A GENERAL METHOD TO STUDY THE MOTION IN A NON-INERTIAL REFERENCE FRAME

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1 The Inenaional onfeence on opuaional Mechanics an Viual Engineeing OME 9 9 OTOBER 9, Basov, Roania A GENERAL METHOD TO STUDY THE MOTION IN A NON-INERTIAL REFERENE FRAME Daniel onuache, Vlaii Mainusi Technical Univesiy Gheoghe Asachi, Iasi, Roania, aniel.conuache@gail.co Technical Univesiy Gheoghe Asachi, Iasi, Roania, vlaainus@gail.co Absac: The pape offes a geneal eho o suy he oion of a ass paicle wih espec o a non-ineial efeence fae. By using an aequae enso insuen, we obain a siplifie fo of he iniial value poble ha oels he non-ineial oion. The suy of he oion in a non-ineial efeence fae in a cenal posiional foce fiel is hen euce o he suy of he classic oion in a cenal foce fiel. The applicaions o his eho ae in solving Keple s poble in non-ineial efeence faes, solving he elaive obial oion poble an eiving he explici soluion o a classic Theoeical Mechanics poble: The Foucaul Penulu. The avanage of using his eho in he suy of he oion in a non-ineial efeence faes is ha is significanly euces he aoun of copuaions an i offes, in soe ipoan paicula cases, close fo analyical soluions. Keywos: Non-ineial fae, Ohogonal ensos, Keple s poble, Foucaul Penulu, enal foce fiel.. INTRODUTION The oion in a non-ineial efeence fae has ipoan applicaions, boh in heoeical an applie scienific pobles. The pesen appoach offes a geneal eho o suy such ypes of oion, wih successful applicaions in soe elevan classical an oen pobles: he Keple poble in a oaing efeence fae, he Foucaul Penulu poble, he elaive obial oion poble. The enso insuen ha siplifies he coplex iniial value poble ha oels he oion was inouce fo he fis ie in 995 by D. onuache [] an i was use o appoach seveal pobles of lassical an elesial Mechanics. The enso opeao which is pesene in his pape is inouce by he Daboux equaion [], wien hee in is ensoial fo []. In os siuaions iscusse hee, his opeao has a ie-explici foula an i allows eeining explici o close fo vecoial expessions fo he elaive law of oion an he elaive velociy. The oion in a cenal posiional foce fiel wih espec o a oaing fae is biefly iscusse, an an algoih fo a poenial appoach is popose. The suy of such ype of oion leas o he hee applicaions which ae pesene in he pape: he Keple poble in oaing efeence faes, he Foucaul Penulu an he elaive obial oion in a gaviaional foce fiel. The equaions of he elaive oion of saellies ae pesene hee in a vecoial cooinaefee fo. a = Q a Q, epesens he acceleaion of he oigin of he nonineial efeence fae. PROBLEM FORMULATION The oion of a paicle wih espec o a oaing efeence fae is escibe by he iniial value poble [4]: && + & + ( ) + & + aq = F(, &, ), ( ) =, & ( ) = v, () whee is he iniial oen of ie, = ( ), epesens he insananeous angula velociy of he nonineial efeence fae, wih espec o an ineial efeence fae an (,,) In he siuaion when he foce is cenal an i epens only on he posiion veco agniue, i.e. = ( ) F & is he foce ha acs on he paicle ha has he ass.. F F, hen he iniial value poble () ay be wien as: && + & + + & + f + a =, =, & = v, () Q 4

2 whee f is a coninuous eal value ap, f = F, epening only on he agniue of he posiion veco, f = f. The veco value funcion is suppose o be known. The poble is o eeine he law of oion ( ) =, which is he soluion o he iniial value poble () o, in he paicula case, ().. A TENSOR INSTRUMENT In his Secion we pesen he enso insuen ha allows eucing he poble of he oion in a non-ineial efeence fae o he poble of he oion in an ineial efeence fae. The key eleen of he eho is epesene by he Daboux equaion, which has as soluion he pope ohogonal enso value ap ha oels he oaion wih a known specific insananeous angula velociy [5,6]. =, be a coninuous veco value ap an le % be he associae skew-syeic enso value Le funcion. Tenso % is also known as he coss-pouc enso, since: x % = x, x V, whee V enoes he hee-iensional linea space of fee vecos. This popey igh also be exene o he se of veco funcions of eal vaiable V. The aix coesponence beween enso % an veco, in a igh hane ohonoae base, is given by [7]: = % = (4) One paicula fo of he Daboux equaion is he iniial value poble: Q= Q%, Q = I, (5) whee I epesens he secon oe uni enso. The soluion o he iniial value poble (5) will be enoe by 4 () F. This enso value funcion igh be eae as an opeao on he se of veco value funcions of eal vaiable, F : V V The popeies of his enso value funcion of eal vaiable ae lise below; hei poof ay be foun in Ref. [6] F a F b= a b, a, b V (i peseves he o pouc).. F a = a, a V.. F a F b= a b, ( ) a, b V (i is an isoey). &. 4. F = F ( + ),( ) V (i peseves he coss pouc). F = + + +, V F && & &. F = v+, V, v = &, = = The enso funcion of eal vaiable enoe: T R F =. whee ( ) F is inveible, since i is pope ohogonal, an is invese is is anspose. We The enso funcion R oels he oaion wih insananeous angula velociy. I is he soluion o an iniial value poble siila o Eq. (5): R& + R % =, R = I (7) In he siuaion when he insananeous angula velociy has a fixe iecion, i.e. = u, = wih u a consan uni veco, hen we ay explicily wie he expession of he enso value funcion R wih he help of a Roigues-like foula (see Ref. [7]): R sin( ( s) s ) cos( ( s) s = I u% + ) u% (9) The auhos of he pesen pape have pove ha he Daboux equaion (5) has close fo soluions in a uch lage nube of cases. Fo insance, when he angula velociy has unifo pecession, i.e. hee exiss a pope ohogonal enso value funcion of eal vaiable R such as: (6) (8)

3 = R () = + sin ( ) + cos ( ) R I u% u% whee is a consan veco, u is a consan uni veco an is a eal nube, hen he soluion o he iniial value poble (9), which offes he oaion enso funcion associae o he insananeous angula velociy, is given by (also see Ref. []): { sin } % cos % { I+ sin ( )( ) ( u% u% ) + cos ( )( ) ( u% u% ) } R = I + u + u () 4. A GENERAL APPROAH TO THE MOTION IN A NON-INERTIAL FRAME By using he enso insuen inouce in he pevious Secion, we will offe a geneal eho o eal wih he iniial value poble (). Fis, le us apply opeao F o he iniial value poble (). By using he popeies enione in he pevious Secion, we igh sae ha Eq. () is equivalen wih: ( F ) + F aq = F (,, ), ( ) =, ( ) = +, F & & v () The fo of Eq. () suggess he inoucion of a change of vaiable in Eq. (): ρ : = F,. () ( ) We now ay sae he ain esul of his pape. Theoe The soluion o he iniial value poble () is obaine by applying he enso opeao R o he soluion o he iniial value poble: ρ&& + F aq = F '( ρ, ρ&, ), ρ( ) =, ρ& ( ) = v+, (4) ',, F, &, by: whee F ( ρ ρ& ) is linke o F '( ρ, ρ, ) = F{ F[ R ρ, R ρ R ρ, ] } & & (5) Poof. The fo of he e in he lef of he iffeenial equaion (4) is euce fo Eqs. () an (). The iniial coniions of he iniial value poble (4) ae euce fo he popey (4.) of he enso opeao F. We have only o jusify he exisence of he e R ρ& R ρ in he expession (5) of he foce. Fo Eq. () an fo he popey (4) of he enso opeao F i follows ha: ρ& = F & + (6) By using Eqs. (6) an (), we euce: & = R ρ& R ρ ; his finalizes he poof. Reak Theoe offes a eho o suy he oion of a ass paicle wih espec o a non-ineial efeence fae. he oiginal iniial value poble ha oels he oion is ansfo, by an aequae change of vaiable, ino a siple oel. This eho is useful o be use in uliple pobles elae o he non-ineial oion: Keple s poble in oaing efeence faes, he Foucaul Penulu, elaive obial oion.. (7) 5. MOTION IN A ENTRAL FORE FIELD WITH RESPET TO A ROTATING FRAME This ype of oion efes o uliple Theoeical Mechanics an Asoynaics pobles. The iniial value poble is eive fo Eq. () by eliinaing he anspo acceleaion a Q : f ( ) ( ) ( ) && + & + + & + =, =, & = v, (8) By applying Theoe, i follows ha he soluion o he iniial value poble (8) is obaine by applying he enso opeao R o he soluion o he iniial value poble [5,6]: ( ρ) ρ&& + f ρ=, ρ =, ρ& = v +, (9) This saeen leas ieiaely o a vey ineesing geoeical inepeaion of he oion in oaing efeence fae in he pesence of a cenal foce fiel: he oion akes place in a plane, as if i was escibe by he iniial value poble (9). This plane is foe a he iniial oen of ie an i oaes wih angula velociy. 4

4 One of he classic ways of ealing wih he iniial value poble (9) is by using pola cooinaes in he plane escibe above. If one chooses he pola axis such as is oienaion coincies wih ha of an he chooses he inceasing sense of he pola angle θ having he oienaion given by he iniial angula oenu of he plana oion, h = v + (), hen he algoih fo solving he iniial value poble (8) has he following seps:. onsie he syse of scala iffeenial equaions fo ρ an θ : && ρ ρθ& + f ( ) = v, wih he iniial coniions ρ( ) =, & ρ =, θ = () ρ & θ = h. Solve he iniial value poble fo he agniue of he posiion veco : h && ρ + f ( ) =, ρ( ) =, & ρ( ) = v, () ρ whee h is he agniue of he veco efine in Eq. ().. Obain he expession of he eal funcion θ fo he secon equaion in (). 4. Expess veco ρ, he soluion o he iniial value poble (9). 5. Deeine veco, he soluion o he iniial value poble (8), by using: = R ρ () Noe ha he above algoih ay be consiee jus a suggesion o appoach he iniial value poble (9). If veco ρ is obaine by whaeve eans, only he las wo seps of he above algoih us be execue, which in fac ansfe he poble fo he ineial fae back o he oaing fae. The nex subsecions pesen soe ipoan applicaions o his eho. 5.. The Keple poble in oaing efeence faes The iniial value poble ha oels he oion has he fo: µ && + & + ( ) + & + =, ( ) =, & ( ) = v, (4) whee µ is he gaviaional paaee of he aacion cene. Following Theoe, he soluion o he iniial value poble (4) is obaine by applying he enso opeao R o he soluion o he iniial value poble: µ ρ&& + ρ=, ρ ( ) =, ρ& ( ) = v+, (5) ρ Noe ha Eq. (5) oels he Kepleian oion wih espec o an ineial fae, an is soluion is consiee o be known. A copehensive appoach o he Keple poble in oaing efeence faes ay be foun in [5]. 5.. The Foucaul Penulu The iniial value poble ha oels he Foucaul Penulu oion is: && + & + + & + =, =, & = v, (6) whee is he pulsaion of he penulu (i epens on is lengh an he gaviaional acceleaion a he expeien place). Hee veco is consan an i oels he Eah oaion aoun is pole axis. Following Theoe, he soluion o he iniial value poble (6) is obaine by applying he enso opeao R o he soluion o he iniial value poble: ρ&& + ρ=, ρ =, ρ& = v +, (7) The aazing qualiy of his eho is ha in he paicula case of he Foucaul Penulu oion (which in os Theoeical Mechanics exbooks is consiee o lack a close fo soluion) offes a siple way o solve i by eucing i o he eleenay poble of a haonic oscillao escibe by he iniial value poble (7). The soluion o he iniial value poble (7) is: + ρ( ) = cos ( ) + v sin ( ), (8) so he soluion o he iniial value poble (6) is: v+ ( ) = cos ( ) ( R ) + sin ( ) R (9) By aking ino accoun ha he angula velociy of he oaing fae is consan, hen he enso opeao expesse like: { } ( ) ( ) R = I sin u% + cos u% () 44 R is

5 whee u is he uni veco associae o veco. The explici soluion o he Foucaul Penulu oion escibe by he iniial value poble (6) ay be wien as: { % { }% } ( ) cos ( ) sin ( ) cos ( ) = u + u + v+ v+ v+ () + sin ( ) sin ( ) + { cos ( ) } u% u% A copehensive appoach of he Foucaul penulu like oion ay be foun in [ ]. 5.. The elaive oion of saellies onsie wo spacecafs obiing aoun he sae aacion cene une he influence of an unpeube gaviaional fiel. One of he will be efee as he hief, an he ohe one he Depuy. The efeence fae whee he oion of he Depuy saellie is suie is nae LVLH (Local-Veical-Local-Hoizonal) an is axes ae oiene as i follows: he Ox axis has he sae oienaion as he ineial posiion veco of he hief saellie, he Oz axis has he sae iecion an sense as he angula oenu of he ineial obi of he hief saellie; he Oy axis coplees a igh hane ohogonal fae. This fae has he oigin in he hief saellie cene of ass. Denoe by he posiion veco of he aacion cene wih espec o he LVLH fae an enoe by he angula velociy of he LVLH fae. The iniial value poble ha oels he oion of he Depuy saellie wih espec o LVLH is: µ µ && + & + ( ) + & + ( + ), = ( ) =, & ( ) = v () + The poble is of gea inees fo his appoach, since i is he sae ype as Eq. (). The foce in he igh han e epens only of ie an he posiion veco. A iec appoach by using he enso insuen inouce in Secion 4 oes no lea o saisfacoy esuls, bu hee exis a vey ingenuous way o solve he poble. The iniial value poble () ay be seen as he iffeence beween wo iniial value pobles of he sae ype, having iffeen iniial coniions. One is: µ v && + & + ( ) + & + =, ( ) = +, & ( ) = + v () an he ohe one is: µ && + & + ( ) + & + =, ( ) =, & ( ) = v (4) whee veco epesens he posiion veco of he aacion cene wih espec o LVLH a he iniial oen of ie an v epesens he iniial velociy of he hief saellie wih espec o an Eah cenee ineial fae. The soluion o Eq. (4) is, an i oels a ecilinea oion. Fo a geoeical poin of view, i oels he oion of he hief saellie wih espec o a LVLH fae oiginae in he aacion cene. The ineial obi of he hief saellie is associae o a Kepleian oion, so veco igh be expesse as: whee p = +, e cos f 45 p is he seilaus ecu, e is he ecceniciy an f f ( ) (5) = is he ue anoaly, all associae o he ineial obi of he hief saellie. Also enoe by h he specific angula oenu of he ineial obi of he hief saellie. The angula velociy of he LVLH efeence fae ay be expesse as: h = ( + e cos ) f (5) p By applying he enso insuen pesene befoe in his pape, we ay sae ha he soluion o he iniial value poble () is: p = R, (6) + e cos f whee is he soluion o he iniial value poble: µ && + =, ( ) = +, & ( ) = v + v+, (7) The esul expesse in Eq. (6) allows us o eeine he elaive velociy of he Depuy saellie wih espec o LVLH. I is: e h sin f v= R & R % (8) p A copehensive analysis of he elaive obial oion by using his poceue ay be foun in [8,9,]. The soluion fo he elaive obial oion pesene hee is a genealizaion of seveal paicula soluions fo he siuaions when he efeence ajecoy is cicula [] an when he efeence ajecoy is ellipic [-4]

6 6. ONLUSIONS The pape pesens a geneal eho fo he suy of he oion in a non-ineial efeence fae. This eho is base on pope ohogonal an skew-syeic enso value funcions, which ae inouce by he Daboux equaion. The case of he oion in a cenal foce fiel wih espec o a oaing efeence fae is suie an an algoih fo eeining he law of oion in his siuaion is popose. Thee applicaions ae pesene: he Keple poble in oaing efeence faes, he Foucaul Penulu an he elaive oion of saellies. The soluion offee fo he elaive obial oion is a genealizaion of he soluions offee by lohessy an Wilshie fo he cicula efeence ajecoy an by Lawen an Tschaune Hepel fo he ellipic efeence ajecoy. REFERENES [] D. onuache, New Sybolic Mehos in he Suy of Dynaic Syses (in Roanian), PhD hesis, Technical Univesiy "Gheoghe Asachi" Iasi, 995. [] Daboux, G., Leçons su la Théoie Généale es Sufaces e les Applicaions Géoeiques u alcul Infiniesial, Gauhie-Villas, Pais, 887, Live, hap.. [] onuache, D., Mainuşi, V., A Tensoial Explici Soluion o Daboux Equaion, The n Inenaional onfeence Avance onceps in Mechanical Engineeing, Iasi,5-7 June 6. [4] Teooescu, P.P., Mechanical Syses, lassical Moels. Volue : Paicle Mechanics, Spinge, Belin, 7. [5] onuache D., Mainusi, V., Keple s Poble in Roaing Refeence Faes. Pa I : Pie Inegals, Vecoial Regulaizaion, AIAA Jounal of Guiance, onol an Dynaics, Vol., no., 7, pp. 9-. [6] onuache D., Mainusi, V., Foucaul Penulu-like Pobles: A Tensoial Appoach, Inenaional Jounal of Non-linea Mechanics, vol. 4, issue 8, 8, pp [7] Angeles, J., Funaenals of Roboic Mechanical Syses: Theoy, Mehos, an Algoihs, Spinge, 6. [8] onuache D., Mainusi, V., Relaive Spacecaf Moion in a enal Foce Fiel, AIAA Jounal of Guiance, onol, an Dynaics, vol., no., 7, pp [9] onuache, D.; Mainusi, V., Exac soluion o he elaive obial oion in a cenal foce fiel, IEEE/AIAA n Inenaional Syposiu on Syses an onol in Aeospace an Asonauics, Shenzhen, hina, - Dec. 8, DOI:.9/ISSAA [] onuache, D., Mainuşi, V.,. An Exac Soluion o Saellie Relaive Obial Moion, s Inenaional onfeence opuaional Mechanics an Viual Engineeing OME, Başov, 5. [] W. H. lohessy an R. S. Wilshie, Teinal Guiance Syse fo Saellie Renezvous, Jounal of he Aeospace Sciences, Vol. 7, No. 9, Sep. 96, pp [] D. F. Lawen, Opial Tajecoies fo Space Navigaion, Buewoh, Lonon, 96. [] J. Tschaune, P. Hepel, Opiale Beschleunigeungspogae fu as Renezvous-Manove, Aca Asonauica, 964, pp [4] J. Tschaune, The ellipic obi enezvous, AIAA 4h Aeospace Sciences Meeing, June 7-9, Los Angeles, A,

Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Sharif University of Technology - CEDRA By: Professor Ali Meghdari Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai Pupose: o exen he Enegy appoach in eiving euaions of oion i.e. Lagange s Meho fo Mechanical Syses. opics: Genealize Cooinaes Lagangian Euaion