machine design, Vol.3(2011) No.2, ISSN pp

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1 ache desg, Vol.(0) No., IN 8-59 pp Prelary ote PLANETARY GEAR TRANMIION OPTIMIZATION WITH EQUAL PRIORITY FUNCTION Jelea TEFANOVIĆ-MARINOVIĆ, * - Mloš MILOVANCEVIĆ, Uversty Nš, Faculty o Mechacal Egeerg, Nš, erba Receved (7.0.0); Revsed (.05.0); Accepted (7.06.0) Abstract: Desg o plaetary gear trasssos eas successul applcato o paraeter optato. Optato task s deed by the varables, objectve uctos ad codtos. The ollog crtera are selected atheatcal odel preseted ths paper: volue, ass, ececy ad producto costs. For optato varables are adopted: teeth ubers, uber o plaetary gears, gear odule ad gear dth. Codtos requred or the proper syste uctog, codtos the scope o geoetry ad stregth are expressed by the uctoal costras. Deterato o the Pareto optal solutos set s a rst step optal soluto dg. Next step s optal soluto choce ro Pareto solutos. I ths paper optal soluto dg s cosdered the case o equal prorty uctos. ce that, eghted coecets ethod ad dstace ethod are applcated or choosg optal soluto ro Pareto solutos. Key ords: ultcrtera optato, eghted coecets ethod, dstace ethod. INTRODUCTION Multcrtera optato probles are very coo ay scetc ad techcal probles. I ths type o optato t s ot alays possble to express the soluto qualty by oe crtero, but several uharoed crtera ust be dscussed. The uctos hch express these crtera caot all have optal values at the sae te. uch probles are called otrval ultple crtera (or ultple objectve, ultcrtera) optato probles. There are a lot o deret ethods or solvg such probles []. Optato o echacal systes supposes very coplex atheatcal odel hch have to descrbe operatg o real syste real crcustaces. Besde that, optato o techcal systes o-lear objectve uctos ad/or uctoal costrats are preseted. Gear tras as cocrete echacal systes are the subject o ultcrtera optato, too. Plaetary gear tras are portat kd o gear trasssos. It s possble all types o plaetary gear trasssos clude the sae paper, specally takg to cosderato that they have ther o geoetrcal codtos ad also they ca be oe stage ad ultstages. I ths paper, the optato o the basc type o plaetary gearg s cosdered.. BAIC THEORY OF MULTICRITERIA OPTIMIZATION The atheatcal odel o olear ultcrtera proble ca be orulated as ollos: ax { ( x), ( x),..., k ( x)} subject to x Fuctos ( ),, ( ) x x x x x K are objectve uctos ad k = (, K, ) s vector o decso varables. These varables ust satsy gve costrats hch are expressed as cluso x here s the set o easble solutos (or easble set). The otato ax eas the sultaeous axato o all the objectve uctos. I soe objectve ucto eeds to be ed a sple act that ato o the ucto ( x ) s equvalet to the axato o the ucto ( x) ca be used. Accordg to the structure o the set, there exst dscrete ad cotuous ultcrtera optato probles, depedg o hether the set s te or cotuous []. Every pot x s apped to the pot ( ( x), ( x),..., k ( x )) k - desoal objectve space. Thereore t ca be troduced as the objectve set: { ( x), ( x),..., ( x)) x } F = k ( It s ote useul to ko the best possble values or each objectve ucto. These values or a so-called deal * * * pot = (, K, k ) the objectve space. Its copoets are coputed as: * = ax ( x), or all =, K, k. x As t ca be see ro the deto, ultcrtera optato probles are atheatcally ll-deed. Ths eas that they have a set o atheatcally equally good optal solutos the objectve space. The ost portat crtero or selectg these equally good solutos s Pareto optalty cocept: *Correspodece Author s Address: Uversty Nš, Faculty o Mechacal Egeerg, Aleksadra Medvedeva, 8000 Nš, erba, jeleas@asak..ac.rs

2 Jelea teaovć-marovć, Mloš Mlovacevć: Plaetary Gear Trasssos Optato th Equal Prorty Fuctos; Mache Desg, Vol.(0) No., IN 8-59; pp oluto x s Pareto optal there s o soluto y such that holds ( x) ( y) or all =, K, ad or at least oe dex holds strct equalty,.e. ( x) < ( y). Thus, soe addtoal orato s eeded order to be able to select oe o the as a al soluto. Ths al decso s usually ade ether by decso aker (hua expert) or by the correspodg scalared proble. I the latter case, oe or ore sgle crtero optato scalared probles have to be costructed ad solved. The ollog ethods based o the costructo o scalared proble are suggested or usage ths atheatcal odel: eghted coecets ethod ad dstace ethod [5]. These ethods are sutable or applcato the case o equal prorty uctos... Weghted coecets ethod I the eghted coecets ethod the ollog scalared proble s costructed: M 0 0 ax ( x) = ( x) + K+ ( x) s.t. x Here, eghted coecets (or eghts) are postve real ubers =,,,. ad 0 0 ( ) ( ) x = ( x) are 0 oraled objectve uctos, here are oralg coecets. I ths approach, the * * * * * copoets o deal pot = (,,, ) are used 0 * as oralg coecets,.e. = or =,,,. Thereore, absolute values o all objectve uctos are betee 0 ad, hch sples the choce o the eghted coecets. All solutos obtaed by ths ethod are Pareto optal [5]. The put paraeter or ths ethod s oly the sequece o eghts... Dstace ethod The a dea dstace ethod s the ato o dstace betee oe gve (easble) reerece pot ad the objectve set F. The ollog scalared proble ca be orulated ths ay: Usually a deal pot * s used as a reerece pot.. OPTIMIZATION TAK DETERMINATION The basc type o plaetary gearg (Fg.),.e a desg hch has a cetral su heel (exteral gearg - ), oble cetral toothed r (teral gearg - ), ad plaetary gears (satelltes - ), s the subject o ths paper, lted to geared pars. atelltes are the sae te esh th exteral ad teral gear. Ths type o plaetary gear trassso s ote used as a sgle stage trassso, ad thus as a buldg block or hgher copoud plaetary gear tras. A optato task s deed by the varables, objectve uctos ad codtos requred or the proper uctog o a syste expressed by the uctoal costrats. I ths paper, the ollog varables are cosdered: teeth uber o cetral su heel, teeth uber o plaetary gears (satelltes), teeth uber o toothed r, uber o plaetary gears, gear odule ad gear dth b. The optato varables are o xed type: ubers o gear teeth (,, ) are tegers, postve ad egatve, uber o plaetary gears ( ) s a dscrete value, odule ( ) s a dscrete stadard value (acc. to DIN 780), hle gear dth (b ) s a cotual varable. Nubers o gear teeth ad uber o plaetary gears are o-desoal values, hle odule ad gear dth are gve lleters. I the orulated proble, sx decso varables exst, correspodg to the basc desg paraeters: x = ( x, x, x, x, x, x ) = (,,,,, b). 5 6 d( ( x), ) s.t. x Here dxy (, ) ca be ay etrc ucto. The ost used etrcs are: rectagular, Eucldea ad Tchebyche etrcs. I ths atheatcal odel Eucldea dstace s used, thus dstace obtas the shape: dxy (, ) = ( x y) = Here are gve postve real ubers, eght coecets. Uder the codto > 0 or all =, K, t ca be prove that a soluto obtaed by ths ethod ad usg Eucldea etrc s Pareto optal [8]. Fg.. Plaetary gear trassso th oble teral gear I ths odel, the ollog characterstcs are chose or objectve uctos o a plaetary gear tra: volue, ass, ececy ad producto cost o gear pars. 00

3 Jelea teaovć-marovć, Mloš Mlovacevć: Plaetary Gear Trasssos Optato th Equal Prorty Fuctos; Mache Desg, Vol.(0) No., IN 8-59; pp ce ass, volue ad producto costs should be ed, ad ececy should be axed, the ollog s deoted ths odel: ( x) = V( x), ( x) = ( x), ( x) = η ( x), ( x) = T( x) p F T = T + T + T Producto tes are detered accordg to the techologes o Fette [7], Lorec [8] ad Höler [9]. I ths paper, the volue o gear pars s used. The approxato o gear volue by cylder volue th daeter equal to ptch daeter ad heght equal to gear dth, ad havg d that satelltes are sde the toothed r, ake possble or the gear volue to be expressed by: V π cosα t = b cos β cosα t here α t s the pressure agle at ptch cycle, α t s the orkg trasverse pressure agle or the par - ad β s the helx agle at ptch daeter. Mass s detered as su o all gear asses trassso. ce the ass o a partcular gear s detered as gear volue ultpled by the desty o gear ateral, = ρ V, ths crtero receves the or = ρ V + ρ V + ρ V The al expresso o ths ucto s: cos αt = 0.5 π b ρ k + cos β cos αt cos α cos α k + k ] t t cos αt cos αt To detere the gear ass, the actor o devato o real gear shape ro cylder (k) has to be take to accout also. For purposes o optato,.e. the coparso o gears th deret paraeters, ths actor does ot have a great sgcace, sce t s gve advace due to the shape o gear heel hub ad t s a costat the process o optato. The calculato o gear trasssos ececy s geerally coed to losses depedg o rcto o tooth sdes,.e. o calculato o esh poer losses. Bearg those cosderatos d, e cosder the ollog expresso or ececy [6,]: η = + + Ecooc deads ust also be take to cosderato he dealg th techo-ecoocal optato. Frst, these deads are related to producto costs. These costs cosst o costs or producto ateral ad costs o the producto process tsel. The te eeded or the producto o gears s take as a easure o producto costs ad as a ecoocal actor. Ths ucto s the detered as a su o tes eeded or the producto o cetral su heel (T ), satelltes (T ) ad toothed r (T ),.e.: Fg.. horteed algorth or optato procedure Plaetary gears represet a specc group o gear trasssos. Thereore, there are uerous exceptos that eed to be take to accout or these trasssos to ucto correctly copared th classcal gear trasssos. The exceptos cosdered ths artcle are related to outg codtos, geoetrcal codtos ad stregth codtos. The outg codtos coprse the codto o coaxalty, the codto o adjacecy ad the codto o cojucto [,]. 0

4 Jelea teaovć-marovć, Mloš Mlovacevć: Plaetary Gear Trasssos Optato th Equal Prorty Fuctos; Mache Desg, Vol.(0) No., IN 8-59; pp Geoetrcal codtos relate to udercuttg ad prole tererece, rato o pressure agle to orkg trasverse pressure agle, tooth thckess ad space dth, trasverse cotact rato value, sldg speeds, rato o gear acedth to reerece daeter o the drvg gear, etc. These codtos are esured accordace th the actual stadards (IO TC 60 lst o stadards 09095). As stregth codtos, saety actors or bedg stregth ad surace durablty o each gear are provded [0]... Optato procedure The shorteed algorth or the coplete optato procedure s sho Fg.. For the gve put data (put uber o revoluto, put torque, servce le hours, applcato actor, accuracy grade [Q-DIN96], al saety actor - lak, al saety actor - root, gears aterals, alloed devato o gear rato, rage o varato), all 6-tuples o desg paraeters (,,,,, b ) satsyg the uctoal costrats are geerated ad the values o the goal uctos or every 6-tuple are coputed. Next, t s ecessary to choose oly oe optal soluto aog all the geerated solutos. Methods or solvg the ultcrtera optato probles are preseted the prees secto. The coplete optato procedure s pleeted the PlaGears sotare. The sotare s rtte the prograg laguage Delph 7.0 ad cotas the coplete 9 step procedure. The sotare also cotas varous other eatures. It s possble to track gear pars paraeters the phase o desg by applyg ths progra. It gves useul drectves or experetal research, hch drastcally reduce ther cost ad coplexty. It s applcable the gear pars luetal paraeters research, ad the process o plaetary trasssos aly developet.. NUMERICAL EXAMPLE Nuercal exaples obtaed usg ths sotare are preseted ths secto. I ths paper or exaple o optato ethod applcato, the ollog put data s cosdered: = 6.5, = 000 -, T = N, T = 000 h, K =., IT 6 or all gears, =., A F =., ateral /ateral /ateral =8CrN8/8CrN8/ CrNMo6, = %, = 5 6 These put data s accordace th a realed plaetary trassso. Accordg to the cosderato secto, the uber o Pareto solutos s detered rst. et o easble solutos cossts o eleets. The uber o Pareto solutos s 6. Ideal values o uctos are detered ad sho Table. H Table. Ideal pot coordates d d kg d d Eucldea dstace ethod s appled th deal pot as a reerece pot. Ths ethod gves the soluto, ro Pareto set, sho Table. The set o objectve uctos values or soluto obtaed by Eucldea dstace ethod s sho Table. Table. oluto obtaed by Eucldea dstace ethod Varable values b Table. Objectve ucto or soluto sho table kg The prorty o uctos s eeded to detere or the ext ethod atcpated here. Weghted coecets values have sgcat luece o optal soluto selecto. ce, ths optato task, all objectve uctos have the equal prporty, eghted coecets have the sae values,.e., = = = = 05.. The obtaed soluto s gve Table, th the set o objectve uctos Table 5. Table. oluto obtaed by eghted coecets ethod Varable values b Table 5. Objectve ucto or soluto sho table kg The derece betee soluto sho table ad soluto sho table s eglectable. The dstcto betee these solutos s oly oe varable value. Ths derece plcate derece objectve uctos. The derece or the ourth ucto s cosderable-6.6%, hle dereces betee rst, secod ad thrd ucto copoets are %,.%, ad 0.0%. oluto obtaed by Eucldea dstace ethod s charactered by a larger uber o teeth o plaetary gear tha other atcpated ethod. Ths s due to the ature o the thrd ucto. The crease ececy s olloed by the crease gear trassso desos (daeter). 0

5 Jelea teaovć-marovć, Mloš Mlovacevć: Plaetary Gear Trasssos Optato th Equal Prorty Fuctos; Mache Desg, Vol.(0) No., IN 8-59; pp olutos obtaed by these ethods or ultcrtera optato are coordated. The ethods, although they start ro deret prepostos ad have deret atheatcal bases, lead to haroed results, hch provdes the a physcal eag. Based o the above exaples a very strog correlato betee ethods appled or obtag the optal soluto accordg to these crtera ca be observed The soluto obtaed by eghted coecets ethod has three ucto copoets closer to deal values o ucto tha the soluto obtaed by Eucldea dstace ethod. Ths s the reaso or adoptg ths soluto or optal soluto o the optato task. I ths ay optal soluto regardg desg paraeters s obtaed. Beore adoptg soluto or ext step product developet, techologcal deads should be cosdered. Qualty o solutos obtaed by ths progra s cored by coparso th selected realed plaetary trassso. For ths coparso the trassso auactured by MIN-FITIP-Nš (erba) s selected. Realed trassso has desg paraeters sho table 6 th set o objectve uctos values sho Table 7 Table 6. oluto o realed plaetary trassso Varable values b Table 7. Objectve ucto or soluto sho table 6 kg soluto obtaed by progra realed trassso Correlato betee optal soluto obtaed by developet progra ad soluto o realed gear trassso chose or coparatve trassso s cosderable. Coparg the selected realed plaetary trassso ad soluto obtaed by ths progra, t ca be cocluded that by optato progra ore copact costructo tha realed costructo s obtaed. Coparatve reves o optato varables ad objectve uctos are sho Fg. ad Fg.. Thus, results obtaed ths exaple dcate a good choce o establshed atheatcal odel ad appled ethods. 5. CONCLUION I ths paper, a orgal odel or ultcrtera optato o plaetary gear trasssos s preseted. The basc type o plaetary gear s the subject o the paper. Foudg ultcrtera odel s a sgcat step toards the plaetary gear trastters realty. The atheatcal odel cossts o objectve uctos, varables ad uctoal costrats. Related th the rst part, uctos descrbg plaetary gear ro the techo-ecoocal aspect are detered. Tha, desg paraeters are observed ad adopted as optato varables. Codtos requred or the proper syste uctog, codtos the scope o geoetry ad stregth are expressed by the uctoal costras. Besdes the deterato o the set o Pareto optal solutos, the preseted orgal approach cludes ethods hch select a optal soluto ro the Pareto solutos set: eghted coecets ethod ad Eucldea dstace ethod. Accordg etoed exaples t ca be cocluded that these ethods although start ro deret prepostos ad have deret atheatcal bass have very strog correlato. Results obtaed ths ay are accordace th the realed gear trassso ad lterature o techcal syste optato. It s sho that the establshed optato odel gves good results ad ca be used or ths type o plaetary gear trassso. Procedure appled here ould be the sae or other types o plaetary gear trasssos, oly atheatcal expressos ould be deret. b Fg.. Coparatve reve o optato varables soluto obtaed by progra realed trassso Fg.. Coparatve reve o objectve uctos REFERENCE [] ARNAUDOV K., KARAIVANOV, D.: Rau ud Massesparede Zahradgetrebe, I thrth Iteratoal yposu about Desg Mechacal Egeerg -KOD 00, 00, Nov ad, erba, Proceedgs, pp [] DEL CATILLO, J.M., The Aalytcal Expresso o the Ececy o Plaetary Gear Tras, Mechas ad Mache Theory 7 (00) pp 97-. [] MIETTINEN, K.: Nolear Multobjectve Optato. Kluer Acadec Publshers, Bosto (999) 0

6 Jelea teaovć-marovć, Mloš Mlovacevć: Plaetary Gear Trasssos Optato th Equal Prorty Fuctos; Mache Desg, Vol.(0) No., IN 8-59; pp [] NIEMANN G., WINTER H.: Mascheeleete, Bad II, prger-verlag Berl, 989. [5] TEFANOVIĆ-MARINOVIĆ J.: Všekrterjuska optacja upčasth parova plaetarh preoska, doktorska dsertacja, Mašsk akultet Uverteta u Nšu, Nš, 008 [6] VOLMER, J.: Getrebetechk, Ulaurädergetrebe, Verlag Techk, Berl, 990. [7] FETTE-Techologe-Ihr Prot, De chttbedguge be Wälräse, WILHELM FETTE, GMBH [8] Veraherkeuge, E Hadbuch ür Kostrukto ud Betreb. Aulage, LORENZ GbH&Co Ettlge [9] BH HÖFLER operatg structos, /./ /./66, /./ /./67 [0] IO 66. Calculato o load capacty o spur ad helcal gears: IO 66-, IO 66-, IO 66-, 996 0