Upload
buithuan
View
338
Download
8
Embed Size (px)
Citation preview
Int. J. Mech. Eng. Autom. Volume 2, Number 10, 2015, pp. 425-441 Received: August 6, 2015; Published: October 25, 2015
International Journal of Mechanical Engineering
and Automation
Random Vibration Fatigue Analysis of a Notched
Aluminum Beam
Giovanni de Morais Teixeira
Research and Development, Dassault Systemes Simulia, Sheffield S10 2PQ, UK
Corresponding author: Giovanni de Morais Teixeira ([email protected])
Abstract: The purpose of this paper is to present a case study where the fe-safe random vibration fatigue approach has been successfully employed. It describes the FEA (finite element analysis) preparation (an aluminum beam) and the necessary steps in fe-safe® to perform a fatigue analysis entirely in frequency domain. The method behind fe-safe combines generalized displacements obtained from SSD (steady state dynamic) finite element simulations to modal stresses to get FRF (frequency response functions) at a nodal level, where stress PSDs are evaluated in order to get spectral moments, which are the building blocks of the PDF (probability density function) used to count cycles and evaluate damage. The loading PSDs are then converted into acceleration time histories that allow fatigue to be evaluated in the time domain likewise. Results show a very good agreement between time and frequency domain approaches. Keywords: Fatigue, random vibration fatigue, high cycle fatigue, multiaxial fatigue, power spectral density, frequency domain fatigue.
Nomenclature
A Von Mises quadratic operator
b Fatigue curve exponent
D Fatigue damage
E[P] Expected number of peaks (peaks per second)
f Frequency (Hz)
F Force (N)
G Gravity of Earth (m s-2), approximately 9.81 m s-2
geqv PSD von Mises equivalent stress (MPa2 Hz-1)
gij Components of the input PSD matrix (G2 Hz-1)
G Input PSD matrix (G2 Hz-1)
h Stress vector (MPa G-1)
k Fatigue curve coefficient
Mn n-th spectral moment (Hzn MPa2 Hz-1)
Nf Number of cycles
p, PDF Probability density function
PSD Power spectral density (MPa2 Hz-1)
0 Standard deviation (MPa1/2)
S Stress component (MPa)
Sa Stress amplitude (MPa)
SR Stress range (MPa)
dSR Stress range step (MPa)
T Time (s)
Z Normalized stress range
Xm Mean frequency
1. Introduction
The random vibration fatigue or frequency domain
fatigue is a new approach in fe-safe®. It is based on
the vibration theory for linear systems subjected to
random Gaussian stationary ergodic loadings [1].
When a structure responds dynamically to an input
excitations there are two possibilities in terms of FEA
(finite element analysis): transient and SSD (steady
state dynamic) analysis [2]. Both can take advantage
of the MSUP (modal superposition) technique
provided the system is linear or any present
non-linearity does not affect the regions of interest.
The SSD analysis is much faster than the Transient
Analysis and it is one of the building blocks of the
random vibration fatigue analysis in fe-safe®, shortly
called PSD analysis. PSD stands for power spectrum
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
426
density. Fig. 1 shows the PSD Analysis flowchart that
describes the analysis procedure in fe-safe. Finite
element modal analysis and SSD analysis are
combined to get the FRF (frequency response
functions) in terms of stresses for every node in the
component or structure. These FRFs are scaled by the
input PSDs to get either PSD projections on critical
planes or von Mises equivalent PSDs. Whatever the
choice, these obtained PSDs are used to evaluate the
first four spectral moments to compose the Dirlik’s
PDF (probability density function) that is integrated to
get damage.
This paper is organized as follows: Section 2
describes the computer model (discretization in terms
of finite element mesh), the loading and boundary
conditions; Section 3 shows the modal and steady
state dynamic analyses used to obtain the modal
stresses and generalized displacements, also known as
modal participation factors; Section 4 give the finite
element dynamic results which are combined to the
loading PSDs to evaluate fatigue damage; in Section 5,
we use the modal superposition technique and
acceleration time signals equivalent to the given PSDs
to perform a transient analysis equivalent to the SSD
analysis in Section 3; in Section 6, we apply the scale
and combine technique in fe-safe to match modal
participation factors and modal results and get stress
tensors to evaluate fatigue using a standard time
domain algorithm; Section 7 gives conclusions.
2. Finite Element Modelling
The performed simulations and fatigue analysis here
Fig. 1 Frequency domain fatigue analysis flowchart.
are inspired on actual experiments [3] for the notched
beam sketched in Fig. 2. In the experiments the region
outlined as restrained nodes in Fig. 2 is attached to a
vertical rod (Z direction) which is the source of the
vibration.
The vibrational experiment in the present paper is
performed in time and frequency domain so that a fair
comparison can be established. It is important to keep
the FEM (finite element model) small because the
correspondent time domain transient analysis is
computationally very expensive. In this study, the
mesh contains 1793 second order hexahedral elements
and 10036 nodes. Fig. 3 shows the von Mises stresses
for the beam under 1G of vertical loading.
The maximum von Mises stress is 8 MPa, on the
edge of notch 1. Static structural analysis is not a
requirement for the random vibration fatigue approach.
However, they provide useful information about the
expected level of stresses as the loading frequency
tends to 0 Hz, an information that can be used to
calibrate the SSD analysis, also known as harmonic
analysis.
There are several ways of performing a harmonic
analysis. Common types of harmonic loads include
forces, moments, pressures, velocities and accelerations.
A typical situation in a dynamic analysis is when
accelerations are prescribed at the supports of a
structure or component. Some finite element packages
Fig. 2 Finite element model used in the studies.
Fig. 3 Static structural analysis—1G of vertical loading.
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
427
offer the possibility of defining local acceleration, but
usually acceleration is the kind of loading defined
globally in a finite element model, i.e., specified at all
nodes. Then, to keep the generality, the LMM (large
mass method) is employed here. The idea is to attach a
very large concentrated mass (the order of 1e7 to 1e10
times the mass of the whole structure) to the supports
where the accelerations are supposed to be applied in
the model. Examples of lumped masses in finite
element packages are Mass21 (ANSYS), *MASS
(ABAQUS) and CONM2 (NASTRAN). Fig. 4 shows
the large lumped mass linked to the region of interest
using RBEs (rigid body elements).
According to LMM principle [4] forces can be used
rather than accelerations, with the same effect on the
component.
The magnitude of the force must be equal to the
product of the large mass and the desired acceleration
(Fig. 4). In ANSYS® Workbench, the user can define a
remote point, set its behavior (rigid or deformable) and
create a point mass attached to it. Remote Forces and
Remote Displacements can be defined at remote points.
3. Frequency Domain FE Analysis
The first step in the random vibration fatigue
approach is the modal analysis. It is fundamentally
important to have the most accurate modal analysis as
possible. In this study, an artificial large mass is
employed; therefore it is necessary to limit the
frequency search range in order to avoid rigid body
modes. Finite element packages usually offer the option
Fig. 4 Large mass approach: preparing the modal analysis.
of defining the number of modes to find and
frequency search range.
In this simulation, 10 modes were requested and the
frequency range was set to 0.1-1e8 Hz. The node
associated with the large mass must have all its
degrees of freedom removed, except UZ
(displacement at Z vertical direction). All the other
displacements and rotations are set to 0 (UX = UY =
ROTX = ROTY = ROTZ = 0). The reason for not
constraining the displacement at Z direction is that
this is the loading direction, i.e., in the harmonic
analysis the beam will be excited by a harmonic
acceleration at Z direction. Stresses are requested as
output and no damping is required at this point.
Table 1 and Fig. 5 show the results of the modal
analysis. The lowest frequency found is 10.95 Hz. The
highest frequency in the searched interval is 510.6 Hz.
The stress results in the modal analysis do not mean
anything until the harmonic analysis is performed.
There is no special requirement for the number of
modes that needs to be evaluated in the modal analysis.
They vary from case to case, depending on the loading
and boundary conditions. Usually, the first 3 or 4
modes are enough to well represent a dynamic
response.
Fig. 6 shows the influence or participation of modes
1, 2 and 4 on the response of the notched beam
subjected to a vertical acceleration. The 4th mode is 2
orders of magnitude lower than the 1st mode. The 2nd
mode is more than 1 order of magnitude lower than the
1st mode. The 3rd mode can be neglected in this case.
The second step in the random vibration fatigue
Table 1 Modal analysis results.
Fig. 5 Mode shapes from the modal analysis.
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
428
Fig. 6 Modal participation factors magnitudes.
approach is the harmonic analysis. There are
essentially two ways of performing a harmonic
analysis: (1) through MSUP (modal superposition)
analysis and (2) through full harmonic analysis. The
modal superposition harmonic analysis is the chosen
approach in fe-safe® for the following reasons:
(1) MSUP harmonic analysis is faster than full
harmonic analysis.
(2) It provides MPFs (modal participation factors),
which can be scaled and combined to the modal
results to get the steady state response. The MPF
weight the contribution of each mode shape included
in the analysis.
(3) The results files are much smaller and easier to
manipulate than the ones generated by full harmonic
analysis.
It is not necessary to prescribe boundary conditions
for the harmonic analysis. The frequency range is set
to 0.1-300 Hz. A constant damping ratio is assumed to
be 1.8e-2. Clustering the results around the resonant
frequencies is requested and the cluster number is set
to 20. Stresses are requested at all frequencies as
output. A harmonic force is defined as 9800e10 N (Z
direction), which produces the same effect as an
acceleration of magnitude 1G (9800 mm s-2). The
command “HROPT, MSUP, nModes, 1, YES” tells
ANSYS to output the modal coordinates to a text file
named “jobname.mcf”, which is an ASCII file as Fig.
7 shows. For every frequency, there is a complex
number (rectangular format) representing the
contribution of each mode. It is recommended to
rename these files to match input channels numbers
when the analysis involves multiple channels. For
instance, file_1.mcf corresponds to channel 1;
file_2.mcf corresponds to channel 2; and so on. This
Fig. 7 Modal participation factor file: file_1.mcf.
example is a single channel analysis; acceleration at Z
direction on the remote point shown in Fig. 4.
When the harmonic analysis is finished FRF
(frequency response functions) as the one in Fig. 8 can
be evaluated for every node and every stress
component in the FEM (finite element model). At this
point, the FRFs can be combined to the loading PSD
matrix (input) to get stress PSDs in order to evaluate
the spectral moments at the nodal level. Premount [5]
describes in greater detail how to evaluate von Mises
PSDs out of frequency response functions and his
method is briefly presented in the next section.
Fig. 8 shows the expected magnitudes for the stress
component Sx at every frequency in the range 0.1-300
Hz. The peaks correspond to the resonant frequencies.
The response at the frequency 11 Hz is the highest (Sx
= 271 MPa), confirming the dominance of the first
mode in this situation. The magnitude of the responses
also depends on the assumed (or measured) damping.
The response at the frequency 54.6 Hz is the second
highest (Sx = 66.9 Hz). Some modes may not be
excited in a Harmonic Analysis (as the 3rd mode in
this example) or its contribution is so small (compared
to the other modes) that it is not perceived in the
dynamic response. Obviously, the FRFs change from
node to node within the finite element model.
4. Frequency Domain Fatigue Analysis
The third step in the random vibration fatigue
approach is to define the input PSDs file, named
“psd_file.psd” in this example, Fig. 9. This file must
follow the convention described in the fe-safe®
manual. Number of channels is set to 1 and there are
no cross PSDs. The only PSD defined is the auto PSD,
characterized only by its magnitudes (9 points in the
psd_file.psd file, 10-300 Hz).
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
429
Fig. 8 Dynamic response, stress component Sx at the critical node.
Fig. 9 Input PSD file: psd_file.psd.
Fig. 10 shows the histogram of the PSD magnitudes
in the PSD file.
Notice that this input PSD scales mode 2 more than
mode 1, meaning that the output stress PSD must
show the highest peak around the frequency 54.6 Hz.
Fig. 11 corresponds to the SN fatigue curve for the
Aluminum 6061 T6, the material used in the
simulations.
Having all the input files (FE results and input PSD)
prepared, fe-safe® can be launched and the project
folder created, Fig. 12. In this folder, we have the
following files: modal.rst (containing the FE modal
results), file_1.mcf (containing the modal participation
factors for the unit load harmonic analysis) and
psd_file.psd (containing the input PSD for the single
channel).
In fe-safe® interface, go to File > Open Finite
Element Model for PSD analysis to get the dialog
window shown in Fig. 13. Click on “Source FE
model” to select the file that contains the modal
results. Click on “Files that provide Modal
Participation Factor (MPF) data” to select the file(s)
generated by the harmonic analysis. Click on “Files
that provide Power Spectral Density (PSD) data” to
select the PSD input file described in Fig. 9.
The steps 3 to 6 in Fig. 14 load the necessary files
in fe-safe®. After hitting the “OK” button a dialog
window pops up to ask about pre-scanning. Select
YES to get to the window shown in Fig. 15.
Be sure only Stresses are selected and click on
“Apply to Dataset List”. In this window, 10
increments are shown, corresponding to the results for
the 10 modes defined in the modal analysis. After
clicking “Okay” (step 8) the Units Window appears,
Fig. 16. Select MPa as the working unit. The fatigue
curve for the Aluminum 6061 T6 is defined by the
two points in Fig. 17 (Nf = 1e4, Sa = 207 MPa; Nf =
1e6, Sa = 112 MPa).
Fig. 10 Input PSD representing the acceleration loading.
Fig. 11 Fatigue curve for aluminum 6061 T6.
Fig. 12 Fe-safe project directory dialog box.
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
430
Fig. 13 Open finite element model for PSD analysis.
Fig. 14 Selecting the files for the PSD analysis.
Fig. 15 Selecting datasets in the pre-scan operation.
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
431
Fig. 16 Selecting the appropriate units in fe-safe.
Fig. 17 Definition of the fatigue SN curve.
Select the Aluminum 6061-T6 (that has just been
created) and double click “Material” in the Analysis
Settings (step 12 in Fig. 18) to assign the material
property to all groups.
Go to Loading Settings (Fig. 19) and define the
exposure time, setting 60 to “Length per repeat in
seconds”. Then the PSD block is a 60 s loading block,
meaning that fe-safe® lives represent minutes in the
Results File.
Go to exports, Fig. 20, and at the Tab “List of
Items” type the element numbers that needs to be
further investigated. The critical node belongs to
element 1069 shown in Fig. 20.
Check “Export PSD Items*” at Tab “Log for Items”,
as in Fig. 21, step 16. fe-safe® is requested to output
the spectral moments (m0, m1, m2, m4) and stress PSDs
in the log file during the fatigue analysis for the
defined elements and nodes.
Obviously, the critical nodes and elements are not
known at the time of the fatigue analysis setup, unless
Fig. 18 Assigning materials to element groups.
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
432
Fig. 19 Defining the loading block.
real tests are performed prior to the simulations. Eq. (1)
evaluates the fatigue damage in Dirlik’s method [6, 7].
The summation represents the integration of the PDF,
Eq. (2), over the range of stress ranges, SR.
bDirlik R R R
E P TD = S p S dS
k
0
(1)
2 2
21 2 2 232
0
1
2
-Z -Z -ZQ R
R
D D Zp S = e + e + D Ze
Q RM
(2)
Fig. 20 Creating a list of items to be analyzed.
Fig. 21 Requesting PSD items to be exported.
02RS
Z =M
, 2
1 2
2 mx - γD =
1+ γ,
21 1
2
1-
1-
γ - D + DD =
R
3 1 21-D = D - D , 4
2
ME P =
M, 1 2
0 4m
M Mx =
M M
21
21 1
mγ - x - DR =
1- γ - D + D, 3 2
1
1.25 γ - D - D RQ=
D (3)
The diagram in Fig. 22 represents the PDF
described in Eq. (2). Ideally the integration of the PDF
(equivalent to the area A) should result 1, meaning
that 100% of the possibilities in the process were
accounted for. But this would imply ad infinitum
summation of Eq. (1). In practice, a good number for
the summation upper limit is a number between 10
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
433
and 20, that provides a result (PDF integration) close
to 0.995 or higher. This upper limit in fe-safe® is
defined by the “RMS stress cut-off multiple” in Fig.
23 (step 20). Go to FEA Fatigue > Analysis Options
and choose the PSD Tab. Make sure the PSD
Response is “von Mises” (step 19 in Fig. 23) and the
“RMS stress cut-off multiple” is 10 for the present
experiment. The integration domain is the integration
upper limit minus integration lower limit, Fig. 22.
Then the field “Number of stress range intervals”
(which defaults to 1000) controls the integration steps
(dSR in Eq. (1)) by dividing the integration domain in
even segments.
These two numbers (upper limit and number of
intervals) have an impact on accuracy and
computation speed. The bigger they are the slower the
calculation and the more accurate the fatigue results. It
is recommended to start with fe-safe® defaults (Fig.
23) and gradually change these values when needed.
The PSD Response in this investigation is von Mises,
evaluated according to Eq. (4). The symbol * stands
for the complex conjugation. A is the quadratic von
Mises operator. hi is the frequency response function
for channel i. gij are terms of the input PSD matrix G.
N is the number of channels. geqv is a scalar
representing the von Mises equivalent stress.
=1 =1
N Nj* i
eqv iji j
g = h Ah g (4)
where Ti
x y z xy yz xzh ,
1 0.5 0.5 0 0 0
0.5 0 0.5 0 0 0
0.5 0.5 1 0 0 0
0 0 0 3 0 0
0 0 0 0 3 0
0 0 0 0 0 3
A
,
11 1
1
N
N NN N N
g L g
G M O M
g L g
.
Next click on Analyze (step 22 in Fig. 24) and
continue (step 23). When the Analysis is finished,
click on “open results folder” (Fig. 25, step 24) to get
the results file. The worst life-Repeats shown in Fig.
25 correspond to 201.72 s (3.363 min).
Fig. 26 shows the life contour plot for the notched
beam. Node 227 (that belongs to element 1069) is the
critical, where life is the lowest. In the output location
there is a file named “modalResults.log”, where detailed
Fig. 22 Dirlik’s probability density function.
Fig. 23 Fe-safe PSD analysis options.
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
434
Fig. 24 Running the fatigue analysis.
Fig. 25 Analysis completed dialog.
Fig. 26 Fatigue life contour plot.
information about element 1069 can be found.
Spectral moments and equivalent stress PSD are the
essential additional information related to node
element 1069 in the Log File. As the exports (Fig. 19)
do not specify the node, all the nodes attached to
element 1069 are exposed in the diagnostics. The
worst node is shown in Fig. 26. The 0th spectral
moment corresponds to the variance of the stress PSD
at node 227. The RMS (root mean square) of the
variance is the standard deviation represented by 0.
In a normal distribution the probability of finding a
stress amplitude within 3 times the standard deviation
(in this case 3 0 = 294 MPa) is 99.73%. SQRT
(M2/M4) corresponds to the expected number of peaks
per second and SQRT (M2/M0) corresponds to the
upward mean crossing per second. The equivalent
stress PSD for node 227 is plotted in Fig. 27. It is
worth mentioning that the frequency range in the
diagram is the intersection of the ranges in the
following files: “file_1.mcf” and “psd_file.psd”. An
interesting aspect of this particular Stress PSD is that
its highest peak occurs at the second resonant
frequency, despite the fact the first mode is dominant
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
435
(Fig. 8). The four spectral moments are evaluated
from this PSD curve, according to Eq. (5):
=1
ΔN
nn k
k
M = f PSD k f (5)
It is important to emphasize that the information
provided in the log file (written
in …\jobs\job_01\fe-results\jobname.log) is enough to
build Dirlik’s PDF. Spectral moments can be
extracted from the PSD in Fig. 28. The PDF, Eq. (2),
can be evaluated from the spectral moments and from
Dirlik’s derived constants Eq. (3).
In his Ph.D. thesis, Benasciutti [8] discusses in
great detail the available frequency domain
approaches and proposes a new method which is
based on a combination of level crossing and range
count PDFs, balanced by a factor that weights the
narrow band and broad band contribution to the
fatigue damage. His work opened the door to a more
comprehensive approach were mean and residual
stresses could then be incorporated by using a
multi-variate distribution concept.
Fig. 27 Von Mises PSD for Item e1069.1.
Fig. 28 Fatigue life results.
5. Time Domain FE Analysis
In order to check the results obtained by the random
vibration fatigue approach, the notched beam is also
analyzed in the time domain. The challenge is to
guarantee the time domain approach is equivalent to
the frequency domain, otherwise the comparison is
useless. The first step in this direction is to get an
acceleration history that is compatible with the
prescribed PSD (Figs. 9 and 10). The problem can be
stated as the generation of random time series with
prescribed power spectra and there are several ways of
solving it [9]. In general lines, the procedure can be
summarized as follows:
(1) Choose the frequencies fi in the PSD
periodogram (Fig. 10);
(2) Choose random phase angles i to match those
frequencies;
(3) Evaluate the amplitudes from the given PSD
2 Δi i iA = G f ,
where Gi represents the PSD amplitudes and fi is the
frequency bandwidth (constant);
(4) Sum the individual spectral components for
every time t. The sampling rate should be at least ten
times the highest spectral frequency. In the equation
below Y is the resultant time vector. If the PSD units
are (G2 Hz-1), for example, the time history units are G
(multiples of the standard gravity acceleration).
=1
sin 2πn
i i ii
Y t = A f t +φ
(5) Assess the quality of the statistical
distribution of the obtained acceleration history.
Check its Gaussianity by evaluating skewness,
kurtosis, standard deviation, etc. Compare the
variance of both PSD and time series and check if the
number of peaks and zero crossings are coherent with
the spectral moments.
Fig. 29 shows the first 3 seconds of the synthetized
acceleration history that corresponds to the PSD in Fig.
10. The length of this signal is 10 s.
The analysis in the time domain needs to be based
on the MSUP technique and LMM approach. The finite
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
436
Fig. 29 Synthetized acceleration time history.
element model is the same (in terms of mesh
definition) and the forces exciting the transient
analysis have the magnitudes of (ACC x 9800e10 mm
s-2). ACC are the acceleration magnitudes in Fig. 29.
The acceleration file contains 32767 acceleration
records. This is the number of transient simulations
that need to be performed. The result of the MSUP
transient analyses is the file msuptrans.mcf. It contains
scale factors to be multiplied by the modal stresses in
order to evaluate the stress history for every node in
the model. Fig. 30 shows the components of the stress
history for node 227 at element 1069. This node is
referred in fe-safe® as item 1069.1.
This example is practically a uniaxial fatigue problem
since the component Sx (stress in the X direction) is
much larger than all the other stress components. Sx
magnitudes are in the range -300 to 300 (Fig. 31).
If the loadings are narrow band there is a good
chance to get sensible results using Bendat’s approach
[10], which tends to be conservative. Dirlik’s solution
can be used for narrow and broad band processes,
therefore chosen to be the approach used in this study.
Lalanne has also developed an arbitrary bandwidth
approach [11] that has served as the foundation to the
latest TB method (Tovo & Benasciutti method). Both
TB and Lalanne Methods are as robust as Dirlik, with
the advantage of being less empirical.
6. Time Domain Fatigue Analysis
The time domain analysis starts with the creation of
a project direction, Fig. 32, where the following files
need to be copied to:
modal_factors_for_msup_analysis.txt (containing the
modal factors for the transient analysis) and
msuptrans.rst (containing the FE modal results).
Fig. 30 Large mass approach for MSUP transient analysis.
Fig. 31 Stress components history at element 1069.
Fig. 32 Fe-safe project directory dialog box.
In fe-safe® interface, right click on Current FE
Models (Fig. 33, step 1) and choose “Open Finite
Element Model”. Select the “msuptrans.rst” file and
click on “YES” when asked about pre-scanning. Make
sure only Stresses are selected and check whether 10
increments are found in the file, Fig. 34. They
correspond to the 10 modes requested in the modal
analysis.
Select MPa as the units for the stresses, Fig. 35, and
keep the default for the other units.
Right click on “Loaded Data Files” and select
“Open Data Files” (Fig. 36) and choose the file
“modal_factors_for_msup_analysis.txt”. This file
contains 103617 rows and 10 columns. Each column
scales a particular modal result. Column 1 scales modal
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
437
Fig. 33 Opening finite element model for transient analysis.
Fig. 34 Pre-scanning the finite element model.
Fig. 35 Defining units in fe-safe.
Fig. 36 Loading the modal participation factors.
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
438
stresses in dataset 1, column 2 scales modal stresses in
dataset 2, and so on.
Click on the first item under the “modal factors for
MSUP analysis” in the Loaded Data Files and on the
“fe-safe plot” shown in Fig. 37 (steps 7 and 8) to see
the diagram for the scale factors in column 1.
The material properties must be defined next. It is
the same Aluminum 6061-T6 shown in Figs. 11 and
17. Choose von Mises algorithm (no mean stress
correction) by following the steps 9 to 13 in Fig. 38.
This study is using von Mises as the fatigue method
for both frequency and time domain analyses.
Right click on Loading Settings panel and clear all
loadings according to Fig. 39.
Click on Dataset 1 and on load file 1 (steps 15 and
16 in Fig. 40). In loading settings click on add (step 17)
and load history (step 18). Follow these steps for
Datasets 1 to 10 and load files #1 to #10 to create the
block displayed in Fig. 41. This procedure
corresponds to the scale and combine technique in the
time domain.
Click on Analyze and continue (Fig. 42) after
checking the fatigue setup displayed.
In Fig. 43, the worst element and node is being
reported as 24.629, which is equivalent to 246.21 s.
Click on “Open results folder” to see the life contour
plot on the notched beam.
Fig. 44 shows the life contour plot for the notched
beam. Node 227 (that belongs to element 1069) shows
a fatigue life of 253 s, since the loading block is
equivalent to 10 s. Compare the contour plots in Figs.
26 and 44 (Time and Frequency Domain) and check
Fig. 37 Plotting participation factors for mode 1.
Fig. 38 Choosing the fatigue algorithm.
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
439
Fig. 39 Clearing the loading definitions.
Fig. 40 Creating a loading block.
Fig. 41 Defining the loading block.
Fig. 42 Running the fatigue analysis.
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
440
Fig. 43 Worst life-repeats result.
Fig. 44 Fatigue life contour plot.
how close the results are. For node 227 the difference
in the reported lives (253 s and 202 s) is 20.2%.
7. Conclusions
This paper has shown how to perform a Fatigue
Analysis in the frequency domain using the software
fe-safe®. It also presented a counter example in the
time domain for comparison. The predicted life at the
failure location differs by 20.2%. Considering the
differences in the FE modelling and in the Fatigue
Methodologies a bigger difference (in terms of fatigue
lives) can be expected between time and frequency
domain approaches. The task of synthetizing time
signals compatible with a given PSD can introduce
noise and undermine the equivalence of the modal
superposition transient analysis. A probability density
function in the frequency domain replaces the
traditional rainflow counter in the time domain.
Therefore there is no such thing as definite number of
cycles in the frequency domain, but the probability of
finding cycles of given amplitude. Moreover, the
random vibration fatigue approach is based on linear
vibration theory and statistical assumptions that may
not be present in some circumstances. It is relevant to
mention that mean and residual stresses have not been
discussed in this study, despite its importance.
Plasticity correction is another subject that deserves
more attention and needs to be addressed separately.
Accuracy is the central theme of this paper, but
actually speed is what makes frequency domain so
attractive. A frequency domain implementation that
solves a problem 1000 times faster than an equivalent
time domain implementation brings the opportunity to
solve much larger problems. Fe-safe® can handle
multiple channels and therefore addresses multiaxial
fatigue problems. If non-proportionality is expected is
recommended to switch from von Mises to Critical
Plane approach.
In summary, the fe-safe® random vibration fatigue
is a powerful and fast approach that can provide
accurate results when compared to equivalent
approaches in the time domain. It can also be used to
design accelerated tests that may be of high economic
importance or used to perform a quick scan on very
large problems that would take weeks to be solved in
the time domain. The tool allows such problems to be
solved faster and allows important adjusts to be made
before either a more detailed time domain
investigation takes place or prototypes are
manufactured.
References
[1] A. Nieslony, E. Macha, Spectral Method in Multiaxial Random Fatigue, Lecture Notes in Applied and Computational Mechanics, Vol. 33, Springer Berlin Heidelberg, 2007.
[2] G.M. Teixeira, R. Hazime, J. Draper, D. Jones, Random vibration fatigue: Frequency domain critical plane approaches, in: ASME International Mechanical Engineering Congress and Exposition, San Diego, California, Nov. 15-21, 2013.
[3] V.K. Nagulapalli, A. Gupta, S. Fan, Estimation of fatigue life of aluminium beams subjected to random vibration, in: 2007 IMAC-XXV: Conference & Exposition on Structural Dynamics, Orlando, Florida, Feb. 19-22, 2007.
[4] Y.W. Kim, M.J. Jhung, Mathematical analysis using two modelling techniques for dynamic responses of a structure subjected to a ground acceleration time history, in: ASME 2010 Pressure Vessels and Piping Division/K-PVP Conference, Washington, USA, Jul. 18-22, 2010.
[5] A. Preumont, Random Vibration and Spectral Analysis, Kluwer Academic Publishers, 2009.
Random Vibration Fatigue Analysis of a Notched Aluminum Beam
441
[6] G.M. Teixeira, Random vibration fatigue—A study comparing time domain and frequency domain approaches for automotive applications, SAE Technical Paper 2014-01-0923, Detroit, April 2014.
[7] T. Dirlik, Application of computers in fatigue analysis, Ph.D. Thesis, University of Warwick, 1985.
[8] D. Benasciutti, Fatigue analysis of random loadings, Ph.D. Thesis, University of Ferrara, Italy, 2004.
[9] M. Giuclea, A.M. Mitu, O. Solomon, Generation of stationary Gaussian time series compatible with given power spectral density, in: Proceedings of The Romanian Academy, Series A, Vol. 15, 2014, pp. 292-299.
[10] J.S. Bendat, A.G. Piersol, Measurement and Analysis of Random Data, Wiley, New York, 1966.
[11] C. Lalanne, Mechanical Vibration and Shock, Volume V, Hermes Penton Ltd, London, 2002.