A method to design hybrid hydrostatic/hydrodynamic journal bearings, with the criterion of optimized self-compensation under misaligning loads, is presented. An analysis considering laminar and turbulent flow of a Newtonian incompressible lubricant between the bearing and a misaligned shaft, with restricted lubricant supply to each recess, is discussed. The mathematical model considers the modified steady-state Reynolds lubrication equation, an exact function for the local bearing radial clearance with a misaligned shaft, the continuity integral–differential equations at the recess limits, and boundary conditions at the cavitation zone and outer limits. The finite-difference method was used, and a modular computer program was developed. The procedure follows a univariate search to determine the optimum size and position of recesses and therefore obtain the design with the maximum reactive moment under misaligning loads. A validation of the model was obtained comparing the results with experimental and calculated data from the literature. Results for a 4 + 4 LBP hybrid bearing design are presented.

References

1.
Kimball
,
A. J.
,
1924
, “
Internal Friction Theory of Shaft Whirling
,”
General Electr. Rev.
,
27
, p.
244
.
2.
Gunter
,
E. J.
,
1966
, “
Dynamic Stability of Rotor-Bearing Systems
,” Office of Technical Utilization, U.S. Government Printing Office, Washington, DC, Report No. NASA SP-113, p.
228
.
3.
Bassani
,
R.
,
2011
, “
Lubricated Hybrid Journal Bearings
,”
ASME J. Tribol.
,
133
(
3
), p.
034501
.
4.
Kang
,
Y.
,
Yang
,
D.-W.
,
Hu
,
S.-Y.
,
Hung
,
Y.-H.
,
Peng
,
D.-X.
, and
Chen
,
S.-K.
,
2014
, “
Design for Static Stiffness of Hydrostatic Bearings: Double-Action Variable Compensation of Spool-Type Restrictors
,”
Ind. Lubr. Tribol.
,
66
(
1
), pp.
83
89
.
5.
Charki
,
A.
,
Diop
,
K.
,
Champmartin
,
S.
, and
Ambari
,
A.
,
2014
, “
Reliability of a Hydrostatic Bearing
,”
ASME J. Tribol.
,
136
(
1
), p.
011703
.
6.
Vance
,
J. M.
,
1988
,
Rotordynamics of Turbomachinery
,
Wiley
,
New York
.
7.
Huzel
,
D.
, and
Huang
,
D.
,
1992
,
Modern Engineering for Design of Liquid-Propellant Rocket Engines
, Vol.
147
,
Progress in Astronautics and Aeronautics
,
Washington, DC
.
8.
Anderson
,
W. J.
,
Coe
,
H. H.
,
Fleming
,
D. P.
, and
Parker
,
R. J.
,
1971
, “
Series-Hybrid Bearing: An Approach to Extending Bearing Fatigue Life at High Speeds
,” NASA Technical Brief No. 71-10173.
9.
Verma
,
S.
,
Jadon
, V
. K.
, and
Gupta
,
K. D.
,
2011
, “
Analysis of Capillary Compensated Hydrostatic Journal Bearing Operating With Micropolar Lubricant
,”
Ind. Lubr. Tribol.
,
63
(
3
), pp.
192
202
.
10.
Cavdar
,
K.
,
2006
, “
Computer Aided Optimization of the Hydrostatic Radial Journal Bearings
,”
Ind. Lubr. Tribol.
,
58
(
3
), pp.
129
134
.
11.
Solmaz
,
E.
, and
Oztürk
,
F.
,
2006
, “
Optimization of Hydrostatic Journal Bearings With Parameter Variations Based on Thermodynamic Effects
,”
Ind. Lubr. Tribol.
,
58
(
2
), pp.
118
122
.
12.
Salem
,
F.
,
El-Sherbiny
,
M.
, and
El-Hefnawy
,
N.
,
1983
, “
Optimum Design of Hydrostatic Journal Bearings. Part II: Minimum Power Losses
,”
J. Eng. Appl. Sci.
,
2
, pp.
171
184
.
13.
Feng
,
H.
,
Xu
,
C.
, and
Wan
,
J.
,
2014
, “
Mathematical Model and Analysis of the Water Lubricated Hydrostatic Journal Bearings Considering the Translational and Tilting Motions
,”
Math. Prob. Eng.
,
2014
, p.
353769
.
14.
Lu
,
D.
,
Zhao
,
W.
,
Lu
,
B.
, and
Zhang
,
J.
,
2014
, “
The Maximum Allowable Misalignment in Hydrodynamic Rolling Hybrid Bearings
,”
ASME J. Eng. Gas Turbines Power
,
136
(
8
), p.
082501
.
15.
San Andres
,
L.
,
1993
, “
The Effect of Journal Misalignment on the Operation of a Turbulent Flow Hydrostatic Bearings
,”
ASME J. Tribol.
,
115
(
3
), pp.
355
363
.
16.
Buckholz
,
R. H.
, and
Lin
,
J. F.
,
1986
, “
The Effect of Journal Bearing Misalignment on Load and Cavitation for Non-Newtonian Lubricants
,”
ASME J. Tribol.
,
108
(
4
), pp.
645
654
.
17.
Qiu
,
Z. L.
, and
Tieu
,
A. K.
,
1995
, “
Misalignment Effect on the Static and Dynamic Characteristics of Hydrodynamic Journal Bearings
,”
ASME J. Tribol.
,
117
(
4
), pp.
717
723
.
18.
Reason
,
B. R.
, and
Siew
,
A. H.
,
1982
, “
A Numerical Solution for the Design and Performance Evaluation of Journal Bearings With Misalignment
,”
Inst. Mech. Eng.
,
C9/82
, pp.
77
85
.
19.
Kurtin
,
A. K.
,
Childs
,
D.
,
San Andres
,
L.
, and
Hale
,
K.
,
1993
, “
Experimental Versus Theoretical Characteristics of a High-Speed Hybrid (Combination Hydrostatic and Hydrodynamic) Bearing
,”
ASME J. Tribol.
,
115
(
1
), pp.
160
168
.
20.
Hirs
,
G. G.
,
1972
, “
A Bulk-Flow Theory for Turbulence in Lubricant Films
,” Paper No. 72-LubI2.
21.
Constantinescu
,
V. N.
,
1962
, “
Analysis of Bearings Operating in Turbulent Regime
,”
ASME J. Basic Eng.
,
84
(
1
), pp.
139
151
.
22.
Hélène
,
M.
,
Arghir
,
M.
, and
Frêne
,
J.
,
2005
, “
Combined Navier–Stokes and Bulk-Flow Analysis of Hybrid Bearings: Radial and Angled Injection
,”
ASME J. Tribol.
,
127
(
3
), pp.
557
567
.
23.
Smith
,
G. D.
,
1969
,
Numerical Solution of Partial Differential Equations
,
Oxford University Press
,
Oxford, UK
.
24.
Morton
,
P. G.
,
1971
, “
Measurement of the Dynamic Characteristics of a Large Sleeve Bearing
,”
ASME J. Lubr. Technol., Ser. F
,
93-1
, pp.
143
150
.
25.
Someya
,
T.
, ed.,
1989
,
Journal-Bearing Databook
,
Springer-Verlag
,
Berlin/Heidelberg, Germany
, p.
179
.
26.
Martínez Báez-Esparza
,
L. F.
,
1998
, “
Metodología de Diseño de Cojinetes Hidrostáticos Cilíndricos con Capacidad de Carga Autocompensada ante Desalineamientos de la Flecha
,” Tesis Doctoral, División de Estudios de Posgrado, Facultad de Ingeniería, Universidad Nacional Autónoma de México, Ciudad Universitaria, D.F., México (in Spanish).
27.
Pierre
,
D. E.
,
1986
,
Optimization Theory With Applications
,
Dover Publications
,
New York
.
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