-
Nonlinear Time Transient Stability Analysis of Rotors
Mounted on Gas Foil Bearings (GFBs)
Thesis Submitted for the Partial Fulfillment
of the Requirements for the Degree
of
Master of Technology
by
Fapal Anand Mohan
(Roll No. 08410305)
Under the Guidance
Of
Dr. S. K. Kakoty
DEPARTMENT OF MECHANICAL ENGINEEERING
INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI
-
ii
ABSTRACT
Gas foil bearings (GFB) have been considered as an alternative
to the traditional gas bearings
with the increasing need for high-speed, high temperature
turbo-machinery. However,
predictions of performance characteristics of GFBs are found to
be not exhaustive. Therefore, an
attempt has been made to study steady state as well as dynamic
characteristics of GFBs in the
present work. Simple model considering only deflection of bump
foils and 1D finite element
model considering deflection of bump as well as top foil has
been developed. Reynolds equation
for hydrodynamic lubrication has been solved by finite
difference scheme with successive over-
relaxation technique. With the help of developed models for foil
deflection, nonlinear time
transient stability analysis of rigid rotors supported on GFBs
has been carried out, besides
finding out the steady-state characteristics such as load
carrying capacity, minimum film
thickness and attitude angle. The steady state results are
compared with the experimental and
theoretical results available in literature. An attempt has been
made to evaluate the critical mass
parameter (a measure of stability and a function of speed) for
various values of eccentricity ratios
and bearing numbers. Equations of motion of rigid rotor are
solved by using fourth order Runge-
Kutta method and trajectories of journal centre are obtained to
determine critical mass parameter.
Stability maps are plotted for various values of eccentricity
ratios and bearing numbers. It has
been observed that the GFBs are more stable than conventional
plain gas bearings at lower
eccentricity ratios vis-a-vis for lightly loaded bearings.
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CERTIFICATE
It is certified that the work contained in the thesis entitled
"Nonlinear Time Transient Stability
Analysis of Rotors Mounted on Gas Foil Bearings (GFBs)", by
Fapal Anand Mohan, a
student of the Mechanical Engineering Department, Indian
Institute of Technology Guwahati,
India, has been carried out under my supervision for the award
of the degree Master of
Technology and that this work has not been submitted elsewhere
for a award of degree.
Dr. S. K. Kakoty
Professor,
Department of Mechanical Engineering,
Indian Institute of Technology Guwahati.
July 2010.
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ACKNOWLEDGEMENT
First of all, I would like to express my sincere gratitude to my
honourable guide, Dr. S.K.
Kakoty, for his valuable guidance, support and constant
encouragement during this work, his
support at every stage proved very helpful in successful
completion of this work.
I would like to express my sincere thanks to Dr. D. Chakraborty,
Head of the Mechanical
Engineering Departmernt, IIT Guwahati and the staff of the
department for providing the needful
facilities in this work.
Also thanks to all my fellow M.Tech friends for their support
and good company, especially to
Sudarshan Kumar for helpful discussions during this project
work.
I am indebted a lot to my parents for their whole hearted moral
support and constant
encouragement towards the fulfilment of the degree and
throughout my life.
July 2010 Fapal Anand Mohan
IIT Guwahati
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CONTENTS
Abstract
...................................................................................................................................
ii
Contents
....................................................................................................................................
v
Nomenclature
.......................................................................................................................
viii
List of figures
...........................................................................................................................
xi
List of tables
.........................................................................................................................
xiii
1. Introduction and literature review
......................................................................................
1
1.1 Introduction to rotor bearing system
...............................................................................
1
1.2 Stability analysis
.............................................................................................................
1
1.3 Gas foil bearing
..............................................................................................................
2
1.3.1 Advantages of GFB
................................................................................................
3
1.3.2 Usage of GFBs
........................................................................................................
4
1.4 Literature review
............................................................................................................
4
1.5 Scope of the present work
...............................................................................................
9
2. Bump type GFB and formulation of the problem
.............................................................
10
2.1 Introduction
..................................................................................................................
10
2.2 Description of bump type GFB
.....................................................................................
10
2.3 Basic Equations
............................................................................................................
11
2.4 Finite difference scheme
...............................................................................................
12
2.5 Static analysis
...............................................................................................................
14
2.6 Dynamic analysis: Stability analysis
.............................................................................
15
2.6.1 Equations of motion of a rigid rotor on plain journal
bearings .............................. 15
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vi
2.6.2 Implementation of Runge-Kutta method to the equations of
motion ..................... 17
2.7 Summary
......................................................................................................................
19
3. Simple elastic foundation model for foil structure
............................................................ 20
3.1 Introduction
.................................................................................................................
20
3.2 Simple elastic foundation model
..................................................................................
20
3.3 Results and Discussion
................................................................................................
21
3.3.1 Static performance analysis
.................................................................................
21
3.3.1.1 Comparison with published theoretical results
....................................... 21
3.3.1.2 Comparison with published experimental results
................................... 22
3.3.1.3 Pressure distribution, Film thickness and Top foil
deflection ................. 24
3.3.2 Nonlinear stability analysis
..................................................................................
26
3.3.2.1 Effect of eccentricity ratio on mass parameter
........................................ 27
3.3.2.2 Effect of Bearing number on mass parameter
............................................ 27
3.3.2.3 Effect of compliance coefficient on mass parameter
.................................. 27
3.4 Summary
.....................................................................................................................
28
4 1D FE model for foil structure
............................................................................................
29
4.1 Introduction
.................................................................................................................
29
4.2 1D FE model for top foil
.............................................................................................
29
4.3 Results and Discussion
................................................................................................
30
4.3.1 Comparison with published experimental results
.................................................. 30
4.3.2 Nonlinear stability analysis
..................................................................................
32
4.3.2.1 Effect of eccentricity ratio on mass parameter
........................................ 33
4.3.2.2 Effect of bearing number on mass parameter
......................................... 33
4.4 summary
......................................................................................................................
34
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vii
5. Conclusions and Future work
............................................................................................
35
5.1 Introduction
....................................................................................................................
35
5.2 Conclusions
....................................................................................................................
35
5.2.1 Static performance analysis
..................................................................................
36
5.2.2 Nonlinear stability analysis
..................................................................................
37
5.3 Scope of Future Work
.....................................................................................................
38
Appendix A
..............................................................................................................................
39
Appendix B
..............................................................................................................................
41
References................................................................................................................................
42
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viii
NOMENCLATURE
c Bearing radial clearance (m)
C Top foil structural coefficient:
4
t t
a
E I cC
R Lp
D Diameter of journal (m)
e Bearing eccentricity (m)
bE Youngs modulus for bump foil (2N/m )
tE Youngs modulus for top foil (2N/m )
XF , YF Vertical and horizontal components of hydrodynamic
forces (N)
XF , YF Non-dimensional vertical and horizontal components of
hydrodynamic forces : 2
X
a
F
p R,
2
Y
a
F
p R
0XF , 0YF vertical and horizontal steady state components of
hydrodynamic forces (N)
0XF , 0YF Non-dimensional vertical and horizontal steady state
components of hydrodynamic forces :
0
2
X
a
F
p R, 0
2
Y
a
F
p R
F , F Hydrodynamic forces in , co-ordinate system (N)
F , F Non-dimensional hydrodynamic forces in , co-ordinate
system : 2
a
F
p R
,2
a
F
p R
h Film thickness (m)
eh Elemental length in FE formulation
minh Minimum film thickness (m)
H Non-dimensional minimum film thickness
minH Non-dimensional minimum film thickness
,i j
Grid location in circumferential and axial directions of FDM
mesh
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ix
bI Second moment of area of bump foil (4m )
tI Second moment of area of top foil (4m )
fK Bump foil structural stiffness per unit area (3N/m )
0l Half bump length (m)
L Bearing length (m)
m Number of divisions along j direction of FDM mesh
M Mass of the rotor per bearing (Kg)
M Non-dimensional mass of the rotor per bearing :
2Mc
W
n Number of divisions along j direction of FDM mesh
O Center of bearing
'O Center of journal
p Hydrodynamic pressure in gas film ( 2N/m )
ap Atmospheric pressure (2N/m )
p Arithmetic mean pressure along bearing length ( 2N/m )
P Non-dimensional hydrodynamic pressure
P Non-dimensional arithmetic mean pressure along bearing
length
R Radius of journal (m)
s Bump foil pitch (m)
S Compliance coefficient of bump foil : a
f
p
cK
t Time (s)
bt Bump foil thickness (m)
tt Top foil thickness (m)
tw Top foil transverse deflection (m)
W Non-dimensional top foil transverse deflection
0W Steady state load carrying capacity (N)
0W Non-dimensional steady state load carrying capacity
, ,x y z Coordinate system on the plane of bearing
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Z Non-dimensional axial coordinate of bearing :
z
R
Compliance of the bump foil ( 3m /N ) :
1
fK
Eccentricity ratio
Bearing number :
26
a
R
p c
Gas viscosity ( 2N-s/m )
Attitude angle (rad)
Angular coordinate of bearing (rad) : /x R
Poissons ratio
Non-dimensional time : t
Rotor angular velocity ( rad/s )
Time step in Runge-Kutta method
, Z Non-dimensional mesh size of FDM mesh
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xi
LIST OF FIGURES
Figure 1.1: Schematic views of two typical GFBs
.....................................................................
3
Figure 2.1: Schematic view of bump type GFB
.......................................................................
10
Figure 2.2: A developed view of a bearing showing the mesh size
( ) .......................... 13
Figure 2.3: Rotor-Bearing configuration
...................................................................................
15
Figure 2.4: Plain circular journal bearing
..................................................................................
16
Figure 3.1: Foil structure
..........................................................................................................
20
Figure 3.2: Minimum film thickness Vs static load for shaft
speed 45,000 rpm ......................... 23
Figure 3.3: Minimum film thickness Vs static load for shaft
speed 30,000 rpm ........................ 23
Figure 3.4: Journal attitude angle Vs static load for shaft
speed 45,000 rpm .............................. 24
Figure 3.5: Journal attitude angle Vs static load for shaft
speed 30,000 rpm ............................. 24
Figure 3.6: Pressure distribution
................................................................................................
25
Figure 3.7: Top foil deflection
...................................................................................................
25
Figure 3.8: Film thickness
.........................................................................................................
25
Figure 3.9: Stable (L/D=1, =0.4, S=1,=2, M =9)
.................................................................
26
Figure 3.10: Critically stable (L/D=1, =0.4, S=1,=2, M =20.9)
............................................ 26
Figure 3.11: Unstable (L/D=1, =0.4, S=1,=2, M =30)
.......................................................... 26
Figure 3.12: Effect of eccentricity ratio on critical mass
parameter for L/D=1, =1. ............... 28
Figure 3.13: Effect of bearing number and compliance coefficient
on critical mass parameter for
L/D=1, =0.3.
.........................................................................................................................
28
Figure 4.1: 1D structural model of top foil
................................................................................
29
Figure 4.2: Minimum film thickness versus static load for shaft
speed 45,000 rpm .................. 31
Figure 4.3: Minimum film thickness versus static load for shaft
speed 30,000 rpm .................... 31
Figure 4.4: Journal attitude angle versus static load for shaft
speed 45,000 rpm ......................... 31
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xii
Figure 4.5: Journal attitude angle versus static Load for shaft
speed 30,000 rpm....................... 31
Figure 4.6: Stable ( =0.3, L/D=1, S=1, C=1, =5, M =15)
.................................................... 32
Figure 4.7: Critically stable ( =0.3, L/D=1, S=1, C=1, =5, M
=25.2) .................................. 32
Figure 4.8: Unstable ( =0.3, L/D=1, S=1, C=1, =5, M =35)
................................................. 32
Figure 4.9: Effect of eccentricity ratio on mass parameter for
L/D=1, =1 .............................. 33
Figure 4.10: Effect of bearing number on mass parameter for
L/D=1, =0.3 ............................. 33
Figure 5.1: Minimum film thickness versus static load for shaft
speed 45,000 rpm .................... 37
Figure 5.2: Journal attitude angle versus static load for shaft
speed 45,000 ................................ 37
Figure 5.3: Minimum film thickness versus static load for shaft
speed 30,000 rpm .................... 37
Figure 5.4: Journal attitude angle versus static load for shaft
speed 30,000 ................................ 37
Figure 5.5: Critical mass parameter versus eccentricity ratio
(L/D=1, =1). ............................. 38
Figure 5.6: Critical mass parameter versus bearing number
(L/D=1, =0.3). ............................ 38
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xiii
LIST OF TABLES
Table 3.1: Steady state characteristics for L/D=1.0, S=0
............................................................. 1
Table 3.2: Steady state characteristics for L/D=1.0, =1.0
.......................................................... 2
Table 3.3: Geometry and operating conditions of GFB in Ruscitto
et al. [12] ............................. 3
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CHAPTER 1
Introduction and literature review
1.1 Introduction to rotor bearing system
Rotor bearing systems are widely used assemblies in diverse
engineering applications such as
power stations, marine propulsion systems, automobiles, aircraft
engines etc. Power
machinery, such as compressors and turbo machines, usually
transmit power by means of
rotor bearing systems. With the increase in performance
requirements of high speed rotating
machineries in various fields, the engineer is faced with the
problem of designing a unit
capable of smooth operation under various conditions of speed
and load. In many of these
applications the design operating speed is well beyond the first
critical speed. The design
trend of such systems in modern engineering is towards lower
weight and operating at higher
speeds. Under these circumstances, for different machineries, it
is difficult to perform with
stable low level amplitude of vibration; therefore accurate
prediction of dynamic
characteristics of such systems is important in the design of
any type of rotating machinery.
A rotor of a rotating machine is a very important element in
power transmission. It is
of intricate design and may have various elements such as gears
or turbine wheels. In many
applications it is supported by bearings that are not passive
and contribute to critical speeds
and stability. When the bearings are operating at high speeds,
there is possibility of whirl
instability; this limits the operating speed of the journal.
Therefore, it is important to know
the speed above which the bearing system will be unstable.
1.2 Stability analysis
The stability analysis can be done in any one of the following
ways
Linearized stability analysis.
Non-linear transient analysis.
In the first method of stability analysis, a small perturbation
of the journal center
about the line of centers and its perpendicular direction from
the equilibrium position are
given. Eight stiffness and damping coefficients are estimated
from the resulting differential
equations and these coefficients are used to determine the mass
parameter (a measure of
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2
stability) with the help of the equations of motion of the
rotor, the critical mass represents the
minimum mass of the rotor, which leads to a stable behaviour of
the bearing. The linear
theory does not give any information on the journal motion once
the instability sets in. This
theory does not provide post whirl orbit details. Although
linear analysis for the estimation of
dynamic coefficients and stability analysis is relatively easy
to apply, it is sometimes
criticized because it utilizes linearized film coefficients,
which are only valid for small
displacements of the journal away from its initial static
equilibrium position. Since oil whirl
implies large amplitude vibration, it may be argued that any
analysis based upon an
assumption of small vibration amplitudes is invalid. For this
reason, where resources permit,
a different approach to oil-whirl instability analysis, which
does not assume a linear film, is
preferred. The non-linear transient analysis, however, removes
theses shortcomings.
1.3 Gas foil bearing
In order to reach high rotation speeds in turbo machinery, gas
bearings are widely used due to
low viscosity of their lubricant. Despite this significant
advantage, low viscosity leads to
smaller load and damping capacity. Gas foil bearing (GFB)
appeared to overcome these
limitations. GFBs fulfill most of the requirements of novel
oil-free high speed turbo
machinery by increasing their reliability in comparison to
rolling elements bearings [1].
GFBs are made of one or more compliant surfaces of corrugated
metal and one or more
layers of top foil surfaces. The compliant surface, providing
structural stiffness, comes in
several configurations such as bump-type, leaf-type and
tape-type. GFBs operate with
nominal film thicknesses larger than those found in a
geometrically identical plain gas
bearing, since the hydrodynamic film pressure generated by rotor
spinning pushes the GFB
compliant surface [2,3]. Fig. 1.1 depicts two typical GFB
configurations; one is a multiple-
leaf type bearing and the other is a corrugated-bump strip type
bearing. The published
literature notes that multiple leaf GFBs are not the best of
supports in high performance
turbo machinery, primarily because of their inherently low load
capacity[4], on the other
hand a corrugated bump type GFB fulfills most of the
requirements of highly efficient oil free
turbo machinery [5,6].
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3
(a) Multiple Leaf GFB (b) Corrugated Bump GFB
Figure 1.1: Schematic views of two typical GFBs
1.3.1 Advantages of GFB
The use of GFBs in turbo machinery has several advantages as
outlined below
Higher reliability GFB machines are more reliable because there
is no lubrication
needed to feed the system. When the machine is in operation, the
air/gas film
between the bearing and the shaft protects the bearing foils
from wear. The bearing
surface is in contact with the shaft only when the machine
starts and stops. During this
time, a coating on the foils limits the wear.
No scheduled maintenance - since there is no oil lubrication
system in machines that
use GFB, checking and replacing of lubricant is not needed. This
results in lower
operating costs.
Soft failure - Because of the low clearances and tolerances
inherent in GFB design
and assembly, if a bearing failure does occur, the bearing foils
restrain the shaft
assembly from excessive movement. As a result, the damage is
most often confined
to the bearings and shaft surfaces. The shaft may be used as it
is or can be repaired.
Damage to the other hardware, if any, is minimal and repairable
during overhaul.
High speed operation - Compressor and turbine rotors have better
aerodynamic
Housing
Rotor spinning
Leaf foil
Top foil
Bump foil
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4
efficiency at higher speeds. GFBs allow these machines to
operate at the higher
speeds without any limitation as with ball bearings. In fact,
due to the hydrodynamic
action, they have a higher load capacity as the speed
increases.
Low and high temperature capabilities - Many oil lubricants
cannot operate at very
high temperatures without breaking down. At low temperature, oil
lubricants can
become too viscous to operate effectively. GFBs, however,
operate efficiently at
severely high temperatures, as well as at cryogenic
temperatures.
Process fluid operations - Foil bearings have been operated in
process fluids other
than air such as helium, xenon, refrigerants, liquid oxygen and
liquid nitrogen. For
applications in vapor cycles, the refrigerant can be used to
cool and support the foil
bearings without the need for oil lubricants that can
contaminate the system and
reduce efficiency.
1.3.2 Usage of GFBs
GFBs are currently used in several commercial applications, both
terrestrial and aerospace.
Aircraft air cycle machines (ACMs), auxiliary power units (APUs)
and ground-based
microturbines have demonstrated histories of successful
long-term operation using GFBs [1].
For over three decades GFBs have been successfully applied in
ACMs used for aircraft cabin
pressurization. These turbomachines utilize gas foil bearings
along with conventional
polymer solid lubricant [7]. Based on the technical and
commercial success of ACMs; oil-
free technology moves into gas turbine engines. The first
commercially available oil-free gas
turbine was the 30 kW Capstone microturbine conceived as a power
plant for hybrid turbine
electric automotive propulsion system [7]. Industrial blowers
and compressors are becoming
more common as well. In addition, small aircraft propulsion
engines, helicopter gas turbines,
and high speed electric motors are potential future
applications.
1.4 Literature review
An extensive part of the literature on GFBs relates to their
structural characteristics, namely
structural stiffness, dry friction coefficient and equivalent
viscous damping. The compliant
structural elements in GFBs constitute the most significant
aspect on their design process.
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5
With proper selection of foil and bump materials and geometrical
parameters, the desired
stiffness, damping and friction forces can be achieved. Heshmat
et al. [2] first present
analysis of bump type GFBs and details of the bearings static
load performance. The
predictive model couples the gas film hydrodynamic pressure
generation to a local deflection
(wt) of the support bumps. In this simplest of all models, the
top foil is altogether neglected
and the elastic displacement, ( )t aw p p is proportional to the
local pressure difference
( )ap p through a structural compliance () which depends on the
bump material, thickness
and geometric configuration. This model is hereby named as the
simple elastic foundation
model.
Ku and Heshmat [8] first develop a theoretical model of the
corrugated foil strip
deformation used in foil bearings. The model introduces friction
force between the bump
foils and the bearing housing or top foil, and the effect of
bump geometry on the foil strip
compliance. Theoretical results indicate that bumps located at
the fixed end of a foil strip
provide higher stiffness than those located at its free end.
Higher friction coefficients tend to
increase bump stiffness and may lock-up bumps near the fixed
end. Similarly, the bump
thickness has a small effect on the local bump stiffness, but
reducing the bump pitch or height
significantly increases the local bump stiffness.
In a follow-up paper, Ku and Heshmat [9] present an experimental
procedure to
investigate the foil strip deflection under static loads.
Identified bump stiffnesses in terms of
bump geometrical parameters and friction coefficients support
the theoretical results
presented in [8]. Through an optical track system, bump
deflection images are captured
indicating that the horizontal deflection of the segment between
bumps is negligible
compared to the transversal deflection of the bumps. The
identification of bump strip
stiffness, from the load-versus-deflection curves, indicates
that the existence of friction forces
between the sliding surfaces causes the local stiffness to be
dependent on the applied load and
deformation.
Rubio and San Andrs [10] further developed the structural
stiffness dependency on
applied load and displacement. An experimental and analytical
procedure aimed to identify
the structural stiffness for an entire bump-type foil bearing. A
simple static loader set up
allows observing the GFB deflections under various static loads.
Three shafts of increasing
-
6
diameter induce a degree of preload into the GFB structure.
Static measurements showed
nonlinear GFB deflections, varying with the orientation of the
load relative to the foil spot
weld. The GFB structural stiffness increases as the bumps-foil
radial deflection increases
(hardening effect). The assembly preload results in notable
stiffness changes, in particular for
small loads. A simple analytical model assembles individual bump
stiffnesses and renders
predictions for the GFB structural stiffness as a function of
the bump geometry and material,
dry-friction coefficient, load orientation, clearance and
preload. The model predicts well the
test data, including the hardening effect. The uncertainty in
the actual clearance upon
assembly of a shaft into a GFB affects most the predictions.
Lee et al. [11] introduce a viscoelastic material to enhance the
damping capacity of
GFBs. The rotordynamic characteristics of a conventional GFB and
a viscoelastic foil bearing
are compared in a rotor operating beyond the bending-critical
speed. Experimental results for
the vibration orbit amplitudes show a considerably reduction at
the critical speed by using the
viscoelastic foil bearing. Furthermore, the increased damping
capability due to the
viscoelasticity allows the suppression of nonsynchronous motion
for operation beyond the
bending critical speed. In term of structural dynamic stiffness,
the viscoelastic GFBs provide
similar dynamic stiffness magnitudes in comparison to the
conventional foil bearings.
Ruscitto et al. [12] perform a series of load capacity tests of
bump type GFBs. The
test bearing, 38 mm in diameter and 38 mm in length, has a
single top foil and a single bump
strip layer. The authors note that the actual bearing clearance
for the test bearing is unknown.
Thus, the journal radial travel was estimated by performing a
static load-bump deflection
test. The authors installed displacement sensors inside the
rotor and measure the gap between
the rotor and the top foil at the bearings center plane and near
the bearing edge. As the static
load increases, for a fixed rotational speed, the minimum film
thickness and journal attitude
angle decrease exponentially. The test data for film thickness
is the only one available in the
open literature.
Lee et al. [13] performed the static performance analysis of
GFBs considering three-
dimensional shape of the foil structure. Using this model, the
deflections of interconnected
bumps are compared to those of separated bumps, and the minimum
film thickness is
compared to those of previous models. In addition, the effects
of the top foil and bump foil
thickness on the foil bearing static performance are evaluated.
The results of the study show
-
7
that the three-dimensional shape of the foil structure should be
considered for accurate
predictions of GFB performances and that too thin top foil or
bump foil thickness may lead to
a significant decrease in the load capacity. In addition, the
foil stiffness variation does not
increase the load capacity much under a simple foil
structure.
Lez et al. [14] studied nonlinear numerical prediction of GFB
Stability and
unbalanced response. The stability analysis has evidenced that
the structural deflection itself
renders the bearing much more stable than a rigid bearing of
same initial clearance. The
introduction of dry friction then allows doubling this stability
gain. The unbalanced responses
show a nonlinear step when the unbalanced load exceeds a certain
limit. This jump can lead
to the contact between the shaft and the top foil and hence can
lead to the destruction of the
system. It evidenced that the foil bearing can support higher
mass unbalance before this
undesirable step occurs.
Iordanoff et al. [15] considered Effect of internal friction in
the dynamic behavior of
aerodynamic GFBs. A non-linear model, coupling a simplified
equation for the rotor motion
to both Reynolds equation and foil assembly model is described.
Then the dynamic behavior,
for a given unbalance is studied. For different values of
friction coefficient, the rotor
trajectory is studied, when velocity is increased. For low and
high friction coefficient, the
dynamic behavior shows critical speeds. For medium values
(between 0.2 and 0.4), these
critical speeds disappear. This work outlines that it is
possible to optimize the friction
between the foils in order to greatly improve the dynamic
behavior of foil bearings.
Lee and Park [16] studied operating characteristics of the bump
GFBs considering top
foil bending phenomenon and correlation among bump foils. This
analysis verifies that the
stiffness at the fixed end where the friction forces between the
bearing housing and bump foil
superpose is more than that at the free end.
Kim and San Andres [17], in comparisons with test data [12]
validate a GFB model
that implements the simple elastic foundation model with
formulas for bump stiffnesses taken
from [18]. Predicted journal eccentricities versus static load
show a nearly constant static
stiffness coefficient for heavily loaded conditions and
independent of shaft speed. Predictions
of minimum film thickness and journal attitude angle show
excellent agreement with
experimental data.
-
8
Kim and San Andres [19] did analysis of GFBs integrating 1D and
2D FE top foil
models. 2D FE model predictions overestimate the minimum film
thickness at the bearing
centerline, but slightly underestimate it at the bearing edges.
Predictions from the 1D FE
model compare best to the limited tests data, reproducing
closely the experimental
circumferential profile of minimum film thickness.
Kim and San Andres [20] studied forced nonlinear response of
rigid rotors supported
on GFBs. Predicted rotor amplitudes replicate accurately the
measured responses, with a
main whirl frequency locked at the system natural frequency. The
predictions and
measurements validate the simple GFB model, with applicability
to large amplitude rotor
dynamic motions.
Xiong et al, [21] developed aerodynamic foil journal bearings
for a high speed
cryogenic turbo-expander; they found that foil stiffness plays
an important role in the
dynamic performance of this new type of foil journal
bearing.
Majumder and Majumdar [22] studied theoretical investigation of
stability using a
non-linear transient method for an externally pressurized porous
gas journal bearing.
Yang et al. [23] studied the non-linear stability of finite
length self-acting gas journal
bearings by solving a time- dependent Reynolds equation using
finite difference method.
Two threshold values are discovered instead of one through which
the self-acting gas journal
bearings are changed from stable to unstable state.
GFBs require solid lubrication (coatings) to prevent wear and
reduce friction at start-
up and shut-down prior to the development of the hydrodynamic
gas film. Earlier
investigations have revealed that with proper selection of solid
lubricants the bearing
rotordynamic performance can be significantly improved. Della
Corte et al. [24] present an
experimental procedure to evaluate the effects of solid
lubricants applied to the shaft and top
foil surface on the load capacity of GFBs.
-
9
1.5 Scope of the present work
In view of the above discussion on available literature, it has
been observed that very little
work has been done with regards to GFBs; therefore it has been
proposed to study dynamic
and stability characteristics along with steady state
characteristics of GFBs.
Besides it has been observed that in stability analysis, mostly
linear approach is used,
therefore an attempt has been made to study nonlinear time
transient stability analysis with
simple model considering deflection of bump foils only and
another model considering
deflection of bump as well as top foil.
-
10
CHAPTER 2
Bump type GFB and formulation of the problem
2.1 Introduction
In this chapter, general description of bump type GFB and the
equations which govern the
rotor bearing system is given and solution schemes used for
their solution are discussed.
Equations of motions of rigid rotor on plain journal bearing
used for nonlinear time transient
stability analysis are derived.
2.2 Description of bump type GFB
Figure 2.1: Schematic view of bump type GFB
Figure 2.1 shows the configuration of a typical bump type GFB.
The GFB consists of a thin
(top) foil and a series of corrugated bump strip supports (bump
foil). The leading edge of the
thin foil is free, and the foil trailing edge is welded to the
bearing housing. Beneath the top
foil, a bump structure is laid on the inner surface of the
bearing. The top foil of smooth
surface is supported by a series of bumps acting as springs,
thus making the bearing
compliant. The bump strip provides a tunable structural
stiffness [2].
Bump foil
Top foil
Y
0
e O
O
-
11
The bump foil layer gives the bearing flexibility that allows it
to tolerate significant
amount of misalignment, and distortion that would otherwise
cause a rigid bearing to fail. In
addition, micro-sliding between the top foil and bump foil and
the bump foil and the housing
generates Coulomb damping which can increase the dynamic
stability of the rotor-bearing
system [6]. The bearing stiffness combines that resulting from
the deflection of the bumps
and also by the hydrodynamic film generated when the shaft
rotates. During normal operation
GFB supported machine, the rotation of the rotor generates a
pressurized gas film that
pushes the top foil out in radial direction and separates the
top foil from the surface of the
rotating shaft. The pressure in the gas film is proportional to
the relative surface velocity
between the rotor and the GFB top foil. Thus, the faster the
rotor rotates, the higher the
pressure, and the more load the bearing can support. When the
rotor first begins to rotate, the
top foil and the rotor surface are in contact until the speed
increases to a point where the
pressure in the gas film is sufficient to push the top foil away
from the rotor and support its
weight. Likewise, when the rotor slows down to a point where the
speed is insufficient to
support the rotor weight, the top foil and rotor again come in
contact. Therefore, during start-
up and shut down, a solid lubricant coating is used, either on
the shaft surface or the foil, to
reduce wear and friction [21].
2.3 Basic Equations
Bearing studied here is a bump type GFB. The Reynolds equation
describes the generation of
the gas hydrodynamic pressure (p) within the film thickness (h).
For an isothermal,
isoviscous ideal gas this equation is given by [25],
3 3 ( ) ( )6 12
p p ph phph ph R
x x z z x t
(2.1)
with film thickness
cos( ) th c e w (2.2)
boundary conditions required for the solution of Eqn. 2.1
are
p=pa at = 0 and 2
p=pa at z = 0 and L
-
12
By using the substitutions
a
pP
p ,
e
c ,
x
R ,
zZ
R ,
hH
c , t
wW
c , t
non-dimensionl Reynolds equation is given by,
3 3 ( ) ( )2
P P PH PHPH PH
Z Z
(2.3)
For steady state condition this equation reduces to
3 3 ( )P P PHPH PH
Z Z
(2.4)
where is Bearing number given by,
26
a
R
p c
(2.5)
Non dimensional film thickness H becomes
1 cos( )H W (2.6)
Boundary conditions required for solution of Eqn. 2.3 are
P=1 at = 0 and 2
P=1 at Z = 0 and L/R
where and Z are the circumferential and axial coordinates in the
plane of the
bearing respectively.
2.4 Finite difference scheme
Equation 2.3 is difficult to solve analytically, various
approximation methods are employed
for the solution. However, there are numerical solution schemes,
finite element method and
finite difference methods, which provide results in close
proximity with the experimental
findings. In the present approach finite difference method has
been used.
Equation 2.3 is solved numerically by FDM with central
differences. A developed
view of bearing is shown in Fig. 2.2. The area is divided into a
number of meshes of size
Z . A mesh of m nodes along circumferential direction and n
nodes along axial
direction is created.
-
13
Figure 2.2: A developed view of a bearing showing the mesh size
( )
Where, ,i jP is the pressure at any point (i,j).
iH is the film thickness at any point (i,j).
1,i jP , 1,i jP , , 1i jP , , 1i jP are pressures at four
adjacent points of ,i jP .
Now using central differences,
Eqn. 2.3 for dynamic state simplifies to,
2
, 1 , 2 4 3 5( ) ( ) 0i j i jP K P K K K K (2.7)
Eqn. 2.4 for steady state simplifies to,
2
, 1 , 2 3 0i j i jP K P K K (2.8)
Where 1K , 2K , 3K , 4K and 5K are given in Appendix A.
Eqns. 2.7 and 2.8 are nonlinear systems of the form
( ) 0F P (2.9)
Newton-Raphson method has been employed for their solution,
1 '
n
n n
n
F PP P
F P (2.10)
,i jP
, 1i jP
, 1i jP
1,i jP
1,i jP
, i
,Z j
(0, 0)
-
14
Where nP is pressure obtained after nth iteration and '( )F P is
first derivative of ( )F P
with respect to P.
To start with Newton-Raphson method, initially pressures and
foil deflections at all
the mesh points are assumed and Eqn. 2.10 is solved for all the
mesh points. Once the
pressure distribution is obtained, foil deflections are
calculated by the GFB deflection model
considered using appropriate equation then new film thickness H
is updated using Eqn. 2.6
and the new pressure distribution is estimated by solving Eqn.
2.10 and so on.
For low eccentricities, first iteration is carried out assuming
foil deflection equal to
zero and the non-dimensional pressure field equal to unity and
the Newton-Raphson method
has no convergence problem. For higher eccentricities, a first
calculation has to be made with
lower eccentricities, then pressures and foil deflections
obtained are taken as first iteration
values for higher eccentricities and so on.
As the pressures are assumed in the beginning, Eqns. 2.7 and 2.8
are not satisfied. The
iterative process is repeated until the following convergence
criterion is satisfied.
, , 61
,
10i j i jn n
i j n
P P
P
(2.11)
2.5 Static analysis
To obtain steady state characteristics of GFB, it is required to
obtain pressure distribution by
solving Eqn. 2.7. Once pressure is obtained, steady state
characteristics of GFB can be
dtermined. Non-dimensional horizontal and vertical steady state
load components can be
obtained by following equations.
/ 2
0/ 0
cosL D
XL D
F P d dZ
(2.12)
/ 2
0/ 0
sinL D
YL D
F P d dZ
(2.13)
Finally total non-dimensional steady state load is obtained
by,
2 2
0 0 0X YW F F (2.14)
To obtain attitude angle for a chosen value of eccentricity
ratio, initially attitude angle
is assumed and load components are determined. For steady state
equilibrium, horizontal load
-
15
component should become zero theoretically, though in actual
numerical solution it is not
exactly zero but negligible with respect to vertical load. For
given eccentricity ratio, assumed
attitude angle is varied till horizontal load component
approximately becomes zero. Finally,
we get load capacity and attitude angle.
2.6 Dynamic analysis: Stability analysis
An attempt is being made here to determine the mass parameter (a
measure of stability) of
GFBs with the help of solution of dynamic Reynolds equation and
the equations of motion of
the rigid rotor under the unidirectional load at every time
step. A non-linear time transient
method is used to simulate the journal center trajectory and
thereby to estimate the mass
parameter which is a function of speed.
2.6.1 Equations of motion of a rigid rotor on plain journal
bearings
Consider a symmetric rigid disc of mass 2M, and supporting a
static load 2W0 along X-axis as
shown in Fig. 2.3. The disc is mounted on two identical plain
cylindrical hydrodynamic
journal bearings.
Above rotor-bearing system may be fully described for nonlinear
transient simulation
of GFB by the coordinate system shown in Fig. 2.3. Where x(t)
and y(t) are the co-ordinates
of the rotor mass centre, and FX, FY are the fluid film bearing
reaction forces. Since the rotor
is rigid, the centre of mass displacements is identical to those
of the journal centre.
Figure 2.3: Rotor-Bearing configuration
X
Y
2M
Disc
Shaft
Bearing
2W 0
-
16
Figure 2.4: Plain circular journal bearing
The equations of motion of the rotating system at constant
rotational speed are
given by,
0XMx F W (2.15)
YMy F (2.16)
Displacements of rotor centre along x and y directions in terms
of e and are given
by,
( ) ( )cos ( )x t e t t (2.17)
( ) ( )sin ( )y t e t t (2.18)
Fluid film bearing reaction forces in terms of e and are given
by
cos sinXF F F (2.19)
sin cosYF F F (2.20)
Substituting the values of x(t), y(t), FX and FY in Eqns. 2.15
and 2.16 we get,
F Bearing Journal
O'
X
Y
h
O
FX
F
FY
e
-
17
2 2 20 cos 0M e Me F W
(2.21)
2 2
02 sin 0Me M e F W (2.22)
By using the substitution,
e
c
we get the non-dimensional equations of motion in the following
form
2
0 0 cosMW F W
(2.23)
0 02 sinMW F W (2.24)
where
2McM
W
, 0
0 2
a
WW
p R ,
2
a
FF
p R
, 2
a
FF
p R
Reynolds equation for dynamic state 2.3 and equations of motion
2.23 and 2.24 are
solved successively at every time step for obtaining the values
of , , and . Once these
values are calculated, the motion trajectories are obtained by
plotting the attitude angle and
eccentricity ratio at every time step showing position of
journal orbit at various time steps. By
observing these trajectories it can be ascertained whether the
rotor system is stable, unstable
or at critical condition. It is observed that at a certain value
of mass parameter journal centre
ends in a limit cycle and above that there is transition in
rotor motion from stable to unstable
state. The corresponding value of the mass parameter at this
transition is known as critical
mass parameter.
2.6.2 Implementation of Runge-Kutta method to the equations of
motion
Equations of motion of rigid rotor 2.23 and 2.24 are second
order ordinary differential
equations; these equations are solved by using fourth order
Runge-Kutta method.
Implementation of fourth order Runge-Kutta method for the
solution of these equations is
given as,
-
18
Step 1: Deducing the second order Eqns. 2.23 and 2.24 into first
order equations,
1f
2f
203
0
cosF Wf
MW
0
4
0
sin 2F Wf
MW
Step 3: Calculating the values of k1, k2, k3, k4, l1, l2, l3,
l4, m1, m2, m3, m4, n1, n2, n3 and n4 by
using the following expressions,
1 1.k f
1 2.l f
1 3 0. , , , , , , ,m f M W F F
1 4 0. , , , , , , ,n f M W F F
2 1 1
1.
2k f m
2 2 1
1.
2l f n
2 3 1 1 1 1 0
1 1 1 1. , , , , , , ,
2 2 2 2m f k l m n M W F F
2 4 1 1 1 1 0
1 1 1 1. , , , , , , ,
2 2 2 2n f k l m n M W F F
3 1 2
1.
2k f m
3 3 2
1.
2l f n
3 3 2 2 2 2 0
1 1 1 1. , , , , , , ,
2 2 2 2m f k l m n M W F F
3 4 2 2 2 2 0
1 1 1 1. , , , , , , ,
2 2 2 2n f k l m n M W F F
-
19
4 1 3.k f m
4 2 3.l f n
4 3 3 3 3 3 0. , , , , , , ,m f k l m n M W F F
4 4 3 3 3 3 0. , , , , , , ,n f k l m n M W F F
Step 4: Calculating , , , and with the help of following
expressions,
1 2 3 41
2 26
k k k k
1 2 3 41
2 26
l l l l
1 2 3 41
2 26
m m m m
1 2 3 41
2 26
n n n n
Step 5: Finding the values of , , and for each and every time
step with the following
expressions.
1i i
1i i
1i i
1i i
Step 6: By plotting the values of and in polar graph, trajectory
of journal centre can be
achieved.
2.7 Summary
In this chapter, a general description of bump type gas foil
bearing is given. Numerical
solutions of steady state and dynamic Reynolds equation are
obtained by using finite
difference method, the nonlinear system of equations obtained by
FDM are solved by
Newton-Raphson method. Nonlinear stability analysis is discussed
in detail with derivation
of equations of motion of rigid rotor on plain journal bearing;
finally Runge-Kutta solution
scheme used for solution of equations of motion has been
given.
-
20
CHAPTER 3
Simple elastic foundation model for foil structure
3.1 Introduction
Compliant foil structure which gives flexibility to GFB can be
modeled in several ways from
simple model considering only deflection of bump foils to more
complex models considering
deflection of bump as well as top foil. Here, in this chapter
simple elastic foundation model
has been considered.
3.2 Simple elastic foundation model
Most published models for the elastic support structure in a GFB
are based on the simple
elastic foundation model which is the original work of Heshmat
et al. [2], same model is
considered in the present analysis.
Foil structure used in simple elastic foundation model is given
in Fig. 3.1
Figure 3.1: Foil structure
This model relies on several assumptions:
1) The stiffness of a bump strip is uniformly distributed
throughout the bearing surface,
i.e. the bump strip is regarded as a uniform elastic
foundation.
2) Bump stiffness is constant, independent of the actual bump
deflection, not related or
constrained by adjacent bumps.
3) The top foil does not sag between adjacent bumps. The top
foil does not have either
bending or membrane stiffness, and its deflection follows that
of the bump.
4) Film thickness does not vary along the bearing length.
s
2lo
tbBump foil
Top foil
-
21
With these considerations, the local deflection of the foil
structure (wt ) depends on the
bump compliance () and the average pressure across the bearing
length,
( )t aw p p (3.1)
The compliance () is given by,
3 2
0
3
2 (1 )
b b
l s
E t
(3.2)
By using following substitutions
a
pP
p , t
wW
c
non-dimensional foil deflection equation is given by
( 1)W S P (3.3)
Coupling of the simple model equation for the foil deflection
with the solution of
Reynolds equation is straightforward for the prediction of the
static and dynamic performance
of GFBs [2,6].
3.3 Results and Discussion
The performance characteristics of GFB have been determined
using the analysis described in
the previous chapter with simple elastic foundation model. Here
static performance
characteristics and nonlinear time transient stability analysis
have been studied.
.
3.3.1 Static performance analysis
3.3.1.1 Comparison with published theoretical results
The validity of the present analysis and computational program
is assessed by comparison of
steady state results with published data available in
literature.
Table 3.1 compares attitude angle and load capacity with the
published results Yang
et al, [23], for L/D=1.0 and S=0. GFB reduces to ordinary gas
bearing for S=0.
-
22
Table 3.1: Steady state characteristics for L/D=1.0, S=0
(ref) W W (ref)
0.6 0.2 79.639 #79.080 0.18.05 #0.1806
0.6 0.4 74.171 #74.020 0.4050 #0.4020
0.6 0.6 61.768 #61.450 0.7540 #0.7555
3.0 0.2 48.020 #47.730 0.7097 #0.6916
3.0 0.4 40.951 #40.690 1.5340 #1.5230
3.0 0.6 30.520 #30.040 2.870 #2.8631
# Yang et al. [23]
Table 3.2 compares attitude angle and load capacity for L/D=1.0
and =1 with the
published results Peng and Carpino [3] and Heshmat et
al.[2].
Table 3.2: Steady state characteristics for L/D=1.0, =1.0
S (ref.1) (ref.2) W W (ref.1) W (ref.2)
0 0.6 35.90 *36.50 #35.70 0.964 *0.961 #0.951
0 0.75 24.51 *24.70 #24.10 1.926 *1.922 #1.894
0 0.9 12.69 *12.90 #12.80 5.150 *5.073 #5.055
1 0.6 35.94 *34.00 #32.10 0.5489 *0.567 #0.568
1 0.75 29.55 *27.70 #26.30 0.7523 *0.778 #0.783
1 0.9 24.24 *22.40 #21.40 0.9882 *1.020 #1.020
*Peng and Carpino[3]
# Heshmat et al.[2].
From the above comparisons in Tables 3.1 and 3.2, it has been
observed that the
present results are in good agreement with those from
references, hence computational model
and present analysis may be considered as valid.
3.3.1.2 Comparison with published experimental results
Minimum film thickness and attitude angle are compared with
experimental results available
in Ruscitto et al. [12]. Table 3.3 provides geometry and
operating conditions for the test
GFB in Ruscitto et al. [12].
-
23
Table 3.3: Geometry and operating conditions of GFB in Ruscitto
et al. [12]
Geometry
Bearing radius, R=D/2 19.05 mm
Bearing length, L 38.1 mm
Bearing radial clearance, c 20 m
Top foil thickness tt 101.6 m
Bump foil thickness, bt 101.6 m
Bump pitch, s 4.572 mm
Half bump length, 0l 1.778 mm
Bump foil Youngs modulus, bE 214 GPa
Top foil Youngs modulus, tE 214 GPa
Bump foil Poissons ratio, 0.29
Operating conditions
Atmospheric pressure, ap
510 N/ 2m
Gas viscosity, 52.98 10 N-s/ 2m
Minimum film thickness versus applied static load
Figures 3.2 and 3.3 present minimum film thickness versus
applied static load for operation
of shaft speeds 45,000 rpm and 30,000 rpm respectively. It has
been observed that present
results overestimate minimum film thickness by 0% to 26% and 18%
to 38% for shaft speeds
45,000 rpm and 30,000 rpm respectively.
Figure 3.2: Minimum film thickness Vs static load
for shaft speed 45,000 rpm
Figure 3.3: Minimum film thickness Vs static load
for shaft speed 30,000 rpm
0 25 50 75 100 125 1501500
10
20
30
40
Static load [N]
Min
imu
m f
ilm
th
ick
nes
s [m
icro
met
er]
Prediction (simple model)
Test point - mid plane (Ruscitto, et al.)
0 25 50 75 100 125 1500
10
20
30
40
Static load [N]
Min
imu
m f
ilm
th
ick
nes
s [m
icro
met
er]
Prediction (simple model)
Test point - mid plane (Ruscitto, et al.)
-
24
Attitude angle versus applied static load
Figures 3.4 and 3.5 present attitude angle versus static load
for operation of shaft speeds
45,000 rpm and 30,000 rpm respectively. It is observed that
present results overestimate
attitude angles by 2% to 6% and 29% to 36% for shaft speeds
45,000 rpm and 30,000 rpm
respectively.
Figure 3.4: Journal attitude angle Vs static load
for shaft speed 45,000 rpm
Figure 3.5: Journal attitude angle Vs static load
for shaft speed 30,000 rpm
3.3.1.3 Pressure distribution, Film thickness and Top foil
deflection
Figures 3.6, 3.7 and 3.8 present the non-dimensional pressure
distribution, top foil deflection
and film thickness respectively from the proposed model for =
0.6, L/D=1, S=1, =1. The
surface of journal moves from the left to right in the direction
of increasing . As shown in
Fig. 3.6, the maximum pressure occurs in the bearing center with
pressure becoming ambient
at both sides. Under the action of this pressure, the foil
structure is deflected as shown in Fig.
3.7 and corresponding film thickness profile has been shown in
Fig. 3.8.
From Fig. 3.6 through Fig. 3.7, it has been observed that
maximum value of averaged
non-dimensional pressure is 1.1720 which occurs at0200 from the
fixed end of top foil, at the
same point maximum deflection of top foil takes place and its
maximum value is 0.1720.
Non-dimensional minimum film thickness is 0.5450 at 0236 from
the fixed end of top foil.
0 25 50 75 100 125 1500
10
20
30
40
50
60
Static load [N]
Jou
rnal
att
itu
de
ang
le [
deg
]
Prediction (simple model)
Test point (Ruscitto,et al.)
0 25 50 75 100 125 1500
10
20
30
40
50
60
Static load [N]
Jou
rnal
att
itu
de
ang
le [
deg
]
Predictions (simple model)
Test point (Ruscitto, et al.)
-
25
Figure 3.6: Pressure distribution
Figure 3.7: Top foil deflection
Figure 3.8: Film thickness
0.5 1 1.5 2
30
210
60
240
90 270
120
300
150
330
180
0
0 360 0
2 0.4 0.6 0.8
1 1.2 1.4 1.6 1.8
2
L/R
H
Circumferential location (theta)
0.05 0.1 0.15 0.2
30
210
60
240
90 270
120
300
150
330
180
0
0 360 0
2 0
0.05
0.1
0.15
0.2
L/R
W
Circumferential location (theta)
0.5 1 1.5
30
210
60
240
90 270
120
300
150
330
180
0
0 360 0
2 1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
L/R
P
Circumferential location (theta)
-
26
3.3.2 Nonlinear stability analysis
The motion trajectories have been obtained by plotting polar
graphs showing position of
journal center at various time steps. By observing these
trajectories it can be ascertained
whether the rotor system is stable, unstable or at critical
condition. It is observed that above a
certain value of mass parameter there is a transition in rotor
motion from stable to unstable
state. This value is the critical mass parameter.
As an example polar plots has been given for stable, critical
and unstable condition
from Fig. 3.9 through Fig. 3.11 for =0.4, L/D=1, S=1, =2. It has
been observed that for
M < 20.9 system is stable and at M =20.9 there is transition
in rotor motion from stable to
unstable state, for M >20.9 system is unstable. This value M
=20.9 is the critical mass
parameter.
Figure 3.9: Stable
(L/D=1, =0.4, S=1, =2, M =9)
Figure 3.10: Critically stable
(L/D=1, =0.4, S=1, =2, M =20.9)
Figure 3.11: Unstable
(L/D=1, =0.4, S=1,=2, M =30)
0.2 0.4 0.6 0.8 1
30
210
60
240
90 270
120
300
150
330
180
0
Unit circle
0.2 0.4 0.6 0.8 1
30
210
60
240
90 270
120
300
150
330
180
0
Unit circle
0.2 0.4 0.6 0.8 1
30
210
60
240
90 270
120
300
150
330
180
0
Unit circle
-
27
3.3.2.1 Effect of eccentricity ratio on critical mass
parameter
In Fig. 3.12, plot of critical mass parameter versus
eccentricity ratio is shown for plain gas
bearing, S=0 and GFB, S=1 for L/D=1, =1. It is observed that
critical mass parameter
increases with increase in eccentricity ratio for both plain gas
bearing and GFB; from this it
appears that bearings operating under highly loaded condition
are more stable. Fig. 3.12 also
shows that at low eccentricity ratios GFBs are more stable than
same configuration plain gas
bearings, here for L/D=1, =1, up to eccentricity ratio =0.4517
GFB is more stable and
beyond that plain gas bearing has better stability.
3.3.2.2 Effect of bearing number on critical mass parameter
In Fig. 3.13, plot of critical mass parameter versus bearing
number is shown for plain gas
bearing, S=0 and GFBs, S=1 and S=2 for L/D=1, =0.3, it is
observed that critical mass
parameter increases with increase in bearing number for both
plain gas bearing and GFBs.
For higher values of bearing number, critical mass parameter
almost remains same in case of
GFBs therefore it appears that plain gas bearings have better
stability at higher values of
bearing numbers. Fig. 3.13 also shows that GFBs are more stable
than plain gas bearings at
low bearing numbers, here for L/D=1, =0.3, up to bearing numbers
=2.34 and =2.28
GFBs with S=1 and S=2 have better stability than plain gas
bearing respectively and beyond
that plain gas bearing has better stability.
3.3.2.3 Effect of compliance coefficient on critical mass
parameter
In Fig. 3.13, plot of critical mass parameter versus bearing
number is shown for plain gas
bearing, S=0 and GFBs, S=1 and S=2 for L/D=1, =0.3, it is
observed that increasing
compliance coefficient improves stability up to a certain value
of bearing number and beyond
that GFBs with less compliance coefficient have better
stability, here for L/D=1, =0.3, up
to bearing numbers =2.34 and =2.28 GFBs with S=1 and S=2 have
better stability than
plain gas bearing respectively and beyond that plain gas bearing
has better stability. GFB
with S=2 has better stability than GFB with S=1 up to bearing
number 2.15 and beyond that
GFB with S=1 is more stable.
-
28
Figure 3.12: Effect of eccentricity ratio on critical
mass parameter for L/D=1, =1.
Figure 3.13: Effect of bearing number and
compliance coefficient on critical mass parameter
for L/D=1, =0.3.
3.4 Summary
In this chapter, simple elastic foundation model for GFB has
been developed and present
steady state results are compared with published theoretical and
experimental data. Nonlinear
stability analysis of GFB has also been carried out. Model for
GFB considering deflection of
bump as well as top foil has been provided in the next
chapter.
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60
5
10
15
20
25
30
35
40
45
50
Eccentricity ratio
Cri
tica
l m
ass
par
amet
er
Plain gas bearing
GFB (S=1)
Stable
Unstable
1 2 3 4 5 6 7 8 9 1010
15
20
25
30
35
40
45
50
55
Bearing number
Cri
tica
l m
ass
par
amet
er
Plain gas bearing
GFB (S=1)
GFB (S=2)
Unstable
Stable
-
29
CHAPTER 4
1D FE model for foil structure
4.1 Introduction
In this chapter, compliant foil structure which gives
flexibility to GFB has been modeled
considering deflection of bump as well as top foil. Top foil is
considered like an Euler-
Bernoulli beam, 1D finite element model has been developed to
calculate deflections of foil
structure.
4.2 1D FE model for top foil
The top foil is modeled like an Euler-Bernoulli type beam with
elastic support of bump foils
having one end fixed and other end free as shown in Fig.
4.1.
Figure 4.1: 1D structural model of top foil
This model relies on assumptions:
1. The stiffness of a bump strip is uniformly distributed
throughout the bearing
surface, i.e. the bump strip is regarded as a uniform elastic
foundation.
2. Bump foil stiffness is constant, independent of the actual
bump deflection, not
related or constrained by adjacent bumps.
3. Axially averaged pressure causes a uniform elastic deflection
along the top foil
width (L).
( )ap p L
Top foil
x R
Fixed end
fK L
tw
-
30
Transverse deflection ( tw ) of top foil is governed by the
fourth order differential equation,
22
2 2( )tt t f t a
d wdE I K Lw p p L
dx dx
(4.1)
Using following substitutions,
x
R ,
a
pP
p , t
wW
c
Equation 4.1 has been converted to non-dimensional form as,
2 2
2 2( 1)
d d W WC P
d d S
(4.2)
This equation is solved by finite element method; elemental
equation for this type of problem
is given by Reddy [26],
[ ]{ } { }e e eK u F (4.3)
where,
[ ]eK = Elemental stiffness matrix
{ }eu = Vector of primary nodal variables or generalized
displacements
{ }eF = Force vector
[ ]eK ,{ }eu ,{ }
eF are given in Appendix B.
Elemental equations are assembled as usual and solved to get
deflections.
4.3 Results and Discussion
4.3.1 Comparison with published experimental results
Predicated minimum film thickness and attitude angle are
compared with experimental data
available in Ruscitto et al. [12]. Table 3.3 provides geometry
and operating conditions for
the test GFB in in Ruscitto et al. [12].
-
31
Minimum film thickness versus applied static load
Figures 4.2 and 4.3 present minimum film thickness versus
applied static load for operation
of shaft speeds 45,000 rpm and 30,000 rpm respectively. It has
been observed that present
results overestimate minimum film thickness by 0% to 26% and 18%
to 38% for shaft speeds
45,000 rpm and 30,000 rpm respectively.
Figure 4.2: Minimum film thickness versus
static load for shaft speed 45,000 rpm
Figure 4.3: Minimum film thickness versus
static load for shaft speed 30,000 rpm
Attitude angle versus applied static load
Figures 4.4 and 4.5 present attitude angle versus static load
for operation of shaft speeds
45,000 rpm and 30,000 rpm respectively. It is observed that
present results overestimate
attitude angles by 2% to 6% and 29% to 36% for shaft speeds
45,000 rpm and 30,000 rpm
respectively.
Figure 4.4: Journal attitude angle versus static
load for shaft speed 45,000 rpm
Figure 4.5: Journal attitude angle versus static
Load for shaft speed 30,000 rpm
0 25 50 75 100 125 1500
10
20
30
40
Static load [N]
Min
imu
m f
ilm
th
ick
nes
s [m
icro
met
er]
Prediction (1D FE model)
Test point - mid plane (Ruscitto, et al.)
0 25 50 75 100 125 1500
10
20
30
40
Static load [N]
Min
imu
m f
ilm
th
ick
nes
s [m
icro
met
er]
Prediction (1D FE model)
Test point - mid plane (Ruscitto, et al.)
0 25 50 75 100 125 1500
10
20
30
40
50
60
Static load [N]
Jou
rnal
att
itu
de
ang
le [
deg
]
Prediction (1D FE model)
Test point (Ruscitto,et al.)
0 25 50 75 100 125 1500
10
20
30
40
50
60
Static load [N]
Jou
rnal
att
itu
de
ang
le [
deg
]
Predictions (1D FE model)
Test point (Ruscitto, et al.)
-
32
4.3.2 Nonlinear stability analysis
As an example, polar plots have been presented for stable,
critical and unstable condition
from Fig. 4.6 through Fig. 4.8 for =0.3, L/D=1, S=1, C=1, =5. It
has been observed that
for M < 25.2 system is stable and at M =25.2 there is a
transition in rotor motion from stable
to unstable state, for M >25.2 system is unstable. This value
M =25.2 is the critical mass
parameter.
Figure 4.6: Stable
( =0.3, L/D=1, S=1, C=1, =5, M =15)
Figure 4.7: Critically stable
( =0.3, L/D=1, S=1, C=1, =5, M =25.2)
Figure 4.8: Unstable
( =0.3, L/D=1, S=1, C=1, =5, M =35)
0.2 0.4 0.6 0.8 1
30
21
0
60
240
90 270
120
300
150
330
180
0
Unit circle
0.2 0.4 0.6 0.8 1
30
210
60
240
90 270
120
300
150
330
180
0
Unit circle
0.2 0.4 0.6 0.8 1
30
210
60
240
90 270
120
300
150
330
180
0
Unit circle
-
33
4.3.2.1 Effect of eccentricity ratio on critical mass
parameter
In Fig. 4.9, plot of critical mass parameter versus eccentricity
ratio is shown for plain gas
bearing, S=0 and GFB S=1, C=1 for L/D=1, =1. It is observed that
critical mass parameter
increases with increase in eccentricity ratio for both plain gas
bearing and GFB; from this it
appears that bearings operating under highly loaded condition
are more stable. Fig. 4.9 also
shows that at low eccentricity ratios GFBs are more stable than
same configuration plain gas
bearings, here for L/D=1, =1, up to eccentricity ratio =0.37
GFBs are more stable and
beyond that plain gas bearings have more stability.
4.3.2.2 Effect of Bearing number on critical mass parameter
In Fig. 4.10, plot of critical mass parameter versus bearing
number is shown for plain gas
bearing, S=0 and GFB S=1, C=1 for L/D=1, =0.3, it is observed
that critical mass parameter
increases with increase in bearing number for both plain gas
bearing and GFB. For higher
values of bearing number, critical mass parameter almost remains
same in case of GFB
therefore it appears that plain gas bearings have better
stability at higher values of bearing
numbers. Fig. 4.10 also shows that GFB is slightly more stable
than plain gas bearings at low
bearing numbers, here for L/D=1, =0.3, up to bearing number
=2.11 GFB has better
stability and beyond that plain gas bearing is more stable.
Figure 4.9: Effect of eccentricity ratio on mass
parameter for L/D=1, =1
Figure 4.10: Effect of bearing number on mass
parameter for L/D=1, =0.3
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.65
10
15
20
25
30
35
40
45
50
Eccentricity ratio
Cri
tica
l m
ass
par
amet
er
Plain gas bearing
GFB (S=1,C=1)
Stable
Unstable
1 2 3 4 5 6 7 8 9 1010
15
20
25
30
35
40
45
50
55
Bearing number
Cri
tica
l m
ass
par
amet
er
Plain gas bearing
GFB (S=1,C=1)
Stable
Unstable
-
34
4.4 Summary
In this chapter, 1D FE model for GFB which considers deflections
of both bump foils and top
foil has been developed and present steady state results are
compared with published
experimental data. Nonlinear stability analysis of GFB has also
been carried out. Conclusion
and scope of future work have been provided in the next
chapter.
-
35
CHAPTER 5
Conclusions and Future work
5.1 Introduction
In the present study, scope of the present work has been
determined by studying available
literature on GFBs (chapter 1). Solution schemes for the
Reynolds equation and equations of
motion of rigid rotors have been outlined (chapter 2). For the
deflections of foil structure two
models have been considered, one simple model which considers
deflection of bump foils
only (chapter 3) and another model which considers deflections
of bump as well as top foil
(chapter 4). Using these models, steady state results have been
compared with the available
theoretical and experimental results. Successively nonlinear
time transient stability analysis
of rigid rotors mounted on GFBs considering both the models has
been carried out.
5.2 Conclusions
From the study of nonlinear time transient stability analysis of
GFB models developed in
chapters 3 and 4, following conclusions have been drawn,
Critical mass parameter increases with increase in eccentricity
ratio for both plain gas
bearing and GFB; from this it appears that bearings operating
under highly loaded
conditions are more stable
At low eccentricity ratios GFBs are more stable than same
configuration plain gas
bearings, in view of this, it may be concluded that GFBs have
better stability than
plain gas bearings of similar configuration at lightly loaded
conditions only.
Critical mass parameter increases with increase in bearing
number for both plain gas
bearing and GFBs; from this it appears that bearings operating
with higher values of
bearing numbers are more stable.
Critical mass parameter almost remains same in case of GFBs for
higher values of
bearing numbers; therefore it appears that plain gas bearings
have better stability than
-
36
GFB at higher values of bearing numbers.
Critical mass parameter of GFB is slightly more than that of
plain gas bearings at
lower values of bearing number, therefore it may be concluded
that at lower values of
bearing numbers GFBs do not help much to improve stability.
Increasing compliance coefficient improves stability of GFB up
to a certain value of
bearing number and beyond that GFBs with less compliance
coefficient have better
stability; therefore it may be concluded that GFBs with more
compliant foil structure
have better stability but up to certain value of bearing number
and beyond that less
compliant GFBs have better stability.
Present results for steady state load capacity and attitude
angle have been compared
with experimental results available in Ruscitto et al [12] and
nonlinear stability analysis
results of GFBs have been studied with those of the plain gas
bearings in chapters 3 and 4. In
an attempt to compare results of both the developed models,
steady state results of both the
models are compared together with experimental results of
Ruscitto et al [12] and stability
curves for both the models of GFB and plain gas bearing have
been plotted together.
5.2.1 Static performance analysis
In an attempt to find out the model with better static
performance, results of both the models
are compared along with available experimental results as shown
in Figs. 5.1 through 5.4.
Fig. 5.1 gives minimum film thickness versus applied static load
and Fig. 5.2 gives journal
attitude angle versus applied static load for the shaft speed
45,000 rpm. It is observed that
both the models overestimate minimum film thickness by 0% to 26%
and attitude angle by
2% to 6%.
Figure 5.3 gives minimum film thickness versus applied static
load and Fig. 5.4 gives
journal attitude angle versus applied static load for shaft
speed 30,000 rpm. It is observed that
both the models overestimate minimum film thickness by 18% to
38% and attitude angle by
29% to 36%.
It has also been observed that steady state results of both the
models are matching
with each other, except attitude angles obtained by 1D FE model,
which are more than those
obtained by the simple model up to the applied static load 32N
for both the shaft speeds
-
37
45,000 rpm and 30,000 rpm; therefore it may be concluded that
for steady state analysis both
the models produce same results.
Figure 5.1: Minimum film thickness versus static
load for shaft speed 45,000 rpm
Figure 5.2: Journal attitude angle versus static
load for shaft speed 45,000
Figure 5.3: Minimum film thickness versus static
load for shaft speed 30,000 rpm
Figure 5.4: Journal attitude angle versus static
load for shaft speed 30,000
5.2.2 Nonlinear stability analysis
In an attempt to compare results of both the developed models of
GFB, stability curves for
both the models of GFB and plain gas bearing have been plotted
together.
Figure 5.5 shows combined plot of critical mass parameter versus
eccentricity ratio
for simple model (S=1) and 1D FE model (S=1, C=1) of GFB along
with the same
configuration plain gas bearing for (L/D=1, =1). It is observed
that, simple model and 1D
FE model of GFB shows better stability than plain gas bearing up
to the eccentricity ratios
0.45 and 0.37 respectively and beyond that plain gas bearing has
more stability. It is also
0 25 50 75 100 125 1501500
10
20
30
40
Static load [N]
Min
imu
m f
ilm
th
ick
nes
s [m
icro
met
er]
Prediction (simple model)
Test point - mid plane (Ruscitto, et al.)
Prediction (1D FE model)
0 25 50 100 125 1500
10
20
30
40
50
60
Static load [N]
Jou
rnal
att
itu
de
ang
le [
deg
]
Prediction (simple model)
Test point (Ruscitto,et al.)
Prediction (1D FE model)
0 25 50 75 100 125 1500
10
20
30
40
Static load [N]
Min
imu
m f
ilm
th
ick
nes
s [m
icro
met
er]
Prediction (simple model)
Test point - mid plane (Ruscitto, et al.)
Prediction (1D FE model)
0 25 50 75 100 125 1500
10
20
30
40
50
60
Static load [N]
Jou
rnal
att
itu
de
ang
le [
deg
]
Predictions (simple model)
Test point (Ruscitto, et al.)
Predictions (1D FE model)
-
38
observed that simple model gives better stability than 1D FE
model.
Figure 5.6 shows combined plot of critical mass parameter versus
bearing number for
simple model (S=1) and 1D FE model (S=1, C=1) of GFB along with
the same configuration
plain gas bearing for (L/D=1, =0.3). It is observed that, simple
model and 1D FE model
have better stability than plain gas bearing up to bearing
numbers 2.34 and 2.11 respectively
and beyond that plain gas bearing has better stability. In view
of this, it may be concluded
that GFBs have better stability than plain gas bearings of
similar configuration at lightly
loaded conditions only. It is also observed that simple model
gives slightly better stability
than 1D FE model at lower bearing numbers and at higher values
of bearing numbers both the
models produce same results, here both the models produce same
results beyond the bearing
number 6.2.
Figure 5.5: Critical mass parameter versus
eccentricity ratio (L/D=1, =1).
Figure 5.6: Critical mass parameter versus
bearing number (L/D=1, =0.3).
5.3 Scope of Future Work
Present study can be extended to incorporate the following,
More realistic models of GFB can be developed by
considering,
2D, 3D shape of foil structure.
Friction between bearing housing and bump foil and bump foil and
top foil.
Deflection dependency of bump foils stiffness.
Effect of temperature on foil structure can be considered.
Time transient stability analysis of flexible rotors mounted on
GFBs can be done by
extending the present work.
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.65
10
15
20
25
30
35
40
45
50
Eccentricity ratio
Cri
tica
l m
ass
par
amet
er
Plain gas bearing
GFB (simple model)
GFB (1D FE model)
Stable
Unstable
1 2 3 4 5 6 7 8 9 1010
15
20
25
30
35
40
45
50
55
Bearing number
Cri
tica
l m
ass
par
amet
er
Plain gas bearing
GFB (Simple model)
GFB (1D FE model)
Stable
Unstable
-
39
APPENDIX A
3
1 1 2( )iK H C C
3
2 3 4( )iK H C C
3 5 6 7K C C C
1 14
( )2 sin( ) cos( )
2
i iW WK
1, 1,
5 22
i j i j
i
P PK H
1 2
2
( )C
2 2
2
( )C
Z
1, 1,
3 2( )
i j i jP PC
-
40
, 1 , 1
4 2( )
i j i jP PC
Z
3 3
1, 1, 1, 1 1, 1
5 2
( )( )
4( )
i j i j i j i i j iP P P H P HC
3 3
, 1 , 1 , 1 , 1
6 2
( )( )
4( )
i j i j i j i i j iP P P H P HC
Z
1, 1 1, 1
7
( )
2( )
i j i i j iP H P HC
-
41
APPENDIX B
Elemental stiffness matrix,
2 2 2 2
2 2
6 156 3 22 6 54 3 13
2 4 3 13 3
6 156 3 22
2 4
e
e e e
e e e e e e e
e e
e e
A B Ah Bh A B Ah Bh
Ah Bh Ah Bh Ah BhK
A B Ah Bh
sym Ah Bh
Where 3
2
e
CA
h ,
420
ehBS
Vector of generalized displacements,
1
2
3
4
{ }
e
e
e
e
e
u
uu
u
u
{ }eF = Force vector
2
1
2
6 9
2
6 2112 60
3
e
e ei ie ei e
e
e e
h
h hq qq hF Q
h
h h
Where
1i iq P , 1 1 1i iq P
Vector of generalized forces,
1
2
3
3
e
e
e
e
e
Q
QQ
Q
Q
-
42
REFERENCES
[1] Agrawal, G. L., 1997, Foil Air/Gas Bearing Technology -An
Overview, International Gas
Turbine & Aeroengine Congress & Exhibition, Orlando,
Florida, ASME paper 97-GT-347.
[2] Heshmat, H., Walowit, J., and Pinkus, O., 1983, Analysis of
Gas-Lubricated Compliant
Journal Bearings, ASME Journal of Lubrication Technology, 105
(4), pp. 647-655.
[3] Peng, J.-P, and Carpino, M., 1993, Calculation of Stiffness
and Damping Coefficient for
Elastically Supported Gas Foil Bearings, ASME Journal of
Tribology, 115 (1), pp. 20-27.
[4] Braun MJ, Choy FK, Dzodzo M, Hsu J. 1996, Two-dimensional
dynamic simulation of a
continuous foil bearing, Tribol Int, 29(1):618.
[5] DellaCorte, C., and Valco, M. J., 2000, Load Capacity
Estimation of Foil Air Journal
Bearings for Oil-Free Turbomachinery Applications,
NASA/TM2000209782.
[6] Heshmat, H., 1994, Advancements in the Performance of
Aerodynamic Foil Journal
Bearings: High Speed and Load Capacity, J. Tribol., 116, pp.
287-295
[7] DellaCorte, C., and Valco, M., 2003, Oil-Free Turbomachinery
Technology for Regional
Jet, Rotorcraft and Supersonic Business Jet Propulsion Engines,
American Institute of
Aeronautics and Astronautics, ASABE 1182.
[8] Ku, C.-P, and Heshmat, H., 1992, Complaint Foil Bearing
Structural Stiffness Analysis
Part I: Theoretical Model -Including Strip and Variable Bump
Foil Geometry, ASME
Journal o