stability analysis of rotors mounted on gas foil bearings (gfbs)

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.

iii

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.

iv

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

v

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

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


4.3.2.1 Effect of eccentricity ratio on mass parameter ........................................ 33

4.3.2.2 Effect of bearing number on mass parameter ......................................... 33

4.4 summary ...................................................................................................................... 34

vii

5. Conclusions and Future work ............................................................................................ 35

5.1 Introduction .................................................................................................................... 35

5.2 Conclusions .................................................................................................................... 35

5.2.1 Static performance analysis .................................................................................. 36


5.3 Scope of Future Work ..................................................................................................... 38

Appendix A .............................................................................................................................. 39

Appendix B .............................................................................................................................. 41

References................................................................................................................................ 42

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

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

x

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

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

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 5.2: Journal attitude angle versus static load for shaft speed 45,000 ................................ 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

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

1

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

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].

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

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.

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


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]



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


Figure 3.5: Journal attitude angle Vs static load


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

]


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


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


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


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


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)


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]



0 25 50 75 100 125 1500

10

20

30

40

50

60

Static load [N]

Jou

rnal

att

itu

de

ang

le [

deg

]



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)


32


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


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

Figure 5.3: Minimum film thickness versus static



load for shaft speed 30,000


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]




0 25 50 100 125 1500

10

20

30

40

50

60

Static load [N]

Jou

rnal

att

itu

de

ang

le [

deg

]




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]




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)


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

stability analysis of rotors mounted on gas foil bearings (gfbs)

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