Mathematical model of drive system for metallurgical roller table unit with rotation of rollers transmitted by chain transmission

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1 Anrze RUSEK Anrze POPENDA Częsohowa Unversy of ehnoogy ahemaa moe of rve sysem meaurga roer ae un wh roaon of roers ransme y han ransmsson Asra. he aer susses he rve sysem wh he han ransmsson rven y nuon moor hrough he mu-sage gear-reuer. he mahemaa moe s muae ang no aoun he oa a oh enngs of eah roer of rve se. he euaons esrng he rve sysem are erve on he ass of varaona Hamon s rne an Lagrange s euaon of he seon ye. he fna sysem of euaons was ransme o he near fferena euaons. he aemenone sysem of euaons aows muang he omuer moe ga smuaons. Sreszzene: W aryue zarezenowano uła naęowy z łańuhowym naęem roe z zasosowanem weosonowego reuora zęaego naęzanego snem nuynym. oe maemayzny smułowano z uwzgęnenem oążeń na ou ońah aże ro zesawu naęowego. W oaru o warayną zasaę Hamona oraz równane Lagranea rugego rozau wyrowazono równana osuąe uła naęowy. Osaezną osać ułau równań srowazono o ułau nowyh równań różnzowyh umożwaąego smułowane moeu omuerowego o ozeń symuaynyh. Uła naęowy z łańuhowym naęem roe z zasosowanem weosonowego reuora zęaego naęzanego snem nuynym Keywors: han ransmsson roer-ae mahemaa moeng ehnooga oas. Słowa uzowe: naę łańuhowy samooowa na ransorowa moeowane maemayzne oążena ehnoogzne. Inrouon he rve sysems roer-ae ses wh roaon of he roers ransme y han ransmsson are exame of a onsruona souon of he rve sysems roer-ae ransorng nes. he aemenone rve sysems are ae n ransorng nes n whh ue o ehnooga roems he nsamen of gear-moors wh sef-ransorng onsruon on eah roer s no osse. I onerns aso hese ases where ransmsson of moon among roers hrough he shaf wh gears s no osse. hese ases aso aear n ehnooga nes n whh eemens have seons ransore erenuar o he roer-ae y snge shfers or shfer ses aar from moon of hese eemens aong he roer-ae [] []. he goa of he aer s o eveo a mahemaa moe of rve sysem roer-ae se wh han ransmsson among roers. he worng orues have o e eermne n he moe. he han ransmsson an have varous souons eenng on ehnooga reuremens. In he aer an exemary way of ransmsson of roers moon s resene. he rve onsss of hree gears fxe on roers shafs wo gears han enson an a rvng gear fxe on he enng of he musage gear-moor. he nema agram of he rve hree roers of ransorng roer-ae wh han ransmsson overe roers of he rve s ee n Fgure. In he resene nema agram of he aovemenone rve sysem he gears of he han enson are shown as eemens wh nera an. hese gears n he oher onsruona souons may e reae y egh gears of he han enson n suh a way ha he gears of he han enson shown as eemens wh nera an are no n he axs of he han gears fxe on he enngs of he ransor roers shown as eemens wh nera. he avanage of he resene rve sysem s he ossy of whhever mng of he roers san eenng on he onsruona reuremens. o o o o o o Fg.. he nema agram of he rve hree roers of he ransorng roer-ae wh han ransmsson he nuon moor wo-sage reuon gear he uh he han rve se roers he ransorng roer PRZEGLĄD ELEKROECHNICZNY Eera Revew ISSN - R. NR /

2 Fg.. he smfe omuae nema agram of he sysem ee n Fgure Deermnaon of he ouu uanes ang no onseraon a nema agram of he rve sysem usrae n Fgure hs sysem was ransme o he reue omuae nema agram gven n Fgure. he srue oa orues shown n Fgure reseve roers aear n he smfe omuae nema agram Fg. as foows: - he frs roer he oa orues o o a reue she wh momens of nera - he seon roer he oa orues o o a reue she wh momens of nera - he hr roer he oa orues o o a reue she wh momens of nera. he gear raos of he gear ransmssons shown n Fgure are eermne y eenenes : he momens of nera reue o he shes shown n Fg. reseve eemens of he sysem are exresse y eenenes : he rvng moor he reuon gear he han se of he rven roers he ransorng roers s In he nema agram Fg. he roang masses are ransme o he shes wh momens of nera =. he oson anges = an oeffens of easy = were assgne o he shes where s an nerna orue resung n orson of any eas eemen y ange. Assumng aon of he momens of nera of he haves he uh o he momens of nera of he shafs wh he gears as we as onserng sera onneon of he shafs wh oeffens of easy an he euvaen oeffen of easy - s efne as foows: For he hans of he rve se he oeffen of easy F = s efne as he rao of he e F an he han eongaon. he foowng exerna orues are n he rve sysem: - he eeromagne orue of rvng moor s - he oa orues a he -h ransorng roer ourrng n he aes of s seemen. In he rve sysem resene as he nema agram Fg. he foureen eemens wh he eermne momens of nera are nae. here are egrees of freeom eermne as neenen oornaes of anges of shafs roaon n he sysem. hese oornaes are exresse as he oumn marx : т [ ] In he rve sysem resene as he nema agram Fg. he ransme momens of nera gven y he agona marx orreson wh he onsere neenen oornaes gven y eeneny : D agon[ ] o o... PRZEGLĄD ELEKROECHNICZNY Eera Revew ISSN - R. NR /

3 PRZEGLĄD ELEKROECHNICZNY Eera Revew ISSN - R. NR / ahemaa moe he mahemaa moe of he rve sysem Fg. s ase on Lagrange s euaon of a seon ye ang no onseraon he Rayegh s ssaon funon [] []. In genera m he Lagrange s euaon of a seon ye s exresse y he foowng eeneny: Q Ф n... s oa ne energy of he sysem s oa oena energy Ф s oa ower of energy ssaon n he sysem Rayegh s ssaon funon s he -h neenen generaze oornae Q s he non-oena generaze e affeng on a ar of he sysem wh -h oornae n s a numer of neenen generaze oornaes of moon. he oa ne energy of he anayze rve Fg. s gven as foows: In he han rve se he roers he emaon oena energy of he srehe han eween rvng han gear an rven han gear s oee. he emaon eween hese gears may y exresse as foows: he vaue of he oena energy oee n he srehe han eween he rvng han gear an rven han gear s gven y he eeneny : a a Dfferenaon of he oena energy gven y over generaze oornae resus n he eeneny : where he un se funon: he orue whh ems srehes he han n he han rve se he roers ransme o he shaf of he rvng han gear may e exresse y eeneny : s he genera amng oeffen of eme han. Conserng he vaue of oena energy oee n he eme han he oa oena energy of he anayze rve sysem Fg. may e exresse y eeneny : he Rayegh s ssaon funon of he anayze rve sysem Fg. s gven y he eeneny : Ф he generaze amng oeffen. Susuon n he euaon an eua an resevey as we as fferenaon of he euaon over a neenen oornaes eermne y he oumn marx an over generaze veoes orresonng wh hese oornaes resu n he sysem of he fferena euaons esrng he rve sysem Fg.. s

4 PRZEGLĄD ELEKROECHNICZNY Eera Revew ISSN - R. NR / o o o o o o. he fna near euaons gven y he eenenes were oane as a resu of he suessve ransmaons of he euaons gven y. s R o

5 o o o o o. are engh nremens of he han n he rve sysem among he han gears wh he momens of nera. he sysem of euaons exresse y eeneny aows seng he rue oas on he reseve roers worng n he ransor ne. In many sues he oa of rve sysems s assume o e a ehnooga orue eermne as a rao of he rae orue of he moor n he rve sysem. he muae sysem of euaons aows onseraon of he rve sysem onssng of he assume han gears a eermne numer of roers varous onsruona souons an mensons of rvng han ae n he han rve se. arous souons wh regar o: he sane eween roers as we as he shae of he han roue may e assume hs rve se eeneny on onsruona reuremens. Aonay he sysem of euaons exresse y eenenes aows onseraon of he meanngfu ehnooga ase onernng rreguar aemen of he ransore eemens agans he suors of he ransorng roers. Irreguar aemen of he ransore eemen may e onsere n he sysem of euaons usng he symmery faor eermne y eeneny : w ax L a a s he wh of ransore eemen measure aong he roer axs a x s he sane eween he ege of ransore eemen an he roer enng L s he roer engh. he sysem of euaons aes no aoun he eenenes eang wh free amng of eas eemens of he anayze rve sysem an aows anayss of hs sysem fferen maeras ae n rue rve sysems. I exans sgnfany he ossy of raa aaon of he mahemaa moe n anayss of he rve sysems a he sage of her esgnng or seng u he rooya sysems. I was no onsere n revous mahemaa moes [] []. he sgnfan raa meanng has he roer seeon of he rvng han n he sysems wh rese osonng of he ransore eemen whh s he mos amage eemen of he sysem an s exose o he sgnfan emaon urng he oeraon of he rve sysem n freuen ynama saes. Conusons he anayss of he rve sysem resene as omuae nema agram Fg. resus n he sysem of euaons. hs sysem of euaons aows he we an oona anayss of he ynama saes of he rve sysem. he oane mahemaa moe aows anayzng ner aa he foowng ases: - he omuae anayss of he sysem n ynama saes onserng or no he ssaon energy - he omuae anayss of he rve sysem varous onfguraon of he han gears onane n he han rve se he roers - he omuae anayss of ynama saes of he rve sysem fferen sors an enghs of he han worng n he han rve se he roers. he anayss resene n he aer an eveoe mahemaa moe oes no onser an nfuene of he aemen an easy of he ransore eemen on he oeraon of he ransorng roers n anayze seon of oeraon of he ransorng nes. REFERENCES [] P u hała A. Dynams of he mahnes an eero-ynam sysems n Posh Warszawa: PWN [] Szars L. a r az K. Seee roems of ynams of eer rves n Posh Warszawa: PWN [] R u s e A. ahemaa moe of he gear-moor ase rve sysem he roer-ae ransorng ne onserng he henomenon of earanes n Posh Eeroehna Revew s - [] R u s e A. Fasza. he nfuene of he rue oa orue on he varay of eeromagne orue n roer-ae rve sysem n Posh Eeroehna Revew - [] F a sza. he ynama saes of he roer-ae rve sysem wh he seay esgne gear-moors onserng he rue oas n Posh oora hess Częsohowa: ehna Unversy of Częsohowa [] Poena A. Ruse A. ahemaa moe an seee ransen saes of he man rve oymerzaon reaor onserng oerang arameers of he mxer hamer n Posh Eeroehna Revew - Auhors: rof. r ha. Anrze Ruse E -ma: ruse@e.z.zes.; r Anrze Poena E-ma: oena@e.z.zes. Poehna Częsohowsa Insyu Eeroehn Przemysłowe A. Arm Kraowe - Częsohowa Poan PRZEGLĄD ELEKROECHNICZNY Eera Revew ISSN - R. NR /

DYNAMICS MODELING OF A TORQUE CONVERTER WITH TWO LINKAGE AND INERTIAL MASS

DYNAMICS MODELING OF A TORQUE CONVERTER WITH TWO LINKAGE AND INERTIAL MASS UNIVRSIY OF PISI FACULY OF MCHANICS AND CHNOLOGY SCINIFIC ULLIN AUOMOIV series ear XIX no. DYNAMICS MODLING OF A ORQU CONVRR WIH WO LINKAG AND INRIAL MASS Ion ION-GUA * Universit of Pitesti Roania Artice