ZBORNÍK VEDECKÝCH PRÁC
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1 ZVÄWK23 ZBORNÍK VEDECKÝCH PRÁC STROJNÍcKEJ FA KULTY SLOVENSKEJ TECHNCKEJ UNVERZTY V BRATSLAVE 1995 Vplyv pólového usporiadania magnetov na magnetické ložisko J. VLNKA. D. ŠEVČOVČ. W. LABABNEH Anotácia V č l ánku sú zhrnuté teoretické pojednania o možnom usporiadaní pčlovania magnetov v magnetickom ložisku a ich vplyv na vytváranie harmonických spektier. Zoz na m použitých oz načeni bn f(x) ao. an - Fourierov koeficient rozvoja - jednotkov á funkcia - Fouriero v koeficient rozvoja - uhol V súčasnos ti l. Úvod pri hľadaní nových techni ckých riešeni v strojárstve je jedna z ciest vývoj nekonvenčných ložisk. Napriek vývoju klasických ložisk. ktorými je neustále venovaná veľká pozorno sť. nie sú v dostupnej literatúre k dispozicii dostatočn é inform ácie týkajúce sa perspektívných magnetických ložisk. V literatúre chýbajú presnejšie údaje o konštrukcii. o usporiadani magnetov magnetického ložiska a presnejšfch numerických metódach. Tiež možno v našich podmienkách konštatovať nedostatok praktických skúseností so stavbou a prevádzkou magnetických ložisk. Cieľom tejto práce je poukázať na možné konštrukčné usporiadanie magnetov v mag netickom ložisku. Konštrukčné usporiadanie mag netických pólov statora mag netického ložiska môže byť nasiedovné: SSJJ. JSJ. SSS a JJJJ, obr.l. Magnetické póly môžu byť usporiadané striedavo SSJJSSJJ. tento typ nazývame skrátene SSJJ. resp. SJSJSJSJ. nazývaného skrátene SJSJ. Ak sú magnetické póly usporiadané identicky. hovoríme ng. Jáo Vlnka. Katedra strojárskej aplikovanej elektrotechniky. Strojnícka fakulta sro.nám. Slobody Bratislava. RNDr. Daniel Ševčov ič, CSc. KMA Matematicko - fyzikálna fakulta UK Bratislava. PhD. Wasef Lababneh, Amman University. College for Applied Engineering. Jordan 203
2 8 J 8 J J 8,-, 8 J J t \8 -s\ r--- ".. J 8 L...J tj a b 8 8 J 8 8 J c d Obr.1 Typy magnetick-ého usporiadania pólov magnetického ložiske. Striedavé usporiadanie pólov: a) typ SSlJ. b) typ SJSJ. Monop6lové usporiadanie pólov : c) typ SSSS. d) typ JJJJ Fig.1 Types of Magnetic poles arrangement. Altemating pcles errangement: a) type SSNN. b) type N5NS. Monopole (identic) poles anangement: c) type NNNN. d) type SSSS o monop6lovom usporiadaní magnetických p6lov, usporiadanie môže byť monop61ne SSSSSSSS, tento typ skr átene nazývame SSSS, resp. monop , je to monop61 skrátene nazývaný monop61 typu J11J obr.1. Vzniklé magnetické polia z týchto p6lov, môžeme analyzov ať pomocou Fourierovho radu. Fourierovu analýzu robíme pre prípad, ak nie je braný do úvahy rozptylový tok. 204
3 Z rozloženia magnetických polí za predpokladu, že neuvažujeme rozptylový tok a magnetickú indukciu uvažujeme za jednotkovú, je vidieť, že funkcia f(x) je definovaná ako tvar signálu s jednotkovými skokmi. Ak poznáme napr. analytické vyjadrenie periodickej funkcie y = f(x), ktorá má periodu t, potom platí vzťah f(x)=f- >" sin n x 11=1 (l) kde >,, =~J f(x) sin nx dx 1t o (2) Poznamenajme, že horeuvedený rad konverguje bodovo k funkcii f(x) s výnimkou bodov nespojitosti funkcie f(x) (pozri [7)). Ak hľadáme koeficienty Fourierovho rozvoja pre usporiadanie pólov v našom prípade, môžu nastať nasledovné prípady, ktoré je vidi eť na obr. 2.. '1('"18 ''m.'1' /4 " o "'18 'lt<... :z': - 511".l![. s; bd sn o, SSJJ -e /1"121 ~ SJSJ.'1 -'li T...-.,.,. '1",.- io e.ssss <.. pt'l"l2l -~ ~!:, "io T ~.".....,., d, JJJJ pl'l"r21..,. 'TK- T -CilS "O ".'f.:jr.,..'f... T io iô 16 n nepómo - '1'1"12n :..''11 "'PÓrN: _..-,t. tli "...14 pá.-no.'1'"",....../ Obr. 2 Rozloženie magnetických pou nacelom intervale. Striedavé usporiadanie pólov: a) typ SSlJ, b) typ SJSJ. Monop6lové usporiadanie pólov: c) typ SSSS. d) typ JJJJ Fig. 2 Magnetic Field Distribution within the Whole Range. Altemating poles arrangement: a) type SSNN, b) type NSNS. Monopole pola arrangement: c) type NNNN, d) type SSSS 205
4 2. Prípad striedav ého usporiadania pólov typu SSJJ Počltame koeficien ty Fourierov čho rozvoja v opakovateľnom intervale <-lt/2 -lt/2>. V prípade, že tvar magnetického poľa je nepáma funkcia, ao a a" sú O. To znamená, že koeficient Fourierovho rozvoja možno vyj adriť ako integrál ~ b, = J.. r f(x) sin 2nx dx lt -", (3) (n =1, 2,3...) kde f(x)=! bo sin 2nx (4) vyplýva, že f(x) = O = 1 = l = - = - X E <O, a> <a, rrl4- a> <rrl4 - a, rrl4 + a> <rrl4 + a, lt/2 - a> <lt/2 + a, lt/2 + a> <lt/2 + a, 3/411 - a> <3/411 - a, a> <3/411 + a, 11- a> <1t- a. 7t> Potom koeficient ~ 11bo =r f(x) sin 2nxdx +rf(x) sin 2nxdx =- -!.. sin lll/4. 2 sin «a - 1l/8)n). ros n1l/8 -", o 2n (5) potom 4 bo = - - 1ln sin nlt/2. cos nrrl4. sin(a -11/8) 2n (6) 206
5 3. Pre prípad striedavého usporiadania pólov typu SJSJ Počítame koeficienty Fourierovho rozvoja v rozsahu <-ll/4 -ll/4>. Keď je tvar signálu nepáma funkcia au a an sú O. Koeficient Fourierovho rozvoja možno vyjadriť, ako integrál % b" = ~ r f(x) sin 4nx dx 11 -"/. (7) Potom f(x)=! b" sin 4nx n=1 (8) vyplýva, že: f(x) = O = l = -l X E <O. o» <a,ll/4 - a> <ll/4, ll/4> <O, - a> <- a, - (ll/4 - a» <-(ll/4 - a), - ll/4> Potom koeficient b n rozpíšeme: ~ b"=[ r (-)sin4nxdx+ 4 {,-:a),-4«j a 1 mt mt sin 4nx dx] = - - sin - sin (4na - -) n 2 2 (9) potom 4 nn nn b" = - - sin - sin (4na - -) n 1l 2 2 (O) 207
6 4. Prípad monopólového usp oriadania pólov typu SSSS Počítame Fourierové koefi cienty rozvoja v rozsahu <- 1t/4... 7t/4>. V prípade, že tvar magnetického poľa je párna funkcia. bo. bn sú O. Koeficient Fourierovho rozvoja možno vyjad riť ako integrál an = ~ 1t % rf(x) cos 4nx dx -'1. ( l) pren >!. Funkciu periodického opakovania možno vyjadriť pre párnu funkciu f(x) = 51.+! an cos 4nx 2 n=1 (12) kde 51. %r 8a f(x)dx = '1. 1t (13) vyplýva. že funkcia má priebeh: f(x) X E <O. a> = l <a.1t/4 - a> <1t/4 - a. 1t/4> <O, - a> = l <- a. - (1t/4 - a» <- (1t/4- a ). -1t/4 > Potom pre koeficient an dostávame 'r l n/ nn M 4 -n / / 4+a a n 2 2 r -r--a., 1t = + cos 4nx dx + cos 4nx dx = - sin( ) cos - (14) 208
7 potom -4 mt mt an= - cos - sin (4na - - ) ml 2 2 (15) 5. Prípad monopólového usporíadania pólov typu JJJJ Počítame Fourierové koeficienty rozvoja v rozsahu <- ll/4 - ll/4>. V prípade. že tvar magnetického poľa je párna funkcia. bu. b, sú O. Potom vyplýva. že funkcia f(x) má priebeh f(x) = O = - = - X E <O. ll> <a. ll/4 - ll> <ll/4 - a. ll/4> <O. - ll> <- a. -(ll/4 - a» <- (ll/4 - a), -ll/4> "o =_rf(x) dx =_8a_ '1. (16) Potom pre koefi cient dostávame -an = 4 r cos 4nx dx - -%+a 'ý. r cos 4nx dx = 2. sin (nll/2-4na) cos nll/2 a n (17) potom 4 mt mt on = - cos - sin (4na - -) 7ln 2 2 (18) Pri zo h ľadne ní pomeru šírky pólu a šírky drážky vyjadrenú koeficientom a dostávame upravené rovnice (6), (O). (15), (18). Pre hodnotu a = / 16 pre prípad SSJJ 209
8 4 mt mt 1t 1t 4 n1t n1t n1t b, ==-- sin- C05- sine- - - ) 2n == - sin- C05- sinnn mt (19) pre prípad SS 4 mt mt mt 4 mt mt b n ==-- sin- sin(- - -) = - sin- sinmt 2 4 2mt 24 (20) pre prípad SSSS -4 a n == mt mt mt n 4 nn mt C05- sin(- - - ) == - cos- sin mt 24 (21) pre prípad ljjj 4 mt an== - cos - mt 2 nn nn 4 nn nn sin(- - - ) =-- cos- sin-r-r- 4 2 mt 2 4 (22) 6. Záver Matematickou analýzou vzťah u (20) dostávame tvar funkcie, ktorý je v idieť na obr. 3, spektrum tejto funkcie je na obr. 4. Pre vzťah (21) je to obr. 5 a obr. 6 a pre vzťah (22) je to obr. 7 a obr. 8. Rozhodujúce harmonické sú v tab. 1. Tabu ľka 1 Usporiadanie magnetov Rozhodujúce harmonické SSJJ 2, 10, 22, 38 SS 4,12, 20, 28, 36 SSSS 8,24,40,52 l JJJ 8,24, 40,
9 cc l l , l uhol 1.5 (rad) - l Obr. 3 Rozloženie funkcie magnetických polí. pomocoumatematického výpočtu Fig.3 Distribution of MagneticFields Funclionby meansof MathematicCalculus 0,4 SSJJ Výpoč ílan é hodnoty 0,3 0,2 0, n- t áharmonická Obr. 4 Spektrum funkcie(19) Fig.4 Function 's Spectrum (19) 211
10 ,-. >AA..fi l V 'V ~ eg 0,5.,75 0, ,25 O,5 v 0,75 -O uhol (rad) -0,5!tv'-,, - l Obr. S Rozloženie funkcie magnetických poupomocou matematického výpočtu Fig. 5 Distribution of Magnetic Fields Function by meansof Mathemetic Calculus ~ ~ 0.4 SJSJ -e " --- Vypočitané hodnoty l ~O.3 CL E oo ~O.2 o s: -e o s: ~O.l.-.'" -ọ..q < la n-tá h.rmoni cká Obr. 6 Spektrum funkcie(20) Fig. 6 Function's Spectrum (20) 212
11 B(-) li'- vj l \..fvv'-v\'v 0,8 0,6 0,4 0,2 -o, 75-0,5 O,5V' O, 75 uhol (rad) Obr. 7 Rozloženie funkcie magnetických polí pomocou matematického výpočtu Fíg. 7 Distribution of Magnetic Fields Function by means of Mathematic Calculus >, "tj," +' ;:0. 3 " E oo oo ~O,2 "-e O s: ol " -o'" 5555 V ypo č ít ané hodnot y.q < 10 ZO n-tá harmon ická Obr. 8 Spektrum funkcie (2 1) a (22) Fig.8 Function's Spectrum (21) and(22) 213
12 Tabuľka l Usporiadanie magnetov Rozhodujúce harmonické ssn 2, 10,22,38 SJSJ 4,12,20,28,36 SSSS 8,24,40,52 JJJJ 8,24,40,52 Z uvedeného je vidno, že z predložených usporiadaní magnetických pólov v magnetickom ložisku, sa nám javilo ako najvýhodnejšie usporiadanie typu ssn, lebo tu sa premagnetováva rotor magnetického ložiska o polovic u menej, ako pri usporiadaní SJSJ. Ďalej je v idieť, že aj spektrálne rozloženie je rovno mernejšie, pri usporiadaní ssn, ako pri SJSJ (pri usporiadaní typu ssn ide po ve ľkej kladnej amplitúde okamžite veľká záporná amplitúda, čo nemožno povedať o usporiadaní typu SJSJ). Je vidieť, že "strafiť sa" pri prípade ssn do zlej frekvencie je ťaž šie, než pri SJSJ. Naviac absolútne hodnoty amplitúd pre ssn sú nižšie ako v SJSJ, resp. SSSS (nn) a rozhodujúce sú iba prvé dve l. j. 2, O. Obálku utlmenia harmonických nám robí vo vzťahoch člen /n. Obrázky a výpoč ty boli robené v programe MATHEMA nea. Literatúra ll) KLUVÁNEK,.-MšfK, L.-ŠVEC. M.: Matematika. Bratislava. Alfa VASLU, M. - JONEL, D. M.: Eddy-current Losses due to Altemating Magnetic Pulses. Rev. Roum. Sei. Techn. Électrotechn. et Énerg., Bucerest. p VASLU, M. - ENACHE. R. - KRATOCHVL, L. S.: Asymptouc evaluation of the penetration oľ periodic magnetic pulses into a condueting halfspace. Rev. Roum. Sei. Techn. - Électrotechn. et Énerg., 35, 1990,4, Bucarest, p (4] DABROWSK. M. - HEBENSTRET. J.: Analiza zjawisk elktromagnetycznych w plyce Stalowej przemieszczajacej sie w stacionamym pólu magnetycznym. Archiwumelektrotechnikt, XX XV, 1985, zeszyt 1/2, s ) MATSUMURA, F.: Methods of construction and ts Consideration on Magnetic Circuits in Magnetic Bearing. Prac. The 2nd Electromagnetics Symposium, 1990 p MATSUMURA, F. - HATAKE, K.: Relation between Magnetic Pole Arragement and Magnetic Less in Magnetic Bearing. n Magnetic Bearings, 2nd.ntemational Symposium on Magnetic Bearing, July , Tokyo, Japan, p ARDY, G.- ROGOSNSK, W.: Fourierovy lady. Praha, SNTL ALFA ) pfrko, Z.: Fourierova transformace.praha, SNTL The influence of pole distribution magnets on a magnetics bearing VLNKA, J., ŠEVČOV Č, D., LABABNEH, W. This paper analyses a periodic nonsinusoidal pulses and analyses ínrerrelauons of magnetic pole arrangements within radial magnetic bearing. 214
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