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Professor Darrell F. Socie

Department of Mechanical Science and Engineering

University of Illinois at Urbana-Champaign

© 2001-2010 Darrell Socie, All Rights Reserved

Frequency Based Fatigue

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Deterministic versus Random

Deterministic – from past measurements the future positionof 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 andstatistical averages

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 Time Domain

 Time (Secs)

   S   t  r  a   i  n   (  u  s   t  r

  a   i  n   )

Bracket.sif-Strain_c56

-750

750

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Frequency Domain

Frequency (Hz)

0 20 40 60 80 100 120 140

102

   S   t  r  a   i  n   (  u  s   t  r  a

   i  n   )

ap_000.sif-Strain_b43

101

100

10-1

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Histogram Domain

1500

-750

750

0

750

   C  o  u  n   t  s

rf_000.sif-Strain_b43

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Outline

Statistics of Time Histories

 Time Domain Nomenclature

Frequency Analysis

PSD Based Fatigue Analysis

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Statistics of Time Histories

Mean or Expected Value

Variance / Standard Deviation

Root Mean Square

Kurtosis

Skewness

Crest Factor

Irregularity Factor

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Mean or Expected Value

Central tendency of the data

( )N

xXExMean

N

1ii

x∑====µ=

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Variance / Standard Deviation

Dispersion of the data

( )N

)xx(

XVar

N

1i

2

i∑=

−

=

)X(Varx

=σ

Standard deviation

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Root Mean Square

N

x

RMS

N

1i

2i∑

==

 The rms is equal to the standard deviationwhen the mean is 0

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Skewness

Skewness is a measure of the asymmetry of the dataaround the sample mean. If skewness is negative, thedata are spread out more to the left of the mean thanto the right. If skewness is positive, the data are spread

out more to the right. The skewness of the normaldistribution (or any perfectly symmetric distribution) is zero.

( )3

N

1i

3

i

N

)xx(

XSkewnessσ

−

= ∑=

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Kurtosis

Kurtosis is a measure of how outlier-prone a distribution is. The kurtosis of the normal distribution is 3. Distributions thatare more outlier-prone than the normal distribution have

kurtosis greater than 3; distributions that are lessoutlier-prone have kurtosis less than 3.

( ) 4

N

1i

4

i

N

)xx(

XKurtosis σ

−

=

∑=

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Crest Factor

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

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Irregularity Factor

Positive zero crossing

Peak 7

4

)P(E

)0(EIF ==

+

IF → 1 is narrow band signal

IF→

0 is wide band signal

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 Time Domain Nomenclature

Random

Stochastic

Stationary

Non-stationary

Gaussian

Narrow-band

Wide-band

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Random

 The instantaneous value can not be predicted at anyfuture time.

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Stochastic

Stochastic processes provide suitable models forphysical systems where the phenomena is governedby probabilities.

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Stationary

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

∆t

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Non-stationary

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Gaussian

Normally distributed around the mean

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Narrow-band

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Wide-band

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Frequency Domain Analysis

Fourier

FFT

Inverse FFT

Autospectral density Transfer Function

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Fourier 1768 - 1830

Fourier studied themathematical theory of heatconduction.He established the partialdifferential equationgoverning heat diffusion andsolved it by using infiniteseries of trigonometric

functions.

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Fourier Series

∑∞

=

ω+ω+=1k

okoko )tkcos(b)tksin(aa)t(X

∑∞

=

θ+ω+=1k

koko )tkcos(cc)t(X

∑∞

∞−=

ω=k

t jk

Soe]k[X)t(X

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

frequency, magnitude, and phase are all describedby the coefficients

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Fourier Transform

 Time

   M  a

  g  n   i   t  u   d  e

   F   F   T   M  a  g  n   i   t  u   d  e

Frequency

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

∆f 

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

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Fast Fourier Transform - FFT

Frequency (Hz)

0 20 40 60 80 100 120 140

16

14

12

108

6

4

2

0

      

   F   F   T      

   S   t  r  a

   i  n   (  u  s   t  r  a   i  n   )

∑=

θ+ω+=2/N

1kkoko )tkcos(cc)t(X

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Fast Fourier Transform - FFT

Frequency (Hz)

0 20 40 60 80 100 120 140

150

100

50

0

-50

-100

-150

   P   h  a  s  e   (   D  e  g  r  e  e  s   )

∑=

θ+ω+=2/N

1kkoko )tkcos(cc)t(X

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Inverse FFT

∑=

θ+ω+=2/N

1kkoko )tkcos(cc)t(X

ck, θk, and kωo are all known

 Time (Secs)

   S   t  r  a   i  n   (  u  s   t  r  a   i  n   )

-750

750

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Autospectral Density

Function Units

 Time History EU X(n)

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

AutoPower EU 2̂ AP(n) = S(n) · S(n)

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

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

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Calculating Autospectral Density

 T 2T 3T nD TND

( ) ( )∑=∆

=Dn

1i

2

k

DD

k f XtNn

1f S

ND Block size

Magnitude only no phase information

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Power Spectral Density - PSD

Frequency (Hz)

   S   t  r  a   i  n   (  u  s   t  r  a   i  n   ^   2   )   /   H  z

Log Magnitude-Power-Strain_c56

100 101 102

105

104

103

102

101

100

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

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Linear Spectrum

Frequency (Hz)

102

   S   t  r  a   i  n   (  u  s   t  r  a   i  n   )

Log Magnitude-Linear-Strain_c56

100

101

100 101 102

Sometimes called Amplitude Spectral Density

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Comparison

Frequency (Hz)

   S   t  r  a   i  n   (  u  s   t  r  a   i  n   ^   2   )

100 101 102

105

104

103

102

101

100

10-1

PSD

Linear Spectrum

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Frequency Domain Limitations

 Time (Secs)

800

600

400

200

0

-200

-400

   S   t  r  a   i  n   (  u  s   t  r  a   i  n   )

EASE1.EDT-Strain_c56

Non-stationary signals

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Linear Spectrum

Frequency (Hz)

0 20 40 60 80 100 120 140

103

102

101

100

10-1

   S   t  r  a   i  n   (  u  s

   t  r  a   i  n   )

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 Transfer Function

m

input

output

Frequency (Hz)

100 101 102

104

103

10

2

input

output

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Frequency Based Fatigue

Stationary LoadingWind

Sea State

Vibration

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Assumptions

RandomGaussianStationary

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Fatigue Analysis

Structuralmodel

Stresstime

history

RainflowMinerlinear

damageDurability

Loadtime

history

AccelerationPSD

 Transferfunction

StressPSD

?

 Time Domain

Frequency Domain

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Dynamics Model

y(t)

m

k cx(t)

F(t)

2

2

2

2

dt

xdm)t(Fzk

dt

dzc

dt

zdm −=++

z(t) = y(t) – x(t)

2

22

nn2

2

dt

xdz

dt

dz2

dt

zd−=ω+ζω+

m

F(t)

y(t)

)t(Fdt

ydm

2

2

=

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Nondimensional Amplitude

2

n

22

n

o

21

1

F

Xk

ωω

ζ+

 

  

 ωω

−

=

0

1

2

3

4

5

0 1 2 3 4

oF

Xk

nω

ω

ζ = 1

ζ = 0.5

ζ = 0.25

ζ = 0.15

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PSD Moments

frequency

   M  a  g

  n   i   t  u   d  e   2   /   H  z

Gk( f )

f k

δf 

∑=

δ=N

1kkk

n

kn f )f (Gf m

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Expected Values

0

2

m

m)0(E =

2

4m

m)P(E =

40

2

2

mm

m

)P(E

)0(EIF ==

zero crossings

peaks

Irregularity factor

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Probability Density Function

p( ∆Si )

∆Si

δS

   P  r  o   b  a

   b   i   l   i   t  y   d  e  n  s   i   t  y

Stress range

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

S)S(p)S(Pis

2

SSand

2

SS iiii δ∆=∆

δ+∆

δ−∆

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Fatigue Damage

Cycles at level i ni = p( ∆Si ) δS N T

 Total cycles N T = E( P ) T

 Total time

b

1

'f 

iif 

S2

S)S(N

 

  

 ∆=∆Fatigue life

Fatigue Damage

( )∑∑ == ∆∆δ=∆=

N

1i b

1

i

ib

1

'f 

N

1i if 

i

S)S(p)S2(S T)P(E

)S(NnD

Fatigue damage is determined by p( ∆Si )

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Narrow Band Solution

∆−

∆=∆

0

2

0 m8

Sexp

m4

S)S(p

Rayleigh distribution

 This wide band signal hasthe same peak distribution

as this narrow band signal

IF → 1 is narrow band signal

IF → 0 is wide band signal

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Dirlik Solution

0

2

32

2

2

21

m2

2

ZexpZD

R

Zexp

R

D

Q

Zexp

Q

D

)S(p

 

  

 −+

 

  

 −+

 

  

 −

=∆

0m2

SZ

∆=

40

2

mm

mIF=

4

2

0

1m

m

m

m

mX =

2

2

m1

IF1

IFX2D

+

−=

R1

DDIF1D

2

112 −

+−−=

2

11

2

1m

DDIF1

DXIFR

+−−

−−=

( )

1

21

D4

RDDIF5Q

−−=

213 DD1D −−=

p( ∆S ) =f ( m0, m1, m2, m4 )

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Loading History

Frequency (Hz)

0 20 40 60 80 100 120 140

102

   S   t  r  a   i  n 

   (  u  s   t  r  a   i  n   )

101

100

10-1

Frequency (Hz)

0 20 40 60 80 100 120 140

102

   S   t  r  a   i  n 

   (  u  s   t  r  a   i  n   )

101

100

10-1

 Time (Secs)

   S   t  r  a   i  n

    (  u  s   t  r  a   i  n   )

-750

750

 Time (Secs)

   S   t  r  a   i  n

    (  u  s   t  r  a   i  n   )

-750

750

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Rainflow Ranges

0.01

0.1

1

10

100

1000

10000

0 250 500 750 1000

PSD 

 Time History

Stress Range

   N  u  m   b  e  r  o   f   C

  y  c   l  e  s

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Relative Fatigue Estimates

SN Slope TH

PSD

N

N

10 371

5 6.6

3 1.8

 TH

PSD

N

N

109

1.9

0.5

Dirlik Rayleigh

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Fatigue Data

1

10

100

   A  m  p   l   i   t  u

   d  e

100

Cycles101 102 103 104 105 106 107

1000

n = 10

n = 5

n = 3nSDamage ∆∝

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Loading History

 Time (Secs)

50 100 150 200 250 300

750

500

250

0

-250

-500

-750

   S   t  r  a   i  n   G  a  g  e   (  u  s   t  r  a   i  n   )

Bracket.sif-Strain_b43

0

1500

-750

750

0

750

   C  o  u  n   t  s

rf_000.sif-Strain_b43

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Slope = 3

0

1500

-750

750

3.15

   %

   d  a  m  a  g  e

Damage

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Slope = 5

0

1500

-750

750

5.14

   %

   d  a  m  a  g

  e

Damage

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Slope = 10

0

1500

-750

750

20.78

   %

   d  a  m  a  g  e

Damage

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Frequency Based Fatigue