Electromagnetic circulatory forces and rotordynamic instability in electric machines

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Hilary Miles
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1 In: Hhn, E. J. & Rndll, R. B. (eds.). Poceedings of the 6th Intentionl Confeence on Roto Dynmics. Sydney, Austli, Vol.. Sydney: Univesity of New South Wles Pinting Sevices,. P ISBN APPENDIX I Publiction P Electomgnetic cicultoy foces nd otodynmic instbility in electic mchines Holoinen TP, Tenhunen A b nd Aio A b VTT Industil Systems, Technicl Resech Cente of Finlnd, P.O. Box 75, FIN-44 VTT, Finlnd b Lbotoy of Electomechnics, Helsini Univesity of Technology, P.O. Box 3, FIN-5 HUT, Finlnd The electomechnicl intection in electic mchines induces dditionl foces between the oto nd stto. To study this intection, simle electomechnicl model ws develoed. The mechnicl behviou ws modelled by the Jeffcott oto. The electomgnetic foces wee descibed by simle metic model including two electomgnetic vibles. The im of the study ws to investigte the effects of electomechnicl intection on otodynmic instbility in electic motos. If the new electomgnetic vibles e inteeted s qusi-dislcements, the intection tuns u in the equtions of motion s dditionl dming, stiffness nd cicultoy tems. The electomgneticlly induced dming nd stiffness effects in electic motos hve been studied eviously. Howeve, the effects of cicultoy tems hve been oveided in electic motos. It is well-nown tht the cicultoy, i.e. coss-couled stiffness, tems e mjo souce of instbility in otting mchines. Thus, the esented model offes new nd simle exlntion fo the otodynmic instbilty in electic motos. The numeicl exmles ised nothe souce of instbility. The system metes my yield negtive-definite stiffness mtix (symmetic t), which destbilize the system without stbilizing foces. Keywods: Rotos, stbility nlysis, cicultoy foce, electomechnicl intection, electic mchines.. Intoduction An electic moto convets electic enegy to mechnicl one. The mgnetic field in the i g of the mchine genetes the tngentil foces equied fo the toque genetion. In ddition, the field oduces othe foce comonents tht intect with the mchine stuctues nd my excite hmful vibtions. At eltively low fequencies, the vibtion mlitudes my be lge enough to coule the electomgnetic system with the mechnicl one. This electomechnicl intection chnges the vibtion chcteistics of the mchine, e.g., it my induce dditionl dming o cuse otodynmic instbility. Füchtenicht, Jodn nd Seinsch [] develoed n nlytic model fo the electomechnicl foces between the oto nd stto, when the oto is in cicul whiling motion. Using this model nd ssuming synchonous whiling motion, they detemined the stiffness nd dming coefficients induced by the electomgnetic field. Belmns, Vndenut nd Geysen [] investigted nlyticlly nd exeimentlly the flexible-shft induction motos. Thei clcultion model esembled to tht of Füchtenicht et l. [], but they focused on the two-ole mchines. They concluded tht one otentil eson fo the otodynmic instbility esults fom the electomgnetic dming coefficient which my be negtive. Subov nd Shumovich [3] develoed n nlytic electomechncil model nd studied the otodynmic instbility. They found out tht the tngentil comonent of the electomgnetic totl foce my be the eson fo the unstbility. Aio et l. [4] esented simle metic foce model fo the electomgnetic foces cting between the oto nd stto when the oto is in whiling motion. The model metes of n electic moto wee detemined by numeicl simultions including the non-line stution of mgnetic mteils. The numeicl esults wee vlidted by extensive mesuements. Pevious esech on this issue hs not been conclusive, in t becuse it hs genelly been bsed on the ssumtion of synchonous whiling motion [,], nd in t becuse the effects of stution of mgnetic mteils e not included in the nlyticl foce models used eviously [,,3]. In esent study, we ejected the ssumtion of synchonous whiling motion nd used s the stting oint the numeicl simultions of electomgnetic fields. The im of this study ws to investigte the effects of electomechnicl intection on otodynmic instbility in electic motos. In this study, we develo n nlyticl model to study the electomechnicl intection nd stbility. The Jeffcott I/

2 oto model is combined with the metic foce model. The couled equtions of motion e tnsfomed into nondimensionl fom. Using this simle model, numeicl esults descibing the effects of electomechnicl intection e clculted. The vibtion chcteistics of the system e investigted by the eigenvlue nlysis. The otodynmic stbility is studied by using the modified Bilhz schem [5]. This yields fou necessy nd sufficient citei fo the stbility of the system. Some of the citei cn be studied in nlyticl fom. The est of the citei cn be studied numeiclly.. Nomencltue ± = electomgnetic foce metes = coefficients of chcteistic olynomil d = mechnicl dming coefficient f = excittion foce j = imginy unit = shft stiffness coefficient e = effective stiffness coefficient = electomgnetic foce mete ± = electomgnetic foce metes m = mss of the oto = numbe of ole is of the moto q ± = electomgnetic field vibles ij = tem ij in ecusive schem s = Llce vible s = mchine sli t = time x, y = co-odintes z = comlex vible D i = Bilhz deteminnt i F em = electomgnetic totl foce K = tnsfe function c i q = vecto of vibles in second ode model u = eigenvecto x = vecto of vibles in fist ode model A = system mtix B = coefficient mtix fo inut C = dming mtix H = cicultoy mtix I = identity mtix K = stiffness mtix M = mss mtix Q = loding vecto δ = dil i-g length δ = decy constnt of eigenvecto τ = non-dimensionl time ω = ngul fequency ω n = ntul fequency of mechnicl system ω = suly fequency Ω = oto ngul fequency % = non-dimensionl fom of Im = imginy t of Re = el t of = comlex conjugte of & = diffeentition of with esect to time, = diffeentition of with esect to τ τ 3. Method of nlysis 3. Roto model To model the mechnicl behviou of the system, we ly the Jeffcott oto which is thin unblnced dis locted t the middle of unifom, mssless, flexible shft. The shft is simly suoted t its ends by igid fictionless beings. We ssume dming foce fom the suounding medium of viscous tye. In Jeffcott model, the disc is ssumed to move only in its own lne, o moe ecisely in xy-lne. The oigin of this lne is ssumed to coincide with the ottion xis nd define the cente osition by comlex vible () () j () z t = x t y t () The eqution of motion of this oto using the comlex co-odintes och cn be witten [6] () mz && dz& z = f t () whee m is the mss of the disc, d is the coefficient fo the non-otting dming, is the shft stiffness coefficient, nd f is the excittion foce, e.g., due to the mss unblnce. 3. Foce model Aio et l. [4] esented low ode line model fo the electomgnetic foces between the oto nd stto. The oosed tnsfe function model cn be witten in comlex fom s I/

3 whee em ( ) ( ) ( ) F s = K s z s (3) F em is the comlex vlued electomgnetic foce exeted on the oto, z is the comlex vlued dislcement of the oto cente, s is Llce vible, nd K ( s ) is second ode tnsfe function Foce [N] 6 4 P d P tn Simu d Simu tn K( s) = s s whee,,,, nd e the metes of the model. The subscits of the metes nd efe to the esective eccenticity hmonics of the electomgnetic fields. The mete efes to the numbe of ole is of the mchine. The metes ± nd ± e genelly comlex vlued. The imginy ts of the ltte i of metes cn be witten (4) Whiling fequency [Hz] ω ± = ±,Re s( ± ) (5) Figue. The dil nd tngentil foces s function of the whiling fequency. The discete oints eesent the simultion esults nd the cuves e obtined by cuve fitting ocedue togethe with the metic model. [4] whee s is the sli of oto with esect to the fundmentl hmonic, nd ω is the ngul fequency of fundmentl hmonic. Aio et l. [4] detemined the foce metes of n electic moto by clculting the electomgnetic foces induced by cicul whiling motion ( s = jω ). Figue shows the dil nd tngentil foce comonents s function of whiling fequency. The 5 W fou-ole cge induction moto is loded by the ted toque nd sulied by the ted voltge. The whiling dius is % of the dil i-g length. The dil comonent is defined in the diection of the shotest i g nd the tngentil comonent eendicul to the dil one. The line second ode tnsfe function (4) coesonds to the equtions = = () () () = () q& q z t q& q z t (6) F t q q z t em whee q nd q e new vibles elted to the eccenticity hmonics nd of the i-g field. 3.3 Combined model The electomechnicl oto equtions e obtined by combining the mechnicl equtions of motion () with the equtions of electomgnetic foces (6). Thus, the couled system of equtions cn be witten ( ) ( ) mz && dz& z q q = f t q& = q z q& = q z (7) 3.4 Model in non-dimensionl fom The system of equtions (7) cn be tnsfomed into non-dimensionl fom emloying new non-dimensionl vibles z tω % (8) z = τ = q% = q δ ± δ ω m ± I/3

4 whee δ is the dil i-g length. Emloying these vibles, the system equtions cn be witten in non-dimensionl fom z%, d% z%, % z% q% q% = f% ττ τ e q%, = % q% % z% τ q%, = % q% % z% τ ( τ ) (9) whee % e = % % is the effective stiffness coefficient, nd the subscit τ fte comm efes to the diffeentition with esect to non-dimensionl time. The new non-dimensionl metes e 3 ± ± 3 mω d d% = % = % = % = mω mω mω Ω ω % s % f f t () ± n ± = Ω% = = ωn = % τ = ω ω ω mδ ω ( ) () whee ω n = mis the ntul mechnicl fequency of the oto, nd Ω is the ngul fequency of the oto. 3.5 Mtix eesenttion Using the comlex fomultion, Equtions (9) cn be witten in mtix fom ( ) Mq&& Cq& K H q = Q () whee q is the vecto of comlex vibles, M, C, K, nd H e the mss, dming, stiffness, nd cicultoy mtices, esectively, nd Q is the loding vecto. The mtices cn be witten exlicitly d% M = C= % - % % % % e % - % K = %,Re H = j s ( ) % % %,Re j s ( ) () whee the comlex conjugte of comlex numbe z is denoted by z. The mss, dming nd stiffness mtices e Hemitin mtices, nd the cicultoy mtix is sew-hemitin mtix. The Hemitin nd sew-hemitin mtices e geneliztions of the el vlued symmetic nd sew-symmetic mtices, esectively. If the new electomgnetic vibles q % ± e inteeted s qusi-dislcements, the electomechnicl intection tuns u in the equtions of motion s dditionl dming, stiffness nd cicultoy tems. The cicultoy, i.e. coss-couled, stiffness tems induce cicultoy foces, which cn stbilize o destbilize the system [7]. In otting mchines, the cicultoy foces e the mjo souce of instbility cting tngentilly to the shft whil obit nd my consequently feed enegy into the whiling motion [8]. 3.6 Fist ode model To exloe the system chcteistics, the equtions e tnsfomed into the fom of stte equtions. These equtions cn be witten in the mtix fom I/4

5 z% % e d% z &% x, τ = Ax Bf% ( τ ), A =, =, = % x B % q% % q % % (3) The homogeneous t of these equtions comise the eigenvlue oblem [ λ ] A I u = (4) whee I is the identity mtix. The solution consists of 4 eigenvlues λ nd ssocited eigenvectos These eigenvlues cn be exessed in the genel fom u (,, 3, 4) =. λ = δ jω (5) whee δ is the decy constnt, nd ω the fequency of the th eigenvlue. The system is symtoticlly stble, if ll the eigenvlues hve ositive decy constnt. 3.7 Stbility citeion To study the stbility of the system, we ly the citeion esented by Ps nd Hhn [5] fo line diffeentil equtions with comlex coefficients. This citeion esembles the citeion develoed by Bilhz [9], which is geneliztion of the bette-nown Routh citeion fo equtions with comlex coefficients. The chcteistic eqution of the system (4) cn be witten [ ] 4 3 det A λi = λ c λ c λ cλ c = (6) 3 whee c i e the comlex coefficients, nd λ is n eigenvlue of the system. Fo this chcteistic eqution, scheme cn be witten in mtix fom c c c c 3,Im,Re,Im,Re c c c c 3,Re,Im,Re,Im M M M (7) whee the suffix Re o Im in the subscit denotes the el o imginy t of the comlex coefficients, esectively, nd the ecusive fomul fo the dditionl tems is,,,, i i j i i j ij = (8) i, The Bilhz deteminnts cn be clculted fom the elements of the fist column D = D = D 3 4 D3 = D 5 6 D4 = D (9) nd the citeion fo symtotic stbility of the system is Di >, i =,,3, 4 () The fist citeion cn be witten exlicitly D d% % % () =,Re,Re > This citeion sets combined condition fo the mechnicl dming nd fo two electomgnetic metes, which e ssocited with the ttenution of the oto cuents. If we ssume tht the mechnicl dming is zeo, i.e. d % =, we obtin fom the second citeion I/5

6 ( ) % % s % % %,Re %,Re > % %,Re,Re 4,Re,Re,Re,Re () This citeion sets the lowe limit fo the sum of metes % in tems of metes ±,Re % ±, s,re nd. The thid nd fouth citeions e moe comlicted in exlicit fom, nd thei imlictions e moe difficult to undestnd. 4. Results 4. The studied moto The studied moto ws 5 W fou-ole cge induction moto. Aio et l. [4] used this moto s n exmle fo extensive numeicl simultions of electomgnetic foces. Thei clcultion esults e veified by mesuements, whee the oto is susended by mgnetic beings. The electomgnetic foce metes e identified by cuve fitting ocedue ssuming eithe el o comlex ± metes. The comlex fitting ocedue yields slightly bette esults. Howeve, fo the se of simlicity, we lied hee only el vlued metes fo ±. We studied the moto oeting t its ted owe (5 W), sulied by the ted voltge (38 V), nd the oto is unning t the ted sli (.3). All the metes used in the clcultions e given in Tble. The stiffness, mss nd dming oeties of the oto e oughly estimted. 4. Vibtion chcteistics of the studied moto The ntul bending fequency of the oto without electomgnetic effects is Hz. Tble shows the eigenvlue fequencies nd decy constnts of the electomechnicl oto system without mechnicl dming. The two lowest modes e ssocited with the eccenticity hmonics of electomgnetic fields, nd the two highe modes e ssocited with the fowd (FW) nd bcwd (BW) whiling modes. Howeve, ll modes hve electomgnetic nd mechnicl contibutions. The effect of electomechnicl couling on the whiling fequencies nd thei decy constnts is mino. The fequencies of electomgnetic modes coesond to the fequencies of the eccenticity hmonics of electomgnetic fields. The decy constnts of these modes e ssocited with the ttenution of oto cuents. We note tht in the studied moto the tio of fundmentl fequency, without electomgnetic effects, to the oeting shft seed is lge ( ωn Ω 5 ). This is tyicl fo smll electic motos. The electomechnicl intection is ently moe onounced in motos hving the tio close to unit. This is usully the cse in lge electic motos. To study the electomechnicl intection we etined the electomgnetic oeties of the moto but ssumed tht the shft stiffness is lowe thn in the ctul moto. A simil och ws emloyed by Füchtenicht et l. [] nd Belmns et l. [] in thei exeimentl investigtions. Tble Electomgnetic foce metes in non-dimensionl fom of 5 W induction moto [4] Dimensionl Non-dimensionl Symb. Vlue Unit Symb. Vlue ω π 5 d/sec m 3 g δ 3.45 m s.3 s N/m % N-d/m-sec % N-d/m-sec % d/sec,re %.,Re.6 d/sec,re %,Re N/m % 3 d 3 g/sec d %.4 Tble The eigenvlues of the studied moto Mode Feq. [Hz] Decy constnt Descition.6.68 Mode 5.8. Mode FW whiling BW whiling I/6

7 6 4 FW mode BW mode P mode P mode Without EMF. ω / Ω ( /m).5 / Ω Figue. Fequencies of eigenvectos s function of shft stiffness. δ FW mode BW mode P mode P mode Without EMF ( /m).5 / Ω Figue 3. Decy constnts of eigenvectos s function of shft stiffness. 4.3 Vibtion chcteistics s function of shft stiffness Figue nd 3 show the fequencies nd decy constnts of the electomechnicl oto system without mechnicl dming. Fo comison, the oto fequency is esented lso without electomgnetic foces (EMF). Figue shows tht the whiling fequencies e somewht lowe thn the fundmentl fequency without electomechnicl couling. This esults fom the so-clled negtive-sing effect of electomgnetic foces. Futhe, the electomgnetic modes e indeendent of the shft stiffness within high stiffness nge. As the shft stiffness is decesed, the whiling modes nd electomgnetic modes intect stongly with ech othe s. Figue 3 shows tht the decy constnts e ll slightly ositive within the high stiffness nge. As the shft stiffness is decesed, Mode nd theefoe the whole system become unstble. The citicl non-dimensionl vlue fo the shft stiffness is % = 8.9. A moe detiled study of the electomgnetic foces eveled tht below this citicl shft stiffness, the electomgnetic foces feed enegy into the whiling motion. In ddition, it cn be noted tht the stiffness mtix K is negtive-definite when the shft stiffness is less thn the limit vlue % = Rotodynmic stbility Ou system model is defined by 7 el vlued metes (Eq. 3 & 5). Figues 4 7 show fou stbility chts, in which thee of the system metes e vied nd fou held constnt. The efeence oint fo ll the chts is defined by metes: % =.35, %.644 =, %,Re =., %,Re =.67, d % =.4, nd s =.3. The effective shft stiffness % e is vied in ll figues Stble Unstble Figue 4. Stbility bodelines fo non-dimensionl = % nd % e. metes ( = % ), ( ),Re Stble Unstble ,Re Figue 5. Stbility bodelines fo non-dimensionl = % nd % e. metes,re ( = %,Re ),,Re (,Re ) I/7

8 Unstble. Unstble Stble.5 Stble e e Figue 6. Stbility bodelines fo non-dimensionl metes e ( = % e), ( ) = % nd %,Re. Figue 7. Stbility bodelines fo non-dimensionl metes e ( = % e), ( ) = % nd %,Re. Figues 4 7 show tht the citicl vlue of effective stiffness % e deends clely on the min system metes. In ddition, Figues 4 nd 5 indicte tht n incese of metes % nd decese of ± % ± my destbilize the system.,re 5. Discussion nd conclusions A simle electomechnicl oto model of n electic moto ws geneted. In ddition to the Jeffcott oto vible, the model included two dditionl comlex vibles descibing the stilly line effects of electomgnetic field. The obtined esults confim tht the electomgnetic fields nd the oto vibtions my intect stongly. If the new electomgnetic vibles e inteeted s qusi-dislcements, the intection tuns u in the equtions of motion s dditionl dming, stiffness nd cicultoy tems. The cicultoy tems, i.e. coss-couled stiffness tems, is mjo souce of instbility in otting mchines. Thus, the esented model offes new nd simle exlntion fo electic motos, which is comtible with the exeience of otodynmic instbilities. The numeicl exmle ised nothe exlntion fo otodynmic instbility. The system metes my yield negtive-definite stiffness mtix (symmetic t), which destbilize the system without stbilizing foces. These exlntions must be used s hyotheses fo moe detiled studies including the numeicl simultion of electomechnicl oto systems nd the vlidtion ocedues by exeimentl methods. 7. Refeences [] Füchtenicht J, Jodn H & Seinsch HO. (98) Exzentizitätsfelde ls Usche von Lufinstbilitäten bei Asynchonmschinen, Teil I und II. Ach. fü Electotechni, 65, 7 9. [] Belmns R, Vndenut A & Geysen W. (987) Clcultion of the flux density nd the unblnced ull in two ole induction mchines. Ach. fü Electotechni, 7, 5 6. [3] Subov D & Shumovich IV. (999) Stbility of the oto of n induction moto in the mgnetic field of the cuent windings. Mechnics of solids, 34(4), 8 4. (Tnslted fom Mehni Tvedogo Tel, 4, 36 5, 999) [4] Aio A, Antil M, Poi K, Simon A & Lntto E. () Electomgnetic foce on whiling cge oto. IEE Poc.-Elect. Powe Al., 47, [5] Ps PC & Hhn V. (99) Stbility theoy. New Yo: Pentice-Hll. [6] Gent G. (998) Vibtion of stuctues nd mchines. Pcticl sects. 3 d ed. New Yo: Singe-Velg. [7] Ziegle H. (968) Pinciles of stuctul stbility. Wlthm, Msschusetts: Blisdell Publishing Comny. [8] Vnce JM (987) Rotodynmics of tubomchiney. New Yo: John Wiley & Sons. [9] Bilhz H. (944) Bemeungen zu einem Stze von Huwitz. Z. ngew. Mth. Mech., 4, Acnowledgments The uthos gtefully cnowledge the finncil suot of the Ntionl Technology Agency of Finlnd (Tees). Secil thns e due to D. E. Lntto, High Seed Tech Ltd., nd M. P. Klinge, VTT Industil Systems, fo vluble discussions. I/8

International Journal of Pure and Applied Sciences and Technology

International Journal of Pure and Applied Sciences and Technology Int. J. Pue l. Sci. Technol. () (0). -6 Intentionl Jounl of Pue nd lied Sciences nd Technology ISSN 9-607 vilble online t www.ijost.in Resech Pe Rdil Vibtions in Mico-Isotoic Mico-Elstic Hollow Shee R.