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• Random Vibration Kurtosis Control

John G. Van BarenJohn G. Van Baren

jvb@vibrationresearch.com

www.VibrationResearch.com

Random Kurtosis Page 1October 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 Page 2October 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

Random Kurtosis Page 3October 2007 - Detroitwww.vibrationresearch.comwww.vibrationresearch.com

• 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

Random Kurtosis Page 4October 2007 - Detroitwww.vibrationresearch.com

• 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

Random Kurtosis Page 5October 2007 - Detroitwww.vibrationresearch.com

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

Random Kurtosis Page 6October 2007 - Detroitwww.vibrationresearch.comwww.vibrationresearch.com

• 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.

Random Kurtosis Page 7October 2007 - Detroitwww.vibrationresearch.comwww.vibrationresearch.com

• The Missing Knob

Random Kurtosis Page 8October 2007 - Detroitwww.vibrationresearch.comwww.vibrationresearch.com

• 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 peakinessp

Solution is use a non-Gaussian vibration and control the Kurtosiscontrol the Kurtosis

This is what we call KurtosionTMMethod to simultaneously control Spectrum, RMS,and KurtosisPatent-pending

Random Kurtosis Page 9October 2007 - Detroitwww.vibrationresearch.comwww.vibrationresearch.com

p g

• PDF Varies with Kurtosis

Random Kurtosis Page 10October 2007 - Detroitwww.vibrationresearch.comwww.vibrationresearch.com

• Increased Kurtosis = More Time at Peaks

Kurtosis = 3 is > 30.27% of time

Kurtosis = 4 is > 30.83% of time

K t i 7 i > 3Kurtosis = 7 is > 31.5% of time

Note: 1.5% of a 1 hour test is nearly a full minute above 3

Random Kurtosis Page 11October 2007 - Detroitwww.vibrationresearch.comwww.vibrationresearch.com

• 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

Random Kurtosis Page 12October 2007 - Detroitwww.vibrationresearch.com

test.

• Waveform Comparison

Random Kurtosis Page 13October 2007 - Detroitwww.vibrationresearch.comwww.vibrationresearch.com

• 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.

Random Kurtosis Page 14October 2007 - Detroitwww.vibrationresearch.com

• 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.

Random Kurtosis Page 15October 2007 - Detroitwww.vibrationresearch.com

• Resonating Bar Test Setup

Random Kurtosis Page 16October 2007 - Detroitwww.vibrationresearch.com

• 10,000 Hz Transition Frequency

-21x10

-11x10

01x10at

ion

(G/

Hz) Acceleration Profile

ity Probability Density Function

200010 100 1000

-41x10

-31x10

Frequency (Hz)

Acc

eler

a

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

0 1 2 3 4 5 6123456

Time (Min)

Kur

tos

ch2-arm (2537)

Random Kurtosis Page 17October 2007 - Detroitwww.vibrationresearch.com

• 1,000 Hz Transition Frequency

-21 10

-11x10

01x10tio

n (G

/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

0 1 2 3 4 5 61234567

Kur

tosi

s

Random Kurtosis Page 18October 2007 - Detroitwww.vibrationresearch.com

• 100 Hz transition Frequency

-21x10

-11x10

01x10tio

n (G

/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

0 1 2 3 4 5 6123456

Time (Min)

Kur

tos

Random Kurtosis Page 19October 2007 - Detroitwww.vibrationresearch.com

• 10 Hz Transition Frequency

-21x10

-11x10

01x10on

(G/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

0 1 2 3 4 5 602468

10

Time (Min)

Kur

tos

Random Kurtosis Page 20October 2007 - Detroitwww.vibrationresearch.com

• 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).

Random Kurtosis Page 21October 2007 - Detroitwww.vibrationresearch.com

• Effect of Transition Frequency on Kurtosis

5.5Kurtosis vs. Transition Frequency on Brass Bar