INTEGRATING SPUR GEAR TEETH DESIGN AND ITS ANALYSIS WITH G2 PARAMETRIC BÉZIER-LIKE CUBIC TRANSITION AND SPIRAL CURVES.

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YAHAYA1_{, S.H., ALI}2_{, J.M., YAZARIAH}3_{, M.Y., HAERYIP SIHOMBING}4_{, and YUHAZRI}5_{, M.Y.} 1, 4, 5

Faculty of Manufacturing Engineering Universiti Teknikal Malaysia Melaka

Hang Tuah Jaya, 76100 Durian Tunggal, Melaka, MALAYSIA

[email protected]

iphaery@ utem.edu.my

[email protected]

2, 3

School of Mathematical Sciences University of Science Malaysia Minden, 11800, Penang, MALAYSIA

[email protected]

1.0 INTRODUCTION

Gear s ar e the most w idely used elements in both applications such in consumer and industr ial machiner ies. As w e know , gear types may be gr ouped into five main categor ies, namely, spur , helical, r ack and pinion, w or m and bevel. As r efer r ed to in (Babu and Tsegaw , 2009; Bradfor d and Guillet, 1943; Higuchi and Gofuku, 2007), these ar e the most common cur ves used for the gear tooth pr ofiles. These cur ves ar e developed based on the appr oximation theor y, for instance; the development of involute cur ves has used a Chebyshev appr oximation (Higuchi and Gofuku, 2007). Besides, the tr acing point method has also been applied along the path (shape) design of involute cur ves (Mar galit, 1995; Reyes et al., 2008; Yeung, 1999). Ther efor e, sever al methods and concepts have been employed in the gener ation of involute cur ves. With r efer ence to the above evidence, it show s that this involute cur ve is not dir ectly pr oduced and is show n as the appr oximated (inexact) cur ves. For these r easons, w e pr opose the theor etical developments of the exact (or know n as the tr ansition) cur ves using the par ametr ic function.

Mathematically, parametr ic function is a method to define a r elation betw een the independent (fr ee) var iables. Pr eviously, Ali (1994) and Ali et al., (1996) explor ed the par ametr ic of Bézier -like cubic cur ve using Her mite inter polation. They ar e, how ever , only focused on the function developments, w hile the scope of designs thr ough the pr oposed cur ves is not touched upon. Ther efor e, in this study, w e use the Bézier -like cubic cur ve appr oach as the degr ee thr ee (cubic) polynomial cur ves that allow the inflection points. This is due to the appr oach is suitable for G2_{(cur vatur e) blending application cur ves and also contains the shape par ameter s}

w hich can contr ol the shape of the cur ve (Ali, 1994; Ali et al., 1996; Walton and Meek, 1999). As compar ed to the cubic Bézier cur ves, the shape par ameter s ar e not automatically included (Rashid and Habib, 2010; Habib and Sakai, 2008). Figur e 1 show s the method of designing the tr ansition cur ves w ill follow the five cases of clothoid templates as w as identified by Baass (1984) and successfully used in highw ays or r ailw ays design. These templates ar e cr ucial to deter mine the design par ameter s in or der to ensur e the comfor t and safety of r oad user s (Baass, 1984). The templates ar e 1) str aight line to cir cle, 2) cir cle to cir cle w ith C tr ansition, 3) cir cle to circle w ith an S tr ansition, 4) straight line to str aight line and 5) circle to cir cle w her e one cir cle lies inside the other w ith a C tr ansition (Walton and Meek, 1999; Baass, 1984; Walton and Meek, 1996). Figur e 2 show s the pr ofiles

A BSTRA CT

An involute curve (or known as an approximated curve) is mostly used in designing the gear teeth (profile) especially in spur gear. Conversely, this study has the intention to redesign the spur gear teeth using the transition (S transition and C spiral) curves (also known as the exact curves) with curvature continuity (G2_{) as the degree of smoothness. Method of designing the transition curves is}

adapted from the circle to circle templates. The applicability of the new teeth model with the chosen material, Stainless Steel Grade 304 is determined using Linear Static Analysis, Fatigue Analysis and Design Efficiency (DE). Several concepts and the related examples are shown throughout this study.

Key w ords: Spur Gear Profile, S-Transition Curve, C-Spiral Curve, FEA, DE.

INTEGRATING SPUR GEAR TEETH DESIGN AND ITS ANALYSIS

WITH G

_{PARAMETRIC BÉZIER-LIKE CUBIC TRANSITION AND}

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design w her e the thir d and fifth templates ar e similar to the involute cur ves. In this study, these tw o templates w ith the application of Bézier -like cubic cur ve function ar e ther efor e chosen to r edesign a spur gear teeth pr ofile. By using the new method, the cur ve is dir ectly gener ated w ith the significant incr eases in accuracy, and also the actual spur gear pr ofile can be pr oduced thr ough this method.

This study w ill continue to find out the applicability of the new pr ofile gener ation and the gear mater ial thr ough Str ess-Str ain Analysis, Fatigue Analysis and DE. FEA is the most common tools for the analysis w hile Stainless Steel Gr ade 304 is chosen for the selection of mater ial in this study. This gear mater ial is often selected for gear application since it has an aesthetic appear ance, ease of fabr ication and good in impact r esistance. The scheme in Str ess-Strain Analysis know n as a Linear Static Analysis is used to deter mine many of the physical str uctur es such in the str ess, for ce and displacement distr ibutions (Westland, 2006; Sapto and Safar udin, 2008). Many of the studies use this linear scheme to deter mine the str uctur al applicability betw een the involute pr ofiles and the gear mater ial (Feng, 2011; Gur umani and Shanmugam, 2011; Reagor , 2010). How ever , ther e is no r elated study for the pr oposed cur ves. The r etr ieved values fr om Linear Static Analysis w ill be fur ther explor ed in Fatigue Analysis and DE w her e both ar e str ongly connected to the safety factor (Moultr ie, 2009; Khai

et al., 2007; Nieder stucke et al., 2003; Fir th and Long, 2010). Incidentally, the descr iptions of DE in the design ar ea ar e still less-decr ypted. Figur e 3 show s one of the applications using Linear Static Analysis. The next section w ill explain the nomenclatur e used in this study.

Figure 1: Railways route design modeling using transition curves application.

Figure 2: Third (left) and fifth (right) cases of circle to circle templates. (Baass, 1984; Walton and Meek, 1996; Meek and Walton, 1989).

Figure 3: Automotive crankshaft in Linear Static Analysis.

2.0 NOMENCLATURE

Consider the notations such as symbol and unit in the nomenclatur e, as show n in Table 1. This nomenclatur e is the basic foundation of Linear Static Analysis (M Br itto, 2005).

It is noticed that the unit of these input and output par ameter s w ill be based on the system of units that ar e applied in the solid model. Table 1 is extensively used for the analysis pur poses. In the next section, w e w ill explor e the descr iption method of the designing of spur gear using S tr ansition and C spir al cur ves.

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Table 1: Some of the parameters that are commonly used in the system of units

System of Units

I nput Par ameter s Output Par ameter s Lengt h, ζ For ce, ƒ Mass, η Density, ρ Displacement, δ For ce, ƒ Str ess, υ

1 m N kg kgm-3 _m _N _Pa

2 mm N mg mgmm-3 _mm _N _MPa

3 ft Ibf slug slugft-3 _ft _Ibf _psf

4 in Ibf Ibf.s2_{i n}-1 _Ibf.s2_{i n}-4 _in _Ibf _psi

3.0 METHOD OF DESCRIPTIONS

3.1 Bézier-Like Cubic Curve as a Function

Bézier -like cubic cur ve (also know n as the cubic alter native cur ve) is a new basis function in the field of Computer Aided Geometr ic Design (CAGD). Bézier -like basis function simplifies the pr ocess of contr olling the cur ve since it has only tw o shape par ameter s, λ0 and λ1 to contr ol or change the shape of the cur ve

such in Figur e 4, compar ed to the cubic Bézier cur ve for w hich the contr ol points need to be adjusted (Farin, 1997). Bézier -like cubic cur ve is for mulated by applying the for m of Her mite inter polation given by (Ali, 1994; Ali et al., 1996; Ahmad, 2009).

1 0 , ) ( ) ( ) ( ) ( )

(t  ₀ t P₀ ₁t P₁ ₂ t P₂ ₃ t P₃ t

z     (1)

w ith, )) 1 )( 2 ( 1 ( ) ( , ) 1 ( ) ( ) 1 ( ) ( ), ) 2 ( 1 ( ) 1 ( ) ( 1 2 3 2 1 2 2 0 1 0 2 0 t t t t t t t t t t t t                     (2)

w her e P₀,P₁,P₂,P₃ ar e the contr ol points, and ()

3 ) ( 2 ) ( 1

0(t), t, t, t

 ar e Bézier -like cubic basis

functions.Hence, Bézier -like cubic cur ve can be w r itten as

3 1 2 2 2 1 1 2 0 0 0 2 )) 1 )( 2 ( 1 ( ) 1 ( ) 1 ( ) ) 2 ( 1 ( ) 1 ( )

(t t t P t tP t t P t t P

z             (3)

Figure 4: Distribution of Bézier-like cubic curves with the different values of λ0 and λ1.

In the next subsection, w e w ill explain the contr ol point s identification in Bézier -like cubic cur ve based on the cir cle to cir cle template (Figur e 2)

3.2 Third Case of Circle Templates as an S-Shaped Transition Curve

The cubic Bézier cur ve had been pr edominantly used in cur ve design, as show n in studies by Habib and Sakai (2003) and Walton and Meek (1999). The similar ity betw een this function and Bézier -like cubic cur ve is that both functions have four contr ol points. By r efer r ing to Habib and Sakai (2003), the contr ol points ar e stated as

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} cos , sin { * }, sin , {cos * } cos , sin { * }, sin , {cos 3 2 1 1 3 0 1 * 0 0 0                   k P P r c P h P P r c P (4)

w her e, c0 and c1are the centre points of circles ψ0and ψ1, r0 and r1are the radii of circles ψ0 and

ψ1, h and k ar e the length or nor m of P₁, P₀ and P₃, P₂ w hile α is an angle's cir cles measur ed

anticlockw ise. These contr ol points ar e unique only in the case of the S-shaped tr ansition cur ve. The impr ovements ar e made in Eq. (4) to incr ease the degr ee of fr eedom and its applicability such

} cos , sin { * }, sin , {cos * } cos , sin { * }, sin , {cos 3 2 1 1 3 0 1 * 0 0 0                   k P P r c P h P P r c P (5)

with the angles in the circle {ψ0, ψ1} denoted as α and β w hich ar e measur ed anticlockw ise. The

other par ameter s r emain the same as in Eq. (4). The value of h and k w ill be calculated by using the cur vatur e continuity (G2_{continuity) show n as}

1 0 1 ) 1 ( , 1 ) 0 ( r t r

t    

 (6) By applying these theor ies, an S-shaped cur ve is demonstr ated in Figur e 5.

Figure 5: An S-shaped transition curve produced by using G2_{Bézier-like cubic curve.}

3.3 Fifth Case of Cir cle Templates as a C-Shaped Spir al Curve

The fifth case w ill pr oduce a type of C-shaped tr ansition cur ve as a single spir al (Baass, 1984; Walton and Meek, 1996; Habib and Sakai, 2005). The cur ve ar chitectur e in tr ansition and spir al cur ves is totally differ ent fr om the segmentations used. As show n in Figur e 6, thr ee segments ar e needed to design C-shaped tr ansition cur ve w her eas only tw o segments ar e used in C-C-shaped spir al cur ve.

Figure 6: Transition (left) and spiral (right) curves architecture applied in cubic Bézier curve (Walton et al., 2003).

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} cos , {sin * }, sin , {cos * } cos , sin { * }, sin , {cos * 3 2 1 1 3 0 1 0 0 0         k P P r c P h P P r c P           (7)

Wher e the notations ar e the same as in Eq. (4). The definitions in Eq. (7) w ill pr oduce a C-shaped tr ansition cur ve w hich is r elated to the second case (Walton and Meek, 1999; Baass, 1984 ; Walton and Meek, 1996). Modifications of Eq. (7) ar e needed for fifth case template, and ther efor e, w e have

} cos , sin { * }, sin , {cos * } cos , {sin * }, sin , {cos * 3 2 1 1 3 0 1 0 0 0                   h P P r c P k P P r c P (8)

w her e {c0, r0,β} is the centre point, radii and angle of the circle, ψ0 w hile the centr e point, r adii and

angle in the circle, ψ1 r epr esented as {c1, r1,α}. Par ameter h and k ar e the length or nor m of P₃, P₂

and P₁, P₀ . In or der to design C spir al cur ve, the segments used in Eq. (8) should be r educed (Walton et al., 2003) by assuming that

1 P

P  (9) Then, in Eq. (9) either h or k can be eliminated by using a dot pr oduct. If the par ameter chosen is k, the expr ession w ill be dotted w ith a vector ,

{cos



,

sin



}

w her e

] sin[ ] cos[ * }) sin , {cos )

((₁ ₀ ₁ ₀

          

 c c



r r

k (10)

The modified contr ol points that satisfy the fifth case template (C spir al cur ve) ar e given by

} sin , {cos * } cos , {sin * } sin , {cos * 1 1 3 0 1 0 0 0       r c P k P P r c P        (11)

In or der to join C spir al cur ve w ith the circles, w e w ill need to apply the cur vatur e continuity such

1 0 1 ) 1 ( , 1 ) 0 ( r t r

t    

 (12) This continuity is also used to deter mine the shape par ameter s, λ0 and λ1 in Eq. (3). A gener ated

C-shaped spir al cur ve is visualized in Figur e 7.

Figure 7: C-shaped curve in the fifth case of circle templates using G2_{Bézier-like cubic curve.}

Both cur ve theor ies w ill be applied to r edesign a pr ofile (tooth) of the spur gear . Cur r ently, an involute or evolut e cur ve is alw ays used in designing this spur tooth pr ofile.

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4.0 SPUR GEAR DESIGN

4.1 Modeling of Spur Gear

Tr aditional spur gear s have the teeth w hich ar e str aight and par allel to the axis of the shaft that conveys the gear (Mott, 2003). Nor mally, the teeth have an involute for m w her e this for m can be acted as in contacting the teeth (Mott, 2003). Figur e 8 show s a schematic and r elated ter minology for spur gear

Figure 8: Some of the terms in spur gear (Price, 1995).

Based on Figur e 8, an inner cir cle and an outer cir cle can be dr aw n, and smaller cir cles can be fitted w ithin their boundar ies, as show n in Figur e 9. This r esulting str uctur e w ill be used to design a pr ofile (tooth) of the spur gear .

Figur e 9: A new basis model of desi gning a spur gear tooth.

4.2 Single Tooth Design uses an S-Shaped Transition Curve

By applying the segmentation of divisions, tw o segments ar e divided to cr eate this single tooth. This is because an S-shaped cur ve has the beginning point at the tangent of base cir cle and ending at the tangent of the outside circle. For the fir st segment, the inputs ar e c0 = {-0.398, 0.689}, c1 = {0, 0.795}, α= 0.6667π

r adian, β = 0.5π radian, r0 = r1 = 0.206 ar e used, w hile for segment tw o, the inputs ar e c0 = {0.398, 0.689}, c1 = {0, 0.795}, α= 0.3333π radian, β = 0.5π radian and other parameters remain the same as in segment

one. Then, h and k ar e deter mined to be 0.2779 and 0.3780 in accor dance w ith the inputs and theor ies above. Notice that, the value of h and k is similar in both segments. Figur e 10 show s the design of the tooth and its segmentation using an S-shaped tr ansition cur ve.

Figur e 10: Single tooth of spur gear using an S-shaped tr ansit ion cur ve (left) and i ts segment ation (r ight).

4.3 Single Tooth Design uses a C-Shaped Spiral Curve

In this case, Figur e 9 is modified to suit the elements i n designing a C-shaped spir al cur ve (Subsection 3.3). The small cir cles ar e dr aw n at the inter section of tw o circles. For this cur ve, it needs four segments to design a single tooth of the spur gear . Segment one consists of c0 = {-0.398, 0.689}, c1 = {-0.247, 0.725},

α= 0.05556π radian, β = 0.6667π radian, r0 = 0.206 and r1 = 0.05. The values of k and shape par ameter s,

λ0, λ1 ar e computed by using the Eqs. (3), (10), (11), and (12). We found that k = 0.1431 w her eas λ0 and λ1

ar e equal to 2.7695 and 0.3619, r espectively. For segment tw o, the inputs ar e c0 = {0, 0.795}, c1 = {-0.247,

0.725}, β = 1.5π radian while the r est is exactly the same as in segment one. This time, k and shape par ameter s, λ0, λ1 ar e found to be 0.2449, 2.1279 and 0.5343, r espectively. These tw o segments have a

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CATI A V5 (Develop t he solid mo del of t he spur

gear ) CATI A V5

(All co ord inat es ar e joi ned using spl ine package) CATI A V5

( Coor dinat es defin it ion in accor d ance t o th e t oo th

pro file in Wo lfr am Mat hem at ica 7.0) W ol f ram

M athemat ica 7 .0 (Ap ply t he cu rve th eo ries

in d esigni ng t he t oo th pr ofi le o f th e spur gear )

symmetr ical cur ve w hich can be defined as mir r or ing or balancing to segment thr ee and four . Thus, segment thr ee and four have the similar shape to segment one and tw o, r espectively (Du Sautoy, 2009). Figur e 11 show s the design of the tooth and its segmentation using this C spiral cur ve. Next, w e w ill apply the single tooth designs using both developed cur ves in or der to design a solid model of the spur gear .

Figur e 11: Single tooth of spur gear using a C-shaped spir al cur ve (left) and it s segmentat ion (r ight).

4.4 S Transition and C Spiral as the Shaped Curves in Solid of Spur Gear

In Computer Aided Geometr ic Design (CAGD), the sur face design technique is commonly used w her e this technique has sever al design and sur face pr oper ties to be consider ed (Bloor et al., 1995), difficult to extr ude as a solid model and unsuitable in Computer Aided Analysis (CAE) pur poses. To solve these pr oblems, the integr ation of Mathematical and Computer -Aided Design (CAD) may be applied. Wolfr am Mathematica 7.0 and CATIA V5 ar e both selected as the integr ated softw ar e of the designing of the solid model of the spur gear . The follow ing pr ocess flow diagram is used to show the integr ation of both softw ar e and its applications in spur gear design (Figur e 12). Tw o models of six gear teeth ar e developed w ith the outside and shaft diameter s as w ell as the gear thickness ar e 40.04 mm, 5.672 mm and 4 mm (Figur e 13).

Figur e 12: I ntegr ated pr ocess flow diagr am used in designi ng a soli d model of t he spur gear .

Figur e 13: Solid of spur gear usi ng S tr ansition (left) and C spir al (r ight) cur ves.

We w ill extend these solid design str uctur es to analyze its applicability of the new tooth pr ofiles w ith the gear mater ial using Linear Static Analysis.

5.0 LINEAR STATIC ANALYSIS

5.1 Fundamental of Linear Static Analysis

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Patr an

Pr ocesses: I mp ort 3D m odel , meshin g pr ocess, mat er ial select ion,

lo cat e t he lo ad and bo und ar y conditi ons.

Na str an

Processes: Cr eat e the St iff ness or Squ are Mat rix, calculat e the

displacement , st r ess an d for ce.

Pa tr an

Pr ocesses: Plot t he cont ou r gr aph of th e st r ess, di splacement an d fo rce

d ist rib ut ions.

assumptions ar e made such as linear behavior , elastic mater ial and static loads (Ziaei Rad, 2011). The scenar io of the assumptions is show n in Figur e 14. One of the applicabilities of Linear Static Analysis is to find out the str uctur al r esponse of the bodies' spinning w ith the effect of the velocities or acceler ations ar e constants since the applied loads ar e static (do not change w ith time). This analysis measur es the r esponses of the displacement, str ain (defor mation), stress and r eaction for ce. The linear static pr oblem can be simplified using the expr ession of the linear matr ix system such

} { } ]{

[K x  F (13) w her e,

[K] is the str uctur al (assembled) stiffness matr ix or squar e matr ix of or der (n x n), {x} ar e the unknow n par ameter s (displacement, strain and str ess) of or der (nx 1) w hile {F} is the loading in the system r epr esents the matr ix of the or der (nx 1). For t he solution of {x}, the matr ix solver that w ill be used is FEA. The next subsection w ill explain the FEA solver and its str uctur al modeling.

Figur e 14: Li near i ty and elastic mat er ial assumptions in the st atic analysis (Hambur ger , 2000).

5.2 FEA Solver and I ts Modeling

MSC Nastran & Patr an is the FEA softw ar e chosen as the pr efer r ed solution for Linear Static Analysis. The follow ing pr ocess flow diagram is the FEA pr ocesses that w ill be applied using this softw ar e, as show n in Figur e 15 (Sapto and Safar udin, 2008; Dar yl, 2006).

Figur e 15: Pr ocess flow diagr am of MSC Nastr an & Patr an and it s FEA application.

5.2.1 CAD models and its meshes

The CAD or solid models (Figur e 13) w ill be impor ted to MSC Patr an softw ar e as the star ting pr ocess of the Linear Static Analysis (FEA). The selection of the element model in FEA is ver y cr ucial and w ill be deter mined the over all pr ocess of the analysis. The gear str uctur es (Figur e 13) ar e the Thr ee-Dimensional (3D) geometr y solid models that r eveal the complex str uctur es. Solid element models ar e w idely used to analyze the complex elements such as str uctur al components and loading conditions and also the estimation of the stress levels (Fellipa, 2009; Entr ekin, 1999). The element topology that is chosen for the str uctur e (Figur e 13) is Tetr ahedr al-10 (Tet-10) that r epr esents a 3D solid tr iangle w ith four planar faces and ten nodes (Fellipa, 2009; Fellipa 2011), as show n in Figur e 16. The Tet-10 or second or der tetr ahedr al element is commonly used for its ability to mesh almost any solid r egar dless of its complexity (Entr ekin, 1999; Said et al., 2012). The r esulting of the applied Tet-10 is show n in Table 2 and Figur e 17.

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Table 2: Tet-10 and its mesh data structures

Solid of the spur gear (S transition) Solid of the spur gear (C spiral)

Global Edge Length Nodes Elements Global Edge Length Nodes Elements

1.0551234 34897 22448 1.011111 34854 22668

Figur e 17: Tet-10 and its spur gear meshi ng of the appl ied cur ves, S tr ansit ion (left) and C spir al (r ight).

5.2.2 Boundary and loading conditions

The next pr ocess is the setting of the boundar y and loading conditions on the solid model (Figur e 17). Boundar y condition can be r efer r ed as the exter nal load on the bor der of the str uctur e (Feng, 2011). As this case, the boundar y condition is applied on the gear shaft (Figur e 17) w her e in this shaft, the displacement does not happen (is equivalent to zero) (Nikolić et al., 2012; Lee, 2009). This is the impor tant setting to simulate the gear tr ansmission (Feng, 2011). Many of the r efer ences used Tor que (T) as the applied load (loading condition) for their analysis and application (Feng, 2011; Nikolić et al., 2012; Lee, 2009). In or der to find out the new tooth pr ofiles and its applicability (Figur e 13), the suitable load to be applied is Pr essur e (P) w hich this load is set on the contact str ess of both spur gear models (Figur e 17) (Gilber t Gedeon, 1999; Wang et al., 2011). The cr itical r egion of the contact str ess is show n in Figur e 18 (Rameshkumar et al., 2010). Figur e 19 show s the setting of both conditions in the models (Figur e 17). One of the conditions, P is applied as 50 MPa

Figur e 18: The cr itical r egion (tw o dots along the blue li ne) of t he contact str ess i n t he tooth pr ofile (Rameshkumar et al., 2010).

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5.2.3 Material selection for spur gear

Stainless Steel Gr ade 304 also know n as AISI 304 is w idely used in the r ange of applications such as a gear mater ial (Zehe et al., 2002). This gr ade of mater ial is the type of Austenitic Stainless Steels for the highly contained of Chr omium and Nickel. These Steels have high ductility and ultimate tensile str ength and also low yield str ess. Figur e 20 show s one of the cases of Austenitic Stainless Steels w hen compar e to a Typical Car bon Steel. It can be seen clear ly that the stability r egion is occur r ed in the types of Austenitic w hile the Car bon type has an instability diagram (Figur e 20). The mechanical pr oper ties of the AISI 304 ar e pr esented in Table 3. These pr oper ties w ill be used in this static analysis.

Figur e 20: Str ess-str ain behavior of the Austenitic and Car bon Steels.

After applying the above settings on the models (Figur e 19), it is r eady to analyze using MSC Nastran & Patr an softw ar e (Figur e 15). The pr ocessor of Intel ® Cor e ™ 2 Duo CPU w ith the RAM, 3.49 GB is used to oper ate the softw ar e. The r esulting of this pr ocess w ill be discussed in the next subsection.

Table 3: AISI 304 and its characteristics (Peckner and Bernstein, 1977) Modulus of

Elasticity

Yield Strength

Ultimate Strength

Poisson

Ratio Density

Damping Coefficient

195 GPa 215 MPa 505 MPa 0.29 8 gcc-1 _0.003

5.2.4 Simulation results and its safety factor

The discussion began w ith the pr esentation of the gener ated r esults using Linear Static Analysis. These r esults of the pr oposed models w ill be compar ed w ith the existing model. The existing model (EM) is dir ectly designed using the gr aphical pr oper ties i n CATIA V5 w ith the dimensions r emained the same. Table 4 show s the maximum Von Mises str ess (υmax) and maximum displacement (δmax)

distr ibutions among the models thr ough the r epeated loads. Hence, the safety factor (Sf) or also

know n as a design margin (Michalopoulos and Babka, 2000) can be der ived as in Eq. (14) (Cliffor d

et al., 2008; Shenoy, 2004) and w her e the yield str ength of AISI 304 can be found in Table 3.



max

304 AISI of Strength Yield

S  (14)

Table 4: The result distributions of δmax, υmax and Sf in the applied models

P (MPa)

EM S Transition C Spiral

δmax

(mm)

υmax

(MPa) Sf

δmax

(mm)

υmax

(MPa) Sf

δmax

(mm)

υmax

(MPa) Sf

10 5.93E-3 61.3 3.507 5.99E-3 61.9 3.473 5.77E-3 62.5 3.440 20 1.19E-2 123 1.748 1.20E-2 124 1.734 1.15E-2 125 1.720 30 1.78E-2 184 1.168 1.80E-2 186 1.156 1.73E-2 187 1.150 40 2.37E-2 245 0.878 2.40E-2 248 0.867 2.31E-2 250 0.860 50 2.97E-2 307 0.700 3.00E-2 309 0.696 2.89E-2 312 0.689

Figur e 21 depicts the contour plot of deter mining the pr esent ed values of υmax and δmax

(Table 4) for the applied P = 30 MPa. It is clear ly show n in the plot legend that δmax appr oximates to

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(r ed color ) occur s in the gear shaft because of the model (Figur e 17) is analyzed in the steady state (static) behavior (Boulos et al., 1997). The resulting of υmax (P = 30 MPa) is far below the yield

str ength of AISI 304. Ther efor e, Sf appr oximates to 1.15 (Table 4). Failur e occur s if and only if Sf < 1

or Sf has the negative sign (Michalopoulos and Babka, 2000). All gear models in Table 4 ar e safe to

be used if P 30 MPa w her eas the models ar e in failure modes if P > 30 MPa. These inequalities show that the new teeth pr ofiles (S tr ansition and C spir al cur ves) have the same applicability (str ength) w hen compar e to the EM. The findings in Table 4 w ill be used to pr edict w hen the failur e (fatigue) modes star t to occur among the gear models. This fatigue pr ediction w ill be discussed in the next section.

Figur e 21: The example of findings {δmax, υmax} for the appl ied P = 30 MPa in C spir al gear model.

6.0 FIRST-ORDER NEWTON INTERPOLATING POLYNOMIAL AS A FATIGUE PREDICTOR

Fatigue can be descr ibed as a failur e mode that happened on the str uctur e design w hich this str ucture is affected by the implications of r epeated or var ying loads, fluctuating loads and r apidly applied loads (Pun, 2001). Fatigue also can be occur r ed by the differ ent physical mechanisms such as low -cycle and high-cycle fatigue (Pun, 2001). As r efer r ed to in failur e mode above, let assume that the fatigue star ts to occur if Sf = 1. Due to this Sf, the r elated

load w ill be pr edicted using the New ton’s Inter polating Polynomial. The gener al for m of this polynomial such (Chapra and Canale, 2010).

) )...( )( ( ... ) )( ( ) ( ) ( 1 1 0 1 0 2 0 1 0

n x b b xx b xx xx  bn xx xx xx_n_

f (15)

Wher e, n is r efer r ed to as the nth-or der polynomial, b0, b1, b2 and bn ar e the polynomial coefficients and

the set of (n+1) data points r epr esent ed by {xi, f(xi)} w ith i = {0, 1, 2, 3,…, n}. The polynomial coefficients also can

be show n as (Chapr a and Canale, 2010).

] , , [ ]; , , [ ]; , [ );

( ₀ ₁ ₁ ₀ ₂ ₂ ₁ ₀ ₁ ₁ ₀

0 f x b f x x b f x x x b f x x ,...,x x

b    n  _n _n_ (16)

And the br acketed functions ar e evaluated using the Finite Divided Differ ences w her e (Chapr a and Canale, 2010). 0 0 2 1 1 1 0 1 1 0 2 0 1 1 2 0 1 2 0 1 0 1 0 1 ] ,..., , [ ] ,..., , [ ] , , [ ; ] [ ] , [ ] , , [ ; ) ( ) ( ] , [ x x x x x f x x x f x x ,..., x x f x x x , x f x x f x x x f x x x f x f x x f n n n n n n n _             (17)

We can see clear ly in Table 4 that the fatigue of the models star ts to occur in the r ange of P [ 30, 40] that allow s only tw o data points. Ther efor e, the Fir st-Or der New ton Inter polating Polynomial or know n as Linear Inter polation is the suitable polynomial to be applied. Let P is denoted as x-axis w hile Sf is equal to f(x) or y-axis.

The fir st-or der or linear for m (n = 1) is given as

) ( )

( ₀ ₁ ₀

1 x b b x x

f    (18) With b0 and b1 ar e calculated using Eqs. (16) and (17). The algor ithms of using this linear inter polation

(12)

for f1(x) = 1. All calculated values of x among the models (Table 4) ar e fully depicted in Table 5. It show s that the

models ar e in safety mode if P 35 MPa w hile the fatigue may star t occur r ing w hen P > 35 MPa. Table 5: The Initial loads of P for the fatigue mode

EM S Transition C Spiral

35.7931 MPa 35.3979 MPa 35.1724 MPa

With the r efer ence of Table 5, the values in this table w ill be used to deter mine the Design Efficiency (DE) for the pr oposed models. Fur ther discussion of DE w ill be explained in the follow ing section.

7.0 DESIGN EFFICIENCY OF THE PROPOSED DESIGN MODELS

Design Efficiency (DE) or Sensitivity Analysis (SA) r elates to the design quality (Aas, 2000; Azarian et al., 2011) that can achieve the stated objectives of the cer tain studies such in this study, to find out the applicability of the new teeth design w ith AISI 304 as the selected mater ial. Gener ally, the levels of DE in the engineer ing field ar e almost 85 % and above. Hence, 85 % can be assumed as the benchmar k of the impr ovements that w ill be made (Gupta et al., 2006). Let make the assumptions in Table 5 that EM equivalents to the tr ue value, xt (input)

w her eas S Tr ansition and C Spiral ar e r epr esented as t he appr oximation values, xa (output). Thus, DE can be

expr essed as (Khai et al., 2007).

% 100 ) (

a _

 

x

Input Output

DE (19)

After applying Eq. (19), DE of an S Tr ansition appr oaches to 99% w hile for C Spir al method appr oximates to 98%. Both r esults have show n that DE almost achieved 100% of the design effectiveness. With r efer ence to the above mentioned benchmar k (85%), the pr oposed designs ar e the acceptable methods of designing the spur gear teeth w ith AISI 304 as the selected mater ial.

8.0 CONCLUSIONS AND FUTURE WORK

This study has pr oposed the designing of spur gear teeth using S-shaped tr ansition and C-shaped spir al cur ves (dir ectly pr oduced). Cir cle to cir cle templates have successfully been applied in both cur ves design. The solids of spur gear have been gener ated by the integr ation betw een Mathematical and CAD softw ar e. The applicability of the pr oposed design and the mater ial, AISI 304 is measur ed using Linear Static Analysis, Fatigue Analysis and DE. Fir st-or der New ton inter polating polynomial can be employed as a fatigue pr edictor for all design models. As r efer r ed to in Table 4, all models ar e in safety mode if P 30 MPa w hile in failur e mode if P > 30 MPa. Gener ally, fatigue star ts to occur w hen P > 35 MPa in all design models w her eas the models ar e safe to use in the r elated applications if P 35 MPa. The applicability (str ength) of the pr oposed design, S and C cur ves ar e the same w hen compar e to EM as show n in the above analyses. The new design methods, S and C cur ves ar e the acceptable methods of designing the spur gear teeth as both methods have pr esented DE gr eater than 85 % of the design effectiveness. In futur e, this study w ill continue in the analysis of dynamic and acoustic featur es such as nor mal, fr equency and tr ansient modes and also in the noise test.

ACKNOWLEDMENT

This r esearch w as suppor ted by Univer siti Teknikal Malaysia Melaka under the Fundamental Research Grant Scheme (FRGS). The author s gr atefully acknow ledge ever yone w ho contr ibuted helpful suggestions comments.

REFERENCES

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[ 2] Ali, J.M., Said, H.B. and Majid, A.A. (1996): Shape Contr ol of Par ametr ic Cubic Cur ves. SPIE, Four t h Int er nat ional Confer ence on Comput er -Aided Design and Comput er Gr aphics, Zhou, J. (Ed), Vol.2644, pp.128-133.

[ 3] Ahmad, A. (2009): Parametr ic Spir al and its Application as Tr ansition Cur ve. Ph.D Thesis, Univer siti Sains Malaysia, pp.19-22.

[ 4] Aas, E.J. (2000): Design Quality and Design Efficiency; Definitions, Metr ics and Relevant Design Exper iences.

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[ 5] Azar ian, A., Vasseur , O., Bennaï, B., Jolivet, V., Claeys-Br uno, M., Ser gent, M. and Bour don, P. (2011): Sensitivity Analysis of Inter fer ence Optical Systems by Numer ical Designs. Jour nal de la Sociét é Fr ançaise de St at ist ique, Vol.152, No.1, pp.118-130.

[ 6] Babu, V.S. and Tsegaw , A.A. (2009): Involute Spur Gear Template Development by Par ametr ic Technique using Computer Aided Design. Afr ican Resear ch Review, Vol.3, No.1, pp. 1-6.

[ 7] Br adfor d, L.J. and Guillet, G.L. (1943), Kinemat ics and Machine Design, John Wiley & Sons, New Yor k. [ 8] Baass, K.G. (1984): The Use of Clothoid Templates in Highw ay Design. Tr anspor t at ion For um, Vol.1,

pp.47-52.

[ 9] Bloor , M.I.G., Wilson, M.J. and Hagen, H. (1995): The Smoothing Pr oper ties of Var iational Schemes for Sur face Design. Comput er Aided Geomet r ic Design, pp.381-394.

[ 10] Boulos, P.F., Wood, D.J. and Lingir eddy, S. (1997): Pr essur e Vessels and Piping Systems-Shock and Water Hammer Loading. Encyclopaedia of Life Suppor t Syst ems (EOLSS), Amer ican Water Wor ks Association, UNESCO.

[ 11] Cliffor d, K.H., Bibeau, T.A. and Quinones, A.Sr. (2008): Finite Element Analyses of Continuous Filament Ties for Masonr y Applications: Final Repor t for the Ar qui n Cor por ation. SAND2006-3750, Sandia National Laborator ies Albuquer que, Califor nia, USA.

[ 12] Chapr a, S.C. and Canale, R.P. (2010): Numer ical Met hods for Engineer s, 6t h_Edn_{. McGr aw -Hill, New Yor k.} [ 13] Du Sautoy, M. (2009): Symmet r y:A Jour ney int o t he Pat t er ns of Nat ur e, Har per Collins, UK.

[ 14] Dar yl, L. (2006): A Fir st Cour se in Fini t e Element Analysis, Br ookscole, Singapor e.

[ 15] Entr ekin, A. (1999): Accur acy of MSC/ Nastran Fir st and Second Or der Tetr ahedr al Elements in Solid Modeling for Str ess Analysis. MSC Aer ospace User s’ Conference Pr oceedings.

[ 16] Far in, G. (1997): Cur ves and Sur faces for Comput er Aided Geomet r ic Design, A Pr act ical Guide, 4t h_{Edit ion,} Academic Pr ess, New Yor k.

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[ 19] Feng, X. (2011): Analysis of Field of Str ess and Displacement in Pr ocess of Meshing Gear s. Int er nat ional Jour nal of Digit al Cont ent Technol ogy and it s Applicat ions, Vol.5, No.6, pp.345-357.

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Eur ope’s Pr emier Wind Ener gy Event, 20-23 Apr il, War saw , Poland.

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Engineer ing Mechanics, Vol.18, No.1, pp.65-78.

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[ 25] Habib, Z. and Sakai, M. (2005): Cir cle to Circle Tr ansition w ith a Single Cubic Spir al. The Fift h IASTED Int er nat ional Confer ence Visualizat ion, Imaging and Image Pr ocessing, 7-9 Sept ember , Benidor m, Spain, pp. 691-696.

[ 26] Habib, Z. and Sakai, M. (2008): G2_{Cubic Tr ansition bet w een Tw o Cir cles w ith Shape Contr ol.}_{Jour nal of} Comput at ional and Applied Mat hemat ics, Vol.223, pp.133-144.

[ 27] Hambur ger , M. (2000): Str ess, Str ain and the Physics of Ear thquake Gener ation. G141 Ear t hquakes and Volcanoes, 25 October , Indiana Univer sity, USA.

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Engineer ing w it h Comput er s, Vol.23, pp.207-214.

[ 29] Khai, N.M., Ha, P.Q. and Obor n, I. (2007): Nutr ient Flow s in Small-Scale Per i-Ur ban Vegetable Far ming Systems in Southeast Asia-A Case Study in Hanoi. Agr icult ur e, Ecosyst ems and Envir onment, Vol.122, No.2, pp.192-202.

[ 30] Lee, C.H. (2009): Non Linear Analysis of Spur Gear using Matlab Code. M.Sc. Thesis, Califor nia Polytechnic State Univer sity.

[ 31] Mar galit, D. (1995): Hist or y of Cur vat ur e. Concept of Cur vat ur e, Br ow n Univer sity, USA.

[ 32] M Br itto, A. (2005): Patr an Beginner ’s Guide. Depar t ment of Engineer ing, 15Apr il, Univer sity of Cambr idge, UK.

[ 33] Meek, D.S. and Walton, D.J. (1989): The Use of Cor nu Spir als in Dr aw ing Planar Cur ves of Contr olled Cur vatur e. Jour nal of Comput at ional and Applied Mat hem at ics, Vol.25, pp.69-78.

[ 34] Michalopoulos, E. and Babka, S. (2000): Evaluation of Pipeline Design Factor s. GRI 00/ 0076, The Har tfor d Steam Boiler Inspection and Insur ance Company, Har tford, USA.

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[ 36] Nieder stucke, B., Ander s, A., Dalhoff, P. and Gr zybow ski, R. (2003): Load Data Analysis for Wind Tur bine Gear boxes. Final Repor t of Enhanced Life Analysis of Wind Pow er Syst ems, Hamburg, Ger man.

[ 37] Nikolić, V.,Dolićanin, Ć. and Dimitrijević, D. (2012): Dynamic Model for the Stress and Strain State Analysis of a Spur Gear Transmission. St r ojniški Vest nik-Jour nal of Mechanical Engineer ing, Vol. 58, pp.56-67.

[ 38] Peckner , D. and Ber nstein, I.M. (1977): Handbook of St ainless St eels, McGr aw -Hill, New Yor k. [ 39] Pr ice, D. (1995): Spur Gear s. Mechanical Engineer ing Desi gn Topics, 9 August, GlobalSpec. [ 40] Pun, A. (2001): How to Pr edict Fatigue Life. Design New s, 17 December , Lexington, USA.

[ 41] Rameshkumar , M., Sivakumar , P., Sudesh, S. and Gopinath, K. (2010): Load Shar ing Analysis of High-Contact-Ratio Spur Gear s in Militar y Tr acked Vehicle Application. Gear Technology, pp.43-50.

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[ 45] Said, M.R., Yuhazr i, M.Y. and Lau, S. (2012): The Finite Element Analysis on Hexagonal Ring under Simple Later al Loading. Global Engineer s and Technologist s Review, Vol.2, No.2, pp.1-7.

[ 46] Sapto, W.W. and Safar udin, M. (2008): Finite Element Analysis of Wide Rim Alloy Wheel. Nat ional Confer ence on Design and Concur r ent Engineer ing, 28-29 October , Melaka, Malaysia, pp.407-411.

[ 47] Shenoy, P.S. (2004): Dynamic Load Analysis and Optimization of Connecting Rod. M.Sc. Thesis, Univer sity of Toledo.

[ 48] Walton, D.J. and Meek, D.S. (1999): Planar G2_{Transition betw een Tw o Circles w ith Fair Cubic Bézier Cur ve.} Comput er Aided Design, Vol.31, pp.857-866.

[ 49] Walton, D.J. and Meek, D.S. (1996): A Planar Cubic Bézier Spir al. Jour nal of Comput at ional and Applied Mat hemat ics, Vol.72, pp.85-100.

[ 50] Walton, D.J., Meek, D.S. and Ali J.M. (2003): Planar G2_{Tr ansition Cur ves Composed of Cubic Bézier Spir al}

Segments. Jour nal of Comput at ional and Applied Mat hemat ics, Vol.157, pp.453-476.

[ 51] Wang, P.Y., Fan, S.C. and Huang, Z.G. (2011): Spir al Bevel Gear Dynamic Contact and Tooth Impact Analysis.

Jour nal of Mechani cal Design, Vol.133, No.8, pp.084501-1-084501-6.

[ 52] Westland, A. (2006): Aer ospace St r uct ur al Analysis, Jesmond Engineer ing Limited, East Yor kshir e, UK. [ 53] Yeung, K.S. (1999): Analysis of a New Concept in Spline Design for Tr ansmission Output Shafts. BEASY

Publicat ions, Southampton, UK.

[ 54] Ziaei Rad, S. (2011): Bar Linear Static Analysis. Finit e Element Met hod, Isfahan Univer sity of Technology, Ir an.

[ 55] Zehe, M.J., Gor don, S. and McBr ide, B.J. (2002): CAP: Computer Code for Gener ating Tabular Ther modynamic Functions fr om NASA Lew is Coefficients. NASA/ TP–2001–210959/ REV1, Glenn Resear ch Center , Ohio, USA.

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Table 2: Tet-10 and its mesh data structures

Solid of the spur gear (S transition) Solid of the spur gear (C spiral)

Global Edge Length Nodes Elements Global Edge Length Nodes Elements

1.0551234 34897 22448 1.011111 34854 22668

Figur e 17: Tet-10 and its spur gear meshi ng of the appl ied cur ves, S tr ansit ion (left) and C spir al (r ight).

5.2.2 Boundary and loading conditions

shaft, the displacement does not happen (is equivalent to zero) (Nikolić et al., 2012; Lee, 2009). This is the impor tant setting to simulate the gear tr ansmission (Feng, 2011). Many of the r efer ences used Tor que (T) as the applied load (loading condition) for their analysis and

application (Feng, 2011; Nikolić et al., 2012; Lee, 2009). In or der to find out the new tooth pr ofiles and its applicability (Figur e 13), the suitable load to be applied is Pr essur e (P) w hich this load is set on the contact str ess of both spur gear models (Figur e 17) (Gilber t Gedeon, 1999; Wang et al., 2011). The cr itical r egion of the contact str ess is show n in Figur e 18 (Rameshkumar et al., 2010). Figur e 19 show s the setting of both conditions in the models (Figur e 17). One of the conditions, P is applied as 50 MPa

Figur e 18: The cr itical r egion (tw o dots along the blue li ne) of t he contact str ess i n t he tooth pr ofile (Rameshkumar et al., 2010).

(2)

5.2.3 Material selection for spur gear

Figur e 20: Str ess-str ain behavior of the Austenitic and Car bon Steels.

Table 3: AISI 304 and its characteristics (Peckner and Bernstein, 1977)

Modulus of Elasticity

Yield Strength

Ultimate Strength

Poisson

Ratio Density

Damping Coefficient 195 GPa 215 MPa 505 MPa 0.29 8 gcc-1 _0.003

5.2.4 Simulation results and its safety factor

the same. Table 4 show s the maximum Von Mises str ess (υmax) and maximum displacement (δmax)

distr ibutions among the models thr ough the r epeated loads. Hence, the safety factor (Sf) or also

know n as a design margin (Michalopoulos and Babka, 2000) can be der ived as in Eq. (14) (Cliffor d

et al., 2008; Shenoy, 2004) and w her e the yield str ength of AISI 304 can be found in Table 3.



max

304 AISI of Strength Yield

S  (14)

Table 4: The result distributions of δmax, υmax and Sf in the applied models

P (MPa)

EM S Transition C Spiral

δmax

(mm)

υmax

(MPa) Sf

δmax

(mm)

υmax

(MPa) Sf

δmax

(mm)

υmax

(MPa) Sf

Figur e 21 depicts the contour plot of deter mining the pr esent ed values of υmax and δmax

(Table 4) for the applied P = 30 MPa. It is clear ly show n in the plot legend that δmax appr oximates to

(3)

str ength of AISI 304. Ther efor e, Sf appr oximates to 1.15 (Table 4). Failur e occur s if and only if Sf < 1

or Sf has the negative sign (Michalopoulos and Babka, 2000). All gear models in Table 4 ar e safe to

Figur e 21: The example of findings {δmax, υmax} for the appl ied P = 30 MPa in C spir al gear model.

6.0 FIRST-ORDER NEWTON INTERPOLATING POLYNOMIAL AS A FATIGUE PREDICTOR

load w ill be pr edicted using the New ton’s Inter polating Polynomial. The gener al for m of this polynomial such (Chapra and Canale, 2010).

) )...( )( ( ... ) )( ( ) ( ) ( 1 1 0 1 0 2 0 1 0

n x b b xx b xx xx  bn xx xx xx_n_

f (15)

Wher e, n is r efer r ed to as the nth-or der polynomial, b0, b1, b2 and bn ar e the polynomial coefficients and

the set of (n+1) data points r epr esent ed by {xi, f(xi)} w ith i = {0, 1, 2, 3,…, n}. The polynomial coefficients also can

be show n as (Chapr a and Canale, 2010).

] , , [ ]; , , [ ]; , [ );

( ₀ ₁ ₁ ₀ ₂ ₂ ₁ ₀ ₁ ₁ ₀

0 f x b f x x b f x x x b f x x ,...,x x

b    n  _n _n_ (16)

The fir st-or der or linear for m (n = 1) is given as

) ( )

( ₀ ₁ ₀

1 x b b x x

f    (18) With b0 and b1 ar e calculated using Eqs. (16) and (17). The algor ithms of using this linear inter polation

(4)

for f1(x) = 1. All calculated values of x among the models (Table 4) ar e fully depicted in Table 5. It show s that the

models ar e in safety mode if P 35 MPa w hile the fatigue may star t occur r ing w hen P > 35 MPa. Table 5: The Initial loads of P for the fatigue mode

EM S Transition C Spiral

35.7931 MPa 35.3979 MPa 35.1724 MPa

7.0 DESIGN EFFICIENCY OF THE PROPOSED DESIGN MODELS

w her eas S Tr ansition and C Spiral ar e r epr esented as t he appr oximation values, xa (output). Thus, DE can be

expr essed as (Khai et al., 2007).

% 100 ) (

t a _  

x

Input Output

DE (19)

8.0 CONCLUSIONS AND FUTURE WORK

ACKNOWLEDMENT

REFERENCES

[ 1] Ali, J.M. (1994): An Alter native Der ivation of Said Basic Function. Jour nal of Sains Malaysiana, Vol.23, No.3, pp. 42-56.

[ 3] Ahmad, A. (2009): Parametr ic Spir al and its Application as Tr ansition Cur ve. Ph.D Thesis, Univer siti Sains Malaysia, pp.19-22.

[ 4] Aas, E.J. (2000): Design Quality and Design Efficiency; Definitions, Metr ics and Relevant Design Exper iences.

(5)

[ 6] Babu, V.S. and Tsegaw , A.A. (2009): Involute Spur Gear Template Development by Par ametr ic Technique

using Computer Aided Design. Afr ican Resear ch Review, Vol.3, No.1, pp. 1-6.

pp.47-52.

[ 9] Bloor , M.I.G., Wilson, M.J. and Hagen, H. (1995): The Smoothing Pr oper ties of Var iational Schemes for Sur face Design. Comput er Aided Geomet r ic Design, pp.381-394.

[ 10] Boulos, P.F., Wood, D.J. and Lingir eddy, S. (1997): Pr essur e Vessels and Piping Systems-Shock and Water

Hammer Loading. Encyclopaedia of Life Suppor t Syst ems (EOLSS), Amer ican Water Wor ks Association, UNESCO.

[ 11] Cliffor d, K.H., Bibeau, T.A. and Quinones, A.Sr. (2008): Finite Element Analyses of Continuous Filament Ties

for Masonr y Applications: Final Repor t for the Ar qui n Cor por ation. SAND2006-3750, Sandia National Laborator ies Albuquer que, Califor nia, USA.