AbstractInthispaper,weemploytheFourierseriesto designthepitchcurveofnoncirculargearswithfunction generation.WiththespeedratioisrepresentedasFourier series,thenonlinearfunctionofangulardisplacementscanbe acquired through integration.And then, based on known finite separatedangulardisplacementsoftwogears,theorderof Fourierseriesisdecidedandasetofsimultaneousequationis obtained.Bysolvingthesimultaneousequations,theFourier series function regarding angular displacement and speed ratio canbeobtained.Finally,synthesizingthesetwoformulas createsthepitchcurvesforthenoncircularpinionandgear, respectively.Inthisresearch,wedemonstratetwodesigns examplestointerpretthedesignprocessofthepitchcurvesof noncirculargears.Inaddition,theresultsderivedfromthis paper can be taken as a reference to design the pitch curves of noncircular gears with function generation. IndexTermspitchcurve,noncirculargears,function generation. I.INTRODUCTION The applications of noncircular gears are often applied in the transmission systems which may need variable speed ratio, nonlinear motion function, or variable input and output loadings, respectively.Basically, most of those applications are using elliptic gears therefore the elliptical pitch curve is morearresting[1-2].Inthefieldofresearch,moststudies focusontheellipticandthemodifiedellipticgears[3-11].Forinstance,EmuraandArakawa[12-15]employedthe ellipticandthemodifiedellipticgearsatadesignofthe steeringmechanismforautomobilesandships.Takingthe linkage mechanism with a sliding slot as an example, Danieli [16] concluded a relationship of the speed ratio between drive and driven links, and further constructed the pitch curves of theequivalentnoncirculargearsaccordingtothis relationship.Dooner[17]usedthenoncirculargearsto eliminate unnecessary loading in transmission and analyzed therelationshipoftheloadingandthespeedratioofgears with input and output on the basis of the law of conservation of energy.With this relationship, he designed the outline of two pitch curves of gears and calculated the reduced torque on noncircular gears. Usually, the pitch curves of the noncircular gears have tomatchaspecificinput/outputrelationship.Thecommon procedure is to employ a known functional relationship, tooth number,andmodulestosynthesizethepitchcurvesofthe noncircular gears which fit this relationship.Sometimes, as ManuscriptreceivedNovember27,2007.Thisworkwassupportedin partbytheNationalScienceCouncilofRepublicofChinafortheir financiallysupportingthisresearchunderContractNo.NSC 96-2221-E-150-048. Jen-YuLiuiswiththePowerMechanicalEngineeringDepartment, National Formosa University, 64 Wen-HwaRoad,HuWei63208,Yunlin, TAIWAN.(correspondingauthor,phone:886-5-6315420fax: 886-5-6312110; e-mail: davidliu@ nfu.edu.tw).Yen-Chuan Chen is with the Engineer, Accton Technology Corporation, TAIWAN. (e-mail: t124395.chung@msa.hinet.net). regards a design for the pitch curves of noncircular gears, the requirements are to satisfy finite separated angular positions, i.e.,inputandoutputangulardisplacementregardingthe finiteseparatedpositions.Therelevantliteraturesfor designingnoncirculargearswithsuchfunctiongeneration are rare. Owing to the periodicity in the Fourier series, its first order derivative, its second order derivative or above [18-19], the demandonperiodicalfeaturesforangulardisplacement, angular velocity, and angular acceleration of the noncircular gear,theFourierseriesisanadequatetooltogeneratethe finiteseparatedangulardisplacementfunctionofthe noncircular gears.Inthispaper,basedontheknownfinite separatedangulardisplacementposition,weconstruct angulardisplacementfunctionusingFourierseriesand deducethepitchcurveequationsofnoncirculargears meetingthisfiniteseparatedpositionaccordingtothis function.In addition, some examples are created to interpret themethodfordesigningthepitchcurvesofnoncircular gears with function generation. II.SYNTHESIZING PRINCIPLES OF NONCIRCULAR PITCH CURVES In Figure 1, the noncircular gears 1 and 2 with angular velocity1and2andangulardisplacement1 and2, respectively, reversely rotate around the fixed axes O1 and O2.Coordinate systems x1y1 and x2y2 are the moving coordinate attached to gears 1 and 2, respectively. Figure 1Pitch curves of noncircular gears The rotating-speed ratio m21(1) is: 2 2 221 11 1 1d dt dm ( )d dt d = = = (1) when 1=0 then 2 also be zero. IntegratingEquation(1)withrespectto1the relationship of 1 and 2 is obtained: ( )12 210m d = (2) The distance from the contacting point P of gear 1 or gear 2 to the rotating center of each gear, and the center distances for two gears are r1(1), r2(2) and D, respectively.Based on the A Design for the Pitch Curve of Noncircular Gears with Function GenerationJen-Yu Liu and Yen-Chuan ChenProceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol IIIMECS 2008, 19-21 March, 2008, Hong KongISBN: 978-988-17012-1-3 IMECS 2008 synthesizing principle [20],r1(1) and r2(2) are acquired as follows: ( )( )( )21 11 121 1Dmr1 m =+ (3) ( )( )2 121 1Dr1 m =+ (4) Then the Equations of pitch curves of gears 1 and 2,R1(1) and R2(2) respectively, are: R1(1) ( )1 1 1r cos = i1 ( )1 1 1r sin + j1 ( )( )21 1121 1Dmcos1 m= + i1( )( )21 1121 1Dmsin1 m+ + j1 (5) R2(2) ( )2 1 2r cos = i2 ( )2 1 2r sin + j2 ( )221 1Dcos1 m=+ i2( )221 1Dsin1 m+ j2(6) III.FOURIER SERIES FUNCTION FOR FINITE SEPARATED ANGULAR DISPLACEMENTS Table 1 shows N finite separated angular displacement positions of 1 and 2.As gear 1 rotates with a constant speed, theangulardisplacementofgear2displaysanonlinear variation.Thus,therotatingspeedratioisnotaconstant.Science gear 1 and gear 2 are periodical rotating, the rotating speed ratio displays a periodical variation.For an arbitrary integraln,Equation1shouldsatisfythefollowing relationship: 21 1 21 1m ( ) m (2n ) = + (7) Let the period time of the noncircular gear is T, Equation (7) can be represented with M order Fourier series [18-19]: ( )M221 1 0 n 1n 11d 2nm A 2 A cosd T= = = + Mn 1n 12n2 B sinT= + (8) Table 1N finite separated angular displacement 12 11,12,1 21,22,2 31,32,3 N-11,12,1 N1,2, Substituting Equation (8) into Equation (2), the relationship between 1 and 2 is: ( )12 210m d = Mn0 1 1n 1TA 2nA 2 sin2n T= = + Mn1n 1TB 2n2 1 cos2n T= + (9) From Equation (7), the period of angular displacement ofnoncirculargearsare2,Equations(8)and(9)canbe modified as: ( ) ( )M21 1 0 n 1n 1m A 2 A cos n= = + ( )Mn 1n 12 B sin n=+ (10) ( )Mn2 0 1 1n 1AA 2 sin nn= = + ( )Mn1n 1B2 1 cos nn=+ (11) Equation(11)mustsatisfyconditionssuchas1=0 togetherwith2=0and1=2togetherwith2=2.SubstitutingtheseconditionsintoEquation(11),wecan obtainA0=1.InEquation(11),letAn/n=anBn/n=bn, An=nan, and Bn=nbn.Equations (10) and (11) can be written as: ( ) ( ) ( )M M21 1 n 1 n 1n 1 n 1m 1 2 na cos n 2 nb sin n= = = + + (12) ( ) ( )M M2 1 n 1 n 1n 1 n 12 a sin n 2 b 1 cos n= = = + + (13) Inordertofindthevaluesofvariouscoefficientsof Fourierseries,suchasa1,b1,a2,b2,,aM,andbM, respectively, Equation (13) can be modified as : ( ) ( )M Mn 1 n 1 2 1n 1 n 1a sin n b 1 cos n= = + = (14) Substituting various separated positions listed in Table 1 into Equation (14), we can obtain the simultaneous formulas to solve various coefficients of Fourier series: 1 1,1 1 1,1 2 2,1 2 2,1M M,1 M M,1 2,1 1,11 1,2 1 1,2 2 2,2 2 2,2M M,2 M M,2 2,2 1,21 1,3 1 1,3 2 2,3 2 2,3M M,3 M M,3 2,3 1,31 1,N 1 1 1,3 2 2,N 1 2 2,3M M,Na b a ba ba b a ba ba b a ba ba b a ba + + + + + + = + + + + + + = + + + + + + = + + + + + LLLML1 M M,N 1 2,N 1 1,N 11 1,N 1 1,N 2 2,N 2 2,NM M,N M M,N 2,N 1,Nba b a ba b + = + + + + + + = L(15) where ( )i, j 1, jsin i = (16) ( )i, j 1, j1 cos i = (17) InEquations(16)and(17),1 i M and1 j N .Further, only with the total number for unknown coefficients of a1, b1, a2, b2, , aM, and bM been equal to N will there be Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol IIIMECS 2008, 19-21 March, 2008, Hong KongISBN: 978-988-17012-1-3 IMECS 2008 onlyonesetofsolutionsforthesimultaneousformulasof Equation (15).Obviously, if N is an even number, M is N/2; however,foranoddN,thesituationthatoneofthose unknown coefficients of a1, b1, a2, b2, , aM, and bM must be 0inordertomatchtheabovecondition.Througha numerical computation and analysis, we find when bM= 0 for the case of an odd N, the solutions of Equation (15) have a better effect for transmission and facilitates processing of the shape of gears.Thus, Neven. N/ 2 , N isModd, b 0. (N 1) / 2, N is= = +(18) Equation (15) can be expressed as the following matrix form: ANN aN = VN(19) where VN2,1 1,12,2 1,22,3 1,32, N 1 1, N 12, N 1, N = M(20) For an even N: ANN1,1 1,1 N/ 2,1 N/ 2,11,2 1,2 N/ 2,2 N/ 2,21,3 1,3 N/ 2,3 N/ 2,31, N 1 1, N 1 N/ 2, N 1 N/ 2, N 11, N 1, N N/ 2, N N/ 2, N = LLLM M O M MLL(21) aN11N/ 2N/ 2abab = MM(22) For an odd N: ANN( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )1,1 1,1 N 1 / 2,1 N 1 / 2,1 N 1 / 2,11,2 1,2 N 1 / 2,2 N 1 / 2,2 N 1 / 2,21,3 1,3 N 1 / 2,3 N 1 / 2,3 N 1 / 2,31, N 1 1, N 1 N 1 / 2, N 1 N 1 / 2, N 1 N 1 / 2, N 11, N 1, N N 1 / 2, N N 1 / 2, N N 1 / 2, N + + + + + = LLLM M O M M MLL (23) aN11(N 1) / 2(N 1) / 2(N 1) / 2ababa+ = M(24) Therefore, aN =A1N NVN(25) ThesolutionsforcoefficientsofFourierseriesunder even and odd known conditions are displayed from Equation (19) to Equation (25). Design example 1 (N is even) Forapairofnoncirculargears,thefiniteseparated angular displacements for gears 1 and 2 are shown in Table 2.Thepitchcurveofnoncirculargearssatisfythesespecified angular displacements are designed as what follows. FromTable2,owingto6finiteseparatedangular displacements,Nequalsto6.ByEquation(18),M=3.Substituting data shown inTable2intoEquations(20)and (22), we can obtain: Table 2Even finite separated angular displacement 12 1 /9/9 2 4/95/9 3 5/6 4 11/925/18 5 25/183/2 6 5/331/18 V60.0000000.3490660.5235990.5235990.3490660.174533 = (26) A6 60.342 0.060 0.643 0.234 0.866 0.5000.985 0.826 0.342 1.940 0.866 1.5000.500 1.866 0.866 0.500 1.000 1.0000.643 1.766 0.985 0.826 0.866 0.5000.940 1.342 0.643 1.766 0.500 0.1340.866 0.500 0.866 1.500 0.000 2.000 = (27) Employ Guass Method [21] to get the inverse matrix of Equation (25): A16 60.022 0.387 0.187 0.208 0.103 0.3300.157 0.607 0.338 0.296 0.010 0.1530.645 0.022 0.256 0.334 0.063 0.0960.269 0.322 0.102 0.411 0.516 0.0550.445 0.181 0.142 0.242 0.213 0.0000.530 0.066 0.038 0.288 0.45 = 7 0.394 (28) SubstituteEquations(26)and(28)intoEquation(25)to acquire: a6112233a 0.030686b 0.278672a -0.005610=b 0.014658a -0.041404b 0.017464 = (29) By Equation (13), it can be obtained: Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol IIIMECS 2008, 19-21 March, 2008, Hong KongISBN: 978-988-17012-1-3 IMECS 2008 ( ) ( )3 32 1 n 1 n 1n 1 n 1a sin n b 1 cos n= = = + + ( )1 1 10.030686sin 0.278672 1 cos = + + ( )1 10.005610sin 2 0.014658 1 cos 2 + ( )1 10.041404sin3 0.017464 1 cos3 + (30) DifferentiateEquation(30)withrespectto1, weobtain rotating speed ratio m21(1) as: m21(1) ( ) ( )3 3n 1 n 1n 1 n 1na cos n nb sin n= == + 1 11 0.030686cos 0.278672sin = + + 1 10.011220cos 2 0.029316sin 2 + 1 10.124212cos3 0.052392sin3 + (31) With Equations (30) and (31) substituted into Equations (5) and (6), the equations of pitch curve of gear 1 and 2 are: R1(1)( )( )21 1121 1Dmcos1 m= + i1( )( )21 1121 1Dmsin1 m+ + j1 (32) R2(2)( )221 1Dcos1 m=+ i2( )221 1Dsin1 m+ j2 (33) Figure2showsFourierseriessatisfyingthefinite separatedangulardisplacementsbysolvingEquation(30) wherein the black bold curve is representing Equation (30), andthedottedlineisthecorrespondingvaluesoffinite separatedangulardisplacements.Figure3indicatesthe drawnpitchcurvesofgears1and2byEquations(32)and (33). Figure 2Fourier series function satisfy Table 2 Figure 3Pitch curve of gears 1 and 2 by Equations (32) and (33) Design example 2 (N is odd) Thefiniteseparatedanulardisplacementsforgears1 and 2 are shown in Table 3.Thedesignofpitchcurvesof noncirculargearsmustsatisfytheseoddfiniteseparated angular displacements.In Table 3, 7 known finite separated angular displacements are listed, N equals to 7.By Equation (18), M = 4.Substituting data in Table 3 into Equations (20), (23), and (24), respectively, we have: Table 3Odd finite separated angular displacement 2 1 /9/9 2 5/8/3 3/95/9 4 5/6 5 11/925/18 6 25/183/2 7 5/331/18 V70.0000000.1745330.3490660.6981320.5235990.3490660.174533 = (34) A7 7 =0.342 0.060 0.643 0.234 0.866 0.500 0.9850.766 0.357 0.985 1.174 0.500 1.866 0.3420.985 0.826 0.342 1.940 0.866 1.500 0.6430.500 1.866 0.866 0.500 1.000 1.000 0.8660.643 1.766 0.985 0.826 0.866 0.500 0.3420.940 1.342 0.643 1. 766 0.500 0.134 0.9850.866 0.500 0.866 1.500 0.000 2.000 0.866 (35) The reverse matrix of Equation (35) is: A17 7=0.456 0.290 0.508 0.156 0.147 0.186 0.2530.129 0.192 0.012 0.318 0.336 0.064 0.1010.002 0.433 0.202 0.209 0.243 0.060 0.2130.456 0.486 0.523 0.154 0.308 0.377 0.0760.351 0.063 0.207 0.148 0.255 0.231 0.0170.468 0.66 9 0.344 0.034 0.147 0.266 0.2150.617 0.450 0.187 0.048 0.095 0.129 0.121 (36) SubstituteEquations(34)and(36)intoEquation(25), we obtain: a71122334a 0.049878b 0.328853a -0.030062= b -0.025730a -0.013766b 0.042001a -0.020946 = (37) Thus, from Equation (13), we have: ( ) ( )4 42 1 n 1 n 1n 1 n 1a sin n b 1 cos n= = = + + ( )1 1 10.049878sin 0.328853 1 cos = + + ( )1 10.030062sin 2 0.025730 1 cos 2 ( )1 10.013766sin3 0.042001 1 cos3 + Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol IIIMECS 2008, 19-21 March, 2008, Hong KongISBN: 978-988-17012-1-3 IMECS 2008 10.020946sin 4 (38) Differentiate Equation (38) with respect to 1, the speed ratio m21 (1) is: m21 (1) ( ) ( )4 4n 1 n 1n 1 n 1na cos n nb sin n= == +

1 11 0.049878cos 0.328853sin = + + 1 10.060124cos 2 0.051460sin 2 1 10.041298cos3 0.126003sin3 + 10.083784cos 4 (39) Substituting Equations (38) and (39) into Equations (5) and (6), the equations of pitch curves of gears 1 and 2 are as follows: R1(1) ( )1 1 1r cos = i1 ( )1 1 1r sin + j1 ( )( )21 1121 1Dmcos1 m= + i1( )( )21 1121 1Dmsin1 m+ + j1(40) R2(2) ( )2 1 2r cos = i2 ( )2 1 2r sin + j2 ( )221 1Dcos1 m=+ i2( )221 1Dsin1 m+ j2 (41) Figure4showsFourierseriesmeetingthefinite separatedangulardisplacementsbysolvingEquation(38).In the figure, the black bold curve is related to Equation (38) and the dotted line is corresponding values of finite separated angulardisplacements.Figure5showsthepitchcurvesof gears 1 and 2 according to Equations (40) and (41) when the center distance D is 100mm. Figure 4Fourier series function satisfy Table 3 Figure 5Pitch curve of gears 1 and 2 by Equations (40) and (41) IV.CONCLUSION Inthispaper,weprovidetheangulardisplacement curves of the noncircular gears by employing Fourier series tosynthesizefiniteseparatedangulardisplacementsand further synthesize the pitch curves of thenoncirculargears.Based on the known finite separated angular displacements, simultaneousequationsforthecoefficientsofMorder Fourier series which satisfy these odd as well as even finite separated angular displacements respectively are derived.By solvingthesimultaneousequations,weobtainMorder Fourier series.And the pitch curves of noncircular gears by meansofthesynthesizingprincipleforthepitchcurvesof noncirculargears.Twodesignexamplesinthisstudy interpret the method to synthesize the functions and the pitch curves of the noncircular gears with finite separated angular displacementsthroughFourierseries.Asareferencefora research of designing the pitch curve of the noncircular gear, this study result can contribute to design a noncircular gear, whichmeetsthefunctionoffiniteseparatedangular displacement. ACKNOWLEDGMENT TheauthorswouldliketothanktheNationalScience Council of Republic of China for their financially supporting this research under Contract No. NSC 96-2221-E-150-048. REFERENCES [1]Chironis, N.P. (Ed.), 1967, Gear Design and Application, McGraw-Hill, New York. [2]Chang, S. 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C.H., 1998, Generation of Identical Noncircular Pitch Curves, Transactions of the ASME, Journal of Mechanical Design, Vol.120, No.2, pp.337-341.[21]Fraleigh, J.B., and Beauregard, R.A., 1995, Linear Algebra, 3rd edition, Wesley, New York. Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol IIIMECS 2008, 19-21 March, 2008, Hong KongISBN: 978-988-17012-1-3 IMECS 2008