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

viii

To My Family

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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.

x

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

xi

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)

3

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

6

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.

7

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.

10

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

11

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.

12

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

40

Figu

re 4

.4: F

light

pro

file

follo

wed

dur

ing

oper

atio

nal t

ests

[27]

.

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

45

Figure 4.12: X-Y-axis cross PSD plot

Figure 4.13: X-Z-axis cross PSD plot

46

Figure 4.14: Y-Z-axis cross PSD plot

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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Analizi” SAVTEK-2002 Savunma Teknolojileri Kongresi, Orta Doğu Teknik Üniversitesi, Ankara, 2002, 321-332.

[2] Farlex Inc., “Metal fatigue - encyclopedia article about Metal fatigue”,

“http://Encylopedia.TheFreeDictionary.com”, 2004. [3] Bishop NWM, Sherratt, F., Finite Element Based Fatigue Calculations,

NAFEMS, Germany, 2000. [4] Ewing, M., Aircraft Structures Design and Analysis / Component Design,

Course notes, The University of Kansas, Seattle, April 2004. [5] Esin, A., Properties of Materials for Mechanical Design, Middle East

Technical University, Gaziantep, 1981. [6] Bishop, NWM., “Vibration Fatigue Analysis in the Finite Element

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

109

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

115

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.

123

APPENDIX-E

Figure E.1: Technical drawing of the analyzed locations on the dispenser bracket