Random Vibration Kurtosis · PDF fileRandom Vibration Kurtosis Control John G. Van BarenJohn...

Random Vibration Kurtosis Control

John G. Van BarenJohn G. Van Baren

[email protected]

www.VibrationResearch.com

Random Kurtosis October 2007 - Detroitwww.vibrationresearch.comwww.vibrationresearch.com

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?

Random Kurtosis October 2007 - Detroitwww.vibrationresearch.com

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


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


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


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


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.


The Missing Knob


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


p g

PDF Varies with Kurtosis


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


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


test.

Waveform Comparison


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.


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.


Resonating Bar Test Setup


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)


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


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


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


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)


( )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


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)


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).


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)


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.


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


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.


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.


New Light Bulb Test


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.


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


What is the Kurtosis of the Real World?


Kurtosis Values of All Tests


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


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


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


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)


Gravel Road Spectrum


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)


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.


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


♦ www.sandv.com has PDFs of Sound and Vibration articles.

October 2005 Sound and Vibration Magazine


Oct/Nov 2005 Test Magazine


August/September 2007 Test Magazine


Thank You

John G. Van BarenJohn G. Van Baren

[email protected]

www.VibrationResearch.com


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