Motor development 7/1 Mechanical motor design

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1 oo developen 7/ ehanial oo design 7. ehanial oo design ehanial oo design is as vial as he eleoagnei oo design o build good oos. This opises quesions onening he - oo balaning, - oo beaing syses, - ooling and hea ansfe, - ehanial sess of oo pas and hei vibaion behaviou, - onsuive design of oo pas wih espe o insulaion o-odinaion of elei pas. 7. Roo balaning The oaing oo body is usually no an ideal ehanial syse wih he ene of gaviy loaed on he oaional ais, bu unde eal ondiions ene of gaviy is disloaed fo oaional ais by a eain displaeen e. Thus enifugal foe of oo ass e (7.- a oaional angula fequeny n, n: oaional speed (7.- will ou as adial foe in dieion of disloaion of ene of gaviy fo oo ais, ying o bend he elasi oo and aing as addiional foes a he oo beaings. As dieion of his foe oaes wih oaional speed, i ay eie ehanial vibaions of he whole oo wih fequeny f = n, whih when hiing naual vibaion fequeny of oo syse auses esonane wih ineased vibaion apliude. o, ibalane ay lead o - ineased beaing sess, - addiional oo loading and - ineased ahine vibaions. In ode o quanify ibalane, usually he eal syse wih elasi oo and elasi beaings is odelled by sep-wise by idealized odels. All effes of oo vibaion ae inluded in he heoy of oo dynais, whih is a aheaial disipline of heoeial ehanis. oo odel oo beaings igid oo odel elasi oo odel oo body ows no defoaion unde foe load (= geoey does no hange ape unde foe load oo body is defoed unde foe load (= Young s odulus of elasiiy E of oo aeial is no infinie igid beaings ow no defoaion unde foe load igid beaings elasi beaing odel elasi oo body beaing geoey is defoed unde foe load (defoaion is uled by Young s odulus of elasiiy E of beaing aeial and end ields Table 7.-: Diffeen aheaial odels fo oo syse (= oo body and beaings Elasi oo is bending unde adial foe load, hus aing like a sping wih a naual bending fequeny f b. If eiing fequeny f = n says below 70% of f b, no esonane effe will ou and oo bending defoaion will usually be sall, and heefoe is negleed. Thus wih aiu speed n a below 0.7 fb, usually he igid oo odel is used fo alulaion of ehanial vibaion. Please noe, ha in addiion o ehanial ibalane also T Dasad Insiue fo Eleial Enegy Convesion

2 oo developen 7/ ehanial oo design unbalaned agnei pull ay be aive, whih in ase of -pole ahines wih eeni oo, as eplained in Chape 5, ay eie adial foes wih f = n. Then of ouse lii fo igid oo odel is na fb. Calulaion of oo dynais and ehnique of balaning oos is oday fine a, whih is augh in speial leues. Inenaional sandadizaion IO95, IO940 and IO5460 and he naional Gean guide line "VDI-Rihlinie 060" apply. Hee, only he basis will be given in o o ge an oveview. 7.. Ibalane of igid oo bodies a Dis oo: We assue he oo body o be igid as well as he beaings. Theal influene on oo, whih leads e.g. also o aial oo epansion (z-dieion in ig. 7..-, whih ay be diffeen fo heal epansion of sao housing, is oped wih one of he wo beaings being a loose beaing. Tha eans, ha he oo and his beaing ay ove in aial dieion (usually aial oveen is well below unde epansion, wheeas he ohe beaing is fied in aial dieion. The os siple odel is a dis-like oo (hikness l e uh salle han diaee d a and beaing disane wih oo ass, ouned on an infinie siff, bu vey hin af (af lengh beween beaing is beaing disane (ig o siplifiaion we assue dis o be ened beween he beaings. ig. 7..-: Rigid body dis oo on siff af wih ene of gaviy disloaed fo oaional ais by disane e (lef: loose beaing A, igh: fied beaing. Disloaion e of ene of gaviy fo oaional ais, whih is ened a geoeial ene of dis, is e.g. given by an addiional ass a adius d a /, wheeas he ass is evenly disibued on he dis. Thus ene of gaviy of ass is loaed on oaional ais. A sandsill gaviy will dag addiional ass down igh below oaional ais, hus uning he dis in ha posiion, so one an eognize he ibalane also a sandsill, hene alling i sai ibalane. The aiu oque eeed by gaviy on he dis a sand sill ous a dis posiion whee addiional ass is in hoizonal plane: g d g e e ( / ( d / (7..- a / a Denoing he displaeen as a veo, dieed fo ene of oaion o ene of gaviy, we ge unde oaion he adial enifugal foe veo, whih defines he sai ibalane veo, whih is NOT a foe: T Dasad Insiue fo Eleial Enegy Convesion

3 oo developen 7/3 ehanial oo design e e (7..- Conlusions: ai ibalane is he popoional oeffiien beween squae of angula ehanial fequeny and enifugal foe. I is independen of speed. Eaple 7..-: Roo ass = 60 kg, oo oue diaee d a = 00, = g, e ( / ( d / (/ (0./ 3.33 a 6 e g. Cenifugal foe a n = 000/in: e ( 000/ N Noe: Gaviy foe of oo is g N. Aleady a a sall displaeen of ene of gaviy by 3.3, a enifugal foe of 8.8/589 =.5% of oo gaviy foe ous a 000/in. A 4000/in enifugal foe is aleady 6% of gaviy foe. As aleady suh sall displaeens e ause onsideable enifugal foes, i is easily undesood, ha i is paially ipossible o build oos wih no ibalane. A balaning poess is always neessay. The oaing foe e os( e sin( ay be deoposed ino an - and y-oponen aoding o ig As dis is ened beween beaings, he load beaing foes in A and ae idenial due o syey: In veial dieion (y-dieion gaviy load and ibalane load ou, wheeas in hoizonal dieion (-dieion only ibalane is aing. A, A, y,, y / os( g / / sin( y (7..-3 A, ~ A, y~, ~, y~ / os( / sin( (7..-4 Conlusions: Due o sai ibalane an addiional osillaing beaing foe ous. Wih he squae of ineasing speed his osillaing oponen of he beaing load ineases due o sai ibalane, wih osillaion fequeny being oaional fequeny. In boh beaings his addiional beaing foe is IN PHAE (oon ode foe. b inie lengh oo: Due o oo ion sak lengh l e elei oo oos ae usually no dis-like (ig e us assue an ideal ylindial oo wih ylinde lengh l e, being plaed ened beween beaings A and. Ideal ylinde has is ene of gaviy on ylinde ais, whih is hee also oaional ais. If wo addiional asses ae plaed on he ylinde ends in opposie adial posiion a ylinde iufeene, hey will no if ene of gaviy fo oaional ais. o a sand sill oo will NOT ove unde gaviaional foe. u if oo is oaing, on eah of he wo asses a enifugal foe wih opposie dieion will ou T Dasad Insiue fo Eleial Enegy Convesion

4 oo developen 7/4 ehanial oo design ( d a /, (7..-5 esuling due o his foe pai in an addiional oque ( d a da le / le. (7..-6 ig. 7..-: Rigid body ylindial oo wih ene of gaviy loaed on oaional ais, bu uneven disibued ass along oo ais, epesened hee by wo asses ", whih lead o ibalane oque ", when oo is oaing. As he effe of his ass aangeen is only aive a oo oaion, wheeas a sand sill no effe is visible, we all his a dynai ibalane, whih is defined as oque pe squae of oaional angula fequeny. Noe, ha no oal enifugal foe ous, as he wo foe oponens ae opposie, anelling eah ohe: 0. The addiional oque leads o eaion foes in he wo beaings as a foe pai wih opposie dieion of in he wo beaings, whih oune-balane his oque: l ( l / (7..-7 A, ~ A, y~ e, ~, y~ ( ( / / e os( sin( (7..-8 Conlusions: Dynai ibalane is he popoional oeffiien beween squae of angula ehanial fequeny and ibalane oque. I is independen of speed. I leads o addiional osillaing beaing foes wih opposie sign in boh beaings (= 80 phase if = diffeenial ode foe, again wih osillaion fequeny equal o oaional speed. In eal oos boh effes, sai ibalane due o disloaion of ene of gaviy fo oaional ais, and dynai ibalane due o uneven ass disibuion along oo ais, ou. Thus beaing osillaing foe ae onaining a oon-ode oponen due o sai ibalane and a diffeenial ode oponen due o dynai ibalane. In ig eaples fo ibalane ae given: a Dis oo wih sai ibalane: The ene of gaviy uns a sandsill beneah oaional ais unde he influene of gaviy. Thus effe of sai ibalane an be epeiened aleady a sand sill (a sai ondiions. T Dasad Insiue fo Eleial Enegy Convesion

5 oo developen 7/5 ehanial oo design b Dynai ibalane if dis oo: Ineia ais of dis oo (whih is geoei of dis, if dis ass is hoogenous disibued is no aligned wih oo ais. Thus ene of gaviy of uppe and lowe half dis is displaed aially by z. nde oaion, he enifugal foes lead o ibalane oque z. upeposiion of sai and dynai ibalane: A ylindial oo wih ineia ais unaligned wih oaional ais in ha way, ha also he ene of gaviy, loaed on ineia ais, is disloaed fo oaional ais. ig : Eaples fo igid body ibalane: a sai ibalane, b dynai ibalane, sai and dynai ibalane If we deopose he ylindial oo of ig by hough ino ylindial hin dis slies ( l 0, eah of hese oo diss will be hin enough o podue no ibalane e, dis oque ( ~, 0. Eah dis has heefoe only sai ibalane of diffeen dis l e dis apliude and dieion. In ig his is own fo eaple fo oo deoposed in 3 dis slies. The sai ibalane e.g. of dis 3 3 leads o enifugal foe 3 a speed n. When dis 3 is plaed a o-odinae z 3, we ge he eaion foes in he beaings A = (lef and = R (igh: foe equilibiu: 3 3 R3, oque equilibiu: 3 3 z3 R3, yielding o z3 z3 3 3, R3 3. (7..-9 upeposiion of 3, R R R R3 leads o esuling beaing foes, R, oesponding wih /, R R / in ig eaing foes and oesponding ibalane ay AWAY be deoposed in oon ode (: syeial, esulan sai oponen and diffeenial ode oponen (A: anisyeial, esulan dynai oponen. T Dasad Insiue fo Eleial Enegy Convesion

6 oo developen 7/6 ehanial oo design ( /, ( / (7..-0 A, R A A R Conlusions: In geneal ase addiional beaing foe in lef and igh beaing ae no dieed in he sae dieion, bu in abiay one. Neveheless eah beaing foe ay always be wien as a su of oon and a diffeenial ode oponen. This eah igid oo ibalane is given as a supeposiion of a oal sai and dynai ibalane. a b ig : a A ylindial oo wih sai and dynai ibalane is deoposed ino hee dis slies. o eah dis slie he dynai ibalane ay be negleed. o he oal ibalane is given by he su of he sai ibalanes of eah dis. Is oponens in he beaing planes (: lef, R: igh ae sued up as veos fo oal beaing eaion. b The beaing eaion ibalane ay be deoposed ino a sai and dynai oponen by aking oon ode (syeial: and diffeenial ode (ani-syeial: A oponen. This deoposiion is always possible fo any abiay ibalane oponen in he R and beaing plane. 7.. alaning equaion fo igid oo bodies ig. 7..-: alaning of oo wih sai and dynai ibalane by wo addiional asses fied in he planes E and ER o opensae he addiional ibalane beaing foes If wo addiional asses and R would be added in he beaings opposie o he beaing addiional foes a a eain disane (ig , hei enifugal foes would anel he beaing addiional foes. This is he basi idea of balaning a oo by addiional asses. As in he beaings hee is no spae fo addiional asses, he idea is o use addiional balaning planes a he o-odinaes z and z, whee he addiional asses ae fied. In any ases he end planes of he oo body iself ae used as hese balaning planes (ig y fiing he wo balaning asses, a he adii, in hose wo balaning planes E and ER (,, hei addiional enifugal foes, us be suh, ha hei eaion foes, R in he beaings opensae he ibalane beaing foes, R whih ae aused by he oiginal oo ibalane (ig Assuing fo siplifiaion, dieed in he sae plane, we ge R T Dasad Insiue fo Eleial Enegy Convesion

7 oo developen 7/7 ehanial oo design foe equilibiu: R, oque equilibiu: R z z, hus z z z z z z R z z z R Wih balane equaions ( z / z z z z R ( / z z / z, R R R z R ( / z / z z / z (7..- we ge fo he balaning asses he (7..- Eaple 7..-: ediu sized elei oo of 75 kw, 500/in, oo ass = 60 kg, oo oue diaee 00, balaning in wo planes a / z 3/, z / z /, balaning adii: d / 00. a easued ibalane beaing foes a 500 /in:. 6 N, R. 0 N. We assue foe dieion in boh beaings o be he sae. How big us be he opensaing balaning asses, o opensae opleely oo ibalane? R R / /.6 /( (500 / 60.0 /( (500 / 60 R ( / z z / z g g (3/ / 00 8g ( z / z R ( / z z / z (/ (3/ /.46g z / z / 00 alaning asses ae 8 g and.5 g, whih due o negaive sign us be fied opposie o dieion of easued beaing foes, o his aoun of ass us be eoved fo he oo. Conlusions: o balaning equaions one onludes ha balaning asses fo given ibalane ae salle, if he fiing adii, fo hese asses ae big, so usually oue oo diaee is hosen. They ae also sall, if he disane beween he wo balaning planes z - z is big alaning of igid oos Roo is pu on o easuing beaings in he balaning ahine (ig and is diven by a sall oo on o balaning speed n. The easuing beaings have a eain elasiiy, whih ay be epessed by he siffness. Due o ibalane he oo geneaes n-fequen foe osillaion ˆ os( in he beaings, whih vibae. In hoizonal plane no gaviy foe is aing, so we ge T Dasad Insiue fo Eleial Enegy Convesion

8 oo developen 7/8 ehanial oo design ˆ os( (7..3- oluion of his seond ode linea diffeenial equaion wih onsan oeffiiens is ˆ ( Xˆ os(, Xˆ, n. (7..3- A esonane fequeny n = f apliude of his (hee assued loss-fee osillaion vibaion apliude ends o infinie, hus denoinao is zeo, yielding fo f 0 n f. ( If oo speed n is opeaed well below o well above esonane fequeny, esonane is avoided and we ge n n f f : : Xˆ Xˆ ˆ ˆ ˆ Xˆ ( In ode o opeae a he sae speed wih n f, one has o build siff beaings wih high siffness as own in ig a. A pessue senso (e.g. piezo-elei senso beween a siff baseen onsuion and he V-aped beaing diely easues he hoizonal foe apliude ˆ, whih is diely popoional o ibalane (siff easuing beaing. If opeaion ould be done a n f fo he sae speed, one has o build "sof" beaings wih low siffness as own in ig b. The V-aped beaing is fied by long spings o he baseen onsuion. The longe he sping, he lowe is he value. An e.g. induive posiion sensos is diely easuing he hoizonal oveen of he easuing beaing X ˆ /, whih is diely popoional o ibalane (low siffness easuing beaing. oh ehods ae used nowadays o easue he ibalane. ig : easuing oo ibalane a wih beaing foe easueen, b beaing posiion easueen easueen signal of beaing foe R (, ( o beaing posiion sr(, s( onains no only oaional fequeny, bu due o ohe eiaions like he balls of he beaings addiional haonis and also soe noise due o EI (ig o signal is fileed wih low pass T Dasad Insiue fo Eleial Enegy Convesion

9 oo developen 7/9 ehanial oo design file, hen only geing he n-fequen signals R (, (, whih have sae fequeny, bu diffeen apliude and phase if. If he signals ae aken as phasos, oaing wih in he ople plane aoding o ˆ ˆ e Rˆ Rˆ e j j R ( ˆ os( Re ˆ e R( Rˆ os( R j, ( one an ake he diely as he ibalane veos, R, whih ae used as inpu fo he balaning equaion (7..-. Wih he balaning asses, alulaed fo (7..-, hese asses ae fied o he oo and easueen of beaing foes o posiion is epeaed. Roo on easuing beaings and R easued beaing foe o posiion s ileed foe o posiion signal Phasos of fileed foe o posiion signal ig : easuing of oo ibalane by deduing esulan ibalane in he beaings If balaning is ideal, now no n-fequen beaing foes ould ou any longe. In ealiy of ouse sill soe sall ibalane is lef. This esidual ibalane us say below eain liis, wheeby hese liis ae defined depending on he pupose of he oo by sandad IO 940. This esidual ibalane is no given as ibalane value, bu as iufeene speed of ene of gaviy G. ( e In ideal ase wih ene of gaviy on oaional ais G e 0 0, so G an also be aken as easue fo esidual ibalane. Eaple 7..3-: ii of esidual ibalane aoding o IO 940 fo elei oo, 000/in, 00 kw, oo ass 00 kg. Table 7..3-: G.5/ s e G / /( 000 / Residual ibalane: g. T Dasad Insiue fo Eleial Enegy Convesion

10 oo developen 7/0 ehanial oo design e Eaples /s G low uning big Diesel engines fo ips G ig wo soke obusion engines G ig fou soke obusion engines G as uning fou soke pison engines G Cobusion engines fo as and loooives G Wheel ses fo as G 6 6 Cadan ansission afs G ans, pup oos, sandad elei oo oos G.5.5 Roos of sea and gas ubines, big elei geneaos, high speed elei oos, ubo pop fo ai af G la high speed sall oos, ginding spindle dives G Gyosopi oos, speial high speed ginding spindle dives Table 7..3-: iis fo esidual ibalane aoding o IO 940 ehods fo balaning: Wih wound oos of sall up o ediu sized oos of DC ahines, univesal oos, wound oo induion ahines he balaning asses ae a speial fas-hadening een, pu ino he winding ovehangs of he oo winding. Cage induion oos ofen have speial ylindial noses inegaed ino he aluiniu end ings of he age. On hese noses eal ings ae fied as balaning asses. o high speed ahines hese noses igh ause addiional fiion in ai, heefoe ofen soe age ass is u off (" negaive asse balaning". igge ahines ge speial addiional diss on he oo, whee he balaning asses ae fied alaning of oplee oo syse Afe ouning he oo ino he sao housing and he end-ields, he addiional ibalane igh be indued by inauay of beaing seas in he end ields. o high speed oos o oos fo vey low vibaion heefoe addiional balaning is neessay e.g. by uing of soe ass fo oo af end o by uing off soe ass of speial oo diss. Thus balaning of he oplee oo is done. This has o be done a old oo, bu has o be epeaed a ho oo, beause due o uneven heal epansion of oo addiional ibalane igh ou (heally indued ibalane. o eaple, he ion ees of oo ion sak usually do no have ealy paallel sides, so hikness vaies a lile bi along he ee plane. If all ees ae saked on he oo af wih bigge hikness on one side, heal epansion hee is bigge, ausing he oo o bend, when ho. Thus ene of gaviy is displaed, ausing ibalane. y saking he oo wih he oo ees ifed by 90 eah, his effe is equalized, hus avoiding his eason fo heal ibalane. u how o hek he saus of ibalane a he opleed oo? oo is pu on an elasi es bed, e.g. beneah oo fee hik ubbe pads ae pu (ig a. Anohe possibiliy is o hang he oo ino spings. This ehod is of ouse only possible fo salle oos (ig b. Vibaion easueen in adial pependiula dieion (, y-o-odinae a he beaing loaion (= end ields and in aial dieion (z-o-odinae is done wih espe o oaional T Dasad Insiue fo Eleial Enegy Convesion

11 oo developen 7/ ehanial oo design T Dasad Insiue fo Eleial Enegy Convesion fequeny f = n, e.g. by easuing vibaion veloiy v o aeleaion a. If oo housing sufae posiion in hoizonal dieion is, we ge os( ˆ ( (, sin( ˆ ( (, os( ˆ ( X a X v X (7..4- a b ig : Vibaion easueen fo opleed oo: a oo pu on ubbe pads, b oo hung in spings oo wih ass, onsising of sao ass s and oo ass is pu on ubbe pads o hung in spings (wih sping o pad siffness G. oo winding is onneed o invee o dive oo a no-load wih diffeen speed n. Roo ibalane leads o osillaing beaing foes, whih also eie he sao housing o vibae. Vibaion ( of sao housing is easued wih espe o oaional fequeny. We assue sai ibalane, geing he - oponen of enifugal foe os( ( e (7..4- fo eiing he vibaion of he whole oo ass in -dieion: os( ( G s. ( Due o he low siffness G he esonane fequeny f G is uh lowe han aed speed. oluion of his linea seond ode diffeenial equaion is aoding o (7..3- o G G o G f X X, ˆ, os( ˆ (. ( Due o he low siffness G he esonane fequeny f G is uh lowe han aed speed. o o G X f n ˆ :. ( s o e v v ˆ sin( ( ( ( Conlusions: Vibaion veloiy ises linea wih speed, is inlinaion being diely popoional o ibalane iself. Thus i is possible by elasi ouning of oo o ge aess o saus of ibalane by vibaion easueen.

12 oo developen 7/ ehanial oo design Vibaion veloiy is diely popoional o iufeene veloiy of ene of gaviy: vˆ e G ( s s The ibalane lii G is onneed o he..s. lii of vibaion veloiy by v.. s G. ( s Eaple 7..4-: Wih a ass aio / o / 3 and G = 6.3 /s a ypial value fo..s. vibaion 6.3 veloiy lii is v.. s. 5 /s. 3 o balaning oplee oo syse in IO373 liis. s ae given, whih aoding o ( ise wih speed. Depending on ineased pefoane of dive syse, whih is defined by appliaion suh as oos fo ooling ahines, diffeen vibaion levels ae given: N: noal use (sandad, R,, R: deeased vibaion level is given by levels R, and R (ig These values ae given of ouse no fo eeely big oos, as fo he i is diffiul o pu he on elasi ouning (ig fo easueen. Values ae defined fo AC- and DC-ahines wih fae size 80 o 400 (af heigh, oesponding wih aed powe 0.5 kw up o 450 kw a 500/in. Vibaion levels ae defined in IEC34-4 and oe deailed in IO373. v. ig : Vibaion levels (..s. vibaion veloiy, depending on aiu speed of oo, aoding o IO373, fo oo size 60 o 80 Above fae size 400 he oos ae oo big o be easued on elasi ouning. Theefoe only liis of values v..s. fo siff ouning ae given. In ha ase ahine and T Dasad Insiue fo Eleial Enegy Convesion

13 oo developen 7/3 ehanial oo design ahine bed, whee ahine is ouned, ae oupled, so vibaion of ahine auses also he bed o vibae. This easueen ehod heefoe does no allow aess o he vibaion behaviou of he ahine iself, bu only of he whole dive syse wih ahine bed. o easueen values ay be quie diffeen, when opaed wih easueen on elasi ouning. Eaple 7..4-: iis of..s. vibaion level aoding o IO37 fo fae size 60 o 80 : evel N:.8 /s fo vaiable speed dives wih aiu speed 600/in /in evels R,, R ae depied in ig fo eended speed ange up o 7000/in Elasi oo syse - Vibaions of oos The oo syse is onsiued by he oo body and he beaing syse, whih opises he beaings and he end ields o beaing seas. Elasiiy of he oo af and of he beaing syse leads o naual vibaions of he oo syse, whih akes i ipossible o balane he oo only in wo balaning planes, as i is possible fo igid oo body syse. a Influene of elasi beaings : ig : Rigid body vibaion on elasi beaings: a oon ode vibaion (.., b diffeenial ode vibaion (d.. The beaing siffness of ball o olle beaings is uh highe han he siffness of he oo body. u in ase of agnei beaings, oil sleeve beaings o ai beaings he beaing siffness is salle. o, if a igid oo body is plaed on elasi beaings, hese beaings ay be desibed by he beaing siffness (ig We assue idenial beaings, so equivalen sping syses fo lef and igh beaing ae idenial: R 0, yielding fo beaing defoaion e.g. in veial dieion y along wih veial foe pe beaing = R = : 0 y. (7..5- If boh beaings vibae IN PHAE (oon ode vibaion e.g. in veial dieion, he oo body is also osillaing veially (ig a. As boh beaing spings a ehanially in paallel, boh beaing foes add fo he AE veial defoaion, leading o esuling veial siffness 0. 0 y y. (7..5- This oon ode beaing vibaion has been aleady desibed in eion 7..3, leading o oon ode (.. naual fequeny T Dasad Insiue fo Eleial Enegy Convesion

14 oo developen 7/4 ehanial oo design f, ( If boh beaings vibae wih 80 PHAE HIT (diffeenial ode vibaion e.g. in veial dieion due o phase if of 80 beween lef and igh beaing foe R, he oo body is no only osillaing veially (ig b, bu ows also angula oveen aound -ais wih angle, whih aoding o d / d is uled by he oen of ineia of oo body aound -ais and by he oque diffeenial ode naual fequeny (given wihou poof hee J. The esuling f, d.. J f,.. ( is usually highe han he oon ode naual fequeny, if is big enough. Conlusions: In he oplee oo syse elasiiy of beaings and end ields leads o wo igid oo body vibaions, whih ay influene (disub he balaning poess of he oplee oo syse, if oaional speed f = n oinides wih on of hese wo naual fequenies, leading o esonane. The balaning of he oo alone on a balaning ahine is no influened by hese beaing vibaions, as on he balaning ahine he oo uns in speial easuing beaings. b Elasi oo af wih luped ass assupion: The oo ion sak, onsising of he saked ion ees, ows no big siffness, opaed o assive ylindial oo body of he sae diensions. o i is ainly he assive oo af (ass, whih deeines oo siffness ( =. The oo ion sak (ass sak as ainly as addiional ass. Hee fo sipliiy he odel of dis oo (hikness l e uh salle han diaee d a and beaing disane is onsideed. This odel is vey appliable o a one sage sea ubine oo and was fis sudied by de aval, hus being alled aval oo. ass of af is added as luped ass o sak ass as oal oo ass = sak +, whih is ouned on elasi ass-less af (hee fo sipliiy in he iddle beween boh beaings a z = /. Elasiiy of af is desibed by Young s odulus E, whih fo 9 seel is E 0 N/. haf iula oss seion wih diaee d esiss o bending 4 wih aea oenu of ineia I d / 64. haf lengh beween beaing is beaing disane (ig o siplifiaion we assue oo dis o be ened beween he beaings. A sand sill due gaviy he af is ben wih aiu defoaion y in he iddle (ig g y ( wih 48 E I. ( Eaple 7..5-: Elei oo wih 75 kw a 500/in: haf lengh / diaee = 0.7, d = 80, sak lengh l e = 350, oue diaee d a = 90, ion ass densiy = 7850 kg/ 3 : T Dasad Insiue fo Eleial Enegy Convesion

15 oo developen 7/5 ehanial oo design d / kg, l ( d d / kg, kg, sak 4 e a 6 I d / , ai oo bending due o gaviy: y g / 5 48 E I N/ Naual bending fequeny is given by ig : Dis oo on elasi af (aval oo y y 0 ( wih naual bending fequeny f b. ( Conlusions: o geing siff oo he disane beween he beaings us be sall and he af diaee ould be big. This and sall oo ass allow inease of oo bending naual fequeny. Eaple 7..5-: Elei oo wih 75 kw a 500/in (Dae of Eaple 7..5-: 9.7 kg, N/ Naual bending fequeny is fb 8. 3Hz 9.7 A speed n = f b = 7700/in an ibalane will eie big oo vibaions due o esonane. Theefoe aiu oo speed ould say below n f /in. Elasi oo af wih disibued ass: In ealiy he af has o be onsideed as ylindial bea of diaee d and lengh wih DITRITED ass along he bea. ike a vibaing guia sing, his bea has no only one bending naual fequeny, bu a whole speu of naual bending odes, whih depend on he elasiiy of he beaings. Assuing igid beaings, whee he - and y-o-odinae of he bea is always zeo, bu allowing he bea o hange is inlinaion also a he beaing s loaion, he naual bending fequenies of he bea (wih oss seion aeas A ae deived as i E I,, i, A fb i, 3,... a b ( T Dasad Insiue fo Eleial Enegy Convesion

16 oo developen 7/6 ehanial oo design As he ion sak ass ineases he oal ass, his onsideed aoding o ( by fb, i, o fb, i, i,, 3,... ( sak ig : odel of af as ylindial bea, owing a speu of bending naual fequenies, whih depend on he influene of beaing siffness Eaple : Elei oo wih 75 kw a 500/in (daa of Eaple : = 0.7, d = 80, = 7850 kg/ 3, 7. 6 kg, 64. kg, / 64./ , sak sak 3 A d / i 3 f b,i,o / Hz Copaed wih he siplified luped ass odel (7..5-6, he oe deailed odel ( yields 40% highe fis naual bending fequeny and in addiion also highe fequeny eigen-odes. igge oos have ahe big disane of beaings and ahe big oo ass. Hene, o onside influene of ahine size on naual bending fequeny, we ake geoei diensions ~, d ~ in (7..5-9, owing ha fequeny deeases invese wih ineasing fae size: fb, i ~ /. Conlusions: Calulaion of naual bending fequeny is ahe opliaed, as disibued ass and geoey of oo has o be onsideed. The siffening influene of oo ion sak is also diffiul o alulae. Elasiiy of beaings ay edue he naual bending fequenies by abou 0 % o 0 %. uhe, big ahines have low naual bending fequeny. d Influene of unbalaned agnei pull on naual bending fequeny: When he oo af is ben by y, his leads o an ai gap eeniiy e = -y. The elei ahine ai gap is edued on side by e and ineased on he ohe side by e. Thus adial agnei flu densiy, whih is eied by..f. in ai gap V, in ase of ahines wih pole nubes p 4 is ineased on side by 0 V /( e and edued on he opposie side by V /(. Thus adial agnei pull pe uni oo sufae A oo p e 0 e 0 V / p l wih as ai gap flu densiy apliude of non-eeni T Dasad Insiue fo Eleial Enegy Convesion

17 oo developen 7/7 ehanial oo design 0 fn fn ( is no longe in equilibiu, bu esidual ( unbalaned agnei pull in dieion of deeased ai gap eains (foula given wihou poof: p p le e ( fn fn A, p 4 ( o wo-pole ahines his unbalaned agnei pull is only 50%. Noe: A zeo eeniiy e = 0 his esidual pull is also zeo. ig : nbalaned agnei pull in fou pole oo due o ai gap eeniiy e This esidual adial agnei foe ends o deease he edued ai gap fuhe. Copaing wih he elasi bending of he af, whee a bending by value y ineases elasi foe y, aing agains he bending foe, he unbalaned agnei pull as in dieion of bending foe, hus opposie o elasi foe: y. ( Consideing y = -e, he "agnei siffness" p p le ( has o be onsideed in he odel of elasi oo. I edues he esuling siffness by es, ( leading o deease of naual bending fequeny by f b, fb. ( Eaple : 4-pole elei oo wih 75 kw a 500/in (daa of eaple : = 0.7, d = 80, l e = 350, d a = 90, 9. 7kg, ai gap flu densiy apliude = 0.9 T, ai gap =.0, pole pih: 49 p T Dasad Insiue fo Eleial Enegy Convesion

18 oo developen 7/8 ehanial oo design Roo gaviy foe: g 900 N nbalaned agnei pull a 0% eeniiy: e / 0. : p p le e N Depending on ai gap eeniiy, he unbalaned agnei pull an eah onsideable values (hee: 5 % of oo gaviy foe. Calulaed fis naual bending fequeny wihou influene of agnei pull: fb 83Hz: equivalen af siffness: ( f b.4 0 N/, p p le 6 agnei siffness: N/ 0 Calulaed fis naual bending fequeny wih influene of unbalaned agnei pull: f f / / Hz: b, b Conlusions: nbalaned agnei pull leads o a onsideable deease of naual bending fequeny by abou 0%... 0%, depending on uilizaion of agnei iui alaning of elasi oos ig : ai ibalane due o disloaion of ene of gaviy by e fo oaional ais. Addiional disloaion of ene of oaion fo geoeial ais of unben af (hee: Eaple of dis oo on elasi af (aval oo We assue sai ibalane due o disloaion of ene of gaviy by e fo oaional ais (ig Negleing he bending due o gaviy, we hee onside only bending due o enifugal foe aused by ibalane, whih auses in y-dieion addiional disloaion y of ene of oaion fo geoeial ais of unben af. In -dieion disloaion ous. Wih esuling adial disloaion gaviy is displaed fo geoeial ais by y due o bending he ene of e, ausing enifugal foe ( e. (7..6- The af bends unil equilibiu beween enifugal foe and elasi foe is eahed, ( e, (7..6- T Dasad Insiue fo Eleial Enegy Convesion

19 oo developen 7/9 ehanial oo design yielding fo he displaeen of ene of oaion fo geoeial ais in dependene of speed b e, b fb ( and fo he displaeen of ene of gaviy fo geoeial ais in dependene of speed (ig b b e. ( ig : Displaeen of ene of gaviy fo geoeial ais in dependene of speed Afe passing hough he esonane egion, he ene of gaviy is ening iself on he geoeial ais a high speed ( 0 ; hene he enifugal foe will vani a high speed. Cenifugal foe is depending no only on and / b, so igid body balaning is no possible any longe., bu also on he aio of b b e ( ( / b In ode o iniize he inease of enifugal foe and he elaed beaing foes A, ~ A, y~, ~, y~ / os( / sin( ( nea esonane b, speial balaning ehniques fo elasi oos have been developed. In a hid plane addiional balaning asses ae fied. This hid plane ould be loaed a he oo nea he loaion of aiu oo bending. The enifugal foe of his added ibalane all a opposie o he enifugal foe of he ben af. all and ediu sized oos ae opeaed even wih high speed appliaion usually below fis naual bending fequeny. o he igid body balaning is suffiien. u bigge oos T Dasad Insiue fo Eleial Enegy Convesion

20 oo developen 7/0 ehanial oo design have hei fis naual bending fequeny aleady below hei aed speed, so elasi balaning is neessay. ig -pole ubo geneaos fo sea powe plans wih beaing disanes of seveal ees, being opeaed a 3000/in = 50 Hz, have ypially o 3 naual bending fequenies below 50 Hz. Elasi balaning needs fo eah naual bending fequeny, whih is below aiu ahine opeaing speed, an addiional balaning plane o fi fuhe balaning asses. The enifugal foes of hese added ibalanes all oune-a o he bending odes of he oo a eah of he naual bending fequenies. Eaple 7..6-: a -pole sandad induion oo, 50 Hz, 500 kw: s naual bending fequeny a f b = 35 Hz. Elasi balaning wih 3 balaning planes is neessay. b -pole lage synhonous ubo geneao, 50 Hz, 000 W (powe plan ippendof, Geany: 3 naual bending fequenies lie in he fequeny ange Hz: Elasi balaning wih 5 balaning planes is neessay. T Dasad Insiue fo Eleial Enegy Convesion