0
9 8 7 6 5 4 3 2 1
Kurtosis v s. Time K ch1: 7.29 K ch2: 4.13
0
1
2
3
4
5
6
Time ((Min))
Random Kurtosis Page 19 October 2007 - Detroit
Acceleration (G²/Hz)
10 Hz Transition Frequency Acceleration Profile 1x10 0 1x10-1 1x10-22 1x10-3 1x10-4
Prrobability Density
ch2-arm (2537) ch1-head (124) Control Demand
10
100
2000
Frequency (Hz)
Probability Density Function 0.100000 0.010000 0.001000 0.000100 0.000010 -150
-100
-50
Kurtos sis ch2-arm (2537) ch1-head (124)
0
50
100
150
4
5
6
Acceleration (G)
ch2-arm (2537) ch1-head (124)
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1000
16 14 12 10 8 6 4 2 0
Kurtosis v s. Time K ch1: 6.38 K ch2: 5.98
0
1
2
3
Time (Min)
Random Kurtosis Page 20 October 2007 - Detroit
Effect of Transition Frequency on PDF 0
Probability Density Function for Brass Bar ch.2 (Arm)
10
Gaussian 10000 Hz Transition 10 Hz Transition
-1
Probability Densitty
10
-2 2
10
-3
10
-4
10
-15
-10
-5
0 Multiple of Sigma
5
10
15
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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Random Kurtosis Page 21 October 2007 - Detroit
Effect of Transition Frequency on Kurtosis Kurtosis vs. Transition Frequency on Brass Bar 5.5
Shaker head Brass bar
5
Kurtosis
4 4.5
4
3.5
3 1 10
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2
3
10 10 Transition Frequency (Hz)
4
10
Random Kurtosis Page 22 October 2007 - Detroit
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 Brass Bar B
10 2000 Hz 10-2000 H
10 H Hz
5
5 78 5.78
50 5.0
Results from Brass Bar Vibrations where the end of the bar was controlled and the head responded. responded Lower Transition Frequency allows controller to easily produce the desired kurtosis value at resonance.
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Random Kurtosis Page 23 October 2007 - Detroit
Light Bulb Test - revisited ♦ 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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Random Kurtosis Page 24 October 2007 - Detroit
Previous Test Kurtosis Results Kurtosis Comparison
Consider the Kurtosis Comparison Chart shown here for the 22 bulbs at each kurtosis level.
200 180 160 140 120
Minutes to Failure 100 80 60
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40 20 0 K=7 K=5 K=3
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 Light Bulb Number
16
17
18
19
20
21
22
K=7 K K=5 K=3
The time it took to complete a test decreased dramatically as the kurtosis level increased.
Random Kurtosis Page 25 October 2007 - Detroit
Previous Test Kurtosis Results (cont) Kurtosis Comparison 70
Consider the graph of the Mean Minutes to Failure vs. Kurtosis Level as shown here
60
50
40 Mean Minutes to Failure 30
20
The increased kurtosis level dramatically decreased the mean time it took for the light bulbs to fail.
10
0 K=7 K=5 Kurtosis Level
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K=3
Random Kurtosis Page 26 October 2007 - Detroit
New Light Bulb Test
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Random Kurtosis Page 27 October 2007 - Detroit
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 62 93 102 41 41 45 52 18 36 44 45
7 25 30 37 11 13 20 27 19 21 28 32
12 12 13 21 10 15 17 19 11 14 15 20
6 15 20 24 5 12 13 14 8 8 9 14
2 6 8 10 2 2 4 11 2 3 4 10
50 9 50.9
22 5 22.5
14 9 14.9
12 3 12.3
53 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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Random Kurtosis Page 28 October 2007 - Detroit
Minutes to Failure
Average Time for Light bulb Failure 60
50
40
Minutes to Failure
30
20
10
0 Gaussian
K 5 K=5 TF=10 kHz
Varied Transition Frequency
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K=5 TF=100 Hz
K=5 TF=10 Hz
K=5 TF=1 Hz
Random Kurtosis Page 29 October 2007 - Detroit
What is the Kurtosis of the Real World?
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Random Kurtosis Page 30 October 2007 - Detroit
Kurtosis Values of All Tests
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Random Kurtosis Page 31 October 2007 - Detroit
PDF of a Real Life Environment Oldsmobile Bravada, dashboard vibration As vehicle travels down I-196.
Probability Density Function for Gaussian, and actual, and controlled kurtosis
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Random Kurtosis Page 32 October 2007 - Detroit
Random Test with PDF of a Real Life Environment Spectrum defined by field measured data Traditional random test with 3 sigma clipping
Same spectrum defined by field measured data Now with Kurtosis Control set to 5
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Random Kurtosis Page 33 October 2007 - Detroit
Time Waveform Illustrations 60 seconds of some actual field data K=5 K 5.7 7
60 seconds of some Traditional random K=3
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60 seconds of K=5.7 “Kurtosion” random
Random Kurtosis Page 34 October 2007 - Detroit
Gravel Road Vibration Waveform Gravel road 1.5
1
Acceleration (G A G)
0.5
0
-0.5
-1
-1.5 0
50
100
150
Time (sec)
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Random Kurtosis Page 35 October 2007 - Detroit
Gravel Road Spectrum
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Random Kurtosis Page 36 October 2007 - Detroit
Effect of Resonance on Kurtosis
Filtered Gravel Road Data, Frequency = 300 Hz 6.5
6
Ku urtosis
5.5
5
4.5
4
3.5 -3 10
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-2
10
-1
0
10 10 Filter Bandwidth (Hz)
1
10
2
10
Random Kurtosis Page 37 October 2007 - Detroit
Conclusions ♦ Papoulis rule, while true, only forces pure Gaussian random on an infinitely narrow resonance 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 test, real-world world measured events. ♦ To significantly g y accelerate the failure of yyour p product during testing, you should increase the kurtosis past the standard of k = 3 that is used today.
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Random Kurtosis Page 38 October 2007 - Detroit
References ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦
Van Baren, John “How to get Random Peaks Back to 120G,” TEST Engineering & Management, August/September 2007. Van Baren, Baren John “Kurtosis: Kurtosis: The Missing Dashboard Knob 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 Smallwood, d D David id O O.: “G “Generating ti N Non-Gaussian G i Vibration Vib ti ffor T Testing ti Purposes,” Sound and Vibration, October 2005, pp 18-23. Connon, W.H., “Comments on Kurtosis of Military Vehicle Vibration , Journal of the Institute of Environmental Sciences,, Data,” 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 1997, pp 33-43 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. www.sandv.com d h PDF has PDFs off Sound S d and d Vib Vibration ti articles. ti l
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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 Baren
[email protected] www.VibrationResearch.com
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Random Kurtosis Page 43 October 2007 - Detroit