Guidelines for the Verification of Frequency Domain
Fatigue Analysis by its Corresponding Representation in
the Time Domain
MSc. G. Teixeira, Eng. G. Zan
(Dassault Systèmes, UK, Marcopolo, BR).
Abstract
The use of frequency domain approaches in durability design requires the
combined knowledge of dynamics, vibration, fatigue and fracture mechanics. It
may take years for an expert to master all these subjects, and many mistakes are
made along the way, regardless of the analysis domain (time or frequency). A
large number of these mistakes, however, are avoidable, as they are related to
the fact that engineers often find it difficult to understand how analysis methods
and parameters in the frequency domain translate into the time domain. The
present work aims at providing the essential guidelines to assist design engineers
in the verification task of frequency domain fatigue results from finite element
analysis (FEA). It is assumed that most engineers have the basic knowledge of
fatigue, dynamics, vibration, and finite element modelling (FEM) in the time
domain. Therefore, the time domain is taken as the reference for verification.
Once the time domain analyses are validated (i.e. the simulated fatigue results
are within admissible error) the frequency domain analyses are likewise
considered acceptable. The paper starts by showing two common types of FEA
available for transient and steady state dynamic analyses, and how they compare.
In the frequency domain the loads are represented in the form of power spectral
densities (PSD). So the paper describes what a PSD is and the derivation of
equivalent time domain histories that share the same statistical properties.
Fatigue approaches in the frequency and time domain are discussed and the
results compared in the framework of stationary, ergodic and Gaussian
processes. The multiaxial reduction method employed is Mises in both domains
in order to avoid discussing the peculiarities of critical plane approaches. When
these assumptions are observed the differences between fatigue results obtained
in the frequency and time domain are around 10%. Following the procedures
outlined in the present paper engineers and companies can benefit from
frequency domain approaches more often and more confidently.
1. Introduction
Performance is one of the most compelling reasons for the adoption of frequency
domain over time domain, particularly when dynamic scenarios are involved.
However, engineers may still prefer to evaluate fatigue in the time domain since
the process is more intuitive and can be more easily verified.
The first step in order to demystify the frequency domain fatigue approach is the
understanding of power spectral density (PSD) and how it relates to time signals
(PSD realization). The second step involves the understanding of frequency
response functions (FRF) and how they transform input PSDs into output PSDs,
which are usually the PSD of stresses when the SN approach is adopted. The
third step involves the concept of spectral moments and how they are combined
into probability density functions (PDF), from which damage and fatigue life
are calculated.
2. Power Spectral Density
The power spectrum of a time signal describes how power is distributed along
the frequency range that composes the signal, i.e. it tells where the energy lies
and what the power density is [1]. The amplitudes of the PSD are evaluated from
the Fourier coefficients of the transformed signal and are normalized by the
frequency resolution used in the measurement (Figure 1).
Figure 1:
Explaining the terms in the PSD acronym.
Figure 2a shows a sample of an acceleration signal (in units of Gravity)
measured on a truck chassis during a proving ground excursion. The acceleration
amplitudes alone cannot tell whether structural damage is expected or not [2].
However, Figure 2b is closer to providing that information. It tells the power is
mostly distributed in the range from 20Hz to 40Hz, with a peak around 35Hz.
Therefore, components attached to the chassis could be compromised if any of
their mode shapes vibrate at a frequency that falls in that particular interval
(20Hz to 40Hz), especially about 35Hz. Then Figure 2b makes it clear which
frequency range should be avoided when designing components mounted on the
chassis.
Figure 2:
An acceleration signal represented as a Power Spectral Density (PSD).
The time signal and the PSD (Figure 2a and Figure 2b) are telling the same
story, however concealed the information seems to be in the PSD. In order to
unravel the information hidden in a PSD periodogram, the concept of spectral
moments shall be introduced.
3. Spectral Moments
The moments of a distribution are the building blocks, the measures that
uniquely determines the distribution and provide valuable information about its
characteristics [3]. The moments of a PSD (represented by the function G) are
evaluated by the following expression:
mn 

f
n
 G  f  df
(1)
0
Similarly to the moments of a normal distribution, that provide information
about its mean, variance, skewness and kurthosis, the spectral moments describe
the family of signals represented by the PSD [4]. For example, the standard
deviation is the square root of the 0th spectral moment:
  m0
(2)
The expected number of peaks per second in the signal is evaluated from the
ratio between the forth and second spectral moments:
E P  
m4
m2
(3)
The expected number of upward crossings per second in the signal is evaluated
from the ratio between the second and 0th spectral moments:
E  0 
m2
m0
(4)
4. PSD Realization
Random vibration fatigue approaches start with the loads described in terms of
PSD (as in Figure 2b), which are obtained from time signal samples as the one
represented in Figure 2a. The translation from either time to frequency or
frequency to time both can be done by means of a Fourier Transform (FT),
usually the Fast Fourier Transform (FFT) [5].
For the purpose of verification the next steps provide the recipe for obtaining a
time signal from a given PSD [6], [7].
1) From the PSD periodogram (e.g. Figure 2b) obtain the maximum
frequency fmax represented (in the present case fmax=100Hz).
2) The time interval t is evaluated in accordance to the Nyquist’s theorem:
fmax  1  2 t  .
3) The length of the time series (T) is related to number of time instances (N)
and time interval (t). The FFT algorithm requires that N=2n, being n a
positive natural number: T  N  t .
4) An interpolation method (usually log-log) is selected to extract the
magnitudes from the PSD periodogram. The frequency resolution is
determined from: f  1 T .
5) The magnitudes of the spectrum function are calculated as:
A  f   S  f   f .
6) Randomly generate phase angles for the spectrum function   f  .
7) Perform the IFFT (Inverse Fourier Transform) transform on the spectrum
function A  f  in order to get A  t   IFFT  A  f   , which is one
realization of the given PSD.
Now taking the PSD shown in Figure 2 as an example, and following the steps
(1-7) above, the signal shown in Figure 3 is obtained.
Figure 3:
The realization of the PSD shown in Figure 2b.
The signals represented in Figure 3 and Figure 2a obviously look very different
to each other, although both share the same PSD. Figure 2a shows a nonGaussian signal (Skewness=0.12, Kurthosis=5.99) and Figure 3 shows a
Gaussian signal (Skewness=0, Kurthosis=3). Actually the realization process
could not render anything other than a Gaussian signal as those information
(skewness and kurthosis) are not present in the PSD. The non-Gaussian signal
represented in Figure 2a is expected to cause more damage than the signal shown
in Figure 3, although both belong to the same PSD and share the same RMS (in
the present case RMS=1.43) [8].
As non-Gaussianity is not the scope of the present paper, the time signals here
(used in the comparison) will refer only to the Gaussian signals, obtained from
a known PSD. Often in the literature the signal in Figure 3 is refered to as the
conditioned signal (in this case conditioned from the signal in Figure 2a).
5. Dynamic Analysis in the Frequency Domain
One of the pre-requisites for evaluating fatigue in the frequency domain is
stationarity, and therefore the steady state dynamic (or harmonic) analysis is
where the numerical studies should begin.
The objective of a steady state dynamic (SSD) analysis is to determine the
frequency response functions (FRF) of the system subjected to vibrating loads.
The FRFs represent the relationship between input and output of a dynamic
system [9]. In the present case, the input are accelerations (but it could be forces,
displacements, etc.) and the output are stresses (it could also be strains). The aim
is to resolve the stresses as a function of the frequency range of interest.
Figure 4 shows the mesh (comprised of first order shell elements) of a finite
element model of an L-Beam that will be used as an example [10]. The loads
excite the two channels indicated by the highlighted edges. The nodes of these
edges are kinematically coupled to the reference points shown (RP1 and RP2),
which have all their DOFs (degree of freedom) constrained (primary boundary
conditions), apart from UZ (displacement at the vertical direction). This
direction is secondary base constrained, (technology founded on the large mass
method) [11]. The beam is made of steel (E=210MPa, nuxy=0.33,
density=7.85e-9kg/mm3) and its thickness is 0.5mm. The material’s fatigue
 
curve is described by the following equation: Sa  1056 N f
Figure 4:
0.1018
.
Finite Element Model of an L-Beam subjected to two sets of loading
(channels 1 and 2).
As this is a two loading channel model, two SSD simulations must be performed.
The modal superposition approach is going to be used for both time and
frequency domain simulations. Hence, a modal analysis (aka frequency
extraction analysis) must precede the dynamic simulations. Figure 5 below
shows the first 4 mode shapes (and respective frequencies) of the L-Beam. The
only output requested was stress. But the modal stresses themselves have no
meaning unless combined with the modal coordinates (aka generalized
displacements). Therefore stress legends can be dismissed in Figure 5.
Figure 5:
The first 4 mode shapes of the L-Beam model (12 modes requested in total).
It is important to stress that this modal analysis is common to the subsequent
time and frequency domain dynamic analyses. No special adjustments are
required for one in relation to the other.
Unit accelerations (1G) are defined to drive the two SSD simulations (applied
via Acceleration Base Motion). The loads are not strictly required to have unit
magnitudes in any unit system. However, if they are defined this way (1G for
the entire frequency range) the PSD of loads don’t need to be scaled, therefore
simplyfing the workflow.
Figure 6 below shows the modal coordinates for all the relevant modes. In order
to retrieve the stress tensor function (across the frequency spectrum) for a given
node, the stress tensors for each mode (from the modal analysis) must be scaled
by the corresponding modal coordinate function (from the SSD analysis) and
finally all tensor functions (the contributions of every mode) are then added.
Figure 6:
The modal coordinates for SSD1.
Figure 7 shows the Von Mises Stress distribution corresponding to the first mode
(15.1Hz). At the location named “POINT-A” (node 48396 indicated) the modal
analysis (aka frequency extraction) results in those stresses shown on Table 1.
Every line in the table presents the mode frequency and the stress tensor
associated with that particular mode. The two first modes (highlighted in orange)
are associated with a rigid body mode and an artificial mode that results from
the use of the large mass method [12].
The combination of the modal stresses (shown on Table 1) with the modal
coordinates (shown in Figure 6) results in a tensor function (4 stress components
for a shell element) for the interval from 0Hz to 500Hz. This function is called
Frequency Response Function (FRF). Figure 8 shows the Von Mises Stress
function (Von Mises FRF) for this same frequency range (0-500Hz).
Table 1:
Modal Stresses (MPa) for Node 48396, indicated in Figure 7
Figure 7:
Von Mises stress distribution corresponding to the first relevant mode
(frequency=15.17Hz).
Figure 8:
Von Mises stress function (FRF) for node 48396 (First Peak Stress at
15.17Hz), due to 1G load at Channel 1.
The FRF shown in Figure 8 describes how loads applied to master point 1 (see
Figure 4) translates into stresses measured at the location A. In other words, it
describes the input/output relation between loads at channel 1 and stresses at
node 48396. Similar FRF can be evaluated for channel 2 and any other node in
the structure.
In a more general way, a Dynamic Response Matrix can be defined (see Figure
9) as a collection of FRFs for all the channels involved [13]. The ultimate goal
is to translate PSD of loads into PSD of stresses, which are used for the
evaluation of spectral moments.
Figure 9:
The Dynamic Response Function for a N number of Loading Channels.
Once the Dynamic Matrix is assembled, the PSD of loads ( GL below) are turned
into the Stress PSD, through the operation indicated in Figure 10. The Stress
PSD is a 6x6 matrix function of frequency.
Figure 10:
Obtaining the Stress PSD from the Input PSD (or PSD of Loads)
It is worth highlighting that Q† above refers to a Hermitian conjugate (i.e.
transpose and complex conjugation). Also, it must be noticed that this equation
is really a set of equations, one per frequency.
The input PSD matrix G (which is the PSD of the applied loading) is shown in
Equation 5. GL is a function of frequency, so GL and GL(f) refers to the same
matrix.
 PSD11


GL  


CPSDN1
CPSD1 N 





PSDNN  NxN
(5)
The matrix set G  must be reduced to a scalar function of frequency in order for
the spectral moments to be calculated (Equation 1). There are several ways to
achieve this [14], including critical plane approaches, but here only Von Mises
will be used [15]. Equation 6 shows how the quadratic Von Mises operator

(matrix A in Equation 7) transforms G  into Gvm
.

GVM
 TraceA G 
(6)
Where:
1
A



 1
 2

 1
 2








1
2
1
2
1
1
1
1
2
2
3
3
3
















(7)
6. Fatigue Analysis in the Frequency Domain
The acceleration PSD shown in Figure 11 is adopted as the input load for both
channels of the L-Beam (see Figure 4). As mentioned previously, it may not be
entirely obvious what the PSD is really saying, therefore its realization is
presented in Figure 12, i.e., the acceleration history that corresponds to the PSD
displayed in Figure 11. In other words, the PSD in Figure 11 and the time signal
in Figure 12 both describe the same phenomenon.
Figure 11:
Figure 12:
Power Spectral Density (PSD).
The acceleration history that corresponds to the PSD presented in Figure 11.
Once the FRFs are obtained the Stress PSD can be calculated for every node in
the structure (see Figure 13).
Figure 13:
The PSD of Stresses for Node #48396.
The spectral moments are evaluated from these Stress PSDs. In this case (node
48396) m0  2124.8 , m1  32883.8 , m2  584105 , m4  1.69757e  09 . The Stress
PSD in Figure 13 can also be realized (by following the same guidelines
aforementioned) resulting in:
Figure 14:
The stress history that correspond to the stress PSD presented in Figure 13
In other words, the acceleration history from Figure 13 results in the stress
history in Figure 14. The largest stress amplitude is approximately 185MPa.
Comparing this value to the provided SN fatigue curve it becomes evident the
component is operating in the high cycle fatigue regime.
The spectral moments are the building blocks of the Probability Density
Function (illustraded in Figure 14) [16], which is used to evaluate damage and
fatigue life in the frequency domain. There is no rainflow counting in the
frequency domain, but rather an integration of damage caused by each event,
weighted by the probability of its occurrence. Therefore total damage is
calculated from the integration of Equation 8:
E P T  b
DDIR 
0 Sr p  Sr  dSr
k
Figure 15:
The probability density function of stress amplitudes
(8)
When Dirlik’s method is adopted the PDF is written as:
1
p  Sr  
2 m0
 Z2 
 D  Z D Z  Z2
 1 e Q  22 e 2R  D3 Ze 2 
R
 Q

2
(9)
Where all the parameters are calculated from some combination of spectral
moments [17]:
For a single slope SN curve of the form NSab  K there is a closed-form available
for the evaluation of fatigue damage [18]:
DDIR 
E[P]  T
K

m0

b

b
D1 Q  1  b  

 2
b


b

b
  1   D2 R  D3 

2
(10)
The use of Equation 10 may lead to considerably conservative results, although
being the exact solution. The alternative to the closed-form solution is the
integration of Dirlik’s PDF up to a cut-off limit, which is often taken as a function
of the RMS value of the Stress PSD (it is common to use a cut-off limit of about
6*RMS).
Figure 16:
LogLife (base 10) Results evaluated in the Frequency Domain.
Figure 16 above is showing the distribution of Log10(Life). The grey areas
correspond values bigger than 10 (i.e. fatigue life bigger than 1e10). The Node
showing minimum life in the figure refers to Life= 3.251e8 seconds.
A quick verification on the stress history presented in Figure 14 yields Fatigue
Life = 4.269e+08 seconds, which is 31.31% more optmistic (as expected) than
the frequency domain result. Since the realization process can render time
histories of any length, it is worth noting that the longer the history (obtained
from the PSD) the closer the fatigue results (calculated in the time domain) get
to the ones obtained in the frequency domain. And the reason is that the
probability of finding large amplitude cycles increases over time.
7. Dynamic Analysis in the Time Domain
The time domain approach will make use of the modal dynamic analysis, which
is also refered as modal superposition transient analysis. The modal behavior of
the L-Beam has already been investigated (see Figure 5). Both modal dynamics
and steady state dynamics analyses make use of the same modal analysis (the
setup is identical). However, instead of using unit acceleration (as in the steady
state dynamics) the modal dynamics use directly the acceleration histories
(shown in Figure 12) obtained from the realization of the input PSD (shown in
Figure 11).
In the present case the two input accelerations (signal a and b shown in Figure
17) are identical, i.e. the loads are fully correlated. The loads are 327.7 seconds
long. Damping is 0.01 across the entire frequency range (0-500Hz).
Figure 17:
L-Beam excited at both ends by the same acceleration history.
Unlike the SSD scenario, where two simulation steps were required (for a two
loading channel component), in the modal dynamics analysis all the loads are
applied in just one step and, as a consequence, only one set of modal coordinates
are obtained, as illustrated in Figure 18. Each modal coordinate is an array of
values that scale the corresponding modal stress (GU1 scales Modal Stresses 1,
GU2 scales Modal Stresses 2, …, etc.).
Figure 18:
The modal coordinates related to each vibration mode
For example, the contribution of the third mode to the stress history at node
48396 is found by multiplying the third line on Table 1 by the modal coordinate
GU3 (Figure 18), resulting in the tensor history shown in Figure 19.
Figure 19:
The contribution of GU3 to the stress history at node 48396
The contribution of all modes shall be added together in order to obtain the tensor
history that takes place at every node in the component.
8. Fatigue Analysis in the Time Domain
Since Von Mises was used as a multiaxial reduction method in the frequency
domain, signed Von Mises is the method to be used in the time domain. The sign
is given by the first principal stress 1 as shown in Equation 11 below [14].
Seqv

 1
1
 1   2    2   3    3   1 
2
2
2
2
(11)
The first step in the fatigue evaluation is the transformation of a tensor history
to a signed Von Mises history, using Equation 11 at every time point in the
history. For example, at node 48396 the equivalent stress history becomes:
Figure 20:
The signed Von Mises equivalent stress history at node 48396
The rainflow technique is employed to count the events so that life (and
consequently damage) can be calculated for each of them, using the SN approach
[19] as in Equation 12:
Sa ,vm   f 2N f 
b
(12)
The damage caused by the several variable amplitude events are linearly
summed up (Miner’s rule). As Von Mises belongs to the class of invariant
methods, there is no need for the critical plane strategy. Figure 21 below shows
the Log10(Life) distribution. At node 48396 Life is 3.791e8 seconds, which is
16.61% more optimistic than the life evaluated in the frequency domain.
Figure 21:
The probability density function of stress amplitudes
9. Case Study of a Bus Seat
An application example taken from the passenger transport industry was used to
investigate how well the two simulation domains compare to each other under
field operating conditions. The tested component is a bus seat used mostly in the
road transport segment. These road buses and seats are manufactured in Brasil
and are responsible for the transport of thousands of people daily.
Figure 22:
The seat model employed in the present study
These buses drive along several different types of pavements on the whole
Brazilian territory, almost most countries in South and Central America, some
African countries, Middle East and Asia. Most part of the pavements are asphalt
(Figure 23a), but unpaved roads are also common, mainly in the country side of
Brasil (Figure 23b).
Figure 23:
Typical asphalt (a) and countryside (b) roads in Brazil
Fatigue cracks nucleate under operating conditions on the lateral support bracket.
In a few occasions cracks would initiate right after the forming process as shown
on Figure 24.
Figure 24:
Lateral support bracket after forming
Accelerations were measured by accelerometers placed on the seats in the
locations indicated in Figure 25 below.
Figure 25:
Measurement (accelerometers) locations inside the bus
The data (accelerometers, strains, etc.) were collected on Randon’s proving
ground (Figure 26), which is composed of many different types of roads that
provide moderate to extreme driving conditions. Pot Holes, Body Twist, Ripples,
River Stones, Paving Stones, Belgium Blocks, Off-Road and Asphalt compose
the loading scenarios available to buses, trucks and small passenger vehicles that
are daily tested on the proving ground.
Figure 26:
Sketch of Randon’s Proving Ground
The passengers were represented by bespoke water gallons strapped on the seats
as shown in Figure 27. Triaxial rosettes are installed at the seat front support as
indicated in Figure 28a, on the location where fatigue failure has been reported.
The finite element simulations were calibrated in accordance to the results
obtained at the rosettes. The PSDs were obtained from the accelerations
measured with the accelerometers placed at the 8 positions indicated on Figure
28b. The data acquisition took place at paved and unpaved roads in order to best
represent the driving conditions of the bus in operation. The number of cycles to
failure obtained in the tests are not disclosed here as the focus of the paper is in
the comparison between analysis domains (verification) and not validation.
Figure 27:
Figure 28:
Water galons representing 80kg weight passangers
a) Strain gages assembled at the seat front support and b) 8 accelerometers
were placed at the indicated positions on the seat
10. Frequency Domain Simulation
The bus seats are subjected to the loading shown on Figure 22, that excite the
three indicated directions X, Y, Z. The three beams on the model belong to the
bus floor. The component is made of LN380, whose fatigue curve is shown on
Figure 23. The accelerations measured at the beams are transformed into PSDs,
displayed on Figure 24. As already explained, the durability investigation starts
at the modal analysis of the assembly, which detects the following first six
frequencies: 11.59Hz, 14.19Hz, 26.76Hz, 48.42Hz, 57.96Hz and 58.27Hz.
Figure 29:
The Loading Directions and Load Distribution on the lower beams
Figure 30:
LN380 SN Fatigue Curve
The passengers weigh 80kg each and are represented by point masses connected
to the seat by kinematic constraints (not shown on Figure 22). The three load
channels require three independent harmonic analyses, by using unit
accelerations (1G) at the three indicated directions. It is clear from Figure 24
that there is a substantial contribution from the vertical acceleration (Y direction)
in the range from 0Hz to 15Hz, exciting two of the first resonant frequencies.
Figure 31:
Acceleration PSDs in the X,Y,Z directions.
Figures 25 and 26 below show the fatigue life contours obtained by the fatigue
analysis in the frequency domain, using Dirlik’s approach. From the 5 appointed
locations the lowest life corresponds to 1044 repeats of the loading block, which
is a 1 second loading block. The size of the loading block is not really important
– a 10 second loading block could be repeated only 104.4 times for example.
The nodes connected by rigid links were not considered in the analysis. The
comparison between frequency and time domains will focus on the 5 nodes
chosen below.
Figure 32:
Life results from the frequency domain fatigue analysis.
Figure 33:
Life results from the frequency domain fatigue analysis.
11. Time Domain Simulation
The realization of the acceleration PSD at the Y direction leads to the
acceleration history shown on Figure 27, where the acceleration magnitudes can
be anywhere in the interval between -0.63G to +0.58G.
Figure 34:
Sample of the acceleration history at the Y direction.
The realization process is identical for the other directions (X and Z). The time
domain approach, similarly to the frequency domain, starts at modal analysis.
The modal analyses are identical for both domains.
However, instead of harmonic analysis, the time domain relies on the modal
superposition transient analysis, driven by the acceleration histories at the three
directions (X, Y, Z).
Figures 35 and 36 show the fatigue life contours obtained by the fatigue analysis
in the time domain, by using Signed Von Mises approach. The lowest life (from
the 5 chosen locations) corresponds to 1097.8 seconds, which is only 5.12%
higher than the fatigue life result obtained in the frequency domain.
Figure 35:
Life results from the time domain fatigue analysis.
Figure 36:
Life results from the time domain fatigue analysis.
Table 2 presents some of the fatigue results obtained in both frequency and time
domains. The largest difference among the comparisons is 14.24%, and occurred
in most critical node of the L-Beam model.
Table 2:
Comparison between fatigue life results obtained in the frequency and
time domain
12. Conclusion
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 when certain conditions are observed: Stationarity, Ergodicity,
Gaussianity. The load histories used in the time domain simulations were
generated from the PSDs (which ensures Gaussianity). Transient effects at the
beginning of the time domain analyses were removed in order to not violate
stationarity and ergodicity. RMS, number of peaks per second and number of
upward zero crossings per second were matched between the acceleration
histories (time domain) and acceleration PSDs (frequency domain) so that the
same load input was used in both domains. The simulations involving the LBeam used white noise PSDs, exciting two vertical load channels. The
simulations involving the Bus Seat instead used measured PSDs to excite three
different directions (3 loading channels). The probability density function (for
the frequency domain analyses) chosen was Dirlik, assembled from the spectral
moments calculated from the Von Mises Stress PSDs at every node in the model.
The time domain approach employed Signed Von Mises. Plasticity was not
included and SN fatigue approach was used in both domains. It is expected, from
these examples, the engineers feel more confident about the fatigue results
obtained in frequency domain and feel empowered to verify them with the
guidelines provided in the present paper.
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