PERPUSTAKAAN KUiTTHOeprints.uthm.edu.my/id/eprint/790/1/24_Pages_from... · untuk penentuan secara tepat halaju kritikal bagi suatu galas dan juga untuk mcngkaj i kestabilan galas

PERPUSTAKAAN KUiTTHO U\'12')b4~

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KOLEJ UNIVERSITI TEKNOLOGI TUN HUSSEIN ONN

BORANG PENGESAHAN STATUS TESIS

JUDUL: SUCCESSIVE OVER RELAXATION TECHNIQUE FOR STEADY STATE AND DYNAMIC CHARACTERISTICS OF A CYLINDRICAL BORE BEARING

SESI PENGAJIAN: MEl 2005/2006

Saya ZAIHARBIN YAACOB

mengaku membenarkan tesis Sarjana ini disimpan di Perpustakaan dengan syarat syarat kegunaan seperti berikut:

I. Tesis adalah hakmilik Kolej Universiti Teknologi Tun Hussein Onn. 2. Perpustakaan dibenarkan membuat salinan untuk tl(juan pengajian sahaja. 3. Perpustakaan dibenarkan membuat salinan tesis ini sebagai bahan pertukaran

antara institusi pengajian tinggi.

D SULiT

D TERHAD

IIv II TlDAK TERHAD

(TA~ULIS) Alamat Tetap:

419 Jln Springhill 10/24

Bandar Springhill Prof. Madya Dr. Ing. Ir. Darwin Sebayang

71100 POl1 Dickson, N.S. Nama Penyelia

Tarikh: Tarikh:

Pcngcsahan Pcnyclia

"Saya aklli bahawa saya telah membaca karya ini dan pad a pandangan

saya karya ini adalah memadai dari segi skop dan kllaliti llntllk tlljllan

penganugerahan ijazah Smjana Kejllruteraan Mekanikal"

Tandatangan

Nama Penyelia

Tarikh

SUCCESSIVE OVER RELAXATION TECHNIQUE FOR STEADY STATE AND

DYNAMIC CHARACTERISTICS OF A CYLINDRICAL BORE BEARING

ZAIHAR BIN Y AACOB

A thesis submitted in fulfillment of the requirement for the award of the

Degree of Master of Mechanical Engineering

Faculty of Mechanical and Manufacturing Engineering

Kolej Universiti Teknologi Tun Hussein Onn

DECEMBER, 2005

11

"I declare that the work presented in this thesis, except where otherwise stated, is based

on my own research, and has not been submitted previously for a degree in this or any

other university .

Signature

Name

Date

. ~ ................. . ZAIHAR BIN Y AACOB ·13/;i/200 .. ··················

..... /.! .. / ......... 'S ............... .

III

DEDICA nONS

TO MY FAMILY,

THANK YOU FOR BEING THERE FOR ME.

1 \

AKNOWLEDGEMENTS

First of all I would like to express my gratitude to the most Gracious and i\!OS!

Merciful ALLAH S.W.T. Praiseworthy to Almighty Allah for giving me the will and

strength to complete this thesis.

Thank you to my family and friends for their love and support. To my

supervisors Prof. Madya Dr. Ing. Ir. Darwin Sebayang and Prof. Ir. Dr. ShahNor I3asri

thank you for their ideas, support and guidance.

Special thanks to my colleagues at Kolej Uniti. staff of the faculty and graduate

school in Kuittho for their help and support.

May ALLAH bless you all.

v

ABSTRACT

The knowledge of static and dynamic characteristics of journal bearings is

crucial for the accurate detennination of the critical speed of a shaft and also for

studying the stability of the rotating shaft against self-excited vibrations. These

characteristics are detennined from the solutions of Reynolds equation numerically

using finite difference methods with successive over relaxation technique (SOR). In

order to implement SOR effectively, the optimum value for over relaxation factor Q had

to be found first. In this thesis, the exact value of Q was calculated by using a fornmla

proposed by G.D. Smith. Khonsari and Booser (K&B) found the value of Q, by trial

and error which is not exact and time consuming. While Orcutt and Arwas (O&A) used

Gauss-Seidel technique which has a much slower convergence rate compared to SOR,

also they used two convergence limits which had to be satisfied before tenninations of

the iteration procedure. This thesis is intended to improve both works by calculating the

exact value for Q and employed the SOR technique using only one convergence limit.

The dynamic coefficients were then used as an input data for studying the stability

characteristics of the rotor-bearing system and the threshold of instability were also plot.

The computer program was written using FORTRAN 95 programming language and run

in the Microsoft Developer Studio environment. Method in this thesis shows that the

time taken for a complete solution for the steady state and dynamic characteristics of a

cylindrical bore bearing were greatly shortened in tenns of number of iterations (about

90%) and the automatic calculation of Q. The accuracy of the results were good with

less than 10% in difference when compared to results from both K&B and O&A. It is

then concluded that the finite difference method and successive over relaxation

technique used in this thesis can predict accurately and effectively the static and

dynamic characteristics of a cylindrical bore bearing.

\'}

ABSTRAK

Pengetahuan mengenai ciri-ciri statik dan dinamik galasjumal adalah pcnting

untuk penentuan secara tepat halaju kritikal bagi suatu galas dan juga untuk mcngkaj i

kestabilan galas yang berpusing terhadap getaran. Ciri-ciri ini ditentukan melalui

penyelesaian berangka persamaan Reynolds menggunakan kaedah beza terhingga serta

teknik santaian secara berturutan (SOR). Untuk menggunakan SOR secara cfektif. nilai

optimum faktor santaian Q harus dicari terlebih dahulu. Dalam tesis ini nilai Q dicari

menggunakan formula yang diperkenalkan oleh G.D. Smith. Khonsari dan Booser

(K&B) mencari nilai Q dengan kaedah cuba jaya yang memakan masa dan tidak tepa!.

Orcutt dan Arwas pula menggunakan teknik Gauss-Seidel yang mempunyai kadar

penumpuan yang jauh lebih perlahan berbanding SOR, juga dua had penumpuan

terpaksa dipenuhi sebelum prosedur lelaran ditamatkan. Tesis ini bertujuan untuk

memperbaiki kedua-dua kerja tersebut dengan mengira nilai Q yang tcpat dan

menggunakan teknik SOR dengan hanya satu had penumpuan diperlukan. Pemalar

pemalar dinamik yang diperolehi digunakan sebagai data input untuk mengkaji

kestabilan sistem rotor-galas dan juga memplotkan kemasukan ketidakstabilan. Program

komputer ditulis menggunakan bahasa pengaturcaraan FORTRAN 95 dan dilarikan di

dalam persekitaran Microsoft Developer Studio. Kaedah di dalam tesis ini mcnunjukkan

masa yang diambil untuk penyelesaian penuh ciri-ciri statik dan dinamik suatu galas

bergerek silinder dapat dikurangkan dengan ketara dari segi bilangan lelaran (kira-kira

90%) dan pengiraan Q secara automatik. Ketepatan keputusan adalah baik dcngan

kurang dari 10% perbezaan apabila dibandingkan dengan kcdua-dua keputusan K&B

dan O&A. Maka dapat disimpulkan bahawa kaedah unsur tcrhingga dan tcknik santaian

secara berturutan yang digunakan di dalam tesis ini dapat mengagak dcngan tcpat dan

berkesan ciri-ciri statik dan dinamik suatu galas bergerek silinder.

TABLE OF CONTENTS

ITEM

THESIS STATUS APPROVAL

SUPERVISOR APPROVAL

TITLE

DECLARATION

DEDICATION

AKNOWLEDGEMENT

ABSTRACT

ABSTRAK

TABLE OF CONTENTS

LIST OF FIGURES

NOMENCLATURE

LIST OF APPENDIX

CHAPTER I INTRODUCTION

l.1

1.2

1.3

1.4

1.5

Introduction

Literature Survey

1.2.1 Experimental Studies

1.2.2 Theoretical Studies

Objective of Study

Scope

Overview

vii

PAGE

11

III

IV

v

VI

VII

XI

XV111

xxi

4

4

6

9

9

10

VIII

ITEM PAGE

CHAPTER II THEORETICAL BACKGROUND

2.1 Introduction II

2.2 Hydrodynamic Lubrication II

2.3 Reynolds Equation 12

2.4 Journal Bearing 15

2.5 Oil film Thickness in Journal Bearing 18

2.6 Steady State Characteristics 22

2.6.1 Load Carrying Capacity 24

2.6.2 Lubricant Flow Rate 25

2.6.3 Friction Coefficients 26

2.7 Dynamic Characteristics 27

2.8 Chapter Summary 31

CHAPTER III NUMERICAL SCHEMES

3.1 Introduction 32

3.2 Finite Difference Method 33

3.3 Nondimensionalization 33

3.4 Finite Difference Schemes 35

3.5 Pressure Boundary Conditions 39

3.6 Evaluation of the Main Parameters 41

3.6.1 Pressure Distribution 41


3.6.3 Lubricant Flow Rate 42


3.6.5 Stiffness and Damping 44


IX

ITEM PAGE

CHAPTER IV COMPARATIVE STUDY

4.1 Introduction 48

4.2 Convergence Characteristics 49

4.3 Steady State Characteristics 51

4.3.1 Pressure Distribution 51


4.3.3 Maximum Pressure 53

4.3.4 Attitude Angle 55

4.3.5 Sommerfeld Number 57

4.3.6 Side Leakage 59

4.3.7 Inlet Flow Rate 61




CHAPTER V RESULTS AND DISCUSSIONS

5.1 Introduction 72

5.2 Pressure 73

5.3 Load Capacity and Sommerfeld Number 77

5.4 Attitude Angle and Friction Coefficients 80

5.5 Inlet Flow Rate and Side Leakage 82



CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions 91

6.2 Recommendations 93

REFERENCES 94

ITEM PAGE

APPENDIX A. REYNODS EQUATION

B. SAMPLE OUTPUT

C. FLOW CHART & SOURCE CODE

FIGURE

2.1

2.2

203

2.4

205

206

207

208(a)

208(b)

209

3.1

302

303

401

402

403

4.4

405

xi

LIST OF FIGURES

TITLE PAGE

Journal bearing geometry and nomenclature

Schematic view of journal bearing

15

15

Schematic view of a journal bearing with radial (r) and tangential (t) 16

coordinate systems

Triangle connecting bearing and journal center with some point

on the bearing surfaces

Schematic view of a journal bearing with radial x-y coordinate

system and components of eccentricity

Pressure distribution around a journal bearing

Shear stresses acting on the journal and bearing

Idealized rotor of weight 2W supported on two journal bearings

Stiffness and damping coefficients of a journal bearing

Small amplitude journal motions about an equilibrium position

Unwrapped journal bearing

Finite difference grid for the oil film mesh

Typical boundary conditions on the Reynolds equation

Convergence characteristics for LID = 2

Convergence characteristics for LID = loS

Convergence characteristics for LID = 1

Variations of dimensionless pressure with angular position.

c = 006. LID =1

Variations of load capacity with eccentricity ratio, LID = 1

18

21

26

28

29

30

35

36

40

49

50

50

51

52

FIGURE

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

4.16

4.17

4.18

4.19

4.20

4.21

4.22

4.23

4.24

4.25

4.26

4.27

4.28

4.29

4.30

4.31

xii

TITLE PAGE

Variations of maximum pressure with eccentricity ratio, LID = 2 53



Variations of attitude angle with eccentricity ratio, LID = 2 55

Variations of attitude angle with eccentricity ratio, LID = 1 56

Variations of attitude angle with eccentricity ratio, LID = 1/2 56

Variations of Sommerfeld number with eccentricity ratio. LID = 2 57

Variations of Sommerfeld number with eccentricity ratio, LID = 1 58

Variations of Sommerfeld number with eccentricity ratio, LID = 112 58

Variations of side leakage with eccentricity ratio, LID = 2 59

Variations of side leakage with eccentricity ratio, LID = 1 60

Variations of side leakage with eccentricity ratio, LID = 1/2 60

Variations of inlet flow rate with eccentricity ratio, LID = 2 61

Variations of inlet flow rate with eccentricity ratio, LID = 1 62

Variations of inlet flow rate with eccentricity ratio, LID = 1/2 62

Variations offriction coefficients with eccentricity ratio. LID = 2 63

Variations of friction coefficients with eccentricity ratio, LID = 1 64

Variations of friction coefficients with eccentricity ratio, LID = 112 64

Variations of feu with eccentricity ratio, LID = 1 67

Variations of k,!' with eccentricity ratio, LID = 1 67

Variations of KXl' with eccentricity ratio, LID = 1 68

Variations of - k n with eccentricity ratio, LID = 1 68

Variations of B with eccentricity ratio, LID = 1 69 .a:

Variations of B with eccentricity ratio, LID = 1 69 xy

Variations of BJ)' with eccentricity ratio, LID = 1 70

Variations of Byx with eccentricity ratio, LID = 1 70

FIGURE

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

5.10

5.11

5.12

5.13

5.14

5.15

5.16

5.17

5.18

5.19

5.20

TITLE

Variations of midplane pressure profile with angular position.

& = 0.05


&=0.5


& = 0.8

Variations of maximum pressure with eccentricity ratio

Variations of maximum pressure location with eccentricity ratio

Variations of cavitation location with eccentricity ratio

Variations ofload carrying capacity with eccentricity ratio

Variations of Sommerfeld number with eccentricity ratio.

LID = 2, 3/2, 1

Variations of Sommerfeld number with eccentricity ratio,

LID = 3/4, 1/2, 1/3

Variations of Sommerfeld number with eccentricity ratio,

LID = 1/4, 1/6, 1/8

Variations of attitude angle with eccentricity ratio

Variations of friction coefficient with eccentricity ratio,

LID = 2, 3/2, 1, 3/4

Variations offriction coefficient with eccentricity ratio,

LID = 112, 1/3, 1/4, 1/6, 1/8

Variations of inlet flow rate with eccentricity ratio

Variations of side leakage with eccentricity ratio

Variations of K with eccentricity ratio ,or

Variations of Kyt with eccentricity ratio

Variations of Kty with eccentricity ratio

Variations of k,y with eccentricity ratio

Variations of iJ with eccentricity ratio .u

XIII

PAGE

74

75

75

76

76

77

78

78

79

79

80

81

81

82

83

85

86

87

87

88

FIGURE

5.21

5.22

5.23

5.24

\1\'

TITLE PAGE

Variations of fl." with eccentricity ratio 88

Variations of EX) with eccentricity ratio 88

Variations of fl,) with eccentricity ratio 89

Non-dimensional critical speed (w) versus eccentricity ratio k) 90

C

D

e

F

F Fr

f h

K xx ' KxyKyx, KJ)' ,... ,... ,.. ,...

K xx ' KxyKyx, KJ~'

L

NOMENCLATURE

Finite difference parameters

Damping coefficients

Dimensionless damping coefficients

Radial clearance

Journal diameter (= 2R )

Eccentricity

Component of eccentricity ratio in x direction

Component of eccentricity ratio in y direction

Convergence factor

Eccentricity ratio used in flowchart

Component of eccentricity ratio in x and y direction used

in flowchart

Oil film force

Dimensionless oil film force

Journal frictional force

Dimensionless oil film forces in x and y direction

Friction coefficients

Oil Film thickness

Dimensionless oil film thickness

Stiffness coefficients

Dimensionless stiffness coefficients

Bearing axial length

xv

M,N

p

T

U

v

w W

f).B, f1z

Q

f.1

/l.

Number of finite difference mesh in circumferential and

axial direction

Dimensionless critical mass of the rotor-bearing system

Centre of the bearing

Centre ofthe journal

Rotational speed (rpm)

Oil Film pressure

Maximum oil film pressure

Atmospheric pressure

Dimensionless oil film pressure

Maximum of P Side leakage

Dimensionless side leakage

Journal radius

Sommerfeld number

Time

Variable in quadratic equation defined in text

Surface speed of shaft

Dimensionless squeeze film velocity

XVI

Dimensionless velocity of rotor centre in x and y coordinates

Load carrying capacity

Dimensionless load carrying capacity

Fluid film mesh in circumferential and axial direction

Difference of oil film forces in x and y direction

Optimum over relaxation factor

Lubricant viscosity

Length over diameter ratio

ax. ~}' =-,y=-al al

p

z

e e

r

x.y

ij

a

a

111

cav

in

0111

L

Journal center velocities

Film density

Eccentricity ratio

Component of eccentricity ratio in x and y direction

Dimensionless length of bearing

Deflections measured in x and y direction

Attitude angle

Angular velocity of journal/shaft = 2m7

Dimensionless angular velocity

Angle measured from vertical

XVll

Angle between position of maximum film thickness and some

point around the bearing circumference

SUBSCRIPTS

Radial coordinate

Coordinate tangential to radial coordinate

Cartesian coordinate

Coordinate of mesh point in e and Z direction

Equilibrium position

Atmospheric

Maximum

Cavitation

Inlet

Outlet

Leakage

xviii

LIST OF APPENDICES

TITLE

A. REYNOLDS EQUATION

B. SAMPLE OUTPUT

C. FLOW CHART & SOURCE CODE

CHAPTER I

INTRODUCTION

In this chapter, the motivation of the research described in this thesis will be

summarized and a brief survey of some literature on journal bearings and rotordynamics

wiII be given. The objectives will be presented and an overview will explain the

organization of the rest of the thesis.

1.1 Introduction

If two bodies are in contact and in relative motion to each other, a tangential

force, the force of sliding friction, results on the surfaces of contact. These surfaces will

wear out rapidly which is the major cause of material wastage, loss of mechanical

performance of machine elements and shortened the life of the machines used.

Reduction in wear can be achieved by improving friction control and any reduction in

wear can result in considerable savings. Lubrication is an effective means of controlling

wear and reducing friction, and it has wide applications in the operation of machine

element such as bearings.

2

Bearings are representative mechanical elements used in many classes of rotatin1!

machinery. They are classified into rolling element bearings and plain bearings.

depending on whether they are in rolling contact or in sliding contact. A sliding bearing

typically uses a lubricant to reduced friction between the sliding surfaces. A shaft and

bushing bearing are known as ajournal bearing. Cylindrical bore bearing is a journal

bearing with plain cylindrical sleeve (bushing) wrapped around the journal (shaft). The

journal is rotating inside the bore of the sleeve with a thin clearance. In journal

bearings, the bearing surfaces are parallel to the axis of rotation.

The journal and bearing surfaces are separated by a film of lubricant that is

supplied to the clearance space between the surfaces through a hole or a groove. When

there is a continuous fluid film separating the surfaces we speak of fluid film lubrication.

When the journal bearing begin rotating there is very little lubricant between the bearing

and shaft at some contact point and rubbing occurs. After the bearing has reached

sufficient speed, the lubricants begins to wedge into the contact area and the relative

motion of the surfaces causes the fluid pressure to support the load without metal to

metal contact. This lubrication phenomenon is known as hydrodynamic lubrication.

The understanding of hydrodynamic lubrication began with the classical

experiments of Beauchamp Tower in 1883 in which the existence of a film was dctected

from measurements of pressure within the lubricant, and ofNikilay Petroff in 1883 who

reached the same conclusion from friction measurements (Hamrock, 1994). Tower's

works was closely followed by Osborne Reynolds celebrated analytical papcr in 1886 in

which he used a reduced form of the Navier-Stokes equations in association with the

continuity equation to generate a second-order differential equation for the pressure in

the narrow converging gap between bearing surfaces. He derived and published not

only the descriptive differential equation that today bears his name but also certain

solutions to this equation that agree well with the experimental measurements ofTowcr

(Gross, et aI., 1980). Since then Reynolds equation has become the foundation of

hydrodynamic analysis of bearing performance.

Documents

PERPUSTAKAAN KUiTTHOeprints.uthm.edu.my/id/eprint/790/1/24_Pages_from... · untuk penentuan secara tepat halaju kritikal bagi suatu galas dan juga untuk mcngkaj i kestabilan galas