1983-Analysis of Gas-Lubricated Foil Journal Bearings

H. Heshmat J. A. Walowit 0. Pinkus Mechanical Technology Inc., Latham, N.Y. 12110 Analysis of Gas-Lubricated Foil Journal Bearings This work is concerned with an evaluation of the performance of a gas journal bearing using a spring supported compliant foil as the bearing surface. The analysis, conducted for both single and multipad configurations, is concerned with the effects that the various structural, geometric, and operational variables have on bearing behavior. Following the solution of the relevant differential equation, tabular or graphical solutions are provided for a range of relevant geometric and operational parameters. The solutions include values of the colinear and cross-coupled spring coefficients due to both structural and hydrodynamic stiffness. Desirable design features with regard to start of bearing arc, selection of load angle, number of pads and degree of compliance are discussed. 1.0 The Compliant Foil Bearing This work is concerned with an evaluation of the per-formance of a gas journal bearing using a spring supported compliant foil as the bearing surface. The configuration of such a journal bearing, shown in Fig. \\(a), consists of a hollow cylinder, the inside of which contains strips of corrugated or bump foils lined on top with a very thin foil; the basic element of such a strip is shown in Fig. \\(b). The foil is anchored at its leading edge but is free at the trailing end. The bumps act as springs and the foil, when loaded, deflects producing (as in any elastohydrodynamic bearing) a film thickness higher than that in an equivalent rigid surface. The actual appearance of such a bearing is shown in Fig. 2. As compared to a conventional gas bearing, the compliant foil bearing has the following advantages: • Higher Load Capacity. In terms of the usual load capacity - film thickness relation, the foil bearing can, for a given nominal film thickness (this term will be discussed more fully later on), carry a higher load. • Lower Power Loss. The existence of larger films is also responsible for yielding lower power losses. • Stability. Due to the ability of the top and bump foils to yield and deflect, the bearing can tolerate excursions of the runner or misalignment of the mating surfaces. In addition, it permits the maintenance of a preselected stiffness by proper choice and distribution of the bumps. • Endurance of High Temperatures. The bearing exhibits superior qualities in operating at high temperatures, primarily in that it is not subject to thermal distortion. 9 Endurance of Foreign Matter. Due to its higher film thickness and ability to conform, the present bearing can tolerate the incursion of foreign matter without undue harm to the mating surfaces. Contributed by the Lubrication Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS and presented at the ASME/ASLE Joint Lubrication Conference, Washington, D.C., October 5-7, 1982. Manuscript received by the Lubrication Division, November 2,1981. Paper No. 82-Lub-40. These bearings have been built and used in high-speed equipment exhibiting all of the above features. Some of this experience [1, 2, 3], with both journal and thrust bearings using air as a lubricant, offers information on the structural features of these bearings, as well as experimental results over a range of operating parameters. 2.0 Analysis 2.1 Assumptions. In formulating the equations governing the elastohydrodynamics of the foil bearing, the following assumptions are made: (a) The stiffness of the foil is taken to be uniformly Top Foil (Lifted) Bump Foil a) Construction of Bearing Shaft Bump Foil b) Baalc Element of Bearing Fig. 1 The compliant foil bearing OCTOBER 1983, Vol. 105 / 647 Copyright © 1983 by ASME

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Journal of Lubrication Technology --.....'....,\",\\\\\\\\II-I---~::\"\"\"'~--++----+­SPLITBUMPFOILFig.2CompliantfoilJournalbearingJIdistributedandconstantthroughoutthebearingsurface.Thestiffness,KB,islinearandisthusindependentoftheamountofbumpdeflection.(b)Thefoilisassumednotto\"sag\"betweenbumps,butrathertofollowthedeflectionofthebumpsthemselves.(c)Thedeflectionofthefoilinitsresponsetotheactingforcesisdependentonthelocaleffectonly.i.e.,ontheforceactingdirectlyovertheparticularpoint.(d)Thefluidinthefilmisisothermalandbehaveslikeaperfectgas.2.2TheDifferentialEquation.WiththenomenclatureofthejournalbearingasgiveninFig.3,therelevantReynoldsequationcanbewritten(4)as:R2aoFig.3Nomenclatureforfalljournalbearingweobtain-a[_ph-ajj]aoao-3+-aai[-\"3a_-ph-ajj]=A-(jjh)aiao(2)Thefilmthicknessvariationh(0)isthatduebothtoec\"centricityeandtothedeflectionofthefoilundertheimposedhydrodynamicpressures.Sincethelatterisproportionaltothelocalpressure,wehaveh=C+ecos(O-cf>o)+K\\(P-Pa)~~[Ph)~J+~[Ph3~J=aoazaz6J1,UR~(ph)ao(I)whereK)isaconstantreflectingthestructuralrigidityofthebumps.Itwasshown[5]thatthisK1isgivenbyK,Writing:i=(zIR)jj=(pIPa)h=(hIC)=(~~)andusingforAtheconstantwhere2_6J1,UR_6J1,w(R)A-Pc2-PCaa----Nomenclature-----CDEF...;...._F(FIPaR2)radialclearancediameterofshaftorbearingmodulusofelasticityforcePPa[JPmaxpressureambientpressuremaximumpressurepitchofbumpfoilthicknessofbumpfoilhorizontalcoordinateverticalcoordinateaxialcoordinate(zlR)(piPa)ONkKLspringconstant(KCIPaR2)zPRTtlengthofbearing(indirection)unitloading,(WILD)radiusofjournalbearingtorquelinearvelocityloadonbearingWIPaR2T!PaCR2=(RIC)!xYzist000,O2J1,vcf>cPocPLWexbearingcompliance,UWIt'hNh10!eeccentricityfrictioncoefficientnominalfilmthickness(hIC)(3I'0halflengthofbumpin0directionOsO£eccentricityratio,(elC)angularcoordinatestartofpadendofpadO£)tangularextentofpad,(Oss2PaCE(~)(I-v2)loadangle=angularvelocity(00angularpositionofhNangularpositionofhminstartofhydrodynamicfilmendofhydrodynamicpressureabsoluteviscosityPoissonratioattitudeangle,(cf>L+cf>o)-1r)A6J1,w(~rPaCSubscriptsmaxmaximumminminimumBbumpfoilnominalNTransactionsoftheASME648/Vol.105,OCTOBER1983Downloaded 19 May 2011 to 202.119.79.4. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

a) Commencement of Hydrodynamic Film h = h. = const. Fig. 4 Configuration of bump foil is the compliance of the bearing. The quantities s, l0, and t are defined in Fig. 4. Consequently, the normalized film thickness is given by h=(-\\=\\.+ecos(B-4>0) + a(p-l) (4) Bumps e„ \"min \"2 \"E 2.3 Boundary Conditions. The construction of foil Fig. 5 Boundary conditions in foil Journal bearing journal bearings essentially does not permit the generation of subambient pressures. Whenever diverging portions of the (• (Z./2) j. 92 film tend to produce subambient pressures in the fluid film, - - cos t -RdBdz i.e., on top of the foil, the prevailing ambient pressure pa -sin 8 underneath the foil will lift it up until the pressures on both given in dimensionless form as follows: sides of the foil are equalized. This fact has different im-plications for the start and end of the hydrodynamic film. - dddz (6) Figure 5 shows two relevant cases of the pad having a y' ppaR2 J-u/D) Je •) ~{L/D) Jes diverging film thickness either at the start, case (a), or at the end of the arc, case (b). In the first case, the tendency of the The dimensionless load is then given by diverging film to generate negative pressures on top of the foil (7) will lift it off the bumps. The foil will continue to be lifted off pK- \\L>/ \\aPa even when, nominally, with respect to the bumps, the film thickness starts to decrease; the foil will then merely start to and the load angle approach the bumps again, maintaining a constant film (8) tanL = (Fx/Fy) thickness and constant ambient pressure. When the foil again contacts the bumps, hydrodynamic pressures will start to The torque on the journal is (L/2) reErRh / dp \\ /^?3C0 form again. The locations of recontact 0! is that point where the film thickness hx equals the film thickness at 6S, the start +of the pad. Thus, the entire portion 0s0j is ineffective as a J-(i/2) }gs I 2 V 30 / hi and, in normalized form: bearing surface. ized form: T f (L/D> CeE(h / dp For case (b), with a diverging film occurring at the trailing (9) +portion of the pad, at some point where negative pressures P„C7?T\"J-(L/D) )eIDS s U \\~dd) 6 ~h) would have occurred if the surface were rigid, the foil lifts off Above and aligns itself parallel to the shaft, with h = h2 = constant. h = Equation (4) for 0S < 6 < 02 In essence, this situation is similar to the trailing edge con-dition in cavitating, liquid lubricated journal bearings. Here, h = h2 = constant for 62 < 6 < 9E as with cavitation, the film ends at an unknown angular position 02 which from continuity requirements must fulfill 2.5 Spring Coefficients. For small displacements from both the zero pressure and zero pressure gradient boundary the bearing equilibrium position (e, 60) the spring coefficient conditions. Thus, the boundary conditions for the solution of is given by the general term equation (2) are: dF 1 dF -- B p = (p/p) = l (5a) saat K= — sin0o + — —-cos0o de e dd0 (.5b) P = (P/Pa)=l Properly normalized, the colinear and cross-coupled spring coefficients are thus given by at 0 = BFX 1 dFx (5c) Kxx = — sm0o + - — cos 0o = o (10fl) 30 oe e ou0 w'^-A-o)0'^-4¥r=H r\\™(^)>^]dedz at z = ± (i) (P/Pa)=i (5d) dFx 1 dFx . Kxy = - — cos0o + - — sin 0o de e 30o dFv 1 dFy Kyx = —^ sin0o + - —- cos 0o oe e 30o Kyy-dFy 1 BFr . —- cos0o -I sm 0o de e 30o (10b) (10c) (lOd) 2.4 Performance Parameters. With the solutionp(B, z) accomplished, the performance quantities of the bearing can then be obtained by proper integration. The load capacity is from Journal of Lubrication Technology OCTOBER 1983, Vol. 105/649 Downloaded 19 May 2011 to 202.119.79.4. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

(L/D) •1+1, '•J\"1 [l 3j I.J+1 rO-fUl'^U .6, Fig. 7 Minimum and nominal film thicknesses 0 -(L/D) 0 Fig. 6 Grid network for finite difference solution fi = 120°, t = 0.6, 90= 220° (L/D) =y^= 1, a = 5 where PaR2 2.6 Method of Solution. For the numerical solution of the Reynolds equation, the dependent variable was represented by a finite number of points located at in-tersections of a grid mesh. Particular attention was devoted to regions of the bearing film in which boundary-layer phenomena may occur at high values of A (trailing edges of pad). Thus a variable grid in they direction, was employed. With the grid network, as shown in Fig. 6, the Reynolds equation (2) can be written in finite difference form as AT= KC •((-*•SUt^)]Fig. 8 Film thicknesses in a 120 deg bearing pad + ((-/*-^+*!*).,_,„ ,-_,}+ [Dj] {P, ) = [Rj) j+approximated by central difference formulas as: y=l, 2 N 1 /Adj + Adj_ ]Pij- 1 >PiJ+ l) =0 which is linearized by the Newton-Raphson method as An) / \\ ;>/•(\") AyThe foil lift off (the boundary conditions given by equation (5) was satisfied by setting negative pressures to zero during the back sweep in the column method. This is the same technique used for satisfying the cavitation boundary con-ditions in liquid lubricated bearings. 3.0 Nature of Solution Several peculiar features of the compliant foil journal bearing should be brought to light because, to some degree, they are a matter of definition. This pertains first to the notion of film thickness, in terms of which load capacity is Transactions of the ASME dPl+lJ \\P'-'-J P-l-'J dPu \\P'-J PhJ) 3^+i,y i(ni\"t)-r>(\"\\ -^ \\PI+1,J Pl + ],jj (14) where 650/Vol. 105, OCTOBER 1983 Downloaded 19 May 2011 to 202.119.79.4. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

usually defined, as well as the relation between the effective smallest. The situation is represented in Fig. 7. The axial film and physical arcs of a journal bearing pad. thickness at 6 = d0 has a value of hmin at the edges (where/? = /?„), but only there; elsewhere along 6 = 6a, the film 3.1 The Nominal Film Thickness. In rigid journal thicknesses are larger than along another angular position 6 = bearings, the minimum film thickness is a clear and fixed dN where, because the pressures are lower, the film thickness, quantity. It occurs at the line of centers and its value is on the average, is smaller than at the line of centers. Figure 8 constant across the axial width of the bearing. Also, shows a 3-dimensional film thickness plot for a 120 deg pad in generally, the film thickness anywhere is constant in the z which, while film thickness at the edge (z = 1/2) is small over direction. Since in our case pressures cause proportional most of the pad area, the surface has been deflected into much deflections of the bearing surface, the film thickness in the larger values of h. interior of the bearing, where pressures are highest, will be For the purposes of the present paper, a nominal film larger than at the edges (z = ± L/2); also since the maximum thickness hN will be defined as the minimum film thickness pressures occur near the line of centers, the film thickness in that occurs along the bearing centerline, i.e., at z = 0. In Fig. the interior of the 0 = d0 line will not necessarily be the 9 this central film thickness is plotted for a centerline z = 0 at various values of a. While hmin for the rigid case occurs at 6 = 1.4 180 deg, with increasing values of a the value of this hmi„, or our hN, shifts downstream and increases in value; at a = 5, it is twice the value of the rigid case and has shifted downstream 1.2 by nearly 100 deg. This should be kept in mind later on, when 10 load capacity, i.e., the W-hN relation is plotted; an increase in 1.0 load while increasing e may also produce an increase in the nominal film thickness. 0.8 0.6 0.4 L/D = 1 A= 1.0 90 = 180° (3 = 360° 1 = 0 «= °-6 0.2 40 _L J_ 160 200 240 _L 80 120 6 , degrees 280 320 360 Fig. 9 Location of nominal film thickness 3.2 Actual and Effective Bearing Arc. As discussed in Section 3.1, compliant foil bearings, suffer a penalty in their ability to generate hydrodynamic pressures whenever the pad arc commences in a diverging region. Thus, for example, in a full bearing that starts at 6S = 0, the following two typical cases may arise: (a) 60 = 180 deg: the film is convergent from the start and pressures commence at 6 = 0. (b) d0 - 220 deg: the foil lifts off and remains parallel to the shaft; film convergence and hydrodynamic pressures do not commence until 6 = 80 deg (twice the value of 6^ in a rigid bearing). This is shown graphically in Fig. 10 where there are no pressures over the first 80 deg of bearing arc. From a hydrodynamic standpoint, case (b) is equivalent to a bearing with 8S = 80 deg. Not only is its physical start at 6 = 0 not adding much in the way of extending the pressure L/D = A = a = 1 = 0.6 9. = 220° L/D = A = a = 1 t = 0.6 0O = 220° Fig. 10 Pressure profile in a 360 deg journal bearing Fig. 11 Pressure profile in a 3-pad (120 deg) journal bearing Table 1 Effect of load angle in 360 deg bearing e = 0.6; (L/D) = K=a=\\\\ 6S = 0, 0£ = 36Odeg 00 0 L 20 deg 45 deg 60 deg 90 deg 180 deg 220 deg 245 deg 270 deg 163.1 deg 141.6 deg 129.8 deg 107.2 deg 33.1 deg -11.9 deg -41.9 deg -72.9 deg -0 _ 4 deg 11.6 deg 20.2 deg 27.2 deg 33.1 deg 28.1 deg 23.1 deg 17.1 deg h Pmax hN _ 32 deg 74 deg 95 deg 142 deg 240 deg 272 deg 300 deg 325 deg 1.018 1.087 1.133 1.201 1.253 1.245 1.228 1.201 0 0.40 0.41 0.48 0.52 0.58 0.62 0.62 0.61 0.58 WxlO1 0 rxio - . 43.9 38.1 33.9 27.2 24.6 23.7 23.7 25.5 1.04 11.5 21.4 40.0 56.8 54.9 50.0 40.1 Journal of Lubrication Technology OCTOBER 1983, Vol. 105/651 Downloaded 19 May 2011 to 202.119.79.4. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Table 2 Performance of a 360 deg journal bearing A = 1.0; L =0 c a 10 r-0 * - e s (e2 - -o p tnax L/D - 0.5 S. 0.70 0.72 V x 102 T x 10 \\\\ W = 1 i \\X 0.3 0 1 5 10 20 63.5 59.0 48.5 40.0 34.0 40.0 36.0 32.0 30.0 27.0 12.0 19.0 21.0 21.0 21.0 81.5 87.0 100.5 114.0 114.0 62.0 76.0 97.0 105.0 114.0 52.0 71.0 91.0 99.0 108.0 1.046 1.043 1.037 1.025 1.017 1.23 1.144 1.073 1.05 1.033 3.73 1.33 1.12 1.077 1.048 L/D - 1.0 4.4 4.2 3.2 2.9 2.1 9.38 9.15 8.53 8.09 7.54 15.85 13.93 8.19 7.33 6.54 26.1 13.8 9.8 8.5 7.3 o.ao 0.85 0.94 0.40 0.51 0.68 0.795 0.90 0.10 0,41 0.64 0.76 0.91 \\ \\ \\ \\ \\ \\ \\ \\ \\ ii_ Nos. reler to a 0.5 (L/D) = 1.0 1.5 i\\1 0.6 0 1 5 10 20 17.9 13.7 8.3 6.1 4.2 W 0.5 v\\\\V* V .5 \\ \\K \\ \\ \\ i \\ \\ 1 \\ \\ 0.9 0 1 5 10 20 157.3 34.7 14.8 9.8 6.3 0.1 ^ 0.02 0.70 0.77 0.94 1.04 1.14 0.40 0.62 0.90 1.055 1.22 0.10 0.32 0.86 1.05 1.26 27.9 23.7 14.8 10.3 6.37 94.9 56.8 28.8 19.4 12.2 504.5 102.8 42.9 27.8 17.2 22.7 21.2 18.6 17.5 16.5 31.1 24.6 19.1 16.9 13.0 58.6 28.1 19.5 16.7 14.4 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.3 0 1 5 10 20 57.0 49.0 36.0 28.0 20.0 36.0 33.0 28.5 23.0 20.0 13.0 21.0 23.0 21.5 19.0 97.0 104.0 117.0 120.0 132.0 77.0 95.0 112.0 117.0 130.0 59.0 86.0 98.0 108.5 127.0 1.137 1.107 1.061 1.041 1.025 1.539 1.253 1.114 1.074 1.046 4.830 1.434 1.164 1.103 1.063 L/D - 1.5 0.6 0 1 5 10 20 0.9 0 1 5 10 20 0.3 0 1 5 10 52.0 43.0 29.0 21.0 35.0 32.0 26.0 22.0 14.0 23.0 23.0 21.0 103.0 113.0 U9.0 141.0 88.0 104.0 120.0 137.0 68.0 95.0 112.0 122.0 1.218 1.152 1.076 1.048 1.731 1.311 1.135 1.086 5.300 1.485 1.164 1.116 0.70 0.82 1.03 1.13 J 70.0 53.2 23.5 18.3 33.4 30.4 26.4 24.8 45.6 34.3 26.3 23.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 85.1 39.2 26.7 22.8 0.6 0 1 5 10 0.40 I 208.9 0.68 | U2.0 1.00 : 52.8 1.18 ' 34.1 0.10 . 898.9 0.56 , 179.7 0.96 ! 74.2 1.17 ' 47.5 0.9 0 1 5 10 Fig. 13 Relation between t and /) N for a 360 deg bearing profile, but it actually imposes a penalty. Had the bearing arc started at 6 = 40 deg instead of at 6 = 0, the pressures would have started there instead of at & = 80 deg. A similar situation for the case of a 3-pad design is shown in Fig. 11, where, due to the ineffectiveness of 80 deg of the first 120 deg pad, it contributes little to the hydrodynamics of the bearing. This effect can be seen in Table 1 which shows that, by shifting in a 360 deg bearing the line of centers from 180 to 270 deg, there was a loss in load capacity of nearly 30 percent as well as a reduction in hN. In designing a foil bearing, if the eccentricity is fixed for the particular application, it is best to start the bearing at 6S = (for a vertical load); if the eccentricities are liable to vary, some compromise value of ds > 0 can be chosen. parameters relevant to a foil journal bearing. These are /3, a, (L/D), A, 4>L, and number of pads. There is also the ec-centricity ratio and the attitude angle 0, the latter tied to the load angle 4>L. In the parametric study to follow, a set of standard conditions consisting of {L/D)=A=a=l; e = 0.6 will be used, and any parametric variation will commence from this set of reference values. 4.1 The Full Bearing. Table 2 gives a detailed listing of the performance of a vertically loaded (4>L = 0) full 360 deg bearing as a function of (L/D), a, and e. Note should be taken of the fact that the start of the bearing, that is 6S is so chosen as to avoid idle (p = pa) regions at the upstream portion of the bearing. In effect, this requires that 9S = <$>. The case of nonvertically loaded foil bearings 4>L ^ 0, is given 4.0 Performance Characteristics in Table 1. Some of the noteworthy points emerging from There are six geometric, structural, and operational these tabulations are: 652/Vol. 105, OCTOBER 1983 Transactions of the ASME Downloaded 19 May 2011 to 202.119.79.4. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Table 3 Performance of a 3-pad bearing (L/D) = A = 1; B = 120 deg each 35 ^V= 1.0 (L/D) = 1 Noa. refer to a 30 40 210 217 220 225 245 275 29 180 208 220 245 270 15 196 210 230 245 260 -140.0 7.2 -2.3 -2.3 -5.2 -10.0 -17.4 -145.0 44.1 -2.6 -10.6 -17.0 -25.5 -145.0 0.0 -10.8 -16.1 -14.7 -26.2 79.2 37.2 37.0 37.7 39.8 55.0 77.6 69.2 44.1 30.6 29.4 48.0 64.5 50.9 16.3 19.2 33.9 50.3 53.8 1.073 1.072 1.075 1.079 1.082 1.088 1.077 1.188 1.133 1.197 1.215 1.284 1.185 1.497 1.340 1.375 1.572 1.412 1.463 12.1 12.7 13.7 14.1 14.6 15.0 12.6 24.0 25.2 37.2 39.8 36.9 28.7 59.3 69.5 76.9 74.3 52.4 55.3 21.4 21.8 21.9 22.0 22.0 21.5 21.4 20 27.8 18.6 27.7 28.8 29.0 27.4 62.9 36.7 45.7 71.8 77.3 56.4 15 J 1 I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 10 25 Fig. 14 Variation of torque in a full journal bearing 38 210 214 220 225 245 275 32 ISO 205 220 245 270 25 200 203 210 230 245 260 143.0 5.1 0.0 -5.3 -8.6 -13.3 -21.1 145.0 55.0 3.6 74.9 35.1 34.3 34.7 36.4 51.7 73.9 67.4 55.5 28.6 30.3 48.7 66.9 61.0 23.5 23.0 23.7 33.7 46.1 1.049 1.038 1.040 1.043 1.045 1.052 1.050 1.104 1.048 1.077 1.103 1.121 1.106 1.178 1.122 1.119 1.158 1.198 1.202 1.186 8.01 7.52 7.87 8.31 8.60 9.07 8.18 15.7 12.6 16.1 18.3 18.4 15.9 24.4 25.3 26.3 28.1 29.7 28.2 25.2 20.7 21.1 21.1 21.2 21.2 20.8 20.7 25.4 16.5 25.3 26.7 27.2 25.4 46.5 34.4 36.2 41.6 67.1 72.1 53.7 -9.7 -16.3 -23.1 144.0 3.5 0.0 -6.3 -16.3 -18.9 -21.5 sa.5 Fig. 15 Effect of A in full bearings Table 4 Mode of loading of 3-pad bearing (L/D) = A=1; /3= 120 deg each Central'loading:. = 0 0L = W 0 13.8 37.5 28.5 36.0 16.0 . 68.0 35.0 7.8 30.0 17.0 23.5 26.2 Optimum loading 0L -10 -10 -10 -14 -14 -14 a 1 1 1 5 5 5 e 0.3 0.6 0.9 0.3 0.6 0.9 0 55.0 29.0 18.5 53.0 41.0 30.0 W 15.0 39.8 78.0 9.0 18.6 29.8 • Effect of a. While in terms of e there is a drastic drop in load capacity with a more compliant bearing, in terms of hN there is actually an increase in load capacity. This is illustrated Journal of Lubrication Technology in Fig. 12. At large values of a, a > 10, the load capacity the bearing can support is low, due to the fact that the flexible foil deflects sufficiently to maintain high film thicknesses even at large eccentricities. Thus from a design standpoint, it may be advisable to use high compliance bearings at low loads; high loads, however, can be supported only with bearings of low values of a. The relation between e and hN is given in Fig. 13 where again we see that in highly compliant bearings (par-ticularly at high (L/D) ratios) an increase in eccentricity may produce an increase in hN, a phenomenon opposite to rigid bearings where hmin is the inverse of e. The variation of torque with a is shown in Fig. 14 where we see that at high values of a. there may be a decrease in torque with eccentricity. This, of course, is tied to the fact that, as shown in Fig. 12, at high a's an increase in e produces also an increase in film thickness. OCTOBER 1983, Vol. 105/653 Downloaded 19 May 2011 to 202.119.79.4. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Table 5 Values of spring coefficients (L/Z?) = A=1;0L = O W 0.6 0.75 0.9 0.6 0.75 0.9 0 0 0 1 1 1 35.7 24.1 12.8 32.1 26.3 21.4 0.951 1.894 5.055 0.568 0.7833 1.028 Kr i3 = 360 deg 1.920 3.416 7.202 1.129 1.231 1.268 3-pad-120 deg each Kr -0.125 -1.166 -6.024 0.174 0.0254 -0.098 K, -2.345 -3.989 -10.151 -0.693 -0.686 -0.627 K„ 3.237 8.981 44.593 1.130 1.378 1.602 0.6 0.75 0.9 0.6 0.75 0.9 0.80 0 0 0 1 1 1 26.0 17.4 8.6 25.5 20.5 16.3 0.635 1.321 3.695 0.359 0.511 0.689 1.123 2.102 4.728 0.5702 0.673 0.759 -0.092 -0.752 -3.344 0.0451 -0.017 -0.057 -2.05 -3.710 -8.768 -0.758 -0.821 -0.855 2.635 7.432 37.103 0.801 1.051 1.274 L/D 1 W= 1 1 i. = 0.6 - 1 Pad — 360° - 3 Pads — 120° ••• 5 Pads — 72° 90 = 180° 60 = 220° 0.50 0.40 0.30 0.20 0.10 -80 Fig. 16 Performance of multipad bearings • Effect of A. The performance of a foil bearing as a function of A conforms to the familiar pattern of com-1 3 5 pressible lubrication. After an initial rise in W with an in-No. of Pads crease in A, the load capacity, both in terms of an increase in W as well as a rise in hN, tends to flatten off and approach an Fig. 17 Relative performance of multipad bearings asymptotic value, as shown in Fig. 15. The torque, however, rises almost as a linear function of the increase in A. The more compliant bearing shows lower power losses due to the • Variation With Number of Pads. Figure 16 shows the prevailing higher film thicknesses. variation of 1-, 2-, and 3-pad bearings as a function of load angle. The plot shows clearly a drop in load capacity with the 4.2 The Multipad Bearing. Two multipad bearings are number of pads, i.e., with a drop in extent of bearing arc j3. examined in this section. The 3-pad design consists of three As seen, the optimum for the 360 deg bearing occurs at 4>L = 120 deg arcs; the 5-pad design has five 72 deg arcs. In each 0, at which point the torque also reaches its minimum value. case the vertical line of symmetry disects the bottom pad, so The 3-pad bearing, as said previously, reaches an optimum at that L = Q represents a load passing through the midpoint of (f>L = - 10 deg; whereas, the 5-pad bearing reaches an op-the bottom pad. Table 3 and 4 give a spectrum of solutions for timum at 4>L = - 15 deg. In Fig. 17 where the loads for the 3 the performance of the 3-pad bearing and these results show bearings are all plotted for a fixed shaft position, the effect of the following: a shorter /3, is seen to be less at high values of a than at low • Variation With Load Angle. Because of the cyclic nature ones. The torque seems to be at a minimum for the 3-pad of this bearing (symmetry for each 120 deg) there is much less configuration. variation in either W or T with a shift in load angle. In particular, there is no acute loss of load capacity when the line 4.3 Stiffness. Table 5 gives the values of the four spring of centers passes between pads. The optimum load angle for a coefficients for two values of compliance, the limiting case of = 1 is L = -10 deg; for a = 5 it is 4>L = - 14 deg. The a = 0, and a = 1. The a = 0 case differs from a rigid gas improvement in load capacity over that of central loading (L bearing in that the subambient pressures are eliminated from = 0) is of the order of 10 to 15 percent. the pressure profile. A comparative evaluation of the stability 654/Vol. 105, OCTOBER 1983 Transactions of the ASME Downloaded 19 May 2011 to 202.119.79.4. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

characteristics of the 1- and 3-pad bearings is, of course, best done in a study of a rotordynamic system, particularly when the cross coupling components vary not only in magnitude but also in sign. However, the following items can be deduced from the tabulated K data: • When plotted against fl^the Kn\\ are about the same for both designs, whereas the Kxx's are lower for the 3-pad configuration. • With the more compliant case, the K's tend to level off with a rise in eccentricity, the values of the coefficients ap-proaching the structural stiffness of the system. In general, the advantage of the compliant bearings in the area of stability lies in that levels of stiffness can be selected by the designer via a proper combination of structural and hydrodynamic stiffnesses. Thus instead of making his inertias and supports suit the inherent stiffnesses of purely hydrodynamic bearings, the designer may try to tailor and adjust bearing stiffness to the demands of his rotordynamic system. References 1 Gray, S., Heshmat, H., and Bhushan, B., 8th Int. Gas Bearing Sym-posium, Apr. 1981. 2 Heshmat, H., Shapiro, W., and Gray, S.,\"Development of Foil Journal Bearings for High Load Capacity and High Speed Whirl Stability,\" ASME-ASLE Lubrication Conference, New Orleans, Oct. 1981. 3 Heshmat, H., and Shapiro, W., \"Advanced Development of Air-Lubricated Foil Thrust Bearings,\" ASME-ASLE Lubrication Conference, New Orleans, Oct. 1981. 4 Walowit, J. A., and Arno, J. N., Modern Developments in Lubrication Mechanics, Applied Science Publishers, Ltd., London, 1975. 5 Heshmat, H., Walowit, J. A., and Pinkus, O., \"Analysis of Compliant Foil Gas Thrust Bearings,\" to be published. Journal of Lubrication Technology OCTOBER 1983, Vol. 105/655 Downloaded 19 May 2011 to 202.119.79.4. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm