Multi-parameter Analysis of a Rigid Body. Nonlinear Coupled Rotations around

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1 Adv. Theo. Appl. Mech., Vol. 6, 3, o., 9-7 HIKARI Ltd, Multi-paamete Aalysis of a Rigid Body Noliea Coupled Rotatios aoud No Itesectig Axes Based o the Vecto Method Katica R. (Stevaović Hedih Mathematical Istitute SANU, Belgade, Sebia ad Faculty of Mechaical Egieeig Uivesity of Niš, Niš, Sebia khedih@euet.s, khedih@sbb.s Daga Milosavljević Faculty of Mechaical Egieeig Uivesity of Kagujevac, Sebia dmilos@kg.ac.s Ljljaa Veljović Faculty of Mechaical Egieeig Uivesity of Kagujevac, Sebia veljovicljilja@yahoo.co.uk Copyight 3 Katica R. (Stevaoviċ Hedih, Daga Milosavljeviċ ad Ljiljaa Veljoviċ. This is a ope access aticle distibuted ude the Ceative Commos Attibutio Licese, which pemits uesticted use, distibutio, ad epoductio i ay medium, povided the oigial wok is popely cited.

2 5 Katica R. (Stevaoviċ Hedih et al. Abstact Multi-paamete aalysis ad coespodig visualizatio of a igid body oliea dyamics with coupled otatios aoud o itesectig axes based o vecto method ae peseted. Fo oliea dyamics of a gyo-oto with oe degee of feedom ad coupled otatios, seies of thee paametes phase tajectoy tasfomatios ae peseted ad aalyzed. Also, some gaphical visualizatio of the vecto otato popeties is peseted. Agula velocity of kiematical vecto otato is expessed by usig both agula acceleatio ad agula velocity of compoet coupled otatios of gyo-oto. Keywods: Coupled otatios, o itesectig axes of otatio, oliea dyamics, mass momet vectos, deviatioal mass momet vecto, otato, otato agula velocity, gyo-oto, phase tajectoy, ecceticity, agle of icliatio, kietic pessue, fixed poit, thee paamete tasfomatio, gaphical pesetatios. Itoductio Vecto method based o mass momet vectos coupled fo pole ad oieted axes is used fo obtaiig vecto expessios fo both liea mometum ad agula mometum ad thei coespodig deivatives with espect to time (see Refeeces [.[ ad [3. This method is suitable to descibe igid body dyamic with coupled otatios aoud o itesectig axes. Mass ietial momet vectos ad coespodig deviatioal vecto compoets fo pole ad oieted axis ae defied by K. Hedih i 99 (see Refeeces [[8, [9 ad [. Based o itoduced vecto method ad mass momet vectos a seies of the vecto expessios of kietic paametes of a igid body simple o coupled otatios aoud o itesectig axes ae deived ad peseted i peviously published papes [83, but o complete aalysis of obtaied vecto expessios, especially accodig oliea dyamics.. A additioal aalysis of obtaied vecto expessios, fo cosideed examples of mechaical system with coupled otatios, is ecessay to complete eseach esults which ae possible to obtai. Also, it is possible to show lage possibilities of the powe of use mass momets vectos coupled to poit ad axis fo vecto aalysis multi-body dyamics with coupled otatios. Results of a additioal vecto ad umeical multi-paamete aalysis of obtaied vecto expessios i cited efeece [3 is give i this ou pape. The kiematical vecto otatos as well as thei agula velocities ad itesity ae defied fo a system with oe degee of feedom, ad fo coupled otatios aoud two o itesectig axes i efeece [3 by which i this pape we peset multi-paamete tasfomatio thei oliea popeties. Reseach i the aea of igid body coupled otatios ad seies of the gyo dyamics dates back about oe huded yeas. Gyoscope is a attactive ad evelastig subject of dyamics, which has bee studied by may authos. Cuet

3 Multi-paamete aalysis of a igid body oliea coupled otatios 5 eseach i aea of multi-body coupled otatios is vey impotat i egieeig applicatios especially i obotics, but based o umeical methods. Mass momet vectos ad vecto otatos i ou opiio ca be ew ope way fo applicatios i this aea. Fom time to time it is useful to pay attetio agai to classical models of dyamics of mechaical systems ad to fid possibilities fo ew appoaches to these classical esults by usig methods that ae ot jet used i the classical liteatue. New vecto appoach gives us ew kowledge about oliea pheomea i classical models of mechaical system. A oveview of gyoscopes theoy with theoetical, pactical aspects, applicatios ad teds is possible to obtai by eadig the followig efeeces: [-, [3-7, [6 ad [8. The oigial eseach esults of dyamics ad stability of gyostats was give i 979 by Achev ad Rumyatsev [. The dyamic behavio of a symmetic gyo with liea-plus-cubic dampig, which is subjected to a hamoic excitatio, is studied i the Refeece [6 by Che, H.-K., (. The Lyapuov diect method has bee used to obtai the sufficiet coditios of the stability of the equilibium poits of the system. By applyig umeical esults, time histoy, phase diagams, Poicae maps, Lyapuov expoets ad Lyapuov dimesios ae peseted to obseve peiodic ad chaotic motios. Also, attetio is shifted to the cotol chaos ad o the sychoizatio of chaos i the two idetical chaotic motios of symmetic gyos. The classical book [3 by Adoov, Vitt ad Hayki cotai a classical ad vey impotat elemetay dyamical model of a fasciate, a heavy mass paticle elative oliea dyamics alog otate cicle aoud vetical axis though it s cete. Noliea dyamics ad sigulaities lead to pimitive model of the simple case of the gyo-oto, which epeset a useful dyamical ad mathematical model of a igid body oliea dyamics by coupled otatios about o itesectig axes. Ogaizatios of this pape is based o both vecto method applicatios with use of the mass momet vectos, ad vecto otatos fo which vecto expessios fo liea mometum ad agula mometum ad thei deivatives of the igid body coupled otatios aoud two o itesectig axes ae peseted i the pevious published Refeece [3.. These obtaied expessios ae used fo aalysis ad umeical expeimet i multi-paamete tasfomatio of the vecto otato popeties ad shapes of the coespodig phase tajectoies. The seies of coclusios ae poited out. Model of a igid body coupled otatios aoud two o itesectig axes Let us to coside igid body coupled otatios aoud two o itesectig axes, as peseted i Refeece [3 ad i Figue. Fist axis is oieted by uit vecto with fixed positio ad secod is oieted by uit vecto, which is otatig

4 5 Katica R. (Stevaoviċ Hedih et al. aoud fixed axis with agula velocityω = ω. Rigid body is positioed o the movig otatig axis oieted by uit vecto. Rigid body otates aoud otatig axis with agula velocity ω = ω ad aoud fixed axis oieted by uit vecto with agula velocity ω = ω. The igid body is scew (iclied positioed o the self-otatio axis. The agles β i, i =, ae agles of scew positio of igid body mai ietia axes to the self otatio axis. Whe cete C of the mass of igid body is ot o self otatio axis of igid body self otatio, we ca say that igid body is eccetically positioed i elatio to the self otatio axis. Ecceticity of positio is omal distace betwee body mass cete C ad axis of self otatio ad it is defied by e = [,[ ρc,. Hee, ρc is vecto positio of mass cete C with oigi i poit O, ad positio vecto of mass cete with fixed oigi i poit O is = + ρ. Fo detail see Refeece [3 ad Figue. C O C C =, [, ρc [ ρc [, [, ρc [ [, ρ v C =, C R C u v v v = [, [, ρc [ [,, ρ C R ζ z v v C C R R C u v = [ [,, ρc u [ [,, ρ ζ ω C = [, ρc = C [, ρ C ω v R R C B η ρ N dm R u O R u v y u C ω α ρc O α B u u v η x ξ ξ R R R Figue. Abitay positio of igid body coupled otatios aoud two o itesectig axes. System is with two degees of mobility (oe degee of feedoms ad oe degee of heoomic costait whee ϕ is heoomic ad ϕ is geealized coodiates. Some of vecto otatos, R, R ad R ae peseted.

5 Multi-paamete aalysis of a igid body oliea coupled otatios 53 3 Vecto stuctues of the deivatives of liea ad agula mometum of s igid body coupled otatios aoud two o itesectig axes By usig vecto expessios fo liea ad agula mometum expessed i the Refeece [3 by use vecto method based o the mass momet vectos (see Appedix I* ad II* ad vecto otatos fo a igid body coupled otatios aoud o itesectig axes descibed i pevious pat ad peseted i Figue, the vecto expessio of coespodig deivatives of liea mometum is peseted i the followig fom: dk ( O ( O ( O = R [, M + R S + R [, S + ω ω S ( dt ( O whee = [ S, ρdm ad ( O = [ S, ρdm. Fom the stuctue of liea V mometum deivative tems, it is visible that expessio cotai tems with poduct of the pue kiematic vectos otatos R (ad R ad coespodig mass momets vectos coupled fo pole ad axis. We ca see that i pevious vecto expessio (, fo deivative of liea mometum, ae itoduced the thee vectos otatos i the followig fom (fo detail see Ref.[3: R ( O ( O S ( ( = + S i i & ω i ωi i,, =, O S i S i ii O V i ( with itesities R = & ω + ω, i =,, depedig of coespodig compoet ii i i agula velocity ad agula acceleatios, with compoet othogoal to the coespodig axis of compoet otatio, ad i diectios depedig of mass liea momet vecto coupled fo coespodig axis of compoet otatio ad fo pole at self-otatio axis. Lets itoduce otatios γ, γ ad γ, which deote diffeece betwee coespodig compoet agles of otatio ϕ ad ϕ, of the igid body compoet otatio, ad coespodig absolute agles of kiematics vecto otatos about axes oieted by uit vectos ad. Agula velocity of elative kiematics vecto otato R, R ad/o R, which otate about coespodig axes i elatio to the compoet agula velocities of the igid body compoet otatios, ae: & ϕi( && ϕi & ϕi&&& & γ i = & γ ii = i =, && ϕi + & ϕi, (3 Also, i same efeece [3 the secod seies of the vecto otatos ae idetified i the expessio fo deivative of agula mometum. These two vectos

6 5 Katica R. (Stevaoviċ Hedih et al. otatos: R ad R ae itoduced i the followig vecto foms (fo detail see Ref.[3: ( O ( O D = + D i i R &, i ω i ω ( O i i, i =, ( ( O D D i i itesities same as i pevious set, R ii = & ω i + ωi, i =,, depedig of coespodig compoet agula velocity ad agula acceleatios, with compoet othogoal to the coespodig axis of compoet otatio, ad i ( O diectios depedig of the deviatioal compoet D i of mass ietia momet vecto coupled fo coespodig axis of compoet otatio ad fo pole at self-otatio axis. Relative agula velocity of body otatio aoud self-otatio axis is detemied by same expessio i the fom (3, but diectios detemied by coespodig expessio (. Noliea dyamics paametes of a igid body coupled otatios aoud two othogoal o itesectig axes ad with oe degee of feedom We ae goig to take ito cosideatio special case of the cosideed heavy igid body with coupled otatios about two othogoal o itesectig axes with oe degee of feedom, ad i the gavitatio field. This is example fom Refeece [3. Fo this case coodiate ϕ is geealized idepedet, ad coodiate ϕ is pogammed. I that case, we say that coodiate ϕ is heoomic coodiate ad system is with kiematical excitatio, pogammed by foced suppot otatio by costat agula velocity. Whe, the agula velocity of shaft suppot axis is costat, ϕ& = ω = cost, it is obvious that heoomic coodiate is liea fuctio of time, ϕ = ωt + ϕ, ad agula acceleatio aoud fixed axis is equal to zeo, ω& =. Special case is, whe the suppot shaft axis is vetical ad the gyo-oto shaft axis is hoizotal, ad all time i hoizotal plae, ad whe axes ae without itesectio ad have omal distace a. The agle of self otatio aoud moveable self otatio axis oieted by the uit vecto is ϕ whose agula velocity isω = & ϕ. Diffeetial equatio of elative self-otatio of igid body is i the fom: ϕ& & + Ω ( λ cosϕsiϕ + Ω ψ cosϕ =, (5 whee coespodig coefficiets ae i the fom: ( ε si β ( ε si β + e g Ω = ω, ε = +, ( ε si β easi β λ =, ψ = (6 eω ( ε si β e ( ε si β ad it is cosideed a eccetic disc (ecceticity is e, with mass m ad adius, which is iclied to the axis of its ow self otatio by the agle β.

7 Multi-paamete aalysis of a igid body oliea coupled otatios 55 5 Sigula poits ad elative equilibium positios of the heavy gyo-oto disk coupled otatios aoud othogoal o itesectig axes ad coditios of thei stability ad istability Let tasfom pevious oliea equatio (5 ito system of two fist ode oliea diffeetial equatios i the followig way: dϕ = v dt dv = Ω ( λ cosϕsiϕ Ω ψ cosϕ dt Fo obtaiig statioay values fo the pevious system (7, we put that ight had sides of both diffeetial equatios of that system ae equal to zeo. The, fom obtaied tascedet oliea equatio, Ω ( λ cosϕ siϕ Ω ψ cosϕ =, as coditio of the statioay values fo geealized agula coodiate ϕ ad agula velocity ϕ& = v, we obtai the followig oots: ϕ s, s =,,3,,.... Fo each of this oots, system have statioay values ( ϕ s, vs =, s =,,3,,..., which coespod to the elative equilibium positios of the disk o the self otatio axis, ad that is: Ω ( λ cosϕs siϕs Ω ψ cosϕs =, s =,,3,,.... Fo obtaiig evaluatio of the stability of each of elative equilibium positios ( ϕ s, vs =, s =,,3,,..., we must apply Lyapuov s citeia of stability by use lieaizatio of the system of diffeetial equatios aoud statioay values ad coespodig gyo-oto-disk elative equilibium positios ( ϕ s, vs =, s =,,3,,..., ad to evaluate stability of oots of chaacteistic equatio. By use of geealizatio of the pevious system (7, we ca wite it i the followig fom: dϕ = v = f ( ϕ, v dt, (8 dv = Ω ( λ cosϕsiϕ Ω ψ cosϕ = g( ϕ, v dt ad thei coespodig lieaizatio aoud statioay values ad coespodig gyo-oto-disk elative equilibium positios ( ϕ s, vs =, s =,,3,,... i the fom: dϕ f ( ϕs, vs f ( ϕs, vs = ϕ + v dt ϕ v (9 dv g( ϕs, vs g( ϕs, vs = ϕ + v dt ϕ v By use of pevious system (9 of lieaized diffeetial equatios, the lieaizatio of the oliea diffeetial equatio (5 aoud statioay values ad coespodig elative gyo-oto-disk equilibium positios ( ϕ s, vs =, s =,,3,,..., of the gyo-oto- disk dyamics, we ca wite i the followig fom: ϕ& + Ω λ cosϕ cosϕ + Ω si ϕ Ω ψ si sϕ ϕ. ( [ ( = s s s s Depedig of the coefficiet sig, [ Ω λ cosϕ cosϕ + Ω si ϕ Ω ψ si sϕ (7 ( s s s s of pevious lieaized equatio (, we ca coclude that, system have stable o

8 56 Katica R. (Stevaoviċ Hedih et al. ustable elative equilibium positio i coespodig statioay poit fom the set ( ϕ s, vs =, s =,,3,,..., ad coespodig type of sigula poit cete type o saddle type: a* Fo the case that coefficiet ω s = Ω ( λ cosϕs cosϕs + Ω si ϕs Ω ψ si sϕs > ( is positive, sigula poit is stable cete type ad elative equilibium positio of the heavy gyo-oto-disk o the self otatio axis is stable. Appoximate solutio fo elative agle of otatio aoud that stable elative equilibium positio is i the followig fom: ϕ ( t As cosωst + Bs si ωst ( b* Fo the case i which coefficiet k s = Ω ( λ cosϕscosϕs + Ω si ϕs Ω ψ si sϕs < (3 is egative, sigula poit is o stable ad o stable saddle poit type, ad elative equilibium positio of the heavy gyo-oto-disk o the self otatio axis is o stable. Appoximate solutio fo elative agle of otatio aoud that o stable elative equilibium positio is i the followig fom: kst kst ϕ ( t AsChskst + BsShkst o ϕ ( t Ase + Bse ( whee A s ad B s ae itegal costats depedig of iitial coditios. 6 Phase potait of the heavy gyo-oto disk coupled otatios about two o itesectig axes ad thei thee paamete tasfomatios Let coside phase potait of the elative oliea dyamics of the heavy gyo-oto-disk aoud self otatio shaft axis ad phase tajectoies tasfomatio by chagig disk positio o the self-otatio axis. Foms of phase tajectoies ad thei tasfomatios by chages of iitial coditios, ad fo diffeet cases of disk ecceticity ad agle of its skew, as well as fo diffeet values of othogoal distace betwee axes of compoet otatios ae obtaied. Fo that easo it is ecessay to fid fist itegal of the diffeetial equatio (5. The o-liea equatio of the phase tajectoies of the heavy gyo oto disk elative dyamics fo the followig iitial coditios, at t =, petubatio ϕ = & ϕ t = &, is i the followig fom: coodiate ( t ϕ, agula velocity ( ϕ ϕ = & ϕ + Ω cosϕ cos ϕ + ψ siϕ Ω cosϕ cos ϕ + ψ siϕ & (5 As the aalyzed system is cosevative, it is also the system eegy itegal. Fo that pevious esticted case, oe ca sepaate pat of expessios i the equatio (5 i the followig fom: ~ E p λ = Ω ( β, ε, ω ( β, ε, e, ω ( cosϕ cosϕ + ( cos ϕ cos ϕ + Ψ( β, e, a, ( siϕ siϕ (6

9 Multi-paamete aalysis of a igid body oliea coupled otatios 57 as a aalog simila to the potetial eegy i this heoomic system, o a expessio of the potetial eegy of the coespodig cosevative system to the cosideed heoomic system. I Figue., the tasfomatio of the gaphical pesetatio of the potetial eegy aalog cuve E ~ p of the heavy gyo oto-disk, with coupled otatio about two o itesectig axes fo diffeet values (d* of the disk ecceticity e ad (a*, b* ad c* of the agle β of disk icliatio to the self shaft axis of otatio ae peseted. 5 E ~ p E ~ p 5 5 ϕ ϕ a* b* [ ad ; & ϕ = π [ ad / sec ϕ = π [ ad E p E p E p E5 p E7 p ϕ = π E ~ p π ϕ E p φ ; ϕ& ( ( ( ( ( E p φ E p φ E p3 φ E p φ = Potecial eegy c* φ d* Figue. The tasfomatio of the gaphical pesetatio of the potetial eegy aalog of the heavy gyo oto-disk with coupled otatio aoud o itesectig axes fo diffeet values (d* of the ecceticity e ad (a*, b* ad c* of the agle β of disk icliatio to the pope shaft axis otatio. 6 π ϕ& π φ t φ t.5 π φ t ϕ φ t φ t φ t φ 3t φ 3t φ t φ t.5.5 φ 5t φ 5t 6 a* b* φ Figue 3. Tasfomatio of a phase tajectoy of the heavy gyo-oto with otatig o itesectig axes,, fo diffeet values of disk icliatio agle β to the axis of self otatio ad fo two diffeet iitial coditios: (a* ϕ = π [ ad ; & ϕ = π [ ad / sec ad (b* ϕ = π [ ad ; ϕ& =. I Figue 3, tasfomatio of a phase tajectoy of the heavy gyo-oto-disk with otatig o itesectig axes fo diffeet values of disk icliatio agle β to the axis of self otatio ad fo two diffeet iitial coditios: (a* ϕ = π [ ad ; & ϕ = π [ ad / sec ad (b* ϕ = π [ ad, ϕ& = is peseted. It is possible to see appeaace of the homocliic obits i the fom umbe eight ad tigge of coupled sigulaities. By tasfomatio of the phase tajectoy it is possible to idicate disk

10 58 Katica R. (Stevaoviċ Hedih et al. icliatio agle β to the axis of self otatio have bifucatio values at which homocliic phase tajectoy i the fom umbe eight appea. I Figue. a* tasfomatio of a closed phase tajectoy pesetatio of the heavy gyo-oto-disk oliea dyamics with otatig o itesectig axes fo diffeet values of omal distace betwee axes ad fo the coespodig iitial coditios is peseted. I Figue.b* tasfomatio of a closed phase tajectoy pesetatio of the heavy gyo-oto-disk oliea elative dyamics with otatig o itesectig axes fo diffeet values of disk ecceticity i positio to the self otatio axis ad fo the coespodig iitial coditios is peseted. φ t φ t 8 6 ϕ& π φ t 3 φ t φ t φ t.35 φ t φ t.5 φ t φ3 t φ3 t φ t φ t ϕ φ t φ t φ t φ5 t φ t3 φ5 t φ6 t φ6 t φ t3.5 φ t a* φ7 t φ7 t φ b* φ t Figue. (a* Tasfomatio of a closed phase tajectoy pesetatio of the heavy gyo-oto-disk with otatig o itesectig axes fo diffeet values of omal distace betwee axes ad fo the coespodig iitial coditios; (b* Tasfomatio of a closed phase tajectoy pesetatio of the heavy gyo-oto-disk with otatig o itesectig axes fo diffeet values of disk ecceticity i positio to the self otatio axis ad fo the coespodig iitial coditios. φ a* b* Figue 5. Tasfomatio of two diffeet closed phase tajectoies pesetatios of the heavy gyo-oto-disk self otatio with otatig o itesectig axes fo diffeet values of omal distace betwee axes ad fo the coespodig iitial coditios. (a* without appeaace of the homocliic obit i the fom umbe eight ad (b* with appeaace of the homocliic obit i the fom umbe eight followed by bifucatio of sigula poits. I Figue 5. tasfomatio of the two diffeet closed phase tajectoies pesetatios of the heavy gyo-oto-disk self otatio with otatig o itesectig axes fo diffeet values of othogoal distace betwee axes ad fo the coespodig iitial coditios is visible. I Figue 5.a* tasfomatio of a closed phase tajectoy without appeaace of the homocliic obit i the fom umbe eight is peseted. I Figue 5. b* tasfomatio of a closed phase tajectoy with appeaace of the homocliic obit i the fom umbe eight followed by bifucatio of sigula poits is

11 Multi-paamete aalysis of a igid body oliea coupled otatios 59 peseted. By this phase tajectoy tasfomatio, we ca see that elative equilibium positio of the disk o the self otatio axis lose stability ad appea two othe stable elative equilibium positios ad also, stable cete type sigula poit sepaates ito thee sigula poits - ito set of thee coupled sigulaities, oe o stable saddle type ad two stable cete type aoud o stable saddle type sigula poit. These coupled, thee, sigula poit build a tigge of coupled sigulaities. φ t φ t φ t φ t φ t.5 φ t φ t φ t φ t φ t φ t.5.5 φ t φ3 t φ3 t φ t φ t.5 φ 3 t φ 3 t φ t φ t φ 5 t φ 5 t φ 5 5 φ a* b* Figue 6. (a* Tasfomatio of a closed homocliic phase tajectoy i the fom of umbe eight of the heavy gyo-oto-disk oliea dyamics with coupled otatios aoud o itesectig axes fo diffeet values of paamete λ, with tigge of coupled sigulaities ( ϕ = ± accosλ, ϕ& = ; (b* Tasfomatio of a closed homocliic phase tajectoy aoud homocliic obit i the fom of umbe eight of the heavy gyo-oto-disk oliea dyamics with coupled otatios aoud o itesectig axes fo diffeet values of paamete λ, ad cotaiig iside tigge of coupled sigulaities I Figue 6.a* tasfomatio of a closed homocliic phase tajectoy i the fom of umbe eight of the heavy gyo-oto-disk oliea elative dyamics with coupled otatios aoud o itesectig axes is peseted fo diffeet values of paamete λ. I this case homocliic phase tajectoy pass though o stable saddle type homocliic sigula poit ad a tigge of coupled sigulaities ( ϕ = ± accosλ, ϕ& = is peset. I Figue 6.b* tasfomatio of a closed homocliic phase tajectoy aoud homocliic obit i the fom of umbe eight of the heavy gyo-oto-disk oliea dyamics, with coupled otatios aoud o itesectig axes is peseted fo diffeet values of paamete λ. I this case, phase potait of oliea dyamics cotai iside a tigge of coupled sigulaities. Paamete λ is bifucatio paamete. I Figue 7. tasfomatio of a closed phase tajectoy of the heavy gyo-oto-disk oliea dyamics with coupled otatios aoud o itesectig axes fo diffeet values of paamete λ is peseted. This closed phase tajectoy, iside, cotai a tigge of coupled sigulaities ( ϕ = ± accosλ, ϕ& =. Fom this figue a layeig of the phase tajectoy with chagig paamete λ is visible.

12 6 Katica R. (Stevaoviċ Hedih et al. φ t.5 φ t φ t φ t φ t.5 φ t φ 3 t φ 3 t φ t φ t.5 φ 5 t φ 5 t φ Figue 7. Tasfomatio of a closed phase tajectoy of the heavy gyo-oto-disk oliea dyamics with coupled otatios aoud o itesectig axes fo diffeet values of paamete λ cotaiig iside a tigge of coupled sigulaities ( ϕ = ± accosλ, ϕ& =. I Figue 8.a* tasfomatio of a phase tajectoy of the heavy gyo-oto-disk oliea elative dyamics with coupled otatios aoud o itesectig othogoal axes fo diffeet values of paamete β of disk icliatio to the self otatio axis, o cotaiig iside a tigge of coupled sigulaities is peseted. This phase tajectoy tasfomatio cotais few fixed poits. I Figue 8.b* tasfomatio of a phase tajectoy of the heavy gyo-oto-disk oliea dyamics with coupled otatios aoud o itesectig axes fo diffeet values of paamete β of disk icliatio to the self otatio axis is peseted. Fo this case phase potait cotai a tigge of coupled sigulaities; ( ϕ = ± accosλ, ϕ& =. Fom this figue ad homocliic phase tajectoy tasfomatio ad its layeig it is possible to coclude that thee ae values of the paamete β of disk icliatio to the self otatio axis fo which disappea homocliic obit i the fom umbe eight togethe with disappeaace of the tigge of coupled sigulaities. Also, i this case appea oe stable type cete sigula poit. Fom this figue ad homocliic phase tajectoy tasfomatio ad its layeig it is possible to coclude that thee ae values of the paamete β of disk icliatio to the self otatio axis fo which disappea homocliic obit i the fom of umbe eight togethe with disappeaace of the tigge of coupled sigulaities i the pocess of the two homocliic phase tajectoy fusio ito oe homocliic tajectoy. Opposite chage pesets appeaace of the tigge of coupled sigulaities ad sepaatios a homocliic obit i the fom of umbe eight ad peset bifucatio. We ca coclude that paametic aalysis of the phase tajectoy potait by diffeet values of paamete β of disk icliatio to the self otatio axis povoke bifucatio of the equilibium positios ad oe equilibium positio lose stability ad appea two ew equilibium positios, but elative, coespod to the stable cete type sigula poits.

13 Multi-paamete aalysis of a igid body oliea coupled otatios φ 8 ϕ&(ϕ φ t ( φ ϕ& φ φ φ φ φ 3 φ 3 φ φ 6 ϕ φ t ( φ qt ( φ qt ( φ wt ( φ wt ( φ t ( φ ϕ φ t ( φ φ 5 φ 5 φ 6 tt ( φ tt ( φ φ 6 a* π φ π b* φ Figue 8. (a* Tasfomatio of a phase tajectoy of the heavy gyo-oto-disk oliea dyamics with coupled otatios aoud o itesectig axes fo diffeet values of paamete β of disk icliatio to the self otatio axis o cotaiig iside a tigge of coupled sigulaities; ad (b* Tasfomatio of a phase tajectoy of the heavy gyo-oto-disk oliea dyamics with coupled otatios aoud o itesectig axes fo diffeet values of paamete β of disk icliatio to the self otatio axis cotaiig iside a tigge of coupled sigulaities; ( ϕ = ± accosλ, ϕ& =. 7 Paametic aalysis of vecto otatos of the heavy gyo-oto disk coupled otatios about two o itesectig othogoal axes ad thei thee paamete tasfomatios I cosideed case, fo the heavy gyo-oto-disk oliea dyamics, i the gavitatioal field with oe degee of feedom ad with costat agula velocity about fixed axis, we have thee sets of vecto otatos fistly peseted i Refeeces [ ad [ ad fo cosideed case i Refeece [3 i vecto ad aalytical fom. Some gaphical pesetatios ae doe i cited efeeces. I this pat a detailed thee paametic aalysis is doe. By use, esults i vecto ad aalytical fom fom Refeece [3 based o the vecto expessios peseted by expessios ( ad ( ad by use odiay diffeetial equatio (5 ad its coespodig fist itegal (5 ad coespodig defiitios of the deviatioal pat of the mass momet vectos (see Appedix I. we ca list the two expessios fo two vecto otatos appea i the cosideed system dyamics i followig foms: ( [ ( [ ρ, C R ϕ = Ω λ cosϕ siϕ + ψ cosϕ + [, ρc [ [ ρ,, C & + Ω ϕ + Ω cosϕ cos ϕ + ψ siϕ Ω cosϕ cos ϕ + ψ siϕ [, ρc (7

14 6 Katica R. (Stevaoviċ Hedih et al. R ( ϕ = Ω [ ( λ cosϕ siϕ + ψ cosϕ + Ω & ϕ D D ( O ( O + Ω cosϕ cos ϕ + ψ siϕ Ω cosϕ cos ϕ with coespodig itesity: + D + ψ siϕ, D = ω & ϕ + Ω cosϕ cos ϕ + ψ siϕ Ω cosϕ cos ϕ + ψ ϕ = R ( ϕ = si ( O ( O (8 R (9 R = Ω [ ( λ cosϕ siϕ + ψ cosϕ + & ϕ + Ω cosϕ cos ϕ + ψ siϕ Ω cosϕ 5 cos ϕ + ψ siϕ ( R R R R R 5 R R 3 R R3 R 5 a* φ b* 5 5 φ Figue 9. Itesity of the vecto otatos, R ad R, fo diffeet values of some paametes; (a* fo ecceticity e = ;,5;3;5; ad (b* fo diffeet iitial coditios h =,5; ;5;7; These vecto otatos ae coected to the pole O, othogoal to the axis oieted by uit vecto ad elative otate about this axis. Itesity of this vecto otato, expessed by geealized coodiate ϕ, agle of self otatio of heavy disk, takig ito accout fist itegal (5, of the diffeetial equatio (5, ae obtaied the fom (9 ad (. Paametic equatios of the tajectoy of the vecto otatos R ad R ae simila, ad may be expessed i the followig fom [3: v R R ( ϕ = Ω [ ( λ cosϕ siϕ + ψ ϕ u, cos ( ϕ Ω & ϕ + Ω cosϕ cos ϕ + ψ siϕ Ω cosϕ cos ϕ + ψ ϕ = si ( It is ecessay to take ito cosideatio that all vecto otatos, it is ot i same diectios, but is i the same plae othogoal to the self otatio axis oieted by uit vecto ad though pole O.

15 Multi-paamete aalysis of a igid body oliea coupled otatios 63 χφ ( χ χ χ3 χ a* b* φ tt, φ tt, φ tt, φ 3tt, φ tt Figue. (a* Itesity of the vecto otatos, R ad R, fo diffeet values of othogoal distace betwee two axes of heavy gyo-oto-disk coupled otatios about o itesectig axes. (b* Tasfomatio of the tajectoy of the vecto otato R (ad R i the plae though pole O ad othogoal to the self otatio axis fo diffeet values of paameteψ Relative agula velocity γ& of both vecto otatos R ad R, i plae othogoal to the axis oieted by uit vecto ad though pole O, i elatio o agula velocity of self otatio, ω = & ϕ, is possible to expess by usig expessio (3, leadig to [3: h + Ω ( λ cosϕ cos ϕ ( Ω ( λ cosϕ si ϕ && ϕ& ( Ω ( λ cosϕ siϕ + ( h + Ω ( λ cosϕ cos ϕ ± & γ =. ( Use of peviously deived expessio (7 ad (8, fo itesity of the vectos otatos, R ad R, coected fo the pole O which otate aoud self otatio axis, oieted by uit vecto i the othogoal plae though pole O, ad by chagig some paametes of heavy gyo-oto stuctue, as it is ecceticity e, agle of disk icliatio β, othogoal distace betwee axes a, as well as paamete ψ cotaied i the coefficiets of the oliea diffeetial equatio (5 ad give i expessios (9 ad (, we obtai seies of the gaphical pesetatio ad thei thee paamete tasfomatios show i Figues 9 ad. Vaiatio with paamete ψ is peseted i Figue 9, ad vaiatio with paamete λ is peseted i Figue. Diffeet foms of the tajectoy (hodogaph of the vecto otato R (ad R i the plae though pole O ad othogoal to the self otatio axis, fo diffeet pais ( λ, ψ of the values of paametes of gyo-oto-disk coupled compoet otatios ae peseted i Figue.

16 6 Katica R. (Stevaoviċ Hedih et al χ t5 χ t5 χ t5 φ tt φ tt φ tt5 λ =, ; ψ =, λ = ; ψ =, λ =,; ψ =, χ t5 χ t5 5 χ t φ tt φ tt φ tt5 λ =,9; ψ =, λ = ; ψ =, λ = ; ψ =,.8.3 χ t χ t χ t φ tt φ tt φ tt5 λ =, ; ψ = λ =, ; ψ =, λ =, ; ψ =,5..3 χ t φ tt χφ ( φ tt χ φ tt λ =, ; ψ = λ =,; ψ =, λ = ; ψ = χ φ tt.75 5 χφ ( φ tt χφ ( λ =, 35; ψ =, λ =,5; ψ =,93 λ =,; ψ =, Figue. Diffeet foms of the tajectoy (hodogaph of the vecto otato R (ad R i the plae though pole diffeet pais ( ψ otatios. φ tt O ad othogoal to the self otatio axis, fo λ, of the values of paametes of gyo-oto-disk coupled compoet.638

17 Multi-paamete aalysis of a igid body oliea coupled otatios 65 I Figue seies of the elative agula velocity γ& of the vecto otato R (ad R, fo diffeet values of paamete a, othogoal distace betwee axes of gyo-oto-disk coupled compoet otatios ae peseted. 5 γ t γ t γ t γ3 t γ t γ5 t 5 γ6 t γ7 t γ t γ t γ t γ3 t 5 γ t γ5 t γ6 t γ7 t Figue. Relative agula velocity γ& of the vecto otato R (ad R i the plae though pole O ad othogoal to the self otatio axis, fo diffeet iitial coditios of the statig motio of gyo-oto-disk coupled compoet otatios. φ 8 Cocludig emaks Fist mai esult peseted i this pape is successful thee paametic aalysis of the aalytical ad vecto expessios obtaied by vecto method i pevious published efeeces [ 9-3 as geealized applicatio of the vecto method based o the itoduced ew mass momet vectos coupled fo pole ad axis fo ivestigatio of the igid body coupled otatios aoud two o itesectig axes ad vecto decompositio of the dyamic paamete stuctue ito seies of the vecto paametes useful fo aalysis of the coupled otatio kietic popeties. Itoducig, i peviously published papes, mass momet vectos ad vecto otatos coupled fo pole ad axis ad used fo thee paamete aalysis ad tasfomatios of the o-liea dyamics popeties, i this pape, we show suitability of vecto expessios, ad especially vecto otatos fo to show ew appoach to vecto decompositios of the coupled otatios of a igid body vey peset i egieeig system dyamics.

18 66 Katica R. (Stevaoviċ Hedih et al. We use vecto expessios of the kietic paametes of the oliea dyamics of a igid body coupled otatios aoud two o itesectig axes, ad o the basis of the thee paametic aalysis of the vecto otatos, ad tasfomatios of the phase tajectoies, we show that vecto method as well as applicatios of the mass momet vectos ad vecto otatos give a simplest way ad expessios fo aalysis chaacteistic vecto stuctues of coupled otatio kietic popeties, especially agula velocities of the vecto otatos which ae i diectios of the kietic pessues o shaft beaigs o thei eactios. Usig the deived aalytical expessios of the gyo-oto-disk coupled otatios ad by stadad softwae tools, the umeous visualizatios of phase tajectoies ae peseted. Special attetios ae focused to the paametic aalysis of the vecto otato tasfomatios, as well as to the absolute ad elative agula velocities of thei otatios. These kiematical vecto otatos of the heavy gyo-oto-disk coupled otatios about two o itesectig axes ad thei thee paamete tasfomatios ae doe as a esult of this pape. At ed, we ca coclude that combiatios of the vecto expessios obtaied i efeece [3 by vecto method based i the mass momet vectos defied i Refeeces [8- with vecto otatos, ope a ew possibility fo applicatio this vecto ad aalytical appoach to the seies of the oliea dyamics of system of multi-body dyamics with multi-coupled otatios. Ackowledgmet: Pats of this eseach wee suppoted by the Miisty of Scieces ad Techology of Republic of Sebia though Mathematical Istitute SANU Belgade Gat ON7 Dyamics of hybid systems with complex stuctues., Mechaics of mateials ad Faculty of Mechaical Egieeig Uivesity of Niš ad Faculty of Mechaical Egieeig, Uivesity of Kagujevac. Refeeces [ S. Adi, A Oveview of Optical Gyoscopes Theoy, Pactical Aspects, Applicatios ad Futue Teds, May 6, 6. [ A. Achev ad V. V. Rumyatsev, (979, O the dyamics ad stability of gyostats. (Russia Adv. i Mech. (979, o. 3, 3-5. [3 A. Adoov, A.A. Vitt, S.E. Hayki, (98, Teoiya kolebaiy, Nauka, Moskva, pp [ F. Ayazi, K. Najafi, (, A HARPSS Polysilico Vibatig Gyoscope, Joual of micoelectomechaical systems, vol., o., Jue.

19 Multi-paamete aalysis of a igid body oliea coupled otatios 67 [5 A. Bashchikov, Aalysis of dyamics fo a satellite with gyos with the aid of to softwae Li Model, Istitute fo System Dyamics ad Cotol Theoy, SB of R.A.S., Russia. [6 H. K. Che, (, CHAOS AND CHAOS SYNCHRONIZATION OF A SYMMETRIC GYRO WITH LINEAR-PLUS-CUBIC DAMPING, Joual of Soud ad Vibatio ( 55(, 79}7, doi:.6/jsvi..86, available olie at [7 W. Flaely, J. Wilso,(965, Aalytical Reseach o a Sychoous Gyoscopic Vibatio Absobe, Nasa CR-338, Natioal Aeoautic ad Space Admiistatio, Decembe 965. [8 K. Hedih (Stevaović, (99, O some itepetatios of the igid bodies kietic paametes, XVIIIth ICTAM HAIFA, Apstacts, pp [9 K. Hedih (Stevaović, (993, Some vectoial itepetatios of the kietic paametes of solid mateial lies, ZAMM. Agew.Math. Mech. 73(993-5, T53-T56. [ K. Hedih (Stevaović, (993, The mass momet vectos at -dimesioal coodiate system, Teso, Japa, Vol 5 (993, pp [ K. Hedih (Stevaović, (, Vecto Method of the Heavy Roto Kietic Paamete Aalysis ad Noliea Dyamics, Uivesity of Niš,Moogaph, p. 5. (i Eglish, YU ISBN [ K. Hedih (Stevaović, (998, Vectos of the Body Mass Momets, Moogaph pape, Topics fom Mathematics ad Mechaics, Mathematical istitute SANU, Belgade, Poceedigs 8(6, 998, pp. 5-. Published i 999. (i Eglish, (Zetalblatt Review. [3 K. Hedih (Stevaović, (, Deivatives of the Mass Momet Vectos at the Dimesioal Coodiate System N, dedicated to memoy of Pofesso D. Mitiović, Facta Uivesitatis Seies Mathematics ad Ifomatics, 3 (998, pp (998, published i. Edited by G. Milovaović. [ K. Hedih (Stevaovic ad Lj. Veljovic, Vecto Rotatos of Rigid Body Dyamics with Coupled Rotatios aoud Axes without Itesectio, Hidawi Publishig Copoatio, Mathematical Poblems i Egieeig, Volume, Aticle ID 3569, 6 pages, doi:.55//3569. [5 Ya-zhu IU, Yu XUE: Dift Motio of Fee-Roto Gyoscope with Radial MassUbalace, i Applied Mathematics ad Mechaics, Eglish Editio, Vol. 5, No 7, pp , Jul, ISSN 53-87, ID: 53-87(

20 68 Katica R. (Stevaoviċ Hedih et al. [6 Yu. A. Kapachev ad D. G. Koeevskii: Sigle-Roto Apeiodic Gyopedulum, i Iteatioal Applied Mechaics, Volume 5, No, pp. 7-76, Spige New Yok, DOI.7/B:88365, ISSN [7 R.M. Kavaah, (7 Gyoscopes fo Oietatio ad Ietial Navigatio Systems, KIG 7, Special issue, p [8 D. Rašković, (97, Mehaika III Diamika (Mechaics III Dyamics, Nauča kjiga, 97, p.. [9 TONG X.ad MRAD N., (, Chaotic motio of a symmetic gyo subjected to a hamoic base excitatio, Tasactios of Ameica Society of Mechaical Egiees, Joual of Applied Mechaics 68, Appedix I. Mass momet vectos fo the axis though the pole The moogaph [, IUTAM exteded abstact [8 ad moogaph pape [ cotai defiitios of thee mass momet vectos coupled to a axis passig though a cetai poit as a efeece pole. Now, we stat with ecessay defiitios of mass mometum vectos. Defiitios of selected mass mometum vectos fo the axis ad the pole, which ae used i this pape, ae: (O * Vecto S of the body mass static (liea momet fo the axis, oieted by the uit vecto, though the poit poleo, i the fom: def ( O S = [, ρ dm = [, ρc M, dm = σdv, (A. V whee ρ is the positio vecto of the elemetay body mass paticle dm betwee pole O ad mass paticle N, ad M mass of the body (fo detail see moogaph [7. (O * Vecto J of the body mass ietia momet fo the axis, oieted by the uit vecto, though the poit poleo, i the fom: def ( O J = [ ρ, [, ρ dm. (A. V I the case whe the pole O is the cete C of the body mass, the vecto C (the positio vecto of the mass cete with espect to the pole O is equal to zeo, wheeas the vecto ρo tus ito ρ C so that the last expessio (3 ca be witte i the followig fom: ( O ( C J = J + [ ρ,[, ρ M (A.3 C C

21 Multi-paamete aalysis of a igid body oliea coupled otatios 69 (O Mass ietia momet vecto J fo the axis to the pole is possible to decompose ( O (O ito two pats: fist (, J colliea with axis ad secod D omal to the axis. So, it may be witte: ( O ( O ( O ( O ( O J = (, J + D = J + D (A. ( O Colliea compoet (, J to the axis coespods to the axial mass (O ietia momet J (O of the body. Secod compoet, D, othogoal to the axis, ( we deote by the D O, ad it is possible to obtai by both side double vecto poducts by uit vecto (O with mass momet vecto J i the followig fom: O O O ( O O ( [ [ ( ( ( ( ( = = = D, J, J,, J J JO (A.5 I case whe igid body is balaced with espect to the axis the mass ietia (O momet vecto J is colliea to the axis ad thee is o deviatioal pat. I this case axis of otatio is picipal axis of body ietia. Whe axis of otatio is ot picipal axis the mass ietial momet vecto fo the axis cotais deviatio (O pat D. I that case the oto is ubalaced to the axis ad a body is skew positioed to the axis of self otatio. II. Liea ad agula mometum of a igid body coupled otatios aoud two o itesectig axes By usig basic defiitio of liea mometum ad expessio fo velocity of elemetay body mass, fo the igid body coupled otatios aoud two o itesectig axes, we ca wite i a vecto fom (fo detail see Refeece [3: ( O ( O K = [ ω, M + ω S + ωs (A.6 ( O whee = [ ( O S, ρdm ad = [ S, ρdm ae coespodig body V mass liea momets of the igid body fo the axes oieted by diectio of compoet agula velocities of coupled otatios though the movable pole O o self otatig axis. By usig basic defiitio of agula mometum ad expessio fo velocity of otatio of elemetay body mass, expessio fo agula mometum fo the igid body coupled otatios aoud o itesectig axes ca be witte i a fom: LO = ω M + ω[ ρ,[, C M + ( O O ( O ( O (A.7 + ω, S + ω, S + ω J + ω J def [ [ ( ( O whee ( O J = [ ρ [, ρ dm ad = ρ [, ρ V, V def J [ dm ae coespodig igid body mass ietia momet vectos fo the axes oieted by diectios of compoet otatios though the pole O o self otatig axes. Coespodig liea V,

22 7 Katica R. (Stevaoviċ Hedih et al. O mass momet vecto S [, M is defied as if complete igid body mass = O M is put ito pole O o the self otatio axis, fo the axis oieted by diectio of pecessio otatio, though the pole O. Received: July, 3