Analysis of an Aerodynamic Compliant Foil Thrust Bearing: Method for a Rapid Design

Transcription

1 I. lordanoff Laboratoire de mecanique des contacts UMR INSA-CNRS 554, 20 avenue Albert Einsten, 6962 Villeurbanne Cedex, France Analysis of an Aerodynamic Compliant Foil Thrust Bearing: Method for a Rapid Design A very simple design method for an aerodynamic compliant foil thrust bearing is presented in this paper. It is based on 3D modeling (called: complete direct calculation) of the elastoaerodynamic problem. In this approach, the structural analysis has been simplified. This enables the calculation to be carried out faster. However this model, based on the resolution of the Reynolds equation, only gives the performance of a thrust bearing for a given geometric profile. An efficient method for solving the inverse problem for predicting the desired bearing performance parameters is presented. The complete direct calculation is only used to improve the profile geometry thus found. Finally, the proposed method has been applied for the design of a 80 mm outer diameter 40 mm inner diameter thrust bearing operating between 20,000 and 50,000 rpm. It is shown that the thrust bearing designed by this approach has a high load capacity (300 kpa) at a speed of 50,000 rpm. It is also shown that the predicted performance of the bearing agrees well with the complete direct calculation. Introduction The compliant pad thrust bearing consists of two parts: a smooth top foil that provides the bearing surface and a flexible bump foil formed by a series of bumps that provides a resilient support for the surface (Fig. ). The high load capacity of such bearings was shown in previous studies (Heshmat et al., 982, 983, Gray et al, 98). A numerical calculation of bearing performances has been proposed by Heshmat for this type of thrust bearing (983). This paper first presents a theoretical calculation of compliant composite profile thrust bearings, based on the first analysis of Heshmat (983) and on a bump stiffness analysis realized by Ku and Heshmat (992, 993). This type of direct modeling requires a long calculation time, as do all the problems where structural behavior is linked with aerodynamic behavior. A recent report (lordanoff, 997) has shown that a simple 2D model enables the optimum profile for a noncompliant thrust bearing sector to be rapidly determined. This profile is a composite profile, i.e., one in which the leading portion has a constant slope following by a surface parallel to the runner. This profile is defined by a transition angle 9\ and an entrance film thickness HI (see Fig. 2). In this study, it is shown that for a constant local compliance S, the composite profile is preserved during the sector deformation. During this deformation HI is the only varying parameter. Based on this behavior, a very simple method that allows the best choice for the initial profile is presented. Finally, this method is applied to a real case. Experimental results are given and a comparison with theoretical predictions are made. 2 Elastoaerodynamic Calculations 2. Elastoaerodynamic Problem: Calculation. For a thrust bearing sector, the dimensionless flow characteristic equation is: 9 i.dp\ PH^9-\^±(RPH^^-^ d f,dp\ kph Rde de) dr\ dr 89 where the dimensionless parameters are (Fig. ): () H = h/h, A = 6/x ajro Pcfil R = r/ro A very complete study on the structural stiffness of a compliant foil bearing has been presented by Ku and Heshmat (992, 993). In the present study a simplified but realistic structural model is developed which enables a direct resolution of the elastoaerodynamic problem to be carried out. Specifically, The top foil follows the bump foil in its deformation, has no deflection between two bumps, and does not interact with the bump deflection. Thus the deflection is supposed to be only dependent on the bump foil. The bump foil is slotted parallel to its circumference to improve the independence of each element (Fig. 3). So, it is assumed that the deflection at one point of the pad only depends on the pressure applied on this point. Then the dimensionless film thickness is linked to pressure by the relationship H = S{P - ). with S = spjh The compliance of a free bump is higher than the compliance of the welded bump. Previous studies (Ku et al, 992, 993) have shown a quasi-linear compliance distribution between the welded bump and the free bump for this type of technology. Thus, for the complete calculation, a linear distribution of compliance will be assumed (Fig. 4). The linear structural behavior can be directly written in the Reynolds equation. For the numerical solution, the variables are represented by a finite number of points located at intersections on a grid mesh (Fig. 5). Derivatives are calculated by finite differences. The nonlinear system obtained F{P) = 0 is solved by a Newton-Raphson method (see Appendix ). The resulting pressure is integrated to yield the load capacity and the power loss, namely: W = pa P{\-R]) Jo JR^ PRdRdd Contributed by the Tribology Division for publication in the JOURNAL OF TRIBOLOGY. Manuscript received by the Tribology Division August, 998; revised manuscript received February 5, 999. Associate Technical Editor: M. M. Khonsari. Pf PaKrl r" r \HR_dP_ AR^ Jo J«, L 2 39^ 6 H drde 86 / Vol. 2, OCTOBER 999 Copyright 999 by ASME Transactions of the ASIVIE

2 Top foil A-A D0=p/(M^) QMN P *^ Fig. Pad description DR=(-R)/(N-) H(R=constant,t H(R,9 = constant) = constant Fig. 5 Grid network for finite difference solution Fig. 2 Best profile for air thrust bearings I Weld No deflection between two bumps wmax Breaking Of the lubricating ibricating film i 2n.m<hn<3^imj ^ T w>wmax Fig. 6 Relationship between the nominal film thickness and the maximum load capacity semi-welded bump. This first has its leading edge welded to the base Fig. 3 Sector description free bump Slottining H.i / / Rotor eii Initial profile (P=0, A=()) -.-r-jlll H= deform ation = Rotor translation t m H BH=I _ ":;Jilil ei Deformated profile (PTK), A^O) Fig. 7 Profile after deformation *inax - Pressure field compliansc s S2 free bump Fig. 4 Simplified structural model parallel to the runner is nearly constant; so this portion remains parallel to the runner after deformation. Heshmat (983) proposed chosing the film thickness in the deformed parallel portion of the sector as the nominal film thickness /i for evaluating bearing performances. As a matter of fact, the maximum load capacity is obtained when the minimal film thickness (in this case, /z ) is about 2 or 3 microns (Fig. 6). Thus, in the calculation, a rotor translation is made until the nominal value ft,, is reached in this parallel part of the foil (i.e. dimensionless film thickness H equals in this part (see Fig. 7). The composite profile, after deformation remains very close to a composite profile, defined by: 2.2 First Results With a Constant Compliant Factor S. As has been shown in previous studies (Heshmat, 983, lordanoff, 997), the pressure field in the portion of the sector that is Qx = 9\i //, = W - - S(P «- ) (2) where 6^ and Hu are the initial profile, 0, and i/, define the Nomenclature A = thrust area, m^ E = young modulus. Pa H = dimensionless film thickness: h/h H, = dimensionless entrance film thickness P = dimensionless pressure: p/p Pf = dimensionless power loss: Pf/Pah rl R = dimensionless radius: r/rn S = dimensionless local compliance: spjh,, WA = dimensionless load capacity: w/paa dhc = converging height, m hn = nominal film thickness, m Pa = ambient pressure. Pa Pf = power loss. Watts To = outer thrust radius, m r\ = inner thrust radius, m w = load capacity, N P = angular extent of sector, degrees A = compressibility number: 6iJ,aUirl/p hl Pa = ambient absolute viscosity. Pa- s /Lti = friction coefficient Qi = transition angle, degrees oj = angular velocity, rad s" Subscripts c = for compliant profile i = for initial profile r = for rigid profile Journal of Tribology OCTOBER 999, Vol. 2 / 87

3 ^^., -^R:cc:ci::ccq:[::p,.^^^' r^crf^*' Fig. 8 Results for the best rigid profile profile after deformation and Pmax the quasi constant maximum pressure on the second part of the sector (see Fig. 7). As 9i remains constant, Oi and du will be noted 9i in the following. These properties will lead to a fast definition of the best initial profile geometry. 3 Definition of the Best Profile Geometry for a Compliant Thrust Bearing For this study, results are given for a sector defined by: -/?, = ri/ro = 0.5 ~/3 = First Approach: Best Geometry at a Given Operating Point. In the dimensionless Reynolds equation, the compressibility number defines the conditions under which the thrust bearing operates (speed, thrust dimensions and ambient pressure). A previous study (lordanoff, 997) has shown that for each compressibility number, the optimum entrance film thickness Hiopt and the optimum transition angle ^lop,, which give the maximum load capacity, can be found. For this compressibility number and this profile, the maximum pressure P^ax is known (Fig. 8). Thus, applying relationships 2, for a compliant profile, the initial optimum profile is defined by Hu «iopt + S(P^ ) 6i 0\«[>t (3) 3.2 Second Approach: Best Geometry for a Given Operating Range. In real cases a thrust bearing has to operate with the best performance as possible not just at a single operating condition but over a specified operating range. So, the optimum operating range will be defined arbitrarily as follows: For a given A, if a thrust bearing develops a load capacity of 95% of the maximum load capacity, it is working at its optimum operating range. 95% is an arbitrary value, chosen in order to make a comparison between different profile geometries possible. The influence of the transition angle on the optimum operating range is first studied for a rigid profile: for each compressibility number value, and with H, = H^opt, ^imax and ^i^in (with ^imin < ^iiop, < ^imax) that givc 95% of the maximum load capacity are calculated. Graphs in Fig, 9 show the evolution of ^imax and 9umn in terms of A. These graphs show that a good choice of 6, allows a very large optimum operating range. In the example given, a 7 deg angle can be used. This angle, that Fig. 0 Optimum operating range for the rigid sector gives an extended optimum operating range will be called 6ir (subscript r is taken as rigid). Now, for each compressibility number, and with 9, = 9^^, /i/imax and Hi, m (with i/limin < -f^iopt < //limax) that give 95% of the maximum load capacity are calculated. Graphs of Fig. 0 show that the choice of//, defines the optimum operating range for a given profile. This entrance film thickness will be called Hir- In this example, with Hi^ = 6, the optimum operating range is between A = 200 and A = 800. Now, applying the relationship 3 between the initial entrance film thickness and the entrance film thickness after deformation gives: Hu = Hu + S(P, ), where Pmaxi corresponds to the maximum pressure for the rigid profile defined by 9ir and ^ max Humin = Hunin + S(P,ax2 " ), whcre Pmax2 corresponds to the maximum pressure for the rigid profile defined by 6u- and ^Imin In this fashion, the variation of Huimx and Hii^i against A can be drawn rapidly. Graphs in Fig. (drawing for 5 = 3) show that the choice of Hu define the optimum operating range for a given initial profile. This initial entrance film thickness will be called //) (subscript c as compliance). Table gives the different optimum operating ranges for three //, values. For the following application: ihlci i -^ ' <>.n _r,'i'-,^..j-,c^''''' ro 40 mm h = 2 yum - h,.oi2^ ^"^ Loo*^ -^^'^'-f-::^ -Ao-,,J'- _LJU _7^ -~r-,<^ cy-..ni^ ^, t^,-ql ^oj \ ^ I Hliniin ; :! ", i^^ -J^ r.oo^ n - A Fig. Optimum operating range for the compliance sector Fig, 9 Definition of dlr *-! A 000 Table fi\ci A, ti'min (rpm) '^m.ix (rpm) Optimum operating range for the compuant sector PI F2 P / Vol. 2, OCTOBER 999 Transactions of the ASME

4 Hlci=2 '^Tf.c-can Hid=6 Hlci= Fig. 2 Load capacity versus compressibility number Fig OO Results for the complete direct calculation Table 2 Best profile PI, H,,, = 2 P2, H,,, = 6 P3, //,,, = 2 operating range (A) /Lt =.8 0"' Pa-s P = W Pa Then A = 0.02aJ (uj speed in rpm), and in few seconds, with these graphs, the best operating speeds can be obtained for different profiles (Table ). 3.3 Direct Calculation: Validation of the Inverse Method. The direct calculation described in Section 2 is then appued to the three composite profiles found in Section 3.2 (see Table ). The results, in terms of dimensionless load capacities are given in Fig. 2: each of the three profiles has a better behavior in a given operating range. Table 2 gives the best profile for three operating ranges in terms of compressibility number: the results given by direct calculation are close to the results given by the inverse method. The proposed inverse method yields very good results. However, the sectors studied in Section 3 are rather simplified and do not exactly represent the real sectors; these results with significant saving in computational time are next applied to a real thrust bearing design. 4 Design of a Thrust Bearing 4. Description. Air Lubricated Foil Thrust Bearings have been analysed by Heshmat (982) in depth. Here only the differences between the real sector and the sector previously described will be underlined. Bumps are parallel: so, the profile is not constant radially (Fig. 3); Compliance is not the same for the welded bump and the free bump (a linear compliance distribution between the two bumps is assumed). In order ito apply the previous study, it is proposed to imposed the optimum values {Hi^ and 6\) at the mean radius of the sector. A five bump foil was chosen. The first three bumps define the convergent part and the last three the parallel part (Fig. 3). with the chosen dimensions, the transition angle at the mean radius is 7 deg. The height difference between bump and 3 gives the initial entrance film thickness. With hn = 3 yum, the entrance film thickness corresponding to the previous dimensionless values 2, 6, and 2 are 36 fim 48 nm and 63 fim. In order to obtain this film thickness at the mean radius, three height differences dhi, have been chosen: 42 /xm (thrust bearing Tl), 60 /xm (72) and 84 Mm (73). With a foil thickness of 0 fim, and the bump geometry, the free bump has a compliance of.8 0"'" mpa~' and the welded bump has a comphance of "" m-pa"' (see Appendix 2). So, the dimensionless values are: 5 = 6 and 55 = 3.6. The compliance, in the parallel part of the sector (between bump 3 and 5) is close to the value chosen in Section 3. Complete direct calculation is finally made with these three sector definitions. The results are given for thrust bearings of eight sectors. Graph in Fig. 4 gives the load capacity and the dissipated power in terms of rotational speed with hn = 3 fim. The required operating range is between 20,000 rpm and 50,000 rpm. Thrust bearing 72 has the best results for load capacity and thrust bearing 73 is better for power loss. Thrust bearing 7 has the lowest performance. Finally, thrust bearing 73 has been chosen because of its high load capacity and its low dissipated power. 5 Experimental Results 5. Test Setup and Instrumentation. The test setup used in this study is very close to the test setup presented by Heshmat in 982 (the schema is shown in Fig. 5). It consists of two main parts: A high speed rotor supported by two precision ball bearings with a steel end plate called a runner. runner precision ball bearings.,, ^ Drsplacement probe ;dl (dl aercstatic dl2-dll) journal rotational f..-l^^^^^s-^^^ / bearing speed probe p ^^^ p'^^^^^ ;iorodynanii(; flow Trailing " edge mean radius dhb nnn Pressure supply Fixed plate ^ Displacement probe : dl2 Flexible sensor for aerodynamic torque measurement Fig. 3 Sector description Fig. 5 Test rig schematic draw Journal of Tribology OCTOBER 999, Vol. 2 / 89

5 Fig Load, N Load capacity determination CO, rpm Fig. 8 Tlieoretical nominal film thickness when the theoretical load capacity equals the experimental load capacity A loading piston supported by an aerostatic journal bearing. This carries the thrust bearing at one end, which is pushed onto the rotating runner. The air pressure behind the loading piston provides the thrust load. A flexible sensor, in the vertical plane prevents the floating piston from rotating. Flexible sensor deflection provides the measurement of aerodynamic friction torque. Rotational speed is measured by a probe which catches the light reflection of a radial reflective band pasted on the back of the rotor. Two capacitance type displacement probes measure the relative motion "dl" between thrust piston and runner. 5.2 Test Procedure. Tests have been carried out for four rotor rotational speeds between 20,000 rpm and 50,000 rpm. For each rotor rotational speed, the piston is loaded until a slight increase in load causes a sudden large increase in frictional torque (Fig. 6). In this manner, maximum load capacity has been reached. Each test enables the evolution of power loss pf (the product of aerodynamic torque by rotation speed) in terms of the axial load to be obtained. In addition to that, the piston is loaded statically (when rotor speed equals 0 rpm) in order to obtain the experimental thrust bearing static stiffness. Thus, experimental maximum load capacity, power loss and thrust bearing stiffness can be compared with theoretical results. 5.3 Results and Analysis. Figure 7 gives the unit load capacity (i.e., maximum piston load divided by the sector areas) against the rotational speed. The high load capacities obtained confirm the validity of the design method. The advantage, in terms of load capacities, is significant compared to the last results published on thrust bearings in the 980's. It can be compared to the advances made by Heshmat on journal bearings in 994 (Heshmat, 994). The analytical load capacities are equal to the experimental load capacities if calculations are carried out with nominal film thickness values between.9 jim and 2.25 /^m (Fig. 8). It is of the same order of magnitude as previous studies: 2.5 jiva for Dayton (976), 8 ^m for Heshmat (982) or Zolton (979). Figure 9(fl) shows the graphs of static compliance against load. For weak loads, experimental compliance than the theoretical one. This could be explained by small manufacturing faults. When the load is weak, the rotor disc is not in contact with all the sectors: the model assumes that the alignment between the rotor disc and the thrust plate is perfect, and that the top foil is perfectly in contact with the bump foil. This is not the case of I -I '^ 0,5 Theo '^m \M!S^^miMmm\T\^gcjsms^^mlQi a -f Fig. 9 Static compliance against load a real thrust'bearing (Fig. 20). When the load increases, theoretical and real contact areas are the same. When this happens theoretical and experimental compliance agreement better. Figure 2 shows the comparison between theoretical and experimental aerodynamic power loss. Differences occur when normal load increases. This is due to a very low film thickness area on the sector inner radius calculated by the model (Fig. 22). Asp/=/sector ((w/j/2)(ap/5s) + (/^ rv/!))rf5, the value Fig. 20(a) Contact areas at weak loads Fig. 20(&) Contact areas at high loads Fig. 2 Aerodynamic power loss w.n 300 T 998 (present study) pimensionless film thickness evolution during theoretica! loading increasing load high load (X) rotation speed ; rpm Fig. 7 Load capacity improvement between 970 and / Vol. 2, OCTOBER 999 Fig. 22 Theoretical development of a very small film thickness area Transactions of the ASME

6 of the term fj, r^/h becomes high. It can be considered that in reality this area does not appear because of the influence of the neighboring points due to the upper leaf. The structural model in this study considers all the influence coefficients equal to 0 (the structural stiffness matrix is assumed to be diagonal in the theoretical study). 6 Conclusions This study presents a very simple method enabling aerodynamic thrust bearings to be designed rapidly. It is based on the results obtained with rigid profiles. Without any elastoaerodynamic calculations, the optimum initial compliant profile can be determined. A comparison with a direct complete calculation agrees well with the previously published resuhs. This elastoaerodynamic model has been simplified in order to reduce the calculation time. An experimental study has shown the good results of the model with respect to load capacities and thrust bearing compliance. The method presented has been applied successfully for the design of a 80 mm o.d, 40 mm i.d thrust bearing with greatly improved load capacities. It could therefore certainly be applied to other operating conditions with success. Equation (3) leads to a nonlinear system which has the following form F(P) = 0 where equation, written at node L is: PLHI, + R "L+M ~ "L-M 2AR (PI.+MHI+M P L-MHI~M){PL+M PL~M) 4AR^ P,Hl{P,,^M + P,,-M - 2P,,) AR' KRHt^i 2A9 + RA9' PL^M + J,.. + P,Hl {Hi,, (4) + HU) References Heshmat, H., 982, "Advanced Development of Air-Lubricated Foil Thrust Bearings," ASME-ASLE Joint Lubrication Conference, Washington, 5-7 October 982, preprint NO. 82-LC-6B-, ASLE Trans. Heshmat, H., Walowit, J. A., and Pinkus, O., 983, "Analysis of Gas-Lubricated Coinpliant Thrust Bearings," ASME JouRNAi, OF LUBRICATION TECHNOL OGY, Vol. 05, No. 4, pp Heshmat, H., 994, "Advancement in the Performance of Aerodynamic Foil journal Bearings: High Speed and Load Capability," ASME JOURNAL ot' TRinOLOOY, Vol. 6, pp iordanoff,., 997, "Maximum load capacities profiles for gas thrust bearings working under high compressibility number conditions," ASME JOURNAL OF TRIBOLOGY. Roger Ku, C. P., and Heshmat, H., 992, "Compliant Foil Bearing Structural Stiffness Analysis; Part T Theoretical Model Including Strip and Variable Bump foil Geometry." ASME JOURNAL OF LUBRICATION TECHNOLOGY, Vol. 4, No. 2, pp Roger Ku, C. P., and Heshmat, H., 993, "Compliant Foil Bearing Structural Stiffness Analysis; Part II Experimental Investigation." ASME JOURNAL OF LUBRICATION TECHNOLOGY, Vol. 5, No. 3, pp Gray, S., Heshmat, H., and Bhushan, B., 98, "Technology Progress on Compliant Foil Air Bearing Systems for Commercial Applications," Proceedings of the 8e Gas Bearing Symposium, 8-0 Avril 98, pp Dayton, DR., 976, "Gas Lubricated Foil Bearing development For Advanced Turbomachines," Technical Report AFAPL-TR-76-II4, Garett, Nov. Neineth, Z. N., 979, "Operating Characteristics of a Cantilever-Mounted Resilient-Pad Gas-Lubricated Thrust Bearing," NASA Technical Paper 438. APPENDIX Numerical Solution of the Reynolds Equation Jt.k -'St.kPiHi, PL+M ~ PL~M, R{Pi,+M + Pi.~M 2AR AR' PL,, + P,_, - 2P,, RAO' ^L-M^kPt-AfH I_^-M} + (SJ^+M,IIPI.+MH 2 L+M ^P(PL+M ~ PL-M) 4AR 2Pt) 2 X 3(Pi.+ i PL--]) 4RA9' AR [P,^,S(L- Ik) - P,,,S(L+ l,k)] 2A9 For compressibility numbers under 00, the dimensionless pressure field is taken as equal to for iteration and the Newton Raphson method has no convergence problem. For compressibility numbers over 00, a first calculation has to be made with A = 00. Then, the pressure obtained is taken as the first iteration pressure for A = 200 and so on until the compressibility number required is reached. Rde PH de ^(RPH^'^]^A''" dr V dr de (3) fixed surface (case I) or sliping surface (case 2) For the numerical solution of this equation, a sector mesh of N radial nodes and M angular nodes is made as in Fig. 5. Thus, the derivative functions are calculated with finite differences: dr'' dq 39 dq dr {qt+x ^ql+m (* + ^;.-i -- 2q,) M' +?/,-«Ai?'.+ -?z.- 2M («L+M - «/,- M) 2AR - 2g,J where q is ix funcrion of the polar coordinates 9 and R..) projected area: i*l Algorithm for the numerical solution APPENDIX 2 Local Bump Compliance Calculation The Welded Bump: It is assumed to be fixed at one end and simply touching at the other end. Coulomb friction forces between bump and plate is taken into account. Local compliance is given by: Journal of Tribology OCTOBER 999, Vol. 2 / 82

7 With J = yc cos 2 6 l2llij{] - vl) F*iL ' Ee^ sin^ (a/2). a a sin V Uf cos 2 '^' 2 The Free Bump: It is assumed to be simply applied at both ends. Coulomb friction forces between bump and plate are taken into account. The local compliance is then: _ 8 _ 6llil(l - vl) F*iL Ee^ sin' (a/2) IXfjl cos a) 4 + ( -yu) a sin a /= U' + +M/ ( - ///) sin (a) ya 3 sin a a.a sm a. a a a sm sin I- «/ cos 2 \ 2 '^ 2 Ijif sin^ hr + T a sm a With: ///(cos (g) - ) - 2A[V ~ cos (a/2) + //; sin (a/2)] A = sin (a/2) + HfCos {all) 822 / Vol. 2, OCTOBER 999 Transactions of the ASME