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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

  • 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

    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

    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)

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

    Time (Min)ch1-head (124)( )

  • 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

    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

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

    Time (Min)ch1-head (124)ch2-arm (2537)

  • 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

    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)

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

    ( )ch1-head (124)

  • 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

    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)

    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

    Shaker headBrass bar

    4

    5

    4

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