Professor Darrell F. Socie - publish.illinois.edupublish.illinois.edu/.../06/4-Frequency-Based-Fatigue.pdfFrequency Based Fatigue © 2001-2011 Darrell Socie, All Rights Reserved 1

Professor Darrell F. SocieDepartment of Mechanical Science and Engineering

University of Illinois at Urbana-Champaign

© 2001-2010 Darrell Socie, All Rights Reserved

Frequency Based Fatigue

Frequency Based Fatigue © 2001-2011 Darrell Socie, All Rights Reserved 1 of 57

Deterministic – from past measurements the future position of a satellite can be predicted with reasonable accuracy

Random – from past measurements the future position of a car can only be described in terms of probability and statistical averages


Time (Secs)

Bracket.sif-Strain_c56

-750

750


Frequency (Hz)

0 20 40 60 80 100 120 140

102

ap_000.sif-Strain_b43

101

100

10-1


1500

-750

750

0

750

rf_000.sif-Strain_b43


Statistics of Time HistoriesTime Domain NomenclatureFrequency AnalysisPSD Based Fatigue Analysis


Mean or Expected ValueVariance / Standard DeviationRoot Mean SquareKurtosisSkewnessCrest FactorIrregularity Factor


Central tendency of the data

N

xXExMean

N

1ii

x


Dispersion of the data

N

)xx(XVar

N

1i

2i

)X(Varx

Standard deviation


N

xRMS

N

1i

2i

The rms is equal to the standard deviation when the mean is 0


Skewness is a measure of the asymmetry of the data around the sample mean. If skewness is negative, the data are spread out more to the left of the mean than to the right. If skewness is positive, the data are spread out more to the right. The skewness of the normal distribution (or any perfectly symmetric distribution) is zero.

3

N

1i

3i

N

)xx(XSkewness


Kurtosis is a measure of how outlier-prone a distribution is. The kurtosis of the normal distribution is 3. Distributions that are more outlier-prone than the normal distribution have kurtosis greater than 3; distributions that are less outlier-prone have kurtosis less than 3.

4

N

1i

4i

N

)xx(XKurtosis


The crest factor is the ration of the peak (maximum) value to the root-mean-square (RMS) value. A sine wave has a crest factor of 1.414.


Positive zero crossingPeak 7

4)P(E)0(EIF

IF 1 is narrow band signalIF 0 is wide band signal


RandomStochasticStationaryNon-stationaryGaussianNarrow-bandWide-band


The instantaneous value can not be predicted at any future time.


Stochastic processes provide suitable models for physical systems where the phenomena is governed by probabilities.


The properties computed over short time intervals, t + t, do not significantly vary from each other

t



Normally distributed around the mean




FourierFFTInverse FFTAutospectral densityTransfer Function


Fourier studied the mathematical theory of heat conduction. He established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions.


1kokoko )tkcos(b)tksin(aa)t(X

1kkoko )tkcos(cc)t(X

k

tjkS

oe]k[X)t(X

ak, bk, ck, XS[k] are Fourier coefficients

frequency, magnitude, and phase are all described by the coefficients


Time Frequency

The area under each spike represents the magnitude of the sine wave at that frequency.

f

The magnitude of the FFT depends on the frequency window f.


Frequency (Hz)0 20 40 60 80 100 120 140

16141210

86420

2/N



Frequency (Hz)

0 20 40 60 80 100 120 140

150100500

-50-100

-150

2/N



2/N


ck, k, and k o are all known

Time (Secs)-750

750


Function Units

Time History EU X(n)

Linear Spectrum EU S(n) = DFT(X(n))

AutoPower EU^2 AP(n) = S(n) · S(n)

PSD (EU^2)/Hz PSD(n) = AP(n) / ( Wf · f )

ESD (EU^2*sec)/Hz ESD(n) = AP(n) · T / ( Wf · f )


T 2T 3T nDTND

Dn

1i

2k

DDk fX

tNn1fS

ND Block size

Magnitude only no phase information


Frequency (Hz)

Log Magnitude-Power-Strain_c56

100 101 102

105

104

103

102

101

100

Average power associated with a 1 Hz frequency window centered at each frequency, f . Phase information is lost.


Frequency (Hz)

102

Log Magnitude-Linear-Strain_c56

100

101

100 101 102

Sometimes called Amplitude Spectral Density


Frequency (Hz)100 101 102

105

104

103

102

101

100

10-1

PSD

Linear Spectrum


Time (Secs)

800

600

400

200

0

-200

-400

EASE1.EDT-Strain_c56

Non-stationary signals


Frequency (Hz)0 20 40 60 80 100 120 140

103

102

101

100

10-1


m

input

output

Frequency (Hz)100 101 102

104

103

102

inputoutput


Stationary LoadingWindSea StateVibration


RandomGaussianStationary


Structuralmodel

Stresstime

history

RainflowMinerlinear

damageDurability

Loadtime

history

AccelerationPSD

Transferfunction

StressPSD

Time Domain

Frequency Domain


y(t)

m

k cx(t)

F(t)

2

2

2

2

dtxdm)t(Fzk

dtdzc

dtzdm

z(t) = y(t) – x(t)

2

22

nn2

2

dtxdz

dtdz2

dtzd

m

F(t)y(t)

)t(Fdt

ydm 2

2


2

n

22

n

o

21

1FXk

oFXk

n

= 1= 0.5

= 0.25

= 0.15


frequency

Gk( f )

fk

f

N

1kkk

nkn f)f(Gfm


0

2

mm)0(E

2

4

mm)P(E

40

22

mmm

)P(E)0(EIF

zero crossings

peaks

Irregularity factor


p( Si )

Si

S

Stress range

The probability P( Si ) of a stress range occurring between

S)S(p)S(Pis2SSand

2SS iiii


Cycles at level i ni = p( Si ) S NT

Total cycles NT = E( P ) T

Total time

b1

'f

iif S2

S)S(NFatigue life

Fatigue DamageN

1i b1

i

ib1

'f

N

1i if

i

S

)S(p)S2(ST)P(E

)S(NnD

Fatigue damage is determined by p( Si )


0

2

0 m8Sexp

m4S)S(p

Rayleigh distribution

This wide band signal has the same peak distribution as this narrow band signal

IF 1 is narrow band signal

IF 0 is wide band signal


0

2

32

2

221

m22ZexpZD

RZexp

RD

QZexp

QD

)S(p

0m2SZ

40

2

mmmIF

4

2

0

1m m

mmmX 2

2m

1 IF1IFX2D

R1DDIF1D

211

2211

21m

DDIF1DXIFR

1

21

D4RDDIF5Q

213 DD1D

p( S ) = f ( m0, m1, m2, m4 )



PSD

Time History

Stress Range


SN SlopeTH

PSD

NN

10 371

5 6.6

3 1.8

TH

PSD

NN

109

1.9

0.5

Dirlik Rayleigh


1

10

100

100

Cycles101 102 103 104 105 106 107

1000

n = 10

n = 5

n = 3nSDamage


Time (Secs)

50 100 150 200 250 300

750

500

250

0

-250

-500

-750

Bracket.sif-Strain_b43

0

1500

-750

750

0

750

rf_000.sif-Strain_b43


0

1500

-750

750

3.15

Damage


0

1500

-750

750

5.14

Damage


0

1500

-750

750

20.78

Damage

Frequency Based Fatigue

Documents

Professor Darrell F. Socie - publish.illinois.edupublish.illinois.edu/.../06/4-Frequency-Based-Fatigue.pdfFrequency Based Fatigue © 2001-2011 Darrell Socie, All Rights Reserved 1