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ANALYSIS OF AN ARCHERY BOW USING FINITE ELEMENT METHOD AND
THE DEVELOPMENT OF AN ARCERY BOW
SHARVEENESH A/L VATHIVELLU
Report submitted in partial fulfillment of the requirements for the awards of Bachelor of
Mechanical Engineering
Faculty of Mechanical Engineering
UNIVERSITI MALAYSIA PAHANG
JULY 2013
vii
ABSTRACT
In terms of Archer, it has been a well known sport worldwide. Basically, a bow consists
of 3 main parts which is the riser, limb and the string. In improving the performance of
the bow, it is crucial to research each and every part of the bow. There are many
mathematical modelling has been done to improve the performance of the bow but only
few analysis and experiments are done. Therefore, the objective of the study is to
analyse the Archery Bow using Finite Element Method and to develop the Archery
Bow. Two different methods were used before fabricating the bow which is to analyse
the stress and strain of the bow drawn in CAD software using FEM software and to test
experimentally using the strain gauge method. Validation between the results was used
as a reference for fabrication. The best material was chosen by doing a simulation
analysis on the composite materials which shows that E fibreglass have the best
properties compared to the other fibreglass. Therefore, as for the results, E fibreglass is
used to fabricate the limb of the bow so that it will be highly in strength and also in
elasticity which is up to 72.4 GPa. The results shows that the fabricated bow have a
lower strain which are 0.00244 compared to the current bow which is 0.004694. This
shows that the fabricated bow have a higher potential to withstand much higher force
compared to the current bow which makes the new bow to have a better quality. As for
the error analysis, it shows that the higher force applied to the bows makes the error to
increase to the increase in vibration and the damping force.
viii
ABSTRAK
Dari segi sukan memanah, ia telah menjadi salah satu sukan yang terkenal di seluruh
dunia. Pada asasnya, sebuah anak panah terdiri daripada 3 bahagian utama iaitu riser,
limb dan tali. Dalam meningkatkan prestasi pada sesebuah busur, ia adalah penting
untuk mengkaji untuk mengkajinya dengan keseluruhan. Terdapat banyak kajian telah
dijalankan dalam bentuk kiraan matematik tetapi hanya beberapa analisis dan
eksperimen telah dijalankan untuk menbuktikanya. Oleh itu, objektif kajian ini adalah
untuk menganalisa anak panah dengan menggunakan kaedah simulasi dan untuk
membuat sebuah anak panah. Dua kaedah yang berbeza telah digunakan sebelum
membuat panah tersebut yang terdiri daripada kaedah menganalisis tekanan pada anak
panah tesebut yang dilukis di perisian lukisan dan simulasinya menggunakan perisian
FEM dan untuk menguji kaji menggunakan kaedah tolok ketegangan. Pengesahan
antara keputusan yang telah digunakan sebagai rujukan untuk fabrikasi. Bahan yang
terbaik telah dipilih dengan melakukan analisis simulasi bahan-bahan komposit yang
menunjukkan bahawa gentian kaca E mempunyai ciri-ciri terbaik berbanding dengan
gentian kaca yang lain. Oleh itu, sebagai untuk keputusan, kaca gentian E digunakan
untuk membina anggota badan bagi anak panah tersebut supaya ia mempunyai kekuatan
dan juga keanjalan yang baik sehingga 72.4 GPa. Keputusan menunjukkan bahawa
busur direka mempunyai tekanan yang lebih rendah iaitu 0.00244 berbanding busur
semasa yang 0.004694. Ini menunjukkan bahawa busur yang baru direka mempunyai
potensi yang tinggi untuk menahan kuasa yang lebih tinggi berbanding dengan anak
panah semasa yang membuat anak panah baru untuk mempunyai kualiti yang lebih
baik. Bagi analisis ralat, ia menunjukkan bahawa kuasa yang lebih tinggi memjadikan
nilai ralat semakin meningkat kerana peningkatan dalam getaran dan daya ketegangan.
ix
TABLE OF CONTENTS
Page
EXAMINAR’S DECLARATION ii
SUPERVISOR'S DECLARATION iii
STUDENT'S DECLARATION iv
DEDICATION v
ACKNOWLEDGEMENTS vi
ABSTRACT vii
ABSTRAK viii
TABLE OF CONTENTS ix
LIST OF TABLES xiii
LIST OF FIGURES xiv
LIST OF SYMBOLS xvi
LIST OF ABBREVATIONS xvii
CHAPTER 1 INTRODUCTION
1.1 Project Background 1
1.2 Problem Statement 2
1.3 Objective 3
1.4
1.5
Scopes of Project
Organisation of the Thesis
3
3
CHAPTER 2 LITERATURE REVIEW
2.1 Characteristics of a Bow 4
2.2 Types of Bow 5
2.2.1 Longbow 6
2.2.2 Recurve bows
2.2.2.1 Non-recurve bow
2.2.2.2 Static-recurve bow
2.2.2.3 Working-recurve bow
6
7
8
9
x
2.2.3 Compound bows 10
2.3 Composite Materials 11
2.3.1 Composites 11
2.3.2 Characteristics of composite materials 11
2.3.3 Reinforcements
2.3.3.1 Fibreglass
2.3.3.1 Aluminium
12
12
14
2.4 Fundamental Theory and Equations 14
2.4.1 Engineering stress and strain 14
2.4.2 Modulus of elasticity 15
2.4.3 Poisson's ratio 15
2.4.4 Equation of elastic curve 15
2.4.5 Deflection at the limb tip 15
2.5 Mathematical Formulas Used For Bow Fabrication 16
2.6 Literature Review Summary 19
CHAPTER 3 METHODOLOGY
3.1 Cad Modelling 21
3.1.1 Overview 21
3.1.2 SolidWorks modelling steps 22
3.1.3 Archery bow riser modelling 23
3.1.4 Archery bow limb modelling
3.1.4.1 Fibreglass limb
3.1.4.2 Wood limb
24
24
26
3.2 Finite Element Method Analysis 27
3.2.1 Overview 27
3.2.2 Model meshing 28
3.2.3 Setting up the boundary conditions and parameters 31
3.3 Experimental Setup 32
3.3.1 Overview 32
3.3.2 Strain gauge set up 32
3.3.3 National Instrument - Data Acquisition set up 34
3.3.4 DASYlab software setup 35
3.4 Fabrication of the Archery Bow 38
3.4.1 Fabrication of Archery Riser 38
3.4.2 Fabrication of Archery Limb 40
3.5 Numerical Analysis 43
3.5.1 Calculation for archery limb 43
xi
3.5.2 Calculation for strain gauge
3.5.3 Galerkin method analysis
44
45
3.6 The Process Flow of the Study
53
CHAPTER 4 RESULTS AND DISCUSSION
4.1 Simulation result 55
4.1.1 Simulation result for Fibreglass limbs 55
4.1.2 Simulation result for Wood limbs 59
4.2 Experimental result 60
4.3 Comparison btween experimental and simulation result 64
4.4 Numerical calculation 67
4.4.1 Numerical validation using Galerkin analysis 67
4.4.2 Numerical Validation using Error Analysis 68
CHAPTER 5 CONCLUSION AND RECOMMENDATIONS
5.1 Conclusion 70
5.2 Recommendation 72
REFERENCES 73
APPENDICES 75
A
B
C
Solidworks Drawings and Simulation
Fabrication Flow
Gantt Chart for Final Year Project 1 & 2
76
82
87
xii
LIST OF TABLES
Table No. Title Page
2.1 Fiberglass and its properties 13
2.2 Terms and description of formula used 18
2.3 Literature review summary table 19
3.1 Mechanical properties of fiberglass 32
3.2 Strain gauge details 33
4.1 Data used for calculation of validation 68
4.2 Amount of error for current and fabricated limb for 50 N 69
4.3 Percentage error analysis for current and fabricated limb 69
xiii
LIST OF FIGURES
Figure No. Title Page
2.1 Main parts in an Archery bow
5
2.2 Non-recurve bow in three different situations
7
2.3 Static recurve bow in three different situations
8
2.4 Working recurve bow in three different situations
9
2.5 Compound bow
10
2.6
2.7
2.8
2.9
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
Free body diagram of a drawn bow
Free body diagram of a drawn bow limb
Free body diagram of a riser
Literature review summary in percentage
Modelling CAD geometry
2-Dimensional drawing of riser
3-Dimensional drawing of riser
Sketching of limb
2-Dimensional drawing of limb
3-Dimensional drawing of riser
3-Dimensional drawing of wood limb
Analysis in FEM
The effect of mesh size towards time of analysis and number of
errors
The different size and ways of meshing
Limb meshed model
Riser meshed model
16
17
17
20
22
23
24
25
25
25
26
27
29
29
30
31
xiv
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
3.26
Strain gauge wire placed on the limb
Location of strain gauge on the bow
Connecting strain gauge wires to NI-DAQmx
Ni-DAQmx
NI-DAQmx and DASYLab setup
Fabrication of Bow riser
Bow riser before surface finishing
Bow riser after surface finishing and varnishing
Chopped Strand Mat (CSM) fiberglass
Woven fiberglass
Layers of fiberglass pasted
The limb which is grinded to the desired size
Fabrication of Bow limb
The process flow of the study
33
33
34
35
36
38
39
39
40
40
41
41
42
53
4.1 Maximum strain versus force applied for fiberglass limb 55
4.2 Maximum stress versus force applied 56
4.3 Maximum deflection versus force applied 57
4.4 Maximum strain versus force applied for wood limb 59
4.5 Strain versus time for fiberglass limb 60
4.6 Strain versus time for wood limb
62
4.7 Strain versus time for fabricated E glass limb 63
4.8 Comparison graph of strain versus time for simulation result and
fiberglass limb
64
4.9 Comparison graph of strain versus time for simulation result and
fabricated limb
65
4.10 Percentage error graph for current and fabricated limb 69
xv
A
P
E
a
m
t
v
b
h
N
R
gS
LIST OF SYMBOLS
Cross sectional area
Pressure
Modulus of Elasticity
Acceleration
Mass
3-Dimensional drawing of riser
Engineering stress
Engineering strain
Change in length
Poisson's ratio
Width
Height
Newton
Difference in resistance
Ohm ( Unit for current )
Gauge factor
xvi
2D
3D
CAD
FEM
FEA
CSM
LIST OF ABBREVATIONS
2-Dimensional
3-Dimensional
Computer Aided Design
Finite Element Method
Finite Element Analysis
Chopped Strand Mat
CHAPTER 1
INTRODUCTION
1.1 PROJECT BACKGROUND
In the early days, the invention of bow plays an important role in the history of
mankind where it is mainly used for the purpose of war and hunting. Besides that, it is
also mentioned as a tool which is used to engage enemies even in a long distance. The
bow is a stringed projectile weapon designed to shoot a long and round sharp edged
arrow to a target. It consists of a slender stave and a cord fastened to it at each end
under a certain amount of tension. The string and the arrow are pulled back until the
limbs is bent and then released so that the impelling force of the bow string will shoot
the arrow. A bow can store energy as deformation energy in its elastic arms or limbs. Its
special feature is that this energy, delivered by the relatively slow human body, can be
quickly released to a light arrow in a very effective way. Where, the deformation energy
if easily transferred into kinetic energy which couples the bow and arrow. But in the
15th century, archery was superseded in battle fields due to the existence of firearms.
Today in the modern world, archery is mostly used as a competitive sport. It is
an important sport which is being played in the summer Olympics. Many countries have
taken the initiative to improve the archery equipments so that their athletes could
perform better in that tournament.
2
1.2 PROBLEM STATEMENT
It is true that, in terms of archery bows, there are many improvements that had
been made compared to ancient times such as improving the materials specifically the
increase in the usage of composites.
''The mechanical properties of the materials of these bows, however are much better
than those of the ancient bows. Indeed, the modern bow holds now the longest flight
shooting record" (B. W. Kooi, 1991).
"The use of composite materials has allowed this (aerodynamic drag) optimisation while
still retaining the required dynamic stiffness for archery performance" (M.Rieckmann,
J.Codrington and B.Cazzolato, 2011).
However, there are also many criteria's that need to be considered such as the
balancing and the efficiency of the limbs, the archer and the whole performance of the
recurve bow itself.
"It can be said very definitely that the smoothness of action and absence of kick in a
bow, depend primarily on two factors. The first is a dynamic balance of the limbs and
the second is that the bow be highly efficient" (B. W. Kooi, 1993).
Many mathematical calculations and experiments have been done regarding the
archery bow but only few analysis or computer simulation is done.
3
1.3 OBJECTIVE
To analyse an Archery bow using Finite Element Method and to fabricate the
Archery bow.
1.4 SCOPES OF PROJECT
i ) Limited to 3-dimensional (3D) modeling of the Archery bow.
ii) Fiberglass is used as the composite material.
iii) The weight of the bow is limited from 10 to 20 Newton.
iii) Type of Bow used is a Recurve Bow by right handed archers.
1.5 ORGANISATION OF THE THESIS
First chapter of the study shows the basic explanation about archery including
with the objective and the scopes of the study. Second chapter is where the literature
review is done with doing some comparison between the previous studies which has
been done. The third chapter tells the method which is used to conduct the study and the
fourth is the part where all the results are shows with discussion. As for the fifth and the
final chapter, the conclusion of the total study is shown with some recommendation to
improve the current study.
CHAPTER 2
LITERATURE REVIEW
In this chapter, basic knowledge which is related to archery bow will be
described. The common types of bow which can be found such as the traditional long
bow recurve bow and also the compound bow is described. Besides that, the material
which is widely used such as the composites materials has also been explained.
Moreover, the fundamental theory which will be used for the fabrication of the bow is
also discussed.
2.1 CHARACTERISTICS OF A BOW
A bow is mainly made up of 3 main parts which is the riser, limbs and the string.
Riser is the part which is located at the middle which is also called as the grip due to its
function. The part which is located on both sides of the riser is called as the limbs.
Where, when it is hold vertically, the upper part is called the upper limb whereas the
lower part is called the lower limb. A string is fastened at both ends of the upper and
lower limbs at a certain amount of force (Kooi, 1983).
The distance between the riser and the string is called as a fistmele. This
distance can be changed by changing the length of the string. The belly side is the part
of the limbs which faces away from the archer. Normally, there will be nocks at the end
of the limb which is used to tie the string at the ends. Nowadays, there is an extra part
which is placed in the middle part of the string which is called the nocking point where
the arrow can be correctly placed at the point to give an easy and accurate shooting.
As shown in Figure 2.1, a common recurve bow is made of riser, upper and
lower limbs, bowstring and the nocking point.
5
Figure 2.1: Main parts in an Archery bow
Source: Archery resource (2010)
2.2 TYPES OF BOW
Today in the modern world, there are mainly three types of bow which is the
traditional longbow, recurve bow and the modern compound bow.
Bow string Upper limb
Sight Window
Riser
Lower limb
Knocking
point
Mounting for
arrow rest
6
2.2.1 Longbow
Longbow was invented at late thirteenth century. This bow had draw weight of
about 445 Newtons to 535 Newtons. Longbows do not put as much energy into the
arrow as other bows, which is why they have to be more powerful. In medieval times,
draw weights could reach up to 150 pounds. But today longbows generally rarely
exceed 50 pounds. They are less accurate and more difficult to shoot than a modern
bow. And there are no any rests with a longbow where the archer has to support the
arrow by themselves (elyarchers, 2012).
2.2.2 Recurve bows
The side-view profile of the bow which looks like a curvature makes it to be
called as a recurve bow. This style of bow stores more energy than an equivalent
straight-limbed bow, and therefore gives a greater amount of force to the arrow. By
contrast, the traditional straight longbow tends to pull back the extra force, but as for the
recurve bow, as the string is drawn further back, the required draw force increases
rapidly. The curved limbs also put greater strain on the materials used to make the bow,
and they may make more noise as they are shot. A typical modern recurve bow as used
by archers in the Olympics and many other competitive events will employ advanced
technologies and materials and will have been made by a professional. The limbs are
usually made from layers of fibreglass, carbon or wood. Carbon limbs will shoot the
arrow faster for a given draw weight when compared with wooden limbs, but they are
much more expensive. The riser is generally separate and is normally constructed from
aluminium or magnesium alloy. The modern recurve is the only form of bow allowed in
the Olympic Games. Recurve archers often have many other pieces of equipment
attached to their recurve bows, such as stabilisers which is for balancing the bow and
absorbing some of the vibration, sights for improving the accuracy and pressure buttons
for a fine tuning of the arrow's flight. A piece of leather called a tab is worn when
shooting to ensure a smooth release and save wear on the fingers. The high technology
materials of a modern bow allow the manufacturers to make a bow with a better
efficiency and also permit the easy attachment of modern aids to accuracy such as the
7
stabilisers. The greater weight of a modern bow is in itself an aid to stability in the grip,
and therefore to accuracy (elyarchers, 2012).
When comparing the recurve bow, it can be divided into three which is non-
recurve bow, static-recurve bow and the working-recurve Bow (Kooi, 1983).
2.2.2.1 Non-recurve bow
(a) (b) (c)
Figure 2.2: (a) unbraced non-recurve bow (b) braced non-recurve bow
(c) fully drawn non-recurve
Source: Kooi (1983)
Figure 2.2 shows the non-recurve bow in three different situations which is in
unbraced, braced and fully drawn situations respectively. It will only have its curve after
a string is attached to the ends of the limbs. The bend tip of the limbs makes it to be
curved which makes it to be a recurve bow.
limb
grip
string
8
2.2.2.2 Static-recurve bow
Figure 2.3: (a) unbraced static recurve bow (b) braced static recurve bow
(c) fully drawn static recurve
Source: Kooi (1983)
As shown in Figure 2.3, static recurve bow is also shown is three different
situations which is unbraced, braced and in the fully drawn situations respectively. The
outermost parts of the limbs are stiff. These parts are called ears or rigid-end pieces. The
elastic part of a limb between grip and ear is called the working part of the limb. In the
braced situation the string rests on the string-bridges, situated at the bend of the ears.
These string bridges are hollowed out sometimes, to receive the string and retain it in its
place. This prevents the string from slipping beside the limb and giving it a fatal twist.
When these bows are about half drawn, the string leaves the string-bridges and has
contact with the limbs only at the tips. After releasing at a certain moment before the
arrow exit, the string touches the string-bridges again.
ear
grip
string
working part
string bridge
9
2.3.2.1 Working-recurve bow
Figure 2.4: (a) unbraced working-recurve bow (b) braced working -recurve bow
(c) fully drawn working -recurve
Source: Kooi (1983)
Figure 2.4 shows the working recurve bow in three situations which is in the
unbraced, braced and fully drawn situations respectively. In the case of a working-
recurve bow the parts near the tips are elastic and bend during the final part of the draw.
When drawing the bow the length of contact between string and limb gradually
decreases until the point where the string leaves the bow coincides with the tip of the
limb and remains there during the final part of the draw. After release the phenomena
happen in reversed order to prevent the possibility of a twist of the limbs in the case of a
working-recurve bow, grooves are present on the belly side of the limbs starting at the
notch and extending sufficiently far in the direction of the grip. We note that bows
belonging to each of the three types may be symmetric or more or less asymmetric.
grip
string
working limb
10
2.2.3 Compound bows
Figure 2.5: Compound bow
Source: METU (2009)
Figure 2.4 shows the picture of a compound bow and the parts in it. A
compound bow is a modern development of bow which use a lever system of cables,
wheels and cams to draw the limbs back. The limbs of a compound bow are usually
much stiffer than those of a recurve bow or longbow. This limb stiffness makes the
compound bow more energy efficient than other bows, but the limbs are too stiff to be
drawn comfortably with a string attached directly to them. The compound bow has the
string attached to the pulleys, one or both of which has one or more cables attached to
the opposite limb. When the string is drawn back, the string causes the pulleys to turn.
This causes the pulleys to pull the cables, which in turn causes the limbs to bend and
thus store energy. The use of this levering system gives the compound bow a chance to
rise to a peek force when releasing the arrow but with only a slight force which is about
60 Newton to give force up to 260 Newton. The compound bow is slightly affected by
11
changes of temperature and humidity but gives superior accuracy, speed, and distance in
comparison with other bows (elyarchers, 2012).
2.3 COMPOSITE MATERIALS
2.3.1 Composites
According to Camphell (2010), a composite material can be defined as a
combination of two or more materials that results in better properties than those of the
individual components used alone. Whereas, Roylance (2000) states that composite can
be mean by everything since all materials composed of dissimilar subunits if examined
at close enough detail.
Most composites have strong, stiff fibres in a matrix which is weaker and less
stiff. The objective is usually to make a component which is strong and stiff, often with
a low density. Commercial material commonly has glass or carbon fibres in matrices
based on thermosetting polymers, such as epoxy or polyester resins.
Sometimes, thermoplastic polymers may be preferred, since they are mouldable
after initial production. There are further classes of composite in which the matrix is
a metal or a ceramic. For the most part, these are still in a developmental stage, with
problems of high manufacturing costs yet to be overcome. Furthermore, in these
composites the reasons for adding the fibres are often rather complex. This software
package covers simple mechanics concepts of stiffness and strength, which, while
applicable to all composites, are often more relevant to fibre-reinforced polymers.
2.3.2 Characteristics of composite materials
Composites materials are well known for its better qualities compared to other
single materials. Normally composite materials have high specific strength and modulus
which allows them to withstand high impact together high fatigue strength and fatigue
damage tolerance. Composites can also be told as anistropic due to its property which is
directionally dependent. Besides that, composites also have some extra properties such
12
as designable or tailor able materials for both microstructure and properties, production
of both material and structure or component in a single operation such as manufacturing
flexible, net-shape, complex geometry and also known for its corrosion resistance and
durable properties (Gou, 2012).
2.3.3 Reinforcements
Reinforcements comprises of fibrous materials used to strengthen cured resin
system. The most common reinforcements are glass, carbon, aramid and boron fibres.
The reinforced material that is widely used for archery purpose is aluminium and
fibreglass.
2.3.3.1 Fibreglass
Fibreglass is widely use in composite materials. Fiberglas is material made from
extremely fine fibres of glass. It is used as a reinforcing agent for many plastic products.
The fibreglass ingredients are silicon oxide with addition of small amounts of other
oxides. Glass fibre is formed when thin strands of silica based or other formulation glass
is extruded into many fibres with small diameters suitable for textile processing. Glass
is unlike other polymers in that, even as a fibre, it has little crystalline structure. The
properties of the structure of glass in its softened stage are very much like its properties
when spun into fibre. One definition of glass is an inorganic substance in a condition
which is continuous with, and analogous to the liquid state of that substance, but which,
as a result of a reversible change in viscosity during cooling, has attained so high a
degree of viscosity as to be for all practical purposes rigid. Fibreglass is widely use
because of its desirable characteristic such as high strength, good temperature and
corrosion resistance, and low price (Salleh, 2006).
Nowadays, many types of fibreglass could be found. Each fibreglass has its own
properties and benefits. Table 2.1 shows few types of fibreglass which is commonly
found and the properties of the each fibreglass.
13
Table 2.1: Fibreglass and its properties
Fibreglass Properties
A-glass Made up of soda lime silicate glasses used
where the strength, durability, and good
electrical resistivity needed.
C-glass Made up from calcium borosilicate glasses
used for chemical stability in corrosive
acid environments.
D-glass Made up from borosilicate glasses with a
low dielectric constant for electrical
applications.
E-glass Made up from Alumina-calcium-
borosilicate glasses with a maximum alkali
content used as general purpose fibres
where strength and high electrical
resistivity are required.
ECR-glass Made up from calcium aluminosilicate
glasses with a maximum alkali content
used where strength, electrical resistivity,
and acid corrosion resistance are desired.
AR-glass
R-glass
S2-glass
Made up of alkali resistant glasses
composed of alkali zirconium silicates
used in cement substrates and concrete.
Made up from calcium aluminosilicate
glasses used for reinforcement where
added strength and acid corrosion
resistance are required.
Made from magnesium aluminosilicate
glasses used for textile substrates or
reinforcement in composite structural
applications which require high strength,
modulus, and stability under extreme
temperature and corrosive environments.
Source: Hartman, Greenwood and Miller (1996)