A Refined Tenique for te Automted Determintion of Frition Losses in te Tooting of ultitreded Trnsmissions wit Differentil enisms nd Plnetry Gers Volontsevi D., Ie. Veretennikov, I Kostinyk, S Psenyi To ite tis version: Volontsevi D., Ie. Veretennikov, I Kostinyk, S Psenyi. A Refined Tenique for te Automted Determintion of Frition Losses in te Tooting of ultitreded Trnsmissions wit Differentil enisms nd Plnetry Gers. enis, terils Siene Engineering Journl, gnolite, 207, 2, <ttps://mmse.xyz/en/-refined-tenique-for-teutomted-determintion-of-frition-losses-in-te-tooting-of-multitreded-trnsmissions-witdifferentil-menisms-nd-plnetry-gers/>. <0.242/mmse.3.52.575>. <l-0578908> HAL Id: l-0578908 ttps://l.rives-ouvertes.fr/l-0578908 Sumitted on 30 Aug 207 HAL is multi-disiplinry open ess rive for te deposit nd dissemintion of sientifi reser douments, weter tey re pulised or not. Te douments my ome from teing nd reser institutions in Frne or rod, or from puli or privte reser enters. L rive ouverte pluridisiplinire HAL, est destinée u dépôt et à l diffusion de douments sientifiques de niveu reere, puliés ou non, émnnt des étlissements d enseignement et de reere frnçis ou étrngers, des lortoires pulis ou privés. Distriuted under Cretive Commons Attriution - NonCommeril 4.0 Interntionl Liense
enis, terils Siene & Engineering, Septemer 207 ISSN 242-5954 A Refined Tenique for te Automted Determintion of Frition Losses in te Tooting of ultitreded Trnsmissions wit Differentil enisms nd Plnetry Gers D. Volontsevi,, Ie. Veretennikov, I. Kostinyk, S. Psenyi Trnsport ine Building Fulty, Ntionl Tenil University "Krkov Polyteni Institute", Krkov, Ukrine vdo_kpi@ukr.net DOI 0.242/mmse.3.52.575 provided y Seo4U.link Keywords: frition losses in te tooting, multitreded trnsmissions, differentil menisms, plnetry gers, mtrix metod. ABSTRACT. ultitreded trnsmissions, wi using differentil menisms for seprting nd summing up power flows, ve een widely used in te trnsmissions of trnsport veiles nd speil drives. Te qulity of su trnsmissions nd te mgnitude of fritionl losses in te tooting ifintly depend on te dopted kinemti seme nd te rtio of powers pssing troug te orresponding links of te differentil menisms. Wit te existing utomted nlysis nd syntesis of su trnsmissions, mtrix ppro s reently een widely used. Te proposed work provides refined tenique for te utomted determintion of frition losses in te tooting of multitreded trnsmissions wi ontining differentil menisms nd plnetry gers. It mkes it possile to more urtely determine frition losses in te tooting nd t te sme time preserves te liner struture of te system of equtions, wi permit to ontinue using te mtrix ppro to nlyze nd syntesize su trnsmissions. Exmples re given for te formtion of equtions systems desriing te fore intertion etween te elements of te seme for ll vrints of te links ommuttion of plnetry rows operting s tree-link differentil menisms nd plnetry gers wit one stopped link. Te work is sed on te nlysis of existing metods for determining frition losses in te tooting nd personl prtil experiene in te de nd study of multitreded stepped nd stepless trnsmissions. Te reeived results llow to pply te offered tenique in te modern progrm omplexes foused on te utomted nlysis nd syntesis of multitred trnsmissions wit use of te mtrix ppro. Tis mkes it possile to ifintly redue te time to develop new multitreded trnsmissions nd inrese teir tenil rteristis. Introdution. In modern trnsmissions of veiles nd oter drives of mines, plnetry gers re widely used s differentil menisms. It is, first of ll, te summtion or seprtion of power flows in dul-treded nd multitreded trnsmissions, s well s teir use in plnetry geroxes. A feture of differentil trnsmissions is te essentil dependene of te mgnitude of te losses in te menism on te kinemti trnsmission seme nd te rtio of te powers pssing troug te orresponding links of te differentil. Terefore, te study nd refinement of te tenique for determining losses on differentil menisms in multitreded trnsmissions is n tul tsk. Anlysis of reent ievements nd pulitions. In te modern tenil literture, s differentil trnsmission, it is ustomry to onsider n utomotive inter-weel differentil: differentil is ger trin wit tree sfts tt s te property tt te ngulr veloity of one sft is te verge of te ngulr veloities of te oters, or fixed multiple of tt verge [Wikipedi] (ess mode: ttps://en.wikipedi.org/wiki/differentil_(menil_devie)) 207 Te Autors. Pulised y gnolite GmH. Tis is n open ess rtile under te CC BY-NC-ND liense ttp://retiveommons.org/lienses/y-n-nd/4.0/ SE Journl. Open Aess www.mmse.xyz
enis, terils Siene & Engineering, Septemer 207 ISSN 242-5954 differentil ger is devie tt is fitted to te xle of veile tt llows te weels to turn t different rtes wen going round orner. However, we will tlk out te use of lssil 2k- plnetry gers s differentil menisms for te summtion nd seprtion of power flows in multitred trnsmissions. Te issue of determining te effiieny of plnetry gers nd differentil menisms s lwys een of interest to engineers nd sientists wo rete veile trnsmissions nd tenologil drives. Te foundtions of te teory of power losses in tese gers were lid in te writings of su sientists s R.H. illn, J.H. Glover, P.W. Jensen [ 4]. In te USSR tis prolem were solving wit y A.D. Vsets, K.I. Zlonsky, A.N. Ivnov, Yu.N. Kirdysev, V.I. Krsnenkov,.A. Kreines,.K. Kristi, V.N. Kudryvtsev nd oters [5], [6], [7]. Wit te introdution of te mtrix metod in te nlysis of plnetry menisms nd trnsmissions in generl [8], [9], new er in te utomtion of tis proess egn. Tere ws n opportunity to ompletely formlize nd utomte te proess of determining te effiieny of plnetry nd differentil trnsmissions. However, te proposed vrints re eiter not very onvenient for utomted mtrix nlysis [0], [], [2], [3], or re onvenient, ut tey give lrge error in speil zones [4]. In modern pulitions on plnetry gers, utors, s rule, solve speifi pplied prolems: onsider te prtiulr onstrutions of onventionl nd stepless trnsmissions onsider te influene of lurints nd toot geometry on te effiieny of plnetry gers, nd study te issues of inresing teir lod pity [5 2]. Te im nd prolem sttement. Terefore, it ws deided to return to te prolem of reting universl lgoritm tt llows to orretly tke into ount te menil losses in ritrry kinemti semes of trnsmissions ontining plnetry trnsmissions nd differentil menisms. To ensure te possiility of using te mtrix ppro in nlyzing nd syntesizing plnetry gers, tis lgoritm must e ompleted y ompiling si system of equtions, wi ontin ll te neessry informtion for determining te losses in e of te rnes of multitreded gers wit differentil menisms. terils reser. Te lssil mtrix ppro to te nlysis of plnetry trnsmissions ssumes te deomposition of te kinemti seme into elementry struturl elements (plnetry rows, stop rkes, loking lutes, et.) nd reording of set of rteristi equtions for e of tem. To te resulting system of liner equtions, te onstrint equtions tt desrie te onnetion seme of te elements, nd te equtions of inlusion of te seleted ger re dded. So, for exmple, for te lssil 2k- plnetry row (differentil menism), te kinemtis desription uses te Willys eqution: k 0 k, were ngulr veloity of te sun ger ngulr veloity of te epiyli ger ngulr veloity of te rrier (lever or rm) internl (si) ger rtio of plnetry row, orresponding to te ger rtio from te sun ger to te epiyli ger wen te rrier is stopped, tking into ount te diretion of rottion. k To desrie te fore intertion witout tking losses into ount, use equtions SE Journl. Open Aess www.mmse.xyz
enis, terils Siene & Engineering, Septemer 207 ISSN 242-5954 k 0, were te torque input or output to te sun ger witout tking losses into ount te torque input or output to te epiyli ger witout tking losses into ount te torque input or output to te rrier (lever or rm) witout tking losses into ount. Wen determining te effiieny of plnetry trnsmission omposed of plnetry nd differentil menisms, it is neessry to rry out n nlysis of te kinemtis nd te fore nlysis of te iruit witout tking losses into ount. As result, t e ger on ll links, we will ve te vlues of ngulr veloities nd torques, wi will llow us to nlyze te diretions of power flows nd determine te reltive mgnitudes of te powers trnsferred in reltive nd figurtive motion. Just s in te ook [6], we divide nd onsider seprtely te modes of opertion of te plnetry row in te role of plnetry or differentil trnsmission. Plnetry trnsmission is otined y stopping one of te entrl links of te differentil menism (plnetry row). Consider plnetry row s seprte element of te generl struturl seme of plnetry or omined trnsmission. In tis se, it is possile to distinguis tree omintions of elements ommuttion for plnetry row operting in te plnetry trnsmission mode (fig. ), end six for plnetry row operting in differentil trnsmission mode (fig. 2). in (out) out (in) in (out) out (in) out (in) in (out) ) ) ) Fig.. Options of elements ommuttion for plnetry trnsmission: ) te epiyli ger is stopped ) te sun ger is stopped с) te rrier (lever or rm) is stopped in in 2 out in out in 2 out in 2 in ) ) ) out out 2 in out in out 2 in out 2 out d) e) f) Fig. 2. Options of elements ommuttion for differentil trnsmission: wit one output in te form: ) te rrier (lever or rm) ) epiyli ger с) sun ger d)wit one input in te form: d) te rrier (lever or rm) e) epiyli ger f) sun ger. SE Journl. Open Aess www.mmse.xyz
enis, terils Siene & Engineering, Septemer 207 ISSN 242-5954 For exmple, for te seme in Fig. () wit te internl ger rtio k = 3 fter te kinemti nd preliminry fore nlysis in reltive units, we will ve te results given in Tle.. Tle. Te power rteristis of plnetry trnsmission in Fig. witout losses. Input Element Element Element Element Output (stellite) 0 0,25 0,75 0,25 М - -3 4 0-4 N - 0 0 - In Tle. N is te power on te orresponding element. It sould e noted tt te power entering te element is negtive, nd te output from te element is positive. Aording to te proedure desried in [4], te torqueses on te elements, tking into ount trnsmission losses, re determined s follows: k n 0,5 n 0,5 d d 0,5 n 0,5 n d d 0, () were,, torque on te links in te ligt of losses reltive ngulr veloity on te erings of stellites oeffiient of frition losses in te gering sun ger stellite wit te rrier (lever or rm) stopped (for externl gering 0, 02 ) oeffiient of frition losses in te gering epiyli ger stellite wit te rrier (lever or rm) stopped (for internl gering 0, 0) d oeffiient of frition losses in te gering stellite stellite 2 wit te rrier (lever or rm) stopped if te plnetry row ontins severl suessively mesed stellites (for externl gering 0, 02, in te sene of suessively mesed stellites 0 ) te frtion of power trnsmitted in reltive rottion troug te gering, wi is lulted from formul. 0,75 In tis exmple 0, 75, internl ger rtio k = 3. In ordne wit 0,75 0,25 Tle., ssuming tt rottion wit ngulr veloity (+) nd torque ( ) is pplied to te sun ger, system () n e written in te form: d SE Journl. Open Aess www.mmse.xyz
enis, terils Siene & Engineering, Septemer 207 ISSN 242-5954 0,750,02 0 0,750,0 0 3 0,750,02 0 0,750,0 0 0,985,0075,0075 3 0,985 0, 0, werefrom 2, 933 nd 0,985,0075 2,933 3, 94. i / 3,94/( ) Aordingly, te effiieny of te seme will e 0, 985. Here i i / / 0,25 power rtio, i kinemti rtio. It sould e noted tt te tenique [4] presries, wit power on te link equl to zero due to te zero speed of rottion, to tke te power s positive. Te first drwk of tis metod is tt in te presene of one stopped entrl ger nd te pssge of power y single flow from te oter entrl ger to te rrier (or vie vers), te overll effiieny does not tke into ount te sre of losses t te stopped element. So, in ft, te effiieny otined from te pssge of 75% of power troug te gering of te sun ger - te stellite, wi is equl to 0,985, s eome te overll trnsmission effiieny. Tis ft is onfirmed y lultion using metod [6] repetedly tested experimentlly, wi gives smller vlue of te effiieny: 3 0,02 0,0 k 0,9775. k 4 Te mgnitude of te error tt we get wen using te tenique [4] for te nlysis of losses in plnetry gers is sown in Fig. 3 nd Fig. 4.,2 Error in determining te menil effiieny, % 0,8 0,6 0,4 0,2 0-5 -4,5-4 -3,5-3 -2,5-2 -,5 Bsi speed rtio Fig. Fig. Fig. Fig. 3. Dependene of te mgnitude of te error in determining te menil effiieny of te plnetry row y te tenique [4]. SE Journl. Open Aess www.mmse.xyz
enis, terils Siene & Engineering, Septemer 207 ISSN 242-5954 Error in determining te menil energy losses, % 50 45 40 35 30 25 20 5 0 5 0-5 -4,5-4 -3,5-3 -2,5-2 -,5 Bsi speed rtio Fig. Fig. Fig. Fig. 4. Dependene of te mgnitude of te error in determining te menil losses of te plnetry row y te tenique [4]. A similr piture rises wen te plnetry row works s differentil menism, wen one of te tree entrl elements is te mover, nd te oter two re driven. Te power on te differentil menism is divided into two flows, wit one prt of te power pssing ot gering, nd te seond prt only one. Te tenique given in [4] works ompletely dequtely only for te se of summtion of two power fluxes, if ot of tem enter te differentil menism troug te sun ger nd epiyli ger, nd not te rrier (lever or rm). To eliminte te desried inuries in te metodology [4], wile retining te possiility of using it in te mtrix ppro to te nlysis nd syntesis of plnetry gers, te following is proposed. For te vrint in Fig. (), te system of equtions desriing te lod lne n e written in te form: k 0, (2) were te totl effiieny t te pssge of power y single flow in ordne wit [6]: k. k For te vrint in Fig. () te system of equtions desriing te lod lne n e written in te form: SE Journl. Open Aess www.mmse.xyz
enis, terils Siene & Engineering, Septemer 207 ISSN 242-5954 SE Journl. Open Aess www.mmse.xyz 0, k (3) were te totl effiieny t te pssge of power y single flow in ordne wit [6]: k. For te vrint in Fig. (), te system of equtions desriing te lod lne n e written in te form: 0, k (4) were effiieny in ger sun ger stellite effiieny in ger stellite epiyli ger. For semes wit differentil menism te system of equtions tt desriing te lod lne will ontin 4 equtions. Te first two equtions of te system represent te lne of torque t te differentil menism. And two dditionl equtions llow you to distriute losses long te rnes of te input (output), oserving te overll lne of losses in ordne wit [6]. It is ssumed tt te loss on te externl tooting of te sun ger stellite is twie s lrge s on te internl tooting of te stellite epiyli ger. So for te vrint in Fig. 2(): 2, k were te overll effiieny in ordne wit [6] k, nd te loss vlues t e input (output) nd re determined from te lst two equtions on te sis of preliminry lultion witout tking losses into ount efore solving te generl system of equtions. Wit te reverse power flow in te sme seme (Fig. 2(d)), te system will look like:
enis, terils Siene & Engineering, Septemer 207 ISSN 242-5954 SE Journl. Open Aess www.mmse.xyz 2, k wi llows reording it for te semes in Fig. 2() nd Fig. 2(d) in te generlized form: 2. k (5) Here it sould e kept in mind tt if te seme relly orresponds to Fig. 2() or Fig. 2(d), ten for simpliity it is possile to equte ) ( ) ( ) (. (6) Similrly for te vrint in Fig. 2(): 2, k were te overll effiieny in ordne wit [6], nd te loss vlues t e input (output) nd re determined from te lst two equtions on te sis of preliminry lultion witout tking losses into ount efore solving te generl system of equtions.
enis, terils Siene & Engineering, Septemer 207 ISSN 242-5954 SE Journl. Open Aess www.mmse.xyz Wit te reverse power flow in te sme seme (Fig. 2(e)), te system will look like 2, k wi llows reording it for te semes in Fig. 2() nd Fig. 2(e) in te generlized form: 2. k (7) Similrly for te vrint in Fig. 2() 2, k were te overll effiieny in ordne wit [6], nd te loss vlues t e input (output) nd re determined from te lst two equtions on te sis of preliminry lultion witout tking losses into ount efore solving te generl system of equtions. Wit te reverse power flow in te sme seme (Fig. 2(f)), te system will look like
enis, terils Siene & Engineering, Septemer 207 ISSN 242-5954 k 2, wi llows reording it for te semes in Fig. 2() nd Fig. 2(f) in te generlized form: k 2. (8) Summry.. Kinemtis nlysis, lod distriutions nd loss determintion in modern multitreded trnsmissions ontining differentil menisms nd plnetry rows re performed sequentilly on e step of ger or operting mode. At te sme time, te sme plnetry rows on some trnsmissions n ply te role of differentil menisms, nd on oters simple plnetry gers wit one stopped link. Te lssil lgoritm for nlysis of kinemtis nd distriution of lods witout loss is universl nd does not require te use of different formuls for one or te oter se. Wen performing n urte loss nlysis in su semes, te lgoritms for nlyzing simple plnetry gers wit one stopped link nd differentil menisms differ ifintly from e oter. 2. Wen nlyzing losses in multitreded trnsmissions, first, using te mtrix nlysis, lulte te kinemtis nd lod distriution witout tking into ount te losses to determine te diretion of te power flows nd te preliminry vlues of te torque on te links long ll rnes of te iruit. 3. Furter, for ll plnetry rows wit one stopped link, equtions (2), (3) or (4) re written into te system of equtions, depending on te type of te stopped element. 4. For ll differentil menisms, te lst two equtions from systems (5), (7) or (8), depending on te diretion of te power flows, re solved efore omposing generl liner system of equtions. After tis, te first two equtions from te sme systems re written in te generl system. Su sequentil solution of te equtions llows solving nonliner equtions for given differentil menisms t te stge of preprtion of lrge generl liner system of equtions tt desries te entire seme. Tis mkes it possile to solve lrge liner system y mtrix metod nd to sve te omputtion time. Tis is espeilly importnt in rrying out te struturl-prmetri syntesis of multitreded trnsmissions, wi is performed y multiple nlysis of te generted iruits nd sets of teir prmeters. Referenes [] illn, R.H. (949). Epiyli ger effiienies. Te Engineer, 23, 727 728. SE Journl. Open Aess www.mmse.xyz
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