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SAID-BALL CUBIC TRANSITION CURVE AND ITS APPLICATION TO SPUR GEAR DESIGN

SAIFUDIN HAFIZ BIN YAHAYA

UNIVERSITI SAINS MALAYSIA

2015

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SAID-BALL CUBIC TRANSITION CURVE AND ITS APPLICATION TO SPUR GEAR DESIGN

by

SAIFUDIN HAFIZ BIN YAHAYA

Thesis submitted in fulfillment of the requirements for the degree of

Doctor of Philosophy (Mathematics)

September 2015

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DEDICATION

One of the joys of this completion is to look over the journey past and remember the most important people in my life, my family, to whom this thesis is dedicated to, who have helped and supported me along this long but fulfilling road.

To my wife, Hafizah, who has been a source of motivation and strengths during moments of despair and discouragement. Thank you for making the time to read, comment and proof-read the thesis. For all that, thank you!

To my sons: Muhammad Iqram Danial and Muhammad Haziq Haiqal. You were not yet born when I started this journey and I always questioned whether there would ever be a completion date for this and now I have come this far. I have to apologize for the most time that I spent on my thesis rather than spending it with you. Thank you for your understanding and for teaching me the meaning of patience.

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ACKNOWLEDGEMENT

In the name of Allah, the Most Gracious and the Most Merciful

Alhamdulillah, all praises to Allah for the strengths and His blessing in completing this thesis. It is a pleasure to thank the many people who made this thesis possible. I would like to express my special gratitude and appreciation to my supervisor, Professor Dr. Jamaludin Md Ali, for his supervision, sound advice and constant support. I am extremely fortunate to have him as a supportive and knowledgeable supervisor. My appreciation also goes to my co-supervisor, Dr. Yazariah Mohd Yatim for her ideas, guidance and support in this study.

I am indebted to the Dean of the School of Mathematical Sciences, Professor Dr. Ahmad Izani Md Ismail and also to all the academic staff and support staff of School of Mathematical Sciences for providing a stimulating and fun environment in which to learn and grow. I wish to acknowledge and thanks all my friends and colleagues. I am especially grateful to Shukor, Hazman, Pak Ip, Yuz and others for their kindness and moral support and for sharing their ideas and experiences with me.

Last but not least, my deepest gratitude goes to my beloved parents, Yahaya Hj. Haron and Siti Aisah Hamid for their endless prayers, encouragement and love and also to my siblings: Ibnu, Habi and Amir for their support.

S.H. Yahaya September, 2015

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TABLE OF CONTENTS

Page

Dedication ii

Acknowledgment iii

Table of Contents iv

List of Tables vii

List of Figures viii

List of Publications xii

Abstrak xiii

Abstract xv

CHAPTER 1: INTRODUCTION 1.1 History of CAGD 1

1.2 Early Applications 4

1.3 Problem Statement 6

1.4 Significance of the Study 8

1.5 Research Objectives 9

1.6 Thesis Organization 10

CHAPTER 2: LITERATURE REVIEW 2.1 Introduction to Differential Geometry of Curve 13

2.2 Cubic Bézier Curve 18

2.3 Transition and Spiral Curves 22

2.4 Introduction to Clothoid Templates 24

2.4.1 Straight line to circle 27

2.4.2 Circle to circle with a C-shaped transition 27

2.4.3 Circle to circle with an S-shaped transition 29

2.4.4 Straight line to straight line 30

2.4.5 Circle to circle with a C-shaped spiral 31

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2.5 Gear Descriptions 36

2.5.1 History of gears 36

2.5.2 Terminology of gears 39

2.5.3 Classification of gears 41

2.5.4 Method of designing gears 43

2.6 Introduction to the Fabrication Tools 46

2.6.1 Wire-cut machine 46

2.6.2 Turning machine 47

2.7 Summary 49

CHAPTER 3: DEVELOPMENTS OF THEORY 3.1 The Review of Said-Ball Cubic Curve 50

3.2 The Characteristics of Said-Ball Cubic Curve 53

3.3 Circle to Circle Templates Design 54

3.3.1 S-shaped transition curve 54

3.3.2 C-shaped curve 55

3.4 Curvature Analysis 57

3.4.1 Curvature analysis in an S-shaped transition curve 57

3.4.2 Curvature analysis in a C-shaped curve 61

3.5 Summary 64

CHAPTER 4 : APPLICATION TO THE SPUR GEAR DESIGN 4.1 Spur Gear Tooth Design using an S-Shaped Transition curve 65

4.2 Spur Gear Tooth Design using a C-Shaped Curve 66

4.3 The Solid of Spur Gear 67

4.4 Summary 69

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CHAPTER 5: STRUCTURAL RESPONSE ANALYSIS

5.1 Strain-Stress Analysis 70

5.2 FEA Solver and its Modelling 71

5.1.2 CAD model and its meshes 72

5.2.2 Boundary and loading conditions 73

5.2.3 Material selection 74

5.2.4 Simulation results and its safety factor 75

5.3 First-Order Newton Interpolating Polynomial as a Fatigue Predictor 77

5.4 Design Efficiency of the Models 78

5.5 Summary 79

CHAPTER 6: DYNAMIC AND ACOUSTIC RESPONSE ANALYSES 6.1 Dynamic Response in All Models 80

6.1.1 Normal modes analysis 80

6.1.2 Frequency response analysis 81

6.1.3 Transient response analysis 83

6.1.4 Result and discussion 84

6.2 Acoustic Response in All Models 85

6.2.1 Acoustic experiment in all models 86

6.2.2 Acoustic results and its discussion 90

6.3 Summary 93

CHAPTER 7: CONCLUSION AND RECOMMENDATIONS 7.1 Conclusion 94

7.2 Recommendations 96

References 98

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LIST OF TABLES

Page

Table 5.1 Some of the parameters that are frequently used in the

system of units 71 Table 5.2 Meshing data components 73 Table 5.3 AISI 304 and its characteristics 75 Table 5.4 The result distributions of δmax, υmax and Sf in all models 76

Table 5.5 The initial loads of P amongst the models in fatigue

analysis 78 Table 6.1 The statistical computation in all models 84

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LIST OF FIGURES

Page Figure 1.1 (a) The cover of the first journal on CAGD published by

Barnhill and Boehm in 1984; (b) is an illustration of

“Uccello’s Chalice” used for the cover 2

Figure 1.2 An example of curve design with GC1 continuity 3

Figure 1.3 Both curves apply C1 continuity, (a) non-positivity and (b) positivity preserving 4

Figure 1.4 Conics as represented in cockpit development 4

Figure 1.5 An example of CATIA’s logo 5

Figure 1.6 The making of tiger in the film of ‘Life of Pi’ 6

Figure 1.7 The illustration of the family of gears: (a) rack and pinion, (b) spur and (c) helical 7

Figure 1.8 Generation of an involute curve 7

Figure 2.1 The example of (a) open and (b) closed curves 14

Figure 2.2 A circle defined parametrically and implicitly 15

Figure 2.3 The art of Bézier curve (red colour) 21

Figure 2.4 Cubic Bernstein basis functions in 2D plot 22

Figure 2.5 The example of road design using transition curves 23

Figure 2.6 Relationship between curvature and arc length in clothoid 23

Figure 2.7 Horizontal alignment in highway design 25

Figure 2.8 The use of clothoid templates in highway designed alignment model 26

Figure 2.9 First case of circle to circle templates using CBC 27

Figure 2.10 Concept of spiral connection in the second case template using CBC 28

Figure 2.11 The design of third case template 29

Figure 2.12 Straight line to straight line with an angle, Ω∈(0,π) 30

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Figure 2.13 The architecture of case 4 using G3 continuity 31

Figure 2.14 Circle to circle with a C spiral curve 32

Figure 2.15 Transition loss differs greatly between clothoid and circular curves 34

Figure 2.16 The first prototype of roller coaster model using the clothoid loop approach 35

Figure 2.17 The Chinese chariot model using a wooden gear set 36

Figure 2.18 The greatest invention by da Vinci known as helical screw helicopter 37

Figure 2.19 Pfauter hobbing machine 38

Figure 2.20 The pair of helical gears via parallel axis 40

Figure 2.21 Spur gear nomenclature 40

Figure 2.22 Bevel gear with its spiral curved 42

Figure 2.23 Involute curve and its philosophy 44

Figure 2.24 Wire-cut EDM and its schematic diagram 47

Figure 2.25 The components in turning machine 48

Figure 3.1 Graph of SBC basis functions when λ0 and λ1 equal to 2 52

Figure 3.2 The influence of shape parameters in SBC design 52

Figure 3.3 Polar coordinate system in circle 54

Figure 3.4 (a) Transition and (b) spiral curves architecture in CBC 56

Figure 3.5 An S-shaped transition curve using SBC 58

Figure 3.6 (a) Curvature profile, k(t) and (b) its derivative, k(1) (t) in S-shaped transition curve for an interval, 0 § t § 0.61507 60

Figure 3.7 C-shaped curve in the fifth case template 62

Figure 3.8 (a) Curvature profile, k(t) and (b) its derivative, k(1) (t) in C-shaped curve for an interval, 0 § t § 0.2583 63

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Figure 4.1 Geometry definition 65

Figure 4.2 Spur gear tooth design using an S-shaped transition curve 66

Figure 4.3 Spur gear tooth design using a C-shaped transition curve 67

Figure 4.4 The process flow diagram in CATIA V5 68

Figure 4.5 The solid models of spur gear using (a) S- and (b) C-shaped curves 69

Figure 4.6 One of the existing models in oil pumps 69

Figure 5.1 Process flow diagram in MSC Nastran & Patran 72

Figure 5.2 Tet-10 and its element’s topology 72

Figure 5.3 An example of meshing process in S transition gear 73

Figure 5.4 The critical region (two dots along the blue line) in the tooth profile 74

Figure 5.5 The setup of boundary and loading conditions in C spiral gear 74

Figure 5.6 The example of findings {δmax, υmax} when P = 30 MPa in C spiral gear 76

Figure 6.1 The natural frequency amongst the models 81

Figure 6.2 Modal frequency analysis in all models 83

Figure 6.3 Transient frequency analysis in all models 84

Figure 6.4 (a) The manufacturing of outside and internal diameters using lathe machine; (b) Shape profiles of the spur gear are split from the work piece using wire-cut machine 87

Figure 6.5 Gear shaft with its hole is drilled using lathe machine 87

Figure 6.6 Process flow diagram in fabricating the models 88

Figure 6.7 Spur gear using the (a) S- and (b) C-transition profiles 88

Figure 6.8 The EM model 88

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Figure 6.10 Process flow diagram in an acoustic experiment 90

Figure 6.11 Sound pressure in all models 91

Figure 6.12 Sound level in all models 92

Figure 6.13 The probabilistic simulation amongst the models 93

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LIST OF PUBLICATIONS

a. S. H. Yahaya, J. M. Ali, M. Y. Yazariah, Haeryip Sihombing and M. Y. Yuhazri (2013). G2 Bézier-Like Cubic as the S-Transition and C-Spiral Curves and its Application in Designing a Spur Gear Tooth. Australian Journal of Basic and Applied Sciences, 7(2), pp. 109-122.

b. S. H. Yahaya, J. M. Ali, M. Y. Yazariah, Haeryip Sihombing and M. Y. Yuhazri (2012). Integrating Spur Gear Teeth Design and its Analysis with G2 Parametric Bézier-Like Cubic Transition and Spiral Curves, Global Engineers &

Technologists Review, Vol. 2, No. 8, pp. 9-22.

c. S. H. Yahaya, J. M. Ali and T. A. Abdullah, (2010). Parametric Transition As A Spiral Curve and Its Application In Spur Gear Tooth With FEA, International Journal of Electrical and Computer Engineering, Volume 5, Number 1, 2010, pp.

64-70.

d. Saifudin Hafiz Yahaya, Jamaludin Md Ali and Mohd Shukor Salleh (2009), Spur gear design with an S-shaped transition curve application using Mathematica and CAD tools. Proceedings of the In Computer Technology and Development, 2009. ICCTD'09, IEEE Computer Society Proceeding, Vol. 2, pp.

426-429.

e. Saifudin Hafiz Yahaya, Jamaludin Md Ali and Wahyono Sapto Widodo (2009), Spur gear design using an S-shaped transition curve with finite element analysis. Proceedings of the 5th Asian Mathematical Conference, pp.346-352.

f. Saifudin Hafiz Yahaya, Jamaludin Md Ali and Muhammad Hafidz Fazli Md.

Fauadi (2008), A product design using an S-shaped and C-shaped transition curves. Proceedings of the Fifth International Conference on Computer Graphics, Imaging and Visualization, IEEE Computer Society Proceeding, pp. 149-153.

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LENGKUNG PERALIHAN KUBIK SAID-BALL DAN PENGGUNAANNYA DALAM REKA BENTUK GEAR BINTANG

ABSTRAK

Tidak dinafikan, gear adalah merupakan antara elemen yang paling banyak digunakan dalam pemesinan dan industri. Kajian lepas telah menunjukkan bahawa lengkung berbentuk involut adalah profil yang sering digunakan di dalam mereka bentuk gigi gear bintang, yang dibangunkan berdasarkan teori-teori penghampiran seperti penghampiran Chebyshev dan kaedah menyurih titik. Walau bagaimanapun, kaedah yang digunakan ini tidak jitu dan hanya menumpukan kepada konsep-konsep penghampiran sahaja. Pengurangan bunyi gear dan kekuatan gigi gear sentiasa menjadi tumpuan kajian dan eksperimen terutamanya dengan pengubahsuaian bentuk gigi atau profilnya. Oleh itu, kajian ini adalah untuk mereka bentuk lengkung peralihan S dan C dengan menggunakan lengkung kubik Said-Ball berdasarkan templet bulatan bagi kes ketiga dan kelima dengan beberapa pembuktian matematik.

Salah satu objektif kajian ini adalah untuk menyiasat keupayaan model lengkung berbentuk S dan C dalam mengurangkan tahap bunyi atau bising melalui eksperimen akustik. Dalam kajian ini, gear bintang dipilih sebagai kajian kes kerana ia adalah gear asas dan hakikatnya ia mudah untuk dibina dan dibuat. Berdasarkan eksperimen dan simulasi yang dijalankan, keputusan menunjukkan bahawa dengan menggunakan lengkung kubik Said-Ball, teori-teori bagi lengkung peralihan S dan C telah berjaya dibangunkan. Lengkung-lengkung ini telah terbukti secara matematik, dengan menggunakan ujian terbitan kedua, ujian kelekukan dan teorem Kneser. Ia juga mendedahkan bahawa lengkung peralihan S dan C telah berjaya digunakan dalam

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mereka bentuk gigi gear bintang. Ini membuktikan juga bahawa model pepejal bagi gear bintang dapat dibangunkan melalui penggunaan berintegrasi antara perisian matematik dan CAD. Apabila diukur melalui analisis statik linear, analisis kelesuan dan DE, kebolehan reka bentuk yang dicadangkan beserta bahan, AISI 304, menunjukkan bahawa polinomial Newton interpolasi peringkat pertama boleh digunakan sebagai peramal kelesuan bagi semua model reka bentuk. Kaedah reka bentuk gigi baru iaitu lengkung-lengkung berbentuk C dan S adalah kaedah yang boleh diterima pakai di dalam mereka bentuk gigi gear bintang di mana kedua-dua kaedah ini telah membentangkan DE yang lebih besar daripada 85% keberkesanan reka bentuk. Semua model juga telah berjaya diukur melalui analisis dinamik dan akustik. Model berbentuk C telah terbukti mempunyai sesaran yang paling rendah berbanding model peralihan S dan EM. Dengan menggunakan model ini, kebisingan gear atau bunyinya terbukti boleh dikurangkan secara konsisten. Model berbentuk C juga lebih dipercayai daripada model-model lain yang menepati PS. Pengubahsuaian profil gigi terbukti sebagai faktor utama dalam mengurangkan kebisingan gear atau bunyi secara signifikan dan konsisten. Sumbangan kajian ini akan memberi manfaat kepada pereka atau pembuat dalam mereka bentuk profil gear bintang di mana lengkung yang disebutkan di atas boleh digunakan sebagai kaedah alternatif bagi profil gear ini. Untuk penyelidikan masa depan, keupayaan lengkung peralihan C boleh lagi diterokai dalam mereka bentuk model-model aerodinamik contohnya, kereta, kereta api berkelajuan tinggi, peluru dan lain-lain. Kajian mengenai reka bentuk gear juga boleh diteruskan lagi dengan jenis gear yang lain seperti gear heliks herringbone.

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SAID-BALL CUBIC TRANSITION CURVE AND ITS APPLICATION TO SPUR GEAR DESIGN

ABSTRACT

Undoubtedly, gears are some of the most widely used elements in consumer and industrial machineries. Past studies have shown that an involute curve is the most common profile used in designing the spur gear tooth, developed based on the approximation theories such as Chebyshev approximation and the tracing points method. However, these employed methods are not accurate (or inexact) and only focusing on the approximation concepts. Gear noise reduction and tooth strength are continually being the focus of exploration and experimentation particularly the modification of the tooth shape or tooth profile. Therefore, this study is to design the S and C-shaped transition curves using Said-Ball cubic curve based on the third and fifth cases of circle to circle templates with some mathematical proofs. One of the objectives is to investigate the capability of this proposed S and C-shaped model in reducing sound or noise level through an acoustic experiment. In this study, spur gear is chosen as a case model due to its fundamental gear and the fact that it is simple to construct and manufacture. Based on the conducted experiment and simulation, results show that by using Said-Ball cubic curve, the theories of S and C-shaped transition curves have successfully developed. These curves have been mathematically proven, by using the concavity and second derivative tests and also Kneser’s theorem. It is also revealed that S and C-shaped transition curves can be applied successfully in designing spur gear tooth. This proves that the solid model of spur gear can also be developed through the integrated use of mathematical and CAD

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software. When measured through linear static analysis, fatigue analysis and DE, the applicability of the proposed design and the material, AISI 304 shows that First-order Newton interpolating polynomial can be employed as a fatigue predictor for all design models. The new teeth design methods, S and C-shaped curves are the acceptable methods in designing the spur gear teeth where both methods have presented DE greater than 85% of the design effectiveness. All models have also been successfully measured via dynamic and acoustic response analyses. C-shaped model has been proven to have the lowest displacement when compared to S-shaped (transition) and EM models. By utilizing this model, it is proven that gear noise or sound can be reduced consistently. C-shaped model is more reliable than other models in accordance to PS. It is proven that tooth profile modification is the main factor in reducing sound or noise in a very significant and consistent way. The contribution of this study will be beneficial to the designers or manufacturers in designing the spur gear profiles where the above-mentioned curves can be applied as an alternate method of these profiles. For future research, the capability of C transition curve can be further explored in designing aerodynamic models for example, car, high-speed train, bullet etc. Study on gear design can also be further explored on other type of gears such as helical herringbone gear.

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CHAPTER 1

INTRODUCTION

1.1 History of CAGD

Lately, Computer-Aided Geometric Design (CAGD) plays a major role in the world of design. CAGD or in another term called, Geometric Modelling is a research field geared toward the development and representation of freeform curves, surfaces or volumes (Làvička, 2011). CAGD is a new field, originally created to bring some great benefits of computers to industries such as automotive, aerospace, shipbuilding or in various applications. Historically, CAGD emerged in the middle of 1970s.

Barnhill and Riesenfeld (1974) can be claimed as the early pioneers in this field as they have organized a conference on this field which was held at the University of Utah, USA, in 1974. The main objective of this conference was primarily to discuss the aspect of CAGD that has attracted a great number of international researchers around Europe and USA to participate, for the first time, in this conference.

Today, CAGD becomes a sovereign discipline in its own right (Làvička, 2011). After the success of this first conference, research findings from the conference have been produced in various forms including the first textbook on CAGD entitled “Computational Geometry for Design and Manufacture” by Faux and Pratt (1988) and also first journal published focusing on the field of CAGD (Figure 1.1). Both publications have been used as the major references for many students and

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young researchers who want to understand further or to study CAGD. Earlier, de Casteljau (1959) and Coons (1964) have constructed the fundamental aspects of CAGD before a conference which was related to the use of CAGD in automotive industries was organized by a French man, Bézier, in 1971 (Farin, 2002).

(a) (b)

Figure 1.1: (a) The cover of the first journal on CAGD published by Barnhill and Boehm in 1984 (Farin, 1992); (b) is an illustration of “Uccello’s Chalice”

used for the cover (Talbot, 2006)

CAGD deals with mathematical expressions to control the shapes when designing curves and surfaces. Several essential mathematical concepts are fully utilized in this control such as geometry, vector, coordinate system and some basic knowledge of calculus. Shapes or profiles are typically produced by related parametric equations (functions). Abstractly, a parametric equation can be defined as a method to determine the relationship amongst equations or functions using independent variables (parameters) (Thomas et al., 1988). One of the most common functions that is always used in this field is Bézier function, normally in cubic but can be represented either in quadratic or in any degree. As for an intention of smoothness or visually pleasant curves and surfaces, the idea of continuity is then applied.

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Hence, control points (in coordinate form) are highly needed to design curves or surfaces completely. Rockwood and Chambers (1996) explained that control points are points in two or more dimensions, which can define the behaviour of the resulting curve. Figure 1.2 shows a generated curve using four control points with the incorporation of cubic Bézier function and GC1 continuity (Sarfraz, 2008).

Hazewinkel (1997) concludes that GC1 continuity can be classified as tangent (G1) continuity. This GC1 continuity has been utilized in designing an airplane wing (Brakhage and Lamby, 2005).

Figure 1.2: An example of curve design with GC1 continuity

Nth-order parametric continuity (Cn) with n=0 ,1 ,2 ,3,...,k are the well-known smoothness properties in shape preserving or in interpolation problem. Figure 1.3 depicts the use of C1 continuity in preserving a shape between curves. These concepts of CAGD will be further discussed in chapter 2.

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(a)

(b)

Figure 1.3: Both curves apply C1 continuity, (a) non-positivity and (b) positivity preserving

1.2 Early Applications

Over the past decades, CAGD has been expanded rapidly in the fields of automobile, aircraft, aerospace or in ship industries. Nowacki (2000) found that the ship’s ribs, introduced during Roman Empire in 13th century, were the earliest geometry application in free-form shapes. Several curves, such as splines have been recorded in these ribs. The revolution is then continued by Liming (1944) and Coon (1947) who proposed conic construction in aeronautics manufacture (aircraft) design.

Farin (1992) defined the conic as a perspective projection of a parabola in Euclidean three space into a plane. Figure 1.4 depicts these conics in aircraft.

Figure 1.4: Conics as represented in cockpit development (Liming, 1944)

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(Rockwood and Chambers, 1996; Prince, 1996). In 2012, Life of Pi is the latest film that applies this legacy technique successfully (Figure 1.6). Several concepts have been applied in the film for instance; geometric shapes and computational method which are strongly connected to this field. The applicability of CAGD will be continuously used throughout this study such as in spur gear design.

Figure 1.6: The making of tiger in the film of ‘Life of Pi’

(http://99designs.com/designer-blog/2013/02/15/oscars-best-visual-effects/, accessed 20 March 2013)

1.3 Problem Statement

Gears are some of the most widely used elements in both applications such as in consumer and industrial machineries. The family of gears also includes spur, helical, rack and pinion, worm and bevel (Figure 1.7). In this study, spur gear is chosen as the case model due to its fundamental gear and the fact that it is simple to construct and manufacture. Babu and Tsegaw (2009); Yoon (1993); Bradford and Guillet (1943) and Higuchi et al. (2007) have identified an involute curve (Figure 1.8) as the most common profiles used in designing the spur gear tooth. This curve is developed based on the approximation theories such as Chebyshev approximation (Higuchi and Gofuku, 2007) and the tracing points method (Margalit, 2005; Reyes et al., 2008). However, these employed methods are not accurate (or inexact) and only focus on the approximation concepts (Babu and Tsegaw, 2009). Furthermore, the

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gear noise reduction and tooth strength are continually viewed as the main issues for consideration, with emphasis on the tooth shape (profile) modification (Yoon, 1993;

Sweeney, 1995; Sankar et al., 2010; Åkerblom, 2001; Beghini et al., 2006).

(a) (b) (c)

Figure 1.7: The illustration of the family of gears: (a) rack and pinion, (b) spur and (c) helical (www.gearsandstuff.com, accessed 20 March 2013)

Figure 1.8: Generation of an involute curve

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1.4 Significance of the Study

This study will be a significant endeavour on the construction of the parametric (or known as an exact) curve theories namely, S and C-shaped transition curves with some related mathematical proofs. This study will also be beneficial to the designers or manufacturers in designing the spur gear profiles where the above- mentioned curves can be applied as an alternative method of these profiles. In addition, the shape of these profiles is preserved exactly over the curvature continuity (G2 continuity).

The applicability of the proposed designs is now measured using linear static, fatigue, normal modes, frequency and transient analyses with the material selected is Stainless Steel Grade 304 (AISI 304). These analyses covered all static and dynamic behaviour. At present, first-order Newton interpolating polynomial is used as a fatigue predictor to predict the fatigue mode in the design. Continuously, acoustic analysis is also carried out through the related experiment to perceive the sound or noise level in the proposed design with the material (AISI 304) remained. The comparison is made between the proposed and existing designs in all analyses. This study uses design efficiency (DE), probabilistic simulation (PS) and coefficient of variation (CV) as the lens of either the proposed designs are acceptable or otherwise by setting and computing the benchmark improvements and the design consistency amongst all models.

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This study provides future recommendations on the function used in designing the curves, the use of integrated software as a significant technique and also to explore new applications.

1.5 Research Objectives

The objectives of this thesis are:

a. To study the characteristics of the circle to circle templates together with the applied function, Said-Ball cubic curve.

b. To design the S and C-shaped transition curves in accordance to the third and fifth cases of circle to circle templates and Said-Ball cubic curve with some mathematical proofs.

c. To apply the S and C-shaped transition curves in spur gear design.

d. To analyze the proposed and existing models using appropriate engineering analyses such as linear static, fatigue and frequency analyses.

e. To find out the sound or noise level of the proposed and existing models throughout an acoustic experiment.

f. To validate between the proposed and existing models using DE, PS and CV.

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1.6 Thesis Organization

This thesis begins by introducing the history of CAGD, its first ideas and involvements, contribution of this field, the early applications of CAGD technique and concept in various areas or industries. Chapter 1 also focuses on the explanation on the problem statement, significance of the study, research objectives and thesis organization.

Chapter 2 reviews the differential geometry of the curves, which will be extensively used throughout this study. The present review deals with parameterized and plane curves, degree of smoothness (continuity) and also some notations and convections. Cubic Bézier curve in Bernstein form representation will be discussed followed by the discussion on the introduction of transition and spiral curves.

Methods of designing these curves will be touched such as straight line to circle, circle to circle with an S-shaped transition and circle to circle with C-shaped spiral.

The application of this curve design in highway and railway designs or in path planning will also be described in details. Chapter 2 also focuses on the investigation of gears by exploring its history, terminology and classification. The general description of spur gears is then discussed together with current curve that has been applied in design process. Finally, a brief overview of fabrication tools such as turning and wire-cut machines will be done.

Theory development will be the focus of discussion in chapter 3. It consists of a review of Said-Ball cubic curve and its curve characteristics and the designing of S and C-transition curves. A method of designing these curves will be dictated by the

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circle to circle templates (as described in chapter 2). Relatively, S and C-shaped curves will be analyzed by examining their curvature profiles and will be concluded with mathematical proofs.

Chapter 4 further elaborates the use of S and C-transition curves in designing spur gear tooth profiles. These tooth profiles will be converted into spur gear solid models using the integrated software, Wolfram Mathematica 7.0 and CATIA V5.

Meanwhile, the existing model is also explained in this chapter.

By using structural response analysis, chapter 5 focuses on the evaluation of outcomes resulting from chapter 4. One of common schemes used is static strength analysis with the tool, FEA. FEA includes CAD model, meshing process with several conditions such as displacement, boundary and loading needed in this tool. This scheme will determine the stress distributions and safety factor amongst the models.

In addition, fatigue analysis will be highlighted in this chapter and finally, the computation of DE in all spur gear models will be discussed and shown.

Chapter 6 discusses the measurement of spur gear models by using the dynamic and acoustic response analyses. Dynamic response comprises of the schemes of normal modes, frequency and transient response analyses with the influence of the damping factor. These schemes will to find out for instance; natural frequency, displacement and stress distributions in real-time computing or in frequency domain amongst the models. Simultaneously, this analysis will be completed by using a simulation method. Conversely, the noise or sound levels of the models are evaluated experimentally. An experiment on the gear acoustics is initially

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intended to study the effect of tooth shape (profile) amongst the proposed and existing models. This experiment involves fabrication of spur gear (model) and experimental setup as well as fundamental features which are mostly applied in this analysis. This chapter ends by briefly discussing the results obtained throughout the presentation of PS and CV. An approximate normal distribution is the method of representing PS.

Chapter 7 discusses and summarizes the findings and highlights suggestions for further study.

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CHAPTER 2

LITERATURE REVIEW

2.1 Introduction to Differential Geometry of Curve

Differential Geometry (DG) or known as the mathematical disciplines are the most fundamental properties in CAGD with its concentration on the shape of the objects (curves and surfaces). The disciplines for instance, calculus and linear algebra have been identified as a main contributor to DG. Since the 18th and 19th centuries, DG has been developed using such a theory of curves and surfaces in Euclidean (real vector) space (Schlichtkrull, 2011). Euler (1707-1783), Monge (1746-1818) and Gauss (1777-1855) are the early mathematicians involved in expanding this theory. For example, modern theory of plane curves is developed by Euler while in the year 1825, Gauss contributed his work on DG of surfaces (Schoen, 2011). Schoen (2011) also explained that DG always begins with the plane curves.

Plane curve is basically a special curve or profile (two-dimensional) situated along the plane (Lawrence, 1972). It can also be defined parametrically, explicitly or implicitly such as in (2.2)-(2.4) below. In general, a standard notation of this curve is depicted as

) 2

, (

: a b →ℜ

z (2.1) with (a,b) an open interval (Mare, 2012). Plane curves may visualize in closed or open region (Figure 2.1). Closed region means the curve is without endpoints

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(enclosed area), whereas vice versa in open curve (Berger and Prior, 2006). On top of that, equation (2.2) below shows that the parametric form in a set of Cartesian coordinates with the relationships between dependent and independent variables, as mentioned previously in chapter 1. Hence, the form expressed as

)), ( ), ( ( )

(t x t y t

z = (2.2) where x=x(t) and y= y(t) with t is in the real interval (non-negative) either open or closed range for instance, t∈[0,1] while an explicit form is represented by

).

(x g

y = (2.3) It can be seen clearly that both forms, parametric and explicit have the same structures regarding their representation types. Accordingly, the following property (2.4) controls an implicit appearance. In addition, explicit and implicit functions are also identified as the non-parametric forms:

. 0 ) , (x y =

f (2.4)

(a) (b) Figure 2.1: The example of (a) open and (b) closed curves

Every form has its own advantage and strength which depends on the application used (Du and Qin, 2007). However, the degree of freedom (DOF) (or a set of independent parameters) can be increased once the parametric form (curve) is employed (Martinsson et al., 2007). This advantage is highly desired in controlling

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such shapes. Agarwal (2013) in his work clarified that the smooth curves can be generated using this form. Thus, it is mostly preferred in representing the plane curves such as in CAD or in geometric modelling. Alpers (2006) also described that the flexible CAD model will be constructed together with the model modifications can be changed easily and rapidly after applying this parametric form. Presently, one of the parametric forms is

), sin , cos ( )

(t r t r t

z = (2.5) where x(t)=rcost and y(t)=rsintwith ras the radius. Equation (2.5) will produce a closed curve as a circle. Conversely, this curve can also be generated implicitly using

. 0 )

,

(x y = x2 + y2r2 =

f (2.6) The simple closed curve (circle) using (2.5) or (2.6) is displayed with requals to 2 as shown in Figure 2.2.

Figure 2.2: A circle defined parametrically and implicitly

Relatively, the represented forms (definition) are always delivered along with some general notations and convections. These rules (calculus and linear algebra) are

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useful in DG and to generate the smooth plane curve. Consider the Euclidean system consisting of the vectors, A=< Ax,Ay > and B=<Bx,By >. The dot and cross products of these vectors are symbolized as, AB andA^B,respectively. Hence, these products can be expanded to (Juhász, 1998; Artin, 1957)

, )

sin(

^

, )

cos(

x y y x

y y x x

B A B A

B A B A

=

=

+

=

=

θ θ B A B A

B A B A

(2.7)

and where, θ (or angle) is normally measured in anti-clockwise direction. Let z(t)be as defined in (2.2), thus its velocity (tangent) will be denoted by z(t) and followed by the norm (speed) equivalents to

, )) ( ( )) ( ( )

(t x t 2 y t 2

z′ = ′ + ′ (2.8) Equation (2.8) is essentially associated to compute the arc length of a curve as depicted by

. ) ( )

(t z t dt S

b

a

= (2.9)

Jia (2014) and Hagen et al. (1995) claimed that the curve is regular (smooth) if the parametric form is employed and z′(t)≠0. In addition, these velocities and speeds are fully dependent on this form. Due to z′(t)≠0 consequently, the studies such as by Hoschek and Lasser (1993) and Faux and Pratt (1988) have discovered the existence of curvature along the curve (the regular characteristic will begin to form). Curvature can be prescribed as a local measure (set of measurement) of the curve shape (Sullivan, 2008). This is the best approach of describing the curves which are said to be entirely beautiful (Margalit, 2005). It is also agreed by Struik (1931) who said that the curvature is the major property in DG.

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Initially, this curvature theorized through mathematicians such as Aristotle (384-322 BC) and Proclus (412-485 C.E.) are the Greeks developed the notations of curvature through classical Geek curves while Pergaeus (ca. 262 BC-ca. 190 BC) in his significant works about the proposed method to identify the radius of curvature and the integration between conic section and normal line (Margalit, 2005). Hence, the study of curvature is then indispensable since this exploration. Fermat (1601- 1665) and Descartes (1596-1650) expanded theory of curvature with some algebraic equations whereas Newton (1642-1727) in his remarkable contribution to conclude that the curvature is inversely proportional to the radius in all circles:

1. ) (t = r

κ (2.10) This relationship is mainly used as the basis form in constructing smoothness of the curve namely, second order geometric (G2) continuity. The shapes in CAGD or in CAD are the current studies which apply this continuity. Euler (1707-1783) has mentioned that the parameterized curves should be lens to DG. He was responsible to modify the definition of curvature by including the tangent concept (Kline, 1972).

These mathematicians are known as the father of DG. Hence, curvature, κ(t) and its derivative have been referred to as

. ) (

) ( )^

) (

( z t 3

t z t t z

′′

= ′

κ (2.11)

, ) (

) ) (

( z t 5

t t

= ′

′ φ

κ (2.12)

where

)}.

( ) ( )}{

( )^

( { 3 )}

( )^

( { ) ( )

( 2 z t z t z t z t z t z t

dt t d z

t = ′ ′ ′′ − ′ ′′ ′ • ′′

φ

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Equation (2.12) is engaged as an indicator to recognize the plane curve either spiral or transition feature if certain conditions have been fulfilled (will be further discussed in chapter 3). An aesthetic (the concept of beauty and interactive) appearance amongst the curves can be found once this recognition is truly finished (Jacobsen et al., 2006). Due to the order derivatives which are applied in (2.12), as explained by Costa (2002) that the curve will be produced smoothly. Besides, Lin (2009) has confirmed that the order derivatives give an influence to the shape of curves becomes smoother and visually pleasing. After knowing these several theories, the descriptions of this chapter will be continued on cubic Bézier curve.

2.2 Cubic Bézier Curve

Cubic Bézier Curve (CBC) is a well-known form in the fields of CAGD, CAD and Geometric Modelling. Walton and Meek (1999 and 2001) described that CBC is frequently chosen as a function due to the properties such as one of the parametric curves (classification); the lowest degree polynomial to permit the inflection points (related to curvature extrema and stability reason); have the geometric and numerical properties that satisfy CBC suitable for use in CAGD or in CAD (flexibility) and ease to handle and implement when compare to other degree.

Historically, Bézier representation is used as the basis form in CBC. This representation has been introduced to the world by Bézier (1910-1999) and de Casteljau (1930-1999), the French engineers to overcome the problems in representing and preserving smooth curves and surfaces in automobile company (Farin, 2002). For example, Citroën and Renault use this curve completely.

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Throughout their life, Bézier (1910-1999) and de Casteljau (1930-1999) are also known as the pioneers in many areas such as solid, geometric and physical modelling. In general, Bézier curve of degree n can be depicted as

∑ ∈

= = n

i PiBin t t t

Z

0 , ( ), [0,1] )

( (2.13) where Pi are defined as the control points and;

otherwise

0 ,

0

, ) 1

!( )!

(

! )

, (

n t i

i t i n

n t

B

i i n n

i

⎪⎩

⎪⎨

⎧ −

= − (2.14)

as the Bernstein polynomials or the Bernstein basis functions of degree n (Qian et al., 2011). These polynomials are indispensable as the core of Bézier curves and have different form when compared with the rational Bézier curves. Statistically,

! )!

(

! i i n

n

− or ⎟⎟⎠

⎜⎜ ⎞

i

n is also classified as the binomial coefficients (Sury et al., 2004).

Farouki (2012) in his review describes several properties of Bernstein polynomials:

a. Symmetry

The basis functions, Bni,n(t)= Bi,n(1−t) for i=0,...,n (mirroring).

b. Non-Negativity (Positivity)

The basis functions, Bi,n(t)≥0 in all t∈[0,1].

c. Partition of Unity

The total of binomial expansion, =

= n i Bin t

0 , ( ) 1for t∈[0,1].

d. Recurrence Relation (de Casteljau’s algorithm)

For example, the basis polynomial of degree n+1 can be produced using the basis polynomial of degree n where

) ( )

( ) 1 ( )

( , 1,

1

, t t B t tB t

Bin+ = − i n + i n since t∈[0,1].

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Sánchez-Reyes and Chacón (2005) state the basic properties of Bézier curves as follows:

a. Endpoint interpolation expressed as ) 0

0

( P

Z = & Z(1)= Pn

b. Geometric continuity (Tangent) for instance, )

( ) 0

( n P1 P0

Z′ = − & Z′(1)=n(PnPn1) c. Convex hull (Polygon)

This property always exists in the control points of Bézier curve. It is also crucial for numerical stability.

d. Invariant under affine transformations (Geometric mappings)

This property engages with any blending of translations, reflections, stretches or rotations (original form remains) such that

∑ ≡

=

=

n

i i in

n

i PiBin t PB t

0 ,

0 , ( )) ( )

( γ

γ

e. Variation diminishing (VD)

This property has verified that a Bézier curve alternates less than its control polygon (point) due to the influence of the segments. Moreover, VD is widely applied in the algorithms for example, intersection and fairness.

These properties are well-suited for interactive design environments and are especially useful in path planning (Ho and Liu, 2009). Figure 2.3 shows the terminology of Bézier curve.

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Figure 2.3: The art of Bézier curve (red colour) (Novin, 2007)

If n = 3, (2.13) and (2.14) become

) ( )

( )

( )

( )

(t P0B0,3 t P1B1,3 t P2B2,3 t P3B3,3 t

Z = + + + (2.15)

where

. ) ( ), 1 ( 3 ) (

, ) 1 ( 3 ) ( , ) 1 ( ) (

3 3

, 3 2

3 , 2

2 3

, 1 3 3

, 0

t t B t t t B

t t t B t

t B

=

=

=

= (2.16)

Both equations are characterized as CBC. CBC consists of four control points symbolized by P0, P1, P2 and P3 while the visualization of Bernstein functions (2.16) is displayed in Figure 2.4. The exploration of this chapter will be continued by introducing the most common types of parametric curves, namely transition and spiral curves.

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Figure 2.4: Cubic Bernstein basis functions in 2D plot

2.3 Transition and Spiral Curves

Shen et al. (2013) explicate that transition curve is a segment with varied radius, gradually increasing or decreasing. This increment happens during the connection between two curves with different radius for instance, circular arc (curve) and tangent track (straight line). The idea of connecting curves is utilized to enable the gradual change (smooth) amongst the curvature and its acceleration or speed (Lindahl, 2001). Therefore, this idea becomes crucial since it has been widely used in civil and transportation engineering particularly; in highway or in railway design (Figure 2.5).

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Figure 2.5: The example of road design using transition curves (Myers, 2001)

In contrast, spiral curve defines as a plane curve with the curvature varies monotonically either increasing or decreasing (only in one sign) (Kurnosenko, 2009;

Leyton, 1987). Since the denominator in (2.12) is always positive, thus monotonic curvature (MC) must satisfy the following spiral condition (SC) such as

⎩⎨

′ <

′ >

= If ( ) 0meansMCisdecreasing increasing is

MC means 0 ) ( SC If

t t κ

κ (2.17)

Clothoid or also known as Euler or cornu spiral is one of the basic spirals which has the curvature changes linearly with its arc-length (Figure 2.6) (Yates, 1974). This review shows that the curvature and arc length have the same identity (identical) since the clothoid has its own function.

Figure 2.6: Relationship between curvature and arc length in clothoid (Séquin, 2005)

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However, in real situation, this curvature plot (Figure 2.6) might be difficult to achieve when using Bézier curve. The clothoid function is formulated with the use of parametric form and Fresnel integral (Abramowitz and Stegun, 1964; Meek and Walton, 2004) where

0 ) , (

) ) (

( ⎟⎟⎠ ≥

⎜⎜ ⎞

= ⎛ t

t y

t t x

H β (2.18) and β is the scaling factor, the Fresnel integrals are

. 2 ] cos[

) ( and 2 ] sin[

) (

0

2 0

2

u du t

y u du

t

x =t∫ π =t∫ π

Some scholars believed that spiral curves can be recognized as aesthetic curves (Ziatdinov et al., 2013; Harary and Tal, 2012; Yoshida and Saito, 2006).

Besides, several researchers also figured out that transition and spiral curves contain an equivalence relation for example, geometric smoothness has been used effectively to design these curves and both are highly useful for the same engineering fields (Levien, 2008; Perco, 2006; Kimia et al., 2003). Nevertheless, spiral curve requires some extensions on its curvature profile (to ensure either MC or not).

2.4 Introduction to Clothoid Templates

Clothoid or circle to circle templates have been introduced to the world by Baass (1984). These templates are firstly utilized in highway design to obtain two major outcomes for instance, to enhance the quality, comfortable and safe driving to the users as well as to design more natural alignments such as in highways that are suitable for its surrounding area while traditional approaches consist of straight line and circular arc (also known as horizontal alignment) (Figure 2.7) are difficult to

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Figure 2.7: Horizontal alignment in highway design (Kang et al., 2012)

According to the design standards of modern highway, it greatly needs a smooth connection (transition) in such cases for example, straight lines to circles or circles with different radius (AASHTO, 1984; RTAC, 1976). Based on this reason, clothoid is preferred due to its curvature uniqueness (Figure 2.6) and to ensure the transition can be generated. The five cases of clothoid templates in highway design are then proposed with the related procedure manual (Baass, 1984; Meek and Walton 1989). Figure 2.8 shows one of the highway designed alignment models using clothoid templates.

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Figure 2.8: The use of clothoid templates in highway designed alignment model (Walton and Meek, 1989)

The description of the cases is shown as follows: (1) To connect between straight line and circle; (2) to connect between two circles with a C-transition (broken back); (3) to connect between two circles with an S-transition; (4) to connect between two straight lines and finally, (5) to connect between two circles with a C- transition (spiral) (Baass, 1984). Extensively, these templates are also explored using Bézier curve such as in (Cai and Wang, 2009; Walton and Meek, 1996a; Walton and Meek, 1999; Sakai, 2000; Li et al., 2006; Habib and Sakai, 2009).

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2.4.1 Straight line to circle

A single spiral (with two segments applied) is produced throughout this case.

Vector concepts for instance, unit tangent and normal vectors have been used to complete this case with CBC and Pythagorean hodograph quintic curve (PHQC) as the functions employed (Walton and Meek, 1996a; Walton and Meek, 1996b). Habib and Sakai (2004) simplified the Walton’s method using polar coordinates together with the notations expressed as

⎩⎨

=

=

=

= =

0 ) 0 (

&

1 ) 0 ( if decreasing is

curvature

1 ) 1 (

&

0 ) 0 ( if increasing is

curvature

case κ κ

κ κ

r

r (2.19)

where an angle, θ falls in the first quadrant and r is the circle’s radius. Moreover, this case is also demonstrated using the application of clothoid (2.18) as described by (Meek and Walton, 2004). Figure 2.9 depicts an example of this case.

Figure 2.9: First case of circle to circle templates using CBC (Habib and Sakai, 2004)

2.4.2 Circle to circle with a C-shaped transition

The second template is initially designed by involving a pair of curves whose first appearance is transition. This case also occurs due to the extension of case 1 (circles are placed on the first and second quadrants) (Meek and Walton, 1989). The

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use of fair curves in this template has been discussed completely by Walton and Meek (1999). Fair curves must satisfy the condition such that only one interior extrema (minimum or maximum) exists in the curvature plot. Besides, Walton et al.

(2003) studied to produce a spiral connection between the circles with the reduction for segmentations used (Figure 2.10). Previously, this template applies three segments to generate a curve. The following continuity is fully employed in designing this template or documented as a C-shaped transition curve (Habib and Sakai, 2005a)

1 0 & (1) 1 1

) 0

( = r κ = r

κ (2.20) where r0 and r1 are the radii of the circles.

Figure 2.10: Concept of spiral connection in the second case template using CBC (Walton et al., 2003)

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2.4.3 Circle to circle with an S-shaped transition

This case has been effectively formed by the properties of case 1 (circles are situated in the first and third quadrants) (Figure 2.11). It possesses the identical characteristics when compared with the second case for example, the involvement of pair curves and the associated segmentation process in an S-shaped curve design.

Many studies have found that this circle template has no extrema (either minimum or maximum) along the curvature plot (Cai and Wang, 2009; Habib and Sakai, 2003;

Walton and Meek, 1999). The common functions employed in designing an S-curve are CBC and quintic Bézier curve (QBC). The existence of curvature extrema in this case 3 also explored using Said-Ball cubic curve (SBC). These descriptions have also been agreed upon by Kneser’s theorem and Rashid and Habib (2010). This case utilizes the following curvature continuity (Habib and Sakai, 2009), denoted by

1 0 & (1) 1 1

) 0

( = r κ =− r

κ (2.21)

Figure 2.11: The design of third case template

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2.4.4 Straight line to straight line

Both lines in this case are originally placed with an angle measured between the first and second quadrants (Figure 2.12). The problem occurs when connecting them. Walton and Meek (1996a) depict that case 4 has no unique (several) solution for the connection. This explanation is also supported by Meek and Walton (1989) and Brezak and Petrović (2011) in their respective work. Curvature (G2) continuity is the common degree of smoothness used for this case. The continuity is expanded to

⎩⎨

=

=

=

= =

0 ) 0 (

&

1 ) 0 ( if decreasing is

curvature

1 ) 1 (

&

0 ) 0 ( if increasing is

curvature Property

κ κ

κ

κ (2.22)

Figure 2.12: Straight line to straight line with an angle, )

, (0π

Ω (Meek and Walton, 1989)

The extension of case 1 has been applied to demonstrate this template as shown in the studies by Ahmad and Ali (2008a) and Habib and Sakai (2005b). The third-order geometric (G3) continuity is employed to join between the lines. McCrae and Singh (2009) have concluded that G3 is a type of curvature continuity with constant changing rate. Figure 2.13 illustrates the use of G3 continuity in designing case 4 completely.

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Figure 2.13: The architecture of case 4 using G3 continuity (Habib and Sakai, 2005c)

2.4.5 Circle to circle with a C-shaped spiral

Walton and Meek (1996a) have highlighted that case 5 is sometimes difficult to seek out a solution due to its complicated form. In fact, there is no evidence to support the existence and uniqueness of this case until Habib and Sakai (2005d) have fully depicted it. This depiction applies the blends of G2 and G3 contact at the big and small circles (Figure 2.14). The G2 contact is usually referred to as (Dimulyo et al., 2009)

1 0 & (1) 1 1

) 0

( = r κ = r

κ (2.23)

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Figure 2.14: Circle to circle with a C spiral curve (Habib and Sakai, 2006)

Continuously, Habib and Sakai (2008) enhanced their studies over other existing methods such as in Dietz and Piper (2004) and Goodman and Meek (2007) by including some positional, tangential and curvature conditions in the template (case 5). Numerous functions namely CBC, PHQC, SBC and rational cubic curve (RCC) are often used in a C-shaped spiral design (Dietz et al., 2008; Ahmad and Ali, 2008b; Chan and Ali, 2012). RCC is a suitable function for this case resulting from its flexibility amongst them (polynomials) as pointed out by Dietz et al. (2008).

2.4.6 The application of clothoid templates

Kommer and Weidner (2007) admit that clothoids are well-established curves in roads, highways, subways or in railway designs since 19th century. Presently, these usage still continue to play a vital role in many countries such as United Arab Emirates (Department of Transport, 2013). Tanzania’s Ministry of Works (2011) emphasized that the use of these templates gives a comfortable and safety to the road

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users (the accident frequencies are significantly reduced as claimed by Elvik et al.

(2004)) as well as an aesthetic appearance in the design. This is related to the occurrence of road shape preservation since the tangent line and circular curve are joined by clothoid templates (Figure 2.8).

The sustainable behaviour and less optical flow (visual motion) phenomena may emerge once these templates are fully employed in road or highway development (Koorey, 2009; Yin and Mourant, 2009). German Autobahn for instance, is an outstanding road network design with its unique attribute “no speed limits” deploys the clothoid templates effectively (Zeller, 2007). Meanwhile, Vermeij (2000) and Shen et al. (2013) remarked that clothoid curves are also usable in designing a high-speed track particularly for railways. The curves have an ability to identify and suggest a suitable superelevation rate.

Path planning (virtual computing) is another example of utilizing the clothoid templates for many years. High speed machining (HSM), unmanned aerial vehicles (UAV) plus steering and navigation models are the existence studies engaged with these templates (Yao and Joneja, 2007; Dai and Cochran, 2010; Mohovoć et al., 2012). The findings of these studies have been used during the developments of computer numerical control (CNC) machine, robot, ship, drone and global positioning system (GPS) navigation device. Brezak and Petrović (2011) verify that the suitability of clothoid curve in this field is due to its property (Figure 2.6) in producing a smooth path. This property is easily controlled when compared to the properties of spline and Bézier forms. In optical communications, clothoid curves are applicable to reduce the transition (change) loss between straight and curve

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waveguides (Figure 2.15). This reduction preserves the stability of optical power (Lin et al., 2009).

Figure 2.15: Transition loss differs greatly between clothoid and circular curves (Lin et al., 2009)

Besides, Harary and Tal (2012) together with Bertails-Descoubes (2012) have asserted that clothoids or Euler and cornu spirals are the indispensable curves in shape completion fields. These fields relate to complete or modify or repair the shape model and in conserving the product originality such as an artifact. The broken of Helenistic and oil lamps are the recent artifacts exploited through this spiral curve.

Kimia et al. (2003) and Cao et al. (2011) stress that the main reason of this exploitation is due to its natural and extensibility features. The use of clothoid curves is also expandable via the field of optical physics. An edge diffraction pattern analysis in such ultrasound system is one of the studies applying these curves (Hitachi, 2012; Born and Wolf, 1999). These curves have the capability to yield a visually pleasant display of the diffraction effects. Besides, Avila and Castano (2010)

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identify that clothoid curves are useful in this physics field because of their geometry and Fresnel properties.

The use of clothoid curve is therefore, being continued in roller coaster loop shape design. This design is firstly invented by Stengel in 1975 through the project entitled “Klothoide-shape loop” (Figure 2.16) (Baine, 2004). A beauty of mathematics and physics lives on this innovation. Good fit and the constant gravity force (G) may remain the same (suitable for heavyweight user) are the cause of using clothoid curve (Pendrill, 2005).

Figure 2.16: The first prototype of roller coaster model using the clothoid loop approach (Wild, 2012)

Simultaneously, the spur gear tooth profiles are also formed by utilizing a cornu spiral or clothoid curve (Kanehiro et al., 2012); however, this invention distinguishes from our studies (chapters 3 and 4). The clothoid properties are generally applied in Kanehiro’s patent while chapters 3 and 4 consist of a combination between parametric Bézier curves and circle to circle (clothoid)

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templates or in short, the use of Fresnel integral is excluded. Hence, the gear descriptions, in particular its background will be further explained below.

2.5 Gear Descriptions 2.5.1 History of gears

Gear is always referred to as a crucial mechanism used to transmit a mechanical power amongst the machines. This mechanism is commonly designed either in pair or in single teeth structure depending on its usage characteristics.

Historically, gear mechanisms have been utilized for many centuries by the Greek, Egyptian, Babylonian and Chinese (Radzevich, 2012; Ugural, 2003). For example, Chinese employs a wooden gear set since 260BC (Figure 2.17).

Figure 2.17: The Chinese chariot model using a wooden gear set (Paz et al., 2010)

According to Sorge (2011) and MacNeil (2013), the precision gear wheels embellished with the corroded bronze and wood was discovered by a Greek archeologist on May 17, 1902. This discovery is then recognized as Antikythera mechanism, an oldest complex scientific instrument.

Rujukan

DOKUMEN BERKAITAN