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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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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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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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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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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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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Kurtosis Values of All Tests

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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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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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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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