Hydrostatic Lubrication (Tribology Series)

HYDROSTATIC LUBRICAT10N TRIBOLOGY SERIES Advisory Board W.J. Bartz (Germany, F.R.G.) R. Bassani (Italy) B. Briscoe (...

Author: Roberto Bassani | B. Piccigallo

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HYDROSTATIC LUBRICAT10N

TRIBOLOGY SERIES

Advisory Board W.J. Bartz (Germany, F.R.G.) R. Bassani (Italy) B. Briscoe (Gt. Britain) H. Czichos (Germany, F.R.G.) D. Dowson (Gt. Britain) K. Friedrich (Germany, F.R.G.) N. Gane (Australia)

Vol. 1 Vol. 2 Vol. 3 VOl.

4

5 Vol. 6 Vol. 7 Vol. 8 VOl.

VOl. 9 VOl. 10 VOl. 11

VOl. Vol. Vol. Vol. Vol. Vol. Vol. VOl. VOl. Vol. VOl.

12 13 14 15 16 17 18 19 20 21 22

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.)

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) Friction and Wear of Polymers (Bartenev and Lavrentev) Microscopic Aspects of Adhesion and Lubrication (Georges, Editor) 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 et 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 LUBRICAT10 N

R. Bassani Dipartimento di Construzioni Meccaniche e Nuclear; Facolta di lngegneria Universita di Pisa Pisa, Italy

B. Piccigallo 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. All 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 A M 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 of 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 conjugated 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 friction, high load capacity (even when the relative velocity of the lubricated surfaces is low or nought), a high degree of stiffness and the ability to damp vibrations. As compared 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 researchers 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 parameters (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 LUBRICATlON

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 compressibility) may reduce the margin of stability. Chapter 11 deals with the problem of the optimization of bearings, aimed a t 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 matter of the experimental testing of hydrostatic bearings are given in chapter 13. Finally, a number of examples of actual applications of hydrostatic lubrication are to be found in the last chapter. We wish to express our gratitude to 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 to thank the firms (namely: FAG Kugelfischer, 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 to thank Dr. Paola Forte, who helped us in a number of ways, Mr. Sergio Martini and Mr. Aldo del Punta, who carried out part of the graphical work.

Contents

List of main symbols Chapter 1

XiV

HYDROSTATIC BEARINGS

1.1 INTRODUCTION ..... 1.2 WORKING PRINCIP 1.3

........................................................... ................................................................ ADVANTAGES AND DRAWBACKS .............................................................

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

1

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

3

1

1.4 APPLICATIONS ........................................................................................... 1.5 TYPES OF BEARINGS ........................................................ 1.5.1 Thrust bearings .................................................................. ........................... 7 ...... ................................................. 9 1.5.2 Radial bearings ....................................... ................................................................................. 11 1.5.3 Multidirectional bearings 1.5.4 Bearing arrangements .......... ..........................................................................

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

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

Chapter 2

14

COMPENSATING DEVICES

2.1 INTRODUCTION .................................................................................................

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

16

2.3 COMPENSATED SUPPLY SYSTEM ............................................................................... ......................................... 2.3.1 Fixed restrictors ................................................................. 2.3.2 Variable restrictors ........................................................... .................................. ..................................................... 2.3.3 Inherently compensated bearings ......... .......................................................... 2.3.4 Reference bearings ....

17 18 19 25

2.2 DIRECT SUPPLY SYSTEMS

2.4 THE COMMONEST SUPPLY SYSTEMS ................................................................. 2.4.1 Direct supply ................................................... .................................................... ............................................................... 2.4.2 Compensated supply ...................................... 2.5 HYDRAULIC CIRCUIT ...................................... REFERENCES ............................................................

Chapter 3

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

LUBRICANTS .......................................................................................

3.1 INTRODUCTION

30 30 31 33

...........35

3.2 MINERAL LUBRI ..................................................................................... .................... 3.2.1 Types ....................................................... ......................................................

36 36

viii

HYDROSTATIC LUBRlCATlON 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7

Viscosity ................................................................................................................................... Oiliness .................................................................................................................................... Density ... Thermal properties Other properties ...................................................................................................................... Additives ..................................................................................................................................

36 41 42 49 50 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 THE REYNOLDS EQUATION ............................................................................................................ 4.3.1 Rectangular coordinates ......................................................................................................... 4.3.2 Cylindrical coordinates ........................................................................................................... ....................................................... 4.3.3 Spherical coordinates

58 58 61 64

4.4 THE LAPLACE EQUATION ...............................................................................................................

65

4.5 LOAD CAPACITY, FLOW RATE, FRICTION ................................................................................... 4.5.1 Load capacity .......................................................................................... 4.5.2 Flow rate .................. .................................. 4.5.3 Friction ...............................................

66 66 66

4.6 THE ENERGY EQUATION 4.7 LAMINAR FLOW THROUGH CHARACTERISTIC CONFIGURATIONS ..................................... 4.7.1 Parallel surfaces ...................................................................................................................... 4.7.2 Infinite-length rectangular pad ....................................................... 4.7.3 Flow recirculation inside recess .......................................... 4.7.4 Annular clearance. .................................................................................................................. 4.7.5 Circular pad .............................................................................................................................

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

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

69 69 71 73 75 76 77

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

79

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

80

4.10 TURBULENT FLOW .........................................................................................................................

80

4.11 THE FLOW IN ORIFICES ................................................................................................................

83

REFERENCES ..

85

Chapter 5

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

PAD COEFFICIENTS

5.1 INTRODUCTION .............................. 5.2 GENERAL STATEMENTS ................ 5.3 CIRCULAR RECESS PAD ....... 5.3.1 Basic equations ............. 5.3.2 Design chart ............................................................................................................................. 94 5.3.3 Effects of errors in parallelism ............................................................................................... 95

ix

CONTENTS

5.3.4 5.3.5 5.3.6 5.3.7

Effects ofthe loss of pressure at the inlet .............................................................................. 98 Turbulent flow ....................................................................................................................... 101 Effects of the inerti ............................................................. 103 Thermal effects ..... ................................................ 107

5.4 ANNULAR RECESS PADS ................................... .......... 112 ................................................ 112 5.4.1 Basic equations ..... 5.4.2 Effects of errors in parallelism ............................................................................................. 117 5.4.3 Effects of pressure losses at the inlet ................................................................................... 118 5.4.4 Turbulent flow ....................................................................................................................... 119 5.4.5 Effects of the inertia forces ........................................ 5.4.6 Thermal effects ........................................................... 5.5 TAPERED PADS ..................................................................... ................................ 5.5.1 Basic equations ...................................................................................................................... 5.5.2 Effect of the inertia forces ................... ........................................................... 5.5.3 Effect of misalignment ........................ ...........................................................

123 123 125 126

.................................................................. 128 5.7 RECTANGULAR PADS ..................................................................................................................... 133

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

5.8 CYLINDRICAL PADS........................................................................................................................

138

5.9 HYDROSTATIC 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 Temperature and viscosity ........................................................

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

6.3 COMPENSATED SUPPLY................................................................................................................ 6.3.1 Laminar flow restrictors (capillaries)................................................................................... 6.3.2 Orifices .... 6.3.3 Constant flow valves .............................. 6.3.4 Spool valves ............................................................... 6.3.5 Diaphragm-controlled restrictors .............................. 6.3.6 Infinite-stiffness devices ....................................................................................................... 6.3.7 Inherently compensated bearings ................................................................ 6.4 DESIGN OF SINGLE-PAD THRUST BEARINGS ... 6.4.1 Direct supply (constant flow) .................................... 6.4.2 Compensated supply (constant pressure) ............................................................................

Chapter 7

153 155 160

169 172

180

OPPOSED-PAD AND MULTPAD BEARINGS

7.1 INTRODUCTION 7.2 OPPOSED-PAD B 7.2.2 Capillary compensation............................................. 7.2.3 Orifices .......... ...........................

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

197

HYDROSTATIC LUBRICA TlON

X

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

213

8.1 INTRODUCTION ...............................................................................................................................

236

7.2.6 Design of opposed-pad bearings ................... 7.3.1 Direct supply ............ 7.3.2 Constant pressure supply ....... 7.4.1 Direct supply .................................................

7.6 MULTIPAD JOURNAL BEARINGS

Chapter 8

MULTIRECESS BEARINGS

8.2 ANALYSIS ..........................................................................................................................................

236

8.3 MULTIRECESS JOURNAL BEARINGS ......................................................................................... 8.3.1 Bearing performance 8.3.2 Effect of geometrical ................................................................ 8.3.3 Design of multirecess journal bearings. 8.3.4 Design procedure ..................................

239

8.4 ANNULAR MULTIRECESS THRUST BEAR1

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

249

260

8.5 TAPERED MULTIRECESS BEARINGS.......................................................................................... 263 ............................................................................... 265 8.5.1 Single-conejournal bearings 8.5.2 Opposed-cone assemblies ............................................................................... 210 8.1 YATES BEARINGS

8.7.5 Design procedure ...

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

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

Chapter 9

283 285

HYBRID PLAIN JOURNAL BEARINGS

9.2 PERFORMANCE OF THE HYBRID PLAIN JOURNAL BEARINGS .....

REFERENCES

Chapter 10 DYNAMICS 10.1 INTRODUCTION ............................................................................................................................. 301 10.2 EQUATION O F MOTION ............................................................................................................... 302

CONTENTS

Xi

10.3 PAD COEFFICIENTS ............................................................... ........................................ 305 ........................................ 305 10.3.1 Circular-recess pads ................... 10.3.2 Annular-recess pads ................... ....................................................... 307 ........307 10.3.3 Tapered pads .............................................................................................. 10.3.4 Screw and nut assemblies ................................................................................................... 308 .................... 308 10.3.5 Other pad shapes .........................................................

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

10.4.1 Direct supply (c

311

.................314 10.4.4 Spool or diaphragm valves.. 10.4.5 Infinite stiffness devices

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

320

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

322

10.5.1 Transfer hnction 10.5.3 Frequency response ...................................................... 10.6 OPPOSED-PAD BEARINGS ....................................................

10.6.1 Direct supply (constant flow) .......................................

10.7 SELF-REGULATING BEARINGS ......................................................... 10.7.1 Constant flow feeding .......................................................................................................... 10.7.2 Constant pressure feeding ..................................................................................................

339 341 342

10.8 MULTIPAD BEARING SYSTEMS 10.8.1 Hydrostatic slideways ......................................................................................................... 10.8.2 Multipad journal bearings ..................................................................................................

344 346

10.9 MULTIRECESS JOURNAL BEARINGS ....................................................................................... 10.9.1 Analysis ..................................................................... 10.9.2 Non-rotating bearings, incompressible lubricant 10.9.3 Rotating bearing, incompressible lubricant ......... 10.9.4 Compressible lubricant ....................................................................................................... REFERENCES ...............

349

358

......................................................... 360

Chapter 11 OPTIMIZATION 11.1 INTRODUCTION ............................................................................................................................. 362 11.2 GENERAL PROCEDURE ............................................................................................................... 362 11.3 CONDITIONS OF 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

11.6.3 Given load ........

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

385

xii

HYDROSTATIC LUBRICATION

11.7 REAL PADS ...................................... ...... ............... 11.7.1 Rectangular pad .................................................................................................................. 11.7.2 Other types of pads .............................................................................................................. 11.7.3 Circular pad ......................................................................................................................... 11.7.4 Annular pad ......................................................................................................................... 11.8 COMPENSATED SUPPLY .............................................................................................................. 11.8.1 Capillary tubes .................................................................................................................... 11.8.2 Steady pad ...........................................................................................................................

.................................................. 11.8.4 Dissipated power and efficiency losses...............................................................................

415 419 420 422 425 425 426 431 432

11.9 OPl'IMIZATION

11.10 OTHER TYPES OF COMPENSATING ELEMENTS ....... ........................ 11.10.1 Orifices ............................................................................................................................... 11.10.2 Flow-control valves ............................................................................................................

443 443 444

11.11 REAL PADS ....................................................................................................................................

444

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

446

Chapter 12 THERMACFLOW 12.1 INTRODUCTION ............................................................................................................................. 447 12.2 TEMPERATURES IN THE BEARING ...... 12.2.1 Temperatures in the film .................................................................................................... 12.2.2 Temperatures a t the film outlet .........................................................................................

447 449

12.3 SUPPLY PIPELINE .........................................................................................................................

456

12.4 COMPENSATING ELEMENTS......................................................................................................

458

12.5 PUMP ......................................

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

459

12.7 SELF-COOLING CAPILLARY TUBE ............................................................................................

461

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

463

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

464

12.6 COOLING PIPELINES

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

Chapter 13 EXPERIMENTAL TESTS

13.3.1 Electric analog field plotter

................................................ 13.3.5 Screws and nuts REFERENCES ......

469

xiii

CONTENTS

Chapter 14 APPLICATIONS 14.2 MACHINE TOOLS .........

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

...................................................................... 14.2.3 Feed drives ........ 14.2.4 Guideways and r 14.4 OTHER APPLICATIONS ......................

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

14.5 HYDRAULIC CIRCUITS............................................. 14.5.1 Simple layout ............................ 14.5.3 Multiple pumps REFERENCES .......

483 483 491 492 496 511 513

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

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

APPENDICES A.l SELF-REGULATED PAIRS AND SYSTEMS.....................................

..................................................................... A.3.1 Resistances ..........

.............................................. ............................................... ...................527

REFERENCES ..

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

Author index

533

Subject index

537

List of main symbols

An inverted comma (') generally indicates that 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 or concentric configuration whenever it is applicable; in chapter 11 generally indicates a suitable (although arbitrary) value used as 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 or diaphragm (controlled restrictors) land length a l L or ul(r4-rl) characteristic parameter of flow dividers (Eqn 7.49) pad width; squeeze coefficient (pad bearings) or damping coefficient (journal bearings)

B'

BO

b b' C C

D e F

Ff

f

BIL; B I Bo (single pad and selfregulating bearings); BI2Bo (opposed-pad b.) reference squeeze coefficient of a pad or reference damping coefficient of a journal bearing recess width blB radial clearance specific heat diameter displacement loading force friction force friction coefficient

LIST OF SYMBOLS

recess friction factor Qpc (chapter 12) see Eqn 12.7 actual value of axial play (opposed-padbearings) friction power nondimensional friction parameter reference friction power of a Pad H f I H f o(single pad and selfregulating bearings); H f l 2Hf0 (opposed-padb.) land friction power recess friction power pumping power (dissipated in a pad) pumping power total power see Eqn 6.58 or 8.32 film thickness normal film thickness (tapered pads) recess depth stiffness KIKo reference stiffness reference stiffness for capillary compensation stiffness of lubricant (Eqn 10.19) equivalent bulk modulus tilting stiffness stiffness of spring or diaphragm (controlled restrictors) 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

Pr Ps

Q R R*

R’ RO Ri Re Rr r r’

6 Sh

Si T Ta Tt? Ti

U V W wf2

wz X

a

8 8” Y

xv

recess pressure supply pressure flow rate hydraulic resistance; thermal resistance (chapter 12) nondimensional resistance parameter RIR, (pad bearings) reference hydraulic resistance of a pad RIRo (self-regulating bearings) Reynolds number hydraulic resistance of compensating restrictor radius inner to outer radius ratio rllr2 (self-regulating bearings) velocity parameter (Eqn 8.8) inertia parameter temperature ambient temperature temperature at the outlet of a land (chapter 12) temperature at the inlet of a land (chapter 12) sliding speed volume of recess and relevant supply pipe (chapter 10) load capacity hydrodynamic load capacity axial load capacity (tapered or spherical journal bearings; Yates bearings) displacement of spool or diaphragm (controlled restrictors) half-cone angle; thermal conductance (chapter 12) reference pressure ratio valve parameter load angle (spherical journal bearings); reference pressure ratio (selfregulating bearings)


area ratio (infinite-stiffness valves) temperature rise 2(C&f-C,) pitch error non-dimensional pitch error (Eqn 7.67) eccentricity: e l h o or e l C axial eccentricity (tapered journal bearings; Yates bearings) damping factor clearance error (Eqn 7.20) circumferential length of inter-recess lands (multirecess bearings); flank angle (screws) helix angle; transfer function of a block (chapter 10); thermal conductivity coeficient (chapter 12) transfer function of the feedback block transfer function of the feedback block in the case of fixed restrictor

dynamic viscosity see Eqn 6.60 see Eqn 6.62 or 8.33 kinematic viscosity; Poisson ratio non-dimensional displacement 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


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 engineering. In particular, it can be used instead of hydrodynamic lubrication when this last proves to be not very effective. The main advantages of externally pressurized 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 kinematic pair, two important conditions have to be met: - the mating surfaces must not be parallel; - a sufficient relative velocity must exist. When one, or both, of these conditions cannot be satisfied, a n “external pressurization” 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


Fig. 1.1contains an outline of the principle of "externally pressurized lubrication", which is most commonly referred t o as "hydrostatic lubrication". The recess (of which the projected area is A) of the bearing pad [ l l 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 to run, the pressure in the recess grows (Fig. l.l.b), until the "lifting pressure" p = W I A is reached (Fig. 1.1.~);at 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],as 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

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 the single pad. When two, o r more, recesses exist, as in Fig. 1.2, it is clearly necessary to feed each of them by means of separate pumps; alternatively, a common source of lubricant may be used, but each recess must be fed through an adequate compensating device (restrictor). We shall deal extensively with this point in the following chapters. -b-

-a-

ri

tttlt t tr

Pr,

Fig. 1.2 Hydrostatic 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 externally 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 lubricant 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 a t 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 to 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 assemblies 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 Frenchman L. D. Girard, who, in the second half of the last century, built a water hydrostatic 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 equations 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),and 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 manufacture) Cost (to install) Life Lubrication circuit Cost of lubrication circuit Supply pressure and pumping power

Axial pairs Radial pairs Helicoidal pairs (Slides) (Journals) (Screw-nuts) Boun Hyst Rol Hydy Hyst Rol 3oun Hyst Roll

3 3 4 4 2 5 1 4 2 5 3 4 4 1 4 3 2 2 4 2 2 2 2 2 2 3 2 3 3 2 3 3 2 3 3 3 2 3 2 3 3 2 4 4 2 2 4 3 4 3 3 2 4 3 2 3 2 4 3 2 3 2-31 4 3 2-31 Load 3-51 2-32 2-41 3 2 2-41 Stiffness 2-32 2-4' 3 3-51 3 2-41 Vibration damping 2-33 4 2 3 4 5 Friction coefficient and friction power 3-52 3 3-52 4 2 3-52 Stick-slip 5 4 5 5 1 5 Wear 5 3 5 3 1 5 (Rating 5 is best or more desiderable. Boun: Boundary lubrication; Hyst: Hydrostatic lubrication; Hydy: Hydrodynamic lubrication; Roll: Rolling elements). It depends on supply type. It depends on speed. 3 Whirl. &@: 5 4 2 4 3 4 3 3 3 3 4 4 3 3 4 3 2 1 2

4 3 4 3 3 3 3 2 3 5 2 2 2-41

3 4 3 2 2 2 2 2 2 3 3 3 4 3 3 2 3 5 4

3 4

3 2 2 2 2 2 3 3 3 3 4 3 3 2 3 5 3

Externally pressurized lubrication is a t present used in the entire field of mechanical 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 mentioned: 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 or slags; thermal problems exist here, too.

6


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 carriages are supported by hydrostatic slideways and are sometimes driven by hydrostatic 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 to 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 to 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 for hydrostatic slideways, hydraulic cylinders for flight simulators. -a-

-b-

-c -

i-

w

t-----+--

. _

I

I

Fig. 1.3 Circular pad bearings: a- circular recess pad; b- annular recess pad; c- multirecess pad.

7


TYPES OF BEARINGS

1.5

Hydrostatic bearings may be classified on the basis of the direction of the load that may be carried. So we have: - thrust bearings; - radial bearings (journal bearings); - multidirectional bearings. Let us examine briefly the most common shapes.

1.5.1

Thrust bearings

(i) Circular p a d bearings. Figure 1.3 shows: -a- a central-recess pad; -b- an annular recess pad; -c- a multirecess pad. As rotary speed becomes very high, the behavior of pad -a- becomes "hybrid": hydrodynamic pressure, caused by centrifugal 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 directions, or when greater stiffness is needed, two circular pads may be assembled a s 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 to 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 p a d bearings. 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


shown in Fig. 1.5. If the pads are moving a t 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 multirecess rectangular pad in Fig. 1.5.f can also sustain tilting moments. -a-

- b-

-c-

-e-

-f-

+I

-d-

I

Fig. 1.5 Rectangular pad bearings: a- equal sill width pad; b- different sill width pad; c-, d-, e- pads with rounded comers; f- rnultirecess pad.

(iv) Opposed-pad 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 pad bearings. The conical pads shown in Fig 1.7 are similar to 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 pad bearings. They are shown in Fig. 1.8. (vii) Screw-nut assemblies. 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.~).In general, however, a double effect

9


-b-

-a-

Fig. 1.6 Opposed-pad rectangular bearings: a- equal pads; b- unequal pads.

I

- a -

'

I

-b-

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 screwnut (Fig. 1.lO.b); in this case, a double-thread screw must be used. Note its similarity with the self-regulating bearing in Fig. 1.4.c.

1.5.2

Radial bearings

Fig. 1.ll.a shows a cylindrical pad; since it can sustain loads only in one direction, 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 to 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.

10


-b-

- a -

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 better 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 are reduced to a minimum to enhance hydrodynamic lift. They are designed to sustain the load by means, in the main, of the hydrodynamic effect a t the regime velocity, while the hydrostatic pressure field is used, in the main, to prevent any contact of the surfaces i n the startstop phases.

11


-a-

-b-

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

Multidirectional bearings

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, to 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-

I

-

i

-a-

I

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

-a-

I

Fig. 1.13 Multidirectionalbearings: a- conical bearing; b- spherical bearing.

Fig. 1.14 "Yates"bearing.

-c-

13


1.5.4

Bearing 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 Hydrostatic spindle: a- with a journal bearing and a combined journal and thrust bearing; b- with two conical bearings.

Fig. 1.16 Hydrostatic slideway.

14


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 combined with a hydrostatic one (the centrifugal oil feed is also shown). This arrangement allows the ball-bearing inner ring to rotate a t a lower speed than the shaft.

Fig. 1.17 Combined rolling-hydrostatic bearing.

REFERENCES

1.1 1.2 1.3 1.4 1.5 1.6 1.7

1.8 1.9

Girard L. D.; Noueau Systkme de Locomotion sur Chemin de Fer; Bachelier, Paris, 1852. Vogelpohl G.; Betriebssichere Gleitlager; Springer Verlag, 1958; 315 pp. Lord Rayleigh; A Simple Problem in Forced Lubrication; Engineering, 104 (1917),617-697. 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 Hydrostatic Bearings, Pt. lt10; Machine Design, Aug.+Nov. 1963. Opitz H.; Aufbau und Auslegung Hydrostatischer Lager und Fuhrungen und Konstruktive Gesichtspunkte bei der Gestaltung von Spindellagerungen mit Walzlagern; VDW-Konstrukteur-Arbeitstagung, 1969 Stansfield F. M.; Hydrostatic Bearings for Machine Tools and Similar Applications; The Machinery Publishing Co. Ltd., 1970; 227 pp. Rowe W. B.; Hydrostatic and Hybrid Bearing Design; Butterworth & Co, 1983; 240 pp.

Chapter 2


2.1

INTRODUCTION

It has been already pointed out that two bearings are necessary to bear loads in reverse direction. Two bearings are also needed if load is not coaxial with the bearing as i n Fig. 2.1.a (that 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 to happen, the supply system must allow for these different

-a-

- b-

tt'tttlLv Fig. 2.1 Eccentric load on hydrostatic pads: a-single pad; b- two-pad arrangement.

16

HYDROSTATIC LUBRlCATlON

pressures. In practice this may be accomplished in two ways: - by using a separate pump to feed each recess directly; this is commonly referred to a s the "constant-flow supply system"; - by using a common source of pressurized lubricant, which is carried to 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 t h a t 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 i t 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.

DIRECT SUPPLY SYSTEMS

2.2

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, the film thickness h is related to 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 because the load is displaced toward the pad [12]), pressure grows in [12], and decreases in [11]. Since Pr=Ps,because no restrictor can dissipate power, the system dissipates the smallest pumping power.

h

Fig. 2.2 Constant flow supply system: one pump for each bearing.

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 a n 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 a n arrangement makes it possible to reduce the pressure step in the pumps, and the related problems, especially in the case of gear pumps.

-

--

Hydrostatic Bearings

-.

--

-.--

j

Fig. 2.3 Constant flow 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) Ri. The pressure p s upstream from the restrictors 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 consequence 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 restrictors

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] is squeezed, and so its hydraulic resistance increases. Hence, the total resistance of the series constituted by the restrictor R2 and the relevant clearance h2 also increases. Since pressure p s is held constant, the rate of flow must decrease, so the pressure step ps-pr2 must decrease a s 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 a s compensating devices. Unlike the laminar restrictors, their hydraulic resistance is no longer a constant. This leads to a slightly better 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 restrictors

Many kinds of restrictor have been proposed that are able to vary their own resistance, depending on the pressure step: more precisely, the hydraulic resistance should grow (often in a non-linear fashion) as the pressure step ps-pr increases; the contrary happens if ps-pPrdecreases. With reference to Fig. 2.5, it follows that, as thickness h, is reduced by the load, the recess pressure pr2 grows faster than in the case of the fixed restrictor; the contrary occurs in the case of the other recess. Consequently, 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 to that is found in the case of constant restrictors. The flow-pressure (load) and film thickness-load characteristics of certain typical 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 deliver a flow rate which is proportional to load. Indeed, in this case, the film thickness 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) Elastic capillaries. The simple device in Fig. 2.7 is made up of a small diameter pipe, filled with a suitable rubber-like material, in which a capillary hole is

20


-a-

-b-

h

Q

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 its balance due to the opposite thrusts exerted by the spring and recess pressure p r on area A,. As Pr varies, the length x of the restrictor varies too, and so does its hydraulic resistance. The shape of the valve may also be that seen in Fig. 2.8.b.

The tapered-spool valve in Fig. 2.9 works in a similar way (note that the aperture angle is very small). However, since its hydraulic resistance varies faster with x as compared to the preceding device, its 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 to p r , thus

21

COMPENSATlNG DEVICES

- b-

-a-

t

I

1

PS

1

ps

pr

Fig. 2.8 Cylindrical-spoolvalves.

Ips

\

Fig. 2.9 Tapered-spoolvalve.

t

pr

Fig. 2.10 Diaphragm-controlledvalve. approaching the "infinite stiffness" behavior for a certain range of loading conditions (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

22


made to deliver a constant flow if it is properly tuned. To improve its performance, a reference restrictor RU can be added as in Fig. 2.11 (ref. 2.7). In order to fix the flow rate a t a certain constant value, it is necessary to ensure that the pressure step across RU does 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, until a new equilibrium 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 restrictor Ru (which in general is an orifice) it is possible to adjust the rate of flow.

Fig. 2.11 Constant-flow valve

The performance of this supply system is similar to that of the direct supply systems 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 to work properly when Ps-Pr reaches a characteristic minimum value; - efficiency is much lower, due to the great loss in pressure Ps-pr in the valve itself. (v) Infinite-stiffnessualue. 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 Pr. The differential piston is in equilibrium due to pressures P r and p,, that act on different areas. Consequently, the piston searches for its 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 is a constant, the rate of flow across it is proportional to p r . An infinite static stiffness is therefore obtained, until p,, approaches supply pressure p s .


t

23

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 a n 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 or lubricant temperature. (vii) Flow dividers. When two opposite pads have to be fed, as in Fig. 1.12.a, a flow divider may prove to be more effective than the use of a separate controlled restrictor 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 recesses 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 performance is better (ref. 2.10). Figure 2.15 shows the lay-out of a diaphragm flow divider (ref. 2.11). The restrictors 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

I

I 1

r

I

I

Fig. 2.14 Tapered-spool flow divider.

Fig. 2.15 Diaphragm-controlledflow 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

Inherently 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 to the peculiar recess shape (Fig. 2.171, o r to the presence of a n 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, that is comparable with clearance h; consequently, the pressure drop in the recess is no longer negligible 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 pressure 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 i t 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 comparable with clearance. These types of bearings create considerable manufacturing problems (precision machining of a very shallow recess); to overcome these, the bearing depicted in Fig.

26


2.18.a (ref. 2.14) has a flat layer of elastomer [l] bonded to a rigid base. Since pressure decreases from the center to the pad boundary, deformation obviously causes the development of a "recess" (similar to 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, it is free from problems like rubber-oil compatibility. -b-

-a-

Fig. 2.17 Inherently compensated bearings: a- shallow recess bearing; b- tapered-recess bearing. -b -

-a-

m

,I

\,

, , , , I,\, /

/

,

-

\ /

/

, ,, /

/

I

A 1

Fig. 2.1 8 Inherently compensated bearings: a- elastomeric bearing; b- flexible-plate bearing.


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). It is supplied at a constant pressure p s . Before entering the bearing gap, the lubricant flows through the clearance b around the tapered plug C , retained by the perforated elastic diaphragm D. Clearly, the bearing simply performs as if it were supplied by means of a diaphragm valve of the type already examined.

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 restrictor (HCR). The bearing i s made up of an inner [ l l and an outer [23 sleeve fixed to a flange [3], and of a moving ring [4] between the sleeves. Oil, supplied at pressure p s , is restricted when i t 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 supported hydrostatically with oil supplied by another pump a t pressure p:. By adjusting 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 to 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 always Rs=Ri, whatever the displacement of the moving member, or 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, a s 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 are 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 that 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.

29

COMPENSATlNG DEVICES

Restrictor lands

No3

No4

No2

NO1

A,

I

I

Main pocket

I

No2

No1

I

No3

I

I

I

Gmove restrictor pocket

Supply hole

1

~ o 4

Fig. 2.22 Developed view of a hydrostatic journal bearing having integral variable restrictors. 2.3.4

Reference bearings

A small bearing r in Fig. 2.23, compensated by a diaphragm valve, has been proposed (ref. 2.6) to 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 accurately manufactured and positioned "reference" bearing, and its 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.

30


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 to the central hole. The bearing may be supplied by a diaphragm valve, and this feature allows it to operate at large gaps with great stiffness and accuracy. Especially if external controlling force is operated by a solenoid, the bearing becomes an effective reference bearing.

2.4 2.4.1

THE COMMONEST SUPPLY SYSTEMS

Direct supply

Systems provided with a separate motor-pump (usually axial or radial pistonpumps) for each recess (Fig. 2.2) are not very often encountered, because of the relevant 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), or 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) a s 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 restrictors, 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 a s 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 latter 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. Furthermore, in the case of the orifices, the system is more sensitive to any change in the temperature (i.e. in the viscosity) of the lubricant. (ii) Variable 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 VlCES

31

For almost all the other controlled restrictors experimental applications are known (chapter 14). Diaphragm flow dividers, in particular, have been widely studied in connection with opposed-pad bearings, journal bearings, hydrostatic slideways 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 (or 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 a t the pads through the restrictors R, which should be as near a s 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 accumulator, preceded by a a check valve Vn, can supply the system during its inertia movements. Pressure gauges may be put after the restrictors to indicate the pressures 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, a n 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, a s a consequence of viscous friction. So, if natural cooling in the reservoir is not sufficient, a cooler may be inserted a t point [5], or a t point [l] if a close temperature control is required. The cooler may also be put in parallel to 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 admissible value.

If pads are provided externally with low-friction seals, a low pressure a t the film exit may pump the return lubricant directly to reservoir S m , thereby simplify-

32


-a-

- b-

%& I! I

1

W a

I

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 bearing in Fig. 2.24.a, versus eccentricity E=(ho-hi)lho,for certain values of the ratio P=p,dp,, ho being the film thickness and pro 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 a n indicative 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 26 2.3 2.4 2.6 2.6 2.7 2.8 29 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 Sliding 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 Hydrostatic 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, 532541. Mayer J. E., Shaw H. C.; Characteristics of an Externally Pressurized Bearing Having Variable External Flow Restrictors; ASME Trans., J . of Basic Engineering, 86 (19631,291-296. Bassani R.; Divisore di Flusso a Spola Conica: sua Applicazione a Supporti Idrostatici; Fluid-Apparecchiature Idrauliche e Pneumatiche, lS,17 1-172 (19781,31-37. 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 Machine 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 Bearings; Proc. Instn. Mech. Engrs., Tribology Convention, 1969. Hirs G. G.; Partly Grooved Externally Pressurized Bearings; Proc. Instn. Mech. Engrs., Lubrication and Wear Convention, 1966; paper 21. Dowson D., Taylor C. M.; Elastohydrostatic of Circular Plate Thrust Bearings; ASME Trans., J. of Lubrication Technology, 89 (19671,237-262. Davies P. B.; Investigation of an All-Metallic Flexible Hydrostatic Thrust Bearing; ASLE Trans., 17 (19741, 117-126. Tully N.; Static and Dynamic Performance of an Infinite Stiffness Hydrostatic Thrust Bearing; ASME Trans., J. of Lubrication Technology, 99 (19771, 106112. Brzeski L., Kazimierski 2.; High Stiffness Bearing; ASME Trans., J. of Lubrication 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, 4954.

34


2.19 Bassani R.; A New Opposed-Pad Hydrostatic Bearing: the Flow Self-Regulating Bearing; Meccanica, 10 (1979, 107-113. 2.20 Geary P. J.; Fluid Film Bearings, A Survey on their Design, Construction and Use; SIRA, 1962. 2.21 Yates S.;Combined Journal and Thrust Bearings; UK Patent 639293, 1950. 2.22 Wong G. S. K.; Interface Restrictor Hydrostatic Bearings; Advance in Mach. Tool Des. and Res., Proc. of the 6th Int. MTDR Conf., 1975; pp. 271-290. 2.23 Bassani R.; Self-Regulated Hydrostatic Pads; Wear, 61 (19801, 49-68. 224 Ohsumi T., Mori H., Ikeuchi K.; Zmprovement of Characteristics of Externally Pressurized Bearings; Proc. of the Japan Int. Tribology Conf., Nagoya, 1990;pp. 1791-1796.

Chapter 3

LUBRICANTS

3.1

INTRODUCTION

Lubricants are put between two surfaces to 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 experiments (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.

36

HYDROSTATICLUBRICATION

3.2

MINERAL LUBRICANTS

3.2.1

Types

Hydrocarbons, which mineral lubricants are mainly made up of, have three basic 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 lubricant; 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-

H

-C

-

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 du ldy, 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 6zZx/&duJ d y )is necessarily constant throughout the fluid or during the time of flow.

LUBRICANTS

37

Fig. 3.2 Laminar shear 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)

du=O dY

when z = O

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 - pseudoplastic material.

38


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/cm2 or poise. Mineral lubricants and synthetic lubricants of low molecular weight are Newtonian 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) and now coming into wide use. Viscosity systems establish a series of definite viscosity levels a s a common basis for specifying the viscosity of industrial fluid lubricants. Reference viscosities are measured in mmVs or cSt (centistokes) a t 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 SAE (Society of Automotive Engineers) classification is also shown. The reference temperature of the SAE classification is lOO"C,and 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 Classification and Identification

Mid-Point viscosity Kinematic viscosity SAE Classification cSt (m2/s)at 40°C limits cSt (mm%)at 40°C Min Max

I S 0 VG 5 IS0 VG 7 ISOVG 10 IS0 VG 15 IS0 VG 22 I S 0 VG 32 I S 0 VG 46 I S 0 VG 68 I S 0 VG 100 IS0 VG 150 I S 0 VG 220

4.6 6.8 10 15

22 32 46 68 100 150 220

4.14 6.12 9.0 13.5 19.8 28.8 41.4 61.2 90.0 135 198

5.06 7.48 11.0 16.5 24.2 35.2 50.6 74.8 110 165 242

5w low 20w 30

50

LUBRICANTS

39

(i) Viscosity-temperature. The viscosity of liquid lubricants decreases with increasing temperature. Variations in temperature may be due to 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; its dimension is that of the inverse of temperature. 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 classified 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 to a Viscosity Index =lo0 or 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 it possible to give a numerical value to such behavior. The viscosity index is based on two groups of oils. In one group, that is naphthenic in nature, VZ=O because of its sensitivity to 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 a t 100°C as the oil to 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

u vz=L----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 = lW0.00715 + 100 ~

(3.7)

where

N=

logH - l0gU logy

and Y is the kinematic viscosity in cSt a t 100°C for the oil being considered. The influence of different viscosity indexes on oils with the same reference viscosity is taken from Fig. 3.5 (ref. 3.4).

(ii) Viscosity-pressure. The viscosity of lubricants increases with pressure. A widely-used model of the viscosity-pressure relationship is

41

LUBRICANTS

-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 to 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.

42


3.2.4

Density

As is well known, the density of a liquid is the mass of a unit volume, generally calculated a t 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 approximately 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-1for p=0.22+0.01 Ns/m2. In hydrostatic lubrication thermal expansion is often negligible.

(ii) Compressibility. The compressibility of a liquid can be defined a s the property 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 )of liquid, then from Eqn 3.9 c=---

1 dV

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 i t is a physical property of the base liquid. Very often, instead of compressibility, its reciprocal is used: the bulk modulus

KL.Figure 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 at any other pressure and at the selected temperature.

43

LUBRlCANTS

-b-

-a-

E

5 cn-

2n

3450 3105 2760 2415 2070

p

1725

5

1380

3

m

1035

690 0 80 160 240 320 400 480 560 (-18 27 71 116 160 204 249 293)

/ / /'

p= 5 M

N

~

TEMPERATUREFF, (OC)

Fig. 3.6 Bulk modulus of mineral oils

For instance, let us consider a mineral oil with density p,=0.90 kg/dm3 at ambient pressure and at temperature T=40°C:its bulk modulus a t pressure -5 MPa and a t 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].109 + 5.6 p , Nm-2

(3.11)

where v is the kinematic viscosity in cSt a t 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=1570 MN/m2. Values of K1 obtained from Fig 3.6 or by means of Eqn 3.11 are fit for high pressures; nevertheless they can also be used, approximately, at mean and low pressures (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 pressure of the gas and at the given temperature.

44


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 may affect lubricant properties, such as viscosity which grows worse. The air dissolved in an oil comes out of solution when temperature and pressure decrease and may produce air bubbles and foam. - b-

-a-

MINERAL OIL

0

2

4

6

8

1

0

PRESSURE, M P a

0

25

50

75

100 125

TEMPERATURE PC

Fig. 3.7 Solubility of gases versus: a- pressure; b- temperature.

(iv) Air entrainment. Common causes of entrained air in a liquid are, for example, 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 lubricating 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 reduced, after a fairly long time, to about 1.5%, and after a very long time to 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.

45

LUBRICANTS

40

7 7

0

1

2

3

4

5

TIME , 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, po 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 assimilated) 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 cv are the specific heats a t constant pressure and constant volume, respectively. If Eqn 5.13 is introduced into Eqn 5.10, the bulk modulus Ka of air is obtained (3.14)

A t temperature T=40"C and a t 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


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 /cv versus 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 to 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 a t ambient pressure (it must also be taken into account that bubbles of other gases may also exist, which may be dissolved in oil in a greater proportion than that of air, as is shown in Fig. 3.71,for 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 to the compressibility of the fluid when evaluating effective compressibility (Eqn 3.10) and hence the apparent bulk modulus. The equivalent bulk modulus of a metallic pipe is, approximately,

(rs is the internal radius of the pipe, ra is the external one, E is the modulus of elasticity 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 a s 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 disregarded. The same is not true for flexible pipes (also for high-pressure pipes made of hard rubber or FTFE with an interwoven sheet of stainless steel) as transpires from existing experimental results. Figure 3.10.a (ref. 3.9) shows the considerable increase in the inner volume of certain flexible hoses. Obtaining realistic design values of the apparent bulk modulus of oil in hydraulic hoses is quite difficult. Values of Kla in the 70t350 MN/m2 range can be found in the literature. Some results are shown in Fig. 3.10.b (ref. 3.5)for a woven hose with rs=6.4 mm: the experimental data are clearly scattered. Recent design practice in relation to equipment dynamic noise reduction has tended to encourage the use of hydraulic hoses in fluid power systems. This does not always seem convenient 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 effective value of the apparent bulk modulus.

48


-a-

"

48

n QJ 40 32:

I - 24 -

Y

/ 0

0

o o o

16-

o

e

810

15

PRESSURE, MNrn-'

20

0

I

I

I

I

4

8

12

16

20

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 that 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, vibrations, 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 to 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 lubricants). 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; a t 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 6.10-4mbar at 4OOC; 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 microscopic 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

49

LUBRICANTS

io'r

/ 1

0)

I E

E

m

Ba

> 40

80

(494

26,7

120 160 200 49 71 93

TEMPERATURE,

OF,

280 360 138 1821 (OC1

Fig. 3.11 Vapor pressure versus temperature..

conventional systems contain sufficient nuclei to 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 increase 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 properties

(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 to 100OC; it also varies linearly with temperature.

50


3.2.6

Other properties

Pour-point is the temperature a t which a n 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 a t which the vapors given off by an oil ignite momentarily on the application of a small flame. The flash-point of naphthenic oils is a t about 170"C, that of paraffinic oils a t about 190°C. Acidity. Low acidity is advantageous €or reducing corrosion. Oxidation. High stability to oxidation is advantageous, because one cause of deterioration in lubricant oils is the formation of oxidation products. This also leads to a reduction of the life of the lubricant and to corrosive effects. Thermal decomposition. In the presence of oxygen, high temperatures may produce the thermal decomposition of mineral lubricants, which shortens their life. Figure 3.12 gives the approximate time-temperature characteristics of refined mineral lubricants, including oxidation (ref. 3.8).

1

10

10 Life, h

Fig 3.12 Approximate life-temperature characteristics of a mineral oil: A - oil without anti-oxidant; B - oil with anti-oxidant. 3.2.7

Additives

Nowadays lubricants often have chemical compounds added to 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 a t 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 a t one end and a -C02H group a t the other. This latter group would be adsorbed on metal surfaces. Actually, owing to surface motion, and in the

51

LUBRICANTS

10

I

1 -

.In

0.003-

0

50

100 120

Temperature,%

Fig. 3.13. Viscosity-temperature characteristics of 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 decomposers 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 paraffinic crystals, thereby preventing them from increasing. Oxidation inhibitors, such as certain phenols, amines and olefines, prevent or reduce the formation of oxidation products. They also prolong the life of the lubricant and act as corrosion inhibitors. Corrosion inhibitors. Rust, a hydrate iron oxide, is a widespread form of corrosion. Corrosion inhibitors, such as sulphonates, generally form a protective coating on metal surfaces. Many others additives, such as detergent, dispersant and extreme-pressure 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.

52

3.3


SYNTHETIC LUBRICANTS

The performance of synthetic lubricants is better than that of mineral lubricants, 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, which have very good fluidity a t low temperatures, and a very good VZ. Organic esters; those of dibasic acids in particular, which also have very good fluidity a t low temperatures, and good thermal stability. Phosphatic esters, whose oiliness is very good and whose thermal stability is fair. Polyglicols, with very good oiliness, a high VZ and also fluidity a t low temperatures. Silicones (with a polymer-like structure, in which the carbon is replaced by silicon). They have a high VZ (see also Fig. 3.131, a high flash-point, a low pourpoint, high thermal stability and oxidation stability and a good anti-foam performance. On the other hand, their oiliness is poor. 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 Chemistry in Lubrication; Elsevier, 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., Chichester, 1980; 505 pp. Liste des Caractkristiques Exigkes pour les Fluides Olkohydrauliques; CETOP (ComitB Europben des Transmissions Olbohydrauliques e t Pneumatiques), London, 1971. Fowle T; Aeration in Lubricating Oils; Tribology International, 14 (19811, 151157.

LUBRICANTS

53

Raznyevich K.;Tables et Diagrammes Thermodynamiques; Eyrolles, Didion, 1970. 3.9 Speich H., Bucciarelli A.; L’Oteodinamica; Techniche nuove, 1971; 727 pp. 3.10 Neale J. M.;Tribology Handbook; Butterworths, London, 1973. 3.11 Lansdown A. R.; Lubrication. A Practical Guide to Lubricant Selection; Pergamon Press, Oxford, 1982; 252 pp. 3.8

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 for viscous fluid mechanics and ref. 4.2 and ref. 4.3 for lubrication theory. Those primarily interested in the results applied to the most common types of bearings, may prefer to omit this chapter.

dx

Fig. 4.1 Equilibrium of a fluid element along direction x. 4.2 4.2.1

NAVIER-STOKES AND CONTINUITY EQUATIONS Rectangular coordinates

Let us consider a fluid element (Fig. 4.1).Along each coordinate axis we can write a n 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 a s follows:

BASlC EQUATlONS

Dv p E =pf+Vo

55

(4.1)

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 linear 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 a s the coefficient of proportionality 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 4.1and Eqn 4.2:

where u, u, w and X,Y, 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 a n infinitesimal control volume:

P+a

(PU) +

a

ay (pv)+

a

(pw) = 0

(4.4)

56

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 speaking, functions of pressure and of temperature. When the dependence on temperature cannot be disregarded, a further equation is required to 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 NavierStokes equations simply become:

where the Laplace operator V2 is defined as:

The continuity equation is reduced to

vv =o

4.2.2

(4.6)

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,y=r sin6, the following is obtained:

(4.7)

and (4.8)

In the equations above, u, u and w are the radial, tangential and axial components of the velocity. The operators DDt and V2 become

57

BASIC EQUATIONS

~a = - + + - + -a- +vw - a ~t

at

a

az

ar rat9

-b-

Fig. 4.2 Coordinate systems: a- cylindrical coordinates; b- spherical coordinates. Spherical coordinates

4.2.3

In t h e spherical coordinate system of Fig. 4.2.b (i.e.: x = r s i n q cos19, y=r sinq sinI9, z=r cosq) the Navier-Stokes and continuity equations become:

(4.9)

and

_i _ a r2

i

a

(U r2)+ - at9 (v r sinq

where

1

aw

sinq) + -r sinq 329 =

(4.10)

58


The symbols u , u , and w now indicate the components of the fluid velocity in the r, Q and 0 directions, respectively.

4.3 4.3.1

THE REYNOLDS EQUATION Rectangular coordinates

In the field of fluid film lubrication we are involved in most cases with the study of thin films (Fig. 4.3).In this connection, the complexity of the Navier-Stokes equations may be greatly reduced, thanks to the following considerations: - the thickness of the fluid film (in the y direction, in Fig. 4.3)is small, compared to its size in the other directions; - consequently, the pressure, as well as the density and the viscosity, may be averaged alongy: i.e. it is stated that aplay=O,arlay=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; - the flow is laminar: no turbulence nor vortex exists; - the body forces are negligible compared to the viscous forces; - the 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:

/-

2

Fig. 4.3 Thin fluid film.

t

59

BASIC EOUATlONS

14.11)

az The first of Eqns 4.11 may now be integrated twice, with the boundary conditions u=U1 for y=O and u=U2 for y=h, to obtain the component of the fluid velocity in the x direction:

u = L*y 2 Dax 0, - h) + (1 - f ) Ul + f U2

(4.12)

In the same way the component (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)to indicate the squeeze velocity of the surfaces, it is easy to find:

$(?$) + $(?$)

= 6 $ [ p h (U1+U2)] - 12 p U 2ahz+ 12 p V +12

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 as

equation 4.14 is simplified to

60


(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 surfaces. 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 it 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, i t transpires that, because the film is very thin, the tangential and radial components of the journal velocity may be written as:

(4.16)

Fig. 4.4 Thin fluid film between cylindrical surfaces.

When we have stated p=const., p=const., U1=0,Eqn. 4.14 becomes: (4.17) From Eqns. 4.16, it is clear that V=dUla&&?ahlals; since hh (i.e. a is selected as the larger dimension):

and if a>>h:

If a=h (square section, Fig. 4.11.~):

For a triangular equilateral section (Fig. 4.11.d):

For an annular duct, with Cc l , moderate/high speed bearings. 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 a s the limit of the ratio of a small change A F of the applied load to the consequent displacement Ae of the moving 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 to 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=-dW dh

It is important to note that 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 bearing, it may be worthwhile writing the expressions of each of the foregoing parameters 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-dimensional displacement E=-

h - ho h0

(5.9)

90


It should be noted that fqr certain types of pad a "natural" reference configuration exists (e.g. the concentric configuration for the cylindrical or spherical pads). In non-dimensional form, hydraulic resistance R and friction power H f can now be rewritten as:

R=RoR'

(5.10)

Ro and Hfo 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=KoK

(5.14)

but now both KO and 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 to 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: to 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 "direct"supply system) and section 6.3 (for the "compensated" supply systems).

5.3

5.3.1

CIRCULAR RECESS PAD

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:

n A, = 4 Dz A:

(5.20)

R=:$R*

(5.21)

A*, and R*, which are plotted in Fig. 5.l.b, are: (5.22)

R* = In llr'

(5.23)

The pumping power dissipated in the bearing is given by Eqn 5.4. It may prove useful to express it as a function of the load capacity: (5.24) where

92

HYDROSTATlC LUBRlCATlON

-a-

-b10

r

5

h

3

Fig. 5.1 Circular recess pad: a- pad geomc y; b- pad coefficients. 1

r*=AT

(5.25)

If we now want to 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. Furthermore, 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 section 4.7.4;Eqn 4.60may be rewritten here as:

where

PAD COEFFICIENTS

Fig. 5.2 Nomograph for a circular pad.

93

94


is plotted in Fig. 5.1.b. The friction power is:

H f = Q M =EED4Q2Hf* f 32h

(5.28)

The friction coefficient, referred to the mean radius of the land P=(rl+r2)/2, is (5.29)

It can be shown (ref. 5.1) that, for moderate speeds, f 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, is p=O.l Ns im2. For the ratio re, the "optimal" value is chosen: rAPt=0.53, because of low velocity. For the film thickness the value h=50 p m is chosen i n 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 effective 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) is 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 to 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 pressures. 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 &=4.6.10-6m%.

5.3.3

Effects of errors in parallelism

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

96


and

ih:

Ah'=-=-

tanly

(5.31)

As usual, it has been assumed that the mean film thickness than the diameter of the pad, and hence the angle ylis small.

K

is much smaller

This problem has been solved in ref. 5.2, by means of a finite-difference technique. Once the pressure field has been obtained, the load capacity can be calculated (by integration), as well as the velocity of the lubricant (from Eqn 4.20 and Eqn 4.21) and then the rate of flow (from Eqns 4.33). In Fig. 5.4 pressure p'=pIpr on the bearing lands and the flow velocity

a t the inner edge (r=rl)areplotted (for several values of 0)for r'=0.53, Ah'=l. Finally, on the basis of these results, certain approximate correction factors may be proposed. The effective area could now be expressed as: -a-

- b1.0

0.8

0.6 2 h

0.4

0.2

0 0.5

0.6

0.7

-r

0.8

0.9

1.0

0

r2

Fig. 5.4 Misaligned circular pad: a- pressure field; b- velocity profile.

0.2

0.4 U'

0.6

97

PAD COEFFICIENTS

(5.32)

where: (5.33) Since, however, the correction introduced is of the order of 1%, even for Ah’=l, it can be disregarded.

res 0.53

0.60

0.65

0.m 0.75

0.80 085

090

0.8

0.9

1

Fig. 5.5 Misaligned circular pad: correction factor CRversus parallelism error Ah‘.

98

HYDROSTATC LUBRlCATlON

The hydraulic resistance of the pad becomes:

(5.34) where: cR=

3

1

(5.35)

l+g(l +r’)ZAh‘2

(a more accurate but rather complicated equation for CR is given in ref. 5.2). It is easy to verify that, when Ah’=@ the correction factors are reduced to 1. The correction factor C,, which may also be interpreted a s the decrease in load capacity when Q a n d x 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 of M’=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)is 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 r b 0 .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 loss of pressure at the inlet

The equations expressing the performance of the bearing should be modified if pressure losses, other than those due to viscous forces, have to be taken into account. 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 p for circular pads.

where the correction factor Cp is given by the equation:

(5.37)

and Re 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

100


used to accelerate the fluid in the inlet zone; the third term represents the contraction loss. Equation 5.38 of the inlet length is valid for Ar=r2-r121ir; for greater values of lir, one can simply substitute Ar for li, in Eqn 5.37, unless (in which case the bearing gap would virtually behave as 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

1 Cp=l+--

0.0313

1 (l+l/rY 1 + 0.00525 l n p

2 + 0.00979

(5.39)

_-

h Re

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 . 1values ) of Arl(h.Re), which are, however, unusual in actual fact. The coefficient Cp may also be explained as a n increase in load capacity, for any given flow rate, or 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 M v rotating at 750 rpm (i2=251rradls). Furthermore, the recess pressure 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 Nslm2 and p=880Kglm3 must be used. From the condition p,6.94.10-3 m2 (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 Mf1.5.Sg;smaller values may be used if ~ ~ c 0 .but 5 , wider clearance 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 to modify the geometrical parameters (namely the recess width) or the design film thickness or the supply pressure. However, if the speed is very low, i t will not be possible to get n=1;in 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 to 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 ho cannot be further reduced, it will be necessary to accept a power ratio of l7>l (e.g. by increasing p or by reducing

OPPOSED-PAD AND MULTIPAD BEARINGS

217

p s ) , 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 expected 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 of 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 KN range, with a stiffness that is always greater than 108 N l m , and rotating at 157 radls (1500 rpm); the supply system to be used delivers lubricant at 7 MPa. Further 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 . 5Since ) . laminar-flow restrictors are selected, the load parameter is readily calculated (Eqn 7.22) as W=0.66; this means that an effective 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 m m 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 60t90 p 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 in order to obtain n=l. From Eqn 5.68 and Eqn 5.72 we obtain Ro=161.109Nslm5 and Hf0=87 W. The calculations are now straightforward. When g=2ho=90pm, stiffness at W=O is K0=O.43.1O9N l m , flow rate is Q=26.10-6 m3/s,pumping power and friction power are Hp=182 W a n d Hf=175 W, respectively. The design load can be sustained with an eccentricity that is slightly smaller than 0.5;the relevant stiffness is again K=O.43.1O9N l m (however, stiffness is greater for intermediate loads); the flow rate and pumping power are smaller, while the friction power is 30% greater. When g=2ho-fig=60pm, the actual pressure ratio becomes (Fig. 7.8) 8=0.59; in order to sustain 10 KN, an eccentricity of ~ = 0 . 5 is 3 now required; the smallest film thickness is therefore h,=14 pm. Stiffness at W=O is much greater than before (K=O.75.1O9N l m ) , while at the maximum load we get K=O.41.1O9 N l m . The pump-

21 8

HYDROSJATlC LUBRICATION

ing power is obviously smaller; on the other hand, it is Hf=262 W at W=O,and H f-365 W at W =10 KN. The hydraulic resistance of the restrictor must be Rr=Ro(l-p)/p=376.1@ Nslm5; this may be easily obtained, say, with a 1=30.75 mm length of a 1 mm-bore pipe. The value of viscosity selected above may seem too high in certain applications. It is clear that to select a lower value would have no effect on bearing stiffness, but flow rate and pumping power would become much greater. For instance, selecting p=O.1 Nslrn2, for g=90 p m 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. I n this case, it might prove suitable to slightly change the size of the pad, increasing both a and D in order to reduce Hp.

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 Hfo (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);this allows us to obtain precisely 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 commercially 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, &I 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 contact occurs a t a n eccentricity

(7.66) where we have introduced the non-dimensional pitch error

(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, provided that A,, Ro, and Hfo 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

w

1

1

= ( 1 4 3 + &i2 (I-&) (1+43 + &'2 (I+&)

(7.68)

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, a t E=O, we get (7.69) where KOis the design value, given by Eqn 7.11.

220


7.3.2

Constant pressure supply

The effect of pitch errors on recess pressures, and then on load capacity and flow rate, can be studied by substituting R’=ll(l+d3with (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 capacity 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 rate is still given by Eqn 7.23, where (7.72)

-a1.I

0

-b-

1.o

0

E

O.!

0.5

W Ae PS

1.o

0.0

0.5

W -

1 .o

Ae PS

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 KOis given by Eqn 7.26. In conclusion it may be noted that, to make manufacturing easier, the continuous recess is often substituted by a sequence of small pockets; in this case, the effective 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 i s fed independently through a separate restrictor, the effect of the pitch error on load capacity, 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 1 5 KN, with a rigidity W / e greater than 109 N l 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 psAe must be substantially greater than the maximum load W ~ = 1 KN; 5 initially, one may state that psAe>1.7.W~=25.5KN: since p s 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 of 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 film thickness, it is necessary to know the maximum value of the pitch error; assuming SpllO pm, it follows that to obtain 6p'% pm (Eqn 7.67). On the other hand, Fig. 7.24 shows that, assuming @=0.3,at the maximum load WM=15KN, we have a n eccentricity ~ 0 . 4 (for 5 Sp3=0.4);thus, in order to have a displacement that is smaller than e~=15.103/109 m=15 pm, we must have h0133 pn. Finally, h0=30 pm may be chosen. 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 r a d l 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 Nslmz.

222

HYDROSTATIC LUBRlCATION

The flow rate depends on the actual value of viscosity (i.e. it is inversely proportional to it); for instance, if we select p=O.l Nslmz, Eqn 5.116gives Ro=185.109 Nslm5 and, hence, at W=O, Q=22.7.10-6 m3ls and Hp=159 W. Both the flow rate and the pumping power change little when the load is applied. The power loss due to friction at E=O and p=O.l N s l m z is Hf=104 W a n d proves to be proportional to viscosity.

7.4

SELF-REGULATING BEARINGS

A self-regulating bearing (SRB) may be regarded as a n 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 a s spools o r diaphragms). On the other hand, the "infinite stiffness" which could, in theory, be achieved with the aforesaid devices, can no longer be obtained,

/

/

II W '

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(Fig. 7.25). In this ances are used, the latter condition is satisfied if rllr2=r31r4=r~ 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.It 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 hydraulic 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, it is easy to see that the hydraulic resistance of the assembly becomes:

R=RoRA

(7.75)

where 1

1

Rk = 2 [ m

I-+

1

(7.76)

The load capacity is obtained by integrating the lubricant pressures on the bearing surfaces; it may be expressed in the following form:

W = p r A , W'

(7.77)

where: ( 1 + 4 3 - ( 1 - €13 + (1- 4 3

W' = ( 1 + €13

For flat circular clearances the effective bearing area is

(7.78)

224


(7.79)

It should be noted that this expression coincides with the one obtained for a n annular-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=riIr', a value of r; exists that minimizes pumping power. This "optimum" 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 is not negligible, it is convenient to use wider recesses, because of the presence 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 (7.81)

H f = Hfo H j ( E ) where:

H

A 0 4

---a2

fo-16h0

pH?

(7.82)

(7.83) 1 1 1 Hi. =z(i-,+1+E) A*,,R* and H? are plotted in Fig. 7.26.b.

(7.84)

225


-b-

-a-

0.0' 0.5 f

'

.

"

'

'

0.7

"

0.9

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

When the bearing is directly fed by a volumetric pump, we obviously have &=const., and the supply pressure is PS

= p r = R Q =RoR;1(4 Q

(7.85)

The load capacity, for Eqn 7.77, is W = A e R o Q R ; l W'

(7.86)

The pumping power is

Hp =Hi=Ro R;1 Q2

(7.87)

Since the friction power is still given by Eqn 7.81, the reference power ratio is:

n=HfO Ro Q2

(7.88)

The average temperature step in the lubricant (supposing, a s usual, that we have adiabatic flow) is:

226


-a-

0.E

-b-

a

!O

E

K

6

0.4

I0

0.2

Hi

AT AT0 2

0.c

2.5

I

5.0 W

AeQRo

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 displacement, stiffness is obtained: (7.91) where (7.92)

K=:[&

1 +ml

In Fig. 7.27, E, ps,K, H p , H f , and AT are plotted as functions of the load.

(7.93)


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 two bearings have similar load-displacement characteristics when the total flow delivered to 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 opposed-pad; this means that, when the load is applied, the maximum supply pressure 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 a t 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 independent 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 constantpressure source, without any compensating devices, i.e. fixed restrictors or 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: (7.95)

and the temperature step: (7.96)

The bearing stiffness is given by Eqn 7.91, in which now (7.97)

228

HYDROSTATIC L UBRICATION

[(I + 4 3 - (1 K ' = (1 + &)2 + (1 - Ep [(I + E~ - (1 (1 + &I3 + (1 - &)3 [(I + &)3+ (1 - &)312 E,

4

(7.98)

K,Q, Hp, H f , and A T are plotted against the load in Fig. 7.28.a.

-a-

.. A, Ps

0.0

0.5

w

1.o

A, Ps

Fig. 7.28 Self-regulating bearing. Constant pressure supply. a- Eccentricity, flow rate, stiffness, pumping power, friction power, and temperature step versus load, for y=l; b- eccentricity and flow rate versus load, for certain values of y

Let us compare the performance of the SRB with that of the usual opposed- pad annular-recess bearing, with the same radii, fed a t the same constant pressure p s , through laminar-flow restrictors. It is clear that, whatever value of /l is selected, KO is a t least double for the SRB, whereas the limiting load psA, is the same. The flow rate a t W=O proves to be the same if a pressure ratio /l=0.25 is selected for the opposed-pad, while it is smaller for the SRB for higher values of 8. When the load is applied, the comparison is even more favourable for the SRB. When a laminar-flow restrictor R , is put between the lubricant supply and the bearing inlet (or when the pressure losses in the supply pipes are not negligible) we have, obviously, pr(L/2-a)as plain indefinite strips (the diameter of the holes is assumed to be small, compared to 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)to 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.

290

HYDROSTATIC L UBRICATlON

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 a s 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: (9.3)

where

E o = s and 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-R 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 a1into 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.

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 i t seems reasonable to evaluate the actual hydraulic resistance R, of the restrictors at E=O and then to consider i t as a constant value, disregarding the variation in hydraulic resistance due to the inherent orifice. 1.2

I

-rt ,

- 0.5

R, .I 0-12 [Ns/ms]

0.9 25

100

0.6 0.6'

'

0.0

0.5

-

q 106

I

1.o [rn3/s]

Fig. 9.2 Effect of flow rate 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 considerable 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 to be quite different from its design value. It is easy to verify, from Eqn 9.2 and Eqn 9.5, that for small-clearance bearings (e.g. C=25 pm) 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 in the concentric configuration is obviously:

x D C 3 P? Hp = p s Q =--B 6 a

(9.6)

whereas the friction power, at E=O, is the following: (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 constant, a n approximate solution to Eqn 9.1 has to 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) or finite element methods. Such methods prove to be more accurate, especially a t high eccentricity, and can also take into account the effect of cavitation that 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 PSLD

(9.9)

is plotted, a s a function of the eccentricity, for certain values of the geometric parameter 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

W

I, /

0.1

'

1-

/'

1-I.-.--//,-0.25 __

1.o

0.5

0.0

&

Fig. 9.3 Load W' versus eccentricity power ratio 9 (ref. 9.2).

E

(forJ=0.5 and L/D=l) for certain values of ratio alL and of


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 .the 9 ) 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 ; a t the highest eccentricities, the performance of this last type of when ~ 0 . 7 while, bearing seems worse (ref. 9.2). In particular, a t zero speed, slot-entry bearings show negative stiffness a t high eccentricity: that is, the lifting capacity at ~ =is l 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=0.5in the case of a single row of feeding ports). There is clearly a sharp increase in load capacity as a 1L 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 to the outer edges of the bearing) are advisable in most cases, for hybrid operation. The effect of the concentric pressure ratio 4,3 for hydrostatic operation (n=O> is 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; a t higher values of eccentricity, W' grows constantly as fi grows (see Fig. 9.6, ref. 8.19).

294


-a-

- .

4

8

I 3

6

W'

W' 2

4

1

2

I

0

0

0.1

0.3

02 a -

0.4

0.5

0 0

L

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;b- II=12; (ref. 9.4).

(ii) Flow rate. Non-dimensional flow rate a t p=0.5 appears, in practice, to 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 pressureinduced 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 -costlt) ~ and U=DNP, we obtain:

HYBRID PLAIN JOURNAL BEARINGS ~~

3

-

~~~

295

-_

~

1

W'

-0

0.2

0.4

B

0.6

0.8

1

Fig. 9.6 Load W' versus pressure ratiofl, for L / D = l , alL=O.l, I7=1 and for certain values of eccentricity E.

M

nLD3 f-4

c

(9.11)

and the friction power is: 1

H f = M f 0= Hi0 -

(9.12)

At high values of s h and E, however, the friction may be lower, due to the onset of cavitation. A t high speed, the torque required to 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 (I7 depends 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.

DESIGN OF HYBRID BEARINGS

9.3

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 repeated 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); to allow greater eccentricities than for the recessed bearings b 0 . 6 ) ; to 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 stiffness and overload capacity are ensured even when ~>0.9.On the other hand, an upper constraint to E may derive from other factors: e.g., at the higher eccentricities, a tilting error of the journal could cause the performance of the bearing to deteriorate 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 3K8),Klcan 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 and ( are, respectively, the undamped natural frequency and the damping factor of the system: (10.51) Parameter t , ~is defined, beai -.ig a ,o in mind Eqn 10.9 and Eqns 10.28, -y the following equation: (10.52)

w*

Since is, in general, much smaller than unity, the squeeze film effect may be disregarded when evaluating the damping factor in the case of incompressible lubricant. If we want to include compressibility, Eqn 10.50 must be substituted by the following transfer function

where

322

10.5.2


Stability

In the previous section it has been shown that a single-effect passively compensated 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 stiffness 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 are 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 to 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 consists in checking if the coefficients of the characteristic equation of the system satisfy or 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 coefficients ai of the characteristic equation, as well as the parameter

b=a,--

a3 a0 a2

(10.55)

(each ai indicates the coefficient of the relevant power of s in the characteristic equation) 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 Kd, although the effective damping of the system may prove to be very poor, for low lubricant 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 to increase the margin of stability when the lubricant compressibility is not low enough to ensure that 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 w1 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:a clear symptom of instability! As before, a more detailed analysis of stability can be carried out by applying 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 a s shown by 1 then, the condition for Eqn 10.41). It is interesting to note that often ~ 3 > > ~and, stability becomes Ko/Kocc{=w~/wl, that 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 control. A detailed analysis of such methods is clearly beyond the scope of the present work: we shall confine ourselves to 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 GH(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 GH; 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 infiniteradius half circles (n being the number of poles in the origin). For the types of function 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.

-b-

Re(GH) I

Fig. 10.9 a- Nyquist path; b- Nyquist diagram for restrictor-compensated bearings with negligible squeeze coefficient B.

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 if the turns are clockwise (bear in mind that the diagram is oriented) and nt1, the Nyquist diagram takes on a shape that is similar to the lower curve in Fig. 10.9.b, whatever the values of and %. Namely, the limit of GH(io) for o+O is - e i Z and the diagram is closed by a n 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.

c

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 and 2 a number of values of 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 bearing, respectively. Since too many parameters are involved, general diagrams cannot be given here, except for passively compensated systems. In the simplest case, when lubricant 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 plotted in Fig. 10.11. 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 practice the effectiue damping proves to be greatly lowered, when 5 and y are small.

327

DYNAMICS

0

1

LL

2

an

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 that 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 constants; 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 in example 6.1. As will be remembered, the main bearing parameters fixed there were: D=O.l m, r’=O.75, p=O.1 Ns 1m2 and, under a load W=40KN, ho=30 pm. Let the moving mass be M=3061 Kg, and the equivalent bulk modulus of the lubricant be Kla=109Nlm2; we have:

328


-a-

10

-b-

10

c= 1 ly = 0.02

6h GFIK,

6h -

6WK,

5

5

0

0

0

2

2

0

0,

0,

Fig. 10.12 Frequency response for a direct-supply or resmctor-compensated bearing for certain values of parameter 5 and for two values of damping factor

c.

0.1

1

r

10

Fig. 10.13 Maximum vibration amplitude (in the full range of frequencies) versus damping factor for certain values of parameters 6 and w.

329

DYNAMICS

ahto check the stability of the bearing for static loads between 30 and 40 m; b)-to asses the frequency response of the system. In order to carry out these verifications, it is first necessary to asses, for both the greatest and the least values of load, the relevant values of film 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 time constant 1 lol (bear in mind that the effective bearing area is A,=5.97.103 ma) and the values of %, & and as shown in table below: h

F

(KN) 30 40

R . I O - ~ Z~,.10-9B . I O - ~ w1

urn)

(Ns/rn5)

(N/m)

33 30

1.53 2.03

2.73 4.00

Eqn 5.21

Eqn 6.5

(Ns/m) 1.49 1.99

(s-1)

(sl)

50.1 55.1

944 1143

Eqn 10.8 Eqn 10.25

c

w

9.7 10.7

0.027 0.027

w,

Eqns 10.51

Eqn 10.52

By means of Eqn 10.14 it is now easy to verify that the system is stable for every value of the ratio &IQ I Kw 3

2 6h -

6WKS

1

1

0 0

100

200

Fig. 10.14 Example 10.1:frequency response at W&

300

400

500

KN.

b) The frequency response has been plotted in Fig. 10.14,for a number of values of (, in the case of W=40 KN (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-

330


sponse from the presence of peaks, which, on the contrary, may be notable for values of 5 lower than 1. This means that supply pump should be very close to the bearing. For instance, in order to have 1,lubricant stiffness Kd should be greater than 4.109 N l m : it follows from Eqn 10.19 that the volume of lubricant i n the recess and supply pipes should be smaller than 8.9.10-6 m3. It is easy to see that, in order to increase >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

Multipad journal bearings

Let us now consider a journal bearing made up of n cylindrical pads (Fig.10.24),

Fig. 10.24 Multipad journal 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: w1). Actually, when the turning speed is high, self-excited vibrations may set in

(ref. 10.9); these are due to the hydrodynamic effects (disregarded in the foregoing statements), which may cause entrainment of air in a recess, when the relevant

DYNAMICS

3 49

recess pressure falls below the atmospheric pressure, and even instability, above a critical speed (due to nonlinearity, instability is transformed into self-excited finiteamplitude 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 problems are likely t o occur only if the design of the bearing is far from commonly accepted practice (namely for n>3)and 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 to analyze than the types of hydrostatic bearings examined above, mainly because of the interdependence of the recesses, which compels us to treat the bearing a s 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

Analysis

The dynamic behaviour of the journal is described by the equations of motion, which in vector form are:

MC

{f } - W = F

(10.105)

350


where F is the external force and W is 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, its 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 displacements are involved, they may be considered to be constants. In order to study small displacements around any steady-state equilibrium point ( C ~ , $ ~ ) = ( ~ ~a , convenient ~J, procedure is to linearize Eqn 10.106, that leads to write the perturbation of load capacity as:

(10.107) Note that we have omitted the subscript 's'in the last equation, but i t 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 second 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 accounts 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 to 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 continuity of flow in and out of each recess. The flow rate reaching the lands from each recess may be obtained (by numerical computing or other approximate calculations) a s 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 lubricant 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 recesses 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 10.103 for 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

HYDROSTATICLUERICATION

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 (10.112)

where:

AA

A = q +a + - I s

(10.113)

Kd

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 negligible). Finally, the equations of motion (Eqns 10.105) become:

( M I s2 + B s + K)

=

SF(s)

(10.115)

In spite of the formal simplicity of Eqns 10.115, their coefficients would quite difficult and tedious to 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 ~ ~ ' Furthermore, 0 . the coefficients may be calculated by means of some

DYNAMICS

353

simplification (typically, the thin lands assumption), which may even lead to general closed-form equations.

As 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, instability may occur due to one of the following reasons: i) - The lubricant stiffness Kd is too low, due to excessive compressibility or to excessive 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 hydrostatic bearings is often very great, the supporting structure may not always be regarded as being rigid, and thus Eqn 10.115 would become quite more complicate.

10.9.2

Non-rotating bearings, incompressible lubricant

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 displacement direction, and Eqns 10.115 may 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 KOand Bo are known. By the way, since Eqns 10.116 are second-order equations with positive coefficients, stability is ensured. The coefficient KO is nothing but the static stiffness already examined in chapter (W, E ) characteristic of the bearing. In section 8.3.1 a n approximate equation (namely, Eqn 8.7) is reported in which KOis 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 or Eqn 8.6). A similar equation may also be obtained for Bo (see ref. 8.12): 8. It may be deduced from the slope of the

HYDROSTATIC 1UBRlCATlON

354 D L3

(10.117)

B o = 1 2 p ~ u ' (-u')2A 1

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 uncoupled (the off-diagonal terms of the K and B matrices are small). In chapter 8 it has been shown that the attitude angle I$ often has only a small influence on the performance of the bearing; hence it is an acceptable loss of generality to take &=O (i.e. {a&). Several plots of the coefficientsB and K are given in figures from 10.26 to 10.28 taken from ref. 10.10 and ref. 10.11. -a-

-b-

1

08

0.8

0.6

0.6 K L DPJC

B 3 &4L( D/CP

0.4

0.4

0.2

0.2

0

0 0

Q2

04

B

0.6

0.8

1

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

Rotating bearing, incompressible lubricant

When the journal rotates a t 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 orthogonal to it. Limiting ourselves to the simplest case of vibrations around E=O and incompressible lubricant, Eqns 10.116 can be completed as follows:

355

DYNAMICS

-a1

~~

LD:/C/

-b-

1.6 I

0.8 Es= 0 c

w F

0.4

"

/ 0

0.5

1.5

1

2

0

2.5

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,a'=0.2, 8=36", J=0.6).

2

0.8

1.5

K

B ~-

L Dps/C

3pL(D/C13

0.6

1

0.4 0.5

0.2 0

0

02

0.6

0.4 Es

0.8

1

0

0.2

0.6

0.4

0.8

1

6s

Fig. 10.28 Multirecess journal bearings: stiffness and damping versus eccentricity, for various a' ( 1 ~ 4 O=na', . LID=l ,J=0.6).

( 10.118)

The coefficient Ku is proportional to the rotating speed R and needs to be calculated by numerical means, or on the basis of suitable simplifying assumptions. An approximate 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.

At the critical speed, corresponding to K,*, the shaft oscillates in a n undamped mode a t the natural frequency 0, (whirl instability). From Eqn 10.119 follows that the critical speed is: l2*=2%

(10.121)

This confirms the well-known fact that, when the turning speed goes beyond the critical value, the shaft oscillates a t 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 that the maximum displacement is small enough (~O must be true. Figure 11.15 contains the results of the optimization of Hi in the OSb'Sl, OSh'S2 range, for OSkg14 and with the constraint b'S0.975. HI, is plotted with the corresponding 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. It should be noted that, for any value of k, bApt has always taken the boundary value b'=0.975;therefore hApt is again given by Eqn 11.60,that is (11.103)

Figure 11.15 also shows Hb,HL,Q' coinciding with HL,W' and K', given by the following equations

397

OPTlMlzATlON

0 < 6 < 0.975.or 0.025 < ~ ' 9 1or , 0.025 H;Jis 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 11.121 obtaining k with which 6kpt is evaluated from Fig. 11.4.b or 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, a lower limit can be imposed on it, that is

K 2 y ,therefore

1 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 present, 6' is chosen a s large as possible and assumed to be equal to 6hpt;i t is introduced 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 can also be expressed as

It must be noted that, apart from the coefficient 4/3, the above expression ofH,'(p',h') is similar to that given by the first of Eqns 11.109; therefore, as regards its variation, we can refer qualitatively to Fig. 11.14 and, as regards Hirn,to Fig. 11.15 and the following figures. Consequently, to obtain Hirn,p' is chosen as small as possible: i t is pAPt;hbpt is then determined &om Eqn 11.60, that 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 0


11.6.3.3 Ht=f(L,h). Equation 11.68 can also be expressed as

The considerations regarding Eqn 11.123 and Hi, are also valid in this case, so long a s pApt is replaced by LApt. 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)

The considerations regarding Eqn 11.123 and Him are also valid in this case, providing p' is replaced by g'. 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 HI, in the Ol+ const.

1.4 H;,F'

14

14

1.2 8

,

Hp,Q 1 4

H'

p"

HPb

0.8-

HP, Rro/Rr

0.6 -

w'

\

\1.

0.4 -

4

K' 0.2 -

2 0

0

0.2

0.4

0.6

0.8

1

02

0

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.

d4/l is constant) as h vanes, h is substituted by the following equation obtained fiom Eqn 11.176

(11.177)

11

Hpc = 3 (1 -PI2

1 L empf ,

and in dimensionless form

11

Hpb = 3 p (1 -

1 L

P)P c

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 characteristics of a bearing with given dimensions, supplied a t 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 that for fi=0.5 and 0.815 i t 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 HP decreases linearly. Figure 11.21.b also shows h '

from which it is possible to 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@is 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 design, 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 following dimensionless equations can be written

430

HYDROSTATIC 1UBRICATION

(11.182)

(11.183)

while H;,, is still identical in form to 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 to be adopted.

As 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

18

1.6

1E

-b-

1.8

1.6

H't

1.4

14

I.4

Hi. F;f' 1.2 K' 1

12

1.2

K'

H;, 10

1

3.8

3.6 3.4

3.2

3 0

02

0.4

B

0.6

0.8

1

Fig. 11.22 Supply pressure pi, 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=0.5and increases further but slowly as p decreases t o /3,,=0.261; but a s 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 or values close t o it, are to 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 Friction. 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 to 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 ji' 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 coefficient can be obtained by substitutingp,=/hp, in Eqn 11.37, thus f=2-p--U 11B-b Ps h B B + b

432


I f p varies with the capillary dimensions, f' can be expressed as the following h n c 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. It does not vary with p if p 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

Dissipated power and efficiency losses

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 11 L Ht = 3 ~ p!p h3 ~ - +bp U z L (B - b )

(11.184)

If B vanes with the dimensions of the capillary tube, with h then equal to a constant value, Eqn 11.184 can be written, in dimensionless form, as the following function of 8: (11.185)

H i is plotted in Fig. 11.21.a, for k=l. I f p varies with h according to Eqn 11.176,Hi 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 also been multiplied by l/$. For k=l, the curves of Hi are presented in Fig. 11.3.a, b and d. are still constant when they are considered to be The efficiency losses rk and 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 is given by Eqn 11.67 and is plotted in Fig. 11.7.

11.8.4.2 Given load. The total dissipated power is W2 - 62) Ht = 43 i1B1h 3 L ( B+ 6)(@

+

2-1L

h

(B - 6 )

(11.187)

Ifp varies with the dimensions of the capillary tube, with h then equal to a constant value, Eqn 11.187 can be written, in dimensionless form, a s the following h n c 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 decreases slowly as k increases, small values of p should not be adopted $20.5). 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 expressions of k having been multiplied by $. For k = l , the curves of Hi are presented in Fig. 11.4.a, b and d. The efficiency loss Fig. 11.8.

ri is given, as a function of h', by Eqn 11.83 and is plotted in

434

HYDROSTATIC LUBRICA TlON

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 a s 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 equations that are formally identical to those obtained in the case of direct supply. As far as the selection of 8 is concerned, the numerous remarks made in section 6.3, 6.4.2 and 11.8 may be briefly summarized as follows: - from the point of view of efficiency, it is best to select a high value for the pressure ratio (perhaps 8=0.7); - a low value of 8 (perhaps 8=0.3)is 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

11.9.1.1Ht=f(b,h) 11.9.1.1.1. Equation 11.184 can also be expressed as

It is easy to see that the above equations are identical to those obtained in the case of direct supply (Eqns 11.102),except that is used in place of recess pressure p r .

.lisps,

In Fig. 11.14 Hi(b’,h’)is plotted in the O2.5.1@ N l m , friction force is F16 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: h APt=0.4, P Apt* bhp,pt=0.545, from which, in dimensional form, Lopt=0.3m, bopt=0.163m, hopt=0.4-104m, kpt=0.12Nslm2 Htm=2.07 Nmls, p,=3.06*106Nlm2, K=2.70*1@N l m , f=1.58.104. The friction coefficient is clearly too high. Since the film thickness cannot be increased too much due to the constraint on stiffness, the friction can be easily reduced by reducing viscosity. For instance, stating pApt=l,we obtain bApt=0.406and Lopt=0.3m, bopt=0.122m, kpt=0.O6Nsl m2 hopt=0.4.104m, Htm=3.56Nmls, p,=4.11.106 Nlm2, K=2.70.1@ N l m , 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.lO-4. I n 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 hpt=0.051Nslm2 bopt=0.127m, hopt=0.4.1f34m, Lopt=0.4m, Htm=2.47 Nmls, p,=2.53.106 Nlmz, K=2.70.1@ N l m , 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 a n annular-recess pad, compensated by means of a capillary tube, with an inner radius r,=0.05 m and a n outer radius r4

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