Durham E-Theses

Friction in journal bearings

Bennett, James

How to cite:

Bennett, James (1981) Friction in journal bearings, Durham theses, Durham University. Available atDurham E-Theses Online: http://etheses.dur.ac.uk/7523/

Use policy

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission orcharge, for personal research or study, educational, or not-for-pro�t purposes provided that:

• a full bibliographic reference is made to the original source

• a link is made to the metadata record in Durham E-Theses

• the full-text is not changed in any way

The full-text must not be sold in any format or medium without the formal permission of the copyright holders.

Please consult the full Durham E-Theses policy for further details.

Academic Support O�ce, Durham University, University O�ce, Old Elvet, Durham DH1 3HPe-mail: e-theses.admin@dur.ac.uk Tel: +44 0191 334 6107

http://etheses.dur.ac.uk

FRICTION IN JOURNAL BEARINGS

BY JAMES BENNETT B.Sc.

A b s t r a c t

F r i c t i o n i n f l u i d Journal Bearings i s u s u a l l y i n v e s t i g a t e d by

measuring the torque which appears on a bear i n g w h i l s t a s h a f t i s

r o t a t e d i n s i d e i t . The bearing i s supported h y d r o s t a t i c a l l y which

would a l l o w i t t o r o t a t e f r e e l y i f otherwise u n r e s t r i c t e d . Motion

o f the b e a r i n g i s r e s t r i c t e d by t e n s i o n gauges which i n d i c a t e the

f r i c t i o n a l f o r c e which they are r e s i s t i n g .

The work described i n t h i s t h e s i s i s concerned w i t h a new method

o f measuring f r i c t i o n a l torque i n j o u r n a l bearings. The method c o n s i s t

o f d r i v i n g a s h a f t up t o a given speed i n s i d e a f i x e d b earing and then

p e r m i t t i n g i t t o decelerate f r e e l y . The d e c e l e r a t i o n s are timed

e l e c t r o n i c a l l y and almost instantaneous d e c e l e r a t i o n r a t e s are

obtained. From these d e c e l e r a t i o n r a t e s and known p o l a r moment o f

i n e r t i a , the f r i c t i o n a l torque i n the b e a r i n g can be found f o r given

speeds. The method has been found t o be ve r y r e l i a b l e and also has

wider a p p l i c a t i o n w i t h o t h e r r o t a t i n g systems.

FRICTION IN JOURNAL BEARINGS

BY JAMES BENNETT B.Sc.

Thesis submitted f o r the Degree o f Doctor o f Philosophy a t the U n i v e r s i t y o f Durham Department o f Engineering Science 1981.

The copyright of this thesis rests with the author.

No quotation from it should be published without

his prior written consent and information derived

from it should be acknowledged.

CONTENTS

No t a t i o n

I n t r o d u c t i o n

1. Previous Work

2. F r i c t i o n i n A i r Bearings

3. L i q u i d Bearings

4. Design o f the 2" Diameter Bearing Ri

5. I n s t r u m e n t a t i o n

6. Experiments w i t h the 2" Diameter Rig

, 7. Design o f the 3" Diameter Bearing Ri

8. Results and Discussion

9. Conclusions

10. Results

Appendices

References

ILLUSTRATIONS

F i g . ( l a ) P l o t s o f Cf versus Re from e a r l y experimental 18

r e p o r t s

F i g . ( l b ) V a r i a t i o n o f c r i t i c a l T a y l o r number w i t h 22

e c c e n t r i c i t y r a t i o

F i g . (2a) V a r i a t i o n o f V t c w i " t h mass f l o w r a t e 30

F i g . (3a) P l o t o f Cf versus Re 40

F i g . (4a) Sketch o f the 2" diameter r i g 44

F i g . (4b) Photograph o f the 2" diameter r i g 45

F i g . (5a) Schematic sketch o f the d e c e l e r a t i o n t i m i n g c i r c u i t 50

F i g . (5b) C i r c u i t diagram o f the d i v i d e / c o n t r o l u n i t 54

F i g . (6a) A i r bearing c o r r e c t i o n p l o t l o g g N/NQ versus t 63

F i g . (6b) Disc drag. P l o t o f dN/dt versus N 64

F i g . (6c) 2"" diameter r i g , mass f l o w r a t e versus N 66

F i g . (6d) P l o t o f £ versus N, c = 2.62 10 _ 5m 67

F i g . (6e) P l o t o f E versus N, c = 5.92 10~5m 68

F i g . ( 6 f ) P l o t o f £ versus N, c = 7.90 10"5m 69

F i g . (6g) P l o t o f Cf versus Re f o r the 2" diameter r i g , 72

supply pressure = 1 . 0 bar

F i g . (6h) P l o t o f Cf versus Re f o r the 2" diameter r i g , 73

supply pressure = 2 . 0 bar

F i g . ( 6 i ) P l o t o f T e x p / T t h versus Re, c = 2.62' lCr 5m 74

F i g . ( 6 j ) P l o t o f T e x p / T t h versus Re, c = 5.92 10 _ 5m

F i g . (6k) P l o t o f T e x p / T t h versus Re, c = 7.90 10 5m

75

76

F i g . (61) P l o t o f Cf versus Re f o r the 2" diameter r i g . 77

(Results c o r r e c t e d f o r mass f l o w r a t e torque)

F i g . (7a) Sketch o f the 3" diameter r i g 80

F i g . (8a) D e c e l e r a t i o n o f the 3" diameter s h a f t i n an a i r 83

bear i n g w i t h o u t d i s c s .

F i g . (8b) Disc Drag. P l o t o f dN/dt versus N 84

F i g . (8c) Mass f l o w r a t e o f l u b r i c a n t versus N f o r the 3" 85

diameter r i g .

F i g . (8d) P l o t o f £, versus N f o r the 3" diameter r i g 86

F i g . (8e) P l o t o f Cf versus Re f o r the 3" diameter r i g , 87

supply pressure = 1 . 0 bar

F i g . ( 8 f ) P l o t o f Cf versus Re f o r the 3" diameter r i g , 89

supply pressure = 2.0 bar

F i g . (8g) P l o t o f Cf versus Re f o r the 3" diameter r i g , 90

summary

F i g . (8h) P l o t o f T e x p / T t h versus Re, supply pressure = 1.0 bar 91

F i g . ( 8 i ) P l o t o f T e x p / T t h versus Re, supply pressure = 2 . 0 bar 92

F i g . ( 8 j ) P l o t o f experimental r e s u l t s and t h e o r e t i c a l 102

p r e d i c t i o n s f o r a t u r b u l e n t l u b r i c a t i n g f i l m

A c k n o w l e d g e m e n t s

I should l i k e t o express my thanks t o the members o f s t a f f o f

the Engineering Science Department a t Durham U n i v e r s i t y f o r t h e i r

h e l p and advice given t o me throughout my research, w i t h s p e c i a l

thanks t o Prof. H. Marsh, Dr. V.G. Endean and Mr. G. Daley.

D e c l a r a t i o n

None o f the m a t e r i a l contained i n t h i s t h e s i s has been p r e v i o u s l y

submitted by me f o r a degree i n t h i s o r any o t h e r U n i v e r s i t y and the

work represents my own o r i g i n a l c o n t r i b u t i o n .

C o p y r i g h t

The c o p y r i g h t o f t h i s t h e s i s r e s t s w i t h the author. No

q u o t a t i o n from i t should be published w i t h o u t h i s p r i o r w r i t t e n

consent and i n f o r m a t i o n d e r i v e d from i t should be acknowledged.

7

NOTATION

A B constants

c r a d i a l clearance

Cf non-dimensional bearing f r i c t i o n a l torque

Cf^ uncorrected non-dimensional b e a r i n g f r i c t i o n a l torque

D b e a r i n g diameter

F f r i c t i o n a l f o r c e

h f i l m t h ickness

h* f i l m t h ickness a t the p o i n t o f maximum pressure

I i n e r t i a

I p r o t o r p o l a r i n e r t i a

lip r o t o r transverse i n e r t i a

k^ constant

k c b e a r i n g angular s t i f f n e s s

k^ b e a r i n g l i n e a r s t i f f n e s s

k x , ky constants i n the t u r b u l e n t Reynolds equation

L l e n g t h o f the bearing

1 2 L/2

M r o t o r mass

m l u b r i c a n t mass f l o w r a t e

r f i c non-dimensional c o r r e c t i o n f o r angular momentum torque

N r o t a t i o n a l frequency

N i n t e g e r v a r i a b l e

N Q i n i t i a l r o t a t i o n a l frequency

s e l f d r i v e steady r o t a t i o n a l frequency

N o n s e t onset speed o f i n s t a b i l i t y

p pressure

time mean pressure

pressure f l u c t a t i o n

b e a r i n g r a d i u s

r a d i u s o f the i n n e r c y l i n d e r

r a d i u s o f the out e r c y l i n d e r

r o t a t i o n a l - Reynolds number

a x i a l Reynolds number

f r i c t i o n a l torque

angular momentum torque

Taylor Number

ec c e n t r i c c r i t i c a l Taylor Number

concentri c c r i t i c a l Taylor Number

t o t a l f r i c t i o n a l torque

r o t o r torque

bearing torque

experimental f r i c t i o n a l torque

t h e o r e t i c a l f r i c t i o n a l torque

time

time f o r N pulses

time f o r 2N pulses

d e c e l e r a t i n g time constant

surface speed

v e l o c i t i e s i n coordinate d i r e c t i o n s

a x i a l f l o w v e l o c i t y

b e aring load

coordinate i n the d i r e c t i o n o f motion

coordinate _L t o the d i r e c t i o n o f motion

coordinate across the f i l m

d e c e l e r a t i o n r a t e

CXJJ t o t a l d e c e l e r a t i o n r a t e o f the r o t o r 'in -the a i r bearing

<XC c o r r e c t i o n f o r the a i r bearing drag

D di s c drag d e c e l e r a t i o n r a t e

cV t 2 - 2 t 1

6 e c c e n t r i c i t y

0 angle o f r o t a t i o n

9 angular displacement "^rom the l i n e o f centres

y- dynamic v i s c o s i t y

V kinematic v i s c o s i t y

^ l u b r i c a n t d e n s i t y

C shear s t r e s s on any l a y e r o f f l u i d

C c Couette surface shear s t r e s s

\y a t t i t u d e angle

UJ angular v e l o c i t y

c o Q i n i t i a l angular v e l o c i t y

Uim angular v e l o c i t y a f t e r time t-^

ujf angular v e l o c i t y a f t e r time t 2

OJ„ angular v e l o c i t y a t time t = 0

dN d e c e l e r a t i o n r a t e f o r the bearing torque + mass fl o w r a t e torque d t

dN t o t a l d e c e l e r a t i o n r a t e i n c l u d i n g windage and mass f l o w r a t e dt-L

e f f e c t s

INTRODUCTION

A j o u r n a l b e a r i n g can be described as a c i r c u l a r s h a f t passing

through a s l i g h t l y l a r g e r hole i n a housing. The two components o f a

f l u i d j o u r n a l b e a r i n g are separated under normal o p e r a t i n g c o n d i t i o n s by

a f l u i d f i l m .

I n h y d r o s t a t i c bearings, the surfaces are f o r c e d apart by f l u i d

s u p p l i e d a t h i g h pressure i n t o the be a r i n g clearance.

Hydrodynamic bearings operate by v i r t u e o f the entrainment o f a

f l u i d between two moving surfaces which form a converging wedge. This

produces h i g h pressures i n the f l u i d which f o r c e the surfaces a p a r t .

I n a hydrodynamic j o u r n a l bearing the s h a f t i s dis p l a c e d i n t o an

e c c e n t r i c p o s i t i o n by r a d i a l loads thus forming a convergent-divergent

clearance and hence a suppo r t i n g pressure system.

A r e s i s t a n c e t o the motion o f the p a r t s i s caused by the shearing

o f the f l u i d f i l m i n the clearance. This r e s i s t a n c e , o r i n o t h e r words

the f r i c t i o n i n the b e a r i n g , v a r i e s w i t h the v e l o c i t i e s o f the moving

surfaces, t h e i r s e p a r a t i o n , the pressure g r a d i e n t i n the f i l m and the

nature o f the l u b r i c a t i n g f l u i d . L u b r i c a n t s o f a very v a r i e d nature are

used i n j o u r n a l bearings, such as a i r , water and o i l s .

Reynolds* (1) i n 1886 c a r r i e d out an extensive a n a l y s i s o f the

hydrodynamic phenomenon and t h i s has provided the b a s i s f o r a l l the work

t h a t f o l l o w e d . He d e r i v e d a d i f f e r e n t i a l equation f o r the pressure

d i s t r i b u t i o n and presented two a n a l y t i c a l s o l u t i o n s . The f i r s t was f o r

a squeeze f i l m between two e l l i p t i c a l p l a t e s and the second f o r a plane

s l i d e r b e a r i n g o f i n f i n i t e w i d t h . Reynolds a l s o presented approximate

s o l u t i o n s f o r the j o u r n a l bearing problem. For p a r t i a l - arc and t o t a l

arc j o u r n a l bearings he sought the s o l u t i o n by the use o f t r i g o n o m e t r i c

* References are l i s t e d a t the end o f the t h e s i s

s e r i e s . The s e r i e s i f l i m i t e d t o ten o r twelve terms and e c c e n t r i c i t i e s

l e s s than 0.6 converged w i t h adequate accuracy. Reynolds found t h a t f o r

a t o t a l arc j o u r n a l b e a r i n g w i t h an e c c e n t r i c i t y r a t i o o f 0.5 the best

agreement between theory and experiment occurred.

Reynolds equation assumes laminar f l o w , constant v i s c o s i t y and

constant pressure across the f i l m t h i c k n e s s , n e g l i g i b l e c u r v a t u r e s , an

incompressible Newtonian f l u i d , n e g l i g i b l e body f o r c e s , n e g l i g i b l e f l u i d

i n e r t i a and steady s t a t e c o n d i t i o n s . As w i t h f l u i d f l o w i n a p i p e , a

non-dimensional group c a l l e d the Reynolds number can be used t o p r e d i c t

whether o r not a b e a r i n g l u b r i c a t i n g f i l m i s laminar o r t u r b u l e n t i n

nature. The Reynolds number i n d i c a t e s the r e l a t i v e importance o f the

i n e r t i a f o r c e s and the viscous f o r c e s i n a f l u i d f i l m . As mentioned

p r e v i o u s l y , Reynolds solved the d i f f e r e n t i a l e quation f o r pressure

d i s t r i b u t i o n by a n a l y t i c a l means and one assumption he made was t h a t

laminar f l o w predominated. When bearings are o p e r a t i n g w i t h h i g h Reynolds

numbers the f l u i d f i l m changes from laminar f l o w i n t o t u r b u l e n t f l o w and

i n e r t i a f o r c e s dominate. I n f u l l y e s t a b l i s h e d t u r b u l e n t f l o w the i n e r t i a

f o r c e s predominate over the viscous f o r c e s . This i s the case, except very

near the surfaces o f the b e a r i n g , where a laminar sublayer e x i s t s .

Attempts have been made t o solve a m o d i f i e d form o f the Reynolds

equation f o r t u r b u l e n t l u b r i c a t i n g f i l m s . They are g e n e r a l l y based on

a p e r t u b a t i o n type theory, i . e . c o n s i d e r i n g mean parameters and the

f l u c t u a t i o n s about the mean, s u b s t i t u t i n g these i n t o the m o d i f i e d

Reynolds equation and then s o l v i n g i t t o f i n d the pressure d i s t r i b t i o n .

The m o d i f i e d Reynolds equation and i t s s o l u t i o n w i l l be discussed l a t e r

i n Chapter 8.

The change from a laminar t o a t u r b u l e n t l u b r i c a t i n g f i l m i s preceded

by a t r a n s i t i o n r e g i o n , sometimes known as superlaminar f l o w . The va r i o u s

12

stages i n t h i s t r a n s i t i o n r e g i o n w i l l be discussed i n d e t a i l i n the

f o l l o w i n g chapter.

The f r i c t i o n which appears i n a t u r b u l e n t f i l m b e aring i s much g r e a t e r

than t h a t o f a laminar b e a r i n g o p e r a t i n g a t the same Reynolds number.

The modem t r e n d i s t o use l a r g e r bearings, t o run them f a s t e r and t o

l u b r i c a t e tnem w i t h low v i s c o s i t y l u b r i c a n t s . The combination o f these

f a c t o r s leads t o bearings now commonly being run w i t h a t u r b u l e n t

l u b r i c a t i n g f i l m and consequently h i g h e r f r i c t i o n a l losses. I n steam

t u r b i n e s manufacted i n B r i t a i n bearings up t o 0.76 metres i n diameter are

commonly used, running a t 1800 r.p.m. A t y p i c a l steam t u r b i n e may have

up t o twelve bearings v a r y i n g i n s i z e from 0.25 metres t o 0.76 metres i n

diameter s u p p o r t i n g the r o t o r . The power l o s s i n these bearings

can be i n the reg i o n o f 0.1 megawatts f o r the smal l e r

bearings t o 0.52 megawatts f o r the 0.76 metre bearings. The t o t a l power

l o s s from the bearings f o r a complete machine can be i n the order o f

5 megawatts.

The above s t a t i s t i c s demonstrate the importance o f the a v a i l a b i l i t y

o f accurate i n f o r m a t i o n on f r i c t i o n t o the designer o f bearing systems t o

enable him t o minimise the t o t a l power consumption o f a machine. The

o b j e c t o f the research r e p o r t e d i n t h i s t h e s i s i s t o i n v e s t i g a t e the

f r i c t i o n i n j o u r n a l bearings the importance o f which i s i n d i c a t e d above.

Most i f not a l l o f the experimental work which has been c a r r i e d out t o

date has con s i s t e d o f measuring the f r i c t i o n a l torque which appears on

the s t a t i o n a r y member o f a be a r i n g . The present work c o n s i s t s o f measuring

the f r i c t i o n a l torque which appears on a r o t a t i n g s h a f t . The method

used t o o b t a i n measurements o f the f r i c t i o n a l torque has not been used

b e f o r e , whereas previous work i n v o l v e d measuring f o r c e s which appeared

on a s t a t i o n a r y bush, the present work i n v o l v e s the t i m i n g o f the

d e c e l e r a t i o n s o f a r o t a t i n g s h a f t and then r e l a t i n g the measurements

obtained t o the f r i c t i o n a l torque r e s i s t i n g the motion o f the s h a f t .

The experimental work reported here was c a r r i e d out on bearings

w i t h clearance r a t i o s which are commonly found i n i n d u s t r y whereas

most research work reported p r i o r t o t h i s t h e s i s has been c a r r i e d out

w i t h l a r g e clearance r a t i o s not i n common i n d u s t r i a l use.

CHAPTER 1

•PREVIOUS WORK

I n t r o d u c t i o n

L u b r i c a t i n g f i l m s can be best described w i t h the a i d o f a

non-dimensional group c a l l e d the Reynolds number. The Reynolds number

used i n t h i s t h e s i s i s d e f i n e d by the f o l l o w i n g : -

Re = co Rc v

where uo = angular v e l o c i t y i n radians/second

R = s h a f t r a d i u s

c = r a d i a l clearance

^) = kinematic v i s c o s i t y

The general o u t l i n e o f the s t a t e o f a f i l m a t v a r i o u s Reynolds numbers

i s as f o l l o w s . At low Reynolds numbers the f i l m i s laminar i n nature.

As the Reynolds number i s increased the laminar f l o w gains secondary

f l o w nature and a t even h i g h e r Reynolds numbers the f l o w breaks down

i n t o f u l l y e s t a b l i s h e d t u r b u l e n t f l o w . The r e g i o n o f secondary f l o w

i s o f t e n r e f e r r e d t o as the t r a n s i t i o n r e g i o n and t h i s w i l l be described

i n more d e t a i l s h o r t l y .

Much research has been c a r r i e d out on bearings running w i t h a

l u b r i c a t i n g f i l m i n the t r a n s i t i o n s t a t e . The p r e d i c t i o n o f the onset

o f t r a n s i t i o n , i t s delay and i t s complete suppression are some o f the

p o i n t s which are c u r r e n t l y under study. Also under study i s the

p o s s i b i l i t y o f secondary f l o w o c c u r r i n g i n the l u b r i c a t i n g f i l m a t

very low Reynolds numbers.

G.I. T a y l o r , ( 2 ) , ( 3 ) , (1923,1936), i n v e s t i g a t e d the s t a b i l i t y

o f a f l u i d f i l m between two c y l i n d e r s , the i n n e r o f which was r o t a t i n g

and the o u t e r a t r e s t . The i n v e s t i g a t i o n s were both experimental and

t h e o r e t i c a l i n nature and were c a r r i e d out on c y l i n d e r s w i t h clearance

15

r a t i o s q u i t e l a r g e i n magnitude compared t o bearing r a t i o s . The r a t i o s

used were 0.06 t o 0.34. Although as mentioned, these are r e l a t i v e l y

l a r g e , the work o f f e r s i n f o r m a t i o n which can be used t o i n t e r p r e t

b e a r i n g f i l m s i n the t r a n s i t i o n r e g i o n . Taylor found t h a t laminar

f l o w p r e v a i l e d a t low Reynolds numbers bu t broke down i n t o a secondary

f l o w o f t o r o i d a l v o r t i c e s as the Reynolds number approached a c r i t i c a l

value.

I n general, i n the f l o w between two c o n c e n t r i c r o t a t i n g c y l i n d e r s

the t r a n s i t i o n from laminar t o t u r b u l e n t f l o w i s s t r o n g l y i n f l u e n c e d

by c e n t r i f u g a l f o r c e s . The e f f e c t on the t r a n s i t i o n depends on whether

v e l o c i t y increases o r decreases w i t h i n c r e a s i n g r a d i u s . When the o u t e r

c y l i n d e r i s a t r e s t and the i n n e r i s r o t a t i n g , i . e . v e l o c i t y decreasing

w i t h i n c r e a s i n g r a d i u s , the e f f e c t o f c e n t r i f u g a l f o r c e s i s a d e s t a b i l i s i n g

one. The f l u i d p a r t i c l e s near the i n n e r c y l i n d e r experience a h i g h e r

c e n t r i f u g a l f o r c e and show a tendency t o be p r o p e l l e d outwards.

G.I. Taylor was the f i r s t t o prove t h a t when a c e r t a i n Reynolds number

has been exceeded, there appear i n the f l o w , v o r t i c e s whose axes are

l o c a t e d along the circumference and which have a l t e r n a t e ^ , opposite

d i r e c t i o n s . The t o r o i d a l v o r t i c e s have a h e i g h t and w i d t h approximately

equal t o the r a d i a l clearance o f the annulus.

The c r i t i c a l Reynolds number f o r the onset o f Taylor v o r t i c e s can

be p r e d i c t e d from the equation,

where Re = R o t a t i o n a l Reynolds number

R = Radius o f the s h a f t

c = Radial clearance o f the b e a r i n g

Measurements i n d i c a t e d t h a t the c e l l u l a r v o r t i c e s remain s t a b l e i n

a f a i r l y wide range o f Reynolds numbers above the value p r e d i c t e d i n

Re V R / c 41.1 (1:1)

16

( 1 : 1 ) . At h i g h e r Reynolds numbers the f l o w breaks down i n t o t u r b u l e n t

f l o w . A g e n e r a l l y agreed value o f Reynolds number f o r the onset o f

t u r b u l e n t f l o w i s approximately 2000. When the i n n e r c y l i n d e r i s a t

r e s t and the o u t e r i s r o t a t i n g , i . e . v e l o c i t y i n c r e a s i n g w i t h i n c r e a s i n g

r a d i u s , the e f f e c t o f c e n t r i f u g a l f o r c e s i s a s t a b i l i s i n g one. Taylor

v o r t i c e s do not occur w i t h t h i s s i t u a t i o n , b ut the t u r b u l e n t onset i s

s t i l l apparent a t Reynolds numbers i n the r e g i o n o f 2000. As the

r a d i a l clearance i s increased, w i t h the o u t e r c y l i n d e r r o t a t i n g and

the i n n e r s t a t i c , the s t a b i l i s i n g i n f l u e n c e o f the c e n t r i f u g a l f o r c e s

causes the c r i t i c a l Reynolds number t o become l a r g e r than the value

o f 2000.

The f r i c t i o n a l torque which appears i n a b e a r i n g as i t passes

through the above-mentioned t r a n s i t i o n s v a r i e s g r e a t l y . I n the laminar

regime the torque i s d i r e c t l y p r o p o r t i o n a l t o the speed o f r o t a t i o n .

I n the t r a n s i t i o n and t u r b u l e n t regimes the torque i s a more complex

f u n c t i o n o f speed such as ( s p e e d ) n f o r t u r b u l e n t f l o w . The value o f the

exponent 'n* w i l l be discussed i n the f o l l o w i n g s e c t i o n .

Reported Experimental and T h e o r e t i c a l Observations

A j o u r n a l b e a r i n g o p e r a t i n g over a wide speed range may d i s p l a y

the f o l l o w i n g c h a r a c t e r i s t i c s . At low Reynolds numbers the f l u i d f i l m

w i l l be laminar i n n a t u r e . As the Reynolds number i s increased w i t h

i n c r e a s i n g speed, the f i l m w i l l go through a ' t r a n s i t i o n r e g i o n 1 and

i f the Reynolds number ± s increased s t i l l f u r t h e r the f i l m w i l l become

t u r b u l e n t .

Much work has been c a r r i e d out on bearings o p e r a t i n g i n these three

regimes. The e a r l y work was c a r r i e d out by P e t r o f f (1883) who produced

an expression which i s now commonly known as P e t r o f f ' s law. The law

can be used t o c a l c u l a t e the f r i c t i o n a l torque which appears i n a

17

journal bearing with i t s members running concentrically and with the

bearing clearance f i l l e d with lubricant. 3

The expression i s : - T = 2-rr R L p.uu c

where T = The f r i c t i o n a l torque

L = Bearing length

u. = Dynamic vi s c o s i t y

R = Bearing radius

oo = Angular v e l o c i t y

This expression f o r f r i c t i o n a l torque i s probably that most commonly used

i n l u b r i c a t i o n and provides a good estimate of the f r i c t i o n a l torque

which a bearing w i l l display w h i l s t operating i n a laminar regime.

Wilcock, (1950), ( 4 ) , carried out the f i r s t detailed investigation

i n t o the characteristics of loaded journal bearings operating i n the

turbulent regime. Wilcock used four bearings i n h i s experimental work,

three of 0.2 m diameter and one 0.1 m diameter. The bearings were fed

wi t h o i l through two longitudinal grooves. The clearance r a t i o s used

by Wilcock were i n the range (1.7 - 3.8) x 10~^. The housing was

stationary and loaded by a hydrostatic ram and the shaft was driven

by an i n l i n e motor. F r i c t i o n a l torque measurements were taken from the

housing by using a torque arm arrangement.

Wilcocks f r i c t i o n a l torque results are p l o t t e d i n Fig. ( l a ) . At

low Reynolds numbers the p l o t has a gradient o f -1 indicating that the

f r i c t i o n a l torque i s proportional to r o t a t i o n a l speed. Wilcock explains

that as the Reynolds number i s increased a d e f i n i t e t r a n s i t i o n occurs at

about the speed predicted by Taylor f o r vortex formation and a f u r t h e r

increase i n speed produced f u l l turbulence. The gradients of Wilcock's

curves varied from -1 to nearly zero when turbulence was f u l l y established.

18

IS 5 0 Wilcock C Icai'ifi^ groove fee)). (2^)

Cf T 0 ro^uE

3 -

ID'3. L

Pchroff C f = 2 /R&.

Re 2nN K c

act! Fu Pre-d'\tVeJl "T& ^ \ o r 0 <\ s e V

R e = "75^

R L c

N

= b «aniM^ Co.

raa\c»l c\ea.r-a.r\ce . k~,i>e WACA (c v"\s c o s i l y 11< b r "i c-a.«\ V" cAe^i'1^

a<-vc\w\c».r ve\oc\V^ rc\d , / sec . r3

C f = O - Q"7^ R £ C 4 3

—1 1 V

S 6 7 7 % I 10 10:

Cf ver sus Re frorrs e a r \ ^ 6* per^ewYo.\ re^orta .

I n the l i g h t of more recent work, Wilcock's explanation of his

re s u l t s may be incorrect. Wilcock's bearings were operated with large

e c c e n t r i c i t y r a t i o s and i t has been shown by Mobbs (5) and others, that

a bearing run i n t h i s way can suppress the vortex formation i n the

l u b r i c a t i n g f i l m to such an extent that the onset of turbulence w i l l be

experienced f i r s t .

The next reported experimental work was by Smith and F u l l e r i n

1956, ( 6 ) . The apparatus used by Smith and Fuller consisted of a

0.076 m x 0.076 m bearing with a clearance r a t i o of 2.93 10~ 3. The

lubr i c a n t , which was water and not o i l as Wilcock had used, was fed in t o

the bearing through a single 0.006 m hole. Again the bush was stationary

and loaded by a hydrostatic ram. The shaft was supported by two r o l l i n g

element bearings, one at e i t h e r side and was driven by a pulley system.

The f r i c t i o n a l torque was measured by a torque arm f i t t e d to the

stationary bush. The apparatus used i n the current experiments and

which w i l l be described i n a l a t e r chapter was s i m i l a r i n dimensions

to the bearing detailed above and also i n the use of water as a lubricant

although the method used to measure the f r i c t i o n a l torque was t o t a l l y

d i f f e r e n t .

The results obtained by Smith and F u l l e r are p l o t t e d i n Fig. ( l a ) .

The results follow the Petroff l i n e very closely u n t i l almost exactly

the predicted Reynolds number of 759 f o r the onset of Taylor vortices.

The p l o t then indicates a well defined t r a n s i t i o n region followed by

the onset of turbulence. The expression obtained f o r the f r i c t i o n a l

torque i n the turbulent regime was,

Cf = 0.078 Re - 0• 4 3

.Huggins (7) i n 1966 investigated the f r i c t i o n a l torque i n a large

turbogenerator bearing. The apparatus which Huggins used consisted of

20

a bearing 0.61 m i n diameter. His work was connected with a p r a c t i c a l

s i t u a t i o n , that of bearings used i n large steam turbogenerators. The

r i g was designed around a f i x e d bearing because t h i s arrangement i s

dynamically more r e a l i s t i c than the f i x e d shaft usually chosen f o r

experimental equipment. At that time no method existed of taking

direct, f r i c t i o n measurements on the apparatus chosen. Consequently

Huggins resorted to a heat balance calculation.

This method i s obviously not as satisfactory as others and therefore

leads to a ce r t a i n amount of doubt about the r e s u l t s . The results

obtained from these experiments are p l o t t e d i n Fig. ( l a ) . With reference

to t h i s p l o t , there i s a deviation from the expected results i n the low-

Reynolds number region and also the t r a n s i t i o n to turbulence at high

Reynolds numbers appears to happen without the p r i o r onset of vortices.

Recent investigations i n t o the regimes of a l u b r i c a t i n g f i l m have

concentrated on the change from a laminar to a turbulent f i l m and the

suppression of the early onset of the t r a n s i t i o n region between these

regimes. The onset of the t r a n s i t i o n region has been found to be

affected by the following:-

1. e c c e n t r i c i t y , e

2. a x i a l flow, m

3. clearance r a t i o , c/R

4. annulus length, L a

Much of the experimental work concerned w i t h the above ef f e c t s has been

carried out on r i g s i n which the r o t a t i n g parts are separated by clearance

r a t i o s which are large i n comparison to bearing clearance r a t i o s commonly

found i n industry. I t has been found that care must be exercised when

the large clearance results are related to p r a c t i c a l bearings. Onset

of f i l m changes i n the above-mentioned experiments i s usually detected

21

by the use of visual or torque measuring techniques.

I t has been found that when a bearing i s running eccentrically

the f i l m usually tends to be more stable and the laminar regime exists

at higher Reynolds numbers than those predicted f o r the onset of Taylor

vortices. Conversely i t has also been found i n some cases that the

onset of secondary flow can be experienced at lower Reynolds numbers

than expected.

A useful summary of e c c e n t r i c i t y e f f e c t findings was presented i n

a paper by Mobbs (8) and t h i s i s i l l u s t r a t e d i n Fig. ( l b ) . The p l o t

i s the r a t i o of Taylor numbers, (a non-dimensional group derived by

G.I. Taylor and defined i n Fig. ( l b ) ) , f o r the onset of vortices, i n

a l i q u i d contained between, eccentric and concentric cylinders, versus

cylinder radius r a t i o . I t i s apparent, from the p l o t s , that the re i s a

large degree of d i s p a r i t y over the e f f e c t of e c c e n t r i c i t y . A possible

explanation f o r the spread i n the r e s u l t s was suggested i n the same

paper by Mobbs. He observed secondary flow between r o t a t i n g cylinders

at s u b - c r i t i c a l Taylor numbers f o r both concentric and eccentric

cylinders and explained that t h i s could sometimes be responsible f o r the

increased f r i c t i o n a l torque observed and interpreted wrongly as the

onset of Taylor vortices.

The table i n Fig. ( l b ) shows that a wide range of radius r a t i o s

has been used i n experimental work. In another paper Mobbs (9)

reported an investigation i n t o the e f f e c t of cylinder radius r a t i o s

on the c r i t i c a l Taylor number. He found that the c r i t i c a l Taylor number

was v i r t u a l l y unaffected by radius r a t i o but that the breakdown of the

Taylor vortices i n t o a wavy mode, which i s found to precede turbulence,

did show a marked dependence when the r a t i o was less than 0.8 and much

less of an e f f e c t when greater than 0.8. This then throws some l i g h t

22

T o i l e r M u ^ W r ( T a " ) = R C ^ 2

T CO

1. Co le .

2. VoUr.

Ulcere R = s U a f V r e n i n s . C — ra rA\C*\ clto.rc«Ace

6 -

5 -

4 -

3 -

2 -MoVe,: 5 o,r, Ji ^ are

5 is fro<v> l"Ue. no ft-loccA VU S >5 f

3 15 V

3 l o c a l ^ tori

O ©•2. 0-4- O b 0-S

E.cce.ft l"r'ic\V>^ r&Vio £

3. C & 5 V l e e V c , \ . T o r c ^ e 0.°|

*V • Vers^-ee^en eV 0-5" 5. D i Prm-x^ oir*c SYwoirV.

6 . G.sV\e eV a\ • O^e W^ecV i o n .

1' Yot^nes.

^ . Do Pr\fT\ a .

0- ^ 5

T c o ~ Cr\V»cov\ U ^ l e i f N w m U r a V z e r o eccev\V<-»t\V^.

* H o V e ; T U e C r v V l t ^ \ T&^\of ^».v^W^'is l W l<*^\or Klu.w,Wr ^ V c ^ U ' c U Vot V i c e s ^i«-5 V <xj>pea.r.

Y a / t o A i o * \ o ? Cr\Viccv\ TtA.^or Hukw^\>e<- w i H\ aVtic'>i-^ rc< V i o ^ M o t b s v ^ } | ,

23

on the l i m i t a t i o n s of applying large clearance results to bearing situations.

Cole (10), (11) reported the effects of a x i a l flow and annulus

length on Taylor vortex c r i t i c a l speeds. At a x i a l Reynolds numbers

related to moderate a x i a l flows i t was found that a x i a l flow had

ne g l i g i b l e e f f e c t on the onset of Taylor vortices. I t was pointed out

that at a x i a l Reynolds numbers approaching 400 the e f f e c t of a x i a l

flow was becoming more s i g n i f i c a n t , so i t would seem that caution

must be exercised w i t h large clearance r i g s when investigating the

onset of Taylor vortices.

Annulus length was found by Cole to have v i r t u a l l y no e f f e c t on

the onset of Taylor vortices. I t d i d however a f f e c t to some degree the

onset of a wavy mode to the vortices but only at L a/c r a t i o s less than

40.

Jackson (12) used a large clearance r i g to investigate the onset

o f wavy modes superimposed on Taylor vortices. He found that three

d i s t i n c t wave modes occurred as the Reynolds number was increased and

called these the Primary wave mode, the Transitional' state and the

Secondary wave mode. Visual and torque changes are found to occur at

the t r a n s i t i o n from one mode to another. I t was found also that i f the

Reynolds number i s f i r s t increased d i s t i n c t torque changes could be

detected as the modes were encountered but i f the 'Reynolds number i s

then decreased the torque changes are not apparent at the same Reynolds

numbers. The torque changes are only detectable at zero e c c e n t r i c i t y

and f o r a small increase i n e c c e n t r i c i t y the d i s t i n c t torque

d i s c o n t i n u i t i e s as the wavy modes are traversed become undetectable.

Two approaches have been used to predict the onset of Taylor r i n g

24

v o r t i c e s f o r eccentric operation. The f i r s t i s a local i n s t a b i l i t y theory based on considering the s t a b i l i t y of the v e l o c i t y p r o f i l e at any section of the flow taken i n i s o l a t i o n . The most unstable section i s found to be at the widest gap position. The second approach i s a non-local s t a b i l i t y theory which considers the l u b r i c a t i n g f i l m on a macroscopic scale. I n Fig. ( l b ) the local s t a b i l i t y theory produces a curve close to the lower set of curves and the non-local theory produces a curve which l i e s close to the higher set of curves. I t has been suggested that these two d i f f e r e n t approaches could provide an explanation f o r the spread i n results obtained i n experiments i n that some workers have detected l o c a l i n s t a b i l i t i e s and others more macroscopic i n s t a b i l i t i e s .

I f we consider the t r a n s i t i o n from Taylor vortices to f u l l y

established turbulence, visual reports indicate that as the Reynolds

number i s increased the vortices gain a wavy mode superimposed on

them. At higher Reynolds numbers the f i l m breaks down fu r t h e r u n t i l

random eddies are formed. Very l i t t l e appears to be known about the

t r a n s i t i o n to f u l l turbulence as few reports have been published on

t h i s subject. Onset of f u l l turbulence has been predicted to occur

at a Reynolds number = 2 R e o n s e t o f T a y i 0 r v o r t i c e s .

I t would appear great care has to be exercised when the onsets

of the wavy mode and turbulence are investigated using large clearance

r i g s . Clearance r a t i o and annulus length can have deceptive effects

on the r e s u l t s obtained by t h i s method i f the onsets are to be related

to small clearance bearing s i t u a t i o n s .

I n most of the experimental investigations described, the onset of

the various states i n the l u b r i c a t i n g f i l m i s detected either by visual

or torque measuring techniques. The torque measuring technique i s

25

generally regarded as the most r e l i a b l e as the onset of the f i l m change of state i s usually marked with a d i s t i n c t change i n displayed f r i c t i o n a l torque. With the onset of Taylor vortices d i f f i c u l t i e s can occur when experiments are being carried out with large e c c e n t r i c i t y r a t i o s as the torque change i s less d i s t i n c t . Visual techniques tend to be less accurate as a method of determining onsets. The d i f f i c u l t i e s are associated with i l l u m i n a t i o n or seeing where i n the bearing clearance the t r a n s i t i o n i s s t a r t i n g .

I t i s apparent that the behaviour of the l u b r i c a t i n g f i l m i s of

a very complex nature and i n f a c t s t i l l requires much furt h e r

investigation.

26

CHAPTER 2 FRICTION IN AIR BEARINGS

The f r i c t i o n a l torque i n an unloaded aerodynamic journal bearing

can be expressed by the equation,

T = 2 TT R 3 L quo ( f o r the concentric case) c

This i s the Petroff f r i c t i o n a l torque mentioned previously. I f the bearing

i s running eccentrically the Petroff torque i s modified i n the following

manner,

T = 2 T T R 3 L ^ L U ( 1 ) v

c ~ ^ ~ ( 1 - £ 2 ) 7*

For most cases the Petroff equation provides a good estimate of the

f r i c t i o n a l torque f o r £ < 0.4. I n the experiments reported here a very

l i g h t l y loaded externally pressurised a i r bearing was used operating w i t h

£ & 0.1 so that the correction was n e g l i g i b l e .

For an externally pressurised journal bearing, the f r i c t i o n a l torque

gains a term related t o the spinning of the lubricant as i t passes through

the bearing clearance. This e f f e c t was reported reference (13) by the

author and Professor H. Marsh.

I t was found that the f r i c t i o n a l torque was made up of two components.

The f i r s t being the usual Petroff f r i c t i o n and the second a torque from

providing the lubricant with angular momentum.

i. e . T t o t a l = (shear torque) + (angular momentum torque)

= 2 TTR 3 L JLUJ + (angular momentum torque) (2:1) c

The Angular Momentum Term

When the lubricant enters the bearing i t has no net angular momentum,

but on passing through the bearing clearance i t acquires angular momentum.

A r e p r i n t of t h i s paper can be found i n Appendix (1)

27

This change i n angular momentum requires a torque which i s supplied by

the rotating shaft as explained i n Ref. (13). This i s , therefore, a

res t r a i n i n g torque on the rotor.

For laminar flow the mass averaged tangential v e l o c i t y of the

lubricant ex i t i n g from the bearing clearance i s UJR/^- A point to note

i s that for any asymmetric v e l o c i t y p r o f i l e the mass averaged tangential

v e l o c i t y i s alsooJR/2. As the lubricant enters the bearing clearance

with no net angular momentum, the change i n angular momentum per unit mass

isujR2/2. The lubricant acquires the tangential v e l o c i t y quickly a f t e r

leaving the feed holes, t h i s i s borne out by markings on the shaft caused

by the a i r flow which show the flow pattern. As the tangential v e l o c i t y

i s obtained soon a f t e r entry to the bearing clearance the change i n

angular momentum of the lubricant i s caused by viscous forces i n the

neighbourhood of the supply holes. The provision of angular momentum

for the lubricant i s thus a f r i c t i o n a l e f f e c t near the supply holes.

The torque required to change the angular momentum of the lubricant

i s given by,

T 2 = mtuR 2/2

where m i s the mass flow r a t e . This torque i s the sum of the additional

torques supplied by the rotor and bearing

T 2 = T R + T B

where, TR _ Torque on the rotor

Tg = Torque on the bearing

A hypothesis, which i s borne out by experiments with a i r bearings

having t i l t e d j e t s , reference Marsh (14), i s that the two torques are

approximately i n the same r a t i o as the r e l a t i v e tangential v e l o c i t i e s

between the incoming a i r and the two moving surfaces. With r a d i a l feed

28

holes the incoming a i r has no net tangential v e l o c i t y . Hence,

TB _ 0

Therefore the change i n angular momentum of the lubricant requires an

additional torque supplied by the r o t o r which can be expressed by the

equation,

T 2 = T R = niu)R 2/ 2

This torque i s a f r i c t i o n a l force because although i t i s connected with the

change i n angular momentum of the lubr i c a n t , i t i s caused by viscous forces

i n the neighbourhood of the supply holes. The t o t a l f r i c t i o n a l torque experienced by the roto r i s obtained by

sub s t i t u t i n g i n t o equation (2:1) T t Q t a l = 2 TT R3 L a. (jo + 'mojR2 (2:2)

c " 2

Experimental Support f o r Equation 2:2: To check the v a l i d i t y of equation 2:2, that i s the expression f o r the

t o t a l f r i c t i o n a l torque i n an a i r journal bearing, experiments were carried

out, these have been reported i n reference (13).

The apparatus consisted of a bearing w i t h the following dimensions: R = 25.4 mm L = 66.0 mm c: = 45.2 |xm Ip - 4.237 x 10-4 Kgm2

The bearing was supplied with a i r at 20°C through two rows of 12 p l a i n

j e t s situated at 0.25 L from each end of the bearing.

The experiments consisted of accelerating the roto r up to the speed required i n the stationary bearing and then permitting i t to decelerate f r e e l y under the r e s t r i c t i o n of only the f r i c t i o n a l torques i n the bearing. Data of speed, to, versus time, t , was obtained by using stop watches and a stroboscope. The mass flow rate through the bearing was varied by using d i f f e r e n t supply pressures. The v a r i a t i o n o f the mass flow rate with supply pressure was measured by the weighing method

29

described i n reference (14). A cylinder of compressed a i r together with

the whole apparatus was mounted on a balance and by using a n u l l i n g

technique, the time to consume a known mass of a i r was determined.

The deceleration of the roto r i s character].sed by the time constant

t c . The time constant f o r a given supply pressure v/as determined by

sett i n g a stroboscope to a f i x e d frequency, d r i v i n g the rotor to a

higher frequency and then measuring the time taken to decelerate from

the f i x e d frequency to various f r a c t i o n s of t h i s frequency, the

stroboscope frequency remaining unchanged throughout the deceleration

of the ro t o r .

The equation of motion f o r the decelerating shaft i s ,

Ip do) = - T t o t a l

dt

where, Ip = The polar i n e r t i a of the ro t o r

or s u b s t i t u t i n g f o r T t o t a l

Ip duj + (2 t t R3 L u. + m R~) to = 0 dt (~~ c 2 )

The solution of t h i s equation i s of the form

UJ =UD-O e ~ t / t c

whereUJ 0 i s the i n i t i a l angular v e l o c i t y and tc i s the deceleration time

constant. The reciprocal of the time constant f o r the deceleration i s

given by, 1 / t c = 1 / l p (2 TT R3 L u + mR 2/) (2:3).

( c ^ l )

The res u l t s of the deceleration experiments are sketched i n Fig. (2a)

which i s a p l o t of V-t c versus m. I t can be seen from Fig. (2a) that there

i s close agreement between predicted and experimental res u l t s .

I f Petroff's prediction i s correct then the p l o t i n Fig. (2a) would

be a s t r a i g h t l i n e p a r a l l e l to the mass flow rate axis. However the

n

o

8

THEORV

E*Pt.R\h\£.MT

P £ T R O F F 6

hi s e c

V \oo O rcae. C UO

31

experimental p l o t shows a dependence on the mass flow rate of the lubricant

and hence the angular momentum.

These experiments have shown that the angular momentum torque can

be greater than the torque predicted by the Petroff formula.

Design Considerations f o r Minimum F r i c t i o n

As stated previously the t o t a l f r i c t i o n a l torque i n an externally

pressurised bearing i s given by T t o t a l = 2 R3 Lu-co + mojR2 (2:2)

c" 2

For a bearing with a given supply pressure, gauge pressure r a t i o , radius '

and length, the mass flow rate of the lubricant i s proportional t o the

cube of the clearance. Equation 2:2 can then be w r i t t e n i n the form

T t o t a l = A / c + Be 3

where A and B are constants. The minimum t o t a l f r i c t i o n a l torque i s

obtained when the clearance i s given by,

dT = 0 = ~k/J- + 3 Be 2

dc or

/

For t h i s value_ the t o t a l torque i s given by T t o t a l - A / c (1 + B c f )

( A )

= 4/3 A/ ' c

which may be compared to the Petroff equation

Tl = A / c

From t h i s analysis i t can be seen that i f a bearing i s to be designed

f o r minimum f r i c t i o n then a clearance should be chosen such that 25% of

the t o t a l torque i s required f o r changing the angular momentum of the

lubr i c a n t .

The Turbo-Bearing

I f the supply j e t s i n an externally pressurised bearing are not

r a d i a l , but are t i l t e d the incoming a i r w i l l enter the bearing with high

angular momentum. Instead of requiring angular momentum from the shaft

causing a f r i c t i o n a l torque, the incoming a i r w i l l a ctually drive the

shaft. . This e f f e c t was investigated f u l l y by the author and

Professor H. Marsh * (15).

Considering the Turbo-Bearing, gas enters the bearing clearance

wi t h high angular momentum and leaves the two ends of the bearing w i t h

low angular momentum. This change i n angular momentum requires a torque

applied to the lub r i c a n t , t h i s being provided by the r o t o r and bearing.

There i s , therefore, a d r i v i n g torque on the r o t o r and the roto r

accelerates u n t i l the d r i v i n g torque i s equal t o the f r i c t i o n a l torque

predicted from the Pe t r o f f analysis. The angular momentum torque i n

t h i s instance i s not a f r i c t i o n a l torque, but a d r i v i n g torque.

A r e p r i n t of t h i s paper can be found i n Appendix (2)

33 CHAPTER 3

LIQUID BEARINGS

The performance of a hydrodynamic journal bearing can be predicted

by solving Reynolds equation f o r the pressure d i s t r i b u t i o n . The method

of solution i s to f i r s t s i mplify the equation and then apply a set of

boundary conditions. I n t h i s way the form of the pressure d i s t r i b u t i o n

i n the bearing can be obtained and hence the load carrying capacity.

Sommerfeld considered a bearing of i n f i n i t e length with no a x i a l

flow. Reynolds equation i n t h i s case corresponds t o ,

dp_ = 6 u. UR ( 1 + h * ) d9 c2 ( ( 1 + e c o s e ) 2 c ( l + ecosB j 3 )

where, U = tangential surface speed of the bearing

6 = angular displacement from the l i n e of centres

p = pressure

h* = f i l m thickness at point of maximum pressure

The boundary conditions used were that the pressure i s equal to zero at

both 0 = rr and 2 IT . Sommerfeld used the above assumption and boundary

conditions w i t h the aid of a s u b s t i t u t i o n known as the Sommerfeid

s u b s t i t u t i o n t o solve the Reynolds equation and thus obtain an expression

f o r the pressure d i s t r i b u t i o n .

by,

The r a d i a l v e l o c i t y gradient i n a journal bearing can be expressed

du = U - 1 ( h _ 2 z ) dp_ dz h 2 p. dx ( 3 : 1 )

where, du = r a d i a l v e l o c i t y gradient of the lubricant dz

h = f i l m thickness

z = coordinate across the f i l m

x = tangential coordinate

34

The f i r s t term arises from the shearing of the lubricant and the second

from a pressure induced flow caused by the v a r i a t i o n i n pressure around

the bearing.

To evaluate the f r i c t i o n a l forces i n a journal bearing, the d e f i n i t i o n

of Newtonian v i s c o s i t y i s used,

i . e . c = n du ^ dz (3:2)

where,C = shear stress

At the shaft surface z = 0, therefore, using equation (3:1),

du = U - h dp (3:3)

dz h 2 |x dx

At the bearing surface z = h, again using (3:1),

'du = U + h dp_ (3:4)

dz h 2 p. dx

Combining equations (3:3) and (3:4) and s u b s t i t u t i n g i n t o equation

(3:2) gives,

C h rt =,, du = + h dp + u U (3:5) h , 0 = ± h + u U

' dz 2 dx " h

The t o t a l f r i c t i o n a l forces on the bearing and shaft are given by,

F = L [ 2 i r R C dx dy (3:6) J o J o

Substituting the expression f o r the shear stress from equation (3:5)

i n t o (3:6) and int e g r a t i n g using the Sommerfeld integ r a t i o n s u b s t i t u t i o n

mentioned previously gives, F

h j 0 = ± c 6 W sin Oi + 2 T T U U R L ; , • (3:7) 2R c (1 - S 2 )

where, w = bearing load

= a t t i t u d e angle

which i s the t o t a l f r i c t i o n a l force on the bearing and shaft. The positive

sign i s f o r the shaft and the negative f o r the bearing.

35

I t can be seen from equation (3:7) that the f r a c t i o n a l forces on

the shaft and bearing d i f f e r by the amount c £ W s i n /R. Equilibrjum

of the o i l f i l m i s maintained by a couple which arises because of the

displacement of the centres of the shaft and bearing with respect to

each other. The integrated o i l forces on the shaft and bearing act

through t h e i r respective centres, which are, i n the d i r e c t i o n normal to

the load, a distance c e sin apart. The couple set up because of

the displacement i s of magnitude W c g. s i n dj , which corresponds to

a f r i c t i o n a l force of (W c 6 sin ^ )/R a t the surface of the journal.

Considering the moments f o r the f i l m ,

R F Q = R F N + Wc £ s i n (3:8)

From equation (3:8) i t can be seen by s u b s t i t u t i n g values f o r FQ and

Fh from equation (3:7) that equilibrium of the f i l m i s maintained.

For a short bearing that i s where L/D ^. 1 the i n f i n i t e bearing solution

i s not very s a t i s f a c t o r y . When solving the Reynolds equation f o r an

i n f i n i t e bearing, a x i a l flow i s neglected and only circumferential flow

i s considered. This i s unacceptable with a short bearing i n which

lubricant i s continually being supplied and from which lubricant flows

at each end. A method of solving the Reynolds equation f o r a short

bearing was formulated by Ocvirk and DuBois. The Reynolds equation i s

f i r s t s i m p l i f i e d by assuming that the pressure gradients around the

circumference are very small compared w i t h those along the length. The

s i m p l i f i e d equation i s ,

b_ (h 3^p_) = 6 u U 6h by ( dy) « <JQ

By inte g r a t i n g twice and applying the boundary condition that p = 0 at

y = - L where y i s measured from the centreline of the bearing, an 2,

expression can be obtained f o r the pressure d i s t r i b u t i o n and hence the

load carrying capacity.

36

From the Ocvirk considerations, the f r a c t i o n a l forces on the shaft

and bearing are,

F h „ = 2 T T R 2 LVXOJ c ( i - e ^ ) X

so that these forces are equal i n magnitude, unlike those obtained from

the i n f i n i t e bearing theory. This i s because the pressure induced shear

flow i s considered to be ne g l i g i b l e .

I t i s found that the Ocvirk solution i s generally more acceptable

to 'real' bearings than the i n f i n i t e bearing solution. The Ocvirk solution

gives good predictions of load carrying capacity when compared with

experimental results f o r moderate e c c e n t r i c i t y r a t i o s , i . e .

(£ < 0.6)

For an i n f i n i t e bearing the t o t a l f r i c t i o n a l force on the shaft

i s from equation (3:7)

F 0 = c & W sin KSJ + 2 i r a U R L 2R c (1 - &2) y2

The magnitude of the f i r s t term, the pressure flow term, f o r a l i g h t l y

loaded bearing i s very small compared to the second term, the shearing

term. For the bearings used i n the experimental work reported i n t h i s

thesis the maximum value of the f i r s t term i s approximately 3% of the

t o t a l f r i c t i o n a l force. As discussed, f o r a bearing of f i n i t e length,

because of large a x i a l pressure gradients the pressure induced

circumferential flow i s n e g l i g i b l e . Therefore the value f o r a f i n i t e

bearing i s l i k e l y to be less than the 3% f o r an i n f i n i t e bearing.

F r i c t i o n i n Liquid Bearings:

From the Ocvirk considerations, the f r i c t i o n a l torque i n a journal

bearing operating i n the laminar regime i s , f o r both bearing and shaft,

T = 2 TT R 3 L_ X_LO c (1 - £ 2) %

37

From t h i s equation i t can be seen that no account has been taken of the

extra f r i c t i o n a l torque experienced by the journal bearing i n connection

with the spinning of the lubricant.

I n the previous chapter mention was made of the extra f r i c t i o n a l

torque experienced by an a i r bearing caused by giving the lubricant flow

angular momentum. The magnitude of t h i s torque i s m R cu

2

The nature of t h i s extra torque has not been reported before f o r

l i q u i d bearings.

For a l i q u i d bearing the s i t u a t i o n i s a l i t t l e more complex than

f o r an a i r bearing, because of the nature of the lubricant. With an

a i r bearing the a i r on reaching the ends of the bearing, escapes f r e e l y

i n t o the atmosphere whereas with a l i q u i d bearing the l i q u i d has to be

dribbled over the bearing end face or thrown o f f the shaft. Under normal

operating speeds the l i q u i d i s thrown o f f the shaft w i t h the surface

v e l o c i t y of the shaft. With an a i r bearing the lubricant at e x i t has

an average circumferential v e l o c i t y oftoR/2, but with a l i q u i d bearing,

a l l of the l i q u i d on leaving the bearing and being thrown by the shaft

gains the shaft v e l o c i t y , to R. The extra f r i c t i o n a l torque i s then 2

mcuR, that i s twice the magnitude of the torque found with an a i r

bearing.

The s i t u a t i o n with a l i q u i d bearing can be fu r t h e r complicated by

v a r i a t i o n of shaft diameter near the e x i t from the bearing. I f the

diameter i s greater at t h i s point than i n the bearing the l i q u i d can

be thrown from the larger diameter, thus making the extra torque greater

i n magnitude. The extra f r i c t i o n a l torque then being, mwR (thrown).

From the Ocvirk theory i t can be shown that the mass flow rate from

38

the ends of the bearing i s given by,

m = ^ R L c e c o

The t o t a l f r i c t i o n a l torque experienced by the shaft assuming that the

lubricant i s thrown from the shaft radius and that the bearing i s running

f u l l i s , T t Q t a l = 2TT R3 W c ^ + L c £ R 3 W 2

c (1 - £ 2) y2

The f r i c t i o n a l torque experienced by the bearing depends on whether

or not the lubricant when thrown o f f the shaft i s caught by 'catchers'

attached to the bearing. I f they are not attached t o the bearing the

f r i c t i o n a l torque on the bearing i s only the f i r s t term of the above

equation. I f attached then the second term i s added.

For c l a r i t y , the modification of the shear torque because of

eccentric running w i l l be neglected i n the following consideration. I n 3

f a c t f o r small e c c e n t r i c i t i e s the Petroff f r i c t i o n a l torque, 2 -rr R L |AUJ c

i s a good approximation. For the purpose of comparison the t o t a l

f r i c t i o n a l torque can conveniently be expressed i n a non-dimensional

form.

For small e c c e n t r i c i t i e s , 3 3 ?

T t o t a l = 2 T R L ^ + ^ L c £ R UJ" c transforming,

T t o t a l = 2 y. + c £ co

TT R3 L OJ C ? ^

d i v i d i n g through byojR,

T t o t a l = 2 V- + c £ TTR 4Loo 2^ c e u j R R i r

s u b s t i t u t i n g the r o t a t i o n a l Reynolds number Re which equals Rojpc T t o t a l = 2 . c £ ?4 r 2

= Cf -rrR^ L UJ ^ Re R-rr

39

The l e f t hand side of the expression i s a non-dimensional f r i c t i o n a l

torque which i s equated to Cf. I f the correction f o r the throwing of

the lubricant i s neglected the value of Cf reduces to 2/Re.

The magnitude of the second term, that from the throwing of the

lubric a n t , can be assessed from Fig. 3(a). This graph was p l o t t e d by-

considering a bearing w i t h a clearance r a t i o of 0.003 running with

various e c c e n t r i c i t i e s . Using equation (3:9) p l o t s of the non-dimensional

f r i c t i o n a l torque versus the Reynolds number can be obtained f o r a given

e c c e n t r i c i t y and these are sketched i n Fig. 3(a). Fig. 3(a) indicates

that the angular momentum e f f e c t i s pronounced at high Reynolds numbers

and becomes less s i g n i f i c a n t at low Reynolds numbers. Typically at

a Reynolds number of 2000 wi t h an e c c e n t r i c i t y of 0.3, 23% of the t o t a l

f r i c t i o n a l torque can be a t t r i b u t e d to the throwing of the lubricant.

The magnitude and nature of the angular momentum torque can lead

to errors i n the torque measuring technique f o r detecting the onset of

Taylor vortices i n a bearing. The onset of vortices i s indicated by

the departure of the measured f r i c t i o n a l torque from that predicted by

Petroff, indicated by the p l o t of Cf = 2/Re i n Fig. 3(a). The departure

of the f r i c t i o n a l torque from that predicted f o r laminar flow, which i s

i n f a c t a t t r i b u t a b l e to an extra torque from the throwing of the lubricant,

can be wrongly interpreted as the onset o f vor t i c e s . Corrections f o r

t h i s angular momentum torque should therefore be made before the true

onset can be determined.

As discussed i n Chapter 1 a bearing running with a large Reynolds

number (> 2000 ) w i l l usually have a turbulent l u b r i c a t i n g f i l m . The

f r i c t i o n a l torque f o r a turbulent f i l m i s greater than that f o r a

laminar f i l m , so the angular momentum torque w i l l be less s i g n i f i c a n t

i n t h i s case. A point t o note i s that irrespective of the turbulent

40

10

6

v -•

10 0-"8 0- (o z

C r = 2. 0-3 R i r 6>

o- \ 1000 K

10 li i i 1 1 I I «—!—I 1 1

* 5 b 7 ? 9 I 5 la 7 H 1 I 0 »0

Re,

41

v e l o c i t i e s present i n a bearing, the lubricant i s s t i l l thrown from the

shaft w i t h the shaft v e l o c i t y .

The suppression of turbulence and the running of a laminar f i l m

bearing at high Reynolds numbers with the corresponding reduction i n

f r i c t i o n a l torque has been attained by using additives i n the lubricant.

This e f f e c t was investigated at Durham by Hampson (16) who used

high molecular weight polymer additives i n a water lubricated bearing.

Hampson did i n fact correct h i s measured f r i c t i o n a l torque f o r the

momentum torque, the importance of which had already been i d e n t i f i e d

i n the work on a i r bearings.

Solutions of Reynolds equation predict a region of negative pressure

i n a journal bearing and as a l i q u i d cannot maintain negative pressures,

the l u b r i c a t i n g f i l m cavitates. I t has been assumed so f a r that a

bearing runs w i t h a f u l l f i l m and that c a v i t a t i o n i s not present. I f

a l u b r i c a t i n g f i l m cavitates, the area over which the lubricant i s

sheared i s reduced w i t h a corresponding reduction i n f r i c t i o n a l torque.

Without the use of special bearings the extent of c a v i t a t i o n i n a

bearing cannot be easily investigated.

Cavitation can be suppressed by using a pressurised supply of

lubricant thus producing a bearing running w i t h a complete f i l m .

Pressurised supplies are commonly used i n bearings so that spent

lubricant can be replaced and also to keep the lubricant from

overheating.

I n the experiments described i n t h i s thesis a pressurised supply

was used to si m p l i f y the experiments by suppressing c a v i t a t i o n .

Another advantage of a pressurised supply i n these experiments i s

that the flow rate can be varied at w i l l by varying the supply

pressure thus permitting useful information to be obtained i n

connection w i t h the angular momentum torque.

CHAPTER 4

DESIGN OF THE 2" DIAMETER BEARiNG'RIG

The experiments were spread over two bearing r i g s , the f i r s t r i g

being a prototype to te s t ideas and to provide low Reynolds number

f r i c t i o n a l torque results and a second r i g to provide results of the

f r i c t i o n a l torque of a bearing operating i n both laminar and turbulent

regimes.

Details of the Prototype Design

Three main problems existed with the design of the prototype r i g ,

i ) To have s u f f i c i e n t polar i n e r t i a i n the decelerating shaft so that

i t would slow down at a reasonable rate to permit measurements to

be obtained.

i i ) To have an easily detachable drive system so that the shaft could

be accelerated up to the required speeds and then allowed to

decelerate f r e e l y .

i i i ) To have a system with a single bearing which could operate at

reasonably high speeds without becoming unstable i n a conical mode

A l l three of these problems were overcome by using discs attached t

the decelerating shaft.

The layout of the prototype r i g can be seen i n Fig. (4a) and

photograph Fig. (4b).

With respect to problem ( i ) , the d i f f i c u l t y of obtaining a

s u f f i c i e n t l y large deceleration time constant to permit experimental

re s u l t s to be taken was overcome by the following means. F i r s t l y , by

the use of discs as mentioned above, which provide the polar i n e r t i a so

that the shaft decelerates slowly. Secondly, by the choice of a low

vi s c o s i t y lubricant. Water was chosen as the lubricant because of i t s

0) 44 —d —6

ft)

10 a 3 M 3

ffl m 7T\ 4J cT7

c CO

ITSl

a)

s-

0^

X

XX>XSX^X^XX •« CO a)

a v a i l a b i l i t y , i t s ease of handling and i t s r e l a t i v e l y low vi s c o s i t y .

The choice of water as a lubricant has other advantages over the use

of o i l s etc. Water has a higher specific heat capacity than o i l s and

also a lower v a r i a t i o n of viscosity with temperature. These two facts

combined with the low v i s c o s i t y of water mean that the temperature r i s e

i n the bearing i s kept to a minimum. Therefore the accuracy w i t h which

the v i s c o s i t y of the K*W«wA i s known i s o f a high order. The choice of

water as a lubricant made i t necessary f o r the bearing and shaft members

to be made from materials which did not rust. The shaft was made from

stainless steel and the bearing from phosphor-bronze.

The second d i f f i c u l t y , that of an e a s i l y detachable drive, was

overGome by the use of d i s c s large enough t o enable an a i r turbine system

to be used f o r the drive mechanism. A l l that was needed to be done to

disconnect the drive from the shaft, was t o switch o f f the a i r supply t o

the turbine. Because the a i r turbine operated on a disc of large radius

i t was possible to obtain a large drive torque to accelerate the shaft

without excessive consumption of compressed a i r .

The t h i r d d i f f i c u l t y , that o f operation of the bearing at high speed

was overcome i n the following manner. Single bearing systems are

p a r t i c u l a r l y susceptible to self-excited conical i n s t a b i l i t i e s . These

tend to occur at low speeds, hence severely r e s t r i c t i n g the top speed

at which a shaft can be driven. Whirl onset speed depends on bearing

design, r o t o r loading and i n e r t i a . Bearings are also susceptible to a

second i n s t a b i l i t y of the t r a n s l a t i o n a l kind, the onset speed of which

i s usually higher than the conical w h i r l speed. Both i n s t a b i l i t y modes

tend t o occur at a frequency close to h a l f the r o t a t i o n a l speed. The

onset of conical w h i r l can be estimated by the equation,

. N onset = 60 (2 k c 1 2 2 ) %

conical ™ ( ( l T - 2Ip))

47

The onset of t r a n s l a t i o n a l w h i r l can be estimated by the equation,

N onset = 60 (\<h)^-t r a n s l a t i o n a l (WL)

where, ^ onset - speed of onset of i n s t a b i l i t y

kj^ = bearing l i n e a r s t i f f n e s s

k c = bearing conical s t i f f n e s s

M = mass of the ro t o r

12 = length of the bearing L/2

Ip - polar i n e r t i a of the ro t o r

Irp = transverse i n e r t i a of the ro t o r

I n the prediction f o r onset of conical w h i r l , the 2 Ip term i n

the denominator i s a correction f o r the gyroscopic e f f e c t when the

shaft i s performing conical vibrations w h i l s t r o t a t i n g . From the

prediction, i f 1^ approaches 2 Ip then the conical w h i r l speed can

be raised above that of the onset f o r t r a n s l a t i o n a l w h i r l . This being

the case the shaft system i s gyroscopically stable and cannot perform

conical vibrations at one h a l f of the r o t a t i o n a l speed. The use of

discs also overcame t h i s problem by designing them such th a t I T < 2 Ip.

The design of the bearing r i g i s therefore a series of compromises,

i . e . having discs large enough to provide the necessary i n e r t i a and

bucket radius f o r the turbine drive and positioned such that I T < 2 i p .

Further Design Considerations

The supply of water to the bearing was at a pressure which could be

varied i n order to suppress c a v i t a t i o n . The bearing would then run f u l l

and allowances would not have to be made f o r the extent of the ca v i t a t i o n

region.

The bearing was designed so that the bush could be turned thus permitting the use of various l u b r i c a t i o n feed directions r e l a t i v e to the d i r e c t i o n of

loading.

48

The drainage of water from the bearing was made easier by discarding

the use of thrust bearings and instead using carbon points on the ends

of the shaft. The carbon points ran against brass pads to provide a x i a l

location. The shaft extended out of each end of the bearing so that

water exhausting from the bearing could escape easily, be collected i n

catchers and drained under gravity away from the system as i l l u s t r a t e d

i n Fig. (4a). The discarding of thrust bearings also meant that the

measured f r i c t i o n a l torque of the bearing was purely journal f r i c t i o n a l

torque.

Three clearance r a t i o s were used i n the prototype bearing to permit

the control of operational e c c e n t r i c i t i e s and to permit a wide v a r i a t i o n

of r o t a t i o n a l Reynolds numbers and mass flow rates.

Bearing Parameters

R = 25.4 mm

L = 50.8 mm

ra d i a l clearances 1. 2.62 10" -5 m

2. 5.92 10" -5 m

3. 7.90 10' -5 m

Disc Diameter = 176 mm

Ip = 1.38 10~ 2 Kg m2

1 2 .3 Surface Finish of Shaft = o.25 micron 0.25 micron 0.25 micron

C.L.A. C.L.A. C.L.A.

Surface Finish of Bush = 0.35 micron C.L.A.

49

INSTRUMENTATION

I n Chapter 2 mention was made of the deceleration experiments carried

out w i t h a i r bearings. The equation of motion f o r the decelerating shaft

can be expressed as,

Ip dy = - T t Q t a l

dt

from equation (2:2),

IP duj = - 2 i r R3 Lpuo - m u j R2

d t c r 2

The solution of t h i s equation i s of the simple form,

lo = w 0 e - t / t c

The timing of the decelerations i n t h i s case was easily measured using

stop watches.

I f the equation of motion o f the shaft i n the sel f - a c t i n g l i q u i d

bearing i s now considered,

again, Ip du> = - T t o t a l (5:1) dt

With the present experimental investigation, the t o t a l torque experienced

by the shaft i s made up of the f r i c t i o n a l torque from the bearing

and an extra torque from the windage losses from the discs attached to

the shaft. When the shaft i s decelerating i n the laminar regime,

t o t a l ~~ n — v ' + mass flow torque + disc drag torque c U l - £ 2 ) / 2 )

When the shaft i s decelerating i n the non-laminar regime,

T t o t a l = T r o t o r non-laminar + m s s f l o w t o r c l u e + d i s c d r a § t o r ^ u e

I f the respective T t o t a l s substituted i n t o equation (5:1), i t can

be seen that the solutions of these equations are not of a simple form but

of a complex nature. This then makes i t necessary to obtain point

deceleration rates at known speeds. To obtain the point deceleration rates accurately electronic timing equipment was used.

T I M E R / C O U N T E R T I M E R , / C O U N T E R

S T O P P U L S E .

S T k R T P U U S E .

C\£Cv)\T.

7 . H S T O P

I N P U T P U L S E S

M ^ G N C T x C PRO 6 E .

Vi**"**^^ c.ircmV.

51

The Timing Equipment In B r i e f

The idea of the electronic deceleration c i r c u i t was to measure the

change i n frequency of e l e c t r i c pulses being emitted from a magnetic

transducer, positioned close to the machined buckets on the discs. The

deceleration can be used to determine the torque on the rotor. Hence the

fr a c t i o n a l torque which appears i n the bearing can be i d e n t i f i e d .

With reference to the sketch, Fig. (5a), pulses from the magnetic

probe are passed i n t o the Control/Divide Unit. When the s t a r t button i s

pressed the Timer/Counters are started simultaneously by a pulse. The

Control/Divide Unit divides the following incoming pulses i n the r a t i o

of 2:1. Timer/Counter 1 i s stopped a f t e r N pulses and Timer/Counter 2

i s stopped a f t e r 2N pulses.

In t e r p r e t a t i o n of the Measurements

From the description of the c i r c u i t operation, the time recorded

f o r 2N pulses i s greater than twice the time recorded f o r N pulses,

because the shaft i s decelerating.

Consider the Sketch

TIMER COUNTERS START t = 0

period t2 TIMER 2 STOP

(N) (N)

to m

Period t± TIMER 1 STOP

52

where, c< = deceleration rate (assumed .constant f o r small change i n w)

ou s = i n i t i a l angular v e l o c i t y

u)f = f i n a l angular v e l o c i t y

U ) M = angular v e l o c i t y a f t e r time t ^

N = number of pulses

9 = angle of r o t a t i o n f o r N pulses

h = t 2 - 2 t x

6 = uostx - % cxt-L2 (5:2)

2 0 = Go st 2 - X * V 2 — * t , 2 equating, w

s t 2 _ % = 2 t ^ t 1 - ^ ^

but, co m = cos ~ o< t i (5:3) 2 p

su b s t i t u t i n g , (ufo + ot t i ) t 2 - X t 2 = 2(w m + a t i ) t x - «t t

2 5 sim p l i f y i n g , c o m t 2 - 2u^ti = o t t i + c* t 2

w - o(. t i t 2

s u b s t i t u t i n g , 6 - t 2 - 2t - j _ ,

° = (5:4) ( t x

2 + *>tx + % h 2) and as ^ i s small compared to t-^

t l 2

From equations, (5:2), (5:3), (5:4)

© = (ufo + c* t i ) t i - ^ cxtx 2

= ^ m t l + c < t l 2

transforming and s u b s t i t u t i n g f o r ,

^m = e t i + ^ ( ^ 2 )

( t i 2 + o t l + 6 2/ 2 }

again f o r small ^ c j A 0 m "5* -

t x + j£d

53

The average uo over the period t 9 = 2 6 = 6

Summarising, c*. = GJ m ^ "2 t 1

t 2

Although these expressions are only approximations they greatly aid the

arithmetic. The maximum error introduced by using these approximations

has been calculated to be at worst less than 0.2%

By determining t-^ and t 9 the point speeds and deceleration rates can

be found for 1 a given deceleration of a shaft i n a bearing. From the

equation,

Ip cko = ( F r i c t i o n a l Torque) + (Angular Momentum Torque) + (Disc Torque) dt

i f dto i s determined and the mass flow rate and Disc Torques are known, then dt

the f r i c t i o n a l torque i n the bearing can be calculated f o r a given speed of

r o t a t i o n .

C i r c u i t Operation i n D e t a i l , Reference Fig. (5b)

I n the i n i t i a l state, that i s without the time s t a r t switch operated,

pulses from the magnetic probe are fed i n t o the Schmitt Trigger (2) and are

shaped i n t o logic square wave pulses. The t r i g g e r also acts as a f i l t e r

i n that by having a signal pass l e v e l , the noise from the probe i s f i l t e r e d

out and not shaped as random pulses.

The pulses are passed onto components (1) and (3) and do not proceed

fu r t h e r i n the c i r c u i t u n t i l the s t a r t switch i s i n an operated state. The

s t a r t c i r c u i t Coo^vsVs of two components, that i s aNand gate (1) and

a J-K f l i p f l o p ( 3 ) . When the s t a r t switch i s operated the f i r s t pulse

received by component (1) i s not permitted to enter the d i v i s i o n part of the

c i r c u i t , but the second and subsequent pulses are. The reason f o r t h i s i s

54 T I M E R

C O U N T E R ^ ) S T O P

T t M £ T 2 . C O U M T t R CO

S T O P

SN7492N Divide-by-12 cevnttr

lhui* Mint r S W O T

nn fin nn rTi! IWI rrir T I

_ 5 _ ! 5

5 S N 7 4 1 3 H SN7400N

Quadruple 2-input fiAND gite Dual 4-Input riAND gate sKt Schmitt Trigg

Srf74SG?l

10 :j 9 jj 11 rcinriilpinriMrTi re?-© B E H "

1 ; ; 2 |" , J : ; 4 , j 5 i ] ( ; j W j

S T A R T puv.se.

SN7470N SN7413N J-K 'lip-flop Dui I 4-ln lut NAND pate

Set milt trigger

SN740CN Quldrjptt 2-input HAND j j te

Vcc "=T S W I T C H .

= 1 — t - J J

GND.

F i ^ . C S b " ^ C i r c u ' . V 0

55

to ensure that the d i v i s i o n components are started at the leading edge

of the second pulse and t h i s excludes the p o s s i b i l i t y of s t a r t i n g i n the

middle of the f i r s t pulse.

The pulses are then passed on to components ( 4 ) , (5) and ( 6 ) , the

purpose of these chips i s to s t a r t the Timer/Counters and to pass on pulses

to the d i v i s i o n chips. Component (4) ensures with component (6) that the

counters receive a simultaneous s t a r t pulse when required and also that the

s t a r t pulse does not registe r w i t h the following d i v i s i o n components as a

period pulse. Component (5) i s used as a c i r c u i t lock so that the Timer/

Counters cannot be restarted before the c i r c u i t reset cycle i s put in t o

operation, thus permitting w r i t t e n recordings of deceleration time periods

to be made.

The pulses following the s t a r t pulse are fed i n t o component (9) which

divides t h e i r number by ten, and then passes a pulse to component (8)

which again divides by ten. The output pulse from (8) i s s p l i t i n t o two

parts, the f i r s t stops the f i r s t Timer/Counter and the other i s fed into-

component (7) which divides the input by 2 and then passes a pulse to

stop the second timer counter. I t can be seen from the above explanation

that the Timer/Counters would then display the relevant times f o r 100 and

200 pulses.

The p a r t i c u l a r c i r c u i t described measures the times f o r 100 and 200

pulses but the number of pulses timed can be easily varied by adding or

subtracting additional d i v i s i o n chips, to or from the c i r c u i t . At the

'outset of a series of experimental runs the number of pulses to be timed

was selected and the c i r c u i t altered accordingly. The method of selection

was based on having enough pulses to obtain measurable times, t 1 and t 2

and a measurable difference, t 2 - 2t]_. Also the difference had not to be

56

so large as to invalidate the assumption that the deceleration rate i s

constant over the time period t 2 .

The c i r c u i t can be quickly reset with the aid of the s t a r t and reset

switches and made ready f o r f u r t h e r deceleration timings.

Comments on C i r c u i t Operation

The c i r c u i t was found t o be very accurate i n operation, so much so

that a f t e r test runs i t was found necessary to s t a r t and stop the Timer/

Counters on pulses from the same machined bucket on the disc. This was

necessary because i f s t a r t i n g and stopping were done with d i f f e r e n t buckets,

the c i r c u i t easily showed up the machining error of the buckets.

Position Checks of the Shaft and Bearing

The e c c e n t r i c i t y of the shaft and i t s mode of operation were continually

monitored using capacitance transducers i n conjunction with Wayne-Kerr

Distance/Vibration meters and an oscilloscope. The capacitance probes were

positioned near the shaft on the opposite sides of the discs to the bearing.

The reason f o r t h i s being t o prevent water from the bearing i n t e r f e r i n g with

the operation of the probes. Small droplets of water which may have f a i l e d

to have been contained by the catchers were prevented from reaching the

probes by the throwing action of the discs.

57 CHAPTER 6

EXPERIMENTS WITH THE 2" DIAMETER RIG

The experimental work with t h i s r i g consisted of d r i v i n g the shaft up

to a predetermined speed and then l e t t i n g i t decelerate f r e e l y . The

decelerations were measured accurately with the aid of the instrumentation

described i n Chapter 5 and the f r i c t i o n a l torque acting on the rotor could

then be calculated.

The equation of motion of the decelerating shaft i s ,

Ip dy = - T t o t a l

dt

w h e r e ' T t o t a l = T r o t o r + T d i s c d r a g

The form of T total depends on the regime i n which the bearing i s operating,

i . e . laminar, t r a n s i t i o n or turbulent.

I t i s f i r s t necessary to determine the disc drag torque before the

decelerations can be interpreted to y i e l d information about the bearing

torque.

Disc Drag Torque Determinations

The method by which t h i s was determined was as follows. The shaft and

discs were removed from the water bearing and assembled i n an externally

pressurised a i r bearing. The drag torque was determined by the deceleration

technique.

I f we consider the equation of motion of the assembly using the a i r

bearing we have ,

Ip , . i du> = - T' t o t a l — t o t a l

Using the a i r bearing the f r i c t i o n a l torque T t o t a l i s rna<ie UP o f '^wo

components, f i r s t l y the drag on the discs and secondly the f r i c t i o n a l torque

i n the a i r bearing. Hence,

IP t o t a l ^ 7 = " T discs ~ T a i r bearing ^ 6 : 1 )

From Chapter 2, T a i r bearing = 2 TT R3 L |A m + m u R2

c 2

= o j say

I f we consider equation (6:1), the drag on the discs determined by t h i s

method has to be corrected by a small amount because of the f r i c t i o n a l

torque from the a i r bearing. Because of the low v i s c o s i t y of a i r , the

correction f o r the f r i c t i o n a l torque from the a i r bearing i s very small.

The f r i c t i o n a l torque from the a i r bearing was also determined by the

deceleration technique. The shaft w i t h the discs removed was decelerated.

i n the a i r bearing and the decelerations were timed using stop watches and

stroboscope. The electronic method was not used because of the simple

nature o f the deceleration and also because of the low v i s c o s i t y of a i r

which provided a s u f f i c i e n t l y large time constant such that timing

measurements could be easily taken during the deceleration of the roto r .

Also the shaft i t s e l f did not have any buckets machined i n i t , i t would

not have been easy to obtain pulses from i t to t r i g g e r the electronic

c i r c u i t r y . A possible solution would have been to use a photo e l e c t r i c

pick-up to produce pulses, but one of these was not available. The

equation of motion f o r the shaft alone decelerated i n the a i r bearing i s

su b s t i t u t i n g ,

JP shaft ^ = - k l w

From Chapter 2,

!P shaft - k

t e

59

The value of tc was found experimentally as described above and hence the

value of k- . The necessary information to correct the disc drag torque

results was then available.

The deceleration of the assembly, shaft plus discs, was carried out.

using the electronic method. The reason why the electronic method was

used being the complex nature of the torque from the discs. From the

results obtained i t was possible to p l o t the disc drag torque as a function

of speed which could be used to i n t e r p r e t the decelerations of the shaft

plus discs i n the l i q u i d bearing. The complex nature of the torque i s

borne out by the change i n gradient of the experimental p l o t Fig. (6b),

at about 2300 r.p.m., thus confirming the need f o r electronic timing of

the decelerations.

The a x i a l location of the shaft was achieved with the aid of carbon

points on the end of the shaft. The points ran on brass pads which were

f i x e d to the base pl a t e . The shaft was not nipped between the pads, but

was permitted a few microns end f l o a t . The f r i c t i o n which appears on the

shaft because of the rubbing o f the carbon bushes on the brass pads,

although small, i s corrected f o r i n the disc drag torque measuring

determination because t h i s appears as an extra component i n the measured

torque.

To i n t e r p r e t the bearing f r i c t i o n a l torque f u l l y , i t was also necessary

to obtain information about e c c e n t r i c i t y , mass flow rates, v i s c o s i t y and

polar i n e r t i a of the r o t a t i n g shaft.

Mass Flow Rate Determination

The mass flow rate was determined by using a glass beaker to c o l l e c t

the lubricant and a stop watch. The beaker weight, being known before i t was

placed i n the bearing o u t l e t flow and the stop watch started, was subtracted

60

from the f i n a l weight of the beaker plus water and hence a flow rate was

determined. A p l o t of the speed of r o t a t i o n versus the mass flow rate f o r

each of the lubricant supply pressures was obtained.

Eccentricity Determination

Plots of ec c e n t r i c i t y versus speed were obtained by using capacitance

probes i n conjunction with Wayne Kerr distance v i b r a t i o n meters. The probes

were f i r s t 'zeroed' by moving the shaft around the bearing clearance so as

to establish i t s e f f e c t i v e clearance c i r c l e , as i s the usual manner. I t

was then easy to determine the eccentric running conditions of the shaft

at various speeds. The measured displacements were f i r s t corrected f o r

shaft curvature before being transposed i n t o e c c e n t r i c i t y . The correction

curves were supplied by Wayne Kerr.

Viscosity of the Lubricant

The lubricant used i n a l l of the experiments reported i n t h i s thesis

was d i s t i l l e d water. The d i s t i l l e d water was obtained from a laboratory

s t i l l . The v i s c o s i t y of the d i s t i l l e d water, which can vary w i t h degree

of p u r i t y , was checked using a laboratory viscometer. The determination

of the v i s c o s i t y versus temperature data i s described i n Appendix ( 3 ) .

The experimentally obtained v i s c o s i t y was used throughout t h i s reported

research.

Polar I n e r t i a

Knowledge of the polar i n e r t i a of the shaft plus discs i s required

so as to rel a t e the t o t a l torque and deceleration rates of t h i s r o t a t i n g

mass. The r o t a t i n g discs and shaft are the energy store which must be

dissipated before the shaft w i l l come to res t . The rate of dissipation

i s a r e f l e c t i o n of the f r i c t i o n a l torque appearing on the r o t a t i n g mass.

Unfortunately, the equipment available f o r use to determine the polar

i n e r t i a experimentally was reported to be very inaccurate. I t was decided

61

that a more accurate method of obtaining the polar i n e r t i a of the

r o t a t i n g members was to measure each member accurately and with the

use of an experimentally determined value f o r density, to calculate a

value f o r polar i n e r t i a . The density was determined by making a sample

block of the materials used and measuring i t to obtain the volume and

then weighing i t to obtain the mass.

Total Torque Measurement

The decelerations with the shaft i n the l i q u i d bearing were carried

out i n the following manner. The shaft was driven up to a predetermined

speed and then the a i r drive was t o t a l l y disconnected, by removing the

supply pipe from the drive system. The shaft was then permitted to

decelerate f r e e l y . Point deceleration rates were obtained, with the

electronic instrumentation previously described w h i l s t the shaft was

decelerating.

With these results and a knowledge of the polar i n e r t i a i t was then

possible to determine the t o t a l torque by calculation.

Experimental Procedure Comments

Before each deceleration of the shaft was timed a 'dummy' signal was

fed i n t o the electronic c i r c u i t from a laboratory o s c i l l a t o r to check

whether the c i r c u i t was d i v i d i n g exactly i n the r a t i o of 2:1. This was

done quickly several times before each deceleration. Agreement of the

two s i x f i g u r e displays of the counters was, when the c i r c u i t was operating

s a t i s f a c t o r i l y , to the l a s t d i g i t . I n b u i l t i n t o the c i r c u i t was an

important feature which ensured that i t s divide sequence was not started

i n the middle of a pulse and that the f i r s t pulse only started the two

timer counters simultaneously and did not also provide the f i r s t count

pulse. This made ce r t a i n that the c i r c u i t divided exactly i n the r a t i o of

2:1 to the l i m i t set by the accuracy of the counters, - 1 microsecond.

62

The t o t a l pulse d i v i s i o n of the c i r c u i t was varied when required

by the addition and removal of various 'chips'. This permitted the

difference i n times recorded f o r N and 2N pulses during a deceleration

run to be of a reasonable value, i . e . neither too large nor too small.

I n a l l experiments the shaft was driven up to speed using the a i r

turbine drive. The a i r supply to the turbine was t o t a l l y disconnected :

a f t e r the required speed of r o t a t i o n had been reached. This ensured that

the decelerations were carried out without any drive torque being delivered

to the shaft.

The accuracy of the timing equipment was found to be so good that i t

was generally only necessary to time one deceleration. Additional

measurements were taken over a small speed range when i t was desired

to investigate a range of Reynolds numbers of p a r t i c u l a r i n t e r e s t .

Results and Comments

The determination of the disc drag torque was carried out i n the

manner described e a r l i e r . The correction f o r the a i r bearing torque was i

determined f i r s t , the r e s u l t s of which are tabulated i n Table (10.1) and

sketched i n graph Fig. (6a). From Fig. (6a) a time constant of 83 seconds

was found. Hence using the calculated value of Ip S h a f t tc was -6 2 - 1

calculated to be 5.87 10 Kgm sec.

The deceleration of the shaft w i t h discs attached i n the a i r bearing

was next carried out. The results of these decelerations are tabulated

i n Table (10.2). From these results Fig. (6b) was drawn, showing dt

versus N, where i s the deceleration rate a f t e r correcting f o r the a i r dt

bearing torque. The data has been l e f t i n the form f o r easier dt

appreciation and f o r more convenient use l a t e r i n the experiments. The

63

{ " ( s e c o n d s ) 50 100 ISO 0

0-5

o n „ K

N

1-0

l»5

Fit,. (GO. A.r 6 C o r r e c - V P\oY \oouN /M D versus r e a r i n

ID I i

A M

AY

sec

2 0 ; Ve, 5 l a m p 3 3 3 rft

1 ( I 1 1 1 i l l i

* 3 * 5 fe l -8 o, , 2

N ( I O O O ^ R . P . H .

Fi«v-UV>} 0 ; 5 c D

65

results p l o t t e d i n Fig. (6b) show very l i t t l e spread and i t was found

that they were very repeatable.

From past work on windage losses on r o t a t i n g discs i t i s apparent

that an a i r j e t directed towards the discs i n question can cause early

onset of turbulent flow over the discs and also increases i n f r i c t i o n

losses. This i s relevant i n the disc drag determination i n the work

reported here since the a i r escaping from the a i r bearing can act as

a j e t to a l t e r the flow over the discs, thus producing incorrect data

f o r windage loss correction i n the l i q u i d bearing experiments. This

was overcome by using the l i q u i d bearing drain catchers attached to

the a i r bearing to deflect the escaping a i r from impinging on the

r o t a t i n g discs. This also simulates the experimental layout f o r the

decelerations with the water bearing as closely as possible.

The mass flow rates of water and the e c c e n t r i c i t i e s f o r given

speeds and pressure supplies were obtained as described previously

f o r each bearing clearance. The results obtained can be found i n tables

(10:3) and (10:4) respectively. These were p l o t t e d versus r o t a t i o n a l speed _5

i n Figs. (6c), (6d), (6e) and ( 6 f ) . For the 7.90 x 10 m clearance

the p l o t s of the mass flow rates, Fig. (6c), show a change i n gradient

i n d i c a t i n g the onset of turbulence which causes increased flow r e s t r i c t i o n s

to the lubricant thus decreasing i t s rate of flow through the bearing.

Three bearing clearances were used as described i n Chapter 4, these

corresponded roughly to clearance r a t i o s o f 0.001, 0.002 and 0.003. For

0.001 clearance r a t i o , a supply pressure of 2.0 bar gauge was used. For

the other two clearance r a t i o s two supply pressures were used, i . e . 1.0

bar and 2.0 bar gauge.

66

•5 2-0 6 ck r

C - 1 •So* lo

Kg. / sec X 0

1-0

1*0 B a r Sue pi

2*0 g>ar S

C = 5-S*Z* 10

1-0 Bor 5 •

C = <10 z Z-0 B a r S 4=

•

13

R.P.M. ( \ 0 O O ) N

Fi<y (fac^ '2" Diar^eVcr R",o j Mas5 Flooo Rcklt VtrSaS Nl

67

V 5 <

r o B> e a 2 • G2 x io

2 - 0 6 r e s $ a r e o r

1

N R.P. K\ . (

l r i<^. (.b t i ^ P\oV o-f £ v e r s u s VJ

I f

1 DiixtofcVer P> Co. r i r> 3

l-O

o-s

0-6

2 - 0 6 <xr

0-4

W Pressure

i i i

M R. P . M . ( l o 3 )

F'icy (GO P\o\- of £ versos N

69

l - o j

0-<5J

e

2. -0 B

/ 1-0 6 t * r

0-2 SWPD Io . P r e s s u r e

N R.P. tA . C 1 0 3 )

F.'cyCfcf.y P I* W of c versus .M .

70

For each of the clearance r a t i o s t o t a l torque decelerations were

carried out. The maximum i n i t i a l speed of each deceleration, and hence

Reynolds number, was l i m i t e d only by the amount of d r i v i n g torque

available from the laboratory a i r supply. The deceleration times were

taken f o r 10 revolutions and 20 revolutions i n each deceleration run

at various speeds throughout the run. For each deceleration the i n l e t

and o u t l e t temperature of the water was measured so that the experimental

v i s c o s i t y of the water would be known f o r a given run. The results

obtained from these runs can be found i n tables (10:5) - (10:2l(a)).

Additional shorter runs were made at the high speeds because as the shaft was

decelerating quickly the number of points which one could obtain i n a given

run was l i m i t e d by the time needed f o r recording the data.

Discussion of Results

Only a b r i e f discussion of the results w i l l be presented here and

a f u l l e r discussion w i l l be presented i n Chapter 8. The deceleration

runs are p l o t t e d i n Figs. (6g) and (6h). Fig. (6g) was pl o t t e d from the

1.0 bar results and (6h) from the 2.0 bar r e s u l t s . The p l o t s have already

been corrected f o r disc drag. I n each graph the upper set of points has

not been corrected f o r momentum torque, but the lower points have been

corrected.

With the 2.0 bar supply about 14% of the t o t a l r o t o r torque can be

a t t r i b u t e d to the throwing of the lubricant. The corresponding percentage

f o r the 1.0 bar supply i s around 10%. The angular momentum torque i s

therefore a s i g n i f i c a n t p o r t i o n of the t o t a l r o t o r torque.

The departure of the corrected p l o t s from the theoretical P e troff

torque can be explained by the eccentric running of the shaft i n the

bearing. Three graphs Fig. ( 6 i ) f ( 6 j ) and (6k), which are p l o t t e d from the

71

T e x p/T th tables i n Chapter 10, section 10 ( i ) , -indicate the r a t i o of the

experimentally determined torque to that of the theo r e t i c a l torque f o r

laminar flow corrected f o r e c c e n t r i c i t y . From these graphs, close

agreement i s apparent between theory and experiment.

I t would appear from these results that the use of a pressurised

lubricant supply has been successful at suppressing c a v i t a t i o n as there

i s no difference between the 1.0 and 2.0 bar results and both l i e close

to the f u l l f i l m t h e o r e t i c a l p l o t .

Decelerations were carried out with two d i f f e r e n t lubricant feed

directions. The f i r s t feed p o s i t i o n was from the top o f the bearing

opposite the load and the second 90° displaced i n the d i r e c t i o n of

ro t a t i o n of the shaft. I t was found that the torque measurements for

second supply p o s i t i o n , did not d i f f e r from the measurements reported

here, w i t h the supply opposite the load.

The departure of T e x p/T from the predicted p l o t s of u n i t y i n

graphs Fig. ( 6 j ) and (6k) at higher Reynolds numbers indicates the onset

of Taylor vortices. The Taylor prediction of onset i s indicated on

the graphs, using Re = 41.1 J R/Cf equation (1:1). Fig. (61) i s a

p l o t of the deceleration r e s u l t s , which have been corrected f o r the

angular momentum torque, f o r the three clearance r a t i o s with 1.0 and

2.0 bar supplies. The onset of turbulence i s indicated by the turning

of the graph onto a new gradient at the higher Reynolds numbers. I t

was not possible to investigate t h i s f u r t h e r as the a i r to drive the

turbine was not available to reach higher speeds.

,Vec\ f correct

- 5 '

C = 7-no

C f = > o re^ w e ^ 2 L t t

6 0 v

|0%> CarrecWo c,f ^ C - K V0"5n-.

f A A

Suppl^Pfensure — y 'OB

—r— — i — 4

- I 1 1 1 1 r -

5 6 T 8 °\ t i r 1-

5" b 1

V>er.

F l c y ( 6 ^ P U H o f C f v e r s o s R e . n ^ W -for 2 "

0 it»rr.e.V-e<_ r i ^ .

73

Cor r£o reA f-o t

C = 7.-1,7 * l o " 5 ^

C = 5^1 * I 0 ~ 6

u e . NoV Co r r t? c\ <? c), f o i

-f ©

\ ©I -f© E3 \ 4 ©

0

C f

Re - 2-n N R c

GO v

1 4 % Cnrre iV>oo (or ^

®

S»Apf\> Pressure = 2-0 B<

r— 6 7

Re, Wer.

74

Te*r O W Cv.

C o r r e c o r e-ccevs 3'

3 Re. ( l O 3 )

—i 4-

v e r s u s Re. r\vjk r»V>er.

75

T e y p .

S u p p l y P r e s s u r e .

*Z-0 B e n + t • 0 ?>o,c »

Te>< >

corfe.t_V«-A for e.cc-e>A

« J- + • +

P r e d t c V e A T a ^ \ o r O o s e V .

0 4 - - r -3 4-

" i r- —r-1

i — lo

i — i3

Re. ( l o a )

P ^ . C & j " ) P l o h o-f T e / . p . / T ( . ^ v e r s u s R e . A u v ^ b e r .

76

2 Dicx<Y\e.V r B>ea.r\f\cx

S a p p ^ Pressure.

1*0 B c r • 2-0 Gcxr +

+ +

+

4 ^ A

- * H — I — h -

Pre<\«cVe,dl Ta«^\or O n s e V

R e . = 7 3 7 .

—r-•7 10 I 12. i 5 lb 17 I *

Re. O 2 )

F i ^ . (Gk ' ) ?\oY o$ Te^^/Tj-Vv versus Re . nm«%V>er.

Su^>f>\<^ Pres"S>u,re.

1 0 - ,

i 74

C - 2 CI* 10 5 r ^

C - 5 - ° i 2x I C

C = 1 • °»0 x \Q - 5

I 6c

©

a

Z 6

as

Re

To r cy e

2 t t N R c

Tex»\or Pre<WV 0 Re = 737.

C= 7-SO x \ 0 ~ 5 , n

( X u i \ o r Pre<i"it.Vior\ Re = S51 .

7 « <? I 10' 10" 10"

Re.

fi^fei) Piof- of vtt^ws Re MAwter for Hie 2 DI<x»»->eVe> (TU J "

78 CHAPTER 7

DESIGN OF THE 3" DIAMETER BEARING RIG

The design of the larger r i g took the same format as that of the

small r i g , that was to have discs on the decelerating shaft. The same

compromises again had to be made. The discs had to be large enough both

to provide s u f f i c i e n t i n e r t i a to produce a long deceleration time constant

and to ensure that the buckets were at a radius which would enable the

shaft to be driven up to speed with an a i r turbine. Consideration also

had to be made of the size and positioning of the discs w i t h a view to

shaft s t a b i 1 i t y .

The design of the large r i g i s i l l u s t r a t e d i n Fig. (7a). I t i s

simila'r to the small r i g i n layout, but a few small changes were made to

f a c i l i t a t e easier manufacture and better operation.

The choice of diameter f o r the bearing of t h i s r i g was aided by a

computer program. The program i s given i n Appendix ( 4 ) . The program i s

of a f a i r l y simple nature, but because of the large number of variables i n the design, i t was found to be a great aid to the arithmetic.

Summary of the Computer Program

With reference to Appendix ( 4 ) : -

The basic parameters of the shaft are fed i n t o the program, i . e . the

various r a d i i , lengths and the clearance r a t i o of the proposed shaft.

From these, the program f i r s t calculates the polar i n e r t i a of the shaft.

I t then calculates the required diameter of the discs such that the

transverse i n e r t i a of the r o t o r i s equal to twice the polar i n e r t i a i n

order to i n h i b i t the s e l f excited conical w h i r l i n s t a b i l i t y of the rot o r .

A fter t h i s operation, the program calculates the time constant f o r

deceleration of the shaft plus discs with the bearing operating i n the

79

laminar regime. This gives an estimate of how feasible i t i s to take

measurements from the proposed shaft w h i l s t i t i s decelerating. The

mass of the rot o r i s then calculated and by using the Ocvirk solution f o r

a short bearing, estimates are made of the e c c e n t r i c i t y of the bearing

versus speed of r o t a t i o n of the shaft. The program then proceeds to

calculate an estimate of the tr a n s l a t i o n a l w h i r l onset speeds, based on an

approximate bearing s t i f f n e s s obtained from the Ocvirk solution to

Reynolds equation, the mean ro t a t i o n a l Reynolds numbers, the flow rates,

the momentum/shear r a t i o s and the stresses set up i n the discs during

r o t a t i o n . The disc stresses were calculated i n order to determine whether

special steel should be used f o r t h e i r manufacture.

The program does not give an exact solut i o n to the bearing design

problem but provides a guide to bearing choice f o r experiments.

Bearing Parameters

R = 38.1 mm

c = 11.48 10~5m

L = 76.2 mm

Ip = 3.93 10" 2 Kgm2

Diameter of Discs = 225.4 mm

Surface Finish of Shaft = 0.3 micron C.L.A.

Surface Finish of Bush = 0.8 micron C.L.A.

0> 80 -C5

V) — 5 6 -*o

0

J- J?

i f f

a: ] t C Z3CLZDG <T7

H 1

•

00 1

\ \ \ CO

81

CHAPTER 8

RESULTS AND DISCUSSION

The experiments with the 3" diameter bearing r i g took the same

form as those reported i n Chapter 6 f o r the 2" diameter r i g . Again the

t o t a l torque was determined using the deceleration technique with electronic

timings and t h i s was corrected f o r windage losses etc., as before. The

mass flow rates were measured with beakers and a stop watch and the

ec c e n t r i c i t i e s measured using the Wayne-Kerr capacitance probes plus

meters. The polar i n e r t i a v/as calculated and the v i s c o s i t i e s of water

used were those determined experimentally, which are p l o t t e d i n Appendix (3). •-F>

Only one ra d i a l clearance was used i n t h i s series of experiments, 11.48-10 m,

which corresponds roughly to a clearance r a t i o of 0.003. Again the equation

of motion i s given by, Ip du = - T t Q t a l

dt The form of T t o t a l t h i s time can be of a very variable nature.

As before:

T t o t a l - T ro t o r + ^ mass flow torque + ^ disc drag (8:1)

With the 3" diameter r i g the design was such that over the experimental

speed range and therefore Reynolds number, the bearing would, at lower speeds,

have a laminar f i l m and a t higher, a turbulent f i l m . The nature of T & rotor

with t h i s r i g therefore depends on the state of the l u b r i c a t i n g f i l m .

T ro^Qp was obtained i n the same manner as described previously. That was

by measuring the t o t a l torque and then subtracting from i t corrections f o r

disc drag and mass flow rate torques.

Experimental Comments

In the Disc Drag determination the f r i c t i o n a l torque on the shaft i n

the a i r bearing had to be corrected f o r a s e l f drive torque which the feed

j e t s i n the bush imparted to the shaft. This form of torque was discussed

82

i n Chapter 2. Again during the disc drag determination deflectors were

used between the bush and discs to prevent the escaping a i r from the bearing

impinging on the discs and upsetting the boundary layer flow. As wi t h the

experiments with the 2" diameter r i g , the shaft was driven up to speed by

using an a i r turbine. This was t o t a l l y disconnected during each deceleration

run. Before each deceleration was timed, the satisfactory operation of

the electronic timing c i r u i t was checked by feeding a dummy signal i n t o

the c i r c u i t from an o s c i l l a t o r . This ensured that the c i r c u i t was di v i d i n g

exactly i n the r a t i o of 2:1.

Results and Comments

The measurements taken during the disc drag determination can be

found i n Chapter 10, tables (10:22) and (10:23). Graphs, Figs. (8a) and

(8b) were p l o t t e d from these r e s u l t s . The experimental p l o t of the disc

drag torque, Fig. (8b), i s of the same form as that obtained f o r the

2" diameter r i g . The p l o t shows very l i t t l e scatter and also a sim i l a r

d i s t i n c t gradient change, confirming again the complex nature of the

drag torque and the need f o r electronic timing of the decelerations.

The mass flow rates and e c c e n t r i c i t i e s f o r the three lubricant supply

pressures used, i . e . 1.0 bar 2.0 bar and 2.5 bar, are p l o t t e d i n Figs. (8c)

and (8d). These were p l o t t e d from the data i n tables (10:24) and (10:25).

The change of gradients i n these p l o t s indicates a change i n l u b r i c a t i n g

f i l m regime i n the bearing. The flow through the bearing i s decreased

because of the extra flow r e s t r i c t i o n caused by the turbulent f i l m .

The measurements taken during the deceleration runs can be found

i n tables (10:26) to (10:34(a)). Fig. (8e) i s a p l o t of the 1.0 bar

supply results both corrected and uncorrected f o r the mass flow rate

torque. The angular momentum torque accounts f o r approximately 14% of

the t o t a l torque from the bearing when operating w i t h a laminar f i l m .

83

R. 3 Pio«v^eVe,r B ecK T 1 / \

h ( s e c o n d s ; t o o

0 1

N +N

O-'R

3 r o, 1 5

5 . 7 \Ac x i o 1-2

»-3

\V1

B e a r 3 xscS

to

R.P.I A

sec 10

it VecB •3 D; 3

o f r»

" I I 1 1 1 i 1 1 t |

NOOOCf) R.P. H.

F k y C S t O D U c D r a a . P l o V o - f / A t versus M .

85

2 - 5 6 SupoW Prepare

m 2 - 0 6 o r

sec

U P

i

NCIOOCO R.P. M.

F i ^ . ( < ? c ' ) McxS'S f U v o ro.Ve o f \w.V)<-\c«».vv\" vjersuk*. N f o r vW

86

0-«

\ \«a P r e s s u r e

\ \

• 0 6 o.r.

P f ' A

O-

I

M C V O O O ) R . P. M

87

10 |

v or roro,ue C o r f £C n o

V Cor\ e c

I"rex«"\"S\V">6 r€<\»oy\ |r\v«sVi & 3 >0 A

lord, we 1 L-rr

Re t 60 v

V 10 i O f c*"%

V o r or

0 r b c o r r e c t - ) ^ C f f * \ ( O f

© a

S u p p l y P r e * s u r € \ * 0 B<*r

Perrotf

10 I •7 ? 1 I

0 o

R e . f i « » ^ o € - r

Versus Rc for iW 3>" D'«\.were - Ee&.nftA Rio

88

Fig. (8f) i s a p l o t of the 2.0 bar supply r e s u l t s . This time the angular

momentum torque i n the laminar regime accounts f o r approximately 19%

of the t o t a l f r i c t i o n a l torque from the bearing. For both supply pressures

the angular momentum torque accounts f o r approximately 4-5% of the torque

when the bearing i s running with a turbulent f i l m . Fig. (8g) i s a summary

p l o t of the r e s u l t s obtained f o r the three supply pressures used. Again

the p l o t s show very l i t t l e scatter.

The 2.5 bar supply pressure was used at high Reynolds numbers so as

to extend the range of running of the bearing with a turbulent f i l m as

f a r as possible.

The deceleration times recorded f o r the l i q u i d f i l m bearing are f o r

10 revolutions and 20 revolutions.

The departure of the 'corrected' r e s u l t s p l o t s i n Figs. (8e), ( 8 f ) ,

and (8g) from the Petroff theoretical l i n e can be explained at low Reynolds

numbers by the e c c e n t r i c i t y e f f e c t and at high Reynolds numbers by

t r a n s i t i o n and turbulent f i l m s causing increased r o t a t i o n a l resistance.

The p l o t s i n Figs. (8h) and (8i)are the r a t i o s of,

T experimental rn t h e o r e t i c a l (laminar torque at a given e )

i. e . T experimental 2 rr R3

L ^ ^ c ( l - e 2 ) y*

versus r o t a t i o n a l Reynolds number. For both supply pressures, the laminar

regime, low Reynolds number res u l t s , agree well w i t h the theo r e t i c a l l i n e .

The marked departure of the results from the theo r e t i c a l l i n e show f i r s t

the onset of Taylor vortices and then the onset of turbulence. I t i s also

evident that the use of a pressurised lubricant supply appears to have

suppressed c a v i t a t i o n as the laminar regime results l i e s l i g h t l y above

the t h e o r e t i c a l p l o t and not below as would be expected with c a v i t a t i o n .

89

10 \ 1

C * II- 4 $ M O - 5

©

(\oY cc. rrecVe^i f o f ^orcyu<2.

C o r t ^ . c l ' R c l f o r t > Vof - cyue •

Re = 2TI N R C

c o r r e c V > q i ^ i f o r ry> l ~ o r c < i u & .

2 - 0 B a r 5%cpfrgiVigyN f ° f ••' oro.v<p

—V—

—v « 1 1 1 1 — r -

10

—r -

3 —r-4- 5 t>

IO

R e . fm*v\t>er,

90

10 l

U P nor correc re s u r e O f iVv Co r r e t V o r c v i \ e 1

l - O 6 cv r

a - 0 B a r 2-S 6

C T 1 RALxr P UJ

Re 2n N R

o i GO v

W i U o c k "Ta>A\or Prfidl'icV

7 4 ^

\ S m i ("U cmcl Pull

9

PeUofC

o 1 i

0 0

P U P o f C f

91

3 J

T e x p. T - rn .

+-*o4 <*—

Poci'ieVeJl To.^\or Onset". 3 Dicn^e,Veo B e o r ' m ^

S «jf>p\vj Pressure - \-O 6ar\

D e c e W r o , V>oo f

F ; o ( % 0 Put- o ? T e K P . / T m . v e r s u s Ke. r\u<v\to€ r.

92

P r e s s u r e =• 2 - O B a r .

D&ce\er<xVior\ ( -rom Vre».r\s\ Vl or..

Dec«,\ero.V\or\ f ro<^ -<ArV>vA\e«V f ^ O ^ J .

P r e < h c r e c l T Q ^ \ o r Ot>seV,

R e = 7 4 - ^ .

Re. (lO*) 3

— l 4-

F!a. Ploi- of Te^.p. / T\y,, >««rs\jis Re - ^ w A x ^ r .

93

I n Fig. (8g), the experimental results l i e very close to those found

by Smith and Fuller (6) i n the turbulent regime. As mentioned previously,

the 3" diameter bearing used i n the experimental work i n t h i s thesis was

very s i m i l a r i n dimensions to the bearing used by Smith and Fuller. The

departure of the r e s u l t s from the Smith and Ful l e r p l o t could possibly

big explained by the f a c t that the Smith and Ful l e r r e s u l t s were not

corrected f o r angular momentum torque. As no d e t a i l s of the drainage

system or flow rates were published by Smith and Fulle r , i t i s not

possible to discuss t h i s f u r t h e r . However, i t was commented that a

pressurised feed was used to suppress c a v i t a t i o n which could mean

r e l a t i v e l y high flow rates and high angular momentum torque. I t i s

also i n t e r e s t i n g to note that the correction of the results from the

present 3" r i g also changes the gradient to a small degree i n the

turbulent regime. This shows that care must be exercised i n the

extrapolation of turbulent p l o t s which have not been corrected f o r

angular momentum torque.

Taylor Vortices

I n Chapter 1 the occurence of Taylor vorti c e s i n bearings was

discussed i n d e t a i l . Mention was made of t h e i r occurrence, t h e i r

suppression and delay i n formation. The factors which were found to

influence the onset of Taylor vortices were e c c e n t r i c i t y , clearance

r a t i o , a x i a l flow and annulus length. The only factor of importance as

f a r as the present experimental work i s concerned i s e c c e n t r i c i t y . The

ef f e c t of a x i a l flow i s discounted i n these experiments as the maximum

Axial Reynolds number encountered i s considerably lower than the value

of 400 above which the a x i a l flow i s found to have significance (Cole

(10) (11)). For the 3" bearing r i g the order of magnitude of the

maximum Axial Reynolds number i n the region o f t r a n s i t i o n i s given by:

94

R e a x i a l = g V x 2 0

V x = E 2TTC^DC

R e a x i a l = 21 TT D JJL

where Re a x i a l = Axial Reynolds number

c = r a d i a l clearance

p. = dynamic v i s c o s i t y

D = bearing diameter

m = lubricant mass flow rate

^ = lubricant density

V x = lubricant a x i a l flow v e l o c i t y

For the 3 " bearing r i g w i t h a 2 bar lubricant supply pressure, operating

i n the region of t r a n s i t i o n :

D = 7.62 x 10~ 2

p. - 10~ 3 N sec/m2

m - 5 x 10 Kg/sec

su b s t i t u t i o n gives Re a x i a l = 209 (<400)

The other factors, clearance r a t i o and annulus length, are related

to large clearance experimental r i g s , such as those used i n visual studies

of Taylor vortices and are not relevant to the present experimental work.

I t i s found that there i s large scale disagreement i n the published

experimental results concerned with the e f f e c t of e c c e n t r i c i t y .

Explanations have been presented to account f o r the spread i n the results ,

but to date no sa t i s f a c t o r y solution has been put forward. Generally, but

not always, i t i s found that e c c e n t r i c i t y has a s t a b i l i s i n g e f f e c t on a

bearing f i l m and the Reynolds number at which Taylor vortices occur i s

95

found to be greater than the value predicted by the Taylor expression,

equation (1:1),

Re = 41.1 y R/c The methods commonly used by experimentalists to investigate Taylor

vortices are f r i c t i o n a l torque measurements and visual observations.

The d i f f i c u l t i e s encountered with these methods are discussed i n Chapter 1.

The f r i c t i o n a l torque measurement technique i s generally considered as

the most r e l i a b l e method. With reference to the f r i c t i o n a l torque results

presented i n t h i s thesis, the e f f e c t of e c c e n t r i c i t y appears to increase

the Reynolds number at which Taylor vortices appear or more exactly at

which they disappear, when decelerating. With reference to Figs. ( 6 i ) ,

( 6 j ) , (6k) and Figs. (8h) and ( 8 i ) , the onsets of Taylor vortices occur

at higher Reynolds numbers than the theor e t i c a l Taylor vortex predictions

indicated on the graphs. The onsets are marked by the departure of the

experimental p l o t s from the horizontal zero e c c e n t r i c i t y laminar flow

l i n e s . The onsets are increased by between 9% and 22% f o r e c c e n t r i c i t i e s

i n the range 0.14 to 0.26. I t i s d i f f i c u l t to determine accurately where

the onsets occur as the p l o t s meet the laminar p l o t tangentially. The

di s c o n t i n u i t i e s of the p l o t s i n the t r a n s i t i o n region'could possibly be

accounted f o r by various 'wavy modes' superimposed on the Taylor vortices.

To investigate Taylor vortices more f u l l y would require a mixture of visual

and torque determinations over a short deceleration range. This could be

achieved w i t h the aid of an a i r drive system which could be disconnected

and reconnected very quickly w i t h deceleration timings being taken w h i l s t

the bearing i s not being driven. From graphs, Figs. (6g), (6k) and Figs.

(8e), (8f) and (8g) i t i s evident that i f account i s not taken of the

angular momentum torque then the onset of Taylor vortices detected by

f r i c t i o n a l torque methods can be confused by the increase caused by the

throwing of the lubricant. This possible confusion i s probably greater

96

w i t h large clearance r i g s where there i s a p o s s i b i l i t y of large flows

and therefore a large angular momentum torque. Also with a large

clearance r i g , t h i s i s made worse by the f a c t that the Petroff torque

w i l l be r e l a t i v e l y smaller with the angular momentum torque possibly

dominating.

Transition Investigations

I t was reported i n Chapter 1 tho^Jackson (12) had investigated a

t r a n s i t i o n d i s c o n t i n u i t y e f f e c t i n bearing f r i c t i o n a l torque. He noticed

that f r i c t i o n a l torque measurements taken i n the t r a n s i t i o n region as

the speed of the shaft was increased, were d i f f e r e n t from those taken i n the

same region as the bearing was slowed down from the turbulent regime.

The differences were that d i s c o n t i n u i t i e s observed on a torque Reynolds

number p l o t as the speed was increased were not found as the speed was

decreased. However i t was pointed out by Jackson that t h i s e f f e c t was

only noticeable at zero e c c e n t r i c i t y and that f o r a small increase i n

ecc e n t r i c i t y the e f f e c t disappears,

i. e . f o r £ > 0.1

During the experimental work with the 3" diameter r i g a series of

decelerations was timed w i t h the s t a r t i n g Reynolds number at various

positions i n the t r a n s i t i o n region. The shaft was not f i r s t run i n t o

the turbulent regime, but only to a selection of Reynolds numbers i n the

t r a n s i t i o n region. The purpose of these experiments was to investigate

the relationship of the f r i c t i o n a l torque i n the bearing to the h i s t o r y

of the flow. The results obtained are tabulated i n Tables (10-'35) to

(10:40) and are p l o t t e d i n Figs. (8e), ( 8 f ) , (8h) and ( 8 i ) . These plo t s

indicate that there i s no apparent difference i n the rotor torques,

whether the r o t o r i s f i r s t driven i n t o the turbulent regime, or well

i n t o the t r a n s i t i o n region, or only s l i g h t l y i n t o the t r a n s i t i o n region.

I n o t h e r words i t does not appear t o matter where the d e c e l e r a t i o n i s

s t a r t e d , the torque measurements are j u s t the same. This i s an important

r e s u l t f o r the experimental i n v e s t i g a t i o n s o f the t r a n s i t i o n r e g i o n

f r i c t i o n a l torque r e p o r t e d i n t h i s t h e s i s . The s t a r t i n g p o i n t o f a

d e c e l e r a t i o n run, e i t h e r i n the t u r b u l e n t o r t r a n s i t i o n r e g i o n does not

a f f e c t the torque measurements taken.

D i s c o n t i n u i t i e s i n the p l o t s o f f r i c t i o n a l torque i n the t r a n s i t i o n

r e g i o n are i n f a c t apparent i n d i c a t e d as 'A' i n F i g . ( 6 k ) , (8h) and ( 8 i ) .

These may i n d i c a t e the onset o f a wavy mode on the Tayl o r v o r t i c e s . The

appearance o f these d i s c o n t i n u i t i e s i s not a f f e c t e d by the s t a r t i n g p o i n t •

o f the d e c e l e r a t i o n , whether i n the t r a n s i t i o n r e g i o n o r n o t , and cannot

be a t t r i b u t e d t o experimental s c a t t e r as they are repeatable.

The Angular Momentum Torque

The experiments w i t h the 2" and 3" bearing r i g s c l e a r l y i n d i c a t e the

magnitude o f the angular momentum torque. As much as 19% o f the t o t a l r o t o r

torque can be a t t r i b u t e d t o the throwing o f the l u b r i c a n t from the s h a f t .

This e f f e c t has not been re p o r t e d f u l l y before i n connection w i t h l i q u i d

bearings. I t was p o i n t e d out i n Chapter 3 t h a t the angular momentum torque

f o r a l i q u i d b e a r i n g f o r a given s i z e and f l o w r a t e i s twice t h a t f o r an

a i r b e a r i n g , i . e . the expression f o r an a i r b e a r i n g i s ,

T = m R 2ou 2

and f o r a l i q u i d b e a r i n g i s , T = m w R .

When the t o t a l l i q u i d b e a r i n g torque measurements are c o r r e c t e d f o r the

angular momentum torque, the experimental p l o t s i n Chapters 6 and 8, agree

f a v o u r a b l y w i t h the t h e o r e t i c a l p r e d i c t i o n s . This i n d i c a t e s t h a t the

t h e o r e t i c a l p r e d i c t i o n f o r the angular momentum torque i s o f the r i g h t

order and i s s i g n i f i c a n t .

98

Turbulence: General Discussion

Very l i t t l e has been published concerning the change from a t r a n s i t i o n

t o a t u r b u l e n t f i l m . I t i s g e n e r a l l y agreed t h a t turbulence occurs a t a

Reynolds number equal t o twice the c r i t i c a l Reynolds number f o r the onset

o f Taylor v o r t i c e s . That i s w i t h the s h a f t r o t a t i n g and the bush s t a t i c .

With reference t o the experimental work reported i n t h i s t h e s i s , the

p r e d i c t i o n does not appear t o be a general c r i t e r i o n f o r t u r b u l e n c e .

Fu r t h e r disagreement i s apparent when the 2" r i g r e s u l t s are compared

w i t h those from the 3" r i g . Although the clearance r a t i o f o r b o t h r i g s

i s approximately 0.003, the 2" r i g i n d i c a t e s an onset t o turbulence a t

a Reynolds number o f approximately 1650 and the l a r g e r i g , around 2000.

These^values do not appear t o be a f f e c t e d by supply pressure, f l o w r a t e

o r e c c e n t r i c i t y .

Turbulence: T h e o r e t i c a l Considerations

For laminar f i l m l u b r i c a t i o n , Reynolds equation i s o f the form,

6__ ( h 3 £p_) + ( h 3 £p_) = 6 UL U oh

a>X ( Sx) by ( by) 1 Sx"

The equation i s d e r i v e d from the Navier-Stokes equations o f motion and

the c o n t i n u i t y o f f l o w c o n d i t i o n . For the x coordinate d i r e c t i o n , the

Navier-Stokes equation o f motion i s , f l u i d i n e r t i a pressure viscous

n hu + p u hu + p v du + p w ^u = _ 6p + no ^u K Jt K K Zy~ K 5^ ^ i x

where u, v and w are the v e l o c i t i e s i n the coordinate d i r e c t i o n s . Body

f o r c e s have been o m i t t e d and the sources o f the o t h e r terms r e p r e s e n t i n g

f o r c e s per u n i t volume are i n d i c a t e d . I n the d e r i v a t i o n o f Reynolds

equation the main assumption i s t h a t the f l o w takes place i n a t h i n f i l m .

The f l o w i s a l s o considered t o be laminar and the f l u i d i n e r t i a f o r c e s

are neglected compared w i t h viscous f o r c e s .

99

At h i g h values o f Reynolds number the laminar f i l m can degenerate i n t o

t u rbulence. A t r a n s i t i o n r e g i o n e x i s t s between the two modes o f f l o w .

The Navier-Stokes equations can be adapted f o r t u r b u l e n t f l o w by co n s i d e r i n g

each parameter i n them, as having a time mean value, w i t h a f l u c t u a t i o n

about t h i s mean value. As such, laminar f l o w i s one i n which the

f l u c t u a t i o n s are n e g l i g i b l e compared w i t h the mean o v e r a l l f l o w . The

t u r b u l e n t pressure f o r example can be represented by,

P = P + P'

where p = pressure, p = time mean pressure and p' i s the pressure

f l u c t u a t i o n .

For the x coordinate the equations become,

mean f l u i d i n e r t i a pressure viscous t u r b u l e n t

^ O b u + p u bu_ + o v hu + o w b u = ' - 6p + ix^> ~u ' + u'w /) ' X bt ^ bx X bv V S¥ bx bz2 h z

I n the above equation, the term g i v i n g r i s e t o the t u r b u l e n t stresses d e r i v e s

from the f l u c t u a t i o n s connected w i t h f l u i d i n e r t i a . For laminar f l o w the

t u r b u l e n t stresses and the f l u i d i n e r t i a e f f e c t s can be neglected i n

comparison t o the viscous stresses. With i n c r e a s i n g Reynolds number the

f l u i d i n e r t i a s tresses s t a r t t o become s i g n i f i c a n t . I n a j o u r n a l bearing

as discussed i n Chapter 1, c e n t r i f u g a l i n e r t i a can lead t o a Taylor v o r t e x

f l o w . For a f u r t h e r increase i n Reynolds number the onset o f turbulence

w i l l be experienced. I n t h i s r e g i o n o f t r a n s i t i o n , l i t t l e i s known about

the balance o f str e s s e s . I n f u l l y e s t a b l i s h e d t u r b u l e n t f l o w i t i s g e n e r a l l y

assumed t h a t t u r b u l e n t stresses dominate although t h i s i s by no means c e r t a i n .

At very h i g h Reynolds numbers, the mean f l u i d i n e r t i a term becomes more

s i g n i f i c a n t i n comparison w i t h the t u r b u l e n t s t r e s s e s . No t h e o r e t i c a l

a n a l y s i s o f t u r b u l e n t f l o w w i t h the e f f e c t o f mean f l u i d i n e r t i a included , has as y e t been pu b l i s h e d .

Three t u r b u l e n t l u b r i c a t i o n t h e o r i e s are a v a i l a b l e a t the present

time. The e a r l i e s t approach was developed by Constantinescu (1959) ( 1 7 ) ,

100

who used the concept o f P r a n d t l ' s mixing length.- L a ter, Ng and Pan (1965)

( 1 8 ) , used the concept o f eddy v i s c o s i t y , t o represent the t u r b u l e n t stresses,

i n terms o f the mean v e l o c i t y g r a d i e n t . H i r s (1972) ( 1 9 ) , used a more

novel approach; he used what he c a l l s a b u l k f l o w approach, which r e q u i r e s

no p h y s i c a l r e p r e s e n t a t i o n o f the t u r b u l e n t t r a n s p o r t mechanism. The

eddy v i s c o s i t y approach i s g e n e r a l l y considered t o be more p r e f e r a b l e .

The f i r s t two o f the above t h e o r i e s lead t o the same form o f Reynolds

equation, which i s , 3 3 JL (h ^jp_) + h_ (h hp) = U hh

^ X ( k x ^ bx) £y ( k y u . dy) 2 h

where b o t h k x and ky are constants r e l a t e d t o the Reynolds number by,

k = 12 + f(Re) x y

The two t h e o r i e s p r e d i c t d i f f e r i n g values o f k x and ky. The values

obtained by Constantinescu are, , 0.8265 k x = 12 + 0.0260 (Re)

and k y = 12 + 0.0198 ( R e ) 0 ' 7 4 1

Constantinescu a l s o d e r i v e d a r e l a t i o n s h i p f o r the shearing s t r e s s a c t i n g

on the surfaces,

t = - ix U C + h dp 2 bx

— , ,0.855 where Cc = 1 + 0.0023 (Re) ( C c = C c h / p. U)

where C = shear s t r e s s

Cc = Couette surface shear s t r e s s

The values obtained by Ng and Pan f o r k x and k y are,

k x = 12 + 0.00136 ( R e ) 0 ' 9

k y = 12 + 0.0043 ( R e ) 0 ' 9 6

The values obtained f o r c c are,

Re > 10,000

Y n = 1 + 0.00232 ( R e ) 0 ' 8 6

101

R e onset o f turbulence < Re < 10,000

0.96 C c = 1 + 0.00099 (Re)

Turbulence: Comparison o f Experimental Results and T h e o r e t i c a l P r e d i c t i o n s

F i g . ( 8 j ) i s a p l o t o f the experimental r e s u l t s obtained w i t h the 3"

diameter r i g o p e r a t i n g w i t h a t u r b u l e n t f i l m . Also i n d i c a t e d on the p l o t

are the t h e o r e t i c a l p r e d i c t i o n s o f Constantinescu and Ng and Pan. Close

agreement i s apparent between the experimental r e s u l t s from the 3" diameter

bearing and the Ng and Pan p l o t . From the r e s u l t s a value f o r Cc was

obtained which i s ,

Cc = 1 + 0.001643 ( R e ) 0 ' 9 0 8 8

An important p o i n t t o note i s t h a t even though the constant and exponent

have values roughly midway between the values obtained by Constantinescu

and Ng and Pan, the p l o t o f crc i s above b o t h o f t h e i r s and not a mean value

as may be expected.

The t u r b u l e n t t h e o r i e s r e l y on the d e t e r m i n a t i o n o f constants, from

experiments w i t h f l o w both i n pipes and between s t a t i o n a r y and s l i d i n g

p l a t e s , t o be used i n t h e i r equations. The experimental work from which

these constants are obtained appears t o be l i m i t e d . However the t u r b u l e n t

theory constants could be obtained from b e a r i n g f r i c t i o n a l torque measurements

de r i v e d i n the manner described i n t h i s t h e s i s . An advantage o f t h i s would

be t h a t the i n f o r m a t i o n would be taken d i r e c t l y from bearings and not have

t o be d e r i v e d from general f l o w s i t u a t i o n s . This more accurate method o f

d e t e r m i n a t i o n would then g i v e more r e l i a b l e t h e o r e t i c a l p r e d i c t i o n s .

I t was intended a t the o u t s e t o f the experimental work w i t h the 3"

bearing r i g , t o i n v e s t i g a t e the f r i c t i o n a l torque i n the t u r b u l e n t regime

a t h i g h e r Reynolds numbers than those a c t u a l l y achieved. The range o f

Reynolds number i n v e s t i g a t e d was foreshortened, t o a degree, by the e a r l y

102

if S D i a feeler 6 3

S \ADp\iA P 0 ress ute

6 s 2 B c r

2-5 B

N

C o *\S Vtxy-At<\€.S

0 I I 1 i V « S I

10 0

103

onset o f what appeared t o be h a l f speed w h i r l , a t a speed s i g n i f i c a n t l y -

lower than p r e d i c t e d . The w h i r l o f the r o t o r was not o f a c o n i c a l nature

as t h i s had been s u c c e s s f u l l y prevented by gyroscopic s t a b i l i z a t i o n from

the d i s c s , reference-Chapter 4, but o f a t r a n s l a t i o n a l form. Attempts

t o suppress t h i s i n s t a b i l i t y were u n f o r t u n a t e l y unsuccessful. My more

recent s t u d i e s o f disc v i b r a t i o n lead me t o b e l i e v e t h a t the i n s t a b i l i t y

may a c t u a l l y have been o f a more complex form, w i t h p o s s i b l y a combined

i n s t a b i l i t y between d i s c , s h a f t and b e a r i n g f i l m .

104

CHAPTER 9

CONCLUSIONS

I t has been demonstrated i n the experimental work r e p o r t e d i n the

previous chapters t h a t the e l e c t r o n i c t i m i n g system, described i n

Chapter 5, i s a very u s e f u l t o o l f o r i n v e s t i g a t i n g the f r i c t i o n a l torque

i n bearings. The c i r c u i t c o n s i s t s o f on l y two good t i m e r counters and

les s than £10 (1976 p r i c e s ) worth o f e l e c t r o n i c 'chips'. Although the

c i r c u i t was designed, b u i l t and used between 1974 and 1976, because o f

the s i m p l i c i t y o f the c i r c u i t , more recent advances i n microprocessors,

would not make i t any cheaper.

From the experimental work r e p o r t e d , i t has been shown t h a t w i t h the

a i d o f the t i m i n g c i r c u i t r y , the f r i c t i o n a l torques experienced by a

r o t a t i n g s h a f t i n a j o u r n a l b earing, can be i d e n t i f i e d e a s i l y and

a c c u r a t e l y . D i r e c t measurements o f r o t o r torque, such as these, have

no t been p o s s i b l e i n the past.

The usual method o f i n v e s t i g a t i n g f r i c t i o n a l torque i n j o u r n a l

bearings, i s t o measure the torque which appears on the s t a t o r . The

apparatus g e n e r a l l y c o n s i s t s o f a r o t o r , which i s mounted i n two slave

bearings, one a t each end, and a c e n t r a l l y p o s i t i o n e d 360° l i q u i d b e a r ing.

The c e n t r a l b e a r i n g i s loaded by means o f a h y d r o s t a t i c pad under the bush.

The bush, which i s otherwise f r e e t o r o t a t e , i s r e f r a i n e d from doing so

by a t e n s i o n gauge on i t s p e r i p h e r y . As the r o t o r i s revolved measurements

are taken from the r e s t r a i n i n g t e n s i o n gauge a t d i f f e r e n t speeds. From

these r e s u l t s the b e a r i n g f r i c t i o n a l torque can be obtained as a f u n c t i o n

o f the r o t a t i o n a l speed. Sources o f e r r o r i n t h i s method o f de t e r m i n a t i o n

stem from r e s t r i c t i o n s t o b e a r i n g motion, caused by the l u b r i c a n t supply

pipework and misalignments i n the h y d r o s t a t i c pad l o a d i n g system. Both

o f these decrease the accuracy o f the t e n s i o n gauge readings. The author's

105

experience w i t h t h i s form o f r i g i s t h a t i t i s d i f f i c u l t t o o b t a i n

r e l i a b l e r e s u l t s from i t .

An important f e a t u r e o f the present experimental work i s t h a t i t

was c a r r i e d out w i t h bearing r i g s having clearance r a t i o s which are

commonly found i n i n d u s t r y .

The r i g s were designed such t h a t c a v i t a t i o n o f the l u b r i c a n t f i l m

was suppressed and the bearings would r un f u l l . Suppression was achieved

w i t h the a i d o f a p r e s s u r i s e d supply o f l u b r i c a n t . The purpose o f t h i s

was t o s i m p l i f y the experimental work and t o e l i m i n a t e the n e c e s s i t y f o r

c o r r e c t i n g the f r i c t i o n a l torque measurements f o r a c a v i t a t e d r e g i o n .

The close agreement between the experimental p l o t s o f the present work

and the t h e o r e t i c a l p r e d i c t i o n s f o r a bea r i n g running f u l l , i n d i c a t e s

t h a t suppression has been su c c e s s f u l .

I n the d i s c u s s i o n o f previous experimental work (Chapter 1) no

mention was made o f the magnitude o f the angular momentum torque. This

i s because the e f f e c t has not been r e p o r t e d i n connection w i t h l i q u i d

bearings. The e f f e c t has been r e p o r t e d o n l y i n connection w i t h experimental

work w i t h a i r bearings, reference Bennett and Marsh ( 1 3 ) . The magnitude

o f the torque f o r an a i r b e a r i n g i s , f o r a gi v e n f l o w r a t e and speed,

h a l f t h a t f o r a l i q u i d b e aring w i t h the same exhaust r a d i u s . The reason

f o r t h i s b eing t h a t the a i r , on l e a v i n g the a i r b e a r i n g , has a mass

averaged t a n g e n t i a l v e l o c i t y oftoR, whereas a l l o f the l u b r i c a n t e x i t i n g 2

from a l i q u i d b e a r i n g , f l o w s along the s h a f t and gains the s h a f t v e l o c t y ,

toR, before being thrown o f f . The momentum torque f o r a l i q u i d b e a r i n g i s 2 2 / t h e r e f o r e , m oo R , as opposed t o t h a t f o r an a i r bearing, m to R / T h e

v e l o c i t y o f the l i q u i d being thrown from the r o t o r depends on the rad i u s

o f the r o t o r and not on l y on i t s r o t a t i o n a l frequency. With an a i r bearing

106

the exhaust r a d i u s o f the bearing i s o f prime importance whereas w i t h a

l i q u i d b e aring, the radius a t which the l i q u i d i s thrown from the r o t o r ,

which may be g r e a t e r than the bearing r a d i u s , i s o f more s i g n i f i c a n c e .

The 'throwing' r a d i u s o f the r o t o r can be several times the magnitude

o f the bearing r a d i u s , f o r example, i f d i s c s are present. The magnitude

o f the angular momentum torque can t h e r e f o r e be g r e a t l y magnified by the

l u b r i c a n t being thrown from these l a r g e r r a d i i . Throughout the present

experimental work steps were taken i n i n i t i a l design, by stepping down

the s h a f t r a d i u s , as i t leaves the bearing, t o ensure t h a t the l u b r i c a n t

was thrown a t the bearing r a d i u s .

For the 3" diameter r i g , o p e r a t i n g w i t h a laminar f i l m , approximately

19% o f the t o t a l f r i c t i o n a l torque o f the r o t o r was a t t r i b u t a b l e t o the

throwing o f the l u b r i c a n t . For a bearing o p e r a t i n g a t low Reynolds numbers,

w i t h a laminar f i l m , the r o t o r torque p l o t s , when c o r r e c t e d f o r momentum

torque and e c c e n t r i c i t y , l i e v e r y close t o t h e o r e t i c a l p r e d i c t i o n s . The

agreement i s good f o r both o f the r i g s , t h i s i n d i c a t e s t h a t the p r e d i c t e d

magnitude o f the c o r r e c t i o n f o r angular momentum torque i s o f the r i g h t

o rder. The magnitude o f the angular momentum torque has a l s o been

confirmed by experiments w i t h a i r bearings as discussed i n Chapter 2.

The r e l a t i v e importance o f the angular momentum torque i s l e s s f o r

o i l bearings than i t i s f o r water bearings, the reason being t h a t the

viscous torque i s s i g n i f i c a n t l y g r e a t e r f o r the h i g h e r v i s c o s i t y o f o i l s .

Considering the t o t a l torque expression, f o r a gi v e n laminar f i l m bearing,

the viscous torque i s p r o p o r t i o n a l t o the v i s c o s i t y o f the l u b r i c a n t and

the momentum torque i s p r o p o r t i o n a l t o i t s d e n s i t y . The d e n s i t y o f o i l s

v a r i e s l i t t l e f o r a l a r g e range o f d i f f e r e n t v i s c o s i t y o i l s . Therefore

f o r a given f l o w r a t e the magnitude o f the viscous torque can increase

107

many f o l d t o t h a t o f the momentum torque. V/ith low v i s c o s i t y l u b r i c a n t s

the angular momentum torque c o n t r i b u t e s more s i g n i f i c a n t l y t o the t o t a l

torque on the r o t o r . Such f l u i d s are being i n c r e a s i n g l y used i n i n d u s t r y .

The use o f water as a l u b r i c a n t i s a l s o becoming more o f an accepted

p r a c t i c e and manufacturers o f l a r g e steam t u r b i n e s are known t o be

co n s i d e r i n g i t s use. Water i s a l s o a popular choice f o r bearing research

r i g s . When u s i n g low v i s c o s i t y l u b r i c a n t s i n bearings omission o f the

c o n s i d e r a t i o n o f the angular momentum torque can lea d t o l a r g e e r r o r s

i n the p r e d i c t e d r o t o r torque.

From the r e s u l t s obtained from the 2" and 3" r i g s the f o l l o w i n g

observations can be made. For a laminar, low Reynolds number f i l m ,

close agreement i s apparent, w i t h the t h e o r e t i c a l P e t r o f f p r e d i c t i o n s ,

which are i n d i c a t e d on the torque p l o t s by the l i n e , Cf = 2/Re. For

the t u r b u l e n t , h i g h Reynolds number f i l m , t h e r e i s good agreement between

the experimental r e s u l t s o f Smith and F u l l e r and those o f the present

work. I n the t r a n s i t i o n r e g i o n , the experimental p l o t s i n d i c a t e t h a t

the onset o f Taylor v o r t i c e s i s delayed by the e c c e n t r i c running o f the

s h a f t i n the bea r i n g . With reference t o Chapter 1, many experimental

i n v e s t i g a t i o n s o f t h i s phenomenum have been r e p o r t e d and much disagreement

i s apparent i n the pub l i s h e d r e s u l t s . Some o f t h i s disagreement can be

explained by the f a c t t h a t no account has been taken o f the angular

momentum torque i n the f r i c t i o n a l torque d e t e r m i n a t i o n s o f the onset o f

v o r t i c e s . I f the f r i c t i o n a l torque i s measured from the s t a t o r as

described p r e v i o u s l y i n t h i s chapter, the r e s u l t s obtained w i l l be i n

e r r o r f o r the ne g l e c t o f the angular momentum torque only i f the exhausting

l u b r i c a n t i s c o l l e c t e d by 'catchers* which are attached d i r e c t l y t o the

bear i n g . I f the spent l u b r i c a n t from the b e a r i n g i s thrown from the

r o t o r and dr a i n e d from catchers attached d i r e c t l y t o the bearing, the

angular momentum, o f the l u b r i c a n t , i n t h i s case would be t r a n s f e r r e d t o

the catchers and appear as an increased measured t o t a l f r i c t i o n a l torque.

I f the catchers, however, are not attached d i r e c t l y t o the bearing, the

momentum torque w i l l not be t r a n s f e r r e d t o the bearing. This f a c t was

demonstrated by a f i n a l year p r o j e c t student a t Durham U n i v e r s i t y who

c a r r i e d out experiments using Hampson's r i g ( 1 6 ) . This r i g was o f the

type mentioned above w i t h f r i c t i o n a l torque measurements being taken from

the s t a t o r . Hampson's measurements were taken w i t h catchers attached t o

the s t a t o r . The student i n v e s t i g a t e d the e f f e c t o f mounting the catchers

independently from the s t a t o r and i d e n t i f i e d a change i n the magnitude

o f the f r i c t i o n a l torque.

A s e r i e s o f experiments has been r e p o r t e d i n Chapter 8 i n connection

w i t h the s t a r t i n g p o i n t o f a d e c e l e r a t i o n t i m i n g run. I t was found t h a t

whether the d e c e l e r a t i o n s t a r t e d i n the t u r b u l e n t r e g i o n o r i n the

t r a n s i t i o n r e g i o n , the measured f r i c t i o n a l torque was unchanged. The

f r i c t i o n a l torque on the r o t o r was independent o f the previous o p e r a t i n g

h i s t o r y o f the bearing. A s i m i l a r i n v e s t i g a t i o n was r e p o r t e d by Jackson

(12 ) . He r e p o r t e d t h a t i n the t r a n s i t i o n r e g i o n , d i s c o n t i n u i t i e s i n the

p l o t o f measured torque versus Reynolds number, which were apparent w i t h

i n c r e a s i n g Reynolds number, were not found w i t h decreasing Reynolds number

Jackson explained the occurrence o f these d i s c o n t i n u i t i e s by the appearanc

o f the onset o f a wavy mode superimposed on the Taylor v o r t i c e s . I t was

p o i n t e d o u t , however, t h a t t h i s phenomenon was o n l y observable f o r a

c o n c e n t r i c a l l y running b e a r i n g and t h a t f o r o n l y a small increase i n

e c c e n t r i c i t y the e f f e c t disappeared.

I n Chapter 8 the a p p l i c a t i o n o f f r i c t i o n a l torque measurements t o

t e s t t h e o r i e s f o r t u r b u l e n t l u b r i c a t i o n was discussed. Close agreement

109

was apparent between the t h e o r e t i c a l p r e d i c t i o n s - o f Ng and Pan and the

r e s u l t s from the 3" diameter r i g , f o r the f r i c t i o n a l torque f o r a bearing

o p e r a t i n g w i t h a t u r b u l e n t l u b r i c a t i n g f i l m . The t u r b u l e n t t h e o r i e s r e l y

on the experimental d e t e r m i n a t i o n o f constants f o r s u b s t i t u t i o n i n t o

t h e o r e t i c a l p r e d i c t i o n s . The accuracy o f d e t e r m i n a t i o n o f these constants

g r e a t l y e f f e c t s the v a l i d i t y o f the t h e o r e t i c a l p r e d i c t i o n s . The accuracy

o f the present method o f de t e r m i n a t i o n o f f r i c t i o n a l torque constants,

can t h e r e f o r e be o f s i g n i f i c a n t help i n f u t u r e t u r b u l e n t l u b r i c a t i o n

i n v e s t i g a t i o n s .

Although the work i n t h i s t h e s i s describes the use o f the d e c e l e r a t i o n

t i m i n g c i r c u i t a p p l i e d t o f r i c t i o n a l torque d e t e r m i n a t i o n i n j o u r n a l

bearings, i t s use can be e a s i l y extended t o o t h e r experimental f i e l d s .

Since 1977, when the experimental work r e p o r t e d i n t h i s t h e s i s was

terminated, the d e c e l e r a t i o n method has been s u c c e s s f u l l y used i n

turbomachinery experiments performed a t Durham U n i v e r s i t y .

I t i s hoped t h a t the work reported i n t h i s t h e s i s , w i l l a i d the

designer and the researcher t o more accurate measurement and knowledge

o f f r i c t i o n i n j o u r n a l bearings.

CHAPTER 10

110

EXPERIMENTAL RESULTS: TABULATIONS

Not a t i o n :

h N

No NoO

ReN°

m

m Q

c e

r a t i o e xP Tth

time i n seconds f o r N pulses

time i n seconds f o r 2N pulses

( t 2 - 2 t ] _ ) , seconds

speed, R.P.M.

i n i t i a l speed, R.P.M.

s e l f d r i v e e q u i l i b r i u m speed

r o t a t i o n a l Reynolds number

l u b r i c a n t mass f l o w r a t e

non-dimensional c o r r e c t i o n f o r angular momentum torque

bearing r a d i a l clearance, metres

e c c e n t r i c i t y

e x p e r i m e n t a l l y measured f r i c t i o n a l torque t h e o r e t i c a l f r i c t i o n a l torque c o r r e c t e d f o r £

For purposes o f convenience the d e c e l e r a t i o n r a t e s are recorded i n

R.P.M./sec

For the d i s c drag d e t e r m i n a t i o n :

C * J J t o t a l d e c e l e r a t i o n r a t e i n c l u d i n g support a i r b e a r i n g

drag, f o r windage losses, R.P.M./sec

c o r r e c t i o n f o r a i r b e a r i n g drag, R.P.M./sec

D c* T - c< c R.P.M./sec ( d i s c drag)

For the l i q u i d bearings:

dN/dti t o t a l d e c e l e r a t i o n r a t e i n c l u d i n g windage and mass f l o w r a t e

e f f e c t s R.P.M./sec

DRAG c o r r e c t i o n f o r windage losses R.P.M./sec

dN/dt d e c e l e r a t i o n r a t e f o r be a r i n g torque + mass f l o w r a t e

torque R.P.M./sec

C f i non-dimensional f r i c t i o n a l torque uncorrected f o r mass

f l o w r a t e torque

Cf non-dimensional f r i c t i o n a l torque c o r r e c t e d f o r mass

f l o w r a t e torque

General l y ,

<X = N(average) b

N(average) = 60 x Number o f r e v o l u t i o n s i n time t 2

Cf = Torque ^ c o 2 R 4 L rr

m c = m RQJ 2 4 ^cu RH L T T

112

10. i ) 2" Diameter Rig

10:1 A i r Bearing c o r r e c t i o n s

10:2 Disc Drag d e c e l e r a t i o n s

10:3 Mass f l o w r a t e s

10:4 E c c e n t r i c i t i e s

10:5 D e c e l e r a t i o n o f 2 " Rig c = 2.62 10"5m

supply pressure = 2.0 bar gauge

10:5(a) T e x p / T t h t a b u l a t i o n f o r run 10:5

10:6 - 10:11(a) Decelerations o f the 2" Rig c = 5.92 lO'^m

supply pressures = 1 . 0 and 2.0 bar gauge

10:12 - 10:2l(a) Decelerations o f the 2" Rig c = 7.90 10 - 5m

supply pressures = 1 . 0 and 2.0 bar gauge

113

Table (10:1) M r Bearing Correct ion:

N R.P.M.

t sees

t sees

t sees

t sees

t ave

sees l o g e

4000 18.5 18.6 18.6 18.7 18 .6 -0 .2231 3000 42.3 42.7 42.8 42.8 42.7 -0 .5108 .2000 76.4 76.1 76.7 76.5 76.4 -0 9160 1000 132.5 133.3 133.6 132.8 133.1 -1 6094

N Q = 5000 R.P.M. Zero running speed 12 R . P . M .

Supply pressure = 4.0 bar gauge

A l l o f the above times were measured w i t h a stop watch. The

speeds were measured w i t h a stroboscope which was set to a p a r t i c u l a r

speed, 5000 R.P.M. (accuracy ± Vz%) .

The theory concerned w i t h the above a i r bearing cor rec t ion

determination can be found i n Chapter 6.

114

Table (10:2) Disc Drag Decelerations

DISC DRAG (50 revs) t i (25 revs) N c< c D

10 ^ sees 10~6 sees 10"6secs R.P.M. R.P.M./sec R.P.M/sec R.P.M/sec

390506 195110 286 7682 57.72 3.27 54.45 414044 206871 302 7246 51.13 3.08 48.05 440373 220027 319 6812 44.89 2.90 41.99 464302 231980 342 6461 41.06 2.75 38.31 499575 249604 369 6005 35.57 2.55 33.02 508845 254235 375 5896 34.21 2.51 31.70 544435 272016 403 5510 30.01 2.34 27.67 585229 292394 441 5126 26.44 2.18 24.26 626120 312818 484 4791 23.70 2.04 21.66 658997 329244 509 4552 21.38 1.94 19.44 694263 346864 535 4321 19.22 1.84 17.38 727214 363326 562 4125 17.56 1.76 15.80 770717 385060 596 3893 15.65 1.66 13.99 812004 405686 632 3695 14.19 1.57 12.62 851204 425268 668 3524 13.02 1.50 11.52 892862 446078 706 3360 11.92 1.43 10.49 929863 464559 745 3226 11.14 1.37 9.77 970655 484934 787 3091 10.34 1.32 9.02

1017980 508571 838 2947 9.55 1.25 8.30 1066794 532952 890 2812 8.81 1.20 7.61 1111203 555130 943 2700 8.26 1.15 7.11 1213880 606408 1064 2471 7.15 1.05 6.10 1333863 666326 1211 2249 6.14 0.96 5.18 1418620 708648 1324 2115 5.58 0.90 4.68 1490826 744702 1422 2012 5.16 0.86 4.30 1567905 783189 1527 1913 4.76 0.81 3.95 1613159 805786 1587 1860 4.55 0.79 3.76 1667293 832808 1677 1799 4.35 0.77 3.58 1710153 854212 1729 1754 4.17 0.75 3.42 1752566 875387 1792 1712 4.00 0.73 3.27 1803763 900946 1871 1663 3.83 0.71 3.12 1849263 923662 1939 1622 3.69 0.69 3.00 1964640 981259 2122 1527 3.37 0.65 2.72 2013493 1005642 2209 1490 3.26 0.63 2.63 2071040 1034365 2310 1449 3.13 0.62 2.51 2140965 1069275 2415 1401 2.96 0.60 2.36 2225083 1111259 2565 1348 2.80 0.57 2.23 2365056 1181119 2818 1269 2.56 0.54 2.02 2441219 1219126 2967 1229 2.45 0.52 1.93 2516495 1256691 3113 1192 2.35 0.51 1.84 2656231 1326424 3383 1129 2.17 0.48 1.69 2735800 1366128 3544 1097 2.08 0.47 1.61 2818601 1407441 3719 1064 2.00 0.45 1.55

115

CM o I CD o to

cc;

CM o o to

o o o i c n a n o m n n t M

i X ) O O Q O O O O O O O O W H O o n n o i

co O O) O) <-< o o

CD C O C O

< i o o

O O O ir> o o o IT) tO CD CM CM ID O < H <-H < H o O o in H CM CO N ID U )

CO CD

CO

to

2 •H §i CD

m CD 4->

00 o I CD O CO

N l O CM (O H CM CO CO CO CO CD

O 00 CO O t> [>

o o o o o o CO ti") CM LO CM If)

CM O O O <H <-H CM CO M" LO CO

CO CO O CO CO LD

o rH <H o O C D

^ "%r co

O Q O O O O o CM LO co m O CM O CM rH CO

i-i CM co 'sf in co

Q

CM

C O

6

• a

O CO Lf) CD O "3" IX) CO CD O O <H

[> CO t> O tN

co O in L D oo C D ' H CM Q O C D O O Q O

CM co in

116

m CD -p •H o •H U -P c CD o o w

2 •H CD

m u -p

o CM

o

CD

iH •8

E CD I D

1 ur

O to i-l es

X ( 0

a, 3 ( 0

C M C Q C O >>

• rH C M C M O H

&

•

II

O H

& s II 3 C O

o

10

10

a;

cM^^^ tNCMLOco M n n w H H O O d d d d d d d d

o o o - I CM 00

§ o o o O Q O o o o

CD *t CM IN CO ( O ^ W H H O O d d d d d d d

o o o o 8 8 8 8 H w n v

8 8 8 O O m m to to

io H I D H en m C O C O CM H H

d d d d d d d

o o o o o o o iH co o o o o b O cn O O O O O !—) i—I CO "3" LO lO N

t> in co en CD in ncvi H H H o d d d d d d d

o o o H f f l O O C D O H H C O

o o o m to [>

co co <H O t \ l H r ) H H

d d d d d

8 8 8 8 8 o o o o o cH CM CO LO tO

CO

1 <H S o ^

CO

1 o 2 • E ^

CO rH 1 <H O U

a C O

% CVI II II

( X CD ( X

u o CO <D CO (0

O H

-P S >, ^ O H rH

a § cc a CO o

CD

to E in

i s 1 o Q <H

< O H X

CVJ CO o • CD CVJ

CO - \ II

o

\ a. CO O

CO • H O -p CD CO Sn

/O to CD r—1 1 CD o O

rH CD Q

• bfl s C

2 OH' 'r-i % O H CD

C O

; v

CO CD > -P CD CO & u o CD Cu

O CO •H C\J O — C O _

1 CM O

CM -P rH

t v

^ ^ CO LO > CD CO 6 (H O rH

CD O CO rH CD ^- to r-H

1 rH O P rH £ H

r H C M C M C O ^ C O C O r H C O H H H H H H r i W W

(DCDOOOOOOO

C M C O C O r H ^ T i n C O ^ r H ^ r i r ) i > r H t > L n i > c o c o L n c O t ^ C D O C M ^ - O O

rH rH rH rH CM

H W C M c \ i c \ i n n ^ i n o o o o o o o o o 6 6 6 6 6 6 6 6 6

9 oomcocooocoooco m c ^ r H O - i n o c o c o

L O i o o c n o c M ' ^ o o H H H H CM

C M C O C O C M O C D C D C O C D

C M L D r H r O r H C O O O C X J C - v O l O C \ J C D C £ > < 3 " C M O CO CO CVJ CVJ <—I *— H H H

i n o r - c D i n o c o c o c o C O r H O i L O ^ r r H O O O ^ r

C O O r H ^ T C O C O C O ^ r r H

[Ncom^-cococMCMCM

c o ^ r i > o i i > L n o c O r H C O C T i W O C O C O H d t ^ t o c \ i c T ) c O L n ' s r c o c \ j CO rH rH

^ - L D L n o o r H i n c o c o ^ r Lnocvicoco^rcooco C O t> C O m 00 CM co C O C D r> co in CO CO CM CM

CM

o CD

CVJ t> C O co o o m CO m CO

in Lf) CO in in CO ts CM C O co 00 rH in o 00 S rH rH CVJ CM

C O 1 1 o rH

C O C O t-> o C O O [> o C D CO m rH co CO co C O C O o O rH CM co rH co CM CD CO mo CM in CO CO CM CM rH rH O CM O .

* • CD

o rH • CM CM O

II II II O rH C O •sr CO o l > CO CO Lfi CO i > C O O CD m o rH C O •sT C D O Q o CM C O O rH 00 <q- O O rH in CD U C O 00 CM 00 in CM rH rH C O 3 CVJ C\J C O C O in CO t> C O 3 -p

cfl In CD CD & •

9-E CD > , CD E H -P

E H •H CD CM Q <tf CO O CD CD CD -P CO C D L D I N CM m in rH -P CD O CVJ m rH in CO CM in CM CD rH O m o CM rH C O -P CO rH CO C D CM co o in rH d ,3 r< rH rH rH rH CM CM C O C O 1—1 C D >

c3 P Q

8 O ) O H ( \ l ( M 0 J C 0 C l rooooo o -o o o

0 u g CO < - 1 < - l T - H < - l t - < < - H < - I C \ J C \ J

6 6 6 6 6 6 6 6 6

a £

CO

LD

6

o o o o o o o o o

C M C O U D T V J O C T i C n o O C D

,Ci O J i n H O H C O O O l M

ai [ > O t D r \ j c D < n ^ c \ j o

U) o j ^ i n i o o n i n c o o j ^ o

d o o d o d d d o d o

00 i

<H O o r H

c o r - i c o n n n o O M » f ) H r o f f l o i c o c o o T O i M ^

C Y j c M c v i c x j c o ^ m m c o i N o o

00 I o

c o o o o c D r H o j ^ r m t N O c o O O O O H r H r l d r l W W d d d d d d d d d d d

00 rH I

<H o

0

-p T 3

O

CO

o 0 co

SS °:

o 4) CO

T3

cO CQ

0

CO 0 u

I— I

a CO

E in I

o

C D

in

c \ i o j o j c o c o ^ l " L n i n c D i > - c o

^ r O C £ ) H N ( D r O ( £ ) C O O > l O

8 i > m c D o o i > c o c n ^ O t > c o i N i D i n - s r ^ r r o o o o o c A j

^ C O I D W O ^ t t f i H O O C D t [ N r H C D O O O t > C 7 > l D

COOOJCDCOvHOCOLO^rCO ' f n r o w w w o j H H H H

r H ^ t - m ^ r c \ i o c o o o i n t N O toi>-CDCDOco<3-rOrHrH<3" i > - c o < - i i > > - i c o i > c D L n ' ^ o o CO OJ O) H H

m o o o o m c \ j ^ r t N ^ " c D i > o l / ) W N O ( D C 0 t ^ C 0 O O

c o c D i n ^ c o c M r M o i c u H H

CO o 0 CO

xO to I o

rH

S

CO > CD U o o j

CO o

CO

. CD I

CM O -p 1 t

CO > CO

o CD CO

rH <£) ^ I •P* rH

-P fi CD i—l 0 o 0

Q

M C

• H

0 CO.

0 -P 0

ID

d

•8 E - i

00 CD rH rH OJ C\] CD 00 CO 00 in rH 00 IN o O in 00 CD CD o rH rH rH C\l 00 00 LO CD CO rH

rH

CO \r OJ CD [> 3 CO O l> in ai CD CO m CD 00 00 CD CD m CO

9 O o in OJ CD CD m 9 00 00 CM OJ OJ rH rH

ID OJ in i> CO OJ CD m CO 00 CO O OJ 00 00 o CD 00 CJ> CD CO CO m 00 m CD o 00 ai in J- CD 00 t> tH O rH CD OJ CO CO rH r - cn CO 00 o rH OJ OJ OJ 00 CO in CD CD

CD CO CO CO m O rH CO i n CD CO CO f - rH CD rH i n rH rH 00 CO rH OJ CO i n m o i n [> O CD OJ [> CO CD CO i n m m 1 > rH 00 CD o OJ CO t> CD rH CO CD O CO

rH rH rH rH rH C\J C\J OJ CO 00

o 0 CO

o o

d o OJ OJ

CO I o

o

II II

0

CtJ f-l 0

E H - H -P CO

-P 0 O 0 rH O

rH -P CO

u)

CO O O

CO I O

CO rH I

<H O O rH

CD

o CD CO

T-H ^ to O T H T H T—1 < 1

P

CL," a?

o <L) CO

a]

o CD CO

P

a]

co o CD CO

CO

o

2

K

CO >

s o OJ

CO

o CD CO

CD I

c\j O - P r H

CO

>

o

CO

o CD CO

C O

CD SH

CO

K O H

>> rH

a C O

B L O

I o CM

C D

L O

II o

co C o

• H - P

'g CD

rH CD O CD O

• H SH

aJ CD

C O

fn CD

• P CD

•H Q

CM

6

o O O O O cn CO CD CO CD rH rH CM co CM CM CM CM CM

CO CO CO CO

o O O o O • • • • •

O o o o o CD LO co •si-

CM rH CO LO CD

CM CM CM CM CM

r H CO CD CO CO CO CM r H CM rH O CD CO

t> C D L O CD 1>

in o H w ( D CM C D CD <tf

C O O CO C O

[> co co [> "sT Ifl H H H

co o r H C D t co CM r H

^ CO rH t> rH rH CM CO "3" rH

rH CD rH O rH CO l > LO LO

CO 00 CD ^ O CD O CM LO CD

CO C O CD O CO rH CO

^ CO 00 r H CO

r-- co L O L O

CM CM CD CO O CM rH CM O l > CO CO CO "sT rH H C D in v co co o O co co rH rH CM CM CM

v_J. co l r H O

- P r H -9 E H

CM CM O t> O CO LO LO CM CD CO CO O rH •sT O CD CM IN rH CO CO O rH CO

rH rH rH

CM

O CD CO

CO I o C O 00 C D

d

•<H CO o o CO

• H

>

o o r H C O

CM CM

f J U CD

CD a & £ 53 E H

E H P

- P CD CD r H

r 8

CD U s CO CD f->

O H

>) rH Q

w C O

S LO I o

CM C D

in II o

CO c o • H

-p g CD

rH CD o CD o $

•H % CD

PQ fn CD

- P CD

P

CM

C O

6

•8 E H

O t <tf L O co co r H r H r H r H r H r H

6 6 d 6 d d ^ CD O CO CO 01 rH CM CO LO CO CM

CM CM CM CM CM CO

LO CO I N CO CD O

O O O O O H

d d d d d d CD LO !>• <3" O CD rH CO CO CO CD CO

CM CM CM CM OJ CO

rH t> CM CD CD CO 00 CO CD CM O CD CO [>- CD CO

CM O CO CO LO CO CO CD CO rH

rH CO 00 O CO CO LO ^ CO CO CM CM

CD rH CM LD CO LO 00 t S [N rH CM CO CD r H !N •rj-f CO CM CM r H r H

<CJ" CM O CO CD CD CM CO CO O CM

CO I > CO CM CO C CD f-« CO LO •q-

LO ^ CO CO Q O w n I D O t H H H H C \ l C V

CM CM ^ r H CO tN CO [ > O CO LO r H CO O O 00 CO CD co -X) L O ^ co

LO CM CD CO O rH CO CM CO rH CO CO rH CO CO ^ CO CO LO O O CO LO CO C - CD rH S j" t> O rH rH CM CM CM CO

LO CD 00 LO O O rH CM CM CO CO LO f v rH rH LO O t> CO LO <3" O- CO 00 CD O CM CO LO

CM

O CD CO

O O O O CO

rH CO CO • • CD

rH rH • CM CM O

II II II

s CD

JJJ ?H £ CD CD a & § >> 53 E H - P

E H -H P CO

P CD O CD rH O

r—j P CO

H O >

-p E H

& CD

E H

CD U

cd CD E

6 Lf)

I o

c3 CQ

U

a 9*

CO

C J ) < M I D C D C O W H O O i O

rH rH rH O rH

N rH i n O rH rH rH rH d d 6 d d

CM

MH

o CD

K

CM C D

i n 05

IN d

C D co I D O ^

CM rH O O O

CD

-p

E H

&

a )

CD

cd CD

E

CM

o CD

(0 c o

• H - P

-2 0)

I—I CD O CD

Q

C •H cd CD

CO

U CD - P 0)

•H Q

CM

•8 E H

CO

a,

a

co

"co d rH CD

r H

•8 E H

rH co C D co CO tO CM rH CM H O CT) C O N

CO Q O CD rH O O O O i O

in co o in to O o o o o r- iOr- irHrHr- lrHrH

c M ^ J - i n t o O c o m c o c M t o o rHrHrHrHCMCMCMCMCOCO^ d d d d d d d d d d d

CDrHrHOCOCOCOINtOCOtO O O O O O O O O O r - l r H

^ f O t O H N i o c o i D c o c n t n

8 N . m o o ( o t N c o c D < 3 " O t N coiNcOLn^rvcocococM

u cd

CO

CD

3 CO CD

OH

a w

CO

cd CO

d rH CD I—I

•8 E H

I D O O M J I r i rH rH CO CD CD O rH rH rH O O rH

• r m to co <-< r-\ <~t rH r-\ ^-t 6 6 0<66d

[N rH rH 00 O CO rH rH O CD O O

rH IN CM CD ^ rj-CD tO 00 tO CD CM O CD CO IN CO tO

00

CO <H O O rH

CO I o

• E

co

<H O U r H

(D

•P

o CO

S OH'

a!

o CD CO

- P

O CD CO

OH'

O H

CO O CD CO

A ) ^ I O

s ol

CO > 0

fn O C\J

CO o 0 CO

CO - I cvj O

P rH

CO

>

o

CO o 0 CO

I rH O

- P rH

0

'3 CO 0 u

O H

>) r H a a 5

LD I o

c\j a i

i n

o

co C o • H P>

g 0

r H 0 O 0

a

• H

0 C O

u 0 -p 0

CVJ

C D

6

i N O c o o o o ^ r c o r H i n o r H C V J C V J C O C O C O C O ^ r ^ i n o d d d d d d o d d

^rcococvjincOrHoococTi c v j ^ o c o ^ r o c o c o m c o c v j c v j c o c o ^ i n i n c o i N o o

O C \ J C O r H < 3 " O O C O t N C \ l O r H r H r H C X J C M C V I C O C O ^ L O

d d d d d d d d d d

^ • L O f t j c o o i ^ r ^ L n i f i C T ) c o m c v j O c o c o r H O c n c o

c \ i c \ j c o ^ r ^ l " i n c o t N C N C D

C V J r H C M C O C O O C n ' ^ ' r H ' ^ -r H c n O L n c r ) C 0 i > ^ " r H i > o c o i N L n ^ r ^ r c o c o c o c v j

i n r H O i n [ > i n i > o o c D c o r H ^ C 0 0 0 ^ H C 0 ^ " ^ l " r H l N

O c v j c o i n ^ - r H c o o c o ^ j -i n - ^ C O C V J C V J C V J r H r H r H r H

O O i O ' v f o c T i c n f f i i n H co N~ CM I N i n i n cr> O CM r j -

00 O CD CO CO rH

H en o- i n i n co

m o m c D C D ^ t i o i N r H i N N c o i o i n i o o i V i n ^ H

C 0 C M O J I > C 0 C 0 ^ - C \ l O C 0 c o i N i n c o c o c v j c v j c v j c v j r H

I N C V J I O O O C V J C D C O C D [ N C O O C D C O C O C D C O C D I N r H C M C V J C O ^ i n C ^ C O r H

rH O CD CO CO CO CVJ CD co CO o- O CVJ CVJ rH CO

m t> i n i n CVJ CO CVJ O IN CO m CO CO CVJ C\J CVJ CVJ rH

C O C O C V J C O C V J O O C D C V J C D C D s f i n o o c O r H i o c o ^ c o c o M - o o c o o i n o c M c n c o c v j c o c o ^ - L n ^ - i n o c n c o c o COOcOCOtNCvJCDCOCDIN r H C M C V J C O C O ^ ^ t L O i n C O

rH t> 00 o i n CVJ CO m CVJ m rH CO o CD [ N o CO O CO CO CO rH O o O CO oo cO I N i n

0 rH CVJ I N N CVJ CD I N 00 r H CD O CO co CO rH CO CD CO •9 r H rH r H rH CVJ cw CvJ CVJ CO

CVJ

o 0 CO

CO I o r H

o o o o q d o OJ CVJ

i n r H o

0

il g 0

0 E H

- P 0

c!

• H CO o o CO

• H >

CO

C O

1 <H O O rH

C O

" 1 • E o rH

C O

rH 1 < H O O- rH h % 3

C O % C M

CD

re II

CD

o CD CO CO

CO •

•p s CH

rs O H

>> *o re f—1 Q.

a o CD C O

to E

- 2 lO O I < 0-, o re rH Q re"

C M

en o •

CD CO

II rH •P S O

T3 \ OH

• •

T3 re !0

o •H o -P cti

CO £H

'O to CD

i—i i CD

o O rH Q

bO C • • H

S OH CH

as re CD C O

CD

J ? -P CD

?J o k CD TO ^ to • H

o Q C M to

1 C M O C M

•P rH

o CO

5) w 6 & ° rH ^ CD

O W

rH ~ CD

i-H o rH O •9 -P rH E H

I N O rH.CO rH CM CM CM CM 6 d d d 6 CO CO CH O rH H CM lO 0 0 H CM CM CM CM CO

O CM <tf CO I N

rH rH rH rH rH 6 6 6 6 6

( O L O C O CO CO

CM co I N en CM

CM CM CM CM CO

H n o H C O CM CM CO N O cn oo tN co

tN m CD co CO CO CO CM

O O) ID H CTl LO CO CO CO CM

E N C D in co rH O I N J" en co O O cn co t n OJ H H

CM O CM I N LO ^ CO CD CO O O O rH in cn I N to in

O O O en O rH CO CD 00 CM rH rH rH rH CM

C D O C O CD C O C D I D rH rH CD tN rH CD CM

CD in m "sj" ^H"

O ^ rH CD IN CM Q CM CM O O O rH CD 00 O CO CD 00 rH CM CM CM CM

CM

O CD 10

ro i

o rH

CO CO CD

•rH to o o to

• H

>

O tN CM CD CO CO t CM ID O CD M" CD CM CD CO CD CD CM 0 0 O rH CM x-\ r-\ <H r-\

O O rH C O

C M C M

II II

CD CD k

it U CD CD &

& 1) CD E H

E H •P

-P CD CD rH

rH -P .C 2

c3 C Q

CM

a 9 CO

in l o

C M

cn in

II

o

o • H -P g CD rH CD O CD

Q

bo c •H CH cci (D

QQ

CD 4-> CD

• H

O

C M

O

CD rH

•9 E H

I D co o CM co in rH rH CM CM CM CM 6 6 6 6 6 6 o m IN CM CO CM rH CM <3" CD CD CO CM CM CM CM CM CO

O rH CO 1" tO 00

rH rH rH rH rH rH 6 6 6 6 6 6

O CO O CO CD o CM CO CD tN o in CM CM CM CM CO CO

m co I N co m C O C O CD C O O C O O C D C O IN tN C O

rH C D CD CM CD tN C O C O C D O CO C O

rH C O C O O IN in co co co CM

in C D o m o rH CD C O CO CO CO (N rH C O tN CM CO <tf C O CM CM rH rH

C O m CD tN O tN CM m C D co tN in C O tN in CO 0 0 CM C D IN CO m ^J" "xT

CO ^ tN CO CM CO O CM ^ CD O rH rH rH rH CM CM

rH in o m co rH CD in C\J IN 0 0 O en CD CD CD ID <d- CO

CM 0 0 rH CO CO rH rH CO o CO tN CO rH CO ^ rH CO CO IN o CM rH rH IN CD CM tN O rH rH CM CM CM CO

C M

O CO

CO I o

X o o

O O CO rH CO 00 • CD rH rH •

CM CM O

II II II

CD

U CD

I f . 8

tN IN <3" tN O CM co 00 CD CD rH rH CD CO O rH m 0 CD rH CM CO m

rH rH rH rH

CD E H

•P CD

-P CO CD O

rH O -P to 3=5

<D E H

CD

e v '

OJ

<H

o CD

ca

CD

H

& CD

E H

Cti CD

e.

OJ

o CD

«

CD K

I O O J

in ii o

C o

-H -P g CD

f-H CD O CD O

rf •H SL, Oj 0)

ra CD -P CD

Q

O J

5H ctf

C Q

OJ II

CD U 3 CO

2

w cn

"o rH

6 r H

CD rH

•9

m O J

II

CD SH

§ CO CD f-,

O H

>> i—l

a w

3 en

6 r H

CD rH

•8 E H

CO rH r j " rH O O O

l > O rH 00 t rH OJ O J O J O J

d 6 d d 6

m oo CD oo ^ rH O O O O

H n o t H CD OJ OJ CO ts O 00 00 CD

cD^rojLn^-ojuoLncD o o o o o o o o o r H r H r H r H r H r H r H r H r H

O - O O O O O O ^ C O r H i n O r H o j o j o j c o c o c o ^ ^ i n

6666606666

c o o o t N c o O r H o m t N o j

r H O O O r H r H r H r H r H O J

r H r H r H r H r H r H r H r H r H r H

W H O J C O C O N O I ^ H en o in cr> co o O c o i > i n ^ r , T c o c o c o o j

IN

CP

a w

CO

cC

o

CD r H

• a E H

CO 0 LO rH o O J r H O O O O O

CD 00 o O J 00 m rH rH OJ OJ OJ OJ d d d d d d

CD t s co 00 m r H O O O O O

LO CO C ^ CO if) M' CO CD CD CO O CO O CD CO O- IN ID

n c o H i D o i w c o w L n o i w i n c o H i x ) O O H H H w w n o n t ^ ' J i n i n

d d o o d d o o d d d d o o o

ro

o o c^cocM^fcoooooi>cMr--'3-cDt>coco cMCMCMCMCMCMCoco tsTLnLDcot -oo

,7 o

c M L O C o ^ i N c j i ^ r i N r H L n O t D r o O O H H H w w w n c o ^ ^ i / i m i O N o o

d o d o d o o o o o o o o o o

CO rH I

C H O O rH

c3 CO

C O C D ^ C ^ O C D C D C D C D C D C M C O O O C D

cMCMCMCMojcoco,st"^i"LOiDcO!>oocr>

a>

- P

•a

o CD co

a]

o CD CO

C5 S

o PS

o CD CO

- P

-a p.. PS

CO o CD CO

CD I

o

S pj PS*

CO >

o CM

CO o CD CO

CD I

CM, O • P rH

CO >

o

CO o CD CO

CO I

o

t CO CD

a

LD I o

o CD t >

II O

a o •iH - P

CD rH CD O CD

O M C •H fn Ctf CD CQ

CD - P 4>

•H Q

CM

CD rH

•9 E H

O i D H O o i o r o c o L n ^ w c o c o c o ^ C D C ^ C O C O C ^ C M O L D O C O C M C O I O C M C D < r ) r H O c o i > t > c D i D L n ^ i - ' s r c o o o c o c \ j

O O O C M C D ^ T C O ' r t L n r H r H r H C D O O C D r H C V l C D C D ^ C D ^ C X J r H O J C O L n c O r H L n O O

rH rH CM CD

C M C O O C D C O C O ^ r c O C M CO CM CVJ rH *—I rH rH rH rH

OICDWHCONHOlN'LnHiDOCKH LnocD^j-ojcocMoocM^rcococo^r C D O ^ C D C O r H Q O I N C D L n ^ r c O C O C M C M 00 CO CM rH rH rH

CO CO CO 00 O CM CO CM CO

C M C D ^ L D C D O O U O C O CM CM I D CD L O N ^ J "

rH CM

rHrHt>CD^t"O^CMOOOlOlD\t"COCM OI>lDCOCOCOC\JCVIOJrHrHrHrHrHrH

c O c O C D L O L O t N C O C O C D C O C M C O L O r H Q C M C O ^ C D O J L O C O C D C O I N O O v J C D C D C O r H r H r H r H O J O J C O O O ^ T t D I N C O C D r H N

CD C O

CD CD CD LD CO CO 00 O CM rH CO ID CM CM CD CO CD O CO rH co CD 10 CD CO CD CD rH rH ID CD t> CM O 00 I > ID CD L O LD CO CO CM CM CM CM CM rH rH rH

CM 00 Q LD t N o CO LD o CD !>- CD CVJ CD I D LO rH < j CD 00 CO CD CM rH r H r H rH LD rH CM CM o 00 LO 00 CO CO CM o rH CD CM CD CD CM LD rH I N o CM CM LD CO O CO CO rH o CO CO CO co CD LD CO rH CM CM CM CO CO 4 LD LD CO CO CO

00 rH O o CO LD rH CO l > CD rH CD rH rH CD CO CD CD CD 00 CD O CM CD CD r H r H LD t > CD rH CD CD CO LD CO CD 00 00 CD LO rH O LD I N O CM o <2 rH CD LD LO CD o rH LD [>• o CM CD CD rH -sT rH

rH rH rH rH rH CM CM CM CM CM CO 00 CO •3"

CM

O CD CO

00 I o rH

O O X

o o CD CD CM

• • O CD CD •

II II II

5 t U CD CD Q. 6 S >> S E H - P

H .H - P CO

• P CD O CD rH O rH - P CO a 3$

n c o H ^ a c D w i o c o c M i n Q c n i D O c o

o o d d d o o d d d d d d d o d

o o rH^rHCDrHiocooicDcDCOLncDOcMtN N c o o j o j r ^ i o c o o j ^ r o o o j c M i n c o o j r H ojcMOJCMojojojcocooo^inincotNOO

CO I o

C M i n O O r H ^ I N C D O J L n o O r H l O O J O O l O m H H H C u r \ i c \ j r v i c o c o o i ^ ^ L n L n i D i >

d d d d d d d d d d d d d d d d

. CO rH I

<H O CJ> rH

i n ctf CO

c o o i o i O i n n N M ^ r ^ ^ H H C O c o i M ao^rcoincDCDrHincoojcoiNrHcococD c v j o j o j o j o j o j c o c o c o s r ^ i n c o c o i N c o

0)

4-> T 3

-p

O

a.

o CD CO

S OH'

o CP CO

oj

CO o CD CO SO ID

I

O rH

CO

> CD fn

O CVJ

CO o CD CO

CD - I Cvj o

P rH

CO >

E o

co o CD CO

Zl CD I

rH O •P r H

CD

CO CD U

D H

>, i-H

& CO

I o rH O CD tN

O CO c o

• H -p

CD rH CD O CD Q bO C

• H SH CTj a> CQ u CD -p

Q

CM

CO

r H

d r H CD

rH

•9 E H

O Q C O C D I N r H ^ t N C O C O C O ^ t C O I N r H C M C D O ^ ^ ' t £ > C 0 H r | c n r J ; O L n H N r J - H c o o j o c D c o N i N i o i n i n i n ^ r ^ c o c o c o rH r H rH

I N C D r H O J C O C D r H L n c O C O I N C O C O C M C O t N O J O J C O C O r H C D

i n co co C D CD co

C V J I A J r H C D C O r H C D l N C D L n C O ^ C V J r H O O CD *vj" CO CVJ CVJ CVJ rH rH rH rH rH rH <—I rH rH \—!

c v i c Q i n c o i n c ^ L n c D i n o o c n ^ i n L n c D L n C O C v J C D C O t O C O r H C V J C n C O C D O C V J C D O c D

CD r H CO CD CD CO CO CO CVJ rH rH rH

r H C D t N C D i n L n ^ r c o c o c v j

C T i l N C D C T l C O C y t D H C O l D O ^ ' O ' J C V i a i C D C D r H ^ C O C D C O C D C V J O O O t N C O O C D

r H C O L n c D O ^ O C O ^ C V J O C D C O m ^ c V J O N m ' J T C O n C V I C V I C M C V I H H H H H

o CD !N CD i n IN CO CVJ CD O rH CO co CO CO CD CO CVJ CO CO CO N CO o

rH

CO r H

IN 00 CD rH m CVJ o CD CO o CO CO CM 00 CVJ CD O CVJ O IN i n rH CD rH rH CO CD rH CD C0 o CD CVJ CO rH CD CD CM O CO CD m CO i n m •sr "r CO 00 CO CM CM CM CM CM r H rH r H

CO CO OJ CO CO CD OJ r H o rH CO CM CO CO o CO rH m rH CO CM CD I N CO i n CD O rH CD CD i n o CVJ i n i n CVJ CM I N CO CD CM rH o CD o rH i n i n O "vf m i n o CM CD o CO CO CM CM o 00 o CO co CO rH CO rH 3: CD CD i n CM CD rH CM CM CM CM CO CO CO 3- m m CD IN IN

rH rH r H i n Q CD rH 00 I N OJ CD 00 CD CO o I N O rH CO 00 rH O co CO CO 00 i n

i n I N rH o CD i n CM CD CD CD CM CD o ^ - i n o CM t N o OJ N - CVJ O m ^ rH t N CD o <3-CD o r H CO < t i n N - CD O OJ I N CD CM CD CD

rH rH rH rH rH rH rH CM OJ OJ CM CM CO CO CO

CM

O CD CO

CO I o

O O X o o CD CD CM • • o CD CD • rH r H r H

8

CD

E CD CD E H

E H -P

- P CD CD i-H

3 3

• H CO o o CO

• H >

127

p

& CD

E H

W> to

E LO

I o

to O H

>> rH

a CO

CO LO O 'rj- C\l rH O rH O O O O O 8 0 cn a> •sr to t>- co

O CT> o o o o o

o

n t O H ^ i O C T l C J ^ C O t M l O O C O t D O C O

O O H H H H c \ ] w w r o n ^ ^ " j i X ) i r ) 0 6 6 6 0 0 6 6 6 6 6 6 6 6 0 6

<H o

cn

t--

11

o

co rH

6

00 t o c o L O ^ c n ^ ^ t o c o c r i L O c n r o c o H O O O O O O O O H H H W W

rH rH rH rH rH rH

CD

& CD

E H

- a

**> to ctf

OJ

O

CD K

CO C O

•H - P

CD rH CD O CD

Q

£ •H fn Cfl 0

CO

SH CD

- P CD

Q

CM

• 9 E H

CC CQ

CD J H

CO CD

O H

>> rH D H

00

v5 CM rH

6

O O t D C D C v - r H ^ - [ > C O C O C O ' ^ C 0 1 > r H C M C O O ^ ^ t O C O r H ^ - C D T O L O r H I N ' ^ - r H C ^ ) C M O C T > C O C ^ ( > C O L O L O L O ^ r ^ r c O C O C O

\ r co CTI *t CO CO rH O

8 8 C O C D r H r H C O ^ C O C O t N c n c n O O O O O O O

r H r H r H r H O O r H r H r H r H r H r H r H

C O C O r H ^ O T C M C C C M L O C T l C M L O C O r H L O O O rH rH r H C M C M C O C O C O ' ^ ' ^ ^ L O L O

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

^cooLOCMcocM^rcoo^r C O C O C M O O O O O O r H r H

CO rH LO CO CM CM CM

H d H H H H d d H

O H

•8 E H

O C O r H O C D O C O C O L O ' r J - C M C O C O C O ' ^ C O r ^ C D C O t > C M O L O O t O C M C O L O C M C J i C T l r H O C X J C ^ I N C O L O L O ' s r ' S j ' C O C O C O C M

CO CO CD CO U D H I D H C O I D O I C O O O H H W C M C O ^ T L O L O L O L n t O

d 6 o o d 6 6 6 6 0 0 0 0

O O

^ c o i D i n c o c n O i D O c o c o o n Q o o c o o j r M r O L O O c o u T r H [ > c o ^ -

c v i c x i c M C M C v i c M c o ^ r L O c o t o t N c D

co o l

• E o

ro r H l

< H O U rH

SH

CQ

C\J

i n i D c n i n w M D w H O O W H H H w n ^ - ^ L D c o a i O o J O i j v i 6 6 6 6 6 6 6 O 6 H H H H

C D v C l O O O t O L O C O r H C O C O L O r H O l O l O l D f f l O i n i D ^ H O l O f f l

C M C O C M C M C M C O C O L O C O I N I N C D O

CD O S

CD £H

• r r c O C M O J C D ^ C D O O t O I N C D C D • q - O r n c D c o c n i N t o c M C O L O C M C O C O C O < - H C T ) C 0 1 > - C O ^ r ' s f C O C O C O C \ i

-p

o CD CO

O S

o CD CO

P O S

o CD CO

-p

OH'

OS*

CO 0 CD CO

CQ

>

O O J

o r H

s OH'

O S

CO o CD CO

O J O P r H

CO

> s

o

CO o CD CO

. CO I

1 o

CO

t O H

>, rH a

s* CO

E LO

I o

o CD

I N

CO c o

•rH p>

CD rH CD

o

p

c • H

c3 0

CQ

SH 0

- P 0

• H P

O

0 r H

• 8 E H

H CO H H O l <J H H C\| LO Vf H CO

CO LO CD CO OJ O CO ID O ^

C O O C O C D ^ r C v J C O C O C O C V J r H r H O O O I C O W W W H H H H H H H

C O ^ C O C O C O I N r H C O O J t O O C O C O ( T J t O ^ O ^ O r H l N C D C V I t O C M i N C M

i f i a i i n O i D C o o i ^ ^ c o c o c v j c y LO LO OJ OJ rH rH

C D C M C M C D C D C O C O O C N r H C O L D C O " J c o o ^ ^ c o m c o a j o c o c o i o

COCMCMOOLOCOOONlD'JCOCM ^ " ^ I D i n ^ C O W H r i H H r i H

r H LO O C\J 'rj- LO CO r H r H LO I N O CO O

CM rH CM CD CO CN CM CO CM CO

co I N CD O CO t N r H r H r H

LO I N CO CO CM LO r H Q CD •sr CO CO CD CM CO CO CD 0 CD LO I N 00 CO CM I N r H I N CM r H 0 CO CO LO CO I N 0 LO •sr co CO CM CM r H r H r H r H

N . r H O L O CO CM r H CM CO CO CO CO r H CO co CO O I N 00 0 CO L O CO r H r H O L O CD CO CO CO CO CD O CM CM CD CO CN CO CO CD O 0 I N CO CM CO O J O LO LO CM LO 00 r H CN CD LO O I N I N r H r H CM CM CM CO CO L O LO CO I N I N CO

CO 00 CM rH CO LO LO O LO O CM rH CM CD CO CO CO 00 CM CO CN CO 00 LO rH 00 LO LO CM LO O LO LO CO CO CO CO 3" CO LO CO CD LO CM LO I N I N rH CM 3" LO CO I N CD CM LO CO

rH rH rH rH rH CM CM CO CO ro

5

9

CM

O 0 CO

7 O

X o o

O O 00 CO CO CD

• • CD o o • CM CM O

II

0 £

U 0 0

& 55 >> 0 E H -P

E H P

- P 0 0

M

CO O O CO

• H >

CO

CO

o o

CO I o

CO rH I

< H O O rH

p X )

o

to

Q

o CD to

- p

o CD CO

to o 0) CO

tO I o

PS

0)

>

o CvJ 10 — I CvJ O

P rH

CO

> CO O 0

o CO

rH ID |

ID | o

P rH

3 PQ

CVJ

II

W 0

PL,

>> i—t

CO

LO I o

o 00

o • H - p

CD I—I <D O CD

Q

bO C

• H

t3 CD

CD

0) P CD

CM

ID rH 6

t D r H i n o o c v j u o c o c o c o

O c o t D c o r - c o c \ j ^ ) - | > -0 0 L D C v i c v i r 0 L n i > - o , : 3 "

C M C M C M C M C M C \ J C M C 0 C 0

00 CO 00 rH CM CM CO

LO O ID rH LD LD tO

0 0 0 0 0 0 0 0 0

0 ) l D C \ l t \ l H C O t \ ) 0 ) C O c n o - L O t o o a i c M L O O

C M C V I C M C M C M C M C O C O ' ^ r

^ l > C M 0 0 O r H ^ I > O • ^ - C M r H 0 0 0 0 C 0 l > t D l O

C M I > - O L O L O O O I > C O C M c o t D o c D c n o o o ^ T c M C O

O r H t D c n u o c o O o o » M o n w w w o j H H

^ C O ^ C ^ - L O r H C M C O C O r H i - i C O t A J ^ C M t O C O C T )

C O ^ t O r H C ^ ^ T r H O O O V CO CM CM rH rH rH

• s T r H I ^ C M C O C O O r H C M

C 0 t D C 0 r H C 0 I > C M C J ) L O r H C O t D L O ^ r c O C O C M C M

C O L O t D O C M C M O r H O C M 0 0 ^ r t > 0 0 C M L 0 O I >

CM rH 0 co t o CM CM 00 to CM CM LO tD 0 00 rH CO CO 00 LO CM 00 to to LO CO CO CO CM

00 CO CO <3" co CM CO ID ^ -to to LO CO CO t o t N 00 CO 0 t > CM 00 t o tD 0 CO t o CM rH O LO l > t > CO t > 00 CM LO 0 co tD rH rH rH CM CVI CM CO CO CO

— CO "3" t D CM CO LO CO CM CM t o LO CO CM O 0 00 t o to O) rH [> ID CO t o CO

tu t D O rH LO CO CM CO CO t o rH CO OO rH CM CO LO t o CO 0 •9 rH rH rH rH rH rH CM

CM

O CD to

CO I o

to o o

^ - 9 r

d d CM CM

II II II

s B CD

CD E - i

P CD

8

CD

E-H

p <D

• H CO o o CO

• H >

130

CO

u o

CO CJ I

E O

CO

^ 1

C H O O rH

CD

- p

O <D 10

o

03

Q K

-P T3

O 0) to

a] PC

to o CD to

/ O CO I o

5

o CM

to o 0) to

ID c\ j O

- P rH

>

o

to o CD to

I rH O

- P rH

0) U

to <u ?H a, >, i—i a

CO

£ i f )

I O

o cn

N

to C o

•H •P a) f-, CD

rH 0) o <D

Q

M |

cr

a) CO

rH

- p CD

a

CM

"5

O rH

oo CM co O O r H rH

d d 6 d

CO rH CO 00 tD CO CM

CM CM CM CM

i n o i a ) rH rH CM CM

d d d d

CM I N I D L O O oo m i n

CO CM CM CM

i n i n oo co •vj- rH CM CD CD ^ CM O

co CM oo m co m cn O

co m co i n 0") CD "sT CO

co cr> n r CM

^ O CO CM

I D H H LO i n co CM

CO rH CM CO I N I N CO CM

co t o i n o < f O M D

H m i n H rH CM 00 Lf)

CD m i n co m co CO I N 00 CM I N t D i n m

co m <-i cn I N O O < H O cn O CM co I N m cn i n o CM H H t \J C\J

rH O 00 CD ^ 00 t D 00

CO i n

o CD to

co I o

00 CJ) CD

d

i H to o o to

•rH

>

o o CO

d CM

g 0

CD E H

- P

U O 00

d CM

c3

CQ

CM

c3 CD m tD

- P

(N 00 O < H

0 I N

CD rH

& d tl) r H 00 cn

E H -—x CO o tN 00

-P <u CD I N CD r H I N 00

r-H X ) c ctf

M E H

3 q d d

rH Vj" cn I N

CM CM

t o cn <H rH

d d

I N CO

O cn • •

CO CVJ

I N CO I N CO

co o N- N-

rH CM i n oo I N tO

CM CM en tD I N < D m i n I N

o tp to

CO I o

00 cn cn

i n I N r H 00 t o >> a -p a •rH 3 to

CO o o to

E •H •) i n rH > I rH rH O • • r H rH CM

i n

O o

O o CJ) 00

L N d CM

II CO rH

O cn 00 II

CO cn • > CO O CD to rH rH u c p o - p •H - P CT> CD SH & (1)

r H t o •^r <D CD i H CM EH

u rH rH CD - P

O CD r H - P

c • H i n CO o

o 00

d CM

CD E H

-P CD r-H C

M

c3 CQ

CM

w cn

i n I o

o CD

I N

I I o

to c o

•H -P

g CD

r H CD O CD

Q

gJ •H

tD

pa u CD - p CD

O

OJ

CO

rH d rH CD

rH

•a E H

00 CM CD O rH rH

o d d

I N CO CM' i n CM CM

CM CM CM

o < t o CM CM CO

o d d

N - CM CM I N i n i n

CM CM CM

CO CM CM cn cn CO CM O rH r H r H

t o co 'cr O I N CO

CM ^ -sT CO CO

O cn ^ co c n cn

a ) rH CO CO CM

i n co co oo I N m

r H co cn O M n

m co CM CM co i n

< H cn ^ ^ H O t o cn CM co i n i n

I N i n co CO CM CD co I N i n

S CM O

O CO rH CM CM

r H CO CO co c n CM CM CM CM o rH i n CD O rH

0\J

o CD to

O

CO CD CD

O

•H to o o to

•H

> o o CO

d CM

O o 00

d CM

131

CO

o o

O I • 6 o

CO t H I

< H O O t H

CD K

P

o CD CO

o 0 CO

-p •a

o

CO

CO o CD CO

I o

CO

I o cvj

CO o CD CO

CD I

o j O -P t H

CO

>

o t H

CO o CD CO

CO I

t H O P t H

a} CQ

II

CD

CO CO

e a, >>

r H a % m

I o

o CD

o

s o

• H -P

CD i—i 0 O CD

Q

CD cq

f-, CD

p 52

Q

o j

i H

6 t H

CD i—l

•8

CO CD CO CO O J L D CO t N CM q q t—H q q H O O H

d d d o d d d o d d

Q ^ CD CO CD O J O J 00 CD O J CO I D CO O J 00 C N CO

O J O J O J O J O J O J O J O J O J O J

CO O CD O J co o ^ CD O J i n CD co t H O J O J t H O J O J O J H H W

o o o

CO "st LO CD N m

O J O J O J

CO CD CD CD OJ Lf)

O J <H O CD 3

i n O t H O J 00 CO

O J CD 00 CD CO O J

00 CD t H t H O

• • • i n i H CD CD o CD t H <H

CO ^ CM O CM t j

CO CM t H OJ o CO CD CD 00 CD m

CO CO t H

t H

"vl - OJ I N CD O J O 00 t H t H O J

Q i n o ^ I N m O O J o O J o I N I N CD O

o CD CO

CO I o

CD m CD N - X CO t H

t H t H t H CO CD CD

O

II

• H CO o o CO

• H

>

o o

00

d O J

0 U B g 0

0

P 0

i H P

3

o o

00

d O J

a

w m

O CD

o

CO C o

• H p

0 1—1 0 o 0

p

2 • H c3

O J

O O J

d

0 i—i

E H

O O O O

CD CD CD t H CD I N i n m

O J O J o j o j

CO I N CD 00 00 CO t H

t > CO OJ t H t H t H t H t H

O 00 CD O J 00 I N O t H

CO t H L O CD O CD ^ CO

CD t H O J CD O ST I N O

O J CD t H CD CD CO CO O J

00 CD t H t H CO t H 00 O J

i n t H cD O J CO O I N CD

t j co m co O CM CO "3" t H rH t H t H

00 CD O J N -O O CD O J CO CO CO CO co co i n m

td" O J t > CD ^ - CD CD CO \ - m CD O J < r t H co m N CO O O J t H t H CVJ O J

O CO CO CD I N t H CD CO t H o o i n O J O t H O J O- CD O t H

t H t H

o 0 CQ

CO I o

00 CD CD

d

• H CO o o CO

• H

>

o o CO

d CVJ

0

u B g 0

0 E H .

p 0

f-H p

o o 00

d O J

8 B g 0

0 EH

P 0

c3 PQ

O J

II

a

9 co E

i n l o

o CD

t N

CO c o

• H P

g 0

i H 0 O 0

O

sl • H

c3 0

CQ

U 0

P 0

O J

O J

d

•8 E H

O O O

CD CD CD CD CO m

O J O J O J

CO O <H i n co co CO O J

O J t H O CD O CO

oo I N CD co

m ^ o t H CD CD

CD t H O J i n ^ j - co

CD CO N -O CD O J

t H CD O i n o co

O m co t H O J CO

CO CO CD O t H O CD CO O I N CO CO

"3-

s t>- m CO CD i H CO CO CD

i n I N CD

CVJ CO t H CD Q 00 00 O I N i n co CD N . 00 CD

CJ 0 CO

CO I o

00 CD CD

d

• H CO o o CO

• H

>

o o 00

d CVJ

g 0

0 EH

p 0

o o CO d CVJ

0 EH

P 0

-p

EH

0)

as

O

CD K

+ J EH

cu E H

X)

CD E

O J

^ H

u

e I

o CD

w c o • H - P

g 0) I—t <D O 0)

o C

•rH

c3 <D m u <L>

- P 0)

OJ

c3

m OJ

II

8 S

to

>>

rH

CO

0) <—I X ) cd

EH

W O N H O O N M B o a w H O O a i m o )

O J r H r H r H r H r H O O O

c D r H m c D C M i n c o c o c D

6 6 6 6 6 6 6 6 6

w H o o n a i O H n i n O c D C M r H O O O O O

C M r H r H r H r H r H r H r H r H

• q " C M < - < C D C D C 0 I > t O l D

Bar

Bar

Bar

CO rH I D O Q O O CD rH O CO CD CD O x—i CM CM CO CO CM rH o O CD O CD o O o O CM CO CD OJ

II CM CM rH rH rH rH o rH O rH rH rH rH II CM rH rH rH

CO m 1H

W

w <u

PH co t n CD CO CO rH CD rH CO CD CD CO CO CM CD PH o O ' ' rH CM CM CO i n m m i n CD q o rH rH

>, o O ' '

>> • • • o O O O O O O o O O O O o rH 6 o o o a a P .

to CO

^ ^ CO rH CD CM CO CO CM CM i n CD rH CO CD .—• CD o CM

as CO CO CM rH o o O rH rH rH CM CM CO OS CO CD CM v s • • • •

CM CM rH rH rH rH rH rH rH rH rH rH rH I D CM rH rH rH <H rH

6 6 rH rH s ' V '

CO CM CM CD CD O O CD [> CD CD i n i n CD CO

le

3 - O *H CD CD CD i > CD CM CO m CM CO CD rH CM CD le

CD ID rH CD CD !> ID CO CO CO CM r-H CD CM O

•8 rH rH rH •9 rH rH rH rH CO

E H

133

a -

-a

^ 3 a) 4>

o $2 cr;

a) cr;

6

a)

-a

< * > 3 cd CO E

O J

< H O

Si

cr;

E i n

I o

o CD

L N

II

o

to C O

•H + J

t CD i—I CD O CD

Q

CD CO

u CD -p 92

a O J

CO

O J

3 to

a, >>

i—i a a ! r CO

CO rH

<D r-H .Q CO

CO rH CD I N ^ ' rH

d H H

CO O l CD O rH rH

d 6 d

CD CM rH i>. CM

CO CM CM CD CD CO CM O

CO

CM

II

3 to

a, >>

r-H a g j

CO

o CM

d

. q co

EH

u CO co­

CO O CD co CO CD

CM • • CM CM rH

II II

<D CD U U D CO to CO CO

t N s OH q q OH

>, d d >> r-H rH a D

a a 3 CO CO

- o CD -

crj CO CD cd • • •v '

I N CM rH CD rH rH

d d rH rH V ^ »—'

I N CO CD I N £2 CD

rH L O rH X2 rH rH £> Cd (ti

E H EH

I N O CM CO CO O J

CM rH rH rH

CO 00 CM LO

q o rH rH

d d d d

t N o co CO < T CM

CO I N CD 00 CO CO rH

I N CO OvJ rH rH rH rH rH

3 O J

CD "3-I N CO

CO CO

q q rH

d d d

3 CM

t N L O I N CO

U CC

CO

CM

3 to

u OH >i

rH

a &

CO

3 r H O J

d

CD I D CO CD I N CD L D

I N CO rH r-H CD rH rH rH -9 rH

L D CO CO CO CD

CM rH rH

CO I N CM

q q rH

d d d

i f ) CO CD CO CD

O J rH rH

9 CO CM

134

10 . i i ) 3" D iameter R i g

10:22

10:23

10 :24

10 :25

10:26 - 1 0 : 3 4 ( a )

10 :35 - 1 0 : 4 0 ( a )

A i r B e a r i n g c o r r e c t i o n s

D i s c Drag d e c e l e r a t i o n s

Mass f l o w r a t e s

E c c e n t r i c i t i e s

D e c e l e r a t i o n s o f the 3" R i g c = 11 .48 1 0 _ 5 m

s u p p l y p r e s s u r e s = 1 . 0 , 2 . 0 and 2 . 5 b a r gauge

D e c e l e r a t i o n s o f the 3" R i g . D e c e l e r a t i o n s

s t a r t e d i n the t r a n s i t i o n r e g i o n ,

c = 11 .48 10~^m s u p p l y p r e s s u r e s = 1 . 0 and

2 . 0 b a r gauge.

135

T a b l e ( 10 :22 ) 3 " D iameter S h a f t D e c e l e r a t i o n i n the A i r B e a r i n g :

N N + l o g e N + Noo t R .P .M. R .P .M. R .P .M. NQ + Noo s e e s

3000 346 3346 - 0 .2615 4 5 . 4 2000 346 2346 - 0 .6165 107 .75 1333 346 1679 - 0 .9511 166 .6 1000 346 1346 -1 .1721 205 .8

Supply P r e s s u r e = 4 . 0 b a r gauge

A l l Runs s t a r t e d a t 4000 R . P . M .

S h a f t d r i v e n a g a i n s t s e l f d r i v e torque

The above t i m e s were measured w i t h a s top w a t c h . The s p e e d s were

measured w i t h a s t r o b o s c o p e s e t a t a p a r t i c u l a r s p e e d , 4000 R .P .M.

( a c c u r a c y i yz%) . The a i r b e a r i n g was found b e c a u s e o f a mach in ing e r r o r to

p o s s e s s a s e l f d r i v e t o r q u e . The e q u a t i o n s o f mot ion f o r the s h a f t

d e c e l e r a t i n g a f t e r b e i n g d r i v e n a g a i n s t the s e l f d r i v e torque a r e ,

I p dN = - kN - k N ^ d t

where k = 2 -w R3 L m R"

dN

( N + N ^ )

dN

k d t I P

k d t IP

l o g e (N + N ^ ) = - k t + c o n s t a n t I p

when t = 0

logg ( N + N ^ ) = c o n s t a n t

l O g e ( N + N o p )

( N Q + N ^ ) = - k t

I p

From t h i s e x p r e s s i o n the f r i c t i o n a l to rque from the a i r b e a r i n g was

found and t a k e n i n t o a c c o u n t i n the d i s c d r a g d e t e r m i n a t i o n .

136

T a b l e ( 10 :23 ) D i s c Drag D e c e l e r a t i o n s

DISC DRAG t 2 (50 r e v s ) t i (25 r e v s ) b N Ok c D

10~6 s e e s 1 0 - 6 s e e s 10~6 s e e s R .P .M. R . P . M . / s e c R . P . M / s e c R . P . M . / s e c

215590 431508 328 6952 48 .99 3 . 3 6 4 5 . 6 3 223945 448232 342 6693 4 5 . 5 7 3 . 2 3 4 2 . 3 4 244426 489226 374 6132 38 .39 3 . 7 6 3 4 . 6 3 259145 518689 399 5784 3 4 . 3 6 3 . 5 5 30 .81 274880 550184 424 5453 3 0 . 6 0 2 . 8 0 27 .80 291393 583236 450 5144 27 .26 2 . 6 5 24.61 306618 613713 477 4888 24 .80 2 . 5 3 22 .27 320753 642006 500 4673 22.71 2 .42 20 .29 335943 672413 527 4462 20 .83 2 . 3 2 18.51 352992 706540 556 4246 18 .95 2 . 2 2 16 .73 370450 741488 588 4046 17 .34 2 . 1 3 15.21 391040 782702 622 3833 15.59 2 .02 13 .57 406901 814452 650 3684 14 .46 1 .95 12.51 423204 847083 675 3542 13 .35 1.88 11 .47 439227 879160 706 3412 12 .49 1.82 10 .67 456266 913268 736 3285 11.61 1 .75 9 .86 472363 945488 762 3173 10 .84 1 .70 9 . 1 4 488683 978156 790 3067 10 .15 1 .65 8 .50 507650 1016119 819 2952 9 . 3 8 1.59 7 .79 529272 1059395 851 2832 8 .60 1 .53 7 .07 554762 1110426 902 2702 7 .92 1 .47 6 . 4 5 577232 1155412 948 2597 7 .39 1 .42 5 .97 600288 1201572 996 2497 6 .90 1 .37 5 .53 620958 1242955 1039 2414 6 . 5 0 1 .33 5 .17 645683 1292463 1097 2321 6 .11 1 .29 4 .82 671143 1 3 4 3 4 4 7 1161 2233 5 .76 1 .25 4.51 697444 1396113 1225 2149 5.41 1 .20 4.21 719090 1439457 1277 2084 5 .15 1 .18 3 .97 743327 1487996 1342 2016 4 . 9 0 1 .14 3 .76 770981 1543380 1418 1944 4 . 6 4 1.11 3 . 5 3 789671 1580819 1477 1898 4 . 4 9 1 .08 3 .41 821214 1643996 1568 1825 4 . 2 4 1 .05 3 . 1 9 852237 1706134 1660 1758 4 . 0 2 1 .02 3 . 0 0 879691 1761129 1747 1704 3 . 8 5 0 . 9 9 2 .86 925073 1852042 1896 1620 3 . 5 9 0 . 9 5 2 . 6 4 962851 1927724 2022 1556 3 . 3 9 0 .91 2 .48 991373 1984858 2112 1511 3 . 2 5 0 . 9 0 2 .35

1036212 2074704 2280 1446 3 . 0 7 0 . 8 7 2 . 2 0 1119303 2241198 2592 1339 2 .77 0 .81 1.96 1159679 2322118 2760 1292 2 . 6 5 0 . 7 9 1.86 1208861 2420678 2956 1239 2 .51 0 . 7 7 1 .74 1246161 2495439 3117 1202 2.41 0 . 7 4 1.67 1313174 2629742 3394 1141 2 . 2 5 0 . 7 2 1.53 1370620 2744915 3675 1093 2 . 1 4 0 . 7 0 1.44 1475778 2955748 4192 1015 1.95 0 . 6 5 1.30

137

to o p

&

CO CO

• H

cH CD pq

!H CD

-P CD

cb

O J

6

CD rH

•a

8

CO fn CD ccj U m

i n >> •

rH CM

Q, I I

CO

CD

p a CO CD u CD

DM CD

o >> rH rH a

I I a I I

CO

CM O 1 CD rH LO CO o CO CM (D CM rH • • •

w CO CO • £

S o o o O o CM CD i > 00 CD LD

0 5

<D CM o

1 CD CO CO o CO [> CO U rH CD Cti - * a t LO k CQ - E

PH

o >> • rH CM a • a ii s rH

CD CO 2 o] 00

CM O I CD O CO rH \

• E ^

L D L O O O r H r H t D C M C O

L O L D L O L O ^ r ^ r ^ r c o c o c o c M

O O O O O O O O i D C O C M C O O C O O C D O O O O L O O C O I N C M L O \ r O H c v i a j c n n ^ ^ c n i o

t N C O C M O O C M L O C D C M O O t O C M ^ MjiwOOoicoinnwajH

CM OO

C O C O C O C O C O C M C M C M C M C M C M C M r H

o o o p g c ^ C D O C D C M O ^ C 0 I > C M C M C D O O O O l D r H L O O C D r H O O C O L O C O L D

C M C M C M C M C 0 C O ^ r ^ ^ J " ' 5 3 " L O

138

to CD

• H -P • H O

• H f n -P c CD o o w

S1

c3 CD

PQ SH CD P 92

•r-t Q

CD

i n CVJ

6

CD i - l

E-t

8 U)

l£) O I s O CO CVJ CVJ t H t H O

6 6 6 6 6

Q O Q O Q O O O O O CO N O CD CD "3" LD CO CO CD

O ^ ^ C O ^ r i N M - t H C O O I N C D L O ^ J ' C Q r v J C v J C V J t H r H

6666666666

Q O O O Q O O O O O O O Q Q O O O Q O O J c v j c o ^ O c D c v j c o ^ o ^ r • H t H c v j c D c o ^ ^ m L n c o

• S j - C V J t H ^ C O C V J r H C O

( O i n ^ n c v i c v i a j H

6 6 6 0 6 6 6 6

Q Q Q O O Q O O o o g o o o w g C V J C O ^ O C D C V J C D ^ H H c v i n n ^ ^ L n

139

CO

o o rH

CO O 1

- E o CH rH OB

m

CO r-1 rH 1

<H O O rH II

O flJ s U 0 to

cr; to CD

. U CJ PH

to > i rH

• a CX

PH to

E LO

CJ 1 0 o to rH

X

CO < P . CC • Q cr; rH

rH

o CO II to

rH O

^ PH • • to

§ cr; C o

•H to - P o ai 0 SH to (D

/ ° CD rH CD

1 O o <D rH P

bX • C

s •H f .

2 PH' c3 • CD

cr; CP

' £H CD

to - P

> 0 0 to

O CO* <U •H

O to Q CM ^ CO

1 CO CM O

• P rH

-~—. CD to CM > 0 10 6 u o rH

0 ^ ^ O to rH 0 ^ CD rH

| rH O •8

- P - rH

CO rH CO rH 00 rH CM CM CO CO

i C M L O i > c r ) r H x j - i > . o o c M ^ t c o ^ ^ ^ ^ m i f i L n i / i c D c D c D

o o o o o o o o o o o o o o o o o

O C D C r ) r H ^ - O O L O O t > O I > l > - C O C O r H C O < D L O C D C O C M C O C M ^ C D C D C O C O C M C O C J l C O C O C O

C M C M C M C M C M C M C M C M C M O J C M C O C O C O ^ T L O L n

O L O O L O r H C O C N r H C M C D C n c O C O C O C O C O C O

H , H t M C M r o c o n ^ ^ ^ ^ a i ' O c D h c o o o

6 d 6 6 6 6 6 d 6 6 d d 6 d d 6 d O ^ C D C D L n H O I H a i l D c D C O H C O C R C D ^ C D C O L O ^ ) - C D C D C O O O C M C O O O C O C D L O r H I >

C M C M C M C M C M C M C M C O C O C O C O C O < 3 " ^ t " L O C O C O

rocMcn^OTco^cM^cMLOcoiNOcDCDCM C D t N C N C n c D O C M L O C V J C O C M C O C O C O L O C M O C O C O L O C O r H r H O C n C T l C O C 0 1 > C D C O L O L O L O CM r H rH rH rH rH rH

CD

CM o

C O C D r H O ^ C D C O O r H r H C O r H L O C M C O C O r H S r t N L O L n C D C M C O r H r H C D L n O C O r H C J )

O L O r O L O C M O C T i C O I N C D ^ r c O C O C M C M r H N " J P ) ( \ I W W H H H H H H H H H H

O L O O O L O C O L O O O O O L O C O L O L O O O C M C O O C T i C M O C O ' T r H ^ r C M C O C D L r )

[ N [ N fyj CD CO U~) ^ r ^ r C O C O C O C M C M r H r H r H CM rH rH

C O C D r H L O ^ " C D C 0 C 0 c O r H C 0 < - I L O t N C O C 0

^ r H ^ c n c o c M C N ^ r c o t N L O t N c r i c M r H C O L O

O t N i > . c M r H c o i o r o c M O C T i ! > L n L O ' q _ c o c o C O O O L O ^ C O C M C M C M C M C M r H r H r H r H r H r H r H

CD en CM CO 3

CD CM CO o CN L O CO CD L O CM cn CM rH rH LT) CO CO CM CJ) rH rH CO L O CO L N CO CO cn o CO i n CO CN rH

rH rH rH r H CM CM CO

t N rH CO t N CD [ N CO CO f N rH CD CO •d" rH CO rH CM CJ) rH CT) I N CO CO rH CN CJ) CO CM CO rH LO

CM CO rH CD LO CO rH rH CJ) CO CO LO CM CM rH LO CO CO CM CM CM CM CM rH rH rH rH rH rH rH rH

LO CO CD t N o CO CM CO CM I N 00 LO co CJ) o I N CO

rH rH CD LO (N CO o CO O CM CJ) LO CM CO CO CD rH CO O I N CM CM CM CO CO CD LO CO 00 CO CM rH CJ) rH LO CO CO rH CJ) CO CO rH LO O CO CD CM LN CO I N 3 - I N rH CO O CO rH 00 CO CO CO O CM CM CO CO 3 " LO LO LO CO CD I N | N CO CD CD rH

rH CO Q t N I N CO o CD CD CD CO co CO LN CM CD LO rH CM CM CM CO CD CO CD CD CO CM O rH o CO CD CD 00 CO CM CO CD CD rH CD LO CM CD o CD LO I N CO CO LO CM CM CD LO ^ 1 I N CO CD rH CO CO CO CM CO LO I N CO O rH LO CD rH CD CD rH rH rH rH rH CM CM CM CM CM CO CO CO CO •sT LO

CM

N ! O 0 CO

CO I o

o o o o CM [ N

00 CO

I I I I

LO o

0 E H

•P CO

r H l D C O t O C ^ C D ^ r H C X > L n C \ ! r H r H C U L n O O O O l ' t l U ^ P J H r H r i O O O O O O O O H

( M W H H H r l d d H H H H H H r i H H

C O r H C O r H C O O C ^ i n [ > C T ) r H ^ ! > - C O C \ ] ^ a )

0 0 ^ 6 6 6 0 0 6 0 0 6 6 6 6 6 6

< £ ) C \ J C T ! ^ t N C O C D ^ ( T ) r H C O O r O L r > ' ^ " < - I N . C D L n c O L n c o c v j c v j c M C M t A J v H r v j c ^ c M c r ) ^ ^

O J C V J r H r H r H r H r H r H r H r H r H r H r H r H r H r H r H

C D C V I C J ) ' N r o O ( X ) ^ r c \ J

C O C O L O D H H O O I OJ rH rH rH rH rH rH

^ T C M L n t O t N O C D C n o j C M < X > C Y I 0 0 t O 0 0 i n c V I O C D C O C O O - C D I D L O L O L O

I D

co u o

CO O I

>e o

CO r H I

<H O

o

os

-p

o <D to

OS o 0 CO

o

o CD to

t H

£ s

to o CD CO

/ 0 CD I O t H

CH'

OS ^ ^

CO

> CO

o CD O CO CVJ *—*

O J o •p t H

CO

> 8

CO

o CD CO

CO I

I o

u CO O cc5 O t H t H PQ • • t H 6 o o

Q LO

II <f in CVJ O J CVJ

CD U p O t H CD W CD O m CO O- O CD CVJ CVJ O J f-.

OH

>> i-H O J CD CD a O co CD a • • •

CO CO CO t H O

<H t H

E in i I

o r H

X Q o o l > CO

co • • • CO CO

• CO CVJ t H t H t H

CD i—I CD O CD

P

bX C

U CD

P 0

l > O J

6

0 rH • 9 E H

O J t N t H CO O rH t H O J i H O J

6 6 d d d m m o m CO CO si" CD CO O J O J O J O J O J

CD CO CD co co co co m CD 00 CD CD O CO

0 EH -P EH - H

P CO •P 0 O 0 i H O H P CO

H ( 5 >

a to

I o

co

CVJ CD CD o CO O CO O J CO O J

• • • o • • CVJ CO O J in CO

in CO CD o t H vH • • t H t H

CO C O

• H P

O J Cti SH

0 t H CO I N o rH co o in 00 CO 0 0 CM CM CVJ CD CO o CM CO

0 s p CD M

1 C rH CO o • H in O J <H CO 00 t H U CO m o in 3 CO CD CD in X 0 in •3-o o CQ

o O CO CO CO CD u

• • CD 0 o O • p O J CM O 92 CD CD CD CO co

m CD O CM

m CD *r II II II •H CD CD CD CD I N Q CD CM CD t H co

Q

O t H CVJ O J P ~ O J CM

ire u

3 CD

ral

0 CO O J

0 I—I

- 8 EH

CD O J

q t-n

d d 5 8 CM CM

t H CO O !> I N CM CM O J

O CD CD t~»

CD in t H 00

o o m O J O J CO CM

O J CD CD Q CO O r H rH O O J

CM

-o 0 10

CO I o rH

u ° „ O O 00 00 00 CD

• CD o o • CM CM O II 0 SH

05 U U 0 0 0

E H P

P 0 0 r H

a s

CO O O CO

• H >

142

CO

ar

CM O CM ar

uo 00 CM CD CO 00 rH

CO CM CM rH CO CM

I I I I

CD 9 U

p P to CO in CO

o> 9 o u CM l>- rH U CO o CM rH rH CM OH rH CM

>> o d d >> d d r-H rH a a & & p

CO CO

.— v . CT3

CO v s rH t> CM cn CO CO rH CO CV1 • • • CM • • • • CO CM CM • • CO CM

O o rH rH —

CU r-H rH n -Q Cti o rH CT> a rH CO

EH CO o IT ) o O -O O l > CM CM CM CM CM CM

CO

CO <H 1 o o rH

CO O 1

• £=. O rH CQ

CM CO

rH 1 < H O II O rH

CD

% p 1 % CO

CD CO SP in

DH o CD >1 CO r H

a • a

- p s CO

e m

o I 0) o CO rH

X s o • CO

rH rH

o CD II CO

rH o • p s •o

\ DH • • CO

§ C c o •H

CO - p o 0) SH CO CD

/ 0 to i—1

- CD 1 O o CD rH Q

b£ r ;

s •H c. a CD

ci cq

u • N CD CO - p > CD CO U o

0) • H O to Q CM

CD 1 1 CO

CM O - P rH

CD CO CM > • • CD CO o £ o rH

0) v •

O CO rH CD

CD rH 1 n

rH O CTj 4-> rH EH

I D ^ O C D N C O l A J l f i l D W ^ t D C O C O O r H C M C 0 C 0 s r ^ " L . n m m c D c D c D C D C D I >

6 6 6 6 0 6 6 0 6 0 0 0 0 0 0

5 C D r H C D O C D C 0 O C M C \ l M " r H t n t > C D i D ^ r r H c o c o c D c o c D i n c n c o o c v j o

C M C M C M C M C M C M C M O J C M P O C O ^ m L n c D

< j H c o i r ) H ^ c \ i c O H c D m c D O ) r o ^ r H C M C M C o m i n i D c D i N c o c n O r H C M C O

O O O O O O O O O O O r H r H r H r H

O O t N C D r H r H C O i n c O C O O C O f - ^ r O O i n o o c D i n c o c D o j ^ t c D ^ J ' c n c o c x j i n ^ r

C M C M C M C M C M C v ) C 0 C 0 C 0 ^ 1 " ^ l " i n t O c D I > -

rococ\)Oincoc\iO)NHtDaioic\jH C D O I N C D C O c D C D C O L n r H ' s r O ^ t C O C D < J O c o , j H O o i c o a ) M D i o i n i f i ' t CM CM rH r H rH rH

I f ) CO H ^ H CM O rH CO CD W O C M i n r H C M ^ r ^ r o O C O C O C D c D r H C O O c D

c M ! N t > r H ! > i n c M O O c D i n m ^ r ^ r c o C M C O L D ^ C M C M C M C M C A J r H r H r H r H r H r H

O O L O C M L O O O J O O O m o m o ^ c o i r ) 0 ( » c o v j w o ) c o i o O O r H ^ r r H l D i n ^ r ^ - C O C M C M C M r H r H r H CO CM rH rH

L n C O r H ^ r r H O O C M C M C D C O C O O C M O C ^ i n c O C M C 0 C M ^ l - c D r H t > O C 0 C M C 0 C M

L n c o r H C M c o r H i > ^ r ^ r c D c o t > - c D i n m m O M o n n w o j w H r i H H H H

CD m CM co

CD O CM rH i n CD CM CO CM ST CM CD CO CM CO CM rH CD CO O CO CD

CD CD CO CD 0 O- 0 CD I S CO rH rH rH CM CM CM CO

CD i n CD CD m CM CO i n Q CM CD O CD O O CD CD CO rH CD CO i n O cn t S

CD I N CD CD m CM rH O CD i n CO CM rH m CO CO CM CM CM CM CM rH rH rH rH rH rH

O CO CD rH 00 00 rH CD CO CM 0 CD CO CD

rH O 0 CO CO 00 m CD 00 CD CO CM co [ s CD 00 CO CO CM CM O 3; CO rH O rH LO 0 CD i n rH rH rH £2 i n CM [> CO CD rH IS. CD CD CM O i n O I N CM CD CO O CO CM rH 0 CM CM CO CO 3- i n i n i n I N 00 CD CD rH

C M i n O c D C D 0 0 m C M C M O C M C D i n i > . r H t S l S O C D O O r H r H C O O C D C O r H C D O [ > ' ^ t > . C 0 C 0 1 > . 0 0 C M L n C O C D I > C M O C 0

8 i n O r H C M L n r H C O C O C O C D C M O O C M r H c M i n i > - c M c n c n c o c D L n c O r H i n i > r H

r H r H r H r H C M C M C M C M C M C O C O ' ^ ^ J ' ^ i n

CM

o CD CO

CO I o

o o * o o CD CD CD

. . o CD CD •

I I I I

B g CD

B g CD

CD EH

- p CD

rH • p

3 S

• H CO o o CO •H >

144

i EH a

r

rH 00 CO CO LO CM CO •sT CO 00 CO oo CO CO LO m O LO oo rH <-H O O o 00 CD oo o o c

& CM CO CM rH rH r H rH rH rH rH o o o rH rH rH 0

ii

CO CD c U o 3

^—^ •H CO X) - p CO 9 co 0 u fn co o CO I N oo CM U0 CO CM CO 00 00 o

(0 CD a , r H CvJ 00 CO -N LO LO uo CO CD CO CO CO I N (0 to a ,

0 >> o o O o o o o O o o o o o o o CD O

CD a Q a

60 CO c • H

& CM

CO CD m -m CD LO I N CM o co N LO LO LO I N rH oo o oo

O CM O co O LO 00 CM CM CM CM CM CM CO 00 CD

CD P O CO CM CM r H rH r H rH rH rH r H r H r H rH r H rH CS 0 rH

§ * — '

• H 0 O (-H O

43 CO 00 l O CM 00 CO 00 CM 00 O r H CO OO oo CM <H CD CO EH CD Q I N CO CO co CO CO LO rH o CO 00 K O CO rH o 00 CO 00 I N co CO LO LO

CvJ CM rH rH r H r H

o) co m i> o O CM <M CM CM 00 CO

*t CO "3" "sT

o o o o o o o o o o

co <H I O O

r ^ O r H L O c o c M O c o ^ r c r )

f O l D l D l D L D ^ t a i H H P J

C M C M C M C M C M O J C M C M C M C V )

CO

CO rH I

<H O O rH

as

H t c o i w i n c o c o Q i n r o

d o d o o d o d d o

c o c o O M n O r o r o c n w ^ • M C c o c o M n i n m c o

C M C M C M C M C M C M C M C M C M C M

CU

C ^ r H O C M O L O O O C D C O r H c o o t D O t ^ ^ r c v j c M ^ H r ^ C O r H O C T i O c O L n c O C M O ( M O J W H H H H H H H

-P

8

s OH"

DS

o

GO

a]

rH a

C/3

I o

00

O i > c o c o n t D c O i - ( r H ^ r

s r o ~ > c r > c o c o L n ^ - c o c o c M

o o o o L O O L O CO

o o in o co IT) O CM C O O)

< T ) ^ l " C M C 7 > t O ^ J ' C M C T > O L n CO CM W H H H W

o CD w

rH -p s X ) •

§ as co o 0) CO

CD I o rH

/O

s a]

to > s

o CVJ

to o CD

CM O •P rH

co >

8 o

p

co o CD CO

o

II

o

CO c o •H p

o CD p

Hf •H aJ CD

CP

u CD p CD

Q

CO

o CO

6 rH

CD 1-1 •8 E H

^ O O H N i n C V J C O C D N l T ) I > - C O ^ r C ^ < H 0 O C O C T ) ^ I -

o o c o < H c o c v j c D ^ r i > ' - i c o c M H O c o M n ^ n c o

L n o o c M c v j c o c n L D O c o c o C O C M ^ I N O O C M O O C O ^ c M c o c o c o ^ r ^ r t ^ m i o

o CM CO 00 o CO L O CD CD H cr> rH o H CO CO iH CD m O C D CD CM C D C D CM CD ID CD i_n ^ - •=T 00 CO CO CM C\J

9

9

CO C D Q CO [> o CO CO CM CO CD CD o o CO o o o rH CO CO CO L O o m <-t o CO CVJ CO CO CO o CO o» CO CM CM CM CM CO CO CO •st"

CM ^ in a i CO IX) IN rH m L O I—i rH iH CO L O CO CO o C D 00 Q CO CO oo CO CO CM rH O CM O CD CO o

rH CM CO ^ I D CD 00 o CO rH rH rH rH rH rH rH CM CM

CM 4 =

O CD CO

CO I o

O CJ> X o o m L O rH • • rH

CD CD •

II II

B

g (D

-P CD

H

EH -p"1

•H -P CO CD O rH O -P W

<3>!

CO i

<H O a <H

CO I o

CO <H O

0

-P

O 0 CQ

OH"

o 0 m

a os

C5

o a) co

rH

\ - a ,

CO o 0) to

to I o

Y

co >

co u o

0 o CO O J • — CO 1

1 o •p 1—1

CO > s

o

CO o 0) CO

w to I rH O •P rH

[3 CQ

0

CO

a .

> i—i a

W CO E

• in l o

00

g CD i—I CO o 0

Q

2 •H ft CD PQ

0 -P 0

•H a

CO

CO

6

CD

E H

L O H CO O H CM

6 d 6

<H CVl CD O J CO ^

CM CVJ O J

O O J I N H H H

6 6 6

<n t co

CO M " I D

O J O J O J

i n w H O O Q CM CD CO O J O J

<H m CD ^ CO ^ H O tO LD CO C D

O Q O CO ^ UO

L D I N L D CO O J

H m o i O J O J cn N CO H CX) CO O J

CO CD CD O CO O O J O J CO

S CO rH O J CO

CD CO O J C D C D in

C D CO O J

H (Jl Ol IN CO CO H H a i

CD O J 00 « H I N CO CN " in 00 CD

in

CM

o CD CO

CO I o

o o O O CO 00 CO CD • • CD

O O • O J cvi O

u CQ

O J

CD u CO CD U

PH

>> rH

a

I CO E

m I o

00 • 3 -

o

(0 C o

•H

CD rH CD O CD

a

ft CD CQ SH 0 -P CD

CM CO

• s EH

I N CO o H w 6 6 6

oo m oj CM co in CM CM CM

rH ^ O H H (\J

6 6 6

I N cn oj co ^ I N

CM CM CM

I N CO CD l O O O l O N H CO CM CM

^ °S ^ o en I N ^ 6 CO

<-H co

o o o C D IN CM H CM H •rj/ CO CM

H CO H O CM CD

OJ CD ^J" co o

CM O J CD CM O J CO

5

8 S o CO CO CD I N C D in *t

O CD CO 00 00 CO H CO CO O CO o co o in H CM CM

C D CD *fr I N CD CD in CM CD vH in 00 O CM

<H <H

O J

o CO

CO I o iH

u ° _ O O CO 00 CO CD

• • CD o o • CM CM O

0 ^

U 0

0 a 0 EH -P

EH -H •P CO

-P 0 O 0 rH O rH -P CO C H O >

n 0 U

">C0 a 0

CM

<H O

0

0

0

(0

T3

g

3 0

e C\J

U

0

m C c

CD i—I CD O CO ft be c •H ?H 03 cu

CQ SL, CD

co

PQ

CM

CO

g a ,

>> i-H ex

CM

II

g

g a ,

>, i—i a %

o CO

I N LO CO CO

CO CO CM

i n iH CO O iH CM

6 6 6

i - l IN CO IT) CO Ol

6 CO CO CM rH '

0 r-1 LO CM H

O O o aj CM O)

CO CM CM

C M C D C M K T O O C O C D t N C J O H i o i c ^ o j o i i n o j H O r T M C M W C J H H H H H

a i c o ^ L T i i ^ O ' ^ ' O ^ r c o O C O C O C v l W C O C O ' J W

o o d o o d o d o d

C n C O O C M C O C D C O i H O C O • ^ I N t N L D C M I N C O ^ C O C M

C O f t l l M W C M H H H H H

CO H O CM CM CM

a

CM

II

g 3 CO

g a ,

a w

CM o LO 00 CD CD iH 00 CD o IN CM CM 0 Lf) o CT)

CD LD CO CM o • - r o IN iH iH iH 1-1 tH <H 1-1 -9 CO CM CM

CD

CO CO CM

CD q iH CM

d d d

00 00 o iH [>

CO CO CM

CO CH 1 O O

rH

CO O 1

• 6 O rH

CO rH 1

<H O O rH

% <D

O <u

\

- P £ •o \ a ,

o 1

0)

\

g

P (K

o

to rH

-p s \ cu

CO O 0) CO

/ 0 CO 1 1 O rH

s s a !

(0 > 0) CO u o

0) O (0 O J

CO 1 O J o

- P rH

CO > CD CO £ O

cu O CO rH ^ CD

| rH O • P rH

cu

^ CO CD fn

a ,

>> i-H en

E in I o

o •H -p

2 0

rH cu O <D

Q tc

00 00

6

1) rH 3

rH (£) C D O rH CM

O 6 O

O J o j O rH co in

O C D ^J" t—It—I O J o d d

OJ co OJ ^ [> O J O J O J

CO 00 rH IN tN

[> oi co oj O J

O O CO CO LO uo

O rH [ > O J CO

o o o o m o ^ w m co OJ

o co

CO m

^3" t£) 01 O J L O O O J <—I T—!

C O ^ CO co un co rH o j co

CO O J O J CO ^ 00 m o r a t> <o

O J O J CD rH rH O J CD CD CO CO CO CD LO CD 'Sf rH rH O J

00 CD rH t > O J rH O CD CD O J C - CD O J

O J

o CD CO

CO I

O O O O 00 00 CO CD

o o O J O J

CD

II II

CU EH

t

CO

8 OH

a

I o

§ •H - P

g CU

rH CU O CU O

bd CT •H c3 CU m cu -p

CO

00

•8 EH

H in I D

O rH O J o d d

o o CO

rH 00 L O

O J O J O J

CD I D I D O rH O J

o d d

C D L D CO

rH ^

O J O J O J

rH O CO O J rH rH CO 00 O J CO O J O J

CO ^1" O CD O J CD

CM 00 O CO O J CO

O O Q O oj co CD L D rH ID 00 O J

00 3 r -O ) v N

rH 00 00 ^ L D O O J rH rH

[N 00 O r-. ^ - v j -rH CM CO

CD O J 00 CO O J O J CO rH 00 t>- co

rH O rH O J CO CO rH O I D CM CO 00 LO CD ST rH rH O J

O J [> C D 00

CD O

CM

o cu CO

CO I o rH

o o o o co 00 00 CD

• • CD o o • o j oj O

II II II

B

I D ( N [> CD O J

CD EH

- P CU

CU EH

- P CU rH - P

3

• H CO o o CO

•H >

149

E-<

CD

CD

« " CO aS cu

J 5

O J

< H O

CD

CD

4->

CD E H

CD

CD

£

O J

o CD Pi

CD

m C o •H - P

g CD

r—! CD a CD

Q

bO C u cci CD

CQ

SH CD

- P CD

CO

Bar

O en o OJ CO <H t>

II CO CO OJ

CD

p w V) CD CH

CM o L O CD q rH OJ

>, • • rH 6 o O a a

CO

as o CO O co OJ 00

co • • • • • CO CO OJ o vH

CD <H o CD •—1 Ol T—i rH £> CD CO OJ cd CO OJ CVI

EH

£H cd

CQ CO 00 o CD rH

OJ « • • CO CO OJ

II

CO

t

>, I— I a

00

8 . . o d d

CD CD <H OJ

00 OJ O CO OJ CO

V ' • • • CO CO CO CM CO

d -

CO CO <H CD tN IN

rH IN OJ

-8 co OJ OJ

EH

150

to CO

<H O O rH

CO U I

• E O

CO rH I

<tH O U rH

0

- P

o CD CO

PH

o CD CO

2

o CD W

rH

CO O CD CO

/ O ^ o

s OH"

CO > p CO

u O CD

o CO O J — C D i

CM 1 o - P rH

CO >

g o

CO o CD CO

^ ID I

rH O - P rH

CO CD

a ,

a

% CO

s Lf> I o

C O

CO c o •H -p g CD

rH CD O CD Q

CJ:

•H

CD

CD - P CD

LO CO

6

CD rH

• s E H

CO O CO LO LO LO CD CD CD

6 O 6 O O

ID CD LO CJ) LO ID CD CM CO rH

CM OJ CO CO ^

OJ OJ CM CM CO CD IN CO CD O 6 6 d 6 H

CO H N H CO CM O O CD <H

co co ^ L O

L O cr co co CD L O 00 CO CD L O 'sf L O CO rH CD CO IN CO CD

<H CO IN N ^ LO rH ID CO

CO O 00 CD LO O J O J rH rH rH

LO LO CO I N CO CO I N CO

CO CO CM OJ

CD M" 00 IN O <H CO ^ CO O

CO <H CD IN CM OJ OJ rH rH

o O J CO O CM LO CD

CD ^ ^ < T I N I N CO

^ O CM rH I N o co O J co co o co CD O J O J rH rH rH

co co N -LO LO O LO I N rH I N OJ O H C O H OJ CD LO CO LO LO CO I N

O O OJ IN

CD CD LO CD LO H CO CD O IN CO CD CD LO CO vj" OJ LO CD OJ CD O OJ CM CO CO N

CM

O CD CO

CO I o

•H CO o o CO

•H

>

o o o

g

B g CD

CD EH

- P CD

rH -p 3

o o q [ N

g

B g 0

0 E H

- P 0

c3 PQ

g

CO

g a ,

a w o E

l o rH

X CO M "

rH rH

II

O

CO C o •H - P

g 0

rH 0 O 0

o

c r

% 0

PQ SU 0

- P 0

CD CO

d

0 rH

•8 EH

O LO CO * t "St" "sT

o d d

LO rH CD rH OJ CO

O J O J O J

o co ^ in

o d d

LO CD CO LO CD CD

CM CM CM

CO CD CO I N LO CO H O

•H- CO LO rH I D

LO CO LO CO CM CM

Ol OJ 00

CD I N LO

C O L O CO CO CM

LO OJ CO CO

"N CD CO 00 I N 00 ^ i n CD

•St CD OJ

CM CO LO CO CM OJ

CO OJ CO O CO If) H H

CO CM N . CM I N CO "st" ^

S 1* O J cH CO OJ O I N N -I N OJ LO 00 rH CO rH OJ OJ

O J

o 0 CO

CO I O

•rH CO O O CO

•H >

o o O

g

B g 0

0 E H

- P 0

rH - P

5

o o o

g

CO

g

w co

LO i o rH

00

CO

g •H - P

g 0 rH 0 O 0 Q

C? •H

c3 0

CQ

SU 0

• P 0

Q

CO

CN CO

d

0 rH

•9 EH

O IN CO CO

d d

N- CD CO <H

OJ OJ

00 CD CM CO

u • OJ 3 o d

CQ o OJ LO OJ 0

CD LO CO

II OJ OJ 2

CD (D CM CO CD

CO I N CO <H

LO

q I N

d

00 IN CO CO

oo d ID LO

O CD rH "H-

OJ I N

co co

<r CM rH CM I N rH CD CO o CO CO

CM CO in co rH oc5 CO CO LO C"-

CO I o rH X

•H CO o o CO

•H >

o o o

g

B g 0 & 0 EH

- P 0

o o q I N

0 U

B g 0

0 EH

- P 0

151

CO

o o

CO I o

CO H I

< H O O rH

0

-p

o 0 CO

o 0 CO

OH"

P

O 0 CO

S a] a! CO o <D CO

/O CD

o H

2 OH"

OS

CO >

CO o 0

o CO O J

CD 1

O J 1 o p rH

CO

>

o

CO o CD CO

CO I

< o

c3 OH­

IO CD U

OH

N, rH a a

CO

L O I o rH x CO

w c o

•H P> g CD r H CD O CD

Q

$ •H

cU CD

CP

U 0 -P 0

O

CO

00 CO 6

CD O J O J CO rH cn N C D O oj m • " CM CM CO ^ L O L O in

O O d • • CO ^ L O

o o o o o O O

L O rH CO L O O J O J rH rH ^ N N CO • • • CO O J oj co co O ^

O J O J O J • • O J O J O J O J O J C O CO

CO CO N <H C\J CM CO O J 'H- C D C O CO C D • • • rH O J CM CO m m

O O O CM

4 = co in O N 0 • CO

CM CM CM 2

i n O J o

co 6 co

I N CO LO o CO m CO CO N co •sr CO

L O CD CM O J 5J- o •sT m

o C D L O C D C D C O co C D CO CO O J

t—I r—I CC CM CD rH CD C D IN O CO O H in o co co ^

C O rH CO

CM o O J rH rH in CD O m N o <—I \—I CM

CO o CD 1 o IN rH o IN CO H H H rH

q

rH

IN CO m LO CO m II

CO CD m CO CM

CO o o CO

• H >

o

C D

P> g 0

& 0 LH -P 0 r—i -P 3

o o

C O

00

B g 0

0 LH - P 0

CO 0 U

OH

N, i—I a

B CO

LO E

I o

C O

CO c o *H P 1

g 0 rH 0 o 0 Q

• H

0 CP

SH 0 -p 0

Q

C O

o o

CO N <sT

CO C D CO CD C D ^

m co CO CD CO N N CO

O N

d d CM rH

in oo CO CD 00 CO CD <tf

CO CD in co co

oo m CM H co <NT CO

O m 00 CM CM C D rH m in CM C O

o 0 co

CO I o

CO o

•H CO o o CO •H

>

u o

JN

00

0 ?H a g 0

0 &H

-p 0

o o

CM

00

C D u C O 0

d & rH <H CO 0 — - C D EH

0 CD L O P I—1 CM I N 0 XJ rH H rH 03 C

6H H

II

2 to CO

OH

N. r—I

f CO

i r i 5

I o

00 <H"

CO c o

• H P> g 0 rH 0 o 0

Q

g« • H a3 0

CO

0 p> 92

Q

CO

d rH

0 rH

•9

O O O O O

co co m co N R O C O C O

CM CO CO CO

co C D in ^j- o CD O H L O CD rH CD C O N CO

I D C O r l H 1 }

CO Lf) lO ^ CD 00 CD m O J rH H rH H

LO O J H L O

in in

I N ^ co co CM

C D co in N

<—I CD L O CD

CO N O C O CO CO CM CM H H

rH C D I N CM N C O H C O H

m oo o rH

N CM CO CD rH CO CM I N CM 00 rH C D N CD O J O J rH rH H

rH CO H ( D N H O J O C O H in m m o CD C D H in I N CM CO CM N CO •^J" in CO CD N

in m CM N o oo O J in C D CO N C D O J CM O rH C D CO H 00 rH C O CO CM CM C O C O CO

CM

o 0 co

CO I o

00 o

•H CO o o CO •H

>

tN

00

0 SH

B g 0

s F H

-P 0

o 3 CM

CO

s

B g 0

0 EH P> 0

CD

T3

«>io

C\J

«H O CD

K

CD

& CD

EH

CD

W g CD

^H o

CD

C o

•rH W ) <D

g •H - P •H CO

X ) CD

- P

c3 - P CO

CO c o •H • P

CD r-H CD O CD

Q

OJJ c •H [3 CD en

CD -p CD

•H Q

CD

CO

>> rH a w

oo

CO CD

Ear

rH rH CD t > I I rH LD CO I I

OJ rH rH

o

• a

CD U

CO CD SM

r-H a a o r

I D CO

6

• a EH

LD CO

6 6 6

05 CO o [> v • CO O J CD • • • CO rH rH rH

CO CD CO LN LO CO rH o rH rH rH

CD CO CO CO O O CD CD CD

rH rH O O O

co IN O cn in LD LD CD CD CD

6 6 6 6 6

00 CD CO CO CD O J oj O J oj oj ^ ^ r-i ^-i r-{

LD ^ 00 \ T CO

CD LO 00 CO CD LO LD CO rH CD CO t > CD CO

00 CO

0 3

O J

II

CO

cu >> rH a 9

00.

CD rH •8 EH

CO CD U

CD ^ 0-i O J O J CO

. . . > . O O O rH

a w oo

I N CO CD H CD ^

8 ^r CD

I N rH f- co rH rH rH

C0 -H-

O I N CO CO

6 6

00 ID CD in

CD CO O J CO CD <T

Ctf

6

rH

CD r-H

•9 EH

a w

00

CD 00

6

CD rH •8

L O N r l O ) O N O O O i O rH rH rH O rH

I N CD O OJ ID CO LD LD LO

6 6 6 6 6

^ J O M D O CO OJ rH rH OJ

CO CO LD ^ O CD O r-i I D CD rH CD CO I N CD

o I D ID

CM rH

rH 00 OJ OJ

6 6

CD O LD CO

OJ rH

00 CD CO 3 CD

153 APPENDIX 1

6th International GAS BEARING SYMPOSIUM March 27-29,1974 University of Southampton

THE FRICTIONAL TORQUE IN EXTERNALLY PRESSURISED BEARINGS

J.Bennett, B.Sc. and

H.Marsh, S. M. , M. A. , Ph. D.

University of Durham, U.K.

Summary

The total frictional torque in an externally pressurised bearing consists of two parts, one being that predicted from the Petroff formula and the other being associated with the change in the angular momentum of the lubricant. The analysis is fully confirmed by exper­iments with an air-lubricated journal bearing. It is shown that if all other parameters are specified, then the minimum total frictional torque is obtained when the clearance is chosen such that 25 per cent of the total torque is required for changing the angular momentum of the lubricant.

ield at the University of Southampton, England. Symposium organised and sponsored by BHRA Fluid Engineering, Cranfield, Bedford n conjunction with the Gas Bearing Advisory Service of the University of Southampton. BHRA Copyright

Al -1

154

Nomenclature

A) 5 c o n s t a n t s ,

B)

c r a d i a l clearance,

D diameter,

I p o l a r i n e r t i a , P

L bearing l e n g t h ,

m l u b r i c a n t mass flo w r a t e ,

p supply pressure, s

p^ ambient pressure,

R bearing r a d i u s ,

T torque,

t t irae , t time constant f o r d e c e l e r a t i n g r o t o r , c |j, v i s c o s i t y ,

UJ angular v e l o c i t y ,

<D i n i t i a l angular v e l o c i t y .

Al-2

I n t r o d u c t i o n

Although tho v i s c o s i t y of a gas i s low i n comparison w i t h l i q u i d l u b r i c a n t s , there are many s i t u a t i o n s i n which i s i s necessary to estimate the f r i c t i o n a l torque and power requirements of a gas bearing. The si m p l e s t method f o r e s t i m a t i n g the f r i c t i o n a l torque i n a j o u r n a l bearing i s t h a t due to P e t r o f f and i s based on approximating the j o u r n a l bearing to two co n c e n t r i c c y l i n d e r s , so t h a t f o r laminar f l o w ,

3 = 2 n R L u, tu ( 1 )

c

This formula f o r the f r i c t i o n a l torque, o f t e n w i t h a m o d i f i c a t i o n f o r e c c e n t r i c i t y r a t i o , i s to be found i n any textbook on l u b r i c a t i o n theory.

In the course of research w i t h a l i g h t l y loaded . e x t e r n a l l y pressurised j o u r n a l bearing, i t was found t h a t the f r i c t i o n a l torque was dependent on the supply pressure. The research bearing operated w i t h a very small load and the v a r i a t i o n of f r i c t i o n a l torque was not caused by a change i n the e c c e n t r i c i t y r a t i o . Further i n v e s t i g a t i o n showed t h a t the f r i c t i o n a l torque of an e x t e r n a l l y pressurised j o u r n a l bearing could be regarded as having two components, one being the f r i c t i o n a l torque T p r e d i c t e d from the P e t r o f f e quation, and the other torque T being associated w i t h the change of angular momentum f o r the l u b r i c a n t passing through the bearing.

In an e x t e r n a l l y pressurised j o u r n a l bearing of conventional design, the l u b r i c a n t enters the bearing clearance through small r a d i a l supply j e t s , which may be p l a i n or recessed, and i t leaves from the two ends of the bearing. At i n l e t to the bearing clearance, the l u b r i c a n t has no net angular momentum, but on passing through the bearing clearance, i t acquires angular momentum. This change of angular momentum f o r the l u b r i c a n t r e q u i r e s a torque and the experimental r e s u l t s reported i n t h i s paper show th a t t h i s torque i s su p p l i e d by the r o t o r . This a d d i t i o n a l torque on the r o t o r can lead to a s i g n i f i c a n t increase i n the t o t a l f r i c t i o n a l torque and power loss i n both gas and o i l l u b r i c a t e d bearings.

The p r e d i c t i o n of the f r i c t i o n a l torque

I f the l u b r i c a n t enters the bearing clearance through r a d i a l supply holes, then i t has no i n i t i a l angular momentum. At the two ends o f the j o u r n a l bearing, the l u b r i c a n t leaves the bearing svith a mean t a n g e n t i a l v e l o c i t y , so that i t has acquired angular momentum w h i l e f l o w i n g through the bearing clearance. For laminar f l o w , the mass averaged t a n g e n t i a l v e l o c i t y o f the l u b r i c a n t l e a v i n g the bearing i s ( I J JR / 2 ) , so t h a t the change o f angular momentum per u n i t mass i s ( U JR ^ / 2 ) . The l u b r i c a n t acquires the t a n g e n t i a l v e l o c i t y p r o f i l e soon a f t e r l e a v i n g the supply holes and the change of angular momentum i s t h e r e f o r e caused by viscous forces i n the neighbourhood o f the supply holes.

The torque which i s r e q u i r e d to produce the change of angular momentum i s

T„ = rfi OJ R 2 ( 2 ) * 2

where m i s the mass f l o w r a t e of the l u b r i c a n t . This torque i s the sum of the a d d i t i o n a l torques s u p p l i e d by the r o t o r and the b e a r i n g ,

Experiments w i t h gas bearings having t i l t e d supply j e t s have shown t h a t the .torques on the r o t o r and bearing are approximately i n the same r a t i o as the r e l a t i v e t a n g e n t i a l v e l o c i t i e s between the incoming gas and the two surfaces, so that-

Al-3

T = l a 9-

On t h i s basis, the change o f angular momentum f o r the l u b r i c a n t r e q u i r e s an a d d i t i o n a l torque on the r o t o r which i s given by

TD = T o = A UJ R2 < 5> R 2 2

Although t h i s torque i s associated w i t h the change of angular momentum, i t i s caused by viscous forces and i t i s a f r i c t i o n a l f o r c e .

The t o t a l f r i c t i o n a l force a c t i n g on the r o t o r i s the sum of th a t given by the P e t r o f f formula and t h a t r e q u i r e d to change the angular momentum of the l u b r i c a n t ,

3 2 T

T O T A L = 2 TT R L u, (jj + m u) R < 6>

c 2

The t o t a l f r i c t i o n a l torque a c t i n g on the bearing remains t h a t given by the P e t r o f f formula, since from equation ( 4 ) , there i s no a d d i t i o n a l bearing torque associated w i t h the change of angular momentum.

Experiments w i t h a d e c e l e r a t i n g r o t o r

I f a r o t o r i s supported i n a j o u r n a l b e a r i n g , then the governing equation f o r d e c e l e r a t i o n w i t h no appli e d torque i s

I dcjj = ~ T (7) p - t o t a l

o r , s u b s t i t u t i n g from equation ( 6 ) , 2 r 3

I djjj + 2 TT R L u, + m R p d t L c

( 8 )

The angular v e l o c i t y of the r o t o r i s then r e l a t e d to the i n i t i a l angular v e l o c i t y , UJ , by

0) = u) e X° ( 9 ) o where the r e c i p r o c a l of the time constant i s given by

[ 3 2" 2 TT R L u, + rh R

c 2 (10)

This r e l a t i o n s h i p between the mass flow r a t e of the l u b r i c a n t and the time constant, t , suggests t h a t experiments w i t h a d e c e l e r a t i n g r o t o r may provide a simple t e s t f o r t h i s new approach to bearing f r i c t i o n .

A s e r i e s of experiments have been performed w i t h a r o t o r and bearing having the f o l l o w i n g dimensions and p o l a r i n e r t i a ,

R = 25.4 mm, c = 45.2 y, m 2 L = 66.0 mm, I = 4.237, -4 kg m p 10

The bearing was supplied w i t h a i r at 20°C through two rows of 12 p l a i n j e t s s i t u a t e d at 0.25L. from each end of the bearing.

Al-4

The v a r i a t i o n of the mass f l o w r a t e w i t h supply pressure was measured by the weighing method described i n Reference ( 1 ) . The whole bearing apparatus and a c y l i n d e r of compressed a i r were mounted on a balance and by using a n u l l i n g technique, the time to consume a known mass of a i r was determined. Figure 1 shows the mass flo w r a t e as a f u n c t i o n of pressure.

The d e c e l e r a t i o n o f the r o t o r i s c h a r a c t e r i s e d by the time constant t . The time constant was determined by s e t t i n g a stroboscope to a f i x e d frequency, d r i v i n g the r o t o r to a higher frequency and then measuring the time taken to dec e l e r a t e from the f i x e d frequency to various f r a c t i o n s of t h i s frequency. These d e c e l e r a t i o n t e s t s were c a r r i e d out at se v e r a l supply pressures. The r e c i p r o c a l of the time constant i s shown as a f u n c t i o n o f the a i r mass flo w r a t e i n Figure 2 where the experimental r e s u l t s may be compared w i t h those p r e d i c t e d from equation ( 1 0 ) . I t i s seen t h a t the experimental r e s u l t s f u l l y c o n f i r m the theory. For t h i s simple bearing system, there i s a large increase i n the f r i c t i o n a l torque w i t h supply pressure. With a supply pressure of 0.7 MN/ra the t o t a l f r i c t i o n a l torque i s 65% g r e a t e r than t h a t p r e d i c t e d from the P e t r o f f formula.

A minimum f r i c t i o n design

The t o t a l f r i c t i o n a l torque i n an e x t e r n a l l y p r e s s u r i s e d j o u r n a l bearing i s given by

3 . 2 = Z TT H

tot T. > t a l = 2 rr R L u (jj + rh IJJ R

c 2

For a given supply gauge pressure and gauge pressure r a t i o and a bearing of given radius and l e n g t h , the mass flow r a t e of the l u b r i c a n t i s p r o p o r t i o n a l to the cube o f the r a d i a l clearance. The t o t a l f r i c t i o n a l torque can then be w r i t t e n as

T t o t a l = ^ + B c 3

c where A and B are co n s t a n t s . The minimum t o t a l f r i c t i o n a l torque i s obtained when the clearance i s given by

dT = 0 = -A_ + 3 B c 2

dc 2 c 4

or c = A_ (12) 3B

For t h i s minimum f r i c t i o n design, the t o t a l f r i c t i o n a l torque on the r o t o r i s

T t o t a l " -C

= 4 A (13) 3 C

which may be compared w i t h the P e t r o f f equation,

T = A (14) C

I f a bearing i s designed f o r minimum f r i c t i o n a l torque, then t h i s a n a l y s i s shows t h a t one q u a r t e r of the t o t a l torque i s re q u i r e d f o r changing the angular momentum of the l u b r i c a n t . I t f o l l o w s t h a t the torque p r e d i c t e d by the P e t r o f f equation i s

Al-5

then three times t h a t r e q u i r e d to change the angular momentum o f the l u b r i c a n t , 3 2

2 n R L u, vo = 3 m OJ R (15) c 2

rh = 4_ TT R L u, 3 c

2 or mL = 2_ TT u L (16)

D 3 c This v a r i a t i o n o f mass f l o w r a t e w i t h clearance i s shown i n Figure 3 f o r s e v e r a l values of \? , the l u b r i c a n t being a i r . Figure 3 also shows the design curves f o r mass flow r a t e f o r a gauge pressure f a c t o r of 0.4 and zero e c c e n t r i c i t y r a t i o , these curves being taken from the work of Shires (Reference 2 ) . The i n t e r s e c t i o n of the l i n e s of constant gauge pressure and those given by equation (16) defines the r a d i a l clearance of a bearing designed f o r minimum t o t a l f r i c t i o n a l torque. For example, i f the bearing parameters are

L = 66 .0 mm R = 25.4 nun 2 p - p =0.35 MN/m o a

then the corresponding clearance f o r minimum t o t a l f r i c t i o n i s c = 45 , t h i s being the bearing used i n the experiments.

Conelus ions

The a n a l y s i s presented i n t h i s paper shows th a t i n an e x t e r n a l l y p r e s s u r i s e d bearing, the t o t a l f r i c t i o n a l torque on the r o t o r c o n s i s t s of two p a r t s , the f i r s t being t h a t p r e d i c t e d by the P e t r o f f formula and the ot h e r being associated w i t h the change of angular momentum f o r the l u b r i c a n t . The a d d i t i o n a l torque required to change the angular momentum of the l u b r i c a n t i s supplied by the r o t o r and i t i s caused by viscous e f f e c t s close to the supply holes. Experiments w i t h an e x t e r n a l l y p r e s s u r i s e d a i r bearing (and also w i t h an o i l bearing) have confirmed the an a l y s i s and i t has been shown t h a t the change i n the angular momentum of the l u b r i c a n t can s i g n i f i c a n t l y increase the power requirements o f a gas bearing.

The equation d e r i v e d f o r the t o t a l f r i c t i o n a l torque en the r o t o r has been used to develop a method f o r designing bearings w i t h the minimum t o t a l f r i c t i o n a l torque. A simple a n a l y s i s shows th a t i f the bearing l e n g t h , the r a d i u s , the gauge pressure f a c t o r and the gauge pressure are s p e c i f i e d , then the minimum t o t a l f r i c t i o n a l torque on the r o t o r i s obtained when the r a d i a l clearance i s chosen such t h a t 25% of the r o t o r torque i s r e q u i r e d f o r changing the angular momentum of the l u b r i c a n t . By combining t h i s r e s u l t w i t h the design curves of Shire s , a simple design method i s obtained f o r bearings w i t h the minimum t o t a l f r i c t i o n a l torque, or minimum power requirement.

Acknowledgement

The authors would l i k e to thank the Science Research Council f o r a g r a n t t o support research on gas l u b r i c a t e d bearings.

References

1. Marsh, H., Drive systems f o r gas bearings, Proc. 5 t h Gas Bearing Symposium, Southampton, 1971

2. Shires, G.L., The design of e x t e r n a l l y p ressurised bearings, Chapter 4 i n Gas Lubricated Bearings, ed. by N.S. Grassam and J.VV. Powell, B u t t e r w o r t h s , 1964

Al-6

159

4 4

to

© ExPEd.ltAE.HT.

I 0 1

F i g . 1 Mass flow rate vs supply pressure.

4 1.o2

THEORY, EQUATION (IO)

PETftOFF

2 _ 3

i £ 5 =

6 i d * 10 JO

Fig . 2 Variation of 1/ with mass flow rate.

A l - 7

to

APPARATUS

32.r 4-10

2 -*

10 lOO

SHIRES DATA Foft Ai£ A T 15 0-4-. £ • O O, 6 * 0 - 2 5 L . 3

F i g . 3 Data for minimum friction design.

A l - 8

161

APPENDIX 2

6th International GAS BEARING SYMPOSIUM March 27-29,1974 University of Southampton

P A P E R C4

T H E STEADY S T A T E AND DYNAMIC BEHAVIOUR O F THE TURBO-BEARING

J.Bennett, B.Sc. and

H. Marsh, S.M. ,M.A. ,Ph.D.

University of Durham, U . K .

Summary

The turbo-bearing is an externally pressurised gas bearing with inclined supply holes, so that the gas enters the bearing clearance with a high angular momentum. There is a change of angular momentum for the gas and this produces a driving torque on the rotor. The paper describes a method for designing the turbo-bearing and the theory is confirmed by the exper­iments. It is shown that the inclined supply holes have no adverse effect on the load carrying capacity or stiffness of the bearing. The onset of half speed whirl has been investigated by driving the rotor to high speeds and it has been shown that the whirl onset speed is depen­dent on the direction of rotation.

Held at the University of Southampton, England. Symposium organised and sponsored by BHRA Fluid Engineering, Cranfield, Bedford in conjunction with the Gas Bearing Advisory Service of the University of Southampton. BHRA Copyright

C4-45

162

N o m e n c l a t u r e

c r a d i a l c l e a r a n c e ,

c end c l e a r a n c e , e

d d i a m e t e r o f s u p p l y h o l e s ,

I p o l a r i n e r t i a o f r o t o r , P

I t r a n s v e r s e i n e r t i a o f r o t o r , T

L l e n g t h o f b e a r i n g ,

m mass f l o w r a t e o f g a s ,

n number o f s u p p l y h o l e s ,

p a m b i e n t p r e s s u r e , a

p s u p p l y p r e s s u r e , s

i n n e r r a d i u s o f t h r u s t p l a t e s ,

R r a d i u s o f j o u r n a l b e a r i n g , o

a n g u l a r s t i f f n e s s ,

S_ t r a n s l a t i o n a l s t i f f n e s s , T

t t i m e c o n s t a n t , c '

T t h r u s t b e a r i n g t o r q u e , t h

T b e a r i n g t o r q u e , B

r o t o r t o r q u e ,

T j d r i v i n g t o r q u e ,

f r i c t i o n a l t o r q u e i n j o u r n a l b e a r i n g ,

V v e l o c i t y i n t h e s u p p l y h o l e s ,

- i n c l i n a t i o n o f t h e s u p p l y h o l e s ,

u v i s c o s i t y ,

o d e n s i t y ,

i) a n g u l a r v e l o c i t y ,

s t e a d y s t a t e a n g u l a r v e l o c i t y

C4-46

I n t r o d u c t i o n

The t u r b o - b e a r i n g i s a s e l f d r i v i n g e x t e r n a l l y p r e s s u r i s e d j o u r n a l b e a r i n g i n w h i c h t h e s u p p l y j e t s a r e i n c l i n e d so t h a t t h e gas e n t e r s t h e b e a r i n g w i t h a h i g h t a n g e n t i a l v e l o c i t y . I t has been known f o r many y e a r s t h a t s m a l l e r r o r s i n t h e m a n u f a c t u r e o f c o n v e n t i o n a l e x t e r n a l l y p r e s s u r i s e d b e a r i n g s c a n l e a d t o n o n - r a d i a l s u p p l y h o l e s and t h i s u s u a l l y r e s u l t s i n a s y s t e m where t he r o t o r r o t a t e s , even when t h e r e i s no e x t e r n a l l y a p p l i e d t o r q u e . The t u r b o - b e a r i n g uses t h i s same e f f e c t , t h e s u p p l y h o l e s b e i n g i n c l i n e d a t a l a r g o a n g l e , t y p i c a l l y 6 0 ° , t o t h e r a d i a l d i r e c t i o n . The gas e n t e r s t h e b e a r i n g c l e a r a n c e w i t h a h i g h a n g u l a r momentum and l e a v e s a t t h e two ends o f t h e b e a r i n g w i t h low a n g u l a r momentum. T h i s change o f a n g u l a r momentum r e q u i r e s a t o r q u e a p p l i e d t o t h e l u b r i c a n t , t h i s b e i n g p r o v i d e d by t h e r o t o r and b e a r i n g . T h e r e i s , t h e r e f o r e , a d r i v i n g t o r q u e on t h e r o t o r and t h e r o t o r a c c e l e r a t e s u n t i l t he d r i v i n g t o r q u e i s e q u a l t o t h e f r i c t i o n a l t o r q u e p r e d i c t e d f r o m the P e t r o f f a n a l y s i s . T h i s method o f d r i v i n g t h e r o t o r i s e x t r e m e l y s i m p l e and i t i s c o m p l e t e l y s i l e n t . The gas s u p p l i e d t o t h e b e a r i n g b o t h s u p p o r t s and d r i v e s t he r o t o r . I n an e a r l i e r p a p e r , r e f e r e n c e ( 1 ) , t h i s f o r m o f d r i v e was compared w i t h a b e a r i n g s y s t e m where t h e r o t o r was d r i v e n by a s m a l l i m p u l s e t u r b i n e .

The p r e d i c t e d p e r f o r m a n c e o f t h e t u r b o - b e a r i n g

I n t h e t u r b o - b e a r i n g , t h e gas e n t e r s t h e b e a r i n g c l e a r a n c e t h r o u g h s m a l l s u p p l y h o l e s w h i c h a r e i n c l i n e d a t an a n g l e 9 t o t h e r a d i a l d i r e c t i o n , as shown i n F i g u r e 1 . The e x p e r i m e n t s d e s c r i b e d i n t h i s pape r have a l l been p e r f o r m e d w i t h a b e a r i n g h a v i n g i n c l i n e d p l a i n j e t s and a n n u l a r o r i f i c e s , s i n c e t he use o f s i m p l e o r i f i c e s may r e s u l t i n l a r g e r e c e s s vo lumes and p n e u m a t i c hammer. The gas f l o w s a l o n g t h e s u p p l y h o l e s w i t h a v e l o c i t y V and e n t e r s t h e b e a r i n g c l e a r a n c e w i t h an a n g u l a r momentum V R s i n 9. A t t h e two ends o f t h e b e a r i n g , t h e gas l e a v e s w i t h a mass a v e r a g e d t a n g e n t i a l v e l o c i t y o f m R / 2 . I f t h e mass f l o w -r a t e o f t h e gas i s m, t h e n t h i s change o f a n g u l a r momentum r e q u i r e s a t o r q u e T 1

w h i c h i s g i v e n b y ,

T = m( V R s i n 9 - u R 2 \ ( 1 ) 1 V 0 — ° )

1

T h i s t o r q u e i s p r o v i d e d b y t h e r o t o r and t h e b e a r i n g ,

T = T + T ( 2 ) 1 B R

The mean t a n g e n t i a l v e l o c i t y o f t h e gas d e c r e a s e s t o uuR / 2 s o o n a f t e r l e a v i n g t h e s u p p l y h o l e s and t h e change o f a n g u l a r momentum i s t h e r e f o r e caused by v i s c o u s f o r c e s i n t he n e i g h b o u r h o o d o f t h e s u p p l y h o l e s .

The t a n g e n t i a l v e l o c i t y r e l a t i v e t o t h e b e a r i n g and assumed t h a t t h e t o r q u e s on t a n g e n t i a l v e l o c i t i e s o f t h e

T B _ V s i n 9

T R V s i n 9 - u)R0

o f t h e gas l e a v i n g t h e t (V s i n 9 - cuR ) r e l a t i v e t h e b e a r i n g and r o t o r a r e

gas and t h e two s u r f a c e s

i l t e d s u p p l y h o l e s i s V s i n 9 t o t h e r o t o r s u r f a c e . I t i s

i n p r o p o r t i o n t o t h e r e l a t i v e

( 3 )

From e q u a t i o n s ( 1 ) , ( 2 ) , and ( 3 ) , t h e d r i v i n g t o r q u e o n t h e r o t o r i s g i v e n by

T R = mR o (V s i n 9 - u ^ ) (4) 2

T h i s d r i v i n g t o r q u e i s opposed by t h e f r i c t i o n a l t o r q u e T^ w h i c h may be o b t a i n e d

C4-47

f r o m the P e t r o f f e q u a t i o n ,

T 2 = 2 n u L u, ( 5 )

The n e t t o r q u e on t h e r o t o r i s e q u a l t o t h e d i f f e r e n c e b e t w e e n t h e d r i v i n g t o r q u e T R and t h e f r i c t i o n a l t o r q u e T .

The e q u a t i o n d e s c r i b i n g t h e a c c e l e r a t i o n o f t h e r o t o r i s

I du) = mR (V s i n 8 - a) R ) - 2 n u R L oo p — o o o P d t _ _ _ ( 6 )

I dui + p d T

3 . 2 2 n u , R L + m R m = m R V s i n 9

* o ( 7 )

When r e l e a s e d f r o m r e s t , t h e r o t o r a c c e l e r a t e s and t h e a n g u l a r v e l o c i t y i s g i v e n by

( l - e ' V t c ) ( 8 )

where i s t h e u l t i m a t e s t e a d y a n g u l a r v e l o c i t y . The r e c i p r o c a l o f t h e t i m e c o n s t a n t i s r e l a t e d t o t h e mass f l o w r a t e o f t h e gas by t h e e q u a t i o n

= 1 TT H R L + m R (9)

I t i s i n t e r e s t i n g t o n o t e t h a t t h i s e q u a t i o n shows t h a t t h e t i m e c o n s t a n t i s i n d e p e n d e n t o f t h e a n g l e 9, t he i n c l i n a t i o n o f t h e s u p p l y h o l e s . The t i m e c o n s t a n t f o r t he t u r b o - b e a r i n g i s i n d e p e n d e n t o f t h e a n g l e 9, so t h a t a l l t u r b o -h e a r i n g s have t he same t i m e c o n s t a n t and t h i s i s e q u a l t o t h e t i m e c o n s t a n t f o r a c o n v e n t i o n a l b e a r i n g w i t h r a d i a l s u p p l y h o l e s .

The r o t o r o f t h e t u r b o - b e a r i n g a c c e l e r a t e s and a p p r o a c h e s a s t e a d y a n g u l a r v e l o c i t y w h i c h i s g i v e n by

OS m R V s i n 9

o 1

3 . 2 2 T U , R L + m R

o o

( 1 0 )

V s i n 9 R 1 + 4 T-

mc

( 1 1 )

The f o r m o f e q u a t i o n ( 1 1 ) c o n f i r m s t h a t t h e s u r f a c e v e l o c i t y o f t h e r o t o r c a n n o t exceed t he t a n g e n t i a l v e l o c i t y o f t h e gas e n t e r i n g t h e b e a r i n g c l e a r a n c e .

The s t e a d y s t a t e a n g u l a r v e l o c i t y , m , i s d e p e n d e n t on t h e v e l o c i t y o f the gas , V , and t h i s i s r e l a t e d t o t h e mass f l o w r a t e by t h e c o n t i n u i t y e q u a t i o n f o r the s u p p l y h o l e s ,

m = n n d 2 p V ( 1 2 )

C4-48

w h e r e n i s t h e number o f s u p p l y h o l e s o f d i a m e t e r d . I n on a n n u l a r o r i f i c e t y p e o f b e a r i n g , t h e t o t a l o r i f i c e a r e a i s n rr d c , so t h a t f o r a g i v e n c l e a r a n c e and t o t a l o r i f i c e a r e a ,

nd = c o n s t a n t ( 1 3 )

From e q u a t i o n s ( 1 2 ) and ( 1 3 ) , i t i s c l e a r t h a t a h i g h v e l o c i t y i n t h e s u p p l y h o l e s i s o b t a i n e d by d e s i g n i n g a b e a r i n g w i t h many s u p p l y h o l e s o f s m a l l d i a m e t e r , t h e c o n d i t i o n f o r an a n n u l a r o r i f i c e b e a r i n g b e i n g d >> 4 c . The Mach number o f t he f l o w i n t he s u p p l y h o l e s i s t y p i c a l l y 0 . 3 t o 0 . 5 and t h e e f f e c t o f c o m p r e s s i b i l i t y must be c o n s i d e r e d when d e t e r m i n i n g t h e v e l o c i t y V . The c a l c u l a t i o n s r e p o r t e d i n t h i s pape r a r e based on t h e a s s u m p t i o n t h a t t h e gas e n t e r s t h e s u p p l y h o l e s r e v e r s i b l y and a d i a b a t i c a l l y , an i s e n t r o p i c f l o w .

E x p e r i m e n t s w i t h a t u r b o - b e a r i n g

E x p e r i m e n t s have been c a r r i e d o u t on a t u r b o - b e a r i n g h a v i n g a c e n t r a l row o f 18 p l a i n j e t s i n c l i n e d a t 6 0 ° t o t h e r a d i a l d i r e c t i o n . The t e s t b e a r i n g has t h e f o l l o w i n g d i m e n s i o n s and r o t o r i n e r t i a s ,

L = 6 6 . 0 mm. R = 2 5 . 4 mm o

2 c = 45 u jn , p - p = 0 . 3 MN/m

s a

i = 4 - 2 3 7

1 0 " 4 k g m 2 ' l

T

= 3 - 5 4 2 i o - 3 k g ™

The d i a m e t e r o f t h e s u p p l y h o l e s was d e t e r m i n e d by f o l l o w i n g t he d e s i g n method o f S h i r e s , r e f e r e n c e ( 2 ) , w h i c h was m o d i f i e d t o a l l o w f o r t h e i n t e r s e c t i o n o f t h e s u p p l y h o l e s and b e a r i n g b e i n g a p p r o x i m a t e l y e l l i p t i c a l . The i n c l i n e d p l a i n j e t s have an a n n u l a r r e s t r i c t i o n w i t h an a r e a g i v e n by

( e l l i p s e p e r i m e t e r ) x ( r a d i a l c l e a r a n c e ) ( 1 4 )

The d e s i g n i s based on a gauge p r e s s u r e f a c t o r o f 0 . 4 . The n o m i n a l d i a m e t e r o f t h e i n c l i n e d s u p p l y h o l e s i s 0 . 3 8 rnm.

The mass f l o w r a t e o f t h e a i r s u p p l i e d t o t h e b e a r i n g was d e t e r m i n e d f o r s e v e r a l s u p p l y p r e s s u r e s by u s i n g t h e w e i g h i n g t e c h n i q u e d e s c r i b e d i n r e f e r e n c e ( 1 ) . F i g u r e 2 shows t h e mass f l o w r a t e as a f u n c t i o n o f s u p p l y p r e s s u r e and i n F i g u r e 3 , t h e c o r r e s p o n d i n g a i r v e l o c i t y V i s a l s o g i v e n as a f u n c t i o n o f s u p p l y p r e s s u r e . From F i g u r e s 2 and 3 , i t i s s een t h a t s i n c e t h e mass f l o w r a t e v a r i e s a l m o s t l i n e a r l y w i t h t h e a b s o l u t e s u p p l y p r e s s u r e , t h e v e l o c i t y V i s a l m o s t c o n s t a n t f o r s u p p l y p r e s s u r e s ( p - p ) g r e a t e r t h a n 0 . 2 M N / m 2 . A t t h e h i g h e r p r e s s u r e s , t h e Mach number o f t h e f l o w i n t h e s u p p l y h o l e s i s 0 . 4 2 .

The a n a l y s i s f o r t h e b e h a v i o u r o f t h e t u r b o - b e a r i n g i s based on t h e a s s u m p t i o n t h a t t h e t o r q u e s on t h e b e a r i n g and r o t o r a r e i n p r o p o r t i o n t o t h e r e l a t i v e t a n g e n t i a l v e l o c i t i e s o f t h e j e t and t h e two s u r f a c e s , e q u a t i o n ( 3 ) . I f t h i s a s s u m p t i o n i s v a l i d , t h e n t h e r e c i p r o c a l o f t h e t i m e c o n s t a n t i s g i v e n by

1 _ = 1_ t I

3 2 2 n u R L + m R

^ o o

The t e s t b e a r i n g c o n s i s t s o f a j o u r n a l b e a r i n g w i t h two t h r u s t b e a r i n g s w h i c h a r e f e d by the a i r l e a v i n g t h e t w o ends o f t h e j o u r n a l b e a r i n g . The d i m e n s i o n s o f t h e two p l a i n t h r u s t b e a r i n g s a r e

R = 2 5 . 4 mm R. = 1 7 . 2 mm

C4-49

c = 90 u.m e

( c l e a r a n c e a t e ach e n d )

and t he f r i c t i o n a l t o r q u e caused by each t h r u s t b e a r i n g i s

4 t h 2c

(R ( 1 5 )

I n a d d i t i o n , t h e r e i s a f u r t h e r t o r q u e on t h e t h r u s t f a c e s o f t h e r o t o r w h i c h i s caused by t h e change o f a n g u l a r momentum o f t h e a i r as i t f l o w s t h r o u g h t h e t h r u s t b e a r i n g . I t i s n o t p o s s i b l e t o g i v e an a c c u r a t e e s t i m a t e f o r t h i s a d d i t i o n a l t o r q u e , s i n c e i t i s n o t known w h a t p r o p o r t i o n o f t h e a n g u l a r momentum o f t h e a i r l e a v i n g t h e j o u r n a l b e a r i n g i s c o n s e r v e d as t h e h"igh speed f l o w t u r n s t h e s h a r p c o r n e r t o e n t e r t h e t h r u s t b e a r i n g .

The s m a l l t o r q u e a s s o c i a t e d w i t h t h e change o f a n g u l a r momentum o f t he a i r i s t h e r e f o r e n e g l e c t e d .

The r e c i p r o c a l o f t h e t i m e c o n s t a n t f o r t h e e x p e r i m e n t a l t u r b o - b e a r i n g i s r e l a t e d t o t h e mass f l o w r a t e o f t h e a i r ,

1 = t

3 4 TT H R L + u (R R 4 ) l

m R ( 1 6 )

The e x p e r i m e n t a l v a l u e s o f t h e t i m e c o n s t a n t , t , w e r e measured d u r i n g a c c e l e r a t i o n f r o m r e s t f o r s e v e r a l s u p p l y p r e s s u r e s and F i g u r e 4 shows t h e v a r i a t i o n o f t h e r e c i p r o c a l o f t h e t i m e c o n s t a n t w i t h t h e mass f l o w r a t e . The c l o s e ag reemen t b e t w e e n t h e e x p e r i m e n t s and t h e t h e o r y p r o v i d e s s t r o n g s u p p o r t f o r t h e o r i g i n a l h y p o t h e s i s c o n c e r n i n g t h e r a t i o o f t h e t o r q u e s on t h e r o t o r and b e a r i n g , e q u a t i o n ( 3 ) .

H a v i n g measured t h e mass f l o w r a t e and c a l c u l a t e d t h e v e l o c i t y o f t h e a i r i n t h e s u p p l y h o l e s , i t i s now p o s s i b l e t o e s t i m a t e t h e s t e a d y s t a t e a n g u l a r v e l o c i t y o f t h e r o t o r . F o r t h e j o u r n a l b e a r i n g w i t h t h e two t h r u s t p l a t e s , t h e a n g u l a r v e l o c i t y app roaches t h e v a l u e

iu = m R V s i n 0 CO o

TT H R L + U ( R 4 . 2

R. ) + m R l o

( 1 7 )

The t h e o r e t i c a l and e x p e r i m e n t a l v a l u e s f o r t h e s t e a d y s t a t e a n g u l a r v e l o c i t y a r e compared i n F i g u r e 5 . I t i s seen t h a t t h e e x p e r i m e n t a l v a l u e s a r e a l w a y s l e s s t h a n t h e p r e d i c t e d v a l u e s , t h e maximum e r r o r b e i n g 19%. T h i s d i f f e r e n c e may be caused by t h e s u p p l y h o l e s b e i n g s l i g h t l y l a r g e r i n d i a m e t e r t h a n t h e d i a m e t e r o f t h e d r i l l , 0 . 3 8 mm and F i g u r e 5 shows t h a t t h e e x p e r i m e n t a l r e s u l t s a r e c o n s i s t e n t w i t h a s u p p l y h o l e d i a m e t e r o f 0 . 4 1 mm.

The t r a n s l a t i o n a l and a n g u l a r s t i f f n e s s e s o f t h i s t u r b o - b e a r i n g a r e shown i n F i g u r e 6 as f u n c t i o n s o f t h e s u p p l y p r e s s u r e . The t w o s t i f f n e s s e s a r e t y p i c a l o f t h e c o r r e s p o n d i n g c o n v e n t i o n a l b e a r i n g w i t h r a d i a l s u p p l y h o l e s , t h u s s h o w i n g t h a t t h e i n c l i n a t i o n o f t h e s u p p l y h o l e s has no a d v e r s e e f f e c t on e i t h e r t h e l o a d c a r r y i n g c a p a c i t y o r s t i f f n e s s .

I n t h e e x p e r i m e n t s w i t h t h e t u r b o - b e a r i n g i n t h e s e l f d r i v i n g mode o f o p e r a t i o n , i t r e m a i n e d s t a b l e a t a l l speeds up t o t h e s t e a d y s t a t e a n g u l a r v e l o c i t y . However , i t i s i m p o r t a n t t o know w h e t h e r t h e s y s t e m i s o p e r a t i n g c l o s e t o t h e o n s e t o f a s e l f e x c i t e d i n s t a b i l i t y . The r o t o r was t h e r e f o r e d e s i g n e d w i t h two s m a l l a i r t u r b i n e s so t h a t i t c o u l d be d r i v e n t o h i g h speeds i n e i t h e r d i r e c t i o n

C4-50

o f r o t a t i o n . The r o t o r c o u l d t h e r e f o r e o p e r a t e w i t h a s u r f a c e v e l o c i t y w h i c h was i n t h e same d i r e c t i o n o r i n t h e o p p o s i t e d i r e c t i o n t o t h e a i r l e a v i n g t h e s u p p l y h o l e s . F i g u r e 7 shows t h e v a r i a t i o n i n t h e w h i r l o n s e t speeds as a f u n c t i o n o f s u p p l y p r e s s u r e f o r t h e two d i r e c t i o n s o f r o t a t i o n . When t h o r o t o r i s d r i v e n i n t h e same d i r e c t i o n as t h e a i r e n t e r i n g t h e c l e a r a n c e , t h e n t h e w h i r l o n s e t speed f o r t h i s s y s t e m i s w e l l beyond t h e s t e a d y s t a t e a n g u l a r v e l o c i t y The s t e a d y s t a t e a n g u l a r v e l o c i t y i s d e t e r m i n e d by t h e b e a r i n g p a r a m e t e r s , whereas f o r a c o n i c a l h a l f speed w h i r l , as f o u n d i n t h e s e e x p e r i m e n t s , t h e w h i r l o n s e t speed i s a l s o a f u n c t i o n o f t h e r o t o r t r a n s v e r s e and p o l a r i n e r t i a s . I f t h e t r a n s v e r s e i n e r t i a o f t h e r o t o r were i n c r e a s e d , t h e n t h e c o n i c a l w h i r l o n s e t speed w o u l d d e c r e a s e , b u t t h e s t e a d y s t a t e a n g u l a r v e l o c i t y w o u l d r e m a i n unchanged As w i t h any b e a r i n g s y s t e m , t h e t u r b o - b e a r i n g must be d e s i g n e d so t h a t t h e w h i r l o n s e t speed i s g r e a t e r t h a n t h e s t e a d y s t a t e a n g u l a r v e l o c i t y .

From F i g u r e 7 , i t i s s e e n t h a t when t h e r o t o r i s d r i v e n i n t h e o p p o s i t e d i r e c t i o n t o t h e a i r l e a v i n g t h e s u p p l y , t h e w h i r l o n s e t speed i s s i g n i f i c a n t l y i n c r e a s e d . The mode o f w h i r l r e m a i n e d c o n i c a l and t h e f r e q u e n c y o f t he s e l f e x c i t e d m o t i o n was a b o u t one h a l f o f t h e r o t o r f r e q u e n c y . These e x p e r i m e n t s s u p p o r t t h e e a r l i e r w o r k o f T o n d l , r e f e r e n c e ( 3 ) , who showed t h a t t h e w h i r l o n s e t speed was i n c r e a s e d when t h e j o u r n a l r o t a t e d i n t h e o p p o s i t e d i r e c t i o n t o t h e i n c o m i n g a i r . The r e a s o n f o r t h i s i m p r o v e m e n t i n s t a b i l i t y i s n o t y e t f u l l y u n d e r s t o o d and r e s e a r c h on t h i s t o p i c i s c o n t i n u i n g ,

•t

C o n c l u s i o n s

I n t h e t u r b o - b e a r i n g , t h e gas e n t e r s t h e b e a r i n g c l e a r a n c e t h r o u g h i n c l i n e d s u p p l y h o l e s so t h a t i t a c t s as a l u b r i c a n t and a l s o p r o v i d e s a d r i v i n g t o r q u e on t h e r o t o r . T h i s me thod o f d r i v e i s c o m p l e t e l y s i l e n t and i n some a p p l i c a t i o n s , i t c a n l e a d t o a r e d u c t i o n i n t h e t o t a l amount o f gas r e q u i r e d , s i n c e t h e r e i s t h e n no need f o r a d r i v e t u r b i n e . T h i s p a p e r d e s c r i b e s t h e method o f d e s i g n f o r t h e t u r b o - b e a r i n g and t h e e x p e r i m e n t s show good ag reemen t w i t h t h e p r e d i c t e d p e r f o r m a n c e . The t r a n s l a t i o n a l and a n g u l a r s t i f f n e s s e s a r e t y p i c a l o f a b e a r i n g w i t h p l a i n j e t s , t h u s s h o w i n g t h a t t h e t i l t i n g o f t h e s u p p l y h o l e s has no a d v e r s e e f f e c t on s t i f f n e s s o r l o a d c a r r y i n g c a p a c i t y . The e x p e r i m e n t s w i t h t h e d r i v e n r o t o r i n d i c a t e t h a t t h e w h i r l o n s e t speed o f t h e t i l t e d j e t b e a r i n g i s d e p e n d e n t o n t h e d i r e c t i o n o f r o t a t i o n . When t h e r o t o r i s d r i v e n i n t h e same d i r e c t i o n as t h e a i r e n t e r i n g t h e b e a r i n g , t h e n f o r t h i s a p p a r a t u s , a c o n i c a l h a l f speed w h i r l o c c u r s a t a speed w h i c h i s f a r beyond t h e s t e a d y s t a t e speed o f t h e s e l f d r i v i n g t u r b o - b e a r i n g . When t h e d i r e c t i o n o f r o t a t i o n i s r e v e r s e d , so t h a t t h e r o t o r s u r f a c e moves i n t h e o p p o s i t e d i r e c t i o n t o t h e a i r e n t e r i n g t h e b e a r i n g , t h e n t h e r e i s a s i g n i f i c a n t i n c r e a s e i n t h e w h i r l o n s e t s p e e d , as r e p o r t e d i n t h e w o r k o f T o n d l , r e f e r e n c e ( 3 ) .

I t i s hoped t h a t t h e r e s e a r c h d e s c r i b e d i n t h i s p a p e r may e n c o u r a g e t h e d e s i g n e r s o f gas b e a r i n g s t o i n v e s t i g a t e t h e p o s s i b i l i t y o f u s i n g t h e l u b r i c a n t gas t o p r o v i d e a d r i v i n g t o r q u e .

A c k n o w l e d g e m e n t s

The a u t h o r s w o u l d l i k e t o t h a n k t h e S c i e n c e R e s e a r c h C o u n c i l f o r a g r a n t t o s u p p o r t r e s e a r c h o n gas l u b r i c a t e d b e a r i n g s .

R e f e r e n c e s

( 1 ) M a r s h , H . , D r i v e s y s t e m s f o r gas b e a r i n g s , P r o c . 5 t h Gas B e a r i n g Sympos ium, S o u t h a m p t o n , 1971

C4-51

S h i r e s , G . L . , The d e s i g n o f e x t e r n a l l y p r e s s u r i s e d b e a r i n g s , C h a p t e r 4 i n Gas L u b r i c a t e d B e a r i n g s , e d . N . S . Grassam and J . W . P o w e l l , B u t t e r w o r t h s , 1964

T o n d l , A . , B e a r i n g s w i t h t a n g e n t i a l gas s u p p l y , 3 r d Gas B e a r i n g Sympos ium, S o u t h a m p t o n , 1967

C4-52

169

/ tXtPPVI OAS

RoToR

66.ARJN£»

F i g . l The turbo - bearing

O EXPERIMENT, >'% SHIRES> Ktr.Z..

O l O l 0-3 0-4- 0-5 °<> 07

F i g . 2 Mass flow rate vs supply pressure .

ISO

V

loo

SO

0 1 o-b 0 7 0-2. 0-3 0-4 o-S

( K - h u ) M N / m l

F i g . 3 A i r velocity in supply holes.

C4-53

ft -3 ° l o

6io"J

7 -5 H o

THEoRy, EQUATION ( i t ) .

2 , 0 " i

i n kg/s.

F i g . 4 Time constant vs .mass flow rate.

12,000

10,000

— T H E O R Y , i ' O ' J f i w n , — „ d « 0-4-1

EXPERIMENT.

Gooo

Aooo

2 0 o o

0 | 0 -2 . 0 - 3 0 - 4 O S 0 - 6 O f

({34-(>«. ) WW/* 2

F i g . 5 The steady state angular velocity.

C4-54

6 ST

* O EKPeR.IME>Jj.

_J I 1_ 0 1 o i o-3 o + o s 04) 0 7

I4oo

I2oo

looo

Zoo -

_ l l l 1_

25,ooo

2o,ooo a )

Ktpoo

Sood-

O-l O Z O S O-* O S 0 £ > OJ

F i g . 6 Stiffness vs. supply pressure .

X AGAlUST -m& TET3 0 WITH THE TETS.

_ l l _ O-l OZ o-5 0 4 OS o-fc o-j

F i g . 7 Whir l onset speeds vs supply pressure

C4-55

172

APPENDIX 3

VISCOSITY DETERMINATION

D i s t i l l e d water was used as the lubricant. The v i s c o s i t y of the

water as a function of temperature was measured experimentally and the

r e s u l t s obtained were used throughout the experimental work described

i n t h i s t h e s i s .

The apparatus used for determining the v i s c o s i t y was a miniature

U-tube viscometer. A U—tube viscometer c o n s i s t s of a glass c a p i l l a r y

tube formed into a 'U' shape. Each side of the 1U 1 has a bulb l e t into

i t and these are separated by a fine c a p i l l a r y . A sample of water i s

put into one bulb between specified etched marks and the time i t takes

to flow through the c a p i l l a r y from one bulb to another can be related

to the v i s c o s i t y . The U-tube i s suspended i n a thermal water bath so

that a plot of v i s c o s i t y versus temperature can be obtained. The density

of the water as a function of temperature i s also measured using a

s p e c i f i c gravity bottle. From the time period measurements the v i s c o s i t y

can be obtained from,

V = Bt + C/t (A3)

and V = ^ / (3

(B and C are constants of the viscometer)

For large time periods the constant C i n equation (A3) above can be neglected

The r e s u l t s obtained from these experiments can be found i n Figs. (A3a) and

(A3b) and table (A3c).

I • ooo

3 cc

0 - ^ ^ 7 4 -n

—i—

1*8

— i — I — to

—r~ — I — i — —i— at n

—i— —i— 5L«l

"Te<vNperokVa.f e ° C

Fi«V ( n l a ) P l o t o f V U Vio*^ o f c \ e » w A ^ ^ J \ l U

174

i

- i

IX 10

Nsec/ ^ r t x \ p \ o V I • o

ro v>-\

o o J

15 20 I S

Tem^pet-cxV a r t ° C

of A v s U l U A w .aVe^.

u <D

P CU s O CO

•H >

o CO

CD iH

•8

175

CO o o ^ CO M

•H W > ^

CO I

OCM

o CO

•p p o • H O

O CO o ° £ ">„ O TJ

0) bfl i-H -•p 4-> o

+ bfj CU

I—I CO - P CO

u cu -p p O 05

X) s

? CO o

0) nS co k <D O CD £ CU > - H CO

< EH —

CU CO e o •H CU EH CO

CM

.CU CO • § CU EH CO

o o EH

CO CO

rH bO C

•H

73

CO Cti

W l O H O H O l O C O C O l D c D t ^ i n c o O c o u n c v i o i i x " ) O O O O O o o i o i c c i c o < - l < - l < - < i - l < - ( 0 0 0 0 0

H H w in O J C D C M ^ - C v l O ^ r C M t D r H g a i O l O L n ^ f c M O h - ^ o i o i o i a i o i a i o i o o c o o i o i a i o i o i a i a i o i o i 0 0 ) 0 ) 0 1 0 1 0 1 0 ) 0 1 0 ) 0 )

H 6 6 6 6 6 6 6 6 6

If) H H H C\J O LO H f l l C O M O O O H C O O 8 0 ) 0 i c o t ^ t > t o m c o c \ ]

0 1 0 ) 0 1 0 1 0 ) 0 1 0 1 0 ) 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

OJ 01 CO IN Ol CO

O J I N C O C O C O ^ t O r H l O O a i o ^ L n ^ f o i O L n t o c o c o o o c o r - ~ c o i i - > L r > c o c \ j o

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 c o c o c o c o c o r a c o c o c o c o

C V l C 0 r H O < - l ! - l t N 0 ) 0 ) f ^ o i t N t n c o o o o i n o j a i L f i O O O O O o i o i a i c o o o

rH tH T H ^—I O O O O O

C v J C O C O O l O O O C M C O O l a i w w H H i o a i H i n c O c o ^ H i D n o i i / i o ^ ocrjcDcOLDLn'sr^r^co

c o c v j L n ^ - i c o o j o i ' ^ o i O C O H H w o m 01 o ( D < -O c o ^ H t D c i o i L n o ^ t r ^ c o c D c o i i i L n ^ r ' T ^ c o r-H rHTHT-H*HT-H*—IT-HT-H* 1

o t O f f l H o o H O n c o o c o t H H i o c o a j t c H C t ) H - H ( 0 n C j , t f 1 g < c iNcDcocDLnLD'T^r'^co

L O O O L D O O O O L D O c v c o i n ^ c o i o i o c o H f o tNINOOOlO^COCOLnO-H H H H O j w a J w a j M

CO V cu CO

8

-p m

c o o CD CO

o 0 CO

CM

Ol ai co CO

o O O

o o c o

•H -p

•H

•a

PQ

176

APPENDIX 4

C FRICTION BEARING (DESIGN)

1 REAL L I L2 L3 L4 L5,LB > » i >

2 REAL MR, MRi,MD,MASS

3 DIMENSION ECC(2,1C0)

4 READ(5,3215)((ECC(I,M),1=1,2),M=l,10O) , 5 3215 FORMAT(9X,F4.2,8X,F9.2)

C TO FIND DISC SIZE

6 DO 9876 M=l,5

7 READ(5,1000)C,R,R1,L1,L2,L3,L4,L5,LB

8 1000 FORMAT (F10.7,8(F8.4))

9 R0=7900

10 VIS=0.001

11 RIPM=(R0*3.142*Ll*R**4)/2

12 RIPS=R0*3.142*L2*R1**4

13 RIPE=R0*3.142*L3*R1**4

14 RIPDH=R0*3.142*L4*R**4

15 RIPl= RIPM+RIPS+RIPE+RIPDH

16 A=((Ll+L2)/2)**2

17 B=(((L3+Ll)/2)+L2+L4+L5)**2

18 D=(((Ll+L4)/2)+L2)**2

19 G=(((Ll+L5)/2)+L2+L4)**2

20 RITC=R0*3.142*L1*R**2*((R**2)/4+(Ll**2)/12)

21 RITS=2*R0*3.142*L2*R1**2*((Rl**2)/4+(L2**2)/12+A)

22 RITE=2*R0*3.142*L3*R1**2*((Rl**2)/4+(L3**2)/12+B)

23 RITDH=2*R0*3.142*L4*R**2*((R**2)/4+(L4**2)/12+D)

24 RITl+RITC+RITS+RITE+RITDH

25 DO 2000 N=l,1000

26 RD=N*0.001

27 RIPD=R0*3.142*L5*RD**4

28 RITD=2*R0*3.142*L5*RD**2*((RD**2)/4=(L5**2)/12+G)

177

29 RIP=RIPD+RIP1

30 RIT=RITD+RITl

31 AA=RIT-RIP*2

32 IF(AA.EQ.0.0.0R.AA.LT.0.0.) GO TO 3000

33 2000 CONTINUE

C TIME CONSTANT

34 TC=(RIP*C)/(2*3.142*R**3*LB*VIS)

C THE MASS

35 MR=R0*3.142*R**2*(L1+2*L4)

36 MR1=RC*3.142*R1**2*(2*L2+2*L3)

37 MD=2*R0*3.142*RD**2*L5

38 MASS=MR+MR1+MD

39 WRITE(6,7779)

40 7779 FORMAT (1H1//3X,'****')

41 WRITE(6,60OO)

42 6000 FORMAT(6X, 'RADIUS OF SHAFT(M) ' ,3X, 'LENGTH OF BUSH(M) • ,3X, 'RADIAL CL

*EARANCE(M)',3X,'RADIUS OF DISCS(M) 1 )

43' WRITE (6,7000)R,LB,C,RD

44 7000 F0RMAT(9X,F8.4,11X,F8.4,15X,F9.7,10X,F8.4)

45 WRITE(6,8000)

46 8CO0 FORMAT(6X,'Rl(M)',6X,'Ll(M)1,6X' ,'L2(M)',6X,'L3(M)',6X, 'L4(M)1, 6X

*,'L5(M)')

47 WRITE(6,9000)R1,L1,L2,L3,L4,L5

48 9000 FORMAT (4X,F8.4,5(3X,F8.4))

49 WRITE(6,9100)MASS

50 9100 FORMAT (6X,'MASS OF ROTOR(KG)=',F8.4)

51 WRITE(6,9898)TC

52 9898 FORMAT(/6X,'TIME CONSTANT=',F7.2,IX,'SECS')

53 WRITE(6,1212)

178

54 1212 F0RMAT(//6X,'SAFE WORKING STRESS FOR MILD STEEL=1.5*10**8 N/M**2•)

55 WRITE(6,9200)

56 9200 FORMAT(////6X,'ECCENTRICITY',4X,'RPM»,4X,'REYNOLDS NO',4X,'MASS FL

*0W RATE (KG/SEC) ' , 4X , ' EFFECT ' , 4X, ' ONSET (RPM ) ' , 2X,

*'SOLID STRESS(N/M**2)',1X,'HOLE STRESS(N/M**2)»)

57 DO 4000 N=l,100

58 E=ECC(1,N)

59 A=ECC(2,N)

60 IF(A.EQ.O.O) GO TO 4000

61 REV=(MASS*9.81*30*C**2)/(R*3.142*VIS*LB**3*A)

62 RON=(2*30/3.142)*SQRT(9.81/(C*E))

63 , REN=(2*REV*3.142*R*C)/((VIS/1000)*60) 64 FL0W=(R*3.142*REV*10**3*C*LB*E)/30

C ASSUMING THROWN

65 RP=0.28

66 WR=3.142*REV/30

67 STH=((3+RP)/4)*R0*WR**2*(RD**2+Ri**2*((1-RP)/(3+RP)))

68 STS=((3+RP)/8)*R0*WR**2*RD**2

69 STH=STH/10**8

70 STS=STS/10**8

71 RATIOP=(FLOW*C*10**2)/(2*3.142*R*LB*VIS)

72 WRITE(6,9300)E,REV,REN,FLOW,RATIOP,RON,STS,STH

73 9300 F0RMAT(10X,F4.2,6X,F7.1,5X,F7.1,9X,F9.6,13X,F7.3,4X,F8.1,

*6X,F6.3,1X,'X10**8',12X, F6.3,IX,'X10**8')

74 4000 CONTINUE

75 9876 CONTINUE

76 STOP

77 END

Input to the program:

c bearing r a d i a l clearance

R radius of the shaft

Rl = radius of the shaft stubbs and recesses

LB = length of the bearing

L I = length of the centre portion of the shaft

L2 - length of the shaft recesses

L3 = length of the shaft stubbs

L4 = length of the disc mounts

L5 = disc thickness

ECC = e c c e n t r i c i t y array

The array ECC (2,ICO) consists of e c c e n t r i c i t y values ranging

from 0.01 to 0.99 and the corresponding values of the expression:

TT £ ((16 - 1) & 2 + 4 ( i - e 2 ) 2 ( ( T T 2 ) )

This expression arises from the Ocvirk s o l u t i o n f o r Reynolds

equation and i s used i n the program to predict the bearing load

carrying capacity.

Output from the program:

The bearing and shaft dimensions

The calculated disc diameter

The mass of the r o t o r

The deceleration time constant

For each value of e c c e n t r i c i t y i n the range 0.01 to 0.99:

The r o t a t i o n a l speed

The r o t a t i o n a l Reynolds number

The lubricant mass flow rate

The r a t i o of the lubricant momentum torque to the shear torque

The predicted onset speed f o r t r a n s l a t i o n a l w h i r l

The disc stresses

181

REFERENCES

1. Reynolds, 0. On the theory of l u b r i c a t i o n and i t s a pplication to Mr. Beauchamp Towers' experiments, including an experimental determination of the v i s c o s i t y of o l i v e o i l . P h i l . Trans. R. Soc., 177, 1886, pp. 157-234.

2. Taylor, G.I., S t a b i l i t y of a viscous l i q u i d contained between two r o t a t i n g cylinders, P h i l . Trans., A223, 1923, pp. 289-343.

3. Taylor, G.I., Fluid f r i c t i o n between r o t a t i n g c y l i n d e r s , Proc. R. Soc. of London, v o l . 157A, 1936, pp. 546-564.

4. Wilcock, D.F., Turbulence i n high speed journal bearings, Trans. A.S.M.E., v o l . 72, 1950, pp. 825-834.

5. Castle, P., Mobbs, F.R., and Markho, P.H., Visual observations and torque measurements i n the Taylor vortex regime between eccentric r o t a t i n g cylinders, Journal of Lubrication Technology, Trans. A.S.M.E., v o l . 93, No. 1, Jan. 1971, p. 121.

6. Smith, M.I, and Fuller, P.P., Journal bearing operation at super-laminar speeds, Trans. A.S.M.E. v o l . 78, 1956, pp. 469-474.

7. Huggins, N.J. , Tests on a 24 i n . diameter journal bearing: t r a n s i t i o n from laminar to turbulent flow, Proc. I n s t . Mech. Engrs., v o l . 181, part 3B, 1966-1967, pp. 81-88.

8. Jackson, P.A., Robati, B., Mobbs, F.R., Secondary flows between eccentric r o t a t i n g cylinders at s u b - c r i t i c a l Taylor Numbers, Proc. 2nd Leeds -Lyon Symposium on Tribology, 1975, pp. 9-14.

9. Z a r t i , A.S., Jones, CP. and Mobbs, F.R., The influence of cylinder radius r a t i o on the v a r i a t i o n of the c r i t i c a l Taylor Number with e c c e n t r i c i t y r a t i o , Proc. 2nd Leeds-Lyon Symposium on Tribology, 1975, pp. 23-27.

10. Cole, J.A. and Hughes, C.J., O i l flow and f i l m extent i n complete journal bearings, The Engineer, 201, 1956, pp. 255-263.

11. Cole, J.A., Experiments on the flow i n r o t a t i n g annular clearances, Proc. Conf. on Lub. and Wear, I n s t . Mech. Engrs., London, 1957, paper 15, pp. 16-19.

12. Short, M.G., Jackson, J.H., Wavy mode super-laminar flow between concentric and eccentric r o t a t i n g cylinders: stroboscopic flow v i s u a l i z a t i o n and torque measurement, Proc. 2nd. Leeds-Lyon Symposium on Tribology, 1975, pp. 28-33.

182

13. Bennett, J. and Marsh, H., The f r i c t i o n a l torque i n externally pressurised bearings, Proc. 6th Gas Bearing Symposium, Southampton 1974.

14. Marsh, H., Drive systems f o r gas bearings, Proc. 5th Gas Bearing Symposium, Southampton 1971.

15. Bennett, J. and Marsh, H., The steady state and dynamic behaviour of the turbo-bearing, Proc. 6th Gas Bearing Symposium, Southampton, 1974.

16. Hampson, L.G., The effects of f r i c t i o n reducing polymers on the operation of Journal Bearings, Ph. D. Thesis, March, 1973, Durham University.

17. Constantinescu, V.N., On turbulent l u b r i c a t i o n , Proc. I n s t . Mech. Engrs., 1959, Vol. 173, No. 38, pp. 881-900.

18. Ng, C.W. and Pan, C.H.T., A linearized turbulent l u b r i c a t i o n theory, Journal of Basic Engineering, Trans. A.S.M.E., 1965, v o l . 87, No. 4, p. 675.

19. Hirs, G.G., A bulk-flow theory f o r turbulence i n l u b r i c a t i o n f i l m s , A.S.M.E. - A.S.L.E. Conference, New York, 1972, Paper 72 - Lub -12.