GAS FOIL BEARING ANALYSIS AND THE EFFECT OF BUMP FOIL THICKNESS ON ITS PERFORMANCE CHARACTERISTICS USING A NON-LINEAR MATRIX EQUATION SOLVER

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1 GAS FOIL BEARING ANALYSIS AND THE EFFECT OF BUMP FOIL THICKNESS ON ITS PERFORMANCE CHARACTERISTICS USING A NON-LINEAR MATRIX EQUATION SOLVER T. Moasunp. Jamir 1)*, S. K. Kakoty 1), Karuna Kalita 1) 1) Dpartmnt of Mchanical Enginring, Indian Institut of Tchnology Guwahati. Guwahati , India. * Corrsponding author: sunpbokdi@gmail.com ABSTRACT Gas foil barings (GFBs) satisfy many of th ruirmnts notd for novl oil-fr turbomachinry. Howvr, GFBs hav a limitd load carrying capacity. This papr prsnts a numrical modl in ordr to assss th prformanc charactristics of gas foil barings. Th finit diffrnc schm has bn usd to discrtiz th govrning Rynolds uation and th prssur is calculatd by solving non-linar matrix uation using Nwton-Raphson tchniu. Th static prformanc analysis has bn carrid out. Th computational analysis hav bn compard with th xprimntal and thortical rsults availabl in th litratur and th ffcts of bump foil thicknss, numbr of bumps and bump complianc cofficint on th load carrying capacity at diffrnt rotor spd hav bn invstigatd. Th rsults of th study show that too thin bump foil thicknss may lad to a significant dcras in th load capacity. Howvr for accurat prdictions of th foil baring prformancs, mor dtails foil structur of 1D and 2D finit lmnt modl should b considrd. KEYWORDS Gas Foil Barings (GFBs), Nwton-Raphson tchniu, bump foil thicknss, bump foil complianc 1. INTRODUCTION Gas foil barings hav bn succssfully usd in various typs of rotating machinry, such as turbochargrs, auxiliary powr units, and gas turbin ngins. It offrs svral advantags ovr th convntional rigid surfac barings including highr load capacity, lowr powr loss and bttr stability. Ths advantagous charactristics of foil barings hav bn dmonstratd through many xprimntal invstigations [1 4]. Th foil structur of a gas foil baring consists of two parts: A smooth top foil that provids th baring surfac and a corrugatd bump foil that provids rsilint support for th top foil. Undr th action of th hydrodynamic prssur, th foil structur dforms. This yilds an apprciabl chang in th lubricant-film profil. Thrfor, th fluid film prssur must b coupld to th dformation of th foil structur in ordr to ascrtain th charactristics of th foil baring DOI : /ijmch

2 prformanc. From this point of viw, many analytical studis hav bn conductd basd on a rang of structural modls. Walowit and Anno [5] first prsntd an lastic modl for a singl bump, whr in th bnding and tnsil stiffnss of th bump wr considrd. This structural modl was usd by Hshmat t al. [6] in an invstigation of th stady stat bhavior of a foil baring. Ku and Hshmat [7, 8] also dvlopd a thortical modl of th corrugatd foil strip dformation introducing friction forc btwn th bump foils and th baring housing or top foil, and th ffct of bump gomtry on th foil strip complianc. An xprimntal procdur has bn prsntd to invstigat th foil strip dflction undr static loads. Through an optical track systm, bump dflction imags ar capturd indicating that th horizontal dflction of th sgmnt btwn bumps is ngligibl compard to th transvrsal dflction of th bumps. Rubio and San Andrés [9] also prsntd an xprimntal and analytical procdur aimd to idntify th structural stiffnss for an ntir bump-typ foil baring. Ruscitto t al. [10] prformd a sris of load capacity tsts of bump typ GFBs. Th static prformanc analysis of GFBs considring thrdimnsional shap of th foil structur was prformd by L t al. [11]. Using this modl, th dflctions of intrconnctd bumps ar compard to thos of sparatd bumps, and th minimum film thicknss is compard to thos of prvious modls. In addition, th ffcts of th top foil and bump foil thicknss on th foil baring static prformanc ar also valuatd. L t al. [12] studid oprating charactristics of th bump GFBs considring top foil bnding phnomnon and corrlation among bump foils. Kim and San Andr s [13, 14] in comparisons with tst data [10] validat a GFB modl that implmnts th simpl lastic foundation modl with formulas for bump stiffnss takn from [15]. Thy did analysis of GFBs intgrating 1D and 2D FE top foil modls. 2D FE modl prdictions ovrstimat th minimum film thicknss at th baring cntrlin, but slightly undrstimat it at th baring dgs. Prdictions from th 1D FE modl compar bst to th limitd tsts data, rproducing closly th xprimntal circumfrntial profil of minimum film thicknss. Svral approachs hav bn put forward to calculat th stady stat charactristics of gas foil barings by diffrnt rsarchrs. Kospl [16] dvlopd a forward itration mthod. Finit diffrnc mthod has bn usd for solving th fluid modl and finit lmnt mthod has bn usd for th structural modl. Hshmat t al. [6] has dvlopd an invrs mthod whr finit diffrnc mthod has bn usd for solving fluid modl and th structural modl. A Nwton- Raphson mthod has bn usd to solv th finit diffrnc mthod. A modifid forward itration mthod has bn put forwardd by Carpino and Png [17] whr finit lmnt mthod has bn usd for both fluid modl and structural modl. Subsuntly, this modifid itration mthod has bn usd by Carpino and Png [17] for calculating stady stat charactristics of gas foil barings whil calculating stiffnss and damping cofficints for lastically supportd gas foil barings. L t al. [11] dvlopd an itrativ mthod for calculating th static prformanc analysis of foil journal barings. A thr dimnsional modl has bn considrd for th foil structur and finit lmnt mthod has bn usd for solving fluid modl and th thr dimnsional foil modl. In this papr finit diffrnc mthod has bn usd to discrtiz th govrning Rynolds uation. Th discrtizd uation has bn writtn in th form of a uadratic function for vry nod and Nwton-Raphson tchniu hav bn usd to solv th non-linar matrix uation. Th stady stat charactristic of th GFB ar thn assssd and th stady stat charactristics rsults hav bn compard with th xprimntal and thortical rsults availabl in th litratur. Th 16

3 ffcts of various baring paramtrs on th load carrying capacity of th baring hav bn nvisagd 2. GAS FOIL BEARING DESCRIPTION 2.1 Govrning Euations Figur 1 shows a schmatic viw of bump typ GFB. Th Rynolds uation dscribs th gnration of th gas prssur (p) within th film thicknss (h) and for an isothrmal, isoviscous idal gas this uation is, 3 p 3 p ( ph) ( ph) ph ph + = 6µω R + 12µ x x z z x t (1) whr, ( x, z ) ar th circumfrntial and axial coordinats on th plan of baring,. Th prssur taks ambint valu p on th sid boundaris of th baring. a Fig.1 Schmatic viw of bump typ GFB Th boundary conditions for th solution of En. (1) ar p = pa at θ = 0 and 2π p = p at z = 0 and L (2) a Th film thicknss ( h ) for a prfctly alignd journal configuration is givn by h = c + cos( θ φ) + wt (3) Whr,c, and wt ar th assmbld claranc, journal ccntricity and th lastic dformation of th foil structur rspctivly. 17

4 Thrfor En. (1) and En. (3) ar coupld through th hydrodynamic prssur and lastic dformation of th foil structur as a function of th hydrodynamic prssur. Th abov govrning uations can b normalizd by using th following substitutions as follows: α p x S = a H = h z = Z = θ C C R R Y p w y = P = ε = W = t L p C C a (4) Hnc stady stat non-dimnsional Rynolds uation as givn in En. (1) is givn by whr 3 P 3 P PH PH PH + Λ = 0 Z Z θ θ θ Λ 6µω = p a ( R ) 2 C (5) And similarly th non-dimnsional film thicknss is givn as, H = 1 + ε cos( θ φ ) + W (6) whr ε is th ccntricity ratio. 2.2 Stady Stat Formulation of Rynolds Euation Th non-dimnsionalizd Rynolds En. (5) has bn discrtizd using a finit diffrnc approach. For all th drivativs prsnt in En. (5) cntral diffrnc formula hav bn usd. Figur 2 shows an xfoliatd viw of a baring showing th msh siz ( θ Z ). m and n ar th divisions along θ and Z rspctivly. Th dimnsion of th prssur in th domain in matrix m + 1 n + 1 rprsnt by P. Th vctor rprsntation of P is p and its lngth form will b ( ) ( ) is( m 1)( n 1) + +. Similarly foil dflction and film thicknss in matrix forms ar rprsntd by W and H rspctivly. If (i, j) is any arbitrary point in th matrix rprsntation, its position in th = j 1 m i. Th four adjacnt nods associatd vctor rprsntation can b writtn as ( )( ) with nod (i, j) ar ( i 1, j), ( i 1, j), ( i, j 1) and ( i, j 1) + + as shown in Fig 2. Th discrtizd form of uation (5) for (i, j) th nod or th nod can b writtn in a uadratic form as ( ) T p A p + b p = 0 (7) whr = { P( i, j 1) P( i 1, j) P( i, j) P( i+ 1, j) P( i, j+ 1) } p and its siz is 5 1, T 18

5 A H 3 3 ( i, j 1) ( i, j 1) ( Z ) 4( Z) 3 3 H( i 1, j) H( i 1, j) ( θ ) 4( θ ) H H 2H 2H H H ( i, j) ( i, j) ( i, j) ( i, j) ( i, j) ( i, j) = ( Z) ( θ ) ( θ ) ( Z) ( θ ) ( Z) 3 3 H( i+ 1, j) H( i+ 1, j) ( θ ) 4( θ ) 3 3 H( i, j+ 1) H(, 1) i j + 2 4( 2 Z) 4( Z ) H A is a 5 5 non-symmtric matrix, ΛH ΛH ( i 1, j) ( i+ 1, j) b = 0 0 0, 2( θ ) 2( θ ) b is a row vctor of siz 1 5. Fig. 2: An xfoliatd viw of a baring showing th msh siz ( θ Z ) Th forc of th gas film acting on th journal can b computd by intgrating th prssur ovr th baring surfac. According to th coordinat systm illustratd in Fig. 3, this intgration can b writtn as: L D 2π Wx Pcosθdθ dz = L D 0 19

6 L D 2π Wy PsinθdθdZ = (8) L D 0 For numrical intgration, Simpson s on third rul has bn usd. Finally th total non-dimnsional load is givn by 2 2 W = Wx + Wy (9) dθ θ 180-dθ 1. Y P sinθ rdθdz P rdθdz -P cosθ rdθdz X Fig. 3: Th coordinat systm and th sign convntion of th journal forcs 3. SOLUTION METHOD In th prsnt algorithm, a fixd coordinat systm has bn considrd (X and Y). Thrfor, for stady stat uilibrium, th horizontal load componnt should bcom zro. Th flowchart of th solution procss is shown in Fig. 4. For a givn ccntricity ratio, th attitud angl is varid till th horizontal load componnt approximatly bcoms zro. Bisction mthod has bn usd for brackting and also to calculat th corrct attitud angl. Th load capacity is simply ual to th vrtical load componnt. Initially, th non-dimnsional top foil dflction W is assumd as zro. To start with th itration, th non-dimntionalisd prssurs at all th msh points ar assumd as 1 (i.. ambint prssur) and thos at th boundaris ar also st to th ambint prssur. Nwton-Raphson mthod has bn usd to solv th discrtizd Rynolds En. (7) for 20

7 air prssur p in this papr. Onc th prssur is found out th non-dimnsional top foil dflction W is calculatd. Aftr that, substituting th nw valu of non-dimnsional top foil dflction W in th uation of film thicknss, th uadratic uation is solvd for all th msh points to stimat th prssur at all ths points. This uadratic uation would not b satisfid for th prssurs which ar assumd to b constant in th bginning. Hnc, th itrativ procss is carrid out until th following convrgnc critrion is satisfid. 3.1 Nwton-Raphson Mthod ( k 1) ( k ) ( p ) p ( ) ( k ) ( p ) 10 In ordr to apply Nwton-Raphson mthod w dfin th rsidu vctor r in En. (7) as ( T ) whr 1,2,..., ( 1)( 1) r = p A p + b p = m + n + (11) whr r is th rsidu vctor for th nod and its lngth is 5 1. Th lngth of th rsidu vctor r is ( m 1)( n 1) ( ) ( ) ( ) + +. Th rsidu vctor of th th nod for th k-th itration is ( k ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) whr 1,2,..., ( 1)( 1) r k T k k k k = + = m + n + p A p b p (12) Th th row of th snsitivity matrix, G is calculatd as ( ) ( k ) ( ) ( k) 6 { } ( k ) ( ) ( ) whr 1,2,..., ( 1)( 1) G T ( ) ( k ) T k = + ( ) + = m + n + p A A b (13) Th siz of th matrix G is ( m + 1)( n + 1) ( m + 1)( n + 1) ( k + 1) ( k ) ( k ) ( 1) ( ) ( k ) k. Th prssur is updatd as p = p λ G r (14) whr ( k +1) p and ( k ) p ar th prssurs at ( k + 1) th and k th itrations rspctivly. ( k) G (10) is th snsitivity matrix at k th ( k) itration. λ is th k th stp lngth. Th computation is stoppd whn th 6 rror btwn two succssiv itration stps is smallr than

8 Fig. 4: Flowchart of solution algorithm 4. MODELLING OF FOIL SUPPORT STRUCTURE Th lastic support structur considrd in th prsnt analysis is a simpl foundation modl, th original work of Hshmat t al [6] which most of th publishd modls ar basd on. This analysis rlis on svral assumptions which othr rsarchrs hav also rproduc: 22

9 (1) Th stiffnss of a bump strip is uniformly distributd throughout th baring surfac, i.. th bump strip is rgardd as a uniform lastic foundation. (2) A bump stiffnss is constant, indpndnt of th actual bump dflction, not rlatd to or constraind by adjacnt bumps. (3) Th top foil dos not sag btwn adjacnt bumps. Th top foil dos not hav ithr bnding or mmbran stiffnss, and its dflction follows that of th bump With this considrations, th dformation ( w t ) dpnds on th bump complianc (α ) and th avrag prssur across th baring width as shown Fig. 5, w = α( p p ) (15) t whr, p and α ar th arithmtic man prssur in th axial dirction and complianc of th bump foil. a Fig.5 Singl sgmnt of Bump foil [18] Th dimnsionlss dflction of th bump is givn by W = S( P 1) (16) Coupling of th simpl modl uation (16) with th solution of Rynolds uation (5) is straightforward for th prdiction of th static prformanc of GFBs. 5. VALIDATION, RESULTS AND PREDICTIONS OF THE MODEL 5.1. Validation Th validity of th prsnt analysis and computational program ar assssd by comparison of prdictions with publishd data availabl in opn litratur. As shown in figur 6 and 7, th 23

10 prsnt computation analysis ar assssd with Hshmat t al [6] and Png and Carpino[5]. From th comparison, it has bn obsrvd that th prsnt rsults ar fairly in good agrmnt with thos from th rfrncs. Not that, a baring rducs to a ordinary gas baring for complianc cofficint S=0. Figur 8 shows th prssur profil of th GFB for L/D =1.0, Λ=1.0, S=1 and ε = 0.6. Fig. 6: Validation of load carrying capacity with Hshmat t al [6] and Png and Carpino[5]. 24

Fig. 7: Validation of attitud angl with Hshmat t al [6] and Png and Carpino[5]. Fig 8: Prssur Profil of GFB for L/D =1.0, Λ=1.0, S=1 and ε = 0.6 5.2.

11 Fig. 7: Validation of attitud angl with Hshmat t al [6] and Png and Carpino[5]. Fig 8: Prssur Profil of GFB for L/D =1.0, Λ=1.0, S=1 and ε = Configuration of Tst GFB Tabl 2 provids th paramtrs for th tst foil baring givn in [10] and Figur 5 dpicts th configuration of th bump foil strip. Th foil baring is a first gnration typ with on 360º top foil and on bump strip layr, both mad of Inconl X-750. Th top foil and bump layr ar spot wldd at on nd to th baring slv. Th othr nd of th top foil is fr as wll as th nd of th bump strip layr. Th journal rotational dirction is from th fr nd of th top foil towards its fixd nd and all tsts in wr prformd with air at ambint condition. Tabl 2: Dsign dtails of foil baring, rfrnc [10] Baring radius, R=D/2 Baring lngth, L Foil arc circumfrntial lngth, l x mm 38.1 mm 120 mm Radial journal travl, c ( ~ Claranc, C ) 31.8 µm J Top foil thicknss t t µm Bump foil thicknss, t b µm Bump pitch, s mm Half bump lngth, l mm Bump hight, h b mm Numbr of Bumps 26 Bump foil Young s modulus, E 214 GPa b 25

12 Bump foil Poisson s ratio, υ Rsults Effct of Bump Foil Thicknss on th Static Prformancs Th GFB computational tools for simpl lastic foundation modl hav bn dvlopd which prdict th static prformanc of th GFB. Th prdiction for th simpl lastic modl uss a msh of 90 and 10 lmnts in th circumfrntial and axial dirctions rspctivly. Th sam msh siz wr usd for th finit diffrnc numrical schm solving Rynolds En. (5) and calculating th hydrodynamic gas film prssur. Th bump foils which is a masur of th foil structur has bn varis and its charactristics has bn obsrvd on th prformanc of load capacity of th baring. (a) 26

13 (b) Fig 9: Effct of bump foil thicknss on th load carrying capacity of GFBs at diffrnt ccntricity whn th rotating vlocity is (a) 30,000 rpm and (b) 45,000 rpm Figur 9 (a) and (b) shows th prdiction of th load carrying capacity for various bump foil thicknss at diffrnt ccntricity ratios and at th rotating vlocity of 30,000 rpm and 45,000 rpm rspctivly. As th ccntricity incrass, th load capacity incras linarly for all valus of bump foil thicknss, which bcoms significant at highr ccntricity ratios and at highr rpm. Th incras though is not so prominnt for smallr bump foil thicknss at tb = 0.05µ m compard to highr foil thicknss at tb = 0.20 µ m oprating at diffrnt spd which is minnt from figur 9 (a) and (b). Ths rsults hnc indicat that too thin a bump foil thicknss can lad to a significant dcras in th load capacity du to th xcssiv foil structur dflction. In addition, a larg incrmnt of th load capacity from variation of th foil stiffnss may not b xpctd undr a simpl foil structur Effct of numbr of bumps and bump complianc cofficint on th load carrying capacity of GFBs Fig 10: Numbr of bumps vrsus load carrying capacity of GFBs 27

14 Fig 11: Complianc cofficint vrsus load carrying capacity of GFBs Figur 10 dpicts th variation of th load capacity of th baring as th numbr of bump foils varis. Th numbr of bumps ar calculatd by dividing th foil arc circumfrntial lngth ( l x ) by th bump pitch (s), and as th bump numbr incrass th bumps bcom mor crowdd and stiffr which rsultd in an incras in th load capacity of th baring as dpictd from fig. 10. Th ffct of complianc cofficint (S) which is a function of th bump complianc, on th load capacity has bn portrait in figur 11. Th load capacity gnrally dcrass as th complianc cofficint incrass, which prdicts that as th bumps complianc incrass th stiffnss tnds to dcras, as bump complianc is th invrs of stiffnss hnc a dcras in th load prformancs of th baring is minnt. 6. CONCLUSION In this study, a simpl lastic foundation modl of th foil has bn considrd whr in th stiffnss of th bump strip ar uniformly distributd throughout th baring surfac, i.. th bump strip is rgardd as a uniform lastic foundation and constant, indpndnt of th actual bump dflction and ar not rlatd to or constraind by adjacnt bumps. Also th top foil dos not sag btwn adjacnt bumps and dos not hav ithr bnding or mmbran stiffnss and its dflction follows that of th bump. A numrical modl is thn dvlopd in ordr to find out th prformanc charactristics of th gas foil barings. Th finit diffrnc schm has bn usd to discrtiz th govrning Rynolds uation and th prssurs ar calculatd by solving a nonlinar matrix uation using Nwton-Raphson tchniu and th ffcts of bump foil thicknss, numbr of bumps and bump complianc cofficint on th foil baring static prformancs wr valuatd. Th rsults of th study show that too thin bump foil thicknss may lad to a significant dcras in th load capacity. Howvr for accurat prdictions of th foil baring prformancs, th prsnt computational analysis can b furthr xtndd by incorporating 1D and 2D finit lmnt modl of th foil considring th dflction of th top foil as wll as th bump foils. 28

15 REFERENCES [1] Hshmat, H., Shapiro, W., and Gray, S., 1982, Dvlopmnt of Foil Journal Barings for High Load Capacity and High Spd Whirl Stability, ASME J. Lubr. Tchnol., 104, pp [2] Hshmat, H., 1994, Advancmnts in th Prformanc of Arodynamic Foil Journal Barings: High Spd and Load Capability, ASME J. Tribol., 116, pp [3] Hshmat, H., 2000, Opration of Foil Barings Byond th Bnding Critical Mod, ASME J. Tribol., 122, pp [4] L, Y. B., Kim, T. H., Kim, C. H., L, N. S., and Choi, D. H., 2004, Dynamic Charactristics of a Flxibl Rotor Systm Supportd by a Vis-colastic Foil Baring, Tribol. Int., 37, pp [5] Walowit, J. A., and Anno, J. N., 1975, Modrn Dvlopmnt of Lubrication Mchanics Applid Scinc, London, Chap. 7. [6] Hshmat, H., Walowit, J. A., and Pinkus, O., 1983, Analysis of Gas Lubricatd Foil Journal Barings ASME J. Lubr. Tchnol., 105, pp [7] Ku, C. P., and Hshmat, H. 1992, Complaint Foil Baring Structural Stiffnss Analysis Part I: Thortical Modl Including Strip and Variabl Bump Foil Gomtry ASME J. Tribol pp [8] Ku, C. P., and Hshmat, H., 1993, Complaint Foil Baring Structural Stiffnss Analysis Part II: Exprimntal Invstigation ASME J. Tribol. 113, pp [9] Rubio, D., and San Andrés, L., 2004, Bump-Typ Foil Baring Structural Stiffnss: Exprimnts and Prdictions ASME Papr GT [10] Ruscitto, D., McCormick, J., and Gray, S., 1978, Hydrodynamic Air Lubricatd Compliant Surfac Baring for an Automotiv Gas Turbin Engin I-Journal Baring Prformanc. NASA CR [11] L, D., Kim, Y., and Kim. K., 2008, Th Static Prformanc Analysis of foil journal barings considring thr-dimnsional shap of th foil structur ASME J. of Tribol. 130/ [12] L, Y., Park, D., Kim, C., Kim, S., 2008, Oprating charactristics of th bump foil journal barings with top foil bnding phnomnon and corrlation among bump foils Tribol. Int., 41, pp [13] Kim, T. H., and San Andrés, L., 2008, Havily Loadd Gas Foil Barings: A Modl Anchord to Tst Data ASME J.Eng Gas Turb Powr pp [14] San Andrés, L., and Kim, T., 2009, Analysis of gas foil barings intgrating FE top foil modls. Tribol. Int., 42, pp [15] Iordanoff, I., 1999, Analysis of an Arodynamic Compliant Foil Thrust Baring: Mthod for a Rapid Dsign J. Tribol., 121, pp [16] Kopsl, W., 1977, Gas Lubricatd Foil Baring Dvlopmnt for Advancd Turbomachins, U. S. Air Forc Aro Propulsion Laboratory, Rp. AFAPL-TR , Vol. 1, 2. [17] Carpino, M., and Png, J. P, 1993, Thortical Prformanc Foil Journal Barings, prsntd at th AIAA/SAE/ASME/ASME. 27th Joint Propulsion Confrnc, papr no. AIAA [18] Andr San Luis, Kim Ho Ta, 2008, Forcd nonlinar rspons of gas foil baring supportd rotors Tribology Intrnational, 41, pp NOMENCLATURE C Baring radial claranc (m) D Diamtr of journal (m) Baring ccntricity (m) 2 E Young s modulus for bump foil ( N/m ) b 2 E Young s modulus for top foil ( N/m ) t h H i, j Film thicknss (m) Non-dimnsional minimum film thicknss Grid location in circumfrntial and axial dirctions of FDM msh 29

16 3 K Bump foil structural stiffnss pr unit ara ( N/m ) l 0 f Half bump lngth (m) L Baring lngth (m) m Numbr of divisions along j dirction of FDM msh n Numbr of divisions along j dirction of FDM msh O Cntr of baring O ' Cntr of journal p 2 Hydrodynamic prssur in gas film ( N/m ) 2 p a Atmosphric prssur ( N/m ) p 2 Arithmtic man prssur along baring lngth ( N/m ) P Non-dimnsional hydrodynamic prssur R Radius of journal (m) s Bump foil pitch (m) S pa Complianc cofficint of bump foil : CK t b t t w t W W x, y, z Z α Bump foil thicknss (m) Top foil thicknss (m) Top foil transvrs dflction (m) Non-dimnsional top foil transvrs dflction Non-dimnsional stady stat load carrying capacity Coordinat systm on th plan of baring Non-dimnsional axial coordinat of baring : z R 3 1 Complianc of th bump foil ( m /N ) : ε Eccntricity ratio Λ 2 6µω R Baring numbr : pa C µ 2 Gas viscosity ( N-s/m ) φ Attitud angl (rad) θ Angular coordinat of baring (rad) : x / R υ Poisson s ratio ω Rotor angular vlocity ( rad/s ) θ, Z Non-dimnsional msh siz of FDM msh f K f 30

17 AUTHORS 1. T. Moasunp Jamir Rsarch Scholar (PhD) Dpartmnt of Mchanical Enginring Indian Institut of Tchnology Guwahati [ sunpbokdi@gmail.com ; t.jamir@iitg.rnt.in 2. Dr. S. K. Kakoty Profssor Dpartmnt of Mchanical Enginring Indian Institut of Tchnology Guwahati [sashin@iitg.rnt.in] 3. Dr. Karuna Kalita Assistant Profssor Dpartmnt of Mchanical Enginring Indian Institut of Tchnology Guwahati [karuna.kalita@iitg.rnt.in] 31

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