THE GENERALIZED LAGRANGE'S EQUATIONS OF THE SECOND KIND AND THE FIELD METHOD FOR THEIR INTEGRATION UDC Ivana Kovačić

Transcription

1 FCT UNIVERSITTIS Seres: Mechancs uomac Conrol and Robocs Vol. N o 5 00 pp. - 8 THE GENERLIZED LGRNGE'S EQUTIONS OF THE SECOND KIND ND THE FIELD METHOD FOR THEIR INTEGRTION UDC 5. Ivana Kovačć Faculy of Techncal Scences Trg Doseja Obradovca 6 Deparmen of Mechancs Nov Sad Serba and Monenegro bsrac. Ths paper presens he complee process of obanng moon of a mechancal sysem wh varable mass subjec o non-holonomc consrans whch are non-lnear or of he frs or hgher order. Frsly he generalzed Lagrange's equaons of he second nd are eended o a non-holonomc sysem wh varable mass by nroducng generalzed reacve forces. Then he feld mehod s appled o hese equaons of moon o fnd her soluon. Fnally an llusrave eample showng he use of hs algorhm s gven.. INTRODUCTION The feld mehod prmary developed as a mehod for negrang he equaons of moon of holonomc non-conservave sysems [ ] perans o he Hamlon-Jacob heory lmed n applcaons o conservave holonomc sysems. Laely was shown ha he feld mehod s also suable for negrang he equaons of moon of nonholonomc sysems [ ] whle he Hamlon-Jacob heory has very src resrcons for her sudy [5 6]. The applcaon of he feld mehod o non-holonomc problems comprses he generalzaon o non-holonomc sysems whose confguraons are deermned by generalzed coordnaes and moons are modeled by Lagrange's equaons wh mulplers beng subjeced o he non-holonomc consran equaons: lnear of he frs order [] or hgher order of Cheaev's ype []. Snce he process of obanng mulplers can be consderable dffcul s more suable o model moon of nonholonomc sysems wh generalzed Lagrange's equaons of he second nd. Wha s more generalzed Lagrange's equaons of he second nd enable us o ae no consderaon sysems subjec o non-lnear consrans and hose of a hgher order. Therefore n hs paper he feld mehod s appled and eended o he sudy of moon of sysems wh varable mass subjec o such nd of consrans and modeled by generalzed Lagrange's equaons of he second nd. Receved prl 0 00

2 I. KOVČIĆ. THE GENERLIZED LGRNGE'S EQUTIONS OF THE SECOND KIND Le he poson of a mechancal sysem s defned by n generalzed coordnaes q j (j= n) whle s moon s subjec o m non-holonomc consrans: q ( ) s ϕ j j j = ( q q!... q q ) () where =... m ; s s he number of degrees of freedom s = n m ; =... s ; ( ) () = () ; s me;. The consrans () are of Chaplygn's ype [7] snce q are ndependen coordnaes and q sv are dependen ones. On he bass of he resuls of he paper [8] he equaons of moon of hs sysem can be wren n he form: ( ) ϕ ( ) = Λ ( ) ( ) ( ) ( ) s s = Λ () where L s a Lagrangan of he sysem and λ are unnown Lagrange's mulplers. Eendng hem o he sysem of N pon wh varable mass one obans: s ( ) ( ) ( ) ( ) s = Λ = Λ ϕ R s R N Generalzed reacve forces " " " dr I r I R j = M! I VI appear as a consequence of I = d q j mass varaon M! " " I of I-h pon and relave velocy of her change ( VI dri d) where V " I s he absolue velocy of an added or separaed parcle [9]. Noe ha he assumpon of he reacve force n hs form requres reang mass as a consan durng he dfferenaon of he Lagrange's funcon. fer elmnang he mulplers he prevous sysem can be ransformed no: * * ( ) ( ) ϕ = ( ) ( ) R R s () q q. * () where ( ) * denoes he erms obaned afer ecludng () s q and q s from ( ).. THE GENERLIZTION OF THE FIELD METHOD The sysem () consss of m dfferenal equaons of he second order whch ogeher wh m consrans () enable us o fnd moon q j (). Ths sysem s of he general form:

3 The Generalzed Lagrange's Equaons of he Second Knd and he Feld Mehod for Ther Inegraon 5 ( u) ( u ) q j = X j ( q j q! j... q j ) u. (5) In order o wre down n he form suable for applyng he feld mehod.e. n he form of he frs order dfferenal equaons he new varables are nroduced: ( u ) q j j j n j j n j j = u n j = q! = q! =... q ( ) (6) and he sysem becomes:! j = n j! n j = n j...! ( u ) n j = X j ( j n j... ( u ) n j ). (7) So for he non-holonomc sysem of Chaplygn's ype whch s modeled by generalzed Lagrange's equaons of he second nd he number of sae varables s s = u n. The value of u for he non-holonomc consrans of he frs and second order corresponds o he order of he lef sde of he equaon () whch s equal o wo. In he case of non-holonomc consrans of an order u u s equal o he order of consrans. Sysem (7) can be consdered as an "eended" holonomc problem whose nal condons sasfy he consrans (). Furher accordng o he basc supposon of he feld mehod one of he sae varables can be epressed as a funcon of me and he res of varables: = Φ( ) =... u. (8) n By dfferenang (8) wh respec o me and usng (7) he basc equaon s obaned: u n = X ( Φ ) X( Φ ) = 0. The feld mehod does no loo for he ased soluon drecly bu fnds hrough a complee soluon of hs quas-lnear paral equaon of he frs order. The soluon of he basc equaon can be represened n he form: u n Φ = f ( ) f ( ) (0) = where f and f are unnown funcons of me whch wll be deermned by subsung (9) no (0) and collecng and equang o zero free erms and erms conanng. I leads o soluon for he feld whch depends on he arbrary consans C C : (9) Φ = Φ C C ). () ( In accordance wh he nal condons ( 0) = 0 (0) = 0 one of he consans say C can be epressed n erms of he nal condons and he res of consans. Consequenly he condoned form soluon s obaned: Φ = Φ C (0 C ) C ). () ( 0 0

4 6 I. KOVČIĆ The fac ha he condoned form soluon should no depend on he value of he addonal consans C produces: = 0 () assumng ha de( Φ /( C B )) 0 B =... u n. So he soluon for moon of he orgnal non-holonomc problem follows from () ( u n ) algebrac equaons () and he consrans equaons for he nal values of sae varables and conans ( u n m) consans.. EXMPLE pon whose mass vares eponenally M = M 0 ep( α) where M 0 and α are posve consans moves on a plane whle s moon s subjec o rheonomc nonholonomc lnear consran: q! = q!. () Snce he Lagrangan of hs sysem s L = M ( q! q! ) he equaon () becomes: () () * * = R R (5) () () where accordng o () q s ndependen coordnae and q s dependen one. Supposng ha " he absolue abandoned velocy of he parcle s zero whch means ha V = r "! calculang necessary dfferenals and usng () and s dfferenal s obaned: q!! = αq! q! q!! = αq! q!. (6) Inroducng he subsuons q = q = = q! = q! follows:! =! =! = α! = α. (7) The basc equaon (9) for he feld Φ = Φ ) s as follows: ( α α In accordance wh (0) s soluon has he form: = 0. (8) Φ = f ) f ( ) f ( ) f ( ). (9) ( fer subsung no (8) and collecng he free erm and he erms conanng he followng sysem s derved:

5 The Generalzed Lagrange's Equaons of he Second Knd and he Feld Mehod for Ther Inegraon 7 Is negraon gves: f f! = 0 f! f α f! = 0 f! f f α f = 0. = 0 C C C f = ep( α) dτ dτ ( ) τ α τ = C C f = C f = C ep( α). () α ccordng o he nal condons 0) = 0 =... one fnds: ( C 0 = C C0 f0( 0 C C C ) 0 C 0. () α whch enable us o epress he consan C as a funcon of he res of consans and he nal condons. Fnally he applcaons of () yeld: C = 0 C 0 = 0 ep( α) 0 ep( α) ep( α) = 0 = 0 (0) C = τ α d ( τ ) α α 0 = 0. () fer solvng hese equaons he feld (9) gves he equaon of moon: = 0 0 dτ τ whle he consran () mposes he resrcon: 0 () 0 = 0. (5) REFERENCES. B. Vujanovc (979) On a graden mehod n nonconservave mechancs ca mechanca Vol. pp B. Vujanovc (98) On he negraon of he nonconservave Hamlon's dynamcal equaons Inernaonal Journal of Engneerng Scences Vol. 9 No. pp

6 8 I. KOVČIĆ. F. X. Me (99) Ob odnom meode negrrovana uravnenj negolonomnh ssem so svjazam vjsego porjada Prladnaja maemaa mehana Vol. 55 pp (n Russan).. F. X. Me (000) On he negraon mehods of non-holonomc dynamcs Inernaonal Journal of Non- Lnear Mechancs. Vol. pp Rumyansev and.s. Sumbaov (978) On he problem of generalzaon oh he Hamlon-Jacob mehod for non-holonomc sysems ZMM Vol. 58 pp R. Van Dooren (976) Generalzed mehods for non-holonomc sysems wh applcaons n varus felds of classcal mechancs Theorehcal nd ppled Mechancs Procdeengs Of he h IUTM Congress pp J. I. Nejmar and N.. Fufaev (967) Dnama negolonomnh ssem Naua Moscow. 8. Dj. Djuc (97) Ob obobscenom vde uravnenj Lagranza voroga rjada Prladnaja maemaa mehana Vol. 7 No. pp L. Cvecann (998) Dynamcs of machne wh varable mass Gordon and Bread Scence Publsher mserdam. GENERLIZCIJ LGRNGE-OVIH JEDNČIN DRUGE VRSTE I METOD POLJ Z NJIHOVU INTEGRCIJU Ivana Kovačć Ovaj rad prezenuje omplean proces dobjanja rešenja reanja mehančog ssema sa promenljvom masom neholonomnm vezama oje su nelnearne e prvog l všeg reda. U radu su najpre Lagranževe jednačne druge vrse prošrene na neholonomne sseme sa promenljvom masom uvođenjem generalsanh reavnh sla. Zam je na ove jednačne prmenjena meoda polja u clju nalaženja njhovog rešenja. Konačno da je prmer oj lusruje prezenovan algoram rešavanja.