ANALYSIS OF TRANSMISSION CHARACTERISTICS OF COSINE GEAR DRIVE

CHINESE JOURNAL OF MECHANICAL ENGINEERING●94 ●Vol.2I, No. 3, 2008ANALYSIS OF TRANSMISSIONWANG JianCHARACTERISTICS OF COSINELUO ShanmingGEAR DRIVE*CHEN LifengCHEN Lei北the contact atio. thesliding coffcient,andthe contact and bending steses. ofthis dniveareanvgzedAcomnarison shuadwofthese charaeteristieswithethe involuteeardriveisalsocariedoutHU HuarongThe influences of design parameters including the number of teeth and the pressure angle on theSchool of Electromechanical Engineering,contact and bending stresses are studied. The following conclusions are achieved: the contact ratio ofHunan Univerity of Sciencethe cosine gear drive is about 1.2 to 1.3, which is reduced by about 20% in comparison with that ofand Technology,the involute gear drive. The sliding coefficient of the cosine gear drive is smaler than that of theXiangtan 411201, Chinainvolute gear drive. The contact and bending stresses of the cosine gear drive are lower than those ofthe involute gear drive. The contact and bending stresses decrease with the growth of the number ofteeth and the pressure angle.Key words:Gear driveCosine profle Contact ratio Sliding coefficient Stresscharacteristics, including contact ratio, sliding coefficient, contact➊INTRODUCTIONand bending stresses, of the cosine gear drive are analyzed, and acomparison study of these characteristics with the involute gearCurrently, the involute, the circular are and the cycloiddrive is also carried out. The influences of design parameters,profiles are three types of tooth profiles that are widely used in theincluding the number of teeth and the pressure angle, on contactgear design'". All of these gears used in different fields due toand bending stresses are studied. Finally, a conclusion summarytheir different advantages and disadvantages. With theof this study is given in section 3.development of computerized numerical control (CNC)technology, a large amount of literature is presented ininvestigations onhanisms and methodstooth profilegeneration. ARIGA， et al4, used a cutter with combinedcircular -arc and involute tooth profiles to generate a new type ofWildhaber-Novikov gear. This particular tooth profile can solvethe problem of conventional W-N gear profile, that is, the profilesensitivity to centre distance variations. TSAY, et alb), studied ahelical gear drive whose profiles consist of involute anccircular-arc. Tbe tooth surfaces of this gearing contact with eachatever yy instant at a point instead of a line. KOMORI, et aM',(a) Cosine gear(b) lnvolute geardeveloped a gear with logic tooth profiles which have zero relativeFig 1 Cosine gear and involue gearcurvature at many contact points. The gear has higher durability andstrength than involute gear. ZHAO, et als, introduced thegeneration process of a micro segment gear. ZHANG, et al，1 MATHEMATICAL MODEL OF tHE COSINEpresented a double involute gear. The tooth profile of the gearGEAR DRIVEconsists of two involute curves, which are 1inked by a transitioncurve and form the ladder shape of tooth.According to Ref. [8], the equation of the cosine tooth profile,LUO, et al", presented a cosine gear drive, which takes the the conjugate tooth profile and the line of action can be expressedzero line of cosine curvee as the pitch circle, a peniiod of the cure as fllwsas a tooth space, and the amplitude of the curve as the toothaddendum. As shown in Fig. 1, the cosine tooth profile appears[ x =[mZ,12+ hcos(Z0)]sin0very close to the involute tooth profile in the area near or above只=[mZ,/2+ hcos(Z,)]cosθ(1)the pitch circle, i.e, the part of addendum. However, in area ofdedendum, the tooth thickness of cosine gear is greater than thatof involute gear.| x =[mZ/12+ hos<(Z.)]sin+ asinThe mathermatical models, including the equation of thecosine tooth profile, the equation of the conjugate tooth profile(2)and the equation of the line of actin, have been established basedp =[mZ/2+ ho.IC.-asnoon the meshing theory, The solid model of cosine gear has beenbuilt, and the meshing simulation of this drive has also beeninvestigated8l. The aim of this work is to analyze the[x =[mZ.12+ hcos(Z0)]sin(0 -q)characteristics of the cosine gear drive. The remainder i(3)organized in three sections. In section 1, the mathematical modelsq)- mZ12中国煤化工，of the cosine gear drive are introduced. In section 2, the where rI the number of teeth,respecti:MHC N M H Genote the contat●Tbis project is supprted by National Natural Science Foundation of Chinaand the center dlstance, respecuvely, 0 is the rotation angleNo 50575071), Narural Science Foundatio ofHunan Povince, China (No. relative to system i(O,期川) as shown in Fig.2.β is the angle01J1000 S&T Programs of Hunan Province, China (No. 2007FJ4047)，between x-axis and the tangent of any point on the cosine profile,and Program for New Century Excellent Talents in University, China. φ1 is the rotational angle of gear 1 which can be given as followsReceived June 20, 2007; recived in revised from March 19, 2008; acpiedApril 10, 2008CHINESE JOURNAL OF MECHANICAL ENGINEERING|o = arcsin, [mZ.12+ hcos(Z0)sin(0+f) _ BThree examples as shown in Table 1 have been caried out byusing program Matlab. The contact ratios of the involute gearmZ,12drives with the same parameters are also shown in Table 1 for the-mZ12 + hcos(Z9)]tanθ - hz, sin(Z.)purpose of comparison. According to Table 1, the contact ratio ofβ = arctan[mZ,/2+ hcos(Z,0)]- hZ, tanOsin(Z,)the cosine gear drive is about 1.2 to 1.3, which is about 20% lessthan that of the involute gear drive. According to Refs. [10-11],the contact ratio of gears applied in gear pump is about 1.1 to 1.3,therefore, such cosine gear drive can be applied in the field ofgear pump.2Table 1 Contact ratio of cosine gear driveTooth number Tooth number Modulus CosineInvolutem/mm__ gear drive__ gear drive_321-3641.6144012401.6772.2 Sliding cofficientSliding cofficient is a measure of the sliding action duringthe meshing cycle. A lower coefficient will have greater powertransmission efficiency because of the less friction. The slidingceofficient is defined as the limit of the ratio of the sliding arclength to the corresponding arc length in plane meshing. Thesliding cofficients U, and U2 can be expressed as followsl2lu=1-5hFig.2 Principle of the cosine gear drive(5)|U,=1-5hi2 CHARACTERISTICS OF THE COSINE GEARz-LDRIVEwhere r and 生denote the radius of the pitch circle, respectively,Based on the mathematical model of the cosine gear drive,L represents the vertical coordinate of point H io coordinate systemthree characteristics, contact ratio, sliding coefficient, and stresses,E(P,x ) H is the intersection point of the normal line of theare analyzed. In addition, all these characteristics are comparedcontact point and the line 002 , as shown in Fig. 4.with those of the involute gears.2.1 Contact ratioThe contact ratio could be considered as an indication ofaverage teth-pairs in mesh of a gear-pair and naturally is ought tobe defined according to the rotation angle ofa gear from gear-inJorto gear-out of a pair of teeth!9. As shown in Fig. 3, the contactratio of the cosine gear can be expressed as followsz=-4(4口where P。and Pr are the values of rotation angle 中as = x .and x =xp respectively, which can be calculated by Eq. (3).aE,0|D. t/onFig. 4 Relative sliding of the cosine gear drive(h2 =1/21 =n/n): PH can be epesede\%as fllowu中国煤化工1YHCNMHG(6)ayFig. 3 Contact matio of the cosine gear driveSubstituting Eq. (3) into Eq. (6) gives●96●WANG Jjian, et al: Analysis of transmission characteristics of cosine gear drive[ mZl+ hcos(Z,) (a-*q)c0s(0-q)-As !「2ok=mz+ h.s(so.(a.(g)isn()-)+B7)-1L2where a' and B' are the differential coeficients ofa andβ toθ,←Cosine gear driverespectively, which can be expressed as- + - Involute gear drive+hcos(Z,9) (1+ B)os(θ+ B)-C量49550551.5525 S35 s54.5 55sp'=Engaged radius of drivinggear n(m2Z?- m:+hcos(Z29)| sn(@+A)(2) Driving gear.[D+Eβ=-- + Cosine gear drivemZ+一Involute gear drive+ hcos(Z0)|- hZ tan0sin(Z)}0.3-hz;"| m+ hcs(,) [sin(Z,)+t tan2 0cos(,)]0.1-0.1{[叭+ hcos(Z,.)|- hZ, ansin(,.)}-0.3where A= hZ sin(@ - q)sin(ZB)B= hZ cos( - q,)sin(Z,.O)”101010405106107 108C = 2hZ, sin(Z0)sin(θ+ B)Engaged rdius of driven gear ne「mZ.(b) Driven gearD=-1+ hrcos(Z,0) ise' θ- 2h'Z? sn2(Z,;se20QFig. 5 Sliding coficients of the cosine gear driveE= h'Z' tan[ sin'(Z,9)- sn(Z,)cos(Z,]2.3 Contact and bending stressesTherefore, the vertical coordinate of the point H inIn general, an FEA model with a larger number of elementscoordinate system 2(P, x, y) can be expressed as followsfor finite element stress analysis may lead to more accurate results.L=-xo+Yo8) However, an FEA model of the whole gear drive is not preferred,where (xo, yo,) denotes the coordinate of the contact point inespecially considering the limit of computer memories and thecoordinate system E(P, x, y). Substituting Eq. (3) and Eq. (7) intoneed for saving computational time. This paper establishes anFEA model of three pairs of contact teeth for the cosine gear drive.Eq. (8) givesTwo models of contacting teeth based on the real geometrty of theF-Gpinion and the gear teeth surfaces created in Pro/Engineer areLexported as an IGES file which is then imported into the sofware"m"4 + hcos(.0) ( -q)si(0- 2) + hZ,cos(O - a)sin(Z.U)Ansys for stress analysis.The numerical computations have been performed for the-mZ, + h.(.Ccos(O-2)-2cosine drive with the following design parameters: =25, Z= 40,m=3 mm, a=22°, a width of b=75 mm. The basic mechanicalproperties are modulus of elasticity E = 210 GPa, and Poisson'sratio pr 0.29. The torque is 98 790 N●mm. Two sides of eachwhere G=| + hcos<(Z,O)| (-q")sin(0- q,)cos(θ-q)model sufficiently far from the illet are chosen to justify the rigidconstraints applied along the boundaries. A large enough part ofF=hZ，mZ. + hcos(Z,8) sin'(@ - q,)sin(Z,6)the wheel below the teeth is chosen for the fixed boundary. Areasare meshed by using plane-82 elements. The finite elementmodels are shown in Fig. 6, and there are 3 373 elements and 10Given nu, rno and θ can be obtained from053 nodes. Two options related to the contact problem, smallsliding and no friction have been selected. Fig. 7 shows the[r = mZ,/2 + hcos(Z,0)contour plot of Von-Mises stress. Tbe poumerical results are listed[a=Vr2n+a2-2rgacos0in Table 2.Substituting θ and Eq. (9) into Eq. (5), the sliding coefficients can be obtained.The gears are designed to have a module of m=3 mm, anumber of teeth of Z=35, and a transmission ratio of i=2. Thepressure angle of the involute gear is 20, while it is 22° for thecosine gear. According to Eqs. (5)-(9), a computer simultion toplot the graphs of sliding coefficients for the driving and the中国煤化工driven gears of the cosine gear drive is developed as sbown inFig. 5. The sliding cofficients of the involute gear drivel13) areYCNMHGalso listed in Fig. 5 for the purpose of comparison. According toFig. 5, the sliding coefficients of the cosine gear drive is smallerthan that of the involute gear drive, which can help to improve thetransmission performance.Fig 6 Models pplied for finite element alysisCHINESE JOURNAL OF MECHANICAL ENGINEERING●97●With the same material parameters as aforementioned, thecontact and bending stresses of three sets of cosine gears are641x10"analyzed by using program Ansys. Results are shown in Fig. 9,53.462 |106 924 |Fig.7 and Fig. 10, and the values of the contact and bending10698stresses are shown in Table 4. According to Table 4, both the160.386contact and bending stresses decrease with the growth of the213.848267.310number of teetb. For instance, the contact stress, tension and320.771compression bendingstresses are 569.76 MPa, 117.51 MPa and374.233124.98 MPa, respectively, as the number of teeth Zj=20, while427.695410.61 MPa, 64.52 MPa and 74.41 MPa as the number of teethZ1=30.498.978 i660x10*1Fig.7 Stess distributio of the cosine gear drive (MPa)75.969151.937Table2 Maximum bending and contact steses MPa227.905Contact stress Bending stressBending stressGe303.873(tension). u__ (compressio) &Cosine gear498.9886.095.59379.841- lavolute gcar641.58115.24134.00455.810Under the samne parameters, stress distribution of an involute569.762 I Igear drive sbown in Fig. 8 is also analyzed for the purpose ofcomparison. The bending stress obtained in the fllet of thecontacting tooth side are considered as tension stresses, and thosein the fllet of the opposite tooth side are considered as compress-ion stresses.Fig.9 Stress dsribution of the cosine gear drive (Zr -20) (MPa)321x10*80.197207*10*160.39461.591240.591123.182320.788184.772400.984246.363481.181307.954561.378369.544641.575 ■410.605Fig. 8 Stress distribution of the involute gear drive (MPa)Fig. 10 Stress dstributio of cosine gear drive (Z: -30) (MPa)From the obtained numerical results, the followingconclusions can be made: the maximum contact stress of theTable 4Stresses of the cosine gear undercosine gear is reduced by about 22.23% in comparison with thedifTerent number of teethMinvolute gear. The tension bending stress of the cosine gear isTooth oumberContact stress Bending stress Bending stress25.34% less than that of the involute gear, and the compression。(tension) ou (compression) obending stress is reduced by about 28.67% in comparison with the0569.76124.98involute gear. An application of a cosine tooth profile allows586.0495.59 .reducing both, contact and bending stresses.410.6164.5274.412.4 Infuences of design parameters on stressesExample 2: the gears are designed to have a module of m=3Based on the finite element models, two examples are usedmm, number of teeth =25, a width of b=75 mm. The other mainto clarify the influences of design parameters including thepararneters are sbown in Table 5.number of teeth and the pressure angle on contact and bendingstresses.Table5 Main parameters of the calculated gears (example 2)Example 1: the gears are designed to have a pressure angle ofPressure angle a/(Transmission ratio ia = 22°, at the pitch circle, a module of m =3 mm, a width of b=75、22.6mm. The other main parameters are shown in Table 3.中国煤化工Table3 Main parameters of the calculated gears (example 1)二 NoTooh Rumber zTransmission ratio[HWitn me same matenal paramerers as aforementioned, the20:6contact and bending stresses are also computed by using program25Ansys. Results are shown in Fig 7,Fig. 11 and Fig.12, and the30.6__values of the contact and bending stresses are shown in Table 6.●98●WANG Jjian, et al: Analysis of transmission characteristics of cosine gear driveunder the given parameters as shown in section 2, the maximumcontact stress of the cosine gear is reduced by about 22.23% in242-10*comparison with the involute gear, and the compression bending59.862stress is 28.67% less than that of the involute gear.119.723(4) Both the contact and bending stresses decrease with the179.585growth of the number of teeth and the pressure angle according tosimulation results of the example FE model.239.447(5) The cosine gear drive is a new type of gear drives.299.308Therefore, other characteristics such as inspection, sensitivity of359.170center distance error of this drive and its manufacturing should beresearched further.448.962 1References1] THOMAS Y. DANTELC H Y, SHIH-HIS T. Design of Dew toothprofiles for high-load capacity gears[] Mechanical and Macbine Theory,Fig. 11 Stress dstibution of the cosine gear drive (a =23°) (MPa)ARIGA Y, NAGATA s. Load capacity ofa new W_N gear with basic .2 ARCOACONACATASLoadcairyofanwourngeywrbasreTasmissions. and Automation in Desigm, 1985, 107(4): 565-572.TSAY C B, FONG z H. Computer simulation and stress analysis986*10helical gear with pinion circular arc teeth and gear involute teeth[J.49.429Mechanism and Machine Theory, 1991, 26(2): 145-154.98.8584] KOMORI T, ARIGA Y, NAGATA S. New gears profle having zerorelative curvature at many contact points (logix tooth profile)[J]. Joumal148.287of Mechanisms, Transmissions, and Automaion in Design, 1990, 112(3):197.716430 436247.1455] ZHAO H, LIANG J H LIU H Y, et al. The constructing principle andcharacteristics of tooth profiles with micro-segments[]. Chinese Jourmal296.573of Mechanical Engineering, 1997, 33(5): 7-11. (in Chinese)346.0026] ZHANG G H, XU H B, LONG H. Double involute gear with395.431■ladder-shape teeth[]. Chinese Joumal of Mechanical Engineering, 1995,31(6): 47- 52. (in Chinese)7] LUO s M, WANGI, YU Y D, a al. A mechanism of cosine gear drive:China, 200710034766.1[P]. 2007.8] WANG J Sudy on the principle and characteistics, of cosine gearFig. 12 Stress dstributio of tbe cosine gear drive (a -249) (MPa)drive[D] Xiangtan: Hunan University of Science and Technology, 2007.(in Chinese)9] TAN W M. Common definition for end-surface contact ratio of gears andTable6 Stresses of the cosine gear underits applications[] Chinese Joumal of Mechanical Engineering, 2004,different pressure anglesMPa14):595-597.Pressure angle Contact stesBending stress[10] CUlJK QIN SVEN B. Analysis of mesbing properties for the straightrre s8m. (tension)_ 0m__ (compressionconjugate internal gear pair[]. Jourmal of Mecbanical Transmission,498.9823448 9680.8991.02LIUZM, HOU D H, WANG x C, et al. Reverse design and analyses of395.4386.32compound tothmrofileofoearmumpin. JoumalofMechanicalTransmission. 2000. 240): 13-24. (in Chinese)According to Table 6, the contact and bending stresses [12] HU L R Spatial gear meshing principle and aplicatio[M. Beijing:ecrease with the growth of the pressure angle. For instance, theCoal Industry Press, 1987. (n Chinesecontact stress, tension and compression bending stresses are[13] YANST, HUCB, WU Z x, et al. The calculating method of slidingwear on tooth outline[J]. Machine Design and Manufacturing498.98 MPa, 86.04 MPa and 95.59 MPa, respectively, as theEngineering, 2001, 30(4): 9- 10. (in Chinese)pressure angle of a= 22", while 395.43 MPa, 71.81 MPa, and86.32 MPa as the pressure angle of a=249.Biogrpbical notesWANG Jjian is currently a teacher in Hunan University of Science and3 CONCLUSIONSTechnology. China. His research interests include mechanical transmission.Tel: +86-13467915575; E-mail: wangjian421 1@yahoo.com.cnA new type of gear drives- a cosine gear drive isLUO Shanming is currently a professor in Hunan University of Science andiestiatede which eakes a coie cuve as the lh profle LuO Ssomii crr。porse i Hturuo loer :1f Siere dBased on the mathematical model, the characteristics includingTechnolosy.Chin His researcbthe contact ratio, the sliding coefficient and stresses are studied.E-mail: s.luo@hotmail.comThe efects of gear design parameters, such as the number of teeh,pressure angle at pitch circle, on stresses of cosine gears have also CHEN Lifeng is currently a teacher in Hunan University of Science anbeen analyzed. The results of performed research allow the Technology, and a PhD candidate in. Chongqing University, China. Hifollowing conclusions to be drawn,research interests include mechanical design and theory, CAD/CAE/CAM.(1) The contact ratio of the cosine gear drive is about 1.2 toE-mail: lfchen@hnust.edu.con1.3, which is about20% less than that of the involute gear drive CHEN Lei is crrendly a gaduate student in Hunan University of Science andaccording to Table 1Technology, China. His research interests include mechanical trans mission.(2) The sliding coefficient of the cosine gear drive is smaller E-mail: chenlei037l@yaboo com.cnthan that of the involute gear drive according to Fig. s.(3) The contact and the bending stresses of the cosine gear ad Te th中国煤化工inan Unvrst of Scicncenan Opiverste mechanical ransmission,drive are lower than that of the involute gear drive. For instance, E mai:hMHCNMHG