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)