Post on 12-May-2023
VIBRATION FATIGUE ANALYSIS OF EQUIPMENTS USED IN AEROSPACE
A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF MIDDLE EAST TECHNICAL UNIVERSITY
BY
MURAT AYKAN
IN PARTIAL FULFILLMENT OF THE REQUIRMENTS FOR
THE DEGREE OF MASTER OF SCIENCE IN
MECHANICAL ENGINEERING
JUNE 2005
Approval of the Graduate School of Natural and Applied Sciences Prof. Dr. Canan ÖZGEN Director I certify that this thesis satisfies all the requirements as a thesis for the degree of Master of Science. Prof. Dr. Kemal İDER
Head of Department This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Master of Science. Assoc. Prof. Dr. Mehmet ÇELİK Assoc. Prof. Dr. F. Suat KADIOĞLU Co-Supervisor Supervisor Examining Committee Members Prof. Dr. Metin AKKÖK (METU, ME)
Assoc. Prof. Dr. F. Suat KADIOĞLU (METU, ME)
Assoc. Prof. Dr. Mehmet ÇELİK (ASELSAN)
Prof. Dr. R. Orhan YILDIRIM (METU, ME)
Prof. Dr. H. Nevzat ÖZGÜVEN (METU, ME)
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I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Last Name: Murat AYKAN
Signature :
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ABSTRACT
VIBRATION FATIGUE ANALYSIS OF EQUIPMENTS USED IN AEROSPACE
AYKAN, Murat
M.Sc., Department of Mechanical Engineering
Supervisor: Assoc. Prof. Dr. F. Suat KADIOĞLU
Co-Supervisor: Assoc. Prof. Dr. Mehmet ÇELİK
June 2005, 123 Pages
Metal Fatigue of dynamically loaded structures is a very common phenomenon in
engineering practice. As the loading is dynamic one cannot neglect the dynamics of
the structure. When the loading frequency has a wide bandwidth then there is high
probability that the resonance frequencies of the structure will be excited. When
this happens then one cannot assume that the structures response to the loading will
remain linear in the frequency domain. Thus to overcome such situations frequency
domain fatigue analysis methods exist which include the dynamics of the structure.
In this thesis, a Helicopters Self-Defensive System’s Chaff/Flare Dispenser Bracket
is analyzed by Vibration Fatigue Method as a part of an ASELSAN project. To
obtain the loading (boundary conditions), operational flight tests with
accelerometers were performed. The obtained acceleration versus time signals are
analyzed and converted to Power Spectral Densities (PSD), which are functions of
frequency. In order to obtain the stresses for fatigue analysis, a finite element
model of the bracket has been created. The dynamics of the finite element model
was verified by performing experimental modal tests on a prototype. From the
verified model, stress transfer functions have been obtained and combined with the
loading PSD’s to get the response stress PSD’s. The fatigue analysis results are
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verified by accelerated life tests on the prototype. Also in this study, the effect of
single axis shaker testing for fatigue on the specimen is obtained.
Keywords: Vibration Fatigue, Operational Flight test, Finite Element Method,
Accelerated Life Testing
vi
ÖZ
HAVACILIK ENDÜSTRİSİNDE KULLANILAN CİHAZLARIN
TİTREŞİM KAYNAKLI YORULMA ANALİZLERİ
AYKAN, Murat
Yüksek Lisans, Makine Mühendisliği Ana Bilim Dalı
Danışman: Doç. Dr. F. Suat KADIOĞLU
Eş-Danışman: Doç. Dr. Mehmet ÇELİK
Haziran 2005, 123 Sayfa
Mühendislik uygulamalarında, dinamik olarak yüklenen yapılarda sıklıkla Metal
Yorulması görülmektedir. Yükleme dinamik olduğundan yapının dinamiği
görmezlikten gelinemez. Yüklemenin frekans bandı geniş ise büyük olasılıkla
yapının rezonans frekansları tahrik olacaktır. Böyle bir durum söz konusu
olduğunda yapının yüklemeye tepkisi frekans alanında lineer kalmayacaktır.
Dolayısıyla bu durumlarda frekans düzleminde kullanılan ve yapının dinamiğini
dikkate alan yorulma analizi yöntemleri mevcuttur.
Bu tezde, bir ASELSAN projesi kapsamında, Helikopter Kendini Koruma
Sisteminin Chaff Fırlatıcı Braketinin Titreşim Kaynaklı Yorulma Analizi
gerçekleştirilmiştir. Yükleme sınır koşullarının elde edilmesi için, ivmeölçerler
kullanılarak operasyonel uçuş testleri gerçekleştirilmiştir. Elde edilen ivme-zaman
sinyalleri analiz edilmiş ve frekansa bağlı fonksiyonlar olan Güç Spektrum
Yoğunluk (GSY) fonksiyonlarına çevrilmiştir. Yorulma analizleri için gerekli olan
gerilme değerlerini elde etmek için yapının sonlu elemanlar modeli
oluşturulmuştur. Sonlu elemanlar modelinin dinamik davranışlarının doğruluğu
braketin prototipi üzerinde Deneysel Modal Analiz yapılarak gerçekleştirilmiştir.
Doğrulanmış olan modelden Gerilme Transfer fonksiyonları elde edilmiş ve bu
fonksiyonlar yükleme GSY fonksiyonları ile birlikte kullanılarak tepki gerilme
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GSY fonksiyonları elde edilmiştir. Yorulma analiz sonuçları prototip üzerinde
gerçekleştirilen hızlandırılmış ömür testleri ile doğrulanmıştır. Ayrıca bu çalışmada
incelenen yapı için eksen-eksen yapılan sarsıcı testlerinin yorulma üzerindeki
etkileri elde edilmiştir.
Anahtar Kelimeler: Titreşim Kaynaklı Yorulma, Operasyonel Uçuş Testi, Sonlu
Elemanlar Yöntemi, Hızlandırılmış Ömür Testi
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ACKNOWLEDGEMENTS
I am grateful to my thesis supervisor Assoc. Prof. Dr. Suat KADIOĞLU and co-
supervisor Assoc. Prof. Dr. Mehmet ÇELİK for their guidance and suggestions
during the preparation of this thesis.
I would like to thank my colleagues at ASELSAN for their support.
I would also like to thank my friends Ercenk AKTAY from FİGES A.Ş. and
Aydın KUNTAY, Ender KOÇ from BİAS A.Ş. for their help.
Finally, many thanks to my wife Fatma Serap AYKAN for her continuous help and
encouragement.
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TABLE OF CONTENTS
PLAGIARISM………………………………….………………..………...…........iii
ABSTRACT............................................................................................................. iv
ÖZ ............................................................................................................................ vi
ACKNOWLEDGEMENTS ..................................................................................... ix
TABLE OF CONTENTS.......................................................................................... x
CHAPTER
1. INTRODUCTION ................................................................................................ 1
1.1. Metal Fatigue ................................................................................................. 1
1.2. History Of Fatigue ......................................................................................... 1
1.3. Classification of Fatigue mechanisms............................................................ 5
1.4. Overview of the Study ................................................................................... 7
2. LITERATURE SURVEY ................................................................................... 10
3. FATIGUE THEORY .......................................................................................... 15
3.1. Stress Life Approach.................................................................................... 15
3.2. Strain Life Approach.................................................................................... 26
3.3. Crack Propagation Approach ....................................................................... 28
3.4. Vibration Fatigue Approach......................................................................... 28
4. OPERATIONAL DATA ACQUISITION AND SIGNAL ANALYSIS OF THE
DISPENSER LOCATIONS ON THE HELICOPTER........................................... 38
5. FINITE ELEMENT ANALYSIS OF THE DISPENSER BRACKET............... 47
5.1. Finite Element Analysis Methods For Fatigue Analysis ............................. 47
5.2. Finite Element Mesh Generation Of The Dispenser Bracket ...................... 48
5.2.1. Contact Modeling for Fasteners ............................................................ 51
5.2.2. Application of the Mass Element for Component Simulation .............. 53
5.2. Numerical Modal Analysis Of The Dispenser Bracket ............................... 55
5.3. Harmonic Response Analysis Of The Dispenser Bracket............................ 58
6. EXPERIMENTAL MODAL ANALYSIS OF THE DISPENSER BRACKET. 64
7. FATIGUE ANALYSIS OF THE CHAFF DIPENSER BRACKET .................. 68
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7.1. Analysis Parameters ..................................................................................... 68
7.2. Fatigue Analysis Results .............................................................................. 70
7.3. Case Studies for Different Design Parameters............................................. 71
8. FATIGUE TESTING OF THE DISPENSER BRACKET ................................. 76
8.1. Test Setup..................................................................................................... 76
8.2. Test Results .................................................................................................. 78
8.3. Test-Analysis Comparison........................................................................... 79
9. DISCUSSION AND CONCLUSIONS .............................................................. 88
REFERENCES........................................................................................................ 92
APPENDICES
APPENDIX-A......................................................................................................... 96
A.1. Identification of the System and Analysis Parameters................................ 96
A.2. Finite Element Analysis Method Selection............................................... 100
A.2.1. Transient Dynamic Analysis .............................................................. 100
A.2.2. PSD Spectrum Analysis ..................................................................... 100
A.2.3. Harmonic Analysis............................................................................. 101
APPENDIX-B....................................................................................................... 105
APPENDIX-C....................................................................................................... 110
APPENDIX-D....................................................................................................... 114
APPENDIX-E ....................................................................................................... 123
1
CHAPTER 1
1. INTRODUCTION
1.1. Metal Fatigue In real life, machine parts, mechanical systems etc. are rarely under static loading.
Most of the time dynamic loadings are encountered and again most of the time they
are repeating themselves in time. These repeating loadings need not necessarily
stress the components above yield point to cause a mechanical failure. If enough
cycles are encountered by the component there will be a failure. This phenomenon
is called as Fatigue Failure.
According to ASTM E-206 Fatigue is, “The process of progressive, localized,
permanent structural change occurring in a material, subjected to conditions which
produce fluctuating stresses and strains at some point or points, and which may
culminate in cracks or complete fracture after a sufficient number of fluctuations”.
Fatigue analysis (numerical and experimental) is applied as a Design Life Cycle. In
this Life Cycle, at various steps analyses are made with accompanying tests. If
required, a prototype is built and tested and the numerical models are updated
according to the test results. An example of a Fatigue Design Life Cycle is given in
Figure 1.1.
1.2. History of Fatigue The Fatigue phenomenon has been recognized early in the 19th century and many
researches have been done on this topic. The Timeline of Fatigue history is given
below [2].
2
Figure 1.1: Fatigue Design Life Cycle applied in ASELSAN INC. [1].
• 1829: Wilhelm Albert first discusses the phenomenon on observing the
failure of iron mine-hoist chains in Clausthal mines.
• 1839: The term fatigue becomes current when Jean-Victor Poncelet
describes metals as being tired in his lectures at the military school at Metz.
• 1843: William John Macquorn Rankine recognizes the importance of stress
concentration in his investigation of railroad axle failures following the
Versailles accident.
• 1849: Eaton Hodgkinson is granted a small sum of money to report to the
UK Parliament on his work in ascertaining by direct experiment, the effects
of continued changes of load upon iron structures and to what extent they
could be loaded without danger to their ultimate safety.
• 1860: The first systematic investigations of fatigue life by Sir William
Fairbairn and August Wöhler. Wöhler's study of railroad axles leads him to
the idea of a fatigue limit and to propose the use of S-N curves in
mechanical design.
• 1903: Sir James Alfred Ewing demonstrates the origin of fatigue failure in
microscopic cracks (Figure 1.2)
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Figure 1.2: Fatigue cracks from Ewing & Humfrey (1903) [2]
• 1910: O. H. Basquin clarifies the shape of a typical S-N curve.
• 1939: Invention of the strain gage at Baldwin-Lima-Hamilton catalyses
fatigue research.
• 1945: A. M. Miner popularizes A. Palmgren's (1924) linear damage
hypothesis as a practical design tool.
• 1954: L. F. Coffin and S. S. Manson explain fatigue crack-growth in terms
of plastic strain in the tip of cracks.
• 1961: P. C. Paris proposes methods for predicting the rate of growth of
individual fatigue cracks in the face of initial skepticism and popular
defense of Miner's phenomenological approach.
• 1968: Tatsuo Endo and M. Matsuiski devise the rainflow-counting
algorithm and enable the reliable application of Miner's rule to random
loadings.
• 1970: W. Elber elucidates the mechanisms and importance of crack closure.
• 1975: S. Pearson observes that propagation of small cracks is sometimes
surprisingly arrested in the early stages of growth.
4
Typically fatigue failures start at the highest stress zones by forming cracks and
then propagating under cyclic loading, where the stress state can be still under the
yield point of the material. When a limit is reached for the number of cycles of
loading the component fails at a fatigue failure surface. A typical fatigue failure
surface is given in Figure 1.3.
Figure 1.3: Fatigue failure surface [3] Unfortunately most of the time fatigue failures cannot be detected until
catastrophic accidents occur. In the past many accidents due to the fatigue failures
in metals have occurred. Figure 1.4 shows some of them.
a) b)
Figure 1.4: a) Crane failure resulting from crack growth [3] b) Railway accident due to failure of the axle [3] c) Aloha Airlines Boeing 737 fuselage failure due to multiple cracks at rivet holes [4] d) Crack growth on fuselage [4]
5
c) d)
Figure 1.4 (cont’d): c) Aloha Airlines Boeing 737 fuselage failure due to multiple cracks at rivet holes [4] d) Crack growth on fuselage [4]
1.3. Classification of Fatigue mechanisms
Fatigue mechanism can be divided into three stages:
1) Crack initiation
2) Crack propagation
3) Fracture
Crack initiation can be defined as: “The progressive, localized, permanent
structural change which initiates fatigue failure is the microplasticity; which, in
other words, is the onset of plasticity at microscopic level whilst a material is
nominally elastic” [5]. Then “the crack propagates in a zigzag transgranular path
along slip planes and cleavage planes from grain to grain and maintains a general
direction perpendicular to the maximum tensile stress” [5]. Finally when the crack
has grown to such an extend that the remaining cross section can no longer hold the
component together, the component fails and fractures.
Here it should be mentioned that failure definitions in fatigue are subjective. It
could be a predetermined crack length, fracture of the component, malfunction of a
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system or loss of stiffness etc. With this definition fatigue can be divided into two
sub-groups according to the time required to have a failure. First group is called
High Cycle Fatigue, and the second group is Low Cycle Fatigue.
High cycle fatigue gets its name from the number of cycles required for failure,
which are relatively higher than low cycle fatigue. Approximately 54 1010 − cycles
[5] or more are required for high cycle fatigue. The reason for this is that, if the
loading to the component is such that the yield is not exceeded and is much below,
than the component will remain mostly in the elastic region and will require higher
number of cycles. But if the loading is such that the yield is exceeded by little
amounts, then because the component is forced plastically, lesser cycles will be
required for failure and thus a low cycle fatigue failure will occur.
Three methods were developed to investigate the fatigue phenomenon.
The first one is the Stress-Life method. In this method no crack initiation or
propagation effects are considered. Stress versus Cycle (S-N) curve of the material
is used for analysis. The failure criterion is that when the damage value is, e.g. for
Palmgren-Miner’s damage rule, equal to one the component fails.
The second method is the Strain-Life method. In this method crack initiation is
checked as a failure criterion using Strain versus Cycle curves.
The final method is the Crack Propagation method, which assumes the existence of
a crack and then analyzes the propagation of it using Linear Elastic Fracture
Mechanics.
From these definitions it is evident that for High cycle fatigue analysis, the Stress-
Life method and for Low cycle fatigue analysis, Strain-Life method should be
used.
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Finally, there are two domains for fatigue analysis. First one is the time domain and
the second one is the frequency domain methods (Figure 1.5). The details of these
methods will be discussed in the following chapters.
Figure 1.5: Time and Frequency domain methods [3].
1.4. Overview of the Study
In this study the application of the Vibration fatigue method for a Helicopters Self-
Defensive System’s Chaff/Flare Dispenser Bracket will be performed. The
components of a Chaff/Flare Dispenser system are given in Figure 1.6.
The use of Flare is to misguide missiles by creating very hot spots at distant
locations to the helicopter. Whereas Chaff is used to hide the signature of the
helicopter by disposing electromagnetic particles to the air, which cover the
surroundings of the helicopter and hides it from radars (Figure 1.7).
FREQUENCY DOMAINFATIGUE
MODELLER
BLACKBOX
M0
M1
M2
M4
TransferFATIGUE
LIFEFunction
PSD PDF
Transient Analysis
RAINFLOWCOUNT
TIMEHISTORY
TIME DOMAIN
FATIGUELIFE
PDFSteady state
or
8
Figure 1.6: Chaff/Flare dispenser, ammunition and control unit
Figure 1.7: Application of Flare
The design of a bracket for such a device has many requirements to fulfill. Static,
modal, transient and fatigue analyses must be performed to complete the design
with necessary safety factors. First of all a preliminary drawing is created, which
fulfills the manufacturing, cabling and usage requirements with engineering
judgments for plate thickness, bolt diameters etc. Then the mentioned analyses are
performed to check and verify the design. Except fatigue analysis, all of the other
analysis types for the bracket model (Figure 1.8) had been completed in other
studies. Therefore the scope of this thesis is limited to the fatigue analysis.
9
Figure 1.8: Chaff/Flare dispenser bracket
The source of fatigue for this bracket is the vibration being transmitted from the
helicopter. In order to apply Vibration Fatigue analysis, a finite element analysis of
the bracket is also required.
As it can be seen from Figure 1.8 the bracket has many bolts and rivets as
connection elements. Especially rivets pose a difficulty for finite element modeling
due to the strong nonlinearities, which are created during the riveting process. This
topic will be discussed in the Finite element modeling section.
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CHAPTER 2
2. LITERATURE SURVEY Bishop [6] investigated time and frequency domain based fatigue techniques and
performed some studies in finite element environment. It is pointed out that, when
the loading excites the natural frequencies of the structure, time domain approach
lacked the dynamics of the structure if the analysis is performed by assuming that
the loading is statically applied. Also in order to include the dynamics of the
structure in the time domain, a transient dynamic analysis has to be performed
which is very time consuming and sometimes practically impossible. Instead of the
time domain methods, spectral methods using the random vibration theory can be
used. The benchmarks in the paper show that spectral methods and transient
dynamics method results were comparable and accurate enough for numerical
analysis. However studies show that when mean stress is present in the structure
Vibration Fatigue analysis predictions loose accuracy.
Giglio [7] performed a Finite element method based fatigue analysis for the upper
and lower folding beams on the rear fuselage of a naval helicopter. A sub-modeling
technique was applied to obtain accurate stresses under flight and folding loads.
Modeling of rivets and pins was performed with three dimensional line elements
with the assumptions of; no prestressing of the rivets and bolts, no contact load
distribution to the hole and no stress distribution resulting from localized plastic
behavior of the material. Furthermore tests utilizing strain gages were performed to
verify the numerical model. Finally fatigue analysis with Palmgren-Miners rule
was used to obtain fatigue damage values.
Wu et al. [8] examined the applicability of methods proposed for the estimation of
fatigue damage and life of components under random loading. Palmgren-Miner and
Morrow’s rule were investigated and verified by low cycle fatigue tests. Test
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results showed that Morrow’s plastic work interaction damage rule worked much
better than the widely used Palmgren-Miner’s damage rule. Furthermore the
fatigue damage, and also from this information fatigue life could be easily
computed from random vibration theory.
Park Jeong Kyu et al. [9] performed static and fatigue analysis for a primary
structure element of an aircraft. In the analysis the fasteners and their connections
to the fastener holes were modeled with gap elements using commercial finite
element analysis codes. Tests had been performed to verify the fastener modeling.
Finally fatigue analysis was performed and Fatigue/Fracture critical locations were
identified. In the analysis it was assumed that no resonant modes of the structure
were excited.
Pitoiset et al. [10] proposed a frequency domain method to estimate the high cycle
fatigue damage for multiaxial stresses caused by random vibrations, directly from a
spectral analysis. In this study a new definition of von Mises stress as a random
process was used. Also the method was generalized to include a frequency domain
formulation of the multiaxial rainflow method for biaxial stress states. The
comparison of both methods showed that instead of the time consuming multiaxial
rainflow method, the new von Mises definition could be used to have a coarse look
at the damage distributions. It is stated that, with this method the life was not
predicted very accurately. Therefore this method should be used mostly to compare
design alternatives rather than to find the life of components.
Liou et al. [11] studied damage accumulation rules and fatigue life estimation
methods for components subjected to random vibrations. In the study, random
vibration theory was used to estimate the fatigue life and damage with Morrow’s
plastic work interaction damage rule. From fatigue tests the damage results were
compared with the traditional cycle by cycle counting methods. The results showed
that the application of random vibration theory to estimate fatigue life worked very
accurately when used with Morrow’s plastic work interaction damage rule. But the
iterative process required finding the material constants for this rule was one of the
reasons why Palmgren-Miner’s damage rule is more preferred.
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Doerfler [12] performed some benchmark analysis between an in-house fatigue
software, LOOPIN, and MSC Fatigue. The aim was to find out whether MSC
Fatigue could be used for preliminary design analysis of airframe structures. The
usage of LOOPIN required the initial crack size and crack type which made the
analysis difficult. But after these inputs the Linear Elastic Fracture method was
used to obtain the life of the component, which gave very accurate results. Whereas
MSC Fatigue, which utilizes high cycle fatigue methods with rainflow cycle
counting methods, was far easier to use and obtain a solution. The results of the
benchmarks showed that MSC Fatigue was a much powerful tool for predicting the
crack initiation location than predicting the life of component.
Langrand et al. [13] developed a numerical procedure based on FE modeling and
characterization of material failure models to limit the costs of experimental
procedures, which are necessary to analyze riveted structures for crashworthiness
of aircraft structures. Quasi-static and dynamic experiments were performed to
identify the Gurson damage parameters of each material by an inverse method. The
aim was to obtain a numerical database for riveted structures.
Eichlseder [14] developed a model for calculating the S/N curves which took into
account the stress gradients to define the local stress limit, the number of fatigue
cycles at the fatigue limit and the slope. In finite element methods, for
geometrically complex components the nominal stresses and stress concentration
factors couldn’t be calculated. Therefore Eichlseder used the stress gradients to
calculate the Local S/N curves of simple components. It is mentioned that by
further developing this model, fatigue analysis of complex components can be
performed.
Hawkyard et al. [15] investigated fatigue crack growth of a component, using the
Wheeler’s model, which is subjected to vibrations caused by the transition from
one flight stage to the next and the steady flight stages. Experiments have been
performed with standard specimens where the loading was applied sinusoidally. It
was found out that crack formation occurred mostly at take off, propagation and
13
fatigue failure was during the cruise stage. Thus the transient effects encountered in
flight stage changes (take off to climb and climb to cruise) were investigated using
models like the Wheeler’s model.
Shang et al. [16] developed a new theory for the application of local stress-strain
field intensity to the fatigue damage at a notch. This theory took into account the
effects of the local stress-strain gradient on fatigue damage at notches. The local
stress-strain intensity parameters required for the fatigue analysis were calculated
from an incremental elastic-plastic finite element analysis under random cyclic
loading. The method was verified with tests on a U-shaped notched specimen.
Liao et al. [17] investigated the fatigue life distribution of fuselage splices by
modeling three-dimensional lap joints and performing a nonlinear finite element
analysis. The squeezing force and coefficient of friction at the contact surfaces
were modeled as random variables with assumed mean values and deviations.
Monte Carlo simulations were performed and it was found out that squeezing force
had a bigger effect on fatigue life than the coefficient of friction. Furthermore, both
parameters had a much bigger effect on fatigue life than the material properties
uncertainties.
Conle et al. [18] performed a research on the currently used fatigue analysis tools
and their applications. In this paper three parameters, which are the material
properties, the loading histories and the damage model were discussed that affect
the fatigue life. It is stated that material properties should be obtained according to
the loading type. Also Bauschinger effect and multiaxial deformation models
should be considered in obtaining the material properties. For loading histories it is
mentioned that one should test as many load histories as possible to obtain the most
damaging one. The final point addressed is to find out an appropriate damage
model to characterize the component under analysis. Approaches like Critical
plane, which search for the most damaging plane should be used.
Urban [19] performed a literature survey and some analysis on fatigue failures of
riveted airframe sheet metal joints. It was found out that different views on the
14
effect of friction between rivets and rivet holes for riveted joint were present. Some
literature in finite element models assumed no friction effect and verified it by test
whereas some literature showed that the opposite was true. The author performed a
number of static and fatigue tests to verify a finite element model. It is found that
in numerical analysis if the effect of friction was not included the results did not
correlate well with test results. Also it is claimed that the correct modeling of
elastic-plastic deformations of the rivet and rivet hole play a major role in fatigue
analysis.
Szlowinski et al. [20] studied the residual stress fields and squeeze forces of
riveting process. The finite element analysis of a riveting process has been
performed and compared to experimental results. The effect of high residual
stresses on the rivet and rivet holes for the fatigue life has been investigated. Also
the effect of rivet head and squeezing forces on fretting damage has been studied.
From researches it has been found out that with high squeeze forces a zone of
compressive residual hoop stress dominated the area close to the hole, whereas a
zone of tensile residual hoop stress was present on areas far from the hole. These
stress fields were found out to be the major reason of the propagation of fatigue
cracks at rivets/rivet holes.
15
CHAPTER 3
3. FATIGUE THEORY In this chapter the types of fatigue analysis will be discussed briefly and finally the
application of the method used in this thesis will be covered. As mentioned earlier
there are three categories in fatigue analysis. These are Stress Life, Strain Life and
Crack propagation approaches. The Stress life approach is used where the stress
levels are much below the yield stress so that high number of cycles is required for
failure. Strain life approach is used where the stress levels are at or above yield
such that high levels of plastic work is present. Crack propagation is used when the
crack size and shape is pre-known and the presence of it cannot be avoided. Then
the propagation of the crack can be monitored. Stress life approach and some basic
concepts of fatigue will be covered in detail whereas the Strain life and Crack
propagation approaches will be briefly discussed. Finally the approach used in this
thesis, Vibration Fatigue approach, will be discussed.
3.1. Stress Life Approach For fatigue phenomena the loading should change with time and the simplest of
this type of loading is a sine wave as given in Figure 3.1.
Figure 3.1: Stress cycle example, a) Fully reversed, b) Offset
16
Some basic information from these figures can be defined as follows:
Sa: Alternating stress amplitude
Sr: Stress range
Sm: Mean stress
Smax: Maximum stress amplitude
Smin: Minimum stress amplitude
R: Stress ratio, Smin/Smax
This type of loading is found mostly in rotating shafts but is not a generalized
loading type encountered. Most of the time, random type loading is present in
mechanical systems (Figure 3.2).
Figure 3.2: Random time history of loading [16] In order to predict life for a component, another information is required. Because
the levels of stress are below yield one cannot judge whether the part will fail from
tensile tests, thus another type of test is necessary. Fatigue tests are usually
performed on a cylindrical test specimen loaded in axial tension, which doesn’t
have any sudden changes of geometry and with a well-polished surface at the
critical section, where failure is expected. A number of identical specimens are
tested to total separation with different levels of loading for each test and the
number of cycles to failure is recorded, as N. Load, not stress, is kept constant
during the test. For each specimen a nominal stress, S, is calculated from elastic
17
stress formulas and the results are plotted as the un-notched S-N diagram [3]
(Figure 3.3).
Figure 3.3: S/N data of AL7050-T7451 [9] The first point to mention is that, as it is seen from Figure 3.3 the Aluminum
material does not have infinite life. If enough cycles are applied it will eventually
fail. This is not the case for steel (Figure 3.4). The limiting stress level below
which the material has infinite life is called fatigue limit. For Aluminum and such
metals, which don’t have a fatigue limit, the stress level at about 810 cycles is
called endurance limit [5].
Figure 3.4: S/N data of steel [21]
18
From these S/N curves the following formula can be obtained.
CSN b =⋅ (3.1)
where;
N : The number of cycles
S : Stress amplitude
b : Basquin exponent
C : Intercept on stress axis
Now from equation (3.1) the fatigue sensitivity to stress can be identified. For
example if b=10, which is approximately for Aluminum, 7% change in stress level
causes 50% change in life. This sensitive behavior is critical in fatigue analysis
because stress concentration zones (notches, holes etc.) present in every design
create a big problem for calculation of stresses. These stress concentration effects
are handled using a stress concentration factor called TK . TK can be defined as
follows [3]:
_ _ _ _ _ _ __ _ _ _ _T
Maximum Stress in the region of the notchKNominal Stress remote from the notch
= (3.2)
Although many closed-form solutions are present in the literature it is impractical
to analyze complex structures with these solutions. Fortunately if modeled
correctly in the finite element environment, the stresses obtained take the stress
concentration effects into account.
There is one more factor that should be considered which is related to the material.
As previously explained the fatigue test specimens are prepared such that they do
not have any sudden changes in geometry like notches. But if the geometry has a
notch then the S/N curve obtained would be different. Then one should define
another factor to take into account this effect [3].
specimennotchedainfailurecausetoneededStressspecimennotchedunaninfailurecausetoneededStressK f ________
________ −= (3.3)
19
The effect of fK is difficult to obtain. There exists literature on this topic trying to
relate fK to TK but it is limited. However it is stated that for low ductility
materials fK will be close to TK . If it perfectly brittle they will be equal [5].
One final parameter to consider is the mean stress effect on fatigue life. The mean
stress present in the system reduces the alternating stress that the material can
sustain. So if this reduction is assumed to be linear than the Goodman’s Rule can
be established. As it is not necessarily linear there are other methods with different
assumption which better fit to experimental results. In Figure 3.5 some of these
methods can be seen.
All of these methods can be used in the fatigue analysis. By using these methods, a
stress cycle with a mean stress and an alternating stress is changed to a different
alternating stress and zero mean stress. Each of these methods gives a different
confidence level where Soderberg’s model is the most conservative one. Also there
is another method, which is the modified Goodman’s model as it can be seen in
Figure 3.6. When the stress state is below these lines then the component is safe.
Modified Goodman’s model assures that the stress state does not exceed yield
stress.
Figure 3.5: Mean stress effects on the alternating stress by different methods [22].
20
Figure 3.6: Modified Goodman line (Blue line) [22]. The formulations used for the above methods are as follows:
For Goodman’s model:
))/(1/( utmaa SSSS −=′ (3.4)
For Soderberg’s model:
))/(1/( ytmaa SSSS −=′ (3.5)
For Gerber’s model:
))/(1/( 2utmaa SSSS −=′ (3.6)
where; ′
aS : Equivalent alternating stress
aS : Alternating stress
mS : Mean stress
utS : Ultimate tensile strength
ytS : Tensile yield strength
21
One final remark on mean stress is that, these models are obtained for tensile
region. It is assumed that it also applies the same way in the compressive region.
But there is one point that cannot be taken into account with these models. When
the mean stress is compressive, this compressive stress tries to close the cracks,
which are the reason of fatigue failures. Thus the models used will be conservative.
It is common practice to cold work or shot peen surfaces of components to gain
some extra fatigue life by inducing compressive residual stresses.
If the stress state is uniaxial then the above formulas can be easily used. But if the
stress state is multiaxial then one has to decide which equivalent stress to use.
Figure 3.7 shows different stress representations of a cylindrical notched specimen
with an axial sine loading.
Figure 3.7: Stress representation types for an axially sine loaded cylindrical notched specimen [23].
22
As it can be seen from the graphs, the best representation is the Absolute
Maximum Principal stress, which does not under-estimate or over-estimate the
stress state.
Using the information up to this point one can obtain the fatigue life of a
component if the loading is cyclic and with constant amplitude and mean stress.
But this is, as stated, nearly never the case. The loading changes its characteristics
(mean and amplitude) in time. First the simple case where the loading type is
sinusoidal which changes its amplitude and/or mean in time will be considered.
Then the general case where the loading is random type will be covered.
Assume that during the flight of an airplane the following (Figure 3.8) stress cycles
are encountered by a component on this airplane.
Figure 3.8: Stress cycles encountered by an airplane for different flight regimes [15] To handle such loadings a damage concept must be introduced. Each of these
cycles causes some damage to the structure and when the damage is sufficiently
high the component fails. Miner and Palmgren investigated this phenomenon and
came up with the following equation:
23
∑=
=k
i i
i
Nn
11 (3.7)
where;
n : The number of stress cycles applied at a fixed stress amplitude
N : The number of cycles the material can withstand at applied fixed stress
amplitude
k : The number of stress cycle blocks with different amplitudes and/or means
This ratio is called the damage fraction and according to Palmgren-Miner when this
fraction reaches unity the material will fail. The assumption of this rule is that
damage accumulation for each stress cycle is linearly summed and is independent
of the application order. Many other theories can be found in the literature like
Shanley’s theory, Marco-Starkey theory, Corten and Dolan theory [5] and
Morrow’s plastic work interaction theory [11]. These theories assume different
damage accumulation types where material plasticity, stress history effects etc. are
considered. Therefore the damage value at failure need not be unity which conflicts
with Palmgren-Miner’s rule. However experimental results ([8], [11]) show that
Palmgren-Miner’s rule sometimes overestimates the life of the component.
This method works fine for cases as shown in Figure 3.8 but if the stress history is
random like in Figure 3.2 then one cannot know the n value directly. One way to
identify n has been proposed by Tatsuo Endo and M. Matsuiski in 1968 by
introducing the Rainflow-counting algorithm. There are many other cycle counting
methods like Peak counting method, Zero-crossing method, Range counting
method, Range-mean counting method, Range-pair counting method and Level-
crossing method [5]. But Rainflow cycle counting method gives all the necessary
information for fatigue analysis, which is the mean, amplitude and, the number of
repetitions.
24
The Rainflow-counting algorithm searches the time signal for cycles having the
same amplitude and mean, and then groups these into pairs. A procedure for
Rainflow-counting is given below [3]:
• Extract peaks and troughs from the time signal so that all points between
adjacent peaks and troughs are discarded.
• Make the beginning, and end, of the sequence have the same level. This can
be done in a number of ways but the simplest is to add an additional point at
the end of the signal to match the beginning.
• Find the highest peak and reorder the signal so that this becomes the
beginning and the end. The beginning and end of the original signal have to
be joined together.
• Start at the beginning of the sequence and pick consecutive sets of 4 peaks
and troughs. Apply a rule that states, if the second segment is shorter
(vertically) than the first, and the third is longer than the second, the middle
segment can be extracted and recorded as a Rainflow cycle. In this case, B
and C are completely enclosed by A and D (Figure 3.9).
Figure 3.9: Presentation of Rainflow cycles [3].
• If no cycle is counted then a check is made on the next set of 4 peaks, i.e.
peaks 2 to 5, and so on until a Rainflow cycle is counted. Every time a
Rainflow cycle is counted the procedure is started from the beginning of the
sequence again.
25
Another approach for Rainflow cycle counting from literature is given in Figure
3.10 to visualize the procedure.
After the application of this procedure to time history, all cycles will be counted
and utilizing a damage rule like the Palmgren-Miner’s damage rule with this cycle
information a fatigue analysis can be performed. Figure 3.11 shows a rainflow
cycle count performed for a random stress time history.
Figure 3.10: Application of Rainflow cycle counting method [23].
Figure 3.11: 3-D visualization of Rainflow cycle count of a random time history [23].
26
3.2. Strain Life Approach Fatigue can be considered as cracks initiating at highly stressed regions, which can
be notches, holes etc., then one should take into account the plasticity effects
occurring during this phenomena. As discussed in the previous topic, in the Stress
life approach the material remained elastic. But now it is seen that in fact this is
never the case. The explanation why Stress life approach works is that when the
stresses are so low that it may be assumed that the material behaves elastically. But
when dealing with high stresses one should always use the Strain life approach.
For this method, the Stress-Cycle curve is no more used. A more meaningful
Strain-cycle curve should be obtained. This can be achieved again by cyclic tests
where the strain is held constant via closed loop control test equipment. A similar
method to Stress life approach is used but with different considerations like
plasticity effects and hysteresis loops.
One of the problems in strain life approach arises in Rainflow cycle counting, when
the strain history is counted. When cycle counting a strain history, it may be
possible that equal strain ranges to have different mean stress levels in the Stress-
Cycle plane. This causes different fatigue lives and can not be detected from
Strain-Cycle plane. So the counting should be performed in a two-dimensional plot
where stress cycles are also checked for mean stress (Figure 3.12).
Also when the strain concentration factor is plotted it is seen that it is different than
the stress concentration factor as shown in Figure 3.13. In order to handle such a
case Neuber proposed that the concentration factor to be used in fatigue analysis
should be the geometric mean of stress concentration factor and strain
concentration factor.
There are some more slight variations in the methods used but the main approach is
the same as Stress Life approach.
27
Figure 3.12: Cycles with identical strain range (3-4, 6-7) but with different stress levels due to different means stresses [3].
Figure 3.13: Schematic representation of Stress concentration and Strain concentration factors for different stress values [23].
28
3.3. Crack Propagation Approach In the strain life approach it was stated that the main reason for fatigue is the
initiation of cracks and then propagation of them. If the initiation of the crack is
obtained from strain life approach then at that location one can assume a known
type of crack and calculate the propagation of it. The analytical calculation of crack
growth is covered in Linear Elastic Fracture Mechanics (LEFM) theory.
3.4. Vibration Fatigue Approach
All of the previous methods discussed were time domain methods. On the other
hand vibration fatigue approach, which is a frequency domain approach, is used
when the input loading or the stress history obtained from the structure is random
in nature and therefore best specified using statistical information about the
process.
The random time histories are usually expressed in frequency domain as Power
Spectral Density (PSD) functions. In order to obtain PSD’s first it is necessary to
discuss Fourier series and Fourier transform.
Every periodic time domain signal can be approximately expressed as the
summation of sines and cosines. This is called Fourier series of the time signal
(Figure 3.14).
∑∞
=
++=1
0 )sincos()(l
llll twbtwaatf (3.8)
where;
∫=T
ll tdtwtfT
a0
cos)(2 ....2,1,0=l (3.9)
∫=T
ll tdtwtfT
b0
sin)(2 ....2,1,0=l (3.10)
29
Tlwlπ2
= (3.11)
T : Period of the function ( )f t
lw : Forcing frequency of the function ( )f t
As it can be seen from the equations above as the number of sines and cosines are
increased the approximation gets better.
The signal is still in the time domain but now it is discretized by sines and cosines.
By mathematically manipulating the Fourier series expression (Equation 3.8) one
can get an extension of Fourier series, which is called the Fourier transform.
Figure 3.14: Fourier series presentation [3] The transformation between the time and frequency domain is accomplished using
the Fourier transform pair. The Fourier transform pair equations are given below.
∫+∞
∞−
−= dtetfwF jwt)()( (3.12)
∫+∞
∞−
= dtewFtf jwt)(21)(π
(3.13)
30
From this transformation a complex random signal can be converted into the
frequency domain and back to the time domain easily (Figure 3.15).
Figure 3.15: Fourier and inverse Fourier Transform presentation [3] The Fourier transform is applied to continuous time signals. In practice however,
the time histories will be digitally recorded in a discrete format by a computer. So a
discrete version of the Fourier transform pair that can be applied to real, digitally
recorded data is needed. A very rapid discrete Fourier transform algorithm was
developed in 1965, by Cooley and Tukey, known as the ‘Fast Fourier Transform’
(FFT).
The discrete transform pair does the same job as the Fourier transform pair but
operates on digitally recorded data. This algorithm is used to transform measured
data between the time and frequency domains. Equation 3.14 and Equation 3.15
show how this is performed.
Fast Fourier Transform to frequency domain
∑⋅
⋅−
⋅⋅=k
kN
ni
kn etyNTfy
)2()(2)(
π
(3.14)
31
Fast Inverse Fourier Transform to time domain
∑⋅
⋅
⋅⋅=n
nN
ki
nk efyT
ty)2(
)(1)(π
(3.15)
where,
T : Period of the function ( )ky t
N : Number of data points for Fourier transform
PSD is a statistical way of representing the amplitude content of a time signal in
the frequency domain. PSD presents this information as a statistical spectrum
where the area under the curve represents the mean square amplitude of the wave
other than the amplitude.
Using this definition PSD is obtained by taking the modulus squared of the FFT
(Equation 3.16). The FFT outputs a complex number given with respect to
frequency but in a PSD only the amplitude of each sine wave is retained (Figure
3.16). All phase information is discarded.
21 ( )2
defnPSD y f
T= (3.16)
frequency
PSD
Figure 3.16: Conversion of FFT to PSD [3]. Figure 3.17 shows the general representations of different of time histories and
their PSD forms.
32
Finally, the units of the PSD should be investigated because its units are not
straightforward due to taking the square of the FFT. If a time history of
acceleration in g’s is recorded, then the PSD will have units of Hzg 2 .
Figure 3.17: Time histories and corresponding PSD graphs [3]. In order to obtain the response of the structure to random loading, transfer
functions are used. That is, when the system is linear then the response of the
system will be the input multiplied by a linear transfer function.
( ) ( ) ( )Response f H f Input f= × (3.17)
where,
( )Response f : Response FFT of the system
( )H f : Transfer function of the system in frequency, f, domain
( )Input f : Input FFT to the system
If the PSD of response is obtained by using Equation 3.16,
* *1( ) ( ( ) ( ) ( ) ( ))2responseG f H f Input f H f Input fT
= × × × (3.18)
33
where,
( )responseG f : Reponse PSD
*( )H f : Complex conjugate of the transfer function
*( )Input f : Complex conjugate of the input FFT
And if the input is also written as a PSD function in Equation 3.18 by using again
Equation 3.16,
*( ) ( ) ( ) ( )response inputG f H f H f G f= × × (3.19)
where,
( )inputG f : Input PSD function
Then, if the system is excited by more than one input signals which are partially
correlated, the response PSD becomes,
*( ) ( ) ( ) ( )response i i input
i j ijG f H f H f G f= × ×∑∑ (3.20)
Thus by using Equation 3.20 one can obtain the response (in the fatigue analysis
case the stress response PSD) PSD to any random input PSD by using the transfer
function of the system. In this equation it is imported to note that the correlation
between the loadings is also considered.
Furthermore in Vibration Fatigue analysis, a method is required to extract the
Probability Density Function (PDF) of Rainflow ranges for damage calculations.
By using such a method, the classical Rainflow method will not be needed and the
PDF’s obtained will be used for fatigue calculations. This PDF can be directly
obtained from the PSD of stresses. The characteristics of PSD that are used to
obtain this information are the spectral moments of the PSD function. Before
proceeding to the fatigue analysis some definitions should be discussed. First if
( )G f is defined as the PSD then one can also define the nth spectral moment of the
PSD as shown in Figure 3.18 and expressed in Equation 3.21.
34
10
( ) ( )m nn k k kk
m G f df f G f fδ∞
== ⋅ = ⋅ ⋅∑∫ (3.21)
where,
fδ : The frequency increment
From Equation 3.21 it is seen that for a PSD function there exists infinite number
of spectral moments. But it has been found out that up to the fourth spectral
moment is enough for fatigue analysis.
Figure 3.18: Spectral moment calculation of a PSD function [3]. The first effort for providing a solution for estimating fatigue damage from PSD’s
was undertaken by S.O. Rice in 1954 [24]. Rice developed a very important
relationship for the number of upward mean crossings per second, E[0], and peaks
per second, E[P], in a random signal expressed in terms of their spectral moments.
[ ]2
4
mm
PE = (3.22)
[ ]0
20mm
E = (3.23)
Also another equation that relates the moments to an irregularity factor can be
obtained as:
35
[ ][ ] 40
220mm
mPE
E==γ (3.24)
This number can theoretically only fall in the range 0 to 1. For a value of 1 the
process must be narrow band (e.g. Sine wave). As the divergence from narrow
band increases then the value for the irregularity factor tends towards 0 (Pure
White noise).
Figure 3.19 summarizes the calculation process of expected number of zeros, peaks
and the irregularity factor.
Figure 3.19: Calculation of Expected Zeros, Peaks and Irregularity factor [3]. For fatigue damage analysis, in traditional time domain methods, mostly Palmgren-
Miner’s rule is used as given in Equation 3.7. This equation can be manipulated in
order to use in the frequency domain. To derive the fatigue damage equation for a
random process, let ( )p S to be defined as the Probability Densitiy Function (PDF)
of the random process.
For a resonant system subjected to a broad band excitation, in time T there will be
on average [ ] TPE ⋅ stress cycles of which dSSp )( will have peak values in the
stress range S to dSS + .
36
In the above definition, [ ]PE gives the number of peaks per second and when
multiplied with T , the number of peaks will be obtained in time T .
When dS , i.e. incremental stress, is multiplied with the PDF, )(Sp , dSSp )( will
give the fraction of the number of cycles which will be in the range of S to
dSS + .
Then, if )(SN is the number of cycles of stress level S that cause failure, then one
cycle at level S will cause a damage of )(
1SN
.
When the number of peaks in time T , i.e. [ ] TPE ⋅ , is multiplied with the fraction
of the number of cycles in the range of S to dSS + , i.e. dSSp )( , one obtains
cycles of stress level S expected in time T as [ ] dSSpTPE )(⋅⋅ .
So the fractional damage done at this stress level is found as
[ ])(
1)(SN
dSSpTPE ⋅⋅⋅ .
Thus the average damage from all stress cycles is [ ] dSSpSN
TPE )()(
1
0
⋅⋅ ∫∞
.
By substituting Equation 3.1 into this damage equation, Equation 3.25 can be
obtained [25].
[ ] [ ] ∫∞
=0
)( dSSpSCTPEDE b (3.25)
where;
[ ]DE : Expected value of damage
[ ]PE : Expected number of peaks per second
T : Life in seconds
C : Material constant from Equation 3.1
37
b : Basquin exponent
S : Stress amplitude
)(Sp : Probability density function of Rainflow stress ranges
From this equation one can find the life of a component by setting the [ ]DE equal
to unity and obtain the life,T in seconds. For the integrations given above a cut-off
value for the upper limit is necessary. This value is typically given in terms of Root
Mean Square (RMS) values of stress. It is common to set the cut-off value to 3
RMS in amplitude or 6 RMS in range, but practice has shown that it should be at
least set to 4.5 RMS in amplitude in order not to miss fatigue damage.
There are many empirical solutions for the Probability density function of
Rainflow stress ranges, )(Sp , like Tunna, Wirsching, Hancock, Chaudhury and
Dover [3] but the best correlation was obtained by Dirlik [26] after extensive
computer simulations to model random signals using the Monte Carlo method.
0
23
22
21
2)(
2
2
2
m
eZDeR
ZDe
QD
Sp
iii Z
iRZ
iQZ
i
−−−
++= (3.26)
where;
02 mS
Z ii = , [ ]
[ ]PEE 0
=γ , [ ]0
20mm
E = , [ ]2
4
mm
PE = , 4
2
0
1
mm
mm
xm =
2
2
1 1)(2
γγ
+−
= mxD
RDD
D−
+−−=
11 2
112
γ
213 1 DDD −−=
1
23 )(25.1D
RDDQ
−−=
γ
211
21
1 DDDx
R m
+−−
−−=
γγ
38
CHAPTER 4
4. OPERATIONAL DATA ACQUISITION AND SIGNAL ANALYSIS OF THE DISPENSER LOCATIONS
ON THE HELICOPTER In order to perform a fatigue analysis it is necessary to have the stress histories. For
this purpose one can experimentally test the structure for stress information with
strain gages, which is practically not possible because of not knowing the fatigue
critical locations beforehand. Another way is to obtain a finite element model of
the bracket and then applying the load boundary conditions to get the stresses. This
approach is very time and money saving when design iterations (geometric changes
of the model) require the stress results more than once. The acquisition of the load
boundary conditions is very important because slight changes in loading may result
in large changes in the stress results.
The location of the bracket was chosen to have the maximum dispersion of
Chaff/Flare particles (Figure 4.1). In order to find the load histories military
standards (MIL-STD-810E/F, GAM-EG-13 etc.) are investigated for the AH-1W
helicopter at the given location but unfortunately very coarse information is
obtained. Therefore operational flight tests are performed to obtain the loading
information [27] (Figure 4.2).
Figure 4.1: Location of the Chaff/Flare dispenser bracket on the AH-1W Helicopter
39
Figure 4.2: A view of operational flight tests Flight tests are performed according to a flight profile that was created by the
pilots, which is a composition of normal flight, and attack maneuvers performed
during the operational life of the helicopter (Figure 4.4).
Tri-axial Integrated Circuit Piezoelectric (ICP) type accelerometers are used to
obtain the loads encountered during the test at the location of the bracket (Figure
4.3).
For data acquisition, a 32-channel data acquisition system (ESA Traveller Plus)
[28] is used. Figure 4.5 shows the data acquisition system and its location on the
AH-1W helicopter.
Figure 4.3: A view of accelerometer location
Z
Y
X
41
a) b)
Figure 4.5: a) ESA Traveller plus system, b) Location of ESA Traveller plus system on AH-1W helicopter
The sampling frequency for the accelerometer is set to 5000 Hz and the flight
profile given in Figure 4.4 is flown two times. For the first one, the data is stored
for each flight regime separately and then for the whole flight profile the data is
stored continuously. In the analysis, the whole flight profile data is used. The time
histories for the accelerations obtained are given below.
Figure 4.6: X-axis acceleration versus time plot, Longitudinal axis of the Helicopter
42
Figure 4.7: Y-axis acceleration versus time plot, Transverse axis of the Helicopter
Figure 4.8: Z-axis acceleration versus time plot, Vertical axis of the Helicopter In order to obtain the required statistical information for fatigue analysis, the time
histories have to be converted into PSD’s (Figure 4.9-Figure 4.11) and for the
correlations between the three axes time histories Cross PSD’s (Figure 4.12-Figure
43
4.14) are also required. Cross PSD is defined as the Fourier transform of
corresponding cross-correlation function for two random processes. It can also be
defined as the comparison of two signals as a function of a time shift between
them. The importance of cross PSD is that when the vibration of a point is
measured in three axes, the obtained signals of the three axes are not independent
from each other. So in order to identify the correlation between them such a
function is required.
These functions are obtained in MSC Fatigue [29] software and given below. The
block size used for the PSD’s and cross PSD’s was 4096 samples. Also linear
averaging for FFT blocks and Root Mean Square (RMS) stress scaling is used. For
the cross PSD functions only X-Y, X-Z and Y-Z axis plots are given because Y-X,
Z-X and Z-Y axis cross PSD functions are symmetrical.
Figure 4.9: X-axis PSD plot, Longitudinal axis of the Helicopter
Also a PSD matrix is created which is to be loaded in MSC Fatigue in order to take
into account all correlation between the three axes as follows;
44
X-axis PSD X-Y axis cross PSD X-Z axis cross PSDX-Y axis cross PSD Y-axis PSD Y-Z axis cross PSDX-Z axis cross PSD Y-Z axis cross PSD Z-axis PSD
Figure 4.10: Y-axis PSD plot, Transverse axis of the Helicopter
Figure 4.11: Z-axis PSD plot, Vertical axis of the Helicopter
47
CHAPTER 5
5. FINITE ELEMENT ANALYSIS OF THE DISPENSER BRACKET
5.1. Finite Element Analysis Methods for Fatigue Analysis
After the loading information is obtained by operational flight tests, it is necessary
to create a finite element model and obtain the stress values due to the loading for
the fatigue analysis.
For this purpose a number of different analysis types can be used in finite element
environment. The finite element methods which are available in ANSYS [30] for
fatigue analysis are as follows:
1. Transient Dynamic Analysis
2. Power Spectral Density Analysis
3. Harmonic Analysis
Each of these methods has advantages and disadvantages over each other. For
vibration fatigue analysis, the harmonic analysis stress results are used. The details
of finite element analysis method selection are given in APPENDIX-A.
In order to perform a harmonic analysis, a finite element model is created. Then a
numerical modal analysis is performed for harmonic analysis. So an appropriate
element type has to be chosen to generate a finite element mesh.
48
5.2. Finite Element Mesh Generation of the Dispenser Bracket
In order to find a suitable element type for the chaff dispenser bracket, the structure
is investigated considering the geometry, mechanical connections and structural
dynamics. Then it is modeled using SOLID45 type solid element that is present in
ANSYS, which is an eight-node hexahedral element (Figure 5.1). Detailed
information on the selection of this element is present in APPENDIX-A.
Figure 5.1: Solid45 Hexahedral element [30]. The solid model of the bracket is created using IDEAS v10 [31]. Then the model is
converted into a Parasolid [31] type model file. This transformation is necessary in
order to keep the details of the model while transferring it into ANSYS. The model
transferred into ANSYS can be seen in Figure 5.2.
In order to reduce the number of elements, distortions of elements and time
required for solution due to the complex geometries of bolts, rivets and nuts, these
components are recreated in ANSYS. A sample rivet model and its finite element
mesh created are shown in Figure 5.3. Bolt meshes are also created similar to rivets
but the geometric differences present like nuts and bolt heads are considered. As it
can be seen from Figure 5.2 and Figure 5.3 the geometry had to be divided into
many sub parts in order to obtain a hexahedral mesh.
49
Figure 5.2: Chaff dispenser Bracket model in ANSYS
a) b)
c)
Figure 5.3: a) Rivet model created in ANSYS, b) Finite element model of the rivet, c) Finite element model of the bolt
Having a hexahedral mesh is very important when one seeks to find the stresses of
a structure. Tetrahedral elements tend to have artificial stress concentrations due to
their shape. The triangular faces of the tetrahedral element causes stress
concentrations in the structure if the mesh is coarse. So in order to have less
number of elements and better stress results (i.e. no stress concentrations due to the
mesh are present), hexahedral elements are used. From the following figures the
model, divided into sub parts shown with different colors, with bolts and rivets can
be seen.
50
a)
b) c)
d) e) Figure 5.4: a) Layout of the bolts and rivets b) Chaff dispenser bracket (Isometric view), c) Interior bolts of the upper casing, d) Interior stiffener plate, e) Interior stiffener plate (Close-up)
Bolts Rivets
51
Then each part is meshed with SOLID45 elements and the finite element model is
obtained (Figure 5.5).
a) b)
c) d)
Figure 5.5: Finite element mesh of the Chaff Dispenser bracket a) Isometric view, b) Right side view, c) Bottom isometric view, d) Bottom isometric, different angle of view
5.2.1. Contact Modeling for Fasteners As mentioned above the rivets and bolts are modeled as shown in Figure 5.3. For
the interaction between the bolt/rivets and the bolted/riveted plates, a special type
contact mesh is generated. By this way the assembly effects are simulated. Due to
the linear theory of modal analysis and harmonic analysis, it is not possible to use a
nonlinear type of contact for this purpose. ANSYS supports a special type of
52
contact, which is called Bonded Contact that can be solved linearly. The properties
of this contact are:
• No relative motion between the contacting surfaces
• Gaps are supported but once contact occurs the contacting surfaces cannot
separate.
While performing an analysis, the contact is checked once and then the stiffness
matrix is created. To eliminate the penetrations and gaps forming the stiffness
matrix is not updated. By this way a linear type contact can be solved which is a
good assumption when the deflections of the structure is small. This can also be
taught as gluing the contact surfaces or using lots of rivets/bolts for the assembly.
To determine whether the deflections are small or not, a static analysis has been
performed for the mentioned coarse model (Figure 5.6).
Figure 5.6: Total deflection of the Chaff dispenser bracket under gravity loading From the results, a maximum of 0.06 mm static deflection (red contours) is
obtained. This shows that the deflection of the structure is really small and relative
motion can be assumed to be small.
Maximum deflection point
53
The elements used for this contact are CONTA174 and TARGE170 [30]. Figure
5.7 shows a sample rivet assembly created using Bonded Contact in ANSYS.
Figure 5.7: Rivet assembly using Bonded Contact
5.2.2. Application of the Mass Element for Component Simulation
Finally, it can be seen from Figure 1.8 the magazine and the breech plates are not
modeled. This is because they are not expected to fail from fatigue. Also one more
reason is that the magazine is a composite structure for which no fatigue material
data is available. For this reason they’re modeled as a mass element with moments
of inertias calculated at the center of gravity of the magazine and breech plate.
The element used for this purpose is MASS21 element with rotary inertias [30].
For the calculation, the weight of the ammunitions has also been considered and
was simulated by changing the density of the magazine to give the correct weight,
which is the ammunitions plus the weight of the magazine itself. The total weight
of the mass element is found out to be 7.6 kg. Because the breech plate is
connected to the magazine, the mass element is connected to the bracket, with rigid
elements, from the bolt holes which are used to connect the magazine to the
bracket (Figure 5.8).
54
Figure 5.8: Mass element and its connections to the bracket In this model 81235 SOLID45 elements with 128034 nodes were used. Totally
including the contact elements (18 contact locations) and the mass element, the
number of elements becomes 202242.
The material properties are taken from the design documents [32]. For all of the
plates of the bracket material properties are taken as E: 73.1 GPa, ν: 0.33, ρ: 2780
kg/m3.
For the bolts it is taken as E: 205 GPa, ν: 0.3, ρ: 7850 kg/m3, and for the rivets E:
71.6 GPa, ν: 0.33, ρ: 2750 kg/m3.
Finally for the base connection parts E: 69 GPa, ν: 0.33, ρ: 2780 kg/m3 are used.
After material assignment the displacement boundary conditions must be applied.
The bracket is fixed to the helicopter by four bolts. Thus the real situation is that
the bottom surface is fixed around the bolts and the connecting surfaces are in
contact with the helicopter body. But due to the fact that the connecting surfaces do
not move relative to each other or separate, fixed displacement boundary
conditions are applied to the bottom face of the bracket as shown in Figure 5.9.
Mass element
55
Now the model is ready for numerical modal analysis which is required for
Harmonic Analysis. At this point the model, with above-mentioned modeling
techniques, is assumed to represent the real structure. This assumption should be
checked whenever there is the possibility to produce a prototype. Thus a prototype
of the Chaff dispenser bracket is produced and Experimental modal analysis has
been performed on the structure for finite element model verification. The details
of this study are given in EXPERIMENTAL MODAL ANALYSIS chapter.
Figure 5.9: Fixed displacement boundary conditions
5.2. Numerical Modal Analysis of the Dispenser Bracket
In order to perform a Harmonic analysis in ANSYS, one can use three methods.
The first one is the Full method which inverts the complete stiffness matrix and
therefore requires a longer solution time. The second method is the Reduced
method which uses the reduced DOF’s to solve a reduced modal matrix. Finally
Mode superposition method is available. This method is very fast compared to the
Full method and it uses all of the DOF’s but not the complete modal solution. It
uses a number of modes, which are chosen from the lowest mode to a specified
mode.
In the Eigen solution performed no damping is defined. The reason is that, in the
subsequent Harmonic analysis a proportional damping value will be defined. Thus
56
this will not change the natural frequencies and mode shapes of the structure. Also
when an undamped modal analysis is performed in ANSYS, the solution is very
fast compared to damped modal analysis. The model prepared, as in Figure 5.9
with fixed displacement boundary conditions applied to the bottom face of the part,
has been solved in ANSYS using Block Lanczos method [30] to give the resonant
frequencies and mode shapes up to 2000 Hz. The first three modes are shown
below (Figure 5.10-Figure 5.12).
The results of this analysis are also used for the verification of the finite element
model in the EXPERIMENTAL MODAL ANALYSIS chapter.
Figure 5.10: First mode shape of the Chaff Dispenser Bracket (first natural frequency 47.19 Hz)
Figure 5.11: Second mode shape of the Chaff Dispenser Bracket (second natural frequency 79.884 Hz)
57
Figure 5.12: Third mode shape of the Chaff Dispenser Bracket (third natural frequency 128.424 Hz) The Natural frequencies up to 2000 Hz are given in Table 5.1. Table 5.1: Natural Frequencies of the Chaff Dispenser Bracket from numerical modal analysis up to 2000 Hz.
SET FREQ (Hz) SET FREQ (Hz) SET FREQ (Hz) 1 47.19 25 729.69 49 1340.3 2 79.884 26 778.48 50 1358.2 3 128.42 27 853.08 51 1391.5 4 186.81 28 881.24 52 1450.3 5 223.53 29 909.95 53 1467.4 6 226.21 30 915.51 54 1472.8 7 249.67 31 920.5 55 1529 8 274.3 32 927.4 56 1535.3 9 279.58 33 961.12 57 1565.3
10 330.8 34 1013.4 58 1587.1 11 362.15 35 1057.6 59 1633.6 12 373.84 36 1063.4 60 1640 13 417.26 37 1078.6 61 1716.5 14 442.59 38 1092.8 62 1718.5 15 466.23 39 1098.4 63 1739.2 16 506.28 40 1132.8 64 1755.5 17 517.57 41 1133.9 65 1756.6 18 564.93 42 1157.7 66 1790.1 19 580.49 43 1198.8 67 1804 20 588 44 1239.2 68 1825.4 21 679.52 45 1247.9 69 1882.3 22 694.95 46 1276.8 70 1897.4 23 702.64 47 1298.5 71 1928.7 24 704.71 48 1321.7 72 1936.6
73 1962
58
5.3. Harmonic Response Analysis of the Dispenser Bracket
After numerical modal analysis, Harmonic Reponse Analysis is performed. A unit
loading of 1g is applied to the structure, which varies sinusoidally. Also it should
be noted that the loading is applied in each axis separately. Then the combination
of each axis is achieved in MSC Fatigue through the PSD matrix of accelerations.
It is important that the loading should be applied at an angle according to the
alignment of the bracket on the helicopter. As it can be seen from Figure 4.1 the
bracket is approximately 45 degrees to the x-axes of the helicopter (Figure 4.2).
Moreover a structural damping value has to be defined. From the experimental
modal analysis results the damping ratio has been chosen as %2. The reason of this
selection will be explained in the EXPERIMENTAL MODAL ANALYSIS
chapter. Also by choosing a lower damping ratio, compared to the experimental
result (%6.5), the fatigue analysis results will be on the safer side.
At this point the frequency range of the Harmonic analysis should be defined.
According to the Military Standards [33] it is required to analyze the structures
used on Helicopters up to 2000 Hz. The reason for this is that unlike Composite
Wheeled Vehicles like Cars, Trucks etc., the helicopters are excited up to 2000 Hz.
This can also be seen from Figure 4.9-Figure 4.11.
The analysis in ANSYS is performed using Mode Superposition technique [30]
where all 73 modes (from 0 to 2000 Hz), obtained from Modal analysis, are used.
The calculation resolution is set to 4 Hz, which means that the sine frequency will
increase by 4 Hz. Finally the stress values at specific frequencies have to be
extracted and written to the results file. The best solution would be to extract all the
frequencies (0-2000 Hz) but practically for large models the results file size
wouldn’t be feasible. So it is required to select the frequencies that do the most
damage to the system.
59
This has been achieved by investigating the loading frequencies and amplitude
content from PSD graphs and the resonant modes of the structure. The highest
levels of stress occur at resonance frequencies. So if the loading is also at the
resonant frequency then this frequency is a critical frequency and should be
included into the fatigue analysis. Also some high loading-no resonance
frequencies have been included into the analysis.
With the above mentioned considerations the extracted frequencies are 44, 80, 124,
180, 216, 248, 280, 364, 588 Hz. The transfer function stress results, with 1g
harmonic loading applied, for the first three frequencies are given in Figure 5.13-
Figure 5.21.
Figure 5.13: Harmonic Analysis Equivalent von Mises stress results (MPa), X-axis (according to the helicopter) unit loading at 44 Hz.
60
Figure 5.14: Harmonic Analysis Equivalent von Mises stress results (MPa), X-axis (according to the helicopter) unit loading at 80 Hz.
Figure 5.15: Harmonic Analysis Equivalent von Mises stress results (MPa), X-axis (according to the helicopter) unit loading at 124 Hz.
61
Figure 5.16: Harmonic Analysis Equivalent von Mises stress results (MPa), Y-axis (according to the helicopter) unit loading at 44 Hz.
Figure 5.17: Harmonic Analysis Equivalent von Mises stress results (MPa), Y-axis (according to the helicopter) unit loading at 80 Hz.
62
Figure 5.18: Harmonic Analysis Equivalent von Mises stress results (MPa), Y-axis (according to the helicopter) unit loading at 124 Hz.
Figure 5.19: Harmonic Analysis Equivalent von Mises stress results (MPa), Z-axis (according to the helicopter) unit loading at 44 Hz.
63
Figure 5.20: Harmonic Analysis Equivalent von Mises stress results (MPa), Z-axis (according to the helicopter) unit loading at 80 Hz.
Figure 5.21: Harmonic Analysis Equivalent von Mises stress results (MPa), Z-axis (according to the helicopter) unit loading at 124 Hz.
64
CHAPTER 6
6. EXPERIMENTAL MODAL ANALYSIS OF THE DISPENSER BRACKET
Experimental modal analysis is a method to determine the dynamic behavior of
structures. In this method the structure is excited with a known force value and the
responses from various locations of the structure is measured. Usually the force
value is measured via a force transducer and the responses are measured by
accelerometers. The force transducer (Figure 6.1-a) is attached to an impact
hammer (Figure 6.1-b) for lightweight structures or to an electro-dynamic shaker
(Figure 6.2) when the structure is big and heavy enough so that it cannot be exited
with a hammer where the excitation frequency is approximately limited to 400 Hz.
Figure 6.1: a) Modal Hammer, b) Force transducer
Figure 6.2: Electro-dynamic shaker with stinger attachment
65
The accelerometers are positioned on the structure such that they are not located on
the nodal points of the structure. Here nodal point means that, when the structure’s
resonant modes are excited, portions of the structure move but there are also some
deflection points, which are seen as stationary. It is advised that no accelerometer
is positioned on such locations, called nodes, in order not to miss a mode shape of
the structure.
Considering these effects, the accelerometers are placed as shown in Figure 6.3.
The structure tested usually rests on soft cushions or hung with elastic cords. The
reason for applying such a boundary condition, which is called a Free-Free
boundary condition, is that it can be easily obtained compared to the fixed
boundary condition. Free-Free boundary condition is as if the structure is free in
space. But due to the stiffness of cushions/elastic cords it is not so. This is the
reason why one gets the rigid body modes of the structure not at 0 Hz but close to
it. Still it is a much easier boundary condition to obtain than fixed boundary
condition. Then also a finite element modal analysis of the model is performed
with Free-Free boundary condition. And if the results are close then it is assumed
that the finite element model simulates the physical structure correctly. Then
further analysis can be made using the finite element model with actual boundary
conditions, which in this study is fixed from the bottom face of the bracket. This is
called model verification.
Note that the experiment is performed with the magazine, breech plate and
ammunition as shown in Figure 6.3-a.
After this step, the input loading and response accelerations are stored and
analyzed to give the Frequency Response Functions (FRF). The frequency
response function is the ratio of the output response of a structure to an applied
force. The response of the structure due to the applied force can be measured as
displacement, velocity or acceleration. Usually measurement of acceleration is
performed due to its simplicity. Then the measured time data is transformed to the
frequency domain by the application of Fourier Transform.
66
a) b)
Figure 6.3: a) Layout of accelerometer on the magazine housing, b) Excitation of the system with a modal hammer when there is no magazine present
FRF calculations for all of the accelerometers are performed and curves are fitted
to these functions in order to obtain the resonant frequency, damping and the mode
shape of the structure. In this analysis Least Squares Complex Exponential method
in LMS Test Lab was used for curve fitting [34].
By evaluating the results, the following comparison (Figure 6.4) is performed for
the first mode of the Chaff dispenser bracket. It should be noted that the following
results are obtained when the magazine is present so that the exact characteristics
of the structure can be identified. This is also important to verify that the mass
element used for the magazine, breech plate and the ammunition correctly
simulates the dynamics of the structure.
As is can be seen from the results (Figure 6.4, 125.738 Hz from finite element, and
128.0391 Hz from experiment) the natural frequencies and mode shapes are close.
Also a damping ratio of % 6.25 is obtained for the first mode. In theory for
aluminum, a damping ratio less than % 1 is defined. The high damping ratio may
be due to application of exponential decay window on the response signal (when
the time frame is insufficient) and experimental errors. However in the numerical
analysis a damping ratio of % 2 (this value is obtained by experience from previous
studies of aerospace platforms) is used in order to be in the safe side and to
67
compensate for experimental errors. So the dynamic behavior of the structure can
be simulated using the finite element model.
a) b) Figure 6.4: a) Finite element Modal Analysis for the first mode (125.738 Hz) of the Chaff Dispenser bracket with Free-Free boundary condition, b) Experimental modal Analysis for the first mode (128.0391 Hz) of the Chaff Dispenser bracket (including magazine, breech plate, ammunition as a mass element) with Free-Free boundary condition
68
CHAPTER 7
7. FATIGUE ANALYSIS OF THE CHAFF DIPENSER
BRACKET
7.1. Analysis Parameters
The transfer functions obtained from harmonic analysis (also called Frequency
Response Function in MSC Fatigue) and the loading PSD matrix are the inputs to
the MSC Fatigue software. The Loading Table is shown in Table 7.1.
Table 7.1: Transfer functions and PSD input Loading Table Frequency Response Cases
# of Frequency Response PSD matrix (i,1) PSD matrix (i,2) PSD matrix (i,3)
1 9 (01:09) X axis PSD X-Y axis cross PSD X-Z axis cross PSD2 9 (10:18) X-Y axis cross PSD Y axis PSD Y-Z axis cross PSD3 9 (19:27) X-Z axis cross PSD Y-Z axis cross PSD Z axis PSD
After the transfer functions and the loading PSD matrix are entered then the
material information is assigned to the elements. In this analysis the material type
is set to 2024-HV-T3 Aluminum (Figure 7.1). The analysis type is chosen as
Vibration fatigue due to the excitation of resonant modes by the loading. As for the
solution method Dirlik’s approach is used.
The mean stress correction method is chosen to be none. The reason for this choice
is that, for Vibration fatigue analysis the mean stresses cannot be defined directly
which is one of the main disadvantages of the technique. Because the stresses input
to the fatigue software are transfer functions, they oscillate around zero stress level.
It is possible to find the static stress level (the mean stress levels due to gravity or
pretensions in the structure) for the structure and then shift the S-N curve by
applying any mean stress correction theories (Goodman, Soderberg). But due to the
69
variation in the mean stress levels throughout the structure, this can lead to
erroneous results. Nevertheless it is a better solution to find the life under only
oscillating stresses and then recalculate the life of the most critical point with the
mean stress added. Such an analysis is performed in this study.
Figure 7.1: 2024 HV T3 S-N curve
Furthermore, due to riveting processes, there is large amount of compressive
residual stresses on and around the rivets. In APPENDIX-B a sample riveting
process is shown. Nevertheless, due to having mostly compressive residual stresses
on the rivet and rivet hole, the mean stress improves the fatigue life as shown in
Figure 7.2. There exists also tension zones a little away from the rivet holes but
their effect is neglected in this study.
Finally, the equivalent stress levels are calculated according to the Absolute
Maximum Principal Stresses, as shown in Figure 3.7, due to the reasons explained
in FATIGUE THEORY chapter. Some sample input windows from MSC Fatigue
can be examined in APPENDIX-D.
70
Figure 7.2: Mean stress effect on the S-N diagram [23].
7.2. Fatigue Analysis Results
For the parameters chosen above the fatigue analysis is performed and the life of
the structure is found to be 6.6e7 seconds (18333 flight hours). Life distributions of
the structure are as given below.
Figure 7.3: Life distribution on the Chaff dispenser bracket (life of the dispenser is 6.60e7 sec).
6.60e7 sec
71
a) b) Figure 7.4: Life distribution on the Chaff dispenser bracket a) Overall representation, b) Critical location, Zoomed view (life of the dispenser is 6.60e7 sec).
a) b)
Figure 7.5: a) Overall representation of life distribution on the Chaff dispenser bracket, different angle of view b) Life distribution of the interior stiffener plate edge, Zoomed view.
7.3. Case Studies for Different Design Parameters
After this analysis, some case studies for different design parameters such as
analysis methods, surface finish effects etc. are performed for the most critical
location as shown in Figure 7.3. First of all, the statistical parameters of the fatigue
analysis are obtained and given in Table 7.2.
6.60e7 sec
72
Table 7.2: Statistical parameters of the stress PSD for fatigue analysis
Statistics Values Expected Number of Mean Crossings 206,6 Expected Number of Peaks 235,3 Irregularity Factor 0,8784 Root Mean Square 43,34 MPa 0th Moment 1879 (MPa)2 1st Moment 3,771E+05 (MPa2)*(Hz) 2nd Moment 8,021E+07 (MPa2)*(Hz)2
4th Moment 4,439E+12 (MPa2)*(Hz)3
For all of the case studies these statistical values are used. Then the effects of the
surface finish and the surface treatments are considered as shown in Table 7.3 and
Table 7.4.
Table 7.3: Surface finish effects on the fatigue life
Surface Finish Fatigue life (sec) Polished 6,619E+07 Ground 2,670E+07 Good Machined 1,286E+07 Average Machined 5,765E+06 Poor Machined 3,025E+06 Hot Rolled 2,192E+06 Forged 1,871E+05 Cast 1,543E+05 Water corroded 3,628E+05 Seawater corroded 2,573E+04 Table 7.4: Surface treatment effects on the fatigue life Surface treatment Fatigue life (sec) No treatment 6,619E+07 Cold Rolled 3,786E+09 Shot Peened 2,670E+08
As expected as the surface finish becomes poor, than the fatigue life dramatically
decreases. Also from Table 7.4 it is evident that, the surface treatments which
produce compressive zones increase the fatigue life of the structure.
73
Furthermore the Vibration Fatigue analysis methods are discussed and the results
are given in Table 7.5.
Table 7.5: Effects of Vibration Fatigue analysis methods on fatigue life
Analysis method Fatigue life (sec) Dirlik 6,619E+07 Narrow Band 5,493E+07 Tunna 2,004E+08 Wirsching 9,205E+07 Hancock 6,254E+07 Kam & Dover 6,376E+07 Steinberg 7,419E+07 The difference between the estimation methods is due to different assumptions
considered. However as explained in the FATIGUE THEORY chapter, the best
method is the Dirlik’s method.
The effect of materials on fatigue life is directly related to the S-N characteristics
of the material. Therefore as shown from Table 7.6, 7075 HV-T6 having a better S-
N characteristics has a higher life than 2024 HV-T3
Table 7.6: Effect of materials on fatigue life
Material Fatigue life (sec) 2024 HV-T3 6,61998E+07 7075 HV-T6 2,64015E+08 Furthermore one can investigate the limiting mean stress value for a given design
life. The limiting mean stress is the mean stress which can be present with the
alternating stresses for the specified design life. Such a study is carried out and for
a design life of 12000 hours (4.32e7 seconds) a mean stress level of 14.06 MPa is
obtained by re-performing the fatigue analysis with previously defined alternating
stresses. Using this means stress and the means stress correction theories the
fatigue life results are given in Table 7.7.
74
Table 7.7: Effect of means stress on fatigue life
Mean Stress Correction for a mean of 14.06 MPa stress Fatigue life (sec) No correction 6,61998E+07 Goodman 4,24667E+07 Gerber 6,46128E+07
One more analysis result obtained for the most critical location is the absolute
maximum principal stress’s power spectral density graph as shown in Figure 7.6.
From Figure 7.6, for the most critical location the Root Mean Square (RMS) stress
value (i.e. as average) encountered can be calculated by taking the integral of the
PSD curve and then taking the square root of that value.
The stress RMS is found to be approximately 44.5 MPa. If the response is assumed
to be Gaussian, then for the 3σ− to 3σ+ distribution ( 6σ in range) the maximum
stress value encountered can be calculated by multiplying the RMS stress value by
three. Thus, the maximum stress is approximately 133.5 MPa.
Figure 7.6: Absolute maximum principal stress power spectral density graph of the most critical location on the Chaff dispenser bracket
75
The conclusion of the fatigue analysis is that, a life of 6.60e7 seconds (18333
hours) has been obtained for this design of dispenser. Also the most damaged areas
are on the left side of the bracket. In order to verify this result component fatigue
tests are performed.
76
CHAPTER 8
8. FATIGUE TESTING OF THE DISPENSER BRACKET
8.1. Test Setup
The physical prototype of the Chaff Dispenser bracket (Figure 8.1) is produced
according to the virtual design model.
Figure 8.1: A view of the physical prototype of the chaff dispenser bracket The acceleration versus time data obtained from helicopter flight tests, as
mentioned in Chapter 4, are used to accelerate the vibration profiles in order to test
the prototype in practical durations. For this purpose LMS Mission Synthesis [35]
software is used.
In this software a design life is set to be 12000 hours, which is required by the
customer. Then the vibration profile is accelerated to be 4 hours that is the
suggested laboratory time given in the military standards [33] in each axis. As the
77
vibration test equipment (Figure 8.2) is capable of performing each axis alone, the
tests are performed in three steps (Figure 8.3).
Figure 8.2: A view of LDS electromagnetic vibration test equipment [36].
a) b)
c)
Figure 8.3: A view of dispenser bracket test, a) X-axis vibration test, b) Y-axis vibration test, c) Z-axis vibration test.
+X -X
+Y -Y
+Z
-Z
78
8.2. Test Results
The tests are performed for the design life (12000 hours) first and then extended to
another 12000 hours, which doubled the design life. The reason for increasing the
design life is to satisfy the Fedaral Aviation Regulations (FAR) [37].
The failure criterion for these tests is defined as the formation of a visible crack. In
the second 12000 hours testing, the x and y axes are tested for 4 hours first (12000
hours of simulated flight test). Then at the third hour, which corresponds to the
9000th hour of simulated flight test due to the lineer characteristics of Palmgren-
Miner’s rule, of the z axis the upper part of the bracket started to losen.
Due to the fact that the x and y axis were completed one can not identify the flight
hour when the three axes would be applied at the same time. To be on the safe side
it is assumed that these deformations occurred at the 9000th flight hour for three
axes loading.
Thus when the first 12000 hours is also considered, the deformations occurred at
21000 flight hours. At the end of the tests, the prototype is disassembled and each
part is examined. The following plastic deformations were observed.
a) b) Figure 8.4: a) Plastic deformation on the bolt hole of the magazine housing, b) Plastic deformation on the bolt hole of the magazine housing, Zoomed view.
79
a) b) Figure 8.5: a) Plastic deformation on the rivet/rivet hole of the side carrier legs, b) Plastic deformation on the rivet/rivet hole of the side carrier legs, Zoomed view. However, these deformations cannot be referred as fatigue failures. The damage
type that can be analyzed with Vibration Fatigue method is only fatigue damage.
Other types of damage can not be identified with this approach. But due to the
plastic deformations occurred, there is high probability that fatigue failures will
occur at these locations.
8.3. Test-Analysis Comparison
As mentioned above the fatigue tests are performed on each axis separately. This
however is not the same as applying all of the vibration profiles at the same time as
occurring in real case. To investigate this effect the following fatigue analysis is
performed.
For test-analysis correlation, fatigue analyses are repeated for each axes which are
solved separately. As it can be seen from these results X-axis (longitudinal axis of
the helicopter) has nearly no effect on the fatigue life of the bracket. Y-axis
(transverse axis of the helicopter) has the largest effect and Z-axis (vertical axis of
the helicopter) has a lesser effect compared to the Y-axis.
80
Figure 8.6: Fatigue life (1.56e17 sec) of the Chaff Dispenser Bracket under X-axis Loading
Figure 8.7: Fatigue life (8.80e7 sec) of the Chaff Dispenser Bracket under Y-axis Loading
1.56e17 sec
8.80e7 sec
81
Figure 8.8: Fatigue life (2.06e12 sec) of the Chaff Dispenser Bracket under Z-axis Loading As for each axis a different critical location can exist, 8 critical locations are
obtained from the first fatigue analysis, in which all the three axis loading is
applied at the same time, and then the damage analyses are performed for these
locations. First of all, the most critical location found under 3-axis loading as
shown in Figure 7.3 is analyzed (Figure 8.9).
When the life of the Chaff dispenser bracket is multiplied by the damage-per-
second-value, the damage value at failure is obtained as 1 according to Palmgren-
Miner’s rule. From the analysis, the damage per second value for the most critical
location is found to be 1.513e-8.
Also it is known that the life of this location is 6.60e7 seconds (18333 flight hours).
Then if the damage per second value and this life is multiplied;
1
(sec)760.6)sec
(8513.1
=⇒
×−=
Damage
edamageeDamage
2.06e12 sec
82
a) b)
Figure 8.9: Damage analysis of the Chaff dispenser bracket under 3-axial loading a) Overall view, b) Location-1, zoomed (1.513e-8 damage per second). So if the test results showed that no fatigue damage was obtained after 24000 hours
(8.64e7 sec) of flight when each axis is applied separately then the damage values
of the axis-by-axis fatigue analysis should be examined. First of all, the damage
values for 3-axial loading are analyzed for different flight hours and are given in
Table 8.1.
Table 8.1: Damage values for different flight hours of Location-1 when 3-axis loading is applied.
Damage per second
Damage at 12000 hours
Damage at 18333 hours
Damage at 24000 hours
3-axis Loading 1,51E-08 6,54E-01 9,99E-01 1,31E+00
Here it can be seen that the failure occurs at 18333 flight hours and at 24000 flight
hours the damage value is 1.31. Then the same location is analyzed for axis-by-axis
loading (Table 8.2).
Table 8.2: Damage analysis of most critical location (Location-1) for axis-by-axis loading.
Damage per second
Damage at 12000
hours
Damage at 18333
hours
Damage at 24000
hours X-axis Loading 1,00E-18 4,32E-11 6,60E-11 8,64E-11Y-axis Loading 1,15E-08 4,96E-01 7,58E-01 9,92E-01Z-axis Loading 7,94E-18 3,43E-10 5,24E-10 6,86E-10 Summed Damage (X-Y-Z axis) 1,15E-08 4,96E-01 7,58E-01 9,92E-01
83
From the summed damage values, it is found out that at 18333 flight hours the total
damage is 0.758, and at 24000 flight hours it is 0.992. Therefore even at 24000
flight hours there is no failure according to the axis-by-axis loading and there is a
%23.8 deviation of damage value from 3-axis loading.
The same analysis is performed for the other locations as shown in Figure 8.10 and
Figure 8.11. The coordinates of these points are shown in APPENDIX-E.
a) b)
c) d) Figure 8.10: Damage analysis locations of the Chaff dispenser bracket, a) Overall view, b) Location-2 zoomed, c) Location-3 zoomed, d) Location-4 zoomed.
84
a) b)
c) d)
e) f)
Figure 8.11: Damage analysis locations of the Chaff dispenser bracket, continued, a) Overall view, different angle, b) Location-5 zoomed, c) Overall view, different angle, d) Location-6 zoomed, e) Location-7 zoomed, f) Location-8 After the damage values per second are obtained a table is generated for each of the
locations analyzed, including Location-1, as given in Table 8.3.
85
Table 8.3: Damage analysis performed for eight locations shown in Figure 8.10 and Figure 8.11.
Damage per second of X-Axis Loading
Damage per second of Y-Axis Loading
Damage per
second of Z-Axis
Loading
Summed damage per second of axis-by-axis
loading (X-Y-Z axis)
Damage per second of 3-Axis loading
Location-1 1,00E-18 1,15E-08 7,94E-18 1,15E-08 1,51E-08
Location-2 1,00E-18 1,22E-10 1,00E-18 1,22E-10 1,33E-10
Location-3 1,00E-18 2,33E-13 6,31E-17 2,33E-13 1,22E-12
Location-4 1,00E-18 8,69E-10 7,94E-16 8,69E-10 1,51E-09
Location-5 1,00E-18 3,30E-13 1,00E-18 3,30E-13 4,30E-13
Location-6 1,00E-18 1,00E-18 1,00E-15 1,00E-15 4,65E-15
Location-7 1,00E-18 1,00E-18 6,31E-15 6,31E-15 2,80E-14
Location-8 1,00E-18 1,00E-18 1,00E-15 1,00E-15 6,31E-15
Using these values a graph is obtained for summed damage per second of axis-by-
axis loading and damage per second of 3-axis loading versus analyzed locations as
given in Figure 8.12.
Damage per second - Location
1.00E-16
1.00E-14
1.00E-12
1.00E-10
1.00E-08
1.00E-06
1.00E-04
1.00E-02
1.00E+00
0 1 2 3 4 5 6 7 8 9Location
Dam
age
per s
econ
d (L
og s
cale
)
3-Axis Loading
Summeddamage for axis-by-axis loading
Figure 8.12: Summed damage per second of axis-by-axis loading and damage per second of 3-axis loading versus analyzed locations
86
Also the damage values at 18333 flight hours are plotted to emphasize the
difference between 3-axial loading and axis-by axis loading (Figure 8.13). Finally
the percent deviation values of axis-by-axis loading from 3-axis loading are plotted
in Figure 8.14.
Damage at 18333 flight hours - Location
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
0 1 2 3 4 5 6 7 8 9Location
Dam
age
(Log
sca
le)
3-Axis Loading
Summeddamage for axis-by-axis loading
Figure 8.13: Damage values at 18333 flight hours versus analyzed locations
Damage percent deviation of axis-by-axis loading from 3-axis loading
0.0010.0020.0030.0040.0050.0060.0070.0080.0090.00
0 1 2 3 4 5 6 7 8 9Location
Perc
ent d
evia
tion
Figure 8.14: Percent deviation values of axis-by-axis loading from 3-axis loading
87
From these analyses it is evident that more test time, i.e. more excitation of the
structure in order to increase the fatigue damage, is required to fully obtain the 3-
axis effect when performing an axis-by-axis test procedure. However the increase
in test time can not be the same for all locations due to having different percent
deviations of damage as shown in Figure 8.14. To be on the safe side, the test time
for axis-by-axis testing can be increased by %85 when location 8 is considered.
This location has the largest deviation from the 3-axis results and covers the other
points. Nevertheless, this increase is valid only for this structure and further studies
should be made in order to obtain a single value for simulating the 3-axis effect
when using axis-by-axis testing of structures.
Also from the results of Table 8.2 it can be concluded after the tests the bracket is
close to failure, for the most critical location, but still has some life left which is
also seen from the examinations on the tested prototype. The damage which
occurred at the holes in the tests can be prevented with local and global
modifications of the thickness. However caution must be made that when the
thickness of the structure changes also the dynamics of the structure changes. So
the fatigue analysis must be repeated.
Furthermore, from Table 8.3 it is seen that the x-axis has a much lesser damage
effect compared to the y and z axis. In some cases, for design qualification tests
performed in industry test tailoring is required. Then only the y and z-axis tests can
be performed.
88
CHAPTER 9
9. DISCUSSION AND CONCLUSIONS
In this study a vibration induced fatigue analysis has been performed for a Chaff
dispenser bracket used on helicopters. A design cycle, which consists of building
the finite element model, verification of the model, fatigue analysis and finally
testing the prototype, has been performed.
Structures used on helicopters are exposed to a wide frequency band of excitation
due to the random aerodynamic loadings and periodic loadings caused by the rotors
and transmissions of the helicopter. This situation makes structural analysis to be
dynamic in nature.
Structures containing dynamic behavior as resonance are indeed difficult to analyze
for fatigue. The response of the structure is not linear when the forcing frequency is
close to a resonance. It is a nonlinear function of the forcing frequency. Due to this
phenomenon one cannot simply use the traditional time domain fatigue methods,
where the forcing frequency is far away from any resonance so that the structure
response can be obtained from static analysis as generally used in automotive
industry. One more point to discuss is that, although the loading is multiaxial, due
to the dynamics of the loading (i.e. simulataneous excitation of many natural
frequencies of the structure) Vibration Fatigue method is used instead of multiaxial
fatigue methods. If stress data for the multiaxial loading can be obtained, which for
large and complex structures in FEM is impractical (requires transient dynamic
analysis), then multiaxial fatigue theories can be used. But this method, without
using FEM, requires experimental methods.
As seen from the fatigue analysis, a life of 6.60e7 sec (18333 flight hours) has been
obtained. However many assumptions has to be made in order to obtain fatigue
89
results. In fact the vibration fatigue analysis is very efficient in solving fatigue
analysis of components. But when assemblies are considered then assumptions due
to contact of parts and elastic-plastic deformations which cause residual stresses by
the assembly elements as bolts/rivets are neglected.
The above problems are also a disadvantage for time domain analysis. The
nonlinear effects can only be implemented via a transient dynamic nonlinear
analysis, which for large models is practically impossible to perform. The
important point is to judge the accuracy of the results required. If the structure is
subject to highly nonlinear effects, than in order to obtain a correct result one has to
perform a non-linear analysis.
The assumptions made have to be verified by tests in order to obtain reasonable
results. For this purpose during the preparation of the finite element model and
after the fatigue analysis tests have been performed. Experimental modal analysis
has been performed to verify the finite element model such that the model
simulates the dynamic characteristics of the structure. Furthermore a fatigue test
has been performed on a prototype to verify the fatigue analysis results.
It should be noted that the accurate stress results obtained from the finite elements
analysis play a critical role for fatigue life of the structure. Approximately 7%
increase of stress causes 50% reduction of life. Therefore it may not be enough that
the finite element model simulates the dynamic characteristics correctly. Stress
concentrations occurring in the finite element is the main reason for inaccurate
stress results, which can only be validated by tests. In the analysis performed, a
more valuable information than the life is the damage distribution of the structure.
Regarding this distribution most damaged areas can be modified to be stronger.
Also it should be noted that the critical location found from a static analysis is
different from the critical location obtained from the dynamic analysis. This is due
to the fact that when the dynamics of the structure are active, i.e. the natural
frequencies of the structure are excited, then the response of the structure is
controlled by the mode shapes.
90
The fatigue tests performed also contain some assumptions. First of all, the loading
is applied axis-by-axis and cross correlations are not considered. Secondly, the
moments (i.e. rotations) are not applied. Finally the acceleration of the test from
12000 hours to 4 hours also contains assumptions for damage conservation. As it
can be seen from Chapter 9, the damage values obtained for axis-by-axis loading
are different from all-three-axes applied. The best is to have a 6-DOF vibration test
equipment and to apply all loadings at the same time with cross correlations and
with no acceleration of loading to reduce the test time. When axis-by-axis testing
has to be performed, one has to do a numerical fatigue analysis in order to find out
the damage differences between 3-axis loading and axis-by-axis loading. Then a
corresponding safety factor for the testing time to compensate the damage
deviation from the 3-axis testing should be applied.
Furthermore during the data acquisition phase for the location of the chaff
dispenser bracket only one accelerometer was used. Increasing the number of
accelerometers at this location would give more accurate loading boundary
conditions.
A better way of obtaining fatigue life is obtained by using strain gages at critical
locations, which can be obtained by the fatigue analysis performed in this work,
and then performing operational flight tests to get the stress history. Then this
stress history can be used to get the fatigue life of the strain gage locations. Due to
the fact that the stress values obtained will be more accurate than the finite element
stress results a more accurate life value can be achieved.
One more point is that when the dispenser fires its ammunition, there exist shock
effects which are not considered in this study. However, the shock forces can be
measured by using force transducers and then they can be entered into the finite
element software. Thus, the stresses generated from the shocks can be calculated
(using transient analysis) and then analyzed for fatigue, in fatigue software’s such
as MSC Fatigue, nCode, LMS Falancs etc.
91
The residual stresses from riveting process are neglected. However the residual
stresses from bending of the plates are stress relieved by heat treatment.
The fatigue analysis performed in this work can be also optimized by performing
mesh sensitivity analysis to reduce the stress concentrations due to a coarse mesh,
increasing the number of frequency steps to increase the response resolution and
using strain gages on the prototype to verify the stress results obtained during the
analysis.
92
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[2] Farlex Inc., “Metal fatigue - encyclopedia article about Metal fatigue”,
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NAFEMS, Germany, 2000. [4] Ewing, M., Aircraft Structures Design and Analysis / Component Design,
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Environment” XVI Encuentro Del Grupo Español De Fractura, Spain, 1999. [7] Giglio, M., “Fem Submodelling Fatigue Analysis of a Complex Helicopter
Component”, International Journal of Fatigue, 1999, Vol.21, pp.445-455. [8] Wu, W. F., Liou, H. Y., Tse, H. C., “Estimation of Fatigue Damage and Fatigue
Life of Components Under Random Loading”, International Journal of Pressure Vessels and Piping, 1997, Vol.72, pp.243-249.
[9] Kyu, P. J., Han, L. D., Jae, L. C., Cheul, C. B., Tae, C. K., Finite Element
Method Analysis and Life Estimation of Aircraft Structure Fatigue/Fracture Critical Location, Technical Report, Korea Aerospace Industries, 2001.
[10]Pitoiset, X., Preumont, A., Kernilis, A., “Tools for Multiaxial Fatigue Analysis
of Structures Submitted to Random Vibrations” Proceedings European
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Conference on Spacecraft Structures Materials and Mechanical Testing, Germany, November 1998.
[11]Liou, H.Y., Wu, W.F., Shin, C.S., “A Modified Model for The Estimation of
Fatigue Life Derived from Random Vibration Theory”, Probabilistic Engineering Mechanics, 1999, Vol.14, pp.281-288.
[12]Doerfler, M. T., “An Evaluation of Service Life Analysis of Metallic Airframe
Structure with MSC Fatigue”, Technical Report, Lockheed Martin Corporation, USA, 1997.
[13]Langrand, B., Deletombe, E., Markiewicz, E., Drazétic, P., “Identification of
Non-Linear Dynamic Behavior and Failure for Riveted Joint Assemblies”, Journal of Shock and Vibration, 2000, Vol.7, pp.121-138.
[14]Eichlseder, W., “Fatigue Analysis by Local Stress Concept Based on Finite
Element Results”, Journal of Computers and Structures, 2002, Vol.80, pp.2109-2113.
[15]Hawkyard, M., Powell, B.E., Stephenson, J.M., McElhone, M., “Fatigue Crack
Growth from Simulated Flight Cycles Involving Superimposed Vibrations”, International Journal of Fatigue, 1999, Vol.21, pp.S59-S68.
[16]Shang, D. G., Wang, D. K., Li, M., Yao, W. X., “Local Stress-Strain Field
Intensity Approach to Fatigue Life Prediction Under Random Cyclic Loading”, International Journal of Fatigue, 2001, Vol.23, pp.903-910.
[17]Liao, M., Shi, G., Xiong, Y., “Analytical methodology for predicting fatigue
life distribution of fuselage splices”, International Journal of Fatigue, 2001, Vol.23, pp.S177-S185.
[18]Conle, F.A., Chu, C.-C., “Fatigue analysis and the local stress-strain approach
in complex vehicular structures”, International Journal of Fatigue, 1997, Vol.19 No: 1, pp.S317-S323.
[19]Urban, M.R., “Analysis of the Fatigue Life of Riveted Sheet Metal Helicopter
Airframe Joints”, International Journal of Fatigue, 2003, Vol.25, pp.1013-1026.
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[20]Szolwinski, M. P., Farris, T. N., “Linking Riveting Process Parameters to The Fatigue Performance of Riveted Aircraft Structures”, Journal of Aircraft, 2000, Vol.37 No: 1, pp.130-137.
[21]Candan, B., Yorulma Teorisine Giriş, Eğitim Notları, Figes, Bursa, 2003. [22]Shigley, J. E., Mechanical Engineering Design, First Metric Edition, McGraw-
Hill, 1986. [23]Bennebach, M., Integrated Durability Management, Course Notes, NCODE
International, Tekno Tasarım, Bursa, 2004. [24]Rice S.O., “Mathematical Analysis of Random Noise. Selected Papers on
Noise and Stochastic Processes”, Dover, New York, 1954. [25]Lalanne C., Mechanical Vibration & Shock, Fatigue Damage, Vol IV, Taylor
& Francis Books, London, 2002. [26]Dirlik, T., Application of computers in Fatigue Analysis, Ph.D. Thesis,
University of Warwick, 1985. [27]Çelik, M., Aykan, M., AH-1W Helikopteri Operasyonel Titreşim Profili
Oluşturma Testleri, Teknik Rapor, ASELSAN, 2003. [28]Traveller Plus and ESAM Software Manual, Measurements Group Inc.,
Munich, 2000. [29]MSC Fatigue Version 2003 User’s Manual, MSC Software Inc., USA, 2003. [30]ANSYS Release 8.1 User’s Manual, ANSYS Inc., USA, 2003. [31]I-DEAS NX Release 10 User’s Manual, Electronic Data System Inc., USA,
2003. [32]Eryürek H., Montaj Fırlatıcı Tasarım Dokümanı, ASELSAN Inc., Ankara,
2003.
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[33]MIL-STD-810F, “Environmental Engineering Considerations and Laboratory Tests”, Department of Defense Test Method Standard, USA, 2000.
[34]LMS Test Lab Release 4A Software Manual and Training Notes, LMS
International, Belgium, 2003. [35]LMS Mission Synthesis Release 3.5.D Software Manual and Training Notes,
LMS International, Belgium, 2003. [36]LDS V895-440 HBT 900C Model Vibration Shaker User’ Manual, Ling
Dynamics Inc., England, 2004. [37]FAR-25, “Federal Aviation Regulations Subchapter C, Section 25.571, Fatigue
Evaluation”, Federal Aviation Administration, Department of Transportation, USA, 2002.
[38]Ciğeroğlu, E., “Non-Linear Vibration Analysis of Bladed Disks with Dry
Friction Dampers”, M.S. Thesis, Middle East Technical University, Ankara, 2002.
[39]MSC Marc Version 2003 User’s Manual, MSC Software Inc., USA, 2003. [40]Rodamaker, M., “Large Mass Method”, “http://xansys.org”, 2004.
96
APPENDIX-A
A.1. Identification of the System and Analysis Parameters
First of all, the structure must be investigated to find out whether its natural
frequencies are excited or not. If the structures natural frequencies are excited then
the response of the structure to the loads will not be linear. It will be a nonlinear
function of the forcing frequency. That is, for linear static loading if x units of
loading is applied than one gets y units of stress and for 2x units of loading, 2y
units of stress is obtained and so on. But when the loading is dynamic and is
around the resonant frequency of the structure then the response will not be linear
as in the static case due to the nonlinear transmissibility characteristics of the
structure. The response will be much higher than the static case. At the start point
for the preliminary study, the structure is assumed to behave linearly, that is no
contact, no material non-linearity and no geometric non-linearity is present. For
this purpose a coarse model (Figure A.1) of the structure is built with tetrahedral
elements in ANSYS v8.1 [30] to investigate the natural frequencies. The element
chosen for this purpose is SOLID92, which has 8 nodes and is designed especially
for generating free meshes Figure A.2.
a) b) Figure A.1: a) Chaff Dispenser bracket (point mass element used for the breech plate, magazine and ammunition), b) Coarse tetrahedral mesh of Chaff Dispenser bracket
97
Figure A.2: Solid92 Tetrahedral element [30]. From this analysis is has been found out that natural frequencies exist at very low
frequencies starting from approximately 46 Hz (Figure A.3). For dynamically
loaded structures this frequency is low and due to the frequency range of excitation
of the helicopter the system cannot be assumed to behave static. Therefore the
vibration fatigue approach, which considers the resonant behavior of the structure,
is required for fatigue analysis.
Figure A.3: First natural frequency (46 Hz) and mode shape of the coarse model.
Furthermore as it can be seen from Figure 1.8 the bracket consists of plates
assembled together using rivets and bolts, which should be modeled with shell
98
elements. But shell modeling has some disadvantages for assembly type structures.
First of all there are difficulties in modeling bolts and rivets. As shells have to be
given thickness at the top or middle plane for simulating the length of the bolt/rivet
one has to define contact between these separate surfaces and the software should
include the thickness effects for contact search. For bolt/rivet modeling, one should
model the bolts/rivets as solid elements or beam elements. As shell elements have
six degrees of freedom (translation in x axis (ux), (translation in y axis (uy),
(translation in z axis (uz), (rotation in x axis (rotx), rotation in y axis (roty), rotation
in z axis (rotz)) at each node and solid element have three degrees of freedom (ux,
uy, uz) these two element types cannot be used together unless special
considerations are undertaken. Thus when shell plates are used bolts and rivets
should be modeled using beam elements (which have 6 DOF as shell elements) as
the bolt/rivet body and for the connection to the bolt/rivet holes rigid links should
be used to transfer the loads to the hole (Figure A.4).
However this type of modeling has no use unless contact between the plates is
modeled. If this contact is not modeled a relative motion as shown in Figure A.5
occurs.
Figure A.4: Modeling bolt/rivet connection in shell assemblies
99
Figure A.5: Bending motion in two direction of shell assemblies For modal analysis and other frequency-based analyses this motion results in lower
natural frequencies for the structure. This cannot be avoided since modeling the
contact, which is non-linear and therefore requires iterations, is not possible yet due
to the linear characteristic of frequency-based analyses that are performed in finite
elements. However, improvements to the frequency domain solution algorithm for
nonlinear models as friction damping are present in literature [38]. Also with some
modifications of the stiffness matrix, material nonlinearities can be handled. But
contact nonlinearities are based on status detection which is iterative and has to be
iterated for each frequency step and this is not achieved yet.
In order to overcome the problems encountered in shell type modeling another
approach to model such a structure can be used. In this approach, the structure can
be modeled using SOLID45 type solid elements present in ANSYS instead of shell
elements. But this time another problem arises. If a thin plate is meshed with solid
elements then there should be at least two elements for the thickness of the plate.
Otherwise the structure will be over stiff. In fact it is better to have as much as
possible elements for the thickness but as the model gets larger and larger then one
has to optimize the number of elements.
By applying solid modeling to the structure the loss of stiffness problem which
occurs for shell modeling (due to the contactless modeling) is solved. Also the
bolts and rivets can be easily modeled as solids and no rigid connection elements
are required for bolt/rivet to hole interaction. But the problem of nonlinear contact
is still present. This problem can be handled up to a certain amount by a special
type of contact model which will be discussed in the following topics.
100
A.2. Finite Element Analysis Method Selection Up to this point only the modeling techniques were discussed. Now the available
methods for stress calculations and their advantages and disadvantages will be
discussed.
A.2.1. Transient Dynamic Analysis
First of all there is the Transient Dynamic Analysis. In this analysis the stress
calculations are performed in the time domain. All nonlinearities as material non-
linearity, geometric non-linearity, damping and contact can be modeled in the
limits of the finite element solver capabilities. Dynamic effect as resonance is also
included. But this method requires the time history of the loading and then for each
loading step performs an iterative solution. When the structure is small and the
time history is short then this approach is practical. But when this is not the case
then either the computer resources will not be enough to solve or the time required
will be practically impossible to allocate.
A.2.2. PSD Spectrum Analysis
Secondly PSD Spectrum Analysis exists in dynamic analysis. In this analysis
Random Vibration theory is used to obtain the statistically σ1 stress values for a
given random loading which can be directly used for fatigue analysis. Dynamic
effect as resonance is included. Structural damping can be modeled either being
constant over the whole frequency range or as discrete values at known
frequencies. No non-linearity effect is allowed. The σ1 stress values have a very
small significant meaning for the fatigue analysis. As it can be seen from Figure
A.6, only a portion of the stress values are present in the σ1− to σ1 range.
However common practice has shown that for an accurate fatigue analysis up to
σ5.4− to σ5.4 stress range should be included. Furthermore PSD analysis
requires mode combination operation in order to obtain the response of the
structure to the random loading. This mode combination requires modal analysis to
101
be performed first. In ANSYS the mode combination method involves squaring
operations causing the component stresses to lose their signs. Hence deriving
equivalent or principal stresses from these unsigned components will be non-
conservative and incorrect [30]. It is more meaningful to perform a fatigue analysis
by using PSD analysis results to compute the fatigue life for a single principal
stress component. This is due to, as mentioned above, the stress response results
are not resolved onto a desired stress axis system by the finite element analysis [3].
Figure A.6: Gaussian distribution of stress values [23].
A.2.3. Harmonic Analysis
Harmonic Analysis method is also used in dynamic analysis. In this method the
stress calculations are also performed again in the frequency domain. Dynamic
effects like resonance are again included and also damping can be defined as in
PSD Spectrum Analysis. No non-linearity effect is allowed. This method is very
fast compared to the Transient Dynamic Analysis method. In this analysis the
structure is excited at specific frequencies harmonically and under these loadings
the responses are obtained. By this way the transfer functions can be obtained.
Because the transfer functions are obtained from the Harmonic analysis, which
102
causes no sign loss for the component stresses as in PSD analysis, the principal
stresses can be easily calculated. This method will be used for stress calculations
for the vibration fatigue analysis.
However the results of the harmonic analysis cannot be directly used for the fatigue
analysis. Harmonic analysis must be discussed before the results of this analysis
can be used.
In harmonic analysis, the loading to the structure is a harmonically varying load,
which can be real (cosine), imaginary (sine) or complex (Equation A.1). The
general equation of motion (Equation A.2) is solved in order to obtain
displacements and then the strains and finally stresses.
)sin(cos)( wtiwtAtF +⋅= (A.1)
where;
)(tF : Harmonic forcing vector
A : Amplitude of harmonic forcing
w : Harmonic forcing frequency
t : Time
Fkxxcxm =++ &&& (A.2)
where;
m : Mass matrix
c : Damping matrix
k : Stiffness matrix
x : Displacement
However the loadings are in the form of PSD of acceleration and not simple sine or
cosine waves. Therefore it is necessary to obtain the transfer functions of the
structure to some known input loading and then obtain the actual response to the
real loading.
103
This is achieved by applying 1g ( 29810 smm ) of loading to the structure and then
obtaining the stresses due to this loading. The load is called as a unit loading
because of its magnitude. Also the unit of the loading is chosen such that it is
compatible with the real loading (PSD having the units of Hzg 2 ). The unit of the
transfer function comes out as gMPa which is obtained at discrete frequencies of
loading. That is, 1g of loading is applied harmonically at discrete frequencies to the
structure and the response stress values are recorded. After this step one can obtain
the actual response stress PSD by simply applying Equation A.3.
2_ _ ( _ ) _ _Input PSD loading Transfer Function Response PSD stress× = (A.3)
The Transfer function is squared in order to obtain the correct units of the
Response PSD of stress as shown in Equation A.4.
HzMPa
gMPa
Hzg 2
22
)( =× (A.4)
At this point the type of the applied loading can be discussed. When 1g of loading
is applied to the structure, all of the elements of the model experience 1g of
loading. At first glance this may not seem to be what happens in reality where the
loading is applied as force from the base of the structure. In fact both situations are
exactly the same unless the loading is an impact type loading where the physics of
loading and the response of the structure are completely different. A benchmark
has been performed to show that both cases give identical results [40]. In this
study, the first analysis was performed by applying a known g value to the whole
structure and the stress results were stored. In the second analysis, in order to
simulate the base to which the structure is attached, a heavy mass element was
created. This mass element had a very large mass compared to the structure. Then
this mass element was connected to the structures base by rigid elements. And then
from Newton’s second law, for the given acceleration and mass, the required force
value was obtained. This force was applied to the mass element harmonically
without constraining the structure in space. As a result of this the structure moves
104
in space but still solution can be obtained. And when the stresses are examined
they are found out to be nearly the same as the first analysis. The reason for this is
that the free movement in space does not produce any stresses, and the base
excitation or application of g directly gives identical results.
Therefore in this study acceleration will be applied to the structure. A sample
benchmark input file for ANSYS can be found in APPENDIX-C.
105
APPENDIX-B
The formation of a rivet is a highly nonlinear process due to permanent
deformations and contact involved. In order to investigate this process and the
significance of the residual stress levels a rivet forming process is simulated using
MSC Marc [39].
The application of rivets for fastening plates is shown in Figure B.1.
a) b)
c) d)
Figure B.1: a) A view of the air gun used for deforming the rivet head, b) Rivet types, c) Metal block used to form the rivet, d) Application of the air gun
106
This simulation is similar to the work of Szlowinski et al. [20]. The materials
properties used are as follows:
Table B.1: Material properties used for the analysis [20], [13].
Material Young's Modulus
Poission's ratio Yield stress UTS
Hardening parameters ( mC εσ ⋅= )
2024-T3 73 GPa 0.33 305.2 MPa 485 MPa C=305.3 MPa, m=0.1461 2117-T4 71.7 GPa 0.33 172 MPa 295 MPa C=544 MPa, m=0.23
2024-T3 is used for the plates and 2117-T4 is used for the rivet. An elasto-plastic
analysis with contact, for which a friction coefficient of 0.20 [20], was chosen.
Adaptive re-meshing was applied to improve highly distorted elements during
formation. The squeezing amount is obtained from tests as 1.05mm for the rivets,
which are used in the bracket. The squeezing plate is moved with a constant speed
of 0.5mm/sec.
A 2-D axisymmetric mesh of 2824 elements and 3051 nodes was created as shown
in Figure B.2.
Figure B.2: Finite Element Mesh and boundary conditions of the riveting model.
Squeezing, rigid plate
Fixed Bondary condition on the rivet head
Riveted Plates
Rivet
107
After the deformation process the following Equivalent von Mises stress levels and
Equivalent Plastic Strain levels are observed (Figure B.3-Figure B.4).
Figure B.3: Equivalent Stress distribution after the riveting process (Max Equivalent von Mises stress; 397.6 MPa)
Figure B.4: Equivalent Plastic strain distribution after the riveting process (Max Equivalent Plastic strain; 0.7).
108
Finally, the principal stress ranges are given below.
Figure B.5: Maximum Principal Stress distribution of the rivet assembly
Figure B.6: Middle Principal Stress distribution of the rivet assembly
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Figure B.7: Minimum Principal Stress distribution of the rivet assembly From the analysis results it is seen that the stress state is mostly compressive on
and around the rivet/rivet hole. However there exist also some tensile zones close
to the rivet hole.
110
APPENDIX-C As discussed in APPENDIX-A, in order to compare force excitation and
acceleration excitation methods and to show that they both give the same results a
finite element analysis was performed in ANSYS.
Below is a series of ANSYS commands that run a transient analysis with acel input
first and then with a large mass. The model is a 2d mesh of the letter "C".
Integration time step is set to a constant 0.00025 for both runs. After the acel run,
uy for a node at the tip and sy for a node are stored and copied to arrays. Then a
large mass is added and the uy constraint is removed. The model is rerun with
applied forces instead of acel input. Then post26 is executed and uy is put in
variable 2 and sy in variable 3. If uy is plotted, one would notice that it is a lot
different from the previous uy but it has the rigid body component in, so uy is put
in variable 4. Next the arrays are copied from the acel run into variable 5 for the uy
and 6 for the sy. A math operation is performed, which is variable 2 minus 4 and
put that in variable 7. This should be the relative displacement since the base
displacement has been subtracted out. If now variables 5 and 7 plotted, one
finds out that they are one curve i.e. they are right on top of each other so the
relative displacements are exactly the same. Also, if variables 3 and 6 are plotted,
which are the stresses for the two runs; they are also exactly the same. Notice that
only beta damping is used. If two different methods produce the exact same result,
then one can say that the methods are equivalent [40].
/BATCH
/PREP7 ! use 8.1
et,1,42
mp,ex,1,10e6
mp,dens,1,.1/386
mp,prxy,1,.3
PCIRC,5,4.5,45,315,
111
esiz,.5
MSHAPE,0,2D
MSHKEY,1
ames,1
nsel,s,node,,40,44 ! base nodes
d,all,all
alls
fini
/solu
anty,tran
TRNOPT,FULL
LUMPM,0
delt,.00025
outres,basic,all
betad,.0001
time,.001
solve
time,.002
acel,0,386.4
solve
time,.008
solve
time,.009
acel,0,0
solve
time,.015
solve
FINISH
/POST26
*DIM,uy_acel,ARRAY,60
*DIM,sy_acel,ARRAY,60
NSOL,2,51,U,Y, UY_2
VGET,uy_acel(1),2, ,
112
ANSOL,3,75,S,Y,SY_3
VGET,sy_acel(1),3, ,
fini
/PREP7
nsel,s,node,,40,44
ddel,all,uy
cp,1,uy,all
ET,2,MASS21
KEYOPT,2,1,0
KEYOPT,2,2,0
KEYOPT,2,3,2
type,2
real,2
r,2,1e6
e,40
alls
fini
/solu
anty,tran
TRNOPT,FULL
LUMPM,0
delt,.00025
outres,basic,all
betad,.0001
time,.001
solve
time,.002
f,40,fy,1e6*386.4
solve
time,.008
solve
time,.009
f,40,fy,0
113
solve
time,.015
solve
FINISH
/POST26
NSOL,2,51,U,Y,UY_2
ANSOL,3,75,S,Y,SY_3
NSOL,4,40,U,Y,uy_base
VPUT,uy_acel,5,,
VPUT,sy_acel,6,,
/show,jpeg
plva,3,6
ADD,7,2,4,,,,,1,-1,1,
plva,5,7
fini
/exit
114
APPENDIX-D Some sample input windows from MSC Fatigue are given below. As it can be seen
from these figures many analysis parameters can be defined in MSC Fatigue and
their effects to the fatigue life can be analyzed.
First of all, the result files from ANSYS have to be imported into MSC Fatigue as
shown in Figure D.1. In this window, some parameters, as shown, are selected and
the Job name is given as “ah1w_chaff_prototip_fat_1”.Then the results file is
selected.
Figure D.1: Window for importing finite element results into MSC Fatigue
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In the second step, the loading acceleration versus time data is imported. For this
purpose under the “\tools\fatigue utilities\file conversion utilities\” option “Convert
ASCII .dac to Binary” is selected. This window is shown in Figure D.2.
Figure D.2: Text to Dac file conversion window
In this window the text file (ASCII file) of the loading is selected and converted
into the file format (Dac file) of MSC Fatigue.
Then the components of the loading PSD matrix should be obtained for X, Y and Z
axis loading. For this the PSD’s and Cross PSD’s have to be obtained. To obtain
the PSD’s, under “\tools\fatigue utilities\advanced loading utilities\” option “auto
spectral density” is selected. Then from input windows, as shown in Figure D.3,
the time data is transformed to PSD for X axis, where other axes are converted
consecutively.
116
Figure D.3: Input windows for obtaining PSD from time data.
To obtain the Cross PSD’s, under “\tools\fatigue utilities\advanced loading
utilities\” option “frequency response analysis” is selected. Then from input
windows, as shown in Figure D.4, the time data is transformed to Cross PSD for X-
Y axis, where other axes (X-Z; Y-Z) are converted consecutively.
117
Figure D.4: Input windows to obtain the Cross PSD’s from time data.
After the PSD’s and Cross PSD’s are obtained the Loading PSD Matrix is created.
For this purpose, under “\tools\fatigue utilities\” option “load management” is
selected. From this menu, “\Add an entry\PSD matrix” is chosen and the steps as
shown in Figure D.5 are followed.
118
Figure D.5: Creating a PSD matrix Then the analysis type, result units and location are selected as shown in Figure
D.6. Then from this menu, the “Specific Setup Forms”, Loading Info is selected
and another window appears which is called the “Loading Vibration Information”
(Figure D.7).
1
2
3
119
Figure D.6: Selection of analysis type and finite element result units/locations The Loading Vibration Information table is created using the transfer functions
from ANSYS and multiaxial loading PSD matrix as shown in Figure D.7.
120
Figure D.7: Construction of multiaxial loading matrix in MSC Fatigue
Then the material properties are entered from a window (Figure D.8) which is
reached from Figure D.6, with the name of “Material Info”. In this window, the
“Standard Database” of MSC Fatigue is used. Finally the “Solution Parameters”
are entered from a window (Figure D.9) which is again reached from Figure D.6,
with the name of “Solution Parameters”.
121
Figure D.8: Material properties selection in MSC Fatigue.
Figure D.9: Selection of analysis method and stress combination method.
122
When all the parameters for the fatigue analysis are set, again from the window
shown in Figure D.6, the “Job Control” option is selected. From Figure D.10 the
“Action” is set to “Full Analysis” and then “Apply” button is pressed.
Figure D.10: Job Control menu
This operation evaluates the fatigue life for the analyzed structure with MSC
Fatigue using Vibration Fatigue Analysis.