AN EFFICIENT INTEGRAL METHOD FOR THE COMPUTATION OF THE BODIES MOTION IN ELECTROMAGNETIC FIELD

AN EFFICIENT INTEGRAL METHOD FOR THE COMPUTATION OF THE BODIES MOTION IN ELECTROMAGNETIC FIELD GEORGE-MARIAN VASILESCU, MIHAI MARICARU, BOGDAN DUMITRU VĂRĂTICEANU, MARIUS AUREL COSTEA Key wods: Eddy cuen inegal equaion, Moving bodies evoluion, Eleco-mechanic coupled poblem A mehod fo compuing he moion of conducing bodies in elecomagneic field is pesened. The field poblem is a 3D eddy cuen poblem and is solved by means of an inegal mehod. The inegal eddy cuen fomulaion equies only he disceizaion of he cuen souces domains and conducing bodies and conains no explici velociy ems. The mechanical poblem is solved by means of a pedico-coeco mehod. Boh poblems ae coupled ogehe in ode o obain he ime evoluion of he conducing body.. INTRODUCTION This pape pesens a mehod fo solving he eleco-mechanic coupled poblem in which he moion of bodies akes place unde he influence of elecomagneic foces. An efficien field compuaion pogam is vey useful fo he fas analysis of a gea numbe of eleco-mechanical devices, such as ail launches, oaing elecical machines, eddy cuen beaking sysems, magneic leviaion sysems. The majo difficulies in solving such poblems aise fom he fac ha he soluion of he 3D eddy cuen poblem depends on he speed and he posiion of he moving bodies while he moion of he bodies depends on he magneic foces, hus on he soluion of he field poblem. As i will be shown nex, his soluion is obained in an ieaive manne. A each ieaion he posiion of he moving body is eevaluaed and a new field poblem is solved. This can lead o subsanial Poliehnica Univesiy of Buchaes, Depamen of Elecical Engineeing, 33, Spl. Independenei, Buchaes, 642, Romania, E-mail: vasilescu@elh.pub.o Rev. Roum. Sci. Techn. Élecoechn. e Éneg., 57, 2, p. 44 53, Bucaes, 22

2 An inegal mehod fo compuaion of bodies moion 45 execuion imes. I is impoan ha he mehods used fo solving such poblems obain as possible an accuae soluion in a mos efficien manne. Vaious mehods fo solving eddy cuen poblems in which he moion of he bodies is imposed aleady exis. The finie elemen mehod (FEM) populaiy sems fom he fac ha i woks wih spase maices and is implemenaion is elaively easy. Applying his mehod fo poblems wih moving bodies, howeve, poses some difficulies. The main poblem is ha, wih each new posiion of he analyzed bodies, a new paial o full emeshing is equied []. This pocess is ofen ineffecive and can lead o subsanial execuion imes, especially in he case of 3D poblems. In ode o avoid he cosly pocess of emeshing, vaious ohe FEM echniques have been developed. Seveal of hese echniques ae oulined in [2]. The hybid mehod FEM-BEM has gea advanages when dealing wih field poblems wih moving bodies. Because i can wok wih unbounded domains he ai egion is no disceized and he bodies can move feely a any disance. The field poblem in ai domain is solved by using he inegal bounday elemen (BEM) mehod. The conducing egion field poblem is solved employing FEM. The esuled wo soluions ae hen coupled ogehe by aking ino accoun he bounday condiions on he suface of he bodies [3]. The mehods descibed in [3 4] do no impose he ype of he moion and he conducing bodies move feely unde he influence of he magneic foce. In [4] he bodies ae eaed wih FEM in hei own local sysem of coodinaes, while he suounding ai is eaed wih BEM in a global sysem of efeence. This mehod, howeve, seems o be laboious due o he fac ha he enie sysem of equaions conains explicily he velociy em. Fuhemoe, he implemenaion of he BEM echnique fo 3D domains migh be poblemaic. The inegal mehod fo 3D eddy cuen poblems fis descibed in [5] has seveal impoan advanages. This mehod only equies he disceizaion of he conducing bodies and of he souces and also eas unbounded domains. The Maxwell-Hez equaion ae wien in he local sysem of coodinaes of each body, hus no explici velociy ems ae equied. Anohe impoan advanage of his mehod is ha i does no equie any emeshing fo each new posiion of he bodies. In [6] he mehod is used fo solving a poblem wih moving bodies, bu in his case he moion is imposed. Alhough he mehod pesened in [5] does no deal wih nonlinea media i can be exended by using he polaizaion fixed poin mehod [7 8]. This ieaive mehod always insues he convege [9 ]. Alhough is speed of convegence is no as fas as ohe simila mehods, like Newon- Raphson, hee ae vaious echniques ha can be used o incease i []. The mehod descibed in his pape couples he inegal mehod s field soluion wih he soluion of he moion poblem. The moion poblem is calculaed by means of a pedico-coeco mehod.

46 Geoge-Maian Vasilescu e al. 3 2. THE INTEGRAL METHOD 2.. PROBLEM FORMULATION Le hee be a numbe of N conducing bodies, each moving wih a ceain unknown v k velociy. In he local fame of each conducing body we can wie A E + gadv, () whee E is he elecic field inensiy, A is he magneic veco poenial, and V is a scala poenial. The consiuive elaionship of he magneic field is given by B µ H, (2) whee B is he flux densiy, H he magneic field inensiy and µ is he pemeabiliy of fee space. Fom (2) and Ampèe s law we obain: ( J J ) o o A + µ, (3) whee J is eddy cuen densiy and J is imposed cuen densiy. The soluion fo his equaion is given by he Bio-Sava fomula µ J A d + A 4π v, (4) Ω C whee A is he magneic veco poenial given by he imposed cuens in he coils. Fom () and (4) he eddy cuen inegal equaion is obained as µ d J d ρj + dv + gadv A 4π, (5) Ω C whee ρ is he elecic esisiviy. The elecic scala poenial T is defined as o T J. (6) The cuen densiy nomal componen is null on he conducing domains Ω C : J n n o T. (7)

4 An inegal mehod fo compuaion of bodies moion 47 2.2. NUMERICAL APPROACH The cuen densiy J is expessed in ems of coee edge shape funcions N k [5]: n J ot i k () on k. (8) k By using he Galekin echnique on (5) he weak fomulaion esuls µ d d ρo o kdv + A T N on k otdvdv on k dv. 4π (9) Ω ΩΩ Ω The final equaion can be wien as d R I + ( L I ) φ, () d whee he vecos I and φ ae I ( i i i ) T, ( φ φ φ ) T, 2,..., n φ. (), 2,..., n The field souces have been chosen as hin conducing wies wih an imposed cuen i. Fo hese he magneic veco poenial is given by he Bio-Sava law iµ l A d 4π. (2) Assuming ha we have M disinc coils, he φ enies can be wien as φ k Ω Γ M iµ iµ v dl k v dl d on d on k π π d 4 4 Ω Γ ν Ω Γ on k A v. (3) The enies of he maices R and L ae given by L ik Rik ρ o i on kdv Ω ΩΩ N, (4) µ on i on kdvdv. 4π One of he main advanages of he poposed mehod is ha fo each new posiion of he bodies, only ceain pas of L and φ will change. The R maix will emain unchanged. ν (5)

48 Geoge-Maian Vasilescu e al. 5 Fom (5) we can see ha he only L ik coefficiens ha need o be updaed ae he ones belonging o conducing bodies ha have changed hei elaive posiion. The φ k ems (see (3)) have simila popeies. If he conducing bodies do no change hei elaive posiions he L maix will emain unchanged. In his case he execuion ime will be dasically deceased, due o he fac ha compuing he L ik coefficiens is a cosly opeaion. Equaion () is solved by inegaing i ove he ime ineval ( LI ) R I + d dφ. (6) Assuming ha I has a linea vaiaion beween and, (6) becomes R 2 ( I + I ) + L I L I φ + φ. (7) The pevious elaion holds fo any consecuive n and n+ momens and can be ewien in he following fom R + Ln+ In+ R + Ln In φ n+ + φn, (8) 2 2 whee he I n+ veco is he sysem s unknown. In ode o solve (8) fo he cuen momen n+ we need o know he soluion I n fom he pevious momen n. 3. THE PREDICTOR-CORECTOR METHOD In his secion he pedico-coeco algoihm fo inegaing Newon s equaions of moion is pesened. Only anslaion is analyzed. By using his mehod we can deemine he ajecoy of a igid body whose iniial velociy and posiion ae known and which moves unde he influence of foces which vay in ime and space. Only he Oz diecion will be analyzed, he esuls can be easily exended o any ohe diecion. Le hee be a body of mass m fo which we know is iniial posiion and speed. Le hee be F he vaying foce ha acs upon i. We seek o deemine he ajecoy of ha body afe a ceain ime T has elapsed. Le hee be he cuen ime sep. We assume ha a he momen he body is locaed a he z posiion, has he velociy v and is unde he influence of a foce F. Having known all his daa we ae equied o compue he posiion z of he body a he momen >. Bu if he foce ha acs upon he body vaies wih posiion, in ode o compue z, we need o know befoehand he value of ha foce a z, which is no always possible.

6 An inegal mehod fo compuaion of bodies moion 49 The pedico-coeco mehod solves his pedicamen by calculaing he posiion a he cuen ime sep in muliple sages. Fisly, i assumes ha he foce F acing a is consan and based on his pedics a empoay z. Then eevaluaes he foce ha acs upon he body a z and coecs his posiion ino a new z, by assuming, his ime, ha he foce has a linea vaiaion beween z and z. The coecion ieaions ae epeaed unil he disance beween wo consecuive coeced posiions become sufficienly small. Once he newly posiion has been deemined we can move fuhe o ime sep 2. In ode o deemine he expessions needed fo applying his mehod we sa off fom Newon s second law of moion and he definiion fo acceleaion and speed F ma, (9) dv a, (2) dz v, (2) whee F is he foce ha acs upon he body of mass m on he Oz diecion. Le he iniial ime fo which we know z and v and be any momen in ime wih >. Replacing (2) in (9) and inegaing we obain v m. (22) () F() τ dτ + v By solving (2) we can compue he posiion a he momen as We denoe wih I m z () v( τ) F d τ + z. (23) () τ and analyically evaluae he inegals fo he wo cases. If he foce does no vay in ime (24) becomes I F τ τ F m d m dτ and J I() τ d τ, (24) m and J F( τ ) 2 2 dτ F, (25) m

5 Geoge-Maian Vasilescu e al. 7 whee is he ime sep. If he foce has a linea vaiaion in ime (24) becomes I m 2m 2 F() τ dτ ( F + F ) and J [ F( ) + F( )] n+ 2 n. (26) 6m Relaionships (22) and (23) hold fo any consecuive n and n+ momens and can be wien, consideing (24), as: vn + vn + I and zn + zn + vnh + J, (27) whee he n+ index denoes he cuen momen and n he pevious momen. In ode o compue he speed and he posiion fo he cuen momen n+ we need o know hei values a he pevious momen n. 4. THE COUPLED PROBLEM. THE ALGORITHM In his secion we descibe he way he eddy cuen poblem is coupled wih he mechanical poblem. The algoihm is he following: ) Fo he ime n he eddy cuen poblem (8) is solved. 2) The magneic foce F fo he cuen posiion z is compued. 3) The posiion z is pediced, using (25), (27), by assuming F is consan. 4) The body is anslaed o z : 4.) The eddy cuen poblem (8) is solved fo he cuen posiion z. 4.2) The magneic foce F fo he cuen posiion z is compued. 4.3) The posiion z is coeced, using (26), (27), by assuming a linea F F vaiaion. 4.4) The body is anslaed o z. 4.5) The execuion coninues again fom 4. unil he diffeence beween wo consecuive coeced z values is small enough. 5) Nex sep (n+). The numbe of ieaions he pedico-coeco mehod needs o obain a soluion is usually small, ypically wo. The magneic foce acing upon he bodies is calculaed by inegaing Maxwell sess enso on a close suface aound he body. 5. NUMERICAL RESULTS ELECTROMAGNETIC LEVITATION Elecomagneic leviaion occus when he lifing foce, caused by eddy cuens inside a conducing body due o vaying magneic field, balances he foce of gaviy. This is he case fo he device pesened in Fig..

8 An inegal mehod fo compuaion of bodies moion 5 Fig. Device fo magneic leviaion. A cylindical aluminum (σ 34 MS/m, m.7 kg) plae is locaed above wo hin cicula wies hough which a ceain cuen passes. All hee objecs ae aligned coaxially. In his case he wo hin wies ac as an appoximaion of some coils wih ceain windings. Boh coils cay a.c. cuens in opposie diecions. The ms value of he oue coil cuen is I ou 8 ka, while fo he inne coil hee diffeen values have been chosen I in 3 ka, I in ka, and I in 8 ka, especively. The cuen fequency is f 5 Hz. The body s iniial disance o he coils is δ 9.8 mm. The coils ae locaed a posiion z. A he iniial momen ( ) he plae is eleased and, unde he influence of he gaviaional foce, descends owads he coils. As he plae is geing close o he field souce, eddy cuens will be induced inside i due o is moion and he vaying imposed cuens fom he coils. The magneic foce will oppose he gaviaional foce and will y o elevae he plae. In Fig. 2 we can obseve he oscillaing behavio of he plae. As he inne cuen inceases, he displacemen of he cylinde will be geae, and he necessay ime fo is sabilizaion will incease. The conducing body will have a highe aliude a which i will each equilibium as he inne cuen will incease. I is woh menioning ha in addiion o he oscillaing movemen epesened in Fig. 2, he conducing body also exhibis a vibaion moion a a fequency of f 2 Hz.

52 Geoge-Maian Vasilescu e al. 9 Posiion [mm] 2 8 6 4 I in 3 ka; I ex 8 ka I in ka; I ex 8 ka I in 8 ka; I ex 8 ka 2.2.4.6.8.2.4 Time [s] Fig. 2 The vaiaion of he posiion. 6. CONCLUSIONS The pesened mehod consiss in coupling he eddy cuen poblem wih he mechanical poblem. The eddy cuen poblem is solved by means of an inegal mehod, while he soluion fo he mechanical poblem is obained wih a pedico-coeco mehod. The mehod does no equie he disceizaion of he ai egion and allows fo an easy eamen of unbounded domains. Alhough i opeaes wih full maices, only ceain pas of hem and he fee ems will be updaed fo each new posiion of he bodies. The mehod woks especially fas when he conducing bodies don change hei elaive posiion and move only in elaion wih he field souces. Seveal numeical esuls have been obained and inepeed fo an illusaive example epesening a leviaion device. ACKNOWLEDGEMENTS The wok has been funded by he Secoal Opeaional Pogamme Human Resouces Developmen 27-23 of he Romanian Minisy of Labou, Family and Social Poecion hough he Financial Ageemen POSDRU/6/.5/S/6. Received on Januay, 22

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