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Guidelines for the Verification of
Frequency Domain Fatigue Analysis
by its Corresponding Representation
in the Time Domain.
Giovanni de Morais
SIMULIA Durability R&D
Technology Senior Specialist
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What is Power Spectral Density?
Spectral denotes the
frequency domain
X k* X k
G f   2
f
Density is related to the contribution of a
band of frequencies in the frequency bin
The size of the frequency bin
is the frequency resolution
2
Power quantifies the
contribution of every
frequency
f 
fs

N
Those magnitudes are the Fourier
Coefficients, determined by Fast
Fourier Transform.
Sampling Freq
Buffer Size
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Two Representations of an Acceleration History
3
Time Domain
Frequency Domain
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Obtaining a Representation of a PSD in the Time Domain
4
1
From the PSD periodogram obtain the maximum frequency fmax.
2
The time interval Dt is evaluated in accordance to the Nyquist’s theorem:
fmax =
3
The length of the time series (T) is related to the number of time
instances (N) and the time interval (Dt).
T  NDt
4
Select an interpolation method, bearing in mind that the frequency
resolution is determined from:
Df = 1 T
5
The magnitudes of the spectrum function are calculated as:
fmax
1
2 Dt
N = 2n
A f   S  f 
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Obtaining a Representation of a PSD in the Time Domain
5
6
Randomly generate phase angles for the spectrum funcion:
 f 
7
Perform the IFFT transform on the spectrum function A(f):
A  t   IFFT  A  f  
8
Scale the obtained history A(t) by:
Scale  N Fs
N is the number of points in the discrete PSD
Fs is the sampling rate (two times the max frequency of the discrete PSD)
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The steps of a Simulation in the Frequency Domain
Two Loading Channels
White noise PSDs exciting the
both (fully correlated) channels
6
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The Frequency Extraction Step
The load directions must not be constrained.
7
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The Steady State Dynamic Analysis
8
This technique is also known as modal
superposition harmonic or steady state
dynamics.
Frequency Response Function
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Fatigue Analysis Flowchart (Frequency Domain)
Modal
Analysis
9
Material
Properties
Frequency Response
Functions (FRF)
Stress PSD
Input General
PSD Matrix
Analysis Setup:
exposure time,
loads scaling, events
definition.
Mode-based SSD
with Unit
Response
Spectral
Moments
Probability
Density
Function
LIFE
DAMAGE
The Building Blocks
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
mn   f nPSD  f  df
0
m0
Standard deviation
E P  
m4
m2
Peaks per second
E 0 
m2
m0
Upward crossings
per second
E  PI  
10
m6
m4
Points of Inflection
per second [R6308]
m0  2124.8
m1  32883.8
m2  584105
m4  1.698e9
The moments are telling a story, which is summarized
(or represented) by a Probability Density Function
(PDF).
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Dirlik’s Method
11
DDIR
E  P T  b

Sr p  Sr  dSr

k
0
Z
Z
 Z2 
1  D1 Q D2 Z 2R2
p  Sr  
 e  2 e  D3 Ze 2 
R
2 m0  Q

2
Sr
Z
2 m0
E P 
m4
m2
D1 
2  xm   2 
1
m
xm  1
m0
2
m2
m4
1    D1  D12
D2 
1 R
  xm  D12
R
1    D1  D12
D3  1  D1  D2
Q
1.25    D3  D2R 
D1
m22

m0m4
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Fatigue Life Results
Frequency Domain Approach
12
Obtaining a Representation of a PSD in the Time Domain
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The PSD realization
13
It is worth noting that a PSD
is an envelop to a family of
time series that share the
same statistical content.
Therefore there are infinite
possibilities of signals that could
be used in the time domain, with
the *Modal Dynamics Solution
PSD Realization
(in accordance to the guidelines)
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The Modal Dynamic Analysis
14
This technique is also known as modal
superposition transient or modal dynamic
analysis.
Stress History
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Fatigue Life Results Comparison
Difference: 14.2%
15
Frequency Domain
Time Domain
An Application Example (Bus Seat)
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How Bus Seats are tested on the proving ground
16
Cracks found in the testing
Accelerometers and strain gages have
been used in the measurements.
Loading channels on a Bus Seat
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Frequency Domain
Loads defined as Acceleration PSDs
17
Time Domain
Loads defined as Acceleration Histories
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Fatigue Life Results Comparison
Frequency Domain
18
Difference: <5%
Time Domain
Conclusion
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It has been demonstrated that fatigue life results obtained in the frequency domain can
be very close (differences lower than 15%) to those obtained in the time domain.
For a fair comparison the time histories need to be Gaussian, Stationary and Ergodic.
Dirlik’s Distribution (PDF) has shown to be adequate to approach fatigue in the
frequency domain when the conditions aforementioned are observed.
Plasticity has not being addressed, but plays an important role. Future work is going to
discuss the possibilities.
This guideline is intended to make the engineers feel more confident in evaluating
fatigue in the frequency domain.
19
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20
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