Phase Space Research of One Non-autonomous Dynamic System

Transcription

1 Pocdngs of h 3d WSEAS/IASE Innaonal Confnc on Dynacal Syss and Conol, Acachon, Fanc, Ocob 3-5, 7 6 Phas Spac sach of On Non-auonoous Dynac Sys ANTON V. DOOSHIN Faculy of acaf consucon Saaa Sa Aospac Unvsy Thocal chancs cha, SSAU, 34, oscovsko shoss, Saaa, 44386, ussa USSIAN FEDEATION doan@nhp:// Absac: - Th non-auonoous dynac sys consdd n hs pap psn a sys of dffnal quaons of vaabl ass coaxal bods (gyosa) oon aound a fxd pon. Th odfcaon of ass paas of coaxal bods (ass, on of na) causs nonval changs of sys angula oons. Th spcal qualav sach hod of a sys phas spac s dvlopd. Th hod s basd on an valuaon of a phas ajcoy cuvau. On h bass of hs hod s possbl o dn h phas ajcoy shap. Also s possbl o synhs condons of alaon of oon spcal cass (fo xapl, a onoon dnuon o agnfcaon of an angl of a nuaon). Th applcaon valu of sachs s conncd wh h analyss of spac vhcls angula oon aound a cn of ass a alaon of acv n-obal anuvs. Ky-Wods: - Non-auonoous Dynac Sys, Coaxal Bods, Spac Vhcls, Vaabl ass, Angula oon, Qualav hod, Phas Tajcoy Cuvau Inoducon Th non-auonoous dynac sys consdd n hs pap psns dffnal quaons of angula oons of vaabl ass gd coaxal bods sys (also calld unbalancd gyosa). sach of angula oons of gd bods syss and gyosas wh consan and vaabl sucu sll ans on of h poan pobls of hocal and appld chancs. Dsp of odn dvlopn h ndcad pobl sll s fa fo coplon. In pacula concns sudy of dynacs of vaabl sucu syss. Th analyss and synhss of condons of pcsson oon wh dnshd aplud of nuaon oscllaons s lad n hs pap. Such oon has h poan applcaon valu o spac flgh chancs asks. Espcally concns gyoscopc sablaon of spac vhcls (SV) a alaon of acv anuvs. Fo undsandng of an ssnc of hs pobl s poan o dscb bfly h an consdd ngnng sngulas of alaon of SV acv anuvs. Fo a alaon of acv anuvs, fo xapl n-obal passag, s ncssay o ca hus of a j ngn fo acclang o bakng puls ( ΔV ). Ths puls s poducd n a ncssay calculad dcon. Engn hus s usually dcd along SV dc-axs. Thfo s ncssay o conduc sablaon of a dc-axs fo sablaon of an puls oupu dcon. Sablaon of a dc-axs can b cad ou n a gyoscopc od whn SV spns aound a dc-axs ond n h calculad dcon. Shapng of puls s no nsan and dands opaon of h j ngn whn sval sconds (o nus). Dung ngn opaon SV aks wo oons: ajcoy oon of a cn of ass and angula oon aound h cn of ass. SV angula oon obvously changs dc-axs dcon and, hnc, a hus dcon. Th hsoy of a hus dcon s a ason of agnud dvaon and dcon dvaon of h n-obal passag puls fo h copuaonal valus. Consqunly, h passag happns o h ob, whch dffs fo h calculad on. Th s say "spung" of hus (fg. ). Thfo, s vy poan o ak no accoun SV angula oons on h acv lg. Fg. I s ncssay o supply such angula oons a whch SV dc-axs (and h hus vco) aks pcsson oon wh onooncally dnshd

2 Pocdngs of h 3d WSEAS/IASE Innaonal Confnc on Dynacal Syss and Conol, Acachon, Fanc, Ocob 3-5, 7 6 nuaon angl. Thus h dc-axs gos nsd of an nal con of a nuaon and h hus vco naually cos na o an axs of a pcsson whch s a calculad dcon of oupu of ansonal puls ("s focusd" along a ncssay dcon). A an angula oons whou a onoon dnuon of nuaon angl h dc-axs can ak ah coplcad oon. In hs cas h hus vco also aks coplcad oon and "spus" ansonal puls. Spung a ansf ob n hs cas aks plac. Gyoscopc sablaon can b cad ou a h xpns of fas oaon no all SV, bu only hs pas - h sablng un. In hs cas a SV wll n ssnc conss as a nu of wo coaxal bods. Such SV can f o h "SV wh doubl oaon o SV-gyosa ( Sall gyosa ). On of h poan consuconal sngulas s possbly of usng of h sablng un as a pow plan []. Th sudy of angula oons of such SV wll b cad ou on h bass of chancal odl of coaxal bods of a vaabl ass. I s ncssay o ak ha h angula oons dynac of vaabl ass coaxal bods s poan no only fo applcaon spac flgh chancs pobls, bu also psns lag ns whn h fawok of basc sachs of gd bods syss dynac. Pobl Foulaon L's consd oon of coaxal bods (unbalancd gyosa) of vaabl ass und an opaon of h dsspav and boosng xo ons dpndng on coponns of angula vlocs. L gyosa wll conss of dynacally sycal an body (coaxal body ) of a consan ass and a oo (coaxal body ) of h vaabl ass anng dynacally sycal dung odfcaon of a ass (fg. ). Th fxd pon O concds a pal gocal poson of cn of asss of a sys. W wll us followng coodna syss: OXYZ - fxd, Ox y - conncd o h an body, Ox y - conncd o h oo. Th oo oas along dc-axs O. Th unbalanc gyosa has vacllang lav angula vlocy of oo oaon concnng h an body. I s possbl n conncon wh no on opaon bwn coaxal bods. L h s a on of j focs aound of dc-axs O. I s possbl o no followng quaons of sys oons []: A() p + ( C() A() ) q+ C() q x A() q ( C () A ()) p C() p y () C () + C() + C() ( + ) + +, H A() A + A() ρ (), C () C + C(), A, A( ), C, C( ) a quaoal and longudnal ons of na of h coaxal bods an body and oo (ndxs and, accodngly);, +, a xo focs ons ( x, y, ) ; s on of j focs; s no on opang bwn coaxal bods (a fcon oqu o a on of h unwsng ngn); ρ ( ) s a whch dscb "gocal" anson of cn of asss concnng a fxd pon [, ] n conncon wh a odfcaon of a ass-na sucu of a sys, ρ ( ) s a vayng dsanc bwn fxd pon and sys asss cn along dc-axs O ; () + () s a vayng ass of h gyosa; p, q, a an body angula vlocy pojcons on axs of sys Ox y ; s lav angula vlocy of h oo. A pon abov a nual ans opaon of dvaon on. I s ncssay o add knac laons (fg. ): γ psnϕ+ qcos ϕ, ψ ( pcosϕ qsnϕ) cosγ () snγ ϕ ( pcosϕ qsn ϕ), δ cosγ Angl δ ( Ox, Ox) s angl of lav oaon of h oo. L's psn h nw vaabls cospondng o agnud of vco of ansvsal angula vlocy G and angl F bwn hs vco and axs Oy : p () G ()sn F (), q G ()cos F () (3) Equaon () wll b nod n nw vaabls as Fg.

3 Pocdngs of h 3d WSEAS/IASE Innaonal Confnc on Dynacal Syss and Conol, Acachon, Fanc, Ocob 3-5, 7 63 F ( C () A ()) + C() + ff ( GF, ) A () ( G, F) fg c, G, A () C (4) C () +,, + C() C C() C In quaons (4) h followng dsubng funcons dscbng xposus ak plac: f G, F snf + cosf (, ) ( cos sn ) G x y ff GF x F y F G W wll consd a cas whn h odul of a ansvsal angula vlocy of an body s sall as conasd o lav longudnal oaon a of h oo: ε p + q << (5) W wll assu angls γ and ψ as sall quany ( γ O ( ε ), ψ O( ε )). Thn h angl of a nuaon θ (an angl bwn axs OZ and O ) wll b dfnd by h followng appoxad foula: θ γ + ψ (6) Wh h hlp of laons (3) knac quaons () can b nod as (s of h scond od of a sallnss a jcd): γ Gcos Φ( ), ψ Gsn Φ( ), ϕ, δ (7) Φ () F() ϕ() Funcon Φ() s a phas of spaal oscllaons. Pcsson oon of h gyosa wh sall nuaon angls s obvously dscbd by a phas spac of vaabls { γ, ψ }. Th phas ajcoy n hs spac coplly chaacs oon of h dc-axs O (an apx of h dc-axs). Thfo ou fuh sachs wll b conncd o h analyss of hs phas spand chancs of bhavos of phas cuvs n hs spac. 3 Pobl Soluon W wll dvlop a spcal qualav hod of h analyss of a phas spac. an da of h hod s h valuaon of a phas ajcoy cuvau n h γ, ψ. phas plan { } On h ndcad plan h phas pon wll hav followng coponns of a vlocy and acclaon: Vγ γ, Vψ ψ, Wγ γ, Wψ ψ Wh h hlp of laons (7) h cuvau of a phas ajcoy (k) s valuad as k 3 γψ ψγ γ + ψ Φ G (8) If cuvau agnud wll ncas, h s a oon on a wsd spal ajcoy sla o a sady focal pon (fg. 3, cas a ) and f dcass - on unwsd. On wsd spal ajcoy oon condon can b nod as: k kk > ΦΦ G G Φ > (9) Fo h analyss of h condon alaon s ncssay o sudy a dsposon of o of a followng funcon: P () ΦΦ G G Φ () Funcon () w wll na as funcon of phas ajcoy voluons. Fg. 3 Dffn qualav cass of phas ajcoy bhavos a possbl dpndng on o of funcon P() (fg.3). In h fs cas (fg. 3, cas a ) h funcon s posv and has no o on a consdd slc of [, T], hus h phas ajcoy s spally wsng. In h scond cas (fg. 3, cas b ) s psn on o and h s on odfcaon n a onooncy of h ajcoy cuvau. In h hd cas (fg. 3, cas c ) s psn so o and h ajcoy has alnaon of unwsd and wsd sgns of oon; also h a pons of slf-nscon. As an xapl w wll consd oon of coaxal bods of vaabl ass a an opaon of h consan no focs on ( cons ) and consan on of j focs ( cons ). Th analyss of a phas spac w wll conduc wh h hlp of dvlopd hod of cuvau valuaon. ons of na w wll coun lna funcons of, and agnuds ρ w wll nglc. In a consdd cas quaons (4) wll b nod as ( C C c A A a) ( C c) F A + A a ( C C c) G +, + () ( C c) C ( C c) / C Analycal soluons fo angula vlocy () and follow fo quaons ():

4 Pocdngs of h 3d WSEAS/IASE Innaonal Confnc on Dynacal Syss and Conol, Acachon, Fanc, Ocob 3-5, 7 64, + s + sln( c ) C () s Cn, s ( + ), c c/ C c Usng soluons () s possbl o cv an xpanson n a ss of a gh pa of an quaon fo phas F() (): F F F (3) + Followng valus appa n xpanson (3): D F A + A 3 k+ j F Dka + Ej+, j+ a A + A k j D B + C, B C + C A A D C s sc c b B C 3 C a a, b c a A + A k l kl, D b c s s c C c s E csc / k Csc / l Th oband xpanson (3) unfoly convgs on h nval,,. Fo knac a c quaons s possbl o cv h soluon fo angl ϕ : ϕ / ϕ + (4) C laons fo dvav fo phas of spaal oscllaons ( Φ) follow fo (4):, Φ F ϕ f + f Φ f + f f F, f F+ / C (5) f j Fj ( j.. ) On h bass of xpansons (5) n a lna appoxaon w wll g a polynoal of h fs dg fo phas ajcoy voluons funcon (): P () f( f + f ) f + ff (6) Th s a unqu o of a polynoal f / f. Fo plnaon of a condon (9) of wsd spal oon s ncssay ha h polynoal was sady ( < ) and posv a. I s possbl only n cas of followng condon fulfln: ff > (6) W wll consd a cas whn followng conngncs a coc:, <, >, > In hs cas valu f wll b posv: f C / A+ A > Fo h posvnss of valu f s ncssay fulfln of a condon: c( A + A) Ca C f > (7) A + A C A + A ( ) Fo condons (7) h goup of condons <, < (8) C A + A o wo oh sla goups of condons ) <, > C A + A ( A + A ) c A + A C a > ) >, < C A + A ( A + A ) c A + A Ca < (9) In fgu (fg. 4) suls of nucal calculaons of phas ajcoy a shown. Fg. 4 Cass a and b (fg. 4) cospond o fulfln of condons (8) and (9), cass c and d (fg. 4) - o h nonfulfln. Paas of sys and nal condons fo calculaons a lsd n abl.

5 Pocdngs of h 3d WSEAS/IASE Innaonal Confnc on Dynacal Syss and Conol, Acachon, Fanc, Ocob 3-5, 7 65 Tabl Cas (fg. 4) a b c d A, kg A, kg C, kg C, kg а, kg /s.... с, kg /s...., N 3, N , ad/s G, ad/s.... Condons (8) (o (9)) can b usd fo synhss of spac vhcls paas. Fo naually ncas of vhcl dc-axs posonng xacud s ncssay o al pcsson oon wh a dcasng nuaon angl. Ths oon wll b ald a fulfln of condons (8) o (9). Fo alaon of o xac sachs, canly, s ncssay o ak no accoun of hgh dgs polynoals P() (). Howv s shown ha alady lna analyss (wh h hlp of a polynoal of h fs dg (6)) allow qu adqualy dscb of angula oons voluons of vaabl ass coaxal bods. Exand abov cas of nvsgaon dos no ak no accoun any poan aspcs of vaabl ass coaxal bods oon. Howv h noducd xapl has wll llusa h appoach o sach of non-auonoous dynac syss of ndcad yp. 4 Concluson W also psn suls of wo hypohcal cass of oon of coaxal bods of vaabl ass. No addssng o any foulas w wll show n fgus a suaon whn h polynoal () has n os (fg. 5) and nublss quany of os (h s a cas whn h funcon conans any haonc ) (fg. 6). Such nsng cass of oon can qu a sach of vaabl ass coaxal bods and unbalancd gyosas oon dynac. In h acl sach of a phas spac of a nonauonoous dynac sys of oon of coaxal bods and unbalancd gyosas of vaabl ass cad ou. Nw hod of nvsgaon bhavo of non-auonoous dynacal sys s dvlopd. suls of sachs hav poan applcaon valu n spac flgh chancs pobls. Fg.5 Fg.6 Th followng gans and pogas suppo h sach: Th Poga of h Psdn of ussan Fdaon fo suppo of ladng scnfc schools of h ussan Fdaon and young canddas of scncs (hp://gans.xch.u). Pojcs No. K , K ; Th ussan Foundaon of Basc sach (hp:// Pojcs No , fncs: [] V. S. Aslanov, A. V. Dooshn and G. E. Kuglov, Th oon of Coaxal Bods of Vayng Coposon on h Acv Lg of Dscn, Cosc sach, Vol.43, No. 3, 5, pp.3-. [] V. S. Aslanov, A. V. Dooshn, Th oon of a Sys of Coaxal Bods of Vaabl ass, Jounal of Appld ahacs and chancs, Vol.68, 4, pp (