Journal of System Design and Dynamics

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1 Journl of Systm Dsign nd Dynmics Vol. 1, No. 3, 7 A Numricl Clcultion Modl of Multi Wound Foil Bring with th Effct of Foil Locl Dformtion * Ki FENG** nd Shighiko KANEKO** ** Dprtmnt of Mchnics Enginring, Univrsity of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo, , Jpn E-mil: kfng@fiv.t.u-tokyo.c.p knko@mch.t.u-tokyo.c.p Astrct Foil rings r supposd to on of th st cndidts of supporting componnt for turo-mchinris cus of thir dsign simplicity, rducd wight nd siz, high spd nd tmprtur cpility, nd sy mintnnc. Among vrious typs of foil rings, multi wound foil ring (MWFB), which hd n dsignd nd frictd in our l, is sy to nlyz sttic chrctristics vn though lod cpility of which is smll comprd with othr typs of foil rings. In this study, thorticl modl of MWFB tking ccount of th ffct of th foil dformtion is dvlopd to prdict its sttic prformnc. Rynolds qution is solvd using Finit Diffrnc Mthod (FDM) to yild ir prssur distriution, whil th lstic dformtion qution is solvd y Finit Elmnt Mthod (FEM) to prdict th dformtion of th foil. Thn, th ov two qutions r coupld y svrl itrtions until th convrgnc critrion is rchd. Bsd on such clcultions, sttic chrctristics of MWFB such s lod cpcity, torqu r prsntd. Ky words: Multi Wound Foil Bring (MWFB), Foil Dformtion, Sttic Prformnc 1. Introduction *Rcivd 11 Apr., 7 (No ) [DOI: 1.199/sdd.1.648] Foil rings, which cn simply dfind s kind of complint, slf-cting hydrodynmic fluid film ring y functioning mint ir s luricnt, r considrd to th ky tchnology for oil fr turo-mchinry ppliction. Thy stisfy most of th rquirmnts of oil fr turo-mchinry, including dsign simplicity, rducd wight nd siz, high spd nd tmprtur cpility, nd rducd mintnnc (1). Thy r supposd to th st sustitutions of th rolling lmnts rings nd oil-lurictd rings. Howvr, t prsnt, th pplictions of foil rings minly rly on n xprimntl uild nd tst dvlopmnt squnc, which rsults svr tim consuming nd mony cost. So, thr is n urgnt rquirmnt to find out n ccurt prdictiv prformnc nlysis mthod. H. Hshmt () wrot crucil ppr in 1983, in which th lstic qution for foil dflction ws sustitutd into Rynolds qution nd finit diffrnc formuls for ths qutions wr drivd nd vrious structurl, gomtric nd oprtionl vrils wr discussd including th strt of ring rc, slction of lod ngl, numr of pds nd dgr of complinc. Thn, h nd C.P. Rogr Ku (3,4) nlyzd th dflction of singl 648

Journl of Systm Dsign nd Dynmics Vol. 1, No. 3, 7 ump, th friction forcs twn ump foils nd th housing or th top foil nd th structurl stiffnss nd dmping cofficints.

2 Journl of Systm Dsign nd Dynmics Vol. 1, No. 3, 7 ump, th friction forcs twn ump foils nd th housing or th top foil nd th structurl stiffnss nd dmping cofficints. In 1994, sd on th prvious modl, thy modifid th modl of ump foil y tking th curvtur ffct nd fixd nd ffct (th moving dirction of ump foil) into considrtion. Mnwhil, Png nd Crpino (5,6) pid ttntion to th ffct of mislignmnt nd ffcts of mmrn strsss, nd finit lmnt mthods wr usd to prdict oth th structurl dflctions nd th prssurs distriution. In 4, sd on th lsticity qution of foil ring y Hshmt (1983), ZC Png, MM Khonsri (7) improvd th mthod to dl with th ffct of foil dformtion y tking th motion of shft into ccount. Howvr, y prsnt, th ffct of foil dformtion on sttic prformnc is still not wll undrstood, prticulrly for th MWFB. This ppr tris to find wy to prdict th sttic prformnc of MWFB with considrtion of foil lsticity nd gs comprssiility. A thorticl modl of MWFB is dvlopd. Rynolds qution is solvd using Finit Diffrnc Mthod (FDM) to yild ir prssur distriution, whil th lstic dformtion qution is solvd y Finit Elmnt Mthod (FEM) to prdict th dformtion of th foil. Thn, th ov two qutions r coupld y svrl itrtions until th convrgnc critrion is rchd. Bsd on such clcultions, sttic chrctristics of MWFB such s lod cpcity nd torqu r prsntd.. Structur of Multi Wound Foil Bring Th sktch of th Multi Wound Foil Bring is shown in Fig.1.. Th componnts of ring r including th housing, th shft nd th wound foil with hmisphricl proctions distriutd on on sid y dqut intrvl. It is shown in Fig.1.. Th foil is md from phosphor ronz plt of.1mm in thicknss, mm in width with mny hmisphricl proctions whos hight is.mm. This plt is mnufcturd y wt tching procss nd proctions r md y spcilly dsignd ig. Th foil is wound triply nd fixd y th housing with th plt dg nt into housing t th position of 3mm. Th ring rdil clrnc twn shft surfc nd ring surfc is dsignd to µm (8). 3. Solution of Rynolds Eqution. Structur of MWFB. th gomtry of Foil Fig. 1. Th sktch of Multi Wound Foil Bring 3.1 Rynolds Eqution Th ir prssur distriution of foil ring is yildd from th Rynolds Eqution y tking th ir s th idl comprssil gs flow. Th dimnsionlss comprssil Rynolds Eqution undr isothrml condition is givn s () 3 p 3 p ph + ph =Λ ( ph) θ θ z z θ whr p h z 6µω R =, =, =, Λ= p C R p C p h z Thn, ccording to Fig., th film thicknss cn givn s following: (1) () 649

3 Journl of Systm Dsign nd Dynmics h 1 ε cos( θ θ ) S ( θ, z) + S ( θ, z) C Vol. 1, No. 3, 7 1 = + + (3) θ is th ngulr position of th minimum film thicknss, whil S ( θ, z) 1 nd S (, z) θ r th dformtions of foils (shown s th dshd lin in Fig. ), clcultd from lstic dformtion qution nd xplind in th nxt sction. And ε indicts th ccntricity rtio of ring, which is dfind s ε = / C. 3. Boundry Condition Th oundry conditions for th solution of Rynolds Eqution r: t θ= θs p = ( p / p) = 1 p = ( p/ p ) = 1 (4) t θ = θ p = θ t z =± ( L / D) p = ( p / p ) = 1 Th foil ring ssntilly dos not Fig.. Th sktch of th foil dformtion nd ir prssur gnrt sumint prssurs nd thr is no ring lod t th position of θ S, so, th ir prssur t th top of foil quls to p. In circumfrnc dirction, th film prssur will incrs du to th rltiv motion of shft nd foil. At n unknown ngulr position θ, th film prssur will dcrs to mint, clld Rynolds oundry condition. Hr th prssur must fulfill oth th mint prssur nd zro prssur grdint oundry condition. For oth sids of th ring r connctd with mint, th prssurs t sids r lso p. 3.3 Numricl Procdurs Finit Diffrnc Mthod (FDM) is usd to dl with this prtil diffrntil qution. Firstly, th ir film is dividd into grid, s shown in Fig. 4. Thn, th prssur grdint of point is dfind with th prssurs of th four points round it. Thn, Eq. (1) cn writtn in diffrnc formuls s: p 3 i+ 1, p i 1, h pi+ 1, p i 1, p 3 i+ 1, + pi 1, p h + 3p h + p h θ θ θ θ (5) p p 1 h p + 1 p 1 + h + 3p h z z z p p 1 p pi+ 1, p i 1, h + p h h =Λ + p z θ θ Th ov qution is nonlinr nd sustitutd s ( + + ) f pi 1, + p + pi 1, + p 1 + p 1 = (6) Th Nwton-Rphson mthod is usd to linriz Eq. (5) (), ( n) ( n) ( n) ( n) f ( n+ 1) ( n) f ( n+ 1) ( n) f ( n+ 1) ( n) f + ( p p i 1, i 1, ) + ( p p ) + ( p p i+ 1, i+ 1, ) pi 1, p pi+ 1, (7) ( n) ( n) f ( n+ 1) ( n) f ( n+ 1) ( n) + ( p p ) + ( p p ) = p p 1 1 i+ 1, i+ 1,

4 Journl of Systm Dsign nd Dynmics whr, n stnds for th old vlu, whil n+1 stnds for th nw vlu. Vol. 1, No. 3, 7 4. Solution of Elstic Dformtion Eqution 4.1 Gomtry Modl of Foil nd Assumptions Th foil is wound triply with ottom lyr contcting with th housing nd top lyr supporting th shft. So, th foil cn dividd into thr prts, which r nmd y thir position s top foil, mid foil nd ottom foil. For sy undrstnding, forc distriution nlysis is don with th cs tht th thr foils r unwrppd s shown in Fig. 3. Th foils r sprtd y proctions on th surfc of ottom foil nd mid foil nd th proctions r plcd t th middl position of two proctions on th othr lyr in circumfrntil dirction. Fig. 3 Gomtry modl nd Forc nlysis of foils tm m In Fig. 3, p-p is th ir prssur forcing on th top foil, F nd F stnd for th mt forc cting on mid foil from ottom foil nd top foil t th proction ( ), whil F tm stnds for th countrforc of F on top foil. In solving th dformtion of foils, th following ssumptions r md: For th ottom foil contcts with th housing, it is ssumd to hv no dformtion. Th proctions r considrd to rigid points, which mns th proctions hv no siz nd dformtion. In Fig. 3, th proctions r nlrgd for sir undrstnding. In fct, thy r smll proturncs on foil without siz. Th strt dg nd nd dg of th mid foil r fixd, whil th strt dg of top foil is fixd nd nd dg is fr, s shown in Fig. 3. Th foil is plt without thicknss, which mns th displcmnts t two surfcs qul to th displcmnt of middl sction. Th foil hs only ndd dformtion, for th ffct of foil xtnsion on th film thicknss cn nglctd compring to th ffct of ndd dformtion. 4. Elstic Dformtion Eqution Bsd on th principl of virtul work, virtul strin nrgy quls to virtul work don y xtrnl forcs { F }. Aftr knowing, w cn formult th strin nrgy y multiplying th strin with strss of lmnt, if w impos n ritrry (virtul) nodl * κ *. displcmnt { δ } nd rlvnt virtul strin { } * ({ } ) T * T T δ { F} = { κ } [ M ] dθdz = ([ B]{ δ} ) [ D][ B]{ δ} dθdz (8) For th ritrrinss of { δ * }, th ov qution cn trnsformd into { F} [ K] { δ} = (9) with th stiffnss mtrix for lmnt (s ppndix) T [ K ] = [ B] [ D][ B] dθdz (1) 4.3 Msh Gnrtion nd Boundry Condition With th gomtry of foil nd ssumptions givn ov, th foil is mshd y using 651

Journl of Systm Dsign nd Dynmics Vol. 1, No. 3, 7 rctngulr lmnt with 4 nods nd 1 dgrs of frdoms s shown in Fig. 4. It must nsurd tht thr is nod on vry proction during mshing, sinc th proctions suffrd th intrction forcs twn foils.

And vry sction my hv n m lmnts, whos vlus cn chngd to rch vrious clcultion prcisions. Th intgrs in th figur stnd Fig.

5 Journl of Systm Dsign nd Dynmics Vol. 1, No. 3, 7 rctngulr lmnt with 4 nods nd 1 dgrs of frdoms s shown in Fig. 4. It must nsurd tht thr is nod on vry proction during mshing, sinc th proctions suffrd th intrction forcs twn foils. Fig. 4 Th msh gnrtion of foil modl In Fig. 4, th solid points stnd for th proctions. To mk sur th proctions r ll t th cornrs of th lmnts, th positions of nods, w msh th foils s ov. And vry sction my hv n m lmnts, whos vlus cn chngd to rch vrious clcultion prcisions. Th intgrs in th figur stnd Fig. 5 A smll sgmnt of top foil supportd y for th sril numr of lmnt nd nod four proctions (th ons in rckts) undr th condition of n= m= 1. Th oundry conditions of th foil FEM modl r s following w = θ= θs, For fixd dgs : w = (11) θ θ= θs, For proctions : w = θ= θ z= z whr, w indicts th displcmnts of nods in vrticl dirction of foil surfc. 4.4 Film Thicknss with th Effct of Foil Dformtions Th film thicknss ws givn y Eq. (3), which mns th film thicknss is du to th ccntricity s wll s th dformtion of foils undr th imposd hydrodynmic prssurs. Sinc th top foil contcts with th gs film, th dformtion of it, S ( θ, z) 1, cn ddd to film thicknss dirctly, whil th dformtion of mid foil ffct th film thicknss y chnging th position of proctions, which support th top foil. To otin th vrity of film thicknss ( S ( θ, z) ) cusd y th dformtion of mid foil, w should find rltionship twn th displcmnts of th top foil nd th proctions on mid foil. Fig.5 shows smll sgmnt of top foil supportd y four proctions. Th displcmnts of four proctions r ssumd to y 1, y, y 3, nd y 4. A nd B r th distncs twn vry two proctions in θ nd z dirction. Thn, th vrity of film thicknss cusd y th dformtion of mid foil cn writtn s z θ θ θ S( θ, z ) = ( y4 y1) + ( y3 y4) ( y y1) + ( y y1) + y (1) 1 B A A A whr, ( θ, z ) is th position rltd to th first proction shown s th proction with displcmnt of y 1 in Fig

6 Journl of Systm Dsign nd Dynmics 5. Coupling Solution nd Rsult Vol. 1, No. 3, Coupling Solution To otin th ffct of th foil dformtions on ir prssur distriution, n pproprit itrtiv mthod is usd to coupl th Rynolds Eqution producing th ir prssur distriution with th lstic dformtion qution yilding th foil dformtion. Th logic chrt is shown in Fig.6. Th convrgnc critrion is (p-p old )/p <.1, whr th p nd p old stnd for th nw vlu nd old vlu of th ir prssur. Fig.6. MWFB progrm logic chrt with th ffct of foil dformtion 5. Clcultion Rsult To dtct th ffct of foil dformtion, w compr th ir prssur distriution with nd without foil dformtion undr th sm oprtion condition. Tl 1 shows th gomtricl nd luricnt dt of ir film, whil Tl shows th gomtricl nd mtril prmtrs of foils. For sy undrstnding, th ir film is cut t th strt position θ s nd unwrppd in circumfrntil dirction.fig. 7 shows th dimnsionlss ir prssur distriutions with nd without th ffct of foil dformtion undr oprting conditions of th rottionl spd ( w ) t 5 Krpm, th ccntricity rtio (ε ) t.8 nd th ngulr position of minimum film thicknss ( θ ) t 4 dgr. Tl 1 Th gomtricl nd luricnt dt of ir film Shft rdius (R) 1 mm Bring Lngth (L) mm Rdil clrnc (C). mm Asolut viscosity (µ ) 1.73*1 11 N*s/mm Amint prssur (p ) 11.3*1-3 N/mm Tl Th gomtricl nd mtril prmtrs of foils Numr of proctions in xil 4 Numr of proctions in circumfrntil dirction 15(top) or 16(mid) Young s Modulus (E) 9.8*1 4 N/mm Poisson s rtio (ν).43 Thicknss of foil (t).1 mm 653

Journl of Systm Dsign nd Dynmics Vol. 1, No. 3, 7 As shown in Fig.

Th prssur rchs its mximum vlu littl for th minimum clrnc tks plc. Th mximum vlu will dcrs from 3.145 to 1.9734.

Th dformtions of top foil nd mid foil undr this oprting condition r shown in Fig. 8.

7 Journl of Systm Dsign nd Dynmics Vol. 1, No. 3, 7 As shown in Fig. 7, th ir prssur in ovr mint rgion coms lowr if th ffct of foil dformtion is tkn into ccount. Th prssur rchs its mximum vlu littl for th minimum clrnc tks plc. Th mximum vlu will dcrs from to Th chng of ir prssur is du to th foil dformtions, which incrs th film thicknss. Th dformtions of top foil nd mid foil undr this oprting condition r shown in Fig. 8. () Air prssur distriution of rigid foil () Air prssur distriution with th ffct of foil dformtion Fig. 7 Th dimnsionlss ir prssur distriutions with nd without th ffct of foil dformtion () Th dformtion of top foil () Th dformtion of mid foil Fig. 8 Th dformtions of top foil nd mid foil 654

8 Journl of Systm Dsign nd Dynmics Vol. 1, No. 3, 7 For top foil, th dg conncting with mid foil hs no displcmnt, shown s th fixd dg in Fig. 8(). Th positions of th proctions supporting th top foil r lso ssumd to hv no displcmnts in th clcultion, for th displcmnts of thm r not contind in th top foil nd considrd in th modl of mid foil nd ddd to film thicknss y Eq. (1). In Fig. 8(), th qully spcd points with no displcmnt indict th proctions. Bsid ths points, ll of th ls rgions of top foil r found to hv minus dformtions, for th dirction of gs prssur is downwrds in Fig. 8(). Th dformtions of th sction nr th position of minimum film thicknss r notd to iggr compring to th ons of othr rs. Tht is cus th gs prssur rchs th mximum vlu t this sction s shown in Fig. 7. For mid foil, th two dgs ointing with th othr foils r oth ssumd to fixd, s pointd out in 4.1, which r dnotd s fixd dgs in Fig. 8(). It is sily dtctd tht th dformtions t th positions of proctions r scrggy with th ulgs nd hollows ppring ltrntly, in which th ulgs indict th proctions supporting mid foil, whil th hollows show th ons supporting top foil. Th rson of this fct cn xplind s tht th mid foil suffrs two sts of forcs with contrry dirction compring to ch othr from th proctions on th mid foil nd utton foil s shown in Fig. 3 nd th ction of thm mk th mid foil dform towrds opposit dirctions t th position of proctions, shown in Fig. 8(). Actully, th dformtions of mid foil t vry proction r not chnglss, nd th mximum vlu of dformtions pprs t th position of mximum ir prssur, which is not shown clrly in Fig. 8() for th diffrncs r too smll. Th rson cn givn s tht th countrforc of th supporting forc from th top foil cus mximl t this position following ir prssur s known in th discussion of Fig. 8(). Aftr knowing th dformtions of oth top foil nd mid foil, w now cn vlut th film thicknss with th ffct of foil dformtion y using Eqs. (3), (1). Th dimnsionlss film thicknss undr th mntiond oprting condition is shown in Fig. 9. Th foil dformtions ffct th film thicknss y not only shifting it in thicknss dirction ut lso forming som locl pks. Fig. 9 Th dimnsionlss film thicknss Fig. 1 Th dimnsionlss minimum film thicknss 5.3 Clcultion of Sttic Prformnc Prmtrs According to rfrnc (), th sttic prformnc prmtrs cn otind y. Th dimnsionlss lod cpcity is givn s following: intgrl of ir prssur p( θ, z) W Fx W = = F,tn x + Fy φl = (13) pr F y 655

9 Journl of Systm Dsign nd Dynmics Vol. 1, No. 3, 7 whr, F x nd F y r th dimnsionlss lods in horizontl dirction nd vrticl dirction, which cn clcultd from F L/ D 36 x x F = = ( p 1) sin( ) d dz pr θ θ L/ D i (14) Fy L/ D 36 F y = = ( p 1 ) ( cos( θ )) dθ dz pr L/ D i Th dimnsionlss torqu on th ournl y th viscous friction forc of ir cn writtn T L/ D 36 h p Λ T = = d dz + pcr L/ D θ 6h θ (15) whr S1( θ, z) + S( θ, z) h = 1 + ε cos( θ θ) + for < θ < θ C h = h for θ < θ < Clcultion Rsult of Sttic Prformnc Prmtrs nd Discussion Fig. 11 Th dimnsionlss lod cpcity Fig. 1 Th dimnsionlss torqu Fig.1, Fig.11 nd Fig.1 r plottd to show th dimnsionlss minimum film thicknss, th dimnsionlss lod cpcity nd th dimnsionlss toqu with nd without th ffct of foil dformtion t diffrnt rottionl spd nd ccntricity rtios with ssumd θ of 4 dgr. Th dshd lins indict th rsults with th ffct of foil dformtion, whil th solid ons show th rsults with th rigid foil. In Fig.1, thr is only on solid lin. Tht is cus h min is ust function of ccntricity rtio if th lsticity of foil is ignord, s shown in Eq. 3. Howvr, th film thicknss will incrs for th ffct of foil dformtion, s shown in Fig. 1 nd lrgr incrs is producd y highr rottionl spd. This is du to th fct tht highr rottionl spd rings lrgr foil dformtion y cusing highr hydrodynmic prssur, which finlly mks lrgr h min. Rfrring to Fig. 1, nothr fct cn noticd. Tht is, for th lstic foil rings, th chng of th minimum film thicknss with th rottionl spd will ris with th incrs of ccntricity rtio. Tht mns th lsticity of foil will mk th foil rings rott mor sfly t high rottionl spd nd high ccntricity rtio, for its lrgr minimum film thicknss, which dcrss th possiility of contct twn shft nd foil during rottion. As shown in Fig. 11, lik th norml rings, th lod cpcity of foil rings riss 656

10 Journl of Systm Dsign nd Dynmics Vol. 1, No. 3, 7 with th incrs of th rottionl spd nd th ccntricity rtio. Th lod cpcity of rings coms smllr t crtin rottionl spd nd ccntricity rtio if th ffct of foil dformtion is considrd. Th diffrnc twn thm coms lrgr with th incrs of ccntricity rtio t th sm rottionl spd, nd lso iggr dcrs in lod cpcity is cusd y highr rottionl spd t crtin ccntricity rtio. This cn xplind y th fct tht th incrs of foil dformtion with lrgr hydrodynmic prssur producd y highr rottionl spd or highr ccntricity rtio will uplift th film thicknss, which finlly dcrs th ir prssur nd th lod cpcity of foil ring. In Fig. 11, it is notd tht th rgion covrd y dshd lins is smllr thn tht covrd y solid lins. It mns tht th chngs in lod cpcity with rottionl spd nd ccntricity rtio dcrs if th foil is trtd s lstic. In nothr word, th lod cpcity of foil rings coms lss snsitiv to th chngs of rottionl spd nd ccntricity rtio du to th complincy of foil. Fig.1 prsnts th vrition of dimnsionlss torqu for th foil rings. Thr is dcrs in torqu if th foil dformtion is considrd, which mns tht th lsticity of foil will incrs th film thicknss nd finlly rduc th torqu of ring. And th highr ccntricity rtio is, th iggr th dcrs will. This, of cours, is du to th chng of ir prssur with th foil dformtion, nd highr ccntricity rtio will cus lrgr dformtion of foils nd iggr dcrs in ir prssur. 6. Conclusion Numricl clcultion modl of MWFB tking ccount of th ffct of th foil locl dformtion nd gs comprssiility is dvlopd to prdict its sttic prformnc. Th ir prssur distriution nd foil dformtion r yildd from th coupling solution of Rynolds Eqution nd lstic dformtion qution. And th sttic prformnc is clcultd from th pproprit intgrl of ir prssur. Th following conclusions cn otind from th rsults. i) Th ir prssur in ovr mint rgion dcrss if th ffct of foil dformtion is tkn into ccount. ii) Th film thicknss will ris for th ffct of foil dformtion nd lrgr incrs is producd y highr rottionl spd. iii) Th lod cpcity nd torqu com smllr y th ffct of foil dformtion. And th diffrncs will incrs with th ccntricity rtio nd rottionl spd. Rfrncs (1) Christophr DllCort, Lod Cpcity Estimtion of Foil Air Journl Brings for Oil-Fr Truomchinry Applictions, NASA/TM--978, Octor () H. Hshmt, J.A. Wlowit nd O. Pinkus, Anlysis of Gs-Lurictd Foil Journl Brings, ASME Journl of Luriction Tchnology, Vol. 15, 1983, pp (3) C.P. Rogr Ku, H. Hshmt, Complint foil ring structurl stiffnss nlysis: Prt1 thorticl modl including strip nd vril ump foil gomtry, Trnsctions of ASME, Vol.114, April 199, pp (4) C.P. Rogr Ku, H. Hshmt, Structurl stiffnss nd coulom dmping in complint foil ournl rings: Thorticl considrtions, Triology Trnsctions, Vol. 37, no.3, 1994, pp (5) M. Crpino, L. Mdvtz, JP Png, Effcts of mmrn strsss in th prdiction of foil ring prformnc, Triology Trnsctions 37(1): pp. 43-5,1994. (6) M. Crpino, JP Png, L. Mdvtz, Mislignmnt in complt shll gs foil ournl ring, Triology Trnsctions 37(4): pp , 1994 (7) ZC Png, MM Khonsr Hydrodynmic nlysis of complint foil rings with 657

11 Journl of Systm Dsign nd Dynmics Vol. 1, No. 3, 7 comprssil ir flow, Journl of Triology ASME 16 (3), JUL 4, pp: (8) Kitzw, S., Knko, S., nd Wtn, T., 3, Prototyping of Rdil nd Thrust Air Bring for Micro Gs Turin, Proc. of IGTC Tokyo 3. Appndix Th stiffnss mtrix for lmnt: 3 Et [ K] = 36 1 µ ( ) k1 k4 k k5 k6 k3 sym. k7 k1 k11 k1 k1 k8 k4 k k11 k9 k5 k6 k3 k1 k15 k16 k17 k k1 k1 k15 k13 k k18 k4 k k16 k14 k1 k19 k5 k6 k3 k17 k k1 k1 k5 k16 k7 k1 k11 k1 k k18 k15 k13 k1 k8 k4 k k1 k19 k16 k14 k11 k9 k5 k6 k3 k1 = 1 6µ , k = 8 8µ + 4 k3= 8 8µ + 4, k4= 3+ 1µ + 3 k5 = 3+ 1µ + 3, k6 = 3µ k7 = 1+ 6µ , k8 = 8 + 8µ + k9= + µ +, k1= 3 1µ + 15 k11 = 3 3µ + 3, k1 = 1 6µ k13 = µ + 1, k14 = µ + 1 k15 = 3+ 3µ + 15, k16 = 3+ 3µ + 15 k17 = 1+ 6µ , k18 = + µ + k19 = 8 + µ +, k = 3 3µ + 3 k1 = 3 1µ + 15 Nomncltur C = rdil clrnc D = dimtr of shft or ring t = thicknss of ump foil w = displcmnt in vrticl dirction E = modulus of lsticity x, y = horizontl nd vrticl coordint F = forc z, θ = xil nd ngulr coordint 658

12 Journl of Systm Dsign nd Dynmics F = dimnsionlss forc z = z / R Vol. 1, No. 3, 7 L = th lngth of ring θ = ngulr position of h min R = rdius of ournl ring θ = nd of hydrodynmic prssur S1( θ, z), S( θ, z) = th dformtion θ s = strt of th foil of top foil nd mid foil T = torqu υ = Poisson rtio U =linr vlocity ε = ccntricity rtio W = lod on ring Λ = ring numr W = W / pr ω = shft ngulr spd = ccntricity { δ * } = ritrry (virtul) nodl displcmnt h = film thicknss { κ * } = virtul strin h = h/ C [ B ] = th gomtry mtrix of lmnt mn, = numr of lmnt [ D ] = th lsticity mtrix p = ir prssur { F } = xtrnl forcs p = mint prssur [ K ] = th stiffnss mtrix of lmnt M = strss of lmnt p = / p p [ ] 659

CIVL 8/ D Boundary Value Problems - Rectangular Elements 1/7

CIVL 8/ D Boundary Value Problems - Rectangular Elements 1/7 CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS - In som pplictions, it m mor dsirl to us n lmntl rprsnttion of th domin tht hs four sids, ithr rctngulr or qudriltrl in shp. Considr