Theoretical and experimental analysis of hybrid aerostatic bearings. Mihai ARGHIR Professor, Fellow of the ASME Université de Poitiers, France

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1 Theoretical and experimental analysis of hybrid aerostatic bearings Mihai ARGHIR Professor, Fellow of the ASME Université de Poitiers, France

2 Lille Strasbourg Nantes Paris Poitiers Grenoble Lyon Bordeaux Toulouse Marseille Université de Poitiers: students on three campuses Institute Pprime (P ): 270 permanent employees and as many PhD students, National Research Council s second largest institute

3 Why (still) analyzing the aerostatic bearing? Hybrid journal bearings Ball bearings Fayolle, et al. «Manufacturing and Testing of TPX LH2-Turbopump Prototype, 46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 2010.

4 OUTLINE Static characteristics Dynamic characteristics Combined «bulk flow» and CFD analysis Experimental analysis (validations) Whirl/whip stability «Pneumatic hammer» hamer instabilty Other candidates: the foil bearing Conclusions

5 Hydrostatic bearing Aerostatic bearing Supply pressure Restrictor Recess surface (10 20)% bearing surface Recess depth < 10 radial clearance Recess surface (50 70)% bearing surface Recess depth 100 radial clearance

6 Static characteristics of the aerostatic bearing W O S R O R e x Y e y φ X Good load capacity Responds like a linear spring up to 40 50% eccentricity with limited cross-coupling effects The static characteristics raise no peculiar problems

7 Dynamic characteristics of aerostatic bearings y x C c c C y x K k k K W W F F y x y x 0 0 y x M M M M y x C C C C y x K K K K W W F F yy yx xy xx yy yx xy xx yy yx xy xx y x y x e e C c c C e e K k k K F F t r 0 0 II. Stability (self-sustained vibrations) 1. Whirl/whip 2. «Pneumatic hammer» I. Linear rotordynamic coefficients Meω 2 Ke Destabilizing effect, ke Stabilizing effect, Ceω r t Circular orbit ωt K xx =K C xx =C K yy =K C yy =C K yx =-k Kxy=k X Y ω O

8 THEORETICAL ANALYSIS -The «bulk flow» system of equations -CFD models

9 Thin film (lubrication) models Classical lubrication assumptions for the thin film: The flow characteristics are considered constant over the film thickness. The curvature is neglected. Reynolds equation for compressible fluid flow 3 h P x 12 x 3 h P z 12 z Uh x 2 t h inappropriate if Re C/R>1 The Bulk Flow system of equations (high Reynolds number): Continuity Momentum + an appropriate turbulence model if needed + Hir s assumption Energy Eq. of state or another equivalent thermodynamic equation The numerical solution of the «bulk flow» system of equations is not always an easy task

10 Boundary conditions & local (inertia) effects Restrictor: compressible orifice flow Thin film boundary conditions Local inertial effects Recess/Film interface generalized Bernoulli equation: Local inertia effects cannot be predicted by thin film flow models therefore we will use CFD

11 What is the place of CFD (nowadays) in aerostatic bearing analysis? A point of view: CFD does a good job for (almost) everything; why not use it for the full size problem? Another point of view: Use CFD only when the thin film flow models fail: -In the recess -At the interface between the recess and the thin film -In the orifice restrictor - Link CFD results to «bulk flow» analyses by using lumped parameters

12 Steady CFD analysis: methodology 1. Simplify the geometry taking into account working conditions feeding chamber fluid inlet Thin film boundary conditions thin film recess orifice fluid exit symmetry 2. Generate the grid: for aerostatic bearings it is the hardest point 3. Solve: understand the numerical solver, tune relaxation parameters 4. Validate: don t immediately accept any result, for example check at least for mass balance 5. Look for relevant information, i.e. results that can be cast as boundary conditions of the «bulk flow» equations: pressure patterns, rapid pressure variations, mass flow rates, etc.

13 Axial pressure variation stemming from CFD analysis Jet effect Local inertia effect A constant pattern is sufficient to describe the pressure in the recess zone

14 Sensitivity analysis ( bulk flow calculations): parameters influencing the dynamic coefficients Parameter list A B C D E F G Supply pressure Recess pressure Supply temperature Clearance Surface roughness Pressure loss in the axial direction Pressure loss in the circumferential direction A B C D E F G The recess pressure has a major influence on the stiffness coefficients; its value is controlled by the restrictor area and geometry

15 Recess pressure and orifice modeling Which surface should be used and when?

16 The equivalent surface of the restrictor [1] Rieger, 1967, Design of Gas Bearings Notes, supplemental to the RPI-MTI course on gas bearing design, RPI-MTI, 2 Volumes

17 Mass flow rate in a deep recess (A orif /A inh =0.1) Mass flow rate in a shallow recess (A orif /A inh =2) C d =const. C d =const. The restriction is ensured by the surface of the orifice The restriction is ensured by the inherent surface

18 Comparison CFD vs. Bulk Flow (Ps=7bar, Ω=0rpm)

19 How good is the equation of the mass flow rate for unsteady working conditions? CFD analysis of the unsteady flow feeding chamber fluid inlet recess thin film Time variation of the film thickness: h t h h 2ft 0 h 10 h 1 % 0 1 sin fluid exit symmetry orifice

20 Orifice mass flow rate: m th t m m sin2ft th0 th1 m 1 Recess pressure: P r t P P sin2ft r r 0 1 r1 P Magnitude of m 1 [kg/s] C d Magnitude of P alv1 [Pa] Excitation frequency [Hz] Excitation frequency [Hz] Phase of m 1 [deg] Phase of P alv1 [deg] Excitation frequency [Hz] Excitation frequency [Hz] Up to 1kHz excitation frequency the orifice mass flow rate equation can be used. For very excitation high excitation frequencies the recess pressure and the film thickness have a phase difference larger than 90.

21 EXPERIMENTAL ANALYSIS -Rotordynamic coefficients

22 The rotor/air bearings test rig Accelerometer Force Impact transducer hammer Displacement probes Air blower Force transducer Piezoelectric force transducer Accelerometer Displacement Accelerometers probe Displacement transducer (inductive probe) Pedestal hybrid aerostaticbearings Pelton (impulse) turbine Rotordynamic coefficients are identified by using impact hammer excitations

23 Measured dynamic coefficients

24 Measured vs. calculated dynamic coefficients Are the predicted rotordynamic coefficients good enough? Unbalance responses Stability analyses

25 Measured vs. calculated unbalance response Dynamic coefficients obatined with the «equivalent» restrictor are injected in a 4DOF rigid rotor model

26 STABILITY ANALYSIS The whirl/whip instability

27 Linear stability of a 2DOF rotor t Critical mass estimated with measured dynamic coefficients Destabilizing effect, ke Meω 2 ωt r Circular orbit Ke Stabilizing effect, Ceω Rotation speed [krpm] Critical mass M cr K k c C C² k² Whirl frequency cr k C Supply pressure [bar]

28 Whip at P supply =2.5 bar and 52 krpm Rotation frequency [Hz] <0.5X 1X Response frequency [Hz] (Very) localized wear of 5µm depth Very rapid impact that caused damages!

29 STABILITY ANALYSIS (2) The «pneumatic hammer» problem

30 A simplified problem: the circular aerostatic thrust bearing Pression Feeding d alimentation Pressure, P s P s y Thin Pression film dans le film pressure mince P(x,z) Orifice Feeding d alimentation orifice (restricteur) (restrictor) Alvéole Recess (poche) de pression pressure, P r P r x Film Thin mince d épaisseur film h(x,z) z Dynamic analysis: first order solution (small perturbation) P t P P t t h h t r r 0 r1 t W W 0 W1 t h 0 1 t M M 0 M 1 t Recesss mass flow rate balance eq. M t b M o t M p t Pr 1 Results of the lumped parameter, static analysis (quasi-analytic solution, Licht, 1967): W1 s 1 1s s K Z 0 h b K jc s 1 s Z b 1 2 Static load: W 0 R 2 r Pr P 2 a Ce 2 Pr C erf P C erf P C r a K K C K Mass flow rate M b h0 Pr Pa 12rT ln R R s b r M C o d A P o s

31 Characteristics of the aerostatic thrust bearing Static stiffness Damping iso-lines for ω 0 (i.e. mass ) recess depth (µm) Damping (Ns/m) Zone of pneumatic hammer instability recess pressure/ supply pressure Negative damping and the accompanying «pneumatic hammer» instability are controlled by: -Recess depth -The ratio P recess /P supply Fuller (1984, pg. 534) suggests operating at high Precess/Psupply by using large size orifice restrictors

32 Theoretical investigation of the «pneumatic hammer» instability in the aerostatic bearing -Centered bearing -Zero rotation speed -Different diameters of the restrictor orifice: 50%, 68%, 84% and 100% of Dorifice Theoretical analysis: -1st step: steady CFD analysis -2 nd step: «bulk flow» calculations of K and C (M )

33 2nd step: «bulk flow» calculations of dynamic coefficients Centered bearing and zero rotation speed Mass i.e. (almost) zero excitation frequency Direct stiffness, K [N/m] Direct damping, C [Ns/m] Zone of pneumatic hammer instability

34 Experimental analysis of a single aerostatic bearing: the «floating bearing» test rig Shaker Hydrostatic double Lomakin bearing Dynamometer (static load) Misalignement stinger Spindle Aerostatic hybrid bearing A new aerostatic bearing with dismounting orifices of different diameters was designed and manufactured.

35 Experimental dynamic coefficients -Centered bearing -Zero rotation speed -Excitations applied by a single shaker in the range Hz -Results obtained with three diameters of the restrictor orifice: 68%, 84% and 100% of Dorifice Direct stiffness, K [N/m] Direct damping, C [Ns/m]

36 Results obtained for the smalest restrictor, 50%Dorifice «Pneumatic hammer» instabilities have a specific signature with a dominant frequency and its entire multiples

37 CONCLUSIONS Is the work on hybrid aerostatic bearing finished? - The accuracy of the predictions is satisfactory but can be still criticized; not all configurations are yet tested. - Theoretical predictions are not very time consuming and enable parametric studies but very extensive theoretical analyses are still a problem (for example analyzing the impact of manufacturing tolerances) - The experimental activity is still going on and is continuously consolidated. Will the hybrid hydro/aerostatic bearing finally replace ball bearings? Acknowledgements SNECMA Space Propulsion Division, Centre National d Etudes Spatiales Laurent RUDLOFF, Amine HASSINI, Franck BALDUCCHI, Lassad AMAMI, Sylvain GAUDILLERE

38 Thank you for your attention