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Experimental High Cycle Fatigue Testing and Shape
Optimization of Turbine Blades
by
Mohamad Ahmadi Tafti
A thesis submitted in conformity with the requirements
for the degree of Masters of Applied Science
Graduate Department of Mechanical and Industrial Engineering
University of Toronto
© Copyright by Mohamad Ahmadi Tafti 2013
ii
Experimental High Cycle Fatigue Testing and Shape
Optimization of Turbine Blades
Mohamad Ahmadi Tafti
Master of Applied Science, 2013
Mechanical and Industrial Engineering, University of Toronto
Abstract
An accelerated high cycle fatigue testing approach is presented to determine the fatigue
endurance limit of materials at high frequencies. Base excitation of a tapered plaque driven into
a high frequency resonance mode allows the test to be completed in a significantly shorter
time. This high cycle fatigue testing is performed using the tracked sine resonance search and
dwell strategy. The controller monitors the structural health during the test. Any change in the
dynamic response indicates crack initiation in the material.
In addition, a shape optimization finite element model is conducted for the design of the
tapered plaques. An integrated neural (Neural-Network) genetic (NSGA_II) optimization
technique is implemented to carry out the shape optimization for this component. This process
results in a significant reduction in the computational cost. A Pareto set is then produced that
meets the designer’s requirements and provides the decision maker several alternatives to
choose from.
iii
Acknowledgments
The author would like to proclaim sincere gratitude to Professor Kamran Behdinan for his invaluable
support and guidance during the project that made its successful completion possible.
The author also acknowledges the sincere appreciation to Professor Jean Zu for her tireless effort and
continuing encouragement.
The author is also thankful to Vincent Iacobellis for his advice in this work.
The author also appreciates strong support of Pratt & Whitney Canada in this work.
iv
Table of Contents
Abstract ........................................................................................................................................ ii
Acknowledgment ........................................................................................................................ iii
Author’s Decleration ................................................................................................................. vii
List of Tables ............................................................................................................................ viii
List of Figures ............................................................................................................................. ix
Nomenclature .............................................................................................................................. xi
Chapter 1: Introduction ................................................................................................. 1
1.1 Background ...................................................................................................................... 1
1.1.1 Crack Initiation ................................................................................................... 2
1.1.2 Crack Growth ...................................................................................................... 2
1.2 Fatigue Test and Stress-Life Approach ........................................................................... 4
1.3 Stress-Life Curves ........................................................................................................... 5
1.4 High Cycle Fatigue (HCF)............................................................................................... 6
1.4.1 HCF Design Considerations ................................................................................. 6
1.4.2 Causes of HCF ....................................................................................................... 7
1.5 Factors Affecting Fatigue Behaviour............................................................................... 8
Chapter 2: Literature Review ...................................................................................... 10
2.1 High-Cycle Fatigue Testing .......................................................................................... 10
2.1.1 Early Methods ..................................................................................................... 13
v
2.1.2 Step Testing Method ............................................................................................ 13
2.1.3 Staircase Method ................................................................................................. 13
2.1.4 Resonance Fatigue Tests...................................................................................... 13
2.2 Multi-Objective Optimization Methods ....................................................................... 18
2.2.1 Weighted Global Criterion Method ..................................................................... 19
2.2.2 Weighted Sum Method ........................................................................................ 19
2.2.3 Lexicographic Method ......................................................................................... 20
2.2.4 Exponential Weighted Criterion .......................................................................... 21
2.2.5 Weighted Product Method ................................................................................... 21
2.2.6 Methods with No Articulation of Preferences ..................................................... 22
2.3 Surrogate-Based Design Optimization .......................................................................... 32
2.3.1 Classification of Surrogate Models ..................................................................... 32
2.3.2 Main Types of Surrogate Models ....................................................................... 34
Chapter 3: Fatigue Testing .......................................................................................... 40
3.1 Equipment for HCF Testing ......................................................................................... 40
3.1.1 Hardware............................................................................................................... 41
3.1.2 Software ................................................................................................................ 44
3.2 Resonance Fatigue Test ................................................................................................. 44
3.2.1 Experimental Testing ............................................................................................ 45
3.2.2 Stress-Displacement Calibration Test ................................................................... 51
LMS configuration .......................................................................................................... 55
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3.3 Test Results .................................................................................................................... 61
Plaque 1 .......................................................................................................................... 62
Plaque 2 .......................................................................................................................... 63
Chapter 4: Multi-Objective Shape Optimization ...................................................... 68
4.1 Finite Element Modeling and Validation ...................................................................... 70
4.1.1 Geometry ............................................................................................................. 70
4.1.2 Modal Analysis .................................................................................................... 72
4.2 Surrogate Modeling ...................................................................................................... 76
4.3 Multi-Objective Optimization ....................................................................................... 77
4.4 Fast and Elitist Multi-Objective Genetic Algorithm: NSGA_II ................................... 80
4.4.1 Non-Dominated Sorting Genetic Algorithm ....................................................... 80
4.4.2 Crowding Distance ............................................................................................. 82
4.4.3 Main Loop ........................................................................................................... 83
Chapter 5: Summary and Conclusion ........................................................................ 86
References.................................................................................................................................. 88
vii
Author’s Declaration
I hereby declare that I am the sole author of this thesis and no part of this thesis has been
published or has been submitted for publication.
I authorize University of Toronto to lend this thesis to other institutions or individuals for the
purpose of scholarly research.
I further authorize University of Toronto to reproduce this thesis by photocopying or by other
means, in total or in part, at the request of other institutions or individuals for the purpose of
scholarly research.
I understand that my thesis may be made electronically available to the public.
viii
List of Tables
Table 2-1 List of multi-objective genetic algorithms ................................................................. 26
Table 3-1 Results summary ........................................................................................................ 66
Table 4-1 Mechanical properties of PEEK ................................................................................. 71
Table 4-2 Mechanical properties of coating ............................................................................... 72
Table 4-3 FEM solution settings ................................................................................................ 72
Table 4-4 Finite element model validation ................................................................................. 73
ix
List of Figures
Figure 1-1 Phases of fatigue failure ............................................................................................. 1
Figure 1-2 Grain boundary effect on crack growth in AL-alloy ................................................. 3
Figure 1-3 Fatigue testing machines ............................................................................................. 4
Figure 1-4 Constant amplitude loading ........................................................................................ 5
Figure 1-5 Schematic of fatigue diagram .................................................................................... 7
Figure 2-1 Staircase testing results on SAE 4340 ..................................................................... 13
Figure 2-2 Cell-based method ................................................................................................... 29
Figure 2-3 Classification of data generation techniques ............................................................ 33
Figure 2-4 Neural Network concept .......................................................................................... 36
Figure 2-5 Simple neuron .......................................................................................................... 36
Figure 2-6 Network’s Transfer Functions ................................................................................. 37
Figure 2.7 Multi-layer Neural Network ..................................................................................... 37
Figure 3-1 Signals in the vibrometer .......................................................................................... 41
Figure 3-2 VB8 module .............................................................................................................. 42
Figure 3-3 BDS4E module ......................................................................................................... 42
Figure 3-4 Modal shop shaker system ........................................................................................ 43
Figure 3-5 HCF testing configuration......................................................................................... 45
Figure 3-6 High cycle fatigue testing configuration and procedure ........................................... 48
Figure 3-7 Strain gage wiring ..................................................................................................... 49
Figure 3-8 Strain gage setup ....................................................................................................... 50
Figure 3-9 Strain gaged plaque ................................................................................................... 50
Figure 3-10 Test Setup................................................................................................................ 51
Figure 3-11 Strain Vs. G level .................................................................................................... 52
Figure 3-12 Center point displacement Vs. G level ................................................................... 52
Figure 3-13 Tracking algorithm ................................................................................................. 54
Figure 3-14 Channel setup .......................................................................................................... 55
Figure 3-15 Sine Control ............................................................................................................ 57
Figure 3-16 Edit sweep profile ................................................................................................... 58
Figure 3-17 Self-check ............................................................................................................... 59
Figure 3-18 Sine control ............................................................................................................. 59
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Figure 3-19 Dwell setup ............................................................................................................. 60
Figure 3-20 Dwell control .......................................................................................................... 61
Figure 3-21 Resonance frequency and the frequency shift for the first plaque .......................... 62
Figure 3-22 Strain measurements for the first plaque ................................................................ 63
Figure 3-23 Resonance frequency and the frequency shift for the second plaque .................... 64
Figure 3-24 Maximum strain measurements from the center point gage ................................... 65
Figure 3-25 Phase angle from sine sweep test ............................................................................ 66
Figure 4-1 Tapered plaques cross section ................................................................................... 70
Figure 4-2 Tapered plaque model ............................................................................................... 71
Figure 4-3 Impact testing ............................................................................................................ 73
Figure 4-4 First bending mode FEM model ............................................................................... 74
Figure 4-5 Third bending mode .................................................................................................. 74
Figure 4-6 Second bending mode ............................................................................................... 74
Figure 4-7 Displacement contour ............................................................................................... 76
Figure 4-8 Plot of observed and predicted regression approaches. ............................................ 77
Figure 4-9 Dang Van diagram criterion ..................................................................................... 80
Figure 4-10 NSGA-II procedure ................................................................................................ 81
Figure 4-11 Crowding distance ................................................................................................. 83
Figure 4-12 First Pareto Front .................................................................................................... 85
xi
Nomenclature
Network output
Fatigue strength exponent
Bias matrix
Euclidean distance [-]
External archive [-]
Evolutionary algorithms
Transfer function [-]
Frequency Response Function
Acceleration due to Gravity [m/s2]
High Cycle Fatigue [-]
[ ]I i Crowding distance
Radial function [-]
Low Cycle Fatigue [-]
Number of cycles [-]
Predefined number of cycles [-]
Number of cycles to failure [-]
Niche count [-]
Network input
Parent generation [-]
Frequency shift factor
Offspring generation [-]
Stress ratio [-]
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Sine tracking dwell factor
Alternating nominal stress [N/m2]
1, 2 3,a a aS S S Alternating principal stress [N/m2]
Maximum stress [N/m2]
Minimum stress [N/m2]
, ,mx my mzS S S Coefficient of mean stress [N/m2]
NfS The uniaxial fully reversed fatigue strength [N/m2]
Domination count
qaS Equivalent stress [N/m2]
Random number between 0 and 1 [-]
Velocity [m/s]
Displacement [m]
Gain amplitude
Mean response
Weight factor [-]
Intercept coefficient [-]
Error [-]
Shape parameter [-]
Micro strain [-]
Alternating stress [N/m2]
Stress variable [N/m2]
f Fatigue strength coefficient [N/m2]
Hydrostatic Stress [N/m2]
xiii
Principal stress [N/m2]
Mean stress [N/m2]
f Fatigue limit in shear stress [N/m2]
Weights [-]
Frequency [Hz]
Resonance frequency
Electrical resistance
1
Chapter 1: Introduction
1.1 Background
When a specimen is subjected to a cyclic load, fatigue crack can nucleate in microscopic scale,
followed by macroscopic cracks and eventually failure would occur in the last cycle. This
phenomenon is only partly learned and understood. To get a better understanding of fatigue
failure, a brief review over the fatigue history is discussed to show the ideas and the
developments by the effort of many researchers. Fatigue life is divided into two stages; crack
initiation and crack growth. Crack initiation involves microscopic crack formation where the
cracks are not visible with naked eye and the crack growth involves the crack propagation until
the failure occurs. It is important to consider crack initiation and growth separately because
various conditions influence on crack initiation but limited number of these conditions might
affect the crack growth. Investigation of microscopic cracks has shown that invisible
microcracks nucleate in slip bands. Microcracks generally start very early in the fatigue life
and they remain invisible for a significant part of the fatigue life. When the cracks are visible,
the remaining life is usually small portion of the total life. Differentiating between two phases
is of great importance. Several surface conditioning or corrosive environment can influence on
crack initiation period while having negligible or no influence on crack growth. Figure 1.1
shows the different phases of fatigue life. [1]
Figure 1.1 Phases of fatigue failure [1]
2
1.1.1 Crack Initiation
Fatigue occurs at stress levels below the yield stress and the plastic deformation is limited to a
number of grains of the material. It is evident that continuous loading and unloading produce
microplasticity on the surface because of lower constraints on slip.
The plane of crack nucleation is in the maximum shear stress plane that has an angle of 45
degrees to the normal stress direction. If the load cycle continues, cracks will propagate. The
crack growth begins and propagates in two stages. First, it starts in the plane with maximum
shear stress (45 degrees) and then grows significantly into the direction perpendicular to the
applied normal stress. These cracks could grow intercrystalline along the grain boundaries or
transcrystalline across the grains. Shear stress on slip planes differs from one grain to another
relying on Size, shape, crystographic orientation and anisotropy of the material. Upon
unloading, although the strain hardening occurs in the same slip band, reverse slip happens on
adjacent planes. [1]
It is not easy to schematize the fatigue failure. Fractography is the science that investigates the
appearance of the failure. What laboratories have been able to present is typical fatigue failure
on well-known materials under certain conditions such as stress or strain controlled tests with
specific type of stress (bending, torsion or tension-compression) on smooth or notched test
specimens at various stress levels. By conducting these tests, some points could be
comprehended: origin of cracks, fatigue breakage points, direction of crack propagation and so
on. What makes the fatigue interpretation more complicated involves different factors such as
multi-axial and variable loading condition, mean stress effects, environmental effects and
strain rate, frequency and so forth. [1]
1.1.2 Crack Growth
After the micro cracks formation, inhomogeneous stress distribution is produced as a result of
stress concentration at the tip of the microcracks. Consequently, more slip bands are activated
and the crack growth occurs on more slip planes to accommodate slip dislocations in adjacent
slip planes. The microcrack growth direction deviates gradually from the initial orientation and
tends to grow perpendicularly from the loading direction.
3
Figure 1.2 shows that crack growth rate in microscopic scale highly depends on crack length. It
has been observed that the crack growth decreases when it reached the grain boundary, after
penetrating through the grains the rates rises until it reaches the next grain. After overcoming
all boundaries, the crack grows in steady rate. [2]
Figure 1.2 Grain boundary effect on crack growth in AL-alloy [2]
Where c is the crack length and N is the number of load cycles. Transition from crack initiation
to crack growth can be qualitatively expressed in the following definition:
“The initiation period is supposed to be completed when microcrack growth is no longer
depending on the material surface conditions.” It indicates that after the crack initiation stage
is completed, not only crack resistance of the material is no longer governed by surface
conditions but it is controlled by crack growth rate. [1]
In the 50’s, Frost et al. [3] observed that the cracks stopped propagating by a type of crack
growth barrier at the length of a grain size. Initiation of the cracks may appear on the surface
where the constraint on the slip bands is small. The crack tip stress field changes the plane
stress from the surface to the inner layers. As a result, the cracks are arrested implying an
increasing restrain on the slip bands.
4
1.21.2 Fatigue Test and Stress-Life Approach
Structures and components are subjected to diverse loading histories. Histories might be simple
and repetitive or even, they can be random and complex. Several standard tests have been
developed to establish fatigue life prediction techniques.
Constant amplitude fatigue testing was first done by Wohler on railway axles in 1850. His
work on fatigue behaviour of materials marks the first methodical investigation of S-N curves.
Figure 1.3 shows some of the conventional fatigue testing machines. Figure 1.3(d) shows a
uniform bending along the specimen while figure 1.3(a) shows no uniform bending testing
machine. Figure 1.3(b) shows an axial loaded fatigue test machine with tension a compression
capability. Figure1.3(c) shows a modern servohydraulic testing machine with its own computer
capable of applying stress or strain by the hydraulic actuator. Several standard fatigue test
procedures are available from ASTM standard. Also, there are certain tests specimens used for
fatigue life prediction. [4]
Figure 1.3 Fatigue testing machines. (a) Cantilever rotating bending. (b) Axial loading. (c) Servo-hydraulic
test system. (d) Pure bending machine [4]
5
1.31.3 Stress-Life Curves
Figure 1.4 shows a cyclic stress curve obtained from a stress control fatigue test. is the
alternating nominal stress and N is the number of cycles to failure. Different materials show
distinct behaviours in S-N plots. A common stress-life (S-N) curve is depicted in figure 1.5
which includes a discontinuity or knee after 107
cycles while many other materials have a
continuing sloping curve. Fatigue consists of crack initiation, growth and ultimate failure;
however, S-N curve does not separate these two stages and only consider total life to failure.
Fatigue strength has an enormous range depending mostly on mean stress, surface finish,
component size, residual stress, corrosive environment and stress concentration. Most S-N data
found in literatures are fully reversed uniaxial tests on highly polished or notched test pieces in
laboratory environment. Therefore, the fatigue limits must be reduced substantially due to
liability concerns and considering the real service condition where the components are used
for. [4]
Figure 1.4 Constant amplitude loading [5]
Where is the mean stress, is the alternative stress, and are the minimum and
maximum stresses, respectively.
6
1.41.4 High Cycle Fatigue (HCF)
HCF generally involves high frequency, low amplitude, elastic cyclic behaviour and large
number of cycles usually tested under load control condition. In S-N (stress-life) curves, HCF
failure occurs on the right side of the curve where the number of cycles is too large to obtain
sufficient data to characterize material behaviour. It is probably one of the most difficult
phenomenons to handle. Fatigue failure comes with a crack or damages that initiate in cyclic
stress well below the yield stress. The repeated cyclic loading induces micro stresses with
dissipation of energy by microplastic strains and permanent dislocation in slip bands.
High cycle fatigue becomes significant when the stress is low but yet close to the yield stress
and the number of cycles is larger than 105. The major cause of HCF is large amplitude
vibration with zero mean stress and the only way to prevent failure is to reduce the vibration
level. Stress concentration is also another cause of HCF due to the fact that the material
behaviour is elastic and there is no way to reduce the stress plastic shakedown. Therefore,
avoiding the surface roughness by polishing and careful machining can be a remedy to
minimize the failure possibility. Corrosion is also a phenomenon that largely increases the
fatigue damage by passivation-dipassivation at the crack tips. Understanding of the state of
stress in a component is essential in fatigue analysis. This state can be described by six stress
and strain components acting on orthogonal planes. In quick design, a safe design is to have the
stress rate below the fatigue limit where no failure would take place. The idea of fatigue limit
is not ideal because nobody has waited for an infinite number of cycles. Thus, the fatigue limit
can be defined objectively, as the maximum stress at large number of cycles (107,10
8). [5]
1.4.1 HCF Design Considerations
Many HCF issues have influenced engine safety, reliability and sustainability over the past
decade. The phenomenon has been studied by so many researchers but still there are so many
issues which need to be understood. The diagram that is mostly used is called stress-life or
Wohler diagram (figure 1.5) which plots the stress as a function of number of cycles to failure.
The diagram is drawn for data at different stress ratio levels. However, in the proposed work, it
is mainly dealing with the model only as they pertain to the values of stress close to fatigue
endurance limit corresponding to failure occurring in large number of cycles.
7
Figure 1.5 Schematic of fatigue diagram [5]
Generally, it is difficult to produce statistically sufficient data in HCF regime. This is a
potential problem that designers face while transferring data from laboratories to components
for which the data is to be used for. In addition, the product manufacturing process, machining
technique and surface finish are not always identical. Hence, the component condition does not
always conform to laboratory size testing state.
The terminology which is used to characterize the fatigue behaviour is the “fatigue strength” or
“fatigue limit”. This limit corresponds to the stress level which the material fails after large
number of cycles typically 106 or 10
7. Recently, a new field has been introduced called the
gigacycle regime that encompasses higher number of cycles (109) as an extension to
conventional testing. The term fatigue endurance limit is referred to materials which do not
experience failure or have “infinite life”. [5]
1.4.2 Causes of HCF
There is a great concern for the case where Lower Cycle fatigue (LCF) damage or crack, can
change the HCF characteristic of the material. It is not clear if LCF or HCF could be the cause
of crack initiation or whether the interaction between them. Fretting fatigue has been also
experienced in the dovetail and has been accounted for several incidents. The solution requires
a comprehensive knowledge over the residual stress and how the stress level or state would
affect the contact region. Manufacturing or handling damages such as machining marks,
inclusions or flaws could affect the HCF strength.
8
In some instances, fan compressor blades experience HCF due to fretting in service under
resonance loading. The failure is not solely due to the stress concentration in the contact region
but also it may be as a result of uneven distribution of stress caused by uneven wear. [5]
1.51.5 Factors Affecting Fatigue Behaviour
Microstructure
In solid mechanics, metals are categorized as isotropic, homogenous and elastic. At the micro
scale, fatigue characteristic are broadly influenced by microstructure effects such as heat
treatment, grain size, inclusions, voids and imperfections. It is not easy to account for these
influences in fatigue behaviour but generalities could be formulated for some microstructural
aspects. Surface finish has a substantial influence on fatigue behaviour since most of the
fatigue failures occur on the surface. This effect is even more critical in long life fatigue where
most of the cycles are involved with crack nucleation. [4]
Size Effect
Under unnotched bending condition, if the diameter of the specimen is less than 10mm, the
fatigue behaviour is reasonably independent of the diameter. If the diameter of the test piece
goes higher, the fatigue limit will decrease by a factor of 0.8 or 0.7. The larger the diameter or
thickness, the smaller the gradient stress is. Hence, the governing stress for fatigue life is the
average stress. For axial loading scenario, gradient stress does not exist; resulting in lesser
effect of size than bending. [4]
Frequency
A number of complicated influences on fatigue behaviour can be listed as corrosive
environment, test temperature, stress rate and frequency. Elevated temperature is usually
detrimental to fatigue resistance. Generation of heat due to cyclic loading at high frequencies
could be accounted for a prime reason of frequency effect on fatigue life. Frequency ranges
less than 200Hz has been reported not having a large impact on fatigue life [3] , [6]. However,
in KHz frequency range, greater changes in fatigue resistance have occurred because the
temperature control is more difficult. In most cases, fatigue resistance also increases in KHz
9
range. However, generalizing the fact that fatigue resistance increases as the frequency grows
is not credible due to the complexity of the testing and material variables involved in fatigue
behaviour.
10
Chapter 2: Literature Review
The objective of this work is to develop a High Cycle Fatigue testing procedure for two
different specimens. The main goal is to establish the HCF endurance limit of two tapered
plaques under a cyclic fully reversed bending load. The plaques are subjected to a sinusoidal
force during a shaker test and the fatigue endurance limit is subsequently identified. Any
change in the natural frequency implies a reduction in stiffness as a result of crack initiation.
The excitation frequency is constantly retuned to track the shifted resonant frequency
throughout the test. Section 2.1 overviews common fatigue testing methods as well as recent
works done by a number of researchers in the field.
Furthermore, a comprehensive shape optimization technique is applied to modify the fatigue
life and total mass of the tapered plaques. The model is simulated using finite element method
and the dynamic behaviour of the plaques is obtained to predict the fatigue life. Multi-objective
optimization method was adopted as the optimization approach to minimize the mass and
maximize the fatigue performance. Furthermore, the Pareto front is introduced from which
results are proposed in order to assist the designer in choosing the best option out of multiple
configuration based on the design compliances. Section 2.2 explains multi-objective
optimization algorithms, while, the adopted approach for solving this problem is elaborated in
chapter 4. Based on the complexity and high computational cost of the finite element
simulation, the surrogate modeling or function approximation method is integrated to the
optimization technique. Accordingly, the surrogate modeling approach is introduced in section
2.3.
1.6 2.1 High-Cycle Fatigue Testing
In recent years, unprecedented fatigue failures have led to the development of various
experimental methods for measuring high cycle fatigue properties of materials. Since the
beginning of fatigue research in the 1800’s, failures associated with high cycle fatigue have
been initiated the need for producing fatigue data in long life regimes. Turbine blades
undergoing resonances, rail road wheels being in contact with the rails in every revolution,
11
bridges carrying moving loads on daily basis and rotating components in machines are just a
few examples where materials experience a large number of cyclic loads. Many researchers
have devoted significant effort to develop equipment that can operate at high frequencies to
conduct the conventional time consuming fatigue tests in such a short period of time. [5]
2.1.1 Early Methods
Standard methods for determination of fatigue behaviour require several fatigue tests with a
significant amount of fatigue data for statistical analysis. The number of testing specimens and
testing time might be too large for practical applications; therefore, numerous accelerated
testing methods have been proposed by many researchers. Among all those early accelerated
methods, Moore and Wishart [7] developed an “overnight” test where they could determine the
tensile strength as well. They claimed that testing below the endurance limit increases the
tensile and endurance strength. Later Gough commented on this article that: “no fundamental
reason exists why any short-time test can be expected to prove reliable.” [8]
Prot [9] proposed an accelerated technique that could reduce the testing time by 90%. He
started the test at a stress level below the endurance limit and increased the stress at a constant
rate. Each test required a single test piece. Although his testing method is currently not
commonly used, but it is considered as the standard reference for accelerated testing methods.
Ward [10] validated Prot’s work on welded SAE4340 and confirmed that it is applicable to
ferrous metals. Hempel [11] observed that at stress levels 22% below the endurance limit slip
bands were not developed. However, at higher stress rates, slip bands were formed in a few
crystallites. The effect of “under stressing” proposed by Prot [8] does not appear to have any
scientific basis. Consequently, step-loading technique is introduced in the next section 2.1.2 for
determining the fatigue limit. [5]
2.1.2 Step Testing Method
Another accelerated testing method was developed by Nicholas and Maxwell [12] where a
specimen was fatigued at a fixed stress ratio, to a limit of typically 107 cycles. After the failure
occurs, the stress level is increased by 5% until the failure occurs at less than 107 cycles. Then
the fatigue limit is determined using the following equation:
12
(2.1)
Where is the fatigue strength corresponding to cycles, is the previous fatigue stress,
is the step increase in the stress level, is the number of cycles to failure and is
the predefined fatigue life. There are some advantages using this technique other than saving
testing time. In conventional fatigue testing, only a fraction of all specimens were fatigued.
Although, in the step-wise loading strategy, all of the specimens fail after a specific number of
cycles. Such tests require a machine to produce the stress at high frequency to accelerate the
time and reduce the cost. The proposed technique developed by Nicholas and Maxwell [12]
involved step-wise loading could save the time and cost of testing to a great extent. An
alternative approach to step-loading test is to conduct various tests at different stress values up
to the point of failure. Although some of the components will not fail and reach cycles.
These components are denoted as run outs. [5]
2.1.3 Staircase Method
Mehl and Ransom [13] introduced a new method known as staircase testing. In this approach,
the next stress level is determined using the current step testing stress value. If the specimen
survives after 107 cycles, the stress in increased one step and if the component fails, the stress
is decreased one step consequently. Statistical methods are then used to estimate the endurance
limit. Figure 2.1 shows the results of the staircase testing on SAE 4340 which includes the
mean stress and the range 2 that 95% of the data would fall in. [5]
13
Figure 2.1 Staircase testing results on SAE 4340 [12]
The primary advantage of this technique is that over 30-40% could be saved in the number of
testing samples. Another advantage of this method is the simpler statistical analysis under
certain conditions. The statistical analysis of staircase method can be found in the article by
Mood and Dixon [14]. This method is based on maximum likelihood estimation and assumes
that the fatigue limit follows a normal distribution.
2.1.4 Resonance Fatigue Tests
Uniaxial fatigue test are types of tests conducted on conventional devices to obtain the required
data points to construct the Haigh diagram. Even with servo-hydraulic testing machines, it
requires several days to achieve a single data point on the fatigue curve. Besides, several tests
have to be done in order to find sufficient data to plot the stress-life curve in different stress
ratios. Therefore, large amount of time is needed to extract all the required data points for a
single material. In high cycle regimes, employing ultrasonic fatigue testing equipment had
been a remarkable way to investigate the fatigue properties of materials. Xue et al. [15]
proposed a method to find the flexural fatigue strength of materials in 105 to 10
10 cyclic ranges.
A non-contact optical sensor was used to measure the displacement up to 10 KHz.
However, these machines often produce uniaxial fatigue data while most of the components are
subjected to biaxial loads or a combination of bending and twisting modes. Among the early
14
tests that were working based on the resonance principle, the Schenck [16] machines used a
spring attached to a small specimen where they were able to generate a constant force at 30Hz.
Later, these machines were equipped with a controller to monitor the failure. Nowadays,
electromagnetic testing equipment is being used on the same fundamentals but lower power
consumption.
In order to identify the fatigue endurance limit of turbine blades, a combination of step-testing
procedure and resonance fatigue testing was utilized to develop a new testing methodology.
George et al. [17] used step-testing to determine the fatigue strength under fully reversed
loading. The cantilever plate was mounted on an electrodynamic shaker and the shaker was
operating close to one of the resonance frequencies of the specimen. The control sensor was
also mounted on the shaker to control the shaker force and a non-contact laser vibrometer was
measuring the response of the particular plate. A strain gage was attached near the high-
stressed region of the test piece and the calibration test was conducted to find the velocity-
stress correlation. The fatigue test was done at the resonance frequency. The fatigue limit was
determined from the frequency shift caused by fatigue cracks during the resonance.
In the vibration-based techniques, there is no phenomenon such as sudden change in the
structural dynamic response. The response of the structure is recorded from the beginning of
the test and any changes in the material stiffness indicate the crack formation and development
into the material. After the fatigue endurance was identified, the test could be preceded in two
different ways. If the amplitude and frequency of the driving force was maintained at a
constant level, a crack would grow until it was arrested. This technique could produce cracks
of 1-2mm. On the other hand, to produce long cracks, the shaker force was re-tuned after the
failure to continue the crack propagation. This implies that for a constant level of frequency
and force, once the cracks are formed, the strain and stress level reduce. Therefore, the crack
can self-arrest and the component needs to meet new frequency shift in order to continue to
fatigue. In the case of turbine blades, since the excitation force is broadband, the cracks would
start to propagate since the driving force is strong enough in a wide range of frequencies.
However, the testing method they used did not have an automated test termination controller
and required persistent monitoring all over the test.
15
Wang et al. [18] investigated the fatigue behaviour of solid films. A piezo shaker was
implemented to generate a vibratory force to excite a clamped cantilever beam. The control
system which maintains the output amplitude had to keep the test piece resonates at the natural
frequency and a 10Hz frequency shift was also the criterion indicating the fatigue failure or
crack initiation. The main drawback of this experiment was the stress evaluation using the
photographs they captured from the oscillating film during the test that introduces considerable
errors in stress estimation. Furthermore, Kim et al. [19] developed a fatigue testing
methodology to assess the fatigue behaviour of thin films. An electrodynamic shaker produced
the required force and a displacement gauge measured the deformation of the film all over the
test. In order to minimize the influence of gravitational force, the specimens were mounted
vertically.
Vanlanduit et al. [20] performed a fatigue test with an online health monitoring technique as a
crack indicator. The main purpose of this work was to develop a novel structural health
monitoring technique without interrupting the fatigue test. They used a shaker to apply the
force and a vibrometer measured the velocity and displacement during the test. The control and
measurement was done by Matlab. Moreover, this technique gives rise to reduce the traditional
extensive fatigue-experiment time.
F. T. Joaquim et al. [21] presented a prototype of a machine for torsional fatigue testing. The
prototype was designed to generate constant or variable loading. The test was conducted below
the first resonant frequency. Any changes in frequency response function (FRF) indicated the
crack nucleation and torsional rigidity reduction. A free vibration test was also performed to
assess the damping ration. This value was then used as an input to the Finite element analysis.
Özsoy et al. [22] performed an accelerated life testing for a prototype of a helicopter’s mission
system sensor cowling assembly in multi-axial vibration induced testing machine. A closed-
loop control system was utilized to maintain the amplitude of the oscillation. In addition,
numerical model of the component was created in ANSYS and locations with maximum stress
were identified.
Since the major problem associated with fatigue testing is related to cost and time demand of
such test, rise of the multi-specimen testing devices has attracted the attention of so many
16
researchers. Kim HY et al. [23] designed a multi-specimen fatigue testing device which could
test ten test pieces at a time. The required stress was applied to each specimen by attached
weight to the other end of the specimens. The results obtained by the new fatigue testing
equipment were compared with those extracted from the commercial testing machines and the
new apparatus seemed to be reliable for reducing the overall test periods. Ay et al. [24]
improved the multi-specimen fatigue testing device which was capable of testing sixteen
specimens simultaneously. Modifications applied on the new machine showed a better balance
being able to adjust the stress ratio for each test. Also the results were validated and the new
machine was claimed to match the real testing condition.
Onome et al. [25] developed an integrated computational-experimental approach for fatigue
life estimation. A series of vibration based bending fatigue tests were carried out to estimate
the fatigue limit under fully reversed bending load. A life prediction technique was also
implemented to calculate the effective fatigue cycles.
Rotem [26] employed a short accelerated testing method by increasing the stress level
gradually. Three specimens were used to determine the fatigue strength; two were tested above
the fatigue limit to characterize the S-N curve behaviour and one below it to find the fatigue
endurance limit. An electro-hydraulic servo-controlled system tested the test pieces confirming
the rupture associated with fatigue failure. This test had the advantage of using a few number
of samples and carrying out the test in a short time.
Panis et al. [27] explored the gigacycle fatigue domain on an accelerated fatigue testing
method. An electrodynamic shaker was used to excite a plate clamped on one end. The plate
was excited near the fourth natural frequency and the stress level was measured by the strain
gages attached to the plate. A laser sensor was also used to measure the displacement to
correlate the stress with the displacement. This methodology showed its capability for
exploring the fatigue behaviour in giga-cycle regime by increasing the frequency to more than
800Hz
G. Yun et al. [28] presented a vibration-based testing method capable of testing multiple
specimens simultaneously. An electrodynamic shaker provides the excitation force at the
resonance mode and a laser vibrometer measures the specimen’s response. A test coordinator
17
was also developed to synchronize the shaker controller and the vibrometer and continuously
monitors the specimen’s health throughout the test. The test pieces are clamped on one end and
a fixture was designed to attach them to the shaker. The stress levels are measured and
recorded in an orderly manner for all specimens. The experimental methodology was validated
with Al 6061-T6 aluminium specimens subjected to fully reversed bending stress test results.
An important feature of this test was testing ten specimens in a single run. However, moving
the laser during the test would need an automated system that moves the sensor head along one
axis in a single pass. This process is repeated while the frequency drop exceeds the
predetermined criterion. One disadvantage of this type of testing is the interval between each
pass. Although increasing the number of samples accelerates the testing time, but on the other
hand, it could cause a gap between a series of measurements while the laser is measuring other
objects response.
Fatigue testing methods and the historical development of the fatigue testing procedure was
discussed. Resonance fatigue testing or so called tracked sine dwell testing approach using
non-contact laser vibrometer was utilized to develop a testing procedure for jet engine turbine
blades. Due to the limitations of the shaker, multi apparatus testing setup could not be used for
the current test since the shaker is not able to operate at the desired level of excitation force.
Following the work done by A. Abdullah et al. and Gerge et al. [17, 28], a high cycle fatigue
testing methodology was established that is capable to test variety of components within a
short time with high accuracy using similar approach.
18
1.72.2 Multi-Objective Optimization Methods
Multi-objective optimization is the process of optimizing a set of objective function
simultaneously. General multi-objective optimization problem formulation is as follows
1 2 , , , T
kMinimize F x F x F x F x (2.2)
0, 1,2, ,jSubjected to g x j m
0, 1,2, , h x l e
Where k is the number of objective functions, is the number of inequality constraints, is
the number of equality constraints and is the vector of variables. is the cost function or
the value function, is the equality constraints and is the inequality constraints. is
the solution that minimize the objective function . The feasible design space is defined as
| 0, 1,2, , ;& 0, 1,2,j ix g x j m h x i
Contrary to a single objective optimization, there is no single solution in multi-objective
optimization and it is necessary to introduce a set of optimum solutions and the trade-off is
made by the decision maker according to the design priorities. [40]
Definition 1. Pareto Optimal: A point is Pareto optimal if there does not exist another
point such that and for at least one function. [40]
Definition 2. Non-Dominated and Dominated Points: A vector of objective function
is non-dominated if there does not exist another vector , such that with
at least one . [40]
Multi-objective methods in this section allow the user to specify preferences according to the
relative importance of different objectives. Most of the methods include parameters constraints
that can be directly formulated to reflect decision maker’s opinion or altered to represent
Pareto optimal set. The most common approach is to define a utility function to impose
constraints and develop a Pareto set. [40]
19
2.2.1 Weighted Global Criterion Method
One of the most common approaches to multi-objective optimization is the global criterion
method in which all the objectives are united to form a single function. The global criterion
method is a utility function that the method parameters are about to model the preferences and
decisions. A simple example of this method that is widely used is weighted exponential sum:
[40]
1
, 0k
p
i i i
i
U F x F x
(2.3)
1
, 0k
p
i i i
i
U F x F x
(2.4)
Where F(x) is the objective function and is the vector of weights ( ∑ ) that is
set by the decision maker and the relative importance of the objective is reflected in the value
of . Altering almost leads to small number of Pareto solutions in a certain neighbourhood,
therefore, typically the user set as a varying parameter to yield a set of Pareto points. Athan
et al. [29] proved that the first equation is a necessary condition for Pareto optimality which
means that for every solution point there exists and that becomes a feasible solution.
2.2.2 Weighted Sum Method
A very popular approach for minimizing several nonlinear functions simultaneously is
converting multi-objective problem to single scalar optimization problem by weighted sum
technique.
1
k
i i
i
U F x
(2.5)
If , minimum of U is sufficient condition for Pareto optimality; some literatures had an
accurate look at the two primary deficiencies of the weighted sum method. Dennis et al. [30]
discussed major drawbacks of this method. They claimed that this standard technique
succeeded only when Pareto curve is convex. Moreover, uneven distribution of weights fails to
produce even solutions for all parts of the Pareto set. Stadler et al. [31], Stadler [32], Athan et
20
al. [29], Huang et al. [33] demonstrated the inability of this method to capture non-convex
parts of the Pareto solution. Many authors have developed systematic approaches to select the
weights efficiently [34] [35]. By using ranking method, the least important criteria receive a
value of one, and other objectives are being assigned an integer with higher value according to
their importance. Pair-wise comparison provides a more accurate rating by the decision maker
by means of comparing two objectives at a time [36]. Weighted sum method has been
extensively presented over the past few years in various applications. For example, Koski et al.
[37] used the weighted sum method to systematic change to obtain a minimal volume and
nodal displacement of a four-bar truss. Proos et al. [38] applied this method to maximize the
first mode of resonance frequency and minimizing the compliance.
Although the weighted sum method is easy to use, the linear approximation of the preferences
may not consider the decision makers solution comprehensively. The solution is highly
dependent on the relative magnitude of the objective functions. It would be helpful to use
function transformation as a priori articulation of preferences. In this way, weights are used to
represent the functions relative importance. [39]
To achieve a consistent comparison between objective functions, it is advantageous to
transform the objectives to a non-dimensional function with an upper limit of one. One of the
most common approaches is given by Proos et al. [38].
max
itrans
i
i
F xF
F (2.6)
The denominator in the former equation can be determined by maximizing a single function
2.2.3 Lexicographic Method
In this method the objective functions are sorted in the order of importance and the following
problem is formed:
21
Minimize F x
* , 1,2, , 1, 1j j jSubjected to F x F x j i i
1,2, , i k
( ) is the optimum of the objective function. It is worth noting that the independent
minimum of ( ) is not the same as because the constraints are applied and in each
iteration new constraints are introduced. [40]
2.2.4 Exponential Weighted Criterion
Athan [29] proposed this method to compensate the deficiency of weighted sum method in
capturing the non-convex parts of the Pareto curve. Using the similar notation, P is the altering
parameter ranging within a specific bound.
1
1F xi
i
kp p
i
U e e
(2.7)
Minimizing this summation provide both necessary and sufficient condition for Pareto
optimality.
2.2.5 Weighted Product Method
This method has not been extensively used and the weighting parameters indicating the
objective significance is still unclear. Gerasimov and Repko used this method to optimize the
weight and maximum displacement of a truss and referred it to a valid compromise [41]. Main
feature of this method is void of having transformation objective function and has the
following formulation:
1
i
k
i
i
U F x
(2.8)
It is difficult for the decision maker to choose between number of methods and solution
settings. Apparently, methods providing both necessary and sufficient condition for optimality
22
are preferable because it includes all potential solution points and allow the decision maker to
reflect his preferences. [40]
2.2.6 Methods with No Articulation of Preferences
Most of the times, the designer or decision maker is not able to concretely identify his
preferences. The methods for multi criteria optimization presented involve specific
optimization engines. However, other approaches can be implemented to solve multi-objective
problems. Holland in 1975 first introduced the Genetic Algorithm. Because GA does not
require gradient calculations contrary to gradient based methods, it’s been widely used in real
life application. Rather than a single point at a time, it is based on random initial population
and improving the random numbers to obtain the potential solution. Another feature of this
technique is its convergence to a global solution rather than a local one. [42]
In multi-objective optimization, the objectives are usually conflicting therefore it is not
possible to optimize all objectives at the same time. Traditional GA is adapted to suit multi-
objective problems. There are two approaches, one is to combine all objectives to form a single
objective with methods described above, but the problem lies in the best selection of weights
and utility function by the decision maker to represent the preferences. Practically, even for
someone familiar with the problem it is difficult to exactly select these weights and slight
perturbations can lead to different solutions. The second approach is to introduce a Pareto set
solution. In this approach, the author is dealing with a set of solutions that are non-dominated
with respect to each other. It means that by moving from one solution to another, there are
specific amounts of sacrifice in one objective while there are some improvements in the others.
Therefore, Pareto optimality due to its trade-off idea, is more practical in real life problems
since allows the decision maker to choose between multiple options. [42]
Consider a decision maker has no obvious preference of objectives. In general, it forms the
following formulation:
In the solution space , variable vector with objective functions
and a set of constraints that restricts the solution.
23
In reality, it is impossible to optimize several objectives simultaneously and optimizing one
objective results in unacceptable results in other objectives. Hence, the acceptable results
would be a set of non-dominated solutions with an acceptable level of satisfaction. A Pareto
optimal set is a solution which it is not dominated by any other solution in the solution space.
In other words, Pareto optimal solution says that a solution can not improve one objective
without worsening other objectives. Practically it is impossible to identify the entire Pareto set
due to its size and complexity. Hence, the best solution is achieved by identifying the best-
known Pareto set close enough to Pareto optimal set. In addition, diversity of the solution is
also of great importance, providing a clear uniformly distributed set of solutions for the
decision maker to make a reasonable trade-off. [42]
In this part, popular approaches to multi-objective optimization (MOO) using GA are
presented for solving multi criteria optimization problems.
Genetic Algorithm
The genetic algorithm optimization was first introduced by Holland in 1960 [43]. GA is
inspired by the evolution theory in the nature. The stronger species have greater opportunity to
last longer and pass genes to the next generation. In long term, those genes with distinguished
and superior traits pass along the generations and become dominant. During evolution, random
changes may occur and provide additional benefits to the next generation. However, unfit
changes will be eliminated from the nature. In GA, the term chromosomes are representing the
solution variable vector x and it is composed of distinct units called genes. Genes are usually
assumed to be binary but later, some other types have been implemented.
GA works with a randomly initialized collection of chromosomes called population. Each
generation is created by two major operators: crossover and mutation. During the evolution
process, the solutions become fitter and closer to the optimal point meaning that it is
converging to the final solution. In crossover, two parents are combined to form the offspring
or the child. Parents are selected from current population so the offspring are expected to
inherit good genes and fitter traits are expected to appear in the population along the evolution.
Mutation plays a crucial role in GA. It operates as a random change in a gene and it is not
24
expected to observe a sharp adaption in the characteristics of the chromosomes. Mutation
introduces diversity all over the space and prevent from local optimum point search. In other
words, it is a tool for increasing the probability of finding a global optimal point and escape
from local optima [42].
Selection of chromosomes for the reproduced generation depends on the selection procedure.
Tournament, ranking and proportional selection are of the most common methods. Fitness
values mainly determine which genes have a larger possibility of survival and transmission to
the next generation.
The GA algorithm is as follows: [42]
1. N solutions over the solution space X are created randomly as the first population .
Fitness value is calculated for each individual.
2. Produce the offspring : two solutions A and B are selected based on fitness values
and operate the crossover to generate the next generation and add them to .
3. According to the predefined mutation rate, apply the mutation to each solution in .
4. Assigned the fitness value to each solution in based on objective function and
constraints.
5. Select N solutions for the next generation
6. Check the stopping criteria. If it is satisfied terminate the search. Otherwise, t=t+1 and
go the second step.
Multi-Objective GA
GA has been the most generic approach to multi-objective optimization due to its ability for
solving non-convex discontinuous problems. The ability for investigating a diverse set of
potential solutions simultaneously in different areas of solution space has made it well suited
for various engineering applications. The prime advantage of this population based approach is
that it does not require any scale or weight for the user to prioritize the preferences.
Vector evaluated GA (VEGA) was proposed first by Schaffer [44] and later on, multi-objective
evolutionary algorithms were developed such as MOGA [45], Niched Pareto Genetic
25
Algorithm (NPGA) [46] and Weighted based Genetic algorithm (WBGA) [47]. Non dominated
sorting genetic algorithm (NSGA) [48], Strength Pareto Evolutionary Algorithm (SPEA) [49],
improved SPEA(SPEA2) [50], Pareto Archived Evolution strategy (PAES) [51], Pareto
Enveloped based selection Algorithm (PESA) [52], Fast non-dominated sorting genetic
algorithm (NSGA-II) [53], Multi-objective evolutionary algorithm (MEA) [54],Rank density
based Genetic Algorithm (RDGA) [55] and dynamic multi-objective optimization evolutionary
algorithm (DMOEA) [56]. These are a few credible well-known methods that have been
implemented extensively in so many applications and their performance has been studied. [42]
Table 2-1 lists the most well-known multi-objective methods comparatively giving the
advantage and disadvantage of each algorithm.
Algorithm Fitness assignment Diversity
mechanism
Elitism Advantages Disadvantages
VEGA Each population is
evaluated with respect to
a different objective
No No First MOGA
straightforward
implementation
Tends coverage to the
extreme of objectives
MOGA Pareto ranking No No Simple extension of
single objective GA
Usually slow Convergence
Problems related to niche
size parameter
WBGA Weighted average of
normalized objectives
No No Simple extension of
single objective GA
Difficulties in nonconvex
objective function space
NPGA No fitness assignment,
tournament selection
No No Very simple selection
process with
tournament selection
Problems related to niche
size parameter Extra
parameter for tournament
selection
RWGA Weighted average of
normalized objectives
Yes Yes Efficient and easy
implement
Difficulties in nonconvex
objective function space
PESA No fitness assignment Pure elitist Yes Easy to implement
Computationally
Performance depends on cell
sizes Prior information
needed about objective space
26
PAES Pareto dominance is used
to replace a parent if
offspring dominates
Yes Yes Random mutation hill
climbing strategy Easy
to implement
Computationally
efficient
Not a population based
Approach Performance
depends on cell sizes
NSGA Ranking based on non-
domination sorting
No No Fast convergence Problems related to niche
size parameter
NSGA-II Ranking based on non-
domination sorting
Yes No Single parameter (N)
Well tested
Crowding distance works in
objective space only
SPEA Raking based on the
external archive of non-
dominated solutions
Yes Yes No parameter for
clustering
Complex clustering
algorithm
SPEA-2 Strength of dominators Yes Yes Improved SPEA Computationally expensive
fitness and density
calculation
RDGA The problem reduced to
bi-objective problem with
solution rank and density
as objectives
Yes Yes Dynamic cell update
Robust with respect to
the number of
objectives
More difficult to implement
than others
DMOEA Cell-based ranking Yes No Includes efficient
techniques to update
cell densities Adaptive
approaches to set GA
parameters
More difficult to implement
than others
Table 2-1 List of multi-objective genetic algorithms [42]
Weighted Sum Approach
The most common approach to solve multi-objective optimization problem is to assign a
weight to each objective function so the problem is converted to single objective
problem [42]. A vector is selected for each run. WBGA-MO was proposed by Hajela and
Lin [57] which each solution uses a different weight vector. Moreover, the weights can be
adjusted to promote distribution of the population.
27
Another interesting approach by other researchers is based on generating random weight
vectors for each solution in each generation. The procedure for RWGA is as follows:
1. Generate a random population.
2. Fitness assignment:
Generate a random parameter [ ] for each objective.
Random weight is calculated using this equation ( ⁄ )∑
Calculate the fitness value using ∑ where is the kth
objective function.
3. Calculate the selection probability of solution using:
1
) |t
min min min
t
y p
p x f x f f y f where f min f x x p
4. Parents are selected by selection probability and crossover and mutation is applied to
create offspring. Children are moved to and update E (external archive for non-
dominated solutions).
5. Randomly n (number of elitist solutions) solutions from are replaced by n solutions
in E.
6. Set is the stopping criterion is not satisfied otherwise return to step 2.
Although there are a few drawbacks discussed in the literatures in using this method, efficient
computational time and straightforward implementation of weighted sum method, has made it
one of major classical multi-objective optimization approaches. [42]
In this part, components of the multi-objective GA are introduced and the methods are
discussed briefly.
28
Components of Multi-Objective GA
Pareto Ranking Approaches
Pareto ranking method is based on the concept of Pareto dominance in assigning fitness to
solutions. Each solution is ranked according to dominance rule and the best solutions are
copied to the first Pareto front [42]. Goldberg [58] was the first who proposed the Pareto
ranking technique. This method classifies the Pareto fronts to non-dominated fronts according
to each solution rank. NSGA also creates a set of Pareto fronts using a fitness sharing function.
NSGA_II uses a fast non-dominated sort algorithm [53] to form the Pareto fronts.
Diversity
One important consideration in MOGA is diversity of solutions. In order to obtain solutions
uniformly distributed all over the design space, several approaches have been developed to
prevent cluster formation in a local area [42].
Fitness Sharing
Fitness sharing allows the solver to explore all sections and to achieve this goal, it penalize the
solutions located in such dense areas. Fonseca and Fleming [45] used the same idea to
penalized solutions which are clustered in a certain area. They used the following procedure to
find the fitness function:
1. Calculate the Euclidean distance between each pair of solutions (x,y)
2
1
,k
k k
max mini k k
z x z ydz x y
z x z x
(2.9)
Where is the cost function.
2. Calculate niche count for each solution as
29
, ,
,, ,0
share z
r y t r x t share
d x ync x t max
(2.10)
Where is the niche size.
3. The fitness function of each solution is as follows
,,
,
f x tf x t
nc x t (2.11)
A solution in a crowded neighbourhood will take a higher niche count and reduces the
possibility of being selected for the next generation. Therefore, niche count limits the
reproduction of solutions in a dense area. One disadvantage of using niche count is that the
user has to select the parameter. Another disadvantage of this method is the extra
computational cost for finding the niche count. [42]
Crowding Distance
Crowding distance approach was developed to obtain the Pareto fronts without fitness sharing
parameter. NSGA_II uses the crowding distance method which will be demonstrated in detail
later. Hence, there is no need to predefine any parameters by the user to calculate the niche
count. In NSGA_II two solutions are randomly selected. If they are in the same front, the
solution with higher crowding distance is the winner. Cell based approach divides the objective
space into K cells and the number of solution in each cell is defined as the cell density. Similar
to fitness sharing function, the solution with lower density has a priority to be passed to the
next generation. Figure 2.2 schematically shows how the cell based method is implemented.
Figure 2.2 Cell-based method [53]
30
PESA [51] uses the cell based density factor to choose between two solutions. The procedure is
as follows: [42]
1. Generate initial population and set external archive
2. Divide the objective space into cells where n is the number of grids for each
objective.
3. Add or remove any new solution from the archive using don-dominated sorting. Update
the cube member in each step.
4. If a solution is not dominated by or is not dominating any other solution, add it to the
archive and remove a solution with the maximum number of members in the cubes.
5. If stopping criteria is satisfied stop and return Et
6. Select solution in and apply operators to generate offspring.
7. Set and go to step 3.
PESA_II was developed using region-based selection where cells with lower density are
selected instead of individuals and the solutions inside a cell are chosen to take part in mutation
and crossover.
RDGA [55] also utilize cell-based density approach to convert k objective problem to a by-
objective one. It is worth noting that, one of the most advantages of cell-based density
approach is that it pushes the solutions in high density region to lower ones and substantially
reduce the computational time in comparison to niching approach.
Elitism
Elitism means that the best solution found survives and passes to the next generation. Early
MOGA did not implemented elitism but recently it is widely used in several approaches.
NSGA_II uses a constant population size of N. The procedure is given below and it
demonstrates how the elitism is implemented without using an external population.
NSGA_II procedure
1. Create an initial population of size N.
31
2. Apply mutation and crossover operators to create offspring if size N.
3. Check if the stopping criterion is satisfied, stop and return .
4. Combine and to form (
5. Identify the Pareto fronts ( using fast non-dominant sorting.
6. Calculate the crowding distance in
7. Create as follows:
If | | | |
Else if | | | | then copy the least crowded | | solutions from
8. Parents are selected from by tournament selection. Cross over and mutation is
applied to generate the offspring.
9. Set and go to step 3.
Elitism Using External Population
Several issues have been addressed regarding the external elitism approach. The elitism list E
is updated each time adding solutions that have not been dominated so far. This imposes
excessive computational cost and high demand for data storing as the E might grow extremely
large according to large number of Pareto sets. Some examples of this approach are SPEA
[59], PESA [52], RWGA [60] and DMOEA [56].
Constraint
Several techniques particularly focusing on numerical non-linear optimization for constraint
handling have been developed over the past few years can be classified in the following
categories: [61]
Using penalty function
Maintaining a feasible population by genetic operators
Separation of objectives and constraints
Hybrid methods
Jimenez et al. [62] proposed a niched method for constraint handling as follows:
32
Select two solutions randomly in the population. Compare two solutions, if one of them is
feasible and the other one is infeasible, the feasible is the winner. In case both solutions are
infeasible, Choose a random infeasible solution C and compare both with C. measure the
relative infeasibility with respect to C and choose the one with less infeasibility value.
All issues with contemporary multi-objective optimization algorithms have been addressed. In
regard to the work done Deb and coworkers [53], an improved version of NSGA called
NSGA_II is introduced in chapter 4.
1.82.3 Surrogate-Based Design Optimization
Direct coupling of finite element modeling with optimization algorithms shows a number of
disadvantages. Therefore, surrogate models are introduced as a remedy for such drawbacks and
they are replaced with high computational modeling. In optimization problems, the structural
FEM solvers proved to be reliable and efficient but the main drawback is high computational
cost and they are often time consuming. With the possibility of several local optima, the use of
analysis codes in shape optimization problems requires more computation and the calculations
become more complex. [63]
The basic idea of using the surrogate or approximation models is to replace the high fidelity
and expensive FEA analysis with a less expensive yet accurate model.
2.3.1 Classification of Surrogate Models
Approximation methods can be classified in to two main categories, black-box and physic-
based approaches. Generally black-box models use a set of design parameters, which is called
the training set and the high fidelity code is applied to the inputs to evaluate the respective
outputs. The surrogate manages to approximate the relation between inputs and outputs such
that it can predict the response at new points. Usually, there are some new points other than
training points which examine the model. Surrogate models can be either parametric or
nonparametric.
Parametric models use the initial training set to find the unknown parameters of the network.
The new points do not change the estimated parameters and consequently, they are no longer
33
used in making decisions for the response. Polynomial and multivariate regression splines are
examples of parametric modeling.
Nonparametric models uses the initial set to find unknown parameters and still continue to use
training points for predicting the response. Hence, the response depends both on parameters
and training points. Kriging (KG), Artificial neural network, Radial basis function (RBF) are
examples of non-parametric methods.
Basic Steps of Surrogate Modeling
The surrogate modeling involves following steps:
Data generation
Model selection
Parameter estimation
Validating the model
Number and location of training point is crucial for designing a surrogate model. In fact,
increasing the number of points (accuracy) and the computational cost are two major
conflicting targets. Generated points for training set can have a random, classical or space-
filling fashion; moreover, these points could be generated at once or stage-wise. The difference
between these two approaches is that, in one shot procedure, no additional data is added to the
iterative process and the model is constructed in the beginning. While, in stage-wise approach,
it continuously adds points in the process and shows more flexibility during the convergence.
[63]. Figure 2.3 shows the classification of surrogate models.
Figure 2.3 Classification of data generation techniques [63]
34
2.3.2 Main Types of Surrogate Models
Here, the following models have been found to be the main models for design optimization
applications [63]:
Polynomial Regression
Polynomial regression also known as response surface model (RSM) was basically used for
physical experiments. In this model, the points in the training set are fit by a polynomial and an
error.
y y (2.12)
Where is the error with zero mean and variance . The polynomial can have ay order.
However, it is mostly of the first or second order.
0
1
ˆk
i i
i
y x
(2.13)
0
1 1
ˆk k k
i i ij i j
i i j i
y x x x
(2.14)
Where is the intercept, is the interaction coefficients and k refers to number of variables.
Kriging (KG)
Kriging is a nonparametric interpolation model based on Guassian stochastic process. The
response of this model is expressed in the following form:
y x f x z x (2.15)
Where is a low-order polynomial and is the Gaussian Stochastic function. It was
found that a constant value can model the input-output relation. Since, a random number can
replace the polynomial
y x z x (2.16)
35
Radial Basis Functions (FBR)
Radial Basis function is a non-parametric regression modeling technique that uses linear
combination of radially symmetric functions based on Euclidean distance from a certain
distance. This model can be expressed as:
1
ˆn
i
i
i
y K x x
(2.17)
Where I refers to the training points, K is the radial function and is the weight factor. The
function K can take several forms such as
clinear K x x (2.18)
3 cCubic splines K x x (2.19)
2
cx xGaussian K exp
(2.20)
is called the shape parameter and is evaluated by solving this equation
K y (2.21)
Where K is matrix ( ‖ ‖)
Artificial Neural Network
This method is a nonparametric regression method which utilizes the concept of neurons in the
human brain. A neural network consist of an input layer, one or more hidden layers that
transform the result from previous layer to an output layer. These elements are inspired by
biological nervous system. The neural network can be trained in such a way to find the
relation between inputs and outputs by changing the values of weights. The network continues
to adapt until the target and the output matches [64]. Figure 2.4 schematically depicts the
neural network concept.
36
Figure 2.4 Neural Network concept [64]
Simple Neuron
A neuron with single input and a bias is shown in figure 2.5
Figure 2.5 Simple Neuron [64]
The input p is multiplied by a weight and transmitted through a transfer function , which
produces the output . Here, as a transfer function, can be in the form of step function or
sigmoid function. The idea of neural network is that the parameters can be adjusted
such that the network shows a desired behaviour. Figure 2.6 schematically plots the transfer
functions [64]
37
Networks Architecture
Several neurons could be combined in a layer and a network can contain two or more layers.
Each layer has a weight matrix , a bias matrix and an output . Number of layers is
appended to each parameter as a superscript. Below, in figure 2.7, a three-layer network is
shown. [64]
Figure 2.7 Multi-layer Neural Network [64]
Multiple-layer networks are powerful tools to be used and trained for any function
approximation.
Figure 2.6 Network’s Transfer Functions [64]
38
Training and Learning Fundamentals
A learning system changes itself and thrives to adapt to meet the desired criteria. In principle,
neural network components changes are conceived as the learning process. A network can
learn by
Developing new connections or removing the current connections
Changing the weights
Changing the neurons thresholds or adding and deleting neurons
Some of these changes are difficult to implement. Thus, let the network learn by modifying the
connection weights according to formulated algorithms.
Unsupervised learning is the most credible method but it is not suitable for all kinds of
problems. On the other hand, in supervised learning for every training set that is fed to the
network, the output can be compared directly with the correct results and the network weights
can be changed with respect to the difference. Two different styles of training are described
below. Incremental training updates the biases and weights each time an input is presented in
the network. In batch training, all the parameters are updated after all inputs are presented.
Once all network weights and biases are initialized, the network is prepared for training. The
network requires proper inputs and targets to approximate the function that relates the
inputs and outputs. During the convergence, all weights and biases are updated to minimize the
error to excel the performance. [65]
Back Propagation Algorithm
There are many different approaches for back propagation algorithms. One of the simplest
learning algorithms is that the parameters update in the direction which the performance
function decreases more rapidly. For instance, for a single iteration:
1k k k kx x g (2.22)
39
Where represents the current vector of parameters, refers to the current gradient and is
the learning rate. [65]
Model Validation
Some techniques for surrogate modeling were introduced. The fitness of a technique is
evaluated using new points other than training points. There are some error-measuring methods
such as:
21
ˆiRMS y y
q (2.23)
i iMAE Max y y (2.24)
Where is the mean of exact response and indicates the surrogate response value. The
smaller values of these error estimators indicate better accuracy. [63]
40
Chapter 3: Fatigue Testing
A High cycle fatigue testing system has been developed to test a specimen and monitor the
health throughout the test. It is required to excite the test piece at the first resonant frequency
while keeping the stress to its maximum level. The frequency starts to shift as the cracks
nucleate and the test terminates when a substantial change in the frequency is detected. Figure
3.1 schematically depicts the configuration of the HCF test.
1.93.1 Equipment for HCF Testing
The HCF test comprises the following equipment:
An electrodynamic shaker for generating excitation force.
An amplifier to amplify the controller output signal and running the shaker.
A controller to control the shaker at the required level.
A data acquisition system to read the measurements from all sensors.
A single point vibrometer to measure the vibration of the specimen.
Other supplementary tools such as accelerometers and strain gauges.
A laptop is connected to the controller to run the LMS.12A dwell and sine sweep
application.
41
3.1.1. Hardware
I. Polytec Vibrometer
Model OFV-5000, Polytec Scanning laser vibrometer, expands the ability to handle high
vibrational velocities. This laser head sensor is capable of measuring velocities up to .
The controller can be configured to with several decoders to meet variety of applications. The
voltage output is transmitted through a standard BNC jack and also the laser head provides
autofocus and a focus lock to stand the test integration.
The sensor head is mounted on a tripod in an optimum stand-off distance and the laser beam is
pointed at the specimen’s surface. Velocity amplitude of the vibrating object generates a
frequency or phase modulation of the laser light. The signal is decoded in the OFV-5000
decoder and the measurements could be displayed with a PC or the vibrometer software. Figure
3.1 schematically shows the layout of signals path.
Figure 3.1 Signals in the vibrometer [66]
42
II. LMS Controller
The LMS controller has a number of input channels on the front end that all have BNC to
LEMO convertors that read the signal from all sensors during the test. VB8 module is the most
complete and versatile member of LMS SCADAS module family, supporting a wide range of
transducers. VB8 modules are basic input channels on the front-end, but with their capability to
accommodate AC and DC voltage sources such as ICP sensors, strain gages in full, half or
quarter bridge mode, potentiometers and DC accelerometers has made them suitable for wide
variety of applications including the current test. In addition, another module that was used is
XSI-V which has various functionalities, but the basic one is transmitting the data and control
signals as fast as possible to the amplifier. BDS4-E is the other module used for recording all
data from strain gage readings. What makes this module ideal for our work is that it covers
exceptionally high dynamic range strain measurements along with supporting multiple
channels for signal conditioning and signal processing. Additionally, it allows measurements of
dynamic strain with optimum signal to noise ratio. [67]
Figure 3.2 VB8 module [67]
Figure 3.3 BDS4E Module [67]
43
III. Electrodynamic Shaker
Electrodynamic shaker model 2110E: Its 1” (25.4 mm) stroke, wide frequency range
(useable to 6500 Hz), an air-cooled shaker with a maximum capability of 110lbf sine-
peak force and maximum acceleration of 150g. This shaker is designed for general
purpose vibration testing of mechanical components.
2050E09-FS amplifier is a high power Linear Amplifier for driving small to mid-size
vibration systems. The amplifier front panels can be turned to voltage or current mode.
This allows the user to work in either a highly damped voltage source or as a high
impedance current source. The amplifier is connected to the field power supply and it
provides sufficient driving voltage for the shaker.
Cooling package is required at all times and should be installed prior to the testing.
Figure 3.4 shows the electrodynamic shaker package as it was mentioned earlier.
Figure 3.4 Modal shop shaker system. (a) Shaker, (b) Amplifier, (c) Cooling package
(a) (b) (c)
44
IV. Supplementary Tools
ICP accelerometers are used as a control sensor in order to maintain a constant level of
acceleration during the test. PCB 352C65 was mounted on the fixture and its dynamic range
covered our required acceleration for this test.
EA-06-032CE-120 Vishay strain gages are widely used for all purpose experimental stress
analysis. Based on its size and fatigue life endurance, these strain gages were selected as being
the best choice for this experiment.
3.1.2 Software
LMS Test.Lab 12.A offers a complete engineering solution for many applications such as
sine dwell testing and modal analysis. This software enables all users to set up their test
configurations and meet various types of testing needs. Also, it offers the possibility to define a
specific profile with fluctuating g levels thus allowing the user to run a test that better reflects
real world behavior. The applications that were used for the current experiment are “Modal
Analysis” and “Tracked Sine Dwell” worksheets. The resonance search and track sine dwell
test in LMS is designed to perform the fatigue test by tracking the resonance frequency and
dwelling on for a certain frequency range. Sine sweep testing is also required to find the
resonance frequency prior to the dwell test.
Two channels are used to control and dwell the shaker. An ICP sensor is mounted on the
fixture which is called the control sensor. Control sensor’s task is to maintain the shaker
operating force equal to the predefined value. Also, the vibrometer signal is sent back to the
controller to allow the controller continuously track the phase angle difference and shift the
driving frequency to the new resonance frequency of the test piece.
1.10 3.2 Resonance Fatigue Test
Figure 3.5 schematically depicts the configuration of the HCF testing procedure. In this test, it
is required to run a single blade dwelling at the first bending mode for maximum cycles to
establish the endurance limit for the coated tapered peek plaque. This RT (request of the test
45
provided by Pratt & Whitney Canada) requires 2 specimens subjected to 35g force on shaker to
achieve the targeted cycles. If substantial change in the frequency is monitored, the test has to
be terminated and the data is stored for post processing.
Figure 3.5 HCF testing configuration
3.2.1 Experimental Testing
To obtain data point for fatigue endurance limit, two tests has to be conducted and the cycles
are counted by LMS controller. The stress level is constantly measured by strain gages. In
order to obtain significant fatigue data, more parts are required but according to the agreement
and test requirements, two points are sufficient for extracting the fatigue limit data. In HCF
test, 1% frequency shift from the resonant frequency is assumed to imply that the crack has
already been initiated and the part has been fatigued.
Test Sequence
The test is required to be conducted at room temperature. Shaker has to operate at the first
bending mode frequency which was identified by tap or sine sweep test. The fixture torque
needs to be fixed at 15in-lbs. To avoid slippage due to loss of torque, the bolts are secured by
spring washers. Before starting the test, a sweep test is performed to obtain the natural
46
frequency. To minimize the influence of sweep tests on the fatigue cycle counts, the tap test
was performed to exactly find the first mode frequency. Consequently, before executing the
dwelling, frequency range is limited in the sweep test close to the resonance frequency. After
completion of the sweep sine, resonance dwell test is performed using the FRF from the sweep
test.
Step 1
The shaker is bolted down to the table, supported by mounting isolators and vibration pads.
The amplifier is plugged to the power supply. The amplifier output cable is connected to the
shaker. Cooling unit needs to work during the test. The unit is plugged to the rear power outlets
of the amplifier. Turning on the amplifier, automatically starts the cooling unit. The fixture and
plaques are weighed precisely for LMS software settings. Accelerometer (control sensor) is
mounted on the fixture base and then it is firmly attached to the shaker armature. The fixture
torque holding the plaque is set to 15in-lbs. The measure sensor or vibrometer measures the
oscillation of the blade. LMS controller is equipped with 8 input channels and 2 outputs. One
output is directly connected to amplifier AC input terminal and sensors are connected to the
controller by LEMO/BNC convertor front panel as well. An Ethernet cable is also required to
control the controller by the LMS Test.Lab software.
Step 2
An initial shakedown at 35g at the first bending mode is performed to confirm that the 15in-lbs
torque is kept constant throughout the test.
Step 3
Accelerometer and strain gage readings are measured continuously throughout the test and if
the frequency shifts by 1%, test is stopped. Number of cycles is recorded and the torque level
at the fixture is measured. If the torque is shifted from predetermined value of 15, re-torquing
is necessary and the test is continued. If no torque shift is noticed, crack inspection is
performed and the pictures from the cracks are captured. Figure 3.6 clearly illustrates the
testing procedure and the step-wise dwell test is demonstrated later in this chapter.
47
Figure 3.6 depicts a schematic configuration of HCF test procedure. Signal to noise ratio is
evaluated in self-check block. If this ratio lies within an acceptable range, the test is ready to
proceed. Sine sweep test is performed to identify the resonance frequency. FRF signal is used
as a reference for estimating the phase difference between the control and measure signals.
Later, dwell setup is adjusted to set the test criteria. Termination time associated with
frequency drop or run-out cycles are applied. Finally, the test is ready to start. Continuous
inspection of the signal qualities is necessary during the test. After the test is stopped by the
controller, number of cycles to failure is saved and all the required data is recorded for data
processing.
48
Figure 3.6 High cycle fatigue testing configuration and procedure
49
Test Setup
The HCF testing procedure was demonstrated earlier. In this section, the LMS.Test.Lab
application and settings particularly for testing the tapered plaques are discussed. Strain-
displacement correlation test is initially performed as a back-up for strain measurements in the
event that any of the strain gages failed during the fatigue test.
Strain gages used in this test (Vishay- CEA-06-032UW-120) were attached in a quarter bridge
mode on the designated locations. SCL-VB8 module supports two types of strain gages:
“Quarter Bridge mode” and “Quarter Bridge B mode”. Note that they require different wirings;
for both types, pin2 is optional and generally not to be used in particular cases. In figure 3.7 the
strain gage wiring and corresponding channel is shown.
Figure 3.7 Strain gage wiring [67]
The plaque is strain gaged as shown in figure 3.8. All the tapered plaques are required to be
strain gaged as well.
50
Figure 3.8 Strain gage setup
Strain values are used to measure the strain level under the first bending mode of vibration.
Later, dynamic stress is calculated via strain measurements by Hooke’s law. Strain gage
specifications are as follows. As you can see in figure 3.9, the plaque is strain gaged at five
locations.
Gage factor:
Gage resistance:
Voltage supply:
Sensitivity:
Figure 3.9 Strain gaged plaque
Specimen
Fixture
Strain
gages
Strain gages
51
3.2.2 Stress-Displacement Calibration Test
Vibrometer is used to measure the velocity at the center point and the gages to measure
the strain level in multiple G levels from 5g to 40g in steps of 5g. The purpose of this test is to
act as a back-up to predict the stress level in the event strain gages failed due to high G levels
endured during the test. Note that the temperature of the specimen is inspected during the test.
If a substantial temperature increase was noticed, record the temperature and the corresponding
number of cycles.
To obtain the strain-displacement curve, the plaques were subjected to seven different g levels.
To get each data point, the plaques are swept in a certain range close to the resonant frequency.
The maximum strain is then recorded at the resonant frequency and the maximum stress could
be calculated by multiplying the Young’s modulus by the strain value. The displacement is
also computed by a single integration from the velocity signal. Figure 3.10 shows the test setup
and the vibrometer stand-off distance from the test object. The control sensor can be observed
on the left, sitting on the fixture.
Figure 3.10 Test Setup
52
The following figures show the relation between the displacements/Strain vs. the g level.
Figure 3.11 Strain gage 3 Vs. G level
Figure 3.12 Center point displacement Vs. G level
53
3.2.3 High Cycle Fatigue Test
After the completion of the calibration tests, the specimens are prepared for the fatigue testing.
In case of any failure during the calibration test in the strain gages, the plaques are removed
from the fixture and the broken strain gages are replaced with new gages.
Vibration sine dwell can cause an extreme situation for a component by dwelling on specific
frequency. Sine Dwell tests are implemented to accelerate failures and tracking of resonant
frequency. Excitation force for this type of test is sine waves and the frequency and amplitude
could be either fixed or tracked at a specific phase or amplitude according to the testing
criteria. Objective of the test is to subject the blade to a sustained level of excitation force for a
predetermined number of cycles or time. Dwell test is based on the same principles involved in
sine control test (sweep-sine). Resonance frequencies could be based on data acquired in the
tap test or “self-check” or even a complete sweep test performed in the dwell application. For
each selected frequencies, dwelling could be fixed or tracked for a desired number of cycles.
Dwell Frequency Identification
Dwell frequencies can be defined once a “self-check” or a normal sweep sine test has been
executed. A number of resonance frequencies can be identified. Resonance frequencies are
selected from the frequency response function (FRF) between any two selected channels.
Resonances can be determined by matching them to a set of criteria or by manual selection. It
is necessary to make sure that the mode which the specimen is dwelling on is the targeted
mode. Therefore, tap testing is a good practice to identify the first few mode shapes and the
corresponding frequencies. Dwell time or cycles are calculated by Test.Lab software according
to the frequency they are dwelling at. Any sharp rise or drop from the fixed amplitude is
controlled by the software and sends an alarm or abort signal to the controller to stop the
excitation. As it was mentioned earlier, dwell can be in the fixed, amplitude or phase tracking
mode. In fixed mode, frequency remains fixed throughout the dwell at the selected resonance
frequency. In case of the phase track mode, frequency varies in a certain band about the
defined frequency in order to maintain the excitation level at the resonance amplitude. The
54
bandwidth is defined by the resonance frequency and the computed Q factor for the
resonance.
n
Q
(3.1)
A continuous iteration process is used to search for the local maximum and for this purpose a
frequency step interval is defined.
sS
(3.2)
Where is the sine tracking dwell factor. The drive output signal is first set to the initial
resonance frequency ( ) and the corresponding gain amplitude ( ) is measured from
the response and reference channels. The drive voltage steps to a new frequency
and the corresponding gain amplitude ( ) is calculated. If is less than , the drive
frequency is decreased by . If is more than the drive frequency is increased by .
In the case of phase tracking mode, the frequency varies over a specific band in order to
maintain the excitation at the targeted phase value ( ). The bandwidth ( ) and the step
frequency ( ) is defined for the amplitude criterion above. The drive signal is initially set to
a defined frequency and the phase between two channels is measured. Unless the phase is on
target, the frequency starts to adapt to maintain the phase difference constant. This process
continues as the structure fatigues. Updated frequency corresponding to target phase requires
continuous data acquisition from the sensors in order to follow the new target. [67] Figure 3.13
schematically depicts the tracking algorithm.
Figure 3.13 tracking algorithm [67]
Ga
in a
mp
litu
de
Ga
in a
mp
litu
de
y0
y1 y0
y1
55
LMS Configuration
To perform sine dwell test:
1. Click on start menu>test Lab12.A>environmental testing> Tracked sine dwell
application.
2. Channel Setup
The Channel Setup worksheet enables you to specify the channel and transducer characteristics
used for the test. These channel characteristics can be either entered manually or be read from a
file or set manually. The status is given at the top of worksheet. Two channels are identified in
this tab. Measure channel for specifying the sensor that measures the specimen vibration and
control sensor sits on the fixture that feeds the measure channel for the closed loop control.
Adjusting the input mode and sensitivity turns the flag light to green. Figure 3.14 is a screen
capture from the channel setup worksheet.
Figure 3.14 Channel setup
56
The input range of data acquisition for both measure and control sensors are 10V. Actual
sensitivity of the vibrometer is 1V/(m/s) which is set on the vibrometer controller. The
accelerometer also has a sensitivity of 10.54 mv/g. The tracking filter can be used to improve
the signal to noise ratio all over the measurements. High pass filters are utilized to attenuate
lower frequencies and let the higher frequencies pass through the filter. Since the natural
frequency is around 400 Hz, filter with cut-off frequency of 300 Hz was used to improve the
measurements. Sampling rate was set five times higher than the resonant frequency (2000Hz)
to prevent any aliasing problem for the frequency range of interest.
Experimental measurements are never devoid of errors. Even with the aid of modern
equipment, errors are always an inherent part of every experiment. In experimental
measurements, the true signal amplitude can rather change from one point to another. In this
situation, it is useful to reduce the noise level by smoothing operator. In smoothing, data points
are modified in each individual point where a sharp change is distinguished relative to the
adjacent points. As long as the pure signal is smooth, the smoothing would not distort the real
data and the noise is reduced significantly. Smoothing removes the noise from the signal to
reveal the unadulterated data obtained from the sensors. Smoothing works as a low-pass filter
remove high frequency noise from the signal. Due to the speckle nature of the light scattered
from the moving object, smoothing is a substantial conditioning to be applied on the velocity
signal. Therefore, on all the signals retrieved from the vibrometer, smoothing is performed to
reduce the white noise.
3. Sine Control
In this part, the sweep rate and measurement estimators are set. The objective of sine sweep is
to sweep in a specific range and identify the frequency that the plaques are supposed to dwell
at. The actual frequencies used by the control loop will not depend on this setting but follow
from the selected sweep rate and the time it takes to perform a control loop. The data acquired
by the frontend will be averaged on the appropriate frequency lines which are defined through
the entered resolution. The lowest and highest frequencies to be used in the control test specify
the starting/ending frequencies. This value depends on the limitations of the shaker and the
sensors. From this menu, you can choose the sweep rate mode. When the mode is linear, the
frequency is varied in Hz per second. If the sweep mode is logarithmic, then the frequency is
57
varied by the specified octaves per minute. The compression factor determines how quickly the
control system corrects for errors before updating the drive level. It can take any value between
1 and 20. A compression factor of 1 represents an immediate correction of the control loop
transfer function while the higher compression factor means that more account of previous
levels will be taken resulting in a smoother, slower but more stable control system.
Figure 3.15 Sine control
4. Edit Reference Profile
Figure 3.16 shows the specified acceleration level in g. The software calculates the maximum
force, velocity and displacement respective to the shakers specifications. If the control system
fails for any reason, the profile manages the alarm or abort points in which the shaker stops
working with a warning making the user has to fix the problem before running it again.
58
Figure 3.16 Edit sweep profile
5. Self-Check
Self-check worksheet (figure 3.17) enables a quick check on the overall status of the test
configuration. Drive signal can be changed manually through the amplifier or by adapting it in
this worksheet. The software measures the sensors output voltage and in the event of high level
of excitation force, the input status indicates an overload for any channel exceeding the channel
limits. Global status also indicates that whether the driving signal is sufficient or not. After a
successful check, procedure is directed to the next tab.
59
Figure 3.17 Self-check
6. Sine Control
Figure 3.18 is a snapshot of the sine control worksheet. This worksheet performs a complete
sweep according to “sine setup” settings. By arming the run and loading all settings, the shaker
starts to operate. Sweep rate and compression factor can be changed in any instance during the
run.
Figure 3.18 sine control
60
7. Dwell Setup
Figure 3.19 shows the dwell setup worksheet. The main frequency is selected to start the dwell.
In the data source drop down menu the signal which is used to identify the input channels
spectrum is being selected. Since the first bending mode is of interest, number of resonances is
set to “1” and the peaks can be selected manually or by the software. Amplitude threshold and
gate values can be adapted in a way to precisely locate the cursor on the peak value. The X axis
can show either time or number of cycles.
Figure 3.19 Dwell setup
8. Dwell Control
The configurations in figure 3.20 are similar to the sine control worksheet. First, the dwell run
is armed to start the shaker. Shaker starts to sweep up to the main frequency. Control algorithm
can be set to track the amplitude or phase. “Progress” shows the number of cycles completed
so far. Controller starts to track the phase/amplitude. 1% frequency shift is required for this
test. Thus, the test terminates when the frequency exceeds the 1% limit.
61
Figure 3.20 Dwell control
1.11 3.3 Test Results
To obtain the fatigue endurance limit of tapered plaques, the test was conducted at 35g force at
the resonance frequency. Each plaque was tested to find the stress level vs. completed cycles.
The test was conducted by sine excitation and dwelling at the first bending mode frequency as
long as the frequency shift did not exceeded 1%. Hence, the test comprises these standard
procedures:
Search for resonance frequencies.
Dwell at detected frequency for specific length of time until the part is fatigued
(frequency shift)
Resonance frequency makes use of the fact that under resonance, the input and response
channel signals are out of phase by 180 degrees in the case that there is no damping in the
system. This relationship was used as a basis to develop the feedback control system to enable
the phase tracking control to perform the test. In this work, the plaques are subjected to 35g
sine excitation force for 1% frequency shift. However, 2% frequency shift was also
investigated for the first plaque.
62
Plaque 1
Figure 3.21 shows the first resonance frequency obtained from the sweep test. Therefore, 393.8
Hz is the resonance frequency which the plaque is settled to dwell at until the material starts to
fatigue resulting in crack initiation and propagation. The aborting criterion is set to be 390Hz
and phase dwell tracking control loop is chosen to be our dwell mode to determine the
resonance frequency during the test.
Figure 3.21 Resonance frequency and the frequency shift of the first plaque
Figure 3.22 also shows the strain measurements during the dwell test. Using this data, it is not
necessary to use the correlation plots since the strain gages endured throughout the fatigue
testing.
63
Figure 3.22 Strain measurements of the first plaque
It can be seen in figure 3.21 that the dwelling control system has terminated the test when a
substantial frequency drop from the previous measurements is observed. A frequency shift of
1% indicated that the test should be terminated at 390 Hz at which point the number of cycles
before failure was recorded.
Plaque 2
The second plaque was tested same as the first one. The following figures (figures 3.23 and
3.24) show the results obtained from this test.
64
Figure 3.23 Resonance frequency and the frequency shift of the second plaque
The above figure depicts the velocity signal measured by the vibrometer during the test. It is
important to note that the signal quality should be constantly monitored and inspected on the
sensor head. This indicator denotes how well the signal is reflected from the laser. Figure 3.24
shows the maximum strain value for the dwell test on the second plaque. Since the first sample
was strain gaged at five different locations and the point with the maximum strain value was
determined, only a single strain gage was used for the second sample to measure the strain
level at the center point. Maximum strain and corresponding stress could be calculated via
these measurements.
65
Figure 3.24 Maximum strain measurements of the center point gage
Two samples were tested in the same condition and the test controller was set to abort the test
when 1% frequency drop was monitored. For further investigation, 2% frequency drop was
also inspected.
This frequency shift implies a fatigue failure or crack initiation within the plaques surface.
Figure 3.25 clearly shows the frequency shift phenomenon and the crack nucleation effect on
the response after the failure takes place. At the resonance frequency before failure, the
response lagged the excitation force 174 degrees which drops to 138 degrees after the failure
occurred. Fatigue crack development in the material results in stiffness degradation in the
testing object while it was excited under fully reversed bending mode. The dynamic response
of the plaque changes as a result of the fatigue cracks and the shaker is re-tuned to the
specimen resonance frequency to maintain the maximum stress level. Throughout the vibration
test, changes in the material stiffness and damping ratio cause a slight increase in the amplitude
of vibration and a decrease in the phase difference as a result of crack formation.
66
Figure 3.25 Phase angle from sine sweep test
Table 3-1 summarizes the test results for both plaques.
Plaque ID Max Strain(µε) Max stress(MPa) Frequency shift% Number of
Cycles
Plaque 1
943 206 1%(394-390Hz)
2%(390-386Hz)
132473
235497
Plaque 2
851 187 1%(387-383Hz)
2%(383-379Hz)
175883
34132
Table 3- 1 Results summary
Experimental fatigue data were obtained from the resonance testing. Due to the high excitation
force at 35g, the plaques did not survive more than a few hundred thousand cycles.
Furthermore, the dissimilarities observed in the test results are believed to be due to flaws on
the plaque surface as well as the possibility of internal defects producing a local stress
concentration in critical areas. Measurement error is another source of disparity in the
results. Strain gage misalignment from the intended direction could result in strain
67
measurement errors. Another significant source of error arises when the plaque is mounted into
the fixture, as the boundary conditions applied by the fixture will have a significant impact on
the dynamic behaviour of the plaques. Therefore, placing the plaques exactly at the centre of
the fixture plays an important role in ensuring the accuracy of the test results. In addition to the
initial preparation, relative slippage of the plaque along the fixture during the vibration could
have changed the resonance frequency to some extent. These systematic errors are an inherent
part of the experiment even with each test being performed with the same instrumentation in
the same setup. Although the test was carried out with careful considerations, a subtle error can
lead to a significant variation in the results. Moreover, good practice in fatigue testing requires
a sufficient number of test pieces to provide a proper representation of the whole population.
Since the test included only two replications, this test needs to be conducted on more samples
in order to provide more reliable statistical data.
68
Chapter 4: Multi-Objective Shape
Optimization
In recent years, shape optimization of mechanical components due to its capability to improve
the quality of the products has been emerged. Practically, shape optimization is an endeavour
to enhance the performance and behaviour of the component by refining the geometrical
boundaries. The ultimate goal of mechanical design is to create components that meet the
application requirements such as operating lifetime, stress level and vibration behaviour.
However, it often leads to opposite designs. For instance, improving the structural strength
would cause an increase in the mass.
Since the shape or geometry of mechanical components directly affects their performance in
real life service, it is crucially important to take into account several considerations and aspects
of design process including cost, manufacturing process and mechanical performance. By
development of powerful numerical methods and progresses in modeling, finite element tools
have largely attracted the attention of so many manufacturer and designers. One of the primary
concerns of any finite element analysis is the computational cost required for convergence.
Typically, constrained optimization problems require large number of function evaluation.
Each function evaluation could be time-consuming and costly. Therefore, an efficient
optimization algorithm should be utilized to minimize the function evaluation time. Therefore,
it might be beneficial to perform a multi-objective optimization by using a surrogate model
during the optimization. This model due to the reduced degree of non-linearity requires
significantly less computational time.
Generally, surrogate modeling methods such as Artificial Neural Network (ANN) are powerful
tools in solving design optimization FE problems. ANN is capable of approximating complex
input-output relationships while displaying acceptable accuracy. This is done by generating a
set of training points and modeling them in FEM commercial package. The results (maximum
stress, displacement and Natural Frequency) are then used to train the network. Integration of
ANN with traditional costly approaches will result in inexpensive yet accurate results.
69
Realistic design problems in engineering involve several considerations such as cost, stability
and vibration. Reduction of vibration is attainable by separating natural frequencies from the
excitation frequency or rotor speed. This would avoid large displacement amplitude or
resonance that could damage the structure severely. The chosen design variables for this
problem were the coating thickness of the plaque in each side and more importantly, the
boundary condition and the plaque chord. The present work considers the optimization strategy
to place the resonance frequency within a certain range (370-410). Higher natural frequencies
are also measured to validate the numerical model with the real experimented plaques.
Additional objectives are also addressed in the proposed work. Maximizing the fatigue life and
minimizing the total mass are the two objectives of this problem. Hence, a multi-objective
approach would be a much more realistic approach to optimize the plaque shape. Moreover, it
is worth noting that improving a component based on one aspect usually deteriorates other
aspects. Therefore, employing the idea of Pareto optimality allows the designer to compare
several objectives with respect to each other and choose from according to the design priorities.
The proposed approach includes the following steps:
1. Creating the model and validation: The model is created in ANSYS and the
validation of the model is carried out. Design variables are chosen according to the
constraints of the problem. Initially, 200 random variables are generated within the
predefined range.
2. Numerical Analysis: The maximum stress and the natural frequency are evaluated in
ANSYS by a code developed for solving the harmonic and dynamic analysis.
3. Surrogate modeling/ function approximation: The features and all training points
generated in step one are used to train the network. This network is later used to
approximate the dynamic behaviour for the optimization process.
4. Multi-objective optimization: two distinct objectives are formulated to be used in
finding the Pareto optimal solution. This approach gives the designer the opportunity to
evaluate a set of viable solutions to choose the best solution that outperforms other
ones.
70
1.12 4.1 Finite Element Modeling and Validation
4.1.1 Geometry
A finite element model was performed in ANSYS. The stress analysis enables to achieve the
stress distribution and the maximum stress in the critical area by the generated force.
Numerical simulations were conducted for cyclic load of 35 g close to natural frequency.
First step towards creating the model is the geometry generation. The model is created by
tracing the cross section of the plaque and extruding the area along the out of plane axis.
Volumes, areas and lines are the basic entities used to build the geometry. The given plaque
consists of VICTREX PEEK with thin alloy coating. PEEK is a polyaryletherketones regarded
as one of the highest performing materials. PEEK polymer is capable of withstanding harsh
environments and they have replaced metals and traditional composites due to high strength,
corrosion and ease of fabrication. [68]
Multiple keypoints are used to draw the interconnecting lines to create the cross-sectional area
(Figure 4.1). Since the blade is uniform, the front face is extruded and the volume is generated.
Figure 4.2 shows the generated volume representing the plaque geometry.
On each side of the plaque, different thickness of coating is used. A two layer shell element
represents the alloy coating on the PEEK material. After the model is created, the plaque is
squeezed between the rigid bodies. For ideally representing the boundary conditions, four
contacts are specified to model the contact between the pins and the plaque clamped region.
The contact regions are completely constrained in all degrees of freedom while the vertical
movement of the plaque is allowed. Therefore, the plaque is not allowed to slide in between the
rigid bodies which represent the pins on the fixture.
Figure 4.1 Tapered plaques cross section
71
Figure 4.2 Tapered plaque model
Solid187 element was used to mesh the geometry. Solid187 is a 10-node 3-D element with
quadratic displacement behaviour suitable for modeling irregular shapes.
Shell281 element was also used to model the layered coating. This 9-node element is suitable
for analyzing thin shell structures with large strains and large rotation. As with all finite
element models, the density of the mesh plays an important role in stress prediction and mesh
refinement study needs to be conducted. Typically, in mesh refinement studies, the result of the
finite element model such as stress or strain is compared to previous iteration of element size.
The model is deemed to be refined and accurate when the changes are minor. Therefore, a
mesh convergence study was performed which resulted in element size of approximately
0.00108 and 0.0008 for the solid elements and shell elements respectively. In the following
tables, list of material properties are presented.
Mechanical Properties Value
Density 1.44 g/cc
Tensile Strength, Yield 330 MPa
Poisson's Ratio 0.4
Young's Modulus 45 GPa
Table 4- 1 Mechanical properties of PEEK [68]
72
Mechanical Properties Value
Density 8.89 g/cm3
Poisson's Ratio 0.305
Shear Modulus 72 GPa
Tensile Strength 345 MPa
Young's Modulus 190 GPa
Table 4- 2 Mechanical properties of coating [72]
4.1.2 Modal Analysis
Modal analysis is used to determine the vibration characteristics such as natural frequencies
and mode shapes of the structure. The natural frequencies and mode shapes are important
parameters in the design of components. There are several mode extraction methods: Block
Lanczos, Power Dynamics, Reduced and damped method. The Block Lanczos method is used
due to its capability to faster convergence rate and also it is typically used in complex models
with the mixture of solid/shell elements. Hence, this method is recommended for most
applications. Table 4-3 briefly summarizes the solution settings.
Element Type For The Peek Solid 187
Element Type For The Coating Shell 281
Number Of Elements For The
Peek
198050
Number Of Elements For The
Coating
53289
Problem Dimensionality 3-D
Analysis Type Modal
Extraction Method Block Lanczos
Equation Solver Option Sparse
Number Of Modes To Extract 3
Table 4-3 FEM solution settings
73
Impact Hammer Modal Testing
An ideal impact is a perfect impulse in an infinitely small duration, producing a sharp rise of
force in the frequency domain. This causes the modes of vibration to be excited. A load cell is
attached to the tip of a hammer to measure the applied force. A measuring sensor
(accelerometer) is attached to the plaque surface to record the response of the impact (figure
4.3). Frequency response function (FRF) is then measured to find the mode shapes and the
corresponding frequencies. The FRF describes the relation between the input and output of a
structure as a function of frequency. In table 4-4, the results from the impact testing and FEM
modeling are compared.
Figure 4.3 Impact testing [69]
Mode
number
Frequency(Hz) Frequency(Hz)
(ANSYS)
1 397 401
2 1024 1104
3 1483 1493
Table 4-4 Finite element model validation
The good agreement between the finite element method and the tap testing results confirms
that the model is a reliable representative of the real testing setup. The most important yet
74
challenging part of the simulation is the boundary conditions modeling. As it was indicated in
table 4-3, No Separation contact type was chosen for the contact region. Several contact types
were also tested. However, this form of contact was found to be more reliable and accurate
compared to other types. Figures 4.4, 4.5, 4.6 depict the mode shapes extracted from the FEM
modeling.
Figure 4.4 First bending mode FEM Model
Figure 4.5 Third bending mode Figure 4.6 Second bending mode
75
Sensitivity analysis plays an important role in the design process. It is required to observe how
sensitive the objectives are to a small change in design variables. Referring to figure 4.1,
changing the coating thickness up to on the upper and lower surfaces altered the
resonance frequency about 1.5% (6Hz) and an increase of 1% (3 Hz) was observed as a result
of the same amount of change on the left-side coating thickness. The right-side coating
thickness did not show to have a considerable influence on the modal analysis response. The
boundary condition location or the clamped area had a significant impact on the natural
frequency. According to the simulation results, a change of 0.2 inch in this parameter shifted
the frequency over 10Hz. Based on the sensitivity analysis conducted on the model and
evaluating the component’s behavior, the plaque is modeled using six variables for the coating
thickness (two-layer coating); one variable for the chord length and a single variable
representing the boundary condition location. The first step towards the optimization approach
is generating random training points within the defined range. Considering the design
constraints, a range of was adopted for the coating thickness, for the chord
length and for the boundary condition location. ANSYS code was developed to generate
random inputs within the specified ranges and creating the model in each iteration. The modal
analysis is initially performed and the resonance frequency is saved in a separate file. Once the
modal analysis simulation is completed, dynamic harmonic analysis is implemented to excite
the structure at the resonance frequency. Fatigue life and dynamic strength are considered as
the two main factors influencing the structural performance. After applying the load, the
response of the model is tracked and the Full Method is selected to calculate the maximum
stress in the system. Since this method generally allows all types of nonlinearities, it is selected
as the processing solver. All the stress and displacement data are then used to feed the neural
network. The displacement amplitudes are also used to add constraints on the optimization
problem. According to the fixture design, the plaque maximum displacement is limited to two
inches. As a result, three constraints can be defined for the current problem. As it was
mentioned earlier, frequency is limited to 370-410Hz according to the in service operating
frequency and resonance phenomenon. The stress level has to be kept below the yield stress at
all times. Third, displacement amplitude is also restricted to 2 inches in the worst case
scenario. Figure 4.7 shows the contour for the displacement during dynamic excitation.
76
Figure 4.7 Displacement contour
1.13 4.2 Surrogate Modeling
High computational cost and complexity of the model essentially require an alternative
technique to reduce the costly function and constraints evaluations. Surrogate approximations
could be extremely helpful and an effective strategy to asses large number of functions in a
short time.
In this work, three multi-layer perceptron networks were used to approximate frequency,
maximum stress and maximum displacement of the structure. Later, the Von-Mises equivalent
stress is used to predict the life of the plaques. The input or training points of each network
consist of 16 variables representing the design input vector, and a single output layer. 200
samples are generated to train the network by Levenberg-Marquardt method.
77
Figure 4.8 plots the statistical measures of performance of the neural networks.
4.3 Multi-Objective Optimization
The two objectives of the problem were to minimize the component’s mass and maximize the
fatigue life while limiting the frequency range. The mass is easily evaluated by multiplying the
materials density by the volume and the resonance frequency is obtained from Modal analysis.
Several researchers have examined the effect of coating on the mechanical fatigue behaviour of
the materials. Coatings have shown to have significantly decreased the fatigue life. The cracks
open up at the surface and run through the interface and the substrate.
It is assumed that the total life is divided into two stages: crack initiation and crack
propagation. Under high cycle fatigue, fatigue behaviour where the significant controlling
parameter is elastic stress or strain, the stress life relationship can be expressed by the Basquin
relation: [70]
(2 )b
a f fN (4.1)
For the special case where the mean stress is zero , the notation is employed for
the stress amplitude. Such condition is called fully reversed cyclic loading where The
Figure 4.8 Plot of observed and predicted regression approaches. Performance of Neural network of the models: Frequency
(right), stress (middle) and displacement (left)
78
fatigue life could be estimated using Smith, Watson and Topper (SWT) equation which
assumes that the stress life curves follow a power relationship. [71]
maxar a (4.2)
Thus, the initiation life due to HCF cycles is obtained from equation 4.1 by replacing
with .
1max
(2 )
1 1( )
2 2
b
ar f f
bf
f
N
RN
(4.3)
Where fN is cycles to failure, and the stress variable is ar , and f (fatigue strength
coefficient) and (fatigue strength exponent) are determined from the zero mean stress tests.
is the stress ratio. The coating material used here has a f of 500 MPa with fatigue
strength exponent . [72]
SWT criterion is one of the most often used methods based on mean stress effect on fatigue
behaviour. This equation is generally applicable to cases of long lives where stresses generate
elastic strain amplitudes. Various models have been developed to account for mean stress
effect on fatigue life of materials. Earlier approaches were used to for correcting the fatigue
limit in the high cycle fatigue regime. SWT was found to be superior to other approaches and
slightly conservative compared to Goodman [73] or Morrow [74] correction model.
Additionally, it is consistently giving excellent correlation for nonferrous materials. [71]
Fatigue Life Prediction (Stress-Based Criteria)
Equivalent Stress Approaches
The most popular approach to fatigue analysis is maximum principal stress theory, Tresca
Theory (maximum shear stress theory) and Von Mises theory (Octahedral shear stress theory)
that can be computed by:
1maximum principal stress theory qa aS S (4.4)
79
1 3maximum shear stress theory qa a aS S S (4.5)
maximum shear stress theory
2 2 2
1 2 2 3 3 1
1
2qa a a a a a aS S S S S S S (4.6)
Where are the principal alternating stresses. The Von Mises criterion is the
most common one and it is widely used for fatigue life prediction for ductile materials and the
maximum principal stress criterion is better for brittle materials [75]. Therefore, Von Mises
criterion was used in order to estimate the fatigue life of the tapered plaques.
Sine Method
This method uses alternating octahedral shear stress for cyclic stresses and the hydrostatic
stress for the mean stress. It can be presented by:
2 2 2
1 2 2 3 3 1 2a a a a a a mx my mz NfS S S S S S m S S S S (4.7)
Where is the coefficient of mean stress influence and is the uniaxial fully reversed
fatigue strength. The coefficient can be calculated experimentally under zero mean stress
level of stress. This method should be used for those cases where the alternative stress does not
change relative to proportional stress. [75]
Dan Van Criterion
If is the fatigue limit and is the fatigue limit is shear stress, is the hydrostatic stress,
is the principal stress, the criterion is formulated as: [75]
I,Jmax
1
2
3
)
2
(I J
ff
f Hf
tmax
t
t
(4.8)
80
Figure 4.9 Dang Van diagram criterion [75]
The implementation of the Pareto optimal solution technique requires a multi-objective
optimization tool. The optimization strategy used in this work is NSGA-II due to its efficiency
and accuracy for engineering problems. It is a powerful tool for searching through global
minimal areas and a quit fast convergence to the solution.
1.14 4.4 Fast and Elitist Multi-Objective Genetic Algorithm:
NSGA_II
The non-dominated sorting genetic algorithm proposed by Srinivas and Deb [76] was one of
the first evolutionary algorithms. High computational complexity of (Where M is the
number of objectives and N is the population size) makes NSGA computationally expensive.
In addition, lack of elitism is another disadvantage of NSGA where it can speed up the
convergence significantly and prevents from loss of good solutions. NSGA_II was developed
as an improved NSGA and it is discussed by some researchers that this method outperforms
several other approaches.
4.4.1 Non-Dominated Sorting Genetic Algorithm
First, an initial set of solutions called population is randomly generated and GA operators are
applied to the solution to create the next generation. After certain number of generations, the
final set of solution is obtained. By performing the Pareto non-dominated sort, the Pareto or
trade-off set of solutions are identified where no solution is strictly better than other solutions.
Solution x dominates solution y in a minimization problem with m objectives if
81
: : i i j jx y i f x f y and j f x f y
The above definition means that for all objectives associated with x, smaller or equal to
objectives associated with y, there exists at least one solution x that is smaller than the one for
y. If the solutions are not dominating each other, it is said to be non-dominated solutions. A set
of such solutions form the Pareto front. Each front is assigned a unique number and in the case
of minimization, a front with higher rank has a smaller front number which means that the
solutions with higher ranks have higher preference of selection. [77] The GA operators are
applied to the initial population to create the children. The current population is used to
generate the non-dominated fronts. Therefore, the population of non-dominate sorting is twice
the size of parent population. The population for the next generation are selection from non-
dominant fronts according to the assigned ranks. Solutions within a front are sorted by
crowding distance and only a portion of the last front can be selected for the next generation.
Figure 4.10 schematically illustrates how the NSGA_II algorithm works.
Figure 4.10 NSGA-II procedure [77]
For All solutions in the first non-dominated front the non-dominated count would be set to zero
( ). Now, for all solutions with zero non dominated count, each member of the
domination count will be reduced. To form the second front, each member q with zero count is
placed in a separate list Q. [53]
82
4.4.2 Crowding Distance
As it was mentioned earlier, to obtain a good spread of solutions, NSGA_II uses a different
approach from the well-known NSGA due to the difficulties with the sharing function
approach. [77] To get a good estimate about the density of a certain solution, the distance of
two points along each of the objectives is calculated. serves as a parameter to
represent the crowding distance of the solution. This crowding distance sorting requires
sorting the whole population in ascending order and assigns an infinite distance value to the
smallest and largest objective function value. All other solutions located in between the
boundary solutions are assigned a value equal to the absolute value of the two adjacent
solutions. Here, the outline of this procedure is shown at the bottom of the page. [53]
83
Figure 4.11 Crowding distance [53]
distance
distance tan
Crowding distanceassignment
| | number of solutions in I
for each i,set [ ] 0 initializedistance
for each objective m
( , ) sort using each objective value
[ ] [ ]dis ce
l I
I i
I sort I m
I l I l
for i
distance distance m
2 ( 1)
[ ] [ ] ( [ 1] [ 1] )m
to l
I i I i I i I i
[ ] refers to the objective function of the solution in the set All the solutions can
be compared with other solutions by the assigned distance metric. In some sense, more
crowded solutions take a higher value of distance measure. Among some solutions in the same
front (same rank) it is preferred to select the one in a lesser crowded region. [53]
4.4.3 Main Loop
In the following procedure, the minimization of the objectives is assumed. Thus, the best front
will be assigned the rank 1 and the next levels incrementally are ranked. First, the parents are
used to create the offspring. Mutation, crossover and tournament selection are applied to do so.
NSGA_II is a simple and straightforward algorithm. First a combined population
of size 2N is formed and it is then sorted by non-dominate sorting. The solutions from the first
set are of the best solutions. Therefore, if is smaller than N, all the members are directly
moved to the next generation. The remaining members of are chosen from the best
solution from the subsequent non-dominate solutions. All the population slots are
accommodated until there is no more space for any members. In other words, assume the size
of the population from is larger than N, the individuals from last front are sorted using
84
crowding distance operator. The member with higher rank (better solutions) is chosen to fill the
remaining slots. The new population is used to create the new offspring . [53]
The main loop procedure is as follows: [53]
1
1 1
1
1 1
min (R ) min
| P |
Calculatecrowding distance
( , )
[0 : ]
t t t
t t
t
t t i
t
t t
R P Q Combine parent and children population
F fast nondo ated sort all nondo ated front of R
until N
P P F
Sort P n sort in desending order
P P N choosethe fi
1
1 1
1
( ) ,
1
t
t t
t
srt N elements of P
Q make new population P use selection mutation and crossover
t t a new populationQ
The current problem could be formulated as:
Minimize (Mass) & Maximize (Fatigue life)
Subjected to: Maximum stress Yield stress & 370 Resonance frequency 410
The maximum stress has to be remained below the yield stress keeping the component in the
elastic region.
NSGA_II Solution setting:
Population: 200
Mutation rate: 0.1
Crossover: 0.9
85
Figure 4.12 First Pareto Front
The optimization problem was solved using the proposed method. The Pareto set is presented
in figure 4.12. The x axis shows the total mass of the component and the y axis is the inverse of
fatigue life. Hence, as it is shown in figure above, as the mass is increased so does the fatigue
life. Each data point in this figure corresponds to a specific design configuration. Therefore,
there is no particular point which could be denoted as the optimal solution. The original model
had a fatigue life of cycles with 0.048Kg of mass. Theoretically, this method was able
to enhance the fatigue performance to cycles with the same weight. Focusing on the
distribution of non-dominated solutions in figure 4.12, Pareto solutions do not necessarily
correspond to the best design variables. However, there should be a compromise between all
objective functions and choose the one which outperforms other solutions according to the
relative importance of each objective function. This approach in some sense provides the
designer multiple alternative opportunities to choose from in order to meet the design criteria.
86
Chapter 5: Summary and Conclusion
A vibration based testing methodology based on the sine dwell technique was implemented to
investigate the high cycle fatigue failure of tapered plaques. A complete testing procedure was
prepared along with the set-up of high level instrumentation used to conduct the testing. The
first step conducted was a calibration test performed in order to obtain the strain-displacement
relationship at key locations on the test piece. Following the calibration, the tapered plaques
underwent a sine dwell fatigue test until crack initiation was observed. The use of such a
method for determining the fatigue behaviour allowed for the point at which failure occurred to
be identified while the crack was still in the initiation stage. It was shown that the vibration
based method to fatigue testing was capable of generating the required stresses for a fully
reversed bending mode at high frequencies. The testing performed also showed the potential to
reduce both test time and overall experimental costs in the design of new structures.
Furthermore, a finite element model was generated to validate the experimental results. Despite
the simulation complexity, good agreement between the simulation and experimental results
was observed. In addition to finite element analysis, an integrated Neuro-Genetic multi-
objective shape optimization approach was used to enhance the plaque performance. A set of
Pareto non-dominated solutions were presented allowing the designer to choose between
various design states. This approach has been proven to be applicable to many components due
to its fast convergence and flexibility. The results were compared with reference geometry and
considerable reduction in blade mass and vibration stability was observed.
Turbine blade design is highly interdisciplinary due to the interaction between aerodynamic
and structural forces. Aerodynamic and heat transfer design criteria should be integrated along
with various constraints on the geometry. Therefore, a multidisciplinary design optimization
technique would be a good practice involving several disciplines in the design process.
Overall, it was found that the proposed high cycle fatigue testing method was successful in
generating accurate fatigue data for the tapered plaques tested. Future testing may look into the
fatigue behaviour of other structures as the testing method outlined could be widely adapted to
a number of applications most notably where high cycle fatigue is a significant issue such as
turbine blades.
87
This study has been the first step to develop a methodology to estimate the fatigue endurance
limit of mechanical components. The focus of this work has been on turbine blades and jet
engine turbine components. The effect of aerodynamic forces in a high temperature
environment needs to be studied as well. All parts that have been tested were excited at the first
bending mode with zero mean stress. The effect of mean stress is an important factor in fatigue
life. Moreover, the second or third bending modes could be also investigated since the second
resonance frequency mode is also located within the operating range of the engines. In
addition, the stress-life curve could be presented by incorporating a step-testing method
through the proposed fatigue testing procedure. T. J. George [17] has used the similar setup to
develop the fatigue limit strength of a Ti-6Al-4V plate. Using this setup, high cycle fatigue
testing data could be generated in a short time contrary to the traditional fatigue test machines.
The solution obtained from the Pareto front was based on the finite element simulation, thus it
is important to modify the FEM model. Mesh type and density are the most important elements
in numerical approaches. Therefore, using different element types with various sizes could be a
good practice in further developing the finite element model.
As it was discussed earlier, the surface approximation and neural network is significantly
governed by the number of neurons used to form the network. As a result, a study needs to be
conducted to find the suitable number of neuron in each network.
88
References
[1] J. Schijve, "Fatigue of structures and materials", Springer, 2009.
[2] A. Blom, A. Hedlund, W. Zhao, "Short fatigue crack growth in Al 2024 and Al 7475. Behaviour of
Short Fatigue Cracks," MEP (1986), pp. 37–66, September 1985.
[3] P. Forrest, "Fatigue of Metals," London, Pergamon Press, 1962.
[4] R. l. Stephens, A. Fatemi, R. R. Stephens, H. O. Fuchs, "Fatigue Tests and The Stress-Life
Approach," in Metal Fatigue in Engineering, 2nd Edition, John Wiley & Sons, INC., 2001, pp. 59-93.
[5] T. Nicholas, "High Cycle Fatigue, A Mechanics of Materials Perspective," Elsevier Ltd., 2006.
[6] T. Adams, "Cumulative Damage of a Reported High Frequency Vibration Loading," University of
Iowa, 1991.
[7] H. F. Moore, H. B. Wishart, "An ‘Overnight’ Test for Determining Endurance Limit," ASTM, pp.
334–347, 1933.
[8] H. J. Gough, "The Fatigue of Metals," Ernst Benn Limited, no. London, 1926.
[9] M. Prot, "Fatigue Tests Under Progressive Load.A New Technique for Testing Materials," Rev.
Métall, vol. 45, no. 12, pp. 481-489, 1948.
[10] S. D. Ward, "An Investigation of the Prot Accelerated Fatgue Test," ASTM, vol. 53, pp. 885-891,
1953.
[11] M. Hempel, "Performance of Steel under Repeated Loading," Fatigue in Aircraft Structures, Vol.
A. M. Freudenthal, Academic press, pp. 83-103, 1956.
[12] T. Nicholas, D. C. Maxwell, "Evolution and Effects of Damage in Ti-6Al-4V under High Cycle
Fatigue, "in Eighth International Conference on the Mechanical Behaviour of Materials, pp. 1161–
1166, 1999.
[13] R. Mehl, J. Ransom, "The Statistical Nature of the Endurance Limit," Metals Trans, vol. 185, pp.
364–365, 1949.
[14] A. M. Mood, A. J. Dixon, "A Method for Obtaining and Analyzing Sensitivity Data," J. Amer.
Stat. Assn., vol. 43, pp. 109-126, 1948.
89
[15] H. Q. Xue, H. Tao, F. Montembault, Q.Y. Wang, C. Bathias, "Development of a three-point
bending fatigue testing methodology at 20 kHz frequency," International Journal of Fatigue, pp. 2085-
2093, 2007.
[16] A. Herzog, "Six ton Schenck Fatigue Testing Machine," Tech. Rep. U.S. Army Air Force, 1947.
[17] T. J. George, J. Seidt, M.-H. Herman Shen, T. Nicholas, C. J. Cross, "Development of a novel
vibration-based fatigue testing methodology," International Journal of Fatigue, pp.477-486, 2004.
[18] Y. C. Wang, T. Hoechbauer, J. G. Swadener, A. Misra, R. G. Hoagland, M. Nastasi, "Mechanical
fatigue measurement via a vibrating cantilever beam for self-supported thin solid films," Experimental
Mechanics, pp. 503-517, 2006.
[19] C.-Y Kim, J.-H Song, D.-Y Lee, "Development of a fatigue testing system for thin films,"
International Journal of Fatigue, pp. 736-742, 2009.
[20] S. Vanlanduit, E. Parloo, P. Guillaume, "On-line detection of fatigue cracksusing an automatic
mode tracking technique," Journal of Sound and Vibration, pp. 805–814, 2003.
[21] F. T. Joaquim, R. Barbieri, N. Barbieri, "Investigating torsional fatigue with a novel resonant
testing fixture," International Journal of Fatigue, pp. 1271-1277, 2009.
[22] Ş. Özsoy, M. Celik, F. Suat Kadıog˘lu, "An accelerated life test approach for aerospace structural
components" Engineering Failure Analysis, pp. 946-957, 2008.
[23] H. Y. Kim , T. R. Marrero , H. K. Yasuda , O. A. Pringle, "A simple multi-specimen apparatus
for fixed stress fatigue testing" Journal of Biomedical Materials Research, pp. 297-300, 1999.
[24] I. Ay, R. Sakin, G. Okoldan, "An improved design of apparatus for multi-specimen bending
fatigue and fatigue behaviour for laminated composites," Materials & Design, pp. 397-402, 2008.
[25] S.-E Onome, H. shen, T. George, C. J. Charles, J. Calcaterra, "Development of an Improved High
Cycle Fatigue Criterion," Eng. Gas Turbines Power , pp. 162-169 , Mar 01, 2004.
[26] A. Rotem, "Accelerated fatigue testing method," International Journal of Fatigue, pp. 211-215,
1981.
[27] M. Panis, C. Bordes, M. Fressinet, "Accelerated fatigue testing on hydraulic shaker," ICAF
symposium, 2009.
90
[28] G. Yun, A. B. M Abdullah, W. Binienda, "Development of a Closed-Loop High-Cycle Resonant
Fatigue Testing System," Experimental Mechanics, pp. 275–288, 2012.
[29] T. P. Athan, "A note on weighted criteria methods for compromise solutions in multi-objective
optimization," Eng. Optim, 1996.
[30] I. Das, J. Dennis, "A closer look at drawbacks of minimizing weighted sums of objectives for
Pareto set generation in multicriteria optimization problems," Structural Optimization, 1997.
[31] D. J. Stadler, "Multicriteria optimization in engineering a tutorial and survey," American Institute
of Aeronautics and Astronautics, Washington, DC, 1992.
[32] W. Standler, "Caveats and boons of multicriteria optimization," Microcomput Civ Eng, 1995.
[33] J. Galuski, C. L. Bloebaum, C. H. Huang, "Multi-objective Pareto concurrent subspace
optimization for multidisciplinary design," AIAA, 2007.
[34] H. Voogd, "Multicriteria Evaluation for Urban and Regional Planning," twenty-third European
congress of the regional science association, 1983.
[35] R. Eckenrode, "Weighting multiple criteria," Manage. Sci, 1965.
[36] T. L. Saaty, "A scaling method for priorities in hierarchies, multiple objectives and fuzzy sets,"
Journal of Math Psychol, 1997.
[37] S. R. Koski, "Norm methods and partial weighting in multicriterion optimization of structures,"
Int J Numer Methods Eng, 1987.
[38] K. Proos, G. Steven, O. Querin, Y. Xie, "Multicriterion evolutionary structural optimization using
the weighted and the global criterion methods" AIAA, 2001.
[39] R. Marler, J. S. Arora, "The weighted sum method for multi-objective optimization: new insights,
"Struct Multidisc Optim, 2009.
[40] R. Marler, J. Arora, "Survey of multi-objective optimization methods for engineering," Struct
Multidisc Optim, 2004.
[41] G. Repko, "Multi-criteria optimization," Soviet Applied Mechanics, 1978.
91
[42] D. W. Coit, A. E. Smith. A. Konak, "Multi-objective optimization using genetic algorithm: A
tutorial," Reliability Engineering & System Safety, 2006.
[43] J. H. Holland, "Adaptation in natural and artificial systems," University of Michigan Press, 1975.
[44] J. D. Schaffer, "Multiple objective optimization with vector evaluated genetic algorithms," in
international conference on genetic algorithm and their applications, 1985.
[45] F. P. Fonseca, "Multiobjective genetic algorithms," IEE colloquium on ‘Genetic Algorithms for
Control Systems Engineering, 1993.
[46] N. Nafpliotis, D. E. Goldberg, J. Horn, "A niched Pareto genetic algorithm for multiobjective
optimization," in evolutionary computatio, Orlando, 1994.
[47] C. Y. Lin, P. Hajela, "Genetic search strategies in multicriterion optimal design," Struct
Optimization, 1992.
[48] K. Deb, N. Srinivas, "Multiobjective optimization using nondominated sorting in genetic
algorithms," Journal of Evol Comput, pp. 221-248, 1994.
[49] L. Thiele, E. Zitzler, "Multiobjective evolutionary algorithms : a comparative case study and the
strength Pareto approach," IEEE Trans Evol Comput, pp. 257–271, 1999.
[50] M. Laumanns, L. Thiele, E. Zitzler, "SPEA2: improving the strength areto evolutionary
algorithm," Zurich, 2001.
[51] C. D. Knowles JD, "Approximating the nondominated front using the Pareto archived evolution
strategy," Evol Comput, pp. 149-172, 2000.
[52] J. D. Knowles, M.J. Oates, D.W. Corne, "The Pareto envelope-based selection algorithm for
multiobjective optimization," in sixth international conference on parallel problem solving from Nature,
Paris, 2000.
[53] K. Deb, Kanpur, S. Agarwal, T. Meyarivan, "A fast and elitist multiobjective genetic algorithm:
NSGA-II," IEEE Trans Evol Comput, pp. 182-197, 2002.
[54] R. Sarker, K.-H Liang, C. Newton, "A new multiobjective evolutionary algorithm.," Eur J Oper
Res, pp. 12-23, 2002.
92
[55] G. G. Yen, H. Lu, "Rank-density-based multiobjective genetic algorithm and benchmark test
function study," IEEE Trans Evol Comput, pp. 325-343, 2003.
[56] G. G. Yen, H. Lu, "Dynamic multiobjective evolutionary algorithm:adaptive cell-based rank and
density estimation," IEEE Trans Evol Comput, pp. 253-274, 2003.
[57] C. Y. Lin, P Hajela, "Genetic search strategies in multicriterion optimal design," Struct
Optimization, pp. 99-107, 1992.
[58] D. Goldberg, J. H. Holland, "Genetic algorithms in search, optimization, and machine learning,"
Addison-Wesley, 1989.
[59] J. N. Morse, "Reducing the size of the nondominated set: pruning by clustering," Comput Oper
Res, pp. 55-66, 1980.
[60] M. T. Ishibuchi, "Multi-objective genetic local search algorithm," in E international conference
on evolutionary computation, Nagoya, 1996.
[61] C. CAC, "A survey of constraint handling techniques used with evolutionary algorithms," in
Laboratorio Nacional de Informtica Avanzada, Veracruz, 1999.
[62] F. Jimenez, A. F. Gomez-Skarmeta, G. Sanchez, K. Deb, "An evolutionary algorithm for
constrained multi-objective optimization," in computational intelligence, Honolulu, 2002.
[63] N. Qin, M. Y. M. Ahmed, "Surrogate-Based Aerodynamic Design Optimization: Use of
Surrogates in Aerodynamic Design Optimization," in Aerospace Sciences & Aviation Technology,
Cairo, 2009.
[64] M. B. Howard Demuth, "Neural Network Toolbox for Use with MATLAB Version 4," by the
MathWorks Inc, Copyright 1992 - 2002 by the MathWorks, Inc.
[65] D. Kriesel, "A Brief Introduction to Neural Networks", Bonn, Germany: www.dkriesel.com,
2005.
[66] "OFV-5000 Manual," Polytec User Manual, Version 2G, 2012.
[67] "LMS SCADAS Mobile User Manual," LMS Instruments, Belgium, 2009.
[68] VICTREX plc, 2013. [Online]. Available:
http://www.victrex.com/en/industries/aerospace/aerospace.php.
93
[69] B. J. Schwarz, M. H. Richardson, "Experimental Modal Analysis", Vibrant Technology, Inc, CSI
Reliability Week, Orlando, FL, October 1999.
[70] S. Kwofie, "An exponential stress function for predicting fatigue strength and life due to mean
stresses," International Journal of Fatigue, vol. 23, no. Elsevier, pp. 829–836, 2001.
[71] N. E. Dowling, "Mean stress effects in stress-life and strain life fatigue," Society of Automotive
Engineers, Inc, 2004.
[72] DAIDO-SPECIAL METALS Ltd, 06 September 2006. [Online]. Available:
http://www.specialmetals.com/documents/Nickel%20200%20&%20201.pdf.
[73] J. Goodman, “Mechanics Applied to Engineering,” Vol. 1, 9th edition. Longmans Green and Co.,
London.
[74] J. Morrow, ”Fatigue properties of metals,” section 3.2. In: Fatigue Design Handbook, Pub. No.
AE-4. SAE, Warrendale, PA.
[75] R. l. Stephens, A. Fatemi, R. R. Stephens, H. O. Fuchs, "Multiaxial Stresses," in Metal Fatigue in
Engineering, 2nd Edition, John Wiley & Sons INC., 2001.
[76] N. Sirinivas, K. Deb, "Multiobjective function optimization using nondominated sorting genetic
algorithms," Evol. Comput, pp. 221–248, 1995.
[77] H. Fang, Q. Wang, Y.-C Tu , M. F. Horstemeyer, "An efficient non-dominated sorting method for
evolutionary algorithms," Evol Comput, pp.355-384, 2008.
[78] S. Vanlanduit, E. Parloo, P. Guillaume, "An on-line combined linear nonlinear fatigue crack
detection technique," Ndt & E Int, pp. 41-45, 2004.