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7/30/2019 4 Frequency Based Fatigue
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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