DESIGN OF DOUBLE HARMONIC TRANSMISSION

Transcription

1 6th Intrnational DAAAM Baltic Confrnc IDUSTRIAL EGIEERIG April 2008, Tallinn, Estonia DESIG F DUBLE HARMIC TRASMISSI Ianici, S.; Ianici, D. & Potocanu,. Abstract: Th papr prsnts th construction and functioning of a nw typ of harmonic driv namd doubl harmonic transmission (D.H.T.). In th scond part of this papr is prsntd th dynamic analysis of th doubl harmonic transmission, which is basd on th rsults of th xprimntal rsarchs on th D.H.T. This study of th strss status and th forcs distribution is ncssary for to dtrmin th durability on th portant lmnts of th D.H.T. Ky words: doubl harmonic transmission, short flxibl toothd whl, forc, strss.. ITRDUCTI In th last 3-4 dcads, rsarchrs paid mor attntion to mchanical transmissions through ngaging du to multipl advantags that this typ of driv has looing to improv thm as much as possibl but not sufficint nough. Th main trnd during th prsnt stag of dvlopmnt and modrnization of tchnical industry is constitutd by th automation of production procsss and thir intgration within systms controlld and co-ordinat by computrs. In this rspct it bcoms ncssary to improv and crat nw mchanic transmissions abl to prform prcis motions and to transmit big loads. This nw catgory includs also th toothd harmonic transmission, mainly usd in th dsign and manufacturing of drivs for industrial robots [ ],[ 2 ], [ 3 ],[ 5 ]. Th toothd harmonic transmission is ssntially diffrnt from th classic mchanic transmissions as it transmits and transforms inmatics and dynamic paramtrs of th rotation motion through lastic dformation propagatd according to a harmonic law of on of its lmnts namd flxibl toothd whl. Through constructiv and functional divrsifications of toothd harmonic transmissions on has cratd a nw variant of th doubl harmonic transmission. This transmission prsnts a sris of advantags such as: grat prcision of positioning and rpatability, grat transmission ratio (i = 40 50), rducd dad cours, xtrmly small loos motions, small wight and siz, small inrtia momnts, coaxial and modular construction, offring thm a largr and largr rang of applicability in th machin-building fild. Th prsnt papr prsnts th construction, functioning and som dynamic aspct of th D.H.T. 2. CSTRUCTI AD FUCTIIG F A D.H.T. It is basd on th sam woring principl as th simpl harmonic transmission, but on must corrlat th numbr of tth of toothd whls from th mounting condition [ 2 ],[ 4 ]: / z = z 2 ; z = z 2 () Fig. prsnts th structural schm of a D.H.T. with short flxibl whl undr th form of a circular tub with thin wall, opn at both nds and having at ach nd a toothd crown (xtrior z 2 and intrior z ¹ 2). A doubl harmonic transmission (Fig. 2) is mad of: a wav s gnrator () as

2 lading lmnt, a short flxibl toothd whl (2). wavs gnrator of th D.H.T. th ld rigid whl will rotat in th opposit sns with four tth compard to fixd on. Fig. Th structural schm of a D.H.T. with th rspctiv xtrior and intrior toothd crowns, th fixd rigid whl (3) and th mobil rigid whl (4) as ld lmnt. Fig. 2 Th doubl harmonic transmission Th wavs gnrator bing in sliding contact along th ntir priphry of th flxibl whl, it dforms this toothd whl so that it will hav four quidistant driving zons: two with th fixd rigid gar having intrior tooting and two with th mobil rigid gar having xtrior tth. Btwn th two pairs of opposd zon of driving (I-I and II-II rspctivly) thr is a 90 o angl (Fig.3, a). As ths zons ar at a 90 o angl from ach othr th transmission of loads is don through bnding and torsion strsss. At th bginning of functioning on finds in contact on th opposd vrtical positions (I-I) th xtrior tth (z 2 ) of th flxibl whl whit th intrior tth ( z 3 ) of th fixd rigid whl, and in th position disposd at 90 o angl to th vrtical axis (II-II) on can find in contact th intrior tth (z ¹ 2) of th flxibl whl whit th xtrior tth (z 4 ) of th ld rigid whl. Thus at a full 360 o rotation (Fig.3, c) of th Fig. 3 Th inmatics modl a. = 0 ; b. = 80 ; c. = 360 Th transmission ratio a D.H.T. is dtrmind with Willis s rlation, [ 2 ]: 3 ω i 34 z 2 z 4 i 4 = = = ω 4 i 34 z 2 z 4 z z 2 3 (2) whr: ω th angular spd of th wavs gnrator; ω 4 th angular spd of th ld rigid whl; z numbr of tth of th transmission whls. 3. DETERMIIG THE STRESS STATUS WITHI THE SHRT FLEXIBLE TTHED WHEEL Th study of th strss status within th short flxibl toothd whl of a D.H.T. by using th finit lmnt mthod (F.E.M.) is basd on th variational formulation and

3 nds th following stag: laborating th computation pattrn, constituting th quations of th finit lmnts, assmbling th quations within th systm of quations of th structur and th solving this systm. Fig.4 prsnt th analysis modl rprsntd by a quartr of th short flxibl toothd whl of th D.H.T., considring only th ara at th bottom of th tth at a lngth of mm, which is dformd undr th form of an llips by th wavs gnrator. As tin cas of th two-wav gnrator th lastic dformations and th strsss within th flxibl whl ar symmtric compard to th cntr of th wav gnrator shaft, for th study it is sufficint to ta into considration only a quartr of th flxibl whl. y 0 x Fig. 4 Th dynamic analysis modl In ordr to apply F.E.M., [ 2 ],[ 6 ], on must prform a dcomposition of th analysis modl in 36 finit lmnts of th triangular typ with thr nots. W considr a triangular finit lmnt (Fig.5) with th nots i, j and numbrd in th trigonomtric dirction, rportd to th global systm of co-ordinats xy having rprsntd on its tops th u and v componnts of th unnown function (shift). For ach componnt of th motion (shift) it is accptd as approximation function th intrpolation polynom of th first grad: u = α + α 2 x + α 3 y (3) v = α 4 + α5 x + α 6 y whr: α i ; i =, 6 ar gnralizd coordinats dtrmind according to th shifts of th nots of th finit lmnt. y Fig. 5 Th finit lmnt Thn th shift vctor of a point situatd in th intrior of th finit lmnt, which dtrmins th strss and dformation status may b xprssd trough, th rlation: v j u j v j u v u {} d = = i = [ ] { d } i j (4) whr:{ d } is th shift vctor of a point within th intrior of th finit lmnt; [] - th matrix of th function of form; { d } - th matrix of nodal shift of th rspctiv lmnt. Aftr som assuming and substitutions w hav obtaind th rlations for th computing th shift vctor of th finit lmnt: u i v i { } = [ L] { d} = [ B] { d } ε (5) whr: [L] is th drivation oprator and th matrix is mad of constants dtrmind through th coordinats of th nots of th finit lmnt. 0 x [ L] = 0 (6) y y x x

4 yj y 0 y yi 0 yi yj B = [ L] [ ] = 0 x xj 0 xi x 0 2A x xj yj y xi x y yi xj xi [ ] 0 xj xi y i yj (7) whr: A is th surfac of th finit lmnt. According to Hoo s law, on stablishs th connction btwn strsss and dformations for th plan stat, taing into account d conditions of compatibility: {} σ = E] {} ε [ (8) whr: {σ} is th vctor strsss; {ε} - th vctor of th spcific dformations; [E]- th matrix of th lastic constants. Th dynamic bhaviour of th finit lmnt undr considration is dscribd trough lmntal quations which ar obtaind by applying th thorm of th minimum potntial nrgy, following in th nd th matrix of th rigidity for th triangular lmnt undr considration: [K] = h A [B] T [E] [B] (9) whr: h is th thicnss of th lmnt; A is th surfac of th lmnt; [B] T is th transposd of th matrix [B]. Assmbling th lmnt quations within th systm of th quations of th lastic structur consists of th assmbling th rigidity matrics [K] of th finit lmnts within th rigidity matrix of th structur [K] as wll as of th vctor of th loads on lmnts {P} within th vctor of load on th ntir structur {P}. Th rigidity matrix of th structur with finit lmnts is formd according to th principl of suprposition of th influnc of ach lmnt, so that in th common nots of th finit lmnts th shift function has th sam valu. For th modl of th runion of all th lmnts w obtain th systm of quations from which w can dduc th shifts in th nots of th modl: [K] {d}={p} (0) whr: th vctor of th xtrior forcs {P} has th sam siz ran as th rigidity matrix [K]. Basd on th abov rlations on has concivd a softwar program in Turbo Pascal for th study of th strss status within th short flxibl toothd whl for th ral cas of a D.H.T. with th following functional paramtrs: i = 48,2; m = 0,3mm; z 2 = 200; z 3 = 202; z ¹ 2 = 90; z 4 = 88 tth; w 0 = 0,3mm; M t4 = 50 m. Th rsults obtaind through th procssing of data on a computr ar prsnts in Fig.6. Fig. 6 Th strss status within short flxibl whl 4. EXPERIMETAL RESULTS As in th cas of long flxibl toothd whls, th wall of th short whls situatd at bottom of tth is th most strssd and xprimnts hav provd that th first fissurs appar in this ara which in th nd lads to bras that put th short flxibl whl out of us. Th xprimntal rsarchs hav bn don on th doubl harmonic transmission with th following functional paramtrs (fig.7): th transmission ratio (i = 48,2), th wav gnrator with ccntric dis, th maximum radial dformation (w 0 = 0,3mm), th numbr of tth of th rigid whls (z 3 = 202 tth, z 4 =88 tth), th numbr of th flxibl short whls tth (z 2 = 200 tth, z ¹ 2 = 88 tth), th tth modulus (m = 0,3mm). For an xprimntal dtrmination of th tangntial forcs which actuat against th flxibl whl's tth, which ar in procss of garing with th rigid whl's ons (th two of doubl harmonic driv), th lastic

5 dformation of th control tth on th rigid whls wr masurd. Fig.7 prsnts th control tth of th two rigid whls and th possibility of soldring th two tnsiomtric dtcting lmnts on ach control tooth, which wr xamind by xprimntal tst. Fig. 7 Th lmnts of th D.H.T. For th xprimntal dtrmination of th tangntial forcs (F t ) and of th strss status ( σ ϕ, σ x, τ xϕ ) from th two zons of harmonic driving of th D.H.T. has bn usd th nown mthod [ 7 ], [ 8 ], [ 9 ], basd on th rsistiv lctrical tnsiomtry. In Fig.8 has bn prsntd th distribution of th tangntial forcs which act on th tth of th flxibl short whl, within th two zons of th harmonic driving, and in Fig.9 has bn prsntd th variation diagram of th tnsions within th flxibl whl wall for th cas M t = 50m and n = 50rpm. forc from th tth situatd in th harmonic driving (for sction I-I) is analogical with th on from th simpl harmonic transmission [ 7 ],[ 9 ]; th zon of harmonic driving II-II (ϕ II ), at M t = 50 m, is divrtd with 90 0 towards th driving zon I-I (ϕ I ) and ovrtas as xtnt with 25 0 ; in th sction I-I th shap of priphrical tnsion curv (-σ ϕ ) is clos to th on nown from th simpl harmonic transmission, and in sction II-II th priphrical tnsion curv(2-σ ϕ ) has mor pas, but th ultimat tnsions don t ovrta th maximum valus of th priphrical tnsions from th sction I-I; th variation charactr of th tangntial tnsions (τ xϕ ) and its xtnts, in both sctions, ar approximatly idntical. Fig.9 prsnts th distribution's of th tnsions on th short flxibl coggd whl's tth in two sctions: curv - for sction I-I; curv 2 - for sction II-II; for th cas M t = 50m and n = 50 rpm. Fig. 8 Th distribution of th tangntial forcs From th analyss of th xprimntally dtrmind graphics w can notic th following: th distribution curv of th tangntial Fig. 9 Th strss status within flxibl whl

6 4. CCLUSIS In th thsis was prsntd dynamic analyss of doubl harmonic transmission basd on thin wall shlls thory and on rotating charactr of distributd forcs that actuat on flxibl toothd whl. An important contribution in solving this problm hav paprs [ 7 ],[ 8 ], which prmittd th idntification on th distort forcs from th wavs gnrator with dis and cam upon th flxibl toothd whl as wll as th forcs from th harmonic ngagmnt. Th diffrncs in th variation graphics of th tangntial forcs on th tth within th zon of harmonic driving I-I and II-II is xplaind through th fact that: in sction I-I th chang of th dformation shap of th short flxibl toothd whl is limitd from two parts (on on part by th fixd rigid whl and on th othr part by th wav gnrator), and in sction II-II it is limitd only by th mobil rigid whl, and thr is a grat possibility of changing th shap of th flxibl whl for th incrasing of th loading charg (M t > 50m). From th study of dynamic analysis it has bn dtrmind that th priphrical tnsion (σ ϕ ) in sction I-I has a gratr valu and it has a dtrminant influnc on th stability of th short flxibl toothd whl. Th apparanc of th additional pas on th priphrical tnsion curv from sction II-II dnots a lss progrssiv shap of dformation of th flxibl whl in this sction, following th diffrnt charactr of th strss distribution on th tth situatd in harmonic driving. From th analysis of th rsults on can notic that th strsss (σ ϕ and σ x ) hav clos maximum absolut valus and th maximum tangnt strss (τ xϕ ) is approximatly four tims smallr. Th strsss σ ϕ and σ x ar annulld in mor nots that in th cas of th simpl harmonic transmission with th long flxibl whl and this is du to th xistnc of four contact and harmonic driving zons in th cas of th doubl harmonic transmission. From th analysis of th xprimntally obtaind rsults and of th on obtaind thortically [ 6 ] w can affirm that on a grat amount thy confirm, th dviations bing in th tolrabl limits (undr 5% at th tangntial forcs and undr 8% at tnsions). 5. REFERECES. Ianici, S., Anghl, Şt. Dsign of Mchanic Transmission, UPT, Timişoara, Ianici, S. Contributions to th synthsis of th transmission with dformabl lmnts, Doctorship dissrtation, Univrsity of Timişoara, 998, Ivanov, M.. Volnovi zubciati prdaci, Visşaia şola, Mosva, Ianici, S. Kinmatics of th doubl harmonic transmission usd in driving th industrial robots, Robotics & Managmnt, 997,, Ianici, S. Structural analysis of th harmonic toothd transmission usd in driving industrial robots, Robotics & Managmnt, 997,, Ianici, S. Study of th strss status within th short flxibl toothd whll of doubl harmonic driv using th mthod of th finit lmnt, Proc. Intrnational Symposium Aurl Vlaicu Univrsity of Arad, 997, Ianici, S. Contributions on dtrmining th forcs distributions in th toothd harmonic driv, Proc. Mchanisms and Mchanical Transmissions, Timişoara, 2000, 2, Ianici, S. Contributions au calcul ds dformations t tnsions dans l'lmnt flxibl d la transmission harmoniqu hrmtiqu, Proc. IFToMM Syrom Bucharst, 200, 3, Ianici, S. Doubl harmonic transmission usd in industrial robotic drivs, Proc. Intrnational Confrnc on Robotics, Timişoara, 2004,,