AERODYNAMIC ANALYSIS OF COMPLIANT JOURNAL FOIL BEARINGS

AERODYNAMIC ANALYSIS OF COMPLIANT JOURNAL FOIL BEARINGS
AERODYNAMIC ANALYSIS OF
COMPLIANT JOURNAL FOIL
BEARINGS
Thesis Submitted in Partial Fulfilment of the
Requirements for the Degree of
BACHELOR OF TECHNOLOGY (B. Tech.)
In
MECHANICAL ENGINEERING
By
Shubhendra Nath Saha (109ME0301)
Under The Guidance of
Prof. Suraj Kumar Behera,
National Institute of Technology,
Rourkela, 2009-2013
i
CERTIFICATE
NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA
This is to certify that this thesis titled by “Aerodynamic Analysis of
Compliant Journal Foil Bearings‟‟ submitted by Shubhendra Nath Saha
(Roll No. : 109ME0301) in the partial fulfilment of the requirements for the
degree of Bachelor of Technology in Mechanical Engineering, National
Institute of Technology, Rourkela, is an original and authentic work carried
out by him under my supervision.
To the best of my knowledge the data or matter used in this thesis has not been
submitted to any other University/ Institute for the award of any degree or
diploma.
Place:
Prof. Suraj Kumar Behera
Date:
Department of Mechanical Engineering
National Institute of Technology
Rourkela, Osidha
2009-13
ii
ACKNOWLEDGEMENT
It gives me immense pleasure to express my sincere gratitude to my guide Suraj Kumar
Behera, Asst. Professor of Department of Mechanical Engineering, National Institute of
Technology, Rourkela, Odisha for giving me this wonderful opportunity to work and learn
under his able guidance. His invaluable guidance and constant support has made this research
possible. I am also indebted for his precious time spent in guiding as well as teaching the
fundamentals at each and every stage of this project and also for all the fruitful discussions
we had in the past one year.
I would also like to mention a note of thanks for Prof. S.C.Mohanty, Department of
Mechanical Engineering, National Institute of Technology, Rourkela for providing me with a
Final Year B.Tech. Project in the Design field as well as constantly evaluating our work from
time to time during the course of this project. I would also take this opportunity to thank Prof.
S.K.Sahoo, Project Coordinator (U.G.) all the other faculties of Department of Mechanical
Engineering for their constant support and encouragement.
I would also express my thanks to my friend Mr. D. Jaswant Kumar, another undergraduate
project student under my guide for helping to overcome various difficulties which I faced
during the coding part of the project.
Finally I would like to express my heartfelt gratitude to my parents and sister for their
constant motivation towards successful completion of the project.
SHUBHENDRA NATH SAHA
DATE: 12/5/13
Place: Rourkela
(109ME0301)
Department of Mechanical Engineering
National Institute of Technology,
Rourkela, Odisha
iii
ABSTRACT
This thesis provides a detailed study about the aerodynamic analysis of compliant journal foil
bearings. In high speed turbomachinery elements, conventional bearings can‟t be used since
at such a high speed (50000-80000) rpm these bearings get worn and thus they fail. In this
project the analysis of a compliant bearing, which has bump foils to enhance the load
carrying capacity, has been done. The thin air film between the rotating shaft and the bearing
creates the pressure to support the load. Reynolds Equation was first devised for the given
compliance system. Due to its nonlinear nature iterative methods are required to solve it. For
calculating the pressure distribution and hence the load carrying capacity of the bearings
Finite Difference Approximations were used and a unique method, method of quadratic
equations was used to find out several parameters. Using MATLAB several codes have been
written to find out the pressure profile and the minimum film thickness whose 3 dimensional
graphs have been plotted.
After the pressure profile has been generated the load carrying capacity has been evaluated
for the bearing under the given input parameters. Thereafter several comparisons have been
made between the foil and the rigid bearing on the basis of the plots. Finally two different
materials, one used for high temperature applications-Inconel X-750 and another used for low
temperature applications Aluminium Bronze have been compared to find out their
compatibility in different commercial applications based on their load carrying capacity.
iv
NOMENCLATURE
e
Θ,θ
:
:
Eccentricity of shaft (mm)
Coordinate along the circumferential direction (degrees)
ω
ϕ
h
W
s
t
l
u
v
w
C
D
E
Le
R
α
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
υ
:
Angular speed of rotation (rpm)
Attitude angle (degrees)
Fluid film thickness (mm)
Load carrying capacity (N)
Bump pitch (mm)
Thickness of the foil (mm)
Half bump length (mm)
Velocity component in x direction (mm/s)
Velocity component in y direction (mm/s)
Velocity component in z direction (mm/s)
Radial clearance (mm)
Diameter of the shaft (mm)
Modulus of elasticity (N/mm2)
Length of bearing (mm)
Radius of shaft (mm)
Compliance Number or Bearing Number
of bearing (dimensionless)
Poisson‟s Ratio of Lubricant (dimensionless)
εf
:
Eccentricity ratio of bearing (dimensionless)
ρ
:
Density of gas used in bearing (kg/m3
η
ef
Λ
pi,j
τ
z
:
:
:
:
:
:
:
:
:
:
:
:
Coefficient of viscosity of bearing (N-s/mm2)
Eccentricity of foil bearing (mm)
Bearing or compressibility number (dimensionless)
Pressure acting on any node(i,j)
Shear stress acting on the control volume (N/mm2)
Non dimensional length along z direction (dimensionless)
Non dimensional pressure (dimensionless)
Non dimensional film thickness (dimensionless)
Non dimensional coefficient of viscosity (dimensionless)
Non dimensional temperature (dimensionless)
Atmospheric pressure (N/mm2)
Coefficients of Quadratic Equation (dimensionless)
p
h

T
pa
A, B, C
v
LIST OF FIGURES
Fig. No.
1.1
1.2
Name of figure
Page
Number
Schematic of Compliant Journal Bearing
Schematic of Compliant Journal Bearing showing thickness,
length, and pitch
Two Converging Plates in Hydrodynamic Lubrication
Two plates which are non-conversing but velocity of plate
changes
Forces applied on a control volume in the bearing
Pressure Profile of Journal Foil Bearing for Inconel X-750
3
4
5.02
5.03
Non-Dimensional Film Thickness for Inconel X-750
Comparison of film thickness of foil and rigid bearing for
Inconel X-750
28
29
5.04
Comparison of non-dimensional pressure of foil and rigid
bearing for Inconel X-750
Variation Non-Dimensional Pressure with eccentricity Ratio
Variation of Load Carrying Capacity with Angular Speed
Pressure Profile of Journal Foil Bearing for Aluminium
Bronze
Non Dimensional Film Thickness of Journal Foil Bearing for
Aluminium Bronze
Comparison of film thickness of foil and rigid bearing for
Aluminium Bronze
Comparison of non-dimensional pressure of foil and rigid
bearing for Aluminium Bronze
30
Variation of Non-Dimensional Pressure with Eccentricity
Ratio
Variation of Non-Dimensional Pressure with Angular Speed
35
3.1
3.2
3.3
5.01
5.05
5.06
5.07
5.08
5.09
5.10
5.11
5.12
vi
13
13
14
27
30
31
33
33
34
34
35
LIST OF TABLES
Table Number
Title of Table
Page Number
5.1
Input properties with Inconel
X-750 as bump Material
25
5.2
Input properties with
Aluminium Bronze as bump
Material
32
vii
CONTENTS
Certificate
--------------------------------------------------------------------------------
(i)
Acknowledgement --------------------------------------------------------------------------------
(iii)
Abstract
--------------------------------------------------------------------------------
(iv)
Nomenclature
--------------------------------------------------------------------------------
(v)
List of figures
--------------------------------------------------------------------------------
(vi)
List of tables
-------------------------------------------------------------------------------- (vii)
CHAPTER 1.
INTRODUCTION
:
1-5
1.1 Use & Applications
1.2 Types of Bearings
1.3 Compliant Journal Bearings
1.4 Advantage of Compliant Nature of Bearing
CHAPTER 2.
LITERATURE SURVEY
:
6-12
CHAPTER 3.
THEORETICAL ANALYSIS OF BEARINGS
:
13-20
:
21-24
:
25-34
3.1 Derivation of Reynolds equation
3.2 Reynolds equation in Polar Coordinates
3.3 Non-Dimensionalization
3.4 Expansion of Reynolds equation
3.5 Final Assumptions
CHAPTER 4.
NUMERICAL METHODS
4.1 Approach to solution
4.2 Mathematical conversion
4.3 Relations for film thickness
4.4 Flowchart
CHAPTER 5.
RESULTS & DISCUSSIONS
5.1 Analysis of material of bump foil Inconel X-750
5.2 Analysis of material of bump foil Aluminium Bronze
CHAPTER 6.
CONCLUSIONS
:
35
CHAPTER 7.
SCOPES AHEAD
:
36
REFERENCES
:
37-38
ROADMAP
:
39
viii
CHAPTER-1
INTRODUCTION
Gas bearings have been used to support rotors in machines since the 1960. They have been
designed for several applications including gyros, supports for magnetic heads in computer
hard disks and special machine tools. Gas bearings enable extremely low frictional torque to
be obtained and are used in gyroscopes and other precision instruments.
1.1 USE AND APPLICATIONS
Gas bearings are particularly valuable when they are used to support oil free high speed
rotors in precision machines as process gas is used as lubricant. Gas lubricated films are
particularly isothermal because the ability of the bearing materials to dissipate heat is greater
than the heat generating capacity of gas films which have very low friction losses so no
thermal effects appear during gas beating operation. These advantages of gas bearings are due
to the fact that the surfaces of journal and bush are separated by a gas (mainly air) layer
characterized by a very low (compared with oil) viscosity. Gas bearings retain their
advantages at high rotational velocities admissible for oil bearings and rolling bearings.
One of the major problems of developing turbo-expander system for gas liquefaction plants is
the instability of the rotor at high rotational speed. For stability of rotor system at high
rotational speed, better bearings are required. Gas bearings are one of the solutions of
maintaining stability and prevent contamination of working fluids in these plants. Gas
bearings of various types can be used in miniature turbines like Aerostatic gas bearings
(externally pressurized gas bearings) and Aerodynamic gas bearings (self acting).
The advantages of gas bearings have been full established in the following areas:

Machine Tools: Use of gas lubrication in grinding spindles allows attainment of high
speeds with minimal heat generation.

Metrology: Air bearings are used for precise linear and rotational indexing without
vibration and oil contamination.

Dental drills: High speed air bearing dental drills are now standard equipment in the
profession.
1

Airborne air cycle turbo machines: Foil type have been successfully introduced for air
cycle turbo machines on passenger aircraft. Increased reliability, leading to reduced
maintenance costs, is the benefit derived from air bearings.

Computer peripheral devices: Air lubrication makes possible high packing density
magnetic memory devices, including tapes, disks and drums. Read write heads now
operate at sub micro meter operation from the magnetic film with practically no risk
of damage due to wear.
1.2 TYPES OF BEARINGS
Gas bearings of categorised into the two following types:
a. Aerostatic bearings (Externally pressurized gas (air) bearings).
b. Aerodynamic bearings. (Self-Acting gas (air) bearings).
AEROSTATIC BEARINGS
Aerostatic Gas Bearings supports its entire designed load at zero speed. This effect results
from its principal disadvantage: it requires an external pressure source to create the air film.
In principle, gas is supplied to the bearing clearance at a certain gauge pressure. The pressure
differential causes the air to flow from the supply point through the bearing gap and out the
periphery to atmosphere. The pressure inside the gap creates the load carrying properties
which are limited only by the available supply line pressure and material strength. The
aerostatic bearing does not suffer from friction induced wear and in addition it has no starting
and stopping friction.
AERODYNAMIC/HYDRODYNAMIC GAS BEARINGS
In the case of aerodynamic or self-acting bearings, the air film is created by the relative
motion of two mating surfaces separated by a small distance. From rest, as the speed
increases, a velocity induced pressure gradient is formed across the clearance. The increased
pressure between the surfaces creates the load carrying effect. The load capacity is dependent
on the relative speed at which the surface moves and therefore at zero speed, the bearing
supports no load. In general, aerodynamic bearings suffer from decreased load carrying
capacity. In addition, the zero loads at zero speed effect causes starting and stopping friction
2
and results in some wearing of the bearing surfaces, however this can be treated with
application of film of solid lubricant on the bearing and/or journal surface.
1.3 COMPLIANT JOURNAL BEARINGS
Compliant journal bearings are bearings with compliant bearing surfaces. Foil bearings is a
compliant bearing having a unique type of operation with several different kinds of
applications. As compared to the conventional rigid bearings they have higher load carrying
capacity, better stability as well as greater amount of enduring strength. Moreover these are
self-acting bearings and can operate with ambient air or any other gas as the lubricating fluid.
Since no hard-core lubrication system is present the result is lesser amount of weight and
lower maintenance cost. The most common lubricant used is air since it is abundant in nature
and can operate at higher temperatures while oil based lubricants tend to fail at higher
temperatures since their viscosity is inversely proportional to rise in temperature.
PHYSICAL LOOK
FIGURE 1.1: Schematic of Compliant Journal Bearing (Source- Reddy Amith
Hanumappa, (2005), [16])
Figure 1.1 shows the overall look of a compliant journal bearing. As opposed to a
conventional journal bearing, it consists of an outer bearing sleeve or an outer housing which
contains a round series of bumps over a thin foil strip. A thin smooth top foil sheet is placed
3
over the bump foil. Now these foils are connected mostly by welding at one of the ends
known as the leading edge and are open at the other end called the trailing edge. The
uniqueness of a compliant journal bearing over any other conventional bearing is the fact that
the bump supports the top foil and act as a spring, hence the bearing is known as compliant.
The journal has an interference fit with very negligible clearance, as a result the journal and
the foils are in metal to metal contact when the shaft is stationary but as soon as a critical liftoff speed is achieved the journal rotates on a thin gas film due to the hydrodynamic pressure
developed. As a result of this hydrodynamic pressure the top foil deforms and forces it away
from the shaft towards the bump strip.
The application of forces, which act perpendicular to direction of rotation of shaft, is shown
in figure 1.2. The hydrodynamic pressure is a function of operating speed and thus influences
the deformation of foils. Thus in a nutshell we can say that the film thickness varies with the
hydrodynamic pressure as well as the physical, rather elastic properties of the foils. The
clearance in radial direction in this type of bearings also represents important criteria which
influences the overall performance of the bearing.
FIGURE 1.2: Schematic of Compliant Journal Bearing showing thickness, length, and
pitch (Source- Reddy Amith Hanumappa, (2005), [16])
1.4 ADVANTAGE OF COMPLIANT NATURE OF BEARING
A unique advantage of compliant materials is that they have a certain degree of elastic selfaligning. The elastic deformation compensates for misaligning or other manufacturing errors
4
of the bearing or sleeve. In contrast metal bearings are very sensitive to any deviation from a
perfect roundness of the bearing and journal. For hydrodynamic metal bearings high
precision as well as perfect surface finish is essential for successful performance with
minimum contact between the surface asperities. In comparison plastic bearings can be
manufactured with lower precision due to their compliance characteristic. The advantage of
surface compliance is that it relaxes the requirement for high precision which involves high
cost.
Moreover the compliant surfaces usually have better wear resistance. Elastic deformation
prevents removal of material due to rubbing of rough and hard surfaces. Compliant materials
allow the rough asperities to pass through without any wear. In addition it has better wear
resistance in the presence of abrasive particles in the lubricant, such as dust, sand etc. Rubber
sleeves are often used with slurry lubricant in pumps. Embedding of the abrasive particles in
the sleeve is possible by means of elastic deformation.
For all these advantages bearings made up of compliant plastic materials is widely used
However their application is limited to light loads and moderates speeds.
5
CHAPTER - 2
LITERATURE SURVEY
Christopher Dellacorte, Antonio R. Zaldana, Kevin C. Radil [1]
The authors made an exclusive research to find out the load capacity of air foil bearings and
thus concluded that adequate design of geometry, smoothness of surface finish; proper
lubrication and wear resistance are three important criteria for achieving good load capacity.
If any of the three criteria is not met the life of the bearing is shortened. Their research paper
showed an approach where the solid lubrication is given by a blend of self-lubricating shaft
coatings and other wear resistant and lubricated foil coatings. Due to the use of more than one
material with each one of them performing different functions, is designed according to oil
lubricated hydrodynamic sleeve bearing technology where different types of coatings and
treatments of the surfaces along with the lubricants to achieve the best working performance.
The authors of this research work performed the various experiments on journal foil air
bearings operating at 1400 rotations per minute. Plasma sprayed composites, polymer,
ceramic, inorganic lubricants etc. were some of the coating technologies used for foil bearing
lubricants. Their results indicated that through testing of various treatments and use of
different lubricants the performance of the bearings is improved. The unique feature of this
research is that the combination of several solid lubricants produced better performance of
the bearings along with higher life than is achieved by any of the lubricants working alone.
By Stefano Morosi, Ilmar F.Santos [2]
The authors published a paper to lay the theoretical basis for application of active control to
gas lubricated journal bearings. The present motivation behind the paper is to demonstrate the
degree to which application of active lubrication to gas journal bearings is feasible. The basic
principle lies in generation of active forces. This is obtained by regulating the injection of
lubricants in radial direction with the help of piezoelectric actuators which are clamped on the
back of bearing sleeves. The authors also developed a mathematical model which couples the
dynamics of rotor bearing system with the dynamics and mechanics of the actuators through a
simple proportional derivative feedback system. They also proved through various numerical
examples that this kind of bearing offers several advantages in comparison to conventionally
lubricated bearings. The main advantages are the notable reduction in the synchronous
6
vibration, effective addressal of the half frequency whirling motion and extending of the
range of application of gas bearings. It was also prove that this type of bearing improves
transient response characteristics, and thus has a better capability of responding to sudden
shocks and excitations which the entire system may be subjected to. Due to the addition of
active lubrication several new parameters and variables are added to the analysis of the
system. The working of the system depends mostly on the proper choice of the control gains,
which are particularly different and depends on the goal of the controller. In the future efforts
both numerically and experimentally can be aimed for confirming the theoretical findings of
this research and thus identify the parameters for tuning the active system at the optimum
level.
By Christopher Della Corte, Kevin C. Radil, Robert J. Bruckner & S. Adam Howard
[3]
The general lag in familiarization of the design and manufacturing of the foil bearings has led
to lack of widespread knowledge and application of gas foil bearings. This research work
makes a study on the available literature to show the design; fabrication and testing of the
working of both the first and second generation of foil bearings with bump style. This paper
could be taken as a standard for beginning new developmental activities for foil bearing
technology. The load carrying capacity in terms of coefficient tested at 25degrees was found
to be 0.27+/- 0.03 for Generation I and 0.54+/-0.05 for Generation II bearings. The error
represents the standard deviation of data and also the results were found to be within the
range of expected values of bearings basted upon the historical data available. However this
research is not considered for the design, manufacturing of Generation III bearings, though
the Generation III bearings have been found to have load carrying capacity coefficient nearly
double and triple of Generation II and Generation III bearings respectively.
A unique tooling technique was employed for doing modifications in the bump foil design
without actually manufacturing all the ne tooling. All the bearings which were manufactured
through this technique were compared with the data already available and were even
predicted by modern models which are based on „rule of thumb‟ law. A very good similarity
was found between the experimental values obtained and the data present in literature and the
values predicted through the model. The authors found a new thing through this paper that
there exists a direct relationship between the complexity of the elasticity of the support
system and the working of the bearings. The load carrying capacity, which is one the most
7
important and most widely uses criteria for comparison of bearings, for bearings which have
been modelled simply was the lowest. However for the bearings in which the foundation was
modelled according to the system level phenomenon like the misalignment of shafts provided
load capacities came out nearly twice than that of the simple bearings. The results of this
research should provide for the need and future study of commercially available bearings.
Also further study in the direction of advanced geometries of bearings should be done. This is
particular to meet the demands of ever growing turbo machinery systems.
By D.Kim, S.Park [4]
The author started by recognizing one of the most critical issues surrounding the reliability of
the foil bearings which is a coating wear on the top foil and rotor during the starting and
stopping of bearings. Also cooling is sometimes very important under certain applications
because the foil bearings can lead to generation of considerable amount of heat depending
particularly upon the operating conditions. The space between the top foil and the bearing
sleeve is filled with axial flow in most of cases. This thesis introduces a hybrid foil bearing
with external pressurized. The advantage of the hybrid nature leads to elimination of coating
wear during start/stop of bearings. Also it leads to reduction of the drag torque during startup. Further a hybrid system does now require any cooling system. The author demonstrated
the high potential of this hybrid air foil bearings under hydrostatic mode since its comparison
with that of hydrodynamic foil bearings in terms of load capacity was measured at 20000
rpm. Thus it was found that hybrid bearing has a much higher load carrying capacity than
hydrodynamic bearing. The author used a simple analytical model to find out the deflection
of the top foil under hydrostatic pressurization. Orbit simulations were used to simulate the
hybrid air foil bearings and was thus predicted that it has a much higher critical speed and
starting speed of instability than its hydrodynamic counterpart. For achieving a realistic top
foil deflection system in hybrid operation, sagging effect on the top foil was used through the
1-dimensional beam model, while the orbit simulation model was implemented to predict the
imbalance responses which are basically the critical speed and onset speed of unstableness.
The simulations show that these properties can have much higher values as compared to the
hydrodynamic bearings. As compared to the hydrodynamic bearings, the smaller clearance
leads to higher onset speed of instability in hybrid operations. The author also found that
higher starting speed of instability could be obtained with the help of increase in stiffness of
the spring and correspondingly increase in the air supply pressure. Also a much higher load
8
carrying capacity was observer with rather less air consumption. This lesser amount of
consumption of air proves to be a noticeable advantage of hybrid air foil bearing both in
terms of efficiency as well as cooling capacity. The injection of air directly can lead to
cooling of both rotor and air film, also the distortion of rotor thermally can be minimised.
This factor is really important since the expansion of rotor is one of the bearing failure causes
in a variety of applications. Also the starting torque being relatively small implies that the
hybrid nature of air foil has the ability to eliminate wear problems, one of the chronic failure
modes in several commercial bearings.
By Daejong Kim, Soongook Park [10]
This research paper throws light on the design; construction and testing of the first air foil
bearing abbreviated as HAFB. It was compared with its hydrodynamic counterpart. The hybrid
case was noticeable with much higher load capacity and much less air consumption. These two factors
are particularly advantageous in terms of efficiency and cooling capacity. Also the direct injection of
air leads to minimization of thermal distortion of the rotor. Another factor was that the starting torque
in hybrid bearings was much lower than the friction torque under steady state operation of
hydrodynamic bearings which helps to eliminate the wear problem.
By V.Arora, P.J.M. Van Der Hoogt, R.G.K.M. Aarts, A. De Boer [5]
In the present research work, stiffness and damping are the two main properties of radial air
foil bearings which have been identified. In air foil bearings the initial torque required is
large, hence in the experimental setup single air foil bearing is employed instead of the usual
two foil bearing. The entire set of experiments is performed at 60000 rpm. For identification
of stiffness and damping properties the approach used is sub structuring in nature. The
developed procedure is found to accomplish the task of identification of the properties. The
authors found that one of the major factors which determine the efficiency of a system with
air foil bearings is the torque loss. It usually occurs in the air layer between the shaft and the
top foils. During operation at high speeds the shearing of air film would cause a torque loss
which comes out to be proportional to the speed of rotation. It is actually possible and quite
easy to measure the spindle‟s output torque in different steady running conditions but because
of the presence of roller bearings in motor and also the electrical losses in coils contribute in
the error in the measured/estimated toque. Hence a torque measurement through mechanical
means is devised using the existing experimental setup. Thus the structural properties were
identified individually and then were used in developing the finite model of the rotor bearing
9
support system. The author has also developed a new identification algorithm for the air foil
bearing. This algorithm depends on the inverse Eigen sensitivity method. It uses the Eigen
data for identifying the other structural characteristics of the bearings. From the results from
identification, the author has concluded that during operation at very high speeds, the
hardening effects are usually observed. Also it is noticed that the stifnesses calculated from
the curve of force vs. deflection are very much comparable with the stiffness identified from
the algorithm.
By Howard Samuel A., Dykas Brian [6]
At NASA Glenn Research Centre, during exhaustive testing of air foil journal bearings
shaft failure was seen again and again at high ambient temperature and rotational speed at
moderate amount of radial load. It is found that the cause of failure is large amount of nonuniform shaft growth which leads to localized viscous heating of the gas film. This eventually
leads to rubbing at high speed and finally the destruction of the bearing and the journal. The
main result of this research was that it was concluded that centrifugal loading of imbalance
correction weights and axial temperature gradients inside the journal (due to the
hydrodynamic nature of the foil) were mainly responsible for the non- economic growth of
shaft. Since the nature of foil bearings is complex and they operate at extreme conditions it
would be inappropriate to directly measure the effect of journal cross-sectional shape
distortion at the conditions where actual failure has happened. The author has proposed a
failure mechanism which includes two separate effects whose combined effect is solely
responsible for shaft failure at extreme operating conditions. Finite element analysis (FEA) is
used to represent the observed failure though it is also shown through experiments. The
conditions where journal actually fails usually lie outside the typical operating range. The
bearings were tested at extremely high ambient temperature with absolutely no cooling air
along with a thin walled journal, bearing destruction is possible. Since the drive is much more
efficient designs using lightweight foils push the Turbomachine toward thinner lighter shafts
and much higher bearing temperatures. Thus with the help of this mechanism and through
careful design, failure can be prevented.
By C. DellaCorte, M.J. Valco [7]
The two renowned author introduce a standard ‟Rule of Thumb‟ to find out the load capacity
of air foil journal bearings considered for Turbomachinery applications. The Rule of thumb is
based on the data already available in the literature and it relates the load capacity to the
10
bearing size and speed through a load capacity coefficient – D. The load capacity is a linear
function of bearing surface velocity and the projected area. It also depends on the bearing
compliant members and the operating conditions, speed and ambient temperature. The earlier
bearing designs or popularly known as first generation bearings have comparatively lower
load capacity. The second generation compliant support elements are tailored to have much
higher load capacity. The limited amount of experimental data available is found to match
with the data obtained from linearly approximated Role of Thumb equation. Thus with the
help of this equation, the performance that is the load capacity coefficients at different
bearing operating conditions can be compared. First generation on bearings have load
capacity coefficients of 0.1 to 0.3, second generation has these coefficients from 0.4 to 0.8
while the third generation capacity coefficients are found to be greater than 0.9. The other
important bearing parameters which must be considered during assessment of foil bearings
for Turbomachinery are the damping and the stiffness. These parameters are required for
rotodynamic stability. The stiffness of the bearing is obtained from a combination of the
compression of fluid film and the elastic deformation of the foil structure. As compared to the
rigid bearings where only limited damping can occur, considerable amount of damping
occurs in both the fluid film and in the foil structure of the air foil bearings. The author has
found that a bearing with larger load capacity for a given application can be design properly
to give way for better stiffness and damping. Thus in the future development of bearing load
capacity remains to be an important research goal.
By Hou Yu, Chen Shuangtao, Chen Rugang, Zhang Qiaoyu, Zhao Hongli [12]
In this paper a numerical model was developed which coupled the hydrodynamic pressure of
the lubricant film with the deformation of the foil structure. In this model the lubricant was
taken as isothermal and isoviscous. And these properties were put to linearize the Reynolds
Equation. The top foil was designed in the form of a strip of rectangular thin plate which is
supported at a rigid point. The pressure distribution, film thickness and deformation of foil
were solved with the help of Finite Element Method. Finally the influence of several
parameters like eccentricity ratio, bearing number and number protuberances on performance
and life of bearing was studied. The results showed that bearing characteristics are change
significantly with increase in the bearing number and eccentricity ratio. Also the
hydrodynamic pressure as well as the load capacity as found to be enhanced for this new type
of bearing since the top foil with considerable lines of protuberant support tends to warp
upwards at the bearing sides due to the bending stresses. This upward warp may lead to
11
negative film thickness for large eccentricity ratio which represents ill-working of the
bearing. Also very few lines of protuberant support can result in the decrease of
hydrodynamic pressure, capacity load and frictional torque. On the other hand increasing the
lines of protuberant support increases the stiffness of the foil structure and thus reduces its
warp.
By Oscar De Santiago, Luis San Andres [14]
Absence of lubricant introduces the possibility of designing the bearings within the flow path
of thermal machines. The process gas could also be used as the working fluid for the bearing.
Gas foil bearing technologies are found in small compressors for aircraft pressurization, in
microturbines and other forms of smaller Turbomachinery. Recent work in the area of foil
bearing technology shows that there are calibrated models which help us to accurately predict
the performance of the bearings of non-conventional sizes. The main focus of this research
work is to make a note of all the parameters of foil bearings which affect the static and
dynamic performance and calibrate them for the industrial applications. This is done through
a computational model, as a result of which a study on the use of gas foil bearings in a
centrifugal compressor for industrial purposes has been done. The first area of investigation
suggests that the nonlinear effect of contact between the bumps and the top foil as well as the
bumps and carrying sleeve could overtake the response of the bearing thus making the
analysis inadequate. The second area of investigation throws light on the fact that this type of
bearing has only a limited dynamic stiffness which when combined with the minimum film
thickness provides with a load capacity of a few Newton. It is necessary to keep a note on the
deflection of bumps behind the top foil at the place where film thickness is minimum. The
dynamic load capacity is directly proportional to the linear response of the bearing. One of
the areas of future experimental work is related to the use of foil gas bearings in the form of
support elements for industrial applications like compressors. The macro mechanism which
has been provided determines the load capacity according to the overheating of the top foil.
Aging, fretting, pitting, wearing, fatigue etc. of the bump foil are some of the other
characteristics which put a limit on the applicability of the foil bearings.
12
CHAPTER-3
THEORETICAL ANALYSIS OF BEARINGS
The two major needs of hydrodynamic lubrication is
 Requires fluid lubricant: lubricant must be viscous in nature.
 Requires relative motion: movement of one surface over the other must produce a
convergent wedge of liquid/gas.
Based upon the needs fluid film lubrication is classified as:
1. Hydrodynamic or Aerodynamic :The requirements are
1.1 Converging edge shape
1.2 Presence of viscosity
Figure 3.1: Two Converging Plates in Hydrodynamic Lubrication
2. Squeeze Film: The requirements are
2.1 Load or Speed variation
2.2 Presence of viscosity
Figure 3.2: Two plates which are non-conversing but velocity of plate changes
3. Hydrostatic or Aerostatic: The requirements are
3.1 External pressure (through a pump etc.)
3.2 Independent nature to support higher loads
13
3.1 DERIVATION OF REYNOLDS EQUATION
Considering a finite volume inside the bearing, we apply and equate the forces in different
directions:
Figure 3.3: Forces applied on a control volume in the bearing
Note that, in the above diagram,
p


(p+ (delp/delx) dx) dydz is representation for  p +
* dx  * dy * dz
x





(tau+ (deltau/dely) dy) dx) dz is representation for  +
* dy  * dx * dz
y


tau*dx*dz is representation for  *dy*dz
Balancing the forces in x direction we get, (Assumption 1: inertia terms are negligible)



p


p * dy * dz +  +
* dy  * dx * dz =  p +
* dx  * dy * dz +  *dy*dz
y
x




 p
i.e
 .............................................................................3.1
y x
For la min ar flow of Netonian Fluid , ( Assumption 2 : fluid is Newtonian)
 =*
u
...................................................................................3.2 .
y
From equations 3.1 and 3.2, we get

u
p
( * )  ......................................................................3.3
y
y
x
Assumption 3: Negligible pressure gradient in direction of film thickness, y direction
Assumption 4 : Coefficient of vis cos ity,  is cons tan t
 *
14
 u
p
( )  ..................................................................3.5
y y
x
p
 2u
i.e
  * 2 ......................................................................3.6
x
y
Similarly force balance in z direction we get
p
2w
  * 2 ...........................................................................3.7
z
y
Integrating equation 3.6 we get
*
u p

y  C1...................................................................3.8
y x
p y 2
 C1y+ C2..........................................................3.9
x 2
At y=0, u=u 2 and
 *u 
at y=h, u=u1
Assumption 5: Assu min g no slip at liquid solid boundary,
i.e. liquid has same boundary as the solid plate
 *u 2  C 2...............................................................................3.10
 *u1 
p h 2
 C1h+  *u 2 ....................................................3.11
x 2
Solving the above we get
 y 2  yh  p
y
u 
+  u1 -u 2  + u 2 .......................................3.12

h
 2*  x
 y 2  yh  p
Here 
is the pressure term.................................3.13

 2*  x
y
while  u1 -u 2  + u 2 are the two velocity terms, u1 and u 2 being the max imum and
h
min imum velocity respectively.
Similarly in z direction we get ,
 y 2  yh  p
y
w 
+  w1 -w 2  + w 2 ..................................3.14

h
 2*  z
Assumtion 6: Incompressible fluid
The equation of continuity or conservation of mass for three dim ensional
incompressible flow is given by,
u
v
w
+

=0..................................................................3.15
x
y
z
15
Solving the above we get
 y 2  yh  p
y
u 
+  u1 -u 2  + u 2 .....................................3.16

h
 2*  x
 y 2  yh  p
Here 
is the pressure term

2*


x


y
+ u 2 are the two velocity terms, u1 and u 2 being the max imum and
h
min imum velocity respectively.
Similarly in z direction we get ,
while  u1 -u 2 
Assumption 7: Neflecting the angle of inclination for coordinate system
Integrating the continuity equation in y direction from y  0 to y  h,
U sin g Leibnitz Rule,

b
a
u (y, x)

db
da
dy =
(  udy ) - u(b,x)
+ u(a,x) .................3.17
x
x a
dx
dx
b
In our case the Leibnitz equation becomes,

h
0
u (x, y)

dh
d0
dy =
(  u (x, y)dy ) - u(x,h)
+ u(x,0) ........3.18
x
x 0
dx
dx
h
u(x,0)

h
0
d0
 0...........................................................................3.19
dx
u

h
dy =
(  udy ) - u1 ...............................................3.20
x
x 0
x
h
h
  1 y 3 y 2 h  p

u1  u2 y 2
udy

(

)

*
+ u 2 y  .............3.21


0
2  x
h
2
  2 3
0
h


h
0

h3 p h
  u1  u 2  .........................................................3.22
12 x 2
u

h3 p h
h
dy 
(
  u1  u 2  - u1 .......................3.23
x
x 12 x 2
x

  h3 p h
  u1  u 2   u1 h  ......................................3.24

x  12 x 2


  h3 p 1 
 
)
(u1  u 2 ) h  .......................................3.25
x  12 x 2 x

16
Similarly in z direction:
w
  h3 p  1 
dy
=
* 
 (w1  w 2 ) h  .......................3.26

0 z
z  12 z  2 z
h
Hence on int egrating the continuity equation 3.16:
u
v
w
0 x dy + 0 y dy + 0 z dy = 0.............................................3.27
h
h
 -
h
  h3 p  1 
  h3 p  1 

(u

u
)
h
+
(v
-v
)
 1 2 
 (w1  w2 ) h   0.............3.28




1
2
x  12 x  2 x
z  12 z  2 z
  h3 p 
  h3 p  1 
1 
+
 (u 2  u1 ) h  + (v1 -v 2 ) +
 (w 2  w1 ) h  .................3.29



 =
x  12 x  z  12 z  2 x
2 z
In the above equation :

Left Hand Side Terms are the PRESSURE TERMS
Right Hand Side Terms are the SOURCE TERMS
h
h
and
are the WEDGE TERMS
x
z
u
w
and
are the STRETCHING ACTION TERMS
x
z
(v1 - v2 ) is the SQUEEZE ACTION terms
Also the squeeze action term can be written as :
(v1 - v2 ) 
h
..........................................................................3.30
t
Thus the Re ynolds equation comes out to be :
  h3 p    h3 p  1 
h 1 
(
 (u 2  u1 ) h  + +
 (w 2  w1 ) h  .........3.31

+
=
x  12 x  z  12 z  2 x
t 2 z
However for perfor min g more in a real world situation, our research work consists of
compressible lub ricant that is density is not a cons tan t , thus assumption 7 is cancelled .
Hence the final Re ynolds equation under compressible condition comes out to be,
   h3 p 
   h3 p  1 
h 1 
+
(
(  (u 2  u1 ) h) +
+
(  (w 2  w1 ) h).................3.32


=
x  12 x  z  12 z  2 x
t 2 z
Assumption 8 : Re lative tan gential velocity only in x direction and not in z direction

17
1 
(  (w 2  w1 ) h)  0.......................................................3.33
2 z
Assumption 9 : Both surfaces are rigid or in other words no stretching action
1 
1  (  h)
(  (u 2  u1 ) h)  (u 2  u1 ) * *
.........................3.34
2 x
2
x
Assumtion 10 : The top surface is stationary or u1  0

 u2  u
Assumption 11: Steady state operation of the bearing
h
 0.....................................................................................3.35
t
Thus the final Re ynolds equation taking int o account all the assumptions is :
   h3 p 
   h3 p  u  (  h)
+
..............................3.36



=
x  12 x  z  12 z  2 x
3.2 REYNOLDS EQUATION IN POLAR COORDINATES
U sin g polar coordinates the two impor tan t transformation equations are:
1. x=R  x=R .............................................................3.37
2. u=R..................................................................................3.38
Thus we get
1    h3 1 p 
   h3 p  R  (  h)
*
..............3.39

+

=
R   12 R   z  12 z 
2 R
p
Considering ideal gas equation i.e.  
Rg T
1   ph3
1 p 
  ph3 p     ph
*

 +

=

R   12 Rg T R   z  12 Rg T z 
2   RgT

 .....3.40

Multiplying by R 2 Rg
3
  ph3 p 
p   R 2   ph 
2   ph
+
R



=

 .........3.41
  12T  
z  12T z 
2   T 
18
3.3 NON-DIMENSIONALIZATION
With the help of normaliztion or non dim ensionalization
z
p
h

T
z ;p
; h  ;  ; T  .....................................3.42
L
pa
C
0
T0
2


3 3

  R 2   ppa hC 

pp
h
C
  ppa hC 3 


a

( ( ppa ))   R 2
(
(
pp
))


 ........3.43

a
L  120 TT0 L
  120 TT0 
2


TT
0





z
z
2 
2

Assumtion : U sin g Isothermal and Isoviscous lub ricant
   1 and T  1....................................................................3.44
  3  p  pa 2C 3 R 2   3  p  120T0 120T0  R 2 pa C  ( ph)
*
 2 3*
*
...........3.45
 ph ( )  
 ph
*
 
  120T0 ( L ) 2  z 
pa C
2
T0 
 z  pa 2C 3
2
Taking Compressibility or Bearing Number as :
60 R 2

( ) .......................................................................3.46
pa C

we get the STANDARDCOMPRESSIBLE REYNOLDS EQUATION IN 2 DIMENSIONS as:
  3  p  2R 2   3  p 
 ( ph)
...................3.47
 ph ( )   ( )
 ph

 
 
L z 

z 
3.4 EXPANSION OF REYNOLDS EQUATION
Expanding the Re ynolds Equation term by term we get ,
L.H.S.= ph
3

2
3 p 2
2
3 p
2 p
 p h
2R 2  3  2 p
 p h
+
p
*3
h
*
*
+
h
(
)
+
(
)  ph
+ p *3h *
*
 h ( )2 .
2
2

 

L 
z z
z 
z
..............................................................................................3.48
 h
p
R.H.S.    p
+h
 ...................................................3.49
 
 
Assu min g h to be a fuction of only  and not z
h
 0...............................................................................3.50
z
Expanded Re ynolds Equation comes out to be :

2 p
2R 2  2 p
3   p h  1   p 2
2R 2  p 2   h
 p
+
(
)

*
(
)
+
(
) ( )  3
+
...............3.51



2
2
2

L z
L
h     p  
 z  h 
ph 
19
3.5 FINAL ASSUMPTIONS
 Negligible inertia terms
 Negligible change in film thickness in z direction i.e. h varies with only θ
 Newtonian Fluid
 Constant coefficient of viscosity (η=constant)
 No slip of liquid solid Boundary
 Neglecting angle of inclination for the coordinate system
 Compressible but ideal gas flow
 Relative tangential velocity in x – direction
 Both the surfaces are rigid, i.e. no stretching action
 Top surface is stationary
 Film Thickness, h is a function of only pressure ‘p’, θ but not z
 Steady state operation of bearing or in other words film thickness does not change
with time.
20
CHAPTER-4
NUMERICAL METHODS
Our next target is to solve the Reynolds Equation:
2 p
2R 2  2 p
3   p h  1   p
2R  p   h
 p
+
(
)
 -  *  - ( ) 2 + ( ) 2 ( ) 2   3
+
..............4.1
2
2
2

L z
L
h     p  
 z  h 
ph 
Thus we can see that the above Reynolds Equation is:
 2nd order.
 Non Linear.
 Partial Differential Equation.
4.1 APPROACH TO THE SOLUTION
Hence the Reynolds Equation can be solved only by Numerical Methods. In our research
work we have used Finite Difference Method for solving the Reynolds Method. The
fundamental difference between Finite Difference Method and Finite Element Method is that
in finite element method we tend to compare the forces, stresses and strains developed in the
system with the help of equations writing them in matrix form while in Finite Difference
Method we change the derivatives or gradients by simple difference formula. Basically these
are just the two alternate or optional methods for discretization process. In this method the
entire bearing space is divided into an M*N grid system. The pressure values at every node is
initialised with certain value, subsequently on every iteration the pressure values are updated.
For finding the pressure at any node, pressure values at all the surrounding nodes must be
known from the previous iteration. Thus a pressure v/s θ v/s z plot is obtained. From the
pressure values the load carrying capacity is obtained.
4.2 MATHEMATICAL CONVERSION
The Reynolds equation is converted into the finite difference form using certain Finite
Difference Approximations:
 2 p pi 1, j  2 pi , j  pi 1, j

.....................................................4.2
 2
 2
 2 p pi , j 1  2 p i , j  pi , j 1

.....................................................4.3
2
2
z
z
21
 p pi 1, j  pi 1, j

..................................................................4.4

2
h hi 1, j  hi 1, j

....................................................................4.5

2
 p p i , j 1  p i , j 1

..................................................................4.6
z
2 z
h hi , j 1  hi , j 1

....................................................................4.7
z
2 z
Thus we need to convert the Re ynolds Equation int o the finite difference form

2 p
2R 2  2 p
3   p h  1   p
2R  p   h
 p
+
(
)
 -  *  - ( ) 2 + ( ) 2 ( ) 2   3
+
.........4.8
2
2
2

L z
L
h     p  
 z  h 
ph 
p i 1, j  2 p i , j  p i 1, j
L.H.S. =

R.H.S.= 

1
pi, j
+(
2 R 2  p i , j 1  2 p i , j  p i , j 1 
) 
 .....................................................4.9
2
L 
z

 p i 1, j  p i 1, j 2
 p i , j 1 2  3
2R p
)  ( ) 2 ( i , j 1
) 
(
2
L
2 z

 hi , j
 hi 1, j  hi 1, j p i 1, j  p i 1, j 
*


2
 2

p i 1, j  p i 1, j
 hi 1, j  hi 1, j


........................................................................4.10
3
2
hi , j
2
2
p i , j hi , j
p i 1, j  2 p i , j  p i 1, j

2
2
+(
2 R 2  p i , j 1  2 p i , j  p i , j 1 
1  p i 1, j  p i 1, j 2
2 R p i , j 1  p i , j 1 2 
) 
)  ( )2 (
) 

(
2
L 
2
L
p i , j 
2 z
z


3  hi 1, j  hi 1, j p i 1, j  p i 1, j 

*


2
hi , j  2


p i 1, j  p i 1, j
 hi 1, j  hi 1, j


.......4.11
hi , j 3
2
2
p i , j hi , j 2
 p i 1, j  p i 1, j
2 R 2 p i , j 1  p i , j 1
3 hi 1, j  hi 1, j 

(
)

*
*

2
 2
L
2
hi , j
2 
2
2R 2 2 
z


pi, j 
 ( ) * 2   pi, j 
2

L
 ( )
z 
 p i 1, j  p i 1, j   hi 1, j  hi 1, j



2
hi , j 3
2
 p
 p i 1, j 2
 p i , j 1 2 
2R p
 p i 1, j  p i 1, j
 ( i 1, j
)  ( ) 2 ( i , j 1
)  +
 0...........................................4.12
2
2
L
2
2 z

 hi , j
The above equation can be converted int o the quadratic form such that ,
2
A p i , j + B p i , j + C =0........................................................................................................................4.13
22
 2
2R
2 
where A= 
 ( ) 2 * 2  .........................................................................................................4.14
2
L
 ( )
z 
 p
p
p
2R p
3 hi 1, j  hi 1, j p i 1, j  p i 1, j
 hi 1, j  hi 1, j 
B=   i 1, j 2 i 1, j  ( ) 2 i , j 1 2 i , j 1 
*
*

 ......4.15

L
2
2
hi , j 3
2
hi , j
z


 p
 p i 1, j 2
 p i , j 1 2 
2R p
 p i 1, j  p i 1, j
and C=  ( i 1, j
)  ( ) 2 ( i , j 1
)  +
..........................................4.16
2
2
L
2
2 z

 hi , j
The process of iterations will be repeated till the specified accuracy is attained by a
convergence criterion as:
M N

M N

p
  i , j 
  p i , j 
 i 1 j 1
 k th iteration  i 1 j 1
 k 1th iteration

M
N


  p i , j 
 i 1 j 1
 k th iteration
where  can be used according to the desired level of accuracy such as 0.01, 0.001 or 0.0001 etc.
4.3 RELATIONS FOR FILM THICKNESS
Now the film thickness is unknown to us. Normally the film thickness is given by the
relation:
h = C + ef*cosθ
But, since our foil bearing is of compliant nature, the film thickness is also dependent upon
the pressure
h = C + ef*cosθ + α (p-1)
Or
hi,j=C + ef*cosθi,j + α(pi,j-1)
Again normalizing the above relation we get,
h = 1 +  f *cos  +  ( p -1)
where h 
ef
h
p
, f 
and p =
C
C
pa
Also  is known as the Compliance Number of the Bearing and is given by the following relation
3
2* pa * s  l 
=
*   *(1- 2 )
C*E  t 
23
4.4 FLOWCHART
Start
Give the input parameters to the
MATLAB Code
Generation of grid system with M*N nodes in theta and zbar direction
Aa at every grid point
Calculate the film thickness
Solve Reynolds Equation to find pressure at all nodes
Update pressure at all nodes and check for convergence
Plot pressure profile, film thickness and load carrying
capacity
Stop
24
Chapter 5
RESULTS AND DISCUSSIONS
For comparison purposes the MATLAB code was prepared and by giving the various inputs,
the results of two different commercially used materials was obtained in terms of



Pressure profile.
Load carrying capacity.
Film Thickness.
Sl. No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Input Property
Radial Clearance (C)
Radius of Shaft (R)
Length of bearing (Le)
Eccentricity (e)
Bump pitch (s)
Atmospheric Pressure (pat)
Half Bump length (l)
Thickness of bump foil (t)
Poisson‟s ratio (υ)
Viscosity of lubricant(η)
Angular speed of Rotation (ω)
Eccentricity ratio
eps = e/C
Modulus of Elasticity of bump
foil (Et)
No. of grid points in theta
direction (M)
No. of grid points in zbar
direction (N)
No. of iterations
13.
14.
15.
16.
Value
Unit
0.03
25
50
0.02
3.17
0.1
1.125
0.1
0.29
17.8*10-12
30000
0.67
mm
mm
mm
mm
mm
N/mm2
mm
mm
N-s/mm2
rpm
-
212000
N/mm2
50
-
50
-
1000
-
Table 5.1: Input properties with Inconel X-750 as bump foil material
5.1 ANALYSIS OF MATERIAL OF BUMP FOIL: INCONEL X-750
The first material to be selected was Inconel X-750 since it is one of the bump foil materials
used in manufacturing the bump foils of bearings chiefly employed in high temperature
applications. Inconel X-750 basically is a Nickel-Chromium alloy made of additions of
Aluminium and Titanium, having creep-rupture strength at high temperatures to about 700°C.
Some of the areas of application of Inconel X-750 are:
25

Nuclear reactors

Gas Turbine

Rocket Engines
With the help of the properties as mentioned in table5.1, the MATLAB Code is solved for
certain fixed number of iterations. Inconel X-750 has Poisson‟s Ratio of 0.29 and Modulus of
Elasticity of 212GPa, these being the two most important mechanical properties of the bump
foil material.
5.1.1 PRESSURE PROFILE
The three dimensional pressure profile under the given input parameters for bump foil made
of Inconel X-750 comes out to be as shown in the next page. As we can see that theta varies
from 0 to 2  i.e. from 0 to 6.28. Similarly the zbar axis varies from 0 to 1. The pressure
profile is represented in the form of non-dimensional pressure (p/pa).
Upto a little distance from the leading edge, and considerable more distance from the trailing
edge the pressure profile is atmospheric in nature. This is due to the detachment of the foils.
Thus the pressure is generated at a certain angle, keeps on increasing to maximum value and
then settles down to atmospheric value.
Figure 5.01: Pressure profile of journal foil bearing for Inconel X-750
26
5.1.2 Non Dimensional Film Thickness
The 3D plot of non-dimensional film thickness under the given input parameters for bump
foil made of Inconel X-750 comes out to be as shown in the next page. The film thickness
varies only in theta direction and not in z direction. During initial rotation of shaft i.e. at
initial value of shaft the film thickness is higher but then the air film becomes lesser and
lesser in thickness. Again after a certain degree of rotation, the film thickness again starts
increasing.
Figure 5.02: Non-Dimensional Film Thickness for Inconel X-750
5.1.3 Comparison of Foil Bearing with Rigid Bearing
The film thickness in case foil bearing differs with that of rigid bearing. Due to the difference
in the film thickness the pressure profile differs. We can see that lower film thickness can be
obtained in case of foil bearing. The film thickness of foil bearing is close to the rigid bearing
and higher than expected total roughness of two surfaces.
27
Figure 5.03: Comparison of film thickness of foil and rigid bearing for Inconel X-750
Another comparison is made between the foil and rigid bearing in terms of pressure profile.
We can see that the higher pressure is generated in case of foil bearing than is obtained in
rigid bearing.
Figure 5.04: Comparison of non-dimensional pressure of foil and rigid bearing for Inconel
X-750
28
5.1.4 Variation of Load Carrying Capacity with Eccentricity Ratio
Figure 5.05: Variation Non-Dimensional Pressure with eccentricity ratio
As the eccentricity ratio is increase the load carrying capacity is also increased. But there is
limitation to increase eccentricity ratio as higher eccentricity will cause larger vibration.
5.1.4 Variation of Load Carrying Capacity with Angular Speed of Rotation
\\\
Figure 5.06: Variation of Load Carrying Capacity with Angular Speed
29
As we can see as the speed of rotation increases from 5000 rpm, the load carrying capacity
also increases. However it becomes almost a constant after certain speed, (from 80000 rpm to
1, 30,000 rpm load carrying capacity remains a constant.
5.1 ANALYSIS OF MATERIAL OF BUMP FOIL: Aluminium Bronze
Next material to be selected was Aluminium Bronze since it is one of the bump foil materials
used in manufacturing the bump foils of bearings chiefly employed in low temperature
applications. Some of the areas of application of Inconel X-750 are

Cryogenic Centres

Refrigerator Compressors
Considering the properties of this material we give the following inputs:
Sl. No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Input Property
Radial Clearance (C)
Radius of Shaft (R)
Length of bearing (Le)
Eccentricity (e)
Bump pitch (s)
Atmospheric Pressure
(pat)
Half Bump length (l)
Thickness of bump foil
(t)
Poisson‟s ratio (υ)
Viscosity of
lubricant(η)
Angular speed of
Rotation (ω)
Eccentricity ratio
esp. = e/C
Modulus of Elasticity of
bump foil (Et)
No. of grid points in
theta direction (M)
No. of grid points in
zbar direction (N)
No. of iterations
Value
Unit
0.03
25
50
0.02
3.17
0.1
mm
mm
mm
mm
mm
N/mm2
1.125
0.1
mm
mm
0.34
17.8*10-12
N-s/mm2
30000
rpm
0.67
-
120000
N/mm2
50
-
50
-
1000
-
Table 5.2: Input properties with Aluminium bronze as bump foil material
30
With the help of the above properties, the MATLAB Code is solved for certain fixed number
of iterations. Aluminium Bronze has Poisson‟s Ratio of 0.33 and Modulus of Elasticity of
120GPa, these being the two most important mechanical properties of the bump foil material.
Similarly giving the inputs in the MATLAB Code we get the following plots for Aluminium
Bronze as shown in the next pages:
Figure 5.07: Pressure Profile of Journal Foil Bearing for Aluminium Bronze
31
Figure 5.08: Non Dimensional Film Thickness of Journal Foil Bearing for Aluminium
Bronze
Figure 5.09: Comparison of film thickness of foil and rigid bearing for Aluminium Bronze
32
Figure 5.10: Comparison of non-dimensional pressure of foil and rigid bearing for
Aluminium Bronze
Figure 5.11: Variation of Non-Dimensional Pressure with Eccentricity Ratio
33
Figure 5.12: Variation of Non-Dimensional Pressure with Angular Speed
34
CHAPTER 5
CONCLUSIONS
The ever growing needs of the aerospace rotating machinery, microturbines and other forms
of turbomachinery requires aerodynamic rather hydrodynamic analysis of the compliant foil
bearings to critically analyse their operating parameters. Their main principle of operation is
governed by Reynolds Equation. But to solve this non-linear partial differential equation a lot
of computational time and effort is consumed. In the current project the Finite Difference
Method is used to solve the Reynolds Equation. Use of non-Dimensionalization and using the
method of Quadratic Solution are the unique features of this project. In the current project an
attempt was made to find the performance parameters like pressure profile, film thickness,
load carrying capacity etc. with several assumptions to simplify modified Reynolds equation.
The analysis was performed on two types of bump foil materials:
1. Inconel X-750
2. Aluminium Bronze
Both these materials are found to be commercially used in turbines and generators etc. Due to
protected (copyrighted) technology of gas foil journal bearings, very less or rather very small
amount of work is found in open literatures. The results of the current project may be useful
to the researchers to work more on further analysis of compliant foil journal bearings.
35
CHAPTER 6
FUTURE SCOPES
Although a large amount of information on the various parameters of the bearings can be
predicted for any foil bearing by following through my research work but certainly there is
always scope for future research both experimentally and analytically to be done on foil
bearings. The following studies about bearings can be done in the near future:

In Turbomachine laboratories various types of experimental work can be performed
for firstly designing the journal foil bearings and then comparing it with the already
available theoretical results.

As already shown investigating the non dimensional load carrying capacity, one can
use various optimisation algorithms to find out the various parameters like
eccentricity ratio, angular speed of rotation for most efficient performance of the
bearings.

We can also look forward to finding out the temperature rise of the bearings through
its thermodynamic analysis and thus improve its performance.

There is also a scope for finding out the stiffness and damping characteristics of the
journal foil bearing, thus one can have knowledge about the vibrations of the bearing.
36
CHAPTER 7
REFERENCES
1. DellaCorte, Christopher R. Zaldana, Antonio C. Radil, Kevin (2004), “A Systems
Approach to
the Solid
Lubrication of Foil
Air Bearings
for
Oil-Free
Turbomachinery”, Journal of Tribology, Vol. 126, 200-207.
2. Morosi Stefano, IlmarF.Santos (2011), “Active lubrication applied to radial gas
journal bearings”, Tribology international, Vol. 42, 1949-1958.
3. DellaCorte,
Christopher, C. Radil, Kevin, J. Bruckner, Robert, S. Adam,
Howard (2008), “Design, Fabrication, and Performance of Open Source Generation I
and II Compliant Hydrodynamic Gas Foil Bearings” , Tribology Transactions, 51:3,
254-264
4. D. Kim, S.Park, (2009), “A Preliminary study of the load bearing capacity of a new
foil thrust gas bearing”, Tribology International42 (2009) 413–425
5. Arora V., Hoogt P. J. M. van der, Aarts R.G.K.M., De Boer A.Der, (2010),
“Identification
of
dynamic
properties
of
radial
air-foil
bearings”,
DOI
10.1007/s10999-010-9137-z
6. Howard Samuel A., Dykas Brian, (2004), “Journal Design Considerations for
Turbomachine Shafts Supported on Foil Air Bearings”, Tribology Transactions, 47:
508-516, 2004
7. Dellacorte C., Valco M. J., (2000), “Load Capacity Estimation of Foil Air Journal
Bearings for Oil-Free Turbomachinery Applications”, Tribology Transactions, 43:
795-801, 2000
8. Kumar Manish, Daejong Kim, (2010), ”Static performance of hydrostatic air bump
foil bearing”, Tribology International 43(2010)752–758
9. Zywica Grzegorz, (2011), “The Static performance analysis of the foil bearing
structure”, Acta mechanica et automatica, vol.5 no. 4
10. Daejong Kim, Soongook Park (2009), “Hydrostatic Air foil bearings, Analytical and
experimental Investigation”, Tribology International 42(2009)413-425
11. Lee Y.B., Kim T.H., Kim C.H. (2004), “Unbalance Response of a Super Critical
Rotor supported by Foil Bearings-Comparison with test results”, Tribology
Transactions 47:54-60, 2004
37
12. Hoy Yu, Chen Shuangtao, Zhang Qiaoyu, Zhao Hongli, (2011), ”Numerical study
on foil journal bearings with protuberant foil structure”, Tribology International,
44(2011)1061-1070
13. Dong-Hyun Lee, Young Cheol Kim, Kyung Woong Kim, (2010), “The effect of
Coulomb friction on the static performance of foil journal bearings”, Tribology
International, 43(2010) 1065-1072
14. Santiago Oscar De, Andres Luis San, (2011), “Parametric Study of Bump Foil Gas
Bearings for Industrial Applications”, Proceedings of ASME Turbo Expo 2011,
Canada
15. Feng Kai, Kaneko Shigehiko, (2010), “Parametric studies on static performance and
non linear instability of bump type foil bearings”, Journals of System Design and
Dynamics, Vol.4 No.6, 2010.
16. Reddy Amith Hanumappa, (2005), “Hydrodynamic Analysis of Compliant Journal
Bearings”, Thesis for the degree of Master of Science, Lousiana State University
17. Chengchun Peng, (2003),”Thermodynamic Analysis of Compressible Gas Flow in
Compliant Foil Bearings”, Thesis for the degree of Master of Science, Louisiana State
University.
38
ROAD MAP
Work done in 7th Semester
Schedule of work
Aug-12
Sep-12
Oct-12
Nov-12
Literature survey
Status
Completed
Derivation of
Reynolds
Completed
equation for
Thrust bearings
Compressible
Reynolds
Completed
Equation in Two
Dimensions
Finite difference
analysis of 2D
equations
Completed
Work done in 8th Semester
Schedule of work
Finite difference
analysis of 2D
equations
Writing program
Jan-13
Feb-13
Mar-13
Apr-13
Status
Completed
Completed
in MATLAB
Results and
Discussions
39
Completed
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