Dynamic Simulation of Harmonic Gear Drives Considering Tooth Profiles Parameters Optimization*

Transcription

1 JOURNAL OF OMPUTERS, VOL. 7, NO. 6, JUNE Dynamic Simulation of Harmonic Gar Drivs onsidring Tooth Profils Paramtrs Optimization* Huimin Dong School of mchanical nginring, Dalian Univrsity of Tchnology, Dalian 11604, PR hina Zhngdu Zhu, Widong Zhou, Zhi hn Abstract onsidring th fact that load distribution of th flxsplin undr altrnating dynamic loads is complicatd, and it s difficult to masur th strss of th flxsplin by xprimnt, a dynamic simulation modl of harmonic gar drivs is stablishd to study th intnsity of th flxsplin in th oprating procss of harmonic gar drivs basd on finit lmnt mthod. In ordr to dcras th forc on flxsplin, by mans of conjugat mthod, th paramtrs of flxsplin and circular splin s tooth profils ar optimizd. A D-100 harmonic driv is simulatd basd on th contact thory and th xplicit algorithm of th finit lmnt mthod. Th paramtrizd finit lmnt grid of th whol harmonic gar driv modl is ralizd by Matlab, and thn th assmbly procss of th wav gnrator is simulatd by ABAQUS. Manwhil, th distribution of th flxsplin strss and th load charactristics of th flxsplin tooth during dynamic transmission procss of a harmonic gar with paramtrs, bfor and aftr optimization, ar analyzd. Th mshing charactristics of th tooth pairs with optimizd paramtrs ar improvd significantly. Th fatigu fractur position on th flxsplin is forcastd according to th simulation rsults. Indx Trms dynamic simulation; harmonic gar drivs; finit lmnt mthod; fatigu fractur; flxsplin. I. INTRODUTION Sinc Mussr invntd a nw typ of transmission calld harmonic gar drivs in th lat 1950s, harmonic drivs hav bn widly usd in many industrial filds, spcially in high prformanc control systm for thir high prcision, compactnss, light in wight proprtis and high rduction ratio charactristics compard with th convntional spd rducr. A harmonic driv works with priodic lastic dformation of a flxsplin which is a spatial dformation lmnt. Obviously, th dformation of th flxsplin is of utmost importanc, Manuscript rcivd April 15, 011; rvisd Jun 1, 011; accptd Jun 1, 011. opyright crdit, corrsponding author: Huimin Dong *Th financial support of th National Natural Scinc Foundation of hina No and th rsarchrs mad many fforts in this rspct. Photolastic mthod was adoptd to study th concntratd root strss of flxsplin in [1], th strss distribution of flxsplin was masurd by xprimnt. Howvr, for th complx of harmonic drivs, th author simplifid flxsplin tooth and structur. Thus, som paramtrs ar difficult to b obtaind by xprimnt mthod, which has grat limitation and high cost. Latr scholars mostly adoptd finit lmnt mthod to analyz th intnsity of th flxsplin. Du to th difficulty of modling and th grat amount of calculation, thy simplifid th modl and boundary conditions in [-4], and only conductd static contact analysis without taking into account th dynamic charactristics of th whol harmonic drivs undr th ffct of variabl load which lads to biggish dviation in th analysis rsults. 3D FEM [7] is adoptd to chck th flxsplin strngth, abl to prdict th flxsplin failur position undr th static stat. Dynamics mthod has bn xtnsivly applid in th structur analysis. Th structural dynamic problms oftn hav two typs, on is structur or mchanism in th movmnt, such as turbin, which possss th inrtia of itslf and dynamic load producd by intraction of mdium surrounding and structur; th othr is nginring structur that bars dynamic loading. A harmonic driv can transform th motion and momnt from input to output du to th flxsplin priodic lastic dformation, which blongs to th first kind of dynamics problms. Th main failur form of harmonic drivs is th flxsplin fatigu fractur du to th flxsplin priodic dformation. This papr taks a D-100 typ harmonic driv as an analyzing xampl. Th tooth profil paramtrs ar optimizd in ordr to fit th conjugat numrical rsults obtaind by conjugat modl [16]. A simulation modl is stablishd, which is clos to th actual situation in a harmonic drivs oprating to th largst xtnt. In considration of th contact, dynamic load impact, mshing intrfrnc btwn th flxsplin and th circular splin tooth, and th strss, strain of th flxsplin varying along with motion of th harmonic doi: /jcp

2 1430 JOURNAL OF OMPUTERS, VOL. 7, NO. 6, JUNE 01 driv undr th nominal load, th dynamic simulation is conductd using Abaqus/Explicit in Abaqus. Th simulating rsults with paramtrs bfor and aftr optimization ar compard. Finally, th position whr flxsplin is pron to aris fatigu fractur is to b prdictd, manwhil, th circumfrntial mshing rul of th flxsplin tooth is studid. II. OPTIMIZATION OF TOOTH PROFILE PARAMETER A. Th dtrmination of th conjugat tooth profil A harmonic gar driv with flxsplin cup consists of thr componnts, a rigid circular splin with intrnal tth, a wav gnrator and a flxsplin with xtrnal tth. Th flxsplin is th ky lmnt in a harmonic driv, bcaus of its priodic lastic dformation causd by th wav gnrator. Th profil curv of th wav gnrator is: R( ϕ) = R + w cosϕ (1) 0 0 whr R0 is th radius of flxsplin nutral circl; w0 is a constant cofficint of th wav gnrator. In Rf. [16], th nutral layr dformation of th flxsplin cup on ach cross sction undr no-load condition has bn discussd, which can b xprssd in trms of ϕ (th indpndnt argumnt) and z as ar z z u = mcos ϕ,v = msinϕ L0 L0 3z msinϕ ϕ ff = 1/ L0 (( R0 + u) + v ) () whr u, v and ϕ ff ar radial and circumfrnc and angular displacmnt rlativ to th nutral circl rspctivly; L0 rfrs to th midsction of th fac width; z is th lngth masurd from th closd nd of th cup to th cross sction along th rotating axis;. Lt th circular splin b a spur gar with an involut profil. Th tooth profil at th right sid and th continu dirction of th profil ar writtn as [8, 16]: rb π ri =, δ = αi + μ cosαi 1 μ = ( π + 4 ξ tan α ) +, αi [ αa, αd ] N invα invα fi whr rb is th bas circl radius; and αa, αd is th addndum and ddndum prssur angl rspctivly; ξ is th modification cofficint. Th conjugat condition of th harmonic driv satisfy th convntional garing conjugat principl : V f i n = 0. Whn th wav gnrator is fixd and th circular tooth profil is givn, th conjugat condition in ach cross sction can b xprssd as [16] (3) ( f ) ( ) v + R 1 ω sin ψ + u v 1 ω f cos ψ + r(ω ω )sin(δ μ ) = 0. u f whr R u =R 0 +u, ψ = δ + ϕ ϕ. Basd on th abov analysis, if th dsird transmission ratio and th cup dformation ar givn, with a prscribd circular splin tooth profil, th conjugat point ( x, y ) and its position angl ϕ can b obtaind by (4). Thus, th complt flxsplin profil corrsponding to th mshing point can b obtaind by th coordinat transformation from circular splin coordinat systm flxsplin coordinat systm [16]. B. Th optimization of tooth profil paramtr According to th mthod of dtrmining th conjugat tooth profils, with wav gnrator paramtr and involut tooth profil of circular splin as known paramtrs,th corrsponding conjugat flxsplin tooth profil can b obtaind. Howvr, th numrical solution of mathmatical modl basd is a sris of discrt points. onsidring manufacturing procss charactristic, th involut is adoptd as actual tooth profils. Thus, involut is usd to fit thos thortical discrt points. An optimal modl with smallst fit rror is stablishd to confirm th bst involut paramtrs. Lt x1 and x (modification cofficints of th flxsplin and circular splin tooth profil) rprsnt th optimization variabls. Th involut quation is usd to approximat to th discrt data of th conjugat tooth x = F x, λ, which profil, it can b dscribd as { } ( ) inv 1 1 is rlatd to x 1 ; th discrt data of th conjugat tooth f profil is:{ x M } = G( x, λ ),which is rlatd to x. Th optimal objctiv function indicating th shortst distanc f btwn th point { xm } and i { x inv} on involut can b writtn as: ( ( ) i ) f ( 1,, λ1, λ ) min distanc { },{ } D = H x x = x x i i M inv i = 1,, 3,, N TABLE I. THE BASI PARAMETERS OF D-100HARMONI GEAR DRIVE Flxsplin ircular splin Numbr of tooth Modul 1 1 Prssur angl 0 0 Initial modification cofficint Addndum cofficint Bottom claranc cofficint A calculation xampl basd on th conjugat thory will b conductd, th basic paramtrs of harmonic gar is givn in Tabl I. (4) (5)

3 JOURNAL OF OMPUTERS, VOL. 7, NO. 6, JUNE Th tooth profil of th circular splin can b solvd basd on (3); for ach cross sction, u, v, ϕ ff and thir drivativ ( u,v and ϕ ff ) ar found from (). Thn, substituting thm into (4) and solving th quation yilds th position angl ϕ ; thus, th flxsplin profil corrsponding to th mshing point can b obtaind. Fig. 1 shows th thortical conjugat tooth profil of flxsplin on th 1 st cross sction. Hnc, th optimization modl can b stablishd: { i ( 1 λ1 λi) } min max D = H x, x,,, i = 1,,3,, N Di f 1 1 F st.. { xm} = G( x, λ) = B ( BF { xm} { xof} ) { xinv} = F( x1, λ1) x 1,x >0 whr N is th numbr of th conjugat point. (6) lb, ub ar limit optimizing variabls, in th rang of [ lb, ub ] to nsur x1, x > 0; th rst of th paramtrs is st to null. With th function itration compltd, th optimal solution x and rror fval can b obtaind, x = [3.0494, 3.090]. Th approximating curv is shown in Fig. and th approximating rror of th maximum rror point is 0.007mm. Th flxsplin tooth along th rotating axis is dividd into 1 cross sctions. For ach cross sction, th radial dformation cofficint w i ( i = 1,,...1) hav diffrnt valus, which is a linar function along th rotating axis, th xprssion is: w wl = 0 i + i L B Whr w0 is th radial dformation cofficint of th wav gnrator; l i is th distanc btwn th i cross sction and th first cross sction (th lft sid tooth profil); L is th lngth of th cup flxsplin. Basd on w i, th dformation function of diffrnt cross sction can b solvd. According to (4), th thortical conjugat spac tooth profil of flxsplin is obtaind shown in Fig. 3 w 0 (8) Figur 1. Th thortical conjugat tooth profil of flxsplin. Figur3. Th 3D viw of thortical conjugat profil of flxsplin Figur. Th approximating curv of flxsplin tooth profil. Th optimization modl (6) nds to b solvd by numrical calculating mthod. In this papr, th optimization mthod solving minimization of maximum valu is adoptd. Th xprssion is: fun, x0, A, b, Aq, bq,, = fminimax lb, ub, nonlcon, options [ x fval] whr fun is th objctiv function; x 0 is th initial valu of optimization variabls, x 0 = [ x 1, x ] = [3.05,.9]; (7) III. THE FINITE ELEMENT MODEL AND THE DYNAMI SIMULATION METHOD A. Finit Elmnt gomtric Modl A harmonic gar driv consists of thr componnts, a TABLE II. STRUTURE PARAMETERS OF THE FLEXSPLINE UP d L 100 δ 1.1 δ1 0.9 B 0 r 5

4 143 JOURNAL OF OMPUTERS, VOL. 7, NO. 6, JUNE 01 rigid circular splin with intrnal tth, a wav gnrator and a flxsplin with xtrnal tth. Th flxsplin is ky lmnt in a harmonic driv, bcaus of its priodic lastic dformation causd by th wav gnrator, whos structur is shown in Fig. 4, th paramtrs of th structur is listd in Tabl Ⅱ. Th profil curv of th wav gnrator is xprssd in (1) flxsplin axis, which is convnint for dfining boundary conditions in ABAQUS. Figur6. Th modl of harmonic gar driv Figur4. Structur of th flxsplin Figur5. Th profil of th wav gnrator whr R 0 is th radius of flxsplin nutral circl; w0 is a constant cofficint of th wav gnrator in Fig. 5; ϕ is th angl masurd from th major axis of th wav gnrator. In ordr to msh th harmonic gar driv componnts rasonably, th following simplifications ar takn on th modl: (1) Th conncting flang btwn th bottom of flxsplin cup and output axis is rmovd. Bcaus it is far from th opn nd of th cup and gar ring, th dformation of th flxsplin is not snsitiv to it. Th nods on th ring surfac xposd whn th conncting flang is rmovd ar coupld to th point on th () Th roundd cornrs btwn gar ring and cup of flxsplin ar simplifid. Howvr, ral tooth profil is adoptd for th flxsplin tooth to idaliz th ngagmnt btwn th flxsplin and circular splin tooth in th procss of th dynamic simulation. (3) In ordr to install th wav gnrator into th flxsplin convnintly, th wav gnrator is cut into two qual parts. Th outlin of wav gnrator in th axial sction is dsignd to b an arc with a largr radius to simulat th dformation of th outr ring of th flxibl baring. Th simplifid modl is shown in Fig. 6,th grid is mshd in an intllignt mthod so that th modl can b changd with profils of th tth asily. B. Th dynamic simulation mthod. To raliz th dynamic simulation, th modl is discrtizd, and thn th dynamic quation can b usd as follows: () + () + () = () Mat at Kat Q t (9) whr: a () t and a () t ar th acclration vctor and vlocity vctor of nods in th systm; M,, K and Q(t) ar th mass matrix, damping matrix, stiffnss matrix and joint load vctor rspctivly, which ar constitutd by unit matrix and vctor, T T M= ρn NdV = μn NdV V V (10) K = B DBdV Q= N fdv + N Tds ( ) T T T V V Sσ Whr ρ is dnsity of matrial, μ is damping cofficint, B is lmnt strain matrix, D is lastic matrix, N is unit shap function matrix, f is unit volum forc, T is forcs on bordrings. To solv dynamic diffrntial quation (9), cntral diffrnc mthod is adoptd to raliz numrical intgration stpwis. In th cntral diffrnc mthod, acclration and spd can b xprssd by displacmnt, 1 a t = ( at Δ t at + at+δ t) (11) Δt

5 JOURNAL OF OMPUTERS, VOL. 7, NO. 6, JUNE a t = ( at Δ t + at+δt) (1) Δt Diffrntiating (11) and (1) to (9) yilds th rcursiv formula 1 1 M+ a = Q K M a Δt Δt Δt 1 1 M a t Δt Δt Δt t+δt t t (13) If at Δ t and a t hav bn acquird, a t +Δ t can b calculatd by (13), thus (13) is rcursiv formula which can solv th displacmnt at ach discrt tim. Whn t=0, a 0 and a Δ t should b dtrmind to calculat a Δ t. Through (11) and (1) w can obtain: Δt a Δ t = a0 Δ ta 0 + a 0 (14) Aftr gtting th displacmnt of all points at diffrnt tim by (13), it is convnint to us gomtrical quations and th physical quations to obtain strss and strain at all discrt points. IV. THE DYNAMI SIMULATION OF A HARMONI DRIVE Fatigu fractur of th flxsplin is th main rason of harmonic gar drivs failur, so it is quit important to study th dynamic rspons of flxsplin undr altrnating loads. In this sction, th dynamic simulation of a harmonic driv is conductd, and th strss charactristic and dformation distribution of th flxsplin changing with tim is xtractd. Finally, th rason for th fatigu fractur of th flxsplin is analyzs and find out, and th location whr th fatigu fractur may happn is prdictd. A. Finit Elmnt Analysis Modl. 3D8R hxahdral rduction lmnt providd by ABAQUS is choosd to msh th flxsplin, wav gnrator and rigid splin. Sinc simplifid modl has bn considrd according to th sction A, th numbr of lmnt can b rducd as lss as possibl to gt a bttr grid quality. Th papr obtains th finit lmnt msh modl basd on Matlab, which xports input fil (*.inp) for ABAQUS calculation. Th mshd modl is shown in Fig. 7, whr is th initial stat for solving. Th numbr of lmnts is 01000, and numbr of nods is Th procss of solving th dynamic simulation modl is dividd into two stps in ABAQUS. Th first stp is simulating th procss that th wav gnrator b installd into th flxsplin. Th scond stp is basd on th first stp, and add a ratd rotating spd n=3000rpm, and ratd load (torqu) T=300Nm is applid on th clos nd of th flxsplin cup. Thn th contact of wav gnrator and flxsplin and th contact of flxsplin tooth and circular splin tooth ar dfind, and th harmonic gar driv in practic oprating is simulatd. To nsur th stability of cntral diffrnc mthod, st th itration tim Δt L /, whr L is th smallst lngth in th minimum lmnt, ( / ) 1/ = E ρ is th vlocity of sound in th mdium, and total simulation tim is ms (about th tim for th wav gnrator to turn on circl), to prvnt th shock of suddn loading, th first ms of th procss is st to load gradually. Th calculation of modl is so complx that w hav to us DELL 8PU-16G srvr to solv it. B. Strss rspons of th flxsplin cup. Fig. 9 shows th curvs of Von Miss strss of th 3 nods on th nutral layr of th flxsplin cup changing with tim, th positions of th thr nods from 1 to 3 succssivly lay along th axial of th cup from th opn nd to th clos nd of th cup as in Fig. 8, in which nod 1 and nod 5 locat at th opn nd and clos nd of th flxsplin cup on th nutral layr rspctivly, th othrs locat btwn thm. As th dformation of flxsplin changs priodically with tim, all nods of ach cross-sction which vrtical to th axis hav th sam charactristics of strss and strain in a cycl, th charactristic of ach point can rprsnt all nods of th corrsponding sction. Figur8. Th distribution map of analysis sction of flxsplin cup Figur7. Finit lmnt modl of harmonic gar Figur9. Th curv of strss-tim of flxsplin

6 1434 JOURNAL OF OMPUTERS, VOL. 7, NO. 6, JUNE 01 to b obtaind by xprimnt, th papr obtains th contact forc by finit lmnt analysis. Figur10. Th curv of strss-tim of Fatigu point In Fig. 9, th nod corrsponding to curv 1 locats at th opn nd of th cup, th strain bcoms largr whn it is against th minor and th major axis of th wav gnrator, and thr ar four paks in on circl of th wav gnrator. Towards th clos nd of th cup, th influnc of th minor axis of th wav gnrator on th flxsplin dformation rducs quickly, whil th influnc of th major axis on that dformation rducs slowly, thus th curv of th flxsplin cup strss gradually has two paks. Th nods corrsponding to th thin shll parts of th flxsplin ar far from th installation position of th wav gnrator. Hnc th strss is small and wavs littl with tim. Th maximum strss of nod 1 is 3.7Mpa, which is 4.16 tims as big as th maximum strss of nod 6, which mans fatigu failur is pron to occur in th front part of th flxsplin cup. urv 1 and curv 3 in Fig. 9 ar in th gar ring of th flxsplin, th amplitud of altrnating strss of its corrsponding nods is 96.1MPa and 403MPa rspctivly. Obviously, th gar ring of flxsplin bars hug altrnating strss. But curv 1 has four tims in altrnating in a circl compard with two tims in curv 3, which may lad to fatigu failur. Th thicknss and dpth of gar ring is far biggr than th thicknss of thin shll of th flxsplin, thrfor, th rigidity of th gar ring is grat and th strss concntration is pron to occur at th thin shll of th flxsplin. Th Miss strss of nod at th thin shll is xtractd, shown in Fig. 10, th altrnating amplitud of strss is 30.6MPa, which provs that th transition position of thin shll and th gar ring of th flxsplin is th most vulnrabl to b fatigu fractur. B. Distribution of ngaging contact forc of th flxsplin gar ring. Du to th diffrnt radial dformations of th flxsplin along th axial dirction, th flxsplin tooth bcoms skw along th axis dirction. Sinc th spur involut tooth rplacd th spatial conjugat tooth form, th conjugat movmnt of tooth pair is not accurat, and th strss condition of th flxsplin bcoms quit complicatd. Th contact forcs of Tth ar impossibl Figur11. Th curv of addndum contact forc-tim of un-optimizd tooth profils with (x 1 =3.05, x =.9) Distribution charactristics of contact forc of th flxsplin gar ring. Fig. 11 shows a curv of contact strss on 3 nods changd with tim in on cycl, th 3 nods ar on top of th flxsplin tooth in a mshing sid of th tooth. Thy locat along th tooth from nod 1 to nod 3, nod 1 is clos to th front of flxsplin gar ring, nod is in th middl sction, nod 3 is in th rar of th flxsplin gar ring (Fig. 8). In th initial stat th tooth is against th minor axis of th wav gnrator. Figur1. Th curv of addndum contact forc-tim of optimizd tooth profils (x 1 =3.0494, x =3.090) In th Fig. 11, thr ar two contact forc ara in a cycl of th 3 nods, and contact tim is vry short, which consists with th charactristics of harmonic gar driv. For th tim appar contact forc, curv 3 always appars arlir than curv and curv 1 and th amplitud is biggr, bcaus intrfrnc at th position of th flxsplin tooth bgins to ngag with th circular splin. urv 1 and curv show th ngagmnt in th continu mshing ara, which mans that th main strss locats in th middl part of th profil. Th contact forcs of th nods that distribut in th 1 cross-sctions in on tooth that in th mshing condition ar xtractd, th contact forcs ar shown in Fig. 13, th maximum contact forc of th tooth surfac (86.7N) happns at th middl sction, and rducs gradually to th nds. Thr ar som sctions nar th rar of th tooth that do not ngag, this phnomnon consists with that in Fig. 11.

7 JOURNAL OF OMPUTERS, VOL. 7, NO. 6, JUNE Figur13. Th distribution curv of contact forc of singl un-optimizd tooth profil Fig. 15 shows th distribution of contact forcs of all un-optimizd flxsplin tth at a crtain momnt, th lngth and dirction of th rd arrows rprsnt th siz and th dirction of contact forc rspctivly. Fig. 15 tlls us th flxsplin and circular splin against th long axis of wav gnrator xist ngagmnt and contact forc ara. A big contact forc xist on ach nd, this ariss from th intrfrnc of tth top of flxsplin and circular splin at th ngaging momnt. Th numbr of tooth pairs participat ngagmnt at th contact forc ara is 16. Aftr optimizd Fig. 16, th numbr of tooth pairs is 18, manwhil, th contact forc rducs to 46.99N, 41.9% lss than that bfor optimization (81.01N). Figur14. Th distribution curv of contact forc of singl optimizd tooth profil Fig. 1 shows a curv of contact strss aftr optimizd on 3 nods changd with tim in on cycl, th thr curvs all rach pak in th contact forc ara and th strss is zro at othr tim, which mans th intrfrnc phnomnon in Fig. 11 has disappard. Th contact forcs of th nods that distribut in th 1 cross-sctions in on tooth aftr optimizd that in th mshing condition ar xtractd in Fig. 14. ompard with Fig. 13, th contact forc ara movs to th back-nd in gnral trnd, and ngagmnts happn in all th tooth profil. Th maximum contact forc of th tooth surfac rducs 41.9% compard with un-optimizd (86.7N). Distribution charactristics of circumfrntial load of th flxsplin gar ring. Distribution of circumfrntial load of th flxsplin gar ring is th load cas of all tth in th gar ring at a crtain tim whil harmonic gar driv is oprating. This can rflct th siz and distribution of load on th flxsplin at any tim. Figur15. Th circumfrntial load distribution of un-optimizd flxsplin tooth Figur16. Th circumfrntial load distribution of optimizd flxsplin tooth As a mattr of fact, only 18% of th total tth participat in ngaging, not rach 40%~50% as [, 4] stat so. Howvr, with th incras of th load, th ngagd tooth pairs may incras rlvantly. V. SUMMARY (1) Th strss on th flxsplin cup dcrass gradually from th opn nd to th clos nd of th cup in th axis dirction; Frquncy and amplitud of th altrnating strss at th opn nd of th cup ar larg. onsquntly, fatigu failur is pron to occur at th junction of gar ring and thin shll of cup whr th strss is concntratd. () In a instant of th oprating procss of th harmonic driv, th numbr of tth which participat ngagmnt is only taking up 18% of th total tth numbr practically, which is much smallr than th rsults of othrs [,4] do not considr dynamic charactristics. (3) Th distribution of contact forc on th flxsplin ring with optimizd tooth profil covrs th whol tooth profil along th axial dirction, and intrfrnc of th tooth pair on th tooth tip at rar of th ring is liminatd; th numbr of tooth pairs participat ngagmnt at th contact forc ara incrass 4%, at th sam tim, th contact forc rducs 41.9%, mshing charactristics of th tooth pairs ar improvd gratly. AKNOWLEDGMENT

8 1436 JOURNAL OF OMPUTERS, VOL. 7, NO. 6, JUNE 01 Authors wish to acknowldg th financial support of th National Natural Scinc Foundation of P.R. hina undr ontract No REFERENES [1] Y. W. Shn and Q. T. Y, Thory and dsign of Harmonic Driv Garing, Bijing, hina Machin Prss, [] Д.П.Волков, А.Х.Крайнев: Harmonic gar transmission, Elctronic industry prss, Bijing, [3] H., Dong, Study of Kinmatics and Mshing haractristic of Harmonic Gar Drivs Basd on th dformation Function of th Flxsplin. E., Ph.D Thsis, Dalian Univrsity of Tchnology, hina, 008. [4] Bown ui, Y.W.Shn Study th distribution of contact strss by finit lmnt analysis [J]. Mchanical transmission, 1997, 1(1): 7-9. [5] H. M. Dong and X. Q. Zhang, Exprimnt basd finit lmnt analysis on flxsplin of harmonic driv, Journal of Machinry Transmission, hins, (001) , 001. [6] J., Fu, H., Dong and Y., Shn, Strss Analysis of th Flxsplin in Harmonic Garing by Using FEM. hins Mchanical Enginring Socity, 18(18):10-14, 007 [7] H. Kazrooni: Dynamics and ontrol of Instrumntd Harmonic Drivs. Journal of Dynamic Systms[J], Masurmnt, and ontrol, 1995, Vol.117: [8] H., Dong, K. L.,Ting, and D., Wang, 011, Kinmatic Fundamntals of Planar harmonic drivs, ASME J. Mich. Ds, , pp [9] W. Z. Liu, N. R. Zhang,. L. Zhang, 006, Finit lmnt analysis of flxsplin cup in harmonic driv, hins Journal of Mchanical Enginring, 006, Vol.4: pp [10] K., Oguz, and E., Fhmi, 007, Shap optimization of tooth profil of a flxsplin for a harmonic driv by finit lmnt modling, Matrials and Dsign, 8, pp [11] H., Dong, and X., Zhang, 001, Exprimnt basd Finit Elmnt Analysis on Flxsplin of Harmonic Driv hins J. Mach. Trans., 5(), pp [1]., Mussr, 1955, Strain Wav Garing, Unitd Stats Patnt Numbr,906,143. [13] S., Ishikawa, 1967, Th Gar Gomtry of Tooth Engagmnt in Harmonic Driv, Proc. of th JSME Smi-Intrnational Symposium, pp [14] K., Kondo, and J.,Takada, 1990, Study on Tooth Profils of th Harmonic Driv, Transaction of th ASME, 11 (3), pp [15] Y., Fan, H., Wang, and D., Song, 00, Rsarch on omputr Simulation for onjugat Tooth Profils of th Harmonic Driv, hins J. Nanjing Univrsity of Scinc and Tchnology, 6(4), pp [16] H., Dong, D., Wang, and K. L.,Ting, 011, Kinmatic Effct of th ompliant up in Harmonic Drivs, ASME J. Mich. Ds, ,Vol. 133 / Huimin Dong was born in Yuto an, Hbi, hina, ; Bachlor Dgr in 198, Mastr Dgr in 1985, Ph.D in 008 at Dalian Univrsity and Tchnology hina. Th author s major fild of study is on th conjugat thory and strngth dsign of mchanical transmission and harmonic drivs. Sh has bn working at School of Mchanical Enginring, Dalian Univrsity and Tchnology hina sinc In 1997, sh was promotd as Associat Profssor. Rprsntativ publications: Kinmatic Fundamntals of Planar harmonic drivs, ASME J. Mich. Ds, , pp ; Kinmatic Effct of th ompliant up in Harmonic Drivs, ASME J. Mich. Ds, ,Vol. 133 / Dr. Associat Prof. Huimin Dong is a mmbr of SME Zhngdu Zhu was born in E zhou, Hubi, hina, ; H is studying at Dalian Univrsity and Tchnology hina and will gt bachlor dgr in 01. Widong Zhou was born in Hunan, hina; Bachlor Dgr in 008 at Dalian Univrsity and Tchnology hina..