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9
YAHAYA
1
, 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
1
saifudinutem.edu.my
4
iphaery utem.edu.my
5
yuhazriutem.edu.my
2, 3
School of Mathematical Sciences University of Science Malaysia
Minden, 11800, Penang, MALAYSIA
2
jamalumacs.usm.my
3
yazariahmycs.usm.my
1.0 I
NTRODUCTION
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 G
2
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 G
2
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
2
PARAMETRIC BÉZIER-LIKE CUBIC TRANSITION AND SPIRAL CURVES
© 2012 GETview
Limit ed. All right s reserved
10
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
