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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

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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.


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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.


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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).


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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.


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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.


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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.


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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


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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.


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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


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(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


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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


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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.


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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.


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Fig. 39 Clearing the loading definitions.

Fig. 40 Creating a loading block.

Fig. 41 Defining the loading block.

Fig. 42 Running the fatigue analysis.


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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

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[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.


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[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.