hydrostatic lubrication 1992

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HYDROSTATIC LUBR

I

CAT10N



TRIBOLOGY SERIES

Advisory Board

W.J. Bartz (Germany, F.R.G.)

R. Bassani (I taly)

B. Briscoe (Gt. Britain)

H. Czichos (Germany, F.R.G.)

D. Dowson (Gt. Britain)

K. Friedrich (Germany, F.R.G.)

N. Gane (Australia)

W.A. Glaeser (U.S.A.)

M. Godet (France)

H.E. Hintermann (Switzerland)

K.C. Ludema (U.S.A.)

T. Sakurai {Japan)

W.O. Winer (U.S.A.)

Vol.

1

Vol. 2

Vol. 3

V O l .

4

V O l . 5

Vol. 6

Vol.

7

Vol.

8

VO l . 9

VO l . 10

VOl.

11

VOl. 12

Vol.

13

Vol. 14

Vol. 15

Vol.

16

Vol. 17

Vol. 18

VOl. 19

VOl. 20

Vol. 21

VO l . 22

Tribology - A Systems Approach to the Science and Technology

of Friction, Lubrication and Wear (Czichos)

Impact Wear of Materials (Engel)

Tribology of Natural and Artificial Joints (Dumbleton)

Tribology of Thin Layers (Iliuc)

Surface Effects in Adhesion, Friction, Wear, and Lubrication (Buckley)

Frict ion and Wear of Polymers (Bartenev and Lavrentev)

Microscopic Aspects of Adhesion and Lubrication (Georges, Edi tor)

Industrial Tribology - The Practical Aspects of Friction, Lubrication

and Wear (Jones and Scott, Editors)

Mechanics and Chemistry in Lubrication (Dorinson and Ludema)

Microstructure and Wear of Materials (Zum Gahr)

Fluid Film Lubrication

-

Osborne Reynolds Centenary

(Dowson et al., Editors)

Interface Dynamics (Dowson et al., Editors)

Tribology of Miniature Systems (Rymuza)

Tribological Design of Machine Elements (Dowson et al., Editors)

Encyclopedia of Tribology (Kajdas et al.)

Tribology of Plastic Materials (Yamaguchi)

Mechanics of Coatings (Dowson e t al., Editors)

Vehicle Tribology (Dowson et al., Editors)

Rheology and Elastohydrodynamic Lubrication (Jacobson)

Materials for Tribology (Glaeser)

Wear Particles: From the Cradle to the Grave (Dowson et al., Editors)

Hydrostatic Lubrication (Bassani and Piccigallo)



TRIBOLOGY SERIES,

22

HYDROSTATIC

LU

BR

I

CAT1

0

N

R. Bass ani

Dipartimento di Construzioni Meccaniche e Nuclear;

Facolta di lngegneria

Universita di Pisa

Pisa, Italy

B. Piccigal lo

Gruppo Construzioni

e

Tecnologie

Accademia Navale

Livorno, ltafy

ELSEVIER

Amsterdam London New

York

Tokyo

1992



ELSEVIER SCIENCE PUBLISHERS B.V.

Sara Burgerhartstraat

25

P.O. Box 21

1,

1000 AE Amsterdam, The Netherlands

ISBN

0 444 88498

x

0 1992 ELSEVIER SCIENCE PUBLISHERS B.V. Al l rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in

any form or by any means, electronic, mechanical, photocopying, recording or otherwise,

without the prior written permission of the publisher, Elsevier Science Publishers B.V.,

Copyright

&

Permissions Department,

P.O.

Box

521, 1000

AM Amsterdam, The Netherlands.

Special regulations for readers in the U.S.A. - This publication has been registered with the

Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained

from the CCC about conditions under which photocopies of parts of this publication may be

made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A.,

should be referred to the publisher.

No responsibility is assumed by the publisher for any injury and/or damage to persons or

property as a matter of products liability, negligence or otherwise, or from any use or

operation o f any methods, products, instructions or ideas contained in the material herein.

Printed

in

The Netherlands



Preface

Hydrostatic lubrication is characterized by the complete separation of the conju-

gated surfaces of a kinematic pair, by means of a film of fluid, which

is

pressurized by

an external piece of equipment. Its distinguishing features are lack of wear, low fric-

tion, high load capacity (even when the relative velocity of the lubricated surfaces is

low o r nought), a high degree of stiffness and the ability to damp vibrations.

As

com-

pared with the other types of lubrication, it may have clear advantages against one

main disadvantage: the lubricant supply system is, generally, more complicated.

This book deals with the study of externally pressurized lubrication, both from the

theoretical and the technical point of view, thereby claiming to be useful for re-

searchers as well as for students and technical designers. In this connection, design

suggestions for the most common types of hydrostatic bearings have been included, as

well

as

a number of examples. The substantial and up-to-date lists of references may

constitute a further aid.

The first chapter, after a very brief historical note, describes the principal types of

hydrostatic bearings, while the second describes the principal types of supply systems

and compensating restrictors. The third chapter briefly reviews lubricants and their

main properties, including viscosity, that plays the most important role in lubrication,

and compressibility, that may considerably affect the dynamic behaviour

of

bearings.

The fundamental equations on which the study of lubrication is based are given in

chapter 4 and are used in chapter

5

in order

to

obtain certain characteristic parame-

ters (e.g. effective area, hydraulic resistance, friction force)

for

the most common pad

bearings. The principal types of hydrostatic bearings (single-pad and opposed-pad

thrust bearings, slideways, journal bearings and so on) are then examined in detail, in

combination with the principal supply systems, in the subsequent four chapters, with



vi HYDROSTATIC LUB RICA TlON

a view to providing a full description of their behaviour in the case

of

static loads.

Afterwards, the dynamic behaviour of the same bearings is considered (chapter 10):

the thin viscous film, typical of hydrostatic lubrication, generally makes them stiff,

stable and well damped, although certain phenomena (chiefly lubricant compressibil-

ity) may reduce the margin of stability. Chapter 11 deals with the problem

of

the

optimization of bearings, aimed at obtaining the minimum waste

of

total power (that

is, pumping power plus friction power). The thermal balance of the lubricant flowing

in a pad bearing is also investigated (chapter 121, taking into account the thermal

flow through the bearing itself and the relevant supply ducts. Some brief notes on the

important matte r of the experimental testing of hydrostatic bearings are given in

chapter 13. Finally, a number of examples of actual applications of hydrostatic lubri-

cation are to be found in the last chapter.

We wish to express

our

gratitude t o the authors, all quoted in the author index

and in the lists of references, whose work we have widely used and whom we have not

been able to thank directly. We also wish

t o

thank the firms (namely:

FAG Kugel-

fischer,

INNSE Machine Tools,Pensotti Machine Tools, SKF)

which kindly provided

us part of the graphical material that we used in the final chapter. Lastly, we wish t o

thank Dr. Paola Forte, who helped

us

in a number of ways, Mr. ergio Martini and

Mr. Aldo del Pun ta, who carried out part of the graphical work.



Contents

List

of

main symbols

XiV

Chapter 1 HYDROSTATICBEARINGS

..... ................................. 1

.1

INTRODUCTION

...........................................................

1.2 WORKING PRINC IP

................................................................

................................... 1

1.3

ADVANTAGES AND DRAWBACKS

.............................................................

..............................

3

1.4 APPLICATIONS ...........................................................................................

1.5 TYPES OF BEARINGS ........................................................

1.5.1

Thru st bearings

.................................................................. ........................... 7

1.5.2 Radial bearings ....................................... ......

.................................................

9

1.5.3

Multidirectional bearings

.................................................................................

11

1.5.4

Bearing arrangeme nts .......... ..........................................................................

REFERENCES

...............................

................................................................................. 14

Chapter 2 COMPENSATINGDEVICES

2.1

INTRODUCTION

.................................................................................................

2.2

DIRECT SUPPLY SYSTEMS

................................................................................................. 16

2.3 COMPENSATED SUPPLY SYSTEM ............................................................................... 17

2.3.1 Fixed restrictors ................................................................. ......................................... 18

2.3.2 Variable restrictors ...........................................................

..................................

19

2.3.3

Inherently compensated bearings

.........

.....................................................

25

2.3.4

Reference bearing s

.... ..........................................................

2.4

TH E COMMONEST SUPPLY SYSTEMS

.................................................................

2.4.1

Direct supply

................................................... .................................................... 30

2.4.2

Compensated supply

......................................

...............................................................

30

REFERENCES ............................................................ ................................ 33

2.5

HYDRAULIC CIRCUIT ......................................

................................ 31

Chapter 3 LUBRICANTS

3.1

INTRODUCTION

.......................................................................................

...........

35

3.2 MINERAL LUBRI .....................................................................................

....................

36

3.2.1 Types ....................................................... ...................................................... 36



viii

HYDROSTATIC

LUBRlCATlON

3.2.2 Viscosity ................................................................................................................................... 36

3.2.3 Oiliness .................................................................................................................................... 41

3.2.4

Density ... 42

3.2.5

Therm al propert ies

49

3.2.6

Othe r propert ies

......................................................................................................................

50

3.2.7

Additives

.................................................................................................................................. 50

3.3

SYNTHETIC LUBRICANTS

...............................................................................................................

52

REFERENCES ...... .......................................... .........

............ 52

Chapter

4

BASIC EQUATIONS

4.1

INTRODUCTION ...........................................................

4.2

NAVIER-STOKES AND CONTINUITY EQUATIONS

4.2.1

Rectangular coordinates .......................................

4.2.2

Cylindrical coordinates

.....................................

4.2.3 Spherical coordinates

..............................................................................................................

57

4.3

TH E REYNOLDS EQUATION

............................................................................................................ 58

4.3.1 Rectangular coordinates ......................................................................................................... 58

4.3.2

Cylindrical coordinates ........................................................................................................... 61

4.3.3

Spherical coordinates .......................................................

64

4.4

TH E LAPLACE EQUATION

...............................................................................................................

65

4.5 LOAD C APACITY , FLOW RA TE, FRICTION ...................................................................................

66

4.5.1

Load capacity

..........................................................................................

66

4.5.2

Flow rate .................. ..................................

66

4.5.3

Friction

...............................................

4.6

TH E ENERGY EQUATION

4.7

LAMINAR FLOW TH ROUGH CHARACTERISTIC CONFIGURATIONS .....................................

69

4.7.1

Parallel surfaces ......................................................................................................................

69

4.7.2

Infinite-length rectangular pad

....................................................... 71

4.7.3 Flow recirculation inside recess

..........................................

73

4.7.4 Ann ular clearance. .................................................................................................................. 75

4.7.5 Circular pad

.............................................................................................................................

76

..........................................................................................

77

.......................................................................................... 79

4.9 INLET LOSSES..................

........................................................................................................

80

4.10

TURBULENT FLOW

......................................................................................................................... 80

4.11

TH E FLOW IN ORIFICES ................................................................................................................

83

REFERENCES

.. ......................................................................................................................... 85

Chapter 5 PAD COEFFICIENTS

5.1

INTRODUCTION ..............................

5.2

GENERAL STATEMENTS ................

5.3

CIRCULAR RECE SS PAD .......

5.3.1

Basic equations

.............

5.3.2

Design ch art

............................................................................................................................. 94

5.3.3

Effects

of

errors in parallelism

............................................................................................... 95



CONTENTS ix

5.3.4 Effects of t he loss of pressure at the inlet .............................................................................. 98

5.3.5 Turbu len t flow

.......................................................................................................................

101

5.3.6 Effects of th e ine rti ............................................................. 103

5.3.7 Therm al effects ..... ................................................ 107

5.4 ANNULAR RECESS PADS ................................... ..........112

5.4.1 Basic equation s

.....

................................................ 112

5.4.2

Effects of er ror s in parallelism .............................................................................................

117

5.4.3

Effects of pressu re losses

at

the inlet ...................................................................................

118

5.4.4

Turbu len t

flow

.......................................................................................................................

119

5.4.5

Effects of th e ine rti a forces

........................................

5.4.6

Therm al effects ...........................................................

5.5

TAPERED PADS .....................................................................

................................ 123

5.5.1

Basic equation s ......................................................................................................................

123

5.5.2

Effect of the iner tia forces

...................

...........................................................

125

5.5.3

Effect of misa lignm ent

........................

...........................................................

126

5.6 SPHERICAL PADS ........... .................................................................. 128

5.7

RECTANGULAR PADS .....................................................................................................................

133

5.8

CYLINDRICAL PADS

........................................................................................................................ 138

5.9

HYDRO STATIC LIFI'S ...........................................................

5.10

SCREW AND NUT ASSE

REFERENCES

............................ .......................

Chapter

6

SINGLE PAD BEARINGS

6.1

INTRODUCTION

...............................................................................................................................

149

6.2

DIRECT SUPPLY ...................................................................

..................

6.2.1

Bearing performance .................................................

6.2.2

Tem pera ture and viscosity ........................................................

6.3

COMPENSATED SU PPLY................................................................................................................

153

6.3.1

Laminar flow restrictors (capillaries)................................................................................... 155

6.3.2 Orifices....

6.3.3 Cons tan t flow valves .............................. 160

6.3.4 Spool valves ...............................................................

6.3.5

Diaphragm -controlled restrictors

..............................

6.3.6

Infinite-stiffness devices .......................................................................................................

169

6.3.7

Inherently compensated bearings

................................................................ 172

6.4 DESIGN OF SINGLE -PAD THRUST BEA RINGS

...

6.4.1

Direct supply (constant

flow)

....................................

6.4.2

Compensated supply (constant pressure)

............................................................................ 180

Chapter 7

7.1

INTRODUCTION

OPPOSED-PADAND M U L T P A D BEARINGS

7.2

OPPOSED-PAD B

7.2.2

Capillary compensation

............................................

7.2.3

Orifices

.......... ...........................

..............................................................

197

..................



X

HYDROSTATIC LUBRICA TlON

7.2.6 Design

of

opposed-pad bearings ................... ................................................ 213

7.3.1

Direct supply ............

7.3.2 Co nstan t pressure supply

.......

7.4.1 Direct supply .................................................

7.6 MULTIPAD JOUR NAL BEARINGS

Chapter

8

MULTIRECESS

BEARINGS

8.1 INTRODUCTION

...............................................................................................................................

236

8.2

ANALYSIS..........................................................................................................................................

236

8.3

MULTIRECESS JOURNAL BEARINGS

......................................................................................... 239

8.3.1

Bearing performance

8.3.2

Effect of geometrical

................................................................ 249

8.3.3

Design of multirecess journal bearings.

8.3.4

Design p rocedure ..................................

8.4

ANNULAR M ULTIRECESS THRUST BEAR1 .....................................

260

8.5

TAPERED MULTIRECESS BEARINGS..........................................................................................

263

8.5.1 Single-cone ournal bearings

...............................................................................

265

8.5.2 Opposed-cone assemblies ............................................................................... 210

8.1

YATES BEARINGS

8.7.5

Design p rocedure

...

..................................

283

REFERENCES

........................................................................................................................................... 285

Chapter

9

HYBRID PLAIN

JOURNAL BEARINGS

9.2

PERFORMANCE O F TH E HYBRID PLAIN JOURNAL BEARINGS .....

REFERENCES

Chapter

10

DYNAMICS

10.1 INTRODUCTION

.............................................................................................................................

301

10.2

EQUATION O F MOTION ............................................................................................................... 302



CONTENTS Xi

10.3 PAD COEFFICIENT S...............................................................

........................................

305

10.3.1 Circular-recess pads

...................

........................................ 305

10.3.2 Annular-recess pad s ................... ....................................................... 307

10.3.3 Tapered pads ..............................................................................................

........

307

10.3.4 Screw and nu t assemblies ................................................................................................... 308

10.3.5 Other pad shapes

.........................................................

.................... 308

10.4.1 Direct supply (c ......................................................... 311

.................314

10.4.4 Spool or diaphragm valves. .

10.4.5 Infinite stiffn ess devices

.......................................................

320

....................................................................................................................

322

10.5.1

Transfer hnct ion

10.5.3 Frequency response

......................................................

10.6

OPPOSED-PAD BEARINGS ....................................................

10.6.1 Direct supply (constan t flow)

.......................................

10.7 SELF-REGULATING BEARINGS ......................................................... 339

Co nstan t flow feeding .......................................................................................................... 341

Co nstan t pressure feeding

..................................................................................................

342

10.7.1

10.7.2

10.8

MULTIPAD BEARING SYSTEMS

10.8.1

Hydrostatic slideways .........................................................................................................

344

10.8.2

10.9

MU LTIREC ESS JOURNAL BEAR INGS .......................................................................................

349

10.9.1 Analysis .....................................................................

10.9.2 Non-rotating bearings, incompressible lubricant

10.9.3

Multipad journal bearings

..................................................................................................

346

Ro tating bearing, incompressible lubricant .........

10.9.4 Compressible lubricant ....................................................................................................... 358

REFERENCES

............... .........................................................

360

Chapter

11

OPTIMIZATION

11.1 INTRODUCTION

.............................................................................................................................

362

11.2 GENERAL PROCEDU RE ............................................................................................................... 362

11.3 CONDITIONS O F MINIMUM ........................................................................................................ 365

11.4 EFFICIENCY.................................................................................................................................... 365

11.5 DIRECT SUPPLY

............................................................................................................................

366

11.5.1 Steady pad

..........................................................................................................

11.5.2 Moving pad

...........................................................................................................................

373

11.6

OPTIMIZATION

.........................................................

385

11.6.3 Given load ........ ................................



x i i HYDROSTATIC LUBRICATION

11.7

REAL PADS

...................................... ...... ...............

11.7.1

Rectangular pad

.................................................................................................................. 415

11.7.2

Oth er types of pads..............................................................................................................

419

11.7.3

Circular pad .........................................................................................................................

420

11.7.4 Annular pad .........................................................................................................................422

11.8

COMPENSATED SUPPLY..............................................................................................................

425

11.8.1

Capillary tubes ....................................................................................................................

425

11.8.2

Steady pad

...........................................................................................................................426

.................................................. 431

11.8.4

Dissipated

power

an d efficiency losses

............................................................................... 432

11.9

OPl'IMIZATION

11.10

OTHER TYPES O F COMPENSATING ELEMENTS

.......

........................

443

11.10.1

Orifices ...............................................................................................................................

443

11.10.2

Flow-control valves............................................................................................................

444

11.11 REAL PADS

.................................................................................................................................... 444

REFERENCES ...........................................................................................................................................

446

Chapter 12 THERMACFLOW

12.1

INTRODUCTION .............................................................................................................................

447

12.2

TEMPERA TURES IN THE BEARING......

12.2.1

Tem peratures in the f ilm ....................................................................................................

447

12.2.2

T e mp e r a tu r e s a t t h e film outlet

.........................................................................................

449

12.3

SUPPLY PIPELINE .........................................................................................................................

456

12.4

COMPENSATING ELEMENTS......................................................................................................

458

12.5

PUMP ......................................

12.6

COOLING PIPELINES ................................................

459

12.7

SELF-CO OLING CAPILLARY TUBE ............................................................................................

461

12.8

VISCOSITY AND TEMPERATURE ...............................................................................................

463

REFERENCES ........................................................... ...........................................................................

464

Chapter

13

EXPERIMENTAL

TESTS

13.3.1

Electric analo g field p lotter

................................................

469

13.3.5

Screws and n u t s

REFERENCES ......



CONTENTS

xiii

Chapter 14 APPLICATIONS

14.2 MACHINE TOOLS ......... .......................................................................... 483

......................................................................

.......................

483

............................................. 491

14.2.3 Feed drives ........

.....................................................................

492

14.2.4

Guideways and

r

.....................................................................

496

14.4

OTHER APPLICATIONS ...................... ..............................................

511

............................................. 513

4.5 HYDRAULIC CIRCU ITS.............................................

14.5.1

Simple layout ............................ ..............................................

513

14.5.3

Multiple pum ps

.....................................................................

REFERENCES

.......

.....................................................................

APPENDICES

A.l SELF-REGULATED PAIRS

AND

SYSTEMS

.....................................

.....................................................................

..............................................

A.3.1

Resistances

.......... ...............................................

...................

527

REFERENCES ..

................................................

Author index 533

Subject index 537



Lis t

of main symbols

An inverted comma

(')

generally indicates tha t the relevant quantity has been

made non-dimensional by dividing it by a suitable reference value (often the value in

the reference configuration, when it is applicable); since the reference values may

depend on the bearing type and on the type of supply system, only a few have been

indicated below.

Meaning of most frequently used subscripts:

generally means 'in the reference configuration', that is, in particular, the

centred o r concentric configuration whenever it is applicable; in chapter 11

generally indicates a suitable (although arbitrary) value used a s reference in

the optimization process;

means 'land';

mean 'maximum' and 'minimum', respectively;

means 'restrictor' or 'recess';

in chapter 10 refers

to

the static equilibrium configuration, except for p s

(supply pressure), whereas in chapter

12

refers t o the land area of a pad;

refers to controlled restrictors.

effective area

nondimensional effective

area

effective friction area

land area

recess area

effective area of spool o r di-

aphragm (controlled restric-

tors)

land length

a l L

or

u l ( r 4 - r l )

characteristic parameter of

flow dividers (Eqn 7.49)

pad width;

squeeze coefficient (pad bear-

ings) or damping coefficient

(journal bearings)

B'

BO

b

b'

C

D

e

F

f

C

Ff

B I L ;

B

I

Bo (single pad and self-

regulating bearings);

BI 2Bo

(opposed-pad b.)

reference squeeze coefficient

of a pad or reference damp-

ing coefficient of a journal

bearing

recess width

b l B

radial clearance

specific heat

diameter

displacement

loading force

friction force

friction coefficient



LIST OF SYMBOLS

xv

recess friction factor

Qpc (chapter

12)

see Eqn 12.7

actual value of axial play

(opposed-pad bearings)

friction power

nondimensional friction

parameter

reference friction power of a

Pad

H f I H f o

single pad and self-

regulating bearings);

H f l 2 H f 0

(opposed-padb.)

land friction power

recess friction power

pumping power (dissipated

in a pad)

pumping power

total power

see Eqn

6.58

o r

8.32

film thickness

normal film thickness

(tapered pads)

recess depth

stiffness

reference stiffness

reference stiffness for capil-

lary compensation

stiffness of lubricant (Eqn

10.19)

equivalent bulk modulus

tilting stiffness

stiffness of spring or di-

aphragm (controlled restric-

tors)

speed enhancement factor

speed parameter (chapter 11)

bearing length

recess length

IIL

moment;

mass

friction moment

number of recesses;

number of active turns

(screw-nut assemblies)

pitch

KIKo

P r

P s

Q

R

R*

R’

RO

R i

Re

Rr

r

r’

6

Sh

Si

T

Ta

Tt?

Ti

U

V

W

wf2

wz

X

a

8

8”

Y

recess pressure

supply pressure

flow rate

hydraulic resistance;

thermal resistance (chapter

12)

nondimensional resistance

parameter

RI R, (pad bearings)

reference hydraulic resis-

tance of a pad

R I R o (self-regulating bear-

ings)

Reynolds number

hydraulic resistance of com-

pensating restrictor

radius

inner

to

outer radius ratio

rllr2

(self-regulating bear-

ings)

velocity parameter (Eqn

8 . 8 )

inert ia parameter

temperature

ambient temperature

temperature

at

the outlet of

a land (chapter

1 2 )

temperature at the inlet of a

land (chapter 12)

sliding speed

volume of recess and rele-

vant supply pipe (chapter 10)

load capacity

hydrodynamic load capacity

axial load capacity (tapered

or spherical journal bear-

ings; Yates bearings)

displacement of spool o r

diaphragm (controlled re-

strictors)

half-cone angle;

thermal conductance

(chapter 12)

reference pressure ratio

valve parameter

load angle (spherical journal

bearings);

reference pressure ratio (self-

regulating bearings)



HYDROSTATIC LUB RICA TlON

area ratio (infinite-stiffness

valves)

temperature rise

pitch error

non-dimensional pitch error

(Eqn

7.67)

eccentricity:e l h o o r e l C

axial eccentricity (tapered

journal bearings; Yates bear-

ings)

damping factor

clearance error (Eqn 7.20)

circumferential length of

inter-recess lands (multire-

cess bearings);

flank angle (screws)

helix angle;

transfer function of a block

(chapter 10);

thermal conductivity coefi-

cient (chapter 12)

transfer function of the feed-

back block

transfer function of the feed-

back block in the case of

fixed restrictor

2(C&f-C,)

dynamic viscosity

see Eqn 6.60

see Eqn

6.62 o r 8.33

kinematic viscosity;

Poisson ratio

non-dimensional displace-

ment of spool or diaphragm

(controlled restrictors);

stiffness ratio

w 2 / w 1

(chapter 10)

reference power ratio

density

restrictor parameter (Eqns

10.29

or

Eqn 10.40)

shear stress

tilt angle;

squeeze parameter (Eqn

10.52)

squeeze parameter (Eqn

10.9)

angular speed

angular frequency

characteristic frequencies

undamped natural frequency



Chapter

1

HYDROSTATIC BEARINGS

1.1 INTRODUCTION

Hydrostatic lubrication consists in pushing a lubricant between the surfaces of

a kinematic pair by means

of

an external pressurization system. This lubrication

mechanism has now a well-defined collocation in the large field of lubrication engi-

neering. In particular, it can be used instead of hydrodynamic lubrication when

this last proves t o be not very effective. The main advantages of externally pressur-

ized lubrication are very low friction and negligible wear, whereas the only actual

drawback is a certain complexity of supply circuits. Applications thus vary from

large, generally slow, machines to small, generally fast, machines: this is also

made possible by the wide range of kinematic pairs to which hydrostatic lubrication

can be applied.

1.2 WORKING PRINCIPLE

It is well known that, to ensure the setting up and the persistence of a steady

hydrodynamic pressure field in the lubricant separating the surfaces of any kine-

matic pair, two important conditions have to be met:

- the mating surfaces must not be parallel;

-

a sufficient relative velocity must exist.

When one,

o r

both, of these conditions cannot be satisfied, an “external pressur-

ization” of the lubricant may be the solution: the pressurized field allows the lift and

the bearing of the moving member on the fixed member of the pair.



2

HYDROSTATIC LUBRICATION

Fig.

1.1

contains an outline of the principle of "externally pressurized lubrica-

tion", which is most commonly referred

t o

as "hydrostatic lubrication". The recess

(of which the projected area is

A)

of the bearing pad [ll

of

the pair is fed by

a

pump;

the bearing runner [21 is loaded by a force

W

(Fig. 1.l.a). When the pump begins

t o

run, the pressure in the recess grows (Fig. l.l.b), until the "lifting pressure"

p = W I A

is reached (Fig. 1.1 .~ ); t this point member

121

is lifted,

a

lubricant film

builds up to separate the surfaces, and a

flow

Q

is delivered, due

t o

the pressure step

along the clearance (Fig. 1.l.d). Different loads lead to different values of the recess

pressure and of the film thickness

h

(Fig. l.l.e, f).

Fig.

1.1 Hydrostatic lubrication: pressure diagrams and

fluid

film formation in

an

axial single-pad

bearing.

To

also sustain loads in the reverse direction, member

[21

is put between two

pads

[l],

s shown in Fig.

1.2. Now

flows

Q

cause recess pressures

p>O

even for

W=O:

the system is preloaded (Fig. 1.2.a.). When a load

W

is applied, the pressure in



HYDROSTATIC BEARINGS 3

the lower recess [11] increases, and pressure in the upper recess decreases (Fig.

1.2.b). Consequently, a greater stiffness is obtained, compared with th e single pad.

When two, o r more, recesses exist, as in Fig. 1.2, it is clearly necessary

t o

feed

each of them by means of separate pumps; alternatively, a common source of lubri-

cant may be used, but each recess must be fed through an adequate compensating

device (restrictor). We shall deal extensively with th is point in the following

chapters.

- a - - b -

r i t t t l t t trPr,

Fig. 1.2Hydrostatic opposed-pad axial bearing: pressure diagrams.

Hydrostatic lubrication can be applied to every type

of

elementary pairs with one

degree

of

freedom: prismatic pairs (Fig. 1.1; 1.2), rotating pairs (Fig. 1.12.a), helical

pairs (Fig.l.1O.a); with two degrees of freedom: rotating pairs without side supports;

with three degrees of freedom: prismatic pairs without side supports, spherical

pairs (Fig. 1.13.b). Every type of motion can be carried out: plane (Fig. 1.1, 1.2, 1.3.a),

spherical (Fig. 1.3.b), and general (Fig. 1.3.c).

1.3

ADVANTAGES AND DRAWBACKS

All contact between the surfaces of the two members is prevented by the exter-

nally pressurized lubrication; this produces several favourable effects:



4


- no wear practically exists;

- friction can be very low, especially when the relative velocity of the surfaces is low;

- no stick-slip exists;

-

stiffness can be considerable: i.e. a very slight variation in the thickness of the lu-

bricant film may be obtained, for any given variation of the load;

- the lubricant film produces an "averaging" of the roughness and the other defects

of the mating surfaces;

-

no localized overpressure exists: the pressure field is uniform in the recesses

(which are generally large) and decreases in the clearances;

- the pressurized fluid film has high damping characteristics;

-

the effectiveness of the lubrication is hardly influenced at all by thermal problems

and by a n y variation in the speed regime;

- every fluid may, in principle, be used as a lubricant;

-

the performance of the hydrostatic bearings is simpler

t o

evaluate than in the case

of hydrodynamic bearings, since the boundary conditions are, generally, well

defined.

The main drawback consists in the need for a supply system, a t medium

o r

high pressure, with the relevant control and safety devices. However, some sort of

supply system is generally required in hydrodynamic lubrication, and even for

rolling bearings.

In Table 1.1 the hydrostatic slideways, journal bearings, and screw-nut assem-

blies are compared with analogous usual pairs, to give some rough direction for the

effective use of externally pressurized lubrication.

1.4 APPLICATIONS

The first application of hydrostatic lubrication was carried out by the French-

man

L.

D. Girard, who, in the second half of the last century, built a water hydro-

static journal bearing (ref.

1.1)

and, subsequently, thrust bearings, too.

In the second decade of this century, a hydrostatic annular thrust bearing was

applied in a hydraulic turbine, see ref. 1.2, and Lord Rayleigh worked out the equa-

tions for calculating load capacity, lubricant flow rate, and friction moment for a

circular thrust bearing (ref.

1.3).

In the third decade an interesting and spectacular application of hydrostatic

lubrication was that of the Hale telescope of Mount Palomar (ref. 1.4).

Several authors, of whom D. D. Fuller (ref. 1.51, H.

C.

Rippel (ref. 1.6), nd H.

Opitz (ref.

1.7)

deserve special mention, subsequently contributed to the development

of hydrostatic bearings.




5

TABLE

1.1

Considerations on types of bearings.

Types

Characteristics

Design

Availability of standard parts

Finish and hardness of surfaces

Space required

Positioning accuracy

Assembly

Guard

Cost (to m anufacture)

Cost (to install)

Life

Lubrication circuit

Cost of lubrication circuit

Supply pressure and pumping power

Load

Stiffness

Vibration damping

Friction coefficient and friction power

Stick-slip

Wear

~~

Axial pairs

(Slides)

Boun Hyst Rol

5 4 3

4 3 4

2 4 3

4 3 2

3 3 2

4 3 2

3 3 2

3 2 2

3 3 2

3 5 3

4 2 3

4 2 3

3 2-41 4

3 3-51

3

4 3-51 3

3 5 2

2 3-52 3

1 5 5

2 5 4

Radial pairs

(Journals)

Hydy Hys t Rol

3 3 4

4 2 5

3 4 4

3 2 2

2 2 2

3 3 2

3 3 2

3 2 3

3 2 4

3 4 3

3 2 4

3 2 4

3 2-31 4

2-32 2-41 3

2-32 2-4' 3

2-33 4 2

3 3-52 4

4 5 5

3 5 3

Helicoidal pairs

(Screw-nuts)

3oun Hys t Roll

4 2 3

5 1 4

1 4 3

4 2 2

2 2 2

3 2 2

3 3 2

3 2 2

4 2 3

2 4 3

3 2 3

3 2 3

3 2-31 4

2 2-41 3

3 2-41 3

3 4 2

2 3-52 3

1 5 5

1 5 3

(Rating

5

is best or more desiderable. Boun: Boundary lubrication; Hyst: Hydrostatic lubri-

cation; Hydy: Hydrodynamic lubrication; Roll:

Rolling

elements).

&@:

It depends on supply type. It depends on speed.

3

Whirl.

Externally pressurized lubrication is at present used in the entire field of me-

chanical engineering, from large machines, where speed

is

in general low,

t o

small high-velocity machinery. Certain characteristic applications will now be

briefly listed.

(i) Large machines

Telescopes, radio-telescopes, big radar antennas, which must move slowly and

accurately.

A

well-known example is the Mount Palomar telescope already men-

tioned: a 500 ton mass, which makes one revolution every 24 hours, supported by

hydrostatic thrust bearings.

Air preheaters for boilers

of

electric power plants:

in

this case the hydrostatic

bearings are exposed to high temperatures.

Rotating mills for ores

o r

slags; thermal problems exist here, too.



6

HYDROSTA TIC LUBRICA

TlON

Machine tools (ref. 1.8),where medium or high precision is required t o move

great weights; for instance large boring or milling machines. The moving car-

riages are supported by hydrostatic slideways and are sometimes driven by hydro-

static screw and nut assemblies.

Hydrostatic steady rests for large lathes.

Assembly lines, where the component parts are carried along on hydrostatic

slides; this allows a very accurate positioning of the components.

Structures, even of very large ones, which can be easily moved on hydrostatic

bearings.

(ii) Medium size machines

Grinding machines, numerical control machine tools, machining centers, which

require very accurate positioning and freedom from vibration. Due to the absence of

stick-slips, and

t o

the high degree of stiffness and damping of the pressurized fluid

film, hydrostatic lubrication is particularly suitable for such machines.

High velocity spindles; in this application hydrostatic bearings often prove to be

better than hydrodynamic ones (particularly in the start and stop stages) as well as

being better than the rolling bearings (where some problems are encountered, due

to the effects of wear and

t o

the high centrifugal forces on the rollers).

(iii) Small machines

Precision balances, dynamometers: hydrostatic bearings are better than the usual

ones, because friction practically vanishes at very low speeds, even in the case of

alternate motion. Their use is particularly advisable for electrical rotating-field

dynamometers.

Vibration attenuators for measuring instruments.

Frictionless oil seals; these seals may be useful in certain cases e.g.: distributors

of lubricant f o r hydrostatic slideways, hydraulic cylinders for flight simulators.

- a -

- b -

. _

I

- c -

i-

t-----+--

w

Fig. 1.3 Circular

pad

bearings:

a-

circular recess

pad; b-

annular recess

pad; c-

multirecess

pad.



HYDROSTATIC

BEARINGS

7

1.5 TYPES

OF BEARINGS

Hydrostatic bearings may be classified on the basis of the direction

of

the load

that may be carried.

S o

we have:

- thrust bearings;

- radial bearings (journal bearings);

- multidirectional bearings.

Let us examine briefly the most common shapes.

1.5.1 Thrust bearings

(i) Circu lar p a d bearings . Figure 1.3 shows: -a- a central-recess pad; -b- an

annular recess pad; -c- a multirecess pad.

A s

rotary speed becomes very high, the

behavior of pad -a- becomes "hybrid": hydrodynamic pressure, caused by centrifu-

gal force, joins the hydrostatic pressure (see section 6.3.l(vi)). Provided each recess

is independently fed, pad -c- may also sustain tilting moments.

(ii) Opposed-pad circular bearings. When the load may act in two opposite di-

rections, o r when greater stiffness is needed, two circular pads may be assembled

as shown in Fig. 1.4.a. If the load has a prevalent direction, it may be found useful

to select two different pads (Fig. 1.4.b).

The bearing in Fig. 1.4.c is a "self-regulating" one; i.e., thanks

t o

its shape, the

flow rates supplied to the two halves of the bearing are always equal to each other.

Consequently, only one supply device is needed.

(iii) Rectangular pad bear ings .

A

number of shapes of rectangular pads are

- a -

- b -

- c -

Fig. 1.4 Opposed-pad circular bearings: a- equal pads; b - unequal pads; c- "self-regulating"

bearing.



8 HYDROSTATIC LUBRICATION

+-

shown in Fig. 1.5. If the pads are moving at very high speed and have a fixed tilt,

their behavior becomes "hybrid": a hydrodynamic effect (in the clearances) is added

to the hydrostatic effect. As for the similar multirecess circular pad, the multire-

cess rectangular pad in Fig. 1.5.f can also sustain tilting moments.

- a -

I - d -

I

-

b -

- e -

- c -

- f -

Fig.

1.5

c-, d- , e- pads with rounded comers; f- rnultirecess pad.

Rectangular pad bearings:

a -

equal

sill

width pad; b - different sill width pad;

(iv) Opp osed-pa d rectangular bearings. Some examples of this kind of bearings

are given in Fig.

1.6:

in case -a- the two pads are equal to each other; different pads

are used in case -b-.

(v)

Tapered pa d bearings.

The conical pads shown in Fig 1.7 are similar

t o

the

circular pads in Fig. 1.3. They require less pumping power (but larger friction

power) for the same load and radial size.

(vi) Spherical pa d bearings. They are shown in Fig. 1.8.

(vii) Screw-nut as sembl ie s . In Fig. 1.9.a a hydrostatic screw-nut is shown,

which may sustain loads in only a direction: the recess may be continuous (Fig.

1.9.b) or, more often, discontinuous (Fig. 1.9.~) .n general, however, a double effect




9

- a - - b -

Fig. 1.6

Opposed-pad rectangular

bearings:

a- equal pads; b- unequal pads.

- a -

'

- b -

I

Fig.

1.7

Tapered pad bearings: a- circular recess pad; b- annular recess pad.

device

is

needed, as in Fig. l.lO.a. It is also possible to have a self-regulating screw-

nut (Fig. 1.lO.b); in this case, a double-thread screw must be used. Note

its

similar-

ity with the self-regulating bearing in Fig. 1.4.c.

1.5.2

Radial

bear ings

Fig. 1 .ll .a shows a cylindrical pad; since it can sustain loads only in one direc-

tion, it should be considered to be a thrust bearing, in spite of its shape; the same

goes for the opposed-pad bearing in Fig. l.ll.b. The assembly in Fig. l.ll.c, on the

other hand, is able t o sustain loads in all the radial directions including a certain

angle; this angle is expanded to the whole turn for the multipad journal bearing in

Fig. 1.12.a.




- a -

- b -

Fig.

1.8

Spherical pad bearings:

a-

circular recess pad

b-

annular recess

pad.

- a - - b -

. _

- c

Fig. 1.9

Screw-nuts:

b-

continuous recess; c- multirecess.

If the drainage grooves separating the pads are eliminated, the "multirecess"

journal bearing is obtained (Fig. 1.12.b), which in general proves to work bette r than

the multipad bearing. In this kind of bearing, if the turning velocity of the shaft is

high enough, a hydrodynamic pressure field is superimposed on the hydrostatic

field, shown in Fig. 1.12.b. This fact is exploited in the case of so-called "hybrid"

bearings (Fig. 1.12.c, ref. 1.9), in which the recesses a re reduced to a minimum

t o

enhance hydrodynamic lift. They are designed

t o

sustain the load by means, in the

main, of the hydrodynamic effect a t the regime velocity, while the hydrostatic pres-

sure field

is

used, in the main, to prevent any contact

of

the surfaces i n the start-

stop phases.




- a -

- b -

11

t t 1

Fig.

1.10

Double effect screw-nuts:

a-

conventional;

b-

self-regulating.

- a -

-

b -

- c -

Fig.

1.11

Radial bearings:

a-

cylindrical pad;

b-

opposed-pad;

c-

double-pad partial journal bearing.

1.5.3 Mult id i rect ional bear ings

The bearings shown in Fig. 1.13 are able to sustain loads in the axial direction

as well as in any radial one. Type -a- is made up of a tapered journal sustained by a

multirecess sleeve; in type -b- the surfaces are spherical. In both cases, t o sustain

reversible axial loads, o r to ensure greater stiffness, two opposite bearings must be

used.

In the peculiar bearing in Fig.

1.14

the same lubricant supplied t o the recesses

of the radial bearing is then used in the annular recesses of the axial pads; such an

arrangement produces a reduction in pumping power.



12


c

-

- a -

I

Fig. 1.12Journal bearings:a- multipad; b- multirecess;

c-

hybrid.

- a -

i

- c -

I

I

Fig. 1.13 Multidirectionalbearings:

a-

conical bearing; b- spherical bearing.

Fig.

1.14

"Yates"bearing.



HYDROSTATIC

BEARINGS

13

1.5.4 Bear ing arrangements

Hydrostatic bearings are variously combined to build hydrostatic bearing

systems.

Figure 1.15.a shows a spindle which is sustained by a journal bearing on the

right-hand side, and by

a

combined axial and radial bearing

on

the other side;

whereas, in Fig. 1.15.b, the spindle is sustained by a pair of conical bearings.

- a -

- b -

Fig.

1.15

Hyd rostatic spindle: a- with a journal bearing and a com bined journal and thrust bearing;

b-

with two conical bearings.

Fig. 1.16 Hydro static slideway.



14

HYDROSTATIC

LUBRICATION

Figure 1.16 shows a slide sustained by a system of opposed-pad bearings in the

vertical and horizontal directions.

Figure

1.17

shows a special hybrid bearing in which

a

rolling bearing is com-

bined with a hydrostatic one (the centrifugal oil feed is also shown). This arrange-

ment allows the ball-bearing inner ring

to

rotate a t a lower speed than the shaft.

Fig.

1.17 Combined rolling-hydrostatic

bearing.

R E F E R E N C E S

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Girard

L.

D.;

Noueau S ystkme de Locomotion

sur

Ch e m in d e F e r;

Bachelier,

Paris, 1852.

Vogelpohl G.;

Betriebssichere Gleitlager;

Springer Verlag,

1958; 315

pp.

Lord Rayleigh; A

Simple Problem

in

Forced Lubrication;

Engineering, 104

Karelitz M. B.;

Oil Pad Bearings and Driving Gears of 200-Znch Telescope;

Mech. Eng., 60 19381,541-544.

Fuller

D.

D.;

Theory and Practice

of

Lubrication

for

Engineers;

Wiley

&

Sons,

1956; 432 pp.

Rippel H. C.;

Design

of

Hydros tat ic Bear ings ,

Pt. lt10; Machine Design,

Aug.+Nov.

1963.

Opitz

H.; Aufbau und Auslegung Hydros tat ischer Lager und Fu hrungen und

Konstruktive Gesichtspunkte bei der Gestaltung von Spindellagerungen

mit

Walzlagern;

VDW-Konstrukteur-Arbeitstagung, 1969

Stansfield F. M.;

Hydrostatic Bearings

for

Machine Tools and Sim ilar Appl i -

cations; The Machinery Publishing Co. Ltd., 1970; 227 pp.

Rowe

W. B.; Hydrostatic and Hybrid Bearing Design;

Butterworth

&

Co,

1983;

240

pp.

(1917),617-697.



Chapter 2

COMPENSATING DEVICES

2.1

INTRODUCTION

It has been already pointed out that two bearings a re necessary to bear loads in

reverse direction. Two bearings are also needed if load

is

not coaxial with the bear-

ing

as

i n Fig. 2.1.a (t hat

is

equivalent to a centered load

plus a

moment). Bearing

runner [2] inclines on bearing pad [l], and may touch it on one side while flow leaks

from the other side, This does not occur if member [2] is supported by two pads (or

more, and not necessarily equal) and different pressures occur in the two recesses

(Fig. 2.1.b). For this t o happen, the supply system must allow for these different

- a - - b -

tt'tttlLv

Fig. 2.1 Eccentric load on hydrostatic pads:

a-

ingle pad;

b-

wo-pad arrangement.



16

HYDROSTATIC LUBRlCATlON

pressures. In practice this may be accomplished in two ways:

- by using a separate pump t o feed each recess directly; this is commonly referred t o

as the "constant-flow supply system";

-

by using a common source of pressurized lubricant, which is carried

t o

each pad

through compensating devices (restrictors); since the pressure is generally held

constant upstream from the restrictors, this is commonly referred to as the

"constant pressure supply systems".

Furthermore, certain particular types of bearings are proposed that are

"inherently compensated"; i.e. they have a built-in compensating device. In this

way, they can be fed directly by a lubricant source (in general, a t constant pressure).

From the foregoing considerations it is clear that the proper working of the

hydrostatic bearings depends on the correct selection of the devices which make up

the supply system, as well as on the correct design of the bearing itself.

2.2 DIRECT

SUPPLY

SYSTEMS

Figure 2.2 shows a direct supply system. If the losses in the supply pipes are

negligible, the pressure

ps

of the lubricant in each pump is the same as the recess

pressure

Pr.

For any given flow rate Q,

of a

lubricant of viscosity

p,

he film thick-

ness

h

is related t o the recess pressure

Pr

(e.g. see Eqn 4.39).Since &=const., when

load W grows, h decreases, while Pr increases. If a tilting moment exists (say be-

cause the load is displaced toward the pad [12]), pressure grows in [12], and de-

creases in [11]. Since Pr=Ps, ecause no restrictor can dissipate power, the system

dissipates the smallest pumping power.

h

Fig.

2.2 Constant

flow

supply system: on e pump

for

each

bearing.



COMPENSATING DEVICES 17

In theory, the only limitation to the increase ofp,, and hence of the load capacity

(and stiffness) of the bearing, comes from the power of the motor, and from the

maximum allowable pressure in the supply system.

In the last analysis, the constant flow system proves to be quite efficient. Its

limit

is

of an economic nature, due to the need for a pump, with the relevant motor,

for each recess. The problem can be partially overcome, particularly when all the

pumps are equal, all being driven by means of a single motor. It is worth noting

that this method also makes it possible to reduce the power required, lower than the

sum of the peak power required for each pump.

Figure

2.3

shows a particular arrangement (ref.

2.1),

in which a motor drives

a

main pump (which steps up pressure to an intermediate value) and at the same

time

a

series of smaller pumps feeding the recesses. The delivery of the main pump

is a little greater than the sum of the flow rate of the other pumps. Such an ar-

rangement makes it possible t o reduce the pressure step in the pumps, and th e re-

lated problems, especially in the case of gear pumps.

j

y d r o s ta t i c B e a r i n g s

- - - - .

- -

- . - -

Fig.

2.3

Constant f l o w

supply system: double pressure step.

Another arrangement

is

shown in Fig.

2.4

(ref.

2.2).

The main pump supplies

the two (or more) bearings thorough a "flow divider" made up of the same number

of equal gear pumps, connected by a shaft.

A n

eccentric load

W

tends to decrease

the film thickness hz

of

bearing

1121

and to increase the film thickness of bearing

[111;

so the flow rate of gear

[32],

if disconnected, should tend to decrease, while the

flow rate of gear [31] should tend to increase. The connection, forcing them to rotate

at the same speed, make them produce the same flow rate.

2.3

COMPENSATED SUPPLY SYSTEM

The general layout of a compensated supply system is shown in Fig.

2.5.

The

lubricant delivered by the pump is sent to the recesses of the bearings through the



18


u

Fig.

2.4 Constant flow supply system: flow

divider.

compensating devices (restrictors)

R i .

The pressure

p s

upstream from the restric-

tors is kept constant by means of

a

suitable regulating system (pressure reducing

valve).

Of

course, the pressure in every recess is always less than

p s

as a conse-

quence of the losses in the restrictors. Many types

of

device can be used, with a fixed

or defonnable geometry.

In the following sections we shall see how the compensating devices work.

2.3.1 Fixed restr ictors

Let

us

assume that the compensating devices in Fig. 2.5 are fixed laminar-flow

restrictors (e.g. capillary tubes). When an eccentric load is applied, the clearance

h2

of the pad

[12]

s squeezed, and so its hydraulic resistance increases. Hence, the

total resistance

of

the series constituted by the restrictor

R2

and the relevant clear-

ance

h 2

also increases. Since pressure

p s

is held constant, the rate

of

flow must de-

crease, so the pressure step

p s - p r 2

must decrease as the rate of flow, until

pr2

reaches a value that balances the load. The contrary happens in the case

of

the

lesser loaded pad

[111.

Orifices can also be used as compensating devices. Unlike the laminar restric-

tors, their hydraulic resistance is no longer a constant. This leads to a slightly bet-

ter performance of the bearing. This point will be dealt with further elsewhere

(chapter

6).




19

Fig.

2.5 Constant

pressure supply system:

one

restrictor for each

bearing.

2.3.2

Variable rest r ic tors

Many kinds of restrictor have been proposed that are able to vary their own re-

sistance, depending on the pressure step: more precisely, the hydraulic resistance

should grow (often in a non-linear fashion) as the pressure step

p s - p r

increases; the

contrary happens if p s - p P r ecreases. With reference t o Fig. 2.5,

it

follows that, as

thickness

h ,

is reduced by the load, the recess pressure p r 2 grows faster than in the

case of the fixed restrictor; the contrary occurs in the case of the other recess. Con-

sequently, a greater static stiffness of the bearing is obtained, i.e. any variation in

the thrust

is

accomplished with a variation in the thickness of the film which is

smaller

as

compared

t o

that is found in the case of constant restrictors.

The flow-pressure (load) and film thickness-load characteristics of certain typi-

cal restrictors are compared in Fig. 2.6 (ref.

2.3).

It

is

easy to understand that the

ideal restrictor (from the point of view of the bearing stiffness) should be able to de-

liver a

flow

rate which

is

proportional t o load. Indeed, in this case, the film thick-

ness remains constant. Certain controlled restrictors virtually behave in this way

("infinite stiffness") at least in certain loading conditions.

Let us now examine some typical variable restrictors.

(i)

Elast ic capi l laries .

The simple device in Fig.

2.7

is made up of

a

small diam-

eter pipe, filled with a suitable rubber-like material, in which a capillary hole is



20


- a -

- b -

Q

h

w

w

Fig.

2.6

Flow rate (a) and

film thickness

(b) versus load for different supply systems:

1-

constant

flow system;

2-

capillary;

3- orifice; 4-

constant

flow

valve; 5- diaphragm-controlled restrictor;

6- infinite stiffness (h=const.).

Fig. 2.7 Plastic throttle.

drilled (ref. 2.4). With any increase in recess pressure p r the hole clearly expands

further, and the hydraulic resistance decreases. Elastic orifices have also been

proposed.

(ii)

Spool-controlled restrictors.

An outline of a

cylindrical-spool valve

is given

in Fig. 2.8.a. The lubricant flows into the small clearance surrounding the spool

[s],

which keeps it s balance due

to

the opposite thrusts exerted by the spring and recess

pressure p r on area

A,.

A s

P r

varies, the length

x

of the restrictor varies

too,

and

so

does it s hydraulic resistance. The shape of the valve may also be th at seen in Fig.

2.8.b.

The tapered-spool valve in Fig.

2.9

works in a similar way (note that the aper-

ture angle is very small). However, since its hydraulic resistance varies faster with

x as compared to the preceding device, it s performance is better (ref. 2.5).

(iii) Diaphragm-controlled restrictors (DCR). In the device shown in Fig. 2.10,

the lubricant

is

drawn through the annular clearance between the inlet duct and

the elastic diaphragm [ml. The device may be tuned by means of the adjustable

spring [s] in such a way that the flow rate becomes almost proportional

t o

p r , thus



COMPENSATlNG DEVICES

- a -

21

- b -

t

PS

Fig. 2.8 Cylindrical-spoolvalves.

I

s

I

ps

pr

\

Fig.

2.9

Tapered-spool valve.

t pr

Fig. 2.10 Diaphragm-controlledvalve.

approaching the infinite stiffness behavior for a certain range of loading condi-

tions (ref. 2.6).

(iv)

Constant-flow

valves. Many kinds of devices able

to

produce a constant flow

rate are widely used in oleodynamic plants. The spool valve in Fig. 2.8 may also be



2 2 HYDROSTATIC LUBRICATION

made to deliver a constant flow if it is properly tuned. To improve its performance, a

reference restrictor

RU

an be added as in Fig. 2.11 (ref. 2.7). In order to fix the flow

rate at a certain constant value, it is necessary t o ensure that the pressure step

across

RU

oes not vary when the load varies. Pressure

p , ,

acting on the left side of

the spool is balanced by pressure Pr and the thrust of the spring on the other side. If

p r varies, the spool is displaced, changing the inlet resistance RU1,ntil a new equi-

librium point is reached. Since displacements of the spool are small, compared to

the compliance of the spring, Pu-Pr practically does not vary. By setting up the re-

strictor Ru which in general is an orifice) it is possible to adjust the rate of flow.

Fig. 2.1 1 Constant-flow valve

The performance of this supply system is similar to that of the direct supply sys-

tems examined in sect.

2.2,

except that:

- the maximum value of the recess pressure (i.e. of the load) is limited by the supply

pressure; indeed the device ceases t o work properly when Ps-Pr reaches a character-

istic minimum value;

-

efficiency is much lower, due to the great loss in pressure Ps-pr in the valve itself.

(v) Infinite-stiffness alue.The device shown in outline in Fig.

2.12

(ref.

2.8)

is

able to deliver a flow rate that is proportional to recess pressure P r . The differential

piston is in equilibrium due to pressures P r and p , , that act on different areas. Con-

sequently, the piston searches for it s own position of equilibrium, adjusting the inlet

resistance until ratio pJp , , is equal to the inverse ratio of the relevant piston areas.

Since the hydraulic resistance of restrictor RU s a constant, the rate of flow across it

is proportional to p r . An infinite static stiffness is therefore obtained, until p , , ap-

proaches supply pressure p s .




23

t

ps

Fig. 2.12 Infinite stiffness valve.

(vi)

Electronic compensators.

All the controlled restrictors cited above are

driven by the recess pressure; new types of variable restrictors have been recently

proposed (ref. 2.24) which are controlled by means of an electromagnetic actuator.

A

feedback signal, drawn by a displacement probe which measures the film thickness

of the bearing (or else by a load cell sensing the force which loads the bearing), is

elaborated by a combination

of

proportional, integral and differential operators and

then amplified in order to drive the actuator.

The benefits of electronic control, confirmed by experimental evidence, consist

in infinite static stiffness, very large dynamic stiffness and short settling time.

moreover the system is not affected by fluctuations

of

supply pressure

o r

lubricant

temperature.

(vii) Flow dividers. When t w o opposite pads have to be fed, as in Fig. 1.12.a, a

flow divider may prove t o be more effective than the use of a separate controlled re-

strictor for each recess.

Most flow dividers are really based on the foregoing controlled restrictors. As an

example, consider the device in Fig. 2.13. When the pressure in any of the two re-

cesses increases, the spool is clearly displaced; hence the lengths of the restrictors

vary, thereby increasing bearing stiffness (ref. 2.9).

The behavior of the tapered-spool flow divider (Fig. 2.14)

is

similar, but its per-

formance is better (ref. 2.10).

Figure 2.15 shows the lay-out of a diaphragm flow divider (ref. 2.11). The restric-

tors are made up of the annular clearances between the diaphragm and the outlet

ducts. If the stiffness of the elastic diaphragm

is

properly selected, a very high

degree of bearing stifiess can be obtained.



24


pi

Fig. 2.13 Cylindrical-spoolflow divider.

I

r

I

I

I 1 I

Fig. 2.14 Tapered-spoolflow divider.

Fig.

2.15

Diaphragm-controlled low divider.

Infinite stiffness dividers may also be proposed

(Fig.

2.16),

which are clearly

based on the valve in Fig. 2.12.




25

Fig. 2.16 Infinite-stiffness

flow

divider.

2.3.3 Inherent ly compensated bearings

The inherently compensated bearingsmain drawback of constant pressure

supply systems is that a considerable fraction of the available pressure step is

dissipated in the external compensating restrictors. To overcome

this

disadvantage

certain inherently compensated bearings have been proposed, all of which are based

on the same principle: the shape of the pressure profile in the bearing clearance is

subject to change with load, due

t o

the peculiar recess shape (Fig. 2.171, o r to the

presence

of

an elastic element, such as a layer of elastomer or a flexible metallic

plate (Fig. 2.18).

Let us take

a

closer look, for instance, a t how the bearing in Fig. 2.17.a (ref. 2.12)

works .

Its distinguishing feature is to have a recess depth

h ,

th at is comparable

with clearance h; consequently, the pressure drop in the recess is no longer negligi-

ble and the shape of pressure profile is that represented by a dashed line. If a higher

load is applied, h is reduced and the ratio

hlh,

increases; it is clear that the pres-

sure drop now tends to concentrate in the clearance, and the pressure profile begins

to take on the typical shape of deep-recess pads, characterized by a constant recess

pressure, making it possible to sustain a higher load without increasing recess

pressure.

Similar considerations could be made in the case of the tapered-recess bearing

in Fig. 2.17.b (ref. 2.13):obviously, in this case, too, the recess depth must be compa-

rable with clearance.

These types of bearings create considerable manufacturing problems (precision

machining of a very shallow recess);

t o

overcome these, the bearing depicted in Fig.




\, ,

,

,

/ / , -

2.18.a (ref. 2.14) has a flat layer of elastomer [ l] bonded to a rigid base. Since pres-

sure decreases from the center to the pad boundary, deformation obviously causes

the development of a "recess" (similar t o the one in Fig. 2.17.b) whose shape varies

with

load.

The flexible-plate bearing in

Fig.

2.18.b (ref. 2.15)works

on

the same principle;

since it is all-metallic, i t is free from problems like rubber-oil compatibility.

, I

\

I,\, ,, A

/ / / / I

- b -

- a -

Fig.

2.17

Inherently compensated bearings:

a-

shallow recess bearing;

b-

tapered-recess bearing.

- a -

-b

-

m

Fig.

2.1

8

Inherently compensated bearings:

a-

elastomeric bearing;

b-

flexible-plate bearing.



COMPENSATING DEVICES 27

Certain bearings have been proposed which have a built-in controlled restrictor.

Let us consider the bearing in Fig. 2.19 (ref. 2.16). I t is supplied at a constant pres-

sure p s . Before entering the bearing gap, the lubricant flows through the clearance

b around the tapered plug

C ,

retained by the perforated e lastic diaphragm

D.

Clearly, the bearing simply performs as if

aphragm valve of the type already examined.

it were supplied by means of a di-

Q ?

Fig.

2.19 Diaphragm

bearing.

Inherent compensation has also been proposed for journal bearings.

The bearing

in

Fig. 2.20 (ref. 2.17) is fitted with a bush [13 mounted by means of

elastic rings [21 and baffles [31 in the casing [41. Before entering bearing gap

h ,

the

lubricant flows through variable clearance b.

Fig. 2.20 High

stiffness journal bearing.

The bearing in Fig. 2.21 (ref. 2.18) is fitted with a hydrostatically-controlled re-

strictor (HCR). The bearing i s made up

of

an inner [ l l and a n outer [23 sleeve fixed

to a flange [3], and of a moving ring [4] between the sleeves. Oil, supplied at pres-

sure

p s ,

is restricted when it passes through the gaps between the moving ring and



28


Fig.

2.21

Infinite stiffness

journal

bearing

with

a

hydrostatically-controlled resmctor.

the inner sleeve. The gaps can be changed because the outside of the ring is sup-

ported hydrostatically with oil supplied by another pump at pressure

p : .

By adjust-

ing p t the stiffness of the bearing can be made infinite.

The self-regulating bearing in Fig. 1.4.c (ref. 2.191, on the other hand, is based

on a different principle. It does not need any external compensating device and does

not increase the number of degrees of freedom of the system. The lubricant supplied

t o

the bearing divides into two parts, one flowing through the hydraulic resistance

Rs=Rsl+Rs2and the other through

Ri=Ril+Riz.

Due to the particular geometry of the

bearing, it is a lways

R s= R i ,

whatever the displacement of the moving member, o r

load W. Consequently, the flow-rates in the two half bearings are always equal.

This kind of bearing may be directly supplied by a pump, at a constant flow rate,

as well as by a constant pressure supply system. In the first case,

it

behaves just

like a n opposed-pad bearing fed by two pumps. In

the

second case, it behaves better

than the corresponding opposed-pad bearing fed through fixed restrictors. Its

dynamic behavior

is

also very good.

The self-regulating bearing idea

is

not the first to use a bearing clearance as a

restrictor: a journal bearing with these restrictors was presented in ref. 2.20. As

can be seen from Fig. 2.22 (ref. 1.81, the restrictors a re the clearances of the small

pockets, supplied at constant pressure, essentially diametrically opposite to the

main pockets. Owing to the fact that lubricant through these variable restrictors

may, in part, flow out of them (or vice-versa), the behavior of the bearing is only a

little better than th at of bearings supplied through capillaries and orifices.

In the combined journal and thrust bearing in Fig. 1.14 (ref. 2.21) the journal

bearing acts as a pair of compensating devices for the thrust bearing.



COMPENSATlNG DEVICES

29

Restrictor

lands

N o 4 N o 3 N O 1

N o 2

A ,

I I

I I

I I

I

1

Main

pocket

N o 2 No3 Gmove Supply

~ o 4

restr ictor hole

pocket

o 1

Fig.

2.22

Developed view of a hydrostatic journal bearing having integral variable restrictors.

2.3.4

Reference bear ings

A

small bearing

r

in Fig. 2.23, compensated by a diaphragm valve, has been

proposed (ref. 2.6)

t o

control the behavior of the large main bearing m by means of a

spool relay s. The positioning accuracy of the main bearing

is

the same as the accu-

rately manufactured and positioned "reference" bearing, and it s stiffness is very

high.

- a - - b -

Fig.

2.23

Reference bearing:

a-

beitring r controls the main bearing m by means of a diaphragm

valve v and a relay s; b- the reference bearing

is

an interface restrictor bearing ri. and the diaphragm

valve is controlled by a solenoid.



3 0 HYDROSTATIC LUBRICA TlON

A bearing which seems particularly suitable for use

as

a reference bearing is

the "interface restrictor bearing" (ref. 2.22) shown in Fig. 2.23.b. It is enclosed by

flat washer (made from low friction material) that prevents leakage of the lubricant,

which flows radially

t o

the central hole. The bearing may be supplied by a di-

aphragm valve, and this feature allows it to operate at large gaps with great stiff-

ness and accuracy. Especially if external controlling force is operated by a solenoid,

the bearing becomes an effective reference bearing.

2.4 THE COMMONEST SUPPLY SYSTEMS

2.4.1 Direct supply

Systems provided with a separate motor-pump (usually axial

or

radial piston-

pumps) for each recess (Fig. 2.2) are not very often encountered, because of the rele-

vant plant (and maintenance) costs. It is more common to find plants in which the

pumps are driven by a single motor (as in Fig. 2.3), o r are linked together to form a

flow-divider, as in Fig. 2.4. The cost of these systems are lower, while at the same

time maintaining the same performance, i.e. high load capacity and stiffness, and

greater eficiency than in the case of the other types of supply systems.

A

certain use may be foreseen for the self-regulating bearings (Fig.l .4) as well

as of

systems of self-regulating pads (ref.

2.23)

and self-regulating screw and nut

assemblies (Fig. 1.lO.b) for their high load capacity and stiffness, and not high cost.

2.4.2 Compensated supply

(i)

Fired restrictors. The most common fixed restrictors are laminar-flow re-

strictors, especially capillary tubes. They are easy t o find o r to build, and hence

cheap. Hypodermic needles are sometimes used as capillaries, as well as many

types of small-diameter pipes of adequate length. Furthermore, their design is very

simple, since their hydraulic resistance is proportional

to

their length (provided the

lat ter is some tens times greater than the diameter).

On the contrary, there are bigger problems with the use of orifices, because

their diameter is generally very small: apart from the fact they tend to be easily

obstructed, this means a great sensitivity to manufacturing tolerances. Further-

more, in the case of the orifices, the system is more sensitive t o any change in the

temperature (i.e. in the viscosity) of the lubricant.

(ii) Var iable restrictors. Constant flow valves are sometimes used, even though

they are somewhat complicated and costly devices. The reason is that they are easily

available and widely used in many oilhydraulic plants.



COMPENSATING

DE

lCES 31

For almost all the other controlled restrictors experimental applications are

known (chapter

14).

Diaphragm flow dividers, in particular, have been widely stud-

ied in connection with opposed-pad bearings, journal bearings, hydrostatic slide-

ways and screw-nut assemblies.

A

wider use of such devices

is

recommended,

since they are simple to build and their performance is excellent.

(iii) Inherently compensated bearings. For almost all inherently compensated

bearings only experimental applications are known.

2.5 HYDRAULIC CIRCUIT

Figure 2.24.a contains the layout of the hydraulic circuit of a hydrostatic system

supplied at constant pressure. The system in the figure is made up of one

( o r

more)

opposed-pad bearing.

A

pump P driven by a motor

M

pushes the lubricant from the reservoir Sm in

the intake line, in which pressure is kept constant by a relief-pressure valve V,

which discharges the surplus lubricant. The lubricant, passed through a filter F,

arrives at the pads through the restrictors R, which should be as near as possible to

the pads

for

good dynamic behavior of the bearing. The lubricant then flows from

the bearing to the collector S and from this to the reservoir S m .

Besides filter

F,

a strainer may be put a t point [l].After filter

F

a pressure

gauge measures supply pressure

p s . A

pressure-sensing emergency switch is

generally put a t the same point

[ Z ] ,

which may, for instance, switch off the motor

driving the moving members of the bearings if a pressure-drop occurs. An accumu-

lator, preceded by a a check valve Vn, an supply the system during

its

inertia

movements. Pressure gauges may be put after the restrictors to indicate the pres-

sures in the recesses, but they should be switched

off

during running for good

dynamic behavior of the bearing. If the gravity discharge is not sufficient, ever in

large pipes, an exhaust pump may be inserted a t point [4].

As will be explained more extensively later, lubricant is heated in the circuit,

and especially in restrictors and in the bearing pads, as a consequence of viscous

friction.

So,

if natural cooling in the reservoir is not sufficient, a cooler may be in-

serted at point

[5],

or a t point [l] f a close temperature control is required. The

cooler may also be put in parallel t o the reservoir. Thermometers T are generally

inserted in the circuit, in particular after the cooler [5]. A temperature transducer,

located a t point [3], may switch off the driving motor if temperature exceeds the ad-

missible value.

If pads are provided externally with low-friction seals, a low pressure at the

film exit may pump the return lubricant directly t o reservoir

S m ,

thereby simplify-



32

HYDROSTATIC LUB RICATlON

- a -

&

I

I

1

W

a

-

b -

a5

&

Fig. 2.24 Schematic diagram of a typical hydrostatic system supplied at constant pressure.

a-

Hydraulic circuit.

b-

Load versus eccentricity

for the

opposed-pad bearing, supplied:

(A)

through

capillary tubes,

(B)

through constant-flow valves. For comparison diagram C is also presented of an

opposed-pad bearing supplied at constant flow.

ing the return circuit. Further details in thus connected are to be found in chapter

14.

Figure 2.24.b shows the non-dimensional load capacity W'=W/Ag, of the bear-

ing in Fig. 2.24.a, versus eccentricity

E=(ho-hi)lho,

or certain values of the ratio

P=p,dp,, ho

being the film thickness and p r o being the recess pressure for W=O; A,

is the "effective" area of a pad. Diagrams have been drawn for the supply through

capillary restrictors (A) and flow control valves (B). In the latter case,

W'

increases

much more quickly than in the former case. For comparison in Fig. 2.24.b an in-

dicative diagram is also given for a bearing directly supplied a t constant flow

( C ) :

this bearing clearly behaves better, but in this case there is a pump

for

each recess,

instead

of

a restrictor.




33

REFERENCES

2.1

2 6

2.3

2.4

2.6

2.6

2.7

2.8

2 9

2.10

2.11

2.12

2.13

2.14

2.16

2.16

2.17

2.18

Siebers

G.; Hydrostatische Lagerungen und Fiihrungen; Verlag Technische

Rundschau, Bern, 1971; 75 pp.

Kundel K., Arsenius T.; Cuscinetti Idrostatici; Rivista dei Cuscinetti-SKF',47

(19721, 1-8.

Opitz H.; Pressure Pad Bearings; Proc. Instn. Mech. Engrs., 182,3A (1967-681,

100-115.

Wiener H.;

The Plastic Throttle - a Novel Component for Hydrostatic Slid ing

Bearings; Ball and Roller Bearing Engineering-FAG, 1974, N. 2; pp. 41-44.

Morsi S. A.; Tapered Spool Controller for Pressurized Oil Film Bearings;

Proc. Instn. Mech. Engrs., 184,l (1969-701,387-396.

Mohsin M. E.; The Use of Controlled Restrictors for Compensating Hydro-

static Bearings;

Advances in Mach. Tool Des. and Res., Proc. 3rd Int. MTDR

conf,, Birmingham, 1962; pp. 429-442.

Merritt H. E.; Hydraulic Control Systems;

J o h n

Wiley &

Sons,

N. Y., 1967; 350

PP.

Royle J. K., Howarth R. B., Casely A. L.; Applications of Automatic Control to

Pressurized Oil Film Bearings; Proc. Instn. Mech. Engrs., 176,22 (19621, 532-

541.

Mayer J. E., Shaw

H.

C.; Characteristics

of

a n Externally P ressurized B ear-

ing Having Variable External Flow Restrictors; ASME Trans.,

J . of

Basic

Engineering,

86

(19631,291-296.

Bassani R.;

Divisore di Flusso a Spola C onica: sua Applicazione a Su pp orti

Idrosta t ic i ; Fluid-Apparecchiature Idrauliche e Pneumatiche, l S,17 1-172

De

Gast

J. G. C.; A New Type of Controlled Restrictor (M.D.R.) for Double

Film Hydrostatic Bearings and its

Application to High-Precision M achine

Tools;

Advance in Mach. Tool Des. and Res., Proc. of the 7th

Int.

MTDR Conf.,

Birmingham, 1966; pp. 273-298.

O'Donoghue J. P., Hooke C. J.; Design of Inherently Stable Hydrostatic Bear-

ings; Proc. Instn. Mech. Engrs., Tribology Convention, 1969.

Hirs G. G.; Partly Grooved Externally Pressurized Be arin gs; Proc. Instn.

Mech. Engrs., Lubrication and Wear Convention, 1966; paper 21.

Dowson D., Taylor C. M.; Elastohydrostatic of Circular Plate Thrust Bear-

ings; ASME Trans., J. of Lubrication Technology,89 (19671,237-262.

Davies P. B.; Investigation of a n All-Metallic Flexible Hydrostatic Th ru st

Bearing; ASLE Trans., 17 (19741, 117-126.

Tully N.; Static and Dynamic Performance of a n Infin ite Stiffness H ydrostatic

Thrust Bearing; ASME Trans.,

J. of

Lubrication Technology, 99 (19771, 106-

112.

Brzeski L., Kazimierski

2.;

High Stiffness Bearing;

ASME Trans.,

J.

of Lu-

brication Technology,

101

(19791,520-525.

Mizumoto H., Kubo M., Makimoto

Y.,

Yoshimochi

S.,

Okamura

S.,

Matsubara T.;

A

Hydrostatically-Controlled Restrictor for Infinite Stiffness

Hydrostatic Journal Bearing; Bull. Japan Soc. of Precision Eng., 21 (19871, 49-

54.

(19781,31-37.



Chapter 3

LUBRICANTS

3.1

INTRODUCTION

Lubricants are put between two surfaces t o prevent direct contact. They may be

subdivided into solid lubricants and fluid lubricants.

Fluid lubricants are used in hydrostatic lubrication; these may be subdivided

into liquid lubricants and gaseous lubricants. Of the two, liquid lubricants are more

frequently employed, including water, also utilized in the first hydrostatic experi-

ments (ref.

1.1),

and liquid metals, especially sodium. But the liquid lubricants most

often employed are mineral lubricants. Nowadays synthetic lubricants are also

used.

Mineral lubricants are obtained from the distillation and refining processes of

crude petroleum, which is separated into fractions of progressively decreasing

volatility, with the elimination of the unwanted ones. Mineral oils are made up of

hydrocarbons, i.e., compounds

of

hydrogen and carbon.

Synthetic lubricants are produced by the substantial chemical modification of

raw materials, which may also be obtained from crude petroleum.

Mineral lubricants are mainly used in hydrostatic lubrication.




3.2 MINERAL LUB RICANTS

3.2 .1 Types

Hydrocarbons, which mineral lubricants are mainly made up of, have three ba-

sic structures: paraffinic, naphthenic, and aromatic. Figure 3.1 shows their typical

configurations.

Paraffinic hydrocarbons generally predominate in mineral lubricants, followed

by naphthenic hydrocarbons. Aromatic hydrocarbons are usually few in number.

If the percentage of carbon present in paraffinic chains is considerably higher

than the percentage in naphthenic rings, the lubricant is called a paraffinic lubri-

cant; otherwise, it is called a naphthenic lubricant. Even a small amount of carbon

in aromatic rings helps boundary lubrication, owing

to

the presence

of

unsaturated

bonds.

- a -

- b -

-C -

H

I

I

Fig.

3.1

Typical hydrocarbon configurations:a- paraffinic chain;

b-

the so-called naphthenic ring;

c- aromatic ring.

3.2.2 Viscosity

Viscosity represents the internal friction of a fluid. Consider two layers in a

fluid, a distance

dy

apart (Fig. 3.2) . If we apply a tangential stress z along one of

these layers and observe

a

shear rate

d u d y ,

with u as the velocity

d x l d t ,

then we

may define the differential viscosity

as

p~ (ref. 3.1)

Note that Eqn 3.1 does not imply that the ratio

6 zZ x /& d u d y )

s necessarily con-

stant throughout the fluid

or

during the time

of flow.



LUBRICANTS

37

Fig.

3.2

Lam inar shea r between parallel planes in

a

fluid.

If

Sz,,/

& du l d y ) is constant and the shear rate is zero when the shear stress is

zero, then a flow is said to be Newtonian. The conditions for a Newtonian flow are:

(3.2)

d u = O

when z = O

dY

A fluid which conforms to Eqn 3.2 is called Newtonian. Indeed, we owe Eqn 3.2

to Newton.

Figure

3.3

illustrates a number of ideal shear rate curves against the shear

stress of a Newtonian fluid (the straight line through the origin),a pseudoplastic

fluid, a dilatant fluid, and a pseudoplastic material (for example a grease) with

a

yield

stress

(ref.

3.1, 3.2).

I

Shear

stress,Z

Fig.

3.3

Shear rate - shear stress characteristics of materials: A

-

Newtonian fluid;

B

- pseudoplastic

fluid;

C

-

dilatant fluid;

D

- pseudop lastic material.




Parameter p, defined by Eqn 3.2, is called dynamic viscosity. In system SI its

unit is Ndm2 or Pas, in system c.g.s. it is dynes/cm2o r poise.

Mineral lubricants and synthetic lubricants of low molecular weight are Newto-

nian in many practical working conditions.

In many fluid flow problems the ratio

(3.3)

is used, where p is the density of the fluid, v is the kinematic viscosity,

its SI

unit is

m2/s and

its

c.g.s. unit is cm2/s or Stoke

(St).

In selecting a n oil for a given application, viscosity

is

a primary consideration,

especially from the point of view of its change with temperature. Various systems

are used to classify and identify oils according

t o

viscosity ranges, including the

"Viscosity system for Industrial Fluid Lubricants", devised by IS0 (Std 3448) nd

now coming into wide use. Viscosity systems establish a series of definite viscosity

levels as a common basis for specifying the viscosity of industrial fluid lubricants.

Reference viscosities are measured in mmVs

o r

cSt (centistokes) at the reference

temperature of 40°C.The viscosity ranges and the corresponding marks t o classify

oils are shown in Table 3.1, for

v=5.06+242

cSt. For comparison, the partial S A E

(Society of Automotive Engineers) classification is also shown. The reference tem-

perature of the SAE classification

is

lOO"C, nd sUmx W is intended for use in cases

where low ambient temperature is encountered.

TABLE 3.1

Viscosity System

for

Industrial Fluid Lubricants.

I

Viscosity System Grade Mid-Point viscosity Kinematic viscosity

Classification cSt (m 2 /s )a t 40°C limits

and Identification cSt (mm%)at 40°C

Min

Max

I S 0

VG 5 4.6 4.14 5.06

IS0 VG 7 6.8 6.12 7.48

ISOVG 10 10 9 .0 11.0

I S 0

VG 15

15

13.5 16.5

I S 0 VG 22 22 19.8 24.2

I S 0 VG 32 32 28.8 35.2

I S 0

VG 46 46 41 .4 50.6

I S 0

VG 68 68 61 .2

74.8

I S 0

VG

100

100 90.0

110

I S 0 VG 150 150 135 165

I S 0

VG

220 220 198 242

S A E

Classification

5w

low

2 0 w

30

50



LUBRICANTS 39

(i) Viscosity-temperature. The viscosity of liquid lubricants decreases with in-

creasing temperature. Variations in temperature may be due t o external causes

and to energy dissipated because of viscous friction and changed to heat.

The following is an equation of the viscosity-temperature relationship, which is

simple but fairly accurate:

(3.4)

where po is the viscosity at reference temperature

To

and

p

is a constant determined

from measured values of the viscosity; it s dimension is that of the inverse of tem-

perature.

Another widely-used equation is

log log(v+ a)= a -

b

logT

(3.5)

where a and b are constants, and a varies with the viscosity level. For viscosities

over 1.5 cSt, a is

0.8;

above 1.5 cSt, a is

0.6.

Using this type of log-log relationship,

charts have been worked out in which viscosity is represented by straight lines. In

Fig.

3.4

Eqn

3.5

is plotted for certain typical trade lubricants, which have been clas-

sified in conformity with IS0 (in actual fact IS0 VG

46

and

IS0 VG 68

fall a little

outside the kinematic viscosity limits at

40°C).

The diagrams refer

t o

a Viscos i ty

Index =lo0

o r

a little higher (ref. 3.3).Note that the log-log relationship compresses

the scale for high values of viscosity,

so a

graphic error of 1% may produce an error

of as much as 10 cSt.

Ever since the Thirties, the viscosity index ( V n has been of practical use for the

approximate estimation of the behavior of kinematic viscosity with temperature. It

makes i t possible to give a numerical value t o such behavior.

The viscosity index is based on two groups

of oils.

In one group, that is naph-

thenic in nature, VZ=O because of its sensitivity t o temperature; in the other, that is

paraffinic in nature, VZ=100 because of its lower sensitivity.

Two oils are selected, one for each group, with the same viscosity at

100°C

as the

oil

t o be tested. The viscosities of the three oils at

40°C

are then evaluated. Taking L

as the value of the oil with VI=O,

H

as that of the oil with VI=lOO, and

U

as that of

the oil being tested, the viscosity index

is

given by the equation

L -

u

vz= ---loo

L - H

(3.6)

A t

present the

VZ

of mineral oils is often larger than

100,

and

as

Eqn

3.6

gives

largely inexact results for VZ>lOO, an empirical equation can be used:



40


Fig.

3.4

Viscosity-temperature chart for certain typical lubricants.

VI = lW -

+

100

0.00715

(3.7)

where

logH

-

l0gU

N = logy

and Y is the kinematic viscosity in cSt at 100°C for the oil being considered.

The influence of different viscosity indexes on oils with the same reference vis-

cosity is taken from Fig. 3.5 (ref. 3.4).

(ii)

Viscos i ty -pressure .

The viscosity of lubricants increases with pressure. A

widely-used model of the viscosity-pressure relationship is



LUBRICANTS

41

-40 -20 0 20

40 60

80

100

120

140 160 180

200

TEMPERATURE .Dc

Fig. 3.5 Viscosity-temperaturebehavior for oils with different viscosity index

where

po

is the viscosity at atmospheric pressure and

y

is a constant determined

from measured values of viscosity; its dimension is

that

of the inverse of a pressure.

Indeed, pronounced deviations from the above relation are often encountered.

Naphthenic oils are more sensitive to pressure than paraffinic ones.

A t

hydrostatic pressures

viscosity may be considered

t o

be constant with

pressure.

3.2.3 Oiliness

Oiliness may be defined as the capacity of a fluid to adhere t o the surfaces of

materials. In usual conditions, especially if pressures are not high, the forces of

molecular adhesion are sufficient. If pressures increase, adsorption of the fluid on

the surfaces is then necessary. Adsorption occurs especially if polar molecules are

present in the fluid, i.e. molecules in which a permanent separation exists between

the positive and negative electric charges.

Mineral lubricants are not very oily, which is particularly unfavorable in

boundary lubrication.




3.2.4

Densi ty

As is well known, the density of a liquid is the mass of a unit volume, generally

calculated at 15°C. In paraffinic mineral oils p=0.85-0.89 kgfdm3; in naphthenie

mineral oils p=0.90-0.93 kg/dm3. Density varies with temperature and with

pressure.

(i) Thermal expansion. For a liquid, thermal expansion can be defined as the

property of being changed in density with temperature.

It

can be stated approxi-

mately by the equation

p(T) = po 11- a ( T -To)]

(3.8)

where

a

increases as p decreases; approximately: a=4.1.10-4

+

8.2.10-40C-1 or

p=0.22+0.01 Ns/m2.

In hydrostatic lubrication thermal expansion is often negligible.

(ii) Compressibil i ty . The compressibility of a liquid can be defined a s the prop-

erty of being changed in density with pressure:

(3.9)

Compressibility can also be expressed as a change in volume with pressure; indeed,

if V is the volume (of a mass

M)

f liquid, then from Eqn 3.9

1 d V

c = - - -

v dP

(3.10)

Compressibility changes with pressure and temperature; it also changes with

molecular structure, but cannot be changed by means of additives, since it is a phys-

ical property of the base liquid.

Very often, instead of compressibility, its reciprocal is used: the bulk modulus

KL. igure

3.6

(ref. 3.4) shows

a

method for predicting the bulk modulus of mineral

oils:

1) with density pT calculated a t ambient pressure (105 Pa) and a t the desired

temperature

T,

Fig. 5.6.a defines the bulk modulus at pressure 1380.105

Pa;

2) with this bulk modulus enter into Fig. 3.6.b: a vertical line

at

the intersec-

tion with the 1380.105 Pa line gives the modulus a t any other pressure

and

at

the selected temperature.



LUBRlCANTS 43

- a - - b -

690

0 80 160 240 320 400 480 560

(-18 27 71 116 160 204 249 293)

TEMPERATUREFF, (OC)

Fig. 3.6 Bulk modulus of

mineral

oils

3450

3105

E

5

2760

cn-

2415

2

2070

n

p 1725

5 1380

1035

3

/

/

/'

p =

5

M N ~

For instance, let us consider a mineral oil with density p,=0.90 kg/dm3

at

ambi-

ent pressure and at temperature

T=40°C:

ts bulk modulus a t pressure -5 MPa and

at the same temperature is

K1=1680

MN/m2.

Alternatively, bulk modulus KLcan be calculated by the semi-empirical equation

(ref.

3.6)

K1= (1,44 + 0,15

logv)

[10°.00235(20-n].1095.6p ,

Nm-2

(3.11)

where

v

is the kinematic viscosity in cSt at a temperature of

20°C

and at ambient

pressure; T

is

the temperature in

"C;p is

the pressure in Pa. So, if the dynamic

viscosity of the oil in the previous example is

p 0 . 0 6

Ns/m2, from Eqn

3.11

the bulk

modulus is

K2=1570MN/m2.

Values of

K1

obtained from Fig

3.6

or by means of Eqn

3.11

are

fit

for high pres-

sures; nevertheless they can also be used, approximately,

at

mean and low pres-

sures (the equation is preferable), as seen in the examples.

(iii)Gas

solubility.

Solubility of gases in liquids is a physical phenomenon which

can be evaluated by the ratio

(3.12)

where Vgis the gas volume and Vi is the liquid volume, at the given partial pres-

sure of the gas and at the given temperature.




Solubility varies with pressure and temperature. In Fig. 3.7.a solubility of air

versus pressure is shown in a mineral oil (Mil-h-5606A) and in other liquids (ref.

3.5). Also at higher temperatures air solubility varies almost linearly. Figure 3.7.b

shows solubility versus temperature in the case of certain gases in

a

mineral oil

with p=850 kg/m3

The air dissolved m a y affect lubricant properties, such as viscosity which grows

worse. The air dissolved in an oil comes out of solution when temperature and pres-

sure decrease and may produce air bubbles and foam.

- a -

- b -

MINERAL

OIL

0 2 4 6 8 1 0

0 25 50 75 100 125

PRESSURE, MP a TEMPERATURE PC

Fig. 3.7

Solubility

of

gase s versus: a- pressure; b- temperature.

(iv)Air entrainment. Common causes

of

entrained air in a liquid are, for ex-

ample, leaks in the pump suction or when the return line discharges liquid above

its surface level in the reservoir.

In any case, air is inevitably taken into a mineral oil as it passes into a lubricat-

ing system

so

that the oil in the reservoir may contain as much as 15%of dispersed

air (ref. 3.7). In a large and suitable reservoir this air should be given up and re-

duced, after a fairly long time, t o about 1.5%, and after a very long time t o about

0.5%. But the air bubble content is rarely reduced to the desired levels.

Air bubbles, when compressed, go into solution, but not immediately. In Fig.

3.8, the percentage of air bubbles dissolved in a hydraulic oil is shown as a function

of time, for certain pressures (ref. 3.5). We see,

for

example, that for p=3 MPa, after

1

second, less than 10%of the air bubbles go into solution. Thus, as a result

of

the oil

velocity in the supply lines of hydrostatic systems (0.5t50

m/s

and even more), these

percentages are generally

low.



LUBRICANTS

45

40

0

1

2 3

4

5

T I M E

, s

Fig 3.8 Rate

of solution

of

air bubbles

in

a mineral oil.

Air viscosity is low, i.e. p=1.78.10-6 Ns/m2 a t l0C and 0,981.105 Pa; therefore,

entrained air affects the viscosity

of

oils. Fortunately, the effect

is

relatively slight,

and can be expressed by the empirical relation

clp = (1+ 0.015 B )

where

B is

the percentage of bubble content,

p o

the viscosity of

oil

and pb the effective

viscosity of bubbly oil.

The air bulk modulus is also low, so the entrained air affects the actual bulk

modulus

of

oils. The equation of state of a perfect gas (to which air may be assimi-

lated)

for

an adiabatic process

( to

which the compression of air bubbles in mineral

oils may be compared) is

p VCP'~V

= const. (3.13)

where cp and c v are the specific heats a t constant pressure and constant volume,

respectively.

If

Eqn 5.13 s introduced into Eqn 5.10, the bulk modulus Ka

of

air

is

obtained

(3.14)

A t temperature T=40"C and at pressure p=0.981.105 Pa, we have cp=1.0048~103

J/kg"C, and cv=0.717.103 J/kg"C; thus we have cplcv=1.401,and Ka=1.37.105 Pa. In



46 HYDROSTATIC L UBRICA

TlON

the context of hydrostatic lubrication, the variations of cp with temperature and

pressure are slight, while those of cv are negligible. Figure 3.9 shows cp/cvversus p

for certain values of

T

(ref. 3.8), and the bulk moduli of the air at

the

pressures in

Table

3.2

become those given in the same table.

0 1,96 3,92

5,aa

7,a5 9,ai

p ,

MNrn-'

Fig. 3.9 Ratio of specific heats of air versus pressure,

for

certain temperature values.

TABLE 3.2

(v) Apparent

bulk modulus. Air

entrainment affects the properties

of

mineral

oils, especially bulk modulus, which greatly decreases. Indeed bulk moduli of

mineral oils are clearly much higher (even more than lo* times) than those of air.

It

is

possible

t o

evaluate the apparent bulk modulus of a volume

Vi

of oil a s follows:

(3.15)

in which

Va

is the volume of bubbly air uniformly entrained in oil, and whose bulk

modulus is

K,

a t working pressure.

If the lubricant contains

5% of

bubbly air at ambient pressure (it must also be

taken into account that bubbles

of

other gases may also exist, which may be dis-

solved in oil in a greater proportion than that

of

air, as

is

shown in Fig.

3.71, or

Eqn

3.13

at

pressures given in Table

3.2

that percentage

is

reduced to the values given in



LUBRlCANTS 47

the same table.

So,

for Eqn 3.15, the apparent bulk moduli given in Table 3.2 may be

assigned to the oil a t those pressures.

The elastic deformation of the supply line may also influence the bulk modulus

of mineral oils. For a circular pipe, the internal pressure ps causes a change in

volume, which should be added to the change in volume due t o the compressibility of

the fluid when evaluating effective compressibility (Eqn 3.10) and hence the appar-

ent bulk modulus. The equivalent bulk modulus of a metallic pipe is, approxi-

mately,

(rs

is the internal radius of the pipe,

ra

is the external one,

E

is the modulus of elas-

ticity and v is the Poisson ratio). Equation 3.15 may then be completed as follows:

(3.16)

For instance, let

us

consider a copper pipe, with ra=6 nun, rs=5 mm, and E=118

GPa, v=0.25; the equivalent bulk modulus of the pipe may be calculated as

Ks=10.2.109 Nlm2. For a steel pipe of the same dimensions, with E=206 GPa, v=0.3,

we have KS=17.6.1O9N/m2.

Generally, values of Ks for metallic pipes are much higher than the apparent

bulk modulus Kla of mineral oils (see Table

3.2) so

their influence may be disre-

garded. The same is not true for flexible pipes (also for high-pressure pipes made of

hard rubber

o r

FTFE with an interwoven sheet of stainless steel) as transpires from

existing experimental results. Figure 3.10.a (ref. 3.9) shows the considerable in-

crease in the inner volume of certain flexible hoses.

Obtaining realistic design values of the apparent bulk modulus of oil in hy-

draulic hoses is quite difficult. Values of Kla in the 70t350 MN/m2 range can be

found in the l iterature. Some results are shown in Fig. 3.10.b (ref. 3.5) or a woven

hose with rs=6.4 mm: the experimental data are clearly scattered. Recent design

practice in relation t o equipment dynamic noise reduction has tended to encourage

the use of hydraulic hoses in fluid power systems. This does not always seem con-

venient in hydrostatic systems, as a way of preventing possible dynamic instability.

Elastic deformations of instruments (manometers), pressure reservoirs

(accumulators) and other elements in the supply line may also influence

the

effec-

tive value of the apparent bulk modulus.



48

48

nJ 4 02:

I

24

-

Y

16-

8 -


"

/

e

-

0 0

o o o

I I I I

- a -

10 15

20

0 4 8 12 16 20

PRESSURE,

MNrn-'

PRESSURE,

MPa

Fig.

3.10

Flexible pipes:

a-

Inner volume variation

(for

unit length) versus pressure; (i) internal

radius

rs=5

mm, rated pressure

pN=26

MPa; (ii)

rs=6.5

mm,

pN=26

MPa; (iii)

rs=5

mm, p ~ = l l

MPa. b- Apparent bulk modulus of lubricant versus pressure: rs=6.4 mm, SAE R2 Hose.

(vi)Foaming. The foaming of a liquid is due to the

air

bubbles tha t collect above

its surface. Common causes of foam are the same, but even greater, as in the case

of entrained

air.

Foam in a lubricating system can cause a decrease in pump efficiency, vibra-

tions, and above all inadequate lubrication.

(vii) Cavitation. In fluid systems "gaseous cavitation" refers to the formation in

the liquid of cavities that may contain air or other gases. "Vaporous cavitation"

refers

t o

the fact that, if pressure is reduced far enough, the liquid will vaporize and

will form vapor cavities (mineral oil vapors may contain volatile fractions

of

lubri-

cants). The vapor pressure of a liquid depends on

its

temperature and decreases

with

it.

A t atmospheric pressure water boils a t 100°C,

so

its vapor pressure is 1.0128

bar; at 21.1"C its vapor pressure is reduced to 0.025 bar. The vapor pressure of

mineral oils is much lower than that of water, typically 6.10-4 bar at 4O O C ; hence

cavitation is less likely to occur in the case

of

these liquids. In Fig. 3.11 the vapor

pressure of certain liquids

is

given as a function of temperature (ref.

3.5).

Cavities are well known to be associated with nucleation centers such as micro-

scopic gas particles

(or

microscopic solid particles which gases join to), and their

development is caused by the rapid growth of these nuclei. Hydraulic liquids used in



LUBRICANTS

49

i o ' r / 1

0

I

E

E

m

B

a

>

40

80 120 160 200 280 360

(494 26,7 49 71 93 138 1821

TEMPERATURE, OF,

( O C 1

Fig. 3.11 Vapor pressure versus temperature..

conventional systems contain sufficient nuclei t o ensure that cavitation will occur

when pressure

is

reduced to vapor pressure.

Cavitation damages hydraulic machinery and systems. Wear rate in particular

can be greatly accelerated if cavitation erosion develops. Cavitation may also in-

crease viscosity and reduce the bulk modulus of

oils.

In hydrostatic systems cavitation may also occur in the

sills

and in the recesses

where depression occurs, and in the recesses where turbulence occurs, which also

favours the formation of gases.

3.2.5 Thermal propert ies

(i)

Specific heat.

Specific heat in mineral oils varies linearly with temperature;

it is:

- for

naphthenic oils, c=1850-2120 J/kg"C from 30 to 100OC;

- for paraffinic oils, c=1880-2170 J/kg°C from 30 to 100°C.

(ii)Thermal conductiuity.

Thermal conductivity in mineral oils is:

0.133-0.123

Wm/m"C from 30

o

100OC; it also varies linearly with temperature.




3.2.6

Other propert ies

Pour-po in t

is the temperature at which an oil ceases to flow freely. This is

caused by the formation of crystals, mainly of a paraffinic type. The pour-point of

paraffinic oils is a t about

-1O"C,

that

of

naphthenic oils a t about

-40°C.

Flash-point is the lowest temperature at which the vapors given off by an oil

ignite momentarily on the application of a small flame. The flash-point of naph-

thenic oils is a t about 170"C, that of paraffinic oils at about 190°C.

Acidi ty .

Low acidity is advantageous €or reducing corrosion.

Oxida t ion . High stability to oxidation is advantageous, because one cause of

deterioration in lubricant oils is the formation of oxidation products. This also leads

t o

a reduction

of

the life of the lubricant and

to

corrosive effects.

Thermal decomposition.

In the presence of oxygen, high temperatures may pro-

duce the thermal decomposition of mineral lubricants, which shortens their life.

Figure 3.12 gives the approximate time-temperature characteristics of refined min-

eral lubricants, including oxidation (ref. 3.8).

10

10

Li fe , h

Fig 3.1 2 Approximate life-temperature characteristicsof a mineral oil: A - oil without anti-oxidant;

B -

oil with anti-oxidant.

3.2.7

Addi t ives

Nowadays lubricants often have chemical compounds added t o them to improve

them.

Viscosity index improvers are generally organic polymers which are soluble in

oils, with a high molecular weight, such as polymethylmetacrilates. They cause a

decrease or a small increase in viscosity a t

low

temperatures, and a substantial

increase at high temperatures. See also Fig. 3.5 and Fig. 3.13 (ref. 3.10).

Oiliness improvers are, for example, fatty acids. They have polar molecules,

with a -CH3 group at one end and a

-C02H

group at the other. This latter group

would be adsorbed on metal surfaces. Actually, owing to surface motion, and in the



LUBRICANTS

5 1

10

1 -

.-

In

0.003

0 50

100

120

I

.-

In

0.003

0 50

100

120

Temperature,%

Fig. 3.13. Viscosity-temperature characteristics o f A -

a

plain mineral oil;

B -

a mineral oil with a

viscosity index improver; C - a silicone fluid.

presence of a metal acting as a catalyst, oiliness improvers seem to change into

more complex compounds.

Foam additives

decompose and, therefore, reduce foam. Common foam decom-

posers include, for instance, silicones and polyacrylates, but the best way to reduce

foam is

a

suitable mechanical design.

Pour-point depressants are generally complex polymers which coat the paraf-

finic crystals, thereby preventing them from increasing.

Oxidation inhibitors, such as certain phenols, amines and olefines, prevent o r

reduce the formation of oxidation products. They also prolong the life of the lubri-

cant and act as corrosion inhibitors.

Corrosion inhibitors.

Rust, a hydrate iron oxide,

is

a widespread form of corro-

sion. Corrosion inhibitors, such as sulphonates, generally form a protective coating

on metal surfaces.

Many others additives, such as de te rgen t, d i spe rsan t and ex treme-pre ssure

additives, are used in lubrication; but they are of little importance in hydrostatic

lubrication.

More than one additive may be used a t the same time. It must be borne in mind,

however, that indiscriminate mixing can produce undesired interactions.



5 2 HYDROSTATIC LUBRlCATlON

3.3 SYNTHETIC LUBRICANTS

The performance of synthetic lubricants is better than that of mineral lubri-

cants, but the former are much more expensive. They are often used in extreme

conditions, for instance in cases of high pressure or temperature.

Synthetic lubricants include the following:

Synthetic hydrocarbons; the polyolefins and hydrobenzene in particular,

Organic esters; those of dibasic acids in particular, which also have very good

Phosphatic esters, whose oiliness is very good and whose thermal stability is

Polyglicols,

with very good oiliness, a high VZ and also fluidity a t low temper-

atures.

Silicones (with a polymer-like structure, in which the carbon is replaced by

silicon). They have a high VZ (see a lso Fig. 3.131, a high flash-point, a low pour-

point, high thermal stability and oxidation stability and a good anti-foam perfor-

mance. On the other hand, their oiliness is poor.

which have very good fluidity a t low temperatures, and a very good VZ.

fluidity a t low temperatures, and good thermal stability.

fair.

Various synthetic lubricants may be used

as

additives.

On lubricants see also ref. 3.11.

REFERENCES

3.1

39

3.3

3.4

3.6

3.6

3.7

Dorinson A., Ludema

K.

C.; Mechanics and Chem istry in Lubrication ; Else-

vier, Amsterdam, 1985; 634 pp.

O'Connor

J.,

Boyd

J.;

Standard Handbook

of

Lubrication Engineering; Mc

Graw-Hill, New York, 1968.

Wills J. G.; Lubrication Fundamentals; M. Dekker Inc., New York, 1980; 465

PP.

Booser E.

R.;

Handbook

of

Lubrication, 2nd

Vol. ; CRS

Press, Boca Raton

(Florida), 1984; 689 pp.

McCloy D., Martin H. R.; Control of Fluid Power; Ellis Horwood Ltd., Chich-

ester, 1980;505 pp.

Liste des Caractkristiques Exigkes pour les Fluides Olkohydrauliques; CETOP

(ComitB Europben des Transmissions Olbohydrauliques et Pneumatiques),

London, 1971.

Fowle T;

Aeration in Lubricating Oils;

Tribology International,

14

(19811, 151-

157.



LUBRICANTS

53

3.8 Raznyevich K.;

ables et Diagrammes Thermodynamiques;

Eyrolles, Didion,

1970.

3.9

Speich

H.,

Bucciarelli

A.;

L’Oteodinamica; Techniche nuove, 1971; 727 pp.

3.10 Neale

J.

M.; ribology Handbook;

Butterworths, London, 1973.

3.11 Lansdown

A. R.;

Lubrication. A Practical Guide to Lubricant Selection;

Pergamon Press, Oxford, 1982; 252 pp.



Chapter 4

BASIC

EQUATIONS

4.1 INTRODUCTION

This chapter contains the equations which constitute the basis for calculating

the performance of hydrostatic bearings, which will be the subject of the following

chapters. Specialized publications may be consulted for a more detailed analysis,

such a s ref.

4.1

or viscous fluid mechanics and ref.

4.2

nd ref. 4.3

or

lubrication

theory. Those primarily interested in the results applied to the most common types

of bearings, may prefer to omit this chapter.

d x

Fig.

4.1

Equilibrium

of

a fluid element along direction x.

4.2 NA VIER-STOK ES AND CONTINUITY EQUA TIONS

4.2.1 Rectan gu lar co ord inates

Let us consider a fluid element (Fig. 4.1).Along each coordinate axis we can

write an equilibrium equation, expressing the balance of the relevant components of

the body force, the external action on the element surfaces, and the inertia force. A

set of equations is obtained, which may be written in vector form as follows:



BASlC

EQUATlONS

55

(4.1)

v

p E = p f + V o

In the preceding statement (generally called the momentum equation),

f

is the body

force per unit mass, V o is the divergence of the stress tensor, v

is

the velocity vector.

D/Dt means the total (also called "material") time derivative:

In order to reduce the number of unknowns, one must resort to the constitutive

equations of the medium. Here we shall consider isotropic fluids alone, with a lin-

ear relationship between stresses and strain rates (Stokes law of friction) and with

no bulk viscosity (Stokes approximation). On these assumptions, the constitutive

equations may be written in the following form:

The tensors

I

and Dq are the unit tensor and the so-called strain rate deviator

respectively. With these equations, we have implicitly defined "pressure" p as the

opposite

of

the mean principal stress and "viscosity"

p

as the coefficient of propor-

tionality between the shearing stresses and the shearing strain rates ( S i j is the

well-known Kroneker delta).

The "Navier-Stokes" equations are obtained by combining Eqn 4.1 nd Eqn 4.2:

where

u,

u, w and

X, ,

Z are the components of the velocity vector and of the body

force (per unit mass), respectively.

Since four unknowns are involved (namely, the velocity components and the

pressure), another equation is needed: this is the "continuity" equation, expressing

the balance of the mass flowing through an infinitesimal control volume:

(4.4)

a a a

ay

+

(PU) +

-

pv)+ (pw)=

0



5 6

HYDROSTATICLUBRICATION

In the field of hydrostatic lubrication, the lubricant parameters

p

and p can, in

the large majority of cases, be regarded as constants, but they are, generally speak-

ing, functions of pressure and of temperature. When the dependence on tempera-

ture cannot be disregarded,

a

further equation is required

t o

define the problem

completely, i.e. the equation which expresses the energy balance. When, on the

other hand,

p

and p are constant, and the body forces are negligible, the Navier-

Stokes equations simply become:

where the Laplace operator V 2 is defined as:

The continuity equation is reduced to

vv

=o

(4.6)

4.2.2 Cylindrical coordinates

In many cases it proves to be convenient to use cylindrical coordinates (Fig.

4.2.a). Transforming Eqn 4.5 and Eqn 4.6 for

z=r

COSB, =r sin6 , the following is

obtained:

and

(4.7)

(4.8)

In

the equations above, u , u and

w

are the radial, tangential and axial components

of the velocity. The operators

DDt

and V 2 become



BASIC EQUATIONS

57

~a a v a a

~t

at

ar rat9 az

-++-+--+w-

- b -

Fig. 4.2 Coordinate systems:a- cylindrical coordinates; b- spherical coordinates.

4.2.3 Spherical coordinates

y=r si nq sinI9,

z=r

cosq) the Navier-Stokes an d continuity equations become:

In the spherical coordinate system

of

Fig. 4.2.b (i.e.:

x = r

s i n q cos19,

(4.9)

and

1

aw

v sinq)

+--

=

2 U

r2)

+

--

r

sinq

at9

_ _a

i a

r

sinq

329

where

(4.10)




The symbols

u ,

u , and w now indicate the components of the fluid velocity in the r , Q

and

0

directions, respectively.

4.3

THE REYNOLDS EQUA TION

4.3.1 Rectangular coordinates

In the field of fluid film lubrication we are involved in most cases with the study

of thin films (Fig. 4.3). n this connection, the complexity of the Navier-Stokes equa-

tions may be greatly reduced, thanks to the following considerations:

-

he thickness

of

the fluid film (in the

y

direction, in Fig.

4.3)

s small, compared to

it s size in the other directions;

-

consequently, the pressure, as well as the density and the viscosity, may be aver-

aged alongy: i.e. it

is

stated that

aplay=O,

rlay=O, apJay=O;

- compared to aulay and

awldy

all the other velocity gradients are negligible; this is

justified because u and

w

are generally much greater than u , and the film thickness

along y is small;

- he flow is laminar: no turbulence nor vortex exists;

- he body forces are negligible compared to the viscous forces;

-

he inertia terms, too, are negligible compared to the viscous forces, i.e.

Du/Dt=Du/Dt=Dw/Dt=O;

-

on the surfaces bounding the fluid film, the velocity

of

the lubricant coincides with

the velocity of the surfaces.

Accepting the foregoing assumptions, the second of Eqns 4.3 can be released,

while the others become:

t

2

/-

Fig. 4.3 Thin fluid film.



BASIC EOUATlONS

59

az

14.11)

The first of Eqns 4.11 may now be integrated twice, with the boundary conditions

u=U1for y=O and u=U2

for

y=h, to obtain the component of the fluid velocity in the

x

direction:

u = L*yDax 0,

-

h)+

(1

- f Ul + f U2

In the same way the component

(4.12)

(4.13)

is found, from the second of Eqns 4.11, with the boundary conditions

w=O

for y=O and

y = h (in other words, we have assumed that the surfaces of the pair do not slide in

the z direction).

Let us now integrate the continuity equation (Eqn 4.4) along the thickness of the

fluid film:

h

h h

h

Substituting the above expressions for u and w and using V=u(h)-v(0)o indicate the

squeeze velocity of the surfaces, it is easy to find:

$(?$) +

$(?$) = 6 [ p h (U1+U2)]

12

p U 2 zh + 12 p

V +

2 h

(4.14)

Equation 4.14 is the generalized Reynolds equation, which is characteristic of

hydrodynamic lubrication (see also

ref.

4.4).

Concerning plane hydrostatic bearings, velocities

U1

and

U2

of the surfaces

often do not depend on the coordinates; furthermore,

it

is generally assumed that

the density and the viscosity of the lubricant do not appreciably vary in the film.

Since velocity

V

may be written a s

equation 4.14 is simplified t o



6 0 HYDROSTATIC LUBRICATlON

(4.15)

where we have stated U=Ul+U2.

Equation 4.14 is still valid when the lubricant film is not flat, on the condition

that

its

thickness

is

much smaller than the curvature radius of the bounding sur-

faces. Let us consider a journal rotating in a sleeve (Fig. 4.4): D is the diameter

of

the bearing and C c c D

is

the radial play. If x=6D/2 and

y

are the tangential and

radial coordinates, respectively, Eqn. 4.14 is valid; however,

it

is important

to

note

that, in general,

U

and

V

may not be considered independent from

6.

The velocity of

any point

of

the journal surface is the vectorial sum

of

the velocity of its axis (we

assume that i t always remains parallel to the axis

of

the sleeve) and of the turning

velocity R=const. around the same axis. Referring

to

Fig.

4.4,

it transpires that,

because the film is very thin, the tangential and radial components of the journal

velocity may be written as:

Fig. 4.4 Thin fluid film between cylindrical surfaces.

When we have stated p=const., p=const.,

U1=0,

qn. 4.14 becomes:

(4.16)

(4.17)

From Eqns.

4.16,

it

is

clear that V=dUla&&?ahlals;ince h<cD, the right hand

side of Eqn.

4.17

may be rewritten as



BASIC EQUATIONS

61

After having substituted Eqns 4.16 and the film thickness:

h

=

C

[l-

COS(~$)] ;

we finally obtain

In the last equation, the term proportional to sin2(6-@)may also be disregarded,

since, on average, it is much smaller than the others.

4.3.2

Cylindrical coordinates

Let us consider a thin plane clearance: when we state the usual simplificative

hypotheses above, the Navier-Stokes equations (Eqns 4.7), are reduced to the

following:

(4.19)

In the first equation the inertia term pv2/r has been retained, since it may prove not

to be negligible

at

high turning velocity, as will be shown in sect. 6.2.1.

Integrating the second of Eqns 4.19, with the boundary conditions

v(O)=O,

v(h)=V,

he tangential velocity of the fluid is obtained:

(4.20)

Substituting it in the &st of Eqns 4.19 and integrating for u(O)=O,

u(h)=U,

he radial

velocity turns out to be (after further simplification):

(4.21)

Let us now integrate the continuity equation (Eqn

4.8)

along the film thickness:



6 2

h


Substituting in it Eqn

4.20

and Eqn

4.21,

the Reynolds equation i s obtained:

+ 1 2 p W + ~ ; z ( h 3 V 2 )~ 3 (4.22)

where

W=h=w(h)-w(O)

s the squeezing velocity of the surfaces.

If the upper surface does not slide, but rotates around the z axis

at

a constant

speed

0,

t is

U=O

nd V=r; Eqn 4.22 now becomes:

Equation 4.22 may also be used for a conical film, provided the thickness of the

film is much smaller than the minimum curvature radius, i.e. all the points of the

domain are far enough from the vertex

of

the cone.

With reference to the coordinate system

in

Fig. 4.5, Eqn 4.22 may be rewritten

as:

If we now assume that the axes of the conical surfaces bounding the lubricant

film always remain parallel (i.e. the thickness of the film does not depend on r) ,and

that the turning velocity of the journal is Ckconst., it is possible to obtain simple

expressions for U, , W. The velocity of each point of the surface of the journal is the

vectorial sum of the velocity of its axis, plus the turning velocity around the same

axis; hence, the velocity components are:

u E c sina cos(++) + E c sina sin(+@) R g a s V, cosa

ah

V =

E

C sin(+@

+

E ~ Cos(+$) +

0 r

ina

w = -

E

C cosa cos(99) - E C cosa sin(+$) + O x V , sina

(4.25)

ah,



BASIC EQUATIONS

63

Fig.

4.5 Thin

fluid

film between conical surfaces.

In Eqns 4.25V,

is

the axial velocity

of

the journal (squeeze velocity) and

C&

tga is

the radial play.

Let

us

now introduce Eqns

4.25

n Eqn

4.24.

Taking into account tha t yl=&ina,

hn=C cosafl-e cos(&$)], and that for all the domain being considered it is C<<r sina,

the Reynolds equation finally becomes:

+ 12

p V, sina + 3 P; hnlZ sina [lZ r sina - C

&

sin(+@)

C

E 4 cos(+$)]

(4.26)

A s

for the journal bearings, the term that is proportional to sins(&-$)may be dis-

carded, since, on average, i t is much smaller than the others.



BASIC EQUATIONS 65

diameter that

is

assumed

to

remain parallel to the axis

z=(p=O),

whatever the dis-

placement of its center is. Hence the components of the velocity of

a

point of the

journal are:

Substituting them in Eqn 4.29,

it is

reduced to:

(4.30)

4.4 THE LAPLACE EQUATION

It is well known that, if no external pressurization of the lubricant

is

provided,

the film of lubricant cannot sustain any load when one of the following circum-

stances occurs:

-

no squeeze nor sliding relative velocity of the surface exists

(V=U=O);

-

a sliding velocity exists, but thickness h

is

a constant all over the film of lubricant.

In this case, Eqn

4.15 is

reduced

to:

- a - -

- b -

(4.31)

Fig. 4.6 Hydrostatic bearings: a- without recess; b- with recess. Pressure profiles are also

represented



6 6


which is the so-called Laplace equation, for which, as is well known, only one

solu-

tion exists when the value of p on the boundary is assigned. Furthermore, if p is a

constant on the entire boundary (e.g. the atmospheric pressure), it takes on the

same value on all the inner points as well.

So,

if the boundary pressure i s the atmo-

spheric pressure, a source of pressurized lubricant must be provided to ensure a

lubricant film between the surfaces, as in Fig. 4.6.a; this is precisely the principle of

hydrostatic lubrication. In Fig. 4.6.b a recess is widened around the source, in

which the pressure is constant. Of course, both the hydrostatic and the hydrody-

namic pressure fields may exist at the same time, as in the so-called

"hybrid"

bearings.

4.5

4.5.1 Lo ad capacit y

LOAD CAPACITY, FLOW RATE, FRICTION

The load capacity W of a film of lubricant is defined as the resultant of the pres-

sure field. Hence, if A is the surface on which the pressure acts, and n s its normal

direction. it is:

4.5.2

Flow

rate

Integrating the velocity components (Eqn 4.12 and Eqn 4.13) between y=O and

y=h,

the components of the flow rate per unit length q are obtained:

(4.32)

(it should be borne in mind that Eqn 4.12 and Eqn 4.13, and consequently Eqns 4.32,

refer

t o

the case of two planes sliding one on the other in the

x

direction).

For a system of polar coordinates, Eqns 4.32 are best substituted by the following:

(4.33)

where it has been assumed that the upper surface

(z=h)

of the pair rotates around

the z axis.



BASIC

EQUATIONS

67

To obtain the volume

flow

rate of lubricant crossing a certain contour S, whose

external normal direction is n, t is necessary to integrate the scalar product qn:

Q =

q.n

dS

S

4.5.3 Frict ion

In connection with bearings, it is important to evaluate the friction forces on the

surfaces.

The shearing stresses in the lubricant can be found by means

of

the constitutive

equations (Eqn

4.2).

On the basis of the assumptions stated in section

4.3,

they are

reduced to:

i.e. t o the well-known Newton formula of the shear stress (see also sect. 3.2). Substi-

tuting Eqn 4.12 and Eqn 4.13 in these equations, we obtain:

(4.34)

When the pressure field is known, (that is after the relevant Reynolds equation

has been solved), Eqns 4.34 directly give the shearing stresses in the film of lubri-

cant between two surfaces sliding with a relative velocity U=U2-U1 in the x direc-

tion. The stresses on the surface

y=O are, therefore,

On the opposite surface

( y = h )

we have:




Integrating shear stress

z,

one obtains the tangential force on the surfaces.

Hence in the direction of the sliding velocity, the total drag is:

(4.35)

the upper sign referring to the fixed surface

(y=O),

the lower to the moving surface

( y = h ) .

In

a

very similar way, it

is

possible t o evaluate the friction moment which acts

on two surfaces rotating around a common axis, with a relative velocity

Lk

(4.36)

4.6

THE ENERGY EQUATION

A s has been already pointed out, the characteristics of a fluid depend on tem-

perature T. n particular, for the most common lubricants, the variation in viscos-

ity is notable, even for temperature changes of few degrees. Thus, in order to carry

out a rigorous analysis, ,u should be considered as a function of the temperature,

and a new equation is required to compensate for the introduction of the new

unknown.

The energy equation i s obtained by imposing the energy balance for an infinites-

imal control volume. Assuming that heat radiation

is

negligible and tha t no heat is

generated in the fluid (except for the dissipation of mechanical work due to shear-

ing), we obtain (ref.

4.2):

(4.37)

In the equation above, the left-hand side is the rate of change in internal energy of

the fluid (c being its specific heat). On the right-hand side, the first term accounts

for the power due t o the expansion in volume of the fluid; the second term accounts

for the heat conduction in the fluid

(k

being the coefficient of thermal conductivity);

the last term is the dissipation of mechanical power.

Since the energy equation must be applied

t o

the thin films typical of hydrostatic

and hydrodynamic lubrication, some further simplification can be introduced:

-

he specific heat

is

a constant (since no great temperature change

is

expected),

as

well

as

coefficient k;



BASIC EQUATIONS

69

-the fluid is considered

to

be incompressible (hence Eqn 4.6

is

valid);

-only steady-state conditions are considered (i.e.

aT/dt=O).

Introducing Eqn 4.2 in Eqn 4.37, we obtain:

A s in section 4.3, it is possible to disregard all the velocity gradients, except for au&

and awlay . Furthermore, heat conduction in the x and z direction is small, com-

pared to the convective term; thus Eqn 4.38 becomes:

(4.39)

Heat conduction may sometimes be totally disregarded (i.e. k=O) as well as the

heat transfer to the boundaries (adiabatic flow).

A similar equation may be obtained in cylindrical coordinates. Limiting our-

selves to axial-symmetric configurations, we find:

(4.40)

4.7 LAMINAR FLOW THROUGH CHARACTERISTIC CONFIGURATIONS

4.7.1 Parallel surfaces

Let us consider a plane surface of infinite length moving on another with veloc-

ity U Fig. 4.7.a).

The velocity of the lubricant is obtained from Eqn 4.12:

(4.41)

When

U=O,

the velocity profile is a parabola, whose axis of symmetry is y = h / 2

(pressure flow, or "Poiseuille

flow").

In general, the linear term

Uy

/ h (shear flow

o r

"Couette flow") has to be added

t o

the parabola. In Fig. 4.7.b the velocity profile is

plotted for certain values of the non-dimensional pressure gradient

P* =& (-g)

When p*<O, the two terms on the right-hand side of Eqn 4.41 are opposite in sign,

and when p*<-1 an inversion of the flow occurs. The lat ter condition is determined



70

- a -

Y t

HYDROSTATICLUBRICATlON

- b -

L a

J

Fig. 4.7 Flow between infinite-length parallel surfaces: a- pressure profile;

b-

velocity profiles.

by imposing aufay=O on the lower surface. The point of inversion y*

is

easily found

by stating u(y*)=O, i.e.:

The Reynolds equation (Eqn 4.15), for h=const.,

is

simply reduced to:

which, integrated twice with the boundary conditions p = p 1 for

x=O

and p = p 2 for x=a

gives the l inear relationship:

(4.42)

P =P 1 -

@ 1 -P2) ,

The flow rate per unit length

is,

from the

first

of Eqns

4.32

and

from

Eqn

4.42:

For a length L, the

flow

rate is:

The "hydraulic

resistance"

of the clearance may be defined as the ratio of the

pressure drop to the flow rate in the Poiseuille flow.

So,

from Eqn

4.43,

for

U=O,

we

obtain:



BASIC

EQUATIONS

71

(4.44)

The tangential force on the moving surface is given by Eqn

4.35:

It should be stressed that, in order that

flow

is laminar as presumed, Reynolds

number

R e Z P u h (4.45)

P

should be lower than the critical value

Re=2300.

In

Eqn

4.45

u

s the mean velocity

4.7.2

Infinite-length rectangular

pad.

Let us consider a segment

L

of a hydrostatic pad of infinite length, as in Fig.

4.8.

The recess is bounded by

two

clearances. If we now use

p

t o indicate the relative

pressure in the clearances, and

p r t o

indicate the relative recess pressure (which

can be considered

to

be constant, since the recess depth is much greater than the

thickness of the film), from Eqn 4.42 we get:

(4.46)

The load capacity of the pad is found by integrating the pressure on the

two

land

areas and adding the contribution p r L b

of the recess:

~ r 2

The total

flow

of the pad is found from Eqn 4.43:

1 1 L

& = - -

h3

~ - b

r

3 P

The hydraulic resistance may therefore be defined as

(4.47)

(4.48)



72


Fig.

4.8

Hydrostatic

pad

of infinite length.

Note that , in Eqn 4.48, the term tha t i s proportional to the velocity (shear flow) no

longer exists. Indeed, it is easy to establish that the flow rates crossing each clear-

ance are Ql 2UL Ul 2 , and hence their

s u m

does not depend on

U .

Similar

considerations hold good for the tangential force. In

this

case, the terms

due to the pressure flow are opposite in sign for the two clearances, and the drag on

the whole pad is null when U=O. The friction force on the lands is hence propor-

tional

to

the velocity:

(the contribution of the recess area is often negligible because the recess depth

h ,

is

much greater than h). In the equation above we have released the minus sign, since

it is implicit tha t the friction force on the moving member is opposite in direction to

the velocity.



74


The friction force over the moving member is (Eqn 4.35)

FC, .= 4UA

IL

' h r

For a sliding hydrostatic pad this force should be added t o the friction force on the

lands, obtaining a total friction force

where A1 and

A,.

are the land and recess area, respectively, and factor

fr

takes on a

value

h

f , .=4-

h r

under the simplifying assumptions made above.

More detailed and thorough studies of recess flow recirculation can be found in

ref. 4.5. and ref. 4.6. Better approximations for critical velocity and recess friction

factor are to be found in ref. 4.6:

h h

f,.

= (4

- 3

- -

hr hr

(4.52)

(4.53)

All the foregoing equations are based on the assumption that recess flow is

laminar, i.e. that the Reynolds number

is

smaller than

1000.

Actually, the problem is more complex because of the pressure

head build-up a t the inlet of the clearance (ref. 4.7).

For turbulent flow it may be stated that:

f =--fe h

r 2 h , P

(4.54)

and

f,

may be read in Fig.

4.9.b

(proposed in ref. 4.5 on the basis of experiments on

journal bearings) or calculated (ref.

4.8)

from the equation

(4.55)



BASIC

EQUATIONS

75

4.7.4 Annular clearance.

Let u s examine the flow between two parallel ring-shaped walls (Fig.4.101, o-

tating around the z-axis with a relative velocitya,which, however, is not

so

great as

to require the effects of the inertia of the lubricant t o be considered. Since the thick-

ness of the film is small, compared to width r2-r-1,Eqn

4.23

(the Reynolds equation)

is valid, which, due to the radial symmetry, is now reduced to

Integrating it twice, with the boundary conditions

p = p 1 at r = r l , p = p 2 at r = r 2 ,

the pressure field is obtained:

In rlr2

P

' P 2 +

( P 1

- P 2 ) & &

i.e. p decreases on the land area with a logarithmic trend.

The flow rate per unit length is, from the first of Eqns 4.33and Eqn

4.57,

(4.56)

(4.57)

Fig.

4.10

Annu lar clearance.



76


Consequently, the total flow across any circle

of

radius

r1esr-2

is

(4.58)

Hence, the “hydraulic resistance“ of the clearance may be defined

as:

(4.59)

P 1 - p 2 =6

In

r 2 h

Q n p 7

The friction moment, from Eqn 4.36, is:

The Reynolds number is again given by Eqn 4.45. Since the mean velocity at

mean radius

7=(rl+r2)/2

s

8

-

.

2 x F h ’

it follows that:

1 P 1

R e = - - Q

” 2 ( 1 + 3

(4.61)

4.7.5

Circular pad

The simple hydrostatic bearing, shown in outline in Fig.

4.6.b, is

made up of a

circular recess bounded by an annular clearance, like the one examined

in

the

previous section. Assuming that

p 2 = 0

(ambient pressure), pressure

p r

in the recess

is related to the ra te of

flow by Eqn

4.58,

which can be rewritten as

The pressure in the clearance decreases with a logarithmic trend,

(4.62)

given by Eqn

4.57,

in which

p l=pr

and

p2=0.

The hydraulic resistance i s still given by Eqn

4.59.

The load capacity of the pad is found by integrating the pressure on the land

area and adding the term

m$pr

due to the recess pressure:

(4.63)



BASIC

EQUATIONS

If Eqn

4.62

is introduced in Eqn

4.63,

we obtain

W = 3 - Q$ ( - - )

77

(4.64)

The friction moment is given by Eqn 4.60, since the friction in the recess is

negligible.

4.7.6 Pipes

In the case of laminar flow in a straight circular pipe, whose length

is

much

greater than the diameter, certain simplifying hypotheses,

similar to

those stated in

section

4.3,

may be applied

t o

the Navier-Stokes equations. In particular, the pres-

sure can be considered t o vary with the z coordinate alone. The third

of

Eqns 4.7 is

now reduced t o the following:

Integrating it with the boundary conditions w=O for r=d/2 and aw/&=o for r=O (due

to matters of symmetry), the velocity field

is

obtained as a function of the pressure

gradient:

(4.65)

which is the equation of a paraboloid. The maximum velocity of the fluid is at

r=O:

while

its

mean velocity

is:

Since the continuity equation (Eqn 4.8)

is

reduced t o dpldz=const., the pressure drop

along the pipe is linear. If a length 1 is considered, the flow rate is related t o the

pressure drop by the law:

Equation

4.66

is used

to

evaluate the hydraulic resistance:

(4.66)

(4.67)



BASIC

EQUATIONS 79

4.8

FLOW

INTO THE INLET LENGTH

inlet lengthIn section 4.7, a number

of

cases of laminar flow have been

examined; the equations which have been obtained expressing the flow as a

function of the pressure step are not strictly accurate, since certain distributed

energy losses have been disregarded. A s an example, let us consider a circular

pipe. When the fluid begins t o flow into the pipe, the velocity w is a constant for the

entire section (Fig. 4.12); the velocity of the fluid on the wall of the pipe, however, is

null, and this layer exerts a great shearing stress on the inner layers, whose

velocity must become greater than the mean value

W

in order to satisfy the

continuity requirement. The thickness

of

the

"boundary layer"

grows until, a t a

distance l i from the inlet, the whole section of the pipe is involved; a t this point, the

fluid velocity reaches the parabolic profile already seen in sect.4.7.6, and remains

so

thereafter.

\

-

'.

,

I

I

I

I

I

I

1 2

Fig. 4.12 Velocity profiles in the inlet length of pipes.

The length

Zi

is named

"inlet length"

(ref. 4.9) and its value is found to be

li=0.0575

d

Re.

When the length of the pipe is greater than

li,

which

is

generally true

for the capillaries used as compensating devices for hydrostatic bearings, Eqn 4.65

should be substituted by the following:

(4.69)

1 1

128

'

d 1

= - - d 4

1+0.0356i

Re

In other words, the value of the flow rate found in sect. 4.7.6 should be multiplied for

a correction coefficient, which accounts for the pressure loss required to accelerate

the inner fluid layers in the inlet length.



8 0

HYDROSTATIC LU5RlCATlON

Similar

considerations hoId good for pipes of different shapes,

and

for the clear-

ances of hydrostatic pads (see,

for

example, sect. 6.2.14~)).

4.9 INLET LOSSES

In the preceding section, the pressure loss in the inlet length of a circular pipe

was examined; other pressure losses occur in the tank near the pipe inlet and a t the

inlet itself.

The loss in the

tank

may be considered to be the sum of two terms: the first is

used

to

accelerate the fluid in the tank (in the conoid bounded by dotted lines in Fig.

4.12) to the inlet velocity, the other is due to the viscous dissipation in the same area.

The loss

at

the inlet, on the other hand, is due to the

"uenacontructu",

which

is

generally present, especially when the inlet edge is sharp.

The overall pressure loss is (ref 2.7):

(4.70)

The values of resistance coefficient k are given in Fig. 4.13 for the cases of (a) a

well-rounded entrance, (b) a slightly rounded entrance, (c) a sharp-edged entrance.

The pressure loss a t the inlet, given by Eqn 4.70, should be added to the pressure

loss in the pipe, given by Eqn 4.69.

- a - - b -

- C -

-

K

= 0.05

Fig. 4.13 Resistance coefficients

due to

the geometry

of

pipe entrances.

4.10 TURBULENT

FLOW

The flow of the lubricant in the gap between two close surfaces is , in most cases,

of a laminar kind, i.e. the Reynolds number is lower than 1000. Nevertheless, in

certain circumstances,

it

may be not so. For instance, with reference to

a

circular

pad and to Eqn

4.61,

if high values are selected for p r and

r h

due to the need to

sustain a heavy load) and if the viscosity of the lubricant

is

low (e.g. when water or



BASIC EQUATIONS

81

liquid metals such as Sodium have to be used), and, above all, if the gap is higher

than the values commonly adopted, Re may be greater than 1000.

When

Re is

greater than

1000,

a transition regime sets in; a t

Rez2300

the flow is

prevalently turbulent, and it is totally turbulent for higher Reynolds numbers (see

ref. 4.10). When the fluid gets into the clearance, the boundary layer is at first lami-

nar but becomes turbulent (except for a thin laminar sublayer) at

a

distance from

the inlet of 25+40 times the clearance. The velocity profile is similar to the one plot-

ted in Fig. 4.14.b and the maximum fluid velocity is about 1.2 times it s mean value.

- a

-

- b -

Fig.

4.14

Velocity distribution in

a

pipe: a- laminar f l ow; b- turbulent f l o w .

In turbulent flow, the velocity components and the pressure in the clearance a t

a given time may be written as a sum of t w o terms:

u=u*+u1;

u=u*+u1; w=w*+wl;

p=p*+pl: the first terms are the average values, the others ar e the fluctuations due

to turbulence. These expressions should be introduced into Eqns

4.7

and Eqn

4.8

(i.e.

the Navier-Stokes and continuity equations). Certain assumptions can now be

made: the viscosity and density of the lubricant are considered to be constants; the

average flow is stationary and merely radial (i.e. u*=u*(F,z) ;u*=w*=O); the turbu-

lent components are considered to depend solely on

r

and z , due t o the radial sym-

metry. The

first

of Eqns 4.7 becomes (ref. 4.2,4.4)

(4.71)

in which z,=p(au*/az) and zt=-p(ulwl)* are the viscous and turbulent components of

the shear stress. It may be assumed that

z,<<q;

applying the Prandtl theory of the

mixing length , the shear stress is

where

l=z

or 02&/2,1=X(h-z)for h/2Q<h; the constant x should be determined by

means of experiments.

Equation 4.71 may be further simplified (ref. 4.11) and then integrated with the

following boundary conditions



8 2


where zo is the thickness of the laminar sublayer on the surfaces. We thus obtain

(4.72)

where

z '=z lh ,

zb=zdh, and

The blunt profile of u* (Fig. 4.14.b) is rather different from the parabolic profile (Fig.

4.14.a) pertaining

t o

laminar flow.

Equation 4.8 becomes

u* au*

- + -=o

r ar

(4.73)

Introducing Eqn 4.72 into Eqn 4.73, and integrating with the following boundary

conditions

we obtain

1

- rz/r

P*

-PI =PI[-- 11

(4.74)

For circular pipes, whose Reynolds number is given by Eqn 4.68, the drop in

pressure is given by the well-known equation (ref. 2.7)

where the friction coefficient

f

can be evaluated, for ReelO5, by means of the experi-

mental relation:

For pipes of different sections, the pressure drop can be approximately evalu-

ated using the same Eqn. 4.75, in which the "hydraulic diameter"

d=4AIS

is

intro-

duced:

A

and

S

are the area and the perimeter of the cross section of the pipe,

respectively.



BASIC EQUATlONS 85

discharge coefficient

Cd

is now given in Fig.

4.16.

The initial value of

Re

has to be

calculated for Cd=0.6.

The sharp-edge orifices are sometimes substituted by less costly orifices (Fig.

4.17)

whose length

1

is no longer negligible. In such cases, the discharge coefficient

can be calculated by means of the following equations (ref.

2.7):

1 m -m

Cd = [ .5 + 13.74 a)

for

?<50

;

(4.77)

the relevant values of

Cd

also being plotted in Fig.

4.16.

Re

has to be initially obtained

from Eqn 4.68, with Q being taken from Eqn 4.76 for Cd=0.6.

Since Cd is now a function of the Reynolds number, the flow rate for any certain

pressure step is no longer completely independent from variations in temperature

and viscosity.

0 s

1

o,61

.4

c d

a2

I I

I

1

lo*Re

/

I lo3

.14 10

Fig. 4.17

Discharge coefficient for a short-tube

orif ice.

4

R E F E R E N C E S

4.1

4 3

4 5

4.4

Goldstein S.

D.; Modern Deuelopments in Fluid Dynamics;

Claredon Press,

Oxford, 1952; 702 pp.

Pinkus O., Sternlicht B.; Theory of Hydrodynamic Lubrication; McGraw-Hill

Book Co., N.Y.,

1961; 465

pp.

Tipei

N.;

Theory

of

Lubrication;

Standford Univ. Press,

1962; 690

pp.

Cameron A,; The Principle of Lubricatioq; Longmans, Green and Co. Ltd.,

1966; 591

pp.




4.6 Shinkle J. N., Hornung V . G.;

Frictional Characteristics

of

Liquid Hydro-

static Bearings;ASME Trans.,J . of Basic Eng., 87 (19651,163-169.

4.6 Ettles

C.

H. M., O'Donoghue

J .

P.; Lam inar Recess Flow in Liquid Hydro-

static Bearings; Instn. Mech. Engrs., C27 (19711,215-227.

4.7 Tipei N.;

Flow Characteristics and Pressure Head Bu ild -up at the Inlet

of

Narrow Passages;

ASME Trans.,J . Lubr. Tech., 100 (1978), 47-55.

4.8 El-Sherbiny M., Salem F . , El-Efnawy N.; Op timum Design of Hydrostatic

Journal Bearings;

Tribology International, 17 (19841, 155-166.

4.9 Langhaar H. L.;

Steady Flow i n the Transition Length

of

a straight Tube;

J . o f

Applied Mech.,9,2 (19421, A55-A58.

4.10 Streeter V.;Handbook

of

Fluid Dynamics; McGraw-Hill, 1961.

4.11 Bassani R.;

Su l Regime Turbolento nei Cu scinetti Zdrostatici d i S pin ta M olto

Caricati;Atti 1st. Mecc Appl. e Costr. Macch., 1968-69,

N

15; 36 pp.



Chapter

5

PAD

COEFFICIENTS

5.1

INTRODUCTION

In this chapter, generalized equations for load capacity W, stiffness

K, low

rate

Q, pumping power

H p ,

friction power

H f

and total power

H t

are proposed, which are

valid for every type of pad. In order t o simplify the representation of the behavior

of

bearings and make their design easier, use is made

of

characteristic parameters,

such as the

effective area A,

and the

hydraulic resistance

R , as well as of the power

ratio ll and the pressure ratio /J

For

the same purpose, certain pad coef f ic ients

(A*,

,R * ,

H F ) are introduced, which are characteristics of the actual pad shape.

They are also considered the effects on the bearing behavior of misalignment

and of certain phenomena, such as turbulence, inlet pressure losses and lubricant

inertia, which are not typical

of

plain hydrostatic lubrication.

5.2

GENERAL STATEMENTS

Whatever the shape of the bearing pads and the supply system, certain general

expressions may be stated. The load capacity of a pad (the resultant of the lubricant

pressure in the bearing) may be put in the following form:

W = Pr A,

(5.1)

p r

is the recess pressure;

A,

is the

“effective“

pad area. This last is a fraction of the

projected area and, in general, depends on the operating conditions; in the most

common cases, however, A, may be considered as a constant, related solely t o the

shape and size of the pad.



PAD

COEFFlClENTS 89

For a fast (and safe) evaluation of AT it is often assumed that all the dissipated

power H , remains in the lubricant (adiabatic flow). As a consequence, we have

simply

(5.7)

where c is the specific heat of the lubricant, and

p

is i ts density (for further discus-

sion on thermal running see chapter 12). The term

H f I H ,

is commonly called the

"power ratio and has also been used (ref. 1.9) for the following grading of hydro-

static bearings: H f / H p = O , zero-speed bearings (purely hydrostatic bearings);

H f I H p < l , lowlmoderate speed bearings; and H f / H p > l , moderate/high speed bear-

ings. In the last case, for certain bearings (e.g. journal bearings), hydrodynamic

load capacity may become important (hybrid bearings).

The static "stiffness" K may be generally defined as the limit of the ratio of a

small change A F of the applied load t o the consequent displacement Ae

of

the mov-

ing member in the direction of the load. In steady-state conditions the external load

F

is balanced by the resultant

W

of the pressure of the lubricant,

so

the stiffness

of

the bearing may be defined as:

For a single-pad thrust bearing, if we use h t o indicate the component of the film

thickness along the direction of the load, we clearly have Ah=Ae, and hence Eqn 5.8

becomes

K=--W

dh

It is important to note tha t the derivative on the right-hand side of Eqn 5.8 is a

total derivative. Indeed,

W

is proportional to recess pressure; this is linked to the

flow rate and to the thickness of the film by a relationship which depends on the

supply system and the compensating devices. For this reason this point will be

examined more closely in sections 6.2 and 6.3.

Once a design value ho has been selected for the clearance of any given pad bear-

ing, it may be worthwhile writing the expressions of each of the foregoing parame-

ters as the product of the value of the parameter in the reference configuration h=ho

times a function of the non-dimensional film thickness h l h , or of the non-dimen-

sional displacement

h - ho

h0

E=-

(5.9)




It should be noted that fqr certain types of pad a "natural" reference configura-

tion exists (e.g. the concentric configuration for the cylindrical o r spherical pads).

In non-dimensional form, hydraulic resistance

R

and friction power

H f

can

now be rewritten as:

R = R o R ' (5.10)

R o and

Hf o

being the relevant reference values. For the pads whose clearances have

an uniform thickness the results obtained in Chapter 4 allow

us

to write:

(5.12)

(5.13)

The stiffness may also be written in the form:

K = K o K

(5.14)

but now both KOand K' are dependent on the supply system, and will be defined

later.

One parameter which will be widely used in the following chapters is the

"pressure ratio" in the reference configuration:

(5.15)

Another important parameter t o take into consideration

is

the "referencepower

ratio", that is the power ratio in the reference configuration:

The importance of this parameter consists in the fact that it provides a simple but

effective way

of

optimizing the selection of the lubricant and of the clearance, as will

be shown later.

In order to evaluate the actual bearing performance, it is now necessary:

t o

evaluate the foregoing parameters for the main types of pad; this will be

done in the remainder of this chapter;

to establish the relationship between the

flow

rate, the recess pressure and the

supply pressure for the most common supply systems. This will be done in section



PAD

COEFFICIENTS

91

6.2 (for

the

"direc t"

supply system) and section

6.3

(for the

"compensated"

supply

systems).

5.3

CIRCULAR RECESS PAD

5.3.1

Basic equations

A set of equations, with which the parameters

of

the simple bearing in Fig.

5.1.a

may be calculated, have been already obtained in section

4.7.5.

In particular,

it

was

found there that the

flow

rate Q and the load capacity

W

(written as functions

of

the

recess pressure

p r )

are

(5.17)

(5.18)

where

is the ratio of the inner radius

of

the land surface to the outer one. Comparing these

equations

to

Eqn

5.1

and Eqn

5.2,

the effective pad area and the hydraulic resistance

are easily obtained:

(5.20)

A,

=

4 Dz A

R=:$R*

A*,and R*, which are plotted in Fig. 5.l.b, are:

(5.21)

(5.22)

R* = In llr'

(5.23)

The pumping power dissipated in the bearing is given by Eqn 5.4. It may prove

useful t o express it as a function of the load capacity:

(5.24)

where



92

HYDROSTATlC LUBRlCATlON

- b -

a -

h

10

r

5

3

Fig. 5.1 Circular recess pad: a- pad geomc y;

b-

pad coefficients.

(5.25)

r*=AT

If we now want

t o

assess the value of the ratio r’, which minimizes the pumping

power (for any given load and film thickness), we have to solve the equation

It

is

easy to obtain

r&,=0.5291.

Note that this optimal value

is

not

a

critical value; in

many cases i t may be advisable to increase

r’

in order, for example, to decrease

recess pressure, and hence the increase in temperature of the lubricant. Further-

more, when velocity Q is not too low, it proves to be convenient to use high values of

r‘, in order to reduce the total power loss, as we shall also see later.

The moment of the friction forces Mi due

to

an angular velocity R has already

been calculated, in section4.7.4; qn 4.60may be rewritten here as:

where



PAD

COEFFICIENTS

93

Fig. 5.2

Nomograph for

a circular

pad.



94


is

plotted in Fig. 5.1.b. The friction power is:

H f = Q M = E E D 4 Q 2 H f * (5.28)

The friction coefficient, referred

to

the mean radius of the land P=(rl+r2)/2, is

f

32h

(5.29)

It can be shown (ref. 5.1) that, for moderate speeds, can be as little

as 0.01

times the

friction coefficient of the rolling bearings.

EXAMPLE 5.1

A pad bearing, which has to carry a load W=20,000 N, rotates at a speed Q=2n

radls (i.e. 60 rpm); its external diameter is D=O.l m. The lubricant viscosity, at a

working temperature

T=4OoC,

s p=O.l Ns im2. For the ratio

re ,

the optimal value

is chosen: rAPt=0.53, because of low velocity. For the f i lm thickness the value h=50

pm is chosen in order to avoid the influence of errors in parallelism (see section

5.3.3).

Let us calculate the main bearing parameters. From Eqn 5.22

or

from Fig. 5.1.b

it is easy to obtain

Af

=0.566 and hence the ef fective area is (from Eqn 5.23)

A,=4.45.10-3 mz. As a consequence, the recess pressure which is needed to sustain

the design load is (from Eqn 5.1) pr=4.5 MPa.

From Eqn 5.23,

or from Fig. 5.1.b, we find R*=0.635, and a hydraulic resistance

(Eqn 5.21) RO=970.10sNsIm5. From Eqn 5.2 it immediately follows that the flow rate

which is required to sustain the load with the planned clearance is &=4.63.10-6

m3Is.

The pumping power dissipated in the bearing is Hi=prQ=20.8 W : for a direct

supply system (i.e. &=const,ps=pr) this is also the actual pumping power. Equation

5.27, or Fig. 5.1.b, gives Hf*=0.921. The friction power is obtained from Eqn 5.28:

Hf=O.714

W;

this low value justifies the choice of

r'=rApt.

The friction moment is

Mf=Hf/LkO.114N m and the friction coefficient (Eqn 5.29) i s f=0.149.10-3

5.3.2

Design chart

For the calculus of a circular-recess hydrostatic thrust bearing, it may prove

worthwhile to use, as a

first

approach, the design chart in Fig. 5.2.) obtained from

Eqn 5.17 and Eqn 5.18 rewritten in logarithmic form. It allows us t o obtain the main

design parameters of a bearing in graphical form. By way of example, let us con-



PAD

COEFFICIENTS 95

sider the bearing in Example 5.1.Starting from point

r’=0.53

on the scale in the

lower left corner, trace

a

horizontal line

t o

meet the curve

p,;

then a line is drawn

vertically t o the assigned value of

W,

and again horizontally, until the external

radius

r2

is met;

a

value of

4.5

MPa can now be read on the scale

of

the recess pres-

sures. To calculate

Q a

similar path has to be traced, except that the curve

pq

has to

be used instead

of p,;

moreover, after meeting

r2 ,

a vertical line is drawn to reach

the selected viscosity, and then a horizontal line (after a rotation of 90 degrees) leads

to the selected value of h and hence to

&=4.6.10-6

m%.

5.3.3 Effects of errors

i n

paral lel ism

As has been noted above, it is generally expedient to select a low value for the

film thickness, but the minimum admissible value of h can be limited by a lack of

parallelism between the pad surfaces (see Fig. 5.3).

Fig.

5.3

Misaligned circular pad.

Since the radial symmetry is lost, the one-dimensional Reynolds equation (i.e.

Eqn 4.56) can no longer be used. Resorting to the complete equation (Eqn 4.23) but

disregarding the effects of velocities R and A we may write:

(5.30)

where the clearance a t any point (r , ts) is



PAD

COEFFICIENTS

where:

Since, however, the correction introduced is of the order of 1%,

be disregarded.

0.8

0.9

1

97

(5.32)

(5.33)

even for Ah’=l, it can

res

0.53

0.60

0.65

0.m

0.75

0.80

085

090

Fig. 5.5 Misaligned circular pad: correction factorCR ersus parallelism error Ah‘.



98 HYDROSTATC

LUBRlCATlON

The hydraulic resistance of the pad becomes:

where:

1

l+g l

r ’ )ZAh‘2

c R = 3

(5.34)

(5.35)

(a more accurate but rather complicated equation for CR s given in ref.

5.2).

It is

easy t o verify that , when Ah’=@ the correction factors are reduced to 1.

The correction factor C,, which may also be interpreted as the decrease in load

capacity when Q an dx are given, is plotted in Fig. 5.5 for certain values of r ’ . The

decrease in W may be compensated by increasing Q.

The friction moment, power and coefficient may still be calculated as in the case

ofM’=O.

The effect of the turning velocity R (which is not considered in the equations

above) is studied in ref. 5.3,where the Reynolds equation (Eqn 4.23) s solved by

means of a prediction-correction method, in order to get the pressure field. Load

capacity, flow rate, and friction coeficient are then obtained. The variability of the

pressure with

19,

already seen for

R=O,

is increased by the turning speed, and even

cavitation may occur at the higher values of R and Ah’. This effect is counteracted

by using higher values for r ‘ . The load capacity is found to increase with R. Note

that the lubricant inertia has not been taken into account; actually,

it

should lead to

a further increase in load capacity (as will be seen below) when rb0.5 . The flow rate

proves to be unaffected by

R

(since the forces of inertia have been disregarded). The

friction coefficient decreases with

Ah

’ and obviously increases with the rotation

speed.

5.3.4

Effects of the l oss of pressure at the in let

The equations expressing the performance of the bearing should be modified if

pressure losses, other than those due

t o

viscous forces, have to be taken into ac-

count. The effects of these losses, which have already been examined in sections 4.8

and 4.9 with regard t o the lubricant pipes, are evaluated in ref. 5.4 and 5.5. While

the consequences on the effective area are negligible, the hydraulic resistance

should be corrected as follows:

(5.36)



PAD COEff /C/ENTS

99

Fig.

5.6 Pressure losses

at

the

inlet

of clearance: correction factor

C pfor

circular

pads.

where the correction factor

C p

is given by the equation:

(5.37)

and R e is the Reynolds number (Eqn 4.61); F=(Fl+F2)/2

is

the mean radius of the

land surface; ti,. is the inlet length:

(5.38)

The first term of the numerator in Eqn 5.37 takes into account the pressure

losses due to the viscous forces in the bearing gaps; the second stands for the energy




used to accelerate the fluid in the inlet zone; the third term represents the contrac-

tion loss. Equation 5.38 of the inlet length is valid for

Ar=r2-r121ir;

or greater values

of

l ir,

one can simply substitute

Ar

for l i , in Eqn 5.37, unless (in which case

the bearing gap would virtually behave a s

an

orifice ). Coefficients

k l , k 2 ,

and

k3

are

obtained in ref. 5.5 from the experimental data given in ref. 5.4, with the following

values: k1=0.75, k2=0.235,and k3=0.021;thus the correction factor for

R

becomes

C p = l + - -

0.0313 2 + 0.00979

1

1 (l+l/rY

1 +0.00525l n p

_ -

h Re

(5.39)

It is plotted in Fig.

5.6

as a function of

Arl(h.Re)

for a number of values of

r'.

Unlike the case of error in parallelism, the inlet losses have favorable effects.

These effects would be rather important for very small (~0 .1 )alues of Arl(h.Re),

which are, however, unusual in actual fact.

The coefficient

C p

may also be explained as a n increase in load capacity, for any

given flow

rate,

o r as a decrease in flow rate, for any given load.

It

should be noted

that C, becomes considerable in the case of ArchaRe.

EXAMPLE 5.2

A circular pad bearing, whose diameter is D=O.l m, has been selected to sustain

a load W=50 Mv rotating at 750 rpm (i2=251r radls). Furthermore, the recess pres-

sure needs to be smaller than 7.2 MPa and the friction torque needs to

be

smaller

than 0.05 Nm. Let us assess the flow rate, assuming that a lubricant with p=O.O18

Ns lm2 and p=880Kglm3 must be used.

From the condition p,<7.2 MPa, it follows that Ae>6.94.10-3m2(Eqn 5.1), and

Az>0.884 (Eqn 5.20). From Fig. 5.1.6 a radius ratio of about 0.9 seems to be neehd. I f

we select r'=0.9,we easily obtain A50.902, R*=0.105, Hf*=0.344. The effective area is

then

Ae=7.08.103

m2,

and hence the recess pressure will be p,=7.06 MPa.

Since necessarily Mf<0.05 Nm, it follows from Eqn 5.26 hat

h95 .5

pm. If a

clearance h=100 pm is selected, it follows that R=3.62.109 NslmS (Eqn 5.21) and

Q=1.95.10-3 m31s. Since Q is high, it is advisable to verify the effect of the pressure

losses at the inlet. From Eqn 4.61, the Reynolds number is Re=319 and, from Fig.

4.6, Cp=1.26. Hence, the new value of the flow rate is calculated: Q=1.54.10-3 m3/s.

This is still a high value; it should be borne in mind, however, that a very low fric-

tion was required, the friction coefficient being of the order of 2.10-5.



PAD COEFFICIENTS 101

5.3.5

Turbulent f low

Although the

flow

in the clearances

of

hydrostatic bearings is, generally, of a

laminar type, in certain circumstances (as already noted in section 4.10) the

Reynolds number (Eqn 4.61) may become greater than 2300, a t which value the flow

is prevalently turbulent. Equations giving pressure and lubricant velocity in the

clearance of a circular pad in turbulent regime have been obtained in section 4.10.

Integrating the mean pressure (Eqn

4.74)

on both the recess and land area, the

load capacity proves

to

be:

(5.40)

W

=

4

D2

r'p ,

while, if we integrate the radial velocity field (Eqn

4.72),

the following flow rate

is

obtained:

where (ref. 4.11):

and

(5.42)

(5.43)

The nondimensional quantities x and A (which depends on zb and accounts for the

rate

of

flow

in the laminar boundary sublayer along the land surfaces) are evalu-

ated in ref. 4.11, for Re>2300:

Comparing Eqn 5.40 and Eqn 5.41 with those obtained for the laminar flow, two

correction factors are easily obtained for the effective area as well as the hydraulic

resistance

of

the pad. After stating that:

(5.44)

(5.45)



102

it

follows that:

HYOROSTATIC LUB RICA

TION

(5.47)

When r’ is greater than 0.80, we have 0 .9 9 <C ~ ~< l,nd hence

CAT

may be disre-

garded; furthermore, for r’>0.5, we have 6 % ~ ) l n ( l / r 7 , hus

(5.48)

‘RT

O X 5 ( l + A ) zb F(zb)

and, hence, CRT may be considered t o depend on zb alone. Both CAT and CRT re

plotted in Fig. 5.7.

- a - - b -

1

oo

CAT

0.95

0.90

0.50

0.75

1

oo

f

10

CRT

5

0

0.00 0.05

0.10

zb

Fig.

5.7

Turbulent flow: a- correction factor

CAT

or effective area; b- correction factor C,, for

hydraulic resistance.

Since CRT roves t o be greater than 1,we can see that, at high Reynolds numbers,

turbulence brings about a considerable decrease in flow rate or, if

flow

rate remains

constant, a considerable increase in load capacity. On the other hand, turbulence

may cause a large increase of foam in the lubricant; the persistence of the foam



PAD COEFFICIENTS

103

depends on the physico-chemical properties

of

the oils, and may be counteracted by

means of suitable anti-foam additives (see chapter

3).

EXAMPLE

5.3

Let us re-examine the hydrostatic pad in Exumple 5.2,

in

order to evaluate how

the flow rate would vary if the clearance and the viscosity were changed to h=200

pm and p=0.005 Ns

Im2

( I S 0 VG5), espectively.

From

Eqn 5.21 it follows that R=O.126.1@ NslmS, and hence Q=56.1.10-3 m31s.

Since the Reynolds number proves to be greater than 2300 (Eqn 4.61), a correction

factor should clearly be evaluated. From Eqn 5.42 we obtain z&0.0112, and hence

C~,=7.78, R=0.979.109

Nslm5,

Q=7.23.103 m3/s (in spite

of

the considerable reduc-

tion due to turbulence, this is a very high value). The Reynolds number proves to be

Re54260.

Bearing in mind the above results, it may be concluded that, for values

of

Re

lower than 10, which is the most common case for these hydrostatic thrust bear-

ings, the whole pressure loss in the clearance is due t o viscous forces; when

Re

is

greater than 100, the pressure losses examined in section (v) become considerable;

when Re is greater than 1000, the effect of turbulence should be taken into account.

In both these cases, for any given load capacity and

film

thickness, the flow rate

proves

t o

be lower than that expected

on

the basis of the laminar theory.

5.3.6 Effects of the inertia forces

When the angular speed of the bearing in Fig. 5.1.a is high, the forces of inertia

in the fluid can no longer be disregarded. Supposing h=const, but retaining the

inert ia term, Eqn 4.23 becomes:

where the “inertia parameter”

(5.49)

(5.50)

has been introduced.

This kind

of

problem has been solved in ref. 5.6. When

r o o l ,

he depth h,. of the

recess should be

at

least 5 imes greater than the film thickness h , while hr=20h is

considered t o be a n optimal value.



104


Fig. 5.8 Pressure distribution for certain values

of

inertia parameter (ref. 5.6).

In Fig.

5.8

the pressure field

is

plotted

for

a number

of

values

of

the inertia

parameter Si; t is assumed that r1=0.5r2and that the radius of the supply duct

is

ro=0.05r2.A pressure peak can be seen at the inner edge of the bearing land. When

no recess exists (i.e.

r o = r l ) ,

negative pressures, and hence cavitation phenomena,

may occur. Note that in Fig. 5.8 the depth of the recess is h,=5h. For higher values

(i.e. h,.220h, as

is

common practice) the depth of the recess does not affect pressure

distribution.

When h,/h>>5, 0 .05~ , / r2<0 .5, 0.534'10.9 i.e. in the large majority of cases),

simplified equations are available

for

the pressure in the recess

fro<rlrl):

(5.51)

and in the clearance

( r l < r < r 2 ) :

(5.52)

Integrating

p

on the entire pad surface,

we

obtain the load capacity, and hence

the effective pad area. Equation 4.20 gives the tangential velocity of the lubricant:



PAD

COEFFICIENTS

105

Since

u

a n d p are now known functions, the radial flow velocity, an d hence the

volume flow rate, may now be easily obtained from Eqn

4.21.

The effective area and the hydraulic resistance may thus be expressed

as

R = : $ R * cRI

where:

c , = l + s i [(

1- - )

+

(1--

1

CRI

=

1

+

si1

-3)

It is clear, however,

that

c,=1+si 1---

(

2

1

CRI

=

(5.53)

(5.54)

(5.55)

(5.56)

(5.57)

(5.58)

are very good approximations in the large majority of cases. In Fig. 5.9 CAI,is given

as a function of Si for certain values of F'. In the same

figure

CRI is also given as a

function of

Si.

The increase in flow rate (the inverse of C R I )may be appreciable even for mod-

era te values of the inertia parameter. I t is moderately compensated by the increase

in load capacity (i.e. C,). Since

C

increases with Si, oad capacity W is increased

by inertia effects if inlet pressure is held constant; thus these bearings are often

named hybr id. However, as CRI simultaneously decreases more quickly, flow rate Q

increases more greatly than W, so an efficiency loss of the bearing may occur.

The Reynolds number, as given by Eqn

4.61, is

usually very low. Nevertheless,

turbulence could set in, due

to the tangential flow velocity. In ref. 5.10, in which the

circular pad

is

compared

t o

the case of

a

couple of faced rotating

disks

(ref.

5.71,

the

Reynolds number R e , = a r ; /

v

is introduced, and the flow is proved to be laminar for




Re,<lO5. In ref.

5.8,

the “mixing length” theory is used to prove that the flow is

laminar when

L?Fh

Re,

=

(5.59)

is smaller than lo3.The latter result agrees closely with the former, since

r2

Re,

=: Re,

and r2/h>102. Reference

5.8

also shows that, when the flow is turbulent, for any

given flow

rate,

the load capacity and the stiffness of the bearing increase notably

with

Re,;

it

should however be noted tha t values of

Re,

greater than

1000

are rarely

reached.

i

CAI

CRI

1

0

0

f=0.9

0.8

0.7

0.6

0.5

1

2

Si

Fig.

5.9

Correction factor

CA,

versus inertia parameter

Si

for certain values of ratio

r‘.

Correction

factor CRI ersus inertia parameter

Sk

EXAMPLE 5.4

A pad bearing, with D=O.l m, r’=0.9, sustains a load W=lO

Mv.

It is fed by a

constant flow source at a rate &=0.3.1O3 m 3/ s with a lubricant with the following

characteristics: p=0.05Nslm2,

p=880

Kglm3, c=1900 JIK&jC Let

us

evaluate how

bearing performance is affected

as

speed becomes

G30Oz

rad Is (9000 rpm).

Proceeding as usual, we f ind Az=0.902, R*=0.105, Hf*=0.344. Thus, at Q=O,

Ae=7.08.10-3m2, and p,=1.41 MPa, the hydraulic resistance o f the pad must be

R=p,l&=4.70.1O9 Nslm5 and hence h=129 pm. At the highest speed, inserting the

value of

pr

calculated above into Eqn

5.50,

we find Si=O.208, and (from Eqn 5.57’

C~ =l .0 9. rom Eqn 5.53 it follows that Ae=7.73.10-3m2 and hencep,=1.29 MPa. For




a more accurate calculation it is advisable to evaluate Si again. We now obtain:

S i=0 .227 , CN=l . lO , C~1=0.816;onsequently, A,= 7.79.1 0-3 m2, p,=1 .28 M Pa. The

hydraulic resis tance

of

the pad becomes R=4.27,1O9 N sl m 5 an d, from Eqn 5.54,

h=124 pm . The pumping power is Hi=384 W a n d the fr ic tion power is Hf= 120 9

W

(disregarding the lubricant inertia, Hi=423

W

and Hf=1167 W would

be

obtained,

respectively).

It

is easy to verifv that the temperature step is small and the flow is

laminar .

5.3.7 Thermal

effects

When velocity L? f the pad is high, the temperature increase, due to the viscous

friction in the lubricant, should also be taken into account, as well as the inertia

forces. This kind of problem has been studied in ref. 5.9 (see also ref. 5.10).The en-

ergy equation must also be associated to the customary Navier-Stokes and continuity

equations, as well as a viscosity-temperature relationship; this last may be Eqn 3.4,

where it can be stated that

p=0.04

or most mineral oils.

For

an axial-symmetric configuration, applying the simplifying hypotheses

stated in section 4.3 o Eqns 4.7 (except that p is no longer considered

a

constant),

the Navier-Stokes equations may be written as:

+$)=o

For the same reasons, Eqn 4.8 is reduced

to:

(5.60)

(5.61)

The energy equation has already been obtained in section

4.6

Eqn

4.40).

Suitable boundary conditions should be associated to this set of equations €or

pressure, velocities and temperature. In particular, as far as temperature is con-

cerned, adiabatic flow is often considered (ref.

5.11,5.12).

In ref.

5.13

hermal ex-

change

is

considered between the lubricant and the pad surfaces held

at

constant

temperature. In ref. 5.14, nstead, an approximate evaluation of the boundary tem-

peratures is considered, obtained by means of experiments (ref. 5.15). The above

equations can now be integrated by means

of

an iterative procedure, based on the

finite difference method, to give

p , T , u ,

and

v ,

from which the load capacity, the

flow rate, and the friction moment can be calculated. Let us now examine a few

results.



108

HYDROSTATIC

LUBRICATION

The temperature field plotted in Fig. 5.10 refers to a lubricant, with a n inlet

temperature T=35'C, in the clearance of a bearing with the following characteris-

tics: r2=0.0657m, r1=0.0348m (r'=0.53),r0=0.006m, h=100 pm, &419 rads, pr=0.294

MPa; the data concerning the lubricant, at

2OoC,

are:

p=0.0645

Ns/m2,

p=857

Kg/m3,

c=1890

J/Kg"K,

~=0.144

W/m°K. The maximum temperature, at r=r2 and

z=0.54h,

is

T=5OoC

with an increase

AT=15OC.

In comparison with the case of isothermal flow,

taking into account inertia effects, pressure in the clearance proves to be slightly

lower. Moreover we have:

W=2190

N,

Q =13.8.10-6

m3/s,

Mf=2.22

Nm. So, compared

to the case of isothermal flow, in which we should have

W=2054

N,

Q =18.3.10-6

m3/s,

Mf=3.2

Nm, the load capacity proves to be

6.2%

lower, the flow rate

24.5%

greater, and the friction moment 30.6%lower. Moreover, in the case of isothermal

flow, from Eqn 5.7, we should have AT=45OC.

34 35 37 3 9 41 43 45 47 49 50

T

( O C )

Fig.

5.10 Temperature distribution in the clearance

of

a circula r pad: r l= 0.0348

m, r2=0.0657

m,

h=lO-4 m,

-19

r d s , r= 0.294 MPa.

As

parameter

Si,

iven by Eqn

5.50,

increases, the deviation from the isothermal

case becomes prominent. For example, for the pad considered above, let us now

suppose f2~628.3ads, pr=98.1 KPa, and hence Si=2.196. The maximum tempera-

ture increase now proves to be AT=45 Ca t r=r2 and z/h=0.8; along the upper pad

surface we have AT=31°C and AT=25.5 Calong the lower surface. The calculated

pressure field is shown in Fig. 5.11, where a negative (relative) pressure (confirmed

experimentally, ref. 5.15) indicates the possibility of cavitation phenomena. More-

over, we have

W=642

N,

Q

=14.6.10-6

m3/s,

Mf=2.25

Nm, and in comparison with

the isothermal case, the load capacity proves to be

33%

lower, the flow rate

68%

greater and the friction moment

55%

lower.



110


maximum A T proves t o be lower than 8°C; he pressure pattern (also shown in Fig.

5.11) is very close to the isothermal case, and so obviously occurs for the load capac-

ity W=642 N, while the flow rate Q=17.7.10-6m3/s is greater by 17.4%. In compari-

son with the previous pad, the load capacity

is

similar, but the flow rate

is

even

greater. A possible countermeasure may be a reduction in film thickness. If we take

h=50 pm, the maximum temperature increase proves to be AT=23'C, but the pres-

sure pattern is still close to the isothermal case (Fig. 5.11) and we have W=618 N,

Q=2.43.10-6 m3/s, Mf=0.821 Nm.

So,

due to the thinner clearance, the actual flow

rate proves to be about one 7th of what i t was before.

We must point out it is best to use pads with a small radius and a large recess,

when Q is large. The convenience of large recesses will be demonstrated in section

6.4

and in chapter 11.

All the above results agree closely with those obtained in ref. 5.11 and 5.12 in the

simplifying hypothesis of adiabatic flow. Reference 5.11 suggests that the possibility

of adopting the isothermal theory (ref. 5.6), instead of the adiabatic theory, depends

on the

"viscosity param eter"

where g is the acceleration of gravity. By way of example, for water-lubricated bear-

K '

Fig. 5.12 Correction parameters

kw. k~

and k~ for thermal regime.



PAD COEFFICIENTS

111

ings, V takes on values of about

compared

t o

the mineral oils) and the isothermal theory

is

fairly valid.

(due to the low viscosity and high specific heat,

Although the iterative procedure mentioned above can be quite accurate, it is

still very complicated and does not lead t o any direct expression of pressure, load

capacity, flow rate, and friction moment as functions of the parameters of the pad.

A simpler method is described in ref. 5.16,which is based on using the results con-

tained in ref. 5.14, o obtain an approximate expression of viscosity as a function of

r ,

in the thermo-fluid-dynamic regime. The Navier-Stokes and the continuity equa-

tions are then solved

to

give the pressure field; finally we obtain the following:

(5.62)

(5.63)

(5.64)

5.16

as functions of

(5.65)

and are plotted in Fig.

5.12.

n Eqn

5.65,

l = p ( r l ) ,

Q

, and H f are the values that

these parameters take on in the isothermal case.

The method has been tested for a number of bearings, with k* in the 0.211.3

range, and good agreement was found with the numerical results referred to above

(deviations were smaller than

5%

for

W

and M, nd smaller than 10% for

Q

1. It

should be pointed out that, for

k*<0.2,

the isothermal theory may be applied, while

k*=1.3 s

quite a high value.



112


4

li

Fig. 5.13 Annular recess pad.

5.4 ANNULAR RECESS PADS

5.4.1 Basic

equations

Figure

5.13

shows an annular recess pad; the inner hole is, in most cases,

crossed

by

a shaft: hence, in general, it is large.

From

Eqn

4.57,

the pressure in the pad is easily expressed:

In rlrq

p = p r

P = P r



PAD

COEFFICIENTS

113

In

r l r 2

P = P r

l-w)

(rl I

r

I rz)

Integrating the pressure field on the whole bearing area, we obtain load capac-

ity

W.

Bearing Eqn 5.1 in mind, we may express the effective pad area

as

follows:

where

(5.66)

(5.67)

A*,

is plotted in Fig. 5.14.a as a function of u ' = ( r 4 - r 3 ) / ( r 4 - r l ) ,

or

certain values of

r ' = r l l r 4

and for the most common case

r4-r3=r2-r1.

The hydraulic resistances of the outer and inner clearances are (Eqn 4.59):

Since the flow ra te

is

the

s u m

of Q3-4 and Q2-l, bearing Eqn 5.2 in mind, the total

hydraulic resistance of the pad is:

where:

1

In

r 4 f r 3

In

r 2 / r l

R*

=

1

+-

(5.68)

(5.69)

R*

is also plotted in Fig. 5.14.b.

The pumping power lost in the pad (Eqn 5.4), written as a function of the load, is

still

given by Eqn

5.24.

Given the inner and outer

radii

of the pad,

it

is now possible

to assess its "optimal" geometry, looking for the minimum of r*=l/(A*,'R*). f we

state that r4-r3=r2-r1,we find that the optimal value of a' is very close to 113 (see Fig.

5.14.d). However, this is not a critical value, and

a

larger recess (perhaps a'=0.2) is

often preferred in order

to

decrease supply pressure and friction power.



114

HYDROSTATIC

LUBRICAT/ON

0.0

' '

0.00

0.25

0.50

a'

- c -

1.01

0.

R'

0

0

0.25 0.50

a'

- d -

a'

Fig. 5.14 Coefficients of annular recess pad.



PAD COEFFlClENTS 115

-

r'

0.4

0.5

0.6

0.7

0.8

0.9

It should be pointed out tha t to configure the pad with equal length of the lands

is

by

far

the most common practice, since it is simple and give very good results, but

it is not, strictly speaking, the best choice from the point of view of pumping power.

Indeed, looking for the absolute minimum of

r*

t

is

possible to find the values of

ratios r3Jr4 nd rl lr2 that ensure the least loss of pumping power for any value of r';

the results contained in Table 5.1 show that the equal-length configuration practi-

cally gives the same results as the true optimal configuration. Other authors (ref.

1.8) prefer pads with equal radius ratios, i.e.

r3/r4=r1/r2,

n such a way that flow

rates crossing the inner and outer clearances are equal; Table 5.1 shows that this

practice also gives good results, especially a t the highest values of

r'.

The friction moment is, from Eqn 4.60,

General case

rllr2 r3/r4 r*

0.680 0.791 22.1

0.758 0.827 35.8

0.823 0.863 65.7

0.877 0.898 146.8

0.924 0.933 468.5

0.965 0.967 3552

where

r2-rI=r4-r3

a'

rl /r2 r3/r4 r*

0.337 0.664 0.798 22.1

0.335 0.749 0.833 35 .8

0.334 0.818 0.866 65 .8

0.334 0.875 0.900 146.9

0.333 0.923 0.933 468.5

0.333 0.964 0.967 3552

which

is

plotted in Fig. 5.14.c. The friction power is, hence,

r1/r2=r3/r4

rl11-2 r3/r4 r*

0:746 0.746 23.5

0.798 0.798 37 .1

0.845 0.845 67 .0

0.889 0.889 148.3

0.928 0.928 470.3

0.966 0.966 3555

(5.70)

(5.71)

(5.72)

The friction coefficient, referred t o the mean radius F=(r4+r1)/2f the pad, is

therefore:

(5.73)




Hydrostatic thrust bearings with annular recesses are frequently used, very

often coupled to hydrostatic journal bearings. When the turning velocity is not high,

pumping power is much greater than friction power; on the basis of the above re-

sults,

it

is

easy

t o

make certain elementary suggestions:

r' should be low. In practice, due t o design constraints, we almost always

find that

r 5 0 . 5 .

Values greater than r'=0.8 should be avoided;

a'

should be in the

0.2+0.4

range;

D, i.e. the bearing area, should be as large as is allowed by the other design

constraints;

The film thickness

h

should be

low;

The lubricant viscosity

p

should be high.

EXAMPLE 5.5

An annular recess bearing must sustain a load W=20 KN, rotating at n=2n

radls (60 rpm); its outer diameter is D=O.l m, the inner one is 2rI=0.04 m (i.e.

r'=0.4). For clearance h=50 pm is chosen, and for a' the near-optimal value a'=l/3.

The lubricant data at T=40OC are:p=O.l Nslmz, p=9ooKglm3, ~ 1 9 0 0lKgSC.

From Eqn 5.66 we get A,=4.40.10-3 m2 and hence a recess pressure pr=4.55

MPa is needed to sustain the load. The hydraulic resistance of the pad (Eqn 5.68) is

R=22O,lO9

Nslm5

nd the flow rate proves to be Q=20.7.10-6m31s. The pumping

power is Hi=94 W. For friction we f ind: MfzO.0857 Nm, Hf=0.54 W, f=O.l22.lO-3,

respectively. The temperature step of the lubricant, from the inlet to the outlet of the

pad, is (Eqn 5.7) AT=2.7%, which is practically negligible.

Comparing these results to those already obtained for a circular recess pad with

the same diameter (Example 5.1), it may be noted that the friction of the annular

recess pad is lower; slightly greater pressure is required, but the flow rate and

power consumption are more than 4.5 times greater.

EXAMPLE 5.6

Let us change the inner radius of the bearing in the former example to rI=0.03

m (i.e. r'=0.6), and the angular speed to R=4n rad ls (120 rpm). We obtain p,=5.97

MPa, Q=46.8.10-6m3/s , Hi=279 W, Mf=0.147 Nm, Hf=1.85

W,

f=0.18.10-3,AT=3.5oC.

In comparison with the former example, a small increase in pressure may

again be noted, while for flow rate and pumping power the increase is much larger.

EXAMPLE 5.7

I f the inner radius is again increased to rl=0.04 m,

r'

becomes

0.8;

et the angu-

lar speed now be Q=6n rad ls (180 rpm). We find: pr=10.6 MPa, Q=187.10-6 m 3 / s ,

Hi=1989 W, Mf=0.146 Nm, Hf=2.76 W, f=0.16.10-3 and AT=6.2 C. Temperature in -



PAD

COEFFlClENTS 117

crease may no longer be negligible: hence, it will be best to p u t a heat exchanger

in

the oil reservoir.

When this bearing is compared to the one in Example 5.5, one may note that

pressure

is

more th an dou bled, while a greater increase occurs fo r the flow rate an d

pumping power (the latter increases more than 10 t imes). It may be useful to in-

crease the viscosity of the lubricant: if p is doubled, Q and Hi are halved;

Mf

nd

H f are doubled,

too,

but they still remain quite low. Another way to reduce

Q

is to

use a lower value for a' (i.e. to reduce the w idt h of the recess); on th e contrary, a

greater a' causes a decrease in pressure. It should be pointed out , however, tha t, fo r

the highest values of

r',

a slight displacement of

a'

fro m its optimal value ca n lead

to a notable increase in Hi (see Fig. 5 .14.d).

5.4.2 Effec ts o f

errors i n

parallelism

The bearing pad

in

Fig. 5.15 is affected by a lack

of

parallelism

of

the surfaces of

the bearing components. We may state:

a h ; = % & * (5.74)

r4

Fig. 5.15

Misaligned

annular

recess

pad.



118


1

oo

0.95

CR

0.90

r'=0.4

0.6

0.8

0.85

0.0 0.5 1 o

Ah'

Fig. 5.16 Misaligned annular recess pad: correction factor

C,

versus parallelism error

Ah'

for two

values

of the land width ratio a'.

where

Ah'

is given by Eqn

5.31.

Bearing the results of section

5.3.3

in mind, we find

that, for a certain recess pressure

p r

and mean film thickness E , the only conspicu-

ous effect of misalignment is an increase in

flow

rate; such an increase may be

expressed as l/CR(Ah';

3 / r 4 )

or the external gap, and 1/cR(Ahi;

l / r z )

or the inter-

nal one. Thus, while the effective area

A,

may be considered to be practically unaf-

fected by a misalignment Ah'<1, the hydraulic resistance R may be expressed in the

form of Eqn 5.34; the correction factor

CR,

calculated as above, is plotted in Fig. 5.16

as a function of

Ah',

for certain values of r' and a'.

5.4.3

Effects

o f

pressure losses

at

the inlet

When the Reynolds number is greater than 100, the pressure losses at the inlet

should be taken into account. Applying the results of section 5.3.4 to both the gaps in

the annular recess pad, the whole hydraulic resistance of the pad should be cor-

rected as in Eqn

5.36,

where C p is given approximately by Eqn

5.39

(or by Fig.

5.61,

but the Reynolds number is

(5.75)

and

r ' ,

r

and A r must be substituted by the following:



PAD

COEFFICIENTS

119

-

r =

rl

+

r2

+

r3

+

r4

4

r2 rl + r4

-

r3

2

r

=

EXAMPLE 5.8

A pad bearing with an annular recess has

an

outer diameter D=O.l m; the

inner one is 2rI=0.08 m (r'=0.8).It has to sustain a load W=15 KN, with

a

recess

pressure pr lower than

8

MPa. The angular speed is Q=6.28 radls (60 rpm); the

main characteristics of the lubricant are:

p=0.03

Nslm2,

885

e l m 3 .

Let us select a'=0.3; h=100 pm. We immediately find A,=1.98.10-3 m2 and,

hence, pr=7.58 MPa, which is safely smaller than the maximum pressure allowed.

If the pressure losses at the inlet are disregarded, we also find R=1.91.109 Nslm5,

and Q=3.97.1

0-3

3I s.

From the above equations, we find Re=207, F'=0.935,

r=0.045

m, Ar=3 mm. A

correction factor Cp=1.28 can be obtained from Fig.

5.6.

Since the flow rate is in-

versely proportional to the hydraulic resistance, a new value is found: Q=3.10.10-3

ma ts , which is still very high. On the other hand, the friction power is very small,

compared to the pumping power (Hf=O.O4

W,

is23.5 Kw). I f such a low friction is

not a design constraint, it is clearly advisable to reduce the selected clearance and to

increase lubricant viscosity, in order to greatly reduce flow rate and pumping

power.

5.4.4

Turbulent

f low

When the Reynolds number

is

high, the

flow

becomes turbulent. The results

already obtained in section 5.3.5 can be extended

t o

the annular recess pad. In par-

ticular, since in most cases both

r l l r 2

and

r31r4

are greater than 0.8, the influence

on the effective bearing area can be disregarded. The hydraulic resistance of the

pad, on the other hand, can now be written as:

(5.76)

Correction coefficient C R ~s still given by Eqn

5.48

(also plotted in Fig.

5.7.b)

n

which parameter

zb

is now defined as follows:



120


The flow rate may prove to be much lower than expected on the basis of the

laminar theory, as in the case of the central recess pads, but the problems con-

nected with turbulence also remain the same.

5.4.5

Effects of the iner t ia forces

When the turning velocity of the pad is high, the forces of inertia in the lubri-

cant should be taken into account. The Reynolds equation (Eqn 5.49) can be solved on

the land surfaces to give the relevant pressure field, while the recess pressure is

hypothesized to be a constant (see ref.

5.17).

ntegrating the pressure field, we find

the load capacity of the pad, while the flow rate

is

obtained by integrating qr, as ob-

tained from Eqns

4.33,

along the recess boundaries.

Fig. 5.17 Annular recess pad: correction factorCR, ersus inertia parameterSi for certain values of

ratio

r'

and

of

land width ratio

a'.

As usual, Eqn 5.66 and Eqn 5.68 have to be substituted by the following

equations:

(5.77)

(5.78)

where



PAD

COEFFICIENTS

121

- a -

- b -

I.€

1.1

1.2

1.0

P

Pro

-

0.8

0.6

0.4

0.2

0

0.6

2

-

1

-

SI'O

~

11

I?

1.:

1c

P

Pro

a8

0.6

0.4

0.2

0

r =0.95

0.8

1

0.6

0.8 1

r

r

-

'4

r4

Fig.

5.18

Annular reces s pad: pressure distribution for one value or ratio r'=r1/r4nd certain

values

of inertia parameterS,. Note thatpm=(3/7r)pQ( /h3)ln(r2/r,)].

(5.80)



122


For the correction factor CAI,however, we always have 0.99<CA1<1 for the most

common range of pad parameters: r50.5 , a'<0.3, Si<l . The correction factor CRI s

plotted in Fig. 5.17. From this diagram, it is clear that the effect of the iner tia forces

is

negligible, except for the lowest values of

r'

(consider, moreover, that

Si=l

is

a

high value). The friction parameters are not affected by the inertia forces.

I t should be pointed out that the pressure variations in the recess a re not always

negligible. In Fig. 5.18 (ref. 5.18) the pressure in the pad is plotted a s a function of

the radius, for two pads directly supplied a t constant flow, in the case

of

r'=0.6 and

r,Jr2=r3/r4=0.845and in the case of r'=0.6 and r11r2=r3/r4=0.95. nertia effects pro-

duce considerable differences in pressure a t the edges of the recess. I t is clear that

the hypothesis

of

constant recess pressure is not adequate. Nevertheless, when W

and

Q

have

to

be evaluated, the above results prove

to

be accurate enough.

ExluMPLE 5.9

An annular recess thrust bearing with D=O.l m and rI=0.03 m (thus r'=0.6)

sustains a load W=12KN and rotates at Lk200n radl s

(6000

pm); the lubricant data

are (at T=4OoC) =0.05 Nslm2, p=890 Qlm3, c=1920J I Q " C . Let us choose a'=0.25

and h=100pm.

If we disregard the turning velocity, we easily find a recess pressure pr=3.18

MPa and

a

flow rate Q=533.10-6m31s. We have Si=0.041, thus from Eqn 5.79 and Eqn

5.80, it is easy to verifv that the effect of

Si

is, in this case, totally negligible.

The pumping power is Hi=1695 W; the friction parameters are Mf=1.40 Nm,

Hf=881 W, and f=0.0029; this last value is comparable with that of the rolling bear-

ings at this speed.

The temperature increase is, in the case of adiabatic flow, about 2.8 c, and may

be disregarded.

EXAMPLE 5.10

Let

us now consider a bearing sustaining a load W=2

K N

at a=600nra dls (i.e.

18000 rpm, a value which cannot be reached by usual rolling bearings). We have

D=0.06 m, rl=0.02 m (r '=2/3),a'=0.25, h=50pm. The characteristics of the lubricant

are p=O.Ol Nslm2, p==70IQlm3, andc=1930JlK&"C,at T=40 C.

The recess pressure for Si=O is p,=1.70 MPa, and hence Si=0.246. It is easily

established that the effect of the turning velocity is still negligible on pr and Q. The

flow rate proves to be Q=O.222.lO-3m3/ s, and

the

pumping power Hi=377 W. The

friction power is Hf=373 W, and f=0.0040. The increase in temperature is about

2.0 C. The Reynolds number in outer clearance is Reu=205 see Eqn 5.59), and hence

the

flow is still laminar.




5.4.6 Thermal effects

It has been shown for the central recess pads that , a t very high turning speeds,

temperature, and hence lubricant viscosity, vary considerably in the bearing clear-

ances. In the annular recess pads, however, the ratio of the inner t o the outer ra-

dius of each clearance is, generally speaking, higher, and this leads

t o

lower in-

creases in temperature. This is confirmed by the examples above.

5.5

TAPERED PADS

5.5.1

Basic equat ions

Figure 5.19 contains a sketch of two tapered pads fitted with central recess (a)

and annular recess (b).

- a - - b -

*

Fig. 5.19 Tapered pads: a- central recess; b-

nnular

recess.

The Reynolds equation for a conical clearance was obtained in section 4.3.2;

assuming steady sta te operation] and uniform clearance h , on all the land surface,

Eqn

4.26

is reduced to:

(5.81)




The term on the right-hand side accounts for the inertia forces due to angular veloc-

ity0 n the simplest case, it i s negligible, and hence Eqn 5.81 is reduced to Eqn 4.56,

which does not depend on the aperture angle

a;

thus the pattern of the pressure

field

is

exactly the same

as

for a plane clearance

( a = d 2 )

of identical radius ratio

r'.

Stating pr=l ,and integrating p.sina on the recess and land surfaces, we obtain

the effective area

of

the pad. Again it turns out

to

be the same as for the plane bear-

ings, i.e. Eqn

5.20,

where A: is given by Eqn 5.22,or Fig. 5.l.b, for

the

central recess

pad, and by Eqn 5.67, or Fig. 5.14, for the annular recess pad.

The flow ra te a t radius r is obtained by integration of the radial velocity:

(5.82)

Let us disregard velocity R for the time being. The hydraulic resistance of the ta-

pered pads may be written as follows:

(5 .83)

where R* is given by Eqn 5.23 o r Fig. 5.l.b, and by Eqn

5.69

or Fig. 5.14, respectively.

It follows that the flow rate through a conical clearance will turn out to be propor-

tional

to

sina. The moment of the friction forces for an axial-symmetric conical land

of length dr is

C r Q

dMf = 2 x- r3 sin3a dr

hn

which may be easily integrated to give the friction moment:

and the friction power:

(5.84)

(5.85)

HF is exactly the same as that already calculated for the plane bearings: i.e. Eqn

5.28 for the central recess pad, and Eqn 5.71 for the other.

When

a

tapered pad is compared to a plane bearing of the same diameter, with

equal values of the normal film thickness and the same radius ratios, it is easy to

see that:



PAD

COEFFICIENTS

125

the load capacity for any given recess pressure is the same (or ,

for

an equal

flow rate, is greater for the conical pads: actually, it is proportional t o

llsina);

the flow rate, a t any given load, is proportional t o sinq i.e. i t is lower for the

conical pads. The same Consequently, the same occurs for the pumping

power;

on the other hand the axial stiffness, which proves t o be inversely propor-

tional t o h=h,lsina, for any given load capacity, is lower (proportional to

sina) for the conical pads;

the friction power (proportional to llsina) is greater for the conical pads.

Furthermore, tapered pads have a greater axial size than the plane pads and

may prove to be more sensitive t o assembling misalignments.

EXAMPLE 5.11

An

annular recess tapered bearing has to sustain a load W=20KN, rotating at

Q=4a radls (120 rpm). The main geometrical parameters are: D=O.l m; L=0.02 m,

a=45'(i.e. rt=l-(2Ltga)lD=0.6).The viscosity of the lubricant is p=O.l N slmz at oper-

ating temperature.

I f

we select a'=113 and hn=50 pm

(or

h=70.7 pm), we easily obtain A50.427,

A,=3.35.103 m2, and hence p,=5.97 MPa; moreover, R*=O.0835, R=180.109 Nslm5,

and hence &=33.10-6 m3/ s . The pumping power is thus Hi=197 W. The friction

moment is easily calculated as Mf=0.208 Nm, while Hf=2.6 W. ompare these re-

sults with those in Example 5.6.

5.5.2

Effect of the inertia forces

As has been pointed out before, when Q is large, the inertia term should be re-

tained in Eqn

5.81.

This kind of problem is examined in ref.

5.17

for

the annular

recess pad, and i n ref. 5.19 for the central recess pad. In the latter, thermal effects

are also considered, as in ref. 5.13.

If we solve Eqn 5.81, and integrate the pressure field, we finally find that the

effective area

A,

is again given by Eqn

5.53

and Eqn

5.77

for the circular-recess and

the annular-recess pads, respectively.

In the same way, the hydraulic resistance will be:

(5.86)



PAD

COEFFlClENTS 127

essentially radial. In this assumption, the effective area does not vary; the flow rate

on a sector dfi of the conical surface is (see Fig.

4.5):

The clearance may be written in the following form:

h ,

=

C

o s a 1

cos

( 1 9- @]

[ "

where C&tga is the mean radial play. The flow rate is, hence:

x

x 3

21r

Q =

J d Q = G z p r sin4a(l+t$)

0

Thus correction factor C R x

for

the hydraulic resistance can be evaluated as:

(5.89)

Friction is also affected by a radial displacement x . Proceeding as above, we can

assess a correction factor

Cfxfor

M f

and

Hf,

hich may be written as:

1.21

1.1

-

1

o

0.9

0.8

-

0.7

-..

0.0 0.1 0.2 0.3 0.4 0.5

X/C

Fig. 5.21 Non coaxial tapered pad: correction factors

C R ~

or the hydraulic resistance and Cfx

for

friction versus radial displacement

xlC.



128

Cfr

=

1

<


(5.90)

It is clear tha t a radial displacement, comparable to the design value of the film

thickness, greatly affects the hydraulic resistance of the pad, and consequently the

performance of the bearing. This point should be carefully considered when design-

ing. For example, if a conical thrust bearing sustains a shaft which is also loaded

in a radial direction, the stiffness of the radial bearings will interact with the axial

load capacity.

The case of Fig. 5.20.a (tilting error) is examined in ref. 5.20 by means of numer-

ical integration (finite differences) of Eqn

5.87.

The effect of a relative tilt

1y'=1yDl(2h,)10.6

on the hydraulic resistance

(at

low speed) is found to be always

smaller than for a flat bearing (a=90 ).n particular, it is negligible for

a=60 +75 ,

while a=45 is an intermediate case.

When the speed

is

high, the hydrodynamic effect, due to the uneven clearance,

may in part counterbalance the effect of the tilting error, and

a

tilted bearing may

even prove to perform better than

a

centered one

5.6

SPHERICAL

PADS

To overcome the problems connected with the tilting errors,

a

spheri-

cally-shaped pad (Fig. 5.22) could be used, instead of the flat or tapered types.

A s

usual, to calculate the performance of the pad, the relevant Reynolds equa-

tion must be solved to find the pressure in the clearances. Such an equation has

been obtained in section

4.3.3.

For an axial-symmetric spherical clearance, Eqn

4.30

is reduced to:

The above equation is easily solved when E=O (concentric configuration), giving:

Pr

tan (~112

In

(5.91)

for the central recess pad, with p=pr for p q l andp=O for p q 2 . A general solution

may however be obtained (ref.

5.21,

5.22), although the relevant equations are quite

cumbersome.



PAD COEFFICIENTS

129

- a -

- b -

Fig. 5.22 Spherical pads: a- central recess; b- annular recess.

The load capacity is calculated by integrating the pressure on the recess and

land area.

For

E=O, we obtain:

(5.92)

From the first of Eqns 4.28; the velocity

of

the fluid in the tangential direction is:

u = -

* ( @ - h , @

D P a P

and hence the rate

of flow,

for

E=O,

is:

(5.93)

Load and flow

rate

can be written in the usual

form

(i.e. Eqn 5.1 and Eqn

5.21,

where the effective area is:

A,

=

x 0 2 sin2q2A*,

(5.94)

and A*,,calculated from Eqn 5.92, is plotted in Fig. 5.23.a.



130

2

A',

Hi

1

t

HYDROSTATIC LUBRICA

TlON

0.8

0.6

0.4

I"

60"

90"

(92

- c -

0.8

0.4

0.0

60" 90"

(92

30'

- d -

0 . 1 5 - 7

0.05

1

0.00' ' '

.

' . ' '

30"

60" 90"

(92

Fig. 5.23 Spherical pads. Pad coefficients

A*,,

R *

and

H,?

(in the case of

E=O)

versus included

angle (p2

for certain values of ratio (p1/(p2:

a, b-

central recess pad; c,

d-

annular recess pad with

(pa=O.25'((p2-(pl).



PAD

COEFFlClENTS

131

The hydraulic resistance may be written as:

(5.95)

Again R* is plotted in Fig. 5.23.b.

The friction moment is:

in which the shear stress

should be substituted. As usual, we may write the friction moment and the friction

power in the following form:

M f o =

2

$

f2 4

in4q2H,F

(5.96)

where HT (which is also plotted in Fig. 5.23.a) is:

(5.98)

Unlike for the plane pads, A*,cannot be considered as a constant] bu t depends on

E especially when

cp2 is

great and q1/cp2 is small.

Also

Eqn 5.12 and Eqn 5.13 cannot

be considered valid for large included angle q2. For instance] Fig. 5.24 (ref.

5.22)

shows how load capacity and flow rate are subjected

to

change with E for a pad with

cp2=85". It is easily seen tha t for negative displacements the effective area may expe-

rience a severe reduction] while the increase of hydraulic resistance is smaller

than the cubic trend predicted on the basis of Eqn 5.12. Depending on the supply

system, even negative stiffness may occur, and the bearing may prove to be unstable

(ref. 5.23). However, this kind of problem may be prevented by using q1/q2 ratios in

the order of 0.5 and values of q2S75".

A peculiar type of central-recess spherical pad is the "fi tted' type, characterized

by having a null clearance in the concentric configuration.

It

is obvious that most of

the foregoing equations now become meaningless, and it is necessary to write load



132

1.0

0.8

w

2

.6

0

2

n

a

5 0.4

0.2

0


- a -

-

b -

0.2

0.4

0.6

a8 t o

?

1

0 0.2 0.4 0.6 0.8

4 ’%

Fig. 5.24 Sphe rical pad with cenh-a1 recess: a- load factor W.4/(zDzprsin&p) and b-

flow

factor

Q-pl[(l+~)%?p,]

ersus ratio q1/(p2or a number of values

of

eccentricity

E

(ref.

5.22).

capacity and flow rate as functions of the displacement h=h,(cp=O). We obtain (ref.

5.24):

x tan2q2 - tan2cpl

W

=4 D2

p r

tan 472

tan2472

-

tan2ql + 2 In-

tan

471

& = - -

h3 p

1

tan472

tan2472

-

tan2cpl + 2 In

tan471

3 P

Torque is found to be (ref. 5.21)

Mr g f

RD4 ( ~ 0 ~ 2 4 7 ~cos2ql -

2

ln-

Spherical bearings have also been studied from the point

of

view

of

the effects of

fluid inertia (ref.

5.21)

and of change in viscosity due

t o

adiabatic flow (ref. 5.25). It



PAD

CO€FF/CENTS 133

has been shown that high values of the inertia parameter Si defined in Eqn

5.50

(perhaps Sp0.25) are detrimental for divergent film shapes, i.e. for negative eccen-

tricities: load capacity may be notably reduced, and even cavitation is likely

to

occur.

On the contrary, for converging film shapes (e.g. for the

fitted

type of bearing) load

capacity is improved and, due to the large pressures developed near the edge, bear-

ings with a wide included angle

(p2

also feature a notable radial load capacity.

The annular recess pads (Fig. 5.22.b) feature an effective area

W l p ,

that does not

vary much with eccentricity, especially when cp z is small. Proceeding

as

above, we

can calculate the relevant values of parameters A:

R*

and HF

to

be inserted in the

equations from Eqn 5.94 to Eqn 5.97. They are plotted in Fig. 5.23.c and Fig. 5.23.d.

For

A*,it

may be taken approximately (see also ref. 5.26):

(5.99)

at least when (pa is smaller than 0.25((p2-cpl).

5.7 RECTANGULAR

PADS

For a rectangular pad, such as in Fig. 5.25, the two-dimensional Reynolds equa-

tion Eqn 4.15 has to be solved, which, if h=const or

U=O,

is

simply reduced to Eqn

4.31 (Laplace equation). The boundary conditions are, as usual,

p = p r

at the inner

boundary of the clearance, and p=O

at

the outer one. Once the pressure on the land

area has been evaluated, the load capacity

is

obtained by integration. The rate of

flow, on the other hand, is given by the following:

(5.100)

where

S

is the inner boundary

( o r

any closed contour including it), and

n

is

the

outer normal direction of

S

(see also section 4.5.2).

In spite of

its

simplicity, Eqn 4.31 may be solved, in the general case, only by

numerical methods.

A first

rough approximation can be very easily obtained by

partitioning the land area as in Fig. 5.26 (ref.

1.8).

The pressure

is

assumed

to

vary

with a linear trend (see section 4.7.1) along the

.z coordinate, for parts [ l l and 121,

and along z for parts

131

and [41. In the corners, a logarithmic trend is assumed (see

section 4.7.4); if we have ri=O, the corners may be neglected altogether.

Calculations for load and flow rate are now straightforward: on the analogy

with the other pad types, the effective area

A,

and the hydraulic resistance R may be

written as:



134


4

J

Fig. 5.25 Rectangular pad.

A , =

L B

A

R=$R*

with

a ' l B ' +

2

rib'

In l+-

a'

4 ri 6'

(a'

+ r ib'

B')

+

Za'

( r i b

B )

R*

= 6

n:

In

( +-

r l E ' B )

where

(5.101)

(5.102)

(5.103)

(5.104)

a'

-a b

B ' = -

ri

and

q'=Z

- L ; b ' = B ;



PAD

COEFFlClENTS

135

Fig.

5.26 The assumed shape of

a

rectangular pad.

A

more refined approach

is

explained in ref. 5.27, in which an approximate

solution of the Laplace equation is searched for as a linear combinationp=Xc jq j of n

harmonic functions qJx, z ) ; coefficients

cj

are determined by imposing that the

mean square error in certain points

( x k ,

Zk) of the boundaries (on which pressure

is

known) is the least possible. In other words, the following system of n equations

must be solved

If a certain skill

is

used in selecting the harmonic functions, only a few terms

are needed (perhaps n=4) to obtain a good approximation of the pressure field: the

relevant error can be easily checked, since it can be shown that it is maximum on

the boundary.

Once a closed form solution has been obtained forp , load capacity is calculated

by integration, while flow rate

is

simply proportional to coefficientc 1 (this is another

consequence of the appropriate selection of functions

9).

A t present, the solution of Eqn 4.31 is based on numerical methods, such as

finite-difference or finite-element methods (for instance see ref. 5.28, 5.29, 5.30): in

particular, the charts in Fig. 5.27, 5.28 and 5.29 (taken from ref. 5.31 and ref. 5.32)

are calculated by means of a finite-difference method.

The pumping power dissipated in the bearing is still given by Eqn5.4. As for the

circular pad, it is useful to write

Hi

as a function of the load capacity:

(5.105)

r*also is plotted in Fig. 5.29; it appears that an “opt imal”value exists for

a‘,

n

connection with the minimum of the r*curves. Concerning the effect of the inner



136 HYDROSTATIC LUBRICATlON

3

R'

2 -

1

0

- a -

F]

.

0.5

0.25

0.

0.

0.0

0.2 0.4

a'

- b -

0.3'

. ' ' ' . . '

0.0

0.1

a'

Fig. 5.27 Rectangular pads: pad coefficient

keversus

land width a'.

0.2

- b -

1.5

1 o

0.5

0.5

0.0

0.1 0.2

a'

Fig. 5.28 Rectangular pads: pad coefficient

R*

versus

land width a'.

fillet radius ri, although this yields a lower load capacity,

it

appears to be effective

from the point

of

view of power. On the contrary, the outer radius

r,,

which is not

considered in the charts above, has negative effects on both A,

and H i , and should

be avoided (however, when r,+r,lBs0.25, it has no practical influence on the per-

formance of the pad).

The friction force on the pad, due

to

a

sliding velocity

U,

s (from

Eqn

4.35):



PAD

COEFFICIENTS 137

r

r

- b -

8

0.25

0.5

6 -

5 -

4 -

.J

0

o

0.1

0.2

a'

Fig. 5.29 Rectangular pads: pad coefficient r*

ersus

land width

a'.

(5.106)

and, consequently, the friction power is:

The effective friction area

A f

coincides, as a rough approximation, with land area

Al;

as a matter of fact, the contribution of recesses

to

friction is often small, since

recess depth h, is much greater than h. However, this is not always true; the

flow

recirculation in the recess can make recess friction comparable to land friction (and

even greater,

for

turbulent recirculation). Moreover, h, should not be too high for

reasons connected with the dynamic behavior (see chapter 10) and, when the speed

is high, the lands should be narrow and, hence, the recess area greater than the

land area.

A better approximation

for

the effective friction area may be written in the fol-

lowing form:

A t = Al

-I-,

A , =

= L € ? { l - ( l - f , ) ( l - 2 ~ ' ) + B ' ( 4 - d [ ( l - f r ) ( 1 - 2 8 ; )'

2 r i 2 - r i 2 ] ]

(5.108)

Coefficient f, can be calculated as indicated in section 4.5.3.



138 HYDROSTA TIC LUBRICA TlON

EXAMPLE 5.12

A rectangular pad has to be designed, able to carry a 45 KN load. The width of

the pad should be B=O.I m; the recess pressure should be lower than 4 MPa and the

friction coefficient lower than 10-4 at a speed U=0.2 mls . The viscosity of the lubri-

cant to be used is p=0.03 Nsl m2 .

Due to the constraint on pr, the effective area (Eqn 5.1) must be greater than

11.3.10-3 m2. Figure 5.27 shows that, for values of a’ near to the “optimal”,A: takes

on values greater than 0.55; consequently, we may take LB=0.02 m2, i.e. L=0.2 m

and B’=0.5. If

we

select a’=0.125 (i.e. b’=0.5) and ri=0.5, in Fig. 5.27 and Fig. 5.28 we

read Az=0.6, R*=0.77; hence A,=0.012

m2,

pr=3.75 MPa. The land area is easily cal-

culated as A1=0.013 m2; from Eqn 5.106 it follows that, for f d O - 4 , it must be h>17.3

pm: a very small value. A more suitable selection may be h=30

pm;

now Eqn 5.104

gives R=0.86.1012Nslm5 and hence &=4.4.10-6mats; the pumping power dissipated

in the bearing clearaAce is Hi=16.5 W . A more accurate calculation o f the effective

friction area (Eqn 5,108) gives, for hr=0.5 mm, Af=0.015 ma; the friction power is,

hence, Hf=O.6 W (Eqn 5.107) and the friction coefficient is f=0.67.lfY4. It should be

stressed that in th is case a relatively small recess has been selected since the speed

was quite low: indeed, the friction power turns out to be much smaller than the

pumping power. For higher speeds, a larger recess would have been a more suit-

able selection. This kind of problem, extensively dealt with in chapter 11, will also be

considered in section 6.4.

5.8

CYLINDRICAL PADS

Cylindrical pads may be used to support a shaft in a radial direction. They may

be thought of as extensions of any flat pad shape (e.g. the pad with a circular pro-

jected area in Fig. 5.30.a), although the rectangular shape is by far the most com-

mon (Fig. 5.30.b). It must be pointed out that a single-recess cylindrical pad is prac-

tically unable to sustain side loads; this greatly reduces the usefulness of such pads,

unless they are a part

of

a multipad bearing system (see section

7.5).

The study of these pads is made, as usual, by finding a solution for the relevant

Reynolds equation, i.e. of Eqn 4.18. n that equation, the right-hand side accounts

for the effects of the turning velocity of the journal and for the squeeze effect under

dynamic loading. Film thickness h depends on the displacement of the journal

center:

h = C [I- E C O S ( ~ - ~ ) ]

(5.109)

in which

C

s the film thickness in the concentric configuration

E=O

(see also Fig.

4.4).Once the pressure field on the pad lands is known, both the load capacity and

the rate of flow are easily evaluated, as in the preceding sections.



I W

PAD COEFFICIENTS

- a -

139

- b -

l Q

.i

Q

Fig.

5.30

Cylindrical

pads: a-

circular

recess; b- rectangular recess.

Even in the simplest case (E=O,

&=O,

h = C) , in which the right-hand side of Eqn

4.18

vanishes, a numerical solution is needed. In Fig.

5.31, 5.32

and

5.33,

he pad

coefficients are plotted, for certain rectangular shapes, obtained by means

of

a finite

element approximation. Other data are available in the literature: e.g. in ref.

5.33,

where they are obtained by means

of

the electric analogue technique (ref.

5.34).

As

in the case of the flat pads, one may write the effective bearing area and the

hydraulic resistance of the pad (for E=O) in a form:

a

A,

=

L D sinTA*,

(5.110)

R ~ = & R *

(5.111)

Certain notable differences should, however, be stressed; first, the reference config

uration

is

no longer arbitrary: a natural reference point exists, in which the centers

of curvature

of

the

two

pad surfaces coincide, and

h=ho=C.

Coefficient

A*,

is not in-

dependent from the journal displacement (however, this dependence is in most

cases negligible). The hydraulic resistance is no longer exactly related

t o

the dis-




1.o

A ' , '

0.6

0.4

o . 8 #

placement by Eqn 5.12,which, however, remains approximately valid for small in-

cluded angles

a

an d small values of displacement E = e / h o n the normal direction

rp=O.

As for other types of pad, it is convenient to write th e pumping power Hi=p,Q as

a function of the load, in order to try to obtain an optimization of the pad design:

60"

75

\

- a - - b -

0.4' . ' ' ' . . '

0.0

0.2

0.4

a'

Fig. 5.31 Cylindrical pads. Pad coefficient Heversus land width a'.

- a -

R'

0.0 0.1 0.2 0.3

a'

a'

0.0 0.2 0.4

a'

Fig. 5.32 Cylindrical pads. Pad coefficientR* versus land width

a'.




In particular, given L,

D,

and a, he relevant "optimal" value of a ' may be readily

evaluated by means of the plots of T*=lfR*A*,n Fig. 5.33.

The friction moment, in the concentric configuration E=O, s:

The friction power is, consequently,

(5.112)

(5.113)

The effective area Af may be calculated as for rectangular pads; namely, as a rough

approximation,

Af

coincides with the developed land area; if instead the recess fric-

tion is also taken into account, Afmay be written as:

(5.114)

The friction factor

f ,

is evaluated as in section 4.7.3; owever, in most cases one may

simply take f,=4C/h,.

- a -

u=90"

2

0.0 0.1 0.2 0.3

a'

- b -

6 -

r*

4

t

2 '

0.0 0.2 0.4

a'

Fig.

5.33

Cylindrical pads. Pad coefficient r*versus land width a .

5.9

HYDROSTATIC

LIFTS

A particular application (ref. 1.5) of externally pressurized lubrication consists

in relieving large hydrodynamic bearings in certain critical operations, such as

starting or stopping, in which, due to insufficient velocity, the lubricant film would

be ruptured, leading

t o

high friction and fast wear. In certain applications, in



142

HYDROSTATIC LUBRICATlON

which the torque available is not much greater than the normal regime torque,

starting can even prove t o be impossible without the aid of a hydrostatic lift. This

kind of bearings (Fig. 5.34) are often referred to as "hybrid" journal bearings; it

seems, however, more appropriate to define them as hydrodynamic bearings with a

hydrostatic lift, since external pressurization is generally confined to low speed

running, while it is removed during normal running.

)Wa-

Li

IQ

IQ

II

I - b -

lw

Fig. 5.34

Hydrostatic

lift:

a-

axial recess; b- circumferential recess.

Behavior of such a "hybrid" bearing (studied for instance in ref. 5.35-5.37) is

beyond the scope of a work on hydrostatic lubrication; we shall therefore present

results relevant to the hydrostatic lift alone.

Due to the large included angle (we may assume a=180°) and to the smallness of

the recess, Eqn 5.12 is clearly far from representing the actual increase of hydraulic

resistance with eccentricity, and even the effective area can no longer be considered

as

a

constant. In Table 5.2, values are reported of the nondimensional load capacity

for eccentricities from E=O &=0.8,assuming constant-flow direct supply, obtained

by means of a finite element program (ref. 5.38). Several sizes of recesses have been




considered, developed in the axial direction (small a,,Fig. 5.34.a), as well as in the

circumferential direction (small

I ,

Fig. 5.34.b).

Table

5.3

also contains an evaluation of the ratio of recess pressure

p ,

t o mean

pressure W I L D . It should be noted, however, that the actual recess pressure can be

notably higher; in particular in the case of axial recess (Fig. 5.30.b), at starting the

recess may be shut

off as

a result of elastic deformation and hence the pressure can

be very high until the journal rises: values up to five times the normal running

pressure are reported (ref. 1.5). It is good practice t o use a pressure relief valve in

order to protect the pump, limiting the maximum pressure.

Concerning the dimensions of the hydrostatic pocket,

it

is generally suggested

that i ts area LDl'sina, is 2 .54% of the projected area LD. Indeed, from Table 5.3 it

is clear that larger recesses have no practical advantage from the point of view of

ratio

W/Q,

hile the advantage in terms

of

pocket pressure is small; furthermore,

the interference of recess with the hydrodynamic pressure pattern should be borne

in mind.

T A B L E 5.2

Iydro:

VD

1

1

1

1

1

0.5

0.5

0.5

0.5

0.5

1

1

1

1

-

-

tic lift:

-

1'

0.1

0.1

0.1

0.2

0.2

0.1

0.1

0.1

0.2

0.2

0.5

0.5

0.75

0.75

-

-

a

L

2 4 O

4 8 O

72"

24"

48"

24

48

72"

24"

48

6"

12O

6'

12"

-

on-dimensional load capacity.

W

D L p Q / C 3

E = O

~~

1.03

0.99

0.93

1.01

0.96

0.66

0.64

0.61

0.65

0.62

0.88

0.87

0.65

0.61

E

= 0.3

2.36

2.19

1.95

2.31

2.14

1.73

1.60

1.44

1.68

1.56

2.08

2.05

1.57

1.54

E =

0.6

7.88

6.67

5.17

7.73

6.50

7.18

6.07

4.74

6.98

5.90

7.34

7.22

5.82

5.67

E

=

0.8

29.6

20.3

12.3

29.1

19.8

35.2

24.1

14.3

34.3

23.3

31.0

30.0

26.1

25.1

E =0.9

88.2

44.6

20.6

87.3

43.5

130.8

64.7

28.4

127.6

62.4

115.5

106.6

102.3

94.3



144

T A B L E

5.3

[ydrol

UD

1

1

1

1

1

0.5

0.5

0.5

0.5

0.5

1

1

1

I

-

-

itic lift:

1'

0.1

0.1

0.1

0.2

0.2

0.1

0.1

0.1

0.2

0.2

0.5

0.5

0.75

0.75

-

-

HYDROSTATIC

LUBRICATION

-o

non-dimensional

recess pressure.

a

24

48"

720

24"

48"

2 4 O

48"

720

Z"

48

6

12

6

12

-

E = O

3.38

2.73

2.36

2.97

2.42

3.97

2.95

2.41

3.55

2.65

2.76

2.56

2.40

2.23

E = 0.3

3.75

2.95

2.49

3.26

2.58

4.15

3.06

2.47

3.72

2.74

3.05

2.81

2.63

2.42

PI

W I D L

E

=

0.6

4.53

3.38

2.73

3.85

2.90

4.55

3.25

2.58

4.05

2.90

3.66

3.31

3.12

2.82

E = 0.8

5.85

4.02

3.06

4.81

3.34

5.21

3.52

2.7 1

4.55

3.11

4.72

4.13

3.92

3.43

E = 0.9

7.35

4.67

3.38

5.85

3.77

5.88

3.80

2.82

5.08

3.30

5.99

5.03

4.83

4.05

EXAMPLE 5.13

The following are the data of a hydrodynamic journal bearing: D=0.5 m, L=0.4

m, C=0.3 mm, W=300 N. The lubricant is a

SAE

30 oil (p=O.l Ns lm 2 at operating

temperature). Let us try to design

a

suitable hydrostatic lift.

First, one may establish the dimensions of the recess.

For

instance, we may

select

1=60

m m (i.e. l'=0.15) and a circumferential width b=100 m m (ar=23?. The

flow rate and pressure

o f

the pump can be calculated once the working eccentricity

has been selected. If

we

choose

&=0.8

corresponding to a minimum film thickness

h=60

pm), by interpolating data in the relevant column

of

Table

5.2,

he flow rate

turns out to be

Q=WC31(32~DLp)=0.013~103

31s. From Table 5.3 we may obtain a

recess pressure pr=Wl(0.19.LD)=7.9 MPa; as noted above, the l ifting pressure will be

much greater, and the value above should be considered as only a rough indication.

5.10 SCREW AND NUT ASSEMBLY

This particular kind of bearings (ref.

5.39),

shown in Fig. 5.35, is dealt with in

the usual way, i.e. by writing down the Reynolds equation in a suitable coordinate

system, and solving it for the pressure field. Then, to find the load capacity and the

rate of flow is straightforward: e.g. see ref.

5.40

and ref.

5.41.



PAD

COEFFlClENTS 145

From the point of view of load capacity, every turn of a screw flank may be re-

garded, with a very good approximation, as a tapered annular-recess pad (section

5.51, i.e. the effective bearing area

is:

x

A, =

n 4 2

A*,

(5.115)

Fig. 5.35 Hydrostatic screw and nut.

where n is the number of active turns and A*,is given by Eqn 5.67,or Fig. 5.14.

In evaluating flow rate, on the other hand, the effect of the helix angle, which

actually increases the length of the recess boundaries in relation

t o

the analogous

tapered pad, should be taken into account as follows:

(5.116)

where

is the mean value of the helix angle of the flank

2. = atan(p*/2xr)

and R* is given by Eqn 5.69, or Fig. 5.14. Note that the actual film thickness is

h,=h

cose cod , which, strictly speaking, varies with r; however, Eqn 5.116 retains

a very good approximation for all the usual values of ratio r l / r 4 and of the screw-

pitch (i.e. r1/r4<0.6, c l O ) . Another approximation made in obtaining Eqn 5.116 is

that no lubricant leaks out through the clearance at the ends of the recess: this

amount is generally negligible (ref. 5.421, except in certain special cases, e.g.

partial-arc hydrostatic nuts.

In addition to the misalignments already mentioned in the case of tapered pads,

an error

of

pitch between screw and nut may considerably affect the performance of




Ting L. L., May er

J.

E.; The Effect of Temperature and Inertia on Hydrostatic

Thrust Bearings Performance; ASME Trans.,

J. of

Lub rication Technology,

Kapur V. K., Verma K.; The Simultaneous Effects of Inertia and Tempera-

ture on the Performance of a Hydrostatic Thrust Bearing;

Wear,

54

(19791,

Kennedy

J. S.,

Sinha P., Radkiewicz Cz. M.; Thermal Effects i n Externally

Pressurized Conical Bearings With Variable Viscosity; ASME Trans. ,

J.

of

Ba ssa ni R.; I Cuscinetti Idrostatici di Spinta in Regime Termofluidodinami-

co;

1st AIMETA Congr., Udine, 1971;p. 25-68.

Bassani R.; Ricerca Sperime ntale sul Regime Term ofluido dinam ico nei

Cuscinetti Idrostatici di Spinta;Atti 1st. Mecc. Appl. Costr. Macch., Univ.

di

Pisa,

Anno Acc. 1968-69,

N.

9; 40 pp.

Bassani R.;

Proporzionamento dei Cuscinetti Idrostatici di Spi nta Ro tant i a

Velocitd Anche Elevate; Att i 1st. Mecc. Appl. Costr. Macch., Univ. di Pisa,

P r a b h u T .

J.,

Ganesan

N.;

Characteristics of Conical Hydrostatic Thrust

Bearings under Rotation;Wear,

73

1981), 95-122.

Bassani R.; Cuscinetti Zdrostatici di Spinta a Recess0 Anulare; Atti Dip.

Co str. Mecc. Nucl., U niv. di Pisa, DCMN 003(87), 1987; 37 pp.

Sa l em E., Khali l F.; Thermal and Inertia Effects i n Externally Pressurized

Conical Oil Bearings;

Wear,

66

1979), 251-264.

Prabhu T . J., G ane san N.; Non Parallel Operation of Conical Hydrostatic

Thru st Bearings;

Wear,

86

(1983),29-41.

Dowson

D.,

Taylor C. M.;

Fluid Inertia Effects

in

Spherical Hydrostatic

Thru st Bearings; ASLE Trans.,

10

1967), 316-324.

Ragab

H.;

Pe4ormance

q f

Spherical T hrust Bearings;Wear,

29

(1974)) 11-20.

O'Donoghue

J. P.,

Lewis G.

K.;

Single Recess Spherical Hydrostatic Bear-

ings;Tribology,

3

19701,232-234.

Sa sak i T., Mori H., Hirai A.; Theoretical Study of Hydrostatic Thrust Bear-

ings; Bull. JSME,

2,5

19591, 75-79.

Salem E. , Khal i l

F.;

Variable-Viscosi ty Effects in Externally Pressurized

Spherical O il Bearings;

Wear,

50

(19781,221-235.

O'Donoghue

J.

P.; Design of Spherical Hydrostatic Bearings; Mach. Prod.

Eng ineering , Oct. 21, 1970;p. 660-665.

M assa E.;

Su lla Determinazione della Forza d i Sostentamento e della Portata

nella Lubrificazione Idrostatica; Apparecch ia tu re Id rau l i che e Pneuma-

tiche, 3,16 (1963),31-39.

Caste l l i V., Sha piro

W.;

Improved m ethod for Numerical Sol utio ns of th e

General Incompressible Fluid Film Lubrication Problem; ASME Trans. ,

J.

of

Lubrication Technology,89 (19671,211-218.

Reddi M.

M.;

Finite Element Solution of the Incompressible Lubrication Prob-

lem;

ASME Tran s,

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Lub rication Technology,

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1975),72-78.

93

19711,307-312.

113-122.

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A n n o

ACC.1968-69,N. 12;

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pp.

6.11

6.12

6.13

6.14

6.16

6.16

6.17

6.18

6.19

6.20

6.21

5 s

6.23

624

5.26

526

527

6.28

5.29

6.30

6.31

Bass ani R.; Rectangular Hydrostatic Bearings;

Ann.

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19

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148

532

5.33

5.34

5%

5.36

537

5.38

539

5.40

5.41

SA2

5.43


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73, N. 45; 29 pp.

Rippel H. C.; Hydrostatic Bearings. Part

8:

Cylindrical-pad Performance;

M ach ine Design, Nov. 7, 1963; pp. 189-194.

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Throu gh the Use of the Electric Analog Field Plotter; ASLE Trans. , 19 (19751,

Hel le r

S.,

Shapiro W.; A Numerical Solution for the Incompressible Hybrid

Journal Bearing with Cavitation; ASME Trans. , J . of Lubrica tion Technol-

S o H., Chen C. R.; Characteristics of a Hybrid Journal Bearing with one

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1:

Dyn amic Considerations; Tribology In t., 18

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331-339.

S o

H. , Chang T.

S.;

Characteristics of a Hybrid Jou rna l Bea ring w ith one

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2:

Thermal Analysis;

Tribology Int. ,

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St racc ia

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Hydrostatic Lead Screws; Machine Des ign ,

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Ingegn eria, 1981; pp. 213-224.

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1,7 (19811,62-72.



Chapter 6

SINGLE PAD BEARINGS

6.1 INTRODUCTION

Pad coefficients, obtained in chapter 5 for several types of hydrostatic pad bear-

ings, are not enough to completely define the static behaviour of a bearing, since it

also depends on the behaviour of the supply system. In the simplest case, the pad is

directly fed by

a

pump at

a

constant

flow

rate

Q

and supply pressure practically

coincides with recess pressure (that is

p s = p r ) ;

he performance of the bearing is

then easily assessed by means of equations from Eqn 5.1 to Eqn

5.8.

On the other

hand, in the case of

a

compensated supply,

pr

is smaller than

p s

and a further rela-

tionship between these pressures is needed, which depends on the characteristics of

the compensating restrictor.

In this chapter the steady-state performance of thrust bearings is studied, when

these are directly fed by a pump as well as by means of several types of compensat-

ing devices. Finally,

a

number of remarks on the optimum design of these bearings

are made and design procedures are proposed.

6.2

DIRECT

SUPPLY

6.2.1 Bearin g perfo rm ance

When a pad bearing is directly fed by a volumetric pump, flow rate Q may be

said

t o

be constant; moreover, if the hydraulic resistance of the lubricant supply

ducts is

low

compared to the hydraulic resistance

R

of the clearances

of

the pad, the

recess pressure

is

nearly equal

to

the pressure a t the pump: p r = p s . The equations

examined in section

5.2

can now be rearranged as follows:



150


w(t”

A;

R~R’

p =

ps

Q =

p,.

Q = Q2

R

=-

(6.1)

(6.3)

H f

=

Hfo H i

(5.11rep.)

In the above equations, the subscript “0“ refers to the “design configuration“ in

which the film thickness is

h=ho

for all the land surfaces.

R ’

and

H i

are non-di-

mensional functions of the non-dimensional displacement and are given by Eqn

5.12

and Eqn 5.13. These equations are, strictly speaking, only valid for the plane pads,

in which the lands bounding the lubricant film always remain parallel to each

other; however, they c a n also be used as an approximation for the other pads when

small eccentricities are involved. In the rest of this chapter we shall assume that

Eqn

5.12

holds good.

Combining the foregoing equations, it is possible to study the behaviour of the

clearance and of the power consumption when the load varies in relation t o the ref-

erence value

W o = A, Q Ro .

Figure

6.1

shows

a

plot on non-dimensional film thick-

ness, pumping power and friction power as functions of the non-dimensional load.

The bearing stiffness is easily obtained from Eqn 5.9. Since Q and A , do not de-

pend on the clearance, while R is proportional to h-3, it follows that:

The stiffness KO in the reference configuration is, of course,

(6.5)

(6.6)

and, hence,

(6.7)



SINGLE PAD BEARINGS

151

0

1 2

-

w,

3

Fig.

6.1

Direct supply. Film thickness, pumping power, friction power

and

stiffness versus

load.

As has already been observed above, Eqn

6.5

(and consequently Eqn

6.6

and Eqn

6.7)

is only valid

as

an approximation for the pads whose film thickness is not uniform

(e.g. cylindrical and spherical pads).

6.2.2 Temperature and viscosity

5.7; in the reference configuration we have:

The temperature step of the lubricant (assuming adiabatic flow)

is

given by Eqn

When the load increases, the temperature step also increases:

AT

Ht

R ' + n H j -

ATo

Hto

l + n

=-=

(6.9)

In Fig.

6.2

it is plotted as a fhction of the load.

Careful consideration of

AT

is of great importance when

a

constant-flow supply

system is selected, since a change in lubricant temperature involves a variation in

its

viscosity, which, in turn, directly affects the bearing performance. Let us con-

sider Eqn

6.2,

which gives the load capacity: the hydraulic resistance of the pad

(evaluated in chapter 5 for several types of pad) proves to be proportional to viscosity



152

HYDROSTATIC L

UBRlCATlON

,LL;

consequently, if the lubricant, say, warms up, Ro mus t be substituted in Eqn 6.2

by the lower value ROpIpO and a higher value of R ’ is needed to sustain the same

load. In other words, this will result in a smaller clearance. The influence

of

the

viscosity

on

the clearance-load relationship

is

shown in Fig.

6.3.a.

3

2

-T

AT0

1

n

0

1 2

3

Fig.

6.2 Direct supply. Temperature step versus load, for certain values of reference power ratio.

2

- t l

h0

1

0

- a -

6

-

KO

4

2

- b -

Fig.

6.3

Direct supply. Clearance

(a)

and stiffness

(b)

versus

load

for certain values of viscosity.



SINGLE PAD BEARINGS 153

On the other hand, it is easy t o verify that

p

has no effect on pumping power

(indeed, Q s fixed and the recess pressure depends on the load alone, since

A,

is not

affected by

p);

friction power, for any given load, decreases if the temperature goes

up:

( H f is proportional to p, for a certain clearance).

Because of the reduced film thickness, stiffness is increased if viscosity is re-

duced; on the contrary, a cooler lubricant (e.g. during the starting phase) may

cause

a

considerable reduction in stiffness:

The effect of p on bearing stiffness is shown in Fig. 6.3.b.

6.3 COMPENSATED

SUPPLY

Let us now consider a pad bearing, supplied through one of the compensation

devices seen in section 2.3, rom a lubricant source, which is maintained a t a con-

stant pressure

p s .

Whatever the kind of compensation system we have, Eqn 5.1 and

Eqn 5.2 remain valid for any given pad. The recess pressure is then a fraction,

proportional t o the load, of the supply pressure:

The relationship between the hydraulic resistance of the pad (which in its turn

is closely connected to clearance

h )

and the load is easily evaluated by equating the

flow rate crossing the restrictor

Ps

P r

R r

Q

=-

to the flow rate crossing the clearances

of

the pad

We obtain:

(6.10)



154


The reference pressure ratio is commonly named

8 :

(6.11)

it

also represents the ratio of the design load to the limiting value

A g , .

From Eqn 6.10 and Eqn 6.11we obtain:

(6.12)

, B WIWO

R'=R pW/Wo

The hydraulic resistance of the compensation device is, in general, dependent

on flow rate and on recess pressure. Once such a relationship

is

known, as well as

the relationship between

R '

and

h ,

Eqn

6.8

allows us

t o

obtain the clearance a s a

function of the load. For the plane pads, Eqn 5.12 and Eqn 6.12 give:

The required

flow

rate,

at any given value of the load, is

Q= ----P W

1

W

h 3

(G)

o

WoR' -

Q0

Consequently, the relevant pumping power is:

(6.13)

(6.14)

(6.15)

The friction power

is

given by Eqn 5.11:

(values of Iffo or several types of pad are calculated in chapter 5 ) . Hence, the refer-

ence power ratio is

(6.16)

The temperature step, in the design configuration, is still given by Eqn 6.8,

while, for different values of the load,

(6.17)



SlNGLE

PAD BEARINGS

155

The bearing stiffness is, on the basis

of

Eqn 5.8,

W

1

1 w a,

= 3 r

l - p w / w o + K

dW

(6.18)

It is worth noting that in order

to

obtain explicit equations connecting the bear-

ing parameters (h,

K,

etc.) to the load, knowledge

is

needed of the actual value

of

R,..

In other words, the foregoing equations need to be specialized for the various com-

pensating devices.

6.3.1

Laminar f low restrlctors (capil laries)

The recess is supplied by a source of lubricant, kept at a constant pressure

ps,

by

means of a capillary tube o r some other laminar-flow fixed restrictor. The hydraulic

resistance R, of such devices proves

t o

be a constant for any given lubricant viscosity

(see section 4.7.6). It may be expressed in terms of Ro and 8:

1.0

-

- .

h o

0.5

t

(6.19)

-

b -

3

2

K

KO

1

0

0

1

2 3 4

5

0 1 2 3 4 5

w W

OAe

PS

PS

Fig. 6.4 Capillary compensation. Bearing clearance (a) and stiffness (b) versus load.



156


The equations describing the stat ic behaviour

of

the bearing may be easily ob-

tained from Eqn 6.10 and the following, up

t o

Eqn 6.18. The reference values

of

flow

rate and pumping power are:

(6.20)

The main non-dimensional bearing parameters are also plotted in Fig. 6.4 and Fig.

6.5 against the non-dimensional load W/ Wo=W/(jlpsAe).In particular, it follows

that:

n

(6.21)

Bearing stiffness comes from Eqn 6.18 and Eqn 6.19. In the reference position, it

is:

thus

(6.22)

(6.23)

KO is plotted in Fig. 6.6 as a hnction

of

8.

- b -

0

1 2 3

4

0

1 2 3

4

W W

BA,

Ps

0 4 s

Fig. 6.5 Capillary compensation . Flow rate, pumping power and friction power (a), and tempera-

ture step

(b)

versus load.




0.0 0.5

1

o

f3

Fig. 6.6

Reference values of stiffness

versus design

pressure ratio for capillary

and

orifice

compensation.

When the temperature of the lubricant is different from its design value, both

the hydraulic resistances of the restrictor and of the bearing gaps vary as does lu-

bricant viscosity

p.

The effect of such a viscosity change is easily evaluated by substi-

tuting

Rop po

and € € f o p /

o

for

Ro

and H f o , respectively, in the foregoing equations.

It

is

easy

t o

see that the clearance, as well as the stiffness, do not depend on the

actual value of

p .

Flow rate and pumping power are proportional

t o

l / p . Friction

power

is

proportional

to

p and the power ratio is therefore proportional to p2.

6.3.2 Orif ices

When orifices are used instead of laminar restrictors, it should be borne in

mind that the hydraulic resistance R , of such devices is no longer independent from

the recess pressure. The flow rate through these devices is proportional

t o

the

square root of the pressure drop ps -pr (see section 4.11).Since in the reference con-

figuration we must have

the hydraulic resistance

of

a sharp-edge orifice may be written in the following

form:

(6.24)



158


By introducing Eqn

6.24

into Eqn

6.13,

the relationship between clearance and load

may be explicitly rewritten as:

(6.25)

The other bearing parameters can now be obtained straightforwardly (Eqn

6.14

t o Eqn

6.18)

and are also plotted in Fig.

6.6,

Fig

6.7

and Fig.

6.8.

In particular, stiff-

ness becomes:

where

(6.26)

(6.27)

is its reference value. The orifices are clearly characterized by greater stiffness than

the laminar-flow restrictors (see Fig.

6.6).

On the other hand, a more careful control

of the lubricant temperature

is now

required.

L

1.0

-

- .

ho .

0.5

-

- a -

3

2

K

KO

1

a

- b -

0

1

2 3 4 5 0 1

2 3 4 5

W

OA,

P s

W

OA,

Ps

Fig.

6.7

Orifice com pensation. Bearing clearance (a) and stiffness (b) versus load.



SINGLE PAD BEARINGS

159

4

9

QO

1

L

HfO

c

- a -

I 1

0.7 / 0.5 /'"=0.3

1

2

3

4

0

1

2

3

4

W W

flAe PS flAe

PS

Fig. 6.8 Orifice compensation. Flow rate, pumping power and friction power (a), and temperature

step (b) versus load.

0

1 2 3 4

JL

PO

Fig.

6.9

Orifice compensation. Effect of viscosity (temperature) on clearance and stiffness.

Unlike laminar restrictors, the temperature of the lubricant affects the load

capacity of the bearing. The hydraulic resistance of the pad

is

proportional to the

viscosity of the lubricant, which in turn depends on temperature; this last, on the

contrary, has no practical effect on

R,.

Consequently, if

a

viscosity

p

occurs, differ-

ent from the design value po, each value

W

of the load will be sustained with a

clearance which does not tally with the value calculated by means of th e preceding



160


equations. It is easy to see that the actual film thickness is proportional to the cubic

root of the viscosity; the opposite applies for stiffness (see Fig. 6.91,whereas the flow

rate does not vary. Actually, we have

P s P r

Rr

Q

=-

but R, depends on pressure drop p s - p r alone; this last depends on the applied load

and is not related to viscosity. In brief, the consequences of a change in lubricant

temperature are the following: a cooler lubricant causes lower stiffness; a warmer

lubricant causes smaller clearance and, consequently, higher friction and higher

temperature step.

6.3.3 Constant f l ow valves

A simple constant flow device is shown in Fig.

2.11.

The spool compares force

F ,

exerted by the spring with the action of the differential pressure p v - p r on spool area

A,. If restrictor

R V 1

aries much more rapidly with spool displacement than spring

force

F,,

differential pressure p v - p r = F , A , is practically kept constant; the same

occurs for flow rate Q, which depends on pressure drop p v - p r across reference re-

strictor R,.

If

R,

s a true sharp-edged orifice, and its bore diameter

is

small enough

t o

ensure that the discharge coefficient

Cd

is not dependent on the Reynolds number

(see F'ig 4.161,

Q

does not actually depend on any change in lubricant viscosity. In-

deed, we have (Eqn 4.76):

Q can be adjusted by setting up the spring force. However, since the orifice sec-

tion

A,

cannot be too small

( to

avoid an excessive drop in pressure in the device),

and true sharp-edged orifices are difficult

t o

build (and to maintain), a certain

dependence on p often exists (see Fig. 4.16 and Fig. 4.17), at least near the lower end

of the flow ra te range. If, on the other hand, R, is a laminar-flow restrictor, its

hydraulic resistance becomes a constant, and thus, for a capillary restrictor, Eqn

4.66 gives:

i.e. Q becomes inversely proportional t o viscosity (in the last equation, 1 and d are the

length and equivalent diameter of the restrictor). It should be noted that, when a



SINGLE

PAD EEARINGS

I61

hydrostatic pad has

to

be supplied, the proportionality

of Q

and

p

ensures that the

bearing stiffness

is

independent from lubricant temperature.

As

long

as

recess pressure

p r

is

lower than

p s - F v / A , ,

the bearing simply be-

haves like a constant flow system:

Ps

B

Q = & o = R ,

P2

H~ =

H~~

=&-

(6.28)

(6.29)

(6.30)

(6.31)

(5.11rep.)

(6.32)

(6.6 rep.)

(6.7

rep.)

Once Fv is fixed, pressure drop Ap=pV-pr=Fv

A v

is a characteristic parameter

of

the device (in most cases, a few bars). Of course, it mu s t always be pv<ps; thus,

when the load increases to approach the value

(6.33)

the device ceases

t o

produce a constant flow rate, with a sharp drop in stiffness.

In Fig.

6.10

clearance and stiffness are plotted versus load. Dashed lines repre-

sent indicatively the limit behaviour

of

the valve

for

a number

of

values of @, for the

sample case Ap=O.lps.



SINGLE

PAD

BEARINGS

163

R,.=C,x (6.34)

On the other hand, for the equilibrium of the spool, the force exerted on it by the

spring

is:

where FvS is the spring force for complete spool displacement ( x = O ) and K, s the

stiffness

of

the spring. Under the design load, we have

Pr=/3ps

and

Rro=Ro(1-a)/a;

thus F 0 s ,

J,

and

Ro

are connected together by the relationship:

(6.35)

If the device is such as

t o

allow us to adjust the spring force, it is possible to achieve

any design value of

Ro

(i.e. of film thickness ho)under the design load

p s A $ .

Note

that, since

R,.

is

a

laminar-flow restrictor, both Ro and C, are proportional to viscos-

ity; F,s does not then depend on lubricant temperature. The displacement of the

spool under the design load is

Combining the foregoing equations, the hydraulic resistance of the controlled

restrictor may be written as follows:

(6.36)

where

For any given value of p, low rate Q, is a characteristic parameter of the device.

Introducing Eqn 6.36 into Eqn 6.13,

it

is possible t o obtain clearance as a function of

the load:

1 a

w/wo

1

--;I

P W I W ,

(

W ) & + U

l - w o P o a

(6.37)

where

(6.38)



164

HYDROSTATIC L UBRlCATiON

- a -

- b -

1.0

-

- .

ho .

0.5

-

1.5

0 1

2

W

fi

Ae PS

- c -

2

0

1 2

W

fi

Ae

PS

4

0.0

0

2

W

fiAe PS

- d -

10

K

-

Koc

c

c

.c

Op0.25 0.3 0.4

. .

, , 0 . 2 ; g j (

1

2

W

fi Ae PS

Fig. 6.11

Cylindrical

spool

valve. Bearing clearance (a,

b),

flow rate

(c)

and stiffness

(d)

versus

load for certain values of valve parameter.



SINGLE

PAD BEARINGS

165

A plot of hlho as a function of W/

o

is given in Fig. 6.11.a,b for certain values of 8

and 8,. It should be pointed out that, when we have P u = / 3 , the system produces a

flow rate which does not depend on load. The spool displacement will generally be

limited

t o

a certain range

X,+XM:

outside the relevant load range

W,+W,,

Eqn 6.37

is no longer valid and the bearing would merely perform like a capillary compen-

sated one; furthermore, for very small values of x,, the flow is no longer laminar.

The other bearing parameters (flow rate, power, etc.) are still given by Eqn 6.14

and the following equations; the flow rate is also plotted in Fig. 6.1l.c.

The bearing stiffness is obtained by combining Eqn 6.18 and Eqn 6.36. In partic-

ular, for the reference load,

it

is

(6.39)

where K O ,

s

the stiffness of the same pad supplied at the same pressure ratio 8

through a fixed restrictor (Eqn 6.22).

For

a generic load we have:

1 Pl8v

El

1-8 WOPV

W

1 WIWO p

WO 1-8

P v

1-8-

1 +

(6.40)

Nondimensional stiffness K/Ko c s plotted in Fig. 6.11.d as a function of WlWO it has

been chosen to show K / K o c instead of K in order to display the gain in stiffness of

the system, compared t o a capillary-compensated one). By tuning the elastic con-

stant of the spring, a very great, and even negative, stiffness could clearly be ob-

tained. On the other hand, the stiffness would be much smaller than KO for loading

conditions different from the reference one, and the flow rate would increase

sharply for loads greater than Wo. Furthermore, the behaviour of the bearing under

dynamic loading must be carefully considered.

Concerning the effects of a change in lubricant temperature, i t has already been

pointed out that, provided R ,

is

a laminar-flow restrictor, the ratio C, Ro does not

depend on

p ;

thus, only flow rate and power are affected, Q and

H p

being propor-

tional to Up and

H f

being proportional to

p .

Other types

of

devices have been proposed, with better performances than the

cylindrical spool. For example, let us consider the tapered spool valve (ref.

2.5)

in

Fig.

2.9.

Its hydraulic resistance may no longer be expressed by

a

linear relation-

ship such as Eqn 6.34. Provided aperture angle

a:

is

very small, we have:

(6.41)



166


- a -

0.0' ' ' . ' .

'

' .

0

1 2

W

BAe

Ps

- c -

2

9

00

1

- b -

0

2 4

W

DA.9

Ps

- d -

1 2

10

K

Koc

5

0

-5

0 1 2

W

B A e

PS

Fig.

6.12

Tapered spool valve. Bearing clearance (a,

b), flow

rate

(c)

and stiffness (d) versus load

for certain values of valve parameterg, and R,=Ro.



SINGLE

PAD

BEARINGS

167

where

& x a l

c is the non-dimensional spool displacement. Proceeding as in the

former case, it is easy to verify that F,s (spring force for {=O), to spool displacement

in the design configuration), p and

Ro

are connected by the following equations:

The hydraulic resistance of the device may thus be rewritten as:

where

1 c Ro

Ku

2 a A ,

RJ

P s

u

=

---(i-

The clearance for any load

W

now proves t o be

(from

Eqn

6.13):

3

h - 1 1

p WlWo

Ro 1

q-

(6.42)

(6.43)

(6.44)

(6.45)

The rate of flow is still given by Eqn 6.14.The stiffness for the reference load is

again given by Eqn 6.39,whereas, for different loads, the relevant stiffness may be

evaluated by introducing Eqn 6.43 nto Eqn 6.18.Film thickness, the flow rate and

stiffness are plotted in Fig. 6.12 s functions of the load.

Again no effect follows a change in viscosity, except for different values of flow

rate and pumping power.

6.3.5 Diephragm-contro l led rest ric tors

The device shown in Fig. 2.10 ref. 2.6)works like the spool valves examined in

the previous section, but i ts hydraulic resistance varies according to a cubic law:

(6.46)




where <=x /

l o

is the non-dimensional displacement from an arbitrary configuration

(say,

lo

may be assumed as the restrictor clearance when no force

is

exerted on the

diaphragm, Fig. 6.13).

Fig. 6.13 Diaphragm-controlled restrictor.

For the equilibrium of the membrane, the elastic force has to be:

A , is the effective area of the membrane, F v s is the elastic force exerted by the

spring, for x=O , from which the constant effort exerted on the membrane by the

supply pressure

p s

has been subtracted. To have a hydraulic resistance

Ro

of the

pad when the design load Wo=ppsA, s applied, F,s must be:

F v s = P P s A ,

-k

5OlOKu

(6.47)

where

Note that we should have Fus cKv lo o prevent a complete shut-offof the restrictor for

small loads.

By rewriting R , in the form

1-8

1

1

- P

(6.48)



SINGLE

PAD

BEARINGS 169

where

(6.49)

and by inserting it into Eqn

6.13,

ne can obtain the relationship between load and

clearance:

(6.50)

A plot of the film thickness , as well as of the flow rate (obtained as usual from

Eqn

6.14))

s a function of the load, is given in Fig.

6.14.

The stiffness in the reference configuration is again given by Eqn 6.391,where

Eqn 6.49 has now t o be used for

Po.

For different loads, the non-dimensional stiff-

ness is:

K W

0 1 -PIP,

h

W P

1 p

wo P u

w

11-w/wop

wo

3

1 - p

P

-8-

1--

(6.51)

Comparing Fig.

6.14

o

Fig.

6.11

and Fig.

6.12,

we see that the performance of the

diaphragm-controlled restrictor is better than that of the spool-controlled devices; in

particular, the load range is larger. Furthermore, due to the small mass of the

membrane, the dynamic behaviour should also be better.

As

happens in the case of the spool valves, any change in the temperature of the

lubricant does not affect the set-up of the device, but only flow rate and pumping

power.

6.3.6

Infinite-stiffness

devices

Certain controlled restrictors, such as the one shown in Fig.

2.12

(ref.

2.81,

are

able

to

keep bearing film thickness constant for a wide range

of

loading conditions.

For

the equilibrium of the spool we must have:

(6.52)

When the load, and therefore p r , vary, the spool is displaced, changing the inlet

restrictor

of

the device, until Eqn

6.52

s again fulfilled. The flow rate is:



170


- a -

- b -

0

1 2

W

DAe P S

- c -

2

9

Qo

1

0

\

1

2

W

flAe

PS

1.5

1 o

h

h0

-

0.5

OS

1C

K

-

Koc

(

-1

2 4

W

OAe PS

- d -

0.5

1

p z q

0.25

n.

1 2

W

flA, Ps

Fig. 6.14 Diaphragm -controlled restrictor. B earing clearance (a, b),

flow

rate

(c)

and stiffness (d)

versus load for certain values of valve parameter.



SINGLE PAD BEARINGS

171

1 o

0.5

*

\ \ \

.

y p 0 . 2 5

\ 0 . 5 \

0.75\

\ \ \

" \

\ \

'

' \

\

\'

\\

\

1

o

0.0

0.0

0.5

- &

Ae PS

- b -

0.5

1 o

W

-

Ae

PS

Fig. 6.15

"Infinite-stifSness"

valv e. Bearing clearance (a)

and flow

rate (b) versus load, for certain

values

of

the

valve parameter.

P v - P r

Q

=-

R ,

This means that the hydraulic resistance of the pad clearances is pegged t o the

constant value

(6.53)

v

R = R o = R ,

l-~

and thus the film thickness is kept constant as well. ho can be adjusted by setting

restrictor R, . It is noteworthy that, if R , is laminar in kind, the system is not sensi-

tive

t o

any change in lubricant viscosity, except as far as

flow

rate and pumping

power are concerned.

The f low rate proves

t o

be proportional t o load W=p,.Ae:

W

Q

= A x

(6.54)

We obviously must have p u < p s , i.e. the system cannot maintain infinite stiffness

after the spool has been completely displaced. When pr>yvps, the whole hydraulic



172

HYDROSTATC

LVBRlCATlON

resistance of the device will be reduced

t o

the fixed value R,, and the system will

merely work like a capillary-compensated bearing with a pressure ratio b=y, (see

Fig.

6.15).

6.3.7 Inherent ly compensated bearings

The simplest

of

the inherently compensated bearings described in Chapter 2 is

represented in Fig. 2.17.a (see also ref. 2.13). From Eqn

4.58,

low rate is:

then pressure at radius

r l

is:

Ps

lnrz/rl 1

+E 1

+

hp/h)3

Load capacity is:

Inherently compensated bearings are generally used

w i t h

compressible

Iubricants.

6.4 DESIGN OF SINGLE-PAD THRUST BEARINGS

The specifications of a pad bearing consist, in most cases, in

a

range

of

loads

W,+WM t o be sustained with a certain stiffness at any given sliding o r turning

velocity. A further set of constraints is generally specified, involving important

parameters such as:

size and shape

of

the pad

film thickness

overload capacity

lubricant viscosity

flow rate

supply pressure

friction.

The first step in design often consists in establishing suitable pad dimensions,

in such a way that the effective area of the pad be greater than the ratio

of

the

maximum expected load

t o

the maximum available pressure. The shape of the pad




is often constrained. However, a few remarks on the most convenient land width to

choose may be useful.

In the foregoing sections it has been shown that, for any pad shape, the land

width ratio (e.g. parameter a' for annular-recess pads) may be chosen in such a

way as t o minimize pumping power. It is clear that when pad velocity is null, or

very low, total power consumption practically coincides with pumping power, and

then such a choice of the land width may be considered to be the optimum one.

However, this is not a critical point: for instance,

it

may be convenient to select

narrower lands to increase load capacity without affecting the external size of the

pad. On the other hand, when the speed is high enough, it is generally more conve-

nient to select narrow lands

to

minimize the total power H t = H p + H p

The selection of the thickness of the film depends on certain contrasting factors:

it must be small enough to ensure sufficient stiffness at any load to meet the

specifications;

it

must be small enough to ensure small flow rate and pumping power (of course,

this point, as well as the next, is also connected with the choice

of

lubricant

viscosity);

it must be large enough

to

avoid excessive friction;

it must be much larger than the geometric errors of the mating surfaces

(roughness, planarity, parallelism).

As

far as the problem of a suitable choice of h and

p

is concerned, it

is

possible to

demonstrate that, for any given pad and any given viscosity,

an

"optimum" clear-

ance may be calculated (from the point of view of total power) on the condition th at

H f = 3 H p .

Conversely, if

h

is given, "optimum" viscosity is the one which gives

Hf-&lp

(ref.

2.3) .

Let us now briefly examine this problem applied

to

rotating pads.

Total power in the reference configuration may be written in the form:

valid for constant-pressure supply, as well as for direct constant-flow supply (in

this

case it must be stated that

/j'=l).

or axisymmetric pads, such

as

plane or tapered

circular pads, the effective area can be written as in Eqn

5 .20 ,

and the hydraulic

resistance as in Eqn

5 .83 :

of course, the relevant equations must be used for

A*,

and

R*,

and a=d2 for plane pads. For these pads total power may hence be rewritten as:

(6.55)



SlNGLE PAD BEARlNGS

175

where

or alternatively as:

where

(6.60)

(6.61)

(6.62)

These equations are valid for any supply system (for direct supply

p=1

and ps=pro) .

Both

p$

and p; are plotted in Fig.

6.16

for central-recess circular pads and Fig.

6.17

€or annular-recess pads.

A problem now arises, since in practice i t is not always possible to select a lubri-

cant with the theoric "optimum" value of viscosity given by Eqn 6.59 or Eqn 6.61 for

n=1. n particular, a t low speed, this optimum viscosity may prove t o be impractica-

bly high. Although one may t r y t o increase land width or pad size, sometimes a

value of l7 smaller than 1 must be accepted: indeed, for Q=O, an infinite viscosity

would be the optimum On the other hand, a t high speed, the optimum viscosity

H i

Fig. 6.16 Circular recess pad. Total power H : and optimum viscosities p$ and p; versus

radius

ratio

r'.



176


- a - - b -

4

H i

2

40

20

400

GI

200

0

0.0

0.2 0.4

"

a'

r'=O.8

\

O

0.0

0.2

0.4

a'

Fig.

6.17

Annular recess pad. Total power H: and optimum viscosities

p$

and p; versus land

width

ratio

a'.

would be the optimum On the other hand, a t high speed, the optimum viscosity

might prove to be very low, even after having reduced the land length to a minimum

and having increased recess pressure (reducing the pad diameter)

as

much

as

possible. In these cases it is recommended t o select a suitably low viscosity

(sometimes, however, p is explicitly prescribed) increasing ho in such a way as to

have n13.Actually, it can be shown that, once p is fixed, the optimum clearance

is the one which leads to n=3, ut the problem is complicated by the fact tha t

h

and p

may not be considered

to

be constant values, since they are subject t o change accord-

ing to load and temperature.

The following is a design criterion which generally produces good results:

take as reference load Wo the one (in the normal load range) which gives the

minimum value of power ratio H f / H p :namely, Wo=WM or constant-flow supply and

W,=W, for constant pressure supply;

select h,, p, and the land width ratio in such a way as

to

obtain n=1. he value of

p selected in this way should be the one which tallies with the maximum lubricant

temperature allowed by the supply system. However, when velocity

is

too low

o r

too

high,

it

is necessary to accept values of ll that are smaller than

1

or greater than

3,

respectively.



178 HYDUOSTA TIC LUBUlCATlON

the land width ratio) or h o and start again from point

(ii)

o r (iii). If p o p t

cannot be brought into the admissible range, because the velocity is too

small, select the highest available viscosity. On the other hand, a very high

speed would require

too

low viscosities to be optimized with

n=1,

nd conse-

quently a greater value of the power ratio must be selected; values of

n

greater than

3

should, however, be avoided, or else excessive temperature

steps could follow (Eqn

6.8);

if the supply system is not able

t o

hold the temperature of the lubricant

within narrow limits, the designer should take a certain viscosity range into

account, instead of a single value. A

simple method consists in considering

popt

as the lower limit of the viscosity range, which tallies with the highest

lubricant temperature.

For

a cooler lubricant viscosity will be greater, as

well as film thickness and, above all, bearing stiffness will be lower. Conse-

quently, it is necessary to check K for the whole temperature range;

v )

calculate the required

flow

rate;

calculate pumping power

H p o

and friction power

H f o

(for circular bearings

Eqn

6.57

and Fig.

6.16 o r

Fig 6.17 give total power

Hto:

now

Hpo=HtoI(l+17)

and

Hfo=Hto17).

For loads smaller than

Wo,

both the powers will be lower;

check the maximum temperature step in the lubricant (Eqn 6.9);

if

a

range of viscosities is to be allowed in the plant, check friction power and

temperature step for the maximum viscosity too;

vi)

check the Reynolds number;

for bearings rotating at high speed also check the effect of inertia forces

(parameter

S i )

and the circumferential Reynolds number;

vii) check the dynamic behaviour of the system (see chapter

10).

In certain cases it may be necessary to have a

flow

rate which differs from the

value calculated at point (v) above (for instance, because the pump which has to be

used does not feature a n adjustable

flow

rate); clearly, if this happens

it

is

necessary

to repeat the calculations (in particular, as far

as

film thickness and stiffness are

concerned) using the actual value

of

the flow rate. If the difference is large, it may

be considered convenient to modify viscosity

o r

geometrical parameters in order to

approach the

opt imum

value of the power ratio again.

EXAMPLE

6.1

Design a circular-recess pad to sustain a thrust varying in the

W,=30

K N

t

W ~ = 4 0

v range at a speed G=lO a d f s , w i t h a friction m oment

lower

than 1 N m .

Further constraints are the following: the outer diameter must be DSlOO mm; the

lubricant viscosity can be chosen in a range of 0 . 0 2 4 1 N s lm 2 ; th e d isp la cem en t a s




the load varies (slowly) from the minimum to the maximum value must be smaller

than

10 pm; the recess pressure must always be smaller than 7 MPa.

i) Since the turning velocity is low, a first trial value for r' could be r'=0.5, and

hence A2=0.541. In order to sustain the maximum load WO=wM=40

KN,

the effective

area must be greater than w 0 / 7 MPa, i.e. A ,~5 .7.10-~2; consequently, due to the

limited diameter, it must be (Eqn 5.20) A*,>0.727, and a considerably larger value is

needed for r' (Fig. 5.1. b).

ii) Let

us

select D=lOO mm, r'=O.75; it is easily obtained (from Fig. 5.1.6 or by

means of the relevant equations) A50.760, R*=0.288, Hf*=0.684; he effective area is

then A,=5.97.10-3 m2 and the maximum recess pressure is pro=WolA,=6.70 MPa.

iii) A suitable value for minimum film thickness may be ho=30pm. The maxi-

mum film thickness, which tallies with the minimum load Wm=30

KN,

is (Fig.

6.1)

h~=1 .10.ho=33 m. Note that the displacement h ~ - h o = 3m is much smaller than

required.

iv) In

Fig.

6.16 we may read &=8.72, i.e. bpt=0.31Nslm2. This last value is too

high,

hence the

maximum

allowable ualue p=O.1 Ns lm2 should be selected, accept-

ing a power ratio smaller than 1 (the friction power will prove to be much smaller

than the pumping power).

v) The reference hydraulic resistance is easily calculated as Ro=2.03.1012Nslm5

(Eqn 5.21), and hence Q=prolRo=3.29.10-6m3/s , Hpo=proQ=22W. As far as friction is

concerned, we have Mf0=0.22Nm, Hf0=2.2 W. Obuiously, for loads smaller than Wo,

both Hp and H f will be smaller. The temperature step is maximum in correspon-

dence to the max imum load: from Eqn 6.8, assuming c.p=l.7.10-3 J lm2 C,

dTod.3"c.

EXAMPLE 6.2

Design an annular-recess bearing to sustain a load W=75

K N

with stiffness

K0>5.109N lm , at a speed l2=628 radl s (6000 rpm). The outer diameter of the pad

must be smaller than 200 mm, and the inner one greater than 120 mm. The maxi-

mum recess pressure can be as high as 12 MPa, but the bearing should be able to

sustain, exceptionally, a maximum overload dW~=75 N . the data of the lubricant

to be used are: p=0.02+0.03 N s lm2 (depending on the actual operating temperature),

p

=go0

e l m 9 c=1890JlQOC.

i) Since the speed is high, let us start with a small value of a', say a'=0.1. I f the

greatest allowable pad area is chosen, it follows that r'=120/200=0.6.

ii)

It follows immediately that: A*,=0.576 and A,=0.0181 m2. The maximum load

WE=150

KtV

hence tallies with a recess pressure p~=8.29MPa; for the normal load

Wo we have pr0=4.14 MPa.

iii) Since Ko=3W0Ih, o obtain the required stiffness we must have hoc45 pm.

However, since a wide viscosity range is expected, to obtain hc45p or the highest




viscosity value (0.03 Nslm2) we must have ho<39

pm

for the lowest viscosity p=0.02

Ns I m2 (see Fig. 6.3).

iv) From Fig. 6.1 7, p:=15.2: to get

I l = l

with p=0.02 Ns lm 2 a clearance ho=60pm

is needed (Eqn

6.60).

Since ho must be

much

lower to ensure adequate stiffness, fric-

tion power would be much greater than pumping power. To reduce total power it is

possible to reduce the land width. For instance, a=2.5 mm (a'=0.0625) may be se-

lected. From equations of section

5.4

it follows that A:=0.60, R*=0.0156, Hf*=0.119;

hence Ae=0.019 m2, pro=WolAe=3.98MPa. In Fig 6.17 we may read pz=25.2, thus for

l7=3 and p=0.02 Nslm2 a clearance ho=36.9 pm is needed, whereas we may select

ho=39pm in order to have Il=2.4 for p=0.02

Ns

m2 and n=3or

p=0.03

Ns

I

m2.

v )

Flow rate is &=0.395.10-3 mats. The power consumptions are Hp0=l.6

KW,

Hf0=3.8KW and Ht0=5.4KW. When the lubricant is cool (p=0.03 Ns lm 2), clearance

increases to 44.7 pm (Fig. 6.3.a) and friction power increases to 5.0 KW. Note that if

viscosity is further increased, stiffness may fall under the designed min imum

value. The maximum temperature step is (Eqn 6.8)AT=6.61(0.395.0.9.1.890)=9.7

C.

ui) The Reynolds number is given by Eqn 5.75: Re=17.7; the circumferential

Reynolds number in the outer clearance is (from Eqn 5.591 Re,=109; consequently,

no turbulence is expected. Also, the inertia forces in the lubricant not need be taken

into account since we have Si=O.13 (Eqn 5.50).

6.4.2 Compensated supply (constant pressure)

In this case we suggest taking the smallest load in the normal load range as

reference. The design procedure is then similar, in its main aspects, to the one out-

lined in the previous section.

Although the procedure below refers t o bearings compensated by means

of

cap-

illary restrictors,

it

may be easily extended to other compensating devices, including

controlled valves:

i)

choose a trial set of geometric parameters. If the pad velocity

is

very low,

select a land width ratio close to the one which gives the minimum pumping

power; else select narrower lands. For instance, for a central-recess circu-

lar pad, one may select

r'=0.53

in the former case, and a greater value

(perhaps r'=0.9)in the latter;

dependent on load ratio WMIWo (for instance, with

the aid of Fig. 6.4): i t is advisable for to be considerably smaller than

Wo1WM e.g.

8 ~ 0 . 8 .

o/W M )o ensure enough overload capacity and to avoid

the stiffness corresponding to W M being too small. In particular, as far as

overload is concerned, if a certain overload capacity

AWE is

given, we must

obviously have PCWOIWE,where WE=WM+AWE.t should, however, be

stressed that the total power will turn out t o be proportional to l/fi (Eqn

choose pressure ratio



SINGLE

PAD

BEARINGS

181

6.57), but, on the other hand, stiffness KO is proportional to (1-p).When the

load range is wide (i.e. when W o is much smaller than W , ) a very small

value is needed for p ; in this case, an opposed-pad bearing may be the most

suitable choice (see chapter 7);

ii) choose suitable values for effective area A , and supply pressure p s , in such a

way as to obtain Wo=AJ?p,;

evaluate the size of the pad, and check it is compatible with the specifica-

tions;

iii) select the minimum clearance h,, tallying with the maximum load WM;

evaluate the reference clearance ho (e.g.

for

capillary compensation from

Eqn 6.21 or Fig. 6.4.a). The thinner

ho

is, the smaller the power losses

(provided a sufficiently low viscosity is available) and the greater the stiff-

ness will be; on the other hand, greater manufacturing accuracy will be

required;

calculate KO (e.g. from Eqn 6.22) and stiffness

K=KoK'

tallying with load W,

(e.g. from Eqn 6.23 or Fig. 6.4.b). For controlled restrictors, the relevant

parameters (e.g. 8 , ) must also be selected;

check that stiffness is great enough for the whole load range: if it is not, one

may try to decrease p, starting again from point (i), or to decrease ho, f pos-

sible, starting again fkom point (iii);

iv) calculate the optimum viscosity, that is the value popt that makes

I7=1

(Eqn

6.4). For circular bearings, this may be done quickly with the aid of Fig. 6.16

or Fig. 6.17;

check that

k p t

is a plausible value (sometimes

p

may be directly imposed by

the specifications). If it

is

not, modify the geometrical parameters (namely

the land width ratio) or ho and start again from point (ii) o r (iii). If

p o p t

cannot be brought into the admissible range, because the velocity is too

small, select the highest available viscosity. On the other hand, a very high

speed would require too low viscosities t o be optimized with n=1, nd conse-

quently a greater value of the power ratio must be selected; values of Il

greater than 3 should, however, be avoided: otherwise excessive temperature

steps could follow;

if the supply system is not able t o hold the temperature of the lubricant

within narrow limits, the designer should take a certain viscosity range into

account, instead of a single value.

A

simple method consists in considering

popt as the lower limit of the viscosity range, which tallies with the highest

lubricant temperature.

For

a cooler lubricant viscosity will be greater: for

many compensating devices this will have no effect on clearance and stiff-



182 HYDROSTATIC L UBRICA TION

ness; instead, the power ratio will be greater (proportional to

p2

for laminar-

flow devices);

v) calculate the flow rate;

calculate pumping power Hpoand friction power Hfo (for circular bearings

Eqn 6.57 and Fig. 6.16 o r Fig 6.17 give total power Hto: now Hpo=Hto / ( l+~)

and Hfo=Hto17).For loads greater than WO,he pumping power is smaller,

but Hf increases (Eqn 5.11);

check the maximum temperature step in the lubr icant (Eqn 6.8 and Eqn

6.17);

if a range of viscosities is to be allowed in the plant, check friction power and

the temperature step for the maximum viscosity,

too;

vi)

design the restrictor: its hydraulic resistance

RrO

must be given by Eqn

6.19;

if capillary pipes are used, select length and diameter by means of Eqn 4.67;

check the Reynolds number in the restrictor to ensure laminar flow (Eqn

4.68);

vii) check the Reynolds number;

for bearings rotating a t high speed, also check the effect of inert ia forces

(parameter

S i )

and the circumferential Reynolds number;

viii) check the dynamic behaviour of the system (see chapter 10).

EXAMPLE

6.3

A large slide is sustained by a set o f rectangular pads (supplied through capil-

lary restrictors) whose dimensions are

B=0.3

m and L=0.4 m. Each pad must

sus-

tain a load

W=60

N

with a stiffness K>2.5.109 N l m and a film thickness not

smaller than 40

pm. The friction force should not be greater than 6 N at a speed

U=0.05

m

I

s.

i) Since velocity is low, it is convenient to have large lands; if we select a'=0.15

and rL: =0.25, from Fig. 5.27 we obtain Az=0.645. Since no great load variations are

expected, we may try to select a relatively high value, such as 8=0.6, for the pressure

ratio.

ii ) The pad dimensions are given: the effective area is A,=LBA:=0.077 m2. The

supply pressure should hence be p,=1.29 Pa.

iii) I f we select ho=40pm, Eqn 6.22 immediately gives K0=1.8.109 N l m , which is

not enough. Hence it is necessary to select a smaller value for

p,

such as p=0.4.

Repeating the above calculations, we now get: ps=1.95 MPa and Ko=2.7.1@ N l m.

iv) From Fig. 5.28 we obtain R*=0.75, and from Eqn 5.108 (stating h,=l mm)

Af=0.079 m2. Since the friction force must be smaller than

6

N,

Eqn 5.106 gives an

upper boundary for viscosity: p=0.0607 Nslm2. For such a viscosity Eqn 6.16 gives

n=0.14 (of course, the pumping power will prove to be much greater than the fric-



SINGLE

PAD

BEARINGS 183

tion power). A lubricant is therefore selected, whose viscosity is

p=0.06

Nsl m2 at the

lowest operating temperature.

v ) Calculations of flow rate and power are now straightforward, leading to:

Qo=1.1.10-6 m3/ s; Hp=2.1 W, H ~ 0 . 3 . Temperature step proves to be slightly

greater than 1

C

assuming c.p=1.7.106 Jlm3 C). For lower viscosities H f decreases

while Q and Hpprove to be proportional to 1I p.

vi) The hydraulic resistance

of

the restrictor

can

be calculated by means of Eqn

5.102 and Eqn 6.19, which lead to R,/p=17.6.1012 m-3. From Eqn 4.67 it follows that

such a hydraulic resistance may be obtained with a pipe 27

mm

in length and with

a 0.5

mm

bore. The Reynolds number, calculated by assuming p=0.06 Nslm2 and

p 9 0 0 Kglm3,

is

Re=42, considerably less than its critical value.

EXAMPLE 6.4

Design a capillary-compensated annular-recess bearing to sustain a load which

varies i n the 2 0 4 5 KN range, with a displacement Ah smaller than 20

pm.

The

outer pad diameter must be smaller than 0.2 m, and the inner diameter must be

equal to 0.1 mm. A further requirement is that the friction moment needed to rotate

the bearing at e 6 2 8 radls (6000 pm) must be smaller than 5 Nm. The supply sys-

tem to be used delivers lubricant at pressure ps=4 MPa; viscosity is expected to vary

in the p=0.0154.020 Ns 1m2 range, depending on actual operating conditions; den-

sity and specific heat are p=920KgIm3andc=1850 JIKgOC.

i) Since the speed is high, choose narrow lands, say a=2 mm. The load ratio

i s

WMIWo=1.75,hence p=0.45 should be a proper selection (see Fig. 6.4).

i i )

Wi th the give n supply pressure, the effecti ve area should be

Ae=2~/(O.45.4.1O6)0.0111 m2. For the sake of simplicity, let us choose D=0.16 m

(i.e. r'=O.625; a'=0.067), obtaining Az=0.569, R*=0.0154, Hf*=0.122;hence, A,=0.0114

m2 and p=0.437.

iii) From Eqn 6.21 (or Fig. 6.4.a), for any certain design clearance ho the mini-

mum clearance (which tallies with the maximum load) is hm=0.62.ho. To ensure

Ah=ho-hm<20 pm we must therefore have ho152.7 pm. On the other hand, to have

Mf<5 N m for p=0.02 Ns lm 2, the minimum film thickness cannot be smaller than

19.8 pm, i.e ho must be not smaller than 31.8 pm (see Eqn 5.70).

iv) From Fig. 6.17, pJ=119; to have ll=1 for p=0.015 Nslm2 we should have (Eqn

6.59) ho=41.4 pm. Selecting ho=40 pm, the reference power ratio will vary in the

1.1411712.04 range, depending on the actual value of the viscosity. Flow rate, powers

and temperature step depend on load and viscosity and are summarized in the fol-

lowing table.

v) The hydraulic resistance of the restrictor must be R,=1.29.Ro (Eqn 6.19), i.e.

R,.=8.87.109 Ns lm5 for p=0.015 Nslm2. Note that, due to the high value of the flow

rate, such a restrictor cannot be made with a small pipe without accepting high



184


W

(KN)

20

35

Reynolds numbers. A more suitable approach may consist in using a narrow annu-

lar clearance as a restrictor (Fig. 4.11.e). A further solution may consist in substi-

tuting the laminar-flow restrictor with an orifice, which works well at a high

Reynolds number (section 4.11).

p h Q.103 H p H f AT

(Ns/m2) (m3/s) (KW) (KW) ("(2

0.015 0.25 1.02 1.16 5.0

0.020 0.19 0.76 1.55 7.1

40.

0.015

0.11 0.42

1.87 12.8

0.020 0.08 0.32 2.49 20.7

24.8

vi) It is easy to verifv that the radial Reynolds number of the bearing (Eqn 5.75) is

suitably low, as well as the tangential Reynolds number (Eqn 5.59) and the inertia

parameter (Eqn 5.50).

EXAMPLE 6.5

The stiffness

o f

the bearing designed in Example 6.4 needs to be improved, with-

out changing the dimensions

o f

the pad, nor the supply system (except for the

compensating device). Namely, Ah needs to be reduced to a value of 10 pm or

smaller.

i) First, one may try to reduce the design value of clearance. In the previous

example it is shown that ho cannot be smaller than 32 pm, in order to avo& exces-

sive friction; consequently, Ah cannot be smaller than 32.(1-0.62)=12

pm.

It

is

clear

that another type of restrictor must be used to emure adequate stiffness.

W One test may consist in substituting the laminar f low restrictor with a sharp-

edge orifice. In this case, Eqn 6.25 gives a minimum film thickness hm=O.717.h0;

assuming ho=30 pm, it follows that h,=21.5 pm and Ah=8.5 pm. For orifice com-

pensation, however, clearance depends on the actual value of viscosity, as shown in

Fig.

6.9: i f p

is increased from 0.015 NsIm2 to 0.02 Nslm2, clearance experiences a

10% increase, and Ah becomes greater than 9 pm. By repeating the calculations, the

flow

rate, powers and temperature step are obtained, as shown in the following

table.

The stiffness constraint has clearly forced us to select values of the power ratio

greater than 3, leading to large temperature steps in the lubricant. The diameter of

the orifice can be calculated by means of Eqn 4.76, in which Q=Qo=O.ll m3/s ,

Ap=(l p)p,=2.25 MPa. The discharge coefficient depends on practical considera-



SlNGL

E

PAD

BEARlNGS

185

W

(KN)

20

35

tions (see Fig. 4.16 and Fig. 4.17), a typical value being Cd=0.65. d=1.72 mm is read-

ily obtained.

p

h

Q.103

H p

Hf

AT

(Ns/m2) (pm)

(m3/s) (KW) (KW)

(T)

0.015 30. 0.11 0.43

1.55

10.9

0.020

33. 0.11

0.43

1.88

12.7

0.015 21.5

0.07 0.28 2.16 20.8

0.020 23.7

0.07 0.28 2.61 24.7

iii) In the last case examined, it is clear that power, flow rate and temperature

step are subject

to

large variations when the load and viscosity change. A great

improvement from this point of view may be obtained by means of a constant flow

valve. Figure. 6.10 shows that now h,=0.830.ho; it is therefore possible to select a

greater value for clearance, such as ho=40 pm. Assuming that a temperature

compensated valve is used, calculations now give:

It may be verified that by reducing the flow rate, say to 0.1 l O - 3 m3/s, clearance

hoproues to be close to 30 pm, with a higher friction power, but the total power ex-

pense will not be notably affected (it will be slightly reduced).



Chapter

7

OPPOSED-PAD AND MULTIPAD BEARINGS

7.1

INTRODUCTION

A multipad bearing system is made up of a certain number of pads, with a

common moving member. The pads may be, in general, quite different, and they,

too,

might be fed by means of differing devices.

All

these bearings may be studied by applying

to

each pad the equations already

developed for the single pads in chapter 5 , and by summing together the effects of all

the pads, after having expressed all the pad clearances as functions of the dis-

placement of the moving member.

The case of the opposed-pad bearings is of particular importance.

7.2

OPPOSED-PAD BEARINGS

For

any opposed-pad bearing, such as the one in Fig. 7.1, the total load capacity

is, obviously,

The

flow

rate is, from Eqn

5.2

and Eqn 5.10,

(7.2)



188


7.2.1 Direct supply

(i)Basic equations. When the two opposite recesses are fed by separate pumps,

providing the constant volume flows Q1

and

Q2,

respectively, the recess pressures

are:

The load capacity is given by Eqn 7.1. For a symmetrical bearing, provided that

Q1=Q2=Q/2(see Fig 7.21, we have:

(7.7)

W = i j R o A , Q W ( E )

where

A,

and

Ro

are the relevant values, for each type of pad, calculated in chapter

5.

When

Eqn 5.12 is

valid,

i.e.

for plane pads, and, approximately for many other pads,

Eqn 7.8

becomes

Fig.

7.2

Opposed-pad bearing: direct supply.




189

In the rest of this chapter, Eqn 5.12 will be considered to be valid.

The bearing stiffness, calculated by means of Eqn 7.5, may be put, as usual, in

the following form:

K = KO K ( E ) (7.10)

where

is the stiffness in the unloaded configuration (E=O) , and

The pumping power is now expressed by the equation:

where

For the friction power, we have:

H f

= 2

H fo H i

( E )

where

1 1

1

Hi.

' Z ( E +F )

Again,

H f o

s the relevant value of the single pads (see chapter 5).

The power ratio in the reference configuration is, quite obviously,

Hfo

Ro Q2

n=4

and the relevant temperature

step is

(7.11)

(7.12)

(7.13)

(7.14)

(7.15)

(7.16)

(7.17)



OPPOSED-PAD AND MUL TIPAD BEARINGS

191

(ii) Working tolerances. Once film thickness

ho

has been chosen on the basis of

the required stiffness of the bearing, of the available rate of flow and pumping

power, and of the moment of friction developed, it has to be borne in mind that, due

to the working tolerances, the actual axial play might be quite different from the

designed value.

In order to study the consequences of this fact, we shall assume th at the refer-

ence value

2ho

is also the maximum allowable value of the axial play. Let the actual

axial play be

g12ho;

clearance error may be then defined as:

(7.20)

If the flow rate is assumed to be equal to design value

Qo,

we can now plot the lesser

film thickness, the higher recess pressure and the stiffness as functions of the load,

as in Fig. 7.4.

- a -

- b -

0.4' ' ' ' '

'

' ' . .

0

5 10

0

5

10

-0

5

10

W

W

W

& Q W

&QRo/2

A,Q Fb/2

Fig. 7.4 Direct supply; working tolerances. Effects of clearance error q.

The bearing stiffness may clearly be much greater when

g c 2 h o ,

whatever the

load. On the other hand, the clearance of the more loaded pad is smaller, and the

relevant recess pressure (and therefore the power required, too) are higher.

Both the pumping power and the friction power are greater when the axial play

is reduced, but the pumping power (which depends on

h-3)

is more notably affected.

The power ratio (for W=O)proves to be proportional t o (g/2ho)2.




In order to avoid excessive pressures, one could adjust the flow rate on field to a

value

Q0(l -9)3 ,

depending on the actual axial play. In this case, the eccentricity and

the recess pressures are equal to the design values, whatever the load. A s a result of

this, however, the minimum film thickness for any given load proves to be reduced

even further. The pumping power also proves

t o

be notably reduced, thanks t o the

smaller flow rate, and the power ratio rises to I7( l-q)?

(iii)

Lubricant temperature.

When a constant-flow supply system is adopted,

adequate control of lubricant viscosity is very important, i.e. control of its tempera-

ture. Indeed, since the hydraulic resistance of the pads proves to be inversely pro-

portional

to p,

t is obvious from Eqn 7.7 that any given load will require a greater

bearing displacement if the viscosity is lower than the design value

po.

On the con-

trary, higher viscosity will lead

t o

smaller displacements, but also to a greater

waste of power.

In Fig.

7.5

eccentricity, pressure and stiffness are plotted as functions of the

load, for several values of the actual viscosity p.

- b - - C -

20

10

0

0

5 10

-0

5 10

-0

5 10

W W

W

&,Q5/2

&gab/: & Q 5 / 2

Fig.

7.5 Direct supply. Effects of

viscosity

on bearingperformance.

7.2.2

Capi l lary compensat ion

(i)Basic equations. When laminar-flow restrictors are used

as

compensating

devices, the recess pressures are (see Eqn 6.10 and Eqn 5.12):



193

where

P1

and

P2

are the relevant pressure ratios for h1=h2=h0.

The load capacity is given by Eqn 7.1. For a symmetrical bearing (Fig. 7.61, it

may be written as follows:

W

=

ps

A, Wc(1;~)

where

p=P1=p2

nd

W' =

(7.21)

1

1

R ' ( 4

P

RYE)

1 - 1 3 1

l uL

Provided Eqn 5.12

is

valid, the last equation may be rewritten as

The total flow rate is:

(7.22)

Fig. 7.6 Opposed-pad bearing: capillary compensation.



194


- a -

0.6

0.8

1

o

w

A,

Ps

- b -

0.6

0.5

0.0I.... ....

0.0 0.5

1 o 0.0

0.5

1

o

Ae PS Ae

PS

Fig.

7.7

Capillary compensation:

a-

eccentricity, b-

flow

rate and c- stiffness versus load, for

certain values of the pressure ratio.




"opt imal"pressure ratio. However, lower values of

p

lead to greater stiffness for

medium and high loads and, furthermore,

to

less waste of flow

rate

and power (both

are halved for p=0.25). On the contrary, no benefit ensues for higher pressure ratios,

which, consequently, should be avoided.

(ii)

Working

tolerances. Let us now examine, as in section 7.2.1.W, the conse-

quences of clearance error q , defined in Eqn 7.20. If the restrictors are fixed, the

bearing simply behaves as an assembly with an actual pressure ratio

1

1+

(1 q ) 3

Bq =

(7.32)

and an actual hydraulic resistance R o q = R 0 / ( l - q ) 3 .he actual power ratio becomes

- a -

1

(7.33)

- b -

8

6 -

n

4 -

0.4

0.3

0.2

6.00

0.25 0.50

rl

Fig.

7.8

Capillary compensation:

a-

actual pressure ratio, and

b-

actual power ratio versus clear-

ance error.

In Fig. 7.8,

pq

and l7,,are plotted as functions of q for certain values of p . Figure 7.9

shows how the tolerances may affect lesser film thickness and bearing stiffness, for

certain values of load. It is clearly advisable to carry out the design in such a way as



198


the flow rate (Eqn 4.76).With the help of the results obtained in sect. 6.3.2, it is pos-

sible t o obtain a set of equations describing the static behaviour of such bearing

systems.

For a symmetrical bearing, the load capacity may be written in the form of Eqn

7.21, but W now stands for the more complicated equation

1

w’=2[l+

- 1 + 4+1 + 4

p2

(1 + &)6 ]

(7.34)

In the same way, the total flow rate, the pumping power, the friction power, the

power ratio and the temperature step are given by Eqn 7.23, Eqn 7.28, Eqn 7.12, Eqn

7.29

and Eqn

7.30,

respectively, in which

Q’

stands for the following equation:

(1

+

&)3

+

1 - &)3

&’=

l + + q z - 1 + 4 +

Bearing stiffness may still be put in the form of Eqn 7.25, where

(7.35)

(7.36)

(1 + &)5

+

4- [

++ q G z 1”

(7-37)

The main parameters are plotted against the load in Fig.

7.10.

It

is easy

t o

see

that the orifices yield greater stiffness than the capillaries (the maximum stiffness

is now obtained for

p=0.586).

(ii) Lubricant temperature and working tolerances. One drawback to orifices is

that the system now proves

to

be highly sensitive to the viscosity

of

the lubricant,

which directly affects stiffness and load capacity.

If /3 is the design value of the pressure ratio for design axial play 2ho and viscos-

ity

p o ,

the actual pressure ratio p,, proves to depend on the clearance error q and on

the actual value of viscosity p:



OPPOSED-PA D AND MULTIPAD BEARINGS 199

- a -

0.0 0.2 0.4

0.6

0.8

1

o

- b -

0.5

0.4

w

Ae

PS

- c -

0.0'

. . ' . ' . . . '

0.0 0.5 1 o

0.0

0.5 1 o

w

Jv

Ae PS Ae PS

Fig.

7.10

Orifice compensation:

a-

eccentricity,

b- flow

rate and c- stiffness versus load, for

certain values of the pressure ratio.



200


- a -

1 C

h

0.5

0.C

C

15

?l

n

10

5

- b -

1 0.25 0.50 0.00 0.25

0.50

1 (1-v m 11

Fig. 7.11 Orifice compensation. Effects of clearance error and lubricant viscosity on: a- pressure

ratio; b- power ratio.

- a -

1 o

- =0.3&,P,

K

2

..

1

- b -

r

0 1 . ’

” ’ . . ‘ .

0.00 0.25

0.50

11

Fig.

7.12 Orifice compensation . Effects of clearance error and lubricant viscosity

on:

a-

lesser film

thickness; b- stiffness.




2

a m =

In Fig.

7.11;Pq

is plotted as well as the actual value of the power ratio:

20

1

(7.38)

(7.39)

Fig.

7.12

shows certain effects of

7

nd p / p 0 on the lesser film thickness and on

stiffness. Clearly, the wider the temperature range allowed

for

the lubricant, the

smaller the tolerances on the axial play should be.

7.2.4 Constant f l ow valves

(i)Basic equations. An opposed-pad bearing can be fed by delivering a flow

Q/2

to

both recesses by means of constant-flow devices, of the type examined in section

6.3.3.

Obviously, the assembly behaves like a constant-flow system until the maxi-

mum pressure p s -Ap is reached in the more loaded recess, i.e. until

(7.40)

I& lI f = l -

3/(l-.4p/ps)

(Ap

is

a minimum pressure drop, characteristic of the device). For larger displace-

ments, the more loaded pad is, in practice, fed through a fixed restrictor, and the

stiffness collapses.

In the normal working range, the equations describing the performance of the

system are written straightforwardly; for the main bearing parameters we may

retain Eqn

7.21,

Eqn

7.23,

Eqn

7.25,

Eqn

7.28,

Eqn

7.15,

Eqn

7.21,

and Eqn

7.30,

in

which we have:

1 1

w ' = p

[m

(7.41)

(7.42)

(7.43)

In Fig.

7.13,

E,

Q,

and

K

are plotted against the load for certain values of j3.



202


- a -

0.0'

' ' . .

'

'

0.6

0.8

1

o

W

-

A, Ps

0.0

0.5 1

o 0.0 0.5

1

o

W

-

-

A, Ps A, Ps

Fig.

7 .13

Constant-flow values: a- eccentricity, b- flow rate and c- stiffness of the bearing

versus load, for certain values

of

the pressure ratio and Ap/p,=O.I.



OPPOSED-PAD AND MULTIPAD B EARINGS 203

(ii) Working tolerances. Let us now consider the effect of the working tolerances.

If the axial play g is a fraction

(1-9)

of design value 2h0, the actual hydraulic resis-

tance of the pad clearances becomes Roq=Ro/(l-q)3 nd hence the actual pressure

ratio rises to

p,=p/(

1-9)s.

The main consequences are:

stiffness increases: e.g. K(W=O)proves t o be proportional to

V(1-9);

on the other hand, the load range + W ( E M ) ,n which both the feeding devices act

correctly, is reduced (substitute

pq

fo rp in Eqn 7.40) as shown in

Fig.

7.14;

friction power and power ratio a re proportional t o 1/(1-9).

- a -

n

v .v

0.00

0.25 0.50

n

Fig. 7.14

Constant-flow valves: maximum displacement and relevant load versus clearance error,

for certain values of the pressure ratio and Apip,=O.l.

Clearly, narrow tolerances and small pressure ratios are required to ensure

high load-capacity; thus, the designer may not make the most of the potentiality of

such a supply system; furthermore, the stiffness of the actual system would be

much higher than necessary. This kind of problem may be overcome by planning

for an on-field adjustment of the flow rates delivered by the valves, in such a way as

to restore the planned pressure ratio whatever the actual value

of

the play

is: i.e.

the total flow rate must be Q(l-9I3 nstead of Q. n

this

way, a n error concerning the

play no longer affects load capacity. Stiffness in the unloaded configuration

is

now

proportional

t o l/(l-q).

The friction power is also proportional to

l/(l-q),

whereas the

power ratio becomes

nq=n/(l-q)4.



204


(iii) Lubricant temperature. The effect of a change in lubricant viscosity

is, of

course, connected with the behaviour of the flow-control valve.

It

has been already

noted that the "ideal" device (for this kind of application) should be able to provide a

flow rate that is independent from recess pressure, but inversely proportional

t o

lubricant viscosity. In this case, the loading performance of the system would not be

affected, while the pumping and friction powers would be proportional to l / p and t o

p, espectively. A

"true"

constant flow valve, on the other hand, provides the same

flow rate, whatever lubricant viscosity is. The main consequences are very similar

to those already seen in the case of the bearings directly fed by positive-displacement

pumps; in particular, the actual pressure ratio is proportional to p, thus a warmer

lubricant leads to less stiffness, while a cooler lubricant leads to greater stiffness,

counterbalanced by a narrower load-range (it is easy to see, from the equations

above, that if viscosity is too high, the valves cannot deliver the flow rate planned,

even

at

W=O ).

7.2.5

Flow dividers

(i)

Basic equations. Flow dividers may be used

to

improve the performance of the

opposed-pad bearings, as compared to the fixed-restrictor supply systems. Some of

these devices are introduced in section 2.3.2.

To s ta rt with, let us consider the

cylindrical-spool valve

in Fig.

2.13.

The hy-

draulic resistance

of

the two branches of the device depends directly on the dis-

placement x of the spool:

(7.44)

The displacement in its turn depends on the recess pressures, i.e. on the load. In

non-dimensional form it may be written as follows

(7.45)

Bearing in mind Eqn

7.2

and Eqn

7.21;

displacement may be rewritten

as

follows:

<=-'-A

i o K u ps w'

(7.46)

Since we have



OPPOSED-PAD AND MULTIPAD BEA RINGS

205

it follows that:

As

usual, the reference pressure ratio p has been introduced above; in the pre-

sent case we have

(7.47)

In the same way

p 2 is

also obtained, and, finally, the non-dimensional load W'

(to be introduced in Eqn 7.21):

(7.48)

Equation

7.48

may be solved together with Eqn

7.46

in order

t o

obtain the non-

dimensional bearing displacement

E

as a function of the non-dimensional load

W'

and of parameters p and

A , 5

a,

=-

lo K, p S =

w'

(7.49)

The flow rate is obtained by adding Q1 nd

Q2

together; bearing Eqn 7.23 in

mind, the non-dimensional flow rate is:

(7.50)

In

Fig.

7.15, E

and

Q'

are plotted against the non-dimensional load, for certain

values of

a,,

n the typical case of p=O.5. It should be stressed th at high values of a,,

may lead

t o

a negative stiffness, coupled with a relatively high flow-rate. In Fig. 7.16

E is plotted against p, for W=0.6and W=0.3.

Pumping power, friction power, the power ratio, and the temperature step are

still given by equations Eqn

7.28,

Eqn

7.15,

Eqn

7.29,

and Eqn

7.30,

respectively.

The above equations are clearly no longer valid after the maximum displace-

ment of the spool has been reached. Since we have {=1 for W'=lla,,

it

follows that



206 HYDROSTATIC LUBRICA TlON

a,<l prevents any out-of-range of the device. If a higher value of a, is selected, to

gain greater stiffness, Eqn 7.49 gives the maximum load for which the system

works properly:

A

w

M - a ,

e p s

(7.51)

For greater loads, the spool cannot displace further and the stiffness strongly de-

creases (in practice, the spool-travel allowed should be even smaller).

- a -

- b -

1 . 0 1 I

1

o

Q

2k'

0.5

0.0

0.0

0.5

1 o 0.0

0.5

1

o

W

- -

A e PS Ae PS

Fig. 7.15 Cylindrical-spool flow divider: a- eccentricity and

b-

flow rate of the bearing versus

load,

for

certain values

of

the valve parameter.

The stiffness of the assembly may be calculated by considering that , after substi-

tuting Eqn

7.46

for 5, Eqn

7.48

gives

W'

as an implicit function of

E;

thus, i t may be

differentiated locally. In particular, in

t h e

centered position, it is easy to find that

(7.52)

where Koc is the reference stifmess of the same bearing fed a t the same value of the

pressure ratio p through fixed restrictors, i.e. Eqn 7.26. t would seem possible to

obtain any stiffness whatsoever by adjusting the parameters of the device; very great

stiffness would, however, require values of a,, hat exceed the limit

aU=1

uoted

above.



OPPOSED-PAD AND MULTlPAD BEA RINGS

207

Better performances may be obtained with the

tapered-spool value

in Fig. 2.14.

The hydraulic resistances of the two branches

of

the device may be expressed in the

following form:

0.d

E

0.

- a -

(7.53)

- b -

0.1

0.0

0.5 1

o

0

0.0 0.5 1 o

B

Fig.

7.16 Cylindrical-spool flow divider: eccentricity of the bearing versus pressure ratio,

for

cer-

tain values

of


The position

of

the spool depends on the load, according to Eqn 7.46. Proceeding

as

before, we again obtain Eqn 7.21 and Eqn 7.23, in which:

1

(7.54)

'=

1



208 HYDROSTATIC LUBRlCATfON

- a -

1.01

0.0

0.5

1o

-

A, PS

1 a

Q

-

24’%

0.5

0.c

I

- b -

1 o

w

A, PS

0.5

Fig.

7.17 Tapered-spool flow divider:

a-

eccentricity and b- flow rate of the bearing versus load,

forcertain values of the valve parameter.

- a -

0.E

E

0.4

0.C

I

I3

- b -

0.5 1

o

I3

Fig. 7.18 Tapered-spool flow divider: eccentricity of the bearing versus pressure ratio, for certain

values

of




OPPOSED-PAD AND MUlTlP AD BEA RINGS

209

It

is now possible to plot the displacement of the bearing and the flow

rate

as

functions of the load and of parameter

a , ,

as in Fig. 7.17,

o r

E as a function

of 8 ,

as

in Fig. 7.18.

The reference stiffness now becomes:

and a very high stiffness may now be obtained, even for a,<l.

Let us now consider the simple, but effective,

diaphragm

v a lv e shown in Fig.

2.15. The relationship between the hydraulic resistances and the displacement of

the diaphragm

is

of

the following type:

Displacement

x

depends on the differential pressure

p 2 - p l ,

according to Eqn

7.45, in which A, is understood as a n "effective"area of the diaphragm. Coefficient

RvO

or diaphragms may be calculated from Eqn

4.59

(substituting h with l o ) , while

the effective area is

(7.58)

Stiffness K , may be calculated as

- b -

r2

I

I

Fig. 7.19 Diaphragm flow-divider.




(7.59)

for diaphragms as in Fig.

7.19.b,

whereas for diaphragms

as

in Fig.

7.19.a

he fol-

lowing may be taken:

(7.60)

Proceeding as above, the non-dimensional load and

flow

rate are easily found

(see also Fig.

7.20

and Fig.

7.21):

1

1

- a -

1 . 0 r I

(7.61)

(7.62)

- b -

1

.o

w

*e

PS PS

0.0 0.5 1 o 0.0

0.5

-

Fig.

7.20

Diaphragm flow-divider:

a-

eccentricity and

b-

flow rate

of

the

bearing

versus load, for

certain

values

of the valve parameter.



OPPOSED-PAD AND MUL TIPAD BEA RINGS

211

- a -

0.4

E

0.:

0.1

- b -

0.8 I

E

0.4

0.0

0.5

1

o 0.0

0.5 1

o

D D

Fig. 7.21 Diaphragm flow-divider: eccentricity of the bearing versus pressure ratio, for certain

values

of


The bearing stiffness, in the unloaded configuration,

is:

(7.63)

Great stiffnesses may be obtained for smaller values of

a,

than with th e preceding

devices. Moreover, thanks

t o

the small mass of the diaphragm, the dynamic be-

haviour is generally better.

(ii)

Workin g tolerances.

Once a device has been selected, with a parameter

a,,

to

obtain a given performance with a given opposed-pad bearing, it has to be borne in

mind that, due to working tolerances, the actual axial play g may differ from the

design value

2 4 .

Consequently, the actual hydraulic resistance of the pads becomes

Roq=RoI(l-q)3n the unloaded configuration, and the actual pressure ratio is now

given by Eqn 7.32.

As

in the preceding sections, q is given by Eqn

7.20.

The actual

performance of the assembly may thus be evaluated again with these new data. I t is

also necessary to take into account the working tolerances of the valve (see Example

7.1).



OPPOSED-PAD AND MU1TlPAD BEARINGS

21

3

Of course, it is necessary to also verify the behaviour of the system for different

loads. For small loads, Eqn 7.63 gives K0=3.26.K,=6.5S.1@ N l m (bear in mind that

KO, s the stiffness tallying to simple capillary restrictors, Eqn 7.26), which is much

higher than required. At the maximum load W=35

K N

(i.e. W'=0.626), eccentricity

may be calculated as ~=0.22, good result compared with capillary compensation.

Flow rate at W=O can be easily calculated: for p=0.05 Nslm2, &0=41.7.10-6m3/s.

The relevant pumping power is, hence, Hp0=188

W.

Equation 7.62 shows that Q and

H p are smaller when the bearing is loaded.

Concerning the design of the controlled restrictor, the reader should verify that

the selected value of the pressure ratio may be obtained (for instance) with r2=6 mm,

r,=3 mm, and 10=76 pm (see Fig. 7.19) and that the relevant Reynolds number is

quite low. Parameter A, is easily calculated: A,=0.061~10-3m2. From Eqn 7.49 it

follows that the stiffnessof the diaphragm should be K,=6.59.106 N l m; for instance,

using a steel diaphragm (El(l-v2)=226GPa) with a thickness s=l mm, the required

value of K, may be obtained selecting rv=12.6 mm, for the diaphragm in Fig. 7.19.a.

It is clear that the designed values of a, and

jl

can only be obtained with a cer-

tain approximation, but

they are not critical, in this case, since limited variations

can be allowed, as may be seen by examining Fig. 7.22. In particular, it may seem

difficult to obtain exactly the calculated value of clearance lo; nevertheless a greater

clearance would clearly lead to quite small changes in stiffness,

For

instance, to

increase

lo

by

20%

would give av=0.458 and /3=0.425, and hence the displacement at

W=20

KN

remains practically unchanged. Of course, flow rate would increase as

the pressure ratio increases.

7.2.6

Design

of

opposed-pad bear ings

In most cases the specifications of thrust bearings consist in a certain value

W,

of the load to be sustained a t a displacement smaller than a certain value eM. Cer-

tain obvious constraints generally exist for the size of the bearing, a s well as for the

supply system (max. pressure and flow rate; viscosity of the lubricant) and for the

friction (max. torque).

Since

too

many parameters and constraints are involved, a general design pro-

cedure cannot be given, except

for

certain main aspects: let

us

briefly examine how

the choices of the designer may affect the final result, and try t o assess optimal

values of certain parameters, on the usual basis of the least power loss. For the sake

of brevity, the following considerations will refer mainly to the case of annu-

lar-recess pads fed through laminar-flow restrictors, but they could easily be ex-

tended

to

other pad shapes and supply systems.

A s

a

first

design step, one may

try

to select

a

maximum value

EM

of the eccen-

tricity, commonly & ~=0 . 5 ;owever, it

is

not advisable t o allow eccentricities greater



21


than 0.6. When a maximum value of the displacement is given, this immediately

leads to an upper constraint

gk=2

eM/&M for the axial play

g=2ho.

When high stiff-

ness is required, g),,may prove to be too small; in this case i t may be useful to select a

smaller value for the eccentricity.

A

set of geometrical parameters can now be stated: as for the single-pad bear-

ings, it is generally convenient to select large lands for low-speed bearings, the con-

trary for high-speed bearings.

The pressure ratio should remain in the 0.3+0.6 range; it is advisable

to

start

fixing a low value for the design pressure ratio j? in order to allow greater margins

for the working tolerances: this point is clarified below.

The load parameter

W ' ( E M $ ) = A ~ ,

an now be calculated, or read in the appro-

priate chart. Supply pressure is often prescribed, and then the effective pad area

(i.e. the pad size) is readily obtained; otherwise, both A, andp, have to be suitably

chosen. Caution is necessary in selecting high values for the supply pressure, since

an excessive temperature step in the lubricant may follow: see Eqn 7.30.

The problem of a suitable selection

of

the axial play and of the lubricant viscosity

may be tackled on the usual basis of minimization of the total power consumption:

Proceeding as in the case of the single-pad bearings (sect. 6.21, it may be found

that, for plane or tapered circular pads, rotating at a speed a, he total power may

be written as

the relevant value

of

the viscosity being

(7.64)

(7.65)

Equations for H; and

p i

are given in Chapter 6 (namely, Eqn 6.58 and Eqn 6.62; see

also Fig. 6.16 and Fig. 6.17). Equations 7.64 and 7.65 are valid for all constant-pres-

sure supply systems, provided the relevant values for W' are selected.

Examining Eqn

7.64, it

should be noted that,

as

for the single-pad bearings,

it

is

convenient to select large recesses and small clearances, provided that

a

lubricant

with sufficiently low viscosity is available, in order to get

n=1.

f, instead,

l

is

fixed,

the optimal value for axial play is the one which gives n=3.



OPPOSED-PADAND MULTlPA D BEARINGS 215

Further observations are, however, necessary. In particular, i t must be stressed

that the choice of axial play is meaningless if the relevant working tolerances are

not specified. As a matter of fact, a whole range ( 2 h 0 - 6 ~ & < 2 h o should be selected,

instead of

a

single value, for the axial play

g.

To

reduce manufacturing costs,

it

is

advisable for tolerance

Sg

to be as large as possible, but i t has been shown th at large

tolerances can notably affect the performances of the bearing (e.g. see Fig. 7.8 and

Fig. 7 . 9 ) .For instance, selecting e ~ = 0 . 5 ,

3=0.5,

and

Sg=h0,

it may be seen that the

actual pressure ratio may prove t o be as great as 0.89 (take

q=0.5

in Eqn 7 . 3 2 ) ; he

main consequence is that load capacity is strongly reduced and the load designed t o

be sustained at

~ ~ = 0 . 5

annot be reached even at

e= l

(this may be verified by means

of Eqn 7 . 2 2 ) .It should therefore be clear that smaller tolerances must be selected.

The actual value of the power ratio will also be dependent on the actual clearance

(see Eqn 7 . 3 3 ) ; or instance, forfi=0.3, 6g=ho/1.5and

I7=1,

he actual power ratio may

prove t o be as great as 2.6.

A convenient design practice consists in first selecting a suitable value for toler-

ance

4;

xial play may then be chosen (e.g. with the aid of Eqn 7 . 3 2 ) n such a way

as

to

keep the actual pressure ratio within a reasonable range.

For

instance, we

may take

ho=l.6.6g

iffi=O.3,

or ho=2.2.6g

f/3=0.4. Smaller values for ratio ho/Sg can

be selected, when small eccentricities (e.g. ~ 0 . 3 )

re expected.

In order t o free the loading performances from the adverse effects of manufac-

turing tolerances, it is possible to select adjustable restrictors, that allow us to re-

store the design value of the pressure ratio on field, whatever the actual value

of

axial play. Nevertheless, for high-speed bearings, 6 should not be greater than h o / 2

to avoid excessive values of the power ratio. Instead, for low-speed bearings, toler-

ances may be larger.

Concerning viscosity of the lubricant,

it

should be borne in mind that, in gen-

eral, it may not be considered a constant, since it largely depends on lubricant

temperature, and hence on the working conditions and on the effectiveness of the

cooling of the reservoir. Unless a temperature control system is provided, it is ad-

visable to verify the performances of the bearing for all the expected range of viscos-

ity. In particular, when laminar-flow restrictors are used, f i is not affected by lubri-

cant temperature, nor load capacity. For low-speed bearings i t is convenient at the

design stage

t o

consider

p

as the lower end of the expected viscosity range (in this

way flow rate and power loss should always turn out to be smaller than calculated).

Otherwise, optimization may be looked for, for a center-of-range value of viscosity: it

is then necessary to check flow rate and power for all the viscosity range

(it

is con-

venient, however, for the supply system to be able to keep lubricant temperature

within a narrow range).



21


A s a result of the above remarks, a simple procedure may be proposed that

should lead to a quick and "optimized"design of the opposed-pad bearings. Let us

assume a certain load W M s given, to be sustained with a displacement smaller

than

e M ;

urther constraints may be required (e.g. viscosity

o r

supply pressure may

be imposed): the device may then be designed following the steps listed below.

i)

Choose a trial set of geometric parameters: select large recesses unless low

speeds are expected.

Choose a trial value of the maximum eccentricity, such as ~ ~ = 0 . 5 .

Choose a trial value of the design pressure ratio: perhaps p=0.3.

ii) Evaluate coefficient W' (e.g. from Eqn 7.22, for capillary compensation).

Decide supply pressure

p s ,

when it is not prescribed, and effective area A,,

in such

a

way

to

satisfy Eqn

7.21.

Calculate the pad size, in order to get the above value ofA,.

iii) Select a suitable value for the manufacturing tolerance on axial play: bear in

mind that the narrower

Sg

is, the lower the power losses can be.

Select the design film thickness ho (the actual value

g

of the axial play must

consequently lie in the 2hot2ho-Sg range); for capillary compensation, it

should be

ho>1.5.Sg;

smaller values may be used if ~ ~ c 0 . 5 ,ut wider clear-

ance should be selected for p>0.3.

Check if stiffness is large enough (Eqn

7.26,

for capillary compensation).

Otherwise reduce Sg and ho, or select a smaller value for EM, starting again

from point (ii),

iv) Calculate the

"optimal"

value of viscosity, i.e. the value k p t that leads to

l7=1

(e.g. by means of Eqn 7.65, for rotating thrust-bearings).

Check that

b p t

is a plausible value (sometimes p may be directly imposed by

the specifications);

if

it is not, try

t o

modify the geometrical parameters

(namely the recess width) or the design film thickness

or

the supply pres-

sure. However, if the speed is very low, it will not be possible to get

n=1;

n

this case select the highest allowable viscosity, and the recess width that

ensures the least consumption of pumping power. On the contrary, a very

high speed would lead t o values for b p t which are too small; in this case

it

is

necessary to accept values of l7 greater than 1.

v) Calculate flow rate (Eqn 7.231, pumping power (Eqn 7.281, friction force (or

friction torque) and friction power (Eqn 7.15), with reference to the design

configuration @=hotE=O).

Check the same parameters for different values of play and eccentricity.

Should flow rate seem

too

great, and

h o

cannot be further reduced, it will be

necessary t o accept a power ratio of l7>l (e.g. by increasing p

or

by reducing



OPPOSED-PAD AND MULTIPAD BEA RINGS 217

ps ) , consequently increasing friction. Conversely, friction may be easily

reduced with a larger expenditure of flow rate and pumping power.

Check the temperature step in the lubricant (Eqn 7.30).

If the supply system is not able to keep lubricant viscosity within a narrow

range, check the friction and the temperature step for the maximum ex-

pected viscosity, too, and check the flow-rate for the minimum viscosity.

vi) Design the compensating restrictors (see section 4.7.6).

vii) Check the Reynolds number in the bearing clearances.

For bearings rotating at high speed, check the effect o f inertia forces in the

lubricant (see sections 5.3.4, 5.4.5 and

5.5.2).

viii) Check the dynamic behaviour

of

the system (see Chapter

10).

EXAMPLE 7.2

Design an annular-recess opposed-pad bearing to sustain a load varying in the

210

K N

range, with a stiffness that is always greater than 108 Nl m , and rotating at

157 radls (1500 rpm); the supply system to be

used

delivers lubricant at 7 MPa. Fur-

ther constraints consist in the inner pad diameter 2r, being required to be greater

or

equal to 45 mm and manufacturing tolerances to be in the order of 30

p

To start with, one may state the trial values for the pressure ratio (8=0.3) and for

the maximum eccentricity ( ~ ~ = 0 . 5 ) .ince laminar-flow restrictors are selected, the

load parameter is readily calculated (Eqn 7.22) as W=0.66; this means that a n effec-

tive area A, not smaller than 2.16.10-3 m2 is required (Eqn 7.21): for instance, D=72

mm, r’=O.625, and a=1.5 mm may be selected; hence Eqn 5.66gives A,=2.2.10-3 m2.

Since the value of tolerance on axial play is given, the maximum film thickness

can be selected: ho=1.5.&=45 pm; consequently the actual axial play g will be in the

60t90p range. From Eqn 6.62 we may obtain $=13.7; from Eqn 7.65 it follows that

a lubricant should be selected with a viscosity of p 0 . 3 Nslm2 n order to obtain

n=l.

From Eqn 5.68 and Eqn 5.72 we obtain Ro=161.109 Nslm5 and Hf0=87 W. The

calculations are now straightforward.

When g=2ho=90pm, stiffness at W=O is K0=O.43.1O9 N l m , flow rate is Q=26.10-6

m3/s,

pumping power and friction power are Hp=182 Wand Hf=175 W, respectively.

The design load can be sustained with an eccentricity that is slightly smaller than

0.5;

he relevant stiffness is again K=O.43.1O9 N l m (however, stiffness is greater for

intermediate loads); the flow rate and pumping power are smaller, while the fric-

tion power is 30% greater.

When g=2ho-fig=60 m, the actual pressure ratio becomes (Fig. 7.8) 8=0.59;

in

order to sustain 10

KN,

an eccentricity of ~ = 0 . 5 3s now required; the smallest film

thickness is therefore h,=14 pm. Stif fness at W=O is much greater than before

(K=O.75.1O9 Nlm), while at the maximum load

we

get K=O.41.1O9 N l m . The pump-



218 HYDROS

JATlC LUBRICATION

ing power is obviously smaller; on the other hand, i t is Hf=262

W

a t W=O, nd

H f-365

W at W= 0 KN.

Th e hyd rau lic resistance of the restrictor m us t be Rr=Ro(l-p)p=376.1@N s l m 5 ;

this m ay be easily obtained, say, w ith a

1=30.75

mm

length o f a

1

mm -bore pipe.

The value of viscosity selected above may seem too high

in

certain ap plications.

It is clear th at to select a lower value would have no effect on bearing stif fness, but

flow rate and pumping power would become much greater.

For

instance, selecting

p=O.1 Nslrn2,

for

g=90 pm and

W=O

we would have

Hp=547 W

and

Hf-58 W ,

i.e.

I l = O . l ,

with a considerable increase

in

total power. In this case,

it

might prove

suitable t o slightly change the size

of

the pa d, increasing both

a

and

D

in order to

reduce H p .

7.3 LEAD

SCREWS

Hydrostatic lead screws may be treated, a t the beginning, in exactly the same

way as opposed-pad bearings, using the relevant values for

A,,

Ro, and

H f o

(see

section 5.10). Indeed it is possible to obtain the recessed nut by injection of resins,

using the treaded shaft itself as a mould (see. ref,

7.1);

his allows us to obtain pre-

cisely controlled values of axial play, and prevents large lead errors between screw

and nut, that may be responsible for a serious decline in performance, especially

when hydrostatic lubrication concerns several turns of the thread. However, the

pitch may not generally be considered to be constant for the whole length of the

screw, since local pitch variations exist, whose maximum value depends on the

degree of accuracy and overall length. Precision screws may show lead variations

in the order of 10+20 pm, although screws of even greater precision are commer-

cially available (ref. 7.2).

Let us now consider a nut working in a zone where the pitch of the threaded

shaft differs from the pitch of the nut by a quantity 6p; of course,

& is

meant

as

an

average error, since the pitch varies continuously. The main consequence is that

the available axial displacement between screw and nut is no longer

ho,

since con-

tact occurs at a n eccentricity

where we have introduced the non-dimensional pitch error

(7.66)

(7.67)

This fact may considerably reduce the load capacity of the system. Furthermore, the

hydraulic resistance of clearances no longer varies according to Eqn 5.12, as was




219

assumed for opposed-pad bearings; instead, Eqn

5.117

should be used, which may

lead

to

a reduction in stiffness.

7.3.1 Direct supply

When c5p is negligible, the equations presented in Sect. 7.2.1 are applicable, pro-

vided that A,, Ro, and H f o are calculated as in section

5.10.

In general, however,

load capacity is given by Eqn

7.7

in which Ro is calculated from Eqn

5.116,

and

(7.68)

1

w

=

( 1 4 3+

&i2 (I-&)

(1+43

+ &'2 (I+&)

Plots of eccentricity are given in Fig. 7.23, versus the non-dimensional load.

Fig.

7.23 Lead screw. Eccentricity versus

load (constant-flow

direct

supply).

Of course, stiffness also is affected by pitch variations; it is obtained, as usual,

from Eqn 7.5. In particular, at E=O,we get

(7.69)

where

KO s

the design value, given by Eqn

7.11.



220


7.3.2

Constant

pressure supply

flow rate , can be studied by substituting

R’=ll(l+d3

with

The effect of pitch errors on recess pressures, and then on load capacity and

(7.70)

For the sake of brevity, we shall discuss the results for only the simplest case,

i.e. capillary compensation, even

if

a new type of floating controlled restrictor (ref.

7.3)

has been proposed, especially for hydrostatic lead screws (ref.

7.4).

Load capac-

ity is obtained from Eqn

7.21,

where

(7.71)

In Fig 7.24 a plot of the load-eccentricity relationship is given for certain values of

the design pressure ratio and relative pitch error.

Flow ra te is still given by Eqn

7.23,

where

(7.72)

- a -

- b -

1

.I

E

O .

0

0

o

1

o

W

o 0.0

0.5

W

-

A e PS Ae PS

0.5

-

Fig.

7.24

Lead screw. Eccentricity versus

load

(constant-pressure supply, capillary restrictors).




221

Stiffness at

W=O

may be calculated by means of the following equation

(7.73)

where K O s given by Eqn 7.26.

In conclusion it may be noted that, to make manufacturing easier, the continu-

ous recess i s often substituted by a sequence of small pockets; in this case, the effec-

tive area (and hence the load capacity) is considerably reduced, but the hydraulic

resistance is greater: hence the flow rate is also reduced. If each pocket is fed inde-

pendently through a separate restrictor, the effect of the pitch error on load capac-

ity, flow rate, and stiffness is less pronounced than in the case of continuous recess

(see also ref.

7.5).

EXAMPLE 7.3

A hydrostatic lead screw has to carry loads as great as f 15 KN, with a rigidity

W / e greater than 109 Nl m , when supplied at a pressure ps=7 MPa; the friction

torque must be smaller than 5 N m at the maximum turning speed R=52 radls ( ~ 5 0 0

rpm). Moreover the overall length of the nut is required to be smaller than 200 m m

and its external diameter must be equal to 150 mm; the lead must be p*=20 mm.

Figure 7.24 shows that the product psA e must be substantially greater than the

maximum load W ~ = 1 5

N ;

initially, one may state that psAe>1.7.W~=25.5

N :

since ps is given, it follows that Ae23.64.10-3 m2. Let us select a thread with flank

angle 0=15 degrees, external diameter

D=90

mm and the root diameter of the nut

2r,=65 mm; taking a land width of 3.5 mm (i.e. a'=0.28) we get Az=0.34, and thus

lubrication should be extended to two turns o f the nut; indeed, introducing n=2 into

Eqn 5.115, it follows that A,=4.38.103 m2.

In order to properly select the design value of the fi lm thickness, it is necessary

to know the maximum value of the pitch error; assuming SpllO pm, it follows that

to obtain 6p'<0.4 we must have hO>%

pm

(Eqn 7.67). On the other hand, Fig. 7.24

shows that, assuming @=0.3,at the maximum load WM=15KN, we have an eccen-

tricity ~ 0 . 4 5for Sp3=0.4);hus, in order to have a displacement that is smaller than

e~=1 5.103/109 =15 pm, we must have h0133 pn . Finally, h0=30 pm may be cho-

sen. It is worth noting that i f the threaded shaft were built with greater accuracy, it

would be possible to select smaller values for h , with notable benefits in stiffness,

flow rate and pumping power.

Equation 5.71

(or

Fig. 5.14) gives HF=0.41 and then the friction power is (from

Eqn 7.15 and Eqn 5.118) Hf=(0.38 rn3).H;p@. The friction torque Mf=HfIRmustbe

smaller than

5

Nm: sincesat R=52 radl s and ~ = 0 . 4 5 i.e. Hj=1.25) it is

Mf=(25m31s).p, it follows that viscosity must be lower than 0.2 Nsl mz .



222

HYDROSTATIC

LUBRlCA

TION

Th e flow rate depe nds on th e actual value o f viscosity (i.e.

it

is inversely propor-

tional to

it);

for instance,

if

we select p=O.l N sl m z, E qn 5 .116gives Ro=185.109 N sl m 5

and, hence, at W=O, Q=22.7.10-6

m3ls

and

Hp=159

W. Both th e f low rate and the

pum pin g pow er change lit t le wh en the load is applied.

The power loss due to friction at E=O and p=O.l N s l m z i s Hf=104 W a n d prov es to

be proportional to viscosity.

7.4

SELF-REGULATING BEARINGS

A self-regulating bearing (SRB) may be regarded as an opposed-pad bearing

with a built-in

flow

divider, which is made up of another couple of pads. Compared

to conventional flow dividers, however, the important advantage of the SRB, under

dynamic loading, is that

it

has no other moving part (such as spools o r di-

aphragms). On the other hand, the

"inf ini te st i f fness"

which could, in theory, be

achieved with the aforesaid devices, can no longer be obtained,

/ I

W

/ I

Fig.

7.25

Self-regulating bearing.

Pressure

distribution.




223

The principle on which flow self-regulation is based has already been explained

in section

2.3.3.

In order to have R,=Ri, whatever the displacement of the moving

member (Fig. 1.4.c), the four clearances must have the same film thickness

ho

and

the same hydraulic resistance

Ro

in the centered position. When circular clear-

ances are used, the latter condition is satisfied if rllr2=r31r4=r~Fig. 7.25). In this

case, Ro is given by Eqn 5.21, i.e.

(7.74)

If different types of pads are used (e.g. rectangular pads or double-thread

screws), the relevant expression of Ro should be used instead of Eqn 5.21: see ref. 7.5

and ref. 7.6.

In the centered position, the resultant of the pressure on the bearing surfaces

vanishes, and hence the load capacity is null. The overall hydraulic resistance of

the assembly is simply Ro. t should be remembered that , if we wish to compare this

bearing with an opposed-pad bearing, made up

of

two annular-recess pads, with the

same values of the radii and of the axial play 2ho,Ro i s not the same in both cases.

Indeed, each of the pads of the opposed-pad assembly is characterized by a hy-

draulic resistance (at

E=O)

which is half the hydraulic resistance of the SRB.

Under an external load

W,

the bearing

is

displaced: if

h

is the thickness of

clearances s l and z2, the thickness of s2 and Ll is 2ho-h.After the usual assumption

E=(h-hO)lhO,t is easy

t o

see tha t the hydraulic resistance of the assembly becomes:

R=RoRA (7.75)

where

1 1 1

Rk = 2 [ m

(7.76)

The load capacity

is

obtained by integrating the lubricant pressures on the bear-

ing surfaces; i t may be expressed in the following form:

W = p r A , W' (7.77)

where:

( 1

+ 4 3 ( 1

€13

W'

=

( 1+ €13 +

(1

4 3

(7.78)

For

flat

circular clearances the effective bearing area is



224


(7.79)

It should be noted that this expression coincides with the one obtained for a n annu-

lar-recess pad with the same radii.

The pumping power lost in the clearances of the bearing is given, as usual, by

(7.80)

it may also be written as a function of the load:

This allows us to look for an optimization of the shape of the bearing, from the point

of view of the pumping power. In particular, for annular clearances, it can be

demonstrated that, given the inner and outer radiuses r l and r4=r i I r ' , a value of r;

exists that minimizes pumping power. This "opt imum"value is obviously obtained

by solving the following equation:

and is plotted in Fig. 7.26.a.

As

for conventional bearings, it can be shown that

when R s not negligible, it is convenient

to

use wider recesses, because of the pres-

ence

of

friction power, which is easily found by summing together the effects

of

the

four lands. Bearing Eqn 4.60 in mind, we find it is

H f = H fo H j E )

(7.81)

where:

A 0 4

H

---a2

fo -16h0

p H ?

1 1 1

Hi. =z(i-,+1+E)

(7.82)

(7.83)

(7.84)

A*,,

R*

and

H?

are plotted in Fig.

7.26.b.



OPPOSED-PAD AND MULTIPAD BEARINGS 225

- a -

- b -

0.0' '

.

'

0.5

0.7

0.9

f r'

Fig. 7.26 Self-regulating bearing. a- optimum value of ratio

r;

versus ratio

r'; b-

relevant values

of &e,

R* and HT

versus

r'.

The foregoing equations need to be completed by a relationship, which depends

on the supply system, between the rate of flow Q and the recess pressure

pr.

7.4.1 Direct

supply

&=const., and the supply pressure is

PS

= p r

= R

Q =RoR;1(4 Q

When the bearing is directly fed by a volumetric pump, we obviously have

(7.85)

The

oad capacity, for Eqn 7.77, is

W = A e R o Q R ; l

W '

The pumping power is

H p

=Hi

Ro R;1 Q2

Since the friction power is still given by Eqn 7.81, the reference power ratio

is:

n = H f O

Ro

Q2

(7.86)

(7.87)

(7.88)

The average temperature step in the lubricant (supposing, as usual, tha t we have

adiabatic flow) is:




0.E

E

0.4

0.2

0.c

I

- a -

2.5 5.0

W

AeQRo

a

O

K 6

H i

0

-

T

AT0

2

- b -

1

0'

0.0

2.5 5.0 7.5

W

&Qb

Fig.

7.27 Self-regulating bearing. Direct supply: a- eccentricity and stiffness versus load;

b-

recess pressure, pumping power, friction power and temperature step versus load.

(7.89)

where

Differentiating load capacity in relation to displacemen t, stiffnes s is obtained:

where

1

K = :[ &

+ml

(7.91)

(7.92)

(7.93)

In Fig. 7.27,

E, ps,

K, H p , H f , and AT are plotted

as

functions of

the

load.



OPPOSED-PA D AND MULTIPAD BEARINGS 227

Let us now compare the SRB

to

the annular-recess opposed-pad bearing, with

the same radii, fed by two pumps (see section

7.2.1

and section 5.4 for the relevant

equations). It is easy to see that the t w o bearings have similar load-displacement

characteristics when the total flow delivered

t o

the opposed-pad

is

twice the flow

required by the

SRB

bearing in mind that

Ro

is not the same in the two cases). The

supply pressure for the SRB equals the sum of the two recess pressures of the op-

posed-pad; this means that, when the load is applied, the maximum supply pres-

sure is approximately the same (slightly greater in the case of the SRB). The total

power expense is the same for both bearings.

The effects of any change in lubricant viscosity are very similar to those already

seen in the case of the opposed-pads fed at a constant flow.

The effect of the working tolerances in the case of the SRB is

a

more complicated

matter to study than in the case of the opposed-pad bearings, since three indepen-

dent axial plays are now involved.

For

a full discussion of this, see ref. 5.43.

7.4.2 Constant pressure supply

Unlike the other hydrostatic bearings, the SRBs can be fed from a constant-

pressure source, without any compensating devices, i.e. fixed restrictors

o r flow

control valves. However, such devices may be useful for modifying the performance

of the bearings.

In the simplest case no restrictor is used, and then p,=p,=const. Load capacity

is given by Eqn 7.77 and the rate of flow is:

(7.94)

The pumping power is still given by Eqn 7.80 and the friction power by Eqn 7.81.

Consequently, the power ratio is:

and the temperature step:

The bearing stiffness is given by Eqn 7.91, in which now

(7.95)

(7.96)

(7.97)



OPPOSED-PAD AND MULTIPAD BEAR INGS

229

and

- Y

Y

R , . = - R o

thus, bearing in mind Eqn 7.75:

The main bearing parameters now become:

W = p s p ' A e W '

P

P C

AT,, = S i m

n = U

H

P? Y

(7.99)

(7.100)

(7.101)

R' H'

l + p

AT = AT^

--&+

(7.102)

The main effects of the inlet restrictor are also shown in Fig. 7.28.b.

What is stated a t the end of the preceding section, concerning the effects of the

working tolerances, also holds good here (see also ref. 7.7).self-regulating bearing

7.5

HYDROSTATIC SLIDEWAYS

One of the most common uses

of

hydrostatic bearings is to ensure tha t the heavy

carriages of large machine tools move smoothly. In th is case, four, o r more, pads

are interposed between the carriage and the guides; for instance, in Fig. 1.16, all the

loads and moments acting on the carriage, except the axial component of the load,

are supported by an assembly of 12 pads, which are so arranged as to form 6 op-

posed-pad bearings. The axial load may, in its turn, be supported by a hydrostatic

lead screw, which ensures a smooth and frictionless drive of the carriage, with a

high degree of stifiess.

Such assemblies are studied by examining how the displacement of the carriage

affects the mean film thickness of each pad (the lack of parallelism of the pad sur-

faces, deriving from tilting displacements, can, in general, be disregarded).

By way of example, let us consider the very simple four-pad assembly in Fig.

7.29. When no external force is acting on the carriage, w e will have, for each pad,

hi=ho ,p i= p r o = W d 4 Ae ,Q i= p r d Ro .W o is the weight of the carriage itselc the coeffi-



230


cients A, and R o depend on the type of pad: for rectangular pads, for instance, see

Eqn 5.101 nd Eqn 5.102.

Under an external load, the carriage is displaced, until a new equilibrium

is

reached:

(7.103)

My = a Ae

(P1

+ P4 - P2 -P3)

Fig. 7.29

Four-pad

assembly.

If each pad is directly fed

by

a separate pump, the

flow

rates Qi do not vary;

hence, from Eqn 5.2,Eqn 5.10 nd Eqn 5.12,we obtain

(7.104)

By

introducing Eqn

7.104

nto Eqns

7.103,

set of equations is obtained, which re-

lates the external force to the mean film thickness of the pads. Each hi can be ex-

pressed

in

terms of the angular displacements

( O x ,

0,)

of the carriage and of the

vertical displacement of its center Ahc.



OPPOSED-PAD AND M UiTlPA D B EAM NGS 231

In Fig. 7.30 the displacements are plotted against the component W of the load

along axis z , for a few values of moments M, nd My.

tors, Eqns 7.104must be substituted by

If the pad is supplied a t a constant pressure p s , through fixed laminar restric-

(7.105)

where

A plot of the relevant displacements, for certain loading conditions, is given in Fig.

7.31.

The total flow rate is the s u m of the pad flows

1.25

k

h0

1

oo

0.75

0

- a -

0.4

b

e x

-

ha

0.2

1

2

rc

wl

0.0

0

- b -

1 2

JY

M6

Fig.

7.30

Load-displacement performance

of

a four-pad slideway (constant flow supply).

M y = 0 . 2 5 * a d o .

(1): M,=M =O; ( 2 ) :MX=O.25 .bWo,M,=O; (3): M,=O.S.bWo, My=O; (4): M X = O . 2 5 * b W o ,



OPPOSED-PAD AND MULTIPAD B EARINGS

233

QiRoi

P i l = ( l - E i ) s

;

Q2

Ro2

Pi2

=

(7.107)

The eccentricities may be written as functions of the average value E=(E~+E~) /~=

and

so

on. It is then possible to evaluate the performance of

=(%+e4)/2 and

of

the angular displacements 6, and

Sy

of the carriage:

e1=E+Syu/hO-6xb/h0,

the assembly.

Greater details concerning an analytical treatment of complex pad assemblies

may be found in ref. 7.9; ref 7.10 contains results concerning a vertical boring

machine.

7.6

MULTIPAD JOURNAL BEARINGS

The multipad journal bearings may be regarded as a se t of cylindrical pads

acting on the same journal (Fig.

1.12.a).

Consequently, the load capacity of such

bearings can be evaluated, for any given displacement of the axis of the shaft from

the axis of the bearing, by calculating the load sustained by each pad, as in section

5.8, and then summing all of them together vectorial.

Although the working of the multipad bearings is simpler to understand and to

calculate (ref. 7.111, compared to multirecess ones, they are less effective and (due to

the drainage grooves separating the pads) more expensive to build. Besides, a t high

speed, the grooves may lead t o the inlet of air in the clearances.

For

these reasons,

multirecess bearings are generally preferred.

Here we shall confine ourselves to showing the load-displacement performance

of a typical four-pad bearing, fed through laminar restrictors (see Fig. 7.32). Of

course, all the other supply systems could, in principle, be used.

The reference hydraulic resistance Ro of each pad, which is necessary to design

the supply restrictors, may be calculated from Eqn 5.111 and Fig. 5.32. Hence, the

flow rate in the concentric configuration is

:

P s

Q = n P R O

(7.108)

( n is the number of pads). When the bearing is loaded, Q proves to be slightly

smaller. The pumping power and the friction power are (for E=O)

(7.109)

(7.110)



234

- a -


- b -

0.4

.

0.0

0.4 0.8

E

Fig.

7.32

Multipad

journal

bearing: load

versus eccentricity (LID=O.5,llL=O.5,=85",ai=70.7O).

Af is the equivalent friction area of each pad, i.e. the sum of the land area Al and of

a fraction of the recess area A,. (Eqn 5.114).The reference power ratio is:

(7.111)

and the temperature step (from Eqn 5.7):

For

remarks on the design, the reader may refer to the multirecess bearings

(section 8.2.3). n particular, we wish to stress the advisability of planning for low

values of p and

l7

when the value of clearance C s at the maximum allowed by the

tolerances. Finally,

it is

worthwhile

to

mention that the complete and optimized

design of multipad journal bearings is the subject of a German industry standard

(see ref. 7.12 and 7.13).



OPPOSED-PAD AND MlJ lTlPA D BEARINGS

235

R E F E R E N C E S

7.1

7.2

7.3

7.4

7.6

7.6

7.7

7.8

7.9

7.10

7.11

7.12

7.13

Mueller-Gerbes

H.,

E r n s t

P.;

Maschinenelemen te z u m Um form en Drehen-

der

in

Geradlininge Bewegung;

W erkstat t und Betrieb,

11,2 (1978),65-77.

Mizumoto H., Matsubara T. , Makimoto

Y.; A

Hydrostat ical ly Control led

Restriction Sy stem for a Hydrostat ic Lead Screw;

Bull . Ja pa n

SOC.

f Preci-

sion

Eng., 0

(19861, 195-196.

Mizumoto

H.,

Okasaki

S.,

Matsubara T., Usuki M.;

An

Active Restriction

Syste m for a Hydrostatic Lead Screw; 7th Word Congr. IFToMM, Seville, 1987;

4 P.

Mizumoto H., Matsubara T., Kubo M.;

Effective Improvements

in

the Design

of Hydr ostatic Lead Sc rew;

Proc. 24th In t. MTDR Conf., 1983;p. 369-374.

Bass ani R.; Z

Pa ttini Zdrostatici Contrapposti Autoregolatori della Po rtata d i

Lubrificante;

Ingeg neria Meccanica, 24,12

(1975)) 11-18.

Bassan i R.;

The Flow Sel f -Regula t ing Hydrosta t ic Screw and N u t;

ASME

Trans. , J. Lu br. Tech., 101 (19791,364-375.

Bassani R., Piccigallo B.; The Dynamic Performance o f the Sel f -Regula ted

Hydrostatic Opposed Pad Bearing;

AGARD Conf. Proc. 323, 1982, pape r 21;

12

p.

Bassan i R., Recchia L.;

Sli t ta Zdrostat ica Alim enta ta Tr am ite Divisori di

Flusso a Spo la Conica; Ingegneria, 1979,p. 129-137.

Decker

O.,

Shapi ro

W.;

Computer-Aided Design of Hydrostat ic B earings for

Machine

Tool

Applicat ions. Part

1:

Analyt ical Foundation; Proc. 9th Int.

Decker

O.,

Sha piro W .; Computer-Aided Design of Hydrostat ic B earings for

Machine

Tool

Applications. Part

2:

Applications;

Proc. 9th Int. MTDR Conf.,

Rippel H. C.;

Hydrostatic Bearings. Part

9:

Single and Mult iple-Pad Journal

Bearings; M achine D esign, Nov. 12, 1963;p. 199-206.

Hydrostatische Radial-Gleitlager

im

Stat ionaren Betrieb (Berechnung von

Olgeschmierten Glei t lagern mit Zwischennuten);

DIN 31656, Teil 1, Nov.

1984; 29 p.

Hydrostatische Radial-Gleitlager

im

Stat ionaren Betrieb (Kenngroben f u r die

Berechnung von O lgeschmierten Glei t lagern

mit

Z w i s c h e n n u t e n ) ;

DIN

31656,Te il2 , Nov. 1984; 9 p.

MTDR Cod .,

1968;

p.

797-818.

1968;p. 819-834.



Chapter 8

MULTIRECESS

BEARINGS

8.1

INTRODUCTION

A

multirecess journal bearing may be thought of as being derived from a multi-

pad bearing, in which the drainage grooves separating the pads disappear (Fig.

1.12.b). Even though these bearings are simpler

t o

build, more effective and there-

fore much more widely used than the multipads, they are rather more difficult

t o

calculate, since a certain amount of

flow

is now exchanged between the adjacent

recesses.

In Chapter

1 t

has already been noted tha t multirecess bearings may be classi-

fied according to the direction

of

the load that they are able

to

bear; namely, we have

axial bearings (Fig. 1.3.c), radial bearings (Fig. 1.12.b), conical and spherical bear-

ings (Fig.

1.13).A

typical application of combined radial and axial bearings, as well

as

of conical bearings,

is

in the spindles of machine tools.

8.2

ANALYSIS

The problem consists in solving the relevant Reynolds equation on the developed

land surface. For example, in the case

of

cylindrical journal bearings (Fig. 8.11, Eqn

4.18 should be used, that, in static loading conditions,

is

reduced to the following:

which may be solved by means of numerical methods, such as finite differences or

finite

element methods.



MULTIRECESS BEA RINGS

237

Fig. 8.1

Multirecess

journal

bearing:

a- cross

section;

b-

developed surface of sleeve.

A problem now arises, since the boundary conditions are not explicitly known.

Indeed we have p=O on So andp=pi on Si, but the recess pressures

pi

depend on the

supply system and are related to the (unknown) flow ra tes Qi.

For

instance, in the

common case of compensation by means of fixed laminar-flow restrictors, we have:

A

solution can be found (ref.

B.l),

thanks

t o

the linearity of the %operator, split-

ting the differential problem

a8

follows:

Q & o ) = G p ~ C L ? s i n p(O)=O on So

;

p( 0)= 0 on

Sj

(j=l ...n)

These n + l differential problems can be solved

t o

find the pressure fields p(i)(note

that, after discretization, this may be accomplished by means of only one matrix

inversion; see, for instance, ref.

8.2).

One can now evaluate the rates of flow across

the boundaries

Si:



239

The pi's are now easily obtained, by schematizing the bearing as

a

network of

hydraulic resistances (Fig. 8.2.b). Now, from the Reynolds equation, the pressure

field is known approximately over the entire land surface of the bearing; conse-

quently, i ts load capacity can be calculated.

- a -

Fig. 8.2 Multirecess journal bearing: a- axial and circumferential flow paths; b- equivalent hy-

draulic circuit.

The method can be extended to account for the effect of the velocity

R

ref. 8.7)

and for different compensation systems. Good results are obtained, except for large

displacements ( ~ > 0 . 6 ) r large lands, that, in any case, should be avoided in

practice.

Most of the numerical results reported in the following sections were obtained

by means of finite-element computer codes (ref.

8.8, 5.38).

8.3 MULTIRECESS JOURNAL BEARINGS

In the following section we shall introduce plots of the non-dimensional load

and of the non-dimensional flow rate, as functions of the displacement and of the

concentric pressure ratio /J, for a few typical geometries. Only the use of laminar




restrictors as compensating devices will be considered in detail (plots for orifice

compensation may be found, for instance, in ref.

8.9

and for constant-flow valves in

ref.

8.10).

Parameter

p

is defined,

as

usual, as the ratio of recess pressures pi, in the

concentric configuration,

t o

the constant supply pressure

p s .

A

few approximate

expressions will also be given for load capacity, f low rate and stiffness.

The influence of the main geometrical parameters will then be explained. Fi-

nally, a number

of

remarks will be made about the design of such journal bearings.

8.3.1 Bearing performance

In Fig. 8.3, the non-dimensional load capacity

W

w'=-

P S L D

(8.3)

of a four-recess bearing is plotted as a function of eccentricity E for certain values of

ratio

L I D

and of pressure ratio p .

It

is assumed that laminar restrictors are used as

compensating devices. In the next section, we shall show how load capacity may be

affected by the varioue geometrical parameters.

In ref.

8.11

a systematic algorithm is used to obtain the load capacity, at small

0.4

W

-

LDPS

0.1

- a -

0.4

W

-

LDPS

0.2

0.L

0.0 0.4 0.8

E

- b -

0.0

0.0 0.4 0.8

E

Fig. 8.3 Multkecess

journal

bearings: load versus eccentricity, for certain values of pressure ratio3

and

of

ratio

LID ( n 4 , a'4.25, 8=30', $4').



MULTIRECESS B EARINGS 241

eccentricities, €or bearings with any number

of

recesses and different supply sys-

tems. The result, when expressed in the nomenclature of the present book, is:

W

=

6 p

~ ( 1

a ’ ) A

(8.4)

where the non-dimensional parameter A depends on the geometry

of

the bearing

and on the characteristics

of

the supply system. Unfortunately, the equations that

allow us to calculate A are rather complicated. In the particular case of n=4, how-

ever, we have simply

n 1

where

0=0,

1,2,

for

constant flow, orifice compensation and capillary compensation,

respectively. For n=6, we have plotted

A

in Fig. 8.4.

-.- a=const

- apillaries

---

Orniies

0.0

0

1

2

rn

Fig.

8.4

Multirecess journal bearings: bearing parameter A versus geometric parameter

rn=2d(1-2a’)lBD2.

In ref.

8.12

another approximate equation is given, valid for any number

of

recesses; using the same notation we have:



242


It is easy

t o

see tha t

it

does not coincide, sometimes significantly, with the results of

ref.

8.11.

Although the load capacity increases with E , even when ~ - 0 . 8 , t is advisable in

designing not t o consider values of E greater than 0.5. Indeed, at high eccentricities,

bearing stiffness rapidly falls (see Fig. 8.51, and then there is not guarantee of a

good safety margin against overloads or against geometric errors of the mating

surfaces.

- a -

- b -

0.6

0.6

0 .4 - 0.4

K K

L

D

p,/C

0.2

L D P X

0.2 -

0 I

I I

0

0 0 . 2 0.4 0.6 0.8 0 . 6 0. 8

E

Fig.

8.5

Stiffness versus:

a-

eccentricity, and b- pressure ratio, for a journal bearing with

n=4,

LID=l ,

a'=0.25,

0=30°.

The effect of the pressure ratio p and of the atti tude angle

$

on load capacity is

easier to see in Fig. 8.6.a. For low values of and high eccentricities, the load capac-

ity is clearly considerably affected by $. This may lead to small oscillations of the

journal when the load rotates in relation to the recesses. In the most common

cases, the variation of W with

$

creates no serious problems; however, i t may be ef-

fectively counterbalanced by using a greater number of recesses

n

(see section

8.3.2).

The choice of

p

has a great influence on load capacity (and hence on stiffness:

see Fig.

8.5)

as well as on flow rate. Load capacity and stiffness have a maximum

which depends on E and $.

/3=0.5

is often referred

t o

as an "optimal"value, while it is

advisable to avoid values outside the

0.3c~c0.7

ange. It should be borne in mind that

the actual value of will depend on the actual value of the radial clearance C

(unless adjustable restrictors are used) and then the working tolerances have a

direct influence on the behaviour of the bearing; this point will be examined more

closely later.

Stiffness may be deduced from the slope

of

the ( W - E ) haracteristics of the bear-

ing. In Fig. 8.5 the stiffness of a typical four-pad bearing is plotted as a function of



MULTIRECESS

BEARINGS 243

the displacement and of the pressure ratio. It may be seen that, in the most useful

range of 8 , stiffness does not vary much with E , thus the reference stiffness

Ko=K(&=O) ay be used with sufficient approximation up to ~ = 0 . 5 . n approximate

value for

KO

mmediately follows from Eqn 8.4:

(8.7)

D

KO

6 B p ,7 1

u ' ) A

- a -

0.4

-9 =

0

' I / / - \ \

Fig. 8.6 a- load and b- flow rate, versus pressure ratio for certain

values

of eccentricity,

for a

journal bearing with n=4, L / D = l , ~ ' 4 . 2 5 ,=30°.

Up to now we have assumed that the turning velocity of the journal

is

small

enough to have no practical effect on load capacity. Actually, if both E and R are not

null, a hydrodynamic pressure field (whose resultant is perpendicular t o the direc-

tion of eccentricity) is superimposed on the hydrostatic one. Consequently, 4 no

longer coincides with the loading angle

+L

(strictly speaking, even when R=O, we

have exactly +=$L only when

+L=O

or when it

is

a multiple of

d2n).

The hydrodynamic effect is particularly marked at high eccentricities, but, in

the most common cases ( ~ < 0 . 5 nd moderate velocities),

it

may be disregarded in

designing, especially as the bearing stiffness turns out to be increased. On the other

hand, at high velocities the increase in static stiffness may become important . The

influence of the velocity parameter




- a -

2.0

W '

-

0.0

0.2

0.4 0.0 0.2 0.4

Sh Sh

Fig. 8.7 a- load and b- attitude angle, versus speed parameter S (Sommerfeld hybrid number)

for

two

journal bearings with different ratio LID and

n=4, a'd.25, b=3Oo, #=Oo,J?=0.5.

(often called the

"Somm erfeld hybrid number")

on

W

and on the angle

#L-#,

s

shown in Fig.

8.7

for a few typical cases.

Hydrodynamic load capacity is calculated in ref. 8.11 as

Total load capacity is the vectorial sum of the hydrostatic and hydrodynamic terms;

therefore, a "speed enhancement factor" may be calculated:

(8.10)

which is not dependent on the number of recesses. For small values of E, the actual

load capacity may be approximately evaluated by multiplying W(a=O) y the same

factor K , . Other workers suggest different equations. For instance, in ref. 8.12

similar results have been obtained, except for (1 2 a ' ) in Eqn 8.9 and Eqn 8.10 is

substituted with (1 a').

In consequence

of

the above, it seems profitable to design in such a way

as

to

have high values of S h , in order t o take advantage of the great increase in load ca-

pacity. However, it must be stressed that the increase in the speed parameter cre-

ates certain undesirable consequences. In particular, in certain recesses, the pres-



MULTIRECESS

BEARINGS

245

sure becomes greater than the supply pressure, causing a reverse flow in the re-

strictor; in other recesses, the pressure tends t o become lower than the ambient

pressure, causing cavitation and entrainment of air in the lubricant. These and

other factors, such as non-linear behaviour of restrictors and thermal effects, may

notably affect bearing performance; indeed, experimental evidence (ref.

8.13

and

ref.

8.14)

has shown that, in certain cases, load capacity may even be worsened by

rotational speed. There is also great friction a t high speed, especially when

a

turbulent flow recirculation star ts up in the recesses. Moreover, if a bearing is

designed for high-speed operation, the alternative choice of a hybrid o r purely

hydrodynamic bearing should be considered. Finally, for high speeds, the damping

progressively decreases, and the dynamic behaviour of the system gets worse, until,

a t a critical speed, instability sets in (see Chapter 10).

When the speed is high i t is necessary

t o

check that the Reynolds number

1 P

R e = - - C D R

2 P

is smaller than

1000:

above this value the effects of turbulence in the lubricating

film can no longer be disregarded. Turbulent flow is unusual in oil-lubricated hy-

drostatic bearings: on the contrary, it is often present in certain particular applica-

tions, in which low-viscosity fluid (such as water, liquid metals o r cryogenic fluids)

are used. A considerable amount of work has been done on this matter and com-

plete numerical solutions have been obtained in which turbulence a s well as the

effects of inertia, cavitation and shaft misalignment are taken into account: see, for

instance, ref. 8.15.

The flow rate

of

the lubricant is not greatly affected by eccentricity (at least when

&<0.6,as

shown in

Fig. 8.6.b),

nor by the attitude angle

9 ,

whereas in practice it is

proportional to

p.

The non-dimensional flow rate

-&§

Q ' - p s

c3

may be read in the plots given in Fig.

means of the approximate equation

(8.11)

8.6.b and

in

the next section, or evaluated by

(8.12)

Since Eqn 8.12 is a limiting value, calculated a t &=O, disregarding the effect of the

axial lands,

it

proves to be slightly overestimated.

The pumping power is clearly



246

HYDRWTA

T1C

LUBRlCATlON

(8.13)

and hence the variations of

H p with E

and I$ are also small.

The friction power may be considered to be the

s u m

of two terms:

which represent the contributions of the lands and

of

the recesses. Land friction Hfi

should be calculated from the integral

extended to the whole land surface, whose area is:

(8.14)

The shear stress

z

is given by the Newton formula (see sections 3.2 and 4.5.3). The

velocity field is made up of the superposition of a "Couette" flow (due to the turning

velocity 52) and of a "Poiseuille" flow (due to the pressure gradient). In the centered

configuration, however, the shearing stress due t o the pressure-induced flow con-

stitutes a symmetric field, whose integral, therefore, vanishes. The velocity gradi-

ent may be substituted by its average value LlDJ2C, and, hence:

(8.15)

Even when the bearing is displaced, the contribution of the Poiseuille-type flows

t o

the friction is, in general, small, and the equation

(8.16)

gives a good approximation, while more accurate equations may be found in ref.

8.16

o r in ref.

8.17.

The power loss in the recess may be comparable to H f l , and

it

may even become

dominant if the fluid recirculation in the recess is turbulent. The friction power lost

in the recess may be expressed in the following form:

(8.17)



MULTIRECESS BEARINGS

where A,.

is

the total recess area:

A,. =

R

D

L

(1 - 2a') 1 -%

(

" 7

247

(8.18)

The coefficient

f,

depends on the ratio

Clh,

and on the Reynolds number

(8.19)

P

Re, = -- D 52 h,

2P

If we have Rer<lOOO (laminar flow)and C<<h,, i t may simply be taken that fr=4Clh,,

or better (ref.

4.5):

(8.20)

In the case of turbulent flow, f, may be calculated from Eqn 4.54.

Lastly, the friction moment and the friction power may be written as follows:

(8.21)

D2

M f = p R

c

f

H f = R M f (8.22)

where the equivalent friction area

A f

may be calculated from the following equation:

(8.23)

Plots of A; are given in Fig. 8.8.

A t

the highest speeds the torque required t o accelerate the fluid entering the

bearing (momentum torque)may become notable. It may be calculated (ref. 8.19) as:

(8.24)Q

= ; I P Q

c Re

D

where the Reynolds number is

1 P

R e = - - C D R

2 P

The relevant power loss

is

obviously

H Q = N Q .

It is interesting to compare H Q with the pumping power:



248


- a -

E-0.5

0.3

0

0.5'

' ' ' ' '

.

. '

. '

0.1

0.2

0.3

a'

- b -

20000

0000

0.8

0.1 0.2 0.3

a'

Fig. 8.8 Multirecess

values of eccentricity

(~4,, ./C=50).

journal bearings: equivalent friction area

A?

versus ratio

a'; a-

for certain

'

E and of ratio h,JC

(ReS1000);

b- or certain values of the Reynolds number

If we assume, for instance, that supply pressure

is ps= 4

MPa,

it

is easy to see

that

a

velocity of the journal

as

high as 10m/s

is

needed for H g to be 1%

of Hp.

Since

in common design practice we generally have Hpmf,t follows that momentum

torque may be disregarded in the vast majority of cases.

The power ratio can be evaluated, from the foregoing results, as follows:

(8.25)

Finally, it is easy to evaluate the temperature step in the lubricant. If we as-

sume, as usual, that flow is adiabatic, for E=O we have:

(8.26)

Since both

Hf

nd

H p

undergo only small variations when the bearing

is

displaced,

AT may also be considered, roughly speaking, not to be dependent on the loading

conditions.



MULTIRECESSBEARINGS

249

8.3.2 Effect of geometr ical parameters

The geometrical ratios which may affect both load capacity and flow rate are

LID, a ' = a l L , 8 (see Fig. 8.1). Besides, the number of the recesses should also be

considered.

The value of

n

could, in theory, be anything, provided

it

is greater than

3,

but

values greater than 6 are seldom encountered; indeed, the benefit of further in-

creasing

n

diminishes and hardly counterbalances the greater manufacturing

costs and the greater complexity of the supply system.

In brief, the main consequences of increasing the number of recesses are that

load capacity and stiffness are higher and, moreover, are less dependent on the atti-

tude angle

$.

The rate of flow a t any given eccentricity is practically unaffected. The

load capacity of a 6-recess bearing is plotted in Fig.

8.9,

which should be compared

with the analogous Fig. 8.6.

- a -

- b -

I

I

I

4

3.2

Q21A

P,C P

1.6

0.8

0

0 Q 2 0.4

0.6

0.8 1 0 0.2

0.4

0.6 0.8 1

B B

Fig. 8.9

a-

load and

b-

flow rate, versus pressure ratio for certain values

of

eccentricity,

for

a

journal bearing with

n=6, L / D = l , ~ ' 4 . 2 5 ,=30°.

The effect of

a l L

and

L I D

is shown in Fig.

8.10.

To increase the parameter

a l l

means to increase the land area to the detriment of the recess area; consequently,

the load capacity decreases with a quasi-linear trend. On the other hand, the flow-

rate is approximately proportional t o

L

1a; ince the pumping power is proportional

to Q ' I W'2 (once load W, eccentricity

E ,

and the other geometrical parameters are

fixed), a value of a' clearly exists a t which the required pumping power is a mini-

mum. This occurs, roughly speaking, when a'=0.3; however, this value is not criti-

cal and, in general, it proves t o be advisable to design a larger recess to step up load

capacity and reduce friction, which, indeed, mainly depends on the developed land



250


- a - - b -

0.0

I

"

0.1

0.2

0.3 0.1

0.2 0.3

a' a'

Fig.

8.10

Multirecess journal bearings:

a-

load and

b- f l ow

rate, versus

a', for

certain values of

ratio LID ( n 4 , =30 , = 4 5 O , J = O . 4 ) .

area Al. Most authors propose "optimal" values for a' in the 0.2510.1 range, o r even

smaller (see ref. 8.17

for

a review); some maintain that a recess area

A,.=O.hDL

is

a

good compromise between high load capacity and low power loss (see ref. 8.20).

Actually, a more detailed analysis (ref. 8.21) would show tha t even a t moderate

speeds (n<l)t is convenient

to

select small values of a' (a'=0.1) n order to reduce H t

and

AT.

At higher speeds, since the hydrodynamic effect becomes appreciable,

slightly greater values of a' may be convenient from the point of view of power loss

(but not from the point of view of temperature). Indeed, in ref. 8.23, it is suggested

that a'=0.15 be selected (independently from the number of recesses and from the

LID

ratio)

for

a

power ratio

n=2;

his

is

no doubt a good choice for the entire most

usual range of power ratios, that is

n=l-t3.

inally, a t very high speeds n>3),he

power loss

is

the smallest for vanishing recesses: this explains the peculiar design

of

hybrid

bearings (see chapter

9).

Concerning LID, since W' decreases as LID grows,

it

is clearly more convenient

to increase D instead of L in order to enlarge the projected area and hence increase

load capacity. On the other hand, flow rate and pumping power are approximately

proportional to D JL,and do not depend on the projected area. Furthermore, Giction

power is proportional to D4L (Eqn

8.22) .

Thus, to reduce power loss, it seem advis-

able to have high L

1D

ratios. L /D=1 s often proposed as a good compromise.



MULTIRECESS B EARINGS 251

The effect of the land angle 8 is shown in Fig. 8.11.

A s

can be seen, the flow-rate

does not vary s o much with 0, whereas the load capacity improves, increasing

8,

o

a certain extent, which depends on

E,

4, a' and

p.

This is not surprising, because the

separation of the recesses is improved even though the recess area is reduced.

However, values of 8 greater than 2 d 3 n (i.e. 30 degrees for n = 4 ) provide no great

benefit, while friction power always grows with 8. Also

H ,

and

AT

are influenced by

the recess angle 8. n ref. 8.21 it

is

shown that the temperature step decreases when

8 s increased up to a certain value, which depends on ll and a', and then remains

practically constant. For instance, for L I D = l , n = 4 , 8 = 0 . 5 , a '= 0 .1 and I7=1, 8 should

not be smaller than 18". Greater values may be suggested for higher values of I7 and

a'. From the point of view of total power an optimal value

of 8

may be determined:

for n=4 and moderate speed, this vanes in the 18"+30"range, when

a'

is between

0.1

and 0 . 2 . In ref. 8.23 i t

is

suggested that8

=0.5

rad be selected

for

n=4 and L I D = l and

smaller values when n is greater and L I D is smaller (e.g. 8 =0.5 for n = 1 2 and

L I D 4 . 3 ) .

0.4

W

-

LDP,

0.2

0.0

- a -

a'=0.1

@

- b -

0.2

0.25

"

10" 20 30"

40

10" 20" 30" 40

Fig. 8.1

1

Multirecess journal bearings: a- load and

b-

concentric

flow

rate, versus land angle

8,

for certain values of ratio a'; ( n 4 , /D=l,J=O.4).

8.3.3

Design of mul t i recess journa l bear ings

Design criteria for journal bearings are not so different from those proposed in

section

7 . 2 . 6 ,

concerning opposed-pad bearings. As usual i t will be assumed that the

main objective consists in obtaining a certain load capacity

W,

with a smaller dis-



252 HYDROSTATIC LUBRlCATlON

placement than

eM=EMC.

Of course, several further constraints may be added, such

as given supply pressure o r viscosity.

Optimization may be tried by means of the usual criterion of minimum power

consumption (see also chapter 5). Since total power

H ,

does not generally experience

great variations for usual eccentricities, analysis may be limited to the centered

configuration ( E=O) .

Total

power is the sum of pumping power and friction power:

Equation 8.25 gives

and, hence,

(8.27)

(8.28)

Shaft diameter and supply pressure cannot be regarded

as

independent vari-

ables, since their selection is connected with the load capacity of the bearing (Eqn

8.3)

D W M

P s D2 =L W'(EM;rO)

(8.29)

Finally, total power and the relevant value of viscosity may be written in the form

where

(8.30)

(8.31)

(8.32)

(8.33)

The non-dimensional quantities

H:

and

1

can be calculated approximately (e.g. by

means of Eqn 8.4, Eqn 8.12 and Eqn 8.23)or read directly in Fig. 8.12.



MULTIRECESS

BEARINGS

253

Examining Eqn 8.30, one may easily draw the same conclusions as those

reached in the case of Eqn 6.57 o r Eqn 7.64, concerning thrust bearings.

For

given

values of clearance and speed, the "op t imal" value of viscosity is the one which

ensures

a

power ratio

n=1.

However this value is not critical, since

H t

experiences

only small variations for values of the power ratio in the range 1/3t3.

Concerning clearance, i t is clearly convenient to reduce i t a s far a s possible, in

order

t o

obtain small power and high stiffness. Limits to the selection of very small

clearances follow from two types of considerations. Firstly, Eqn 8.31 shows that a

very small clearance may require an impracticably low viscosity to obtain n=1, ven

after a small land width has been selected, in order t o reduce pp*. In certain cases

the value of viscosity is a design constraint, and i t may be shown that when p is

given, the lowest value of

Hto

is obtained by selecting

C

n such a way as

to

have

n=3.

I

0.1

0.2 0.3

a'

- b -

1.5

- =0.5

- - - €10.3

0.0'

. I . . . .

0.1

0.2 0.3

a'

Fig. 8.12 Multirecess journal bearings: a- total power H : and b- relevant viscosity p; versus

ratio a'

for

certain values of ratio LID and of working eccentricity E; (n=4, 8=3# ,$=45 ,J=#.4,

f,=#.2)

Another limit

to

the selection of C is connected with manufacturing tolerances

(ref. 8.241. Indeed, the designer must often select a whole range CmlCIC, instead

of a single value and, if the tolerance 6g=2(CM-Cm)

s

large, the actual performance

of the bearing may differ notably from the one calculated. Let us assume that the

maximum radial clearance CM.has been selected in such

a

way as to satisfy all the

design constraints with a certain value B of the concentric pressure ratio. When



254

HYDROSTATIC

LUBRICATION

E=O, the flow rate is proportional to

C-3:

it is now clear tha t the actual pressure ratio

will be

1

f i l l =

1-p

c

3

l +

P

(G)

(8.34)

Since it is convenient to impose an upper limit /lM on the pressure ratio, to avoid a

sharp decrease in load capacity (see Fig. 8.6.a)) we readily obtain

(8.35)

For example, if

p=0.4

and

8 ~ = 0 . 7 ,

e must have

c , / c~>0 .66 ,

.e.

6g<2/3'cMM.

on-

sequently,

CM

cannot be too small, otherwise an excessively small tolerance would

be required. The actual value of the power ratio also depends on the actual value of

clearance. Namely, if n s the designed power ratio tallying with the greatest clear-

ance C for any other clearance

w e

get

which, strictly speaking, is not valid when recess friction makes a predominant

contribution to the total power loss (e.g. in the case of turbulent flow in the recess).

I t is worth noting that if

p=0.4, n=1

and

c,=2/3'cM,

the greatest relevant value of

the concentric power ratio proves

to

be close to

3.

We wish to stress that the foregoing considerations concerning the optimal

design of such bearings are based on a simplified analysis, which nevertheless

gives good results. In particular, the increase in load capacity due to the hydrody-

namic effect has not been taken into account, and total power has been evaluated for

E=O instead of a t a working eccentricity. It is also worth noting that there is a turbu-

lent recess flow in the case of high-speed bearings. Consequently, coefficient

f,

s

much greater than the value assumed in Fig.

8.12

and, above all, H t and p; cannot

be considered t o be virtually independent from p,D and

C .

Several authors express

different points of view, obtaining somewhat different results (see, for example, ref.

8.17

and

8.18

for a review). In particular, i t must be remembered that the optimized

design of multirecess journal bearings is the subject of a German industrial stan-

dard (ref.

8.22

and

8.23).

Direct optimization methods have also been proposed,

based on well-known optimization algorithms such as the

complex method

or the

f lexible polyhedron method, in which the object function (e.g. the

H ,

J W ratio) is

found by solving the Reynolds equation by means of finite-difference or finite-

element methods (ref. 8.25).



MULTIRECESS BEARINGS 255

Besides the effects

of

the error on the clearance value, to be controlled by means

of

size tolerancing

as

indicated above, other kinds of manufacturing errors may

affect the bearing performance. For instance, ovality may cause a loss of stiffness

and greater directionality. In ref. 8.26

it

is reported tha t a 10% ovality (with respect

to the mean clearance) causes a loss of stiffness in the order of 20% in a typical

bearing.

The effects of journal misalignment on load capacity have been studied in ref.

8.27, by means of thin-land analysis, and the results have been verified by means of

experiments. It has been concluded that misalignment reduces load capacity and

causes directionality

of

stiffness. The effect

is

most marked near the optimum pres-

sure ratio

@=0.5);

t is rather small for values of slope

6

lower than 0.2, but becomes

quickly appreciable for higher values (a loss of stiffness smaller than 10% may be

assumed when

6=0.4).

Slope 6 used in ref. 8.27 may be practically defined as

6=(L/2C)tg~,

y

being the angle a t which the shaft is tilted.

It

may be concluded that clearance should be carefully selected, also bearing in

mind that a very small clearance enhances the effects of journal misalignment;

moreover, it is recommended to design for moderate eccentricities (e.g.

~ ~ = 0 . 5 ) ,

thus allowing considerable safety margins

t o

compensate for the effects of manufac-

turing errors.

Finally, for the sake

of

completeness, a number

of

remarks may be made on the

roughness

of

land surfaces and on the shape of recesses. Roughness is not gener-

ally a problem. Indeed,

it

is widely accepted that the roughness of the land surfaces

should be smaller than 0.1 times the lowest operating film thickness, which often

leads to values of R , greater than

1

pm, that are quite easy to obtain by grinding.

The roughness of pockets may even be much greater, except in the case of the high

speed bearings, where it may cause turbulence.

For recesses, a shape of type (a) in Fig. 8.13 requires a dismountable construc-

tion of the bearing sleeve,

so

a

shape

of

type (b)

is

often selected, because it may be

obtained by milling; of course, the convergent recess outlet may involve some differ-

- a - - b -

/----\.

i

\

Fig. 8.13

Shapes

of

recesses

in

journal bearings.




ences in the behaviour of the bearing. Particular recesses have also been proposed

in order to reduce recess friction in high-speed bearings (ref. 8.28).

8.3.4

Design procedure

Following the remarks made above, a simple procedure may be proposed, that

should lead to an "optimized"design of opposed-pad bearings. As before

it

will be

assumed that a certain load W M is given, to be sustained with a displacement

smaller than eM at a speed iz; some further constraints may be required: the design

may then be carried out following the steps below.

i) Choose a trial set of geometric parameters; perhaps

L I D = l ,

alL=0.15,

0=30°,

n=4;

it is convenient to select large recesses (i.e. small values of

a )

unless

low speeds are expected.

Choose a trial value of the maximum eccentricity, such as ~~=0.5.

Take

p=0.4

as the design pressure ratio: a smaller value can be accepted if

n>4or

&&0.5.

ii ) Evaluate coefficient

W'

by means of figures given in the preceding sections.

Select a suitable supply pressure

p s ,

if it is not given.

Calculate a first value for diameter D , by means of Eqn 8.29.

Check the temperature step (Eqn 8.26); since for many mineral oils we have

~ ~ - 1 . 6 . 1 0 ~/m3"C, and i t should be

n < 3 ,

the maximum temperature step is

expected

t o

be AT<pS.2.5"CIMPa.

iii) Select a suitable value for the manufacturing tolerance Sg=2(C~-Cm):ee

also Fig. 8.14; bear in mind that the narrower Sg is, the lower the power

losses can be.

Take Crn+ and cMM=cm+6g/2.

Check if stiffness is great enough. Otherwise reduce

Sg

or select a smaller

value for EM, restarting from point

(ii).

iv) Calculate the "optimal"value of viscosity, i.e. the value k p t that leads

to n=1

when C=C, (Eqn 8.31).

Check that

kpt

s a plausible value (sometimes p may be directly imposed by

the specifications); if it is not, try

t o

modify the geometrical parameters

(namely the recess width), the clearance or the supply pressure. However, if

R is very low it will not be possible

to

get

n=1

indeed, for 8=0, n "infinite"

viscosity would be required ); in this case select the highest allowable viscos-

ity, and

a

larger land width, such

as

a'=0.25, On the contrary,

a

very

high

speed would lead to values too small for

h p t ;

in this case

it

will be necessary

to accept values of the maximum power

ratio

(tallying with the least clear-

ance

C , )

greater than 3 : the ensuing hydrodynamic effect will enhance



MULTIRECESS B EARINGS

i o o l

IT=7

"1

0 18

30 50

80

120 180

250

D [mml

257

Fig.

8.14

Values of parameter

6g=2(C,+,-C,),

for certain international tolerance (IT) grades

(ISOIR286).

bearing stiffness, but any of the problems already noted might arise

(cavitation, excessive warming of lubricant, and even dynamic instability).

v) Calculate flow rate, pumping power, friction torque and friction power, with

reference to the design configuration (C=CM, =O) .

Check the same parameters for different values of

C

and

E.

If

flow

rate seems too great, and

6

cannot be further reduced, it will be

necessary to accept a power ratio

n>l

(e.g. increasing p or reducing

p s )

consequently increasing friction. Conversely, friction may be easily reduced

with a larger expense of flow rate and pumping power.

I f the supply system is not able

t o

maintain lubricant viscosity within a

narrow range, check friction and the temperature step for the maximum

expected viscosity, and check the flow rate for the minimum viscosity.

vi)

Design the compensating restrictors.

vii) Check the Reynolds number in the bearing clearances and in the recess.

viii)

Check the dynamic behaviour of the system (see Chapter 10).

EXAMPLE 8.1

Design a m ultirecess bearing to sustain a journal, w hose diam eter is D =80 m m,

rotating at a speed i2110.5

r a d l s

(-1000

pm); the operating load W l tO

KN

must be

susta ined wi th a d isplacement smaller than e ~ = 2 0m; manufacturing tolerances




should not be smaller t ha n IT5 grade for both journ al an d s leeve; supply pressure

m ay not be greater tha n 5 MPa.

i ) Let us select a trial set of para me ters: L I D = l , a'=0.15, n=4, 9=30°, ~ ~ = 0 . 5 ,

8=0.4.

i i) F r om F ig , 8 .1 1 w e m a y read W ( ~ ~ ) = 0 . 2 6 9 ;t is clear (E qn 8.29) that , d ue to the

cons tra in ts on pressure and shaf t d iameter , the m ax im um operat ing load cannot

be sustained w it h a n eccentricity ~ ~ = 0 . 5actually it sh ou ld be W p 8 . 6 KN) . I n order

to increase load capacity, one ma y try to increase the a xial len gth o f the s leeve and

to reduce the width of the circumferential lands (to increase the number of recesses

would also have a notable effect on load capacity). Selecting LID=1.25 and a'=O.1, a

sim ple interpolation of available da ta gives W'(&=0.274, and hence wM=ll.o

kW

wi th a supply pressure ps=5 MPa.

i i i) Since the tolerance grade is given, the w idt h of th e tolerance range for di -

ametral clearance must be Gg=2(C~-C,,J=26m; hence, a suitable choice for radial

c learance m ay be C=40+27 pm. Note tha t , assuming h o = C ~ = 4 0 m as the reference

c lear ance , a t the eccen t r i c i t y EM =0.5 w e h a v e a n a c t u a l d i s p la c e m e n t

e=CMEM=20pm; fo r sm all er clearances, of course, stiff ne ss is grea ter.

iv) From Fig. 8 .12.b one may obtain pi=1.12, thus to have I l=1 a viscosity p 4 0 2 0

N s 1m2 is required ( E qn 8.31).

v ) Non-dimensional f low rate may be evaluated by means of Fig. 8 .10.6 as

Q'(O)=1.57; fo r the "optimal" viscosity calculated above, i t fol low s th at Q=25.10-s

m 3 / s , w he n

C= CM ,

and Hp0=126 W , S ta t ing hr=l mm,it follows that frZO.155 (Eqn

8.20); E qn 8.23gives A;=0.549 and hence Hf0=122 W , and Mf=1 .16 N m ;

at

th e eccen-

tricity ~ = 0 . 5

t

is easy to calculate: Aj-=O.622, Hf =1 38 W, a n d M f = 1 . 3 1 N m .

I L

i n -

stead, the actual clearance is the lowest value allowed C,=27 pm , the actual pres-

sure ratio becomes 8,,=0.684; nevertheless flow rate and pumping power will be

nearly halved; on the contrary friction will be notably increased, giving Aj=0.523,

H f = 1 7 2 W , M f= 1 .6 4 N m a t

E=O;

Aj=0.595, Hf=195 W , Mf=1.86 N m at ~=0.5 .l l these

results are subject to change when actual viscosity does not tal ly with the value

assumed above;

in

particular during s tart ing-up operations , with

cool

lubricant,

p

m ay be m uc h greater , and hence fr iction wil l also be proportionally highe r; in this

condit ion, however , the temperature s tep is also greater , contr ibuting to bringing

the lubricant to th e normal operating temperature.

vi) Since f lo w rate i s low, it is possible to use sm all bore pip es a s compe nsating

restr ictors; i ts hydraulic res is tance in the design condit ions ( i .e . C=cM=40 pm,

8=0.4, p=0.02 N s l m z ) should be

Rr=(l-8)p,l(Q/n)=480.109

s lm 5 . Fr om E qn 4 .67 i t

fol lows that , us ing a pipe with a 1 m m internal bore, a length

1=589 mm

is neces-

sary for each restrictor; the relevant Reynolds num ber proves to be low eno ugh. It is

obviously possible to select a smaller bore to reduce the pipe leng th; it is , how ever,

necessary t o check the relevant Reynolds n umb er.




259

vi i ) I t

is

easy to veri fy by m eans of Eq n 8.19 that t he Reynolds number in t h e

recesses i s quite low (Rer=189 ass um ing p=900

@In3 .

he Reynolds numb er

in

the

clearance is obviously much smaller.

A

concluding remark concerning the velocity parameter

S h :

this should be

in

the 0 .0 6 7 4 1 4 7 r ange , depending o n the ac tua l va lue o f C, nd hence a n enhance-

m ent of load cap acity of between 13% and 21% m ay be expected ( Eq n 8.10).

E X A M P L E 8.2

Design a journ al bearing able to sustain a load W ~ = 2 5 0 N rotating

at

0 = 1 5 7

r a d l s (1500 rpm). T otal power consumption should be smaller th an 100 KW, supply

pressure should not be greater th an

8

MPa.

Let

us

se lect L /D = l , a lL = O. l , &SO0,

~ ~ = 0 . 5 ,

=0.4. It follows that W’=0.306,

Q’=l.86, A t thi s poin t, diame ter and pressure ratio m ay be selected

in

such a way as

to satis fy E qn 8.29. Since this is clearly a hig h speed bearing, E qn 8.31 suggests tha t

supply pressure should not be low, to avoid too small a value o f th e optimal viscosity.

Let us select D=0.32

m

an d, hence, p,=7.97 MP a.

The next s tep consis ts in selecting the value of radial clearance. If we as sum e

6=50 p m (tallying w ith I T 5 quali ty grade) , we should select th e greates t radial

c l e a r a n c e a s C ~ = h o = 7 5m: i n this case, however, this does not prove t o be a suit-

able selection. Stating h,=5 mm (and hence fr=0.06), we fin d Air0.5 (Eqn 8.23) and

pt=1 .34 (Eq n 8.33); consequently,

in

order to get n = 1 the optima l viscosity should be

p=0.006 N sl m 2 , a rather low value. Furthermore, it is easy to see f ro m E qn 8 .19 that

th e hig h velocity of the jou rna l w ould cause a turbu lent recess flow (Rer=19000), an d

a much higher recess fr ict ion factor f ,

than

the value considered above (namely,

E q n 4.54 gives fr=0.78). Th is leads to a higher value for power ratio

n.

f course, if

th e actu al clearance is smaller t ha n cM’75 pm , the pow er ratio proves to be even

greater (e.g. see table below for C=50 pm) .

100

1

00

100

125

125

P

(Ns/m2

0.006

0.010

0.006

0.010

0.013

0.010

0.013

0.016

0.0

13

0.016

0.63

0.48

-

Mf

0

-

Nm)

115

168

91

128

155

108

129

149

113

130

-

-

Hf0

(KW)

18.1

26.5

14.3

20.1

24.3

17.0

20.2

23.4

17.8

20.4

-

-

n

7.36

17.90

1.73

4.04

6.34

1.44

2.22

3.17

1oo

1.41

-

-

AT0

(“0

41.6

94.0

13.6

25.1

36.5

12.1

16.0

20.7

9.9

12.0

-



260


In order to avoid excessive friction power a nd too great a temperature step, one

may use very narrow tolerances or a lower viscosity (the advisability

of

selecting

viscosity on the basis of the actual value of clearance may also be considered); an -

other suitable approach may consist

in

increasing clearance, which also makes it

possible to select a lubricant with a greater viscosity. Calculations for certain values

of C and p are summa rized in the above table.

8.4

ANNULAR MULTIRECESS

THRUST

BEARINGS

The thrust bearing shown

in

Fig. 8.15 works in a

very

similar way to the annu-

lar-recess pads, seen in section 5.4, from the point of view of axial load capacity.

Furthermore, since each recess

is

supplied through an independent compensating

Fig. 8.15 Annular rnultirecess plane bearing.



MULTIRECESS

BEARINGS 261

device, it is able t o react t o tilting moments. In the present work, we shall confine

ourselves t o considering constant pressure supply systems which feed the recesses

through laminar-flow restrictors (capillaries).

Calculation of the performance of such bearings is based on the approximate

solution of Eqn 4.23 (the Reynolds equation in polar coordinates),

for

example, by

means of the finite-element method (ref. 8.29).

For y=O, he bearing simply behaves like an annular-recess pad, and hence an

effective area A, may be defined (Eqn 5.1) as the ratio of the load t o the recess

pressure.

A,

turns out t o be independent from the film thickness, and

it

may be

calculated approximately using Eqn 5.66, which proves

to

be accurate especially

when the angle

8 is

small (however, i t is advisable for the lands separating the

recess

to

be at least twice as wide as the annular lands,

to

improve tilting stiffness).

The values plotted in Fig. 8.16.a, on the other hand, were calculated by means of a

finite-element computer code (ref. 5.38).

- a - - b -

0.:

0.5 0.6

0.7

0.8

r'

0.00

'

0.5

0.6 0.7 0.8

r'

Fig. 8.16 Annular

multirecess thrust

bearings: a-

Effective

area

#,and b- hydraulic

resistance R*

versus

ratio r1/r4

for certain values

of

landwidth

ratio

cd(r4-t-I).[ n d , =(1-r?.25 ].

The

flow

rate and the film thickness, for any given axial load,

as

well as the

axial stiffness and the pumping power, can be calculated as in section 6.3.1, taking

into account that

R ,

is

now the equivalent resistance of

n

restrictors in parallel. The

hydraulic resistance of the pad may be written as in Eqn 5.68, in which R* may be

read from Fig. 8.16.b o r roughly calculated from Eqn 5.69.




In calculating the friction power, the contribution of the recess, mainly caused

by the damming effect of the radial lands, should be taken into account. Evaluating

it as in section 4.7 .3 , friction moment and friction power may still be expressed by

Eqn

5.70

and Eqn

5.72 ,

where:

Assuming h,.>>h and laminar flow, i.e. a Reynolds number

Re,. =$: O h ,

D

( 1 + r')

(8.36)

(8 .37)

less than

1000,

it may be stated that

f r=4h/h , .

A t high speed the effects of lubricant inertia become appreciable. Certain

numerical calculations are reported in ref. 8.30; although the data are presented in

a completely different form, a detailed analysis would show close agreement with

the results presented in section

5.4.5

for the continuous-recess pad (i.e. Eqn

5.77

to

Eqn

5 .80 ) .

Centrifugal forces cause an increase in flow rate, for a given recess pres-

sure, and, conversely, a decrease in recess pressure and load capacity when supply

pressure or flow rate are given.

As

has already been noted, the main distinguishing characteristic of these bear-

ings is their ability

t o

sustain tilting moments. Tilting stiffness, defined as the ratio

of the tilting moment M t t o angle

ry,

may be written as:

(8 .38)

The non-dimensional factor K ; is plotted in Fig. 8.17 (taken from ref. 8.31) as a func-

tion of the non-dimensional axial load WI A,ps and of the radius ratio r4/rl .A pres-

sure ratio /3=0.5 is clearly advisable, since it brings maximum stiffness both for ax-

ial and tilting loads.

Axial load capacity proves, in practice, t o be unaffected by a limited tilting an-

gle, at low speed; the problem may be quite different when til t is coupled t o high

speed. This kind of problem is dealt with in ref. 8.32 , in which Eqn 4.23 is solved, for

certain bearings. I t transpires that load capacity slightly increases with ti lt angle v

(thanks to the hydrodynamic effect), until cavitation occurs and load capacity breaks

down. For the bearings considered in the reference cited above [a '=0 .25 ; 8=30";

/3=0.5; il=(3/8)pLU I2/@,h2 )=50]his occurs a t y'=vD/ hz0.4 when

r'=0.4.

Higher val-

ues of

y'

could be allowed

for

r 5 0 . 5 .

However, the bearings considered i n ref.

8.32

are characterized by high values of power ratio

n,

hus, it may be concluded that, if

the bearing is designed on the usual basis

of

a power ratio Z7=Hf/Ht smaller than 3 ,




263

025

0.2

0.15

K;

0.1

0.05

0

\

0.2 0.3 0.4 0.5

0.6

0.7

0.8

4

A,P,

Fig.

8.17 Annular multirecess thrust bearings: tilting stiffness Ki versus load

WIA,p,

for certain

values of ratio

r&,.[n=6; a'=0.25].

and large tilting angles are not allowed, no problem should arise. If, on the other

hand, the speed is very high, more careful numerical calculations should be car-

ried out.

Tilting stiffness also seems

t o

be enhanced by the hydrodynamic effect (ref.

8.291,

although the direction of tilt no longer coincides with the direction of the tilting

moment.

Concerning the design

of

a multirecess thrust bearing, all the remarks made

about the annular-recess pad could be repeated, adding the obvious constraint of the

minimum tilting stiffness that may be required

by

the specifications.

8.5 TAPERED MULTIRECESS BEARINGS

The conical bearing in

Fig.

8.18 may be considered as a generalization of the

cylindrical journal bearings examined in section

8.3.

The main advantage of such

bearings is their ability to support loads in the axial a s well as in the radial direc-

tion. For this reason, they may be used

to

substitute assemblies made up of a jour-

nal plus a th rust bearing when the thrust has a constant direction.



264


Fig. 8.18 Tapered multirecess bearing.

Unlike cylindrical bearings, the radial play C s not fixed but depends on axial

loading, and hence the single-cone bearings may not be suitable when great varia-

tions of the axial thrus t are planned. The opposed-cone assemblies, on the other

hand, are able t o sustain a wide range of loads in every direction (Fig. 8.191, with

greater stiffness, thanks to the effect of the preload.

In order to assess the performance of tapered multirecess bearings, it

is

neces-

sary, as outlined in section 8.2,

to

solve, at least approximately, the relevant

Reynolds equation on the conical surface. From Eqn

4.26,

if the inertia of the lubri-

cant and the squeezing components of the surface velocity are disregarded, we

obtain:

In the following pages

will

be found results (load capacity,

flow rate)

obtained by

means of a finite-element technique (ref. 5.38). Other results can be found else-

where, for instance in ref, 8.33.

Most

of the remarks made in section 8.3 could be repeated here, but with certain

important differences. First of all, the concentric radial clearance

C

s no longer a



MUL

TIRECESS

BEARINGS 265

constant, being related to the axial film thickness, i.e. t o the axial load. Conse-

quently, the actual value of the concentric pressure ratio

p

also depends on the

axial

load.

As for the other multirecess bearings, we shall concentrate on the most com-

mon constant-pressure supply systems, with capillary compensation.

- a -

_-

- b -

- c -

--

Fig. 8.19 Arrangements

of

tapered multirecess bearings.

8.5.1 Slngle-cone journal bear ings

Let us begin by examining the performance of the single-cone bearings, fed by

means of laminar-flow restrictors. If no radial load is present (i.e. it is E=O ) , axial

load capacity may be written in the following form:

Wz =A,

pr

= A,

P Ps

(8.40)

where, as usual, A,

is

the effective bearing area and

p

is the concentric pressure

ratio. A, may be considered to be a constant. At least when no radial load is applied,



266 HYDROSTATIC LUBRlCATlo N

the conical journal bearing merely behaves like the single-pad thrust bearings

examined in chapter

6.

Furthermore, even after radial displacement, axial load

capacity experiences variations of a few percent (at least for ~<0.6).n other words,

radial loads have practically no effect on axial stiffness (whereas the opposite is not

true ). In what follows, for the sake of simplicity, we shall consider axial load

W, o

be independent from radial eccentricity

E.

Axial stiffness

is

clearly the same as for single-pad thrust bearings; hence

it

is

given by Eqn

6.22,

for laminar-flow restrictor compensation. Nor does the

flow

rate

greatly vary with E (at least for the usual values of E and

B) ;

consequently, it may be

written in the following form:

P B w,

Q=*=m

and

(8.41)

(8.42)

may be considered a constant. On the other hand,

Q

clearly depends greatly on the

axial load, just as in the case of single-pad bearings (see section

6.3

and Fig. 6.5.a).

An initial approximation for

(8.43)

and R* an be obtained from the results given in section 5.5, i.e. by considering the

multirecess cone as an annular-recess pad with geometrical parameters of the

same value, while values obtained by numerical calculation are given in Fig. 8.20,

as functions of the semi-cone angle, for certain values of ratio LI D.

The pumping power

is

directly calculated from:

(8.44)

The friction power developed on the land surface is calculated, when no radial

load is applied, by integrating the elementary power

and we easily find:

(8.45)




0.0

0.e

0.6

A',

0.4

0.2

0.00

- a -

0.15

R'

0.10

0.05

a

- b -

UD=1 I I 0.75

I

/

1 0 . 5

15" 30" 45"

a

Fig.

8.20

Tapered multirecess bearings: a- Effective area A*,

and

b- hydraulic resistance R* ver-

sus

semi-cone

angle a

or

certain values of ratio LID;

(n=4,

=30°).

Friction power, of course, varies with the eccentricity, but

it

is

often enough for

the designer t o consider the value above alone.

When the recesses are large, their contribution

t o

friction should also be taken

into account. Proceeding as in the case of the journal bearings (sect. 8.3.11,we find:

(8.46)

where

it

may be stated that fr=4h,lh, if h,>>h, and the Reynolds number in the

recess

I P

L

2 P

e =

-

-

, D 2

(1

-

tga)

is

smaller than

1000.

The friction power may now be written as follows:

(8.47)



MULTIRECESS

BEARINGS

269

journal bearings, the pressure ratio, which now depends on the axial load, should

not fall outside the 0.4t0.7 range. Beyond these limits, the bearing stiffness rapidly

falls, both in the axial and radial directions. Moreover, at low pressure ratios (small

axial loads) the performance is largely dependent on the load angle, whereas, a t the

higher values of p, the limit load capacity (i.e. the load which corresponds to the

higher eccentricities,

~>0.8)

s greatly reduced, as in the case of cylindrical bearings

(see Fig. 8.3).

A

greater range could be allowed for p by using 6-recess bearings.

- a -

0.4

0.0

0.4 0.8

E

0.3

W

-

L D Ps

0.2

0.1

0.0

- b -

0.0 0.5 1 o

0- J h-

A, Ps

Fig. 8.22 Tapered multirecess bearings: load W': a- versus eccentricity

E

and b- versus pressure

ratio4 for four bearings

with

n=4,

8=30°,

a'=0.2. (1)LID=l . a=lOO; 2) LID=0.75, a=20°;

(3)

LID=O.5,

a=3Oo;

4) LID=O.25,

a=40°.

In Fig. 8.23 the non-dimensional radial load at

~ = 0 . 3

nd ~ = 0 . 5s plotted for

certain values of the geometric parameters. Since the radial load, for small eccen-

tricities, is virtually proportional to E, Fig. 8.23.a may also be used to evaluate the

radial stiffness

D

L

p SW ( E ) D

L

p S

WYE)

KO =

-

C

E h t g a E

(8.50)

The design of a single-cone bearing could be based on the following procedure:

state the minimum axial load

W z m = W z o

and the relevant maximum radial

load WM. tate a value for the maximum eccentricity EM and the minimum pres-

sure ratio /3 (perhaps eM=0.5and

p=0.4).

Note that the maximum axial load should



270

HYDROSTATIC LUB RlCAT/ON

- a -

I

- b

0.4 I

0.2

-

w

0.1

-

- '=0.1

- - - a'=0.3

0.0

0"

1

5" 30" 45"

a

0.3

w

0.2

0.1

\

\0 .75

\ 1 '

\

V .

0" 15 30" 45

a

Fig.

8.23

Tapered multirecess bearings: load W versus semi-cone angle

a

for: a- eccentricity

~ = 0 . 3 ,nd b- ~ 4 . 5 ;n=4, 9=30°,J=0.4,

4 4 5 ) .

not exceed

1.75.W,,,

otherwise the pressure ratio under the highest load may prove

to be too high to give good radial stiffness;

assume a/L=O.l (a greater value is, however, suitable for slowly rotating bear-

ings). Select (for example with the help

of

Fig. 8.241 the aperture angle a and the

length to diameter ratio LID;

the bearing can now be calculated just like a single-pad thrust bearing

(section 6.4.21,except that, in selecting the film thickness, the required radial stiff-

ness must also be considered. In other words, it must be

eM

C

=

h t g a s -

EM

where eM is the maximum allowable value

of

the radial displacement. "Optimal"

values

of

the viscosity and pumping power are given by Eqn

6.56

and Eqn

6.57.

Figure

8.25

contains plots of

HF

and

p; for

certain bearing shapes.

8.5.2

Opposed-cone assembl ies

Let us now consider an opposed-cone assembly, such as those in Fig. 8.19. The

axial play 2ho is established during manufacturing (although plans can be made to



M U L

TIRECESS

B EARlNGS 271

adjust it during assemblage); hence, the radial play at

W,=O

s Co=hotga

or

both

cones. The concentric pressure ratio is

1

R ,

p

1+-

n

R* h: sin3a

(8.51)

where

R,

is the hydraulic resistance of each restrictor and n the number of recesses

of each cone: the two cones are assumed t o be identical.

1

.I

-

wz

02

- a -

0.1

0" 15 30 45

a

- b -

0.c

0 15 30 45

a

Fig.

8.24

Tapered multirecess bearings: ratio of radial load W to axial load

W,

versus semi-cone

angle a for: a- eccentricity

~ = 0 . 3 ,

nd b- ~ = 0 . 5 ;n=4, 0=30",&0.4, 4=45 ).

In this case, the response to an axial load is the same as tha t of the opposed-pad

bearings, that is:

wz

= A, Ps w;v;

(8.52)

K

=KO

K&3;

&,I

where:

(8.54)



272


1 1

w; =

I+? ( 1 - ~ 3 +? ( 1 + . ~ 3

(8.55)

(8.58)

The radial stiffness of each half bearing is greatly influenced by the axial load:

when the shaft

is

displaced in the z direction, the more loaded cone actually works

with a radial play

c, = c, (1

-

4)

and a concentric pressure ratio

5

4

3

";

2

1

- a -

1 0.75

-

'=0.1

_ _ -

' 4 . 3

0

0

15

30 45

a

3

2

6

1

0

- t

UD=1

0.25

\,

0.5,

15 30'

45

a

Fig. 8.25 Tapered multirecess bearings:

a-

total power H: . and b- viscosity

p;

versus semi-cone

angle

a

or certain values of ratios alL and Y D ; n 4 , 9=30°,&0.4,

$ 4 5 ' ,

fr=0.2) .




273

1

8 2

=

I + $ - @

( 1 - ~ 3

Whereas for the other cone we shall have:

c1=

c, 1 + &,I

Figure 8.26 contains a plot of G,

1,

and &, as functions of the axial Ioad.

- a -

0=0.6

I

0.0 0.4

0.8

w

A, Ps

0 . 0 " ' " . ' '

0.0

0.4 0.8

w

A, Ps

Fig.

8.26

Opposed cone bearings:

a-

axial eccentricity

&

and b-pressure ratiosJ1 and&, versus

axial load W;

or

certain values of reference pressure ratioj.

Bearing Fig. 8.22 in mind, it should be clear that the radial load capacity and

the stiffness of each cone are greatly affected by the axial load. An example is given

in Fig. 8.27, in which the non-dimensional radial load capacities W i and

W;

are

plotted, for a typical bearing, against the axial load Wl. When large axial displace-

ments are allowed

to

sustain a certain load W,, rrangements such as in Fig. 8.19.a

and Fig. 8.19.b are clearly not able

to

sustain high radial load components on both

the half bearings a t the same time.



2 7 4

HYDROSTATIC LUB RICATION

The opposed-cone arrangement shown in Fig. 8.19.c may be regarded as a sin-

gle bearing, able

t o

react to external forces in every direction. The main difference

as compared to the arrangements examined above

is

that the eccentricities and

~2

of

the two cones may no longer be considered independently, since we have:

0.0 0.2

0.4

0.6

w

Ae

PS

Fig.

8.27 Radial loads

W ;

and

W ;

versus axial load

W; for

certain values

of

pressure ratio$, for a

bearing with

L I D 4 . 7 5 ,

a=2Oo,

alL=0.2,

n=4,

0=30°.

0.4

-

.-

0.0

0.2 0.4

w

0.6

Ae

PS

Fig.

8.28

Radial load versus axial load. for certain values

of

pressure r a t i o j , for an opposed-cone

bearing

with

L I D 4 . 7 5 .

a=20°,

a/L=0.2,

n=4.

8=30°.



MULTIRECESS

BEARINGS

275

The radial load capacity is,

of

course,

W=W,+W2.

When an axial load is applied,

becomes smaller than

E,,

and hence the radial load capacity is mainly due to W,.

Figure 8.28 shows, by way

of

example, how the radial load capacity of a given bear-

ing is affected by a radial component

W,.

8.6

SPHERICAL JOURNAL BEARINGS

Spherical multirecess bearings (Fig. 8.29) have the advantage,

as

compared to

similar conical types, of being intrinsically insensitive

to

the tilting misalignment of

the journal. On the other hand, they are more difficult and more expensive

t o

build.

Fig. 8.29 Spherical journal bearing.



276 HYDROSTATICLUBRICATlON

To evaluate the performance of such bearings, the relevant Reynolds equation

(Eqn 4.30) should be solved numerically. Due t o the large number of geometrical

parameters, general charts are not given, but in Fig. 8.30 (obtained from ref. 8.34,

as are the following figures in this section) the main performance of

a

typical 6

recess bearing is shown. In more detail, in Fig. 8.30.a, the axial and radial compo-

nents of the load, i.e. W, and W, are plotted as functions of the eccentricity

E

and the

relevant angle whereas the effect of the attitude angle is small. In Fig. 8.30.b the

flow rate for W=O:

(8.59)

is given as

a

function of E. The pressure ratio in the concentric position is assumed

to be p=0.5. As usual, this value is considered t o be the optimal one, when fixed

laminar restrictors

a r e

used as compensating devices, from the point of view of both

axial and radial stiffness. Furthermore, for different values of p, higher interaction

occurs between the loads in the axial and radial directions, whereas from Fig. 8.30

it is clear that the axial load has little influence on radial stiffness, a t least when yis

between 60"and 90 . In Fig. 8.31.a the effect o fa on load capacity a t ~= 0 . 5nd on the

concentric flow rate is shown for the same bearing as in Fig. 8.30.

Concerning the geometrical parameters, ref. 8.34 shows that

t o

decrease rp, t o

below 30" has a detrimental effect on radial load capacity.

It

may be advisable

t o

use

- a -

-

b-

12

Q'

8

4

0.04 0.1

-

Ps

D'

0

Fig. 8.30

a-

Load-displacement characteristics;

b-

flow rate versus eccentricity, for a spherical

bearing

with 'pl=5Oo, 2=850 , qa=8=3S0,n=6.J=0.5 (ref. 8.34).




larger lands than for the bearing considered above; radial load capacity is virtually

unaffected by an increase

t o

0.25-((p4-(pl)f the width of the axial and radial lands,

while the reduction in axial load capacity is largely compensated by the decrease in

flow

rate. With reference

t o

Fig. 8.31.b, the non-dimensional axial load capacity

is

clearly reduced by a factor

0.9

when passing from a land angle of 3.5"

t o a

land

angle

of 7".

This may be compensated by a reciprocal variation in supply pressure.

On the other hand, the non-dimensional

flow

rate

is

halved; consequently, taking

the pressure increase into account, the actual

flow

rate will be reduced by almost

44%, and pumping power by 38%. Furthermore, thanks

t o

the small increase in

supply pressure, the radial stiffness will also actually be better.

- b -

0.2 I I

W (0.5, 0°)

W

D2P s

20

10

0.0

0.10 0.15 0.20 0.25

'Pa

92-91

Fig.

8.31

Axial load, radial load and flow rate versus a- pressure ratio and b- land width ratio,

for a spherical bearing

with (pl=500,p2=850, n=6

(ref.

8.34).

Friction power may be roughly calculated, in the concentric position E=O, by the

equation:

This last equation was obtained on the assumption that the recess depth

hr is

much greater than the radial clearance C. For the friction coefficient f, i t may be



278 HYDROSTATIC

1

UBRICA TION

assumed, by way of approximation, that f r = 4 C l h r ,provided that the flow in the re-

cess is laminar.

From Eqn 8.60 and

H p = p s Q

the concentric power ratio 17=HfdHpo can be easily assessed.

As

usual,

it

is advis-

able for

II

to be in the li-3 range.

The performance of the opposed spherical bearings i s easily obtained by sum-

ming together the effects of the two halves. In Fig. 8.32 the axial and radial load

capacities are plotted for a typical symmetrical opposed bearing.

03

0.18

0 6

0.03

0.09 015

-

Z

Ps D2

Fig. 8.32 Load-displacement characteristics for an opposed spherical bearing arrangement

with

'pl=5O0, 'p2=85",

( ~ , = 8 = 3 . 5 ~ ,

=6,J=0.5

(ref. 8.34).

For the design of spherical bearings, similar remarks to those concerning the

other multirecess types can be made. As usual, the designer should select bearing

size and supply pressure in such a way that, in the worst loading conditions, the

eccentricity is smaller than

a

prefixed maximum value

EM;

it is advisable to choose

a relatively small value for EM (e.g.

~ ~ = 0 . 5 )

o allow for sufficient stiffness and over-

load capacity. A pressure ratio lower than the optimal value (e.g. 8=0.5) should be

selected for these initial calculations.

The minimum and maximum values of the radial clearance can now be se-

lected, in such a way that

p

remains within a suitable range. As for cylindrical



MULTIRECESS

BEARINGS

279

bearings, the smaller the working tolerances, the smaller the minimum value of

the clearance, and, hence, the smaller the power expenditure and the higher the

bearing stiffness.

A

lubricant should now be selected whose viscosity leads

to

a power ratio

n = 1

for

the maximum value of the radial clearance. Of course, if the turning velocity is low

o r even null, this last condition will be impossible to satisfy, in which case the

maximum allowable viscosity must be selected.

8.7

YATES

BEARINGS

Figure 8.33.a contains a sketch of the so-called "Yates" bearing, which in prac-

tice consists of a multirecess journal bearing with, in series, two opposite thrust

pads, fed by the lubricant flowing out of the circumferential lands of the journal

bearing. For the sake of precision, it must be said that in the original proposal (ref.

2 .21 )

there was no side recess, i.e.

Di=D.

The same operating principle may be ap-

plied

t o

various other types of multidirectional bearing arrangements (ref. 8.35).

8.7.1 Axia l load

When no load is applied (fig. 8.33.b), in all the recesses of the journal bearing

pressure is

p j o = p l p , ,

and in the side recesses

p z 0 = P 2 p s .

When an axial thrust is

applied (Fig. 8.33.c), the side pads perform just like an opposed-pad bearing, sup-

plied at a pressure p j o through the circumferential lands of the journal bearing,

that consequently act as compensating restrictors. Their hydraulic resistance is

(8.61)

Values of Q'(O)/jl may be found in section 8.3 (e.g. Fig. 8.10.b); as a preliminary

approximation we may take

The axial load capacity may now be calculated as

wz = A , Pjo w;(a,; ) (8.62)

In the last equation,

W;

s given by Eqn

7 . 2 2 ,

~~=(h~-hO)lhos the axial eccentricity,

and the effective pressure ratio

8,

is

Pz =

Pzo

lPj0 =

82

18 1

(8.63)



280

HYDROSTA

- a -

t+zi

- c -

m

,TIC

LUBRICATION

- b -

Fig

8.33.

a- Yates bearing, b- pressure profile in the no-load condition,

c-

pressure profile in the

case of axial load, d - pressure profile in the case of radial load .

The effective area of the thrust pads is equal to the effective area of analogous

circular pads, from which the cross-section area of the shaft must be subtracted:

A, = 2 [DO"

A*,(r') 0 2 1

(8.64)

In Eqn

8.64

it is

r'=Di/D,;

the non-dimensional value A*,is given by Eqn

5.22.

Pressure

pjo

in Eqn

8.62

is not constant, due to the existence of the restrictors

R,

in series with the recesses of the journal bearing. However, Fig. 7.7.b shows tha t the



282

HYDROSTATIC L UBRICATlON

of the opposed-cone assemblies seen in section 8.5.2.An example is given in Fig.

8.35, aken from ref. 8.36.Loss of axial load capacity due to radial displacement and

(to a lesser degree) loss of radial load capacity due t o axial displacement are consid-

erable at high eccentricities; on the other hand, such losses are fairly small when

E

and

E,

are lower than 0.5.

Fig. 8.34 Pressure ratiosfl, andJ2 versus pressure ratioJz, for two values

of

pressure ratioJj,

0.3

-

0.2-

0

0.1

w 0.2

0.25

..

L

DP,

Fig. 8.35 Load-displacement characteristics for a typical Yates bearing (ref. 8.36).

8.7.4

Other

bearing

parameters

Let us now examine briefly the other bearing parameters (see also ref. 8.37 for

further details).



MULTIRECESS

BEARINGS 283

Flow rate in the reference configuration E=G=O is easily calculated from Eqn

8.11

or Eqn

7.23

(8.69)

where R* is given by Eqn 5.23.When any load

is

applied, flow rate should be slightly

smaller, if both Pj and P, are lower than 0.5. Pumping power is

H p o

=~s QO

(8.70)

Friction may be evaluated by summing the effects due to the journal and to the

side pads. In the unloaded configuration we find with no difficulty that

(8.7

)

Factors + nd H; may be calculated by means of Eqn

8.23

and Eqn

5.27,

respec-

tively. It should be noted that C and ho cannot be chosen in a completely indepen-

dent fashion, since in order to obtain the selected values of the pressure ratios we

must have

(8.72)

The power loss due to friction is easily calculated by multiplying the friction

moment by the angular speed

HfO = MfO a

(8.73)

8.7.5 Design procedure

The design

of

a Yates bearing is not very different from the design of journal

and opposed-pad bearings.

Once the maximum values of the load components are given, the first step con-

sists in selecting the values of the pressure ratios (see Fig.

8.34);

t should be taken

into account that PjeO.5 is the optimum for the radial stiffness (however, values

smaller than

0.4

should be avoided). as far as p , is concerned, selecting too small a

value would give poor axial stiffness; on the contrary, too high values lead to poor

radial load capacity (since it is proportional to

1-PZ)

nd even the axial stiffness

breaks down when is close

to

unit.

A

typical selection might be

&=0.2,

and

p1=0.6;

possibly, both may be increased in order to increase the axial load capacity; the

opposite may be done when the radial load is more critical.




The next step may consist in dimensioning the journal bearing (see

also

section

8.3.3).After suitable trial values for a / Land L I D (say a/L=0.1and L I D = l )have been

stated, diameter D and supply pressure p s may be calculated by means of Eqn 8.68.

In designing

it

may be best to slightly increase loads (perhaps by

10%)

o

account for

the losses in load capacity due to interactions between thrust and radial load. The

radial clearance C may then be selected; as usual, the narrower

C

s the higher

radial stiffness will prove t o be and the lower the total power consumption (provided

that

it

is possible t o select a lubricant with a viscosity close to the relevant

"optimal"

value).

The thrust bearing may now be designed (see also section 7.2.6). First, one may

choose the radius ratio

r'

(say r'=0.9); n general, thin lands are more convenient,

unless

R

is very slow. The axial gap

ho

may not be chosen at

will

but must be calcu-

lated by means of Eqn 8.72; should i t prove t o be

too

narrow, it might be necessary

t o

reduce r'; on the contrary, if ho seems too thick, causing poor axial stiffness, it may

be necessary to reconsider the journal bearing, increasing a IL . The effective area

needed

t o

sustain the axial load may be calculated by means of Eqn

8.65,

and then

the outer diameter Do by means of Eqn 8.64. Should the axial load be small, it could

turn out that

Di=r'D@:

in this case a smaller value can be chosen for

E ~ ,

s well as

for 81 nd

8 2 .

The selection of the "optimal" value of viscosity will be based, as usual, on the

power ratio

17=HfdHpo.

From Eqn 8.70 and Eqn 8.73 one may obtain

(8.74)

in which one should state l7=1 o obtain the optimal viscosity.

As

for the other bear-

ing types considered in previous sections, it may happen than this calculated viscos-

ity is

too

low or too high; the remedies are, of course, similar: increase film thick-

ness and

I l

in the former case (avoid, however, values of

Il

greater than

3);

increase

the land width and decrease I7 in the latter case. When the viscosity value is a con-

straint, the same Eqn 8.74 allows us to select clearance C in such a way as

t o

get a

power ratio in the l t 3 range.

Once the main bearing parameters have been established,

it is

possible to calcu-

late

flow

rate, friction torque, and the relevant power losses. It

is

also advisable

to

check the Reynolds numbers and the temperature steps, as in the case of the other

bearings. Finally, the hydraulic resistance of each of the

n

restrictors R, which

compensates recess pressures in the journal bearing has t o be



MULTIRECESS

BEARINGS 285

where Rj may be calculated by means

of

Eqn 8.61.

One final remark: it should be noted that the actual values of

C

and

ho

have a

considerable influence of the loading performance of the assembly. For instance, let

us assume that radial clearance proves

t o

be greater than the selected value

C ,

while all the other remain the same; consequently, the resistance of the lands

of

the

journal bearing (proportional to

C - 3 )

will be lower, and we actually get a smaller

value

of P I ,

and a greater value of

Icj2,

i.e. both axial and radial load capacities have

been lowered. Of course, if clearance is narrower than designed, and p 2 change

in the opposite way; nevertheless, stiffness may become very poor, since Bj and 0,

differ from their optimal values.

R E F E R E N C E S

8.1

82

8.3

8.4

8.5

8.6

8.7

8.8

8.9

8.10

8.11

8.12

O'Donoghue

J. P.,

Hooke C. J., Rowe

W.

B.; A Solution Using Superposit ion

Technique for Externally Pressurized Multirecess Jour nal Bearings Includ -

ing Hydrodynamic Effects;Proc. Instn. Mech. Engrs.; 185,5 (1970-71), 57-61.

Colsher R., Anwar I., Katsumata S.;

An Advanced Method fo r Predict ing

Hybrid Bearing Performance; AGARD Conf. Proc. 323, 1982, paper 28;

13

pp.

Ghai R. C., Singh D.

V.,

Sinhasan R.; Load Capacity and Flow Characteris-

t ics of a Hydrostatically Lubricated Four-Pocket Journ al Bearing by Finite

Element Method;

Int.

J .

Mach. Tool Des Res, 16 (19761, 233-240.

Shapiro W.;

Computer-Aided Design of Externally Pressurized Bearings;

Instn. Mech. Engrs., C

10/71

(1971); 22 pp.

Raimondi

A.

A., Boyd J.; An

Analysis of Orifice an d Capillary Com pensated

Hydrostatic Journa l Bearings; Lubr. Eng., 13 (1957), 28-37.

Kher A. K., Cowley A,; Th e Design and Performance C haracteristics of a

Capillary Compensated Hydrostatic Journal Bearing;

Proc. 8th Int. MTDR

Davies P. B.; A

General Ana lysis o f Multirecess Hydrostatic Journ al Bear-

ings;

Proc.

Instn. Mech. Engrs.; 184,l (1969-701, 827-836.

Bettini B.;

Calcolo d i Cuscinetti Ra dia li Zdrostatici; Progetto d i una Attrez-

zatura d i Prova; Doct. Thesis, 1985; 269 pp.

Singh D. V., Sinhasan R., Ghai R. C.; Finite Element Analysis of Orifice

Compensated Hydrostatic Jo urnal Bearings;

Tribology Int. ,

9

(1976), 281-284.

Sinhasan R., Sharma S. C., Jain S. C.;

Performance Charac ter ist ics o f a

Constant Flow Value Compensated Multirecess Flexible Hydrostatic Journal

Bearing;

Wear, 134 (19891,335-356.

Davies P. B.; Th e Dynamic Behaviour of Passively Compensated, Hydrostatic

Journal Bear ings w i th Var ious Nu mb er of Recesses;

J. of Mech. Eng. Sci-

ence,

18

(19761,292-302.

Rowe

W. B.; Dynamic and Static Properties of Recessed Hydrostatic Journal

Bear ings by Small Displacement Analys is ; ASME Trans, J . of Lubrication

Technology, 102 (1980), 71-79.

Co d. (1967), pt. 1,397-418.




Ho Y. S.,

Chen N. N.

S.; Performance Characteristics of a Capillary-Con-

pensated Hydrostatic Journal Bearing;

Wear, 52 19791,285-295.

Lingard

S.,

hen N. N.

S.,

Kong Y. C.;

Aspects of the Performance of Exter-

nally Pressurized Journal Bearings; Wear, 78 19821,343-353.

Ar tiles A., Walowit J. , S hap iro W.;

Analysis of Hybrid, Fluid-Film Journal

Bearings with Turbulence and Znertia Effects;

Advances in Computer-Aided

Be arin g Design, ASME-ASLE Lubrication conf., Washington D.C., 1982;

p.

Manea G.; Machine Elements; Technical Publishing House, Bucharest; 1970.

D um bra w a M. A.;

Review of Principles and Methods Applied to the Optimum

Calculation and Design of Externally-Pressurized Bearings. Part 1:

Low /Moderate Speed Bearings;

Tribology Int., 18 19851, 149-156.

Dumbrawa M. A.;

Review of Principles and Methods Applied to the Optimum

Calculation and Design of Externally-Pressurized Bearings. Part 2: Moder-

atelHigh-Speed Bearings;

Tribology Int.,

18

1985), 223-228.

Rowe W. B., Koshal D., Stout K. J.; Investigation of Recessed Hydrostatic and

Slot-Entry Journal Bearings for Hybrid, Hydrodynamic and Hydrostatic

Operation;

Wear, 43 (19771,55-69.

El She rb iny M., Sa lem F., El Hefnawy N.;

Optimum Design of Hydrostatic

Journal Bearings. Part 2: Minimum Power;Tribology

Int., 17 19841, 162-166.

Vermeulen M.; Optimisation of Hydrostatic Journal Bearings Zncluding

Hydrodynamic Effects;

Proc. Euro trib 81, Varsaw, 1981;V. 2, pp. 371-388.

Hydrostatische Radial-Gleitlager im Stationaren Betrieb (Berechnung von

Olgeschmierten Gleitlagern ohne Zwischennuten);

DIN 31655, Teil 1, Nov.

1984; 29

p.

Hydrostatische Radial-Gleitlager im Stationaren Betrieb (Kenngropen fur die

Berechnung von Olgeschmierten

Gleitlagern ohne Zwischennuten);

D I N

31655,Te il2 , Nov. 1984; 11 p.

Rowe W. B., Stout K.

J.; The Design of Externally Pressurized Bearings for

Reliability with Particular Reference to Manufacturing Errors; Instn. Mech.

Engrs.; C310 19731,421-429.

X u S.,

Chen B.;

Optimum Design and Automatic Drawing

of

Recessed Hy-

drostalCc Bearings;

Tribological Design

o f

Machine Elements , p roc .

15 th

Leeds-Lyon Sym p. on Tribology, Leeds, 1988;p. 411-418.

O'Donoghue

J.

P., Rowe W. B., Hooke C.

J.;

Some Tolerancing Effects in

Hydrostatic Bearings;Proc. 11th In t . MTDR Co d. (1970);pp. 317-322.

Sa to Y., Ogiso

S.;

Load Capacity and Stiffness of Misaligned Hydrostatic

Recessed Journal Bearings;

Wear, 92 1983), 231-241.

Moshin M. E.;

A Hydrostatic Bearing for High Speed Applications;

Tribology

Int., 14 1981),47-54.

Aoyama T., Inasaki I., Yonetsu

S.; Friction and Tilting Characteristics of

Hydrostatic Thrust Bearings;

Bull . Japan

SOC.

f Precision Eng., 10 19761, 68-

70.

P r a b h u

T.

J. , Ganesan N.;

Analysis

of

Multirecess Conical Hydrostatic

Thrust Bearings under Rotation;

Wear,

89

19831, 29-40.

O'Donoghue

J.

P.;

Design

of

Annular Multi-Recess Hydrostatic Thrust

Bearings;

Mach. a n d Prod. Eng., Nov. 18, 1970; p. 830-834.

25-51.

286

8.13

8.14

8.15

8.16

8.17

8.18

8.19

820

8.21

a22

824

8.25

8.26

827

82%

829

8.30

8.31



MULTIRECESS

BEARINGS

287

P r a b h u T. J . , G a n e s a n

N.;

Behaviour o f Mul t i recess Plane Hydrosta t ic

Thru st Bearings und er Conditions of Ti l t and Rotat ion; Wear,

92

(1983), 243-

251.

Hessey M. F., O'Donoghue

J.

P.; The Performance of a Four-Pocket Conical

Hydrosta t ic Bearing;

Externa l ly Pressur ized Bear ings

/

Jo in t Conf . Ins tn .

Mech. Engrs .

-

Ins tn. Prod. Engrs., 1971;p. 133-145.

Rowe W.

B.,

Stout K.

J.;

Design data and Manufacturing Technique for

Spherical Hydrostatic Bearings

in

Machine Tool Applications; Int .

J.

Mach.

Tool Des Res, 11

(1971), 293-307.

O'Donoghue J. P., Wearing R. S., Rowe W . B.; Multirecess Externally Pres-

surized Bearings Using the Y ates Principle;

Proc. Instn. Mech. Engrs,

C44

W earing R.

S.,

O'Donoghue

J. P.,

Rowe

W .

B.; Design of Combined Journal

Thrust Hydrosta t ic Bearings ( the Yates Bearing); Mach. and Prod . Eng. ,

Lund J. W.;

Stat ic St i f fness and D ynamic Angular St i f fness of th e Com bined

Hydrostat ic Journal-Thrust bearing; Mechanical Technology Inc., Rept.

No.

8.32

8.33

8.34

8.36

(19711,337-351.

8.36

Aug.

19, 1970;

p.

301-308.

8.37

MTI-63TR45, 1963.



Chapter

9

HYBRID PLAIN JOURNAL BEARINGS

9.1

INTRODUCTION

When a comparison is made between hydrostatic and hydrodynamic (self-

acting) bearings, the performance of the former at low speed is clearly unique,

while the latter may show a better load-to-power ratio a t high speed. Although the

turning velocity increases the load-carrying capacity

of

the recessed hydrostatic

journal bearings, too, the hydrodynamic effect can only develop on a fraction of the

total area (i.e. on the circumferential lands). Furthermore, the "hydrostatic" con-

tribution to the load capacity

of

a recessed bearing can be negatively affected, at high

velocity, by a number of factors such as cavitation in the inter-recess lands and even

in the recesses, lubricant recirculation in the recesses, reversal of the flow in cer-

tain restrictors, local antagonistic superposition of the Couette and Poiseuille

flows

(ref.

8.18).

The hybrid plain (i.e. non-recessed) bearings may be seen as special hydrostatic

bearings whose shape has been optimized t o take the maximum advantage from the

hydrodynamic effect on the load-carrying capacity; in particular, the recesses are

reduced t o a minimum, since they are made up of one o r two rows of 8t16 entry

ports (holes or slots). Consequently, these bearings are well suited to support jour-

nals t ha t may rotate a t different velocities, especially when the load grows with the

velocity (due, for example, t o the existence of some unbalanced mass).



HYBRID PLAIN JOURNAL B EARINGS 289

9.2 PERFORMANCE OF THE HYBRID PLAIN JOURNAL BEARINGS

To evaluate the performance of the hybrid journal bearings, shown in Fig. 9.1,

the relevant Reynolds equation (Eqn 4.18) should be solved, which for stat ic loading

may be rewritten as follows:

(9.1)

where

Sh

is the so-called Somm erfeld hybrid num ber (Eqn

8.8).

An

approximate solution to

Eqn

9.1 is easily obtained in the trivial case E=O, in

which the film thickness is a constant h&. Now, in the whole portion of the bearing

surface included between the two rows of entry holes (i.e. for

l z II(L/2-a)),

he rela-

tive pressure takes on a constant value

p0=Dps

which depends on the hydraulic re-

sistance of the inlet restrictors and of the outer clearances. This last resistance is

easily calculated by considering the clearances in the zone l z

I

>(L/2-a)

s plain

indefinite strips (the diameter of the holes is assumed to be small, compared t o a).

Bearing in mind the results obtained in section 4.7.1, pressure clearly decreases

with a linear trend from p=@ps at Iz

I

=(L/2-a)

o

p=O at I z I= L /2 .The volume rate of

the lubricant, flowing out

from

each side of the bearing, is

The total

flow

rate of the bearing is obviously:

I

Fig. 9.1 Hybrid plain journal bearings.




The hydraulic resistance of the restrictors could be calculated in the usual

ways. However, since the restrictors are generally made up of short drilled holes,

they can hardly be considered as indefinite-length capillaries. Furthermore, the

pressure

loss

at

the inlet losses of the bearing clearance may prove not to be

negligible.

In ref. 9.1 the overall pressure loss due to each restrictor is written as follows:

where

E o = s

and

(9.3)

are the mean ve-xity of the lubricant in the capillary anL the mean velocity at the

inlet of the clearance, respectively;

q

is the flow rate in the restrictor. With the aid of

reasonable simplifying hypotheses, the

a

coefficients are evaluated as follows:

@=1.16[1 exp (-250-

L d ) ] + 6 4 &

; a,= 0.4 + 1.54 ( T

In the foregoing equations, n is the number of holes for each row,

d

and 1 are the

diameter and the length of each capillary,

h

is the local thickness

of

the clearance

and

Re

is the Reynolds number in the capillary:

Introducing the above values of ar and a1 nto Eqn 9.3, we obtain:

In most cases, however, since

D>>d>>h,

Eqn

9.4

is reduced

to

the following:

A feeding hole may clearly be regarded as the series

of

a laminar restrictor and an

orifice, whose coefficient vanes with local film thickness.

As an example, let us consider a typical restrictor made of a drilled hole with

d=0.5 mm, k10 mm.

If

a lubricant with p 0 . 1 Ns/m2 and

p=900

Kg/m3 is used, the

influence of

q

and

h

on its hydraulic resistance

R,=ApIq

is that shown in Fig. 9.2.



HYBRID

PLAINJOURNAL BEARINGS

291

The increase in hydraulic resistance may become considerable, a t the higher

eccentricities, for the most heavily loaded restrictors; fortunately, the flow rate in

these restrictors

is

only a fraction

of

the flow rate in concentric conditions, and

hence it seems reasonable

t o

evaluate the actual hydraulic resistance

R,

of the re-

strictors at E=O and then to consider it as a constant value, disregarding the varia-

tion in hydraulic resistance due

to

the inherent orifice.

1.2

R, .I 0-12

[Ns/ms]

0.9

0.6

0.0 0.5 1 o

q 106 [rn3/s]

-rt , 0.5

.2

I

-

25

100

0.6'

'

I

0.0 0.5 1 o

q 106 [rn3/s]

Fig.

9.2

Effect of flow ate

q

and of film thickness h on the hydraulic resistance of a restrictor

(p=O.

1

Ns/m*, p 9 0 0 Kg/m3).

Another important problem connected with the use of drilled holes is the con-

siderable dependence of R, on the hole diameter. As can be clearly seen from Eqn

9.5, a 5% error on d causes an error on R, of about 20%, and so the actual value of B

may turn out t o be quite different from its design value. I t is easy t o verify, from Eqn

9.2 and Eqn

9.5,

that for small-clearance bearings (e.g. C=25pm) a very small bore

(0.1+0.2 mm) is required for the restrictors, and hence a bore accuracy of a few km

should be ensured.

The total pumping power lost in the bearing and in the restrictors i n the concen-

tric configuration is obviously:

x D C 3 P?

6 a

p =p s Q

=--B

whereas the friction power, at

E=O, is

the following:

(9.6)

(9.7)

and hence the concentric power ratio is:



292 HYDROSTATIC

L

UBRlCATlON

(9.8)

When a load is applied, the journal is displaced, and, since

h

is no longer a con-

stant, an approximate solution

to

Eqn 9.1 has

t o

be looked

for.

For example, in ref.

9.1 a perturbation method is proposed in which a row of feeding holes is replaced by

a continuous band source. However, a more versatile way of solving this kind of

problem is by discretization methods, such as finite difference (ref. 9.2, 9.3) o r finite

element methods. Such methods prove

t o

be more accurate, especially a t high ec-

centricity, and can also take into account the effect of cavitation th at may set in a t

high velocities and eccentricities.

(i)

Load capacity .

The results obtained in ref. 9.2 for hybrid bearings a t P=0.5 are

summarized in Fig. 9.3, in which the non-dimensional load capacity

w'=- W

P S L D

(9.9)

is plotted, as a function of the eccentricity, for certain values of the geometric pa-

rameter

a

lL and of the power ratio Il (i.e. of the Sommerfeld hybrid number

Sh).

Figure 9.4 contains plots of the attitude angle

$.

Since the hydrodynamic effect is

a/L=O.l

/

0.1

I ,

W '

1 -

/'

/--,---I.-.---

0.25

_

0.0

0.5 1

o

&

Fig. 9.3 Load

W'

ersus eccentricity

E

(forJ=0.5 and

L / D = l )

or certain values of ratio

alL

and of

power ratio 9 ref. 9.2).



HYBRID PLAIN JOURNAL B EARINGS

293

predominant, the direction of the eccentricity is different from that of the load, in as

far as

E

is greater, until cavitation occurs. For higher velocities

(17>3),

as is pointed

out in the same ref. 9.2, a t high eccentricity ( ~ ~ 0 . 9 )he load capacity is proportional

to

the square

root

of

Z7,

i.e.

it

is proportional to the velocity. The same plots may be

used for an approximate evaluation of the performance of the slot-entry bearings

when ~ 0 . 7 ;hile, a t the highest eccentricities, the performance of this las t type of

bearing seems worse (ref. 9.2). In particular, at zero speed, slot-entry bearings

show negative stiffness a t high eccentricity: that is, the lifting capacity

at

~ = ls

smaller than the apparent maximum load capacity at ~ ~ 0 . 8 .

0 0.5 1

0

Fig.

9.4

Attitude angle

4

versus eccentricity

E,

forfl=0.5,

L / D = l ,

alL=0.25

and

for two values of

power ratio

ZT

(ref.

9.2).

Figure

9.5,

taken from an earlier work (ref.

9.41,

shows the effect of parameter

a l L on load capacity, for two values of the power ratio

(a/L=0.5

n the case of

a

single

row

of feeding ports). There is clearly a sharp increase in load capacity as

a

1

L

becomes smaller. Although the flow rate (and hence the pumping power) increases

too,

it has been shown (ref. 9.4) that small values of a l L (perhaps

alL=0.1,

since the

feeding holes cannot be too close t o the outer edges of the bearing) are advisable in

most cases, for hybrid operation.

The effect of the concentric pressure ratio

,43

for hydrostatic operation

(n=O>s

quite similar

to

that already seen for the recessed journal bearings, and, hence,

p

should be within the 0.4+0.7 range. In hybrid operation, a t moderate velocities and

eccentricities, the load capacity becomes virtually independent from

/3

when /3>0.4;

at higher values of eccentricity, W'

grows

constantly as

f i

grows (see Fig. 9.6, ref.

8.19).



294

4

3

W'

2

1

0


- a -

- .

I

0

0.1

02 0.3

0.4 0.5

a

L

-

8

6

W'

4

2

0

I

0 0.1 0.2 0.3 0.4 0.5

a

L

-

Fig. 9.5 Load W'=WlLDp, versus ratio alL. for certain values of eccentricity E and for: a-

power

ratio

n=3;

-

II=12;

(ref.

9.4).

(ii)

Flow rate.

Non-dimensional flow rate a t p=0.5 appears, in practice, t o be

independent from the loading conditions; consequently, the flow rate can always be

evaluated using Eqn 9.2.

(iii)

Power.

Like the flow rate, the pumping power may also be considered not t o

depend on the load, and hence it may be calculated from Eqn

9.6.

The friction power may be approximately calculated from Eqn 9.7, even when

E>O.

A better approximation is obtained by integrating the shear stress on the bear-

ing surface. Bearing Eqn

4.36

in mind, disregarding the term due to the pressure-

induced flow, and assuming that no cavitated region exists in the clearance, the

friction torque on the journal is:

2a

Since we have

h = C ( l - ~

ostlt) and U = D N P ,we obtain:



W '

HYBRID PLAIN JOURNAL BEARINGS

-

~

-_

~ ~~~

3 1

0 0.2 0.4 0.6 0.8 1

B

295

Fig. 9.6 Load W' versus pressure

ratiofl, for

L / D = l , alL=O.l, I7=1 and for certain values

of

ec-

centricity E.

M n L D 3

f - 4 c

-

nd the friction power is:

1

H f = M f

0

Hi0

(9.11)

(9.12)

A t

high values of

s h

and

E,

however, the friction may be lower, due

t o

the onset

of

cavitation.

A t high speed, the torque required t o accelerate the fluid entering the bearing

(momentum torque), which is usually negligible, may become perceptible and may

be evaluated,

as

well

as

the relevant power, as indicated in section

8.3.1

(Eqn 8.24).

(iv)

Temperature .

The temperature step in the lubricant, for a single pass in

the bearing, may be evaluated with the aid of the usual assumption of adiabatic

flow:



296


(9.13)

It must be pointed out that careful control of lubricant temperature

is

more

important for the hybrid bearings than for the hydrostatic ones. If the temperature

rises, the consequent decrease in viscosity causes a considerable decrease in the

load capacity of the bearing (I7depends on the square of the viscosity). Hence it is

very important for the cooling system to be able

to

maintain viscosity below the value

used when designing.

9.3

DESIGN

OF

HYBRID BEARINGS

An optimization

of

a hybrid plain journal bearing may be based on the same

criteria already seen for the recessed bearings, i.e. mainly on the minimization of

the total power. Most of the remarks made in section

8.3.3

could, therefore, be re-

peated here, but there are certain differences, due to the fact that the hydrodynamic

effect plays an important role in sustaining the load. In particular, it is advisable

(ref. 9.4):

to use a small value for a / L e.g.a/L=O.l);

t o

allow greater eccentricities than for the recessed bearings b 0 . 6 ) ;

t o

select larger values for the concentric power ratio

(n>3).

Indeed, in the case of purely hydrostatic bearings the maximum eccentricity

must be limited because the stiffness is

very

poor when 00.6; in hybrid operation,

on the contrary, the load capacity increases sharply when

E

increases, and so stiff-

ness and overload capacity are ensured even when ~>0.9 . n the other hand, an

upper constraint to E may derive from other factors: e.g., at the higher eccentrici-

ties, a tilting error of the journal could cause the performance of the bearing

t o

dete-

riorate and even produce a localized contact.

The advisability

of

using higher values of the power ratio (as compared

t o

the

recessed hydrostatic bearings) is based on the fact that turning velocity not only

affects power expense, but also increases load capacity (this fact was disregarded in

the optimization of the hydrostatic bearings, section

8.3.3).

Consequently,

a 3<n<9

range is suggested for the power ratio (ref.

9.21,

although higher values could be

used.

The following is a possible procedure for the initial design of an hybrid bearing:

i) Select a set of geometrical parameters, with such typical values as

L / D = l ,

a

/L=O.l,

n=12

(this last parameter has little influence in hybrid operation,

provided

n>8).



HYBRID PLAIN JOURNAL BEARINGS 297

ii) Select a suitable range for 8, and a maximum value for eccentricity, perhaps

0.4</3<0.7,and ~ ~ = 0 . 7 .igher values for

p

and

E

can be allowed in hybrid

operation, but not a t low velocities.

State

l7=3

and calculate the relevant

W'

coefficient from Fig. 9.3

or

Fig. 9.5.

Using Eqn 9.9, select the values of the diameter of the bearing and of feeding

pressure, keeping in mind that it is generally advantageous to have larger

bearings and lower pressures.

iii) Select a tolerance range 6, for the clearance (see Fig. 8.14). The maximum

and minimum values

C M

and C, are then calculated in such a way as to

ensure that p always remain in the proper range: e.g. if it has been stated

that 0.4<p<0.7, t must be

c m 2 0 . 6 6 - c ~

see also section 8.3.31, and hence

C ~ > 1 . 5 . 6 ~ ;

n the other hand, radial clearance should be small, since i t is

more suitable from the point

of

view of the power expense. The choice of the

tolerance range clearly follows from a reasonable compromise between

manufacturing and running costs. Furthermore, an upper constraint e M

may be required for the displacement e=&Cof the journal. In this case, if it

turns out that E&M>eM and Sg cannot be reduced, an attempt can be made

to select a smaller value for

EM

after which one must start again from point

(iii).

iv) From Eqn 9.8 calculate the optimal value of the viscosity:

If this value is impracticably great, and it is not possible to further reduce

C M , one may try to reduce

ps ,

increasing the bearing dimensions. If, on the

other hand, p is too small, one may choose t o increase the clearance, but it

might be more advisable to select a higher value for l7,and restar t the calcu-

lations from point (iii).

v) Calculate flow rate, pumping power and friction power.

Check the rise in temperature, using Eqn 9.13.

vi)

Design the compensating restrictors.

vii) Check the Reynolds number in the bearing clearance.

viii) Check the dynamic behavior of the system, especially as far as whirl insta-

bility is concerned (see Chapter 10).

If different loads have

to

be sustained at different velocities, the performance

of

the bearing should be checked in all the situations foreseen. In particular,

at

low

velocities (hydrostatic operation) stiffness is strongly reduced and, moreover, great

eccentricities

(&>0.6)

hould not be allowed.



298 HYDROSTATIC L UBRICA TlON

9.4 CONCLUDING

REMARKS

Certain problems connected with hybrid bearings do not seem

t o

have been dealt

with yet in any depth. In particular, the isothermal flow model may be far from

representing the actual temperature profile in the clearance. Due to the great val-

ues of the power ratio needed

t o

get a high hydrodynamic load capacity, the temper-

ature step (Eqn 9.13) may prove to be considerable; furthermore, this is only an

average value, obtained by dividing the total power by the total flow rate . I t is easy t o

see that, actually, most

of

the lubricant flows directly outward (the entry ports are

close

to

the bearing edge) while the heat due to friction mainly develops in the lubri-

cant enclosed between the rows of supply ports, in the area of minimum clearance.

The temperature in this area is clearly much higher than the average value. Fur-

thermore, this hot lubricant, instead of being readily flushed out, may be recircu-

lated round the bearing, due

t o

the presence of the rows of entry ports (especially of

the slot type, that occupy practically the entire circumference with no interecess

land) and of large cavitated regions (ref. 9.5). Although other circumstances (heat

transport through the bearing sleeve, smaller friction power due to cavitation, back

flow

through the most heavily-loaded restrictors) may attenuate the temperature

peaks, it is clear t ha t bearing performance may be considerably affected.

Other problems involving the hybrid bearings may be (ref. 9.5) the existence

of

large cavitated regions (which may cause severe starvation problems), back flow

through the restrictors in the high pressure area (which occurs at high speed and

eccentricity, and thwarts the build up of the hydrodynamic pressure profile) and the

possibility of whirl instability. Instability is connected with the existence of a hydro-

dynamic side thrus t on the journal, revealed by the att itude angle (Fig. 9.4). In this

sense, cavitation may be considered beneficial and, hence, eccentricities larger than

&=0.6

hould be selected for hybrid operation.

Lastly, a peculiar type of hybrid bearing exists, characterized by the asymmetri-

cal configuration of its entry ports (ref. 9.5). Namely, each row is made up of a few

slots clustered symmetrically a t the bottom dead center, while a t the top dead center

there is an axial groove supplied a t low pressure (Fig. 9.7.b). This bearing presents

certain advantages, as compared with a usual slot-entry hybrid bearing (Fig. 9.7.a),

such as higher load capacity at small eccentricity, reduced cavitated regions,

smaller flow ra te and easily flushed-out hot fluid. On the other hand, a t high eccen-

tricity, load capacity is smaller, and (a t low speed) great negative stiffness occurs.

A

similar bearing (Fig. 9.8) is described in ref. 9.6 a s an alternative

to

conven-

tional generator bearings. The latter usually have small high-pressure pockets to

provide hydrostatic lift during start-up and shut-down operations, whereas in the

case of the hybrid bearing two rows of slot-type entry ports are also pressurized



HYBRID PLAIN

JOURNAL BEARINGS 299

0

3

2

W

LDPS

1

0

a=I

I

I

I

- a -

*

n=9

a2

04 0.6 aa

I

&

-

b -

Fig.

9.7

Load versus eccentricity for a hybrid journal bearing with:

a-

twelve symmetrical slots per

row;

b-

fiv e slots per row and

an

axial groove.

[a=a,dRD is

the slot width ratio].

during normal running. The load support arc is

120

degrees wide and i s bounded by

large low-pressure inlet recesses. This type

of

hybrid bearing is claimed to be more

efficient than conventional generator bearings.

I

---

I

O I

l o

I ---

I

1

Fig. 9 .8 Slot-entry generator bearing.



300

REFERENCES


9.1

9.2

9.3

9.4

9.5

9.6

Ichikawa A.;

A

study of High Speed Hydrostatic Bearings (Part 1, Theoretical

Ana lysis of Static and Dynam ic Characteristics of Hybrid Pl ain Journ al Bear-

ings; Bull. JSME, 20 (19771,652-660.

Rowe W. B., Xu S. X., Chong F.

S.,

Weston

W.; Hybrid Journal Bearings;

Tribology Int., 15 (19821,339-348.

El Kayar A., Salem E. A., Khalil M. F., Hegazy A. A.; Two-Dimensional Fi-

nite Difference Solution for Externally Pressurized Journa l Bearings o f Finite

Length; Wear,

84

(1983),1-13.

Rowe W. B., Koshal D.; A New Basis for the O ptimization of Hybrid Journal

Bearings; Wear, 64 (1980), 115-131.

Ives D., Rowe W. B.; The E f fect of Multiple Supply Sources on the Perfor-

mance

of

Heauily Loaded Pressurized High-speed Journal Bearings;

Proc.

Inst. Mech. Engrs., C199 (19871, 121-127.

Ives

D.,

Weston W., Morton P. G . , Rowe W. B.;

A Theoretical Znuestigation of

Hybrid Jou rna l Bearings Applied to H igh-s pee d Heauily Loaded Conditions

Requiring Jacking Capabil i t ies;

Tribological Design o f Machine Elements,

proc. 15th Leeds-Lyon Symp. on Tribology, Leeds, 1988; p. 425-433.



Chapter 10

DYNAMICS

10.1 INTRODUCTION

In previous chapters we have examined the behaviour of hydrostatic bearings

loaded by static forces. We now intend to evaluate the bearing response to time-

dependent loads; this means:

studying bearing stability; i.e. is the bearing able to return to its previous

equilibrium when excited by a small perturbation?

assessing bearing behaviour under given loads (impulsive and periodical, in

particular1.

A s a first step, the lubricant film may be assimilated to a mechanical system

formed by a (nonlinear) spring and

a

viscous damper, whose coefficient depends (in

a nonlinear way) on film thickness (Fig. 1O.l.a). Stiffness and damping can easily

be evaluated, without taking inertia and the compressibility of the lubricant into

account. On such assumptions, hydrostatic bearings always prove to be stable, but

in practice instability can sometimes arise. Actually, the compressibility of the oil

volume contained in bearing recesses and supply pipes and the compliance of the

tubing itself can play an important part in lowering the dynamic stiffness of bear-

ings and in causing instability. Moreover,

it

has

to

be borne in mind that, in practi-

cal cases, lubricant compressibility may be greatly increased by aeration phenom-

ena. Finally, the influence of the dynamic behaviour of the supply system (e.g. con-

trolled valves) must also be considered.

For a better approach

t o

bearing dynamics, more elaborate mechanical models

are often used (ref. 10.1, 10.21, like the ones in Fig. 10.l.b and Fig. lO.l.c, which




allow compressibility to be accounted for. Also electrical analogies have been pro-

posed. A more effective and more general approach may be based on the results of

control theory (ref. 10.3, 10.4).

Fig. 10.1 Equivalent mechanical systems

of

hydrostatic thrust bearings.

10.2 EQUATION

OF

MOTION

As will be shown in the following sections, in a thrust bearing the lubricant

fluid exerts a force W on the moving member; this force may be written in the form:

where A,

is

the effective area, p r is the fluid pressure in the recess and

B

is a

squeeze coefficient, which depends on clearance

h. For

the pads whose clearance is

not the same for the whole land surface (e.g. spherical pads),

h is

understood as

being the clearance in a reference point of the surface. In every case,

h

does not

indicate the

fiam thickness

along the normal to the surface, but the

clearance

mea-

sured along the direction of the displacement.

Let us now consider a reference configuration in which

h=ho,

and let us take

E=e

1ho

to define the non-dimensional displacement of the "moving member", whose

mass is

M,

rom this reference position. As noted in chapter 5, the reference con-

figuration is completely arbitrary for plane and tapered pads, whereas it

is

conve-

nient to assume as reference the

centred

configuration for cylindrical and spherical

pads. In certain circumstances, e.g. when studying the performance of the bearing

as a vibration attenuator, it is also necessary to consider the displacement <=z / ho of

the "foundation" (Fig. 10.2).

Hence in genera1,we have:

h - ho

= e

z

=

ho E -

0 (10.2)



DYNAMICS

303

Prs

1

Fig. 10.2 Dy n am ic pressure profile for a pad bearing.

Thereafter,

it

will be generally assumed that

z=O,

i.e.:

Recess pressure will be related t o the position of the moving member by a differ-

ential equation f ( p r ; , ;

h;

h)=O,which, in general terms, is nonlinear and depends

on the supply system.

Hence, the law of motion is found by solving the differential system:

(10.4)

In Eqns 10.4,M is the moving mass and F the external force acting on the bear-

ing. F can be put in the form

F

=

F,

+

6F ( t )

where F, is a constant force and 6F a time-dependent perturbation, which is as-

sumed not to be dependent on the configuration

of

the bearing.

In studying system stability, the bearing can be assumed to make small vibra-

tions

~ E = E - E ,

about the static equilibrium point E,, at which F , =- W , =- A , ~ , ( E , ) ;t is

thus possible to linearize Eqns 10.4 and then apply the Laplace transformation.

Equations 10.4 are transformed as follows:

(10.5)



304


where s is the Laplace operator and B,=B(hL,).Hereafter subscript S will mean "in

the static case". Since the reference configuration is often arbitrary, it could be

assumed hs=hO and, hence, E,=O and

~ E = E .

By examining Eqns

10.5 it

may be stressed (ref. 10.1) that the damping capacity

of the system (that is its ability to change the pressure distribution to react to the

squeeze velocity) is associated with two factors. The first one is the change in recess

pressure

6p,

caused by pressing away the lubricant from the pad through bearing

gaps and inlet restrictors (see Fig. 10.2). The second is the change

of

the shape

of

pressure distribution across the lands (squeeze film effect represented by coefficient

B ) . This factor is often much less important than the former, but because it is prac-

tically unaffected by lubricant compressibility, it may even become dominant when

lubricant stiffness

is

poor

(e.g. due

t o

a low bulk modulus caused by air entrain-

ment: see section 3.2.4).

Introducing the second

of

Eqns 10.5 into the first, we obtain:

W -Ae jlp 6E

=

ho

s

(B , + M S)8~ (10.6)

The vibrating systems described in Eqn 10.6 may be represented by the block

diagram in Fig. 10.3. Block &(s) depends on the lubricant feeding system, whereas

coefficient

B

depends on the pad shape: in the following sections will be shown how

they can be obtained in several cases.

Fig. 10.3 Block diagram for hydrostatic pad bearings.

The transfer function of the system is easily obtained from Eqn 10.6:

8 E 1

W

-=

h ,

M

~2

+

ho B ,

s +Ae;lp

(10.7)

By means of Eqn 10.7 one may study bearing stability and assess the linearized

frequency response for small vibrations.



DYNAMICS

305

10.3 PAD

COEFFICIENTS

The aim of this section

is

ta show how Eqn 10.1 may be written for a pad bearing.

In particular, the value of squeeze coefficientB must be calculated.

10.3.1

Circular-recess pads

Let us star t with the plane circular pad (Fig. 5.1.a) whose static behaviour has

already been examined in section 4.7.5 and section 5.3.

As

usual, we can start from

the Reynolds equation, in

this

case Eqn 4.23 which, assuming that film thickness is

uniform, becomes:

The above equation may be integrated twice to obtain the pressure pattern as

a

func-

tion of the boundary pressures, as well as of the angular and squeeze velocities L2

and

h:

The pressure

at

a certain radius proves to depend on the boundary pressures,

on the square of the rotation speed and on the squeeze rate in

a

linear fashion; this

will be true for every shape of pad, since it is

a

consequence of the linearity of the

Reynolds equation. The load capacity of the pad may therefore be calculated (as in

Eqn 10.1) by adding the term A,p,, already known from chapter

5,

o the term

Wd=-Bh obtained by integrating the following squeeze overpressure on the land

surface:

The followingis

easily

found

(10.8)

CoefficientB is plotted in Fig. 10.4.a, in which it must obviously be taken that

a=d2.

Another parameter related

t o

the squeeze effect, that will prove to be useful in

the following section, may be introduced a t this point; it is defined by the following

equation:

(10.9)



306


It can be noted that, for any given pad shape,

B

is proportional

t o

the square of the

pad area, t o viscosity, and t o

h-3

(see for example Eqn 10.8);R

is

proportional to p

and to

h-3 ,

hus ay * depends on the pad shape alone, and not on its actual size, nor

on any other parameter like

p

and

h .

For central-recess pads, parameter

a y *

has

also been plotted in Fig. 10.4.a.

- a - - b -

r'

a'

Fig.

10.4

Squeeze coefficient

B* = B

.32h3sin4a/(3qfD4)and squeeze parameter

a y *

for plane

( a = x / 2 ) and tapered circular pads:

a-

central recess:

b-

annular recess.

It should be pointed out that

A,pr is

not actually the

"static"

load capacity, be-

cause recess pressure

p r

is also affected

by

squeeze velocity

h.

Indeed,

p r

is related

to the flow rate by a law which depends on the supply system; the flow rate, in its

turn, depends on the squeeze velocity and may be written as the sum of the usual

term p r IR

and of the squeeze term (obtained, like the former, by the integration

of

Eqns 4.33):

(10.10)

Note that Q d ( r 2 ) - Q d ( r l ) = - & ( r ~

r : ) ;

that is, the volume of lubricant squeezed gut

from the land area because of a reduction in clearance

(or ,

conversely, flowing back

when

h

increases) is partly added to the pressure-induced flow p r I R leaving the



DYNAMICS 307

outer boundary of the land, and partly subtracted from the same flow p , l R entering

at th e inner boundary. In particular, it may be easy to see that the flow rate crossing

the recess boundary (i.e. a t radius

r = r l )

is:

where A, is the projected area of the recess (in this case, simply Ar=zr;) and A, is

the effective bearing area of the pad, viz. the ratio of the static load capacity to the

recess pressure.

Equation 10.11 may be considered to be of general validity, whatever the shape of

the pad.

10.3.2 Annular-recess pads

recess pads of the type shown in Fig. 5.13:

Proceeding a s above, the squeeze coefficient is easily obtained for the annular-

I

(10.12)

A plot of

B

and w* (Eqn 10.9) is given in Fig. 10.4.b, both for tapered and flat (a=x/2)

annular-recess bearings.

10.3.3 Tapered pads

For the pads in Fig. 5.19, we may assume that h=h,lsina, where h, is the film

thickness over the land surface. The following

is

easily obtained (see ref. 10.5) for

a

central recess pad:

(10.13)

that is, it equals the coefficient

B

already obtained for the flat pad (Eqn 10.81, divided

by sin4a. Equation 10.13 is plotted in Fig. 10.4.a.

Coefficient B for the annular-recess pad may be obtained in the same fashion,

namely by dividing Eqn 10.12 by sin44 and

it

may be found in Fig. 10.4.b.



308 HYDROSTATIC LUBRlCATlON

10.3.4

Screw and nut assemblies

Equation 10.13 and Fig. 10.4may also be used for the hydrostatic screw and nut

assemblies (see Fig. 5.35), substi tuting sina with cose, i.e, with the cosine of the

flank angle. If hydrostatic lubrication is extended over more than one turn of the

screw,

B

must also be multiplied by n (the number of active turns).

10.3.5 Other pad shapes

In general, the Reynolds equation cannot be explicitly integrated, for a generic

pad shape, and approximate solutions should be looked for.

For

the sake of clarity,

let

us

consider a plane pad of any shape; the relevant Reynolds equation will be

(from Eqn

4.15):

a a a a

zh3 Z P )

+ ( h 3

Z P ) =

12P h

(10.14)

to which the inner and outer boundary conditions p = p r on

ri

and

p=O

on

ro

have to

be added. If the film thickness h is not uniform (e.g. tilted pads), another term

should be added, depending on the bearing velocities in directions

x

and z .

Since Eqn

10.14 s

linear, the differential problem can be split into two parts:

$(ha ,PO)+

g h3

P O )

=0

;

p O = p r

on

;

p O = O on

To

(10.15)

(10.16)

where

P'Pa+Pd is

the solution being sought.

It

is

easy to see that p a is the static pressure distribution for the given boundary

conditions, while P d is the dynamic overpressure due to the squeeze effect. If tan-

gential velocities have to be accounted for, another term

p u

could be evaluated in the

same way. Similar considerations hold good for pads of any shape, cylindrical,

spherical, and

so

on.

The solution of the static problem (Eqns

10.15)

as been dealt with in Chapter

5 ,

leading to the evaluation of the effective area A,, and Eqns

10.16

may be solved with

the aid of the same numerical methods. Whatever the method (for instance finite

differences, finite elements, as well as boundary elements), a pressure distribution

pd,

proportional

to

the squeeze

rate h ,

will be obtained. If we integrate the pressure

Pd on the whole land area, we may evaluate the contribution

Sh

of squeezing to the

load capacity of the land itself, which has

to

be added to the static load capacity and,

if necessary, to a hydrodynamic term depending on tangential velocity.



DYNAMICS 309

In many cases, however, coefficients B and

ty*

may be totally disregarded.

Indeed, it will be shown in section

10.5

hat, when the compressibility of the lubri-

cant

is

negligible, the damping of the bearing is little affected assuming

B=O.

On the

contrary, when the lubricant stiffness

is

poor, as compared to the bearing stiffness,

the squeeze film effect may become important in assessing bearing stability. How-

ever, B may be evaluated roughly by dividing the land area into parts with simple

shapes, whose contribution to squeeze may be easily obtained.

For

instance, the land

of a rectangular pad may be split up in the way shown in Fig. 5.26; or parts 1 o 4

the expression of the squeeze load of an indefinite rectangle can be used:

while each corner can be approximated by an arc

of

a circular ring, for which

The flow rate can be found

for

a given pressure distribution by integrating the

flow vector q,

whose components are given in Eqns

4.32:

Q = q v dT

r

where

I- is

a closed contour of which v

is

the normal external direction. Again, the

flow rate will be the sum of the static and dynamic terms. If Tis the inner contour

of

land area, the result will be an equation such as Eqn

10.11.

10.4 SUPPLY

SYSTEMS

In order to write the second of Eqns 10.4,we may state the continuity between

the flow rate

Q

delivered by the supply system and the flow rate entering the

bearing.

With reference to Fig.

10.5,

he difference between flow rate Q and flow rate Qi

entering the bearing clearance is equal to the volume variation of the recess and

relevant tubing, to which the effect of the lubricant density variation hasto be added:

(10.17)

P

Qi Q

=

- V

-

-

The total volume

V

is the sum of recess volume

V,.

and of tubing volume

Vt;

hence:



31

0


Fig. 10.5 Flow rates in a pad bearing: Q flow rate delivered by the supply system; Qi f low rate

entering the bearing clearance (in static conditionsQ=Qi).

av, .

V =

V,.

+ V, = A ,

h

+

-pr

apr

(10.18)

(here we assume that pressure

p ,

is uniform in the recess and pipes up to the com-

pensation devices); A, is the projected area of the recess. The term

aV,lap,

is due to

tubing compliance and, in general, may be regarded as a constant. Density p may

be considered

to

depend solely on pressure, hence

where Kla is the apparent bulk modulus of the lubricant (see section 3.2.4).

Let us now define the lubricant stiffness

Kd

as:

Bearing in mind Eqn 10.11, Eqn 10.17 becomes:

(10.19)

(10.20)

From the point of view of flow continuity, the bearing may be represented by the

hydraulic system in Fig. 10.6; Q is the volume

flow

rate delivered by the supply de-

vices,

R

is the hydraulic resistance of the bearing clearances. In dynamic condi-

tions, the flow rates of a spring accumulator (whose spring stiffness and section

area are Kd and A,, respectively) and of a piston (section area A,) fastened to the

moving member of the bearing have to be added to flow Q delivered by the supply

system.

It should be pointed out that a number of simplifications have been introduced

in developing the foregoing equations. In particular, we have assumed that the

pressure is uniform in the recess and in the ducts connecting the recess to the com-

pensation device (o r to the pump), that the compressibility of the lubricant in the



DYNAMICS

31

1

Fig. 10.6 Equivalent hydraulic system of

a pad

bearing.

clearance may be disregarded (since the volume of lubricant in the clearances is

very small), and tha t the inertia of the lubricant is negligible. On the other hand, in

certain circumstances (such as long supply pipes and low viscosity), other dynamic

phenomena may occur, such as pressure waves in the supply pipes, that will not be

considered here.

In the following sections, several supply systems will be examined, for each of

which an expression of the flow rate Q will be found and introduced into Eqn 10.20.

The second of Eqns 10.4 will thus be obtained and then linearized in order to write

an equation for block

A,

which appears in the block diagram (Fig.

10.3)

and in the

relevant transfer function (Eqn 10.7). n applying small perturbation linearization,

one should remember that, as a general rule, the hydraulic resistance of the bear-

ing is proportional to h-3 (a t least in the case of small amplitude vibrations), hence

(10.21)

is

the perturbation of

1lR.

10.4.1

Direct supply (constant

f low)

The supply system is constituted simply by

a

positive displacement pump,

which delivers a flow rate Q which is not dependent on recess pressure

pr.

There-

fore, the equivalent hydraulic system shown in Fig.

10.6

may be completed by substi-

tuting the "supply" block with a constant-flow pump.

Equations 10.4 can now be written in the form:

(10.22)



31


W,=-F,

is the static load capacity of the bearing in the position of equilibrium

es=O,

and R , the relevant hydraulic resistance of bearing clearances.

Equations 10.22 constitute a third order nonlinear system, which may be inte-

grated by numerical means or linearized (see also appendix A.2). Applying the

small perturbation method and the Laplace transformation, we obtain:

(10.23)

Bearing Eqn 10.21 in mind, and comparing Eqns 10.23 with Eqns 10.5, it is easy

to find the following expression of block

&:

The characteristic frequencies

w1

and w 2 take on the following values:

Kd

0 2 = A x ;

Ks

01=A2,R, ;

(10.24)

(10.25)

K, is the static bearing stiffness

(see Eqn

6.51,

and Kd is the lubricant stiffness. In this connection, in evaluating Kd

(by means of Eqn 10.191, it is important

to

take into account the volume of lubricant

contained not only in the recess but also in the entire length of the feeding pipes

back

to

the pump.

We can now resort to Eqn 10.6 governing the dynamic behaviour of the bearing

system in the case

of

small-amplitude vibrations; the relevant transfer function

(Eqn 10.7) will be examined in section 10.5 in order to assess stability and frequency

response.

10.4.2

Compensated supply (constant pressure)

The bearing

is

fed by a system which is able to maintain a constant pressure

p s

over a compensation device, whose hydraulic resistance may be constant (capillary

restrictor or similar devices) or depend on the pressure step (in most case orifices,

though elastic restrictors have also been proposed). The overall hydraulic scheme of

the bearing is, then, the one to be seen in Fig. 10.7.



DYNAMICS

313

Fig.

10.7

Equivalent hydraulic system of a pad bearing supplied at constant pressure, compensated

by:

a-

a fixed restrictor;

b-

a

spool

valve; c- an infinite-stiffness valve.

The volume rate of flow passing through a laminar-flow restrictor

R ,

is easily

written (see Eqn 6 .19 )as:

whereas

for

an orifice we find (Eqn

6.24):

P s c E z p

& = Ro

(10.26)

(10.27)

As

usual,

p

is the pressure ratio

p r I p s

in the reference configuration

h=ho.

Equating the right-hand members

of

Eqn 10.20 and Eqn 10.26, the desired differ-

ential relation between

p ,

and E is obtained



314

HYDROSTATIC

LUBRlCATlON

In order to study stability and low-amplitude frequency response, one may lin-

earize and then apply Laplace transformation, thus obtaining:

from which

it is

easy to find the transfer function of block $ in Fig.

10.3. A

similar

equation may clearly be obtained in the case of orifice restrictors:

In both cases Ap may just be written as in Eqn

10.24,

but the characteristic fiequen-

cies are now:

The value of parameter

0 s:

(capillary)

(orifice)

(10.28)

(10.29)

As usual, Kd indicates lubricant stiffness (Eqn

10.19),

while Ks

s

the static stiffness

of the bearing (Eqn

6.23

and Eqn

6.26)

that may

a l s o

be written in the following form:

(10.30)

10.4.3

Control led restr ictors (constant pressure)

The bearing

is

now fed by

a

constant pressure system through

a

compensation

device whose hydraulic resistance will depend on the degree of freedom of a moving

element (e.g. a membrane) which, in its turn, will depend on the pressure step and

sometimes on the rate

of

flow being supplied, too. The transfer function of the over-

all system will obviously depend on a larger number of time constants than in pre-

vious cases, since the parameters describing the dynamics of the valve have to be

accounted for. This section will show the way the dynamic study of such a device

may be planned; in the following sections a few particular kinds of valve will be ex-

amined in

a

little more detail.

In order to evaluate

A

for passively compensated systems, Eqn

10.20,

express-

ing inlet flow rate Q as a function of the recess pressure and the position of the bear-



DYNAMICS

315

ing, has been put together with another equation expressing the rate of flow from

the supply system as a function ofp, (e.g. Eqn 10.26); the block ;lp was easily ob-

tained by eliminating Q and performing linearization. In the case of controlled

devices, the supply flow rate may be expressed as a function of

x

(the degree of free-

dom of the valve) and of its time derivative, as well as of p,:

Since another degree of freedom has been introduced in the system, another

equation is required, which can be obtained by writing down the balance of forces

acting on the moving member of the pressure-compensating device; in general, this

relation can be written as:

Substituting Eqn

10.20

for Q in both Eqn

10.31

and Eqn

10.32,

after performing

small perturbation linearization and Laplace transformation, the two equations

will take the following form:

(10.33)

The relevant block diagram in is shown in Fig. 10.8. Note that

Ap0

is the value

that the transfer function

;lp

would have for &=O (i.e. if the moving member of the

controlled valve was "blocked" a t it s static position); and

&&

must be null func-

tions when valve control is not sensitive to pressure or to flow rate (and hence to

E ) ,

respectively.

A n expression for block

is easily obtained from Eqns 10.33:

(10.34)

The bearing transfer function will, formally speaking, remain Eqn 10.7, pro-

vided the above value is substituted for %. In the following sections, we shall exam-

ine a few examples of controlled devices.

10.4.4

Spool or diaphragm

valves

The static behaviour of spool and diaphragm devices has already been dealt with

in chapter 6: in both cases the moving member reaches

a

position of balance due to

the opposing thrusts of an elastic element and recess pressure pr. The hydraulic

resistance of the valve therefore depends on the degree

of

freedom (namely

" x " ) of



31

6


Eo,

Fig. 10.8 Block diagram for a hydrostatic pad bearing compensated by means of a controlled valve.

the device, with a law R,=R,(x), which, in general cases,

is

a non-linear one. If the

valve restrictor

is

not laminar, a more complicated law R,=R,(x,

pr )

will emerge.

As indicated above, the flow rate delivered by the valve in dynamic conditions

must be written as a function of x and p,.. From Fig. 10.7.b it follows that Q is the

sum of the flow rate passing through the variable hydraulic resistance R , and of the

volume of lubricant displaced by the moving spool:

Q

=-Ps

- Pr + A, x

Rr

(10.35)

Note that , in the case of the diaphragm valve, x will be the displacement of the cen-

tre of the membrane and A, will be an "effective" area. Substituting Eqn 10.20 or

Q,

Eqn 10.35 becomes:

(10.36)

Equation 10.32 is,

in

this case:

M , X + K , X + A , ~ , = F , ~

(10.37)

where M u ,

K ,

and

A,

are the spool mass, the spring stiffness and the spool section

area, respectively; is a constant (namely, the spring force for x=O) .

The set of differential equations made up by Eqn 10.36, Eqn 10.37 and the first of

Eqns 10.4 describes the dynamic behaviour of the bearing system.

As

usual, in the

case of small amplitude vibrations, linearization may be carried out. For the sake of

simplicity, let us limit ourselves to examine the case

of

small vibrations around the



DYNAMICS 317

reference configuration

h,=ho

(hence, E,=O and B E = ) . In the neighbourhood of

E=O,

R,

may be written as:

R,

=

R,,

+

m,

=

+

C,

6e

+ c u p

p ,

(10.38)

P

The coefficients

C,

and

C u p

should be evaluated by considering the constructive

details of the device. If R, is a laminar-flow restrictor, i t is simply C,=O; whereas,

if the restrictor is a true orifice, it is

1 Ro

C u p = - @ p,

For the devices considered in sections 6.3.4 and 6.3.5, coefficientC,may be writ-

ten as

c,=---

Ku

Ro

P” A , Ps

The reader may refer to the relevant sections for the meaning of the symbols used

above. It is easy

to

see that the complex operators which appear in Eqns 10.33, and

hence in Eqn 10.34, are in this case the following:

(10.39)

In the equations above, KO,is the static stiffness of the bearing supplied through a

fixed restrictor Rr=RrOwith the same pressure ratio

P :

w, 1

ho

0

K w = 3 - -

Coefficient

CT

is given by

(10.40)



318


(the reader may easily check that, when the restrictor is a pure laminar-flow device

or an orifice, Eqn

10.40

give the same result as the

first or ,

respectively, the second

of Eqns 10.29).The characteristic frequencies o re given by the following equations:

(10.41)

Note that q, s the natural frequency of the valve spool, whose internal damping has

been taken to be negligible; should this last assumption not be applicable, the term

(l+s2/a$) could be substituted by a more comprehensive one in the form

(1+2CVS

%+S2/

o&.

Equation

10.34

can now be rewritten as

(10.42)

The overall transfer function will, as usual, be provided by Eqn 10.7. By the way,

the static stiffness of the bearing compensated by a controlled device (i.e. Eqn 6.39)

may again be easily obtained from Eqn

10.7;

for a laminar-flow valve:

10.4.5

Infinite stiffness devices

Let us now try to obtain the transfer function of a bearing, supplied by the con-

trolled valve in Fig. 2.12. The static behaviour of the valve has already been exam-

ined in section 6.3.6.

As

has been outlined in previous sections, equations express-

ing the flow and force balance for the valve should be written down; by examining

the

flow

paths in Fig. 10.7.c, we can obtain the following equation:

(10.43)

where Q i is the flow rate that crosses the inlet restrictor R i and y,=(Av-A,)lAv.

Since pU=pr+(Q-AvX)Rv , e have:

(10.44)



DYNAMICS

319

Force balance on the spool gives:

M , X + A , P ~ - ~ ~ ~ - A ~ ~ ~ , = O (10.45)

and hence

The laminar restrictor

Rd,

which has no effect on static behaviour, may be added to

increase the damping of the movements

of

the spool.

As

usual, the compressibility

of the lubricant in the device has been disregarded.

The hydraulic resistance

Ri

is, in general, dependent on the position of the spool

and on pressure p u .For small-amplitude vibrations

it

may be stated, that

Of course, it would be C=O

for

plain laminar flow: for the sake of simplicity, let us

limit to this las t case.

Eliminating Q from Eqn 10.44 and Eqn 10.46 by means of Eqn 10.20, two differen-

tial equations in E, x and p r are obtained, which, added

to

the first of Eqns 10.4, allow

the problem to be solved numerically. For small vibrations around the reference

configuration E=O, by linearizing and Laplace-transforming, a set of equations like

Eqns 10.33 is obtained. After a number of manipulations, the following equations

may be written:

s

1 + -

1 + -

" 2

1- -

Kocho 1

& 0 = 7

-

S

PSCV " 3

;1 =p2--

s

Ro

1 + -

A? 1 -

;1 =- 1 Kocho 1 0 1

Rd

s

1 + -

"2

& P = - A X Rd 1 + -

l+-

Y V R V @4

s

1 + -

V& l - p A v A e R o

l . + -u Rv

(

i4

(10.47)



320 HYDROSTATl C LUBRICATION

A s

before, Koc and A are, respectively, the static stiffness and the value of

A,

for

Cv=O that is, the values th at would be obtained if the spool was blocked). We have

stated

10.5 DYNAMICS OF SINGLE-PAD BEARINGS

10.5.1 Transfer function

The block diagram representing the dynamics of a thrust bearing has been

presented in section 10.2 (Fig. 10.3), as well as the relevant transfer function (Eqn

10.7). The block A which appears in Eqn 10.7 stands for the feedback of the system:

namely it shows how the system reacts, dynamically changing the recess pressure,

to displacements induced by load variations.

In section 10.4 it has been shown how $ depends on the lubricant supply system

and on the lubricant itself (namely, on its compressibility). In particular, in the

case of passive compensation (e.g. a constant flow pump or fixed restrictors)

Ap

is

given by Eqn 10.24. The characteristic frequencies o1 nd

02,

on which mostly

depends the dynamic behaviour of the system, turn out to be proportional to bearing

stiffness

K ,

and to lubricant stiffness

Kd,

respectively (of course l / w 2 = 0 when com-

pressibility

is

negligible); they may generally be written in the form of Eqn 10.28, in

which the value of constant

Q

depends on the type of supply system (namely, we

have to take o=l in the case of direct supply by a constant-flow pump, or a value

calculated by means of Eqns 10.29 for capillary and orifice compensation).

In the case of an "active" supply system (e.g. constant flow valves

or

"infinite

stiffness" devices) block

$

s obviously more complicated (see a l s o ref. 10.4 and 10.6):

recess pressure

p,.,

in fact, depends also on another variable

x

(the degree

of

free-

dom of the supply device), which, in turn, depends on recess pressure and some-

times on the hydraulic resistance of the bearing, and hence on E . This argument

has been treated in sections 10.4.3 to 10.4.5.

In the case of passive compensation it may be interesting

to

return to the equiva-

lent mechanical system of Fig. 10.l.b, and search the values to be assigned

to

opera-

tors K 1 and

B , .

We can write the linear differential equations which describe the

mechanical system and then obtain the relevant transfer function; having estab-

lished tha t it must be identical to Eqn 10.7,it turns out that:



32

(10.49)

It

is

obvious that K, and B, turn out to be negative when &<K8 and, conse-

quently, the system may prove to be unstable. On the contrary, when fluid com-

pressibility

is

very

poor

(that is Kd>>K8),

Kl

an be substituted by

a

rigid connection

and the model in Fig. 10.l.b

is

reduced to the second order model in Fig. 10.l.a,

where the spring stiffness

is

K=K,, and the damping constant

is

B,+K8101. Thus, in

the case of

a

passive supply system and negligible fluid compressibility, Eqn 10.7

takes on the usual appearance of the transfer function of second order vibrating

systems:

(10.50)

where

wn

nd ( are, respectively, the undamped natural frequency and the damp-

ing factor of the system:

(10.51)

Parameter

t , ~

s defined, beai -.ig a

ing equation:

,o

in mind Eqn 10.9 and Eqns 10.28,-y the follow-

(10.52)

Since

w*

s, in general, much smaller than unity, the squeeze film effect may be

disregarded when evaluating the damping factor in the case of incompressible lu-

bricant. If we want to include compressibility, Eqn 10.50 must be substituted by the

following transfer function

where



322


10.5.2 Stabi l i ty

In the previous section it has been shown that a single-effect passively compen-

sated bearing, making small amplitude vibrations, behaves in exactly the same way

as the mechanical system in Fig. 10.l.b)provided that condition

Kd>Ks

is satisfied;

that is , the stiffness of the lubricant in the recess and relevant tubing (and the stiff-

ness of the tubing itself ) must be greater than the static stiffness of the bearing.

This proves to be a sufficient condition

for

stability (the mechanical systems in Fig.

10.1

are always stable when spring and damping constants are greater than zero).

Condition Kd>K, is often easily satisfied due to the great bulk modulus of lubricants,

while in gas bearings stability is often an important factor

to

be dealt with. Problems

may, however, arise when:

- the recess and relevant tubing contain a large volume of lubricant;

-

rubber hoses ar e used to connect compensation devices to the recess;

-

the lubricant may hold a great amount of air.

This last factor is the most dangerous, because, in practical applications, it is

not easy

t o

forecast quantitatively the compressibility increase due to aeration.

A less restrictive condition for the stability of Eqn 10.53 may be obtained by

means of the well known Routh or Hurwitz criteria (ref.

10.7).

These methods con-

sists in checking if the coefficients of the characteristic equation of the system sat-

isfy o r not certain conditions. In our case, the characteristic equations is:

For a third-order system

to

be stable, the Routh criterion requires that all the coeffi-

cients ai of the characteristic equation, as well as the parameter

a3

a0

b = a , - -

a2

(10.55)

(each ai indicates the coefficient of the relevant power of

s

in the characteristic equa-

tion) must have the same sign. Since all the ai are greater than zero, the system

proves to be stable when:

(10.56)



DYNAMICS 323

All the parameters in Eqn 10.56 are positive, and then the right-hand

side

is

always less than unity; this confirms that {>1 is a sufficient condition, tallying with

the limit case of

B=O.

On the other hand, if

B

and are great enough, the right-hand

side becomes negative and stability is clearly ensured whatever the value of

K d,

although the effective damping of the system may prove to be very poor, for low lu-

bricant stiffness, in spite of high values of

c.

Bearing in mind that ty depends on the shape of the bearing (see section 10.31,

it

may be concluded that , from the point of view

of

dynamic behaviour, i t is advisable

to design bearing with large lands in order t o increase the margin of stability when

the lubricant compressibility is not low enough

t o

ensure tha t Kd

is

safely greater

than K,.

When controlled devices are used for pressure compensation, Eqn 10.7 proves to

be of a higher degree and depends on a larger number of time constants. Instability

could now occur even when lubricant compressibility is negligible. For instance, let

us consider a diaphragm-controlled restrictor and assume that the mass of the

diaphragm is very low: in other words we assume that w, is much greater than

w 1

and w 3 , Equation 10.42 may, therefore, be simplified as follows:

(10.57)

We can now substitute Eqn 10.57 into Eqn 10.7, draw the characteristic equation

(which is again of degree 3) and examine

its

coefficients: i t is easy to see that the

coefficient of s2may become negative for certain values of@2/PU: clear symptom of

instability As before, a more detailed analysis of stability can be carried out by ap-

plying the Routh criterion t o the coefficients of the same characteristic equation.

Furthermore, for the sake of simplicity, we may disregard the squeeze coefficient

B,

and thus it is easy to see that instability is likely to occur when

(the last term on the right-hand side does not actually depend on But as shown by

Eqn 10.41). It is interesting t o note that often ~ 3 > > ~ 1nd, then, the condition for

stability becomes K o / K o c c { = w ~ / w l ,hat is Kod(d.

The problem is rather more complicated when the parameters disregarded

above need to be taken into account. Stability should be carefully studied in these

cases with the valuable aid

of

the methods developed in the theory of automatic con-

trol. A detailed analysis of such methods is clearly beyond the scope of the present

work: we shall confine ourselves t o briefly recalling how the Nyquist method may be



324


used to assess system stability (the reader may consult specialized works, such

as

ref.

10.8,

for further details).

The first step consists in tracing the Nyquist diagram, that is mapping the

Nyquist path

onto the plane of the

open-loop transfer function

G H ( s ) .This last is, in

our case:

(10.58)

The Nyquist path (shown in Fig. 10.9.a) is an oriented closed contour in the plane of

the complex variable

s,

embracing the entire right half-plane. The half circle with

vanishing radius

is

due to the need to exclude the origin, which is

a pole

(namely, a

point of singularity) for the complex function

G H ;

if other poles should exist on the

imaginary axis

s=io,

they must be excluded in the same fashion.

It

may be shown

that the whole infinite-radius half circle is mapped onto the origin of the plane of

G H , while the vanishing half circle around the origin is mapped onto

n

infinite-

radius half circles

( n

being the number of poles in the origin). For the types of func-

tion we are considering, the Nyquist diagram proves to be symmetric around the

real axis, and hence i t is enough to plot GH for s=iw, where w goes from 0 to -.

The second step consists in counting the number

np

of poles of

G H ( s )

included

in the Nyquist path (i.e. belonging

to

the right half plane);

this

may be done with the

aid of the Routh criterion, applied

to GH.

I

- b -

Re(GH)

Fig.

10.9 a-

Nyquist path;

b-

Nyquist diagram for restrictor-compensated bearings with negligi-

ble squeeze coefficientB.



DYNAMICS

325

Finally, the number nt of turns that the diagram makes around the point GH=-1

need to be counted. We have n p O f the turns are clockwise (bear in mind tha t the

diagram is oriented) and nt<O if they

are

counter-clockwise.

For the system to be absolutely stable, it must

be:

(10.59)

nP

t =

-

Since

np20,

he system clearly cannot be stable if the turns are clockwise, th at is

By way of example, let us consider the typical case of a restrictor-compensated

bearing,

for

which the block

Ap

takes

on

the simple form of Eqn 10.24, where

01

and

0 2 = 6 0 1

are proportional

to

the static bearing stiffness and to the lubricant stiffness,

respectively. For the sake of simplicity, let us first consider th at the squeeze coeffi-

cient B is negligibly small, which gives us:

if (-1,O)is an internal point for the Nyquist diagram.

A s long as {>1, the Nyquist diagram takes on a shape that is

similar

to the lower

curve in Fig. 10.9.b, whatever the values of

c

and

%.

Namely, the limit of GH(io) for

o+O is

- e i Z

and the diagram is closed by an infinite circle (the origin is a double

pole). The point GH=-1 is outside the Nyquist contour (nt=O)and, since the poles are

the origin and

s=-50n/2c ,

we have np=O. The system is, therefore, stable. If, on the

other hand, we have &1, we get a plot like the upper one in Fig. 10.9.b. Now the

point GH=-1 is inside the Nyquist diagram (n,=l ) and the system

is

unstable,

as

predicted in Eqn 10.56.

The problem becomes slightly more complicated when we introduce the squeeze

coefficient B. The open-loop transfer function now becomes:

(10.61)

which has a single pole in the origin and two poles in the left half-plane. Figure

10.10 contains sample Nyquist diagrams, obtained for c=2, ~ 0 . 0 2nd

a

number of

values of <.The limit of GH(iw) for o+O is now

=-eid2

and there is only one infinite

half circle.

A s

pointed out above, if

B

and are large enough, the stability of the sys-

tem

is

ensured, whatever the value of compressibility; in other words, all the dia-

grams of the family obtained by varying

5

do not cross the

real

axis,

or

cross

it

at a

point on the right of GH=-l. Whereas, in the case shown in Fig. 10.10, the system

proves to be unstable for the lowest values of 5. The crossing frequency onmay be



326


Fig. 10.10 Nyquist diagrams for restrictor-compensated bearings ( B S ) .For the sake of clarity, the

drawing is out of scale.

found by solving the real equation Im[GH(io,)]=O. Provided a finite real solution

exists, the system will be unstable if IGH(iw,)l>l.

10.5.3 Frequency response

The frequency response of the system (i.e. the amplitude and phase shift

of

the

steady vibration of the bearing when the force perturbation has a sinusoidal shape

with unitary amplitude and frequency f=wl2x) can be found by substituting s=iw in

the transfer function (Eqn 10.7). A complex number is obtained, whose modulus

and argument represent the amplitude and phase shift of the vibration of the bear-

ing, respectively.

Since too many parameters are involved, general diagrams cannot be given

here, except for passively compensated systems. In the simplest case, when lubri-

cant compressibility is negligible, Eqn

10.7

may be written in the simpler form of

Eqn

10.50

and the relevant frequency response, typical of second order systems, is


When the effects of lubricant compressibility have to be evaluated, one can use

Eqn

10.53

instead of Eqn

10.50:

a number of sample plots are given in Fig.

10.12.

It

may be seen that, when the lubricant stiffness is comparable to the stiffness of the

bearing, a resonant peak is present even for high values of 6. In order to visualize

better the effect of lubricant compressibility, in Fig. 10.13 we have plotted against

[

the values of the peaks of the frequency response for certain values of 4 and w (bear

in mind that this last parameter is proportional to the squeeze coefficient

B

and

is

therefore a sign of the intrinsic damping capacity of the lands of the pad). In prac-

tice the effectiue damping proves to be greatly lowered, when 5 and y are small.



DYNAMICS 327

0

1 2

L L

a n

Fig.

10.11

Frequency response

for

a direct-supply or restrictor-compensated bearing

(incompressible lubricant).

However, the influence of w is insignificant when

5>5;

since

w

is usually much

smaller than

1,

it follows that it may simply be taken that

B=O

and

w=O

when the

lubricant is

stiff

enough.

It should be borne in mind tha t the considerations above are only valid for small

vibrations around a point of equilibrium. Actually, when the amplitude of vibration

exceeds 20-30%of h,, stiffness and damping may no longer be considered to be con-

stants; thus if we wish to forecast the behaviour of a bearing with large amplitude

vibrations, we must integrate the nonlinear equations

10.4

by means of numerical

methods; the second of these depends on the supply system (for instance one should

use equations 10.22 for constant-flow feeding).

EXAMPLE 10.1

Let

us

consder again the simple pad bearing, directly fed at a constant flow

rate, whose static calculations were performed i n exam ple 6.1. As will be remem-

bered, the main bearing parameters fixed there were: D=O.l m,

r’=O.75,

p=O.1

Ns

1

m2 and, u nder a load

W=40

N , ho=30 pm.

Let th e moving

mass

be M=3061

Kg,

nd the equivalent bulk modulus of the

lu-

bricant be Kla=109N lm 2; we have:



DYNAMICS

F h R . I O - ~ Z ,.10-9

. I O - ~ w1

w,

(KN)

urn) (Ns/rn5)

(N/m)

(Ns/m) (s-1)

( s l )

30

33

1.53

2.73 1.49

50.1

944

40 30 2.03 4.00 1.99

55.1 1143

329

c

w

9.7 0.027

10.7 0.027

ah to check the stability of the bearing for static loads between 30 and 40m;

b)-to asses the frequency response o f the system.

In order to carry out these verifications, it

is

first necessary to asses, for both th e

greatest and the least values of load, the relevant values of fi l m thickness, static

stiffness, hydraulic resistance and squeeze coeffEient (see the synoptic table below).

a) From Eqn 10.36 it

is

now possible to calculate the t im e constant

1

ol (bear in

min d tha t the effective bearing area isA,=5.97.103ma) and the values of %,

&

and

as

shown

in table below:

Eqn 5.21 Eqn

6.5

Eqn 10.8

Eqn 10.25

Eqns

10.51

Eqn 10.52

By means of Eqn 10.14 it

is

now easy to verify tha t the system

is

stable fo r every

value

of

the ratio &IQ I Kw

3

2

6h

6WKS

-

1

0

1

0 100

200

300

400 500

Fig. 10.14 Example 10.1:requency response

at

W& KN.

b)

Th e frequency response has been plotted i n Fig. 10.14, or a number of values

o f

(,

i n th e case of

W=40

N

(for smaller

loads

a similar diagram would have been

obtained, with slightly greater amplitudes). If the compressibility of the lubricant

were negligible, the high value

of

the damping would prevent the frequency re-




F

(KN)

20

35

sponse from the presence of peaks, which, on the contrary, may be notable for val-

ues of 5 lower than 1. This means that supply pump should be very close to the bear-

ing. For instance, in order to have <>1, lubricant stiffness Kd should be greater than

4.109 Nl m : it follows from Eqn 10.19 that the volume of lubricant in the recess and

supply pipes should be smaller than 8.9.10-6 m3.

It is easy to see that, in order to increase

<,

one could try to increase slightly the

fi lm thickness or the effective area of the bearing. In the first case greater flow rate

and pumping power would be required, while stiffness and damping factor would

turn out to be lowered. In the latter case, on the contrary, a great increase in lubri-

cant stif fness (namely, parameter

5

proves to be proportional to the fourth power of

D )

would match a consistent reduction in flow rate, supply pressure and total power

(because turning speed is

low); damping factor, too, would be greater.

P h prlps d Ks.10-9

<

w,,

C

Y

(Ns/m*) urn) (N/m)

@-I)

0.015 40

0.437 1.78

0.845 3.87 650

0.196 0.003

0.015 24.8

0.765

4.25 0.995

3.29

705 0.315 0.007

EXAMPLE 10.2

The pad bearing already considered in Example 6.4 bears a load W0=20

K N ,

with

a clearance ho=40 pm, when fed at

a

constant pressure ps=4 MPa through a com-

pensating restrictor, with a pressure ratio p=0.437. Under a load W ~ = 3 5N , clear-

ance is reduced to 24.8 ,um and pressure ratio rises up to 0.765.

Assuming that moving mass is M=2000h that the equivalent bulk modulus of

lubricant is Kla=500 MNI m2 and that the lubricant volume comprised between the

restrictor and the pad clearances

is

V=20.10-6m4

we

have to check for stability and

to assess the frequency response

o f

the system.

From Example 6.4 we get: D=0.16 m2, r'=O.625, a'=0.067, A,=O.0114 m2,

R*=0.0154. Hydraulic resistance R may

be

calculated from Eqn 5.68, squeeze coeffi-

cient

B

from Eqn 10.12, parameter

w*

rom Eqn 10.9 (or

from

Fig. 10.4.b), static stiff-

ness

K,

from Eqn 10.30, while lubricant stiffness Kd is given by Eqn 10.19, in which

the contribution o f the compliance of supply pipe may be disregarded. The main

dynamic parameters can now be evaluated, as shown in table below.

Eqn 10.29 Eqn 10.30 Eqn 10.54 Eqns 10.51 Eqn 10.52

Stability is clearly out of question,

since

5>1; the linearized frequency response

has been plotted for both the greatest and the least loads in Fig. 10.15. Comparing

this diagram with the frequency response in the case of incompressible lubricant

(Fig. 10.11) it should be evident that compressibility produces a noticeable decrease



DYNAMICS

4

331

Fig. 10.15

Example

10.2:

frequency response.

in damping, that

is

the amplitude

of

oscillation in the neighborough

of

resonance is

greater. In order to reduce the peaks of vibration one should increase 5 and, above

all,

c.

This may be obtained reviewing the design and using,

if

possible,

a

slightly

larger pad.

10.6 OPPOSED-PAD BEARINGS

Opposed-pad thrust bearings (Fig. 7.1)may be regarded as a set of two single-

effect pad bearings. For the sake of convenience, the position of the moving member

which divides the axial playg into two equal parts may be taken as reference;

hence,

hlo=h20=ho=g12.or

the two pads, we have (see Eqns 7.4):

hl

-ho

& 1 = h = &

,

0

(10.62)

If each pad is supplied independently (e.g. by two pumps or through two restric-

tors), the relevant block diagram can be obtained by summing the effects of both

components (Fig. 10.16).The equation of motion may be obtained from the balance of

forces acting on the moving member:

By linearizing, Laplace-transforming and bearing in mind t ha t (as follows from

previous sections):



332

HYDROSTATIC LUBRlCA

TION

Fig. 10.16 Block diagram for opposed-padbearings.

equation 10.63 becomes:

M

ho S ~ S E

(B1, +

B a )ho

s

BE+

(Ae, &I +Ae2 &2)

S E =

6F

(10.64)

(10.65)

Squeeze coefficients

B1,

and

Bas

can be calculated from the results in section 10.4,

while blocks jLP have been the matter of section 10.5.

In what follows, symmetrical bearings alone (i.e.

Ael=Aez=Ae

and

R ~ ( E ) = R ~ ( - E ) )

will be studied in a greater detail.

For

such bearings the block scheme in

Fig.

10.16

may be substituted by the one in Rg. 10.17; the latter is valid even when the two re-

cesses are not supplied by independent devices (e.g. when a flow divider

is

used),

after the relevant expression for

&

has been found. In the case of symmetrical bear-

ings, Eqn 10.65 may be rewritten as:

M

ho

s2&+ B,

ho

s

& + A e

jLP

8 ~ =

F

(10.66)

Fig. 10.17 Block diagram for symmetrical opposed-padbearings.



DYNAMICS

333

10

B

5 -

0

and, thus, the relevant transfer function is formally identical to Eqn 10.7.Block

&

is

clearly the sum of AP1 and APz. The squeeze coefficient may be calculated from the

following equation:

-

1

where Bo has the same value as each pad (for example, for circular bearings, see

Fig. 10.41,while B' is given (for pads having uniform film thickness) by

A plot of B' is also given in Fig.

10.18.

(10.68)

10.6.1 Direct supply (constant f low)

flow Q / 2 , we have:

When each recess is supplied independently by a pump delivering a constant

(10.69)

S

1+ (1 &)-

5P2

Koho 1

w1

(1-EY 2

42'

-= -

E

2Ae

-

+ 1 s

where:



334


(10.70)

(hereafter we shall omit the subscript s, taking i t for known that in linearized equa-

tions all the parameters take a value corresponding to the steady part of the load).

A s usual, KO s the static stiffness of the bearing for E,=O (Eqn

7.11).

The equations

above have been obtained from Eqn

10.24

and Eqns

10.25,

taking into account tha t

static stiffness and hydraulic resistance of each pad depend on

E ,

as indicated in

Eqn

6.7 and

Eqn

5.12.

The transfer function of block

&

is easily obtained adding &1 and h2ogether:

(10.71)

In the particular case of small amplitude vibrations around cS=O, Eqn

10.71

takes the simpler form of Eqn

10.24.

10.6.2

Compensated supply, passive compensation (constant pressure)

We may obtain the feedback functions ;Ipl and Ap2 for an opposed-pad bearing

compensated by laminar restrictors (Fig.

7.6)

from Eqn

10.24

and Eqns

10.28;

pro-

ceeding

as

in the previous section, namely bearing in mind Eqn

6.22, 6.23

and

5.12,

we find

(10.72)

where function

0

s given by

=

(1

-P)

A

(1

+

(10.73)

and

(10.74)

KO s the static stiffness of the bearing in the centre (unloaded) position (Eqn

7.26).

Block

4 2

s obtained changing the sign of

E

in Eqn

10.72.

As

before, the

total

feedback

,$

will be the sum of

jZpl

and ;Lpa

(i.e.

Eqn

10.71).

In

the particular case of

eS=O,

we have

0=1;

thus, an equation similar

to

Eqn

10.24

will

again be obtained (the relevant values must of course be used for KO, 1 ,2 ) .



DYNAMlcS

335

P,.10-'2

(NsIm5)

0.093

0.185

0.277

0.370

Eqn 5.116

The reader may obtain similar equations in the

case

of orifice-type restrictors.

B,-10-6

o1 w2 on 5 w

(Ns/m)

(s-1) (s-1) (s-1)

0.18 518 1545 0.68

0.35 259 772 1.35

0.53

173 515

655

2.03

0.71 130 386 2.7 1

0.07 1

Eqn

10.67

Eqns

10.74

Eqns 10.51

Eqn

10.52

E W P L E 10.3

Let us examine, from the point

of

view

o f

dynamics, the hydrostatic lead screw

considered

in

Example 7.3, assuming that the reduced mass

of

moving members is

M=3000I@. For what concerns lubricant, we must consider an effective bulk modu-

lus Kla=109N lm z an da viscosity varying in the range p=0.05+0.2 Nslm2, depending

on the actual temperature; the volume of recess and relevant tubing is V=10-5

m3

for

each side.

Let us consider first the unloaded case

(E~=O).

We can calculate the static s ti ff -

ness from Eqn 7.26 and the stiffness of lubricant (for each side) from Eqn 10.19, thus

obtaining K0=1.29.1# N lm , Kd=1.92.1@ Nlm; rom Fig. 10.4 weget Bol(np)=0.89.106

m. The other

main

parameters, which in general depend on viscosity, may be cal-

culated as shown in the following table.

- b -

1 . 5 1

1

o

0.5

0.0

I 1 I

0 100 200

300

Fig. 10.19 Example 10.3: frequency response for two values

of

eccentricity.



336

HYDROSTATIC L UBRlCATiON

I n practice,

in

centred position, the system

is

the s um of two identically behav-

ing pad s and , therefore, the considerations m ade about stability o f thru st bearings

can be used again. Name ly, s tabi l i ty depends on the value of param eter

t=%/

l=2KdlK&

ince it

is

greater tha n unity, a good margin o f stability is ensured.

Frequency response is obtained substituting s=iw in the tra nsfer funct ion (Eqn

10.71,

n

which B, s calculated by means o f Eq n 10.67, nd ;\p

is

t he sum o f ;\pl and

App2 (Eqns 10.72). I n Fig. 10.19 we have plotted the amplitude of frequency response

for

E,=O and for

~,=0.37

which is reached under a load W=l5 P& when pitc h error

is null).

10.6.3 Flow

dividers

If the bearing recesses are fed by means of two independent valves, one may

proceed

as

in the above cases by evaluating Ap for both recesses and then summing

their effects. If, on the other hand, a flow divider is used (see sections 2.3.2 and

7.2.51, the block diagram in Fig. 10.16 is no longer valid, while the diagram in Fig.

10.17

s still useful (it

is

assumed that the pads are symmetrical). In the latter case,

block may be substituted by the one in Fig. 10.20, if the valve

is

controlled by the

recess pressures alone.

Proceeding in the same way

as

in section

10.4.4,

we can write two equations

connecting the recess pressures, the displacement of the bearing and the degree of

freedom of the controlled device:

I

I

I

I

I

I

I

I

I

I

I

I

I

: t i €

I

I

I

I

I

I

I

I

I

I

I



DYNAMICS

337

(10.75)

to which we may add another equation expressing the balance of the forces acting

on the moving member of the valve:

In the above equations

K ,

is the stiffness of the elastic member of the valve

(spring

o r

membrane),

M ,

and

A ,

are, respectively, the reduced mass and the effec-

tive area of the spool

o r

membrane (see section 7.2.5). The hydraulic resistances R,1

and R v 2 of both sides of the valve depend, very often in a nonlinear way, on the de-

gree of freedom

x .

For the sake of clarity let us consider the case of

a

diaphragm valve (Fig.2.15):

the hydraulic resistance of each side proves to be inversely proportional to the third

power of the relevant gap, as can be seen in Eqns

7.57,

n which we introduced the

non-dimensional diaphragm displacement <=x / l o ;as usual, p is the ratio of static

recess pressures at

E=O

to supply pressure

p s .

After linearization and Laplace-

transformation, Eqns

10.75

and Eqn

10.76

take on the following form:

In

the case of

~~'0,

he A blocks are given by the following equations:

(10.77)

S

1 -

"3

1+-

"2

S

(10.78)

The value of KW is given by Eqn

7.26;q,=

w u s clearly the natural frequency of

the diaphragm; for the other characteristic frequencies we have:




(10.79)

Finally, the transfer function of block

,Ip

can be easily obtained:

Once & has been evaluated, the dynamic behaviour of the relevant bearing can

be examined as in section 10.5.

EXAMPLE 10.4

T h e opposed-pad thrust bearing already examined in Example

7 . 1

reaches a

hig h s t i f fness because i t s supply pressure i s compensated

by

m e a n s o f a d i -

aphragm-controlled

flow

divider. Let

us

now examine

it

for wh at concerns stability

and frequency response, assuming that moving mass is

M=lOOO @.

From data reported in Example

7 .1

it is easy to find:

A,=12.4.10-3 m2

( E q n

5.66),

- a -

R e ( G H )

\

- b -

0 '

I

0

200 400 600

$$ ( H a

Fig.

10.21

Example

10.4:

Nyquist diagrams (a) and frequency response curves (b) for certain

values of

{=w2/w,=2Kd/KOc

dotted line represents the frequency response o f the sam e

bearing

with

capillary compensation and <=S).



DYNAMICS 339

Ro=64.8.109Ns lm5 (Eqn5.68), Kk=2.01.109 N l m (Eqn 7.26),B0=0.158.106Ns lm (Eqn

10.12) and B,=2Bo=0.315.106 Nslm. Also the parameters of the controlled restrictors

were selected in Example 7.1 (in particular 8=0.3 and a,=0.55), therefore, from Eqns

10.79

we can obtain the relevant characteristic frequencies:

W1=144

s-1

and

A first rough assessment of stability could be made, as indicated in section

10.5.2, disregarding the squeeze coefficient and the mass o f the diaphragm; the

Routh criterion applied to the relevant third-order characteristic equation would

indicate that the system is stable when

5 = w 2

/

o1

1/[1- @ 1-P)a,

(1 + w 1 / 03 1

3.34 ,

that is when Kd>l.7~Koc=3.36~109lm . A more detailed analysis, however, shows

that

w,

is large enough to have no practical effect, whereas squeeze parameter con-

tributes to increase the margin of stability. In Fig. 10.21 Nyquist diagrams and

frequency response curves have been plotted

for

a number of values

of

parameter 4.

It may be seen that the controlled supply device may very easily enhance static stiff-

ness at will, but this gain is rapidly lost as the frequency of exciting force increases.

03=13.5.1@s'.

10.7 SELF-REGULATING BEARINGS

The dynamic behaviour of SRBs (Fig. 7.25) can be studied in a similar way to

usual bearings.

For

the sake

of

simplicity, the theoretical case alone will be consid-

ered here, in which all bearing clearances are equal to ho when the external load is

F=O.

A

quantitative evaluation of the consequences of working tolerances may be

found in ref. 7.7.

Dynamic load capacity is, as usual, the sum of

a

term proportional to recess

pressure and of another one due to squeeze:

W = A ,p,. W ' ( E ) B(E)o

&

(10.81)

where

W

and A, are given by Eqn 7.78 and Eqn 7.79, respectively, and (ref. 10.7):

B =

Bo

B'(E)

(10.82)

Bo and B' are plotted in Fig. 10.22 for certain values of r', while

rh

is assumed to take

on the relevant "optimal" value as in section 7.4 (see Fig. 7.26). Note that , in evaluat-

ing

B,

lubricant compressibility in gaps and in secondary recesses has not been

taken into account.

Comparing Fig 10.22 with Fig.

10.4.b,

the intrinsic damping of the SRB proves

t o

be much greater than for conventional opposed-pad bearings. This occurs because

the SRB has a built-in pressure-compensating system, whose damping effects are



340


- b -

B*

0

2

B

1

.

0.5

0.7

0.9

I

I"

0.5

1 o

E

Fig. 10.22 Self regulating bearings: a- damping coefficient

B*=B0.32h30/(3x/d)4)

versus radius

ratio

r ' ; b-

damping coefficient B'=

BIB,

versus eccentricity

E ; ( r i=r i , op t ) .

reflected in the high value of coefficientB. Namely, for the opposed-pad bearings, B

may even be disregarded, since the damping relies mainly on the external compen-

sating system; the contrary happens in the case of the SRB.

The balance of the forces applied

to

the moving member

of

the bearing is ex-

pressed by:

In studying small amplitude vibrations around position E = E ~ (static displace-

ment under load

Fs),

e may apply the small perturbation method and Laplace

transformation to

Eqn 10.83,

which becomes

The lubricant flow rate delivered by the supply system is:

(10.85)

which may be transformed into:



D WA

MlCS 341

(10.86)

(Equations

for

Ro

nd

R i

are given in section

7.4).

10.7.1

Constant f low feeding

The behaviour of the system is represented by Eqn

10.83

and Eqn

10.85

in which

Q is

assumed to be a constant. In the case of small amplitude vibrations, Eqn

10.84

and Eqn

10.86

may be used instead, and SQ=O. Equation

10.86

leads to:

where

Kd

e O

" 2 = = ;

KO

1 = - 2 '

3A,Ro '

(10.87)

(10.88)

KO

s

the bearing stiffness in the unloaded configuration (Eqn

7.92),

Kd is the

stiff-

ness of lubricant contained in the central recess and in the relevant supply pipes

(Eqn

10.19);

coefficients G1 nd

G2

are non-dimensional functions of the displace-

ment, given in Fig. 10.23. It should be noted that if the static part

of

the load is null,

;lp

also vanishes.

The transfer function

of

the whole system is (from Eqn 10.84):

where

(10.89)

K

is given by Eqn

7.93

(and Fig.

7.27),

coefficients

G2

and G3 re plotted in Fig.

10.23

as functions of the static displacement. It is easy to see that, for static loads

(s=O),

Eqn 7.91 is again obtained.



342

100

Gl

10

1

0.'


- a -

0

o

0.5 1

o

E

- b -

11

G2

G3

O

0.1

0.5

1

o

E

Fig. 10.23 Self-regulating bearings: coefficients G , , G 2and G 3 versus eccentricity.

When the sta tic load is null, we get G3=0 nd

K'=l;

hence, the bearing behaves

like a second order system with undamped natural frequency 0 ~ 1 2 ~nd a damping

factor

c.

In general cases, it will be a three-pole system.

For

what concerns stability, it is easy to see that no problem generally exists,

even when { = w , / q is small, since the damping properties of the system rely mostly

on the bearing itself (namely on coefficient B ) rather than on the supply system.

10.7.2 Constant pressure feeding

When the SRB is directly fed by a hydraulic network at constant pressure p s , we

shall obviously have

p r = p s

and

6 p r = 0 .

Hence, the behaviour of the system is de-

scribed by Eqn

10.83

or, in the case of the linearized model, by Eqn

10.84

which may

be written as follows:

(10.91)

where KO s the static stiffness in the centre position (Eqn 7.97) and K is given by

Eqn

7.98.

Note that, under the simplifying assumptions we have made, lubricant

compressibility has no influence on the dynamic behaviour of the bearing, which is

reduced to a second order system.



DMvAMlCS 343

When the

SRB

is fed through a laminar-flow restrictor

R,

(see section 7.4.21, the

variation in flow rate due to a change in recess pressure is

which can be substituted in Eqn 10.86 t o obtain the feedback function

Ap=6p,./8&.

Then, the transfer function may be obtained from Eqn 10.84. In this case, too, we get

a second-order system when the static part of the load is null.

10.8

MULTIPAD

BEARING SYSTEMS

This section deals with the hydrostatic bearing systems consisting of a certain

number of independent pads, which are able to sustain loads in multiple directions.

The multipad journal bearing in Fig. 1.12.a and the hydrostatic slideway in Fig.

1.16 are examples of such systems. In the first example the bearing is able to SUS-

tain loads along any radial direction; hence, its displacement may be described by a

set of two coordinates. In the second case, the 12 hydrostatic pads take the loads in

every direction, except along the x: axis: five coordinates (and five equations) will

therefore be required to describe the static and dynamic behaviour of the carriage.

In every case the system may be studied along the following lines:

i) first, we must fix a suitable set of n generalized (Lagrangian) coordinates,

able

to

describe any displacement of the system: obviously,

n

is the number of

degrees of freedom constrained by the bearing system;

ii) we must obtain an equation for each pad, giving

its

dynamic load capacity

Wj as

a

function of the parameters which characterize the supply system, of

the generalized coordinates and of their time-derivatives;

iii) we must write down a set of n independent differential equations, express-

ing the balance of the external forces, of the inertia forces and of the load

capacities (D'Alembert

o r

Lagrange equations). In general,

a

complicated

set of nonlinear equations will be obtained, requiring numerical simulation

to trace the system response to large-amplitude loads. In most case, how-

ever, it will be enough to linearize these equations and to examine them to

judge the stability of the system and

t o

obtain its response t o dynamic loads.

A number of other considerations must be borne in mind in certain cases. For

instance, when the tangential speed is high and the thickness of the film of lubri-

cant

is

not uniform,

as

happens in journal bearings, the forces due to the hydrody-

namic effect should be taken into account.



344 H YDRCSTATIC LUBRICATION

10.8.1

Hydrostatic slideways

Let us now see, with the aid of a practical example, how the dynamic study of a

system of pads may be stated. The simple carriage sketched in Fig. 7.29 is supported

by four hydrostatic plane pads. In any given steady configuration, the mean film

thickness of the j-th pad

is hjo

and

e&-hjo)/hj0

is the relative variation of the gap.

Each pad exerts on the carriage a force

Wj,

that may be written in the following

form:

W j

=

Aj pj

-

BjC&j)hjo

Ej

Cj= 1 ...m )

(10.92)

The effective area

Aj

may be considered to be a constant; actually certain displace-

ments of the carriage may make the bearing surfaces out of parallel, and conse-

quently may affect the coeficient

Aj:

however, such changes are generally negligi-

ble. The squeeze coefficient

Bj

is greatly affected by the actual value of the film

thickness; on the other hand, as already noted in section 10.5, initially

it

can be to-

tally disregarded; alternatively, it may be substituted by

its

reference value

Bjo=Bj(0),

hen small displacements from the steady configuration are considered.

The equations of motion of such systems can be obtained equating the external

forces to the inertia forces; their general form is, therefore:

(z = 1

..

n)

where m is the number of pads, n the number of generalized coordinates

(10.93)

x i

6.e. the

degrees of freedom constrained by the pads); the terms auWj are the generalized

components of the pad forces, i.e. aii=hjo.(dej/dxi),nd

Fi

are the generalized com-

ponents of the external forces. In the case of the system in Fig. 7.29, it is clearly

m=4, n=3; the xi are the axis z and the tilt angles 8, and 8,; therefore Eqns 10.93

become:

-M +

C,Wj)

+

Fz

= 0

-J, 8, + b ( W , +Wz -W3 -W4)+M, = 0

J

I-J,

8

-

a (w,

w,

-w3 w 4 )+ M,

= o

(it is assumed that x , y and z are principal axes). The Ji are the inert ia moments,

the Fi and Mi are the components of the resultant and of the resultant moment of

the external forces. The forces

Wj

are given by Eqns 10.92, in which the recess pres-

sures

p j

are still unknown. If each pad is fed by a n independent supply device, each

pressure

p , is

related

to

the relevant displacement 5 by an equation



DYNAMICS 345

like the ones already examined in section

10.4.

Of course, if the supply devices are

not independent (for instance when flow dividers are employed), more complicated

relations are needed, the general form being:

(10.95)

The pad displacements ~j may be written as functions of the Lagrangian coordi-

nates:

For

instance, for the system in Fig.

7.29,

we have:

1

E~ = (Z

+ b

0, - a eY)

Q

=g

(z

+ 6 0,

+

a OY)

1

and so on.

If we introduce Eqns

10.92

and Eqns

10.96

into Eqns

10.93,

hese last , together

with Eqns

10.94 (or,

more generally, Eqns

10.95))

onstitute a set of non-linear dif-

ferential equations that are clearly difficult to handle. As usual, a very great simpli-

fication is obtained by limiting ourselves to studying the system for the case of small

vibrations. It is possible, therefore, to linearize and Laplace-transform the foregoing

equations. ARer the transformation, Eqns

10.94

may be written in the form:

spj =

-&jW

6Ej

(10.97)

Functions Apj of the complex variable s may be written as in section

10.4,

de-

pending on the type of supply device. Typically, each

&j

can be written in the form of

Eqn

10.24,

n which

w 1

and w 2 are given by Eqns

10.25

when multiple pumps are

used, or by Eqns

10.28

or restrictor compensation. A couple of opposite pads may be

treated also as a single opposed-pad bearing, especially when compensated by

means

of

a controlled restrictor: in this case, h j will be obtained as indicated in sec-

tion

10.6.

Equations

10.92

now become

SWj = - [Aj APj(s)

+

Bj hjo

S]

&j

(10.98)

and, thanks to Eqns

10.96,

may further be written in terms of the coordinates

x i ,

instead of the

~ j .




Finally, by linearizing Eqns 10.93, a set of n linear equations

is

obtained which

constitutes a model of the dynamic behaviour of the carriage. For instance let us

consider again the system in Fig. 7.29, with some further simplifications: all the

pads a re equal ( the same values for the effective area

A,,,

and the same clearance

ho

in the steady configuration) and are fed through capillary restrictors with the same

pressure ratio 8 . Equations 10.98 now give (see section 10.4.2):

Bo

1+s/w1

KO

here

A = - s + l + s / o z

and

w1

and

0 2

are given by Eqns 10.28. Proceeding as outlined above, the following

set of equations is obtained:

M s ~ &

4

KO

A & =

SF,

J , ~ 2 6 0 ~4 b2Ko

A

60,

=

W x

J Y s 2

66, +

4a2 KO 6ey

= SM,

These equations are completely uncoupled: obviously this is only a consequence

of the simplicity of the system we have considered and will not be generally ob-

tained. If the lubricant is sufficiently stiff (w2>>w1), the operator A

becomes

A=1+ (l/q+Bo/&)s. No stability problem should hence exists: indeed, this kind of

systems often feature a great damping.

10.8.2 Mult ipad journal bearings

Let us now consider a journal bearing made up of

n

cylindrical pads (Fig.10.24),

Fig.

10.24 Multipadjournal bearing.



DYNAMICS 347

that may completely surround the shaft. The actual position of the shaft axis, with

reference to the centred configuration, may be defined by two non-dimensional

coordinates: < = x / C and q = y / C . The general problem is quite difficult: the hydraulic

resistance

Ri

and the load capacity

Wi

of each pad show a nonlinear dependence on

the shaft displacement and on its velocity; Wi

is

not always directed toward the cen-

tre of the bearing, but a tangential component may exist; furthermore, the Reynolds

equation may not be directly solved, and hence numerical computing should be used

to obtain load and flow rate.

However, if the displacement is not

too

great

( E c O . ~ ) ,

if the arc taken by each pad

is smaller than

90°

and the turning speed of the journal is not so high as to give

appreciable hydrodynamic effects, great simplifications can be introduced.

First

the

tangential component of the load capacity may be disregarded:

Wi

is directed as

~ i .

Then it may be assumed that the load capacity and hydraulic resistance of the i-th

pad depends only on the relevant components

of

the shaft displacement and velocity,

namely on

~i

and

E i .

The load capacity of each pad may be written in the usual form

The effective area A, may be considered a constant (see section

5.8).

The coefficient

Bi

depends mainly on the clearance:

a

rough evaluation

is

often enough; it may

even be totally disregarded.

The relevant perturbation is, therefore,

SWi

=A,

Spi -

Bi,

C

s

6 ~ i

(10.100)

The perturbation of each recess pressure may be written in the usual form (Eqn

10.97) and each operator

$(s)

obtained as shown in the preceding sections. For

instance, in the classical case of capillary compensation (see also section

10.6.2)

we

obtain:

where KO (reference stiffness of each pad) is given

by

Eqn 6.22,

w 1

and

o2

re given

by Eqn 10.28 and O(E) y Eqn

10.73

(hereafter we shall omit the subscript "s",by

which we mean that all the parameters are calculated in the point of sta tic equilib-

rium). If

n

is an even number,

it

may be preferable to consider the multipad bearing

as a set of

n/2

opposed-pad bearings, obtaining the operators as in section

10.6.

In

any case, Eqn

10.100

becomes



348


(10.101)

The equations of motion for the journal may now be easily written:

(10.102)

where

W t

and 6Fq are the components of the external perturbation.

A t this point the problem is completely defined, for we have a set of differential

equations connecting the displacement of the journal to the external excitation. In

facts, it is easy to see that the pad displacements S E ~ epend on the journal dis-

placement:

(10.103)

Introducing Eqns 10.101 and Eqn 10.103 into Eqn 10.102, the equations of motion

become:

(10.104)

Apparently, this is a complicated set of equations; in particular cases, however con-

siderable simplifications can be introduced: for instance, if we assume that n=4,

&=O, and that

5

is directed toward the centre of a pad, i t is very easy to see that the

above set splits into two independent equations:

Very simple equations are obtained in the case of small vibrations around the

centred configuration E ~ O ,ince for all the pads we have:

Clearly no stability problem should occur when the lubricant is sufficiently stiff

(w2>w1). Actually, when the turning speed is high, self-excited vibrations may set in

(ref. 10.9); these a re due t o the hydrodynamic effects (disregarded in the foregoing

statements), which may cause entrainment of air in a recess, when the relevant



DYNAMICS 349

recess pressure falls below the atmospheric pressure, and even instability, above a

critical speed (due to nonlinearity, instability is transformed into self-excited finite-

amplitude oscillations of the shaft axis around the rest point: the well-known

"whirl", which we shall go into further in the next section). However, such prob-

lems are likely t o occur only if the design

of

the bearing is far from commonly ac-

cepted practice (namely for n>3) nd can be effectively counteracted by increasing

the supply pressure o r by selecting a less viscous lubricant.

10.9 MULTIRECESS JOURNAL BEARINGS

The dynamic behaviour of multirecess bearings (Fig.

10.25)

is more complicated

t o analyze than the types of hydrostatic bearings examined above, mainly because of

the interdependence of the recesses, which compels us to treat the bearing as a

whole, rather than as a set of simple pads. Furthermore, the hydrodynamic effect

due to the turning velocity of the journal should not be disregarded: indeed, i t may

be shown that , above a certain critical speed, instability problems may occur.

In the following sections, we shall first examine the general statement of the

problem and then particular cases of loading will be considered.

Fig.

10.25

Multirecess journal

bearing.

10.9.1

Analys is

The dynamic behaviour of the journal i s described by the equations of motion,

which in vector form are:

M C

f

} - W = F

(10.105)



350


where F s the external force and

W

s the resultant of the lubricant pressure on the

journal.

The pressure distribution can be found by solving the Reynolds equation,

namely Eqn 4.18, by numerical computing. In section 8.2

it

has already been pointed

out that, thanks

to

the linearity of the Reynolds equation in the absence of cavitation,

it s solution can be obtained as the superposition

of

n+2 pressure fields, which are

proportional to the

n

recess pressures p i , to & and to

4

-n/2, respectively. The same

may be done for the boundary flow rates. By integrating the pressure fields, we find

that, for any given displacement, the load capacity is

a

linear function of the recess

pressures and of the shaft velocities:

(10.106)

The components of the array

p

are the n recess pressures. The 2xn coefficients Aij

are the contributions of the i-th recess pressure to the load capacity along

5

and q ;

they are functions of the displacement of the journal, although, when small dis-

placements are involved, they may be considered t o be constants.

In order to study small displacements around any steady-state equilibrium

point ( C ~ , $ ~ ) = ( ~ ~ , ~ J ,convenient procedure is to linearize Eqn 10.106, that leads

t o

write the perturbation of load capacity as:

(10.107)

Note th at we have omitted the subscript 's' n the last equation, but it goes without

saying that all the finite parameters are calculated in the equilibrium point; the

same will be done for all the following linearized equations. In Eqn 10.107, the sec-

ond term on the right-hand side accounts for the squeezing effect of lubricant on the

bearing lands (it is analogous to coefficient

B

of pad bearings);

it

is often much

smaller than the first term and may be disregarded, unless recesses are small, or

compressibility is high. The transformation matrix X is defined by the equation

The

2x2

matrix Uw ccounts for the changes of the hydrodynamic load capacity due

to the shaft displacements; in practice, its elements may be obtained by means of

repeated numerical computing, namely considering how much the components of

the hydrodynamic load capacity vary after small displacements S{ and tiq from the

equilibrium point.

A

further term (calculated in the same way) could be added to the



DYNAMICS 351

right-hand side of Eqn 10.107 t o account for the fact that the elements of A are not

exactly constants.

The variations in recess pressure 6pi can be calculated by introducing the con-

tinuity of flow in and out of each recess. The flow rate reaching the lands from each

recess may be obtained (by numerical computing o r other approximate calculations)

as a linear combination of the recess pressures and of the shaft velocities. The flow

rate Qi delivered by the supply system to each recess must be equal to the flow rate

entering the bearing clearance, except for the variation in the density of the lubri-

cant and the variation in the volume of the recess due to the displacement of the

journal; sometimes the variation in volume of the supply pipes (due to the change in

pressure) should also be considered. In other words it may be written as follows (see

also section

10.4):

(

10.108)

In the equation above, A,.i is the area of each recess (we have assumed that all re-

cesses are equal). "Lubricant stiffness" K d is defined as

where

V,

is the volume of a recess,

V,

the volume of the relevant supply ducts and

Kl, the equivalent bulk modulus of the lubricant. In the large majority of cases

Kd

may be considered as a constant.

The components of vector V are the rates of change in each recess volume and

clearly depend on the speed of the journal axis; in the case of equal recesses we

have:

(see Eqn 10.103for the meaning

of

$).

Equations 10.108 may be linearized, after which the variations in the n recess

flow rates may be written in the form

(10.109)

(as for Eqn 10.107, the coefficients of the nx2 matrices q (k ) may be obtained by means

of numerical computing). On the other hand, the flow rates Qi delivered by the



352 HYDROSTATIC LUERICATION

supply system depend on the recess pressures pi, the relationship being connected

with the type of supply system; linearizing, we have:

s&=-Ct6p

(10.110)

For a constant-flow system

we

clearly have ac=O,while for capillary compensation

(see Eqn 10.26) we have:

(10.111)

More complicated statements can be obtained for other supply devices, in particular

in connection with controlled restrictors.

Introducing Eqns

10.110

into Eqns

10.109

and Laplace-transforming, we obtain

where:

AA

Kd

A

= q +

a + - I s

(10.112)

(10.113)

That is, we may obtain a set of n complex equations which establish a relationship

between journal displacements and variations in the recess pressures. We may now

left-multiply Eqn 10.112 by A-1 and substitute it to Sp in Eqn 10.107, in order

to

obtain

the components of the load capacity in the following form:

(10.114)

(it is worth noting that in general the 2x2 matrices K and B depend on the complex

variable

s

except when

A

is real, that is when lubricant compressibility is negligi-

ble). Finally, the equations of motion (Eqns 10.105) become:

( M I s2 + B + K)

= SF(s)

(10.115)

In spite of the formal simplicity of Eqns 10.115, their coefficients would quite

difficult and tedious t o obtain and, since they depend on too many parameters, they

would need to be calculated case by case. In practice, however, great simplifications

may be introduced, especially when particular cases are considered such as, say,

f2=0 or ~ ~ ' 0 .urthermore, the coefficients may be calculated by means of some



DYNAMICS

353

simplification (typically, the

thin

lands assumption), which may even lead to gen-

eral closed-form equations.

A s in the case of the other types of hydrostatic bearings, the journal bearings

also usually prove to be stable and well damped; in certain circumstances, instabil-

ity may occur due to one of the following reasons:

i)

-

The lubricant stiffness Kd is too low, due to excessive compressibility

or

to exces-

sive volume (or low stiffness) of the supply ducts. As for the other types of bearings

examined above, care should be taken to ensure that

Kd

is greater than the static

stiffness

K,

in order to avoid problems of this kind.

ii) - Cross coupling exists in Eqns 10.115, due to the turning speed of the journal. If

f2 and the reduced mass of the journal are great enough, the system may prove

to

be

unstable (whirl instability).

iii)

-

In certain circumstances the off-diagonal terms of K may not be negligible (and

hence cross-coupling exists) even when Q=O; thus, for great values of mass M and

low damping, instability could set in. However, this does not seem likely to occur in

practical applications.

Another important consideration to be made is that, since stiffness of hydro-

static bearings is often very great, the supporting structure may not always be re-

garded

as

being rigid, and thus Eqn

10.115

would become quite more complicate.

10.9.2 Non-rotat ing bearings, incom pressib le lubr icant

Let us first consider the simplest case of small vibrations around the point E=O.

Stiffness and damping may now be considered to be independent from the dis-

placement direction, and Eqns 10.115may be rewritten as:

(10.116)

The equations of motion are now uncoupled, and the response of the system to

any exciting load is easily obtained once the coefficients

K O

and

Bo

are known. By the

way, since Eqns 10.116are second-order equations with positive coefficients, stability

is ensured.

The coefficient KO is nothing but the static stiffness already examined in chapter

8.

It may be deduced from the slope of the

(W,

) characteristic of the bearing. In

section 8.3.1 an approximate equation (namely, Eqn

8.7)

is reported in which

KO

s

considered proportional to

a

parameter

A;

this last depends on geometrical factors

and on the type of supply system (for instance see Fig.8.4

o r

Eqn

8.6).

A similar

equation may also be obtained for

Bo

(see ref. 8.12):



354

D

L3

B o = 1 2 p ~ u ' ( 1u ' ) 2 A

HYDROSTATIC 1 BRlCATlON

(10.117)

A slightly different equation may be drawn from ref. 8.11.

Even when a static load is applied, Eqns 10.115 may be considered to be uncou-

pled (the off-diagonal terms of the K and B matrices are small). In chapter 8 it has

been shown tha t the a ttitude angle

I$

often has only a small influence on the perfor-

mance of the bearing; hence it is an acceptable loss of generality to take &=O (i.e.

{a ). Several plots of the coefficients

B

and

K

are given in figures from 10.26 t o 10.28

taken from ref. 10.10 and ref. 10.11.

1

a -

0.8

0.6

K

L DPJC

0.4

0.2

0

0

Q2

04 0.6 0.8 1

B

0 8

0.6

B

3&4

L (

D/CP

0.4

0.2

0

-b -

o a2

04

0.6 0.8

I

B

Fig. 10.26 Multirecess journal bearings: stiffness and damping versus the pressure ratio

(n=4,

a'=0.2,

8=36",

L/D=l).

10.9.3

Rotat ing bear ing, incompressible lubr icant

When the journal rotates at high speed, a hydrodynamic load capacity is added

to the hydrostatic one; the sum is clearly intended in the vectorial mode, because the

direction of the resultant of the hydrostatic pressure is close to the direction of the

journal displacement, while the hydrodynamic load capacity is in practice orthogo-

nal to it.

Limiting ourselves to the simplest case of vibrations around

E=O

and incom-

pressible lubricant, Eqns 10.116 can be completed as follows:



DYNAMICS

355

LD:/C/ 0.8

0.4

-b-

1.6

I

- a -

~

Es= 0

-wF

/

0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5

L

D

-

L

D

-

Fig. 10.27 Multirecess journal bearings: stiffness and damping versus LID (n=4, '=0.2, 8=36",

J=0.6).

0.8

K

L Dps C

0.6

0.4

0.2

0

2

1.5

B

3pL(D/C13

~-

1

0.5

0

02

0.4

0.6

0.8 1

E s

0

0.2

0.4 0.6 0.8 1

6 s

Fig. 10.28 Multirecess journal bearings: stiffness and damping versus eccentr icity, for various a

( 1 ~ 4 .

=na', ID=l , =0.6).

(10.118)

The coefficient

Ku

s proportional to the rotating speed

R

and needs to be calculated

by numerical means, or on the basis of suitable simplifying assumptions. A n ap-

proximate evaluation is given in ref. 8.11 and ref. 8.12, in which it is found that:



356

HYDROSTATIC LUBR/CAT/ON

(10.119)

Examination of the characteristic equation of the differential system of Eqns

10.118, shows that instability arises when

K,

reaches the critical value

where w, and rare the undamped natural frequency of the shaft and a damping

factor, respectively (Eqns 10.901.

A t the critical speed, corresponding to K,*, the shaft oscillates in an undamped

mode a t the natural frequency

0

(whirl instability). From Eqn 10.119 follows that

the critical speed is:

l 2 * = 2 %

(10.121)

This confirms the well-known fact that, when the turning speed goes beyond the

critical value, the shaft oscillates at a frequency equal to half the critical turning

speed (ref. 10.12).

Equations 10.118 may be used also when the static load is not null, on condition

th at the maximum displacement is small enough (~<0.5) . or a better approxima-

tion we must return

t o

the general form of Eqns 10.115 and calculate the four stiff-

ness coefficients (tha t is the elements of matrix K) nd the four damping coefficients

(matrix

B)

or given values of static displacement kS, and turning velocity a.

simple approximate way is the thin land assumption, that is to assume that, when

land width is small and eccentricity is not great, pressure variation over the lands

is linear (ref. 10.13). Axial and circumferential lands may be treated separately and

flow calculations may be performed using the simple Poiseuille and Couette flow

equations. In practice, the lumped resistance method (see section 8.2) has been

extended to account for turning velocity and for the additional flow rates caused by

the shaft movements around its static equilibrium position (the squeeze effect of the

lands is disregarded, since it is negligible when lands are narrow and fluid is

incompressible). Once the force W due to the pressure of the lubricant has been ob-

tained as a function of the displacement and of the shaft velocity, the dynamic coef-

ficients can be calculated by differentiating the components W, nd Wy ith respect

tox,y,x andy.

A more general approach is based on the numerical solution of the Reynolds

equation, which also allows us to account for possible effects of cavitation (see, for

instance, ref. 10.14, 10.15, 10.16).

Equations 10.118 also allow us t o foresee the dynamic response of the bearing to

external actions. In particular, let us examine the classical case of a harmonic ex-



DYNAMICS

357

citing force of a given angular frequency

o.

aking W,=O (since for small eccentric-

ities the influence of the load angle

is

negligible, there is no loss of generality), Eqns

10.118give:

We may now substitute iw for s, obtaining two complex numbers, whose moduli

and arguments give the amplitude and phase of the oscillation in the 5 and q direc-

tions. The shaft axis is seen to cover an elliptical orbit, the maximum displace-

ments in the

5

and

17

directions being:

I" =& l-w2/w;4 16 ( a 2 - w 2 / w ;2 + 8 c2

1-w2/o;

)2 (R'2+~2 /o :

ISF I

(1-02/w,2

2

+ 4 o /O,

(10.122)

16F I 4 pa2

'" =

[ l-02/c$4 16

c4

( R r 2 - 0 2 / o i2 + 8

(1-w2/02

2 (R.2+q2/o$

in which

R'

is the ratio of the actual turning speed to the critical one:

Examination of Eqns 10.122 mmediately leads to a number of considerations.

First,

the vibration amplitude in the normal direction, with respect to the load, van-

ishes for non-rotating bearings. Actually, this is due to our simplified approach,

since a certain cross-coupling generally exists (namely,

B

and K n Eqns 10.115 are

not diagonal matrices even when a=O,although off-diagonal terms are often quite

small). Then it is easy t o see that the amplitude of vibration becomes infinite when

eu,, nd a=R*,which agrees with the above remarks concerning stability limits.

If we now plot the dynamic flexibility

as

a function of the exciting frequency for

a given value of the damping coefficient, and certain values of the turning speed

(Fig. 10.29),we see that, at high values of R', the hydrodynamic effect makes a con-

siderable (indeed dominant) contribution to the static load capacity. On the other

hand, the damping progressively decreases: at low speed we get an overdamped

behaviour, then a resonant peak occurs, whose amplitude increases, until it be-

comes infinite a t

S'=1.

In conclusion, we see that whirl instability requires particular care during the

design stage, since the actual behaviour of the bearing might prove to be vastly dif-




- a -

0.0

0.5 1

o 1.5

w

Wn

0.0 05

1 o

1.5

Wn

0

-

Fig. 10.29 Multirecess journal bearings: typical frequency response for various turning speeds

(

c=a.

ferent from what foreseen when the turning speed was not taken into consideration.

Fortunately, however, in most cases, the turning speed in actual applications is far

from dangerous limits.

10.9.4

Compressib le

lubricant

Let us consider a four-recess bearing, rotating at low speed; if the axis of the

journal is assumed to undergo small vibrations along the

z

axis, directed toward

the centre of a recess, the pressure in the side recesses may be considered

t o

remain

practically constant. In this case, a simplified approach may be attempted and the

bearing may be studied just like an opposed-pad bearing. Even, i t is easy to consider

particular types of compensating devices, such as diaphragm-controlled flow di-

viders (ref.

10.4).

A more general approach is delineated in ref. 10.17. The main simplifications

introduced in that work consist in assuming that eccentricity is not too high ( ~ < 0 .5 ) ,

in order

t o

avoid cavitation and large attitude angles, and that, when a harmonic

dynamic load is superimposed

to

the steady load, the journal centre executes plane

small-amplitude harmonic vibration around its steady state position, namely



DYNAMICS

359

G&=QeiWt.The local film pressure may be expressed as the s u m of a static term plus

a dynamic one, proportional to the vibration amplitude:

The dynamic film pressure can be obtained by numerical (finite differences)

so-

lution of the first-order perturbation of the Reynolds equation, with suitable bound-

ary conditions. In particular, the boundary conditions at the recess edges (that is

the dynamic recess pressures) are obtained imposing continuity

of

recess flow and

prove to be complex functions of the compressibility of the lubricant and of the fre-

quency of vibration, besides

of

usual parameters, as eccentricity, pressure ratio, etc.

K0.5

LDPslC

- a -

- C -

4 I

-

A

- 2

I

- b -

0.17

J

*lo4

10’ l o3 u - l o 4 2 x ~ 0 4

-d-

A = 5

A

= 2

Fig.

10.30

Dynamic coefficients (for a multirecess journal bearing with r r 4 ,

L / D = l , a ‘ = 0 .2 5 ,

0=45q

J

.6,

~ ~ 4 . 5 )ersus frequency parameter

a=3ptw’[pS(CID)z] ,or

certain values of com-

pressibility parameter

y=psAri /CK,

and of speed parameter

.4=12xSh.




By integration of the dynamic pressure field acting on the journal we can obtain

the radial and tangential components of the dynamic load and, hence, the relevant

stiffness and damping coefficients:

K , + i w B ,

K4&+ i

w

Be

Obviously, the four coefficients ar e functions of the frequency of vibration and

also depend on the geometrical parameters, on eccentricity, on lubricant compress-

ibility and on pressure

ratio;

moreover the cross coefficients

K@&

nd

B @

depend on

angular speed, too, whereas the direct coefficients are practically not affected by

0.

A

lot of calculations are therefore required in order to give a full description of the

dynamic characteristics of

a

bearing with given geometrical ratios: in ref. 10.17, for

instance, the calculations of the coefficients for a typical bearing a re condensed in

several plots, some of which are shown in Fig. 10.30.

REFERENCES

10.1

10.2

10.3

10.4

10.6

10.6

10.7

10.8

10.9

Opitz H., Bottcher R., Effenberger W.;

Znuestigation on the Dynamic Be-

hauiour of Hydrostatic Spindle-Bearing Systems;

10th

Int. MTDR Conf., Uni-

versity of Manchester, 1969, pap.

MS-21;

15pp.

Masuko M., Nakahara T.; The Influences o f th e Fluid Capacitance in the Oil

Feed Li ne S yste m on the Transient Response of Hydrostatic G uideways;

Int.

J .

Mach. Tool Des. Res.,

14

(19741,233-244.

Wilcock D.

F.; Externally Pressurized Bearings as Servomechanisms. - The

Simple Thrust Bearing; ASME Trans., J . of Lubrication Technology, 89

(1967),418-4 24.

Chen K. N., Yang G. P., Wang

X.,

Yang H. H.; A System Approach to the

Dynamic Characteristics of Hydrostatic Bearings Used on Machine Tools;

Int. J. Mach. Tool Des.

Res.,

20 (19801, 287-297.

Prabhu T. J., Ganesan

N.;

Characteristics of Conical Hydrostatic Thrust

Bearings under Rotation;

Wear, 73 (1981),95-120.

Moshin M. E., Morsi S. A.;

The Dynamic Stiffness of Controlled Hydrostatic

Bearings; ASME Trans., J . of Lubrication Technology,

91

(19691, 597-608.

Wylie C. R., Barrett L. C.;

Advanced Engineering Mathematics;

MacGraw

&

Hill, 1985; 1103 pp.

Ogata

K.; Modern Control Engineering;

Prentice-Hall, 1970; 836 pp.

Inasaki I.; Stability o f Hydrostatic Journal Bearings; Eurotrib 81, proc.

3rd

Int. Tribology Congr., Warszawa, 1981, vol.

2;

pp. 116-122.

10.10

Ghosh

M.

K.;

Dynamic Characteristics of Multirecess Externally Pressurized

Oil Journal Bearing; ASME Trans., J. of Lubrication Technology, 100 1978),

467-471.



DYNAMICS 361

10.11 Ghosh M . K., Majumdar B. C.; Stiffness and Damping Characteristics of

Hydrostatic Multirecess Oil Journal Bearings; Int.

J .

Mach. Tool Des. Res.,

18 (19781,139-151.

10.12 Leonard R., Rowe

W . B.; Dynamic Force Coefficients and the Mechanism of

Znstability i n Hydrostatic Journal Bearings;

Wear,

23

(19731,277-282.

10.13 Vermeulen M., e Shepper M.; Theoretical and Experimental St ud y of the

Dynamic Behaviour

of

Hydrostatic Radial Bearings;

Eurotrib 89,

proc.

5th Int.

Congr. on Tribology, Helsinki, 1989, vol. 3;p. 180-185.

10.14 Chen

Y.

S.,Wu H. Y., Xie P. L.; Stability of Multirecess Hybrid Operating O il

Journal Bearings;

ASME Trans.,

J. of

Tribology, 107 (19851,115-121.

10.16 Rowe W. B., Chong F.

S.;

Com putatio n of D ynam ic Force C oefficients for

Hybrid (Hydrostatic JHydrodyn amic) Journal B earings by th e Finite D istu r-

bance and Perturbation Techniques; Tribology International, 19 (19861, 260-

271.

10.16 Lund

J.

W.;

Review

of

the

Concept of Dynamic Coefficients for F luid Fil m

Journal Bearings;

ASME Trans.,J.

of

Tribology, 109 (19871,37-41.

10.17 Ghosh M.

K.,

Viswanath

N. S.; Recess Volume Fluid Compressibility Effect

on th e Dynamic Characteristics of Multirecess Hydrostatic Jou rna l Bearings

w ith Journal Rotation;

ASME Trans.,J. of Tribology, 109 (19871, 417-426.



Chapter

11

OPTIMIZATION

11.1

INTRODUCTION

In this chapter an important aspect of the study and design of hydrostatic bear-

ings, already presented in previous chapters, especially in chapter 6, will be dealt

with. The "optimum" conditions corresponding to the minimum power dissipated

by the bearing will be identified and a general procedure for the solution of the prob-

lem will be described.

Our investigation is carried out on an infinitely long pad directly supplied by a

pump. The same procedure

is

then applied t o real bearings. Afterwards, the inves-

tigation is extended to an infinitely long pad and real bearings supplied by means of

compensators. The results will also show that a direct supply system, as compared

with a compensated supply system is more efficient (that is, less power is dissipated

and stiffness is greater).

This subject may be dealt with using only some of the elementary formulae of

hydrostatic lubrication presented in chapters

4

and 5 .

11.2

GENERAL PROCEDURE

First of all we shall s tart with the study of the static behaviour of an elementary

hydrostatic bearing: the infinitely long hydrostatic pad, Fig. 11.1,

of

which we shall

consider a finite part and afterwards we shall go on t o determine its optimum

dimensions.



OPTlMIZ4TlON 363

The pad is

first

considered as being supplied directly (Fig. 1l .l .a) and then as

being supplied by a compensating element (Fig. ll .l .b), and in both cases first when

the pad is still and then when

it

is in motion.

Therefore, we shall start with a pad that is supplied directly and still. Consider-

- a -

- b -

Fig. 1 1 . 1 Hydrostatic pad of infinite length. A finite length L

of

it

is

considered. a- Direct supply;

b- Compensated supply (supply

through

compensating elements).



364


ing tha t its behaviour is described by the fundamental flow rate-pressure relation,

Eqn 4.48, we assume that:

the lubricant

flow

rate Q is constant; we then verify the variation of recess

pressure p r and of the other quantities: load capacity

W,

stiffness K and dis-

sipated pumping power H p , as functions of the characteristic variables:

length L , recess width b, film thickness h , and lubricant viscosity p ; the pad

width B is generally assigned.

Afterwards we assume that:

recess pressure

p,. is

constant; starting from

Q

we continue as above.

Finally, we assume that:

load

W is

constant and we continue as above.

It

should be noted that al-

though W is a derived quantity it is fundamental for the determination of the

dimensions of any hydrostatic system, as it is almost always assigned and

often the most important required characteristic.

In the investigation described above, particular care is taken in the determina-

tion of the constrained minima of H p , that is of the corresponding optimum values

o f L , b , h and

p ,

considered individually.

Afterwards a moving pad is considered. Its performance

is

described by the

fundamental relation Eqn

4.49,

and, assuming that speed

U

s

constant, the varia-

tions of friction F, of dissipated friction power H f and of friction coeficient f as func-

tions of L,

b, h,

and p , are verified.

Finally the variation of the total dissipated power H,, that is the s u m of H p and

H f , is considered and its minima and the corresponding optimum values of

L,

b , h ,

and

p

in the three above-mentioned cases are determined: Q constant,

pr

constant,

W

constant and assuming in all cases that

U

s constant. Optimization is first car-

ried out as regards one variable, then two, three and finally all four variables. The

concept of pad emciency is also introduced.

The optimization procedures obtained for the infinite pad, that are already in-

dicative for any real pad, are transferred in the end to specific pads: the rectangu-

lar, the circular and the annular pads.

Afterwards the pad supplied by means of

a

capillary tube is considered. In such

a case the ratio

/3 (Eqn

5.15)

between recess pressure

p r

and supply pressure p s also

plays

a

role. As done before, the pad is first studied when

it

is still and assuming

that there is

a constant supply pressure

p s

and then

a constant load

W.

In both cases

p

is obviously taken into account.

Afterwards the moving pad is studied.



OPTlMlZATlON

365

Finally the variation of total dissipated power H t is studied, determining its

minima and the corresponding optimum values of

L, b, h,

and

p .

The procedures of optimization obtained for the infinitely long pad supplied by a

capillary tube are transferred in the end to other types of compensating elements:

orifices and constant flow valves, and to the above-mentioned real pads.

The single cases of optimization, with one

or

more variables, are all widely

discussed to make their application easy and examples with three and four vari-

ables are given.

11.3 CONDITIONS OF MINIMUM

The evaluation of the condition corresponding to minimum pumping and fric-

tion power and, more generally, t o the minimum total dissipated power of a bear-

ing, requires, as is well known, the solution of equations o r systems of equations

such as

0

,

i = 1, 2,

...

n

H

a x i

-=

(11.1)

where

x i

can be a dimension of the bearing, film thickness, lubricant viscosity, etc.

Thus the "optimum" values

of

xi are obtained so as to make

H

a minimum.

Sometimes, in addition

t o

the above-mentioned condition, others are imposed:

for example that film thickness, or load capacity or stiffness should not be lower

than an assigned value In such cases we have

t o

deal with problems of "constrained

optimization".

11.4 EFFICIENCY

Useful indications for optimization can be obtained from the ratios of total power

and load and

of

total power and stiffness:

(11.2)

r w and

r,

will be defined afterwards as "efficiency losses".

It should be noted that psl W can be interpreted as a "pressure coefficient" o n the

analogy of the friction coefficient

F l

W

which is interpreted as an index

of

the em-

ciency loss for friction of a kinematic couple.



366

HYDROSTATIC LUBRICATIOW

11.5

DIRECT SUPPLY

Let us consider the infinitely long pad in Fig. l l . l . c ,of width B , directly supplied

by a pump (Fig. 5.11.a),and let us study the performance of a portion

of

length L.

11.5.1 Steady pad

11.5.1.1

Given

flow rate. Flow rate Q is assumed to be constant. This is easily ac-

complished in the case of direct supply with a constant flow supply pump. Film

pressure which is determined using Eqn 4.46, replacing z with x, considering Eqn

4.48 and disregarding the sign of absolute value, is

while recess pressure is

1

B - b

~r = 3~

Q 7

If the losses in the

supply pressure p s . In

(11.3)

supply line are assumed to be equal to zero, p r equals the

this chapter the notation p r = R Q , where R is the hydraulic

resistance of the bearing (section

4.7.21,

will be rarely used, whereas, in previous

chapters, i t was used for the sake of synthesis. The diagram of the recess and film

pressures of the bearing is showing in Fig. 11.1.~.

The dissipated pumping power

H p ,

given by Eqn 5.3, with p s = p r ,

is

1

B - b

H p = 3p

Q 2

-

h3

L

Load capacity

W,

3 1

W = p Q

p B2 b2)

(11.4)

obtained from Eqn 4.47 with Eqn 11.3,is

(11.51

Therefore stiffness

K,

given by Eqn

5.8,

is

(11.6)

Let

us

now consider the influence

of

the dimensions

of

the bearing, t ha t is its

length L, idth

B

and recess width b, on the above-mentioned quantities and then

on the performance of the bearing. Using B as the “reference“ dimension we

introduce

L

L ’ = B

(11.7)



OPTIMIDITION

367

and so pressure p r may be expressed as a dimensionless function of

L'

where

p,.o=pQ(B-b)

Bh3

is the reference pressure. Pumping power

H p

becomes

(11.8)

(11.9)

where Hpo*=pQ2(B-b) /Bh3 s the reference power. It should be noted that H i = p ; .

Such equalities between dimensionless quantities will often be found from here on.

In Fig. 11.2.a

p ;

and

H i

are presented as functions of

L' .

They are inversely propor-

tional t o L'.

Introducing

b

b ' = B

the following dimensionless functions of b' are obtained:

3

p ; = H i = 3 ( 1 - b ' ) ,

W'

= = (1 b'2)

(11.10)

(11.11)

In Fig. 11.2.b W',

K ,

; ,

H i

are presented. They decrease as

b'

increases.

Let us now consider the influence of film thickness expressed in the usual

dimensionless form

h ' = -

h0

(11.12)

where ho is the reference film thickness. The following relationships are obtained:

(11.13)

In Fig. 11.2.c W',

K ' , p ;

and H i are given. They decrease rapidly as h' in-

creases. These diagrams show the performance of the bearing working with vary-

ing loads. It should be noted that, as film thickness reaches zero, the bearing would

bear infinite loads (with infinite stiffness). This is obviously not possible, since the

pump should yield an infinite pressure and the supply system should bear it: in

practice the maximum pressure is limited by a relief-valve placed downstream

from the pump (Fig. 1l.l .a).




12

10

8

6

4

2

0

H i

Hb. ~ ' r

Hi, F'

1'

W '

K '

- a -

6

5

4

4, P

4

H i , F '

3

2

1

0

f '

W '

K '

0

L'opt

x

/3

1

H

3

- -

HD

0 1 h'

- c -

- b -

0

0.5

b'

6

5

4

3

2

I

H;

H9

Pr

H i , F'

W '

K'

a

0

- d -

Fig.

11.2

Load

W ,

stiffness K , recess pressure

p;,

pumping power H', friction force F', friction

power

H j ,

friction coefficientf' and total power

H ;

(for speed factor k=4) versus:

a-

bearing length

L'; b-

recess width b';c-

film

thickness h';

d-

viscosity p', or c'=(l-b')p'.

Fina l ly le t

us

consider

the

influence of

the

viscosity

p

of

the

lubricant , expressed

in

the d imens ionless form

=

JL (11.14)

PO

w h e r e a g a i n

po

is the

referenc e viscosity. T he following relation ship s

are

obtained



OPTiMiZ4 TION

369

(11.15)

In Fig. 11.2.d W',

K',

i, Hi are presented. They vary linearly with p'. Such

diagrams may be employed

t o

study the performance of the bearing in operating

conditions, as p varies with temperature. So, if viscosity decreases because temper-

ature increases, load also should decrease. Since that is generally impossible in

operating conditions, film thickness must decrease.

11.5.1.2

Given

pressure. Supply pressure

p s

is assumed to be constant; therefore if

there are no friction losses in the supply line, recess pressure

p r = p s

is a l s o constant.

The following formulas ar e especially interesting for the study and design of a bear-

ing operating

at

a given pressure which, for the sake of safety, is generally much

lower than the maximum pressure the pump can supply or the supply circuit can

bear.

Load capacity is given by Eqn 4.47 which may be rewritten as

1

(4.47 rep.)

=

j p r

L ( B + b )

The product L B is the pad area. It is often defined as "projected pad area" to distin-

guish

it

from the product

L[ (B+ b) / 2 ]

efined as the "effective pad area".

Stiffness is

3 P

2 h

= - ' L (B + b )

Flow-rate is

Pumping power is

(11.16)

(4.48 rep.)

(11.17)

In dimensionless form the following expressions, functions of L ' , are obtained:

Figure 11.3.a shows the linear variations of W , K , Q' and H i with L' . Consider-

ing their dependence on

b'

they can be written as:



370


- a -

3

H i

2

H b

, Q '

H i

,

F'

1

W ' ,

K '

0

3

H i

l ib

, Q '

2

H i

,

F '

1.15H;,

~

Hi, __-

I

f '

W ' , K '

0

- b -

7

0 1 2 3 4 5

L'

pr

=

const.

u

P

3

H i

I

Q'

Hi

,

F'

2

1.15H;, ___

Hi, ~-

I

f '

0

10

3

1

-H t

HP

l/s 1

3

h-

"P

- c - - d -

Fig. 11.3 Load W', stiffness

K',

low rate Q', pumping power H i , friction force F', friction

power

Hi ,

friction coefficientf' and total powerH; for speed factor k = l ) versus:

a-

bearing length

15';b- recess width 6'; c- film thickness h';

d-

viscosity p ' , or c'=( 1-6')p'.

1 1 (11.19)

'=H' --

1

w = K ' = 2 ( 1 + b ' ) ,

P

3 1 - 6 '

W'

and

K'

are often called "load and stiffness factors", respectively, while Q ' and

H i may be called "flow and pumping power factors", respectively. In Fig. 11.3.b

W',



OPTlMlzATlON

371

K , Q'

and

H i

are plotted against

b'.

While

W'

and K' vary linearly with

b', Q'

and

H i vary exponentially.

Considering the dependence on

h'

the following expressions are obtained:

(11.20)

In Fig. 11.3.c

K ' ,

Q'

and

H L

are plotted against

h'.

The

first

quantity is inversely

proportional to

h'

while the others increase according to

a

cubic law.

Finally, considering the dependence on

p ' ,

the following expressions are

obtained:

(11.21)

Figure 11.3.d shows the inversely proportional variations of

Q'

and

H i

with

p ' .

11.5.1.3 Given

load.

The load

W

carried by the pad is assumed to be constant and

equal t o an assigned value, as commonly occurs in practical applications and there-

fore in design. Recess pressure is

(11.22)

Stiffness is

(11.23)

K = 3 h

Flow-rate is

Pumping power is

(11.24)

(11.25)

In dimensionless form the following expressions, functions of L', re obtained:

4 1

p;=L' I

H ; = j L ' (11.26)

In

Fig. 11.4

p;

and

H b

are plotted against

L',

with both quantit ies inversely propor-

tional

to

L'.


b

the following expressions a re obtained:



372

HYDROSTATIC

L

UBRlCATlON

- a - -b

-

1

0.8

0.6

b'opt

0.4

0.2

0

bopt

u

2

4 6 8 k 1 0

3

H;

H6

Him-

2

1.15H;,

Hi,

, f

'

1

Q'

K '

C

~

1/5 1 3

--

i

HP

I h' 2

I h'opt 1

- C -

3

/3

1

3

P

- d -

Fig.

11.4

Stiffness K'. recess pressure

p;,

flow rate

Q',

umping power H' friction force

F' ,

friction power

H i ,


f'

and total power

H ;

(for speed &;tor

k = l )

versus:

a-

bearing length

L',

or

g'=L'p';

b- recess width

b'; c-

film thickness

h';

d- viscosity

p'.

In case

b

optimal recess width

b&

versus speed factor

k is also

represented.

(11.27)

2 1 4 1

Q ' = - -

i== 9

3

1

- b"

'

Hb

=

3

(1 +

b')

(1

- b")

In Fig.

11.4.b

p;,

Q'

andH i are plotted against b':p; decreases,

Q'

increases,

H i

has

a minimum. Such minimum value

is

important

for

the optimization

of a

bearing



OPTlMlzATlOfU 373

and it can be determined by solving Eqn 11.1 for i = l and

x l = b ' ,

thus yielding b'=1/3.

It should be noted tha t around that value Hb is not critical so that greater values of

6' may also be used. The value of 0.5 is often suggested for b', with a n increment of

H i

lower than

6%,

with respect to its minimum value.

Considering the dependence on h ' the following expressions a re obtained:

1

K= r

Q'=3h'3 ,

(11.28)

In Fig. 11.4.c K', Q' and Hi, re plotted against h'. The first quantity is inversely

proportional to h' while the others increase with a cubic law.


p',


obtained:

(11.29)

Figure 11.3.d shows the inversely proportional variations of Q' and H i with p'

11.5.2

Moving

pad

If the pad shown in Fig. 11.1 should move in the z direction, perpendicular to its

length, the influence

on

it s performance of the inertia forces acting on the lubricant

in the recess as well as the effect of lubricant recirculation in the recess cannot be

disregarded. On the other hand if the pad moves in the direction of its length, the

influence is nil. In this case the expressions relevant t o the motionless pad still hold

good, as well as the following expression of the friction force in the recess.

11.5.2.1

Friction. The friction force in the film is given by Eqn

4.49

F f =

p

U L

( B

b )

(4.49

rep.)

The friction force in the recess is

F f p= p U

1

P

L

B

Friction power is given by Eqn

5.5,

where &O. As regards the power dissipated in

the film, this is

H f Z p i F i L ( B - 6 )

while in the recess it is

(11.30)




If the recess is not too wide, a s often happens, andfor its height is much greater that

the thickness of the film,

H f p

s negligible compared to

H f ,

as will be assumed from

here on.

The power necessary to accelerate the lubricant in the film should also be added

to I f f , hat is:

However, even for high values of U ,Hl is generally negligible compared to H f , as

will be assumed from here on.

Ff

and

H f

may be expressed as dimensionless functions of

L':

=

L' (11.31)

H H

= L' f=

5

F f

F i . - F f o

- 1U K B ( B - b ) ' Hi- - H f o p u 2 K i ( B - b )

where F f o = p U ( l f h f B ( R - b )nd H f o = p U 2 ( l f h ) B ( R - b )re reference values for the fric-

tion force and power. Figures 11.2.a, 11.3.a and 11.4.a show how F j and H i vary

linearly with

L' .

Considering the dependence of

Ff

and

H f

on

b'

they can be written as:

F j = H i = 1 - b '

(1 .32)

F i

may be called the "friction factor" while

H i

is sometimes called the "power fac-

tor". Figures 11.2.b, 11.3.b and 11.4.b show how

F j

and

H j

decrease linearly as

b'

increases.

Considering the dependence on h' the following expressions are obtained:

1

F j

= H i

= r

(11.33)

Figures 11.2.c, 11.3.c and 11.4.c show the inversely proportional variations of F j and

H j with h' .


p'


obtained:

F j = H j = p' (11.34)

Figures 11.2.d, 11.3.d and 11.4.d show how

F j

and

H j

vary linearly with p'



OPTIMIZATION

375

11.5.2.2

Friction coefficient.

If the flow rate is assigned and then load capacity is

expressed by Eqn 11.5, the friction coefficientf=Ff lW becomes

2 u

L

3 Q B + b

f

=--

h2

-

If expressed in dimensionless form, as a function of

L' ,

6' and h', i t becomes

(11.35)

(11.36)

In Fig. 11.2.a and Fig. 11.2.b,f is plotted against

L'

and b': its variations are simi-

lar to those of F j and

H i ;

in Fig. 11.2.c, f ' is plotted against h': it s variation differs

from those of F j and H i as i t increases with h'.

If pressure

is

assigned and load capacity

is

given by Eqn 4.47, the friction coefi-

cient is

f = 2 - p - -1 B - b

p r h B + b

(11.37)

When expressed in dimensionless form,

as

a function of b', h' and p', it becomes

(11.38)

In Fig.

11.3.b,

11.3.c and 11.3.d f is plotted against b', h' and p': its variations are

similar to those of

F j

and

H i .

Finally, if load is assigned, f ' may be expressed as

u 1

f

= p

w k

(B - b)

(1

.39)

In dimensionless form, as a function of

L ' ,

b ' , h' and p', it becomes

Figures 11.4.a, 11.4.b, 11.4.c and 11.4.d show how the variations off' are identical to

those of

F i

and H i .

11

25.3

11.5.3.1 Given f low rate. In the case of direct supply and constant flow rate, pad total

dissipated power, given by Eqn 5.6, is obtained by adding Eqn 11.4 to Eqn 11.30, that

is

Minimum dissipated power and ef f ic iency losses



376 HYDROSTATIC

LUBRICA

TION

(11.41)

Introducing a reference pumping power, total power H t may be expressed in

dimensionless form, as

a

function

of

L':

where

U B h

k

=-

Q

k

may be called a "speed parameter".

H i

is

minimum for

d3

LhPt

=

-

k

and for such a value of L' it is

(11.42)

(11.43)

(11.44)

(11.45)

where

H f I H ,

is

the often mentioned "power ratio". Equation

11.45

and Eqn

11.60

below were given for the first time in ref. 2.3 and have been used many times in

optimization problems in previous chapters. In Fig. 11.2.a H i is plotted for k = l .

Within the range 1<HfIHp<3,Hi differs from its minimum by less than 15%.

The speed parameter is related

to the

power ratio by the following equation:

where Hp0*=pQ2(B-b) lh3Bnd Hfo*=pU2B(B-b)Ih.

Considering the dependence on b' ,

HI

can be expressed as

(11.46)

L h

Q

i = 3 ( 1 - b ' ) + k z ( l - b ' ) , with

k = p

In Fig. 11.2.b H i is plotted for k = l . As b' approaches 1 , H i as well as W' and K ap-

proach

0.

H i can be expressed as the following function of h':

1 1 U L h oH ; = 3 7 + k 2 j 7

,

with

k=-

h Q

( 1 .47)



OPTlMlZATION

377

In Fig. 11.2.c Hi is plotted for k = l . A s h' increases, Hi decreases; W' also decreases

but more rapidly and K even more

so.

Finally considering the dependence on p ' the following expressions are

obtained:

Hi = 3 p' + k2 p'

(11.48)

where K is given by the second of Eqns 11.46. In Fig. 11.2.d Hi is plotted for k = l . H i

decreases linearly with

p'

as do W and K'.

A s regards the variation of H't as a function of

B,

considered a reference quan-

tity until now, it is sufficient to put

(11.49)

B ' = b

and we immediately get the variation of

H&B')

from that of Hi(b').

The study of the variation of total dissipated power Ht becomes more complicated

if it is considered as

a

function of two

or

more variables unless they a re

b (or B )

and

p. Indeed, in such a case the following "compound variable can be introduced:

C'=(l-b')p' (11.50)

and Eqn 11.41 becomes

Hi

=

3

C ' + k 2 ~ '

(11.51)

similar to Eqn 11.48 and Eqn 11.46 and wi th the same

k.

In Fig. 11.2.d Hi(c') is also

plotted, for k = l .

10- In the previous paragraphs i t has been shown tha t if b, h, p

(and B ) vary in such a way as to make

H,

decrease,

W

and

K

also decrease though in

a different way;

this

must generally be taken into account in dimensioning a bear-

ing. In this connection useful informations may be obtained from the "efficiency

loss"

rw,r

from its inverse (ref. 9.41, and from

r K

given by Eqns 11.2. Considering

the dependence on b', dividing Eqn 11.46 by the second of Eqns

11.11

gives us:

(11.52)

Both rb and r k decrease as b' approaches 1 suggesting the choice of a wide recess.

In Fig. 11.5.a

rw

and rk are plotted for k=O, 1,2.


h',

dividing Eqn 11.47 by the second and the

third of Eqns 11.13,respectively, gives



378 HYDROSTATIC LUBRICA

TlON

In Fig. 11.5.b Tb(h') and rk(h ' ) re plotted for h=O, 1, 2. riy decreases

as

h' ap-

decreases

as

h'

approaches

0

in every case.

This

roaches 0 except for k=O, while

suggests the choice of a small film thickness.

- a -

- b -

5

4

3

rh

2

1

0

Q

= c ons t .

E l

0.5 b l 1

2

h '

Fig.

11.5 Efficiency losses

rb

and rK ersus:

a-

recess width

b', b-

film thickness

h',

for cer-

tain values of speed factor

k .

Finally from Eqn 11.48 and from the second of Eqns 11.15 it transpires that the

efficiency loss does not depend on lubricant viscosity.

The choice of the values of the variables B, L, 6, h, and p also depends on various

other conditions which may be encountered in the design of a bearing, regarding its

operation and construction. For example, in actual pads, Fig. 6 .25 , it

is

convenient

to choose BSLSZB.

11.5.3.2

Given

pressure. Total dissipated power is

H t , expressed in dimensionless form, as a function of L', is given by:



OPTIMIZATION 379

(11.55)

where k is similar to the Sommerfeld number

of

plane hydrodynamic pads. In Fig.

11.3.a

Hi

is plotted for k = l .

Hi

decreases linearly with L' as well

as W

and

K'.

Considering the dependence on 6 ' )Hi can be expressed as

(11.56)

1 _l .L .E

P r h2

i = 3 1 + 6 ' 2 ( 1 - 6 ' ) , with k -

Hi is minimum for

1 1

6ipt = 1

3

(11.57)

to which Eqn 11.45 still corresponds. It should be noted that Eqn 11.57 leads to 6Apt<0

for k<l/&; in practice

6Apt=0

up

t o

k=1/fi and afterwards it rapidly approaches

unity. In Fig. 11.3.b Hi is plotted for k = l .

H i

can be expressed as the following function of h':

(11.58)

Hi

is minimum for

hApt =6 (11.59)

to which the following relation corresponds:

It should be noted that from Eqn 5.7 high values of the ratio HfIHpcan lead

to

high

temperature increments. In Fig. 11.3.c

Hi

is plotted for

k = l .

Finally considering the dependence on

p '


obtained:

HI is minimum for

1 1

Pbpt = 3 i

(11.61)

(11.62)

t o which Eqn 11.45 again corresponds. In Fig. 11.3.dH i is plotted for k= l.



380


It should be noted that, within the range 1<H+Hp<3, Hi(b') as well as Hi(h') and

H&p') differ from their minima by less than 15%. As regards the variation of H i as

a function of B', given by Eqn 11.49, it can be immediately deduced from that of

Hi(b').

Therefore we may conclude that high values of B' can be used for

k>O,

too.

For example, with reference to Fig. 11.3.b for k = l , putting

B'=- ,

the value of H i

would not exceed its corresponding value for B&l/b~pt=2.37 by more than 15%.

Considering Eqn 11.50, Eqn 11.54 becomes

H i is minimum for

1 1

c b p t = z i

(11.63)

(11.64)

to which Eqn 11.45 again corresponds. In Fig. 11.3.d Hi(c'1, given by Eqn 11.63,

is

also plotted, for k = l . Therefore we can immediately look for the minimum value of

Hi : ckpt is determined from Eqn 11.64 and then any couple of values of

b '

and p'

which satisfy Eqn 11.50 yield such a minimum. The choice of such values makes

the selection of the bearing dimensions easier.

For couples of variables that are different from b and p, or example b and h, the

search for the minimum value of

Hi

can only be approximate: for a series of values

of one of the two variables, the optimum values of the other variable are searched

for, together with the corresponding minimum values of

H i ,

of which the absolute

minimum i s evaluated (as a check, the trial can be repeated, starting with the other

variable). The case of three variables, if they are b, p and h, can be reduced to that of

two variables using Eqn 11.50.

Efficiencv losses, Equat ions 11.55 and the first of Eqns

11.18

yield

Tw(L')=rk(L')=const., while Eqn 11.56 and Eqns 11.19 give

2 1 1 - 6 '

rw=

+

2

k2

-

r, =

r;,

1 + b ' '

rw and r k are minimum for

(11.65)

(11.66)

In Fig. 11.6.a r i nd r k are plotted for

k=O,

1,2. It should be noted that for k = l ,

while the value of

Hi

for b'=O exceeds that for b'=bApt by only 15% (Fig. 11.3.b)) the

value of for b'=O exceeds that for b'=bAPt by 74%.b&(k) is plotted in Fig. 11.6.b.

Finally, Eqn 11.58 and the first of Eqns 11.20 yield



- a -

OPTlMlzATION

- b -

E

381

Fig.

1 1.6

a-

Efficiency

losses rp

nd rh versus

m e s s

width b’ for certain values o f speed factor

k;

b- optimal values

bLPl

of recess width versus speed factor

k .

In Fig. 11.7 k is plotted as a function

of h’

for k = O , 0.5, 1,1.5, 2, 2.5,

.

rk increases

very slowly in the range of commonly used values of

h’ .

11.5.3.3Given load. Total dissipated power is

4 1

w2

1

Ht =

3 E

h 3 L

B

b)

( B

+ b)2

+

u 2 h

tB b,

H t , expressed in dimensionless form, as a function of

L ’ ,

is given by:

H i

is minimum for

(11.68)

(11.69)

(11.70)



382


0

0.5

1 15 2

h'

Fig.

11.7

Efficiency loss r~ ersus film

thickness

h' for certain values of speed parameter

k .

to which Eq n 11.45 aga in corresponds. HI is plotted for k = l in F ig . 11.4.a.

Considering th e dependence on

b',

Hi can be expressed

as

4

1 I I TIT.

In Fig. 11.4.bH i is plotted for k = l . Th e condition for

its

minimum dH;/db'=O yields a

f if th degree equa tion. In Fig. 11.4.b th e calculated values of

bbpt

ar e a lso presented

a s a function of

k .

A s k increases, b' r ap id ly approaches un i ty an d Eqn 11.71 ap-

proaches the form of Eqn 11.56. The value of bhpt can a lso be calcula ted with the

semiempir ical formulae

1

bbpt =j 0.33745k2

-

0.16818k3

+

0.024475k4

,

for

0 5

k c3

(11

72)

with a max imum error of

0.85%

for k=0.4 n t h e first formula a nd one of 0.042% or

k=3 in t he second.

Hi can be expressed as th e following function of h'

(11.73)



OPTlMlzATlON

383

H i is minimum for

hbpt

=

4

(11.74)

to which Eqn 11.60 again corresponds.

H i

is plotted for K = l in Fig. 11.4.d.

Finally, considering the dependence on p', the following expressions are

obtained:

H i

is minimum for

2 1

P&t

=az

(11.75)

(11.76)

t o

which Eqn 11.45 again corresponds.H i is plotted for k = l in Fig. 11.4.d.

As regards the variation of

H i

as a function of B', given by Eqn 11.49, it can be

immediately deduced from that of

H l ( 6 ' ) .

If we put

g ' = L ' p '

Eqn 11.68

can

be written as

(11.77)

(11.78)

In Fig. 11.4.aH;(g ' ) ,given by Eqn 11.78,is also plotted, for k=1.

Hi

is minimum for

(11.79)

to which Eqn 11.45 again corresponds. Therefore the search for the minimum value

of H i , as a function of L ' and p', is immediate, as already specified in a previous

case (Eqn 11.63).

Putting

4(1-6')

in place of

(l-b')(l+b')z

n the first of Eqns

11.71,

the minimum

occurs for

6'

given by Eqn

11.57

with K given by the second of Eqns

11.71.

The values

of H i , obtained from the first of Eqns 11.71, thus modified, differ from the exact val-

ues by less than

10%

for

k23.

Therefore, putting

q' =

L'

(1 -

6') p'

(11.80)



384


in Eqn 11.68, modified and expressed in dimensionless form,

is obtained.

H i

is minimum for

1 1

qApt =

(11.81)

(11.82)

to which Eqn 11.45 corresponds. So for k23 the approximate minimum of H i of Eqn

11.68, considered as a function of L', b', and

p'

is given by Eqn 11.82; it then allows a

wide choice for the three optimum values of L' , b', and p'. The search for the mini-

mum of

H t

given by Eqn

11.68

as a function of

h

and

L,

o r

b, o r

p

can be carried out

in an approximate way, as already seen in the case of constant pressure.

Efficiencv losses. Since W is now assigned, the variation of

H t

is the same as r,

and r, except for T'(h), for which dividing Eqn 11.73by the first of Eqns 11.28yields

r;, 4

h'4

+ k2

(11.83)

Figure 11.8, which is very similar to Fig. 11.7, shows

as a function of h', for k=O,

0.5,1, 1.5,2,2.5,3.

ri

increases very slowly in the range of commonly used values

of

h'.

0 0.5

1

1.5

h'

L

Fig.

11.8

Efficiency

loss rK ersus film

thickness h'

for certain

values of speed parameterk.



OPTlMlzATlON 385

11.6 OPTIMIZATION

The minima of

total

power H , , previously specified, have been determined con-

sidering H t as a function of only one of the variables

L , b , h

and

p

and having taken

B

as

a

reference quantity. The cases in which

it

was considered

as

a function of two

or more variables was actually reduced to the case of only one variable.

We shall now go on to the determination of the minima of Ht as really a function

of two variables which is a rather frequent case in practice since o h n he values of

two variables are assigned, and as a function of three variables which

is

a frequent

case in practice since a t least the value of one variable is assigned, and finally as a

function of all four variables. Optimization is also carried out in the presence of con-

straints of both the variables and of other quantities such as, for example, stiffness

K.

his

is done for the three cases previously mentioned in section

11.2:

given flow

rate, given pressure and given load. Its

application

t o

this last case, that

is

when

the load is assigned,

is

particularly important since this is the condition most fre-

quently encountered in design.

An outline

of

optimization methods, tha t is for the determination of the uncon-

strained or constrained minima of a function, is presented for example in ref.

11.1.

Here sufice it to say that optimization has been carried out with the techniques of

non-linear optimization of the "objective function"

H t ,

for which the Adaptative

Random Search Method proved to be especially suitable. Constraints involved

"penalty functions", for which the Schuldt's Functions proved

t o

be particularly

suitable.

11.6.1

Given

f low

rate

As shown in Fig.

11.2,

Ht only has a minimum as a function of L' , whereas with

the other variables

it

merely increases or decreases.

11.6.1.1

Ht=f(L,b).Equation

11.41

can be expressed in dimensionless form as

(11.84)

where k is given by Eqn 11.43.

In Fig.

11.9

H;(L' ,b ') s plotted in the range O G ' G and

O<b'<l,

for

K = O . l , 1,

10.

In every case

H i

obviously approaches zero as 6' approaches unity

( W

and

K'

also

approach zero while the efficiency losses

r'w(b')

and

rK

(b ' )

decrease).



HYDROSTATIC LUBRlCA TlON

5 8 8

H

1 .88

8.67

8.33

8.88

1.

H

188.88

.88

66.67

.67

3 3 . 3 3

.33

8.88 588 8.7,

a

5 88

Fig. 11.9 Total power H,' versus pad length L' and versus recess

width b;

or viscosity p ' , or

c'=(l-b')p',for certain values

of

speed factor k.

Figure 11.10 shows the results of the optimization ofHi, or OIk112 with the fol-

lowing constraints: lsL'12, 0.2<b'10.6. The constraint L'12 is mostly due to practi-

cal reasons (maximum size); the constraint b'10.6 is due

t o

the fact that both

W

and

K

must be included in a range of convenient values (see Fig. 11.2.b).

H i m

is plotted

with the corresponding optimum values of Lip t and bhpt. Figure 11.10

also

shows

HFk,

H i , p ; coinciding with H i W' iven by the second

of

Eqns 11.11, K coinciding

with W' and

(11.85)

It must be pointed out that

bbpt

takes the higher of the two boundary values, that

is 0.6. Therefore, if a supply

flow

rate is assigned and the search concerns the val-

ues of LApt and

bbpt

for which dissipated power is minimum, bhpt

is

assumed

to

be

equal to the maximum value of

b'.

As regards

Lbpt

it is determined from Eqn 11.44



60

5c

Him

Hb

p’,

H;

4c

30

20

10

0

0

Q = cons t .

I

2

4 6 8 10

1

4

?

Copt

bopt

P b p t

C’

opt

a‘

K

387

6

5

4

f‘

3

2

1

3

k

Fig.

1

1.10 Values of minimum total power Hi,,,, and corresponding optimal values of pad length

LApf, nd of recess width

b i ,,

or

of

viscosity pA f

or of c&=(I-b&)p+

versus speed factor

k .

Values of pumping power 4, f friction power fib,

of

recess pressure pr,

of

load capacity

W’,

of

stiffness

K‘

and of friction coefficientf’ are also represented (n ote that the values

of

W’,

K’

ndf’

are valid only for the first case, that is for Hi,, ,=f(L&f,b~pl.

where

k

is given by the second of Eqns

11.43,

f it is inside the boundaries (Eqn

11.45

is then verified); otherwise it takes the boundary value nearest to it. The couple L&,

6APt

allows the calculation

of

the other quantities. It should be noted that for these

values the efficiency losses are also minimum.

11.6.1.2Ht=f(L,p).Equation 11.41 can be expressed as

(11.86)

where k is again given by Eqn

11.43.

On the analogy of Eqn

11.86

with Eqn

11.84,

n

Fig.

11.9Him(L&,t,p&)

is also presented, now in the range

OSL’<5

and

OSp’S1.

or

Hj,(LApt,p&,,pt)

he considerations concerning H i ( L

‘,6‘)

are still valid. It must be

borne in mind solely that p ’ is the complement of 6‘ t o 1.




Similarly Fig. 11.10 may be used

t o

obtain the results of the optimization of

H;m(L&,,pt,~&,t) ith the constraints lILY2,0.41~'10.8.Figure 11.10 also shows Hjk,

H i andp; ( which coincides with Hi), whereas W and

K'

are given by the second of

Eqns 11.15 andf 'by the following equation

It is obvious th at p ' has always taken the lower of the two boundary values, that

is

p&0.4.

To obtain

Him(L&,p&),

a procedure similar t o tha t for

H;m(L;Zpt,b&,,pl)

is

followed.

11.6.1.3 H,=frL,p,b). Equation 11.41can be again expressed a8

Hi=--

1 - 6 '

1

Ht

- 3 p ' r @p'

(1 6 ' ) L '

POQ2 jp

(11.87)

where k is again given by Eqn 11.43. After having introduced Eqn 11.50, we can also

express Eqn 11.87 as

(11.88)

'

Hi = 3

L';

+

k2 c'

L'

On the analogy of Eqn 11.88 with Eqn 11.86, H~m(L&,,pt,c~p,pt)s also presented in

Fig. 11.9, now in the O G ' S 5 and O<c'<1 range. Similarly, fig. 11.10 may again be

used to obtain cApt and the optimal values of Hirn,

H@,

H i and

p; (

which coincides

with Hi) in the case in which

X L ' a , 0.4<c'10.8.

Friction f is given by Eqn

11.85. W

is determined with the equation

as

well as

K'

which coincides with

W'.

After Eqn 11.50 has been introduced, Eqn

11.89

becomes

(11.90)

W =

3 (1+ 6')

c'

that for c'=constant increases linearly with

6'.

cAPt has always taken the lower of the two boundary values 0.4 and any couple

of

To

obtain Him(L&&,pt), that is

H~m(L~p,p t ,6~p,p , ,&,p ,p t ) ,

procedure similar

t o

that

values of b ' and

p'

satisfying

(1-6')p'=0.4 is

an optimum couple.

for Him(L',6') is followed.



OPTIMIZATION

389

11.6.1.4 Ht=KL,h). Equation 11.41 can be expressed then a s

U B

o

HI=

1 B - b = 3 s + k 2 , = H i + H j k , with k=-

(11.91)

Ht

1 L'

P Q 2 j p B

h Q

H;(L',h') is presented in Fig. 11.11

,

n the OIL'<5,O<h'<2 range,

for

k=0.1, 1, 10.

In every case Hi obviously approaches zero as h ' increases. On the other hand

as

h'

increases

W

and K' a l s o approach zero, as shown as well in Fig. 11.2.c.

Fig.

1 1 . 1

1

Total power

H;

ersus pad length

L'

and

film

thickness

h', for

certain values

of

speed

factor

k.

In Fig. 11.12 the results

of

the optimization o f H & L ' , p ' )are also presented, with

the constraints 15L.12 and 0.3<h'<0.6. In

the last range,

by

the way, W and

K

are

suitably

high (see also Fig. 11.2.12). Figure 11.12 also shows H)h, H i ,

p r

coinciding



390

250

200

H k l

150

HP

p;

100

H;k

50

0

0

' 2


4

6

8

I 0

12

k

2.5

0

W'

.5

K'

10

.5

1

Fig.

11.12

Values

of

minimum total power

Hi,,,, and

corresponding optimal values of pad length

L &

and of

film

thickness

h&.

versus speed factor

k.

Values of pumping power

H i ,

of friction

power H j k , of recess pressure p, , of load capacity

w',

f stiffness 1y'and of friction coefficientf'

are also represented.

with

Hi,, W'

and

K

given by the second and third of Eqns

11.13,

espectively. The

friction coefficient is given

by

(11.92)

It should be noted that hAp,Pthas always taken the higher of the two boundary

values:

0.6.

Therefore,

to

obtain

Hi,(L&,t,hAp,Pt),

the maximum value of

h'

must be

selected for

h&,

while

LApt is

determined from Eqn 11.44, hat

is

d 5 1

LApt

=--

k h&

(1

1.93)

If this last value is outside the range of possible values for

L',

the nearest boundary

value will be assumed t o be

LApt.



391

11.6.1.5 H,=f(L,b,p,h). Equation 11.41 can be expressed finally as

(11.941

where

k

is again given by the second of Eqns

11.91.

Figure

11.13.a

shows the results

of the optimization of

Hi(L',b',p',h'),

with the constraints

l I L ' i 2

,

0.21b'I0.6 , 0 .31h ' l0 .6

,

0.81p'11.6 (11.95)

Fig.

11.13

a lso shows

H@,H i p;

coinciding with

HbJ

and

w

,

K'T

' = p

h 3

, 1 - b''

(11.96)

(11.97)

It should be noted that

bbpt

as well as

hApt

and PApt always taken boundary val-

ues: the higher for b' and h' , the lower for p ' , that is bApt=0.6,hApt=0.6 and pApt=0.8.

Therefore, t o obtain

H ; , ( L ~ p t , b ~ p t , p ~ p t , h ~ p t ) ,

he maximum value of b' and h' must

be selected for

bbpt

and

hApt,

while the minimum value of

p'

must be selected for

PApt.

LApt

is

determined from Eqn

11.93,

if

it

is inside the boundaries; otherwise it

takes the boundary value nearest to it.

With Eqn 11.50, Eqn 11.94 can also be expressed as

C ' L'

c'

Hi= 3- , h,3 + k 2 F

(11.98)

In Fig.

11.13.a,

therefore,

Hi,,,

also represents

Hi,(LApt,hApt,c~pt),

obtained with the

constraints

l S L ' I 2

,

0.32Ic'11.28

,

0.3Ih'10.6 (11.99)

cApt has always taken the lower of the two boundary values, i.e. 0.32. Any couple of

values of b' and

p'

satisfying

(1 b' )p'

=

0.32

(1 .100)

is an optimum couple.

To obtain

Hi,(L~p,pt,h~pt,cApt)

he procedure is the same as in the previous cases

but now, since we always have

cbpt=chin

and (from Eqn

11.50) c~i n=( l -b ~, , , )p hin ,

we obtain

bApt=b&

and pLpt=phin.



392


- a

-

100

80

H;

m

Hb

p;

60

40

H;k

20

0

k

- b -

2.5

Copt

Gopt

hbpt

Pbpt

cbpt

2

1.5

1

0.5

f '

0

100

2.5

Copt

bopt

H i m

hbpt

PLbpt

HP

p;

H;k

opt

80 2

60 1.5

40

1

C'

2 0 0.5

f '

0 0

k

Fig.

11.13

Values of minimum total power Hirn, nd corresponding optimal values of

Lhpl, of film thickness

h C I ,

of recess width

bhpr,

nd of viscosity &

,,

or of cipl=(l

versus speed factor k . Values of pumping power H i , of friction power

&h,

f recess pre

load capacity

W ,

f stiffness

K

and

of

friction coefficientf

are

also represented.

10

8

w'

6

K'

4

2

0

10

8

W'

6

K'

4

2

0

I

ength

'ep, ,

of

pt)Phpr*



OPTlMlZ4TlON

393

In Fig. 11.13.b the results are presented with the same constraints for L' and

p',

but with

0.2

5

b'

1

.9

,

0.3

5

h'

I0.4 (11.101)

We must point out that in this latter case load capacity is the same as in the former

case, but stiffness is

50%

higher

(this

can be immediately deduced from Eqns 11.961,

even with lower dissipated power and a much lower friction coefficient. The same

figure

also

shows the results obtained with the constraints

l l L ' 1 2 0 . 0 8 l ~ ~ 5 1 . 2 8 0.35h'l0 .4

and now chpt=0.08.

EXAMPLE

11.1

The pad of Fig. 11.1, with width B=0.1

m

and length LQ-B, is directly supplied

by a pump as in Fig. 1 l.l.a . The pump isassumed to supply a constant

f low

Q=5.10-6

m31s, that is a little higher than hydrodynamic flow rates. We want to evaluate the

pad load capacity W and stiffness K in the condition of minimum total dissipated

power Ht, for speed U=0.3, 1 ,3 mls .

H t , can be calculated following the first procedure described

in

section 11.6.1.5.

I f the reference value of the film thickness is assumed to be ho=lO-4

m,

factor k,

given by Eqn 11.91, for the three values of speed, takes the values k=0.6, 2, 6. I f , for

example, Constraints 11.95 are adopted, the following results are obtained (also

directly from

Fig.

11.13.a):

1) k=0.6

Lhpt=2, bApt=0.6, h Apt,pt=0.6,

IJ ApFO.8

W'=3.56, K'=5.93, f'=o.18

LoPt=0.2

m,

bopt=O.06m, hopt=0.6.10-4

m

Ht,=6.52 Nmls, Hp=5.56 Nml s, H ~ 0 . 9 6 mls, p,=11.1.105 Nlm2

W=1.78.104 N, K=8.89.1@ Nlm, f=1.8.104.

H;,=2.61, Hi=p; =2.23, Hb=0.384

from which, in dimensional form,

and, taking

as

reference viscosity &=0.1 Nslm2, hpt=0.08Nslm2 and

It should be noted that the value of pressure is not much higher than the values of

hydrodynamic supply pressure and the very low value of the friction coefficient.

2) k=2

Lopt=O.145 m, bOpt=0.06

m,

hOpt=0.6.1O4m, ~pt=0.08A?slmz

Ht,=15.4 NmIs, Hp=Hf =7.7N m l s , p,.=15.4*1@ Nlm2

W=1.78-104N, K=8.89.1@ Nlm, fd.35.1Q4

It should be noted that assuming, for example, the recess depth to be h,=0.08 m, the

power dissipated in it for friction would be Hfp=8.7.10-aN m l s (section 11.5.2.1), just




a little higher than 1% of Hf. Assuming the lubricant density to be p=900 IQlrn3, the

power dissipated to accelerate it in the film would be H1=5.63.10-4Nmls, less than

0.01 % of Hf.

3)

k=6

Lopt=O.

m, bOpt=0.06m, hOpt=O.6.104m, bpt=0.08Nslm2

Htm=59.1Nm ls , Hp=ll.l Nml s, Hf=48Nmls, pr=22.2-105Nlm2

W=1.78.104N, K=8.89.1@ Nlm, f=9.104

We must point out that the value of load capacity, identical in the three cases, is

suff icien tly high; that o f stiffness, also identical in the three cases, is high

(comparable to that of the deformation of a roller slide-way substituting the pad).

4) I f constraints 11.95 are also considered, though in the form of Eqns 11.99, on the

basis of the second method explained in section 11.6.1.5, or again from Fig. 11.13.a,

we obtain, for example for the case in which k=2:

and for Hirn, Hb=p; and H b and the corresponding dimensional quantities, the

same values obtained in such a case. According to Eqn 11.100 we can now put, for

example, bhpt=0.8 and consequently pbpt=1.6,obtaining

and

Therefore, comparing this case to the second, for the same values of dissipated

power and supply pressure, there is an increase in load capacity and stiffness of

over 12% and a reduction in friction coefficient of over 16%.

5)

Pad efficiency can be improved further by changing the constraints.

So

i f in

Eqns 11.95 new upper boundaries are introduced for b' and he, or example: b'10.9

and h'10.4, that is, i f the constraints 11.101 are considered, according to the second

above-mentioned method in section 11.6.1.5, or from Fig. 11.13.b, we obtain, for

example, for the case in which k=2:

from which

L&=1.45* bhPt=O.32, h

hpt=0.6

W'=4, K=6.67, f'=0.16

W=2*104 , K=1@ Nlm, f=1.6.10-4.

L

Apt=

1, bbpt=0.9,

h

&=0.4,

pApt=O.8

Lopt=O.l m, bopt=0.09m, hOpt=0.4.1O m, kpt=0.08 Ns lm2

Htm=11.38 W, Hpz9.38 W, Hf=2 W, pr=14.1.105 Nlm2

W=1.78.104 N, K=1.34-1@Nlm, f=1.13.104.

Therefore, comparing this case to the second, for the same load capacity, there

is

a

50%

increase

in

stiffness, a 35% reduction in dissipated power, a 9% reduction in

pressure, while the friction coefficient has become four times smaller.

6) It should be noted that without changing constraints the results can be modified

by changing the reference values of the constrained quantities.

So,

i f the dimen-

sional constraints allow it, we can put B=0.2

m.

Therefore, for U=l mls , for exam-

ple, Eqn 11.91 gives k=4 and consequently:



OPTlMlzATlON 395

LoPt=0.2m, bOpt=O.12m, hOpt=0.6.104m, bpt=0.08Nslrnz

Htm=32.4 NmIs , Hp=ll.l Nmls, Hf=21.3 NmIs, pr=22.2.105 Nlm2

W=7.12.1@ N, K-3.56.109Nlm, f=3.104.

We must point out that, as compared to the second case, the dissipated power is

almost doubled, pressure is increased by

50%

but load capacity and stiffness are

four times higher while the friction coefficient is decreased.

IL on the contrary, because of considerable surface roughness and errors in

parallelism, we want to increase fi lm thickness, we can put, for example, ho=2.lO4

m, so that, again for U=l ml s, Eqn 11.91 again gives k=4 and consequently:

Lopt=O.lm, bopt=O.06m, hOpt=l.2.lO4m, kpt=0.08Nslmz

Htm=4.06 Nm ls, Hp=1.39 Nmls , Hr2.67 Nm ls, pr=2.78.105Nlm2

W=0.22-104N, K=0.185.108 lm, f=12.10-4.

It should be pointed out that, again as compared to the second case, total power and

pressure are reduced to almost a fourth and to less than a fifth, respectively, while

load capacity and stiffness are reduced to an eighth and to a little less than a fiftieth,

respectively. The friction coefficient is greatly increased.

Finally i f the reference viscosity is increased, for example, to the value po=0.2

Nslmz, while we still have U = l

mls, k

is

again that of the second case. Therefore all

the dimensionless quantities remain the same, as well as Lop&

bopt and

hopb

while

popt and consequently Ht,

Hp,

Hf, pr, W and K are double. f is the same as

in

the

second case.

11.6.2 Given pressure

As

shown in Fig. 11.3, Ht has a minimum when it i s considered t o be a function

of each variable except forL.

11.6.2.1 Ht=f(b,

h)

11.6.2.1.1.

Equation

11.54

can be expressed as

(1 .102)

In Fig. 11.14 Hi(b‘,h’) s plotted in the O<b’<land OSh’s2 range, for k=O.l, 1, 10.

In all three cases, actually

for

any value of

k,

Hi is minimum for

b’=l

and for h’ in-

creasing with k , starting from zero in the

first

case (actually for

k=O).

Actually, an

investigation in the neighbourhood of point (1,O)has shown that

H i

is equal to zero a t

that point. Hi always approaches zero as 6’ approaches unity and h’ approaches

zero, provided that it happens according t o a suitable law, for example b’=l-ah’p

where a>l and p>l. But investigation has also shown that (1,O)is a singular point:



396


H;

I8

88

6.67

3 . 3 3

8 .8 8 2.m

1

2.80

Fig.

11.14

Total power

Hi

versus recess width

b',

versus film thickness

h'

and versus viscosity

p' ,

or

c'=( 1

-b l)p ' , for certain values of speed factor k.

therefoye i t would not be convenient, not even in theory, to choose

HI

corresponding

to

(1,O)

r to points very close t o it. On the other hand, in practice, for obvious con-

structive reasons,

b'<l

and

h'>O

must be true.

Figure 11.15contains the results of the optimization of H i in the OSb'Sl,O Sh' S2

range, for OSkg14 and with the constraint b'S0.975. HI, is plotted with the corre-

sponding optimum values of bApt and

hipt;

values of hApt corresponding to a number

of small values of k are also given in Table

11.1.

t should be noted that, for any

value of

k , bApt

has always taken

t h e

boundary value b'=0.975; herefore

hApt

is again

given by Eqn

11.60,

hat is

(11.103)

Figure 11.15also shows

Hb,

L, ' coinciding with HL,

'

and K', given by the fol-

lowing equations



OPTlMlzATlON

397

0< 6

<

0.975.or 0.025

<

~ ' 9 1 ,

r 0.025 <c'<l

0

2 4

6 a 10 12 14

k

Fig.

11.15

Values of minimum total power H;,,,, and corresponding optimal values of film thick-

ness

h&,,

and of recess width

b+,

or

of

viscosity &, ,or of

c&,r=(l-b+r)p&,l

versus speed factor

k. Values of pumping power H and

f l o w

rate Q',

of

friction power Hfi of load capacity

W', of

stiffness K 'and

of

friction coe ffh en tf' are also represented (note that the values of

W',

K' andf'

are

valid

only

for the first case, that is for

H~,,,=f(h&,t,b~p,)) .

(11.104)

which yield very high values because the value of

bApt

is high and the relevant val-

ues of hiptare small (for example, for k = l , h& 0.1581 andK '=6.247), and

f " & = zk - - 1 1 - b '

h ' l + b '

which

is

still sm all; for exam ple, for

k = l ,

f'=O.l60.



398

HYDROSTATIC LUBRlCAT/ON

k

0

hbt

0

0.01 0.025

0.05

0.075 0.1 0.2 0.4

0.6

0.8

0.0158

0.0249 0.0353 0.0433 0.0501 0.0707 0.0999 0.1224 0.1415

Therefore it is to be noted that, for the values of bAptand hApt thus calculated, the

condition of minimum total dissipated power is satisfied as well as that of maxi-

mum load capacity with very high stiffness and a low friction coefficient. Therefore,

if pressure in the recess is assigned, it is convenient to design the pad with the re-

cess as wide a s possible.

Obviously

b’=0.975

is

a limit value. Practical values are lower:

b’=0.95

to 0.75 and

bbpt will take on such values. However the corresponding values of hApt and power

increase while load capacity and stiffness decrease; the friction coefficient also in-

creases. This is proved by the results reported in Fig. 11.16.a, obtained with the con-

straints OSb’S0.9 and O<h’12, and in Fig. 11.17.a, obtained with the constraint

b ‘SO.8.

In conclusion, on the basis of the above considerations, for b’ the maximum pos-

sible value

is

chosen: that is bApt; t is then introduced into Eqn 11.103, thus deter-

mining hApt.

11.6.2.1.2. In practice the condition

OSh’

is replaced by

a s h ’ (11.105)

where a is not too small; this is because of surface roughness and errors in pla-

narity and parallelism, etc., even if the present-day technologies make it possible to

achieve increasingly smaller values of film thickness.

Figure 11.16.b contains the results of the optimization of

Hi(b’,h’),

with the con-

straints

OSb’10.9

and

0.9Sh’SZ

(that is

a=O.9).

For k28.1 the diagrams coincide with

those in Fig. 11.16.a. For 8.bk24.68 only

bApt

and consequently

W’

stil l coincide

while

hbpt=0.9

and consequently H b and

K

become constant;

H ik

decreases more

rapidly while

Him

decreases more slowly than in Fig. 11.16.a. For 4.68>k10.468, we

still have hApt=0 .9 , but W‘ approaches

0.5

and

K’

pproaches 0.5. Moreover

H@=Hb, hat is we again find Eqn 11.45. For 0.468>k20, we still have hAPt=0.9and

bbpt=O.

On the basis of the above considerations, in the presence of constraint 11.105, b’

is again chosen as large as possible and is assumed

to

be bApt; hen from Eqn 11.103

h‘ is determined; if h’ satisfies condition 11.105 i t is h&. If this does not occur, let-



OPTIMIZATlON 399

a k

h&,

b k

bbpt

Fig.

1

1.16 Values of minimum total power H i m , and corresponding optimal values of film thick-

ness

h;,,,,

and of recess width

b

[,

or

of viscosity

&

[,

or of

c&,l=(l-bL

versus speed factor

k.

Values

of pumping power HF of friction power

i j k ,

of flow rate

d ,

f load capacity

W',

of

stiffness K' and

of

friction coeff&entf' are also represented (no te that the values of

W', K'

andf '

are valid only for the first case, that

is

for H;m=f(h&t,b~pr)).

0 0.01. 0.025 0.05 0.075 0.1 0.2 0.4 0.6 0.8

0 0.0316

0.0500

0.0707 0.0866 0.1000 0.1415

0.2001

0.2448 0.2830

0.4 0.5 0.6 0.7

0.8 0.9 1

1.1 1.2 1.3

0 0.1125 0.2210 0.3272 0.4151 0.4814 0.5323 0.5750 0.6100 0.6387

ting hApt=a,H i and H@ are calculated and if the condition

H @ > H i

is satisfied, b&

is still the one initially chosen. If this condition is not satisfied, keeping hAPt=a,

bbt

is

determined from Eqn

11.45,

that

is

(1 .106)

T A B L E 1 1 . 2

Optimal values of film thickness for

&&,=0.9 (a)

and of recess width for h&,=0.9 (b) ,

versus speed factor (see also Fig. 11.16,

Eqn 11.103

and

Eqns 11.106).



400

k 0 0.01 0.025 0.05 0.075 0.1

It&

0

0.0447 0.0707 0.1000 0.1225 0.1414


0.2 0.4

0.2001 0.2828

14/

p,

= const.

1 2 1

p

0

2

k

- a -

1.4

f '

1 0

6

14

12

Him

Hb

H i

10

Q'

8

6

K6

4

2

0

- b -

k

I 4

1.2

bopt

tiopt

PLpt

Gpt

1

1.8

1.6

W'

1.4

f l

1.2

1

Fig.

11.17

Values of minimum total power

H b ,

and corresponding optimal values of film thick-

ness hip, , and of recess width bGI , or of viscosity

&

,,or of c&,,=(l-b~ ,)&,, ersus speed factor

k. Values of pum ping power

H p ,

of friction power of flow rate

Q?

of load capacity

W',

of

stiffness

K'

with constraint

K'21

in Fig.

11.1 7.b) and

of friction coefficientf' are also represented

(note that the values of

W ,

K' ndf' are valid only for the first case, that is for

H;,=f(hApl,bLp,)).

11.6.2.1.3. The optimization procedure examined in section 11.6.2.1.1 yields high

stiffness especially for high values of

bhpt

and low values of k . If a higher value of K

is required, for example for functional reasons, a lower limit can be imposed on

it.

T A B L E 11.3

Optimal values of

film

thickness versus

speed

factor, for b&,,=0.8 (see

also

Fig.

11.17

and

Eqn

11.103).



OPTiMiZ4TloN

40 1

For

Eqn 11.104 where W' is already maximum, since bApt always takes the maxi-

mum value, it all comes down to imposing an upper limit on h'. Indeed, letting

W'

K ' 2

y

,

we have

h ' s -

Y

(11.107)

Figure 11.17.b contains the results of the optimization of Hi obtained with the

constraints OIb'10.8, O<h'12 and K > l . Employing Eqns 11.107, the last two condi-

tions are reduced

to

O<h'sW'/

yand since bhpt=0.8and W'=O.9, they are finally re-

duced to O<h'<0.9.The diagrams differ from those in Fig. 11.17.a in the case of

k>4.05: H i remains constant, Hjk approaches Hi increasing more rapidly, and

hipt=0.9 nd

K'=l.

This is true for any other constraint on b' and

K'.

On the basis of the above considerations, in the presence of the further con-

straints 11.107 on stiffness, b' is still chosen as large as possible and is assumed to

be bhpt; then from Eqn 11.103h ' is determined which is hi p t if it satisfies conditions

11.107;otherwise, hipt=W'/y .

11.6.2.1.4.

From the two previous sections, it follows that if, for practical reasons,

film thickness satisfies condition

11.105

and, for functional reasons, stiffness must

satisfy conditions 11.107, the search for the minimum total power is constrained by

the following conditions:

W'

O s b ' < l ; a s h ' s -

Y

(11.108)

sections.

This search is carried out according to the methods described in the foregoing

11.6-2.2Ht=f(uh)

11.6.2.2.1.

Equation

11.54

can also be expressed as

(11.109)

On the analogy of the first

of

Eqns 11.109 with the first of Eqns 11.102, Hi(p',h') is

also presented in Fig. 11.14, in the Osp'S1 and Osh'12 range. As for Hi(p',h') the

considerations regarding H'(b',h')are still valid.

Similarly in Fig. 11.15 Hirn, hPt and hApt are also plotted, with the constraints

0.0251p'sl

and

0 4 ' 1 2 .

Figure

11.15

also shows

H @ ,

H i

and

Q '

coinciding with

Hi .

W'

and

K'

are easily determined, since W'=1/2 and K=W'lh&,t, while f can be ex-

pressed i n the form




PApt has always taken the boundary value p’=0.025; hence, h’ is again given by Eqn

11.60.that is

h&?t

=dmg

(11.110)

Then

Him(p&,t,h&t)

is also presented in Fig.

11.16.a,


O.l<p’<l

and

O<h’12.

Finally

Him(p&,t,hhp,pt)

s presented in Fig.

11.17.a,

with the

constraints

0.25pL1

and

O<h’<2.

In conclusion, on the basis of the above considerations, p’ s chosen

as

small as

possible: it is p&; it is introduced in Eqn 11.110, thus determining h&.

11.6.2.2.2.H;m(p&,h&,t)

is presented in Fig.

11.16.b,


O.lSp’11

and

0.91h’52,

the latter being related to constraint

11.105.

The considerations and conclusions presented in section

11.6.2.1.2

are still

valid; only

bApt

must be replaced by its complement t o one p&, and Eqn 11.103, the

second of Eqns 11.56 and Eqn 11.57 must be replaced by the corresponding Eqn

11.110,the secondof Eqns 11.61and Eqn 11.62.

11.6.2.2.3.H;,(pAPt,h&)

is presented in Fig.

11.17.b,


0.2Sp’11,

(kh’12

and

K‘21,

where the latter two are reduced

t o O<h’10.9,

according to con-

straints 11.107.

only ( l -b&) must again be replaced by p&

The considerations and conclusions deduced in section

11.6.2.1.3

are still valid;

11.6.2.2.4.

Finally, if the search for

H~m(p&,trh&,,pt)

s constrained by the conditions

11.108,

the second of which comes from conditions

11.105

and

11.107,

it is carried out

according to the methods described in the two previous sections.

11.6.2.3Ht=f(b,p,h)

11.6.2.3.1.

Equation

11.54


With

Eqn

11.50

Eqn

11.111

becomes



OPTIMlZ4TION

403

(11.112)

On the analogy ofEqn 11.112with the first of Eqns 11.109, H&‘,h’) is also presented

in Fig.

11.14,

in the

O<c’<l

and

O<h’<2

range.

Similarly,

H;,(chPt,pApt)

is also presented in Fig. 11.15, with the following

constraints:

0.025<c’<l , 0 c h ’ < 2 ( 11.113)

H j k , Hi, ’

coinciding with H i are also plotted. Fig.

11.15

also shows

W’, K

(given

by Eqns 11.104)and f ‘ which is

-

f

-

L X -

C ’

f ‘ - ho/B

’

’1

+

b‘ -

‘

’ (1

+

6 ’ )

in the case of b’=0.975=bApt,hat is phpt=l.It must be pointed out that chp,pt=0.025 nd

any couple of values of b ’ and

p‘

satisfying

(l-b’)p’=0.025

s always an optimum

couple. Since chpt has always taken on a boundary value,

hAPt

is given by

h hpt

=

4 d k

phPt

(1 -

bbpt)

(11.114)

H;,(cApt,hApt) is presented In Fig. 11.16.a, with the following constraints:

O . l < c ’ < l

,

O<h‘<2 (11.115)

W’,

K ‘

and f ‘ are also presented as well as the other quantities, in the case of

b’=0.9=bhpt,hat is phpt=l.

Finally, Hi,(chpt,hhp,t) is presented in Fig. 11.17.a,with the constraints 0.21c‘<l

and O<h‘<2. W’, K’ and f ’ are also presented, as well as the other quantities,

b’=0.8=bApthaving been assumed, that is phpt=l.

In conclusion, t o obtain Ht,(bbpt,p~pt,h~pt),’ is chosen as small a s possible:

it

is

chpt; any couple of values of b’ and p’ atisfying

( l -b ’ )p ’= chp t

an be chosen: it is the

optimum couple bbpt, pbpt; such choices can be made on the basis of specific design

requirements.

bhpt

and pbpt are introduced in Eqn 11.114,thus obtaining h&.

11.6.2.3.2. Figure 11.16.b shows H;,(chpt,hbpt) , that

is H

;m(b~pt,p~pt,h~pt)or

(l-b~pt)p~p,p,=chpt,ith the constraints O.l<c’<l and O.9shf<2,the latter of which is

related to condition 11.105. Considerations similar to those presented above, in par-

ticular i n section 11.6.2.1.2, lead us t o the following conclusions:

To find the minimum of H t , in the presence of condition 11.105, c‘ is chosen as

small a s possible. Letting c’=cbptrt is introduced in Eqn 11.114, thus determining



404


h‘.

If h’ satisfies condition 11.105 it is h&; otherwise, letting h&=a, H i and H)k are

calculated. If H )p H I; , cApt is still equal to the value initially chosen; otherwise, still

letting h&=a, c& is determined from Eqn 11.45, that is

(11.116)

Any couple of values of

b‘

and p ’ satisfying

(l-b’)p’=c&,t

and

hAPt

form the optimum

combination yielding

Hi,.

11.6.2.3.3. H&&t,h&,t) is

presented in Fig.

11.17.b,


0.21c‘<l ,

O<h’<2 and K’21, the last two of which are reduced to O<h’50.9 from conditions

11.107.

Considerations similar

t o

those presented above, in particular in section

11.6.2.1.3, lead us to the following conclusions:

To find the minimum of

H i ,

in the presence of conditions

11.107, c’ is

chosen as

small as possible. Letting

c’=c&,

it is introduced in Eqn

11.114

obtaining h’. If

h’

satisfies conditions

11.107

it is

hAPt;

otherwise

h&=W’ly .

For the set of optimum

values see section

11.6.2.3.2.

11.6.2.3.4. The search for H t , constrained by conditions 11.108, the second o f which

comes from conditions

11.105

and

11.107,

is carried out according to the methods

described in section

11.6.2.3.2

and section

11.6.2.3.3.

11.6.2.4 Ht= f(L,b,p,h). As seen in section 11.5.3.2, from Eqns 11.55 and Eqns 11.18 we

can deduce tha t the efficiency losses are independent from

L ;

nevertheless the val-

ues of

L

cannot be too large o r

too

small, for many practical reasons. As for real

pads, as mentioned in section

3.3.1, L

is chosen

so

that

B < L I 2 B .

cated in section

11.6.2.3.

In conclusion, having chosen

L , H,,

is obtained according t o the methods indi-

EXAMPLE 11.2

The pad in Fig. 11.1 , wi th m axim um dimensions B=O.l m and L=0.15 m, is

directly supplied by a p um p, as in Fig. 1l . l .a . The supply and recess pressure is

assumed to be p,=106 N lm 2 , that is a l i tt le higher th an in hydrodynam ic lubrication.

We want

to

evaluate load capacity W and stiffness

K

i n t h e c o nd i ti on o f m in im u m

total dissipated power Ht , for speed U=0.3, 1 ,3 m l s .

H t , can be calculated following the procedure described

in

section 11.6.2.3.1.

Then i f ho=10-4m, and

po=O.l

Ns lm 2, the parame te r k, given by the second of Eq ns

11.111, for the three values of speed, takes on the values k=0.3, 1,

3.

I f for example



OPTlMlZ4

TlON 405

constraints 11.115 are adopted, the following results are obtained (also directly from

Fig. 11.16.a):

1) k=0.3

h

bp,pt=O.

173, c;pt=O.l and choosing bApt=0.9, pApt=l

Him=0.0693, Hi=Q'=O.O173, Hb30.052

W=0.95,

K'=5.48, f

'=O.

82

bopt=0.09m, hOpt=0.173~1O4, k p t = O . l

Nslm2

Htm=1.04Nmls,

Hp=0.26Nmls, Hf=0.779Nmls, Q=0.260.106m31s

W=1.43.1@N, K=2.47.108 Nlm, f=1.82.10.4

bopt=0.09m,

h0pt=0.316-10-4 , k p t = O . 1 Nslm2

Ht,=6,32 Nmls,

Hp=1.58Nmls, Hf=4.74Nmls, Q=1.58.106m3ls

W=1.43-1@N, K=1.35.108 Nlm, f=3.33.104

bop,=0.09 m, hOpt=0.548.1O4 , kpt=O.lNslm2

Ht,=32.9 Nm s , Hp=8.22 Nmls, Hf324.6 Nm

s,

Q=8.22.106 m31s

W=1.43.1@N,

K=0.78-108Nlm, f=5.77.10-4.

from which

2) k=I

3)

k-3

It must be pointed out that the value of load capacity, identical in the three

cases, is acceptable as well as that of stiffness, even

i f

this decreases from the first

case to the third one. It should also be noted that,

even

in

the third case, flow rate,

though higher than in the second case and thirty times higher than

in

the first

case, is still not much higher than in the hydrodynamic range.

4) Since, for k=0.3, hoptis too small, again letting ho=10-4 m and, for example,

h'20.4, the method presented in section 11.6.2.3.2 can be followed. Furthermore,

considering cipt=O.lr from Eqn 11.112 we obtain H~=O,O255<0,213=Hi. bpt=0.4 is

then introduced into Eqn 11.116 obtaining chpt=0.308.Also letting bbpt=0.9, we obtain

p'=3.08 and

bopt=0.09m, h0pt=0.4.10-4m, kpt=0.308Nslm2

Htm=2.08Nmls, Hp=Hf=l.04Nm ls , &=1.04.106 m3/s

W=1.43.1@ N, K=1.07.108 Nlm, f=2.43.10-4.

However it should be noted that the total power is almost doubled, stiffness is

reduced to less than a ha lf , while viscosity is too high. A s far as viscosity is con-

cerned, we can make up for it by letting bApt=0.7; hen p'=1.03 and we obtain

bOpt=0.07m, hOpt=0.4.1O4 ,

kpt=0.103

Nslm2

Htm=2.08Nmls, Hp=Hf=1.04 mls, &=1.04.1@6m31s

W=1.275.1@N, K=0.956.108 Nlm, f=2.72.10-4.

As already stated in point 6 in example 11.1, the results can be modified without

changing the constraints but by modifying the reference values of the constrained

quantities.



406


11.6.3 Given

load

As

also shown in Fig. 11.4,

H t

has a minimum as a function of each variable.

11.6.3.1

Ht=f(b,h)

11.6.3.1.1.Equation 11.68 can be expressed as

In Fig. 11.18

H;(b',h')

s plotted in the OSb'S1 and OIh'12 range, for k=0.1, 1, 10.

The variation of

Hi

is quite similar around the minimum to tha t in Fig.

11.14,

so the

same considerations

can

be made. That is also proved by the results of the optimiza-

tion of H ; in the range O<b'<l and Oeh'12, as k increases. A s an example, the re-

sults obtained with the constraint 6'20.9 are reported in Fig. 11.19.a.

H i

H t

1 . 0 0 100.60

0.67

66.67

8.33

3 3 .3 3

8.88

2 . w 0.88

w

=

COflSt.

Fig. 11.18

Total power

H;

ersus recess width

b'

and

film

thickness

h', for

certain values

of

speed

factor

k.



OPTlMlzATION

407

a

b

- a -

1.6

1 4

O < t i ~ O . 9 , 0 < h ’ <

2

k

0 0.01 0.025 0.05 0.075 0.1 0.2 0.3 0.4

0.5

h b t

0.0308 0.0487 0.0689 0.0844 0.0975 0.1378 0.1694 0.1949 0.2180

b&

0.3331 0.3333 0.3345 0.3386 0.3447 0.3523 0.4090 0.4844 0.5602 0.6247

0

0 2 4 6

k

8

1

6

Him

HP

5

4

H;k

3

K’

i

1

0

- b-

0

2 ’ 6

k

Fig. 11.19 Values of minimum total power Him, and corresponding optimal values of recess width

bAp,

and of film thickness

h&,

versus speed factor

k .

Values of pumping power

H’,

of friction

power

Hh,

of flow rate Q’, of recess pressure

p;, of

stiffness K’ nd

of

friction coehicientf’ are

also shown.

T A B L E 11.4

Optimal values of film thickness

for b ,,=0.9

(a) and recess width

for

h&=0.6

(b),

versus

speed

factor

(see also

Fig. 11.19).




Hirn is plotted with the corresponding optimum values of bbpt and hhpt.Figure

11.19

also shows H @ and H i , as well asp;,

K', Q '

and

f

which are given by the fol-

lowing formulae:

(11.118)

(11.119)

It should be noted that, for any value of

I t ,

bhPt has always taken the boundary

value

b'=0.9;

therefore

hhpt

is found

to

be the one which satisfies Eqn

11.60,

that is

(11.120)

From the above results and with reference to section 11.6.2.1, we deduce that the

optimization with an assigned load must be carried out as follows: b' is chosen as

large as possible: it is bhpt;it is introduced into Eqn

11.120,

thus determining

h&

The couple

bbpt

and

hhpt

makes it possible

to

evaluate the other quantities.

It should be noted that since

bLpt

is as high as possible, consistently with any

other design constraint,

p ;

is the lowest and that since

hbpt

is small, especially for

small

k ,

K is high.

11.6.3.1.2.

If

hhpt

is too small, condition

11.105

can be introduced. Figure

11.19.b

shows the results of the optimization of

H i ,

with the constraints OIb'I0.9, 0.65h'12.

For k23.79 the diagrams are coincident with those in Fig. 11.19.a.For 3.79<kS2.19

bhpt andp; are still coincident but now hhpt=0.6,

o

Q', H i and K become constant;

Hik and H i rn decrease more slowly. For k 2 . 1 9 bbpt decreases, so p; increases, but

we still have

hbpt=0.6, o

K'

remains constant and

Q',

H i

and

Hirn

don't approach

zero as

k

approaches zero. bApt can be evaluated from Fig. 11.4.b o r from Eqns 11.72,

where k is given by the second of Eqns

11.71

into which

hhpt

has been introduced,

that is

(11.121)

In conclusion, as in the case

of

optimization with an assigned load and satisfy-

ing condition 11.105, bhpt is chosen as large as possible, it is introduced into Eqn

11.120, giving h'. Then, if condition 11.105 is satisfied, such a value of h'

is

an opti-

mum value with the chosen

b'.

As a proof, Eqn 11.60 must be satisfied. If condition



OPTlMlZ.4

TlON 409

11.105 is not satisfied,

hbpt=a

and if condition H h > H ; J s satisfied,

6bpt

is still equal to

the value initially chosen. If that is not true,

hApt

is still assumed to be equal to

a

and is introduced into Eqn 11.121obtaining

k

with which 6kpt is evaluated from Fig.

11.4.b

o r

from Eqns

11.72.

If, on the other hand, the value of 6' thus calculated is

greater than the upper limit selected for

6',

this last has to be chosen as 6&

11.6.3.1.3. For small values of k , the optimization described in section 11.6.3.1.1

yields small values of

h';

consequently, from the first of Eqns 11.28, we obtain high

values of stiffness K . If high values of K' are required even for higher values of

k,

lower limit can be imposed on it, that is

1

K

2

y

,

herefore

h'

-Y

(11.122)

On the other hand, for higher values of k ( k 2 3 ) , with reference to section

11.5.3.3,Eqn 11.117 is reduced to Eqn 11.102.Consequently, the results are similar to

those obtained in section 11.6.2.1.3.

Therefore we deduce that if the further constraints 11.122 on stiffness are pre-

sent, 6' is chosen as large as possible and assumed to be equal to

6hpt;

t is intro-

duced in Eqn

11.120

giving

h',

Then, if constraints

11.122

are satisfied, such

a

value

of h' is h&; otherwise hbpt=lly.

11.6.3.2H,=f(p,h).

Equation

11.68


It must be noted tha t, apart from the coefficient4/3, the above expression o f H , ' ( p ' , h ' )

is

similar

to

that given by the first of Eqns

11.109;

therefore, as regards it s variation,

we can refer qualitatively t o Fig. 11.14 and, as regards H i r n , o Fig. 11.15 and the

following figures.

then determined &om Eqn 11.60,that is

Consequently,

to

obtain

Hirn ,p'

is chosen as small as possible: it is

pAPt;hbpt

is

hbpt

=

@-zp

(11.124)

If this value does not satisfy the constraint 11.105 on minimum film thickness, we

shall assume

h&a

and

(11.125)



41


11.6.3.3 Ht=f(L,h). Equation 11.68 can also be expressed as

long as pApt is replaced by LApt.

The considerations regarding Eqn 11.123 and H i , are also valid in this case, so

11.6.3.4 H,=f(L,jl,h). Equation 11.68 can also be expressed

as

If

we substitute Eqn 11.77 in Eqn 11.127, it becomes

(11.128)

providing p ' is replaced by g'.

The considerations regarding Eqn 11.123 and H i m are also valid in this case,

11.6.3.5 Ht=fi'b,h,/d

11.6.3.5.1. Equation 11.68 can also be expressed as

(11.129)

Figure 11.20.a contains the results of the optimization of H I , in the O<b'<0.9,

O<h'<2,

0.1<p<2 range, for k=O to 6. H i , and the corresponding optimum values

bApt,

hApt

and

phpt

are plotted. Figure 11.20 also shows H @ , H i , p ; and

K'

given by

Eqns 11.118, and

(1 .130)

It must be noted that for any value of

k, bhpt

and

PA pt

have always taken on the

boundary values

0.9

and

0.1,

respectively.

Therefore optimization is carried out as follows:

b'

is chosen as large as possible

while

p '

is chosen as small as possible; they are bhpt and PApt, respectively. The

optimal value of the film thickness is then given by the following equation:



- b -

-

I

t

-

-

-

c

-

41 1

-.a45

0.4

0.35

030

1'

0.2

hT4

0.20

015

:o.lo

0.05

-0

6

"(o.91 40.35

10666

,Q14~<2,or0.l<g<21J

6

k 4

15 45

10 4

35

3.5

H i m

Hb

30 3

5 2:

Q'

<'

p;

20

2

k

15

l.

POP

%

10 1

5

0

3 0

0 ' * k '

Fig. 11.20 Values of minimum total power Hirn, and corresponding optimal values of recess w idth

b&,

of film thickness

h;

, nd of viscosity

&,,. or

of

gAPl=L;,+hp,

versus speed factor k. Values

of

pumping power H;,offriction power ~ h ,f flow rate Q', f recess pressure p;,

of

stiffness K'

and

of

friction coefficient7 are also shown (note that

in

the last case, i.e.

H;,= f (h ; , , , b ~ , , . g L , ) ,

diagrams of

p i

and Q'are valid only when

LAPl=l) .




a

b

k 0 0.01 0.025 0.05 0.075

0.1

0.2 0.3 0.4

0 .5

h b t

0.0098 0.0154 0.0216 0.0267 0.0309 0.0434 0.0533 0.0619 0.0689

b& 0.3331 0.3365 0.3593 0.4235 0.5125 0.5939 0.7751 0.8477 0.8848

0.9000

0

11.6.3.5.2. If

hhpt

is too small, condition 11.105 can be considered. Figure 11.20.b

shows the results of the optimization of

H i ,

with the constraints 056'50.9, 0.4Sh'S2

and 0.3Sp52. For k25.61, bApt takes on the upper boundary value 0.9 while PApt takes

on the lower boundary value 0.3 and H@=3Hi; or 5.6bk23.24

hApt

also takes on the

lower boundary value

0.4;

for

3.24>k>0.487 PApt

increases to the upper boundary

value

2

and H+=Hi.

Therefore, optimization is carried out as follows: b' is chosen

as

large as possi-

ble while p ' is chosen as small as possible; they are bApt and PApt, respectively. They

are introduced into Eqn 11.131, obtaining

h' .

Then, if constraint 11.105 is satisfied,

h'=hhptand, as a check, relation H@=3Himust be satisfied; otherwise, h&=a and

phpt

is calculated from Eqn 11.45, that is

(11.132)

It must be checked that

PApt

is compatible with the range assigned; in particular,

when it is too high, the highest possible value is assumed t o be

PApt

and bApt

is

eval-

uated from either Fig. 11.4.bor Eqns 11.72, in which

k

is given by the second of Eqns

11.71,that is

11.6.3.5.3. f for higher values of

k ,

K is too low, we can introduce the further con-

straints 11.122 on stiffness and go on as in section 11.6.3.1.3,now choosing boundary

values for both

bApt

and

pApt,

that are the uppermost and the lowest, respectively.

11.6.3.6

Ht=flz,b,h).

Equation 11.68can also be expressed as

T A B L E

11 .5

Optimal values of film thickness for

b ,=0.9

(a) and of recess width for

hAPl=0.4

(b),

versus speed factor (see also Fig.

11.20).



OPTlMlZATlON 413

Eqn

11.134

is similar

to

Eqn

11.129, so

it s characteristics can be deduced from all the

considerations regarding Eqn

11.129

to be found in section

11.6.3.5

and from corre-

sponding Fig.

11.20.

11.6.3.7 H,=f(L,b,h,p)

Finally Eqn

11.68

can be expressed as

(11.135)

In this dimensionless form,

Q'

is given by the first of Eqns

11.130, K'

by the

second of Eqns

11.118

and

Substituting Eqn

11.77

in Eqn

11.135,

it becomes

(1 .136)

(11.137)

Eqn

11.137

is similar

to

Eqn

11.129,

so

its characteristics can be deduced from all the

considerations regarding Eqn

11.129

t o be found in section

11.6.3.5

and from corre-

sponding Fig.

11.20.

In particular, Fig.

11.20.a

shows the results of the optimization

of

H i , in the following ranges:

Ol b ' 10 .9 , O c h ' 5 2 , O. lS g' 12 (1 .138)

while Fig.

11.20.b

shows those in the following ranges:

O< b ' 1 0 . 9 , 0 . 4 1 h ' 5 2

,

0. 31 g' 52 (1 .139)

Once the set of optimum values gApt,bLpt,

h i p t , s

determined, any couple

L' ,

i '

satisfying

gApt=L'p'

forms the required combination of four optimum values, to-

gether with

bhpt

and

h&.

Obviously, Eqn

11.77

increases the number of possible

design choices. If the further constraints

11.122

are present, the reader is referred

t o

section

11.6.3.1.3.

EXAMPLE 11.3

The pad of Fig. 11.1,with size constraints

on

width B10.2

m

and length

L12.B,

must be designed

so

as to carry fa oad

W=40000

,

f or

speed

U=O.S,

2.4,

7.2

m

Is .

We

want to evaluate pad stiffness in the condition of minimum total dissipated power.

Ht,

can be calculated following the indications in section 11.6.3.7 bout Eqn 11.137,



41

4 HYDROSTATIC LUB RICATION

and then with the procedure described in section 11.6.3.5 (but replacing p' with g').

So,

i f

B=0.2 m, ho=2.10-4m and &=0.2 Nslm2, parameter

k ,

given by the second of

Eqns 11.135, takes on the values k=0.8, 2.4, 7.2. Then, with constraints 11.138, for

example, the following results are obtained (also directly from Fig. 11.20.a):

1) k=0.8

So letting for example LLpt=l, p~pt=O.l,

bLPt=O.9, hbp;p,=0.0872, gAp;p,=O.l.

Him=O.0979, Hb=0.0245, Hjk=0.0734, Q'=0.0232, pi=1.05

K = 1.5, f'=0.0918

from which,

in

dimensional form,

LoPt=0.2m, bopt=0.18m, hOpt=O.174.104m, kpt=0.02Nslm2

Htm=3.92 Nmls , Hp=0.979 Nm Is, Hf=2.94 Nm

s,

Q=0.930.106 m3 /s

pr=1.05.1@Nlmz, K=68.8.l@ Nlm, f=0.918.104;

LoPt=0.2

m,

bopt=0.18m, h0pt=0.303~10-4, kpt=0.02 Ns I

mz

Htm=20.4Nmls, Hp=5.09Nmls, Hr 15 .3 Nm ls , Q=4.83.106 m3/s

pr=1.05.1@ Nlm2, K=39.7.1@ N Jm, f=1.59.104;

LoPt=0.2m, bopt=0.18m, hOpt=0.523

lO- 4

m, kpt=O.02 Nsl m2

Htm=105.7 Nmls, Hp=26.4Nmls, Hr79.3 Nmls, Q=25.1.106 m31s

pr=l.05.1

06

N l

m2,

K=22.9-108N l m, f=2.75.104;

2) k=2.4

3) k=7.2

It must be pointed out that in this case and even more so

in

the previous ones,

the value of stiffness is very high.

4) Since in the first case, that is for k=0.8, hoptis too small, the lower limit of h' can

be increased. There would be no benefit in increasing the reference value ho because

we would obtain the same results. Then, with constraints 11.139, for example, the

following results are obtained (also directly from Fig. 11.20. b):

k=0.8

b&O.9, h hp,pt=0.4, g bpt=1.22

so

letting LApt=2

(so

as not to have too high viscosity), pApt=0.608

H im=0.389, Hb=O.194, Ha=O.l94, Q'=0.37, p;=0.527

K=2.5, f'=0.243


LoPt=0.4m, bopt=0.18m, hOpt=0.8.104m, kpt=0.122Nslm2

Htm=15.6Nmls, Hp=7.78Nmls, H ~ 7 . 7 8 m/s, Q=14.8.106 m3 /s

pr=0.526.106 N l m2, K=15.1@ N l m, f=2.43.104.

This value of h certainly makes the influence of roughness negligible, and

reduces that of errors on parallelism, while stiffness is still quite high. It should

also be noted that flow rate and supply pressure are not much higher than those



found in hydrodynamic lubrication, while the friction coefficient is at least one

hundred times lower.

11.7 REAL

PADS

11.7.1 Rectangular

pad

We refer to the pad with continuous film and characterized by the relation

(B-b)/2=(L-l)/2=a,

hown in

Fig.6.25,

whose outer corner radius is

r,=O,

as i t gener-

ally is in actual practice. For the sake

of

simplicity and considering that often the

inner radius is also small,

ri=O

is assumed.

11.7.1.1 Given flow rate.

In this case, referring to sections

5.2

and

5.3.5:

3 1 2 B L - ( B + L ) ( B - b )

K = 3 h

W

(L B

+ 2

b)(B

+

b)

'

= 3

/

Q Q (B2-b2)

(1 .140)

(

11.141)

If the pad is in motion, the fluid is subjected to inertia forces; nevertheless,

since the variation of average pressure is negligible (Fig.

6.25),

that

is

also true for

W;

as regards

flow

rate, its decrease on one side is partially compensated by its

increase on the other. The fluid is also subjected to recirculation in the recess but

the phenomenon is negligible at fairly low speed. Therefore, from Eqns

5.106

and

5.108

we obtain

(1 .142)

F =

p

U

Total power is

2'(L

+

b)(B b)

H t = 3 P Q 2 P ~ - ~ + 2 b + C L U

( L

+

b)(B b)

(

11.143)B - b

In dimensionless form,

as

a function of

L'

and

b',

Eqn

11.143

becomes

where k is given by Eqn

11.43. As

a function of

L', Hi

is minimum for

(11.144)

Lbpt

=

1

-

2

b'

+

d ( 1 -2 b')2

-

4 b'(1-b') 1

+

3/12 ( 1 .145)



41 6


but obviouslyLhpt21-b’must be satisfied. Lbpt,given by Eqn 11.145

as

a hnction of

b’,

approaches zero as b’ approaches unity.

A s a function of h’,

H i

can also be expressed as in Eqn

11.47

(with obviously a

different reference power) and it approaches zero as h’ approaches infinity. k is

given by the second of Eqns

11.47

multiplied by

(L+ b)(L - B + 2 b)

LZ

(11.146)

As a

function of

p’,

H i can also be expressed as in Eqn

11.48

(with obviously a

different reference power) and it approaches zero as

p’

approaches zero.

k

is given

by the second of Eqns

11.46

multiplied by factor

11.146.

As

a function of all four variables,

HI

can also be expressed a s

(11.147)

where

k

is still given by the second of Eqns

11.91.

9

Its optimization, that is the determination of the values of the independent vari-

ables that make it minimum, can be carried out as in section 11.6.1.5, taking into

account the changes described in this paragraph as regards the above-mentioned

variables. Therefore, the maximum value of

b’

and h’ must be chosen as bhpt and

hhpt,

while the minimum value of p’ must be chosen as

p&. Lhpt

is given by Eqn

11.145

(where

b’=bhpt

and k is given by Eqn

11.43

with h=hhptho)f it is inside its

boundaries, otherwise it takes on the boundary value nearest

t o

it. Other useful

information concerning the optimization of

H ,

can be found in section

11.6.1.

As regards efficiency losses, as a function of L‘ and b‘, Eqns 11.2 yield

L ’ -1 +2 b ’ (L

+

‘b’)(L’-1

+

2 b’)

‘W” 2L’- (L’+ 1)(1+6‘)’ k2 2L’- (L’+ 1)(1+b‘) ’

rK=

rw

As function of L’ they are minimum for

1 1

LhPt= 1.1+ 2-

(11.148)

which is an approximate relation valid only in the 0.51b’Sl and Osk<lO range. A s a

function of

6 ’

their variations are more complex. We can say, as a rough guide, that

they decrease a s b’ approaches unity only for small values of k and high values of

L’.

A s regards the variations of

rb

and

r k

as a function of h’ and

p’,

what has been

stated in section

11.5.3.1

s still valid.



OPTIMIZATION

11.7.1.2 Giuen pressu re. In this case:

41

7

Moreover,

In dimensionless form, as a function of L’ and

b’,

Eqn 11.149 becomes

(1 .149)

(1

.150)

where k is given by Eqn 11.56.HI decreases with L’, while generally it has a mini-

mum when it is considered to be a function of

b’.

As a function of h‘, HI can also be

expressed as in Eqn 11.58 but k must be multiplied by

(11.151)

A s a function of p ’ , H i can also be expressed as in Eqn 11.61, but

k

must be mul-

tiplied by factor 11.151. As a function of h’ and p’ , H i can finally be expressed

as

in

the

first

of Eqns 11.109, where k is given by the second of Eqns 11.109 multiplied by

factor 11.151.

Its optimization can then be carried out as in section 11.6.2.2.1. Therefore,

p’ is

chosen a s high as possible: it is PApt; it is introduced into Eqn 11.110 (together with

the value of k obtained from the second of Eqns 11.109 and multiplied by factor

11.151), thus determining

hAPt.

Useful informations can generally be found in sec-

tion 11.6.2.

As regards efficiency losses Tiy(L’) and r i (L’) , they decrease as L‘ increases,

while

Th(b’ )

nd Ti((b’) generally have a minimum. A s for T i y ( h ’ )and r k ( h ‘ ) ,

what has been stated in section 11.5.3.2 is still valid.

11.7.1.3 Giuen load. In this case:

W L(B+ b)

2 h 3

W (L-B+2b)(B+b)

Q = 3 F m BL

-

(L + B)(B - 6)

r =

2BL

-

(L + B)(B b) ’

Moreover

In dimensionless form, as a function of L’ and b’, Eqn 11.152 becomes



41


(11.153)

where k=(pUB3)/(h2W).HI(L') always has a minimum, but for much higher values

than those given by Eqn 11.70: for instance, for b'=0.9, the optimal value of

L'

goes

from 6.8 to 1.05 when k goes from

1 o

10.H ; ( b ' ) always has a minimum as well.

Such a value can be approximately evaluated, for L '=l and k>l, from Eqns 11.72 and

increases with L'. As a function of h',

H i

can also be expressed as in the first of

Eqns 11.73 but k now is

(1 .154)

A s a function of p ' ,

HI

can also be expressed as in the first of Eqns 11.75, but

k

must be multiplied by the expression between square brackets in Eqn 11.154.

As

a function of L', b', h' and

p ' , HI

can be expressed as

(11.155)

Po

B4

where

k

is given by the second

of

Eqns 11.135.

Its optimization, for a given value of L ' , can be carried out as in section

11.6.3.5.1, taking into account the changes described in this section. Therefore,

b'

is

chosen as large as possible while p' is chosen as small

as

possible; they are b&,t and

php t , respectively.The optimal value of film thickness follows from Eqn 11.60, that is

(11.156)

When the above value is too small we can

go

on as in section 11.6.3.5.2. Namely, we

assume the minimum possible value to be hApt and obtain phPt from Eqn 11.45, that

is

(1 .157)

If phPtproves

to

be too large it will be necessary to assume the highest possible value

to be pApt and to reduce b';

bhpt

may be approximately evaluated from Eqn 11.72, in

which

k

is given by Eqn 11.133.

As

regards efficiency losses, what has been stated in section 11.5.3.3 is still

valid.



OPTlMZ4TlON 419

EXAMPLE 11.4

Let us consider a pad with width B=O.l m, which must carry a load of 45000 N at

a speed of 0.2 mls. Recess pressure must be lower than 4-106N lm 2 and the friction

coefficient must be lower than 10-4; the lubricant viscosity is

0.03

Nslm2. They are

the same data as in example 5.12 but now

ri=O

is also assumed. Letting ho=0.75.10-4

m and p0=o.03 Nslm2, the speed parameter, given by Eqn 11.135,

is

k=0.0237. For

this value, from Eqn 11.156, we would get

a

ualue of

h

which

is

too small. So let us

impose a lower limit on h and consequently on h': for instance h'10.4. If we select

L'=2 and b'=0.9, Eqn 11.157gives an excessively high value of optimal viscosity; then

we may assume pApt=l and obtain an approximate value of bApt from Eqn 11.72 (after

having calculated k by means of Eqn ll.133), that is bApt=0.36. Then we obtain, in

dimensional form,

and the last three values are coincident with those in example 5.12. Moreover,

bOpt=0.036m, hOpt=0.3O4 m, L=0.2 m, p=0.03 Ns Im2

p,=4.33.1@ Nlm2, &=3.48.106m 3 /s , Hp=15.1Nmls,

Hf=O.6Nm/s, Htm=15.7Nmls, f=0.67.104.

Recess pressure is a little higher than the maximum value given above,

whereas

Q

and Htm are a little lower than the values obtained in example 12.5.

In order to reduce recess pressure one may increase the recess width

or,

more

efficiently, the pad length. For instance, stating L'=2.5, the following values are

obtained: LoPt=0.25m, popt=O.024 s lm2, which

is

a

very

common value and

bOpt=0.037m, p,.=3.21,1@ Nlm2, &=3.44.106 m3/s,

Hp=ll.lNm ls , HrO.7 Nmls, Htm=11.8 Nmls, f=0.80.10-4.

Now pr

is

lower than that in example 5.12.

11.7.2 Other types of pads

The methods presented above are a useful reference for the optimization of other

types of pads, such as, in particular:

cylindrical pads with rectangular recesses (Fig. 5.30)

multipad bearings made of several cylindrical pads, separated by grooves

(Fig. 7.32.a)

multirecess bearings, without grooves (Fig. 8.l .a), in which more complex

phenomena are involved.

The optimization methods described above have shown that often bearings with

wide recesses (high values

of b')

are more convenient. This confirms what is t o be

found in the literature regarding multipad and multirecess bearings. In ref.

11.2

an average value

of

about 0.78

is

proposed while in ref. 8.17 the values suggested by

other authors range from

0.45 t o

0.85.



420


11.7.3

Circular

pad

The reader is referred t o the bearing in Fig. 5.1.a. If the bearing rotates, the

lubricant is subjected t o inertia forces (section 5.3.61, with a pressure peak at the

inner edge of the recess (Fig. 5.8). This causes a loss of flow rate, slightly hindered

by the inlet (section 5.3.4) and turbulence effects (section 5.3.51,but largely compen-

sated by the increase of load capacity. That justifies the assumption of negligible

inertia forces. The friction torque in the recess is also disregarded.

11.7.3.1 Given f l o w rate. Assuming a linear pressure drop in the film (r2-rl)/r2 in

place of the logarithmic drop ln (r2/r l), the pumping power of the bearing can be

written as

(11.158)

which yields lower values with errors decreasing from 30 t o

3%

as r'=r1/r2 in-

creases from

0.55

to

0.95.

The friction power H f can approximately be written as

H -/ 2 E

f

- h Up

(2 r2)

r2

rl) ,

with Up = 1.7 f2r2

which yields values higher than the exact ones with an error of 30 for r'=0.55 but

lower by 3 for r'=0.95.Thus to ta l power is

which is similar

t o

Eqn 11.41.

A s regards its optimization as a function of one or more of the three variables rl,

h

and

p ,

with

r2

as

the reference quantity, what has been stated about the

H t

of

the

indefinite pad, as a function of

6 , h

and p , with

B

as the reference quantity, can be

applied, approximately, replacing L with (rr12)r2 (given datum) and U with

U p .

n

particular, the remarks made in section

11.6.1.5

are still valid, but now L' can no

longer be considered

t o

be an independent variable, since we still have L'=x12.

11.7.3.2

Given pressure. In this case



OPTIMIZA

TlON 421

which now yields higher values than the exact ones, with the same error as in Eqn

11.158.Total power is then

5c

-

H =--h3

r2 + , U 2 'r2)(r2

-

rl) ,

with Up

= 1.7

R r z

t

3 p p F G h

P 2

(11.159)

which is similar to Eqn 11.54.

The remarks regarding the previous case can be extended to this one.

11.7.3.3Given load. In this case

with the same error as in Eqn 11.158. Moreover the minimum of H p now occurs for

rhpt=113,which is much lower than the exact value 0.529. H f can be written as

H f = f Up2 (2n r2)(rz rl) ,

with U p = 0.85 Rrz

Total power then becomes

(11.160)

which is similar to Eqn 11.68.

The remarks regarding the previous cases can be extended t o

this

one, but now

L must be replaced by

2xr2

(that is we have L ' = ~ K )nd U by

Up=0.85Rr2.

Moreover,

after the evaluation of

bhpt

for the equivalent infinite pad, the corresponding value of

rApt can be calculated from the following equations:

159

rbpt = bhpt

,

for k > 1

rhpt = 100 + 59

k0.04

bhpt

,

for r;

(11.161)

which, approximately, take into account the above-mentioned error corresponding

to the minimum of

H p .

EXAMPLE 11.5

Let us consider a circular pad with diameter D=2rz=0.1

m,

which must carry a

load W=50000 at an angular speed R=25n rad

I s

(same data as in example

5.2).

Letting ho=O.

75.10-4

m, 1.(~=0.009s

Jmz,

L=2mz and U=Up=0.85Qrz,

the

speed pa-



422


rameter, given by the second of Eqns 11.129, is k=0.0839. For

it,

from Fig. 11.20.a

or

from Table 11.5.a we would get a value of h which is too small.

So,

imposing a lower

limit to

h

and consequently to h’

(for

example h’20,4), from Fig. 11.20.b

or

by inter-

polation from Table 11.5.b we obtain bApt=0.540; hen from the first of Eqns 11.161

from which, in dimensional form, rlopt=0.028 m; hoPt=0.3.104 m, which is much

lower than the value in example 5.2; popt=O.018Nslm2, equal to the value in example

5.2. Then, utilizing of course exact equations, we find

rhpt=0.560, hhp,pt=0.4, C(hpt=2,

pr=10.8.106 Nlm2, Q=14.6.106m31s, Hp=157Nmls,

Hr32.8 Nml s, Ht,=190 Nml s, f=2.1.10-4.

It should be noted that, as r’ decreases to r’=0.529, Htm remains virtually constant.

EXAMPLE 11.6

Let us again consider a circular pad with diameter D=2r2=0.1 m, but which

must carry a load W=lOOOO N at an angular speed l2=300nradls (the same data as in

example 5.4). Letting ho=1.5.104

m,

p,,=0,05 Nslm2, L=2m2 and U=Up=0.85f2r2, he

speed parameter, given by the second of Eqns 11.129, is k=6.99. For it, with the

method described in section 11.6.3.5.1 with the constraints ~(’20.3 nd b’10.9, we

obtain

Thus

the two latter values are both lower than the values in example 5.4. Moreover,

pr=1.41.106 Nlm2, the same value as in example 5.4, Q=141.10-6m31s, Hp=199Nmls,

Hf=671 Nmls , Htm=870 N m l s and f=1.5.103. The power ratio is HfIHp=3.34, not

much higher than the value 3 indicated in section 11.6.3.5.2. This proves that the

optimization method described is sufficiently valid. Obviously, more approximated

values could be obtained by introducing the inertia corrective factors, as in example

5.4. In any case the values of powers and of the friction coefficient are about half

those in that example.

rApt=bhpt=0.9,

h

hpt=0.4#7,

&=0.3.

rlopt=0.045m; hopt=0.67.10-4 , hpt=O.015Nslm2;

11.7.4 Annular pad

The reader is referred to the bearing in Fig.

5.13.

If the bearing rotates, the lu-

bricant

is

subjected to inertia forces, with pressure variations also occurring inside

the recess (Fig. 5.18).They have little effect on average pressure, on flow rate, espe-

cially

for

common values

of

the inner radius (that are never small) (Fig.

5-17],

nd

on load capacity (section

5.4.5).

Therefore the inertia forces can be assumed to be

negligible. The friction torque in the recess can also be disregarded.



11.7.4.1Given

flow

rate.

The relation

(r2-rl)=(r4-r3)=a

is assumed. The closer rl is to

r4,the closer th e former relation is to

r l /

~ 2 = r 3 /4 corresponding t o equal flow rates

through the films. Linear pressure drops in the film ( r 2 - r 1 ) / r 2nd

(r4-r3)/r4

re also

assumed in place of the logarithmic drops 1n(r2/rl)and ln(r4/r3).Finally

1t(r4+r2)

is

replaced, roughly, by ~ r 4 + r l ) .he pumping power of the bearing can then be writ-

ten as

(11.162)

which yields lower values with errors decreasing from 16% to

3%

as r ’ = r l / r 4 n-

creases from

0.5

to 0.9. In Eqn 11.162

~(r4+rl )

nd 2(r4-r3)correspond to L and (B-b)

in Eqn 11.4. The friction power

H f

can also be written approximately as

which yields values lower than the exact ones with an error decreasing from 8%

t o

0.2%

as

r’

increases from 0.5 t o

0.9. As

regards approximations, it should be noted

that as r l goes

to

infinity the bearing turns into an infinite pad. Then total power is

which is quite similar

t o

Eqn 11.41.

As regards its optimization as a function of one or more of the three variables r3

(o r r2 ) ,h and p , with B=(r4-r1) as a reference quantity, what has been stated con-

cerning the total power of the infinite pad as a function of b ,

h

and p , with

B

as a

reference quantity, can be applied here too, replacing L with x(r4+rl) (given datum)

and

U

with

U p .

As far as

r3,

o r

r2

are concerned, they are correlated in an elemen-

tary manner to 6’; we have for example

r,

+

rl

+

(r4

-

rl)b’

2

3

=

11.7.4.2

Given

pressure.

In this case

which now yields higher values than the exact ones, with the same error as in Eqn

11.162.

Total power is then

(11.163)



4 2 4


previous case (i.e. section

11.7.4.1)

can be extended

t o this

one.

Equation

11.163

is similar to Eqn

11.54

and the considerations regarding the

11.7.4.3 Given load.

In this case

4 h3 W2

H

=--

P 3

p

n

(r4+ r l) r4 r3) [(r4 rl )

+ r3 - r2)12

with the same error as

in

Eqn

11.162. In

i t

n(r4+rl)/2, (r4-r3), r4-rl)

and

( r3 - r~ )

correspond to L , (B-b),B and b in Eqn 11.25, respectively. The friction power can be

written as

which yields values lower than the exact ones, with an error decreasing from

11%

to

3%

as

rl

increases from

0.5

to

0.9,

which is virtually equal

to

that of H p , especially

for high values

of rl.

Total power then becomes

Equation

11.164

is similar

to

Eqn

11.68

and what has been stated in the previous

cases can be extended to this one, but now L must be replaced by

n(r4+r1)/2

nd U by

Up=1.390(r4+r1)/2.

EXAMPLE 11.7

Let us consider an annular pad with an outer radius r4=0.05 m and an inner

radius rl=0.03m (r'=0.6), which must carry a load W=20000 N at an angular speed

0 = 4 n r a d / s the same data as in example 5.6). Letting ho=0.75.10-4

m,

h=0.05

Ns 1m2, L=n(r4+r3 2, B=r4-rl and U=Up=1.390(r4+rJ 2, the speed parameter, given

by Eqn 11.129,

is

k=O.0156. For it, from Fig. 11.20.a

or

from Table 11.5.a, we would

get a value of h

which is

too

small. So imposing a lower limit to

h

and consequently

to h',

for

example that in Fig. 11.20.b, from the same figure or from Table 11.5.b, we

obtain

bhpt=O.345,hApt=0.4 and pApt=2, from which r3,,=0.0435 m and rZoPt=O.0366m;

hopt=0.3.lO-4

m,

lower than the value in example 5.6;

popt=O.l

Nslm2, equal to the

value

in

that example.

Moreover, p,.=5.92.1O6 Nlm2, Q=10.2.10-6m3/s, Hp=60.3Nmls , H ~ 3 . 0 m ls ,

Ht,=63.3 N m / s and f=3.0.10-4. Furthermore, letting p=900 Kglm 3 and c=1900

JIKgOC, AT=3.6

"c.



OPTlMlZATlON

425

Flow rate, pumping and total power are much lower than those in example

5.6,

but this is due to the lower value of h, obtainable with an accurate construction and

assembling of the bearing.

EXAMPLE 11.8

Let us consider an annular pad with an outer radius r4=0.03 m and an inner

radius r1=0.02 m (r'=2/3), which must carry a load W=2000N at an angular speed

Q=GOOzradjs (the same data as in example 5.10). Letting ho=0.75.10-4 m and

pt,=0.005Nsjm2, the speed parameter, given by Eqn 11,129 with the above-mentioned

substitutions of L and U, s k=0.229 and

for

this, what has been stated in example

5.10 can be applied here

too.

Thus, we obtain

bApt=0.8, hApt=0.4, pipt=2, from which r30pt=0.029m, then a'=O.l, r2,t=0.021 m;

hOpt=0.3.10-4m, lower than the value in example 5.10; popt=O.O1 Nslm2, the same

value as in example

5.10.

Moreover, pr=1.41.106 Nlm2, &=100.106 mais, Hp=141 Nmls, Hf=255 Nmls ,

Ht,=397 N m l s and f=2.71-10-3. Furthermore, letting p=870 Qlm3 and c=1930

JIKgV, AT=2.4

"c.

Flow rate, pumping and total power are definitely lower than those in example

5.10. It should be borne in mind that the above results are based on an approximate

equation ( tha t is Eqn 11.164); indeed it could be proved that for a'=0.1 and

hOpt=0.3.10-4m the optimal viscosity is rather lower, being hp,=0.0074 N s lm2; never-

theless, the total power which tallies with this new value of viscosity is only slightly

lower (Ht,=380 Nmls).

11.8 COMPENSATED SUPPLY

Let us consider the infinitely long pad in Fig. l l . l . c , with width B , supplied by a

pump through a compensating element (Fig. 1 l. l .b) across which pressure drops

from the supply value

ps,

kept constant by a relief valve, to that in the recess

pr

The

performance of a portion of length

L

i s

studied.

11.8.1 Capillary tubes

If the compensating element is

a

capillary tube with diameter

d

and length

L,

equating expression

4.66 to

expression 4.48 yields

(11.165)

where



426

HYDROSTATIC

LUERICATiON

R

= - p -

28

1

r

n d4

is the capillary tube hydraulic resistance and

is the hydraulic resistance

of

the pad. From Eqn

153

we obtain

is the characteristic ratio

of

all compensated bearings. We also obtain

(4.66

rep.)

(6.11rep.)

(6.19 rep.)

11.8.2 Steady pad

11.8.2.1Given pressure. Substitutingpp, forp, in Eqn 4.47 yields

With simple operations (see also section 6.31, we also obtain

(

11.167)

Ps

K = 5 (1 - B )

B

L ( B

+

b )

1 1 L

3 P

=

- -P

pSh3

~ - 6

(11.168)

H =--1 L (11.169)

It should be noted that

Hp =Hpc Hpb

(11.170)

where Hpc=Q(ps-pr) is the power dissipated in the capillary tube, which can be wri t -

ten as

(11.171)

and Hpb=&pr s the power dissipated in the bearing, which can be written as



OPTlMlZ4TlON

427

(11.172)

Comparison of the foregoing equations, for the typical value p = 0 . 5 of the pres-

sure ratio, with the analogous equations in section

11.5.1.2

(direct supply) immedi-

ately shows that, when supply pressure is the same, load capacity is halved, as well

as recess pressure, flow rate and pumping power, and stiffness is reduced to one

fourth. When recess pressure (that is load capacity) is the same, supply pressure

and pumping power are double that found in the case of direct supply, whereas

stiffness is halved.

In

our

investigation into the performance of the pad as p vanes because of the

variation in the dimensions of the capillary tube, diameter

d

an do r length

1 ,

we

assume that the dimensions of the pad are given, as well as film thickness

h

and

viscosity p; he above equations may hence be written in a dimensionless form as

follows

The ratio of Hic and Hib may also be of interest:

(11.173)

(11.174)

(11.175)

which coincides with Eqn

6.19.

The results are plotted in Fig. 11.21.a. From it, the characteristics of a bearing

with assigned dimensions, film thickness (equal

to

the reference value) and supply

pressure can be determined as the capillary tube geometry varies. It must be

pointed out that K

s

maximum for

p=0.5

(as clearly follows from

dKldb=O)

and that

for p=0.3 and 0.7 it still reaches 84%of its maximum value. For this reason it is

commonly recommended to adopt p= 0 .5 o r values close to it. R , / R

is

also plotted in

Fig.

11.21.a.

From that curve, for given bearing dimensions and h , the dimensions

of the capillary tube can be determined.

Eqn 6.19 can also be written in the form

(11.176)

Then, in the investigation into the performance

of

the pad as

p

varies, now assum-

ing the dimensions of the pad and capillary tube are constant (or, better, the ratio



428

1.8

-

1.6

-

H;

1.4 -

H;,F'

HpQ

H'

p"

HPb

0.8

HP,

R r o / R r

0.6

-

w

'

K '

1.2-

8 ,

1 4

0.4

-

0.2 -


- a -

d > l +

const.

14 1

\ \1.

4

2

0

0

0.2 0.4

0.6 0.8

1

- b -

0

02

0.4

p

0.6

0.8

1

Fig. 11.21 Load W ' , stiffness

K',

total power Hi for speed parameter

k = l )

and other quantities

versus pressure ratioJ, which varies with:

(a)

capillary dimension

d

and/or I

(b)

film thickness h.

d 4 / l

is constant) as

h

vanes,

h

is substituted

by

the

following

equation obtained fiom

Eqn

11.176

(11.177)

1 1 1 L 1 1 1 L

Hpc

=

3

(1

- P I 2 empf ,

H pb =3 p

1

- P)P

c

and in dimensionless form



429

while W' and

H i p

are identical in form to the previous ones.

The above-mentioned quantities are plotted in Fig. 11.21.b. From it, the charac-

teristics of a bearing with given dimensions, supplied at a given pressure through a

capillary tube with given dimensions, can be determined as the film (reference)

thickness varies. It must be pointed out that K is maximum for 8 = 2 / 3 (also derived

from dKldp=O) and th at for f i =0 .5 and 0.815 it is still 90% of

its

maximum value.

Therefore values of

P2O.5

up to 0.8 seem to be convenient. This is also due to the fact

that

W'

increases linearly with /3 while H P decreases linearly. Figure 11.21.b also

shows h'

from which it is possible

t o

determine h, for given bearing dimensions, capillary

tube dimensions and pressure ratio.

The performance of the bearing as a function of

L ' , b ' ,

h' and p' for any given

value of@ s still represented by Fig. 11.3.a, b, c, d, as transpires from Eqn 11.166

t o

Eqn 11.169.

11.8.2.2 Given load. In this case, which is the most frequently encountered in de-

sign, as stated earlier, p,., Q and HPb are directly given by Eqn 11.22, Eqn 11.24 and

Eqn 11.25. Furthermore

(11.178)

(11.179)

(11.180)

(11.181)

In order to investigate the performance of the pad as

varies as a consequence

of changing the dimensions of the capillary tube, as done in section 11.8.2.1, the fol-

lowing dimensionless equations can be written



430

HYDROSTATIC

1

BRICATION

(11.182)

(11.183)

while H;,,

is

still identical in form

t o

that obtained in section 11.8.2.1.

The results are presented in Fig. 11.22.a. It must be noted that K decreases

linearly as p increases; however H i decreases as well (because H i c decreases) and

this is quite positive. Therefore, on the whole, it can be stated that intermediate

values of 8, that is /3=0.5

or

values close

to

it, are t o be adopted.

A s

for the investigation into the performance

of

the pad as

/3

varies, this time

because

h

varies, Eqn 11.177 is substituted in Eqns 11.178, 11.179, 11.180, 11.181, 11.24

and 11.25, as done in section 11.8.2.1,obtaining

K=3(1-p)2/3pu3Cv3 w

1.8

1.6

1.4

1.2

K'

1

18

1E

H't

14

H i . F;f'

1 2

H;,

10

-b-

0

02

0.4 0.6 0.8 1

B

18

1.6

I

.4

1.2

K '

1

3.8

3.6

3.4

3.2

3

Fig.

11.22

Supply pressure

p i ,

stiffness

K',

total power

H i

(for speed parameter k = l ) and other

quantities versus pressure

ratioJ,

which varies with: (a) capillary dimension d and/or

I (b)

with film

thickness

h.



OPTlMlZATlON

431

In dimensionless form p i is still given by Eqn 11.182 while

whereas

HLp

is still identical in form to that in Eqn 11.175.

These quantities are plotted in Fig. 11.22.b. It should be noted that K is already

high when

8=0.5

nd increases further but slowly as p decreases t o /3,,=0.261; but

as decreases

HI; , Hbb

and

H i c

increase rapidly and this is quite unfavourable.

Therefore, on the whole, it can be stated that intermediate values of p , that is

p=0.5

o r values close

t o

it, are

t o

be adopted.

The performance of the bearing as a function of L', b', h' and p' is clearly still

represented in Fig. 11.4.a, b, c, d.

11.8.3 Moving

pad

11.8.3.1 Frict ion. The friction force and power are given (see section 11.5.2) by Eqn

4.49 and Eqn 11.30. They do not vary withp (Fig. 11.21.a and Fig. 11.22.a) i f i t varies

with the capillary dimensions. On the contrary they vary with /3 if

it

varies with

h

according t o Eqn 11.177. Substituting then

h

with the expression obtained from Eqn

11.177, the following dimensionless relations are obtained.

Hi.

and F j increase with j and very rapidly, too (Fig. 11.21.b and 11.22.b).

In Fig. 11.3.a, b, c, d

F i

and

H i

are plotted against

L ' , b ' ,

h' and

p '

for a given

supply pressure

p s ,

while in Fig. 11.4.a, b, c, d they are plotted for a given load W.

11.8.3.2. Friction coefficient.If the supply pressure p s is assigned, the friction coef-

ficient can be obtained by substitutingp,=/hp, in Eqn 11.37, thus

f=2-p--- 1 1 B - b

Ps

h B B + b



432


I fp varies with the capillary dimensions, f can be expressed as the following hnc-

tion of p

2

f ' = a

f is plotted in Fig. 11.21.a. If@varies with

h

according to Eqn 11.177, f ' can be ex-

pressed as

f ' is now plotted in Fig. 11.21.b. It is minimum forfiOpt=2/3.

If the load is assigned, f is given by Eqn 11.39. I t does not vary with p ifp vanes

with the capillary dimensions (Fig. 11.22.a). If j3 varies with h according to Eqn

11.177,f' can be written as

f ' is plotted in Fig. 11.22.b.

In Fig. 11.4.a, b, c and d,

f '

is plotted against

L',

b' , h ' and

p'.

11.8.4

11.8.4.1 Given pressure. The total power dissipated in the capillary tube and in the

moving pad is obtained by adding Eqn 11.169 to Eqn 11.30, that is

Dissipated power and ef f ic iency losses

(1 .184)

1

L

Ht

= 3 ~ p h3~ - bp U z

L

(B

-

b )

If B vanes with the dimensions of the capillary tube, with

h

then equal to a con-

stant value, Eqn 11.184 can be written, in dimensionless form, as the following func-

tion of

8:

(11.185)

H i is plotted in Fig. 11.21.a, for k=l.

I fp varies with h according to Eqn 11.176,

H i

can be expressed as the following

function of p:



OPTIMIZATION

433

Hi is plotted in Fig. 11.21.b, for k=1.

As

a function of L',

b',

p' and c'=(1-6')p',

Hi

is again given by Eqns 11.55, 11.56,

11.61 and 11.63, respectively, p r having been replaced by p s in the expressions of k;

they have a lso been multiplied by l/$. For k= l, the curves of

H i

are presented in

Fig. 11.3.a, b and d.

The efficiency losses rk and

are still constant when they are considered to be

a fbnction of L'; they are still given by Eqn 11.65 and by the corresponding Fig. 11.6.a

and b, as a function of

6'.

Finally, as a fbnction of

h', r i

s given by Eqn 11.67 and is


11.8.4.2

Given

oad.

The total dissipated power is

4 1 1

W2

2-1

L

(B 6 ) (11.187)

Ht = 3 i B h 3 L ( B

6 ) ( @

-

62)

h

Ifp varies with the dimensions of the capillary tube, with

h

then equal to a con-

stant value, Eqn 11.187 can be written, in dimensionless form, a s the following hnc-

tion ofp:

Hi

is plotted in Fig. 11.22.a, for k= l .

I f p varies with

h

according to Eqn 11.176,Hi can be expressed

as

the following

function of 8:

(11.189)

Hi is plotted in Fig. 11.22.b, for k = l . It is minimum for jIoPt=0.795. Since

Popt

As a function of L',

b',

h',

p',g'=L'p' and q'=L'(l-b')p',

Hi

is again given by Eqn

11.69, Eqn 11.71, Eqn 11.73, Eqn 11.75, Eqn 11.78 and Eqn 11.81,respectively, the ex-

pressions of k having been multiplied by $. For k = l , the curves of Hi are presented

in Fig. 11.4.a, b and d.

decreases slowly as k increases, small values of p should not be adopted

$ 2 0 . 5 ) .

The efficiency loss

r i

s given, as a function of h' , by Eqn 11.83 and is plotted in

Fig. 11.8.




11.9 OPTIMIZATION

What

has been stated in section 11.6 still holds in general, but now the hydraulic

resistance

of

the restrictor or, better, the pressure ratio should also be considered

as an independent variable. On the other hand, it is

generally preferable to select

8

directly and treat it as a constant in the optimization process, which leads to equa-

tions that are formally identical

t o

those obtained in the case of direct supply.

As

far as the selection of

8

is concerned, the numerous remarks made in sec-

tion

6.3,

6.4.2 and 11.8 may be briefly summarized as follows:

-

from the point of view of efficiency, it is best

t o

select a high value for the pressure

ratio (perhaps

8=0.7);

-

a low value of

8

(perhaps

8=0.3)

s needed when the bearing has to sustain loads

which may considerably exceed the design value

o r

when a very high degree of

stiffness is required for any given load;

-

a value near p=0.5 is often a good compromise.

11.9.1

Given pressure

11.9.1.1

Ht=f(b,h)

11.9.1.1.1. Equation 11.184 can also be expressed as

It i s easy

to

see that the above equations are identical to those obtained in the

case of direct supply (Eqns 11.102), except that

.lisps,

is used in place of recess pres-

sure

p r .

In Fig. 11.14

Hi(b’,h’)

s plotted in the

O<b’<l

and

O<h’<2

range, for k=O.l, 1, 10.

Figure 11.16.a shows the results of the optimization of

H i ,

for OIk19, with the

constraints Osb‘s0.9 and

Och’s2 . H Im

is plotted with the corresponding optimum

values of

bbPt and h ip t; a number

of

values of hApt are also given in Table 11.2.a.

Figure 11.16 also shows

H i , Q ’

which coincides with

H i , H@

and

(11.191)

(11.192)



Since the foregoing nondimensional equations are identical to those obtained in

section 11.6.2.1.1, most of the remarks made there still retain their validity for the

compensated supply system, too.

11.9.1.1.2, In practice the condition Osh'

is

replaced by condition 11.105, for the

general reasons mentioned in section 11.6.2.1.2.

The results of the optimization of Hi are presented in Fig, 11.16.b, with the con-

straints OIb'10.9 and 0.91h'<2: we refer the reader to section 11.6.2.1.2 for the rele-

vant remarks.

11.9.1.1.3. Figure 11.17.b shows the influence of adding the further constraints

11.107, namely a minimum value of stiffness, on the optimal values of b' and h' (see

also sections 11.6.2.1.3 and 11.6.2.1.4).

11.9.1.2 Ht=f(uh) Equation 11.184 can also be expressed as

On the analogy of the first

of

Eqns 11.193 with the first of Eqns 11.190 and the

first of Eqns 11.102, Fig. 11.14 also presents Hi(p',h'), in the O<p'<1 and OIh'g2

range.

A s

for H;(p',h') the remarks concerning Hi(b',h') still hold good. Similarly

in Fig. 11.15

Hirn, APt

and hAPt are also plotted, with the constraints 0.0251p'Il and

Osh'12.

Hb, Q '

and

H@

are also shown. The friction coefficient can be expressed in

the form

PApt always takes the boundary value p'=0.025, whereas the optimal value of h' is

given by Eqn 11.110.

H~rn(p~P,pt ,h~Pt)as also been plotted in Fig. 11.16.a with the constraints O.lSp'11,

k h ' a and in Fig. 11.17.a with the constraints O.21pu'11 nd 0 4 ' 1 2 .

Finally, in Fig. 11.16.b the effect is shown of a constraint of type 11.105

(minimum film thickness), whereas Fig. 11.17.b refers to the case of a fur ther con-

straint on minimum stiffness.

11.9.1.3 H&'b,b h)

11.9.1.3.1. Equation 11.184 can also be expressed

as




Substi tuting Eqn

11.50

into Eqn

11.194

leads

t o

an equation which

is

formally

identical to Eqn 11.112; H i(c’,h‘) is plotted in Fig. 11.14, in the O s c ‘ l l and OIh’52

range.

H ~ , ( C ~ ~ ~ , ~ & Js presented in Fig. 11.15, with the constraints 11.113. H i , Q ’ = H i

and H h are also plotted. There are also

W

and K‘ given by Eqns 11.191 and f ’ which

is given by the following equation

f ‘ = f _ = , , g r :

1 ho h l + b

_ _

4 F B

Since

W’,

K’ and f ’ depend explicitly on b‘, they have been plotted in Fig. 11.15, it

having been assumed that b’=b&=0.975, that is p ~ p t = c & l - b ~ p t ) = lbear in mind

that any couple of values of b’ and p’ satisfying ( l-b’)p’=c& is an optimum couple).

Since chpt has always taken on a boundary value, the optimal value of

h‘

is given by

Eqn 11.114.

H;m(c&&,p,pt) and the other related functions are also plotted in Fig. 11.16.a and

11.17.a, with a different choice of constraints (see also section 11.6.2.3.1).

11.9.1.3.2. The effect of introducing a constraint for minimum film thickness is

again shown in Fig. 11.16.b.

As

pointed out in section 11.6.2.3.2, in order to find the

minimum of H , , in the presence of condition 11.105, c’

is

initially chosen as small as

possible. Letting c’=c&, h‘ is calculated from Eqn 11.114; if h‘ satisfies condition

11.105 it is

h&,

otherwise, letting h&=a, H i and H @ are calculated. If H j k > H i , c&t

is still equal

t o

the value initially chosen; if not, still letting

h&,,pt=a,

&,,t is calculated

from Eqn 11.116. Any couple of values of 6’ and

p’

satisfying (l-b’)p’=c&,t and h&,t

form an optimum combination yielding

Hirn .

11.9.1.3.3. The effect of introducing a constraint for minimum film thickness is

again shown in Fig. 11.17.b. In order t o find the minimum of H , , in the presence of

conditions 11.107, c‘ is initially chosen as small as possible. Letting c’=c&, h’ is cal-

culated from Eqn 11.114; if h ’ satisfies conditions 11.107

it

is h&,, otherwise

h & = W /

EXAMPLE

11.9

T h e p a d e x am i n ed in example 11.2 (direct supply) is considered again. This

time, however, it is supplied by means of a capil lary tube, a s show n in Fig . 1 l . l .b ;



the supply pressure, upstream of the capillary tube, is still the same ps=106 Nlm2.

The values of speed are also the same: U=0.3, I , 3 mls. So, assuming h O = l O 4 m, a

pressure ratio @=0.5and

h = O . l Nslm2, from the second of Eqns 11.194, k=0.424,

1.414, 4,24. If constraints O.le'51,

OIh'12

(as

in

the case of direct supply) are

adopted, the following results are obtained (also approximately from Fig. 11.16.a):

1) k=0.424

hbpt=0.206, c bpt=O.1 and choosing bbpt=0.9,

pAp+

Him=O.l17, Hb=Q'=0.0291,

H@=0.0874 and H@lHi=3

W=0.95, K'=4.61, f'=0.217.

From which:

bopt=0.09m, hopt=0.206.10-4m, kPt=O.1Nslmz, p,=0.5.106 Nlm2

Htm=O.874 Nmls, Hp=0,218Nmls, Hf=0.655Nmls, Q=0.218.106 m3/s

W=7125N, K=0.519.109 N

1

m, f=3.07.1@4

It should be noted that stiffness is nearly five times lower than the value relative

to

direct supply and load capacity is halved, wAereas power and flow rate are

slightly smaller.

2) k=1.41

bopt=0.09m, hopt=0.376. 0-4 rn, popt=O.1 NsIm2, pr=0.5.1

06

N Im2

HtmS.32 N m s, Hp=1.33N m

s ,

Hfd.99 Nmls, &=1.33.106 m3 /s

W=7125N, K=0.284.109 Nlm, f=5.60.10-4.

bopt=0.09m, hopt=0.651.10-4m, kpt=0.l NsIm2, p,=0.5.106 Nlm2

Htm=27.6 Nmls, Hp=6.91Nmls , Hf=20.7 Nmls , &=6.91.1@6 m3/s

W=7125N, K=O. 164.1

09

N 1m, f=9.70.10-4.

3) kd.24

For all the cases considered above the efficiency of direct supply as compared to

capillary compensation

is

clearly greater.

4) Load capacity may be improved by selecting a higher value for the pressure ra-

tio, such as p=O.7. For the lowest speed we now have k=0.359 and then

bopt=0.09

n,

hopt=0.189.10-4m, popt=O.l Nslm2, pr=0.7-106Nlm2

Htm=O.95 Nmls, Hp=0.24 Nmls, Hf=O.71Nmls, H@lHb=3

Q=2.38.106 m3Is

W=9975N, K=0.474.108 Nlm, f=2.38.104.

In this way load capacity is 40% greater (but stiffness is slightly smaller); the

efficiency loss rw is decreased from 123.106 m l s to 95.10-6 ml s and friction coeffi-

cient is also lower.

For the highest speed

we

have k=3.59 and then

bopt=0.09m, hopt=0.599.10-4m, kpt=0.l Nslm2, p,=0.7.106 Nlm2

Htm=30.1 Nmls, Hp=7.5Nmls , Hf'22.5 Nmls , Q=7.52.106 mats

W=9975N, K=O. 150.109 N l m, f = 7.53-104.



438

HYDROSTATIC LUBRICA

TlON

11.9.2 Given

load

11.9.2.1 Ht=f(b,h)

11.9.2.1.1. Equation 11.187 can be also expressed as

(11.195)

The analogy with Eqn 11.117 is clear. In Fig. 11.18 Hi(b',h') is plotted in the

Ogb'll

and Osh'12 range, for k=0.1,

1,

10. What has been said

in

section 11.6.3.1.1, concern-

ing Fig. 11.18, still holds good.

Figure 11.19.a shows the results of the optimization of Hi, for OIkI6, with the

constraints Osb's0.9 and Osh's2.

Him

is plotted with the corresponding optimum

values of bApt and h& H@and

H ;

are also plotted, as well as p; (given by the first of

Eqns 11.1181,

Q'

(given by the first of Eqns 11.119) and

(11.196)

(1

.197)

It should be noted that, as in the case of direct supply, bApt has always taken the

boundary value b'=0.9; therefore hAPt can be determined from Eqn 11.120.

11.9.2.1.2. In practice the condition Osh' is substituted with condition 11.105 for the

reasons mentioned in section 11.9.1.1.1. The results of the optimization of Hi, with

the constraints Od~'10.9,0.61h'<2, are presented in Fig. 11.19.b (see section

11.6.3.1.2 for the relevant remarks).

For the optimization with assigned load and constraint 11.105, bAPt is initially

chosen as large as possible, and h ' is calculated by means of Eqn 11.120. Then,

if

constraint 11.105 is satisfied, such a value of h' is the optimum one; otherwise,

a

s

chosen as hhpt and

bhPt

is calculated from Eqns 11.72

(or

Fig. 11.4.b) in which

(1 .198)

If, on the other hand, the value of

b'

thus calculated is greater than the upper limit

selected for b', the latter has to be chosen as

bhPt.



OPTlMlzATION

439

11.9.2.1.3. For high values of k the above procedure may lead

t o

a high film thick-

ness and, hence, to poor stiffness. In the presence of constraints 11.122 on mini-

mum stiffness the optimization procedure should be modified as follows: the upper

limit of

b '

is firstly selected as

bApt

and h ' is calculated from Eqn 11.120. If con-

straints 11.122 ar e not satisfied, i t must be assumed that

h&=l/y.

11.9.2.2 H,=f(Uh) Equation 11.187 can also be expressed as

(11.199)

The nondimensional coefficients

p i , p i

and K can still be calculated by means of

Eqns 11.196, whereas for Q' and

f '

we have:

Optimization may be performed choosing the lowest available viscosity and

calculating h& from Eqn 11.124.

As

usual, for small values of

k ,

film thickness will

be too low; in this case, in the presence of constraint 11.105, we shall assume h&=a

and calculate p& from Eqn 11.125.

11.9.2.3 H,=f&,h) Equation 11.187 can also be expressed

as

The considerations regarding Eqn 11.199 are still valid, if only p ' is replaced by L'.

11.9.2.4 H,=f(L,p,h) Equation 11.187 can also be expressed as

(1 .201)

If

we substitute Eqn 11.77 into Eqn 11.201,

it

becomes

(11.202)




The remarks regarding Eqn 11.199 are still valid, if only

p'

is replaced byg'.

11.9.2.5H,=flb,h,p) Equation 11.187 can also be expressed as

(11.203)

Since HI is identical to that given in Eqn 11.129, the remarks made in section

11.6.3.5 can be repeated word for word. In particular, in Fig. 11.20.a the results

of

the optimization of

HI

are presented, in the O<b'10.9,Och'12, O.1<p1'S2 ange, for

O<k<6.

Him

and the corresponding optimum values

bhpt,

hhpt and

pbpt

are plotted.

Figure 11.20.a also shows Hb, H@,p;=p; andK' (given by Eqns 11.1961,

Q'

(given by

the first of Eqns 11.130) and f given by the following equation:

-f k

$ 1

6')

f - 1 ho

Figure 11.20.b, on the other hand, shows the effect on optimal values of con-

straint 11.105 on minimum film thickness.

Optimization may be carried out as follows: the largest and the smallest possible

values are selected for bhpt and p&, respectively. The optimal value

of

film thick-

ness is then given by Eqn 11.131.

If the film thickness is too small (that is, constraint 11.105 is not satisfied), we

must assume h&a instead of the above value; if we still have H@>Hbno other

change is needed. Otherwise, the optimal value of viscosity needs to be recalculated

from Eqn 11.132. Again it must be checked that this last value is compatible with the

constraints. Should

it

be too high, the maximum allowable value has to be selected

for

p&

and

b& has

t o

be evaluated from either Fig. 11.4.b

or

Eqns 11.72, in which

(11.204)

On the other hand, when speed is high, the film thickness obtained from Eqn

11.131may be too high to ensure sufficient stiffness (constraints 11.122). In this case

we have to select h&,,pt=llyand he greatest and the lowest available values for b&

and

p&,

respectively.

11.9.2.6 H,=f(L,b,h). Equation 11.187 can also be expressed as



OPTIMIZ4TION

441

(11.205)

The analogy of the above equation with Eqn 11.203 is clear and the remarks

made in section 11.9.2.5 can be repeated here, apart from simply substituting

p'

with L'.

11.9.2.7 H,=f(L,b,h,p). Finally Eqn 11.187 can be expressed as

(11.206)

PO U B

with k =

Furthermore, p;

is

given by the first of Eqns 11.136,K

is

given by the second of Eqns

11.196, Q' by the

first

of Eqns 11.130,whereas forpi and

f

we have

Substituting Eqn 11.77 in Eqn 11.206,we again obtain Eqn 11.137, which, on the

other hand, is similar to Eqn 11.129. All the remarks made in sections 11.6.3.5 and

11.6.3.7 may then be repeated.

EXAMPLE 11.10

The pad examined in example 11.3 is considered again. This time, however, it is

supplied by means of a capillary tube, as shown in Fig. 11.1.6; it

must

carry the

same load W=40000 N, for the same values

of

speed U=0.8, 2.4,

7.2

mls.

So,

again

assuming B=0.2 m, ho=2.10-4m, &=0.2 Nslm2 and P=0.5, from the second of Eqns

11.206, k=0.566, 1.70, 5.09. I f constraints 11.139 are adopted, the following results are

obtained :

1) k=0.566

Then, letting LApt=2,as in the 4th case (k=O.8)in example 11.3,

bAp,pt=0.9, h Ap,pt=0.4, gAp,pt=l.2.

~Ap,t=O.859, Him=0.275, Hi=O.138, Hb=O.138,

Q'=0.261, pi=0.526, K=2,5, f'=0.243.

Lapt=0.4m, bapt=0.18m, hapt=0.8.104

m,

hpt=0.172 Nslm2

Htm=22.0 NmJs , Hp=ll.ONmls , Hp11.0 Nmls , Q=10.4.10-6 m3/s

From which,

in dimensional form:



4 4 2 HYDROSTATE

L

UBRlCATloN

p,=1.05.106 Nlm2, p,.=0.526.106 Nlm2, K=7.50.1@ Nlm, f=3.44.104.

Comparing the above values with those obtained

in

the fourth case in example

11.3 (direct supply), it may be noted that stiffness is halved, supply pressure is dou-

bled and power is 32% greater.

2) k=1.7

example 11.3,

Letting now LApt=l and using constraints 11.138, as in the second case in

Lopt=0.2m,

bopt=0.18m, hOpt=0.254.1O4 , kpt=0.02Nslm2

Htm=24.2 N m Is,

Hp=6.1Nm s, Hp18.1 Nmls, Q=2.87.1Q6 m31s

ps=2.10.106 N

1

m2, p,=1.05.106 N l m2,

K=23.6-1@N l m, f=l.89.104.

Compared to the second case in example 11.3, Htm is 20% greater and supply

pressure is doubled, whereas K is 40% lower. Moreover these results require a film

thickness which

is

notably smaller. If the same

f i lm

thickness as in example 10.3

(case 2) were used, stiffness would be further reduced.

3) kd .09

Again letting LAPt=l,as in the third case in example 11.3,

LOpt=0.2m,

bopt=O. 18m, hopt=0.440-104m, kpt=0.02 Nslmz

Htm=126Nmls,

Hp=31.4 Nm ts , Hp94.3 Nml s, Q=14.9.106 m3ts

p,=2.10.106 Nlm2,

p,=1.05.106 Nlm2, K=13.6.108 Nlm, f=3.27.104.

4) In all three cases stiffness is clearly lower than in the case of the pad which is

directly supplied, in spite of the smaller values of optimal f ilm thickness. I t should

be noted that the only way to improve stiffness (apart the obvious solutions of further

reducing clearance

or

increasing load) is to reduce p. In reference to the first case,

still letting L=0.4 m and h'20.4, the following results are obtained, now for p=0.3

(that

is

k =0.438):

bApt=0.89, h Ap,pt=0.4,

g

Apt'2.

From

which, in dimensional form:

Lopt=0.4m,

bopt=O. 79 m, hopt=0.8-104m, kpt=0.2Nslm2

Htm=28.5 Nmls,

Hp=15.0Nm ls , Hp13.5 Nml s, Q=8.55.106 m31s

p,=l.76.106 Ntm2, p,=0.528.106 Nlm2, K=10.5.1@ Nlm, f=4.2-104.

It is clear than the 40% increase in stiffness has been paid for with a notable

increase in power and supply pressure. Compared with the fourth case in example

11.3, K is still much lower, even though Htm is now much higher. A further de-

crease in pressure ratio would lead to a higher stiffness (although the stiffness of

the pad that is directly supplied cannot be reached anyway), but with very high val-

ues

of

supply pressure and power consumption. For instance, for

p=O. l

we would

obtain:

Lopt=0.4 m, bopt=O. 63 m, hopt=0.8. 04 m, kpt=0.2Ns Im2

Htm=51.7 Nm S, Hp=28.3Nmts , Hr23 .4 Nmls , Q=5.14-1O6 m31s

ps=5. 0-106N Im2, p,,=0.550.106 N l

m2,

K=13.5.1OB N Im,

f = 7.3.104.



OPTlMlZ4

TlON

443

Althoug h stiffnes s is still lower than in the case of direct su pply, sup ply pres -

sure is now more than ten times larger and total power is more than three times

larger .

11.10

OTHER

TYPES

OF

COMPENSATING ELEMENTS

11.10.1

Orif ices

If the compensating element is a sharp-edged orifice, equating expression 4.76

t o expression 4.48 again leads t o Eqn 11.165, where now the hydraulic resistance of

the restrictor is

which depends on supply and recess pressures and where R is still the hydraulic

resistance of the clearances of the pad. Solving for p, , we obtain

11.10.1.1

Given pressure.

Proceeding as in section 11.8.2.1, for the capillary tubes,

the expressions of the various quantities are obtained (see also section

6.5.2);

in par-

ticular

W, &, H p , H f

andHt are still given by equations

11.166, 11.168, 11.169, 11.30

and

11.184,

respectively, whereas stiffness is now:

The remarks made in section 11.9.1 regarding the capillary tubes can be re-

peated here. Indeed, the equations

for

all nondimensional parameters except stiff-

ness remain exactly the same.

As

far as R is concerned, all that is needed is to sub-

stitute P(1-p)with 2P(l-p)/(2-p).

11.10.1.2 Given load. Proceeding as in section 11.8.2.2, the same equations can be

obtained, except for stiffness which now is:

and remarks similar t o those regarding the capillary tubes can again be made.



4 4 4

HYDROSTATlC L UBRlCATlON

11.10.2 Flow-cont ro l va lves

If the compensating element is a flow-control valve, the flow-rate Q through the

pad is constant. The hydraulic resistance of the restrictor must hence be

where R is the hydraulic resistance of the pad. Thus

11.10.2.1 Given pressure.

Proceeding as in section

11.8.2.1

the expressions of the

various quantities are obtained (see also section

6.5.3);

in particular Eqn

11.166

and

all the equations from 11.168

to

11.172 are still valid, as well

as

Eqn 11.184, whereas

stiffness

is

now

The remarks made in section 11.9.1 regarding the capillary tubes can be re-

peated here. Indeed, the equations

for

all nondimensional parameters except stiff-

ness remain exactly the same.

As

far as

K

is concerned, all that is needed is

to

sub-

stitute p(1-PIwith

p.

11.10.2.2 Given load. Proceeding as in section 11.8.2.2,we can obtain again the same

equations, except for stiffness which is now given by Eqn 11.23: namely, i t is identi-

cal to th at obtained in the case of direct supply (naturally, within the operating

range of the valve).

As

regards the operating range, the smaller

P

is, the wider the

operating range is, as shown in Fig.

6.10.

11.11 REAL

PADS

The formulae of the total power for the infinite pad in the case of compensated

supply differ from those of the directly supplied pad for p, . replaced by @ps in the

case of constant pressure, and for

W

replaced by W I G n the case of constant load.

As for the rest they are unchanged.

This

also

holds good for the other types of pad: rectangular, circular and annu-

lar, the formulae of which are the following:

Eqns 11.149and 11.152for the rectangular pad,



OPTlMlZATlON

445

Eqns 11.159 and 11.160 for the circular pad,

Eqns 11.163 and 11.164 for the annular pad.

Consequently the optimization procedures in sections

11.9.1

and

11.9.2,

which

are good for the infinite pad, can be followed

for

the above-mentioned pads, bearing

in mind the approximations introduced in sections

11.7.1, 11.7.3, 11.7.4.

When the

optimum values of

L, b ,h

and p for the rectangular pad,

rl, h

and p

for

the circular pad,

r3 (or r z ) ,

h

and p for the annular pad,

have been determined, the values of the other quantities ar e calculated, obviously, by

means of he exact equations to be found in Chapters 5 and 6.

The optimization procedure of the rectangular pad can be a useful reference for

that of the cylindrical pad with a rectangular recess and of the multi-pad and multi-

recess bearings.

Finally, what has been said as regards the capillary-tube supply can be

extended t o the supply by means of orifices, flow-control valves, etc.

EXAMPLE 11.11

Consider a rectangular pad, compensated by means o f a capillary tube, with a

width

3=0.3

m, which must carry a load

W=60000

N at a speed U=0.05ml s. Film

thickness is h20.4.104

m,

stiffness is K>2.5.1@ Nlm, friction force is

F 1 6

N (that is,

the friction coefficient must be lower than 10-4). Therefore these are the same data

as in example 6.3, except for length

L

which is not given and for the corner radius

ri

which is assumed to be equal to zero.

Letting ho=104m,

&=0.06

Nslm2 and 8=0.4, the speed parameter, given by the

second of Eqns 11.206, is k=0.0854.

I f

we search for an optimization with constraints

0.4&'12, 0.35pr52,b'10.9 and for L'=l, we obtain:


bhp,pt=0.545, h APt=0.4,

P

Apt*

Lopt=0.3m, bopt=0.163m, hopt=0.4-104m, kpt=0.12Nslm2

Htm=2.07 Nmls, p,=3.06*106 Nlm2, K=2.70*1@Nlm, f=1.58.104.

The friction coefficient is clearly

too

high. Since the film thickness cannot be in-

creased too much due to the constraint on stiffness, the friction can be easily re-

duced by reducing viscosity. For instance, stating pApt=l,we obtain bApt=0.406and

Lopt=0.3m, bopt=0.122m,

hopt=0.4.104m, kpt=0.O6Nsl m2

Htm=3.56Nmls, p,=4.11.106 Nlm2, K=2.70.1@ Nlm,

f=0.94.104.

Comparing these results with those in example 6.3,

we

observe that power and

supply pressure are much higher, a consequence of the shorter length of the pad. I f

we state L'=4/3 (as in example 6.3) we obtain bopt=0.135 m, Htm=2.18 N m l s and



446

HYDROSTATIC 1UBRlCATlON

p,=2.4l.1O6 Nlm2, but the friction coefficient becomes f = l . l . l O - 4 . In order to reduce

friction we may use a larger recess (for instance b'=0.6, as in example 6.3)

or

select

a lower viscosity. Stating pApt=0.85,we obtain bApt=0.423and

Lopt=0.4m, bopt=0.127m, hopt=0.4.1f34m,

hpt=0.051

slm2

Htm=2.47 Nmls, p,=2.53.106 Nlmz,

K=2.70.1@ Nlm, f=0.97.10 1.

By further increasing the length of the pad we can further reduce power and

pressure ratio.

EXAMPLE 11.12

Consider an annular-recess pad, compensated by means of a capillary tube,

with an inner radius r,=0.05 m and an outer radius r4<0.1

m,

which must carry a

load W=35000N (the highest load in example 6.4) at an angular speed Q=628 rad

Is .

Supply pressure must be p,=4.lO6 Nlm2; friction torque Mf must be smaller than 5

N m .

Putting r4=0.09 m, ho=0.3.104 m, h = O . O l Nslm2, the speed parameter, given by

the second ofEqns 11.194 in which B=r4-rl and U=R(r4+rl)12 see section 11.7.4) and

8=0.5,

s k=6.91. Then from Fig. 11.16

C ~ ~ ~ = O . ~

from which, putt ing bApt=0.9, pAPt=l)

and, also from Eqn 11.114,

h&=0.83.

Therefore,

r30pt=0.088

m, a=0.002 m and

rzopt=O.052m; hOpt=0.25.1O4 ,

,pt=O.Ol

Ns

Im2.

The effective area (Eqn 5.66) is A,=0.0167 m2 and thus Eqn 6.11 shows that a

slightly larger value of the pressure ratio, such as 8=0.524, is needed to obtain the

required load capacity: however the optimal parameters calculated above are still

very

largely valid; thus for 8=0.524, the following results are obtained (from the rele-

vant equations in chapters 5 and 6): p,=2.O9.1O6 Nlm2, W=35000 N, K=2.109 Nlm ,

Q=120.106 m3/s, Hp=479Nmls, M ~ 2 . 6 4 m, Hf=1661Nm ls , Ht,=2140 Nmls . And i f

p=920Kg/mand c=1850JI IQC, AT=10.5%. Note that the power ratio isgreater than

3: this is due to the approximations introduced in section 11.7.4; indeed, i t could be

shown that the true optimal value for fi lm thickness is hOpt=0.259.10-4m; however,

this would lead to a min imum total power of Htm=2136 N m / s which is only slightly

lower.

R E F E R E N C E S

11.1

Siddal J. N.; Optimal Engineering Design; M. Dekker, N.Y.,

1982,

523

pp.

11.2

Michelini C., Ghigliazza R.; Optimum Geometrical Design of Multipad Ex-

ternally Pressurized Journal Bearings;

Meccanica,

3

(19681, 231-241.



Chapter 12

THERMAL FLOW

12.1 INTRODUCTION

In the previous chapters, all the heat produced in the lubricant of hydrostatic

bearings by friction from viscous drag was assumed to remain in the lubricant itself

(adiabatic flow).

In this chapter the thermal flow in a hydrostatic system is studied, taking into

account heat transfer by the various parts: bearing, supply pump, compensating

element, and

so

on, showing how cooling can occur especially in the supply

pipelines and in the reservoir, provided that they are suitably dimensioned.

12.2 TEMPERATURES IN THE BEARING

12.2.1 Temperatures

in

the film

As

already seen, with the assumption

of

adiabatic flow, the elementary rela-

tionship

5.7

holds good, but a further analysis involves the introduction of the en-

ergy equation

4.37

n the mathematical model, as happens in ref.

5.14

in the case of

the circular pad, assuming known temperatures on the facing surfaces (ref.

5.15).

In this chapter the thermal flow in another elementary pad is studied: the in-

finitely long hydrostatic pad in Fig.

11.1, of

width

B,

recess width b and of which a

finite part of length L

is

considered. The mathematical model is quite simple (ref.

12.1),

f the variation of viscosity with temperature is disregarded, a s in this case.

On the other hand, heat transfer between lubricant, pad and ambient is not

disregarded.




Assuming that heat in the lubricant

is

transferred only by conduction, the

energy equation 4.39 becomes

(12.1)

where Al is the thermal conductivity coefficient of the lubricant.

If

Eqns

4.12

and

4.13,

suitably simplified, are introduced into Eqn

12.1,

and the following boundary

conditions are assumed:

T=T, for y=O, T=T2 for y = h

we obtain

Then, if Tl=T2=Tond Td=T-T0, nd taking into account that in the pad

with y'=y

I

h, in dimensionless form, Eqn 12.2 becomes

(12.3)

and k

is

the speed parameter. Figure 12.1 shows the diagrams of Tiand of

T i a = y ' ( l -

~ ' + 4 ~ ' 2 - 2 ~ ' 3 ) Tja=y'(l -y')

- a - -

b -

- c -

0

.2

.4 .6 .8 0 .2 .4

0

.2

T i , T+T& T i

I

ThqT;d

T i I

,Ti&

Fig. 12.1

TemperaturesT' n the clearance

of

a pad bearing, or certain values

of

speed parameter k.



THERMAL

FLOW

449

for k=1/3, 1, 3. As far as the intermediate case is concerned, it should be noted that,

even if Hp=Hf for k = l , Ths s always smaller than Ti8 because of the different pat-

terns of

u

and w , he former being linear, the latter parabolic. This still holds good

for

k=1/3

and obviously for

k=3,

too.

The diagrams in Fig. 12.1 partly correspond to those in Fig.

5.10,

obtained by

numerical analysis for the circular pad, without the simplifying assumptions of

Eqn 4.39.

As may be seen in

Fig.

5.10, which also takes into account the variation of vis-

cosity with temperature, the average temperature of the lubricant increases

(virtually linear) with the radius of the circular pad. Consequently the average

temperature of the film may be assumed to increase with

z

while this cannot be

established from Eqn

12.3.

As

a h r ther approximation the temperature

at

the film

inlet, where

w=O

(see section 4.8 for what concerns the "inlet length), is assumed

t o

be equal to TFs.

12.2.2 Temperatures at

the

f i lm out let

Naturally the temperature variation in the lubricant is related to the heat en-

ergy transferred &om i t to the pad and then to the ambient. The phenomenon can be

described briefly as follows.

H t

is

the total power dissipated because of friction from viscous drag in the film

(it is assumed to be equal

t o

zero in the recess), H , and H s are the heat energies

transferred between the lubricant and the pad in the recess and in the film, respec-

tively;

T,

nd T, are the temperatures a t the recess inlet and at the film outlet,

re-

spectively (Fig.

12.2).

Then

(12.4)

On the other hand, using Ti to denote the temperature at the film inlet and

Tr

and Fs t o indicate the average temperatures in the recess and in the film, respec-

tively, and putting

- T,. + Ti Ti + T,

T,.

- 2 1 Ts=- 2

(12.5)

we have

(12.6)

where



THERMAL FlOW 45 1

where

AT

is given by Eqn

5.7.

Equations

12.5

might yield

Te<Ta;

n such case, of

course, Te=Ta.

Once the operating condition

T,

= Te

is reached, Eqn 12.8 becomes

[ ( ~ + G R ) R , + ( ~ - G ~ ) R , ] T ~ + ( ~ + G ~ ) G R , R , A T

T , = ( l + G ~ ) ( l + G R s ) R , + ( l - G ~ ) ( l - G R r ) R s

(12.9)

and T , is then the maximum temperature reachable by the lubricant: a t that tem-

perature all the heat produced in i t is transferred. Equation 12.8 and especially Eqn

12.9

give useful informations on the thermal flow in the pad.

Resistances Ri, and Ri, are, respectively,

(12.10)

R1,

and R1, have been obtained from the formulae of forced convection with refer-

ence

t o

the laminar flow in the pipelines (Appendix

21.

In the expression of

Rzs,

if

(B-b)/2<h,

Fig.

12.21,

we may put

h,=h,;

therefore, a t least for

(B-b)/2chr,

according

t o the pattern of the thermal streamlines (which, in this case, tend

to

go out from

the sides rather than cross h, entirely), the heat transmission (ref.

12.2)

through

the side walls of the pad is taken into account. Moreover, we also have

A,=bL

,

A,=(B-b)L ) recess and sill areas;

2 1 3 2 9

ac aj

, a=ac+aj

external air transmission, radiation, global conductance

lubricant and pad thermal conductivity coefficients;

coefficients.

The above-mentioned coefficients can reasonably take on the following values

A1 = 0.15 J/ms°C (oil ),

ac=

3+150

J/m2s°C,

& = 45 J/ms°C (steel) ,

a j

= 4*7

J/m2soC

therefore, overall unitary conductance is

a=7+157

J/m2soC.

(12.11)

EXAMPLE 12.1

11 The pad shown in

Fig.

12.2 has the following dimensions: L=0.2

m,

B=O.l

m,

b10.08 m, h,=0.0175 m, h,=0.0025 m. The recess is also carved in the upper element.




The bearing operating conditions are W=31180 N, U=8.66ml s , and the lubricant

properties are: p=0.025 Ns I

m2,

p 8 9 7

Kg

m3, c=1930J I Kg

"y:

(at 35"c).

According to what has been specified in section 11.6.3.1.1, we put bAPt equal to

0.8,

thus bOpt=0.08

m

and putting the reference value ho=104

m,

the second of Eqns

11.73yields k=0.144 which substituted in Eqn 11.74 yields

hhpt=0.5, hus hOpt=0.5.10-4m. Moreover

p,=1.73.106 Nlm2, Q=28.9.10-6 m3/s, Hp=50 NmIs, F=17.3 N , Hf=150 NmI s, Ht=200

N m / s , HflHp=3, K=1.87.109 Nl m , f=5.5.10-4. Then, from Eqn 5.7, AT=4"C. These

values are given in Table 12.1, together with those obtained in the same conditions

but for other values of

U.

'able

b'

-

-

0.8

-

0.95

-

2.1

U

m / S

0.962

16

2.887

5

8.66

15

25.98

0.962

1.6

2.887

5

8.66

15

25.98

HP

W

50

-

24.02

-

F

N

1.925

3.3

5.774

10

17.32

30

51.96

0.924

1.601

2.773

4.804

8.32

14.41

24.96

-

Hf

W

1.852

5.5

16.6

50

150

450

1350

0.890

2.669

8.006

24.02

72.06

216.2

648.5

H ,

W

51.85

55.1

66.6

100

200

500

1400

24.91

26.69

32.03

48.04

96.08

24.02

672.5

Hf%

1/27

1l9

113

1

3

9

27

1/27

1l9

113

1

3

9

27

AT

Oc

1.038

1.112

1.334

2.001

4.002

10.01

28.01

0.958

1.026

1.23 1

1.847

3.694

9.235

25.86

-

f

. l o 4

0.617

1.069

1.852

3.208

5 . 5

9.623

16.67

0.296

0.514

0.889

1.541

2.669

4.623

8.006

-

Figure 12.3 shows, as straight continuous lines, the values of T , obtained from

Eqn 12.8, as a function of T, and for various values of a, for HflHp=1/3and

AT=1.33C (Fig. 12.3.a), for Hf /Hp=land AT=2"y:(Fig. 12.3.b), and for HflHp=3and

AT=4oC (Fig. 12.3.c).

In all three cases T,.=35%', that is the temperature at the pad

inlet is assumed to be constant.

I t should be noted that, for HflHp=l or example, the corresponding adiabatic

temperature rise AT=2oC is reduced, for ambient temperature Ta=300C and for

global conductance a=5 J/mzs°C, to AT=1.6g0C, that is only 16% less, while for

Ta =0 T and -150

Jlm%"y:,

it is reduced to AT=O.O46Y , that

is

98% less.

Figure 12.4 shows the values of T, obtained from Eqn 12.9, that is where T,=T,,

as a function of a and for various values of Ta, for HflHp=l 3 and AT=1.33Y , for



THERMAL FL0

453

- a -

39

8

L

Te I

b'=0.8 , AT

=133OC-

37 t

39

38

Te

OC

37

36

35

34

- b -

Hf /Hp

=

1 , Tr =35 OC

6 = 0 . 8

, A T =

2OC __

b'=0.95

,

~ l T = 1 , 8 5 ~ C -

_

@

//

150

/

39

38

Te

OC

37

36

35

34

- c -

0

10 20 30 0 10 20 30 0 10 20 30

Ta Ta Ta

OC OC OC

Fig.

12.3

Temperature at bearing exit T , versus ambient temperature

T ,

or certain values of global

conductivity

a

and power ratio

HflHp.,:

temperature at recess entry;

A%

adiabatic temperature rise;

b':recess width.

H f I H p = l and AT=2.001 "c, and fo r H f l H p= 3and AT=4.002"c. As may be seen in the

diag ram s, T , can reach excessive values so that some action must be taken on the

system

in

order to reduce it, that

is

to reduce T,.

Th e plotted values of T e are to be considered as examples, since the y have been

determined assuming p=constant. Such a relation is true only for small tempera-

ture increments, this being, on the other hand, one of the objects of the present

investigation.

Figure 12.5 shows th e values of T , obtained from E qn 12.8 a s a fun ctio n

of

T w f o r

a=50 J l m2s "c, for various values of H f l H p , or controlled tem perature s Tr=25OC

(Fig. 12.5.a), Tr=35"C (Fig, 12.5.b), and Tr=45"c (Fig. 12 .5 .~ ).Th e straight line rele-

vant

to

H f I H p = l 9 ha s not been plotted to avoid overlapping. Th e values o f the a dia -

batic rise

in

temp erature AT, corresponding to the various va lues of HfIH ,, are

given

in

Table 12.1. I t should be noted th at , compared

to

the case

of

H f l H p = land

AT=2, in the case of H fI H p = l127 we have a AT which is almost a h al f, while i n the

case H flH p= 27 we have

a

AT which

is

14 times higher.



454

- a -


-b - - c

-

100

Te

80

OC

60

40

20

100

0.8

,dT=1,33OC_

0.95,dT=1.230C--

Te

OC

80

60

40

20

100.

Te

OC

8 0 .

60

4 0 .

0

40 80

120, 160 0

40 80

120,60 0 40 80 120, 160

J m-2s-bC- l J m - 2 s - l ~ - l J m-2S-’oc-1

Fig. 12.4 Tem perature a t bearing exit Te versus global conductivity a, or certain values of ambient

temperature

To

and power ratio Hf/Hp.T,.: emperature at recess entry;

A T

adiabatic temperature rise;

b’xecess width.

- a -

6 7

20

65-

Te

oc

-

55

- b-

1

60 - b’=

0.8

b ‘=095

_ _ _ _ _

_ _ _ - - - -

k;

- -

50

-

3

_ _ - - - -

_ _ - - -

- _ -

1 5

n

0 -

0

10

20 0

10

20

30

0 10

20 30 40

Ta Ta Ta

OC OC OC

Fig. 12.5 Tem perature at bearing exit T , versus am bient temperature Ta,

for

certain values

of

power

ratio

Hf/Hp T,.:

temperature entering recess;

a: global

conductivity: b’:recess width.



THERMAL

FLOW

455

Finally Fig. 12.6 shows the values of T, btained from Eqn 12.9 as a function of

T,, for a=100 /m%oC, for various values of

HfIH,.

It must be noted that in spite of

the high value of a, or HflHp-3, Te would reach excessive values so

that

the system

must be acted upon

in

order to reduce

it.

0 10 20

30

40

50

Ta

OC

Fig.

12.6

Temperature at bearing exit T , versus ambient temperature

To, for

certain values

of

power

ratio H f / H p .

T,:

temperature at recess entry; CE global conductivity; &recess width.

2) ZL for the pad being considered, b10.095 m, put bhpt=0.95, thus b=0.095 m. Again

assuming that ho=10-4m, with the same operations we get

k=0.0677

and

h &=0.26, thus hOpt=0.26.10-4m. Moreover

p,=1.6.106 Nlm2, &=15.10-6m3/s , Hp=24 Nm /s, F=8.32 N, H ~ 7 2 . 1 ml s, Ht=96.1

Nm ls , HflHp=3,K=3.59.109 N l m and f=2.67.104.

Compared

to

the previous case, pr i s a little lower, K is about double and

the

other quantities are about half. Viscous friction should not be disregarded now.

Finally, the adiabatic rise in temperature is AT=3.69"C, 8% lower than in the

previous case. The above-mentioned values are given in Table 12.1, together with

the others obtained for different values of H f l H p that isof U.

Figures 12.3 to 12.6 show, as dashed lines, the values of Te relevant to the pad

with b'=0.95, obtained from Eqns 12.8 and 12.9 (putting h,=hr in R2&, n the same

conditions defined for the pad with

b'=0.8.

Figure 12.3, in particular, shows that the

pad

wi th

b'=0.95 is convenient, especially for high values of

a

and for increasing

H f

IH, while it is not convenient for increasing

T,.

Even in the case of Fig. 12.4, the

pad with b'=0.95 is still conveniently used, especially for low values of

a

now, and




again for increasing HfIH, , . For example, for H f l H p = l ,

e30

lm%"C, Ta=20"C,

n

steady thermal conditions Te=61.60C which is still an acceptable value, while for

b'=0.8, Te=97.6"c which is unacceptable.

Figure 12.5, and Fig. 12.6 even more, confirm the convenience

of

the pad with

b'=0.95 for high values

of

H f IH,. The fact that Ht

is

at a minimum does not necessar-

ily mean that AT is at a minimum as well, i f Q is small. See, for example, the case

where U=8.66 m l s and

Ht is

at a minimum for

Hf lHp = 3 .

Then it may be convenient

to increase

Q,

not by increasing

h

which would mean a decrease of K but by increas-

ing b. This is a further proof

of

the usefulness of pads with wide recesses.

The results presented above can be extended approximately to real pads. For

example, in the case of rectangular pads, in Eqn 5.7 H f increases because of friction

in the frontal sills, but

H p

also increases because of lubricant losses from them.

From the results presented above, it may be deduced that, in general, the cool-

ing of the lubricant in any bearing is moderate, about

25%,

except for very high

values of a,hat i s of R3,. and R3,, often not achievable in common practice. Some

further modest benefit could be achieved by reducing h, (for example to hr=50h),

that is R1,,nd h,, that is

R p ,

and then h, and thus R3, are also reduced.

12.3 SUPPLY PIPELINE

As has been pointed out, it is convenient to control the thermal performance of

the bearing by acting on temperature T,. at the recess inlet.

This

can be achieved by

designing a suitable supply pipeline, i.e. by also using it as a heat dissipating

element.

In this case the mean temperature difference

AT

between the lubricant in the

pipeline and the ambient is (ref. 12.2):

(12.12)

T

m =

where AT is still given by Eqn 5.7 and Ta is the ambient temperature.

On the other hand, the heat power transferred from the lubricant to the ambient

is

where

R

is

the total thermal resistance, that is (Appendix 2)

(12.13)

(12.14)



THERMAL FLOW 457

where

1 ,

D , d are the tube length and the outer and inner diameters. A1, &, and Glj

can take on the values given in Eqns 12.11 while now cl, = 8+320 J/m2s°C.

From Eqns 12.12,12.13 and 12.14, solving for

1

(12.15)

EXAMPLE 12.2

1) Consider the pad in Example 12.1 with

b'=0.8.

The temperature at the pad inlet

is required to remain equal to Tr=350C in the operating conditions HflHp=1/3, ,3

with the corresponding adiabatic temperature increments AT=1.33, 2, 4.

Figure 12.7.a shows the values of 1 as

a

function of T,, and for various values of

AT and of a=cl,+aj or D=O.Ol m, d=0.007m, thus (D-d)l2=s=O.o015m. For example,

for

AT=2 c (Hf IHp=l ) ,Ta=280Canda=80JlmzsoC, ll12.36 m, while for Ta=l4oCand

e l 6 0 JlmzsoC, 1=3.57 m. For AT=4cY: HflHp=3),Ta=14cY:and -160 JlrnzsoC, 1=6.84

m .

For the given values, Rll=0.584 m s V lJ; R21=0.00126 m s v l J; R31=3.18 to

0.0998

ms@lJ, for a=10 to 320 Jlm2sV. Therefore the second term of the third member

of

Eqn 12.14 is quite small compared to the first and also on an average as compared to

- a -

15

1

m

10

5

0

10 20 30

Ta

0

-b -

1 5

I

m

10

T, = 35OC

5

0

0

10

20 30

Ta

OC

A T = 4 O C

f / H p = 3 1 -

--

Fig.

12.7

Supply line length I versus ambient temperature T,, for certain values of power ratio

H Hp .

of adiabatic temperature

rise

AT for a

f ixed

value

of

temperature at recess entry

T ,

and

for:

a-

certain values of global conductivity

a

and a fixed value

of

outer diameter D (and

of

inner

diameterd)of the line;

b-

for certain values of D andd)and a fixed value of a.



458


the

third . Zt certainly becomes negligible for tube s wi th h igh t her m al con ductivity,

for exam ple, for copper tubes for w hich

&=300

J l m s V . There fore the in fluence o f d

is also often negligible an d the results presented are also valid fo r differ en t values

of

d , for lower values ( that

is

for higher values of thickness

s)

and even more

so

f o r

higher va lues (tha t is for lower values of

s).

Figure 12.7.6 shows the values of

1

as a function of T, and f or var ious values of

AT, D and d , for a=100 J I m 2 s T . The values of D and d are: D=0.002, 0.004, 0.008,

0.016

m;

d=O.001, 0.0025, 0.0055, 0.011 m ;

s=0.0005,0.00075,0.00125,

.0025 m ; th e

smaller values correspond to capillary tubes. For example, for AT=2 "c (HflH,,=l),

T=14"c and D=0.016 m, still 1=3.57 m, while for D=0.032 m and d=0.028 m (not plot -

te d) 1=3.11

m. As

regards d an d s th e above considerations still hol d good.

2) Pad w it h b'=0.95. Since AT and especially

Q

are lower t h a n the va lues relevant to

the pa d wi th b'10.8, length 1 is also lower.

For

example, for AT=1.85"C (HfIHp=l),

Tr=35"c , T ,=14"c0, for a d 6 0 JJm2s°C ,D=O.Ol m, d=0.007 m, the length is 1=1.72 m;

for -100 Jlm 2s "c , D= 0.016m , d=0.011 m, still 1=1.72 m ; fo r -100 Jl m 2s "C , D=0.032

m, d=0.028

m,

1=1.50 m.

12.4 COMPENSATING ELEMENTS

The heating of the lubricant occurs in the bearing and in the regulation devices

as well, that i s fixed ones such as capillaries and orifices and variable ones

such

as

flow control valves, etc. The rise in temperature is still given by Eqn

5.7,

putting

H+Hp=O and replacing p s with the pressure drop Apc in the regulator, tha t is

APC

ATc =-

P C

(12.16)

The tube length required t o keep the inlet temperature Tc a t a given value, is

then again given by Eqn

12.15,

replacing

AT

with

ATc

and

T ,

with

Tc.

EXAMPLE

12.3

1) Pad w ith b'=0.8 in examp le 12.1.

Z f

it is compensated by a capillary with p=0.5,

Ap,=1.73.106 N l m 2 , thu s ATc=l"c; in this case, fo r exam ple for Tc=35"c, Ta=14"C,

@lo0 J/m % "C , D=0.016 m, d=0.011 m, the length is 1=1.83 m.

2)

Pad with b'=0.95. In this case for 8=0.5, pc=1.6.106 Nlm2, thus ATC=0.924'C

and in the same conditions as

in

the previous example, the len gth is 1=0.879 m.

12.5 PUMP

The heating of the lubri'cant also occurs in the pump.

If

q

is its

efficiency, the

temperature rise

of

the lubricant may be put in the form



1-71PiT

AT,= --

71 P C

THERMAL FLOW

459

(12.17)

where

pz=Apc+pr.

The tube length required t o keep temperature

T ,

entering the pump at a given

value, is then again given by Eqn 12.15, replacing

AT

with

AT,

and

T,

with

TT

EXAMPLE 12.4

1) Pad with b'=0.8

in

example 12.1. It

is

supplied by

a

pump

with an efficiency (for

example) q=0.8.

I f

the pad

is

supplied directly, p,=p,=1.73.106 N lmz and

ATZ=0.25 c; n this case, for T,=35 c, Ta=14 c,

-100

Jlm% c, D=0.016 m, d=O.011

m, the length is 1=0.464

m. I f

the pad is compensated, for p=0.5, p,=3.46.106 Nlm2,

AT,=0.5

"c

and 1=0.923 m.

2) Pad with b'=0.95. It is supplied by a pump with an efficiency q=0.8. If the pad

is

supplied directly, now p,=16,105 Nlm2 and AT,=0.231 C; n this case, in the same

conditions

as

in the previous example, the length is 1=0.223 m. I f the pad is compen-

sated, for p=O.S, p,=3.2.106 Nlm2, ATz=0.462 c and

1=0.444

m.

12.6 COOLING PIPELINES

Cooling can occur in the supply tubes between the pump and the bearing, as

already seen in section 12.3, as well as in the return tubes between the bearing and

the pump. If the bearing is directly supplied, and cooling occurs in the supply tube,

it s length is still given by Eqn 12.15,where

AT

is substituted by the total temperature

rise

ATT = AT, + AT

(12.18)

If cooling occurs in the return tube, 1 is still given by Eqn 12.15, where again

AT

is substituted by

ATT

in Eqn 12.18,and

T,

by

(T,-AT,).

Consequently I is

a

little larger

than in the previous case because the mean temperature

of

the lubricant in the tube

is a little lower.

If

the bearing is compensated and cooling occurs in the supply tube, 1 is still

given by Eqn

12.15,

where

AT

is substituted by the total temperature rise

ATT

=

ATz

-I-

ATc

-I-

AT

(12.19)

and

T,

by

(T,-AT,).

If

cooling occurs in the return tube, in Eqn

12.15

AT

is

substituted by

ATT

in Eqn

12.19, and

T,

by

(Tr-ATc-AT,).




In common practice, temperature

T ,

at the inlet of the pump

is

often taken as a

reference value. In this case, with compensation and cooling in the supply tube

where

T,=T,

referred to

T,

is

T ,

=

T, + AT.

If, on the other hand, cooling occurs in the return tube

(12.20)

(12.21)

where

T,

referred to

T,

is now

T,= T,

-

ATc

-

AT,

(12.22)

In the case of direct supply, the previous expression without

ATc

still holds good.

If the return tube

is

used as the cooler, larger diameters and small thicknesses

may be selected while on the contrary they must be avoided if cooling occurs in the

supply tube because tube elasticity could have a negative influence on the dynamic

stability of the bearing. If cooling occurs in the return tube, however, a recirculation

pump

is

almost always necessary. Therefore there

is

a further temperature rise

ATnp,

still expressed by Eqn

12.17,

but generally quite small since pressure head

ATnp,even with filters, is small.

Further temperature increments may occur in other elements of the circuit, in

particular in the filters, for which

ATp

can be calculated with Eqn

12.16,

ubstitut-

ing

Apc

with the pressure drop App in the filter, and also in the relief valve, for

which

AT,,is

again given by Eqn

12.16,

ut only for the lubricant escaping from

it.

In the case of compensation and cooling in the supply tube, in Eqn

12.20,AT,,

and ATq are added to

ATT

; f cooling occurs in the re turn tube, in Eqn

12.21 ATnp

is

added to

AT,

as well as

ATv

and

ATp

while i t must be subtracted in Eqn

12.22.

In the case of direct supply what has been said above still holds good but the

expressions lack the term

ATc.

EXAMPLE

12.5

Compensated supply with cooling in the supply tube.

1)

Pad with b'=0.8;

H f I H p = l ,

AT=2OC

(see

example

12.1).

For

8=0.5, dpC=1.73.1O6

N l m 2 a n d ATc=l

"c;

for App=0.3.106N lm 2 (roughly), ATp=0.173"c; he temperature

rise in the valve

i s

assumed (very roughly) to be AT,,=0.05°C. p ,=1.73.106 N l m z ,



THERMAL FLOW

461

therefore, for q=0.8, ATn=0.544 c. The total temperature rise is then dT ~3 .7 7 C.

Putting Tr=35 c, Tn=370C.

For

Ta=14 c, e l 0 0 J/m2s0C,D=0.008 m, d=0.0055

m,

Eqn 12.20 yields 1=8.49 m.

2) Pad with b'=0.95;

H f I H p = l ,

AT=1.85 C. Ap,=1.6,106 Nlm2 and ATC=O.924 C,

AT,=0.173°C, ATv=0.050C;pr=16.106Nlm2 and ATn=0.505V; then A T ~ 3 . 5 c nd

Tn=36.850C.For the former values of T,, a, D and d, L=4.11 m. As seen above, the

length is almost unchanged even for lower values of d.

EXAMPLE 12.6

Cooling in the return tube

1) Pad with b'=0.8. Compared to the previous case

we

still have AT=BOC, AT,=l "C,

AT,=O.l 73 c, ATv=0.O5 C, but ATz=O.5OC and including the recirculation pump

with

q=O.8

and discharge pressure pnP=0.3.1O6N/m2, ATnp=0.0433

"C.

Then

AT~3.77 c nd Tp33.6V. For the former values of T,,

4

whereas D=0.032 m and

d=0.028 m, Eqn 12.21yields 1=6.06 m.

2) Pad with b'=0.95. Now ATn=0.4620C, thus A T ~ 3 . 5 c nd Tn=33.50C.For the

former values of the other quantities, 1=2.93m.

It must be noted tha t heat losses may occur in the bearing as well as in the other

elements of the circuit: in the pumps (for which efficiency losses also depend on

lubricant losses), in the compensator, especially in the capillary tubes, in the filters,

especially at high pressures, in the valves, etc. The lubricant may be cooled further

in the reservoir if this is reasonably sized (Appendix

2)

and finned (ref.

12.2).

More-

over

it

should also be noted that the set of supply tubes of a hydrostatic system can

work as the tube bank of an air-oil heat exchanger. Assuming that, on average,

50%

of the cooling ra te pertains t o all the other elements of the system, the length of the

cooling tube would be reduced by more than 50%' approximately.

12.7

SELF-COOLING CAPILLARY TUBE

Consider a capillary tube through which a pressure drop Ap, occurs. Its length

is easily determined from Eqn 4.66:

(12.23)

On

the other hand, the pressure drop produces heat, therefore there is a rise in

temperature AT, of the lubricant flowing through it, given by Eqn

12.16.

Heat, how-

ever, can be dissipated by a tube of the following length:

(12.24)



462 HYDROSTATIC

LUBRlCA

T/ON

where Tc is the temperature a t the inlet of the tube. The tube may even be the capil-

lary itself. The diameters of the capillary must be related, in accordance with the

typical equation of thick pressurized tubes:

(12.25)

where p is the maximum nominal pressure in the tube, o he material limiting

stress and m the longitudinal deformation modulus. Actually, for construction

requirements, Eqn

12.25

is already satisfied for small diameter tubes.

Solving the system of three equations yields the dimensions of the self-cooling

capillary. As may also be seen in Fig. 12.8, it will be much longer than the length

given by Eqn 12.23 for very small values of d. Anyway, d must be large enough t o

avoid clogging (d>0.5.10-3m).

EXAMPLE 12.7

The capillary is made of drawn steel with the following characteristics:

0=3.5.107 Nlm2, so that elastic deformations are negligible, m=10/3.

1) Pad with b'=0.8 in example 12.1;B=0.5, thus Apc=1.73.106 N lm2 and ATc=l C.

The operating pressure is at a maximum at the inlet of the capillary tube and it is

p~= 3.4 6. 106 lmz ; for the maximum nominal pressure in Eqn 12.25 the value

assumed is p=4pM=13.9'106N l m2, also considering possible dynamic overloading;

Tc=35 c.

Figure 12.8.a shows length 1, capillary tube inner diameter d and outer diame-

ter D, as functions of ambient temperature T,, for a given value of overall conduc-

tance a=lOO Jlm2s C. It can be seen,

for

example, that for T,=lO C, 1=2.6 m,

d=0.0026 m and D=0.0042 m; while for Ta=2O C,1=4.1 m, d=0.0029 m and D=0.0047

m. In any case l l d is always much greater than 100, as usual for capillaries.

In

Fig. 12.8.b

1,

d and

D

are plotted as functions o f

a

for Ta=15V.It may be seen,

for example, that for a=60 Jlm% C, 1=4.2 m, d=0.0029 m and D=0.0047 m; while for

a=120 J/mZs C, 1=2.9 m, d=0.0027 m and D=0.0043 m. Capillaries may be wound i n

large pitch spirals.

2) Pad with b'=0.95;

/3=0.5,

hus Apc=1.6.106 N l m z and ATc=0.9240C; p=12.8.106

Nlm2 is assumed; TC=35 c. n Fig. 12.8 1, d, and D are plotted as functions of Ta a)

as functions of a (b). Length and diameters are smaller than those in the previous

case.

The results obtained apply even more

t o

copper capillary tubes; however,

it

is

advisable to increase the outer diameter by 25% with no detriment to cooling which



THERMAL

FLOW

463

- a -

- b -

~~

-

T = 350C , T = I 5 OC

b‘ =

0.8 ___

- - - _ _

- -

_

- - _

-

-

0.0075

D,

d

m

0.005

3.0025

3

0

10

20 30 0 40 80 120 a 160

Ta

OC

J m-*s-’

o c - 1

Fig.

12.8

Length

I,

outer diameter

D

and inner diameter

d

of a self-cooling capillary tube

of

a

bear-

ing with a recess width b‘,versus: a- ambient temperature T,, for a fixed value of temperature enter-

ing tube

T,,

and global conductivity a; b- a, for fixed va lues of T , and T,.

is still higher than in steel capillaries because of the much higher conductivity of

copper as compared to steel.

In general, the design

of

a self-cooling capillary, without solving the system of

three equations, may be carried out as follows:

assign

d

(for instance, with the aid of Fig. 12.8);

from Eqn 12.23 determine

I

and introduce it into Eqn 12.24 without the generally

negligible term (1/2)(1/&)ln(D/d); thus determine D;

22D/d21.5 should be satisfied, instead of Eqn 12.25, otherwise:

the procedure is repeated until the result is achieved.

12.8

VISCOSITY

AND

TEMPERATURE

Up to now viscosity has been assumed to be constant as temperature varies,

especially in the pad film. This can be considered to be true if heat dissipation in the

various elements of the hydrostatic system keeps the increment in the lubricant

temperature low, especially if the supply and return tubes and the reservoir are

designed as coolers. Otherwise the new value of viscosity in the film must be evalu-

ated according

to

the mean value of temperature, and calculations must be repeated

until the difference between two consecutive values of viscosity is sufficiently small.

In the case of compensated supply the actual value of viscosity in the compensating



464


elements must be taken into account. When lubricant temperature is under control,

the hydrostatic system operates properly and thermal deformations of the bearings,

detrimental

t o

the smooth running of the machine and the accuracy of products in

the case of

a

machine tool, are avoided.

REFERENCES

12.1

Bird

R.

B., Stewart

W. E.,

Lightfoot E.

N.;

Transport Phenomena;

Wiley and

12.2

Kreith

F.;

Principles of Heat Transfer;

Intext Educational Publisher,

N.Y.,

Sons,1960; 780 pp.

1973;

50 pp.



Chapter 13

EXPERIMENTAL TESTS

13.1 INTRODUCTION

An important aspect of the study of hydrostatic bearings

is

testing them, con-

sisting in the measurement of a considerable number of input and output variables

of the hydrostatic system. This, together with the large number of types of bearings,

has led to the assembly of a considerable number of test rigs, only a few of which are

equipped for the testing of more than one type of bearing.

In this chapter, af ter some brief notes on the most important input and output

variables and testing procedures, a number of test rigs are described, chosen

among

a

host of equally good rigs, and details are given of the tests performed on a

few particular types of bearing.

13.2 HYDROSTA TIC SYSTEMS; INPUT AND OUTPUT VARIA BL ES

There a re many variables affecting a tribological system and a hydrostatic sys-

tem in particular, as shown in Fig.

13.1.

The behaviour of the mechanical system

made up

of

a pad and a slide separated by lubricant and placed in the atmosphere,

depends on numerous input variables and it is also characterized by the output

variables; the validity of the tests largely depends on the correct experimental

measurement of such variables. The following input variables are examined:

the type of motion, which is sliding in hydrostatic lubrication;

speed, which can be linear

o r

angular and the control and measurement of which

is

carried out with electronic tachometers, which are especially necessary for very

low or very high speeds;




load. The devices

for

measuring loads or torques can be quite different: leverisms,

springs, hydraulic jacks and electromagnetic systems, the la tter being especially

suitable for dynamic loads. The measurement of the load is now generally obtained

by means

of

strain gauge bridges;

lubricant flow rate measured in volume by variable area and turbine flow-meters;

supply pressure, measured by means

of

manometers and capacitive and piezore-

sistive transducers;

lubricant supply temperature measured by means of common and infrared ther-

mometers, o r thermocouples allowing continuous measurement;

ambient temperature;

the physical and chemical characteristics

of

the lubricant, such as viscosity,

density, specific heat, etc., generally measured with standard test devices and

methods.

Input

variables

Typeof motion

m

elocity v

I

LoadFN

I r t

Supply pressure

D

of

lubricant

T

RIB -SYST EM

utput

variables

1-

First body

2- Second body

3-

Third body: lubricant

i

- Atmosphere

4

ype of lubrication

I

I 1

f T i G m - l

recess and in the

film

{Filmthickness)

1

Friction

Thermal increase

of lubricant

L

Base lubricant I

Modified charatcteri-

haracteristics

of lubricant stics of lubricant

Additives

Fig. 13.1

Hydrostatic system.



EXPERIMENTAL TESTS 467

The following output variables are examined:

type of lubrication. It

is

known that often, together with hydrostatic lubrication,

hydrodynamic lubrication takes place and can become prevalent in radial bearings

so

that hybrid lubrication occurs;

recess and film pressures. Pressure in the recess equals supply pressure in the

case of direct supply while it is lower, sometimes much lower, in the case of com-

pensated supply;

film thickness, measured by contact micrometers and displacement transducers,

the latter being eminently suitable

for

dynamic

tests;

viscous friction, measured by strain gauge load cells or torque meters; these also

are of the inductive type;

rise in temperature

of

the lubricant, due to viscous friction, which can cause

changes in viscosity and in the other characteristics of the lubricant.

For the validity of the experimental results, it is necessary to check, before test-

ing, that the macro and micro geometry of the experimental model complies with

the design requirements (in tolerance), such as the following:

dimensions of the bearing;

dimensions and location of the recess;

planarity and parallelism

of

the sliding surfaces;

cylindricity of the rotating elements;

parallelism of the axes of the rotating elements;

orthogonality of axes, of planes and of axes and planes;

dimensions and location

of

the compensating elements.

All this is due to the fact that, for example, flow rate through a film varies with

the third power

of

its thickness and that flow rate through a capillary tube varies

with the fourth power of its diameter.

If

some of the requirements are not complied

with, the experimental test can still be carried out but the actual values of the geo-

metrical variables mentioned must be introduced in the calculations.

Considering the complexity of the tribo-system presented in Fig.

13.1,

it

is

advis-

able, in testing, t o collect input and output data systematically. The reader is re-

ferred to ref. 13.1.

In general, basic tests are carried out, that is on elementary systems (pin-disc,

slide-way, etc.) and the results may be insufficient for an adequate study of the

phenomenon.

S o

the following categories of tests may be carried out, possibly in

varying combinations:

1-field tests, tha t is tests performed on the machine

2-

stand tests

on

the machine

3-

stand tests only on that part of the machine which the tribo-system is part of

4-stand tests on that par t a t a reduced scale



468


POTENTIOMETER

4

5-tests on the tribo-system taken off the part of machine containing

it

6-basic tests, already mentioned.

- V D C + DETECTOR

-A

0

10

1 0

30--100

r.

,

13.3

EXPERIMENTAL

RIGS

13.3.1 Electr ic analog f ie ld p lot ter

The apparatus shown in Fig. 13.2.a is not an experimental bearing but a device

for the determination of the characteristics of the bearing through the use of an

electric analog field plotter which w a s being used some decades ago in the study of

hydrostatic bearings of various shapes, giving satisfactory results. It is based on the

fact that Reynolds equation, which allows us to calculate pressure distribution in

the bearing clearance, is analogous to the electric field equation which makes it

possible t o calculate voltage distribution in a conducting sheet with the shape of the

clearance. Figure 13.2.a contains an outline of the circuit of the electric analog field

plotter (ref. 5.34).

- a -

- b -

E l

SILVER

.

ECT

RODES-

11OV ]16VAC

I

AC

8

7

6

5

4

3

2

1

0

a

80

70

60

50

40 0

30

20

10

0

n

I

0 0.2 0.4

0.6 0.8

1.0

'1

1'2

Fig.

13.2 Electric analogy: a- Circuit of the electric field plotter; b- Performance factors of a cir-

cular thrust

bearing

Figure 13.2.b gives the values of the pressure in the recess, of the flow rate and

the pumping power as functions

'of

the radius,

as

obtained with the plotter

for

a

circular bearing (ref. 13.2). As can be seen, there is a remarkable agreement be-

tween these and the theoretical results (solid lines).



EXPERIMENTAL

TESTS

469

13.3.2

Axial bearings

i)

Device simulating

a

hydrostatic p a d .

Of the numerous types of experimental

thrust bearings, a circular and particularly simple one (ref. 13.3), shown in Fig.

13.3, is examined. Pad 1, axially free to move in seat 2 , is subjected t o a load pro-

duced by spring 3. The load can be varied by varying the spring tension by means of

screw 4. The load is balanced by the pressure in the recess and in the film, the

thickness of which is measured with micrometer

5 .

The friction between pad

1

and

seat 2 is lowered by the lubricant between them. Though lacking precision mainly

because of friction, the device is quite useful for investigating circular pads with

ratio

r21r2

equal t o the experimental one.

Fig. 13.3

Device simulating a hydrostaticpad.

With devices similar to that in Fig. 13.3, though less precise, differently shaped

pads (rectangular, etc.) can be tested. If pads are supported radially by aerostatic

bearings, there i s almost no friction, thus the results are more precise.

ii) Turn-tab le simulat ing a sl iding table . Figure 13.4.a shows a (tilted) circular

pad tested (ref. 13.4) with the apparatus outlined in Fig. 13.4.b. The test rig consists

of a turn-table substituting a sliding table; this does not imply there are large errors

because the turn-table

is

much larger (about 1.2 m) than the test pad (about 0.1 m)

placed at its periphery. The turn-table is supported and located by three pairs of

opposed hydrostatic thrust pads equally spaced out around the structure. The test

pad is attached to the base of a plunger which is located in a cylinder rigidly

mounted on the base of the test rig. Free vertical movement of the plunger is as-

sured by two rows of air bearings in the wall of the cylinder. Loads are applied t o the

test pad by adding weights

to

the plunger, and the tilt of the test pad with respect to

the table is introduced by shims.

In Fig. 13.4.c the variation of the dimensionless flow rate

Q=Qp/@oh$

s

plotted

as a function of the "dynamic term"

S=6pRz(Ua+2V) l (pohg) ,

where U and V are the

sliding and the squeeze velocities, respectively, for certain values of the angle of ti lt



470


- a - - c -

S

positive

u

- b -

u

SyEbol

0

+ I A A v v

a a2 04 a5

OB

065 07 a75 0.8

Fig. 13.4 Test

r i g with

turn-table: a- tilted pad;

b-

apparatus; c- flow rate versus dynamic term.

-

a=ar2/ho nd for two different values

of

the ratio of radii iZ=rl/r2.The load being

equal, flow increases linearly with the dynamic term.

iii)

Transparen t pad .

In Fig. 13.5 a rectangular pad, studied in ref. 13.5 and

tested in ref. 13.6, is presented. I t is characterized by the relationship

b I B = l I L

and it

has non-rounded corners, that is ri and

r,,

are equal t o zero (Fig. 5.25). The pad is

made of transparent material (Plexiglas)

so

that the fluid streamlines in the film

can be visualized by introducing coloured liquid (ethylene glycol). Obviously, with

Plexiglas, tests have been carried out with relatively low loads.

In Fig. 13.6 lubricant velocities at certain points [

f

the diagonals are plotted.

In Fig. 13.5.b the glycol fluid line is indicated with an arrow a t point

c2

in Fig. 13.6.

The line, almost tangent to velocity

Rz in

the same diagram, is continuous, with no

breaks; this has a l s o occurred a t the other points of the diagonals.

iv)

Flexible p a ds .

In ref. 2.15 an all-metallic flexible hydrostatic thrust bearing

is investigated both theoretically and experimentally. Figure 13.7.a shows the pad

being tested while Fig. 13.7.b contains an outline of the experimental apparatus. In

this rig the flexible bearing being tested (a)

is

fitted into the adaptor (b) which is in



EXPERIMENTAL

TESTS

471

Fig.

13.5

Visualization

of

streamlines:

a-

Plexiglas pad;

b-

ethylene glycol fluid lines.

its turn mounted on a heavy, rigid base plate (c). This base plate is attached

t o

a free

standing frame (d) and loads are applied t o the upper member of the bearing (e), by

the cross-head

(f)

through a ball (g). A lever system connected t o the link. (h) enables

loads to be applied by means of dead weights. To ensure that the lower surface of the

block (e) remains parallel to the outside edge of the bearing, a system of four flexible

restraints c) are fitted between it and the pillars (k). When properly adjusted these

flexible elements make the block (e) move parallel to the bearing edge without offer-

ing any significant resistance

t o

vertical movement. The lubricant is supplied to the

bearing through the flexible pipe (1) and returns to the supply unit through the

drain pipes (m).

In Fig. 13.7.c dimensionless pressure in the bearing film F=p/pr,pr being the

recess pressure, is plotted as a function of the ratio of radii r= r/r2;ratio r1/r2 is 0.25.

The variation of is quite different from the logarithmic variation of the rigid pad.

The diagram is obtained by assuming that film thickness a t the outer edge of the

bearing is h&h&/(p,r$)=0.0045, with D=Et4/[12(1-v2)],where t is the plate thick-

ness,

E

the elastic modulus

of

the material and

v

its Poisson ratio. In the same

diagram the experimental results for t =2 . 08 mm, E=209000 N/mm2 and v=0.3 are



472


Fig. 13.6 Velocity of

fluid at

certain points of a diagonal.

also given.

It

should be noted that the effectiveness of flexible bearings decreases as

the recess increases.

13.3.3

Radial bear ings

i) Test rig

for

static loads. In Fig. 13.8 a test rig for radial bearings subjected to

static loads (ref. 10.9) is shown. A shaft driven by a variable-speed motor, is sup-

ported a t both ends by roller bearings and a block of the hydrostatic bearing floating

on the middle of the shaft. In both sides of the floating block there are hydrostatic

thrust bearings. A static load is applied by pulling up the floating block, using a

ring.

With such an apparatus it is possible

to

investigate the occurrence, in the pres-

ence of static loads, of forced vibrations caused by the fluctuations in recess pres-

sure due to aeration of the working fluid, as well as self-excited vibrations of larger

amplitudes, i.e. oil-whirl. The appearance of recess pressure P,. lower than ambient

pressure is considered as the boundary of the allowable operating range. This

boundary, for the upper pad of a four-pad capillary compensated bearing, is given in

Fig. 13.9, as a function of the eccentricity ratio

E=elho

and of the speed parameter or

Sommerfeld hybrid number S = p o fP J c fD)2 (where c is the diametral clearance and

D s the journal diameter), for certain values of load

F.

ii)

Experimental apparatus

for

static and dynamic loads.

Figure 13.10.a shows a

test rig for journal bearings subjected

t o

both static and dynamic loads (ref. 13.7).A s



EXPERIMENTAL TESTS

473

- a -

- c -

I 4 1

0.3

04

0.5 06 0.7 0.8 0.9 1.0

Dimens ion les s r a d i u s ,

T

Fig. 13.7 Flexible hydrostatic thrust bearing:

a-

pad;

b-

experimental apparatus;

c-

pressure

versus radius.

in the previous case, a shaft

(A)

driven by a motor

(B)

through a n elastic joint (C) is

supported by two roller bearings mounted in a trunnion assembly

(D)

nd the block

(El

of the hydrostatic bearing is floating on the middle of the shaft. Eight symmetri-

cally-placed flexible restraints (F)ensure that the bearing bIock moves perpendicu-

larly t o the shaft without imposing any significant restraint. Steady loads are ap-

plied to the bearing by the screwjack ( G ) hrough the calibrated spring (H). Oscilla-

tory loads are applied to the bearing by the electromagnetic vibrator

(I)

which is

mounted in a space provided in the base (L) mounted on four flexible supports

(M).

A strain gauge force transducer (N) easures the dynamic loads applied to the

bearing. '

The dynamic tests have been carried out on

a

four-recess orifice-compensated

bearing both with static loads and without them. Figure 13.10.b concerns the first

type of tests performed with the bearing at rest and shows the flexibility, or recep-

tance, f u as a function of the forcing frequencyaf=Qf/SZ of a sinusoidal load

(f2

being

the undamped frequency of the bearing) for an equal amplitude of oscillation

6,

around the centre, for two v a l ~ es f the supply pressure and with the pressure ratio

p=0.5. There is a good correspondence between the experimental and theoretical



474

HYDROSTATIC

1

BRlCA

TlON

t

t r ans duc ers

J o u n a l b e a r i n g '

~~~~~~~~~~~~

b l o c k

Fig.

13.8

Experimental apparatus for journal bearings.

1.0

&

0 50 100 150

s

Fig.

13.9

Effeg of eccentricity ratio and speed parameter on the appearance of recess pressure

(dimensionlessP,=P,JP, where P, is supply pressure) lower than ambient pressure.

results up to +0.65, above which they differ because cavitation takes place in the

recess.

iii) Experimental apparatus for static and dynamic magnetic loads. The test rig

(ref. 13.8) shown in Fig. 13.11.a consists of a symmetrical rotor with a flywheel (1)

equipped with transformer steel sheets. Compressed air from a blower drives a

light turbine wheel (2). Symmetrically, there is an equivalent disc

(3)

for triggering

and sampling the signals measured. The rotor is supported by two identical hydro-

static bearings (4). The shaft is axially positioned by two

air

bearings (5 ) . External

vertical radial forces are applied by magnetic shoes

(6).

In order to induce basic

harmonic sinusoidal forces alone, a static component must be superimposed on the

top magnetic shoe. In this way the rotor is lifted up t o the concentric position in the

bearings. A bottom magnetic shoe is used to apply higher static load downwards.



EXPERIMENTAL

TESTS

475

With this magnetic force system it is possible

to

vary the force frequency indepen-

dently of shaft rotation frequency. An appropriate set

of

instruments, such as

dynamometer (7) and eddy current probes (8),makes it possible to measure the vari-

ous variables.

- a -

C

B

N

- b -

supply pressure. lb / in2

o

0

0 66

n .

0

1 2

Dim ens ion les s f o r c ing t r equenc y , a

'

0 4 '

0 8

Fig. 13.10 Experiments on journal bearings:

a-

Experimental apparatus;

b-

Bearing response

curve.

- a -

- b -

Fig.

13.1

1

Experiments on journal bearings:

a-

Experimental apparatus;

b-

orbit

of

the shaft

centre.




Because of the hydrodynamic component, which is almost always also present

in hydrostatic bearings, a bearing subjected

to

both static and dynamic loads moves

on an orbit (characteristic of hydrodynamic bearings) around the static equilibrium

position. This is clearly shown by both the theoretical (solid line) and the experimen-

tal (dashed line) results contained in Fig. 13.11.b for a compensated four-recess-

bearing with 8=0.5, subjected t o

a

static load and t o a sinusoidal dynamic load

giv-

ing rise

t o

elliptical orbits. There is good agreement between the theoretical and the

experimental results. In the diagram E is the static eccentricity, S = p n / p , ( c

/

0 2 ) the

speed parameter, n the angular velocity and

fF

the dynamic load frequency.

13.3.4

Spher ica l bear ings

Figure 13.12. shows an experimental apparatus (ref. 5.25) for spherical bear-

ings (either the fitted or the clearance type). Pad (11, rigidly connected t o the rotating

shaft (2), is supported by the central recess bed (31, which is placed on the bed car-

rier (4). The bed carrier and the base (5) are adjusted by screws. The apparatus is

equipped with a set

of

instruments for the measurement of some of the variables

involved, in particular with mercury-in-glass manometers for the measurement of

pressure in the recess and in the film.



EXPERIMENTAL TESTS 477

-b-

Fig.

13.13

Experiments

on

spherical bearings:

a-

outline

of

the bearing;

b-

Comparison between

experimentaland theoretical pressure distribution.

Fig. 13.14 Test-rig

for

screws and nuts,

In Fig. 13.13.b the theoretical pressures in a directly supplied bearing are plot-

ted, according to the isothermal assumption (solid line; the temperature

of

the fluid

remains constant because of heat transfer to the surrounding environment) and

according to the adiabatic assumption (dash-dot line; no heat transfer); PI s the



478


supply pressure. The agreement between the theoretical and the experimental re-

sults is remarkable in the latter case. The results are ghen for a pad with a sphere

radius R=59.31 mm with a vertical film thickness of 100 pm

for

8=0,

with

81=5",

(&=15O,

3=750

nd with an inertia parameter S=0.15pR2R2/Pl=2.

13.3.5

Screws and nuts

In Fig. 13.14 a test-rig

for

hydrostatic screws and nuts (ref. 13.9) is shown. The

test rig allows the screw (S) axial movement, while the nut

(N)

is at rest and is

- a -

; @

-b -

5'

- c -

4

o

p o = l O x l O ' ~ g r n ~ '

po=20110'Kg

m- '

3 -

P'

2 -

o 0.1 a2

03

04 05 06

&

Fig. 13.15 Test-rig for screws and nuts: a- self-regulated nut;

b-

dynamometer for loads and

torques; c- load capacity P' versus eccentricity E, for two values

of

supply pressurepo.



EXPERIMENTAL TESTS

479

engaged in the dynamometer

(5)

which hides

it.

The base (2) carries the support (41,

then the dynamometer and then the nut; it also carries the four capillary compen-

sated bearings (3), which support the nut.

A

variable speed motor

(6)

moves the

screw, which

is

loaded axially by the hydraulic jack

(12).

The various joints elimi-

nate the effects of construction and assembly errors and those resulting from

strains in the structure under loads. The screw and nut being tested are self-

regulated. Figure 13.15.a shows the nut

(2)

with its two lateral seals (1) and (3)

which cannot entirely prevent leakages. The nut is supplied through the

dynamometer (Fig. 13.15.b) by means of two lateral pipes (5')of small diameter and

thickness equipped with strain gauges. So these pipes are supply lines and also part

of the dynamometer.

In Fig. 13.15.c the screw and nut load capacity

P'

is plotted as a function

of

ec-

centricity E , with the screw

at

rest. For

&=0.5

(and for higher values not shown in the

diagram but very frequent in common practice) the experimental values are re-

Fig.

13.16

Test-rig for slide-way: ( 1 ) slide-way,

(2)

frame, ( 3 ) hydraulic jack.



480 HYDROSTATIC LUBRICAJlON

markably lower than the theoretical ones. This may be due to the leakage mentioned

from the nut seals and to construction and coaxiality errors of the screw and nut.

13.3.6

Slide-way

Figure

13.16

shows a hydrostatic slide-way

(1)

(ref.

13.10)

and

its

frame of

H-beams (2) (ref.

13.11)

loaded statically and dynamically by hydraulic jacks

(3).

It is

a slide-way of a boring machine under which the upper pads are obtained (pads

1

in

Fig.

13.18)

by setting two ledgers, the front one of which can be seen partially in Fig.

13.17.The other two pads, the lower and lateral ones (pads

2

and 3 in Fig. 13.18)are

located in the L-blocks (4) fixed underneath the ledgers. In the illustration the

micrometric screws

(5)

or vertical and horizontal film calibration can be seen.

In Fig.

13.18

the film thickness hl of the upper pads

(1)

is plotted as a function of

the static load

Fz.

ts theoretical and experimental variation is almost linear.

Fig. 13.17 Test-rig for slide-way: ( 1 ) ledger, (2) L-block, (3) micrometric screw s.



EXPERIMENTAL TESTS

481

kg

Fig. 13.18 Film thickness h , versus normal load Fz .

R E F E R E N C E S

13.1

13.2

13.3

13.4

13.6

13.6

13.7

13.8

Czichos H.; Tribology;Elsevie r, 1978; 400 p.

Loeb A. M., Rippel

H.

C.; Determination of O pt im um Proport ion for Hydro-

static Bearings;

ASLE Trans,

1 (1958),

241-247.

Meo F.; La lubri f icazione Zdrostat ica Realizzata con Alimentazione Attra-

verso Resistenze Zdrauliche e le Sue Applicazioni ai Cuscinetti Piani (ZZZ

parte); Lubrificazione Industriale e per Autoveicoli, 1968,N.

8;

p. 19-26.

H o w a r t h

R.

B., Newton

M.

J.;

Invest igat ion on the Effects of Ti l t a nd Sl tding

on the Performance of Hydrostatic Thrust Bearings; Ins tn Mech Engrs , C20

Bassan i

R.; Calcolo Numeric0 del le Grandezze Caratterist iche dei Patt in i

Zdrostatici; Automazione ed Autom atismi, Anno XIV (1970),

N.

3; p. 20-30.

Bassan i R.; Ricerca Sperimentale sui Pattini Zdrostatici; Automazione ed

Autom atismi, Anno XIV (1970), N. 4; . 3-14.

Leonard R., Davies

P.

B.;

An

experimental Invest igat ion of the Dynamic

Behav iour o f a Four Recess Hydrostatic Journal Bearing ;

Ins tn Mech E ngrs ,

V e r m e u l e n M . ;

Dynamic Behav iour o f Hydros ta t i c Rad ia l Bear ings ;

Vibra t ion and Wear Damage in H igh Speed Rota t ing Mach inery ; p roc .

NATO/Adv. Stu dy In st. , Kluwer Acad. Publ., D ordrech t, 1989; 16 p.

(1971), 146-156.

C29

19711,245-261.



482

HYDROSTATIC 1 BRlCATlON

13.9 Bassani

R.;

he Self-Regulated Hydrostatic Screw and Nut;

Tribology

Inter-

national, 12

(19791, 185-190.

13.10 Bassani

R.,

Culla C.; Progetto e Costruzione

di

una Slit ta di Macchina Uten-

sile, a Lubrificazione Zdrostatica; Atti 1st. Mecc.

Appl.

Costr. Macch., Univ.

di Pisa, Anno Acc. 1973-74,

N .

47; 69

pp.

13.11 Bassani R.,Culla C.; Progetto e Costruzione d i una Attrezzatura pe r Prove di

Carico su una Slit ta Idrostatica di Macchina Utensile . Primi Risultati Spe-

r imental i ; Atti 1st. Meccanica, Univ. di Pisa, AIM

7612,

1976;51

p.



Chapter 14

APPLICATIONS

14.1 INTRODUCTION

In the first chapter we have already mentioned that hydrostatic lubrication has

been successfully applied in many branches of mechanical engineering, from

large, slowly rotating machines to small and fast machines.

In this chapter, a number of applications will be briefly described, beginning

with the very important field of machine tools. Certain types of hydrostatic tilting

pads used to build bearings for large machinery, such as telescopes, air preheaters,

ore mills, debarking drums, and so on will then be examined.

Lastly, after having mentioned a few applications of a different kind, a number

of supply systems will be described, with particular reference to constant-flow sys-

tems making use of flow dividers or multiple pumps.

14.2 MACHINE TOOLS

14.2.1 Spindles

Machine tool spindles form one of the most common fields of application of ex-

ternally pressurized lubrication, since a high degree of stiffness and damping (i.e.

precision characteristics) is required.

Hydrostatic spindles may be supported by separate journal and thrust bearings,

as well as by a couple of opposed conical bearings; in certain cases other configura-

tions may prove to be suitable: for instance, conical bearings may be substituted by

spherical bearings,

o r

an opposed-pad bearing and a journal bearing may be com-



484 HYDROSTATC LUBRlCATlON

bined in a Yates configuration. Lubricant may be supplied directly, by means of

multiple pumps, o r at a constant pressure (restrictor-compensated). The latter

method is generally preferred because it is simpler.

As

a matter of fact, the com-

pensating restrictors may be easily incorporated in the spindle housing; i t is there-

fore possible to build compact standardized units with only one inlet and one outlet

port for lubricant: the supply system has merely to deliver lubricant at a given con-

stant pressure and a t a temperature varying in a reasonably narrow range.

Examples of spindles equipped with separate radial and axial bearings are to be

found in Fig. 14.1 and Fig. 14.2. Figure 14.3.a shows how

a

combined journal and

thrust bearing (see also section 8.7) may be used in a spindle, instead of conven-

tional rolling bearings, Fig. 14.3.b.

In this connection it must be remembered that attempts have been made

t o

produce ranges of hydrostatic bearings with outside and inside diameters following

V

6

Fig.

14.1

Hydrostatic spindle with journal and thrust bearings (compensating restrictors are not

shown). (Reference 14.1).

U

Fig. 14.2 Hydrostatic spindle with journal bearing and combined journal and opposed-pad thrust

bearing. (Reference 14.1).



APPL

/CATIONS

- a - -b-

485

Fig 14.3 Hydrostatic spindle with a combined journal and thrust bearing (ref. 14.1).

the

IS0

series for rolling bearings. In particular, the bearings depicted in Fig.

14.4

(ref.

14.21,

mainly intended

for

use in machine tool spindles, follow the

IS0

"0" se-

ries (their main dimensions are given in Table

14.1).

After this

first

experimental

range, another range was produced with similar dimensions and performance, but

without the built-in seals, as shown in Fig.

14.5.

- a - - b -

Fig. 14.4 Standardized hydrostatic bearings: a- journal bearing; b- combined journal and thrust

bearing (ref.

14.2).

In a ll the above units the journal bearings, of the multirecess type, with four

recesses, are characterized by narrow lands: this has been done in order to obtain

the greatest load capacity, while reducing the friction area and rise in temperature

in the lubricant. The thickness of the film can be chosen in a small range of values,

dependending on the stiffness and speed required, while the viscosity of the lubri-

cant should be chosen, as usual, bearing minimum power consumption in mind.

Journal bearings may also

work

without the inner ring.




T A B L E

14.1

Standardized hydrostatic bearing un

(mm)

(mm)

(mm) (mm)

50

80

60 75

60 95 70

90

70 110

80

100

80

125 90

110

90

140

105 130

100 150

115

140

110 170 130 160

120 180

140 170

d

D

dl Dl

- a - -b-

(see Fig.

14.4).

B (mm)

Journal b. Combined b.

68 75

80 85

88 95

98

106

110 115

118 125

126 136

140 152

-C

-

Fig.

14.5

Standardized hydrostatic bearings:

a-

journal bearing; b- hrust bearing; c- combined

journal and thrust bearing (ref.

2.2).

Bearing un its ar e usually fed

at

constant pressure and for this reason can be

provided with laminar-flow restrictors, made up

of

a

stack of special discs fitted in

proper holes in the outer ring, very near the recesses of the bearing (Fig. 14.6.a).

These discs are

of

two types: one

is

plain with

a

hole in

its

centre, whereas the other

has

a

rectangular groove on both sides. Restriction

is

obtained in the grooves, since

they a re shallow (however, not less th an 80 pm). The total hydraulic resistance of

the restrictor may be changed by varying the number of stacked discs.

Another type of variable restrictor

is

shown in Fig. 14.6.b: in this case,

a

set-

screw is used to adjust the hydraulic resistance.

Bearings can also be feed a t

a

constant flow rate: this may be convenient espe-

cially for thru st bearings, in order to increase stiffness,

at

the cost of a slightly more

complicated supply system.



- a -

t

APPLlCATlONS

- b -

487

1

2

3

2

3

4

.c

t

t

Fig. 14.6 Variable restrictors. a- Laminar-flow disc restrictor:

1

-locking ring, 2-spacer disc, 3-re-

strictor disc, 4-bottom disc

(ref.

14.2).

b-

Laminar-flow screw

restrictor.

A typical application of the aforegoing standardized units is shown in Fig. 14.7.

Another example is shown in Fig. 14.8: a spindle for a vertical grinding machine

supported by two journal bearings and an opposed-pad thrust bearing (ref. 14.3). In

the latter example a high degree of axial stiffness was required: for this reason it

was decided to feed the thrust pads at a constant flow rate, by means of a flow di-

vider; the radial stiffness of the spindle, measured a t the nose, was found to be 180

N/pm under a

300 N

oad (the spindle diameter was 80 mm, the supply pressure 5

MPa), whereas axial stiffness was 500 N/pm under a 800 N load; maximum axial

load was 14

KN,

since the maximum supply pressure of the thrust bearings was

limited to 8 MPa.

Figure 14.9 shows a different type of spindle, used in a plane grinding machine,

borne by a journal and an opposed-pad thrust bearing. Note that this type of com-

bined bearing may be made to support large tilting moments using multirecess

thrust pads (section 8.4) instead of the simpler annular-recess pads.

Fig. 14.7 Hydrostatic spindle w i t h a journal bearing and a journal and thrust bearing (ref. 2.2).



488


E

Fig. 14.8 Hydrostatic spindle for a grinding machine; the thrust bearing is fed at a constant f low rate

(ref. 14.3).

Fig.

14.9

Hydrostatic spindle with a journal and a double-effect thrust bearing. (Reference 14.1).

For tapered-bearing spindles the most common configuration seems to be that to

be found in Fig. 14.10, although different types of spindles have been built, for in-

stance with cones arranged as in Fig. 8.19.a. Standardized spindle units are cur-



A

PPLICA

TIQNS 489

Fig. 14 .10 Hydrostatic spindle with conical

bearings.

Ring

R

is used to adjust film thickness.

(Reference 14.1).

rently produced, which are interchangeable with rolling-bearing o r hydrodynamic

units produced by the same firm; of course, the main spindle dimensions comform

with international standards for machine tools (ref

14.4).

A n example of a standard spindle unit is shown in Fig.

14.11

nd the main rele-

vant data are to be found in Table 14.2 ref.

14.4).

L a -

Fig. 14.11 Standard spindle unit with cylindrical housing for boring, turning or milling (ref. 14.4).

Selection of the main hydrostatic parameters (number of recesses and their

dimensions, film thickness, lubricant viscosity and so on) is generally made case-

by-case by the manufacturer, on the basis of the operating conditions for which the

spindle is designed (mainly load and velocity) and also on the basis of particular

requirements, concerning stiffness and damping.

Comparing the data in Table 14.2with data concerning the equivalent spindle

units equipped with ball bearings (ref. 14.41,

t

should be noted tha t the hydrostatic

units show greater radial stiffness (although units equipped with special roller

bearings are much stiffer). It should be borne in mind, however, that the stiffness of



4 9 0

Size

3

4

5

6

8

11

H Y D R O S T A X

LUBRlCATION

D a d CR

t l

t2 nmax

(mm) (mm)

(mm)

(W @m)

(rpm)

120 350 40 130

0.5

0.5 8500

150 450 50 200 0.5 0.5

7000

180 550 70

330

0.5 0.5 5500

230 650

90 550

0.6 0.6 4000

300

850 110

750 0.8 0.8 3000

380 1050 150

1000

1 1 2000

the hydrostatic units is proportional

t o

supply pressure, and may be considerably

affected by large axial loads (see section 8.5.2).A distinguishing feature of hydro-

static spindles is their very good running accuracy: values of t l and t 2 are always

smaller than

1

pm, whereas the values of similar ball-bearing spindles range from

2 t o 4

pm for

t l

and from

1.5 to 2

pm for

t 2

(these values may even double int he case

of

roller-bearing spindles).

Figure 14.12 shows an opposed-cone multirecess bearing that may be used t o

build hydrostatic spindles (ref. 2.11, as in Fig. 14.13. Note that , in this case, only the

right-hand bearing sustains axial loads, whereas the other is used as a pure radial

bearing.

Fig.

14.12

Hydrostatic opposed-cone

bearing.



APPLICATIONS

491

Fig.

14.13

Hydrostatic spindle

with

a pair of opposed-cone bearings.

14.2.2

Steady rests

Mounting of long and heavy rotors (e.g. turbine rotors, steel mill rolls, calen-

ders, etc.) on lathes or other machine tools often requires the use of steady rests in

order

t o

relieve the headstock and tailstock spindles from excessive loads and t o

reduce the bending of the axis of the workpiece. On the other hand, conventional

steadies are characterized by high friction, with the relevant wearing and heating

of the rubbing surfaces: these problems can be completely eliminated by means of

hydrostatic lubrication.

A

steady for a heavy machine tool may easily be built with a couple of self-align-

ing shoes (ref. 14.5) f the type shown in section 14.3. ach shoe must be mounted on

a radially adjustable support to allow exact positioning of the workpiece.

In the application described in ref. 14.6, teadies for sustaining rubber-coated

cylinders (up to 600 KN in weight) on a grinding machine have been built. The

hydrostatic shoe is provided not only with a spherical seat allowing tilt in all direc-

tions, but also with a screw and nu t assembly for easily adjusting the radial position

of the shoe (Fig. 14.14). t should be noted that the intermediate piece of the bearing

is fitted in a hydraulic cylinder which is widened in the base piece; pressure in the

cylinder is the same as in the recess: in this way the fillets of the screw and nu t are

loaded with only a fraction of the force acting on the bearing. In this case the cylin-

der to be machined does not lean directly on the shoe bearings since intermediate

rings are fitted on the necks of the cylinder: the same steadies can hence be used

with different workpieces without needing to change the shoes, but using different

rings, all of which have the same external diameter.



492

HYDROSTATIC 1

UBRlCATION

Fig. 14.14 Adjustable hydrostatic shoe

bearing (ref.

14.6).

14.2.3

Feed drives

Modern high precision machine tools require feed drives with high feeding

accuracy, freedom from backlash and low friction. For these reasons recirculating-

ball lead screws and nuts are widely used. Hydrostatic lead-screw nuts meet the

same requirements and also have other advantages as compared to recirculating-

ball nuts. In particular, they are inherently free from backlash (without the need

for mechanical preload) and from wear (which ensures continuity of performance)

and have better damping properties. This last feature has a certain importance in

machines with roller-bearing or hydrostatic guideways, since the intrinsic lack of

damping in the feed direction of frictionless guides can lead to poor stability against

chatter in the same direction (ref. 14.7). Moreover, construction of the lead screw

should be simpler in the case of hydrostatic nuts, since a very high degree of surface

hardness

is

not required.

Nevertheless, hydrostatic nuts are much less used than recirculating ball nuts

(a t least in small and medium-size machines). The main reasons are , probably, the

following: recirculating-ball nuts are well proven and perform satisfactorily; hydro-

static lubrication requires a high-pressure lubricant source; construction of hydro-

static nuts is much more difficult and critical than other types of hydrostatic bear-



APPLlCATlONS 493

63 36.5

ings. This last is also obviously true for recirculating-ball units, but does not consti-

tute a drawback in this case, since they are easily available in the stock of special-

ized manufacturers. Since some firms have recently begun to produce a wide range

of standardized screw and nut assemblies, this type

of

feed system

is

expected to

spread in the future.

Data concerning a range of hydrostatic screws are to be found in Table 14.3 ref.

14.8).

The nut constitutes a compact unit, with built-in restrictors and seals , an

inlet port and an outlet port, requiring only an adequate but fairly simple supply

system. Feeding accuracy depends mainly on the pitch error

of

the male screw, but

owing to the levelling effect

of

hydrostatic lubrication the manufacturer claims that

actual feeding inaccuracy is less than one third of the pitch fluctuations of the male

screw.

L o

T A B L E 14.3

Hydrostatic screw

and

nuts (ref.

14.8).

D

173 134 2.79

181 1.733

205

166

3.72

241 2.300

108 167 124

20 2.26

147 0.755

207 164 3.39 220 1.133




As already noted, the manufacture of hydrostatic nuts is somewhat difficult,

either because of the relatively inaccessible position of the recesses, o r because a

small pitch difference in relation to the male screw can lead

to

a considerable loss of

loading capacity (see section

7.3).

Both

problems can be easily overcome by means of

a clever technique consisting in coating the inner surface

of

the nut with a thick

layer of plastic, which is cast while the lead screw is held in position; recesses are

obtained by means of patterns temporarily fixed

to

the flanks of the screw with an

adhesive (note that hydrostatic nuts are in general of the multirecess type rather

than of the continuous recesses type). The gap is obtained because of the shrinkage

of cast plastic (ref. 14.9).

In large machine tools it may be preferable to substitute the screw-nut feed drive

with rack and worm systems, which permit runs

of

practically any length, with a

high degree of stiffness; furthermore, stiffness proves t o be independent from run

length and the position of the slide.

These systems can also obviously be assisted with hydrostatic lubrication. An

example is tha t of the so-called hydrostatic "Johnson drive" (ref. 14.9) hown in Fig.

14.15. n this case, a short worm drives a long rack firmly fixed to the slide. The

Fig. 14.15 Hydrostatic Johnson drive (Ingersoll). 1-Slide, 2-rack, 3-pump pressure, 4-capillaries,

5-cells,

6-worm,

7-external gear teeth, 8-oil

supply

for forward flanks, 9-bed.



A PPLICA TIONS

495

worm is supported by means of hydrostatic thrust bearings; its circumference is

toothed and is in mesh with a pinion driven by the feed gear. Recesses are hollowed

in the flanks of the rack. A simple distributing device is needed to deliver lubricant

only

t o

the recesses covered by the worm, hence avoiding a considerable waste

of

power.

In other applications (see, for instance, ref.

7.1

o r ref. 14.11) the rack is fixed t o

the bed, whereas the worm is supported by the slide, together with the relevant feed

gear, which drives it by means of a toothed gear, fitted near the worm on the same

shaft. Lubricant is supplied through ducts drilled in the worm; recesses may be

hollowed in the flanks of the rack (as in Fig. 14.16) s well as in the flanks of the

male screw.

Fig. 14.16 Hydrostatic rack and

worm;

iarnete-270 mm, itch=60 mm (INNSE).

In this case, too, a distributor is needed in order to cut off the high-pressure

supply of lubricant to the ducts not ending on the flanks of the rack. When speed is

high (speeds up to 750 rpm can be used) the centrifugal force may empty inactive

ducts and that may cause aeration of the lubricant: hence the supply distributor

should incorporate a pre-filling device whose task is to pump lubricant

at

low pres-




sure into the inactive ducts, just before they become active again (a similar device is

described in ref. 14.12).

Hydrostatic worms are generally built with a pitch of between

36 and 60

mm and

an outside diameter of between

150

and

300

mm; load capacity may vary between

50

and

180 KN.

The rack may be of virtually any length since

it

is built in sections (for

instance, 1000 mm in length) that are bonded and firmly bolted to the slide bed after

having been adjusted in relation to one another and measured to verify the pitch

error (ref. 7.1). Accuracy may be about 70i-80 pm on a length of 25 m. Owing t o this

accuracy and to the very high degree of stiffness this feed system can also be used

for monitoring the position of the slide during normal operation (by means of elec-

tronic compensation the relevant error can be further reduced to a very small

value).

14.2.4 Guideways and rotat ing tables

Hydrostatic lubrication proves to be particularly suitable

for

guideways

of

mod-

ern high precision machine tools (especially those equipped with numerical con-

trol), because of their intrinsic characteristics:

very low friction (and proportional to speed);

freedom from stick-slip;

freedom from wear (which means constancy of performance for an indefi-

nite time);

thickness of the oil film independent of the sliding speed (whereas for lubri-

cated plain bearings it increases with speed);

high damping capacity fin directions perpendicular to guide);

levelling ability: the fairly high film thickness (commonly a few hundredths

of

a

millimeter) allows the hydrostatic lubrication to compensate, a t least

partially,

for

small geometric inaccuracies and deformations

of

the guides;

possibility of building guides of virtually any length (which is difficult with

roller guides).

On the other hand, it should be noted that the virtual elimination of friction can

enhance the effects of the flexibility of other parts of the machine and in particular

of the feed drive. For instance, consider the experimental diagrams in Fig.

14.17

(see ref.

14.13

for further details): they refer to a milling machine and show that the

displacement due to loading in the direction of the guides (mainly due to the flexibil-

ity of the ball screw and nut and of the relevant thrust bearing) is greatly reduced by

the friction of the sliding ways.

Diagrams in Fig.

11.18

(ref.

14.71,

obtained with a similar experimental rig,

show the reduction in damping connected with the use of frictionless guides (either



APPLlCATlONS

497

0

10000 kN 20000

load

P

Fig. 14.17 Influence of hydrostatic ways on static stiffness, compared with sliding ways.

a-

Hydrostatic system

in

action; b- without the hydrostatic system.

plain or ball screws were used

as

feed drives, without leading to any notably differ-

en t behaviour).

Problems of this kind are easily eliminated by means of simple clamping de-

vices when feed rate

is

null, whereas in other cases they may be solved by stiffening

th e feed drive (for instance, in heavy machines, by selecting a worm and rack feed

drive instead of screw an d nut), by eliminating any backlash a nd increasing damp-

I I I I

I

*

0 200

400 V

mmlmin

Fig. 14.18 Influence of feed rate

V

on maximum vibration amplitudeA (at resonance frequency)

along feed direction for: a- sliding guideways; b- hydrostatic guideways; c - roller guideways.




ing (for instance, introducing hydrostatic lubrication in the feed drive) or by means

of external dampers (ref. 14.13).

A number of different examples of layout for slideway guides are presented in

Fig.

14.19;

type

‘c ’

and

‘d‘

use an opposed-pad design: this

is

necessary when great

stiffness and damping are required for a large range of loading conditions. The

lower pads a re in this case much smaller th an t he upper ones, in order to compen-

sa te for th e weight of the slide.

- a -

- b -

- c - - d -

Fig.

14.19

Sample

layouts

of

hydrostatic guideways.

A compromise, often used in rotary tables, may consist in substituting the

preloading effect of the hydrostatic recesses on the underside of the guide with a

spring force applied by means of rolling bearings (in practice, this is a trick for

increasing the weight of the slide without increasing

its

mass).

At least two recesses must be used on each guide

to

absorb torque, but a larger

number of smaller recesses (each fed independently) provide greater compensating

ability for the geometric inaccuracies of guideways; moreover, since the load is

more evenly distributed on the guides, better results should also be obtained from

the point of view of elastic distortion. Recesses may be either of the conventional

fully-hollowed type,

or

be reduced to narrow grooves,

as

in the guides in Fig.

14.20.

From the point of view of static load capacity both designs perform in the same way,

but the narrow-groove recesses have greater damping ability and a larger bearing

area in the absence of lubrication (hence, they a re less prone t o damage in th e event

of failure of the supply system). On the other hand, friction is also much higher and

this type of recess proves to be adequate only for low-sliding velocities.

The ability of hydrostatic lubrication to even out inaccuracies due to manufac-

turing errors or deformations caused by external forces is limited by the thickness

of the lubricant film. Especially in the case of very large and heavily loaded slides



APPLlCA

TlONS

499

Fig.

14.20

Rototraversing table equipped with hydrostatic lubrication of the guides

(INNSE).

In

a

the thrust bearing of the rotary table

is

shown; the pinions of the feed drive are also visible, a s well as

four clamps that may be used to

fix

the angular position of the table and the lam inar-flow restrictors.

In b the same table is shown from another angle: the linear guideways are visible, as well as two

clamps.



500


and rotary tables (such as the rotary table of a large vertical lathe), elastic deforma-

tion might even force the designer to select an excessively thick film to avoid metal-

to-metal contact. A solution may be to build the guideway with self-aligning tilting

pads, as will be shown in section

14.3.

The geometric inaccuracies of the slideways (e.g. waviness) might be completely

compensated by controlling recess pressure: the principle is outlined i n Fig. 14.21

(see also ref.

14.10).

Pressure in each recess is controlled by a valve, piloted by a

regulator which compares a reference signal with the signal produced by a trans-

ducer. This last is, for instance, a pneumatic sensor which monitors the position of

the slide in relation to

a

reference straight edge,

o r a

photoelectric sensor, which

uses a laser beam as a reference "guide".

1 2

Fig. 14.21 Scheme of compensating bearing

control.

1-Guide, 2-reference guide, 3-distance trans-

ducer, 4-regulator, 5-set value, 6-controlledvalve, 7-supply pressure.

A n example of hydrostatic lubrication applied to guideways is presented in Fig.

14.20,

n which details are shown of a hydrostatic rototraversing table: one of a wide

range of such equipment, suitable for indexing and contour milling (ref.

14.14)

with

a load capacity varying from

400

to

5000

KN.

A

similar range of rotary and roto-

traversing tables

is

also suitable for turning operations, with

a

turning speed of up

to 2565 pm, depending on the diameter of the table (2.5+10m).

The rotary table in Fig.

14.20

has a circular thrust bearing (with a mean diame-

ter of

1400

mm) made up of

12

pads, all fed independently through a set of laminar-

flow restrictors. These are made by cutting small-diameter (111.5mm) pipes t o the

appropriate length and are also visible in the photographs. With

a

supply pressure

of

6

MPa, the table can bear loads of up to 600 KN. The radial forces are sustained by

a tapered roller bearing, which also exerts a preloading force (150KN) on the hydro-

static thrust bearing, in order

t o

increase its stiffness. The photographs also show



APPL

ICATlONS 501

clamps that are able to hold the table firmly in any position, without affecting the

film thickness of the hydrostatic bearings.

Rotary motion is obtained by means of two controlled-preload pinions meshing

with a helical crown gear, whereas a ball screw

is

used for linear axis transmis-

sion (the largest members of the same family of tables use hydrostatic worms and

racks for axial feed drive).

Hydrostatic lubrication is often also applied to the guides of ram-type milling

arms (Fig.

14.22).

The design of the guides is of course different from that of the

guides of horizontal tables: in this case the ram

is

supported by two rows of eight

recesses (two for each side). The recesses in the lower end of the guide are generally

larger since they must support higher loads (in other applications there are three

rows of recesses, two of which are set

at

the lower end of the guide). The supply

system is made up of a set of multiple pumps (each pump directly feeds one recess),

Fig. 14.22.a-Hydrostatically lubricated milling

arm

Pensotti).



502

HYDROSTATC LUBRlCAT/ON

Fig. 14.22.b-

Hydrostatically lubricated milling

arm:

etail showing hydrostatic pads.

which are fed

at

constant pressure

(-2.5

MPa) by a larger pump. In this case, too,

the recesses of the pads (which are made of bronze) are reduced to narrow grooves.

It is

interesting that hydrostatic lubrication has also been used to compensate

for the deflection of the ram due

t o

the cutting force. The geometric adaptive control

system described in ref.

14.15

measures the displacement of the milling head by

means of a laser gun fixed

t o

the milling arm, which emits a laser beam parallel

t o

the undeformed axis of the ram, and a photoelectric scanner attached

t o

the milling

head. The signal produced by the measuring equipment is taken as its input by a

control unit which varies accordingly the speed of a servo-motor driving a further

set of pumps. The

flow

produced by these compensating pumps

is

directed towards

the appropriate recesses and added

t o

the normal flow in order to produce a dis-

placement of the milling head, realigning it with the laser beam.



A

P P l

CA TlONS

503

A particular application of externally pressurized lubrication to the ram guide

of

a

gear-shaping machine is described in ref. 14.10.

A

cross section of the guide is

shown in Fig. 14.23: the ram is shaped like a spur gear with every third tooth re-

moved. The accuracy of the internal bore of the sleeve is obtained by casting with a

plastic material (this technique is briefly described in section 14.2.3).

Fig.

14.23

Hydrostatic ram guide of a gear-shaping machine (Liebherr).

14.3

LARGE

TILTING PADS

Hydrodynamic bearings for very large rotating machine-members have been

equipped fo r many years now with high-pressure hydrostatic pockets, used as jack-

ing devices

at

starting (hydrostatic lifts). More recently, it has been found t o be

expedient to retain the hydrostatic effect in normal running and then t o substitute

the hydrodynamic bearings completely with hydrostatic (or hybrid) bearings, in the

case of slowly rotating machines in particular, or when irregularities in load

o r

speed are expected.

One problem connected with this type of bearing in certain machines (such as

ore mills) is that the elastic deformation of the runner, due to the pressure of the

lubricant, may greatly reduce the effectiveness

of

hydrostatic lubrication (Fig.

11.24.a). This problem may be overcome by foregoing the "optimum" design, ob-

tained by assuming rigid surfaces and uniform film thickness, and displacing the

recesses from the centre of the bearing (Fig. 11.24.b); separate pads may even be

used instead

of a

multirecess bearing (ref. 14.16).

A further improvement in design, able t o eliminate most of the problems con-

nected with elastic deformation, machining tolerance, thermal expansion and so



504

HYDROSTATIC

LUBRICATlON

Fig. 14.24 Trunnion deformation due to bearing pressure: a- bearing

as

designed;

b-

improved

concept; e- most effective concept (ref. 14.16).

on, consists in supporting the large journal by means of a set of self-aligning hydro-

static shoes, as shown in Fig. 14.25 (ref.

14.17).

Each shoe is split up into t w o parts:

the upper part rests on a spherical seat and hence can tilt in all directions. The

underside of the upper part is shaped like a piston which fits into a cylinder in the

base: since the piston area, on which the recess pressure acts, is slightly smaller

than the effective area of the pad, the load on the spherical seat is quite low during

normal operation.

Fig. 14.25 Arrangement of tilting-pad hydrostatic bearings (ref. 14.17).



APPLICATIONS

505

The spherical rest of each inner shoe (slave shoe) is pushed against the runner

by a further piston on which, thanks to a hydraulic connection, the recess pressure

of the relevant outer shoe (master shoe) acts. Clearly, if the sum of the two piston

areas equals the effective pad area, the slave shoe must necessarily have the same

film thickness, and thus the same recess pressure, as the relevant master shoe

(each pad is fed by the same flow rate). Thus when the load direction is vertical all

four shoes have the same film thickness and recess pressure, regardless of the

deviation

of

the runner from the ideal circular shape. When the load deviates from

the vertical direction the

two

shoes on each side have an equal part of the load com-

ponent falling along the line between the two shoes (ref. 14.17).

The shape of the recess is also of particular interest. It is known that when a

cylindrical pad with a simple recess

(as

in Fig.

5.30)

s

tilted from the concentric

configuration the pressure field on the land surface is altered and produces a

moment that tends to realign the pad; however, this self-aligning capacity

is

too

small to ensure the stability of the shoe in all conditions and i t is hence necessary to

use multirecess pads. In Fig. 14.26 he main recess is surrounded by four auxiliary

recesses, situated in the corners of the pad, which are fed with the lubricant which

passes from the central recess over the bearing lands and through small drilled

ducts (this is a compromise aimed at avoiding dependence upon the direction of

rotation: for the greatest stability the auxiliary recesses on the trailing side should

only be supplied over the lands).

The hydrostatic system described in ref. 14.17 upported a large tube mill for

crushing ores: each bearing runner had a diameter of 2700 mm, the maximum

I

I

L

-

Fig. 14.26 Improved recess pattern (ref. 14.17).

W

.

I



506 HYDROSTA

T C

LUBRlCA

T/ON

load was

3500 KN

and the velocity was 0.24 re ds . Each shoe was 640 mm long and

500

mm wide and was fed at 25 Ym with a lubricant whose viscosity was 0.1 Ns/m2

at 50°C. Film thickness in normal operation was 0.1410.15 111111.

A range of hydrostatic shoes based on the foregoing working principles is cur-

rently produced by the same firm (ref. 14.5): a sketch of them i s to be found in Fig.

14.27 and their main dimensions are given in table 14.4. The recess patte rn is simi-

lar t o that shown in Fig. 14.26, but the main recess is now annular in shape, in

order t o increase the bearing area a t rest ( in the absence of hydrostatic lubrication)

virtually without affecting bearing performance during normal operation.

Hydrostatic shoes may be used to support horizontal as well as vertical rotating

equipment. In the first case the rotating drum may lean on the shoes by means of

trunnions (Fig. 14.28.a) or by means of girth rings (Fig. 14.28.b). The lat ter a r-

rangement, which is often inapplicable with rolling bearings due to their size lim-

its, permits large feed openings and a simplified (and less expensive) design. Each

- a - - b -

Fig.

14.27

Bearing shoes:

a-

master shoe; b- slave shoe.

T A B L E 1 4 . 4

410

530

SO0 640

600 756

H

(mm)

Master

Slave

180+190

2601270

300 2951305

320+330

425 4201430



APPLlCA

TlONS

507

ring (or trunnion)

is

supported by two master shoes, to each of which one or two

slave shoes may be added

t o

boost the load-carrying capacity (Fig.

14.29).

The sug-

gested ring diameter D varies between 500 and 5400 mm, with a load capacity F

ranging from

480

t o

12000

KN,

depending on pad size and the total number of pads.

- a - - b -

Fig.

14.28

Horizontal rotating arrangements: a-

trunnion

arrangement;

b-

girth ring arrangement.

D

as ter s hoe: O S l a v e s hoe

Fig.

14.29

Shoe arrangements

for

horizontal rotating cylinders.

In the case of vertical equipment three master shoes are obviously required in

order

t o

obtain a statically determined load distribution; t o each master shoe a slave

unit can be added, thereby doubling the load capacity. A typical arrangement is

shown in Fig. 14.30, in which two alternatives are also proposed for the radial guid-

ance of the runner: a rolling bearing mounted on the shaft, or a set of hydrostatic

guiding pads (see below). For six pad arrangements the load capacity ranges from

1900 t o 14000 KN (depending on the pad size) and correspondingly the minimum

pitch diameter D varies from 800 to 2400 mm.

Besides the hydrostatic shoes described above, the same firm produces a range

of smaller tilting pads of simplified design (see Fig. 14.31 and table 14.5).These still

retain

a

self-aligning capacity, since they have a spherical seat and multiple re-

cesses, but are not equipped with hydraulic cylinders. They are mainly proposed

(ref. 14.5) as guiding pads for the axial location of

a

girth ring (Fig. 14.32) o r for the



508


r

D

ALT

II

I

L T I

Fig.

14.30

Shoe bearing arrangement

for

vertical rotating equipment.

radial guiding of platforms (Fig. 14.30). Compact assemblies are also available con-

sisting in a master shoe bearing with t w o guiding pads (in an opposed-pad configu-

ration) mounted on the fixed part

of

the shoe (Fig. 14.32.b).

Another type of tilting pad (ref. 14.18) can be used t o build spherical th rust bear-

ings with

a

very large diameter. In practical terms, it consists

of

a circular recess

pad laid on a spherical rest whose position can be adjusted by means

of

a wedge. In

Fig. 14.33 a set

of

twenty pads is used

t o

build

a

large bearing (with a mean diameter

of

5000 mm) for a large parabolic antenna. The dimensions of the bearing and angle

a depend on the value of the axial and radial components of the load: bearings with

an external diameter of up t o

8000

mm can be built.



APPLlCATlONS

Fig. 14.31 Guiding pad.

509

T A B L E

14.5

Dimensions of guiding pads (ref. 14.5).

Fig. 14.32 Axial guiding pads: a- separate axial guidance; b- axial guidance integral with a master

shoe.

A further type of tilting shoe is shown in Fig. 14.34 ref. 14.19): t can ti lt around

the cylindrical rib on the underside and align itself thanks to the multiple recesses

(two o r four) which are fed independently through capillary restrictors. These pads

can

a lso

be used to sustain radial loads as well as the axial thrust of a large rotating

platform. In the latter case, bearings with diameters exceeding 5000 mm may be

built, which sustain thrusts greater than 5000 KN and rotating at more than 20

rpm. These pads prove t o be particularly suitable for building rotary tables for large

machine tools (e.g. for vertical lathes): an example is given in Fig.

14.35.

A different

application

is

described in ref.

14.20,

oncerning the supporting ring of a

3.5

m

telescope.



510


P I

Fig. 14 .33 Spherical pad arrangement,

Fig. 14.34 "Hydro-tilt"shoe bearing (ref. 14.19).



APPLlCA TlONS

511

Fig. 14.35 "Hydro-tilt" shoe

arrangement.

Lastly, Fig. 14.36 shows a large-size spherical bearing (ref. 2.2); it has three

recesses fed a t a constant

flow

rate. Bearings like this can sustain heavy loads (up to

10,000 KN) and in general their rotating speed is low.

For

instance, the bearing

depicted

in

Fig. 14.36 was made

to

support the rotor

of

an air preheater weighing

800

KN

and rotating at 2 rpm.

14.4 OTHER APPLICATIONS

Apart from those quoted in the foregoing sections, hydrostatic lubrication has a

number of different applications.

For

instance, let

us

consider the pump in Fig. 14.37 (ref. 14.21): the pistons (1)

lean on the tilted plate (2) by means of the spherical pads (3) which are hydrostati-

cally borne by the same circulating fluid.

Another special application is quoted in ref. 14.22, that is the lower journal bear-

ing of the main pump

of

the Super-PhBnix nuclear power plant. This bearing has a



512

HYDROSTATIC L UBRICATlON

L . L . 1

r ' t ' i

Fig. 14.36 Large sphericalbearing.

Fig.

14.37

Piston

pump.

diameter of 0.85 m and a width

of

0.3m; it has twelwe recesses carved in the shaft.

In this case, too, the lubricant is the fluid circulating in the plant, i.e. liquid

sodium.

Hydrostatic lubrication has been successfully used in a number

of

testing rigs.

An example

is

shown in Fig. 14.38:an experimental rig for testing rolling bearings

(ref. 14.23).The bearing being tested (1) is made t o rotate by a motor (2) y means of a



APPLICATIONS

513

Fig.

14.38 Experimental

rig

for rolling bearings.

belt drive

(3)

and is loaded by a jack (4) through the hydrostatic bearing (5); his last

leans on a cell

(61,

which measures load

Fa,

by means of

a

spherical seat. Friction

moment

MR

s measured by means of dynamometer

(7)

nd the

angular

speed

n

by

means of thacheometer

(8).

14.5 HYDRAULIC

CIRCUITS

14.5.1 Simple layout

A

typical supply system for hydrostatic spindles, such

as

the one in Fig.

2.24,

is

shown in Fig.

14.39 (ref.

2.2).

The bearings are fed

at

a constant pressure, which is

usually in a range between

3 and 7 MPa. A gear pump supplies lubricant

at

a rate

which is 30% greater than the calculated value: the

surplus

flows back to the reser-

voir through the pressure regulating valve. Lubricant is pushed through two filters,

the first of which

is

coarser

(15

pm), while the other

is

narrower

(5-10

m).

A

pres-

sure switch prevents the spindle from running until the pressure reaches the estab-

lished value and stops it when pressure falls: in the last case,

an

oil accumulator



514


Fig. 14.39 Sup ply system for a hydrostatic spindle: 1-oil tank; 2-pum p; 3-motor; 4-pre ssure regulat-

ing valve; 5-pressure filter; 6-pressure switch; 7-check valve; 8-piston accumulator; 9-pressure

gauge; 10-cooler;

1

1-thermostaticsystem; 12-heater.

1 2

Fig. 14.40 Supply system for the hydrostatic bearing of

an

air preheater: 1-pum p; 2-motor; 3-pres-

sure filter; 4-pressure switch; 5-check valve; 6-flow divider; 7-piston accum ulator; 8-pressure-limit-

ing valve; 9-cooler.



APPLICA TlONS

515

keeps on feeding the bearings for the time needed for the spindle to come to a com-

plete stop.

A

thermostatic system keeps the temperature of the lubricant close to the

design value. Sometimes a further pump may also be needed (generally inserted

upstream from the cooler)

t o

bring the lubricant back from the spindle to the

reservoir.

14.5.2

Flow div iders

Figure 14.40 shows the supply circuit for the three-recess preheater bearing in

Fig. 14.36.The flow rate produced by the main pump is divided into three equal

streams by means of a flow divider made up of three equal gear pumps mounted on

\

3

8

1

1 9

17

5

6

4

3

2

15 16 14 1

Fig. 14.41 Supply system for the hydrostatic bearing

of

an ore mill: 1-oil tank; 2-pump; 3-pressure

switch; 4-pressure filter; 5-check valve; 6-pressure limiting valve; 7-pressure gauge; 8-piston accu-

mulator; 9-nitrogen ga s bottle; 10-flow divider; 1 1-shoe bearing; 12-circulation pum p

for

cooling

circuit 13-throttle valve; 14-oil cooler; 15-water flow control valve; 16-temperature-sensing device.



51

6 HYDROSTATIC LUB RICATlON

a common shaft. To ensure continuous operation a second pump is ready to be

started up automatically when supply pressure drops below

a

safe value. A further

spare pump is available for replacement, to permit maintenance operations. In the

case

of

an electric mains failure a diesel generator can provide power for the motors

of the pumps. The last emergency device is a set of oil accumulators which can

supply lubricant to the bearings for a short time.

The hydraulic circuit for the bearing arrangement in Fig. 14.25

is

shown in Fig.

14.41. The flow rate produced by the main pump is divided into four equal streams

by means of a flow divider. To ensure continuous operation a second pump

is

ready

to be started up automatically in case of failure of the other one and

a

set of piston

accumulators (driven by pressurized nitrogen bottles) can feed oil to the bearings for

a certain time in case

of

power failure, allowing the runner

to

stop without damag-

ing the bearings.

14.5.3 Mult ip le pumps

The constant-flow supply circuit

of

the guideway presented in Fig. 14.22.b is

shown in Fig. 14.42. The pre-feeding pump (1)delivers lubricant at

a

pressure of 25

bar to two multiple pumps (2). Each pump can feed ten recesses independently,

each with a

0.33

m3/s flow rate, a t a pressure

of

40 bar.

Fig.

14.42 Supply system, with multiple pumps,

of

a hydrostatic slide.



A PPLlCATlONS 517

Figure 14.43 shows the supply circuit of the pad arrangement in Fig. 14.33.

Each pad is directly fed at a constant flow rate; that is, a set of five multiple pumps

is used and each pump delivers

four

equal streams which

are

supplied to four pads

situated a t

90

degrees from each other. Thanks to the layout mentioned, emergency

operation of bearing system is

possible even if

a

pump fails.

Fig. 14.43 Supply system

for

the bearing system of a large-beam antenna: M-motor; P-multiple

pump; 1-20pads.

REFERENCES

14.1

14.2

14.3

14.4

14.6

14.6

14.7

Die Arbeitsspindel und Hire Lagerung

-

Herzstuck Leistungsfahiger Werk-

zeugmaschinen; FAG publ. 02-113 A (1985); 8pp.

Hallstedt

G.;

tandardized Hydrostatic Bearing Units;Instn. Mech. Engrs.,

C48

(1971),420-430.

Lewinschal L.; Contributo dei Cuscinetti Zdrostatici allXumento di Produt-

tivitb delle Rettificatrici;La Rivista dei Cuscinetti/SKF, 196(19781, 4-27.

FAG Spindeleinheiten fur das Bohren-Drehen-Frasen; FAG publ. 02-1OW3

DA (1985); 2pp.

Hydrostatic Shoe Bearing Arrangements;SKF Publication 3873 E (19881, 8

PP.

Bildtsh C., Htillnor G.; Problema Risolto con 1'Adozione di Pattini Idrosta-

tici;

La

Rivista

dei CuscinettYSKF,

181

(1974),18-20.

Polseck M., avra

Z.;

The influence of different types of guideways on the

static and dynamic behaviour of feed drives;Proc. 8th Int. MTDR Conf. (19671,

pt. 2,

p.

1127-1138.



518

HYDRCSTATIC LUBRICA

TlON

14.8

Catalog B1025E;Nachi Corp., Japan,

1984; 4

p.

14.9

Weck M.; Handbook of Machine Tools, Volume 2 (Construction and Mathe-

matical Analysis);

J.

Wiley & Sons,

1980; 296

pp.

14.10

Rohs H. G.;

Die Hydostatische Bewegungspaarung im Werkzeugmaschinen-

bau;Konstrudion,

22 (1970); 321-329.

14.11

Andreolli C.; Eliminazione dell'Attrito e dei Giochi nelle Macchine Utensili;

Controlli NumericiIMacchine a CN/Robot Industriali, anno XI1 (19791, n. 7,

14.12

Appoggetti P.; Perfezionamento negli Accoppiamenti Vite-Cremagliera a

Sostentamento Zdrostatico;Patent IT

51829 N69;

Bollettino Tecnico RTM n.

9,

14.13

Umbach R., Haferkorn W.; Some Examples and Problems in Zmplementa-

tion of Mwlern Design Features on Large Size Machine Tools;

10th

Int. MTDR

cod., Manchester,

1969;

paper

34; 30

pp.

14.14

Rototraversing Tables for Indexing, Milling and Turning; INNSE Publication

DMU/27 (1985),4

pp.

14.15

Weck M.; Handbook of Machine Tools, Volume

3

(Automation and Controls);

J . Wiley & Sons,

1980; 451

pp.

14.16

Rippel T., Hunt J.

B.;

Design and Operational Experience of 102-Znch Diame-

ter Hydrostatic Journal Bearings for Large Size Tumbling Mills; Instn.

Mech. Engrs.,

C16 (1971), 76-100.

14.17

Arsenius H. C., Goran R.; The Design and Operational Experience

of

a Self-

Adjusting Hydrostatic Shoe Bearing for Large Size Runners;

Instn. Mech.

p.

32-45.

1969;

p.

47-51.

EWS.,

C303 (19731,361-367.

14.18

Supporti idrostatici FAG;FAG Publication

44109

IB

(19711, 8

pp.

14.19

Andreolli C.; Guida Circolare Idrostatica Assiale per Tavola Portapezzo

Rotante;Patent IT

2353CA, 1975; 15

pp.

14.20

Andreolli C.; Sopporto Zdrostatico per 1'Asse Azimutale del 3.5 m New Tech-

nology Telescope (NTT) dell %SO; Convegno AIM-AMME (Tribologia-Attrito,

Usura e Lubrificazione), Sorrento, 1987;p. 421-430.

14.21

Giordano M., Boudet M.;

Thermohydrodynamic Flow of a Piezoviscous Fluid

Between Two Parallel Discs;J. Mech. Eng.,

1980.

14.22

F r h e J ., Nicolas D., Deguerce B., Berthe D., Godet M.; Lubrification Hydro-

dynamique; Edition Eyrolles, Paris,

1990; 488

pp.

14.23

Martin

F.

J.; Prove Funzionali e di Qualificazione nello Sviluppo dei Cusci-

netti Volventi;

La Rivista dei CuscinettiBKF,

224

(1986),28-36.



APPENDICES

A . l SELF-REGULATED PAIRS AND SYSTEMS

The principle of self-regulating flow, applied to circular bearings and screws

and nuts, can also be applied

t o

pads of infinite length. See the pair of pads shown in

Fig. Al.l.a, clearly

similar,

from the functional point of view, to the bearing in Fig.

7.25.

The formulae and the corresponding diagrams are also quite similar (see ref.

7.5).

Naturally the pad of finite length needs lateral seals. Consider, for example,

those shown in Fig. Al.l.b, made of two shaped plates 3, in the peripheral grooves of

which internal (static)

3.1

and external (dynamic)3.2 seals are housed. The former

are elastomeric seals while the latter are made of a material with a very low friction

coefficient (PTFE), which can be lowered even further by allowing small side

leakage.

A fur ther development

of

the principle of self-regulation

is

its application to

mutually orthogonal pairs of opposed pads of infinite length (ref. 2.23). Figure A1.2

schematically shows that application.

A

purely vertical load has almost no effect on

the gaps of the horizontal pairs while

it

makes the vertical ones bearing

it

work

as

self-regulating. Similarly a purely horizontal load makes the horizontal pairs work

as self-regulating bearings. The simultaneous presence of a vertical load and a

horizontal one, each supported by the corresponding self-regulating pair, involves

the self-regulation of

total

flow Q, which

is

subdivided into two equal partial flows

Q12 which are again subdivided into two equal partial flows Ql4. For the formulae

and the relevant diagrams the reader is referred to ref. 2.23. The hydrostatic system

in Fig. A1.2 which has the same advantages

as

self-regulated pads, in particular

very high stiffness, is directly applied in hydrostatic slideways. Obviously, lateral

seals of the type shown in Fig. Al.1.b will be necessary.



520

HYDROSTATIC LUBRICAT I m

- a -

I?

-

b -

Fig.

A l . l

Self-regulating opposed-pad hydrostatic bearing: a- theoretical bearing; b- actual bear-

ing with side

seals.

A

further application of the principle of self-regulation is that of a system made

up of a self-regulating screw and nut in series with the above-mentioned slideway.

Self-regulating circular bearings, screws and nuts, pads and mutually orthog-

onal pairs, can be supplied

at

constant pressure a s well as with constant flow rate.

A s

for circular bearings, the matter has been discussed in section 7.4.2 and in

ref.

A l . 1 ,

where it is pointed out tha t the efficiency of the self-regulating bearing is

comparable to that

of

a conventional one fed through two flow-control valves. For

equal flow rate and pumping power in particular, the load capacity of the self-regu-

lating bearing

is

generally higher than tha t of the conventional bearing, especially

if the latte r is fed through capillaries

o r

orifices.

What has been said above also holds good for self-regulating screws and nuts,

compared in ref. A1.2 with the conventional ones fed through two flow-control



APPENDICES

521

Fig.

A l . 2

System of self-regulating opposed-pad hydrostatic bearings.

valves. What has been said abovealso holds good for self-regulating pads, compared

in ref. A1.3 with conventional pads fed through capillarieso r orifices.

Finally for systems made up of mutually orthogonal pairs of opposed pads,it is

pointed out, ref. A1.4, that the self-regulating system bears higher loads than the

conventional system with fixed compensators and the phenomenon

is more marked

as

the load increases. Again, a further application of self-regulation consists in a

screw and nut assembly in series with a slideway.

A.2

DYNAMICS

In Chapter 10 the dynamic behaviour of hydrostatic bearings has been studied,

using linear mathematical models and Laplace transform. For a bearing with a

circular recess (Fig. 5.11, directly supplied by a pump with constant flow Q, arry-

ing a static load W and subjected to an instantaneous overload LW, he non-linear

mathematical model yields the following equation (ref. A2.1)




(A2.1)

where h ' = h / h , ,h , being the film thickness under the final static load W+AF (which

may be due to gravity),

which can be considered

a

damping constant,

C s = 2 m ,

where M is the bearing

mass and

K,=9pr~(1-r'2)Qlh4,

ts stiffness; C, can then be considered to be the criti-

cal damping;

is another damping constant;

is the fluid stiffness, K, the apparent bulk modulus

c

the fluid in the supply tube and in the recess.

the fluid and

\

the volume of

The initial conditions in Eqn

A2.1,

at the time t ' =O , are the following

where ho is the film thickness under load W and t ' = t / t s ,with

t s = 2 n m s

period).

Film and recess pressures are

(A2.2)

respectively, where p ' = p / p s ,p; =p , lp , and p s is the recess pressure under load

W + A F .

In Fig.

A2.1, A2.2

and

A2.3

the variations of film thickness

h'

and of recess and

film pressures p i and p ' , versus time are plotted. The former are determined by

solving Eqn

A2.1

numerically, the lat ter by introducing the values

of

h',

h'

and

x '

obtained from Eqn A2.1 in Eqns A2.2. The results concern the cases defined in Table

A2.1.



APPENDiCES

case

1

2

3

523

'2 r' p . 1 0 2 W AF h , .104 K , . I o - ~ K ~ . I o - ~ ~ I K ,

m Ns/m2 N N

m

N/m N/m

0.05

0.9 5.4

1000

430 0.9

48 77

1.6

0.05

0.9

5.4

500

1500 0.95

63

77

1 .2

0.025 0.875 5.4 1000

430 0.9

48 7.7

0.16

In the first case the bearing is stable (Fig. A2.l.a).

After

a few oscillations it

stops in the equilibrium position

h'= l .

Figure A2.1.b shows

an

over-pressure in the

inner part of the bearing clearance followed, however, by a

small depression in the

outer part with possible cavitation and development of air bubbles.

- a - - b -

Fig. A2.1 Stable pad: a- oscillation of film thickness h' and recess pressure $; b- recess pressure

p i

and film pressure p'. Start of cavitation in the film.

In the second case the bearing is the same but the initial load is lower while the

instantaneous over-load is higher. Anyhow the bearing is still stable (Fig. A2.2.a)

but the initial oscillations are larger and the bearing comes to a stop after a greater

number of oscillations (interrupted in the diagram). Figure A2.2.b shows a consid-

erable over-pressure followed by a remarkable dangerous depression. Of course, in

such a case, cavitation must be considered in the numerical solution otherwise it

yields meaningless results.

In the third case the bearing

is

smaller but with the same loads

as

in the

first

case. It is unstable (Fig. A2.3): film oscillations ("relaxation oscillations", charac-

teristic of a self-exciting system (ref. A2.2) with positive damping) settle a t very high



524

--

OA

I

I


n

~ 2.0

P\, 1.8

I

:

I ,

I (

5 -1.6

: 1.4

I 1

I

I 1

I i

12

I

I ~ 1.0

- .

--

- -

I

0.8

\ I 06 P'

T

i \

I

0.4

1

8

;

0.2

I

0

I

1 -02

i I

0.9 1

,I

-a4

' ;

\, ;

; ..-0.8

' - 4 6

"

"

-1.0

i'

amplitude values, as well as recess pressure and in the recess initial signs of de-

pression can be seen (ref. A2.3).

-

Putting h'=l+&,with E <el, qn A2.1 becomes the following linear

equation

with the initial conditions

(at t '=O)

.. 4 AF

E = Q

, E = O

, E

= p 2 -

Adopting Routh's elementary criterium for stability, mentioned

and

applied in

section 10.5.2, the following relationship must be satisfied



APPENDICES

525

4

3

h’

2

pd

1

0

0

1

2 3 4 5

6 7

a 9

t‘

Fig. A2.3 Unstable pad: oscillation of film thickness h‘ and recess pressure p i . Start of cavitation

even in the recess.

(A2.3)

where

It must be noted that, but for high values of r’, often required in order

to

have

minimum total dissipated power H , (Chapter 111, p s l p f approaches unity while

(p,-pf)pf

approaches zero (e.g. for

r’=0.6,ps

/pf=1.09and (ps-pf)pf=0.115).f a l s o

(ps-pf)pfc2/(KdM<<1,qn A2.3

is

reduced

to

Kd/Ks>l.

spectively, consistently with the diagrams.

Referring to the cases shown in Fig. A2.1, A2.2, A2.3, KflKs=l .6 ,1.2, 0.16, re-

The viscosity of the fluid has always been assumed to be constant. Actually, the

viscous squeeze of the film causes an increase in fluid temperature, thus

a

reduc-

tion of viscosity and of the effectiveness of the squeeze itself. When the rise in tem-

perature becomes quite high, for high values of film thickness h, a decrease i n load

capacity might even occur as

h decreases (ref. A2.4).

The results given refer

t o

direct supply. In the case of supply with restrictors,

there are differences, some of which

are

considerable.

In

particular:

the volume of fluid influenced by compressibility

is

reduced to the volume down-




stream from the restrictor;

. f the restrictor is rigid, it works as a damper. This does not mean that compen-

sated supply is better than direct supply from this point of view, because, on the

contrary, the lat ter shows higher damping, as can be seen in Chapter

10;

if the restrictor is variable, it may produce a negative effect because the number of

degrees of freedom increases.

As

regards degrees of freedom, a bearing, and a hydrostatic system in general,

is part of a bigger system: a machine made up of various elements (each with its

own stiffness and internal damping), therefore with various degrees of freedom.

It should also be noted that hydrostatic systems, because of the fact that their

films work a s vibration attenuators (ref. A2.5),may be preferred to other low friction

systems in those machines in which forced vibrations are expected, especially if

resonance is possible, and with several degrees of freedom. In this connection, it

must be pointed out that very stiff films are not always convenient because they

would behave in practice as stiff elements with no damping properties.

A.3 THERM AL EXCHANGE

A.3.1

Resistances

In the case of laminar flow with forced convection in a tube

of

diameter

d

and

length 1, that is for Red =V d/ v<2000, the Nusselt number is

where Pr=pccll is the Prandtl number and

I

fluid thermal conductivity. It s thermal

unit conductance is

(A3 .1)

For

fluids with high P r , such

as

lubricating oils, in long tubes, with good approxi-

mation

Nud=3.66,

thus

'yc=3.66(Ild).

The lubricant thermal resistance in the tube is

where A is the wet surface. Considering the recess as a rectangular tube, i ts equiv-

alent diameter is



APPENDICES

for L>>h, (in the examples given L=40h,.).So R 1 , becomes

527

Its value may be increased without causing any problem (almost doubled). R 1 , can

be determined in the same way.

For turbulent flow, that is for

Re&6000,

and the values of NUd can be much higher than those relevant to the laminar flow.

In transient conditions, that

is

for

2000dZedC6000,

the evaluation of

NUd

is very

difficult.

A.3.2 Coeff icients

A.3 .2.1 coefficient

ac

i)

Forced convection on an infinite plate of width B with fluid lapping a t one face,

a t speed V , far from the face.

N U H

=

0 . 6 6 4 Re ~ l"PrIf3

and since

Pr=0.72

for air

N U H

=

0.595

ReBlI2

This formula is true for laminar

that is for

Rt?B=V,B/ ~ 4 5 . 1 0

values, not easily achievable, flow becomes turbulent, and

N U H

=

0.036 (ReBo.8 23,2 00)Pru3

and for P-0.72

NUH = 0 . 0 3 2 R e ~ ~ . ~748

( A 3 . 2 )

for higher

Forced convection on an infinite plate of width

B ,

for

R e ~ C 5 . 1 0 5 ,

ith flow per-

pendicular to one face. The relationship

NuBa = o . 1 5 R f ? ~ ~ ~ ~ ( A 3 .3 )

can be used with a good approximation.

Trail of an infinite plate of width

B ,

for

R e ~ C 5 . 1 0 ~ .



528


(A3.4)

With equations

A3.4, A3.2

and

A3.3,

the conductance

a,

of a prismatic structure

(slideway) of infinite length can be roughly evaluated. For example, for

B = l

m and

thickness

H=0.2

m, and for

V,=50 d s ,

with

v=1.6.10-5

m2ls for air, it is

R e ~ , = 1 7 6 8 ,

Re g = 7 N . 6 ,Re g p 2 1 3 7 0 . ThereforeN U ~ ,= 2 6 5. 2, u ~ = 4 7 0 . 4 , u ~ , = 4 2 7 . 5 ,o % ~ , = 7 . 1 6

J/ms2s°C, a c B= 6 3 . 5 J/ms2s0C, acB,=115.4 J/ms% C and the average value is

a,=(

c g a + 2 a c ~ +& , ) /4=62.5 /ms2s°C.

ii) Natural convection around a square plate of side

B

and thickness

H

much

smaller than B , with flow perpendicular

t o

one face:

N U B= 0.45 (G rg pr)'I4

where

Grg is

the Grashof number which in natural convection replaces

R e ,

where d e q = 2 [ B H / ( B + H ) ] ,? is the coefficient of volume expansion,

T,

surface tem-

perature and

T,

the air temperature far from the surface.

For

Pr=0.72,

N U B

=

0.414 (Gr#4

This formula is true for laminar flow, that is for G r ~ < S . l o ~ ;or higher values of

Grg, that is for turbulent flow,

NUB=

0.083

(Grg

Pt')y3

The corresponding values of spy, obtained from Eqn A3.1, substituting

d

with deq,

are small anyhow. For example, for

B=0.15

m,

H=0.015

m,

T,=5OoC, Tm=2OoC,

since

(gp)/v2=108

/m3 C for air,

G r ~ = 6 . 1 . 1 0 ~ ,

hus

N U B=6.5,

and since

k 0 . 0 2 7

J1ms C for

air , from Eqn

A3.1 in which d

is

replaced by deq,we have a,=6.4 J/ms2s°C.

by (L+B)/2.In the case of a disk of diameterD, is substituted by 0.9D.

In the case of a rectangular plate of width B and length L ,

B

may be substituted

iii) Horizontal rotating disk of diameterD, ith flow perpendicular to a face.

w D 2

v2

NuD

=

0.18

7)



APPENDICES

529

The boundary layer is laminar if R e ~ < 5 . 1 0 6 ;or higher values

where D , is smaller than D and decreases with w , and flow is laminar from 0 to

D J 2 and turbulent from D J 2 to 012 . For example, for w=500 s-1, D p O . 1 2 7 m. For

D=0.2

m flow

is

laminar from

0

to

0.0635

m and then it becomes turbulent. Therefore

N u ~ = 3 . 5 . 1 0 ~

nd

q = 4 7

Jlms2sOC.

In

ref. 5.15 the cooling of the lubricant is due to the high speed(-628

rads)

and

the large diameter of the rotating disk ( D = 0 . 3 m) and mostly to the “ducted fan”

effect.

iv) Horizontal rotating cylinder of diameter

D,

ith flow perpendicular to its axis.

For rotational speed

w

lower than the critical value, that is for R e ~ < 2 . 5 . 1 0 ~ ,ee the

following case of the motionless cylinder; for higher values, the flow becomes turbu-

lent and

This formula can be used for journal bearings.

Natural convection around

a

horizontal cylinder of diameter D , with flow per-

pendicular to its axis.

and for

P r 4 . 7 2

This formula

is

true for laminar flow, that

is

for

G r ~ < 5 . 1 @ ;

or higher values, that

is for turbulent flow,

and forPr=0.72

For

example, for

D=O.OOl

m, T,=50°C, T,=20°C, we have

G r p 3 . 1 0 3 ,

thus

N u ~ = 3 . 6

nd

q = 9 . 6 2

J/m2soC. With the simplified formula



530 HYDROSTATIC LUBRlCATfON

the value q=8.5J/mspsOCwould be obtained.

Forced convection around a horizontal cylinder of diameter D, with flow per-

pendicular to its axis.

where the first term between square brackets concerns the laminar boundary layer

on the front part of the cylinder, the second concerns the partly turbulent trail. The

term

(pU,/p,)O.25

akes into account the influence of temperature variations on the

physical properties of the fluid. This relationship is true for l I R e ~ 1 l O ~ .s for air,

in a hydrostatic system, (p,/&0.25=0.98, and putting also Pr=0.72

Table A3.1 contains the values of ReD, NuD and ac.They have been determined

for D=O.Ol

m

and for increasing values of V, , putting ~ = 1 . 6 . 1 0 - ~2/s for air.

It

should be noted that for

V,=O.l

m the value of q s almost equal to that obtained in

the previous case of natural convection for AT=30°C.

‘ A B L E A 3

v,

m

0.1

0.5

1

5

10

20

50

100

1

Re,

6.25 .10

3.125.102

6.25 .102

3 . 1 2 5 1 0 ’

6 .25 .103

1.25.104

3.125 .104

6 . 25 .

1

04

3.42

8.19

12

29.6

43.3

67 .4

109

164

a,

J/m*s°C

9.23

22.1

32.3

79 .7

117

182

295

442

Forced convection around banks of tubes with perpendicular flow.

For

more

than 10 rows of in-line or staggered tubes,

Nub = 0.33Reb0.6Prv3

This formula is true for turbulent flow, that is for Re>6000, and

where Vm, , is the velocity reached by air in the minimum available cross-section.



APPENDICES

531

For laminar flow (Reb<200) and also for transient flow (200dEeb<6000) the

relationship

NUb

= J

Reb Pru3

is true. This formula is complicated because

j ,

which is Colburn's dimensionless

factor, is a function of R e , of the number of rows of the tubes and of their

arrangement.

A

grid upstream from the bank

of

tubes makes turbulence possible even at low

Reynolds numbers. Barnes in the bank make the air move in a winding way, thus

increasing the actual surface

of

heat transfer.

A.3 .2.2 Coefficient

aj. Coefficient aj for air is:

where F11.2=o.7o 0 . 9 is the geometric form factor, 0=5.7 J/m2s°K is the Stefan-

Boltzman constant,

TI

nd

T2

are body and ambient temperatures, respectively. For

temperatures included in the hydrostatic range, aj=4.5 to 7 . 5 J/m2s°K, approxi-

mately. Such values must be added to ac in order to obtain the global unit conduc-

tance a.

What has been described above can be useful for the design of an air-oil heat

exchanger with the tube bank made of the hydrostatic system supply pipelines, bent

more times to a

U

shape. The approximate value of the coefficient of global heat

transfer for air-oil exchangers is

a=30

to

180

J/m2s°C.

For high lubricant temperature rise it may be advisable t o use water-oil heat

exchangers for which a=120 to 350 J/m2s°C approximately (ref. A3.1) . Water heat

exchangers, as well as air exchangers, can be placed in the return line; however,

the absence of water o r air leakage in the lubricant must be carefully checked.

Finally, when heat exchangers are placed in the supply line, a more effective

control of the temperature of the bearings may be obtained.

REFERENCES

A l . l

Bassani R.;

The Self-Regulated Hydrostatic Oppo sed-Pad Bearing in a Con-

stant Pressure System;

ASLE

Trans.,

25,

(1982) ,95-100.

A1.2 Bassani R., Piccigallo B.; Vite-Ma dreuite Zdrostatica Autoregolata Ali m enta -

ta a Pressione Costante; Tribologia e Lubrificazione, AMO XIV (1979) ,98-109.



532

HYDROSTATlC LUBRlCATfON

A1.3 Bassani

R.;

Pattini Contrapposti Zdrostatici Autoregolati, Al im en tat i a Pres-

sione Costante; Oleodinamica-Pneumatca,24(19831, 8-25.

A1.4 Bassani

R.;

Sistema d i Pattini Zdrostatici Autoregolati, Alim en tat i a Pres-

swne Costante;

Scritti

per L. Lazzarino; Pacini Editore,

Pisa, 1986;

.

235-250.

A2.1 Bassani

R.;

Cuscinetti Zdrostatici di Spinta Sottoposti a Varia zioni Istantanee

del Carico;

2nd

AIMETA Congr., Napoli,

1974;

ol.

3,

p.

225-236.

A2.2

Nayfeh A.

H.,

Mook D. T.;

Non linear oscillation;

J.Wiley & Sons Inc.,

1979;

704pp.

A2.3 Hell H., Savci M.; Bynamische Eigenshaften Hydrostatischer Axiallager bei

Kleistmilglichem Gesamtleistungsaufwad;Konstruction, 27 (19751, 37-144.

A2.4 Pinkus

O.,

Wilcock D. J.; Thermal Effect in Fluid Film Bearing; Mech. Engi-

neering Publ. Ltd,

1980;

.

3-23.

A2.5

Wilcock D. F.,Bevier W. E.; Externally Pressurized Bearings.

-

Vibrat ion

Attenuators; ASME

Trans.,

J.

of

Lubrication Technology,

90(19681,614-617.

A3.1

Wilcock D.

F.,

Booser E.

R.; Lubrication Technique for Journal Bearings;

Machine Design, June 25,1987; 84-89.



Author index

For each author this index shows the relevant reference numbers, as well as

the pages on which each reference is cited. For example

Fuller

D. D.

1.5-

4, 141, 143

The reference is the fifth item in the reference list of chapter

1

and is cited on pages

4,141

nd

143.

Andreolli C.

14.11- 495

14.19- 5@?,510

14.20-

509

Anwar I.

Aoyama T.

AppoggettiP.

Arsenius H.C.

Arsenius T.

Artiles A.

Barrett

L.

C.

10.7-

322,339

Bassani R.

2.10-

23

2.19-

28

2.23- 30,519

4.11- 81,101

5.1-

94

5.2-

%, 98

5.4- 98, loo

5.5- 98,100

5.9-

1 M

5.10-

105,107

5.14-

107,111,447

5.15-

107, aS,

447,529

5.16-

111

5.18- l22

5.31- 135

8.2- 237

8.29- 261,263

14.12- 496

14.17- 504,505

2.2-

17,486,487,512,613

8.15-

245

Bassani R. (continued)

5.32- 135

5.42- 145

5.43-

146,227

7.5- 221,223,619

7.6- 223

7.7-

29,339

7.8- a.2

13.5- 470

13.6- 470

13.9-

478

13.10- 480

13.11-

480

Al.l- 520

A1.2- 520

A1.3- 521

A1.4-

521

A2.1- 521

14.22- 511

Berthe D.

Bettini B.

8.8- 239

Bevier W. .

A2.5-

526

Bil dts h C.

14.6- 491,492

Bird R. B.

12.1- 447

Booser

E. R.

3.4-

40,42

10.1- 301,304

A3.1- 531

Bottcher

R.

BoudetM.

Boyd J.

14.21- 511

3.2-

37

8.5- 238

Brzeski L.

2.17-

27

Bucciarelli A.

3.9- 47

Cameron A.

4.4- 69,81

Casely A. L.

2.8- 22,169

Castelli V.

5.28- 135

Chang T.

S.

5.37-

142

Chen B.

Chen C. R.

Chen K. N.

Chen

N.

N.

S.

8.25- 2-54

5.36- 142

10.4- 302,320,358

8.13- 245

8.14- 245

Chen Y.S.

Chong F. S.

10.14- 356

9.2-

292,233,296

10.15-

356

8.2- 237

Colsher R.




34

Cowley A.

Culla C.

8.6-

238

5.32- 135

13.10-

480

13.11-

480

Czichos H.

Davies P. B.

13.1- 467

2.15- 26,470

8.7- 239

8.11-

240,242,244,354,355

13.7- 472

De Gast

J.

G. C.

2.11-

23

De Shepper M.

10.13- 356

Decker 0.

7.10-

233

7.9- 233

Deguerce

B.

14.22- 511

Dorinson A.

Dowson

D.

3.1- 36,37

2.14-

26

5.6-

lm,

104,110

5.21- 128,132

Dumbrawa M. A.

8.17-

W 50,254,419

8.18-

254,288

Effenberger W.

El Hefnawy

N.

8.20- 250

El Kayar A.

9.3-

292

El

Sayed

H.

R.

5.41- 144

El Sherbiny M.

El-Efnawy

N.

El-Sherbiny

M.

Ernst

P.

Et t les

C .

H. M.

4.6- 74

Fowle

T.I.

3.7- 44

Fr6ne

J.

14.22- 511

Fuller D. D.

Ga ne sa n

N.

10.1-

301,304

8.20- 250

4.8- 74

4.8- 74

7.1-

218,495,496

1.5- 4,141, 43

5.17-

l20,125

5.20- 128

8.30- 262,268

8.32-

262

10.5-

307

Geary

P.

J.

2.20-

28

Ghai R.

C.

8.3-

238

8.9- 240

Ghigliazza R.

Ghosh M.

K.

11.2- 419

10.10- 354

10.11- 354

10.17- 358,360

Giordano M.

14.21- 511

Girard L. D.

1.1- 4,35

Godet M.

14.22- 511

Goldstein S. D.

Goran R.

Haferkorn

W.

Hallnor G.

Hal lstedt G.

14.2- 485,487

Hegazy A.

A.

9.3- 292

Hell H.

A2.3-

524

Heller

S.

5.35-

142

Hessey

M. .

8.33- 264

I i i rai A.

Hirs G. G.

Ho

Y.

S.

8.13- 245

Hooke C. .

2.12-

25

8.1-

237

8.26- 255

4.1-

54

14.17- 504,505

14.13-

496, 98

14.6- 491,492

5.24- 132

2.13- 25,172

Hornung V. G.

Howarth R.

B.

4.5-

74,247

2.8- 22,169

13.4-

469

Hunt

J.

€3.

Ich ikawa A.

Ikeuchi K.

Inasaki

I.

14.16- 503,504

9.1-

290,292

2.24-

23

8.29-

261,263

10.9- 348,472

9.5-

298

Ives D.

Ives D. (continued)

J a i n

S.

C.

Kapur V.

K.

Karelitz

M. B.

Ka t suma t a S.

Kazimiersk i

2.

Kennedy J.

S.

Khalil

F.

9.6-

298

8.10- 240

5.12-

107,110

1.4- 4

8.2- 237

2.17-

27

5.13-

lW,

25

5.19-

125

5.25- 132,476

Khalil

M. F.

9.3- 292

Kha t a a n H . A.

5.41- 144

Kher A. K.

Kong

Y.

.

Koshal D.

8.6-

238

8.14-

245

8.19- 247,293

9.4- 293,294,296, 77

12.2-

451,456,461

Kreith F.

Kubo

M.

2.18- 27

7.4-

220

2.2- 17,486,487,511,513

4.9-

79

3.11- 52

10.12- 356

13.7- 472

Kundel K.

La ngha a r

H.

.

Lansdown A. R.

Leonard R.

Lewinschal L.

14.3-

487,488

Lewis G.

K.

5.23-

131

Lightfoot E . N.

12.1-447

Lingsrd

S.

8.14- 245

Loeb

A.

M.

5.34- 139,468

13.2- 468

Lombard J.

Lord Rayleigh

Ludema

K. C.

Lund J.

W.

5.40- 144

1.3-

4

3.1- 36,37

8.37-

282



Lund J. W. (continued)

10.16-

356

Majumdar B.C.

10.11- 354

Makimoto Y.

2.18-

27

7.2-

218

Manea G.

Mart in

F. J.

Martin H. R.

Massa E.

Masuko M.

Ma t suba r a T.

8.16- 246

14.23- 512

3.5- 44,47,48

5.27- 135

10.2-

301

2.18-

27

7.2-

218

7.3- 220

7.4-

220

Mayer J. E.

2.9-

23

5.11- 107,110

McCloy D.

Meo F.

Merri t t

H. E.

Michelini

C.

Mizumoto H.

2.18- 27

7.2- 218

7.3-

220

7.4-

220

Mohsin M. E.

2.6- 21,29,167

Moisan

A.

Mook D. T.

Mori H.

3.5-

44,47,48

13.3- 469

2.7-

22,78,

o,

2,85

11.2- 419

5.40- 144

A2.2-

523

2.24- 23

5.24- 132

Morsi S. A.

2.5- W,65

10.6- 320

Morton P.

G.

Moshin M.

E.

9.6-

298

8.28- w6

10.6- 320

Mote C. D.

5.3-

98

Mueller-Gerbes

H.

7.1-

218,495.4s

N a k a h a r a T.

10.2- 301

AUTHOR

INDEX

Nayfeh A. H.

A2.2-

523

Neale J. M.

3.10-

50

Newton M.

J.

Nicolas D.

O'Connor J.

ODonoghue J. P.

13.4-

469

14.22- 511

3.2-

37

2.12-

25

4.6- 74

5.23- 131

5.26-

I 3 3

8.1- 2-37

8.26-

255

8.31- 2 M

8.33-

264

8.35-

279

8.36-

282

Ogate K

10.8- 324

o g i s o s.

Ohsumi

T.

Oka mur a S.

2.18- 27

Okasaki

S.

7.3- 220

Opitz H.

1.7- 4

2.3-

19,173,376

10.1- 301,304

8.27- 255

2.24- 23

Piccigallo B.

5.42-

145

5.43- 146,227

7.7- 229,339

A1.2- 5 W

Pinkus

0.

4.2- 54,68,81

A2.4- 525

PolsEeck M.

14.7- 492,496

Prabhu

T.

.

5.20-

128

8.30- 262,268

5.17- 120,125

8.32- 262

10.5- 307

Radkiewicz Cz.M.

Ragab H.

Raimondi A.

A.

Raznyevich

K

Recchia L.

5.13- 107,125

5.22-

l28,131,132

8.5-

2-38

3.8-

46,50

7.8- 232

535

Reddi M. M.

Rippel H. C.

5.29- 135

1.6- 4

5.33- 139

7.11-

233

13.2- 468

Rippel T.

Rohs

H.

G.

Rowe W. B.

14.16- 503,504

14.10- 500,503

1.9-

l0,89

8.1- 237

8.12- 241,244,353,355

8.19- 247,293

8.24-

253

8.26- 255

8.34- 276,277,278

8.35-

279

8.36- B2

9.2- 292,293,296

9.4-

293,294,296,377

9.5- 298

9.6- 298

10.12- 356

10.15-

356

Royle J. K

2.8- 22,169

Rumbarger J.

H.

5.39- 144

Sa f a r Z. S.

5.3-

%

Salem

E.

5.19- 125

5.25- 132,476

Salem E. A.

9.3- 292

Salem F.

4.8-

74

8.20- 250

Sa sa k i T.

Sato

Y.

Savci M.

Shapiro W.

5.24- 132

8.27- 2-55

A2.3-

524

5.28- 135

5.35- 142

7.9-

233

7.10- 233

8.4- 238

8.15-

245

Sha r ma S. C.

Shaw

H.

C.

Shen F. A

8.10- 240

2.9- 23

5.8- 106



536

Shinkle

J.

N.

4.5- 74.247

Siddal

J.

N.

11.1-

385

SiebersG.

2.1-

17,490

Singh D.V.

8.3-

238

8.9- 240

Sinha P.

Sinhasan R.

5.13- 107,

I25

8.3-

238

8.9-

240

8.10-

240

So H.

5.36- 142

5.37- 142

5.7- 105

3.9- 47

1.8- 6 ,B , 115,133

so0

L.

s.

Speich H.

Stansfield F.

M.

SternlichtB.

Stewart W. E.

Stout K. J.

4.2- 54,68,81

12.1- 447

8.19- 247,293

8.24-

253

8.34-

276,277,278

Straccia P. F.

Streeter

V.

Szeri A. 2.

5.38- 142,239,261,264

4.10- 81

5.30- 135


Taylor C. M.

2.14- 26

5.21- rzS,l32

6.11- 107,110

4.3-

54

4.7- 74

2.16- 27

Ting

L. L.

Tipei

N.

Tully

N .

Umbach

R.

Usuki

M.

Vavra 2.

Verma

K

Vermeulen

M.

14.13- 4%, 498

7.3- 220

14.7- 492,4%

5.12- 107,110

8.21- W0,Wl

10.13-

356

13.8- 474

Viswanath N. S.

10.17- 358,360

Vogelpohl G.

1.2-

4

Walowit J.

8.15- 245

Wang X.

Wearing R. S.

10.4- 302,320,358

8.35-

279

8.36- 282

Weck

M.

14.9- 4M

14.15- 502

Wertwijn

G.

5.39- 144

Weston W.

9.2- 292,293,296

9.6- 2M

Wiener H.

Wilcock

D.

F.

2.4- 20

10.3-

302

A2.5-

526

A3.1- 631

Wilcock D.

J.

A2.4-

625

Wills

J.

G.

3.3- 39

Wong G.S.K.

2.22-

30

Wu H. Y.

10.14-

356

Wylie

C.

R.

Xie P. L.

x u s.

xu

s.x.

Yang G.

P.

Yang

H.

H.

Yates S.

Yonetsu S.

Yoshimochi

S.

10.7-

322,339

10.14-

356

8.25- 254

9.2- 292,293,296

10.4- 302,320,358

10.4- 302,320,358

2.21- 28,279

8.29- 261,263

2.18- 27



Subject

index

Bold page numbers indicate that the item

is

the main topic of a section.

Symbol

u->”

means “see”.

additives

50,51, 2

adiab atic flow 89,107,110,132,151,225,248,295

a i r en t ra inmen t 44, 45

air preheaters

511

ann u la r c lea rance 75-’33

a n n u l a r - r e c e s s p a d s 7,112 23,133,907,422-425

an tennas

5,508

appa rent bulk modulus -> equivalent b. m.

a t t i tude angle 292

Bernoulli equation 83

boring machines

6

boundary layer 79

bulk modulus 42,45,46,47

bulk modulus (equivalent) -> equivalent b. m.

capi l lar ies -> res t r ic tors ( laminar-f low r.)

cavitation 48,104,108,133,245,262,288

characteristic equation

322,323

c i rcu la r - recess pads

7,76-77,91-11,123,128,

305-307,4210

422,469,521

clearance -> f i lm th ickness

clearance ( radia l ) -> radia l c learance

compensated supply 16,17- 0,31,88,91,15372,

173,180- 86,192- 13,220- 21,312 20,

334-339,415-433

compensating devices -> res t r ic tors

compressibility

42,47, 26

conical bearings -> tapered bearings

conical pads

->

tapered pads

constant- f low supply

->

direct supply

cons tant- pressure supply -> compensated

constitutive equations

55

continuity equation 54-

58,59,61,64,77,81,107,

111

contraction coefficient 83

cooler 460

correction factors

96,98,99,100, 01,105, 11,118,

Couette flow

69,246,288

critical speed

73,74,356

supply

119,122,126,127

cryogenic fluids

245

cylindrical pa ds 9, 136- 41,151,233

dam ping coefficients (journal bear ings)

354,

356,360

damping factor 321,342,356

density 42-49,310

des ign

hybrid bearings 296- 97

mult i recess journ al bear ings 251-HI

mult i recess thr us t bear ings 263

opposed-pad bearings

213- 18

s ing le -p ad bea r ings 172- %

spherical bear ings

278

tapered bearings

269

Yates bearings 283-285

diaphragm bear ing s 27

direct supply

16,17,

O,

2,88,91,148 53,173,

177-

80,188

92,219, S-227,230,311-12,

333-334,988-381

discharge coefficient

84, 85

displacement (nondimens.) --> eccentricity

dynamic viscosity --> viscosity

dynamics 301- 61,472,474,521-

%

eccentr ic i ty

hybrid bearings

292

mult ipad journal bear ing s 233

mult i recess journal b ear ings 240, 242

opposed- pad bearing s

187

direct supply 192

flow divide rs 205

l am ina r - f low rest ri cto rs

195

screw-nut assemblies

direct supply 219

l am ina r - f low rest ri cto rs 220

s ing le -p ad bea r ings

89,302

Yates bearings 279

annu la r - recess pad 113,118,120

c i r c u l a r -recess pad

91,96,101,105,111

cylindrical pad

139

effective a rea 87




effective area (continued)

infinite-length pad 369

multirecess thrust bearings

261

rectangular pad 133

screw-nut assembly 145

self- regulating bearings

223

spherical pad 129,131,133

tapered pad 124,125,127

Yates bearings 280

efficiency losses365,377,380,384,416,417,433

electric analog field plotter 468

electronic compensators 23

electronic control 23,502

energy equation

CW-69,107,448

equivalent bulk modulus 46,47,310

experimental tests

485-

82

feed drives

492-433

film thickness


261

opposed-pad bearings

direct supply 191

orifices 201

single-pad bearings 89

compensated supply 154

constant-flow valves 161,162

diaphragm- controlled restrictors 169

direct supply 150, 152

laminar-flow restrictors 156, 157

orifices158,159

spool valves 163,167

finite-differencemethod 96,107,128,135,236

finite-elementmethod 135,139,142,236,239,261,

flash-point50,52

flexible-plate bearings

26,470

flow dividers -> restrictors

flow rate 66

-

67,87.309

annular clearance 75

circular-recess pad 76,91,98

hybrid bearings 289,294

infinite-length pad 71,369,371,426

infinite-length strip 70

inherently compensated bearings 172

multipad journal bearings 233

multirecess journal bearings 245


261

opposed-pad bearings 186

flow dividers 205,207,210

laminar-flow restrictors 193, 195

orifices 198

264

orifices 84

pipes 77,78

rectangular pad 133,417

screw- nut assemblies220

self- regulating bearings

compensated supply229

direct supply (constant pressure)227


constant-flow valve8 161

diaphragm- controlled restrictors 169

single- pad bearings

flow rate (continued)

sing1e

-

pad bearings (continued)

infinite-stiffness valves

169,171

laminar-flow restrictors 156,157

spool valves 165,167

slideways

231

spherical bearings 276

spherical pad 129,131,132

tapered bearings 266,271

tapered clearance 124,127

tapered pad 125

Yates bearings 283

foam 44,48,51,52

frequency response 304,328-321,357

friction area

cylindrical pad 141

multirecess journal bearings 247

rectangular pad 137

Yates bearings

283

annular-recess pad

115

circular-recess pad 94

infinite-length

pad375,431

infini te- length pad 72,373,431

infinite-length strip 71

recess 74


annular clearance 76

annular-recess pad 115

circular-recess pad 77,92

cylindrical pad 141

hybrid bearings 294

multirecess journal bearings

247

multirecess thrust bearings 262

spherical pad 131,132

tapered pad 124,127

Yates bearings 283

friction power 87,88,90

annular-recess pad 115,423,424

circular-recesspad 94,111,420,421

cylindrical pad 141

hybrid bearings 291,294,295

infinite-length pad373,431

multipad journal bearings 233

multirecess journal bearings 246, 247

multirecess thrust bearings 262

opposed- pad bearings


flow dividers 205

laminar-flow restrictors 195

orifices 198


friction force

68

friction moment 68

rectangular pad 137

screw

-

nut assembly 146

self-regulating bearings 224

direct supply (constant flow)

225

direct supply (constant pressure) 227

compensated supply

154

constant-flow valves 161

single- pad bearings



SUSJECT

lNDEX

539

friction power (continued)


l am ina r - f low rest ri cto rs 157

s i n g l e - pad bearings (continued)

spherical bearings

277


tapered bearings 266,267

tapered pad

124,125,127

Yates bearings

283

gas solubili ty 43,44

Grashof num ber 528

gr ind ing mach ines

6, 491

hybrid bearings 7,8,10,14,89,105,142,250,288-

hydraulic circuit

31-2

hydraulic circuits

613-617

hydraul ic d iameter 82

hydraulic resistance

87,88,90, 51

288

ann u la r c l ea rance

76

a n n u l a r - r e c e s spad 113,118,119,120

circular-recesspad91,98,101,105,111

cylindrical pad

139

dia ph rag m - controlled res t ric tors

167, 168

inf in i te- length pad 71,426

in f in i t e - l eng th s t r ip

70


155,290,426

mult i recess journal b ear ings 238

mult i recess th ru st bear ings 261

orifices

157,443

pipes 77

rec tangular pad 133

sc re w- nu t a ssembly

145

se l f - regula t ing bear ings

223

spherical pad 131, 133

spool valves

162, 165, 167

tapered pad

124,125,127,128

Yates bearings 279

hydrodynamic load capacity 89,244,254

hydrostatic

lifts 141- 43,298

inertia effects

a n n u l a r - r e c e s s p a d 120- 22

c i rcu la r - recess pad 109-

08

mult irecess journal bear ings

245

mult i recess th ru st bear ings 262

spherical pad 132

tapered pad

125-

28

iner t ia parameter

103,108,133

in f in i t e - l eng th pad 71-72,362,366,425,447

in f in i t e - l eng th s t r ip 69

inhe rently compen sated bearings

16,26- 8,31,

172

inlet length 79- 0,100

inle t losses 80

a n n u l a r - r e c e s s p ad 118

circular- recess pad W-100

hybrid bearings 290

ins tabi l i ty 47

interface restrictor bearings

30

IS0 classification of lubricants -> viscosity

Johnson dr ive

494

system for industr ia l lubr icants

journal bear ings

9-10,13,31,89,472-458

multipad

9,233

234,348

349

multirecess 10, 11,236,239-&O 349- 60,485

kinem atic viscosity 38, 40

Laplace equation

SS-aS,33,135

lathes6,491

load capacity 66,87,281

a n n u l a r - r e c e s spad

113

circular- recess pad

76,91,98

hybrid bearings 292,293

hydro static lift 142

inf in i te- length pad

71,366,369,426

inherently compensated bearings 172

mult ipad journal bear ing s 233

multirecess journ al bearings

240,242,244,249

multirecess thru st bearings

261,262

opposed- pad bearing s 186

constant- f low valves

201

direct supply

188,191,192

flow divid ers 205,207,210

l am ina r - f low rest ri e to r s 193, 195

orifices

198


screw - nu t a s sembl ie s

direct supply

219

l a m in a r - low rest ri cto rs

220

se l f -regula t ing bear ings 223

compen sated supply

229

direct supply (constan t low)

225

direct supply (constant pressure ) 227

compensated supply

153

constant- f low valves

161

d i a p h r a g m - controlled rest rictors 169

d ir ec t s u p ~ l y

50, 152

inf in i te- s t if fness valves

17

l am ina r - f l ow res tr ie to r s 156

orifices

158, 159

spool valves

163,167

direct supply 231

1

a m i n

ar-

low restrictors

231

s ing le pad bea r ings

s l ideways

spherical bearing s

276,277,278


tapered bearin gs 265,268,269,271,273,275

tapered pad

125

Yates bearing s

279,281

mine ral oils

35, 36- 1

synthetic lubricants

35,38,52

lubricants

35

-I

lumped res is tance s -> t h i n - l a n d m e t h od

machine tools6,483-

oc

measur ing ins t rumen ts

6

mechanical models

301,320,322

milling arm 501, 02

mill ing machines 6

mills

5,

505

m i s a l i g n m e n t

a n n u l a r - r e c e s s p a d 117-118

c i rcu la r - recess pad 95-98

mult i recess journal bear ings

245, 255




misa l ignment ( con t inued)

sc re w- nu t a s sembly

145

tapered pad 128- 28

mixing length 81, 106

momentum equat ions 55

momentum torque 247,295

mult ipad journal bear ings --> journal bear ings

multiple pum ps 501,616-617

multirecess bearings 7,8,236-285

mu lt i recess journa l bear ings --> journal

mult i recess thru st bear ings --> thrust bear ings

naphthenic oils 36,39,41,42,49,0

natu ral frequency

321,342,356

N av ier - Stokes equations 54-58,61,64,77,81,107,

Newtonian f lu ids

37

Nusselt number

526,527,528,529,530

Nyquist method 323

oiliness41,50, 2

oils --> lubr icants

opposed- pad be arings 7,8,9,14,31,186-18,331-

optimization

362

446

b e a r i n g s

111

339

a n n u l a r - r e c e s spad

113,445

given flow ra te 423

given load 424

given pressu re

423


given load

421

given pressu re 420

cylindrical pad 140

hybrid bearings 296

inf in i te- ength pad

585-416,434-43

c i rcu la r - recess pad 92,445

given flow rat e 3 8 5 - S

given load 406-415,438,443

given pressure

395-406,434-437

mult ipad journal bear ings

234

multirecess journ al bearings 251,252,253,254

opposed- pad bearing s 213,214

rectangular pad

135,445


given load 418

given pressu re 417

se l f - regula t ing bear ings

224

s ing le -padbea r ings 173,174,175

tapered bearings 270

Yates bear ings

284

orifices --> res t r ic tors

ovality 255

oxidation 50,51,52

paraffin ic oils

36,39,41,42,49,50

para l le l i sm e r ro r --> misa l ignment

pitch error 146,218,219,220

plastic throttl e --> restrictors (elastic

Poiseuille flow 69,246,288

pour-point 50,51,52

power ratio

87,89,90

c a p i l l a r i e s )

hybrid bearings

291,292,296

power ratio (continued)

inf in i te- length pad

376

mult ipad journal be ar ings 234

mult i recess journal b ear ings 248, 254

opposed- pad beari ngs

constant- f low valves 203

direct supply 189,191

flow divide rs

205


195, 196

orifices 198,201

compensated supply

229

direct sup ply (constan t flow)

225

direct supply (constant pressure ) 227

compensated supply

154

direct supply 150

l a m i n a r -flow restrictors 157


tapered bearings

268

Yates bearings 284

se l f - r egu la t ing bea r ings

s ing le -p ad bea r ings 174

Prand t l number 526

pred iction - correction method

98

preheaters 5

pressure 40,42,44,55

pressure ratio

87,90,154,193,196,198,203,205,

211,231,240,242,254,271,272,279,293,426,

427,434,443

pump ing power 87,88

annular-recess pad

113,423,424

circular- recess pad 91,420,421

cylindrical pad 140

hybrid bearings

291,294

inf in i te- ength pad

366,369,371,426,429

muk ipad journal bear ings 233

mult i recess journal b ear ings 245

mult i recess thru st bear ings

261

opposed- pad bear ings


flow divider s

205

l am ina r - f low rest ri c to r s

195

orifices 198

rectangular pad 135

se l f - reg ula t ing bear ings

224

compensated supply

229

direct supply (co nstant flow) 225

direct supply (constant pressu re)

227

compensated supply

154



l am ina r - f low rest ri c to r s

156, 157

s i n g l e - pad bearings


tapered bearings 266

Yates bearings

283

pumps

511

ra c k- worm assemblies 494

radial clearance 253,272,284

recess flow recirculation

73- 4,137,245,246,288

recess pressu re

circula r- reces s pad 76,91



SUBJECT

INDEX

541

recess p ressure (continued)

hydrostatic lift 143

in fi ni te - length pad

71,366,371,429

opposed- pad bear ings

direct supply 188,191,192

l am ina r - f lo w rest ri c to r s 192

rectangular pad

415,417

s e l f - r e g u l a t i n g b e a r i n g s

s i n g l e - pad bear ings

Compensated supp ly

228



Sommerfe ld hybr id num ber --> velocity

specific he at

49

speed enhancem ent fac tor 244

speed param eter --> velocity parameter

spherical bearing s 8,11,275-279,4%-478

spherical pads 8,128 33,151

spindles 6, 3,483-490,513

squeeze coefficient

302,304,306-3EB

a n n u l a r - r e c e s s p a d

307

circula r- reces s pad 305

opposed- pad bearing s

333

paramete r

hydrostatic lubrication 1992

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