Analysis of Stress in PD Front End Solenoids I. Terechkine

Transcription

1 TD Sepembe 0, 005 I. Ioducio. Aalysis of Sess i PD Fo Ed Soleoids I. Teechkie Thee ae fou diffee ypes of supecoducig soleoids used fo beam focusig i he Fod Ed of he Poo Dive. Table 1 gives a idea abou equiemes fo ceai paamees of he soleoids (some quesio maks i he able eflec lack of udesadig of he siuaio ih he available space a he mome. Table 1 DTL MEBT SSR DSR Paamee Boe diamee 5 mm 5 mm 30 mm 30 mm Boe ype am am cold cold Field Iegal FI B dl (T cm Recommeded Bm (T Leff Bm Available eio gap (cm 5 (18? (18? 39 (18? 30 (3? The ceal mageic field is i he age (5 6 T he logiudial field pofile is quasi-ecagula. I eal siuaio, his is fa fom beig ue, ad o each equied Field Iegal (FI ihi available space ad ih sufficie magi fo sable ok, mageic field segh i he coil i some cases mus be as high as ~8.5 T. Wih his field level, sess developed i he sysem leads o defomaio ha ca esul i coil bouday sepaaio ad subseque coil quechig. Tadiioal mehod of solvig his poblem is applyig o he poblem some of ko echiques of sess maageme. Mehods of he sess maageme mus be aalyzed i cojucio ih he soleoid assembly echique befoe pooypes of he devices ae buil. A coveie mea of sess aalysis duig his developme sage ould be vey useful o acceleae he pocess of maeial ad geomey choice. Afe his choice is made, adiioal meas of sess aalysis, like usig ANSIS code, ca be used fo veificaio of sess maageme soluio ad adjusme of desig feaues. Because of elaively simple geomey of he soleoids, hich ae axially symmeical, i is possible o employ geeal mehods of aalysis based o diec soluio of diffeeial equaios ha descibe mechaical behavio of he sysem. I spie of is axial symmey, i he adial diecio sucue of he soleoids ca be quie complex ad usually icludes beam pipe, bobbi, epoxy-impegaed coil (maybe seveal layes, sepaaed by addiioal poecio shells, a colla, ad a seel yoke. Besides, duig idig sage, ieviable ie esioig povides addiioal pe-sess, ha mus be ake io he accou. As a esul, sysem of equaios descibig he sysem becomes quie lage, ad difficul fo diec aalysis. Neveheless, ih some simplificaios, his sysem of equaios ca be solved usig MahCad eviome, visualized, ad aalyzed. This oe povides a descipio of he appoach o he aalysis of sess maageme soluio fo supecoducig soleoids. Examples i his oe do o eflec eal desig of ay of meioed soleoid; hey ahe help o illusae ceai aspecs of he appoach. 1

2 TD Sepembe 0, 005 II. Basic Equaios Descipio of mechaical behavio of ay solid media is based o equaios of equilibium ad elasiciy. Equilibium codiio i a cylidical sysem ih volume foce of ay aue ca be ie as: d ( σ σ F(, /1/ d hee σ is omal sess, σ is ageial sess, ad F( is a exeal foce applied o a ui of volume (e.g. 1.0 m 3 of he maeial. Elasiciy equaios descibe defomaio of maeial subjec o omal ad ageial sess: u σ µσ, E, // du σ µσ d E hee µ is Poisso s aio. Combiig /1/ ad //, geeal equaio of defomaio ca be obaied: d u 1 du u µ + F( /3/ d d E Geeal soluio of he homogeeous pa of his diffeeial equaio (volumeic foce is zeo is ell ko [1]: C u h C1 +, /4/ hee C1 ad C ae cosas ha ca be foud by aalyzig bouday codiios. To fid a paicula soluio of he o-homogeeous equaio /3/, e eed o ko ha he volumeic foce F( looks like. I he case of mageic foce, F( j B( /5/ hee j (A/m is cue desiy i he coil idig. Radial pofile of mageic field ide he body of he coil i soleoids is close o liea, ad i he media plae oly z-compoe exiss, so e ill appoximae i usig he ex fom: (1 αβ (1 β Ro B ( Bi /6/ α hee α R i /R o is he aio of ie ad oue adii R i ad R o, ad β B o /B i is a aio of mageic field levels o he ouside ad ide bode of a paicula laye. As i is easy o check, B(R i B i, ad B(R o β B i. Value of B i mus be foud fo each laye by solvig coespodig mageosaic poblem aalyically (e.g. see [] o by usig a appopiae solve. I ou case, i is coveie o epese his value of mageic field o he ide bode of laye by iig: Bi, K I, /7/

3 TD Sepembe 0, 005 hee I is ie cue. This cue is usually kep cosa eve if ie diamee is diffee fo diffee layes. Ioducig ie coss-secio aea S ad coil compacio faco k S /S c, e ca ie do expessio fo he volumeic foce (fo each cue-cayig laye: kk αβ β F ( I S α R o α /8/ The, o o-homogeeous equaio /3/ looks like d u 1 du u + C01 + C0 d d /9/ µ kki αβ µ kki β ih C 01, ad C 0. E S α E S α Paial soluio of equaio /9/ ca be foud as: u p C01 + C0 /10/ 3 8Ro The he geeal soluio of his equaio is: C u C1 + C01 + C0 /11/ 3 8Ro So, fo each laye of he soleoid (ih o ihou cue, equaio /11/ gives a aalyical soluio fo defomaio. All he coefficies i equaios /9/ /11/ ae uique fo each laye. C0 1 ad C0 become ko he coil dimesios ae posulaed; C1 ad C mus be foud fo each laye by applyig coespodig bouday codiios, hich usually meas koig sess o sess elaioship o boudaies. Koig u, omal ad ageial (hoop sess ca be deived fom //: E du u σ + µ /1/ µ d ad E u du σ + µ /13/ µ d By combiig /11/, /1/, ad /13/, e ge quie geeal aalyical expessios fo omal ad ageial sess i ay laye of he soleoid: E C C0 1 C0 σ ( C1 (1 + µ (1 µ ( + µ + (3 + µ /14/ µ 3 8Ro E C C0 1 C0 σ ( C1 (1 + µ + (1 µ (1 + µ + (1 + 3µ /15/ µ 3 8Ro No e ca use /14 ad /15/ o fid uko C1 ad C fo each laye by solvig a sysem of algebaic equaios ha defie bouday codiios. This sysem is made by combiig coiuiy equaios fo omal sess a he boudaies beee layes m ad m+1 : σ σ /16/ m _ m+1 _ ad iefeece equaios a he same boudaies: R o 3

4 TD Sepembe 0, 005 u m u m +1 δ /17/ Hee δ has sese of a gap beee he o layes, so fo ovelappig layes i mus be egaive: δ < 0. I ohe ods, expessio /17/ meas ha afe assembly, he ie adius of he oue laye is equal o he oue adius of he ie laye: R m+1 + u m+1 R m + u m o R m+1 R m + δ. /18/ Fo ay umbe of layes, e have sufficie amou of algebaic equaios o fid coefficies C1 ad C fo each laye. Oe impoa pacical aspec is ha hile assemblig a mulilaye soleoid, ovelap ca oly be measued elaively o aleady assembled sub-sucue. So, o eadily available ifomaio exiss abou he fee sae ovelap ad meas mus be foud o obai his ifomaio. This evaluaio ill be a muli-sep pocess epeaed as may imes as he umbe of layes is. Defomaio u fo ay laye is defied elaive o he fee o udisubed posiio if he laye, hich ofe ca o be measued diecly. A algoihm o fid he fee sae ovelap is descibed belo. Sep 1: Addig laye # o he laye #1 R1o_0 is he oue adius of he laye #1 befoe addig he laye #. Fo he laye #1 i is he same as he fee sae posiio. Iiial ie adius of he laye # Ri_0 R1o_0 + δ1. /19/ Afe he o layes ae assembled, Ri_1 R1o_1. A his sep, Ri_1 Ri_0 + u(ri R1o_0 + δ1 + u(ri /0/ ad R1o_1 R1o_0 + u1(r1o. /1/ Combiig he las o equaios, e come o ha /17/ equies a he bouday beee he o layes. Afe solvig he sysem of equaios /14/ - /17/, e ca fid Ro_1 Ro_0 + u(ro. // Ro_1 becomes a efeece adius fo deemiig he ovelap beee he layes # ad #3. I he case of oud layes (see belo, koig u(ro allos ecalculaio of a fee sae posiio of he laye s bouday. Sep : Addig laye #3 o he assembly (#1+# Iiial ie adius of he laye 3 R3i_0 Ro_1 + δ3. /3/ Hee ovelap δ3 is measued elaively o he posiio of he oue bouday of he laye # afe i is added o #1. Fo sess evaluaio, oe eed o ko a fee sae ovelap, ha is he ovelap beee he udisubed machig boudaies of he layes: δ3_fee R3i_0 - Ro_0 δ3 + u(ro /4/ No e have hee layes ih o uko coefficies defiig defoma io i eac h of hem. So, e eed six equaios o fid a soluio. Fou equaios ill be made by defiig omal sess a he boudaies (equaios /16/, ad o equaios /17/ ill sae ha boudaies of he iefeig layes ae fused ogehe: Ri_ R1o_ ad R3i_ Ro_. Whe he sysem of equaios is solved e ko defomaios i each layes, icludig he las oe. This allos o fid a efeece adius fo deemiig he ovelap beee he layes #3 ad #4: R3o_ R3o_0 + u3(r3o /5/ 4

5 TD Sepembe 0, 005 The pocedue ca be epeaed as may imes as he umbe of layes is. Alhough mahemaically e ca fid a soluio, i ca be oo vague if e do o exa cly ko popey of all maeials e ae usig o build a soleoid. Some of hem ae composie maeials ih aisoopic popeies depedig o a echique used fo soleoid assembly, so i is ecessay be vey caeful i usig available daa. Hee e ll y o make pelimiay esimae of ha hese popeies ca look like, bu oly pope pooypig ca give us coec umbes. This is also ue fo hemal coacio coefficies of all maeials. This kid of daa is scace ad ofe ueliable fo composie maeials, so ou o judgme is eeded. Oce maeial popeies ae ko, oe ca use he model developed above o make seveal ieaios appoachig a soluio by chagig maeials ad geomeical paamees of a soleoid. III. Composie maeials i soleoid Some aeas of he assembled soleoid have aisoopic popeies ha deped o fabicaio echique. These aeas ae a coil, hich is oud usig NbTi ie ih some esio ad afe idig is impegaed ih epoxy, ad a sepaaio/poecio laye ha is oud usig fibeglass ape above he coil ad also impegaed ih epoxy. I boh cases, oud layes of he coil ae ude esio ad cued epoxy is o sessed. Duig coolig do o he pe-sess is applied, boh oud base ad cued epoxy ok simulaeously o fom a uified composie objec. I his pa, eleva popeies of composie sucue ill be evaluaed based o ko popeies of used maeials. Ho he coil is oud has a big impac of ha popeies o expec. We ill sa ih a egula layeed idig pae sho i Figue 1 belo. Figue 1: Sucue of oud ad impegaed coil. Pae 1 Hee, o ceae a base fo idig each ex laye, a elaively hick film mus be used fo laye sepaaio. This sucue suggess diffee mechaical popeies alog ie ad acoss he layes of idig. Alog he legh of ie (ageial diecio all pas of he coil coss secio ae acig i paallel, so effecive elogaio modulus of he sucue i his diecio is defied by elaive aea of NbTi ie ad ulaig maeial. If (fo simplificaio o accep ha he hee ulaig maeials used i he assembly have simila mechaical popeies (ha of cued epoxy, effecive elogaio modulus ca be foud as 5

6 TD Sepembe 0, 005 E E S + E S /6/ S + S Usig E 100 GPa, E 15 GPa, ad akig ie ad ulaio dimesios fom Figue 1, e ge effecive elogaio modulus i GPa: E E π D 4 bae + E [ D ( D + π Dbae ] D ( D + Hee D bae ad D ae diamees of bae ad ulaed sad. Compacio faco of his coil is 0.57, so epoxy fillig adds abou 10% o ha idig ihou fillig ould sho he seched. Pesece of epoxy is cucially impoa he coil is ude compessio: epoxy fillig ad bodig ues ha he sucue behaves like a solid body. Acoss he layes of idig (i omal diecio hee is moe complicaed combiaio of ieacio paes. The aio of he compoe coss-secios chages coiuously i he omal diecio (coodiae h. We ill evaluae popeies of his composie by aalyzig seial coecio of may hi layes ih coss secio shaed by ie ad ulaio (see Figue. I each laye, effecive elogaio modulus, as befoe, is defied by elaive aea of he o compoes of he sucue ha ac i paallel (simplificaio agai, so his effecive modulus vaies i he omal diecio. /7/ Figue : Illusaio o evaluaio of mechaical popeies of coil sucue i asvese diecio Takig io he accou /6/, ad all vaiable desigaios fom Figue, e ca ie do: h E R h E R + R + (1 R Eeff ( h if h R /8-a/ R + ad E ( h if h > R /8-b/ eff E Global elasiciy modulus ca be foud by akig iegal: R+ + d / 1 1 dh E R + + d / E ( h eσ 0 eff /9/ 6

7 TD Sepembe 0, 005 Evaluaio usig ko popeies of ie ad epoxy ih 46 µm ad d 100 µm gives E eσ 39 GPa. Aohe mode of coil idig is possible he hi ad flexible ulaio is used beee he layes of idig ha ca o peve ies fom fidig hee posiio i oches beee ies of he peviously oud laye (see Figue 3. The oly fucio of his ulaio is o povide addiioal poecio aga shos i he coil duig coolig do: Figue 3: Sucue of oud ad impegaed coil. Pae This ay highe desiy of idig ca be obaied, ha esuls i highe aveage elasiciy modulus. This modulus ca be evaluaed as i as doe fo he pae 1, bu ead of usig +d/ i Figue ad iegal expessio, oe mus use d (see Fig. 4, ha ca be foud geomeically ad depeds o he ulaio hickess, hich is sho as beig 50 µm i Figue 3. Wih his ulaio hickess, d 14.3 µm if he es of he paamees coespod o ha as used fo he pevious case of Pae 1 idig. Figue 4: Pae cell used fo aveage elasiciy modulus evaluaio. Calculaios usig /8/ ad /9/ give i his case elasiciy modulus i asvese diecio of ~58 GPa. I he azimuhal diecio, alog ies, oe mus apply /6/ o fid i GPa: E π R + E [( R + ( R + d π ] 4 bae R 4 bae E 7.6 /30/ ( R + d ( R + Compacio faco i his case is ~ Resuls of his aalysis ca be compaed ih es esuls i [3] o epoxy impegaed coils made of Nb 3 S ad NbTi usig ie ih ecagula coss-secio. Elogaio module as measued i he age GPa. Ohe ok of he same auhos fom NHMFL, [4], also povides some daa elaed o mechaical behavio of oud coils. Wih packig faco of 0.8 (compae ih 0.56 ad 0.68 i ou case, aveage modulus fo coil sechig as ~ 95 GPa. Fo coil compessio i adial diecio, aveage modulus as oly ~ 50 GPa. 7

8 TD Sepembe 0, 005 Resuls of his simple aalysis ad exisig daa sho ha aeio mus be paid o ho coil is fabicaed. Aohe oe: If idig ih glass ape ad subseque impegaio ih epoxy is used above he coil, simila cosideaios ca be used o fid popeies of he compoud maeial. Reliable glass ape (sad segh ad desiy ifomaio is eeded o make his ok havig sese. As a fis appoach, G-10 i he ad diecio daa ca be used. IV. Effecs of coolig do A oom empeaue, dimesios of all pas i he soleoid ca be ell coolled befoe hey ae added o he assembly. Afe seveal coceic pas ae assembled ogehe ih some ovelap, oe does o have immediae cool o he dimesio of pas because of defomaio (alhough dimesios sill ca be calculaed. We ill assume ha he oue suface of each pa is machied afe i is added o he assembly; so ha he as assembled oue diamee is ko evey ime. The he equied ie adius of he ex pa ca be foud usig /18/ if he ovelap δ beee he o layes is posulaed. The equaios /16 ad /17/, ha model he assembly ad ae descibed above, ae based o he diffeeial equaio fo defomaio /3/. As a esul, hee is o eed o calculae exac values of adii; sligh chage of he adius of each bouday adds a coecio o he esul popoioal o u/r. The value of ovelap δ is impoa hough because i defies he iiial defomaio. So, he goal is o fid e values of ovelap fo each couple of laye afe coolig do. Oe of ays of doig i is o use esuls of he sess aalysis a oom empeaue ha povides us ih koledge of he fee dimesios of each laye afe e fid coespodig values of defomaio. Koig he fee o udisubed posiios, e ca apply o hem he ko hemal coacio popeies o fid udisubed dimesios a 4 K. I is saighfoad he o fid ovelap a each bouday ad poceed o solvig he sysem of equaios /14/ - /17/ ih (maybe modified sucual popeies of used maeials. Aohe ay is o eevaluae ovelap ifomaio based o he fee ovelap daa ko fom he oom empeaue case. Fo he iefeece beee he layes m ad, oe ca ie, simila o /17/: (300K (300K δ R R R (1 χ R (1 χ m i mo i _ fee mo _ fee (300K o δ m δ m Rmo ( χ χ m /31/ hee χ m ad χ ae iegaed coacio coefficies of he layes i he empeaue age fom 300 K o 4 K, R mo is he oue adius of he ie pa, ad idexes mo ad i efe o he machig oue ad ie adii of he layes m ad. So, agai, he oly hig ha is eeded hee is daa of fee ovelap a oom empeaue ha ca be foud as as descibed above. Afe e ko ovelap daa, he sysem of equaios /14/ - /17/ mus be solved oly oce o give all he ifomaio e eed. m 8

9 TD Sepembe 0, 005 V. Effec of ie esioig duig idig. Wie esioig is immie duig idig. Quesio aises o ha impac his esioig has o he soleoid mechaical behavio ad hehe his esioig ca be useful hile solvig he sess maageme poblem. Le s coside a laye of ie oud ih esio foce F above he bobbi o peviously oud laye. We ca ea his idig pocess as addig a solid cylidical laye o he exisig sucue ih he ovelap δ a oom empeaue ha coespods o he esio foce: F δ E S, /3/ Whee E is Youg s modulus ad S is coss-secio aea of he sad (ie. Each ex laye ill have diffee iefeece because of diffee adius R ad (maybe esio foce F. Fo each oud laye, he same basic equaio /3/ ill ok ih simila se of bouday codiios. Maeial popeies mus be used caefully hile solvig hese equaios. A his sage, liquid epoxy o o epoxy is used, so oly ie ad ielaye ulaio mus be ake io he accou hile evaluaig sucual popeies of oud layes. Afe epoxy is cued, e values of maeial popeies ad modified hemal coacio coefficie (see belo mus be used o aalyze sess afe coolig do. The appoach o sess calculaio a oom empeaue developed i he pevious chapes is fully applicable because a each idig sep e ko he oue adius of he las oud laye ad he ie adius of he ex laye. Duig coolig do, because hemal popeies of he layes of idig ae ideical, ovelap ill chage oly because of defomaio chage due o sess edisibuio, so agai he same algoihm ca be used. VI. Mechaical behavio of oud ad epoxy impegaed coil Themal coacio of coil sucue is he issue o pay aeio o. If coil s ie is oud ih esio ad he impegaed ih epoxy ad cued, a oom empeaue cued epoxy is usessed hile he coil ie is seched. So, e equilibium ie adius a oom empeaue mus be foud ha coespods o he case he his laye of coil assembly is emoved fom a bobbi (o pevious laye i as assemble o. Koig his adius allos oe o fid iefeece eeded o calculae displacemes ad sess. Aveage popeies of he maeial ca be foud i a ay simila o ha as used i he chape 3 above, so fo each maeial e ill use aveaged value of elasiciy modulus. Afe idig a laye of ie ih esio F, icease of adius is defied by /3/: F F EW SW Whe oud ad cued laye is emoved fom he bobbi, i coacs eleasig he esio uil foce equilibium is eached beee compessed epoxy fillig ad seched ie. Ne equilibium posiio ca be foud fom equaio: eq F eq EW SW ES /33/ 9

10 TD Sepembe 0, 005 Hee eq shos icease of ie adius of he oud coil ih cued epoxy afe i is emoved fom he bobbi i as oud o elaive o ha ihou epoxy; S is ulaio coss-secio pe oe u f he idig. Fom /33/: eq F /34/ EW SW EW SW (1 + ES A his posiio, emaiig ie sechig foce pe ie is F F /35/ EW SW 1+ ES ad adjused ovelap F δ eq F /36/ E S + E S Mechaical behavio of he composie sucue, amely displaceme elaive he e equilibium posiio, ca be deived based o he ko popeies of he compoes makig he sucue ad he e equilibium posiio. If displaceme elaive o his posiio is x, coespodig foce ca be calculaed as: x x + eq F ( x + eq S Eeff EW S Σ W ES /37/ Usig /31/, e ca simplify his expessio o ie: S Σ Eeff EW SW + ES /38/ ha oe could expec jus lookig a /36/. W W VII. Themal coacio of epoxy impegaed sucues Fo supecoducig soleoids, i is also impoa o ko hemal coacio popeies of composie maeials o calculae ovelap a 4K. We ca make a esimae usig he appoach simila o ha as used above. Duig coolig do, due o diffee coefficies of coacio of coil ie ad epoxy, oe ca expec edisibuio of ieal sess ad chage of he equilibium posiio of ie ad oue adii. A oom empeaue, ie adius of he coil i fee codiio is defied by he adius of he bobbi b, ovelap δ, hich is a fucio of ie esio ( b + δ, ad he sechig foce of compessed fillig maeial (epoxy ih he ie adius of b (hich is he oue adius of he bobbi. Afe coolig do, fee sae dimesios of he idig ad of he epoxy fillig chage: 4K (1 χ ( b + δ (1 χ /39/ ad 4K (1 χ, /40/ epo Ne equilibium posiio of he ie adius is commo boh fo idig ad fo epoxy fillig. Also ieal foce mus be compesaed, so b epo 10

11 TD Sepembe 0, 005 eq ep ep eq N i S E W W N i S E hee N is he ieal foce of ieacio beee ie ad epoxy fillig. Usig /39/, /40/, ad /41/, ad akig io he accou /3/, e ge he ex expessio fo he e equilibium adius a 4 K: SW EW χ + S E χ + F eq b 1 /4/ SW EW + S E The secod em i he paeheses above is he effecive iegaed hemal coacio coefficie of a composie maeial. As befoe, S icludes o oly epoxy, bu also ielaye ulaio. Youg s modulus E W of he sad mus be ake fo he logiudial diecio because e deal hee ih laye elogaio ha aslaes io coespodig adius chage. Usig ypical daa oe ca ge: S E N, χ 1.9 * 10-3 S E N, χ 4.3 * 10-3 S E χ + S E χ 10 N, hich is compaable ih he esio foce usually used duig idig. So, ie esio ca modify maeial shikage i he case he epoxy is used as a fille maeial. Fo sue, coil idig pae (compacio faco ill modify umbes sho above. Takig io he accou compacio faco, expessio fo he effecive shikage coefficie ca be ie i he fom: k E F χ + χ + k E E S χ eff /43/ k E 1+ k E Fo k 0.63, ih F 50 N, effecive shikage χ eff Wih F 0, effecive shikage χ eff If he coil spool is made of sailess seel (χ , i ould be a good idea o apply some esio o ie o o lose sess duig coolig do. VIII. Coclusio This oe foms a base fo sess aalysis of supecoducig soleoids: 1. Basic equaios ee deived o aalyze sess ad defomaios;. Expessios ee foud o esimae mechaical popeies of composie maeials like oud ad epoxy impegaed Nb-Ti coil; 3. Algoihm of sess aalysis a LHe empeaue as suggesed; 4. Mehod of sess aalysis he ie esio is used duig idig as suggesed 5. Themal shikage of epoxy impegaed coil as aalyzed o sho ha i depeds o coils idig paes. /41/ 11

12 TD Sepembe 0, 005 This se of algoihms ill be applied ad deived expessios ill be used o make a choice of maeials ad assembly echiques of supecoducig focusig coils i he PD fo ed. IX. Refeeces [1] J.P. De Haog, Segh of maeials, Dove Publicaios, Ic, N-Y, [] D. Buce Mogomey, Soleoid mage desig. The mageic ad mechaical aspecs of esisive ad supecoducig sysems. Wiley-Iesciece, NY, [3] I. Dixo, e al. Axial Mechaical Popeies of Epoxy Impegaed Supecoducig Soleoids a 4. K, IEEE Tas. o Applied Supecoduciviy, v. 10, No. 1, Mach 000 [4] I. Dixo, e al, Mechaical Popeies of Epoxy Impegaed Supecoducig Soleoids, IEEE Tas. o Mageics, v. 3, No. 4, July