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High-Speed Gas Bearings for Micro-Turbomachinery

  • Zoltán S. Spakovszky
Chapter
Part of the MEMS Reference Shelf book series (MEMSRS)

Abstract

The mechanical design and architecture of high-speed rotating machinery, independent of size or scale, are strongly governed by the rotordynamic behavior of the spool and its bearing arrangement. Large-scale gas turbine engines yield multi-spool shaft constructions where the rolling contact bearings are close to the centerline of the engine supporting the shaft and disk assemblies as shown in Fig. 6.1 on the left.

Keywords

Journal Bearing Thrust Bearing Labyrinth Seal Rotor Unbalance Bearing Clearance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

a

rotor imbalance

A, AR

area, area ratio

A0

coefficient

c

angular damping coefficient

C

journal bearing clearance, coefficient

C,\(\tilde {C}\)

damping coefficient

Cf

friction coefficient

d

orifice diameter

D

rotor diameter, orifice diameter

Dh

hydraulic diameter

DN

bearing diameter times rotor speed in mm-rpm

e

dynamic eccentricity

ê

unit vector

F

force, function

G

Green’s function

h

thrust-bearing gap or clearance

I

diametral moment of inertia

Ip

polar moment of inertia

k, K

stiffness

L

journal bearing length, orifice length

\(\dot m\)

mass flow

m

mass of rotor disk

M

Mach number, moment

n, N

harmonic number, number of

p

pressure,

Δp

hydrostatic differential pressure across journal bearing

q, Q

specific flow rate, flow rate

r

radial coordinate, radial location of orifices

R

rotor radius, radius of thrust-bearing pad, specific gas constant

Re

Reynolds number

\(\mathfrak{R}\)

whirl-ratio Ω W N

s

Laplace variable

t

time

T

temperature

u, \({\overline{U}}\)

velocity, mean axial velocity due to hydrostatic flow

v, V

velocity

\(\mathfrak{W}\)

whirl number = k v /k p

z

axial location, parameter

α

plenum circumferential angle

β

reduced frequency ωL/\({\overline{U}}\)

βFD

ratio of axial flow-through time and viscous diffusion time

βx, βy

Euler angles

δ

perturbation

ɛ=e/C0

rotor normalized radial eccentricity

φ

angle

γ

ratio of specific heats

Λ

bearing number

χ

dynamic imbalance

μ

dynamic viscosity

ν

kinematic viscosity

ω

frequency, stagnation pressure loss coefficient

ωinlet

journal bearing inlet stagnation pressure loss coefficient

Ω

rotor speed

ΩN

natural frequency

ΩW

rotor speed at onset of whirl instability

ψ

function

ρ, ρd

fluid density, rotor disk density

τw

wall shear stress

τd

characteristic viscous diffusion time

τf

flow-through time of axial hydrostatic flow

ζ

damping ratio, non-dimensional hydrodynamic moment

{}dp

damping

{}hd

cross-coupled hydrodynamic

{}hs

direct-coupled hydrostatic

{}sys

system

{}p

hydrodynamic pumping action

{}v

hydrodynamic viscous effect

{}*

critical, singular condition

{}a

ambient

{}o

nominal, equilibrium condition

{}r

radial

{}R

rotor

{}t

total or stagnation

{}x

axial

{}θ

tangential

Notes

Acknowledgments

The work described in this chapter has resulted from contributions of a number of individuals, many of whom are co-authors of the referenced papers. It has been a pleasure working with the students and members of the bearing team of the microengine project. In particular the author would like to mention his gratitude to Dr. L. Liu, Dr. C.J. Teo, Dr. S. Jacobson, and Dr. F. Ehrich. The editorial help by Ms. D. Park is gratefully acknowledged.

References

  1. 1.
    Allen DS, Stokes PJ, Whitley S (1961) The performance of externally pressurized bearings using simple orifice restrictors. ASLE Trans 4:181–196CrossRefGoogle Scholar
  2. 2.
    Arghir M, Frene J (2001) A triangle based finite volume method for the integration of lubrication's incompressible bulk flow equations. J Tribol-T ASME 123:118–124CrossRefGoogle Scholar
  3. 3.
    Chen NNS, Ho KW (1981) Performance study of a hydrostatic air thrust bearing. Wear 70:207–217CrossRefGoogle Scholar
  4. 4.
    Childs D (1993) Turbomachinery rotordynamics–phenomena, modeling and analysis. John Wiley and Sons, New YorkGoogle Scholar
  5. 5.
    Constantinescu VN, Galetuse S (1987) On the dynamic stability of the spiral-groove gas-lubricated thrust bearing. J Tribol-T ASME 109:183–188CrossRefGoogle Scholar
  6. 6.
    Diez S (2003) Preliminary performance characteristics of a microfabricated turbopump. M.Sc. thesis, Massachusetts Institute of TechnologyGoogle Scholar
  7. 7.
    Ehrich FE (1989) The role of bearing support stiffness anisotropy in suppression of rotordynamic instability. ASME DTC-12th Biennial Conference on Mechanical Vibration and Noise, MontrealGoogle Scholar
  8. 8.
    Ehrich FE, Jacobson SA (2003) Development of high-speed gas bearings for high-power-density micro-devices. J Eng Gas Turb Power 125:141–148CrossRefGoogle Scholar
  9. 9.
    Frechette LG, Jacobson SA, Breuer KS, Ehrich FE, Ghodssi R, Khanna R, Wong CW, Zhang X, Schmidt MA, Epstein AH (2005) High-speed microfabricated silicon turbomachinery and fluid film bearings. J Microelectromech S 14:141–152CrossRefGoogle Scholar
  10. 10.
    Fuller D (1969) A review of the state-of-the-art for the design of self-acting gas-lubricated bearings. J Lubr Technol 91:1–16CrossRefGoogle Scholar
  11. 11.
    Garner D, Lee C, Martin F (1980) Stability of profile bore bearings: infulence of bearing type selection. Tribol Int 13:204–201CrossRefGoogle Scholar
  12. 12.
    Gunter E, Trumper P (1969) The influence of internal friction on the stability of high speed rotors with anisotropic supports. J Eng Ind:1105–1113Google Scholar
  13. 13.
    Idelchik IE (1994) Handbook of hydraulic resistance, 3rd edn. CRC Press, Boca Raton, FLGoogle Scholar
  14. 14.
    Kim D, Lee S, Bryant M, Ling F (2004) Hydrodynamic performance of gas microbearings. J Tribol-T ASME 126:711–718CrossRefGoogle Scholar
  15. 15.
    Larson R, Richardson H (1962) A preliminary study of whirl instability for pressurized gas bearings. J Basic Eng-T ASME 84:511–520CrossRefGoogle Scholar
  16. 16.
    Licht L, Fuller DD, Sternlicht B (1958) Self excited vibrations of an air-lubricated thrust bearing. Trans ASME 80:411–414Google Scholar
  17. 17.
    Liu L (2005) Theory for hydrostatic gas journal bearings for micro-electro-mechanical systems. Ph.D. thesis, Massachusetts Institute of TechnologyGoogle Scholar
  18. 18.
    Liu L, Spakovszky Z (2007) Effects of bearing stiffness anisotropy on hydrostatic micro gas journal bearing dynamic behavior. ASME J Eng Gas Turb Power 128(1): 177–184CrossRefGoogle Scholar
  19. 19.
    Liu L, Teo CJ, Epstein AH, Spakovszky ZS (2005) Hydrostatic gas journal bearings for micro-turbomachinery. J Vib Acoust 127:157–164CrossRefGoogle Scholar
  20. 20.
    Lomakin A (1958) Calculation of critical number of revolutions and conditions necessary for dynamic stability of rotors in high-pressure hydraulic machines when taking into account forces originating in sealings. Power Mech Eng (in Russian)Google Scholar
  21. 21.
    Lund J (1968) Calculation of stiffness and damping properties of gas bearings. J Lub Tech, Ser. F 90:793–803CrossRefGoogle Scholar
  22. 22.
    Miki N, Teo CJ, Ho L, Zhang X (2000) Precision fabrication of high-speed micro-rotors using deep reactive ion etching (DRIE). In: Hilton Head Solid-State Sensors and Actuators Workshop, Hilton Head Island, SCGoogle Scholar
  23. 23.
    Orr DJ (2000) Macro-scale investigation of high speed gas bearings for MEMS devices. Ph.D. thesis, Massachusetts Institute of TechnologyGoogle Scholar
  24. 24.
    Piekos ES (2000) Numerical simulation of gas-lubricated journal bearings for microfabricated machines. Ph.D. thesis, Massachusetts Institute of TechnologyGoogle Scholar
  25. 25.
    Piekos ES, Orr DJ, Jacobson SA, Ehrich FE, Breuer KS (1997) Design and analysis of microfabricated high speed gas journal bearings. AIAA Paper 97–1966, 28th AIAA Fluid Dynamics Conference, Snowmass Village, COGoogle Scholar
  26. 26.
    Roudebush WH (1957) An analysis of the effect of several parameters on the stability of an air-lubricated hydrostatic thrust bearing. NACA Technical Note 4095Google Scholar
  27. 27.
    San Andres L, Wilde D (2000) Finite element analysis of gas bearings for oil-free turbomachinery. Revue Européenne des Eléments Finis, 10 (6/7): 769–790Google Scholar
  28. 28.
    Savoulides N (2004) Development of a MEMS turbocharger and gas turbine engine. Ph.D. thesis, Massachusetts Institute of TechnologyGoogle Scholar
  29. 29.
    Shapiro AH (1953) The dynamics and thermodynamics of compressible fluid flow, vol I. Ronald Press, New YorkGoogle Scholar
  30. 30.
    Smith D (1933) Motion of a rotor carried by a flexible shaft in flexible bearings. Proc R Soc Lon Ser-A 142:92–118CrossRefGoogle Scholar
  31. 31.
    Spakovszky Z, Liu L (2005) Scaling laws for ultra-short hydrostatic gas journal bearings. ASME J Vibr Acoust 127:254–261CrossRefGoogle Scholar
  32. 32.
    Stowell TB (1971) Pneumatic hammer in a gas lubricated externally pressurized annular thrust bearing. ASME J Appl Mech 93:498–503Google Scholar
  33. 33.
    Teo CJ (2006) MEMS turbomachinery rotordynamics: Modeling, design and testing. Ph.D. thesis, Massachusetts Institute of TechnologyGoogle Scholar
  34. 34.
    Teo CJ, Spakovszky, Z (2006) Modeling and experimental investigation of micro-hydrostatic gas thrust bearings for micro-turbomachines. ASME J Turbomach 128:597–605CrossRefGoogle Scholar
  35. 35.
    Teo CJ, Spakovszky Z (2006) Analysis of tilting effects and geometric non-uniformities in micro-hydrostatic gas thrust bearings. ASME J Turbomach 128: 606–615CrossRefGoogle Scholar
  36. 36.
    Teo CJ, Spakovszky Z (2006) Analysis of annular seals for high-speed power-MEMS devices. Proc Power MEMS conference, Berkeley, CAGoogle Scholar
  37. 37.
    Teo CJ, Liu L, Li HQ, Ho L, Jacobson S, Ehrich F, Epstein A, Spakovszky Z (2006)“High-Speed Operation of a Gas-Bearing Supported MEMS Air Turbine. ASME paper IJTC2006-12173, STLE / ASME International Joint Tribology Conference, San Antonio, TXGoogle Scholar
  38. 38.
    Teo CJ, Spakovszky Z, Jacobson S (2008) Unsteady flow and dynamic behavior of ultra-short Lomakin gas bearings. ASME J Tribol 130(1): 1–9CrossRefGoogle Scholar
  39. 39.
    Vohr JH (1966) An experimental study of flow phenomenon in the feeding region of an externally pressurized gas bearing. Proc ASME Lubrication Symposium, New Orleans, LAGoogle Scholar
  40. 40.
    Yabe H, Watanabe N (1988) A Study on the running accuracy of an externally pressurized gas thrust bearing (load capacity fluctuation due to machining errors of the bearing). JSME Int J Ser III 31:114–120Google Scholar
  41. 41.
    Yabe H, Yamamoto M (1989) A study on the running accuracy of an externally pressurized gas thrust bearing (bearing stiffness and damping coefficient). JSME Int J Ser III 32:618–624Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Zoltán S. Spakovszky
    • 1
  1. 1.Gas Turbine LaboratoryMassachusetts Institute of TechnologyCambridgeUSA

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