DOI: 1435/UBOVGU-17-1 TECHNISCHE MECHANIK, 37, -5, (17), 39-49 utte: June 16, 17 Influene of eletroagnet el Dapng on te Vraton Stalt of oft ounte Inuton Motor t Sleee Bearng, ae on a Multo Moel U Werner Te paper o an analtal raton oel for talt anal of oft ounte nuton otor t leee earng, epeall foung on te nfluene of eletroagnet fel apng on te lt of raton talt Te oel a ulto oel, onerng te eletroagnet nfluene nlung te eletroagnet fel apng effet, tffne an nternal ateral apng of te rotor truture, tffne an apng of te earng oung an en el, tffne an apng of te founaton eleent an tffne an apng of te ol fl of te leee earng Te a of te paper to unte all tee nfluene n a oel an to ere a proeure for alulatng te lt of raton talt, t onerng te eletroagnet fel apng effet Atonall, a nueral exaple preente, ere te nfluene of eletroagnet fel apng on te lt of raton talt on, a ell a te nfluene of te founaton eleent an of te nternal apng of te rotor Te proeure an onluon an alo e aopte nto fnte-eleent anal 1 Introuton or analng rotatng aner, t portant to kno te lt of raton ntalt, eg nfluene te ol fl of te leee earng an nternal ateral apng of te rotor (rotatng apng), ere Rao (1996), Ga () an Genta (5) In eletral ane, alo eletroagnet fel our, ouple te rotor to te tator an nfluene te raton eaor, ere Suk (197), rütent et al (198), Sen (199), St et al (1996), Arkko et al (), Holopanen (4) an Werner (6) Epeall n nuton otor an eletroagnet fel apng effet our, telf nfluene te eletroagnet fore an terefore te raton eaor, on rütent et al (198), Sen (199), Arkko et al (), Holopanen (4), Werner (6) an Werner (16) Soete large nuton otor (1MW-1MW) t leee earng an g pee (3 rp-6 rp) are ounte on oft founaton eleent (ruer eleent) to eouple te otor fro te founaton (g1) Hoeer, a oft ountng nfluene te raton eaor of te ane learl, eontrate Krk at al (1974), Ga et al (1984) an Werner (8) Stator t en el Saft en x Sleee earng Sleee earng En el Stator Rotor En el Sleee earng ounaton eleent ounaton eleent gure 1 Soft ounte nuton otor t leee earng Up to no, eletroagnet fel apng of nuton otor often not onere for raton anal n te nutr Terefore te a of te paper no to preent an analtal raton oel of a oft ounte nuton otor t leee earng an to ere a pratale eto to oner te nfluene of eletroagnet fel apng on te lt of raton talt 39
Eletroagnet el Dapng An eentrt eteen te aft entre pont W of te rotor an te entre of te tator ore lea to atonall eletroagnet eentrt fel, aue a raal agnet fore r n reton of te allet ar gap (g a) If te rotor angular frequen fferent to te angular frequene of te eentrt fel, tee fel nue oltage nto te rotor age, reue te agntue of orgn eentrt fel Terefore, te raal agnet fore r ll e reue an an atonal agnet fore t n tangental reton aue (g a) gure a) Eletroagnet fore; ) Magnet prng an agnet aper Te follong equaton are funaentall ae on te netgaton of rütent et al (198) an Sen (199) Te raal agnet fore an e ere an eletroagnet prng eleent (g ): >1: = p1 p-1 ; =1: = p1 (1) Were p te nuer of pole par of te otor, eletroagnet prng eleent tout eletroagnet fel apng, an p1 an p-1 are te real part of te oplex fel apng alue Wtout eletroagnet fel apng, te real part of te fel apng oeffent get p1 = p-1 =1 or -pole otor (p =1) te oponent p-1 get ero, negletng te oopolar flux >1: = p ; =1: = p () Were te rau of te tator ore, te lengt of te ore, te perealt of ar, te equalent agnet ar gap t epenng on eanal ar gap, aturaton effet an CARTER-fator, ere Sen (199), p te apltue of funaental ar gap fel Te tangental agnet fore an e ere te eletroagnet aper eleent (t ) (g ): >1: = p1 p-1 ; =1: = p1 (3) Were te rlng angular frequen of te rotor an p1 an p-1 are agnar part of te oplex fel apng alue Wtout eletroagnet fel apng, te fel apng oeffent get p1 = p-1 = or -pole otor (p =1) te oponent p-1 get alo ero, negletng te oopolar flux If eletroagnet fel apng a to e onere, te eletroagnet fel apng oeffent p1, p-1, p1, p-1 ae to e ere Wt te ornal nuer = ±1 for an eentrt fel ae, te eletroagnet fel apng oeffent an e alulate: ν =1 ν ν ; ν = ν ν ν (4) t: ν =,ν (,ν σ,ν) an = ν ν Sr,ν K,ν σ,ν,ν Were, te retane of a rotor ar an rng egent, te eletral tator angular frequen,, te an fel nutane of a rotor e,, te leakage nutane of a ar an rng egent,, te reng fator an, te ouplng fator, are all ere n etal Sen (199) 4
Te aron lp an e ere : ν = ν t: = (1 ) (5) Were te funaental lp of te nuton otor, te eletral tator angular frequen an ν/ are te angular frequene of te eentrt fel, epenng on te kn of eentrt: - Stat eentrt: ν = - Dna eentrt a a rular forar rl: ν = ± - Dna eentrt a a rular akar rl: ν = To oner te eletroagnet fel apng effet a agnet prng eleent an a agnet aper eleent, te oproe a to e ae, tat te alulaton of an ere onl ae on rular forar rl Wt t plfaton, te eletroagnet nfluene uppoe to e ger tan t ae n realt If te aolute alue of te aron lp ν g a t for a rular akar rl n onjunton t a all funaental lp uual for tea tate operaton te ape agnet prng get er all a ell a te agnet aper In t ae, te eentrt fel nue trongl n te rotor age an o te eentrt fel get learl reue ue to te aron rotor urrent Terefore, alulatng an ae on rular forar rl preent te ort ae regarng te egt of te eletroagnet nfluene, en onerng eletroagnet fel apng, on Werner (16) 3 Vraton Moel Te oel an enaneent to te oel, ere Werner (8), ere no eletroagnet fel apng a een onere, no nternal apng of te rotor, an no apng of earng oung an enel Te raton oel a plfe plane raton oel (plane, ) It ont of to an ae, te rotor a, onentrate n te aft entre pont W, an te tator a, a te nerta θ x an onentrate n te entre of grat S (g 3) Ol fl tffne C atrx Rotor a = f f Bearng oue an en el tffne an apng atrx: C = D = L f (L: Left e) Note: Magnet prng an aper not pture n tat e fl f fl W,V B S Ψ Rg ar, D = ϕ Boreole f of te ane feet, Stator: a an nerta Θ x R Saft journal a fr Ol fl apng atrx Bearng oung a f f f (R: Rgt e) gure 3 Vraton oel Rotor tffne an apng Rg ar Atonal ae are te a of te aft journal an te a of te earng oung, are aounte eparatel, otl to ao ero at te an agonal of te a atrx Due to ter lo a, ter nfluene onl argnal Te rotor rotate t te rotar angular frequen Te nerta oent of te Rg ar W S fr fl fr ounaton tffne atrx: f Cf = = f L fl, fr a) Ve fro x ) Ve fro V C C B D D Magnet apng atrx: D C R ϕ Magnet prng atrx: =,Θ x Rg ar ounaton apng atrx: f D = f V C C f B D D 41
rotor are not onere an terefore alo no groop effet Te aft journal entre pont V ere te oeent of te aft journal n te leee earng Te pont B potone n te axal le of te leee earng ell an ere te oeent of te earng oung Te rotor a lnke to te tator a te tffne an nternal apng of te rotor, te ol fl tffne atrx C an ol fl apng atrx D of te leee earng, are uppoe to e equal for te re e an te non-re e, an te earng oue an en el tffne an apng atrx C an D, are alo aue to e equal for te re e an te non-re e Te tator truture aue to e rg, opare to te oft founaton Te founaton tffne atrx C f an te founaton apng atrx D f onnet te tator feet, L (left e) an R (rgt e), to te groun Te founaton tffne an apng on te rgt e te ae a on te left e an te founaton tffne alue f an f an te founaton apng alue f an f are te alue for ea otor e Te eletroagnet onere te eletroagnet prng an aper atrx an, ere alo eletroagnet fel apng nlue All ue oornate te are fxe or erng te apng oeffent, t portant to oner ere, tat natural raton t te angular natural frequen ta of te rtal oe at te lt of talt t te rotar angular frequen ta a to e anale Terefore te rlng frequen eoe ta: = ta (6) Te ol fl tffne an apng oeffent j an j (, =, ) of te leee earng are alulate olng te Renol fferental equaton, on Tonl (1965), Glenke (1966) an Lun et al( 1987) = an = (7) Referrng to Ga (), te nternal ateral apng of te rotor ere ere te eanal lo fator tan of te rotor, epenng on te rlng angular frequen ( )= (8) Wt te tffne of te earng oung an en el ( ; ), te apng of te earng oung an en el ( ; ) ere alo ere te eanal lo fator tan : ( )= ; ( )= Te apng of te founaton eleent alo ere te eanal lo fator tan f : f( )= f f ; f( )= f f Te eletroagnet tffne oeffent an apng oeffent are epenng on te aron lp ν, an terefore alo epenng on te rlng angular frequen (ee apter ) = ( ) an = ( ) (11) 4 Mateatal Derpton To alulate te lt of talt, t neear to ere te oogenou fferental equaton Terefore, Aleert eto apple to te raton te n orer to ere te equaton of oton, leang to te part: a) rotor a te, ) journal te, ) earng oue te an ) tator a te (g 4) Beaue of te all plaeent of te tator a (,, ϕ ), opare to te enon of te ane (,,Ψ ), follong lnearaton pole: Te ane feet plaeent on te left e ( fl, fl ) an te ane feet plaeent on te rgt e ( fr, fr ) an e ere te plaeent of te tator (,, ϕ ) : fl = ; fr = ; fl = fr = (1) Wt te equlru of at ea ngle te, te oogenou fferental equaton an e ere: Coornate etor : (9) (1) = (13) [ ; ; ; ; ϕ ; ; ; ] T q = (14) ; 4
43 gure 4 Vraton oel, ut free nto u-te Ma atrx M: (15) = x Θ M Dapng atrx : (16) = f f f f f f D S W V B Rotor a te Journal te Bearng oue te Stator a te a) ) ) ) fl fl fr fr x ϕ Θ f fl f fl f fl f fl f fr f fr f fr f fr L R ϕ Note: Te negate ertal plaeent n, relatng to te oornate, onere te reton of te ertal fore n, o a to e ere n te fferental equaton :
Stffne atrx : (17) C = ( ) f ( ) f f ( ) f f f ( ) ( ) Due to te non-etr tffne atrx aue te ol fl an nternal ateral apng of te rotor an ue to a negate eletroagnet apng oeffent, te raton te ll get ntale, en te lt of talt exeee ( > ) Te lt of raton talt an e ere, nreang te rotar angular frequen, an analng te egenalue Wen a real part of one egenalue get ero, te lt of raton talt reae If te rotar angular frequen nreae furterore, te real part get pote an te raton te get ntale To fn te lt of talt, te oogenou fferental equaton a to e anale To alulate te egenalue, te tate-pae forulaton ue ere (t = ;Inex for oogenou): = Wt te forulaton =, te egenalue an e ere fro: At te lt of talt, te egenalue A (18) et[ ] = (19) of te rtal oe get: = ta =± ta () Knong te egenalue ta, te rtal oe ape at te lt of talt an e ere Te real part of te rtal egenalue ta ero an te rlng angular frequen ten ental to ta, le te rotor rotatng t ta It a to e onere, tat te oeffent,,, f, f, are epenng on te rlng angular frequen, a to e eterne Referrng to Werner (16) ere a rg ounte nuton otor a netgate, an terate oluton a to e eue, aorng to g 5, for a oft ounte nuton otor rt, te oeffent,,, f, f, an, epen on te rlng angular frequen, are et to ero, o tat onl te nonetr tffne atrx of te ol fl ll aue ntalt Te egenalue are alulate aorng to (19), epenng on te rotar angular frequen To ere te lt of talt, te rotar angular frequen nreae, tll te real part of an egenalue get ero At t lt, te rotar angular frequen ta,1 nex 1 for te frt alulaton an te natural angular frequen of te rtal oe ta,1 an e ere fro te egenalue Afterar te oeffent,,, f, f, an are alulate t = ta,1, an te lt of talt an te natural angular frequen are alulate agan, leang to ta, an ta, Ten te ne alulate natural angular frequen ta, ll e opare to te orgn natural angular frequen ta,1 If te rato le tan Δ an artrarl oen alue, te alulaton fne an ta = ta, an ta = ta, If te raton larger a te oen alue Δ, a ne alulaton eue an te oeffent,,, f, f, an are no alulate t = ta, Wt tee ne oeffent, te ne lt of talt ta,n1 an te natural angular frequen ta,n1 are ere Ten agan te ne alue ta,n1 opare to te preou alue ta, If te eaton tll too large, te loop n g 5 ll run troug, tll te eaton le tan Δ Wt t terate proe te lt of talt ta an e ere, a ell a te orreponng natural angular frequen ta of te rtal oe T proeure ueful, f te gget nfluene on te lt of talt aue te non-etr tffne atrx of te ol fl, uuall te ae for oon nuton otor t leee earng 44
,,, Loop pae? Loop gure 5 lo agra to ere te lt of talt for a oft ounte nuton otor 5 Nueral Exaple Te lt of raton talt for a -pole nuton otor (Tale 1), ounte on oft ruer eleent, an ren a onerter t ontant agnetaton, anale Te loa ane a pup an terefore te loa torque a quarat funton of te rotor pee n At rate pee, te loa torque ental to te rate torque of te otor Tale 1 -pole nuton otor, ounte on oft ruer eleent Data of te otor: - Rate poer N = 4 kw - Rate oltage N = 416 V ( ) - Rate frequen N = 615 1/ - Rate pee N = 36 rp - Rate torque N = 6366 N - Rate lp = 5 - Nuer of pole par = 1 - Unape agnet prng ontant = 7 1 kg/ - Mae of te tator = 74 kg - Ma nerta of te tator at te x-ax x = 155 kg - Ma of te rotor = 19 kg - Ma of te rotor aft journal = 1 kg - Ma of te earng oung = 8 kg - Stffne of te rotor = 18 1 kg/ - Hegt of te entre of grat S = 56 - Dtane eteen otor feet = 16 - Horontal tffne of earng oung an en el = 48 1 kg/ - Vertal tffne of earng oung an en el = 57 1 kg/ - Meanal lo fator of te earng oung an en el tan = 4 - Meanal lo fator of te rotor tan = 3 Data of te leee earng: - Bearng ell Clnral - Lurant ot grae ISO VG 3 - Nonal ore aeter / Bearng t = 1 / = 814 45
- Aent teperature / Suppl ol teperature T a = C / T n = 4 C - Mean relate earng learane (DIN 31698) Ψ = 16 Data of te founaton eleent (for ea otor e): - Vertal tffne for ea otor e = 1 kg/ - Horontal tffne for ea otor e = 1 1 kg/ - Meanal lo fator of te founaton eleent tan = 1 Te ol fl tffne an apng oeffent of te leee earng ae een alulate t te progra SBCALC fro RENK AG rt, te oeffent,,,,, an are et to ero aorng to te flo agra n gure 5 an te real part an te agnar part of te rtal egenalue anale Te rtal raton oe te oe, ll get ntale (g 6) gure 6 o, tat at a rotor pee of 4595 rp te real part of te rtal egenalue eoe ero Inreang te rotor pee, ll lea to a pote real part an terefore to ntalt So te lt of talt reae at a rotor pee of, =4595 rp (, =481 ra/) Real part an agnar part of te rtal egenalue[ra/] 315 31 35 3 95 9 85 8 75 7 65 6 55 5 1 5-5 9 1 15 18 1 4 7 3 33 36 39 4 45 48 51 54-1 -15 - -5-3 -35-4 -45-5 gure 6 Real an agnar part of te rtal egenalue, epenng on te rotor pee, t te ounar onton: = = = f = f = = =; Crtal oe ape at te lt of talt At t rotor pee te agnar part, repreent te rlng angular frequen of te rtal natural oe, eoe = ta,1 =553 ra/ Aorng to te flo agra t angular natural frequen ta,1 an no e ue to alulate te oeffent,,, f, f, an (Tale ) Tale Coeffent,,, f, f, an at a rotor pee of n ta,1 = 4595 rp an at a rlng angular frequen of ω = ω ta,1 = 553 ra/ Dapng ontant of te rotor Dapng ontant of earng oung an en el (orontal reton) Dapng ontant of earng oung an en el (ertal reton) Dapng ontant of founaton eleent for ea otor e (orontal reton) Dapng ontant of founaton eleent for ea otor e (ertal reton) Eletroagnet apng ontant Eletroagnet prng ontant Iagnar part Rotor pee n [rp] Real part tale ntale Moe ape of te rtal oe at te lt of talt ( Ort of te tator entre = 1 1 kg/ = 75 1 kg/ = 893 1 kg/ f = 39 1 kg/ f = 783 1 kg/ = 617 kg/ = 48 1 kg/ Te large fferene eteen rotor angular frequen ta,1 =481 ra/ an te rlng angular frequen = ta,1 =553 ra/, lea n onaton t a er all funaental lp =3 to a large aron lp ν = 47, an terefore to a trong eletroagnet fel apng, an te eletroagnet oeffent an get er all g 7 o, tat te eletroagnet oeffent oul e u ger, f te rlng angular frequen oul e loe to te rotor angular frequen L Ort of te aft journal Ort of te left otor feet S B V W Ort of te rotor entre Ort of te earng oung entre Ort of te rgt otor feet R 46
gure 7 Magnet prng ontant a) an ) an agnet aper ontant ) an ) for fferent rlng angular frequene an fferent rotor angular frequene Wt te a oeffent fro Tale, te lt of talt alulate agan aorng to te flo agra To larf t atonal alulaton, te nex ue ntea of nex 1 So te lt of raton talt reae no at a rotor pee of ta, =4554 rp an te rlng frequen = ta, =556 ra/ Aorng to te flo agra a tr alulaton not neear, eaue ta, ffer onl argnal fro ta,1 (1%) Terefore, te alue for te lt of talt are =, =4554 rp; =, =556 ra/ No fferent ae are netgate Tale 3 o, tat te eletroagnet aper an te eletroagnet prng ontant nfluene te lt of raton talt onl argnal (ae an ), eaue te large aron lp at te lt of talt aue a trong eletroagnet fel apng an terefore leang to lo alue for an Wtout onerng eletroagnet fel apng (ae ), te agnet prng alue get axal ( = ) an te agnet aper oeffent get ero ( =) or t ae, te lt of talt ll e learl reue fro 4554 rp to 4377 rp, ean -39% Wen negletng te nternal apng of te rotor (ae e), te lt of talt ll e nreae (%) Cae Derpton Tale 3 Lt of talt for fferent ae [ra/] [rp] a) Data Tale (Ba onton) 556 4554 of to a) [%] ) Data Tale t = 565 4556 4 ) Data Tale t = 565 4566 6 ) Data Tale t = ; = (no eletroagnet fel apng) 481 4377-39 e) Data Tale t = (no nternal apng of te rotor) 557 4645 f) Data Tale t = = (no apng of te earng oung an en el) 565 459-1 g) Data Tale t = = (no apng of te founaton eleent) 561 4537-37 ) Data Tale t = (otor rg ounte) 64 39-139 47
Hoeer, negletng te apng of te earng oung an en-el (ae f) ll loer te lt of talt (-1%), a ell a negletng te apng of te founaton eleent (-37%) Te lo nfluene of te apng of te founaton eleent aue ter lo tffne, lea n onjunton t te lo fator to lo apng alue, an e een n Tale gure 8 o, tat f te founaton eleent tffne oul e eg 4 te ger an te eanal lo fator of te founaton eleent oul e te ae (tan f =1), te lt of talt oul nreae fro 4554 rp to 488 rp (7%) Hoeer, for a lo lo fator (tan f =1) te lt of talt oul onl nreae to 473 rp (39%) Lt of talt 49 485 48 475 47 465 46 455 45 n_lt (D=,1) n_lt (D=,1) for for Rate tffne,5 1 1,5,5 3 3,5 4 4,5 Stffne fator for te founaton eleent gure 8 Influene of te founaton eleent tffne on te lt of talt n gure 9 o te nfluene on te lt of talt an on te rlng angular frequen, for tffne araton of te founaton eleent n a range of =11 41 kg/ ; =11 81 kg/ gure 9 Influene of te artrar tffne of te founaton eleent on a) te lt of talt lt an on ) te rlng angular frequen ta; t a lo fator of tan =1 for te founaton eleent If te otor oul e rg ounte (ae ), ean f = f, te lt of talt oul e reae alrea at 39 rp, an te rlng frequen at te lt of talt ta oul e 64 ra/ ntea of 556 ra/, on Werner (16) Terefore, puttng ere te otor on te oft founaton eleent nreae te lt of talt learl (161%), opare to te rg ounte nuton otor 6 Conluon In te paper a ulto raton oel an a proeure are preente for talt anal of oft ounte nuton otor, t leee earng Te fou of te paper on te nfluene of eletroagnet fel apng, regarng te lt of talt After te ateatal oerene ae een ere, a nueral exaple a preente, ere te lt of talt a anale It oul e on, tat negletng eletroagnet fel apng lea to a loer lt of talt Atonall te nfluene of te founaton eleent an of te nternal apng of te rotor on te lt of talt oul e eontrate Altoug, te raton oel a plfe ulto oel, te proeure an onluon an e aopte nto fnteeleent anal 48
Referene Arkko, A; Antla, M; Pokk, K; Son, A; Lantto, E: Eletroagnet fore on a rlng age rotor Eletr Poer Appl, Vol 147, No 5, (), 353-36 rütent, J; Joran, H; Sen, HO: Exentrtätfeler al Urae on Laufntaltäten e Anronanen Ar für Elektrotenk, Vol 65, Iue 4, Part 1, pp 71-81, Part, (198), 83-9 Ga, R; Maurer J; Sarfel, W: Te nfluene of te elat alf pae on talt an unalane of a ple rotor-earng founaton te, Conferene Vraton n Rotatng Maner, C3/84, IMeE, Enurg, (1984), 1-1 Ga, R; Norann, R; Pfütner H: Rotornak Sprnger-Verlag, Berln-Heelerg, () Genta, G: Dna of Rotatng Ste Sprnger Sene & Bune Mea, (5) Glenke, J: eer- un Däfpungkontanten on Gletlagern für Turoanen un eren Enflu auf a Sngungeralten ene enfaen Rotor Dertaton TH Karlrue, (1966) Holopanen, TP: Eletroeanal nteraton n rotor na of age nuton otor VTT Tenal Reear Centre of nlan, P D Te, Helnk Unert of Tenolog, nlan, (4) Krk, RG; DeCour, P; Gunter, EJ: Te effet of upport flexlt on te talt of rotor ounte n plan lnral earng, IUTAM Spou Dna of Rotor, (1974), 44-98 Lun, J; Toen, K: A alulaton eto an ata for te na of ol lurate journal earng n flu fl earng an rotor earng te egn an optaton, ASME, Ne York, (1978), 1-8 Rao, J S: Rotor Dna Ne Age Internatonal, (1996) Suk, W: Magnet pull n eletral ane ue to te eentrt of te rotor Eletral Reear Aoaton Tran 95, (197), 391-399 Sen, H-O: Oerfelerenungen n Drefelanen Teuner-Verlag, Stuttgart, (199) St, AC; Dorrell, DG: Calulaton an eaureent of unalane agnet pull n age nuton otor t eentr rotor I Analtal oel Eletr Poer Applaton Vol 143, Iue 3, (1996), 193 1 Tonl, A: Soe Prole of Rotorna Capan an Hall, Lonon, (1965) Werner, U: Rotornae Anale on Anronanen t agneten Unetren Dertaton, Tenal Unert of Dartat, (6) Werner, U: Vraton talt of oft ounte anronou ane t flexle aft an leee earng onerng eletroagnet effet ISMA 3r Internatonal Conferene on Noe & Vraton Engneerng, Leuen, Belgu, 15-17 Septeer, (8), 1167-118 Werner, U: Stalt anal of nuton rotor upporte n leee earng onerng eletroagnet fel apng an nternal ateral apng of te rotor Vraton n rotatng aner, Maneter, UK, Sept 13-15, (16), 155-166 Are: Georg Son O Unert of Apple Sene Nureerg, Keßlerplat 1, 9489 Nureerg, Geran eal: ulrerner@t-nuernerge 49